Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZWW and γ^*WW
Stefano Di Vita, Pierpaolo Mastrolia, Amedeo Primo, Ulrich Schubert
PPrepared for submission to JHEP
DESY 17-032
Two-loop master integrals for the leading QCDcorrections to the Higgs coupling to a W pair and tothe triple gauge couplings Z W W and γ ∗ W W
Stefano Di Vita, a Pierpaolo Mastrolia, b,c
Amedeo Primo, b,c
Ulrich Schubert d a DESY, Notkestraße 85, D-22607 Hamburg, Germany b Dipartimento di Fisica ed Astronomia, Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy c INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy d High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We compute the two-loop master integrals required for the leading QCDcorrections to the interaction vertex of a massive neutral boson X , e.g. H, Z or γ ∗ , with apair of W bosons, mediated by a SU (2) L quark doublet composed of one massive and onemassless flavor. All the external legs are allowed to have arbitrary invariant masses. TheMagnus exponential is employed to identify a set of master integrals that, around d = 4space-time dimensions, obey a canonical system of differential equations. The canonicalmaster integrals are given as a Taylor series in (cid:15) = (4 − d ) /
2, up to order four, withcoefficients written as combination of Goncharov polylogarithms, respectively up to weightfour. In the context of the Standard Model, our results are relevant for the mixed EW-QCDcorrections to the Higgs decay to a W pair, as well as to the production channels obtainedby crossing, and to the triple gauge boson vertices ZW W and γ ∗ W W . a r X i v : . [ h e p - ph ] F e b ontents X W + W − z, ¯ z variables 155.2 Internal massive lines: the u, v variables 175.3 Analytic continuation of the master integrals 18 X W + W − B.1 Topologies (a)-(b) 23B.2 Topologies (c)-(d) 23
C dlog-forms 25
C.1 Topologies (a)-(b) 25C.2 Topologies (c)-(d) 30
The calculation of Feynman integrals with massive particles both as external legs and asinternal lines is not an easy task. Indeed, the combination of external kinematic invariantsand internal masses may give rise to physical and non-physical singularities which requirethe use of special functions, with non-trivial arguments, embedding them. Identifying therise of such functions within the parametric representation of Feynman integrals is verychallenging. Therefore, rather than by direct integration, multivariate Feynman integralsmay be more simply determined by solving differential equations (DEs) [1–3].In general, Feynman integrals in dimensional regularization obey relations that can beused, on the one side, to identify a basis of independent integrals, dubbed master integrals – 1 –MIs), and, on the other side, to write special equations satisfied by the MIs themselves.MIs are found to obey systems of linear, first-order partial DEs in the kinematic variables.Solving these equations, provided that their values or behavior at special points is known,becomes a method to completely determine MIs, hence to compute Feynman integralsalternatively to their direct integration (as reviewed in [4, 5]).The entries of the matrix associated to the system of DEs depend, in general, on thekinematic invariants and on the space-time dimension d . Although it is a mathematicallyinteresting problem, finding the expression of the MIs for arbitrary values of d is not alwayspossible and, according to the physical context, it may be sufficient to know the MIs arounda critical dimension d c , with d = d c + (cid:15) and (cid:15) →
0. In a perturbative approach, the solutionof the system around d = d c may admit a representation in terms of iterated integrals (asreviewed in [6, 7]), where the matrix associated to the system constitutes the integrationkernel. Therefore, the structure of such a matrix has a direct impact on the form of thesolutions, namely on the functions required to classify them: simplifying the matrix meanssimplifying the solutions.The idea of finding MIs that obey canonical systems of DEs, i.e. systems with anassociated matrix where the dependence on the space-time dimensions is decoupled from thekinematics [8, 9], has led to a substantial improvement of the system-solving strategy [10–19], and to the availability of many novel results. In the case of Feynman integrals thatdepend on several scales, we have shown that the Magnus exponential [11] is an efficienttool to derive MIs obeying canonical systems starting from a basis of MIs that obey systemsof DEs whose matrix has a linear dependence on the space-time dimension [20–22].In the Standard Model, the coupling between two W bosons and one neutral boson X = H, Z, γ ∗ is present in the tree-level Lagrangian . At one-loop, the X W + W − in-teraction receives electro-weak (EW) corrections, either via bosonic- or via fermionic-loop.Strong (QCD) corrections must proceed through a closed quark-loop so that they can firstoccur at the two-loop level. In this article, we present the calculation of the two-loopthree-point integrals required for the determination of the leading QCD corrections to theinteraction vertex between a neutral boson X with arbitrary mass and a pair of W bosonsof arbitrary squared four-momenta ( X W + W − ), mediated by a fermion loop of a SU (2) L quark doublet, with one massive and one massless flavors. In what follows, we refer to themassive flavor as to the top ( m t = m ), and to the massless one as to the bottom ( m b = 0).Representative Feynman graphs for the considered integrals are shown in figure 1. TheMIs for the case in which only massless quarks propagate in the loops has been previouslystudied in [23–25].Our results represent the full set of MIs needed to compute the O ( αα s ) corrections tothe Higgs decay into a pair of W bosons, and to the triple gauge boson processes Z ∗ W W and γ ∗ W W , with leptonic final states, at e + e − colliders. As for the latter process with Several motivated extensions of the Standard Model feature an extended Higgs sector with Yukawacouplings to the SU (2) fermion doublets. In particular, one or more neutral pseudoscalar bosons might bepart of the spectrum, together with other neutral scalar bosons. While we do not refer explicitly to thispossibility, our results would also be applicable to the case X = S , A , where we schematically indicatewith S the scalars and with A the pseudoscalars. – 2 –emi-leptonic or hadronic final states, our MIs would only be a subset of the needed MIs.They are also a subset of the MIs needed for the computation of the two-loop mixed EW-QCD corrections to the Higgs production cross section either in the W W -fusion channelor in association with a W boson, and to W W production in higher multiplicity processes.The same MIs would also be needed for the computation of a class of diagrams enteringthe NNLO EW corrections. Except for the first-generation quarks (that are approximatelydegenerate), the fermionic one-loop diagrams always involve an SU (2) L doublet with a(nearly or exactly) massless flavor. Is is then clear that the corrections due to photonexchange between the fermionic lines share the same topologies as the ones of the leadingQCD corrections.We distinguish two sets of integrals, according to the flavor that couples to the X boson, i.e. either the massive or the massless one, for which integration-by-parts reductionreturns 24 and 23 MIs, respectively. The calculation of the MIs proceeds according to thefollowing strategy. We identify a set of MIs that obey systems of DEs whose matrix has a linear dependence on d = 4 − (cid:15) , and, by means of the Magnus exponential, we derive acanonical set of master integrals. The matrices associated to the canonical systems admit alogarithmic-differential form ( d log) with rational arguments, therefore, the canonical MIscan be cast in Taylor series around d = 4 with coefficients written as combinations ofGoncharov polylogarithms (GPLs) [26–29]. Boundary conditions are imposed by requiringthe regularity of the solutions at special kinematics points, and by using simpler integrals asindependent input. The analytic expressions of the MIs have been numerically evaluatedwith the help of GiNaC [30] and successfully tested against the values provided by thepublic computer code
SecDec [31]. The package
Reduze [32] has been used throughout thecalculations.The paper is organized as follows. In section 2 we fix our notation and conventions.In section 3 we discuss the general features of the systems of DEs satisfied by the masterintegrals and its general solution in terms of iterated integrals. In section 4 we describe thecomputation of the two-loop MIs for X W + W − in Euclidean kinematics and in section 5we discuss the analytic continuation of our result. Conclusions are given in section 6. Inappendix A we recall the main properties of iterated integrals. In appendix B we list thecoefficients of the linear combinations of MIs that satisfy a canonical system of DEs andfinally, in appendix C, we give the expressions of the d log-form of the matrices associatedto such systems. The analytic expressions of the canonical MIs up to O ( (cid:15) ) are attachedto the arXiv version of the manuscript as ancillary files. In this paper we will consider the two-loop three-point functions of a X boson withmomentum q , and two W bosons with momenta p and p , X ( q ) → W + ( p ) + W − ( p ) (2.1)where s = q = ( p + p ) and p (cid:54) = p (cid:54) = 0 . (2.2)– 3 – W − W + t tgt t b X W − W + tt tg bb X W − W + tt tg t b X W − W + tt b bg b Figure 1 : Representative two-loop Feynman diagrams contributing to the X W + W − interaction, where X = H, Z, γ ∗ . Similar diagrams where t and b quarks are exchangedare also taken into account. The diagrams have been generated using FeynArts [33].The calculation involves the evaluation of Feynman integrals in d = 4 − (cid:15) dimensions ofthe type (cid:90) (cid:103) d d k (cid:103) d d k D n a . . . D n p a p . (2.3)In our conventions, the integration measure is defined as (cid:103) d d k i = d d k i (2 π ) d (cid:18) i S (cid:15) π (cid:19) − (cid:18) m µ (cid:19) (cid:15) , (2.4)where m is the mass of the top quark circulating in the loops, µ the ’t Hooft scale ofdimensional regularization and S (cid:15) = (4 π ) (cid:15) Γ(1 + (cid:15) ) . (2.5) In this section we briefly discuss the general features of the systems of DEs obeyed by theMIs and the properties of the corresponding solutions. The details of the calculations ofthe MIs for X W + W − are described in section 4.The two-loop Feynman diagrams contributing to X W + W − can be reduced to fourparent topologies, which are depicted in figure 2. The integrals belonging to these topologiesdepend on the three external invariants p , p , s, (3.1)as well as on the top mass m . These four dimensionful parameters can be combinedinto three independent dimensionless variables, (cid:126)x = ( u, z, ¯ z ) for topologies (a)-(b) and (cid:126)x = ( v, z, ¯ z ) for topologies (c)-(d), whose explicit definition will be later specified. The MIssatisfy a linear system of partial DEs in these variables, which, if we organize the MIs intoa vector F , can be combined into a matrix equation for the total differential of F , d F = K F . (3.2)In general, the matrix-valued differential form K = K a dx a ( a = 1 , ,
3) depends both onthe kinematic variables and on the spacetime dimension d = 4 − (cid:15) . Since the left-hand– 4 – a) (b) (c) (d) Figure 2 : Two-loop topologies for X W + W − interactions. Thin lines represent masslesspropagators and thick lines stand for massive ones. The dashed external line correspondsto the off-shell leg with squared momentum equal to s whereas the red and blue linesrepresent the two external vector bosons with off-shell momenta p and p respectively.side of the system in eq. (3.2) is a total differential by construction, it is easy to show that K satisfies the (matrix) integrability condition ∂ a K b − ∂ b K a − [ K a , K b ] = 0 , a, b = 1 , , . (3.3)Starting from a basis of MIs associated to a matrix K with a linear dependence on (cid:15) ,one can use the Magnus exponential [11, 20] and apply the procedure outlined in section 2of [21] in order to perform a basis transformation and obtain a canonical set of MIs [8] I enjoying two remarkable features: first, the canonical basis I obeys a system of DEs wherethe dependence on (cid:15) is factorized from the kinematics and, in addition, the kinematicmatrices can be organized into a logarithmic differential form, referred to as canonical d log-form. Thus, the canonical basis I satisfies a system of equations of the form d I = (cid:15) d A I , (3.4)where d A = n (cid:88) i =1 M i d log η i , (3.5)is the d log matrix written in terms of the differentials d log η i , whose arguments η i = η i ( (cid:126)x )solely enclose the kinematic dependence and form the so called alphabet of the problem.The coefficient matrices M i have rational-number entries. Due to the (cid:15) -factorization, theintegrability condition for eq. (3.4) splits into ∂ a ∂ b A − ∂ a ∂ b A = 0 , [ ∂ a A , ∂ b A ] = 0 , a, b = 1 , , . (3.6) At any point (cid:126)x , the general solution of the canonical system of DEs (3.4) can be expressedin terms of
Chen’s iterated integrals [34] as the path-ordered exponential I ( (cid:15), (cid:126)x ) = P exp (cid:26) (cid:15) (cid:90) γ d A (cid:27) I ( (cid:15), (cid:126)x ) , (3.7)– 5 –here I ( (cid:15), (cid:126)x ) is a constant vector depending on (cid:15) only and γ is a piecewise-smooth pathconnecting (cid:126)x to (cid:126)x , γ : [0 , (cid:51) t (cid:55)→ γ ( t ) = ( γ ( t ) , γ ( t ) , γ ( t )) γ (0) = (cid:126)x γ (1) = (cid:126)x. (3.8)In the limit (cid:126)x → (cid:126)x the integration path γ shrinks to a point and I ( (cid:15), (cid:126)x ) → I ( (cid:15), (cid:126)x ). Inthis perspective, the integration constants I ( (cid:15), (cid:126)x ) which, together with d A , completelyspecify the solution, can be thought as the initial values of the MIs, which then evolve toarbitrary points (cid:126)x under the action of the path-ordered exponential. It can be proven that,whenever γ does not cross any singularity or branch cuts of d A (but at its endpoints), thepath-ordered exponential is independent of the explicit choice of the path. By choosing aproper normalization, we can assume all canonical MIs to be finite in the (cid:15) → I ( (cid:126)x ) admits a Taylor expansion in (cid:15) , I ( (cid:15), (cid:126)x ) = I (0) ( (cid:126)x ) + (cid:15) I (1) ( (cid:126)x ) + (cid:15) I (2) ( (cid:126)x ) + . . . (3.9)and, according to eq. (3.7), the n -th order coefficient is given by I ( n ) ( (cid:126)x ) = n (cid:88) i =0 ∆ ( n − i ) ( (cid:126)x, (cid:126)x ) I ( i ) ( (cid:126)x ) , (3.10)where we introduced the weight- k operator∆ ( k ) ( (cid:126)x, (cid:126)x ) = (cid:90) γ d A . . . d A (cid:124) (cid:123)(cid:122) (cid:125) k times , ∆ (0) ( (cid:126)x, (cid:126)x ) = 1 , (3.11)which iterates k ordered integrations of the matrix-valued 1-form d A along the path γ .According to eq. (3.5), each entry of ∆ ( k ) is a linear combination of Chen’s iterated integralsof the type C [ γ ] i k ,...,i = (cid:90) γ d log η i k . . . d log η i = (cid:90) ≤ t ≤ ... ≤ t k ≤ g γi k ( t k ) . . . g γi ( t ) dt . . . dt k , (3.12)being g γi ( t ) = ddt log η i ( γ ( t )) . (3.13)It should be remarked that, as explicitly indicated in (3.12), individual Chen’s iteratedintegrals are, in general, functionals of the path and that only the full combinations ap-pearing as entries of ∆ ( k ) are independent of the particular choice of γ . The most relevantproperties of Chen’s iterated integrals are summarized in appendix A. X W + W − In this section we present the solution of the system of DEs for the MIs associated to thefour topologies (a)-(d). Since topologies (a)-(b) and (c)-(d) belong to two distinct integralfamilies, we discuss their evaluation separately.– 6 – .1 Topologies (a)-(b)
The two topologies (a) and (b) belong to the 7-denominator integral family identified bythe set of denominators D = k , D = k , D = ( k − p ) − m , D = ( k − p ) − m ,D = ( k − p − p ) , D = ( k − p − p ) , D = ( k − k ) , (4.1)where k and k are the two loop momenta. The integrals belonging to this family can bereduced to a set of 29 MIs which are conveniently expressed in terms of the dimensionlessvariables u , z and ¯ z defined by − sm = u , p s = z ¯ z , p s = (1 − z )(1 − ¯ z ) . (4.2)The same parametrization for p and p was used also for the massless triangles consideredin [25]. The following set of MIs obeys a system of DEs which is linear in (cid:15) :F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , (4.3)where the T i are depicted in figure 3. We observe that some of integrals T i are triviallyrelated by p ↔ p symmetry, T ↔ T , T ↔ T , T ↔ T , T ↔ T , T ↔ T , T ↔ T , (4.4)so that the actual number of independent integrals is reduced to 23. However, in orderto determine the solution of the DEs with the method described in section 3, i.e. bysimultaneously integrating the whole system of equations, one has to consider the full setof integrals given in eq. (4.3).The Magnus exponential allow us to obtain a set of canonical MIs obeying a systemof equations of the form (3.4) – 7 – T T T T T T T T T T T T T T T T T T T T T T T T T T T T Figure 3 : Two-loop MIs T ,..., for topologies (a)-(b). Graphical conventions are the sameas in figure 2. Dots indicate squared propagators.I = F , I = − p F , I = − s F , I = − p F , I = − p F , I = 2 m F + ( m − p ) F , I = − s F , I = − p F , I = 2 m F + ( m − p ) F , I = −√ λ F , I = p p F , I = p s F , I = p F , I = p s F , I = s F , I = −√ λ F , – 8 – = c , F + c , F , I = −√ λ F , I = −√ λ F , I = c , F + c , F + c , F , I = −√ λ F , I = c , F + c , F , I = s √ λ F , I = p √ λ F , I = −√ λ F , I = −√ λ F , I = ( p − m ) √ λ F , I = ( p − m ) √ λ F , I = c , F + c , F + c , F + c , F + c , F + c , F + c , F , (4.5)where λ is the K¨all´en function related to the external kinematics, λ ≡ λ ( s, p , p ) = ( s − p − p ) − p p . (4.6)Explicit expressions for the coefficients c i, j are given in appendix B.1. The alphabet of thecorresponding d log-form contains the following 10 letters: η = u , η = z ,η = 1 − z , η = ¯ z ,η = 1 − ¯ z , η = z − ¯ z ,η = 1 + u z ¯ z , η = 1 − u z (1 − ¯ z ) ,η = 1 − u ¯ z (1 − z ) , η = 1 + u (1 − z )(1 − ¯ z ) . (4.7)The coefficient matrices M i are collected in the appendix C.1. It can be easily checked thatall letters are real and positive in the region0 < z < , < ¯ z < z , < u < z (1 − ¯ z ) . (4.8)If one fixes m >
0, this corresponds to a patch of the Euclidean region, s , p , p < (cid:113) − p (cid:113) − p > m , − (cid:0) p − m (cid:1) (cid:0) p − m (cid:1) m < s < p + p − (cid:113) − p (cid:113) − p . (4.9)Since the alphabet is rational, the solution can be directly expressed in terms of GPLswith argument depending on the kinematics variables u , z and ¯ z . The prescriptions forthe analytic continuation to the other patches of the Euclidean region ( s, p , p <
0) andto the physical regions are given in section 5.Imposing the regularity of our solutions at the unphysical thresholds, z, ¯ z = 0 (corre-sponding to p = 0) and z, ¯ z = 1 (corresponding to p = 0) entails relations between theboundary constants. These relations allow us to derive all boundary constants from fivesimpler integrals I , , , , , which are obtained in the following way:– 9 – I is a constant to be determined by direct integration and, due to the normalizationof the integration measure (2.4), it is simply set toI ( (cid:15), (cid:126)x ) = 1 . (4.10) • I can be obtained by direct integrationI ( (cid:15), (cid:126)x ) = Γ(1 − (cid:15) ) Γ(1 − (cid:15) ) u − (cid:15) , (4.11) • Besides being regular in the massless kinematic limit z → p → is reduced,through IBPs, to a two-loop vacuum diagram,I ( (cid:15), z = 1) = − (cid:15) (1 − (cid:15) )(1 − (cid:15) ) m . (4.12)Therefore, by using as an input the analytic expression of the two-loop vacuum graph,= − m Γ( − (cid:15) )Γ( − (cid:15) )(1 − (cid:15) )Γ(1 + (cid:15) ) , (4.13)we can fix the boundary constants by matching the z → obtained from the solution of the DE against the (cid:15) -expansion of eq. (4.12),I ( (cid:15), z = 1) = − − π (cid:15) + 2 ζ (cid:15) − π (cid:15) + O ( (cid:15) ) . (4.14) • I and I can be directly integratedI ( (cid:15), (cid:126)x ) = − Γ(1 − (cid:15) ) Γ(1 + 2 (cid:15) )Γ(1 − (cid:15) )Γ(1 + (cid:15) ) u − (cid:15) , (4.15)I ( (cid:15), (cid:126)x ) = Γ(1 − (cid:15) ) Γ(1 − (cid:15) ) u − (cid:15) . (4.16)The results have been numerically checked, in both the Euclidean and the physical regions,with the help of the public computer codes GiNaC and
SecDec 3.0 , and their analyticexpressions are given in electronic form in the ancillary files attached to the arXiv versionof the manuscript.
The topologies (c) and (d) belong to the 7-denominator family defined by the set of de-nominators D = k − m , D = k − m , D = ( k − p ) , D = ( k − p ) ,D = ( k − p − p ) − m , D = ( k − p − p ) − m , D = ( k − k ) , (4.17)– 10 –here k and k are the two loop momenta. The integrals belonging to this integral familycan be reduced to a set of 31 MIs which are conveniently expressed in terms of the variables v , z and ¯ z , defined by − sm = (1 − v ) v , p s = z ¯ z, p s = (1 − z )(1 − ¯ z ) . (4.18)The following set of MIs obeys a system of DEs which is linear in (cid:15) :F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = − (cid:15) (1 − (cid:15) ) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , F = (cid:15) T , (4.19)where the T i are depicted in figure 4. As for the case of topologies (a)-(b), some of integrals T i are related by p ↔ p , T ↔ T , T ↔ T , T ↔ T , T ↔ T , T ↔ T , T ↔ T , T ↔ T , (4.20)so that the total number of independent integrals is 24. However, as discussed alreadyafter eq. (4.4), we work with the complete set of integrals given in eq. (4.19).The Magnus exponential allows us to obtain a set of canonical MIs obeying a systemof equations of the form (3.4),I = F , I = − p F , I = ρ F , I = − p F , I = ( m − p ) F + 2 m F , I = − p F I = ρ F + 12 ( ρ − s ) F , I = − s F , I = − p F , I = 2 m F + ( m − p ) F , I = p F , I = − p ρ F , I = p p F I = ρ F , I = − p ρ F , I = c , F + c , F + c , F + c , F , – 11 – T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T Figure 4 : Two-loop MIs T ,..., for the topologies (c)-(d). Graphical conventions are thesame as in figure 2. Dots indicate squared propagators.I = −√ λ F , I = −√ λ F , I = c , F + c , F + c , F , I = −√ λ F , I = c , F + c , F + c , F + c , F + c , F + c , F , – 12 – = −√ λ F , I = −√ λ F , I = c , F + c , F + c , F , I = c , F + c , F + c , F + c , F , I = c , F + c , F + c , F + c , F , I = −√ λ F , I = − ρ √ λ F , I = ( p − m ) √ λ F , I = c , F + c , F + c , F + c , F + c , F , I = −√ λ F , (4.21)where ρ ≡ √− s √ m − s and λ is defined as in eq. (4.6). The expression of the coefficients c i, j is given in appendix B.2. The alphabet of the corresponding d log-form contains thefollowing 18 letters η = v , η = 1 − v ,η = 1 + v , η = z ,η = 1 − z , η = ¯ z ,η = 1 − ¯ z , η = z − ¯ z ,η = z + v (1 − z ) , η = 1 − z (1 − v ) ,η = ¯ z + v (1 − ¯ z ) , η = 1 − ¯ z (1 − v ) ,η = v + z ¯ z (1 − v ) , η = v + (1 − z − ¯ z + z ¯ z )(1 − v ) ,η = v + z (1 − v ) , η = v + (1 − z )(1 − v ) ,η = v + ¯ z (1 − v ) , η = v + (1 − ¯ z )(1 − v ) . (4.22)The coefficient matrices M i are collected in the appendix C.2. In this case, all the lettersare real and positive in the region0 < v < , < z < , < ¯ z < z . (4.23)If one fixes m >
0, this corresponds to a patch of the Euclidean region, s , p , p < s < − (cid:18)(cid:113) − p + (cid:113) − p (cid:19) < . (4.24)The solution of the system of DEs is straightforwardly obtained in terms of Chen’siterated integrals. Moreover, since the alphabet is rational, the solution can be convertedin terms of GPLs of argument 1, with kinematic-dependent weights, as we discuss inappendix A. The prescriptions for the analytic continuation to the other patches of theEuclidean region and to the physical regions are given in section 5.The boundary constants can be fixed by demanding the regularity of the basis (4.19)for vanishing external momenta, s = p = p = 0. In particular, if we choose as a base-pointfor the integration (cid:126)x = (1 , , , (4.25)– 13 –hen the prefactors appearing in the definitions (4.21) of the canonical MIs I vanish, withthe only exceptions of I , , , , , . Therefore the boundaries of the former MIs are deter-mined by demanding their vanishing at (cid:126)x → (cid:126)x ,I i ( (cid:15), (cid:126)x ) = 0 , i (cid:54) = 1 , , , , , . (4.26)The boundary constants of integrals I , , can be taken from the previous topologies inequations (4.10,4.14), whereas for integrals I , , the boundary constants are fixed asfollows: • The boundary constants for I and I can be determined by imposing regularity atthe pseudothresholds v → s = p = p = 0) and, respectively, z →
1, ¯ z → p = 0),I , ( (cid:15), (cid:126)x ) = 16 π (cid:15) − ζ (cid:15) + 120 π (cid:15) + O ( (cid:15) ) . (4.27) • Finally, the boundary constants for I can be fixed by observing that, from (4.21),we can derive F ( (cid:15), (cid:126)x ) = lim (cid:126)x → (cid:126)x vm (1 − v ) I ( (cid:15), (cid:126)x ) . (4.28)Therefore, in order for F ( (cid:15), (cid:126)x ) to be regular we must demandI ( (cid:15), (cid:126)x ) = 0 , (4.29)All results have been numerically checked, in both the Euclidean and the physical regions,with the help of the computer codes GiNaC and
SecDec 3.0 , and the analytic expressionsof the MIs are given in electronic form in the ancillary files attached to the arXiv versionof the manuscript.
In this section we discuss in detail the variables used to parametrize the dependence ofthe MIs on the kinematic invariants. In particular, we elaborate on the prescriptions toanalytically continue our results to arbitrary values of s, p , p . Both topologies (a)-(b) and(c)-(d) feature two independent, kinematic structures:1. the off-shell external legs are responsible for the presence in the DEs of the squareroot of the K¨all´en function, (cid:112) λ ( s, p , p );2. the presence of massive internal lines can generate square roots in the DEs, as in thecase of topologies (c)-(d) where one has also √− s √ m − s .In the following we separately discuss the variable changes that rationalize the two typesof square roots. – 14 – .1 Off-shell external legs: the z, ¯ z variables To deal with the square root of the K¨all´en function, we begin by choosing one of theexternal legs as reference, s , and trading the other squared momenta for dimensionlessratios τ , = p , s . (5.1)In the ( s, τ , τ ) variables, the square root of the K¨all´en function is proportional to (cid:112) λ (1 , τ , τ ) = (cid:112) (1 − τ − τ ) − τ τ (5.2)and is rationalized by the following change of variables [25] τ = z ¯ z , (5.3) τ = (1 − z )(1 − ¯ z ) , (5.4)(see eqs. (4.2) and (4.18)), that leads to λ (1 , τ ( z, ¯ z ) , τ ( z, ¯ z )) = ( z − ¯ z ) . (5.5)Without loss of generality, we choose the following root of eq. (5.4) z = 12 (cid:16) τ − τ + (cid:112) λ (1 , τ , τ ) (cid:17) , (5.6)¯ z = 12 (cid:16) τ − τ − (cid:112) λ (1 , τ , τ ) (cid:17) . (5.7)Varying the pair ( τ , τ ) in the real plane, we identify the following possibilities for z, ¯ z ¯ z = z ∗ λ (1 , τ , τ ) < , τ , τ > < ¯ z < z < √ τ + √ τ < , < τ , τ < z < z < √ τ > √ τ , τ > z > ¯ z > √ τ > √ τ , τ > z = ¯ z = ±√ τ τ = (cid:0) ± √ τ (cid:1) , τ , τ > z > , ¯ z < τ , τ < < z < , ¯ z < τ < , τ > z > , < ¯ z < τ > , τ < τ , τ )-plane is shown if fig 5.The variables z, ¯ z are complex conjugates in region I, where λ (1 , τ , τ ) <
0, andreal in all the other regions. In regions I-V one has τ , >
0, which requires that either s, p , p < s, p , p >
0. The former case defines the Euclidean region. The latter case,for λ (1 , τ , τ ) >
0, describes 1 → → λ (1 , τ , τ ) = 0, so that z = ¯ z . Since our expressions are obtained ingeneral for z (cid:54) = ¯ z , the limit ¯ z → z has to be taken carefully. Regions VI-VIII have at leastone of the τ i <
0, which requires either two external legs to be spacelike and the remaining– 15 – �� � λ < ��� ��������� λ = ��� � τ � τ � Figure 5 : Regions of the ( τ , τ )-plane classified in eq (5.8). Region V, which is identifiedby the condition λ (1 , τ , τ ) = 0, corresponds to the blue curve.one to be timelike, or vice versa. The former configuration, in the 2 → z, ¯ z are not in the half of the unit square whereall the letters are real, therefore analytic continuation is required. A consistent physicalprescription is inherited in regions VI-VIII from the Feynman prescription on the kinematicinvariants, and it is naturally extended to the other regions, as we argue below. For themoment we hold s <
0, and we will discuss later the case s > s < p , p >
0, then τ i → −| τ i | − iε , (5.9)so that the vanishing imaginary parts outside the square root in eq (5.7) cancel againsteach other, and only the one stemming from the square root is left: z → z + iε , ¯ z → ¯ z − iε . (5.10)In region VII, p > s, p <
0, then τ → −| τ | − iε , τ → | τ | , (5.11)so that z → z + i ε (cid:32) | τ | + τ (cid:112) λ (1 , −| τ | , τ ) − (cid:33) (cid:39) z + iε , – 16 – z → ¯ z − i ε (cid:32) | τ | + τ (cid:112) λ (1 , −| τ | , τ ) + 1 (cid:33) (cid:39) ¯ z − iε , (5.12)where the approximate equalities are allowed because the factor in the bracket is alwayspositive, and a redefinition of ε is understood.In region VIII, p > s, p <
0, then τ → | τ | , τ → −| τ | − iε , (5.13)so that z → z + i ε (cid:32) τ + | τ | (cid:112) λ (1 , τ , −| τ | ) + 1 (cid:33) (cid:39) z + iε , ¯ z → ¯ z − i ε (cid:32) τ + | τ | (cid:112) λ (1 , τ , −| τ | ) − (cid:33) (cid:39) ¯ z − iε , (5.14)where again the approximate equalities are allowed because the factor in the bracket isalways positive, and a redefinition of ε is understood.We have so far only discussed the case in which s <
0. It is easy to see that, if instead s >
0, the prescription on z, ¯ z is the opposite.In regions I-V there is no physical prescription for the analytic continuation of z, ¯ z .Indeed, if s, p , p >
0, then the vanishing imaginary parts of the Feynman prescriptioncancel out in the ratios τ and τ : τ i → p i (1 + iε ) s (1 + iε ) = τ i . (5.15)This cancellation affects also region I, where (cid:112) λ (1 , τ , τ ) < z ∗ = ¯ z . Indeed, whilethis condition fixes the relative sign of their imaginary parts, the sign of Im z dependson the choice of the branch of the square root in eq. (5.7), which is not fixed. This laststatement holds true also in the Euclidean region.This ambiguity is resolved by the definite iε prescription in regions VI-VIII discussedabove. In order to have a smooth analytic continuation in the Euclidean, in region I, onechooses the branch of the square root that gives Im (cid:112) λ (1 , τ , τ ) >
0, and in regions III-IV,one assigns vanishing imaginary parts for z, ¯ z according to the previous discussion. Theopposite prescription should be used if the three external legs are timelike.Summarizing, according to the sign of s , we choose the following analytic continuationprescriptions for z, ¯ z in the whole real ( p , p ) plane z → z + iε , ¯ z → ¯ z − iε s < , (5.16) z → z − iε , ¯ z → ¯ z + iε s > . (5.17) u, v variables For topologies (a)-(b), the change of variables eq. (5.4) is actually enough to rationalize theDEs completely. In eq. (4.2) we simply rescale s by the internal mass ( m > − s/m = u ,– 17 –o deal with a dimensionless variable. If s < u >
0. If s >
0, the Feynman prescription s → s + iε fixes the analytic continuation for uu → − u (cid:48) − iε , with u (cid:48) > . (5.18)In the case of topologies (c)-(d), the DEs still contain the square roots related to the s -channel threshold at s = 4 m . They are rationalized by the usual variable change (seeeq. (4.18)), − sm = (1 − v ) v , (5.19)of which we choose the following root v = √ m − s − √− s √ m − s + √− s . (5.20)For completeness, we discuss how v varies with s . Holding m >
0, and keeping in mindthe Feynman prescription for s >
0, one finds the following cases • For s < v is on the unit interval, 0 ≤ v ≤ • For 0 ≤ s ≤ m , v is a pure phase, v = e iφ , with 0 < φ < π ; • For s > m , v is on the negative unit interval, and one must replace v → − v (cid:48) + iε , ≤ v (cid:48) ≤ . (5.21) As discussed in section 4, for all the topologies we start in the patch of the Euclideanregion where the alphabet is real and positive (see eqs. (4.8) and (4.23)), and we solve theDEs there. As far as the variables z, ¯ z are concerned, the conditions of positivity of thealphabet are the same for all our topologies,0 < z < , < ¯ z < z , (5.22)i.e. we start from region II (see eq. (5.8)). Regarding the condition on the variables as-sociated to s , i.e. u and v (respectively for topologies (a)-(b) and topologies (c)-(d)), werequire 0 < u < z (1 − ¯ z ) , < v < , (5.23)It is clear from eq. (5.8) that, if these conditions are satisfied, one does not have accesseven to the full Euclidean region. Results in the remaining patches of the latter, as wellas in the physical regions, are obtained by analytic continuation using the prescriptionsdescribed in sections 5.1 and 5.1.In the present work we performed the analytic continuation numerically, i.e. we as-signed to u, v, z, ¯ z the vanishing imaginary parts discussed above choosing sufficiently smallnumerical values. For convenience, we summarize the analytic continuation prescriptionfor the physically interesting cases. – 18 – X → W W : In this region a particle of mass s > p > p >
0, so that √ s ≥ (cid:113) p + (cid:113) p . (5.24)Regarding z, ¯ z , this corresponds to region II (see eq. (5.8)), therefore no analyticcontinuation is needed. Furthermore, for topologies (a)-(b) one must replace u → − u (cid:48) − iε irrespectively of the value of s , according to eq. (5.18). Instead, for topologies (c)-(d),if 0 < s < m then v is on the unit circle in the complex plane, while if s > m ,one has to replace v → − v + iε , according to eq. (5.21). • W → W X : This is again a 1 → (cid:113) p ≥ √ s + (cid:113) p > , (5.25)or (cid:113) p ≥ √ s + (cid:113) p > . (5.26)The former case corresponds to region IV, the latter to region III (see eq. (5.8)).Therefore, in addition to the analytic continuation in u, v discussed already for the X -decay, one must further use the replacement (5.17) z → z − iε , ¯ z → ¯ z + iε . (5.27) • W W → X : Here p , p < , s > , (5.28)corresponding to region VI, so that z, ¯ z inherit the analytic continuation prescriptioneq. (5.17) from s → s + iε z → z − iε , ¯ z → ¯ z + iε . Concerning u, v , the discussion is the same as for X -decay. • X W → W : Here p , s < , p > , (5.29)or p , s < , p > , (5.30)corresponding to region VII and VIII respectively, so that z, ¯ z inherit from p i → p i + iε the prescription (5.16) z → z + iε , ¯ z → ¯ z − iε . Since s <
0, no continuation is due on u, v .– 19 –
Conclusions
In this paper we have computed the two-loop master integrals required for the leadingQCD corrections to the interaction vertex between a massive neutral boson X , such as H, Z or γ ∗ , and pair of W bosons, mediated by a SU (2) L quark doublet composed ofone massive and one massless flavor. We considered external legs with arbitrary invariantmasses. The master integrals were computed by means of the differential equation method.After identifying a set of master integrals obeying a system of equations which dependslinearly on the space-time dimension d , we used the Magnus exponential in order to tofind a novel set of integrals that, around d = 4 dimensions, obey a canonical system ofdifferential equations. The canonical master integrals were finally obtained as a Taylorseries in (cid:15) = (4 − d ) /
2, up to order four, with coefficients written as combination ofGoncharov polylogarithms, respectively up to weight four.In the context of the Standard Model, our results are relevant for the mixed EW-QCDcorrections to the Higgs decay to a W pair, as well as the production channels obtained bycrossing, and to the triple gauge boson vertices ZW W and γ ∗ W W . Acknowledgments
We acknowledge clarifying discussions with R. Bonciani, G. Degrassi, A. Ferroglia, K. Mel-nikov, P. Paradisi, G. Passarino, F. Petriello, F. Tramontano, A. Vicini and A. Wulzer.A. P. wishes to thank the Institute for Theoretical Particle Physics of the Karlsruhe In-stitute of Technology for the kind hospitality during the completion of this work. U. S. issupported by the DOE contract DE-AC02-06CH11357.
A Properties of Chen’s iterated integrals
In this appendix we recall the main properties of Chen’s iterated integrals [34]. We closelyfollow the notation of [22]. Chen’s iterated integrals are defined by C [ γ ] i k ,...,i = (cid:90) γ d log η i k . . . d log η i = (cid:90) ≤ t ≤ ... ≤ t k ≤ g γi k ( t k ) . . . g γi ( t ) dt . . . dt k , (A.1)where γ is a piecewise-smooth path connecting (cid:126)x to (cid:126)x , γ : [0 , (cid:51) t (cid:55)→ γ ( t ) = ( γ ( t ) , γ ( t ) , γ ( t )) γ (0) = (cid:126)x γ (1) = (cid:126)x . (A.2)and g γi ( t ) = ddt log η i ( γ ( t )) . (A.3) • Invariance under path reparametrization.
The integral C [ γ ] i k ,...,i does not depend onthe way one chooses to parametrize the path γ .– 20 – Reverse path formula.
If the path γ − is the path γ traversed in the opposite direction,then C [ γ − ] i k ,...,i = ( − k C [ γ ] i k ,...,i . (A.4) • Recursive structure.
From (A.1) and (A.3) it follows that the line integral of one d logis defined, as usual, by (cid:90) γ d log η = (cid:90) ≤ t ≤ d log η ( γ ( t )) dt dt = log η ( (cid:126)x ) − log η ( (cid:126)x ) , (A.5)and only depends on the endpoints (cid:126)x , (cid:126)x .It is convenient to introduce the path integral “up to some point along γ ”: givensome path γ and a parameter s ∈ [0 , γ s : [0 , (cid:51) t (cid:55)→ (cid:126)x = ( γ ( s t ) , γ ( s t ) , γ ( s t )) . (A.6)If s = 1, then trivially γ s = γ whereas if s = 0 the image of the interval [0 ,
1] is just { (cid:126)x } . If s ∈ (0 , γ s ([0 , γ (0) = (cid:126)x and overlaps with thecurve γ ([0 , γ ( s ), where it ends. It can be easily shown that thepath integral along γ s can be written as C [ γ s ] i k ,...,i = (cid:90) ≤ t ≤ ... ≤ t k ≤ s g γi k ( t k ) . . . g γi ( t ) dt . . . dt k , (A.7)which differs from eq. (3.12) by the fact that the outer integration (i.e. the one in dt k )is performed over [0 , s ] instead of [0 , γ s , we can rewrite (3.12)in a recursive manner: C [ γ ] i k ,...,i = (cid:90) g γi k ( s ) C [ γ s ] i k − ,...,i ds . (A.8)In addition, eq. (A.7) can be used in order to derive the identity dds C [ γ s ] i k ,...,i = g γi k ( s ) C [ γ s ] i k − ,...,i . (A.9) • Shuffle algebra.
Chen’s iterated integrals fulfill shuffle algebra relations: if (cid:126)m =( m M , . . . , m ) and (cid:126)n = ( n N , . . . , n ), with M and N natural numbers, one has C [ γ ] (cid:126)m C [ γ ] (cid:126)n = (cid:88) shuffles σ C [ γ ] σ ( m M ) ,...,σ ( m ) ,σ ( n N ) ,...,σ ( n ) , (A.10)where the sum runs over all the permutations σ that preserve the relative order of (cid:126)m and (cid:126)n . – 21 – Path composition formula. If α, β : [0 , → M are two paths such that α (0) = (cid:126)x , α (1) = β (0), and β (1) = (cid:126)x , then the composed path γ = αβ is obtained by firsttraversing α and then β . One can prove that the integral over such a composed pathsatisfies C [ αβ ] i k ,...,i = k (cid:88) p =0 C [ β ] i k ,...,i p +1 C [ α ] i p ,...,i . (A.11) • Integration-by-parts formula.
The computation of eq. (A.1) requires, in principle,the evaluation of k nested integrals. Nevertheless, we observe that the innermostintegration is always reduced to (A.5), so that one has k − k = 2, we have C [ γ ] m,n = (cid:90) g m ( t ) C [ γ t ] n dt = (cid:90) g m ( t )(log η n ( (cid:126)x ( t )) − log η n ( (cid:126)x )) dt (A.12)and one is left with a single integral to be evaluated, either analytically or numerically.Moreover, one can show that the integration involving the outermost weight g k canbe performed by parts, returning C [ γ ] i k ,...,i = log η i k ( (cid:126)x ) C [ γ ] i k − ,...,i − (cid:90) log η i k ( (cid:126)x ( t )) g i k − ( t ) C [ γ t ] i k − ,...,i dt . (A.13)The combined use of eqs. (A.12) and (A.13) allows, for instance, a remarkable sim-plification in the numerical evaluation of weight k ≥ k − • Conversion to
GPLs formula . If all letters appearing in a Chen’s iterated integral arerational functions with algebraic roots, then the iterated integral can be converted interms of GPLs.Suppose we connect the endpoints (cid:126)x = ( a , a , a ) and (cid:126)x = ( b , b , b ) through apiecewise path of the type γ ( t ) : ( a + t ( b − a ) , a , a ) γ ( t ) : ( b , a + t ( b − a ) , a ) γ ( t ) : ( b , b , a + t ( b − a )) . (A.14)the conversion of the iterated integral to GPLs can be achieved by factorizing allletters such that the dependence on the varied variable x i , becomes linear. Anyweight- k integral can then be transformed by (cid:90) γ i d log( x i − w k ) . . . d log( x i − w ) = G (cid:18) a i − w k a i − b i , . . . , a i − w a i − b i ; 1 (cid:19) , (A.15)where w i are weights which may depend on the constant variables along the path γ i .– 22 – Canonical master integrals for X W + W − In this appendix we give the explicit expression of the kinematic coefficients appearing inthe definition of canonical master integrals defined by eq. (4.5) and eq. (4.21).
B.1 Topologies (a)-(b)
The coefficients of the canonical MIs listed in eq. (4.5) are given by c , = 32 (cid:16) √ λ + s − p − p + 2 m (cid:17) , (B.1) c , = p p − ( p + p ) m + m + m s , (B.2) c , = 32 (cid:16) √ λ + s − p − p + 2 m (cid:17) , (B.3) c , = p p − ( p + p ) m + m + m s , (B.4) c , = m s ( p − m )( p − m ) , (B.5) c , = − m sp ( p − m )( p − m ) , (B.6) c , = − m sp ( p − m )( p − m ) , (B.7) c , = p (cid:0) p − m (cid:1) + p (cid:16) √ λ + m + s − p (cid:17) + m (cid:16) s − √ λ (cid:17) , (B.8) c , = p (cid:16) p − √ λ + m + s − p (cid:17) + m ( √ λ + s − p ) , (B.9) c , = − s (cid:0) p ( p − m ) + m ( s + m − p ) (cid:1) , (B.10)where the K¨all´en function λ is defined in eq. (4.6). B.2 Topologies (c)-(d)
The coefficients of the canonical MIs listed in eq. (4.21) are given by c , = − p (cid:16) − m (cid:16) p − s + √ λ (cid:17) − p (cid:16) p − p + s − m − √ λ (cid:17)(cid:17) ( p − p )( p + m ) − s ( p − m ) , (B.11) c , = √ λ (cid:0) p (2 m − s ) − m (2 p − s ) (cid:1) ( p − p )( p + m ) − ( p − m ) s − ρ, (B.12) c , = − p (cid:16) m (cid:16) p − s + √ λ (cid:17) + p (cid:16) p − p + s − m − √ λ (cid:17)(cid:17) ( p − p )( p + m ) − ( p − m ) s , (B.13) c , = √ λ (cid:0) m ( p − p ) + s ( p − m )( p − m ) (cid:1) ( p − p )( p + m ) − ( p − m ) s , (B.14) c , = (cid:16) p − p + 2 m − s − √ λ (cid:17) , (B.15) c , = 12 (cid:16) p − p + 2 m − s − √ λ (cid:17) , (B.16) c , = m ( p + m − s ) + p ( s − m ) , (B.17)– 23 – , = m ( p − m ) s + ρ , (B.18) c , = − m ( p + m ) s + ρ , (B.19) c , = 2 m ( p + m ) s + ρ , (B.20) c , = − m ( p − m ) s + ρ , (B.21) c , = − √ λ + 12 s ( s + p − p ) ρ, (B.22) c , = ρ, (B.23) c , = p − p − s + 2 m + √ λ, (B.24) c , = 12 ( p − p − s + 2 m + √ λ ) , (B.25) c , = m p − ( p − m )( m − s ) , (B.26) c , = − p (cid:16) m (cid:16) p − s + √ λ (cid:17) + p (cid:16) p − p + s − m − √ λ (cid:17)(cid:17) ( p − p )( p + m ) − s ( p − m ) , (B.27) c , = p √ λ (cid:0) s ( p − m ) − m ( p − p ) (cid:1) ( p − p )( p + m ) − ( p − m ) s + p ρ, (B.28) c , = p p (cid:16) p (cid:16) p − p + s − m − √ λ (cid:17) + m (cid:16) p − s + √ λ (cid:17)(cid:17) ( p − p )( p + m ) − ( p − m ) s , (B.29) c , = − p √ λ (cid:0) m ( p − p ) + s ( p − m )( p − m ) (cid:1) ( p − p )( p + m ) − ( p − m ) s , (B.30) c , = p ρ (cid:16) p (cid:16) p − p + s − m − √ λ (cid:17) + m (cid:16) p − s + √ λ (cid:17)(cid:17) ( p − p )( p + m ) − ( p − m ) s , (B.31) c , = ρ + ρ √ λ (cid:0) p (cid:0) m − s (cid:1) + m (cid:0) s − p (cid:1)(cid:1) ( p − p )( p + m ) − ( p − m ) s , (B.32) c , = − p ρ (cid:16) p (cid:16) p − p − s + √ λ (cid:17) + m (cid:16) p − s + √ λ (cid:17)(cid:17) ( p − p )( p + m ) − ( p − m ) s , (B.33) c , = 8 √ λ (cid:0) m ( p − p ) + s ( p − m )( p − m ) (cid:1)(cid:0) ( p − p )( p + m ) − ( p − m ) s (cid:1) ( s − m + ρ ) ×× (cid:18) s ( s + ρ ) + 2 m (8 s + ρ ) − m s (5 s + 4 ρ ) − m s (11 s + 4 ρ )+ 2 m s (17 s + 10 ρ ) (cid:19) , (B.34) c , = 2( p − p ) − p ( p − p ) p − m − s + ρ , (B.35) c , = 2 p (cid:18) − p − p ) + 2 p ( p − p ) p − m + s − ρ (cid:19) , (B.36)– 24 – , = s ( p + p + 2 m − s ) − ρ √ λ, (B.37) c , = (cid:0) p ( s − m − p ) + p ( p − m ) − m s (cid:1) + ( p − m ) √ λ , (B.38) c , = − m ( p − p ) − ( p − m )( p − m ) s , (B.39)where ρ is defined after eq. (4.21) and the K¨all´en function λ is given in eq. (4.6). C d log -forms In this appendix we collect the coefficient matrices of the d log-forms for the master integrals(4.5) and (4.21). C.1 Topologies (a)-(b)
For the two-loop integrals discussed in section 4.1 we have d A = M d log( u ) + M d log( z ) + M d log(1 − z )+ M d log(¯ z ) + M d log(1 − ¯ z ) + M d log( z − ¯ z )+ M d log(1 + u z ¯ z ) + M d log (1 − u z (1 − ¯ z ))+ M d log (1 − u ¯ z (1 − z )) + M d log (1 + u (1 − z )(1 − ¯ z )) , (C.1)with M = − − − − − − − − − −
12 12
00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − − − , (C.2)– 25 – = − − − − − − − − − − − − − − − − − − − − − −
12 32 − − − − −
10 2 0 − − − − , (C.3) M = − − − − − − − − − − −
12 12 −
12 12 −
12 12 − − − − − , (C.4)– 26 – = −
12 12 − − − − − − − − − − − −
12 12 −
12 12 − − − − − , (C.5) M = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , (C.6)– 27 – = − − − − − , (C.7) M = − − − − − − − − − − − −
12 32 − − − , (C.8)– 28 – = − − − − − − − − − − − − − − − − − − − − − − − − − − − − , (C.9) M = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , (C.10)– 29 – = − − − − − − − − − − − − − − − − − − − − − − − . (C.11) C.2 Topologies (c)-(d)
For the two-loop integrals discussed in section 4.2 we have d A = M d log( v ) + M d log(1 − v ) + M d log(1 + v ) + M d log( z )+ M d log(1 − z ) + M d log(¯ z ) + M d log(1 − ¯ z )+ M d log( z − ¯ z ) + M d log ( z + v (1 − z )) + M d log (1 − z (1 − v ))+ M d log (¯ z + v (1 − ¯ z )) + M d log (1 − ¯ z (1 − v ))+ M d log (cid:0) v + z ¯ z (1 − v ) (cid:1) + M d log (cid:0) v + (1 − z − ¯ z + z ¯ z )(1 − v ) (cid:1) + M d log (cid:0) v + z (1 − v ) (cid:1) + M d log (cid:0) v + (1 − z )(1 − v ) (cid:1) + M d log (cid:0) v + ¯ z (1 − v ) (cid:1) + M d log (cid:0) v + (1 − ¯ z )(1 − v ) (cid:1) , (C.12)with – 30 – = − − − − − − − − − − − − − − − − − − − − − − − , (C.13) M = − − − − − − − , (C.14)– 31 – = − − − − − − − − − − − − − , (C.15) M = − − − − − − − − − − − − − − − − − − − − − − , (C.16)– 32 – = − − − − − − − − − − − − − − − −
12 12 − −
00 0 0 0 0 0 0 0 0 0 0 − − − − − − − − − − − − − − − − − , (C.17) M = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , (C.18)– 33 – = − − − − − − − − − − − − − − − − − − − − −
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