Probing $HZ\gamma$ and $H\gamma\gamma$ Anomalous Couplings in the Process of $e^+e^- \to H\gamma$

We propose to measure the $HZ\gamma$ and $H\gamma\gamma$ anomalous couplings in the process of $e^{+}e^{-}\rightarrow H\gamma$ with the sequential decay of $H\to b{\bar b}$. The discovery potential of observing the anomalous couplings are explored in details. Our study shows that the electron-positron collider has a great potential of testing the $HZ\gamma$ and $H\gamma\gamma$ couplings. Conservative bounds on the two anomalous couplings are also derived when no new physics signal were detected on top of the SM backgrounds.


Introduction
After the discovery of the Higgs boson, precision measurement of the Higgs boson's properties is placed on the agenda, especially the measurement of the rare decay modes of the Higgs boson as the Standard Model (SM) contribution is fairly small. Observing a deviation from the SM prediction would shed light on new physics (NP) beyond the SM. Among the rare decay modes of the Higgs boson, the γγ mode is bounded much tighter than others, and its best-fit signal strength relative to the standard model prediction is 1.17±0.27 obtained by the ATLAS collaboration [1] and 1.14 +0. 26 −0.23 by the CMS collaboration [2], respectively. The H → Zγ decay, however, is loosely constrained. ATLAS collaboration reported an upper limit of 11 times the SM expectation at the 95% confidence level [3]. A similar result is achieved by the CMS Collaboration [4], which sets an upper limit of 9.5 times the SM expectation at the 95% confidence level. Note that the Hγγ and HZγ couplings are sensitive to different kinds of NP and therefore are independent in principle. Ref. [5] pointed out that the HZγ coupling could be sizably modified in certain composite Higgs model while still keeping the Hγγ coupling untouched. On the other hand, the HZγ and Hγγ couplings were highly correlated in the NMSSM or MSSMlike models [6,7]. Thus the NP models can be tested and discriminated by their different expected correction of the HZγ and Hγγ couplings. In this work, we explore the potential of probing the anomalous couplings of HZγ and Hγγ through the Hγ production at the fu-ture electron-positron collider.
In this work we assume the NP resonances are too heavy to be observed directly at the LHC, but they might generate sizable quantum corrections. Such effects are then described by an effective Lagrangian of the form where c i 's are coefficients that parameterize the nonstandard interactions. Note that dimension-5 operators involve fermion number violation and are assumed to be associated with a very high energy scale and are not relevant to the processes studied here. The relevant CPconserving operators O i contributing to the anomalous HZγ and Hγγ couplings are [22] 1) E-mail: qinghongcao@pku.edu.cn 2) E-mail: haoranw@pku.edu.cn 3) E-mail: zhangya1221@pku.edu.cn in which φ T = (0, (v+H)/ √ 2) is the Higgs doublet in the unitary gauge with v = 246 GeV the vacuum expectation value, are the strength tensors of the gauge fields, and the Lie communicators [T a , T b ] = if abc T c define the structure constants f abc .
The O φφ and O BW are constrained strongly by the electroweak precision measurements [12,23] and are neglected in our study. After spontaneous symmetry breaking, the operators yield the effective Lagrangian in terms of the mass eigenstates of photon and Z-boson as follows: Therefore, the other two couplings would exhibit a nontrial relation which could be verified in future experiments.
2 Hγ production at e − e + collider Now we are ready to calculate the Hγ production with the contributions of the HZγ and Hγγ anomalous couplings. There is a subtlety in the calculation. The scattering process of e + e − → Hγ is absent at the treelevel in the SM when ignoring the electron mass, but it can be generated through the electroweak corrections at the loop-level [17][18][19]. The effects of the HZγ and Hγγ anomalous couplings, as suppressed by the NP scale Λ, might be comparable to those SM loop effects. Therefore, one has to consider the SM contributions as well in the discussion of the NP effects in the Hγ production. The loop corrections in the SM can be categorized as follows: (1) the bubble diagrams originating from the external γ wave-function renormalization; (2) the triangle diagrams with the HZγ, Hγγ or the Hee in the external lines; (3) the box diagrams with e + e − Hγ in the external line. Figure 1 displays the representative Feynman diagrams, which also includes the HZγ anomalous coupling. Consider the case of unpolarized incoming beams and ignore the electron mass. Summing over the polarization of the photon, the differential cross section of the scattering of e − e + → Hγ can be written as [19] dσ(e + e − → Hγ) d cos θ where √ s is the energy of center-of-mass (c.m.) and the Mandelstam variables are with p i the momentum of particle i and θ the scattering angle of the photon. The coefficient a i , which sums contributions from all the loop diagrams and the anomalous HZγ and Hγγ couplings, is where a γ i and a Z i denote the contributions of the photon and Z pole vertex diagrams, a e i the t-channel H 0 ee vertex corrections and a box i the contribution of the box diagrams; see Fig. 1. Detailed expression of all the coefficients in the SM can be found in Ref. [19]. The anomalous F Zγ and F γγ couplings contribute only to a Z ± i and a γ ± i as follows: where e is the electric charge, The F Z,W , F γ,W and F t are obtained from the gauge boson (W and Z) and top-quark loops respectively. Only the top-quark loop is taken into account in this work as the contributions from other fermion loops are highly suppressed. The F Z,W , F γ,W and F t are where the three-point functions C t ij and C W ij are defined as and C 0 is the Passarino-Veltman scalar function [24]. We first calculate the SM loop corrections in Form-Calc [25] and LoopTools [26]. Our analytical and numerical results are consistent with those in Refs. [19]. We then incorporate the HZγ and Hγγ anomalous couplings in our calculation to examine their impact on the Hγ production, respectively.
In order to quantify the NP effects, we separate the total cross session of the Hγ production (σ t ) into the following three pieces: where σ SM is the cross section in the SM, σ √ s 2m t , the virtual topquark loop develops an imaginary part and thus contributes maximally. Above the top-quark pair threshold, the cross section drops smoothly with √ s as expected. The interference effect (σ (1,2) IN ) exhibits a similar behavior as the SM contribution and drops with √ s. On the contrary, the NP contributions (σ (1,2) NP ) increase with √ s as induced by a high-dimensional operator.
The interference effects between the SM and NP depend on the sign of the effective HZγ, Hγγ couplings. We plot in Fig. 2(b) the total cross section for F Zγ = ±1. For reference σ SM , i.e. F Zγ = 0, is also plotted. For a large F Zγ , the NP contribution dominates over the interference and SM contributions. We also plot in Fig. 2(c) the total cross section for F Zγ = ±0.1 to illustrate the interference effects. For a small F Zγ , we can ignore the NP contributions as it is proportional to F 2 Zγ . Therefore, the interference effects yield three similar curves. This discussion above is also applied to F γγ displayed in For illustration we list the total cross section (in the unit of femtobarn) for four benchmark of c.m. energies ( √ s) as follows:  In this section, we discuss how to detect the HZγ and Hγγ anomalous couplings at the e + e − collider with various c.m. energies. Firstly we focus on the contribution of HZγ with the bb mode of the Higgs boson decay where F Zγ = 1 and F γγ = 0. The collider signature of interests to us is one hard photon and two b-jets. We generate the dominant backgrounds with MadGraph [27] At the analysis level, all signal and background events are required to pass the following selection cuts: where p i T and η i denotes the transverse momentum and pseudo-rapidity of the particle i, respectively. The separation ∆R in the azimuthal angle-pseudo-rapidity (φ-η) plane between the objects k and l is For simplicity we ignore the effects due to the finite resolution of the detector and assume a perfect b-tagging efficiency.  To compare the relevant background event rates (B) to the signal event rates (S), we assume an integrated luminosity of 1 ab −1 . The numbers of the signal and background events after imposing the above selection cuts are summarized in the second, fourth, eighth, twelfth rows of Table 1. We consider three kinds of the signal: one is induced solely by the SM loop corrections, the other two are generated both by the SM loop correction and by NP effects where F Zγ = 1, F γγ = 0 for one and F Zγ = 0, F γγ = 1 for the other. The former is named as the S SM , shown in the fourth to sixth rows in Table 1, while the latter are denoted as the S Zγ/γγ , shown in the seventh to fourteenth rows. Obviously, the backgrounds are larger than the signals by three or four order of magnitudes. One has to impose other cuts to extract the small signal out of the huge background. As from the Higgs boson decay, the two b-jets in the signal events exhibit a sharp peak around m H in the distribution of their invariant mass (m bb ). Figure 4 displays the m bb distribution of the signal events (red-peak) and the background events (blue) for √ s = 250 GeV. The    two b-jets in the background events are mainly from the on-shell Z-boson, yielding a peak around m Z ∼ 91 GeV. The background events also exhibit a long tail in the region of m bb ∼ m H region owing to the Z-boson width.
The difference of the m bb distribution between the signal and background events remains at other √ s of the e + e − collider. We impose a hard cut on m bb to suppress the background. After the lepton and jet reconstruction, we demand that the invariant mass of the two b-jets is within a mass window of 5 GeV around m H , i.e.
The ∆M cut suppresses the background dramatically; for example, for almost all the c.m. energies, less than 1% of the background survives after the ∆M cut. On the other hand, most of the signal events pass the mass window cut. Unfortunately, the SM contribution alone still cannot be observed owing to the tiny production rate; see the fifth row in Table 1. For F Zγ = 1, F γγ = 0 and F Zγ = 0, F γγ = 1, both the anomalous HZγ coupling and Hγγ coupling lead to a few hundreds of the signal events after the mass window cut separately and thus are testable experimentally. The significance (S Zγ/γγ / √ B) increases with √ s owing both to the non-renormalizable feature of the high-dimensional operators and to the decreasing SM backgrounds.
We now use the results of last section to discuss the potential of testing the HZγ, Hγγ couplings at the electron-positron linear collider. Most attention is paid on the scenario in which only one of HZγ and Hγγ anomalous couplings is nonzero. We first consider the discovery of HZγ and Hγγ anomalous couplings at the electron-positron linear collider. Demanding the 5σ significance, S Zγ/γγ = 5 √ B, yields the discovery potential of the HZγ/Hγγ coupling in the scattering of e + e − → Hγ. Figures 5(a), (d) display the 5σ significance curve (dashed-line). The shade regions are good for the discovery of the anomalous HZγ/Hγγ coupling. Owing to the SM contribution and the interference effects, the discovery regions are asymmetric around F Zγ/γγ = 0. We also plot the CMS exclusion limits of the HZγ/Hγγ coupling. We note that the discovery potential of HZγ coupling at e − e + collider at √ s = 250 GeV is marginally close to the current CMS exclusion limit. With the c.m. energy increased from 250 GeV to 1000 GeV, the e + e − collider could cover the regions of 0.50 < F Zγ < 1.03 and −2.02 < F Zγ < −0.76 which cannot be probed at the 8 TeV LHC; while the discovery potential of Hγγ coupling could cover the nonexclusion red region of F γγ ∼ 0.56 at a high energy e − e + collider.
The CMS limits are derived from the Higgs boson decay as follows. The partial decay width of H → Zγ and H → γγ are given by where F SM Zγ , F SM γγ , induced by the W boson and fermion loops in the SM, are given by [5,28] The , are given in Ref. [29] where τ i = 4m 2 i /m 2 H and λ i = 4m 2 i /m 2 Z . Q t is the top-quark electric charge in units of |e| and T t 3 = 1/2. In the SM F SM Zγ ∼ 0.007, F SM γγ ∼ −0.004 for m H = 125 GeV [30]. The CMS measurement requires which yields the CMS exclusion bounds shown in Figs. 5(a) and (d), one bound on F Zγ as −2.02 ≤ F Zγ ≤ 1.03, two bounds on F γγ as −0.051 ≤ F γγ ≤ 0.013 and 0.55 ≤ F γγ ≤ 0.62; see the horizontal black-dashed curves and red regions. A recent study on projected performance of an upgraded CMS detector at the LHC and high luminosity LHC (HL-LHC) [31] shows that the H → Zγ process is expected to be measured at 14 TeV LHC with ∼ 60 % and ∼ 20 % uncertainties at the 95 % confidence level using an integrated dataset of 300 fb −1 and 3000 fb −1 , respectively, while for the H → γγ process, the uncertainties are ∼ 6 % and ∼ 4 %. We plot the corresponding CMS projection limits in Figs. 5(b), (e) and Figs. 5(c), (f). Future experiments at the LHC and HL-LHC are expected to impose tighter bounds on F Zγ/γγ . For the negative F Zγ and F γγ ∼ 0.56, the e + e − collider has a better performance than the LHC and HL-LHC at a high energy level; see the overlapping regions of the red region and shaded region.
When both the HZγ and Hγγ couplings are considered, Fig. 6 displays the total cross section of Hγ production changing as a function of F Zγ and F γγ with various energies. The allowed discovery regions of F Zγ and F γγ are the red regions outside the black dashed lines. With the c.m. energy increased from 250 GeV to 1000 GeV, more and more red regions can be discovered. When √ s ≥ 500 GeV, the non-exclusive red region of F γγ ∼ 0.56 are entirely allowed. For more detail, see Eqs. 15.

Further Analysis
The HZγ and Hγγ anomalous couplings affect both the Higgs boson decay and the Hγ production, but their interference effects with the SM contributions is different for the two processes. In order to examine the different interference effects, we define a ratio of the cross section of the Hγ production, R σ , a ratio of the width of H → Zγ/γγ decay, R Zγ/γγ , and the relative sign µ Zγ/γγ , as follows: Figure 7 displays the strong correlation between R σ and R Zγ/γγ for several c.m. energies when one anomalous coupling is considered at a time; see the red-dashed curves. There are two values of R σ for each fixed R Zγ/γγ ; the larger value R σ corresponds to µ Zγ/γγ < 0 while the smaller value to µ Zγ/γγ > 0. The two-fold ambiguity in the Γ(H → Zγ/γγ) measurement can be resolved by precise knowledge of R σ if the F Zγ/γγ is large enough to discover the Hγ signal at the e + e − collider. In Fig. 7 we also plot the discovery region of R Zγ/γγ in the scattering of e + e − → Hγ for various c.m.energies; see the shaded bands. One can uniquely determine both the magnitude and sign of F Zγ/γγ in those shaded-band regions. The discrimination power of the two-fold R σ for a fixed R Zγ/γγ increases dramatically with c.m. energy of e + e − collider; for example, for R Zγ = 9, R σ is equal to 8 and 10 at a √ s = 250 GeV collider while it is equal to 40 and 110 at a √ s = 1000 GeV collider. It is worthwhile to mentioning that the partial decay width of H → Zγ is exactly the same as the SM prediction when v 2 /Λ 2 F Zγ = −2F SM Zγ . In that case one can still observe the anomalous HZγ coupling at the e + e − collider when √ s > ∼ 500 GeV. For the R γγ , it is highly limited by the current LHC data and yields two solutions of F γγ : one is v 2 /Λ 2 F γγ ∼ −2F SM γγ which could be detected in the Hγ production when √ s ≥ 500 GeV, the other is F γγ ∼ 0 which cannot be probed. If no NP effects were observed in the Hγ production, one can obtain a 2σ exclusion limits of F Zγ/γγ which are displayed in Fig. 8. The CMS current and projection sensitivities are also plotted for comparison; see the red-shaded region.

Summary
We study the potential of measuring the HZγ and Hγγ anomalous couplings in the process of e − e + → Hγ. Such a scattering process occurs only at the loop level in the SM. After considering the interference of the SM loop effects and the anomalous coupling contributions, we perform a collider simulation of the the Hγ production with H → bb. Even though the SM contribution alone cannot be detected, the anomalous couplings can enhance the production rate sizeably and lead to a discovery at the high energy electron-positron collider with an integrated luminosity of 1 ab −1 .
When considering one anomalous coupling at a time, our study shows that, for negative F Zγ or F γγ ∼ 0.56, the e + e − collider has a better performance than the current LHC and HL-LHC. When both couplings contribute simultaneously to the Hγ production, more parameter regions are allowed and can be fully explored at a high energy e + e − collider.
We also derive exclusion bounds on the anomalous couplings in the case that no NP effects were observed in the Hγ production. The current CMS data indicates a two-fold solution of the anomalous coupling. Resolving such an ambiguity is beyond the capability of the upgraded LHC or High luminosity LHC. But it can be discriminated easily at the e + e − collider.
The work is supported in part by the National Science Foundation of China under Grand No. 11275009.