Higgs boson pair production at a photon-photon collision in the two Higgs doublet model

We calculate the cross section of Higgs boson pair production at a photon collider in the two Higgs doublet model. We focus on the scenario in which the lightest CP even Higgs boson ($h$) has the standard model like couplings to the gauge bosons. We take into account the one-loop correction to the $hhh$ coupling as well as additional one-loop diagrams due to charged Higgs bosons to the $\gamma\gamma\to hh$ helicity amplitudes. It is found that the full cross section can be enhanced by both these effects to a considerable level. We discuss the impact of these corrections on the $hhh$ coupling measurement at the photon collider.

The Higgs sector is the last unknown part of the standard model (SM) for elementary particles. Discovery of the Higgs boson and the measurement of its properties at current and future experiments are crucial to establish our basic picture for spontaneous electroweak symmetry breaking (EWSB) and the mechanism of particle mass generation. The Higgs mechanism would be experimentally tested after the discovery of a new scalar particle by measuring its mass and the coupling to the weak gauge bosons. The mass generation mechanism for quarks and charged leptons via the Yukawa interaction is also clarified by the precise determination of both the fermion masses and the Yukawa coupling constants. If the deviation from the SM relation between the mass and the coupling is found, it can be regarded as an evidence of new physics beyond the SM. The nature of EWSB can be revealed through the experimental reconstruction of the Higgs potential, for which the measurement of the Higgs self-coupling is essential [1,2,3,4,5]. The structure of the Higgs potential depends on the scenario of new physics beyond the SM [6,7], so that the experimental determination of the triple Higgs boson coupling can be a probe of each new physics scenario.
Furthermore, the property of the Higgs potential would be directly related to the aspect of the electroweak phase transition in the early Universe, which could have impact on the problem of the electroweak baryogenesis [8].
It is known that the measurement of the triple Higgs boson coupling is rather challenging at the CERN Large Hadron Collider (LHC), requiring huge luminosity. A study has shown that at the SLHC with the luminosity of 3000 fb −1 , expected accuracy would be about 20-30% for the mass (m h ) of the Higgs boson (h) to be around 170 GeV [1,2]. At the international linear collider (ILC), the main processes for the hhh measurement are the double Higgs boson production mechanisms via the Higgs-stlahlung and the W-boson fusion [3,4]. If the collider energy is lower than 1 TeV, the double Higgs strahlung process e + e − → Zhh is important for a light Higgs boson with the mass of 120-140 GeV, while for higher energies the W-boson fusion process e + e − → hhνν becomes dominant due to its t-channel nature [5]. Sensitivity to the hhh coupling in these processes becomes rapidly worse for greater Higgs boson masses. In particular, for the intermediate mass range (140 GeV < m h < 200 GeV), it has not yet been known how accurately the hhh coupling can be measured by the electron-positron collision.
The photon collider is an option of the ILC. The possibility of measuring the hhh coupling via the process of γγ → hh has been discussed in Ref. [9], where the cross section has been calculated at the one-loop level, and the dependence on the triple Higgs boson coupling constant is studied. In Ref. [10] the statistical sensitivity to the hhh coupling constant has been studied especially for a light Higgs boson mass in relatively low energy collisions.
Recently, these analyses have been extended for wider regions of the Higgs boson masses and the collider energies. It has been found that when the collision energy is limited to be lower than 500-600 GeV the statistical sensitivity to the hhh coupling can be better for the process in the γγ collision than that in the electron-positron collision for the Higgs boson with the mass of 160 GeV [11].
Unlike the double Higgs production processes e + e − → Zhh and e + e − → hhνν in e + e − collisions, γγ → hh is an one-loop induced process. When the origin of the shift in the hhh coupling would be due to one-loop corrections by new particles, it may also affect the amplitude of γγ → hh directly through additional one-particle-irreducible (1PI) one-loop diagrams of γγh and γγhh vertices.
In this letter, we consider the new particle effect on the γγ → hh cross sections in the two Higgs doublet model (THDM), in which additional CP-even, CP-odd and charged Higgs bosons appear. It is known that a non-decoupling one-loop effect due to these extra Higgs bosons can enhance the hhh coupling constant by O(100) % [6]. In the γγ → hh helicity amplitudes, there are additional one-loop diagrams by the charged Higgs boson loop to the ordinary SM diagrams (the W-boson loop and the top quark loop). We find that both the charged Higgs boson loop contribution to the γγ → hh amplitudes and the non-decoupling effect on the hhh coupling can enhance the cross section from its SM value significantly.
We consider how the new contribution to the cross section of γγ → hh would affect the measurement of the triple Higgs boson coupling at a γγ collider.
In order to examine the new physics effect on γγ → hh, we calculate the helicity amplitudes in the THDM. We impose a discrete symmetry to the model to avoid flavor changing neutral current in a natural way [12]. The Higgs potential is then given by where Φ 1 and Φ 2 are two Higgs doublets with hypercharge +1/2. We here include the soft breaking term for the discrete symmetry by the parameter µ 2 3 . In general, µ 2 3 and λ 5 are complex, but we here take them to be real for simplicity. We parameterize the doublet fields where vacuum expectation values (VEVs) v 1 and v 2 satisfy v 2 1 + v 2 2 = v 2 ≃ (246 GeV) 2 . The mass matrices can be diagonalized by introducing the mixing angles α and β, where α diagonalizes the mass matrix of the CP-even neutral bosons, and tan β = v 2 /v 1 . Consequently, we have two CP even (h and H), a CP-odd (A) and a pair of charged (H ± ) bosons. We define α such that h is the SM-like Higgs boson when sin(β − α) = 1. We do not specify the type of Yukawa interactions [13], because it does not much affect the following discussions.
where M(= |µ 3 |/ √ sin β cos β) represents the soft breaking scale for the discrete symmetry, There are several important constraints on the THDM parameters from the data. The LEP direct search results give the lower bounds m h > 114 GeV in the SM-like limit and m H , ∼ 80-90 GeV [14]. In addition, the rho parameter data at the LEP requires the approximate custodial symmetry in the Higgs potential. This implies that m H ± ≃ m A or sin(β − α) ≃ 1 and m H ± ≃ m H . The Higgs potential is also constrained from the tree level 1 For the case without the SM-like limit, see Ref. [7] for example. unitarity [15,16], the triviality and vacuum stability [17], in particular for the case where the non-decoupling effect is important as in the discussion here. For M ∼ 0, masses of the extra Higgs bosons H, A and H ± are bounded from above by about 500 GeV for tan β = 1, when they are degenerated [15]. With non-zero M, these bounds are relaxed depending on the value of M. The constraint from b → sγ gives a lower bound on the mass of H ± depending on the type of Yukawa interaction; i.e., in Model II [13], m H ± > 295 GeV (95% CL) [18].
Recent data for B → τ ν can also give a constraint on the charged Higgs mass especially for large values of tan β in Model II [19,20]. In the following analysis, we do not include these constraints from B-physics because we do not specify type of Yukawa interactions.
The Feynman diagrams which contribute to ∆M are shown in Fig. 1. ∆M is given for each helicity set for sin(β − α) ≃ 1 as and whereŝ,t andû are ordinary Mandelstam variables for the sub processes, and Here we employ the Passarino-Veltman formalism in Ref. [21]. We take the same normalization for these amplitudes as in Ref. [9]. We note that ∆M(+, −, λ hhh ) is independent of λ hhh because of noŝ-channel diagram contribution.
The scalar coupling constants λ hH + H − and λ hhH + H − are defined by The relative sign between M(ℓ 1 , ℓ 2 , λ hhh ) and ∆M(ℓ 1 , ℓ 2 , λ hhh ) has been checked to be consistent with the results for the effective Lagrangian in Eq. (19) in Ref. [22] in the large mass limit for inner particles.
In Eq. (7), λ hhh is the tree level coupling constant. It is known that in the THDM λ hhh can be changed by the one-loop contribution of extra Higgs bosons due to the nondecoupling effect (when M ∼ 0). In the following analysis, we include such an effect on the cross sections replacing λ hhh by the effective coupling Γ THDM hhh (ŝ, m 2 h , m 2 h ), which is evaluated at the one-loop level as [6] Γ THDM is given in Ref. [7], which has been used in our actual numerical analysis. Finally, the cross section for the each subprocess is given by 2 where M 2−loop THDM (ℓ 1 , ℓ 2 ) is defined by We comment on the consistency of our perturbation calculation. One might think that The full cross section of e − e − → γγ → hh is given from the sub cross sections by convoluting the photon luminosity spectrum [9]: where √ s is the centre-of-mass energy of the e − e − system, and where τ =ŝ/s, y = E γ /E b with E γ and E b being the energy of photon and electron beams respectively, and y m = x/(1+x) with x = 4E b ω 0 /m 2 e where ω 0 is the laser photon energy and m e is the electron mass. In our study, we set x = 4.8. The photon momentum distribution function f γ (x, y) and mean helicities of the two photon beams ξ i (i = 1, 2) are given in Ref. [23].
In Fig. 3 that is sensitive to the hhh vertex, we take the polarizations of the initial laser beam to be both −1, and those for the initial electrons to be both +0.45 [9]. The full cross section for m Φ = 400 GeV has similar energy dependences to the sub cross sectionσ(+, +) in Fig. 2, where corresponding energies are rescaled approximately by around √ s ∼ E γγ /0.8 due to We note that the analysis in this letter can be applied to the models [24] in which extra charged scalar bosons appear with a potentially large loop correction in the hhh coupling.
The work of S. K. was supported in part by Grant-in-Aid for Science Research, Japan Note added: After this work was finished, we noticed the paper [25] which studied γγ → hh in the THDM. Our paper includes the additional contribution of the hhh vertex (the leading two-loop effect on γγ → hh), which was not considered in [25]. where we used the fact that the effective γγh and γγhh vertices come from the dimension six operator |Φ i | 2 F µν F µν , so that they are proportial to q 2 /v and q 2 /v 2 at the leading order, respectively, where q 2 ∼ s. Therefore, the effect of m H ± on γγ → hh can be at most log m H ± at the one-loop level. A similar conclusion of power counting can also be obtained for one-loop effects of top and bottom quarks and W bosons to γγ → hh.
Next, let us examine two-loop diagrams shown in Fig. 5. The non-decoupling effect in the diagram (a) in Fig. 5(up-left) is calculated as where momenta of external lines are neglected, and k is the momentum in the loop of the effective hhh vertex, which is replaced by the greatest dimensionful parameter of the system; i.e. m H ± . This result of the power counting is not changed even after the renormalization of the hhh vertex is performed [6]. There are other two loop diagrams which are generated from the s-channel type one-loop diagram, such as the diagram (b) in Fig. 5(up-right) where there is the bridge of h in the H ± triangle type loop. Its non-decoupling effect is evaluated as The dependence on m m ± H is not quartic but quadratic. We have examined all the other two-loop diagrams which are generated from the one-loop s-channel diagram and confirmed that they are the same or less power dependence on m ± H as the diagram (b). A similar counting can also be applied for the diagrams such as the diagram (c) in Fig. 5(down-left) where charged Higgs bosons are running in the both loops, and the diagram