Hypothetical and Concepts 86.Extradimensions(rev.) ...... 889 87. W ′-bosonsearches(rev.) ...... 897 88. Z′-bosonsearches(rev.) ...... 900 89.Supersymmetry:theory(rev.) ...... 905 90. Supersymmetry: experiment (rev.) ...... 923 91. and other similar particles (rev.) ...... 939 92. and compositeness, searches for (rev.) . . . . 953 93. Dynamical electroweak symmetry breaking: ...... 958 implications of the H(0) (rev.) 94.Grandunifiedtheories(rev.) ...... 971 95.(rev.) ...... 986 96.Magneticmonopoles(rev.) ...... 989

86. Extra Dimensions 889

86. Extra Dimensions

Revised August 2019 by Y. Gershtein (Rutgers U.) and A. Po- 5D field with the same mass-dimension as in 4D. This theory is 3 marol (U. Autònoma de Barcelona; IFAE). perturbative for energies E ∼< `5M5/λ5 where `5 = 24π [14]. Let us now consider that the fifth dimension is compact with the topology of a circle S1 of radius R, which corresponds to the 86.1 Introduction identification of y with y + 2πR. In such a case, the 5D complex Proposals for a spacetime with more than three spatial dimen- scalar field can be expanded in a Fourier series: sions date back to the 1920s, mainly through the work of Kaluza ∞ and Klein, in an attempt to unify the forces of nature [1]. Al- 1 X φ(x, y) = √ einy/Rφ(n)(x) , (86.2) though their initial idea failed, the formalism that they and oth- 2πRM5 ers developed is still useful nowadays. Around 1980, string theory n=−∞ proposed again to enlarge the number of space dimensions, this that, inserted in Eq. (86.1) and integrating over y, gives time as a requirement for describing a consistent theory of quan- tum gravity. The extra dimensions were supposed to be compact- (0) (n) ified at a scale close to the Planck scale, and thus not testable S5 = S4 + S4 , (86.3) experimentally in the near future. where A different approach was given by Arkani-Hamed, Dimopou- Z (0) 4  (0) 2 (0) 4 los, and Dvali (ADD) in their seminal paper in 1998 [2], where S4 = − d x |∂µφ | + λ4|φ | , (86.4) they showed that the weakness of gravity could be explained by postulating two or more extra dimensions in which only grav- and, ity could propagate. The size of these extra dimensions should Z  2  range between roughly a millimeter and ∼1/TeV, leading to pos- (n) 4 X (n) 2  n  (n) 2 S = − d x |∂µφ | + |φ | sible observable consequences in current and future experiments. 4 R A year later, Randall and Sundrum (RS) [3] found a new pos- n6=0 sibility using a warped geometry, postulating a five-dimensional + quartic interactions . (86.5) Anti-de Sitter (AdS) spacetime with a compactification scale of order 1/TeV. The origin of the smallness of the electroweak scale The n = 0 mode self-coupling is given by versus the Planck scale was explained by the gravitational redshift factor present in the warped AdS metric. As in the ADD model, λ5 λ4 = . (86.6) originally only gravity was assumed to propagate in the extra di- 2πRM5 mensions, although it was soon clear that this was not necessary in warped extra-dimensions and also the SM gauge fields [4, 5] The above action corresponds to a 4D theory with a massless (0) and SM [6, 7] could propagate in the five-dimensional scalar φ , referred to as the zero mode, and an infinite tower (n) spacetime. of massive modes φ with n > 0, known as KK modes. The The physics of warped extra-dimensional models has an al- KK reduction thus allows a treatment of 5D theories as 4D field ternative interpretation by means of the AdS/CFT correspon- theories with an infinite number of fields. At energies smaller dence [8–10]. Models with warped extra dimensions are related than 1/R, the KK modes can be neglected, leaving the zero-mode to four-dimensional strongly-interacting theories, allowing an un- action of Eq. (86.4). The strength of the interaction of the zero- derstanding of the properties of five-dimensional fields as those mode, given by Eq. (86.6), decreases as R increases. Thus, for of four-dimensional composite states [11]. This approach has a large extra dimension R  1/M5, the massless scalar is very opened new directions for tackling outstanding questions in par- weakly coupled. ticle physics, such as the flavor problem, grand unification, and 86.2 Large Extra Dimensions for Gravity the origin of electroweak symmetry breaking or supersymmetry 86.2.1 The ADD Scenario breaking. The ADD scenario [2, 15] (for a review see, for example, [16]) 86.1.1 Experimental Constraints assumes a D = 4 + δ dimensional spacetime, with δ compactified Constraints on extra-dimensional models arise from astrophysi- spatial dimensions. The apparent weakness of gravity arises since cal and cosmological considerations, tabletop experiments explor- it propagates in the higher-dimensional space. The SM is assumed ing gravity at sub-mm distances, and collider experiments. Col- to be localized in a 4D subspace, a 3-brane, as can be found in lider limits on extra-dimensional models are dominated by LHC certain string theory constructions [17, 18]. Gravity is described results, which can be found on the public WWW pages of AT- by the Einstein-Hilbert action in D = 4 + δ spacetime dimensions LAS [12] and CMS [13]. This review includes the most recent −1 ¯ 2+δ Z Z limits, most of which are published results based on 36 fb LHC MD 4 δ p 4 p SD = − d xd y −g R + d x −gind LSM , (86.7) data collected in 2015-16 at a center-of-mass energy of 13 TeV 2 and legacy results from 20 fb−1of 8 TeV data collected in Run 1. In addition, there are a few published and preliminary results where x labels the ordinary four coordinates, y the δ extra coor- based on the full Run 2 datasets. For most of the models, Run dinates, g refers to the determinant of the D-dimensional metric 2 results surpass the sensitivity of Run 1, even in the cases when whose Ricci scalar is defined by R, and M¯ D is called the reduced the integrated luminosity is smaller. Planck scale of the D-dimensional theory. In the second term of Eq. (86.7), which gives the gravitational interactions of SM 86.1.2 Kaluza-Klein Theories fields, the D-dimensional metric reduces to the induced metric Field theories with compact extra dimensions can be written on the 3-brane where the SM fields propagate. The extra dimen- as theories in ordinary four dimensions (4D) by performing a sions are assumed to be flat and compactified in a volume Vδ. Kaluza-Klein (KK) reduction. As an illustration, consider a sim- As an example, consider a toroidal compactification of equal radii ple example, namely a field theory of a complex scalar in flat R and volume V = (2πR)δ. After a KK reduction, one finds 1 δ five-dimensional (5D) spacetime. The action will be given by that the fields that couple to the SM are the spin-2 gravitational Z field Gµν (x, y) and a tower of spin-1 KK graviscalars [19]. The 4  2 2 4 graviscalars, however, only couple to SM fields through the trace S5 = − d x dy M5 |∂µφ| + |∂yφ| + λ5|φ| , (86.1) of the energy-momentum tensor, resulting in weaker couplings to the SM fields. The Fourier expansion of the spin-2 field is given where y refers to the extra (fifth) dimension. A universal scale by M5 has been extracted in front of the action in order to keep the (0) 1 X i~n·~y/R (~n) Gµν (x, y) = Gµν (x) + √ e Gµν (x) , (86.8) 1Our convention for the metric is η = Diag(−1, 1, 1, 1, 1). Vδ MN ~n6=0 890 86. Extra Dimensions

where ~y = (y1, y2, ..., yδ) are the extra-dimensional coordinates One signature arises from direct emission. By mak- and ~n = (n1, n2, ..., nδ). Eq. (86.8) contains a massless state, ing a derivative expansion of Einstein gravity, one can construct (0) (~n) 2 an effective theory, valid for energies much lower than MD, and the 4D graviton Gµν , and its KK tower Gµν with masses m~n = |~n|2/R2. At energies below 1/R the action is that of the zero use it to make predictions for graviton-emission processes at col- mode liders [19, 27, 28]. produced in the final state would escape detection, giving rise to missing transverse energy (E6 T ). 2+δ M¯ Z Z q The results quoted below are 95% CL lower limits on MD for a (0) D 4 p (0) (0) 4 (0) S = − d x V −g R + d x −g LSM , 4 2 δ ind range of values of δ between 2 and 6, with more stringent limits (86.9) corresponding to lower δ values. At colliders, experimentally sensitive channels include where√ we can identify the 4D reduced Planck mass, MP ≡ 18 the jet (j) + E6 T and γ + E6 T final states. CMS (ATLAS) j + E6 T 1/ 8πGN ' 2.4 × 10 GeV, as a function of the D-dimensional −1 parameters: results with 36 fb of 13 TeV data provide limits of MD > M 2 = V δM¯ 2+δ ≡ RδM 2+δ . (86.10) 5.3 − 9.9 TeV [29] (MD > 4.8 − 7.7 TeV [30]). For these analyses, P D D both experiments are assuming leading order (LO) cross sections. Fixing MD at around the electroweak scale MD ∼ TeV to avoid Since the effective theory is only valid for energies much less than introducing a new mass scale in the model, Eq. (86.10) gives a MD, the results are quoted for the full space, and include the prediction for R: information√ that suppressing the√ graviton cross section by a factor M 4 /sˆ2 for sˆ > M , where sˆ is the parton-level center-of- δ = 1, 2, ..., 6 → R ∼ 109 km , 0.5 mm , ... , 0.1 MeV−1 . (86.11) D D mass energy of the hard collision, weakens the limits on MD by a The option δ = 1 is clearly ruled out, as it leads to modifications negligible amount for δ = 2 (∼3% for δ = 6). Less stringent limits −1 of Newton’s law at solar system distances. However this is not are obtained by CMS [31] from analysis of 36 fb of 13 TeV data the case for δ ≥ 2, and possible observable consequences can be in the γ + E6 T final state. The analogous ATLAS search [32] has sought in present and future experiments. comparable sensitivity but does not quote ADD interpretation of Consistency of the model requires a stabilization mechanism the results. for the radii of the extra dimensions, to the values shown in In models in which the ADD scenario is embedded in a string Eq. (86.11). The fact that we need R  1/MD leads to a new theory at the TeV scale [18], we expect the string scale Ms to hierarchy problem, the solution of which might require imposing be smaller than MD, and therefore expect production of string supersymmetry in the extra-dimensional bulk (for the case of two resonances at the LHC [33]. A Run 2 result from CMS analyzing −1 extra dimensions see for example [20]). the dijet invariant mass distribution for 36 fb of 13 TeV data excludes string resonances that decay predominantly to q +g with 86.2.2 Tests of the Gravitational Force Law at Sub-mm Dis- masses below 7.7 TeV [34]. ATLAS dijet analysis [35] provides tances their results in the context of model-independent limits on the The KK modes of the graviton give rise to deviations from cross section times acceptance for generic resonances of a variety < Newton’s law of gravitation for distances ∼ R. Such deviations of possible widths. are usually parametrized by a modified Newtonian potential of the form 86.2.4.2 Virtual graviton effects m1m2  −r/λ V (r) = −GN 1 + α e . (86.12) One can also search for virtual graviton effects, the calculation r of which however depends on the ultraviolet cut-off of the theory For a 2-torus compactification, α = 16/3 and λ = R. Searches for and is therefore very model dependent. In the literature, sev- deviations from Newton’s law of gravitation have been performed eral different formulations exist [19,28,36] for the dimension-eight in several experiments. Refs. [21–23] give the present constraints: operator for gravity exchange at tree level: R < 37µm at 95% CL for δ = 2, corresponding to MD > 3.6 TeV. 4  1  86.2.3 Astrophysical and Cosmological Constraints µν µ ν L8 = ± 4 Tµν T − Tµ Tν , (86.13) The light KK gravitons could be copiously produced in stars, MTT δ + 2 carrying away energy. Ensuring that the graviton luminosity is low enough to preserve the agreement of stellar models with where Tµν is the energy-momentum tensor and MTT is related to MD by some model-dependent coefficient [37]. The relations observations provides powerful bounds on the scale MD. The with the parametrizations of Refs. [36] and [19] are, respectively, most stringent bound arises from supernova SN1987A, giving 1/4 MTT = MS and MTT = (2/π) ΛT . The experimental results MD > 27 (2.4) TeV for δ = 2 (3) [24]. After a supernova ex- plosion, most of the KK gravitons stay gravitationally trapped in below are given as 95% CL lower limits on MTT , including in the remnant star. The requirement that neutron stars some cases the possibility of both constructive or destructive in- are not excessively heated by KK decays into leads to terference, depending on the sign chosen in Eq. (9). M > 1700 (76) TeV for δ = 2 (3) [25]. The most stringent limits arise from LHC analyses of the dijet D −1 Cosmological constraints are also quite stringent [26]. To avoid angular distribution. Using 35.9 fb of 13 TeV data, CMS [38] obtains results that correspond to an approximate limit of M > overclosure of the universe by relic gravitons one needs MD > 7 TT TeV for δ = 2. Relic KK gravitons decaying into photons con- 9.1 TeV. tribute to the cosmic diffuse gamma radiation, from which one The next most restrictive results come from the analyses of can derive the bound MD > 100 TeV for δ = 2. diphoton (MTT > 6.1 TeV from ATLAS [39] and MTT > 7.0 TeV We must mention however that bounds coming from the decays from CMS [40]) and dilepton mass spectra (MTT > 6.1 TeV from −1 of KK gravitons into photons can be reduced if we assume that CMS [41]). The complete Run 2 (139 fb ) analysis of ATLAS KK gravitons decay mainly into other non-SM states. This could di-lepton data [42] does not quote the limits on ADD. happen, for example, if there were other 3-branes with hidden At the one-loop level, gravitons can also generate dimension-six sectors residing on them [15]. operators with coefficients that are also model dependent. Exper- imental bounds on these operators can also give stringent con- 86.2.4 Collider Signals straints on MD [37]. 86.2.4.1 Graviton and Other Production 86.2.4.3 Black Hole Production Although each KK graviton has a purely gravitational coupling, √ suppressed by 1/MP , inclusive processes in which one sums over The physics at energies s ∼ MD is sensitive to the details the almost continuous spectrum of available gravitons have cross of the unknown quantum√ theory of gravity. Nevertheless, in the sections suppressed only by powers of MD. Processes involv- transplanckian regime, s  MD, one can rely on a semiclassical ing gravitons are therefore detectable in collider experiments if description of gravity to obtain predictions. An interesting feature MD ∼ TeV. A number of experimental searches for evidence of of transplanckian physics is the creation of black holes [43,44] (for large extra dimensions have been performed at colliders, and in- a review see, for example, [45]). A black hole is expected to be terpreted in the context of the ADD model. formed in a collision in which the impact parameter is smaller 86. Extra Dimensions 891

than the Schwarzschild radius [46]: 86.3 TeV-Scale Extra Dimensions 86.3.1 Warped Extra Dimensions

1/(δ+1) The RS model [3] is the most attractive setup of warped ex-  δ (δ−3)/2  1 2 π  δ + 3  MBH tra dimensions at the TeV scale, since it provides an alternative RS = Γ , (86.14) MD δ + 2 2 MD solution to the hierarchy problem. The RS model is based on a 5D theory with the extra dimension compactified in an orbifold, 1 1 S /Z2, a circle S with the extra identification of y with −y. This where MBH is the mass of the black hole, which would roughly corresponds to the segment y ∈ [0, πR], a manifold with bound- correspond to the total energy in the collision. The cross section aries at y = 0 and y = πR. Let us now assume that this 5D for black hole production can be estimated to be of the same or- theory has a cosmological constant in the bulk Λ, and on the two der as the geometric area σ ∼ πR2 . For M ∼ TeV, this gives a boundaries Λ and Λ : S √ D 0 πR production of ∼ 107 black holes at the s = 14 TeV LHC with an −1 Z integrated luminosity of 30 fb [43,44]. A black hole would pro- 4 np h 1 3 i p S5 = − d x dy −g M R + Λ + −g0 δ(y)Λ0 vide a striking experimental signature since it is expected to ther- 2 5 mally radiate with a Hawking temperature TH = (δ + 1)/(4πRS ), p o and therefore would evaporate democratically into all SM states. + −gπR δ(y − πR)ΛπR , (86.15) Nevertheless, given the present constraints on MD, the LHC will not be able to reach energies much above M . This implies that D where g0 and gπR are the values of the determinant of the induced predictions based on the semiclassical approximation could receive metric on the two respective boundaries. Einstein’s equations can sizable modifications from model-dependent quantum-gravity ef- be solved, giving in this case the metric fects. 2 2 µ ν 2 −ky The most stringent limits on microscopic black holes arise from ds = a(y) dx dx ηµν + dy , a(y) = e , (86.16) LHC searches which observed no excesses above the SM back- p 3 ground in high-multiplicity final states. The results are usually where k = −Λ/6M5 . Consistency of the solution requires quoted as model-independent limits on the cross section for new Λ0 = −ΛπR = −Λ/k. The metric in Eq. (86.16) corresponds physics in the final state and kinematic region analyzed. These to a 5D AdS space. The factor a(y) is called the “warp” factor results can then be used to provide constraints of models of low- and determines how 4D scales change as a function of the posi- scale gravity and weakly-coupled string theory. In addition, limits tion in the extra dimension. In particular, this implies that energy are sometimes quoted on particular implementations of models, scales for 4D fields localized at the boundary at y = πR are red- which are used as benchmarks to illustrate the sensitivity. shifted by a factor e−kπR with respect to those localized at y = 0. A CMS analysis [47] of multi-object final states using 36 fb−1of For this reason, the boundaries at y = 0 and y = πR are usually 13 TeV data, excludes semiclassical black holes below masses of referred to as the ultraviolet (UV) and infrared (IR) boundaries, respectively. up to 10.1 TeV for MD = 2 TeV and δ = 6. Analogous Run 2 ATLAS analysis [48], using 3.0 fb−1of 13 TeV data, excludes black As in the ADD case, we can perform a KK reduction and obtain the low-energy effective theory of the 4D massless graviton. In this hole masses up to 9.0 − 9.7 TeV, depending on MD, for δ = 6. Another ATLAS search [49] for an excess of events with multiple case we obtain high transverse momentum objects, including charged and Z πR 3 −1 M jets, using 3.2 fb of 13 TeV data, excludes semiclassical black M 2 = dy e−2kyM 3 = 5 1 − e−2kπR . (86.17) P 5 2k holes below masses of ∼ 8.7 TeV for MD = 2 TeV and δ = 6. 0 A complementary approach is to look for jet extinction at high Taking M5 ∼ k ∼ MP , we can generate an IR-boundary scale of transverse momenta, as we expect hard short distance scattering order ke−kπR ∼ TeV for an extra dimension of radius R ' 11/k. processes to be highly suppressed at energies above MD [50]. The −1 Mechanisms to stabilize R to this value have been proposed [57,58] CMS analysis [51] of inclusive jet pT spectrum in 10.7 fb of that, contrary to the ADD case, do not require introducing any 8 TeV data set a lower limit of 3.3 TeV on the extinction mass new small or large parameter. Therefore a natural solution to the scale. hierarchy problem can be achieved in this framework if the Higgs For black hole masses near MD, the semi-classical approxima- field, whose vacuum expectation value (VEV) is responsible for tion is not valid, and one could instead expect quantum black electroweak symmetry breaking, is localized at the IR-boundary holes (QBH) that decay primarily into two-body final states [52]. where the effective mass scales are of order TeV. The radion field LHC Run 2 results at 13 TeV provide lower limits on QBH masses is generically heavy in models with a stabilized R. Nevertheless, of order 2.3 − 9.0 TeV, depending on the details of the model. it has been recently discussed that under some conditions a nat- Searches that consider interpretations in terms of QBH limits urally light radion can arise [59–62]. In these cases the radion include the CMS multi-object analysis [53], ATLAS dijet analy- is identified with the , the Goldstone associated to sis [35], and different flavor di-lepton analyses at CMS (eµ, 36 fb−1 the spontaneous breaking of scale invariance, and its mass can be at 13 TeV [54]) and ATLAS (eµ, eτ, µτ, 36.1 fb−1 at 13 TeV [55]). naturally below ke−kπR ∼ TeV. Collider bounds on the radion mass and couplings can be found in [63–65]. In weakly-coupled string models the semiclassical description of In the RS model [3], all the SM fields were assumed to be local- gravity fails in the energy range between M and M /g2 where s s s ized on the IR-boundary. Nevertheless, for the hierarchy problem, stringy effects are important. In this regime one expects, instead only the Higgs field has to be localized there. SM gauge of black holes, the formation of string balls, made of highly excited and fermions can propagate in the 5D bulk [4–7] (for a review see, long strings, that could be copiously produced at the LHC for for example, [66, 67]). By performing a KK reduction from the M ∼ TeV [56], and would evaporate thermally at the Hagedorn s 5D action of a , we find [4, 5] temperature giving rise to high-multiplicity events. The same analyses used to search for black holes can be interpreted in the πR 1 Z 1 πR context of string balls. For example, for the case of δ = 6 with = dy = , (86.18) g2 g2 g2 Ms = MD/1.26 = 3 TeV, the ATLAS multiple high transverse 4 0 5 5 momentum object analysis [48] excludes string balls with masses below 6.5 to 9.0 TeV for values of 0.2 < gs < 0.8. The CMS where gD (D = 4, 5) is the gauge coupling in D-dimensions. multi-object analysis [47] studies string ball production in two Therefore the 4D gauge couplings can be of order one, as is the 2 scenarios, both assuming δ = 6. For the constant gs = 0.2 and case of the SM, if one demands g5 ∼ πR. Using kR ∼ 10 and 1 < Ms < 3.5 TeV the string ball masses below 7.2 to 9.4 TeV g4 ∼ 0.5, one obtains the 5D gauge coupling are excluded, while at constant Ms = 3.6 TeV and 0.2 < gs < 0.4 √ masses below 7.2 to 8.1 TeV are excluded. g5 ∼ 4/ k . (86.19) 892 86. Extra Dimensions

Boundary kinetic terms for the gauge√ bosons can modify this re- 86.3.1.2 Constraints from Electroweak Precision Tests lation, allowing for larger values of g5 k. Models in which the SM gauge bosons propagate in 1/TeV-sized Fermions propagating in a warped extra dimension have 4D extra dimensions give generically large corrections to electroweak massless zero-modes with wavefunctions which vary as f0 ∼ observables. When the SM fermions are confined on a boundary exp[(1/2−cf )ky], where cf k is their 5D mass [7,68]. Depending on these corrections are universal and can be parametrized by four the free parameter cf k, fermions can be localized either towards quantities: Sb, Tb, W and Y , as defined in Ref. [81]. For warped the UV-boundary (cf > 1/2) or IR-boundary (cf < 1/2). Since models, where the 5D gauge coupling of Eq. (86.19) is large, the the is localized on the IR-boundary, one can gen- > most relevant parameter is Tb, which gives the bound mKK ∼ erate exponentially suppressed Yukawa couplings by having the 10 TeV [66]. When a custodial symmetry is imposed [75], the zero-modes localized towards the UV-boundary, generat- main constraint comes from the S parameter, requiring m > 3 ing naturally the light SM fermion spectrum [7]. A large overlap b KK ∼ TeV, independent of the value of g . Corrections to the Zb ¯b with the wavefunction of the Higgs is needed for the , in 5 L L coupling can also be important [66], especially in warped models order to generate its large mass, thus requiring it to be localized for electroweak symmetry breaking as the ones described above. towards the IR-boundary. In conclusion, the large mass hierar- chies present in the SM fermion spectrum can be easily obtained 86.3.1.3 Kaluza-Klein Searches in warped models via suitable choices of the order-one parame- The main prediction of 1/TeV-sized extra dimensions is the ters cf [69]. In these scenarios, deviations in flavor physics from presence of a discretized KK spectrum, with masses around the the SM predictions are expected to arise from flavor-changing KK TeV scale, associated with the SM fields that propagate in the couplings [70], putting certain constraints on the parameters extra dimension. of the models and predicting new physics effects to be observed In the RS model [3], only gravity propagates in the 5D bulk. in B-physics processes (see, for example, [71, 72]). Experimental searches have been performed for the lightest KK The masses of the KK states can also be calculated. One finds graviton through its decay to a variety of SM particle- [7] pairs. The results are usually interpreted in the plane of the  α 1  −πkR dimensionless coupling k/MP versus m1, where MP is the reduced mn ' n + − πke , (86.20) 2 4 Planck mass defined previously and m1 is the mass of the lightest KK excitation of the graviton. Since the AdS curvature ∼ k where n = 1, 2, ... and α = {|cf − 1/2|, 0, 1} for KK fermions, KK cannot exceed the cut-off scale of the model, which is estimated gauge bosons and KK gravitons, respectively. Their masses are of √ to be `1/3M [37], one must demand k  2` M . order ke−πkR ∼ TeV (for this reason we refer to these scenarios 5 5 5 P as TeV-scale extra dimensions). The first KK state of the gauge The most stringent limits currently arise from LHC searches for bosons would be the lightest, while gravitons are expected to be resonances in the dilepton and diphoton final states, using 13 TeV the heaviest. collisions. Best sensitivities are obtained in the γγ final state, which is quite powerful since it has a branching fraction twice 86.3.1.1 Models of Electroweak Symmetry Breaking that of any individual lepton flavor. The CMS analysis [40] of 36 Theories in warped extra dimensions can be used to implement fb−1of 13 TeV data excludes KK gravitons below 2.3 to 4.6 TeV, symmetry breaking at low energies by boundary conditions (for depending on the value of the coupling k/MP , which is varied be- a review see, for example, [73]). For example, for a U(1) gauge tween 0.01 and 0.2, while ATLAS [39] provides a lower limit on the symmetry in the 5D bulk, this can be easily achieved by imposing KK graviton mass of 4.1 TeV for the coupling parameter 0.1. The a Dirichlet boundary condition on the IR-boundary for the gauge- CMS [82] dilepton analyses, combining results from the ee and µµ boson field, Aµ|y=πR = 0. This makes the zero-mode gauge boson channels, exclude KK gravitons with masses below 4.25 TeV for p 2 −πkR coupling parameter 0.1. The ATLAS [42] analysis of 139fb−1of get a mass, given by mA = g4 2k/g5 e . A very different situation occurs if the Dirichlet boundary condition is imposed Run 2 data does not include a RS KK graviton interpretation of the results. Less stringent limits on the KK graviton mass come on the UV-boundary, Aµ|y=0 = 0. In this case the zero-mode gauge boson disappears from the spectrum. Finally, if a Dirich- from analyses of the dijet [34], HH [83, 84], and VV [85, 86] final let boundary condition is imposed on the two boundaries, one states, where V can represent either a W or Z boson. obtains a massless 4D scalar corresponding to the fifth compo- In warped extra-dimensional models in which the SM fields propagate in the 5D bulk, the couplings of the KK graviton to nent of the 5D gauge boson, A5. Thus, different scenarios can be implemented by appropriately choosing the 5D bulk gauge sym- ee/µµ/γγ are suppressed [87], and the above bounds do not ap- ply. Furthermore, the KK graviton is the heaviest KK state (see metry, G5, and the symmetries to which it reduces on the UV and Eq. (86.20)), and therefore experimental searches for KK gauge IR-boundary, HUV and HIR, respectively. In all cases the KK bosons and fermions are more appropriate discovery channels in spectrum comes in representations of the group G5. Among the most interesting scenarios are those called gauge- these scenarios. For the scenarios discussed above in which only Higgs unified models, where the Higgs boson appears as the fifth the Higgs boson and the top quark are localized close to the IR- boundary, the KK gauge bosons mainly decay into top , component of a 5D gauge boson, A5. The Higgs mass is protected by the 5D gauge invariance and can only get a nonzero value longitudinal W/Z bosons, and Higgs bosons. Couplings to light p 2 from non-local one-loop effects [74]. To guarantee the relation SM fermions are suppressed by a factor g/ g5 k ∼ 0.2 [7] for the 2 2 2 MW ' MZ cos θW , a custodial SU(2)V symmetry is needed in value of Eq. (86.19) that is considered from now on. Searches have the bulk and IR-boundary [75]. The simplest realization [76, 77] been made for evidence of the lightest KK excitation of the gluon, has through its decay to tt pairs. The searches take into account the natural KK gluon width, which is typically ∼ 15% of its mass. G5 = SU(3)c × SO(5) × U(1)X , The decay of a heavy particle to tt would tend to produce highly HIR = SU(3)c × SO(4) × U(1)X , boosted top (anti-)quarks in the final state. Products of the sub- sequent top decays would therefore tend to be close to each other HUV = GSM . in the detector. In the case of t → W b → jjb decays, the three jets The Higgs boson gets a potential at the one-loop level that trig- could overlap one another and not be individually reconstructed gers a VEV, breaking the electroweak symmetry. In these models with the standard jet algorithms, while t → W b → `νb decays there is a light Higgs boson whose mass can be around 125 GeV, as could result in the lepton failing standard isolation requirements required by the discovered Higgs boson [78]. This state, as will be due to its proximity to the b-jet; in both cases, the efficiency for explained in Sec. 86.3.2, behaves as a composite pseudo-Goldstone properly reconstructing the final state would fall as the mass of the boson with couplings that deviate from the SM Higgs [79]. The original particle increases. To avoid the loss in sensitivity which present experimental determination of the Higgs couplings at the would result, a number of techniques, known generally as “top LHC, that agrees with the SM predictions, put important con- quark tagging”, have been developed to reconstruct and identify straints on these scenarios [78]. The lightest KK modes of the highly boosted top quarks, for example by using a single “wide” model are color fermions with charges Q = −1/3, 2/3 and 5/3 [80]. jet to contain all the decay products of a hadronic top decay. The 86. Extra Dimensions 893

large backgrounds from QCD jets can then be reduced by requir- UV-boundary corresponds in the CFT to a UV cut-off scale at ing the “jet mass” be consistent with that of a top quark, and ΛUV = k ∼ MP , breaking explicitly conformal invariance, while also by examining the substructure of the wide jet for indication the IR-boundary can be interpreted as a spontaneous breaking that it resulted from the hadronic decay of a top quark. These of the conformal symmetry at energies ke−kπR ∼ TeV. Fields lo- techniques are key to extending to very high masses the range of calized on the UV-boundary are elementary fields external to the accessible resonances decaying to tt pairs. The CMS analysis [88] CFT, while fields localized on the IR-boundary and KK states cor- of 36 fb−1 of 13 TeV data combines di-lepton, lepton-plus-jet, responds to composite resonances of the CFT. Furthermore, local and all-hadronic tt decays and excludes KK with masses gauge symmetries in the 5D models, G5, correspond to global below 4.55 TeV. ATLAS uses all-hadronic [89] and lepton-plus- symmetries of the CFT, while the UV-boundary symmetry can jet [90] final states to exclude KK gluons up to 3.4 and 3.8 TeV be interpreted as a gauging of the subgroup HUV of G5 in the respectively with 36fb−1of 13 TeV data. The results are not di- CFT. Breaking gauge symmetries by IR-boundary conditions cor- rectly comparable between the two LHC experiments, since they responds to the spontaneous breaking G5 → HIR in the CFT employ in their respective analyses different implementations of at energies ∼ ke−kπR. Using this correspondence one can eas- the theoretical model. For masses between 3 and 5 TeV, the cross- ily derive the 4D massless spectrum of the compactified AdS5 3 3 2 2 section limits are around 20 fb for CMS and 30 fb for ATLAS. models. One also has the identification k /M5 ≈ 16π /N and 2 2 r A gauge boson KK excitation could be also sought through its g5 k ≈ 16π /N (r = 1 or 2 for CFT fields in the fundamental or decay to longitudinal W/Z bosons. Recent analyses from 36fb−1of adjoint representation of the gauge group), where N plays the role 13 TeV data by ATLAS [91] and CMS [92] searching for heavy vec- of the number of colors of the CFT. Therefore the weak-coupling tor resonances decaying to a W or Z boson and a Higgs in the qqb¯ ¯b limit in AdS5 corresponds to a large-N expansion in the CFT. final state have set a lower limit on the mass of these KK of ∼ 2.5 Following the above AdS/CFT dictionary one can understand TeV (warped models are equivalent√ to the Model B considered the RS solution to the hierarchy problem from a 4D viewpoint. in the analyses with gV ∼ g5 k). The decay to a pair of inter- The equivalent 4D model is a CFT with a TeV mass gap and a mediate vector bosons has also been exploited to search for KK Higgs boson emerging as a composite state. In the particular case gravitons in models in which the SM fields propagate in the 5D where the Higgs is the fifth-component of the gauge-boson, A5 bulk. The analyses typically reconstruct hadronic W/Z decays us- [108], this corresponds to models, similar to those proposed in Ref. ing variants of the boosted techniques mentioned previously. An [109,110], where the Higgs is a composite pseudo-Goldstone boson ATLAS analysis [85] combines leptonic and hadronic final states arising from the spontaneous breaking G5 → HIR in the CFT. from the KK graviton decay G∗ → VV , where V can represent The AdS/CFT dictionary tells us that KK states must behave either a W or Z boson, exclude KK gravitons with masses below as composite resonances. For example, if the SM gauge bosons 2.3 TeV, for a value of k/MP = 1. CMS VV analysis [93] also propagate in the 5D bulk, the lowest KK SU(2)L-gauge boson provides cross section limits in the context of bulk gravitons; how- must have properties similar to those of the Techni-rho ρT√ [98] ever, a maximum value of k/MP = 0.5 is presented, for which no with a coupling to longitudinal W/Z bosons given by g5 k ≈ −1 mass exclusion is possible using the 36 fb of 13 TeV data. 2 p 2 gρ , while the coupling to elementary fermions is g / g k ≈ The lightest KK states are, in certain models, the partners of T 5 g2F /M . the top quark. For example, in 5D composite Higgs models these ρT ρT are colored states with charges Q = −1/3, 2/3 and 5/3 (arising Fermions in compactified AdS5 also have a simple 4D holo- graphic interpretation. The 4D massless mode described in from SU(2)L doublets with Y = 7/6, 1/6), and masses expected to Sec. 86.3.1 corresponds to an external fermion ψi linearly cou- be below the TeV [80]. They can be either singly or pair-produced, ¯ and mainly decay into a combination of W/Z with top/bottom pled to a fermionic CFT operator Oi: Lint = λiψiOi + h.c.. The quarks [94–97]. An exhaustive review of these searches can be dimension of the operator Oi is related to the 5D fermion mass found in Ref. [98]. Of particular note, the Q = 5/3 state de- according to Dim[Oi] = |cf + 1/2| − 1. Therefore, by varying cf cays mainly into W +t → W +W +b, giving a pair of same-sign one varies Dim[Oi], making the coupling λi irrelevant (cf > 1/2), leptons in the final state. An analysis by ATLAS [99] search- marginal (cf = 1/2) or relevant (cf < 1/2). When irrelevant, the ing in the lepton-plus-jets final state for evidence of pair pro- coupling is exponentially suppressed at low energies, and then the duction of the Q = 5/3 state provides a lower mass limit of coupling of ψi to the CFT (and eventually to the composite Higgs) 1.25 TeV. A CMS analysis [100] searching for pair production is very small. When relevant, the coupling grows in the IR and of the Q = 5/3 state using both lepton-plus-jets and same sign become as large as g5 (in units of k), meaning that the fermion is lepton final states excludes masses below 1.3 TeV. Both LHC ex- as strongly coupled as the CFT states [76]. In this latter case ψi periments have searched for pair production of vector-like quarks behaves as a composite fermion. T and B of charges Q = 2/3 and −1/3 respectively, assuming the 86.3.3 Flat Extra Dimensions allowable decays are T → W b/Zt/Ht and B → W t/Zb/Hb. In Models with quantum gravity at the TeV scale, as in the ADD each case, it is assumed the branching fractions of the three decay scenario, can have extra (flat) dimensions of 1/TeV size, as hap- modes sum to unity, but the individual branching fractions, which pens in string scenarios (see, for example, [111]). All SM fields are model-dependent, are allowed to vary within this constraint. may propagate in these extra dimensions, leading to the possibil- Both ATLAS [101] and CMS [102–104] obtain lower limits on the ity of observing their corresponding KK states. mass of the T and B vector-like quarksup to 1.3 TeV. A simple example is to assume that the SM gauge bosons prop- Recent analyses from ATLAS [105] and CMS [106] also search agate in a flat five-dimensional orbifold S1/Z of radius R, with for a single top partner production, the cross section for which is 2 the fermions localized on a 4D boundary. The KK gauge bosons model-dependent [107] but does not carry the kinematic penalty behave as sequential SM gauge bosons with a coupling to fermions for producing two heavy objects. √ enhanced by a factor 2 [111]. The experimental limits on such 86.3.2 Connection with Strongly Coupled Models via the sequential gauge bosons could therefore be recast as limits on KK AdS/CFT Correspondence gauge bosons. Such an interpretation of the ATLAS 7 TeV dilep- The AdS/CFT correspondence [8] provides a connection be- ton analysis [112] yielded the bound 1/R > 4.16 TeV, while a tween warped extra-dimensional models and strongly-coupled the- CMS 8 TeV search with a lepton and missing transverse energy ories in ordinary 4D. Although the exact connection is only known in the final state [113] give 1/R > 3.4 TeV. Indirect bounds from for certain cases, the AdS/CFT techniques have been very useful LEP2 require however 1/R ∼> 6 TeV [81, 114], a bound that can to obtain, at the qualitative level, a 4D holographic description of considerable improve in the future by high-energy measurements the various phenomena in warped extra-dimensional models [11]. of the dilepton invariant mass spectrum from Drell-Yan processes The connection goes as follows. The physics of the bulk AdS5 at the LHC [115]. More recent LHC limits on leptonically decay- models can be interpreted as that of a 4D conformal field the- ing gauge bosons [42, 82, 116, 117] are not interpreted as bounds ory (CFT) which is strongly coupled. The extra-dimensional co- on 1/R by the collaborations, but the published results allow for ordinate y plays the role of the renormalization scale µ of the independent derivation of such bound. CFT by means of the identification µ ≡ ke−ky. Therefore the An alternative scenario, known as Universal Extra Dimensions 894 86. Extra Dimensions

(UED) [118] (for a review see, for example, [119]), assumes that [22] J. Murata and S. Tanaka, Class. Quant. Grav. 32, 3, 033001 1 all SM fields propagate universally in a flat orbifold S /Z2 with (2015), [arXiv:1408.3588]. an extra Z2 parity, called KK-parity, that interchanges the two [23] W.-H. Tan et al., Phys. Rev. Lett. 116, 13, 131101 (2016). boundaries. In this case, the lowest KK state is stable and is a Dark Matter candidate. At colliders, the KK particles would have [24] C. Hanhart et al., Phys. Lett. B509, 1 (2001), [arXiv:astro- to be created in pairs, and would then cascade decay to the lightest ph/0102063]. KK particle, which would be stable and escape detection. The [25] S. Hannestad and G. G. Raffelt, Phys. Rev. D67, 125008 UED mass-spectrum depends not only on the extra-dimensional (2003), [Erratum: Phys. Rev.D69,029901(2004)], [hep- radius R, but also on the cut-off of the 5D theory Λ, since quantum ph/0304029]. corrections sensitive to ΛR induce mass-splittings between the KK [26] L. J. Hall and D. Tucker-Smith, Phys. Rev. D60, 085008 states. Experimental signatures, such as jets or leptons and E6 T , (1999), [hep-ph/9904267]. would be similar to those of typical R-parity conserving SUSY searches. An interpretation of the recent LHC experimental SUSY [27] E. A. Mirabelli, M. Perelstein and M. E. Peskin, Phys. Rev. searches for UED models has been presented in Refs. [120,121]. A Lett. 82, 2236 (1999), [hep-ph/9811337]. lower bound 1/R > 1.4−1.5 TeV was derived for ΛR ∼ 5−35 [120]. [28] T. Han, J. D. Lykken and R.-J. Zhang, Phys. Rev. D59, Finally, realistic models of electroweak symmetry breaking can 105006 (1999), [hep-ph/9811350]. also be constructed with flat extra spatial dimensions, similarly [29] A. M. Sirunyan et al. (CMS), Phys. Rev. D97, 9, 092005 to those in the warped case, requiring, however, the presence of (2018), [arXiv:1712.02345]. sizeable boundary kinetic terms [122]. There is also the possibility of breaking supersymmetry by boundary conditions [123]. Models [30] M. Aaboud et al. (ATLAS), JHEP 01, 126 (2018), of this type could explain naturally the presence of a Higgs boson [arXiv:1711.03301]. lighter than MD ∼ TeV (see, for example, [124–126]). [31] A. M. Sirunyan et al. (CMS), JHEP 02, 074 (2019), References [arXiv:1810.00196]. [1] For a comprehensive collection of the original papers see, [32] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 6, 393 “Modern Kaluza-Klein Theories”, edited by T. Appelquist (2017), [arXiv:1704.03848]. et al., Addison-Wesley (1987). [33] S. Cullen, M. Perelstein and M. E. Peskin, Phys. Rev. D62, [2] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. 055012 (2000), [hep-ph/0001166]. B429, 263 (1998), [hep-ph/9803315]. [34] A. M. Sirunyan et al. (CMS), JHEP 08, 130 (2018), [3] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 [arXiv:1806.00843]. (1999), [hep-ph/9905221]. [35] M. Aaboud et al. (ATLAS), Phys. Rev. D96, 5, 052004 [4] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Lett. (2017), [arXiv:1703.09127]. B473, 43 (2000), [hep-ph/9911262]. [36] J. L. Hewett, Phys. Rev. Lett. 82, 4765 (1999), [hep- [5] A. Pomarol, Phys. Lett. B486, 153 (2000), [hep- ph/9811356]. ph/9911294]. [37] G. F. Giudice and A. Strumia, Nucl. Phys. B663, 377 [6] S. Chang et al., Phys. Rev. D62, 084025 (2000), [hep- (2003), [hep-ph/0301232]. ph/9912498]. [38] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C78, 9, 789 [7] T. Gherghetta and A. Pomarol, Nucl. Phys. B586, 141 (2018), [arXiv:1803.08030]. (2000), [hep-ph/0003129]. [39] M. Aaboud et al. (ATLAS), Phys. Lett. B775, 105 (2017), [8] J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999), [Adv. [arXiv:1707.04147]. Theor. Math. Phys.2,231(1998)], [hep-th/9711200]. [40] A. M. Sirunyan et al. (CMS), Phys. Rev. D98, 9, 092001 [9] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), [hep- (2018), [arXiv:1809.00327]. th/9802150]. [41] A. M. Sirunyan et al. (CMS), JHEP 04, 114 (2019), [10] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. [arXiv:1812.10443]. Lett. B428, 105 (1998), [hep-th/9802109]. [42] G. Aad et al. (ATLAS), Phys. Lett. B796, 68 (2019), [11] N. Arkani-Hamed, M. Porrati and L. Randall, JHEP 08, [arXiv:1903.06248]. 017 (2001), [hep-th/0012148]. [43] S. B. Giddings and S. D. Thomas, Phys. Rev. D65, 056010 [12] ATLAS public results are available on WWW at (2002), [hep-ph/0106219]. https://twiki.cern.ch/twiki/bin/view/AtlasPublic. [44] S. Dimopoulos and G. L. Landsberg, Phys. Rev. Lett. 87, [13] CMS public results are available on WWW at 161602 (2001), [hep-ph/0106295]. https://cms-results.web.cern.ch/cms-results/public- results/publications. [45] P. Kanti, Int. J. Mod. Phys. A19, 4899 (2004), [hep- ph/0402168]. [14] Z. Chacko, M. A. Luty and E. Ponton, JHEP 07, 036 (2000), [hep-ph/9909248]. [46] R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986). [15] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. [47] A. M. Sirunyan et al. (CMS), JHEP 11, 042 (2018), Rev. D59, 086004 (1999), [hep-ph/9807344]. [arXiv:1805.06013]. [16] R. Rattazzi, hep-ph/0607055 (2006); I. Antoniadis, Yellow [48] G. Aad et al. (ATLAS), JHEP 03, 026 (2016), report CERN-2002-002 (2002). [arXiv:1512.02586]. [17] J. D. Lykken, Phys. Rev. D54, R3693 (1996), [hep- [49] M. Aaboud et al. (ATLAS), Phys. Lett. B760, 520 (2016), th/9603133]. [arXiv:1606.02265]. [18] I. Antoniadis et al., Phys. Lett. B436, 257 (1998), [hep- [50] C. Kilic et al., Phys. Rev. D89, 1, 016003 (2014), ph/9804398]. [arXiv:1207.3525]. [19] G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. [51] V. Khachatryan et al. (CMS), Phys. Rev. D90, 3, 032005 B544, 3 (1999), [hep-ph/9811291]. (2014), [arXiv:1405.7653]. [20] N. Arkani-Hamed et al., Phys. Rev. D62, 105002 (2000), [52] P. Meade and L. Randall, JHEP 05, 003 (2008), [hep-ph/9912453]. [arXiv:0708.3017]. [21] E. G. Adelberger et al., Prog. Part. Nucl. Phys. 62, 102 [53] A. M. Sirunyan et al. (CMS), Phys. Lett. B774, 279 (2017), (2009). [arXiv:1705.01403]. 86. Extra Dimensions 895

[54] A. M. Sirunyan et al. (CMS), JHEP 04, 073 (2018), [88] A. M. Sirunyan et al. (CMS), JHEP 04, 031 (2019), [arXiv:1802.01122]. [arXiv:1810.05905]. [55] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 9, 092008 [89] M. Aaboud et al. (ATLAS), Phys. Rev. D99, 9, 092004 (2018), [arXiv:1807.06573]. (2019), [arXiv:1902.10077]. [56] S. Dimopoulos and R. Emparan, Phys. Lett. B526, 393 [90] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 7, 565 (2002), [hep-ph/0108060]. (2018), [arXiv:1804.10823]. [57] W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83, [91] M. Aaboud et al. (ATLAS), Phys. Lett. B774, 494 (2017), 4922 (1999), [hep-ph/9907447]. [arXiv:1707.06958]. [58] J. Garriga and A. Pomarol, Phys. Lett. B560, 91 (2003), [92] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C77, 9, 636 [hep-th/0212227]. (2017), [arXiv:1707.01303]. [59] See talk by R. Rattazzi at Planck 2010, CERN. [93] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1906.00057]. [60] B. Bellazzini et al., Eur. Phys. J. C74, 2790 (2014), [94] R. Contino and G. Servant, JHEP 06, 026 (2008), [arXiv:1305.3919]. [arXiv:0801.1679]. [61] F. Coradeschi et al., JHEP 11, 057 (2013), [95] J. A. Aguilar-Saavedra, JHEP 11, 030 (2009), [arXiv:1306.4601]. [arXiv:0907.3155]. [62] E. Megias and O. Pujolas, JHEP 08, 081 (2014), [96] J. Mrazek and A. Wulzer, Phys. Rev. D81, 075006 (2010), [arXiv:1401.4998]. [arXiv:0909.3977]. [63] K. Blum et al., JHEP 03, 099 (2015), [arXiv:1410.1873]. [97] G. Dissertori et al., JHEP 09, 019 (2010), [arXiv:1005.4414]. [64] A. Efrati et al., Phys. Rev. D91, 5, 055034 (2015), [98] See, for example, PDG review of Technicolor searches in this [arXiv:1410.2225]. volume. [65] F. Abu-Ajamieh, J. S. Lee and J. Terning, JHEP 10, 050 [99] M. Aaboud et al. (ATLAS), JHEP 10, 141 (2017), (2018), [arXiv:1711.02697]. [arXiv:1707.03347]. [66] E. P. H. Davoudiasl, S. Gopalakrishna and J. Santiago, New [100] A. M. Sirunyan et al. (CMS), JHEP 03, 082 (2019), J. Phys. 12, 075011 (2010), [arXiv:0908.1968]. [arXiv:1810.03188]. [67] T. Gherghetta, in “Physics of the large and the small, TASI [101] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 121, 21, 09, proceedings of the Theoretical Advanced Study Institute 211801 (2018), [arXiv:1808.02343]. in Elementary , Boulder, Colorado, USA, 1- [102] A. M. Sirunyan et al. (CMS), Phys. Lett. B779, 82 (2018), 26 June 2009,” 165–232 (2011), [arXiv:1008.2570]. [arXiv:1710.01539]. [68] Y. Grossman and M. Neubert, Phys. Lett. B474, 361 [103] A. M. Sirunyan et al. (CMS), JHEP 08, 177 (2018), (2000), [hep-ph/9912408]. [arXiv:1805.04758]. [69] S. J. Huber and Q. Shafi, Phys. Lett. B498, 256 (2001), [104] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 4, 364 [hep-ph/0010195]. (2019), [arXiv:1812.09768]. [70] A. Delgado, A. Pomarol and M. Quiros, JHEP 01, 030 [105] M. Aaboud et al. (ATLAS), JHEP 05, 164 (2019), (2000), [hep-ph/9911252]. [arXiv:1812.07343]. [71] K. Agashe, G. Perez and A. Soni, Phys. Rev. D71, 016002 [106] A. M. Sirunyan et al. (CMS), Phys. Lett. B781, 574 (2018), (2005), [hep-ph/0408134]. [arXiv:1708.01062]. [72] M. Bauer et al., JHEP 09, 017 (2010), [arXiv:0912.1625]. [107] A. De Simone et al., JHEP 04, 004 (2013), [arXiv:1211.5663]. [73] A. Pomarol, Int. J. Mod. Phys. A24, 61 (2009), [ In Kane, Gordon (ed.) et al.: Perspectives on LHC physics [108] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B671, ,259(2008)]. 148 (2003), [hep-ph/0306259]. [74] Y. Hosotani, Phys. Lett. 126B, 309 (1983). [109] H. Georgi, D. B. Kaplan and P. Galison, Phys. Lett. 143B, 152 (1984). [75] K. Agashe et al., JHEP 08, 050 (2003), [hep-ph/0308036]. [110] D. B. Kaplan and H. Georgi, Phys. Lett. 136B, 183 (1984). [76] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B719, 165 (2005), [hep-ph/0412089]. [111] I. Antoniadis and K. Benakli, Int. J. Mod. Phys. A15, 4237 (2000), [hep-ph/0007226]. [77] For a review see, for example, R. Contino, arXiv:1005.4269. [112] G. Aad et al. (ATLAS), JHEP 11, 138 (2012), [78] See, for example, PDG review of Higgs boson in this Review. [arXiv:1209.2535]. [79] G. F. Giudice et al., JHEP 06, 045 (2007), [hep- [113] V. Khachatryan et al. (CMS), Phys. Rev. D91, 9, 092005 ph/0703164]. (2015), [arXiv:1408.2745]. [80] R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D75, [114] K. Cheung and G. L. Landsberg, Phys. Rev. D65, 076003 055014 (2007), [hep-ph/0612048]. (2002), [hep-ph/0110346]. [81] R. Barbieri et al., Nucl. Phys. B703, 127 (2004), [hep- [115] M. Farina et al., Phys. Lett. B772, 210 (2017), ph/0405040]. [arXiv:1609.08157]. [82] A. M. Sirunyan et al. (CMS), JHEP 06, 120 (2018), [116] G. Aad et al. (ATLAS) (2019), [arXiv:1906.05609]. [arXiv:1803.06292]. [117] A. M. Sirunyan et al. (CMS), JHEP 06, 128 (2018), [83] A. M. Sirunyan et al. (CMS), Phys. Rev. Lett. 122, 12, [arXiv:1803.11133]. 121803 (2019), [arXiv:1811.09689]. [118] T. Appelquist, H.-C. Cheng and B. A. Dobrescu, Phys. Rev. [84] M. Aaboud et al. (ATLAS), JHEP 01, 030 (2019), D64, 035002 (2001), [hep-ph/0012100]. [arXiv:1804.06174]. [119] A. Datta, K. Kong and K. T. Matchev, New J. Phys. 12, [85] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 5, 052008 075017 (2010), [arXiv:1002.4624]. (2018), [arXiv:1808.02380]. [120] N. Deutschmann, T. Flacke and J. S. Kim, Phys. Lett. [86] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1906.05977]. B771, 515 (2017), [arXiv:1702.00410]. [87] K. Agashe et al., Phys. Rev. D76, 036006 (2007), [hep- [121] J. Beuria et al., Comput. Phys. Commun. 226, 187 (2018), ph/0701186]. [arXiv:1702.00413]. 896 86. Extra Dimensions

[122] G. Panico, M. Safari and M. Serone, JHEP 02, 103 (2011), [hep-ph/9806263]. [arXiv:1012.2875]. [125] I. Antoniadis et al., Nucl. Phys. B544, 503 (1999), [hep- ph/9810410]. [123] J. Scherk and J. H. Schwarz, Phys. Lett. 82B, 60 (1979). [126] R. Barbieri, L. J. Hall and Y. Nomura, Phys. Rev. D63, 105007 (2001), [hep-ph/0011311]. [124] A. Pomarol and M. Quiros, Phys. Lett. B438, 255 (1998), 0 87. W -Boson Searches 897

87. W 0 -Boson Searches

Revised November, 2019 by B.A. Dobrescu (Fermilab) and S. 10 ATLAS Willocq (U. Massachusetts). ) [pb] Expected limit ν •1

e s = 13 TeV, 139 fb 0 1 The W boson is a massive hypothetical particle of spin 1 and → W’ → eν Expected ± 1σ W’ electric charge ±1, which is a color singlet and is predicted in 95% CL ± σ → Expected 2 various extensions of the (SM). 10−1 (pp Observed limit 87.1 W 0 couplings to quarks and leptons σ 10−2 SSM The Lagrangian terms describing the couplings of a W 0+ boson to fermions are given by 10−3 0+ Wµ h   i √ u CR P +CL P γµd +ν CR P +CL P γµe . i qij R qij L j i `ij R `ij L j 2 10−4 (87.1) Here, u, d, ν, and e are the SM fermions in the mass eigenstate 1 2 3 4 5 6 7 basis, i, j = 1, 2, 3 label the fermion generation, and PR,L = (1 ± m(W’) [TeV] L R L R γ5)/2. The coefficients C , C , C , and C are complex qij qij `ij `ij 0 0 dimensionless parameters. If CR 6= 0, then the ith generation Figure 87.1: Upper limit on σ(pp →W X) B(W →eν) from AT- `ij LAS [12]. The red line shows the theoretical prediction in the includes a right-handed . Using this notation, the SM W L L R R Sequential SM. couplings are Cq = gVCKM, C` = g ≈ 0.63 and Cq = C` = 0. Unitarity considerations imply that the W 0 boson is associated with a spontaneously-broken gauge symmetry. This is true even ± as doublets under different SU(2) gauge groups [10]. In particular, when it is a composite particle (e.g. ρ -like bound states [1]) the couplings to third generation quarks may be enhanced [11]. if its mass is much smaller than the compositeness scale, or a It is also possible that the W 0 couplings to SM fermions are Kaluza-Klein mode in theories where the W boson propagates in highly suppressed. For example, if the quarks and leptons are extra dimensions [2]. The simplest extension of the electroweak 0 singlets under one SU(2) [13], then the couplings are proportional gauge group that includes a W boson is SU(2)1 ×SU(2)2 ×U(1), to the tiny mixing angle θ+. Similar suppressions may arise if but larger groups are encountered in some theories. A generic some vectorlike fermions mix with the SM fermions [14]. property of these gauge theories is that they also include a Z0 0 0 Gauge groups that embed the electroweak symmetry, such as boson [3]; the W -to-Z mass ratio is often a free parameter. 0 SU(3)W × U(1) or SU(4)W × U(1), also include one or more W A tree-level mass mixing may be induced between the bosons [15]. electrically-charged gauge bosons. Upon diagonalization of their mass matrix, the W -to-Z mass ratio and the couplings of the 87.2 Collider searches observed W boson are shifted from the SM values. Their mea- At LEP-II, W 0 bosons could have been produced in pairs via surements imply that the mixing angle, θ+ , between the gauge their and Z couplings. The production cross section is −2 √ eigenstates must be smaller than about 10 . In certain theories large enough to rule out MW 0 < s/2 ≈ 105 GeV for most pat- the mixing is negligible (e.g., due to a new parity [4]), even when terns of decay modes. the W 0 mass is near the electroweak scale. Note that SU(2) gauge At hadron colliders, W 0 bosons can be detected through reso- invariance suppresses the kinetic mixing between the W and W 0 nant pair production of fermions (f and f 0) or electroweak bosons bosons (in contrast to the case of a Z0 boson [3]). with a net electric charge equal to ±1. When W 0 has a width much 0 The W coupling to WZ is fixed by Lorentz and gauge invari- smaller than its mass (MW 0 /ΓW 0 . 7%), the contribution of the 0 ¯0 ances, and to leading order in θ+ is given by [5] s-channel W exchange to the total rate for pp → ff X, where X is any final state, may be approximated by the branching fraction 0 0 g θ+ i  0+ − νµ −µν  ν −µ 0+ B(W → ff¯ ) times the production cross section Wµ Wν Z + Zν W + Z W Wνµ + H.c., cos θW (87.2) 0  π X L 2 R 2 2  σ pp→W X ' (C ) +(C ) w M /s, M 0 . µν µ ν ν µ qij qij ij W 0 W where W ≡ ∂ W − ∂ W , etc. The θW dependence shown 6s here corrects the one given in Ref. [6], which has been referred to i,j (87.4) as the Extended Gauge Model by the experimental collaborations. The functions w include the information about structure, The W 0 coupling to W h0, where h0 is the SM Higgs boson, is ij and are given to leading order in αs by 0+ µ− 0 − ξh g 0 MW Wµ W h + H.c., (87.3) 1 W Z dx h  z   z i 0 wij (z, µ) = ui(x, µ) dj , µ + ui(x, µ) dj , µ , where g 0 is the gauge coupling of the W boson, and the coeffi- x x x W z cient ξh satisfies ξh ≤ 1 in simple Higgs sectors [5]. (87.5) In models based on the “left-right symmetric” gauge group [7], where ui(x, µ) and di(x, µ) are the parton distributions inside SU(2)L × SU(2)R × U(1)B−L, the SM fermions that couple to the proton at the factorization scale µ and parton momentum the W boson transform as doublets under SU(2)L while the other fraction x for the up- and down-type quarks of the ith generation, fermions transform as doublets under SU(2)R. Consequently, the respectively. QCD corrections to W 0 production are sizable (they W 0 boson couples primarily to right-handed fermions; its coupling L also include quark-gluon initial states), but preserve the above to left-handed fermions arises due to the θ+ mixing, so that Cq factorization of couplings at next-to-leading order [16]. is proportional to the CKM matrix and its elements are much 0 R R The most commonly studied W signal consists of a high- smaller than the diagonal elements of Cq . Generically, Cq does momentum or and large missing transverse mo- not need to be proportional to VCKM. mentum. The signal transverse mass distribution forms a Jaco- There are many other models based on the SU(2)1 × SU(2)2 × bian peak with its endpoint at MW 0 (see Fig. 1 (top) of Ref. [12]). U(1) gauge symmetry. In the “alternate left-right” model [8], Given that the branching fractions for W 0 → eν and W 0 → µν all the couplings shown in Eq. (87.1) vanish, but there are some 0 could be very different, the results in these channels should be new fermions such that the W boson couples to pairs involving presented separately. Searches in these channels often implic- a SM fermion and a new fermion. In the “ununified SM” [9], itly assume that the left-handed couplings vanish (no interfer- the left-handed quarks are doublets under one SU(2), and the ence between W and W 0), and that the right-handed neutrino is left-handed leptons are doublets under a different SU(2), leading light compared to the W 0 boson and escapes the detector. An to a mostly leptophobic W 0 boson: CL  CL and CR = `ij qij `ij example of parameter values that satisfy these assumptions is CR = 0. Fermions of different generations may also transform CR = gV , CR = g, CL = CL = 0, which define a model that qij q CKM ` q ` 0 898 87. W -Boson Searches

W' → Wh → lν bb 35.9 fb-1 (13 TeV) 10 95% CL limit ATLAS 2000 CMS 95% CL upper limits ) [pb]

c Observed (CLs) Observed -1 1000 1l categories ,c s = 13 TeV, 36.1 fb bb) (fb) b 1 Expected (CLs) Expected → b ± ± σ 1 std. deviation → Expected 1 (h 200 ± 2 std. deviation ± σ Β

B(h − Expected 2 100 ⋅ 10 1 HVT model A HVT Model A, g =1 HVT model B

V Wh) Wh) HVT Model B, g =3 20 → → V −2 10 W’ 10 (W' → Β

(pp 2 σ −3 10 (W') 1 σ

0.2 −4 10 0.1 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1000 1500 2000 2500 3000 3500 4000 4500 mW’ [GeV] mW' (GeV)

Figure 87.2: Upper limits on W 0 production cross section times branching fraction into a W and a SM Higgs boson decaying into heavy-flavor quarks, from ATLAS [17] (left) and CMS [18] (right).

preserves lepton universality and predicts the same total cross sec- the fully hadronic decay channel for MW 0  mt with the use of tion as the Sequential SM used in many W 0 searches. However, jet substructure techniques to tag highly boosted top-jets. For a if a W 0 boson were discovered and the final state fermions have detailed discussion of this channel, see Ref. [30]. left-handed helicity, then the effects of W − W 0 interference could Searches for dijet resonances may be used to set limits on be observed [19], providing information about the W 0 couplings. W 0 → qq¯0. ATLAS [31] and CMS [32] provide similar coverage in The effects of the W 0 width on interference are discussed in [20]. −1 the ∼ 1.5 − 8.0 TeV mass√ range with 139 and 137 fb of data, In the eν channel, the ATLAS and CMS collaborations set lim- respectively, collected at s = 13 TeV. Interpretation in terms of its on the W 0 production cross section times branching fraction W 0 decays with 139 fb−1 of 13 TeV data yields a W 0 mass lower (and thus indirectly on the W 0 couplings). These limits are set for limit of 4.0 TeV in the Sequential SM [31]. For masses in the −1 M√ W 0 in the 0.15 − 7 TeV range and are based on 36–139 fb at range ∼ 0.5 − 1.5 TeV, analyses based on jets reconstructed on- s = 13 TeV [12,21], as shown in Fig. 87.1 for the most stringent line provide the best sensitivity because they circumvent trigger 0 limits. ATLAS sets the strongest mass limit MW 0 > 6.0 TeV in bandwidth limitations [33, 34]. For W masses below ∼ 0.5 TeV, the Sequential SM (all limits in this mini-review are at the 95% the best limits are set in novel analyses exploiting boosted tech- CL). The coupling limits are much weaker√ for MW 0 < 150 GeV, nologies and initial state radiation [35–38]. Cross-section limits a range last explored with the Tevatron at s = 1.8 TeV [22]. for W 0 masses below ∼ 1.5 TeV can be derived from the dijet 0 In the µν channel, ATLAS and CMS set rate limits for MW 0 in limits on Z bosons summarized in Ref. [3]. the 0.15−7 TeV range from the same analyses as mentioned above, In some theories [4] the W 0 couplings to SM fermions are sup- 0 with the strongest mass lower limit of 5.1√ TeV in the Sequential pressed by discrete symmetries. W production then occurs in SM set by ATLAS [12] using 139 fb−1 of s = 13 TeV data. When pairs, through a photon or Z boson. The decay modes are model- combined with the eν√ channel assuming lepton universality, the dependent and often involve other new particles. The ensuing upper limit on the s = 13 TeV cross section times branching collider signals arise from cascade decays and typically include fraction to `ν varies between 0.05 and 2.1 fb for MW 0 values in missing transverse momentum. 0 the range between 1 and 6 TeV [12]. Only weak limits on W → µν Searches for WZ resonances at the LHC have focused on the exist for MW 0 < 150 GeV [23]. Note that masses of the order of process pp → W 0 → WZ with the production mainly from ud¯ → the electroweak scale are interesting from a theory point of view, W 0 assuming SM-like couplings to quarks. ATLAS and CMS have while lepton universality does not necessarily apply to a W 0 boson. 0 set the upper limits on the W WZ coupling for MW 0 in the 0.2 − Dedicated searches for W 0 → τν have been performed by CMS 5.0 TeV range with a combination of fully leptonic, semi-leptonic at 8 TeV [24] and both ATLAS and CMS with 36 fb−1 at 13 and fully hadronic channels with ∼ 36 fb−1 at 13 TeV [39,40] (see 0 TeV [25, 26]. Limits are set on σ · B for MW 0 between 0.4 and 4 also Ref. [30]). The strongest lower limits on the W mass are TeV for the former and between 0.4 and 5.6 TeV for the latter. A set by ATLAS [41] and CMS [42] at 13 TeV with 139 fb−1 and mass lower limit of 4.0 TeV is set in the Sequential SM and the 77 fb−1, respectively, in the WZ → (jj)(jj) final state, where upper limit on the cross section times branching fraction to τν at the parentheses represent a resonance. The lower limit on MW 0 is 13 TeV varies between 1.7 and 12 fb for MW 0 values in the range 3.4 TeV in the context of the Heavy Vector Triplet (HVT) weakly- between 1 and 5 TeV [26]. coupled scenario A [43]. A fermiophobic W 0 boson that couples The W 0 decay into a charged lepton and a right-handed neu- to WZ may be produced at hadron colliders in association with a trino, νR, may also be followed by the νR decay through a virtual Z boson, or via WZ fusion. This would give rise to (WZ)Z and W 0 boson into a charged lepton and two quark jets. The CMS [27] (WZ)jj final states [44]. and ATLAS [28] searches in the eejj and µµjj channels have set W 0 bosons have also been searched for in final states with a W limits on the cross section times branching fraction as a function boson and a SM Higgs boson in the channels W → `ν or W → qq¯0 of the ν mass or of M 0 . No requirement is placed on the charge 0 ¯ −1 R W and√h → bb by ATLAS [17, 45] and CMS [18, 46] with 36 fb of the lepton pair. A related W 0 search in the ττjj channel with at s = 13 TeV. Cross-section limits are set for W 0 masses in hadronic τ decays was also performed by CMS [29]. the range between 0.5 and 5.0 TeV. The ATLAS and CMS 13 The t¯b channel is particularly important because a W 0 boson TeV analyses both set the most stringent lower limit on the mass: that couples only to right-handed fermions cannot decay to lep- MW 0 > 2.7 TeV for the HVT weakly-coupled scenario A, as shown tons when the right-handed are heavier than MW 0 . Ad- in Fig. 87.2. ditional motivations are provided by a W 0 boson with enhanced couplings to the third generation [11], and by a leptophobic W 0 87.3 Low-energy constraints boson. The usual signature consists of a leptonically-decaying The properties of W 0 bosons are also constrained by measure- W boson and two b-jets. Recent studies have also incorporated ments of processes at energies much below MW 0 . The bounds on 0 87. W -Boson Searches 899

W −W 0 mixing [47] are mostly due to the change in W properties [18] A. M. Sirunyan et al. (CMS), JHEP 11, 172 (2018), compared to the SM. Limits on deviations in the ZWW couplings [arXiv:1807.02826]. 0 provide a leading constraint for fermiophobic W bosons [14]. [19] T. G. Rizzo, JHEP 05, 037 (2007), [arXiv:0704.0235]. Constraints arising from low-energy effects of W 0 exchange are [20] E. Accomando et al., Phys. Rev. D85, 115017 (2012), strongly model-dependent. If the W 0 couplings to quarks are not [arXiv:1110.0713]. suppressed, then box diagrams involving a W and a W 0 boson contribute to neutral -mixing. In the case of W 0 couplings [21] A. M. Sirunyan et al. (CMS), JHEP 06, 128 (2018), to right-handed quarks as in the left-right symmetric model, the [arXiv:1803.11133]. R limit from KL − KS mixing is severe: MW 0 > 2.9 TeV for Cq = [22] F. Abe et al. (CDF), Phys. Rev. Lett. 74, 2900 (1995). gV [48]. However, if no correlation between the W 0 and W CKM [23] F. Abe et al. (CDF), Phys. Rev. Lett. 67, 2609 (1991). couplings is assumed, then the limit on MW 0 may be significantly relaxed [49]. [24] V. Khachatryan et al. (CMS), Phys. Lett. B755, 196 (2016), W 0 exchange also contributes at tree level to various low-energy [arXiv:1508.04308]. processes. In particular, it would impact the measurement of the [25] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 120, 161802 Fermi constant GF in muon decay, which in turn would change (2018), [arXiv:1801.06992]. the predictions of many other electroweak processes. A recent test [26] A. M. Sirunyan et al. (CMS), Phys. Lett. B792, 107 (2019), of parity violation in polarized muon decay [50] has set limits of 0 [arXiv:1807.11421]. about 600 GeV on M 0 , assuming W couplings to right-handed W [27] A. M. Sirunyan et al. (CMS), JHEP 05, 148 (2018), leptons as in left-right symmetric models and a light νR. There are also W 0 contributions to the neutron electric dipole moment, [arXiv:1803.11116]. β decays, and other processes [47]. [28] M. Aaboud et al. (ATLAS), Submitted to: Phys. Lett. If right-handed neutrinos have Majorana masses, then there are (2019), [arXiv:1904.12679]. tree-level contributions to neutrinoless double-beta decay, and a [29] A. M. Sirunyan et al. (CMS), JHEP 03, 170 (2019), limit on MW 0 versus the νR mass may be derived [51]. For νR [arXiv:1811.00806]. masses below a few GeV, the W 0 boson contributes to leptonic [30] K.M. Black et al., “Dynamical electroweak symmetry break- and semileptonic decays, so that limits may be placed 0 ing” in this Review. on various combinations of W parameters [49]. For νR masses below ∼30 MeV, the most stringent constraints on MW 0 are due [31] G. Aad et al. (ATLAS) (2019), [arXiv:1910.08447]. to the limits on νR emission from supernovae. [32] CMS Collab., CMS PAS EXO-19-012, Jul. 2019. References [33] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 121, 081801 [1] M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (2018), [arXiv:1804.03496]. (1988). [34] A. M. Sirunyan et al. (CMS), JHEP 08, 130 (2018), [2] H.-C. Cheng et al., Phys. Rev. D64, 065007 (2001), [hep- [arXiv:1806.00843]. th/0104179]. [35] M. Aaboud et al. (ATLAS), Phys. Lett. B788, 316 (2019), [3] See the Section on “Z0-boson searches” in this Review. [arXiv:1801.08769]. [4] H.-C. Cheng and I. Low, JHEP 09, 051 (2003), [hep- [36] M. Aaboud et al. (ATLAS), Phys. Lett. B795, 56 (2019), ph/0308199]. [arXiv:1901.10917]. [37] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1909.04114]. [5] B. A. Dobrescu and Z. Liu, JHEP 10, 118 (2015), [arXiv:1507.01923]. [38] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1905.10331]. [6] G. Altarelli, B. Mele and M. Ruiz-Altaba, Z. Phys. C45, 109 [39] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 052008 (2018), (1989), [Erratum: Z. Phys. C47, 676 (1990)]. [arXiv:1808.02380]. [7] R. N. Mohapatra and J. C. Pati, Phys. Rev. D11, 566 (1975). [40] A. M. Sirunyan et al. (CMS), Phys. Lett. B798, 134952 (2019), [arXiv:1906.00057]. [8] K. S. Babu, X.-G. He and E. Ma, Phys. Rev. D36, 878 [41] G. Aad et al. (ATLAS), JHEP 09, 091 (2019), (1987). [arXiv:1906.08589]. [9] H. Georgi, E. E. Jenkins and E. H. Simmons, Nucl. Phys. [42] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1906.05977]. B331, 541 (1990). [43] D. Pappadopulo et al., JHEP 09, 060 (2014), [10] X.-y. Li and E. Ma, J. Phys. G19, 1265 (1993), [hep- [arXiv:1402.4431]. ph/9208210]. [44] H.-J. He et al., Phys. Rev. D78, 031701 (2008), [11] D. J. Muller and S. Nandi, Phys. Lett. B383, 345 (1996), [arXiv:0708.2588]. [hep-ph/9602390]. [45] M. Aaboud et al. (ATLAS), Phys. Lett. B774, 494 (2017), [12] G. Aad et al. (ATLAS), Phys. Rev. D100, 052013 (2019), [arXiv:1707.06958]. [arXiv:1906.05609]. [46] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C77, 636 (2017), [13] A. Donini et al., Nucl. Phys. B507, 51 (1997), [hep- [arXiv:1707.01303]. ph/9705450]. [47] See the particle listings for W 0 in this Review. [14] R. S. Chivukula et al., Phys. Rev. D74, 075011 (2006), [hep- [48] Y. Zhang et al., Phys. Rev. D76, 091301 (2007), ph/0607124]. [arXiv:0704.1662]. [15] F. Pisano and V. Pleitez, Phys. Rev. D46, 410 (1992), [hep- [49] P. Langacker and S. U. Sankar, Phys. Rev. D40, 1569 (1989). ph/9206242]. [50] J. F. Bueno et al. (TWIST), Phys. Rev. D84, 032005 (2011), [16] Z. Sullivan, Phys. Rev. D66, 075011 (2002), [hep- [arXiv:1104.3632]. ph/0207290]. [51] See Fig. 5 of G. Prezeau, M. Ramsey-Musolf, and P. Vogel, [17] M. Aaboud et al. (ATLAS), JHEP 03, 174 (2018), [Erratum: Phys. Rev. D 68, 034016 (2003). JHEP 11, 051 (2018)], [arXiv:1712.06518]. 0 900 88. Z -Boson Searches

88. Z0 -Boson Searches

Revised October, 2019 by B.A. Dobrescu (Fermilab) and S. the form Willocq (U. Massachusetts).  − 0 − 0  µ +ν  0 i Wµ Zν −Wν Zµ ∂ W + H.c. The Z boson is a massive, electrically-neutral and color-singlet (88.5) + − + − µ 0ν hypothetical particle of spin 1. This particle is predicted in many + i Wµ Wν −Wν Wµ ∂ Z extensions of the Standard Model (SM) and has been the object of extensive phenomenological studies [1]. 2 2 0 with a coefficient of order MW /MZ0 [5]. The Z also couples to 0 0 µ 0 88.1 Z boson couplings one SM Higgs boson and one Z boson, ZµZ h , with a coefficient The couplings of a Z0 boson to the first-generation fermions are of order MZ . given by 88.2 Z0 models 0 L µ L µ R µ R µ Zµ gu uLγ uL + gd dLγ dL + gu uRγ uR + gd dRγ dR A simple origin of a Z0 boson is a new U(1)0 gauge symme- L µ L µ R µ  try. In that case, the matricial equalities zL = zL and zL = zL + g νLγ νL + g eLγ eL + g eRγ eR , u d ν e ν e e are required by the SM SU(2) gauge symmetry. Given that (88.1) W the U(1)0 interaction is not asymptotically free, the theory may be well-behaved at high energies (e.g., by embedding U(1)0 in a where u, d, ν, e are the quark and lepton fields in the mass eigen- non-Abelian gauge group) only if the charges are commensurate state basis, and the coefficients gL, gL, gR, gR, gL, gL, gR are u d u d ν e e numbers, i.e. any ratio of charges is a rational number. Satisfying real dimensionless parameters. If the Z0 couplings to quarks and the anomaly equations with rational numbers is highly nontrivial, leptons are generation-independent, then these seven parameters 0 0 and typically new fermions charged under U(1) are necessary. describe the couplings of the Z boson to all SM fermions. More L,R 0 generally, however, the Z couplings to fermions are generation- If the couplings are generation-independent (Vf are then unit dependent, in which case Eq. (88.1) may be written with genera- matrices in Eq. (88.2)) and the mixings of Z˜0 are negligible, then tion indices i, j = 1, 2, 3 labeling the quark and lepton fields, and R R R L there are five commensurate couplings: gu , gd , ge , gq (q = u with the seven coefficients promoted to 3 × 3 Hermitian matrices or d), gL (l = ν or e). Four sets of charges are displayed in L i µ j 2 l (e.g., ge ij eLγ eL, where eL is the left-handed muon, etc.). Table 88.1, each of them spanned by a free parameter x [6]. The The parameters describing the Z0 boson interactions with first set, labelled B − xL, has charges proportional to the quarks and leptons are subject to some theoretical constraints. number minus x times the lepton number. These charges allow all Quantum field theories that include a heavy spin-1 particle are SM Yukawa couplings to a Higgs doublet which is neutral under 0 well behaved at high energies only if that particle is a gauge boson U(1)B−xL, so that there is no tree-level Z˜ − Z˜ mixing. For associated with a spontaneously broken gauge symmetry. Quan- x = 1 one recovers the U(1)B−L group, which is non-anomalous tum effects preserve the gauge symmetry only if the couplings of in the presence of one “right-handed neutrino” (a chiral fermion the gauge boson to fermions satisfy the anomaly equations [2]. that is a singlet under the SM gauge group) per generation. For Furthermore, the fermion charges under the new gauge symmetry x 6= 1, it is necessary to include some fermions that are vectorlike are constrained by the requirement that the quarks and leptons (i.e. their mass terms are gauge invariant) with respect to the get masses from gauge-invariant interactions with the Higgs fields. electroweak gauge group and chiral with respect to U(1)B−xL. The relation between the couplings displayed in Eq. (88.1) and In the particular cases x = 0 or x  1, the Z0 is leptophobic or L R the gauge charges zfi and zfi of the fermions f = u, d, ν, e in- quark-phobic, respectively. L R The second set, U(1) ¯, has charges that commute with volves the unitary 3 × 3 matrices Vf and Vf that transform the 10+x5 the representations of the SU(5) grand unified group. Here x gauge eigenstate fermions f i and f i , respectively, into the mass L R is related to the mixing angle between the two U(1) bosons en- eigenstates. The Z0 couplings also depend on the mixings of the countered in the E → SU(5) × U(1) × U(1) symmetry breaking new gauge boson in the gauge eigenstate basis (Z˜0 ). The main 6 µ patterns of grand unified theories [1, 9]. With these charges, two µν ˜0 ones are a kinetic mixing (−χ/2)B Zµν with the hypercharge µ Higgs doublets are typically required to generate masses for both gauge boson B (χ is a dimensionless parameter), and a mass up- and down- type fermions. This set leads to Z˜ − Z˜0 mass mix- 2 ˜µ ˜0 ˜ mixing δM Z Zµ with the linear combination (Zµ) of neutral ing at tree level, such that for a Z0 mass close to the electroweak bosons that couples as the SM Z boson [3]. Since both the ki- scale, the measurements at the Z-pole require some fine tuning netic and mass mixings shift the mass and couplings of the Z between the charges and VEVs of the two Higgs doublets. Vec- boson, electroweak measurements impose upper limits on χ and torlike fermions charged under the electroweak gauge group and δM 2/(M 2 − M 2 ) of the order of 10−3 [4]. Keeping only linear Z0 Z also carrying color are required (except for x = −3) to make this terms in these two small quantities, the couplings of the mass- set anomaly free. The particular cases x = −3, 1, −1/2 are usu- eigenstate Z0 boson are given by ally labelled U(1)χ, U(1)ψ, and U(1)η, respectively. Under the third set, U(1)d−xu, the weak-doublet quarks are neutral, and the ratio of u and d charges is −x. For x = 1, this is the L L L L† R R g = gzV 0 z V f ij fii f i0 f i0j “right-handed” group U(1)R. For x = 0, the charges are those ! of the E6-inspired U(1)I group, which requires new quarks and 2 2 (88.2) 0 e sW χMZ0 + δM 3 leptons. Other generation-independent sets of U(1) charges are + σ −  Qf , 2 2  f given in [10]. cW 2sW M 0 −M Z Z In the absence of new fermions charged under the SM group, the most general generation-independent charge assignment is R R R R† e g = gzV 0 z 0 V −  Q , (88.3) U(1)q+xu, which is a linear combination of hypercharge and B−L. f ij fii fi f 0 f i j cW Many other anomaly-free solutions exist if generation-dependent charges are allowed. An example is B − xLe − yLµ + (y − 3)Lτ , where gz is the new gauge coupling, Qf is the electric charge of with x, y free parameters. This allows all fermion masses to be f, e is the electromagnetic gauge coupling, sW and cW are the sine and cosine of the weak mixing angle, σ3 = +1 for f = u, ν generated by Yukawa couplings to a single Higgs doublet, without f inducing tree-level flavor-changing neutral current (FCNC) pro- and σ3 = −1 for f = d, e, and f cesses. There are also lepton-flavor dependent charges that allow neutrino masses to arise only from operators of high dimensional- 2 2 2  2 χ MZ0 − cW MZ + sW δM ity [11].  = . (88.4) 0 2 2 If the SU(2)W -doublet quarks have generation-dependent U(1) M 0 − MZ Z charges, then the mass eigenstate quarks have flavor off-diagonal 0 0 L L† The interaction of the Z boson with a pair of W bosons has couplings to the Z boson (see Eq. (88.1), and note that Vu Vd 0 88. Z -Boson Searches 901

140 fb-1 (13 TeV, µµ) 10−4 102 ATLAS s = 13 TeV, 139 fb-1 ] Z CMS Obs. 95% CL limit B ⋅ B [fb] → Preliminary σ

× X ee

Exp. 95% CL limit, median −

fid 5

σ 10 10 Exp. (68%) ] Z' / [ B

⋅ Exp. (95%) σ [ −6 1 10 Z'SSM

Z'ψ 10−1 10−7

Observed limit at Γ/m = 6% Expected limit at Γ/m = 6% Z'χ model −2 Γ Γ 10 /m = 1.2% /m = 0.5% Z'ψ model 10−8 3×102 103 2×103 3×103 1000 2000 3000 4000 5000

mX [GeV] M [GeV]

Figure 88.1: Upper limits on the cross section for Z0 production times the branching fraction for Z0 → e+e− (left panel, set by 0 + − 0 0 ATLAS [7]) or Z → µ µ (right panel, set by CMS [8]) as a function of MZ0 . The lines labelled by Zψ and Zχ are theoretical predictions for the U(1)10+x5¯ models in Table 88.1 with x = −3 and x = +1, respectivxoely, for gz fixed by an E6 unification 0 0 condition. The ZSSM line corresponds to Z couplings equal to those of the Z boson.

Table 88.1: Examples of generation- that forces all couplings of Eq. (88.1) to vanish in the case of the 0 independent U(1) charges for quarks and lightest KK bosons, while allowing couplings to pairs of fermions leptons. The parameter x is an arbitrary rational involving a SM and a heavy vectorlike fermion. There are also 4- number. Gauge anomaly cancellation requires dimensional gauge theories (e.g. little Higgs with T parity) with certain new fermions [6]. Z0 bosons exhibiting similar properties. By contrast, in a warped extra dimension, the couplings of Eq. (88.1) may be sizable even fermion U(1)B−xL U(1)10+x5¯ U(1)d−xu U(1)q+xu when SM fields propagate along the extra dimension. 0 (uL, dL) 1/3 1/3 0 1/3 Z bosons may also be composite particles. For example, in confining gauge theories [17], the ρ-like is a spin-1 uR 1/3 −1/3 −x/3 x/3 boson that may be interpreted as arising from a spontaneously dR 1/3 −x/3 1/3 (2 − x)/3 broken gauge symmetry [18]. (νL, eL) −x x/3 (−1 + x)/3 −1

eR −x −1/3 x/3 −(2 + x)/3 88.3 Non-resonant Z0 signatures at colliders In the presence of the couplings shown in Eq. (88.1), the Z0 is the CKM matrix). These are severely constrained by measure- boson may be produced in the s-channel at colliders, and would ments of FCNC processes, which in this case are mediated at tree- decay to pairs of fermions. The decay width into a pair of level by Z0 boson exchange [12]. The constraints are relaxed if the is given by first and second generation charges are the same, although they are increasingly tightened by the measurements of B meson prop- h 2 2i M 0 Γ Z0 → e+e− ' gL + gR Z , (88.6) erties [13]. If only the SU(2)W -singlet quarks have generation- e e 24π dependent U(1)0 charges, there is more freedom in adjusting the R flavor off-diagonal couplings because the Vu,d matrices are not where small corrections from electroweak loops are not included. observable in the SM. The decay width into qq¯ is similar, except for an additional color The anomaly equations for U(1)0 could be circumvented only factor of 3, QCD radiative corrections, and fermion mass cor- if there is an with certain dimension-5 couplings to the rections. Thus, one may compute the Z0 branching fractions in gauge bosons. However, such a scenario violates unitarity unless terms of the couplings of Eq. (88.1). However, other decay chan- the quantum field theory description breaks down at a scale near nels, such as WW or a pair of new particles, could have large MZ0 [14]. widths and need to be added to the total decay width. Z0 bosons may also arise from larger gauge groups. These may As mentioned above, there are theories in which the Z0 cou- extend the electroweak group, as in SU(2)×SU(2)×U(1), or may plings are controlled by a discrete symmetry that forbids decays embed the electroweak group, as in SU(3)W × U(1) [15]. If the into a pair of SM particles. Typically, such theories involve several larger group is spontaneously broken down to SU(2)W × U(1)Y × new particles, which may be produced only in pairs and undergo 0 0 U(1) at a scale v?  MZ0 /gz, then the above discussion applies cascade decays through Z bosons, leading to signals involving 2 2 up to corrections of order MZ0 /(gzv?) . For v? ∼ MZ0 /gz, addi- missing (transverse) momentum. Given that the cascade decays 0 tional gauge bosons have masses comparable to MZ0 , including at depend on the properties of new particles other than the Z boson, least a W 0 boson [15]. If the larger gauge group breaks together this case is not discussed further here. with the electroweak symmetry directly to the electromagnetic The Z0 contribution to the cross sections for e+e− → ff¯ pro- U(1)em, then the left-handed fermion charges are no longer corre- ceeds through an s-channel Z0 exchange (when f = e, there are L L L L 0 + − √ 0 lated (zu 6= zd , zν 6= ze ) and a Z W W coupling is induced. also t- and u-channel exchanges). For MZ0 < s, the Z appears If the electroweak gauge bosons propagate in extra dimensions, as an ff¯ resonance in the radiative return process where photon 0 then their Kaluza-Klein (KK) excitations include a series of Z emission tunes the effective center-of-mass energy to MZ0 . The boson pairs. Each of these pairs can be associated with a different agreement between the LEP-II measurements and the SM predic- SU(2) × U(1) gauge group in four dimensions. The properties of tions implies that either the Z0 couplings are smaller than or of −2 the KK particles depend strongly on the extra-dimensional theory order 10 , or else MZ0 is above 209 GeV, the maximum energy of [16]. For example, in universal extra dimensions there is a parity LEP-II. In the latter case, the Z0 exchange may be approximated 0 902 88. Z -Boson Searches

CMS Preliminary EPS 2019 95% CL exclusions q g' Γ Γ / M < ~100% ATLAS Boosted Dijet, 13 TeV Z' / MZ' = 100% Z' Z' [arXiv:1801.08769]

Γ CMS Dijet χ, 13 TeV ATLAS Dijet+ISR γ, 13 TeV 1 Z' / MZ' = 50% [arXiv:1803.08030] [arXiv:1901.10917] Γ Z' / MZ' = 30% Γ / M < ~30% ATLAS Dijet TLA, 13 TeV Z' Z' [arXiv:1804.03496] Γ Z' / MZ' = 10% CMS Broad Dijet, 13 TeV ATLAS Dijet, 13 TeV [arXiv:1806.00843] [arXiv:1703.09127] Γ Z' / MZ' = 5% Γ / M < ~10% CMS Dijet b tagged, 8 TeV Z' Z' [arXiv:1802.06149]

CMS Dijet, 8 TeV CMS Dijet Scouting '16, 13 TeV [arXiv:1604.08907] [arXiv:1806.00843]

CMS Dijet, 13 TeV CMS Boosted Dijet, 13 TeV [EXO-19-012] [EXO-18-012] 10−1 UA2 CMS Boosted Dijet+γ, 13 TeV [Nucl. Phys. B 400, 3 (1993)] [arXiv:1905.10331]

CDF Run1 Γ / M < ~5% Z'→qq [arXiv:hep-ex/9702004] Z' Z'

CDF Run2 CMS tt, 13 TeV [arXiv:0812.4036] [arXiv:1810.05905]

8 10 20 30 100 200 1000 2000 Z width (all Γ /M ) Υ width (all Γ /M ) Z' Z' Z' Z' [arXiv:1404.3947] [arXiv:1404.3947] MZ' [GeV]

0 Figure 88.2: Upper limits on the Z coupling to quarks as a function of MZ0 based on various searches performed by the ATLAS, CMS, CDF, and UA2 experiments [19].

2 up to corrections of order s/MZ0 by the contact interactions NLO and the deviations from it induced at NNLO are very small. Note that the wu and wd functions are substantially larger than 2 gz  L R    ¯ µ L R   the wq functions for the other quarks. Eq. (88.8) also applies to eγ¯ µ z PL + z PR e fγ z PL + z PR f , M 2 − s e e f f the Tevatron, except for changing the pp initial state to pp¯, which Z0 2 (88.7) implies that the wq(s, MZ0 ) functions are replaced by some other 2 2 where PL,R are chirality projection operators, and the relation functions w¯q((1.96 TeV) ,MZ0 ). between Z0 couplings and charges (see Eq. (88.2) in the limit It is common to present results of Z0 searches as limits on the where the mass and kinetic mixings are neglected) is used, assum- cross section versus MZ0 (see for example Fig. 88.1). ing generation-independent charges. The four LEP collaborations An alternative is to plot exclusion curves for fixed MZ0 values have set limits on the coefficients of such operators for all possible f f in the cu − cd planes, allowing a simple derivation of the mass chiral structures and for various combinations of fermions [20]. limit within any Z0 model. CMS upper limits in the c` − c` plane L L −1/2 u d Thus, one may derive bounds on (MZ0 /gz)|z z | and the e f (` = e or µ) for different MZ0 are shown in Ref. [23] (for Tevatron analogous combinations of LR, RL and RR charges, which are limits, see Refs. [10, 24]). typically on the order of a few TeV. LEP-II limits were derived [6] The discovery of a dilepton resonance at the LHC would deter- on the four sets of charges shown in Table 88.1. mine the Z0 mass and width. A measurement of the total cross Somewhat stronger bounds can be set on MZ0 /gz for specific section would define a band in the c` − c` plane. Angular distri- 0 u d sets of Z couplings if the effects of several operators (88.7) are butions can be used to measure several combinations of Z0 param- combined. Dedicated analyses by the LEP collaborations have eters (angular distributions were used in Ref. [25] to improve the 0 set limits on Z bosons for particular values of the gauge coupling Tevatron sensitivity). Even though the original quark direction in (see section 3.5 of Ref. [20]). For example, M > 1.76 TeV ZSSM a pp collider is unknown, the leptonic forward-backward asymme- for a “sequential” Z0 of same couplings as the SM Z boson, while ` try AFB can be extracted from the kinematics of the dilepton sys- 0 0 MZχ > 0.785 TeV for the Z associated with U(1)χ assuming a tem, and is sensitive to parity-violating couplings. A fit to the Z unification condition for the gauge coupling. rapidity distribution can distinguish between the couplings to up and down quarks. These measurements, combined with off-peak observables, have the potential to differentiate among various Z0 88.4 Searches at hadron colliders ` 0 models [26]. In some cases, AFB may provide discovery sensitivity Z bosons with couplings to quarks (see Eq. (88.1)) may be that is competitive with the mass distribution [27]. The spin of produced at hadron colliders in the s-channel and would show the Z0 boson may be determined from angular distributions [28]. up as resonances in the invariant mass distribution of the decay Searches for Z0 decays into e+e− and µ+µ− by the ATLAS and products. The cross section for producing a Z0 boson at the LHC, CMS collaborations [7, 8] have set 95% C.L. upper cross-section which then decays to some ff¯ final state, takes the form [21] limits as low as 0.02 fb (see Fig. 88.1), with the mass lower limits in specific models as high as 4.9 TeV in a single channel. Cross 0 ¯  π X f 2  σ pp → Z X → ffX ' c wq s, M 0 (88.8) 6 s q Z section limits in the dimuon channel for low mass regions, below q 200 GeV but not near the Z mass, have been set at the LHC by CMS [29] and LHCb [30] (and also in e+e− collisions by BaBar for flavor-diagonal couplings to quarks. Here, we have neglected [31], assuming a , i.e., a Z0 boson whose couplings ¯ the interference with the SM contribution to ff production, which arise only from the kinetic mixing with the hypercharge gauge 0 is a good approximation for a narrow Z resonance (deviations boson). from the narrow width approximation are discussed in Ref. [22]). In the case of final states with -leptons, the mass lower limits The coefficients obtained at 13 TeV are as high as ∼ 2.4 TeV for the τ +τ − [32] h i decay in the case of a sequential Z0. Limits in the flavor-violating f L2 R2 0 ¯ cq = gq + gq B(Z → ff) (88.9) leptonic final states have also been reported by ATLAS [33] and CMS [34] at 13 TeV, for resonances in the e±µ∓, e±τ ∓ and µ±τ ∓ contain all the dependence on the Z0 couplings, while the func- channels. tions wq include all the information about parton distributions Final states with higher background, tt¯, b¯b and jj, are also and QCD corrections [6, 10]. This factorization holds exactly to important as they probe various combinations of Z0 couplings to 0 88. Z -Boson Searches 903

quarks, see Ref. [35] for further discussion. Besides the improved [6] M. Carena et al., Phys. Rev. D70, 093009 (2004), [hep- sensitivity at masses of several TeV, the LHC searches in the dijet ph/0408098]. channel have been also extended to masses as low as 10 GeV, [7] G. Aad et al. (ATLAS), Phys. Lett. B796, 68 (2019), through the use of new techniques involving boosted topologies [arXiv:1903.06248]. and initial state radiation [36]. Limits from such Z0 searches in hadronic final states are summarized in Fig. 88.2. [8] CMS Collab., PAS EXO-19-019, Aug. 2019. Z0 decays to Zh0 with Z → `+`−, νν¯ or qq¯ and h0 → b¯b have [9] F. Del Aguila, M. Cvetic and P. Langacker, Phys. Rev. D52, been studied by ATLAS [37, 38] and CMS [39, 40] using 13 TeV 37 (1995), [hep-ph/9501390]. data. The most stringent constraint is set in the fully hadronic [10] E. Accomando et al., Phys. Rev. D83, 075012 (2011), channel, with a mass lower limit of 2.65 TeV in the context of [arXiv:1010.6058]. the Heavy Vector Triplet (HVT) model weakly-coupled scenario [11] M.-C. Chen, A. de Gouvea and B. A. Dobrescu, Phys. Rev. A [41]. D75, 055009 (2007), [hep-ph/0612017]. Searches for a Z0 boson lighter than the SM Z and which couples to leptons have been performed in the 4-lepton final state. CMS [12] P. Langacker and M. Plumacher, Phys. Rev. D62, 013006 [42] focused on the Z decays into a muon pair followed by the (2000), [hep-ph/0001204]. radiation of a Z0 boson which decays itself into a muon pair. [13] A. J. Buras, F. De Fazio and J. Girrbach, JHEP 02, 116 ATLAS [43] considered the h0 → ZZ0 and h0 → Z0Z0 processes (2013), [arXiv:1211.1896]. 0 followed by the leptonic decays of both Z and Z . [14] L. E. Ibanez and G. G. Ross, Phys. Lett. B332, 100 (1994), 0 + − The pp →Z X →W W X process has also been searched for [hep-ph/9403338]. at the LHC. The channel where the Z0 boson is produced through [15] See the Section on “W 0 searches” in this Review. its couplings to quarks, and the W bosons decay hadronically, has been explored using boosted techniques to analyze the 13 TeV [16] J. Parsons and A. Pomarol, “Extra dimensions” in this Re- data [44,45] with a mass lower limit of 2.9 TeV in the HVT model view. A. The Z0 boson may also be produced through its couplings to W [17] K.M. Black et al., “Dynamical electroweak symmetry break- bosons [46], which has been explored by ATLAS with the use of ing” in this Review. forward jets consistent with a vector boson fusion event topology. [18] M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 At the Tevatron, the CDF and DØ collaborations have searched (1988). for Z0 bosons in the e+e− [47], µ+µ− [48], e±µ∓ [49], τ +τ − [50], tt¯ [51], jj [52] and W +W − [53] final states. These limits have [19] CMS Collab., https://twiki.cern.ch/twiki/bin/view/ CM- been mostly superseded by the LHC results. SPublic/SummaryPlotsEXO13TeV. [20] S. Schael et al. (ALEPH, DELPHI, L3, OPAL, LEP Elec- 88.5 Low-energy constraints troweak), Phys. Rept. 532, 119 (2013), [arXiv:1302.3415]. Z0 boson properties are also constrained by a variety of low- [21] G. Paz and J. Roy, Phys. Rev. D97, 075025 (2018), energy experiments [54]. Polarized electron- scattering [arXiv:1711.02655]. and atomic parity violation are sensitive to electron-quark con- [22] E. Accomando et al., JHEP 10, 153 (2013), 0 tact interactions, which get contributions from Z exchange that [arXiv:1304.6700]. can be expressed in terms of the couplings introduced in Eq. (88.1) [23] V. Khachatryan et al. (CMS), JHEP 04, 025 (2015), and M 0 . Further corrections to the electron-quark contact inter- Z [arXiv:1412.6302]. actions are induced in the presence of Z˜−Z˜0 mixing because of the shifts in the Z couplings to quarks and leptons [3]. Deep-inelastic [24] A. Abulencia et al. (CDF), Phys. Rev. Lett. 95, 252001 neutrino-nucleon scattering is similarly affected by Z0 bosons. (2005), [hep-ex/0507104]. Other low-energy observables are discussed in [4]. Viable models [25] A. Abulencia et al. (CDF), Phys. Rev. Lett. 96, 211801 with Z0 bosons much lighter than the Z boson have been con- (2006), [hep-ex/0602045]. structed, despite many additional experimental constraints [55]. [26] F. Petriello and S. Quackenbush, Phys. Rev. D77, 115004 In some models, the lower limits on MZ0 set by low-energy (2008), [arXiv:0801.4389]. data are above 1 TeV. For example, M > 1.1 TeV and M > Zχ Zη [27] E. Accomando et al., JHEP 01, 127 (2016), 0.43 TeV assuming that the Higgs sectors consist of electroweak [arXiv:1503.02672]. doublets and singlets only [4], while the gauge coupling is fixed by an SO(10) unification condition for U(1)χ and U(1)η. For [28] P. Osland et al., Phys. Rev. D79, 115021 (2009), more general models, see [1, 6, 56]. The mass bounds from direct [arXiv:0904.4857]. searches at the LHC [7, 8] exceed the electroweak constraints by [29] CMS Collab., PAS EXO-19-018, Aug. 2019. a factor of three or more for the models mentioned here. This [30] R. Aaij et al. (LHCb), Phys. Rev. Lett. 120, 061801 (2018), conclusion could change if the collider bounds are weakened by [arXiv:1710.02867]. exotic decay channels [57]. Although the LHC data are most constraining for many Z0 [31] J. P. Lees et al. (BaBar), Phys. Rev. Lett. 113, 201801 models, one should be careful in assessing the relative reach of (2014), [arXiv:1406.2980]. various experiments given the freedom in Z0 couplings. For ex- [32] M. Aaboud et al. (ATLAS), JHEP 01, 055 (2018), 0 ample, a Z coupled to B−yLµ +(y−3)Lτ has implications for the [arXiv:1709.07242]. muon g − 2, neutrino oscillations or τ decays, and would be hard [33] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 092008 (2018), to see in processes involving first-generation fermions. Moreover, [arXiv:1807.06573]. the combination of LHC searches and low-energy measurements [34] A. M. Sirunyan et al. (CMS), JHEP 04, 073 (2018), could allow a precise determination of the Z0 parameters [58]. [arXiv:1802.01122]. References [35] See the Section on “Dynamical Electroweak Symmetry [1] A. Leike, Phys. Rept. 317, 143 (1999), [hep-ph/9805494]. Breaking” in this Review. [2] T. P. Cheng and L. F. Li, Gauge Theory of Elementary Par- [36] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1905.10331]. ticle Physics (1984), ISBN 9780198519614. [37] M. Aaboud et al. (ATLAS), Phys. Lett. B774, 494 (2017), [3] K. S. Babu, C. F. Kolda and J. March-Russell, Phys. Rev. [arXiv:1707.06958]. D57, 6788 (1998), [hep-ph/9710441]. [38] M. Aaboud et al. (ATLAS), JHEP 03, 174 (2018), [Erratum: [4] J. Erler et al., JHEP 08, 017 (2009), [arXiv:0906.2435]. JHEP 11, 051 (2018)], [arXiv:1712.06518]. [5] B. A. Dobrescu and P. J. Fox, JHEP 05, 047 (2016), [39] A. M. Sirunyan et al. (CMS), JHEP 11, 172 (2018), [arXiv:1511.02148]. [arXiv:1807.02826]. 0 904 88. Z -Boson Searches

[40] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C77, 636 (2017), [49] A. Abulencia et al. (CDF), Phys. Rev. Lett. 96, 211802 [arXiv:1707.01303]. (2006), [hep-ex/0603006]. [50] D. Acosta et al. (CDF), Phys. Rev. Lett. 95, 131801 (2005), [41] D. Pappadopulo et al., JHEP 09, 060 (2014), [hep-ex/0506034]. [arXiv:1402.4431]. [51] T. Aaltonen et al. (CDF), Phys. Rev. D84, 072004 (2011), [42] A. M. Sirunyan et al. (CMS), Phys. Lett. B792, 345 (2019), [arXiv:1107.5063]. [arXiv:1808.03684]. [52] T. Aaltonen et al. (CDF), Phys. Rev. D79, 112002 (2009), [arXiv:0812.4036]. [43] M. Aaboud et al. (ATLAS), JHEP 06, 166 (2018), [arXiv:1802.03388]. [53] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 104, 241801 (2010), [arXiv:1004.4946]. [44] G. Aad et al. (ATLAS), JHEP 09, 091 (2019), [54] V. D. Barger et al., Phys. Rev. D57, 391 (1998), [hep- [arXiv:1906.08589]. ph/9707412].

[45] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1906.05977]. [55] R. Harnik, J. Kopp and P. A. N. Machado, JCAP 1207, 026 (2012), [arXiv:1202.6073]. [46] H.-J. He et al., Phys. Rev. D78, 031701 (2008), [56] E. Rojas and J. Erler, JHEP 10, 063 (2015), [arXiv:0708.2588]. [arXiv:1505.03208]. [57] J. Kang and P. Langacker, Phys. Rev. D71, 035014 (2005), [47] V. M. Abazov et al. (D0), Phys. Lett. B695, 88 (2011), [hep-ph/0412190]. [arXiv:1008.2023]. [58] Y. Li, F. Petriello and S. Quackenbush, Phys. Rev. D80, [48] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 106, 121801 055018 (2009), [arXiv:0906.4132]. (2011), [arXiv:1101.4578]. 89. Supersymmetry, Part I (Theory) 905

89. Supersymmetry, Part I (Theory)

Revised August 2019 by B.C. Allanach (DAMTP, Cambridge U.) less, with some restrictions on the dimension-three terms as eluci- and H.E. Haber (UC Santa Cruz). dated in Ref. [11]. The impact of the soft terms becomes negligible at energy scales much larger than the size of the SUSY-breaking masses. Thus, a theory of weak-scale supersymmetry, where the effective scale of supersymmetry breaking is tied to the scale of 89.1 Introduction ...... 905 electroweak symmetry breaking, provides a natural framework for 89.2 Structure of the MSSM ...... 905 the origin and the stability of the gauge hierarchy [7–10]. 89.2.1 R-parity and the lightest supersymmetric At present, there is no unambiguous experimental evidence for particle ...... 906 the breakdown of the SM at or below the TeV scale. The ex- 89.2.2 The goldstino and ...... 906 pectations for new TeV-scale physics beyond the SM are based 89.2.3 Hidden sectors and the structure of SUSY primarily on three theoretical arguments. First, in a theory with breaking ...... 907 an elementary scalar field of mass m and interaction strength λ 89.2.4 SUSY and extra dimensions ...... 907 (e.g., a quartic scalar self-coupling, the square of a gauge coupling 89.2.5 Split-SUSY ...... 907 or the square of a Yukawa coupling), the stability with respect to 89.3 Parameters of the MSSM ...... 908 quantum corrections requires the existence of an energy cutoff 89.3.1 The SUSY-conserving parameters ...... 908 roughly of order (16π2/λ)1/2m, beyond which new physics must 89.3.2 The SUSY-breaking parameters ...... 908 enter [13]. A significantly larger energy cutoff would require an 89.3.3 MSSM-124 ...... 908 unnatural fine-tuning of parameters that govern the effective low- 89.4 The supersymmetric-particle spectrum . . . . . 908 energy theory. Applying this argument to the SM leads to an 89.4.1 The and ...... 909 expectation of new physics at the TeV scale [10]. 89.4.2 The squarks and sleptons ...... 909 Second, the unification of the three SM gauge couplings at a 89.5 The supersymmetric Higgs sector ...... 910 very high energy close to the Planck scale is possible if new physics 89.5.1 The tree-level Higgs sector ...... 910 beyond the SM (which modifies the running of the gauge couplings 89.5.2 The radiatively-corrected Higgs sector . . . 910 above the electroweak scale) is present. The minimal supersym- 89.6 Restricting the MSSM parameter freedom . . . 911 metric extension of the SM, where masses lie below 89.6.1 mass relations ...... 911 a few TeV, provides an example of successful gauge coupling uni- 89.6.2 Constrained versions of the MSSM: fication [14]. mSUGRA, CMSSM, etc...... 911 Third, the existence of dark matter that makes up approxi- 89.6.3 Gauge-mediated SUSY breaking ...... 912 mately one quarter of the energy density of the universe, cannot 89.6.4 The phenomenological MSSM ...... 913 be explained within the SM of particle physics [15]. Remarkably, 89.6.5 Simplified models ...... 913 a stable weakly-interacting massive particle (WIMP) whose mass 89.7 Experimental data confronts the MSSM . . . . . 913 and interaction rate are governed by new physics associated with 89.7.1 Naturalness constraints and the little hier- the TeV-scale can be consistent with the observed density of dark archy ...... 913 matter (this is the so-called WIMP miracle, which is reviewed 89.7.2 Constraints from virtual exchange of super- in Ref. [16]). The lightest supersymmetric particle, if stable, is symmetric particles ...... 914 a promising (although not the unique) candidate for the dark 89.8 Massive neutrinos in weak-scale SUSY . . . . . 915 matter [17–21]. Further aspects of dark matter can be found in 89.8.1 The supersymmetric seesaw ...... 915 Sec. 27. 89.8.2 R-parity-violating SUSY ...... 915 89.9 Extensions beyond the MSSM ...... 916 89.2 Structure of the MSSM The minimal supersymmetric extension of the SM (MSSM) con- sists of the fields of the two-Higgs-doublet extension of the SM 89.1 Introduction and the corresponding [22, 23]. A particle and its Supersymmetry (SUSY) is a generalization of the space-time superpartner together form a supermultiplet. The corresponding symmetries of quantum field theory that transforms fermions into field content of the supermultiplets of the MSSM and their gauge bosons and vice versa [1]. The existence of such a non-trivial quantum numbers are shown in Table 89.1. The electric charge 1 extension of the Poincaré symmetry of ordinary quantum field Q = T3 + 2 Y is determined in terms of the third component of theory was initially surprising, and its form is highly constrained the weak isospin (T3) and the U(1) weak hypercharge (Y ). by theoretical principles [2]. SUSY also provides a framework for The gauge supermultiplets consist of the gluons and their the unification of particle physics and gravity [3–6] at the Planck fermionic superpartners and the SU(2)×U(1) gauge bosons and 19 energy scale, MP ∼ 10 GeV, where the gravitational interac- their gaugino fermionic superpartners. The matter supermulti- tions become comparable in strength to the gauge interactions. plets consist of three generations of left-handed quarks and leptons Moreover, supersymmetry can stabilize the hierarchy between the and their scalar superpartners (squarks and sleptons, collectively energy scale that characterizes electroweak symmetry breaking, referred to as ), and the corresponding . The MEW ∼ 100 GeV, and the Planck scale [7–10] against large ra- Higgs supermultiplets consist of two complex Higgs doublets, their diative corrections. The stability of this large gauge hierarchy fermionic superpartners, and the corresponding antipar- with respect to radiative quantum corrections is not possible to ticles. The enlarged Higgs sector of the MSSM constitutes the maintain in the Standard Model (SM) without an unnatural fine- minimal structure needed to guarantee the cancellation of gauge tuning of the parameters of the fundamental theory at the Planck anomalies [25] generated by the higgsino superpartners that can scale. In contrast, in a supersymmetric extension of the SM, it is appear as internal lines in triangle diagrams with three external possible to maintain the gauge hierarchy while providing a natural electroweak gauge bosons. Moreover, without a second Higgs dou- framework for elementary scalar fields. blet, one cannot generate mass for both “up”-type and “down”- If supersymmetry were an exact symmetry of nature, then par- type quarks (and charged leptons) in a way consistent with the ticles and their superpartners, which differ in spin by half a unit, underlying SUSY [26–28]. would be degenerate in mass. Since superpartners have not (yet) In the most elegant treatment of SUSY, spacetime is extended been observed, supersymmetry must be a broken symmetry. Nev- to superspace which consists of the spacetime coordinates and ertheless, the stability of the gauge hierarchy can still be main- new anticommuting fermionic coordinates θ and θ† [29,30]. Each tained if the SUSY breaking is soft [11,12], and the corresponding supermultiplet is represented by a superfield that is a function of SUSY-breaking mass parameters are no larger than a few TeV. the superspace coordinates. The fields of a given supermultiplet Whether this is still plausible in light of recent SUSY searches at (which are functions of the spacetime coordinates) are coefficients the LHC (see Sec. 90) will be discussed in Sec. 89.7. of the θ and θ† expansion of the corresponding superfield. In particular, soft-SUSY-breaking terms of the Lagrangian in- Vector superfields contain the gauge boson fields and their gaug- volve combinations of fields with total mass dimension of three or ino partners. Chiral superfields contain the spin-0 and spin-1/2 906 89. Supersymmetry, Part I (Theory)

Table 89.1: The fields of the MSSM and their SU(3)×SU(2)×U(1) quantum numbers are listed. For simplicity, only one generation of quarks and leptons is exhibited. For each lepton, quark, and Higgs supermultiplet (each denoted by a hatted upper-case letter), there is a corresponding antiparticle multiplet of charge-conjugated fermions and their associated scalar partners [24].

Field Content of the MSSM Super- Super- Bosonic Fermionic SU(3) SU(2) U(1) multiplets field fields partners ˆ gluon/gluino V8 g eg 8 1 0 gauge boson/ Vˆ W ± ,W 0 We ± , We 0 1 3 0 gaugino Vˆ 0 B Be 1 1 0 slepton/ Lˆ (ν , e−) (ν, e−) 1 2 −1 eL eL L ˆc + c lepton E e˜R eL 1 1 2 ˆ squark/ Q (euL, deL) (u, d)L 3 2 1/3 quark Uˆ c u∗ uc 3¯ 1 −4/3 eR L ˆ c ∗ c ¯ D deR dL 3 1 2/3 ˆ 0 − 0 − Higgs/ Hd (Hd ,Hd ) (Hed , Hed ) 1 2 −1 ˆ + 0 + 0 higgsino Hu (Hu ,Hu) (Heu , Heu) 1 2 1

fields of the matter or Higgs supermultiplets. A general super- is sufficient for eliminating all B and L-violating operators of di- symmetric Lagrangian is determined by three functions of the mension d ≤ 4. chiral superfields [4]: the superpotential, the Kähler potential, As a consequence of the B−L symmetry, the MSSM possesses a and the gauge kinetic function (which can be appropriately gen- multiplicative R-parity invariance, where R = (−1)3(B−L)+2S for eralized to accommodate higher derivative terms [31]). Mini- a particle of spin S [35]. This implies that all the particles of the mal forms for the Kähler potential and gauge kinetic function, SM have even R-parity, whereas the corresponding superpartners which generate canonical kinetic energy terms for all the fields, have odd R-parity. The conservation of R-parity in scattering are required for renormalizable globally supersymmetric theories. and decay processes has a critical impact on supersymmetric phe- A renormalizable superpotential, which is at most cubic in the nomenology. For example, any initial state in a scattering experi- chiral superfields, yields supersymmetric Yukawa couplings and ment will involve ordinary (R-even) particles. Consequently, it fol- mass terms. A combination of gauge invariance and SUSY pro- lows that supersymmetric particles must be produced in pairs. In duces couplings of gaugino fields to matter (or Higgs) fields and general, these particles are highly unstable and decay into lighter their corresponding superpartners. The (renormalizable) MSSM states. Moreover, R-parity invariance also implies that the light- Lagrangian is then constructed by including all possible super- est supersymmetric particle (LSP) is absolutely stable, and must symmetric interaction terms (of dimension four or less) that sat- eventually be produced at the end of a decay chain initiated by isfy SU(3)×SU(2)×U(1) gauge invariance and B−L conservation the decay of a heavy unstable supersymmetric particle. (where B = baryon number and L = lepton number). Finally, the In order to be consistent with cosmological constraints, a stable most general soft-supersymmetry-breaking terms consistent with LSP is almost certainly electrically and color neutral [19]. Con- these symmetries are added [11, 12, 32]. sequently, the LSP in an R-parity-conserving theory is weakly Although the MSSM is the focus of much of this review, there interacting with ordinary matter, i.e., it behaves like a stable is some motivation for considering non-minimal supersymmetric heavy neutrino and will escape collider detectors without being di- extensions of the SM. For example, extra structure is needed to rectly observed. Thus, the canonical signature for conventional R- generate non-zero neutrino masses as discussed in Sec. 89.8. In parity-conserving supersymmetric theories is missing (transverse) addition, in order to address some theoretical issues and tensions momentum, due to the escape of the LSP. Moreover, as noted in associated with the MSSM, it has been fruitful to introduce one Sec. 89.1 and reviewed in Refs. [20] and [21], the stability of the additional singlet Higgs superfield. The resulting next-to-minimal LSP in R-parity-conserving SUSY makes it a promising candidate supersymmetric extension of the Standard Model (NMSSM) [33] for dark matter. is considered further in Sec. 89.4–89.7 and 89.9. Finally, one is The possibility of relaxing the R-parity invariance of the MSSM always free to add additional fields to the SM along with the (which would generate new B and/or L-violating interactions) will corresponding superpartners. However, only certain choices for be addressed in Sec. 89.8.2. the new fields (e.g., the addition of complete SU(5) multiplets) will 89.2.2 The goldstino and gravitino preserve the successful gauge coupling unification of the MSSM. Some examples will be briefly mentioned in Sec. 89.9. In the MSSM, SUSY breaking is accomplished by including the most general renormalizable soft-SUSY-breaking terms consistent 89.2.1 R-parity and the lightest supersymmetric particle with the SU(3)×SU(2)×U(1) gauge symmetry and R-parity in- variance. These terms parameterize our ignorance of the funda- The (renormalizable) SM Lagrangian possesses an accidental mental mechanism of supersymmetry breaking. If supersymmetry global B−L symmetry due to the fact that B and L-violating breaking occurs spontaneously, then a massless Goldstone fermion operators composed of SM fields must have dimension d = 5 or larger [34]. Consequently, B and L-violating effects are suppressed called the goldstino (Ge1/2) must exist. The goldstino would then d−4 be the LSP, and could play an important role in supersymmetric by (MEW/M) , where M is the characteristic mass scale of the physics that generates the corresponding higher dimensional op- phenomenology [36]. erators. Indeed, values of M of order the grand unification scale However, the goldstino degrees of freedom are physical only in or larger yield the observed (approximate) stability of the proton models of spontaneously-broken global SUSY. If SUSY is a local and suppression of neutrino masses. Unfortunately, these results symmetry, then the theory must incorporate gravity; the resulting are not guaranteed in a generic supersymmetric extension of the theory is called supergravity [5, 37]. In models of spontaneously- SM. For example, it is possible to construct gauge invariant su- broken supergravity, the goldstino is “absorbed” by the gravitino persymmetric dimension-four B and L-violating operators made (Ge), the spin-3/2 superpartner of the graviton, via the super- up of fields of SM particles and their superpartners. Such oper- Higgs mechanism [38]. Consequently, the goldstino is removed ators, if present in the theory, could yield a proton decay rate from the physical spectrum and the gravitino acquires a mass many orders of magnitude larger than the current experimental (denoted by m3/2). If m3/2 is smaller than the mass of the lightest bound. It is for this reason that B−L conservation is imposed on superpartner of the SM particles, then the gravitino is the LSP. the supersymmetric Lagrangian when defining the MSSM, which In processes with center-of-mass energy E  m3/2, one can 89. Supersymmetry, Part I (Theory) 907

employ the goldstino–gravitino equivalence theorem [39], which SUSY breaking scales). In particular, the gravitino is a potential 1 implies that the interactions of the helicity ± 2 gravitino (whose dark matter candidate (for a review and guide to the literature, see properties approximate those of the goldstino) dominate those of Ref. [21]). Big bang nucleosynthesis (see Section 24) also provides 3 the helicity ± 2 gravitino. The interactions of with some interesting constraints on the gravitino and the properties other light fields can be described by a low-energy effective La- of the next-to-lightest supersymmetric particle that decays into 1 grangian that is determined by fundamental principles [40]. the gravitino LSP [57]. The couplings of the helicity ± 2 com- 89.2.3 Hidden sectors and the structure of SUSY breaking ponents of Ge to the particles of the MSSM (which approximate It is very difficult (perhaps impossible) to construct a realistic those of the goldstino as previously noted in Sec. 89.2.2) are sig- model of spontaneously-broken weak-scale supersymmetry where nificantly stronger than gravitational strength and amenable to the supersymmetry breaking arises solely as a consequence of the experimental collider analyses. interactions of the particles of the MSSM. A more successful The concept of a hidden sector is more general than SUSY. scheme posits a theory with at least two distinct sectors: a visible Hidden valley models [58] posit the existence of a hidden sector sector consisting of the particles of the MSSM [32] and a sector of new particles and interactions that are very weakly coupled to where SUSY breaking is generated. It is often (but not always) particles of the SM. The impact of a hidden valley on supersym- assumed that particles of the hidden sector are neutral with re- metric phenomenology at colliders can be significant if the LSP spect to the SM gauge group. The effects of the hidden sector lies in the hidden sector [59]. supersymmetry breaking are then transmitted to the MSSM by 89.2.4 SUSY and extra dimensions some mechanism (often involving the mediation by particles that Approaches to SUSY breaking have also been developed in comprise an additional messenger sector). Two theoretical sce- the context of theories in which the number of spatial dimen- narios that exhibit this structure are gravity-mediated and gauge- sions is greater than three. In particular, a number of SUSY- mediated SUSY breaking. breaking mechanisms have been proposed that are inherently Supergravity models provide a natural mechanism for trans- extra-dimensional [60]. The size of the extra dimensions can be mitting the SUSY breaking of the hidden sector to the particle −1 −1 significantly larger than MP ; in some cases of order (TeV) or spectrum of the MSSM. In models of gravity-mediated super- even larger (see, e.g., Sec. 86 or Ref. [61]). symmetry breaking, gravity is the messenger of supersymmetry For example, in one approach the fields of the MSSM live on breaking [41–45]. More precisely, supersymmetry breaking is me- some brane (a lower-dimensional manifold embedded in a higher- diated by effects of gravitational strength (suppressed by inverse dimensional spacetime), while the sector of the theory that breaks powers of the Planck mass). The soft-SUSY-breaking parame- SUSY lives on a second spatially-separated brane. Two exam- ters with dimensions of mass arise as model-dependent multiples ples of this approach are AMSB [48] and gaugino-mediated SUSY of the gravitino mass m3/2. In this scenario, m3/2 is of order breaking [62]. In both cases, SUSY breaking is transmitted the electroweak-symmetry-breaking scale, while the gravitino cou- through fields that live in the bulk (the higher-dimensional space plings are roughly gravitational in strength [3,46]. However, such between the two branes). This setup has some features in common a gravitino typically plays no direct role in supersymmetric phe- with both gravity-mediated and gauge-mediated SUSY breaking nomenology at colliders (except perhaps indirectly in the case (e.g., a hidden and visible sector and messengers). where the gravitino is the LSP [47]). Since a higher dimensional theory must be compactified to four Under certain theoretical assumptions on the structure of the spacetime dimensions, one can also generate a source of SUSY Kähler potential (the so-called sequestered form introduced in breaking by employing boundary conditions on the compactified Ref. [48]), SUSY breaking is due entirely to the super-conformal space that distinguish between fermions and bosons. This is the (super-Weyl) anomaly, which is common to all supergravity mod- so-called Scherk-Schwarz mechanism [63]. The phenomenology els [48]. In particular, gaugino masses are radiatively gener- of such models can be strikingly different from that of the usual ated at one-loop, and squark and slepton squared-mass matrices MSSM [64]. are flavor-diagonal. In sequestered scenarios, squared- masses arise at two-loops, which implies that gluino and sfermion 89.2.5 Split-SUSY masses are of the same order of magnitude. This approach is called If SUSY is not connected with the origin of the electroweak anomaly-mediated SUSY breaking (AMSB). Indeed, anomaly me- scale, it may still be possible that some remnant of the super- diation is more generic than originally conceived, and provides a particle spectrum survives down to the TeV-scale or below. This ubiquitous source of SUSY breaking [49]. However in the sim- is the idea of split-SUSY [65, 66], in which scalar superpartners plest formulation of AMSB as applied to the MSSM, the squared- of the quarks and leptons are significantly heavier (perhaps by masses of the sleptons are negative (known as the tachyonic slep- many orders of magnitude) than 1 TeV, whereas the fermionic ton problem). It may be possible to cure this otherwise fatal superpartners of the gauge and Higgs bosons have masses on the flaw in non-minimal extensions of the MSSM [50]. Alternatively, order of 1 TeV or below. With the exception of a single light one can assert that anomaly mediation is not the sole source of neutral scalar whose properties are practically indistinguishable SUSY breaking in the sfermion sector. In non-sequestered sce- from those of the SM Higgs boson, all other Higgs bosons are also narios, sfermion squared-masses can arise at tree-level, in which assumed to be very heavy. Among the supersymmetric particles, case squark masses would be parametrically larger than the loop- only the fermionic superpartners may be kinematically accessible suppressed gaugino masses [51]. at the LHC. In gauge-mediated supersymmetry breaking (GMSB), gauge In models of split SUSY, the top squark masses cannot be arbi- forces transmit the supersymmetry breaking to the MSSM. A trarily large, as these parameters enter in the radiative corrections typical structure of such models involves a hidden sector where to the mass of the observed Higgs boson [67,68]. In the MSSM, a SUSY is broken, a messenger sector consisting of particles (mes- Higgs boson mass of 125 GeV (see Sec. 11) implies an upper bound sengers) with nontrivial SU(3)×SU(2)×U(1) quantum numbers, on the top squark mass scale in the range of 10 to 108 TeV [69–71], and the visible sector consisting of the fields of the MSSM [52–55]. depending on the value of the ratio of the two neutral Higgs field The direct coupling of the messengers to the hidden sector gener- vacuum expectation values, although this mass range can be some- ates a supersymmetry-breaking spectrum in the messenger sector. what extended by varying other relevant MSSM parameters. In Supersymmetry breaking is then transmitted to the MSSM via the some approaches, gaugino masses are one-loop suppressed relative virtual exchange of the messenger fields. In models of direct gauge to the sfermion masses, corresponding to the so-called mini-split mediation, there is no separate hidden sector. In particular, the SUSY spectrum [68,72]. The higgsino mass scale may or may not sector in which the SUSY breaking originates includes fields that be likewise suppressed depending on the details of the model [73]. carry nontrivial SM quantum numbers, which allows for the direct The SUSY breaking required to produce such a split-SUSY transmission of SUSY breaking to the MSSM [56]. spectrum would destabilize the gauge hierarchy, and thus would In models of gauge-mediated SUSY breaking, the gravitino is not provide an explanation for the scale of electroweak symmetry the LSP [17], as its mass can range from a few eV (in the case of breaking. Nevertheless, models of split-SUSY can account for the low SUSY breaking scales) up to a few GeV (in the case of high dark matter (which is assumed to be the LSP gaugino or higgsino) 908 89. Supersymmetry, Part I (Theory)

and gauge coupling unification, thereby preserving two of the de- vu (also called v1 and v2, respectively, in the literature) are both sirable features of weak-scale SUSY. Finally, as a consequence of real and positive. 2 2 2 2 2 −1/2 2 the very large squark and slepton masses, neutral flavor chang- The quantity, vd + vu = 4mW /g = (2GF ) ' (246 GeV) , ing and CP-violating effects, which can be problematic in models is fixed by the Fermi constant, GF , whereas the ratio with TeV-scale SUSY-breaking masses, are sufficiently reduced to avoid conflict with experimental observations. tan β = vu/vd (89.3) 89.3 Parameters of the MSSM is a free parameter such that 0 < β < π/2. By employing the The parameters of the MSSM are conveniently described by tree-level conditions resulting from the minimization of the scalar considering separately the supersymmetry-conserving and the potential, one can eliminate the diagonal and off-diagonal Higgs squared-masses in favor of m2 = 1 (g2 +g0 2)(v2 +v2 ), the CP-odd supersymmetry-breaking sectors. A careful discussion of the con- Z 4 d u ventions used here in defining the tree-level MSSM parameters can Higgs mass mA and the parameter tan β, be found in Refs. [74,75]. For simplicity, consider first the case of 2m2 2m2 one generation of quarks, leptons, and their scalar superpartners. sin 2β = 12 = 12 , (89.4) 2 2 2 2 m1 + m2 + 2|µ| mA 89.3.1 The SUSY-conserving parameters 1 m2 − m2 tan2 β The parameters of the supersymmetry-conserving sector con- m2 = −|µ|2 + 1 2 . (89.5) 0 Z 2 sist of: (i) gauge couplings, gs, g, and g , corresponding to the 2 tan β − 1 SM gauge group SU(3)×SU(2)×U(1) respectively; (ii) a super- One must also guard against the existence of charge and/or color symmetry-conserving higgsino mass parameter µ; and (iii) Higgs- breaking global minima due to non-zero vacuum expectation val- fermion Yukawa couplings, λu, λd, and λe, of one generation ues for the squark and charged slepton fields. This possibility can of left- and right-handed quarks and leptons, and their super- be avoided if the A-parameters are not unduly large [42, 76, 77]. partners to the Higgs bosons and . Because there is no Additional constraints must also be respected to avoid the possi- right-handed neutrino/sneutrino in the MSSM as defined here, a bility of directions in scalar field space in which the full tree-level Yukawa coupling λν is not included. The complex µ parameter scalar potential can become unbounded from below [77]. and Yukawa couplings enter via the most general renormalizable Note that SUSY-breaking mass terms for the fermionic super- R-parity-conserving superpotential, partners of scalar fields and non-holomorphic trilinear scalar in- teractions (i.e., interactions that mix scalar fields and their com- ˆ ˆ ˆ c ˆ ˆ ˆ c ˆ ˆ ˆc ˆ ˆ W = λdHdQD − λuHuQU + λeHdLE + µHuHd , (89.1) plex conjugates) have not been included above in the soft-SUSY- breaking sector. These terms can potentially destabilize the gauge where the superfields are defined in Table 1 and the gauge group hierarchy [11] in models with a gauge-singlet superfield. The lat- indices are suppressed. ter is not present in the MSSM; hence as noted in Ref. [12], these 89.3.2 The SUSY-breaking parameters so-called non-standard soft-SUSY-breaking terms are benign. The phenomenological impact of non-holomorphic soft SUSY-breaking The supersymmetry-breaking sector contains the following sets terms has been reconsidered in Refs. [78–80]. However, in the of parameters: (i) three complex gaugino Majorana mass param- most common approaches to constructing a fundamental theory eters, M , M , and M , associated with the SU(3), SU(2), and 3 2 1 of SUSY-breaking, the coefficients of these terms (which have di- U(1) subgroups of the SM; (ii) five sfermion squared-mass parame- mensions of mass) are significantly suppressed compared to the ters, M 2 , M 2 , M 2 , M 2 , and M 2 , corresponding to the five elec- Qe Ue De eL Ee TeV-scale [81]. Consequently, we follow the usual approach and troweak gauge multiplets, i.e., superpartners of the left-handed omit these terms from further consideration. c c − c fields (u, d)L, uL, dL, (ν, e )L, and eL, where the superscript c indicates a charge-conjugated fermion field [24]; and (iii) three 89.3.3 MSSM-124 Higgs-squark-squark and Higgs-slepton-slepton trilinear interac- The total number of independent physical parameters that de- fine the MSSM (in its most general form) is quite large, primarily tion terms, with complex coefficients TU ≡ λuAU , TD ≡ λdAD, due to the soft-supersymmetry-breaking sector. In particular, in and TE ≡ λeAE (which define the “A-parameters”). The nota- the case of three generations of quarks, leptons, and their su- tion TU , TD and TE is employed in Ref. [75]. It is conventional to 2 2 2 2 2 separate out the factors of the Yukawa couplings in defining the perpartners, M , M , M , M , and M are hermitian 3 × 3 Qe Ue De eL Ee A-parameters (originally motivated by a simple class of gravity- matrices, and AU , AD, and AE are complex 3 × 3 matrices. In mediated SUSY-breaking models [3, 6]). If the A-parameters are addition, M1, M2, M3, B, and µ are in general complex parame- parametrically of the same order (or smaller) relative to other ters. Finally, as in the SM, the Higgs-fermion Yukawa couplings, SUSY-breaking mass parameters, then only the third generation λ (f =u, d, and e), are complex 3×3 matrices that are related to f √ A-parameters are phenomenologically relevant. the quark and lepton mass matrices via: Mf = λf vf / 2, where 2 Finally, we have (iv) two real squared-mass parameters, m1 ve ≡ vd [with vu and vd as defined above Eq. (89.3)]. and m2 (also called m2 and m2 , respectively, in the literature), 2 Hd Hu However, not all these parameters are physical. Some of the 2 MSSM parameters can be eliminated by expressing interaction and one complex squared-mass parameter, m12 ≡ µB (the latter defines the “B-parameter”), which appear in the MSSM tree-level eigenstates in terms of the mass eigenstates, with an appropriate scalar Higgs potential [28], redefinition of the MSSM fields to remove unphysical degrees of freedom. The analysis of Ref. [82] shows that the MSSM pos- 2 2 † 2 2 † 2 sesses 124 independent real degrees of freedom. Of these, 18 V = (m + |µ| )H Hd + (m + |µ| )HuHu + (m HuHd + h.c.) 1 d 2 12 correspond to SM parameters (including the QCD vacuum angle 1 † † 1 † 2 0 2 2 2 2 θQCD), one corresponds to a Higgs sector parameter (the analogue + (g + g )(HdHd − HuHu) + g |HdHu| , 8 2 of the SM Higgs mass), and 105 are genuinely new parameters of (89.2) the model. The latter include: five real parameters and three where the SU(2)-invariant combination, H H ≡ H+H− − u d u d CP -violating phases in the gaugino/higgsino sector, 21 squark 0 0 HuHd . Note that the quartic Higgs couplings are related to the and slepton (sfermion) masses, 36 real mixing angles to define 0 gauge couplings g and g as a consequence of SUSY. The breaking the sfermion mass eigenstates, and 40 CP -violating phases that of the SU(2)×U(1) electroweak symmetry group to U(1)EM is only can appear in sfermion interactions. The most general R-parity- possible after incorporating the SUSY-breaking Higgs squared- conserving minimal supersymmetric extension of the SM (without 2 2 2 mass parameters m1, m2 (which can be negative) and m12. After additional theoretical assumptions) will be denoted henceforth as minimizing the Higgs scalar potential, these three squared-mass MSSM-124 [83]. parameters can be re-expressed in√ terms of the two Higgs√ vacuum 0 0 expectation values, hHd i ≡ vd/ 2 and hHui ≡ vu/ 2, and the 89.4 The supersymmetric-particle spectrum CP-odd Higgs mass mA [cf. Eqs. (89.4) and (89.5) below]. One is The supersymmetric particles (sparticles) differ in spin by half a always free to re-phase the Higgs doublet fields such that vd and unit from their SM partners. The superpartners of the gauge and 89. Supersymmetry, Part I (Theory) 909

Higgs bosons are fermions, whose names are obtained by append- To determine the physical states and their masses, one ing “ino” to the end of the corresponding SM particle name. The must perform an Autonne-Takagi factorization [94,96] (also called gluino is the color-octet partner of the gluon Takagi diagonalization [95,97]) of the complex symmetric matrix with mass M = |M3|. The superpartners of the electroweak MN : eg gauge and Higgs bosons (the and higgsinos) can mix T W MN W = diag(M 0 ,M 0 ,M 0 ,M 0 ) , (89.10) due to SU(2)×U(1) breaking effects. As a result, the physical χ˜1 χ˜2 χ˜3 χ˜4 states of definite mass are model-dependent linear combinations of the charged or neutral gauginos and higgsinos, called charginos where W is a unitary matrix and the right-hand side of Eq. (89.10) and neutralinos, respectively (sometimes collectively called elec- is the diagonal matrix of (real non-negative) neutralino masses. The physical neutralino states are denoted by χ0 (i = 1,... 4), troweakinos). The neutralinos are Majorana fermions, which can ei lead to some distinctive phenomenological signatures [84,85]. The where the states are ordered such that M 0 ≤ M 0 ≤ M 0 ≤ χ˜1 χ˜2 χ˜3 superpartners of the quarks and leptons are spin-zero bosons: the 0 Mχ˜0 . The χi are the linear combinations of the neutral gaug- squarks, charged sleptons, and sneutrinos, respectively. A com- 4 e ino and higgsino states determined by the matrix elements of W plete set of Feynman rules for the sparticles of the MSSM can be (which is denoted by N −1 in Ref. [92]). The neutralino masses found in Ref. [86]. The MSSM Feynman rules also are implic- correspond to the singular values of M , i.e., the positive square itly contained in a number of amplitude generation and Feynman N roots of the eigenvalues of M † M . Exact formulae for these diagram software packages (see e.g., Refs. [87–89]). N N masses can be found in Refs. [98] and [99]. A numerical algorithm It should be noted that all mass formulae quoted below in this for determining the mixing matrix W has been given in Ref. [100]. Section are tree-level results. Radiative loop corrections will mod- ify these results and must be included in any precision study of If a or neutralino state approximates a particular gaug- supersymmetric phenomenology [90]. Beyond tree level, the def- ino or higgsino state, it is convenient to employ the corresponding inition of the supersymmetric parameters becomes convention- nomenclature. Specifically, if |M1| and |M2| are small compared to m and |µ|, then the lightest neutralino χ0 would be nearly a dependent. For example, one can define physical couplings or Z e1 running couplings, which differ beyond the tree level. This pro- pure , eγ, the superpartner of the photon. If |M1| and mZ vides a challenge to any effort that attempts to extract supersym- are small compared to |M2| and |µ|, then the lightest neutralino metric parameters from data. The SUSY Les Houches Accord would be nearly a pure bino, Be, the superpartner of the weak (SLHA) [75,91] has been adopted, which establishes a set of con- hypercharge gauge boson. If |M2| and mZ are small compared ventions for specifying generic file structures for supersymmetric to |M1| and |µ|, then the lightest chargino pair and neutralino model specifications and input parameters, supersymmetric mass would constitute a triplet of roughly mass-degenerate pure winos, and coupling spectra, and decay tables. These provide a universal ± 0 We , and We3 , the superpartners of the weak SU(2) gauge bosons. interface between spectrum calculation programs, decay packages, Finally, if |µ| and mZ are small compared to |M1| and |M2|, then and high energy physics event generators. the lightest chargino pair and neutralino would be nearly pure higgsino states, the superpartners of the Higgs bosons. Each of 89.4.1 The charginos and neutralinos the above cases leads to a strikingly different phenomenology. ± The mixing of the charged gauginos (We ) and charged higgsi- In the NMSSM, an additional Higgs singlet superfield is added + − nos (Heu and Hed ) is described (at tree-level) by a 2 × 2 complex to the MSSM. This superfield comprises two real Higgs scalar mass matrix [92, 93], degrees of freedom and an associated neutral higgsino degree of freedom. Consequently, there are five neutralino mass eigenstates 1  M2 √ gvu 2 that are obtained by a Takagi-diagonalization of the 5 × 5 neu- MC ≡ 1 . (89.6) √ gvd µ tralino mass matrix. In many cases, the fifth neutralino state is 2 dominated by its SU(2)×U(1) singlet component, and thus is very To determine the physical chargino states and their masses, one weakly coupled to the SM particles and their superpartners. must perform a singular value decomposition [94, 95] of the com- 89.4.2 The squarks and sleptons plex matrix MC : For a given Dirac fermion f, there are two superpartners, feL ∗ −1 and f , where the L and R subscripts simply identify the scalar U MC V = diag(Mχ˜+ ,Mχ˜+ ) , (89.7) eR 1 2 partners that are related by SUSY to the left-handed and right- 1 where U and V are unitary matrices, and the right-hand side of handed fermions, fL,R ≡ 2 (1 ∓ γ5)f, respectively. (There is no Eq. (89.7) is the diagonal matrix of (real non-negative) chargino eνR in the MSSM.) However, feL–feR mixing is possible, in which χ˜± χ˜± masses. The physical chargino states are denoted by 1 and 2 . case feL and feR are not mass eigenstates. For three generations These are linear combinations of the charged gaugino and higgsino of squarks, one must diagonalize 6 × 6 matrices corresponding to states determined by the matrix elements of U and V [92,93] The the basis (eqiL, eqiR), where i = 1, 2, 3 are the generation labels. chargino masses correspond to the singular values [94] of MC , i.e., For simplicity, only the one-generation case is illustrated in detail † below. (The effects of second and third generation squark mixing the positive square roots of the eigenvalues of MC MC : can be significant and are treated in Ref. [101].) 2 1 n 2 2 2 Using the notation of the third family, the one-generation tree- M + + = |µ| + |M2| + 2mW χ˜1 ,χ˜2 2 level squark squared-mass matrix is given by [102],

q 2 o 2 2 2  2 2  2 2 ∗  ∓ |µ| + |M2| + 2m − 4|µM2 − m sin 2β| , M + mq + Lq mqXq W W 2 Q M = e 2 2 , (89.11) (89.8) mqXq M + mq + Rq Re where the states are ordered such that Mχ˜+ ≤ Mχ˜+ . The rel- 1 2 where ative phase of µ and M2 is physical and potentially observable. X ≡ A − µ∗(cot β)2T3q , (89.12) The mixing of the neutral gauginos (Be and We 0) and neutral hig- q q gsinos (H0 and H0) is described (at tree-level) by a 4×4 complex 1 1 ed eu and T3q = 2 [− 2 ] for q = t [b]. The diagonal squared-masses are symmetric mass matrix [92, 93], governed by soft-SUSY-breaking squared-masses M 2 and M 2 ≡ Q R 1 0 1 0 2 2 e e  M1 0 − g vd g vu M [M ] for q = t [b], the corresponding quark masses mt [mb] 2 2 U D 1 1 ande thee electroweak correction terms: 0 M2 gvd − gvu M ≡  2 2  . (89.9) N  1 0 1  2 2 − g vd gvd 0 −µ  Lq ≡ (T3q − eq sin θW )m cos 2β , 2 2 Z (89.13) 1 0 1 2 2 2 g vu − 2 gvu −µ 0 Rq ≡ eq sin θW mZ cos 2β , 910 89. Supersymmetry, Part I (Theory)

2 1 where eq = 3 [− 3 ] for q = t [b]. The off-diagonal squark squared- (ii) interaction terms related by SUSY to the coupling of the scalar masses are proportional to the corresponding quark masses and fields and the gauge fields, whose coefficients are proportional to depend on tan β, the soft-SUSY-breaking A-parameters and the the corresponding gauge couplings (the so-called “D-terms”). higgsino mass parameter µ. Assuming that the A-parameters are In the MSSM, F -term contributions to the quartic Higgs self- parametrically of the same order (or smaller) relative to other couplings are absent. As a result, the strengths of the MSSM SUSY-breaking mass parameters, it then follows that the first quartic Higgs interactions are fixed in terms of the gauge cou- and second generation eqL–eqR mixing is smaller than that of the plings, as noted below Eq. (89.2). Consequently, all the tree-level third generation where mixing can be enhanced by factors of mt MSSM Higgs-sector parameters depend only on two quantities: and mb tan β. tan β [defined in Eq. (89.3)] and one Higgs mass usually taken to In the case of third generation eqL–eqR mixing, the mass eigen- be mA. From these two quantities, one can predict the values states (usually denoted by q1 and q2, with mq˜1 < mq˜2 ) are deter- of the remaining Higgs boson masses, an angle α that measures e e 2 0 0 mined by diagonalizing the 2×2 matrix M given by Eq. (89.11). the mixture of the hypercharge ±1 scalar fields, Hu and Hd , in The corresponding squared-masses and mixing angle are given the physical CP -even neutral scalars, and the Higgs boson self- by [102]: couplings. Moreover, the tree-level mass of the lighter CP -even Higgs boson is bounded, m ≤ m | cos 2β| ≤ m [27, 28]. This 1 h p i h Z Z m2 = Tr M2 ∓ (TrM2)2 − 4 det M2 , bound can be substantially modified when radiative corrections q˜1,2 2 are included, as discussed in Sec. 89.5.2. (89.14) 2mq|Xq| In the NMSSM, the superpotential contains a trilinear term sin 2θq˜ = . that couples the two Y = ±1 Higgs doublet superfields and the m2 − m2 q˜2 q˜1 singlet Higgs superfield. The coefficient of this term is denoted by The one-generation results above also apply to the charged slep- λ. Consequently, the tree-level bound for the mass of the lightest 1 CP -even MSSM Higgs boson is modified [105], tons, with the obvious substitutions: q → ` with T3` = − 2 and e` = −1, and the replacement of the SUSY-breaking parameters: 2 2 2 2 2 2 2 1 2 2 2 M → M , M → M , and Aq → Aτ . For the neutral sleptons, mh ≤ mZ cos 2β + λ v sin 2β , (89.15) Qe eL De Ee 2 ν does not exist in the MSSM, so ν is a mass eigenstate. eR eL 2 2 1/2 In the case of three generations, the SUSY-breaking scalar- where v ≡ (vu + vd) = 246 GeV. If one demands that λ should squared masses [M 2 , M 2 , M 2 , M 2 , and M 2 ] and the A- stay finite after renormalization-group evolution up to the Planck Qe Ue De eL Ee scale, then λ is constrained to lie below about 0.7–0.8 at the elec- parameters [AU , AD, and AE ] are now 3 × 3 matrices as noted in troweak scale [33] (although larger values of λ have also been Sec. 89.3.3. The diagonalization of the 6 × 6 squark mass matri- considered [106]). ces yields feiL–fejR mixing. In practice, since the feL–feR mixing is The tree-level Higgs-quark and Higgs-lepton interactions of the appreciable only for the third generation, this additional compli- MSSM are governed by the Yukawa couplings defined by the su- cation can often be neglected (although see Ref. [101] for exam- perpotential given in Eq. (89.1). In particular, the Higgs sector of ples in which the mixing between the second and third generation the MSSM is a Type-II two-Higgs doublet model [107], in which squarks is relevant). one Higgs doublet (Hd) couples exclusively to the right-handed down-type quark (or lepton) fields and the second Higgs doublet 89.5 The supersymmetric Higgs sector (Hu) couples exclusively to the right-handed up-type quark fields. Consider first the MSSM Higgs sector [27, 28, 103]. Despite the Consequently, the diagonalization of the fermion mass matrices si- large number of potential CP -violating phases among the MSSM- multaneously diagonalizes the matrix Yukawa couplings, resulting 124 parameters, the tree-level MSSM Higgs potential given by in flavor-diagonal tree-level couplings of the neutral Higgs bosons Eq. (89.2) is automatically CP -conserving. This follows from 0 0 0 2 h , H and A to quark and lepton pairs. the fact that the only potentially complex parameter (m12) of the MSSM Higgs potential can be chosen real and positive by 89.5.2 The radiatively-corrected Higgs sector rephasing the Higgs fields, in which case tan β is a real positive When radiative corrections are incorporated, additional param- parameter. Consequently, the physical neutral Higgs scalars are eters of the supersymmetric model enter via virtual supersymmet- CP -eigenstates (at tree-level). The MSSM Higgs sector contains ric particles that appear in loops. The impact of these corrections five physical spin-zero particles: a charged Higgs boson pair (H±), can be significant [108]. The qualitative behavior of these radia- two CP -even neutral Higgs bosons (denoted by h0 and H0 where tive corrections can be most easily seen in the large top-squark 0 mh < mH ), and one CP -odd neutral Higgs boson (A ). The mass limit, where in addition, both the splitting of the two diago- discovery of a SM-like Higgs boson at the LHC with a mass of nal entries and the off-diagonal entries of the top-squark squared- 125 GeV (see Sec. 11) strongly suggests that this state should mass matrix [Eq. (89.11)] are small in comparison to the geometric 0 mean of the two top-squark squared-masses, M 2 ≡ M M . In be identified with h , although the possibility that the 125 GeV S t t 0 e1 e2 state should be identified with H cannot yet be completely ruled this case (assuming mA > mZ ), the predicted upper bound for out [104]. mh is approximately given by In the NMSSM [33], the scalar component of the singlet Higgs superfield adds two additional neutral states to the Higgs sec- 3g2m4   M 2  X2  X2  m2 m2 cos2 2β + t ln S + t 1 − t , tor. In this model, the tree-level Higgs sector can exhibit explicit h . Z 8π2m2 m2 M 2 12M 2 CP-violation. If CP is conserved, then the two extra neutral W t S S (89.16) scalar states are CP -even and CP -odd, respectively. These states where X ≡ A − µ cot β [cf. Eq. (89.12)] is proportional to the can potentially mix with the neutral Higgs states of the MSSM. t t off-diagonal entry of the top-squark squared-mass matrix (where If scalar states exist that are dominantly singlet, then they are for simplicity, A and µ are taken to be real). The Higgs mass weakly coupled to SM gauge bosons and fermions through their t upper limit specified by Eq. (89.16) is saturated when tan β is small mixing with the MSSM Higgs scalars. Consequently, it is √ large (i.e., cos2 2β ∼ 1) and X = 6 M , which defines the possible that one (or both) of the singlet-dominated states is con- t S so-called maximal mixing scenario. siderably lighter than the Higgs boson that was observed at the A more complete treatment of the radiative corrections shows LHC. that Eq. (89.16) somewhat overestimates the true upper bound of 89.5.1 The tree-level Higgs sector mh. These more refined computations, which incorporate renor- The tree-level properties of the Higgs sector are determined by malization group improvement, the two loop and the leading the Higgs potential given by Eq. (89.2). The quartic interaction three-loop contributions, yield mh . 135 GeV in the region of terms are manifestly supersymmetric (although these are modified large tan β (with an accuracy of a few GeV) for mt = 175 GeV by SUSY-breaking effects at the loop level). In general, the quar- and MS . 2 TeV [109]. tic couplings arise from two sources: (i) the supersymmetric gen- In addition, one-loop radiative corrections can introduce CP - eralization of the scalar potential (the so-called “F -terms”), and violating effects in the Higgs sector that depend on some of 89. Supersymmetry, Part I (Theory) 911

the CP -violating phases among the MSSM-124 parameters [110]. 89.6.1 Gaugino mass relations This phenomenon is most easily understood in a scenario where One prediction of many supersymmetric grand unified models mA  MS (i.e., all five physical Higgs states are significantly is the unification of the (tree-level) gaugino mass parameters at lighter than the SUSY breaking scale). In this case, one can inte- some high-energy scale, MX , grate out the heavy superpartners to obtain a low-energy effective theory with two Higgs doublets. The resulting effective two-Higgs doublet model will now contain all possible Higgs self-interaction M1(MX ) = M2(MX ) = M3(MX ) = m1/2 . (89.17) terms (both CP-conserving and CP-violating) and Higgs-fermion interactions (beyond those of Type-II) that are consistent with Due to renormalization group running, in the one-loop approxi- electroweak gauge invariance [111]. mation the effective low-energy gaugino mass parameters (at the In the NMSSM, the dominant radiative correction to electroweak scale) are related, Eq. (89.15) is the same as the one given in Eq. (89.16). How- ever, in contrast to the MSSM, one does not need as large a boost 2 2 0 2 2 from the radiative corrections to achieve a Higgs mass of 125 GeV M3 = (gs /g )M2 ' 3.5M2 ,M1 = (5g /3g )M2 ' 0.5M2. in certain regimes of the NMSSM parameter space (e.g., tan β ∼ 2 (89.18) and λ ∼ 0.7 [112]). Eq. (89.18) can arise more generally in gauge-mediated SUSY- breaking models where the gaugino masses are generated at the 89.6 Restricting the MSSM parameter freedom messenger scale Mmess (which typically lies significantly below the In Sections 89.4 and 89.5, we surveyed the parameters that unification scale where the gauge couplings unify). In this case, comprise the MSSM-124. However, without additional restric- the gaugino mass parameters are proportional to the correspond- tions on the choice of parameters, a generic parameter set within ing squared gauge couplings at the messenger scale. the MSSM-124 framework is not phenomenologically viable. In particular, a generic point of the MSSM-124 parameter space ex- When Eq. (89.18) is satisfied, the chargino and neutralino masses and mixing angles depend only on three unknown pa- hibits: (i) no conservation of the separate lepton numbers Le, Lµ, rameters: the gluino mass, µ, and tan β. It then follows that and Lτ ; (ii) unsuppressed flavor-changing neutral currents (FC- NCs); and (iii) new sources of CP violation that are inconsistent the lightest neutralino must be heavier than 46 GeV due to with the experimental bounds. the non-observation of charginos at LEP [128]. If in addition For example, the MSSM contains many new sources of CP vi- |µ|  |M1| & mZ , then the lightest neutralino is nearly a olation [113]. Indeed, for TeV-scale sfermion and gaugino masses, pure bino, an assumption often made in supersymmetric parti- some combinations of the complex phases of the gaugino-mass cle searches at colliders. Although Eq. (89.18) is often assumed in parameters, the A-parameters, and µ must be less than about many phenomenological studies, a truly model-independent ap- 10−2–10−3 to avoid generating electric dipole moments for the proach would take the gaugino mass parameters, Mi, to be inde- neutron, electron, and [114–116] in conflict with observed pendent parameters to be determined by experiment. Indeed, an data [117]. The rarity of FCNCs [118–120] places additional con- approximately massless neutralino cannot be ruled out at present straints on the off-diagonal matrix elements of the squark and by a model-independent analysis [129]. slepton soft-SUSY-breaking squared-masses and A-parameters It is possible that the tree-level masses for the gauginos are zero. (see Sec. 89.3.3). In this case, the gaugino mass parameters arise at one-loop and The MSSM-124 is also theoretically incomplete as it pro- do not satisfy Eq. (89.18). For example, the gaugino masses in vides no explanation for the fundamental origin of the super- AMSB models arise entirely from a model-independent contribu- symmetry-breaking parameters. The successful unification of the tion derived from the super-conformal anomaly [48, 130]. In this SM gauge couplings at very high energies close to the Planck case, Eq. (89.18) is replaced (in the one-loop approximation) by: scale [8, 66, 121–123] suggests that the high-energy structure of the theory may be considerably simpler than its low-energy real- 2 bigi ization. In a top-down approach, the dynamics that governs the Mi ' m3/2 , (89.19) more fundamental theory at high energies is used to derive the ef- 16π2 fective broken-supersymmetric theory at the TeV scale. A suitable choice for the high energy dynamics is one that yields a TeV-scale where m3/2 is the gravitino mass and the bi are the coefficients theory that satisfies all relevant phenomenological constraints. of the MSSM gauge beta-functions corresponding to the corre- In this Section, we examine a number of theoretical frameworks sponding U(1), SU(2), and SU(3) gauge groups, (b1, b2, b3) = 33 that potentially yield phenomenologically viable regions of the ( 5 , 1, −3). Eq. (89.19) yields M1 ' 2.8M2 and M3 ' −8.3M2, MSSM-124 parameter space. The resulting supersymmetric par- which implies that the lightest chargino pair and neutralino com- ticle spectrum is then a function of a relatively small number of prise a nearly mass-degenerate triplet of winos, We ±, We 0 (cf. Ta- input parameters. This is accomplished by imposing a simple ble 1), over most of the MSSM parameter space. For example, if structure on the soft SUSY-breaking parameters at a common |µ|  mZ , |M2|, then Eq. (89.19) implies that Mχ˜± ' Mχ˜0 ' M2 high-energy scale M (typically chosen to be the Planck scale, 1 X [131]. Alternatively, one can construct an AMSB model where M , the grand unification scale, M , or the messenger scale, P GUT |µ|, m  M , which yields an LSP that is an approximate hig- M ). These serve as initial conditions for the MSSM renormal- Z 2 mess gsino state [132]. In both cases, the corresponding supersym- ization group equations (RGEs), which are given in the two-loop metric phenomenology differs significantly from the standard phe- approximation in Ref. [124] (an automated program to compute nomenology based on Eq. (89.18) [133, 134]. RGEs for the MSSM and other models of new physics beyond the SM has been developed in Ref. [125]). Solving these equations nu- Finally, it should be noted that the unification of gaugino merically, one can then derive the low-energy MSSM parameters masses (and scalar masses) can be accidental. In particular, the relevant for phenomenology. A number of software packages exist energy scale where unification takes place may not be directly re- that numerically calculate the spectrum of supersymmetric parti- lated to any physical scale. One version of this phenomenon has cles, consistent with theoretical conditions on SUSY breaking at been called mirage unification and can occur in certain theories high energies and some experimental data at low energies [126]. of fundamental SUSY breaking [135]. Examples of this scenario are provided by models of gravity- mediated, anomaly mediated and gauge-mediated SUSY break- 89.6.2 Constrained versions of the MSSM: mSUGRA, ing, to be discussed in more detail below. In some of these CMSSM, etc. approaches, one of the diagonal Higgs squared-mass parameters In the minimal supergravity (mSUGRA) framework [3–6, 41– is driven negative by renormalization group evolution [127]. In 43], a form of the Kähler potential is employed that yields minimal such models, electroweak symmetry breaking is generated radia- kinetic energy terms for the MSSM fields [45]. As a result, the soft tively, and the resulting electroweak symmetry-breaking scale is supersymmetry-breaking parameters at the high-energy scale MX intimately tied to the scale of low-energy SUSY breaking. take a particularly simple form in which the scalar squared-masses 912 89. Supersymmetry, Part I (Theory)

and the A-parameters are flavor-diagonal and universal [43]: conditions in defining the mSUGRA model (in which case the mSUGRA model and the CMSSM are synonymous). The au- 2 2 2 2 M (MX ) = M (MX ) = M (MX ) = m01 , thors of Ref. [140] advocate restoring the original nomenclature Qe Ue De 2 2 2 in which the mSUGRA model is defined with the extra condi- M (MX ) = M (MX ) = m01 , tions as originally proposed. Additional mSUGRA variations can eL Ee (89.20) m2(M ) = m2(M ) = m2 , also be considered where different relations among the CMSSM 1 X 2 X 0 parameters are imposed. AU (MX ) = AD(MX ) = AE (MX ) = A01 , One can also relax the universality of scalar masses by where 1 is a 3 × 3 identity matrix in generation space. As in decoupling the squared-masses of the Higgs bosons and the the SM, this approach exhibits minimal flavor violation [136,137], squarks/sleptons. This leads to the non-universal Higgs mass whose unique source is the nontrivial flavor structure of the Higgs- models (NUHMs), thereby adding one or two new parame- ters to the CMSSM depending on whether the diagonal Higgs fermion Yukawa couplings. The gaugino masses are also unified 2 2 according to Eq. (89.17). scalar squared-mass parameters (m1 and m2) are set equal Renormalization group evolution is then used to derive the val- (NUHM1 [141]) or taken to be independent (NUHM2 [142]) at the high energy scale M 2 . Clearly, this modification preserves ues of the supersymmetric parameters at the low-energy (elec- X troweak) scale. For example, to compute squark masses, one the minimal flavor violation of the mSUGRA approach. Never- should use the low-energy values for M 2 , M 2 , and M 2 in theless, the mSUGRA approach and its NUHM generalizations Qe Ue De are probably too simplistic. Theoretical considerations suggest Eq. (89.11). Through the renormalization group running with that the universality of Planck-scale soft SUSY-breaking param- boundary conditions specified in Eq. (89.18) and Eq. (89.20), one eters is not generic [143]. In particular, effective operators at the can show that the low-energy values of M 2 , M 2 , and M 2 de- Planck scale exist that do not respect flavor universality, and it is Q U D e e e difficult to find a theoretical principle that would forbid them. pend primarily on m2 and m2 . A number of useful approxi- 0 1/2 In the framework of supergravity, if anomaly mediation is the mate analytic expressions for superpartner masses in terms of the sole source of SUSY breaking, then the gaugino mass parameters, mSUGRA parameters can be found in Ref. [138]. diagonal scalar squared-mass parameters, and the SUSY-breaking In the mSUGRA approach, four flavors of squarks (with two trilinear scalar interaction terms (proportional to λf AF ) are de- squark eigenstates per flavor) are nearly mass-degenerate. If tan β termined in terms of the beta functions of the gauge and Yukawa is not very large, ebR is also approximately degenerate in mass with couplings and the anomalous dimensions of the squark and slep- the first two generations of squarks. The ebL mass and the diago- ton fields [48, 130, 134]. As noted in Sec. 89.2.3, this approach nal etL and etR masses are typically reduced relative to the common yields tachyonic sleptons in the MSSM unless additional sources squark mass of the first two generations. In addition, there are six of SUSY breaking are present. In the minimal AMSB (mAMSB) 2 flavors of nearly mass-degenerate sleptons (with two slepton eigen- scenario, a universal squared-mass parameter, m0, is added to the states per flavor for the charged sleptons and one per flavor for AMSB expressions for the diagonal scalar squared-masses [134]. the sneutrinos); the sleptons are expected to be somewhat lighter Thus, the mAMSB spectrum and its interaction strengths are de- 2 than the mass-degenerate squarks. As noted below Eq. (89.11), termined by four parameters, m0, m3/2, tan β and sgn(µ0). third-generation squark masses and tau-slepton masses are sen- The mAMSB scenario appears to be ruled out based on the ob- served value of the Higgs boson mass, assuming an upper limit on sitive to the strength of the respective feL–feR mixing. The LSP 0 M of a few TeV, since the mAMSB constraint on A implies that is typically the lightest neutralino, χ˜1, which is dominated by its S F bino component. Regions of the mSUGRA parameter space in the maximal mixing scenario cannot be achieved [cf. Eq. (89.16)]. which the LSP is electrically charged do exist but are not phe- Indeed, under the stated assumptions, the mAMSB Higgs mass nomenologically viable [19]. upper bound lies below the observed Higgs mass value [144]. Thus One can count the number of independent parameters in the within the AMSB scenario, either an additional SUSY-breaking mSUGRA framework. In addition to 18 SM parameters (exclud- contribution to λf AF and/or new ingredients beyond the MSSM are required. ing the Higgs mass), one must specify m0, m1/2, A0, the Planck- scale values for µ and B-parameters (denoted by µ0 and B0), and 89.6.3 Gauge-mediated SUSY breaking the gravitino mass m3/2. Without additional model assumptions, In contrast to models of gravity-mediated SUSY breaking, the m3/2 is independent of the parameters that govern the mass spec- flavor universality of the fundamental soft SUSY-breaking squark trum of the superpartners of the SM [43]. In principle, A0, B0, µ0, and slepton squared-mass parameters is guaranteed in gauge- and m3/2 can be complex, although in the mSUGRA approach, mediated SUSY breaking (GMSB) because the supersymmetry these parameters are taken (arbitrarily) to be real. breaking is communicated to the sector of MSSM fields via gauge As previously noted, renormalization group evolution is used interactions [53,55]. In GMSB models, the mass scale of the mes- to compute the low-energy values of the mSUGRA parameters, senger sector (or its equivalent) is sufficiently below the Planck which then fixes all the parameters of the low-energy MSSM. In scale such that the additional SUSY-breaking effects mediated by particular, the two Higgs vacuum expectation values (or equiva- supergravity can be neglected. lently, mZ and tan β) can be expressed as a function of the Planck- In the minimal GMSB approach, there is one effective mass scale supergravity parameters. The most common procedure is to scale, Λ, that determines all low-energy scalar and gaugino mass remove µ0 and B0 in favor of mZ and tan β [the sign of µ0, de- parameters through loop effects, while the resulting A-parameters noted sgn(µ0) below, is not fixed in this process]. In this case, the are suppressed. In order that the resulting superpartner masses MSSM spectrum and its interaction strengths are determined by be of order 1 TeV, one must have Λ ∼ O(100 TeV). The origin of five parameters: the µ and B-parameters is model-dependent, and lies somewhat outside the ansatz of gauge-mediated SUSY breaking [145]. m0 ,A0 , m1/2 , tan β , and sgn(µ0) , (89.21) The simplest GMSB models appear to be ruled out based on the and an independent gravitino mass m3/2 (in addition to the 18 observed value of the Higgs boson mass. Due to suppressed A pa- parameters of the SM). In Ref. [139], this framework was dubbed rameters, it is difficult to boost the contributions of the radiative the constrained minimal supersymmetric extension of the SM corrections in Eq. (89.16) to obtain a Higgs mass as large as 125 (CMSSM). GeV. However, this conflict can be alleviated in more complicated In the early literature, additional conditions were obtained by GMSB models [146]. To analyze these generalized GMSB mod- assuming a simplified form for the hidden sector that provides the els, it has been especially fruitful to develop model-independent fundamental source of SUSY breaking. Two additional relations techniques that encompass all known GMSB models [147]. These emerged among the mSUGRA parameters [41,45]: B0 = A0 − m0 techniques are well-suited for a comprehensive analysis [148] of and m3/2 = m0. These relations characterize a theory that was the phenomenological profile of gauge-mediated SUSY breaking. called minimal supergravity when first proposed. In the subse- The gravitino is the LSP in GMSB models, as noted in quent literature, it has been more common to omit these extra Sec. 89.2.3. As a result, the next-to-lightest supersymmetric par- 89. Supersymmetry, Part I (Theory) 913

ticle (NLSP) now plays a crucial role in the phenomenology of in the search. Their interpretation by the experimental collab- supersymmetric particle production and decays. Note that un- oration usually involves only a small number of supersymmetric like the LSP, the NLSP can be charged. In GMSB models, the particles (often two or three). Other supersymmetric particles most likely candidates for the NLSP are χ˜0 and τ ±. The NLSP are assumed to play no role (this may happen by virtue of them 1 eR will decay into its superpartner plus a gravitino (e.g., χ˜0 → γG, being too heavy to be produced). Experimental bounds from 1 e non-observation of a signal are usually presented in terms of the χ˜0 → ZG, χ˜0 → h0G or τ ± → τ ±G), with lifetimes and branch- 1 e 1 e e1 e physical masses of the supersymmetric particles involved. Bounds ing ratios that depend on the model parameters. There are also may be presented on the relevant supersymmetric particle masses GMSB scenarios in which there are several nearly degenerate co- assuming a 100% branching ratio for a certain decay, or as an NLSP’s, any one of which can be produced at the penultimate upper bound on signal production cross-section times branching step of a supersymmetric decay chain [149]. For example, in the ratio as a function of the relevant supersymmetric particle masses. slepton co-NLSP case, all three right-handed sleptons are close For example, consider a search for hadronic jets plus missing enough in mass and thus can each play the role of the NLSP. transverse momentum. One can match such a search to the sim- Different choices for the identity of the NLSP and its decay plified model of squark pair production followed by the subsequent rate lead to a variety of distinctive supersymmetric phenomenolo- decay of each squark into a quark (which appears as a jet) and χ˜0 gies [55,150]. For example, a long-lived 1-NLSP that decays out- a neutralino LSP that produces the missing transverse momen- side collider detectors leads to supersymmetric decay chains with 0 0 tum, i.e. q˜q˜ → (qχ˜1)(qχ˜1). Excluded regions resulting from the missing energy in association with leptons and/or hadronic jets non-observation of a signal may be exhibited in the squark mass (this case is indistinguishable from the standard phenomenology versus LSP mass plane. χ˜0 χ˜0 of the 1-LSP). On the other hand, if 1 → γGe is the dominant Simplified models have the advantage that one makes fewer decay mode, and the decay occurs inside the detector, then nearly assumptions, compared to more complete supersymmetric mod- all supersymmetric particle decay chains would contain a photon. els, where the larger number of free parameters makes it difficult In contrast, in the case of a τ ±-NLSP, the τ ± would either be e1 e1 to present excluded regions in any generality. It is hoped that long-lived or would decay inside the detector into a τ-lepton plus simplified models may be a reasonable approximation over size- missing energy. able regions of parameter space of more complete models, within A number of attempts have been made to address the origins which the simplified model is embedded. On the other hand, of the µ and B-parameters in GMSB models based on the field as stressed in Sec. 90, simplified models have the disadvantage content of the MSSM (see, e.g., Refs. [145, 151]). An alternative that the presentation of negative search limits tends to be overly approach is to consider GMSB models based on the NMSSM [152]. strong when compared to the more complete models, particularly The vacuum expectation value of the additional singlet Higgs su- if 100% branching ratios are assumed. A contrast between su- perfield can be used to generate effective µ and B-parameters persymmetric particle search limits in the context of simplified [153]. Such models provide an alternative GMSB framework for models and the corresponding constraints obtained in the more achieving a Higgs mass of 125 GeV, while still being consistent complete pMSSM is provided in Ref. [162]. As long as one is able with LHC bounds on supersymmetric particle masses [154]. to dispel undue pessimism, simplified models remain an efficient 89.6.4 The phenomenological MSSM vehicle for organizing and presenting the results of supersymmet- ric particle searches. Of course, any of the theoretical assumptions described in the previous three subsections must be tested experimentally and 89.7 Experimental data confronts the MSSM could turn out to be wrong. To facilitate the exploration of MSSM At present, there is no direct evidence for weak-scale SUSY phenomena in a more model-independent way while respecting from the data analyzed by the LHC experiments. Recent LHC the constraints noted at the beginning of this Section, the phe- data have been effectively employed in ruling out the existence nomenological MSSM (pMSSM) has been introduced [155]. of colored supersymmetric particles (primarily the gluino and The pMSSM is governed by 19 independent real supersymmet- the first generation of squarks) with masses below about 2 TeV ric parameters: the three gaugino mass parameters M1, M2 and (see Sec. 90). The precise mass limits are model dependent. For M3, the Higgs sector parameters mA and tan β, the Higgsino mass example, as Sec. 90 demonstrates, regions of the pMSSM param- parameter µ, five sfermion squared-mass parameters for the de- eter space can be identified in which lighter squarks and 2 2 2 2 generate first and second generations (M , M , M , M and below 1 TeV cannot be definitely ruled out. Additional constraints Qe Ue De eL M 2 ), the five corresponding sfermion squared-mass parameters arise from limits on the contributions of virtual supersymmetric E particle exchange to a variety of SM processes [118–120]. fore the third generation, and three third-generation A-parameters In light of these negative results, one must confront the tension (At, Ab and Aτ ). The first and second generation A-parameters are typically neglected in pMSSM studies, as their phenomenolog- that exists between the theoretical expectations for the magni- ical consequences are negligible in most applications (one coun- tude of the SUSY-breaking parameters and the non-observation of supersymmetric phenomena at colliders. terexample is the Aµ dependence of the anomalous magnetic mo- ment of the muon, which can be as significant as other contribu- 89.7.1 Naturalness constraints and the little hierarchy tions due to superpartner mediated radiative corrections [156]). In Sec. 89.1, weak-scale SUSY was motivated as a natural Since its initial proposal, the pMSSM approach has been extended solution to the hierarchy problem, which could provide an un- to include CP-violating SUSY-breaking parameters in Ref. [157]. derstanding of the origin of the electroweak symmetry-breaking A comprehensive study of the 19-parameter pMSSM is compu- scale without a significant fine-tuning of the fundamental param- tationally expensive. This is somewhat ameliorated in Ref. [158], eters that govern the MSSM. In this context, the weak scale where the number of pMSSM parameters is reduced to ten by soft supersymmetry-breaking masses must be generally of the or- assuming one common squark squared-mass parameter for the der of 1 TeV or below [163]. This requirement is most easily first two generations, a second common squark squared-mass pa- seen in the determination of mZ by the scalar potential mini- rameter for the third generation, a common slepton squared-mass mum condition. In light of Eq. (89.5), to avoid the fine-tuning of parameter and a common third generation A parameter. Ap- 2 MSSM parameters, the soft SUSY-breaking squared-masses m1 plications of the pMSSM approach to supersymmetric particle 2 2 and m2 and the higgsino squared-mass |µ| should all be roughly searches, and a discussion of the implications for past and future 2 of O(mZ ). Many authors have proposed quantitative measures of LHC and dark matter studies can be found in Refs. [158–160]. fine-tuning [163–167]. One of the simplest measures is the one ad- 89.6.5 Simplified models vocated by Barbieri and Giudice [163] (which was also introduced previously in Ref. [164]), As Sec. 90 demonstrates, experiments present their searches for supersymmetric particles primarily in terms of simplified models. 2 Simplified models for supersymmetric searches [161] are defined ∂ ln mZ ∆i ≡ , ∆ ≡ max ∆i , (89.22) mostly by the empirical objects and kinematic variables involved ∂ ln pi 914 89. Supersymmetry, Part I (Theory)

where the pi are the MSSM parameters at the high-energy scale gests that the masses of the two neutral higgsinos and charged MX , which are set by the fundamental SUSY-breaking dynamics. higgsino pair (which are governed by |µ|) should not be signifi- The theory is more fine-tuned as ∆ becomes larger. However, cantly larger than mZ to avoid an unnatural fine-tuning of the different measures of fine-tuning yield quantitatively different re- supersymmetric parameters, which would imply the existence of sults; in particular, calculating minimal fine-tuning based on the light higgsinos (whose masses are not well constrained, as they high-scale parameters [as defined in Eq. (89.22)] yields a differ- are difficult to detect directly at the LHC due to their soft decay ence by a factor ∼ 10 to fine-tuning based on TeV-scale parame- products). Nevertheless, it may be possible to avoid the conclu- ters [168, 169]. sion that µ ∼ O(mZ ) if additional correlations among the SUSY One can apply the fine-tuning measure to any explicit model breaking mass parameters and µ are present. Such a scenario of SUSY breaking. For example, in the approaches discussed in can be realized in models in which the boundary conditions for Sec. 89.6, the pi are parameters of the model at the energy scale SUSY breaking are generated by approximately conformal strong MX where the soft SUSY-breaking operators are generated by dynamics. For example, in the so-called scalar-sequestering model the dynamics of SUSY breaking. Renormalization group evolu- of Ref. [184], values of |µ| > 1 TeV can be achieved while naturally tion then determines the values of the parameters appearing in maintaining the observed value of mZ . Eq. (89.5) at the electroweak scale. In this way, ∆ is sensitive to all Finally, one can also consider extensions of the MSSM in which the SUSY-breaking parameters of the model (see e.g. Ref. [170]). the degree of fine-tuning is relaxed. For example, it has already It should be noted that the computation of ∆ is often based on been noted in Sec. 89.5 that it is possible to accommodate the Eq. (89.5), which is a tree-level condition. For example, an anal- observed Higgs mass more easily in the NMSSM due to contribu- 2 2 ysis in Ref. [80] shows that the fine tuning measure can be re- tions to mh proportional to the parameter λ . This means that duced by as much as a factor of two when loop corrections are we do not have to rely on a large contribution from the radiative included [171]. corrections to boost the Higgs mass sufficiently above its tree- As anticipated, there is a tension between the present experi- level bound. This allows for smaller top squark masses, which are mental lower limits on the masses of colored supersymmetric parti- more consistent with the demands of naturalness. The reduction cles [172,173] and the expectation that supersymmetry-breaking is of the fine-tuning in various NMSSM models was initially advo- associated with the electroweak symmetry-breaking scale. More- cated in Ref. [185], and subsequently treated in more detail in over, this tension is exacerbated [174] by the observed value of the Refs. [106, 186]. Naturalness can also be relaxed in extended su- Higgs mass (mh ' 125 GeV), which is not far from the MSSM persymmetric models with vector-like quarks [187] and in gauge extensions of the MSSM [188]. upper bound (mh . 135 GeV) [which depends on the top-squark mass and mixing as noted in Sec. 89.5.2]. If MSUSY character- The experimental absence of any new physics beyond the Stan- izes the scale of supersymmetric particle masses, then one would dard Model at the LHC suggests that the principle of naturalness 2 2 is presently under significant stress [189]. Nevertheless, one must crudely expect ∆ ∼ MSUSY/mZ . For example, if MSUSY ∼ 1 TeV then one expects a ∆−1 ∼ 1% fine-tuning of the MSSM parame- be very cautious when drawing conclusions about the viability of weak-scale SUSY to explain the origin of electroweak symmetry ters to achieve the observed value of mZ . This separation of the electroweak symmetry-breaking and SUSY-breaking scales is an breaking, since different measures of fine-tuning noted above can example of the little hierarchy problem [175, 176]. lead to different assessments [168, 169]. Moreover, the maximal The fine-tuning parameter ∆ can depend quite sensitively on value of ∆ that determines whether weak-scale SUSY is a fine- the structure of the SUSY-breaking dynamics, such as the value tuned model (should it be ∆ ∼ 10? 100? 1000?) is ultimately subjective. Thus, it is premature to conclude that weak-scale of MX and relations among SUSY-breaking parameters in the fundamental high energy theory [177]. For example, in so-called SUSY is on the verge of exclusion. However, it might be possi- focus point SUSY models [166, 178], all squark masses can be as ble to sharpen the upper bounds on superpartner masses based on heavy as 5 TeV without significant fine-tuning. This can be at- naturalness arguments, which ultimately will either confirm or re- tributed to a focusing behavior of the renormalization group evo- fute the weak scale SUSY hypothesis [190]. Of course, if evidence lution when certain relations hold among the high-energy values for supersymmetric phenomena in the multi-TeV regime were to of the scalar squared-mass SUSY-breaking parameters. Although be established at a future collider facility (with an energy reach the focus point region of the CMSSM still yields an uncomfort- beyond the LHC [191]), it would be viewed as a spectacularly ably high value of ∆ due to the observed Higgs mass of 125 GeV, successful explanation of the large gauge hierarchy between the one can achieve moderate values of ∆ in models with NUHM2 (multi-)TeV scale and Planck scale. In this case, the remaining boundary conditions for the scalar masses [174]. little hierarchy, characterized by the somewhat large value of the fine-tuning parameter ∆ discussed above, would be regarded as a Among the colored superpartners, the third generation squarks less pressing issue. typically have the most significant impact on the naturalness con- straints [179], while their masses are the least constrained by the 89.7.2 Constraints from virtual exchange of supersymmet- LHC data. Hence, in the absence of any relation between third ric particles generation squarks and those of the first two generations, the nat- There are a number of low-energy measurements that are sen- uralness constraints due to present LHC data can be consider- sitive to the effects of new physics through indirect searches via ably weaker than those obtained in the CMSSM. Indeed, models supersymmetric loop effects. For example, the virtual exchange with first and second generation squark masses in the multi-TeV of supersymmetric particles can contribute to the muon anoma- range do not necessarily require significant fine tuning. Such mod- 1 lous magnetic moment, aµ ≡ 2 (g − 2)µ, as reviewed in Ref. [192]. els have the added benefit that undesirable FCNCs mediated by The SM prediction for aµ exhibits a deviation in the range of squark exchange are naturally suppressed [180]. Other MSSM 3—4σ from the experimentally observed value [193]. This dis- mass spectra that are compatible with moderate fine tuning have crepancy is difficult to accommodate in the constrained SUSY been considered in Refs. [177] and [181]. models of Sec. 89.6.2 and 89.6.3 given the present sparticle mass The lower bounds on squark and gluino masses may not be bounds [173]. Nevertheless, there are regions of the more general as large as suggested by the experimental analyses based on the pMSSM parameter space that are consistent with the observed CMSSM or simplified models. For example, mass bounds for the value of aµ [194]. An updated value of the fine structure con- gluino and the first and second generation squarks based on the stant has resulted in a new SM prediction [195] for the electron CMSSM can often be evaded in alternative or extended MSSM anomalous magnetic moment ae which is 2.4σ above the mea- models, e.g., compressed SUSY [182] and stealth SUSY [183]. surement [196]. Indeed, it is possible within the pMSSM to find Moreover, the experimental upper limits for the third generation allowed parameter space regions where the observed values of aµ squark masses (which have a more direct impact on the fine-tuning and ae are simultaneously accommodated [197]. measure) are weaker than the corresponding mass limits for other The rare inclusive decay b → sγ also provides a sensitive probe colored supersymmetric states. to the virtual effects of new physics beyond the SM. The experi- Among the uncolored superpartners, the higgsinos are typically mental measurements of B → Xs + γ [198] are in agreement with the most impacted by the naturalness constraints. Eq. (89.5) sug- the theoretical SM predictions of Ref. [199]. Since supersymmet- 89. Supersymmetry, Part I (Theory) 915

ric loop corrections can contribute an observable shift from the light neutrino states that are identified with the light neutrinos SM predictions, the absence of any significant deviation places observed in nature. This is the seesaw mechanism [219]. useful constraints on the MSSM parameter space [200]. It is straightforward to construct a supersymmetric generaliza- + − + − The rare decays Bs → µ µ and Bd → µ µ are especially tion of the seesaw model of neutrino masses [220,221] by promot- sensitive to supersymmetric loop effects, with some loop contri- c ing the right-handed neutrino field to a superfield Nb = (eνR ; νR). butions scaling as tan6 β when tan β  1 [201]. At present, a Integrating out the heavy right-handed neutrino supermultiplet combination of the measurements of these rare decay modes [202] yields a new term in the superpotential [cf. Eq. (89.1)] of the are in slight tension at the 2σ level [203] with the predicted SM form rates [204]. Such a tension can be resolved by the aforementioned f supersymmetric loop effects [201]. Wseesaw = (HbU Lb)(HbU Lb) , (89.23) MR ± ± (∗) − The decays B → τ ντ and B → D τ ντ are noteworthy, since in models with extended Higgs sectors such as the MSSM, where MR is the mass scale of the right-handed neutrino sector these processes possess tree-level charged Higgs exchange contri- and f is a dimensionless constant. Note that lepton number is bro- butions that can compete with the dominant W -exchange. As ken by two units, which implies that R-parity is conserved. The ± ± supersymmetric analogue of the Majorana neutrino mass term in Section 71 shows, experimental measurements of B → τ ντ are currently consistent with SM expectations [205]. The BaBar the sneutrino sector leads to sneutrino–antisneutrino mixing phe- − nomena [221, 222]. Collaboration measured values of the rates for B → Dτ ντ and ∗ − The SUSY Les Houches Accords [75, 91], mentioned at the B → D τ ντ [206] that showed a combined 3.4σ discrepancy from the SM predictions, which was also not compatible with end of the introduction to Sec. 89.4, have been extended to the the Type-II Higgs Yukawa couplings employed by the MSSM. supersymmetric seesaw (and other extensions of the MSSM) in Some subsequent measurements of the LHCb and Belle Collab- Ref. [223]. orations [207] were consistent with the BaBar measurements, al- 89.8.2 R-parity-violating SUSY though more recent Belle measurements using a semi-leptonic It is possible to incorporate massive neutrinos in renormalizable tag are more consistent with SM expectations [208]. The com- supersymmetric models while retaining the minimal particle con- bined difference between the measured and expected values of the − ∗ − tent of the MSSM by relaxing the assumption of R-parity invari- B → Dτ ντ and B → D τ ντ decay rates relative to the cor- ance. The most general R-parity-violating (RPV) model involving responding SM values has a significance of about three standard the MSSM spectrum introduces many new parameters to both the deviations [209]. There are a number of additional anomalies in SUSY-conserving and the SUSY-breaking sectors [75, 224]. Each B decay data that have recently attracted some attention, al- new interaction term violates either B or L conservation. For ex- though at present the observed deviations from SM expectations ample, starting from the MSSM superpotential given in Eq. (89.1) are mostly at the level of about two to three standard deviations [suitably generalized to three generations of quarks, leptons and (see, e.g., Ref. [203]). their superpartners], consider the effect of adding the following In summary, although there are a few hints of possible devi- new terms: ations from the SM in B decays, none of the discrepancies by themselves are significant enough to conclusively imply the exis- c 0 c WRPV =(λL)pmnLpLmEn + (λL)pmnLpQmDn tence of new physics beyond the SM. The absence of definitive b b b b b b (89.24) c c c evidence for deviations in various B-physics observables from their + (λB )pmnUbp DbmDbn + (µL)pHbuLbp , SM predictions places useful constraints on the MSSM parameter space [120,172,210]. In contrast, if one or more of the B anomalies where p, m, and n are generation indices, and gauge group indices referred to above were to be experimentally confirmed, it would are suppressed. Eq. (89.24) yields new scalar-fermion Yukawa require require significant modifications to the supersymmetric couplings consisting of all possible combinations involving two models treated in this review. SM fermions and one scalar superpartner. Finally, we note that the constraints from precision electroweak Note that the term in Eq. (89.24) proportional to λB vio- observables (see Sec. 10) and measurements of the 0/ anomaly lates B, while the other three terms violate L. The L-violating [211], if it persists [212], in the system are easily accom- term in Eq. (89.24) proportional to µL is the RPV generaliza- modated in models of TeV-scale SUSY [213, 214]. Thus, robust tion of the µHbuHbd term of the MSSM superpotential, in which regions of the MSSM parameter space, compatible with the results the Y = −1 Higgs/higgsino supermultiplet Hbd is replaced by the of direct and indirect searches for SUSY, remain viable. slepton/lepton supermultiplet Lbp. 89.8 Massive neutrinos in weak-scale SUSY Phenomenological constraints derived from data on various low- In the minimal SM and its supersymmetric extension, there energy B- and L-violating processes can be used to establish lim- its on each of the coefficients (λ ) , (λ0 ) , and (λ ) are no right-handed neutrinos, and Majorana mass terms for L pmn L pmn B pmn the left-handed neutrinos are absent. However, given the over- taken one at a time [224, 225]. If more than one coefficient is whelming evidence for neutrino masses and mixing (see Sec. 14 simultaneously non-zero, then the limits are in general more com- and Ref. [215]), any viable model of fundamental particles must plicated [226]. All possible RPV terms cannot be simultaneously provide a mechanism for generating neutrino masses [216]. In present and unsuppressed; otherwise the proton decay rate would extended supersymmetric models, various mechanisms exist for be many orders of magnitude larger than the present experimental producing massive neutrinos [217]. Although one can devise mod- bound. One way to avoid proton decay is to impose B or L invari- els for generating massive Dirac neutrinos [218], the most com- ance (either one alone would suffice). Otherwise, one must accept mon approaches for incorporating neutrino masses are based on the requirement that certain RPV coefficients must be extremely L-violating supersymmetric extensions of the MSSM, which gen- suppressed. erate massive Majorana neutrinos. Two classes of L-violating su- One particularly interesting class of RPV models is one in which persymmetric models will now be considered. B is conserved, but L is violated. It is possible to enforce baryon number conservation (and the stability of the proton), while al- 89.8.1 The supersymmetric seesaw lowing for lepton-number-violating interactions by imposing a dis- Neutrino masses can be incorporated into the SM by intro- crete Z3 baryon triality symmetry on the low-energy theory [227], ducing SU(3)×SU(2)×U(1) singlet right-handed neutrinos (νR) in place of the standard Z2 R-parity. Since the distinction be- whose mass parameters are very large, typically near the grand tween the Higgs and matter supermultiplets is lost in RPV models unification scale. In addition, one must also include a standard where L is violated, the mixing of sleptons and Higgs bosons, the Yukawa couplings between the lepton doublets, the Higgs dou- mixing of neutrinos and neutralinos, and the mixing of charged blet, and νR. The Higgs vacuum expectation value then induces leptons and charginos are now possible, leading to more compli- an off-diagonal νL–νR mass on the order of the electroweak scale. cated mass matrices and mass eigenstates than in the MSSM. The Diagonalizing the neutrino mass matrix (in the three-generation treatment of neutrino masses and mixing in this framework can model) yields three superheavy neutrino states, and three very be found, e.g., in Ref. [228]. 916 89. Supersymmetry, Part I (Theory)

Alternatively, one can consider imposing a lepton parity such can be found in Ref. [75]. Although the GNMSSM does not solve that all lepton superfields are odd [227, 229]. In this case, only the µ-problem, it does exhibit regions of parameter space in which the B-violating term in Eq. (89.24) survives, and L is conserved. the degree of fine-tuning is relaxed, as discussed in Sec. 89.7.1. Models of this type have been considered in Ref. [230]. Since L The generation of the µ term may be connected with the solu- is conserved in these models, the mixing of the lepton and Higgs tion to the strong CP problem [243]. Models of this type, which superfields is forbidden. Moreover, neutrino masses (and mixing) include new gauge singlet fields that are charged under the Peccei- are not generated if lepton parity is an exact symmetry. However, Quinn (PQ) symmetry [244], were first proposed in Ref. [239]. The one expects that lepton parity cannot be exact due to quantum breaking of the PQ symmetry is thus intimately tied to SUSY gravity effects. Remarkably, the standard Z2 R-parity and the breaking, while naturally yielding a value of µ that is of order the Z3 baryon triality are stable with respect to quantum gravity ef- electroweak symmetry breaking scale [245]. fects, as they can be identified as residual discrete symmetries It is also possible to add higher dimensional Higgs multi- that arise from spontaneously broken non-anomalous gauge sym- plets, such as Higgs triplet superfields [246], provided a custodial- metries [227]. symmetric model (in which the ρ-parameter of precision elec- The symmetries employed above to either remove or suppress troweak physics is close to 1, see Sec. 10) can be formulated. R-parity violating operators were flavour independent. In con- Such models can provide a rich phenomenology of new signals for trast, there exist a number of motivated scenarios based on flavor future LHC studies. symmetries that can also yield the suppression as required by the All supersymmetric models discussed so far in this review pos- experimental data (e.g., see Ref. [231]). sess self-conjugate fermions—the Majorana gluinos and neutrali- The supersymmetric phenomenology of the RPV models ex- nos. However, it is possible to add additional chiral superfields hibits features that are distinct from that of the MSSM [224]. The in the adjoint representation. The spin-1/2 components of these LSP is no longer stable, which implies that not all supersymmet- new superfields can pair up with the gauginos to form Dirac gaugi- ric decay chains must yield missing-energy events at colliders. A nos [247,248]. Such states appear in models of so-called supersoft comprehensive examination of the phenomenology of the MSSM SUSY breaking [249], in some generalized GMSB models [250] extended by a single R-parity violating coupling at the unifica- and in R-symmetric SUSY [251,252]. Such approaches often lead tion scale and its implications for LHC searches has been given to improved naturalness and/or significantly relaxed flavor con- in Ref. [232]. As an example, the sparticle mass bounds obtained straints. The implications of models of Dirac gauginos on the in searches for R-parity-conserving SUSY can be considerably re- observed Higgs boson mass and its properties are addressed in laxed in certain RPV models due to the absence of large missing Ref. [253]. transverse momentum signatures [233]. This can alleviate some For completeness, we briefly note other MSSM extensions con- of the tension with naturalness discussed in Sec. 89.7.1. sidered in the literature. These include an enlarged electroweak Nevertheless, the loss of the missing-energy signature is often gauge group beyond SU(2)×U(1) [254]; and/or the addition of compensated by other striking signals (which depend on which new (possibly exotic) matter supermultiplets such as vector-like R-parity-violating parameters are dominant). For example, su- fermions and their superpartners [187, 255]. persymmetric particles in RPV models can be singly produced References (in contrast to R-parity-conserving models where supersymmet- [1] The Supersymmetric World—The Beginnings of the The- ric particles must be produced in pairs). The phenomenology of ory, World Scientific, Singapore (2000), edited by G. Kane pair-produced supersymmetric particles is also modified in RPV and M. Shifman, contains an early history of supersymme- models due to new decay chains not present in R-parity-conserving try and a guide to the original literature. SUSY models [224]. In RPV models with lepton number violation (these include [2] R. Haag, J. T. Lopuszanski and M. Sohnius, Nucl. Phys. weak-scale SUSY models with baryon triality mentioned above), B88, 257 (1975); S. R. Coleman and J. Mandula, Phys. both ∆L = 1 and ∆L = 2 phenomena are allowed, leading Rev. 159, 1251 (1967). to neutrino masses and mixing [234], neutrinoless double-beta [3] H. P. Nilles, Phys. Rept. 110, 1 (1984). decay [235], sneutrino-antisneutrino mixing [236], and resonant + − [4] S. Weinberg, The Quantum Theory of Fields, Volume III: s-channel production of sneutrinos in e e collisions [237] and Supersymmetry (Cambridge University Press, Cambridge, charged sleptons in pp¯ and pp collisions [238]. UK, 2000). 89.9 Extensions beyond the MSSM [5] P. Nath, Supersymmetry, Supergravity, and Unification Extensions of the MSSM have been proposed to solve a variety (Cambridge University Press, Cambridge, UK, 2017). of theoretical problems. One such problem involves the µ parame- [6] S. P. Martin A Supersymmetry Primer, [hep-ph/9709356]. ter of the MSSM. Although µ is a SUSY-preserving parameter, it [7] E. Witten, Nucl. Phys. B188, 513 (1981). must be of order the effective SUSY-breaking scale of the MSSM to yield a consistent supersymmetric phenomenology [239]. Any [8] S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150 natural solution to the so-called µ-problem must incorporate a (1981). symmetry that enforces µ = 0 and a small symmetry-breaking [9] N. Sakai, Z. Phys. C11, 153 (1981). parameter that generates a value of µ that is not parametrically [10] L. Susskind, Phys. Rept. 104, 181 (1984). larger than the effective SUSY-breaking scale [240]. A number of proposed mechanisms in the literature (e.g., see Refs. [239–241]) [11] L. Girardello and M. T. Grisaru, Nucl. Phys. B194, 65 provide concrete examples of a natural solution to the µ-problem (1982). of the MSSM. [12] L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 (1990); In extensions of the MSSM, new compelling solutions to the I. Jack and D. R. T. Jones, Phys. Lett. B457, 101 (1999), µ-problem are possible. For example, one can replace µ by the [hep-ph/9903365]. vacuum expectation value of a new SU(3)×SU(2)×U(1) singlet [13] V. F. Weisskopf, Phys. Rev. 56, 72 (1939). scalar field. This is the NMSSM, which yields phenomena that [14] See e.g., N. Polonsky, Supersymmetry: Structure and phe- were briefly discussed in Sections 89.4–89.7. The NMSSM super- nomena. Extensions of the standard model, Lect. Notes potential consists only of trilinear terms whose coefficients are di- Phys. M68, 1 (2001). mensionless. There are some advantages to extending the NMSSM further to the USSM [97] by adding a new broken U(1) gauge sym- [15] G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279 metry [242], under which the singlet field is charged. (2005), [hep-ph/0404175]. Alternatively, one can consider a generalized version of the [16] D. Hooper, “TASI 2008 Lectures on Dark Matter,” in The NMSSM (called the GNMSSM in Ref. [186]), where all possi- Dawn of the LHC Era, Proceedings of the 2008 Theoret- ble renormalizable terms in the superpotential are allowed, which ical and Advanced Study Institute in yields new supersymmetric mass terms (analogous to the µ term Physics, Boulder, Colorado, 2–27 June 2008, edited by Tao of the MSSM). A discussion of the parameters of the GNMSSM Han (World Scientific, Singapore, 2009). 89. Supersymmetry, Part I (Theory) 917

[17] H. Pagels and J. R. Primack, Phys. Rev. Lett. 48, 223 [42] L. Alvarez-Gaumé, J. Polchinski, and M.B. Wise, Nucl. (1982). Phys. B221, 495 (1983). [18] H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983). [43] L. J. Hall, J. D. Lykken and S. Weinberg, Phys. Rev. D27, [19] J. R. Ellis et al., Nucl. Phys. B238, 453 (1984). 2359 (1983). [20] G. Jungman, M. Kamionkowski, and K. Griest, Phys. Re- [44] S. K. Soni and H. A. Weldon, Phys. Lett. 126B, 215 (1983); ports 267, 195 (1996). Y. Kawamura, H. Murayama and M. Yamaguchi, Phys. Rev. D51, 1337 (1995), [hep-ph/9406245]. [21] F. D. Steffen, Eur. Phys. J. C59, 557 (2009), [arXiv:0811.3347]. [45] See, e.g., A. Brignole, L.E. Ibáñez, and C. Munoz, in Per- spectives on Supersymmetry II, edited by G.L. Kane (World [22] H. E. Haber and G. L. Kane, Phys. Rept. 117, 75 (1985). Scientific, Singapore, 2010) pp. 244–268. [23] M. Drees, R. Godbole, and P. Roy, Theory and Phe- [46] A. B. Lahanas and D. V. Nanopoulos, Phys. Rept. 145, 1 nomenology of Sparticles (World Scientific, Singapore, (1987). 2005); H. Baer and X. Tata, Weak Scale Supersymmetry: from Superfields to Scattering Events (Cambridge Univer- [47] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. sity Press, Cambridge, UK, 2006); I.J.R. Aitchison, Super- Lett. 91, 011302 (2003), [hep-ph/0302215]; J. L. Feng, symmetry in Particle Physics: an elementary introduction A. Rajaraman and F. Takayama, Phys. Rev. D68, 063504 (Cambridge University Press, Cambridge, UK, 2007). (2003), [hep-ph/0306024]; J. L. Feng, A. Rajaraman and F. Takayama, Int. J. Mod. Phys. D13, 2355 (2004), [hep- [24] Our notation for the charge-conjugated fields follows the th/0405248]. notation of P. Langacker, The Standard Model and Beyond, 2nd edition (CRC Press, Boca Raton, FL, 2017). [48] L. Randall and R. Sundrum, Nucl. Phys. B557, 79 (1999), [hep-th/9810155]. [25] H. Georgi and S. L. Glashow, Phys. Rev. D6, 429 (1972). [49] F. D’Eramo, J. Thaler and Z. Thomas, JHEP 06, 151 [26] P. Fayet, Nucl. Phys. B90, 104 (1975). (2012), [arXiv:1202.1280]; F. D’Eramo, J. Thaler and [27] K. Inoue et al., Prog. Theor. Phys. 67, 1889 (1982). Z. Thomas, JHEP 09, 125 (2013), [arXiv:1307.3251]; [28] J. F. Gunion and H. E. Haber, Nucl. Phys. B272, 1 (1986), S. P. de Alwis, Phys. Rev. D77, 105020 (2008), [Erratum: B402, 567 (1993)]. [arXiv:0801.0578]; S. P. de Alwis, JHEP 01, 006 (2013), [arXiv:1206.6775]; K. Harigaya and M. Ibe, Phys. Rev. D90, [29] A. Salam and J. A. Strathdee, Nucl. Phys. B76, 477 (1974). 085028 (2014), [arXiv:1409.5029]. [30] J. Wess and J. Bagger, Supersymmetry and Supergravity [50] I. Jack, D. R. T. Jones and R. Wild, Phys. Lett. B535, 193 (Princeton University Press, Princeton, NJ, 1992). (2002), [hep-ph/0202101]; B. Murakami and J. D. Wells, [31] I. L. Buchbinder, S. Kuzenko and Z. Yarevskaya, Nucl. Phys. Rev. D68, 035006 (2003), [hep-ph/0302209]; R. Ki- Phys. B411, 665 (1994); I. Antoniadis, E. Dudas and D. M. tano, G. D. Kribs and H. Murayama, Phys. Rev. D70, Ghilencea, JHEP 03, 045 (2008), [arXiv:0708.0383]. 035001 (2004), [hep-ph/0402215]; R. Hodgson et al., Nucl. [32] D. J. H. Chung et al., Phys. Rept. 407, 1 (2005), [hep- Phys. B728, 192 (2005), [hep-ph/0507193]; D. R. T. Jones ph/0312378]. and G. G. Ross, Phys. Lett. B642, 540 (2006), [hep- ph/0609210]. [33] J. R. Ellis et al., Phys. Rev. D39, 844 (1989); U. Ellwanger and C. Hugonie, Eur. Phys. J. C25, 297 (2002), [hep- [51] S. Asai et al., Phys. Lett. B653, 81 (2007), ph/9909260]; U. Ellwanger, C. Hugonie and A. M. Teixeira, [arXiv:0705.3086]. Phys. Rept. 496, 1 (2010), [arXiv:0910.1785]; M. Maniatis, [52] M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189, Int. J. Mod. Phys. A25, 3505 (2010), [arXiv:0906.0777]. 575 (1981); S. Dimopoulos and S. Raby, Nucl. Phys. B192, [34] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979); S. Weinberg, 353 (1981); S. Dimopoulos and S. Raby, Nucl. Phys. B219, Phys. Rev. D22, 1694 (1980); F. Wilczek and A. Zee, Phys. 479 (1983); M. Dine and W. Fischler, Phys. Lett. 110B, 227 Rev. Lett. 43, 1571 (1979); H. A. Weldon and A. Zee, Nucl. (1982); C. R. Nappi and B. A. Ovrut, Phys. Lett. 113B, 175 Phys. B173, 269 (1980). (1982); L. Alvarez-Gaume, M. Claudson and M. B. Wise, Nucl. Phys. B207, 96 (1982). [35] P. Fayet, Phys. Lett. 69B, 489 (1977); G. R. Farrar and P. Fayet, Phys. Lett. 76B, 575 (1978). [53] M. Dine and A. E. Nelson, Phys. Rev. D48, 1277 (1993), [36] P. Fayet, Phys. Lett. 84B, 421 (1979); P. Fayet, Phys. Lett. [hep-ph/9303230]; M. Dine, A. E. Nelson and Y. Shirman, 86B, 272 (1979). Phys. Rev. D51, 1362 (1995), [hep-ph/9408384]. [37] D.Z. Freedman and A. Van Proeyen, Supergravity (Cam- [54] M. Dine et al., Phys. Rev. D53, 2658 (1996), [hep- bridge University Press, Cambridge, UK, 2012); M. Rausch ph/9507378]. de Traubenberg and M. Valenzuela, A Supergravity Primer [55] G. F. Giudice and R. Rattazzi, Phys. Rept. 322, 419 (1999), (World Scientific, Singapore, 2020). [hep-ph/9801271]. [38] S. Deser and B. Zumino, Phys. Rev. Lett. 38, 1433 (1977); [56] E. Poppitz and S. P. Trivedi, Phys. Rev. D55, 5508 (1997), E. Cremmer et al., Phys. Lett. 79B, 231 (1978). [hep-ph/9609529]; H. Murayama, Phys. Rev. Lett. 79, 18 [39] R. Casalbuoni et al., Phys. Lett. B215, 313 (1988); (1997), [hep-ph/9705271]; M. A. Luty and J. Terning, R. Casalbuoni et al., Phys. Rev. D39, 2281 (1989); A. L. Phys. Rev. D57, 6799 (1998), [hep-ph/9709306]; K. Agashe, Maroto and J. R. Pelaez, Phys. Rev. D62, 023518 (2000), Phys. Lett. B435, 83 (1998), [hep-ph/9804450]; N. Arkani- [hep-ph/9912212]. Hamed, J. March-Russell and H. Murayama, Nucl. Phys. B509, 3 (1998), [hep-ph/9701286]; C. Csaki, Y. Shirman [40] Z. Komargodski and N. Seiberg, JHEP 09, 066 (2009), and J. Terning, JHEP 05, 099 (2007), [hep-ph/0612241]; [arXiv:0907.2441]; I. Antoniadis et al., Theor. Math. Phys. M. Ibe and R. Kitano, Phys. Rev. D77, 075003 (2008), 170, 26 (2012), [Teor. Mat. Fiz. 170, 34 (2012)]. [arXiv:0711.0416]. [41] A.H. Chamseddine, R. Arnowitt, and P. Nath, Phys. Rev. [57] M. Kawasaki et al., Phys. Rev. D78, 065011 (2008), Lett. 49, 970 (1982); R. Barbieri, S. Ferrara and C. A. [arXiv:0804.3745]. Savoy, Phys. Lett. 119B, 343 (1982); L. E. Ibanez, Nucl. Phys. B218, 514 (1983); H. P. Nilles, M. Srednicki and [58] M. J. Strassler and K. M. Zurek, Phys. Lett. B651, 374 D. Wyler, Phys. Lett. 120B, 346 (1983); H. P. Nilles, (2007), [hep-ph/0604261]; T. Han et al., JHEP 07, 008 M. Srednicki and D. Wyler, Phys. Lett. 124B, 337 (1983); (2008), [arXiv:0712.2041]. E. Cremmer, P. Fayet and L. Girardello, Phys. Lett. 122B, [59] M. J. Strassler [hep-ph/0607160]; K. M. Zurek, Phys. Rev. 41 (1983); N. Ohta, Prog. Theor. Phys. 70, 542 (1983). D79, 115002 (2009), [arXiv:0811.4429]. 918 89. Supersymmetry, Part I (Theory)

[60] See e.g., M. Quiros, in Particle Physics and Cosmology: [81] S. P. Martin, Phys. Rev. D61, 035004 (2000), [hep- The Quest for Physics Beyond the Standard Model(s), Pro- ph/9907550]. ceedings of the 2002 Theoretical Advanced Study Institute [82] S. Dimopoulos and D. W. Sutter, Nucl. Phys. B452, 496 in Elementary Particle Physics (TASI 2002), edited by (1995), [hep-ph/9504415]; D.W. Sutter, Stanford Ph. D. H.E. Haber and A.E. Nelson (World Scientific, Singapore, thesis, arXiv:hep-ph/9704390. 2004) pp. 549–601; C. Csaki, in ibid., pp. 605–698. [83] H.E. Haber, Nucl. Phys. B (Proc. Suppl.) 62A-C, 469 [61] V.A. Rubakov, Sov. Phys. Usp. 44, 871 (2001); J. L. Hewett (1998). and M. Spiropulu, Ann. Rev. Nucl. Part. Sci. 52, 397 (2002), [hep-ph/0205106]. [84] R. M. Barnett, J. F. Gunion and H. E. Haber, Phys. Lett. B315, 349 (1993), [hep-ph/9306204]; H. Baer, X. Tata and [62] Z. Chacko, M. A. Luty and E. Ponton, JHEP 07, 036 J. Woodside, Phys. Rev. D41, 906 (1990). (2000), [hep-ph/9909248]; D. E. Kaplan, G. D. Kribs and M. Schmaltz, Phys. Rev. D62, 035010 (2000), [hep- [85] S. M. Bilenky, N. P. Nedelcheva and E. K. Khristova, Phys. ph/9911293]; Z. Chacko et al., JHEP 01, 003 (2000), [hep- Lett. 161B, 397 (1985); S. M. Bilenky, E. K. Khristova and ph/9911323]. N. P. Nedelcheva, Bulg. J. Phys. 13, 283 (1986). [63] J. Scherk and J. H. Schwarz, Phys. Lett. 82B, 60 (1979); [86] J. Rosiek, Phys. Rev. D41, 3464 (1990). J. Scherk and J. H. Schwarz, Nucl. Phys. B153, 61 (1979). [87] J. Alwall et al., JHEP 09, 028 (2007), [arXiv:0706.2334]. [64] R. Barbieri, L. J. Hall and Y. Nomura, Phys. Rev. D66, [88] T. Hahn, Comput. Phys. Commun. 140, 418 (2001), [hep- 045025 (2002), [hep-ph/0106190]; R. Barbieri, L. J. Hall and ph/0012260]. Y. Nomura, Nucl. Phys. B624, 63 (2002), [hep-th/0107004]; [89] A. Pukhov et al., INP MSU report 98-41/542 I. Garcia Garcia, K. Howe and J. March-Russell, JHEP 12, (arXiv:hep-ph/9908288); E. Boos et al. [CompHEP 005 (2015), [arXiv:1510.07045]. Collab.], Nucl. Instrum. Methods A534, 50 (2004); Comp- [65] J. D. Wells, in “11th International Conference on Super- HEP webpage, https://theory.sinp.msu.ru/dokuwiki/ symmetry and the Unification of Fundamental Interactions doku.php/comphep/news. (SUSY 2003) Tucson, Arizona, June 5-10, 2003,” (2003), [90] D. M. Pierce et al., Nucl. Phys. B491, 3 (1997), [hep- [hep-ph/0306127]; J. D. Wells, Phys. Rev. D71, 015013 ph/9606211]. (2005), [hep-ph/0411041]. [91] P. Z. Skands et al., JHEP 07, 036 (2004), [hep-ph/0311123]. [66] N. Arkani-Hamed and S. Dimopoulos, JHEP 06, 073 (2005), [hep-th/0405159]; G. F. Giudice and A. Romanino, Nucl. [92] For further details, see e.g., Appendix C of Ref. [22] and Phys. B699, 65 (2004), [Erratum: B706, 487 (2005)], [hep- Appendix A of Ref. [28]. ph/0406088]. [93] J. L. Kneur and G. Moultaka, Phys. Rev. D59, 015005 [67] G. F. Giudice and A. Strumia, Nucl. Phys. B858, 63 (2012), (1999), [hep-ph/9807336]. [arXiv:1108.6077]. [94] R.A. Horn and C.R. Johnson, Matrix Analysis, 2nd Edition [68] A. Arvanitaki et al., JHEP 02, 126 (2013), (Cambridge University Press, Cambridge, UK, 2003). [arXiv:1210.0555]; N. Arkani-Hamed et al. (2012), [95] H. K. Dreiner, H. E. Haber and S. P. Martin, Phys. Rept. [arXiv:1212.6971]. 494, 1 (2010), [arXiv:0812.1594]. [69] E. Bagnaschi et al., JHEP 09, 092 (2014), [96] L. Autonne, Annals de l’Université de Lyon, Nouvelle Série [arXiv:1407.4081]. I, Fasc. 38, 1 (1915); T. Takagi, Japan J. Math. 1, 83 (1925). [70] J. Pardo Vega and G. Villadoro, JHEP 07, 159 (2015), [97] S. Y. Choi et al., Nucl. Phys. B778, 85 (2007), [hep- [arXiv:1504.05200]. ph/0612218]. [71] B. C. Allanach and A. Voigt, Eur. Phys. J. C78, 573 (2018), [98] S. Y. Choi et al., Eur. Phys. J. C22, 563 (2001), [Adden- [arXiv:1804.09410]. dum: Eur. Phys. J. C23, 769 (2002)], [hep-ph/0108117]. [72] Y. Kahn, M. McCullough and J. Thaler, JHEP 11, 161 [99] M. M. El Kheishen, A. A. Aboshousha and A. A. Shafik, (2013), [arXiv:1308.3490]. Phys. Rev. D45, 4345 (1992). [73] L. J. Hall and Y. Nomura, JHEP 01, 082 (2012), [100] T. Hahn (2006), [arXiv:physics/0607103]. [arXiv:1111.4519]; M. Ibe and T. T. Yanagida, Phys. Lett. B709, 374 (2012), [arXiv:1112.2462]. [101] K.-i. Hikasa and M. Kobayashi, Phys. Rev. D36, 724 (1987); F. Gabbiani and A. Masiero, Nucl. Phys. B322, 235 (1989); [74] H. E. Haber and L. Stephenson Haskins (2018), Supersym- P. Brax and C. A. Savoy, Nucl. Phys. B447, 227 (1995), metric Theory and Models, in Anticipating the Next Dis- [hep-ph/9503306]. coveries in Particle Physics, Proceedings of the 2016 The- oretical Advanced Study Institute in Elementary Particle [102] J. R. Ellis and S. Rudaz, Phys. Lett. 128B, 248 (1983); Physics, edited by Rouven Essig and Ian Low (World Sci- F. Browning, D. Chang and W.-Y. Keung, Phys. Rev. D64, entific, Singapore, 2018) pp. 355-499, [arXiv:1712.05926]. 015010 (2001), [hep-ph/0012258]; A. Bartl et al., Phys. Lett. B573, 153 (2003), [hep-ph/0307317]; A. Bartl et al., Phys. [75] B. C. Allanach et al., Comput. Phys. Commun. 180, 8 Rev. D70, 035003 (2004), [hep-ph/0311338]. (2009), [arXiv:0801.0045]. [103] J.F. Gunion et al., The Higgs Hunter’s Guide (Westview [76] J. M. Frere, D. R. T. Jones and S. Raby, Nucl. Phys. B222, Press, Boulder, CO, 2000); M. Carena and H. E. Haber, 11 (1983); J. P. Derendinger and C. A. Savoy, Nucl. Phys. Prog. Part. Nucl. Phys. 50, 63 (2003), [hep-ph/0208209]; B237, 307 (1984); J. F. Gunion, H. E. Haber and M. Sher, A. Djouadi, Phys. Rept. 459, 1 (2008), [hep-ph/0503173]. Nucl. Phys. B306, 1 (1988); D. Chowdhury et al., JHEP 02, 110 (2014), [Erratum: 03, 149 (2018)], [arXiv:1310.1932]; [104] E. Bagnaschi et al., Eur. Phys. J. C79, 617 (2019), W. G. Hollik, JHEP 08, 126 (2016), [arXiv:1606.08356]. [arXiv:1808.07542]. [77] J. A. Casas, A. Lleyda and C. Munoz, Nucl. Phys. B471, 3 [105] H. E. Haber and M. Sher, Phys. Rev. D35, 2206 (1987). (1996), [hep-ph/9507294]. [106] L. J. Hall, D. Pinner and J. T. Ruderman, JHEP 04, 131 [78] C. S. Ün et al., Phys. Rev. D91, 105033 (2015), (2012), [arXiv:1112.2703]. [arXiv:1412.1440]. [107] L. J. Hall and M. B. Wise, Nucl. Phys. B187, 397 (1981). [79] G. G. Ross, K. Schmidt-Hoberg and F. Staub, Phys. Lett. [108] H. E. Haber and R. Hempfling, Phys. Rev. Lett. 66, 1815 B759, 110 (2016), [arXiv:1603.09347]. (1991); Y. Okada, M. Yamaguchi and T. Yanagida, Prog. [80] G. G. Ross, K. Schmidt-Hoberg and F. Staub, JHEP 03, Theor. Phys. 85, 1 (1991); J. R. Ellis, G. Ridolfi and 021 (2017), [arXiv:1701.03480]. F. Zwirner, Phys. Lett. B257, 83 (1991). 89. Supersymmetry, Part I (Theory) 919

[109] P. Draper and H. Rzehak, Phys. Rept. 619, 1 (2016), [130] G. F. Giudice et al., JHEP 12, 027 (1998), [hep- [arXiv:1601.01890]. ph/9810442]; A. Pomarol and R. Rattazzi, JHEP 05, 013 [110] A. Pilaftsis and C. E. M. Wagner, Nucl. Phys. B553, 3 (1999), [hep-ph/9903448]; D.-W. Jung and J. Y. Lee, JHEP (1999), [hep-ph/9902371]; D. A. Demir, Phys. Rev. D60, 03, 123 (2009), [arXiv:0902.0464]. 055006 (1999), [hep-ph/9901389]; S. Y. Choi, M. Drees and [131] J. F. Gunion and H. E. Haber, Phys. Rev. D37, 2515 (1988); J. S. Lee, Phys. Lett. B481, 57 (2000), [hep-ph/0002287]; S. Y. Choi, M. Drees and B. Gaissmaier, Phys. Rev. D70, M. Carena et al., Nucl. Phys. B586, 92 (2000), [hep- 014010 (2004), [hep-ph/0403054]. ph/0003180]; M. Carena et al., Phys. Lett. B495, 155 [132] H. Baer, V. Barger and D. Sengupta, Phys. Rev. D98, (2000), [hep-ph/0009212]; M. Carena et al., Nucl. Phys. 015039 (2018), [arXiv:1801.09730]. B625, 345 (2002), [hep-ph/0111245]; M. Frank et al., JHEP 02, 047 (2007), [hep-ph/0611326]; S. Heinemeyer et al., [133] J. L. Feng et al., Phys. Rev. Lett. 83, 1731 (1999), [hep- Phys. Lett. B652, 300 (2007), [arXiv:0705.0746]. ph/9904250]; J. F. Gunion and S. Mrenna, Phys. Rev. D62, [111] H. E. Haber and J. D. Mason, Phys. Rev. D77, 115011 015002 (2000), [hep-ph/9906270]. (2008), [arXiv:0711.2890]. [134] T. Gherghetta, G. F. Giudice and J. D. Wells, Nucl. Phys. [112] M. Carena et al., Phys. Rev. D93, 035013 (2016), B559, 27 (1999), [hep-ph/9904378]. [arXiv:1510.09137]. [135] M. Endo, M. Yamaguchi and K. Yoshioka, Phys. Rev. D72, [113] S. Khalil, Int. J. Mod. Phys. A18, 1697 (2003), [hep- 015004 (2005), [hep-ph/0504036]; K. Choi, K. S. Jeong and ph/0212050]. K.-i. Okumura, JHEP 09, 039 (2005), [hep-ph/0504037]; O. Loaiza-Brito et al., AIP Conf. Proc. 805, 198 (2005), [114] W. Fischler, S. Paban and S. D. Thomas, Phys. Lett. B289, [hep-th/0509158]. 373 (1992), [hep-ph/9205233]. [115] A. Masiero and L. Silvestrini, in Perspectives on Supersym- [136] See e.g., G. D’Ambrosio et al., Nucl. Phys. B645, 155 metry, edited by G.L. Kane (World Scientific, Singapore, (2002). 1998) pp. 423–441. [137] C. Smith, Acta Phys. Polon. Supp. 3, 53 (2010), [116] M. Pospelov and A. Ritz, Annals Phys. 318, 119 (2005), [arXiv:0909.4444]. [hep-ph/0504231]. [138] M. Drees and S.P. Martin, in Electroweak Symmetry Break- [117] J. M. Pendlebury et al., Phys. Rev. D92, 092003 (2015), ing and New Physics at the TeV Scale, edited by T. Barklow [arXiv:1509.04411]; V. Andreev et al. (ACME Collabora- et al. (World Scientific, Singapore, 1996) pp. 146–215. tion), Nature 562, 7727, 355 (2018). [139] G. L. Kane et al., Phys. Rev. D49, 6173 (1994), [hep- [118] F. Gabbiani et al., Nucl. Phys. B477, 321 (1996), [hep- ph/9312272]. ph/9604387]. [140] J. R. Ellis et al., Phys. Lett. B573, 162 (2003), [hep- [119] M. J. Ramsey-Musolf and S. Su, Phys. Rept. 456, 1 (2008), ph/0305212]; J. R. Ellis et al., Phys. Rev. D70, 055005 [hep-ph/0612057]. (2004), [hep-ph/0405110]. [120] M. Carena, A. Menon and C. E. M. Wagner, Phys. Rev. [141] H. Baer et al., Phys. Rev. D71, 095008 (2005), [hep- D79, 075025 (2009), [arXiv:0812.3594]. ph/0412059]. [121] M. B. Einhorn and D. R. T. Jones, Nucl. Phys. B196, 475 [142] V. Berezinsky et al., Astropart. Phys. 5, 1 (1996), [hep- (1982). ph/9508249]; J. R. Ellis et al., Nucl. Phys. B652, 259 [122] W. J. Marciano and G. Senjanovic, Phys. Rev. D25, 3092 (2003), [hep-ph/0210205]. (1982). [143] L. E. Ibanez and D. Lust, Nucl. Phys. B382, 305 (1992), [123] R.N. Mohapatra, Unification and Supersymmetry, Third [hep-th/9202046]; B. de Carlos, J. A. Casas and C. Munoz, Edition (Springer Science, New York, 2003). Phys. Lett. B299, 234 (1993), [hep-ph/9211266]; V. S. [124] S. P. Martin and M. T. Vaughn, Phys. Rev. D50, 2282 Kaplunovsky and J. Louis, Phys. Lett. B306, 269 (1993), (1994), [Erratum: Phys. Rev. D78, 039903 (2008)], [hep- [hep-th/9303040]; A. Brignole, L. E. Ibanez and C. Munoz, ph/9311340]; R. M. Fonseca et al., Nucl. Phys. B854, 28 Nucl. Phys. B422, 125 (1994), [Erratum: B436, 747 (2012), [arXiv:1107.2670]; F. Staub, Comput. Phys. Com- (1995)], [hep-ph/9308271]. mun. 182, 808 (2011), [arXiv:1002.0840]. [144] A. Arbey et al., Phys. Rev. D87, 115020 (2013), [125] F. Staub, Comput. Phys. Commun. 185, 1773 (2014), [arXiv:1304.0381]. [arXiv:1309.7223]; F. Staub, Adv. High Energy Phys. 2015, [145] G. R. Dvali, G. F. Giudice and A. Pomarol, Nucl. Phys. 840780 (2015), [arXiv:1503.04200]; The SARAH homepage B478, 31 (1996), [hep-ph/9603238]. is https://sarah.hepforge.org/. [146] P. Draper et al., Phys. Rev. D85, 095007 (2012), [126] B. C. Allanach, Comput. Phys. Commun. 143, 305 [arXiv:1112.3068]. (2002), [hep-ph/0104145]; The SOFTSUSY homepage is http://softsusy.hepforge.org/; A. Djouadi, J.-L. [147] P. Meade, N. Seiberg and D. Shih, Prog. Theor. Phys. Suppl. Kneur and G. Moultaka, Comput. Phys. Commun. 176, 177, 143 (2009), [arXiv:0801.3278]; M. Buican et al., JHEP 426 (2007), [hep-ph/0211331]; The Suspect homepage is 03, 016 (2009), [arXiv:0812.3668]. http://suspect.in2p3.fr/; F. E. Paige et al. (2003), [148] A. Rajaraman et al., Phys. Lett. B678, 367 (2009), [hep-ph/0312045]; Isajet may be obtained from http: [arXiv:0903.0668]; L. M. Carpenter et al., Phys. Rev. D79, //www.nhn.ou.edu/~isajet/; W. Porod, Comput. Phys. 035002 (2009), [arXiv:0805.2944]. Commun. 153, 275 (2003), [hep-ph/0301101]; Spheno may be obtained from https://spheno.hepforge.org/; [149] S. Ambrosanio, G. D. Kribs and S. P. Martin, Nucl. Phys. P. Athron et al., Comput. Phys. Commun. 190, 139 B516, 55 (1998), [hep-ph/9710217]. (2015), [arXiv:1406.2319]; The FlexibleSUSY homepage is [150] For a review and guide to the literature, see J.F. Gunion and https://flexiblesusy.hepforge.org/. H.E. Haber, in Perspectives on Supersymmetry II, edited by [127] L. E. Ibanez and G. G. Ross, Phys. Lett. 110B, 215 (1982). G.L. Kane (World Scientific, Singapore, 2010) pp. 420–445. [128] J. Abdallah et al. (DELPHI Collaboration), Eur. Phys. J. [151] T. S. Roy and M. Schmaltz, Phys. Rev. D77, 095008 (2008), C31, 421 (2003), [hep-ex/0311019]. [arXiv:0708.3593]. [129] H. K. Dreiner et al., Eur. Phys. J. C62, 547 (2009), [152] A. de Gouvea, A. Friedland and H. Murayama, Phys. Rev. [arXiv:0901.3485]. D57, 5676 (1998), [hep-ph/9711264]. 920 89. Supersymmetry, Part I (Theory)

[153] T. Han, D. Marfatia and R.-J. Zhang, Phys. Rev. D61, [173] P. Bechtle et al., Eur. Phys. J. C76, 96 (2016), 013007 (2000), [hep-ph/9906508]; Z. Chacko and E. Ponton, [arXiv:1508.05951]. Phys. Rev. D66, 095004 (2002), [hep-ph/0112190]; A. Del- [174] H. Baer et al., Phys. Rev. D89, 115019 (2014), gado, G. F. Giudice and P. Slavich, Phys. Lett. B653, 424 [arXiv:1404.2277]. (2007), [arXiv:0706.3873]; T. Liu and C. E. M. Wagner, JHEP 06, 073 (2008), [arXiv:0803.2895]. [175] R. Barbieri and A. Strumia, in “4th Rencontres du Viet- nam: Physics at Extreme Energies (Particle Physics and [154] B. Allanach et al., Phys. Rev. D92, 015006 (2015), Astrophysics) Hanoi, Vietnam, July 19-25, 2000,” (2000), [arXiv:1502.05836]. [hep-ph/0007265]. [155] A. Djouadi, J.L. Kneur, and G. Moultaka, Comp. Phys. [176] L. Giusti, A. Romanino and A. Strumia, Nucl. Phys. B550, Comm. 176, 426 (2007); C. F. Berger et al., JHEP 02, 023 3 (1999), [hep-ph/9811386]; H.-C. Cheng and I. Low, JHEP (2009), [arXiv:0812.0980]. 09, 051 (2003), [hep-ph/0308199]; H.-C. Cheng and I. Low, [156] S. P. Martin and J. D. Wells, Phys. Rev. D64, 035003 JHEP 08, 061 (2004), [hep-ph/0405243]; R. Harnik et al., (2001), [hep-ph/0103067]. Phys. Rev. D70, 015002 (2004), [hep-ph/0311349]. [157] J. Berger et al., Phys. Rev. D93, 035017 (2016), [177] H. Baer et al., Phys. Rev. D87, 035017 (2013), [arXiv:1510.08840]. [arXiv:1210.3019]; H. Baer et al., Phys. Rev. D87, 115028 [158] K. J. de Vries et al., Eur. Phys. J. C75, 422 (2015), (2013), [arXiv:1212.2655]; J. L. Feng, Ann. Rev. Nucl. Part. [arXiv:1504.03260]. Sci. 63, 351 (2013), [arXiv:1302.6587]. [159] M. Cahill-Rowley et al., Phys. Rev. D90, 095017 (2014), [178] J. L. Feng, K. T. Matchev and T. Moroi, Phys. Rev. [arXiv:1407.7021]; M. Cahill-Rowley et al., Phys. Rev. D91, Lett. 84, 2322 (2000), [hep-ph/9908309]; J. L. Feng 055011 (2015), [arXiv:1405.6716]; A. Barr and J. Liu, Eur. and F. Wilczek, Phys. Lett. B631, 170 (2005), [hep- Phys. J. C77, 202 (2017), [arXiv:1608.05379]. ph/0507032]; D. Horton and G. G. Ross, Nucl. Phys. B830, [160] G. Bertone et al., JCAP 1604, 037 (2016), 221 (2010), [arXiv:0908.0857]. [arXiv:1507.07008]. [179] M. Drees, Phys. Rev. D33, 1468 (1986); S. Dimopoulos [161] N. Arkani-Hamed et al. (2007), [hep-ph/0703088]; J. Alwall and G. F. Giudice, Phys. Lett. B357, 573 (1995), [hep- et al., Phys. Rev. D79, 015005 (2009), [arXiv:0809.3264]; ph/9507282]; A. Pomarol and D. Tommasini, Nucl. Phys. J. Alwall, P. Schuster and N. Toro, Phys. Rev. D79, B466, 3 (1996), [hep-ph/9507462]. 075020 (2009), [arXiv:0810.3921]; D. S. M. Alves, E. Iza- [180] M. Dine, A. Kagan and S. Samuel, Phys. Lett. B243, 250 guirre and J. G. Wacker, Phys. Lett. B702, 64 (2011), (1990); A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. [arXiv:1008.0407]; D. S. M. Alves, E. Izaguirre and J. G. Lett. B388, 588 (1996), [hep-ph/9607394]. Wacker, JHEP 10, 012 (2011), [arXiv:1102.5338]; D. Alves (LHC New Physics Working Group), J. Phys. G39, 105005 [181] C. Brust et al., JHEP 03, 103 (2012), [arXiv:1110.6670]. (2012), [arXiv:1105.2838]. [182] S. P. Martin, Phys. Rev. D75, 115005 (2007), [hep- [162] F. Ambrogi et al., Eur. Phys. J. C78, 215 (2018), ph/0703097]; S. P. Martin, Phys. Rev. D78, 055019 (2008), [arXiv:1707.09036]. [arXiv:0807.2820]. [163] R. Barbieri and G.F. Giudice, Nucl. Phys. B305, 63 (1988). [183] J. Fan, M. Reece and J. T. Ruderman, JHEP 11, 012 (2011), [arXiv:1105.5135]; J. Fan, M. Reece and J. T. Ruderman, [164] J. R. Ellis et al., Mod. Phys. Lett. A1, 57 (1986). JHEP 07, 196 (2012), [arXiv:1201.4875]. [165] G. W. Anderson and D. J. Castano, Phys. Lett. B347, 300 (1995), [hep-ph/9409419]; G. W. Anderson and D. J. [184] H. Murayama, Y. Nomura and D. Poland, Phys. Rev. Castano, Phys. Rev. D52, 1693 (1995), [hep-ph/9412322]; D77, 015005 (2008), [arXiv:0709.0775]; G. Perez, T. S. G. W. Anderson and D. J. Castano, Phys. Rev. D53, 2403 Roy and M. Schmaltz, Phys. Rev. D79, 095016 (2009), (1996), [hep-ph/9509212]; P. Athron and D. J. Miller, Phys. [arXiv:0811.3206]. Rev. D76, 075010 (2007), [arXiv:0705.2241]; M. E. Cabr- [185] R. Dermisek and J. F. Gunion, Phys. Rev. Lett. 95, 041801 era, J. A. Casas and R. Ruiz de Austri, JHEP 03, 075 (2005), [hep-ph/0502105]; Phys. Rev. D75, 095019 (2007); (2009), [arXiv:0812.0536]; H. Baer et al., Phys. Rev. Lett. R. Dermisek and J. F. Gunion, Phys. Rev. D76, 095006 109, 161802 (2012), [arXiv:1207.3343]. (2007), [arXiv:0705.4387]. [166] J. L. Feng, K. T. Matchev and T. Moroi, Phys. Rev. D61, [186] G. G. Ross and K. Schmidt-Hoberg, Nucl. Phys. B862, 710 075005 (2000), [hep-ph/9909334]. (2012), [arXiv:1108.1284]; G. G. Ross, K. Schmidt-Hoberg [167] D. M. Ghilencea and G. G. Ross, Nucl. Phys. B868, 65 and F. Staub, JHEP 08, 074 (2012), [arXiv:1205.1509]; (2013), [arXiv:1208.0837]. A. Kaminska, G. G. Ross and K. Schmidt-Hoberg, JHEP 11, 209 (2013), [arXiv:1308.4168]. [168] H. Baer, V. Barger and D. Mickelson, Phys. Rev. D88, 095013 (2013), [arXiv:1309.2984]. [187] S. P. Martin and J. D. Wells, Phys. Rev. D86, 035017 [169] M. van Beekveld, S. Caron and R. Ruiz de Austri, JHEP (2012), [arXiv:1206.2956]. 01, 147 (2020), [arXiv:1906.10706]. [188] B. Bellazzini et al., Phys. Rev. D79, 095003 (2009), [170] G. L. Kane and S. F. King, Phys. Lett. B451, 113 [arXiv:0902.0015]. (1999), [hep-ph/9810374]; M. Bastero-Gil, G. L. Kane [189] M. Dine, Ann. Rev. Nucl. Part. Sci. 65, 43 (2015), and S. F. King, Phys. Lett. B474, 103 (2000), [hep- [arXiv:1501.01035]. ph/9910506]; J. A. Casas, J. R. Espinosa and I. Hi- [190] H. Baer, V. Barger and M. Savoy, Phys. Rev. D93, 035016 dalgo, JHEP 01, 008 (2004), [hep-ph/0310137]; H. Abe, (2016), [arXiv:1509.02929]. T. Kobayashi and Y. Omura, Phys. Rev. D76, 015002 (2007), [hep-ph/0703044]; R. Essig and J.-F. Fortin, JHEP [191] M.L. Mangano, editor, Physics at the FCC-hh, a 100 04, 073 (2008), [arXiv:0709.0980]. TeV pp collider, CERN Yellow Report, CERN-2017-003-M (2017). [171] B. de Carlos and J. A. Casas, Phys. Lett. B309, 320 (1993), [hep-ph/9303291]; S. Cassel, D. M. Ghilencea and G. G. [192] D. Stockinger, J. Phys. G34, R45 (2007), [hep-ph/0609168]; Ross, Nucl. Phys. B825, 203 (2010), [arXiv:0903.1115]; P. Athron et al., Eur. Phys. J. C76, 62 (2016), S. Cassel, D. M. Ghilencea and G. G. Ross, Nucl. Phys. [arXiv:1510.08071]. B835, 110 (2010), [arXiv:1001.3884]. [193] F. Jegerlehner, Acta Phys. Polon. B49, 1157 (2018), [172] O. Buchmueller et al., Eur. Phys. J. C74, 6, 2922 (2014), [arXiv:1804.07409]; M. Davier et al., Eur. Phys. J. C80, [arXiv:1312.5250]. 3, 241 (2020), [arXiv:1908.00921]. 89. Supersymmetry, Part I (Theory) 921

[194] M. Ibe, T. T. Yanagida and N. Yokozaki, JHEP 08, 067 [211] A. J. Buras, Acta Phys. Polon. B49, 1043 (2018), (2013), [arXiv:1303.6995]. [arXiv:1805.11096]. [195] R. H. Parker et al., Science 360, 191 (2018), [212] V. Cirigliano et al., JHEP 02, 032 (2020), [arXiv:1812.04130]. [arXiv:1911.01359]. [196] D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett. [213] J. R. Ellis et al., JHEP 08, 083 (2007), [arXiv:0706.0652]; 100, 120801 (2008), [arXiv:0801.1134]. S. Heinemeyer et al., JHEP 08, 087 (2008), [arXiv:0805.2359]; G.-C. Cho et al., JHEP 11, 068 [197] B. Dutta and Y. Mimura, Phys. Lett. B790, 563 (2019), (2011), [arXiv:1104.1769]; E. Bagnaschi et al., Eur. Phys. [arXiv:1811.10209]; M. Endo and W. Yin, JHEP 08, 122 J. C78, 256 (2018), [arXiv:1710.11091]. (2019), [arXiv:1906.08768]; M. Badziak and K. Sakurai, JHEP 10, 024 (2019), [arXiv:1908.03607]. [214] A. J. Buras et al., Nucl. Phys. B592, 55 (2001), [hep- ph/0007313]. [198] A. Limosani et al. (Belle Collaboration), Phys. Rev. Lett. 103, 241801 (2009), [arXiv:0907.1384]; J. P. Lees et al. [215] I. Esteban et al., JHEP 01, 106 (2019), [arXiv:1811.05487]. (BaBar Collaboration), Phys. Rev. Lett. 109, 191801 [216] K. Zuber, Phys. Rept. 305, 295 (1998), [hep- (2012), [arXiv:1207.2690]; J. P. Lees et al. (BaBar Collabo- ph/9811267]; S. F. King, J. Phys. G42, 123001 (2015), ration), Phys. Rev. D86, 112008 (2012), [arXiv:1207.5772]. [arXiv:1510.02091]; S. F. King, Prog. Part. Nucl. Phys. 94, [199] M. Misiak et al., Phys. Rev. Lett. 114, 221801 (2015), 217 (2017), [arXiv:1701.04413]. [arXiv:1503.01789]; M. Czakon et al., JHEP 04, 168 (2015), [217] For a review of neutrino masses in supersymmetry, see [arXiv:1503.01791]. e.g., B. Mukhopadhyaya, Proc. Indian National Science [200] H. Baer and M. Brhlik, Phys. Rev. D55, 3201 (1997), Academy A70, 239 (2004); M. Hirsch and J. W. F. Valle, [hep-ph/9610224]; M. Ciuchini et al., Phys. Rev. D67, New J. Phys. 6, 76 (2004), [hep-ph/0405015]. 075016 (2003), [Erratum: D68, 079901 (2003)], [hep- [218] F. Borzumati and Y. Nomura, Phys. Rev. D64, 053005 ph/0212397]; T. Hurth, Rev. Mod. Phys. 75, 1159 (2003), (2001), [hep-ph/0007018]. [hep-ph/0212304]; F. Mahmoudi, JHEP 12, 026 (2007), [219] P. Minkowski, Phys. Lett. 67B, 421 (1977); M. Gell-Mann, [arXiv:0710.3791]; K. A. Olive and L. Velasco-Sevilla, JHEP P. Ramond, and R. Slansky, in Supergravity, edited by 05, 052 (2008), [arXiv:0801.0428]. D. Freedman and P. van Nieuwenhuizen (North Holland, [201] S. R. Choudhury and N. Gaur, Phys. Lett. B451, 86 (1999), Amsterdam, 1979) p. 315; T. Yanagida, Prog. Theor. Phys. [hep-ph/9810307]; K. S. Babu and C. F. Kolda, Phys. 64, 1103 (1980); R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 84, 228 (2000), [hep-ph/9909476]; G. Isidori Rev. Lett. 44, 912 (1980); R. N. Mohapatra and G. Sen- and A. Retico, JHEP 11, 001 (2001), [hep-ph/0110121]; janovic, Phys. Rev. D23, 165 (1981). G. Isidori and A. Retico, JHEP 09, 063 (2002), [hep- [220] J. Hisano et al., Phys. Lett. B357, 579 (1995), [hep- ph/0208159]. ph/9501407]; J. Hisano et al., Phys. Rev. D53, 2442 [202] S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. (1996), [hep-ph/9510309]; J. A. Casas and A. Ibarra, Nucl. Lett. 111, 101804 (2013), [arXiv:1307.5025]; V. Khacha- Phys. B618, 171 (2001), [hep-ph/0103065]; J. R. Ellis tryan et al. (CMS and LHCb Collaborations), Nature 522, et al., Phys. Rev. D66, 115013 (2002), [hep-ph/0206110]; 68 (2015), [arXiv:1411.4413]; M. Aaboud et al. (ATLAS A. Masiero, S. K. Vempati and O. Vives, New J. Phys. Collaboration), JHEP 04, 098 (2019), [arXiv:1812.03017]; 6, 202 (2004), [hep-ph/0407325]; E. Arganda et al., Phys. R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 118, Rev. D71, 035011 (2005), [hep-ph/0407302]; F. R. Joaquim 191801 (2017), [arXiv:1703.05747]. and A. Rossi, Phys. Rev. Lett. 97, 181801 (2006), [hep- [203] J. Aebischer et al., Eur. Phys. J. C 80, 3, 252 (2020), ph/0604083]; J. R. Ellis and O. Lebedev, Phys. Lett. B653, [arXiv:1903.10434]. 411 (2007), [arXiv:0707.3419]. [204] C. Bobeth et al., Phys. Rev. Lett. 112, 101801 (2014), [221] Y. Grossman and H. E. Haber, Phys. Rev. Lett. 78, 3438 [arXiv:1311.0903]. (1997), [hep-ph/9702421]; A. Dedes, H. E. Haber and J. Rosiek, JHEP 11, 059 (2007), [arXiv:0707.3718]. [205] M. Bona et al. (UTfit Collaboration), Phys. Lett. B687, 61 [222] M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Ko- (2010), [arXiv:0908.3470]. valenko, Phys. Lett. B398, 311 (1997), [hep-ph/9701253]; [206] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. Lett. L. J. Hall, T. Moroi and H. Murayama, Phys. Lett. B424, 109, 101802 (2012), [arXiv:1205.5442]; J. P. Lees et al. 305 (1998), [hep-ph/9712515]; K. Choi, K. Hwang and (BaBar Collaboration), Phys. Rev. D88, 072012 (2013), W. Y. Song, Phys. Rev. Lett. 88, 141801 (2002), [hep- [arXiv:1303.0571]. ph/0108028]; T. Honkavaara, K. Huitu and S. Roy, Phys. [207] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. Rev. D73, 055011 (2006), [hep-ph/0512277]. 115, 111803 (2015), [Erratum: 115, 159901 (2015)], [223] L. Basso et al., Comput. Phys. Commun. 184, 698 (2013), [arXiv:1506.08614]; R. Aaij et al. (LHCb Collaboration), [arXiv:1206.4563]. Phys. Rev. Lett. 120, 171802 (2018), [arXiv:1708.08856]; [224] M. Chemtob, Prog. Part. Nucl. Phys. 54, 71 (2005), [hep- M. Huschle et al. (Belle Collaboration), Phys. Rev. ph/0406029]; R. Barbier et al., Phys. Rept. 420, 1 (2005), D92, 072014 (2015), [arXiv:1507.03233]; Y. Sato et al. [hep-ph/0406039]. (Belle Collaboration), Phys. Rev. D94, 072007 (2016), [arXiv:1607.07923]; S. Hirose et al. (Belle Collaboration), [225] H. Dreiner, in Perspectives on Supersymmetry II, edited by Phys. Rev. Lett. 118, 211801 (2017), [arXiv:1612.00529]; G.L. Kane (World Scientific, Singapore, 2010) pp. 565–583. S. Hirose et al. (Belle Collaboration), Phys. Rev. D97, [226] B. C. Allanach, A. Dedes and H. K. Dreiner, Phys. Rev. 012004 (2018), [arXiv:1709.00129]. D60, 075014 (1999), [hep-ph/9906209]. [208] G. Caria et al. (Belle Collaboration), Phys. Rev. Lett. 124, [227] L. E. Ibanez and G. G. Ross, Nucl. Phys. B368, 3 16, 161803 (2020), [arXiv:1910.05864]. (1992); L. E. Ibanez, Nucl. Phys. B398, 301 (1993), [hep- [209] S. Klaver (LHCb Collaboration), in “Proceedings of the 17th ph/9210211]. Conference on Flavor Physics and CP Violation (FPCP [228] A. Dedes, S. Rimmer and J. Rosiek, JHEP 08, 005 (2006), 2019) Victoria, BC, Canada, May 6-10, 2019,” (2019), [hep-ph/0603225]; B. C. Allanach and C. H. Kom, JHEP 04, [arXiv:1907.01500]. 081 (2008), [arXiv:0712.0852]; H. K. Dreiner et al., Phys. [210] F. Mahmoudi, S. Neshatpour and J. Orloff, JHEP 08, 092 Rev. D84, 113005 (2011), [arXiv:1106.4338]. (2012), [arXiv:1205.1845]; A. Arbey et al., JHEP 11, 132 [229] H. K. Dreiner, C. Luhn and M. Thormeier, Phys. Rev. D73, (2017), [arXiv:1707.00426]. 075007 (2006), [hep-ph/0512163]. 922 89. Supersymmetry, Part I (Theory)

[230] K. Tamvakis, Phys. Lett. B382, 251 (1996), [hep- [243] R. D. Peccei, Lect. Notes Phys. 741, 3 (2008), [hep- ph/9604343]; G. Eyal and Y. Nir, JHEP 06, 024 (1999), ph/0607268]. [hep-ph/9904473]; A. Florez et al., Phys. Rev. D87, 095010 (2013), [arXiv:1303.0278]. [244] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977); R. D. Peccei and H. R. Quinn, Phys. Rev. D16, [231] C. Csaki, Y. Grossman and B. Heidenreich, Phys. Rev. D85, 1791 (1977). 095009 (2012), [arXiv:1111.1239]. [232] D. Dercks et al., Eur. Phys. J. C77, 856 (2017), [245] H. Murayama, H. Suzuki and T. Yanagida, Phys. Lett. [arXiv:1706.09418]. B291, 418 (1992); T. Gherghetta and G. L. Kane, Phys. Lett. B354, 300 (1995), [hep-ph/9504420]; K. J. Bae, [233] B. C. Allanach and B. Gripaios, JHEP 05, 062 (2012), H. Baer and H. Serce, Phys. Rev. D91, 1, 015003 (2015), [arXiv:1202.6616]; M. Asano, K. Rolbiecki and K. Saku- [arXiv:1410.7500]; H. Baer, V. Barger and D. Sengupta, rai, JHEP 01, 128 (2013), [arXiv:1209.5778]; N. Chamoun Phys. Lett. B790, 58 (2019), [arXiv:1810.03713]. et al., JHEP 08, 142 (2014), [arXiv:1407.2248]. [234] J. C. Romao, Nucl. Phys. Proc. Suppl. 81, 231 (2000), [hep- [246] A. Delgado, G. Nardini and M. Quiros, Phys. Rev. D86, ph/9907466]; Y. Grossman and S. Rakshit, Phys. Rev. D69, 115010 (2012), [arXiv:1207.6596]. 093002 (2004), [hep-ph/0311310]. [247] P. Fayet, Phys. Lett. 78B, 417 (1978). [235] R. N. Mohapatra, Phys. Rev. D34, 3457 (1986); K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 75, 2276 (1995), [248] K. Benakli, Fortsch. Phys. 59, 1079 (2011), [hep-ph/9506354]; M. Hirsch, H. V. Klapdor-Kleingrothaus [arXiv:1106.1649]. and S. G. Kovalenko, Phys. Rev. Lett. 75, 17 (1995); [249] P. J. Fox, A. E. Nelson and N. Weiner, JHEP 08, 035 (2002), M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Ko- [hep-ph/0206096]. valenko, Phys. Rev. D53, 1329 (1996), [hep-ph/9502385]. [250] K. Benakli and M. D. Goodsell, Nucl. Phys. B816, 185 [236] Y. Grossman and H. E. Haber, Phys. Rev. D59, 093008 (2009), [arXiv:0811.4409]; K. Benakli and M. D. Goodsell, (1999), [hep-ph/9810536]. Nucl. Phys. B840, 1 (2010), [arXiv:1003.4957]. [237] S. Dimopoulos and L. J. Hall, Phys. Lett. B207, 210 (1988); J. Kalinowski et al., Phys. Lett. B406, 314 (1997), [hep- [251] U. Sarkar and R. Adhikari, Phys. Rev. D55, 3836 (1997), ph/9703436]; J. Erler, J. L. Feng and N. Polonsky, Phys. [hep-ph/9608209]; R. Fok et al., Phys. Rev. D87, 055018 Rev. Lett. 78, 3063 (1997), [hep-ph/9612397]. (2013), [arXiv:1208.2784]. [238] H. K. Dreiner, P. Richardson and M. H. Seymour, Phys. [252] G. D. Kribs, E. Poppitz and N. Weiner, Phys. Rev. D78, Rev. D63, 055008 (2001), [hep-ph/0007228]. 055010 (2008), [arXiv:0712.2039]. [239] J. E. Kim and H. P. Nilles, Phys. Lett. 138B, 150 (1984). [253] K. Benakli, M. D. Goodsell and F. Staub, JHEP 06, 073 [240] J. E. Kim and H. P. Nilles, Mod. Phys. Lett. A9, 3575 (2013), [arXiv:1211.0552]. (1994), [hep-ph/9406296]. [254] J. L. Hewett and T. G. Rizzo, Phys. Rept. 183, 193 (1989). [241] G. F. Giudice and A. Masiero, Phys. Lett. B206, 480 (1988); J. A. Casas and C. Munoz, Phys. Lett. B306, 288 [255] S. F. King, S. Moretti and R. Nevzorov, Phys. Lett. (1993), [hep-ph/9302227]; K. J. Bae et al., Phys. Rev. D99, B634, 278 (2006), [hep-ph/0511256]; S. F. King, S. Moretti 115027 (2019), [arXiv:1902.10748]. and R. Nevzorov, Phys. Rev. D73, 035009 (2006), [hep- ph/0510419]. [242] M. Cvetic et al., Phys. Rev. D56, 2861 (1997), [Erratum: D58, 119905 (1998)], [hep-ph/9703317]. 90. Supersymmetry, Part II (Experiment) 923

90. Supersymmetry, Part II (Experiment)

Revised September 2019 by O. Buchmuller (Imperial Coll. Lon- The discovery of a Higgs boson with a mass around 125 GeV don) and P. de Jong (NIKHEF). imposes constraints on SUSY models, which are discussed else- where [27, 33]. 90.1 Introduction In this review we limit ourselves to direct searches, covering Supersymmetry (SUSY), a transformation relating fermions to data analyses at LEP, HERA, the Tevatron and the LHC, with bosons and vice versa [1–9] is one of the most compelling possible emphasis on the latter. For more details on LEP and Tevatron extensions of the Standard Model of particle physics (SM). constraints, see earlier PDG reviews [34]. On theoretical grounds SUSY is motivated as a generalization of space-time symmetries. A low-energy realization of SUSY, i.e., 90.2 Experimental search program SUSY at the TeV scale, is, however, not a necessary consequence. The electron- collider LEP was operational at CERN Instead, low-energy SUSY is motivated by the possible cancella- between 1989 and 2000. In the initial phase, center-of-mass tion of quadratic divergences in radiative corrections to the Higgs energies around the Z-peak were probed, but after 1995 the boson mass [10–15]. Furthermore, it is intriguing that a weakly LEP experiments collected a significant amount of luminosity at −1 interacting, (meta)stable supersymmetric particle might make up √higher center-of-mass energies, some√ 235 pb per experiment at some or all of the dark matter in the universe [16–18]. In addition, s ≥ 204 GeV, with a maximum s of 209 GeV. SUSY predicts that gauge couplings, as measured experimentally Searches for new physics at e+e− colliders benefit from the at the electroweak scale, unify at an energy scale O(1016) GeV clean experimental environment and the fact that momentum bal- (“GUT scale”) near the Planck scale [19–24]. ance can be measured not only in the plane transverse to the In the minimal supersymmetric extension to the Standard beam, but also in the direction along the beam (up to the beam Model, the so called MSSM [11,25,26], a supersymmetry transfor- pipe holes), defined as the longitudinal direction. Searches at LEP mation relates every chiral fermion and gauge boson in the SM to are dominated by the data samples taken at the highest center- a supersymmetric partner with half a unit of spin difference, but of-mass energies. otherwise with the same properties (such as mass) and quantum Constraints on SUSY have been set by the CDF and D0 experi- numbers. These are the “sfermions”: squarks (q˜) and sleptons (`˜, ments at the Tevatron, a proton- collider at a center-of- ν˜), and the “gauginos”. The MSSM Higgs sector contains two dou- mass energy of up to 1.96 TeV. CDF and D0 collected integrated −1 blets, for up-type quarks and for down-type quarks and charged luminosities between 10 and 11 fb each up to the end of collider leptons respectively. After electroweak symmetry breaking, five operations in 2011. Higgs bosons arise, of which two are charged. The supersym- The electron-proton collider HERA provided collisions to the metric partners of the Higgs doublets are known as “higgsinos.” H1 and ZEUS experiments between 1992 and 2007, at a center- The weak gauginos and higgsinos mix, giving rise to charged mass of-mass energy up to 318 GeV. A total integrated luminosity of −1 eigenstates called “charginos” (χ˜±), and neutral mass eigenstates approximately 0.5 fb was collected by each experiment. Since 0 at HERA collide with leptons, SUSY searches at HERA called “neutralinos” (χ˜ ). The SUSY partners of the gluons are typically look for R-parity violating production of single SUSY known as “gluinos” (g˜). The fact that such particles are not yet particles. observed leads to the conclusion that, if supersymmetry is real- The Large Hadron Collider (LHC) at CERN started proton- ized, it is a broken symmetry. A description of SUSY in the form proton operation at a center-of-mass energy of 7 TeV in 2010. By of an effective Lagrangian with only “soft” SUSY breaking terms the end of 2011 the experiments ATLAS and CMS had collected and SUSY masses at the TeV scale maintains the cancellation of about 5 fb−1 of integrated luminosity each, and the LHCb experi- quadratic divergences of soft SUSY breaking scalar mass squared ment had collected approximately 1 fb−1. In 2012, the LHC oper- parameters. ated at a center-of-mass energy of 8 TeV, and ATLAS and CMS The phenomenology of SUSY is to a large extent determined collected approximately 20 fb−1 each, whereas LHCb collected by the SUSY breaking mechanism and the SUSY breaking scale. 2 fb−1. In 2015, the LHC started Run 2, with a center-of-mass This determines the SUSY particle masses, the mass hierarchy, energy of 13 TeV. At the end of Run 2 in November 2018, ATLAS the field contents of physical particles, and their decay modes. In and CMS had both collected approximately 140 fb−1, and LHCb addition, phenomenology crucially depends on whether the multi- had collected almost 6 fb−1. plicative quantum number of R-parity [26], R = (−1)3(B−L)+2S , Proton-(anti)proton colliders produce interactions at higher where B and L are baryon and lepton numbers and S is the spin, center-of-mass energies than those available at LEP, and cross is conserved or violated. If R-parity is conserved, SUSY particles sections of QCD-mediated processes are larger, which is reflected (sparticles), which have odd R-parity, are produced in pairs and in the higher sensitivity for SUSY particles carrying color charge: the decays of each SUSY particle must involve an odd number squarks and gluinos. Large background contributions from Stan- of lighter SUSY particles. The lightest SUSY particle (LSP) is dard Model processes, however, pose challenges to the trigger and then stable and often assumed to be a weakly interacting massive analysis. Such backgrounds are dominated by multijet produc- particle (WIMP). If R-parity is violated, new terms λ , λ0 ijk ijk tion processes, including, particularly at the LHC, those of top 00 and λijk appear in the superpotential, where ijk are generation quark production, as well as jet production in association with indices; λ-type couplings appear between lepton superfields only, vector bosons. The proton momentum is shared between its par- 00 0 λ -type are between quark superfields only, and λ -type couplings ton constituents, and in each collision only a fraction of the to- connect the two. R-parity violation implies lepton and/or baryon tal center-of-mass energy is available in the hard parton-parton number violation. More details of the theoretical framework of scattering. Since the parton momenta in the longitudinal direc- SUSY are discussed elsewhere in this volume [27]. tion are not known on an event-by-event basis, use of momentum Today, low-energy data from flavor physics experiments, high- conservation constraints in an analysis is restricted to the trans- precision electroweak observables as well as astrophysical data verse plane, leading to the definition of transverse variables, such impose strong constraints on the allowed SUSY parameter space. as the missing transverse momentum, and the transverse mass. Recent examples of such data include measurements of the rare B- Proton-proton collisions at the LHC differ from proton-antiproton + − meson decay Bs → µ µ [28,29], measurements of the anomalous collisions at the Tevatron in the sense that there are no valence magnetic moment of the muon [30], and accurate determinations anti-quarks in the proton, and that gluon-initiated processes play of the cosmological dark matter relic density constraint [31, 32]. a more dominant role. The increased center-of-mass energy of the These indirect constraints are often more sensitive to higher LHC compared to the Tevatron, as well as the increase at the SUSY mass scales than experiments searching for direct sparticle LHC between Run 1 and Run 2, significantly extends the kine- production at colliders, but the interpretation of these results is matic reach for SUSY searches. This is reflected foremost in the often strongly model dependent. In contrast, direct searches for sensitivity for squarks and gluinos, but also for other SUSY par- sparticle production at collider experiments are less subject to ticles. interpretation ambiguities and therefore they play a crucial role The main production mechanisms of massive colored sparticles in the search for SUSY. at hadron colliders are squark-squark, squark-gluino and gluino- 924 90. Supersymmetry, Part II (Experiment)

gluino production; when “squark” is used “antisquark” is also im- plied. Assuming R-parity conservation, the typical SUSY search signature at hadron colliders contains high-pT jets, which are pro- duced in the decay chains of heavy squarks and gluinos, and sig- nificant missing momentum originating from the two LSPs pro- duced at the end of the decay chain, which escape experimental detection. Standard Model backgrounds with missing transverse momentum include leptonic W/Z-boson decays, heavy-flavor de- cays to neutrinos, and multijet events that may be affected by instrumental effects such as jet mismeasurement. Selection variables designed to separate the SUSY signal from miss the Standard Model backgrounds include HT, ET , and meff. miss The quantities HT and ET refer to the measured transverse energy and the missing transverse momentum in the event, re- spectively. They are usually defined as the scalar sum of the transverse jet momenta or calorimeter clusters transverse ener- miss gies measured in the event (HT), or the magnitude (ET ) of the negative vector sum of transverse momenta of reconstructed miss Figure 90.1: Cross sections for pair production of different spar- objects like jets and leptons in the event (~pT ). The quantity meff is referred to as the effective mass of the event and is de- ticles as a function of their mass at the LHC for a center-of-mass miss energy of 8 TeV (solid curves) and 13-14 TeV (dotted curves), fined as meff = HT + ET . The peak of the meff distribution for SUSY signal events correlates with the SUSY mass scale, in par- taken from Ref. [47]. Typically the production cross section of ticular with the mass difference between the primary produced √colored squarks and gluinos, calculated with NLL-FAST [48] at SUSY particle and the LSP [35], whereas the Standard Model s =8 and 13 TeV, is several orders of magnitude larger than backgrounds dominate at low meff. Additional reduction of mul- √the one for electroweak gauginos, calculated with Prospino [49] at tijet backgrounds can be achieved by demanding isolated leptons s =8 and 14 TeV for higgsino-like neutralinos. Except for the ex- or photons in the final states; in such events the lepton or photon plicitly shown pair production of stops, production cross sections transverse momentum may be added to HT or meff for further for squarks assumes mass degeneracy of left- and right-handed u, signal-background separation. d, s, c and b squarks. At the LHC, alternative approaches have been developed to increase the sensitivity to pair production of heavy sparticles with TeV-scale masses focusing on the kinematics of their decays, the largest production cross sections in proton-proton collisions, and to further suppress the background from multijet produc- and thus the LHC searches typically provide the tightest mass tion. Prominent examples of these new approaches are searches limits on these colored sparticles. This, however, implies that using the αT [36–40], razor [41], stransverse mass (mT2) [42], and the allowed parameter space of constrained SUSY models today contransverse mass (mCT) [43] variables. Recently, the topologi- has been restrained significantly by searches from ATLAS and cal event reconstruction methods have expanded with the super- CMS. Furthermore, confronting the remaining allowed parameter razor [44] and recursive jigsaw reconstruction [45] techniques. space with other collider and non-collider measurements, which Furthermore, frequently the searches for massive SUSY particles are directly or indirectly sensitive to contributions from SUSY, the attempt to identify their decay into top quarks or vector bosons, overall compatibility of these models with all data is significantly which are themselves unstable. If these are produced with a signif- worse than in the pre-LHC era (see section II.8 for further discus- icant boost, jets from their decay will typically overlap, and such sion), indicating that very constrained models like the CMSSM topologies are searched for with jet-substructure [46] techniques. are no longer good benchmark scenarios to solely characterize the results of SUSY searches at the LHC. 90.3 Interpretation of results For these reasons, an effort has been made to complement the Since the mechanism by which SUSY is broken is unknown, a traditional constrained models with more flexible approaches. general approach to SUSY via the most general soft SUSY break- One approach to study a broader and more comprehensive ing Lagrangian adds a significant number of new free parameters. subset of the MSSM is via the phenomenological-MSSM, or For the minimal supersymmetric standard model, MSSM, i.e., the pMSSM [59–61]. It is derived from the MSSM, using experimental model with the minimal particle content, these comprise 105 new data to eliminate parameters that are free in principle but have real degrees of freedom. A phenomenological analysis of SUSY already been highly constrained by measurements of e.g., flavor searches leaving all these parameters free is not feasible. For the mixing and CP-violation. This effective approach reduces the practical interpretation of SUSY searches at colliders several ap- number of free parameters in the MSSM to typically 19 or even proaches are taken to reduce the number of free parameters. less, making it a practical compromise between the full MSSM One approach is to assume a SUSY breaking mechanism and and highly constrained models such as the CMSSM. lower the number of free parameters through the assumption Even less dependent on fundamental assumptions are interpre- of additional constraints. Before the start of the LHC, in- tations in terms of so-called simplified models [62–65]. Such mod- terpretations of experimental results were predominately per- els assume a limited set of SUSY particle production and decay formed in constrained models of gravity mediated [50,51], gauge- modes and leave open the possibility to vary masses and other mediated [52–54], and anomaly mediated [55, 56] SUSY break- parameters freely. Therefore, simplified models enable compre- ing. The most popular model was the constrained MSSM hensive studies of individual SUSY topologies, and are useful for (CMSSM) [50, 57, 58], which in the literature is also referred to optimization of the experimental searches over a wide parameter as minimal supergravity, or MSUGRA. space without limitations on fundamental kinematic properties These constrained SUSY models are theoretically well moti- such as masses, production cross sections, and decay modes. vated and provide a rich spectrum of experimental signatures. As a consequence, ATLAS and CMS have adopted simplified However, with universality relations imposed on the soft SUSY models as the primary framework to provide interpretations of breaking parameters, they do not cover all possible kinematic sig- their searches. In addition to using simplified models that describe natures and mass relations of SUSY. In such scenarios the squarks prompt decays of SUSY particles, the experiments are now also are often nearly degenerate in mass, in particular for the first focusing more on the use of simplified models that allow for decays and second generation. The exclusion of parameter space in the of long-lived SUSY particles as they can arise in different SUSY CMSSM and in CMSSM-inspired models is mainly driven by first scenarios (see Section 90.7 for further discussion). Today, almost and second generation squark production together with gluino every individual search provides interpretations of their results in production. As shown in Fig. 90.1 [47–49] these processes possess one or even several simplified models that are characteristic of 90. Supersymmetry, Part II (Experiment) 925

~~ ~ ∼0 ∼0 gg production, B(g → qq χ )=100% Squark-gluino-neutralino model, m(χ )=995 GeV 1800 1 7000 1 ± σ Exp. limit ( 1 exp) ATLAS Preliminary ± σSUSY ATLAS Preliminary ) [GeV] ) [GeV] 1600 Obs. limit ( 1 theory)

0 1 -1 -1 ~ q χ ∼ s=13 TeV, 139 fb s=13 TeV, 139 fb Exp. limits MB 6000 m( m( 1400 0-leptons, 2-6 jets Exp. limits BDT 0-leptons, 2-6 jets -1 All limits at 95 % CL 0L obs. 36 fb All limits at 95 % CL [arXiv:1712.02332] 1200 5000 ± σ Exp. limit ( 1 exp) Obs. limit (±1 σSUSY) 1000 theory Exp. limits MB 4000 Exp. limits BDT 800 -1 Kinematically Forbidden 0L obs. 36 fb [arXiv:1712.02332] 600 3000

400 2000 200

0 1000 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 1000 2000 3000 4000 5000 6000 7000 m(g~) [GeV] m(g~) [GeV]

0 Figure 90.2: Left: lower mass limits, at 95% C.L., on gluino pair production and decay in a simplified model with g˜ → qq¯χ˜1. Right: 95% C.L. mass limits on gluinos and squarks assuming gluino and squark production, and m 0 = 995 GeV. Results of the ATLAS χ˜1 collaboration.

SUSY topologies probed by the analysis. cay may involve the lightest neutralino (typically the LSP) or However, while these models are very convenient for the inter- chargino, but, depending on the masses and couplings of the gaug- pretation of individual SUSY production and decay topologies, inos, may involve heavier neutralinos or charginos. For pair pro- care must be taken when applying these limits to more complex duction of first and second generation squarks, the simplest de- SUSY spectra. Therefore, in practice, simplified model limits are cay modes involve two jets and missing momentum, with poten- often used as an approximation of the constraints that can be tial extra jets stemming from initial state or final state radiation placed on sparticle masses in more complex SUSY spectra. Yet, (ISR/FSR) or from decay modes with longer decay chains (cas- depending on the assumed SUSY spectrum, the sparticle of in- cades). Similarly, gluino pair production leads to four jets and terest, and the considered simplified model limit, this approxi- missing momentum, and possibly additional jets from ISR/FSR mation can lead to a significant mistake, typically an overesti- or cascades. Associated production of a gluino and a (anti-)squark mation, in the assumed constraint on the sparticle mass (see for is also possible, in particular if squarks and gluinos have similar example [66]). Only on a case-by-case basis can it be determined masses, typically leading to three or more jets in the final state. whether the limit of a given simplified model represents a good In cascades, isolated photons or leptons may appear from the de- approximation of the true underlying constraint that can be ap- cays of sparticles such as neutralinos or charginos. Final states are plied on a sparticle mass in a complex SUSY spectrum. In the thus characterized by significant missing transverse momentum, following, we will point out explicitly the assumptions that have and at least two, and possibly many more high pT jets, which entered the limits when quoting interpretations from simplified can be accompanied by one or more isolated objects like pho- models. tons or leptons, including τ leptons, in the final state. Table 90.1 This review covers results up to September 2019 and since none shows a schematic overview of characteristic final state signatures of the searches performed so far have shown significant excess of gluino and squark production for different mass hierarchy hy- above the SM background prediction, the interpretation of the potheses and assuming decays involving the lightest neutralino. presented results are exclusion limits on SUSY parameter space. Unless stated differently, all quoted exclusion limits are at 95% Table 90.1: Typical search signatures at hadron confidence level. colliders for direct gluino and first- and second- generation squark production assuming different 90.4 Exclusion limits on gluino and squark mass hierarchies. masses Gluinos and squarks are the SUSY partners of gluons and Mass Main Dominant Typical quarks, and thus carry color charge. Limits on squark masses Hierarchy Production Decay Signature of the order 100 GeV have been set by the LEP experiments [67], ¯ 0 miss mq˜  mg˜ q˜q,˜ q˜q˜ q˜ → qχ˜1 ≥ 2 jets + ET + X in the decay to quark plus neutralino, and for a mass difference be- ¯ 0 miss mq˜ ≈ mg˜ q˜g,˜ q˜g˜ q˜ → qχ˜1 ≥ 3 jets + ET + X tween squark and quark plus neutralino of typically at least a few 0 g˜ → qq¯χ˜1 GeV. However, due to the colored production of these particles at 0 miss mq˜  mg˜ g˜g˜ g˜ → qq¯χ˜1 ≥ 4 jets + E + X hadron colliders (see e.g. Fig. 90.1), hadron collider experiments T are able to set much tighter mass limits. Pair production of these massive colored sparticles at hadron 90.4.1 Exclusion limits on the gluino mass colliders usually involve both the s-channel and t-channel parton- Limits set by the Tevatron experiments on the gluino mass as- parton interactions. Since there is a negligible amount of bottom sume the framework of the CMSSM, with tan β = 5 (CDF) or and top quark content in the proton, top- and bottom squark pro- tan β = 3 (D0), where tan β is the ratio of vacuum expectation duction proceeds through s-channel diagrams only. In the past, values of the Higgs fields for up-type and down-type fermions. experimental analyses of squark and/or gluino production typi- Furthermore, A0 = 0 and µ < 0 is assumed, and the resulting cally assumed the first and second generation squarks to be ap- lower mass limits are about 310 GeV for all squark masses, or proximately degenerate in mass. However, in order to have even 390 GeV for the case mq˜ = mg˜ [68, 69]. These limits have been less model dependent interpretations of the searches, the exper- superseded by those provided by ATLAS and CMS, and the tight- iments have started to also provide simplified model limits on est constraints have been set with up to approximately 140 fb−1 individual first or second generation squarks. of data recorded at the LHC at a center-of-mass energy of 13 TeV. Assuming R-parity conservation and assuming gluinos to be Limits on the gluino mass have been established in the frame- heavier than squarks, squarks will predominantly decay to a quark work of simplified models. Assuming only gluino pair production, and a neutralino or chargino, if kinematically allowed. The de- in particular three primary decay chains of the gluino have been 926 90. Supersymmetry, Part II (Experiment)

∼ ∼ → ~~ ~ → χ0 → ~ ~ ~ → χ0 pp g g, g bb 1 August 2019 pp g g, g tt 1 August 2019 2000 2000 137 fb-1 (13 TeV) 137 fb-1 (13 TeV) 1800 CMS Preliminary 1800 CMS Preliminary miss 1705.04650 (M ), 36 fb-1 1908.04722, 0-lep (H )

[GeV] T2 [GeV] T 0 1 miss Expected 0 1 Expected

∼ χ 1908.04722, 0-lep (HT ) ∼ χ SUS-19-005, 0-lep (MT2) 1600 Observed 1600 Observed SUS-19-005, 0-lep (MT2) SUS-19-007, 1-lep (MJ) m m SUS-19-008, ≥2-lep (same-sign) 1400 1400 1710.11188, 0-lep (stop), 36 fb-1

1200 1200

1000 1000

800 800

600 600

400 400

200 200

0 0 800 1000 1200 1400 1600 1800 2000 2200 2400 800 1000 1200 1400 1600 1800 2000 2200 2400

mg~ [GeV] mg~ [GeV]

Figure 90.3: Lower mass limits, at 95% C.L., on gluino pair production for various decay chains in the framework of simplified models. 0 0 Left: g˜ → b¯bχ˜1. Right: g˜ → tt¯χ˜1. Results of the CMS collaboration. considered by the LHC experiments for interpretations of their trinos in the W decay and from the two neutralinos. As shown in 0 search results. The first decay chain g˜ → qq¯χ˜1 assumes gluino Fig. 90.3 (right), the CMS search based on the mT2 variable [73] mediated production of first and second generation squarks (on- rules out gluinos with masses below ≈ 2.25 TeV for massless neu- shell or off-shell) which leads to four light flavor quarks in the tralinos in this model. For neutralino masses above ≈ 1.3 TeV, final state. Therefore, inclusive all-hadronic analyses searching no limit can be placed on the gluino mass. The ATLAS multiple miss b-jets search [72] obtains similar limits. for multijet plus ET final states are utilized to put limits on this simplified model. These limits are derived as a function of The ATLAS collaboration also provides limits in a pMSSM- the gluino and neutralino (LSP) mass. As shown in Fig. 90.2 inspired model with only gluinos and first and second generation 0 (left), using the cross section from next-to-leading order QCD squarks, and a bino-like χ˜1 [70]. As shown in Fig. 90.2 (right), corrections and the resummation of soft gluon emission at next-to- assuming mχ˜0 = 995 GeV, gluinos with masses below ≈ 1.6 TeV leading-logarithmic accuracy as reference [48], the ATLAS collab- 1 are excluded for any squark mass. For mq˜ ≈ mg˜, the mass exclu- oration [70] excludes in this simplified model gluino masses below sion is about 3.0 TeV. The dependence of these limits on mχ˜0 approximately 2.3 TeV, for a massless neutralino. In scenarios 1 0 where neutralinos are not very light, the efficiency of the analyses is illustrated in Ref. [70]. For massless χ˜1, gluino masses below is reduced by the fact that jets are less energetic, and there is less 2.2 TeV are excluded for all squark masses. missing transverse momentum in the event. This leads to weaker R-parity violating gluino decays are searched for in a number limits when the mass difference ∆m = mg˜ − m 0 is reduced. For of final states. Searches in multilepton final states set lower mass χ˜1 example, for neutralino masses above about 1.2 TeV no limit on limits of 1 to 1.4 TeV, depending on neutralino mass and lepton 0 the gluino mass can be set for this decay chain. Therefore, limits flavor, on decays mediated by λ and λ couplings [74–78], assum- on gluino masses are strongly affected by the assumption of the ing prompt decays. Searches for displaced vertices are sensitive to neutralino mass. Similar results for this simplified model have non-prompt decays [79–82]. Multijet final states have been used to 00 been obtained by CMS [71]. search for fully hadronic gluino decays involving λ , by CDF [83], ATLAS [79, 84–86] and CMS [87–89]. Lower gluino mass limits The second important decay chain of the gluino considered for range between 600 and 2000 GeV depending on neutralino mass interpretation in a simplified model is g˜ → b¯bχ˜0. Here the de- 1 and flavor content of the final state. cay is mediated via bottom squarks and thus leads to four jets miss from b quarks and ET in the final state. Also for this topol- 90.4.2 Exclusion limits on squark masses ogy inclusive all-hadronic searches provide the highest sensitiv- Limits on first and second generation squark masses set by the ity. However, with four b quarks in the final state, the use of Tevatron experiments assume the CMSSM, and amount to lower secondary vertex reconstruction for the identification of jets origi- limits of about 380 GeV for all gluino masses, or 390 GeV for the nating from b quarks provides a powerful handle on the SM back- case mq˜ = mg˜ [68, 69]. miss ground. Therefore, in addition to a multijet plus ET signature At the LHC, limits on squark masses have been set using up these searches also require several jets to be tagged as b-jets. As to approximately 140 fb−1 of data at 13 TeV. Interpretations in shown in Fig. 90.3 (left), for this simplified model CMS [71] ex- simplified models typically characterize squark pair production cludes gluino masses below ≈ 2.3 TeV for a massless neutralino, 0 with only one decay chain of q˜ → qχ˜1. Here it is assumed that the while for neutralino masses above ≈ 1.5 TeV no limit on the gluino left and right-handed u˜, d˜, s˜ and c˜ squarks are degenerate in mass. mass can be set. Comparable limits for this simplified model are Furthermore, it is assumed that the mass of the gluino is very high provided by searches from ATLAS [72]. and thus contributions of the corresponding t-channel diagrams Gluino decays are not limited to first and second generation to squark pair production are negligible. Therefore, the total squarks or bottom squarks; if kinematically allowed, decays to production cross section for this simplified model is eight times top squarks via g˜ → tt˜ are also possible. This leads to a “four the production cross section of an individual squark (e.g. u˜ ). 0 0 L tops” final state ttttχ˜1χ˜1 and defines the third important simpli- Under these assumptions, ATLAS obtains a lower squark mass 0 fied model, g˜ → tt¯χ˜1, characterizing gluino pair production. The limit of ≈ 1.9 TeV for light neutralinos [70], as shown in Fig. 90.4 topology of this decay is very rich in different experimental signa- (left). The effects of heavy neutralinos on squark limits are similar tures: as many as four isolated leptons, four b-jets, several light to those discussed in the gluino case (see Section 90.4.1), and only flavor quark jets, and significant missing momentum from the neu- for neutralino masses below ≈ 800 GeV can any squark masses 90. Supersymmetry, Part II (Experiment) 927

~~ ~ ∼ ~~ ~ → χ∼0 → → χ0 qq production, B(q q )=100% pp tt, t t 1 August 2019 1600 1 ± σ -1 ATLAS Preliminary Exp. limit ( 1 exp) CMS Preliminary 137 fb (13 TeV) ) [GeV]

0 1 -1 -1 1400 SUSY 1000 χ ∼ s=13 TeV, 139 fb 1711.00752, 0-, 1- and 2-lep (stop), 36 fb ± σ [GeV]

Obs. limit ( 1 ) 0 1 theory miss Expected ∼ χ 1908.04722, 0-lep (HT ) m( 0-leptons, 2-6 jets Observed -1 SUS-19-005, 0-lep (MT2) 1200 0L obs. 36 fb m All limits at 95 % CL [arXiv:1712.02332] SUS-19-009, 1-lep (stop) 800 1000

600 800 Kinematically Forbidden

600 400

400 200

0 200 ∼ χ 1 + m t = m ~ 0 m t 0 200 400 600 800 1000 1200 1400 600 800 1000 1200 1400 1600 1800 2000 ~ ~ m(q) [GeV] mt [GeV]

Figure 90.4: Left: 95% C.L. exclusion contours in the squark-neutralino mass plane defined in the framework of simplified models 0 assuming a single decay chain of q˜ → qχ˜1, obtained by ATLAS. Right: the 95% C.L. exclusion contours in the stop-neutralino mass 0 plane defined in the framework of a simplified model assuming a single decay chain of t˜ → tχ˜1 as obtained by CMS.

be excluded. Bottom squarks are expected to decay predominantly to bχ˜0 miss For the same analysis ATLAS also provides an interpretation of giving rise to the characteristic multi b-jet and ET signature. their search result in the aforementioned pMSSM-inspired model Direct production of bottom squark pairs has been searched for with only gluinos and first and second generation squarks, and at the Tevatron and at the LHC. Limits from the Tevatron are 0 m > 247 GeV for a massless neutralino [91] [92]. The LHC a bino-like χ˜1 [70], as shown in Fig. 90.2 (right). In this model, ˜b squark production can take place with non-decoupled gluinos, en- experiments have surpassed these limits, and the latest results are −1 √ hancing the squark production cross section through gluino ex- based on up to 140 fb of data collected at s = 13 TeV. CMS change diagrams. has set a lower limit of m˜b >≈ 1250 GeV for massless neutralinos in this model [73]. For m 0 ≈ 700 GeV or higher no limit can be If the assumption of mass degenerate first and second genera- χ˜1 tion squarks is dropped and only the production of a single light placed on direct bottom squark pair production in this simplified squark is assumed, the limits weaken significantly. For example, model. Limits from ATLAS are comparable [93]. Further bottom the CMS limit on degenerate squarks of 1750 GeV for light neu- squark decay modes have also been searched for by ATLAS [94,95] tralinos drops to ≈ 1300 GeV for pair production of a single light and CMS [71, 76, 96]. squark, and for neutralinos heavier than ≈ 600 GeV no squark The top squark decay modes depend on the SUSY mass spec- mass limit can be placed [73]. It should be noted that this limit is trum, and on the t˜L-t˜R mixture of the top squark mass eigen- not a result of a simple scaling of the above mentioned mass limits state. If kinematically allowed, the two-body decays t˜ → tχ˜0 assuming eightfold mass degeneracy but it also takes into account ˜ χ˜± (which requires mt˜ − m 0 > mt) and t → b (which requires that for an eight times lower production cross section the analy- χ˜ mt˜−mχ˜± > mb) are expected to dominate. If not, the top squark ses must probe kinematic regions of phase space that are closer to 0 the ones of SM background production. Since signal acceptance decay may proceed either via the two-body decay t˜ → cχ˜ or 0 and the ratio of expected signal to SM background events of the through t˜ → bff¯0χ˜ (where f and f¯0 denote a fermion-antifermion analyses are typically worse in this region of phase space not only pair with appropriate quantum numbers). For mt˜−mχ˜0 > mb the the 1/8 reduction in production cross section but also a worse latter decay chain represents a four-body decay with a W boson, analysis sensitivity are responsible for the much weaker limit on charged Higgs H, slepton `˜, or light flavor squark q˜, exchange. If single squark pair production. the exchanged W boson and/or sleptons are kinematically allowed

For single light squarks ATLAS also reports results of a ded- to be on-shell ((mt˜−mχ˜± ) > (mb+mW ) and/or (mt˜−m`˜) > mb), icated search for pair production of scalar partners of charm the three-body decays t˜ → W bχ˜0 and/or t˜ → bl`˜ will become quarks [90]. Assuming that the scalar-charm state exclusively dominant. For further discussion on top squark decays see for decays into a and a neutralino, scalar-charm masses example Ref. [97]. up to 800 GeV are excluded for neutralino masses below 260 GeV. ˜ Limits from LEP on the t1 mass are mt˜ > 96 GeV in the charm Besides placing stringent limits on first and second generation plus neutralino final state, and > 93 GeV in the lepton, b-quark squark masses, the LHC experiments also search for the produc- and sneutrino final state [67]. tion of third generation squarks. SUSY at the TeV-scale is often The Tevatron experiments have performed a number of searches motivated by naturalness arguments, most notably as a solution to for top squarks, often assuming direct pair production. In the b`ν˜ cancel quadratic divergences in radiative corrections to the Higgs decay channel, and assuming a 100% branching fraction, limits are boson mass. In this context, the most relevant terms for SUSY set as m > 210 GeV for m < 110 GeV and m − m > 30 GeV, phenomenology arise from the interplay between the masses of t˜ ν˜ t˜ ν˜ ˜ χ˜0 the third generation squarks and the Yukawa coupling of the top or mt˜ > 235 GeV for mν˜ < 50 GeV [98] [99]. In the t → c 1 quark to the Higgs boson. This motivates a potential constraint on decay mode, a top squark with a mass below 180 GeV is excluded the masses of the top squarks and the left-handed bottom squark. for a neutralino lighter than 95 GeV [100] [101]. In both analy- ses, no limits on the top squark can be set for heavy sneutrinos Due to the large top quark mass, significant mixing between t˜L ˜ χ˜± and t˜R is expected, leading to a lighter mass state t˜1 and a heavier or neutralinos. In the t → b 1 decay channel, searches for a rela- mass state t˜2. In the MSSM, the lightest top squark (t˜1) can be tively light top squark have been performed in the dilepton final 0 the lightest squark. state [102] [103]. The CDF experiment sets limits in the t˜− χ˜1 928 90. Supersymmetry, Part II (Experiment)

∼ ∼ → χ± χ0 -1 July 2019 ATLAS Preliminary s=8,13 TeV, 20.3-139 fb-1 All limits at 95% CL CMS pp 1 2 35.9 fb (13 TeV) 400 ∼ ∼ 700 Expected limits χ0 χ0 ~ ∼ ∼ ∼ ∼ ± ∼ B( →H ) = 1 (WH) m( l / τ / ν ) = 1 [ m( χ 0 ) + m( χ , χ 0 ) ] ∼2 ∼1 Expected L L 2 1 1 2 B(χ0→Z χ0) = 1 (WZ) Observed limits ∼2 ∼1 ∼ ∼ 350 χ0 χ0 χ0 χ0 Observed [GeV] → →

) [GeV] B( Z ) = B( H ) = 0.5 (WH+WZ) 0 1 2 1 2 1 0

600 ∼ χ 1 ∼ ± ∼ 0 χ χ χ ∼ via Z H

1 2 m 300 +m +m 0 ∼ 0 0 ~ ∼ ∼ 1 ∼ χ 1 χ χ 1

m( l / ν 2l+3l 500 L = m = m = m ± ∼ ± ± arXiv:1509.07152 ∼χ 1 χ 1 ∼χ 1 0 ) ∼ 250 m m m χ 1 arXiv:1803.02762 400 ) = m( ∼ 0 ∼+ ∼ − 200 χ 2 χ χ via 1 1 m( ~ ν∼ 300 lL / 2l 150 arXiv:1509.07152

ATLAS-CONF-2019-008 200 ∼τ ν∼ τ 100 L / τ 2 arXiv:1407.0350 100 arXiv:1708.07875 50

∼ ∼ ∼ ± ∼ χ+ χ −/χ χ 0 via 0 0 1 1 1 2 200 300 400 500 600 700 200 400 600 800 1000 1200 ∼τ ν∼ τ L / τ 2 ∼ ± ∼ 0 ∼ ∼ m( χ , χ ) [GeV] arXiv:1708.07875 mχ0 = mχ± [GeV] 1 2 2 1

Figure 90.5: LHC exclusion limits on chargino and neutralino masses in a number of simplified models. Left: limits on chargino and neutralino masses for pair production of charginos, pair production of heavier neutralinos, or pair production of chargino and neutralino, under the assumption of light sleptons mediating the decays. Right: limits on chargino and neutralino masses for pair production of chargino and neutralino, under the assumption of decoupled sleptons, and chargino/neutralino decay through W ∗, Z∗ or H.

July 2019 mass plane for various branching fractions of the chargino decay 600

1 to leptons and for two value of m ± . For m ± = 105.8 GeV 8 TeV, 20.3 fb− `˜ [e˜ , µ˜ ] arXiv:1403.5294 χ˜ χ˜ ATLAS Preliminary ∈ 1 1 2` compressed `˜ [e˜ , µ˜ ] CONF-2019-014 √ 1 ∈ and mχ˜0 = 47.6 GeV, top squarks between 128 and 135 GeV are 500 s = 13 TeV, 139 fb− 2` `˜ [e˜ , µ˜ ] CONF-2019-008 1 ∈ ) [GeV] `˜+ `˜ `˜ `χ0 2τ hadronic `˜ = τ˜ CONF-2019-018 excluded for W -like leptonic branching fractions of the chargino. 0 1 pp L,R L−,R, ˜ 1 → → LEP µ˜ excluded ˜ R χ The LHC experiments have improved these limits substantially. ( 400 All limits at 95% CL Observed limits m As shown in the right plot of Fig. 90.4, limits on the top squark Expected limits mass assuming a simplified model with a single decay chain of 0 300 ˜ χ˜ 0 ) t → t 1 now surpass 1 TeV. The most important searches for this (χ˜ 1 m < top squark decay topology are dedicated searches requiring zero ) R miss ˜ L, or one isolated lepton, modest E , and four or more jets out of 200 (` T m which at least one jet must be reconstructed as a b-jet [71,73,104– 106]. For example, CMS excludes top squarks with masses below 100 about 1200 GeV in this model for massless neutralinos, while for m 0 > 600 GeV no limits can be provided. χ˜1 0 Assuming that the top squark decay exclusively proceeds via 100 200 300 400 500 600 700 800 ± ± the chargino mediated decay chain t˜ → bχ˜ , χ˜ → W ±(∗)χ˜0 1 1 1 m(`˜L,R) [GeV] yields stop mass exclusion limits that vary strongly with the as- ± 0 sumptions made on the t˜− χ˜1 − χ˜1 mass hierarchy. For exam- Figure 90.6: LHC exclusion limits on slepton (selectron and ple, for m ± = (m + m 0 )/2, a stop mass below ≈ 1150 GeV χ˜ t˜ χ˜ smuon) masses, assuming equal masses of selectrons and smuons, 1 1 ˜ ˜ 0 degeneracy of `L and `R, and a 100% branching fraction for for a light χ˜1 is excluded, while no limit can be placed for 0 `˜→ `χ˜1. m 0 > 550 GeV [104]. These limits, however, can weaken sig- χ˜1 nificantly when other assumptions about the mass hierarchy or the decay of the charginos are imposed [104, 106–108]. bff¯0χ˜0. Here the ATLAS one-lepton [106] and two-lepton [110] ˜ χ˜0 ˜ χ˜± χ˜± ±(∗)χ˜0 If the decays t → t 1 and t → b 1 , 1 → W 1 are searches exclude up to mt˜ ≈ 440 GeV for mχ˜0 below 340 GeV, 0 0 1 kinematically forbidden, the decay chains t˜ → W bχ˜ and t˜ → cχ˜ while the monojet analysis [115] excludes at the kinematic bound- can become important. The one-lepton ATLAS search provides ary top squarks below 400 GeV. As for the t˜ → cχ˜0 decay, CMS for the kinematic region mt˜ − mχ˜± > mb + mW lower limits on uses the zero-lepton searches [111, 113] to also place constraints the top squark mass of ≈ 700 GeV for a neutralino lighter than 0 0 on t˜ → bff¯ χ˜ . Also in this case CMS excludes m˜ ≈ 550 GeV ≈ 570 GeV [109]. Other analyses with zero, one or two leptons t for mχ˜0 below 450 GeV. also target this kinematic region [105, 106, 110–114]. 1 In general, the variety of top squark decay chains in the phase For the kinematic region in which even the production of real ˜ χ˜0 W bosons is not allowed, ATLAS and CMS improve the Tevatron space region where t → t 1 is kinematically forbidden represents a challenge for the experimental search program and more data and limit on t˜ → cχ˜0 substantially. Based on a monojet analysis [115] refined analyses will be required to further improve the sensitivity ATLAS excludes top squark masses below m 0 ≈ 450 GeV along χ˜1 in this difficult but important region of SUSY parameter space. 0 the kinematic boundary for the t˜ → cχ˜ decay. A dedicated anal- R-parity violating production of single squarks via a λ0-type 0 ysis for t˜ → cχ˜ excludes stop masses below 500 GeV for m 0 coupling has been studied at HERA. In such models, a lower limit χ˜1 below 420 GeV [90]. The CMS collaboration uses the hadronic on the squark mass of the order of 275 GeV has been set for 0 searches [111,113] to place constraints on this particular stop de- electromagnetic-strength-like couplings λ = 0.3 [116]. At the LHC, both prompt [75,78,117] and non-prompt [80,118] R-parity cay and excludes mt˜ ≈ 550 GeV for mχ˜0 below 450 GeV. The 1 violating squark decays have been searched for, but no signal was exclusion at mt˜ ≈ m 0 is also about 550 GeV. χ˜1 found. Squark mass limits are very model-dependent. The other decay chain relevant in this phase region is t˜ → R-parity violating production of single top squarks has been 90. Supersymmetry, Part II (Experiment) 929

~ ∼0 ∼0 ∼± ± ± ∼0 0 g (R-hadron) → qq χ ; m(χ ) = 100 GeV March 2019 χ ~ → π χ ~ March 2019 1 1 1[~W ] 1[~W ] -1 -1 RPC 0L 2-6 jets arXiv:1712.02332 ( s=13 TeV, 36 fb ) ATLAS Preliminary 31.6 - 36 fb , s=13 TeV 3000 -1 Expected limits RPC 0L 2-6 jets ATLAS-CONF-2018-003 ( s=13 TeV, 36 fb ) 1400 Disappearing track (pixel-only) arXiv:1712.02118 -1 Expected Displaced vertices arXiv:1710.04901 ( s=13 TeV, 33 fb ) Stable charged arXiv:1902.01636 Observed limits

) [GeV] -1

Pixel dE/dx arXiv:1808.04095 ( s=13 TeV, 36.1 fb ) ) [GeV] ~ g Observed 95% CL limits. -1 ± -1 Stable charged arXiv:1902.01636 ( s=13 TeV, 36.1 fb ) 95% CL limits 1 18.4-20.3 fb , s=8 TeV SUSY -1 ∼ χ σ not included Stopped gluino arXiv:1310.6584 ( s=7,8 TeV, 5.0,23 fb ) 1200 theory 2500 Pixel dE/dx arXiv:1506.05332 Disappearing track arXiv:1310.3675 1000 ATLAS Preliminary 2000 800 Lower limit on m( Lower limit on m( 1500 600

400 1000

200 Prompt Stable 500 Stable −2 −1 2 3 4 − 10 10 1 10 10 10 10 τ [ns] 10 1 1 10 η βγ (r for η=0, βγ=1) Beampipe Inner Detector Calo MS (r for =0, =1) Inner Detector Calo MS τ [ns]

− − − 10 3 10 2 10 1 1 10 102 103 104 10−2 10−1 1 10 cτ [m] cτ [m]

Figure 90.7: Limits at 95% C.L. on the gluino mass in R-hadron models (left), and on the chargino mass in a model where the wino-like chargino is almost degenerate with the LSP (right), as a function of gluino or chargino lifetime, as obtained by ATLAS. searched for at LEP, HERA, and the Tevatron. For example, an 90.5 Exclusion limits on the masses of charginos analysis from the ZEUS collaboration [119] makes an interpre- and neutralinos tation of its search result assuming top squarks to be produced Charginos and neutralinos result from mixing of the charged 0 ± via a λ coupling and decay either to bχ˜1 or R-parity-violating wino and higgsino states, and the neutral bino, wino and hig- 0 to a lepton and a jet. Limits are set on λ131 as a function of gsino states, respectively. The mixing is determined by a limited the top squark mass in an MSSM framework with gaugino mass number of parameters. For charginos these are the wino mass pa- unification at the GUT scale. rameter M2, the higgsino mass parameter µ, and tan β, and for The search for top squark pair production in the context of neutralinos these are the same parameters plus the bino mass pa- R-parity violating supersymmetry has now also become a focus rameter M1. If any of the parameters M1, M2 or µ happened to point for searches at the LHC. CMS and ATLAS have performed be substantially smaller than the others, the chargino/neutralino searches for top squarks using a variety of multilepton final states composition would be dominated by specific states, which are re- 0 [75, 120]. The λ -mediated top squark decay t˜ → b` has been ferred to as bino-like (M1  M2, µ), wino-like (M2  M1, µ), studied by ATLAS for prompt decays [121], and by ATLAS and or higgsino-like (µ  M1,M2). If gaugino mass unification at CMS for non-prompt decays [122–124], setting limits up to 1.4 − the GUT scale is assumed, a relation between M1 and M2 at the 2 1.6 TeV in simplified models for this mode. CMS also searched electroweak scale follows: M1 = 5/3 tan θW M2 ≈ 0.5M2, with 0 for the λ -mediated decay t˜ → b`qq, setting lower stop mass limits θW the weak mixing angle. Charginos and neutralinos carry no of 890 GeV (e) or 1000 GeV (µ) [125]. The fully hadronic R- color charge. parity violating top squark decays t˜ → bs, t˜ → ds, and t˜ → bd, involving λ00, have been searched for by ATLAS [75, 79, 94, 90.5.1 Exclusion limits on chargino masses 126], and CMS [127,128], and lower top squark mass limits up to If kinematically allowed, two body decay modes such as χ˜± → 610 GeV were set. f˜f¯0 (including `ν˜ and `ν˜ ) are dominant. If not, three body decays It should be noted that limits discussed in this section belong χ˜± → ff¯0χ˜0, mediated through virtual W bosons or sfermions, to different top and bottom squark decay channels, different spar- become dominant. If sfermions are heavy, the W mediation dom- ticle mass hierarchies, and different simplified decay scenarios. inates, and ff¯0 are distributed with branching fractions similar Therefore, care must be taken when interpreting these limits in to W decay products (barring phase space effects for small mass the context of more complete SUSY models. gaps between χ˜± and χ˜0). If, on the other hand, sleptons are light enough to play a significant role in the decay, leptonic final 90.4.3 Summary of exclusion limits on squarks and gluinos states will be enhanced. assuming R-Parity conservation At LEP, charginos have been searched for in fully-hadronic, A summary of the most important squark and gluino mass lim- semi-leptonic and fully leptonic decay modes [129] [130]. A gen- its for different interpretation approaches assuming R-parity con- eral lower limit on the lightest chargino mass of 103.5 GeV is de- servation is shown in Table 90.2. rived, except in corners of phase space with low electron sneutrino For gluino masses rather similar limits of about 2.3 TeV are mass, where destructive interference in chargino production, or obtained from different model assumptions, indicating that the two-body decay modes, play a role. The limit is also affected if the LHC is indeed probing direct gluino production at the TeV scale ± 0 mass difference between χ˜ and χ˜ is small; dedicated searches and beyond. However, for neutralino masses above approximately 1 1 for such scenarios set a lower limit of 92 GeV. 1 to 1.4 TeV, in the best case scenarios, ATLAS and CMS searches do not place any limits on the gluino mass. At the Tevatron, charginos have been searched for via associ- χ˜±χ˜0 Limits on direct squark production, on the other hand, depend ated production of 1 2 [131] [132]. Decay modes involving mul- strongly on the chosen model. Especially for direct production of tilepton final states provide the best discrimination against the top squarks there are still large regions in parameter space where large multijet background. Analyses have looked for at least three masses below 1 TeV cannot be excluded. This is also true for charged isolated leptons, for two leptons with missing transverse first and second generation squarks when only one single squark is momentum, or for two leptons with the same charge. Depending ± 0 0 0 considered. Furthermore, for neutralino masses above ≈ 500 GeV on the (χ˜1 − χ˜1) and/or (χ˜2 − χ˜1) mass differences, leptons may no limits on any direct squark production scenario are placed by be soft. the LHC. At the LHC, the search strategy is similar to that at the Teva- 930 90. Supersymmetry, Part II (Experiment)

Overview of CMS long-lived particle searches

CMS preliminary 3 - 137 fb 1 (8, 13 TeV)

1 RPV UDD, g tbs, mg = 2200 GeV g 1808.03078 (Disp. vertices) 0.0006 0.08 m 38 fb (13 TeV) 1 RPV UDD, g tbs, mg = 2200 GeV g 1811.07991 (Displaced dijet) 0.0025 1.2 m 36 fb (13 TeV) 1 RPV UDD, t dd, mt = 1300 GeV t 1808.03078 (Displaced vertices) 0.0004 0.1 m 38 fb (13 TeV) 1 RPV UDD, t dd, mt = 1300 GeV t 1811.07991 (Displaced dijet) 0.0014 1.55 m 36 fb (13 TeV) RPV SUSY 1 RPV LQD, t bl, mt = 600 GeV t 1808.05082 (2 + 2 jets) <0.031 m 36 fb (13 TeV) 1 RPV LQD, t bl, mt = 600 GeV t CMS-PAS-EXO-16-022 (Disp. e + disp. ) 0.0005 0.4 m 3 fb (13 TeV) 1 RPV LQD, t bl, mt = 1300 GeV t 1811.07991 (D. dijet) 0.0045 0.2 m 36 fb (13 TeV)

10 4 10 3 10 2 10 1 100 101 102 103 1 GMSB, g gG, mg = 2100 GeV g 1811.07991 (Displaced dijet) 0.0041 0.81c m [m] 36 fb (13 TeV) 1 GMSB, g gG, mg = 2100 GeV g 1906.06441 (Delayed jet + MET) 0.32 34 m 137 fb (13 TeV) 0 1 Split SUSY, g qq 1, mg = 1300 GeV g 1802.02110 (Jets + MET) <1 m 36 fb (13 TeV) 1 Split SUSY (HSCP), fgg = 0.1, mg = 1600 GeV g CMS-PAS-EXO-16-036 (dE/dx) >0.7 m 13 fb (13 TeV) mGMSB (HSCP) tan = 10, > 0 , m = 247 GeV CMS-PAS-EXO-16-036 (dE/dx + TOF) >7.5 m 13 fb 1 (13 TeV) 0 1 RPC 1801.00359 (Delayed jet) SUSY Stopped t, t t 1, mt = 700 GeV t 60 1.5e+13 m 39 fb (13 TeV) 0 1 Stopped g, g qq 1, fgg = 0.1, mg = 1300 GeV g 1801.00359 (Delayed jet) 50 3e+13 m 39 fb (13 TeV) 0 0 1 Stopped g, g qq 2( 1), fgg = 0.1, mg = 940 GeV g 1801.00359 (Delayed ) 600 3.3e+12 m 39 fb (13 TeV) ± 0 ± ± 1 AMSB, 1 , m ± = 505 GeV 1804.07321 (Disappearing track) 0.15 18 m 38 fb (13 TeV) 0 1 GMSB SPS8, G, m 0 = 400 GeV 0 CMS-PAS-EXO-19-005 (Delayed ( )) 0.2 6 m 77 fb (13 TeV) 1 1 1

10 4 10 3 10 2 10 1 100 101 102 103 1 H XX(10%), X ee, mH = 125 GeV, mX = 20 GeV X 1411.6977 (Displaced ee) c [m] 0.00012 25 m 20 fb (8 TeV) 1 H XX(10%), X , mH = 125 GeV, mX = 20 GeV X 1411.6977 (Displaced ) 0.00012 100 m 20 fb (8 TeV)

Other dark QCD, m = 5 GeV, m = 1200 GeV 1 DK XDK XDK 1810.10069 (Emerging jet + jet) 0.0022 0.3 m 16 fb (13 TeV)

10 4 10 3 10 2 10 1 100 101 102 103 c [m]

Selection of observed exclusion limits at 95% C.L. (theory uncertainties are not included). The y-axis tick labels indicate the studied long-lived particle. July 2019

Figure 90.8: Excluded regions, at 95% C.L., in the lifetimes of long-lived particles in several models, as obtained by CMS.

Table 90.2: Summary of squark mass and gluino of LEP and Tevatron searches for direct chargino/neutralino pro- mass limits using different interpretation ap- duction. With the full LHC Run 1 and Run 2 data sets, ATLAS proaches assuming R-parity conservation. Masses and CMS have surpassed the limits from LEP and Tevatron in in this table are provided in GeV. Further details regions of SUSY parameter space. about the assumptions and analyses from which Chargino pair production is searched for in the dilepton plus these limits are obtained are discussed in the cor- missing momentum final state. In a simplified model interpreta- responding sections of the text. tion of the results, assuming mediation of the chargino decay by light sleptons (e˜ and µ˜), ATLAS [133] and CMS [134] set limits on Model Assumption mq˜ mg˜ the chargino mass up to 1 TeV for massless LSPs, but no limits 0 Simplified model m 0 = 0, mq˜ ≈ mg˜ ≈ 3000 ≈ 3000 on the chargino mass can be set for χ˜ heavier than 480 GeV. χ˜1 1 ¯ Limits are fairly robust against variation of the slepton mass, un- g˜q,˜ g˜q˜ mχ˜0 = 0, all mq˜ - ≈ 2200 1 less the mass gap between chargino and slepton becomes small. mχ˜0 = 0, all mg˜ ≈ 2600 - 1 For decays mediated through τ˜ or ν˜ , limits of 630 GeV are set Simplified models g˜g˜ τ 0 by ATLAS [135] for LSPs not heavier than 200 GeV. The CMS g˜ → qq¯χ˜1 m 0 =0 - ≈ 2300 χ˜1 experiment provides similar limits [136]. ATLAS also sets limits m 0 >≈ 1200 - no limit on charginos decaying via a W boson [133]: chargino masses be- χ˜1 0 low 420 GeV are excluded for massless LSPs, but no limits are g˜ → b¯bχ˜1 m 0 =0 - ≈ 2300 χ˜1 set for LSPs heavier than 120 GeV. m 0 >≈ 1500 - no limit χ˜1 The trilepton plus missing momentum final state is used to set 0 ± g˜ → tt¯χ˜1 m 0 =0 - ≈ 2250 χ˜ χ˜0 χ˜± χ˜0 χ˜1 limits on 1 2 production, assuming wino-like and 2, bino- 0 mχ˜0 >≈ 1300 - no limit like χ˜1, and m ± = m 0 , leaving m ± and m 0 free. Again, 1 χ˜ χ˜2 χ˜ χ˜1 Simplified models q˜q˜ the branching fraction of leptonic final states is determined by χ˜0 the slepton masses. If the decay is predominantly mediated by a q˜ → q 1 mχ˜0 =0 ≈ 1900 - 1 light `˜ , i.e. `˜ is assumed to be heavy, the three charged-lepton m 0 >≈ 800 no limit - L R χ˜1 0 flavors will be produced in equal amounts. It is assumed that u˜L → qχ˜1 m 0 =0 ≈ 1300 - ˜ χ˜1 `L and sneutrino masses are equal, and diagrams with sneutri- m 0 >≈ 600 no limit - nos are included. In this scenario, ATLAS [137] and CMS [138] χ˜1 0 exclude chargino masses below 1140 GeV for massless LSPs; no ˜b → bχ˜1 m 0 =0 ≈ 1250 - χ˜1 limits are set for LSP masses above 700 GeV. If the decay is m 0 >≈ 700 no limit - ˜ χ˜1 dominated by a light `R, the chargino cannot be a pure wino but ˜ χ˜0 needs to have a large higgsino component, preferring the decays t → t 1 mχ˜0 =0 ≈ 1200 - 1 to tau leptons. Limits are set in various scenarios. If, like for m 0 >≈ 600 no limit - χ˜1 ˜ ± `L, a flavor-democratic scenario is assumed, CMS sets limits of t˜ → bχ˜1 m 0 =0 ≈ 1150 - 1060 GeV on the chargino mass for massless LSPs, but under the χ˜1 χ˜± χ˜0 (m ± = (mt˜ − mχ˜0 )/2) mχ˜0 >≈ 550 no limit - assumption that both and 2 decay leads to tau leptons in the χ˜1 1 1 0 final state, the chargino mass limit deteriorates to 620 GeV for t˜ → W bχ˜1 m 0 <≈ 570 ≈ 700 - χ˜1 massless LSPs [138]. ATLAS assumes a simplified model in which (mW < mt˜ − mχ˜0 < mt) staus are significantly lighter than the other sleptons in order to ˜ χ˜0 search for a similar multi-tau final state, and sets a lower limit t → c 1 mχ˜0 <≈ 450 ≈ 550 - 1 on the chargino mass of 760 GeV in this model [135]. The CMS mt˜ ≈ mχ˜0 ≈ 550 - 1 experiment provides similar limits [136]. 0 0 t˜ → bff χ˜1 m 0 <≈ 450 ≈ 550 - χ˜1 If sleptons are heavy, the chargino is assumed to decay to a W mt˜ ≈ m 0 ≈ 550 - 0 χ˜1 boson plus LSP, and the χ˜2 into Z plus LSP or H plus LSP. In (mt˜ − mχ˜0 < mW ) the WZ channel, ATLAS [137, 139] and CMS [140] limits on the chargino mass reach 650 GeV for massless LSPs, but no limits are set for LSPs heavier than 300 GeV. In the WH channel, for mH = tron. As shown in Fig. 90.1, the cross section of pair production of 125 GeV and using various Higgs decay modes, ATLAS [141– electroweak gauginos at the LHC, for masses of several hundreds 143] and CMS [140] set lower limits on the chargino mass up to of GeV, is at least two orders of magnitude smaller than for col- 740 GeV for massless LSPs, but vanish for LSP masses above ored SUSY particles (e.g. top squark pair production). For this 240 GeV. reason a large data sample is required to improve the sensitivity The results on electroweak gaugino searches interpreted in sim- 90. Supersymmetry, Part II (Experiment) 931

plified models are summarized in Fig. 90.5 for the two cases of Table 90.3: Summary of weak gaugino mass light or decoupled sleptons. For both cases, ATLAS and CMS limits in simplified models, assuming R-parity have comparable limits. conservation. Masses in the table are provided In both the wino region (a characteristic of anomaly-mediated in GeV. Further details about assumptions and SUSY breaking models) and the higgsino region of the MSSM, the analyses from which these limits are obtained are ± 0 discussed in the text. mass splitting between χ˜1 and χ˜1 is small. The chargino decay products are very soft and may escape detection. These com- pressed spectra are hard to detect, and have triggered dedicated Assumption mχ ± ± 0 search strategies. ATLAS has performed a search for charginos χ˜1 , all ∆m(χ˜1 , χ˜1) > 92 and neutralinos in a compressed mass spectrum using initial ± χ˜1 ∆m > 5, mν˜ > 300 > 103.5 state radiation [144]. For wino-like charginos, assuming degen- ± χ˜ , m = (m ± + m 0 )/2 ± 0 1 (`,˜ ν˜) χ˜ χ˜ erate χ˜1 and χ˜2, exclusion contours in the chargino-mass versus 1 1 ± 0 mχ˜0 ≈ 0 > 1000 ∆m(χ˜1 − χ˜1) plane are derived. As an example, such charginos 1 ± ± χ˜ χ˜0 χ˜1 , m 0 > 480 no LHC limit are excluded below 200 GeV for ∆m( 1 − 1) = 10 GeV. CMS χ˜1 ± has searched for chargino-pair production through vector-boson- χ˜ , m > m ± 1 `˜ χ˜ fusion [145], also targetting compressed mass spectra. Assuming 1 ± 0 m 0 ≈ 0 > 420 χ˜ χ˜ χ˜1 degenerate 1 and 2, charginos with a mass below 112 GeV are ± ± 0 χ˜1 , m 0 > 120 no LHC limit excluded for ∆m(χ˜1 − χ˜1) = 1 GeV. CMS has published further χ˜1 searches for such compressed spectra with soft leptons [146] or a m ± = mχ˜0 , m`˜ = (m ± + mχ˜0 )/2 χ˜1 2 L χ˜1 1 soft tau lepton [147]. m 0 ≈ 0 > 1140 χ˜1 90.5.2 Exclusion limits on neutralino masses m 0 > 700 no LHC limit χ˜1 In a considerable part of the MSSM parameter space, and in m ± = m 0 , m = (m ± + m 0 )/2 flavor-democratic χ˜ χ˜ `˜ χ˜ χ˜ particular when demanding that the LSP carries no electric or 1 2 R 1 1 0 mχ˜0 ≈ 0 > 1060 color charge, the lightest neutralino χ˜1 is the LSP. If R-parity is 1 0 mχ˜0 > 600 no LHC limit conserved, such a χ˜1 is stable. Since it is weakly interacting, it will 1 typically escape detectors unseen. Limits on the invisible width of m ± = mχ˜0 , mτ˜ = (m ± + mχ˜0 )/2τ ˜-dominated χ˜1 2 χ˜1 1 the Z boson apply to neutralinos with a mass below 45.5 GeV, but mχ˜0 ≈ 0 > 620 depend on the Z-neutralino coupling. Such a coupling could be 1 mχ˜0 > 260 no LHC limit small or even absent; in such a scenario there is no general lower 1 m ± = mχ˜0 , m`˜ > m ± , BF(WZ) = 1 limit on the mass of the lightest neutralino [148]. In models with χ˜1 2 χ˜1 gaugino mass unification and sfermion mass unification at the mχ˜0 ≈ 0 > 650 GUT scale, a lower limit on the neutralino mass is derived from 1 mχ˜0 > 300 no LHC limit limits from direct searches, notably for charginos and sleptons, 1 m ± = mχ˜0 , m`˜ > m ± , BF(WH) = 1 and amounts to 47 GeV [149]. Assuming a constrained model χ˜1 2 χ˜1 like the CMSSM, this limit increases to 50 GeV at LEP; however m 0 ≈ 0 > 740 χ˜1 the strong constraints now set by the LHC increase such CMSSM- m 0 > 240 no LHC limit 0 χ˜1 derived χ˜1 mass limits to well above 200 GeV [150–152]. In gauge-mediated SUSY breaking models (GMSB), the LSP is typically a gravitino, and the phenomenology is determined by the nature of the next-to-lightest supersymmetric particle (NLSP). A multi-lepton one. Searches for events with four or more isolated NLSP neutralino will decay to a gravitino and a SM particle whose charged leptons by ATLAS [74] and CMS [78] are interpreted in nature is determined by the neutralino composition. Final states such models. With very small coupling values, the neutralino with two high pT photons and missing momentum are searched would be long-lived, leading to lepton pairs with a displaced ver- for, and interpreted in gauge mediation models with bino-like neu- tex, which have also been searched for [118, 124, 170]. tralinos [153–158]. Various searches, including searches for multi-lepton and lepton Assuming the production of at least two neutralinos per event, plus jets events, and searches for events with a displaced hadronic neutralinos with large non-bino components can also be searched vertex, with or without a matched lepton, are interpreted in a for by their decay in final states with missing momentum plus any model with R-parity violating neutralino decays involving a non- two bosons out of the collection γ, Z, H. A number of searches zero λ0 coupling [75,80,86,171]. Neutralino decays involving non- at the LHC have tried to cover the rich phenomenology of the zero λ00 lead to fully hadronic final states, and searches for multi- various Z and H decay modes [74, 96, 138, 140, 142, 156, 159–165]. 0 jet events and jet-pair resonances are used to set limits, typically Heavier neutralinos, in particular χ˜2, have been searched for on the production of colored particles like top squarks or gluinos, in their decays to the lightest neutralino plus a γ, a Z boson which are assumed to be the primary produced sparticles in these 0 or a Higgs boson. Limits on electroweak production of χ˜2 plus interpretations, as discussed earlier [79, 84, 86]. χ˜± 1 from trilepton analyses have been discussed in the section on The limits on weak gauginos in simplified models are summa- 0 ± charginos; the assumption of equal mass of χ˜2 and χ˜1 make the rized in Table 90.3. Interpretations of the search results outside 0 limits on chargino masses apply to χ˜2 as well. Multilepton analy- simplified models, such as in the phenomenological MSSM [172– 0 0 ses have also been used to set limits on χ˜2χ˜3 production; assum- 176], show that the simplified model limits must be interpreted ing equal mass and decay through light sleptons, limits are set up with care. Electroweak gauginos in models that are compatible to 680 GeV for massless LSPs [166]. Again, compressed spectra with the relic density of dark matter in the universe, for exam- with small mass differences between the heavier neutralinos and ple, have particularly tuned mixing parameters and mass spectra, the LSP form the most challenging region. which are not always captured by the simplified models used. 0 0 In χ˜2 decays to χ˜1 and a lepton pair, the lepton pair invari- ant mass distribution may show a structure that can be used to 90.6 Exclusion limits on slepton masses 0 0 measure the χ˜2 − χ˜1 mass difference in case of a signal [35]. This In models with slepton and gaugino mass unification at the ˜ structure, however, can also be used in the search strategy itself, GUT scale, the right-handed slepton, `R, is expected to be lighter ˜ as demonstrated by ATLAS [167, 168] and CMS [96, 169]. than the left-handed slepton, `L. For tau sleptons there may be In models with R-parity violation, the lightest neutralino can considerable mixing between the L and R states, leading to a decay even if it is the lightest supersymmetric particle. If the significant mass difference between the lighter τ˜1 and the heavier decay involves a non-zero λ coupling, the final state will be a τ˜2. 932 90. Supersymmetry, Part II (Experiment)

pMSSM CMS −1 Combined, 7 + 8 TeV ATLAS s = 8 TeV, 20.3 fb 1 2000 1 1.4 0.9 0.9 1.2 0.8 0.8 1500

0.7 Mass [GeV] 0.7 1 mass [TeV]

0 1 0.6 0.6 ∼ χ 0.8 0.5 1000 0.5 0.6 0.4 0.4 0.3 0.3

0.4 500 Fraction of Models Excluded 0.2 survival probability 0.2 0.2 0.1 0.1 0 0 0 ∼ ∼ 0 0 0.5 1 1.5 2 2.5 3 ~ ~ ~ ~ ~ ~ ∼0 ∼0 ∼0 ∼0 τ τ ~ χ∼± χ∼± g t t b b q χ χ χ χ 1 2 l ~ 1 2 1 2 1 2 3 4 1 2 g mass [TeV] Sparticle

Figure 90.9: The plot on the left shows the survival probability of a pMSSM parameter space model in the gluino-neutralino mass plane after the application of the relevant CMS search results. The plot on the right shows a graphical representation of the ATLAS exclusion power in a pMSSM model. Each vertical bar is a one-dimensional projection of the√ fraction of models points excluded for each sparticle by ATLAS analyses. The experimental results are obtained from data taken at s = 7 and 8 TeV.

90.6.1 Exclusion limits on the masses of charged sleptons ATLAS and CMS have searched for direct production of selec- The most model-independent searches for selectrons, smuons tron pairs and smuon pairs at the LHC, with each slepton decay- 0 and staus originate from the LEP experiments [177]. Smuon pro- ing to its corresponding SM partner lepton and the χ˜1 LSP. In duction only takes place via s-channel γ∗/Z exchange. Search simplified models, ATLAS [133] and CMS [179] set lower mass ˜ ˜ results are often quoted for µ˜R, since it is typically lighter than limits on sleptons of 700 GeV for degenerate `L and `R, for a 0 µ˜L and has a weaker coupling to the Z boson; limits are therefore massless χ˜1 and assuming equal selectron and smuon masses, as χ˜0 0 conservative. Decays are expected to be dominated by µ˜R → µ 1, shown in Fig. 90.6. The limits deteriorate with increasing χ˜1 mass leading to two non-back-to-back and missing momentum. due to decreasing missing momentum and lepton momentum. As Slepton mass limits are calculated in the MSSM under the as- 0 a consequence, no limits are set for χ˜1 masses above 400 GeV. sumption of gaugino mass unification at the GUT scale, and de- 0 Limits are also derived without the assumption of slepton mass pend on the mass difference between the smuon and χ˜1.A µ˜R degeneracy [133, 179]. A dedicated search for sleptons with small 0 with a mass below 94 GeV is excluded for mµ˜R − m 0 > 10 GeV. ˜ χ˜1 mass difference between ` and χ˜1 is performed by ATLAS [144] The selectron case is similar to the smuon case, except that demanding the presence of ISR jets. an additional production mechanism is provided by t-channel ATLAS and CMS have also searched for τ˜-pair production. In neutralino exchange. The e˜R lower mass limit is 100 GeV for simplified models, ATLAS excludes τ˜ masses between 120 and m 0 < 85 GeV. Due to the t-channel neutralino exchange, e˜Re˜L χ˜0 χ˜1 390 GeV assuming light 1, combining the production of degen- pair production was possible at LEP, and a lower limit of 73 GeV erate left- and right-handed τ˜s [180]. The CMS analysis [181] was set on the selectron mass regardless of the neutralino mass covers lower masses and closes the mass gap with LEP. No limits 0 by scanning over MSSM parameter space [178]. The potentially are set for χ˜1 masses above 130 GeV. large mixing between τ˜L and τ˜R not only makes the τ˜1 light, but In gauge-mediated SUSY breaking models, sleptons can be (co- can also make its coupling to the Z boson small. LEP lower limits )NLSPs, i.e., the next-to-lightest SUSY particles and almost de- on the τ˜ mass range between 87 and 93 GeV depending on the generate in mass, decaying to a lepton and a gravitino. This decay 0 χ˜1 mass, for mτ˜ − m 0 > 7 GeV [177]. can either be prompt, or the slepton can have a non-zero lifetime. χ˜1 At the LHC, pair production of sleptons is not only heavily Combining several analyses, lower mass limits on µ˜R of 96.3 GeV suppressed with respect to pair production of colored SUSY par- and on e˜R of 66 GeV are set for all slepton lifetimes at LEP [182]. ticles but the cross section is also almost two orders of magnitude In a considerable part of parameter space in these models, the τ˜ is smaller than the one of pair production of charginos and neutrali- the NLSP. The LEP experiments have set lower limits on the mass nos. With the full data sets of Run 1 and Run 2, however, ATLAS of such a τ˜ between 87 and 97 GeV, depending on the τ˜ lifetime. and CMS have surpassed the sensitivity of the LEP analyses under ATLAS and CMS have searched for final states with τs, jets and certain assumptions. missing transverse momentum, and has interpreted the results in GMSB models setting limits on the model parameters [183, 184]. Table 90.4: Summary of slepton mass limits CMS has interpreted a multilepton analysis in terms of limits from LEP and LHC, assuming R-parity conser- on gauge mediation models with slepton NLSP [185]. CDF has 0 vation and 100% branching fraction for `˜ → `χ˜1. put limits on gauge mediation models at high tan β and slepton Masses in this table are provided in GeV. NLSP using an analysis searching for like-charge light leptons and taus [186]. Limits also exist on sleptons in R-parity violating models, both Assumption m `˜ from LEP and the Tevatron experiments. From LEP, lower limits χ˜0 µ˜R, ∆m(˜µR, 1) > 10 > 94 on µ˜ and e˜ masses in such models are 97 GeV, and the limits on 0 R R e˜R, ∆m(˜eR, χ˜1) > 10 > 94 the stau mass are very close: 96 GeV [187]. CMS has searched for e˜R, any ∆m > 73 resonant smuon production in a modified CMSSM scenario [188], χ˜0 0 τ˜R, ∆m(˜τR, 1) > 7 > 87 putting limits on λ211 as a function of m0, m1/2. ν˜ , ∆m(˜e , χ˜0) > 10 > 94 e R 1 90.6.2 Exclusion limits on sneutrino masses me˜L,R = mµ˜L,R , m 0 ≈ 0 > 700 χ˜1 The invisible width of the Z boson puts a lower limit on the m 0 >≈ 400 no LHC limit χ˜1 sneutrino mass of about 45 GeV. Tighter limits are derived from mτ˜L = mτ˜R , m 0 ≈ 0 > 390 other searches, notably for gauginos and sleptons, under the as- χ˜1 sumption of gaugino and sfermion mass universality at the GUT mχ˜0 >≈ 130 no LHC limit 1 scale, and amount to approximately 94 GeV in the MSSM [189]. 90. Supersymmetry, Part II (Experiment) 933

It is possible that the lightest sneutrino is the LSP; however, a mary vertex could lead to signatures such as kinked-tracks, or left-handed sneutrino LSP is ruled out as a cold dark matter can- apparently disappearing tracks, since, for example, the in ± ± 0 didate [190, 191]. χ˜1 → π χ˜1 might be too soft to be reconstructed. At the Production of pairs of sneutrinos in R-parity violating mod- LHC, searches have been performed for such disappearing tracks, els has been searched for at LEP [187]. Assuming fully lep- and interpreted within anomaly-mediated SUSY breaking mod- tonic decays via λ-type couplings, lower mass limits between 85 els [210–212]. The right plot of Fig. 90.7 shows constraints for and 100 GeV are set. At the Tevatron [192, 193] and at the different ATLAS searches on the chargino mass-vs-lifetime plane LHC [188, 194–196], searches have focused on scenarios with res- for an AMSB model (tan β = 5, µ > 0) in which a wino-like χ˜± onant production of a sneutrino, decaying to eµ, µτ and eτ final decays to a soft pion and an almost mass-degenerated wino-like states. No signal has been seen, and limits have been set on 0 χ˜ [200,201,211,212]. For a similar model, CMS excludes cτ values sneutrino masses as a function of the value of relevant RPV cou- 1 between 0.15 and 18 m for a chargino mass of 505 GeV [210], see plings. As an example, the LHC experiments exclude a resonant Fig. 90.8. Charginos with a lifetime longer than the time needed tau sneutrino with a mass below 2.3 TeV for λ = λ > 0.07 312 321 to pass through the detector appear as charged stable massive par- and λ0 > 0.11. 311 ticles. Limits have been derived by the LEP experiments [213], by The limits on sleptons in simplified models are summarized in D0 at the Tevatron [209], and by the LHC experiments [200,214], Table 90.4. and such charginos with mass below 1090 GeV are excluded. 90.7 Exclusion limits on long-lived sparticles In gauge mediation models, NLSP neutralino decays need not Long-lived sparticles arise in many different SUSY models. In be prompt, and experiments have searched for late decays with 0 particular in co-annihilation scenarios, where the NLSP and LSP photons in the final state. CDF have searched for delayed χ˜1 → are nearly mass-degenerate, this is rather common in order to γG˜ decays using the timing of photon signals in the calorime- obtain the correct Dark Matter relic density. Prominent exam- ter [215]. CMS has used the same technique at the LHC [216]. ples are scenarios featuring stau co-annihilation, or models of Results are given as exclusion contours in the neutralino mass SUSY breaking, e.g. minimal anomaly-mediated SUSY break- versus lifetime plane, and for example in a GMSB model with a ing (AMSB), in which the appropriate Dark Matter density is neutralino mass of 300 GeV, cτ values between 10 and 2000 cm obtained by co-annihilation of the LSP with an almost degen- are excluded [216]. D0 has looked at the direction of showers in erate long-lived wino. However, in general, also other sparticles the electromagnetic calorimeter with a similar goal [217], and AT- can be long-lived and it is desirable to establish a comprehensive LAS has searched for photon candidates that do not point back search program for these special long-lived cases, which lead to to the primary vertex, as well as for delayed photons [218]. distinct experimental search signatures, including displaced ver- Charged slepton decays may be kinematically suppressed, for tices or disappearing tracks, etc. example in the scenario of a NLSP slepton with a very small Past experiments have performed dedicated searches for long- mass difference to the LSP. Such a slepton may appear to be lived SUSY signatures, but given the absence of any experimental a stable charged massive particle. Interpretation of searches at evidence for SUSY so far, more effort and focus has gone into LEP for such signatures within GMSB models with stau NLSP such searches at the LHC recently. As for the interpretation of or slepton co-NLSP exclude masses up to 99 GeV [213]. Searches more standard searches for e.g. R-parity conserving SUSY, sim- of stable charged particles at the Tevatron [208, 209] and at the plified models are also a convenient tool to benchmark long-lived LHC [200, 204] are also interpreted in terms of limits on stable scenarios (see e.g. [197, 198]). charged sleptons. The limits obtained at the LHC exclude stable If the decay of gluinos is suppressed, for example if squark staus with masses below 430 GeV when produced directly in pairs, masses are high, gluinos may live longer than typical hadroniza- and below 660 GeV when staus are produced both directly and tion times. It is expected that such gluinos will hadronize to indirectly in the decay of other particles in a GMSB model. long-living strongly interacting particles known as R-. In particular, if the suppression of the gluino decay is strong, as in 90.8 Global interpretations the case that the squark masses are much higher than the TeV Apart from the interpretation of direct searches for sparticle scale, these R-hadrons can be (semi-)stable in collider timescales. production at colliders in terms of limits on masses of individ- Searches for such R-hadrons exploit the typical signature of sta- ual SUSY particles, model-dependent interpretations of allowed ble charged massive particles in the detector. R-hadrons decay- SUSY parameter space are derived from global SUSY fits. Typi- ing in the detector are searched for using dE/dx measurements cally these fits combine the results from collider experiments with and searches for displaced vertices. As shown in the left plot of indirect constraints on SUSY as obtained from low-energy ex- Fig. 90.7, the ATLAS experiment excludes semi-stable gluino R- periments, flavor physics, high-precision electroweak results, and hadrons with masses below 1.9 − 2.3 TeV for all lifetimes in a astrophysical data. simplified model where such gluinos always form R-hadrons, and In the pre-LHC era these fits were mainly dominated by in- decay into jets and a light neutralino, by combining a number direct constraints. Even for very constrained models like the of analyses [79, 199–201]. A combination of CMS searches for CMSSM, the allowed parameter space, in terms of squark and long-lived particles, as shown in Fig. 90.8, reaches similar lim- gluino masses, ranged from several hundreds of GeV to a few TeV. its [82, 202–204]. Furthermore, these global fits indicated that squarks and gluino Alternatively, since such R-hadrons are strongly interacting, masses in the range of 500 to 1000 GeV were the preferred re- they may be stopped in the calorimeter or in other material, gion of parameter space, although values as high as few TeV were and decay later into energetic jets. These decays are searched allowed with lower probabilities [219–226]. for by identifying the jets [205–207] or muons [207] outside the With ATLAS and CMS now probing mass scales around 1 TeV time window associated with bunch-bunch collisions. As shown and beyond, the importance of the direct searches for global analy- in Fig. 90.8, the CMS collaboration sets limits on such stopped ses of allowed SUSY parameter space has increased. For example, R-hadrons over 13 orders of magnitude in gluino lifetime, up to imposing the new experimental limits on constrained supergrav- masses of 1390 GeV [207]. ity models pushes the most likely values of first generation squark Top squarks can also be long-lived and hadronize to a R-hadron, and gluino masses significantly beyond 2 TeV, typically resulting for example in the scenario where the top squark is the next-to- in overall values of fit quality much worse than those in the pre- lightest SUSY particle (NLSP), with a small mass difference to the LHC era [150–152, 174, 227–234]. Also the measured value of mh LSP. Searches for massive stable charged particles are sensitive pushes the sparticle masses upwards. Although these constrained to such top squarks. Tevatron limits are approximately mt˜ > models are not yet ruled out, the extended experimental limits 300 GeV [208,209]. ATLAS sets a limit of 1340 GeV on such top impose very tight constraints on the allowed parameter space. squarks [200], the CMS limits are comparable [204]. For this reason, the emphasis of global SUSY fits has shifted to- In addition to colored sparticles, also sparticles like charginos wards less-constrained SUSY models. Especially interpretations may be long-lived, especially in scenarios with compressed mass in the pMSSM [172–176, 214, 227] but also in simplified models spectra. Charginos decaying in the detectors away from the pri- have been useful to generalize SUSY searches, for example to re- 934 90. Supersymmetry, Part II (Experiment)

ATLAS SUSY Searches* - 95% CL Lower Limits ATLAS Preliminary October 2019 √s = 13 TeV −1 Model Signature RL dt [fb ] Mass limit Reference

0 miss q˜ [10× Degen.] χ˜0 q˜q˜, q˜ qχ˜1 0 e,µ 2-6 jets ET 139 1.9 m( 1)<400 GeV ATLAS-CONF-2019-040 → miss q˜ [1×,8× Degen.] χ˜0 = mono-jet 1-3 jets ET 36.1 0.43 0.71 m(q˜)-m( 1) 5 GeV 1711.03301 0 miss g˜ χ˜0 = g˜g˜, g˜ qq¯χ˜ 1 0 e,µ 2-6 jets ET 139 2.35 m( 1) 0 GeV ATLAS-CONF-2019-040 → 0 g˜ Forbidden 1.15-1.95 m(χ˜1)=1000 GeV ATLAS-CONF-2019-040 0 ˜ ˜0 g˜g˜, g˜ qq¯(ℓℓ)χ˜1 3 e,µ 4 jets 36.1 g 1.85 m(χ1)<800 GeV 1706.03731 → ee,µµ miss g˜ χ˜0 = 2 jets ET 36.1 1.2 m(g˜)-m( 1 ) 50GeV 1805.11381 0 miss g˜ χ˜0 g˜g˜, g˜ qqWZχ˜1 0 e,µ 7-11 jets ET 36.1 1.8 m( 1) <400 GeV 1708.02794 → 0 SS e,µ 6 jets 139 g˜ 1.15 m(g˜)-m(χ˜1)=200 GeV 1909.08457 0 0 Inclusive Searches miss g˜ χ˜ g˜g˜, g˜ tt¯χ˜ 1 0-1 e,µ 3 b ET 79.8 2.25 m( 1)<200 GeV ATLAS-CONF-2018-041 → 0 SS e,µ 6 jets 139 g˜ 1.25 m(g˜)-m(χ˜1)=300 GeV ATLAS-CONF-2019-015

0 ˜ χ˜0 = χ˜0 = b˜1b˜1, b˜1 bχ˜1/tχ˜1± Multiple 36.1 b1 Forbidden 0.9 m( 1) 300 GeV, BR(b 1) 1 1708.09266, 1711.03301 → ˜ 0 0 Multiple 36.1 b1 Forbidden 0.58-0.82 m(χ˜1)=300 GeV, BR(bχ˜1)=BR(tχ˜1± )=0.5 1708.09266 ˜ 0 Multiple 139 b1 Forbidden 0.74 m(χ˜1)=200GeV, m(χ˜1±)=300GeV, BR(tχ˜1±)=1 ATLAS-CONF-2019-015

˜ ˜ ˜ 0 0 miss ˜ ∆ χ˜0 χ˜0 = χ˜0 = b1b1, b1 bχ˜2 bhχ˜1 0 e,µ 6 b ET 139 b1 Forbidden 0.23-1.35 m( 2, 1) 130GeV, m( 1) 100 GeV 1908.03122 → → ˜ 0 0 0 b1 0.23-0.48 ∆m(χ˜2, χ˜1)=130GeV, m(χ˜1)=0 GeV 1908.03122 0 0 miss ˜ χ˜0 t˜1t˜1, t˜1 Wbχ˜1 or tχ˜1 0-2 e,µ 0-2 jets/1-2 b ET 36.1 t1 1.0 m( 1)=1 GeV 1506.08616, 1709.04183, 1711.11520 → 0 miss ˜ χ˜0 t˜1t˜1, t˜1 Wbχ˜1 1 e,µ 3 jets/1 b E 139 t1 0.44-0.59 m( 1)=400 GeV ATLAS-CONF-2019-017 → T ˜ miss ˜ m(τ˜ )=800 GeV t˜1t˜1, t˜1 τ˜1bν, τ˜1 τG 1 τ + 1 e,µ,τ 2 jets/1 b ET 36.1 t1 1.16 1 1803.10178 gen. squarks → → χ˜0 χ˜0 0 e,µ Emiss c˜ χ˜0 = 1805.01649 rd t˜1t˜1, t˜1 c 1 / c˜c˜, c˜ c 1 2 c T 36.1 0.85 m( 1) 0 GeV

3 0 direct production → → t˜1 0.46 m(t˜1,c˜)-m(χ˜1 )=50GeV 1805.01649 miss ˜ χ˜0 = 0 e,µ mono-jet ET 36.1 t1 0.43 m(t˜1,c˜)-m( 1) 5 GeV 1711.03301 ˜ ˜ ˜ ˜ + miss ˜ χ˜0 = χ˜0 = t2t2, t2 t1 h 1-2 e,µ 4 b ET 36.1 t2 0.32-0.88 m( 1) 0 GeV, m(t˜1)-m( 1) 180 GeV 1706.03986 → miss 0 0 t˜2t˜2, t˜2 t˜1 + Z 3 e,µ 1 b E 139 t˜2 Forbidden 0.86 m(χ˜ )=360GeV, m(t˜ )-m(χ˜ )= 40 GeV ATLAS-CONF-2019-016 → T 1 1 1 ± ˜ ˜0 miss χ˜ χ˜ 0 χ˜0 = 1403.5294, 1806.02293 χ1±χ2 via WZ 2-3 e,µ ET 36.1 1 / 2 0.6 m( 1) 0 miss ± 0 0 ee,µµ 1 E 139 χ˜ /χ˜ 0.205 m(χ˜±)-m(χ˜ )=5 GeV ATLAS-CONF-2019-014 ≥ T 1 2 1 1 χ˜ χ˜ miss χ˜ ± χ˜0 = 1± 1∓ via WW 2 e,µ ET 139 1 0.42 m( 1) 0 1908.08215 ˜ ˜0 miss χ˜ ± χ˜ 0 χ˜0 = χ1±χ2 via Wh 0-1 e,µ 2 b/2 γ ET 139 1 / 2 Forbidden 0.74 m( 1) 70 GeV ATLAS-CONF-2019-019, 1909.09226 χ˜ χ˜ ˜ miss χ˜ ± ˜ χ˜ χ˜0 1± 1∓ via ℓL/ν˜ 2 e,µ ET 139 1 1.0 m(ℓ,ν˜)=0.5(m( 1± )+m( 1)) ATLAS-CONF-2019-008 0 miss ˜ ˜ ˜ 0 EW τ [τ , τ ] τ˜τ˜, τ˜ τχ˜ 2 τ E 139 L R,L 0.16-0.3 0.12-0.39 m(χ˜1)=0 ATLAS-CONF-2019-018 direct 1 T ˜ ˜→ ˜ ˜0 miss ˜ χ˜0 = ℓL,RℓL,R, ℓ ℓχ1 2 e,µ 0 jets ET 139 ℓ 0.7 m( 1) 0 ATLAS-CONF-2019-008 miss 0 → 2 e,µ 1 E 139 ℓ˜ 0.256 m(ℓ˜)-m(χ˜ )=10 GeV ATLAS-CONF-2019-014 ≥ T 1 ˜ ˜ ˜ ˜ ˜ miss ˜ χ˜0 ˜ HH, H hG/ZG 0 e,µ 3 b ET 36.1 H 0.13-0.23 0.29-0.88 BR( 1 hG)=1 1806.04030 → ≥ miss 0 → 4 e,µ 0 jets E 36.1 H˜ 0.3 BR(χ˜1 ZG˜)=1 1804.03602 T → + χ˜ χ˜ χ˜ miss χ˜ ± Direct 1 1− prod., long-lived 1± Disapp. trk 1 jet ET 36.1 1 0.46 Pure Wino 1712.02118 χ˜ ± 1 0.15 Pure Higgsino ATL-PHYS-PUB-2017-019 Stable g˜ R-hadron Multiple 36.1 g˜ 2.0 1902.01636,1808.04095 0 0 particles χ˜ Multiple 36.1 g˜ [τ(g ˜ ) =10 ns, 0.2 ns] 2.05 2.4 m(χ˜ )=100 GeV 1710.04901,1808.04095 Long-lived Metastable g˜ R-hadron, g˜ qq 1 1 → LFV pp ν˜τ + X, ν˜τ eµ/eτ/µτ eµ,eτ,µτ 3.2 ν˜τ 1.9 λ′ =0.11, λ132/133/233=0.07 1607.08079 → → 311 ˜ ˜ ˜0 miss χ˜ ± χ˜ 0 , , χ˜0 = χ1±χ1∓/χ2 WW/Zℓℓℓℓνν 4 e,µ 0 jets ET 36.1 1 / 2 [λi33 0, λ12k 0] 0.82 1.33 m( 1) 100 GeV 1804.03602 →0 0 χ˜ 0 Large λ g˜g˜, g˜ qqχ˜1, χ˜ 1 qqq 4-5 large-R jets 36.1 g˜ [m( 1)=200 GeV, 1100 GeV] 1.3 1.9 112′′ 1804.03568 → → g˜ [λ′′ =2e-4, 2e-5] 0 Multiple 36.1 112 1.05 2.0 m(χ˜1)=200 GeV, bino-like ATLAS-CONF-2018-003 0 0 ′′ 0 ˜˜ ˜ χ˜ χ˜ Multiple 36.1 g˜ [λ =2e-4, 1e-2] 0.55 1.05 (χ˜ ATLAS-CONF-2018-003 RPV tt, t t 1, 1 tbs 323 m 1)=200 GeV, bino-like → → t˜1t˜1, t˜1 bs 2 jets + 2 b 36.7 t˜1 [qq, bs] 0.42 0.61 1710.07171 → t˜1t˜1, t˜1 qℓ 2 e,µ 2 b 36.1 t˜1 0.4-1.45 BR(t˜1 be/bµ)>20% 1710.05544 → ˜ ′ ′ → 1 µ DV 136 t1 [1e-10< λ <1e-8, 3e-10< λ <3e-9] 1.0 1.6 BR(t˜1 qµ)=100%, cosθt=1 ATLAS-CONF-2019-006 23k 23k →

1 *Only a selection of the available mass limits on new states or 10− 1 Mass scale [TeV] phenomena is shown. Many of the limits are based on simplified models, c.f. refs. for the assumptions made.

Figure 90.10: Overview of the current landscape of SUSY searches at the LHC. The plot shows exclusion mass limits of ATLAS for different searches and interpretation assumptions. The corresponding results of the CMS experiment are similar. design experimental analyses in order to increase their sensitivity 90.9 Summary and Outlook for compressed spectra, where the mass of the LSP is much closer The absence of any observation of new phenomena at the first to squark and gluino masses than predicted, for example, by the √ √run of the LHC at s = 7/8 TeV, and after the second run at CMSSM. As shown in Table 90.2, for neutralino masses above s = 13 TeV, place significant constraints on SUSY parameter 0.5 − 1 TeV the current set of ATLAS and CMS searches, in- space. Today, inclusive searches probe production of gluinos at terpreted in simplified models, cannot exclude the existence of about 2.3 TeV, first and second generation squarks in the range squarks or gluinos with masses only marginally above the neu- of about 1 to 1.9 TeV, third generation squarks at scales around tralino mass. However, as these exclusion limits are defined in 600 GeV to 1.2 TeV, electroweak gauginos at scales around 400 − the context of simplified models, they are only valid for the as- 1100 GeV, and sleptons around 700 GeV. However, depending sumptions in which these models are defined. on the assumptions made on the underlying SUSY spectrum these As an alternative approach, both ATLAS [172] and CMS [173] limits can also weaken considerably. have performed an analysis of the impact of their searches on the With√ the LHC having reached almost its maximum energy of parameter space of the pMSSM. Fig. 90.9 shows graphically the about s = 14 TeV, future sensitivity improvement will have LHC√ exclusion power in the pMSSM based on searches performed to originate from more data, the improvement of experimental at s = 7 and 8 TeV. The plot on the left shows the survival analysis techniques and the focus of special signatures like the probability in the gluino-neutralino mass plane, which is a mea- one arising in long-lived sparticle decays. Therefore, it is expected sure of the parameter space that remains after inclusion of the that the current landscape of SUSY searches and corresponding relevant CMS search results. As can be seen, gluino masses below exclusion limits at the LHC, as, for example, shown in Fig. 90.10 about 1.2 TeV are almost fully excluded. This result agrees well from the ATLAS experiment [235] (CMS results are similar [236]), with the typical exclusion obtained at 8 TeV in simplified mod- will not change as rapidly anymore as it did in the past, when the els for gluino production. However, as shown in the right plot of LHC underwent several successive increases of collision energy. Fig. 90.9, when a similar analysis for other sparticles is performed The interpretation of results at the LHC has moved away from it becomes apparent that exclusions on the pMSSM parameter can constrained models like the CMSSM towards a large set of simpli- be significantly less stringent than simplified model limits might fied models, or the pMSSM. On the one hand this move is because suggest. This is especially apparent for the electroweak sector, the LHC limits have put constrained models like the CMSSM un- where even at rather low masses several of the pMSSM√ test points der severe pressure, while on the other hand simplified models still survive the constraint of ATLAS searches at s = 7 and 8 leave more freedom to vary parameters and form a better repre- TeV. This again indicates that care must be taken when inter- sentation of the underlying sensitivity of analyses. However, these preting results from the LHC searches and there are still several interpretations in simplified models do not come without a price: scenarios where sparticles below the 1 TeV scale are√ not excluded, the decomposition of a potentially complicated reality in a lim- even when considering the most recent results at s = 13 TeV. ited set of individual decay chains can be significantly incomplete. Furthermore, the discovery of a Higgs boson with a mass around Therefore, quoted limits in simplified models are only valid under 125 GeV has triggered many studies regarding the compatibility the explicit assumptions made in these models. The recent ad- of SUSY parameter space with this new particle. Much of it is still dition of more comprehensive interpretations in the pMSSM will work in progress and it will be interesting to see how the interplay complement those derived from simplified models and, thus, will between the results from direct SUSY searches and more precise enable an even more refined understanding of the probed SUSY measurements of the properties of the Higgs boson will unfold in parameter space. the future. In this context, the limit range of 1.5 − 2.3 TeV on generic 90. Supersymmetry, Part II (Experiment) 935

colored SUSY particles only holds for light neutralinos, in the [33] M. Carena et al., Status of Higgs Boson Physics, in this R-parity conserving MSSM. Limits on third generation squarks Review. and electroweak gauginos also only hold for light neutralinos, and [34] K. Nakamura et al. (Particle Data Group), J. Phys. G37, under specific assumptions√ for decay modes and slepton masses. 075021 (2010). The next LHC runs at s = 13 or 14 TeV with significantly larger integrated luminosities (notably the High-Luminosity [35] I. Hinchliffe et al., Phys. Rev. D55, 5520 (1997), [hep- LHC), will provide a large data sample for future SUSY searches. ph/9610544]. As mentioned above, the improvement in sensitivity will largely [36] L. Randall and D. Tucker-Smith, Phys. Rev. Lett. 101, have to come from a larger data set, and evolution of trigger and 221803 (2008), [arXiv:0806.1049]. analysis techniques, since there will be no significant energy in- [37] V. Khachatryan et al. (CMS), Phys. Lett. B698, 196 (2011), crease at the LHC anymore. Although the sensitivity for colored [arXiv:1101.1628]. sparticles will increase somewhat as well, the expanded data set [38] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 107, 221804 will be particularly beneficial for electroweak gaugino searches, (2011), [arXiv:1109.2352]. and for the more difficult final states presented by compressed particle spectra, stealth SUSY, long-lived sparticles, or R-parity [39] S. Chatrchyan et al. (CMS), JHEP 01, 077 (2013), violating scenarios. [arXiv:1210.8115]. References [40] S. Chatrchyan et al. (CMS), Eur. Phys. J. C73, 9, 2568 [1] H. Miyazawa, Prog. Theor. Phys. 36, 6, 1266 (1966). (2013), [arXiv:1303.2985]. [41] S. Chatrchyan et al. (CMS), Phys. Rev. D85, 012004 [2] Yu. A. Golfand and E. P. Likhtman, JETP Lett. 13, 323 (2012), [arXiv:1107.1279]. (1971), [Pisma Zh. Eksp. Teor. Fiz.13,452(1971)]. [42] C. G. Lester and D. J. Summers, Phys. Lett. B463, 99 [3] J.-L. Gervais and B. Sakita, Nucl. Phys. B34, 632 (1971). (1999), [hep-ph/9906349]. [4] D. V. Volkov and V. P. Akulov, Phys. Lett. 46B, 109 (1973). [43] D. R. Tovey, JHEP 04, 034 (2008), [arXiv:0802.2879]. [5] J. Wess and B. Zumino, Phys. Lett. 49B, 52 (1974). [44] M. R. Buckley et al., Phys. Rev. D89, 5, 055020 (2014), [6] J. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974). [arXiv:1310.4827]. [7] A. Salam and J. A. Strathdee, Nucl. Phys. B76, 477 (1974). [45] P. Jackson, C. Rogan and M. Santoni, Phys. Rev. D95, 3, [8] H. P. Nilles, Phys. Rept. 110, 1 (1984). 035031 (2017), [arXiv:1607.08307]. [9] H. E. Haber and G. L. Kane, Phys. Rept. 117, 75 (1985). [46] J. M. Butterworth et al., Phys. Rev. Lett. 100, 242001 (2008), [arXiv:0802.2470]. [10] E. Witten, Nucl. Phys. B188, 513 (1981). [47] E. Halkiadakis, G. Redlinger and D. Shih, Ann. Rev. Nucl. [11] S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150 Part. Sci. 64, 319 (2014), [arXiv:1411.1427]. (1981). [48] W. Beenakker et al., Int. J. Mod. Phys. A26, 2637 (2011), [12] M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B189, [arXiv:1105.1110]. 575 (1981). [49] W. Beenakker et al., Nucl. Phys. B492, 51 (1997), [hep- [13] S. Dimopoulos and S. Raby, Nucl. Phys. B192, 353 (1981). ph/9610490]. [14] N. Sakai, Z. Phys. C11, 153 (1981). [50] A.H. Chamseddine, R. Arnowitt, and P Nath, Phys. Rev. [15] R. K. Kaul and P. Majumdar, Nucl. Phys. B199, 36 (1982). Lett. 49, 970 (1982). [16] H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983). [51] E. Cremmer et al., Nucl. Phys. B212, 413 (1983). [17] J. R. Ellis et al., Nucl. Phys. B238, 453 (1984). [52] P. Fayet, Phys. Lett. 70B, 461 (1977). [18] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. [53] M. Dine, A. E. Nelson and Y. Shirman, Phys. Rev. D51, 267, 195 (1996), [hep-ph/9506380]. 1362 (1995), [hep-ph/9408384]. [19] S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D24, [54] P. Meade, N. Seiberg and D. Shih, Prog. Theor. Phys. Suppl. 1681 (1981). 177, 143 (2009), [arXiv:0801.3278]. [20] W. J. Marciano and G. Senjanovic, Phys. Rev. D25, 3092 [55] G. F. Giudice et al., JHEP 12, 027 (1998), [hep- (1982). ph/9810442]. [21] M. B. Einhorn and D. R. T. Jones, Nucl. Phys. B196, 475 [56] L. Randall and R. Sundrum, Nucl. Phys. B557, 79 (1999), (1982). [hep-th/9810155]. [22] L. E. Ibanez and G. G. Ross, Phys. Lett. 105B, 439 (1981). [57] R. L. Arnowitt and P. Nath, Phys. Rev. Lett. 69, 725 (1992). [23] U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260, [58] G. L. Kane et al., Phys. Rev. D49, 6173 (1994), [hep- 447 (1991). ph/9312272]. [24] P. Langacker and N. Polonsky, Phys. Rev. D52, 3081 [59] A. Djouadi, J.-L. Kneur and G. Moultaka, Comput. Phys. (1995), [hep-ph/9503214]. Commun. 176, 426 (2007), [hep-ph/0211331]. [25] P. Fayet, Phys. Lett. 64B, 159 (1976). [60] C. F. Berger et al., JHEP 02, 023 (2009), [arXiv:0812.0980]. [26] G. R. Farrar and P. Fayet, Phys. Lett. 76B, 575 (1978). [61] H. Baer et al., in “Workshop on Physics at Current Accelerators and the Supercollider Argonne, Illinois, June [27] B.C. Allanach and H.E. Haber, Supersymmetry, Part I 2-5, 1993,” 0703–720 (1993), [hep-ph/9305342], URL (Theory), in this Review. http://lss.fnal.gov/cgi-bin/find_paper.pl?other/ [28] V. Khachatryan et al. (CMS, LHCb), Nature 522, 68 ssc/sscl-preprint-441. (2015), [arXiv:1411.4413]. [62] R. M. Barnett, H. E. Haber and G. L. Kane, Nucl. Phys. [29] R. Aaij et al. (LHCb), Phys. Rev. Lett. 118, 19, 191801 B267, 625 (1986). (2017), [arXiv:1703.05747]. [63] H. Baer, D. Karatas and X. Tata, Phys. Lett. B183, 220 [30] A. Höcker and W.J. Marciano, Muon Anomalous Magnetic (1987). Moment, in this Review. [64] J. Alwall, P. Schuster and N. Toro, Phys. Rev. D79, 075020 [31] G. Hinshaw et al. (WMAP), Astrophys. J. Suppl. 208, 19 (2009), [arXiv:0810.3921]. (2013), [arXiv:1212.5226]. [65] J. Alwall et al., Phys. Rev. D79, 015005 (2009), [32] Planck Collab., Astron. & Astrophys. 594, A13 (2016). [arXiv:0809.3264]. 936 90. Supersymmetry, Part II (Experiment)

[66] O. Buchmueller and J. Marrouche, Int. J. Mod. Phys. A29, [102] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 104, 251801 06, 1450032 (2014), [arXiv:1304.2185]. (2010), [arXiv:0912.1308]. [67] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [103] V. M. Abazov et al. (D0), Phys. Lett. B674, 4 (2009), and OPAL experiments, note LEPSUSYWG/04-02.1, [arXiv:0901.1063]. http://lepsusy.web.cern.ch/lepsusy. [104] CMS Collab., CMS-PAS-SUS-19-009 (2019). [68] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 102, 121801 [105] M. Aaboud et al. (ATLAS), JHEP 12, 085 (2017), (2009), [arXiv:0811.2512]. [arXiv:1709.04183]. [69] V. M. Abazov et al. (D0), Phys. Lett. B660, 449 (2008), [106] M. Aaboud et al. (ATLAS), JHEP 06, 108 (2018), [arXiv:0712.3805]. [arXiv:1711.11520]. [70] ATLAS Collab., ATLAS-CONF-2019-040 (2019). [107] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 3, 032008 [71] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1908.04722]. (2018), [arXiv:1803.10178]. [72] ATLAS Collab., ATLAS-CONF-2018-041 (2018). [108] CMS Collab., CMS-PAS-SUS-19-003 (2019). [73] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1909.03460]. [109] ATLAS Collab., ATLAS-CONF-2019-017 (2019). [74] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 3, 032009 [110] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 12, 898 (2018), [arXiv:1804.03602]. (2017), [arXiv:1708.03247]. [75] M. Aaboud et al. (ATLAS), JHEP 09, 084 (2017), [111] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C77, 10, 710 [arXiv:1706.03731]. (2017), [arXiv:1705.04650]. [76] CMS Collab., CMS-PAS-SUS-19-008 (2019). [112] A. M. Sirunyan et al. (CMS), Phys. Rev. D96, 3, 032003 [77] V. Khachatryan et al. (CMS), Phys. Rev. D94, 11, 112009 (2017), [arXiv:1704.07781]. (2016), [arXiv:1606.08076]. [113] A. M. Sirunyan et al. (CMS), JHEP 10, 005 (2017), [78] CMS Collab., CMS-PAS-SUS-13-010 (2013). [arXiv:1707.03316]. [79] ATLAS Collab., ATLAS-CONF-2018-003 (2018). [114] A. M. Sirunyan et al. (CMS), JHEP 10, 019 (2017), [80] G. Aad et al. (ATLAS), Phys. Rev. D92, 7, 072004 (2015), [arXiv:1706.04402]. [arXiv:1504.05162]. [115] M. Aaboud et al. (ATLAS), JHEP 01, 126 (2018), [81] A. M. Sirunyan et al. (CMS), Phys. Rev. D98, 9, 092011 [arXiv:1711.03301]. (2018), [arXiv:1808.03078]. [116] F. D. Aaron et al. (H1), Eur. Phys. J. C71, 1572 (2011), [82] A. M. Sirunyan et al. (CMS), Phys. Rev. D99, 3, 032011 [arXiv:1011.6359]. (2019), [arXiv:1811.07991]. [117] ATLAS Collab., ATLAS-CONF-2015-018 (2015). [83] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 107, 042001 [118] V. Khachatryan et al. (CMS), Phys. Rev. D91, 5, 052012 (2011), [arXiv:1105.2815]. (2015), [arXiv:1411.6977]. [84] M. Aaboud et al. (ATLAS), Phys. Lett. B785, 136 (2018), [119] S. Chekanov et al. (ZEUS), Eur. Phys. J. C50, 269 (2007), [arXiv:1804.03568]. [hep-ex/0611018]. [85] ATLAS Collab., ATLAS-CONF-2016-057 (2016). [120] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 111, 22, [86] M. Aaboud et al. (ATLAS), JHEP 09, 088 (2017), 221801 (2013), [arXiv:1306.6643]. [arXiv:1704.08493]. [121] M. Aaboud et al. (ATLAS), Phys. Rev. D97, 3, 032003 [87] A. M. Sirunyan et al. (CMS), Phys. Lett. B783, 114 (2018), (2018), [arXiv:1710.05544]. [arXiv:1712.08920]. [122] ATLAS Collab., ATLAS-CONF-2019-006 (2019). [88] S. Chatrchyan et al. (CMS), Phys. Lett. B730, 193 (2014), [arXiv:1311.1799]. [123] A. M. Sirunyan et al. (CMS), Phys. Rev. D99, 3, 032014 (2019), [arXiv:1808.05082]. [89] V. Khachatryan et al. (CMS), Phys. Lett. B770, 257 (2017), [arXiv:1608.01224]. [124] CMS Collab., CMS-PAS-EXO-16-022 (2016). [90] M. Aaboud et al. (ATLAS), JHEP 09, 050 (2018), [125] V. Khachatryan et al. (CMS), Phys. Lett. B760, 178 (2016), [arXiv:1805.01649]. [arXiv:1602.04334]. [91] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 105, 081802 [126] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 3, 250 (2010), [arXiv:1005.3600]. (2018), [arXiv:1710.07171]. [92] V. M. Abazov et al. (D0), Phys. Lett. B693, 95 (2010), [127] V. Khachatryan et al. (CMS), Phys. Rev. D95, 1, 012009 [arXiv:1005.2222]. (2017), [arXiv:1610.05133]. [93] M. Aaboud et al. (ATLAS), JHEP 11, 195 (2017), [128] V. Khachatryan et al. (CMS), Phys. Lett. B747, 98 (2015), [arXiv:1708.09266]. [arXiv:1412.7706]. [94] G. Aad et al. (ATLAS) (2019), [arXiv:1909.08457]. [129] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [95] G. Aad et al. (ATLAS) (2019), [arXiv:1908.03122]. and OPAL experiments, note LEPSUSYWG/01-03.1, http://lepsusy.web.cern.ch/lepsusy. [96] A. M. Sirunyan et al. (CMS), JHEP 03, 076 (2018), [arXiv:1709.08908]. [130] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 and OPAL experiments, note LEPSUSYWG/02-04.1, [97] C. Boehm, A. Djouadi and Y. Mambrini, Phys. Rev. D61, http://lepsusy.web.cern.ch/lepsusy. 095006 (2000), [hep-ph/9907428]. [131] CDF Collab., CDF Note 10636 (2011). [98] T. Aaltonen et al. (CDF), Phys. Rev. D82, 092001 (2010), [arXiv:1009.0266]. [132] V. M. Abazov et al. (D0), Phys. Lett. B680, 34 (2009), [arXiv:0901.0646]. [99] V. M. Abazov et al. (D0), Phys. Lett. B696, 321 (2011), [arXiv:1009.5950]. [133] G. Aad et al. (ATLAS) (2019), [arXiv:1908.08215]. [100] T. Aaltonen et al. (CDF), JHEP 10, 158 (2012), [134] A. M. Sirunyan et al. (CMS), JHEP 11, 079 (2018), [arXiv:1203.4171]. [arXiv:1807.07799]. [101] V. M. Abazov et al. (D0), Phys. Lett. B665, 1 (2008), [135] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 2, 154 [arXiv:0803.2263]. (2018), [arXiv:1708.07875]. 90. Supersymmetry, Part II (Experiment) 937

[136] A. M. Sirunyan et al. (CMS), JHEP 11, 151 (2018), [170] G. Aad et al. (ATLAS) (2019), [arXiv:1907.10037]. [arXiv:1807.02048]. [171] V. Khachatryan et al. (CMS), Phys. Rev. D91, 1, 012007 [137] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 12, 995 (2015), [arXiv:1411.6530]. (2018), [arXiv:1803.02762]. [172] G. Aad et al. (ATLAS), JHEP 10, 134 (2015), [138] A. M. Sirunyan et al. (CMS), JHEP 03, 166 (2018), [arXiv:1508.06608]. [arXiv:1709.05406]. [173] V. Khachatryan et al. (CMS), JHEP 10, 129 (2016), [139] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 9, 092012 [arXiv:1606.03577]. (2018), [arXiv:1806.02293]. [174] P. Athron et al. (GAMBIT), Eur. Phys. J. C77, 12, 879 [140] A. M. Sirunyan et al. (CMS), JHEP 03, 160 (2018), (2017), [arXiv:1705.07917]. [arXiv:1801.03957]. [175] K. J. de Vries et al., Eur. Phys. J. C75, 9, 422 (2015), [141] M. Aaboud et al. (ATLAS), Phys. Rev. D100, 1, 012006 [arXiv:1504.03260]. (2019), [arXiv:1812.09432]. [176] C. Strege et al., JHEP 09, 081 (2014), [arXiv:1405.0622]. [142] ATLAS Collab., ATLAS-CONF-2019-019 (2019). [177] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [143] G. Aad et al. (ATLAS) (2019), [arXiv:1909.09226]. and OPAL experiments, note LEPSUSYWG/04-01.1, [144] ATLAS Collab., ATLAS-CONF-2019-014 (2019). http://lepsusy.web.cern.ch/lepsusy. [145] A. M. Sirunyan et al. (CMS), JHEP 08, 150 (2019), [178] A. Heister et al. (ALEPH), Phys. Lett. B544, 73 (2002), [arXiv:1905.13059]. [hep-ex/0207056]. [146] A. M. Sirunyan et al. (CMS), Phys. Lett. B782, 440 (2018), [179] A. M. Sirunyan et al. (CMS), Phys. Lett. B790, 140 (2019), [arXiv:1801.01846]. [arXiv:1806.05264]. [147] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1910.01185]. [180] ATLAS Collab., ATLAS-CONF-2019-018 (2019). [148] H. K. Dreiner et al., Eur. Phys. J. C62, 547 (2009), [181] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1907.13179]. [arXiv:0901.3485]. [182] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [149] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 and OPAL experiments, note LEPSUSYWG/02-09.2, and OPAL experiments, note LEPSUSYWG/04-07.1, http://lepsusy.web.cern.ch/lepsusy. http://lepsusy.web.cern.ch/lepsusy. [183] M. Aaboud et al. (ATLAS), Phys. Rev. D99, 1, 012009 [150] O. Buchmueller et al., Eur. Phys. J. C74, 6, 2922 (2014), (2019), [arXiv:1808.06358]. [arXiv:1312.5250]. [184] S. Chatrchyan et al. (CMS), Eur. Phys. J. C73, 2493 (2013), [151] C. Strege et al., JCAP 1304, 013 (2013), [arXiv:1212.2636]. [arXiv:1301.3792]. [152] A. Fowlie et al., Phys. Rev. D86, 075010 (2012), [185] S. Chatrchyan et al. (CMS), Phys. Rev. D90, 032006 [arXiv:1206.0264]. (2014), [arXiv:1404.5801]. [153] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [186] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 110, 20, 201802 and OPAL experiments, note LEPSUSYWG/04-09.1, (2013), [arXiv:1302.4491]. http://lepsusy.web.cern.ch/lepsusy. [187] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 [154] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 104, 011801 and OPAL experiments, note LEPSUSYWG/02-10.1, (2010), [arXiv:0910.3606]. http://lepsusy.web.cern.ch/lepsusy. [155] V. M. Abazov et al. (D0), Phys. Rev. Lett. 105, 221802 [188] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 4, 305 (2010), [arXiv:1008.2133]. (2019), [arXiv:1811.09760]. [156] M. Aaboud et al. (ATLAS), Phys. Rev. D97, 9, 092006 [189] DELPHI Collab., Eur. Phys. J. C31, 412 (2003). (2018), [arXiv:1802.03158]. [190] T. Falk, K. A. Olive and M. Srednicki, Phys. Lett. B339, [157] A. M. Sirunyan et al. (CMS), JHEP 06, 143 (2019), 248 (1994), [hep-ph/9409270]. [arXiv:1903.07070]. [191] C. Arina and N. Fornengo, JHEP 11, 029 (2007), [158] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1907.00857]. [arXiv:0709.4477]. [159] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 9, 092002 [192] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 105, 191801 (2018), [arXiv:1806.04030]. (2010), [arXiv:1004.3042]. [160] M. Aaboud et al. (ATLAS), Phys. Rev. D99, 1, 012001 [193] V. M. Abazov et al. (D0), Phys. Rev. Lett. 105, 191802 (2019), [arXiv:1808.03057]. (2010), [arXiv:1007.4835]. [161] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 5, 444 [194] G. Aad et al. (ATLAS), Phys. Rev. Lett. 115, 3, 031801 (2019), [arXiv:1901.06726]. (2015), [arXiv:1503.04430]. [162] A. M. Sirunyan et al. (CMS), JHEP 01, 154 (2019), [195] M. Aaboud et al. (ATLAS), Eur. Phys. J. C76, 10, 541 [arXiv:1812.04066]. (2016), [arXiv:1607.08079]. [163] A. M. Sirunyan et al. (CMS), Phys. Lett. B780, 118 (2018), [196] V. Khachatryan et al. (CMS), Eur. Phys. J. C76, 6, 317 [arXiv:1711.08008]. (2016), [arXiv:1604.05239]. [164] A. M. Sirunyan et al. (CMS), Phys. Rev. D97, 3, 032007 [197] O. Buchmueller et al., JHEP 09, 076 (2017), (2018), [arXiv:1709.04896]. [arXiv:1704.06515]. [165] A. M. Sirunyan et al. (CMS), Phys. Lett. B779, 166 (2018), [198] V. V. Khoze, A. D. Plascencia and K. Sakurai, JHEP 06, [arXiv:1709.00384]. 041 (2017), [arXiv:1702.00750]. [166] G. Aad et al. (ATLAS), Phys. Rev. D93, 5, 052002 (2016), [199] M. Aaboud et al. (ATLAS), Phys. Rev. D97, 5, 052012 [arXiv:1509.07152]. (2018), [arXiv:1710.04901]. [167] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 8, 625 [200] M. Aaboud et al. (ATLAS), Phys. Rev. D99, 9, 092007 (2018), [arXiv:1805.11381]. (2019), [arXiv:1902.01636]. [168] M. Aaboud et al. (ATLAS), Eur. Phys. J. C77, 3, 144 [201] G. Aad et al. (ATLAS), Eur. Phys. J. C75, 9, 407 (2015), (2017), [arXiv:1611.05791]. [arXiv:1506.05332]. [169] C. P. Herzog and M. Spillane, JHEP 04, 124 (2016), [202] A. M. Sirunyan et al. (CMS), Phys. Lett. B797, 134876 [arXiv:1506.06757]. (2019), [arXiv:1906.06441]. 938 90. Supersymmetry, Part II (Experiment)

[203] A. M. Sirunyan et al. (CMS), JHEP 05, 025 (2018), [220] E. A. Baltz and P. Gondolo, JHEP 10, 052 (2004), [hep- [arXiv:1802.02110]. ph/0407039]. [204] CMS Collab., CMS-PAS-EXO-16-036 (2016). [221] B. C. Allanach and C. G. Lester, Phys. Rev. D73, 015013 [205] V. M. Abazov et al. (D0), Phys. Rev. Lett. 99, 131801 (2006), [hep-ph/0507283]. (2007), [arXiv:0705.0306]. [222] R. Ruiz de Austri, R. Trotta and L. Roszkowski, JHEP 05, [206] G. Aad et al. (ATLAS), Phys. Rev. D88, 11, 112003 (2013), 002 (2006), [hep-ph/0602028]. [arXiv:1310.6584]. [223] R. Lafaye et al., Eur. Phys. J. C54, 617 (2008), [207] A. M. Sirunyan et al. (CMS), JHEP 05, 127 (2018), [arXiv:0709.3985]. [arXiv:1801.00359]. [224] M. Shaposhnikov, JHEP 08, 008 (2008), [arXiv:0804.4542]. [208] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 103, 021802 [225] R. Trotta et al., JHEP 12, 024 (2008), [arXiv:0809.3792]. (2009), [arXiv:0902.1266]. [226] P. Bechtle et al., Eur. Phys. J. C66, 215 (2010), [209] V. M. Abazov et al. (D0), Phys. Rev. D87, 5, 052011 (2013), [arXiv:0907.2589]. [arXiv:1211.2466]. [227] E. Bagnaschi et al., Eur. Phys. J. C 78, 256, 1 (2018). [210] A. M. Sirunyan et al. (CMS), JHEP 08, 016 (2018), [228] J. Costa et al., Eur. Phys. J. C 78, 158, 1 (2018). [arXiv:1804.07321]. [229] E. Bagnaschi et al., Eur. Phys. J. C77, 4, 268 (2017), [211] M. Aaboud et al. (ATLAS), JHEP 06, 022 (2018), [arXiv:1612.05210]. [arXiv:1712.02118]. [230] E. Bagnaschi et al., Eur. Phys. J. C77, 2, 104 (2017), [212] G. Aad et al. (ATLAS), Phys. Rev. D88, 11, 112006 (2013), [arXiv:1610.10084]. [arXiv:1310.3675]. [231] L. A. Harland-Lang, V. A. Khoze and M. G. Ryskin, Eur. [213] LEP2 SUSY Working Group, ALEPH, DELPHI, L3 Phys. J. C76, 1, 9 (2016), [arXiv:1508.02718]. and OPAL experiments, note LEPSUSYWG/02-05.1, http://lepsusy.web.cern.ch/lepsusy. [232] E. A. Bagnaschi et al., Eur. Phys. J. C75, 500 (2015), [arXiv:1508.01173]. [214] V. Khachatryan et al. (CMS), Eur. Phys. J. C75, 7, 325 (2015), [arXiv:1502.02522]. [233] O. Buchmueller et al., Eur. Phys. J. C74, 12, 3212 (2014), [arXiv:1408.4060]. [215] T. Aaltonen et al. (CDF), Phys. Rev. D88, 3, 031103 (2013), [arXiv:1307.0474]. [234] M. Citron et al., Phys. Rev. D87, 3, 036012 (2013), [arXiv:1212.2886]. [216] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1909.06166]. [235] Supersymmetry Physics Results, ATLAS experiment, [217] V. M. Abazov et al. (D0), Phys. Rev. Lett. 101, 111802 http://twiki.cern.ch/twiki/bin/view/ (2008), [arXiv:0806.2223]. AtlasPublic/SupersymmetryPublicResults/. [218] G. Aad et al. (ATLAS), Phys. Rev. D90, 11, 112005 (2014), [236] Supersymmetry Physics Results, CMS experiment, [arXiv:1409.5542]. http://cms-results.web.cern.ch/cms-results/ [219] O. Buchmueller et al., Eur. Phys. J. C71, 1722 (2011), public-results/publications/SUS/index.html. [arXiv:1106.2529]. 91. Axions and Other Similar Particles 939

91. Axions and Other Similar Particles

Revised October 2019 by A. Ringwald (DESY, Hamburg), L.J. is the quantity that enters all low-energy phenomena [19]. Non- Rosenberg (U. Washington) and G. Rybka (U. Washington). perturbative topological fluctuations of the gluon fields in QCD in- duce a potential for φA whose minimum is at φA = Θ¯ fA, thereby 91.1 Introduction canceling the Θ¯ term in the QCD Lagrangian and thus restoring In this section, we list coupling-strength and mass limits for CP symmetry. light neutral scalar or pseudoscalar bosons that couple weakly to The resulting axion mass, in units of the PQ scale fA, is iden- normal matter and radiation. Such bosons may arise from the tical to the square root of the topological susceptibility in QCD, spontaneous breaking of a global U(1) symmetry, resulting in a √ mAfA = χ. The latter can be evaluated further [20,21], exploit- massless Nambu-Goldstone (NG) boson. If there is a small ex- ing the chiral limit (masses of up and down quarks much smaller plicit symmetry breaking, either already in the Lagrangian or due √ than the scale of QCD), yielding mAfA = χ ≈ fπmπ, where to quantum effects such as anomalies, the boson acquires a mass mπ = 135 MeV and fπ ≈ 92 MeV. In more detail one finds, to and is called a pseudo-NG boson. Typical examples are axions next-to-next-to-leading order in chiral perturbation theory [22], (A0) [1–4] and majorons [5], associated, respectively, with a spon- taneously broken Peccei-Quinn and lepton-number symmetry.  109 GeV  mA = 5.691(51) meV . (91.3) A common feature of these light bosons φ is that their coupling f to Standard-Model particles is suppressed by the energy scale that A characterizes the symmetry breaking, i.e., the decay constant f. A direct calculation of the topological susceptibility via QCD lat- The interaction Lagrangian is tice simulations finds almost the same central value, albeit with

−1 µ an about five times larger error bar [23]. L = f J ∂µ φ , (91.1) Axions with fA  vEW evade all current experimental lim- its. One generic class of models invokes “hadronic axions” where where Jµ is the Noether current of the spontaneously broken new heavy quarks carry U(1) charges, leaving ordinary quarks global symmetry. If f is very large, these new particles interact PQ and leptons without tree-level axion couplings. The archetype is very weakly. Detecting them would provide a window to physics the KSVZ model [24], where in addition the heavy new quarks far beyond what can be probed at accelerators. are electrically neutral. Another generic class requires at least Axions are of particular interest because the Peccei-Quinn (PQ) two Higgs doublets and ordinary quarks and leptons carry PQ mechanism remains perhaps the most credible scheme to preserve charges, the archetype being the DFSZ model [25]. All of these CP-symmetry in QCD. Moreover, the cold dark matter (CDM) of models contain at least one electroweak singlet scalar that ac- the universe may well consist of axions and they are searched for quires a vacuum expectation value and thereby breaks the PQ in dedicated experiments with a realistic chance of discovery. symmetry. The KSVZ and DFSZ models are frequently used as Originally it was assumed that the PQ scale f was related to A √ benchmark examples, but other models exist where both heavy the electroweak symmetry-breaking scale v = ( 2G )−1/2 = EW F quarks and Higgs doublets carry PQ charges. In supersymmetric 247 GeV. However, the associated “standard” and “variant” ax- models, the axion is part of a supermultiplet and thus inevitably ions were quickly excluded—we refer to the Listings for detailed accompanied by a spin-0 saxion and a spin-1 , which both limits. Here we focus on “invisible axions” with f  v as the A EW also have couplings suppressed by f and are expected to have main possibility. A large masses due to supersymmetry breaking [26]. Axions have a characteristic two-photon vertex, inherited from their mixing with π0 and η. This coupling allows for the main 91.2.2 Model-dependent axion couplings search strategy based on axion-photon conversion in external mag- Although the generic axion interactions scale approximately 0 netic fields [6], an effect that also can be of astrophysical interest. with fπ/fA from the corresponding π couplings, there are non- While for axions the product “Aγγ interaction strength × mass” is negligible model-dependent factors and uncertainties. The axion’s essentially fixed by the corresponding π0 properties, one may con- two-photon interaction plays a key role for many searches, sider a more general class of axion-like particles (ALPs) where the g Aγγ ˜µν two parameters (coupling and mass) are independent. A number LAγγ = − Fµν F φA = gAγγ E · B φA , (91.4) 4 of experiments explore this more general parameter space. ALPs populating the latter are predicted to arise generically, in addi- where F is the electromagnetic field-strength tensor and F˜µν ≡ µνλρ 0123 tion to the axion, in low-energy effective field theories emerging  Fλρ/2, with ε = 1, its dual. The coupling constant from string theory [7–14]. The latter often contain also very light is [27] Abelian vector bosons under which the Standard-Model particles α  E   E  m are not charged: so-called hidden-sector photons, dark photons A gAγγ = − 1.92(4) = 0.203(3) − 0.39(1) 2 , or paraphotons. They share a number of phenomenological fea- 2πfA N N GeV tures with the axion and ALPs, notably the possibility of hidden (91.5) photon to photon conversion. Their physics cases and the current where E and N are the electromagnetic and color anomalies of the constraints are compiled in Refs. [15–17]. axial current associated with the axion. In grand unified models, and notably for DFSZ [25], E/N = 8/3, whereas for KSVZ [24] 91.2 Theory E/N = 0 if the electric charge of the new heavy quark is taken 91.2.1 Peccei-Quinn mechanism and axions to vanish. In general, a broad range of E/N values is possible The QCD Lagrangian includes a CP-violating term LΘ = [28, 29], as indicated by the diagonal yellow band in Fig. 91.1. ¯ µνa ˜a ¯ However, this band still does not exhaust all the possibilities. −Θ (αs/8π) G Gµν , where −π ≤ Θ ≤ +π is the effective Θ a In fact, there exist classes of QCD axion models whose photon parameter after diagonalizing quark masses, Gµν is the color field strength tensor, and G˜a,µν ≡ µνλρGa /2, with ε0123 = 1, its couplings populate the entire still allowed region above the yellow λρ band in Fig. 91.1, motivating axion search efforts over a wide dual. Limits on the neutron electric dipole moment [18] imply range of masses and couplings [30, 31]. |Θ¯| 10−10 even though Θ¯ = O(1) is otherwise completely satis- . The two-photon decay width is factory. The spontaneously broken global Peccei-Quinn symmetry 2 3 U(1)PQ was introduced to solve this “strong CP problem” [1, 2], g m 5 Aγγ A −24 −1  mA  the axion being the pseudo-NG boson of U(1) [3,4]. This sym- ΓA→γγ = = 1.1 × 10 s . (91.6) PQ 64 π eV metry is broken due to the axion’s anomalous triangle coupling to gluons, The second expression uses Eq. (91.5) with E/N = 0. Axions  φ  α A ¯ s µνa ˜a decay faster than the age of the universe if mA & 20 eV. The L = − Θ G Gµν , (91.2) fA 8π interaction with fermions f has derivative form and is invariant under a shift φA → φA + φ0 as behooves a NG boson, where φA is the axion field and fA the axion decay constant. Color anomaly factors have been absorbed in the normalization C f ¯ µ of f which is defined by this Lagrangian. Thus normalized, f LAff = Ψf γ γ5Ψf ∂µφA . (91.7) A A 2fA 940 91. Axions and Other Similar Particles

6 10 In the other extreme of a macroscopic field, usually a large-scale LSW (OSQAR and Others) PVLAS B-field, the momentum transfer is small, the interaction is coher- 10 8 ent over a large distance, and the conversion is best viewed as ABRACADABRA Helioscopes (CAST)

Telescopes an axion-photon oscillation phenomenon in analogy to neutrino

) flavor oscillations [38]. 1 10 10 V HESS Horizontal Branch Stars Photons propagating through a transverse magnetic field, with e G

( incident Eγ and magnetic field B parallel, may convert into ax- Fermi | SN1987A 2 12 gammas ions. For m L/2ω  2π, where L is the length of the B field A 10 A g Chandra | region and ω the photon energy, the resultant axion beam is coher-

Haloscopes KSVZ ent with the incident photon beam and the conversion probability 14 (ADMX and Others) 2 10 is Π ∼ (1/4)(gAγγ BL) . A practical realization uses a laser beam DFSZ propagating down the bore of a superconducting dipole magnet (like the bending magnets in high-energy accelerators). If an- 10 16 10 12 10 10 10 8 10 6 10 4 10 2 100 other magnet is in line with the first, but shielded by an optical Axion Mass (eV) barrier, then photons may be regenerated from the pure axion beam [39, 40]. The overall probability is P (γ → A → γ) = Π2. Figure 91.1: Exclusion plot for ALPs as described in the text. The first such Light-Shining-through-Walls (LSW) experiment was performed by the BFRT collaboration. It utilized two mag- nets of length L = 4.4 m and B = 3.7 T and found |g | < Here, Ψf is the fermion field, mf its mass, and Cf a model- Aγγ 6.7×10−7 GeV−1 at 95% CL for m < 1 meV [41]. More recently, dependent coefficient. The dimensionless combination gAff ≡ A several such experiments were performed (see Listings) [42–48]. Cf mf /fA plays the role of a Yukawa coupling and αAff ≡ −8 −1 g2 /4π of a “fine-structure constant.” The often-used pseu- The current best limit, |gAγγ | < 3.5 × 10 GeV at 95% CL Aff for m 0.3 meV (see Fig. 91.1), has been achieved by the OS- doscalar form L = −i (C m /f ) Ψ¯ γ Ψ φ need not be A . Aff f f A f 5 f A QAR (Optical Search for QED Vacuum Birefringence, Axions, equivalent to the appropriate derivative structure, for example and Photon Regeneration) experiment, which exploited two 9 T when two NG bosons are attached to one fermion line as in axion LHC dipole magnets and an 18.5 W continuous wave laser emit- emission by nucleon bremsstrahlung [32]. ting at the wavelength of 532 nm [48]. Some of these experiments In the DFSZ model [25], the tree-level coupling coefficient to have also reported limits for scalar bosons where the photon E electrons is [33] γ must be chosen perpendicular to the magnetic field B. sin2 β Ce = , (91.8) The concept of resonantly enhanced photon regeneration may 3 open unexplored regions of coupling strength [49, 50]. In this where tan β is the ratio of the vacuum expectation values of scheme, both the production and detection magnets are within the two Higgs doublets giving masses to the up- and down-type Fabry-Perot optical cavities and actively locked in frequency. The quarks, respectively: tan β = vu/vd. γ → A → γ rate is enhanced by a factor FF 0/π2 relative to a For , Cp,n have been determined as [27] single-pass experiment, where F and F 0 are the finesses of the two cavities. The resonant enhancement could be of order 10(10−12), Cp = −0.47(3) + 0.88(3)Cu − 0.39(2)C − 0.038(5)Cs (2.5−3) d improving the gAγγ sensitivity by 10 . The experiment − 0.012(5)Cc − 0.009(2)Cb − 0.0035(4)Ct , ALPS II (Any Light Particle Search II) is based on this concept C = −0.02(3) + 0.88(3)C − 0.39(2)C − 0.038(5)C and aims at an improvement of the current laboratory bound on n d u s 3 gAγγ by a factor ∼ 10 in the year 2020 [51]. − 0.012(5)Cc − 0.009(2)Cb − 0.0035(4)Ct , Resonantly enhanced photon regeneration has already been ex- (91.9) ploited in experiments searching for ‘radiowaves shining through a in terms of the corresponding model-dependent quark couplings −5 shielding’ [52–55]. For mA . 10 eV, the upper bound on gAγγ Cq, q = u, d, s, c, b, t. established by the CROWS (CERN Resonant Weakly Interacting For hadronic axions with Cq = 0, Cn is compatible with zero sub-eV Particle Search) experiment [56] is slightly less stringent whereas Cp does not vanish. In the DFSZ model, on the other than the one set by OSQAR. 1 2 1 2 hand, Cu = Cc = Ct = 3 cos β and Cd = Cs = Cb = 3 sin β, and Cp and Cn, as functions of β, 91.3.2 Photon polarization An alternative to regenerating the lost photons is to use the 2 Cp = −0.435 sin β + (−0.182 ± 0.025) , beam itself to detect conversion: the polarization of light prop- (91.10) 2 agating through a transverse B field suffers dichroism and bire- Cn = 0.414 sin β + (−0.160 ± 0.025) , fringence [57]. Dichroism: The Ek component, but not E⊥, is do not vanish simultaneously. depleted by axion production, causing a small rotation of linearly 2 The axion-pion interaction is given by the Lagrangian [34] polarized light. For mAL/2ω  2π, the effect is independent of mA. For heavier axions, it oscillates and diminishes as mA in- CAπ 0 + − 0 − + + − 0 creases, and it vanishes for mA > ω. Birefringence: This effect oc- LAπ = π π ∂µπ + π π ∂µπ − 2π π ∂µπ ∂µφA , fπfA curs because there is mixing of virtual axions in the Ek state, but (91.11) not for E⊥. Hence, linearly polarized light will develop elliptical where CAπ = (1 − z)/[3(1 + z)] in hadronic models, with 0.38 < polarization. Higher-order QED also induces vacuum magnetic z = mu/md < 0.58 [35, 36]. The chiral symmetry-breaking La- birefringence (VMB). A search for these effects was performed 0 2 0 0 in the same dipole magnets of the BFRT experiment mentioned grangian provides an additional term LAπ ∝ (mπ/fπfA)(π π + − + 0 2π π ) π φA. For hadronic axions it vanishes identically, in con- before [58]. The dichroic rotation gave a stronger limit than the −7 −1 trast to the DFSZ model (Roberto Peccei, private communica- ellipticity rotation: |gAγγ | < 3.6 × 10 GeV at 95% CL, for −4 tion). mA < 5 × 10 eV. The ellipticity limits are better at higher masses, as they fall off smoothly and do not terminate at mA. 91.3 Laboratory Searches In 2006, the PVLAS collaboration reported a signature of mag- 91.3.1 Light shining through walls netically induced vacuum dichroism that could be interpreted Searching for “invisible axions” is extremely challenging due as the effect of a pseudoscalar with mA = 1–1.5 meV and −6 −1 to its extraordinarily feeble coupling to normal matter and ra- |gAγγ | = (1.6–5) × 10 GeV [59]. Later, it turned out that diation. Currently, the most promising approaches rely on the these findings are due to instrumental artifacts [60]. This particle axion-two-photon interaction, allowing for axion-photon conver- interpretation is also excluded by the above photon regeneration sion in external electric or magnetic fields [6]. For the Coulomb searches that were inspired by the original PVLAS result. The field of a charged particle, the conversion is best viewed as a scat- fourth generation setup of the PVLAS experiment has published tering process, γ + Ze ↔ Ze + A, called Primakoff effect [37]. results on searches for VMB (see Fig. 91.1) and dichroism [61]. 91. Axions and Other Similar Particles 941

The bounds from the non-observation of the latter on gAγγ are If axions couple directly to electrons, the dominant emission slightly weaker than the ones from OSQAR. processes are atomic axio-recombination and axio-deexcitation, axio-bremsstrahlung in electron-ion or electron-electron collisions, 91.3.3 Long-range forces and Compton scattering [75]. Stars in the red giant (RG) branch New bosons would mediate long-range forces, which are severely of the color-magnitude diagram of GCs are particularly sensitive constrained by “fifth force” experiments [62]. Those looking for to these processes. In fact, they would lead to an extension of the new mass-spin couplings provide significant constraints on pseu- latter to larger brightness. Reference [76] provided high-precision doscalar bosons [63]. Presently, the most restrictive limits are ob- photometry for the Galactic globular cluster M5 (NGC 5904), tained from combining long-range force measurements with stellar allowing for a detailed comparison between the observed tip of cooling arguments [64]. For the moment, any of these limits are the RG branch with predictions based on state-of-the-art stellar far from realistic values expected for axions. Still, these efforts evolution theory. It was found that, within the uncertainties, the provide constraints on more general low-mass bosons. observed and predicted tip of the RG branch brightness agree In Ref. [65], a method was proposed that can extend the search reasonably well, leading to the bound for axion-mediated spin-dependent forces by several orders of −13 magnitude. By combining techniques used in nuclear magnetic |gAee| < 4.3 × 10 (95% CL), (91.14) resonance and short-distance tests of gravity, this method appears to be sensitive to axions in the µeV – meV mass range, indepen- implying an upper bound on the axion mass in the DFSZ model, dent of the cosmic axion abundance, if axions have a CP-violating m sin2 β < 15 meV (95% CL), (91.15) interaction with nuclei as large as the current experimental bound A on the electric dipole moment of the neutron allows. Experimen- see Fig. 91.2 (left panel). Intriguingly, the agreement would im- tal tests to demonstrate the requirements of ARIADNE (Axion prove with a small amount of extra cooling that slightly post- Resonant InterAction DetectioN Experiment) are under way [66]. pones helium ignition, prefering an electron coupling around −13 2 91.4 Axions from Astrophysical Sources |gAee| ∼ 1.9 × 10 , corresponding to mA sin β ∼ 7 meV. Bremsstrahlung is also efficient in white dwarfs (WDs), where 91.4.1 Stellar energy-loss limits the Primakoff and Compton processes are suppressed by the large Low-mass weakly-interacting particles (neutrinos, gravitons, plasma frequency. A comparison of the predicted and observed lu- axions, baryonic or leptonic gauge bosons, etc.) are produced in minosity function of WDs can be used to put limits on |gAee| [78]. hot astrophysical plasmas, and can thus transport energy out of A recent analysis, based on detailed WD cooling treatment and stars. The coupling strength of these particles with normal matter new data on the WD luminosity function (WDLF) of the Galac- and radiation is bounded by the constraint that stellar lifetimes −13 tic Disk, found that electron couplings above |gAee| & 3 × 10 , or energy-loss rates are not in conflict with observation [67, 68]. 2 corresponding to a DFSZ axion mass mA sin β & 10 meV, are We begin this discussion with our Sun and concentrate on disfavoured [79], see Fig. 91.2 (left panel). Lower couplings can- hadronic axions. They are produced predominantly by the Pri- not be discarded from the current knowledge of the WDLF of makoff process γ + Ze → Ze + A. Integrating over a standard the Galactic Disk. On the contrary, features in some WDLFs solar model yields the axion luminosity [69] can be interpreted as suggestions for electron couplings in the −14 −13 2 −3 range 7.2 × 10 . |gAee| . 2.2 × 10 , corresponding to LA = g10 × 1.85 × 10 L , (91.12) 2 2.5 meV . mA sin β . 7.5 meV [79, 80]. This hypothesis will 10 be further scrutinized by the Large Synoptic Survey Telescope where g10 = |gAγγ | × 10 GeV. The maximum of the spec- trum is at 3.0 keV, the average at 4.2 keV, and the number flux (LSST) which is expected to increase the sample of WDs in the 2 11 −2 −1 Galactic halo to hundreds of thousands [81]. This will allow for at Earth is g10 × 3.75 × 10 cm s . The solar photon lu- minosity is fixed, so energy losses due to the Primakoff process the determination of independent WDLFs from different Galac- require enhanced nuclear energy production and thus enhanced tic populations, greatly reducing the uncertainties related to star neutrino fluxes. The all-flavor measurements by SNO (Solar Neu- formation histories. For pulsationally unstable WDs (ZZ Ceti ˙ trino Observatory), together with a standard solar model, imply stars), the period decrease P /P is a measure of the cooling speed. The corresponding observations of the pulsating WDs G117-B15A LA . 0.10 L , corresponding to g10 . 7 [70], mildly superseding a similar limit from helioseismology [71]. In Ref. [72], this limit was and R548 imply additional cooling that can be interpreted also in terms of similar axion losses [82, 83]. improved to g10 < 4.1 (at 3 σ), exploiting a new statistical anal- ysis that combined helioseismology (sound speed, surface helium Recently, it has been pointed out that the hints of excessive and convective radius) and solar neutrino observations, including cooling of WDs, RGs and HB stars can be explained at one theoretical and observational errors, and accounting for tensions stroke by an ALP coupling to electrons and photons, with cou- −13 −11 −1 between input parameters of solar models, in particular the solar plings |gAee| ∼ 1.5 × 10 and |gAγγ | ∼ 1.4 × 10 GeV , element abundances. respectively [84, 85]. Intriguingly, good fits to the data can be A more restrictive limit derives from globular-cluster (GC) stars obtained employing the DFSZ axion with a mass in the range that allow for detailed tests of stellar-evolution theory. The stars 4 meV . mA . 250 meV [84]. on the horizontal branch (HB) in the color-magnitude diagram Similar constraints derive from the measured duration of the have reached helium burning with a core-averaged energy release neutrino signal of the supernova SN 1987A. Numerical simulations of about 80 erg g−1 s−1, compared to Primakoff axion losses for a variety of cases, including axions and Kaluza-Klein gravitons, of g2 30 erg g−1 s−1. The accelerated consumption of helium reveal that the energy-loss rate of a nuclear medium at the den- 10 14 −3 reduces the HB lifetime by about 80/(80+30 g2 ). Number counts sity 3 × 10 g cm and temperature 30 MeV should not exceed 10 19 −1 −1 of HB stars in a large sample of 39 Galactic GCs compared with about 1×10 erg g s [86]. The energy-loss rate from nucleon 2 4 2 the number of red giants (that are not much affected by Primakoff bremsstrahlung, N+N → N+N+A, is (CN /2fA) (T /π mN ) F . losses) give a weak indication of non-standard losses which may be Here F is a numerical factor that represents an integral over the accounted by Primakoff-like axion emission, if the photon coupling dynamical spin-density structure function because axions couple −11 −1 to the nucleon spin. For realistic conditions, even after consid- is in the range |gAγγ | = (2.9 ± 1.8) × 10 GeV [73,74]. Still, the upper bound found in this analysis, erable effort, one is limited to a heuristic estimate leading to F ≈ 1 [68]. The SN 1987A limits are of particular interest for −11 −1 |gAγγ | < 6.6 × 10 GeV (95% CL), (91.13) hadronic axions where the bounds on |gAee| are moot. Using a proton fraction of 0.3, gAnn = 0, F = 1, and T = 30 MeV, one 8 represents the strongest limit on gAγγ for a wide mass range, finds fA & 4×10 GeV and mA . 16 meV [68], see Fig. 91.2 (right see Fig. 91.1. The conservative constraint, Eq. (91.13), on gAγγ panel). A more detailed numerical calculation [87] with state of 7 may be translated to fA > 3.4 × 10 GeV (mA < 0.2 eV), using the art SN models, again assuming gAnn = 0, found that a cou- 7 −10 E/N = 0 as in the KSVZ model, or to fA > 1.3 × 10 GeV pling larger than |gApp| & 6 × 10 , would shorten significantly (mA < 0.5 eV), for the DFSZ axion model, with E/N = 8/3, see the timescale of the neutrino emission. This result is, not sur- Fig. 91.1. prisingly, rather close to the estimate in Ref. [68]. Improving the 942 91. Axions and Other Similar Particles

10 8 10 8 SN 1987A 9 9 10 10 Neutron Stars

10 10 10 10 ) ) s s s s e e 11

11 l l t t 10 10 i i Solar Searches (LUX and others) n n KSVZ u u ( (

| | 12 12 p e e 10 p 10 A Red Giants A g g DFSZ | White Dwarfs | 10 13 10 13

DFSZ 10 14 10 14 KSVZ 10 15 10 15 10 7 10 5 10 3 10 1 101 10 7 10 5 10 3 10 1 101 Axion Mass (eV) Axion Mass (eV)

Figure 91.2: Exclusion plots for ALPs as described in the text. For the DFSZ range we have taken into account the constraint 0.28 . tan β . 140 [77] arising from the requirement of perturbative unitarity of the Yukawa couplings of Standard Model fermions. calculation of axion emission via nucleon-nucleon bremsstrahlung by the emission of axions from the breaking and re-formation of beyond the basic one-pion exchange approximation appears to neutron triplet Cooper pairs [97], requiring a coupling to the neu- losen the bound [88, 89]. The latter analysis finds a reduction of tron of 2 −19 the axion emissivity by an order of magnitude if one takes into gAnn = (1.4 ± 0.5) × 10 , (91.19) account the non-vanishing mass of the exchanged pion, the con- tribution from two-pion exchange, effective in-medium nucleon corresponding to an axion mass masses and multiple nucleon scattering, leading to a looser bound m = (2.3 ± 0.4) meV/C . (91.20) (Maurizio Giannotti and Alessandro Mirizzi, private communica- A n tion) On the other hand, Ref. [98] considered another hot young NS in the supernova remnant HESS J1731-347. Its high temperature g2 + 0.29 g2 + 0.27 g g < 3.25 × 10−18. (91.16) Ann App Ann App implies that all the neutrino emission processes except neutron- neutron bremsstrahlung must be strongly suppressed, which can However, with the present understanding of SNe (current lack of be realized with a negligible neutron triplet gap and a large proton self-consistent 3D SN simulations) and the sparse data from SN singlet gap. In this setup, the bremsstrahlung from is 1987A, the constraint on the axion-nucleon couplings from SN the dominant channel for axion emission, from which one obtains 1987A should be considered more as indicative than as a sharp a limit bound [87]. g2 < 7.7 × 10−20 , (91.21) If axions interact sufficiently strongly they are trapped. Only Ann about three orders of magnitude in gANN or mA are excluded. see Fig. 91.3 (left panel). For even larger couplings, the axion flux would have been negligi- Finally, let us note that if the interpretation of the various hints ble, yet it would have triggered additional events in the detectors, for additional cooling of stars reported in this section in terms of excluding a further range [90]. A possible gap between these two emission of axions with mA ∼ meV were correct, SNe would lose SN 1987A arguments was discussed as the “hadronic axion win- a large fraction of their energy as axions. This would lead to a dow” under the assumption that gAγγ was anomalously small [91]. diffuse SN axion background in the universe with an energy den- This range is now excluded by hot dark matter (HDM) bounds sity comparable to the extra-galactic background light [99]. How- (see below). ever, there is no apparent way of detecting it or the axion burst There is another hint for excessive stellar energy losses from from the next nearby SN. On the other hand, neutrino detectors the neutron star (NS) in the supernova remnant Cassiopeia A such as IceCube, Super-Kamiokande or a future mega-ton water (Cas A): its surface temperature measured over 10 years reveals Cherenkov detector will probe exactly the mass region of inter- an unusually fast cooling rate. This rapid cooling of the Cas est by measuring the neutrino pulse duration of the next galactic A NS may be explained by NS minimal cooling with neutron SN [87]. superfluidity and proton superconductivity [92, 93]. The rapid 91.4.2 Searches for solar axions and ALPs cooling may also arise from a phase transition of the neutron Instead of using stellar energy losses to derive axion limits, one condensate into a multicomponent state [94]. Recently, Ref. [95] can also search directly for these fluxes, notably from the Sun. analyzed Cas A NS cooling in the presence of axion emission and The main focus has been on ALPs with a two-photon vertex. obtained They are produced by the Primakoff process with a flux given by g2 + 1.6g2 < 1 × 10−18, (91.17) App Ann Eq. (91.12) and an average energy of 4.2 keV, and can be detected which is comparable to the SN 1987A bound. Refs. [96] put a at Earth with the reverse process in a macroscopic B-field (“axion more conservative bound without an attempt to fit a transient helioscope”) [6]. In order to extend the sensitivity in mass towards behavior of Cas A, larger values, one can endow the photon with an effective mass in a gas, mγ = ωplas, thus matching the axion and photon dispersion 2 −17 7 relations [100]. gApp < (1 − 6) × 10 (or fA > (5 − 10) × 10 GeV), (91.18) An early implementation of these ideas used a conventional from the temperatures of Cas A and other NSs. The Cas A NS dipole magnet, with a conversion volume of variable-pressure gas cooling may also be interpreted as a hint for extra cooling caused with a xenon proportional chamber as x-ray detector [101]. The 91. Axions and Other Similar Particles 943

10 8 10 8 10 9 SN 1987A 10 10 Neutron Stars 10 10 10 12 ) ) s 2 s e

11 V l

t 10 e 14 i 10 nEDM n DFSZ G ( u

( |

|

12 n n

A 16 n 10 A

g 10 |

g KSVZ | 13 10 10 18

10 14 10 20

QCD Axion Coupling 10 15 10 22 10 7 10 5 10 3 10 1 101 10 20 10 17 10 14 10 11 10 8 Axion Mass (eV) Axion Mass (eV)

Figure 91.3: Exclusion plots for ALPs as described in the text. conversion magnet was fixed in orientation and collected data Another idea is to look at the Sun with an x-ray satellite when for about 1000 s/day. Axions were excluded for |gAγγ | < 3.6 × the Earth is in between. Solar axions and ALPs would convert −9 −1 −9 −1 10 GeV for mA < 0.03 eV, and |gAγγ | < 7.7 × 10 GeV in the Earth magnetic field on the far side and could be detected for 0.03 < mA < 0.11 eV at 95% CL. [111]. The sensitivity to gAγγ could be comparable to CAST, but Later, the Tokyo axion helioscope used a superconducting mag- only for much smaller mA. Deep solar x-ray measurements with net on a tracking mount, viewing the Sun continuously. They re- existing satellites, using the solar magnetosphere as conversion −10 −1 ported |gAγγ | < 6 × 10 GeV for mA < 0.3 eV [102]. This region, have reported preliminary limits on gAγγ [112]. experiment was recommissioned and a similar limit for masses Direct detection experiments searching for dark matter (DM) around 1 eV was reported [103]. consisting of weakly interacting massive particles, such as The most recent helioscope CAST (CERN Axion Solar Tele- EDELWEISS-II, LUX, and XENON100, have also the capability scope) uses a decommissioned LHC dipole magnet on a track- to search for solar axions and ALPs [113,114]. Recently, the LUX ing mount. The hardware includes grazing-incidence x-ray optics experiment [114] has put a bound on the axion-electron coupling with solid-state x-ray detectors, as well as novel x-ray Micromegas constant by exploiting the axio-electric effect in liquid xenon, position-sensitive gaseous detectors. Exploiting a IAXO (see be- −12 low) pathfinder system, CAST has established the limit |gAee| < 3.5 × 10 (90% CL), (91.23) −11 −1 excluding the DFSZ model with m sin2 β > 0.12 eV, cf. see |gAγγ | < 6.6 × 10 GeV (95% CL), (91.22) A Fig. 91.2 (left panel). However, as obvious from the same fig- for mA < 0.02 eV [104]. To cover larger masses, the magnet bores ure, this technique has not reached the sensitivity of energy-loss are filled with a gas at varying pressure. The runs with 4He cover considerations in stars (RGs and WDs). masses up to about 0.4 eV [105], providing the 4He limits shown 91.4.3 Conversion of astrophysical photon fluxes in Fig. 91.1. To cover yet larger masses, 3He was used to achieve Large-scale B fields exist in astrophysics that can induce axion- a larger pressure at cryogenic temperatures. Limits up to 1.17 eV photon oscillations. In practical cases, B is much smaller than in allowed CAST to “cross the axion line” for the KSVZ model [106], the laboratory, whereas the conversion region L is much larger. see Fig. 91.1. Therefore, while the product BL can be large, realistic sensitiv- Going to yet larger masses in a helioscope search is not well ities are usually restricted to very low-mass particles, far away motivated because of the cosmic HDM bound of m 1 eV (see A . from the “axion band” in a plot like Fig. 91.1. below). Sensitivity to significantly smaller values of g can Aγγ One example is SN 1987A, which would have emitted a burst be achieved with a next-generation axion helioscope with a much of ALPs due to the Primakoff production in its core. They would larger magnetic-field cross section. Realistic design options for have partially converted into γ-rays in the galactic B-field. The this “International Axion Observatory” (IAXO) have been studied lack of a gamma-ray signal in the GRS instrument of the SMM in some detail [107] and its physics potential has been reviewed satellite in coincidence with the observation of the neutrinos emit- recently [108]. Such a next-generation axion helioscope may also ted from SN 1987A therefore provides a strong bound on their push the sensitivity in the product of couplings to photons and coupling to photons [115]. This bound has been revisited and to electrons, g g , into a range beyond stellar energy-loss Aγγ Aee the underlying physics has been brought to the current state-of- limits and test the hypothesis that WD, RG, and HB cooling is the-art, as far as modelling of the supernova and the Milky-Way dominated by axion emission [84, 109]. As a first step towards magnetic field are concerned, resulting in the limit [116] IAXO, an intermediate experimental stage called BabyIAXO is currently under preparation at DESY (for a short introduction, −12 −1 −10 |gAγγ | < 5.3×10 GeV , for mA . 4.4×10 eV, (91.24) see Ref. [108]). Other Primakoff searches for solar axions and ALPs have been see Fig. 91.1. Magnetically induced oscillations between photons carried out using crystal detectors, exploiting the coherent con- and ALPs can modify the photon fluxes from distant sources version of axions into photons when the axion angle of incidence in various ways, featuring (i) frequency-dependent dimming, satisfies a Bragg condition with a crystal plane [110]. However, (ii) modified polarization, and (iii) avoiding absorption by propa- none of these limits is more restrictive than the one derived from gation in the form of axions. the constraint on the solar axion luminosity (LA . 0.10 L ) dis- For example, dimming of SNe Ia could influence the interpreta- cussed earlier. tion in terms of cosmic acceleration [117], although it has become 944 91. Axions and Other Similar Particles

clear that photon-ALP conversion could only be a subdominant permassive and stellar mass black holes. The existence of desta- effect [118]. Searches for linearly polarised emission from magne- bilizing light bosonic fields thus leads to gaps in the mass vs. spin tised white dwarfs [119] and changes of the linear polarisation from plot of rotating black holes. Stellar black hole spin measurements radio galaxies (see, e.g., Ref. [120]) provide limits close to gAγγ ∼ – exploiting well-studied binaries and two independent techniques −11 −1 −7 −15 −13 −11 10 GeV , for masses mA . 10 eV and mA . 10 eV, – exclude a mass range 6 × 10 eV < mA < 2 × 10 eV at 2 σ, 17 19 respectively, albeit with uncertainties related to the underlying which for the axion excludes 3 × 10 GeV < fA < 1 × 10 GeV −13 −1 assumptions. Even stronger limits, gAγγ . 2 × 10 GeV , for [11, 144, 145]. These bounds apply when gravitational interac- −14 mA . 10 eV, have been obtained by exploiting high-precision tions dominate over the axion self-interaction, which is true for measurements of quasar polarisations [121]. the QCD axion in this mass range. Long lasting, monochromatic Remarkably, it appears that the universe could be too trans- gravitational wave signals, which can be distinguished from or- parent to TeV γ-rays that should be absorbed by pair produc- dinary astrophysical sources by their clustering in a narrow fre- tion on the extra-galactic background light [122–126]. The sit- quency range, are expected to be produced by axions or ALPs an- uation is not conclusive at present [127–129], but the possible nihilating to gravitons. Gravitational waves could also be sourced role of photon-ALP oscillations in TeV γ-ray astronomy is tanta- by axions/ALPs transitioning between gravitationally bound lev- lizing [130]. Fortunately, the region in ALP parameter space, els. Accordingly, the gravitational wave detector Advanced LIGO −12 −10 −1 −7 −10 gAγγ ∼ 10 − 10 GeV for mA . 10 eV [131], re- should be sensitive to the axion in the mA . 10 eV region. quired to explain the anomalous TeV transparency of the uni- LIGO measurements of black hole spins in binary merger events verse, could be conceivably probed by the next generation of could also provide statistical evidence for the presence of an ax- laboratory experiments (ALPS II) and helioscopes (IAXO) men- ion [146, 147]. Similar signatures could arise for supermassive −15 tioned above. This parameter region can also be probed by black holes for particle with masses . 10 eV. Gravitational searching for an irregular behavior of the gamma ray spectrum waves from such sources could be detected at lower-frequency ob- of distant active galactic nuclei (AGN), expected to arise from servatories such as LISA. photon-ALP mixing in a limited energy range. The H.E.S.S. col- −11 −1 91.5 Cosmic Axions laboration has set a limit of |gAγγ | . 2.1 × 10 GeV , for −8 −8 1.5 × 10 eV . mA . 6.0 × 10 eV, from the non-observation 91.5.1 Cosmic axion populations of an irregular behavior of the spectrum of the AGN PKS 2155- In the early universe, axions are produced by processes involv- 304 [132], see Fig. 91.1. The Fermi-LAT collaboration has put an ing quarks and gluons [148]. After color confinement, the domi- even more stringent limit on the ALP-photon coupling [133] from nant thermalization process is π + π ↔ π + A [34]. The resulting observations of the gamma ray spectrum of NGC 1275, the cen- axion population would contribute an HDM component in anal- tral galaxy of the Perseus cluster, see Fig. 91.1. A similar analysis ogy to massive neutrinos. Cosmological precision data provide has been carried out in Ref. [134] using Fermi-LAT data of PKS restrictive constraints on a possible HDM fraction that translate 2155-304. into mA . 1 eV [149], but in detail depend on the used data Evidence for spectral irregularities has been reported in Galac- set and assumed cosmological model. In the future, data from tic sources, such as pulsars and supernova remnants, and has a EUCLID-like survey combined with Planck CMB data can de- been interpreted as hints for ALPs [135, 136]. However, the in- tect HDM axions with a mass mA & 0.15 eV at very high signifi- ferred ALP parameters are in tension with the CAST helioscope cance [150]. bounds. It should also be noted that the high signal-to-noise spec- For mA & 20 eV, axions decay fast on a cosmic time scale, tra of Galactic gamma-ray sources are dominated by uncertainties removing the axion population while injecting photons. This ex- of the instrumental systematics rather than statistical errors. Fur- cess radiation provides additional limits up to very large axion thermore, it has been shown that additional care has to be taken masses [151]. An anomalously small gAγγ provides no loophole when deriving confidence intervals on ALP parameters based on because suppressing decays leads to thermal axions overdominat- Wilks’ theorem [137], since it has been shown that it does not ing the mass density of the universe. apply for testing the ALP hypothesis [133]. The main cosmological interest in axions derives from their pos- −12 At smaller masses, mA . 10 eV, galaxy clusters become sible role as CDM. In addition to thermal processes, axions are highly efficient at interconverting ALPs and photons at x-ray en- abundantly produced by the vacuum re-alignment (VR) mecha- ergies. Constraints on spectral irregularities in the spectra of lu- nism [152]. minous x-ray sources (Hydra A, M87, NGC 1275, NGC 3862, The axion DM abundance crucially depends on the cosmolog- Seyfert galaxy 2E3140; taken by Chandra and XMM-Newton) lo- ical history. Let us first consider the so called pre-inflationary cated in or behind galaxy clusters then lead to stringent upper PQ symmetry breaking scenario, in which the PQ symmetry is limits on the ALP-photon coupling [138–143]. In this type of broken before and during inflation and not restored afterwards. studies, the uncertainty in the cluster magnetic field needs to be After the breakdown of the PQ symmetry, the axion field relaxes taken into account. This typically leads to a range of limits on somewhere in the bottom of the “wine-bottle-bottom” potential. the ALP-photon coupling that depend on the modelling assump- Near the QCD epoch, topological fluctuations of the gluon fields tions. Reference [143] recently performed the most sensitive x-ray such as instantons explicitly break the PQ symmetry. This tilting searches for ALPs to date by employing Chandra’s High-Energy of the “wine-bottle-bottom” drives the axion field toward the CP- Transmission Gratings that allow for an unsurpassed spectral res- conserving minimum, thereby exciting coherent oscillations of the olution. New observations of the AGN NGC 1275 then led to the axion field that ultimately represent a condensate of CDM. The bound fractional cosmic mass density in this homogeneous field mode, created by the VR mechanism, is [23, 153–155], −13 −1 |gAγγ | < 8 × 10 GeV (99.7% CL) (91.25)  f 1.165 ΩVRh2 ≈ 0.12 A FΘ2 for light ALPs, see Fig. 91.1. A 9 × 1011 GeV i (91.26) 1.165 91.4.4 Superradiance of black holes  6 µeV  2 ≈ 0.12 FΘi , Light bosonic fields such as axions or ALPs can affect the dy- mA namics and gravitational wave emission of rapidly rotating astro- physical black holes through the superradiance mechanism. When where h is the present-day Hubble expansion parameter in units −1 −1 their Compton wavelength is of order of the black hole size, they of 100 km s Mpc , and −π ≤ Θi ≤ π is the initial “misalign- form gravitational bound states around the black hole. Their oc- ment angle” relative to the CP-conserving position attained in the cupation number grows exponentially by extracting energy and causally connected region which evolved into today’s observable angular momentum from the black hole, forming a coherent axion universe. F = F (Θi, fA) is a factor accounting for anharmonici- 2 or ALP bound state emitting gravitational waves. When accre- ties in the axion potential. For FΘi = O(1), mA should be above tion cannot replenish the spin of the black hole, superradiance ∼ 6 µeV in order that the cosmic axion density does not exceed the 2 dominates the black hole spin evolution; this is true for both su- observed CDM density, ΩCDMh = 0.12. However, much smaller 91. Axions and Other Similar Particles 945

axion masses (much higher PQ scales) are still possible if the ini- ALPs lighter than this bound are allowed if their cosmic energy tial value Θi just happens to be small enough in today’s observable density is small, but they are quite distinct from other forms of universe (“anthropic axion window” [156]). In this cosmological DM [190]. Ignoring anharmonicities in the ALP potential, and scenario, however, quantum fluctuations of the axion field dur- taking the ALP mass to be temperature independent, the relic ing inflation are expected to lead to isocurvature density fluctua- density in DM ALPs due to the VR mechanism is given by tions which get imprinted to the temperature fluctuations of the 1/2 2 CMB [157,158]. Their non-observation puts severe constraints on VR 2  mA   fA  ΩALPh =0.12 × the Hubble expansion rate HI during inflation [159–163], which 4.7 × 10−19 eV 1016 GeV read, in the simplest cosmological inflationary scenario, 3/4 (91.32)  2  −3/4 Ωmh  1 + zeq  2 0.4175 Θi . 8  5 neV  0.15 3.4 × 103 HI . 5.7 × 10 GeV , (91.27) ma An ALP decay constant near the GUT scale gives the correct if axions represent all of DM. relic abundance for ultralight ALPs (ULAs), which we now define. In the post-inflationary PQ symmetry breaking scenario, on the Extended discussions of ULAs can be found in Refs. [191, 192]. other hand, Θ will take on different values in different patches of i The standard CDM model treats DM as a distribution of the present universe. The average contribution is [23, 153–155] cold, collisionless particles interacting only via gravity. Below 1.165 the Compton wavelength, λc = 2π/mA, the particle descrip- VR 2  30 µeV  ΩA h ≈ 0.12 . (91.28) tion of ALPs breaks down. For large occupation numbers we mA can model ALPs below the Compton wavelength as a coherent The decay of cosmic strings and domain walls gives rise to a fur- classical field. Taking as a reference length scale the Earth ra- ther population of CDM axions, whose abundance suffers from dius, R⊕ = 6371 km, we define ULAs to be those axions with significant uncertainties [154, 155, 164–172] which arise from the λc > R⊕, leading to the defining bound difficulty in understanding the energy loss process of topological −13 defects and the generated axion spectrum in a quantitative way. mULA < 2 × 10 eV . (91.33) In fact, in the present state-of-the-art it is still possible that the ULAs encompass the entire Earth in a single coherent field. The CDM contribution from the decay of topological defects is sub- coherence time of the ULA field on Earth can be estimated from dominant or overwhelmingly large in comparison to the one from the crossing time of the de Broglie wavelength at the virial velocity the VR mechanism. Correspondingly, the plausible range of ax- in the Milky Way, τ ∼ 1/m v2 . ion masses providing all of CDM in scenarios with postinflationary coh. ULA vir. PQ symmetry breaking is still rather large, namely We notice that by the definition, Eq. (91.33), an ultralight QCD axion must have a super-Planckian decay constant, fA > 3 × 19 mA ≈ 25 µeV − 4.4 meV, (91.29) 10 GeV and would require fine tuning of θi to provide the relic abundance. Natural models for ULAs can be found in string and for models with short-lived (requiring unit color anomaly N = 1) M-theory compactifications [7–14], in field theory with accidental domain walls, such as the KSVZ model. For models with long- symmetries [193], or new hidden strongly coupled sectors [194]. lived (N > 1) domain walls, such as an accidental DFSZ model In addition to the gravitational potential energy, the ULA field [173], where the PQ symmetry is broken by higher dimensional also carries gradient energy. On scales where the gradient en- Planck suppressed operators, the mass is predicted to be signifi- ergy is non-negligible, ULAs acquire an effective pressure and do cantly higher [168, 173, 174], not behave as CDM. The gradient energy opposes gravitational collapse, leading to a Jeans scale below which perturbations are mA ≈ (0.58 − 130) meV . (91.30) stable [195]. The Jeans scale suppresses linear cosmological struc- ture formation relative to CDM [196]. The Jeans scale at matter- However, the upper part of the predicted range is in conflict with radiation equality in the case that ULAs make up all of CDM stellar energy-loss limits on the axion, cf. Fig. 91.2 (right panel) is: and Fig. 91.3 (left panel). 1/4 In this post-inflationary PQ symmetry breakdown scenario, the −1/4  VR  −1  1 + zeq  ΩALP spatial axion density variations are large at the QCD transition kJ,eq =8.7 Mpc × 3.4 × 103 0.12 and they are not erased by free streaming. Gravitationally bound (91.34) “axion miniclusters” form before and around matter-radiation  m 1/2 equality [175–177]. A significant fraction of CDM axions can re- ULA . −22 side in these bound objects [171,178]. Remarkably, the minicluster 10 eV fraction can be bounded by gravitational lensing [179–181]. On non-linear scales the gradient energy leads to the existence of a In the above predictions of the fractional cosmic mass den- class of pseudo-solitons known as oscillatons, or axion stars [197]. sity in axions, the exponent, 1.165, arises from the non-trivial Cosmological and astrophysical observations are consistent temperature dependence of the topological susceptibility χ(T ) = with the CDM model, and departures from it are only al- 2 2 mA(T )fA at temperatures slighty above the QCD quark-hadron lowed on the scales of the smallest observed DM structures with phase transition. Lattice QCD calculations of this exponent 6−8 M ∼ 10 M . The CMB power spectrum and galaxy auto- [23, 182–186], but also Ref. [187], found it to be remarkably close correlation power spectrum limit the ULA mass to mULA > to the prediction of the dilute instanton gas approximation [188] 10−24 eV from linear theory of structure formation [190,198]. An- which was previously exploited. Therefore, the state-of-the-art alytic models [199] and N-body simulations [200] for non-linear prediction of the axion mass relevant for DM for a fixed initial structures show that halo formation is suppressed in ULA models misalignment angle Θi differs from the previous prediction by just relative to CDM. This leads to constraints on the ULA mass of a factor of order one. −22 mULA > 10 eV from observations of high-z galaxies [200,201], The non-thermal production mechanisms attributed to axions −21 and mULA > 10 eV from the Lyman-alpha forest flux power are generic to light bosonic weakly interacting particles such as spectrum [202]. Including the effects of anharmonicities on struc- ALPs [189]. The relic abundance is set by the epoch when the ture formation with ALPs can weaken these bounds if the mis- axion mass becomes significant, 3H(t) ≈ m (t), and ALP field A alignment angle Θi ≈ π [203]. Cosmological simulations that oscillations begin. For ALPs to contribute to the DM density treat gradient energy in the ULA field beyond the N-body ap- this epoch must precede that of matter radiation equality. For a proximation have just recently become available [204, 205], and temperature independent ALP mass this leads to the bound: show, among other things, evidence for the formation of axion 1/2 stars in the centres of ULA halos (various consequences of axion  2  3/2 −28 Ωm h  1 + zeq  stars are considered in Refs. [206]). These central axion stars have mA 7 × 10 eV . (91.31) & 0.15 3.4 × 103 been conjectured to play a role in the apparently cored density 946 91. Axions and Other Similar Particles

profiles of dwarf spheroidal galaxies, and other central galactic re- 91.5.3 Microwave cavity experiments gions [204, 207–209]. However, the relationship between the halo In a big part of the plausible mA range for CDM, galactic halo mass and the axion star mass [210] leads to problems with this axions may be detected by their resonant conversion into a qua- scenario in some galaxies [211–213]. It should be emphasised that si-monochromatic microwave signal in a high-Q electromagnetic many of the conclusions about the role of ULA axion stars in cavity permeated by a strong static B field [6,233,234]. The cavity galactic dynamics are based on use of simulation results that do frequency is tunable, and the signal is maximized when the fre- not contain baryons (however, see Ref. [214]), and feedback [215] quency is the total axion energy, rest mass plus kinetic energy, of −6 could be important. ν = (mA/2π) [1 + O(10 )], the width above the rest mass repre- Inside DM halos the axion gradient energy causes coherence senting the virial distribution in the galaxy. The frequency spec- on the de Broglie wavelength and fluctuations on the coherence trum may also contain finer structure from axions more recently time [204,216]. These fluctuations can be thought of as short-lived fallen into the galactic potential and not yet completely virial- and lead to relaxation processes that can be de- ized [235, 236]. scribed statistically [192,217] (this relaxation processes also leads 10 10 to the gravitational condensation of axion stars [218]). The typical CAST Limit relaxation time is: 10 11 3 2 4 10  mULA   v   r  t ∼ 10 years , ORGAN −22 −1 12 ADMX 10 eV 100 km s 5 kpc ) 10 1

(91.35) V e

G 13 where v and r are the velocity and radius of the orbit in the host ( 10 HAYSTAC

DM halo. |

A RBF g | Relaxation processes such as these are not observed in galaxies, 10 14 ADMX UF though there are some circumstances where they may be desir- KSVZ able [192]. An absence of observed relaxation can be used to set 15 10 QCD AxionDFSZ Coupling limits on the ULA mass. An absence of Milky Way disk thicken- −22 ADMX ing excludes mULA > 0.6 × 10 eV [219], while stellar streams 10 16 −22 6 5 4 give the stronger bound mULA > 1.5 × 10 eV [220]. The sur- 10 10 10 vival of the old star cluster in Eridanus II [221] excludes the range Axion Mass (eV) −21 −19 of masses 10 eV . mULA . 10 eV [222]. As in the case Figure 91.4: Exclusion plot for ALPs as described in the text. of ULA axion stars, current constraints from heating do not fully account for the possible role of baryons. The feasibility of this technique was established in early ex- Finally, one should note that the beyond-CDM physics of ULAs periments (RBF and UF) of relatively small sensitive volume, (Jeans scale, relaxation, axion star formation) of course also ap- O(1 liter), with HFET-based amplifiers, setting limits in the plies to the QCD axion on smaller length scales. This is of par- range 4.5 < mA < 16.3 µeV [237], but lacking by 2–3 orders ticular interest inside axion miniclusters [175, 176, 218, 223]. of magnitude the sensitivity required to detect realistic axions, 91.5.2 Telescope searches see Fig. 91.4. Later, ADMX (B ∼ 8 T, V ∼ 200 liters) has achieved sensitivity to KSVZ axions, assuming they saturate the The two-photon decay is extremely slow for axions with masses local DM density and are well virialized, over the mass range 1.9– in the CDM regime, but could be detectable for eV masses. The 3.3 µeV [238]. Should halo axions have a significant component signature would be a quasi-monochromatic emission line from not yet virialized, ADMX is sensitive to DFSZ axions over the galaxies and galaxy clusters. The expected optical line intensity entire mass range [239]. The corresponding 90% CL exclusion re- for DFSZ axions is similar to the continuum night emission. An gions shown in Fig. 91.4 are normalized to an assumed local CDM early search in three rich Abell clusters [224] and a recent search density of 7.5 × 10−25 g cm−3 (450 MeV cm−3). More recently, in two rich Abell clusters [225] exclude the “Telescope” range in the ADMX experiment commissioned an upgrade [240] that re- Fig. 91.1. Of course, axions in this mass range would anyway places the microwave HFET amplifiers by near quantum-limited provide an excessive hot DM contribution. low-noise dc SQUID microwave amplifiers [241]. It has reached Very low-mass axions in halos produce a weak quasi- an unprecedented axion DM sensitivity in the mass range be- monochromatic radio line. Virial velocities in undisrupted dwarf tween 2.66 and 3.31 µeV [242,243], down to the DFSZ benchmark galaxies are very low, and the axion decay line would therefore axion-photon coupling, see Fig. 91.4. This apparatus is also sensi- be extremely narrow. A search with the Haystack radio tele- tive to other hypothetical light bosons, such as hidden photons or scope on three nearby dwarf galaxies provided a limit |gAγγ | < −9 −1 chameleons, over a limited parameter space [189,244,245]. ADMX 1.0×10 GeV at 96% CL for 298 < mA < 363 µeV [226]. How- has also done a testbed experiment to probe higher masses. This ever, this combination of mA and gAγγ does not exclude plausible experiment lives inside of and operates in tandem with the main axion models. ADMX experiment, searches in three widely spaced frequency A monochromatic signal is also produced in the conversion ranges (4202–4249 MHz, 5086–5799 MHz and 7173–7203 MHz), of DM axions in the background of slowly varying galactic B- uses both the TM010 and TM020 cavity modes, and demonstrates fields [227]. The signal is, however, sensitive to magnetic field the successful use of a piezoelectric actuator for cavity tuning power on the scale of the axion mass [228]. Present and future [246]. Recently, the HAYSTAC experiment reported on first re- radio telescopes appear to be able to probe ALP DM in the mass sults from a new microwave cavity search for DM axions with −13 −1 range 0.1 − 100 µeV for couplings gAγγ & 10 GeV [228] – masses above 20 µeV. They exclude axions with two-photon cou- unfortunately not reaching down to the benchmark QCD axion −14 −1 pling |gAγγ | & 2×10 GeV over the range 23.15 µeV < mA < sensitivity. 24.0 µeV [247,248], a factor of 2.7 above the KSVZ benchmark, see Resonant conversion of QCD axion DM in neutron star mag- Fig. 91.4. Exploiting a Josephson parametric amplifier, this ex- netospheres may give a detectable signal from individual neutron periment has demonstrated total noise approaching the standard stars for axion masses in the µeV range [229]. Furthermore, stim- quantum limit for the first time in an axion search. A Rydberg ulated ALP decays in high radiation environments may be de- single-photon detector [249], like any photon counter, can in tectable, by next-generation radio telescopes such as the Square principle evade the standard quantum limit for coherent photon −11 −1 Kilometer Array, down to gAγγ & 10 GeV , for masses be- detection. The ORGAN experiment is designed to probe axions tween µeV and 0.1 meV [230]. in the mass range 60 µeV < mA < 210 µeV. In a pathfinding run, −12 −1 Photon propagation on an ULA DM background can induce it has set the limit |gAγγ | < 2 × 10 GeV at 110 µeV, in a birefringence that can be compared with upper limits from the span of 2.5 neV [250]. There are further microwave cavity axion CMB [231] and may also be probed with other sources such as DM experiments recently in operation (CULTASK [251]), under pulsars [232]. construction (RADES [252]) or proposed (KLASH [253]). 91. Axions and Other Similar Particles 947

91.5.4 New concepts for axion DM direct detection axion – established upper limits on the axion-photon coupling in Other new concepts for searching for axion DM are also be- the mass range 3.1 × 10−10 eV − 8.3 × 10−9 eV [274], which are, ing investigated. An alternative to the microwave cavity tech- however, not competitive yet with other limits in this mass range, nique is based on a novel detector architecture consisting of see Fig. 91.1. an open, Fabry-Perot resonator and a series of current-carrying An eventually non-zero axion electron coupling gAee will lead wire planes [254]. The Orpheus detector has demonstrated to an electron spin precession about the axion DM wind [275]. this new technique, excluding DM ALPs with masses between The QUAX (QUaerere AXions) experiment aims at exploiting MR 68.2 and 76.5 µeV and axion-photon couplings greater than 4 × inside a magnetized material [276]. Because of the higher Larmor 10−7 GeV−1. This technique may be able to probe DM axions frequency of the electron, it is sensitive in the classic window. in the mass range from 40 to 700 µeV. Another detector concept exploits the fact that a magnetized mirror would radiate photons 91.6 Conclusions in the background of axion DM, which could be collected like in a There is a strengthening physics case for very weakly coupled dish antenna [255]. Searches for hidden photon DM exploiting this light particles beyond the Standard Model. The elegant solution technique are already underway [256]. The proposed MADMAX of the strong CP problem proposed by Peccei and Quinn yields experiment will place a stack of dielectric layers in a magnetic field a particularly strong motivation for the axion. In many theo- in order to resonantly enhance the photon signal, aiming a sensi- retically appealing ultraviolet completions of the Standard Model axions and ALPs occur automatically. Moreover, they are natural tivity to probe the mass range 40 µeV . mA . 200 µeV [257,258]. Optical dielectric haloscopes with single photon signal detection CDM candidates. Perhaps the first hints of their existence have have been proposed to search for axions in the 50 meV – 10 eV already been seen in the anomalous excessive cooling of stars and mass range [259]. Absorption of axions on molecular transitions the anomalous transparency of the Universe for VHE gamma rays. Interestingly, a significant portion of previously unexplored, but can be sensitive to the axion pseudoscalar coupling gANN to nu- phenomenologically very interesting and theoretically very well cleons or the pseudoscalar coupling gAee to electrons in the 0.5 – 20 eV range [260]. Another proposed axion DM search method motivated axion and ALP parameter space can be tackled in the sensitive in the 100 µeV mass range is to cool a kilogram-sized foreseeable future by a number of terrestrial experiments searching sample to mK temperatures and count axion induced atomic tran- for axion/ALP DM, for solar axions/ALPs, and for light appar- sitions using laser techniques [261]. ently shining through a wall. The oscillating galactic DM axion field induces oscillating nu- References clear electric dipole moments (EDMs) [262], [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 p (1977). dN (t) = gANγ 2ρADM cos(mA t)/mA , (91.36) [2] R. D. Peccei and H. R. Quinn, Phys. Rev. D16, 1791 (1977).

where gANγ is the coupling of the axion to the nucleon EDM [3] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978). operator, [4] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978). i µν LA ⊃ − gANγ A Ψ N σµν γ5ΨN F . (91.37) [5] Y. Chikashige, R.N. Mohapatra, and R.D. Peccei, Phys. 2 Lett. B98, 265 (1981); G. B. Gelmini and M. Roncadelli, For the QCD axion, this coupling is predicted as [263] Phys. Lett. 99B, 411 (1981). [6] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983) and Erratum −3  1  1 gAnγ = −gApγ = (3.7 ± 1.5) × 10 = ibid., 52, 695 (1984). f GeV A (91.38)  m  1 [7] E. Witten, Phys. Lett. 149B, 351 (1984). = (6.5 ± 2.6) × 10−13 A meV GeV2 [8] J. P. Conlon, JHEP 05, 078 (2006), [hep-th/0602233]. [9] P. Svrcek and E. Witten, JHEP 06, 051 (2006), [hep- and plotted as a yellow diagonal band in Fig. 91.3 (right panel). th/0605206]. An analysis of the ratio of spin-precession frequencies of stored ul- tracold neutrons and 199Hg atoms measured by neutron EDM ex- [10] K.-S. Choi et al., Phys. Lett. B675, 381 (2009), periments for an axion-induced oscillating neutron EDM revealed [arXiv:0902.3070]. no signal consistent with axion DM, excluding a sizeable region of [11] A. Arvanitaki et al., Phys. Rev. D81, 123530 (2010), −24 −17 parameter space in the mass region 10 eV ≤ mA ≤ 10 eV [arXiv:0905.4720]. [264], which surpass the limits on anomalous energy loss of SN [12] B. S. Acharya, K. Bobkov and P. Kumar, JHEP 11, 105 1987A [262] by more than seven orders of magnitude, but are still (2010), [arXiv:1004.5138]. a few orders of magnitude above the QCD axion expectations, see Fig. 91.3 (right panel). The oscillating EDMs cause also the [13] M. Cicoli, M. Goodsell and A. Ringwald, JHEP 10, 146 precession of nuclear spins in a nucleon spin polarized sample in (2012), [arXiv:1206.0819]. the presence of an electric field. The resulting transverse mag- [14] J. Halverson, C. Long and P. Nath, Phys. Rev. D96, 5, netization can be searched for by exploiting magnetic-resonance 056025 (2017), [arXiv:1703.07779]. (MR) techniques, which are most sensitive in the range of low os- [15] J. Jaeckel and A. Ringwald, Ann. Rev. Nucl. Part. Sci. 60, cillation frequencies corresponding to sub-neV axion masses. The 405 (2010), [arXiv:1002.0329]. aim of the corresponding Cosmic Axion Spin Precession Exper- iment (CASPEr) [265] is to probe axion DM in the anthropic [16] A. Ringwald, Phys. Dark Univ. 1, 116 (2012), 15 window, fA & 10 GeV (mA . neV), motivated from Grand Uni- [arXiv:1210.5081]. fication [266–269]. Sub-µeV ALP masses can also be probed by [17] J. Jaeckel, Frascati Phys. Ser. 56, 172 (2012), using the storage ring EDM method proposed in Ref. [270] which [arXiv:1303.1821]. exploits a combination of B and E-fields to produce a resonance between the g − 2 spin precession frequency and the DM ALP [18] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006), [hep- field oscillation frequency. This method, however, does not reach ex/0602020]. the sensitivity to probe the QCD axion prediction for gANγ . [19] H. Georgi, D. B. Kaplan and L. Randall, Phys. Lett. 169B, In the intermediate mass region, neV . mA . 0.1 µeV, one may 73 (1986). exploit a cooled LC circuit and precision magnetometry to search [20] R. J. Crewther, Phys. Lett. 70B, 349 (1977). for the oscillating electric current induced by DM axions in a strong magnetic field [271]. A similar approach is followed by the [21] P. Di Vecchia and G. Veneziano, Nucl. Phys. B171, 253 proposed ABRACADABRA [272] and DM-Radio Pathfinder [273] (1980). experiments. Recently, ABRACADABRA-10 cm – a small-scale [22] M. Gorghetto and G. Villadoro, JHEP 03, 033 (2019), prototype for a future detector that could be sensitive to the QCD [arXiv:1812.01008]. 948 91. Axions and Other Similar Particles

[23] S. Borsanyi et al., Nature 539, 7627, 69 (2016), [55] R. Povey, J. Hartnett and M. Tobar, Phys. Rev. D82, [arXiv:1606.07494]. 052003 (2010), [arXiv:1003.0964]. [24] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. A. Shifman, [56] M. Betz et al., Phys. Rev. D88, 7, 075014 (2013), A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B166, 493 [arXiv:1310.8098]. (1980). [57] L. Maiani, R. Petronzio and E. Zavattini, Phys. Lett. B175, [25] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B, 359 (1986). 199 (1981); A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 [58] Y. Semertzidis et al., Phys. Rev. Lett. 64, 2988 (1990). (1980), [Yad. Fiz.31,497(1980)]. [59] E. Zavattini et al. (PVLAS), Phys. Rev. Lett. 96, 110406 [26] J. E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010), (2006), [Erratum: Phys. Rev. Lett.99,129901(2007)], [hep- [arXiv:0807.3125]. ex/0507107]. [27] G. Grilli di Cortona et al., JHEP 01, 034 (2016), [60] E. Zavattini et al. (PVLAS), Phys. Rev. D77, 032006 [arXiv:1511.02867]. (2008), [arXiv:0706.3419]. [28] J. E. Kim, Phys. Rev. D58, 055006 (1998), [hep- [61] F. Della Valle et al., Eur. Phys. J. C76, 1, 24 (2016), ph/9802220]. [arXiv:1510.08052]. [29] L. Di Luzio, F. Mescia and E. Nardi, Phys. Rev. Lett. [62] E. Fischbach and C. Talmadge, Nature 356, 207 (1992). 118, 3, 031801 (2017), [arXiv:1610.07593]; L. Di Luzio, F. Mescia and E. Nardi, Phys. Rev. D96, 7, 075003 (2017), [63] J. E. Moody and F. Wilczek, Phys. Rev. D30, 130 [arXiv:1705.05370]. (1984); A. N. Youdin et al., Phys. Rev. Lett. 77, 2170 (1996); W.-T. Ni et al., Phys. Rev. Lett. 82, 2439 (1999); [30] M. Farina et al., JHEP 01, 095 (2017), [arXiv:1611.09855]. D. F. Phillips et al., Phys. Rev. D63, 111101 (2001), [31] P. Agrawal et al., JHEP 02, 006 (2018), [arXiv:1709.06085]. [arXiv:physics/0008230]; B. R. Heckel et al., Phys. Rev. [32] G. Raffelt and D. Seckel, Phys. Rev. Lett. 60, 1793 (1988); Lett. 97, 021603 (2006), [hep-ph/0606218]; S. A. Hoedl M. Carena and R. D. Peccei, Phys. Rev. D40, 652 (1989); et al., Phys. Rev. Lett. 106, 041801 (2011). K. Choi, K. Kang and J. E. Kim, Phys. Rev. Lett. 62, 849 [64] G. Raffelt, Phys. Rev. D86, 015001 (2012), (1989). [arXiv:1205.1776]. [33] M. Srednicki, Nucl. Phys. B260, 689 (1985). [65] A. Arvanitaki and A. A. Geraci, Phys. Rev. Lett. 113, 16, [34] S. Chang and K. Choi, Phys. Lett. B316, 51 (1993), [hep- 161801 (2014), [arXiv:1403.1290]. ph/9306216]. [66] A. A. Geraci et al. (ARIADNE), Springer Proc. Phys. 211, [35] H. Leutwyler, Phys. Lett. B378, 313 (1996), [hep- 151 (2018), [arXiv:1710.05413]. ph/9602366]. [67] M. S. Turner, Phys. Rept. 197, 67 (1990). [36] C. Patrignani et al. (Particle Data Group), Chin. Phys. [68] G. G. Raffelt, Lect. Notes Phys. 741, 51 (2008), [,51(2006)], C40, 10, 100001 (2016). [hep-ph/0611350]. [37] D. A. Dicus et al., Phys. Rev. D18, 1829 (1978). [69] S. Andriamonje et al. (CAST), JCAP 0704, 010 (2007), [38] G. Raffelt and L. Stodolsky, Phys. Rev. D37, 1237 (1988). [hep-ex/0702006]. [39] A. A. Anselm, Yad. Fiz. 42, 1480 (1985). [70] P. Gondolo and G. G. Raffelt, Phys. Rev. D79, 107301 (2009), [arXiv:0807.2926]. [40] K. van Bibber et al., Phys. Rev. Lett. 59, 759 (1987). [71] H. Schlattl, A. Weiss and G. Raffelt, Astropart. Phys. 10, [41] G. Ruoso et al., Z. Phys. C56, 505 (1992); R. Cameron 353 (1999), [hep-ph/9807476]. et al., Phys. Rev. D47, 3707 (1993). [72] N. Vinyoles et al., JCAP 1510, 10, 015 (2015), [42] M. Fouche et al., Phys. Rev. D78, 032013 (2008), [arXiv:1501.01639]. [arXiv:0808.2800]. [73] A. Ayala et al., Phys. Rev. Lett. 113, 19, 191302 (2014), [43] P. Pugnat et al. (OSQAR), Phys. Rev. D78, 092003 (2008), [arXiv:1406.6053]. [arXiv:0712.3362]. [74] O. Straniero et al., in “Proceedings, 11th Patras Work- [44] A. S. Chou et al. (GammeV (T-969)), Phys. Rev. Lett. 100, shop on Axions, WIMPs and WISPs (Axion-WIMP 2015): 080402 (2008), [arXiv:0710.3783]. Zaragoza, Spain, June 22-26, 2015,” 77–81 (2015). [45] A. Afanasev et al., Phys. Rev. Lett. 101, 120401 (2008), [75] J. Redondo, JCAP 1312, 008 (2013), [arXiv:1310.0823]. [arXiv:0806.2631]. [76] N. Viaux et al., Phys. Rev. Lett. 111, 231301 (2013), [46] K. Ehret et al. (ALPS), Phys. Lett. B689, 149 (2010), [arXiv:1311.1669]. [arXiv:1004.1313]. [77] C.-Y. Chen and S. Dawson, Phys. Rev. D87, 055016 (2013), [47] P. Pugnat et al. (OSQAR), Eur. Phys. J. C74, 8, 3027 [arXiv:1301.0309]. (2014), [arXiv:1306.0443]. [78] G. G. Raffelt, Phys. Lett. 166B, 402 (1986); S. I. Blinnikov [48] R. Ballou et al. (OSQAR), Phys. Rev. D92, 9, 092002 and N. V. Dunina-Barkovskaya, Mon. Not. Roy. Astron. (2015), [arXiv:1506.08082]. Soc. 266, 289 (1994). [49] F. Hoogeveen and T. Ziegenhagen, Nucl. Phys. B358, 3 [79] M. M. Miller Bertolami et al., JCAP 1410, 10, 069 (2014), (1991). [arXiv:1406.7712]. [50] P. Sikivie, D. B. Tanner and K. van Bibber, Phys. Rev. Lett. [80] J. Isern et al., Astrophys. J. 682, L109 (2008), 98, 172002 (2007), [hep-ph/0701198]; G. Mueller et al., [arXiv:0806.2807]; J. Isern et al., J. Phys. Conf. Ser. 172, Phys. Rev. D80, 072004 (2009), [arXiv:0907.5387]. 012005 (2009), [arXiv:0812.3043]. [51] R. Baehre et al. (ALPS Collab.), JINST 1308, T09001 [81] A. Drlica-Wagner et al. (LSST Dark Matter Group) (2019), (2013). [arXiv:1902.01055]. [52] F. Hoogeveen, Phys. Lett. B288, 195 (1992). [82] J. Isern et al., Astron. & Astrophys. 512, A86 (2010); A. H. [53] J. Jaeckel and A. Ringwald, Phys. Lett. B659, 509 (2008), Corsico et al., Mon. Not. Roy. Astron. Soc. 424, 2792 [arXiv:0707.2063]. (2012), [arXiv:1205.6180]; A. H. Corsico et al., JCAP 1212, [54] F. Caspers, J. Jaeckel and A. Ringwald, JINST 4, P11013 010 (2012), [arXiv:1211.3389]. (2009), [arXiv:0908.0759]. [83] A. H. Córsico et al. (2019), [arXiv:1907.00115]. 91. Axions and Other Similar Particles 949

[84] M. Giannotti et al., JCAP 1710, 10, 010 (2017), [113] E. Armengaud et al., JCAP 1311, 067 (2013), [arXiv:1708.02111]. [arXiv:1307.1488]; E. Aprile et al. (XENON100), [85] M. Giannotti et al., JCAP 1605, 05, 057 (2016), Phys. Rev. D90, 6, 062009 (2014), [Erratum: Phys. [arXiv:1512.08108]. Rev.D95,no.2,029904(2017)], [arXiv:1404.1455]. [86] G.G. Raffelt, Stars as Laboratories for Fundamental Phys- [114] D. S. Akerib et al. (LUX), Phys. Rev. Lett. 118, 26, 261301 ics, (Univ. of Chicago Press, Chicago, 1996). (2017), [arXiv:1704.02297]. [115] J. W. Brockway, E. D. Carlson and G. G. Raffelt, Phys. [87] T. Fischer et al., Phys. Rev. D94, 8, 085012 (2016), Lett. B383, 439 (1996), [arXiv:astro-ph/9605197]; J. A. [arXiv:1605.08780]. Grifols, E. Masso and R. Toldra, Phys. Rev. Lett. 77, 2372 [88] J. H. Chang, R. Essig and S. D. McDermott, JHEP 09, 051 (1996), [arXiv:astro-ph/9606028]. (2018), [arXiv:1803.00993]. [116] A. Payez et al., JCAP 1502, 02, 006 (2015), [89] P. Carenza et al. (2019), [arXiv:1906.11844]. [arXiv:1410.3747]. [90] J. Engel, D. Seckel and A. C. Hayes, Phys. Rev. Lett. 65, [117] C. Csaki, N. Kaloper and J. Terning, Phys. Rev. Lett. 88, 960 (1990). 161302 (2002), [hep-ph/0111311]. [91] T. Moroi and H. Murayama, Phys. Lett. B440, 69 (1998), [118] A. Mirizzi, G.G. Raffelt, and P.D. Serpico, Lect. Notes Phys. [hep-ph/9804291]. 741, 115 (2008). [92] D. Page et al., Phys. Rev. Lett. 106, 081101 (2011), [119] R. Gill and J. S. Heyl, Phys. Rev. D84, 085001 (2011), [arXiv:1011.6142]. [arXiv:1105.2083]. [93] P. S. Shternin et al., Mon. Not. Roy. Astron. Soc. 412, L108 [120] D. Horns et al., Phys. Rev. D85, 085021 (2012), (2011), [arXiv:1012.0045]. [arXiv:1203.2184]. [94] L. B. Leinson, Phys. Lett. B741, 87 (2015), [121] A. Payez, J. R. Cudell and D. Hutsemekers, JCAP 1207, [arXiv:1411.6833]. 041 (2012), [arXiv:1204.6187]. [95] K. Hamaguchi et al., Phys. Rev. D98, 10, 103015 (2018), [122] A. Dominguez, M. A. Sanchez-Conde and F. Prada, JCAP [arXiv:1806.07151]. 1111, 020 (2011), [arXiv:1106.1860]. [96] J. Keller and A. Sedrakian, Nucl. Phys. A897, 62 (2013), [123] W. Essey and A. Kusenko, Astrophys. J. 751, L11 (2012), [arXiv:1205.6940]; A. Sedrakian, Phys. Rev. D93, 6, 065044 [arXiv:1111.0815]. (2016), [arXiv:1512.07828]. [124] D. Horns and M. Meyer, JCAP 1202, 033 (2012), [97] L. B. Leinson, JCAP 1408, 031 (2014), [arXiv:1405.6873]. [arXiv:1201.4711]. [98] M. V. Beznogov et al., Phys. Rev. C98, 3, 035802 (2018), [125] G. I. Rubtsov and S. V. Troitsky, JETP Lett. 100, 6, [arXiv:1806.07991]. 355 (2014), [Pisma Zh. Eksp. Teor. Fiz.100,no.6,397(2014)], [99] G. G. Raffelt, J. Redondo and N. Viaux Maira, Phys. Rev. [arXiv:1406.0239]. D84, 103008 (2011), [arXiv:1110.6397]. [126] K. Kohri and H. Kodama, Phys. Rev. D96, 5, 051701 (2017), [arXiv:1704.05189]. [100] K. van Bibber et al., Phys. Rev. D39, 2089 (1989). [127] D. A. Sanchez, S. Fegan and B. Giebels, Astron. Astrophys. [101] D. M. Lazarus et al., Phys. Rev. Lett. 69, 2333 (1992). 554, A75 (2013), [arXiv:1303.5923]. [102] S. Moriyama et al., Phys. Lett. B434, 147 (1998), [hep- [128] J. Biteau and D. A. Williams, Astrophys. J. 812, 1, 60 ex/9805026]; Y. Inoue et al., Phys. Lett. B536, 18 (2002), (2015), [arXiv:1502.04166]. [arXiv:astro-ph/0204388]. [129] A. Domínguez and M. Ajello, Astrophys. J. 813, 2, L34 [103] Y. Inoue et al., Phys. Lett. B668, 93 (2008), (2015), [arXiv:1510.07913]. [arXiv:0806.2230]. [130] A. De Angelis, G. Galanti and M. Roncadelli, [104] V. Anastassopoulos et al. (CAST), Nature Phys. 13, 584 Phys. Rev. D84, 105030 (2011), [Erratum: Phys. (2017), [arXiv:1705.02290]. Rev.D87,no.10,109903(2013)], [arXiv:1106.1132]; M. Simet, [105] E. Arik et al. (CAST), JCAP 0902, 008 (2009), D. Hooper and P. D. Serpico, Phys. Rev. D77, 063001 [arXiv:0810.4482]. (2008), [arXiv:0712.2825]; M. A. Sanchez-Conde et al., [106] S. Aune et al. (CAST), Phys. Rev. Lett. 107, 261302 Phys. Rev. D79, 123511 (2009), [arXiv:0905.3270]. (2011), [arXiv:1106.3919]; M. Arik et al. (CAST), Phys. [131] M. Meyer, D. Horns and M. Raue, Phys. Rev. D87, 3, Rev. Lett. 112, 9, 091302 (2014), [arXiv:1307.1985]; 035027 (2013), [arXiv:1302.1208]. M. Arik et al. (CAST), Phys. Rev. D92, 2, 021101 (2015), [132] A. Abramowski et al. (H.E.S.S.), Phys. Rev. D88, 10, [arXiv:1503.00610]. 102003 (2013), [arXiv:1311.3148]. [107] E. Armengaud et al., JINST 9, T05002 (2014), [133] M. Ajello et al. (Fermi-LAT), Phys. Rev. Lett. 116, 16, [arXiv:1401.3233]. 161101 (2016), [arXiv:1603.06978]. [108] E. Armengaud et al. (IAXO), JCAP 1906, 06, 047 (2019), [134] C. Zhang et al., Phys. Rev. D97, 6, 063009 (2018), [arXiv:1904.09155]. [arXiv:1802.08420]. [109] K. Barth et al., JCAP 1305, 010 (2013), [arXiv:1302.6283]. [135] Z.-Q. Xia et al., Phys. Rev. D97, 6, 063003 (2018), [110] F. T. Avignone, III et al. (SOLAX), Phys. Rev. Lett. 81, [arXiv:1801.01646]. 5068 (1998), [arXiv:astro-ph/9708008]; S. Cebrian et al., [136] J. Majumdar, F. Calore and D. Horns, JCAP 1804, 04, 048 Astropart. Phys. 10, 397 (1999), [arXiv:astro-ph/9811359]; (2018), [arXiv:1801.08813]. A. Morales et al. (COSME), Astropart. Phys. 16, 325 (2002), [hep-ex/0101037]; R. Bernabei et al., Phys. Lett. [137] S. S. Wilks, Annals Math. Statist. 9, 1, 60 (1938). B515, 6 (2001); Z. Ahmed et al. (CDMS), Phys. Rev. Lett. [138] D. Wouters and P. Brun, Astrophys. J. 772, 44 (2013), 103, 141802 (2009), [arXiv:0902.4693]. [arXiv:1304.0989]. [111] H. Davoudiasl and P. Huber, Phys. Rev. Lett. 97, 141302 [139] M. Berg et al., Astrophys. J. 847, 2, 101 (2017), (2006), [hep-ph/0509293]. [arXiv:1605.01043]. [112] H. S. Hudson et al., ASP Conf. Ser. 455, 25 (2012), [140] M. C. D. Marsh et al., JCAP 1712, 12, 036 (2017), [arXiv:1201.4607]. [arXiv:1703.07354]. 950 91. Axions and Other Similar Particles

[141] J. P. Conlon et al., JCAP 1707, 07, 005 (2017), [171] M. Buschmann, J. W. Foster and B. R. Safdi (2019), [arXiv:1704.05256]. [arXiv:1906.00967]. [142] L. Chen and J. P. Conlon, Mon. Not. Roy. Astron. Soc. 479, [172] M. Hindmarsh et al. (2019), [arXiv:1908.03522]. 2, 2243 (2018), [arXiv:1712.08313]. [173] A. Ringwald and K. Saikawa, Phys. Rev. D93, 8, 085031 [143] C. S. Reynolds et al. (2019), [arXiv:1907.05475]. (2016), [Addendum: Phys. Rev.D94,no.4,049908(2016)], [144] A. Arvanitaki and S. Dubovsky, Phys. Rev. D83, 044026 [arXiv:1512.06436]. (2011), [arXiv:1004.3558]. [174] T. Hiramatsu et al., JCAP 1301, 001 (2013), [145] A. Arvanitaki, M. Baryakhtar and X. Huang, Phys. Rev. [arXiv:1207.3166]. D91, 8, 084011 (2015), [arXiv:1411.2263]. [175] C. J. Hogan and M. J. Rees, Phys. Lett. B205, 228 (1988). [146] A. Arvanitaki et al., Phys. Rev. D95, 4, 043001 (2017), [176] E. W. Kolb and I. I. Tkachev, Phys. Rev. Lett. 71, 3051 [arXiv:1604.03958]. (1993), [hep-ph/9303313]. [147] K. K. Y. Ng et al. (2019), [arXiv:1908.02312]. [177] K. M. Zurek, C. J. Hogan and T. R. Quinn, Phys. Rev. [148] M. S. Turner, Phys. Rev. Lett. 59, 2489 (1987), [Erra- D75, 043511 (2007), [arXiv:astro-ph/0607341]. tum: Phys. Rev. Lett.60,1101(1988)]; E. Masso, F. Rota [178] A. Vaquero, J. Redondo and J. Stadler (2018), and G. Zsembinszki, Phys. Rev. D66, 023004 (2002), [hep- [JCAP1904,no.04,012(2019)], [arXiv:1809.09241]. ph/0203221]; P. Graf and F. D. Steffen, Phys. Rev. D83, 075011 (2011), [arXiv:1008.4528]. [179] E. W. Kolb and I. I. Tkachev, Astrophys. J. 460, L25 (1996), [arXiv:astro-ph/9510043]. [149] S. Hannestad et al., JCAP 1008, 001 (2010), [arXiv:1004.0695]; M. Archidiacono et al., JCAP 1310, [180] M. Fairbairn, D. J. E. Marsh and J. Quevillon, Phys. Rev. 020 (2013), [arXiv:1307.0615]; E. Di Valentino et al., Phys. Lett. 119, 2, 021101 (2017), [arXiv:1701.04787]. Lett. B752, 182 (2016), [arXiv:1507.08665]. [181] A. Katz et al., JCAP 1812, 005 (2018), [arXiv:1807.11495]. [150] M. Archidiacono et al., JCAP 1505, 05, 050 (2015), [182] E. Berkowitz, M. I. Buchoff and E. Rinaldi, Phys. Rev. D92, [arXiv:1502.03325]. 3, 034507 (2015), [arXiv:1505.07455]. [151] E. Masso and R. Toldra, Phys. Rev. D55, 7967 (1997), [hep- [183] S. Borsanyi et al., Phys. Lett. B752, 175 (2016), ph/9702275]; D. Cadamuro and J. Redondo, JCAP 1202, [arXiv:1508.06917]. 032 (2012), [arXiv:1110.2895]. [184] R. Kitano and N. Yamada, JHEP 10, 136 (2015), [152] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. B120, [arXiv:1506.00370]. 127 (1983); L. F. Abbott and P. Sikivie, Phys. Lett. B120, 133 (1983); M. Dine and W. Fischler, Phys. Lett. B120, [185] P. Petreczky, H.-P. Schadler and S. Sharma, Phys. Lett. 137 (1983). B762, 498 (2016), [arXiv:1606.03145]. [153] K. J. Bae, J.-H. Huh and J. E. Kim, JCAP 0809, 005 [186] Y. Taniguchi et al., Phys. Rev. D95, 5, 054502 (2017), (2008), [arXiv:0806.0497]. [arXiv:1611.02411]. [154] O. Wantz and E. P. S. Shellard, Phys. Rev. D82, 123508 [187] M. Dine et al., Phys. Rev. D96, 9, 095001 (2017), (2010), [arXiv:0910.1066]. [arXiv:1705.00676]. [155] G. Ballesteros et al., JCAP 1708, 08, 001 (2017), [188] R. D. Pisarski and L. G. Yaffe, Phys. Lett. 97B, 110 (1980). [arXiv:1610.01639]. [189] P. Arias et al., JCAP 1206, 013 (2012), [arXiv:1201.5902]. [156] M. Tegmark et al., Phys. Rev. D73, 023505 (2006), [190] R. Hlozek et al., Phys. Rev. D91, 10, 103512 (2015), [arXiv:astro-ph/0511774]. [arXiv:1410.2896]. [157] A. D. Linde, Phys. Lett. 158B, 375 (1985). [191] D. J. E. Marsh, Phys. Rept. 643, 1 (2016), [158] D. Seckel and M. S. Turner, Phys. Rev. D32, 3178 (1985). [arXiv:1510.07633]. [159] M. Beltran, J. Garcia-Bellido and J. Lesgourgues, Phys. [192] L. Hui et al., Phys. Rev. D95, 4, 043541 (2017), Rev. D75, 103507 (2007), [hep-ph/0606107]. [arXiv:1610.08297]. [160] M. P. Hertzberg, M. Tegmark and F. Wilczek, Phys. Rev. [193] A. G. Dias et al., JHEP 06, 037 (2014), [arXiv:1403.5760]; D78, 083507 (2008), [arXiv:0807.1726]. J. E. Kim and D. J. E. Marsh, Phys. Rev. D93, 2, 025027 [161] J. Hamann et al., JCAP 0906, 022 (2009), (2016), [arXiv:1510.01701]. [arXiv:0904.0647]. [194] H. Davoudiasl and C. W. Murphy, Phys. Rev. Lett. 118, [162] P. A. R. Ade et al. (Planck), Astron. Astrophys. 571, A22 14, 141801 (2017), [arXiv:1701.01136]. (2014), [arXiv:1303.5082]. [195] M. Khlopov, B. A. Malomed and I. B. Zeldovich, Mon. Not. [163] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A20 Roy. Astron. Soc. 215, 575 (1985). (2016), [arXiv:1502.02114]. [196] W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett. [164] S. Chang, C. Hagmann and P. Sikivie, Phys. Rev. D59, 85, 1158 (2000), [arXiv:astro-ph/0003365]; L. Amendola 023505 (1999), [hep-ph/9807374]. and R. Barbieri, Phys. Lett. B642, 192 (2006), [hep- [165] C. Hagmann, S. Chang and P. Sikivie, Phys. Rev. D63, ph/0509257]; D. J. E. Marsh and P. G. Ferreira, Phys. Rev. 125018 (2001), [hep-ph/0012361]. D82, 103528 (2010), [arXiv:1009.3501]. [166] T. Hiramatsu et al., Phys. Rev. D83, 123531 (2011), [197] E. Seidel and W. M. Suen, Phys. Rev. Lett. 66, 1659 (1991). [arXiv:1012.5502]. [198] R. Hlozek, D. J. E. Marsh and D. Grin, Mon. Not. Roy. [167] T. Hiramatsu et al., Phys. Rev. D85, 105020 (2012), [Erra- Astron. Soc. 476, 3, 3063 (2018), [arXiv:1708.05681]. tum: Phys. Rev.D86,089902(2012)], [arXiv:1202.5851]. [199] D. J. E. Marsh and J. Silk, Mon. Not. Roy. Astron. Soc. [168] M. Kawasaki, K. Saikawa and T. Sekiguchi, Phys. Rev. 437, 3, 2652 (2014), [arXiv:1307.1705]. D91, 6, 065014 (2015), [arXiv:1412.0789]. [200] H.-Y. Schive et al., Astrophys. J. 818, 1, 89 (2016), [169] V. B. Klaer and G. D. Moore, JCAP 1711, 11, 049 (2017), [arXiv:1508.04621]. [arXiv:1708.07521]. [201] B. Bozek et al., Mon. Not. Roy. Astron. Soc. 450, 1, 209 [170] M. Gorghetto, E. Hardy and G. Villadoro, JHEP 07, 151 (2015), [arXiv:1409.3544]; P. S. Corasaniti et al., Phys. Rev. (2018), [arXiv:1806.04677]. D95, 8, 083512 (2017), [arXiv:1611.05892]. 91. Axions and Other Similar Particles 951

[202] E. Armengaud et al., Mon. Not. Roy. Astron. Soc. 471, [230] A. Caputo et al., JCAP 1903, 03, 027 (2019), 4, 4606 (2017), [arXiv:1703.09126]; V. Iršič et al., Phys. [arXiv:1811.08436]. Rev. Lett. 119, 3, 031302 (2017), [arXiv:1703.04683]; [231] M. A. Fedderke, P. W. Graham and S. Rajendran, Phys. T. Kobayashi et al., Phys. Rev. D96, 12, 123514 (2017), Rev. D100, 1, 015040 (2019), [arXiv:1903.02666]. [arXiv:1708.00015]. [232] T. Liu, G. Smoot and Y. Zhao (2019), [arXiv:1901.10981]. [203] H.-Y. Schive and T. Chiueh, Mon. Not. Roy. Astron. Soc. 473, 1, L36 (2018), [arXiv:1706.03723]. [233] P. Sikivie, Phys. Rev. D32, 2988 (1985), [Erratum: Phys. Rev.D36,974(1987)]. [204] H.-Y. Schive, T. Chiueh and T. Broadhurst, Nature Phys. [234] R. Bradley et al., Rev. Mod. Phys. 75, 777 (2003). 10, 496 (2014), [arXiv:1406.6586]. [235] P. Sikivie and J. R. Ipser, Phys. Lett. B291, 288 (1992). [205] B. Schwabe, J. C. Niemeyer and J. F. Engels, Phys. Rev. D94, 4, 043513 (2016), [arXiv:1606.05151]; J. Veltmaat [236] P. Sikivie, I. I. Tkachev and Y. Wang, Phys. Rev. Lett. 75, and J. C. Niemeyer, Phys. Rev. D94, 12, 123523 (2016), 2911 (1995), [arXiv:astro-ph/9504052]. [arXiv:1608.00802]; P. Mocz et al., Mon. Not. Roy. Astron. [237] S. De Panfilis et al., Phys. Rev. Lett. 59, 839 (1987); Soc. 471, 4, 4559 (2017), [arXiv:1705.05845]. W. Wuensch et al., Phys. Rev. D40, 3153 (1989); C. Hag- [206] D. G. Levkov, A. G. Panin and I. I. Tkachev, Phys. Rev. mann et al., Phys. Rev. D42, 1297 (1990). Lett. 118, 011301 (2017); T. Helfer et al., JCAP 1703, 03, [238] S. J. Asztalos et al. (ADMX), Phys. Rev. D69, 011101 055 (2017), [arXiv:1609.04724]. (2004), [arXiv:astro-ph/0310042]. [207] D. J. E. Marsh and A.-R. Pop, Mon. Not. Roy. Astron. Soc. [239] L. Duffy et al., Phys. Rev. Lett. 95, 091304 (2005), 451, 3, 2479 (2015), [arXiv:1502.03456]; S.-R. Chen, H.-Y. [arXiv:astro-ph/0505237]; J. Hoskins et al., Phys. Rev. Schive and T. Chiueh, Mon. Not. Roy. Astron. Soc. 468, D84, 121302 (2011), [arXiv:1109.4128]. 2, 1338 (2017), [arXiv:1606.09030]; A. X. González-Morales [240] S. J. Asztalos et al. (ADMX), Phys. Rev. Lett. 104, 041301 et al., Mon. Not. Roy. Astron. Soc. 472, 2, 1346 (2017), (2010), [arXiv:0910.5914]. [arXiv:1609.05856]. [241] S. J. Asztalos et al. (ADMX), Nucl. Instrum. Meth. A656, [208] I. De Martino et al. (2018), [arXiv:1807.08153]. 39 (2011), [arXiv:1105.4203]. [209] T. Broadhurst et al. (2019), [arXiv:1902.10488]. [242] N. Du et al. (ADMX), Phys. Rev. Lett. 120, 15, 151301 [210] H.-Y. Schive et al., Phys. Rev. Lett. 113, 26, 261302 (2014), (2018), [arXiv:1804.05750]. [arXiv:1407.7762]. [243] T. Braine et al. (ADMX) (2019), [arXiv:1910.08638]. [211] V. H. Robles, J. S. Bullock and M. Boylan-Kolchin, [244] G. Rybka et al. (ADMX), Phys. Rev. Lett. 105, 051801 Mon. Not. Roy. Astron. Soc. 483, 1, 289 (2019), (2010), [arXiv:1004.5160]. [arXiv:1807.06018]. [245] A. Wagner et al. (ADMX), Phys. Rev. Lett. 105, 171801 [212] V. Desjacques and A. Nusser (2019), [arXiv:1905.03450]. (2010), [arXiv:1007.3766]. [213] M. Safarzadeh and D. N. Spergel (2019), [arXiv:1906.11848]. [246] C. Boutan et al. (ADMX), Phys. Rev. Lett. 121, 26, 261302 [214] J. H. H. Chan et al., Mon. Not. R. Astron. Soc 478, 2686 (2018), [arXiv:1901.00920]. (2018), [arXiv:1712.01947]. [247] B. M. Brubaker et al., Phys. Rev. Lett. 118, 6, 061302 [215] A. Pontzen and F. Governato, Nature 506, 171 (2014), (2017), [arXiv:1610.02580]. [arXiv:1402.1764]. [248] L. Zhong et al. (HAYSTAC), Phys. Rev. D97, 9, 092001 [216] J. Veltmaat, J. C. Niemeyer and B. Schwabe, Phys. Rev. (2018), [arXiv:1803.03690]. D98, 4, 043509 (2018), [arXiv:1804.09647]. [249] I. Ogawa, S. Matsuki and K. Yamamoto, Phys. Rev. D53, [217] B. Bar-Or, J.-B. Fouvry and S. Tremaine, Astrophys. J. R1740 (1996). 871, 1, 28 (2019), [arXiv:1809.07673]. [250] B. T. McAllister et al., Phys. Dark Univ. 18, 67 (2017), [218] D. G. Levkov, A. G. Panin and I. I. Tkachev, Phys. Rev. [arXiv:1706.00209]. Lett. 121, 15, 151301 (2018), [arXiv:1804.05857]. [251] W. Chung, PoS CORFU2015, 047 (2016). [219] B. V. Church, P. Mocz and J. P. Ostriker, Mon. Not. R. [252] A. A. Melcon et al. (RADES), JCAP 1805, 05, 040 (2018), Astron. Soc 485, 2861 (2019), [arXiv:1809.04744]. [arXiv:1803.01243]. [220] N. C. Amorisco and A. Loeb (2018), [arXiv:1808.00464]. [253] D. Alesini et al. (KLASH) (2017), [arXiv:1707.06010]. [221] T. S. Li et al. (DES), Astrophys. J. 838, 1, 8 (2017), [254] G. Rybka et al., Phys. Rev. D91, 1, 011701 (2015), [arXiv:1611.05052]. [arXiv:1403.3121]. [222] D. J. E. Marsh and J. C. Niemeyer, Phys. Rev. Lett. 123, [255] D. Horns et al., JCAP 1304, 016 (2013), [arXiv:1212.2970]. 5, 051103 (2019), [arXiv:1810.08543]. [256] J. Suzuki et al., JCAP 1509, 09, 042 (2015), [223] B. Eggemeier and J. C. Niemeyer (2019), [arXiv:1504.00118]. [arXiv:1906.01348]. [257] A. Caldwell et al. (MADMAX Working Group), Phys. Rev. [224] M. A. Bershady, M. T. Ressell and M. S. Turner, Phys. Rev. Lett. 118, 9, 091801 (2017), [arXiv:1611.05865]. Lett. 66, 1398 (1991); M. T. Ressell, Phys. Rev. D44, 3001 [258] P. Brun et al. (MADMAX), Eur. Phys. J. C79, 3, 186 (1991). (2019), [arXiv:1901.07401]. [225] D. Grin et al., Phys. Rev. D75, 105018 (2007), [arXiv:astro- [259] M. Baryakhtar, J. Huang and R. Lasenby, Phys. Rev. D98, ph/0611502]. 3, 035006 (2018), [arXiv:1803.11455]. [226] B. D. Blout et al., Astrophys. J. 546, 825 (2001), [260] A. Arvanitaki, S. Dimopoulos and K. Van Tilburg, Phys. [arXiv:astro-ph/0006310]. Rev. X8, 4, 041001 (2018), [arXiv:1709.05354]. [227] K. Kelley and P. J. Quinn, Astrophys. J. 845, 1, L4 (2017), [261] P. Sikivie, Phys. Rev. Lett. 113, 20, 201301 (2014), [arXiv:1708.01399]. [arXiv:1409.2806]. [228] G. Sigl, Phys. Rev. D96, 10, 103014 (2017), [262] P. W. Graham and S. Rajendran, Phys. Rev. D88, 035023 [arXiv:1708.08908]. (2013), [arXiv:1306.6088]. [229] A. Hook et al., Phys. Rev. Lett. 121, 24, 241102 (2018), [263] M. Pospelov and A. Ritz, Nucl. Phys. B573, 177 (2000), [arXiv:1804.03145]. [hep-ph/9908508]. 952 91. Axions and Other Similar Particles

[264] C. Abel et al., Phys. Rev. X7, 4, 041034 (2017), [271] P. Sikivie, N. Sullivan, and D. B. Tanner, Phys. Rev. Lett. [arXiv:1708.06367]. 112, 131301 (2014). [265] D. Budker et al., Phys. Rev. X4, 2, 021030 (2014), [272] Y. Kahn, B. R. Safdi and J. Thaler, Phys. Rev. Lett. 117, [arXiv:1306.6089]. 14, 141801 (2016), [arXiv:1602.01086]. [266] M. B. Wise, H. Georgi and S. L. Glashow, Phys. Rev. Lett. 47, 402 (1981). [273] M. Silva-Feaver et al., IEEE Trans. Appl. Supercond. 27, 4, [267] A. Ernst, A. Ringwald and C. Tamarit, JHEP 02, 103 1400204 (2017), [arXiv:1610.09344]. (2018), [arXiv:1801.04906]. [274] J. L. Ouellet et al. (ABRACADABRA), Phys. Rev. Lett. [268] L. Di Luzio, A. Ringwald and C. Tamarit, Phys. Rev. D98, 122, 12, 121802 (2019), [arXiv:1810.12257]. 9, 095011 (2018), [arXiv:1807.09769]. [269] P. Fileviez Perez, C. Murgui and A. D. Plascencia (2019), [275] L. Krauss et al., Phys. Rev. Lett. 55, 1797 (1985). [arXiv:1908.01772]. [276] R. Barbieri et al., Phys. Dark Univ. 15, 135 (2017). [270] S. P. Chang et al., Phys. Rev. D99, 8, 083002 (2019), [arXiv:1710.05271]. 92. Searches for quark and lepton compositeness 953

92. Searches for Quark and Lepton Compositeness

Revised 2019 by K. Hikasa (Tohoku University), M. Tanabashi Over the last three decades experiments at the CERN Sp¯pS [4,5], (Nagoya University), K. Terashi (ICEPP, University of Tokyo), and the Fermilab Tevatron [6,7], and the CERN LHC [8–12] have searched N. Varelas (University of Illinois at Chicago) for quark contact interactions, characterized by the four-fermion effective Lagrangian in Eq. (92.1), using jet final states. These 92.1. Limits on contact interactions searches have been performed primarily by studying the angular distribution of the two highest transverse momentum, pT, jets (dijets), If quarks and leptons are made of constituents, then at the and the inclusive jet p spectrum. The variable χ = exp( (y1 y2) ) T | − | scale of constituent binding energies (compositeness scale) there is used to measure the dijet angular distribution, where y1 and should appear new interactions among them. At energies much below y2 are the rapidities of the two jets with the highest transverse the compositeness scale (Λ), these interactions are suppressed by momenta. For collinear massless parton scattering, χ is related to inverse powers of Λ. The dominant effect of the compositeness of the polar scattering angle θ∗ in the partonic center-of-mass frame fermion ψ should come from the lowest dimensional interactions with by χ = (1+ cos θ∗ )/(1 cos θ∗ ). The choice of χ is motivated four fermions (contact terms), whose most general flavor-diagonal by the fact that| the| angular− | distribution| for Rutherford scattering, color-singlet chirally invariant form reads [1,2] which is proportional to 1/(1 cos θ∗)2, is independent of χ. In perturbative QCD the χ distributions− are relatively uniform and = LL + RR + LR + RL, only mildly modified by higher-order QCD or electroweak corrections. L L L L L Signatures of quark contact interactions exhibit more isotropic angular with distribution than QCD and they can be identified as an excess at low values of χ. In the inclusive jet cross section measurement, 2 gcontact ij ¯i i ¯j µ j quark contact interaction effects are searched for as deviations from LL = X η (ψ γµψ )(ψ γ ψ ), L 2Λ2 LL L L L L the predictions of perturbative QCD in the tails of the high-p jet i,j T spectrum [11]. 2 gcontact ij i i j µ j = X η (ψ¯ γµψ )(ψ¯ γ ψ ), LRR 2Λ2 RR R R R R i,j s=13 TeV, 37.0 fb-1 ATLAS 2 gcontact ij ¯i i ¯j µ j χ m > 5.4 TeV Data SM LR = X η (ψ γµψ )(ψ γ ψ ), jj L 2Λ2 LR L L R R 0.06 CI η =−1, Λ=22 TeV i,j LL CI η =+1, Λ=15 TeV 2 LL gcontact ij i i j µ j 1/N dN/d = X η (ψ¯ γµψ )(ψ¯ γ ψ ), (92.1) 0.04 Theoretical uncert. LRL 2Λ2 RL R R L L i,j Total uncertainty where i, j are the indices of fermion species. Color and other indices 0.06 4.9 < mjj < 5.4 TeV 4.6 < mjj < 4.9 TeV are suppressed in Eq. (92.1). Chiral invariance provides a natural explanation why quark and lepton masses are much smaller than their ij ji 0.04 inverse size Λ. Note ηαβ = ηβα, therefore, in order to specify the contact interaction among the same fermion species i = j, it is enough to use ηLL, ηRR and ηLR. We will suppress the indices of fermion 0.05 4.3 < mjj < 4.6 TeV 4.0 < mjj < 4.3 TeV species hereafter. We may determine the scale Λ unambiguously by using the above form of the effective interactions; the conventional 0.04 2 2 method [1] is to fix its scale by setting gcontact/4π = gcontact(Λ)/4π =1 for the new strong interaction coupling and by setting the largest 0.03 magnitude of the coefficients η to be unity. In the following, we αβ 3.7 < m < 4.0 TeV 3.4 < m < 3.7 TeV denote 0.05 jj jj

Λ=Λ± for (η , η , η ) = ( 1, 0, 0) , 0.04 LL LL RR LR ± ± 0.03 Λ=ΛRR for (ηLL , ηRR , ηLR )=(0, 1, 0) , ± 1 2 3 4 5 6 7 10 20 301 2 3 4 5 6 7 10 20 30 Λ=Λ± for (η , η , η ) = ( 1, 1, 1) , χ χ V V LL RR LR ± ± ± ± Figure 92.1: Normalized dijet angular distributions in several Λ=ΛAA for (ηLL, ηRR , ηLR ) = ( 1, 1, 1) , ± ± ∓ dijet mass (mjj) ranges. The data distributions are compared Λ=Λ± for (η , η , η )=(0, 0, 1) . (92.2) to PYTHIA8 predictions with NLO and electroweak corrections V −A LL RR LR ± applied (solid line) and with the predictions including a contact interaction (CI) term in which only left-handed quarks Such interactions can arise by interchanging constituents (when the + fermions have common constituents), and/or by exchanging the participate of compositeness scale ΛLL = 15 TeV (dashed line) − binding quanta (whenever binding quanta couple to constituents of and ΛLL = 22 TeV (dotted line). The theoretical uncertainties both particles). and the total theoretical and experimental uncertainties in the predictions are displayed as shaded bands around the SM Fermion scattering amplitude induced from the contact interaction prediction. Figure adopted from Ref. 9. in Eq. (92.1) interferes with the Standard Model (SM) amplitude destructively or constructively [2]. The sign of interference depends on the sign of ηαβ (α, β = L, R). For instance, in the parton level Recent results from the LHC, using data collected at proton-proton ± qq qq scattering cross section in the ΛLL model, the contact center-of-mass energy of √s = 13 TeV, extend previous limits on quark interaction→ amplitude and the SM gluon exchange amplitude interfere contact interactions. Figure 92.1 shows the normalized dijet angular destructively for ηLL = +1, while they interfere constructively for distributions for several dijet mass ranges measured in ATLAS [9] ηLL = 1. In models of quark compositeness, the quark scattering at √s = 13 TeV. The data distributions are compared with SM cross sections− induced from the contact interactions receive sizable predictions, estimated using PYTHIA8 [13] with GEANT4-based [14] QCD radiative corrections. Ref. 3 provides the exact next-to-leading ATLAS detector simulation and corrected to NLO QCD calculation order (NLO) QCD corrections to the contact interaction induced provided by NLO Jet++ [15] including electroweak corrections [16], quark scattering cross sections. and with predictions including a contact interaction term in which only 954 92. Searches for quark and lepton compositeness

+ left-handed quarks participate at compositeness scale ΛLL = 15 TeV the LHC. The most stringent searches for llqq contact interactions are (Λ− = 22 TeV) with destructive (constructive) interference. Over a performed by the LHC experiments using high-mass oppositely-charged LL + − wide range of χ and dijet mass the data are well described by the lepton pairs produced through the qq l l Drell-Yan process. The → SM predictions. Using the dijet angular distributions measured at contact interaction amplitude of the uu¯ l+l− process (l = e or → high dijet masses and √s = 13 TeV, the ATLAS [9] and CMS [12] µ) interferes with the corresponding SM amplitude constructively Collaborations have set 95% confidence level (C.L.) lower limits on (destructively) for ηul = 1 (ηul = +1). The ATLAS Collaboration αβ − αβ the contact interaction scale Λ, ranging from 9.2 to 29.5 TeV for has extracted limits on the llqq contact interaction at √s = 13 TeV different quark contact interaction models that correspond to various for the right-right (η = 1, η = η = η = 0), left-left RR ± LL LR RL combinations of (ηLL, ηRR, ηLR), as summarized in Figure 92.2. (η = 1, η = η = η = 0), and left-right (η = η = 1, LL ± RR LR RL LR RL ± The contact interaction scale limits extracted using the dijet angular ηRR = ηLL = 0) models. Combining the dielectron and dimuon distributions include the exact NLO QCD corrections to dijet channels, the 95% C.L. lower limits on the llqq contact interaction production induced by contact interactions [3]. In proton-proton scale Λ are 35 TeV (28 TeV) for the right-right model, 40 TeV (25 ± ± collisions, the ΛLL and ΛRR contact interaction models result in TeV) for the left-left model, and 36 TeV (28 TeV) for the left-right identical tree-level cross sections and NLO QCD corrections and yield model, each with constructive (destructive) interference [26]. The ± ± −1 the same exclusion limits. For ΛV V and ΛAA, the contact interaction CMS Collaboration, using a 36 fb dataset at 13 TeV, has set 95% predictions are identical at tree level, but exhibit different NLO QCD C.L. exclusion limits on the llqq contact interaction scale that range corrections and yield different exclusion limits. from ΛLL > 20 TeV for the destructive interference to ΛRR > 32 TeV for the constructive interference, for the left-left and the right-right models, respectively [29]. Note that the contact interactions arising from the compositeness of quarks and leptons in Eq. (92.1) can also be regarded as a part ATLAS of more general dimension six operators in the context of low energy Λ+ CMS LL/RR standard model effective theory. For a complete list of these dimension Λ− six operators, see [30,31]. LL/RR Interactions of hypothetical dark matter candidate particles with Λ+ VV SM particles through mediators can also be described as contact Λ− interactions at low energy. See “Searches for WIMPs and Other VV Particles” in this volume for limits on the interactions involving dark Λ+ AA matter candidate particles. Λ− AA Λ+ 92.2. Limits on excited fermions (V•A) Λ− Another typical consequence of compositeness is the appearance (V•A) Observed ∗ ∗ Expected of excited leptons and quarks (l and q ). Phenomenologically, an excited lepton is defined to be a heavy lepton which shares a leptonic 5 10 15 20 25 30 quantum number with one of the existing leptons (an excited quark is ∗ Contact Interaction Scale Limit [TeV] defined similarly). For example, an excited electron e is characterized by a nonzero transition-magnetic coupling with electrons. Smallness of the lepton mass and the success of QED prediction for g 2 Figure 92.2: Observed (solid lines) and expected (dashed suggest chirality conservation, i.e., an excited lepton should− not lines) 95% C.L. lower limits on the contact interaction scale Λ couple to both left- and right-handed components of the corresponding for different contact interaction models from ATLAS [9] and lepton [32–34]. CMS [12] using the dijet angular distributions. The contact Excited leptons may be classified by SU(2) U(1) quantum interaction models used for the dijet angular distributions × include the exact NLO QCD corrections to dijet production. numbers. Typical examples are: The shaded band for the Λ+ model indicates the range of LL/RR 1. Sequential type contact interaction scale that was not excluded in ATLAS [9] due to statistical fluctuation of observed data. ν∗ ! , [ν∗ ], l∗ . l∗ R R If leptons (l) and quarks (q) are composite with common L constituents, the interaction of these constituents will manifest itself ∗ ∗ in the form of a llqq-type four-fermion contact interaction Lagrangian νR is necessary unless ν has a Majorana mass. at energies below the compositeness scale Λ. The llqq terms in the 2. Mirror type contact interaction Lagrangian can be expressed as ∗ ∗ ∗ ν ! [νL], lL, ∗ . 2 l gcontact ij i i j µ j R = X η (¯q γµq )(¯l γ l ), LLL Λ2 LL L L L L i,j 3. Homodoublet type

2 gcontact ij i i ¯j µ j ν∗ ! ν∗ ! RR = 2 X ηRR(¯qRγµqR)(lRγ lR), L Λ ∗ , ∗ . i,j l l L R 2 gcontact ij i i j µ j = X η (¯q γµq )(¯l γ l ), LLR Λ2 LR L L R R i,j Similar classification can be made for excited quarks. Excited fermions can be pair produced via their minimal gauge 2 gcontact ij i i j µ j couplings. The couplings of excited leptons with Z are given by = X η (¯q γµq )(¯l γ l ). (92.3) LRL Λ2 RL R R L L i,j e 2 ¯∗ µ ∗ ( 1 + 2 sin θW )l γ l Zµ Searches on quark-lepton compositeness have been reported from 2 sin θW cos θW − experiments at LEP [17–20], HERA [21,22], the Tevatron [23,24], e ∗ µ ∗ + ν¯ γ ν Zµ and recently from the ATLAS [25,26] and CMS [27–29] experiments at 2 sin θW cos θW 92. Searches for quark and lepton compositeness 955

in the homodoublet model. The corresponding couplings of excited quarks can be easily obtained. Although form factor effects can be CMS Preliminary 77.8 fb-1 (13 TeV) present for the gauge couplings at q2 = 0, they are usually neglected. 6 3 Data Excited fermions may also be produced via the contact interactions 10 Fit Method with ordinary quarks and leptons [35] 2 χ2/NDF = 48.11 / 37 10 Ratio Method

2 [pb/TeV] 2 gcontact ′ ∗ µ ∗ jj χ /NDF = 40.37 / 31 = η (ψ¯ γµψ )(ψ¯ γ ψ ) L Λ2 LL L L L L 10 gg (2.0 TeV)

/dm qg (4.0 TeV) ′′ ¯ ¯∗ µ +(ηLL(ψLγµψL)(ψLγ ψL) + h.c.) +  . (92.4) σ 1 qq (6.0 TeV)

··· d 2 −1 Again, the coefficient is conventionally taken gcontact =4π. It is widely 10 ′ ′′ ′ ′′ ′ ′′ ′ ′′ assumed ηLL = ηLL = 1, ηLR = ηLR = ηRL = ηRL = ηRR = ηRR =0 − in experimental analyses for simplicity. 10 2 In addition, transition-magnetic type couplings with a gauge boson Wide PF-jets − are expected. These couplings can be generally parameterized as 3 10 m > 1.53 TeV follows: jj − 10 4 |η| < 2.5, | η∆ | < 1.1 (ψ∗) λγ e ¯∗ µν 1 γ5 1+ γ5 = ψ σ (ηL − + ηR )ψFµν L 2m ∗ 2 2 4 ψ 3 (ψ∗) 2 λZ e ¯∗ µν 1 γ5 1+ γ5 1 + ψ σ (ηL − + ηR )ψZµν 2 ∗ 2 2 0 mψ −1 − ∗ 2 (l ) −3 λW g ∗ µν 1 γ5 + ¯l σ − νWµν −4 2ml∗ 2 Uncertainty 2 3 4 5 6 7 8

(ν∗) (Data-Prediction) λW g ∗ µν 1 γ5 1+ γ5 † Dijet mass [TeV] + ν¯ σ (ηL − + ηR )lWµν 2mν∗ 2 2 Figure 92.3: Dijet mass distribution measured by CMS using + h.c., (92.5) wide jets reconstructed from two highest transverse momentum jets by adding nearby jets within ∆R = p∆η2 + ∆φ2 < 1.1. where g = e/ sin θ , ψ = ν or l, Fµν = ∂µAν ∂ν Aµ is the photon W − The data distribution is compared to a fit representing a smooth field strength, Zµν = ∂µZν ∂νZµ, etc.. The normalization of the − background spectrum (solid curve). The excited quark signal coupling is chosen such that with mass of 4.0 TeV (labeled as qg) is shown together with other benchmark signals. Shown at the bottom panel is the difference max( η , η )=1. | L| | R| between the data and the fitted parametrization divided by the statistical uncertainty of the data. Figure adopted from Ref. 62. Chirality conservation requires If leptons are made of color triplet and antitriplet constituents, ηLηR =0. (92.6) we may expect their color-octet partners. Transitions between the octet leptons (l8) and the ordinary lepton (l) may take place via the These couplings in Eq. (92.5) can arise from SU(2) U(1)- dimension-five interactions invariant higher-dimensional interactions. A well-studied model× is the 1 = X ¯lαg F α σµν (η l + η l ) + h.c. (92.10) interaction of homodoublet type l∗ with the Lagrangian (see [36,37]) L 2Λ 8 S µν L L R R l a where the summation is over charged leptons and neutrinos. The 1 ¯∗ µν τ a ′ ′ 1 γ5 = L σ (gf Wµν + g f YBµν) − L + h.c., (92.7) L 2Λ 2 2 leptonic chiral invariance implies ηLηR = 0 as before. Searches for the excited quarks and leptons have been performed where L denotes the lepton doublet (ν,l), Λ is the compositeness scale, over the last decades in experiments at the LEP [38–41], HERA [42,43], ′ a g, g are SU(2) and U(1)Y gauge couplings, and Wµν and Bµν are Tevatron [44,45], and LHC [46–71]. Most stringent constraints, which the field strengths for SU(2) and U(1)Y gauge fields. These couplings are described below at 95% confidence level, come from the LHC satisfy the relation experiments. ∗ 2 The signature of excited quarks q at hadron colliders is λ = √2 sin θ (λ cot θ + λγ ) , (92.8) W − W Z W characterized by a narrow resonant peak in the reconstructed invariant mass distribution of the q∗ decay products. The decays via the (ℓ∗) with λW,Z,γ being defined in Eq. (92.5) with λW,Z,γ = λW,Z,γ or transition-magnetic type operator in Eq. (92.9) are considered for (ν∗) excited quarks in LHC searches, and the final states to search for λW,Z,γ = λ . Here (ηL, ηR) = (1, 0) is assumed. It should be W,Z,γ are dijet (qg) [46, 47, 59–62] or a jet in association with a photon noted that the electromagnetic radiative decay of l∗ (ν∗) is forbidden (qγ) [48, 49, 63, 64] or a weak gauge boson (qW , qZ) [65, 66]. if f = f ′ (f = f ′). − All analyses consider only spin-1/2 excited states of first generation Additional coupling with gluons is possible for excited quarks: quarks (u∗, d∗) with degenerate masses, expected to be predominantly a a produced in proton-proton collisions except for the excited b quark 1 ¯∗ µν  λ a τ a ′ ′  searches described below. Only the minimal gauge interactions and = Q σ gsfs Gµν + gf Wµν + g f YBµν L 2Λ 2 2 the transition-magnetic couplings with the form given in Eq. (92.9) are considered in the production process, and hence the contact 1 γ5 − Q + h.c. , (92.9) interactions in Eq. (92.4) are not considered. The compositeness scale × 2 Λ is taken to be the same as the excited quark mass mq∗ . The ′ where Q denotes a quark doublet, gs is the QCD gauge coupling, and transition-magnetic coupling coefficients fs, f and f are assumed to a be equal to 1 (denoted by f). Gµν the gluon field strength. 956 92. Searches for quark and lepton compositeness

With proton-proton collision data recorded at √s = 13 TeV at the in ATLAS [56] further constrains the excited charged leptons and LHC, the excited quark masses are excluded in dijet resonance searches neutrinos. Considering both the transition-magnetic (Eq. (92.7)) and up to 6.7 TeV in ATLAS using 140 fb−1 [47] and 6.0 TeV in CMS contact interaction (Eq. (92.4)) processes, the lower mass limits for the using 77.8 fb−1 [62]. Figure 92.3 shows the dijet mass distribution e∗, µ∗, τ ∗ and ν∗ (for every excited neutrino flavor) are obtained to measured in CMS [62] by using the two highest pT jets reconstructed be 3.0, 3.0, 2.5 and 1.6 TeV, respectively, for Λ = me∗ , mµ∗ , mτ ∗ and with the anti-kT algorithm [72] of a distance parameter of 0.4, and by mν∗ . The rate of pair-produced excited leptons is independent of Λ combining nearby jets within ∆R = p∆η2 + ∆φ2 < 1.1 around the for the minimal gauge interaction processes, and it allows to improve leading two jets. The measured dijet mass spectrum is compared to a search sensitivity with multi-lepton signatures at high Λ, especially for fit with smoothly falling background shape (solid curve) to look for excited neutrinos because the predominant ν∗ l + W decays result l → a narrow resonance; an excited quark signal with mass of 4.0 TeV is in a higher acceptance for 3 charged lepton final states. ≥ shown in the figure (denoted by qg) as one of the benchmark signals Both ATLAS and CMS Collaborations performed searches for considered in the analysis. singly produced excited leptons. The single excited leptons, both The photon + jet resonance searches, targeting excited quarks produced and decayed in contact interaction processes (Eq. (92.4)), decaying into a quark and a photon (q∗ q + γ), have excluded q∗ were searched for in CMS using 77.4 fb−1 at 13 TeV [71] and in ATLAS masses up to 5.3 TeV in ATLAS [49] and→ 5.5 TeV in CMS [64] using using 20.3 fb−1 at 8 TeV [57] using the final states with two leptons collision data at √s = 13 TeV. The W/Z boson + jet final states are and two jets (qq¯ ll∗ llqq¯). The CMS search [71] considered → → examined to look for the q∗ q + W and q + Z signal in CMS [66], both excited electrons and excited muons, using the invariant mass of exploiting jet substructure technique→ designed to provide sensitivity the combination of the two leptons and two jets as a discriminating for highly-boosted hadronically decaying . The lower variable to separate signal from background. The ATLAS search [57] mass limit of 5.0 (4.8) TeV is obtained from the W + jet (Z + jet) considered only excited muons. The single excited electrons, produced search using dataset recorded at √s = 13 TeV. in contact interactions and decayed either in contact interaction or −1 The excited b quarks (b∗) are also considered in the present charged-current processes, were considered in ATLAS using 36.1 fb searches at the LHC. Assuming the similar production processes to at 13 TeV [58] in the final states with two electrons and two jets ∗ the first-generation excited quarks, the b∗ has been searched for in (qq¯ ee eeqq¯) or an electron, missing transverse momentum and a hadronically→ → decaying W -boson candidate (qq¯ ee∗ eνW ). The final states containing at least one jet identified as originating from → → a b quark (b-tagging). The searches using two jets including at least excited electron (muon) mass was excluded up to 5.6 (5.7) TeV in one b-tagged jet have been performed at 8 and 13 TeV [50, 51, CMS [71] at Λ = me∗ (Λ = mµ∗ ), which is the best limit to date on 60], resulting in b∗ lower mass limits of 2.6 TeV in ATLAS using the excited electrons (muons). 36.1 fb−1 at √s = 13 TeV [50] and 1.6 TeV in CMS using 19.7 fb−1 The CMS Collaboration also performed an excited lepton search in at √s = 8 TeV [60]. The CMS Collaboration also performed a search the final states containing a Z boson [69], probing the excited leptons for b∗ b + γ in events with a b-tagged jet in association with a produced in contact interactions and decayed in neutral-current photon→ using data at √s = 13 TeV [64], and excluded b∗ masses processes (l∗ l + Z) with f = f ′ =1 or f = f ′ = 1. The leptonic up to 1.8 TeV. Excited b quarks with charged-current decay into a and hadronic→ decays of Z bosons have been considered− in the search, W -boson and a top quark (b∗ t + W ) were looked for in both and the most stringent limits are obtained from the hadronic Z ATLAS and CMS using the full→ 8 TeV data [52, 67]. ATLAS excluded decay to be 2.08 (2.34) TeV and 2.11 (2.37) TeV for the e∗ and µ∗, ∗ ∗ ′ ′ b masses below 1.5 TeV for the b with left- and right-handed respectively, with f = f = 1 (f = f =1)at Λ= m ∗ . − l couplings [52] while CMS excluded the masses below 1.39(1.43) TeV Figure 92.4 summarizes the most stringent 95% C.L. lower mass for the left(right)-handed couplings [67]. limits for excited quarks and leptons obtained from the LHC experiments. References: 1. E.J. Eichten, K.D. Lane, and M.E. Peskin, Phys. Rev. Lett. 50, 811 (1983). 2. E.J. Eichten et al., Rev. Mod. Phys. 56, 579 (1984); Erratum ibid. 58, 1065 (1986). 3. J.Gao et al., Phys. Rev. Lett. 106, 142001 (2011). 4. G. Arnison et al. [UA1 Collab.], Phys. Lett. B177, 244 (1986). 5. J.A. Appel et al. [UA2 Collab.], Phys. Lett. B160, 349 (1985). 6. F.Abe et al. [CDF Collab.], Phys. Rev. Lett. 62, 613 (1989); F. Abe et al. [CDF Collab.], Phys. Rev. Lett. 69, 2896 (1992); F. Abe et al. [CDF Collab.], Phys. Rev. Lett. 77, 5336 (1996); Figure 92.4: 95% C.L. lower mass limits for the excited quarks F. Abe et al. [CDF Collab.], Erratum Phys. Rev. Lett. 78, 4307 (left) and excited leptons (right) at ATLAS [47,49,55–57] and (1997). CMS [62,64,66] [69–71] experiments. Shown are the most 7. B. Abbott et al. [DØ Collab.], Phys. Rev. Lett. 80, 666 (1998); stringent limits for each final state (denoted in parentheses) of the B. Abbott et al. [DØ Collab.], Phys. Rev. Lett. 82, 2457 (1999); excited fermions from both experiments. Only first generation B. Abbott et al. [DØ Collab.], Phys. Rev. D62, 031101 (2000); quarks (u, d) with transition-magnetic type interactions with B. Abbott et al. [DØ Collab.], Phys. Rev. D64, 032003 (2001); ′ fs = f = f = 1 are considered for the excited quarks. The B. Abbott et al. [DØ Collab.], Phys. Rev. Lett. 103, 191803 excited lepton limits are given for the production via contact (2009). interactions with Λ = ml∗ . 8. G. Aad et al. [ATLAS Collab.], Phys. Lett. B694, 327 (2011); G. Aad et al. [ATLAS Collab.], New J. Phys. 13, 053044 (2011); Searches for excited leptons l∗ are also performed at the LHC using G. Aad et al. [ATLAS Collab.], JHEP 01, 029 (2013); proton-proton collision data recorded at √s = 7 and 8 TeV [54–58, G. Aad et al. [ATLAS Collab.], Phys. Rev. Lett. 114, 221802 68, 69] as well as at 13 TeV [70]. Considering single l∗ production in (2015); contact interactions (Eq. (92.4)) and electromagnetic radiative decay G. Aad et al. [ATLAS Collab.], Phys. Lett. B754, 302 (2016). ∗ to a SM lepton and a photon (l l + γ where l = e,µ), the excited 9. G. Aad et al. [ATLAS Collab.], Phys. Rev. D96, 052004 (2017). → electron and muon masses are excluded for Λ = ml∗ up to 3.9 and 10. V. Khachatryan et al. [CMS Collab.], Phys. Rev. Lett. 105, −1 3.8 TeV, respectively, using 35.9 fb at √s = 13 TeV in CMS [70] 262001 (2010); −1 and 2.2 TeV using 13 fb at √s = 8 TeV in ATLAS [55]. V. Khachatryan et al. [CMS Collab.], Phys. Rev. Lett. 106, With the full 20.3 fb−1 data at √s = 8 TeV, the inclusive 201804 (2011); search on multi-lepton signatures with 3 or more charged leptons S. Chatrchyan et al. [CMS Collab.], JHEP 1205, 055 (2012); 92. Searches for quark and lepton compositeness 957

V. Khachatryan et al. [CMS Collab.], Phys. Lett. B746, 79 A. Abulencia et al. [CDF Collab.], Phys. Rev. Lett. 97, 191802 (2015); (2006); A.M. Sirunyan et al. [CMS Collab.], JHEP 07, 013 (2017). T. Aaltonen et al. [CDF Collab.], Phys. Rev. D79, 112002 11. S. Chatrchyan et al. [CMS Collab.], Phys. Rev. D87, 052017 (2009). (2013). 45. V.M. Abazov et al. [DØ Collab.], Phys. Rev. D73, 111102 12. A.M. Sirunyan et al. [CMS Collab.], Eur. Phys. J. C78, 789 (2006); (2018). V.M. Abazov et al. [DØ Collab.], Phys. Rev. D77, 091102 13. T. Sj¨ostrand, S. Mrenna, and P. Skands, Comp. Phys. Comm. (2008); 178, 852 (2008). V.M. Abazov et al. [DØ Collab.], Phys. Rev. Lett. 103, 191803 14. S. Agostinelli et al. [GEANT4 Collab.], GEANT4: a simulation (2009). toolkit, Nucl. Instrum. Methods A506, 250 (2003). 46. G. Aad et al. [ATLAS Collab.], Phys. Lett. B708, 37 (2012); 15. Z. Nagy, Phys. Rev. Lett. 88, 122003 (2002); G. Aad et al. [ATLAS Collab.], JHEP 1301, 29 (2013); Z. Nagy, Phys. Rev. D68, 094002, (2003). G. Aad et al. [ATLAS Collab.], Phys. Rev. D91, 052007 (2015); 16. S. Dittmaier, A. Huss, and C. Speckner, JHEP 1211, 095 (2012). G. Aad et al. [ATLAS Collab.], Phys. Lett. B754, 302 (2016); 17. S. Schael et al. [ALEPH Collab.], Eur. Phys. J. C49, 411 (2007). M. Aaboud et al. [ATLAS Collab.], Phys. Rev. D96, 052004 18. J. Abdallah et al. [DELPHI Collab.], Eur. Phys. J. C45, 589 (2017). (2006). 47. ATLAS Collaboration, ATLAS-CONF-2019-007 (2019). 19. M. Acciarri et al. [L3 Collab.], Phys. Lett. B489, 81 (2000). 48. G. Aad et al. [ATLAS Collab.], Phys. Rev. Lett. 108, 211802 20. K. Ackerstaff et al. [OPAL Collab.], Phys. Lett. B391, 221 (2012); (1997); G. Aad et al. [ATLAS Collab.], Phys. Lett. B728, 562 (2014); G. Abbiendi et al. [OPAL Collab.], Eur. Phys. J. C33, 173 G. Aad et al. [ATLAS Collab.], JHEP 1603, 41 (2016). (2004). 49. M. Aaboud et al. [ATLAS Collab.], Eur. Phys. J. C78, 102 21. F.D. Aaron et al. [H1 Collab.], Phys. Lett. B705, 52 (2011). (2018). 22. S. Chekanov et al. [ZEUS Collab.], Phys. Lett. B591, 23 (2004). 50. M. Aaboud et al. [ATLAS Collab.], Phys. Rev. D98, 032016 23. F. Abe et al. [CDF Collab.], Phys. Rev. Lett. 68, 1463 (1992); (2018). F. Abe et al. [CDF Collab.], Phys. Rev. Lett. 79, 2198 (1997); 51. M. Aaboud et al. [ATLAS Collab.], Phys. Lett. B759, 229 T. Affolder et al. [CDF Collab.], Phys. Rev. Lett. 87, 231803 (2016). (2001); 52. G. Aad et al. [ATLAS Collab.], JHEP 1602, 110 (2016). A. Abulencia et al. [CDF Collab.], Phys. Rev. Lett. 96, 211801 53. G. Aad et al. [ATLAS Collab.], Phys. Lett. B721, 171 (2013). (2006). 54. G. Aad et al. [ATLAS Collab.], Phys. Rev. D85, 072003 (2012). 24. B. Abbott et al. [DØ Collab.], Phys. Rev. Lett. 82, 4769 (1999). 55. G. Aad et al. [ATLAS Collab.], New J. Phys. 15, 093011 (2013). 25. G. Aad et al. [ATLAS Collab.], Phys. Rev. D84, 011101 (2011); 56. G. Aad et al. [ATLAS Collab.] JHEP 1508, 138 (2015). G. Aad et al. [ATLAS Collab.], Phys. Lett. B712, 40 (2012); 57. G. Aad et al. [ATLAS Collab.], New J. Phys. 18, 073021 (2016). G. Aad et al. [ATLAS Collab.], Phys. Rev. D87, 015010 (2013); 58. M. Aaboud et al. [ATLAS Collab.], Eur. Phys. J. C79, 803 G. Aad et al. [ATLAS Collab.], Eur. Phys. J. C74, 3134 (2014); (2019). M. Aaboud et al. [ATLAS Collab.], Phys. Lett. B761, 372 59. S. Chatrchyan et al. [CMS Collab.], Phys. Lett. B704, 123 (2016). (2011); 26. M. Aaboud et al. [ATLAS Collab.], JHEP 10, 182 (2017). S. Chatrchyan et al. [CMS Collab.], JHEP 1301, 13 (2013); S. Chatrchyan [CMS Collab.], Phys. Rev. D87, 114015 27. S. Chatrchyan et al. [CMS Collab.], Phys. Rev. D87, 032001 et al. (2013). (2013). 60. V. Khachatryan et al. [CMS Collab.], Phys. Rev. D91, 052009 28. V. Khachatryan et al. [CMS Collab.], JHEP 04, 025 (2015). (2015). 29. A.M. Sirunyan et al. [CMS Collab.], JHEP 04, 114 (2019). 61. V. Khachatryan et al. [CMS Collab.], Phys. Rev. Lett. 116, 30. W. Buchmuller and D. Wyler, Nucl. Phys. B268, 621 (1986). 071801 (2016); 31. B. Grzadkowski et al., JHEP 1010, 085 (2010). V. Khachatryan et al. [CMS Collab.], Phys. Rev. Lett. 117, 32. F.M. Renard, Phys. Lett. B116, 264 (1982). 031802 (2016); 33. F. del Aguila, A. Mendez, and R. Pascual, Phys. Lett. B140, A.M. Sirunyan et al. [CMS Collab.], Phys. Lett. B769, 520 431 (1984). (2017); 34. M. Suzuki, Phys. Lett. B143, 237 (1984). A.M. Sirunyan et al. [CMS Collab.], JHEP 1808, 130 (2018). 35. U. Baur, M. Spira, and P.M. Zerwas, Phys. Rev. D42, 815 62. CMS Collaboration, CMS-PAS-EXO-17-026 (2018). (1990). 63. V. Khachatryan et al. [CMS Collab.], Phys. Lett. B738, 274 36. K. Hagiwara, D. Zeppenfeld, and S. Komamiya, Z. Phys. C29, (2014). 115 (1985). 64. A.M. Sirunyan et al. [CMS Collab.], Phys. Lett. B781, 390 37. N. Cabibbo, L. Maiani, and Y. Srivastava, Phys. Lett. B139, (2018). 459 (1984). 65. S. Chatrchyan et al. [CMS Collab.], Phys. Lett. B722, 28 38. D. Decamp et al. [ALEPH Collab.], Phys. Reports 216, 253 (2013); (1992); S. Chatrchyan et al. [CMS Collab.], Phys. Lett. B723, 280 P. Barate et al. [ALEPH Collab.], Eur. Phys. J. C4, 571 (1998). (2013); 39. P. Abreu et al. [DELPHI Collab.], Nucl. Phys. B367, 511 (1991); V. Khachatryan et al. [CMS Collab.], JHEP 1408, 173 (2014). J. Abdallah et al. [DELPHI Collab.], Eur. Phys. J. C37, 405 66. A.M. Sirunyan et al. [CMS Collab.], Phys. Rev. D97, 072006 (2004). (2018). 40. O. Adriani et al. [L3 Collab.], Phys. Reports 236, 1 (1993); 67. V. Khachatryan et al. [CMS Collab.], JHEP 1601, 166 (2016). P. Achard et al. [L3 Collab.], Phys. Lett. B531, 28 (2002); 68. S. Chatrchyan et al. [CMS Collab.], Phys. Lett. B704, 143 P. Achard et al. [L3 Collab.], Phys. Lett. B568, 23 (2003). (2011); 41. G. Abbiendi et al. [OPAL Collab.], Phys. Lett. B544, 57 (2002); S. Chatrchyan et al. [CMS Collab.], Phys. Lett. B720, 309 G. Abbiendi et al. [OPAL Collab.], Phys. Lett. B602, 167 (2004). (2013). 42. C. Adloff et al. [H1 Collab.], Phys. Lett. B525, 9 (2002); 69. V. Khachatryan et al. [CMS Collab.], JHEP 1603, 125 (2016). F.D. Aaron et al. [H1 Collab.], Phys. Lett. B663, 382 (2008); 70. A.M. Sirunyan et al. [CMS Collab.], JHEP 1904, 015 (2019). F.D. Aaron et al. [H1 Collab.], Phys. Lett. B666, 131 (2008). 71. CMS Collaboration, CMS-PAS-EXO-18-013 (2019). 43. S. Chekanov et al. [ZEUS Collab.], Phys. Lett. B549, 32 (2002). 72. M. Cacciari, G.P. Salam, and G. Soyez, JHEP 0804, 063 (2008). 44. D. Acosta et al. [CDF Collab.], Phys. Rev. Lett. 94, 101802 73. T. Sj¨ostrand et al., Comp. Phys. Comm. 135, 238 (2001). (2005); 958 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

Revised August 2019 by K.M. Black (Wisconsin U.), R.Sekhar 93.1.2 The Higgs doublet as a pseudo-Nambu-Goldstone Bo- Chivukula (UC San Diego) and M. Narain (Brown U.). son, v/f < 1, Λ > 1 TeV In technicolor models, the symmetry-breaking properties of the 93.1 Introduction and Phenomenology underlying strong dynamics necessarily breaks the electroweak In theories of dynamical electroweak symmetry breaking, the gauge symmetries. An alternative possibility is that the under- electroweak interactions are broken to electromagnetism by the lying strong dynamics itself does not break the electroweak in- vacuum expectation value of a composite operator, typically a teractions, and that the entire quartet of bosons in the Higgs 0 fermion bilinear. In these theories, the longitudinal components doublet (including the state associated with the H ) are compos- of the massive weak bosons are identified with composite Nambu- ite (pseudo-) Nambu-Goldstone particles [23,24], In this case, the Goldstone bosons arising from dynamical symmetry breaking in a underlying dynamics can occur at energies exceeding 1 TeV and strongly-coupled extension of the standard model. Viable theories additional interactions with the top-quark mass generating sec- of dynamical electroweak symmetry breaking must also explain tor (and possibly with additional weakly-coupled gauge bosons) (or at least accommodate) the presence of an additional composite cause the vacuum energy to be minimized when the composite scalar state to be identified with the H0 [1,2] – a state Higgs doublet gains a vacuum expectation value [25,26]. In these unlike any other observed previously. theories, the couplings of the remaining singlet scalar state would naturally be equal to that of the standard model Higgs boson up Theories of dynamical electroweak symmetry breaking can be to corrections of order (v/f)2 and, therefore, constraints on the classified by the nature of the composite singlet state to be asso- size of deviations of the H0 couplings from that of the standard 0 ciated with the H and the corresponding dimensional scales f, model Higgs [27] give rise to lower bounds on the scales f and Λ.5 the analog of the pion decay-constant in QCD, and Λ, the scale The electroweak gauge interactions, as well as the interactions 1 of the underlying strong dynamics.√ Of particular importance is responsible for the top-quark mass, explicitly break the chiral 2 2 the ratio v/f, where v = 1/( 2GF ) ≈ (246 GeV) , since this ra- symmetries of the composite Higgs model and lead generically tio measures the expected size of the deviations of the couplings to sizable corrections to the mass-squared of the Higgs-doublet of a composite Higgs boson from those expected in the standard – the so-called “Little Hierarchy Problem” [28]. “Little Higgs” model. The basic possibilities, and the additional states that they theories [29–32] are examples of composite Higgs models in which predict, are described below. the (collective) symmetry-breaking structure is selected so as to suppress these contributions to the Higgs mass-squared. 93.1.1 Technicolor, v/f ' 1, Λ ' 1 TeV Composite Higgs models typically require a larger global sym- Technicolor models [7–9] provided the first examples of theories metry of the underlying theory, and hence additional relatively of dynamical electroweak symmetry breaking. These theories in- light (compared to Λ) scalar particles, extra electroweak vector corporate a new asymptotically free gauge theory (“technicolor”) bosons (e.g. an additional SU(2)×U(1) gauge group), and vector- and additional massless fermions (“technifermions” transforming like partners of the top-quark of charge +2/3 and possibly also under a vectorial representation of the gauge group). The global +5/3 [33]. In addition to these states, one would expect the un- chiral symmetry of the fermions is spontaneously broken by the derlying dynamics to yield additional scalar and vector resonances formation of a technifermion condensate, just as the approximate with masses of order Λ. If the theory respects a custodial symme- chiral symmetry in QCD is broken down to isospin by the forma- try [34], the couplings of these additional states to the electroweak tion of a quark condensate. The SU(2)W × U(1)Y interactions and Higgs boson will be related – and, for example, one might ex- are embedded in the global technifermion chiral symmetries in pect a charged vector resonance to have similar branching ratios such a way that the only unbroken gauge symmetry after chiral to WZ and WH. Different composite Higgs models utilize dif- 2 symmetry breaking is U(1)em. The theories naturally provide ferent mechanisms for arranging for the hierarchy of scales v < f the Nambu-Goldstone bosons “eaten” by the W and Z boson. and arranging for a scalar Higgs self-coupling small enough to pro- There would also typically be additional heavy states (e.g. vec- duce an H0 of mass of order 125 GeV, for a review see [35] . If tor , analogous to the ρ and ω mesons in QCD) with TeV the additional states in these models carry color, they can provide masses [13,14], and the WW and ZZ scattering amplitudes would additional contributions to Higgs production via gluon fusion [36]. be expected to be strong at energies of order 1 TeV. The extent to which Higgs production at the LHC conforms with There are various possibilities for the scalar H0 in technicolor standard model predictions provides additional constraints (typi- models. First, the H0 could be identified as a singlet scalar res- cally lower bounds on the masses of the additional colored states onance, analogous to the σ particle expected in pion-scattering of order 0.7 TeV) on these models. in QCD [15, 16]. Alternatively, the H0 could be identified as a In addition, if the larger symmetry of the underlying composite dilaton, a (pseudo-)Goldstone boson of scale invariance in theo- Higgs theory does not commute with the standard model gauge ries of “walking technicolor” [17–21].3 Finally, the H0 could be group, then the additional states found in those models – es- identified as an additional isosinglet state if the chiral symme- pecially those related to the top-quark, which tend to have the try breaking pattern of the technicolor theory provides for such a largest couplings to the electroweak sector – may be colorless. state.4 In all of these cases, however, one expects large deviations For example, in twin Higgs models [37], the top-partners carry no in the couplings of this particle from those of the standard model standard model charges. The phenomenology of the additional Higgs boson. Since the couplings observed for the H0 approxi- states in twin Higgs theories is rather different, since lacking color mate those of the Higgs boson to the 10% level, models of this the production of these particles at the LHC will be suppressed – kind are very highly constrained. and, their decays may occur only via the electroweak symmetry breaking sector, leading to their being long-lived. 93.1.3 Top-Condensate, Top-Color, Top-Seesaw and related 1In a strongly interacting theory “Naive Dimensional Analysis” [3, 4] theories, v/f < 1, Λ > 1 TeV ∗ ∗ implies that, in the absence of fine-tuning, Λ ' g f where g ' 4π is A final alternative is to consider a strongly interacting theory the typical size of a strong coupling in the low-energy theory [5, 6]. This estimate is modified in the presence of multiple flavors or colors [7] . with a high (compared to a TeV) underlying dynamical scale that 2For a review of technicolor models, see [10–12]. would naturally break the electroweak interactions, but whose 3If both the electroweak symmetry and the approximate scale symme- try are broken only by electroweak doublet condensate(s), then the decay- 5In these models v/f is an adjustable parameter, and in the limit constants for scale and electroweak symmetry breaking may be approxi- v/f → 1 they reduce, essentially, to the technicolor models discussed in mately equal – differing only by terms formally proportional to the amount the previous subsection. Our discussion here is consistent with that given of explicit scale-symmetry breaking. there, since we expect corrections to the SM Higgs couplings to be large 4In this case, however, the coupling strength of the singlet state to WW for v/f ' 1. Current measurements constrain the couplings of the H0 to and ZZ pairs would be comparable to the couplings to gluon and photon equal those predicted for the Higgs in the standard model to about the 10% pairs, and these would all arise from loop-level couplings in the underlying level [27], suggesting that f must have values of order a TeV or higher and, technicolor theory [22]. therefore, a dynamical scale Λ of at least several TeV. 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 959

strength is adjusted (“fine-tuned”) to produce electroweak sym- tial which allow for v/f < 1 [49–53]. metry breaking at 1 TeV. This alternative is possible if the elec- Alternatively, one can assume that the underlying flavor dy- troweak (quantum) phase transition is continuous (second order) namics respect flavor symmetries (“minimal” [54, 55] or “next- in the strength of the strong dynamics [38]. If the fine tuning to-minimal” [56] flavor violation) that suppress flavor-changing can be achieved, the underlying strong interactions will produce neutral currents in the two light generations. Additional con- a light composite Higgs bound state with couplings equal to that siderations apply when extending these arguments to potential of the standard model Higgs boson up to corrections of order explanation of neutrino masses (see, for example, [57, 58]). (1 TeV/Λ)2. As in theories in which electroweak symmetry break- Since the underlying high-energy dynamics in these theories ing occurs through vacuum alignment, therefore, constraints on are strongly coupled, there are no reliable calculation techniques the size of deviations of the H0 couplings from that of the stan- that can be applied to analyze their properties. Instead, most dard model Higgs give rise to lower bounds on the scale Λ. For- phenomenological studies depend on the construction of a “low- mally, in the limit Λ → ∞ (a limit which requires arbitrarily energy” effective theory describing additional scalar, fermion, or fine adjustment of the strength of the high-energy interactions), vector boson degrees of freedom, which incorporates the rele- these theories are equivalent to a theory with a fundamental Higgs vant symmetries and, when available, dynamical principles. In boson – and the fine adjustment of the coupling strength is a man- some cases, motivated by the AdS/CFT correspondence [59], the ifestation of the hierarchy problem of theories with a fundamental strongly-interacting theories described above have been investi- scalar particle. gated by analyzing a dual compactified five-dimensional gauge In many of these theories the top-quark itself interacts strongly theory. In these cases, the AdS/CFT “dictionary” is used to map (at high energies), potentially through an extended color gauge the features of the underlying strongly coupled high-energy dy- sector [39, 39–43]. In these theories, top-quark condensation (or namics onto the low-energy weakly coupled dual theory [60]. the condensation of an admixture of the top with additional More recently, progress has been made in investigating strongly- vector-like quarks) is responsible for electroweak symmetry break- coupled models using lattice gauge theory [61]. These calculations ing, and the H0 is identified with a bound state involving the offer the prospect of establishing which strongly coupled theories third generation of quarks. These theories typically include an of electroweak symmetry breaking have a particle with properties extra set of massive color-octet vector bosons (top-gluons), and consistent with those observed for the H0 – and for establishing an extra U(1) interaction (giving rise to a top-color Z0) which cou- concrete predictions for these theories at the LHC [62]. ple preferentially to the third generation and whose masses define the scale Λ of the underlying physics. 93.2 Experimental Searches As discussed above, the extent to which the couplings of the 93.1.4 Flavor H0 conform to the expectations for a standard model Higgs bo- In addition to the electroweak symmetry breaking dynamics son constrains the viability of each of these models. Measure- described above, which gives rise to the masses of the W and ments of the H0 couplings, and their interpretation in terms of Z particles, additional interactions must be introduced to pro- effective field theory, are summarized in the H0 review in this duce the masses of the standard model fermions. Two general volume. In what follows, we will focus on searches for the addi- avenues have been suggested for these new interactions. In “ex- tional particles that might be expected to accompany the singlet tended technicolor” (ETC) theories [44, 45], the gauge interac- scalar: extra scalars, fermions, and vector bosons. In some cases, tions in the underlying strongly interacting theory are extended detailed model-specific searches have been made for the particles to incorporate flavor. This extended gauge symmetry is broken described above (though generally not yet taking account of the down (possibly sequentially, at several different mass scales) to demonstrated existence of the H0 boson). the residual strong interaction responsible for electroweak sym- In most cases, however, generic searches (e.g. for extra W 0 or metry breaking. The massive gauge-bosons corresponding to the Z0 particles, extra scalars in the context of multi-Higgs models, or broken symmetries then mediate interactions between mass op- for fourth-generation quarks) are quoted that can be used – when erators for the quarks/leptons and the corresponding bilinears of appropriately translated – to derive bounds on a specific model the strongly-interacting fermions, giving rise to the masses of the of interest. ordinary fermions after electroweak symmetry breaking. The mass scale of the new particles implied by the interpreta- In the case of “partial compositeness” [46], the additional tions of the low mass of H0 discussed above, and existing studies flavor-dependent interactions arise from mixing between the or- from the Tevatron and lower-energy colliders, suggests that only dinary quarks and leptons and massive composite fermions in the Large Hadron Collider has any real sensitivity. A number of the strongly-interacting underlying theory. Theories incorporat- analyses already carried out by ATLAS and CMS use relevant fi- ing partial compositeness include additional vector-like partners nal states and might have been expected to observe a deviation of the ordinary quarks and leptons, typically with masses of order from standard model expectations – in no case so far has any a TeV or less. such deviation been reported. The detailed implications of these In both cases, the effects of flavor interactions on the elec- searches in various model frameworks are described below. troweak properties of the ordinary quarks and leptons are likely Except where otherwise noted, all limits in this section are 6 √ to be most pronounced in the third generation of fermions. The quoted at a confidence level of 95%. The searches at s = 8 TeV additional particles present in these theories, especially the addi- (Run 1) are based on 20.3 fb−1 of data recorded by ATLAS, tional scalars, often couple more strongly to heavier fermions. −1 and an integrated luminosity√ of 19.7 fb analyzed by CMS. The Moreover, since the flavor interactions must give rise to quark datasets collected at s = 13 TeV during Run 2 of the LHC since mixing, we expect that a generic theory of this kind could give 2015 are based on analyses with varied integrated luminosities rise to large flavor-changing neutral-currents [45]. In ETC theo- ranging from ∼2-140 fb−1. ries, these constraints are typically somewhat relaxed if the theory 0 0 incorporates approximate generational flavor symmetries [47], the 93.2.1 Searches for Z or W Bosons theory has a slowly running coupling constant or “walks” [17–21], Massive vector bosons or particles with similar decay channels or if Λ > 1 TeV [48]. In theories of partial compositeness, the would be expected to arise in Little Higgs theories, in theories of masses of the ordinary fermions depend on the scaling-dimension Technicolor, or models involving a dilaton, adjusted to produce a 0 of the operators corresponding to the composite fermions with light Higgs boson, consistent with the observed H . These parti- which they mix. This leads to a new mechanism for generat- cles would be expected to decay to pairs of vector bosons, or to ing the mass-hierarchy of the observed quarks and leptons that, third generation quarks, or to leptons. The generic searches for 0 0 potentially, ameliorates flavor-changing neutral current problems W and Z vector bosons listed below can, therefore, be used to and can provide new contributions to the composite Higgs poten- constrain models incorporating a composite Higgs-like boson. A general review of searches for Z0 and W 0 bosons is also in-

6Indeed, from this point of view, the vector-like partners of the top- cluded in this volume [63, 64]. In the context of the dynamical quark in top-seesaw and little Higgs models can be viewed as incorporating electroweak symmetry breaking models, we emphasize their de- partial compositeness to explain the origin of the top quark’s large mass. cays to third generation fermions by including a detailed overview, 960 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 while also briefly summarizing the other searches. CMS [78] has carried out a complementary search in the τν final state. As noted above, such searches place limits on models 0 Z → ``: with enhanced couplings to the third generation. No excess was 0 ATLAS [65] and CMS [66] have both searched for Z produc- observed and limits between 2.0 and 2.7 TeV were set on the mass 0 tion with Z → ee or µµ. No deviation from the standard model of a W 0 decaying preferentially to the third generation; a W 0 with prediction was seen in the dielectron and dimuon invariant mass universal fermion couplings was also excluded for masses less than spectra, by either the ATLAS or the CMS analysis, and lower lim- 2.7 TeV. 0 0 its on possible Z boson masses were set. A ZSSM with couplings equal to the standard model Z0 (a “sequential standard model” W 0 → tb: Z0) and a mass below 5.1 TeV was excluded by ATLAS, while Heavy new gauge bosons can couple to left-handed fermions CMS set a lower mass limit of 5.15 TeV. The experiments also like the SM W boson or to right-handed fermions. W 0 bosons 0 place limits on the parameters of extra dimension models and in that couple only to right-handed fermions (WR) may not have the case of ATLAS on the parameters of a minimal walking tech- leptonic decay modes, depending on the mass of the right-handed nicolor model [17–21], consistent with a 125 GeV Higgs boson [67]. neutrino. For these W 0 bosons, the tb (tb + tb) decay mode is For a general review of searches in these channels see the PDG especially important because in many models the W 0 boson is review of Z prime in this volume [63]. expected to have enhanced couplings to the third generation of In addition, both experiments have also searched for Z0 decay- quarks relative to those in the first and second generations. It is ing to a ditau final state [68, 69]. An excess in τ +τ − could have also the hadronic decay mode with the best signal-to-background. interesting implications for models in which lepton universality is ATLAS and CMS have performed searches for W 0 bosons via the not a requirement and enhanced couplings to the third genera- W 0 → tb decay channel in the lepton+jets and all-hadronic final tion are allowed. This analysis led to lower limits on the mass of state. 0 0 a ZSSM of 2.4 and 2.1 TeV from ATLAS and CMS respectively. The CMS lepton+jets search [79–82], W → tb → Wbb → Z0 → qq: `νbb, proceeded via selecting events with an isolated lepton (elec- The ability to relatively cleanly select tt pairs at the LHC to- tron or muon), and at least two jets, one of which is identified to 0 gether with the existence of enhanced couplings to the third gen- originate from a b-quark. The mass of the W boson (Mtb) was eration in many models makes it worthwhile to search for new par- reconstructed using the four-momentum vectors of the final state ticles decaying in this channel. Both ATLAS [70] and CMS [71] objects (bb`ν). The distribution of Mtb is used as the search dis- −1 √ have carried out searches for new particles decaying into tt. criminant. A search [82] using 35.9 fb of data, collected at s 0 Both the ATLAS and CMS collaborations searched for tt in = 13 TeV, led to an exclusion of WR bosons with masses below the all hadronic mode [72] [73] in both the resolved and boosted 3.4 TeV (3.6 TeV) if MW 0 >> MνR (MW 0 < MνR ), where MνR R R regions. No evidence of resonance production were seen and lim- is the mass of the right-handed neutrino. its were produced for various models including the Z’ boson in The CMS search for W 0 →tb decays using the all-hadronic topcolor-assisted technicolor which excludes masses less than 3.1 final state focused on W 0 masses above 1 TeV [81]. In this region, to 3.6 TeV (ATLAS) depending on the details of the model and the top quark gets a large Lorentz boost and hence the three 3.3, 5.25, and 6.65 TeV for widths of 1, 10 and 30 percent relative hadronic products from its decay merge into a single large-radius to the mass of the resonance . jet. Techniques which rely on substructure information of the ATLAS also presented results on the lepton plus jets final state, jets [83] are employed to identify boosted all hadronic W boson where the top quark pair decays as tt → W bW b with one W bo- and top quark jets and compute the mass of the jet. W 0 candidate son decaying leptonically and the other hadronically; CMS used mass was computed from back-to-back boosted top-tagged jet and final states where both, one or neither W decays leptonically and a low mass b-tagged jet. From this all-hadronic search, W 0 bosons then combined the results. The tt¯ invariant mass spectrum was were excluded for masses up to 2.02 TeV. analyzed for any excess, and no evidence for any resonance was 0 ATLAS has searched for WR bosons in the tb final state both seen. ATLAS excluded a narrow (Γ/m = 1.2%) leptophobic top- for lepton+jets [84] and all-hadronic [85] decays of the top. No color Z0 boson with masses between 0.7 and 2.1 TeV and with significant deviations from the standard model were seen in ei- Γ/m = 3% between 0.7 and 3.2 TeV. CMS set limits on lepto- ther analysis and limits were set on the W 0 → tb cross section phobic Z0 bosons for three different assumed widths Γ/m = 1.0% times branching ratio and W 0 bosons with purely right-handed , Γ/m = 10.0%, and Γ/m = 30.0% of 3.9 TeV to 4.0 TeV and couplings to fermions were excluded for masses below 3.15 TeV exclude RS KK gluons up to 3.3 TeV. when the two channels are combined. Both ATLAS [74] and CMS [75] have also searched for reso- In addition, the above studies also provided upper limits on nances decaying into qq, qg or gg using the dijet invariant mass the W 0 effective couplings to right- and left-handed fermions. spectrum. Excited quarks are excluded up to masses of 6.7 TeV In Fig. 93.1 (bottom) the upper limits on W 0 couplings normalized and model-independent upper limits on cross sections with a gaus- to the SM W boson couplings derived by ATLAS [86] are shown. sian signal shape were set. CMS excluded string resonances with The top panel of Fig. 93.1 shows the upper limits for arbitrary masses below 7.9 TeV, scalar diquarks below 7.5 TeV, axigluons combinations of left- and right-handed couplings of the W 0 boson and colorons below 6.6 TeV, excited quarks below 6.3 TeV, color- to fermions set using a model independent approach by CMS [82]. octet scalars below 3.7 TeV, W’ bosons below 3.6 TeV, Z’ bosons with SM-like couplings below 2.9 TeV and between 3.1 TeV and 93.2.2 Searches for Resonances decaying to Vector Bosons 3.3 TeV, Randall–Sundrum Gravitons below 2.6 TeV. and/or Higgs Bosons Both the ATLAS√ and CMS experiments have used the data W 0 → `ν: collected at s = 13 TeV to search for resonances decaying to Both LHC experiments have also searched for massive charged pairs of bosons. Overall no significant excesses were seen in the vector bosons. In this section we include a summary of the re- full datasets that were analyzed and the results are interpreted in sults, with emphasis on final states with third generation fermions, models with heavy vector triplets (HVT) [87], models with strong while the details on other decays are discussed in the mini-review gravity and extra spatial dimensions, as well as setting model in- of W 0 [64]. ATLAS searched for a heavy W 0 decaying to eν or dependent limits as a function of mass. For a full review of models µν and find no excess over the standard model expectation. A se- including extra spatial dimensions including the interpretation of quential standard model (SSM) W 0 boson (assuming zero branch- many of these results in that context please see the review of extra dimensions in this volume [60]. ing ratio to WZ) with mass√ less than 7 TeV was excluded [76] √ using 139 fb−1 dataset at s = 13 TeV, as well as setting model Utilizing data collected at s = 13 TeV, ATLAS [88] and CMS independent cross-section limits as a function of mass. Based on a [89], have both looked for a resonant state decaying into VV (with smaller dataset, the CMS experiment excluded a SSM W 0 boson V = W or Z) , VH ( with H representing the SM Higgs boson), with mass up to 4.1 TeV [77] and presented the upper limits on and HH. Both collaborations have analyzed bosonic decay modes. the production of generic W 0 bosons decaying into this final state ATLAS searches in the qqqq, ννqq, lνqq , llqq , lνlν, llνν, lνll, using a model-independent approach. llll, qqbb ννbb, lνbb, and llbb final states and combined the results 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 961 /g R

1 ATLAS 95% CL limit on g' 95% CL

s = 8 TeV, 20.3 fb -1

Expected limit

Observed limit -1 10 0.5 1 1.5 2 2.5 3

W' R mass [TeV]

0 Figure 93.1: Left panel: Observed limits on the W boson mass as function of the left-handed (aL) and right-handed (aR) couplings. Black lines represent contours of equal W 0 boson mass [82]. Right panel: Observed and expected regions, on the g0/g vs mass of the W 0 boson plane, that are excluded at 95% CL, for right-handed W 0 bosons [86].

(separately within each experiment) . While CMS analyzes the In the 2l2q final state, the CMS collaboration searches for a heavy qqqq, ννqq, lνqq , llqq , llνν, ννbb, lνbb, llbb , bbbb, ττbb , and resonance decaying into ZV [95] looking for events with one large qqττ final states. R-jet consistent with the hadronic decay of a vector boson and a The combined limits are expressed both as limits on the cross- Z boson reconstructed in the charged lepton decay channel (e or section as a function resonance mass as well as constraints on the µ). Limits are set for a HVT W’ with a lower mass of 2270 (2330) coupling of the heavy boson triplet to quarks, leptons, and the for gV = 1 (gV = 3). Higgs boson.

X → WZ: X → WW : ATLAS searches for new heavy resonances decaying into WZ in The ATLAS collaboration searches for a new heavy resonance the channels WZ → qqqq [90] , lνqq [91], and lνll [92]. In the fully decaying into WW in the channels WW → qqqq [90] , lνqq [91], leptonic channel , the invariant mass of the W Z pair is obtained and lνν [96]. In the case where both W s decay leptonically, AT- by considering all possible four lepton permutations in each event. LAS utilizes the transverse of the two lepton and two neutrino The dominant background is Standard Model continuum WZ pro- final state and searches for an excess in this distribution between duction, ZZ production where one lepton is not identified or falls 200 GeV and 5 TeV. No excess is seen and the mass of a HVT is outside the detector acceptance, and top quark plus vector boson excluded below 1300 GeV. Further vector boson fusion is also con- production. No resonant production is seen in data and lower sidered and cross-section limits as a function of mass are placed limits on the mass of a HVT decaying into WZ are set at 2260 ranging from 1.3 pb to 0.006 pb at 200 GeV to 3 TeV, respectively. (2460) GeV assuming a coupling constant of gV = 1 (gV = 3). In the lνqq mode , ATLAS completed a companion analysis to the In the WZ → lνqq mode, ATLAS searches in both the cases that WZ → analysis discussed above and places lower mass limits of the quarks are observed as individual jets (resolved) and where 2850 (3150) GeV for gV = 1 (gV = 3) in the HVT model. ATLAS they merge into one jet in the detector (boosted) which probe the also interprets the all hadronic mode analysis in the hypothesis low and high pT regime of the Z boson. No significant excess is that WW → qqqq and places limits on a HVT boson decaying seen in either channel and combined lower mass limits are placed into WW in the all hadronic mode between 1200 to 2200 (1200 to at 2900 (3000) GeV for g = 1 (g = 3) in the HVT model. In V V 2800) GeV for gV = 1 (gV = 3). the all hadronic decay mode, ATLAS searches for two high pT The CMS collaboration searches for VV → qqqq [93] in the hadronically decaying vector bosons looking for a resonant struc- large R dijet search. The W and Z boson are identified through ture. No excess is seen and limits are placed excluding 1200-3000 the mass of the large R jet and substructure variables. No excess (1200-3300) GeV for gV = 1 (gV = 3) in the HVT model. is seen and limits are set for charged HVT bosons with masses The CMS collaboration searches for VV → qqqq [93] in the lower than 2700 (2800) GeV for gV = 1 (gV = 3). Cross-section large R dijet search. The W and Z boson are identified through limits as function of mass are reported for the uncharged spin-1 the mass of the large R jet and substructure variables. No excess resonance interpretation and are placed at 41.6 fb at 1.4 TeV to is seen and limits are set for charged HVT bosons with masses 0.6 pb at 4 TeV. lower than 3200 (3800) GeV for gV = 1 (gV = 3). Cross-section limits as function of mass are reported for the charged spin-1 X → VH: resonance interpretation and are placed at 44.4 fb at 1.4 TeV to The ATLAS Collaboration searches for a new heavy resonance 0.7 pb at 4 TeV. In the ννqq final state [94], the CMS collaboration decaying into WH and ZH in the qqbb (WH and ZH) [97], lνbb searches for a charged spin 1 resonance decaying into a VZ final (WH), ννbb (ZH), llbb(ZH) [98]. In the all hadronic mode, AT- state with a Z boson decaying into a pair of neutrinos and the LAS searches for boosted VH production looking for two large R other boson decaying into two collimated quarks reconstructed jets where the larger invariant mast large R jet is interpreted as as a large R-jet. The transverse mass of the VZ candidate is the Higgs boson decay products while the lesser the hadronically reconstructed and utilized to search for evidence of resonant VZ decaying vector boson requiring b tagging on the Higgs boson sub- production. No excess is seen and lower mass limits are placed on jets. The invariant mass is reconstructed a and a search is done the charged resonance at 3100 (3400) GeV for gV = 1 (gV = 3). for resonant production of ZH. None is found and limits from 962 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

1100 to 2500 (1300 to 3800 GeV) are placed for gV = 1 (gV = 3). re-interpreted to obtain the sensitivity to a combination with var- ATLAS also searches for ZH where the W or Z boson decays into ied branching fractions to the different decay modes. νν, lν, and ll . The analysis searches for both resolved and merged In the following sections, the results obtained for T (B) quarks (boosted) b jets from the decay of the Higgs boson as well as defin- assuming 100% branching ratio to W b (W t) are also applicable ing signal regions based on the number of reconstructed charged to heavy vector-like Y−4/3 (X5/3) with charge 4/3 (5/3). leptons (0,1,or 2). In the dilepton channel the invariant mass is explicitly reconstructed of the entire diboson system, the single 93.2.3.1 Searches for T quarks that decay to W , Z and H0 lepton channel reconstructs the diboson final state constraining bosons the lepton and missing transverse momentum utilizing the known T/Y → bW : W boson mass, while the 0 charged lepton channel reconstructs CMS has searched for pair production of heavy T quarks that the transverse mass of the diboson system. No excess is seen in decay exclusively to bW [101–103]. The analysis selected events any channel and limits on the production of a HVT are placed at with exactly one charged lepton, assuming that the W boson from 2800 GeV (2930) GeV for gV = 1 (gV = 3). the second T quark decays hadronically. Under this hypothesis, The CMS Collaboration searches for a heavy resonance de- a 2-constraint kinematic fit can be performed to reconstruct the caying into VH [99] searching for resonances decaying into a mass of the T quark as a narrow mass peak with a mass resolu- Higgs boson and either a hadronically decaying W or Z bo- tion of around 7%. In Refs. [102] and [103], the two-dimensional son. The search identifies events with two large-R jets using distribution of reconstructed mass vs ST was used to test for the substructure variables and requires one large-R jets is tagged signal, where ST is the scalar sum of the missing pT and the with a pair of b-hadrons clustered in a single jet. The in- transverse momenta of the lepton and the leading four jets. This variant mass of the VH bosons is reconstructed and evidence analysis, when combined with the search in the fully hadronic fi- for resonance production is sought. No excess is seen and nal state [104] excluded new quarks that decay 100% to bW for limits are placed. With gV = 1 (3) a narrow W’ reso- masses below 0.89 TeV [103]. At times the hadronically-decaying nance with mW 0 < 2470(3150) GeV and mZ0 < 1150(1190). W boson is produced with a large Lorentz boost, leading to the W Summary of Searches with Diboson Final States: decay products merged into a large-radius jet. Algorithms such Both ATLAS [88] and CMS [89] provide plots summarizing the as jet pruning [105] were used to remove contributions from soft, various searches results and limits combining . The results are wide angle radiation, from large-radius jets, leading to better dis- shown in the context of HVT models and models of strong gravity crimination between QCD jets and those arising from decays of with extra spatial dimensions. No excess is seen in any search the heavy particles. If the mass of the boosted jet was compatible and limits on the 4.3 (4.5) TeV (ATLAS) and (CMS). Inclusion with the W boson mass, then the W boson candidate jet and its of decays directly to fermions increase these limits to 5.3 (5.5) subjets were used in the kinematic reconstruction of the T quark. TeV and 5.0 (5.2) TeV from the ATLAS and CMS combinations, No excess over standard model backgrounds was observed. Upper respectively. Both collaborations also place varying limits on the limits on the production cross section as a function of the mass of coupling strength as a function of HVT boson mass as well. T quarks were measured. By comparing them with the predicted cross section for vector like quark pair production, the strong pair 93.2.3 Vector-like third generation quarks production of T quarks was excluded for masses below 1.30 TeV Vector-like quarks (VLQ) have non-chiral couplings to W (1.28 TeV expected) [101]. bosons, i.e. their left- and right-handed components couple in the same way. They therefore have vectorial couplings to W Another “cut-based” search for pair produced T quarks in the bosons. Vector-like quarks arise in Little Higgs theories, top- all-hadronic final state targeting the W b decay mode [106], relies coloron-models, and theories of a composite Higgs boson with on mass reconstruction of two highest pT W b combinations using partial compositeness. In the following, the notation T quark boosted W boson candidates with pT > 200 GeV and b-tagged refers to a vector-like quark with charge 2/3 and the notation B jets. HT is used as the signal discriminator, with selected events quark refers to a vector-like quark with charge −1/3, the same divided into nine categories based on the multiplicity of W and b charges as the SM top and b quarks respectively. The exotic jets in the event. From this search T quarks with pure W b decays are excluded for masses below 1.04 TeV (1.07 TeV expected). vector-like quarks X5/3 and Y−4/3 have charges 5/3, and −4/3 respectively. Vector-like quarks couple with SM quarks with An analogous search has been carried out by ATLAS [107,108] Yukawa interactions and may exist as SU(2) singlets (T , and B), for the pair production of heavy T quarks. It used the lepton+jets final state with an isolated electron or muon and at least four jets, doublets [(X5/3,T ),(T,B),(B,Y−4/3)], or triplets [(X5/3,T,B), including a b jet and required reconstruction of the T quark mass. (T,B,Y−4/3)]. At the LHC, VLQs can be pair produced via the dominant gluon-gluon fusion. VLQs can also be produced Given the mass range of the T quark being explored was from a singly by their electroweak effective couplings to a weak boson 0.4 TeV to a couple of TeV, the W boson from the T quark may and a standard model quark. Single production rate is expected fall in two categories: those with a high boost leading to merged to dominate over the rate of pair production at large VLQ masses. decay products, and others where the two jets from the W boson T quarks can decay to bW , tZ, or tH0. Weak isospin singlets were resolved. In addition, the selection was optimized to require are expected to decay to all three final states with (asymptotic) large angular separation between the high pT W bosons and the branching fractions of 50%, 25%, 25%, respectively. Weak isospin b jets. doublets are expected to decay exclusively to tZ and to tH0 [100] The T → W b candidates were constructed from both the lep- with equal branching ratios. Analogously, B quarks can decay to tonically and hadronically decaying W bosons by pairing them 0 tW , bZ, or bH . The Y−4/3 and X5/3 quarks decay exclusively to with the two highest pT b-tagged jets in the event. The pairing of bW and to tW . While these are taken as the benchmark scenarios, b jets with W bosons which minimizes the difference between the other representations are possible and hence the final results are masses of leptonically decaying T (mlep(T )) and the hadronic T interpreted for many allowed branching fraction combinations. (mhad(T )) was chosen. Finally, mlep(T ) was used as the discrim- 0 0 Given the multiple decay modes of the VLQs, the final state sig- inating variable in a signal region defined by high ST (here ST is natures of both pair produced and the singly produced VLQs are defined as the scalar sum of the missing pT , the pT of the lepton fairly rich with leptons, jets, b jets, and missing energy. Depend- and jets), and the opening angle between the lepton and the neu- 0 −1 ing on the mass of the VLQ, the top quarks and W/Z/H bosons trino√ (∆ R(e, ν)). With the 36.1 fb data collected during Run 2 may be Lorentz boosted and identified using jet substructure tech- at s = 13 TeV, assuming 100% branching ratio to the W b decay, niques. Thus the searches are performed using lepton+jets signa- the observed lower limit on the T mass was 1.35 TeV, and in the tures, multi-lepton and all-hadronic decays. In addition, T or B SU(2) singlet scenario, the lower mass limit was obtained to be quarks with their antiparticles can result in events with same- 1.17 TeV [107]. sign leptons, for example if the decay T → tH → bW W +W − is A targeted search for a T quark, produced singly in association present, followed by leptonic decays of two same-sign W bosons. with a light flavor quark√ and a b quark and decaying into bW , was In the following subsections, while we describe the searches for carried out by CMS at s=13 teV and a dataset corresponding to each of the decay modes of the VLQs, the same analysis can be 2.3 fb−1 [109]. The analysis used lepton+jets events, with at least 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 963

2016 diboson combination 35.9 fb-1 (13 TeV)

Combination 104 ATLAS Observed 95% CL limit CMS WZ → qqqq (PRD 97(2018)072006) s = 13 TeV, 36.1 fb-1 1 WZ → ννqq (JHEP 07(2018)075)

(W') (pb) Expected 95% CL limit

σ WZ → lνqq (JHEP 05(2018)088) 3 WZ → llqq (JHEP 09(2018)101) 10 Expected ± 1σ → WH qqbb (EPJC 77(2017)636) ± σ −1 WH → lνbb (JHEP 11(2018)172) Expected 2 10 → ττ WH qq (JHEP 01(2019)051) 102 HVT model A WZ + WH lv) [fb] → 10−2 10 W’ → 1 −3 95% CL upper limits 10 (pp

Observed σ Median expected −1 DY HVT W’ → WZ + WH + lv HVT model B 10 10−4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1000 1500 2000 2500 3000 3500 4000 4500 mW' (GeV) m(W’) [TeV]

Figure 93.2: Left panel:Observed limits from W 0 to diboson from CMS [89]. Right panel: Observed limits from W 0 to diboson decays from ATLAS [88].

one b-tagged jet with large transverse momentum, and a jet in T → tH0 upto a T quark mass of 1.01 TeV [113]. This analysis 00 the forward η region. Selected events were required to have ST > used a deep neural network technique to identify jets originat- 00 500 GeV, where ST is defined as the scalar sum of the transverse ing from boosted bosons and top-quarks and described further in momenta of the lepton, the leading central jet, and the missing subsection 93.2.3.2. transverse momentum. The invariant mass of the T candidate was The CMS search for T T production, with T → tH0 decays has used as the discriminating variable and was reconstructed using been performed in both lepton+jets, multilepton and all-hadronic the four-vectors of the leptonically decaying W boson and the final states. The lepton+jets analysis [114] emphasizes the pres- leading central jet. No excess over the standard model prediction ence of large number of b-tagged jets, and combined with other was observed. As the VLQ width is proportional to the square of kinematic variables in a Boosted Decision Tree (BDT) for en- the coupling, upper limits were set on the production cross section hancing signal to background discrimination. The multilepton assuming a narrow width VLQ with coupling greater than 0.5. For analysis [114] was optimized for the presence of b jets and the Y/T quarks with a coupling of 0.5 and a 100% branching fraction large hadronic activity. For B(T → W b) = 1, the combined for the decay to bW the excluded masses were in the range from lepton+jets and multilepton analyses led to a lower limit on T 0.85 to 1.40 TeV [109]. A similar search [110, 111] performed by quark masses of 0.71 TeV. A search for T → tH0 in all-hadronic ATLAS singly produced T or Y−4/3 quark decaying to W b using decays [115], optimized for a high mass T quark, and based on a dataset corresponding to 36.1 fb−1. The search was performed identifying boosted top quark jets has been carried out by CMS. using lepton+jets events with a high pT b-tagged jet, and at least This search aimed to resolve subjets within the jets arising from one forward jet. The reconstructed mass of the T /Y−4/3 quark, boosted top quark decays, including b tagging of the subjets. A was used as the discriminating variable and showed no excess likelihood discriminator was defined based on the distributions of above the expectation from SM. Interference effects with the SM HT , and the invariant mass of the two b jets in the events for sig- background are included in the study. This search led to 95% CL nal and background. No excess above background expectations W b was observed. Assuming 100% branching ratio for T → tH0, this upper limits on the mixing angle |sin(θL)| (CL ) in the range of 0.18–0.35 (0.25–0.49) for singlet T quark mass between 0.8– analysis led to a lower limit of 0.75 TeV on the mass of the T 1.2 TeV. This search also provided limits as a function of the quark. √ −1 Y−4/3 quark mass, on the coupling of the Y−4/3 quark to bW , Searches for T quarks at s=13 TeV, based on a 2.6 fb W b dataset [116] have been performed by CMS using the lepton+jets and the mixing parameter |sinθR| (CR ) for a (B,Y−4/3) doublet model [110]. For VLQ masses between 0.08–1.8 TeV, the limits on final state. This search has been optimized for high mass T quarks W b by exploiting techniques to identify W or Higgs bosons decay- |sin(θR)| (C ) is in the range 0.17–0.55 (0.24–0.77), where limits R ing hadronically with large transverse momenta. The boosted on |sinθR| ar around 0.18–0.19 and below the constraints from W channel excluded T quarks decaying only to bW with masses electroweak precision observables for Y−4/3 quark mass between below 0.91 TeV, and the boosted tH channel excluded T quarks 0.9–1.25 TeV. For Y−4/3 quark in the triplets (T ,B,Y−4/3), limits decaying only to tH for masses below 0.89 TeV. on |sin(θ )| (CW b) are between 0.16–0.39 (0.31–0.78) for masses L L A CMS search for T → tH0 with H0 → γγ decays has been between 0.8 GeV–1.6 TeV [110]. performed [117] in pair production of T quarks. To identify the T → tH0: Higgs boson produced in the decay of the heavy T quark, and the ATLAS has performed a search for T T production with T → subsequent H0 → γγ decay, the analysis focused on identification 0 0 tH [108, 112]. Given the dominant decay mode H → bb, these of two photons in events with one or more high pT lepton+jets or events are characterized by a large number of jets, many of which events with no leptons and large hadronic activity. A search for are b jets. Thus the event selection required one isolated electron a resonance in the invariant mass distribution of the two photons or muon and high jet multiplicity (including b-tagged jets). The in events with large hadronic activity defined by the HT variable sample is categorized by the jet multiplicity (5 and ≥6 jets in the showed no excess above the prediction from standard model pro- 1-lepton channel; 6 and ≥7 jets in the 0-lepton channel), b tag cesses. The analysis resulted in exclusion of T quark masses below multiplicity (2, 3 and ≥4) and mass-tagged jet multiplicity (0, 0.54 TeV. 1 and ≥2). The distribution of meff , defined as the scalar sum A search for electroweak single production of T quark decaying 0 of the lepton and jet pT and the missing ET , for each category to tH using boosted topologies in fully hadronic [118] and lep- were used as the discriminant for the final signal and background ton+jets [119] in the final states has been performed by CMS. The separation. No excess of events were found. Weak isospin doublet electroweak couplings of the T quarks to the SM third generation T quarks were excluded below 1.16 TeV. quarks are highly model dependent and hence these couplings de- A search by ATLAS for pair produced VLQs with an all- termine the rates of the single T quark production. In both analy- hadronic final state signature yields an exclusion of pure decays ses, T quark candidate invariant mass was reconstructed using the 964 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 boosted Higgs boson jets and the top quark. Higgs boson jets were Zt have been performed in final state signatures with two or three identified using jet substructure techniques and subjet b tagging. charged leptons [121]. The analysis relies on tagging b jets and For the lepton+jets analysis the top quark was reconstructed high pT large-radius jets originating from top-quarks. Additional from the leptonically decaying W and the b jet, while in the all- selections are devised to reduce the contributions from pair pro- hadronic analysis the top quark jet was tagged using substructure duction of T quarks. For events with dilepton analysis, the dis- analysis. There was no excess of events observed above back- criminating variable is the mass of the T quark formed using the ground. Exclusion limits on the product of the production cross invariant mass of the Z boson candidate and the highest pT top- 0 section and the branching fraction (σ(pp → T qt/b)×B(T → tH ) tagged jet, while for the trilepton analysis, the variable ST is used were derived for the T quark masses in the range 0.70-1.8 TeV. to search for an excess of data over the expected SM background. From the lepton+jets analysis, for a mass of 1.0 TeV, values No excess above the SM expectations is observed. The two final of (σ(pp → T qt/b) × B(T → tH0) greater than 0.8 and 0.7 pb states (dilepton and trileptons) are combined to obtain the final were excluded assuming left- and right-handed coupling of the results. For the coupling parameter κT between 0.1–1.6, the 95% T quark to standard model fermions, respectively [119]. For the CL upper limits on the production cross section times branching all-hadronic analysis, upper limits between 0.31 and 0.93 pb were fraction into Zt is between 0.16–0.18 (0.03–0.05) pb at T quark obtained on (σ(pp → T qt/b) × B(T → tH0) for T quark masses mass of 0.7 (2) TeV. in the range 1.0-1.8 TeV [118]. Combination of T → bW/tZ/tH0: T → tZ: Most of the analyses described above targeted an individual Both ATLAS and CMS searched√ for T quarks that decay exclu- decay mode of the T quark, with 100% branching ratio to either sively into tZ in pp collisions at s = 13 TeV. No excesses were bW , tZ or tH0 and were optimized accordingly. However, they found in either search. have varied sensitivity to all three decay modes and the results ATLAS performed a search [120] for optimized pair production can be interpreted as a function of branching ratios to each of the of vector-like top quarks decaying into tZ where the Z boson sub- three decay modes, with the total adding up to unity (B(tH) + sequently decays into neutrino pairs utilizing 36.1 fb−1 of data. B(tZ) + B(W b) = 1). The search selected events with one lepton, multiple jets, and sig- Combinations of analyses have been performed by both AT- nificant missing transverse momentum. No significant excesses LAS and CMS. The limits set by ATLAS searches in W (`ν)b = were found and lower limits on the mass of a vector like top quark X,H(bb)b + X,Z(νν),Z(``)t/b + X, dileptons with same-sign were placed, excluding masses below 0.87 TeV (weak-isopsin sin- charge, trileptons, all-hadronic final states have been combined glet), 1.05 TeV (weak-isospin doublet), and 1.16 TeV ( pure tZ and the results obtained for various sets of branching fractions for mode). T quark decays to bW , tH0 and tZ are shown in Fig. 93.3 (left). Another search by ATLAS for pair produced T decaying to tZ In the combined analysis, ATLAS sets lower T quarks mass limit has been carried out by reconstructing the high transverse mo- of 1.31 TeV for all possible values of the branching fractions to the mentum Z boson from a pair of opposite-sign same-flavor lepton, three decay modes [107,120,124]. In Fig. 93.3, exclusion is shown using events with two or three charged leptons [121]. The final in the plane of B(T → Ht) versus B(T → W b), for different values analysis is based on three final signatures. In the trilepton events, of the T quark mass. The default branching ratio values for the at least one b-tagged jet is required and ST is used as the discrim- weak-isospin singlet and doublet cases are also shown in Fig. 93.3 inating variable. In events with two leptons, at least 2 b jets are as yellow circle and star symbols respectively. Assuming a weak requested and those with zero or one high pT top-tagged jet, use isospin (T ,B) doublet and |VT b| << |VtB |, T quark mass below HT as the discriminator. The second dilepton analysis with two 1.37 TeV is excluded. top-tagged high pT jets focuses on hadronically decaying heavy CMS analysis for pair production of T , combined three channels resonances and the invariant mass of the Z boson and the highest with lepton final states: single lepton, two leptons with the same pT b-tagged jet if found to be a good discriminating variable. No sign of the electric charge (SS), or at least three leptons (trilep- excess was observed over the background expectations. The com- ton) [125]. For various combinations of branching fractions for T bined analysis yields a lower limit on T quark mass of 1.03 TeV quark decays to bW , tH0 and tZ, the combined results exclude (1.21 TeV) in the singlet (doublet) model or 100% branching ra- T quarks with masses below 1.14–1.3 TeV and are shown in Fig- tio for T → tZ a lower limit on the T quark mass of 1.34 TeV is ure 93.3 (right). Single lepton events are classified into 16 signal obtained. categories and 6 background control regions based on multiplicity 0 ATLAS has subsequently carried out a search [122] for singly of b-tagged, high pT H and W -tagged jets. The discriminat- 0 produced T quark decaying to tZ where the Z boson decays into ing variables are HT for H -tagged and the minimum invariant −1 neutrino pairs. The search is carried out using 36.1 fb of data in mass constructed from the lepton and the b jet, min[Mlb], for events with two different final state signatures: one with jets, and zero H0-tagged events. For the same-sign dilepton and trilep- significant missing pT (0L) and the other with single lepton, jets ton analyses, the non-prompt backgrounds due to misidentified and missing pT (1L). Events are divided into signal and dedicated jets and leptons are derived from data control regions. In the W +jets and tt background control regions. The sensitivity to T trilepton analysis the ST variable is used as the signal discrimi- quark signal is extracted using distributions of missing pT for nator binned in four categories based on the lepton flavor combi- 1L and the distributions of transverse mass T quark constructed nations (eee, eeµ, eµµ, µµµ). The single lepton analysis is most from missing pT and the high pT large-radius top-tagged jet for sensitive for tHbW and W bW b decay modes, within the the SS 0L analysis. The is no excess found over the expected background dilepton analysis tHtH and tHtZ have the best efficiency and for and lower limits on the production of T singlets is obtained as a trileptons the tZtZ and tHtZ decays modes have the highest ef- function of the left- and right-handed couplings cL,W and cR,W ficiency. CMS excludes singlet (doublet) T quark masses below to top quarks and W bosons, where cW above 0.7 are excluded 1.2 (1.28) TeV. Masses below 800 GeV were excluded in previous for T quark mass of 1.4 TeV. The limits on cW are also recasted searches. For T quark masses in the range 0.8–1.8 TeV, cross sec- into expected and observed 95% CL upper limits for the mixing tions smaller than 30.4–9.4 fb (21.2–6.1 fb) are excluded for the angle (θL) of a singlet T with the top quark. singlet (doublet) scenario. CMS searched [123] for single production of T quarks decaying Another inclusive search for pair produced T in all-hadronic into tZ with the Z boson decaying to pairs of charged leptons final state [106] has been performed by CMS using the boosted (electrons and muons) and the top quark decaying hadronically event shape tagger (BEST) neural network technique to classify using 35.9 fb−1 of data. Limits were placed on T quarks with jets in six categories W , Z, H0, t, b, and light. This search does masses between 0.7 and 1.7 TeV excluding the product of cross not focus on a given VLQ mode, but on various combinations of section and branching fraction above values of 0.27 to 0.04 pb. the boson and quark jets in the final state. Anti-kT jets with Additionally, limits on the product of cross section and branching a distance parameter of 0.8 are used. The BEST NN algorithm fractions for a Z0 boson decaying into tZ were set between 0.13 simultaneously classifies jets according to heavy object type. For and 0.06 pb for Z0 boson masses in the range from 1.5 to 2.5 TeV. each of the six particle hypothesis, it boosts each jet constituent Similar searches by ATLAS for singly produced T decaying to into corresponding frame along jet momentum direction, and cal- 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 965

culates event shape and angular variables in the boosted frame, above the expected background and yielded a lower limit on the with the expectation that when boosting to the correct rest frame, B quark mass of 0.73 TeV for BR(B → W t) = 1. jet constituents will be isotropic and show the expected N-prong CMS [116] also searched for pair√ production of both TT and BB structure of the decaying object in its rest frame. A neural net- with collisions from 2.5 fb−1 of s = 13 TeV data. The analysis work is trained using the event shape and angular variables in the searches for events with one high pT lepton , multiple jets, and boosted frame to classify jets according to one of those six possi- highly boosted W or Higgs bosons decaying hadronically. The bilities (W, Z, H, t, b, or light). The analysis bins the events into analysis focuses on pair production and selects events with either 126 categories depending on the number of W, Z, H, t, b, or light a boosted W or Higgs candidate and then proceeds to search for jets in the final state with a maximum of four such objects. For anomalous production in excess of standard model production. AK8 each category HT , the scalar sum of pT of all AK8 jets, is used Seeing no significant excesses CMS then proceeded to set limits as the signal discriminator. A scan over a combination of vari- in many different interpretations. The strongest was from the the ous branching fractions is also performed. This search excludes T B → W t interpretation leading to excluding heavy vector-like B quark masses in the range 0.74–1.37 TeV for the tH decay mode quark less than 0.73 TeV. in the NN analysis. The all-hadronic inclusive analysis [106] performed by CMS us- An inclusive search for VLQs has been carried out by CMS ing the BEST NN technique to classify W/Z/H0/t/b/light jets targeted at heavy T quarks decaying to any combination of bW , also gives exclusion limits on B quark production for various com- tZ, or tH0 is described in [114]. Selected events have at least binations of branching fractions for decays to tW, bZ, bH0. By one isolated charged lepton. Events were categorized according considering categories based on various combinations of the boson to number and flavour of the leptons, the number of jets, and and quark jets in the final state it excludes B quarks with masses the presence of hadronic vector boson and top quark decays that up to 1230 GeV, for B decays to tW with a 100% branching frac- are merged into a single jet. The use of jet substructure to iden- tion. tify hadronic decays significantly increases the acceptance for high Electroweak production of single heavy B+b productions has T quark masses. No excess above standard model backgrounds been studied by CMS in the decay to tW with lepton+jets final was observed. Limits on the pair production cross section of the state [127]. Single lepton events with hadronic jets, including a new quarks are set, combining all event categories, for all com- forward jet, missing pT are selected and divided into 10 different binations of branching fractions into the three final states. For categories based on lepton flavor (e/µ), top-tagged, W -tagged, 0 T quarks that exclusively decay to bW/tZ/tH , masses below and 0/1/2 b-tagged jets. B quark mass mreco is fully recon- 0.70/0.78/0.71 TeV are excluded. structed from lepton, jets, and missing pT , where the neutrino 0 four-momentum is computed using the missing pT and the W 93.2.3.2 Searches for B quarks that decay to W , Z and H mass constraint (assuming massless ν). For events within top- bosons tagged category, the high pT top-tagged hadronic jet and the lep- ATLAS and CMS have performed searches for pair production tonically decaying W boson is used to compute m . The m 0 reco reco of heavy B quarks which subsequently decay to W t, bZ or bH . distribution is used as the signal discriminator. In the absence of The searches have been carried out in final states with single lep- a excess over the expected SM background, the exclusion limits tons, dileptons (with same charge or opposite charge), multilep- on the production cross section is for B quark masses between 0.7- tons, as well as in fully hadronic final states. 2 TeV varies between 0.3 to 0.03 pb. In addition, B quarks with left-handed couplings and a relative width of 10, 20, and 30% are B → bH0X: excluded for masses below 1.49, 1.59, and 1.66 TeV respectively. Using 36.1 fb−1 of data, ATLAS has performed a search for pair produced VLQs with all-hadronic final state signature [113]. B → bZX: While this analysis provides exclusion limits for all third genera- A search by CMS [128] for the pair-production of a heavy B tion VLQs, it provides the strongest results for B → bH0 decay quark and its antiparticle has been performed, where one the mode and excludes pure B → bH0 decays for B masses upto heavy B quark decays to bZ. Events with a Z boson decaying 1.01 TeV. The limits are also casted as a two-dimensional plane to e+e− or µ+µ− and at least one b jet are selected. The signal of branching ratio values of B → bH0 vs. B → W b. This analysis from B → bZ decays are expected to appear as a local enhance- required the presence of high pT jets and multiple b tags. It used ment in the bZ mass distribution. No such enhancement was a multi-class DNN to classify jets arising from W , Z, H0 bosons found and B quarks that decay 100% into bZ are excluded be- and top-quarks. In addition, the matrix element method is used low 0.70 TeV. This analysis also set upper limits on the branching to compute the likelihood for the event to arise from a particu- fraction for B → bZ decays of 30-100% in the B quark mass range lar VLQ final state and used to construct the final discriminator. 0.45-0.70 TeV. A complementary search has been carried out by To increase the performance sensitivity, processes with the same ATLAS for new heavy quarks decaying into a Z boson and a b- number of top quarks, W /Z bosons, and H0 Higgs bosons are quark [129]. Selected dilepton events contain a high transverse combined into a single hypothesis. momentum Z boson that decays leptonically, together with two b jets. If the dilepton events have an extra lepton in addition B → W tX: to those from the Z boson, then only one b-jet is required. No A search for B → tW in B pair produced events has been per- significant excess of events above the standard model expectation formed by the ATLAS experiment [107] using lepton+jets events was observed, and mass limits were set depending on the assumed with one hadronically decaying W and one leptonically decaying branching ratios, as shown in Fig. 93.4. In a weak-isospin singlet −1 √ W utilizing 36.1 fb of data at s = 13 TeV. The search was scenario, a B quark with mass lower than 0.65 TeV was excluded, optimized for T production decaying into W b. Since the analysis while for a particular weak-isospin doublet scenario, a B quark was optimized for T → W b rather than W t decays the analy- with mass lower than 0.73 TeV was ruled out. sis does not reconstruct the full B mass. As discussed earlier, the ATLAS has searched for the electroweak production of single hadronically and leptonically decaying heavy quarks were required B quarks, which is accompanied by a b jet and a light jet [129]. to have similar reconstructed masses (within 300 GeV). The inter- The dilepton selection for double B production was modified for pretation of the T → W b in the context of B → tW production the single B production study by requiring the presence of an led to the exclusion of heavy B like VLQs for masses less than additional energetic jet in the forward region. An upper limit of 1.25 TeV and 1.08 TeV, assuming a 100% branching fraction to 200 fb was obtained for the process σ(pp → Bbq) × B(B → Zb) tW or SU(2) singlet B scenario, respectively. with a heavy B quark mass at 0.70 TeV. This search indicated −1 √ A similar search by CMS [126], using 19.8 fb of s = 8 TeV that the electroweak mixing parameter XBb below 0.5 is neither data, selected events with one lepton and four or more jets, with expected or observed to be excluded for any values of B quark at least one b-tagged jet, significant missing pT , and further cat- mass. egorizes them based on the number of jets tagged as arising from 0 0 the decay of boosted W , Z or H bosons. The ST distributions Combination of B → tW/bZ/bH : of the events in different categories showed no excess of events The ATLAS experiment has combined the various analyses tar- 966 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

35.9 fb-1 (13 TeV) 1 1400 ATLAS Ht) 1 1140 0.9 -1 1380 (bW) CMS → s = 13 TeV, 36.1 fb Β 1300 0.8 VLQ combination 1360 0.8 1160 1150 BR(T 0.7 Expected limit 1340 1200 0.6 1320 SU(2) doublet 0.6 1190 1180 1180 0.5 1300 SU(2) singlet 1100 0.4 1280 0.4 1240 1230 1220 1220 95% CL mass limit [GeV]

1300 1250 0.3 1260 1000 0.2 1240 0.2 1280 1260 1260 1250 1250

1350 0.1 1220 900 0 1300 1290 1290 1280 1270 1270 1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 95% CL observed T quark mass limit (GeV) 0 0.2 0.4 0.6 0.8 1 BR(T → Wb) Β(tH)

Figure 93.3: Left panel: observed limits on the mass of the T quark in the plane of B(T → tH0) versus B(T → bW ) from a combination [124] of all ATLAS searches for TT production. The markers indicate the default branching ratios for the SU(2) singlet and doublet scenarios. Right panel: the observed lower limits on the T quark mass (in GeV), from CMS searches after combining all lepton channels [125], for various branching fraction scenarios. B(T → W b) + B(T → tZ) + B(T → tH0) = 1 is assumed.

geted for specific decay modes to obtain the most sensitive lim- number of b jets, HT , and missing pT in the event, corresponded its on the pair production of B quarks [107, 108, 124]. Various to a lower mass limit on X5/3 of 1.19 TeV [136, 137]. searches (W (`ν)t + X,Z(``)t/b + X, same sign charge dilepton Searches for X5/3 using leptons+jets final state signatures are events, trilepton events, and all-hadronic) are combined to ob- based on either full or partial reconstruction of the T mass from tain lower limits on the mass of the B quark in the plane of the lepton, jets (including b jets) and missing pT . The CMS BR(B → W t) vs BR(B → bH). The searches were optimized for search [133, 138] also utilized jet substructure techniques to iden- 100% branching fractions and hence are most sensitive at large tify boosted X5/3 topologies. The discriminating variable used 0 BR(B → W t), and also at large BR(B → bH ). For all possi- was the mass constructed from the lepton and b-tagged jet, M(`,b), ble values of branching ratios in the three decay modes tW , bZ, which corresponds to the visible mass of leptonically decaying top or bH0, the lower limits on the B quark mass was found to be quark. To optimize the search sensitivity, the events were fur- 1.03 TeV and shown in Fig. 93.4 (left) as a function of both B ther separated into categories based on lepton flavor (e, µ), the mass and branching ratio. number of b-tagged jets, the number of W-tagged jets, and the CMS combined three channels with lepton final states: sin- number of t-tagged jets. In the absence of a signal, the CMS anal- gle lepton, two leptons with the same sign of the electric charge ysis excluded X5/3 quark masses with right-handed (left-handed) (SS), or at least three leptons (trilepton) [125]. For various com- couplings below 1.32 (1.30) TeV [138]. Combining the lepton+jets binations of branching fractions for B quark decays to tW , bH0 with the same-sign leptons analyses leads to a slight improvement and bZ, the combined results exclude b quarks with masses below and excludes X5/3 quark masses with right-handed (left-handed) 0.91–1.24 TeV and are shown in Figure 93.4 (right) the details are couplings below 1.33 (1.30) TeV. provided earlier in subsection 93.2.3.1. The single lepton analy- The ATLAS lepton+jets search for X5/3 utilized events with sis is most sensitive for tHbW and W bW b decay modes, within high pT W bosons and b jets. The search described earlier for the the SS dilepton analysis tHtH and tHtZ have the best effi- T pair production, with T → W b decays, can be reinterpreted as ciency and for trileptons the tZtZ and tHtZ decays modes have a search for X → tW . This analysis excluded X5/3 with masses the highest efficiency. CMS excludes singlet (doublet) B quark below 1.25 TeV [107]. masses below 1.17 (0.94) TeV. Masses below 800 GeV were ex- The single X5/3 production cross section depends on the cou- cluded in previous searches. For B quark masses in the range pling constant λ of the tW X vertex. ATLAS has performed an 0.8–1.8 TeV, cross sections smaller than 40.6-–9.4 fb (101-–49.0 analysis of same-sign dileptons which includes both the single and fb) are excluded for the singlet (doublet) scenario. pair production. This analysis led to a lower limit on the mass of the X of 0.75 TeV for both values of λ = 0.5 and 1.0 [139]. 93.2.3.3 Searches for top-partner quark X 5/3 5/3 Single heavy X +t production has been studied by CMS Searches for a heavy top vector-like quark X , with exotic 5/3 5/3 in the decay to tW with lepton+jets final state [127]. The de- charge ±5/3, such as that proposed in Refs. [131,132], have been scription of the analysis is provided earlier in the discussion of performed by both ATLAS and CMS [107, 133]. B → W tX decays, where the reconstructed mass of X5/3, mreco The analyses assumed pair-production or single-production of distribution is used as the signal discriminator. In the absence of X5/3 with X5/3 decaying with 100% branching fraction to to tW . a excess over the expected SM background, the exclusion limits Searches for X5/3 have been performed using two final state sig- on the production cross section is for X5/3 quark masses between natures: same-sign leptons and lepton+jets. 0.7–2 TeV varies between 0.3 to 0.03 pb, depending on the width of The analysis based on searching for same-sign leptons, from X5/3 between 1–10%. In addition, X5/3 quarks with left-handed the two W bosons from one of the X5/3, has smaller backgrounds couplings and a relative width of 10, 20, and 30% are excluded compared to the lepton+jets signature. Requiring same-sign lep- for masses below 0.92, 1.3, and 1.45 TeV respectively. tons eliminates most of the standard model background processes, leaving those with smaller cross sections: tt, W, ttZ, W W W , and 93.2.4 Heavy resonances decaying to VLQ same-sign WW . In addition, backgrounds from instrumental ef- CMS has performed search for VLQ production in the decay of fects due to charge misidentification were considered. Assuming massive resonances such as Z0 and W 0 bosons. 0 pair production of X5/3, the analyses by CMS using HT as the dis- Z → tT : Specifically searches are presented by CMS in 0 criminating variable restrict the X5/3 mass to be higher than 1.16 Refs. [140] and [141] for massive spin-1 Z resonances decaying (1.10) TeV for a right (left) handed chirality particle [133–135]. to a top quark and heavy VLQ top quark partner T . The results The limits obtained by ATLAS, by classifying the signal region by of this search for a heavy spin-1 resonance are interpreted in the 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 967

35.9 fb-1 (13 TeV) 1 1400 ATLAS (tW) Hb) 1 1240 0.9 1000 -1 1350 CMS s = 13 TeV, 36.1 fb Β → 1300 0.8 VLQ combination 1300 Expected limit 0.8 1220 1200

BR(B 0.7 1250 1200 SU(2) (T,B) doublet 0.6 SU(2) (B,Y) doublet 1200 1100 0.6 1190 1180 1180 SU(2) singlet 0.5 1150 1100 0.4 1100 0.4 1150 1150 1140 1160 95% CL mass limit [GeV] 0.3 1050 1000 1200 0.2 1000 0.2 1080 1080 1080 1070 1080

0.1 1300 950 900 0 960 960 950 920 910 910 900

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 95% CL observed B quark mass limit (GeV) 0 0.2 0.4 0.6 0.8 1 BR(B → Wt) Β(bH)

Figure 93.4: Observed limits on the mass of the B quark in the plane of BR(B → bH0) versus BR(B → tW ) from ATLAS searches [124] on the left panel, and CMS searches [125] on the right panel, for BB production. B(B → Hb) + B(B → bZ) + B(B → Wt) = 1 is assumed. The yellow markers indicate the branching ratios for the SU(2) singlet and doublet scenarios.

-1 -1 5 27 fb & 36 fb (13 TeV) 10 ←→ Low High String 4 [pb] 10 mass mass Excited quark Α Axigluon/coloron 3 CMS Β Scalar 10 2 Color-octet scalar (k = 1/2) s σ 2 10 W' Z' 10 DM mediator RS graviton 1 10−1 10−2 −3 95% CL limits 10 gluon-gluon 10−4 quark-gluon quark-quark −5 10 1 2 3 4 5 6 7 8 9 Resonance mass [TeV]

Figure 93.5: Observed 95% C.L. limits on σ × B × A for string resonances, excited quarks, axigluons, colorons, E6 diquarks, s8 0 0 resonances, W and Z bosons, and Randall-Sundrum Gravitons gKK from [130]. context of two different models. In the G? model which predicts tH0, tZ, W b are characterized by the presence of two top quark ˜ ˜ 0 0 0 ten VLQs (T,B, T, B,T5/3,T2/3,T ,B ,B−1/3,B−4/3) with the decays and a boson (H /Z). A search [141] by CMS, optimized mass relationship M(T5/3) = M(T2/3) = M(T )cos(φL). For the for T → tH/Zt decays is carried out in the lepton+jets final benchmark scenario [142], cos(φL)=0.84 and the branching frac- state using a dataset corresponding to an integrated luminosity tions T → tH0, tZ, W b are 0.25, 0.25 and 0.5 respectively. The ρ0 of 35.9 fb−1. Jet substructure techniques are used to identify (or 0 model predicts a multiple of four new VLQs T,B,X2/3,X5/3, tag) the high pT large-radius jets originating from H , Z bosons and in the benchmark scenario [143], the branching fractions and merged top quarks. The mass of the Z0 boson is used as T → tH0, tZ, W b are 0.5, 0.5 and 0 respectively. the signal discriminator and constructed using H0 or Z-tagged Two of the three different decays of the Z0 → tT with T → dijet, the hadronic and leptonic top quark four vectors. For the 968 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

leptonic top quark reconstruction (t → b`ν`), the neutrino four colorons below 6.1 TeV, and color-octet scalars below 3.4 TeV. vector is obtained from the event missing p using the W boson T 93.3 Conclusions mass constraint. While the high pT hadronic top quark jets from decays of massive Z0 bosons are mostly merged and identified As the above analyses have demonstrated, there is already sub- by top tagging techniques, those from T decays maybe resolved. stantial sensitivity to possible new particles predicted to accom- 0 The reconstructed Z0 candidate events are classified into six dif- pany the H in dynamical frameworks of electroweak symmetry ferent categories requiring the presence of either H0-tagged jet breaking. No significant hints of any deviations from the standard with 2 b-tagged subjets or one b-tagged subjet or a Z-tagged bo- model have been observed, and limits typically at the scale of a son, each with either zero or one top-tagged jet. This search does few hundred GeV to a few TeV are set. 0 not observe any significant deviation in data over the expectation Given the need to better understand the H and to determine in from standard model backgrounds. Within the context of the G? detail how it behaves, such analyses continue to be a major theme model, for a T mass of 1.2 (1.5), this search excludes G? [142] of Run 2 the LHC, and we look forward to increased sensitivity as resonances with masses between 1.5–2.3 (2.0–2.4) TeV. a result of the higher luminosity at the increased center of mass energy of collisions. The search in the all-hadronic final state is based on a 2.6 fb−1 dataset [140], and optimized for T → W b decays. Jet substructure References techniques are deployed for tagging jets from high pT W boson [1] G. Aad et al. (ATLAS), Phys. Lett. B716, 1 (2012), and top quarks. Events are categorized into two groups based on [arXiv:1207.7214]. the presence of b-tagged subjet in the top-tagged jet. Multijet [2] S. Chatrchyan et al. (CMS), Phys. Lett. B716, 30 (2012), background estimation is challenging and determined using side- [arXiv:1207.7235]. bands defined by inverting the b tagging requirement. Upper limits on cross section for Z0 → tT are obtained in the range of [3] S. Weinberg, Physica A96, 1-2, 327 (1979). 0.13–10 pb. With additional data, this search has the potential to [4] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984). exclude scenarios in composite Higgs and extra dimension models. [5] H. Georgi, Nucl. Phys. B266, 274 (1986). 0 0 W → T b/Bt: W bosons are predicted to decay to VLQ third [6] R. S. Chivukula, in “Flavor physics for the millennium. Pro- generation partners T,B quarks within composite Higgs and ceedings, Theoretical Advanced Study Institute in elemen- warped extra dimensional models [144]. In the benchmark sce- tary particle physics, TASI 2000, Boulder, USA, June 4-30, 0 narios of this framework, W decays to T b or Bt are equally dis- 2000,” 731–772 (2000), [hep-ph/0011264]. tributed and the subsequent VLQ decays T → tH and B → bH0 [7] R. S. Chivukula, M. J. Dugan and M. Golden, Phys. Rev. each are assumed to have a branching fraction of 0.5. The search D47, 2930 (1993), [hep-ph/9206222]. for W 0 → T b/bH0 → tbH0 is performed using a sample of 35.9 fb−1 by CMS [145] in the final state with all-hadronic de- [8] S. Weinberg, Phys. Rev. D13, 974 (1976), [Addendum: cays of both the Higgs boson (H0 → bb) and the top quark. Both Phys. Rev.D19,1277(1979)]. the H0 boson and the top quark are expected to be boosted in [9] L. Susskind, Phys. Rev. D20, 2619 (1979). 0 the decay of a heavy W , and hence jet substructure techniques, [10] K. Lane (2002), [hep-ph/0202255]. including subjet b-tagging and double b-tagging are deployed to [11] C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003), identify the H0-tagged and the top-tagged jets. The three par- [Erratum: Phys. Rept.390,553(2004)], [hep-ph/0203079]. ticle mass mtbH0 , is used as the signal discriminant to observe the W 0 resonance. There is no excess observed in data above the [12] R. Shrock, in “The origin of mass and strong coupling expected SM background. This search excludes W 0 production gauge theories. Proceedings, 5th International Workshop, cross section above 0.01–0.43 pb for masses between 1.5–4.0 TeV. SCGT’06, Nagoya, Japan November 21-24, 2006,” 227–241 (2007), [hep-ph/0703050]. 93.2.5 Colorons and Colored Scalars [13] E. Eichten et al., Rev. Mod. Phys. 56, 579 (1984), [Adden- These particles are associated with top-condensate and top- dum: Rev. Mod. Phys.58,1065(1986)]. seesaw models, which involve an enlarged color gauge group. The [14] E. Eichten et al., Phys. Rev. D34, 1547 (1986). new particles decay to dijets, tt¯, and b¯b. Direct searches for colorons, color-octect scalars and other [15] R. S. Chivukula and V. Koulovassilopoulos, Phys. Lett. B309, 371 (1993), [hep-ph/9304293]. heavy objects decaying to qq, qg, qq√, or gg has been performed us- ing LHC data from pp collisions at s =7, 8 and 13 TeV. Based on [16] R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D87, the analysis of dijet events from a data sample corresponding to a 9, 095001 (2013), [arXiv:1211.1083]. −1 √ luminosity of 19.6 fb , at s = 8 TeV the CMS experiment ex- [17] B. Holdom, Phys. Lett. 150B, 301 (1985). cluded pair production of colorons with mass between 1.20 − 3.60 [18] K. Yamawaki, M. Bando, and K.-i. Matumoto, Phys. Rev. and 3.90−4.08 TeV [146]. Analyses of inclusive 8- and 10-jet final Lett. 56, 1335 (1986). states with low missing transverse momentum by CMS [147], set limits in several benchmark models. Colorons (axigluons) with [19] T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, masses between 0.6 and 0.75 (up to 1.15) TeV were excluded, and Phys. Rev. Lett. 57, 957 (1986). gluinos in R-parity violating supersymmetric scenarios were ruled [20] T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D35, out from 0.6 up to 1.1 TeV. 774 (1987). A search for pair-produced colorons based on an integrated lu- [21] T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D36, −1 √ minosity of 5.0 fb at s = 7 TeV by CMS excluded colorons 568 (1987). with masses between 0.25 TeV and 0.74 TeV, assuming colorons [22] E. Eichten, K. Lane and A. Martin (2012), decay 100% into qq [148]. This analysis was based on events with [arXiv:1210.5462]. at least four jets and two dijet combinations with similar dijet mass. Color-octet scalars (s8) with masses between 1.20−2.79 TeV [23] D. B. Kaplan and H. Georgi, Phys. Lett. 136B, 183 (1984). were excluded by CMS [146], and below 2.7 TeV by ATLAS [149]. [24] D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. These studies have now been extended to take advantage of the 136B, 187 (1984). increased center-of-mass energy during√ Run 2 of the LHC. Using [25] M. E. Peskin, Nucl. Phys. B175, 197 (1980). the 35.9 fb−1 of data collected at s =13 TeV, searches for narrow [26] J. Preskill, Nucl. Phys. B177, 21 (1981). resonances have been performed by CMS. An analysis of the dijet invariant mass spectrum formed using wide jets [130, 150, 151], [27] See “Status of Higgs Boson Physics” review in this volume. separated by ∆ηjj ≤ 1.3, led to limits on new particles decaying [28] R. Barbieri and A. Strumia, in “4th Rencontres du Viet- to parton pairs (qq, qg, gg). Specific exclusions on the masses nam: Physics at Extreme Energies (Particle Physics and of colorons and color-octet scalars were obtained and are shown Astrophysics) Hanoi, Vietnam, July 19-25, 2000,” (2000), in Fig. 93.5. Exclusions have been obtained for axigluons and [hep-ph/0007265]. 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0 969

[29] N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. [64] PDG review of Wprime boson in this volume. B513, 232 (2001), [hep-ph/0105239]. [65] M. Aaboud et al. (ATLAS), Phys. Lett. B, 68 (2019), [30] N. Arkani-Hamed et al., JHEP 08, 020 (2002), [hep- [arXiv:1903.06248]. ph/0202089]. [66] CMS Collaboration, Technical Report CMS-PAS-EXO-19- [31] N. Arkani-Hamed et al., JHEP 07, 034 (2002), [hep- 019, CERN, Geneva (2019), URL http://cds.cern.ch/ ph/0206021]. record/2684757. [32] M. Schmaltz and D. Tucker-Smith, Ann. Rev. Nucl. Part. [67] G. Aad et al. (ATLAS), Phys. Rev. D90, 5, 052005 (2014), Sci. 55, 229 (2005), [hep-ph/0502182]. [arXiv:1405.4123]. [33] K. Agashe et al., Phys. Lett. B641, 62 (2006), [hep- [68] M. Aaboud et al. (ATLAS), JHEP 01, 055 (2018), ph/0605341]. [arXiv:1709.07242]. [34] P. Sikivie et al., Nucl. Phys. B173, 189 (1980). [69] CMS Collaboration, JHEP 0217, 48 (2017). [35] B. Bellazzini, C. Csáki and J. Serra, Eur. Phys. J. C74, 5, [70] ATLAS Collaboration, ATLAS-CONF-2016-014 (2016). 2766 (2014), [arXiv:1401.2457]. [71] CMS Collaboration, JHEP 0717, 001 (2016). [36] R. Essig et al., JHEP 09, 085 (2017), [arXiv:1707.03399]. [72] M. Aaboud et al. (ATLAS), Phys. Rev. D. 99, 092004 [37] Z. Chacko, H.-S. Goh and R. Harnik, Phys. Rev. Lett. 96, (2019), [arXiv:1902.10077]. 231802 (2006), [hep-ph/0506256]. [73] A. M. Sirunyan et al. (CMS), JHEP 04, 031 (2019), [38] R. S. Chivukula, A. G. Cohen and K. D. Lane, Nucl. Phys. [arXiv:1810.05905]. B343, 554 (1990). [74] ATLAS Collaboration, ATLAS-CONF-2019-007 (2019). [39] V. A. Miransky, M. Tanabashi and K. Yamawaki, Mod. [75] CMS Collaboration, CMS-PAS-EXO-19-012 (2019). Phys. Lett. A4, 1043 (1989). [76] M. Aaboud et al. (ATLAS), Accepted by Phys. Rev. [40] W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D41, [arXiv:1906.05609]. 1647 (1990). [77] V. Khachatryan et al. (CMS), Phys. Lett. B770, 278 (2017), [41] C. T. Hill, Phys. Lett. B266, 419 (1991). [arXiv:1612.09274]. [42] B. A. Dobrescu and C. T. Hill, Phys. Rev. Lett. 81, 2634 [78] V. Khachatryan et al. (CMS), Phys. Lett. B755, 196 (2016), (1998), [hep-ph/9712319]. [arXiv:1508.04308]. [43] R. S. Chivukula et al., Phys. Rev. D59, 075003 (1999), [79] S. Chatrchyan et al. (CMS), JHEP 05, 108 (2014), [hep-ph/9809470]. [arXiv:1402.2176]. [44] S. Dimopoulos and L. Susskind, Nucl. Phys. B155, 237 [80] V. Khachatryan et al. (CMS), JHEP 02, 122 (2016), (1979), [2,930(1979)]. [arXiv:1509.06051]. [45] E. Eichten and K. D. Lane, Phys. Lett. 90B, 125 (1980). [81] A. M. Sirunyan et al. (CMS), JHEP 08, 029 (2017), [46] D. B. Kaplan, Nucl. Phys. B365, 259 (1991). [arXiv:1706.04260]. [47] T. Appelquist, M. Piai and R. Shrock, Phys. Rev. D69, [82] A. M. Sirunyan et al. (CMS), Phys. Lett. B777, 39 (2018), 015002 (2004), [hep-ph/0308061]. [arXiv:1708.08539]. [48] R.S. Chivukula, B.A. Dobrescu, and E.H. Simmons, Phys. [83] CMS Collaboration, Technical Report CMS-PAS-JME-15- Lett. B401, 74 (1997). 002, CERN, Geneva (2016), URL http://cds.cern.ch/ record/2126325. [49] Y. Grossman and M. Neubert, Phys. Lett. B474, 361 (2000), [hep-ph/9912408]. [84] G. Aad et al. (ATLAS), Phys. Lett. B788, 347 (2019), [arXiv:1807.10473]. [50] S. J. Huber and Q. Shafi, Phys. Lett. B498, 256 (2001), [85] G. Aad et al. (ATLAS), Eur. Phys. J. C75, 4, 165 (2015), [hep-ph/0010195]. [arXiv:1408.0886]. [51] T. Gherghetta and A. Pomarol, Nucl. Phys. B586, 141 [86] G. Aad et al. (ATLAS), Phys. Lett. B743, 235 (2015), (2000), [hep-ph/0003129]. [arXiv:1410.4103]. [52] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B719, [87] D. Pappadopulo et al., JHEP 09, 060 (2014), 165 (2005), [hep-ph/0412089]. [arXiv:1402.4431]. [53] G. F. Giudice et al., JHEP 06, 045 (2007), [hep- [88] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 5, 052008 ph/0703164]. (2018), [arXiv:1601.04027]. [54] R. S. Chivukula and H. Georgi, Phys. Lett. B188, 99 [89] A. M. Sirunyan et al. (CMS), Phys. Lett. B798, 134952 (1987). (2019), [arXiv:1906.00057]. [55] G. D’Ambrosio et al., Nucl. Phys. B645, 155 (2002), [hep- [90] M. Aaboud et al. (ATLAS), Phys. Lett. B, 91 (2017), ph/0207036]. [arXiv:1708.04445]. [56] K. Agashe et al. (2005), [hep-ph/0509117]. [91] M. Aaboud et al. (ATLAS), JHEP 03, 042 (2018), [57] T. Appelquist and R. Shrock, Phys. Lett. B548, 204 (2002), [arXiv:1710.07235]. [hep-ph/0204141]. [92] M. Aaboud et al. (ATLAS), Phys. Lett. B, 68 (2018), [58] K. Sakai, Nucl. Phys. B867, 429 (2013), [arXiv:1207.4057]. [arXiv:1806.10532]. [59] J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999), [Adv. [93] A. M. Sirunyan et al. (CMS), Phys. Rev. D97, 072006, Theor. Math. Phys.2,231(1998)], [hep-th/9711200]. [arXiv:1708.05379]. [60] For a review, see C. Csaki, J. Hubisz, and P. Meade, [94] A. M. Sirunyan et al. (CMS), JHEP 07, 075, hep-ph/0510275 (2005), and “Extra Dimensions” review in [arXiv:1802.09407]. this volume. [95] A. M. Sirunyan et al. (CMS), JHEP 09, 101, [61] C. Pica, PoS LATTICE2016, 015 (2016), [arXiv:1803.10093]. [arXiv:1701.07782]. [96] M. Aaboud et al. (ATLAS), Eur. Phys. J. C, 78 (2017), [62] T. Appelquist et al., Phys. Rev. D93, 11, 114514 (2016), [arXiv:1710.01123]. [arXiv:1601.04027]. [97] M. Aaboud et al. (ATLAS), Phys. Lett. B, 774 (2017), [63] PDG review of Zprime boson in this volume. [arXiv:1707.06858]. 970 93. Dynamical Electroweak Symmetry Breaking: Implications of the H0

[98] M. Aaboud et al. (ATLAS), JHEP B, 174 (2018), [125] A. M. Sirunyan et al. (CMS), JHEP 08, 177 (2018), [arXiv:1712.06518]. [arXiv:1805.04758]. [99] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C, 636, [126] CMS Collaboration, Technical report (2012). [arXiv:1707.01303.]. [127] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 90 (2019), [100] F. del Aguila et al., Nucl. Phys. B334, 1 (1990). [arXiv:1809.08597]. [128] V. Khachatryan et al. (CMS), Phys. Rev. D93, 11, 112009 [101] A. M. Sirunyan et al. (CMS), Phys. Lett. B779, 82 (2018), (2016), [arXiv:1507.07129]. [arXiv:1710.01539]. [129] G. Aad et al. (ATLAS), JHEP 11, 104 (2014), [102] CMS Collaboration, Technical report (2014). [arXiv:1409.5500]. [103] V. Khachatryan et al. (CMS), Phys. Rev. D93, 1, 012003 [130] A. M. Sirunyan et al. (CMS), JHEP 08, 130 (2018), (2016), [arXiv:1509.04177]. [arXiv:1806.00843]. [104] CMS Collaboration, Technical report (2012). [131] R. Contino and G. Servant, JHEP 06, 026 (2008), [105] S. D. Ellis, C. K. Vermilion and J. R. Walsh, Phys. Rev. [arXiv:0801.1679]. D80, 051501 (2009), [arXiv:0903.5081]. [132] J. Mrazek and A. Wulzer, Phys. Rev. D81, 075006 (2010), [arXiv:0909.3977]. [106] A. M. Sirunyan et al. (CMS) (2019), [arXiv:1906.11903]. [133] A. M. Sirunyan et al. (CMS), JHEP 08, 073 (2017), [107] M. Aaboud et al. (ATLAS), JHEP 10, 141 (2017), [arXiv:1705.10967]. [arXiv:1707.03347]. [134] CMS Collaboration, Technical report (2017). [108] G. Aad et al. (ATLAS), JHEP 08, 105 (2015), [135] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 112, 17, [arXiv:1505.04306]. 171801 (2014), [arXiv:1312.2391]. [109] A. M. Sirunyan et al. (CMS), Phys. Lett. B772, 634 (2017), [136] M. Aaboud et al. (ATLAS), JHEP 12, 039 (2018), [arXiv:1701.08328]. [arXiv:1807.11883]. [110] M. Aaboud et al. (ATLAS), JHEP 05, 164 (2019), [137] G. Aad et al. (ATLAS), Phys. Rev. D91, 11, 112011 (2015), [arXiv:1812.07343]. [arXiv:1503.05425]. [111] G. Aad et al. (ATLAS), Eur. Phys. J. C76, 8, 442 (2016), [138] A. M. Sirunyan et al. (CMS), JHEP 03, 082 (2019), [arXiv:1602.05606]. [arXiv:1810.03188]. [112] ATLAS collaboration (ATLAS), Technical report (2016). [139] G. Aad et al. (ATLAS), JHEP 10, 150 (2015), [arXiv:1504.04605]. [113] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 9, 092005 (2018), [arXiv:1808.01771]. [140] A. M. Sirunyan et al. (CMS), JHEP 09, 053 (2017), [arXiv:1703.06352]. [114] S. Chatrchyan et al. (CMS), Phys. Lett. B729, 149 (2014), [141] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C79, 3, 208 [arXiv:1311.7667]. (2019), [arXiv:1812.06489]. [115] V. Khachatryan et al. (CMS), JHEP 06, 080 (2015), [142] C. Bini, R. Contino and N. Vignaroli, JHEP 01, 157 (2012), [arXiv:1503.01952]. [arXiv:1110.6058]. [116] A. M. Sirunyan et al. (CMS), JHEP 11, 085 (2017), [143] D. Greco and D. Liu, JHEP 12, 126 (2014), [arXiv:1706.03408]. [arXiv:1410.2883]. [117] CMS Collaboration, cds.cern.ch/record/1709129 (2014). [144] N. Vignaroli, Phys. Rev. D89, 9, 095027 (2014), [arXiv:1404.5558]. [118] A. M. Sirunyan et al. (CMS), JHEP 04, 136 (2017), [arXiv:1612.05336]. [145] A. M. Sirunyan et al. (CMS), JHEP 03, 127 (2019), [arXiv:1811.07010]. [119] V. Khachatryan et al. (CMS), Phys. Lett. B771, 80 (2017), [arXiv:1612.00999]. [146] V. Khachatryan et al. (CMS), Phys. Rev. D91, 5, 052009 (2015), [arXiv:1501.04198]. [120] M. Aaboud et al. (ATLAS), JHEP 08, 052 (2017), [147] V. Khachatryan et al. (CMS), Phys. Lett. B770, 257 (2017), [arXiv:1705.10751]. [arXiv:1608.01224]. [121] M. Aaboud et al. (ATLAS), Phys. Rev. D98, 11, 112010 [148] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. 110, 14, (2018), [arXiv:1806.10555]. 141802 (2013), [arXiv:1302.0531]. [122] M. Aaboud et al. (ATLAS), JHEP 05, 041 (2019), [149] G. Aad et al. (ATLAS), Phys. Rev. D91, 5, 052007 (2015), [arXiv:1812.09743]. [arXiv:1407.1376]. [123] A. M. Sirunyan et al. (CMS), Phys. Lett. B781, 574 (2018), [150] A. M. Sirunyan et al. (CMS), Phys. Lett. B769, 520 (2017), [arXiv:1708.01062]. [Erratum: Phys. Lett.B772,882(2017)], [arXiv:1611.03568]. [124] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 121, 21, [151] CMS Collaboration, Technical report (2015). 211801 (2018), [arXiv:1808.02343]. 94. Grand Unified Theories 971

94. Grand Unified Theories

Revised August 2019 by A. Hebecker (Heidelberg U.) and J. superpartners are problematic: They also contribute to SUSY- Hisano (KMI, Nagoya U.). breaking Higgs mass parameters and thereby to the Higgs poten- tial, tending to raise the Z mass. As a result, the incarnation of 94.1 The standard model SUSY in terms of the minimal supersymmetric SM (MSSM) is be- The Standard Model (SM) may be defined as the renormalizable coming questionable. Turning the logic around, one may say that field theory with gauge group GSM = SU(3)C ×SU(2)L ×U(1)Y , compared to expectations based on the MSSM with superpart- with 3 generations of fermions in the representation ner masses below about 1 TeV, the measured Higgs mass value of 125 GeV is somewhat too high [10]. Independently, the LHC (3, 2) + (3¯, 1) + (3¯, 1) + (1, 2) + (1, 1) , (94.1) 1/3 −4/3 2/3 −1 2 has disfavored light colored superpartners (which does not imply

and a scalar Higgs doublet H transforming as (1, 2)1. Here and that all superpartners are heavy). These facts represent new hints below we use boldface numbers to specify the dimension of rep- for future work on SUSY GUTs or on GUTs without TeV-scale resentations of non-Abelian groups (in this case fundamental and supersymmetry. antifundamental) and lower indices for U(1) charges. The fields 94.2 Basic group theory and charge quantization of Eq. (94.1) should also be familiar as [Q, uc, dc, L, ec], with Q = (u, d) and L = (ν, e) being the quark and lepton SU(2)- 94.2.1 SU(4)C × SU(2)L × SU(2)R doublets and uc, dc, ec charge conjugate SU(2)-singlets.1 Espe- Historically, the first attempt at unification was the Pati-Salam cially after the discovery of the Higgs, this model is remarkably model with gauge group GPS = SU(4)C ×SU(2)L ×SU(2)R [11]. complete and consistent with almost all experimental data. It unifies SM fermions in the sense that one generation (plus A notable exception are neutrino masses, which are known to an extra SM singlet) now comes from the (4, 2, 1) + (4, 1, 2) of be non-zero but are absent in the SM even after the Higgs acquires GPS . This is easy to verify from the breaking pattern SU(4)C → its vacuum expectation value (VEV). The minimalist attitude is SU(3)C × U(1)B−L together with the identification of SM hy- to allow for the dimension-five operator (HL)2 [1], which induces percharge as a linear combination between B − L (baryon minus (Majorana) neutrino masses. In the seesaw mechanism [2–4] this lepton number) and the T3 generator of SU(2)R. This model ex- operator is generated by integrating out heavy singlet fermions plains charge quantization, that is, why all electric charges are (right-handed (r.h.) neutrinos). Alternatively, neutrinos can have integer multiples of some smallest charge in the SM. Concretely, Dirac masses if light singlet neutrinos are added to the SM spec- the 4 and 4 of SU(4)C identify lepton number as the 4th colour trum. and the tracelessness of the diagonal generator implies that quark Conceptual problems of the SM include the absence of a Dark charges are expressed in terms of 1/Nc fractions of lepton charges. Matter candidate, of a mechanism for generating the baryon However, GPS is not simple (containing three simple factors), and asymmetry of the universe, and of any reason for the observed thus it does not predict gauge coupling unification. smallness of the θ parameter of QCD (θQCD). In addition, the 94.2.2 SU(5) apparently rather complex group-theoretic data of Eq. (94.1) re- Since GSM has rank four (two for SU(3)C and one for SU(2)L mains unexplained. Together with the abundance of seemingly and U(1)Y , respectively), the rank-four group SU(5) is the min- arbitrary coupling constants, this disfavors the SM as a candidate imal choice for unification in a simple group [12]. The three SM fundamental theory, even before quantum gravity problems arise gauge coupling constants derive from a universal coupling αG at energies near the Planck mass MP . at the GUT scale MG. Explicitly embedding GSM in SU(5) is To be precise, there are 19 SM parameters which have to be fit- straightforward, with SU(3)C and SU(2)L corresponding e.g. to 2 ted to data: Three gauge couplings g3, g2 and g1, 13 parameters the upper-left 3×3 and lower-right 2×2 blocks, respectively, in associated with the Yukawa couplings (9 charged fermion masses, traceless 5 × 5 matrices for SU(5) generators of the fundamen- three mixing angles and one CP phase in the CKM matrix.), the tal representation. The U(1)Y corresponds to matrices gener- Higgs mass and quartic coupling, and θQCD. In addition, Ma- ated by diag(−2/3, −2/3, −2/3, 1, 1) and hence commutes with jorana neutrinos introduce 3 more masses and 6 mixing angles SU(3)C × SU(2)L ⊂ SU(5). It is then easy to derive how one SM and phases. As we will see, the paradigm of grand unification generation precisely comes from the 10 + 5 of SU(5) (where 10 addresses mainly the group theoretic data of Eq. (94.1) and the is the antisymmetric rank-2 tensor): values of the three gauge couplings. In many concrete realiza- tions, it then impacts also the other mentioned issues of the SM,  c c   c  such as the family structure and fermion mass hierarchy. 0 ub −ug ur dr dr c c c More specifically, after precision measurements of the Wein- −ub 0 ur ug dg dg  c c   c  berg angle θW in the LEP experiments, supersymmetric GUTs 10 :  ug −ur 0 ub db  and 5 :  db  . c (SUSY GUTs) have become the leading candidates in the search −ur −ug −ub 0 e   e  c for ‘Physics beyond the SM’. Supersymmetry (SUSY) is a sym- −dr −dg −db −e 0 −νe metry between bosons and fermions which requires the addition (94.2) of superpartners to the SM spectrum. If SUSY is motivated as a In addition to charge quantisation this structure explains why the solution to the gauge hierarchy problem (i.e. to the naturalness or l.h. quark and lepton states fall in SU(2)L doublets while the r.h. fine-tuning problem of the electroweak scale) [5], superpartners states are singlets. have to be present near the weak scale. SUSY GUTs [6] then Since SU(5) has 24 generators, SU(5) GUTs have 12 new gauge lead to the prediction of θW , in good agreement with subsequent bosons known as X bosons (or X/Y bosons) in addition to the observations [7]. However, the non-discovery of new particles at SM. X bosons form an SU(3)C -triplet and SU(2)L-doublet. Their the LHC puts into question the presence of new physics at the interaction connects quarks and leptons such that baryon and TeV scale in general and in particular of low-scale supersymmetry. lepton numbers are not conserved and nucleon decay is predicted. Still, SUSY may be present just outside the presently explored en- Furthermore, U(1)Y hypercharge is automatically quantized since ergy domain. it is embedded in SU(5). The measured Higgs mass (125 GeV) is in principle consistent In order to break the electroweak symmetry at the weak scale with this picture, assuming superpartners in the region of roughly and give mass to quarks and leptons, Higgs doublets are needed. 10 TeV. Such heavy superpartners then induce radiative correc- In the minimal SU(5) model, they can sit in either a 5H or 5¯H . tions raising the Higgs mass above the Z boson mass mZ [8, 9]. The three additional states are referred to as color-triplet Higgs However, from the vantage point of the hierarchy problem, heavy scalars. Their couplings also violate baryon and lepton numbers, inducing nucleon decay. In order not to violently disagree with the 1 In our convention the electric charge is Q = T3 + Y/2 and all our non-observation of nucleon decay, the triplet mass must be greater spinor fields are left-handed (l.h.). than ∼ 1011 GeV [13]. Moreover, in SUSY GUTs [6], in order to 2 Equivalently, the SU(2) and U(1) couplings are denoted as g = g L Y 2 cancel anomalies as well as give mass to both up and down quarks, 0 p 2 2 and g = 3/5 g1. One also uses αs = α3 = (g /4π), α = (e /4π) 3 EM both Higgs multiplets 5H and 5¯H are required. As we shall discuss 2 0 2 2 0 2 with e = g sin θW and sin θW = (g ) /(g + (g ) ). later, nucleon decay now constrains the Higgs triplets to have 972 94. Grand Unified Theories

Table 94.1: Quantum numbers of 16- mass significantly greater than MG in the minimal SUSY SU(5) GUT since integrating out the Higgs triplets generates dimension- dimensional representation of SO(10). five baryon-number-violating operators [14]. The mass splitting between doublet and triplet in the 5H (and 5H ) comes from their state Y Color Weak SU(5) SO(10) interaction with the SU(5) breaking sector. νc 0 − − − −− 1 c 94.2.3 SO(10) e 2 − − − ++ While SU(5) allows for the minimal GUT models, unification ur 1/3 + − − −+ is not complete: Two independent representations, 10 and 5¯, are dr 1/3 + − − +− required for one SM generation. A further representation, an ug 1/3 − + − −+ SU(5) singlet, has to be added to serve as r.h. neutrino in the dg 1/3 − + − +− 10 seesaw mechanism. In this case, the r.h. neutrino masses are ub 1/3 − − + −+ not necessarily related to the GUT scale. By contrast, a single db 1/3 − − + +− 16 c 16-dimensional spinor representation of SO(10) accommodates ur −4/3 − + + −− c a full SM generation together with an extra singlet, potentially ug −4/3 + − + −− c providing a r.h. neutrino [15]. This is most easily understood ub −4/3 + + − −− c from the breaking pattern SO(10) → SU(5) × U(1)X and the dr 2/3 − + + ++ 3 c associated branching rule 16 = 10−1+5¯3+1−4. Here the indices dg 2/3 + − + ++ c ¯ refer to charges under the U(1)X subgroup, which is orthogonal to db 2/3 + + − ++ 5 SU(5) and reflects the fact that SO(10) has rank five. From the ν −1 + + + −+ above, it is easy to see that U(1)X charges can be given as 2Y − e −1 + + + +− 5(B − L). Intriguingly, all representations of SO(10) are anomaly free in four dimensions (4d). Thus, the absence of anomalies in an SU(5)-GUT or a SM generation can be viewed as deriving from the case of this ‘direct’ breaking and for the breaking pattern this feature. SO(10) → SU(5) → GSM (with SU(5) broken at MG). In the We now describe in more detail how one family of quarks case of intermediate-scale Pati-Salam or flipped SU(5) models, and leptons appears in the 16. To understand this, recall that gauge coupling predictions are modified. The Higgs multiplets in the Γ -matrices of the 10d Clifford algebra give rise to five inde- the minimal SO(10) come from the fundamental representation, pendent, anticommuting ‘creation-annihilation’ operators Γ a± = 10H = 5H + 5¯H . Note, only in SO(10) does the representation 2a−1 2a (Γ ± iΓ )/2 with a = 1, ..., 5. These correspond to five type distinguish SM matter from Higgs fields. fermionic harmonic oscillators or “spin” 1/2 systems. The 32- dimensional tensor product of those is reducible since the 10d 94.2.4 Beyond SO(10) m n rotation generators Mmn = −i[Γ ,Γ ]/4 (m, n = 1, ..., 10) al- Finally, larger symmetry groups can be considered. For exam- ways flip an even number of “spins”. This gives rise to the 16 as ple, the exceptional group E6 has maximal subgroup SO(10) × displayed in Table 94.1. Next, one also recalls that the natural U(1) [18]. Its fundamental representation branches as 27 = embedding of SU(5) in SO(10) relies on ‘pairing up’ the 10 real 161 + 10−2 + 14. Another maximal subgroup is SU(3)C × 10 5 dimensions to produce 5 complex dimensions, R ≡ C , simi- SU(3)L × SU(3)R ⊂ E6 with branching rule 27 = (3, 3, 1) + m larly to the paring up of Γ s used above. This makes it clear (3¯, 1, 3¯)+(1, 3¯, 3). Independently of any underlying E6, the group 3 how to associate one |± > system to each complex dimension [SU(3)] with additional permutation symmetry Z3 interchanging of SU(5), which explains the labeling of the “spin” columns in the three factors can be considered. This is known as “trinifica- 3 Table 94.1: The first three and last two “spins” correspond to tion” [19]. The E6 → [SU(3)] breaking pattern has been used in SU(3)C and SU(2)L, respectively. In fact, an SU(3)C rotation phenomenological analyses of the heterotic string [20]. However, just raises one color index and lowers another, changing colors in larger symmetry groups, such as E6,SU(6), etc., there are {r, g, b}, or changes relative phases between the three spin states. now many more states which have not been observed and must Similarly, an SU(2)L rotation raises one weak index and lowers be removed from the effective low-energy theory. another, thereby flipping the weak isospin from up to down or Intriguingly, the logic by which GSM is a maximal subgroup vice versa, or changes the relative phase between the two spin of SU(5), which together with U(1)X is a maximal subgroup of states. In this representation U(1)Y hypercharge is simply given SO(10), continues in a very elegant and systematic way up to the by Y = −2/3(P color spins) + (P weak spins). SU(5) rota- largest exceptional group. The resulting famous breaking chain tions corresponding to X bosons then raise (or lower) a color E8 →E7 →E6 → SO(10) → SU(5) → GSM together with the index, while at the same time lowering (or raising) a weak in- special role played by E8 in group and in string theory is a tanta- dex. It is easy to see that such rotations can mix the states lizing hint at deeper structures. However, since all representations c c c c {Q, u , e } and {d ,L} among themselves and ν is a singlet. of E8 and E7 are real and can not lead to 4d chiral fermions, this Since SO(10) has 45 generators, additional 21 gauge bosons are is necessarily outside the 4d GUT framework. introduced including the U(1)X above. The 20 new SO(10) rota- tions not in SU(5) are then given by either raising any two spins 94.3 GUT breaking and doublet-triplet splitting or lowering them. With these rotations, 1 and 5 are connected In the standard, 4d field-theoretic approach to GUTs, the with 10. The last SO(10) rotation changes phases of states with unified gauge group is broken spontaneously by an appropriate weight 2(P color spins) + 2(P weak spins), which corresponds GUT Higgs sector. Scalar potentials (or superpotentials in SUSY to U(1)X . GUTs) exist whose vacua spontaneously break SU(5) or SO(10). SO(10) has two inequivalent maximal subgroups and hence While these potentials are ad hoc (just like the Higgs potential in breaking patterns, SO(10) → SU(5) × U(1)X and SO(10) → the SM), the most naive expectation is that all their dimensionful SU(4)C × SU(2)L × SU(2)R. In the first case, one can carry on parameters are O(MG). In the simplest case of SU(5), the 24 breaking to GSM ⊂ SU(5) precisely as in the minimal SU(5) case (adjoint) GUT Higgs develops a VEV along the GSM -singlet di- above. Alternatively, one can identify U(1)Y as an appropriate rection as hΦi ∝ diag(−2/3, −2/3, −2/3, 1, 1). In order for SO(10) linear combination of U(1)X and the U(1) factor from SU(5), to break to SU(5), the 16 or 126, which have a GSM -singlet with leading to the so-called flipped SU(5) [17] as an intermediate non-zero U(1)X charge, get a VEV. step in breaking SO(10) to GSM . In the second case, we have The masses of doublet and triplet in the 5H (and 5H ) generi- an intermediate Pati-Salam model thanks to the branching rule cally split due to their coupling to the GUT Higgs. In addition, 16 = (4, 2, 1) + (4, 1, 2). Finally, SO(10) can break directly to both the doublet and the triplet masses also get an equal con- the SM at MG. Gauge coupling unification remains intact in tribution from an SU(5)-invariant GUT-scale mass term. With- out any further structure, an extreme fine-tuning between two 3 Useful references on group theory in the present context include [16] large effects is then necessary to keep the doublet mass at the and refs. therein. electroweak scale. Supersymmetry plays an important role in for- 94. Grand Unified Theories 973

bidding large radiative correction to the doublet mass due to the both Wilson lines [34] and non-vanishing field-strength and leads, non-renormalization theorem [5]. However, even in this case we in general, to a reduced gauge symmetry and to chirality in the have to fine tune parameters at tree level. This is the doublet- resulting 4d effective theory. The 4d fermions arise from 10d gaug- triplet splitting problem which, in the SUSY context, is clearly inos. 5 related the µ-term problem of the MSSM (the smallness of the Given an appropriate embedding of GSM in E8×E8, gauge coefficient of µHu Hd). coupling unification is automatic at leading order. Corrections Several mechanisms for natural doublet-triplet splitting have arise mainly through (string)-loop effects and are similar to the been suggested under the assumption of supersymmetry, such as familiar field-theory thresholds of 4d GUTs6 [41]. Thus, one may the sliding singlet [21], missing partner [22], missing VEV [23], and say that coupling unification is a generic prediction in spite of pseudo-Nambu-Goldstone boson mechanisms [24]. Particular ex- the complete absence7 of a 4d GUT at any energy scale. This amples of the missing partner mechanism for SU(5) [25], the miss- absence is both an advantage and a weakness. On the up side, ing VEV mechanism for SO(10) [26, 27] and the pseudo-Nambu- GUT breaking and doublet-triplet splitting [43] are more natu- Goldstone boson mechanism for SU(6) [28] have been shown to rally realized and dimension-five nucleon decay is relatively easy be consistent with gauge coupling unification and nucleon decay. to avoid. On the down side, there is no reason to expect full GUT From the GUT-scale perspective, one is satisfied if the triplets are representations in the matter sector and flavor model building is naturally heavy and the doublets are massless (µ ' 0). There are much less tied to the GUT structure than in 4d. also several mechanisms for resolving the subsequent issue of why Let us pause to explain the beautiful idea behind the advertised µ is of order the SUSY breaking scale [29].4 For a review of the µ solution of the doublet-triplet splitting problem: One starts with problem and some suggested solutions in SUSY GUTs and string a simply connected CY X and mods out the action of a discrete theory, see [30–33] and references therein. group G (say Z2). In the absence of fixed points, X/G is smooth In general, GUT-breaking sectors successfully resolving the and has a non-contractible 1-cycle. Furthermore, let G also act on doublet-triplet splitting problem, dynamically stabilizing all the gauge bundle, according to an embedding G → E8. Now the GUT-scale VEVs and allowing for realistic neutrino masses and parallel transport around the 1-cycle is tied to a gauge rotation Yukawa couplings (including the GUT-symmetry violation in the (one says a non-trivial Wilson-line is present). Moreover, this latter) require a number of ingredients. However, for validity of Wilson line can not be continuously turned off since, e.g. in the the effective theory, introduction of higher or many representa- case of Z2, its square is the unit element of the group. The induced tions is limited, otherwise a Landau pole may appear below the ‘Wilson-line breaking’, which comes on top of the breaking by Planck scale. In addition, GUTs are only effective theories below non-zero field strengths, may remove certain sub-representations the Planck scale in the 4d field-theoretic approach. Since MG is (e.g. the triplet of SU(5) → GSM ) while keeping others exactly close to this scale, the effects of higher-dimension operators are massless. A simpler and, due to fixed points, singular version of not obviously negligible. In particular, operators including the this will appear below in the context of orbifold GUTs. GUT-breaking Higgs may affect low-energy predictions, such as One technical problem of heterotic constructions is the depen- quark and lepton masses. dence on the numerous size and shape parameters of M6 (the Thus, especially in the context of GUT breaking and doublet- so-called moduli), the stabilization of which is poorly understood triplet splitting, models beyond 4d field theory appear attractive. (see [44] for recent developments). Another is the sheer math- While this is mainly the subject of the next section, some ad- ematical complexity of the analysis, involving in particular the vantages can already be noted: In models with extra dimensions, study of (non-Abelian) gauge-bundles on CY spaces [45] (see how- in particular string constructions, GUT breaking may occur due ever [46]). to boundary conditions in the compactified dimensions [34–37]. An interesting aspect of heterotic string constructions is repre- No complicated GUT breaking sector is then required. Moreover, sented by orbifold models [35]. Here the internal space is given boundary conditions can give mass only to the triplet, leaving the by a six-torus, modded out by a discrete symmetry group (e.g. 6 doublet massless. This is similar to the ‘missing partner mecha- T /Zn). More recent progress is reported in [47, 48], including nism’ since the effective mass term does not ‘pair up’ the triplets in particular the systematic exploration of the phenomenologi- from 5H and 5H but rather each of them with further fields which cal advantages of so-called ‘non-prime’ (referring to n) orbifolds. are automatically present in the higher-dimensional theory. This The symmetry breaking to GSM as well as the survival of Higgs can eliminate dimension-five nucleon decay (cf. Sec. 94.6). doublets without triplet partners is ensured by the appropriate 94.4 String-theoretic and higher-dimensional embedding of the discrete orbifold group in E8×E8. String the- ory on such spaces, which are locally flat but include singularities, unified models is much more calculable than in the CY case. The orbifold ge- As noted earlier, the GUT scale is dangerously close to the ometries can be viewed as singular limits of CYs. scale of quantum gravity. It may hence be necessary to discuss An even simpler approach to unified models, which includes unified models of particle physics in the latter, more ambitious many of the advantages of full-fledged string constructions, is pro- context. Among the models of quantum gravity, superstring or vided by Orbifold GUTs [37]. These are (mostly) 5d or 6d SUSY M-theory stands out as the best-studied and technically most de- field theories with unified gauge group (e.g. SU(5) or SO(10)), veloped proposal, possessing in particular a high level of internal, broken in the process of compactifying to 4d. To give a partic- mathematical consistency. For our purposes, it is sufficient to 1,3 1 0 ularly simple example, consider SU(5) on R × S /(Z2 × Z2). know that five 10d and one 11d low-energy effective supergravity Here the compact space is an interval of length πR/2 and the em- theories arise in this setting (cf. [38] and refs. therein). 0 bedding of Z in the hypercharge direction of SU(5) realizes the Grand unification is realized most naturally in the context 2 breaking to GSM . Concretely, 5d X bosons are given Dirichlet of the two ‘heterotic’ theories with gauge groups E8×E8 and BCs at one endpoint of the interval and thus have no Kaluza-Klein SO(32), respectively [36, 39] (see [40] for some of the more re- (KK) zero mode. Their lightest modes have mass ∼ 1/R, mak- cent results). Justified in part by the intriguing breaking path ing the KK-scale the effective GUT scale. As an implication, the E8 → · · · → GSM mentioned above, the focus has historically boundary theory has no SU(5) invariance. Nevertheless, since the largely been on E8×E8. To describe particle physics, solutions of 1,3 SU(5)-symmetric 5d bulk dominates 4d gauge couplings, unifica- the 10d theory with geometry R × M6 are considered, where tion remains a prediction. Many other features but also problems M6 is a Calabi-Yau (CY) 3-fold (with 6 real dimensions) [36]. The background solution involves expectation values of higher- 5 All embeddings of G in one E factor which are consistent with dimensional components of the E ×E gauge fields. This includes SM 8 8 8 a breaking-pattern E8 → SU(5) → GSM are suitable (cf. for example the natural breaking chain from E8 to GSM through maximal subgroups 4 The solution of [29] relies on the absence of the fundamental superpo- mentioned at the end of Sect. 94.3). Other embeddings can change the tential term µH H (or µ 5 5 ). This can be ensured either by a discrete ratios between the three resulting GSM couplings at the GUT scale by u d H H group-theoretic factors. Crucially, due to the single 10d gauge coupling, no R symmetry or by a U(1)R. The latter clashes with typical superpotentials for the GUT breaking sector. However, higher-dimensional or stringy GUTs, continuous tuning is possible. 6 where the triplet Higgs is simply projected out, can be consistent with the Field-theory thresholds of 4d GUTs are discussed in 94.5. 7 U(1)R symmetry. See however [42]. 974 94. Grand Unified Theories

SM MSSM of 4d GUTs can be circumvented, especially doublet-triplet split- (bi = {41/10, −19/6, −7} or bi = {33/5, 1, −3}). Incom- ting is easily realized. plete GUT multiplets, such as gauge and Higgs bosons in the SM With the advent of the string-theory ‘flux landscape’ [49], which and also their superpartners and the additional Higgs bosons in is best understood in 10d type-IIB supergravity, the focus in string the MSSM, contribute to the differences between the β functions. model building has shifted to this framework. While type II string Inverting the argument, one expects that extrapolating the mea- theories have no gauge group in 10d, brane-stacks support gauge sured couplings to the high scale, we find quantitative unification dynamics. A particularly appealing setting (see e.g. [50]) is pro- at µ ∼ MG. While this fails in the SM, it works intriguingly well vided by type IIB models with D7 branes (defining 8d submani- in the MSSM (cf. Fig. 94.1). folds). However, in the SO(10) context the 16 is not available and, The three equations contained in Eq. (94.3) can be used to for SU(5), the top-Yukawa coupling vanishes at leading order [51]. determine the three ‘unknowns’ α3(mZ ), αG(MG) and MG, as- As a crucial insight, this can be overcome on the non-perturbative suming that all other parameters entering the equations are given. branch of type IIB, also known as F-theory [52, 53]. This setting Focusing on the SUSY case and using the MS coupling constants −1 2 allows for more general branes, thus avoiding constraints of the αEM(mZ ) and sin θW (mZ ) from [62], Dp-brane framework. GUT breaking can be realized using hy- −1 percharge flux (the VEV of the U(1)Y field strength), an option αEM(mZ ) = 127.955 ± 0.010 , (94.4) not available in heterotic models. The whole framework combines 2 the advantages of the heterotic or higher-dimensional unification sin θW (mZ ) = 0.23122 ± 0.00003 , (94.5) approach with the more recent progress in understanding moduli as input, one determines α−1(m ), which then gives stabilization. It thus represents at this moment the most active 1,2 Z and promising branch of theory-driven GUT model building (see −1 16 e.g. [54] and refs. therein). αG (MG) ' 24.3 and MG ' 2 × 10 GeV . (94.6) As a result of the flux-breaking, a characteristic ‘type IIB’ or ‘F- Here we have set δ = 0 for simplicity. Crucially, one in addition theoretic’ tree-level correction to gauge unification arises [55]. The i obtains a prediction for the low-energy observable α , fact that this correction can be rather significant numerically is 3 occasionally held against the framework of F-theory GUTs. How- −1 5 −1 12 −1 α (mZ ) = − α (mZ ) + α (mZ ) + ∆3 , (94.7) ever, at a parametric level, this correction nevertheless behaves 3 7 1 7 2 like a 4d threshold, i.e., it provides O(1) additive contributions to the inverse 4d gauge couplings α−1(M ). where i G 5 12 A final important issue in string GUTs is the so-called string- ∆3 = δ1 − δ2 + δ3 . (94.8) scale/GUT-scale problem [56]. It arises since, in heterotic com- 7 7 pactifications, the Planck scale and the high-scale value of the Here we followed the elegant formulation in Ref. [63] of the clas- gauge coupling unambiguously fix the string-scale to about 1018 sical analyses of [7]. Of course, it is a matter of convention which GeV. As the compactification radius R is raised above the string of the three low-energy gauge coupling parameters one ‘predicts’ length, the GUT scale (identified with 1/R) goes down and the and indeed, early works on the subject discussed the prediction of 2 string coupling goes up. Within the domain of perturbative string sin θW in terms of αEM and α3 [64, 65]. theory, a gap of about a factor ∼ 20 remains between the lowest Remarkably, the leading order result (i.e. Eq. (94.7) with δi = GUT scale achievable in this way and the phenomenological goal 0) is in excellent agreement with experiments [62]: of 2 × 1016 GeV. The situation can be improved by venturing into LO EXP the non-perturbative regime [56], by considering ‘anisotropic’ ge- α3 (mZ ) = 0.117 vs. α3 (mZ ) = 0.1181 ± 0.0011 . ometries with hierarchically different radii R [56,57] or by includ- (94.9) ing GUT scale threshold corrections [58, 59]. However, this near perfection is to some extent accidental. To In F-theory GUTs, the situation is dramatically improved since see this, we now discuss the various contributions to the δi (and the gauge theory lives only in four out of the six compact dimen- hence to ∆3). (2) sions. This allows for models with a ‘decoupling limit’, where the The two-loop running correction from the gauge sector ∆3 GUT scale is parametrically below the Planck scale [53]. How- (l) and the low-scale threshold correction ∆3 from superpartners ever, moduli stabilization may not be without problems in such can be summarized as [63] constructions, in part due to a tension between the required large volume and the desirable low SUSY breaking scale. (2) (l) 19  mSUSY  94.5 Gauge coupling unification ∆3 ' −0.82 and ∆3 ' log . (94.10) 28π mZ The quantitative unification of the three SM gauge couplings at the energy scale MG is one of the cornerstones of the GUT The relevant scale mSUSY can be estimated as [66] paradigm. It is obviously of direct phenomenological relevance. !28/19 !3/19 Gauge coupling unification is well understood in the framework m m of effective field theory (EFT) [60]. In the simplest case, the 3/19 12/19 4/19 W l mSUSY → mH m m × e e , relevant EFT at energies µ  M has a unified gauge symmetry H W m m G e e eg eq (say SU(5) for definiteness) and a single running gauge coupling (94.11) α (µ). At energies µ  M , states with mass ∼ M (such as X G G G where mH stands for the masses of non-SM Higgs states and su- bosons, GUT Higgs, color-triplet Higgs) have to be integrated out. perpartner masses are given in self-evident notation. Detailed The EFT now has three independent couplings and SM (or SUSY analyses including the above effects are best done using appropri- SM) matter content. One-loop renormalization group equations ate software packages, such as SOFTSUSY [61] (or alternatively readily allow for an extrapolation to the weak scale, SuSpect [67] or SPheno [68]). See also [61] for references to the underlying theoretical two-loop analyses. −1 −1 bi  MG  α (mZ ) = α (MG) + log + δi , (94.3) To get a very rough feeling for these effects, let us assume i G 2π m Z that all superpartners are degenerate at mSUSY = 1 TeV, ex- (l) (i = 1, 2, 3). Here we defined δ to absorb all sub-leading effects, cept for heavier gluinos: m /m ' 1/3. This gives ∆3 ' i We eg such as threshold corrections at or near the weak scale (e.g. from −0.35 + 0.22 ln(mSUSY/mZ ) ' 0.18. The resulting prediction of superpartners and the additional Higgs bosons in the case of the α3(mZ ) ' 0.126 significantly upsets the perfect one-loop agree- MSSM) and at the GUT scale, and also higher-order corrections. ment found earlier. Before discussing this issue further, it is useful We will discuss them momentarily. to introduce yet another important type of correction, the high or It is apparent from Eq. (94.3) that the three low-scale cou- GUT scale thresholds. plings can be very different. This is due to the large energy range To discuss high scale thresholds, let us set all other corrections mZ  µ  MG and the non-universal β-function coefficients to zero for the moment and write down a version of Eq. (94.3) 94. Grand Unified Theories 975

β SM MSSM: m0=M1/2=2 TeV, A0=0, tan =30 60 60 α α α1 α1 50 α2 50 α2 3 3 40 40 (Q) (Q) 30 30 -1 i -1 i α α 20 20

10 10

0 SOFTSUSY 3.6.2 0 SOFTSUSY 3.6.2 0 5 10 15 0 5 10 15

log10(Q/GeV) log10(Q/GeV)

Figure 94.1: Running couplings in SM and MSSM using two-loop RG evolution. The SUSY threshold at 2 TeV is clearly visible on the MSSM side. (We thank Ben Allanach for providing the plots created using SOFTSUSY [61].) that captures the running near and above the GUT scale more can also be much larger or of different sign if, as is required in correctly. The threshold correction at one-loop level can be eval- many fully realistic 4d GUT models, many additional (and in uated accurately by the simple step-function approximation for particular higher) representations are introduced. Thus, there is the β functions in the DR scheme8 [72], considerable model building freedom. Nevertheless, a significant constraint from getting the right GUT threshold corrections while −1 αi (mZ ) = keeping the triplet Higgs heavy remains. 1 h µ µ µ µ i The above analysis implicitly assumes universal soft SUSY α−1(µ) + b ln + bC ln + bX ln + bΦ ln . G i i i i breaking masses at the GUT scale, which directly affect the spec- 2π mZ MC MX MΦ (94.12) trum of SUSY particles at the weak scale. In the simplest case we have a universal gaugino mass M1/2, a universal mass for squarks Here we started the running at some scale µ  MG, including and sleptons m16 and a universal Higgs mass m10, as motivated the contribution of the minimal set of states relevant for the tran- by SO(10). In some cases, threshold corrections to gauge cou- sition from the high-scale SU(5) model to the MSSM. These are pling unification can be exchanged for threshold corrections to the color-triplet Higgs multiplets with mass MC , massive vec- soft SUSY parameters (see [75] and refs. therein). For example, if tor multiplets of X-bosons with mass MX (including GUT Higgs gaugino masses were not unified at MG and, in particular, gluinos degrees of freedom), and the remaining GUT-Higgs fields and su- were lighter than winos at the weak scale (cf. Eq. (94.11)), then it C,X,Φ is possible that, due to weak scale threshold corrections, a much perpartners with mass MΦ. The coefficients bi can be found in Ref. [73]. Crucially, the bi in Eq. (94.12) conspire to make smaller or even slightly negative threshold correction at the GUT the running GUT-universal at high scales, such that the resulting scale would be consistent with gauge coupling unification [76]. prediction for α3 does not depend on the value of µ. It is also noteworthy that perfect unification can be realized To relate this to our previous discussion, we can, for example, without significant GUT-scale corrections, simply by slightly rais- define MG ≡ MX and then choose µ = MG in Eq. (94.12). This ing the (universal) SUSY breaking scale. In this case the dark gives the high-scale threshold corrections matter abundance produced by thermal processes in the early h i universe (if the lightest neutralino is the dark matter particle) (h) 1 C MG Φ MG is too high. However, even if the gaugino mass in the MSSM is δi = bi ln + bi ln , (94.13) 2π MC MΦ about 1 TeV to explain the dark matter abundance, if the Hig-

(h) gsino and the non-SM Higgs boson masses are about 10-100 TeV, and a corresponding correction ∆3 . To get some intuition for the effective SUSY scale can be raised [77]. This setup is realized the magnitude, one can furthermore assume MΦ = MG, finding in split SUSY [78] or the pure gravity mediation model [79] based C (with bi = {2/5, 0, 1}) on anomaly mediation [80]. Since the squarks and sleptons are much heavier than the gaugino masses in those setups, a gauge 9  M  (h) G hierarchy problem is reintroduced. The facts that no superpart- ∆3 = ln . (94.14) 14π MC ners have so far been seen at the LHC and that the observed Higgs (2) (l) mass favors heavier stop masses than about 1 TeV force one to To obtain the desired effect of −∆3 − ∆3 ' +0.64, the triplet accept a certain amount of fine-tuning anyway. Higgs would have to be by about a factor 20 lighter than the GUT For non-SUSY GUTs or GUTs with a very high SUSY breaking scale. While this is ruled out by nucleon decay in the minimal scale to fit the data, new light states in incomplete GUT multi- model [74] as will be discussed Sec. 94.6, it is also clear that plets or multiple GUT breaking scales are required. For example, threshold corrections of this order of magnitude can, in general, be non-SUSY models SO(10) → SU(4) ×SU(2) ×SU(2) → SM, realized with a certain amount of GUT-scale model building, e.g. C L R with the second breaking scale of order an intermediate scale, de- in specific SU(5) [25] or SO(10) [26,27] constructions. Corrections termined by light neutrino masses using the see-saw mechanism, can fit the low-energy data for gauge couplings [81] and at the 8The DR scheme is frequently used in a supersymmetric regularization [69]. The renormalization transformation of the gauge coupling constants same time survive nucleon decay bounds [82]. Alternatively, one from MS to DR scheme is given in Ref. [70]. For an alternative treatment can appeal to string-theoretic corrections discussed in Sec. 94.4 using holomorphic gauge couplings and NSVZ β-functions see e.g. [71]. to compensate for a high SUSY breaking scale. This has, for ex- 976 94. Grand Unified Theories

ample, been concretely analyzed in the context of F-theory GUTs tain two fermionic components and the rest scalars or products of in [83]. Similarly, one may even wonder whether particularly large scalars. Within the context of SU(5) the dimension-four and five GUT threshold corrections could be sufficient to ensure non-SUSY operators have the form precision unification. Notice here that the gauge coupling unifica- tion predicts just one parameter. When introducing new states, (10 5¯ 5¯) ⊃ (uc dc dc) + (Q L dc) + (ec LL), typically one can fit the data by choice of their masses. This is not the case in SUSY GUTs with low-scale SUSY breaking scale (10 10 10 5¯) ⊃ (QQQL) + (uc uc dc ec) where the masses are constrained by fine tuning. + B- and L-conserving terms, In 5d or 6d orbifold GUTs, certain “GUT scale” threshold cor- rections come from the Kaluza-Klein modes between the compact- respectively. ification scale, Mc ∼ 1/R, and the effective cutoff scale M∗. In The dimension-four operators in (10 5¯ 5¯) violate either baryon string theory, this cutoff scale is the string scale. Gauge coupling number or lepton number. The nucleon lifetime is extremely unification at two loops then constrains the values of Mc and short if both types of dimension-four operators are present in the 9 M∗. Often, one finds Mc to be lower than the 4d GUT scale. SUSY SM since squark or slepton exchange induces the dangerous Since the X-bosons, responsible for nucleon decay, get mass at dimension-six SM operators. Even in the case that they violate the compactification scale, this has significant consequences for baryon number or lepton number only but not both, they are nucleon decay. constrained by various phenomena [91]. For example, the pri- Finally, it has been shown that non-supersymmetric GUTs in mordial baryon number in the universe is washed out unless the warped 5d orbifolds can be consistent with gauge coupling unifi- dimensionless coupling constants are less than 10−7. Both types cation. This assumes (in 4d language) that the r.h. top quark and of operators can be eliminated by requiring R parity, which dis- the Higgs doublets are composite-like objects with a composite- tinguishes Higgs from ordinary matter multiplets. R parity [92] ness scale in the TeV range [85]. or its cousin, matter parity [6, 93], act as F → −F,H → H with F = {10, 5¯},H = {5¯ , 5 } in SU(5).10 In SU(5), the 94.6 Nucleon decay H H Higgs multiplet 5¯H and the matter multiplets 5¯ have identical Quarks and leptons are indistinguishable in any 4d GUT, and gauge quantum numbers. In E6, Higgs and matter multiplets both the baryon (B) and lepton number (L) are not conserved. could be unified within the fundamental 27 representation. Only This leads to baryon-number-violating nucleon decay. In addi- in SO(10) are Higgs and matter multiplets distinguished by their tion to baryon-number violation, lepton-number violation is also gauge quantum numbers. The Z4 center of SO(10) distinguishes required for nucleon decay since, in the SM, leptons are the 10s from 16s and can be associated with R parity [94]. only free fermions which are lighter than nucleons. The lowest- The baryon-number violating dimension-five operators have a dimension operators relevant for nucleon decay are (B+L) violat- dimensionful coupling. They are generated by integrating out ing dimension-six four-fermion-terms in the SM, and all baryon- the color-triplet Higgs with GUT-scale mass in SUSY GUTs such violating operators with dimension less than seven preserve (B−L) that the coefficient is suppressed by 1/MG. Note that both triplet [1, 86]. Higgsinos (due to their fermionic nature) and Higgs scalars (due In SU(5) GUTs, the dimension-six operators are induced by to their mass-enhanced trilinear coupling with matter) contribute X boson exchange. These operators are suppressed by (1/M 2 ) X to the operators. The dimension-five operators include squarks (MX is the X boson mass), and the nucleon lifetime is given by and/or sleptons. To allow for nucleon decay, these must be con- τ ∝ M 4 /(α2 m5) (m is proton mass). The dominant decay N X G p p verted to light quarks or leptons by exchange of a gaugino or mode of the proton (and the baryon-violating decay mode of the + 0 + − Higgsino in the SUSY SM. The nucleon lifetime is proportional to neutron), via X boson exchange, is p → e π (n → e π ). 2 2 5 MG mSUSY/mp, where mSUSY is the SUSY breaking scale. Thus, In any simple gauge symmetry, with one universal GUT cou- dimension-five operators may predict a shorter nucleon lifetime pling αG and scale MX , the nucleon lifetime from gauge boson than dimension-six operators. Unless accidental cancellations are exchange is calculable. Hence, the GUT scale may be directly present, the dominant decay modes from dimension-five opera- observed via the extremely rare decay of the nucleon. Exper- tors include a K meson, such as p → K+ ν¯ (n → K0 ν¯). This imental searches for nucleon decay began with the Kolar Gold is due to a simple symmetry argument: The operators are given Mine, Homestake, Soudan, NUSEX, Frejus, HPW, IMB, and c c c c as (Qi Qj Qk Ll) and (ui uj dk el ), where i, j, k, l (= 1, 2, 3) Kamiokande detectors [64]. The present experimental bounds on are family indices and color and weak indices are implicit. They the modes come from Super-Kamiokande. With 306 kton-years + 0 34 must be invariant under SU(3)C and SU(2)L so that their color of data they find τp/Br(p → e π ) > 1.67 × 10 years at 90% and weak doublet indices must be anti-symmetrized. Since these CL [87]. In addition, Hyper-Kamiokande [88] is planned to reach + 0 35 operators are given by bosonic superfields, they must be totally to τp/Br(p → e π ) ∼ 10 years. The hadronic matrix elements symmetric under interchange of all indices. Thus the first oper- for baryon-number-violating operators are evaluated with lattice ator vanishes for i = j = k and the second vanishes for i = j. QCD simulations [89]. In SUSY SU(5) GUTs, the lower bound Hence a second or third generation member exists in the domi- on the X boson mass from null results in nucleon decay searches 16 nant modes of nucleon decay unless these modes are accidentally is approaching 10 GeV [90], which is close to the GUT scale suppressed [93]. suggested by gauge coupling unification. On the other hand, the The Super-Kamiokande bounds on the proton lifetime severely prediction for nucleon decay in non-SUSY GUTs is hard to quan- constrain the dimension-five operators. With 306 kton-years of tify. The reason is that gauge couplings do not unify with just data they find τ /Br(p → K+ν¯) > 6.61 × 1033 years at 90% CL the SM particle content. Once extra states or large thresholds p [87]. In the minimal SUSY SU(5), τ /Br(p → K+ν¯) is smaller are included to ensure precision unification, a certain range of p than about 1031 years if the triplet Higgs mass is 1016 GeV and unification scales is allowed. mSUSY = 1 TeV [95]. The triplet Higgs mass bound from nucleon In SUSY GUTs there are additional sources for baryon and/or decay is then in conflict with gauge coupling unification so that lepton-number violation – dimension-four and five operators [14]. this model is considered to be ruled out [74]. These arise since, in the SUSY SM, quarks and leptons have scalar Since nucleon decay induced by the triplet Higgs is a severe partners (squarks and sleptons). Although our notation does problem in SUSY GUTs, various proposals for its suppression not change, when discussing SUSY models our fields are chiral have been made. First, some accidental symmetry or acciden- superfields and both fermionic and bosonic matter is implicitly tal structure in non-minimal Higgs sectors in SU(5) or SO(10) represented by those. In this language, baryon- and/or lepton- theories may suppress the dimension-five operators [22,26,27,96]. number-violating dimension-four and five operators are given as so-called F terms of products of chiral superfields, which con- 10 This forbids the dimension-four operator (10 5¯ 5¯), but allows the Yukawa couplings for quark and lepton masses of the form (10 5¯ 5¯H ) and 9 It is interesting to note that a ratio M∗/Mc ∼ 100, needed for gauge (10 10 5H ). It also forbids the dimension-three, lepton-number-violating coupling unification to work in orbifold GUTs, is typically the maximum operator (5¯ 5H ) ⊃ (LHu) as well as the dimension-five, baryon-number- value for this ratio consistent with perturbativity [84]. violating operator (10 10 10 5¯H ) ⊃ (QQQHd) + ··· . 94. Grand Unified Theories 977

Symmetries to suppress the dimension-five operators are typi- are, as before, family indices. Hence, at the GUT scale we cally broken by the VEVs responsible for the color-triplet Higgs have tree-level relations between Yukawa coupling constants for masses. Consequently the dimension-five operators are generically charged lepton and masses, such as λb = λτ in generated via the triplet Higgs exchange in SUSY SU(5) GUTs, which λb/τ are the /τ lepton Yukawa coupling con- as mentioned above. In other words, the nucleon decay is sup- stants [104,105]. In SO(10), there is only one type of independent pressed if the Higgs triplets in 5¯H and 5H do not have a common renormalizable Yukawa interaction given by λij (16i 16j 10H ), mass term but, instead, their mass terms involve partners from leading to relations among all Yukawa coupling constants and other SU(5) multiplets. Second, the SUSY breaking scale may be quark and lepton masses within one generation [106,107] (such as around O(10–100) TeV in order to explain the observed Higgs bo- λt = λb = λτ , with λt the top quark Yukawa coupling constant). son mass at the LHC. In this case, nucleon decay is automatically In addition to gauge coupling unification, the ratio of bottom suppressed [78, 97, 98]. Third, accidental cancellations among di- quark and τ lepton mass has been a central target in the study agrams due to a fine-tuned structure of squark and slepton flavor of GUTs since it was found that this ratio was almost consistent mixing might suppress nucleon decay [99]. Last, we have also with observations after including the QCD correction [104]. Today implicitly assumed a hierarchical structure for Yukawa matrices the quark masses and the gauge coupling constants are known in the analysis. It is however possible to fine-tune a hierarchical precisely such that, as discussed below, the ratio of bottom quark structure for quarks and leptons which baffles the family struc- and τ lepton mass has become a target of precision analyses in ture so that the nucleon decay is suppressed [100]. The upper the GUT context. bound on the proton lifetime from some of these theories is ap- 94.7.1 The third generation, b–τ or t–b–τ unification proximately a factor of 10 above the experimental bounds. Future Third generation Yukawa couplings are larger than those of the experiments with larger neutrino detectors, such as JUNO [101], first two generations. Hence, the fermion mass relations predicted Hyper-Kamiokande [88] and DUNE [102], are planned and will from renormalizable GUT interactions which we introduced above have higher sensitivities to nucleon decay. are expected to be more reliable. In order to compare them with Are there ways to avoid the stringent predictions for proton data, we have to include the radiative correction to these relations decay discussed above? Orbifold GUTs and string theories, see from the RG evolution between GUT and fermion mass scale, from Sec. 94.4, contain grand unified symmetries realized in higher integrating out heavy particles at the GUT scale, and from weak dimensions. In the process of compactification and GUT sym- scale thresholds. metry breaking, the triplet Higgs states may be removed (pro- Since testing Yukawa coupling unification is only possible in jected out of the massless sector of the theory). In such mod- models with successful gauge coupling unification, we here focus els, the nucleon decay due to dimension-five operators can be on SUSY GUTs. In the MSSM, top and bottom quark and τ severely suppressed or eliminated completely. However, nucleon lepton masses are related to the Yukawa coupling constants at decay due to dimension-six operators may be enhanced, since the the scale mZ as gauge-bosons mediating proton decay obtain mass at the com- pactification scale, Mc, which is typically less than the 4d GUT mt(mZ ) = λt(mZ ) vu(1 + δmt/mt), scale (cf. Sec. 94.5). Alternatively, the same projections which m (m ) = λ (m ) v (1 + δm /m ) , eliminate the triplet Higgs may rearrange the quark and lepton b/τ Z b/τ Z d b/τ b/τ states such that the massless states of one family come from dif- √ √ where hH0i ≡ v = sin β v/ 2, hH0i ≡ v = cos β v/ 2, ferent higher-dimensional GUT multiplets. This can suppress or u u d d v /v ≡ tan β and v ∼ 246 GeV is fixed by the Fermi constant, completely eliminate even dimension-six proton decay. Thus, en- u d G . Here, δm /m (f = t, b, τ) represents the threshold correc- hancement or suppression of dimension-six proton decay is model- µ f f tion due to integrating out SUSY partners. For the bottom quark dependent. In some complete 5d orbifold GUT models [63, 103] mass, it is found [108] that the dominant corrections come from the lifetime for the decay τ /Br(p → e+π0) can be near the bound p the gluino-sbottom and from the Higgsino-stop loops, of 1 × 1034 years with, however, large model-dependence and/or theoretical uncertainties. In other cases, the modes p → K+ν¯ and p → K0µ+ may be dominant [63]. Thus, interestingly, the  δm  g2 m µ b ∼ 3 eg tan β and observation of nucleon decay may distinguish string or higher- 2 2 mb g 6π m dimensional GUTs from 4d ones. 3 SUSY (94.16)  δm  λ2 A µ In orbifold GUTs or string theory, new discrete symmetries con- b ∼ t t tan β , 2 2 sistent with SUSY GUTs can forbid all dimension-three and four mb λt 16π mSUSY baryon- and lepton-number-violating operators. Even the µ term where m , µ, and At stand for gluino and Higgsino masses and and dimension-five baryon- and lepton-number-violating opera- g tors can be forbidden to all orders in perturbation theory [33]. The trilineare stop coupling, respectively. Note that Eq. (94.16) only µ term and dimension-five baryon- and lepton-number-violating illustrates the structure of the corrections – non-trivial functional operators may then be generated, albeit sufficiently suppressed, dependences on several soft parameters ∼ mSUSY have been sup- via non-perturbative effects. The simplest example of this is a pressed. For the full one-loop correction to the bottom quark R Z4 symmetry which is the unique discrete R symmetry consis- mass see, for example, Ref. [109]. tent with SO(10) [33]. Even though it forbids the dimension-five Note also that the corrections do not go to zero as SUSY parti- proton decay operator to the desired level, it allows the required cles become much heavier than mZ . They may change the bottom dimension-five neutrino mass term. In this case, proton decay is quark mass at the O(10)% level for tan β = O(10). The total ef- dominated by dimension-six operators, leading to decays such as fect is sensitive to the relative phase between gluino and Higgsino + 0 p → e π . masses since At ∼ −m due to the infrared fixed point nature of eg 94.7 Yukawa coupling unification the RG equation for At [110] in settings where SUSY breaking terms come from Planck scale dynamics, such as gravity media- In the SM, masses and mixings for quarks and leptons come tion. The τ lepton mass also receives a similar correction, though from the Yukawa couplings with the Higgs doublet, but the values only at the few % level. The top quark mass correction, not being of these couplings remain a mystery. GUTs provide at least a proportional to tan β, is at most 10% [111]. partial understanding since each generation is embedded in unified Including one loop threshold corrections at mZ and additional multiplet(s). Specifically, since quarks and leptons are two sides of RG running, one finds the top, bottom and τ pole masses. In the same coin, the GUT symmetry relates the Yukawa couplings SUSY GUTs, b–τ unification has two possible solutions with (and hence the masses) of quarks and leptons. tan β ∼ 1 or O(10). The small tan β solution may be realized In SU(5), there are two types of independent renormal- in the MSSM if superpartner masses are O(10) TeV, as suggested izable Yukawa interactions given by λij (10i 10j 5H ) by the observed Higgs mass [97]. The large tan β limit such as 0 ¯ ¯ + λij (10i 5j 5H ). These contain the SM interactions tan β ∼ 40–50 overlaps the SO(10) symmetry relation [111, 112]. c 0 c c λij (Qi uj Hu) + λij (Qi dj Hd + ei Lj Hd). Here i, j ( =1, 2, 3) When tan β is large, there are significant threshold corrections to 978 94. Grand Unified Theories

down quark masses as mentioned above, and Yukawa unification quire that the fermions of all three generations come from a sin- is only consistent with low-energy data in a restricted region of gle representation of a large gauge group. A somewhat weaker SUSY parameter space, with important consequences for SUSY assumption is that the flavor group (e.g. SU(3)) unifies with searches [111, 113]. More recent analyses of Yukawa unification the SM gauge group in a simple gauge group at some energy after LHC Run-I are found in Ref. [114]. scale M ≥ MG. Early work on such ‘flavor-unified GUTs’, see Gauge coupling unification is also successful in the scenario of e.g. [119, 122], has been reviewed in [123, 124]. For a selec- split supersymmetry [78], in which squarks and sleptons have mass tion of more recent papers see [125]. In such settings, Yukawa at a scale m˜  mZ , while gauginos and/or Higgsinos have masses couplings are generally determined by gauge couplings together of order the weak scale. Unification of b–τ Yukawa couplings with symmetry breaking VEVs. This is reminiscent of heterotic requires tan β to be fine-tuned close to 1 [97]. If by contrast, string GUTs, where all couplings come from the 10d gauge cou- 4 tan β & 1.5, b–τ Yukawa unification only works for m˜ . 10 GeV. pling. However, while the E8 → SU(3)×E6 branching rule This is because the effective theory between the gaugino mass 248 = (8, 1) + (1, 78) + (3, 27) + (3, 27) looks very suggestive scale and m˜ includes only one Higgs doublet, as in the standard in this context, the way in which most modern heterotic models model. As a result, the large top quark Yukawa coupling tends arrive at three generations is actually more complicated. to increase the ratio λb/λτ due to the vertex correction, which is absent in supersymmetric theories, as one runs down in energy 94.8 Neutrino masses below m˜ . This is opposite to what happens in the MSSM where We see from atmospheric and solar neutrino oscillation ob- servations, along with long baseline accelerator and reactor ex- the large top quark Yukawa coupling lowers the ratio λb/λτ [105]. periments, that neutrinos have finite masses. By adding three 94.7.2 Beyond leading order: three-family models c c “sterile” neutrinos νi with Yukawa couplings λν,ij (νi Lj Hu) Simple Yukawa unification is not possible for the first two gen- (i, j = 1, 2, 3), one easily obtains three massive Dirac neutrinos erations. Indeed, the simplest implementation of SU(5) implies with mass mν = λν vu, analogously to quark and charged lepton λs = λµ, λd = λe and hence λs/λd = λµ/λe. This is an RG- masses. However, in order to obtain a τ neutrino with mass of invariant relation which extrapolates to ms/md = mµ/me at the order 0.1 eV, one requires the exceedingly small coupling ratio weak scale, in serious disagreement with data (ms/m ∼ 20 and −10 d λντ /λτ . 10 . By contrast, in GUTs the seesaw mechanism mµ/me ∼ 200). An elegant solution to this problem was given naturally explains such tiny neutrino masses as follows [2–4]: The by Georgi and Jarlskog [115] (for a recent analysis in the SUSY sterile neutrinos have no SM gauge quantum numbers so that context see [116]). there is no symmetry other than global lepton number which for- More generally, we have to recall that in all of the previous dis- 1 c c bids the Majorana mass term 2 Mij νi νj . Note also that sterile cussion of Yukawa couplings, we assumed renormalizable interac- neutrinos can be identified with the r.h. neutrinos necessarily con- tions as well as the minimal matter and Higgs content. Since the tained in complete families of SO(10) or Pati-Salam models. Since GUT scale is close to the Planck scale, higher-dimension opera- the Majorana mass term violates U(1)X in SO(10), one might ex- tors involving the GUT-breaking Higgs may modify the predic- pect Mij ∼ MG. The heavy sterile neutrinos can be integrated tions, especially for lower generations. An example is provided by out, defining an effective low-energy theory with only three light the operators 10 5¯ 5¯H 24H with 24H the GUT-breaking Higgs active Majorana neutrinos with the effective dimension-five oper- of SU(5). We can fit parameters to the observed fermion masses ator with these operators, though some fine-tuning is introduced in doing so. The SM Higgs doublet may come in part from higher 1 − L = cij (Li Hu)(Lj Hu) , (94.17) representations of the GUT group. For example, the 45 of SU(5) eff 2 includes an SU(2) doublet with appropriate U(1) charge [115]. L Y T −1 This 45 can, in turn, come from the 120 or 126 of SO(10) af- where c = λν M λν . This then leads to a 3 × 3 Majorana T −1 ter its breaking to SU(5) [117, 118]. These fields may also have neutrino mass matrix m = mν M mν . renormalizable couplings with quarks and leptons. The relations The seesaw mechanism implemented by r.h. neutrinos is some- among the Yukawa coupling constants in the SM are modified if times called the type-I seesaw model. There are variant models the SM Higgs doublet is a linear combination of several such dou- in which the dimension-five operator for neutrino masses is in- blets from different SU(5) multiplets. Finally, the SM fermions duced in different ways: In the type-II model, an SU(2)L triplet 2 may not be embedded in GUT multiplets in the minimal way. In- Higgs boson Σ is introduced to have couplings ΣL and also 2 deed, if all quarks and leptons are embedded in 16s of SO(10), the ΣHu [117, 126]. In the type-III model, an SU(2)L triplet of fermions Σ˜ with a Yukawa coupling ΣLH˜ is introduced [127]. renormalizable interactions with 10H cannot explain the observed u CKM mixing angles. This situation improves when extra matter In these models, the dimension-five operator is induced by inte- grating out the triplet Higgs boson or fermions. Such models can multiplets, such as 10, are introduced: After U(1)X , which dis- tinguishes the 5s coming from the 16 and the 10 of SO(10), is also be implemented in GUTs by introducing Higgs bosons in the 15 or fermions in the 24 in SU(5) GUTs or the 126 in SO(10) broken (e.g. by a VEV of 16H or 126H ), the r.h. down quarks and l.h. leptons in the SM can be linear combinations of components GUTs. Notice that the gauge non-singlet fields in the type-II and in 16s and 10s. As a result, λ 6= λ0 in SU(5) [119]. III models have masses at the intermediate scale. Thus, gauge To construct realistic three-family models, some or all of the coupling unification is not automatic if these variant mechanisms above effects can be used. Even so, to achieve significant pre- are implemented in SUSY GUTs. dictions for fermion masses and mixing angles grand unification Atmospheric neutrino oscillations discovered by Super- 2 −3 alone is not sufficient. Other ingredients, for example additional Kamiokande [128] require neutrino masses with ∆mν ∼ 2.5×10 2 global family symmetries are needed (in particular, non-Abelian eV with maximal mixing [62], in the simplest scenario of two neu- symmetries can strongly reduce the number of free parameters). trino dominance. With hierarchical neutrino masses this implies p 2 These family symmetries constrain the set of effective higher- mντ = ∆mν ∼ 0.05 eV. Next, we can try to relate the neutrino dimensional fermion mass operators discussed above. In addition, Yukawa coupling to the top quark Yukawa coupling, λντ = λt at sequential breaking of the family symmetry can be correlated with the GUT scale, as in SO(10) or SU(4) × SU(2)L × SU(2)R mod- the hierarchy of fermion masses [120]. One simple, widely known els. This gives M ∼ 1014 GeV, which is remarkably close to the idea in this context is to ensure that each 10i enters Yukawa in- GUT scale. teractions together with a suppression factor 3−i ( being a small Neutrinos pose a special problem for GUTs. The question is parameter). This way one automatically generates a stronger why the quark mixing angles in the CKM matrix are small while hierarchy in up-type quark Yukawas as compared to down-type there are two large lepton mixing angles in the PMNS matrix. quark and lepton Yukawas and no hierarchy for neutrinos, which Global fits of neutrino masses and mixing angles can, for example, agrees with observations at the O(1)-level. Three-family models be found in Refs. [129] and [130]. For SUSY GUT models which exist which fit all the data, including neutrino masses and mix- fit quark and lepton masses, see Ref. [131] for reviews. Finally, for ing [27, 121]. a compilation of the range of SUSY GUT predictions for neutrino Finally, a particularly ambitious variant of unification is to re- mixing, see [132]. 94. Grand Unified Theories 979

94.9 Selected topics monopole density. Other possible solutions to the monopole prob- 94.9.1 Global symmetries lem include: sweeping them away by domain walls [144], U(1) As we discussed, global symmetries are frequently introduced electromagnetic symmetry breaking at high temperature [145] or to control higher-dimension operators in GUT models. This is GUT symmetry non-restoration [146]. Parenthetically, it was also particularly important in the context of nucleon decay but also shown that GUT monopoles can catalyze nucleon decay [147]. A plays a role in GUT-based flavor model building and cosmolog- significantly stronger bound on the monopole flux can then be ical applications, such as baryogenesis and inflation. However, obtained by considering X-ray emission from radio pulsars due we should note that appealing to global symmetries to suppress to monopole capture and the subsequent nucleon decay cataly- specific interactions may not be as straightforward as it naively sis [148]. seems. Indeed, there are two possibilities: On the one hand, the Note that the present upper bound on the inflationary vac- 1/4 relevant symmetry might be gauged at a higher scale. Effects uum energy density is very close to the GUT scale, Vinf = of the VEVs responsible for the spontaneous breaking are then in (1.88 × 1016 GeV) × (r/0.10)1/4, with the scalar-to-tensor ratio principle dangerous and need to be quantified. On the other hand, constrained to r < 0.07 [149]. This guarantees that reheating the symmetry might be truly only global. This must e.g. be the does not lead to temperatures above MG and hence the monopole case for anomalous symmetries, which are then also violated by problem is solved by inflation (unless MG is unexpectedly low). field-theoretic non-perturbative effects. The latter can in principle be exponentially small. It is, however, widely believed that global 94.9.4 Flavor violation symmetries are always broken in quantum gravity (see e.g. [133]). Yukawa interactions of GUT-scale particles with quarks and One then needs to understand which power or functional form the leptons may leave imprints on the flavor violation induced by Planck scale suppression of the relevant interaction has. For exam- SUSY breaking parameters [150]. To understand this, focus first ple, dimension-five baryon number violating operators suppressed on the MSSM with universal Planck-scale boundary conditions (as by just one unit of the Planck or string scale are completely ex- e.g. in gravity mediation). Working in a basis where up-quark and cluded. lepton Yukawas are diagonal, one finds that the large top-quark In view of the above, it is also useful to recall that in string mod- Yukawa coupling reduces the l.h. squark mass squareds in the els 4d global symmetries generally originate in higher-dimensional third generation radiatively. It turns out that only the l.h. down- gauge symmetries [38, 134]. Here ‘global’ implies that the gauge type squark mass matrix has sizable off-diagonal terms in the boson has acquired a Stückelberg-mass. This is a necessity in the flavor basis after CKM-rotation. However, in GUTs the color- anomalous case (Green-Schwarz mechanism [135]) but can also triplet Higgs has flavor violating interactions from the Yukawa happen to non-anomalous symmetries. One expects no symme- coupling λij (10i 10j 5H ), such that flavor-violating r.h. slepton try violation beyond the well-understood non-perturbative effects. mass terms are radiatively generated in addition [151]. In SU(5) Discrete symmetries arise as subgroups of continuous gauge sym- extension of the type-I seesaw model, where r.h. neutrinos are in- troduced as SU(5) singlets with interactions λ00 (1 5¯ 5 ), the metries, such as ZN ⊂ U(1). In particular, non-anomalous sub- ij i j H groups of Stückelberg-massive U(1)s represent unbroken discrete doublet and color-triplet Higgses acquire another type of Yukawa gauge symmetries and as such are non-perturbatively exact (see coupling, respectively. They then radiatively generate flavor- e.g. [136]). Of course, such discrete gauge symmetries may also violating l.h. slepton [152] and r.h. down squark masses [153]. arise as remnants of continuous gauge symmetries after conven- These flavor-violating SUSY breaking terms induce new contri- tional 4d spontaneous breaking. butions to FCNC processes in quark and lepton sectors, such as µ → eγ and K0–K¯ 0 and B0–B¯0 mixing. Note that even if the 94.9.2 Anomaly constraints vs. GUT paradigm SUSY breaking terms are generated at MG, the r.h. neutrino As emphasized at the very beginning, the fact that the SM Yukawa coupling may induce sizable flavor violation in l.h. slep- fermions of one generation fill out the 10 + 5 of SU(5) appears to ton masses due to the running between MG and the right-handed- provide overwhelming evidence for some form of GUT embed- neutrino mass scale. ding. However, one should be aware that a counterargument EDMs are also induced when both l.h. and r.h. squarks/sleptons can be made which is related to the issue of ‘charge quantiza- have flavor-violating mass terms with relative phases, as discussed tion by anomaly cancellation’ (see [137,138] for some early papers for SO(10) in [154] or for SU(5) with r.h. neutrinos in [155]. Thus, and [139] for a more detailed reference list): Imagine we only knew such low-energy observables constrain GUT-scale interactions. that the low-energy gauge group were GSM and the matter con- 94.9.5 From GUT baryogenesis to leptogenesis and B/L- tent included the (3, 2)Y , i.e. a ‘quark doublet’ with U(1)-charge Y . One can then ask which possibilities exist of adding further violating transitions matter to ensure the cancellation of all triangle anomalies. It During inflation, any conserved quantum number is extremely turns out that this problem has only three different, minimal11 diluted. Thus, one expects the observed baryon asymmetry of the solutions [138]. One of those is precisely a single SM generation, universe to originate at reheating or in the subsequent cosmolog- with the apparent ‘SU(5)-ness’ emerging accidentally. Thus, if ical evolution. In detail, the situation is slightly more involved: one randomly picks models from the set of consistent gauge theo- Both baryon number B and lepton number L are global symme- tries of the SM. However, (B +L) is anomalous and violated by ries, preconditioning on GSM and (3, 2)Y , one may easily end up with ‘10 + 5’ of an SU(5) that is in no way dynamically present. thermal fluctuations in the early universe, via so-called sphaleron This is precisely what happens in the context of non-GUT string processes. Moreover, it is violated in GUT models, as is most model building [140]. apparent in proton decay. By contrast, (B −L) is anomaly free and preserved by both the SM as well as SU(5) or SO(10) gauge 94.9.3 Magnetic monopoles interactions. In the broken phase of a GUT there are typically localized clas- Now, the old idea of GUT baryogenesis [156,157] is to generate a sical solutions carrying magnetic charge under an unbroken U(1) (B+L) and hence a baryon asymmetry by the out-of-equilibrium symmetry [141]. These magnetic monopoles with mass of order decay of the color-triplet Higgs. However such an asymmetry, MG/αG can be produced during a possible GUT phase transition generated at GUT temperatures, is washed out by sphalerons12. in the early universe. The flux of magnetic monopoles is exper- This can be overcome [159] using lepton-number violating inter- imentally found to be less than ∼ 10−16 cm−2 s−1 sr−1 [142]. action of neutrinos to create a (B−L) from the (B+L) asymmetry, Many more are however predicted, hence the GUT monopole before sphaleron processes become sufficiently fast at T < 1012 problem. In fact, one of the original motivations for inflation GeV. This (B −L) asymmetry can then survive the subsequent was to solve the monopole problem by exponential expansion af- sphaleron dominated phase. Note that this does not work in the ter the GUT phase transition [143] and hence dilution of the minimal SUSY GUT setting, with the triplet Higgs above the

11 12 Adding extra vector-like sets of fields, e.g. two fermions which only To be precise, if lepton flavor numbers Li (i = 1–3) are nonzero and transform under U(1) and have charges Y and −Y , is considered to violate (Li −Lj ) 6= 0 (i 6= j), one may obtain nonzero values for B and L even if minimality. (B−L) = 0 [158]. 980 94. Grand Unified Theories

GUT scale. The reason is that a correspondingly high reheating part of a larger, unifying gauge symmetry at a high energy scale. temperature would be required which, as explained above, is ruled In most models, especially in the simplest and most appealing out by Planck data. variants of SU(5) and SO(10) unification, the statement is much However, the most widely accepted simple way out of the stronger: One expects the three gauge couplings to unify (up dilemma is to directly generate a net (B−L) asymmetry dynam- to small threshold corrections) at a unique scale, MG, and the ically in the early universe, also using r.h. neutrinos. Indeed, we proton to be unstable due to exchange of gauge bosons of the have seen that neutrino oscillations suggest a new scale of physics larger symmetry group. Supersymmetric grand unified theories of order 1014 GeV. This scale is associated with heavy Majorana provide, by far, the most predictive and economical framework neutrinos in the seesaw mechanism. If in the early universe, the allowing for perturbative unification. Many more details than decay of the heavy neutrinos is out of equilibrium and violates could be discussed in the present article can be found in some of both lepton number and CP, then a net lepton number may be the classic reviews [123,170] and the two books [171] (see also [172] generated. This lepton number will then be partially converted for two recent overviews). into baryon number via electroweak processes [160]. This mecha- Thus, the three classical pillars of GUTs are gauge coupling 16 nism is called leptogenesis. unification at MG ∼ 2 × 10 GeV, low-energy supersymmetry If the three heavy Majorana neutrino masses are hierarchical, (with a large SUSY desert), and nucleon decay. The first of these the net lepton number is produced by decay of the lightest one, may be viewed as predicting the value of the strong coupling – a and it is proportional to the CP asymmetry in the decay. The prediction which has already been verified (see Fig. 94.1). Numer- CP asymmetry is bounded from above, and the lightest neutrino ically, this prediction remains intact even if SUSY partner masses mass is required to be larger than 109 GeV in order to explain the are somewhat above the weak scale. However, at the conceptual observed baryon asymmetry [161]. This implies that the reheating level a continuously increasing lower bound on the SUSY scale is temperature after inflation should be larger than 109 GeV so that nevertheless problematic for the GUT paradigm: Indeed, if the the heavy neutrinos are thermally produced13. In supersymmetric independent, gauge-hierarchy-based motivation for SUSY is com- models, there is a tension between leptogenesis and Big Bang pletely abandoned, the SUSY scale and hence α3 become simply Nucleosynthesis (BBN) if gravitinos decay in the BBN era. The free parameters and the first two pillars crumble. Thus, it is im- gravitino problem gives a constraint on the reheating temperature portant to keep pushing bounds on proton decay which, although 6−10 . 10 GeV though the precise value depends on the SUSY again not completely universal in all GUT constructions, is ar- breaking parameters [163]. Recent reviews of leptogenesis can be guably a more generic part of the GUT paradigm than low-energy found in Ref. [164]. SUSY. One of the important tests of leptogenesis are searches for neu- Whether or not Yukawa couplings unify is more model depen- trinoless double-β (0νββ) decays14. In a 0νββ decay, only two dent. However, irrespective of possible (partial) Yukawa unifi- electrons but no (anti-)neutrinos are emitted by the decaying nu- cation, there certainly exists a very interesting and potentially cleus. This is in contrast to ordinary double-β decay. Thus, 0νββ fruitful interplay between flavor model building and grand unifi- decays are lepton-number-violating with ∆L = 2. At the nucleon cation. Especially in the neutrino sector this is strongly influenced level, this is described by dimension-nine effective operators for by the developing experimental situation. nn → ppee. These operators may in turn come from SM oper- Another phenomenological signature of grand unification is the ators of with dimension less than nine, in combination with SM strength of the direct coupling of the QCD axion to photons, weak interactions. The lowest one is the dimension-five opera- relative to its coupling to gluons. It is quantified by the predicted tor generating the Majorana neutrino mass terms (Eq. (94.17)). anomaly ratio E/N = 8/3 (see [173, 174]). This arises in field- Thus, if the lepton-number violating effective interactions come theoretic axion models consistent with GUT symmetry (such as from physics at energies much above the weak scale, the 0νββ de- DFSZ [175]) and in string-theoretic GUTs [174,176]. In the latter, cay rates are proportional to the Majorana neutrino masses. The the axion does not come from the phase of a complex scalar but latest experimental results are reviewed in [62]. For recent studies is a fundamental shift-symmetric real field, coupling through a of the 0νββ decay including SM operators up to mass dimension higher-dimension operator directly to the product of the GUT nine, see [165] and refs. therein. field-strength and its dual. In addition to L-violation, one can consider (B−L) and B vi- It is probably fair to say that, due to limitations of the 4d ap- olating phenomena. They are interesting in their own right and proach, including especially remaining ambiguities (free parame- may also be relevant to baryogenesis. The relevant operators have ters or ad hoc assumptions) in models of flavor and GUT break- higher mass dimension than the familiar dimension-six (B + L)- ing, the string theoretic approach has become more important violating operators (cf. Sec. 94.6). They may be predicted in in GUT model building. In this framework, challenges include SO(10) GUTs with an intermediate scale, at which baryogene- learning how to deal with the many vacua of the ‘landscape’ as sis is realized, such as in [166]. First, one may have nucleon well as, for each vacuum, developing the tools for reliably cal- − + decays with ∆(B − L) = 2, such as n → e π . This is in- culating detailed, phenomenological observables. Finally, due to duced by dimension-seven effective operators in the SM, which limitations of space, the present article has barely touched on the are suppressed by the SM Higgs VEV or derivatives. Second, interesting cosmological implications of GUTs. They may become there are neutron- (n–n¯) oscillations, which are in- more important in the future, especially in the case that a high duced by ∆B = 2 dimension-nine effective operators in the SM. inflationary energy scale is established observationally. The upper bound on the mean time for n–n¯ transitions is directly derived using free neutrons [167]. It is also constrained from the References lower limit on the lifetime for neutrons bound in 16O, derived by [1] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979). Super-Kamiokande [168]. Their results are very similar. Super- [2] P. Minkowski, Phys. Lett. 67B, 421 (1977). Kamiokande also searches for dinucleon decays with ∆B = 2, such as pp → π+π+ and nn → π±π∓ [169]. [3] T. Yanagida, in Proceedings of the Workshop on the Uni- fied Theory and the Baryon Number of the Universe, eds. O. 94.10 Conclusion Sawada and A. Sugamoto, KEK report No. 79-18, Tsukuba, Most conservatively, grand unification means that (some of) the Japan, 1979; S. Glashow, Quarks and leptons, published SM gauge interactions of U(1)Y , SU(2)L and SU(3)C become in Proceedings of the Carg‘ese Lectures, M. Levy (ed.), Plenum Press, New York, (1980); M. Gell-Mann, P. Ra- 13 This constraint may be avoided in resonant leptogenesis [162], in which mond and R. Slansky, in Supergravity, ed. P. van Nieuwen- the right-handed neutrinos are required to be almost degenerate in mass. huizen et al., North-Holland, Amsterdam, (1979), p. 315 14 Another important test of leptogenesis would be the observation of [arXiv:1306.4669]. CP violation in neutrino oscillations. Strictly speaking, the CP phase in the PMNS matrix does not contribute to 1 in the seesaw model. Nevertheless, the observation of CP violation in neutrino oscillations would suggest that [4] F. Wilczek, eConf C790823, 437 (1979); E. Witten, Phys. the seesaw mechanism is associated with a large CP violation, similarly to Lett. 91B, 81 (1980); R. N. Mohapatra and G. Senjanovic, the quark sector. Phys. Rev. Lett. 44, 912 (1980), [,231(1979)]. 94. Grand Unified Theories 981

[5] G. ’t Hooft, NATO Sci. Ser. B 59, 135 (1980); M. J. G. [20] B. R. Greene et al., Nucl. Phys. B278, 667 (1986); B. R. Veltman, Acta Phys. Polon. B12, 437 (1981); L. Maiani, Greene et al., Nucl. Phys. B292, 606 (1987); B. R. Greene, Gif-sur-Yvette Summer School on Particle Physics, 11th, C. A. Lutken and G. G. Ross, Nucl. Phys. B325, 101 (1989); Gif-sur-Yvette, France, 1979 (Inst. Nat. Phys. Nucl. Phys. J. E. Kim, Phys. Lett. B591, 119 (2004), [hep-ph/0403196]. Particules, Paris, 1979); E. Witten, Nucl. Phys. B188, 513 [21] E. Witten, Phys. Lett. 105B, 267 (1981). (1981). [22] A. Masiero et al., Phys. Lett. 115B, 380 (1982); B. Grin- [6] S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D24, stein, Nucl. Phys. B206, 387 (1982). 1681 (1981); S. Dimopoulos and H. Georgi, Nucl. Phys. [23] S. Dimopoulos and F. Wilczek, Proceedings Erice Summer B193, 150 (1981); L. E. Ibanez and G. G. Ross, Phys. School, ed. A. Zichichi (1981); M. Srednicki, Nucl. Phys. Lett. 105B, 439 (1981); N. Sakai, Z. Phys. C11, 153 (1981); B202, 327 (1982). M. B. Einhorn and D. R. T. Jones, Nucl. Phys. B196, 475 (1982); W. J. Marciano and G. Senjanovic, Phys. Rev. D25, [24] K. Inoue, A. Kakuto and H. Takano, Prog. Theor. Phys. 3092 (1982). 75, 664 (1986). [25] Y. Yamada, Z. Phys. C60, 83 (1993); J. Hisano et al., Phys. [7] U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260, Lett. B342, 138 (1995), [hep-ph/9406417]; G. Altarelli, 447 (1991); J. R. Ellis, S. Kelley and D. V. Nanopoulos, F. Feruglio and I. Masina, JHEP 11, 040 (2000), [hep- Phys. Lett. B260, 131 (1991); P. Langacker and M.-x. ph/0007254]. Luo, Phys. Rev. D44, 817 (1991); C. Giunti, C.W. Kim and U.W. Lee, Mod. Phys. Lett. A6, 1745 (1991); P. Lan- [26] K. S. Babu and S. M. Barr, Phys. Rev. D48, 5354 (1993), gacker and N. Polonsky, Phys. Rev. D47, 4028 (1993), [hep- [hep-ph/9306242]; K. S. Babu and S. M. Barr, Phys. ph/9210235]; M. Carena, S. Pokorski and C. E. M. Wagner, Rev. D50, 3529 (1994), [hep-ph/9402291]; K. S. Babu, Nucl. Phys. B406, 59 (1993), [hep-ph/9303202]; See also J. C. Pati and Z. Tavartkiladze, JHEP 06, 084 (2010), the review by S. Dimopoulos, S.A. Raby and F. Wilczek, [arXiv:1003.2625]. Physics Today, p. 25 October (1991). [27] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 74, [8] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. 2418 (1995), [hep-ph/9410326]; V. Lucas and S. Raby, Phys. Phys. 85, 1 (1991); J. R. Ellis, G. Ridolfi and F. Zwirner, Rev. D54, 2261 (1996), [hep-ph/9601303]; T. Blazek et al., Phys. Lett. B257, 83 (1991); H. E. Haber and R. Hempfling, Phys. Rev. D56, 6919 (1997), [hep-ph/9611217]; S. M. Phys. Rev. Lett. 66, 1815 (1991). Barr and S. Raby, Phys. Rev. Lett. 79, 4748 (1997), [hep- ph/9705366]; K. S. Babu, J. C. Pati and F. Wilczek, Nucl. [9] M. Carena et al., Phys. Lett. B355, 209 (1995), [hep- Phys. B566, 33 (2000), [hep-ph/9812538]; R. Dermisek, ph/9504316]; G. Degrassi et al., Eur. Phys. J. C28, 133 A. Mafi and S. Raby, Phys. Rev. D63, 035001 (2001), [hep- (2003), [hep-ph/0212020]; P. Kant et al., JHEP 08, 104 ph/0007213]. (2010), [arXiv:1005.5709]. [28] R. Barbieri, G. R. Dvali and A. Strumia, Nucl. Phys. [10] J. L. Feng et al., Phys. Rev. Lett. 111, 131802 (2013), B391, 487 (1993); Z. Berezhiani, C. Csaki and L. Randall, [arXiv:1306.2318]; S. Heinemeyer (2014), [arXiv:1405.3781]; Nucl. Phys. B444, 61 (1995), [hep-ph/9501336]; Q. Shafi P. Draper and H. Rzehak, Phys. Rept. 619, 1 (2016), and Z. Tavartkiladze, Phys. Lett. B522, 102 (2001), [hep- [arXiv:1601.01890]. ph/0105140]. [11] J. C. Pati and A. Salam, Phys. Rev. D8, 1240 (1973); For [29] G. F. Giudice and A. Masiero, Phys. Lett. B206, 480 more discussion on the standard charge assignments in this (1988); J. E. Kim and H. P. Nilles, Mod. Phys. Lett. A9, formalism, see A. Davidson, Phys. Rev. D20, 776 (1979) 3575 (1994), [hep-ph/9406296]. and R.N Mohapatra and R.E. Marshak, Phys. Lett. B91, [30] L. Randall and C. Csaki, in “Supersymmetry and unifica- 222 (1980); see also J. C. Pati, Int. J. Mod. Phys. A 32, tion of fundamental interactions. Proceedings, International 1741013 (2017), arXiv:1706.09531 for a recent account. Workshop, SUSY 95, Palaiseau, France, May 15-19, 1995,” [12] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 99–109 (1995), [,235(1995)], [hep-ph/9508208]. (1974). [31] E. Witten, in “Supersymmetry and unification of fun- [13] J. R. Ellis, M. K. Gaillard and D. V. Nanopou- damental interactions. Proceedings, 10th International los, Phys. Lett. 80B, 360 (1979), [Erratum: Phys. Conference, SUSY’02, Hamburg, Germany, June 17- Lett.82B,464(1979)]; E. Golowich, Phys. Rev. D24, 2899 23, 2002,” 472–491 (2001), [hep-ph/0201018], URL (1981). http://www-library.desy.de/preparch/desy/proc/ proc02-02/Proceedings/susy02/special/hertz_pr.ps; [14] S. Weinberg, Phys. Rev. D26, 287 (1982); N. Sakai and M. Dine, Y. Nir and Y. Shadmi, Phys. Rev. D66, 115001 T. Yanagida, Nucl. Phys. B197, 533 (1982). (2002), [hep-ph/0206268]. [15] H. Georgi, Particles and Fields, Proceedings of the APS [32] A. Hebecker, J. March-Russell and R. Ziegler, JHEP 08, 064 Div. of Particles and Fields, ed. C. Carlson, p. 575 (1975); (2009), [arXiv:0801.4101]; F. Brummer et al., JHEP 08, 011 H. Fritzsch and P. Minkowski, Annals Phys. 93, 193 (1975). (2009), [arXiv:0906.2957]; F. Brummer et al., JHEP 04, 006 [16] R. Slansky, Phys. Rept. 79, 1 (1981); H. Georgi, Front. (2010), [arXiv:1003.0084]. Phys. 54, 1 (1982); R. Feger and T. W. Kephart, Comput. [33] H. M. Lee et al., Phys. Lett. B694, 491 (2011), Phys. Commun. 192, 166 (2015), [arXiv:1206.6379]; N. Ya- [arXiv:1009.0905]; R. Kappl et al., Nucl. Phys. B847, 325 matsu, PTEP 2016, 4, 043B02 (2016), [arXiv:1512.05559]. (2011), [arXiv:1012.4574]; H. M. Lee et al., Nucl. Phys. [17] S. M. Barr, Phys. Lett. 112B, 219 (1982); J. P. Derendinger, B850, 1 (2011), [arXiv:1102.3595]. J. E. Kim and D. V. Nanopoulos, Phys. Lett. 139B, 170 [34] Y. Hosotani, Phys. Lett. 126B, 309 (1983). (1984); I. Antoniadis et al., Phys. Lett. B194, 231 (1987); [35] L. J. Dixon et al., Nucl. Phys. B261, 678 (1985), I. Antoniadis et al., Phys. Lett. B231, 65 (1989). [,678(1985)]; L. J. Dixon et al., Nucl. Phys. B274, 285 [18] F. Gursey, P. Ramond and P. Sikivie, Phys. Lett. 60B, 177 (1986); L. E. Ibanez, H. P. Nilles and F. Quevedo, Phys. (1976). Lett. B187, 25 (1987); L. E. Ibanez et al., Phys. Lett. B191, 282 (1987). [19] A. de Rujula, H. Georgi and S. L. Glashow, 5th Workshop on Grand Unification, ed. K. Kang, H. Fried and P. Frampton, [36] P. Candelas et al., Nucl. Phys. B258, 46 (1985). World Scientific, Singapore (1984), p. 88; See also earlier pa- [37] Y. Kawamura, Prog. Theor. Phys. 103, 613 (2000), [hep- per by Y. Achiman and B. Stech, p. 303, “New Phenomena ph/9902423]; Y. Kawamura, Prog. Theor. Phys. 105, in Lepton-Hadron Physics,” ed. D.E.C. Fries and J. Wess, 999 (2001), [hep-ph/0012125]; G. Altarelli and F. Fer- Plenum, NY (1979). uglio, Phys. Lett. B511, 257 (2001), [hep-ph/0102301]; 982 94. Grand Unified Theories

L. J. Hall and Y. Nomura, Phys. Rev. D64, 055003 [55] R. Donagi and M. Wijnholt, Adv. Theor. Math. Phys. (2001), [hep-ph/0103125]; A. Hebecker and J. March- 15, 6, 1523 (2011), [arXiv:0808.2223]; R. Blumenhagen, Russell, Nucl. Phys. B613, 3 (2001), [hep-ph/0106166]; Phys. Rev. Lett. 102, 071601 (2009), [arXiv:0812.0248]; K.- T. Asaka, W. Buchmuller and L. Covi, Phys. Lett. B523, S. Choi and J. E. Kim, Phys. Rev. D83, 065016 (2011), 199 (2001), [hep-ph/0108021]; L. J. Hall et al., Phys. Rev. [arXiv:1012.0847]; C. Mayrhofer, E. Palti and T. Weigand, D65, 035008 (2002), [hep-ph/0108071]; R. Dermisek and JHEP 09, 082 (2013), [arXiv:1303.3589]; G. K. Leon- A. Mafi, Phys. Rev. D65, 055002 (2002), [hep-ph/0108139]; taris and Q. Shafi, Phys. Rev. D96, 6, 066023 (2017), H. D. Kim and S. Raby, JHEP 01, 056 (2003), [hep- [arXiv:1706.08372]. ph/0212348]. [56] E. Witten, Nucl. Phys. B471, 135 (1996), [hep-th/9602070]. [38] L.E. Ibanez and A.M. Uranga, “String theory and particle [57] A. Hebecker and M. Trapletti, Nucl. Phys. B713, 173 physics: An introduction to string phenomenology,” Cam- (2005), [hep-th/0411131]. bridge University Press 2012; K.-S. Choi and J. E. Kim, [58] B. Dundee and S. Raby (2008), [arXiv:0808.0992]. Lect. Notes Phys. 696, 1 (2006); R. Blumenhagen et al., Phys. Rept. 445, 1 (2007), [hep-th/0610327]. [59] G. G. Ross (2004), [hep-ph/0411057]. [39] D. J. Gross et al., Phys. Rev. Lett. 54, 502 (1985). [60] H. Georgi, H. R. Quinn and S. Weinberg, Phys. Rev. Lett. 33, 451 (1974); S. Weinberg, Phys. Lett. 91B, 51 (1980); [40] V. Braun et al., JHEP 05, 043 (2006), [hep-th/0512177]; L. J. Hall, Nucl. Phys. B178, 75 (1981). V. Bouchard and R. Donagi, Phys. Lett. B633, 783 (2006), [hep-th/0512149]; L. B. Anderson et al., Phys. Rev. D84, [61] B. C. Allanach, Comput. Phys. Commun. 143, 305 (2002), 106005 (2011), [arXiv:1106.4804]. [hep-ph/0104145]. [62] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, [41] L. J. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. 3, 030001 (2018). B355, 649 (1991). [63] M. L. Alciati et al., JHEP 03, 054 (2005), [hep-ph/0501086]. [42] G. Aldazabal et al., Nucl. Phys. B452, 3 (1995), [hep- th/9410206]; Z. Kakushadze et al., Int. J. Mod. Phys. A13, [64] See talks on proposed and running nucleon decay exper- 2551 (1998), [hep-th/9710149]. iments, and theoretical talks by P. Langacker, p. 131, and W.J. Marciano and A. Sirlin, p. 151, in The Second [43] E. Witten, Nucl. Phys. B258, 75 (1985). Workshop on Grand Unification, eds. J.P. Leveille et al., [44] S. Gukov et al., Phys. Rev. D69, 086008 (2004), [hep- Birkhäuser, Boston (1981). th/0310159]; G. Curio, A. Krause and D. Lust, Fortsch. [65] W.J. Marciano, p. 190, Eighth Workshop on Grand Unifica- Phys. 54, 225 (2006), [hep-th/0502168]; L. B. Anderson tion, ed. K. Wali, World Scientific Publishing Co., Singapore et al., Phys. Rev. D83, 106011 (2011), [arXiv:1102.0011]. (1987). [45] R. Friedman, J. Morgan and E. Witten, Commun. Math. [66] M.S. Carena et al., in Ref. [7]. Phys. 187, 679 (1997), [hep-th/9701162]. [67] A. Djouadi, J.-L. Kneur and G. Moultaka, Comput. Phys. [46] R. Blumenhagen, G. Honecker and T. Weigand, JHEP 08, Commun. 176, 426 (2007), [hep-ph/0211331]. 009 (2005), [hep-th/0507041]; L. B. Anderson et al., JHEP [68] W. Porod and F. Staub, Comput. Phys. Commun. 183, 01, 047 (2014), [arXiv:1307.4787]. 2458 (2012), [arXiv:1104.1573]. [47] T. Kobayashi, S. Raby and R.-J. Zhang, Phys. Lett. B593, [69] W. Siegel, Phys. Lett. 94B, 37 (1980). 262 (2004), [hep-ph/0403065]. [70] I. Antoniadis, C. Kounnas, and R. Lacaze, Nucl. Phys. [48] S. Forste et al., Phys. Rev. D70, 106008 (2004), [hep- B221, 377 (1983). th/0406208]; T. Kobayashi, S. Raby and R.-J. Zhang, Nucl. [71] M. A. Shifman, Int. J. Mod. Phys. A11, 5761 (1996), Phys. B704, 3 (2005), [hep-ph/0409098]; W. Buchmuller [hep-ph/9606281]; N. Arkani-Hamed and H. Murayama, et al., Nucl. Phys. B712, 139 (2005), [hep-ph/0412318]; JHEP 06, 030 (2000), [hep-th/9707133]; I. Jack, D. R. T. W. Buchmuller et al., Phys. Rev. Lett. 96, 121602 (2006), Jones and A. Pickering, Phys. Lett. B435, 61 (1998), [hep- [hep-ph/0511035]; W. Buchmuller et al., Nucl. Phys. B785, ph/9805482]. 149 (2007), [hep-th/0606187]; O. Lebedev et al., Phys. Lett. B645, 88 (2007), [hep-th/0611095]; J. E. Kim, [72] M.B. Einhorn and D.R.T. Jones in Ref. [6]; I. Antoniadis, J.-H. Kim and B. Kyae, JHEP 06, 034 (2007), [hep- C. Kounnas and K. Tamvakis, Phys. Lett. 119B, 377 (1982). ph/0702278]; O. Lebedev et al., Phys. Rev. D77, 046013 [73] J. Hisano, H. Murayama and T. Yanagida, Phys. Rev. Lett. (2008), [arXiv:0708.2691]. 69, 1014 (1992); J. Hisano, H. Murayama and T. Yanagida, [49] S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. Nucl. Phys. B402, 46 (1993), [hep-ph/9207279]. D66, 106006 (2002), [hep-th/0105097]; S. Kachru et al., [74] H. Murayama and A. Pierce, Phys. Rev. D65, 055009 Phys. Rev. D68, 046005 (2003), [hep-th/0301240]. (2002), [hep-ph/0108104]. [50] R. Blumenhagen et al., Nucl. Phys. B815, 1 (2009), [75] G. Anderson et al., eConf C960625, SUP107 (1996) [hep- [arXiv:0811.2936]. ph/9609457]. [51] R. Blumenhagen et al., Nucl. Phys. B616, 3 (2001), [hep- [76] L. Roszkowski and M. A. Shifman, Phys. Rev. D53, th/0107138]. 404 (1996), [hep-ph/9503358]; S. Raby, M. Ratz and K. Schmidt-Hoberg, Phys. Lett. B687, 342 (2010), [52] R. Donagi and M. Wijnholt, Adv. Theor. Math. Phys. 15, [arXiv:0911.4249]. 5, 1237 (2011), [arXiv:0802.2969]. [77] J. Hisano, T. Kuwahara and N. Nagata, Phys. Lett. B723, [53] C. Beasley, J. J. Heckman and C. Vafa, JHEP 01, 058 324 (2013), [arXiv:1304.0343]. (2009), [arXiv:0802.3391]; C. Beasley, J. J. Heckman and [78] N. Arkani-Hamed, A. Delgado and G. F. Giudice, Nucl. C. Vafa, JHEP 01, 059 (2009), [arXiv:0806.0102]. Phys. B741, 108 (2006), [hep-ph/0601041]. [54] T. Weigand, Class. Quant. Grav. 27, 214004 (2010), [79] M. Ibe, T. Moroi and T. T. Yanagida, Phys. Lett. B644, [arXiv:1009.3497]; J. J. Heckman, Ann. Rev. Nucl. Part. 355 (2007), [hep-ph/0610277]. Sci. 60, 237 (2010), [arXiv:1001.0577]; M. Cvetic, I. Garcia- Etxebarria and J. Halverson, JHEP 01, 073 (2011), [80] G. F. Giudice et al., JHEP 12, 027 (1998), [hep- [arXiv:1003.5337]; A. Maharana and E. Palti, Int. J. Mod. ph/9810442]; L. Randall and R. Sundrum, Nucl. Phys. Phys. A28, 1330005 (2013), [arXiv:1212.0555]; S. Krippen- B557, 79 (1999), [hep-th/9810155]. dorf, S. Schafer-Nameki and J.-M. Wong, JHEP 11, 008 [81] R. N. Mohapatra and M. K. Parida, Phys. Rev. D47, 264 (2015), [arXiv:1507.05961]. (1993), [hep-ph/9204234]. 94. Grand Unified Theories 983

[82] D.-G. Lee et al., Phys. Rev. D51, 229 (1995), [hep- Phys. Rev. D49, 3660 (1994), [hep-ph/9308333]; B. Anan- ph/9404238]. thanarayan, Q. Shafi and X. M. Wang, Phys. Rev. D50, [83] L. E. Ibanez et al., JHEP 07, 195 (2012), [arXiv:1206.2655]. 5980 (1994), [hep-ph/9311225]. [84] K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Rev. Lett. [108] L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D50, 7048 91, 061601 (2003), [hep-th/0210294]. (1994), [hep-ph/9306309]; M. Carena et al., Nucl. Phys. B426, 269 (1994), [hep-ph/9402253]. [85] K. Agashe, R. Contino and R. Sundrum, Phys. Rev. Lett. 95, 171804 (2005), [hep-ph/0502222]. [109] A. Anandakrishnan, B. C. Bryant and S. Raby, JHEP 05, 088 (2015), [arXiv:1411.7035]. [86] F. Wilczek and A. Zee, Phys. Rev. Lett. 43, 1571 (1979). [110] M. Lanzagorta and G. G. Ross, Phys. Lett. B364, 163 [87] K. Abe et al. (Super-Kamiokande), Phys. Rev. D95, (1995), [hep-ph/9507366]. 1, 012004 (2017), [arXiv:1610.03597]; S. Mine (Super Kamiokande), J. Phys. Conf. Ser. 718, 6, 062044 (2016). [111] K. Tobe and J. D. Wells, Nucl. Phys. B663, 123 (2003), [88] K. Abe et al. (Hyper-Kamiokande) (2018), [hep-ph/0301015]. [arXiv:1805.04163]. [112] G. Ross and M. Serna, Phys. Lett. B664, 97 (2008), [89] Y. Aoki et al., Phys. Rev. D96, 1, 014506 (2017), [arXiv:0704.1248]. [arXiv:1705.01338]. [113] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. Lett. [90] J. Hisano, T. Kuwahara and Y. Omura, Nucl. Phys. 88, 111804 (2002), [hep-ph/0107097]; T. Blazek, R. Der- B898, 1 (2015), [Erratum: Nucl. Phys.B907,476(2016)], misek and S. Raby, Phys. Rev. D65, 115004 (2002), [hep- [arXiv:1503.08561]. ph/0201081]; D. Auto et al., JHEP 06, 023 (2003), [hep- ph/0302155]; R. Dermisek et al., JHEP 04, 037 (2003), [91] R. Barbier et al., Phys. Rept. 420, 1 (2005), [hep- [hep-ph/0304101]; R. Dermisek et al., JHEP 09, 029 (2005), ph/0406039]; F. Domingo et al., JHEP 02, 066 (2019), [hep-ph/0507233]. [arXiv:1810.08228]. [114] A. Anandakrishnan, S. Raby and A. Wingerter, Phys. Rev. [92] G. R. Farrar and P. Fayet, Phys. Lett. 76B, 575 (1978). D87, 5, 055005 (2013), [arXiv:1212.0542]; M. Adeel Ajaib [93] S. Dimopoulos, S. Raby and F. Wilczek, Phys. Lett. 112B, et al., JHEP 07, 139 (2013), [arXiv:1303.6964]; Z. Poh 133 (1982); J. R. Ellis, D. V. Nanopoulos and S. Rudaz, and S. Raby, Phys. Rev. D92, 1, 015017 (2015), Nucl. Phys. B202, 43 (1982). [arXiv:1505.00264]; M. Badziak, M. Olechowski and [94] C. S. Aulakh et al., Nucl. Phys. B597, 89 (2001), [hep- S. Pokorski, JHEP 10, 088 (2013), [arXiv:1307.7999]. ph/0004031]. [115] H. Georgi and C. Jarlskog, Phys. Lett. 86B, 297 (1979). [95] T. Goto and T. Nihei, Phys. Rev. D59, 115009 (1999), [hep- [116] S. Antusch and M. Spinrath, Phys. Rev. D79, 095004 ph/9808255]. (2009), [arXiv:0902.4644]. [96] G. D. Coughlan et al., Phys. Lett. 160B, 249 (1985); J. L. [117] G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181, Chkareuli and I. G. Gogoladze, Phys. Rev. D58, 055011 287 (1981). (1998), [hep-ph/9803335]. [118] T. E. Clark, T.-K. Kuo and N. Nakagawa, Phys. Lett. [97] G. F. Giudice and A. Romanino, Nucl. Phys. B699, 115B, 26 (1982); K. S. Babu and R. N. Mohapatra, Phys. 65 (2004), [Erratum: Nucl. Phys.B706,487(2005)], [hep- Rev. Lett. 70, 2845 (1993), [hep-ph/9209215]. ph/0406088]. [119] R. Barbieri and D. V. Nanopoulos, Phys. Lett. 91B, 369 [98] J. Hisano et al., JHEP 07, 038 (2013), [arXiv:1304.3651]. (1980). [99] B. Bajc, P. Fileviez Perez and G. Senjanovic, Phys. Rev. D66, 075005 (2002), [hep-ph/0204311]. [120] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B147, 277 (1979). [100] K.-S. Choi, Phys. Lett. B668, 392 (2008), [arXiv:0807.2766]. [121] R. Barbieri et al., Nucl. Phys. B493, 3 (1997), [hep- ph/9610449]; T. Blazek, S. Raby and K. Tobe, Phys. [101] F. An et al. (JUNO), J. Phys. G43, 3, 030401 (2016), Rev. D60, 113001 (1999), [hep-ph/9903340]; T. Blazek, [arXiv:1507.05613]. S. Raby and K. Tobe, Phys. Rev. D62, 055001 (2000), [102] R. Acciarri et al. (DUNE) (2015), [arXiv:1512.06148]. [hep-ph/9912482]; Q. Shafi and Z. Tavartkiladze, Phys. [103] L. J. Hall and Y. Nomura, Phys. Rev. D66, 075004 (2002), Lett. B487, 145 (2000), [hep-ph/9910314]; C. H. Albright [hep-ph/0205067]; H. D. Kim, S. Raby and L. Schradin, and S. M. Barr, Phys. Rev. Lett. 85, 244 (2000), [hep- JHEP 05, 036 (2005), [hep-ph/0411328]. ph/0002155]; Z. Berezhiani and A. Rossi, Nucl. Phys. B594, 113 (2001), [hep-ph/0003084]; C. H. Albright and S. M. [104] M.S. Chanowitz, J.R. Ellis and M.K. Gaillard, Nucl. Phys. Barr, Phys. Rev. D64, 073010 (2001), [hep-ph/0104294]; B128, 506 (1977); A. J. Buras et al., Nucl. Phys. B135, 66 M.-C. Chen and K. T. Mahanthappa, Int. J. Mod. Phys. (1978). A18, 5819 (2003), [hep-ph/0305088]; R. Dermisek and [105] K. Inoue et al., Prog. Theor. Phys. 67, 1889 (1982); L. E. S. Raby, Phys. Lett. B622, 327 (2005), [hep-ph/0507045]. Ibanez and C. Lopez, Nucl. Phys. B233, 511 (1984). [122] H. Georgi, Nucl. Phys. B156, 126 (1979); P. H. Frampton, [106] H. Georgi and D. V. Nanopoulos, Nucl. Phys. B159, 16 Phys. Lett. 88B, 299 (1979); P. Frampton and S. Nandi, (1979); J. A. Harvey, P. Ramond and D. B. Reiss, Phys. Phys. Rev. Lett. 43, 1460 (1979); J.E. Kim, Phys. Rev. Lett. 92B, 309 (1980); J. A. Harvey, D. B. Reiss and P. Ra- Lett. 45, 1916 (1980), Phys. Rev. D23, 2706 (1981) and mond, Nucl. Phys. B199, 223 (1982). Phys. Rev. D26, 674 (1982); Y. Fujimoto, Phys. Rev. D26, [107] T. Banks, Nucl. Phys. B303, 172 (1988); M. Olechowski 3183 (1982). and S. Pokorski, Phys. Lett. B214, 393 (1988); S. Poko- [123] P. Langacker, Phys. Rept. 72, 185 (1981). rski, Nucl. Phys. Proc. Suppl. 13, 606 (1990); B. Anan- thanarayan, G. Lazarides and Q. Shafi, Phys. Rev. D44, [124] H. Georgi, Conf. Proc. C 820726, 705 (1982). 1613 (1991); Q. Shafi and B. Ananthanarayan, ICTP Sum- [125] S. M. Barr, Phys. Rev. D78, 055008 (2008), mer School lectures (1991); S. Dimopoulos, L. J. Hall and [arXiv:0805.4808]; Y. Goto, Y. Kawamura and T. Miura, S. Raby, Phys. Rev. Lett. 68, 1984 (1992); S. Dimopou- Phys. Rev. D88, 5, 055016 (2013), [arXiv:1307.2631]; J. E. los, L. J. Hall and S. Raby, Phys. Rev. D45, 4192 (1992); Kim, JHEP 06, 114 (2015), [arXiv:1503.03104]; C.H. Al- G. W. Anderson et al., Phys. Rev. D47, R3702 (1993), bright, R.P. Feger and T.W. Kephart, arXiv:1601.07523; [hep-ph/9209250]; B. Ananthanarayan, G. Lazarides and M. Reig, J. W. F. Valle, C. A. Vaquera-Araujo and Q. Shafi, Phys. Lett. B300, 245 (1993); G. Anderson et al., F. Wilczek, arXiv:1706.03116. 984 94. Grand Unified Theories

[126] M. Magg and C. Wetterich, Phys. Lett. 94B, 61 (1980); [150] L. J. Hall, V. A. Kostelecky and S. Raby, Nucl. Phys. B267, J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227 415 (1986). (1980); R. N. Mohapatra and G. Senjanovic, Phys. Rev. [151] R. Barbieri and L. J. Hall, Phys. Lett. B338, 212 (1994), D23, 165 (1981); G. B. Gelmini and M. Roncadelli, Phys. [hep-ph/9408406]; R. Barbieri, L. J. Hall and A. Strumia, Lett. 99B, 411 (1981). Nucl. Phys. B445, 219 (1995), [hep-ph/9501334]. [127] R. Foot et al., Z. Phys. C44, 441 (1989). [152] F. Borzumati and A. Masiero, Phys. Rev. Lett. 57, 961 [128] Y. Fukuda et al. (Super-Kamiokande), Phys. Rev. Lett. 81, (1986); J. Hisano et al., Phys. Lett. B357, 579 (1995), [hep- 1562 (1998), [hep-ex/9807003]. ph/9501407]; J. Hisano et al., Phys. Rev. D53, 2442 (1996), [129] G. L. Fogli et al., Phys. Rev. D84, 053007 (2011), [hep-ph/9510309]; J. Hisano and D. Nomura, Phys. Rev. [arXiv:1106.6028]. D59, 116005 (1999), [hep-ph/9810479]. [130] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys. [153] T. Moroi, JHEP 03, 019 (2000), [hep-ph/0002208]; 13, 109401 (2011), [arXiv:1108.1376]. D. Chang, A. Masiero and H. Murayama, Phys. Rev. D67, 075013 (2003), [hep-ph/0205111]. [131] G. Altarelli, Soryushiron Kenkyu Electron. 116, A29 (2008), [arXiv:0711.0161]; S. F. King and C. Luhn, Rept. [154] S. Dimopoulos and L. J. Hall, Phys. Lett. B344, 185 (1995), Prog. Phys. 76, 056201 (2013), [arXiv:1301.1340]. [hep-ph/9411273]. [132] C. H. Albright and M.-C. Chen, Phys. Rev. D74, 113006 [155] J. Hisano et al., Phys. Lett. B604, 216 (2004), [hep- (2006), [hep-ph/0608137]. ph/0407169]. [133] R. Kallosh et al., Phys. Rev. D52, 912 (1995), [hep- [156] A. Yu. Ignatiev et al., Phys. Lett. 76B, 436 (1978); th/9502069]. M. Yoshimura, Phys. Rev. Lett. 41, 281 (1978), [Erratum: Phys. Rev. Lett.42,746(1979)]. [134] H. Ruegg and M. Ruiz-Altaba, Int. J. Mod. Phys. A19, 3265 (2004), [hep-th/0304245]. [157] D. Toussaint et al., Phys. Rev. D19, 1036 (1979); S. Wein- berg, Phys. Rev. Lett. 42, 850 (1979); M. Yoshimura, Phys. [135] M. B. Green and J. H. Schwarz, Phys. Lett. 149B, 117 Lett. 88B, 294 (1979); S. M. Barr, G. Segre and H. A. Wel- (1984). don, Phys. Rev. D20, 2494 (1979); D. V. Nanopoulos and [136] L. E. Ibanez and G. G. Ross, Nucl. Phys. B368, 3 (1992); S. Weinberg, Phys. Rev. D20, 2484 (1979); A. Yildiz and M. Berasaluce-Gonzalez et al., JHEP 12, 113 (2011), P. H. Cox, Phys. Rev. D21, 906 (1980). [arXiv:1106.4169]. [158] H. K. Dreiner and G. G. Ross, Nucl. Phys. B410, 188 [137] N.G. Deshpande, OITS-107; C. Q. Geng and R. E. Marshak, (1993), [hep-ph/9207221]. Phys. Rev. D39, 693 (1989); A. Font, L. E. Ibanez and [159] M. Fukugita and T. Yanagida, Phys. Rev. Lett. 89, 131602 F. Quevedo, Phys. Lett. B228, 79 (1989); K. S. Babu and (2002), [hep-ph/0203194]. R. N. Mohapatra, Phys. Rev. Lett. 63, 938 (1989). [160] M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 (1986). [138] R. Foot et al., Phys. Rev. D39, 3411 (1989). [161] S. Davidson and A. Ibarra, Phys. Lett. B535, 25 [139] M. Nowakowski and A. Pilaftsis, Phys. Rev. D48, 259 (2002), [hep-ph/0202239]; K. Hamaguchi, H. Murayama (1993), [hep-ph/9304312]. and T. Yanagida, Phys. Rev. D65, 043512 (2002), [hep- [140] T. P. T. Dijkstra, L. R. Huiszoon and A. N. ph/0109030]. Schellekens, Phys. Lett. B609, 408 (2005), [hep- [162] A. Pilaftsis and T. E. J. Underwood, Phys. Rev. D72, th/0403196]; F. Gmeiner et al., JHEP 01, 004 (2006), [hep- 113001 (2005), [hep-ph/0506107]. th/0510170]; B. Gato-Rivera and A. N. Schellekens, Nucl. Phys. B883, 529 (2014), [arXiv:1401.1782]. [163] M. Kawasaki et al., Phys. Rev. D78, 065011 (2008), [arXiv:0804.3745]. [141] G. ’t Hooft, Nucl. Phys. B79, 276 (1974) A.M. Polyakov, Pis’ma Zh. Eksp. Teor. Fiz. 20, 430 (1974) [Sov. Phys. JETP [164] W. Buchmuller, R. D. Peccei and T. Yanagida, Ann. Rev. Lett. 20, 194 (1974)]; For a pedagogical introduction, see Nucl. Part. Sci. 55, 311 (2005), [hep-ph/0502169]; C. S. S. Coleman, in Aspects of Symmetry, Selected Erice Lec- Fong, E. Nardi and A. Riotto, Adv. High Energy Phys. tures, Cambridge University Press, Cambridge, (1985), and 2012, 158303 (2012), [arXiv:1301.3062]. P. Goddard and D. Olive, Rept. on Prog. in Phys. 41, 1357 [165] V. Cirigliano et al., JHEP 12, 097 (2018), (1978). [arXiv:1806.02780]. [142] M. Ambrosio et al. (MACRO), Eur. Phys. J. C25, [166] K. S. Babu and R. N. Mohapatra, Phys. Rev. D86, 035018 511 (2002), [hep-ex/0207020]; S. Balestra et al. (2011), (2012), [arXiv:1203.5544]; K. S. Babu and R. N. Mohapatra, [arXiv:1105.5587]; L. Patrizii and M. Spurio, Ann. Rev. Phys. Lett. B715, 328 (2012), [arXiv:1206.5701]; K. S. Babu Nucl. Part. Sci. 65, 279 (2015), [arXiv:1510.07125]. and R. N. Mohapatra, Phys. Rev. Lett. 109, 091803 (2012), [143] For a review, see A.D. Linde, Particle Physics and Inflation- [arXiv:1207.5771]. ary Cosmology, Harwood Academic, Switzerland (1990). [167] M. Baldo-Ceolin et al., Z. Phys. C63, 409 (1994). [144] G. R. Dvali, H. Liu and T. Vachaspati, Phys. Rev. Lett. 80, [168] K. Abe et al. (Super-Kamiokande), Phys. Rev. D91, 072006 2281 (1998), [hep-ph/9710301]. (2015), [arXiv:1109.4227]. [145] P. Langacker and S.-Y. Pi, Phys. Rev. Lett. 45, 1 (1980). [169] J. Gustafson et al. (Super-Kamiokande), Phys. Rev. D91, [146] G. R. Dvali, A. Melfo and G. Senjanovic, Phys. Rev. Lett. 7, 072009 (2015), [arXiv:1504.01041]. 75, 4559 (1995), [hep-ph/9507230]. [170] K. R. Dienes, Phys. Rept. 287, 447 (1997), [hep- [147] V. Rubakov, Nucl. Phys. B203, 311 (1982) and Institute of th/9602045]; P. Nath and P. Fileviez Perez, Phys. Rept. Nuclear Research Report No. P-0211, Moscow (1981), un- 441, 191 (2007), [hep-ph/0601023]. published; C. G. Callan, Jr., Phys. Rev. D26, 2058 (1982); [171] G.G. Ross, “Grand Unified Theories”, Ben- F. Wilczek, Phys. Rev. Lett. 48, 1146 (1982); S. Dawson jamin/Cummings, 1984.; S. Raby, Lect. Notes Phys. and A. N. Schellekens, Phys. Rev. D27, 2119 (1983). 939, 1 (2017). [148] K. Freese, M. S. Turner and D. N. Schramm, Phys. Rev. [172] P. Nath, Int. J. Mod. Phys. A33, 20, 1830017 (2018), Lett. 51, 1625 (1983). [arXiv:1807.05302]; D. Croon et al., Front.in Phys. 7, 76 [149] P. A. R. Ade et al. (BICEP2, Keck Array), Phys. Rev. Lett. (2019), [arXiv:1903.04977]. 116, 031302 (2016), [arXiv:1510.09217]; N. Aghanim et al. [173] A. Ringwald, L.J. Rosenberg and G. Rybka, Review of ‘Ax- (Planck) (2018), [arXiv:1807.06209]. ions and other Similar Particles’ in Ref. [62]. 94. Grand Unified Theories 985

[174] P. Svrcek and E. Witten, JHEP 06, 051 (2006), [hep- 199 (1981); A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 th/0605206]. (1980), [Yad. Fiz.31,497(1980)]. [176] J. P. Conlon, JHEP 05, 078 (2006), [hep-th/0602233]. [175] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B, 986 95. Leptoquarks

95. Leptoquarks

Updated August 2019 by S. Rolli (US Department of Energy) and M. Additionally, this assumption gives strong constraints on models of Tanabashi (Nagoya U.) leptoquarks. Leptoquarks are hypothetical particles carrying both baryon Refs. [15,16,17] give extensive lists of the bounds on the - number (B) and lepton number (L). The possible quantum numbers induced four-fermion interactions. For the isoscalar scalar and vector of leptoquark states can be restricted by assuming that their direct leptoquarks S0 and V0, for example, which couple with the first- (second-) generation left-handed quark, and the first-generation interactions with the ordinary standard model (SM) fermions are 2 2 left-handed lepton, the bounds λ < 0.07 (MLQ/1 TeV) for S0, and dimensionless and invariant under the SM gauge group. Table 95.1 2 2 2 × 2 shows the list of all possible quantum numbers with this assumption [1]. λ < 0.4 (MLQ/1 TeV) for V0 (λ < 0.7 (MLQ/1 TeV) for S0, 2 × 2 × The columns of SU(3) , SU(2) , and U(1) in Table 95.1 indicate and λ < 0.5 (MLQ/1 TeV) for V0) with λ being the leptoquark C W Y × the QCD representation, the weak isospin representation, and the coupling strength, can be derived from the limits listed in Ref. [17]. + weak hypercharge, respectively. The spin of a leptoquark state is The e e− experiments are sensitive to the indirect effects coming from + taken to be 1 (vector leptoquark) or 0 (scalar leptoquark). t- and u-channel exchanges of leptoquarks in the e e− qq¯ process. The HERA experiments give bounds on the leptoquark-induced→ four-fermion interaction. For detailed bounds obtained in this way, Table 95.1: Possible leptoquarks and their quantum numbers. see the Boson Particle Listings for “Indirect Limits for Leptoquarks” and its references.

Spin 3B + L SU(3)c SU(2)W U(1)Y Allowed coupling Collider experiments provide direct limits on the leptoquark states through limits on the pair- and single-production cross sections. The ¯ c c 0 2 3 1 1/3q ¯LℓL oru ¯ReR leading-order cross sections of the parton processes − ¯ ¯c 0 2 3 1 4/3 dReR − ¯ c q +¯q LQ + LQ 0 2 3 3 1/3q ¯LℓL → − ¯ c µ ¯c µ 1 2 3 2 5/6q ¯Lγ eR or dRγ ℓL g + g LQ + LQ − c µ → 1 2 3¯ 2 1/6u ¯ γ ℓL − − R e + q LQ (95.1) 0 03 2 7/6q ¯LeR oru ¯RℓL → ¯ 0 03 2 1/6 dRℓL may be written as [21] µ ¯ µ 1 03 1 2/3q ¯Lγ ℓL or dRγ eR 2 µ 2αsπ 3 1 03 1 5/3u ¯Rγ eR σˆLOhqq¯ LQ + LQi = β , µ → 27ˆs 1 03 3 2/3q ¯Lγ ℓL 2 αsπ σˆLOhgg LQ + LQi = → 96ˆs

If we do not require leptoquark states to couple directly with 2 2 4 1+ β hβ(41 31β )+(18β β 17) log i, SM fermions, different assignments of quantum numbers become × − − − 1 β possible [2,3]. − 2 πλ 2 Leptoquark states are expected to exist in various extensions of σˆLOheq LQi = δ(ˆs MLQ) (95.2) the SM. The Pati-Salam model [4] is an example predicting the → 4 − existence of a leptoquark state. Leptoquark states also exist in grand for a scalar leptoquark. Here √sˆ is the invariant energy of the parton unification theories based on SU(5) [5], SO(10) [6], which includes subprocess, and β q1 4M 2 /sˆ. The leptoquark Yukawa coupling Pati-Salam color SU(4), and larger gauge groups. Scalar quarks ≡ − LQ in supersymmetric models with R-parity violation may also have is given by λ. Leptoquarks are also produced singly at hadron colliders leptoquark-type Yukawa couplings. The bounds on the leptoquark through g + q LQ+ ℓ [22], which allows extending to higher masses → states can therefore be applied to constrain R-parity-violating the collider reach in the leptoquark search [23], depending on the supersymmetric models. Scalar leptoquarks are expected to exist at leptoquark Yukawa coupling. See also Ref. [24] for a comprehensive the TeV scale in extended technicolor models [7,8] where leptoquark review on the leptoquark phenomenology in precision experiments and states appear as the bound states of techni-fermions. Compositeness particle colliders. of quarks and leptons also provides examples of models which may Leptoquark states which couple only to left- or right-handed quarks have light leptoquark states [9]. are called chiral leptoquarks. Leptoquark states which couple only to Bounds on leptoquark states are obtained both directly and the first (second, third) generation are referred as the first- (second-, indirectly. Direct limits are from their production cross sections third-) generation leptoquarks. at colliders, while indirect limits are calculated from bounds on The LHC, Tevatron and LEP experiments have been searching leptoquark-induced four-fermion interactions, which are obtained for pair production of the leptoquark states, which arises from from low-energy experiments, or from collider experiments below the leptoquark gauge interaction. Due to the typical decay of the threshold. These four-fermion interactions often cause lepton-flavor leptoquark into charged and neutral leptons and quarks, the searches non-universalities in heavy quark decays. Anomalies observed recently are carried on in signatures including high PT charged leptons, high in the RK and RD ratios [10,11] in the semi-leptonic B decays may be ET jets and large missing transverse energy. Additionally searches for explained in models with TeV scale leptoquarks. pair produced LQs are often organized by the decay mode of the pair If a leptoquark couples to quarks (leptons) belonging to more of LQs, via the decay parameter β, which represents the branching than a single generation in the mass eigenbasis, it can induce fraction into a charge lepton vs a neutrino: beta = 1 for both LQs four-fermion interactions causing flavor-changing neutral currents decaying into a charged letpon, beta = 0.5 for one LQ decaying into (lepton-family-number violations). The quantum number assignment a charged lepton and one into a neutrino. The gauge couplings of a of Table 1 allows several leptoquark states to couple to both left- scalar leptoquark are determined uniquely according to its quantum and right-handed quarks simultaneously. Such leptoquark states are numbers in Table 95.1. Since all of the leptoquark states belong called non-chiral and may cause four-fermion interactions affecting to color-triplet representation, the scalar leptoquark pair-production the (π eν)/(π µν) ratio [12]. Non-chiral scalar leptoquarks also cross section at the Tevatron and LHC can be determined solely as a contribute→ to the→ muon anomalous magnetic moment [13,14]. Since function of the leptoquark mass without making further assumptions. indirect limits provide more stringent constraints on these types of This is in contrast to the indirect or single-production limits, which leptoquarks, it is often assumed that a leptoquark state couples only give constraints in the leptoquark mass-coupling plane. to a single generation of quarks and a single generation of leptons in Older results from the Tevatron run can be found here: [26], [27], a chiral interaction, for which indirect limits become much weaker. [28] and [29]. 95. Leptoquarks 987

Since the previous version of this review, both ATLAS and CMS where κ = 0, placing the most stringent constraint to date from pair have updated their results concerning searches for first, second, production of vector LQs. and third generation LQs and leptoquark states which couple only The leptoquark pair-production cross sections in e+e− collisions with the i-th generation quarks and the j-th generation leptons depend on the leptoquark SU(2) U(1) quantum numbers and (i = j) without causing conflicts with severe indirect constraints. The Yukawa coupling with electron [43].× datasets6 were almost all collected at center of mass energy of 13 TeV Searches for first generation leptoquark singly produced were and corresponding to the latest integrated luminosity collected before performed by the HERA experiments. Since the leptoquark single- the shutdown of the LHC occuring in 2019 and 2020. production cross section depends on its Yukawa coupling, the It is worthy to note that organizing LQs by flavor quantum number leptoquark mass limits from HERA are usually displayed in the first before organizing them by gauge quantum number is becoming mass-coupling plane. For leptoquark Yukawa coupling λ = 0.1, early more common and advantageous because it relates more closely to ZEUS Collaboration bounds on the first-generation leptoquarks range some of the experimental searches being performed. The traditional from 248 to 290 GeV, depending on the leptoquark species [45]. The nomenclature for 1st, 2nd, and 3rd generation LQ encourages only ZEUS Collaboration has recently released a new paper [46] where looking for the diagonal elements in a flavor matrix of possibilities, data corresponding to a luminosity of around 1 fb1 have been used which has been the traditional experimental search strategy. in the framework of eeqq contact interactions (CI) to set limits Current results extend previous mass limits for scalar leptoquarks on possible high-energy contributions beyond the Standard Model to > 1435 GeV (first generation, CMS, β =1, √s = 13 TeV) and to electron-quark scattering. The analysis of the ep data has been > 1270 GeV(first generation, CMS, β =0.5,√s = 13 TeV) [30]; based on simultaneous fits of parton distribution functions including > 1400 GeV (first generation, ATLAS, β =1, √s = 13 TeV) and contributions of Contact Interaction (CI) couplings to ep scattering. > 1290 GeV (first generation, ATLAS, β =0.5, √s = 13 TeV ) [31]; Several general CI models and scenarios with heavy leptoquarks > 1530 GeV (second generation, CMS, β =1, √s = 13 TeV) and were considered. As unambiguous deviations from the SM cannot be > 1285 GeV (second generation, CMS, β =0.5, √s = 13 TeV) [32]; established, limits for CI compositeness scales and LQ mass scales and > 1560 GeV (second generation, ATLAS, β =1,√s = 13 TeV) and were set that are in the TeV range. The H1 Collaboration has a > 1230 GeV (second generation, ATLAS, β =0.5,√s = 13 TeV) [31]. comprehensive summary of searches for first generation leptoquarks All limits are presented at 95% C.L. using the full data sample collected in ep collisions at HERA (446 As for third generation leptoquarks, CMS results are the following: pb−1). No evidence of production of leptoquarks was observed in 1) assuming that all leptoquarks decay to a top quark and a τ lepton, final states with a large transverse momentum electron or large the existence of pair produced, third-generation leptoquark up to missing transverse momentum. For a coupling strength λ = 0.3, first a mass of 900 GeV (β =1, 13 TeV) is excluded at 95% confidence generation leptoquarks with masses up to 800 GeV are excluded at level [33]; 2) assuming that all leptoquarks decay to a bottom 95% C.L. [48]. quark and a τ lepton, the existence of pair produced, third-generation At the LHC, the CMS collaboration performed searches for single leptoquark up to a mass of 1020 GeV (β =1, 13 TeV) is excluded production of first and second generation leptoquarks [49], which at 95% confidence level [34]; 3)assuming that all leptoquarks decay is complementary to the HERA searches in the high λ region (for to a bottom quark and a τ neutrino, the existence of pair produced, coupling strength λ = 1.0, first generation leptoquarks are excluded for third-generation leptoquark up to a mass of 450 GeV (β =0, 7 TeV)is masses up to 1.73 TeV and second generation leptoquark are excluded excluded at 95% confidence level [35]. In a recent paper [36], the up to masses of 530 GeV). CMS also recently searched for third ATLAS collaboration has limits on pair production of third generation generation LQ decaying into τ and bottom in [50]. Assuming unit scalar leptoquarks where all possible decays of the leptoquark into Yukawa coupling (λ), a third generation scalar leptoquark is excluded a quark (t, b) and a lepton (τ, ν) of the third generation are for masses below 740 GeV. Limits are also set on λ of the hypothesized considered. The limits are presented as a function of the leptoquark leptoquark as a function of its mass. Above λ = 1.4, the results mass and the branching ratio into charged leptons for leptoquark up d provide the best upper limit on the mass of a third-generation scalar of up-type (LQ3 τν/bτ) and down-type (LQ3 bν/tτ); many leptoquark decaying to a τ lepton and a bottom quark. results are re-interpretation→ of previously published→ ATLAS searches. The collaboration finds that masses below 800 GeV are excluded for Searches for LQ will continue with more LHC data, particularly u d in light of the renewed interest in this type of particle to explain both LQ3 and LQ3 independently of the branching ratio, with masses below about 1 TeV being excluded for the limiting cases of branching violation of letpon flavor universality and other anomalies, which point ratios equal to zero or unity. to explanations laying outside the Standard Model. It is also possible to consider leptoquark states which couple only References: with the i-th generation quarks and the j-th generation leptons 1. W. Buchm¨uller, R. R¨uckl, and D. Wyler, Phys. Lett. B191, 442 (i = j) without causing conflicts with severe indirect constraints. (1987). Such6 couplings have received renewed attention because they may 2. K.S. Babu, C.F. Kolda, and J. March-Russell, Phys. Lett. B408, provide an explanation to anomalies in rare B meson decays and 261 (1997). the anomalous magnetic moment of the muon.− See Ref. [37], [38] 3. J.L. Hewett and T.G. Rizzo, Phys. Rev. D58, 055005 (1998). and [39] and references therein for collider search strategies and limits on the pair production cross sections of this class of leptoquark states. 4. J.C. Pati and A. Salam, Phys. Rev. D10, 275 (1974). In this framework, a novel CMS result [40] presents a non-traditional 5. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974). search for pair production of LQs coupled to a top quark and a muon. 6. H. Georgi, AIP Conf. Proc. 23, 575 (1975); As no deviation from the standard model prediction was observed, H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975). scalar LQs decaying exclusively into top µ are excluded up to masses 7. For a review, see, E. Farhi and L. Susskind, Phys. Reports 74, of 1420 GeV. − 277 (1981). The magnetic-dipole-type and the electric-quadrupole-type interac- 8. K. Lane and M. Ramana, Phys. Rev. D44, 2678 (1991). tions of a vector leptoquark are not determined even if we fix its gauge 9. See, for example, B. Schremp and F. Schremp, Phys. Lett. 153B, quantum numbers as listed in the Table 95.1 [41]. The production 101 (1985). of vector leptoquarks depends in general on additional assumptions, 10. R. Aaij et al. [LHCb Collab.], Phys. Rev. Lett. 113, 151601 where the leptoquark couplings and their pair production cross (2014); sections are enhanced relative to the scalar leptoquark contributions. R. Aaij et al. [LHCb Collab.], JHEP 08, 055 (2017); The most stringent limits on vector LQ production are now from R. Aaij et al. [LHCb Collab.], Phys. Rev. Lett. 122, 191801 CMS [42] where previous searches for squarks and gluinos have (2019). been reinterpreted to constrain models of leptoquark production. LQ 11. Y. Amhis et al., [HFLAV Collab.], Eur. Phys. J. C77, 895 (2017), masses below 1530 GeV are excluded assuming the Yang-Mills case updated results and plots available at hflav.web.cern.ch/. with coupling κ = 1, or 1115 GeV in the minimal coupling case 12. O. Shanker, Nucl. Phys. B204, 375, (1982). 988 95. Leptoquarks

13. U. Mahanta, Eur. Phys. J. C21, 171 (2001) [Phys. Lett. B515, 28. V.M. Abazov et al. [DØCollab.], Phys. Lett. B671, 224 (2009). 111 (2001)]. 29. V.Abazov et al. [DØCollab.], Phys. Lett. B693, 95 (2010). 14. K. Cheung, Phys. Rev. D64, 033001 (2001). 30. [CMS Collab.], Phys. Rev. D99, 052002 (2019). 15. S. Davidson, D.C. Bailey, and B.A. Campbell, Z. Phys. C61, 613 31. [ATLAS Collab.], Eur. Phys. J. C79, 733 (2019). (1994). 32. [CMS Collab.], Phys. Rev. D99, 032014 (2019). 16. M. Leurer, Phys. Rev. D49, 333 (1994); 33. [CMS Collab.], Eur. Phys. J. C78, 707 (2018). Phys. Rev. D50, 536 (1994). 34. [CMS Collab.], JHEP 03, 170 (2019). 17. M. Carpentier and S. Davidson, Eur. Phys. J. C70, 1071 (2010). 35. [CMS Collab.], JHEP 1212, 055 (2012). 18. A. Falkowski, M. Gonz´alez-Alonso and K. Mimouni, JHEP 08, 36. [ATLAS Collab.], JHEP 06, 144 (2019). 123 (2017). 37. B. Diaz, M. Schmaltz, and Y. M. Zhong, JHEP 10, 097 (2017). 19. S. Davidson and A. Saporta, Phys. Rev. D99, 015032 (2019). 38. M. Schmaltz, and Y. M. Zhong, JHEP 01, 132 (2019). 20. M. Chala, J. Santiago and M. Spannowsky, JHEP 04, 014 (2019). 39. D. Muller, EPJ Web Conf. 179, 01015 (2018).. 21. T. Plehn et al., Z. Phys. C74, 611 (1997); 40. [CMS Collab.] Phys. Rev. Lett. 121, 241802 (2018). M. Kramer et al., Phys. Rev. Lett. 79, 341 (1997); and references 41. J. Bl¨umlein, E. Boos, and A. Kryukov, Z. Phys. C76, 137 (1997). therein. 42. [CMS Collab.] Phys. Rev. D98, 032005 (2018). 22. J.L. Hewett and S. Pakvasa, Phys. Rev. D37, 3165 (1988); 43. J. Bl¨umlein and R. Ruckl, Phys. Lett. B304, 337 (1993). O.J.P. Eboli and A.V. Olinto, Phys. Rev. D38, 3461 (1988); 44. G. Abbiendi et al. [OPAL Collab.], Eur. Phys. J. C31, 281 A. Dobado, M.J. Herrero, and C. Mu˜noz, Phys. Lett. 207B, 97 (2003). (1988); 45. S. Chekanov et al. [ZEUS Collab.], Phys. Rev. D68, 052004 V.D. Barger et al., Phys. Lett. B220, 464 (1989); (2003). M. De Montigny and L. Marleau, Phys. Rev. D40, 2869 (1989) 46. H. Abramowicz et al. [ZEUS Collab.]. Phys. Rev. D99, 092006 [Erratum-ibid. D56, 3156 (1997)]. (2019). 23. A. Belyaev et al., JHEP 0509, 005 (2005). 47. A. Aktas et al. [H1 Collab.], Phys. Lett. B629, 9 (2005). 24. I. Dorˇsner et al., Phys. Reports 641, 1 (2016). 48. F.D. Aaron et al. [H1 Collab.], Phys. Lett. B704, 388 (2011). 25. D. Acosta et al. [CDF Collab.], Phys. Rev. D72, 051107 (2005). 49. V. Khachatryan et al. [CMS Collab.], Phys. Rev. D93, 032005 26. V.M. Abazov et al. [DØCollab.], Phys. Lett. B681, 224 (2009). (2016). 27. A. Abulencia et al. [CDF Collab.], Phys. Rev. D73, 051102 50. [CMS Collab.] JHEP 07, 115 (2018). (2006). 96. Magnetic Monopoles 989

96. Magnetic Monopoles

Revised August 2019 by D. Milstead (Stockholm U.) and E.J. The essential ingredient here was the fact that the Higgs fields Weinberg (Columbia U.). at spatial infinity could be arranged in a topologically nontrivial configuration. A discussion of the general conditions under which 96.1 Theory of magnetic monopoles this is possible is beyond the scope of this review, so we restrict The symmetry between electric and magnetic fields in the ourselves to the two phenomenologically most important cases. source-free Maxwell’s equations naturally suggests that electric The first is the standard electroweak theory, with SU(2) × U(1) charges might have magnetic counterparts, known as magnetic broken to U(1). There are no topologically nontrivial config- monopoles. Although the greatest interest has been in the super- urations of the Higgs field, and hence no topologically stable massive monopoles that are a firm prediction of all grand unified monopole solutions. theories, one cannot exclude the possibility of lighter monopoles. The second is when any simple Lie group is broken to a sub- In either case, the magnetic charge is constrained by a quan- group with a U(1) factor, a case that includes all grand unified tization condition first found by Dirac [1]. Consider a monopole theories. The monopole mass is determined by the mass scale with magnetic charge QM and a Coulomb magnetic field of the symmetry breaking that allows nontrivial topology. For Q ˆr example, an SU(5) model with B = M . (96.1) 2 4π r M M SU(5) −→X SU(3) × SU(2) × U(1) −→W SU(3) × U(1) (96.5) Any vector potential A whose curl is equal to B must be singular

along some line running from the origin to spatial infinity. This has a monopole [7] with QM = 2π/e and mass Dirac string singularity could potentially be detected through the extra phase that the wavefunction of a particle with electric charge 4πMX Mmon ∼ , (96.6) QE would acquire if it moved along a loop encircling the string. g2 For the string to be unobservable, this phase must be a multiple of 2π. Requiring that this be the case for any pair of electric and where g is the SU(5) gauge coupling. For a unification scale of 16 17 magnetic charges gives the condition that all charges be integer 10 GeV, these monopoles would have a mass Mmon ∼ 10 – min min 1018 GeV. multiples of minimum charges QE and QM obeying In theories with several stages of symmetry breaking, min min QE QM = 2π . (96.2) monopoles of different mass scales can arise. In an SO(10) theory with (For monopoles which also carry an electric charge, called dyons M M [2], the quantization conditions on their electric charges can be SO(10) −→1 SU(4) × SU(2) × SU(2) −→2 SU(3) × SU(2) × U(1) modified. However, the constraints on magnetic charges, as well (96.7) as those on all purely electric particles, will be unchanged.) 2 there is monopole with QM = 2π/e and mass ∼ 4πM1/g and a Another way to understand this result is to note that the con- 2 much lighter monopole with QM = 4π/e and mass ∼ 4πM2/g [8]. served orbital angular momentum of a point electric charge mov- The central core of a GUT monopole contains the fields of ing in the field of a has an additional compo- the superheavy gauge bosons that mediate baryon number vi- nent, with olation, so one might expect that baryon number conservation L = mr × v − 4πQE QM ˆr (96.3) could be violated in baryon–monopole scattering. The surpris- Requiring the radial component of L to be quantized in half- ing feature, pointed out by Callan [9] and Rubakov [10], is that integer units yields Eq. 96.1. these processes are not suppressed by powers of the gauge boson If there are unbroken gauge symmetries in addition to the U(1) mass. Instead, the cross-sections for catalysis processes such as of electromagnetism, the above analysis must be modified [3] [4]. p + monopole → e+ + π0 + monopole are essentially geometric; −27 2 For example, a monopole could have both a U(1) magnetic charge i.e., σ∆B β ∼ 10 cm , where β = v/c. Note, however, that this and a color magnetic charge. The latter could combine with the catalysis is model-dependent and is not even a universal property color charge of a quark to give an additional contribution to the of all GUT monopoles. phase factor associated with a loop around the Dirac string, so D 96.2 Production and Annihilation that the U(1) charge could be the Dirac charge QM ≡ 2π/e, the result that would be obtained by substituting the electron GUT monopoles are far too massive to be produced in any fore- charge into Eq. (96.1). On the other hand, for monopoles without seeable accelerator. However, they could have been produced in color-magnetic charge, one would simply insert the quark electric the early universe as topological defects arising via the Kibble mechanism [11] in a symmetry-breaking phase transition. Esti- charges into Eq. 96.1 and conclude that QM must be a multiple of 6π/e. mates of the initial monopole abundance, and of the degree to The prediction of GUT monopoles arises from the work of which it can be reduced by monopole-antimonopole annihilation, ’t Hooft [5] and Polyakov [6], who showed that certain sponta- predict a present-day monopole abundance that exceeds by many neously broken gauge theories have nonsingular classical solutions orders of magnitude the astrophysical and experimental bounds that lead to magnetic monopoles in the quantum theory. The sim- described below [12]. Cosmological inflation and other proposed plest example occurs in a theory where the vacuum expectation solutions to this primordial monopole problem generically lead to value of a triplet Higgs field Œ breaks an SU(2) gauge symmetry present-day abundances exponentially smaller than could be plau- sibly detected, although potentially observable abundances can be down to the U(1) of electromagnetism and gives a mass MV to two of the gauge bosons. In order to have finite energy, Œ must ap- obtained in scenarios with carefully tuned parameters. proach a vacuum value at infinity. However, there is a continuous If monopoles light enough to be produced at colliders exist, one family of possible vacua, since the scalar field potential determines would expect that these could be produced by analogs of the elec- only the magnitude v of hŒi, but not its orientation in the in- tromagnetic processes that produce pairs of electrically charged ternal SU(2) space. In the monopole solution, the direction of Œ particles. Because of the large size of the magnetic charge, this is in internal space is correlated with the position in physical space; a strong coupling problem for which perturbation theory cannot i.e., φa ∼ vrˆa. The stability of the solution follows from the fact be trusted. Indeed, the problem of obtaining reliable quantitative that this twisting Higgs field cannot be smoothly deformed to a estimates of the production cross-sections remains an open one, spatially uniform vacuum configuration. Reducing the energetic on which there is no clear consensus. cost of the spatial variation of Œ requires a nonzero gauge po- 96.3 Astrophysical and Cosmological Bounds tential, which turns out to yield the magnetic field corresponding If there were no galactic magnetic field, one would expect to a charge QM = 4π/e. Numerical solution of the classical field monopoles in the galaxy to have typical velocities of the order equations shows that the mass of this monopole is of 10−3c, comparable to the virial velocity in the galaxy (relevant if the monopoles cluster with the galaxy) and the peculiar veloc- 4πMV Mmon ∼ . (96.4) ity of the galaxy with respect to the CMB rest frame (relevant e2 990 96. Magnetic Monopoles

if the monopoles are not bound to the galaxy). This situation is 96.4 Searches for Magnetic Monopoles modified by the existence of a galactic magnetic field B ∼ 3µG.A To date there have been no confirmed observations of exotic monopole with the Dirac charge and mass M would be accelerated particles possessing magnetic charge. Precision measurements of by this field to a velocity the properties of known particles have led to tight limits on the values of magnetic charge they may possess. Using the induction (c, M 1011GeV , method (see below), the electron’s magnetic charge has been found . m −24 D D v ∼ 1/2 (96.8) to be Qe < 10 QM [24] (where QM is the Dirac charge). mag −3  1017 GeV  11 10 c M ,M & 10 GeV . Furthermore, measurements of the anomalous magnetic moment of the muon have been used to place a model dependent lower limit of 120 GeV on the monopole mass 1 [25]. Nevertheless, Accelerating these monopoles drains energy from the mag- guided mainly by Dirac’s argument and the predicted existence netic field. Parker [13] obtained an upper bound on the flux of of monopoles from spontaneous symmetry breaking mechanisms, monopoles in the galaxy by requiring that the rate of this energy searches have been routinely made for monopoles produced at loss be small compared to the time scale on which the galactic accelerators, in cosmic rays, and bound in matter [26]. Although field can be regenerated. With reasonable choices for the astro- the resultant limits from such searches are usually made under the physical parameters (see Ref. [14] for details), this Parker bound assumption of a particle possessing only magnetic charge, most of is the searches are also sensitive to dyons.

( −15 −2 −1 −1 17 10 cm sr sec ,M . 10 GeV , 96.5 Search Techniques F <   Search strategies are determined by the expected interactions 10−15 M cm−2 sr−1 sec−1 ,M 1017 GeV . 1017 GeV & of monopoles as they pass through matter. These would give (96.9) rise to a number of striking characteristic signatures. Since a Applying similar arguments to an earlier seed field that was the complete description of monopole search techniques falls outside progenitor of the current galactic field leads to a tighter bound of the scope of this minireview, only the most common methods [15], are described below. More comprehensive descriptions of search techniques can be found in Refs. [27] [28]. The induction method exploits the long-ranged electromagnetic h M −6 i −16 −2 −1 −1 F < + (3 × 10 ) 10 cm sr sec . (96.10) interaction of the monopole with the quantum state of a super- 1017GeV conducting ring which would lead to a monopole which passes Considering magnetic fields in galactic clusters gives a bound [16] through such a ring inducing a permanent current. The induction which, although less secure, is about three orders of magnitude technique typically uses Superconducting Quantum Interference lower than the Parker bound. Devices (SQUID) technology for detection and is employed for searches for monopoles in cosmic rays and matter. Another ap- A flux bound can also be inferred from the total mass of proach is to exploit the electromagnetic energy loss of monopoles. monopoles in the universe. If the monopole mass density is a frac- Monopoles with Dirac charge would typically lose energy at a rate tion Ω of the critical density, and the monopoles were uniformly M which is several thousand times larger than that expected from distributed throughout the universe, there would be a monopole particles possessing the elementary electric charge. Consequently, flux scintillators, gas chambers and nuclear track detectors (NTDs) have been used in cosmic ray and collider experiments. A fur-  17    −16 10 GeV v −2 −1 −1 ther approach, which has been used at colliders, is to search for Funiform = 1.3×10 ΩM cm sr sec . M 10−3c particles describing a non-helical path in a uniform magnetic field. (96.11) 96.5.1 Searches for Monopoles Bound in Matter If we assume that Ω ∼ 0.1, this gives a stronger constraint than M Monopoles have been sought in a range of bulk materials which the Parker bound for M ∼ 1015 GeV. However, monopoles with it is assumed would have absorbed incident cosmic ray monopoles masses ∼ 1017 GeV are not ejected by the galactic field and can be over a long exposure time of order million years. Materials which gravitationally bound to the galaxy. In this case their flux within have been studied include moon rock, meteorites, manganese the galaxy is increased by about five orders of magnitude for a modules, and sea water [29, 30]. A stringent upper limit on the given value of Ω , and the mass density bound only becomes M monopoles per nucleon ratio of ∼10−29 has been obtained [30]. stronger than the Parker bound for M ∼ 1018 GeV. A much more stringent flux bound applies to GUT monopoles 96.5.2 Searches in Cosmic Rays that catalyze baryon number violation. The essential idea is that Direct searches for monopoles in cosmic rays refer to those ex- compact astrophysical objects would capture monopoles at a rate periments in which the passage of the monopole is measured by an proportional to the galactic flux. These monopoles would then active detector. Searches made assuming a catalysis processes in catalyze proton decay, with the energy released in the decay lead- which GUT monopoles could induce nucleon decay are discussed ing to an observable increase in the luminosity of the object. A va- in the next section. To interpret the results of the non-catalysis riety of bounds, based on neutron stars [17–21] , white dwarfs [22], searches, the cross section for the catalysis process is typically and Jovian planets [23] have been obtained. These depend in the either set to zero [31] or assigned a modest value (1mb) [32]. obvious manner on the catalysis cross section, but also on the Although early cosmic ray searches using the induction tech- details of the astrophysical scenarios; e.g., on how much the ac- nique [33] and NTDs [34] observed monopole candidates, none cumulated density is reduced by monopole-antimonopole annihi- of these apparent observations have been confirmed. Recent ex- lation, and on whether monopoles accumulated in the progenitor periments have typically employed large scale detectors. The star survive its collapse to a white dwarf or neutron star. The MACRO experiment at the Gran Sasso underground laboratory bounds obtained in this manner lie in the range comprised three different types of detector: liquid scintillator, lim- ited stream tubes, and NTDs, which provided a total acceptance  σ β  of ∼ 10000m2 for an isotropic flux. As shown in Fig. 96.1, this F ∆B ∼ (10−18 − 10−29)cm−2sr−1sec−1. (96.12) 10−27cm2 experiment has so far provided the most extensive β-dependent flux limits for GUT monopoles with Dirac charge [32]. Also shown are limits from an experiment at the OHYA mine in Japan [31], It is important to remember that not all GUT monopoles cat- which used a 2000m2 array of NTDs. alyze baryon number nonconservation. In particular, the inter- In Fig. 96.1, upper flux limits are also shown as a function mediate mass monopoles that arise in some GUTs at later stages of mass for monopole speed β > 0.05. In addition to MACRO of symmetry-breaking are examples of theoretically motivated monopoles that are exempt from the bound of the above equa- 1Where no ambiguity is likely to arise, a reference to a monopole implies tion. a particle possessing Dirac charge. 96. Magnetic Monopoles 991

15 101,00E-15 ) OHYA ) -1 -1 sr sr -1 -1 s s

-2 MACRO -2 16 101,00E-16 ANTARES

17 101,00E-17 Flux upper limit (cm limit upper Flux ICECUBE (cm limit upper Flux

18 101,00E-18 101,00E-055 10 1,00E-044 10 1,00E-033 10 1,00E-022 10 1,00E-011 1,00E+001 Series2 Series3 Icecube Series1  Mass (GeV)

Figure 96.1: Upper flux limits for (left) GUT monopoles as a function of β (right) Monopoles as a function of mass for β > 0.05. and OYHA flux limits, results from the SLIM [35] high-altitude gions. A conservative approach with as little model-dependence experiment are shown. The SLIM experiment provided a good as possible is thus to present representative values of the upper 5 12 sensitivity to intermediate mass monopoles (10 . M . 10 cross-section limits as a function of one half the centre-of-mass GeV). In addition to the results shown in Fig. 96.1, limits as energy of the collisions, as shown in Fig. 96.2 for recent results low as ∼ 1.5 × 10−18 cm−2s−1sr−1 were obtained for monopoles from high energy colliders.

with β > 0.51 and β > 0.6 by the IceCube [36] and Antares

[37] experiments, respectively. Stringent constraints on the flux )

푒+푒− pb

of ultra-relativistic monopoles have been obtained at the Pierre ( 푒푝 Auger Observatory [38] which was sensitive to monopoles with 푝 푝 9 12 γ values ranging from 10 to 10 , leading to flux limits in the 푝푝 range 10−15 − 2.5 × 10−21 cm−2s−1sr−1. The RICE [39] and section H1 ANITA-II experiments [40] at the South Pole have also sought E882 7 12 L6-MODAL ultra-relativistic monopoles with γ values of 10 . γ . 10 and 9 13 10 . γ . 10 , respectively, and which produced flux limits as

low as 2.5 × 10−21 cm−2s−1sr−1. OPAL CDF

limit on cross on limit cross

96.5.3 Searches via the Catalysis of Nucleon-Decay MOEDAL

Searches have been performed for evidence of the catalysed Upper decay of a nucleon by a monopole, as predicted by the Callan- Rubakov mechanism. The searches are thus sensitive to the assumed value of the catalysis decay cross section. Searches ATLAS have been made with the Soudan [41] and Macro [42] experi- ments, using tracking detectors. Searches at IMB [43], the un- √푠/2 (GeV) derwater Lake Baikal experiment [44] and the The IceCube ex- periment [45] which exploit the Cerenkov effect have also been Figure 96.2: Upper limits on the production cross sections of made. The resulting β-dependent flux limits from these experi- monopoles from various collider-based experiments. ments typically vary between ∼ 10−18 and ∼ 10−14cm−2sr−1s−1. A recent search for low energy neutrinos (assumed to be pro- Searches for monopoles produced at the highest available ener- duced from induced proton decay in the sun) was made at gies in hadron-hadron collisions were made in pp collisions at the Super-Kamiokande [46]. A model- and β-dependent of limit of LHC by the ATLAS [47, 48] and MoEDAL [49, 50] experiments. 6.3 × 10−24( β )2cm−2sr−1s−1 was obtained. 10−3 The experiments looked for highly ionising particles leaving char- acteristic energy deposition profiles and stopped monopoles with 96.5.4 Searches at Colliders the induction method, respectively. The charge-dependent mass Searches have been performed at hadron-hadron, electron- limits extend up to around 4 TeV. The ATLAS work considers positron and lepton-hadron experiments. Collider searches can D D monopoles with 0.5QM and 2QM while MoEDAL quotes limits be broadly classed as being direct or indirect. In a direct search, D D evidence of the passage of a monopole through material, such as for monopoles with charges from QM to 5QM . MoEDAL con- a charged particle track, is sought. In indirect searches, virtual sidered monopole-pair production via photon fusion along with, monopole processes are assumed to influence the production rates as is commonly used in hadron-hadron collisions, Drell-Yan pro- of certain final states. cesses [51]. Tevatron searches have also been carried out by the CDF [52] and E882 [53] experiments. The CDF experiment used a 96.5.4.1 Direct Searches at Colliders dedicated time-of-flight system whereas the E882 experiment em- Collider experiments typically express their results in terms ployed the induction technique to search for stopped monopoles of upper limits on a production cross section and/or monopole in discarded detector material which had been part of the CDF mass. To calculate these limits, ansatzes are used to model the and D0 detectors using periods of luminosity. Earlier searches at kinematics of monopole-antimonopole pair production processes the Tevatron, such as [54], used NTDs and were based on com- since perturbative field theory cannot be used to calculate the rate paratively modest amounts of integrated luminosity. Lower en- and kinematic properties of produced monopoles. Limits there- ergy hadron-hadron experiments have employed a variety of search fore suffer from a degree of model-dependence, implying that a techniques including plastic track detectors [55] and searches for comparison between the results of different experiments can be trapped monopoles [56]. problematic, in particular when this concerns excluded mass re- The only LEP-2 search was made by OPAL [57] which quoted 992 96. Magnetic Monopoles

cross section limits for the production of monopoles possessing [28] M. Fairbairn et al., Phys. Rept. 438, 1 (2007), [hep- masses up to around 103 GeV. At LEP-1, searches were made ph/0611040]. with NTDs deployed around an interaction region. This allowed [29] J. M. Kovalik and J. L. Kirschvink, Phys. Rev. A33, 1183 a range of charges to be sought for masses up to ∼ 45 GeV. The L6- (1986). MODAL experiment [58] gave limits for monopoles with charges D D [30] H. Jeon and M. J. Longo, Phys. Rev. Lett. 75, 1443 (1995), in the range 0.9QM and 3.6QM , whilst an earlier search by the MODAL experiment was sensitive to monopoles with charges as [Erratum: Phys. Rev. Lett.76,159(1996)], [hep-ex/9508003]. D low as 0.1QM [59]. The deployment of NTDs around the beam [31] S. Orito et al., Phys. Rev. Lett. 66, 1951 (1991). + − interaction point was also used at earlier e e colliders such as [32] M. Ambrosio et al. (MACRO), Eur. Phys. J. C25, 511 + − KEK [60] and PETRA [61]. Searches at e e facilities have also (2002), [hep-ex/0207020]. been made for particles following non-helical trajectories [62] [63]. [33] B. Cabrera, Phys. Rev. Lett. 48, 1378 (1982). There has so far been one search for monopole production in lepton-hadron scattering. Using the induction method, monopoles [34] P. B. Price et al., Phys. Rev. Lett. 35, 487 (1975). were sought which could have stopped in the aluminium beampipe [35] S. Balestra et al., Eur. Phys. J. C55, 57 (2008), which had been used by the H1 experiment at HERA [64]. Cross [arXiv:0801.4913]. section limits were set for monopoles with charges in the range [36] M. G. Aartsen et al. (IceCube), Eur. Phys. J. C76, 3, 133 QD − 6QD for masses up to around 140 GeV. M M (2016), [arXiv:1511.01350]. 96.5.4.2 Indirect Searches at Colliders [37] A. Albert et al. (ANTARES), JHEP 07, 054 (2017), It has been proposed that virtual monopoles can mediate [arXiv:1703.00424]. processes which give rise to multi-photon final-states [65] [66]. Photon-based searches were made by the D0 [67] and L3 [68] ex- [38] A. Aab et al. (Pierre Auger), Phys. Rev. D94, 8, 082002 periments. The D0 work led to spin-dependent lower mass limits (2016), [arXiv:1609.04451]. of between 610 and 1580 GeV, while L3 reported a lower mass [39] D. P. Hogan et al., Phys. Rev. D78, 075031 (2008), limit of 510 GeV. However, it should be stressed that uncertain- [arXiv:0806.2129]. ties on the theoretical calculations which were used to derive these [40] M. Detrixhe et al. (ANITA-II), Phys. Rev. D83, 023513 limits are difficult to estimate. (2011), [arXiv:1008.1282]. References [41] J.E. Bartelt et al., Phys. Rev. D36, 1990 (1987) [Erratum- [1] P. A. M. Dirac, Proc. Roy. Soc. Lond. A133, 821, 60 (1931). ibid. D40, 1701 (1989)]. [2] J. S. Schwinger, Science 165, 757 (1969). [42] M. Ambrosio et al. (MACRO), Eur. Phys. J. C26, 163 [3] F. Englert and P. Windey, Phys. Rev. D14, 2728 (1976). (2002), [hep-ex/0207024]. [4] P. Goddard, J. Nuyts and D. I. Olive, Nucl. Phys. B125, 1 [43] R. Becker-Szendy et al., Phys. Rev. D49, 2169 (1994). (1977). [44] V. A. Balkanov et al. (Baikal), Prog. Part. Nucl. Phys. 40, [5] G. ’t Hooft, Nucl. Phys. B79, 276 (1974). 391 (1998), [arXiv:astro-ph/9801044]. [6] A. M. Polyakov, JETP Lett. 20, 194 (1974). [45] M. G. Aartsen et al. (IceCube), Eur. Phys. J. C74, 7, [7] C. P. Dokos and T. N. Tomaras, Phys. Rev. D21, 2940 2938 (2014), [Erratum: Eur. Phys. J.C79,no.2,124(2019)], (1980). [arXiv:1402.3460]. [8] G. Lazarides and Q. Shafi, Phys. Lett. 94B, 149 (1980). [46] K. Ueno et al. (Super-Kamiokande), Astropart. Phys. 36, 131 (2012), [arXiv:1203.0940]. [9] C. G. Callan, Jr., Phys. Rev. D26, 2058 (1982). [47] G. Aad et al. (ATLAS), Phys. Rev. D93, 5, 052009 (2016), [10] V. A. Rubakov, Nucl. Phys. B203, 311 (1982). [arXiv:1509.08059]. [11] T. W. B. Kibble, J. Phys. A9, 1387 (1976). [48] G. Aad et al. (ATLAS) (2019), [arXiv:1905.10130]. [12] J. Preskill, Phys. Rev. Lett. 43, 1365 (1979). [49] B. Acharya et al. (MoEDAL), Phys. Rev. Lett. 118, 6, 061801 [13] E. N. Parker, Astrophys. J. 160, 383 (1970). (2017), [arXiv:1611.06817]. [14] M. S. Turner, E. N. Parker and T. J. Bogdan, Phys. Rev. [50] B. Acharya et al. (MoEDAL), Phys. Rev. Lett. 123, 2, 021802 D26, 1296 (1982). (2019), [arXiv:1903.08491]. [15] F. C. Adams et al., Phys. Rev. Lett. 70, 2511 (1993). [51] S. Baines et al., Eur. Phys. J. C78, 11, 966 (2018), [Erratum: [16] Y. Rephaeli and M. S. Turner, Phys. Lett. 121B, 115 (1983). Eur. Phys. J.C79,no.2,166(2019)], [arXiv:1808.08942]. [17] E. W. Kolb, S. A. Colgate and J. A. Harvey, Phys. Rev. Lett. [52] A. Abulencia et al. (CDF), Phys. Rev. Lett. 96, 201801 49, 1373 (1982). (2006), [hep-ex/0509015]. [18] S. Dimopoulos, J. Preskill and F. Wilczek, Phys. Lett. 119B, [53] G. R. Kalbfleisch et al., Phys. Rev. D69, 052002 (2004), 320 (1982). [hep-ex/0306045]. [19] K. Freese, M. S. Turner and D. N. Schramm, Phys. Rev. Lett. [54] P. B. Price, G.-X. Ren and K. Kinoshita, Phys. Rev. Lett. 51, 1625 (1983). 59, 2523 (1987). [20] E. W. Kolb and M. S. Turner, Astrophys. J. 286, 702 (1984). [55] B. Aubert et al., Phys. Lett. 120B, 465 (1983). [21] J. A. Harvey, Nucl. Phys. B236, 255 (1984). [56] R. A. Carrigan, F. A. Nezrick and B. P. Strauss, Phys. Rev. [22] K. Freese and E. Krasteva, Phys. Rev. D59, 063007 (1999), D8, 3717 (1973). [arXiv:astro-ph/9804148]. [57] G. Abbiendi et al. (OPAL), Phys. Lett. B663, 37 (2008), [23] J. Arafune, M. Fukugita and S. Yanagita, Phys. Rev. D32, [arXiv:0707.0404]. 2586 (1985). [58] J. L. Pinfold et al., Phys. Lett. B316, 407 (1993). [24] L. L. Vant-Hull, Phys. Rev. 173, 1412 (1968). [59] K. Kinoshita et al., Phys. Rev. D46, R881 (1992). [25] S. Graf, A. Schaefer and W. Greiner, Phys. Lett. B262, 463 [60] K. Kinoshita et al., Phys. Lett. B228, 543 (1989). (1991). [26] Review of Particle Physics 2020 (this paper), listing on [61] P. Musset, M. Price and E. Lohrmann, Phys. Lett. 128B, Searches for Magnetic Monopoles. 333 (1983). [27] L. Patrizii and M. Spurio, Ann. Rev. Nucl. Part. Sci. 65, 279 [62] T. Gentile et al. (CLEO), Phys. Rev. D35, 1081 (1987). (2015), [arXiv:1510.07125]. [63] W. Braunschweig et al. (TASSO), Z. Phys. C38, 543 (1988). 96. Magnetic Monopoles 993

[64] A. Aktas et al. (H1), Eur. Phys. J. C41, 133 (2005), [hep- [67] B. Abbott et al. (D0), Phys. Rev. Lett. 81, 524 (1998), [hep- ex/0501039]. ex/9803023]. [65] A. De Rujula, Nucl. Phys. B435, 257 (1995), [hep- th/9405191]. [68] M. Acciarri et al. (L3), Phys. Lett. B345, 609 (1995). [66] I. F. Ginzburg and A. Schiller, Phys. Rev. D60, 075016 (1999), [hep-ph/9903314]. 994