Addendum: New approach to the resummation of logarithms in Higgs-boson decays to a vector quarkonium plus a photon [Phys. Rev. D 95, 054018 (2017)]

In this addendum to Phys.\ Rev.\ D {\bf 95}, 054018 (2017) we recompute the rates for the decays of the Higgs boson to a vector quarkonium plus a photon, where the vector quarkonium is $J/\psi$, $\Upsilon(1S)$, $\Upsilon(2S)$, or $\Upsilon(3S)$. We correct an error in the Abel-Pad\'e summation formula that was used to carry out the evolution of the quarkonium light-cone distribution amplitude in Phys.\ Rev.\ D {\bf 95}, 054018 (2017). We also correct an error in the scale of quarkonium wave function at the origin in Phys.\ Rev.\ D {\bf 95}, 054018 (2017) and introduce several additional refinements in the calculation.

In Ref. [1], rates for the decays of the Higgs boson to a vector heavy quarkonium plus a photon (H → V +γ) were computed by making use of the light-cone amplitude for the direct decay process. In the direct process, the Higgs boson decays into a heavy-quark-antiquark pair (QQ), which radiates a gluon and evolves into a vector heavy quarkonium.
In Ref. [1], φ V (x), the heavy-quarkonium light-cone distribution amplitude (LCDA) at light-cone momentum fraction x, was evolved from the initial scale µ 0 , which is of order the heavy-quark mass, to the final scale µ, which is of order the Higgs-boson mass m H , by making use of the Abel-Padé method. The Abel summation in the Abel-Padé method defines generalized functions in the nonrelativstic QCD (NRQCD) expansion of the LCDA at the scale µ 0 as limits of sequences of ordinary functions.
Because the evolution matrix in leading logarithmic (LL) order is diagonal, this change affects the expression for the light-cone amplitude only at next-to-leading logarithmic (NLL) order and higher. In Ref. [1], only the LCDA at leading order in α s and v was evolved at NLL order. (Here, α s is the QCD running coupling and v is one-half the velocity of the Q orQ in the quarkonium rest frame.) The correction in Eq. (1) has no significant effect on the numerical results that were reported in Ref. [1].
The light-cone amplitude for the direct process can be written [2], by virtue of NRQCD factorization [3], as a sum of products of short-distance coefficients with NRQCD longdistance matrix elements (LDMEs) [4]. We denote the NRQCD LDME of leading order in v by O V 1 . The quantity O V 1 can be written in terms of the quarkonium wave function at the origin Ψ V (0) (Ref. [3]): where N c = C A = 3 for SU(3) color.
For the computations in Ref. [1], which involve J/ψ and Υ(nS) final states, with n = 1, 2, or 3, values for Ψ V (0) were inferred from values of O V 1 that were given in Refs. [5,6]. In those papers, O V 1 was computed by making use of the NRQCD expansion of the electronic decay rate Γ(V → e + e − ) through orders α s (µ) and v 2 . 1 That expansion can be obtained from Eq. (2), the relation and the NRQCD expansion Here, f V is the decay constant for a longitudinally polarized quarkonium, α(µ) is the electromagnetic fine-structure constant at the scale µ, e Q is the heavy-quark charge, m V is the .) In Refs. [5,6], the scale µ in Eq. (4) was taken to be m V . Hence, the values of O V 1 V that are given in those papers are at the scale m V .
The decay rate for the process H → V +γ depends on the decay constant for a transversely polarized quarkonium f ⊥ V (µ), which has the NRQCD expansion [4,7] f ⊥ where m Q is the heavy-quark pole mass.
In Ref. [1], f ⊥ V (µ) was assumed to be at the scale µ = m Q . However, the values of Ψ V (0) that were used in Ref. [1] were inferred from the values of O V 1 in Refs. [5,6]. Because those values of O V 1 are at the scale µ = m V , it is necessary to correct for the difference in scales. One can do this straightforwardly by making the following replacement in the expression for Instead, we follow a procedure that was suggested in Ref. [8]. We compute f V from the measured quarkonium electronic decay rates, using Eq. (3). Then, we use Eq. (4) to eliminate Ψ V (0) from the amplitude and use Eqs. (4) and (5) to express f ⊥ V in terms of f V : Because the pole mass is subject to renormalon ambiguities and, hence, is ill defined, we have used the one-loop relation between the MS mass m Q and the pole mass [9], namely, to express f ⊥ V /f V in terms of the MS mass in Eq. (7b). As can be seen, if µ is approximately equal either to m Q or to m Q , then the apparent rates of convergence of both the velocity expansions and the α s expansions in Eqs. (7a) and (7b) are better than the corresponding rates of convergence in Eq. (5). Furthermore, the removal of Ψ V (0) from the amplitude reduces some of the theoretical uncertainties. The resulting expression for the direct amplitude, which corresponds to Eq. (37) in Ref. [1], is Here, e(> 0) is the electric charge at momentum scale zero, κ Q is a factor that accounts for deviations of the HQQ coupling from the standard model (SM) value (κ Q = 1 is the SM value), G F is the Fermi constant, p and ǫ * V are the quarkonium momentum and polarization, respectively, and p γ and ǫ * γ are the photon momentum and polarization, respectively. Each of the quantities M (i,j) is a convolution of a component of the light-cone hard-scattering kernel at the scale µ with a component of the LCDA of the quarkonium, evolved from the scale µ 0 to the scale µ. The detailed definitions of the M (i,j) are given in Ref. [1]. As in that work, we carry out the evolution of the LCDAs at next-to-leading-logarithmic accuracy, using the Abel-Padé method that is presented in that paper to sum the resulting series expansions in the eigenfunctions (Gegenbauer polynomials) of the leading-order evolution operator. Our uncertainties are also computed as outlined in Ref. [1].
Note that, because we include the NRQCD expansion for f ⊥ V /f V as an overall factor in the direct amplitude in Eq. (9), this expression contains cross terms of orders α 2 s and α s v 2 that are beyond the accuracy of the present calculation. We structure the amplitude in this way because of the possibility that the perturbation expansion of f ⊥ V /f V may be better behaved in higher orders in α s than the perturbation expansion of f V [10][11][12]. We use Eq. (9) to compute the direct amplitude, taking µ = m H , as in Ref. [1], but taking µ 0 = m Q , rather than µ 0 = m Q because, as we have already remarked, the pole mass is ill defined.
As was pointed out in Ref. [13], there is also a contribution to the decay process from an indirect amplitude, which arises from the decay of the Higgs boson through a W -boson loop or a quark loop into a photon plus a virtual photon, with the virtual photon decaying into a quarkonium. We compute the indirect amplitude as in Ref. [1], but with minor updates to the numerical values of the quarkonium-photon couplings.
We obtain the results that are shown in Table I for the direct and indirect amplitudes.
The results in Table I    and by about 2 % for H → Υ(nS) + γ, relative to the results that were given in Ref. [1]. The correction factor in Eq. (6) would shift the direct amplitude by about 13 % for H → J/ψ + γ and by about 4-5 % for H → Υ(nS) + γ. Apparently, the additional refinements that we have introduced in this calculation work to reduce those shifts in the direct amplitude.
Our results for the decay rates and branching fractions are shown in Table II  for H → Υ(1S) + γ, by about −42 % for H → Υ(2S) + γ, and by about −49 % for H → Υ(3S) + γ, relative to the results that were given in Ref. [1]. The strong sensitivity of the rates for H → Υ(nS) + γ to small changes in the direct amplitude occurs because of the near cancellation of the direct and indirect amplitudes in these processes.
Given the correction and refinements to the calculation of the rates for H → V + γ that we have outlined above, we believe that the numerical results that we have presented in this paper are more accurate and reliable than those in Ref. [1] and should be used in preference to the results in Ref. [1] in future comparisons with experimental measurements.

Acknowledgments
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