New Physics in multi-Higgs boson final states
Wolfgang Kilian, Sichun Sun, Qi-Shu Yan, Xiaoran Zhao, Zhijie Zhao
PPrepared for submission to JHEP
SI-HEP-2017-05
New Physics in multi-Higgs boson final states
Wolfgang Kilian a Sichun Sun b,c
Qi-Shu Yan d,e
Xiaoran Zhao f Zhijie Zhao a a Department of Physics, University of Siegen, 57072 Siegen, Germany b Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, ClearWater Bay, Hong Kong c Department of Physics, National Taiwan University, Taipei, Taiwan d School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China e Center for future high energy physics, Chinese Academy of Sciences, Beijing 100039, China f Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Lou-vain, Chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We explore the potential for the discovery of the triple-Higgs signal in the b l ± j + /E decay channel at a TeV hadron collider. We consider both the StandardModel and generic new-physics contributions, described by an effective Lagrangian thatincludes higher-dimensional operators. The selected subset of operators is motivated bycomposite-Higgs and Higgs-inflation models. In the Standard Model, we perform both aparton-level and a detector-level analysis. Although the parton-level results are encourag-ing, the detector-level results demonstrate that this mode is really challenging. However,sizable contributions from new effective operators can largely increase the cross sectionand/or modify the kinematics of the Higgs bosons in the final state. Taking into accountthe projected constraints from single and double Higgs-boson production, we propose bench-mark points in the new physics models for the measurement of the triple-Higgs boson finalstate for future collider projects. a r X i v : . [ h e p - ph ] J u l ontents b l ± j + /E channel in the SM 9 gg → hhh and Kinematics 184.3 Correlations between gg → hhh and single and double-Higgs production 224.4 Analysis for models 294.4.1 Strongly-interacting Higgs models 314.4.2 The Gravity-Higgs Model 32 gg → h , gg → hh , and gg → hhh After the discovery of the Higgs boson h ( GeV) at the LHC [1, 2], measurements of theHiggs self-couplings become crucial for our understanding of fundamental particle physics.In the Standard Model (SM), the Higgs boson has three types of interaction: (1) the in-teractions with electroweak gauge bosons ( W ± and Z ); (2) the Yukawa interactions withfermions; (3) the triple and quartic self-interactions. A measurement of the last type ofinteraction would complete the phenomenological reconstruction of the Higgs potential [3]and thus should lift our knowledge about electroweak symmetry breaking (EWSB) to a newlevel. Furthermore, Higgs self-interactions could be related to the problems of baryogene-sis [4] and vacuum stability [5–7].In the SM, the Higgs potential is written as V ( H † H ) = − µ ( H † H ) + λ H † H ) , (1.1)– 1 –here H = ( G + , √ ( v + h + iG )) T is the Higgs doublet, and G ± , G are the unphysicalGoldstone bosons associated with spontaneous EWSB in a renormalizable gauge. Thispotential has a minimum for the Higgs-field vacuum expectation value v = 2 | µ | /λ ≈ GeV. After EWSB and switching to unitarity gauge, the Higgs self-interactions take thefollowing form V self = λ vh + 116 λh , (1.2)which corresponds to a triple-Higgs self-coupling g hhh = λv and a quartic Higgs self-coupling g hhhh = λ , respectively. The parameter λ can be determined by measuring theHiggs mass m h , since λ = m h v . In the SM, the Higgs potential is thus completely fixed afterthe measurement of m h ≈ GeV. However, the story could be different if new physics cancontribute to the Higgs self-interactions. Independently measuring the triple and quarticcouplings of the Higgs boson via double and triple-Higgs final states is an essential projectfor future collider experiments.Deviations from the SM that manifest themselves prominently in double and triple-Higgs final-state processes are expected for various new-physics scenarios. In order tostudy the Higgs potential in a largely model-independent way, we will parameterize newphysics beyond the SM (BSM) in terms of an effective field theory (EFT). This systematicmethod captures the essence of a wide class of BSM models. It is well suited to colliderstudies that require exclusive Monte Carlo simulations.For concreteness, we will describe BSM Higgs physics in terms of the strongly-interactinglight Higgs (SILH) version [8] of the EFT approach [9, 10]. The operators in this choice ofbasis are designed to directly correspond to low-energy effects of specific BSM Higgs-sectorrealizations, including composite Higgs models [8, 11–14] and the Higgs inflation model [15].We will consider operators up to dimension 6. Nonvanishing coefficients for some of those,such as ∂ µ ( H † H ) ∂ µ ( H † H ) , can substantially enhance multi-Higgs production rates and/ormodify final-state kinematics.In the SM, the leading order (LO) for the production of one or more Higgs bosonsin gluon-gluon fusion involves one-loop diagrams. The calculation of higher order correc-tions becomes quite a challenge. Most of these calculations [16–25] are based on effective-theory methods, working in the limit of infinite top-quark mass. Regarding effects of finitetop-quark mass, only NLO QCD corrections to single-Higgs production are known analyt-ically [26, 27]. One way to estimate finite top-quark mass effects is series expansion, whichcan work well for single-Higgs production [23, 28, 29] but converges poorly for double-Higgsproduction [30]. Recently, NLO QCD corrections for double-Higgs production with full top-quark mass dependence have been calculated numerically [31, 32]. The results show largedifferences in kinematical distributions compared to the prediction of the infinite top-masslimit.The feasibility of an analysis of double-Higgs production at the LHC has become a hottopic [33–37], because this process probes the triple coupling g hhh . The dominant modefor double-Higgs production is gluon fusion via a box or triangle loop of quarks. Variousdecay channels have been studied in the literature, such as W W W W [38], b ¯ bγγ [39–41],– 2 – ¯ bW W [42], b ¯ bτ τ [43–45], b ¯ bµµ [39], W W γγ [46] and b ¯ bb ¯ b [43, 47, 48]. It has been arguedthat the triple self-coupling can be measured within accuracy at the high luminosityLHC ( ab − ) with TeV energy [49], but recently more detailed studies have shed doubton this estimate [50]. At a future
TeV hadron collider [51, 52], the rate for double-Higgsproduction will be significantly higher. The prospects for a measurement at such a machinehave been investigated in Refs. [40, 53–57].The triple-Higgs self-coupling g hhh can also be measured at a future lepton colliderthrough the double Higgs-strahlung process e + e − → Zhh or the vector-boson fusion process e + e − → ν ¯ νhh . It has been shown that g hhh can be measured within accuracy at theluminosity-upgraded ILC [58]. At a low-energy machine, such as the GeV CEPC, thetriple self-coupling could be determined indirectly via the loop corrections to the
ZZh vertex [59, 60].By contrast, a measurement of the quartic self-coupling g hhhh is a real challenge at theLHC, since at √ s = 14 TeV the cross section of gg → hhh is only O (0 . fb [61, 62].Alternatively, one can consider pp → Zhhh , but that cross section is also tiny [63]. Thisproblem cannot be solved at a lepton collider either, because the cross section for e + e − → Zhhh is only O (0 . ab at a √ s = 1 TeV machine [64], too small for a measurement.The proposals for future pp colliders have motivated the study of the process gg → hhh at high energy. The cross section of gg → hhh at a TeV hadron collider can beestimated to be about fb if NLO corrections are accounted for [65], which makes it atleast possible to observe the final states of this process. The discovery potential of decaychannels hhh → b ¯ bb ¯ bγγ [66, 67] and hhh → b ¯ bb ¯ bτ τ [68] has been explored. It turns outthat the discovery of three-Higgs final state through these channels is challenging, and anextreme high quality detector is needed.In this paper, we investigate the sensitivity of the decay channel hhh → b ¯ bW W ∗ W W ∗ → b l ± j + /E , which has not been carefully analyzed in the literature before. We also ex-amine how new physics can contribute to triple-Higgs production. We will consider theeffects of a set of dimension-6 effective operators to the cross section and kinematics ofHiggs bosons in the final state. Especially, we extend the study of Ref. [15] to the triple-Higgs production case, where the effects of derivative operators on the kinematics of Higgsbosons in double-Higgs production were explored. We also study the projected bounds forall relevant couplings in the EFT at the LHC and at a future 100 TeV pp collider.This paper is organized as follows. In Sec. 2, we briefly introduce the EFT Lagrangianas appropriate for our study and relate our parameterization to particular models that areof interest in the context of new Higgs-sector BSM physics. In Sec. 3, we present a MonteCarlo (MC) analysis of hhh → b ¯ bW W ∗ W W ∗ → b l ± j + /E in the SM, and investigatethe discovery potential and identify challenges of this channel. In Sec. 4, we describethe calculation of triple-Higgs production in the context of the EFT with dimension-sixoperators in detail and present our numerical results. We conclude this paper with adiscussion of our findings in Sec. 5. – 3 – Effective Lagrangian up to dimension-6 operators
It has been accepted for a long time that new-physics effects associated with a characteristicscale higher than the energy of the processes under study, can be conveniently expressed interms of a low-energy EFT. This is a local Lagrangian which includes an infinite series ofoperators of dimension greater than four, constructed as monomials of fields and organizedin terms of the canonical dimension. The operators may incorporate only the unbrokenLorentz, electromagnetic and colour symmetries [69]. However, our knowledge of flavordata, electroweak precision data, and Higgs properties strongly suggests to furthermoreimplement the power counting of EWSB and thus build operators out of classically gauge-invariant combinations under the full electroweak symmetry. Up to dimension four, thisreproduces the SM. The set of operators up to dimension six was introduced in Ref. [9] andhas been reworked to a minimal basis in Ref. [10]. Adopting this as a phenomenologicalmodel implies rather generic assumptions on the flavor and gauge structure of the underlyingfundamental theory.In the present context, we are more specifically interested in the possibility that Higgsself-couplings act as primary probes to new-physics effects, while other SM fields are affectedonly by secondary corrections. This notion is realized by scenarios where the Higgs fieldacts as the only SM field with sizable couplings to a new sector. Specific models with thisproperty have been proposed, e.g., in Refs. [11, 13]. A general discussion can be foundin Ref. [8] where the resulting effective low-energy Lagrangian, expanded up to dimensionsix, has been introduced as the SILH Lagrangian. As expected, and confirmed in Ref. [14],this Lagrangian is equivalent to the basis of Ref. [10], but the assumptions of Ref. [8]on the underlying dynamics suggest a hierarchy between induced tree-level and loop-levelcoefficients that allows for dropping part of the operator set and thus keeping a moreeconomical number of phenomenological parameters. If we follow this line of reasoning, wecan adopt the SILH Lagrangian as the basis of the present phenomenological study. Wesupply a more detailed discussion below in Sec. 2.1.For the actual applications in later sections, we can focus on the interactions of thephysical Higgs field h , after EWSB and expressed in unitarity gauge. The Lagrangianreduces to L EF T = L SM + L t + L h + L ggh , (2.1) L t = − a m t v ¯ tt h − a m t v ¯ tt h − a m t v ¯ tt h , (2.2) L h = − λ m h v h − κ v h∂ µ h∂ µ h − λ m h v h − κ v h ∂ µ h∂ µ h, (2.3) L ggh = g s π (cid:18) c hv + c h v (cid:19) G aµν G a µν (2.4)Here we confine to the CP conserving operators and omit the CP violating operators. Inthe SM, we have a = λ = λ = 1 and a = a = κ = κ = c = c = 0 . It isunderstood that the corresponding terms have been removed from L SM , such that they arenot double-counted. – 4 –nother set of models that couple the Higgs sector to new physics is provided by certainmodels of inflation. As we show below in Sec. 2.2, this effectively results in the same HiggsLagrangian, Eq.(2.1). In Sec. 2.3 we briefly review the relation to the EFT version ofRefs. [9, 10] as it has been applied to the Higgs sector in Ref. [70]. Finally, it can be shownthat in a framework that implements a non-linear realization of electroweak symmetry, theresult is again equivalent to SILH if equivalent assumptions on coefficient hierarchies aretaken [71].In summary, the phenomenological Lagrangian (2.1) provides a robust parameterizationof new physics in the Higgs sector under the condition that no new on-shell states appearin the kinematically accessible range. The relevant part of the SILH Lagrangian [8, 14], including operators up to dimension six,has the form L SILH = c H f ∂ µ (cid:16) H † H (cid:17) ∂ µ (cid:16) H † H (cid:17) + c T f (cid:16) H † ←→ D µ H (cid:17) (cid:16) H † ←→ D µ H (cid:17) − c λf (cid:16) H † H (cid:17) + (cid:18) c y y f f H † H ¯ f L Hf R + h . c . (cid:19) + c g g S π f y t g ρ H † HG aµν G aµν + ic W g m ρ (cid:16) H † σ i ←→ D µ H (cid:17) ( D ν W µν ) i + ic B g (cid:48) m ρ (cid:16) H † ←→ D µ H (cid:17) ( ∂ ν B µν )+ ic HW g π f ( D µ H ) † σ i ( D ν H ) W iµν + ic HB g (cid:48) π f ( D µ H ) † ( D ν H ) B µν + c γ g (cid:48) π f g g ρ H † HB µν B µν . (2.5)It includes all the CP-conserving gauge-invariant operators up to dimension six with pureHiggs interactions and Higgs-gauge boson interactions. Some operators such as H † HW µν W µν are not included here since they can be generated by integration by parts from the otheroperators. There are further operators with fermions coupling to the Higgs, which areomitted here.There is only one dimension-5 operator allowed by the SM gauge symmetry, up toHermitian conjugation and flavour assignments: ( H(cid:96) i ) T C ( H(cid:96) j ) . It gives rise to the neutrinoMajorana mass and violates lepton number, so we do not include it, either.The SM Higgs may appear as a composite pseudo Nambu-Goldstone (NG) boson asso-ciated with some enlarged symmetry beyond the SM. The Lagrangian L SILH then emergesat low energy via spontaneous breaking of that symmetry. Since any terms in the Higgspotential will violate the shift symmetry of this NG-boson Higgs, the coefficients above areall suppressed by the small breaking in relation to the compositeness scale f , i.e., carryinga ξ = v f factor. m ρ , g ρ stand for the characteristic mass and coupling of a strongly coupledsector, respectively, and c i ∼ .We focus on the first five operators in Eq. (2.5), since they are the relevant operatorsfor the hadron-collider processes that we want to study. The first three terms in L SILH contribute to the Higgs potential. They contain only two independent terms, as can be– 5 –arameters SILH with Eq. (2.5) MCHM4 MCHM5 a (1 − c y ξ )(1 − c y ξ ) − (1 + c H ξ ) − / − ξ − ξa − c y ξ (1 − c y ξ ) − (1 + c H ξ ) − − ξa − c y ξ (1 − c y ξ ) − (1 + c H ξ ) − / − ξc c g ξ y t g ρ ξ y t g ρ ξ y t g ρ c c c c κ − c H ξ (1 + c H ξ ) − / − ξ − ξκ − c H ξ (1 + c H ξ ) − − ξ − ξλ (1 + c ξ )(1 + c ξ ) − (1 + c H ξ ) − / ξ − ξλ (1 + c ξ )(1 + c ξ ) − (1 + c H ξ ) − ξ − ξ Table 1 . Parameter relationship between our convention and that in SILH, Eq. 2.5. The MCHM4and MCHM5 models are from Refs. [13, 80]. Some notation is from Ref. [8, 81]. Note that c and c are sensitive to the detailed construction of the models. We consider c g as roughly of order 1. Forthe relation between our conventions and other conventions used in the literature, cf. Appendix A. verified by applying the equations of motion. After EWSB, the SILH potential reduces tothe effective potential of Eq. (2.1). We list the relations between Eq. (2.1) and Eq. (2.5)in Table 1. Note that we have the relation κ = κ , since the associated terms come fromthe same operator c H f ∂ µ (cid:0) H † H (cid:1) ∂ µ (cid:0) H † H (cid:1) . The rest of the operator coefficients can bemeasured at future electron-positron colliders, via W -pair production, Z -pair production,and Z -Higgs production [72, 73].Regarding hadron-collider measurements, the coefficient c g is accessible via the pp → h process at the LHC. Run- data have constrained c g /m ρ ∼ − [74]. Bounds for thecoefficients c H , c y and c are currently much weaker [75]. It is expected that the high-luminosity LHC will yield bounds c H ξ ∈ [ − . , . and c y ξ ∈ [ − . , . for thetop quark [76]. The coefficient c H ξ can be further constrained to O (10 − ) at a future e + e − collider, and the tests for c y can be extended to b , c quarks, and leptons [77]. The coefficient c contributes to the triple and quartic Higgs self-couplings only, so the bounds on c willstay relatively weak for both LHC and a future lepton collider.We may also consider two more specific composite Higgs models [8, 13], dubbed asMCHM4 and MCHM5, respectively. Both models result in the SILH Lagrangian as theirlow-energy EFT. They contain extra fermions, which are in representations 4 and 5 of anassumed global SO (5) symmetry, respectively. We adopt the notation from Ref. [78]. TheSILH coefficient values areMCHM4: c H = 1 , c y = 0 , c = 1 , (2.6)MCHM5: c H = 1 , c y = 1 , c = 0 . (2.7)The current LHC constraints and electroweak precision data imply f ≤ GeV and v /f ≤ . [79]. Later, we will study the projected constraints from the LHC and a 100TeV collider. – 6 – .2 Operators from Higgs inflation In this section, we demonstrate how an equivalent set of dimension-six operators arises fromthe standard Higgs inflation paradigm [82–86]. We incorporate a non-minimal coupling ofthe Higgs field to gravity and work in unitarity gauge where H = (0 , h/ √ . The gaugeinteractions are more complicated in this scenario; we ignore them for now and just focuson the Higgs potential. In the Jordan frame, the Lagrangian has the form S Jordan = (cid:90) d x √− g (cid:40) − M + ξh R + ( ∂h ) − m h h − λ h (cid:41) . (2.8)We consider ξ in the range (cid:28) √ ξ ≪ , in which M (cid:39) M PLanck .We perform a conformal transformation from the Jordan frame to the Einstein frame, ˆ g µν = Ω g µν , Ω = 1 + ξh /M Planck . (2.9)This transformation will give rise to derivative terms in Higgs potentials. We furthermoreredefine dχ = (cid:114) Ω + 6 ξ h /M Planck Ω dh. (2.10)Then the action in the Einstein frame is given by S E = (cid:90) d x (cid:112) − ˆ g (cid:40) − M Planck R + ∂ µ χ∂ µ χ − V ( χ ) (cid:41) , (2.11)where the potential becomes V ( χ ) = 1Ω( χ ) (cid:20) λ h ( χ ) + 12 m h h ( χ ) (cid:21) . (2.12)In the standard Higgs inflation paradigm, h takes large values h (cid:29) M Planck / √ ξ (or χ (cid:29) √ M Planck ) during inflation and plays the role of the inflaton. We have the expressions h (cid:39) M Planck √ ξ exp (cid:18) χ √ M Planck (cid:19) , V ( χ ) = λM Planck ξ (cid:18) (cid:18) − χ √ M Planck (cid:19)(cid:19) − (2.13)This allows the potential to be exponentially flat at large h to produce a viable inflatonpotential.When the value of h is near the origin as today, we can approximate h (cid:39) χ and Ω (cid:39) ,so the potential for the field χ generates a potential for the SM model Higgs field pluscorrections at O ( ξ /M Planck ) . For the purpose of this collider study, we thus replace χ by h . Plugging Eq. (2.10) into Eq. (2.11) and omitting higher order terms, we arrive at S E = (cid:90) d x (cid:112) − ˆ g (cid:26) − M Planck R + ∂ µ h∂ µ h + 3 ξM Planck h ∂ µ h∂ µ h Ω − (1 − ξh M Planck ) (cid:20) λ h ( χ ) + 12 m h h ( χ ) (cid:21) (cid:27) . (2.14)Note that after EWSB, replacing h → h + v yields similar extra terms as in Eq. (2.1).– 7 – .3 Alternative Parameterization of the Higgs boson self-interaction operators Another representation of the set of gauge-invariant dimension-6 operators which can mod-ify the Higgs self-interactions, has been studied in Ref. [70] O = f Λ ( D µ H ) † HH † ( D µ H ) , (2.15) O = f ∂ µ ( H † H ) ∂ µ ( H † H ) , (2.16) O = f ( H † H ) , (2.17) O = f Λ ( D µ H ) † ( D µ H )( H † H ) . (2.18)The operator O was considered in Ref. [75] and can safely be neglected. In the subset ( O , O , O ) , one operator can be eliminated by the equations of motion, so we drop O .Thus we only need to consider the operators ( O , O ) .As mentioned in Ref. [15], the operator O induces a derivative term for the Higgs field O → f ( v + h ) ∂ µ h∂ µ h. (2.19)Therefore the kinetic term of the Higgs field is modified to L kin = 12 (cid:18) f v Λ (cid:19) ∂ µ h∂ µ h. (2.20)This means that the Higgs field should be rescaled by h → ζh , where ζ = (1+ f v / Λ ) − / .After EWSB and choosing unitarity gauge, the Lagrangian reduces to (2.1) as before,where the coefficients ( a , λ , λ , κ , κ ) of Eq. (2.1) can be expressed in terms of just twoindependent parameters: ˆ x = x ζ , (2.21) ˆ r = − x ζ v m h , (2.22)where x i = f i v / Λ ( i = 2 , . With this definition, the rescaling factor ζ can be rewrittenas ζ = (1 − ˆ x ) / . The relations between our parameters and those in Ref. [15] are listed inTable 2.At the time when the measurements that we discuss in the present work can be carriedout, we should expect that data exist that set significant bounds on the parameters ˆ x and ˆ r .1. a ( ˆ x ) is related to the direct measurement of the top Yukawa coupling, and its value isexpected to become determined within precision at the high-luminosity LHC [87],via measuring the tth production rate. At a 100 TeV collider, the Yukawa couplingcan be pinpointed down to a precision [88] by measuring the ratio between the tth and ttZ production rates. – 8 –ur operators Operators in Ref. [15] Relations − m t v a ¯ tth − m t v ζ ¯ tth a = ζ − λ m h v h − ζ v (1 + ˆ r ) m h h λ = ζ (1 + ˆ r ) − λ m h v h − ζ v (1 + 6ˆ r ) m h h λ = ζ (1 + 6ˆ r ) − v κ h ( ∂h ) v ˆ xζh ( ∂h ) κ = − xζ − κ v h ( ∂h ) x v ζ h ( ∂h ) κ = − xζ Table 2 . Parameter relationship between our convention and that in Ref. [15].
2. Another bound on ˆ x is obtained from the measurement of Higgs-gauge couplings [75],since they become universally rescaled by ζ . A future e + e − Higgs factory can constrain | ˆ x | at the level [89]. Since there are many other dimension-6 operators which cancontribute to the gauge-boson kinetic terms, we nevertheless take ˆ x as a free parameterin our later analysis.3. The parameter ˆ r can only be constrained by double-Higgs or triple-Higgs production.Concerning double-Higgs boson production, the bound will be around ∼ atthe HL-LHC at most. At a TeV hadron collider, (ˆ x, ˆ r ) will become more stronglyconstrained by double-Higgs production. As shown in Ref. [15], the bounds on ˆ x and ˆ r will be of the order ∼ and ∼ , respectively. b l ± j + /E channel in the SM We study triple-Higgs production in high-energy proton-proton collisions, pp → hhh , whereone Higgs boson decays into a b ¯ b pair while the two other Higgses decay into W W ∗ . Thesemi-virtual W pairs can subsequently decay semileptonically, h → W W ∗ → (cid:96)νjj .The dominant partonic contribution to the pp → hhh signal is gluon-gluon fusion, gg → hhh . This process involves one-loop diagrams. As we did for our previous work [67],we compute the production matrix element at LO with MadLoop/aMC@NLO [90]. Wetake the parton distribution functions from CTEQ6l1 [91]. For phase-space evaluation andexclusive event generation, we interface the production process with VBFNLO [92–94].Background event samples are generated by MadGraph 5 [95, 96]. Since we require a b ¯ b pair, the dominant background is caused by top-quark pairs in association with electroweakbosons, namely pp → h ( W W ∗ ) t ¯ t and pp → t ¯ tW − W + . Both classes of processes can leadto the same final state as the signal. To veto further background from Z bosons, we restrictthe analysis to same-sign leptons in the final state, l + l + or l − l − .We list the calculated cross sections of signal and backgrounds at
100 TeV in Table 3.In the absence of a complete NLO calculation for the signal, we adopt the K-factor of . that was obtained in Ref. [66] for Higgs pair production. For the H ( W W ∗ ) t ¯ t background,we use K = 1 . [97]. The K-factor for t ¯ tW − W + at TeV is taken . from Ref. [56].In the Ref. [98], a K-factor around 1.2 was obtained while the total cross section σ NLO isgiven as . pb, which is around . times larger than our LO cross section. The derivationis mainly attributed to the choice of renormalisation and factorisation scales, i.e. our choice– 9 –f K-factor equal to . is consistent with the results given in Ref. [98] after taking theseuncertainties into account.We ignore all background from h + jets, hh + jets and W ± W ± + jets, since the crosssections of those processes are negligible compared to the h ( W W ∗ ) t ¯ t background. Fur-thermore, we observe that the total cross section of the background b ¯ bW − W + W − W + isessentially exhausted by the resonant contribution t ¯ tW − W + . Therefore, we approximatethe former process by the latter with subsequent top-quark decay, which considerably sim-plifies the calculation.We have three comments on the background processes hhjj in the SM and new physicsmodels. • In SM, the hhjj final state receives contribution for heavy-quark loop and vector bosonfusion, while the former is dominant. Currently, the cross section of loop-inducedprocesses with 2 jets can be calculated by interfacing GoSam [99] or OpenLoops [100]to Madgraph5 [101] or Herwig7 [103]. We use Madgraph5 to compute the cross sectionof top quark loop induced pp → hhjj at a 100 TeV collider. After imposing the MLMmatching[102] and using cuts P t ( j ) > GeV and η ( j ) < , we obtain an inclusivecross section fb, which is around times larger than the cross section σ ( hhh ) of the signal processes gg → hhh . Meanwhile, by using Madgraph5 [96], we find thatthe cross section of VBF with √ s = 100 TeV is 34 fb. • It is known that when the b tagging efficiency is taken as . , the rejection rate of lightjets can reach 0.1% or so. Since we required one(two) tagged b jets in our preselectioncuts, therefore the background gg → hh + 2 jets is suppressed by a factor − ( − ) or so. After imposing b taggings and the decay branching fraction of h → b ¯ b , we findthat the signal cross section ¯ bbhh is around . . σ ( hhh ) , while the cross sectionof background hh + 2 jets is . . × − ) × σ ( hhh ) or so. Obviously, when n b ≥ is imposed, it is safe to neglect this type of background in the SM. • In the new physics models we will consider below, the background process of hhjj canhave extra contributions from higher dimensional operators. When the cross sectionis ∼ magnitude orders smaller than the signal process, we can neglect it safely.In the cases when such a background is greatly enhanced or in the cases the signalprocess gg → hhh is greatly suppressed by the higher dimensional operators to such adegree that the cross sections of them are comparable, the background of hhjj shouldbe included in the analysis.Table 3 shows a yield of signal events in this final-state channel for ab − inte-grated luminosity. However, without further selection there are ∼ background events.Clearly, it is a challenge to observe triple-Higgs production through this channel. In the fol-lowing subsection we discuss observables and selection methods for suppressing backgroundand raising the signal/background ratio to an acceptable level.– 10 –rocess σ × BR (ab) K-factor Expected number of eventsSignal .
71 2 . h ( W W ∗ ) t ¯ t . × . . × t ¯ tW − W + . × . . × Table 3 . Cross sections of signal and background for the b l ± j + /E final state in the SM. Theexpected number of events corresponds to ab − integrated luminosity. We simulate the Higgs boson decays that lead to the final state b l ± j + /E by using theDECAY package provided by MadGraph 5. Here we do not consider any parton showereffects, which will be discussed in section 3.2. The transverse momentum ( P t ) distributionsof the visible particles and missing transverse energy (MET) are shown in Fig. 1. In thisfigure, the objects are sorted by P t . On the one hand, one can expect that the b quarksare harder than the light quarks, since they originate from a Higgs boson decay directly.On the other hand, the decay chain h → W W ∗ → jjlν leads to soft leptons and light jets,especially when they are coming from the off-shell W bosons.In Fig. 1(b) and Fig. 1(c), we observe that the P t distributions of the softest leptons andjets peak around GeV, which might make it a challenge to successfully reconstruct theseobjects with the currently planned detectors. Since the signal contains only two neutrinos,MET should not be too large. As illustrated by Fig. 1(d), MET peaks around GeV,somewhat below half the Higgs boson mass.Because there are two unobserved neutrinos in the final state, their mothers being eitheron-shell or off-shell W bosons, it is not convenient to fully reconstruct the Higgs bosons. Apartial reconstruction should nevertheless be possible. In order to extract this information,it is crucial to correctly associate the mother Higgs bosons with their decay products. Herewe encounter a problem of combinatorics, which leads to a 12-fold ambiguity. To simplifythe problem, we assume that both b quarks can be tagged correctly, so only the light quarkscan be reassigned and the ambiguity reduces to 6-fold.To find the correct combination of the visible particles from Higgs boson decays, weexamine the following four alternative reconstruction methods at parton level:1. The decay chain h → W W ∗ → jjlν suggests that the lepton and the hadronicallydecayed W boson should have a small angular separation ∆ R ( l, W jj ) . Since there aretwo Higgs bosons with this decay chain, the sum of ∆ R ( l, W jj ) + ∆ R ( l, W jj ) shouldbe minimal. We choose a combination with minimal value of this observable.2. The semileptonic Higgs invariant masses can be computed from the visible parti-cles; we denote them as m vis h ( l, jj ) and m vis h ( l, jj ) . We choose a combination whichminimizes their sum.3. We compute the mT observable as it has been defined in Refs. [104–108], from thevisible particles that originate from semileptonic Higgs decay. The observable can– 11 –a) (b) (GeV) t P F r a c t i on o f E v en t s bottom 1bottom 2 (b) (l) (GeV) t P F r a c t i on o f E v en t s lepton 1lepton 2 (c) (j) (GeV) t P F r a c t i on o f E v en t s jet 1jet 2jet 3jet 4 (d) MET (GeV) F r a c t i on o f E v en t s Figure 1 . Distributions of (a) the transverse momentum of b quarks, (b) the transverse momentumof leptons, (c) the transverse momentum of light quarks (labeled by j ), and (d) missing transverseenergy in the signal events. Methods The percentage of correctness min[∆ R ( l, W jj ) + ∆ R ( l, W jj )] 47 . m vis h + m vis h ) 61 . mT
2) 66 . | mT − m h | . Table 4 . Methods for determining the correct combinations of ( l, j, j ) and their percentages ofcorrectness. set an upper bound on the Higgs mass, so we choose a combination which minimizes mT .4. Since mT should have a value close to the Higgs mass m h = 126 GeV, we choose acombination which minimizes | mT − m h | .– 12 –hese methods and their associated percentages of correct assignment in a simulatedevent sample are listed in Table 4. The effect of realistic b -tagging efficiency will be discussedin the next subsection. We observe that in a parton-level analysis, the method that relieson the quantity | mT − m h | has the best performance, approaching probability forcorrect particle assignment in the reconstruction. To obtain a hadronic event sample, we use the parton-shower and hadronization modulesof Pythia 6.4 [109]. For jet clustering, we use the package FASTJET [110] with the anti- k t algorithm [111] and cone parameter R = 0 . . To veto the large number of soft jets frominitial-state radiation, only jets with P t > GeV are accepted.The multiplicity distribution of jets is plotted in Fig. 2(a). Both signal and backgroundin the MC sample provide six jets at parton level, which explains the peak of n j around in Fig. 2(a). In Fig. 2(b), we show the P t distributions of the six leading jets in thesignal event sample. The 1st to 4th jet exhibit similar distributions as at parton level, butthe 5th and 6th jet P t distributions have different shapes with respect to their parton-levelcounterparts.There are two simple reasons for this result: (1) the softest quark in Fig. 1(c) typicallyhas P t only around 10 GeV while most of the low- P t jets are vetoed by our P t > GeV cut;(2) jets from initial-state radiation can easily be as hard as GeV at a
TeV collider.So the 5th and 6th jet are more likely produced by initial-state radiation than by Higgsboson decays. Fig. 2 illustrates the challenge of reconstructing the soft jets generated bythe multi-Higgs signal.(a) j n F r a c t i on o f E v en t s (b) (j) (GeV) t P F r a c t i on o f E v en t s jet 1jet 2jet 3jet 4jet 5jet 6 Figure 2 . Distributions of (a) the number of jets and (b) P t of the six leading jets of the signal. Another important problem is the reconstruction of leptons. We assume that the futuredetector can reach a better efficiency in reconstructing leptons than possible today ( for P t > GeV), so it becomes feasible to find the soft lepton as shown in Fig. 1(b). Butin order to reject huge QCD background, we need isolated leptons. To find a suitableisolation condition, we investigate the angular separations between two leptons ( ∆ R ( l, l ) )– 13 –a) R(l,l) D min F r a c t i on o f E v en t s (b) R(l,j) D min F r a c t i on o f E v en t s Figure 3 . Distributions of (a) the minimum angular separation between two leptons, and (b) theminimum angular separation between lepton and jet. and between leptons and jets ( ∆ R ( l, j ) ), respectively. The minimum-value distributions ofthese two observables at hadron level are displayed in Fig. 3. On the one hand, min ∆ R ( l, l ) tends to have a large value, and only of the events have min ∆ R ( l, l ) < . . On theother hand, almost of the events have min ∆ R ( l, j ) < . . This makes it difficult toisolate the leptons from the jets.To study the detector effects, we use DELPHES [112, 113] to perform a detector sim-ulation for the generated event samples. The setup of DELPHES is similar as in Ref. [67],with the following modifications:1. The b -tagging efficiency is assumed to be a constant (cid:15) b = 0 . , and mistagging ratesare . and . for charm and light jets, respectively. The pseudorapidity for b ( c ,jet) is required to be η < . , respectively.2. As described above, the jets are clustered by FASTJET with a cut P t ( j ) > GeV.3. The efficiency of lepton indentification is assumed to be when P t ( l ) > GeV and η ( l ) < . .4. Isolated leptons are defined by Ref. [113] I ( l ) = (cid:80) ∆ R
24 9 . × . × mT < GeV
23 9 . × . × | m bb − m h | < GeV
21 6 . × . × m vis h < GeV
21 6 . × . × S/B . × − S/ √ S + B . Table 5 . Efficiencies of cuts as described in the text, for a total integrated luminosity of ab − .. Given the dim prospects for observing triple Higgs production in the pure SM, we may askthe question about SM extensions that enhance the production rate such that the processbecomes observable at a
TeV collider. In that case, such an observation would not justindicate a significant deviation from the SM, but at the same time provide a measurement– 16 –f new BSM parameters.We work in the context of the genuine Higgs-sector BSM models that we have in-troduced above, conveniently parameterized by the SILH Lagrangian with dimension-sixoperators, or, alternatively, by the effective Higgs Lagrangian in unitarity gauge, Eq. (2.1).In the SM, the production process gg → hhh proceeds via top-quark loop diagrams coupledto Higgs bosons and involves triple and quartic Higgs couplings, where we are obviouslymost interested in the quartic-coupling contribution. The various anomalous couplings gen-erated by Eq. (2.1) modify all contributing loop Feynman graphs, and furthermore directHiggs-gluon couplings can appear which are induced either from the underlying theory di-rectly or emerge from loop-diagram renormalization via operator mixing. Therefore, weredo the calculation of the production process at one-loop order with all new parametersincluded.While the observation of the triple-Higgs process ultimately would determine a partic-ular combination of the new parameters, sizable values for those will definitely also affectother, more easily accessible processes such as Higgs production in association with a topquark and double-Higgs production. Any actual measurement of the EFT parameters willinvolve a fitting procedure that takes all available information into account. Nevertheless,the fact that the quartic Higgs coupling appears only in triple-Higgs production indicatesthat the current process will contribute independent, and potentially essential information.If the triple-Higgs final state is to become observable, we have to allow for EFT pa-rameter values that distort the amplitudes rather drastically, at least in the high-energyor high- P t regions of phase space. We should worry about unitarity of the amplitudesand consistency of the EFT. Physically, we expect a dampening effect from strong rescat-tering of intermediate real top quarks into multiple Higgs bosons. While the effects ofstrong rescattering have extensively been studied in the linear EFT context for vector-boson scattering [114, 115], no results are available for processes involving top quarks andHiggs bosons. Power counting suggests that the dimension-six operators that we considerin this work are affected to a lesser extent than the dimension-eight operators considered inRef. [114]. We also note that in the SM as a weakly interacting theory, the Higgs mechanismtends to suppress electroweak production cross sections by orders of magnitude in relationto the bounds enforced by unitarity, so there is a significant margin for enhancing eventyields in BSM models. For the current study, we take the EFT unmodified over the com-plete parameter space and defer a study of constraints and relations imposed by unitarityto future work. In this section, we describe our calculation of the one-loop induced production amplitude inthe presence of the new parameters of the unitarity-gauge effective Lagrangian, Eq. (2.1).We use the package Madgraph5/aMC@NLO [95, 96] for calculating the loop diagrams and,subsequently, evaluating phase space and generating event samples. To this end, we haveimplemented the model described by Eq. (2.1) as a UFO model file. Note that our choiceof unitarity gauge for the electroweak symmetry does not affect the QCD loop calculation,since the color symmetry is implemented in a renormalizable gauge as usual. The program– 17 –educes the one-loop Feynman integrals to scalar integrals in four dimensions, employingtechniques such as the OPP method [116]. The difference between the D -dimensional and -dimensional expressions that arises in the calculation yields additional rational terms [117].These are identified as R1 terms associated with D -dimensional denominators, and R2 termsassociated with D -dimensional numerators. All R1 terms are automatically generated as abyproduct of the reduction method, while the R2 terms must be calculated manually [118].We have performed this calculation and supplied the results as effective tree-level verticesin the UFO model file [119].In particular, we obtain the R2 terms that amount to contact interactions of a pair ofgluons with one to three Higgs bosons: = − i g s m t δ ab g µ µ π v a (4.1) = − i g s m t δ ab g µ µ π v ( a + a ) (4.2) = − i g s m t δ ab g µ µ π v ( a + 3 a a ) (4.3)The coefficients depend on the EFT parameters a , a , and a . Since these terms arerequired to restore the exact QCD symmetries in the calculated amplitude, by themselvesthey manifestly violate gauge invariance. We have verified that the complete renormalizedone-loop result does respect gauge invariance, a convenient cross-check of the calculation.Besides these loop-induced contributions, gluon fusion into Higgs bosons also receivescontribution from contact interactions between gluons and Higgs bosons that do not ex-ist in the SM. As mentioned above, the inclusion of such contact interactions is requiredby loop-induced operator mixing in the EFT, but could also originate from independentBSM contributions. Technically, we implement them as independent R2 terms, so thatMadgraph5/aMC@NLO will sum them together with loop-induced contribution. gg → hhh and Kinematics The amplitudes of the process pp → hhh are constructed from the Feynman diagrams inFig. 4.4. For illustrating the method, we take the terms that depend on a , κ , κ , λ and λ . Complete results are given in the Appendix B.Each of the following top-quark loop diagrams has a different dependency on the top-– 18 – t t t t t t t t t t t t t t t t t t t -18.8 -16.4 -0.63 -3.16 -1.07 -6.47 0.09 1.12 5.61 13.6 t t t t t t t t t t Table 6 . Numerical values of t . . . t for a 100 TeV hadron collider, for use in Eq. (4.6). Yukawa couplings and Higgs-boson self-couplings, ∝ a ∝ a λ /a κ ∝ a λ /a λ κ /a κ ∝ a λ /a κ (4.4)and the corresponding matrix element is proportional to the following terms M ( gg → hhh ) ∝ f a + f a λ + f a κ + f a λ + f a λ κ + f a κ + f a λ + f a κ , (4.5)where f i are form factors, which after loop integration depend on the external momenta,partly in form of Higgs-boson propagators. After squaring the matrix element and inte-grating over phase space, we arrive at the total cross section which can be parameterisedas below σ ( pp → hhh ) = t a + t a λ + t a κ + t a λ + t a λ κ + t a κ + t a λ + t a κ + t a λ + t a λ κ + t a λ κ + t a κ + t a λ λ + t a λ κ + t a κ λ + t a κ κ + t a λ + t a λ κ + t a λ κ + t a λ κ + t a κ + t a λ λ + t a λ κ + t a λ κ λ + t a λ κ κ + t a κ λ + t a κ κ + t a λ + t a λ κ + t a κ . (4.6)To determine the integrated form factors t . . . t , we calculate the total cross section at480 selected points in the space of parameters ( a , λ , λ , κ , κ ) , then obtain the numericalvalues of these coefficients t . . . t via linear regression. The resulting values are shown inTable 6. The complete set of results that accounts for all effective operators is provided inthe Appendix B. – 19 – t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t Table 7 . Numerical values of integrated form factors
To study the parameter dependence of the cross section of gg → hhh in one of themore specific models introduced in Section 2.3, we simply replace ( a , λ , λ , κ , κ ) by (ˆ r, ˆ x ) according to Table 2, so we obtain Eq. (4.7) below. The numerical results for ˆ t . . . ˆ t are listed in Table 7, and the total cross section has the value σ hhhSM = 5 . fb. (This includesa K factor of 2.0, following Ref. [65]). σ ( pp → hhh ) = σ hhhSM (1 − ˆ x ) (1 + ˆ t ˆ x + ˆ t ˆ r + ˆ t ˆ x + ˆ t ˆ x ˆ r + ˆ t ˆ r + ˆ t ˆ x + ˆ t ˆ x ˆ r + ˆ t ˆ x ˆ r + ˆ t ˆ r + ˆ t ˆ x + ˆ t ˆ x ˆ r + ˆ t ˆ x ˆ r + ˆ t ˆ x ˆ r + ˆ t ˆ r ) (4.7)(a) − . − . . . σ hhh ( f b ) ˆ x ˆ r = 0ˆ r = 1 . r = − . (b) − − . . σ hhh ( f b ) ˆ r ˆ x = 0ˆ x = − . x = 0 . Figure 6 . Dependence of the cross section on (a) ˆ x and (b) ˆ r . The other observable is kept fixed,as indicated by the curve labels. A visual representation of the cross-section dependence on the parameters (ˆ x, ˆ r ) isshown in Fig. 6. We read off that the cross section can exceed the SM value by two ordersof magnitude for reasonable variations of (ˆ x, ˆ r ) . In particular, if ˆ r is fixed (Fig. 6(a)), thecross section increases in the ˆ x < region. In this region, all of the dependent parameters λ , λ , κ , and κ have the same sign, and the derivative couplings can greatly enhancethe cross section. By contrast, in the ˆ x > region, the contributions of λ and λ cancelagainst the terms with κ and κ .The complementary plot Fig. 6(b) shows the dependence on ˆ r for fixed ˆ x . The crosssection changes only mildly with ˆ r as long as ˆ x is small or positive, and for ˆ r > itactually undershoots the SM value. We recall that the dominant contribution to triple-Higgs production originates from the diagram with a pentagon top-quark loop [67]. Thispart does not depend on the Higgs self-couplings which enter the parameter ˆ r . Only if thelatter contributions become sizable and the interference is constructive, we expect a largeenhancement of the cross section. – 20 –a) (h) (GeV) t P F r a c t i on o f E v en t s =0x=+0.5 x=-0.5 x (b) (h) (GeV) t P F r a c t i on o f E v en t s =0x=+0.5 x=-0.5 x (c) (h) (GeV) t P F r a c t i on o f E v en t s =0x=+0.5 x=-0.5 x (d) (GeV) hhh M
400 600 800 1000 1200 1400 F r a c t i on o f E v en t s =0x=+0.5 x=-0.5 x Figure 7 . P t distributions of (a) the leading Higgs, (b) the sub-leading Higgs, and (c) the softestHiggs. In (d), we show the distribution of the invariant mass of the triple-Higgs system. We plotresults for three values of ˆ x : − . , and +0 . , where ˆ r is fixed to zero. Beyond the effect on the total cross section, we may look at distortions of kinemati-cal distributions. The P t distributions of the three Higgs bosons are shown in Fig. 7(a),Fig. 7(b), and Fig. 7(c), for three different values of ˆ x : − . , and +0 . , respectively. Weobserve that the distributions change significantly with respect to the SM reference valueif ˆ x = +0 . , especially in the large P t region. The distortion happens in the parameterregion where the total cross section is not enhanced by a large factor, and it is helpful forthe b l ± j + /E channel since it should improve the reconstruction of the softest jet. For ˆ x = − . the distributions do not change that much, but the analysis would benefit fromthe remarkable cross-section enhancement in that region. We also show the invariant massdistribution of the three Higgs bosons (Fig. 7(d)); this is also modified by the derivativeoperator.In Fig. 8, we show the same observables as in Fig. 7; this time ˆ r is varied and ˆ x is fixedto zero. The distributions do not actually depend on ˆ r , since the parameter affects only λ and λ which are not associated with derivative couplings.– 21 –a) (h) (GeV) t P F r a c t i on o f E v en t s =0r=+0.1 r=-0.1 r (b) (h) (GeV) t P F r a c t i on o f E v en t s =0r=+0.1 r=-0.1 r (c) (h) (GeV) t P F r a c t i on o f E v en t s =0r=+0.1 r=-0.1 r (d) (GeV) hhh M
400 600 800 1000 1200 1400 F r a c t i on o f E v en t s =0r=+0.1 r=-0.1 r Figure 8 . P t distributions of (a) the leading Higgs, (b) the sub-leading Higgs, and (c) the softestHiggs. In (d), we show the distribution of the invariant mass of the triple-Higgs system. We plotresults for three values of ˆ r : − . , and +0 . , where ˆ x is fixed to zero. gg → hhh and single and double-Higgs production A measurement of triple-Higgs production would not occur in a vacuum. Apart from thegenuine quartic Higgs couplings, all parameters of the EFT also enter other Higgs pro-cesses, and the discussion in Sec. 2 suggests that in typical strongly-interacting models,all parameters would receive BSM contributions. We can expect that a measurement orexclusion limit on the triple-Higgs process would add information on top of the amount ofHiggs-physics data gained up to that point, and all results should be combined in the inter-pretation towards BSM physics. Therefore, in this section we study correlations between gg → hhh and gg → h , gg → hh in particular. We phrase the problem in terms of thefollowing questions: • To what extent can a and c be determined from gg → h at the LHC (14 TeV) andat a 100 TeV collider? • To what extent can a , c , λ , κ be determined from gg → hh at the LHC and at a– 22 –rocess σ (14 T eV ) (fb) err.[th] err.[exp] σ (100 T eV ) (fb) err.[th] err. [exp] gg → h . × . − . ±
1% 8 . × . − . ± . gg → hh . +7 . − . < f b +5 . − . ± gg → hhh . +8 . − . − . +4 . − . < f b Table 8 . Cross sections of the processes gg → h , gg → hh and gg → hhh at TeV and
TeV hadron colliders, respectively. The TeV cross section of gg → h is taken from Ref. [120]; theother values are taken from Ref. [57]. The cross sections for gg → h and gg → hh are the NNLOresults, while the cross sections for gg → hhh are the NLO results. gg → h gg → hh gg → hhh Parameters a , c a , c a , c involved - a , c , λ , κ a , c , λ , κ - - a , λ , κ Table 9 . Parameters that contribute to the particular Higgs-production processes.
100 TeV Collider? • To what extent can a , λ , κ be determined from gg → hhh at a 100 TeV collider,including channels not considered in this paper?In Table 8 we quote estimates for the theoretical and projected experimental uncer-tainties for the processes gg → h , gg → hh and gg → hhh . For convenience, we list theparameters that enter these processes in Table 9. The theoretical uncertainties are obtainedby summing the squared uncertainties in PDF, renormalisation scales, and α s , based oncurrent knowledge. For the process gg → h , the experimental uncertainties are mainlystatistical ones which pertain to the Higgs decay h → γγ . The projected experimentalbound for gg → hh at the LHC is taken from the studies of the b ¯ bγγ final state [15] and (cid:96) j + MET [55]. We also quote the expected experimental bound for gg → hhh which isexpected from the analysis of b γ final states [67] at TeV. We emphasize that theseestimates are derived from phenomenological studies; full simulation and experience gainedin the analysis of actual data may change the conclusions significantly, such as in the ex-pectations for the observation of Higgs-pair production at the LHC [74].We first consider Higgs-pair production in gluon fusion, the dominant contribution tothe process pp → hh . The dependence of the total cross section on the EFT parameterscan be written as in Eq. (4.8). Explicitly, we have σ ( pp → hh ) = f a + f a λ + f a κ + f a λ + f a λ κ + f a κ + f a a + f a λ a + f a κ a + f a . (4.8)The numerical values for the coefficients f − f at a TeV hadron collider are providedin Table 10. – 23 – f f f f f f f f f Table 10 . Numerical results for f − f (in pb) at a 100 TeV hadron collider. ˆ f ˆ f ˆ f ˆ f ˆ f -3.63 -0.72 4.32 1.72 0.23 Table 11 . Numerical results for ˆ f − ˆ f at a 100 TeV hadron collider. By substituting ( a , λ , λ , κ ) into (ˆ r, ˆ x ) according to Table 2, we obtain σ ( pp → hh ) = σ hhSM (1 − ˆ x ) (1 + ˆ f ˆ x + ˆ f ˆ r + ˆ f ˆ x + ˆ f ˆ x ˆ r + ˆ f ˆ r ) , (4.9)where σ hhSM = 1 . pb. (We insert a NNLO K factor of 2.17 [22].) Table 11 lists thenumerical values for ˆ f i in this expansion.Turning to single-Higgs production, in Fig. 9 we show projections on the bounds of a and c from the process gg → h at the LHC ( TeV) and at a
TeV collider, respectively.Assuming a measurement result equal to the SM prediction, the allowed ranges for a and c are highly correlated and are confined to be two narrow bands, one of which containingthe SM reference point. The other band is centered on a mirror solution a = − , c = 0 .At the 14 TeV LHC, both theoretical and experimental (statistical) uncertainties arerelevant and have to be taken into account. By contrast, at a 100 TeV collider the main un-certainties will come from theory, while statistical uncertainties will become less than . .Compared with the ultimate LHC bounds, the projected accuracy on the determination of a and c from a 100 TeV collider will improve by a factor . If theoretical and parametricuncertainties can also be improved in the future, we can expect the bounds on a and c totighten even further.The correlation of the parameters in Fig. 9a and Fig. 9b indicates an obvious shortcom-ing of a simple cross-section analysis. The degeneracy of solutions can be lifted by examiningthe kinematics of the Higgs boson in the final state, and by adding the measurement of gg → hh , as we will explain later.A future e + e − collider can improve the measurement of a and c beyond the reach ofthe LHC. This is mainly due to QCD entering the processes only beyond leading order, sostatistical uncertainties will likely dominate. For instance, at the CEPC a combination ofthe parameters c and a can be determined within for a collision energy √ s = 240 − GeV and an integrated luminosity 5 ab − [77]. At the ILC, the parameter a can bedetermined within for √ s = 500 GeV and an integrated luminosity of 1 ab − , viathe process e + e − → t ¯ th [121]. We add the result that at a 100 TeV hadron collider withintegrated luminosity 30 ab − , a can be further constrained by the measurement of tth towithin [88].In Fig. 10, we display the expected bounds in the a - c plane for both the LHC 14 TeVand a 100 TeV collider. Adding in gg → hh , the linear degeneracy in the a - c plane that– 24 –a) a-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 c -4-3-2-101234
49 pb 49 pb60 pb 60 pb SM (b) a-1.5 -1 -0.5 0 0.5 1 1.5 c -3-2-10123
762 pb 762 pb842 pb 842 pbSM (c) a-1.5 -1 -0.5 0 0.5 1 1.5020406080100120 SMUpper boundLower bound (d) a-1 -0.5 0 0.5 1 s SMUpper boundLower bound (e) c-2.5 -2 -1.5 -1 -0.5 0 0.5020406080100 SMUpper boundLower bound (f) c-2.5 -2 -1.5 -1 -0.5 0 0.502004006008001000120014001600 SMUpper boundLower bound
Figure 9 . Upper row: correlations between a and c extracted from the process gg → h for theLHC 14 TeV (a) and for a 100 TeV pp collider (b), respectively. Middle row: individual bounds on a (c) and c (e) for the LHC 14 TeV (the total uncertainties are assumed to be ), respectively.Lower row: the analogous results for a 100 TeV pp collider (the total uncertainties are assumed tobe ). – 25 –a) a-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 c -10-50510
140 fb SMMirror (b) a-1.5 -1 -0.5 0 0.5 1 1.5 c -8-6-4-20246810 Figure 10 . Projected exclusion bounds in a - c plane extracted from the process gg → hh forboth (a) the LHC 14 TeV and (b) a 100 TeV collider. In each plot, the straight lines indicatethe solutions for the cross section of gg → h , assuming a measurement consistemt with the SM.The mirror region of the SM point, the solution with a = − and c = 0 , is denoted by a circle,respectively. No. a c σ ( gg → h ) [pb] σ ( gg → hh ) [fb] σ ( gg → hhh ) [fb]1 0.99 -0.01 771 1710 5.902 -0.86 1.94 839.6 1685 29.73 0.78 -1.82 763 1747 6.234 -0.66 -0.37 817.8 1690 5.74 Table 12 . Four representative points in the four parameter regions and the corresponding crosssections for Higgs production at a 100 TeV collider. follows from the measurement of gg → h is cut down to a limited region, even for the LHC14 TeV. With a 100 TeV collider, the degenerate solutions for both gg → h and gg → hh shrink to four small regions, cf. TABLE. 12, and it becomes possible to exclude the mirrorsolution.In this situation, we may include the cross section of the process gg → hhh anddistinguish the second point from the rest. The second point has a production rate for theprocess gg → hhh that is large enough to be observed. (Here we would like to remind thereader that when we examine the correlations of a and c , we set the other free parametersto their SM values.)We now extend the study to the other parameters that enter gg → hh . In Fig. 11 wedisplay the expected LHC bounds for those in four planes, namely a - λ , c - λ , c - λ , and κ - λ . All bounds are derived by requiring the cross section of gg → hh to be smaller than140 fb, so the points inside the exclusion bounds will remain allowed by the LHC 14 TeVdata, if no deviation from the SM is detected. The SM reference points are also shown.When we examine the correlations between two parameters, here and in all later discussionswe set the remaining parameters equal to their SM values.– 26 –a) a-2 -1 0 1 2 3 l -8-6-4-202468
140 fbSM (b) c-10 -5 0 5 10 l -4-2024681012
140 fb 140 fbSM (c) c-3 -2 -1 0 1 2 3 l -4-202468
140 fb SM (d) k -6 -4 -2 0 2 4 6 l -10-50510
140 fb SM
Figure 11 . Projected exclusion bounds in two-parameter planes between ( a , c , c , κ ) and λ ,extracted from the process gg → hh at the LHC 14 TeV. If the coefficient values are equal to theSM prediction, parameter values inside the contours are still allowed by the measurement. Theexclusion bounds correpond to a limit of 140 fb for the cross section. In Fig. 12, we draw the analogous bounds that we can expect from analyzing gg → hh in TeV pp data. We assume that the cross section of gg → hh can be measured to aprecision of . This uncertainty results from combining theoretical and experimentaluncertainties. The allowed parameter regions shrink considerably and become pinchedbetween two contours, in each plot. Comparing Fig. 11 and Fig. 12, we conclude that a 100TeV collider can significantly improve the precision on a , c , κ , and λ .We can also individually project out single-parameter bounds for each of these fourparameters, shown in Fig. 13. In this approach, the parameters a , κ , and λ can bedetermined with a precision close to . The two-fold ambiguities in the solutions couldpossibly be removed by using the kinematics of final states, as demonstrated in Ref. [55].The parameter c can be determined within the range [ − . , . .Finally, we add the exclusion bounds that we can expect from the process gg → hhh .There are three parameters, namely a , κ and λ , which only contribute to the process– 27 –a) a-1 -0.5 0 0.5 1 1.5 l -2-1012345 (b) c-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 l (c) c-1 -0.5 0 0.5 1 l (d) k -3 -2 -1 0 1 2 3 l -4-20246 Figure 12 . Projected exclusion bounds in two-parameter planes between ( a , c , c , κ ) and λ ,extracted from the process gg → hh a 100 TeV pp collider. If the coefficient values are equal to theSM prediction, parameter values between the two contours are still allowed by the measurement.The exclusion bounds correspond to a total error of on σ ( gg → hh ) , theoretical and experimentaluncertainties combined. gg → hhh . We present two-dimensional bounds for all pairs of these three independentfour-point couplings in Figs. 14(a)–14(c). The corresponding one-dimensional bounds aregiven in Figs. 14(b)–14(d). The parameter a can be constrained to the range [-0.8,1.2];the parameter κ can be constrained to the range [-2.3, 1.5]. As shown already in Ref. [67],the parameter λ can only be determined within a quite wide range [-13,20]. Clearly, λ isthe most difficult parameter to measure in this framework.These results have to be combined with the parameter exclusion regions derived from gg → hh . In Fig. 15, we show the correlations between a and a , c , κ , and λ . In these plots,we overlay the limits that follow from Higgs-pair production, presented in Fig. 13, to theexclusion contours that follow from triple-Higgs production. The SM prediction is displayedfor reference. Clearly, the triple-Higgs results yield weaker constraints, but they are nev-ertheless sensitive to a different combination of parameters and thus cut off part of thetwo-parameter exclusion regions. As an example, we note that adding in gg → hhh can– 28 –a) a-0.2 0 0.2 0.4 0.6 0.8 1 1.2 ( pb ) s SM Upper boundLower bound (b) c-0.4 -0.2 0 0.2 0.4 0.6 ( pb ) s SM Upper boundLower bound (c) k -0.5 0 0.5 1 1.5 2 ( pb ) s SM Upper boundLower bound (d) l ( pb ) s SM Upper boundLower bound
Figure 13 . Projected one-parameter exclusion bounds for a , c , κ and λ , extracted from theprocess gg → hh . If the coefficient values are equal to the SM prediction, parameter values betweenthe upper and lower bounds are allowed by the measurement. The exclusion bounds correspond toa total error of on σ ( gg → hh ) , theoretical and experimental uncertainties combined. help in resolving a two-fold ambiguity in the κ - a plane, cf. Fig. 15(c).Similarly, in Fig. 16, we show the correlations between κ and a , c , κ , and λ . InFig. 17, we show the correlations between λ and a , c , κ , and λ . In Fig. 17(c), the κ - λ plane, including triple-Higgs production it becomes possible to separate the κ = 0 and κ (cid:54) = 0 regions. Finally, we can adapt the above results to more specific scenarios, as we have introducedabove in Sec. 2. In any given model, the parameters of the unitarity-gauge Lagrangian (2.1)can be related to the original model parameters. In particular, for a model with a smallset of independent parameters, we can recast the analysis to a concrete prediction for theexpected sensitivity to this model, in a straightforward way.– 29 –a) l -30 -20 -10 0 10 20 30 40 a -3-2-10123
30 fbSM (b) a-1.5 -1 -0.5 0 0.5 1 1.5 2 ( f b ) s SMUpper bound (c) l -30 -20 -10 0 10 20 30 40 k -3-2-101234
30 fb SM (d) k -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ( f b ) s SMUpper bound (e) a-3 -2 -1 0 1 2 3 k -3-2-101234
30 fb 30 fbSM (f) l -20 -15 -10 -5 0 5 10 15 20 25 30 ( f b ) s SMUpper bound
Figure 14 . Projected two-parameter (left) and one-parameter (right) exclusion bounds, extractedfrom the process gg → hhh . The left column shows the two-parameter planes a - λ , κ - λ , and a - κ , while the right column displays a , κ , and λ . – 30 –a) a-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 a -1-0.500.511.5
30 fbSM (b) a-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 c -2-1.5-1-0.500.511.522.53
30 fbSM (c) a-4 -3 -2 -1 0 1 2 k -1.5-1-0.500.511.522.5
30 fb SM (d) a-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 l -3-2-101234567
30 fbSM
Figure 15 . Projected two-parameter exclusion bounds, extracted from the process gg → hhh .The straight-line pairs indicate the exclusion bounds extracted from gg → hh . The plots showtwo-parameter correlations between a and a , c , κ , and λ . In the generic dimension-six SILH Lagrangian, there are four free parameters relevant tothis stody, denoted by C y = c y ξ , C H = c H ξ , C = c ξ , and c . For both MCHM4 andMCHM5 as models which reduce to SILH at low energy, there are only two independentparameters, ξ and c . We apply the above analysis to those and conclude that at a 100TeV collider, data from gg → h and gg → hh significantly constrain the allowed parameterspace. This is demonstrated by Figs. 18(a)–18(b). Data from gg → h will result in anexclusion region bounded by two lines in the ξ − c plane, while data from gg → hh willfurther reduce the allowed parameter space to two small spots in the plane.To illustrate the added value from triple-Higgs production, we consider two benchmarkpoints for MCHM4 and MCHM5 in Table 13. For both benchmark points we obtain alarge cross section for the gg → hhh process, actually 80 and 55 times larger than thatof the SM, respectively. These are examples of benchmark models that can not just bedetected, but also be distinguished from each other at a 100 TeV collider, given the result– 31 –a) k -3 -2 -1 0 1 2 3 4 a -1-0.500.511.5
30 fbSM (b) k -3 -2 -1 0 1 2 3 4 c -2-1.5-1-0.500.511.522.53
30 fbSM (c) k -8 -6 -4 -2 0 2 4 k -1.5-1-0.500.511.522.5
30 fb SM (d) k -4 -3 -2 -1 0 1 2 3 l -202468
30 fbSM
Figure 16 . Projected two-parameter exclusion bounds, extracted from the process gg → hhh .The straight-line pairs indicate the exclusion bounds extracted from gg → hh . The plots showtwo-parameter correlations between κ and a , c , κ , and λ . No. ξ c σ ( gg → h )[ pb ] σ ( gg → hh ) [fb] σ ( gg → hhh ) [fb]MCHM4 0.97 0.48 764 1618 321MCHM5 -0.20 -0.30 817 1854 122GHM ˆ x = 0 .
02 ˆ r = 3 .
816 1786 37.78
Table 13 . Three representative points for the models MCHM4, MCHM5, and GHM, respectively,and the corresponding cross sections for Higgs production. that the threshold cross section value of gg → hhh for being sensitive to new physics is30 fb or smaller [67]. (We add the caveat that the benchmark point of MCHM4 could beindependently excluded by incorporating electroweak precision data due to a large ξ value.) The Gravity-Higgs model has only two free parameters, ˆ x and ˆ r . The analysis is straight-forward, since in this model, single-Higgs production gg → h depends only on a single BSM– 32 –a) l -30 -20 -10 0 10 20 30 40 a -1.5-1-0.500.511.522.5
30 fbSM (b) l -30 -20 -10 0 10 20 30 40 c -1.5-1-0.500.511.522.5
30 fbSM (c) l -30 -20 -10 0 10 20 30 40 k -1.5-1-0.500.511.522.5
30 fbSM (d) l -30 -20 -10 0 10 20 30 40 l -3-2-101234567
30 fbSM
Figure 17 . Projected two-parameter exclusion bounds, extracted from the process gg → hhh .The straight-line pairs indicate the exclusion bounds extracted from gg → hh . The plots showtwo-parameter correlations between λ and a , c , κ , and λ . parameter ˆ x , while double-Higgs production constrains the second parameter ˆ r . The crosssection of gg → hhh is completely determined once ˆ x and ˆ r are constrained.The expected LHC exclusion contours in the ˆ x − ˆ r plane are depicted in Fig. 19(a).From the process gg → h we obtain a narrow band which is cut off by adding in the resultfrom measuring gg → hh . The latter constrains the parameter ˆ r down to the range from-1.8 to 5.0 if we assume that parameter space with a cross section of σ ( gg → hh ) largerthan 120 fb can be excluded.Fig. 19(b) shows the analogous results for a TeV collider. The measurements of gg → h and gg → hh let the allowed parameter space shrink substantially compared to theLHC results. From these measurements alone, the value of ˆ r is extracted with a two-foldambiguity. However, for the solution with the larger value of ˆ r ( ˆ r ≈ ) the cross section of gg → hhh is 5 times larger than near ˆ r = 0 due to the λ dependence in the cross section,cf. the benchmark points in Table 13. A measurement of triple-Higgs production couldtherefore eliminate one of the solutions. – 33 –a) ξ
1 0.5 0 0.5 1 1.5 c SM h → Bounds from g g h h → Bounds from g g =100 TeV, MCHM4s (b) ξ c SM h → Bounds from g g h h → Bounds from g g =100 TeV, MCHM5s
Figure 18 . Projected two-parameter exclusion bounds in the ξ − c plane, extracted from theprocesses gg → h and gg → hh at a 100 TeV collider for the models MCHM4 and MCHM5,respectively. (a) x 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 r SM h → Bounds from g g h h → Upper bound from gg =14 TeVs (b) x 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 r SM h → Bounds from g g h h → Bound from gg =100 TeVs
Figure 19 . Comparison of the projected exclusion bounds on ˆ x − ˆ r from the LHC 14 TeV (left)and a 100 TeV collider (right). In this paper, we have explored the discovery potential for triple-Higgs production via the b l ± j + /E decay channel at a TeV hadron collider. Despite the extremely smallcross section of the signal process in the SM, the parton-level study demonstrates thatthe signal can be detected in principle. We find that the mT variable is useful to findthe correct combinations of the visible objects that originate from Higgs boson decay, andto suppress background efficiently. However, once hadronic events and detector effects areproperly accounted for, extraction of the SM signal becomes a real challenge. The two mainproblems are that (1) the transverse momentum of the softest jet from Higgs boson decayassumes values of about GeV, which makes it difficult to reconstruct; (2) since there aresix jets in the final state, the lepton isolation criteria reject most of the signal events.– 34 –ew-physics effects may enhance the cross section significantly and distort kinematicaldistributions, so if the SM is not the true theory, observation of the triple-Higgs productionprocess becomes more likely. A measurement would then amount to a determination ofBSM parameters. To discuss this possibility in a suitably generic framework, we haveemployed an EFT approach and added a set of dimension-6 operators that is taylored torepresent new physics in the Higgs sector. In particular, we find that a sizable coefficient fora derivative operator can modify the kinematical distributions of the visible objects suchthat a reconstruction of the triple-Higgs signal becomes feasible. Using this information,we have investigated the potential of such a measurement to improve on knowledge whichcan already be gathered from single and double-Higgs production data. It turns out thatwhile those processes are generally more powerful in constraining BSM parameters, thetriple-Higgs signal nevertheless reduces the allowed parameter space and in some cases caneliminate ambiguities in the parameter determination.We would like to point out that our EFT approach does not incorporate any new BSMparticles which may be discovered in the future. Our study also treats the SM particles,especially the Higgs, as elementary degrees of freedom at the energy scale relevant for 100TeV collider. We thus assume that, if the SM Higgs is a composite particle, the compos-iteness scale is higher than a few TeV, at least. If this assumption turns out to be invalid,i.e., qualitatively new phenomena become observable at lower energy, our conservative ap-proach would have to be revised to include explicit model-dependent BSM effects in thecalculation.
A Derivation of parameters relations
Expanding the SILH Lagragian in unitarity gauge and introducing the physical Higgs scalar h , the derivative operator induces the following term c H f ∂ µ (cid:16) H † H (cid:17) ∂ µ (cid:16) H † H (cid:17) → c H f ( v + h ) ∂ µ h∂ µ h. (A.1)In effect, the kinetic term of the Higgs field is modified to L kin = 12 (1 + c H ξ ) ∂ µ h∂ µ h, (A.2)where ξ ≡ v /f . This means that the Higgs field should be rescaled by h → ζh , where ζ = (1 + c H ξ ) − / . Eq. (A.1) induces two further derivative operators c H ξv ζ h∂ µ h∂ µ h, (A.3) c H ξ v ζ h ∂ µ h∂ µ h, (A.4)which translate into the relations for κ and κ in Table 1.To find the relations for λ and λ , we have to consider the Higgs potential amendedby the c term: V (cid:16) H † H (cid:17) = − µ (cid:16) H † H (cid:17) + λ (cid:16) H † H (cid:17) + c λf (cid:16) H † H (cid:17) (A.5)– 35 –n this case, the VEV is given by − µ + 2 λv + 34 c ξλv = 0 , (A.6)and the corresponding Higgs mass is defined by m h = − µ + 32 λv + 158 c ξλv . (A.7)After combining Eq. (A.6) and Eq. (A.7) and rescaling the Higgs field, we obtain a modifiedHiggs mass m h = 2 λv (cid:18) c ξ (cid:19) ζ . (A.8)With these definitions of Higgs field and Higgs mass, we can write down the h and h terms: m h v ζ c ξ/
21 + 3 c ξ/ h , (A.9) m h v ζ c ξ/
21 + 3 c ξ/ h , (A.10)which yield the relations for λ and λ in Table 1.Finally, we consider the operator c y y f f H † H ¯ f L Hf R , which generates a term c y y f f H † H ¯ f L Hf R → c y y f √ f ( v + h ) ¯ f f. (A.11)This term modifies the fermion mass by m f = y f v √ (cid:18) − c y ξ (cid:19) . (A.12)After this redefinition, we obtain the following Higgs-fermion interaction operators − m f v ζ − c y ξ/ − c y ξ/ h ¯ f f, (A.13) m f v ζ c y ξ/ − c y ξ/ hh ¯ f f, (A.14) m f v ζ c y ξ/ − c y ξ/ hhh ¯ f f, (A.15)which yield the relations for a , a and a in Table 1.An alternative way of deriving such relations is performing a non-linear transformation h → h − c H ξ ( h + h v + h v ) [8]. We compare the results of both approaches, up to O ( c H ξ ) terms, in Table 14 (the coefficients of all other effective operators are set to zero).Despite these differences, both transformations necessarily yield the same cross section upto O ( c H ξ ) . However, in the non-linear transformation approach, higher-order terms suchas O ( ξ ) are much more complex than in the rescaling approach, and we have to deal withvertices such as tthh and tthhh even if c y is zero. Therefore, we adopt rescaling rather thanthe non-linear transformation method for defining our phenomenological parameters.– 36 –escaling Non-linear a − c H ξ − c H ξa − c H ξa − c H ξc c λ − c H ξ − c H ξλ − c H ξ − c H ξκ − c H ξ κ − c H ξ Table 14 . Comparison of parameters relations between field rescaling and non-linear transforma-tion. K h C ,h = a C ,h = a c C ,h = c
14 TeV 2.85 F h = 19 . F h = 36 . F h = 17 .
100 TeV 2.24 F h = 357 . F h = 687 . F h = 332 . Table 15 . The numerical value of F h − F h at hadron colliders in Eq. (B.1). B Numerical cross sections of gg → h , gg → hh , and gg → hhh The cross section for gg → h can be put as σ ( gg → h ) = K h × ( (cid:88) i =1 F hi C i,h ) , (B.1)where the integrated form factors F hi and the coefficients C i,h are given in Table 15, and K h denotes the K-factor. The unit of F hi is pb.It is found that values of F hi given in Table (15) do produce a positive definite crosssection of gg → h .The cross section for gg → hh at 14 TeV LHC and a 100 TeV collider can be put as σ ( gg → hh ) = K h × ( (cid:88) i =1 F hi C i, h ) , (B.2)where the integrated form factors F hi and the coefficients C i, h are given in Table 16 andTable 17, and K h denotes the K-factor, which is equal to . for the LHC 14 TeV and . for the 100 TeV collision, respectively. The unit of F hi in these two tables is fb.The largest absolute value goes to the coefficient F h , which is . and . foreither 14 TeV or 100 TeV cases. The minimal absolute value goes to the coefficient F h ,which is . for 14 TeV and . for 100 TeV case.– 37 – h C h C h F h = 36 . a F h = − . a a F h = 44 . a F h = − . a c F h = 43 . a c F h = 23 . c F h = − . a κ F h = 56 . a a κ F h = − . a c κ F h = 29 . a c κ F h = 28 . a c κ F h = 29 . c c κ F h = 18 . a κ F h = 19 . a c κ F h = 9 . c κ F h = − . a λ F h = 24 . a a λ F h = − . a c λ F h = 14 . a c λ F h = 14 . a c λ F h = 11 . c c λ F h = 17 . a κ λ F h = 21 . a c κ λ F h = 8 . c κ λ F h = 4 . a λ F h = 6 . a c λ F h = 2 . c λ Table 16 . The numerical value of F h − F h at the LHC 14 TeV in Eq. (B.2). C h C h C h F h = 1565 . a F h = − . a a F h = 2274 . a F h = − . a c F h = 1790 . a c F h = 2407 . c F h = − . a κ F h = 2781 . a a κ F h = − . a c κ F h = 1174 . a c κ F h = 1202 . a c κ F h = 2651 . c c κ F h = 866 . a κ F h = 797 . a c κ F h = 745 . c κ F h = − . a λ F h = 1014 . a a λ F h = − . a c λ F h = 567 . a c λ F h = 604 . a c λ F h = 510 . c c λ F h = 679 . a κ λ F h = 817 . a c κ λ F h = 342 . c κ λ F h = 172 . a λ F h = 232 . a c λ F h = 88 . c λ Table 17 . The numerical value of F h − F h at a 100 TeV collider in Eq. (B.2). Compared with those of the 14 TeV case, most of coefficients can be enhanced by afactor around 40 or so for the 100 TeV case. Among them, the coefficients F h , F h and F h have the largest enhancements from 14 TeV to 100 TeV collisions, and they are . , . , and . , respectively.In order to guarantee the positive and definite results of the cross section of all pointsin the parameter space, the contribution of b quark should be removed from the diagrams.Otherwise, a more general parameterisation of the cross section should be introduced. Fur-thermore, we have used more than 5,000 points in the parameter space of a , c , κ , and λ to determine these F hi after taking into account the constraints on parameters a and c from the projected precision in the measurement of σ ( gg → h ) . The positivity and def-initeness of the cross sections are examined to be hold in a random scan in the parameterspace of a , c , κ , and λ with a total number of points . If a and c can significantlydeviate from the values of the SM, these results might not be valid anymore.The cross section for gg → hhh at a 100 TeV collider can be put as σ ( gg → hhh ) = K h × ( (cid:88) i =1 F hi C i, h ) , (B.3)– 38 – h C h C h F h = 7 . a F h = − . a a F h = 31 . a a F h = − . a a F h = − . a a a F h = 11 . a F h = − . a κ F h = 38 . a a κ F h = − . a a κ F h = − . a a κ F h = 40 . a a κ F h = − . a c κ F h = 68 . a a c κ F h = − . a c κ F h = 12 . a κ F h = − . a a κ F h = 35 . a κ F h = 28 . a a κ F h = − . a c κ F h = 46 . a a c κ F h = − . a c κ F h = 44 . a c κ F h = − . a c κ F h = 935 . c κ F h = − . a κ F h = 49 . a a κ F h = 30 . a c κ F h = − . a c κ F h = − . a c κ F h = 1244 . c c κ F h = 17 . a κ F h = − . a c κ F h = 414 . c κ F h = − . a κ F h = − . a a κ F h = 12 . a a κ F h = − . a c κ F h = 20 . a a c κ F h = − . a c κ F h = − . a κ κ F h = 21 . a a κ κ F h = 13 . a c κ κ F h = − . a c κ κ F h = − . a c κ κ F h = 609 . c c κ κ F h = 14 . a κ κ F h = − . a c κ κ F h = 406 . c κ κ F h = 3 . a κ F h = − . a c κ F h = 99 . c κ F h = − . a λ F h = 19 . a a λ F h = − . a a λ F h = − . a a λ F h = 11 . a a λ F h = − . a c λ F h = 13 . a a c λ F h = − . a c λ F h = 14 . a κ λ F h = − . a a κ λ F h = 22 . a κ λ F h = 9 . a a κ λ F h = − . a c κ λ F h = 10 . a a c κ λ F h = 0 . a c κ λ F h = 9 . a c κ λ F h = 2 . a c κ λ F h = 73 . c κ λ F h = − . a κ λ F h = 35 . a a κ λ F h = 7 . a c κ λ F h = 3 . a c κ λ F h = 4 . a c κ λ F h = 97 . c c κ λ Table 18 . The numerical value of F − F at 100TeV hadron collider in Eq. (B.3). where the integrated form factors F hi and the coefficients C i, h are given in Table 18 andTable 19, and K denotes the K-factor which is taken as . . The unit of F hi is fb. We haveused more than 12,000 points to determine these F hi .The largest absolute coefficient is F h . In contrast, the smallest absolute coefficientsare F h and F h .After taking into account the constraints on parameters a and c from the projectedprecision data of σ ( gg → h ) and the constraints on parameters a , c , λ and κ from theprojected precision data of σ ( gg → hh ) , the positivity and definiteness of the cross sectionsare examined to be hold in a random scan in the parameter space of a , λ and κ with atotal number of points . Acknowledgments
We would like to thank Ning Chen for contributing to this project at an early stage, and– 39 – h C h C h F h = 13 . a κ λ F h = 4 . a c κ λ F h = 32 . c κ λ F h = − . a κ λ F h = 6 . a a κ λ F h = 0 . a c κ λ F h = 0 . a c κ λ F h = − . a c κ λ F h = 22 . c c κ λ F h = 5 . a κ κ λ F h = 1 . a c κ κ λ F h = 14 . c κ κ λ F h = 4 . a λ F h = − . a a λ F h = 5 . a λ F h = 0 . a a λ F h = − . a c λ F h = 0 . a a c λ F h = 0 . a c λ F h = 1 . a c λ F h = 2 . a c λ F h = 7 . c λ F h = − . a κ λ F h = 11 . a a κ λ F h = 0 . a c κ λ F h = 3 . a c κ λ F h = 3 . a c κ λ F h = 12 . c c κ λ F h = 5 . a κ λ F h = 3 . a c κ λ F h = 5 . c κ λ F h = 0 . a κ λ F h = 0 . a c κ λ F h = 0 . c κ λ F h = − . a λ F h = 1 . a a λ F h = − . a c λ F h = 0 . a c λ F h = 0 . a c λ F h = 0 . c c λ F h = 1 . a κ λ F h = 0 . a c κ λ F h = 0 . c κ λ F h = 0 . a λ F h = 0 . a c λ F h = 0 . c λ F h = 0 . a λ F h = − . a a λ F h = 0 . a a λ F h = − . a c λ F h = 0 . a a c λ F h = 0 . a c λ F h = − . a κ λ F h = 1 . a a κ λ F h = 0 . a c κ λ F h = 0 . a c κ λ F h = 0 . a c κ λ F h = 3 . c c κ λ F h = 1 . a κ λ F h = 0 . a c κ λ F h = 2 . c κ λ F h = 0 . a κ λ F h = 0 . a c κ λ F h = 1 . c κ λ F h = − . a λ λ F h = 0 . a a λ λ F h = 0 . a c λ λ F h = 0 . a c λ λ F h = 0 . a c λ λ F h = 0 . c c λ λ F h = 0 . a κ λ λ F h = 0 . a c κ λ λ F h = 0 . c κ λ λ F h = 0 . a λ λ F h = 0 . a c λ λ F h = 0 . c λ λ F h = 0 . a λ F h = 0 . a c λ F h = 0 . c λ Table 19 . The numerical value of F − F at 100TeV hadron collider in Eq. (B.3). Jiunn-Wei Chen for discussions. Z.J. Zhao is supported by a Nikolai Uraltsev Fellowshipof the Center for Particle Physics, University of Siegen. S.C. Sun is supported by theMOST of Taiwan under grant number of 105-2811-M-002-130 and the CRF Grants of theGovernment of the Hong Kong SAR under HUKST4/CRF/13G. Q.S. Yan and X.R. Zhaoare supported by the Natural Science Foundation of China under the grant NO. 11175251and NO. 11575005. X.R. Zhao is also partially supported by the European Union as partof the FP7 Marie Curie Initial Training Network MCnetITN (PITN-GA-2012-315877).
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