JHEP12(2015)162 Springer September 8, 2015 December 28, 2015 November 11, 2015 : : : 10.1007/JHEP12(2015)162 Received Accepted Published doi: , b Published for SISSA by gian.giudice@cern.ch , and Matthew McCullough b . 3 1509.00834 Gian Giudice The Authors. c Supersymmetry Phenomenology a,b

We study how, as a result of the scanning of supersymmetry breaking during , [email protected] [email protected] Department of Physics andPittsburgh, Astronomy, U.S.A. University of Pittsburgh, Theory Division, CERN, Geneva, Switzerland E-mail: Pittsburgh Particle Physics, Astrophysics, and Cosmology Center, b a Open Access Article funded by SCOAP ArXiv ePrint: field drives the relaxation andother breaks supersymmetric supersymmetry. particles Since bySupersymmetry gauginos a free are from lighter one-loop the than factor, naturalness the problem. the theory isKeywords: a realisation of Split Abstract: the cosmological evolution, a relaxation mechanismtween can the naturally weak determine scale a and hierarchy the be- is masses of determined supersymmetric by particles. QCD Supersymmetry instanton breaking effects, in an extremely minimal setup in which a single Brian Batell, Natural heavy supersymmetry JHEP12(2015)162 15 8 25 24 14 32 13 – 1 – 12 31 18 21 22 10 14 23 27 22 20 24 12 20 4 2 29 8.2.3 Wino NLSP 8.2.1 Gluino NLSP 8.2.2 Bino NLSP 9.3 Non-QCD relaxion 9.1 Inflaton-dependent relaxion9.2 potential Inflaton-dependent instanton barriers 8.1 Relaxion detection 8.2 LHC phenomenology 5.1 Planckian effects 5.2 Generalising the5.3 relaxion potential Supergravity effects 5.4 Effects beyond quantum field theory B Conditions for electroweak breaking 10 Summary A Computation of the soft terms 9 Strong CP problem 8 Phenomenology 6 The structure of supersymmetry breaking 7 The relaxino (alias gravitino) 5 Remarks on the UV completion 3 Relaxation of supersymmetry breaking 4 Inflationary dynamics Contents 1 Introduction 2 The framework JHEP12(2015)162 ], 11 . In summary, Z ] particle (called M 14 – ] for earlier attempts 12 3 , 2 ] (see ref. [ 1 – 2 – ] requires an energy cutoff, which is found to be 1 ] for related work) offers an interesting mechanism to deal with the problem 10 – 4 These considerations lead us to believe that relaxation and supersymmetry could be Supersymmetry offers an elegant solution to the Higgs naturalness problem. From the By construction, the setup of ref. [ theoretical framework that we want to explore in this paper. supersymmetry gives a splendid UV picture, but suffers ina the perfect IR. match.while the Supersymmetry relaxation will mechanismthe deal will separation explain between with the the the scales so-called of grand “little supersymmetry picture and hierarchy electroweak of problem”, breaking. i.e. particle This is physics, the of problems in high-energybreaking. physics Moreover, supersymmetry and appears as cosmology, atheory well necessary field-theoretical and, beyond link with the therefore, string issueUV possibly of picture with electroweak is quantum notpresence gravity. corroborated of by supersymmetry Unfortunately, IRnaturalness up this would information. to magnificent require supersymmetric about Experiments particles the have with masses TeV not around scale, detected while the a resolution of the Higgs Higgs relaxation mechanism is satisfying in the IR, butUV cries point out of for view, the aproblem solution UV is given solved picture. by not supersymmetry only for is thecutoff very Higgs, ambitious. limitation. but for The This any naturalness scalar makes particle the in the supersymmetric theory and framework with very no attractive for a variety properties of the Higgs.expect that However, in more the fundamentalthan broader theories the perspective will Higgs, such we require as wantof the the to presence these inflaton, adopt, GUT-like of scalar states, one other particles orrequire can scalar fields will a related particles introduce solution to their unrelated dark to energy. own the Any naturalness Higgs problem, relaxation. and For each these one reasons, will we claim that the cles that live beyond the regimeproblem of if the effective they theory can coupleHiggs easily (directly naturalness reintroduce up an or to even indirectly) a bigger cutoffcal to scale point the of Λ view is Higgs. but, a in veryhigher a In important scales. broader result other perspective, Moreover, from the is words, the just relaxion phenomenologi- solving a mechanism way is the to a postpone solution the tailored real to problem cure to the quantum considerably smaller than manyin of theories the more new-physics mass fundamentalproblem scales than is that rather the special are Standard because believedsmall. it Model to involves If physics (SM). exist from naturalness However, all is distance the solved scales, naturalness in no an matter effective how theory with cutoff Λ, hypothetical new parti- troweak breaking. Thisrelaxion) situation which, during is the achieved cosmologicalder with evolution parameter at an of the the axion-like inflationary electroweaknon-perturbative [ epoch, phase QCD scans transition. effects the Once give or- a electroweakmuch symmetry further. back-reaction is that broken, prevents the relaxion from rolling The cosmological relaxation of theand electroweak refs. scale [ [ of Higgs naturalness. Instead ofally introducing new done dynamics in at other the weak solutions,in scale, it as which convention- gives the an system explicit is realisation dynamically of self-organised attracted criticality towards [ the near-critical condition for elec- 1 Introduction JHEP12(2015)162 ]. 23 , the – 0 15 µ and the 0 µ ], although a final 24 is not correlated with 0 µ ]. Moreover, the scanning 1 (which is the typical size of the 0 µ ] where it is necessary to introduce a PQ- 1 – 3 – ] appeared in which it was claimed that, within quantum ). Note that the value of 0 10 µ ( for more comments on this point. O 5 However, fields with such unusual properties have been conjectured to 1 In combining relaxation and supersymmetry, we find that the total is much greater than At the beginning of inflation, the relaxion is found very far from its true vacuum, To achieve this goal, we treat the relaxion as a QCD-like axion, except for the un- As this paper was being completed, ref. [ 1 beyond quantum field theory. See section such interactions are needed inviolating the coupling case between of the ref. Higgs [ and the relaxion. field theory, the cutoffscale of and any UV the completion smallgrounds realising the parameter the widespread in relaxion belief the as that relaxion the an non-compact potential axion properties must is of be not the around relaxion natural. the must weak originate These from conclusions physics put on firmer field varies. This isvacuum energy because of the the theory. breaking Wheneverbreaking of the global (and, relaxion slow-rolls, supersymmetry consequently, the is the scalemechanism of associated does order supersymmetry with not parameter the require for anythat electroweak violate special the breaking) interactions shift between scans. symmetry the and relaxion are The not and accounted other for fields by monodromy. On the contrary, individual theories. From theUV point completion, of which view tames of anycould the possible spoil relaxion, contribution Higgs we from have naturalness. physicssoft gained above Indeed, a terms) the the controllable cutoff plays that parameter theof role the supersymmetry of breaking the scale cutoff is in an the automatic setup feature of of ref. any [ theory in which the breaking, while the soft massesthe are weak scale in theweak scale fundamental theory. is not Nonetheless, the the result hierarchy of between a tuning,the but sum of of a the dynamical two relaxation parts. mechanism. New interesting elements emerge, which were not evident in the When the soft termssymmetric become vacuum of of the the Higgs ordertry potential of is is the suddenly spontaneously destabilised supersymmetric broken. andnon-perturbative mass electroweak This parameter QCD symme- triggers effects, a which back-reactionthe stops on Higgs the the vacuum field relaxion expectation potential from value from further is evolution. found As near a the result, critical condition for electroweak and thus the fieldevolution, starts the slow-rolling vacuum towards energy thebreaks associated minimum supersymmetry with of and the the is relaxion potential.particles. the changes. This leading During means This source that this vacuum the of energy soft soft terms terms effectively scan for during the the history partners of of the the universe. SM As the field windsits around couplings its to periodic fluxes. potential,field. its This energy In effectively increases allows our at forsmall context, each large parameter we cycle that super-Planckian assume due generates excursions thatsmallness a to of the of sliding the this shift potential. parameter symmetry From isassessment a of natural requires field-theory the according knowledge point to of relaxion of ’t its is view, Hooft non-field-theoretical broken the criterion origin. by [ a familiar property that thedeparture field from spans the a commona interpretation non-compact compact of space. field the space Goldstone This andtum modes its hypothesis field as realisation is theory. excitations requires a going around exist strong beyond in the the ordinary context rules of of string quan- theory and the underlying mechanism is monodromy [ JHEP12(2015)162 . ) ), S α 2.2 ) are 2 (2.1) (2.2) (2.3) i √ + ), and its s a → a ], as a way to avoid 27 . – 25 ] on supersymmetry-violating 33 – 28 , i + derivative terms Φ i α α F i + 2 iq S e θ – 4 – + ), its scalar counterpart (the srelaxion → → ˜ a a i ), S θ a Φ 2 , in which the only degrees of freedom are the usual is the global transformation parameter. From eq. ( √ f α + to be dimensionless. The transformations under PQ symme- i a 2 S √ + s = S describes the relaxion ( S ] emerging from anomaly mediation, and gives some hope of detection at the 39 – are the PQ charges and 34 and of the quark, lepton and Higgs chiral superfields (collectively denoted as Φ i q S By construction our relaxation mechanism predicts that the supersymmetric particles From the point of view of supersymmetry, we have gained a natural explanation of the where we see that, under PQ transformations, the relaxion changes by a shift ( For convenience, we choose try of Our theoretical setup isthe simple and PQ minimal. symmetry breaking Wefields consider scale of an the effective supersymmetric theoryThe extension valid superfield of below the SMsupersymmetric together partner with (the a relaxino new ˜ chiral superfield LHC and, especially, atdifference future of colliders our operating scenario infairly is the light. that, 100 As TeV a unlike domain. result, the An we case find important of some anomaly very2 characteristic mediation, signatures the at gravitino hadron The is colliders. framework are heavy, with masses parametrically unrelatedfrom to the inflationary weak dynamics scale. imply Nonetheless,of constraints that TeV. their An masses interesting feature mustmass of be our by smaller setup a than is some thatmodels one-loop gaugino hundreds [ gauge masses are factor. smaller than squark This makes the spectrum very similar to Mini-Split absolute minima. In our theory,potential, supersymmetry but is non-perturbative recovered QCD atThe the effects main bottom trap obstacle of the the for field relaxion requires the very some viability far modifications of from of the its the theory true minimal is vacuum. model the and strong we CP present problem. some Its possible resolution ways out. Goldstino. Supersymmetry is brokenbetween the in minute PQ-breaking a effect metastable andsupersymmetry-breaking QCD vacuum instantons. structure generated The by makes essential simplicity the the ofof interplay framework the the interesting, relaxation quite mechanism. independently the general The idea basic of breaking reasonthe supersymmetry for theorems in that this metastable dictate vacua simplicity very [ constraining can conditions be [ traced back to little hierarchy problem, which was the originalalso target discovered of an this work. economical and Butextremely on elegant economical the way in way, we of terms have breaking ofadded supersymmetry. field only The content. one theory Besides chiral is thethe superfield. relaxion, usual whose SM scanning superfields, The value we pseudoscalar breaks have supersymmetry; component the of fermionic this component is supermultiplet the is JHEP12(2015)162 ). W 2+ 3 d / 2.5 (2.7) (2.8) (2.9) (2.4) (2.5) (2.6) 2 H ) † u S H + are generic qS , is not useful S − is technically U e  f S , ) i = ( † Z S  h.c. 0. We have chosen K , ] models, in which + + m K → 43 S  , ( . , any other form of  m . 42 5 d θ U W h.c. H 4 q d + + Z F u  ] and references therein. . H  + q ) only through derivative terms). 45 , s i i a + Yukawa int 2 ( a i ≡ 44 d κ . Φ √ + ] axion models, in which the PQ sector 2 H V  s u e S 41 † i , i a H 2  2 m ), the Lagrangian for the relaxion multiplet 40 )Φ √ qS − m † remain massless too. To obtain a non-trivial − = s , q 6= 0 describes DFSZ [ 2.7 S e – 5 – + 2 d a  q 0 + exactly vanishes because of the shift symmetry, ) in the superpotential H µ F m u f ∗ S a + ( − and ˜ W/f H F (which contains i ) a  s † Z s = ). Solving the equation of motion for the auxiliary field ( iqα W S is the second derivative of the in K¨ahlerfunction eq. ( m s 1 but, as we will show in section e a F ( . − 00 ) + + † S 2 κ O W 2 → K S S S π d a = (with c ) Tr + 16 is generated. 2 H ) can be eliminated by a superfield redefinition, as discussed in appendix A. u S real. At the field-theory level, the hypothesis S a ( m − ) and /f ( 2.5 H a s ) = 1 + 2 m a 2 K C L s π 2 (  Θ √ f κ θ i ( 16 h 2 were approximately canonical at small field value, i.e. 00 runs over the 3 factors of the SM gauge group and in eq. ( d θ − 4 a K qS /K d 2 a Z remain invariant. We assign the PQ charges such that the Yukawa interac- − g 1 e a 2 + ], then Z = 0 belongs to the class of KSVZ [ 3 ) ) = 1 q = † s and ˜ S ( ) = κ S L s + ( With the inclusion of the term in eq. ( At this stage, the potential for The most general Lagrangian, up to dimension-4 interactions invariant under super- a For the effective theory ofThe the factor supersymmetric axion, see [ S 2 3 C [( , we find O F where For instance, if a superpotential quadratic in would lead to thebecause same no conclusions. potential for Of course, a superpotential linearat in zero derivatives is a small mass parameter For simplicity, we take natural because the theory acquires a larger symmetry in the limit Here the index functions of the combination while supersymmetry insures that dynamical evolution of the relaxion wethat introduce mimics an the explicit effect of breaking monodromy. of the We choose shift to symmetry break softly the shift symmetry through is made of heavythe matter, ordinary while Higgs the fields case are charged under PQ. symmetry and PQ, is given by tions are invariant, but wecarries allow a for PQ the charge possibility that the gauge-invariant Higgs bilinear The case while JHEP12(2015)162 ). 1, and = 0. 2.5  given a (2.12) (2.11) (2.10) s and all a s s = 0, a , with ¯ s and the gauge rolls. (1). We will not does not depend = ¯ S a O s s = s ma . 1). Here we are making ∝  a . is displaced far from its minimum ) in the K¨ahlerpotential of eq. ( s a ) is minimised at m , F ( i ma . κ Z ) ∝ a s π 2 2.10 ˜ a α 4 a ), starts at a large field value, while + ≈ and a 2.5 2 s a 2 F ( a π 2 – 6 – c 2 2 a 16 m g 0 (valid in the limit will not change during the cosmological evolution and gauge superfields is determined by the anomaly ma , m s = ≈ = , but the important point is that ¯ S is positive and generic; hence ¯ ∝ 2 s ) a and matter is more model-dependent. For this reason, ˜ g s s κ 4 (¯ . 0 S M V/f κ ]. The complete calculation of the soft terms is presented as an approximate relation, but we suspect that this equation must ma ) has a supersymmetry-preserving minimum at 46 scans very large field values. ≈ ma a 1. 2.10 F ≈ m , m  F 1. As a result, ¯ ∝ a a  acquire masses larger than the Hubble rate, and therefore it is natural m a a in the superpotential of eq. ( background, the potential in eq. ( a W in the limit Soft scalar masses are induced by the functions Since the relaxion background breaks supersymmetry, effective soft terms are gener- Here we have omitted factors of order unity coming from wave-function renormalisa- On the relaxion background, the relaxion, srelaxion, and relaxino masses and the Our assumption that in the early universe On a fixed The potential in eq. ( Here we have derived a 4 However, while the coupling between condition, the coupling between have a more universal(neglected validity. here) However, will we modify expect our that equation. effects suppressed by powers of the Planck mass Gaugino masses are expectedsupersymmetry to be breaking a one-loop factor smaller than the parametric scale of ated, as in axion-mediationin [ appendix A.dependence. Here we Gauginos give acquire onlyfield their strength approximate masses expressions from that the coupling exhibit between the parametric tion. The important pointthe is mass that, of during the theare srelaxion cosmological much and larger evolution the than in supersymmetry-breaking the the scale curvature range scan of linearly the quadratic with potential on which to expect that, at thenecessarily beginning true of for inflation, the they relaxion are field. found at their minimum.auxiliary field This are is proportional not to of the relaxion, as long as scalar fields of the supersymmetricas SM self-consistent. lie As at we thefields will minimum other show of than in the the potential, can following, be on the viewed relaxion background, all scalar by the solution ofthe the assumption equation that theneed function to know theon exact location of ¯ However, we assume that atand the starts beginning at of a inflation value From this we obtain the scalar potential for JHEP12(2015)162 0 . a µ µ U (2.14) (2.15) (2.13) with is given ) breaks µ is chosen, µ , which is 2.5 ∗ 0 B . -problem in , forcing the µ µ F M U . Also a is technically natural. . . 2 f a a 2 . The non-renormalisation  m f 0 . B , with a coefficient that scales i µ c µ and comes from the function Φ † i + are generated for general functions a Φ 2 in the Lagrangian in eq. ( ) ma † 0 -symmetry is explicitly broken by the µ ma bilinear in the potential). The complete S µ ≈ 0 R d if the mediation between the matter and , gaugino masses are the dominant source c + ma . ) that parametrizes the mediation of direct H ∗ ijk f S = and ∗ u ( ) and scales linearly with A f GeV. µ H ≥ M θ U πf/α The second one, parametrized by the coefficient 4 – 7 – f/M ∗ 4 16 2.5 d term in the Lagrangian, the scaling of ≈ 5 . M i 10 a  µ Z ), there are supersymmetry-breaking sources for ˜ m ∗ . 2 ∗ 2 2.5 M f f M in eq. ( ma , B (with µ ∗ U = c . If M f the wave-function renormalisation computed in appendix A. Thus, − ) and L 0 in eq. ( 0 P µ µ 0 M ) much smaller than the PQ scale µ = ) and, as a consequence, by the background value of to vanish. However, the ≈ 2.5 µ ∗ 2.7 B c M is given by the sum of two contributions. The first one is -symmetry. By imposing such a symmetry, one could forbid and ; the other one scales quadratically with in eq. ( µ in eq. ( a R µ 0 c µ m is much larger than , as expected in the most general effective theory, then scalar masses dominate f ∗ (which is the coefficient of the scalar M ≈ µ ∗ B Note that the simultaneous presence of Besides the explicit Since supersymmetry allows for a Soft-breaking trilinear couplings of order Here we have included into 5 M , but could be suppressed by powers of , originates from the function i µ a continuous coefficients parameter should be regarded here as a running parameter. independent of the background value of c by the sum oflinearly two with contributions: one is proportional to and calculation of these termsdependence: is presented in appendix A. Here we only show theWe parametric find that parameter property of the superpotentialThe insures hierarchy between that these the twoour condition masses context. is just a reincarnation of the usual relaxion sector occurs only throughtrilinear heavy states. couplings, scalar However, whatever and value of gaugino masses scale linearly with is different than inrelaxation the mechanism. case For of phenomenological reasons, scalar we and are gaugino interested in masses taking and the this mass will be crucial for our where of supersymmetry breaking ingravity the mediation visible ( sector. The latter case occurs,Z for instance, for If the supersymmetric spectrum, with gauginos lighteruse by of a one-loop the factor. non-renormalisation theorem However, making of supersymmetry, it is possible to imagine setups couplings between matter and the relaxion superfield through the interaction K¨ahler This gives scalar soft masses parametrically equal to we introduce a new mass scale JHEP12(2015)162 , a ). = ≈ µ f the B β F (3.2) (3.3) (3.4) (3.1) ∗ , 2 a µ in terms ∗ c ) for , 2 | larger than , with tan µ 2.15 | ∗ . We require the β are the two mass 4 . In this situation, M a + 2 H ∝ /m d 0 ) 2 H µ . a ( m and 2 |  D µ + h a u B , 2 H have a logarithmic dependence − | m i
c 2 = . Here | µ D | 2 H are model-dependent, but are expected + i , c d ∗ refers to a generic coupling constant. In 2 H c -interactions α 0 β , m m H m 2 µ 2
+ – 8 – 2 = | 2 ) = 0. Parametrically, it is given by ∗ cos µ ∗ | H a a 2 0 ( + g 2 H ), where 2 D u 8 m + 2 H ma 2 ( + m g . The coefficients 4 2 ) and we find the simple scaling f/ , together with the conditions necessary to have solutions for h a = i ≡ 2 c will progressively decrease, as the relaxion evolves. Once 4 ) can flip sign and trigger electroweak breaking. We call 3.1 λ ) ) ln , the Higgs potential for the real scalar components of the two m a a a ∗ ( π ( + a -scaling is violated and, under certain conditions on the coefficients 4 , λ , the energy scale at which heavy modes are integrated out. Below u,d D 2 4 D c ) f is contained in the soft terms, as described in eq. ( h a α/ a 2 H 2 h ( a = 2 m m D u,d , the 2 H = = of the operators responsible for squark and slepton masses remain positive /m m 2 h V 0 i c m µ is a coefficient of order unity. In appendix B we show the expression of ∗ c . The expressions derived in appendix A for the soft terms should be viewed as the In the proximity of We consider an initial condition for the relaxion such that a . We are interested in the situation in which this evolution triggers a non-vanishing can be neglected in eq. ( . The calculation demonstrates that electroweak breaking can be consistently achieved , the order parameter 0 , the soft terms run according to the usual renormalisation-group equations and receive ∗ i where we have kepteigenstates only obtained the from leading-order the terms current eigenstates in by rotation of an angle a in a fairly broad range of parameters. Higgs doublets can be written as value of the relaxion field for which where of the soft-terms coefficients µ proportionality factor to be positive,unbroken. since we The want value thatapproaches electroweak of symmetry is initially c In practice, this is aeasier conservative to assumption, achieve symmetry since breaking thecoefficients by logarithmic dynamical running evolution. makes We it willthroughout only also the assume evolution, that such the that colour or electric charge breaking vacua are avoided. on matching condition at f corrections of order ( our calculation, presented in appendix B, we have neglected this logarithmic dependence. The dependence on while we take to be of order unityAs (unless soft scalar terms masses are come from running a parameters, mediation the scale coefficients During its dynamical evolution, thema relaxion scans the supersymmetryHiggs breaking vacuum scale expectation value. Theis order the parameter determinant for of electroweak the symmetry breaking Higgs mass matrix, defined as 3 Relaxation of supersymmetry breaking JHEP12(2015)162 (3.5) (3.6) (with ∗∗ ). More because, = 0 and a 4 decouples a i = h ∆ H h a 0 has reached the mµ 4 ∼ 2 h m . ]. ) decreases, while the second , the heavy Higgs 1 4 H , a very small variation. In the 3.5 a a , m ) have grown enough to make sure ∆ = 0. As a result, for the relaxation 4 cos m a |  4 is the present lifetime of the universe. ≈ D | U . Adding this interaction term to the PQ ) flips sign again at a value , the SM Higgs gets a vacuum expectation τ + Λ F a a ( D 2 a (local minimum) D 2 changes more rapidly, ∆ – 9 – f 2 2 2 0 m ), where /µ 4 0 Λ D 4 µ / ) = is growing. A local minimum of the relaxion potential 4 2 Λ ∼ a , the first term in eq. ( | f ( f ∗ 2 h D − a V ) with respect to the typical available energy (Λ | ), we obtain m ≈ πf ], the barrier heights grow as we probe smaller field values and m exp( ) can cancel each other. This happens when Λ 1 2.10 4 a scales roughly linearly with the Higgs vacuum expectation value. ) = 0: ( f ). 0 a 4 ( 4 U 0 2 H V ). For a separation of scales τ V 4 H ∼ λm and the soft mass scale by ∆ ( /m / evolves below P π D D a ( ] is that Λ − , the Higgs mass so that = 2 O ∗ 1 and stop the relaxion before it could slide into the region with ∗ a = + a a i u ∗∗ 2 ) and turns back positive, restoring electroweak symmetry. This is consistent with 2 a 2 H . Let us investigate the issue. ∗ h m behaves like the SM Higgs. For negative h 2 ∗∗ /m f h Between two adjacent troughs of the periodic potential, the relaxion changes by an There is an important difference in our setup with respect to the non-supersymmetric So far, our discussion has been purely classical. However, at the first local minimum, After electroweak breaking, QCD instanton effects generate a potential for the relaxion < a d and 2 H ∗ ∗∗ a a < a amount ∆ vicinity of case. In the modelnever of ref. cease [ to exist.the supersymmetry As scale. shown Quite ina generically, appendix B, thisthe is notion not that the amechanism supersymmetric case to vacuum work, for the exists the barriers at have relaxation to grow of sufficiently high during the evolution between uum. Quantum evolution appears toroughly populate the different minima. same Although value allwith of of different the Higgs them weak values. have scale, This potentialmechanism this cosmological and could problem is is result present generic in also of a the in relaxation universe the made original of model patches of ref. [ We estimate that the probability ofable vacuum universe decay during is the pastThis light-cone of is the completely observ- negligible,consecutive vacua as (of a size resultproblematic 2 are of the the quantum considerable fluctuations field at the distance time between the two relaxion is settling into its vac- the barrier height is sufficientlyrelaxion small has to established make itself quantum in fluctuations its important. final Once minimum, the quantum tunnelling is not a problem. one quickly increases because is generated when thethat barrier the heights two (measured terms by in size Λ where Λ is the typical scaletion emerging of from ref. non-perturbative [ effects.Therefore, An important as observa- and value that respects only a discretebreaking shift potential symmetry in for eq. ( m JHEP12(2015)162 . ) is /m 4.3 H (4.5) (4.1) (4.2) (4.3) (4.4) 0 µ ∼ π 2 / such that the ) ∗∗ m . a . Λ. before the relaxion − 0 > ∗ a 0 . 2) is subdominant with mµ µ 2 / 2 = ( containing the inflaton as f a . I ) and here is only treated as 2 ), we find N 2 f . 2 f 2.5 i 2 . Nonetheless, the value of m φ a ( ∗ i H / φ 2 2 P ) has only a modest relative variation, H M = 3.5 i = 2 Φ η † i Φ = (relaxion slow roll) I  – 10 – . † of a chiral superfield ) implies θ I P /m 2 4 P 0 d µ ) is automatically satisfied, as M f P 2 HM Z ≈ 0 (inflaton dominates the vacuum energy) 4.4 H M 2 ∗ µ P = (soft terms from inflaton are subleading) 1 a I M F Λ (potential barriers from QCD) m < f , with P 0 ∗ 0 H < a M µ ≈ , it crosses a huge number of oscillations have the chance to grow up to values of order a ∗∗ H < µ periods of oscillation. 4 a H > . So there is enough freedom to choose a suitable value of /m 0 and 0 µ µ ∗ a happens to be near a zero, its derivative is generic. As the relaxion scans the range D We require that the Hubble rate be smaller than the QCD scale Λ to insure the The requirement that the relaxion potential energy ( The vacuum energy that drives inflation breaks supersymmetry and can be described The first requirement is that the relaxion satisfies the slow-roll condition during evo- formation of the potential barriers from instanton effects This condition implies that eq. ( In order not to spoilmust the relaxation be of subleading the supersymmetry withimplies scale, respect the soft to masses the in eq. contribution ( from the relaxion superfield. This by the auxiliary component scalar component. This sourcethe of SM supersymmetry fields breaking through will interactions feed that into cannot the be soft weaker terms than of gravity, thus giving respect to the inflaton energy (3 lution. Since the slow-roll parameters are inflationary sector is not includeda in spectator the that Lagrangian in provides eq. asubjected ( nearly to constant several Hubble strong rate constraints,the which theory. limit We the list here allowedfield range these constraints, of in which the the must parameters range be in satisfied for values of the relaxion completes 4 Inflationary dynamics The dynamics ofit inflation provides is the an friction essential term element that of stops the the relaxation relaxion mechanism at because a local minimum. However, the between In doing so, the firstwhile term the in the second potential one in0 grows eq. to fast, ( about since thebarrier Higgs heights Λ vacuum expectation value changes from while JHEP12(2015)162 ) in (4.9) (4.6) (4.7) (4.8) π larger (4.10) (4.11) 2 / a 3 H ) ) and expressing , . 3.6 . GeV, the so-called 30 4.2 . 12 10 GeV 42 . ). Finally, note that GeV 2 9 25  10 –10 4.9 9 − . 4 10 GeV  10 10 from eq. ( 0 5  GeV µ  ∼ 5 H 10 0 ). GeV f ) and ( 10 µ 2 MeV and × GeV . 5 f 0  4.7 0 5 GeV 9 4.8 µ 2 10 3 3 5 is derived from eq. ( / / 10   4 1 10 6  GeV, for the same reference values of m     39 < 2 GeV f ) 0 ) and identify the acceptable range of the  , see eq. ( 9 µ GeV GeV f f GeV (classical evolution) f 4.6 m 10 4.6 9 9 of 10 9 – 11 –  GeV 10 10 f )–( ) dominates over the stochastic term (3 10 4 9   a f   fa 3 3 3.6 10 8 2 / / 0 4 4 ) comes from the requirement of a classical relaxion   ), we find m   4 4.9 Λ =  4.5 ). Using the lower bound on a Λ /f ( 300 MeV Λ Λ 0 < m f µ ) a  3 Λ ( 300 MeV 0 ) and ( 300 MeV 300 MeV H = a/V  V ∗   300 MeV ∆ 4.2 300 MeV and 2  < f ) do not add any information, since they are automatically satisfied 2 N > ), we find 0 ≈ a > a H < µ H H < 4.4 3.6 m ∆ = 3 N ]). The value of the PQ-breaking mass ) and ( ). The number of e-folds required for this field excursion, in the slow-roll regime, 48 . This implies , ∗ 4.1 a 47 4.10 A successful relaxation mechanism requires that the relaxion scans a range ∆ We can now put together eqs. ( Finally we impose that the evolution of the relaxion is governed by the classical poten- through eq. ( m This enormous number oftential e-folds caused is by the a tiny consequence value of of the its shallowness mass of the relaxion po- which corresponds toeq. an ( excursion ∆ is given by eqs. ( when the other conditions are met. than The stronger constraint inevolution. eq. Although ( justified,mechanism may this operate also condition in maywe presence of be have sizeable quoted too quantum fluctuations. separately restrictive For and this the reason, the two relaxation bounds in eqs. ( A stronger constraint is obtained from eq. ( Taking together eqs. ( theory parameters. The PQaxion scale window, can satisfying vary presentref. in experimental [ the and range cosmological bounds (for reviews see tial, rather than followingthat a the random classical walk force driven ( the by relaxion quantum equation fluctuations. of motion. This This requires implies JHEP12(2015)162 . ) ), 2 P 4 / 1 ) gM ), the f ]. This 0 µ ∼ 2.7 V < f 59 ( ], for any n E ) ∼ P 4 / 1 always appears V a is present at low characterising the g af/M controls the breaking  m/f )[1 + ( = a  ( V h ]. In our case, the super-Planckian = 66 4 – /f 60 ) [ a ( P . More generally, as the relaxion explores are allowed. The relaxation mechanism re- f V M ] and can lead to Planck-suppressed operators 58 – 12 – – 49 a < f/m . Therefore, the mechanism of relaxation is incompatible P , which is much smaller than one. Thus, a selection rule , but to the relaxion monodromy. The violation of gravity a generic function. In the specific case of eq. ( P P M V h . If gravity respects the selection rule, Planckian operators are /M 0 µ f > M m/f ∼ = P  ), where are ruled to be smaller than , and so the condition is satisfied. For the same reason, we can argue that a in the potential. Using dimensional analysis, our assumption states that a f (  is ultimately related to the super-Planckian displacement of the relaxion. In V af/M /m  h 0 µ = ), casts doubts on the use of the effective field theory, since the Lagrangian in ≈ . In the case we have considered in our paper, the typical value of the expansion ) is valid only up to energies of order 4 n a /f 4.10 2.5 Another concern is related to the conjecture of gravity as the weakest force [ Nevertheless, these considerations are not sufficient to believe that super-Planckian ) a ( strength (or, more precisely, thesmallness weakness) of of the forcesummary, the breaking conjecture the of shift gravity symmetry.the as effective The the theory weakest at force a indicates scale with a the premature conjecture violation of of gravity as the weakest force. This occurs becausestates the are effective added at theory thein becomes cutoff particular scale. prohibiting inconsistent The super-Planckian with conjecturedecay displacements constants gravity may of be axion-like unless extended particles, new toexcursion since non-gauge is their forces, not due to as the weakest force comes from the very small parameter could keep Planckian corrections underselection rule control. may Nevertheless, be we violated will in show realistic later examples. that the conjecture can be statedenergy, then as the follows. effective If field a theory gauge ceases force to with be coupling valid at energies around V small parameter is expected to modify thepower potential in the form parameter is that do not respectin the the super-Planckian shift domain. symmetry,circumvented. giving To From enhanced see a this, contributions low-energyof let to the point us shift the of symmetry assume view, potential multiplied and, that this because by a problem of small a may parameter selection be rule, the relaxion field knowledge of quantum gravity isis needed, much as smaller the than typical potential the energy Planck ( mass. physics is not goingexpected to to violate modify global symmetries the [ relaxion potential. A first concern is that gravity is the super-Planckian regime, theHowever, if quantum we field require that theorywe the approach obtain potential may energy that does seem field not excursionsquires questionable. exceed up the to cutoff scaledescription ( of the relaxion evolution in the context of quantum field theory suffices and no 5.1 Planckian effects The vastly super-Planckianeq. field ( excursion requiredeq. by ( the relaxation mechanism, see 5 Remarks on the UV completion JHEP12(2015)162 ) = and x (5.3) (5.4) (5.5) (5.6) (5.1) (5.2)  ( ˜ m h the form P . and M . n x . m/f ) = = x  ( . V h . is independent of the special , 0 ) 0 µ = 1, and µ a ( ). This means that the relaxation p ˜ m 4.9 for the more general case defined in f h . = 2, 4 p (local minimum) n = n 2 1 P − ˜ and m p M n 3 3 14 p n 8 3 f (classical evolution) Λ , – 13 – 1 ) p − < n a 0 (  (critical condition for EW breaking) (inflaton dominates the vacuum energy) µ V 0 f h µ 3 3 4 / ) coincides with eq. ( / 4 1  f f 4 Λ 5.6 . In this case, the bound on =  ), we find the upper limit on p 1 p p at which electroweak symmetry is first broken is determined , which gives n f 2 Λ V 0  5.5 a µ   H < = 2 0 f 0 µ f n ≈ ≈ µ  )   ∗  1 a ) and ( P 2 ( ≈ f are generic functions. For concreteness, we take M m 5.4 ∗ ˜ m a h H > and the result in eq. ( and n V ). We will recover our previous results for h 5.1 The critical value for . We can now repeat the analysis of section p Supersymmetry implies value of of supersymmetry breaking worksbreaks the independently shift of symmetry. the form of the superpotential that Combining eqs. ( Classical evolution of the relaxion at early times requires The constraint that the relaxion potential is smaller than the inflaton energy gives The local minimum of the relaxion potential is reached for x eq. ( by the condition ˜ analysis in the effective theory then gives where On the positive side,not the require relaxation a mechanism very is specialof rather structure the potential for robust is the in not relaxionoperate the the potential. nonetheless. same sense Even as that To if the explain it one above assume the does we that point, assumed the let at expression us lower of energy, reconsider the the soft our mechanism masses analysis can satisfies in the terms same of selection rule. Dimensional 5.2 Generalising the relaxion potential JHEP12(2015)162 0 , ) 2 P W 5.8 (5.7) (5.8) /M . The 2 | and W W W , preventing with respect ∼ | is the super- ˜  m V W . a , H > ! measures the breaking ) and 2 . |  a 2.5 W | ) has a validity cutoff at the 2 2 P f . 2.5 3 M 2 , where | 2 P − a depends also on W M and the potential is 2 |

2 P needed by the relaxation mechanism W ! W 00 0 2 a 0 /M | K K K ]. In our case, this runaway direction is a 2 2 P P 2 W 2 f 67 f M M ∼ | + − – 14 – 0 m is a function of 3 alone, while

W is typically suppressed by a factor of s

W 0 1 − W ≈ − 00 (1) and the potential develops an unstable direction as ) K O a (

= V K 00 2 2 P 2 f f M /K e 2 , independently of the specific form of the superpotential 0 is a function of 2 P = ) cannot both depend on the single variable K , where the relaxion interactions lead to a violation of perturbative uni- K M f V 2 ˜ m dW/da ∼ = V (1), then 0 is the (dimensionless) K¨ahlerpotential defined in eq. ( O W K = and we obtain i Unfortunately, supergravity brings in a problem that was not manifest in the effective Suppose, as done previously, that s W h grows. This feature is well known in supergravity inflationary models and it usually 5.4 Effects beyondThere quantum is field another theory important issue aboutupon. the The UV effective completion theory of defined therelatively by theory low the we scale Lagrangian want in to eq. remark ( tarity and require a UV completion. However, we are assuming monodromy for the relaxion cosmological constant. Nevertheless, we believethat that one this could example encounter illustrates when violations the ofon difficulties the the selection rule effective modify theory. the(where expectations In based the case of supergravity, this comes about because theory analysis. Since the softwe masses are obtain ˜ constraint that the energy densitythe relaxation is mechanism. inflaton-dominated Of then course, requires to in relaxation order in to supergravity, understand one if should this have is control a over serious the impediment mechanism that cancels the universe. Assuming that thewould relaxion make starts it at slide Planckian alongit values, the the runaway is direction potential stopped deep in into by eq.would the QCD ( super-Planckian not region effects. be until the Thefrom result huge the of dynamical value an evolution. of artificial choice of initial conditions, but would be derived If a requires the addition ofvirtue new because stabiliser it fields could [ naturally explain the initial condition of the relaxion in the early the superpotential, of the shift symmetry. Into this case where potential. Since we have assumed that the breaking of the shift symmetry resides only in Our conclusions about the robustnesseffective of theory. the mechanism However, come a(like from estimates gravity) UV based and completion on violates that the thepoint, introduces selection take new rule the example can dimensionful of modify couplings supergravity, our in conclusions. which the To illustrate scalar the potential is given by 5.3 Supergravity effects JHEP12(2015)162 0, → as de-  f , we expect  and f . We know how to UV- f 0, shutting off the effects of monodromy. →  . In our case, this would be a supersymmetric . We want to claim that this is not the case. f f – 15 – is so large that, although the relaxion potential is PQ is some power characteristic of a given process. In the limit , otherwise the derivative couplings of the relaxion would lead to f k is very small, Λ  , where f k ) f/ ( ∼ . Let us consider a theory where the interactions leading to non-compact behaviour are As all of the relevant relaxation dynamics occurs at energy below the QCD scale, we In conclusion, although a UV-completion of the relaxion monodromy would provide We now introduce the shift-symmetry breaking terms controlled by the small parame- Let us first consider the theory in the limit  PQ a supersymmetry-breaking minimum depends onmay non-perturbative only instanton be effects estimated which from the chiral Lagrangian. wish to determine therelaxino. potential To for begin with the we lightest will degrees construct a of supersymmetric freedom: effective theory the below the relaxion weak and the To understand the supersymmetry breaking mechanismstructure it of is the necessary theory. to The study processsupersymmetry the is vacuum complicated breaking in scenarios comparison to by morein the familiar a tree-level fact metastable that minimum arises themuch at higher dynamics the energy. stabilising QCD Very the scale, differentall whereas energy relaxion of the scales them soft must play masses be a are considered role induced simultaneously, in since at the mechanism. Furthermore, the stabilisation of the relaxion in a UV-completion is not urgentlyenergies required below for the the Planck application scale. of the relaxion mechanism at 6 The structure of supersymmetry breaking high energies. Since exotic from a field theory perspective,until we well do above not the expect Planck it scale, to at exhibit which pathological behaviour point fieldvaluable insight theory and may understanding break of down the in possible any UV case. physics behind the mechanism, such present, while the ordinary axion-likescribed interactions above. are As UV-completed the at onlyany two the pathological parameters scale in behaviour this of simplifiedΛ picture scattering are amplitudes towe become recover the apparent result that, only leaving at gravity aside, a the theory scale is UV complete up to arbitrarily freedom remaining in the low-energy effectivethe theory, axion-like we couplings envisage to such a enter UV-completion ataxion for the model. scale ter parametric growth of scatteringcomplete amplitudes the at axion-like energies interactionswhich above by spontaneously breaks promoting a the globalbe theory PQ-symmetry. identified as In to the this a argument case of renormalisablewill the a enter complex relaxion model scalar scattering will field typically amplitudes, andWith the curing additional this any heavy radial pathological in mode behaviour mind, in although physical processes. in this work we always consider only the light degrees of could then believe that ourmaybe theory involving requires string a theory, drastic at departure the from scale quantum field theory, In this case wecompleted at simply the recover scale the usual axion interactions. This theory must be UV- and it is not clear if a consistent UV completion within quantum field theory exists. One JHEP12(2015)162 . ˜ S a f (6.2) (6.3) (6.4) (6.5) (6.6) (6.1) . , and 4 . In order fs ) q , † m S fa + 4! h.c. , S ( 2 ) + f c ), but retain only the − ˜ aq , ) q 2.5 c 2 s + ˜ Q h.c. † 2 c + ˜ aq m Q 2 c ˜ a 2 q f . ˜ a + (˜ 2 h.c. 2 s 2 a 2 Q S + √ † 2 − ˜ q f Q c ia/
( ˜ e q amf 2 2 2 m | q i 2 ) 2 c † ˜ q √ m + | S − 2 − ia/ ∗ Q q + + c c ) 2 c M | Q S – 16 – ˜ q ( s q S 6 | ˜ aq am e 2 2 e ∗ f q q √ c 2 2 q m − √ ia/ m im 2 ∗ e 2 c + ˜ = f q M 2 Q ⊃ 2 † ˜ aq m √ c a W ∗ (anti-quark). To illustrate the mechanism more clearly, we Q q L (˜ to be dimensionless. This choice makes the dimensions of the interactions c 2 + S ⊃ − ⊃ − Q is a singlet scalar, no tadpole term is induced. The relevant Q † s ) L L amf Q 2 2 ∗ 2.7 + √ M i 2 ) acquire the soft masses † s S ⊃ − (quark) and + 2. As a result, supersymmetry-breaking terms are induced. Squarks and the S Q L ( √ 2 2 f ima/ = The background value of the relaxion gives a non-vanishing value to the auxiliary field Our starting effective theory below the weak scale includes the relaxion multiplet and We recall that we are taking 6 = K look unusual. Ordinary dimensionsMoreover, we are use recovered two-component by Weyl recalling spinors that for the fermions. physical fields are metry preserving) Yukawa interactions between relaxino, quarks, and squarks are and the Yukawa couplings of the srelaxion to quarks and the relaxino are together with a chirality-flipping squark mass term Although the srelaxion chirality-preserving (but supersymmetry breaking) and chirality-flipping (but supersym- F scalar srelaxion In a basis in which thesuperpotential relaxion is given couplings by have the been sum rotatedrelaxion of into term a the in PQ-invariant quark quark eq. mass mass ( terms, term and the the PQ-breaking will not use theessential most terms, which general are K¨ahlerpotential, as done in eq. ( light quarks (together with gluonsThe and Higgs photons, although multiplets we have do beenfrom not integrated the write out Higgs them vacuum and explicitly). expectation the valueto effect is keep of captured track of through chiral the the symmetry relations quarkmake breaking between mass supersymmetry the manifest couplings of and the describe(relaxion), relaxion the and theory the relaxino, in we terms will of the chiral superfields essential to retain thewill correct integrate properties out of the theFinally, heavy relaxion to squarks and go to relaxino below determine couplings.effects the the on Then the effective QCD we relaxion theory scale and at we relaxino. the will QCD include scale. the chiral condensate and study its scale, including light quarks and the relaxion multiplet. At this stage, supersymmetry is JHEP12(2015)162 ) 6.4 from (6.8) 4 ∗ 2. This / 4 /M Λ ), after inte- . ˜ a a f 4 i → ) (˜ q q L c c q q + h drops out from the q ) (6.7)  ∗ m h.c. M h.c. + + ˜ a q c a (˜ q 2  √ h.c. ia/ e + q q m c  q 2 − , we can treat the mixing term in eq. ( ), the induced four-fermion operator is √ q c ) ia/ m aq e – 17 – h.c. q ) (˜ . + s im aq ˜ a  2(˜ a (˜ − 2 ), squark and srelaxion exchange generates four-fermion am f 1 2 6.6 2 factor appearing in the chirality-preserving Yukawa coupling ) = m √ q 2 2 ∗ c )–( − q − 2 /M )( a 6.5 = ˜ 2 a f a f 4 2 2 L m − = L . The Feynman diagrams that generate the four-fermion operator ( . Working at the leading order in 1 To study the theory below the QCD scale, we capture the non-perturbative QCD effects Combining this with the PQ-breaking mass terms and the interactions of the relaxion, After summing the contributions from squark and srelaxion exchange and accounting By construction the soft masses are well above the weak scale, thus to understand the that generate the quark chiral condensate through the replacement the low-energy theory with supersymmetric states integrated out is described by the two Yukawa vertices is cancelledthe by srelaxion the soft double squark mass propagator. also cancels The in dependence the on samefor way. the Fierz identity (˜ figure as a masscoefficients insertion. of the four-fermion The interactions.squark dependence propagators, In the on the 1 two the diagramsis cancelled with messenger by chirality-preserving mass the squark mass. In the diagram with mass insertion, the factor 1 theory at the QCDthe scale interactions the of squarks eqs.interactions ( and involving the the srelaxion relaxino must and be quarks, as integrated described out. by Due the Feynman to diagrams in Figure 1 grating out the squarks and the srelaxion JHEP12(2015)162 2 ∗ /M (7.1) (7.2) (7.3) 2 (6.10) (6.11) f ≈ k , or . f 2 2 a √ 7 ), exactly vanishes . amf i 2 sin for gauginos, while for a 6.10 2 4 h √ . √ Λ π 4 sin − α/ 2 4 m ) that the inflaton dominates Λ √ ≈ 17 keV 4.2 k  = ) = i a ( can be different for each individual a ˜ a h . 2 (6.9) GeV k f ) that f P 9 2 2 M 10 m h.c. ), we obtain the vacuum expectation value 6 F f a k F , + ( √ , once the relaxion is stabilised, it plays the , m   ˜ a V = = a 6 2 – 18 – (˜ a 2 ˜ √ m ⇒ 2 / GeV 3 ˜ f m ) 5 m cos a . In this phase, the Goldstino resides primarily in the ( 4 10 2 ˜ a m  m = 0 + Λ 1 k i − 2 a a = ) h 2 a = 2 ( f / a

3 2 V ) 2 m a m is of order unity if the mediation occurs at the scale − ( 0 k = V ) = a L ( ) exhibits how the periodic term in the relaxion potential is generated from is the typical mass scale of supersymmetric partners. To keep track of the V ) = 0. i 6.10 ma a is the physical mass of the sparticle and h ( ≈ ˜ a m m F This completes our consistency check, as in global supersymmetry we know that spon- For simplicity we have not included the SM pion fields which have a small mixing with the axion. This 7 otherwise. Thus, we can write simplification does not modify our result, which is not affected by the inclusion of the pions. where ˜ sparticle. For instance, wesquarks have and found sleptons in section Here model-dependence of the supersymmetric mass spectrum, we define inflationary sector, as dictatedthe by vacuum the energy. conditionrole As of of eq. shown Goldstino. ( in However, gravityspin-1/2 section insures components that of its the degrees gravitino, of which freedom acquires are a absorbed mass in the 7 The relaxino (aliasDuring gravitino) the cosmological evolution of theremains relaxion, its light, supersymmetric with partner mass (the relaxino) of order On the vacuum, thesince effective mass of the relaxino, defined in eq.taneous ( supersymmetry breaking must beclude accompanied that by supersymmetry a is masslessby Goldstino. spontaneously non-perturbative QCD We broken con- effects at and the the metastable Goldstino can vacuum be generated identified with the relaxino. Equation ( QCD instantons. Minimising theof potential the relaxion leads to the following effective Lagrangian for the relaxion and relaxino JHEP12(2015)162 f ∼ = a (7.5) (7.6) (7.7) (7.4) . QCD 8 ] H 73 . ) , the change . ) ˜ ]) m 3 ˜ m . 71 , relic gravitinos > m < . In the case of a m . Using RH ˜ m RH T RH T T . m on RH (for ∼ T (for 2 / a 3 . ∆ / 4 . P 2 h 7 2 P M . Since this change is insignificant, m 2 10 8 M / 6 − 4 3 2 × T / 5 P 2 3 10 15 2 is ∆ ˜ must be smaller than ˜ πm m − 2 × a keeps on sliding down its potential. This 4  π m 16 10 ) from the relaxation mechanism. Thus, we RH a T 48 × /f is sufficiently smaller than ˜ . As shown at the end of section 3 i ≈ RH GeV 2 ˜ v m  T – 19 – ) ) = 5 RH 2 GeV ˜ a ˜ a ] (for a recent reanalysis, see ref. [ ˜ / T mf lighter than the other supersymmetric particles, it ˜ a 3 10 9 P 70 m m MeV P → (10 →    ∼ always exceeds the measured dark matter density for ) decays into the relaxino with a width 7 2 0 ˜ ∼ ˜ P / 2 P f/M  PP 3 / V ) ( 3 Γ( 2 h m MeV σ h m GeV , the Hubble rate at the QCD phase transition, when barriers RH 1 GeV, we find that the relative change in the Higgs mass is   4 T 4 1 ] QCD ), with ∼ − 10 T QCD 72 4  H GeV f RH 2 QCD ( 8 T ≈ / of the thermal bath produced by inflaton decays. This brings up the H , the thermal relic abundance of gravitinos gives a contribution to the QCD 2 2 P 2 10 T ˜ m f h , gravitinos can still be created in the early universe by pair-production from  M 2 RH 8 / ), we find that Ω ˜ (3 > ) we infer that as soon as m T 3 / = Λ 0 is larger than the typical QCD scale, at the end of inflation the barriers in Ω with < 7.3 2 7.7 V ∼ RH h P 2 T 2 h RH = / RH 3 T T /M Ω /m 1 If For If More interesting for cosmological applications is the relic abundance of relaxinos. Since As the gravitino is a factor 2 h − QCD m 2 QCD ˙ From eq. ( are never overabundant, thanks to the steep sensitivity of Ω scattering of SM particles andrate virtual times sparticle velocity exchange. is The [ thermal-averaged scattering From this, we obtain the gravitino contribution to the universe energy density [ Using eq. ( in the axion window. Thus we conclude that ∆ we conclude that the relaxation mechanism gives no boundenergy on density of the universe today [ continues for a time are restored. DuringaH this time, thein slow-rolling the relaxion Higgs hasT mass travelled parameter a distance for ∆ a variation ∆ temperature concern about possible upperturn bounds to discuss on this issue. the relaxion potential disappear and the field The decay rate isconsiderations, but too can fast play to a role be at of high-energy much colliders,Goldstinos as consequence have discussed for derivative in couplings, cosmological section duced in or at the astrophysical high early universe temperatures. they will Their be relic more abundance easily will pro- then depend on the reheating varies in the keV to GeV range. is the LSP. Any other sparticle ( This means that, depending on the parameter choice, the gravitino (or relaxino) mass JHEP12(2015)162 (8.1) ]). The should be 48 , m 47 is expected to be . ). These extra con-  /µ µ 2 2 H 8.1 µ B m  ] or oscillating atomic and f ˜ g 74 µ ), or else the gravitino energy B ˜ g µ M ) because 8.1 π EM < M α − RH T ˜ W M from Higgs-Higgsino loops. This contribution W θ ˜ . If the relaxion comprises some fraction of the W – 20 – 2 e ˜ G B, M ˜ g aG + sin M ˜ B M W θ . 2 ˜ g ). The same coupling also gives rise to gaugino masses. This . Loop effects are also going to modify eq. ( ) not very useful for phenomenological applications, although the ˜ M W cos 2.5 ]. Thus it may be possible in the future to probe both the relaxion- ˜ 8.1 B, ≈ = 76 M , ˜ ˜ F G RH 75 T aF aG in eq. ( c c a W a W ) Tr , and the coupling to the gluon The relaxion couplings to gauge fields come from the super potential term The relaxion-gluon coupling is directly related to the neutron electric dipole moment. In conclusion, we find that the reheating temperature must be smaller than the mass S e F ( a gauge-mediation-like contributions to is of the same order of magnitudeone-loop of larger the than first term in eq.taminations ( make eq. ( relation is representative of possible links between low-energy and high-energy observables. the results presented in appendix A, we derive that such relation is Unfortunately the link between relaxion couplings and gaugino masses is polluted by the molecular EDMs [ photon and relaxion-gluon couplings. C means that there is aenergy relation physics between observable) the and relaxion gaugino coupling masses to (a photons and high-energy gluons physics (a observables). low- From astrophysical objects such as stars,for compact the stars, production and of supernovae, relaxions may via be the used axion-photon to coupling. If search the relaxion comprisessearches some for of NMR effects the generated dark by matter an this oscillating coupling nEDM [ may be probed through two couplings most relevantaF for relaxion phenomenology aredark the matter coupling the to relaxion-photonments the (also coupling photon, referred can to beis as negligible probed haloscopes). then with relaxion In microwave helioscopes, cavity light-through-wall the experiments, experi- case and that observations of the relic abundance of relaxions at very different energy scalesin and, both in of our these setup, two one seemingly may disparate observe experimental8.1 experimental frontiers. signatures Relaxion detection The relaxion can be detected in usual axion searches (for reviews see ref. [ density is too large.matter On density the if other hand, the gravitino (i.e. relaxino)8 can explain the dark Phenomenology The phenomenology of the relaxion and supersymmetric particles is concerned with physics interpreted as the masslightest of gaugino). the This is lightest becauselight supersymmetric relic gauginos, partner gravitinos even can of in still the SM be limit produced particles of by (e.g. scattering heavy the of scalars. of the the lightest SM supersymmetric partner (e.g. split spectrum (e.g. when gauginos are lighter than scalars by a loop factor), ˜ JHEP12(2015)162 a k are W i a (8.2) (8.3) (8.4) c W ]. The ]. So our (where Tr 79 – S 39 – imply that kF 77 34 4 = ), it is clear than m 8.4 )–( 8.2 , , , 700 GeV 250 GeV 120 GeV    GeV (for a split spectrum) [ 8 GeV GeV GeV ). Their values depend on the PQ completion ˜ ˜ ˜ 5 5 5 m/k m/k m/k – 21 – 10 10 10 2.6    3 2 1 c c c could vanish. Nevertheless, this would not change our ≈ ≈ ≈ ˜ g 1 ˜ ˜ B W c M M M must be non-zero because the QCD anomaly is an essential ) because the contribution to electroweak gaugino masses from and 3 c 2 8.4 c )–( ). This region of masses is favourable for the prediction of the Higgs 8.3 4.9 . While f ) for the gaugino masses and expressing the scalar masses as ˜ 2.12 GeV (for a degenerate spectrum) or 10 Notwithstanding the model dependence inherent in eqs. ( In spite of having supersymmetry broken at such a high scale, all hopes for discovery Nonetheless, due to the connections between relaxion and gaugino physics it may be 10 is parametrically of the same order. µ gauginos could be withinspectrum the that reach we of obtained the isclaim LHC very is similar or, that to at relaxation anomaly-mediated least, provides Mini-Split of a [ future framework for colliders. a The natural mass realisation of Mini-Split. where we have chosen a GUT normalisationthe of the anomaly U(1) coefficients gauge coupling defined constant.at in Here the eq. ( scale element of the story, estimate in eqs. ( deeply rooted in thethe structure relaxion sector of through theeq. the theory, ( quantum since anomaly, the whichparametrizes gauge originates the sector model at dependence), communicates one we with loop. obtain Taking heavy sparticles also eliminateproton problems decay with operators. flavour-violating processes and dimension-5 at the LHC areexpected not to lost. be The lighter key than point squarks for and collider sleptons phenomenology by is a that gauge gauginos loop are factor. This result is masses. Nevertheless,supersymmetric the masses cosmological cannotof constraints be TeV, discussed arbitrarily see eq. large, inmass, ( but section since its must measured lie value10 gives below an upper some bound hundreds on the supersymmetry scale of about dominant source of the bino andthe wino Higgs masses sector. was likely due to gauge-mediated effects8.2 from LHC phenomenology Our relaxation mechanismfrom parametrically the decouples weak the scale, thus supersymmetry-breaking naturally scale predicting that supersymmetric particles have large possible to extract some aspectsFor example, of in the this soft setup massthat if structure at a in least relaxion the some contribution coupling eventcoupling. to of to the axion photons On bino the discovery. were and other observed, wino hand, itwere soft if would observed masses a imply and came relaxion from no coupling the to coupling gluons to (through photons an were oscillating measured, EDM) then this would imply that the JHEP12(2015)162 m µ (8.6) (8.5) . This ˜ a + MET, ˜ a ) is suffi- gg jj 8.2 → ˜ g functions (as in ˜ g ) predicts that a β in eq. ( , the relaxino mass → 8.5 3 greater than 100 c 7 /c , pp /c . NSLP m ` µ meters ) and is given by 1 2 − , the displacement in the decay that vary between 100 microns 7.4 10 10 E × 5 and the displacement for this decay 7  ˜ a . NLSP ` 1 Z ˜ g 5 → M  1 TeV ˜ B  4 NLSP or ). As displacements – 22 –  1 TeV ˜ a M γ 8.5  2 GeV → ˜ m  5 1. Since, as discussed in section ˜ B 2 ]). / 10 − 3 2 80  ) m 1 MeV ≈ ]  ˜ B NLSP ¯ q 81 = q → ˜ g τ E/M ( NLSP τ p NLSP τ . Second, unlike anomaly mediation where the gravitino is heavier than gauginos, with a lifetime [ = f is the speed of light. For an NLSP with energy ˜ B c q could in principle vary between the keV and GeV range, eq. ( q The gluino lifetime is given by eq. ( At hadron colliders, gluino production is the leading discovery process. As R-parity There are two differences with respect to the case of anomaly-mediated Mini-Split. 2 NLSP / ` 3 → ˜ For gluinos in the TeV massit range is and unlikely squarks that a theseciently loop decays small. would factor be or The observably bino less displaced, decays above unless via the gluino mass, searches (for a review, see ref. [ 8.2.2 Bino NLSP If the binog is the NLSP, the gluino decays predominantly through an off-shell squark can be experimentally resolved,distance. gluino If decays the are decay likelywhere occurs to the within be jet the displaced detector, vertices bywould the are an appear signature displaced. observable as would a thus If long-lived be the coloured decay particle occurs and outside would the show up detector, in the dedicated gluino R-hadron If the gluino isat the the NLSP, LHC it would decays leadsignature into to would a two be hard gluon striking gluon andmultiplicity as jets than a the and in relaxino. lack missing of the energy, Gluinothrough a typical pair an cascade Mini-Split off-shell production decay case squark. chain where implies each a gluino much decays lower jet to a neutralino collider-produced NLSP could traveland for a distances journey toconsider the the moon. three possible To cases describe of the NLSP. 8.2.1 collider signatures we Gluino will NLSP now individually where is m is assumed to beHowever, conserved, we all gluino can decays envisagelightest will supersymmetric different eventually particle terminate situations, (NLSP). with A dependingthe the common relaxino on relaxino. feature with is the a that nature lifetime the that of NLSP can decays the be into next-to- derived from eq. ( First, the ratios ofanomaly gaugino mediation) but masses are arethe essentially not scale free rigidly parameters, unless determined onehere by specifies the the the gravitino theory (i.e.collider at phenomenology. relaxino) is the LSP. These two features change completely the JHEP12(2015)162 0 ˜ B ˜ W γ/Z . In . In (8.8) (8.7) + a a  +˜ +˜ +MET   π γγ W W → → → + ]. The conse-    is highly sup- ˜ ˜ ˜ W W 82 W ˜ a ` ` jjjj h   ) wino decay to the 6 cm. If the gluinos → a 0 0 ˜ ˜ ˜ + ˜  B . W W + MET where the + + a 4   +˜ , ! π π  γ/Z ˜ 2 Z → → 2 W 4 W   → m ! m → ˜ ˜ W W 0 ˜  γ/Z γ/Z 2 Z 2 B ` ` − ˜ ˜ W W m m + 1

− 165 MeV at two loops [ < ` 1 W jjjj ≈ m θ

0 µ 2 ˜ W NLSP could be distinguished from the W θ 0 2 M . This will give rise to signals analogous to the ˜ = cot − W 0 ) or neutral ( – 23 – a )  ) ˜ ˜ W ˜ a ˜ a W = tan + ˜ γ Z + ) ) M  ). Due to the absence of mixing with the Higgsinos, the ˜ a  ˜ a → → ]. This leads to a very interesting scenario as the typical γ π W Z 0 0 8.6 82 ˜ ˜ [ → → → W → W ), thus it is observable. The decay ˜  ˜  B Γ( /c B Γ( ) and for the charged wino the branching ratio for this decay 8.5 ˜ ]. ˜ W W Γ( Γ( 8.5 87 – 83 , with a displacement 100 = 6 cm [ a 0  ˜ + ˜ W ˜ + W   π W →  → ˜ W  τ ˜ previous case, with thelong-lived additional feature of charged disappearingOne tracks or from both the gluinosthis regime decay the to majority charged ofW charged winos winos and produced from gluino decays will decay as One or both gluinosthis decay regime to the charged majoritythrough winos of the channel and charged winos produced from gluino decays will decay be displaced, leadingcollider to topologies a are finalNLSP similar, state scenario the through with the four branching ratio displaced prediction vertices. Although the Both gluinos decay toand the neutral collider winos. signaturecould would This be again displaced. would be: look Again, if similar the to squarks the were heavy bino enough NLSP the jet pairs may also In summary, for a bino NLSP the collider signature would be striking: • • • relaxino is given bymay eq. exceed ( or fall short of the decay to pions, leading to a number of distinct signatures. charged and neutral mass splitting is quence is that, if the chargedwhere wino is the produced pions in are a too cascade, softlifetime it to will be decay via reconstructed at thedisplacement LHC. for This charged decay ( would occur with a also be displaced, leading to a final state8.2.3 with four displaced Wino vertices. NLSP If the wino iswino the with NLSP, a the lifetime given gluino by decays eq. ( into a pair of quarks and a charged or neutral which, if measured, could confirm the bino nature ofwhere the the NLSP. photons wouldmodels emerge with from gauge a mediation.be displaced If replaced vertex. kinematically by available, a This one Z-boson. signal or In both can addition, of also if the the occur squarks photons in were may heavy enough the jet pairs may pressed by the negligible bino-Higgsinoare mixing. almost Because pure Higgsinos states, are there heavy is and a gauginos prediction for the branching ratio of the bino decays is also determined by eq. ( JHEP12(2015)162 . ]. 1 10 − (9.1) (9.2) (9.3) 10 towards < θ | θ | +MET, with a  W = 0 at high energy  θ W + , jjjj I . parameter of QCD is a number ϕ 2 I I θ 2) and auxiliary field are given by ) are displaced from their minima, 2 I . / a 2 λm m 2 a a 2 + √ + f 2 ) 2 ) I 2 eff I S 2 (0). On the other hand, the relaxion mass m 2 λϕ λϕ f eff + ) F − ) chiral superfields ) = – 24 – = 0. We find that, on the inflaton background, m a S λI ≈ m ( ( s i ) − V vertices, and potentially also displacement of the jet I ϕ m ) and the relaxion ( ( ) = ( ) = W I + MET and again a modest displacement of the gauge I I eff ϕ ϕ ϕ = ( F ( ( 2 eff eff γ/Z W F ]). Each solution has some drawbacks. m + ), and thus it is necessarily displaced by an amount of order 1 (called  ) and relaxion ( 3.5 I I W . Then, the supersymmetry-breaking scale during and after infla- + m  jjjj I λϕ  I modest displacement of both pair vertices. If the gluinowould decayed to be one charged andboson one neutral vertices. wino the signature both decayed to charged winos the collider signature is Let us suppose that, during inflation, the inflaton lies somewhere in the range In our context, the idea can be realised by adding to the superpotential a small coupling /m between the inflaton ( 2 m tion is always about the same, choose a canonical K¨ahlerpotential.the Our scalar dynamical component assumption of iswhile that, the during srelaxion inflation, sitsthe at effective the relaxion mass vacuum (defined as λ Here, just for illustration, we have taken a simple mass term for the inflaton while we The first class of solutionsevolution employs of the the idea relaxion that towards thedisappear electroweak PQ-breaking breaking after potential can that reheating. be drives present In the to during this inflation, rearrange way, but after itself the veryzero end close with of an to inflation, unsubstantial a the change relaxion minimum of will of the be Higgs the able mass. periodic potential, driving only sketch some ideas ona how complete to model, address leaving theto this issue, tackle task but the to we problemsolutions will future suggested not (two work. in attempt of ref. We to them will [ construct are present three adaptations9.1 possible to ways the Inflaton-dependent supersymmetric relaxion case potential of the experiments, since the limitThe on problem the is neutronwould severe electric be because dipole undone any moment byis hypothetical implies the endemic solution relaxation in mechanism setting all operatingOf relaxation at mechanisms course, low that any energies. employ model the This that QCD difficulty axion claims as to driving be agent realistic [ must solve this problem. Here we will The theory we presented predictsof that order the unity. CP-violating Thisof happens because the the potential relaxion inone is from eq. stabilised the ( at minima one of of the the periodic local potential. minima This result is in blatant contradiction with 9 Strong CP problem JHEP12(2015)162 . θ 2 (9.6) (9.7) (9.4) (9.5) . The 2 m ≈ ]). In figure (0) 89 2 eff , m 88 . GeV 2 become now very constrain- / 1 4  10 ]. As it is not possible to estimate − θ a , . 88 10 1 without affecting the scanning of the  ) to ensure that a quartic term in the )] [ ) cos Λ, we make a simple parametrisation f   a . θ T 0 T ( ) µ ∼ s I ( cos ], otherwise the complete function is not continuous. Θ( / ϕ 0 (0) T GeV 4 ( 2 4 90 µ π/α Λ 2 eff 5 Λ 2 eff – 25 – 2 m 10 ∼ m − ∼  a ) V ≈ λ < m  T is much larger than its value today exp[ ( θ a 4 I V T ϕ 8 GeV I f ]. ∼ 9 a 90 λm 10 V coefficient given in [  ≈ ) (4) 0 ) encodes the finite temperature suppression. This suppression has I d T ϕ ). On the other hand, the condition for classical evolution is not Λ. Previously this regime has been avoided as the instanton effects, ( H > 2 eff 4.5 m H > ) gives a negligible contribution with respect to the mass term. In con- a ( V ), gives Λ, see eq. ( 4.2 Let us first consider the behaviour of a thermal system, as this will lead us to an inter- However, the inflationary conditions discussed in section We flip the sign of the 8 where the function Θ( been estimated in a numberwe of plot different the temperature finite regimes temperatureapproximate (see suppression expressions e.g. factor in [ as [ calculated more recently, by using the to lowest order is giventhe by potential in the cross-over regime where esting analogy. At temperaturesthe well effective below axion Λ potential thepotential may instanton be is effects given calculated are by from unsuppressed the and chiral Lagrangian. The resulting At temperatures far above Λ the potential may be calculated in perturbation theory and, to occur during a periodcoupling of scale, inflation in which theand Hubble hence constant the exceeds axion the potential, QCDmay are strong be exponentially suppressed. advantageous. However, we will see that this satisfied and itreaching the remains correct dubious vacuum. if the relaxion9.2 can have Inflaton-dependent instanton a barriers An significant alternative strategy probability to of deal with the strong-CP problem is to have the field evolution eq. ( For sparticle masses inH the < tens of TeV range, one can barely satisfy the requirement potential clusion, this mechanismsupersymmetry-breaking can scale, efficiently nor reducing reduce its value after inflation. ing. The requirement that the inflaton dominates the vacuum energy, i.e. the analogue of ratio of these twoThus, masses we obtain squared corresponds to the reduction factor in the value of We must also require the condition during inflation, JHEP12(2015)162 ) . /H (9.9) (9.8) 1 H ρ < ), which now 3.5 Λ. We will assume that 5.0 > ]. While it should be kept H T 91 a . [ 2.0 π 2 ) cos . 1.0 H D H/ 0 T ∼ . Since for modes of small size ( mµ Θ( GeV 2 0.5 4 @ H f /H T T 1 ≈ + Λ ) 2 – 26 – Λ then a new cosmological IR cutoff is imposed H a 0.2 ρ < / 2 T 1 f 2 2 Θ( 4 0.1 m will contribute to the path integral. The path integral Λ ]. /H < ) = 90 /H 6 9 1 a 1 12 - - ( - V 10 10 10 0.001 ρ < L Λ, the axion potential is suppressed. The first is that instanton T H Q H > Λ. However, if 1 ). / 1 9.7 ρ < . The suppression of the axion potential at finite temperature, calculated using the Let us now consider the relaxation during an epoch with The effective potential for the relaxion during inflation is given by eq. ( The second perspective is that de-Sitter space may be thought of as exhibiting a Rather than considering bone-fide finite temperature effects, let us instead consider into eq. ( H The relaxion evolution of thisthat model the heights is of identical the tothe barriers our relaxion in original to the model, instanton-induced stop potential with after are the electroweak now exception breaking suppressed. we For need to satisfy the full evolution of the relaxion occurs during anbecomes inflationary period with constant finite-horizon Gibbons-Hawking temperature in mind thatstatistical this mechanics, an temperature estimate of is theT axion physically potential may different be determined from by substituting the usual interpretation in only instantons of radius is IR divergent and adistances cutoff is needed, with theand integration the over integration is the limited radius to extendingthe distances up gauge to coupling is perturbative, the instanton effect is exponentially suppressed. inflationary Hubble scales which exceedstanding Λ. how, There for are twoeffects different will perspectives only for be under- physical for instanton sizes which are contained within the horizon, i.e. Figure 2 approximate expression from ref. [ JHEP12(2015)162 0 c ]. 10 − 92 10 . Θ). This | 4 ) . H (Λ gauge group 3 T / 0 c ∗ a Θ( 2 | f 2 m ). Thus the strong CP ≈ H a T Λ as usual and suppress the ]. A new SU(3) 1 H < Ê 18 GeV) we have Ê & H at the end of relaxion stabilisation will be will remain the same as before. This means θ m 8 GeV ( – 27 – . 2 & H changes from the inflationary epoch (when Θ must be T θ is much smaller, as shown by the example red minima. θ that for ). 2 H T Θ( ≈ ) to late times (when Θ = 1) by a factor of Θ( θ H T . A schematic illustration of the resolution of the relaxion strong-CP problem with ] and our task here is to describe the supersymmetric counterpart. For our scenario Λ. This may be achievable by allowing the inflaton field to decay to visible sector 1 Unfortunately the condition for classical evolution is not satisfied once again. We Quantitatively, the relaxion at the minimum is such that sin We see from figure As before, the effective strong-CP angle axion-like field of a newref. gauge [ group beyond thethis SM. requires This follows supersymmetrizing the theis original non-QCD proposal introduced model of and of vector-likeis matter [ added. in the Some fundamental of and the anti-fundamental matter of fields SU(3) carry the electroweak quantum numbers of the left- fields throughout the inflationary period,The in details a of setup such similar a in scenario spirit remain to warm to inflation be9.3 [ investigated. Non-QCD relaxion Another solution to the strong-CP problem may be found by realising the relaxion as the and the strong-CP problem is resolved. This mechanism isspeculate sketched that in an figure alternativeaxion strategy potential would be during inflation toT take by genuine > finite temperature effects in the visible sector means that the effectiveevaluated value of at angle today is below the QCD scalepotential will and have grown, in while this thethat value at post-inflationary of late epoch times the therelaxion relaxion will amplitude potential evolve is of towards dominated the theto by new the the minimum periodic usual of QCD axion the axion. potential potential, This and appearing would the almost then identical solve the strong-CP problem. dashed line). At latevalue during times inflation the (solid axion-likevalue black potential in line). is which This the unsuppressed change effective and shifts has the grown final relative relaxion to minimum its closer to a of order unity. However, after inflation has ended the Hubble parameter will have dropped Figure 3 inflaton-dependent instanton barriers. During inflation the axion-like potential is suppressed (blue JHEP12(2015)162 . . 0 c a L N M M (9.10) (9.11) further N M -term has been F N. c to enforce that its L 0 c d ] lies in the couplings H 1 d . Hence both the QCD 0 y must couple to the visible + S c . For this reason the non-QCD LN e F u , while the others are electroweak c , which stops the field from rolling. H ). Confinement due to the SU(3) aF L c u a , y ]. e G 9.11 1 + and NN c d aG L = 0, a dominant mass contribution for the L H i 6 u NN M H u,d N d h – 28 – h y M u strong coupling leads to confinement of the matter must have masses at or above the weak scale to y + is the source of supersymmetry breaking. Thus to 0 c c c ∼ S LL , W c and electroweak symmetry breaking would be dominated L L, L 3 0 M Λ L, L + 0 a . The theory is described by the usual K¨ahlerpotential and a i ∼ c W c 0 a must be dominated by the Higgs vev. This mass contribution is 0 c W LL relaxion may be detectable in this setup. h 0 N,N ) Tr S will roll down a shallow potential, scanning the supersymmetry breaking ( a S C = , otherwise L W fields is induced by the interaction in eq. ( c A difference between this setup and the non-QCD model of [ The dynamical evolution of this setup proceeds as before. The pseudoscalar field In order for the relaxation mechanism to work the mass of the lightest Dirac fermion -problem associated with taking a small value for the superpotential parameter < M µ 0 axion, albeit with mass thatmust is also anomalously exist large. the To usualdominant resolve QCD the mass strong-CP axion contribution problem and will there itaxion come must and from not the couple QCD QCD to and SU(3) not QCD of the relaxion.generate In visible our sector soft realisation masses,sector including superfields, gaugino hence masses, thegauge non-QCD fields relaxion via the must usual haverelaxion axion-like couplings may interactions be to detectable the throughits visible the sector dominant usual axion mass experimental from strategies. an As it additional obtains gauge group it would appear much like a QCD N,N strong coupling generates anThis axion-like creates potential a metastable for minimum instabilised which at the supersymmetry a breaking large value and the Higgs vev at a small value. Unfortunately supersymmetry, which is brokenthan at that. a high scale, cannot protect contained in scale as it does so. At some point, when from a supersymmetric seesaw.of In order forHowever this this is mass technically term natural to so dominate long there as it is is a not form more than a loop factor below charged under SU(3) found after we integrate out The electroweak chargedhave fields evaded collider detection. SU(3) fermions. We mustΛ require the condensationby of a these technicolour phase. fields Theretechnical to are naturalness, be additional which restrictions may suppressed on be by the found mass taking in spectrum ref. related [ to neutral superfields superpotential given by handed lepton superfield, thus they are labelled JHEP12(2015)162 . Using an f . 0 µ , with SM superfields and a chiral superfield for – 29 – f the relaxion evolution is determined by the classical force (ii) we explain how the evolution of the relaxion leads to electroweak breaking we explore the limitations and the uncertainties associated with the super- 3 5 describes our theoretical framework, which is based on a supersymmetric describes the conditions under which the inflationary dynamics is compatible . The back-reaction from QCD instantons stops the evolution of the relaxion 0 4 2 µ the relaxion does not dominate the vacuum energy, so that it does not affect the (i) In section Section In section Section the relaxion is in the slow-roll regime. A further requirement is that inflation lasts for gravity-mediated effects on the soft terms from the inflaton sector are negligible and enormously driven by the dynamicsinstantons. of the The runaway vastly directionassumption super-Planckian to on values be initial of stopped conditions, only theto but us by relaxion rather if QCD from would this scenario the not canof dynamical result be the evolution. made cosmological from compatible It constant. an with is realistic not mechanisms for clear the cancellation of the mechanism underthat modifications relaxation of may the survive relaxionation Planckian potential is effects. inconsistent makes with usthat On the more supergravity the conjecture confident could lead of other to gravity hand,along the as interesting which we the situation the show of weakest relaxion an that force.conditions unstable could relax- nearly-flat at slide. We direction early also This times. remark would The give relaxion a would start natural at explanation typical of Planckian the values, initial then grow ably require UV completionsus beyond from the making rules definitive statements. ofto However, quantum emerge we field argue in theory that theeffective such and field-theory completion Planckian this approach, needs domain prevents we only observe andunder that not selection control rules in necessarily could the at keep region Planckian the of effects lower interest of scale the relaxion potential. Moreover, the robustness (ii) an astronomically large numberprobe of a e-folds sufficiently in large order portion to of give its enough shallow time potential. Planckian to excursion the of relaxion the to relaxion. The non-compact properties of the axion most prob- and not by thepotential. quantum random The walk, combinationHubble of so rate these that during two the inflation conditions and slow-rollingof an imply field some upper a tracks hundreds bound strong the of on(i) upper classical the TeV. scale bound Once of these on supersymmetry two the breaking conditions are satisfied, it is guaranteed that and stabilises the scale of supersymmetry breaking at awith value the of relaxation order mechanism.that The most stringent constraintsdynamics come of from inflation, the conditions and show how the background value ofinduced the soft relaxion terms. breaks supersymmetry and we compute the when the scanning softparameter terms become of the order of the supersymmetric higgsino mass For the ease of the reader we summarise hereeffective our theory results. valid below thethe PQ relaxion. scale To mimicshift the symmetry effect through of monodromy a we small introduce mass an term, explicit which breaking generates of the the relaxion potential. We 10 Summary JHEP12(2015)162 . P 7 f/M ] where additional matter is added such that 1 – 30 – ] free from the naturalness problem. As the LSP is the relaxino 97 ]. Unfortunately in the supersymmetric model it is challenging to – 1 95 to small values. For this scenario it also appears difficult to enforce θ we elucidate the mechanism of supersymmetry breaking, which is par- we considered the low-energy relaxion phenomenology and high-energy we investigate three scenarios to address the problematic strong-CP pre- 6 8 9 In section In section Once gravity is turned on, the relaxino plays the role of the spin-1/2 components of In section classical evolution of the relaxion,a although type we speculate of that warmthe modified inflation third scenarios may scenario involving be we adapt arelaxation a promising is model avenue a of for result ref.scenario future of [ satisfies investigation. chiral the symmetry classical Finally, breaking evolution in constraint. due to a non-QCD gauge group. This ated during inflation but dropssetup significantly described afterwards. in This ref. is [ satisfy an the adaptation constraint of that classical aIn dynamics similar the dominates second the scenario evolution we considered ofby a the QCD new relaxion instantons possibility may field. that be thestrong suppressed axion-like CP potential during induced relaxation angle and grow afterwards to force the signatures and a binopossibly or accompanied wino by disappearing NLSP charged would tracks. lead to adiction jets of and the missing relaxionrelaxion energy model. coupling signature, in In which the the slope first of scenario the we potential sketch breaking a the shift supersymmetric symmetry inflaton- is gener- (i.e. gravitino), the modelor predicts even gaugino outside NLSP the decays detector,signatures that giving may depend a be variety on of prompt, characteristicenergy, the displaced, signals. possibly nature two The or specific of more collider If the displaced the NSLP, vertices, NLSP and and additional decays they electroweak outside typically gauge the bosons. involve detector jets, then missing a gluino NLSP would generate R-hadron gaugino phenomenology at colliders. Althoughof the reach scalar at superpartners are the likelymay LHC, to be the be within gaugino out LHC masses andSplit are Supersymmetry possible suppressed [ future by collider an reach. additional So loop relaxation factor gives and a realisation of smaller than the typical softThermal mass. abundance The considerations cosmology require ofshould the that not relaxino the be is larger reheating discussed than temperature incould the section be typical after soft the inflation mass. dark matter. Whena this More combination bound generally, of dark is relaxinos matter saturated, could the and be relaxino relaxions. made of two species, being relaxino below the QCDnon-perturbative scale. QCD chiral The condensateat supersymmetric conspire the relations to metastable among make vacuum.Goldstino couplings the and This relaxino and supersymmetry indicates exactly the is that massless indeed spontaneously the broken. relaxino mustthe be gravitino. identified with In the our scenario, the relaxino is the LSP, since its mass is a factor ticularly simple in termsin of terms field of content dynamics —scales, because a varying of single from the chiral hundreds simultaneousactive superfield of participation role TeV — of to in but vastly hundreds complicated different thephysics of energy process, and MeV. obtain we Since all an devise these effective a scales theory simple play of but an the effective interactions way to between the capture relaxion the and relevant the JHEP12(2015)162 2 a = √ † / ˜ S ) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) + ia ], of the ˜ S ! + 94 s , h.c. 93 + = ( d a ˆ H ˜ S W u a ˆ H W i , 4 4 tr θ θ 2 2 | | . Here ) ), respectively). i ) becomes F F F | 2 | 2 N
00 θ . 0 2.5 i F θ Z 0 + ) Φ i QU  ˜ S , Z ) + 2 d − 2 4 ¯ ˆ θ are functions of the variable ), in presence of the supersymmetry- θ 00 H = F θ ∗ 2 0 . u | U representation under the gauge group ) U F ) ˆ i 2.5 + (ln S H F 2 i | i k Z + 00
is the auxiliary field. On this background, + ˆ Z Φ 2 2 and 2 U j i qF θ ¯ θ F ln ˆ Φ ˆ ∗ Z + Φ i F θ h − † i ( – 31 – ˆ 2 0 F 2 QF θ . Φ ), 0 ˆ 1 + (ln ¯ i Φ θ 0 a π
(1 Z  ) ∗ − U i T 2 ˜ , S k 2 4 h 16 q 0 F 1 + / A.3 0 θ Z ) 1 i 2 − F θ 2 d / | Z U e i Z 2 1 )–( H / X ) ijk F 1 | Z d = = ˜ ) ˜ P S S 00 + (ln H q d i A.1 q ) = ) = ) 0 i ) + † † − − ˆ H Z qS − Φ ) e S Z u e j S S − Z 1 ( + (ln e 0 H u Z a + + 0 µ H Z ) C ( ijk S S u Z ( ( ( ( θ H i 4 1 + (ln + (ln U for a fundamental or an adjoint of SU( θ θ θ Y Z Z d h 0 2 2 2 ) as the Dynkin index of the Φ d d d θ i N Z 4 i a Z d Z Z Z

= T + (ln i + + + q (ln Z + T ≡ ≡ = are the running Yukawa couplings, which include the wave-function renormalisa- Q 2 or L ijk / P ijk Y and primes denote derivatives with respect to this variable. To obtain physical masses, we need to work in a basis in which the kinetic terms for = 1 s i 2 T Here tion. We define ( the SM-sector fields arechiral canonically superfields normalised. This is achieved by defining the rescaled In terms of the rescaled superfields, the Lagrangian in eq. ( On the√ right-hand side of eqs. ( breaking background ofdenotes the the relaxion complex superfield scalarthe component functions and appearing in the Lagrangian can be expressed as A Computation of theIn soft this terms appendix wesoft outline terms the computation, coming based from on the the Langrangian method in of eq. ref. ( [ We would like toWe gratefully thank acknowledge Tevong useful You discussions with forP. A. Draper, collaborating Arvanitaki, N. in J. Craig, the March-Russell, S.is Dimopoulos, initial A. grateful stages to Pomarol, N. of R.relaxion. Craig this Rattazzi, and work. P. P. Schwaller, Draper F. for Zwirner. conversations on M.M. SUSY implementations of the Acknowledgments JHEP12(2015)162 ) + u H (B.1) (A.8) (A.9) q A.14 ( (A.14) (A.10) (A.11) (A.12) (A.13) i , 2 a 2 and of order m ). positive. The , a , B ) 0 c i factor from the 2 3.4 q ) depends on the µ a i a + δ qS T 2.5 . Note that the soft + − q i e 1 X a , and , µ ma δ B 2 ( 0 + 2 c / c | 1 a  , ) c F d = | d 2 2 H c 00 H µ µ → m U Z a ˜ S u  q H f − change by an overall phase exp[ Z F, e µ ( , c  µ i 0 B under the transformation in eq. ( ) ma , B B , − π µ Z a k 9 i 2 µ ) α Z † c Φ , S F µ and S + − +
 i 0 ∗ q S 0 in the Lagrangian in eq. ( µ ) is a gauge-mediation effect from the Higgs ) ( F e i µ d F + (ln ) one can eliminate the q q 0 i 0 0 H – 32 – e ) ) → = U i Z j A.12 i Z → Z + A.13 Φ i ), both 0 real and we can choose µ (ln , µ i i + (ln a 2 , c 0 is the heavy Higgs mass defined in eq. ( A.14 T + (ln a 2 ) -dependent superfield redefinition 2 ,Z | 0 2 / u ). ) S H d i ) is induced by the quantum anomaly of the PQ symmetry. 1 ∗ µ i F H m ) | H X m Z d 00 Z d q ˜ S µB ) c H q − i λ ), − A.14 − Z (ln a 1). Z = m e ) are manifestly invariant  u c u h d − H H + (ln (ln a q A.12 2 H ) with the replacements π ijk Z x q α ( 4 (  − Y A.12 − / in eq. ( 2.5 ) q ) we can immediately read off the soft terms = = = = = a x )–( , m c 2 i a µ µ . It cannot be neglected here because it is parametrically comparable with 2 → ˜ g ln ijk ˜ A.5 B a m A.8 q x 2 M A u,d m H u are the corresponding PQ charges. After this transformation, the Lagrangian is c are model-dependent coefficients that we take to be independent of ) = ( i i x 2]. However, this phase is irrelevant since physical quantities can depend only on the basis- q ( = c √ f u With the transformation in eq. ( Note that the second term in eq. ( It is useful to remark that the value of From eq. ( a/ Under the transformation in eq. ( 2 H ) 9 d m H where unity. For simplicity, we take all q independent combination arg( In this appendix we deriveby the the conditions under dynamical which evolution the ofthe electroweak the symmetry soft-breaking is relaxion, parameters broken as as it rolls down its potential. We parametrize superpotential, thus exhibiting theterms basis in dependence eqs. of ( theand value therefore of are independent of the field basis, asB physical quantities should. Conditions for electroweak breaking The variation of The Yukawa interactions remain invariant because we are assuming that they respect PQ. where obtained from eq. ( superfields the first term. In eq. ( field basis. Let us consider the with JHEP12(2015)162 ) . -flat B.4 (B.5) (B.6) (B.2) (B.3) (B.4)  ), and D 2 B )–( c B.4 − B.3 . d . . c )–( 1 ) u − a c can be written # B.3 + 2 ∗ ) to change sign . /  a 1 a 1 µ (  c ), which is expected − d D # c − 2 / 0 B.1 − is given by 1 , we can easily compute µ ma ∗ u ∗  = 0, in which electroweak c c c  d 0 c B − c c -flat direction for any value of -flat direction for any value of − 2 0 0 in eq. ( implies c c u D D i , and c 2 c d c − /m ) + − u 0 2 d c c 2 B µ  c µ √ + c  (no EW breaking at large u ) decreases. However, an overall rescaling − + 4 c a + 2 a ( 2 0 ( d 2 0 ) c µ c ma d D 2 2 c )  + µ – 33 –
− − ), we find u c 2 0 ) = 0 is achieved. The value of 0 4 0 u c ∗ c c 0 (approach towards EW breaking) c B.1 ( > c a − ( ). This implies ) is automatically satisfied. − + d > 2 0 | , together with the conditions in eqs. ( c q c 2 d µ D B ) B c  /m + c c d B u 0 ( | u 0 c − c 2 µ c > − µ > c > + c
for squarks and leptons are taken such that the vacuum does +  u 2 2. These two examples illustrate how it is always possible to 2 µ
2 4 c 2 B = 0. Electroweak breaking is achieved when eqs. ( i a c 2 0 ( / c c  µ √ c 2 0 , and thus the condition for approaching electroweak breaking is µ B is a function of the coefficients " ). Using eq. ( c + q c c a − 2 ∗ d + 2  + − c c d 3.1 0 for √ d c µ − 0 0, the condition for stability of the Higgs potential along the 2 H c
. µ > = | µ ma + | 2 µ remains positive as the relaxion rolls down its potential, eventually at a 0. Imposing this condition at 0 m c > ) ∗ c B  u c 2 c a | c c )  + | ( 2 + √ a 4 2 u ( + /da the critical condition D " u a ), where ) > 2 H c 4 2 µ > 2 4 D ∗ /da > c d m ) a d m 3.2 √ 2 c 4 /a c ) = = a + µ /a )= ) does not trigger electroweak breaking. Instead, we want + ( c ) a ∗ a a u u a ( ( c c D c The second case is Although we cannot give a general analytic expression of As the relaxion rolls down its potential, The order parameter of electroweak breaking is the determinant of the Higgs mass ( ( D D is given by d is D is ( ∗ The condition for thea stability of the potentialfind along a the range ofcondition parameters for in electroweak breaking. which the relaxion evolution is driven towards the critical a are supplemented by the condition c The condition for the stability of the potential along the as in eq. ( to be of order unity. it in two simple,breaking but is representative, achieved cases. when The first case is d If value As long as direction ( of during the evolution of We require that, duringis the preserved initial ( stage of the relaxion evolution, electroweak symmetry not spontaneously break colour or electric charge. matrix, given by eq. ( corresponding coefficients JHEP12(2015)162 . , ]. ∗∗ (B.7) (B.8) (B.9) , , . (2004) SPIRE 1 1 a < a IN − − [ # # Cosmological 2 2 / / D 70 1 1 . (2015) 384   1 d d ), after exploring − c c a ]. # ( 2 − − C 75 / D u u 1 , and eventually turns with = 0. After becoming (2015) 077 c c  − (2006) 025018 Phys. Rev. d SPIRE a ∗∗ − − B c , 11 c a IN 2 0 2 0 = c c − ][ = arXiv:1506.09217 a D 74 u , c a ) + ) + JHEP d d − c c , Eur. Phys. J. 2 B , + + c , we find that u u ∗ c c , then at ( ( a + 4 Phys. Rev. 2 0 2 0 2 ∗∗ , c c Cosmological relaxation of the electroweak ) a 2 2 d c = − − arXiv:1504.07551 below − 4 0 4 0 a c c [ u – 34 – a c + + ( 2 2 ) ) q d d ].  c c , the potential barriers, besides being modulated by the + − − a µ ]. ]. u u ), which permits any use, distribution and reproduction in c SPIRE c c ]. ]. Relaxing the electroweak scale: the role of broken dS symmetry ( ( 2 ]. (2015) 221801 IN Cosmic attractors and gauge hierarchy , once electroweak symmetry is broken, the relaxion is trapped √ , even if this range is not explored by the dynamical evolution. The half-composite two Higgs doublet model and the relaxion q q ][ ∗ " 3 SPIRE SPIRE −  115 SPIRE SPIRE 0 IN IN   , where ) flips sign first at SPIRE µ IN IN a [ [ ][ ][ + IN 2 m − + ( a < a CC-BY 4.0 ][ √ µ µ D = 0, as we decrease Quantum relaxation of the Higgs mass c c , 2 2 This article is distributed under the terms of the Creative Commons = 0 ∗ a < a c √ √ a ∗∗ " " Electroweak relaxation from finite temperature a = Large hierarchies from attractor vacua 0 0 hep-th/0304043 a µ µ [ 2 2 m m Phys. Rev. Lett. , √ √ = = arXiv:1507.07525 arXiv:1506.04840 hep-th/0410286 [ arXiv:1507.08649 arXiv:1508.01112 [ Higgs-axion interplay for a naturally small electroweak scale scale 063501 [ An even more complicated pattern is found in the case In the case As discussed in section S.P. Patil and P. Schwaller, O. Antipin and M. Redi, A. Kobakhidze, J.R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujol`asand G.E. Servant, Hardy, G. Dvali and A. Vilenkin, G. Dvali, P.W. Graham, D.E. 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