Hypergeometric presentation for one-loop contributing to $H\rightarrow Z\gamma$

In this paper, new analytic formulas for one-loop contributing to Higgs decay channel $H \rightarrow Z\gamma$ are presented in terms of hypergeometric functions. The calculations are performed by following the technique for tensor one-loop reduction developed in [A.~I.~Davydychev, Phys.\ Lett.\ B {\bf 263} (1991) 107]. For the first time, one-loop form factors for the decay process are shown which are valid at arbitrary space-time dimension $d$.


Introduction
Among Higgs (H) decay processes, the decay channel H → Zγ is the most important at the Large Hadron Collider (LHC) [1][2][3]. Because the channel arises at first from one-loop Feynman diagrams. As a result, the decay width of this channel is sensitive to new physics in which we assume that new heavy particles may exchange in one-loop diagrams. For this reason, theoretical evaluations for one-loop and higher-loop decay amplitudes of H → Zγ play crucial roles in controlling standard model (SM) background as well as constraining physical parameters in many beyond standard models (BSM).
There have been many computations for one-loop contributions to H → Zγ within SM and its extensions in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The calculations have performed following the method for tensor one-loop reduction in [22]. When one-loop contributions to H → Zγ are evaluated in unitary gauge, the results may meet large numerical cancellations. This is because higher-rank tensor one-loop integrals appears from Feynman loop diagrams with exchanging by vector bosons. To avoid this problem, many of the above references have considered the calculations in 't Hooft-Feynman gauge. In this gauge, we need to handle more Feynman diagrams with exchanging by Goldstone bosons. As a result, the calculations are rather complicated. Furthermore, when we consider two-loop or higher-loop corrections to H → Zγ, two-loop and higher-loop Feynman integrals may be evaluated by applying methods [23][24][25], the resulting integrals may contain the one-loop integrals in general space-time dimension. These integrals have been not available in previous papers.
In this paper, we apply an alternative approach for evaluating one-loop contributions to H → Zγ. In this calculation, we follow the method for tensor one-loop reduction developed in [26] in which tensor integrals are decomposed into scalar functions with arbitrary propagator indexes and at higher space-time dimension d > 4. Using integration-by-part method (IBP) [23,24], scalar one-loop integrals are then expressed in terms of master integrals which can be solved analytically via generalized hypergeometric series. For instant, analytic formulas for the master integrals which are one-loop one-, two-, three-point functions at general d appearing in H → Zγ are provided in this work. Therefore, our methods are easy to apply for H → Zγ and expect to be numerical stability in unitary gauge. Furthermore, our analytic expressions for the form factors of the decay process are general as well as valid at arbitrary space-time dimension. The layout of the paper is as follows: In section 2, we present a general method for evaluating one-loop Feynman integrals. Using the method, the computations for one-loop contributions to Higgs decay to Z photon are reported in the section 3. Conclusions are shown in section 4. Several useful formulas used in this calculation and detailed calculations for the process amplitudes are given in the appendixes.

Method
In this section, we describe a general approach for evaluating one-loop Feynman integrals. In general, tensor one-loop N -point Feynman integrals with rank M are defined as follows: Where p i (m i ) for i = 1, 2, · · · , N are external momenta (internal masses) respectively. In this convention, q 1 = p 1 , q 2 = p 1 + p 2 , · · · , q i = i j=1 p j , and q N = N j=1 p j = 0 thanks to momentum conservation. The term iρ is Feynman's prescription and d is space-time dimension. One of physical interests is d = 4 + 2n − 2ǫ for n ∈ N.
In the Appendix B, this method is demonstrated in detail for the case of H → Zγ. We show here all analytic results for the master integrals involving the decay process. In particular, scalar one-loop one-point functions with arbitrary propagator index ν are given [27]: In unitary gauge, the decay process H → Zγ consists top loop and W boson loop as shown in Figs. 1, 2. In general, the total amplitude of the decay H → Zγ is expressed in terms of form factors with reflecting the Lorentz invariant structure and the content of gauge symmetry as follows: Where ε µ * 1 and ε ν * 2 are the polarization vectors of the Z boson and the photon γ respectively. ǫ µναβ is the Levi-Civita tensor. Kinematic invariant variables related to this process are We also have ε ν * 2 (q 2 )q 2,ν = 0 for external photon. Following Ward identity, we confirm that and F 12,22 do not contribute to the total amplitude. Summing all the top-loop diagrams, the result shows that F 5 = 0. Detailed calculations for the form factors at general d are presented in the appendix D. The total amplitude for this decay process is then casted in the form of where F H→Zγ (d; are form factors which can be derived from F 00 or F 21 . These form factors are decomposed in terms of W -loop and top-loop (including fermion-loop) contributions as follows: Where θ W is Weinberg angle, I 3 f , Q f and m f are iso-spin, electric charge, mass of fermions f in the loops respectively. N C is a color factor for fermions. It becomes 1 for leptons and 3 4/18   for quarks. We use the symbolic-manipulation Package-X [33] to handle all Dirac and tensor algebra in d dimension.

Form factors
We show two representations for the form factors in terms of 3 F 2 hypergeometric functions in this subsection.

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3.1.1. First representation. We first present the form factors which are derived from F 00 in (9) in terms of 3 F 2 hypergeometric functions as follows: The form factors F  H , M 2 Z , m 2 f ). In the limit d → 4, we confirm that the terms in curly brackets of right hand side results of (14,15) tend to zero It means that the form factors always stay finite in the limit. 6/18 3.1.2. Second representation. Another presentation for the form factors which are obtained from F 21 in (9) are given: In the limit d → 4, we also confirm that the terms in curly bracket of right hand side results of (18,19) tend to zero. It means that the form factors always stay in finite in the limit.

H → γγ reduction
In order to reduce to H → γγ, we take M 2 Z → 0, and λ f 1 = eQ f , λ f 2 , λ f 3 → 0, the total amplitude of the decay H → Zγ is reduced to H → γγ. In detail, the results read Where the form factors are given and F (W ) To arrive at the last line result, we have already used the transformation for hypergeometric functions 3 F 2 in Eq. (31). These agree with the results in [32].

Numerical results
In of hypergeometric functions. We first confirm two representations for the form factors in (14,15) and (18,19) at general d. It means that we verify numerically the Ward identity at general d. In Tables 1, 2, we show numerical checks for the form factors at general d. Two  representations for the form factors are perfect agreement up to last digit for 3 Table 2 Numerical confirmations for two representations for the form factors involving to W -loop diagrams at arbitrary d.
We next perform higher-order ǫ-expansion for the form factors in this work up to ǫ 5 . We also compare our results with [19] (F SM 21,W ) at ǫ 0 -terms. Our numerical results are shown in Eqs. (25,27). We find a perfect agreement between two results at ǫ 0 -expansion. It is important to note that higher-power ǫ-expansions for the form factors in this paper are our 9/18 first results.

Conclusions
In this paper, we have discussed the alternative approach for evaluating one- 10/18 The Mellin-Barnes representation for 3 F 2 is provided that |Arg(−z)| < π. The integration contour is chosen in such a way that the poles of Γ(−s) and Γ(· · · + s) are well-separated. Analytic continuation of 3 F 2 functions: In this work, a useful transformation for 3 F 2 functions is mentioned: Appendix B: Calculating master integrals Tensor one-loop three-point Feynman integrals with rank M appearing in the process H → Zγ are given as follows: Where the inverse Feynman propagators are The related kinematic invariant are q 2 1 = M 2 Z , q 2 2 = 0, and p 2 = (q 1 + q 2 ) 2 = M 2 H . In this paper, p 2 2 = M 2 Z , 0 and internal masses M 2 = m 2 f , M 2 W . After presenting tensor one-loop three-point integrals to scalar functions, we next apply IBP for scalar one-loop functions with the general propagator indexes. We then arrive at the following system of equations Here, the standard notation for increasing and lowering operators is used for j = 1, 2, 3.