Full one-loop electroweak corrections to $e^+e^- \to ZH\gamma$ at a Higgs factory

Motivated by the future precision test of the Higgs boson at an $e^+e^-$ Higgs factory, we calculate the production $e^+e^- \to ZH\gamma$ in the Standard Model with complete next-to-leading order electroweak corrections. We find that for $\sqrt{s}=240$ (350) GeV the cross section of this production is sizably reduced by the electroweak corrections, which is $1.03$ (5.32) fb at leading order and 0.72 (4.79) fb at next-to-leading order. The transverse momentum distribution of the photon in the final states is also presented.


INTRODUCTION
Recently, a Standard Model (SM)-like Higgs boson around 125 GeV was observed by ATLAS and CMS collaborations at the LHC [1,2]. This discovery is a great step towards the understanding of electroweak symmetry breaking of the SM. So far, most measurements of the properties of this new boson are consistent with the SM prediction. The new physics that affects the Higgs couplings has been cornered to a decoupling region [3,4]. Besides, since many extensions of the SM (like the supersymmetric models) contain a SM-like Higgs boson [5] whose properties can be quite similar to the SM Higgs boson, it is difficult for the LHC to verify whether or not this new boson is the SM one. In order to precisely study this newly discovered Higgs boson, an e + e − collider, the so-called Higgs factory, is needed.
In such an e + e − Higgs factory, the properties of Higgs boson can be measured with rather high precisions [6][7][8]. The dominant Higgs production is the Higgs-strahlung process e + e − → ZH, where the ZH events can be inclusively detected by tagging a leptonic Z decay without the assumption of the Higgs decay mode. The individual Higgs decay branching ratios can then be directly measured as the fractions of the total e + e − → ZH cross section by observing the specific states. For √ s ∼ 240−250 GeV with an integrated luminosity of 500 fb −1 , about O(10 5 ) Higgs bosons can be produced per year, which allows to measure the Higgs couplings at a few percent [8]. So the electroweak radiative corrections should be taken into account in the theoretical calculations of the production rate. For the process e + e − → ZH, the leading order calculation was performed in [9] and the one-loop electroweak corrections were calculated with the soft-photon approximation in [10][11][12] (a compact analytical formula for the electromagnetic corrections was given in [11] and a numerical calculation algorithm for the real photon emission was proposed in [13]).
For an e + e − Higgs factory with √ s ∼ 240−250 GeV another possibly important process is e + e − → HZγ. On one hand, it is an important part of the inclusive process e + e − → ZH +X or can be distinguished for a hard photon; On the other hand, since the HZγ vertex occurs at one loop in the SM, the HZγ couplings is particularly sensitive to possible new physics contributions, such as the existence of new heavy particles propagating in the loop [14,15].
In this work we calculate this production in the SM with the complete next-to-leading order electroweak (NLO EW) corrections. In Sec. II we will give a description for the analytic calculations. The numerical results and discussions are given in Sec. III. Finally, we draw

II. A DESCRIPTION OF ANALYTICAL CALCULATIONS
In the SM the process e + e − → HZγ is induced by the electroweak interaction at leading order (LO). Due to the small Yukawa couplings, we ignore the contributions from the Feynman diagrams involving the Yukawa couplings of light fermions. We denote the fourmomenta of initial and final states in the process as The NLO EW corrections (∆σ EW ) include two parts: • Virtual correction (∆σ vir ).
We adopt the dimensional regularization to isolate the ultraviolet divergences (UV) in the one-loop amplitudes. Then we remove the UV singularities by using the on-massshell renormalization scheme [16]. The pentagon Feynman diagrams in the calculation are presented in Fig.1. The reductions of N-point (N ≤ 4) tensor integrals are implemented by using the Passarino-Veltman algorithm [17]. But for the calculation of the 5-point tensor functions, we adopt the Denner-Dittmaier method developed in Ref. [18] to reduce the tensor integrals and use our fortran subroutines to perform numerical study, which has been validated in our previous works [19,20]. We also numerically checked that our results are UV finite.
Due to the exchange of virtual photon in the loops, the infrared (IR) divergences can appear in the virtual correction. According to the Kinoshita-Lee-Nauenberg (KLN) theorem [21], these IR divergences will be canceled by the real photon bremsstrahlung corrections in the soft photon limit. We denote the momenta of initial and final states for the real photon radiation process as We take the phase-space-slicing method [22,23] to isolate the IR singularity in the above process. An arbitrarily small cut-off parameter δ s is introduced to split the phase space into soft region (E 6 ≤ δ s √ s/2) and hard region (E 6 > δ s √ s/2). So the real photon emission correction can be decomposed into the soft and hard parts: In the soft photon approximation [24], we can calculate the soft part of the correction by using the following equation where E 6 = | q 6 | 2 + m 2 γ and we give a small mass m γ to the photon to eliminate the IR divergence (we checked that the dependence on this non-physical mass m γ is exactly canceled when the real radiation correction and the virtual correction are combined).
Since the hard part of the correction is insensitive to this fictitious photon mass, it can be directly evaluated by the numerical Monte Carlo method [25]. We notice that there are two photons in the real emission process and one of them should be tagged as the observed hard photon with p T > 10 GeV and |η| < 2. The phase space integral of these two identical photons in hard part of real emission can be expressed as: where the factor 1 2 is from the identical photons in the final states, and E c is the energy cut that corresponds to the above hard p T cut. In order to improve the numerical stability of Eq.(5), we adopt the method in the Ref. [26] to carry out the integral Eq.(5). Since each of the two photons in the final states can be softer or harder than the other one with an equal probability, the Eq.(5) can be equivalent to: This means that we can technically assume the photon γ(q 5 ) to be the tagged hard photon and impose a transverse momentum cut p T > 10 GeV and pseudo-rapidity cut |η| < 2 on γ(q 5 ) in the numerical calculations [26,27].
We also numerically checked our LO calculations in Feynman gauge(F.G.) with the package

III. NUMERICAL RESULTS AND DISCUSSIONS
In the numerical calculations we take the input parameters of the SM as [34] m t = 171.2 GeV, m e = 0.519991 MeV, m Z = 91.19 GeV, The Higgs mass is taken as m H = 125.66 ± 0.34 GeV [4], which is the combined result of the measurements of the ATLAS and CMS collaborations.
We numerically check the stability of the results versus the soft photon cutoff parameter in  can be seen that the values of ∆σ vir , ∆σ hard and ∆σ sof t depend on the soft cutoff log δ s , while the total NLO EW correction ∆σ tot is independent of log δ s within reasonable calculation errors. Besides, we checked that the total correction is independent of m γ for a fixed δ s . Therefore, in the following calculations we take the δ s = 2 × 10 −3 and m γ = 10 −8 GeV.
In Fig.3   Finally in Fig.4 we show the transverse momentum distribution of the photon in the process e + e − → HZγ at LO and NLO for √ s = 240, 350 GeV. It can be seen that the NLO EW correction can greatly reduce the LO differential cross section at low p T region. The impact of the uncertainty of the Higgs mass on the p T distribution becomes weak as the collider energy increases. For √ s = 240 GeV most of the events are produced in the region of p γ T < 20 GeV due to the center-of-mass energy close to the production threshold; while for √ s = 350 GeV the p T value of the photon gets much harder.

IV. CONCLUSION
In this work we calculated the cross section of e + e − → ZHγ with complete next-toleading order electroweak corrections in the SM. We found that for √ s = 240 (350) GeV the cross section of this production can reach 1.03 (5.32) fb at leading order and 0.72 (4.79) fb at next-to-leading order. In a future e + e − Higgs factory, this process can be measured as a precision test of the SM.