Probing Higgs self-coupling of a classically scale invariant model in $e^+e^- \to Zhh$: Evaluation at physical point

A classically scale invariant extension of the standard model predicts large anomalous Higgs self-interactions. We compute missing contributions in previous studies for probing the Higgs triple coupling of a minimal model using the process $e^+e^- \to Zhh$. Employing a proper order counting, we compute the total and differential cross sections at the leading order, which incorporate the one-loop corrections between zero external momenta and their physical values. Discovery/exclusion potential of a future $e^+e^-$ collider for this model is estimated. We also find a unique feature in the momentum dependence of the Higgs triple vertex for this class of models.


Introduction
The properties of the Higgs boson, now being uncovered through measurements at the LHC experiments, seem to point to an increasingly consistent picture with predictions of the Standard Model (SM) of particle physics. In particular, up to now measured interactions of the Higgs boson with other SM particles agree well with the SM predictions and no sign of significant deviation has been observed [1,2,3,4].
In view of the current status, it would be worth considering a class of models beyond the SM, which possess Higgs portal couplings and at the same time induce the electroweak symmetry breaking via the Coleman-Weinberg (CW) mechanism [5]. On the one hand, in such models anomalies tend to be suppressed in the interactions among the SM particles other than the Higgs boson, as well as in the interactions of the Higgs boson with other SM particles. On the other hand, large anomalies are predicted in the Higgs self-interactions, since the global shape of the Higgs potential [which is given by a φ 4 log φ-type potential at the leading-order (LO)] significantly deviates from that of the SM. Furthermore, this class of models are discussed within the context of dark matter physics [6,7,8,9,10,11,12].
Testability of this class of models has been studied. The CW-type potential predicts a universal value of the three-point Higgs self-coupling from computation of the one-loop effective potential, which is given by λ hhh = 5/3 × λ SM hhh [13,14,15]. The large deviation from the SM value can be probed relatively easily using the process e + e − → Zhh at a future linear collider with the center-of-mass energy around 500 GeV [16].
There is, however, a caveat in these testability analyses. The value of the above three-point coupling is determined at the zero external momentum limit p i → 0, and this constant value has been used to scale the tree-level h 3 -vertex in the analyses. The CW mechanism is unique in that certain one-loop contributions become comparable to tree-level contributions, the very reason why it is called a radiative symmetry breaking mechanism. This feature applies to the Higgs self-interactions, and one needs to include a part of the one-loop corrections even in the LO analyses, if the proper order counting is respected. According to this order counting the one-loop corrections between p i = 0 and p i ∼ O(m t ) become formally the same order as the three-point coupling determined at p i → 0. Most pessimistically before explicit computation, one could be worried if the large deviation predicted at p i = 0 may even be almost canceled at the physical values of the external momenta.
In this paper we compute the total and differential cross sections for e + e − → Zhh in a CW-type Higgs portal model, at the LO of the proper order counting. We take up the minimal model analyzed in ref. [14] which includes N singlet scalar particles and use the order counting developed in ref. [17]. This model has a high predictability because of the small number of model parameters. Let us describe briefly the current status of this model. It is known that this model can be tested by direct dark matter searches [18]. Using the most recent bounds by the XENON1T experiment [19], the model is excluded in the region N ≥ 2, while the case N = 1 is marginal. Nevertheless, this test assumes that the reheating temperature of the Universe exceeds the singlet mass scale, hence the model cannot be excluded without this assumption, or alternatively, we can put bounds on the reheating temperature. It was also pointed that this model may be tested using W W scattering processes in the future [17].
We show that in this model one-loop corrections induce non-trivial kinematical dependences to the e + e − → Zhh cross sections, which cannot be accounted for by the constant scaling of the Higgs triple coupling. The kinematical dependence of the h 3 -vertex reflects characteristic features of the model. We also show a general feature valid for CW-type Higgs portal models with more general non-SM sectors.
In Sec. 2 we describe our model and its order counting rule. In Sec. 3 we define an effective Higgs triple coupling. The total and differential cross sections for e + e − → Zhh are computed in Sec. 4. Conclusion is given in Sec. 5. In Appendix A, loop functions are defined. In Appendix B, a relation between the h 3 -vertex and the Higgs wave function renormalization is derived using derivative expansion of the effective potential.

Lagrangian
We consider a model, which has an extended Higgs sector with classical scale invariance (CSI). Throughout the paper we adopt the Landau gauge and dimensional regularization with d = 4 − 2ǫ space-time dimensions. The bare Lagrangian of the CSI model is given by S = (S 1 , · · · , S N ) T denotes a real scalar field, which is a SM singlet and belongs to the N representation of a global O(N) symmetry. The above Lagrangian is invariant under the SM gauge symmetry and the O(N) symmetry and is perturbatively renormalizable.
H denotes the doublet Higgs field. Subscripts or superscripts "B" in eq. (1) show that the corresponding fields or couplings are the bare quantities. The Higgs interaction terms relevant in our analysis are given by Here we have re-expressed the interaction terms by renormalized quantities and counterterms: H and S i denote the renormalized fields; λ H and λ HS represent the renormalized coupling constants; the terms proportional to δλ H and δλ HS represent the counter-terms; µ denotes the renormalization scale. The Higgs field acquires a non-zero vacuum expectation value (VEV) via the CW mechanism, whereas the singlet field does not [14]. The singlet particles become massive and degenerate (with mass m s = λ 1/2 HS v). We expand the Higgs field about the VEV as where h, G 0 and G + represent the physical Higgs, neutral-and charged-NG bosons, respectively; v denotes the Higgs VEV. Substituting them into eq. (2), one obtains the Feynman rules for the CSI model.

Order counting: ξ expansion
According to ref. [17] we introduce an auxiliary expansion parameter ξ and rescale the parameters of the model as follows: where y t = √ 2m t /v denotes the top-quark Yukawa coupling. Then we expand each physical observable in series expansion in ξ, and in the end we set ξ = 1. If an observable is given as A(ξ) = ξ n (a 0 + a 1 ξ + a 2 ξ 2 + . . . ), we define the LO term of A as a 0 , the nextto-leading order (NLO) term of A as a 1 , etc. From previous experiences we expect the size of the effective expansion parameter in these series expansions to be order 10-30%, depending on the observables. It follows from the above counting that m 2 h ∼ O(ξ 2 ) and m 2 t , m 2 s ∼ O(ξ). The reason for assigning ξ 2 to the other couplings (in particular to the electroweak gauge couplings) will be made clear below.

Determination of parameters
Following Sec. II of ref. [17], we can determine λ H and λ HS from the tadpole condition and the on-shell Higgs mass condition in terms of VEV v = 246.6 GeV, the Higgs mass m H = 125.03 ± 0.27 GeV [20,21] and the top-quark mass m t = 173.34 ± 0.76 GeV [22]. λ H and δλ H depend on renormalization scheme; if the counter-terms are defined by the MS scheme, The on-shell Higgs mass condition is given by   where Σ(q 2 ) denotes the Higgs self-energy, counted as 1 O(ξ 2 ), whose order is the same as m 2 h . The analytic expression of Σ(q 2 ) is given in ref. [17]. The values of λ HS and m s determined by eq. (6) and v = 246.6GeV are summarized in Table 1.

Higgs triple coupling
We define an effective Higgs triple coupling for h * → hh as where Γ hhh (q 2 ) denotes the 1PI three-point vertex of the Higgs boson, see Fig. 1. The two external Higgs bosons corresponding to the final state are taken to be on-shell, while the invariant mass q of the initial (off-shell) Higgs boson is taken as a variable.
The diagrams contributing to Γ hhh (q 2 ) are shown in Fig. 2. Its analytic expression is so that we can ignore Σ h (q 2 ). Since, however, the hierarchy between m 2 h ∼ O(ξ 2 ) and m 2 t ∼ O(ξ) is not large, we prefer to keep this propagator ratio in our computation. given by where B 0 (p 2 , m 2 , m 2 ) and C 0 (p 2 1 , p 2 2 , p 2 3 , m 2 , m 2 , m 2 ) represent loop functions defined in Appendix A. Σ(q 2 ) and Γ(q 2 ) do not depend on the renormalization scale µ 2 , due to cancellation between λ H and loop functions. We can use λ hhh (q 2 )/λ SM,tree hhh to scale the tree-level SM Higgs three-point vertex in the Higgs exchange diagram for e + e − → Zhh. We show the real and imaginary parts of λ hhh (q 2 ) as functions of q in Figs. 3 a,b. For comparison we also show λ hhh as determined from the one-loop effective potential, 5 3 λ SM hhh , which corresponds to setting all the external momenta to zero. By comparison we see non-trivial q 2 dependence in λ hhh (q 2 ). In the same figures we also show λ SM hhh including the tree plus top-quark one-loop contribution, where the q 2 dependence stems from the top-loop contribution. The top-quark loop contribution raises the couplings at q > ∼ 2m t , which is common in the SM and CSI model.
It is useful to see q 2 expansion of λ hhh (q 2 ) for model identification. q 2 expansion is reasonable for singlet contribution, because q 2 /m 2 s ≪ 1 at q ≃ 2m h . We show in Appendix B, using derivative expansion of the effective action, that the coefficient of the q 2 term of this expansion is determined by the divergent part of the wave function renormalization for the Higgs boson in a general CSI-model. Since the singlet-loop does not contribute to the divergent part of the Higgs wave function renormalization, there is no q 2 term from the singlet-loop in λ hhh (q 2 ). The absence of q 2 term in the singlet contribution is also confirmed by an explicit calculation of Γ hhh : See eq.(8). In the third line, m 2 s = λ HS v 2 is used. Thus, the q 2 dependence starts at order q 4 for the singlet contribution. On the other hand, since the top-quark contributes to the divergent part of the Higgs wave function renormalization, the q 2 term of λ hhh (q 2 ) stems solely from the top-quark contribution at LO. This is why N dependence is almost absent until very close to the singlet-pair threshold in Fig. 3 a,b.

e + e − → Zhh cross sections
In the SM there are four tree-level diagrams which contribute to the process e + e − → Zhh. See Fig. 4. One of the diagrams contains the Higgs three-point vertex, while the other three diagrams contribute as irreducible background diagrams for probing the triple Higgs coupling. To compute the cross section for the CSI model, we replace the tree-level λ SM hhh by λ hhh (q 2 ). The background diagrams and the signal diagram are counted as the same order in ξ since we assign ξ 2 to the electroweak gauge couplings. Numerically this order counting is reasonable.
We calculate the total and differential cross sections using MadGraph5 [23], with the initial e ± longitudinal polarizations P (e + , e − ) = (0.3, −0.8). At √ s = 500 GeV, the total cross section is evaluated to be σ CSI tot (e + e − → Zhh) = 0.341 fb. This amounts to +47% deviation compared to the tree-level SM total cross section. We show the √ s dependence of the total cross section in Fig. 5. The deviation of the total cross section from the tree-level SM prediction decreases as √ s increases. We also compute dσ/dq at √ s = 500 GeV for N = 1 and at several √ s between 600 − 1200 GeV for N = 12, where The results are shown in Figs. 6 and 7. The kinematically allowed range is given by 2m h ≤ q ≤ √ s−m Z . In Fig. 6, the enhancement from the SM prediction in the low q region is due to the enhancement of the triple Higgs coupling at low q. The relative enhancement factor decreases as q increases, since the relative weight of the signal diagram is reduced due to rapid decrease of the Higgs propagator 1/(q 2 − m 2 h − Σ h ). In Fig. 7, no peak is visible corresponding to the singlet pair creation at q ≃ 2m s (≃ 600GeV), because of the suppression by the Higgs propagator. The difference between the prediction using λ hhh (q 2 ) and λ hhh = 5 3 λ SM hhh determined from the effective potential, is found in small q region in the N = 12 case.
Integrated luminosity necessary for a discovery at 5σ (an exclusion at 3σ) of this model is estimated to be 710 fb −1 (260 fb −1 ) at √ s = 500GeV. This estimation uses the number of signal and background events from the full simulation of International Linear Collider (ILC) experiment given in ref. [24] for e + e − → Zhh, h → bb process and m h = 120GeV at √ s = 500GeV, with events corresponding to 2ab −1 . We rescaled the number of the signal event N sig by the ratio of the total cross sections and of the branching ratios for h → bb as while the 1σ standard deviation is approximated by N sig / √ N BG . To end this section we give some discussion. The W -fusion process is another process at ILC to measure the triple Higgs coupling and is the dominant process at √ s 1200GeV in SM. It turns out that the total cross section of this process decreases as the triple Higgs coupling increases due to a negative interference [25], while the Zhh cross section are enhanced by positive interference. As a result it is advantageous to analyze e + e − → Zhh rather than W -fusion process for the CSI model. It is pointed that this model has a Landau pole around 3.5 TeV (N = 1), 16 TeV (N = 4) and 28 TeV (N = 12) at LO [14]. The existence of the Landau pole indicates that perturbative expansion of the model does not work around its scale or the model turns into some UV theory. Perturbative validity has been discussed in the leading-logarithmic analyses of the effective potential and W W -scattering amplitude [17]. Since our analysis deals with the energy scale well below the Landau pole, we consider that the perturbative analysis given in this paper is justified, where we regard this model as an effective theory valid around O(100GeV).

Conclusion
In the minimal CSI model, we have incorporated the one-loop corrections between the zero external momenta and physical point in the total and differential cross sections for e + e − → Zhh. We find that the bulk of the large anomaly predicted at the zeroexternal-momentum limit remains, while a non-trivial q 2 dependence of the Higgs triple interaction is induced. We also find that at LO, the effect of the singlet scalar boson has no q 2 term in q 2 expansion of the Higgs triple coupling. This feature does not depend on the number or mass spectrum of the singlet scalars and is a general feature of CW-type models with singlet scalar bosons, as shown in Appendix B. The top loop effects induce a non-negligible q 2 dependence, in accord with the expectation based on order counting.  s = 600(blue), 800(green), 1000(red) and 1200(magenta) GeV for e + e − → Zhh, where q 2 = (p h1 + p h2 ) 2 . Black dotted lines show the same cross sections using λ hhh determined from the effective potential.
In contrast, N-dependent effects by the singlet-loop are found to remain small below the singlet pair threshold due to this unique feature of the q 2 expansion. We have estimated sensitivity of the e + e − → Zhh total cross section to the deviation from the SM and found that it is fairly promising.
The expansion of B 0 (p 2 , m 2 , m 2 ) and C 0 (p 2 , 0, 0, m 2 , m 2 , m 2 ) in p 2 are given by classical field Φ and renormalization scale µ 2 as dimensionful parameters, and since µ 2 always appears in the logarithm, the form of Γ 1,i is given by where a i and b i represent dimensionless constants determined by coupling constants. a i does not contain Φ, only the logarithmic term contributes to eqs. (B.2,B.3). Furthermore, ln(µ 2 ) is associated with the 1/ǫ part of the wave function renormalization. The above argument does not apply to models with dimensionful parameters such as m 2 , since polynominal of Φ † Φ/m 2 contributes to eq.(B.5).
In this paper, we consider a CSI model with SM gauge singlet scalar bosons. Since the VEV of the singlet bosons is equal to zero, the previous argument applies and the contribution of the singlet-loops is expressed by eq.(B.5). Noting that the wave function renormalization by the singlet-loop is finite (a i < ∞, b i = 0), the singlet bosons do not contribute to the q 2 term of the three-point function 2 .
On the other hand, the fermion and vector bosons, e.g. the top, W and Z, contribute to the divergent part of the Higgs wave function renormalization since the mass dimension of a fermion propagator is equal to −1 and V V h has derivative couplings. As a result, fermions and vector bosons generally contribute to the q 2 term of the Higgs three-point function.