Completing the hadronic Higgs boson decay at order $\alpha_s^4$

We compute four-loop corrections to the hadronic decay of the Standard Model Higgs boson which are induced by effective couplings to bottom quarks and gluons, mediated by the top quark. Our numerical results are comparable in size to the purely massless contributions which have been known for a few years. The results presented in this paper complete the order $\alpha_s^4$ corrections to the hadronic Higgs boson decay.


Introduction
In particle physics, one of the most important tasks in the coming years is the precise measurement of the couplings of the Higgs boson to fermions and bosons. An important ingredient in this context is the decay rate of the Higgs boson into bottom quarks, which has the by far largest branching ratio. Together with the decay rate into gluons it constitutes almost 70% of the hadronic decay width and it thus has a major influence on all Higgs boson branching ratios.
One-loop 1 QCD corrections to Γ(H → bb) have been known for a long time, including the full bottom quark-mass dependence [1]. The massless approximation 2 at order α 2 s has been computed in Ref. [2] and the full bottom quark-mass dependence is known from Ref. [3][4][5]. Three-and four-loop corrections, of order α 3 s and α 4 s , have been computed in the massless limit in Refs. [6][7][8]. A summary of further corrections, including top quarkmass-suppressed terms and electroweak effects can be found in recent review articles [9,10] (see also the program HDECAY [11]).
The main aim of this paper is to complete the corrections of order α 4 s to the total decay rate of the Higgs boson into hadrons. In Ref. [8] only the contribution involving the bottom quark Yukawa coupling was considered. We compute the contributions induced by effective Higgs-bottom quark and Higgs-gluon couplings. The corresponding threeloop calculation, which was performed in Ref. [7], produces a similarly-sized contribution to the α 3 s coefficient as that of the purely massless contribution. It is therefore necessary also to evaluate the top quark-induced contributions at order α 4 s . For the calculation performed in this paper the relevant part the Standard Model (SM) Lagrange density is given by the Yukawa terms supplemented by the strong interaction terms. For the production and decay of the SM Higgs boson it turns out that the effective theory in which the top quark is integrated out provides a good approximation to the full theory. This leads to the following effective Lagrangian [12][13][14] In the following we count the number of loops needed for the virtual corrections 2 Here "massless" refers to the bottom quark mass in the propagators; the bottom quark Yukawa coupling remains non-zero. 3 We follow the notation of Ref. [7].
where G 0′ a,µν is the bare gluon field strength tensor andb 0′ is the bare bottom quark field. Further corrections to L eff are suppressed by the inverse top quark mass, contributing terms of order M 2 H /M 2 t to the decay rate. These terms are available to order α 3 s [15][16][17] and are known to be small. For example, at order α 2 s the M 2 H /M 2 t term changes the coefficient by less than 1% and thus induces a correction which is of the same order of magnitude as non-suppressed contributions of order α 4 s . We also restrict ourselves to the leading m 2 b term and neglect higher powers in the bottom quark mass which are numerically even smaller than the 1/M t terms.
On the basis of the Lagrange density of Eq. (1) we define correlators formed by the Sample Feynman diagrams contributing to Π 11 , Π 12 and Π 22 are shown in Fig. 1.
Using the optical theorem, the total decay rate can be obtained from the imaginary part of Π ij . In this context it is convenient to introduce the quantities where Note that for clarity, we restrict ourselves in Eq. (5) to the QCD corrections that we compute in this paper; we neglect both electroweak effects and power corrections suppressed by M 2 H /M 2 t . Furthermore, we concentrate on the decay of the Higgs boson only to bottom quarks and to gluons. The results can easily be extended to include the decay to additional light quark flavours, if necessary. A more complete formula can be found in Eq. (10) of Ref. [7]. Note that in Eq. (6), m b (µ) refers to the MS bottom quark mass evaluated at the renormalization scale µ.
In Ref. [8] Π 22 has been computed to five-loop order, yielding order α 4 s corrections to the Higgs boson decay. For these corrections we have that C 2 = 1 and therefore refer to them in the following as "massless contributions", despite the fact that there is an overall factor of m 2 b from the bottom quark Yukawa coupling. The leading-order term of Π 11 describes the decay of the Higgs boson into gluons. Starting from next-to-leading order (two loops) the gluonic and fermionic decay cannot be separated in the approach based on the optical theorem, since there are diagrams containing both purely gluonic cuts and cuts involving both gluons and quark-antiquark pairs.
The main result of this paper is the extension of [7]. We compute the four-loop correction to Π 12 which contributes to the hadronic Higgs boson decay at order α 4 s , along with the five-loop calculation of Ref. [8]. This is because the leading term of C 1 contains a factor α s . Note that Π 22 has an overall prefactor m 2 b , which comes from the two operators O ′ 2 . Π 12 is also proportional to m 2 b ; one factor arises from O ′ 2 the other from the trace of the bottom quark loop. In the limit m b → 0 the correlator Π 11 has a non-zero contribution. Terms proportional to m 2 b appear for the first time at two-loop order, due to the presence of closed bottom quark loops. We compute such terms up to three loops, which give rise to order α 4 s corrections to the Higgs boson decay. We want to remark that the m bindependent terms of Π 11 have been computed to four-loop order in Ref. [18] leading to corrections of order α 5 s to the hadronic Higgs boson decay. In the next section we provide several technical details of our calculation. In particular, we discuss the computation of the four-loop integrals and explain the operator mixing and renormalization. We additionally provide explicit expressions for the effective couplings C 1 and C 2 . We present analytic results in Section 3 and discuss the numerical impact of our new corrections. Our conclusions are given in Section 4.

Calculation
For the calculation of the Feynman diagrams we use a well-tested automated setup which uses qgraf [19] for the generation of the Feynman amplitudes, and q2e and exp [20][21][22] for the mapping to one of eleven pre-defined four-loop integral families. The Dirac algebra is performed with FORM [23], which also re-writes the amplitude of each diagram as a linear combination of scalar integrals. Next we generate, using FIRE 5.1 [24, 25], tables for the reduction of the integrals of all eleven families to master integrals. We then apply tsort [26], in the form of the FIRE command FindRules, to minimize the number of master integrals among all eleven families and end up with 28 four-loop master integrals, which have been computed in Refs. [27][28][29].
We have re-computed the one-, two-and three-loop corrections to all correlators using both the setup described above and, as a cross check of our approach, MINCER [30]. Both calculations produce identical results, which agree with the literature. As a further check we have performed our calculations using a generic gauge parameter ξ. Our four-loop expressions have been expanded to linear order in ξ which drops out after reducing the master integrals to a minimal set.
We have used this method to compute the four-loop corrections to Π 12 and the three-loop corrections to Π 11 which, after taking the imaginary part, lead to the bare quantities ∆ 0 12 and ∆ 0 11 . At this point we perform the renormalization of the strong coupling constant and the quark mass in the MS scheme where the renormalization constants are introduced via Z αs and Z m are required to third order in α s and can be found in, e.g., Ref. [31].
Afterwards, we have to take into account that the operators O ′ 1 and O ′ 2 mix under renormalization according to [7,32] The renormalization constants Z 11 and Z 12 are obtained from Z αs and Z m as follows, In terms of these renormalization constants, the renormalized correlators ∆ ij are given by Note that the contributions of ∆ 0 12 and ∆ 0 22 are proportional to m 2 b whereas ∆ 0 11 contains both m 2 b and m b -independent terms. Since Z 12 ∝ α s only the (n − 1)-loop terms of ∆ 0 12 and the (n − 2)-loop terms of ∆ 0 22 enter the n-loop renormalization of ∆ 11 . Similarly, the n-loop renormalization of ∆ 12 requires the (n − 1)-loop terms of ∆ 0 22 . For completeness we also provide explicit expressions for the effective couplings C 1 and C 2 , which are available in the literature up to fifth order [33][34][35]. It is convenient to parametrize the perturbative expansion in terms of where the superscript indicates the number of active quark flavours used for the running, and the on-shell top quark mass. To obtain corrections of order a 4 s to the decay rate, C 1 is needed to third order and C 2 to fourth order. In the following we present C 1 to order a 4 s since we include a 5 s corrections when evaluating the decay rate numerically. The analytic results read L t + 1 216 ≈ 1 + 0.494759 a 2 s + a 3 s 2.3946 + 0.2689 n l − a 4 s 6.0125 + 1.1543 n l − 0.05480 where ζ n is the Riemann Zeta function, Li n (z) is the Polylogarithm function and we have defined L t = log(µ 2 /M 2 t ). The numerical expressions are given at the renormalization scale µ 2 = M 2 H , for n l = 5 massless flavours running in fermion loops, and for M H = 125.09 GeV and M t = 173.21 GeV [36]. Since µ is of the order of the Higgs boson mass one generates potentially large logarithms which should be resummed [14]. In practice, however, the numerical effect is small and we have decided to consider only the fixed-order result here.

Results
We use this section to present our results. The new ingredients of Eq.
where L H = log(µ 2 /M 2 H ). As above, n l counts the number of light quarks running in fermion loops. For the numerical evaluation we have set µ 2 = M 2 H and n l = 5. For both ∆ 11 and ∆ 12 we observe a rapid growth of the coefficients, however, we postpone discussion of the convergence properties to the decay rate, since ∆ 11 and ∆ 12 do not themselves represent physical quantities.
For the numerical evaluation of the decay rate it is convenient to cast Eq. (5) in the form where we have chosen A bb as a common prefactor so that we can easily compare the relative sizes of the individual contributions. ∆ light contains all corrections obtained for C 1 = 0 and C 2 = 1. They have already been presented and discussed in Ref. [8]. ∆ top contains the top quark-induced corrections obtained from the contributions proportional to C 1 and (C 2 − 1). For completeness we also list the corrections from ∆ 11 which have no factor m 2 b . They are collected in ∆ m b =0 gg . Note that these terms have already been computed in Ref. [37]. For convenience we provide the formulae which relate the quantities in Eq. (20) to the ones in Eq. (5): For the numerical evaluation we use α where the subscripts in the expression for ∆ top indicate the origin of each term. For ∆ m b =0 gg we have included the corrections of order a 5 s from Ref. [18]. From Eqs. (22), (23) and (24) we observe that the a 2 s term of ∆ m b =0 gg amounts to almost 50% of the a s term in ∆ light . Furthermore, the a 5 s term of ∆ m b =0 gg has the same order of magnitude as the a 4 s term of ∆ light . Note that the latter is only about twice as large as the a 4 s contribution to ∆ top , obtained from the sum of the three numbers in the last line of Eq. (23); this amounts to 0.0006593.
It is a disturbing feature of ∆ light that the a 3 s and a 4 s terms deviate by less than 30%. Furthermore they have opposite signs. Therefore, it is interesting to add ∆ light and ∆ top which leads to 1 + ∆ light + ∆ top ≈ 1 + 0.2033 + 0.04208 + 0.004490 − 0.0007090 , where the different loop orders are kept separate. We observe a reduction by a factor of about six between the three-and four-loop contributions; the convergence of the sum is significantly better than that of the individual expressions. . µ is varied between 10 GeV and 500 GeV which is significantly larger than the usual range spanned between M H /2 and 2M H . Nevertheless, one observes a steady flattening of the curves when including higher order corrections; the result represented by solid line is almost µ-independent.

Conclusions
We complete the corrections of order α 4 s to the hadronic decay rate of the Standard Model Higgs boson by computing the top quark-induced contributions in an effective field-theory framework. This requires the calculation of four-loop propagator-type integrals. Our new corrections are numerically of the same order of magnitude as the purely massless contribution [8], however they have an opposite sign. We provide all analytic results presented in this paper in a computer-readable format [41], making it straightforward to implement the corrections in existing computer codes which evaluate decay rates of the Higgs boson. Finally, we want to mention that Γ(H → hadrons) is one of very few physical quantities for which five terms of the perturbative expansion are known and the perturbative expansion can be studied, see Eq. (25).