Analytic integration of real-virtual counterterms in NNLO jet cross sections I

Home Search Collections Journals About Contact us My IOPscience Analytic integration of real-virtual counterterms in NNLO jet cross sections I This content has been downloaded from IOPscience. Please scroll down to see the full text. JHEP7 http://iopscience.iop.org/-7///7 View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 7..7. This content was downloaded on //7 at : Please note that terms and conditions apply. You may also be interested in: Analytic integration of real-virtual counterterms in NNLO jet cross sections II Paolo Bolzoni, Sven-Olaf Moch, Gábor Somogyi et al. A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of real-virtual emission Gábor Somogyi and Zoltán Trócsányi Non-singlet QCD analysis in the NNLO approximation A N Khorramian and S A Tehrani Higher order predicted terms for an QCD observable, using PMS procedure Bakniehl and A Mirjalili Holographic renormalization of probe D-branes in AdS/CFT Andreas Karch, Andy O'Bannon and Kostas Skenderis A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the subtraction terms I Gábor Somogyi and Zoltán Trócsányi Event shapes and jet rates in electron-positron annihilation at NNLO Stefan Weinzierl A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of doubly-real emissions Gábor Somogyi, Zoltán Trócsányi and Vittorio Del Duca

Published by Institute of Physics Publishing for SISSA Received: July, Accepted: September, Published: September, Analytic integration of real-virtual counterterms in NNLO jet cross sections I Ugo Aglietti Dipartimento di Fisica, Università di Roma La Sapienza and INFN, Sezione di Roma, Roma, Italy E-mail: Ugo.Aglietti@roma.infn.it Vittorio Del Duca INFN, Laboratori Nazionali di Frascati, Via E. Fermi, I- Frascati, Italy E-mail: Vittorio.DelDuca@lnf.infn.it Claude Duhr Institut de Physique Théorique and Centre for Particle Physics and Phenomenology CP, Université Catholique de Louvain, Chemin du Cyclotron, B- Louvain-la-Neuve, Belgium E-mail: claude.duhr@uclouvain.be Gábor Somogyi Institute for Theoretical Physics, University of Zürich, Winterthurerstrasse, CH-7 Zürich, Switzerland E-mail: sgabi@physik.unizh.ch JHEP7 Zoltán Trócsányi University of Debrecen and Institute of Nuclear Research of the Hungarian Academy of Sciences, H- Debrecen P.O.Box, Hungary E-mail: Zoltan.Trocsanyi@cern.ch Abstract: We present analytic evaluations of some integrals needed to give explicitly the integrated real-virtual counterterms, based on a recently proposed subtraction scheme for next-to-next-to-leading order NNLO jet cross sections. After an algebraic reduction of the integrals, integration-by-parts identities are used for the reduction to master integrals and for the computation of the master integrals themselves by means of differential equations. The results are written in terms of one- and two-dimensional harmonic polylogarithms, once an extension of the standard basis is made. We expect that the techniques described here will be useful in computing other integrals emerging in calculations in perturbative quantum field theories. Keywords: Jets, QCD. On leave of absence from INFN, Sezione di Torino. c SISSA

Contents. Introduction. The method. Integrals needed for the integrated subtraction terms. Definition of the collinear integrals. Definition of the soft-type integrals. Iterated integrals. One-dimensional harmonic polylogarithms. Two-dimensional harmonic polylogarithms. Special values. Interchange of arguments. The soft integral J 7. The soft-collinear integral K 7. Analytic result for κ = 7. Analytic result for κ = 7. The collinear integrals I. The A-type collinear integrals for k. The A-type collinear integrals for k =. The B-type collinear integrals JHEP7. The soft R -type J I integrals. The soft-collinear R -type K I integrals. Numerical evaluation of integrated subtraction terms. Conclusions A. Spin-averaged splitting kernels B. The J integrals 7 C. The K integrals C. The K integral for κ = C. The K integral for κ =

D. The A-type collinear integrals D. The A integral for k = and arbitrary κ D. The A integral for k = and arbitrary κ D. The A integral for k = and arbitrary κ D. The A integral for k = and κ = D. The A integral for k = and κ = 7 E. The B-type collinear integrals 7 E. The B integral for k = and δ = 7 E. The B integral for k = and δ = E. The B integral for k = and δ = and d = E. The B integral for k = and δ = E. The B integral for k = and δ = E. The B integral for k = and δ = E.7 The B integral for k = and δ = and d = E. The B integral for k = and δ = F. The J I-type integrals F. The J I integral for k = F. The J I integral for k = F. The J I integral for k = F. The J I integral for k = G. The K I-type integrals G. The K I integral for k = G. The K I integral for k = G. The K I integral for k = G. The K I integral for k = JHEP7. Introduction LHC physics demands calculating physical observables beyond leading order LO accuracy, by including the virtual and real corrections that appear at higher orders. However, the evaluation of phase space integrals beyond LO is not straightforward because it involves infrared singularities that have to be consistently treated before any numerical computation may be performed. At next-to-leading order NLO, infrared divergences can be handled using a subtraction scheme exploiting the fact that the structure of the kinematical singularities of QCD matrix elements is universal and independent of the hard process. This allows us to build process-independent counterterms which regularize the one-loop or virtual corrections and real phase space integrals simultaneously [].

In recent years a lot of effort has been devoted to the extension of the subtraction method to the computation of the radiative corrections at the next-to-next-to-leading order NNLO [ ]. In particular, in ref. [, ], a subtraction scheme was defined for computing NNLO corrections to QCD jet cross sections to processes without coloured partons in the initial state and an arbitrary number of massless particles coloured or colourless in the final state. This scheme however is of practical utility only after the universal counterterms for the regularization of the real emissions are integrated over the phase space of the unresolved particles. The integrated counterterms can be computed once and for all and their knowledge is necessary to regularize the infrared divergences appearing in virtual corrections. That is indeed the task of this work: we analytically evaluate some of the integrals needed for giving explicitly the counterterms appearing in the scheme [, ]. The method is an adaptation of a technique developed in the last two decades to compute multi-loop Feynman diagrams [ ]. To our knowledge this is the first time that these techniques are applied to integrals of the type F z = α dx dy x k ɛ x k ɛ y k ɛ y k ɛ xyz k ɛ fx, y, z,. where fx, y, z = x n x n y n y n xyz n,. with n i being non-negative integers and < α. An alternative method for computing the ɛ-expansion of the integrals is iterated sector decomposition. This approach allows one to express the expansion coefficients of all functions we consider as finite, multidimensional integrals. Integrating these representations numerically, we obtain the expansion coefficients for any fixed value of the arguments. Every integral in this paper was computed numerically as well, with this alternative method for selected values of the parameters. We found that in all cases the analytical and numerical results agreed up to the uncertainty associated with the numerical integration. The outline of the paper is the following. In section we outline the steps of our method. In section we define the integrals of the subtraction terms that we will consider in the paper. Our analytic results will be presented in terms of one- and two-dimensional harmonic polylogarithms. We summarize those properties of these functions that are important for our computations in sections and, respectively. In sections and 7 we calculate analytically the integrals needed for integrating the soft-type counterterms as a series expansion in the dimensional regularization parameter ɛ. In section we calculate some of the integrals needed for integrating the collinear counterterms. In sections and we calculate two sets of convoluted integrated counterterms, which can be obtained from a successive integration of the results obtained in section. In section we briefly discuss the numerical calculation of the integrated subtraction terms. Finally in section we present the conclusions of this work and we discuss possible developments concerning more complicated classes of integrals. Appendix A contains the spin-averaged splitting functions at tree level and at one-loop, which are needed for the evaluation of the counterterms. There are further appendices containing the often rather lengthy expressions of the integrated counterterms. JHEP7

. The method Our method of computing the integrals involves the following steps: Algebraic reduction of the integrand by means of partial fractioning. For each class of integrals, we perform a partial fractioning of the integrand in order to obtain a set of independent integrals. For example, for the integrand in eq.. with n = n = n = n = n = one can perform partial fractioning with respect to the integration variable x first, so that x x xyz = x yz x y z yz xyz.. Note the appearance of the new denominator yz, not originally present in the integrand and coming from x partial fractioning. One then performs partial fractioning with respect to y, by considering the denominator xyz as a constant: that is because the latter was already involved in the x partial fractioning and, to avoid an infinite loop, it cannot be subjected to any further transformation. For example: y y yz xyz = z z z yz xyz y xyz y xyz.. After this final partial fractioning over y, the original integrand f, depending on five denominators, is transformed into a combination of terms having at most two denominators, out of which at most one depends on x. JHEP7 Reduction to master integrals by means of integration-by-parts identities. We then write integration-by-parts identities ibps for the chosen set of independent amplitudes. If the upper limits in the x or y integrals in eq.. differ from one, α <, surface terms have to be taken into account. That is to be contrasted with the case of loop calculations, in which surface terms always vanish. By solving the ibps with the standard Laporta algorithm, complete reduction to master integrals is accomplished. Analytic evaluation of the master integrals. After having identified for each class of integrals a set of master integrals, we write the corresponding system of differential equations. The ɛ-expansion of the master integrals is obtained by solving such systems expanded in powers of ɛ. A natural basis consists of one- and two-dimensional harmonic polylogarithms [, ]; for representing some master integrals, an extension of the standard basis functions has proved to be necessary. By increasing the number of variables, the number of additional denominators grows very fast. Performing first the partial fractioning in y and then in x results in a different basis of independent amplitudes.

. Integrals needed for the integrated subtraction terms The subtraction method developed in refs. [, ] relies on the universal soft and collinear factorization properties of QCD squared matrix elements. Although the necessary factorization formulae for NNLO computations have been known for almost a decade, the explicit definition of a subtraction scheme has been hampered for several reasons. Firstly, the various factorization formulae overlap in a rather complicated way beyond NLO accuracy and these overlaps have to be disentangled in order to avoid multiple subtractions. At NNLO accuracy this was first achieved in ref. []. A general and simple solution to this problem was subsequently given in ref. [], where a method was described to obtain pure-soft factorization at any order in perturbation theory leading to soft-singular factors without collinear singularities. Secondly, the factorization formulae are valid only in the strict soft and collinear limits and have to be extended to the whole phase space. A method that works at any order in perturbation theory requires a mapping of the original n momenta {p} n = {p,..., p n } to m momenta { p} m = { p,..., p m } m is the number of hard partons and n m is the number of unresolved ones that preserves momentum conservation. Such a mapping leads to an exact factorization of the original n-particle phase space of total momentum Q, n d d p i n dφ n p,..., p n ; Q = π d δ p i π d δ d Q p i,. in the form i= dφ n {p} n ; Q = dφ m { p} m ; Q [dp nm;m {p} nm ; Q].. In the context of computing QCD corrections, this sort of exact phase-space factorization was first introduced in ref. [], where only three of the original momenta {p} that of the emitter p µ i, the spectator pµ k and the emitted particle pµ j were mapped to two momenta, p µ ij and pµ k, the rest of the phase space was left unchanged. This sort of mapping requires that both i and k be hard partons, which is always satisfied in a computation at NLO accuracy because only one parton is unresolved. However, in a computation beyond NLO the spectator momentum may also become unresolved unless this is explicitly avoided by using colour-ordered subamplitudes [7, ]. In order to take into account the colour degrees of freedom explicitly, as well as to define a phase space mapping valid at any order in perturbation theory, in ref. [], two types of democratic phase-space mappings were introduced. In this paper we are concerned with the integrals of the singly-unresolved counterterms, therefore, in the rest of the paper we deal with the case when n m =. Symbolically, the mapping i= JHEP7 C {p} ir ir n { p} n = { p,..., p ir,..., p n },. used for collinear subtractions, denotes a mapping where the momenta p µ i and p µ r are replaced by a single momentum p µ ir and all other momenta are rescaled, while for soft-type subtractions, S {p} r r n { p} n = { p,..., p n }.

Q n i r C ir Q ĩr n ir i r n ñ Figure : Graphical representation of the collinear momentum mapping and the implied phase space factorization. denotes a mapping such that the momentum p µ r, that may become soft, is missing from the set, and all other momenta are rescaled and transformed by a proper Lorentz transformation. These mappings are defined such that the recoil due to the emission of the unresolved partons is taken by all hard partons. In both cases the factorized phase-space measure can be written in the form of a convolution.. Definition of the collinear integrals In the case of collinear mapping the factorized phase-space measure can be written as [dp ir ;n p r, p ir ; Q] = dα α nɛ s fir Q π dφ p i, p r ; p ir ΘαΘ α,. where s f ir Q = p ir Q and p µ ir = α pµ ir αqµ. The collinear momentum mapping and the implied factorization of the phase-space measure are represented graphically in figure. The picture on the left shows the n-particle phase space dφ n {p}; Q, where in the circle we have indicated the number of momenta. The picture on the right corresponds to eq.. with n m = and eq..: the two circles represent the n -particle phase space dφ n { p} ir ; Q and the two-particle phase space dφ p i, p r ; p ir respectively, while the symbol stands for the convolution over α, as precisely defined in eq... Writing the factorized phase space in the form of eq.. has some advantages: JHEP7 It makes the symmetry property of the factorized phase space under the permutation of the factorized momenta manifest. For instance, for any function f, [dp ir ;n p r, p ir ; Q] fp i, p r = [dp ir ;n p r, p ir ; Q] fp r, p i,. which can be used to reduce the number of independent integrals. It exhibits the n-dependence of the factorized phase space explicitly. This allows for including n-dependent factors of α d nɛ Θα α with d ɛ= in the subtraction terms such that the integrated counterterms will be n-independent for details see ref. []. Eq.. generalizes very straightforwardly for more complicated factorizations. The formula for the general case when phase-spaces of N groups of r, r,..., r N partons are factorized simultaneously can be given explicitly.

To write the factorized two-particle phase-space measure we introduce the variable v, v = z r z r z r z r..7 In eq..7 z r is the momentum fraction of parton r in the Altarelli-Parisi splitting function that describes the f ir f i f r collinear splittings f denotes the flavour of the partons. This momentum fraction takes values between z r = α α x αx. and z r = z r x = s f ir Q /Q. Using the variables s ir = p i p r, and v the two-particle phase-space measure reads dφ p i, p r ; p ir = sɛ ir π S ɛ ds ir dv δ s ir Q α α αx [v v] ɛ Θ vθv,. where Sɛ = πɛ Γ ɛ.. The integrated collinear subtraction terms involve the integrals of the spin-averaged Altarelli-Parisi splitting kernels over the unresolved phase space [] π S ɛ Q κɛ α dα α s d fir Q π dφ p i, p r ; p ir s κɛ ir P κ f i f r z i, z r ; ɛ, κ =,.. As explained in ref. [], we include harmless factors of α dnɛ Θα α with d ɛ= in the subtraction terms to make their integrals independent of n and to restrict the phase space over which the subtractions are non-zero. Thus α, ] is the cut parameter controlling this restriction α = corresponds to subtracting over the full phase space and d is an exponent which may be fixed freely with the constraint that d ɛ=. For a more elaborate discussion, including the explanation of our eventual choice of d = ε, see appendix A of ref. []. P f i f r and P f i f r denote the average of the tree-level and one-loop splitting kernels over the spin states of the parent parton Altarelli- Parisi splitting functions, respectively. These spin-averaged splitting kernels depend, in general, on z i and z r, with the constraint JHEP7 z i z r =,. The MS renormalization scheme as often employed in the literature uses S ε = π ε e εγ E. It is not difficult to check that in a computation at the NLO accuracy using S ɛ leads to the same expressions as the usual MS definition. At NNLO the different normalizations lead to slightly different bookkeeping of the IR and UV poles at intermediate steps of the computation, but the physical cross section of infrared-safe observables is the same. Our normalization leads to somewhat simpler bookkeeping at the NNLO level. 7

δ Function g ± I z g A g ± B z ±ɛ g ± C z ±ɛ F ±ɛ, ±ɛ, ± ɛ, z ± g ± D F ±ɛ, ±ɛ, ± ɛ, z Table : The values of δ and g ± I z r at which eq.. needs to be evaluated. and are listed in appendix A. Inspecting the actual form of the Altarelli-Parisi splitting functions and using the symmetry property of the factorized phase space under the interchange i r, we find that. can be expressed as a linear combination of the integrals π S ɛ Q κɛ α dα α d s fir Q π dφ p i, p r ; p ir zkδɛ r s κɛ g ± I z r,. ir for k =,,,, κ =, and the values of δ and functions g ± I as given in table. Using eqs..7. and z r expressed with v, z r = we can see that the integrals in eq.. take the form Ix; ɛ, α, d ; κ, k, δ, g ± I = x α dv[v v] ɛ α αxv α αx α αxv α αx,. dα α κɛ α d [α αx] κɛ kδɛ g ± I α αxv.. α αx JHEP7 We compute the integrals corresponding to the first two rows of table in section.. Definition of the soft-type integrals In the case of soft mapping the factorized phase-space measure can be written as [dp r ;n p nɛ Q r; Q] = dy y π dφ p r, K; Q ΘyΘ y,. where the timelike momentum K is massive with K = yq. We show the soft momentum mapping and the implied phase space factorization in figure. The picture on the left shows again the n-particle phase space dφ n {p}; Q, while the picture on the right corresponds to eq.. with n m = and eq..: the two circles represent the twoparticle phase space dφ p r, K; Q and the n -particle phase space dφ n { p} r ; Q respectively. The symbol stands for the convolution over y as defined in eq...

r Q n r S r Q K Q n n ñ Figure : Graphical representation of the soft momentum mapping and the implied phase space factorization. The integrated soft and soft-collinear subtraction terms involve the integral of the eikonal factor and its collinear limit over the factorized phase space of eq.. [], namely the integrals π y Q S κɛ dy y d ɛ π y Q S κɛ dy y d ɛ Q π Q π κɛ sik dφ p r, K; Q, κ =,,.7 s ir s kr z κɛ i dφ p r, K; Q, κ =,.. s ir z r Here again, we included harmless factors of y d nɛ Θy y with d ɛ= in the subtraction terms to make their integrals independent of n and to restrict the phase space over which the subtractions are non-zero. Thus y, ] is the cut parameter controlling the restriciton y = corresponds to subtracting over the full phase space and d is an exponent in principle independent of d, hence the prime which may be fixed freely but with d ɛ=. See appendix A of ref. [] for a more detailed discussion, including the explanation of our eventual choice of d = ε. The computation of these integrals is fairly straightforward using energy and angle variables. In order to write the factorized phase-space measure, we choose a frame in which JHEP7 and Q µ = s,..., p µ i = Ẽi,...,, p µ k = Ẽk,..., sin χ, cos χ,. p µ r = E r,.. angles.., sin ϑ sin ϕ, sin ϑ cos ϕ, cos ϑ.. In eq.. the dots stand for vanishing components, while the notation angles in eq.. denotes the dependence of p r on the d angular variables that can be trivially integrated. Then in terms of the scaled energy-like variable ε r = p r Q Q = E r s. and the angular variables ϑ and ϕ the two-particle phase space reads dφ p r, K; Q = Q ɛ π Sɛ Γ ɛ Γ ɛ dε r ε ɛ r δy ε r dcos ϑ dcos ϕsin ϑ ɛ sin ϕ ɛ,.

where y, ] and the cosines of both angles run from to. To write the integrands in these variables, we observe that the precise definitions of p i and p k as given in ref. [] imply s ik = ε r sĩ k, s ir = sĩr, s kr = s kr,. and From eqs..,.,. and. we find s ik = ε r s ĩ k = Y ĩ k,q ε r s ir s kr sĩr s kr Q and s iq = ε r sĩq sĩr.. ε r cos ϑ cos χ cos ϑ sin χ sin ϑ cos ϕ,. z i = ε r sĩq sĩr = [ s ir z r sĩr s rq Q ε ] r.. ε r ε r cos ϑ In eq.. above we have set Yĩ k,q = Q sĩ k sĩq s kq..7 Using eqs..,. and. we see that the integral of the soft subtraction term in eq..7 may be written as J Yĩ k,q ; ɛ, y, d ; κ = Yĩ k,q κɛ Γ ɛ πγ ɛ Ωκɛ,κɛ cos χ y where Ω i,k cos χ denotes the angular integral dy y κɛ y d κɛ,. JHEP7 Ω i,k cos χ = dcos ϑ sin ϑ ɛ dcos ϕ sin ϕ ɛ cos ϑ i cos χ cos ϑ sin χ sin ϑ cos ϕ k.. Furthermore, from eq.. it is easy to see that cos χ = Yĩ k,q Q sĩ k sĩq s kq.. We compute the soft integrals J Y, ɛ; y, d ; κ in section. The soft-collinear subtraction term in eq.. leads to the integral Kɛ, y, d ; κ = y [ which we compute in section 7. dy y κɛ y d dcos ϑ sin ϑ ɛ y y cos ϑ ] κɛ Γ ɛ πγ ɛ dcos ϕ sin ϕ ɛ,.

. Iterated integrals In an NNLO computation, iterations of the above integrals also appear. In this paper we compute also two of those. The first one is the integration of a soft integral with a collinear one in its argument, J IYĩ k,q ; ɛ, α, d, y, d Γ ɛ ; k = Yĩ k,q πγ ɛ Ω, cos χ y dy y ɛ y d Iy; ɛ, α, d ;, k,,,. which we need for k =,,,. Details of the computation are given in section. The second case is when the collinear integral appears in the argument of a soft-collinear one, K Iɛ, α, d, y, d ; k = Γ ɛ πγ ɛ dcos ϑ sin ϑ ɛ y dcos ϕ sin ϕ ɛ dy y ɛ y d y cos ϑ cos ϑ Iy; ɛ, α, d ;, k,,, needed again for k =,,,. Details of the computation are given in section.. One-dimensional harmonic polylogarithms. As anticipated in the introduction, it is convenient to represent the integrals depending on a single variable x in terms of a general class of special functions called harmonic polylogarithms HPL s introduced in ref. []. The HPL s of weight one, i.e. depending on one index w =,,, are defined as: JHEP7 H; x log x ; H; x logx ; H; x log x.. These functions are then just logarithms of linear functions of x. The HPL s of higher weight are defined recursively by the relation Ha, w; x x fa; x H w; x dx for a and w n,. i.e. in the case in which not all the indices are zero. The left-most index takes the values a =,, and w is an ndimensional vector with components w i =,,. We call n the weight of the HPL s, so the above relation allows one to increase the weight w = n n. The basis functions fa; x are given by f; x x ; f; x ; f; x x x.. In the case in which all indices are zero, one defines instead, H n ; x n! logn x..

The HPL s introduced above fulfill many interesting relations, one of the most important ones being that of generating a shuffle algebra, H w ; x H w ; x = w= w w H w; x,. where w w denotes the merging of the two weight vectors w and w, i.e. all possible concatenations of w and w in which relative orderings of w and w are preserved. The basis of HPL s can be extended by adding some new basis functions to the set in eq..; for our computation we have to introduce the function f; x x.. The HPL s can be evaluated numerically in a fast and accurate way; there are various packages available for this purpose [ 7].. Two-dimensional harmonic polylogarithms To represent integrals depending on two arguments, an extension of the HPL s to functions of two variables proves to be convenient []. Since a harmonic polylogarithm is basically a repeated integration on one variable, a second independent variable is introduced as a parameter entering the basis functions: fi; x fi, α; x. We may say that in addition to the discrete index i, we have now a continuous index α. In ref. [] the following basis functions were originally introduced: where fc i α; x = x c i α,. c α = α or c α = α.. Let us remark that the above extension keeps most of the properties of the one-dimensional HPL s. In this work we have to introduce the following new basis functions, which are slightly more complicated than the ones above, fc α; x = x c α fc α; x = x c α,. with c α = α α, c α = α α.. The explicit definition of the two-dimensional harmonic polylogarithms dhpl s reads: Hc i α, wα; x x fc i α; x H wα; x dx.. In general, the dhpl s have complicated analyticity properties, with imaginary parts coming from integrating over the zeroes of the basis functions. Our computation does not involve such complications because we can always assume x, α. That implies that c k α < for any k: the denominators are never singular and the dhpl s are real. The numerical evaluation of our dhpl s can be achieved by extending the algorithm described and implemented in ref. []. JHEP7

. Special values For some special values of the argument, the dhpl s reduce to ordinary one-dimensional HPL s. It is easy to see that for α = and α = we have From this it follows that fc k α = ; x = f; x, H..., c i α =,... ; x =H...,,... ; x, lim H..., c iα,... ; x =. α lim fc k α; x =.. α.7 Similarly, for x =, the dhpl s reduce to combinations of one-dimensional HPL s in α. This reduction can be performed using an extension of the algorithm presented in []. We first write the dhpl s in x = as the integral of the derivative with respect to α, H wα; = H wα = ; α dα α H wα ;.. In the case where w only contains objects of the type c i, we have H wα = ; x =. Thus, α H wα; = dα α H wα ;.. The derivative is then carried out on the integral representation of H wα ;, and integrating back gives the desired reduction of H wα; to one-dimensional HPL s in α, e.g.. Interchange of arguments Hc α; = H; α, Hc α; = H; α H; α ln.. The basis of dhpl s introduced above selects x as the explicit integration variable and α as a parameter, but an alternative representation involving a repeated integration over α of different basis functions depending on x as an external parameter is also possible. Therefore, we have to deal with the typical problem of analytic computations: multiple representations of the same function. It is well known that a complete analytic control requires the absence of hidden zeroes in the formulae. That means that one has to know all the transformation properties identities of the functions introduced in order to have a single representative out of each class of identical objects. In ref. [] an algorithm was presented which allows one to interchange the roles of the two variables. The algorithm is basically the same as the one presented for the special values at x = : let us just replace everywhere x = by x in eq... Then we have to introduce the following set of basis functions for the dhpl s, where fd k x; α = d k x = α d k x,. x x k.. JHEP7

All the properties defined at the beginning of this section can be easily extended to this new class of denominators. One finds for example: Hc α; x = H; x H; α Hd x; α, Hc α; x = H; x H; α ln Hd x; α... The soft integral J In this section we present the analytic calculation of the soft integral defined in eq.. for κ =, and d = D d ɛ, with D being an integer. The angular integral Ω i,k cos χ was evaluated in ref. []. The integration over y leads to a hypergeometric function, and for the complete soft integral. we obtain the analytic expression J Y, ɛ; y, d ; κ = Y κɛ y κɛ κ ɛ Γ κɛ Γ κɛ F d κɛ, κɛ, κɛ, y F κɛ, κɛ, ɛ, Y, i.e., we only need to find the ɛ-expansion of an integral of the form fx, ɛ; n, n, n, r, r, r =. dt t n r ɛ t n r ɛ xt n r ɛ.. which can be obtained using the HypExp Mathematica package []. Nevertheless, we compute the expansion to show our procedure. The first hypergeometric function on the right hand side of eq.. is of the specific form F a, b, b; x, whose expansion reduces to the expansion of the incomplete beta function B x, which is a simple case to illustrate the steps of our procedure. It involves the integrals βx, ɛ; n, n, r, r = fx, ɛ; n,, n, r,, r = dt t n r ɛ xt n r ɛ = x n r ɛ B x n r ɛ, n r ɛ.. JHEP7 The class of independent integrals can be easily obtained using partial fractioning in x. However, when writing down the integration-by-parts identities for the independent integrals, we have to take into account a surface term coming from the fact that the denominator in xt does not vanish for t =, dt t t n r ɛ xt n r ɛ = x n r ɛ.. Solving the inhomogeneous linear system we find a single master integral which fulfills the differential equation β x, ɛ = βx, ɛ;,, r, r,. x β = r ɛ β xr ɛ,. x x

with initial condition β x = ; ɛ = dt t rɛ = r ɛ = rɛ k k..7 Solving this differential equation, we obtain the expansion of the incomplete beta function in terms of HPL s and thus the expansion of hypergeometric functions of the form F a, b, b; x. Turning to the general case, we note that if we want to calculate the integral. using the integration-by-parts identities, we must require r r r, because the integrationby-parts identities can exhibit poles in r i =. It is also useful to notice that not all of the integrals are independent, but only those where just one of the indices n, n, n is nonzero and where n, n. In fact, all other integrals can be reduced to one of this class using partial fractioning, e.g. fx, ɛ;,,, r, r, r = fx, ɛ;,,, r, r, r xfx, ɛ;,,, r, r, r.. If r r r, we can write immediately the integration-by-parts identities for the independent integrals for f obtained by partial fractioning, dt t k= t n r ɛ t n r ɛ xt n r ɛ =.. Solving the integration-by-parts identities we find that f has two master integrals, f x, ɛ = fx, ɛ;,,, r, r, r, f x, ɛ = fx, ɛ;,,, r, r, r.. The master integrals fulfill the following differential equations with initial condition x f = ɛr x f ɛr x f, x f =f ɛr ɛr ɛr ɛr ɛr ɛr x x, f ɛr ɛ r x ɛr ɛr ɛr x. JHEP7 f x =, ɛ = f x =, ɛ = B r ɛ, r ɛ.. Solving this set of linear differential equations we can write down the ɛ-expansion of the hypergeometric function in terms of HPL s in x. The solution for the integral J can be easily obtained by using the expansion of the hypergeometric function we just obtained. The results for κ =, and D = can be found in appendix B. As representative examples, in figure we compare the analytic and numeric results for the ɛ coefficient in the expansion of J Y, ɛ; y, ɛ; κ for κ =, and y =.,. The agreement between the two computations is seen to be excellent for the whole Y -range. We find a similar agreement for other lower-order, thus simpler expansion coefficients and/or other values of the parameters.

O coeff. of JY, ;y,-; J N -J A /J A J N -J A /J A.. -. -. -. -. -. -. -. - - - - y = analytic A y =. analytic A y = y =. J, = y = numeric N y =. numeric N - - - - - - - - - -7 - - - - - - log Y O coeff. of JY, ;y,-; J N -J A /J A J N -J A /J A.. -. -. -. -. -. -. - - - - y = analytic A y =. analytic A y = y =. J, = y = numeric N y =. numeric N - - - - - - - - - -7 - - - - - - log Y Figure : Representative results for the J integral. The plots show the coefficient of the Oɛ term in J Y, ɛ; y, ɛ; κ for κ = left figure and κ = right figure with y =.,. 7. The soft-collinear integral K In this section we calculate analytically the soft-collinear integral defined in eq.. for κ =, and d = D d ɛ, D being an integer. The ϕ integral is trivial to perform and we find Γ ɛ dcos ϕ sin ϕ ɛ = ɛ. 7. πγ ɛ Putting cos ϑ = ξ, we are left with the integral JHEP7 y Kɛ; y, d ; κ = dy 7. Analytic result for κ = dξ y κɛ y d ξ ɛ ξ κɛ yξ κɛ. 7. For κ =, the integral decouples into a product of two one-dimensional integrals and we get Kɛ;y, d ; = B y ɛ, d B ɛ, ɛ B y ɛ, d B ɛ, ɛ, 7. Using the expansion of the incomplete B-function, carried out in section, we can immediately write down the expansion of K for κ =. The result for D = can be found in appendix C.

7. Analytic result for κ = The integral. for κ = reads y Kɛ; y, d ; = dy dξ y ɛ y d ξ ɛ ξ ɛ yξ ɛ. 7. The analytic solution for this integral cannot be obtained in a straightforward way, due to the presence of the factor yξ ɛ that couples the two integrals. Therefore, we rewrite the integral in the form where Kɛ; y, d ; n, n, n, n, n = dy Kɛ; y, d ; = y ɛ Kɛ; y, d ;, D,,,, 7. dξ y n ɛ y y n d ɛ ξ n ɛ ξ n ɛ y yξ n ɛ. 7. We now calculate the integral K using the Laporta algorithm. The independent integrals can be obtained by partial fractioning in y and ξ, using the prescription that denominators depending on both integration variables are only partial fractioned in ξ, e.g. ξ y yξ ξ y y y yξ, 7.7 y y yξ y y yξ. When writing down the integration-by-parts identities for the independent integrals, we have to take into account a surface term coming from the fact that the denominator in y y does not vanish in y =, dy dy = dξ ξ dξ y y n ɛ y y n d ɛ ξ n ɛ ξ n ɛ y yξ n ɛ y n ɛ y y n d ɛ ξ n ɛ ξ n ɛ y yξ n ɛ = y n d ɛ K S ɛ; y, d ; n, n, n, 7. JHEP7 with K S ɛ; y, d ; n, n, n = K S is just a hypergeometric function, dξ ξ n ɛ ξ n ɛ y ξ n ɛ. 7. K S ɛ; y, d ; n, n, n = B n ɛ, n ɛ F n ɛ, n ɛ, n n ɛ; y, 7. 7

and can thus be calculated using the technique presented in section. Knowing the series expansion for the surface term K S, we can solve the integration-byparts identities for the K integrals, eq. 7.. We find the following two master integrals, K ɛ; y, d = Kɛ; y, d ;,,,,, fulfilling the following differential equations, K ɛ; y, d = Kɛ; y, d ;,,,,, K = ɛ K y d ɛ f, y y y K = ɛ K y d ɛ f, y y y 7. 7. where f denotes the master integral of the hypergeometric function calculated in section and where the initial conditions are given by K ɛ; y =, d = B ɛ, B ɛ, ɛ, K ɛ; y =, d = B ɛ, B ɛ, ɛ. 7. Plugging in the series expansion of f, and expanding y d ɛ into a power series in ɛ, we can solve for the K and K as a power series in ɛ whose coefficients are written in terms of HPL s in y. Knowing the series expansions of K and K, we can obtain the integral Kɛ; y, d ; for any fixed integer D. In appendix C we give the explicit result for D =.. The collinear integrals I In this section, we calculate the collinear integrals defined in eq.. for g I = g A and g I = g B analytically. JHEP7. The A-type collinear integrals for k The collinear integral for g I = g A requires the evaluation of an integral of the form Ax, ɛ; α, d ; κ, k = x Ix, ɛ; α, d ; κ, k,, g A = α dα dv α κɛ α d [α αx] κɛ α αxv k v ɛ v ɛ, α αx. where k =,,,, κ =, and d = D d ɛ with D an integer. For k this two-dimensional integral decouples into the product of two one-dimensional integrals, out of which one is straightforward, Ax, ɛ; α, d ; κ, k = k j= k j x j B j ɛ, ɛ. α dα α kjκɛ α jd [α αx] κɛ [α αx] k.

We will therefore treat separately the cases k and k <. For k the calculation of the A integrals reduces to the calculation of a onedimensional integral of the form A x,ɛ; α, d ; κ; n, n, n, n = α dα α n κɛ α n d ɛ [α αx] n κɛ [α αx] n,. n i being integers. The integration-by-parts identities, including a surface term for the independent integrals, are α dα α nκɛ α n d ɛ [α αx] nκɛ [α αx] n α = α n κɛ α n d ɛ [α α x] n κɛ [α α x] n.. Using the Laporta algorithm we find three master integrals for A, where A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,, = A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,, = A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,, = dµ ɛ α, x = dα α κɛ α d ɛ α α x κɛ, α α α dµ ɛ α; x, dµ ɛ α; x α, dµ ɛ α; x α αx. = dα ɛ dα d ln α κ ln α κ lnα x αx Oɛ, = dα ɛ dα κh; α κh; x d H; α κ Hd x; α Oɛ... JHEP7 where we used the d-representation of the two-dimensional HPL s defined in section, Hd x; α = ln x x α, Hd x, d x; α = ln x x α.7, etc. Notice that all three master integrals are finite for ɛ =. This allows us to expand the integrand into a power series in ɛ and integrate order by order in ɛ, using the defining property of the HPL s, eq... We obtain in this way the series expansion of the master integrals as a power series in ɛ whose coefficients are written in terms of the d-representation

of the two-dimensional HPL s. We can then switch back to the c-representation using the algorithm described in section. Having a representation of the master integrals, we can immediately write down the solutions for Ax, ɛ; α, d ; κ, k for k and fixed D using eq... In appendix D we give as an example the series expansions up to order ɛ for D =.. The A-type collinear integrals for k = For k =, the integral. does not decouple, so we have to use the Laporta algorithm to calculate the full two-dimensional integral. However, for k =, we can get rid of the denominator in α αx in the integrand. So we only have to deal with an integral of the form A x, ɛ; α, d ; κ; n, n, n, n, n, n = = α dα dv α n κɛ α n d ɛ [α αx] n κɛ v n ɛ v n ɛ [α αxv] n,. n i being integers. We write down the integration-by-parts identities for A including a surface term for α, α dα dv α nκɛ α n d ɛ [α αx] n κɛ v v nɛ v nɛ [α αxv] n =, α with dα dv α nκɛ α n d ɛ [α αx] n κɛ α v nɛ v nɛ [α αxv] n = α n κɛ α n d ɛ [α α x] n κɛ A,S x, ɛ; α, d ; n, n, n,. A,S x, ɛ; α, d ; n, n, n = dv v nɛ v nɛ [α α xv] n = α n B n ɛ, n ɛ F n ɛ, n, n n ɛ; α x. α. As in the case of K we are going to evaluate this surface term using the Laporta algorithm, especially to get rid of the strange argument the hypergeometric function depends on, and to get an expression for A,S in terms of two-dimensional HPL s in α and x. JHEP7 Evaluation of the surface term A,S. Because the v integration is over the whole range [, ], we do not have to take into account a surface term in the integration-by-parts identities for A,S, dv v v n ɛ v n ɛ [α α xv] n =..

Using the Laporta algorithm we see that A,S has two master integrals, A,S x, ɛ; α, d = A,S x, ɛ; α, d ;,,, A,S x, ɛ; α, d = A,S x, ɛ; α, d ;,,.. A i,s x, ɛ; α, d, i =,, are functions of the two variables x and α defined on the square [, ] [, ], so in principle we should write down a set of partial differential equations for the evolution of both α and x. However, it is easy to see that in x = we have A,S x =, ɛ; α, d = B ɛ, ɛ, A,S x =, ɛ; α, d = α B ɛ, ɛ,. for arbitrary α. So we are in the special situation where we know the solutions on the line {x = } [, ], and so we only need to consider the evolution for the x variable. In other words, we consider A i,s as a function of x only, keeping α as a parameter. The differential equations for the evolution in the x variable read x A,S =, x A,S = A,S ɛ α x α ɛ A ɛ,s α ɛ, α α x x α x α x x α. and the initial condition for this system is given by eq... As the system is already triangular, we can immdiately solve for A,S and A,S. Notice in particular that the denominator in α xxα will give rise to two-dimensional HPL s of the form Hc α ; x, etc. Evaluation of A. Having an expression for the ɛ-expansion of the surface term, we can solve the integration-by-parts identities for A, eq... We find four master integrals, JHEP7 A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,,,,, A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,,,,, A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,,,,, A x, ɛ; α, d ; κ = A x, ɛ; α, d ; κ;,,,,,.. It is easy to see that all of the master integrals are finite for ɛ =. As in the case of the surface terms, we are only interested in the x evolution, because the master integrals are known for x = for any value of α, and A x =, ɛ; α, d ; κ = B α κɛ, d ɛ B ɛ, ɛ, A x =, ɛ; α, d ; κ = B α κɛ, d ɛ B ɛ, ɛ, A x =, ɛ; α, d ; κ = A x =, ɛ; α, d ; κ, A x =, ɛ; α, d ; κ = A x =, ɛ; α, d ; κ...7

The master integrals A and A form a subtopology, i.e. the differential equations for these two master integrals close under themselves: x A = κɛ A x d ɛ κɛ A x x A α dɛ xα α x κɛ α κɛ x = κɛ A x d ɛ κɛ A x α dɛ xα α x κɛ α κɛ x The two equations can be triangularized by the change of variable à à = A A, A,S, A,S.. = A.. The equations for the subtopology now take the triangularized form xã ɛ = x ɛ à x d x d ɛ x à α d ɛ xα α x ɛ α ɛ α x α x A,S, xã = à ɛ x d x ɛ à α dɛ α ɛ xα α x ɛ a x A,S.. The initial condition for à can be obtained from eq... For à however, eq.. gives only trivial information. Furthermore, the solution of the differential equation has in general a pole in x =, but it is easy to convince oneself that à is finite in x =, which serves as the initial condition. We can now solve for the remaining two master integrals. The differential equations for A and A read x A x A = κɛ A x d ɛ κɛ A x α dɛ xα α x κɛ α κɛ x ɛ = ɛ x x ɛ x A d κɛ x κɛ x A A α dɛ xα α x κɛ α κɛ x A,S, A,S. These equations can be brought into a triangularized form via the change of variable à à = A A,. = A,. JHEP7

and eq.. now reads xã ɛ = ɛ à x x d ɛ d ɛ x x α dɛ α ɛ xα α x ɛ x x ɛ x ɛ A x, xã ɛ ɛ = x à d ɛ ɛ à x x ɛ ɛ A x x The initial condition for A to be finite in x =. A à A,S α dɛ xα α x ɛ α ɛ A,S x.. and A can again be obtained from eq.. and requiring Having the analytic expressions for the master integrals, we can now easily obtain the solutions for A for k = for a fixed value of D. The results for D = can be found in appendix D. In figure we compare the analytic and numeric results for the ɛ coefficient in the expansion of Ix, ɛ; α, ɛ;, k,, g A for k =, and α =., as representative examples. The dependence on α is not visible on the plots. The agreement between the two computations is excellent for the whole x-range. We find a similar agreement for other lower-order, thus simpler expansion coefficients and/or other values of the parameters.. The B-type collinear integrals The B-type collinear integrals require the evaluation of an integral of the form JHEP7 Bx, ɛ;α, d ; δ, k = x Ix, ɛ; α, d ;, k, δ, g B = α dα dv α ɛ α d [α αx] ɛ [α αx] k v ɛ v ɛ [α αxv] kδɛ [α α vx] δɛ,. where k =,,,, δ = ± and d = D d ɛ as before D is an integer. Unlike the A-type integrals, the B-type integrals do not decouple for k, due to the appearance of the ɛ pieces in the exponents, so we have to consider the denominators altogether, and have to deal with an integral of the form Bx,ɛ; α, d ; δ; n, n, n, n, n, n, n 7, n = = α dα dv α n ɛ α n d [α αx] n ɛ [α αx] n v n ɛ v n ɛ [α αxv] n 7δɛ [α α vx] n δɛ..

O coeff. of Ix, ;,-;,-,,g A I N -I A /I A I N -I A /I A..... -. -. - - - - A-type, = analytic A =. analytic A = =. =, k=- = numeric N =. numeric N - - - - - - - - - -7 - - - - - - log x O coeff. of Ix, ;,-;,,,g A I N -I A /I A I N -I A /I A.. -. -. -. -. -. -. -. - - - - A-type, =, k= = analytic A =. analytic A = =. = numeric N =. numeric N - - - - - - - - - -7 - - - - - - log x Figure : Representative results for the A-type integrals. The plots show the coefficient of the Oɛ term in Ix, ɛ; α, ɛ;, k,, g A for k = left figure and k = right figure with α =.,. We use again the Laporta algorithm, and write down the integration-by-parts identities for B, α dα dv v α n ɛ α n d [α αx] n ɛ [α αx] n v n ɛ v n ɛ [α αxv] n 7δɛ [α α vx] n δɛ JHEP7 α dα =, dv α α n ɛ α n d [α αx] n ɛ [α αx] n v n ɛ v n ɛ [α αxv] n 7δɛ [α α vx] n δɛ = α n ɛ α n d [α α x] n ɛ [α α x] n where the surface term is given by B S x,ɛ; α, d ; δ; n, n, n 7, n = = B S x, ɛ; α, d ; δ, k; n, n, n 7, n, dv v n ɛ v n ɛ [α α xv] n 7δɛ [α α vx] n δɛ..

Evaluation of the surface term B S. The surface term B S is no longer a hypergeometric function as it was the case for the K and A-type integrals. It can nevertheless be easily calculated using the Laporta algorithm. The integration-by-parts identities for B S read dv v v n ɛ v n ɛ [α α xv] n 7δɛ [α α vx] n δɛ =. We find three master integrals for B S,.7 fulfilling the differential equations B S x, ɛ; α, d ; δ = B S x, ɛ; α, d ; δ;,,,, B S x, ɛ; α, d ; δ = B S x, ɛ; α, d ; δ;,,,, B S x, ɛ; α, d ; δ = B S x, ɛ; α, d ; δ;,,,, x B S = ɛα α ɛ B S α α x α α ɛ ɛ α δɛ δɛ α α α x α ɛδ δɛ ɛ δɛ ɛ α xɛδ B S B S α ɛ α α x α α ɛ α α x α α δɛ ɛ ɛ δɛ ɛ δɛ ɛ xα α x ɛδ xɛδ α δɛ ɛ δɛ ɛ, α x α ɛδ. x B S = δɛ ɛ B S B ɛ S B α ɛδ S x x x, x B S = B S δ α ɛ ɛ α ɛ α δɛ δɛ ɛ α δ δ α α x α ɛδ δɛ ɛ δɛ ɛ δ α δɛ δɛ α ɛ ɛ α δ δ α x ɛδ α α x α ɛδ B α ɛ ɛ δδ S α α x α ɛδ α ɛ ɛ δδ α α x α ɛδ B δɛ ɛ δɛ ɛ S α δɛ ɛ δɛ ɛ x ɛδ α x α ɛδ α δɛ ɛ ɛ. xα α x ɛδ. JHEP7

The initial conditions for the differential equations are B S x =, ɛ; α, d ; δ = B ɛ, ɛ, B S x =, ɛ; α, d ; δ = B ɛ, ɛ, B S x =, ɛ; α, d ; δ = B ɛ, ɛ, α. The system can be triangularized by the change of variable B S = B S and then solved in the usual way. B S, B S = B S, B S = B S,. Evaluation of the B integral. Solving the integration-by-parts identities for the B integrals, we find nine master integrals B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B 7 x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,, B x, ɛ; α, d ; δ = Bx, ɛ; α, d ; δ;,,,,,,,,. JHEP7 The master integrals B i, i, 7, form a subtopology, i.e. the differential equations for these master integrals close under themselves. Furthermore the differential equations for B, B, B and B have a triangular structure in ɛ, i.e. all other master integrals are suppressed by a power of ɛ. For δ =, the corresponding differential equations are

given by x B = ε B S α ε B d ε ε xα α x x ε d ε ε x εε B ε d ε ε x d ε ε x εε d ε ε x ε x ε ε B εb ε B d ε ε x x d ε ε x ε B d ε ε x εα d ε ε α x α B S, α d ε ε x α α x εα d ε ε x x B = ɛb d ɛ ɛ B α α B S, x x x d ɛ ɛ x B = d ɛ ɛ B d ɛ ɛ x x x α α B S x α x α, ɛ x B = ɛ B x x ɛ ɛ B ɛ d ɛ ɛ x x d ɛ ɛ d ɛ ɛ x d ɛ ɛ ɛ d ɛ ɛ x ɛ ɛ ɛ B d ɛ ɛ x x d ɛ ɛ d ɛ ɛ ɛɛ B d ɛ ɛ x d ɛ ɛ x d ɛ ɛ x x B B ɛ B x ɛ d ɛ ɛ x ɛ d ɛ ɛ x ɛ x ɛ x d ɛ ɛ ɛ d ɛ ɛ x d ɛ ɛ ɛ B d ɛ ɛ x ɛ d ɛ ɛ x ɛ B d ɛ ɛ x α ɛ α d ɛ ɛ α x α α ɛ α d ɛ ɛ x B S ɛ α α d ɛ ɛ α x α ɛα ɛ α α ɛα α α d ɛ ɛ x B S α α B S, x d ɛ ɛ x JHEP7 7

whereas for δ =, the differential equations are x B = x ɛ B S α d ɛ ɛ x α α x ɛb ɛ α d ɛ ɛ xα α x ɛ α d ɛ ɛ x ɛɛ d ɛ ɛ x ɛ B ɛɛ d ɛ ɛ x d ɛ ɛ x ɛ d ɛ ɛ x ɛ B x ɛɛ B ɛb ɛ B d ɛ ɛ x x d ɛ ɛ x ɛ B d ɛ ɛ x ɛ α d ɛ ɛ α x α B S, x B = ɛb d ɛ ɛ B α α B S, x x x d ɛ ɛ x B = d ɛ ɛ B d ɛ ɛ x x x α α B S x α x α, ɛ x B = ɛ B ɛ x x d ɛ ɛ x ɛ d ɛ ɛ x ɛ ɛ d ɛ ɛ x B ɛ x d ɛ ɛ d ɛ ɛ x d ɛ ɛ ɛ d ɛ ɛ x ɛ ɛ d ɛ ɛ x ɛ x d ɛ ɛ d ɛ ɛ ɛ ɛ B d ɛ ɛ x d ɛ ɛ x ɛ B x B ɛ x d ɛ ɛ ɛ ɛ B x x x d ɛ ɛ ɛ d ɛ ɛ x d ɛ ɛ ɛ B ɛ d ɛ ɛ x d ɛ ɛ x ɛ B α ɛ d ɛ ɛ x α d ɛ ɛ α x α B S α ɛ α d ɛ ɛ x α α d ɛ ɛ α x α ɛα ɛα α ɛα α ɛ α d ɛ ɛ x B d ɛ ɛ x S α α B S. x JHEP7

The differential equations for B, B and B read, for δ =, ɛ x B = x ɛ ɛ B x x ɛ B x ɛ B x d ɛ ɛ B ɛb ɛ B x x x α α B S, x d ɛ x B = d ɛ ɛ B d ɛ ɛ B x x x x α α B S x α x α, ɛ x B B S = α d ɛ ɛx α α x d ɛ B x x ɛ d ɛ ɛ B α x x α x α α x α α x ɛα α ɛ α α ɛ ɛxα α x B S α α B S α x α x,. and for δ = x B = ɛ x ɛ B x x ɛ B x ɛ B x d ɛ ɛ B ɛ B x x ɛb x α α B S, x d ɛ x B = d ɛ ɛ B d ɛ ɛ B x x x x α α B S x α x α, ɛ x B B S = α d ɛ ɛx α α x d ɛ B x x ɛ d ɛ ɛ B α x x α x α α x α ɛxα α x α B S α x α α B S α x α x.. JHEP7 Knowing the solutions for the subtopology, we can solve for the remaining two master

integrals B and B 7. They fulfill the following differential equations, for δ =, d ɛ ɛ x B = d ɛ ɛ B d ɛ ɛ ɛ B x x x x α x α B S α x α, d ɛ x B7 = d ɛ ɛ B d ɛ ɛ B 7 x x x x α α B S x α x α,. whereas for δ = the differential equations read d ɛ ɛ x B = d ɛ ɛ B d ɛ ɛ x x x α x α B S α x α, d ɛ x B7 = d ɛ ɛ B x x x α α x α x α The initial conditions are the following. At x =, we have d ɛ ɛ B 7 x ɛ B x B S.. B x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ = B α ɛ, d ɛ B ɛ, ɛ..7 At x =, we have JHEP7 B x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ, B x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ, B x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ.. At x =, we have B x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ, B 7 x =, ɛ; α, d ; δ = B x =, ɛ; α, d ; δ.. It is easy to check that B is finite at x = and x =. The integration constants of B and B can then be fixed in an implicit way by requiring the residues of the general solution for B to vanish at x = and x =. Having the analytic expression for the master integrals, we can calculate the B-type integrals for a fixed integer value of D. We give the explicit results for D = in appendix E.

O coeff. of Ix, ;,-;,-,,g B- I N -I A /I A I N -I A /I A....... -. -. - - - - B-type, =, k=-, = = analytic A =. analytic A = =. = numeric N =. numeric N - - - - - - - - - -7 - - - - - - log x O coeff. of Ix, ;,-;,,,g B- I N -I A /I A I N -I A /I A.. -. -. -. -. -. -. -. - - - - B-type, =, k=, = = analytic A =. analytic A = =. = numeric N =. numeric N - - - - - - - - - -7 - - - - - - log x Figure : Representative results for the B-type integrals. The plots show the coefficient of the Oɛ term in Ix, ɛ; α, ɛ;, k,, g B for k = left figure and k = right figure with α =.,. In figure we show some representative results of comparing the analytic and numeric computations for the ɛ coefficient in the expansion of Ix, ɛ; α, ɛ;, k,, g B for k =, and α =.,. The dependence on α is not visible on the plots. The two sets of results are in excellent agreement for the whole x-range. For other lower-order, thus simpler expansion coefficients and/or other values of the parameters, we find similar agreement. JHEP7. The soft R -type J I integrals In this section we calculate the integral defined in eq... Substituting the result for the angular integral Ω,, we can rewrite eq.. as J IY, ɛ; y, d, α, d ; k = Y Bɛ, ɛ F,, ɛ; Y y dy y ɛ y d Iy; ɛ, α, d ;, k,, g A.. The hypergeometric function can be easily evaluated using the technique described in section. The evaluation of the y integral order by order in ɛ is a little bit more cumbersome because the integrand has two kinds of singularities,. The pole in y =.