Next-to-leading order QCD corrections to Higgs boson decay to quarkonium plus a photon

In this paper, we investigate the decay of Higgs boson to $J/\psi(\Upsilon)$ plus a photon based on NRQCD factorization. For the direct process, we calculate the decay width up to QCD NLO. We find that the decay width for process $H \to J/\psi(\Upsilon)+ \gamma$ direct production at the LO is significantly reduced by the NLO QCD corrections. For the indirect process, we calculate the $H \to \gamma^\ast\gamma$ with virtual $\gamma$ substantially decaying to $J/\psi(\Upsilon)$, including all the SM Feynman diagrams. The decay width of indirect production is much larger than the direct decay width. Since it is very clean in experiment, the $H \to J/\psi(\Upsilon)+ \gamma$ decay could be observable at a 14 TeV LHC and it also offers a new way to probe the Yukawa coupling and New Physics at the LHC.


I. INTRODUCTION
Recently, both ATLAS and CMS collaborations have announced that they observed a new boson with mass around 125 GeV, whose properties are consistent with the Standard Model(SM) Higgs in any measured channel [1][2][3][4]. After discovery of the Higgs boson, the main task is to determine its properties, such as spin, CP, and couplings. The couplings to gauge bosons and the third-generation fermions are measured directly, which are fixed through the well measured diboson decays of the Higgs determined at the 20 ∼ 30% level.
However, we have little information about the Higgs Yukawa couplings to the first-and second-generation quarks at current experiments, since these couplings are predicted to be small in the SM, and the inclusive decays of the Higgs to these states are swamped by large QCD backgrounds. These couplings are indirectly and weakly constrained by the inclusive Higgs production cross section [5,6]. Such constraints only probe the simultaneous deviation of all Yukawa couplings. They do not provide information about the separate Yukawa couplings of the different quarks.
The study of heavy quarkonium is one of the interesting subjects in high energy physics, which offers a good testing ground for investigating the Quantum Chromodynamics (QCD) in both the perturbative and non-perturbative regimes. The factorization formalism of nonrelativistic QCD (NRQCD) [7] as a rigorous theoretical framework to describe the heavyquarkonium production and decay has been widely investigated both at experimental and theoretical aspects. Many experimental data of the heavy quarkonium production and decay are fairly well described by the NRQCD theory [8][9][10][11][12][13].
Recent works showed that the exclusive decays of the Higgs boson to vector mesons can probe the Yukawa couplings of first-and second-generation quarks at future runs of the LHC [14]. These couplings are hard to access in hadron colliders through the direct H → qq decays, owing to the overwhelming QCD background. While the Yukawa couplings Hcc might be probed at the LHC by making use of charm-tagging techniques, its phase must be determined through the processes involving quantum interference effects, such as the decay [15]. Although the branch ratios of Higgs boson to vector mesons are small, it offers complimentary information about Higgs couplings and can serve as searching New Physics (NP) beyond SM. Besides, subsequent decays of J/ψ(Υ) into pair of leptons is a clean channel in experiments. Recently, Higgs rare decay to a vector quarkonium (J/ψ, Υ) received considerable attention [15][16][17][18][19]. The relativistic correction for Higgs boson decay to an S-wave vector quarkonium plus a photon have been calculated in Ref. [35]. A search for the decays of the Higgs and Z bosons to J/ψ and Υ is performed in integrated luminosities 20.3f b −1 with the ATLAS detector at 8 TeV LHC. No significant excess of events is observed above expected backgrounds and 95% CL upper limits are placed on the branching fractions.
As we know, the NLO QCD corrections to quarkonium production are usually significant [21][22][23]. We should generally take the NLO QCD corrections into account in studying heavyquarkonium production processes. In this paper, we will calculate the H → J/ψ(Υ) + γ process up to the QCD NLO within the NRQCD framework by applying the covariant projection method [24]. The paper is organized as follows: we present the details of the calculation strategies in Sec.II. The numerical results are given in Sec.III. Finally, a short summary and discussions are given.

A. LO calculation for direct production
We begin to discuss the decay H → J/ψ + γ. Since the calculation of the Υ decay is identical to the J/ψ, we will not present it explicitly in this section. There are two Feynman diagrams for this process at leading order(LO), which are shown in Fig.1. We calculate the amplitudes by making use of the standard methods of NRQCD factorization [7]. The process H → cc + γ at LO is denoted as: (1) The amplitudes for the two diagrams are given by The relative momentum between the c andc is defined as q = (p 2 − p 3 )/2, and the total momentum of the J/ψ is defined as p = p 2 + p 3 . Then, we obtain the following relations among the momenta: In the cc rest frame, p = (E, 0) and q = (0, q). In the non-relativistic v = 0 limit, p 2 = 4m 2 c , q 2 = 0. In order to produce a J/ψ, the cc pair must be produced in a spin-triplet, color-singlet Fock state. We can obtain the short-distance amplitudes by applying certain projectors onto the usual QCD amplitudes for open cc production. By using the notations in Ref. [24], we get the amplitudes: where the spin-triplet projector is given by The colour singlet state will be projected out by contracting the amplitudes with the following operators : The amplitude M is obtained by calculating the two Feynman diagrams in Fig.1 in QCD perturbation theory. The trace is over both the Lorenz and color indices.
After the application of this set of rules, we obtain the short-distance partial decay widtĥ 1 ] + γ processes: where |p| = and m H represent the Higgs boson mass. dΩ = dφd(cosθ) is the solid angle of particle J/ψ.
The decay width read: where N col and N pol refer to the number of colours and polarization states of the cc pair produced. The color-singlet states N col = 1, and N J = 3 for polarization vectors 3 S [1] 1 state in 4 dimensions. 2N c is due to the difference between the conventions in Ref. [24] and Ref. [7].
The on-mass-shell scheme is adopted to fix the wave function and mass renormalization constant of the external charm quark field, then we obtain . After applying the renormalization procedure the UV divergences in the virtual correction are canceled. The IR singularities are analytically canceled when we added all the virtual Feynman diagrams together. We adopt the expressions in Ref. [25] to deal with the IR divergences in Feynman integral functions, and apply the expressions in Refs. [26][27][28] to implement the numerical evaluations for the IR safe parts of N-point integrals. In the virtual correction calculation, we find that only Fig.2(13) and

C. Indirect Decay Calculation
The direct Higgs decay process to the heavy quarkonium plus photon, is mainly produced through the Higgs and charm quarks Yukawa coupling. While the indirect decay process is mainly produced through Higgs decaying into two photons, then one virtual photon substantially decaying to a cc quark pair. Since Higgs decays into di-photon process is forbidden at tree level in SM, the leading order contribution comes from the one-loop Feynman diagrams, including top quark and W boson triangle diagrams, which are shown in Fig.3. Due to the fact that the coupling strength of Higgs and top(W ) is proportional to the particle mass, the contribution of indirect decay is not small. The process Higgs decays into di-photon at leading order in α s have been calculated in Ref. [30]. The two-loop electroweak and QCD corrections to this process have also been studied in Ref. [31]. In Ref. [14], the authors gave the approximate results for the Higgs decay to J/ψ(Υ) and photon through Higgs decaying into two photon. In our paper, we analytically calculate this process based on NRQCD factorization. In Feynman gauge, there are 28 Feynman diagrams, which include the contribution from not only the top, W-boson loops, but also the ghost and goldstone loops. First we generate the amplitudes of Higgs decay to di-photon, which is given by The expressions of coefficients A and B are listed in the appendix. Then we multiply it to the amplitude of virtual photon decay to cc quarks pair. After the application of the projection operator, we get the short-distance amplitude, Following the Passarino-Veltman(PV) method [14,16], we can expressed the tensor integrals We take two-loop running α s in the calculation, and the corresponding fitted value α s (M Z ) = 0.118 is used for the calculations. The renormalization and NRQCD scales are chosen as µ r = m H and µ Λ = m c (m b ), respectively. The Long Distance Matrix Elements (LDME) for J/ψ and Υ used in this paper are set as: [33] (1) 1 ] >= 9.28 GeV 3 .
Finally, we get the decay widths for H → J/ψ(Υ)+γ for the direct and indirect processes: .
As the results show, for the decay process H → J/ψ + γ, the direct contribution is much smaller than the indirect contribution, so it is difficult to observe direct contribution in the total cross section and not suitable for studying the coupling of Higgs and charm quarks.
For the process H → Υ + γ, the direct and indirect contributions are comparable and the total cross section is sensitive to the direct decay process, so this process can be used to  The main uncertainties for the results of H → J/ψ(Υ) + γ arise from the uncertainties in LDMEs, renormalization scale, and the relativistic corrections. The relativistic corrections and the uncertainties have been discussed in Ref. [35]. In Table.I, we illustrate the renormalization scale dependence of the direct and indirect decay widths for the process H → J/ψ(Υ) + γ. We assume µ = µ r and define µ 0 = m H . When the scale µ running from µ 0 /4 to 4µ 0 , The related theoretical uncertainty for H → J/ψ + γ amounts to +55.0 −83.2 % for direct process and to +4.2 −4.1 % for indirect process, and the related theoretical uncertainty for H → Υ + γ amounts to +40.2 −26.5 % for direct process and to +4.2 −4.2 % for indirect process. The LO direct process is independent of the renormalization scale µ R , because it is pure electroweak channels.

IV. SUMMARY
In this paper, we investigated the decay of Higgs boson to J/ψ(Υ) plus a photon based on NRQCD factorization. For the direct process, we have calculated the decays width up to QCD NLO and found that the LO decay widths are significantly reduced by the NLO QCD corrections. For the indirect process, we calculated the process H → γ * γ with virtual γ substantially decaying to J/ψ(Υ), including all the SM diagrams. The decay width of indirect production is much larger than the direct decay width. Therefore, it is difficult to probe the Yukawa coupling of Higgs and charm quarks using the process H → J/ψ(Υ) + γ.