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

Analytic integration of real-virtual counterterms in NNLO jet cross sections II This article has been downloaded from IOPscience. Please scroll down to see the full text article. JHEP08(2009)079 (http://iopscience.iop.org/1126-6708/2009/08/079) Download details: IP Address: 128.141.30.188 The article was downloaded on 19/03/2010 at 10:37 Please note that terms and conditions apply. The Table of Contents and more related content is available Home Search Collections Journals About Contact us My IOPscience JHEP08(2009)079 Published by IOP Publishing for SISSA Received:June 3, 2009 Revised:July 24, 2009 Accepted:August 4, 2009 Published:August 20, 2009 Analytic integration of real-virtual counterterms in NNLO jet cross sections II Paolo Bolzoni,aSven-Olaf Moch,aG´abor Somogyiband Zolt´an Tr´ocs´anyic aDESY, Platanenalle 6, D-15738 Zeuthen, Germany bInstitute for Theoretical Physics, University of Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland cUniversity of Debrecen and Institute of Nuclear Research of the Hungarian Academy of Sciences, H-4001 Debrecen P.O.Box 51, Hungary E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: We present analytic expressions of all integrals required to complete the explicit evaluation of the real-virtual integrated counterterms needed to define a recently proposed subtraction scheme for jet cross sections at next-to-next-to-leading order in QCD. We use the Mellin-Barnes representation of these integrals in 4 −2ǫdimensions to obtain the coefficients of their Laurent expansions around ǫ= 0. These coefficients are given by linear combinations of multidimensional Mellin-Barnes integrals. We compute the coefficients of such expansions in ǫboth numerically and analytically by complex integration over the Mellin-Barnes contours. Keywords: Jets, QCD ArXiv ePrint: 0905.4390 c SISSA 2009 doi:10.1088/1126-6708/2009/08/079 JHEP08(2009)079 Contents 1 Introduction 1 2 Integrals needed for the integrated subtraction terms 3 2.1 Basic integrals 4 2.2 Nested integrals 5 3 The method of Mellin-Barnes representations 7 4 Collinear integrals I13 5 Nested collinear-type I∗I and I∗J integrals 15 6 Nested soft-type J ∗J integrals 18 7 Nested soft-collinear K∗J integral 21 8 Conclusions 21 1 Introduction Precision predictions in perturbative Quantum Chromodynamics (QCD) at colliders demand calculating physical observables beyond leading order (LO) accuracy and, in the traditional approach to higher order predictions with fully differential kinematics, real and virtual corrections are separately evaluated. Integration over the phase space then requires a consistent treatment of the infrared singularities before any numerical computation may be performed. At next-to-leading order (NLO), infrared divergences can be handled using a subtraction scheme, which exploits the universal structure of the kinematical singularities of QCD matrix elements. The necessary (process-independent) counterterms regularize the virtual corrections at one loop and the real emission phase space integrals simultaneously [1]. At next-to-next-to-leading order (NNLO), the calculation of the radiative corrections to fully differential cross sections is a challenging problem and various extensions of the subtraction method at NNLO have been proposed, see e.g. refs. [2–5]. Currently, the available results for electron-positron annihilation at NNLO include total rates [6–8] and event shapes [9,10] for the process e+e−→3 jets and are all based on the antenna subtraction method [11–13]. On the other hand for colorless final states, such as vector boson or Higgs boson production at hadron colliders dedicated subtraction schemes at NNLO [14,15] have been applied. The infrared structure of scattering processes with three or more colored partons is involved if calculated at NNLO with the antenna subtraction method [16] — a fact – 1 – JHEP08(2009)079 which has motivated the formulation of alternative subtraction schemes. In particular, refs. [17–19] introduce a scheme for computing NNLO corrections to QCD jet cross sections for processes without colored partons in the initial state and an arbitrary number of massless particles (colored or colorless) in the final state. Very recently, following the steps of ref. [17], this subtraction scheme has been extended to cross sections for hadron-initiated processes [20], although yet to NLO accuracy only, but in a way which is NNLO-compatible. Any subtraction scheme is of practical utility only after the counterterms for the regularization of the real emissions are integrated over the phase space of the unresolved partons. In the scheme of refs. [17–19] these counterterms are universal (but complete only for processes without colored particles in the initial state) and, therefore can be computed once and for all. Their knowledge is necessary to regularize the infrared divergences appearing in the virtual corrections. Some of the integrals needed explicitly in the so-called real-virtual counterterms of this scheme have been calculated in refs. [21,22]. In the present paper we complete this task by computing all integrals needed for the the real-virtual counterterms in the subtraction scheme of refs. [17–19] by means of Mellin-Barnes (MB) representations. The use of MB integrals when dealing with Feynman integral calculus has proved powerful in the last years. MB integrals were first applied to Feynman integrals in refs. [23,24] and pioneering work has been performed since then in refs. [25–27] (see also ref. [28] and references therein for many other examples). For a given integral the MB representation replaces the power of a sum in the integrand by a product of the individual terms of the sum raised to some other powers. This leads then to integrals over certain complex contours of Γ-functions. As a crucial point it is then very convenient with this MB representation to resolve all singularities in the limit ǫ= 0 within dimensional regularization, d= 4 −2ǫ. In this paper, we adapt the MB method to derive analytic expressions for all integrals appearing in the real-virtual counterterms of refs. [17–19]. Let us briefly discuss the merits of the analytic approach for the computation of the integrated subtraction terms. First of all, in a higher-order computation, the ǫpoles of the integrated subtraction terms need to cancel the corresponding ǫpoles coming from the loop matrix elements in the virtual corrections. The cancellation of these poles can be demonstrated most convincingly once the pole structure of the integrated subtraction terms is exhibited analytically. Second, in terms of speed and precision of the evaluation, analytic results are very fast and very accurate compared to numerical ones. Moreover, they demonstrate that the final result consists of smooth functions only. Nevertheless also the numerical evaluation of the integrated counterterms has its utility, because it serves as an independent check. Then, there are indeed some cases, where it is very difficult to find the analytic computation of the multi-dimensional MB integral and only the complex numerical integration can be carried out. In these cases, however, the method of MB integrals provides a fast and reliable way to obtain the final results with small numerical uncertainties. From a practical point of view, the combination of both, analytic and numerical evaluations of all MB integrals implies that the final results for the integrated real-virtual counterterms can be conveniently given e.g. in the form of interpolating tables which can be computed once and for all. This suffices for any practical application, because in an actual computation the relative uncertainty associated with the numerical phase space integrations is generally much greater than that of the integrated subtraction terms. – 2 – JHEP08(2009)079 Q 1 b b b i r b b b n nCir Q ˜ 1 b b b e ir b b b en n−1⊗2 (ir) i rQ 1 b b b r b b b n nSr Q r ⊗ KQ 2 ˜ 1 b b b r b b b b b b en n−1 Figure 1. Graphical representations of the momentum mappings and the implied phase space factorization: collinear (left) and soft momentum mapping (right). The outline of the paper is the following. In section 2we briefly review the phase space integrals of the real-virtual corrections at NNLO and we define the integrals of the subtraction terms that we will consider in this paper. In section 3we present a brief explanation of the method of MB representations. We outline the steps of our calculation and we also discuss explicitly an example to display the typical structure of the integrals we are interested in. In section 4we complete the analytic evaluation of all integrals needed for integrated collinear counterterms. Next, in sections 5–7we compute also all different types of the nested integrals. Finally in section 8we present the conclusions of this work. 2 Integrals needed for the integrated subtraction terms The subtraction method developed in refs. [18,19] relies on the universal soft and collinear factorization properties of QCD squared matrix elements. Once the subtraction scheme is defined, one has to integrate the subtraction terms over the factorized phase space of the unresolved parton(s). This is the content of the present work (see also ref. [21]). There are two crucial elements in the formulation of a subtraction scheme beyond NLO. Firstly, the factorization formulae should disentangle the overlaps in soft-singular factors and collinear singularities in order to avoid multiple subtractions and a simple solution to this problem has been given in ref. [30]. Secondly, because the factorization formulae are valid only in the strict soft and collinear limits, they have to be extended to the whole phase space. Typically, this requires a mapping of the original nmomenta {p}n={p1,...,pn}in an n-parton matrix element at any order in perturbation theory to mmomenta {˜p}m={˜p1,...,˜pm}in such a way, that momentum conservation is preserved. Here mdenotes the number of hard partons and n−mis the number of unresolved ones. The original n-particle phase space of total momentum Qreads dφn(p1,...,pn;Q) = n Y i=1 ddpi (2π)d−1δ+p2 i(2π)dδ(d) Q− n X i=1 pi!,(2.1) and, for a given mapping, one obtains the phase-space factorization as dφn({p}n;Q) = dφm({˜p}m;Q) [dpn−m;m({p}n−m;Q)] ,(2.2) which was first introduced in ref. [1] in the context of computing QCD corrections at NLO. In this paper we are concerned with the integrals of the singly-unresolved counterterms – 3 – JHEP08(2009)079 δFunction g(±) I(z) 0gA1 ∓1g(±) B(1 −z)±ǫ 0g(±) C(1 −z)±ǫ2F1(±ǫ, ±ǫ, 1±ǫ, z) ±1g(±) D2F1(±ǫ, ±ǫ, 1±ǫ, 1−z) Table 1. The values of δand g(±) I(zr) for which eq. (2.5) needs to be evaluated. (i.e. the case m= 1), which imply two types of mappings: {p}n Cir −→ {˜p}(ir) n−1={˜p1,...,˜pir,...,˜pn},(2.3) {p}n Sr −→ {˜p}(r) n−1={˜p1,...,˜pn}.(2.4) In the collinear momentum mapping Cir −→in eq. (2.3) the momenta pµ iand pµ rare replaced by a single momentum ˜pµ ir and all other momenta are rescaled, while for soft-type subtractions, Sr −→ in eq. (2.4) the momentum pµ r, that may become soft, is missing from the set, and all other momenta are rescaled and transformed by a proper Lorentz transformation. Both momentum mappings and the corresponding factorization of the phase-space measure are represented graphically in figure 1, where the symbol ⊗stands for the convolution as implied by eq. (2.2). The integration of the singly-unresolved subtraction terms requires three basic types of integrals over the corresponding factorized phase space, as well as iterations of these (nested integrals are denoted by a ∗). All necessary integrals were derived in refs. [21,22]. 2.1 Basic integrals The three basic integrals are those used in the collinear, soft and soft-collinear subtraction counterterms. The collinear integrals have the general form Ix;ǫ, α0, d0;κ, k, δ, g(±) I=xZα0 0 dα α−1−(1+κ)ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−(1+κ)ǫ ×Z1 0 dv[v(1−v)]−ǫα+(1−α)xv 2α+(1−α)xk+δǫ g(±) Iα+(1−α)xv 2α+ (1−α)x.(2.5) These integrals need to be known as a function of x∈[0,1] in a Laurent-expansion in ǫfor k=−1,0,1,2. The necessary values of δand the expressions for the functions g(±) Iare given in table 1. Here κ= 0,1 for the first row and κ= 1 for all other cases. Analytic expressions for all cases corresponding to the first two rows of table 1were derived in ref. [21] and contain the first five terms in the ǫ-expansion. We compute all cases anew and present our results explicitly in section 4. The other parameters α0∈(0,1] and d0in eq. (2.5) will be specified in section 3. Our analytic results for these integrals include all the coefficients of the poles in ǫand the first three terms in the ǫ-expansion. – 4 – JHEP08(2009)079 Next, the soft subtractions require the integral J(Y˜ i˜ k,Q;ǫ, y0, d′ 0;κ) = −(4Y˜ i˜ k,Q)1+κǫ Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Ω(1+κǫ,1+κǫ)(cos χ) ×Zy0 0 dy y−1−2(1+κ)ǫ(1 −y)d′ 0+κǫ ,(2.6) as a function of Y˜ i˜ k,Q ∈[0,1] in a Laurent expansion around ǫ= 0, where Ω(i,k)(cos χ) denotes the angular integral in d-dimensions Ω(i,k)(cos χ) = Z1 −1 d(cos ϑ) (sin ϑ)−2ǫZ1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ ×(1 −cos ϑ)−i(1 −cos χcos ϑ−sin χsin ϑcos ϕ)−k,(2.7) with cos χ= 1 −2Y˜ i˜ k,Q .(2.8) For the present paper the exact definition of the kinematic variables xand Y˜ i˜ k,Q is unimportant, nevertheless we recall their definition to make their physical meaning explicit. The kinematic variable xis given by x=2˜pir ·Q Q2,(2.9) where ˜pir is the momentum of the parent parton in the (ir)→i+rsplitting, which appears on the right hand side of eq. (2.3) above, while Qis the total incoming momentum. We note that in the strict collinear limit we have ˜pir →pi+pr. The kinematic variable Y˜ i˜ k,Q is defined as Y˜ i˜ k,Q =1 2 Q2(˜pi·˜pk) (˜pi·Q) (˜pk·Q)(2.10) Finally, the soft-collinear subtractions lead to the integral K(ǫ, y0, d′ 0;κ) = 2 Zy0 0 dy y−(2+κ)ǫ(1 −y)d′ 0−1Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×1+ 2(1 −y) y(1−cos ϑ)1+κǫ Γ2(1 −ǫ) 2πΓ(1−2ǫ)Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ,(2.11) which does not depend on kinematical variables. The integrals Jand Kin eqs. (2.6) and (2.11) have been computed in ref. [21] for all relevant parameters in y0,d′ 0and κ. We have evaluated these soft and soft-collinear integrals, too, and we have checked that the two results agree numerically. We do not deal with the cases Jand Kin this paper. 2.2 Nested integrals In a NNLO computation, also iterations of the above integrals appear. In this paper we complete the list of nested integrals necessary for the integrated real-virtual counterterms, in particular we cover all cases which have not been addressed in ref. [21]. – 5 – JHEP08(2009)079 Of the nested integrals, which we generally denote by a star ∗, three are collinear integrals with one of the basic types in its argument, I∗Ii(x;ǫ, α0, d0;k, l) = xZα0 0 dαZ1 0 dv α−1−ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−ǫ ×[v(1 −v)]−ǫα+(1−α)xv 2α+(1−α)xk Ixα+(1−α)x(1−v) 2α+(1−α)x;ǫ, α0, d0; 0, l, 0,1,(2.12) I∗Ir(x;ǫ, α0, d0;k, l) = xZα0 0 dαZ1 0 dv α−1−ǫ(1 −α)2d0−1[α+ (1 −α)x]−1−ǫ ×[v(1 −v)]−ǫα+ (1 −α)xv 2α+ (1 −α)xk Ixα+ (1 −α)xv 2α+ (1 −α)x;ǫ, α0, d0; 0, l, 0,1,(2.13) which we need for k, l =−1,0,1,2, and I∗J x;ǫ, α0, d0, y0, d′ 0;k=xZα0 0 dαZ1 0 dv α−1−ǫ(1−α)2d0 −1[α+ (1 −α)x]−1−ǫ(2.14) ×[v(1 −v)]−ǫα+ (1 −α)xv 2α+ (1 −α)xk ×Jα(α+ (1 −α)x)(2α+ (1 −α)x)2 (α+(1−α)xv)(α+(1−α)x(1−v))x2;ǫ, y0, d′ 0,0, for k=−1,0,1,2. Both, I∗I and I∗J are needed as a function of x∈[0,1] in an ǫ-expansion with Iand Jgiven in eqs. (2.5) and (2.6), respectively. A discussion about the choice of the relevant parameters α0,d0,y0and d′ 0is given at the end of section 3and details of the computation are also given in section 5. Three other iterated integrals are defined as soft integrals with other soft integrals appearing in the argument, J∗Jik(Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×Zy0 0 dy y−1−2ǫ(1 −y)d′ 0[2 −(1 + cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×J4(1 −y)Y˜ i˜ k,Q [2 −y(1 + cos ϑ)][2−y(1+cos χcos ϑ+sin χsin ϑcos ϕ)];ǫ, y0, d′ 0,0, (2.15) J∗Jir(Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ)(sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×[2 −(1 + cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×Zy0 0 dy y−1−2ǫ(1 −y)d′ 0J(1 −cos ϑ) 2−y(1 + cos ϑ);ǫ, y0, d′ 0,0,(2.16) – 6 – JHEP08(2009)079 and J∗Jkr(Y˜ i˜ k,Q;ǫ, y0, d′ 0) = −8Y˜ i˜ k,Q Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ(2.17) ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫ(1 −cos ϑ)−1 ×Zy0 0 dy y−1−2ǫ(1−y)d′ 0[2−(1+cos χ) cos ϑ−sin χsin ϑcos ϕ]−1 ×J(1 −cos χcos ϑ−sin χsin ϑcos ϕ) 2−y(1 + cos χcos ϑ+ sin χsin ϑcos φ);ǫ, y0, d′ 0,0, with Jgiven in eq. (2.6). The three integrals in eqs. (2.15)–(2.17) need to be calculated for Y˜ i˜ k,Q ∈[0,1] as expansion in ǫ. Explicit results and details of the computation (and values for the parameters y0and d′ 0) for these integrals are presented in sections 3and 6. The final case is when the soft integral appears in the argument of a soft-collinear one, K∗J(ǫ, y0, d′ 0) = 2 Γ2(1 −ǫ) 2πΓ(1 −2ǫ)Z1 −1 d(cos ϑ) (sin ϑ)−2ǫ ×Z1 −1 d(cos ϕ) (sin ϕ)−1−2ǫZy0 0 dy y−1−2ǫ(1 −y)d′ 0−1 ×2−y(1 + cos ϑ) 1−cos ϑJ1−cos ϑ 2−y(1 + cos ϑ);ǫ, y0, d′ 0,0,(2.18) which is again independent of the kinematics, i.e. the coefficients of the expansion in ǫ are pure numbers. Details of the computation and the parameters y0and d′ 0are given in sections 3and 7. 3 The method of Mellin-Barnes representations In this section we briefly review the essential steps in the derivation of MB representations for the integrals of sections 2.1 and 2.2. The starting point is the well known basic formula, 1 (a+b)ν=1 Γ(ν)Zq+i∞ q−i∞ dz 2πi a−ν−zbzΓ(ν+z)Γ(−z),(3.1) where νand qare real numbers (the case of ν= 0 is trivial) and qsets the asymptotic position of the integration contour. The application of eq. (3.1) to Feynman integral calculus was initiated in refs. [23,24] (see also ref. [28]) and is an algorithmic procedure which can be completely automatized, as e.g. in the Ambre.m package [31] in MATHEMATICA. In general, the contour in eq. (3.1) is not necessarily a straight line and its standard definition is such that the poles of Γ(ν+z) (at z=−i−νwith ibeing non-negative integer) are all to the left and the poles of Γ(−z) (at non-negative integers) are all to the right of it. The condition on the poles of the Γ-functions can be satisfied by such a contour in eq. (3.1) if and only if q < 0 and ν > 0. However, as a key observation, ref. [27] realized straight-line contours parallel to the imaginary axis in an algorithmic way. If ν < 0, we start with a curved contour that fulfills the condition on the pole and then deform it into a straight – 7 – JHEP08(2009)079 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 -2 10-4 0 210-4 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ0) coeff. of I(x, ǫ; 1,3; 1,−1,0, gC+) C-type, κ= 1, k=−1, δ= 0 α0= 1 analytic (A) α0= 1 numeric (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 -1.0 102 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 6.0 102 7.0 102 -2 10-4 0 210-4 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ0) coeff. of I(x, ǫ; 1,3; 1,−1,1, gD+) D-type, κ= 1, k=−1, δ= 1 α0= 1 analytic (A) α0= 1 numeric (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 Figure 3. Representative results for the C-type and D-type integrals. The plots show the coefficient of the O(ǫ0) term for k=−1 in I(x, ǫ; 1,3; 1,−1,0, g(+) C) (left) and I(x, ǫ; 1,3; 1,−1,1, g(+) D) (right) with d0= 3 and α0= 1. Here we introduced the short-hand notation P(m) n,k (x;a(k) m,...,a(k) 0) = Lin(1 −x) (1 −x)m m X i=0 a(k) ixi.(4.7) According to their definition, the limit of the functions given in eqs. (4.3)–(4.6) must be finite in x= 1 even if some terms are separately divergent. Indeed computing the limit at x= 1 we find lim x→1F(x;ǫ, d0= 3,−1) = −8731 3600 −3 2ζ2,(4.8) lim x→1F(x;ǫ, d0= 3,0) = −257 60 ,(4.9) lim x→1F(x;ǫ, d0= 3,1) = −257 120 ,(4.10) lim x→1F(x;ǫ, d0= 3,2) = −1801 90 +80 3log(2) .(4.11) In figure 3we compare the analytic and numeric results for the ǫ0coefficient in the expansion of I(x, ǫ; 1,3; 1,−1,0, g(+) C) and I(x, ǫ; 1,3; 1,−1,1, g(+) D) for k=−1, α0= 1 and d0= 3 as representative examples. The agreement between the two computations is excellent for the whole x-range. The numeric results have been obtained using a sector decomposition [40] and Monte Carlo integration program as explained in detail in refs. [21, 22]. This shows that the expansion coefficients of all the collinear integrals Iand hence also of the collinear subtraction terms are smooth functions of the kinematical variable x. – 14 – JHEP08(2009)079 The complete results for all necessary cases (like in the later sections) are of considerable size, such that we shall not list them here. They are all contained in a MATHEMATICA file provided with the sources of the paper on the archive http://arXiv.org. 5 Nested collinear-type I∗I and I∗J integrals In this section we discuss the analytic computation of the nested collinear integrals defined in eqs. (2.12)–(2.14). As an example we show explicitly the fully analytic result for the case I ∗ Ir(x, ǫ; 1,3; −1,2) for which we were able to compute the complete pole structure analytically. Choosing d0= 3 and α0= 1 we get I∗Ir(x, ǫ; 1,3; −1,2) = −1 12 1 ǫ3+−2 9+1 3log(x)1 ǫ2+1 (1 −x)5−1 3ζ2−25 36 log(x) +1 3log(1−x) log(x)+ 1 3Li2(x)+1 (1−x/2)51 6log x 2+1 (1−x)4−13 36 +1 6log(x) +1/6 (1−x/2)4+1 (1−x)3−7 72 −1 18 log(x)+1/12 (1 −x/2)3+1 (1 −x)2−1 6−2 9log(x) +1/18 (1 −x/2)2+1 (1 −x)−25 72 −7 12 log(x)+1/24 (1 −x/2) +31 216 +1 6log(2) +19 9log(x) + 2 3log(1 −x) log(x)−2 3log2(x) + 2 3Li2(x)1 ǫ+ O(ǫ0).(5.1) This result is representative, because its form is typical of all the collinear nested integrals. The plot of the O(ǫ−1) coefficient of this Laurent expansion for the integral I∗Ir(x, ǫ; 1,3; −1,2) is shown on the right side of figure 4together with the comparison with the numerical evaluation obtained using sector decomposition and Monte Carlo integration. On the left side of figure 4we plot the same coefficient of the Laurent expansion for the integral I∗Ii(x, ǫ; 1,3; −1,2). For both cases we note that the agreement between the numerical evaluation and the analytic result is excellent. These plots show also that the coefficients of the Laurent expansion of the nested collinear integrals I∗I are very smooth functions of x. We note that in the Laurent expansion of I∗Ir(x, ǫ; 1,3; −1,2) in eq. (5.1) there are some terms that are divergent in x= 1. However according to its definition in eq. (2.13) the limit in x= 1 must be finite. To verify this is a further check of the correctness of the result. For the case of eq. (5.1) we obtain that: lim x→1I∗Ir(x, ǫ; 1,3; −1,2) = −1 12 1 ǫ3−2 9 1 ǫ2+3091 675 +2 3ζ2−31 6log(2)1 ǫ+ O(ǫ0).(5.2) The case of I∗Ir(x, ǫ; 1,3; 2,−1) is more difficult. For this integral we are unable to compute the coefficients of the ǫpoles in a fully analytic form. The reason is that in its Mellin-Barnes representation also three-fold MB integrals are involved. For this case the coefficient O(ǫ−3) and O(ǫ−2) are fully analytic but the coefficient of O(ǫ−1) is semi-analytic. This last coefficient is thus written in terms of an analytic expression – 15 – JHEP08(2009)079 -5.0 102 -4.0 102 -3.0 102 -2.0 102 -1.0 102 0 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coeff. of I∗Ii(x, ǫ; 1,3; −1,2) I∗Ii,k=−1, l= 2 α0= 1 analytic (A) α0= 1 numeric (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 -2.5 102 -2.0 102 -1.5 102 -1.0 102 -5.0 101 0 5.0 101 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coeff. of I∗Ir(x, ǫ; 1,3; −1,2) I∗Ir,k=−1, l= 2 α0= 1 analytic (A) α0= 1 numeric (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0= 1 Figure 4. Representative results for the I∗I-type integrals. The plots show the coefficient of the O(ǫ−1) term for k=−1 and l= 2 in I∗Ii(x, ǫ; 1,3; −1,2) (left) and I∗Ir(x, ǫ; 1,3; −1,2) (right) with d0= 3 and α0= 1. to which a three-fold MB integral must be added. The remaining MB integral can be efficiently computed in MATHEMATICA by use of the package MB.m [29]. Explicitly for I∗ Ir(x, ǫ; 1,3; 2,−1) we have: I∗Ir(x, ǫ; 1,3; 2,−1) = −1 6 1 ǫ3+1 (1 −x/2)6−5 12 log x 2+1 (1 −x)51 6log(x) +1 (1 −x/2)5−5 12 +5 12 log x 2+1/6 (1 −x)4+5/24 (1 −x/2)4+1/12 (1 −x)3 +5/72 (1 −x/2)3+1/18 (1 −x)2+5/144 (1 −x/2)2+1/24 (1 −x)+1/48 (1 −x/2) −59 72 +1 2log(x)1 ǫ2 +1 (1 −x/2)625 24 ζ2−21 8log(2) + 5 4log2(2) + 5 3log(2) log(1 −x/2) + 21 8log(x) −5 2log(2) log(x) + 5 12 log(1 −x) log(x)−5 3log(1 −x/2) log(x) + 5 4log2(x) −5 3Li2x 2+5 12 Li2(x)+1 (1 −x)5−1 3ζ2−1 6log2(2) −1 3log(2) log(1 −x/2) +17 24 log(x) + 1 3log(2) log(x) + 1 6log(1 −x) log(x) + 1 3log(1 −x/2) log(x) −1 2log2(x) + 1 3Li2x 2+1 6Li2(x)+1 (1 −x/2)523 24 −25 24 ζ2+71 24 log(2) −5 4log2(2) −5 3log(2) log(1 −x/2) −17 8log(x) + 5 2log(2) log(x) + −5 12 log(1 −x) log(x) + 5 3log 1−x 2log(x)−5 4log2(x) + 5 3Li2x 2−5 12 Li2(x) – 16 – JHEP08(2009)079 +1 (1 −x)47 8−1 4ζ2+2 3log(2) −11 8log(x) + 1 4log(1 −x) log(x) + 1 4Li2(x) +1 (1 −x/2)4−59 48 +1 6log x 2+1 (1 −x)3−1 16 −1 12 ζ2−7 36 log(x) +1 12 log(1 −x) log(x) + 1 12 Li2(x)+1 (1 −x/2)3−29 432 −1 6log(2) + 4 9log(x) +1 (1 −x)2−1 27 +2 9log(2) −23 36 log(x)+1 (1 −x/2)2211 864 −2 9log(2)+ 7 9log(x) +1 (1−x)−31 72 −59 24 log(x)+1 (1−x/2) 139 288 −1 3log(2) + 4 3log(x)−1177 432 +7 8ζ2 −1 3log(2) + 1 6log2(2) + 1 3log(2) log 1−x 2+71 24 log(x)−1 3log(2) log(x) +1 2log(1 −x) log(x)−1 3log 1−x 2log(x)−5 6log2(x)) −1 3Li2x 2+1 2Li2(x) + MBint[x]1 ǫ+ O(ǫ0),(5.3) where MBint[x] is a three-fold Mellin-Barnes integral, which for this case is given by MBint[x] = Zq1+i∞ q1−i∞ dz1 2πi Zq2+i∞ q2−i∞ dz2 2πi Zq3+i∞ q3−i∞ dz3 2πi 2z3−1x−z1−z2−z3(5.4) ×Γ −z1,1+z1,3−z2,−2+z2,5−z1−z2−z3,−z3,2+z3, z1+z2+z3 4,4−z2!, where q1=q2=q3=−1/4. Similarly to the analytic expression of eq. (5.1), also in this case we have many terms that are singular in x= 1 even though the full expression is well defined. Moreover in cases like this where we have a semi-analytic expression we find that the analytic part and the remaining part expressed in terms of a three-fold MB integral are separately well defined in x= 1. In particular for the case of the integral I∗Ir(x, ǫ; 1,3; 2,−1) in eq. (5.3) we obtain the following limit: lim x→1I∗Ir(x, ǫ; 1,3; 2,−1) = −1 6 1 ǫ3+−607 60 +40 3log(2)1 ǫ2+77349 14400 +509 24 ζ2 −3571 45 log(2) + 40 3log2(2) + MBint[1]1 ǫ+ O(ǫ0),(5.5) where MBint[1] is given by MBint[1] = 0.329808.(5.6) This number is the result of the MB integral in eq. (5.4) with the choice x= 1 obtained using the MATHEMATICA package MB.m [29]. Finally we note that this example is representative for a small subset of the collinear nested integrals which have these features. They are I∗ Ii(x, ǫ; 1,3; k, l) and I∗Ir(x, ǫ; 1,3; k, l) with k=−1,1,2 and l=−1 and I∗J(x, ǫ; 1,3,1,3; k) with k=−1. The results for the pole structure of all the remaining cases of nested collinear integrals are fully analytic. – 17 – JHEP08(2009)079 0 5.0 101 1.0 102 1.5 102 2.0 102 2.5 102 3.0 102 3.5 102 4.0 102 4.5 102 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−1) coeff. of I∗J (x, ǫ; 1,3,1,3; 0) I∗J,k= 0 α0=y0= 1 anl. (A) α0=y0= 1 num. (N) (IN− IA)/IA log10(x) 1-σ2-σ3-σ α0=y0= 1 Figure 5. Representative results for the I∗J-type integrals. The plots show the coefficient of the O(ǫ−1) term for k= 0 in I∗J(x, ǫ; 1,3,1,3; 0) with d0=d′ 0= 3 and α0=y0= 1. In table 2we list numerical values for the non-trivial coefficients of the ǫ-poles (i.e. the O(ǫ−2) and O(ǫ−1) coefficients) of the nested collinear integrals I∗Ir(x, ǫ; 1,3; −1,2) and I∗Ir(x, ǫ; 1,3; 2,−1). These numbers have been obtained using the fully analytic expression in eq. (5.1) and the semi-analytic one in eq. (5.3). For this last case using the default options of numerical integration accuracy in MB.m the relative uncertainty is at most of order 10−5, for values of xaround one. For x≪1, we find that the analytic part of the full semi-analytic result contains all contributions that are divergent as x→0 and in fact the numeric contribution decreases as we approach the limit. Thus the relative uncertainties become very small. Numbers for the O(ǫ0) coefficient for the same representative integrals are listed in table 3. In this case they have been entirely obtained evaluating their MB representations. The relative uncertainties reported in table 3were obtained with the numerical integration option MaxPoints set to 5 ·107in MB.m. Finally in figure 5we plot as a further example the fully analytic result for the first order ǫ-pole for I∗J(x, ǫ; 1,3; 0) together with the numbers obtained numerically using sector decomposition and Monte Carlo integration. As for all other cases the agreement is excellent and the coefficient is given by a very smooth function of x. In this section as in the rest of the paper the representative plots are shown for the most complicated integrals for which a full analytic result was obtained. 6 Nested soft-type J ∗J integrals In this section we discuss the analytic computation of the integrals defined in eqs. (2.15)–(2.17). For them we were able to compute a fully analytic result for the coefficient of the Laurent expansion up to O(ǫ−2). The O(ǫ−1) coefficient is computed – 18 – JHEP08(2009)079 log10(x)I∗Ir(x, ǫ; 1,3; −1,2) I∗Ir(x, ǫ; 1,3; 2,−1) O(ǫ−2) an. O(ǫ−1) an. O(ǫ−2) an. O(ǫ−1) semi-an. -10. -7.89751 -374.957 -15.9061 -759.736 ±2.16974E-12 -9.66667 -7.64166 -351.104 -15.3944 -711.688 ±2.92666E-12 -9.33333 -7.38582 -328.036 -14.8828 -665.211 ±4.09941E-12 -9. -7.12998 -305.753 -14.3711 -620.305 ±6.01115E-12 -8.66667 -6.87413 -284.256 -13.8594 -576.969 ±7.79755E-12 -8.33333 -6.61829 -263.544 -13.3477 -535.205 ±1.08953E-11 -8. -6.36245 -243.618 -12.836 -495.012 ±1.54886E-11 -7.66667 -6.10661 -224.477 -12.3243 -456.389 ±1.84206E-11 -7.33333 -5.85076 -206.122 -11.8126 -419.337 ±2.91762E-11 -7. -5.59492 -188.552 -11.301 -383.857 ±3.7663E-11 -6.66667 -5.33908 -171.768 -10.7893 -349.947 ±4.44314E-11 -6.33333 -5.08324 -155.769 -10.2776 -317.608 ±6.83357E-11 -6. -4.82739 -140.556 -9.7659 -286.841 ±8.82413E-11 -5.66667 -4.57155 -126.128 -9.25423 -257.644 ±1.11892E-10 -5.33333 -4.31571 -112.485 -8.74256 -230.019 ±2.04546E-10 -5. -4.05986 -99.628 -8.2309 -203.965 ±4.4044E-10 -4.66667 -3.80402 -87.556 -7.71928 -179.482 ±9.48925E-10 -4.33333 -3.54818 -76.269 -7.20772 -156.573 ±2.0463E-9 -4. -3.29234 -65.7665 -6.69628 -135.237 ±4.4068E-9 -3.66667 -3.03649 -56.0477 -6.18508 -115.476 ±9.49612E-9 -3.33333 -2.78065 -47.1111 -5.6743 -97.2944 ±2.04532E-8 -3. -2.52481 -38.9536 -5.16432 -80.6957 ±4.39917E-8 -2.66667 -2.26896 -31.5702 -4.65576 -65.6862 ±9.46543E-8 -2.33333 -2.01312 -24.9522 -4.14969 -52.2723 ±2.03847E-7 -2. -1.75728 -19.0853 -3.64776 -40.4585 ±4.37447E-7 -1.66667 -1.50144 -13.9478 -3.15236 -30.2408 ±9.3859E-7 -1.33333 -1.24559 -9.50936 -2.66658 -21.5965 ±1.99232E-6 -1. -0.989751 -5.73082 -2.19382 -14.4712 ±4.20263E-6 -0.666667 -0.733908 -2.5675 -1.73699 -8.76877 ±8.68698E-6 -0.333333 -0.478065 0.0265877 -1.2975 -4.35339 ±1.73005E-5 0. -0.222222 2.09462 -0.874704 -1.06702 ±3.28582E-5 Table 2. Numerical values for the O(ǫ−2) and O(ǫ−1) coefficients of I∗Ir(x, ǫ; 1,3; −1,2) (second and third column) and I∗Ir(x, ǫ; 1,3; 2,−1) (last two columns) for various values of log10(x) (first column). These numbers have been obtained evaluating the fully analytic expression in eq. (5.1) and the semi-analytic one in eq. (5.3). In the last column, we also show the numerical uncertainty as reported by MB.m. – 19 – JHEP08(2009)079 log10(x)I∗Ir(x, ǫ; 1,3; −1,2) I∗Ir(x, ǫ; 1,3; 2,−1) -5. -1642.9 ±1.40321 -3380.06 ±1.88480E-1 -4.66667 -1355.91 ±2.483460E-1 -2792.08 ±5.41150E-2 -4.33333 -1104.52 ±4.46988E-2 -2276.68 ±2.74382E-2 -4. -886.641 ±1.16175E-2 -1829.26 ±2.16858E-2 -3.66667 -699.73 ±8.45731E-3 -1444.98 ±1.81844E-2 -3.33333 -541.355 ±7.21400E-3 -1119.05 ±1.53613E-2 -3. -409.039 ±5.95157E-3 -846.659 ±1.28763E-2 -2.66667 -300.296 ±4.91329E-3 -622.987 ±1.08469E-2 -2.33333 -212.588 ±3.96846E-3 -443.178 ±8.21964E-3 -2. -143.342 ±3.13593E-3 -302.315 ±6.39871E-3 -1.66667 -89.9695 ±2.39827E-3 -195.382 ±4.76177E-3 -1.33333 -49.9198 ±1.77836-3 -117.264 ±3.27515E-3 -1. -20.7586 ±1.13024E-3 -62.7773 ±2.08013E-3 -0.666667 -0.267976 ±6.05459E-4 -26.8568 ±1.13023E-3 -0.333333 13.489 ±3.05477E-4 -4.81243 ±5.52177E-4 0. 22.1524 ±5.24078E-4 7.37746 ±4.201149E-4 Table 3. Numerical values for the O(ǫ0) coefficient of I∗Ir(x, ǫ; 1,3; −1,2) (second column) and I∗Ir(x, ǫ; 1,3; 2,−1) (last column) for various values of log10(x) (first column). These numbers have been obtained evaluating their MB representation. Aalso shown are the numerical uncertainties as reported by MB.m. semi-analytically similarly to the nested collinear integral I∗Ir(x, ǫ; 1,3; 2,−1) discussed in section 5. As a representative example we show the structure of the fully analytic part of the result for the nested soft integrals J∗J. For example choosing d′ 0= 3 we have: J∗Jik(Y;ǫ; 1,3) = 1 ǫ4+22 3−2 log(Y)1 ǫ3+H(Y)1 ǫ2+ O(ǫ−1),(6.1) J∗Jir(Y;ǫ; 1,3) = 1 2 1 ǫ4+11 3−log(Y)1 ǫ3+533 36 −22 3log(Y) + log2(Y) +3 2Li2(1 −Y)1 ǫ2+ O(ǫ−1) (6.2) and finally J∗Jkr(Y;ǫ; 1,3)= 1 ǫ4+22 3−2 log(Y)1 ǫ3+H(Y)−ζ2+1 2Li2(1−Y)1 ǫ2+O(ǫ−1).(6.3) The function H(Y) which appears in eqs. (6.1) and (6.3) is given by H(Y)= 497 18 −2ζ2+6−8Y 3(1−Y)2+33Y3−117Y2+126Y−44 3 (1 −Y)3log(Y) + 2 log2(Y) + 4 Li2(1 −Y). (6.4) – 20 – JHEP08(2009)079 Also for this function even if some terms are singular at Y= 1, we still have that the limit is well defined. Indeed we find lim Y→1H(Y) = 97 3−2ζ2.(6.5) For these three soft-type integrals the O(ǫ−2) coefficient has been plotted in figure 6 using its fully analytic expression eqs. (6.1) and (6.4) and its numerical evaluation obtained using sector decomposition and Monte Carlo integration. The agreement is excellent and the analytic result confirms that also the coefficients of the Laurent expansion for the J∗J integrals are smooth functions of Y. The numbers in table 4have been obtained evaluating the nested soft integral J∗Jik(Y;ǫ; 1,3) using the fully analytic expression in eq. (6.1) for the O(ǫ−3) and O(ǫ−2) coefficients. For the O(ǫ−1) coefficient a semi-analytic expression in terms of a MB integral has been used and finally the representation only in terms of MB integrals has been evaluated for the O(ǫ0) coefficient. In this example the relative uncertainty as reported by MB.m with the default options for numerical integration for both the O(ǫ−1) and O(ǫ0) coefficients is at most of order 10−5. For the semi-analyitc result, we see a similar phenomenon as in the case of the I∗Ir(x, ǫ; 1,3; 2,−1) integral: in the Y→0 limit, the analytic part contains all divergent contributions while the numeric part decreases. The relative uncertainties thus become very small. 7 Nested soft-collinear K∗J integral In this last section we discuss the pole structure of the integral defined in eq. (2.18). In this case the result for the Laurent expansion is very simple because the integral has no dependence on the kinematics. The coefficients of the poles in K∗J(ǫ, 1,3) with d′ 0= 3 and y0= 1 read: K∗J(ǫ, 1,3) = −1 2 1 ǫ4−11 3 1 ǫ3−557 36 1 ǫ2+−10825 216 +5 3ζ2−3ζ31 ǫ+ O(ǫ0).(7.1) This completes our discussion of the analytic computation of the fundamental integrals that contribute to the singly-unresolved counterterms. 8 Conclusions In this work we have completed the evaluation of all integrals needed for the computation of the integrated real-virtual counterterms of the subtraction scheme for NNLO jet cross sections proposed in refs. [17–19]. We have discussed representative examples for all types of soft and collinear as well as nested integrals in sections 4–7(the complete results are contained in a MATHEMATICA file). These integrals (i.e. their Laurent expansions in ǫto sufficient depth) have to be computed once and for all and their knowledge is necessary in order to make the subtraction scheme an effective tool. We have achieved this task by deriving MB representations for all integrals under consideration and, in a subsequent step, we have performed analytically the summation of the nested sums over the series – 21 – JHEP08(2009)079 0 2.0 102 4.0 102 6.0 102 8.0 102 1.0 103 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coeff. of J∗Jik(Y, ǫ; 1,3) J∗Jik y0= 1 analytic (A) y0= 1 numeric (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 0 1.0 102 2.0 102 3.0 102 4.0 102 5.0 102 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coeff. of J∗Jir (Y, ǫ; 1,3) J∗Jir y0= 1 analytic (A) y0= 1 numeric (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 0 2.0 102 4.0 102 6.0 102 8.0 102 1.0 103 -1 10-3 0 110-3 -8 -7 -6 -5 -4 -3 -2 -1 0 O(ǫ−2) coeff. of J∗Jkr (Y, ǫ; 1,3) J∗Jkr y0= 1 analytic (A) y0= 1 numeric (N) (IN− IA)/IA log10(Y) 1-σ2-σ3-σ y0= 1 Figure 6. Representative results for the J∗J-type integrals. The plots show the coefficient of the O(ǫ−2) term in J∗Jik(Y, ǫ; 1,3) (left), J∗Jir(Y, ǫ; 1,3) (right) and J∗Jkr (Y, ǫ; 1,3) (bottom) with d′ 0= 3 and y0= 1. of residues. In some cases, this second step of summing the series has not been achieved and we have resorted to a numerical evaluation of the MB integrals in the complex plane. All MB representations for both the numerical and, if available, the analytic results have been checked by an independent evaluation of the integrals using sector decomposition as in ref. [22]. We have shown, that all integrals contributing to the real-virtual counterterms are smooth functions. For practical applications, this means that all integrals (in particular the finite in ǫcontributions) can be used in terms of interpolating tables, which can be computed once and for all. Here we want to stress again that the tables and plots we – 22 – JHEP08(2009)079 log10(Y)J∗Jik(Y;ǫ; 1,3) O(ǫ−3) an. O(ǫ−2) an. O(ǫ−1) semi-an. O(ǫ0) MB -10. 53.385 1430.99 25680. ±1.44332E-11 347094. ±1.69388E-3 -9.66667 51.85 1350.22 23545.6 ±2.90334E-11 309328. ±1.31378E-3 -9.33333 50.3149 1271.81 21533.4 ±5.83683E-11 274744. ±1.09995E-3 -9. 48.7799 1195.75 19639.8 ±1.17829E-10 243157. ±9.10225E-4 -8.66667 47.2448 1122.05 17861.1 ±2.34891E-10 214389. ±9.47250E-4 -8.33333 45.7098 1050.7 16193.8 ±4.73302E-10 188265. ±8.07442E-4 -8. 44.1747 981.714 14634.2 ±9.39828E-10 164617. ±6.22748E-4 -7.66667 42.6396 915.081 13178.6 ±1.86537E-9 143283. ±5.50516E-4 -7.33333 41.1046 850.805 11823.6 ±3.70809E-9 124106. ±5.03212E-4 -7. 39.5695 788.886 10565.3 ±7.23854E-9 106934. ±5.38078E-4 -6.66667 38.0345 729.322 9400.38 ±1.44201E-8 91621. ±5.52685E-4 -6.33333 36.4994 672.116 8325.04 ±2.82069E-8 78027.5 ±5.08049E-4 -6. 34.9644 617.265 7335.7 ±5.49916E-8 66018.2 ±5.08676E-4 -5.66667 33.4293 564.771 6428.75 ±1.05801E-7 55463.9 ±5.09976E-4 -5.33333 31.8942 514.633 5600.58 ±2.03064E-7 46240.8 ±4.63345E-4 -5. 30.3592 466.852 4847.56 ±3.86389E-7 38230.9 ±5.01881E-4 -4.66667 28.8241 421.427 4166.08 ±7.45290E-7 31321.6 ±6.62424E-4 -4.33333 27.2891 378.358 3552.51 ±1.35799E-6 25405.6 ±6.96993E-4 -4. 25.754 337.645 3003.24 ±2.52452E-6 20381.7 ±7.53947E-4 -3.66667 24.219 299.287 2514.63 ±4.73890E-6 16153.6 ±7.61827E-4 -3.33333 22.6839 263.283 2083.06 ±8.67357E-6 12631. ±8.70475E-4 -3. 21.1488 229.632 1704.89 ±1.59261E-5 9728.88 ±8.12022E-4 -2.66667 19.6138 198.33 1376.45 ±2.79802E-5 7367.63 ±8.23619E-4 -2.33333 18.0787 169.37 1094.05 ±4.80260E-5 5473.13 ±9.09435E-4 -2. 16.5437 142.739 853.961 ±8.03383E-5 3976.65 ±1.09059E-3 -1.66667 15.0086 118.417 652.369 ±1.30870E-4 2814.76 ±1.41898E-3 -1.33333 13.4736 96.3641 485.392 ±2.06356E-4 1929.49 ±1.90893E-3 -1. 11.9385 76.5204 349.046 ±3.17701E-4 1268.34 ±2.67185E-3 -0.666667 10.4034 58.7892 239.262 ±4.48758E-4 784.581 ±3.65297E-3 -0.333333 8.86839 43.0286 151.932 ±5.96048E-4 437.509 ±4.88145E-3 0. 7.33333 29.0435 82.998 ±7.61558E-4 192.684 ±6.83182E-3 Table 4. Numerical values for the O(ǫ−3), O(ǫ−2), O(ǫ−1) and O(ǫ0) coefficients of J∗Jik(x;ǫ; 1,3) for various values of log10(Y). The numbers have been obtained from eq. (6.1), the semi-analytic one for the O(ǫ−1) coefficient and MB integrals for the O(ǫ0) coefficient. In the last two columns, we also show the numerical uncertainties as reported by MB.m. – 23 –