Next-to-leading order QCD corrections to the decay of Higgs to vector meson and Z boson

The exclusive decay of the Higgs boson to a vector meson ($ J/\psi $ or $ \Upsilon(1S) $) and $ Z $ boson is studied in this work. The decay amplitudes are separated into two parts in a gauge invariant manner. The first part comes from the direct coupling of the Higgs boson to the charm (bottom) quark and the other from the $ HZZ^{*} $ or the loop-induced $ HZ\gamma^{*} $ vertexes in the standard model. While the branching ratios from the direct channel are much smaller than those of the indirect channel, their interference terms give nontrivial contributions. We further calculate the QCD radiative corrections to both channels, which reduce the total branching ratios by around 20% for both $ J/\psi $ and $ \Upsilon(1S) $ production. These results may help to check the SM predictions of the $ H c\bar{c}(H b\bar{b}) $ coupling and to seek for hints of new physics at the High Luminosity LHC or future hadron colliders.


I. INTRODUCTION
The discovery of the 125 GeV Higgs-like boson by the ATLAS and CMS collaborations [1,2] has attracted much interest from both experimental and theoretical aspects.It is most important to precisely determine the properties of this new boson to check the predictions of standard model (SM).No significant deviation from the SM has been found from measurements of the decays H → γγ, ZZ and W W [3, 4].
In the SM, the coupling of the Higgs to the first and second generation fermions is very weak and thus difficult to measure directly.In Ref. [5], the authors point out that the Hcc coupling may be probed by the measurements of the H → J/ψ + γ decay mode.Their results show that the partial width via the direct decay channel, where the Higgs boson is coupled directly to the charm quark, is too small to be probed.They introduce a new mechanism where the Higgs boson decays through the loop-induced Hγγ * vertex, followed by γ * → J/ψ.This indirect mechanism turns out to dominate the H → J/ψ + γ decay, and the interference between the direct and the indirect channel may provide us with more detailed information about the Hcc coupling.
The proposed High Luminosity LHC (HL-LHC) project is designed to run at 14 TeV, with its integrated luminosity upgraded to 3000 fb −1 .The HL-LHC will allow us to perform precise measurements that are impossible otherwise, which motivates us to make a more detailed study of the rare decays of the SM Higgs boson.The small Yukawa coupling of the light fermions leaves space for theories of new physics beyond the SM and has attracted a lot of interest.In Refs.[6][7][8][9][10], the rare decay H → V + Z (V = J/ψ or Υ(1S)) has been studied.Very similar to H → V + γ, we can separate the decay amplitudes into two parts.The first part comes from the direct coupling of Higgs boson to the charm (bottom) quark, and the other part from the HZZ * or the loop-induced HZγ * vertexes in the SM, followed by Z * (γ * ) → V .Since both the vector meson and Z boson can decay to a pair of charged leptons, this experimentally-clean final state may provide us a chance to probe this rare decay and seek for a hint of new physics.
In Ref. [7], the author shows that the decay rate of H → V + Z via the loop-induced HZγ * vertex is comparable with the contributions from the tree level HZZ * vertex in the SM, by analyzing this decay via the indirect channel at leading-order (LO).In Ref. [8], the authors analyze the decay H → V + Z from all channels, and interference contributions are also included.They further show that this rare decay is sensitive to the anomalous Higgs couplings originating from physics beyond the SM, by observations of the decay of the final-state vector meson and Z boson into charged leptons.In this paper, we revisit this rare decay at LO, and evaluate its QCD corrctions at next-to-leading order (NLO) for the (un)polarized final-state vector meson and Z boson.We also carefully analyze the interference terms between the indirect channel and the direct one at both LO and NLO level.
This paper is organized as follows.In Section II, we introduce the formalism and notations used in this work.In Section III, we revisit the decay H → V + Z at LO, and further we calculate its QCD corrections at NLO.In Section IV, we analyze the helicity amplitudes by projecting the vector meson and Z boson to particular polarized states.We conclude our calculations in Section V.

II. FORMALISM AND NOTATIONS
For the H → V + Z decay, there are several distinct dynamic scales: m H ∼ m Z > m q (m q = m b or m c ).There is another scale in heavy quarkonium production, m q v r , where v r is the relative velocity of the heavy quark pair in the rest frame of the quakonium.For heavy quarkonium, v r ≪ 1 is assumed, making it a non-relativistic system.The scale m q v r characterizes the hadronization process of the heavy quark pair to vector mesons, which is inherently nonperturbative.
We adopt the nonrelativistic QCD (NRQCD) factorization formula as described in Ref. [11].In the NRQCD factorization framework, the production of heavy mesons can be separated into two parts.First, the constituent quarks are produced at an energy greater than the heavy quark mass and thus can be calculated perturbatively.Secondly, the quark pair binds into quarkonium at an energy much smaller than the heavy quark mass, and this process is represented by the non-perturbative long distance matrix elements (LDMEs).In this formula, the LDMEs can be expanded by the relative velocity v r of the heavy quark pair and the short-distance coefficients are expanded perturbatively by the strong coupling α s order by order.As a result, the production rate of the heavy quarkonium can be expressed as the sum of products of the non-perturbative LDMEs and the corresponding perturbative short-distance coefficients.We will expand the short-distance coefficients to NLO of the strong coupling α s and the LDMEs are expanded to the lowest order of v r throughout this work.
We define the momenta of the initial-state Higgs and the outgoing vector meson and Z boson as where p = p q + p q and k represent the momenta of the final-state vector meson and Z boson respectively, with m V is the mass of the vector meson and m V = 2m q at the lowest order of v r .p H = p + k is the momentum of the initial-state Higgs boson and λ V (λ Z ) = 0, ±1 represent the helicities of the final-state vector meson (Z boson).
To project the free quark pair into our desired S 3 1 and color singlet quantum state, we adopt the relatively normalized spin projection operator in Eq. (A9b) of Ref. [12]: where E q in Eq. ( 4) is the energy of the constituent quarks in the rest frame of the heavy quark pair.Since we expand the LDMEs to the lowest order of v r , the relative momentum between the quark pair is thus neglected.As a result, the momenta of the constituent heavy quarks are assigned as and E q = m q .
By Lorentz covariance, the amplitudes for H → V + Z can be decomposed as where ǫ µ V (p, λ V ) and ǫ ν Z (k, λ Z ) are the polarization vectors of the final-state vector meson and Z boson, and their explicit expressions will be given in Section IV.We will show in Section IV that the helicity amplitudes in Eq. ( 6) are free of polar and azimuthal angles.
The decay width of H → V + Z can be expressed as At order-v 0 r , the helicity amplitudes in Eq. ( 7) can be expressed as where O 1 V represents the nonperturbative LDME and we take O 1 J/ψ = 0.44 GeV 3 and O 1 Υ(1S) = 3.07 GeV 3 for the case of J/ψ and Υ(1S), respectively [13].

A. LO results revisited
Feynman diagrams for LO H → V + Z are shown in Fig. 1.The first two diagrams come from the direct channel where the charm (bottom) quark is coupled directly to the Higgs boson.The last two diagrams come from the HZZ * and the loop-induced HZγ * vertexes in the SM, respectively.
In Table I, we list the branching ratios for H → V + Z from the direct, the indirect and the interference channels 1 .The branching ratios are rather small at LO for both J/ψ V in the denominator of Eq. ( 6) in Ref. [8] should be

and Υ(1S).
The contributions from the direct channels of J/ψ and Υ(1S) production are totally negligible, as expected.The branching ratios from the indirect channel dominate the total results and the interference branching ratios amount to -1.60% and 3.93% of the total results for J/ψ and Υ(1S) respectively.

B. QCD corrections
QCD radiative corrections are performed by attaching gluons to the final-state quarks as shown in Fig. 2. Note that the loop-induced HZγ * vertex may also have QCD corrections at NLO.In Refs.[18,19], the authors calculate the QCD corrections to the rare decay H → Zγ at NLO and the relative corrections turn out to be rather small, around 3‰ of the LO decay width.We neglect the tiny QCD corrections to the HZγ * vertex throughout this work and extract the parameters for the effective HZγ * coupling by evaluating this vertex at LO.
We choose the dimensional regulation to regularize the possible UV and IR singularities in d = 4 − 2ǫ dimensions.The renormalization constants Z 2 and Z m are defined as where ψ q (ψ 0 q ) and m q (m 0 q ) represent the renormalized(bare) quark field and mass, respectively.In the on-shell (OS) scheme, the renormalization constants expanded to O(α s ) can be expressed as The singularities are removed after renormalization, and the decay width for H → V + Z at NLO can be expressed as For the QCD corrections to the direct decay channel, δ dir J/ψ = −60.98%and δ dir Υ(1S) = −48.88%for J/ψ and Υ(1S) respectively.We can see that the results for the direct channel are greatly reduced by the NLO QCD corrections, which makes the NLO results for the direct channel of H → J/ψ + Z negative.In Ref. [20], the authors evaluated the NLO corrections to H → V + γ and the large and negative corrections to the direct channel are similar.At scale µ = m H /2, the relative corrections ∆Γ direct /Γ direct defined in Ref. [20] are around -85.38% and -68.38% for J/ψ and Υ(1S) respectively.These large and negative corrections at NLO may be compensated by the NNLO or the relativistic contributions.The QCD radiative corrections to the indirect channel come from the last two diagrams in Fig. 2. The results are trivial and independent of the quark species:  The NLO results for H → V + Z are listed in Table II.Both of the branching ratios from the direct and the interference channels are greatly reduced, which makes these results sensitive to the strong coupling α s .In Fig. 3, we show the dependence of the total branch ratios on the scale by sliding µ from 2m c to m H .
In this work, we generate the decay amplitudes with FeynArts [21], and employ the the package FeynCalc [22] to deal with the traces over the Dirac matrices.The amplitudes are further decomposed with the package Apart [23] and the IBP (integration by parts) reductions are performed with FIRE [24].The master integrals are calculated with Package-X [25] analytically and further checked numerically with LoopTools [26].

IV. RESULTS FOR POLARIZED VECTOR MESON AND Z BOSON
It is sometimes useful to project the final-state vector meson and Z boson to particular helicity states.The corresponding polarization vectors are e ±iφ (0, ∓ cos θ cos φ + i sin φ, −i cos φ ∓ cos θ sin φ, ± sin θ), (12a) e ∓iφ (0, ± cos θ cos φ + i sin φ, −i cos φ ± cos θ sin φ, ∓ sin θ), and where we have introduced θ, φ as the polar and azimuthal angles of p with respect to a fixed z-axis.The Jacob-Wick convention [27] is adopted for the polarization vector of the final-state Z boson in Eq. (12).
Since the initial-state Higgs boson is spin-0, the helicity amplitudes must be proportional to where D J λ i λ f (θ, φ) is the Wigner-D matrix element in the partial wave expansion [28], and λ i = 0, λ f = λ V − λ Z = 0. Thus the helicity amplitudes defined in Eq. ( 6) are free of the polar and azimuthal angles θ, φ, and λ V = λ Z = 0, ±1.The numerical results for the decay of the Higgs to the polarized vector meson and Z boson are shown in Table III.For the case of H → J/ψ + Z, the branching ratios for the longitudinally polarized J/ψ are around 62% of the total branching ratios at LO and NLO.While the branching ratios for the transversely polarized Υ(1S) and J/ψ are very close, the results for the longitudinally polarized Υ(1S) are one order of magnitude larger than those of J/ψ.As a result, the branching ratios for the longitudinally polarized Υ(1S) dominate the total branching ratios of the decay H → Υ(1S) + Z at both LO and NLO.

V. CONCLUSION
In this work, we have revisited the rare decay H → V + Z (V = J/ψ or Υ(1S)) in the SM.We separated the decay amplitudes into two parts: the first part from the direct Yukawa coupling of the Higgs boson to the charm (bottom) quark, and the second from the HZZ * and the loop-induced HZγ * vertexes in the SM.We also analyzed their interference effects carefully.We took a further step by evaluating the QCD radiative corrections to both cases at NLO.The QCD corrections reduce the LO branching ratio by around 20%.Our results show that the contributions from the direct channel are negligible compared with the indirect channel, due to the tiny Yukawa coupling between the Higgs and the charm (bottom) quark.The interference branching ratios turn out to reach several percent level of the total results at both LO and NLO.
The decay of the Higgs to the polarized vector meson and Z boson was studied in Section IV.For the decay H → J/ψ + Z, the production of the longitudinally polarized J/ψ is about twice that of the transversely polarized J/ψ.For the decay H → Υ(1S) + Z, our results show that the decay to the longitudinally polarized Υ(1S) dominates.
Although the branching ratios from the indirect and interference terms give nontrivial contributions at both LO and NLO, the measurements of the Hcc and Hb b couplings seem to be rather difficult via the rare decay H → V + Z.The small Yukawa coupling may be enhanced by new physics beyond the SM, which provide a chance to observe this rare decay and to test the SM predictions in the projected HL-LHC or in future hadron colliders.

FIG. 3 :
FIG. 3:The dependence of the branching ratios for the decay H → V + Z on the scale µ at NLO.The scale µ stems from the QCD running coupling α s (µ), which is taken from 2m c = 3.0 GeV to m H = 125.09GeV.We evaluate the running coupling α s (µ) with the package RunDec and the initial value of α s (µ) is taken as α s (m Z ) = 0.1181.

TABLE I :
Branching ratios for H → V + Z at LO.

TABLE II :
Branching ratios for H → V + Z at NLO.

TABLE III :
The branching ratios for the polarized vector meson and Z boson for H → V + Z.The notations "T" and "L" represent the transversely and longitudinally polarized vector meson and Z boson.