Planar master integrals for the two-loop light-fermion electroweak corrections to Higgs plus jet production

We present the analytic calculation of the planar master integrals which contribute to compute the two-loop light-fermion electroweak corrections to the production of a Higgs boson in association with a jet in gluon-gluon fusion. The complete dependence on the electroweak-boson mass is retained. The master integrals are evaluated by means of the differential equations method and the analytic results are expressed in terms of multiple polylogarithms up to weight four.


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are new and presented in this paper for the first time. They are analitically computed using the differential equations method [67][68][69][70][71]. This method has proved to be very efficient for the computation of the MIs needed for higher-order corrections in the SM. In particular, we adopt the canonical basis approach [71][72][73][74][75][76][77] to the solution of the system of differential equations, which is expressed in terms of Chen-iterated integrals [78] represented as Goncharov multiple polylogarithms [79][80][81] (GPLs) up to weight four. The solutions are evaluated numerically using the software GiNaC [82,83] and tested numerically against the software FIESTA [84] in the Euclidean and Minkowski regions of the phase space. We find agreement in both regions.
The analytic results presented in this paper are given as ancillary files uploaded with the arXiv submission. 2 The paper is structured as follows. In the section 2 we give our notations and we describe the kinematic of the processes studied. In section 3 we describe briefly the method of differential equations and the canonical basis approach, moreover we give the alphabet associated to the solution. In section 4 we present the canonical basis and the transformation among the MIs in canonical form and the pre-canonical ones. Finally, in section 5 we present our conclusion.

Notations
In this paper we consider the partonic processes gg → Hg, gq → qH, qq → gH, and the crossed channels H → ggg and H → qqg. The external momenta corresponding either to gluons or to quarks are on their mass-shell p 2 i = 0, while the external Higgs momentum p 2 4 is regarded as a variable. Therefore, we appoach the solution of the master integrals (MIs) of the topology as a three-scale problem, where, apart from the Higgs momentum p 2 4 , the other two variables are the Mandelstam variables (2.1) For later convenience we define the dimensionless variabiles x, y, z such that where m 2 B is the squared mass of the internal Electroweak boson (W or Z). The physical phase-space region of the kinematic invariants (2.2) is for the decay channel, while for the production channel is The planar corrections can be computed considering the 7-denominator topology shown in figure 1. The MIs of the topology are defined by the two-loop dimensionally regularized proofs JHEP_175P_1018 integrals where d = 4 − 2ǫ, a i with i = 1, . . . , 7 are integer numbers, while a 8 and a 9 are natural numbers and the normalization is such that The D i , i = 1, . . . , 7, are the denominators involved, while D 8 , D 9 the numerators. They belong to the following set: The number of MIs for this topology is 48, considering the different channels and crossings. Apart from two-point functions, 3 the MIs presented in this paper are new.

The system of differential equations
The analytic computation of the master integrals is performed by using the differential equations method [67][68][69][70] applied to a canonical basis for the MIs [71,72]. In order to find the canonical basis several approaches exist [71][72][73][74][75]. We adopt the semi-algorithmic approach described in [76,77]. The canonical basis f ( x, ǫ) satisfies a system of first order partial linear differential equations with respect to the kinematic invariants x, where i ∈ {1, . . . , n}, n is the number kinematic invariants and A i ( x) is the set of matrices defining the differential equations. The linear system of partial differential equations, eq. (3.1), is equivalent to the following differential form, where the matrix elements ofÃ( x) are Q-linear combinations of logarithms, with

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The set of linearly indipendent arguments of the logarithms is called the alphabet of the solution, and its elements are called letters. The matrices A i ( x) satisfy the integrability condition The solution of the differential equations (3.2) can be formally written as a path ordered exponential: where P stands for the path-ordering operator, γ is a path in the space of kinematic invariants and f ( x 0 , ǫ) is a vector of boundary conditions. In practice we are interested in a solution around ǫ = 0. By series expanding f ( x, ǫ), and by parametrizing the integration contour with t ∈ [0, 1], the solution, eq. (3.5), translates to iterated integrals [78]: (3.7) In general, the alphabet letters depend algebraically on the kinematic invariants x i . However, for the problem under consideration, it is possible to perform a variable change such that the alphabet depends only rationally on the new variables. This implies that the solution eq. (3.7) can be directly expressed in terms of Goncharov multiple polylogarithms (GPLs) [79,87], defined recursively as, The recursion ends when n = 0 where we conventionally set Moreover, in order to deal with the divergency at the basepoint 0 when α n = 0, one defines: We remark that also when the alphabet letters are not rational functions it is often possible to find a representation of the solution in terms of polylogarithmic functions. One starts from the symbol [88] of the solution, which is obtained from the differential equations matrix A( x) and the ǫ 0 order of the boundary conditions, by the following recursive formula, The corresponding polylogarithmic expressions are found by using the algorithm of [88,89], and its algebraic generalisation [48].

The alphabet
The system of differential equations depends originally on rational functions of x and y and on the following square root z(1 + z) . (3.12) Exploiting for instance the methods described in [77], it is possible to rationalize the square root (3.12) by means of the change of variables It is then straightforward to define the differential equations with respect to x, y, w defined by the matrices A x (x, y, w), A y (x, y, w), A w (x, y, w) respectively and solve them by using the following iterative formula, where we denoted by, e.g., A x,jk (x, y, w) the matrix element of the matrix A x (x, y, w) and x 0 , y 0 , w 0 is a set of boundary points that in general depend on the master integral (see next section). The recursion above can be directly solved in terms of GPLs by factorizing the matrix elements with respect to the integration variable. By performing the factorization we obtain the (inverse) GPLs integration kernels, 2y, − 2y 4y + 1 , y − y 2 + y, y + y 2 + y, H decay: H + jet: where no branch cuts are present. We checked different points of the phase space against FIESTA [84], finding complete agreement.

Conclusions
In this paper we have computed analytically the planar master integrals relevant to the evaluation of the two-loop four-point amplitudes for Higgs plus three gluons, where one loop is an electroweak-boson loop and the other is a light-quark loop. Those amplitudes contribute to the mixed QCD-electroweak light-fermion corrections to the production of a Higgs boson with an additional jet, as well as to the real radiation of the NLO QCD corrections to the two-loop light-quark contribution to Higgs production via gluon-gluon fusion. The master integrals are evaluated with the differential equations method applied to a canonical set of basis integrals. Since the alphabet of the solution depends on a single square root, it is possible to find a variable change such that the matrices associated to the system of differential equations can be expressed in terms of rational functions. This allows a direct integration of the differential equations in terms of generalized polylogarithms up to weight 4. The expression of the master integrals in terms of generalized polylogarithms is quite flexible and can be evaluated numerically in a fast and precise way.

Acknowledgments
MB and VC thank the Institut für Theoretische Physik of the ETH Zürich for the hospitality and the COST (European Cooperation in Science and Technology) Action CA16201 PARTICLEFACE for the support, during the early stages of this work.

A Routing for the pre-canonical master integrals
In this appendix we give the expression for the pre-canonical master integrals in the form of eq. (2.5). (A.1)

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B The canonical basis