Exact quark-mass dependence of the Higgs-photon form factor at three loops in QCD

We follow up on our discussion of the exact quark-mass dependence of the Higgs-gluon form factor at three loops in QCD [1] and turn our attention to the closely related Higgs-photon form factor. Similarly to our previous work, we intend to examine the form factor for the decay of a Higgs-boson with variable mass into two photons at the three-loop level in QCD. The set of master integrals is known numerically due to prior work on the Higgs-gluon form factor and is exploited to obtain expansions around the threshold as well as in the high-energy limit. Our results may be utilised to derive the photonic decay rate of the Higgs-boson through next-to-next-to-leading order.


Introduction
Since its discovery in 2012 at the Large Hadron Collider (LHC) by the two collaborations ATLAS and CMS [2,3], the Higgs-boson became one of the most promising candidates to study the Standard Model (SM) and physics beyond the SM. Even though the SM passed the most precise tests until now, small deviations between theoretical computations and experimental data could reveal missing pieces of a more complete theory of particle physics. It is therefore necessary to investigate the production and decay modes of the Higgs-boson in great detail. According to theory predictions for the branching-ratios (BR) ref. [4], the decay of a 125 GeV Higgs-boson into a pair of bottom-quarks is favoured, but less significant for experimental studies due to the large background at hadron colliders. Despite the fact that the BR for the decay H → γγ is of O 10 −3 , Higgs-boson decay into a pair of photons belongs to the most relevant decay channels due to the high precision to which the final state particles can be measured.
Moreover, the feature that H → γγ is a loop induced process makes it an appealing channel to determine not only the Higgs-boson mass with excellent resolution, but also to extract Yukawa couplings, since the Higgs-boson couples to all massive particles running in the loops.
Although the process at hand is loop induced and therefore hard to examine within the framework of a multi-loop calculation, the two-loop corrections to the Higgs-decay were computed a long time ago in the heavy-top limit in refs. [5,6] and, subsequently, results covering the region below and even above the top-threshold followed with refs. [7][8][9][10] via numerical integration. A decade later, these results became available in analytical JHEP04(2021)196 form [11][12][13]. However, the three-loop calculation seems to be more involved. Nevertheless, expansions in the regime, where the mass of the mediating quark is considered much larger than the mass of the Higgs-boson, have been employed to determine the three-loop form factor as a series expansion in terms of the fraction of the mentioned masses [14,15]. The only analytical result currently available captures contributions originating from diagrams with one massless fermion loop [16]. Finally, the large logarithms of O αα 2 s L k have been predicted in refs. [17][18][19][20].
The paper at hand was motivated by the authors of refs. [19,20], who kindly requested the availability of the Higgs-photon form factor expanded in the high-energy limit to perform consistency-checks with their own results. Since the diagrams that account for the Higgs-photon form factor form a subset of diagrams contributing to the Higgs-gluon form factor, we closely follow our previous publication ref. [1]. Hence, the reduction table for the simplification to master integrals and their numerical solution, which was obtained via solving a system of differential equations, can be exploited to determine the desired expansions and the form factor itself.
Throughout this publication, we treat the diagrams shown in figure 1 that incorporate two fermion loops as follows: either both fermions are massive quarks or one of them, in particular the one that couples to the Higgs-boson, is massive and the other one massless. In this way, we arrive at the three-loop Higgs-photon form factor in QCD with a single massive quark flavour.
This publication is structured as follows: in the following section, we clarify the notation and conventions used in this paper. Subsequently, we briefly discuss our findings and draw conclusions. Explicit results for the expansions of the missing piece of the three-loop form factor and information on the contents of the supplementary material are given in the appendices. An entire chapter dedicated for a thorough discussion on the technical details is given in ref. [1].

Definitions
In this section, we introduce the notation and conventions used throughout this paper. The process of interest is the decay of a Higgs-boson with arbitrary mass into two photons with momenta p 1 and p 2 and helicities λ 1 and λ 2 . We write the amplitude as follows: Q q denotes the electric charge of the top-quark, α is the electromagnetic coupling constant and v indicates the Vacuum Expectation Value originating from the tree-level Lagrangian term −MQQH/v, which is responsible for the coupling of the quark field to the Higgsboson. For the photon polarisation vectors, the normalisation conditions hold: In accordance with eq. (2.1), the Form Factor C admits a perturbative expansion in terms of the strong coupling constant, α s .
As far as renormalisation is concerned, we stick to the same conventions as in [1] for the sake of convenience. We define the strong coupling constant in MS scheme with massivequark decoupling. The β-function for n l massless quarks gives rise to the dependence on the renormalisation scale: α s ≡ α (n l ) s (µ). Furthermore, the quark-mass and henceforth the Yukawa coupling are renormalised in the on-shell scheme. All relevant constants for renormalisation and decoupling can be taken from [21][22][23][24][25].

JHEP04(2021)196
In contrast to the known one-and two-loop contributions, C (0) and C (1) , respectively, the three-loop coefficient, C (2) , may be subdivided into contributions stemming from different classes of Feynman diagrams: Here, the splitting into the four tree-loop coefficients, C (2,k) , is motivated by the fact that Feynman diagrams with more than one fermion loop contribute at three-loop level for the first time. C (2,0) gathers all diagrams with exactly one closed fermion chain to which the external particles are necessarily attached. Two typical diagrams are shown in figures 1ab. Diagrams that contribute to C (2,1) are those, which embed two massive fermion loops depicted in figures 1c-h. We do not distinguish between diagrams in which one of the fermion loops is neither connected to the Higgs-boson nor to the external photons, as well as those where one of the fermion loops couples to the photons and the other one to the Higgsboson. In this context, n h indicates the number of massive quarks not coupling to the Higgsboson. With the three-loop coefficients C (2,2) and C (2,3) , which are known analytically [16], we associate all Feynman diagrams that involve one massless and one massive fermion loop. One usually differentiates between singlet and non-singlet contributions. Singlet diagrams (figures 1g-h) collected in C (2,3) incorporate one massive fermion loop attached to the Higgs-boson and one massless fermion loop that couples to the external photons. Hence, we have to sum over the electric charges of all massless fermion flavours. In contrast to that, the diagrams displayed in figures 1c-f with a massless fermion loop in the centre account for the non-singlet part, C (2,2) . C (2,0) encompasses non-singlet diagrams only, but as pointed out before, C (2,1) covers both singlet and non-singlet parts.
The form factors and their individual components depend on the fraction of the masses of the Higgs-boson and the mediating massive quark and on the logarithm containing the renormalisation scale: In order to clarify the notation, we state the leading contribution: which in the heavy-top limit takes the value: The form factor is scale-independent implying

JHEP04(2021)196
Thus, the dependence of the form factor on the aforementioned logarithm, L µ , can be expressed with the aid of the coefficients of the QCD β-function: (2.10) For the sake of completeness, we quote the first coefficient of the β-function:

Results
In this section, we briefly present out findings. The scale-dependence is fixed such that L µ = 0 and can easily be restored by applying eqs. (2.10).
We check our results for the light-fermion contributions, in particular, the three-loop coefficients C (2,2) and C (2,3) , numerically against the analytical results in ref. [16]. For the numerical probes as well as for the expansions in the kinematic limits, we find full agreement.
Similar to our previous work, the exact result for the Higgs-photon form factor at threeloop level, C (2) , is stored in the form of a univariate interpolation based on nearly 200.000 numerical probes distributed over the physical parameter space in the variable z. Other than that, we derived high-order large-mass, threshold and high-energy expansions, which cover most parts of the parameter space to sufficient precision. The radii of convergence of the three expansions are limited due to singularities located at z = 0, z = 1 and 1/z = 0. The supplementary material HaaNiggetiedt.m attached to this paper shipped with this publication contains the large-mass expansion with exact coefficients truncated at O z 100 , the threshold expansion truncated at O (1 − z) 20 and the high-energy expansion truncated at O 1/z 8 . The latter ones are expansions with numerical coefficients. We choose the truncation order of the numerical expansions such that we can confidently guarantee the correctness of at least ten digits for every numerical coefficient.
A comparison of the mentioned expansions with the exact numerical result for the sum of three-loop coefficients C (2,0) + C (2,1) is illustrated in figure 2. For those values of z which are not covered by expansions, we provide interpolation tables in tables 1 and 2, where we reshaped the domain of positive z values to the interval (0, 1) by applying the conformal mapping With a relative error of at most 10 −5 , the exact result for the sum of three-loop coefficients C (2,0) + C (2,1) is approximated as follows: 0 < ρ < 1/6 -large-mass expansion, appendix A and figure 3; 1/6 ≤ ρ < 1/4 -threshold expansion, appendix B and figure 4; 3/4 ≤ ρ < 1 -high-energy expansion, appendix C and figure 5.

Conclusions and outlook
Provided the findings of this paper, the Higgs-photon form factor is now known exactly at the three-loop level in QCD with a single massive quark-flavour. Moreover, the longing for the desired expansions has been satisfied. We presented the results with the Yukawa  Table 2. Numerical values of the three-loop coefficient C (2,0) + C (2,1) coupling renormalised in the on-shell scheme, which can be translated to any other scheme due to the fact that the one-and two-loop results are available in analytical form. We finally note that our results may be utilised to obtain the cross section for Higgsboson production via photon-photon fusion and the photonic decay rate of a Higgs-boson through next-to-next-to-leading order in QCD.
Let us again emphasise that the form factor with the most general quark-mass dependence requires additional elaboration of the diagrams with two closed fermion chains. We postpone this analysis to future publications.

Re Exact Im Exact
Re LME Im LME Re HE Im HE Re THR Im THR 0.

A Large-mass expansion
(a n,0 + a n,1 L s ) z n ,  In order to present the high-order LME in a space-saving way, we refrain from showing the analytical result here and refer to HaaNiggetiedt.m attached to this paper for the exact expansion. After conversion to a quark-mass renormalised in MS scheme, we find full agreement with ref. [15] up to O z 20 .  20 with the sum C (2,0) + C (2,1) evaluated numerically (L µ = 0). The absolute difference between the exact result and the expansions is shown in the bottom panel.    The parts of the numerical coefficients of terms proportional to L k s /z for k ∈ {6, 5, 4, 3}, which stem from C (2,0) , comply with the exact coefficients predicted recently in refs. [19,20].

D Supplemental material
The supplemental material, HaaNiggetiedt.m attached to this paper, in form of a single file can be imported in Wolfram Mathematica for subsequent analysis. All variables are explained in the header of this file. The main function returns the form factor as a series in aspi≡ α s /π: CHaa[z, nh, nl, QQsum, Lmu] -C, eq. (2.3); The function CHaa[z, nh, nl, QQsum, Lmu] is entirely based on the aforementioned expansions and interpolations. Hence, the analytical results of ref. [16] are not exploited. For the benefit of the reader, we provide all constituents of the form factor in terms of expansions and interpolation tables. The following functions are evaluated at L µ = 0:

JHEP04(2021)196
One must supply numerical values for z. The large-mass expansion of C (2) L µ = 0 with exact coefficients can be called with C2LME. Its dependence on n h , n l and the sum over electric charges is kept variable.
All functions are evaluated with fixed gauge group constants C A = 3, C F = 4/3, T F = 1/2.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.