Two-loop master integrals for the mixed QCD-electroweak corrections for $H \rightarrow b\bar{b}$ through a $H t \bar{t}$-coupling

We present the two-loop master integrals relevant to the ${\mathcal O}(\alpha \alpha_s)$-corrections to the decay $H \rightarrow b \bar{b}$ through a $H t \bar{t}$-coupling. We keep the full dependence on the heavy particle masses, but neglect the $b$-quark mass. The occurring square roots can be rationalised and the result is expressed in terms of multiple polylogarithms.


Introduction
The precise determination of the properties of the Higgs boson is now a central pillar of the experimental programs at the LHC. As the Higgs boson decays predominately to bb, the partial width for this decay is of central importance [1]. This raises immediately the question, how precise can we predict this partial width within the Standard Model from theory? This requires the computation of quantum corrections.
The state-of-the-art for the partial decay width for H → bb is as follows: In massless QCD the corrections are known to O(α 4 s ) [2]. Keeping the mass dependence, QCD corrections are known to O(α 3 s ) [3][4][5]. The QCD calculation of order O(α 2 s ) for the decay H → bb through a Htt-coupling has become available quite recently [6]. Two-loop QED corrections and mixed QED-QCD corrections have been considered in [7]. For the mixed electroweak-QCD corrections to the decay H → bb of order O(αα s ) the leading term in an expansion in m 2 H /m 2 t has been obtained in [8,9]. This has been improved by including systematically more terms in [10].
As far as the two-loop electroweak corrections of order O(α 2 ) are concerned only the leading term in an expansion in m 2 H /m 2 t is known [11,12]. In recent years, there has been a substantial progress in our abilities to compute Feynman integrals, and Feynman integrals which previously could only be calculated approximatively come into reach. In this article we present the two-loop master integrals relevant to the O(αα s )-corrections to the decay H → bb through a Htt-coupling.
We keep the full dependence on the heavy particle masses (m t , m H and m W ), but neglect the b-quark mass. In Higgs and top physics there are two-loop Feynman integrals related to elliptic curves [13][14][15][16][17]. One might fear that we are in a similar situation here. To some relief this is not the case. All master integrals can be expressed entirely in terms of multiple polylogarithm. The difficulty of these integrals is entirely due to the required simultaneous rationalisation of two square roots.
Large parts of the calculation are based on by now standard techniques: We first derive a system of differential equations for the master integrals [18][19][20][21][22][23][24][25][26][27][28], using integration-by-parts identities [29,30] and the Laporta algorithm [31]. We then bring the system of differential equations into an ε-form [24]. This will introduce two square roots. It turns out that the square roots can be rationalised simultaneously with the methods of [32]. This is the essential new ingredient of our calculation. The resulting system of differential equations involves only dlog-forms and can be solved in terms of multiple polylogarithms. We are able to express all master integrals in terms of multiple polylogarithms with an alphabet consisting of 13 letters. This paper is organised as follows: In the next section we introduce our notation. In particular we define the kinematic variables which will rationalise the square roots. In section 3 we define the master integrals, for which the system of differential equations is in ε-form. The system of differential equations is presented in section 4. The analytic results for the master integrals are given in section 5. In addition, section 6 gives numerical results for the most interesting case p 2 = m 2 H . Finally, our conclusions are contained in section 7. The appendix shows for all master integrals the corresponding Feynman diagrams and describes the content of the supplementary electronic file attached to the arxiv version of this article.

Notation
We are interested in the mixed O(αα s )-corrections to the decay H → bb through a Htt-coupling.
Examples of Feynman diagrams are shown in fig. 1. Not shown are diagrams whose master integrals are related to the master integrals of the diagrams of fig. 1 by symmetry. We will neglect the b-quark mass. However, we will treat the dependence on the top quark mass m t , the W -boson mass m W and the momentum p of the Higgs boson exactly. For an on-shell Higgs-boson we have p 2 = m 2 H . For the two-loop contributions to the Higgs decay we have two independent external momenta p 1 and p 2 , which label the momenta of b-quark andb-quark, respectively. With two independent loop momenta we thus have seven linearly independent scalar products. For each of the four Feynman diagrams G A -G D we introduce an auxiliary topology with seven propagators. These are shown in fig. 2. The master integrals related to diagram G E are a subset of the master integrals related to diagram G A (and likewise a subset of the master integrals related to diagram G D ). We consider the integrals where D = 4−2ε denotes the number of space-time dimensions, γ E denotes the Euler-Mascheroni constant, µ is an arbitrary scale introduced to render the Feynman integral dimensionless, and the quantity ν is defined by The inverse propagators P X j are defined as follows: Topology A: Topology B: Topology C: Topology D: For all topologies our conventions are such that we are interested in the integrals with ν 7 ≤ 0. The Feynman parameter representations for the four topologies are given by where the integration is over The differential form ω is given by where the hat indicates that the corresponding term is omitted. The graph polynomials are given by Let us introduce an operator i + , which raises the power of the propagator i by one, e.g.
In the following we will set After setting µ 2 = m 2 t the master integrals depend kinematically on two dimensionless quantities.
with p = p 1 + p 2 . However, with this choice we will encounter square roots. In particular, the square roots will occur. The Källen function is defined by In order to rationalise the square roots we introduce dimensionless quantities x and y through The first transformation is standard and has occurred in many places before, the second one is easily obtained with the methods of ref. [32]. The Feynman integrals are then functions of x, y and the dimensional regularisation parameter ε. The inverse transformations are given by such that x = 0 corresponds to v = ∞ and y = 0 corresponds to w = 1. Let us also note that the point (v, w) = (0, 1) is blown up in (x, y)-space to the hypersurface x = 1. This motivates our final change of coordinates and we introduce

Master integrals
For the reduction to master integrals we use the programs Reduze [35], Kira [36] or Fire [37] combined with LiteRed [38,39]. Each topology involves a certain number of master integrals. This number is shown in table 1 and corresponds to the number of master integrals if we just consider one topology in isolation. Of course, the various topologies share some master integrals and the final number of master integral which we have to compute is lower. We have the following relations Table 1: The number of master integrals for a given topology.

Topology Number of master integrals
, I A µνρ0σ00 = I B µνσ0ρ00 = I C µν0ρσ00 , I A 0µνρσ00 = I C µ00νσρ00 = I D 0ρσµν00 , I A µνρσκ00 = I C µν0ρκσ0 , I B µνρσκ00 = I C µνσκρ00 , In total we have to consider 39 master integrals, which are grouped into 25 blocks such that one block corresponds to one sub-topology. Some of the master integrals are taken as integrals in D − 2 = 2 − 2ε space-time dimensions. Of course, with the help of the dimensional shift relations in eq. (13) they are easily expressed as (longer) linear combinations of master integrals in D = 4 − 2ε space-time dimensions. A system of master integrals is given by 4 The system of differential equations Let us set J = (J 1 , ..., J 39 ) T for the vector of master integrals. In this basis the system of differential equations is in ε-form [24]. We have where the matrix A is independent of ε. Let us first describe the singularities of the system of differential equations. The singularities are on hypersurfaces and each hypersurface is defined by a polynomial in x and y. There are sixteen polynomials, which are given by p 4 = y, p 5 = y − 1, p 6 = y + 1, p 7 = xy + x − y + 1, p 8 = xy + x − 2y, p 9 = 2xy − y + 1, p 10 = xy 2 + 2xy − 2y 2 + x + 2y, p 11 = xy 2 + 2xy + y 2 + x − 2y + 1, p 12 = xy 2 + 2xy − y 2 + x + 2y − 1, p 13 = 2xy 2 + 2xy − y 2 + 2y − 1, p 14 = 2xy 2 + 2xy − 3y 2 + 2y + 1, p 15 = 3xy 2 + 2xy − 2y 2 − x + 2y, We note that the polynomials p k are maximally of degree 3. The highest degree in the variable y is two, the highest degree in the variable x is one. The entries of the matrix A are Q-linear combinations of dlog-forms of these polynomials: We find that the matrix A contains only fifteen Q-independent linear combinations of dlog-forms. A basis for these is given by The entries of A are therefore of the form By a rescaling of the master integrals with constant factors we may actually achieve For our choice of basis of master integrals J we have c i jk ∈ Z. Equivalently, we may express the matrix A as where the entries of the 39 × 39-matrices C k are integer numbers. The matrix A is given in the supplementary electronic file attached to the arxiv version of this article. On specific hypersurfaces the differential forms simplify considerably. On the hypersurface y = 0 (i.e. for the case m W = m t ) the differential forms reduce to a linear combination of On the hypersurface x = 0 (i.e. for the case p 2 → ∞) the differential forms reduce to a linear combination of On the hypersurface x = 1 (i.e. for the case p 2 = 0 and m 2 W = m 2 t ) the differential forms reduce to a linear combination of dy y , dy y − 1 , dy y + 1 , 2ydy The derivative of the master integrals is given by the product of the matrix A with the vector J: The right-hand side vanishes if J is in the kernel of A. This is what happens on the hypersurface x = 1: Although A = 0 we have It follows that the master integrals are constant on the hypersurface x = 1. This is a significant simplification for solving the differential equations.

Analytical results
The analytic result for the master integrals at a point (x, y) is obtained from the value of the master integrals at a boundary point (x i , y i ) by integrating the system of differential equations along a path from (x i , y i ) to (x, y). The result does not depend on the chosen path, but only on the homotopy class of the path. For the case at hand we have several significant simplification: 1. All master integrals are constant on the hypersurface x = 1 (corresponding to p 2 = 0 and m 2 W = m 2 t ) and we may take the values of the master integrals on this hypersurface as boundary values. We therefore have a boundary line.

Conclusions
In this paper we presented the two-loop master integrals relevant to the O(αα s )-corrections to the decay H → bb through a Htt-coupling. We kept the exact dependence of the masses of the heavy particles (m W and m t ) and the momentum p 2 of the Higgs boson, but neglected the b-quark mass. All master integrals are expressed in terms of multiple polylogarithms with an alphabet of 13 letters. They can be evaluated to arbitrary precision with the help of the GiNaC-library. For the special case p 2 = m 2 H we presented the numerical values.

B Supplementary material
Attached to the arxiv version of this article is an electronic file in ASCII format with Maple syntax, defining the quantities A, J.
The matrix A appears in the differential equation The entries of the matrix A are Z-linear combinations of ω 1 , ..., ω 15 , defined in eq. (26). These differential forms are denoted by omega_1, ..., omega_15.