]>NUPHB13980S0550-3213(17)30034-210.1016/j.nuclphysb.2017.01.020The Author(s)High Energy Physics – PhenomenologyFig. 1The two-loop light-quark electroweak contributions to gg→H. V stands for W±,Z and the fermionic lines represent different quarks, depending on the electroweak boson V.Fig. 1Fig. 2Planar (left) and non-planar (right) topologies. See text for momenta assignments and propagator labels.Fig. 2Table 1Different kinematic regions in s, ω and y.Table 1VariableEuclideanMinkowski, below thresholdAbove threshold

s−∞<s<00<s<4M24M2<s<+∞

ω=−M2s0<ω<+∞−∞<ω<−14−14<ω<0

y=1+4ω−11+4ω+10<y<1eiϑ, 0<ϑ<π−1<y<0

Two-loop electroweak corrections to Higgs–gluon couplings to higher orders in the dimensional regularization parameterMarcoBonetti⁎marco.bonetti@kit.eduKirillMelnikovkirill.melnikov@kit.eduLorenzoTancredilorenzo.tancredi@kit.eduInstitute for Theoretical Particle Physics, KIT, 76128 Karlsruhe, GermanyInstitute for Theoretical Particle PhysicsKITKarlsruhe76128Germany⁎Corresponding author.Editor: Tommy OhlssonAbstractWe compute the two-loop electroweak correction to the production of the Higgs boson in gluon fusion to higher orders in the dimensional-regularization parameter ε=(d−4)/2. We employ the method of differential equations augmented by the choice of a canonical basis to compute the relevant integrals and express them in terms of Goncharov polylogarithms. Our calculation provides useful results for the computation of the NLO mixed QCD-electroweak corrections to gg→H and establishes the necessary framework towards the calculation of the missing three-loop virtual corrections.1IntroductionThe recent discovery of the Higgs boson and the non-observation of any New Physics at the LHC establishes the validity of the Standard Model as the low-energy effective theory of Nature. At the same time, the apparent inability of the Standard Model to explain several experimental facts makes the need for physics beyond the Standard Model (BSM) as strong as ever. Searching for clues about BSM physics is in the focus of contemporary particle physics. The Higgs boson is bound to play an important role in this endeavor. Indeed, the Higgs mechanism in the Standard Model is very simplistic and rather ad hoc. At the same time, there are many extensions of the Standard Model where the Higgs boson is the only particle that is sensitive to rich physics beyond it. More generally, if the Higgs boson is responsible for generating masses not only of the Standard Model but also of some BSM particles, which appears to be necessary for protecting the Higgs boson mass from large radiative corrections, these new particles may affect the couplings of the Higgs boson to gauge bosons and fermions through radiative corrections. A percent modification of the Higgs couplings is a generic consequence of physics beyond the Standard Model at the energy scale of about 1 TeV. Therefore, measurement of the Higgs bosons couplings to Standard Model particles to this level of precision is an important goal of the LHC physics program.The major production channel of Higgs bosons at the LHC is gluon fusion. The recent computation of the three-loop QCD corrections to σgg→H significantly reduces theoretical uncertainty in the predicted cross section. According to Ref. [1], the theory uncertainty in σgg→H is close to 5% and the uncertainties related to imprecise knowledge of parton distribution functions and the strong coupling constant are close to 4%. The theoretical uncertainty has several sources such as the residual scale dependence of the three-loop QCD result, imperfect knowledge of the bottom quark contribution to gg→H and the mixed three-loop QCD-electroweak corrections which are known in the unphysical limit mZ,W≫mH [2]. Each of these sources contributes similar amount to the final uncertainty which implies that a better understanding of all of them is required for reducing the uncertainty to ∼1–2%.In this paper we focus on the computation of the two-loop electroweak correction to the production of the Higgs boson in gluon fusion. This contribution arises because gluons couple to electroweak vector bosons W and Z through a quark loop; a subsequent fusion of the electroweak bosons to the Higgs boson gives rise to electroweak-mediated ggH coupling. The quark loop receives contributions from both light and heavy quarks but the relatively small mass of the Higgs boson leads to a strong dominance of the light quark contributions.11More precisely, about 95% of the full electroweak contribution to ggH is due to the light quark loops.The electroweak contributions to ggH have been evaluated analytically at leading (two-loop) order in Refs. [3–6]. Since the QCD corrections increase the leading, top-quark mediated, contribution to gg→H by almost a factor two, it is essential to understand if a similar enhancement is present in case of the electroweak corrections to gg→H. To clarify this issue, we need a computation of QCD corrections to the electroweak contribution to ggH. However, since the electroweak contribution starts at two loops, calculation of NLO QCD corrections requires dealing with three-loop diagrams with massive internal lines. Given the complexity of the required computation, one can try to simplify it by considering different kinematic limits: mixed QCD-electroweak corrections in the unphysical limit of a vanishingly small Higgs boson mass mZ,W≫mH were calculated in Ref. [2].However, recent progress in the theoretical understanding of QCD effects in gg → H and continuous developments in the technology of multi-loop computations make it worthwhile and interesting to attempt an exact computation of the NLO QCD corrections to the electroweak contribution to ggH. In this paper, we make an important step in this direction by setting up a modern calculational framework for this problem that employs canonical bases for master integrals and differential equations, and computing the two-loop electroweak contribution to ggH to higher orders in the dimensional regularization parameter ε=(4−d)/2. The knowledge of the two-loop amplitude to higher orders in ε is necessary for subtracting infrared and collinear singularities from the electroweak contributions to the gg→Hg inelastic process or, alternatively, for extracting the relevant finite remainder, defined by the Catani formula [7], from the three-loop mixed QCD-electroweak contribution to ggH amplitude.Specifically, we derive the two-loop electroweak correction to gg→H through O(ε2) and show that only GPLs up to weight five appear in this amplitude. We perform our calculation using the method of differential equations [8–10], augmented by the choice of a canonical basis of master integrals, introduced in Ref. [11].22An alternative way to construct and solve differential equations has been investigated in Ref. [12]. A canonical basis of master integrals is presented and the master integrals are calculated in terms of Goncharov's multiple polylogarithms (GPLs) [13–15]. In order to fix analytically all boundary conditions we make extensive use of the large mass expansion the PSLQ algorithm. This allows us to derive the expansion for the master integrals in the dimensional regularization parameter ε through weight six. From our calculation we can easily reproduce the O(ε0) results which have been known for a long time in the literature [5].The paper is organized as follows. We introduce the notation and discuss the structure of the scattering amplitude gg→H in Section 2. We describe the master integrals and the differential equations in Section 3. We explain how the boundary conditions can be fixed using the large mass expansion and outline the analytic continuation of GPLs, required to obtain results in physical kinematics, in Section 4. The gg→H finite part of the amplitude is given in Section 5. We provide constants of integration for the master integrals up to weight six in Appendix A. The explicit expressions for the master integrals up to this weight, and the gg→H amplitude through O(ε2) are available in the ancillary file.2Feynman diagrams and master integralsWe consider the electroweak correction to the gg→H amplitude mediated by a light-quark loop. The relevant contributions are shown in Fig. 1. The fermionic lines represent up, down, strange and charm quarks, that are taken to be massless.33Bottom quarks require a special treatment, together with top quarks. The incoming gluons g1 and g2 are on-shell and carry momenta p1 and p2, with color indices c1 and c2 and polarizations ελ1(p1) and ελ2(p2), respectively. The momentum of the Higgs boson is taken to be p3=p1+p2 with p32=mH2=s.Thanks to gauge-invariance and parity constraints, the gg→H scattering amplitude is expressed in terms of a single form factor(2.1)Mλ1λ2c1c2=F(s,mW2,mZ2)[ημν−p2μp1νp1⋅p2]δc1c2ελ1μ(p1)ελ2ν(p2). It is possible to extract the form factor F by contracting Mλ1λ2c1c2 with the projection operator(2.2)Pc1c2λ1λ2=εμ⁎λ1(p1)εν⁎λ2(p2)δc1c2Nc2−11d−2[ημν−p1μp2ν+p1νp2μp1⋅p2]. We find(2.3)F(s,mW2,mZ2)=∑λ1,λ2,c1,c2Pc1,c2λ1λ2Mλ1λ2c1,c2.The form factor F is a linear combination of integrals which depend on the scalar products between loop and external momenta, and on the scalar products of loop momenta between themselves. All the integrals in F are obtained starting from the two topologies shown in Fig. 2. At variance with Feynman diagrams in Fig. 1, when we consider topologies and master integrals, we use wavy (solid) lines to denote massless (massive) propagators, respectively. We take all momenta to be incoming, i.e.(2.4)p3=−p1−p2,p1,22=0,p33=s=mH2.The planar and non-planar integrals are parametrized as(2.5)IP(a1,a2,a3,a4,a5,a6,a7)=∫ddk1ddk2[iπ2Γ(1+ε)]21[1]a1[2]a2[3]a3[4]a4[5]a5[6]a6[7]a7,(2.6)INP(a1,a2,a3,a4˜,a5,a6,a7)=∫ddk1ddk2[iπ2Γ(1+ε)]21[1]a1[2]a2[3]a3[4˜]a4˜[5]a5[6]a6[7]a7, where(2.7)[1]=k12,[2]=(k1+p1)2,[3]=k22,[4]=(k1−p2)2,[4˜]=(k2+p2)2,[5]=(k1−k2+p1)2−M2,[6]=(k2−k1+p2)2−M2,[7]=(k1+k2)2. In both cases, the propagator [7] is auxiliary; it is only needed for the parametrization of tensor integrals with (otherwise) irreducible numerators. Both planar and non-planar integrals are analytic functions in the complex plane of the variable s with the cut along the real axis starting at s=0. This discontinuity corresponds to massless intermediate states in Feynman diagrams. At s≥4M2, it also becomes possible to produce pairs of vector bosons on the mass shell; this leads to additional contributions to the discontinuities of IP,NP. We use the program Reduze2 [16] to express all integrals that appear in the evaluation of gg→H amplitude through master integrals (MIs). We also use the integration-by-parts reduction identities to derive the differential equations in s and M2 satisfied by the master integrals.3Differential equationsWe denote a vector of master integrals by I, a set of kinematic variables by x∈{s,M2}, and write the differential equations as(3.1)∂I(x,ε)∂xi=Ai(x,ε)I(x,ε). It was conjectured in Ref. [11] that in many physically relevant cases a canonical basis of master integrals I′ exists with the property that the right hand side of the differential equation has a simple, factorized dependence on the regularization parameter ε. While the statement has not been rigorously proved, it is expected to be true at least for those cases that can be expressed in terms of Chen iterated integrals. The differential equations for the canonical basis assume the following form(3.2)∂I′(x,ε)∂xi=εAi′(x)I′(x,ε), so that the iterative construction of I′ as series in ε becomes straightforward. General criteria to find candidate canonical integrals are given in Ref. [17] and, under certain conditions for ordinary differential equations, in Ref. [18]. We do not use this last algorithm in this paper; instead, we begin by constructing canonical bases for the simplest integrals in the set and gradually move to more complex ones, as described extensively in [19]. Since the original matrices Ai are relatively sparse, this approach turns out to be quite practical for finding the canonical basis.It is convenient to choose as independent variables the center of mass energy squared s and the dimensionless ratio ω=−M2/s. Since the dependence of any master integral on s follows uniquely from its mass dimension, we write master integrals as(3.3)I(s,ω)=s−a−2εI(ω), where a is an integer determined by the canonical mass dimension of the integral. The non-trivial information is contained in the functions Ii(ω), which are dimensionless quantities. By choosing these functions to be appropriately re-scaled versions of the master integrals found by Reduze2I1(ω)=ε2(ε−1)(−s)2εI2(ω)=−ε2(−s)2ε+1(ω+1)I3(ω)=−ε2(−s)2ε+1(ω+1)I4(ω)=ε3(−s)2ε+1I5(ω)=ε2(−s)2ε+2ωI6(ω)=−ε2(−s)2ε+2I7(ω)=ε4(−s)2ε+1I8(ω)=ε4(−s)2ε+1(3.4)I9(ω)=ε4(−s)2ε+2(4ω+1)I14(ω)=ε2(−s)2ε+1I11(ω)=−ε2(−s)2ε+2I12(ω)=−ε2(1−2ε)(−s)2ε+2ω, we can cast the system of differential equations for I(ω) in the following form(3.5)dI(ω)dω=[A0(ω)+εA1(ω)]I(ω). The matrices A0,1 are rational functions of ω, and have a block-triangular structure.To construct a systematic expansion of master integrals in ε, it is convenient to change basis of master integrals and transform the system of differential equations into a canonical form. This requires A0 to be removed. We can do that in a symbolic form by first solving the matrix differential equation(3.6)dSˆA0dω=A0(ω)SˆA0→SˆA0=Pωe∫A0(ω′)dω′, where Pω is the path-ordering operator defined as(3.7)Pωe∫A0(ω′)dω′=∑k=0+∞∫ω0ωA0(ω1)∫ω0ω1A0(ω2)…∫ω0ωk−1A0(ωk)dωk…dω2dω1. By defining a new set of master integrals(3.8)F=SˆA0−1I, it is easy to see that F satisfies differential equations in the canonical form(3.9)dF(ω)dω=εSˆA0−1A1SˆA0F(ω). The non-trivial part of this procedure is to find the matrix SˆA0. A systematic way to do that, based on the Magnus exponentiation was suggested in Ref. [20]. Instead, we do that iteratively, using the sparse nature of the matrix A0 and considering different blocks of A0 separately. As an illustration, consider two integrals from the list of master integrals, I2 and I3. Neglecting the matrix A1, we find that they satisfy the system of coupled differential equations(3.10)ddω(I2I3)=1ω+1(0−201)(I2I3). Integrating this equation, we find(3.11)(I2I3)=Sˆ0(C1C2),Sˆ0=(−1−2ω0ω+1), where C1,2 are the two integration constants. Since the above solution satisfies the system of differential equations for arbitrary C1, C2, the matrix Sˆ0 satisfies the original differential equation(3.12)dSˆ0dω=1ω+1(0−201)Sˆ0 and, therefore, can be taken to be a part of SˆA0. Finally, we compute F=Sˆ0−1I and findF2=−I2+2ω1+ωI3=−ε2(−s)2ε+1[(ω+1)+2ω],F3=11+ωI3=ε2(−s)2ε+1. The system of differential equations for the integrals F2,3 is then guaranteed to be in the canonical form. We find(3.13)ddω(F2F3)=ε(−2ω+14ω+11ω+1−1ω−2ω+1−1ω)(F2F3).We apply this procedure block by block, to the block-triangular matrix A0+εA1 and obtain the canonical system of differential equations that we write in the following form(3.14)dF(ω)=εdB(ω)F(ω). The canonical basis of the master integrals F(ω) reads(ε2(ε−1)(−s)2ε−ε2(−s)2ε+1[(ω+1)+2ω]ε2(−s)2ε+1ε3(−s)2ε+1ε2[(1−ε)21+4ω(−s)2ε+ε3(1+2ω)1+4ω(−s)2ε+1−(−s)2ε+2ω21+4ω]ε2[(1−ε)(−s)2ε1+4ω2+(−s)2ε+1(ω+1)1+4ω4+(−s)2ε+1(ω+1)1+4ω2+(−s)2ε+2ω1+4ω]ε4(−s)2ε+1ε4(−s)2ε+1ε4(−s)2ε+24ω+1ε2(−s)2ε+1−ε2(−s)2ε+24ω+1ε2[1−ε2(−s)2ε+(−s)2ε+1ω+14+(−s)2ε+1ω+12+(−s)2ε+2ω+(−s)2ε+2+(1−2ε)(−s)2ε+2ω]).For the matrix B(ω) we obtain44The signs of the arguments of the logarithms are chosen to ensure that for positive ω the logarithms are real.(3.15)B(ω)=B1logω+B2log(1+ω)++B3[log(−1+1+4ω)−log(1+1+4ω)]+B4log(1+4ω). The ω-independent matrix coefficients in the above equation are given by(3.16)B1=(−2000000000000000000000000−1−1000000000000−3000000000000−1000000000000−1000000121412−1200−200000−1−1200000−20000000000000000000000000−1000000000000001121200000000−2),B2=(0000000000000−2400000000001−2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000),(3.17)B3=(000000000000000000000000000000000000000010000000−40030000000012−3400000000000000−12−1000000000000000000630−800−840000000000000000000000000−1000000020000−20),B4=(0000000000000000000000000000000000000000000000000000−1000000000000−100000000000000000000000000000000000000−20000000000000000000000000−10000000000000).It is convenient to remove the square roots from the Eqs. (3.14), (3.15) by changing variables ω→y where55The relations between s, ω and y are summarized in Table 1.(3.18)y=1+4ω−11+4ω+1. The differential equations (3.14) take the following form(3.19)dF(y)=εdC(y)F(y), where the matrix C reads(3.20)C(y)=C1logy+C2log(1−y)+C3log(1+y)+C4log(1−y+y2). The y-independent matrices C1,…,4 are(3.21)C1=(−2000000000000000000000000−1−1000000000000−310000000−4003−1000000012−34000−1000000121412−12−12−1−200000−1−1200000−20000630−800−840000000000000−100000000000−100112120020000−2−2),C2=(40000000000004−8000000000006000000000000600000000000040000000000004000000−1−12−1100400000210000040000000000004000000000000200000000000020−2−1−1000000004),(3.22)C3=(0000000000000000000000000000000000000000000000000000−2000000000000−200000000000000000000000000000000000000−40000000000000000000000000−20000000000000),C4=(0000000000000−2400000000001−2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000).It is straightforward to write the solution of the system of differential equations (3.19) as Taylor series in ε(3.23)F(y)=F0(0)+ε[∫0yC(τ1)F0(0)dτ1+F0(1)]++ε2[∫0yC(τ1)∫0τ1C(τ2)F0(0)dτ2dτ1+∫0yC(τ1)F0(1)dτ1+F0(2)]+..., where F0(i) are integration constants that can not be fixed from the differential equations.Given the iterative structure of the solution, it can be written as a linear combination of the so-called Goncharov's polylogarithms (GPLs), also known as hyperlogarithms, defined as [13–15,21](3.24)G(mw,mw−1;x):={1w!logwxif mw=(0,…,0)∫0xf(mw;τ)G(mw−1;τ)dτif mw≠(0,…,0), where mw indicates the vector (mw,mw−1). The functions f(a;τ) represent the integration kernels; for our system of differential equations they span the following set(3.25)f(0;τ)=1τ,f(1;τ)=1τ−1,f(−1;τ)=1τ+1,f(r;τ)=2τ−1τ2−τ+1. The last term in the set is quadratic in the integration variable; it is possible to re-write it in a usual linear form(3.26)f(r;τ)=2τ−1τ2−τ+1=1τ−r++1τ−r−=f(r+;τ)+f(r−;τ),r±=e±iπ3, at the expense of introducing complex-valued poles. This last step is essential for numerical evaluation of the GPLs but it is not required to integrate the system of differential equations since, thanks to the linearity of the differential equations and the definition of the GPLs,(3.27)G(…,r,…;x)=G(…,r−,…;x)+G(…,r+,…;x). This implies that we can perform the analytic integration using the symbol r and then, for the numerical evaluation of the final result, switch to r± using Eq. (3.27).66The generalization of GPLs studied here has been already considered in detail in [22,23].In Table 1 the relations among s, ω and y, as well as the different kinematic regions, are summarized.4Boundary conditions and analytic continuationLinear differential equations allow us to restore the dependencies of master integrals on ω up to a single constant. The constant can be fixed by computing the required integral at any point ω=ω0 and comparing the result with the solution of the differential equation.It turns out to be convenient to determine the boundary conditions by computing the master integrals in ω→∞ or y→1 limit. Since ω=−M2/s, this corresponds to M2→∞ at fixed s; in this limit the integrals can be easily computed using the so-called large-mass expansion procedure [24]. The large-mass expansion procedure can be formulated as follows: consider two different scalings for each of the loop momenta k1,2∼M and k1,2∼s and, for a chosen scaling, systematically expand the integrand in Taylor series in all small variables. The set of small variables will differ from scaling to scaling; for example, if k1∼k2∼M, one Taylor expands in external momenta but if the scaling k1∼M, k2∼s is considered, one Taylor expands in the external momenta and k2. It is easy to see that, to leading order in s/M2, most of the master integrals are expressed in terms of two-loop tadpole integrals and, in some cases, in terms of products of one-loop three- and two-point integrals and one-loop tadpole integrals.As an illustration, consider the non-planar master integral(4.1)F9(ω)=ε4(−s)2ε+24ω+1. We are interested in determining its behavior in the y→1 limit. By applying the large mass expansion, we find that the non-planar integral scales as(4.2)limM≫s==O(M−4) in the large-M limit. This implies that the large-M limit of the F9 non-planar master integral reads(4.3)F9(ω)∼ε4(−s)2ε+24ω+1M4∼1ω3/2∼(1−y)3, where we used relations among s, ω and y variables summarized in Table 1. F9 vanishes as (1−y)3 as y goes to 1, and this information is sufficient to fix the constant of integration for this master integral.A similar analysis reveals that only three master integrals F1,F2,F10 possess non-vanishing y→1 limit. These limits are(4.4)limy→1F1(y)=−(1−y)4εΓ(1+2ε)Γ(1−ε)2Γ(ε+1),(4.5)limy→1F2(y)=(1−y)4εΓ(1+2ε)Γ(1−ε)Γ(ε+1),(4.6)limy→1F10(y)=(1−y)4εΓ2(1−ε)Γ(1−2ε).To fix the boundary conditions, we need to evaluate the GPLs of the form G(m,y), where m is composed of the elements of the set(4.7){0,1,−1,eiπ3,e−iπ3}, and the limit y→1 is taken where possible. For weights higher than three, not all the Goncharov polylogarithms at y=1 with the arguments from Eq. (4.7) can be analytically expressed in terms of canonical irrational numbers such as π and ζ(n). Nevertheless, we expect that the boundary constants are linear combinations of these irrational numbers; to find them we follow a numerical approach. We use our solution of the differential equations and the boundary conditions discussed above to find the integration constants numerically with high precision.77For the evaluation of the GPLs, the GINAC implementation was used, see Ref. [15]. We then fit the resulting numerical value to a linear combination of π2 and ζ(n) of a well-defined weight. For example, at weight two we only have π2, at weight three ζ(3), at weight four π4, at weight five ζ(5) and π2ζ(3) and at weight six π6 and ζ(3)2. For each of the master integrals, we have achieved the matching of the numerical and the analytic results to at least 750 digits.Our final remark concerns the analytic continuation of the master integrals F(y). So far, we have studied them in the Euclidean region but we need them in the region where s=mH2>0 and, yet, s<4M2. The correct analytic continuation is achieved by replacing s→s+i0 at fixed M2. It is easy to see that this implies ω→ω+i0 and y→y+i0.All the master integrals evaluated here have been compared for at least four different values of s/M2, both in the Euclidean and Minkowski region, to the numerical results obtained with the program Secdec [25]. In all cases we found excellent agreement.5Form factor for gg→HThe gg→H amplitude is described by a single form factor, as explained in Section 2. This form factor receives contributions from loops with W and Z bosons. The form factor is finite in four dimensions (ε→0) and can be written as:(5.1)F(s,mW2,mZ2)=−(4π)4ε(−s)2εΓ2(ε+1)×iαSα24πsin4ϑWv2[4A(yW)+2cos4ϑW(54−73sin2ϑW+229sin4ϑW)A(yZ)], where(5.2)yW=1−4mW2/mH2−11−4mW2/mH2+1,yZ=1−4mZ2/mH2−11−4mZ2/mH2+1.We take the CKM matrix to be an identity matrix. The contributions of W bosons is computed in Eq. (5.1) taking into account first and second generations. The contribution of the Z boson is calculated for five massless quarks (u, d, s, c and b).The function A in Eq. (5.1) can be expanded in ε; we have computed it through O(ε2):(5.3)A(y)=A0(y)+εA1(y)+ε2A2(y)+O(ε3).The function A0(y) reads(5.4)A0(y)=16(y−1)3[−6−6y(y2−y+2)G(0,0,r,y)−6(1−y)(y2−y+1)G(r,y)+(y+1)(y2+1)[18G(−1,0,r,y)+π2G(−1,y)−18G(−1,0,0,y)]+12y(2y2+y+1)[G(0,1,0,y)−G(0,1,r,y)]+2(1−y)(y2+y+1)[6G(1,0,r,y)−12G(1,1,r,y)−π2G(1,y)+12G(1,1,0,y)]+6y(1−y)[G(0,r,y)−2G(1,r,y)+G(0,0,y)−2G(1,0,y)]−6y2(y+1)G(0,0,0,y)−y(3π2y2+12y2+π2y−18y+2π2+6)G(0,y)−12(1−y)(2y2+y+2)G(1,0,0,y)−6(y+3)(y2+1)ζ(3)+(1−y)(12y2−π2y−24y+12)].We see that, although A0 is the finite part of a 2-loop form factor, the highest weight of the GPLs that appears in Eq. (5.1) is three. This happens because none of the master integrals that have 1/ε4 poles contribute to the gg→H amplitude at leading order in the ε→0 limit.The expression for F in Eq. (5.1) has been compared with a previous calculation in Ref. [4] and agreement was found. The terms A1(y) and A2(y) are new. They can be found in the ancillary file provided with this paper.6ConclusionsWe have presented a calculation of the mixed two-loop QCD-electroweak corrections mediated by massless quarks to the production of the Higgs boson in gluon fusion. We extended the known result for these corrections to two higher orders in the dimensional regularization parameter ε. This is one of the ingredients required for the computation of the NLO mixed QCD-electroweak corrections to gg→H amplitudes. We employed the method of differential equations, determined a canonical basis of master integrals and expressed all the relevant functions in terms of Goncharov polylogarithms. Finally, we used a mixed numerical and analytical approach, based on the PSLQ algorithm, in order to fix all necessary boundary conditions. This establishes the necessary framework to successfully address the calculation of the missing three-loop virtual contributions, whose calculation is ongoing.AcknowledgementsThe work of M.B. was supported by a graduate fellowship from Karlsruhe Graduate School “Collider Physics at the highest energies and at the highest precision” GRK 1694. We wish to thank Christopher Wever for many fruitful discussions and help during all steps of this project, and Roberto Bonciani for his assistance with comparing our results with Ref. [4].Appendix AF0 valuesIn this appendix we present the boundary conditions for the master integrals defined in Eq. (3.23). The weights 0, 1 and 2 were determined analytically. For weights 4, 5 and 6 the results were obtained by fitting numerical results to an analytic Ansatz to at least 750 digits.(A.1)F1(y)=ε2(ε−1)(−s)2ε(A.2)F0,1(0)=−12,F0,1(1)=0,F0,1(2)=−π26,F0,1(3)=ζ(3),F0,1(4)=−π420,F0,1(5)=13[π2ζ(3)+9ζ(5)],F0,1(6)=−ζ2(3)−61π63780.(A.3)F2(y)=−ε2(−s)2ε+1[(ω+1)+2ω](A.4)F0,2(0)=1,F0,2(1)=0,F0,2(2)=−π23,F0,2(3)=−10ζ(3),F0,2(4)=−11π490,F0,2(5)=10π2ζ(3)3−54ζ(5),F0,2(6)=50ζ2(3)−121π61890.(A.5)F3(y)=ε2(−s)2ε+1(A.6)F0,3(0)=0,F0,3(1)=0,F0,3(2)=π26,F0,3(3)=8ζ(3),F0,3(4)=7π472,F0,3(5)=48ζ(5)−3π2ζ(3),F0,3(6)=127π62160−48ζ2(3).(A.7)F4(y)=ε3(−s)2ε+1(A.8)F0,4(0)=0,F0,4(1)=0,F0,4(2)=0,F0,4(3)=−2ζ(3),F0,4(4)=−π4180,F0,4(5)=−π2ζ(3)3−12ζ(5),F0,4(6)=9ζ2(3)−37π63780.(A.9)F5(y)=ε2[−2(ε−1)(−s)2ε+ε3(1+2ω)1+4ω(−s)2ε+1+−(−s)2ε+2ω21+4ω](A.10)F0,5(0)=0,F0,5(1)=0,F0,5(2)=−π23,F0,5(3)=−4ζ(3),F0,5(4)=−41π4180,F0,5(5)=π2ζ(3)−30ζ(5),F0,5(6)=21ζ2(3)−97π6756.(A.11)F6(y)=ε2[(1−ε)(−s)2ε1+4ω2+(−s)2ε+1(ω+1)1+4ω4++(−s)2ε+1(ω+1)1+4ω2+(−s)2ε+2ω1+4ω](A.12)F0,6(0)=0,F0,6(1)=0,F0,6(2)=π24,F0,6(3)=6ζ(3),F0,6(4)=5π448,F0,6(5)=36ζ(5)−5π2ζ(3)2,F0,6(6)=77π61440−36ζ2(3).(A.13)F7(y)=ε4(−s)2ε+1(A.14)F0,7(0)=0,F0,7(1)=0,F0,7(2)=0,F0,7(3)=ζ(3),F0,7(4)=5π472,F0,7(5)=7π2ζ(3)6+6ζ(5),F0,7(6)=3ζ2(3)2+13π6270.(A.15)F8(y)=ε4(−s)2ε+1(A.16)F0,8(0)=0,F0,8(1)=0,F0,8(2)=−π26,F0,8(3)=4ζ(3),F0,8(4)=−π49,F0,8(5)=π2ζ(3)3+20ζ(5),F0,8(6)=−16ζ2(3)−173π63780.(A.17)F9(y)=ε4(−s)2ε+24ω+1(A.18)F0,9(0)=0,F0,9(1)=0,F0,9(2)=−π23,F0,9(3)=−24ζ(3),F0,9(4)=13π445,F0,9(5)=46π2ζ(3)3−100ζ(5),F0,9(6)=264ζ2(3)+397π6945.(A.19)F10(y)=ε2(−s)2ε+1(A.20)F0,10(0)=1,F0,10(1)=0,F0,10(2)=−π26,F0,10(3)=−2ζ(3),F0,10(4)=−π440,F0,10(5)=π2ζ(3)3−6ζ(5),F0,10(6)=2ζ2(3)−79π615120.(A.21)F11(y)=−ε2(−s)2ε+24ω+1(A.22)F0,11(0)=0,F0,11(1)=0,F0,11(2)=π26,F0,11(3)=2ζ(3),F0,11(4)=−π4360,F0,11(5)=6ζ(5)−π2ζ(3),F0,11(6)=−6ζ2(3)−47π615120.(A.23)F12(y)=ε2[1−ε2(−s)2ε+(−s)2ε+1ω+14+(−s)2ε+1ω+12+(−s)2ε+2ω+(−s)2ε+2+(1−2ε)(−s)2ε+2ω](A.24)F0,12(0)=0,F0,12(1)=0,F0,12(2)=π212,F0,12(3)=4ζ(3),F0,12(4)=77π4720,F0,12(5)=30ζ(5)−3π2ζ(3)2,F0,12(6)=1711π630240−30ζ2(3).Appendix BSupplementary materialSupplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.nuclphysb.2017.01.020. Appendix BSupplementary materialThe following is the Supplementary material related to this article.Ancillary fileThe ancillary Mathematica file provides: 1) the form factor of the amplitude up to order ε2; 2) the full expression of all the master integrals, up to order ε6; 3) the value of the integration constants for all the master integrals up to order ε6; 4) the matrix of coefficient of the differential equations for the master integrals in the canonical form.Ancillary fileReferences[1]C.AnastasiouC.DuhrF.DulatE.FurlanT.GehrmannF.HerzogA.LazopoulosB.MistlbergerHigh precision determination of the gluon fusion Higgs boson cross-section at the LHCJ. High Energy Phys.052016058arXiv:1602.0069[2]C.AnastasiouR.BoughezalF.PetrielloMixed QCD-electroweak corrections to Higgs boson production in gluon fusionJ. High Energy Phys.042009003arXiv:0811.3458[3]U.AgliettiR.BoncianiG.DegrassiA.ViciniTwo-loop electroweak corrections to Higgs production in proton–proton collisionsTeV4LHC Workshop: 2nd Meeting, Brookhaven Upton, New York, February 3–5, 20052006arXiv:hep-ph/0610033[4]U.AgliettiR.BoncianiG.DegrassiA.ViciniTwo loop light fermion contribution to Higgs production and decaysPhys. Lett. B5952004432441arXiv:hep-ph/0404071[5]U.AgliettiR.BoncianiG.DegrassiA.ViciniMaster integrals for the two-loop light fermion contributions to gg→H and H→γγPhys. Lett. B60020045764arXiv:hep-ph/0407162[6]S.ActisG.PassarinoC.SturmS.UcciratiNNLO computational techniques: the cases H→γγ and H→ggNucl. Phys. B8112009182273arXiv:0809.3667[7]S.CataniThe singular behavior of QCD amplitudes at two loop orderPhys. Lett. B4271998161171arXiv:hep-ph/9802439[8]A.V.KotikovDifferential equations method: new technique for massive Feynman diagrams calculationPhys. Lett. B2541991158164[9]E.RemiddiDifferential equations for Feynman graph amplitudesNuovo Cimento A110199714351452arXiv:hep-th/9711188[10]T.GehrmannE.RemiddiDifferential equations for two loop four point functionsNucl. Phys. B5802000485518arXiv:hep-ph/9912329[11]J.M.HennMultiloop integrals in dimensional regularization made simplePhys. Rev. Lett.1102013251601arXiv:1304.1806[12]C.G.PapadopoulosSimplified differential equations approach for Master IntegralsJ. High Energy Phys.072014088arXiv:1401.6057[13]E.RemiddiJ.A.M.VermaserenHarmonic polylogarithmsInt. J. Mod. Phys. A152000725754arXiv:hep-ph/9905237[14]A.B.GoncharovPolylogarithms in arithmetic and geometryProceeding of the International Congress of Mathematicians1994374387[15]J.VollingaS.WeinzierlNumerical evaluation of multiple polylogarithmsComput. Phys. Commun.1672005177arXiv:hep-ph/0410259[16]A.von ManteuffelC.StuderusReduze 2 – distributed Feynman integral reductionarXiv:1201.4330[17]J.M.HennLectures on differential equations for Feynman integralsJ. Phys. A482015153001arXiv:1412.2296[18]R.N.LeeReducing differential equations for multiloop master integralsJ. High Energy Phys.042015108arXiv:1411.0911[19]T.GehrmannA.von ManteuffelL.TancrediE.WeihsThe two-loop master integrals for qq‾→VVJ. High Energy Phys.062014032arXiv:1404.4853[20]M.ArgeriS.Di VitaP.MastroliaE.MirabellaJ.SchlenkU.SchubertL.TancrediMagnus and Dyson series for master integralsJ. High Energy Phys.032014082arXiv:1401.2979[21]T.GehrmannE.RemiddiTwo loop master integrals for γ⁎→ 3 jets: the Planar topologiesNucl. Phys. B6012001248286arXiv:hep-ph/0008287[22]J.AblingerJ.BlumleinC.SchneiderHarmonic sums and polylogarithms generated by cyclotomic polynomialsJ. Math. Phys.522011102301arXiv:1105.6063[23]A.von ManteuffelR.M.SchabingerH.X.ZhuThe complete two-loop integrated jet thrust distribution in soft-collinear effective theoryJ. High Energy Phys.032014139arXiv:1309.3560[24]V.A.SmirnovApplied asymptotic expansions in momenta and massesSpringer Tracts Mod. Phys.17720021262[25]S.BorowkaG.HeinrichS.P.JonesM.KernerJ.SchlenkT.ZirkeSecDec-3.0: numerical evaluation of multi-scale integrals beyond one loopComput. Phys. Commun.1962015470491arXiv:1502.0659