Determining the Shape of Higgs Potential at Future Colliders
Pankaj Agrawal, Debashis Saha, Ling-Xiao Xu, Jiang-Hao Yu, C.-P. Yuan
DDetermining the Shape of the Higgs Potential at Future Colliders
Pankaj Agrawal a,b , Debashis Saha a,b , Ling-Xiao Xu c , , Jiang-Hao Yu d,e , C.-P. Yuan f . a Institute of Physics, Sainik School Post, Bhubaneswar 751 005, India b Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India c Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing100871, China d CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing100190, P. R. China e School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049,P.R. China f Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Abstract
Although the Higgs boson has been discovered, its self-couplings are poorly constrained. This leavesthe nature of the Higgs boson undetermined. Motivated by different Higgs potential scenarios otherthan the Landau-Ginzburg type in the standard model, we systematically organize various new physicsscenarios – elementary Higgs, Nambu-Goldstone Higgs, Coleman-Weinberg Higgs, and Tadpole-inducedHiggs, etc. We find that double-Higgs production at the 27 TeV high energy LHC can be used todiscriminate different Higgs potential scenarios, while it is necessary to use triple-Higgs production at afuture 100 TeV proton-proton collider to fully determine the shape of the Higgs potential. [email protected] [email protected] [email protected] [email protected], corresponding author [email protected] a r X i v : . [ h e p - ph ] A p r ontents t ¯ thh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2.2 Interference effects with t ¯ thh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Model Discrimination and λ Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 t ¯ thh or t ¯ thhh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2.2 Interference with t ¯ thh and t ¯ thhh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Shape Determination and λ Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Introduction
After a long wait of about half a century, in 2012, the Higgs boson was discovered at the Large HadronCollider (LHC) by the CMS and ATLAS Collaborations. With the discovery of this missing piece, all theparticles of the standard model (SM) have now been discovered. With the measured value of the Higgsboson mass, all the parameters in the SM are now known. The main goal of the LHC machine now is tomeasure the properties and interactions of the Higgs boson, as well as to look for signatures of possiblenew physics beyond the SM. Until now, direct searches for evidence of new physics (NP) have not yieldedanything of significance. This has pushed the new physics scale to be around TeV. On the other hand,precision measurements on various SM processes provide us with an indirect way to probe new physics. TheHiggs boson couplings to the gauge bosons and the SM fermions have been measured at the LHC throughvarious production processes and decay modes. However, the Higgs self-couplings are not yet determined atthe end of the Run-2 of the LHC [1–8].The self-couplings of the Higgs boson, including the trilinear and quartic Higgs couplings, are still mys-teries. Experimentally, the trilinear and quartic Higgs couplings can be directly measured using double- andtriple-Higgs production processes pp → hh and pp → hhh , respectively, at hadron colliders. The ATLASand CMS Collaborations have been looking for the hh production signal in the data collected so far at theLHC. This data can put a very loose bound on the trilinear Higgs coupling. The hhh production signal hasnot yet been investigated with the Run-2 data. It is quite challenging to measure the Higgs self-couplingsat the LHC, and this provides a strong motivation for building future high energy colliders.Theoretically, there are still many unknowns about the Higgs boson, such as the nature of the Higgsboson, the origin of electroweak symmetry breaking (EWSB), the shape of the Higgs potential, and thestrength of the electroweak phase transition, etc. All these questions can only be addressed after the Higgsself-couplings are determined. So far the Higgs self-couplings are not tightly constrained. The Higgs potentialcan be very different from the Landau-Ginzburg type in the SM. In this work, we systematically investigatevarious classes of new physics scenarios based on different types of Higgs potential. To be specific, we considerthe following Higgs scenarios: • Elementary Higgs boson, in which the Higgs boson is taken as an elementary scalar with rescaledself-couplings. The Higgs mass parameter is negative and thus triggers EWSB. • Nambu-Goldstone Higgs, in which the Higgs boson is taken as a pseudo Nambu-Goldstone (PNG)boson [9, 10] emerging from strong dynamics at a high scale (see Refs. [11–13] for comprehensivereviews). • Coleman-Weinberg (CW) Higgs, in which EWSB is triggered by renormalization group (RG) runningeffects [14–16] with classical scale invariance. • Tadpole-induced Higgs, in which EWSB is triggered by the Higgs tadpole [17,18], and the Higgs bosonmass parameter is taken to be positive.In general, the Higgs potentials could be organized according to their analytic structure. The key structure3f the Higgs potential in each scenario is as follows: V ( H ) (cid:39) − m H † H + λ ( H † H ) + c λ Λ ( H † H ) , Elementary Higgs − a sin ( √ H † H/f ) + b sin ( √ H † H/f ) , Nambu-Goldstone Higgs λ ( H † H ) + (cid:15) ( H † H ) log H † Hµ , Coleman-Weinberg Higgs − κ √ H † H + m H † H, Tadpole-induced Higgs (1.1)where f denotes the decay constant of the NG Higgs boson, and µ denotes the renormalization scale incase EWSB is triggered by radiative corrections, m , λ, c , Λ , a, b, (cid:15), κ are dimensionful or dimensionlessparameters in each new physics scenario. The shapes of the Higgs potential are schematically illustrated inFig. 1, respectively. In both the elementary and Nambu-Goldstone Higgs cases, the Higgs potential couldbe expanded in the powers of H † H , which could recover the Landau-Ginzburg effective theory descriptionif a truncation on the series provides a good approximation. The decoupling limit of these two scenarioscorresponds to the case when new physics sets in at a much higher energy scale than the EW scale. However,such kind of decoupling limit does not exist in either the Coleman-Weinberg Higgs or the Tadpole-inducedHiggs scenario. In all the above cases, the trilinear and quartic Higgs couplings could be very different fromthose in the SM.Figure 1: The shapes of Higgs potential for various scenarios studied in this work.All the above mentioned scenarios can be described in an effective field theory (EFT) framework. Oneof the most popular EFT frameworks is the SMEFT [19–21], which assumes new physics decouple at ahigh energy scale, and EW symmetry is in the unbroken phase. The SMEFT is suitable for describing theelementary Higgs and the Nambu-Goldstone Higgs scenarios, when the Higgs non-linearity effect can beneglected [22]. On the other hand, the Coleman-Weinberg Higgs and the Tadpole-induced Higgs scenarioscannot be described within the SMEFT framework due to the existence of non-decoupling effects. Hence,to compare all the four NP scenarios in one theory framework, we utilize the EFT framework in the brokenphase of EW symmetry, which is known as the Higgs EFT [23–29]. Adopting the Higgs EFT framework, wesummarize the general Higgs effective couplings in various scenarios, and parameterize the scaling behaviorof multi-Higgs production cross sections at various high energy hadron colliders.In this work, we study how to utilize the measurements of the hh and hhh production rates in hadroncollision to discriminate the above mentioned scenarios. The hh production process, via gluon-gluon fusion,has been extensively studied in the literature for measuring the trilinear Higgs boson coupling [30–49] andthe t ¯ thh couplings in the EFT framework [50–52], and for probing various new physics models [53–64]. Inparticular, probing the composite Higgs models via studying the hh production process has been studied4n [62–64]. For completeness, we have reproduced some of the results shown in the literature, but withsomewhat different emphasis on its analysis so as to compare the predictions on the hh production ratesfrom the above mentioned Higgs potential scenarios side-by-side.We compare total cross sections, kinematical distributions, and various interference effects at the 14TeV high luminosity LHC (HL-LHC), the 27 TeV high energy LHC (HE-LHC), and a future 100 TeVproton-proton (pp) collider (which may be FCC-hh [65] or SppS [66]) for various NP scenarios. We estimatethe 1 σ uncertainty in the measurement of cross sections in various scenarios by scaling the SM signal-to-background studies available in the literature [43]. We find that different scenarios of the Higgs potential canbe distinguished by measuring the double-Higgs production cross sections at the HE-LHC and the 100 TeVpp collider. We also consider the possibility of constraining the trilinear Higgs coupling in these scenarios,assuming certain accuracies for the measured cross section.Next, we compare total cross sections and kinematical distributions for the process gg → hhh in variousHiggs potential scenarios at the 100 TeV pp collider. This process was first studied in [67], by scaling theSM Higgs boson self-couplings, for exploring the potential of the 100 TeV pp collider for measuring thequartic Higgs coupling. To reduce the backgrounds, the most promising signature has one of the Higgsbosons decaying into the rare decay channel of two photons, while the other two Higgs bosons each decayinto a pair of bottom jets [68, 69]. In this work, we study various interference effects between the diagramsfor gg → hhh production process in order to understand the dependence of these terms on various couplings,including quartic Higgs couplings. The dependence of the cross sections on the quartic Higgs coupling isfound to be weak. In the composite Higgs model, the presence of t ¯ thhh coupling further complicates thesituation. Assuming that the triple-Higgs production cross section can be measured to a certain accuracy atthe 100 TeV pp collider, we could obtain the possible bounds on the strength of the quartic Higgs coupling.We estimate the 1 σ uncertainty in the measurement of cross sections in various scenarios by scaling the SMsignal-to-background studies available in the literature [68–72]. We find that the potential of the 100 TeV ppcollider to discriminate various NP scenarios strongly depends on the tagging efficiency of multiple bottomjets in the data analysis.The paper is organized as follows. In Sections 2 and 3, we lay out the general framework of Higgseffective couplings and discuss various NP scenarios that could yield a different Higgs potential from theSM. In Section 4, we consider the theoretical constraints on the strength of Higgs boson self-couplings, byexamining the conditions of tree-level partial wave unitarity and vacuum stability. In Section 5, we considerthe pp → hh process for its potential to discriminate various Higgs potential scenarios. In Section 6, weexamine the usefulness of the process pp → hhh for measuring the quartic Higgs coupling and for determiningthe shape of the Higgs potential. Our conclusion is presented in Section 7. In an EFT framework, new physics effect in the Higgs sector could be described using Higgs EFT and SMEFTin the broken and unbroken phase of electroweak symmetry, respectively. Higgs EFT could describe all theabove-mentioned NP scenarios, while SMEFT is only suitable for describing NP models with decouplingbehavior, such as the elementary Higgs scenario and the Nambu-Goldstone Higgs scenario with negligibleHiggs non-linearity. 5 .1 Higgs EFT: Higgs in the Broken Phase
In the broken phase of electroweak symmetry, it is convenient to use the Higgs EFT Lagrangian [23–29] todescribe the interactions of the top quark, the Higgs boson, and the Goldstone bosons eaten by the massivegauge bosons W ± and Z . Only the U (1) EM symmetry is manifest (or equivalently, the SM gauge symmetry SU (2) L × U (1) Y is non-linearly realized) in the broken phase. Furthermore, the custodial symmetry SU (2) V should be respected and the Higgs boson h is taken as a custodial singlet, when constructing the effectiveLagrangian. With the nonlinearly-realized symmetry SU (2) L × SU (2) R /SU (2) V , the leading Higgs EFTLagrangian, in the limit of turning off gauge couplings, is [23–29] L = 12 ( ∂ µ h ) − V ( h ) + v ∂ µ U ) † ∂ µ U ] (cid:18) a hv + b h v + · · · (cid:19) − v √ t L , ¯ b L ) U (cid:18) c hv + c h v + c h v + · · · (cid:19) (cid:32) y t t R y b b R (cid:33) + h.c. , (2.1)where V ( h ) is the Higgs potential, U is the Goldstone matrix of SU (2) L × SU (2) R /SU (2) V , and U = e iwaτav . (2.2)Here, v (= 246 GeV) denotes the electroweak scale, τ a are the Pauli matrices (with a = 1 , , w a arethe Goldstone bosons eaten by W ± , Z bosons. In general, the unknown coefficients a, b, c , c are independentfrom each other. The SM corresponds to a = b = c = 1 and other couplings equal to zero. Note that thestandard model gauge symmetry SU (2) L × U (1) Y is a subgroup of SU (2) L × SU (2) R . After turning on thegauge coupling, one needs to replace the usual derivative with the gauge covariant derivative in the aboveequation, as ∂ µ → D µ , and the form in the unitary gauge can be easily obtained by setting U →
1. Forconvenience, we will work with the above effective Lagrangian with the gauge couplings being turned off.To be specific, the general Higgs potential is written as V ( h ) = 12 m h h + d (cid:18) m h v (cid:19) h + d (cid:18) m h v (cid:19) h + · · ·≡ m h h + λ h + λ h + · · · , (2.3)and the Goldstone matrix U can be parameterized as U = (cid:115) − (cid:18) w a τ a v (cid:19) + i w a τ a v , (2.4)where d , are the independent Higgs self-couplings. In the SM, d = d = 1. It is easy to see that the nor-malization of the Goldstone matrix satisfies the condition U † U = 1, thus the above parameterization for theGoldstone bosons is equivalent to the one in the exponential form, cf. Eq. (2.2). With this parameterization,the derivative-coupled interactions of the Goldstone bosons take a relatively simple form asTr[( ∂ µ U ) † ∂ µ U ] = 2 v ∂ µ w a ∂ µ w a + 2 v ( w a ∂ µ w a ) v − w , (2.5)which would give rise to the usual kinetic terms for w a and their derivative-coupled interactions with theHiggs boson h . In this work, we neglect the effects of heavy particles contributing to the contact interactions In this work, we focus on the effects of Higgs boson couplings in the double- and triple-Higgs production processes. Hence,we could take the gauge-less limit, i.e. , taking g, g (cid:48) → h n G ˆ aµν G ˆ aµν , as these effective couplings vanish when the heavy particlesdecouple. For simplicity, we assume these particles are heavy enough and thus the h n G ˆ aµν G ˆ aµν interactionscan be safely neglected. Depending on the nature of the Higgs boson, SMEFT could be a good general framework to parameterizethe Higgs couplings. In the scenarios of Coleman-Weinberg Higgs and Tadpole-induced Higgs, the SMEFTcannot be used because of the non-decoupling behavior of new particles. On the other hand, the elementaryHiggs and Nambu-Goldstone Higgs scenarios could be well described in the SMEFT framework, because ofthe decoupling feature of these new physics models. In the following, we present the SMEFT framework andprovide the correspondence between the SMEFT and the Higgs EFT defined above.If the new physics scale Λ is much higher than the electroweak scale, and can be decoupled as Λ → ∞ ,the SMEFT with higher-dimensional operators will be a useful framework to describe the effects of newphysics at the weak scale. The SM gauge symmetry SU (2) L × U (1) Y is manifested (or linearly-realized) inthis case. Neglecting lepton-number violating operator at the dimension D = 5 (irrelevant to our study),the leading effective operators arise from dimension D = 6. The non-redundant set of D = 6 operatorswas summarized in Ref. [20], known as the Warsaw basis. There are 53 CP-even and 6 CP-odd effectiveoperators at the D = 6 level. In this paper, we will focus on the CP-conserving case. By employing equationsof motion, we can translate the D = 6 operators in the Warsaw basis to the ones in the so-called strongly-interacting light Higgs (SILH) basis [21]. The Rosetta package [73] can be used for translating betweendifferent bases. The main difference between these two bases are the operators involving fermionic currents(in the Warsaw basis) and the ones involving pure bosonic fields (in SILH basis); as (cid:80) ψ Y ψ O Hψ ∼ O T , O B ,and O (cid:48) Hq + O (cid:48) HL ∼ O W , where the sum is over all fermions with Y ψ denoting the corresponding hyperchargeof ψ [74]. When considering the S parameter constraint, it is more convenient to consider O B and O W instead of O Hψ or O (cid:48) Hq , O (cid:48) HL , as the latter operators can induce vertex corrections and modify the Fermiconstant. Furthermore, the operators such as O W W and O BB (in the Warsaw basis) can be reparameterizedby the linear combinations of the operators O W,B,HW,HB,γ (in the SILH basis) [74].For discussing the production processes of multi-Higgs bosons via gluon-gluon fusion, we list the followingrelevant D = 6 operators as L D =6 = c H ∂ µ ( H † H ) ∂ µ ( H † H ) − c Λ λ ( H † H ) − (cid:16) c t Λ y t H † H ¯ Q L H c t R + h . c . (cid:17) + α s π c g Λ H † HG aµν G aµν + α (cid:48) π c γ Λ H † HB µν B µν , (2.6)where λ and y t are, respectively, the SM quartic Higgs coupling and the top Yukawa coupling. α s = g s / π , α (cid:48) = e / π , and c i ( i = H, , t, g, γ ) are unknown Wilson coefficients. It is worth pointing out that there isanother operator O T = (cid:16) H † ←→ D µ H (cid:17) which violates custodial symmetry at tree level, thus we neglect it inthe following discussion. Further complication introduced by the flavor structure of the D = 6 Yukawa termwill not be explored in this paper. 7 .3 Relating SM EFT to Higgs EFT Since the Higgs EFT is formulated at the broken phase of the electroweak symmetry, it is a more generaldescription than the SM EFT. Hence, we could identify the SM EFT Wilson coefficients with the Higgs EFTcoefficients.With appropriate field redefinitions taken into account [75], we can match the Higgs-Goldstone couplings,Higgs-top couplings, and Higgs self-couplings defined in Eqs. (2.1) and (2.3) to the Wilson coefficients inEq. (2.6), as a = 1 − c H v + O ( 1Λ ) , (2.7) b = 1 − c H v Λ + O ( 1Λ ) , (2.8) c = 1 − c H v + c t v Λ + O ( 1Λ ) , (2.9) c = c t v − c H v + O ( 1Λ ) , (2.10) c = c t v − c H v + O ( 1Λ ) , (2.11) d = 1 + c v Λ − c H v + O ( 1Λ ) , (2.12) d = 1 + c v Λ − c H v + O ( 1Λ ) . (2.13)As we will see later, different Higgs couplings in the Higgs EFT are usually correlated for a given NPmodel. For simplicity, we again assume that the heavy particles decouple and the coefficients c g and c γ vanish, though in general these two effective operators can be induced by heavy particles with nontrivialcolor or electric charges circulating in loops. In contrast to the model-independent discussions presented in the last section, we explicitly derive theHiggs effective couplings in some specific NP scenarios, i.e. , the elementary Higgs, Nambu-Goldstone Higgs,Coleman-Weinberg Higgs, and Tadpole-induced Higgs. To identify the Higgs boson’s nature through Higgsself-interactions, we will derive the trilinear and quartic Higgs couplings for each scenario. Since differentHiggs couplings are usually correlated for a specific NP model, we will also present the relevant hV V ( V = W ± , Z ), ht ¯ t and hht ¯ t couplings, when necessary. When the Higgs boson is an elementary scalar, we include in the Ginzburg-Landau potential, as in the SM,the dimension-six operator ( H † H ) to effectively describe the new physics contributions parameterized in theSMEFT, cf. Eq. (2.6). In the NP models with scalar extensions, such as the singlet extension, the two Higgsdoublet model, the real and complex triplets and the quadruplet models [76–78], the ( H † H ) operator can8e induced, which has been classified in Ref. [76] based on group theory construction. Similarly, integratingout new heavy fermions and gauge bosons at the one-loop level could also induce the ( H † H ) operator.To be specific, the Ginzburg-Landau potential considered in this work is V = − µ H † H + λ ( H † H ) + c Λ λ ( H † H ) , (3.1)where the Higgs doublet is H = 1 / √ , v + h ) T in the unitary gauge, the Higgs boson mass term is m h = 2 λv (1 + c v ), and the electroweak scale v is obtained by solving µ = λv (1 + 34 c v Λ ) . (3.2)In the SMEFT description, the trilinear and quartic Higgs couplings are d = 1 + c v Λ − c H v + O ( 1Λ ) , (3.3) d = 1 + c v Λ − c H v + O ( 1Λ ) . (3.4)Here c H , cf. Eq. (2.6), modifies the kinetic term of the Higgs field, which universally shift the Higgscouplings to electroweak gauge bosons. Thus, the coefficient c H is highly constrained by the measurement ofthe couplings of Higgs boson to weak gauge bosons. The coefficient c t is constrained by the measurementsof t ¯ th and Higgs production cross section via gluon-gluon fusion process. To probe the Higgs self-couplings,we assume in this work that the operator ( H † H ) makes the most significant NP contribution and the otheroperators can be safely neglected. The Higgs boson can be a pseudo Nambu-Goldstone boson [9, 10], arising from strong dynamics at the TeVscale. The pseudo Nambu-Goldstone Higgs corresponds to one of the broken generators for some sponta-neously broken global symmetry G / H , based on which all the operators, consistent with Higgs nonlinearity,can be systematically constructed [79, 80].With its PNG nature, the general Higgs potential is approximately V ( h ) = − Af sin (cid:18) hf (cid:19) + Bf sin (cid:18) hf (cid:19) + · · · . (3.5)with higher order terms being neglected, where A and B are the two coefficients whose values are determinedby the specific dynamics responsible for generating the Higgs potential, and 4 πf denotes the NP scale. Withthe above notation, the coefficients A and B are positive. One can further define a ratio between theelectroweak scale v and the NP scale f to denote the Higgs nonlinearity in this scenario. To be specific, theminimization condition of the Higgs potential gives ξ ≡ v f = sin (cid:18) (cid:104) h (cid:105) f (cid:19) = A B . (3.6) It is nontrivial to realize a small ξ (less than about 0 . e.g., Ref. [81–84] for recent attempts to achieve this goal. It is also found experimentallychallenging to extract out small ξ values from measuring the Higgs couplings at the LHC [85]. Note that the parameter ξ ispositive for compact cosets, while it is negative for non-compact cosets [86]. In this work, we only focus on models with compactcosets, as EWSB is difficult to be triggered in the models based on non-compact cosets.
9y expanding the Higgs potential in the powers of h after EWSB, we have V ( h ) = Bf sin (cid:18) (cid:104) h (cid:105) f (cid:19) h + Bf sin (cid:18) (cid:104) h (cid:105) f (cid:19) h + B (cid:18) −
16 + 76 cos (cid:18) (cid:104) h (cid:105) f (cid:19)(cid:19) h + · · · , (3.7)The Higgs boson mass is given by m h = 2 Bf sin (cid:18) (cid:104) h (cid:105) f (cid:19) , (3.8)and the trilinear and quartic Higgs couplings are, respectively, d = Bf sin (cid:16) (cid:104) h (cid:105) f (cid:17)(cid:16) m h v (cid:17) = 1 − ξ √ − ξ , (3.9) d = B (cid:16) − (cid:16) (cid:104) h (cid:105) f (cid:17)(cid:17)(cid:16) m h v (cid:17) = 28 ξ − ξ + 33 − ξ , (3.10)where the ratio of d and d is obviously not one, depending on the parameter ξ .Due to the Higgs nonlinear effects associated with its nature as a PNG, the Higgs couplings in the topsector (the h ¯ tt , hh ¯ tt , hhh ¯ tt couplings) and the Higgs couplings with electroweak gauge bosons can deviatefrom the SM values. Regarding the Higgs couplings in the top sector, the h ¯ tt and hh ¯ tt , hhh ¯ tt couplingsdepend on the representation in which the top quark is embedded. As the two benchmarks, we consider theminimal composite Higgs model (MCH or MCHM) [87, 88], where both the left-handed t L and right-handed t R are embedded in the fundamental representation 5 of the global SO (5) symmetry; and the compositetwin Higgs model (CTH or CTHM) [89–91], where the left-handed t L is embedded in the fundamentalrepresentation 8, while the right-handed t R is a singlet of the global SO (8) symmetry. The Higgs couplingsin these two models are systematically derived in Ref. [22] and collected in Table 1. Another theoretically attractive scenario is the Coleman-Weinberg Higgs, where the Higgs potential at theclassical level is assumed to be scale invariant, i.e. only the quartic Higgs term is present at the treelevel [14–16]. However, with quantum corrections, the Higgs mass term is usually generated at the one-looplevel through the Coleman-Weinberg mechanism [92]. To be specific, the Higgs self-couplings are essentiallydetermined by the β -function of the quartic Higgs coupling λ , and the electroweak scale v is generated atquantum level [14]. The β -function of the quartic Higgs coupling β λ is positive-definite, and accordingly therunning quartic coupling at the EW scale λ ( v ) is negative [14], which corresponds to the minimum of theHiggs potential.The general Coleman-Weinberg Higgs boson h has the following potential V ( h ) = Ah + Bh log h Λ , (3.11)where A = (cid:88) i n i m i π v (cid:18) log m i v − c i (cid:19) , B = (cid:88) i n i m i π v . (3.12)10ere, the masses m i denote the masses of the particles circulating in the loop, which are defined in thevacuum background, n i denotes the internal degrees of freedom, and c i is the renormalization-scheme de-pendent constant . The parameter B is directly related to the β -function of the quartic Higgs coupling β λ .The minimization condition d V ( (cid:104) h (cid:105) )d (cid:104) h (cid:105) = 0 gives [93] v = (cid:104) h (cid:105) = Λ GW exp (cid:20) − − A B (cid:21) , (3.13)which leads to a relation between A and B . At this minimum, the running quartic coupling at the EW scale λ ( v ) is negative. Since v is determined from the dimensionless parameters, this is one specific realization ofthe dimensional transmutation mechanism.After expanding the above Higgs potential in the powers of h , after EWSB, we obtain V ( h ) (cid:39) B (cid:104) h (cid:105) h + 203 B (cid:104) h (cid:105) h + 113 Bh + · · · . (3.14)Here, all the Higgs self-couplings are related to the parameter B (or equivalently β λ ). Note that the higherorder terms, such as h , are neglected here. Therefore, the Higgs mass is m h = 8 B (cid:104) h (cid:105) (3.15)and the trilinear and quartic Higgs couplings are, respectively, d = B (cid:104) h (cid:105) (cid:16) m h (cid:104) h (cid:105) (cid:17) = 53 , (3.16) d = B (cid:16) m h (cid:104) h (cid:105) (cid:17) = 113 . (3.17)We note that the trilinear and quartic Higgs couplings are fixed at the one-loop order. Small corrections tothe above relations of d and d would appear only at the two-loop or higher orders [14, 15]. Another interesting scenario is the Tadpole-induced Higgs. Because of the existence of a non-zero tadpoleterm, the electroweak symmetry is spontaneously broken. As a result, the Higgs self-couplings, both thetrilinear and quartic Higgs couplings, are largely suppressed with respect to the SM prediction. In suchmodels, an additional source of electroweak symmetry breaking other than the SM Higgs mechanism isneeded. One specific realization of this class of models is the bosonic technicolor model [94, 95]. In thetypical technicolor models [96, 97], only the condensate of technifermions (cid:104) ¯ Q i Q j (cid:105) ∼ Λ triggers EWSB,and thus it predicts no Higgs boson. However, this has been ruled out due to the discovery of the Higgsboson at the LHC. On the other hand, in the bosonic technicolor model, an elementary Higgs boson is alsothere to trigger EWSB with vacuum expectation value (VEV) v h such that v ≡ v h + f , (3.18) The specific types of the particles running in the loop is irrelevant at the one-loop order. For example, in the MS scheme, c i = for gauge bosons, while c i = for scalars and fermions. H and Σ, while the red dot denotes the self-couplingsof the auxiliary doublet Σ.where f ≡ Λ tech . As both scalars can contribute to the W ± and Z boson masses, the scale f should besuppressed with respect to the electroweak scale v , so that the hV V (with V = W ± , Z ) couplings can beclose to the SM predictions, as required by the experimental findings. This leads to v (cid:39) v h (cid:29) f .At the low energy, the bosonic technicolor condensate could be parameterized as another effective scalardoublet field with the same quantum numbers as of the Higgs doublet. For convenience, let us name thisauxiliary doublet as Σ, and the Σ field is interpreted as the condensate of technifermions Σ ∼ (cid:104) ¯ Q i Q j (cid:105) / Λ .The simplified Lagrangian [17, 18] for the Tadpole-induced Higgs scenario is L = ( D µ H ) † ( D µ H ) + ( D µ Σ) † ( D µ Σ) − V ( H, Σ) (3.19)where V ( H, Σ) = m h H † H − (cid:0) (cid:15) Σ † H + h.c. (cid:1) − m Σ † Σ + λ S (cid:0) Σ † Σ (cid:1) . (3.20)Note that the mass term of the Higgs doublet H is positive, so EWSB is not triggered by the m h H † H term asin the SM. In order for the Tadpole-induced mechanism to be dominant, the quartic term λ H ( H † H ) shouldbe sub-dominant (thus negligible) in the above Higgs potential. The vacuum structure is then parameterizedas Σ = 1 √ (cid:32) f (cid:33) , H = 1 √ (cid:32) v h (cid:33) , (3.21)where the VEV f is obtained from the sector with auxiliary doublet alone, and v h is obtained from the m h H † H term and the mixing term between the two scalar sectors, such that v h = (cid:15)fm h . (3.22)More interestingly, the self-couplings of the Higgs boson are highly suppressed in this class of models.Let us assume that the Higgs particle in the auxiliary scalar sector is heavy enough ( v (cid:28) m Σ ) such that onecan integrate out the auxiliary scalar field and derive the tree-level effective potential for the Higgs boson.Because of the self-interactions of the auxiliary scalar and the mixing between the auxiliary field and theHiggs boson, trilinear and quartic Higgs couplings are induced, as shown in Fig. 2. To be specific, we havethe tree-level effective Higgs potential as V ( H ) = 12 m h H † H − (cid:15)f √ H † H + (cid:32) (cid:15) m Σ (cid:16) √ H † H (cid:17) + (cid:18) (cid:15)m (cid:19) m f (cid:16) √ H † H (cid:17) b c c c d d relevant couplings hV V hhV V h ¯ tt hh ¯ tt hhh ¯ tt hhh hhhh SM 1 1 1 0 0 1 1SMEFT (with O ) 1 1 1 0 0 1 + c v Λ c v Λ MCH − ξ − ξ − ξ − ξ − ξ − ξ − ξ CTH − ξ − ξ − ξ − ξ − ξ − ξ − ξ CW Higgs (doublet) 1 1 1 0 0 (1 . (4 . (1 . (4 . (cid:39) (cid:39) (cid:39) (cid:39) (cid:39) (cid:18) (cid:15)m (cid:19) m f (cid:16) √ H † H (cid:17) + · · · (cid:33) , (3.23)where, H † H = ( h + v h ) / m Σ is sufficiently large, which in turn requires theself-interactions of the auxiliary scalar field to be strong, all the self-coupling terms of the Higgs field h aresuppressed. Hence, after performing a shift of ( h → h + v h ), after EWSB, to remove the tadpole term inEq. (3.20), we find the self-couplings of the physical Higgs boson yield d (cid:39) d (cid:39)
0. However, if the Higgspotential V ( H, Σ) contains a quartic term λ H ( H † H ) , the Higgs self-couplings can yield non-zero d and d values. In this work, we simply assume that the quartic term λ H ( H † H ) vanishes, cf. Eq. (3.20). We collect all the relevant Higgs couplings in Table 1 for different NP scenarios – the elementary Higgs(both the SM and the SMEFT with the operator O ), Nambu-Goldstone Higgs (MCH and CTH models),Coleman-Weinberg Higgs and Tadpole-induced Higgs. As we will see, these couplings are important forderiving theoretical constraints, from the partial wave unitarity, the tree-level vacuum stability, and thestudy of the phenomenology of the double-Higgs production gg → hh and the triple-Higgs production gg → hhh at the LHC and the future hadron colliders.Below, we summarize the specific assumptions made in each class of NP models, which yield the Higgscouplings listed in Table 1. • In the SMEFT scenario, we only include the O operator, for simplicity, since almost all the otheroperators are (and will be further) constrained by the precision measurements of the Higgs bosoncouplings to gauge bosons or fermions. 13 In the Nambu-Goldstone Higgs scenario, the Higgs self-couplings depend on the Higgs nonlinearityparameter ξ , whose value has been constrained by the precision hV V coupling measurements. It isfound that ξ < . σ level. To be concrete, we restrict ourselves to two specific benchmarkmodels, MCH and CTH . For consistency, we have included the Higgs nonlinear effects inderiving all the Higgs boson couplings. Here, we neglect the contribution of the composite statesto Higgs couplings, by assuming that all the composite particles are heavy enough. The effects ofcomposite particles in Higgs couplings have been systematically discussed in Ref. [22]. • In the Coleman-Weinberg Higgs scenario, we simply take all the other Higgs couplings, except theHiggs self-couplings d and d , to be identical to the SM values. This is the case when the extra scalarparticles do not mix with the Higgs boson after the EWSB. It is found that the Higgs self-couplings d = and d = , at the one-loop order. For completeness, their values at the two-loop order [14,15]are also included in Table 1. • In the Tadpole-induced Higgs scenario, we approximate d = d (cid:39)
0, as they can be highly suppressed,though their exact values would depend on the self-couplings of the auxiliary scalar field. We alsosimply neglect the mixing between the auxiliary doublet and the Higgs doublet, as it is required bythe result of the precision hV V coupling measurement.
In this section, we aim to obtain the unitarity constraints on Higgs couplings defined after the EWSB,especially, on the trilinear and quartic Higgs couplings. We adopt the method of the coupled-channelanalysis to obtain the optimal bound [98, 99], since the most restrictive limit would come from the largesteigenvalue of the matrix for all the coupled scattering processes. For constraining the trilinear and quarticHiggs couplings, we, therefore, consider the electric-neutral channels for the scatterings between the topquark ( t ), longitudinal W ± and Z , and the Higgs boson at the energy √ s (cid:29) m t , m W , m Z , m h . According tothe Goldstone equivalence theorem [98], the longitudinal W ± and Z are equivalent to the Goldstone bosons( w a ) when √ s (cid:29) m W , m Z .To be specific, the following coupled 2 → t λ ¯ t λ → t λ ¯ t λ , t λ ¯ t λ → w b w b , t λ ¯ t λ → hh,w a w a → t λ ¯ t λ , w a w a → w b w b , w a w a → hh,hh → t λ ¯ t λ , hh → w b w b , hh → hh, (4.1)where λ , , , = ± denote the helicity of the initial-state and final-state top and anti-top quarks, while a = 1 , , b = 1 , , w a w a → w b w b occurs only when a (cid:54) = b .In the isospin basis, the 2 → M if ( √ s, cos θ ) can be decomposed into partial waves ( a j )as M if ( √ s, cos θ ) = 32 π ∞ (cid:88) j =0 j + 12 a j ( √ s ) P j (cos θ ) , (4.2)14here P j (cos θ ) are the orthogonal Legendre polynomials. Therefore, partial wave amplitudes are a j ( √ s ) = 132 π (cid:90) π d θ sin θ P j (cos θ ) M if ( √ s, cos θ ) , (4.3)which are bounded at the tree level as | Re( a j ) | < , (4.4)to satisfy partial wave unitarity. For the coupled channels listed above, the s -wave ( j = 0) scattering matrixat high energies, √ s (cid:29) m t , m W , m Z , m h , is explicitly a ( √ s ) = 316 π m t v − ( c + 1) m t − ac ) (cid:112) s − c (cid:112) s − ( c + 1) m t ( − ac ) (cid:112) s c (cid:112) s (1 − ac ) (cid:112) s ( − ac ) (cid:112) s s m t (1 − a ) − s m t ( b − a ) − c (cid:112) s c (cid:112) s − s m t ( b − a ) − d m h m t (4.5)in the basis (cid:26) t + ¯ t + , t − ¯ t − , √ w a w a , √ hh (cid:27) . (4.6)Here the factor √ is due to identical particles in the initial and final states. Note that the states t + ¯ t − and t − ¯ t + do not contribute to the s -wave scatterings. For any given NP model, we can always diagonalize thescattering matrix in Eq. (4.5) numerically. Elementary Higgs, CW Higgs, Tadpole-induced Higgs in → scatterings: The s -wave unitarity boundson d and d , obtained from the above 2 → hV L V L couplings ( V = W ± , Z )equal to the SM predictions. This corresponds to the elementary Higgs, Coleman-Weinberg Higgs andTadpole-induced Higgs. Moreover, many channels would further decouple when the t ¯ thh contact interactionvanishes, and in this case we can solve the s -wave unitarity constraints on d analytically. This leads to theresult lim √ s →∞ | a ( √ s ) | = | d | π m h v < . (4.7)Roughly, | d | < π . The coefficient d is only moderately bounded as | d − | < d can be translated into the bound on the Wilson coefficient c / Λ for the case of the SMEFT,which yields | c | < (cid:16) πv m h − (cid:17) Λ v . Nambu-Goldstone Higgs in → scatterings: When hV L V L couplings ( V = W ± , Z ) deviate from theSM values, the s -wave unitarity bound from 2 → . This applies to thecase of the Nambu-Goldstone Higgs scenario due to the Higgs nonlinearity. The unitarity violating scale isfound to be √ s (cid:39) ξ (cid:39) .
1, which yields d (cid:39) − ξ (cid:39) .
85. However,this unitarity bound could be weakened with appropriate composite resonances in the bosonic sector [101].In this work, we neglect the effects from those composite resonance states, by assuming that they are allvery heavy. In Fig. 3, we recast the unitarity constraints on Higgs self-couplings d and d , with ξ varyingin the range 0 . < ξ < .
15, for the NG Higgs scenario. The unitarity bound mainly results from the deviation of Higgs-Goldstone (eaten by EW gauge bosons in the unitary gauge)couplings. One can explicitly check the eigenvectors after diagonalizing the scattering matrix and find that the w a w a → w b w b ( a (cid:54) = b ) channel contributes the most to the eigenstate that violates the s -wave unitarity. CW Higgs and Tadpole-induced Higgs beyond → scatterings: It is interesting to notice that a relativelystronger unitarity bound on the Higgs self-couplings can be obtained from 2 → n ( n >
2) processes, if theHiggs potential is non-analytical [102, 103]. It corresponds to a pole in the Higgs potential, when H † H → πv ∼ d and d deviate from the SM values [102, 103].Schematically, for the high dimensional operator [103] L int = λ n n ! · · · n r ! φ n φ n · · · φ n r r , (4.8)the 2 → n ( n >
2) scattering process only matters when λ n is an order-one coefficient ( λ n ∼ O (1)), and theunitarity condition requires the energy is bounded roughly as E < (1 /λ n ) /n [103]. The stringent unitaritybound would come in the large n limit. Physically, λ n ∼ O (1) is only possible in non-decoupling theories,because there is no large scale that is responsible for suppressing this coefficient λ n . On the other hand, onecould expect that the coefficient λ n is highly suppressed by the cutoff scale in the NP models with decouplingbehavior, then the unitarity bound from the 2 → n ( n >
2) process will be very loose. Thus, it is sufficientto consider the conventional 2 → Even though the unitarity bound on the Higgs self-couplings is not very tight, the trilinear Higgs coupling d cannot be arbitrary large if the EW vacuum is required to be the global minimum. Based on the Higgs16cenarios unitarity constraintsSMEFT 0 < c < √ s < ξ = 0 . √ s < πv ∼ √ s < πv ∼ → πv ∼ → n ( n >
2) scattering partial-wave amplitudes [103]. Notethat we require c to be positive in the SMEFT, since the Higgs potential should be bounded from below.potential in Eq. (2.3), this requirement isd V ( h )d h = m h h + d m h v h + d m h v h = 0 . (4.9)When 9( d ) − d is positive or zero, the roots of the above equation are explicitly h = 0 ; h = v (cid:112) d ) − d − d d ; h = v − (cid:112) d ) − d − d d . (4.10)In this case, h = 0 corresponds to the EW vacuum, h corresponds to another local minimum of the Higgspotential, while h corresponds to the local maximum. Tree-level vacuum stability requires the EW localminimum to be the global minimum, i.e. V ( h ) < V ( h ). When 9( d ) − d is negative, only one solution, i.e., h = 0, exists for d V ( h ) / d h = 0, which corresponds to the only minimum of the Higgs potential. Asa result, we obtain the tree-level vacuum stability bound on d and d , as shown in Fig. 4 . Consistentwith Ref. [102], the conservative bound on the trilinear Higgs coupling is obtained as 0 < ∆ ≡ d − < d can be slightly relaxed in case when d is much larger than the SM value. Aswe see in Fig. 4, when | ∆ | > d is required to be more than 10 times of the SM value. For comparison,we also mark several benchmark points of various Higgs scenarios discussed in this work. In this section, we utilize the double-Higgs production cross section measurements to discriminate differentHiggs scenarios, since these scenarios predict very different strength of trilinear Higgs boson couplings. At ahigh energy hadron collider, the gluon-gluon fusion channel is the dominant production mechanism for thedouble-Higgs boson production. As mentioned in the introduction, this process has been widely consideredin the literature for validating the SM cross section, measuring the trilinear Higgs coupling [30–49] and the Given a NP model, d and d are usually correlated. Thus, we only focus on the region where both d and d are positive,rather than treating them as independent parameters. d and d . The shaded region isnot favored if the EW vacuum is required to be the global minimum. We note that higher powers of Higgsself-couplings are relevant for stabilizing the EW vacuum for the NG Higgs scenario. Thus it is only anartifact that the NG Higgs scenario is in the shaded region, resulting from the truncation of the full Higgspotential in deriving the coefficient d . t ¯ thh coupling [51, 52], and for probing various NP models [53–61, 63]. It remains to be established whetherthis process can be observed at the 5 σ level at the LHC.The ATLAS and CMS Collaborations have been looking for the hh signal in the data collected so far atthe LHC and have accordingly set upper limits on its production cross section [1–6]. Both collaborationshave also examined the prospects of detecting hh signal at the high-luminosity LHC (HL-LHC) and the high-energy LHC (HE-LHC) [7, 8]. At the HL-LHC, without (with) the systematic uncertainty, the signal can bemeasured at 31% (40%) accuracy relative to the standard model prediction with the confidence level of 3 . σ (3 σ ), and the trilinear Higgs coupling can be constrained in the range − . < λλ SM < . . < λλ SM < . − . < λλ SM < . − of integrated luminosity), the signal can bemeasured at the confidence level of 7 . σ and 11 σ , without the systematic uncertainty, in the b ¯ bγγ and b ¯ bτ τ channels, respectively [8]. A number of the above studies have performed detailed background analysis withoptimized cut-based analysis or with multivariate techniques. In this paper, we do not intend to perform anydetailed signal-to-background analysis. Instead, we utilize the NP cross section, after some basic kinematiccuts, to calculate the confidence level of observing the double-Higgs production, as predicted in the specificNP scenario, by assuming the same background rates as reported in the literature for detecting the SMdouble-Higgs production. As stated above, we mostly focus on the double-Higgs production at the 27 TeVHE-LHC and the future 100 TeV pp collider, and explore the possibility of distinguishing various scenariosand extracting the unknown Higgs couplings, especially, the trilinear Higgs coupling.18 .1 Cross Section and Distributions With the Higgs effective couplings listed in Table 1, the total cross section for the double-Higgs productionat hadron colliders can be written as σ = c σ SM b + c d σ SM t + c d σ SM bt + c σ t ¯ thh + c c σ b, t ¯ thh + c d c σ t, t ¯ thh , (5.1)where σ SM b ( σ SM t ) denotes the SM cross section with only the box (triangle) contribution, σ SM bt the interferencebetween the box and triangle contribution, σ t ¯ thh the new triangle contribution with non-vanishing SM-like t ¯ thh coupling, and σ b, t ¯ thh ( σ t, t ¯ thh ) the interference between new triangle and the box (triangle) contribution.A representative set of Feynman diagrams, including the triangle and box diagrams, is given in Fig. 5 forillustration.Figure 5: Representative Feynman diagrams for the pp → hh production. The third diagram appears onlyif the t ¯ thh coupling is non-vanishing.Our methodology of computing the pp → hh cross section has been discussed in Refs. [104, 105], whichwe follow below. We use the leading-order CTEQ parton distribution functions, CT14LLO [106], with therenormalization and factorization scales chosen as √ ˆ s . The numerical result for each SM-like cross section,defined in Eq. (5.1), at the 14, 27, and 100 TeV proton-proton colliders, respectively, is listed in Table 3.To suppress the large QCD backgrounds, we apply a hard cut on the transverse momentum ( p T ) of theHiggs boson, as discussed in next subsection. Thus Table 3 also includes the results on the SM-like crosssections with a cut p hT >
70 GeV at the 14, 27, and 100 TeV proton-proton colliders, respectively. No furtherkinematic cuts are applied here, as we are not doing detailed signal-to-background study. Although higherorder QCD corrections up to NNLO in the SM [107–110] and the EFT framework [111–116] are known, e.g.the K -factor is about 2.3 (1.7) for the 14 (100) TeV colliders [115]. We do not include the K -factors in ournumerical results presented in Table 3. According to the Table, the SM cross section without cuts (with p hT >
70 GeV cut) is 17.2 (15.4) fb at the 14 TeV collider, and it is 73.6 (66.2) fb at the 27 TeV collider,about 5 (4) times larger. The SM cross sections at 100 TeV collider are 830.1 fb and 756.8 fb, respectively,which are about 50 times larger than the results at the 14 TeV.As seen from Table 3, there are also some interesting patterns for the interference between differentFeynman diagrams. These could help us understand how the cross sections and differential distributionsdepend on various effective Higgs couplings. This will be discussed in more detail in next subsection.Combining the effective Higgs couplings in Table 1 and the SM-like cross section in Table 3, we obtainthe total cross sections in different models using the Eq. (5.1). Take the 27 TeV collider for example. Thetotal cross sections in the Tadpole-induced Higgs model and the Coleman-Weinberg model are 149 . . p hT >
70 GeV cut, they are 44 . . p hT σ SM b σ SM t σ SM bt σ b, t ¯ thh σ t, t ¯ thh σ t ¯ thh
14 TeV no cut 36.1 4.9 -23.8 -147.0 48.9 175.8 p hT >
70 GeV 29.6 2.9 -17.1 -122.4 36.3 151.927 TeV no cut 149.2 18.9 -94.5 -618.9 197.92 777.0 p hT >
70 GeV 124.1 11.6 -69.6 -524.5 151.1 684.5100 TeV no cut 1607.6 184.3 -961.8 -6872 2077.3 9356 p hT >
70 GeV 1370 118.8 -732 -5970 1645 8464Table 3: The SM-like cross section defined in Eq. (5.1), without cuts and with p hT >
70 GeV cut, at the 14TeV, 27 TeV, and 100 TeV proton-proton colliders, respectively.and CTH models, if we take the benchmark value ξ = 0 .
05, the total cross sections without cuts are 97 . . p hT >
70 GeV, they are 87 . . no cut p Th >
70 GeV - -> hhd σ / σ S M s =
14 TeV no cut p Th >
70 GeV - -> hhd σ / σ S M s =
27 TeV no cut, MCH p Th >
70 GeV, MCHno cut, CTH p Th >
70 GeV, CTH -> hh ξ σ / σ S M s =
14 TeV no cut, MCH p Th >
70 GeV, MCHno cut, CTH p Th >
70 GeV, CTH -> hh ξ σ / σ S M s =
27 TeV
Figure 6: Variation of the ratio of the new-physics cross section to that of the SM for hh production, at the14 TeV HL-LHC and the 27 TeV HE-LHC, respectively, as a function of the trilinear Higgs coupling d inthe elementary Higgs, Coleman-Weinberg Higgs and Tadpole-induced Higgs scenarios (upper row), and as afunction of the parameter ξ in the Nambu-Goldstone Higgs scenario (lower row).20n Fig. 6, for illustration, we display the ratio of the new-physics cross sections to the SM cross sectionin various Higgs potential scenarios at the 14 TeV LHC and the 27 TeV HE-LHC, respectively. In the toprow of Fig. 6, the ratio of the cross sections exhibits the quadratic dependence of d with minimum around d = 2 ∼
3. This behavior will be explained below. The bottom row of the figure shows the ratio as afunction of the parameter ξ in the Nambu-Goldstone Higgs scenario. In the case of the CTH model, thecross section ratio slowly increases as the ξ increases, and this behavior does not change much when the p hT >
70 GeV cut is imposed. This ratio could also be presented in the model-independent way using thegeneral Higgs couplings. In Fig. 7, this ratio is plotted in a two dimensional ( c , d ) contour with otherparameters taken as the SM-like values. In this figure, the values in the SM Higgs, Coleman-Weinberg Higgs,and Tadpole-induced Higgs scenarios are marked. The behavior of the cross section ratio in these modelscan be understood based on the interference patterns, as to be discussed in the next section. s =
27 TeVno cut gg -> hh c d - - s =
27 TeV p Th >
70 GeV gg -> hh c d - - Figure 7: Cross section ratio σ/σ SM as a function of c and d : without any cut (left), and with the kinematiccut p hT >
70 GeV (right). The SM cross sections without cuts and with cut, at the 27 TeV HE-LHC collider,are 73.6 fb and 66.2 fb, respectively. The total cross sections in the Tadpole-induced Higgs and Coleman-Weinberg scenarios are 149.2 fb and 124.1 fb, respectively, while with p hT >
70 GeV, they are 44.2 fb and 40.3fb, respectively. The magenta, blue, and cyan dots denote the σ/σ SM ratios in the Tadpole-induced Higgs,the SM, and Coleman-Weinberg scenarios, respectively.In Fig. 8, we display the normalized di-Higgs invariant mass M ( hh ) (left) and p hT (right) distributionsat the 14 TeV LHC and the 27 TeV HE-LHC, respectively. These distributions play an important role indetermining suitable kinematic cuts to reduce the SM backgrounds. The upper row of Fig. 8 shows thenormalized M ( hh ) distribution with a range of values of d . In the case of d = 3, there is an interestingtwo-peak structure in the normalized M ( hh ) distribution, arising from the competition between the triangleand box diagram contributions. We will come back to this in the next subsection, cf. Fig. 9.21 =- = = = = -> hh d σ / σ no cuts =
27 TeV300 400 500 600 700 8000.1110100 M ( hh ) ( GeV ) d σ / d σ S M d =- = = = = -> hh d σ / σ no cuts =
27 TeV100 200 300 4000.10.51510 p Th ( GeV ) d σ / d σ S M SMTadpole HiggsSM - like, d = ξ = ξ = -> hh d σ / σ no cuts =
27 TeV300 400 500 600 700 8000.1110100 M ( hh ) ( GeV ) d σ / d σ S M SMTadpole HiggsSM - like, d = ξ = ξ = -> hh d σ / σ no cuts =
27 TeV100 200 300 4000.51510 p Th ( GeV ) d σ / d σ S M Figure 8: Various normalized distributions on the di-Higgs invariant mass M ( hh ) (left) and the Higgs p T (right) at the 27 TeV HL-LHC with different d couplings (upper) and various Higgs potential models (lower).The case of d = 3 shows an interesting feature, due to the competition between the triangle and box diagramcontributions, as explained in the text. As shown in Fig. 5, the trilinear Higgs coupling is only presented in the triangle diagrams. However, as thebox and triangle diagrams interfere, the contribution of the trilinear Higgs coupling to the cross section alsodepends on the box diagram contribution. Furthermore, their interference effect is destructive. Since someof the new Higgs potential scenarios would allow large deviations of the Higgs couplings from the SM value,22he total cross section and various distributions could change significantly. Moreover , the Nambu-GoldstoneHiggs scenario also predicts non-zero t ¯ thh coupling due to the Higgs non-linearity, and there is correlationamong the hV V , t ¯ thh and t ¯ th couplings [22] for non-zero ξ . Because of this new t ¯ thh interaction, two newtriangle diagrams appear. These diagrams interfere with the SM triangle diagram destructively, and withthe box diagram constructively. This behavior happens as the t ¯ thh coupling has a negative sign relative to t ¯ th coupling in this scenario, as shown in Table. 1. In Table 3, we observe that taking the SM couplings, e.g., d = 1, the pure box contribution is large, while the pure triangle contribution is small, and furthermore,the interference contribution is large but negative, i.e. destructive. This leads to a small total cross sectionfor the pp → hh process in the SM. t ¯ thh Let us first consider scenarios without the t ¯ thh vertex. They are the elementary Higgs, Coleman-WeinbergHiggs, and Tadpole-induced Higgs scenarios. As one can see from Eq. (5.1), the pure triangle contributiondepends quadratically on d , whereas the interference term depends linearly on it. However, the pure boxcontribution does not depend on d . For the negative d , the cross section keeps on increasing with increasingmagnitude of d , cf. Fig. 6, as both σ SM t and σ SM bt contributions increase. For positive d , however, the crosssection first decreases and then keeps on increasing after reaching some threshold value of d , as first σ SM bt dominates which decreases the cross section, then σ SM t dominates which increases the cross section. Thisexplains the feature found in the upper row of Fig. 6.To understand the feature in Fig. 8, let us examine the contribution to the M ( hh ) distribution bydecomposing each class of Feynman diagrams and their interference effect. As shown in Fig. 9, the trianglediagram mostly contributes near the Higgs pair threshold, while the box diagram mainly contributes atthe threshold of the top quark pair system. As the d increases, the contribution to the M ( hh ) and p hT distributions from the triangle diagram keeps increasing and eventually exceeds the box diagram when d becomes very large. When d = 3, both the triangle and box diagrams are sizable. Together with theirinterference effect, they result in the two peaks in the M ( hh ) and p hT distributions, as shown in Fig. 8.Moreover, as we increase the minimum cut on the p T variable of the Higgs boson, which further suppressesthe large QCD background, the contribution from the pure triangle diagrams decreases more than theinterference and the pure box contributions relatively, as shown in Table 3. For the SM case, with the p hT >
70 GeV cut on the Higgs boson, the pure triangle contribution decreases by a factor of around 1 . .
4, and the one of the pure box term by 1 .
2. This explains why,in Fig. 6, the minimum of the curves, in which the pure triangle contribution starts to dominate over theinterference term, shifts to the right with the increase of the p hT cut. For the SM case, since the trianglecontribution is small, the reduction on the total cross section is not that steep with the increase in theminimum p hT cut, e.g., the total contribution decreases by a factor of 1.1 only with the p hT >
70 GeV cut.However, for larger positive d values, the pure triangle contribution does not dominate over the negativeinterference before applying any cuts. Thus, even though the cross section without cuts is large, imposingcertain minimum p hT cut would lead to a larger reduction in the cross section. For instance, for d = 10, the In the elementary Higgs scenario, t ¯ thh can also be induced via integrating out heavy particles, as shown in Eq. (2.10).Here, for simplicity, we take the t ¯ th coupling and the hV V to be the SM ones, which eliminates the t ¯ thh coupling because the t ¯ thh and t ¯ th are correlated in this model. xtrbx - trtotal
200 300 400 500 600 700 800 -
505 gg -> hhM ( hh ) ( GeV ) d σ d M ( f b / b i n ) no cutd = =
14 TeV bxtrbx - trtotal200 300 400 500 600 700 800 -
505 gg -> hhM ( hh ) ( GeV ) d σ d M ( f b / b i n ) no cutd = =
14 TeV
Figure 9: Contribution of various classes of Feynman diagrams and their interference effects to the M ( hh )distribution of hh production, for d = 1 (left) and d = 3 (right). When we increase the value of d from 1 to3, the triangle diagrams contribution and the negative interference term get scaled by 9 and 3, respectively.However, as the box-diagram contribution “bx” does not depend on d , it remains the same. The peak of thetotal distribution gets shifted to the left with the increase in d , as the triangle diagram, being an s-channelprocess, contributes significantly near the threshold of hh production.total cross section is 288 . p T cut; it reduces to 150 . p T >
70 GeV cut is applied, i.e.,a reduction by a factor of 1 .
9. The cross section for any d , before and after cuts, can easily be obtainedusing Table 3.At the 14 TeV HL-LHC, the double-Higgs production cross section is not statistically large. Thus, inthe case of the most promising final state signature ‘ bbγγ ’, there are only few tens of events with 3 ab − luminosity, which could only put very loose constraints on d . Nevertheless, the cross sections at the 27 TeVHE-LHC are about 5 − bbγγ ’,one could have significant constraint on d , and the d value can be determined within around 20% [8].Therefore, we expect at the 27 TeV HE-LHC, it is possible to distinguish different Higgs potential scenarioswhich do not contain the t ¯ thh vertex. t ¯ thh In the Nambu-Goldstone Higgs scenario, e.g. the MCH and CTH models, in addition to the appearance ofa new t ¯ thh vertex, the existing vertices, such as the t ¯ th and hhh couplings, also get modified from the SMones, as displayed in Table 1. In Ref. [22], a global fit on the MCH and CTH parameter space was performedby using the available data from the LHC Run-2 data. The 95% CL limit on ξ is obtained to be ξ < . ξ up to 0 . ξ . With a fixed ξ value, the totalcross section in the MCH model is significantly larger than that in the CTH model. In both models, thetrilinear Higgs coupling remains to be the same because of the universal form of the Higgs potential, however24 ModSM σ tt - hh σ tot -> hh ξ σ ( f b ) s =
14 TeV p Th >
70 GeV, MCH σ ModSM σ tt - hh σ tot -> hh ξ σ ( f b ) s =
14 TeV p Th >
70 GeV, CTH σ ModSM σ tt - hh σ tot -> hh ξ σ ( f b ) s =
27 TeV p Th >
70 GeV, MCH σ ModSM σ tt - hh σ tot -> hh ξ σ ( f b ) s =
27 TeV p Th >
70 GeV, CTH
Figure 10: Variation of different contributions to the SM-like cross sections, cf. Eq. (5.1), as a function of ξ in the MCH and CTH scenarios, at the LHC. The Magenta line, parametrizing the effect of t ¯ thh , crossesthe blue line, parametrizing the effect of t ¯ th and hhh couplings, around ξ = 0 . t ¯ th and t ¯ thh couplings are different due to different fermion embedding in both models. From Eq. (5.1)and Tabel 1, we see that the scaling of the combination σ b, t ¯ thh + σ t, t ¯ thh in the MCH model is larger by afactor of 4 than that in the CTH model. Similarly, σ t ¯ thh in the MCH model is larger by a factor of 16 incomparison to the CTH model. However, this term does not contribute noticeably when ξ is as small as0 .
01 because of the ξ scaling in the cross section. However, for ξ = 0 .
1, this term contributes moderatelyin the MCH model. The above discussion explains the difference in the increase of the rates of the totalcross section in the MCH model and the CTH model as ξ increases. A similar conclusion also holds afterapplying the p hT cut. Finally, in Fig. 10, we show the different contributions to the SM-like cross sectionsas function of the ξ parameter in the MCH and CTH models at the LHC. It shows that the t ¯ thh couplingis important to enhance the cross sections. As the ξ increases, although the contribution from the SM-likediagrams decreases, the total cross section increases due to the dominance of the t ¯ thh contribution, mostnoticeably in the MCH model. λ Extraction
In this subsection, we investigate the possibility to distinguish various new physics scenarios of Higgs poten-tials at the 27 TeV HE-LHC and the 100 TeV pp collider. At the HL-LHC, due to the limited cross section,it is difficult to constrain the trilinear Higgs coupling d . At higher energy hadron colliders, the total crosssection increases significantly, and thus the accuracy of measuring the total cross section, and the constrainton d , improves significantly.It has been shown in the literature [43] that the double Higgs boson production cross section of the SM, at25igure 11: The cross section ratio σ/σ SM in the double-Higgs production at the 27 TeV HE-LHC with anintegrated luminosity of 15 ab − (upper), and the 100 TeV pp collider with an integrated luminosity of30 ab − (lower) for various models. Here, we consider the case that the SM cross section can be measuredwith an accuracy of 13 .
8% and 5%, at the 1 σ level, respectively, at the 27 TeV HE-LHC and the 100 TeV ppcollider. The accuracy for the NP models are obtained using the rescaling procedure described in the text.The blue bars denote the expected accuracy for a given model.the 27 TeV HE-LHC with the integrated luminosity 15 ab − , can be measured with the accuracy of 13 .
8% atthe 1 σ level. This accuracy would be further improved at the 100 TeV pp collider with a 30 ab − integratedluminosity. Accordingly, the SM signal for the double-Higgs production can be measured with the accuracyof 5% at the 1 σ level [43]. We shall use this information as the benchmark point and perform a recast toobtain the signal significance in various NP scenarios. Using the fixed luminosity and the backgrounds fromRef. [43], the significance is obtained using Z = Φ − (1 − / p ) = √ − (1 − p ) [117, 118], where Φ is the26umulative distribution of the standard Gaussian and Erf is the error function. In this case, the Z value is Z = (cid:115) (cid:20) n ln n n + ( n − n ) (cid:21) . (5.2)Here, n is defined as n = n b + n s , and n = n b + n (cid:48) s . The n b denotes the number of background eventsand n s denotes the number of signal events rescaled in each NP scenarios as n s ∼ σ SMafter all cuts σ SMafter PT cuts σ NPafter PT cuts . (5.3)The n (cid:48) s denotes the signal event number that can be constrained at the 1 σ level, which can be obtained bysolving Eq. (5.2) with Z = 1 for a given n . With n s and n (cid:48) s , the relative accuracy for each NP scenario isobtained as | n s − n (cid:48) s | /n s . As expected, the larger cross sections lead to smaller relative errors for differentnew physics models.The results are shown in Fig. 11. At the 27 TeV HE-LHC and the 100 TeV pp collider, based on thetotal cross sections of the double-Higgs production, it is sufficient to distinguish new physics scenarios withdifferent Higgs potentials. The following conclusions can be drawn: • For the SMEFT with non-vanishing O ∼ ( H † H ) operator, the total cross section tends to be smallerthan that of the SM. Because of the tree-level vacuum stability constraint discussed in Sec. 4, theWilson coefficients of the O operator is preferred to be positive, which renders d to be larger thanone and yields a small cross section, as shown in Fig. 6. For the benchmark d = 2, it leads to anaccuracy of being 29 .
4% at the 27 TeV HE-LHC, and 10 .
9% at the 100 TeV pp collider, respectively. • For the Nambu-Goldstone Higgs, the total cross section tends to be larger than the SM predictionbecause of the existence of the contact t ¯ thh coupling. We show that for the benchmark ξ (cid:39) . σ level is about 10% at the 27 TeV HE-LHC,and about 5% at the 100 TeV pp collider, respectively. • The trilinear Higgs coupling in the Coleman-Weinberg Higgs scenario is universally predicted to be d = 5 /
3. So, similar to SMEFT, models of Coleman-Weinberg Higgs also have a smaller cross sectioncompared to the SM one. The 1 σ relative accuracy is about 23% at the 27 TeV HE-LHC, and about4 .
7% at the 100 TeV pp collider, respectively. • The trilinear Higgs coupling in the Tadpole-induced Higgs scenario is highly suppressed. ThereforeTadpole-induced Higgs models can have a much larger cross section compared to the SM value, dueto the enhanced box contribution and small interference. It turns out this scenario could be examinedvery well at both the 27 TeV HE-LHC (relative accuracy of 7 .
4% at the 1 σ level) and the 100 TeVpp collider (relative accuracy of 2 .
7% at the 1 σ level), and it can be well-discriminated from the SMscenario.With the total cross section of the double-Higgs production measured up to certain precision, we wouldlike to extract the information on d . In Fig. 12, assuming the measured accuracy of the double-Higgsproduction cross section is 10% and 20% respectively, we extract the parameter range for the trilinearHiggs coupling d . We use ˜ d to denote the scaled d . As shown in Fig. 12, we find that the ranges are27 caling10 % accur20 % accur0 1 2 3 4 5 6050100150200250 gg -> hh d ˜ / d σ ( f b ) p Th >
70 GeVs =
27 TeVd = scaling10 % accur20 % accur0.0 0.5 1.0 1.5 2.0 2.5 3.0050100150200250 gg -> hh d ˜ / d σ ( f b ) p Th >
70 GeVs =
27 TeVd = scaling10 % accur20 % accur0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5050100150200250 gg -> hh d ˜ / d σ ( f b ) p Th >
70 GeVs =
27 TeVCW Higgs
Figure 12: Constraints on the scaling ˜ d /d , assuming that the cross section can be measured up to 10%and 20% accuracy, respectively. Here, ˜ d denotes the scaled d value. % accur20 % accur0 1 2 3 4 5 6 7012345 gg -> hh d ˜ / d c ˜ / c p Th >
70 GeVs =
27 TeVMCH ( ξ = ) % accur20 % accur0 1 2 3 4 5 6 7012345 gg -> hh d ˜ / d c ˜ / c p Th >
70 GeVs =
27 TeVCTH ( ξ = ) Figure 13: Constraints on ˜ c /c and ˜ d /d , assuming that the cross section can be measured up to 10%and 20% accuracy, respectively, in the MCH and CTH models. Here, ˜ c and ˜ d denote the scaled c and d values, respectively.0 . < ˜ d /d < . ∪ . < ˜ d /d < .
12 (0 . < ˜ d /d < . ∪ . < ˜ d /d < .
25) if the accuracy is 10%(20%) for d = 1, and 0 . < ˜ d /d < . ∪ . < ˜ d /d < .
06 (0 . < ˜ d /d < . ∪ . < ˜ d /d < . d = 2, respectively.In Fig. 13, we show the parameter contour of the general effective couplings c and d (with fixed c ) thatcan be constrained by the double-Higgs production at the 27 TeV HE-LHC, assuming the 1 σ accuracy is10% and 20%, respectively. The scaling factors of the trilinear Higgs coupling and the contact t ¯ thh couplingare denoted as the ratio ˜ d /d and ˜ c /c , respectively. Compared to the CTH model, the constrained regionsin the MCH model are tighter because the absolute value of c in the CTH model is smaller than that in theMCH model, cf. Table. 1. Hence, the cross section does not vary much with the scaling of c . Overall, wesee that the 27 TeV HE-LHC can already set strict bounds on these Higgs couplings. In this section, we investigate the possibility and sensitivity to measure the quartic Higgs coupling, d , byusing the hhh production via gluon fusion, gg → hhh . This process can help in a better understanding ofthe shape of the Higgs potential in different scenarios.As discussed in the literature [67–72, 119, 120], measuring the quartic Higgs coupling in the triple-Higgsproduction channel is not easy even at the 100 TeV pp collider, but may be possible. This is because the28ignal cross section of the triple-Higgs production pp → hhh is small as compared to its SM backgrounds.Furthermore, the contribution of the quartic Higgs coupling is over-shadowed by other Higgs couplings. Thequartic Higgs coupling appears in a very few diagrams which make a very small contribution to the totalcross section. According to literature [68, 69], the quartic Higgs coupling is only constrained in the range of[ − ,
30] (at the 2 σ level) by measuring the triple-Higgs boson production rate at the 100 TeV pp colliderwith a 30 fb − data. In another approach, there have been attempts to measure the trilinear and quarticHiggs couplings indirectly using higher-order loop corrections [44,121,122]. These indirect searches put quiteloose bound on the quartic Higgs coupling at future colliders, such as the double Higgs production at thefuture linear collider (ILC). A partial list of other related studies is included as Refs. [123–125].To further pin down the quartic Higgs coupling, it is straightforward to utilize the triple-Higgs productionchannel at the 100 TeV pp collider with high luminosity run. We calculate the triple-Higgs productioncross sections with general parametrization of new physics effects in different NP scenarios. Five scenarios:independent scaling of SM trilinear and quartic Higgs couplings, the SMEFT models with correlated trilinearand quartic Higgs coupling, the Nambu-Goldstone Higgs, Coleman-Weinberg Higgs and Tadpole-inducedHiggs models are considered. We shall first compute and discuss cross sections and distributions in thesemodels, then we estimate how well the quartic Higgs coupling can be measured, assuming other couplingsare already determined by other experiments. It is expected that one could determine the t ¯ th coupling,trilinear Higgs coupling, and t ¯ thh coupling more precisely before measuring the quartic Higgs coupling. As shown in Fig. 14, there are several basic classes of Feynman diagrams contributing to the process gg → hhh , i.e. the pentagon-class diagrams, box-class diagrams, and triangle-class diagrams. In the pentagon-class diagrams, the Higgs self-coupling does not exist and the relevant coupling is the t ¯ th coupling. In thebox-class diagrams, the trilinear Higgs coupling plays a major role. Only the triangle-class diagrams havea dependence on both the trilinear and quartic Higgs couplings. However, only a few diagrams depend onthe quartic Higgs coupling . Besides, the contribution of the triangle-class diagrams is comparatively small.Because of this, the process gg → hhh is only moderately sensitive to the quartic Higgs coupling. The crosssection could change significantly with large modification of the quartic Higgs coupling and the trilinearHiggs coupling.Figure 14: Different classes of Feynman diagrams for the gg → hhh production in the SM. To be specific, for each quark flavor in the loop, there are 24 pentagon-class diagrams, 18 box-class diagrams, and 8triangle-class diagrams. Out of these 50 diagrams, only two triangle diagrams have a dependence on quartic Higgs coupling. gg → hhh production in the presence of the t ¯ thh and t ¯ thhh vertices.Furthermore, as shown in Fig. 15, several new diagrams would appear if additional t ¯ thh and t ¯ thhh couplings are non-zero. This scenario is realized explicitly, e.g. in the Nambu-Goldstone Higgs case, becauseof the Higgs non-linearity. In these scenarios, there are strong correlations among the t ¯ th , t ¯ thh , and t ¯ thhh couplings. As we will see, given the nonlinear parameter ξ ∼ .
1, the diagrams with t ¯ thh and t ¯ thhh couplingscould have large contributions, which render it more difficult to extract the quartic Higgs coupling.In the pp → hhh process, there are strong destructive interferences between different classes of diagrams.Interference between pentagon, box, and triangle diagrams plays a crucial role in dictating the cross sectionand various distributions. We first parameterize the contribution of each class of diagrams to the total crosssection. To be specific, the total cross section is written as σ = c σ SM p + c d σ SM b + c d σ SM t + c d σ SM t + c d σ SM p,b + c d σ SM p, t + c d σ SM p, t + c d σ SM b, t + c d d σ SM b, t + c d d σ SM t, t + (cid:0) c c σ p, b − t h + c d c σ b, b − t h + c d c σ t, b − t h + c d c σ t, b − t h + c c σ b − t h + c c d σ p, t − t h + c c d σ b, t − t h + c c d σ t, t − t h + c c d d σ t, t − t h + c c d σ b − t h,t − t h + c d σ t − t h (cid:1) + (cid:0) c c σ p, t − t h + c d c σ b, t − t h + c d c σ t, t − t h + c d c σ t, t − t h + c c c σ b − t h,t − t h + c d c σ t − t h,t − t h + c σ t − t h (cid:1) , (6.1)where we separate individual Feynman diagrams, and thus can explicitly read out their dependence onvarious Higgs couplings.We carry out the calculation in the way discussed in Refs. [104, 105]. We use FORM [126] to computethe trace of gamma matrices in the amplitude and to write the amplitude in terms of tensor integrals.These tensor integrals are computed using an in-house package, OVReduce [104], which implements theOldenborgh-Vermaseren [127] technique of tensor integral reduction. Scalar integrals are computed usingthe package OneLOop [128]. We use the leading-order CTEQ parton distribution functions, CT14LLO [106],and set the renormalization (and factorization) scale to be the invariant mass of the hard-scattering process √ ˆ s . The numerical value of each individual SM-like cross section is calculated and summarized in Table 4.Here we do not include the higher-order QCD correction, which may lead to a K -factor (the ratio of thenext-to-leading to the leading-order cross section) of about 2 [129]. Due to the extremely small cross section30 arts p hT Parts p hT Parts p hT no cut > > > σ SMp σ p, b − t h -41310 -20509 σ p, t − t h -9702 -13422 σ SMb σ b, b − t h σ b, t − t h -35207 - 19578 σ SM t σ t, b − t h -3960 -1558 σ t, t − t h σ SM t σ t, b − t h - 3164 -1628 σ t, t − t h σ SMp,b -8026 -2873 σ b − t h σ b − t h,t − t h -228538 -159601 σ SMp, t σ p, t − t h σ t − t h,t − t h σ SMp, t σ b, t − t h -13626 -5906 σ t − t h σ SMb, t -985 -298 σ t, t − t h σ SMb, t -673.3 -266 σ t, t − t h σ SM t, t σ b − t h,t − t h -66447 -36259 σ t − t h Table 4: Numerical values of various SM-like cross sections, cf. Eq. (6.1), at the 100 TeV pp collider.of this process at the HL-LHC and HE-LHC and the large QCD backgrounds at the same time, we directlypresent results at the 100 TeV pp collider. Basic p T cuts are also implemented for each Higgs boson in theHiggs final state. At the 100 TeV collider, the SM cross-sections without cuts and with p T >
70 GeV cutare 2987 ab and 1710 ab, respectively. We summarize the total cross sections of the double- and triple-Higgsproductions for the SM in Fig. 16 at the 14 TeV LHC, the 27 TeV HE-LHC and the 100 TeV pp collider.At the 100 TeV collider, the total cross-sections for the Tadpole-induced Higgs model and Coleman-Weinberg model without any cut are 7796 ab and 1272 ab, while with p hT >
70 GeV cut these are 3579 aband 836 ab, respectively. For the benchmark value ξ = 0 .
05 in the MCH and CTH models, the cross-sectionswithout any cut are 5033 ab and 3479 ab, while with p hT >
70 GeV cut these are 3302 ab and 2057 ab,respectively.Based on these numerical values, we display the cross sections in the ( d , d ) parameter contour in Fig. 17and the ξ dependence in Fig. 18, for different NP scenarios, without and with including the contact t ¯ thh and t ¯ thhh couplings. Fig. 17 shows the total cross section σ as a function of the trilinear and quartic Higgscouplings, i.e. d and d . We see that there is a significant increase in the cross section for the negativevalue of the d because the largest negative interference between the box and pentagon diagrams σ SM p,b , eithervanishes or becomes positive. There is only a marginal increase in the cross section for the negative valueof the d . In this figure, we also mark the SM scenario, the Coleman-Weinberg Higgs scenario, the Tadpole-induced Higgs scenario by blue, cyan, and magenta dots, respectively. The orange line denotes the SMEFTwith nonzero O ∼ ( H † H ) operator in the linear expansion as in Eq. (2.12) and Eq. (2.13). However,since the Nambu-Goldstone boson models contain additional t ¯ thh and t ¯ thhh couplings, and the t ¯ th couplingis different from the SM one, they cannot be directly compared with in this figure. Instead, the result of To be specific, the total cross section is about 44 ab at the 14 TeV LHC, and is only about 218 ab at the 27 TeV HE-LHC,respectively. o cut p Th >
70 GeV14 27 10010 Cross Sections vs. CollidersS ( TeV ) σ ( a b ) gg -> hhgg -> hhh14 27 10010 Cross Sections vs. CollidersS ( TeV ) σ ( a b ) gg -> hhgg -> hhh Figure 16: The total cross sections of the pp → hh and pp → hhh processes for the SM at the 14 TeV LHC,the 27 TeV HE-LHC and the 100 TeV pp collider, respectively. The blue lines denote the cross sectionswithout the cut, and the red lines denote the ones with p hT >
70 GeV. Here, we do not include the QCD K -factors, which are known to be about 1 . pp → hh and around 2 [129] for pp → hhh , respectively.Nambu-Goldstone Higgs scenario is presented in Fig. 18, which only depends on the nonlinearity parameter ξ . To be concrete, we consider two specific models, i.e. the MCH and CTH models, in Fig. 18. Comparedto the MCH, the cross section of the CTH remains close to the SM prediction for the given ξ .To complete discussion of this subsection, we present several basic differential distributions. In Fig. 19,we show the invariant mass M ( hhh ) distribution for various d and d values, and the normalized plots toexamine the modification of the shape of the distributions. We observe the quite distinct behavior near thethreshold of triple-Higgs boson production for different values of the d and d couplings. In the case of d ,there is a larger increase in the cross section near the threshold for its negative and zero value, while decreasefor positive values of d . The behavior is the opposite in the case of d . Most of the increase is for smallervalues of the invariant mass of the triple-Higgs system, up to about 700 GeV, and it is near the thresholdwhere the triangle diagram with quartic Higgs coupling is important. In this subsection, we investigate the interference patterns for the triple-Higgs production pp → hhh process,for better understanding variation of the total cross section and distributions in the different Higgs scenarios. t ¯ thh or t ¯ thhh Let us first consider the scenarios without the t ¯ thh and t ¯ thhh couplings. There are 10 relevant terms inthe total cross section, as shown in Eq. (6.1). The first four terms are always positive, and the rest ofthe six terms are interference terms and can be either positive or negative. As shown in the left panel ofFig. 20, the cross section first decreases and then increases within the range − < d < d increases.32 =
100 TeVno cut gg -> hhh d d - s =
100 TeV p Th >
70 GeV gg -> hhh d d - Figure 17: The cross section ratio σ/σ SM , as a function of the scaling of the trilinear and quartic Higgscouplings with various cuts. At the 100 TeV pp collider, the SM cross section without any cut and with p T >
70 GeV cut are 2987 ab and 1710 ab, respectively. The blue, cyan, and magenta dots denote the SM,CW Higgs and Tadpole-induced Higgs scenarios, respectively. The orange dashed line denotes the SMEFT(with non-vanishing O ) for d in the range of [5/6,2.5]. no cut, MCH p Th >
70 GeV, MCHno cut, CTH p Th >
70 GeV, CTH0.00 0.02 0.04 0.06 0.08 0.101.01.52.02.53.03.54.0 gg -> hhh ξ σ / σ S M Figure 18: The cross section ratio σ/σ SM , as a function of the parameter ξ in the MCH and CTH Models,at the 100 TeV pp collider.In addition, we show the variation of cross section, as the green band in Fig. 20, with the quartic Higgscoupling d varying within 0 < d <
10. In the right panel of Fig. 20, we explicitly see the variation of σ/σ SM as function of d , with d fixed. Although it is theoretically less plausible to have a large d ( d ≤ d values. In this case, there is large degeneracy in the d determination when d is around 5 to 6. 33 = = = = = - -> hhh d σ / σ d = =
100 TeV no cut300 400 500 600 700 8000.51510 M ( h h h ) ( GeV ) d σ / d σ S M d = = = = = - -> hhh d σ / σ d = =
100 TeV no cut300 400 500 600 700 8000.60.81.01.21.41.6 M ( h h h ) ( GeV ) d σ / d σ S M MCH; ξ = = = ξ = ξ = ξ = = = = = = = = = -> hhh d σ / σ no cuts =
100 TeV500 700 900 1100 1300 150002468 M ( h h h ) ( GeV ) d σ / d σ S M Figure 19: Distributions with partonic center-of-mass energy M ( hhh ) for hhh production via gluon-gluonfusion with different benchmark values of d and d at the 100 TeV pp collider. No cut on p T of Higgs bosonshas been imposed. d ∈ [ ] d = - -> hhh d σ / σ S M p Th >
70 GeVSM d = = = = = = =
60 2 4 6 8 100.10.5151050 gg -> HHH d σ / σ S M p Th >
70 GeVSM
Figure 20: Variation of the ratio of the cross section σ/σ SM with respect to d and d at the 100 TeV ppcollider. In the left, we show a band for varying d in the range of [0,10]. In the right, variation with d forvarious fixed d values is shown. The standard model cross section without any cut and with p hT >
70 GeVare 2987 ab and 1710 ab, respectively. t ¯ thh and t ¯ thhh In this subsection, we discuss NP scenarios in which the t ¯ thh and t ¯ thhh couplings are non-vanishing, e.g.the Nambu-Goldstone Higgs scenario, and investigate the interference terms involving these couplings indetails. In this scenario, all the Higgs couplings are only related to the single parameter ξ due to the Higgsnon-linearity.In Fig. 21, we show the interference effect of the t ¯ thh and t ¯ thhh couplings in two specific NG Higgsmodels, i.e. the MCH and CTH models. As expected, in the case of CTH model, the contribution of thesecouplings remains very small, except at large value of ξ , where it is also not that significant. However, inthe case of MCH model, both the t ¯ thh and t ¯ thhh couplings play important role. At a larger value of ξ ,the significant increase in the cross section is induced by these couplings. As ξ increases, the contribution( σ SMMod ) of SM-like diagrams decreases due to the smaller t ¯ th , d and d couplings, but the contribution ofdiagrams with t ¯ thh and t ¯ thhh couplings increases. 34 ModSM σ t t - hh σ t t - hhh σ tot -> hhh ξ σ ( a b ) s =
100 TeVno cut, MCH σ ModSM σ t t - hh σ t t - hhh σ tot -> hhh ξ σ ( a b ) s =
100 TeV p Th >
70 GeV, MCH σ ModSM σ t t - hh σ t t - hhh σ tot -> hhh ξ σ ( a b ) s =
100 TeVno cut, CTH σ ModSM σ t t - hh σ t t - hhh σ tot -> hhh ξ σ ( a b ) s =
100 TeV p Th >
70 GeV, CTH
Figure 21: The cross section [in ab] as a function of the parameter ξ in the MCH and CTH models. Themagenta line shows the effect of t ¯ thh coupling. In the MCH model, it exceeds the ”SM-like” effect ( σ SMMod )around the value ξ = 0 .
05. The blue line shows the effect of t ¯ thhh coupling, which includes the interferenceeffect between t ¯ thhh and t ¯ thh couplings (this interference is destructive for the shown range of ξ ).In Fig. 22, the ratios of the cross sections in the MCH and CTH models with respect to the SM value areshown, and the ratios depend on the single parameter ξ . The green band shows variation of the ratios dueto the scaling of the quartic Higgs coupling, denoted by ˜ d /d , and the dashed line is for d = 1. We see thevariation due to the scaling of the quartic Higgs coupling decreases for the larger values of the parameter ξ . d ˜ / d ∈ [ ] d ˜ / d = -> hhh ξ σ / σ S M MCH p Th >
70 GeVSM d ˜ / d ∈ [ ] d ˜ / d = -> hhh ξ σ / σ S M CTH p Th >
70 GeVSM
Figure 22: Variation of the ratio of the cross section to the SM, with respect to ξ and ˜ d /d , at the 100 TeVpp collider. The band is obtained by varying ˜ d /d in the range of [0,10] for the MCH and CTH models,respectively. 35ith 2 b-tagged jets SM SMEFT NG Higgs CW Higgs Tadpole Higgsluminosity (ab − ) 1 . × . × . × . × . × With 3 b-tagged jets SM SMEFT NG Higgs CW Higgs Tadpole Higgsluminosity (ab − ) 1198 33819 111 4976 277With 4 b-tagged jets SM SMEFT NG Higgs CW Higgs Tadpole Higgsluminosity (ab − ) 51 873 8 . . σ observation of the process pp → hhh in the SM andother new physics scenarios. Here, we take d = 2 (and d = 7, cf. Eqs. (2.12) and (2.13)) for the SMEFT,and ξ = 0 . K -factor, which is known up to NNLO [129–131] and is about a factor of 2 for the SM. The K -factors for hhh production with scaled SM Higgs couplings have also been evaluated in [130]. By including this K -factor,we expect the required luminosity would be slightly reduced for discovering these NP scenarios. λ Extraction
Here in this subsection we investigate how to measure the quartic Higgs coupling at the 100 TeV pp collider todiscriminate various NP scenarios. Similar to the study of the double-Higgs production process in the abovesection, we do not perform any detailed collider analysis, but only utilize the existing collider simulations asthe benchmark point and perform recast to obtain the signal significance in various scenarios. There havealready been several signal-to-background studies to observe the pp → hhh process at the 100 TeV collider.To reduce the backgrounds, the most promising signature has one of the Higgs bosons decaying into the raredecay channel of two photons, while the other two Higgs bosons each decay into a pair of bottom jets [68,69].Different studies tag different number of bottom jets. Tagging more bottom jets reduces the signal events,but the backgrounds reduce very significantly. So it is not surprising that the studies where the four bottomjets are tagged perform better than where two bottom jets are tagged. It has been shown that if two orthree bottom jets are tagged, then the largest background is due to the production of γγb ¯ bjj . However, iffour bottom jets are tagged, then the largest background is due to the production of h ( → γγ ) b ¯ bb ¯ b [69]. Thisis because of the small mistagging rate for the light jets.In Table 5, we have summarized the required luminosity for a 5 σ discovery by rescaling the signal crosssection, cf. Eq. (5.3). The results are presented for two, three or four bottom jet tagging scenarios. In orderto measure the SM cross section of the pp → hhh process to be within 30% accuracy at the 1 σ level, theneeded integrated luminosity is around 50 ab − when we adopt the four bottom jet tagging scenario. Thecorresponding accuracies for other scenarios are obtained through the same procedure as in the section 5.With an increasing luminosity of the 100 TeV pp collider, it is still challenging to extract the quartic Higgscoupling due to the large contribution from the tthh and tthhh couplings to the total cross sections, althoughthe cross sections in the MCH and CTH models are relatively larger than the one in the SM. Furthermore, itwould be difficult to extract the quartic Higgs coupling in the Tadpole-induced Higgs scenario, with d (cid:39) − , the expected accuracy for measuringthe cross section of the triple-Higgs production process for a given model is shown in Fig. 23. To be specific,the SMEFT with d = 2 and d = 7 can be measured with an accuracy of 86% at the 1 σ level, while16% (21%) for MCH with the parameter ξ = 0 . ξ = 0 . . ξ = 0 . ξ = 0 . σ/σ SM in the triple-Higgs production at the 100 TeV pp collider, with anintegrated luminosity of 50 ab − , for various models. Here, we consider the case that the SM cross sectioncan be measured with an accuracy of 30% at the 1 σ level. The accuracy for the NP models are obtainedusing the rescaling procedure described in the text. The blue bars denote the expected accuracy for a givenmodel.More importantly, our goal is to extract out the range of the quartic Higgs coupling from the triple-Higgsproduction process. In Fig. 24, we show the variation of the cross section for the triple-Higgs productionas a function of the scaling factor ˜ d /d , denoted by the dashed line. In these plots, we present the bandsfor the 10% and 20% accuracies on measuring the pp → hhh cross section in each scenario. We considerthe SM ( d = 1 , d = 1), and take the SMEFT with ( d = 2 , d = 1) and ( d = 2 , d = 7), and theColeman-Weinberg Higgs case with ( d = 5 / , d = 11 / d /d , and hence the quartic Higgs coupling, can be measuredfor a given benchmark scenario. In case there are non-vanishing contact t ¯ thh and t ¯ thhh couplings, e.g., in theNambu-Goldstone Higgs scenario, the situation is slightly different. We focus on the MCH and CTH modelswith the nonlinear parameter ξ = 0 .
05 and ξ = 0 .
1, respectively. Assuming the pp → hhh cross section couldbe measured with the accuracy of 10% and 20%, we show the corresponding contours in Fig. 25, in which˜ c /c and ˜ d /d are respectively the scaling factors for the t ¯ thhh coupling and the quartic Higgs coupling,37hen the other couplings are fixed in the given models. We do not include the result for the Tadpole-inducedHiggs scenario ( d (cid:39) , d (cid:39)
0) because it would be difficult to pin down the quartic Higgs coupling in thisscenario due to its tiny value. scaling10 % accur20 % accur0 2 4 6 8 10 12 1401000200030004000500060007000 gg -> hhh d ˜ / d σ ( a b ) p Th >
70 GeVd =
1; d = scaling10 % accur20 % accur0 5 10 1501000200030004000500060007000 gg -> hhh d ˜ / d σ ( a b ) p Th >
70 GeVd =
2; d = scaling10 % accur20 % accur0.6 0.8 1.0 1.2 1.4 1.6 1.80500100015002000 gg -> hhh d ˜ / d σ ( a b ) p Th >
70 GeVd =
2; d = scaling10 % accur20 % accur0 1 2 3 401000200030004000500060007000 gg -> hhh d ˜ / d σ ( a b ) p Th >
70 GeVCW Higgs
Figure 24: Constraints on ˜ d /d in various new physics scenarios, when the cross section can be measuredup to 10% and 20% accuracy, respectively. The parameter ˜ d /d scales the quartic Higgs coupling in a givenmodel. 38 % accur20 % accur0 2 4 6 8 10012345 gg -> hhh d ˜ / d c ˜ / c p Th >
70 GeVs =
100 TeVMCH ( ξ = ) % accur20 % accur0 2 4 6 8 10012345 gg -> hhh d ˜ / d c ˜ / c p Th >
70 GeVs =
100 TeVMCH ( ξ = ) % accur20 % accur0 2 4 6 8 10012345 gg -> hhh d ˜ / d c ˜ / c p Th >
70 GeVs =
100 TeVCTH ( ξ = ) % accur20 % accur0 2 4 6 8 10012345 gg -> hhh d ˜ / d c ˜ / c p Th >
70 GeVs =
100 TeVCTH ( ξ = ) Figure 25: Constraints on ˜ c /c and ˜ d /d , assuming that the cross section can be measured up to 10% and20% accuracy, respectively, in the MCH and CTH models.Fig. 24 shows that for the (SM) case of d = d = 1, the scaling factor is constrained to be within therange of 0 . < ˜ d /d < . ∪ . < ˜ d /d < .
66 (0 < ˜ d /d < . ∪ . < ˜ d /d < . • For the SMEFT scenario, we note that the bound on the quartic Higgs coupling will be generally quiteloose, unless the cross sections can be measured with better than 10% accuracy. However, as shownin Fig. 20, the bounds on the quartic Higgs coupling d could be tight when the trilinear coupling d (cid:39) −
3, in which case the triple-Higgs production cross section shows sizable variation as the d value changes. • For the Coleman-Weinberg Higgs scenario, the bound on the quartic Higgs coupling d is relativelytight, as the trilinear Higgs coupling is 5 / • For the Nambu-Goldstone Higgs scenario, the scaling factor ˜ c /c could be constrained to be withinthe order of 10, but ˜ d /d could only be constrained to the order of much larger than 10. • For the Tadpole-induced Higgs scenario, because the trilinear Higgs coupling d could be highly sup-pressed, the dependence on the quartic Higgs coupling d is very weak. This renders the precisiondetermination of d to be very difficult in this scenario.39 Conclusion
The nature of the Higgs boson is still mysterious, for its potential is not well understood yet. In thispaper, we consider several theoretically compelling new physics scenarios, in which the Higgs self-couplingscan be quite different from the SM prediction. To be specific, we have considered the elementary Higgs,Nambu-Goldstone Higgs, Coleman-Weinberg Higgs, and Tadpole-induced Higgs scenarios, with the trilinearand quartic Higgs couplings being either smaller or larger than the SM ones. Trilinear Higgs couplingis enhanced in the elementary Higgs scenario (with the preferred positive coefficient c for the effectiveoperator ( H † H ) ) and Coleman-Weinberg Higgs scenario, while it is reduced in the Nambu-Goldstone Higgsscenario and Tadpole-induced Higgs scenario. The same pattern also holds for the quartic Higgs coupling.In the Nambu-Goldstone Higgs scenario, we have also considered the Higgs nonlinear effect and explored therelations among the t ¯ th , t ¯ thh and t ¯ thhh couplings.In general, both the SMEFT and the Higgs EFT can be used to describe the Higgs boson’s nature andparameterize Higgs interactions, depending on whether the SM gauge symmetry is linearly or nonlinearlyrealized. The SMEFT is defined in the unbroken phase of the electroweak symmetry, while the Higgs EFTis defined in the broken phase. Comparing these two EFT frameworks, only the Higgs EFT can exhibitthe non-decoupling feature of new physics, this renders the Higgs EFT more general than the SMEFT.Among the new physics scenarios of different Higgs potentials, the SMEFT can only describe the elementaryHiggs and the Nambu-Goldstone Higgs, but the Higgs EFT can describe all the scenarios, including theColeman-Weinberg Higgs and the Tadpole-induced Higgs scenarios.In this work, we study how well the trilinear and quartic couplings of the Higgs boson in various newphysics scenarios can be measured at the 14 TeV HL-LHC, 27 TeV HE-LHC and the future 100 TeV ppcollider. First, we have investigated the theoretical constraints on the Higgs self-couplings using the partialwave unitarity and tree-level vacuum stability analyses. It turns out that the partial wave unitarity bound isnot very tight for the 2 → hW L W L , hhW L W L are the same as those in the SM. The tree-level vacuum stability prefersthe trilinear Higgs couplings to be within 0 < d <
3, while the quartic Higgs coupling can be 10 times largerthan the SM value.Given the unique patterns of the Higgs self-couplings predicted by various new physics scenarios, weexplore the possibility of discriminating various new physics scenarios through the process of double-Higgsproduction pp → hh at the 27 TeV HE-LHC and the 100 TeV pp collider. We have studied in detail the totalcross sections and various differential distributions, including the effects from distinct interference patterns,in each NP scenario. The values of the cross sections are typically smaller, compared to the SM value, for theelementary Higgs, and the Coleman-Weinberg Higgs cases, while they are larger for the Nambu-GoldstoneHiggs and Tadpole-induced Higgs cases. With larger cross sections, the corresponding uncertainties in thedetermination of the Higgs self-couplings are reduced. Thus, one can distinguish different new physicsscenarios at the 27 TeV HE-LHC, given the SM is expected to be measured with the accuracy of 14% at the1 σ level. The discrimination power is further enhanced at the 100 TeV pp collider. For completeness, wehave also extracted the possible range of the allowed trilinear Higgs coupling values for several new physicsscenarios, assuming the cross section is measured with 10% and 20% accuracy, respectively. These are shownin Figs. 11, 12, and 13.To fully pin down the quartic Higgs coupling and thus the shape of the Higgs potential in various scenarios,40e also need to investigate the triple-Higgs production process pp → hhh at future colliders. However, due tothe small rate of the signal event with respect to its backgrounds, it might be quite challenging to determinethe quartic Higgs coupling. It appears, on the basis of current studies, that one needs about 50 ab − , cf.Table. 5, at the 100 TeV pp collider, to discover this process and precisely measure the quartic Higgs couplingin the case of the SM. The integrated luminosity required for the 5 σ observation of the triple-Higgs productionprocess in new physics scenarios is also shown in Table 5. However, using some special techniques, includingmachine learning, one might be able to bring this luminosity to within the proposed luminosity for the 100TeV collider. After investigating the interference patterns of the pp → hhh process, we find the dependenceof the cross section on the quartic Higgs coupling is moderate because other couplings obscure the extractionof the quartic coupling. Thus, even when the total cross section can be relatively well measured with 10%and 20% accuracy, it is still not easy to measure the quartic Higgs coupling, cf. Figs. 23, 24, and 25. Hence,more effort and better techniques are called for a better understanding of the Higgs potential. Acknowledgements
J.-H.Yu is supported by the National Science Foundation of China under Grants No. 11875003 and theChinese Academy of Sciences (CAS) Hundred-Talent Program. C.-P. Yuan is supported by the U.S. NationalScience Foundation under Grant No. PHY-1719914. C.-P. Yuan is also grateful for the support from theWu-Ki Tung endowed chair in particle physics. L.-X. Xu is supported in part by the National ScienceFoundation of China under Grants No. 11635001, 11875072.
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