Testing Naturalness at 100

Home , Little Higgs

JHEP09(2017)129 and t λ Springer July 2, 2017 : August 14, 2017 August 24, 2017 : : September 26, 2017 f : Received -collider, naturalness Revised pp , Accepted Published at leading order, where 2 t and Hao Zhang λ − d,e at a 100 TeV = Published for SISSA by 1 https://doi.org/10.1007/JHEP09(2017)129 [email protected] T − , a Ian Low c Tao Liu, [email protected] , b,c [email protected] , . 3 Jan Hajer, a 1705.07743 The Authors. c Beyond Standard Model, Higgs Physics, Technicolor and Composite Models

Solutions to the electroweak hierarchy problem typically introduce a new sym- , [email protected] are the Higgs couplings to a pair of top quarks and top partners, respectively. Using E-mail: [email protected] Department of Physics andEvanston, Astronomy, IL Northwestern 60208, University, U.S.A. High Energy Physics Division,Argonne, Argonne IL National 60439, Laboratory, U.S.A. Institute of High EnergyBeijing Physics, 100049, Chinese China Academy of Sciences, Department of Physics, NationalTaipei Taiwan 116, Normal Taiwan University, Institute for Advanced Studies, TheClear Water Hong Bay, Kong Kowloon, University Hong ofDepartment Kong Science of and S.A.R., Physics, Technology, China TheClear Hong Water Kong Bay, University Kowloon, of Hong Science Kong and S.A.R., Technology, China T b c e d a f Open Access Article funded by SCOAP Model, we show that,could with be a tested luminosity with a of precision 30 ab of 10 %Keywords: for a top partner mass up toArXiv 2.5 TeV. ePrint: top partners infor little naturalness Higgs and theories initiatebreaking, as a naturalness an study in illustration, on thea testing we top sector naturalness. construct requires a Aftera simplified electroweak multivariate symmetry model method of Boosted Decision Tree to tag boosted particles in the Standard The new symmetry is eitherand broken softly little or Higgs collectively, theories. as forStandard At example low Model in energies fields supersymmetric such which theoriessquared are contain Higgs responsible naturalness mass. partners for Post of canceling the the aforementioned the discovery of cancellation quadratic any divergence by partner-like in particles, measuring we the relevant propose Higgs to test couplings. the Using the fermionic Abstract: metry to stabilize the quadratic ultraviolet sensitivity in the self-energy of the Higgs boson. Testing naturalness at 100 TeV Chuan-Ren Chen, JHEP09(2017)129 ] 4 ], there is a 3 , 2 ], which require the existence of naturalness 6 7 , 5 2 – 1 – ], whose discovery potential at a next-generation ]. However, the discovery of a partner-like particle 10 – 11 7 14 16 10 ], motivated by the smallness of the Standard Model (SM) 1 7 1 11 Among the naturalness partners, the partner of the SM top quark typically plays the 3.1 Simulation of3.2 signal and background Event reconstruction 3.3 Results hadron collider has also beenis studied [ only the firstdiscovered step partner-like in particle confirming isyields theories the not cancellation of a of naturalness.to naturalness the further quadratic impostor, In UV-sensitivity measure but order in the to the indeed naturalness Higgs ensure the sum self-energy, that one rule. one the needs that In this article, we propose a novel way to cancellation of the quadratic divergencetheories in the of squared naturalness Higgs is mass.energy one Confirming collider of or experiments. the refuting most important goals for themost current and prominent future role. high going into Currently, searches a for top tremendous partners amount [ of effort at the LHC has been Typically a new symmetryfrom which quadratic UV-sensitivity protects is the introduced.and squared Two spontaneously eminent mass broken examples global are of symmetry supersymmetry [ the [ partners SM-like coupling Higgs to the boson couplings Higgs with boson. the SM The particles symmetry-protected and their relation partners, among called the the Higgs naturalness sum rule, ensures developments in particlethe physics 125 for GeV several Higgsrenewed boson decades. urgency at to understand the Especially why Largein its after mass particle Hadron is the physics, Collider so the discovery (LHC) much Planckenergy smaller of in scale, of than the given 2012 the Higgs. the next [ known quadratic In scale the ultraviolet past sensitivity decades, in many the theories self- of naturalness have been proposed. 1 Introduction The naturalness problemparticle [ masses in comparison to a UV cutoff, has been the driving force behind theoretical 4 Discussion and conclusion A Higher order corrections 3 Collider analysis Contents 1 Introduction 2 A simplified model for naturalness JHEP09(2017)129 ]. + 18 (1.1) (1.2) , SM

]. We 17 Similar 2 H ] or the 1 m 16 , ]. 19 , 15 14 – 14 [ 12 c 3 u . . Finally we conclude . Then we show that 3 . This suggests that, in = 1 w T a , naturalness | , only two measurements are neces- , into a naturalness parameter 2 T 2 2 T 2 H v m m /m 2 v O + O 2 | . We then proceed to study how this relation can – 2 – t T λ a which enables the cancellation of the top quadratic and a pair of top-partners − | t H and λ , µ = † t T λ H a NP SM

2 H 2 H m m ∆ ∆ − = µ we introduce a simpliﬁed model for naturalness, by extending the SM top 2 . = 0, which relates the diﬀerent contributions to the quadratically divergent 4 is the top partner mass and the relation involves at leading order only the couplings T NP

m In section Note that in the SM the top quarks contribute the strongest to the UV-sensitivity of the 2 H The case of scalar top partners as predicted by e.g. supersymmetry will be pursued in a future work. 1 m ∆ if their masses areallowing a not quartic too coupling heavy. with divergence in These the top Higgs partners self-energy.ners are We carry focus vector-like the on and same the massive, cases EW thus where charges the as vector-like top either part- the right-chiral top quark which, in order of decreasingbosons magnitude, are and the the coupling to Higgsquarks, top boson quarks, EW electroweak itself. gauge gauge bosonsspective In and partner a the particles. typical Higgs naturalparticles, boson In theory, which are this the carry expected work QCD contributions we to colors of consider be and top a canceled could top by be sector their produced with re- copiously fermionic in partner a hadron collider, naturalness of fermionic top partners atin a section 100 TeV collider in section 2 A simplified modelIn for the naturalness SM, the UV-sensitivity of the Higgs self-energy originates from three major sources be measured to quantify thethe quadratic degree divergence of of cancellation the Higgs in self the energy in contribution a of future the 100 TeVsector top hadron collider sector with [ to a pairof of naturalness. vector-like top Based partners, on and this derive discussion relation we from design the a requirement detector study dedicated to test here of the Higgs to aorder pair to of test top the quarks naturalnesssary, sum one rule for each up of to the couplings top-partners, which arethe either naturalness singlets sum or rulebreaking, doublets leads at under leading to order SU(2) an in especially the simple Higgs vacuum form expectation after value electroweak (VEV) symmetry Higgs self-energy. As atop first partners illustration which carry for QCD testingcolored colors, naturalness, as top we predicted partners will by little exist analyzedefine Higgs a fermionic in models simplified the [ model so-called consisting of composite the Higgs SM models top fields as and well a pair [ of vector-like fermionic theories of naturalness. For convenience, we rephrase the naturalness sumquantum rule corrections ∆ to the squared Higgs mass ∆ probe the naturalness sum rule at hadron colliders, in order to facilitate the testability of JHEP09(2017)129 (2.7) (2.4) (2.6) (2.1) (2.2) ) is a (2.5c) (2.5a) (2.3a) (2.5b) (2.3b) 0 and we T ,U . c f . c 0 T U 2 | ) with the re- + h.c. H + h.c. | doublets can be becomes massive 2.4 0 Q 0 c . 3 T w U T ) , q , , 3 m α ... 1 2 c b , 2 c b c b 0 ... . 3 0 ... + 0 c b q U, 2 0 c b + c b ... + 3 . c b 0 + + 0 q c † + = = + t + h.c. † + 3 T 3 + 0 0 1 c q 2 c H † 0 H 2 0 c † c t 0 4 ↔ 2 c 0 0 c 2 | H 2 c | H c c | H 0 2 q 2 H | H | | H together with = = = ( | | = H O 2 0 0 0 c H | 2 0 3 | 3 f T T T c 0 b T f U 2 + c b t 6 2 3 6 0 3 f c 3 ) consists of SU(2) α m c f t c 6 2 u c 6 0 + c 3 and ,Q T u + + + U c † c 3 0 are free and for odd-dimensional couplings 2 2 u t | + Q | U H i c 0 2 c b 2 2 H | H | | | T | – 3 – † H 2 H f H | 2 f | | c b 2 H c b 2 and ,,T t 0 0 2 f c 2 f 0 c i T 2 T 2 c c 3 c 2 ]. In the case where the vector-like pair ( Q + c T c λ u U m β 3 0 20 0 Q + + 6 c , β q c + c † † 3 ) , λ 0 , t )[ , α + t q c c 3 + + 0 3 − 1 c H † 2 H c b † 0 0 c c b 1 T c b 1 0 t t c 3 0 c b c 0 H † c b c H , u u U 0 fc , c c c 1 c 3 1 t 3 0 0 ↔ c + H † c u − c c b b d c 0 − = c 3 − c t 3 0 + H 0 u 1 + λ 2 c = = c T 2 T c fU 0 = ( | 0 f c c 0 0 + c b + 0 0 can be expanded to c 0 b 3 m c b c b the effective Lagrangian for the non-linear coupling of the top sector t fU 0 H c b q f T 0 | c 0 ) becomes T w c H = = = ( . The parameter 0 c c c 0 Q 0 0 0 c 0 2 T t t t t H U 2.1 T c 3 † Q β λ 0 β α m u + 6 T + H + m = = = U = 2 Q | L 0 L T H | L The case in which the vector-like pair ( The rotation into massplacement eigenstates of leads to a Lagrangian identical to ( written as where the couplings are given by the Lagrangian ( Whereas, the orthogonal combination remainsthe massless Higgs and doublet. acquires a In Yukawa coupling terms to of these mass eigenstates The mass dimension in theseimply couplings is given bythey the can symmetry be complex. breaking scale Awith linear a combination top of partner mass of singlet under SU(2) to the Higgs scalar left-chiral top quark JHEP09(2017)129 ), 2.9 (2.8) (2.9) (2.11) (2.10) (2.12) (2.13) (2.14) vanishes 2 M . , 3 ) H only contributes to H ) and subject to the ( 0 , t O term in tr 2 2.4 α M c b ) the mass matrix reads 0 2 † 0 + | ) c b as a background field, and 0 ) it can also be applied to 2 H H − ,T T | | 0 ( H t m H 2.1 = , | 2 . M . 2 0 | ) this naturalness sum rule reads 2 ], although their phenomenology differs ! 1 = c 0 b 3 ≡ . 0 c b t 0 | 24 2 T 2.6 c b ) and ( 2 H α | α c =0 0 0 t M − 0 0 H term to vanish leads to the naturalness λ O

2 |

,T 2 1 | c 2 0 − | | c b t ) and ( + T H 2 quad | H | | 0 V − | H 2.1 2 T ∂ ∂ = ! λ – 4 – 0 0 ) to the Higgs potential can be computed using 2 term contributes to the quadratic divergences with 1 2 c 0 − | 0 0 0 t T 2.4 c T = λ λ = = constant + λ ] and the quadratically divergent part is 0 , +

2 T 2 2 , 21 | α 2 quad + M 2 1 M c c m | ! 0 tr which must have a negative sign. In terms of the parameter 0 c tr ∆ T 0 2 − T m 2 ) are little Higgs models. Some typical examples are presented π α = Λ 0 0 0 16 2 |

2.12 1 = c | ) = T H quad ( ) and our results can easily be adjusted to correspond to that case. V M 2.6 ) is the fermion mass matrix with the Higgs field The collective symmetry breaking in the Yukawa sector of little Higgs models we have also considered mirror twin Higgs models [ 2 H 1 . ( 1 M terms and does not enter into the cancellation of quadratic divergences in the Higgs The best known examples of models described by Lagrangian ( The contribution of Lagrangian ( In table 4 | 2 H yields the relations significantly from the one inthis little section. Higgs theories under consideration. See also the discussion at the end of naturalness sum rule ( in table It is worthwhile pointing out that the the same sign asof the the SM four-point top coupling quark,in and the the Lagrangians cancellation of the hinges gauge entirely eigenstates on ( the existence no linear term and| hence no tadpole contributionmass. arises. Finally, requiring Second, the coefficientssum of rule the is then equivalent to the requirement that the coefficient of the Several comments follow from this equation. First, due to the structure of the matrix ( The absence of quadratically divergent contributions in the Higgs mass where Λ is the cutoff scaleup of to the quadratic theory. order In in the the basis Higgs ( field Lagrangian ( the Coleman-Weinberg potential [ Although the following discussion is based on Lagrangian ( JHEP09(2017)129 t for λ 2 λ λ = 0. i c b 2 − c − 0 (2.16) (2.15) T (2.17a) (2.17b) ∓ λ ]. When = iλ = λ λ 2 -odd, while 1 i 0 0 0 0 0 0 0 0 26 c b c b t , T − √ ). A common α − 25 , = , , 0 2.13 t λ λ 2 2 2 0 T c b λ y 0 0 λ λ − − β ). 2 2 2 T 2 T v v m m 2.6 -parity [ 1 1 1 λ 174 GeV T y λ λ λ λ λ 2 λ 2 0 1 2 2 c − − ≈ O O − − − − ) or ( + + 0 0 2.1 t 1 T 0 1 iλ 0 . 1 iλ T t y v λ λ 2 T 1 λ 2 0 0 3 v 2 c m i c c b iλ − − √ m − 0 √ √ 0 ) for some representative models. In the T − − into mass eigenstates = ∗ T λ λ 0 2.1 0 v t T + 1 1 1 0 − λ 0 λ 0 λ λ λ c y λ λ 0 ] the top partner particles are t T , v = 19 = = 0 – 5 – 0 T = 0, which leads to t T α 1 1 1 2 1 1 1 α ··· , h = + = 0 h 0 . This ensures at one-loop level the cancellation of the t ] ] ] ] ] ] ] 2 T c b β β 1 2 22 23 14 20 19 19 24 [ [ [ [ [ [ [ 1 = ,T √ , t 1 c b 2 + 0 0 2 2 T T 2 2 T ) v v m SU(3) SU(2) SU(3) SU(2) SU(5) SU(5) SO(5) SO(5) m , v SU(3) SU(2) SO(9) SU(4) U(1) SU(3) U(1) O SO(5) SO(4) O -even. Therefore the mixing terms have to vanish = (0 + T + c 0 c H 0 , respectively. t T i = = , similar to terms that have been neglected during the derivation of the naturalness sum rule. 2 T c c Parameter of the couplings between top quarks, top partners and the Higgs scalar in t m T / 2 and hence if this model corresponds to the Lagrangian ( v w Finally, expanding the Higgs field around its VEV In theories where the Higgs arises as a pseudo-Nambu-Goldstone boson from a spontaneously broken -parity invariant -parity invariant 3 -parity is implemented in the top sector [ Model Coset SU(2) Toy model Simplest Littlest Higgs Custodial T T Mirror twin Higgs symmetry, the VEV of theorder Higgs of differs slightly from the SM VEV of 174 GeV. The difference is formally of and rotating the fermion fields to linear order in quadratic divergences from thebut top by sector no means according necessaryto to additional the the restriction Higgs condition is field ( to by couple setting only one of the top partners even and odd the gauge eigenstate basiscolumn as labeled defined SU(2) we in indicateSU(2) Lagrangian if ( the top partners are singlets or embedded into doublets under as can easily be checked in table For phenomenological reasons itT can be desirableSM to particles introduce are Before rotation into the mass eigenstates this corresponds to the condition Table 1. JHEP09(2017)129 ) ). 2.1 . with 0 (2.21) (2.18) (2.20) 2.20 (2.19a) T (2.19b) α . The new + h.c. A T 2 T 2 c v m , hT . (1) TeV. Addition- ) greatly simplifies 3 , 0 ) by measuring the T h T 2 v | m 0 λ T 0 2.20 2 T O 2.12 a T O λ identical to their primed √ | T α + T m + + t − c m 0 0 . In contrast, the naturalness / T T to the remarkable relation T T v c f 2 α β v h ht . = = 2 T T v v T T T m m t 2 T b 2 4 2 a v . m 0 √ + T t f + c m O t ) suggests that three measurements, one for t 0 , and extracting the information of 2 c T + h th λ 2 2 T 2.12 hT | v – 6 – term for example can be eliminated by imposing t t 2 = m → 0 T b lead to terms of higher order in Lagrangian ( λ 0 4 T λ √ T T λ − | . In conclusion the relation ( + α πf , are necessary in order to test naturalness. However, , and involves only two free parameters, the coupling + 0 f 4 T t = T T ) has significant implications on tests of the naturalness c , a , b c m are up to linear order in α ∼ 0 0 T / t T ht v a T 2 ] to test the naturalness relation ( λ 2 ) leads at leading order in t 2.20 0 λ t,T h 0 λ ∗ T t 22 √ and α T ]. Second, it implies that in order to extracting the value of the λ α 0 T 2.12 + m T α 19 + − 4 t λ 0 and c 0 t , t 0 + t α β vt t are not defined in the basis of mass eigenstates after electroweak λ t,T T λ = = c λ 0 t t t , T 2 + b a T h α T T c m t ) has the virtue of being generic for all models which can be described by the T α m it is necessary to measure the decay constant 4 T and 0 T 0 + m 2.20 α T ). Our calculations show that these terms do not influence the relation in ( λ = , 2.6 T 0 t The naturalness sum rule ( coupling via the top partner decay L λ 0 -parity. It is exact up to order -parity in the top sector [ T simplified model under consideration.T Especially it isof the independent Higgs of to the twomeasurement top assumption of quarks on the and to decay twotesting constant top the partners. naturalness Notably, sum this rule. does not involve the However, the proposed strategygeneric is for all limited little by Higgs two models.T observations. The First thiscoupling relation is not sum rule ( symmetry breaking, the separate reconstruction ofPreviously their it values becomes was rather suggested challenging. λ [ the relation and ( sum rule at colliders. Naively,each the relation of ( the couplings as Though the naturalness cancellationhigh-loop in corrections Little to the Higgs sum modelsgibly rule small applies are compared at suppressed to by one-loop the extraally, uncertainty only, loop in in the factors. this the relation They simplified unless arenew model negli- physics at under the consideration cutoff higher scale Λ dimensional operators encoding The naturalness sum rule ( The couplings counterparts, and we haveVEV collected induced the couplings higher are order correction in appendix leads to the Lagrangian JHEP09(2017)129 (3.3) (3.1) (3.2) ] has (2.22) 11 – 7 ], and at a 35 the branching . However, the – equal to one. In v 1 33 . 2 T m / -parity. The analysis 2 f v T 25 % ) are derived for vector- ' collider [ . ) = 0. Note the similarity to − 2.20 0 term in e W b T t + λ e a ) without + h.c. → . = h T ) and ( 0 t 2.18 2 T α 2 T . c v BR( 2.13 2 m t = c b 1 2 v from the T 0 is up to an order of 0 t c b ), ( T a t T m ' O − β µ m ) + / + ], at a future v = 2.12 t – 7 – tZ 2 t c T 2 of the newfound particle has been determined. 32 | λ and we leave the removal of this degeneracy for a a T – 1 → T c T -parity. However, in this case the naturalness sum − T | 27 λ a T T m = ] t 2 µ 1 BR( ) is satisfied 37 √ production at a 100 TeV hadron collider. The naturalness ' ⊃ ) 2.12 ]. We will focus on the measurement of the Higgs coupling to T h T th c L 36 T → T via T a BR( 0, which yields the relations . The equivalence theorem ensures that in the limit ) applied to this analysis reads ≡ 1.1 2 ]. In this case, top quarks carry QCD charge while top partners carry mirror c ) is reduced to , Z, h 24 = 1 2.13 c b W Although, the naturalness sum rules ( = = 0 This relation receives a correction of order 3.1 Simulation ofIn signal little and Higgs background models,B fermionic top partners decay mainlyratios into (BR) heavy relate quarks according and to bosons [ If the naturalness sum rulethe ( following we assumecolored the fermionic particle top content partner and subjectproposed interactions to here of Lagrangian is the ( blind SMfuture to augmented publication. the by sign a of two top partners parameter ( Based on thepair discussions of above, top the quarksthe measurements and naturalness a of sum pair rule. theliterature, of The couplings top for former of partners measurement searches a constitute has at100 TeV been the Higgs the hadron extensively two with collider LHC discussed cornerstones [ a in [ for the testing such as little Higgs models. 3 Collider analysis In this section webeen assume successful that and one of that the the searches mass for fermionic top partners [ rule ( Although, the collider kinematics ofcolored colorless top top partners, partners we differs listcollider significantly twin from analysis Higgs that in models of the as following one section possibility applies in only table to models with colored top partners like top partners inmodels little [ Higgs models,QCD charge, they which can forbids alsoc mixing be between them. appliedthe to This case is mirror of ensured twin little by Higgs the Higgs condition models that with JHEP09(2017)129 ] , 39 3000 (3.4c) (3.4a) (3.4b) 4. We . 2.4 [ and the 0 T MadGraph >

m t 0.1 2500 µ | . The horizontal | t MadGraph µ | ) to Dirac fermions in fb as a function of is not sensitive to the

T b 1 ,T ]. m c 2000 process with T 41 T T h . ) , bb ¯ T T h ) (b) 100 TeV. , 3.3 [ ( ) bb ¯

( 10 bb ¯ 1500 ( and 100 TeV lνbj ¯ b ¯ a ¯ tjbj ¯ ttjj → Delphes → h . →

∓

1 µ | | 1000 h

0.6 0.4 1.6 1.4 1.2 0.8 0 t µ at 14 TeV bW -jets. The decay channels in this notation are tB bW b ) together with the coupling which ensure the 0 0 – 8 – T T h ¯ ¯ tB tB bW ¯ . Light particle decay and showering is done in 2000 2.18 instead of

    

| 0.0001 t . In particular, we neglect the Higgs coupling to two µ | → 1 also include MadSpin

j 0.001 ]. This enables us to generate events with T T h 1500 38 → in ﬁgure

T 2.3 [ pp 0.01 and the absolute value of the naturalness parameter µ m ). T is small, the tree-level LO production rate becomes small and the one- m

3.2 0.1 | t (a) 14 TeV. µ 1000 | ] followed by the detector simulation

FeynRules 1 40 and assume that jets Cross section of the signal process = 1 indicates natural models. Note, that the production of |

, hence all ﬁgures depend on ) in t

6.4 [ for the remaining part of this section. We abbreviate heavy neutral bosons by t 10 µ µ | Z, h 3.2

1 We have calculated the LO production cross section of the We have implemented Lagrangian ( In order to be more consistent with the majority of the literature on collider phe- 500

| =

0.8 0.6 0.4 1.6 1.4 1.2 t µ 0 T,T gluons, which is induced at one-loopHowever, level when and formally a next-to-leading orderloop contribution. contribution arising fromFor the this Higgs reason coupling we to choosehave two to checked that gluons focus in could on this become the case region dominant. the of tree-level production parameter rate space still where dominates. For simplicity BRs ( and decay heavy particlesPythia with and present it fornaturalness 14 TeV parameter and 100 TeV as a function of the top partner mass Here we assume that the associatedjets, Higgs is while unboosted the and other decays bosons intosingle two are separate jet. boosted bottom Furthermore, and we decay require hadronically the within top the quarks to cone decay radius either of fully a or semi-leptonically. B For simplicity we ignore thisvalues in uncertainty in relation ( the current analysis by fixing thenomenology we BRs will switch to the the notation from using Wely fermions ( Figure 1. the top partner mass line at sign of JHEP09(2017)129 (3.5) ) which 3.4a ). isol l ( n 40 GeV. The other two > T p -factor in this analysis. k and have required the jet trans- . ) j B ( m R at 100 TeV in fb. During event generation 2 ∆ takes the value known from the SM, up -like we restrict these jets to come from t ≈ b MadGraph λ ¯ ttjjjj 3150 526 400 GeV. The resulting cross sections are given in – 9 – Cross section [fb] > 400 GeV T p > semi-leptonic full-leptonic ) 5 j . ( 0 ) 1315 111 T p ≈ 400 and 40 GeV for the two leading and the remaining jets, isol l . ( R T > 2 T p n m T / p 2 v . Before the analysis we require the presence of at least one and two isolated . As a cut on the jet number is only a weak discriminator the top-pair T 2 p . We give the cross section after this pre-cut in the row labeled Cross section of the background process 3.1 quarks. Therefore, the main background in this analysis is top pair production in b In order to reduce the hadronic background we focus on the decay channel ( or -like. We have generated this background with quarks in table production with more than fourcoupling jets will constant contribute appears to the inwhich background. leads an Because to the higher a strong powerrespect larger to and phase the space due leading this to contribution. higher the We order do increased background not will number consider be of any suppressed jets with As we require thec Higgs boson signatureassociation to with be four jetsb where the two leading onceverse momentum are to boosted be whilerespectively. the other We show two are the LO cross sections for semi- and full leptonic decays of the top We expect these jetstionally, to a mimic combination the of bosons two originating softer from jets the can top be partner misidentified decay. as Addi- being the Higgs boson. boosted and tend to bewith detected a as low a transverse singlethis momentum jet. study and On we its the consider other decayof only hand, products the Higgs the will top decays Higgs into pair be isto bottom plus produced detected lie quarks. jet separately. within background In a In we order jet require to cone the reduce of two ∆ the leading size jets to be boosted enough is mimicked by the combination oftop one pair top quark production with in one association electroweakwe boson. with consider multiple the However, the quarks mis-identification has ofjets a jets mimic much we reconstructed larger have objects cross to from section. distinguishboosted the the If object. cases cases in On in which the which a one pairs singe hand, jet of is the mistaken decay for products a of single heavy top-partner are generically we have also assumed theto top corrections Yukawa coupling of order involves two top quarks. Thepair dominant electroweak of background top for quarks this produced signal in consists association of with a three bosons. In this case each top partner we require at leastquarks two are quarks required to to bethe be bottom row boosted labeled or charm with leptons quarks for with semi- andof full section hadronic events, respectively. The isolation criterion is given at the end Table 2. JHEP09(2017)129 b as c and ± ± a h l h l and W W t t t t g c h b Z q g c h b Z q a 700 GeV. 1000 GeV. < < T T < p < p 5, we expect the bosons of the Signal acceptance Signal acceptance . = 0 algorithm using a minimal trans- R T k tagger for 400 GeV 400 and 700 GeV, respectively. tagger for 700 GeV Z & 1 3 1 3 Z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 1 1 − − − − − −

T 10 10 10 10 10 10

p (b) Background acceptance Background Background acceptance Background (d) 400 GeV. We require at least one and two > – 10 – T p ± ± l l h h W W t t t t Z g c q Z g c q h b h b . All processes are generated at 100 TeV via pair-production. 700 GeV. 1000 GeV. d < < 3 and a maximal transverse momentum ratio of 0.2. The T . As the jet cone size is ∆ . T and 0 d b < p ]. Finally, we disregard events with less than ﬁve jets. < p 42 and R > c boson and top partners. Such an object can consist of a sub-jet, a single Signal acceptance Signal acceptance Z -boson tagger in Z ROC curves of the signal and background acceptance for the Higgs tagger in 1 3 1 3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 1 1 − − − − − − −

During the analysis we cluster jets with the anti-

10 10 10 10 10 10 10

Background acceptance Background Background acceptance Background (a) Higgs tagger for 400 GeV (c) Higgs tagger for 700 GeV 3.2 Event reconstruction We reconstruct the signalof signatures them with is multiple optimized boosted toquark, Higgs decision tag or or trees reconstruct (BDTs). a Each single physical object, such as bottom or top leptons for the semi- andan full-leptonic analysis, isolation respectively. radius Leptons of musttransverse be ∆ momentum isolated with ratio isthe deﬁned isolation between cone. the Ifrequire lepton the it and lepton to transverse other be momentum isolated cell is [ activity larger within than 100 GeV, we do not SM and the top quarks to lie within a single jet for verse momentum of 40leading GeV and jets a to jet be cone boosted size more of than 0.5. Additionally we require the two Figure 2. well as for the (We are using antagging eﬀective we model have in ensured order thatand the to to transverse produce 700–1000 momenta GeV Higgs are for boson constrained pairs.) to 400–700 GeV During for production and JHEP09(2017)129 ] a 43

3000

T T h

(3.6)

− 1

] 0ab 30 in ﬁgure 50 ) 0.3 [ ] together 1 − 47 2500 .

BoCA

−

1 . ab 3 b T m 2000

in ﬁgure

−

1 5 while the associated Higgs 1

ab 3

. 6.06 framework [ − 0 0 1500 , (b) Signiﬁcance of 5 ) ) 30 ab 5. Finally, we reconstruct the R < . ROOT n n , | | 0 3 x n , ( ( 3

L L

| | R > 1000

0.6 0.4 1.6 1.4 1.2 0.8 t µ 2 ln ] of the − ) for pair production of top partners in association 46 s – 11 – s 3000 + . We show the results for two diﬀerent ranges of b | backgrounds we require the neutral bosons from the 2 ) = b ] between its components. Furthermore, we train a ( n j

Z library [

2 49 x 1 ( + 5 . 1 Z 2500 ] where we also discuss the eﬃciency of the bottom and ¯ tt − with luminosities of 0 TMVA σ 45 , 44 T and the m 2000 j ] for jet clustering. Here we introduce additionally taggers dedicated uses the

48 10

+ 3 5 boson tagging. For boosted bosons we demonstrate their background ¯ tt Z 3.2 [ BoCA 1500 Discovery reach defined as (a) Luminosity of 3 ab observed events the significance can be defined as the log-likelihood ratio [

FastJet

1 n

| | 1000

0.8 0.6 0.4 1.6 1.4 1.2 t µ 3.3 Results For boson has to be un-boostedsignature with by a combining cone two sizeseparated top of from ∆ partner the tagger background with with a a Higgs single boson cut on tagger. the The final BDT signal result. is are the boson massneutral and boson tagger the to pullpartners detect [ from both tops of and theseboostness particles, neutral of which bosons. the allows decay us During productsto to this suppress by reconstruct the reconstruction using top the we availabletop cater substructure partner towards information. decays the to In be order boosted with a cone size of ∆ mis-identification as a functionCharacteristic of (ROC) curves a in giventransverse figure momentum, signal which rate, correspond to theenough the to so cases end called in up which Receiver invertices the one Operating of top and quarks the two are jets, bottom respectively. boosted decay The products. Higgs tagger The utilizes strongest the discriminators displaced within these taggers jet or behas combined been from introduced multipletop jets. in [ taggers. This Boostedwith Collider Analysis ( to Higgs and Figure 3. with one Higgs boson atand 100 TeV. for We a present the fixed reaches significance for of a 5 fixed luminosity of 3 ab JHEP09(2017)129 . 1 ). − 5 3.2 (3.8) (3.7) 5. As = 10 % 30 ab ≥ b , ) / s 3 n | = 1, against , b 3 ( | 6 . t Z µ | with the prediction ] experiments have 2 and for the discov- n 10 ≥ 5 for the discovery of a ) . b n 1 can be excluded up to we require | / . . ≥ 3000 s s b 0 2 5 TeV, with 0 ) . + s 2 t > b

+ ( 0% 10 | 2 δλ b Z 1 | 3 t T , ] and CMS [ b λ 4 a 9 ( 2500 . –

2 . | −

1 Z 0% 30 7 t x µ ∼ − ! || + e

n 0 % 100 2 n T x we require 780 GeV, given BRs compatible with ( m 2000 T s < δa + ) = – 12 – b 2 t n 1 T | which indicates that the contour with λ x m ( 4 − L 1500 s Ratio of signal over background 2 for the exclusion of an unnatural model. = | ≥ t

µ )

1 | | 1000

1.2 0.8 0.6 0.4 1.6 1.4 nat t µ 5 TeV for natural theories. | production channel at 100 TeV. This may not be the most sensitive . s Figure 4. + b T T h | s = 1. The exclusion limits of unnatural theories are presented in figure + | of data, unnatural theories with , we present the discovery reach against the background-only hypothesis of t b µ ( 3 1 | − Z is the number of events predicted by the hypothesis which is tested. For counting 2 TeV. . x 2 Second, we calculate the precision of measuring the naturalness parameter, defined as In the following we introduce two measures for the sensitivity of testing the naturalness In figure ∼ T With 30 ab m the uncertainty at one sigma confidence level could extend to above 2 sum rule. First, we calculatethe the exclusion assumption limit that of the unnaturaltheory theories observation with with at the collider is consistent with the prediction of a coupled to a Higgsof data, boson respectively. canexcluded be In fermionic pushed comparison, top up the partnerThe to ATLAS systematic [ with errors aresignal not over included background in ratio our in analysis. figure Instead, we present the expected model and top partners in the channel for the search of fermionic topfor partners, testing but naturalness. it shows For the natural effectiveness theories of we our see analysis that the discovery reach for top partners For the exclusion of the signalery hypothesis of a signal bywe exclusion are of projecting the to background future onlyfor experiments hypothesis an we replace alternative the hypothesis. event number Hence, we are using where experiments the likelihood function is given by the Poisson probability JHEP09(2017)129 ) 1 −

3.8

and 3000 3000 −

1 (3.9) 1

−

a . Here 0ab 30 4 TeV, 2 TeV, .

1 b

1 0ab 30 at 3 ab − ∼ | t ∼ T

30 ab µ | m in ﬁgure 2500 2500 .

σ 1

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3 ab (b) Signiﬁcance of 2

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δa

1 shows that in natural theories the naturalness param- 0 % 100 can be estimated with the relation shows that with 0 . . 6a 1 1 1 2500 2500

production at 100 TeV. For a luminosity of 3 ab − −

0% 30 2 a 6b

2 T T h % 10 T T m m 2000 2000 = 0. Figure

t % 3 δλ and a ﬁxed signiﬁcance of 2

1 5 − of data. Figure 1500 5 1500 1 Precision of measuring the absolute value of the naturalness parameter we present the precision under the assumption of a perfectly measured top Exclusion limit for unnatural theories deﬁned by (a) Luminosity of 3 ab (a) Luminosity of 3 ab − 30 ab , and the reach of measuring with a precision of 10 % at 0

6 10 10 , a 3 ,

1 1 .

| | | | 1000 1000

0.8 0.6 0.4 1.6 1.4 1.2 0.8 0.6 0.4 1.6 1.4 1.2 2 TeV and above 2.5 TeV, respectively. Note that the definition of the precision ( t t µ µ Yukawa coupling eter could be measured withusing a 3 ab precision of 1010 % % up could to be a achieved∼ top for partner top mass partners of in natural theories with masses up to The precision of the coupling In figure Figure 6. in figure we assume that the top Yukawa coupling can be perfectly measured. Figure 5. top partner mass. Basedfor on 0 JHEP09(2017)129 3000

]. In

0% 50 52

0% 20 , 0% 10 at 100 TeV = 1 %. 51 | t is included. t t

2500 δλ | % 5 λ at 100 TeV and and 1 1 − are approximately to be 10 % and 1 %, − T t 3 m λ 2000 , we show the precision 7 . 1500 6 ]. In ﬁgure 36

| 1000

(b) Luminosity of 30 ab 0.6 0.4 1.6 1.4 1.2 0.8 t µ measurements at the HL-LHC [ 5 TeV, we do see a signiﬁcant impact from . t 2 λ ) = 5 presented in ﬁgure ∼ – 14 – n | 3000 T b ( m Z we assume a luminosity of 3 ab

= 10 % shown in ﬁgure 7a = 10 %. 0 % 100 | t t 2500 µ

δλ and an uncertainty of 1 % potentially reachable at a future 0% 50 / | . Therefore, it is crucial to improve the precision of measuring t 1 µ T |

−

δ and 0% 30 m 1 T − m 2000 ]. Although a precision of 10 % for the measurement of the naturalness of data. We assume the uncertainty of measuring 36 1 − we have neglected the uncertainty on the measurement of the top Yukawa 1500

20 % 6 Precision of measuring the absolute value of the naturalness parameter we consider 30 ab 7b

1 Using the fermionic top sector as an example, we show how the naturalness sum rule In ﬁgure

| | 1000

, particularly for small (a) Luminosity of 3 ab

0.8 0.6 0.4 1.6 1.4 1.2 t t µ , in order to test naturalness with a higher precision. t the Higgs self-energy. can be measured and whatWe developed sensitivity a reach simplified can model be foruralness achieved vector-like sum fermionic at top a rule partners 100 in and TeV hadron this derived the collider. general nat- setup. Remarkably, we found that the naturalness sum The naturalness problem playsWhile, searches an for extremely partner important particlesducted predicted role at by in the theories of driving LHC,particles. naturalness a particle are further physics. Once actively step con- discovered,impostor, is it but necessary a must particle to be with actually the ensured right test that properties the to the naturalness cancel newly the of found quadratic these UV-sensitivity particle of is not an parameter could still beδλ pushed up to λ 4 Discussion and conclusion of measuring the naturalness parameterWe when consider the two uncertainty of cases.an measuring uncertainty In of 10 figure %figure achievable in future 100 TeV collider [ has the consequence thatidentical curves to the for curves for coupling. However, thebe measurement an of important the task top at Yukawa a coupling 100 TeV is hadron non-trivial collider and [ will Figure 7. with 3 and 30 ab respectively. JHEP09(2017)129 : 1 → − T T h and hence T α via a pair of top T T h . The 100 TeV collider f events). This yields an addi- 0, new strategies such as indirect TT < t could be achieved with a top partner 1 could be excluded for a top partner µ | . t 0 µ | > | 1 | − – 15 – t µ || . The derived condition is rather generic, covering deviating from the naturalness assumption by more 2 T | m t / 2 . Additionally, the Higgs boson accompanying the top µ v | h − production in the little Higgs models indicates that at 30 ab bW ¯ 0 5 TeV, given the precision of measuring the top Yukawa coupling 2 TeV. . . tB 2 2 T T h . An accurate estimation of the influence of this factor will be nec- T ∼ ∼ → δα -parity and mirror twin Higgs models. In particular, the necessary mea- production is insensitive to the sign of the naturalness parameter and hence T T h T or T T h h + mass of up to of some percent. Theories which predict a than 10 % which correspondsmass to of up to A precision of 10 % for the measurement of Given the crucial role of the naturalness problem in driving particle physics, it is But, in this analysis we have neglected the precision with which the top partner mass We expect that the reach of this measurement can be further improved. Recall, that bW • • 0 ¯ hand, a complete cancellation ofHiggs quadratic models UV-sensitivity requires in the the introductionbosons, Higgs of which self-energy additional have in partner to little particles fulfillother for the theories gauge corresponding of and naturalness naturalness, Higgs sum mostopposite rules. notably statistics supersymmetry, On in rely the comparison on other to partner hand, their particles corresponding with SM particles, yielding a sum rule causes a degeneracy problem, despitebreak its this important degeneracy role and in excludedetection testing the needs naturalness. theories to with In be order introduced. to worthwhile to extend the analysis pursued in this article to other contexts. On the one essary. Additionally, we didsystematic not errors take could into account receiveQCD systematic multiple background, errors contributions, PDF in uncertainty, e.g., collider the high-order performanceestimation analysis. of is corrections future beyond to The collider, the the etc.would scope postpone A of reliable its this considerations articlethat to due the future to studies. the lack At of last, we input would information. like So to we stress again channels bring non-trivial improvements to the sensitivity of testingcan naturalness. be measured (e.g.,tional uncertainty by on mass the reconstruction relationa in between contribution the to the signal rate and the coupling quarks and require themfraction to originates decay from fully the decay or viatB semi-leptonically. one top However, quark a and largerpartners one branching bottom are quark, required that is todecay decay modes may to lead a to pair a stronger of background bottom suppression. quarks. We would In expect that reality, these some of its other in this analysis presented in this article we only consider decays of and is exact uprepresentative theories to of an naturalness, order includingand for of without example little Higgssurements models do both not with involve theanalyses measurement based of on the the decay constant rule only involves couplings of the Higgs to a pair of top quarks and a pair of top partners, JHEP09(2017)129 (A.1) (A.2) (A.3) (A.4a) (A.4b) DE-AC02- reads № v . 2 c ) . Y4545171Y2. I. Low 4 . , c b ) . 0 ]. A full exploration of № c b 2.1 0 0 16312716. Both the CRF 55 + 4 4 T 4 T 4 + h.c. – 0 v v 4 № m m c 53 + h.c. T 0 c 0 c U c T O O 4 U = ( | HUKST4/CRF/13G. T. Liu is also 4 0 | H ) + ) + T | 0 0 № t H 0 T | MOST-105-2112-M-003-010-MY3. λ DE-SC0011702, as well as by IHEP un- 3 T 0 λ 0 3 0 T ∗ T ∗ t 4 m f № № γ c b λ λ 4! 4! + 2 + 2 + + 0 0 0 t t – 16 – U T α α c 3 ) leads to the following additional terms in La- c ( ( 0 u t 0 0 4 T 2 T 2 4 2.3 | 2 2 | v v , γ m m ) at this order are in terms of gauge eigenstate param- H H 2 | 2 2 | c c c 0 0 0 3 2.5 ) t 4 3 T f 4 0 T c t c b + 4! m γ 0 − c c 0 4! c 0 = t T − = U 4 0 = = c L T 0 c c c b t ∆ L T ∆ DE-SC 0010143. C.-R. Chen was supported in part by the Ministry of = ( ) of the right handed fermion fields up to cubic order in 0 t № Y6515580U1 and IHEP Innovation Grant contract γ 2.17 № ) 2.4 eter is given by The rotation ( The rotation into massgrangian ( eigenstates ( The mass eigenstate parameter ( the mass diagonalization after electroweakthis symmetry omission breaking. we add In the order following to quartic make terms good to for Lagrangian ( A Higher order corrections In the main part of our discussion we have neglected higher order correction originating in was supported by the U.S.der DOE Contract under Contract was supported in part06CH11357 by and the U.S.Science Department and of Technology Energy of Taiwan under under contracts Grant sions, while I. LowTechnology, acknowledges where the part hospitality of this atthe work Hong Collaborative was Kong Research performed. Fund University (CRF) J. of undersupported Hajer Grant Science by and and the T. General Liu Research areand Fund GRF supported (GRF) grants by under are Grant issued by the Research Grants Council of Hong Kong S.A.R.. H. Zhang these possibilities will be deferred to future work. Acknowledgments J. Hajer and T. Liu would like to thank J. Bernon, Y.-Y. Li and J. Shu for helpful discus- as well. Some relevant discussions can be found in, e.g., ref [ JHEP09(2017)129 ) 2.5c (A.6) NATO (A.8a) (A.9a) (A.5a) (A.7a) (A.8b) (A.9b) (A.5b) (A.7b) , . , , . . 0 0 4 4 4 T T 4 0 v v 0 4 0 4 T m m 4 4 T v 4 4 T v m v . m m . O O 0 O + 4 4 T + O O v + 0 m + + 4 4 T ) is v 2 , | 2 m ) | 0 O 2 0 0 , | 2 T 2.2 0 T | T + t 0 0 λ λ t | λ λ 0 O 4 4 T | 0 | ∗ t λ 4 v | 4 4 T + | λ 2 + m + 0 0 2 v 0 t m T T − 2 α − λ 2 | α 2 | 0 O | 0 2 t + 0 | −| T 0 0 λ O 0 T t + ) λ T ) up to quadratic order are 0 T λ | λ | λ T β − | | 0 − λ + 0 . 0 )) + + 0 ∗ T 2 0 5 0 T + | T 0 t T λ 0 T 2.19a 0 α λ T λ H α T 0 0 T α – 17 – t 2 α λ T α | 2 α − 0 α 0 + 2 ( 0 t 0 − 0 2 T 0 2 0 ∗ T 2 λ T 0 | 2 − ]. T t 2 T 0 T | λ v β m 0 0 ) up to this order are λ β T α t ( λ 3 + 2 0 T 3 λ 0 β | 0 ∗ T λ ( + − − T | 0 − 0 ∗ T λ 0 0 0 ), which permits any use, distribution and reproduction in 2.5b λ 0 2 t SPIRE t t 0 2 T ∗ T 0 1 + λ 2 t t + 2 T λ α α 2 IN λ v 0 0 0 λ m + α t ∗ t v 0 [ m 0 ) are altered to 0 T 2 t + t α λ 2 0 T λ 0 α 0 + 2 3 λ 3 0 T t 0 T 3 0 + m ∗ T t ∗ 2.5a T γ + 3 0 − β γ − − + 3 β 0 0 T = ( 0 0 t 0 0 ∗ T + β 0 CC-BY 4.0 T ∗ T + β T T 2 T 0 β 2 2 T 0 λ λ 2 β t T (1980) 135 3 v 0 m 0 0 3 m v This article is distributed under the terms of the Creative Commons 0 m γ γ 0 2 T 2 T T 2 2 T 2 0 v v 2 3 T Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking 3 v 3 T m m m v 3 0 0 + m v 0 0 2 m + v 2 2 0 t 2 2 T 2 T m B 59 6 0 T v v T t 6 0 + m m 0 + + α α ) as they involve terms of order T t 0 − 0 0 − − 0 t T T t = = + − λ λ = = t t T = = T α = = 2.19b a α Sci. Ser. t a t T G. ’t Hooft, T ∆ λ ∆ λ [1] any medium, provided the original author(s) and source areReferences credited. We have refrained from writing down theand quadratic ( corrections to the cubic couplings ( Open Access. Attribution License ( The correction to the VEV induced terms ( While the quadratic couplings ( The Yukawa couplings ( The resulting quadratic correction to the top partner mass ( At the same time the left handed fields are rotated by JHEP09(2017)129 , B , , , , ]. The 07 , ]. ]. 136B , (2015). SPIRE Nucl. Phys. IN Phys. Lett. JHEP , (2003) 057 , SPIRE ][ SPIRE , IN IN 12 ][ ][ Phys. Lett. (2012) 1 preCDR.pdf , TeV with the ATLAS JHEP GeV with the CMS -jets and a pair of same , b ]. B 716 ]. = 13 125 s ]. hep-ph/0105239 √ ]. The littlest Higgs [ SPIRE SPIRE hep-ph/0206020 IN symmetry collisions with the ATLAS detector [ IN arXiv:1207.7235 SPIRE ][ TeV with the ATLAS detector ]. [ SPIRE ][ pp IN Phys. Lett. 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