CP-odd Higgs Boson Production in $e\gamma$ Collisions

We investigate the CP-odd Higgs boson production via two-photon processes in $e\gamma$ collisions. The CP-odd Higgs boson, which we denote as $A^0$, is expected to appear in the Two-Higgs Doublet Models (2HDM) as a minimal extension of Higgs sector for which the Minimal Supersymmetric Standard Model (MSSM) is a special case. The scattering amplitude for $e\gamma\rightarrow eA^0$ is evaluated at the electroweak one-loop level. The dominant contribution comes only from top-quark loops when $A^0$ boson is rather light and $\tan\beta$ is not large. There are no contributions from the $W$-boson and $Z$-boson loops nor from the scalar top-quark (stop) loops. The differential cross section for the $A^0$ production is analysed.


Introduction
After the Higgs boson with mass about 125 GeV was discovered by ATLAS and CMS at LHC [1] and its spin, parity and couplings were examined [2], there has been growing interest in constructing a new accelerator facility, like a linear e + e − collider [3], which would offer much cleaner experimental data. Along with e + e − collider, other options such as e − e − , e − γ and γ γ colliders have also been discussed. See Refs. [4][5][6][7][8] and the references therein. Each option for colliders will provide interesting topics to study, such as the detailed measurement of the Higgs boson properties and the quest for the new physics beyond the Standard Model (SM). An e − e − collider is easier to build than an e + e − collider and may stand as a potential candidate before positron sources with high intensity are available. The e − γ and γ γ options are based on e − e − collisions, where one or two of the electron beams are converted to the photon beams.
In our previous papers [9,10], we have studied the SM Higgs boson (H SM ) production in eγ collisions, focusing on the transition form factor of H SM boson [9] and also on the dependence of polarizations of the initial electron and photon beams [10]. In this paper we investigate the production of the CP-odd Higgs boson ( A 0 ), which appears in the 2HDM or in the MSSM [11], in an e − γ collider (Fig. 1). A originally proposed center of mass energy was 500 GeV for an e + e − linear collider [3]. In the light of an e − γ collider, we study for a case when A 0 boson is rather light. More specifically, we assume that the A 0 mass is less than 500 GeV.
We examine the reaction eγ → e A 0 at the one-loop level in the electroweak interaction. Due to the absence of the tree-level Z Z A 0 and W + W − A 0 couplings, the one-loop diagrams which contribute to the reaction are through the γ * γ -fusion and Z * γ -fusion processes. It turns out that the contribution of the γ * γ -fusion diagrams is far more dominant over the one from the Z * γ -fusion diagrams. Thus the A 0 production in eγ collisions is well-described by the "so-called" transition form factor [9]. We investigate the Q 2 dependence of the transition form factor and the production cross section.
In the next section we briefly outline the CP-odd Higgs boson A 0 in the type-II 2HDM or in the MSSM. In section 3, we calculate the one-loop electroweak corrections to the A 0 production in eγ collisions. We also discuss the transition form factor for the γ * γ -fusion process in eγ scattering. In section 4, we present the numerical analysis of the differential cross section for the A 0 production and its dependence on the A 0 mass. The final section is devoted to the concluding remarks.

CP-odd Higgs boson in 2HDM/MSSM
As a minimal extension of the Higgs sector of the SM, we consider the type-II 2HDM which includes the MSSM as a special case [11]. We denote the two SU(2) L doublets H 1 and H 2 with weak hypercharge Y = −1 and Y = 1, respectively, by the 4 complex scalar fields, φ 0 1 , φ − 1 , φ + 2 , φ 0 2 as follows: where, in the type-II model, H 1 (H 2 ) couples only to down-type (up-type) quarks and leptons. They acquire the following vacuum expectation values after the spontaneous symmetry breaking: Then 3 degrees of freedom out of 8 consisting of the 4 complex scalar fields are absorbed by the longitudinal components of W ± , Z , and the remaining 5 degrees of freedom become the following two charged and three neutral physical Higgs bosons: Here we are particularly interested in the CP-odd Higgs boson A 0 and investigate its production in eγ collisions. We enumerate some characteristics of A 0 couplings to other fields in the type-II 2HDM and the MSSM. with [11] Here g and m W are the weak gauge coupling and the weak boson mass, respectively. In the MSSM, charginos also couple to A 0 . When C P is conserved (which we assume in this paper), the diagonal couplings of A 0 to the chargino mass eigenstates are purely pseudoscalar [12], whose couplings are expressed as gκ i γ 5 with (see Eq. (4.32) of [12]), where U and V are 2 × 2 orthogonal matrices. Thus κ i ∼ O(1).
In the following we deal with two chargino mass eigenstates as a whole and write its coupling to A 0 and mass as κ and m χ , respectively. We put Recently at LHC, ATLAS [13] and CMS [14] excluded chargino masses below 1140 GeV for the case that the lightest supersymmetric particles are massless [15]. The results depend on the various scenarios for the production and decay of charginos and neutralinos. We therefore take m χ = 1 TeV as a benchmark mass for chargino in this paper. 4) In the case of the MSSM, the trilinear A 0 coupling to masseigenstate squark pairs q iqi (i = 1, 2) vanishes [11]. Hence, the scalar top-quark (stop) does not contribute to the A 0 production in eγ collisions at one-loop level.

CP-odd Higgs boson production in eγ collisions
We investigate the production of the CP-odd Higgs boson A 0 in an eγ collision experiment ( Fig. 1): where we detect the scattered electron in the final state. The oneloop diagrams which contribute to the reaction (8) are classified into two groups: γ * γ fusion diagrams and Z * γ fusion diagrams ( Fig. 2). As we will see later, the contribution of the former is far more dominant over that of the latter.
Since p is the momentum of a real photon, we have p 2 = 0. We set q = l − l . Assuming that electrons are massless so that l 2 = l 2 = 0, we introduce the following Mandelstam variables: where p 2

One-loop γ * γ fusion diagrams
Due to the characteristics of A 0 couplings to other fields, we take into account only the loops of three fermions (top (t) and bottom (b) quarks and chargino (χ )) for the γ * γ fusion diagrams ( Fig. 2 (a)). The contribution from the one-loop γ * γ fusion diagrams to the scattering amplitude is expressed as where u(l) (u(l )) is the spinor for the initial (scattered) electron with momentum l (l ) and ν (p) is the photon polarization vector where e is the electromagnetic coupling, N f C is a color factor with N t q χ = 1 and C 0 is a Passarino-Veltman three-point scalar integral [16]: The integral C 0 is expressed as the sum of two functions f (τ f ) and g(ρ f ) as where the dimensionless variables τ f and ρ f are defined as and Similar combinations of functions f (τ ) and g(ρ) as in Eq. (14) with the time-like virtual mass, which are different from our space-like case, appear in the Higgs decay processes H SM → γ * γ and H SM → Z * γ in Ref. [17] (see also Ref. [11] for on-shell decays, H SM → γ γ [18] and H SM → Z γ ).

Transition form factor
Inserting the expressions of λ f given in Eqs. (4), (5) and (7) back to Eq. (12), we see that A f μν is expressed as where we have introduced a transition form factor given by Note that for m A < 2m f , i.e. τ f > 1, f (τ f ) is given by Eq. (16) which is a real function, while for m A > 2m f , i.e. τ f < 1 we have f (τ f ) given by Eq. (17) which is a complex function.
Taking the mass parameters as m t = 173 GeV, m b = 4.3 GeV and m χ = 1000 GeV, we analyze the behaviors of |F f (Q 2 , m 2 A , Fig. 3 as a function of m A for the case Q 2 = (100) 2 On the other hand, we obtain Direct searches for heavy neutral Higgs bosons have been performed at LHC. The results were interpreted in the MSSM benchmark scenarios. In the context of the hMSSM scenario [19], ATLAS data [20] excluded tan β > 1.0 for m A = 250 GeV and tan β > 42.0 for m A = 1.5 TeV at the 95% CL. Here in this paper we are dealing with a rather light A 0 boson with mass m A ≤ 500 GeV. Therefore we consider the case where tan β is not large, e.g. tan 2 β ≤ 10.
The production cross section is proportional to the absolute square of the amplitude. Hence the ratio of the bottomquark (charginos) contribution to the one of top-quark is given as the square of the quantity in Eq. (21) (Eq. (22)) multiplied . Then we find that for the case tan 2 β ≤ 10 we can ignore the contributions from the bottomquark and charginos as compared to the one from top-quark. When tan β 10 we can still neglect the bottom-quark contribution but the chargino's contribution becomes the same order as the topquark contribution.
In the following we proceed with our analysis of the reaction eγ → e A 0 assuming that A 0 boson is rather light and tan β is not large.

One-loop Z * γ fusion diagrams
The one-loop Z * γ fusion diagrams for the A 0 production are obtained from the one-loop γ * γ fusion diagrams by replacing the photon propagator with that of the Z boson with mass m Z (Fig. 2 (b)). The loop contributions from three fermions (top (t) and bottom (b) quarks and chargino (χ )) are expressed in terms of the (20). Since the coupling strengths of Z · t · t, Z · b · b and Z · χ · χ vertices are the same order of magnitude, the argument in the previous subsection again follows: we can ignore the contributions from the bottom-quark and charginos for the case when A 0 boson is rather light and tan β is not large while the chargino mass is around 1 TeV.
We consider the top quark loop contribution to the Z * γ fusion diagrams and obtain e A 0 |T |eγ t where f Ze and f Zt are the strength of vector part of the Z -boson coupling to electron and top quark, respectively, and are given by with θ W being the Weinberg angle. In terms of the function F t given in Eq. (20), A t μν is rewritten as

Differential cross section
Adding two amplitudes e A 0 |T |eγ t γ * γ and e A 0 |T |eγ t Z * γ given in Eqs. (11) and (23), we calculate the differential cross section for the A 0 production in eγ collisions with unpolarized initial beams, which turns out to be the sum of three terms: where each corresponds to the contribution of the γ * γ fusion diagrams, the Z * γ fusion diagrams and their interference, respectively, and α em = e 2 /4π .

Numerical analysis
We analyze numerically the three differential cross sections given in Eqs. (27)-(29). We choose the mass parameters and the coupling constants as follows: The electromagnetic constant e 2 is chosen to be the value at the scale of m Z . From Eqs. (25) and (30), we find f Zt f Ze < 0 and, therefore, Eq. (29) shows that the interference between the γ * γ Fig. 4. Comparison of the contribution among three differential cross sections for √ s = 500 GeV, m A = 400 GeV and cot β = 1. and Z * γ fusion diagrams works constructively and dσ (Interference) dt is positive.
We plot these differential cross sections as a function of Q 2 in Fig. 4 for the case √ s = 500 GeV, m A = 400 GeV and cot β = 1. (In fact, the cross sections are proportional to cot 2 β.) We find that the contribution from the γ * γ fusion diagrams is far more dominant over those from Z * γ -fusion diagrams as well as from the interference term. Actually we observe that at Q 2 = 1000 (5000) GeV 2 , the ratio of dσ (Z * γ ) . Thus the A 0 production in eγ collisions is well-described by the γ * γ fusion diagrams with the top quark loop. This means that the transition form factor of the A 0 boson defined as N t C q 2 Eq. (20) indeed makes sense and may be measurable in eγ collider experiments.
Now we shall focus on the γ * γ fusion process based on the formula for the production cross section given in Eq. (27). In Fig. 5 we plot the differential production cross section of A 0 with mass m A = 200, 300, 400 GeV for the case √ s = 500 GeV and cot β = 1.
We find that for this kinematical region the production cross section for A 0 increases as m A gets larger, which looks somewhat unexpected at first glance. We examine this behavior in more detail by computing the mass dependence of the differential cross section. We plot in Fig. 6 the dependence of the differential cross section dσ /dQ 2 on the A 0 boson mass with Q 2 = (80) 2 , (90) 2 and (100) 2 GeV 2 for the case √ s = 500 GeV and cot β = 1. We see that, in the region m A < 2m t , the differential cross section dσ /dQ 2 with fixed Q 2 increases along with m A . When m A goes beyond 2m t , it turns to decrease. We observe the strong kink structure corresponding to the threshold effect at m A = 2m t ≈ 346 GeV (see Eqs. (16) and (17)).

Summary and discussion
In this paper we have investigated the production of the CP-odd Higgs boson A 0 which appears in the type-II 2HDM and the MSSM through eγ collisions. In contrast to the SM Higgs boson H S M or the CP-even Higgs boson h 0 and H 0 , the A 0 boson does not couple to W + W − and Z Z pairs because of the CP-odd nature. Hence W -boson and Z -boson loop diagrams do not contribute to the A 0 production at one-loop level.
The A 0 production arises via γ * γ fusion or via Z * γ fusion processes. It has turned out that because of the smallness of the e-e-Z and Z -t-t couplings as well as the Z boson propagator, the contribution from the γ * γ fusion diagrams is far more dominant over that from Z * γ fusion. Thus, in effect, we have to consider only the photon-exchange diagrams, and it makes sense to introduce the transition form factor of the A 0 boson.
Up to the electroweak one-loop order, the top quark triangle diagrams are only relevant for the production of the A 0 boson when A 0 boson is rather light and tan β is not large. There is no scalar top-quark (stop) contribution. Thus the production amplitude as well as the transition form factor show much simpler structure compared with those of the SM Higgs boson or the CP-even Higgs bosons.
When the mass of the A 0 boson, m A is smaller than 2m t the transition form factor is a real function of Q 2 , while if m A is larger than 2m t , the transition form factor becomes complex. The production cross section of the A 0 boson is given by the absolute square of the transition form factor together with some kinematical factors.
For a fixed value of m A , the differential production cross section shows a decreasing function of Q 2 . On the other hand, if we fix Q 2 and vary the mass of A 0 , it increases as m A for m A < 2m t and decreases for m A > 2m t . This feature is common with the total cross section.