Calculation of the quark and gluon form factors to three loops in QCD

arXiv:1004.3653v4 [hep-ph] 24 Jun 2010

Preprint typeset in JHEP style - HYPER VERSION

ZU–TH 07/10 IPPP/10/26 SI-HEP-2010-09

Calculation of the quark and gluon form factors to three loops in QCD

T. Gehrmanna , E.W.N. Gloverb , T. Huberc , N. Ikizlerlib , C. Studerusa a

Institut f¨ ur Theoretische Physik, Universit¨ at Z¨ urich, Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland b Institute for Particle Physics Phenomenology, University of Durham, South Road, Durham DH1 3LE, England c Fachbereich 7, Universit¨ at Siegen, Walter-Flex-Strasse 3, D-57068 Siegen, Germany

Abstract: We describe the calculation of the three-loop QCD corrections to quark and gluon form factors. The relevant three-loop Feynman diagrams are evaluated and the resulting three-loop Feynman integrals are reduced to a small set of known master integrals by using integration-by-parts relations. Our calculation confirms the recent results by Baikov et al. for the three-loop form factors. In addition, we derive the subleading O(ǫ) terms for the fermion-loop type contributions to the three-loop form factors which are required for the extraction of the fermionic contributions to the four-loop quark and gluon collinear anomalous dimensions. The finite parts of the form factors are used to determine the hard matching coefficients for the Drell-Yan process and inclusive Higgs-production in soft-collinear effective theory. Keywords: QCD, Multi-loop calculations.

Contents 1. Introduction

1

2. Quark and gluon form factors in perturbative QCD 2.1 Results at one-loop 2.2 Results at two-loops

3 6 7

3. Calculation of the three-loop form factors

10

4. Three-loop form factor master integrals

12

5. Three-loop form factors

14

6. Infrared pole structure

21

7. Effective Theory Matching Coefficients

24

8. Conclusions

27

A. Master Integrals for three-loop form factors

28

B. Form factors in terms of master integrals

34

1. Introduction The form factors are basic vertex functions, and are as such fundamental ingredients for many precision calculations in QCD. They couple an external, colour-neutral off-shell current to a pair of partons: the quark form factor is the coupling of a virtual photon to a quark-antiquark pair, while the gluon form factor is the coupling of a Higgs boson to a pair of gluons through an effective Lagrangian. They appear as virtual higher-order corrections in coefficient functions for the inclusive Drell-Yan process [1–3] and the inclusive Higgs production cross section [3–6]. In these observables, the infrared poles of the form factors cancel with infrared singularities from real radiation corrections. Consequently, it is possible to relate the coefficients of the infrared poles of the form factors to the coefficients of large logarithmic terms in the corresponding real radiation processes [7, 8]. A framework for combining the resummation of logarithmically enhanced terms at all orders with fixed-order results is provided in an effective field theory expansion [9] of QCD, which is systematized by soft-collinear effective theory [10]. In this context, the pole terms of the form factors yield the anomalous dimensions of the effective operators, while their finite terms determine the matching coefficients to a given order [11–13].

–1–

The form factors are actually the simplest QCD objects that display a non-trivial infrared pole structure. As such, their infrared pole coefficients can be used to extract fundamental constants: the cusp anomalous dimensions [14] which control the structure of soft divergences and the collinear quark and gluon anomalous dimensions. While the cusp anomalous dimensions were first obtained to three loops from the asymptotic behaviour of splitting functions [15, 16], it is the calculation [17, 18] of the pole terms of the three-loop form factors (and finite plus subleading terms in the two-loop and one-loop form factors [19– 21]), which led to the derivation of the three-loop collinear anomalous dimensions [17, 22, 23]. An important observation is the agreement (up to an overall colour factor) of the cusp anomalous dimension for the quark and gluon, the so-called Casimir scaling [24]. Casimir scaling has been verified to three loops [15, 16], but it is an open question whether it holds at four loops and beyond [25]. From non-perturbative arguments, the Casimir scaling is expected to break down at some loop order [26]. Based on the observation that infrared singularities of massless on-shell amplitudes in QCD are related to ultraviolet singularities of operators in soft-collinear effective field theory [14, 27], the pole structure of these amplitudes can be analyzed using operator renormalization. The singularity structure of arbitrary multi-leg massless QCD amplitudes is determined by an anomalous dimension matrix. The terms allowed in this anomalous dimension matrix are strongly constrained by relations between soft and collinear terms, from non-abelian exponentiation and from soft and collinear factorization. Independently, Becher and Neubert [23] and Gardi and Magnea [28] have proposed a remarkable all-loops conjecture that describes the pole structure of massless on-shell multi-loop multi-leg QCD amplitudes (generalizing earlier results at two [29] and three loops [30]) in terms of the cusp anomalous dimensions and the collinear anomalous dimensions. In this conjecture, the colour matrix structure of the soft anomalous dimension generated by soft gluons is simply a sum over two-body interactions between hard partons, and thus the matrix structure at any loop order is the same as at one loop. This result builds on the earlier work of Refs. [31, 32] which showed the colour matrix structure of the soft anomalous dimension at two loops is identical to that at one loop. There may be additional colour correlations at three loops or beyond, which cannot be excluded at present [33]. However strong arguments for the absence of these terms are given in Refs. [23]. If the all-order conjecture [23, 28] holds, the calculation of the pole parts of the form factors to a given loop order (and of the finite and subleading parts at fewer loops) would be sufficient to determine the infrared poles of all massless on-shell QCD amplitudes to this order. The calculation of the three-loop form factors requires two principal ingredients: the algebraic reduction of all three-loop integrals appearing in the relevant Feynman diagrams to master integrals, and the analytical calculation of these master integrals. The reduction of integrals to master integrals exploits linear relations among different integrals, and is done based on a lexicographic ordering of the integrals (the Laporta algorithm [34]). Several dedicated computer-algebra implementations of the Laporta algorithm are available [34– 37]. The reduction of the integrals relevant to the three-loop form factors is among the most challenging applications of the Laporta algorithm to date: due to the very large number of interconnected integrals to be reduced, the linear systems to be solved are often containing

–2–

tens of thousand equations with a similar number of unknowns. The master integrals in the three-loop form factors were identified already several years ago [38]. Their analytical calculation proved to be a major computational challenge, which was completed only in several steps. The one-loop bubble insertions into two-loop vertex integrals as well as the two-loop bubble insertions into one-loop vertex integrals were derived using standard Feynman parameter integrals [38], while the genuine three-loop integrals required an extensive use of Mellin-Barnes integration techniques [39–41]. A first calculation of the three-loop form factors (based in part on numerical results for some of the expansion coefficients of the master integrals) was accomplished by Baikov et al. [42] in 2009. The analytical calculation of the last remaining master integrals was only completed recently [41]. It is the purpose of this paper to validate the three-loop form factor results of Ref. [41, 42] by an independent calculation, and to extend them in in part to a higher order in the expansion in the dimensional regularization parameter ǫ = 2 − d/2. These further expansion terms will be needed for an extraction of the quark and gluon collinear anomalous dimensions from the single pole pieces of the four-loop form factors. We define the quark and gluon form factors in Section 2, where we also discuss their UV-renomalization and summarize existing results at one- and two-loops. The reduction of the form factors to master integrals is described in Section 3, and the three-loop master integrals are discussed in Section 4. Explicit analytical expressions for them are collected in Appendix A. Our results for the three-loop form factors are presented in Section 5, and supplemented by Appendix B. The infrared structure of the QCD form factors up to fourloops is analyzed in Section 6. The three-loop hard matching coefficients for Drell-Yan and Higgs production in soft-collinear effective theory are determined from the form factors in Section 7. An outlook on future applications is contained in Section 8.

2. Quark and gluon form factors in perturbative QCD The form factors are the basic vertex functions of an external off-shell current (with virtuality q 2 = s12 ) coupling to a pair of partons with on-shell momenta p1 and p2 . One distinguishes time-like (s12 > 0, i.e. with partons both either in the initial or in the final state) and space-like (s12 < 0, i.e. with one parton in the initial and one in the final state) configurations. The form factors are described in terms of scalar functions by contracting the respective vertex functions (evaluated in dimensional regularization with d = 4 − 2ǫ dimensions) with projectors. For massless partons, the full vertex function is described with only a single form factor. The quark form factor is obtained from the photon-quark-antiquark vertex Γµqq¯ by Fq = −


1 µ , Tr p / Γ p / γ 2 1 µ q q ¯ 4(1 − ǫ)q 2

(2.1)

while the gluon form factor relates to the effective Higgs-gluon-gluon vertex Γµν gg as Fg =

p1 · p2 gµν − p1,µ p2,ν − p1,ν p2,µ µν Γgg . 2(1 − ǫ)

–3–

(2.2)

The form factors are expanded in perturbative QCD in powers of the coupling constant, with each power corresponding to a virtual loop. We denote the unrenormalized form factors by F a and the renormalized form factors by F a with a = q, g. At tree level, the Higgs boson does not couple either to the gluon or to massless quarks. In higher orders in perturbation theory, heavy quark loops introduce a coupling between the Higgs boson and gluons. In the limit of infinitely massive quarks, these loops give rise to an effective Lagrangian [43] mediating the coupling between the scalar Higgs field and the gluon field strength tensor: λ Lint = − HFaµν Fa,µν . 4

(2.3)

The coupling λ has inverse mass dimension. It can be computed by matching [44, 45] the effective theory to the full standard model cross sections [5]. Evaluation of the Feynman diagrams, contributing to the vertex functions at a given loop order yields the bare (unrenormalised) form factors, Fbq (αbs , s12 )

= 1+

Fbg (αbs , s12 ) = λb

 ∞  b n  X −s12 −nǫ α s

Sǫn Fnq , 2 µ 0 n=1 !     n ∞ X αb −s12 −nǫ n g s 1+ Sǫ Fn , 2 4π µ 0 n=1 4π

(2.4) (2.5)

where µ20 is the mass parameter introduced in dimensional regularisation to maintain a dimensionless coupling in the bare Lagrangian density and where Sǫ = e−ǫγ (4π)ǫ , αb

with the Euler constant γ = 0.5772 . . .

(2.6)

The renormalization of the form factor is carried out by replacing the bare coupling with the renormalized coupling αs ≡ αs (µ2 ) evaluated at the renormalization scale µ2 2ǫ 2 αbs µ2ǫ 0 = Zαs µ αs (µ ).

For simplicity we set µ2 = |s12 | so that in the MS scheme [46], "  2  β0 β0  αs  β1  αs 2 −1 + − Zαs = Sǫ 1 − ǫ 4π ǫ2 2ǫ 4π #  3  β0 7 β1 β0 1 β2  αs 3 4 − + O(αs ) , − + ǫ3 6 ǫ2 3 ǫ 4π

(2.7)

(2.8)

where β0 , β1 and β2 are [47–49] 11CA 2NF − , (2.9) 3 3 2 34CA 10CA NF β1 = − − 2CF NF , (2.10) 3 3 2N 3 1415CA 11CF NF2 79CA NF2 2857CA 205CF CA NF F + CF2 NF − − + + .(2.11) β2 = 54 18 54 9 54

β0 =

–4–

The renormalization relation for the effective coupling λb in the MS scheme is given by, λb = Zλ λ

(2.12)

with  2  β0  αs  β0 β1  αs 2 Zλ = 1 − + − ǫ 4π ǫ2 ǫ 4π    3 2β1 β0 β2 αs 3 β0 + O(α4s ) . − + − 3 2 ǫ ǫ ǫ 4π

(2.13)

The i-loop contribution to the unrenormalized coefficients is Fia , while the renormalised coefficient is denoted by Fia where a = q, g. If s12 is space-like, the form factors are real, while they acquire imaginary parts for time-like s12 . These imaginary parts (and corresponding real parts) arise from the ǫ-expansion of ∆(s12 ) = (−sgn(s12 ) − i0)−ǫ

(2.14)

so that the renormalized form factors are given by, q

n ∞  X αs (µ2 )

Fnq , 4π n=1 ! n ∞  X αs (µ2 ) g g 2 2 Fn . F (αs (µ ), s12 , µ = |s12 |) = λ 1 + 4π 2

2

F (αs (µ ), s12 , µ = |s12 |) = 1 +

(2.15) (2.16)

n=1

Up to three loops, the renormalized coefficients for the quark form factor (with µ2 = |s12 |) are then obtained as, F1q = F1q ∆(s12 ), β0 q F ∆(s12 ), ǫ 1   2β0 q β1 β02 F3q = F3q (∆(s12 ))3 − F2 (∆(s12 ))2 − − 2 F1q ∆(s12 ), ǫ 2ǫ ǫ F2q = F2q (∆(s12 ))2 −

(2.17)

while those for the gluon form factor are given by, β0 , ǫ   2β0 g β1 β02 g g 2 F2 = F2 (∆(s12 )) − F ∆(s12 ) − − 2 , ǫ 1 ǫ ǫ     3β1 3β02 β2 2β1 β0 β03 3β0 g F2 (∆(s12 ))2 − − 2 F1g ∆(s12 ) − − + . F3g = F3g (∆(s12 ))3 − ǫ 2ǫ ǫ ǫ ǫ2 ǫ3 (2.18) F1g = F1g ∆(s12 ) −

Unless explicitly stated otherwise, the renormalized form factors are given in the space-like case in the following sections. The one-loop and two-loop form factors were computed in many places in the literature [17–21]. All-order expressions in terms of one-loop and two-loop master integrals are given in [21], and are summarized below.

–5–

2.1 Results at one-loop Written in terms of the one-loop bubble integral, which is normalized to the factor SΓ =

(4π)ǫ , 16π 2 Γ(1 − ǫ)

(2.19)

the unrenormalised one-loop form factors are given by  4 +D−3 , (D − 4)   4 4 g F1 /SR = CA B2,1 − + 10 − D , (D − 4) (D − 2) F1q /SR = CF B2,1



(2.20) (2.21)

where SR =

16π 2 SΓ exp(ǫγ) . = Sǫ Γ(1 − ǫ)

(2.22)

Eqs. (2.20) and (2.21) agree with eqs. (8) and (9) of ref. [21] respectively. Inserting the expansion of the one-loop master integrals and keeping terms through to 5 O(ǫ ), we find that "

    2 3ζ2 14ζ3 2 3 2 47ζ2 = CF − 2 − + (ζ2 − 8) + ǫ + − 16 + ǫ + 4ζ2 + 7ζ3 − 32 ǫ ǫ 2 3 20   2 7ζ2 ζ3 56ζ3 62ζ5 3 141ζ2 +ǫ − + 8ζ2 + + − 64 40 3 3 5   3 47ζ22 7ζ2 ζ3 49ζ32 112ζ3 93ζ5 4 949ζ2 + − − + 16ζ2 + + − 128 +ǫ 280 5 2 9 3 5  3 94ζ22 329ζ22 ζ3 28ζ2 ζ3 31ζ2 ζ5 49ζ32 5 2847ζ2 +ǫ + − − − − 560 5 60 3 5 6 #  248ζ5 254ζ7 224ζ3 (2.23) + + − 256 , +32ζ2 + 3 5 7 "     2 2 14ζ3 g 2 47ζ2 F1 = CA − 2 + ζ2 + ǫ −2 +ǫ −6 ǫ 3 20     3 62ζ5 49ζ32 14ζ3 7ζ2 ζ3 4 949ζ2 3 + ζ2 + − 14 + ǫ − + 3ζ2 + − 30 +ǫ − 3 5 280 9 3 #   2 329ζ22 ζ3 31ζ2 ζ5 254ζ7 5 47ζ2 +ǫ (2.24) − − + 7ζ2 + 14ζ3 + − 62 20 60 5 7

F1q

where the gluon form factor agrees with eq. (7) of ref. [18] through to O(ǫ4 ). Note that at each order in ǫ, the terms of highest harmonic weight are the same for both quark and gluon form-factor. This is guaranteed by the equivalence of the coefficient of the leading pole in eqs. (2.20) and (2.21).

–6–

2.2 Results at two-loops Written in terms of the two-loop master integrals (listed in the appendix), the unrenormalised two-loop gluon form factor is given by "   16 8 q 2 2 2 F2 /SR = CF B4,2 + + D − 6D + 17 (D − 4)2 (D − 4)  2 983D 565 20 28 7D − − − − −C4,1 8 48 32(2D − 7) 9(3D − 8) (D − 4)  40 10693 − + (D − 4)2 288  2 1293D 3955 17 476 27D − + − − +B3,1 8 16 32(2D − 7) 2(D − 3) (D − 4)  456 288 581 − − + (D − 4)2 (D − 4)3 32 # D 3 − 20D 2 + 104D − 176 −C6,2 8(2D − 7) "  2 D 77D 565 12 23 +CF CA −C4,1 + + + + 16 32 64(2D − 7) 5(3D − 8) 15(D − 1)  16 163 8 + + + 3(D − 4) (D − 4)2 64  2 75D 1837D 3955 3 186 −B3,1 − + + − 16 32 64(2D − 7) 4(D − 3) (D − 4)  144 96 3845 − − + (D − 4)2 (D − 4)3 64 # D 3 − 20D 2 + 104D − 176 +C6,2 16(2D − 7) " # (D − 2)(3D 3 − 31D 2 + 110D − 128) +CF NF −C4,1 (2.25) (3D − 8)(D − 4)(D − 1)

2 F2g /SR

=

2 CA

"



32 16 48 + + (D − 2) (D − 4) (D − 2)2  + 100

B4,2 D 2 − 20D −

16 + (D − 4)2  27D 119 75 10 80 +C4,1 + + + + 2 48(2D − 5) 16(2D − 7) 3(D − 1) (D − 2)  32 24 609 103 − + − + 3(D − 4) (D − 2)2 (D − 4)2 8  525 116 96 107 +B3,1 24D + + + + 144(2D − 5) 16(2D − 7) 9(D − 1) (D − 2)

–7–

1175 32 1388 192 2 − − − − 2 2 (D − 3) 3(D − 4) (D − 2) 3(D − 4) (D − 4)3  1955 − 8 # 3(3D − 8)(D − 3) +C6,2 4(2D − 5)(2D − 7) "  7D 119 35 20 40 +CA NF C4,1 + + + − 8 12(2D − 5) 48(2D − 7) 3(D − 1) 3(D − 2)  2 45 − − (D − 4) 16  107 245 232 40 19D − − − + −B3,1 8 36(2D − 5) 48(2D − 7) 9(D − 1) 3(D − 2)  3 8 8 61 − + − − 2(D − 3) 9(D − 4) (D − 4)2 16 # 3 2 (2D − 25D + 94D − 112)(D − 4) +C6,2 8(D − 2)(2D − 5)(2D − 7) " (46D 4 − 545D 3 + 2395D 2 − 4606D + 3248)(D − 6) +CF NF −C4,1 2(2D − 7)(2D − 5)(D − 4)(D − 2)  107 245 8 1 35D − − + − +B3,1 4 18(2D − 5) 24(2D − 7) 3(D − 2) (D − 3)  448 112 333 − − − 9(D − 4) 3(D − 4)2 8 # (2D 3 − 25D 2 + 94D − 112)(D − 4) (2.26) −C6,2 4(D − 2)(2D − 5)(2D − 7) −

which, after re-expressing in terms of N and NF agrees with eqs. (10) and (11) of ref. [21]. Inserting the expansion of the two-loop master integrals and keeping terms through to 3 O(ǫ ), we find that "     2 6 1 41 1 64ζ3 221 q 2 F2 = CF + − − 2ζ2 − − ǫ4 ǫ3 ǫ2 2 ǫ 3 4   1151 17ζ2 + 58ζ3 − − 13ζ22 − 2 8   2 171ζ2 112ζ2 ζ3 213ζ2 839ζ3 184ζ5 5741 −ǫ − − + + − 5 3 4 3 5 16  2 2 3 3401ζ2 2608ζ3 1839ζ2 223ζ2 − + 54ζ2 ζ3 + + +ǫ2 5 20 9 8  6989ζ3 462ζ5 27911 − − + 6 5 32  2 3 5488ζ2 ζ3 29157ζ22 757ζ2 ζ3 184ζ2 ζ5 2434ζ32 768ζ2 + − + + + +ǫ3 7 15 40 3 5 3

–8–

# 58283ζ3 3251ζ5 8942ζ7 133781 13773ζ2 − − + + + 16 12 5 7 64 "     11 1 4129 83 1 11ζ2 +CF CA − 3 + 2 ζ2 − − 13ζ3 + − 6ǫ ǫ 9 ǫ 6 108   89173 44ζ22 119ζ2 467ζ3 − + − + 5 9 9 648   2 1891ζ2 89ζ2 ζ3 6505ζ2 6586ζ3 1775893 +ǫ − − + + 51ζ5 − 60 3 108 27 3888  3 2 2 809ζ2 2639ζ2 397ζ2 ζ3 569ζ3 146197ζ2 −ǫ2 − + + + 70 18 9 3 648  159949ζ3 3491ζ5 33912061 − − + 162 15 23328  2 3 7103ζ2 ζ3 638441ζ22 4358ζ2 ζ3 497ζ2 ζ5 16439ζ32 3817ζ2 − + − − − +ǫ3 140 30 1080 27 5 27 # 3709777ζ3 49786ζ5 632412901 2996725ζ2 + + − 372ζ7 − − 3888 972 45 139968 "     1 14 1 ζ2 353 14ζ2 26ζ3 7541 +CF NF + − + + 2+ + 3 3ǫ 9ǫ ǫ 3 54 9 9 324   150125 41ζ22 353ζ2 364ζ3 − + − −ǫ 30 54 27 1944   2 287ζ2 26ζ2 ζ3 7541ζ2 4589ζ3 242ζ5 2877653 −ǫ2 + − + + − 45 9 324 81 15 11664  2 2 3 14473ζ2 364ζ2 ζ3 338ζ3 127ζ2 − − + +ǫ3 − 14 540 27 27 # 150125ζ2 98033ζ3 3388ζ5 53933309 + , (2.27) − − + 1944 486 45 69984 which agrees through to O(ǫ2 ) with eq. (3.6) of ref. [17] and provides the next term in the expansion. Similarly we find that the two-loop expansion of the gluon form factor is given by "     11 1 67 1 11ζ2 25ζ3 68 2 g 2 − − − + ζ2 + + F2 = CA ǫ4 6ǫ3 ǫ2 18 ǫ 2 3 27   2 21ζ2 67ζ2 11ζ3 5861 − − − − 5 6 9 162   2 23ζ2 ζ3 106ζ2 1139ζ3 71ζ5 158201 77ζ2 − − + − − −ǫ 60 3 9 27 5 972  2 2 3 1943ζ2 55ζ2 ζ3 901ζ3 481ζ2 2313ζ2 − − + + +ǫ2 70 60 3 9 54  26218ζ3 341ζ5 3484193 − + + 81 15 5832

–9–

2057ζ23 1291ζ22 ζ3 28826ζ22 335ζ2 ζ3 313ζ2 ζ5 5137ζ32 + − + − + 60 10 135 9 5 27 # 397460ζ3 5963ζ5 6338ζ7 70647113 4019ζ2 − − + + − 324 243 45 7 34992 "     1 5 1 808 5ζ2 74ζ3 26 +CA NF + 2− + + ζ2 + − 3 3ǫ 9ǫ ǫ 27 3 9 81   23131 51ζ22 16ζ2 604ζ3 + + + −ǫ 10 9 27 486   2 257ζ2 50ζ2 ζ3 28ζ2 3962ζ3 542ζ5 540805 −ǫ2 − − + + + 18 3 27 81 15 2916  2 2 3 103ζ2 380ζ2 ζ3 2306ζ3 253ζ2 − + + +ǫ3 − 210 3 9 27 # 3157ζ2 30568ζ3 854ζ5 11511241 + − − − 162 243 9 17496 "     1 67 16ζ22 7ζ2 92ζ3 2027 +CF NF − + 8ζ3 − + + − +ǫ + ǫ 6 3 3 3 36   2 40ζ2 ζ3 209ζ2 1124ζ3 47491 2 184ζ2 − + + + 32ζ5 − +ǫ 9 3 18 9 216  2 3 22147ζ2 460ζ2 ζ3 176ζ2 + − − 120ζ32 +ǫ3 − 35 270 9 # 15284ζ3 368ζ5 987995 4273ζ2 , (2.28) + + − + 108 27 3 1296 +ǫ

3



which agrees through to O(ǫ2 ) with eq. (8) of ref. [18] and provides the next term in the expansion. Expressions for the renormalized one-loop and two-loop form factors, expanded to the appropriate order in ǫ, can be found in [21].

3. Calculation of the three-loop form factors To compute the three-loop quark and gluon form factors, we evaluate the relevant threeloop vertex functions within dimensional regularisation. At this loop order, there are 244 Feynman diagrams contributing to the quark form factor, and 1586 diagrams contributing to the gluon form factor. We generated these diagrams using QGRAF [50]. After contraction with the projectors (2.1)–(2.2), each diagram can be expressed as a linear combination of (typically hundreds of) scalar three-loop Feynman integrals. The three-loop integrals appearing in the form factors have up to nine different propagators. The integrands can depend on the three loop momenta, and the two on-shell external momenta, such that 12 different scalar products involving loop momenta can be formed. Consequently, not all scalar products can be cancelled against combinations of denominators, and we are left with irreducible scalar products in the numerator of the integrand. We denote the number of different propagators in an integral by t, the total number of propagators by r and the

– 10 –

total number of irreducible scalar products by s. The topology of each integral is fixed by specifying the set of t different propagators and subtopologies are obtained by removing one or more of the propagators. Using relations between different integrals based on integration-by-parts (IBP) [51] and Lorentz invariance (LI) [52], one can express the large number of different integrals in terms of a small number of so-called master integrals. These identities yield large linear systems of equations, which are solved in an iterative manner using lexicographic ordering [34]. To carry out the reduction in a systematic manner, we introduce so-called auxiliary topologies. Each auxiliary topology is a set of 12 linearly independent propagators. Within the auxiliary topology, the integrand of a three-loop form factor integral with (r, s, t) is expressed by r propagators (with exactly t different propagators) in the denominator, and s propagators (with at most 12-t different propagators) in the numerator. All three-loop form factor integrals can be cast into one of three auxiliary topologies, which are listed in Table 1. The first auxiliary topology contains planar integrals only. Three-loop integrals with 4 ≤ t ≤ 9 and t ≤ r ≤ 9 appear in the form factors. These come with up to s = 4 irreducible scalar products for the quark form factor and up to s = 5 for the gluon form factor. For a fixed topology and given (r, s, t), there are in total    r−1 11 − t + s Nr,s,t = t−1 s different integrals. To obtain a reduction, one has to solve very large systems of equations. Already for s ≤ 4, the system for a given auxiliary topology contains 900000 equations, and its solution is feasible only with dedicated computer algebra tools. For this reduction, we used the Mathematica-based package FIRE [36] and the C++ package Reduze [37], which was developed most recently by one of us. AuxTopo 1

AuxTopo 2

AuxTopo3

k12 k22 k32 (k1 − k2 )2 (k1 − k3 )2 (k2 − k3 )2 (k1 − p1 )2 (k1 − p1 − p2 )2 (k2 − p1 )2 (k2 − p1 − p2 )2 (k3 − p1 )2 (k3 − p1 − p2 )2

k12 k22 k32 (k1 − k2 )2 (k1 − k3 )2 (k2 − k3 )2 (k1 − k3 − p2 )2 (k1 − p1 − p2 )2 (k2 − p1 )2 (k1 − k2 − p2 )2 (k3 − p1 )2 (k3 − p1 − p2 )2

k12 k22 k32 (k1 − k2 )2 (k1 − k3 )2 (k1 − k2 − k3 )2 (k1 − p1 )2 (k1 − p1 − p2 )2 (k2 − p1 )2 (k2 − p1 − p2 )2 (k3 − p1 )2 (k3 − p1 − p2 )2

Table 1: Propagators in the three different auxiliary topologies used to represent all three-loop form factor integrals.

– 11 –

B2,1

B3,1

C4,1

B4,2

C6,2

Figure 1: One and two-loop master integrals appearing in the quark and gluon form factors.

With Reduze, the reduction and its performance are as follows. The topologies with more than 4 propagators are reduced after inserting the results of the sub-topologies into the system. With increasing t the number of equations decrease as (in general) does the time taken to solve the system which is in the range of a few days to less than an hour with the program Reduze on a modern desktop computer. The total computing time for all the planar diagrams is more than 2 months. However, the parallelization of topologies with an equal number of propagators reduced the overall reduction time to a few weeks. The three-loop form factors contain in total 22 master integrals, of which 14 are genuine three-loop vertex functions, 4 are three-loop propagator integrals and 4 are products of oneloop and two-loop integrals. They are described in detail in the following section.

4. Three-loop form factor master integrals Our notation for the master integrals follows [38], and we distinguish three topological types of master integrals: genuine three-loop triangles (At,i -type), bubble integrals (Bt,i type) and integrals that contain two-loop triangles (Ct,i -type). In this notation, the index t denotes the number of propagators, and i is simply enumerating the topologically different integrals with the same number of propagators. The one-loop and two-loop master integrals appearing in the form factors at these loop orders are displayed in Figure 1. Their expansions to finite order have been known for a long time, all-orders expressions were derived in [21], they can for example be expanded using HypExp [53]. Bt,i -type and Ct,i -type three-loop integrals are listed in Figure 2. The Bt,i -type integrals were computed to finite order in [51, 54], and supplemented by the higher order terms in [55]. Finally, the genuine three-loop vertex integrals are shown in Figure 3, their expansions to finite order were derived in [38–41]. The calculation of the nine-line three-loop integrals was the last missing ingredient to the form factor calculation for a long time. The full result for A9,1 and most of the pole parts of A9,2 and A9,4 were computed analytically in [40]. Analytical expressions for the

– 12 –

B4,1

B5,2

B5,1

B6,1

B6,2

C6,1

B8,1

C8,1

Figure 2: Three-loop two-point and factorizable three-point integrals.

remaining pieces of the latter two integrals were subsequently obtained in [41]. In [40], it was pointed out that for each of these three integrals one can find an integral from the same topology with an irreducible scalar product, which has homogeneous transcendentality. These integrals were named A9,1n , A9,2n and A9,4n , and are defined in [40]. Compared to [40] we increased the numerical precision of the remaining coefficients, both for A9,2 and A9,4 , by means of conventional packages like MB.m [56]. We reproduce thirteen significant digits of the analytic result of [41] in the case of A9,2 , and fourteen in the case of A9,4 . We also converted our numerical results for these two integrals into the corresponding integrals of homogeneous transcendentality, A9,2n and A9,4n . On the coefficients of these integrals, a PSLQ [57] determination was attempted. For the pole coefficients, the PSLQ algorithm converged to a unique solution in agreement with [41]. For the finite coefficients, the numerical precision that we obtained is yet insufficient for PSLQ to yield a unique solution. An analytic result for A9,2 and A9,4 , derived by purely analytic steps and without fitting rational coefficients to numerical values, is still a desirable task, and remains to be investigated in the future. This goal is definitely within reach in the case of A9,4 , whereas the situation is less clear for A9,2 . Expansions of all master integrals to the order in ǫ where transcendentality six first appears are listed in the Appendix.

– 13 –

A5,2

A5,1

A6,3

A6,1

A6,2

A7,1

A7,3

A7,2

A7,4

A7,5

A8,1

A9,1

A9,2

A9,4

Figure 3: Three-point integrals listed in Refs. [38–40].

5. Three-loop form factors

The unrenormalised three-loop form factors can be decomposed into different colour struc-

– 14 –

tures as follows: 2 3 XCq F3q /SR = CF3 XCq 3 + CF2 CA XCq 2 C + CF CA F

F

+CF CA NF XCq F CA NF

+ CF2 NF XCq 2 N F  2F  N −4 + CF NF,V XCq F NF,V(5.1) N

2 F CA

A

+ CF NF2 XCq

2 F NF

and 3 3 2 F3g /SR = CA XCg 3 + CA NF XCg 2 N + CA CF NF XCg A CF NF + CF2 NF XCg 2 N A

+CA NF2

A

XCg N 2 A F

+

F

CF NF2

F

XCg N 2 F F

,

F

(5.2)

where the last term in the quark form factor is generated by graphs where the virtual gauge boson does not couple directly to the final-state quarks. This contribution is denoted by NF,V and is proportional to the charge weighted sum of the quark flavours. In the case of purely electromagnetic interactions, we find, P q eq . (5.3) NF,γ = eq The coefficient of each colour structure is a linear combination of master integrals, resulting from the reduction of the integrals appearing in the Feynman diagrams. All coefficients are listed in Appendix B. Inserting the expansion of the three-loop master integrals and keeping terms through to O(ǫ0 ), we find that the three-loop coefficients are given by "   6 1 1 100ζ3 4 q 3 − 83 F3 = CF − 6 − 5 + 4 (2ζ2 − 25) + 3 −3ζ2 + 3ǫ ǫ ǫ ǫ 3   1 213ζ22 77ζ2 515 + 2 − + 138ζ3 − ǫ 10 2 2   2 214ζ2 ζ3 467ζ2 2119ζ3 644ζ5 9073 1 1461ζ2 − − + + − + ǫ 20 3 2 3 5 12  2 3 53675 13001ζ2 12743ζ2 9095ζ2 + − − + − + 2669ζ3 + 61ζ3 ζ2 24 12 40 252 #  4238ζ5 1826ζ32 + − 3 5 "     11 6415 1 431 7ζ2 1 2 +CF CA − 26ζ3 + + −2ζ2 + + 3 − 3ǫ5 ǫ4 18 ǫ 6 54   79277 83ζ22 1487ζ2 1 + − 210ζ3 + + 2 − ǫ 5 36 162   2 1 215ζ2 ζ3 38623ζ2 6703ζ3 1773839 9839ζ2 + + + − − 142ζ5 + − ǫ 72 3 108 6 972  2 3 1265467ζ2 18619ζ2 37684115 664325ζ2 + − − + 5832 324 2160 1260

– 15 –

2 +CF CA

"

+CF2 NF

"

+CF CA NF

"

+CF NF2

"

+CF NF,V

# 46ζ2 ζ3 1616ζ32 46594ζ5 96715ζ3 + + − − 18 9 3 45     242 1 88ζ2 6521 40289 88ζ22 553ζ2 1672ζ3 1 − + − − + − + 2 − 81ǫ4 ǫ3 27 243 ǫ 45 81 27 243   12106ζ3 136ζ5 1870564 1 802ζ22 88ζ2 ζ3 68497ζ2 − − + − − + ǫ 15 9 486 27 3 2187  2 3 52268375 767320ζ2 152059ζ2 6152ζ2 + − − + − 13122 729 540 189 # 2 710ζ2 ζ3 1136ζ3 2932ζ5 1341553ζ3 − − + + 486 9 9 9     2 6499 37 1 ζ2 545 133ζ2 146ζ3 1 − 5− 4+ 3 − − + − + 2 − 3ǫ 9ǫ ǫ 3 27 ǫ 18 9 81   2557ζ3 138865 1 337ζ22 2849ζ2 − + − + ǫ 36 54 27 486  # 2 8149ζ2 343ζ2 ζ3 45235ζ2 51005ζ3 278ζ5 2732173 + − − + + − 216 9 162 81 45 2916     1 256ζ3 13679 16ζ2 2254 1 316ζ2 88 + 3 − + − + + 2 81ǫ4 ǫ 27 243 ǫ 81 27 243   1 6436ζ3 623987 44ζ22 11027ζ2 + + − + − ǫ 5 243 81 2187  # 2 1093ζ2 368ζ2 ζ3 442961ζ2 45074ζ3 208ζ5 8560052 + − + + − − + 27 9 1458 81 3 6561     8 188 1 4ζ2 124 1 94ζ2 136ζ3 49900 − − + − + − − + − 81ǫ4 243ǫ3 ǫ2 9 27 ǫ 27 81 2187 #   83ζ22 62ζ2 3196ζ3 677716 + − − + − 135 3 243 6561 # "   2 2ζ22 14ζ3 80ζ5 N −4 . (5.4) 4− + 10ζ2 + − N 5 3 3

The pole contributions of F3q are given in eq. (3.7) of ref. [17] while the finite parts of the NF2 , CA NF and CF NF contributions are given in eq. (6) of ref. [18]. The finite NF,V contribution can be obtained from the δ(1 − x) contribution to the dabc dabc colour factor in eq. (6.6) of ref. [58]. The remaining finite contributions are given in eqs. (8) and (9) of ref. [42]. Similarly, the expansion of the gluon form factor at three-loops is given by F3g

=

3 CA

"

−

11 361 1 4 + + + 3ǫ6 3ǫ5 81ǫ4 ǫ3



−

22ζ3 3506 517ζ2 + − 54 3 243

– 16 –



2 +CA NF

"

+CA CF NF

"

+CF2 NF

"

+CA NF2

"

+CF NF2

"

  1 247ζ22 481ζ2 209ζ3 17741 + 2 + − − ǫ 90 162 27 243   2 85ζ2 ζ3 20329ζ2 241ζ3 878ζ5 145219 3751ζ2 1 − + + − − − + ǫ 360 9 243 9 15 2187  2 3 14474131 307057ζ2 8459ζ2 22523ζ2 + + + − 13122 1458 1080 270 # 77ζ2 ζ3 1766ζ32 20911ζ5 68590ζ3 + − + − 243 18 9 45     2 4280 2 1 47ζ2 1534 425ζ2 518ζ3 1 − 5− + + + + + 2 − 3ǫ 81ǫ4 ǫ3 27 243 ǫ 81 27 243   2 7561ζ2 1022ζ3 92449 1 2453ζ2 − + − + ǫ 180 243 81 2187  # 437ζ22 439ζ2 ζ3 37868ζ2 754ζ3 3238ζ5 10021313 + − − − + − 60 9 729 27 45 13122     20 1 160ζ3 526 1 176ζ22 22ζ2 224ζ3 2783 + + − − + − + − 9ǫ3 ǫ2 9 27 ǫ 15 3 27 81 #   41ζ2 11792ζ3 32ζ5 155629 16ζ22 + 48ζ2 ζ3 − + + − + − 5 3 81 9 486  # 2 296ζ3 304 + − 160ζ5 + 3ǫ 3 9     8 80 1 20ζ2 8 664ζ3 34097 1 200ζ2 − − + 2 + + + + 81ǫ4 243ǫ3 ǫ 27 9 ǫ 81 81 2187 #   797ζ22 76ζ2 11824ζ3 1479109 + + + + 135 27 243 13122    # 8 1 10562 32ζ3 424 112ζ22 16ζ2 704ζ3 . (5.5) + + − − + − + − 9ǫ2 ǫ 3 27 15 3 9 81

The divergent parts agree with eq. (8) of ref. [18] while the finite contributions agree with eq. (10) of ref. [42]. Using our knowledge of the three-loop form factors, we can also write down the O(ǫ) contributions to the NF parts of the quark and gluon form factors. For the quark formfactor we find that, ! 2248 2108 24950 68 3901 2913928 + ζ5 + ζ3 − ζ2 + ζ2 ζ3 − ζ2 F3q |NF = CF NF2 ǫ − 6561 135 27 243 9 810 2 +CF CA NF ǫ

24570881 28156 2418896 10816 2 7137385 − ζ5 − ζ3 + ζ + ζ2 4374 45 729 27 3 4374 ! 2674 352559 2 17324 3 + ζ2 ζ3 − ζ + ζ 27 1620 2 945 2

– 17 –

929587 5771 2 1263505 50187205 5863 + ζ5 + ζ3 − ζ − ζ2 17496 135 243 9 3 972 ! 8515 821749 2 875381 3 − ζ2 ζ3 + ζ − ζ 54 3240 2 7560 2  2  N −4 94 344 2 260 170 752 + ζ5 + ζ3 − ζ + ζ2 ǫ N 3 9 9 3 3 3 ! 196 2 9728 3 +30ζ2 ζ3 − , ζ − ζ 15 2 315 2

+CF2 NF ǫ −

+CF NF,V

(5.6)

and for the gluon form factor F3g |NF

16823771 9368 5440 30283 988 14018 2 = CA NF2 ǫ + ζ5 + ζ3 − ζ2 − ζ2 ζ3 + ζ 26244 135 27 1458 27 405 2

!

48658741 10066 349918 11657 2 904045 − ζ5 + ζ3 − ζ + ζ2 8748 45 729 27 3 4374 ! 791 34931 2 52283 3 + ζ2 ζ3 − ζ − ζ 9 1620 2 1080 2 ! 4208 112 2464 2 196900 800 2 − ζ5 − ζ3 − 54ζ2 + ζ2 ζ3 − ζ +CF NF ǫ 243 9 9 3 45 2

2 +CA NF ǫ −

+CF CA NF ǫ −

+CF2 NF ǫ

10508593 17092 240934 4064 2 8869 + ζ5 + ζ3 + ζ + ζ2 2916 27 243 9 3 54 ! 640 28823 2 23624 3 + ζ2 ζ3 + ζ + ζ 9 270 2 315 2

18613 3080 10552 74 − ζ5 + ζ3 − 272ζ32 − ζ2 54 3 9 3 ! 328 2 35648 3 −16ζ2 ζ3 + ζ − ζ 5 2 315 2

(5.7)

The UV-renormalization of the form factors is derived in Section 2 above. Applying (2.17) and (2.18) yields the expansion coefficients of the renormalized form factors. These are in the space-like kinematics: "   6 1 1 100ζ3 4 q 3 + 83 F3 = CF − 6 − 5 + 4 (2ζ2 − 25) − 3 3ζ2 − 3ǫ ǫ ǫ ǫ 3   515 1 213ζ22 77ζ2 − + 138ζ3 − + 2 ǫ 10 2 2   2 1 1461ζ2 214ζ2 ζ3 467ζ2 2119ζ3 644ζ5 9073 + − − + + − ǫ 20 3 2 3 5 12  2 3 12743ζ2 9095ζ2 53675 13001ζ2 − + − + 2669ζ3 + 61ζ3 ζ2 + − 24 12 40 252

– 18 –

+CF2 CA

"

2 +CF CA

"

+CF2 NF

"

+CF CA NF

"

+CF NF2

"

# 1826ζ32 4238ζ5 + − 3 5     11 1 1 361 27ζ2 1703 − 5 − 4 + 2ζ2 + 3 − − 26ζ3 + ǫ ǫ 18 ǫ 54 2   83ζ22 1 6820 482ζ3 1487ζ2 − + − + 2 ǫ 81 9 36 5   1 374149 215ζ3 ζ2 4151ζ3 31891ζ2 2975ζ22 + − 142ζ5 + − + − ǫ 486 3 6 108 72  11169211 6890ζ5 806ζ3 ζ2 19933ζ3 + − − − 2916 9 3 6 # 723739ζ22 18619ζ23 1616ζ32 537803ζ2 + − − + 3 324 2160 1260     1331 1 2866 110ζ2 88ζ22 1 11669 902ζ3 1625ζ2 − + 3 − − + − + 2 81ǫ4 ǫ 243 27 ǫ 486 27 81 45   2 88ζ3 ζ2 3526ζ3 7163ζ2 166ζ2 139345 136ζ5 1 − − + − − − + ǫ 8748 3 9 27 243 15  51082685 434ζ5 416ζ3 ζ2 505087ζ3 + − − + + 52488 9 3 486 # 2 2 412315ζ2 22157ζ2 6152ζ23 1136ζ3 − + − − 9 729 270 189     2 35 1 139 775 110ζ3 133ζ2 1 + + − 3ζ − − − + 2 ǫ5 9ǫ4 ǫ3 27 ǫ2 81 9 18   2183ζ2 287ζ22 1 24761 469ζ3 + − − + − ǫ 243 27 54 36  # 35ζ3 ζ2 21179ζ3 16745ζ2 8503ζ22 691883 386ζ5 − + + − − + − 1458 9 3 81 81 1080     484 1 476ζ2 752 20ζ2 2068 212ζ3 1 + + + − − − + 81ǫ4 ǫ3 243 27 ǫ2 243 27 81   2594ζ2 44ζ22 8659 964ζ3 1 − + + − + ǫ 2187 81 243 15  # 1700171 4ζ5 4ζ3 ζ2 4288ζ3 115555ζ2 2ζ22 + − + − + + 6561 3 3 27 729 27     8 20 8 1 46 4 1 2417 44 + ζ2 + − ζ3 − ζ2 − + − 81ǫ4 243ǫ3 ǫ2 81 9 ǫ 2187 81 27 #   190931 416 824 188 2 + − − ζ3 − ζ2 − ζ 13122 243 81 135 2

– 19 –

+CF NF,V 3 CA

"

2 +CA NF

"

+CA CF NF

"

+CF2 NF

"

+CA NF2

"

+CF NF2

"

+NF3

"

F3g

=

# 2ζ22 14ζ3 80ζ5 , (5.8) 4− + 10ζ2 + − 5 3 3   4 55 9079 1 5453 22ζ3 77ζ2 − 6− 5− + 3 + + 3ǫ 3ǫ 162ǫ4 ǫ 486 3 54   1393ζ2 247ζ22 4277 2266ζ3 1 + − + + 2 − ǫ 243 27 81 90   1 27301ζ2 12881ζ22 1307704 878ζ5 85ζ3 ζ2 1814ζ3 + − − + − + − ǫ 2187 15 9 9 486 360  23496187 13882ζ5 1441ζ3 ζ2 24893ζ3 + − + − + 26244 45 18 243 # 126071ζ22 22523ζ23 1766ζ32 118165ζ2 + + − − 9 1458 1080 270     10 1780 1 2344 7ζ2 1534 68ζ3 169ζ2 1 + + 3 − + + + 2 − 3ǫ5 81ǫ4 ǫ 243 27 ǫ 243 27 81   2 3536ζ2 941ζ2 1 854467 3002ζ3 + + + + ǫ 4374 81 243 180  # 301ζ3 ζ2 1414ζ3 6440ζ2 527ζ22 2143537 4516ζ5 + − + − + + 13122 45 9 9 729 20     34 13ζ2 176ζ22 1 427 160ζ3 1 13655 2600ζ3 − 3+ 2 − − − − + 9ǫ ǫ 27 9 ǫ 81 27 3 15 #   284929 32ζ5 14398ζ3 118ζ2 928ζ22 + + + 48ζ3 ζ2 − − − 972 9 81 3 15  # 1 304 296ζ3 − + − 160ζ5 + 3ǫ 9 3     170 998 1 92 2ζ2 1 37133 164ζ3 70ζ2 − + − − − + + − 81ǫ4 243ǫ3 ǫ2 27 27 ǫ 4374 81 81 #   20ζ2 157ζ22 125059 952ζ3 + − − + 13122 243 27 135    # 14 2881 152ζ3 1 2ζ2 16ζ22 212 16ζ3 + + − − + − + 9ǫ2 ǫ 27 3 162 9 3 5 # 8 (5.9) 27ǫ3 

N2 − 4 N

"

The O(ǫ) contributions to the NF parts of the UV-renormalized space-like quark and gluon form factors are given by, ! 88 2632 5515 8 952 2 3769249 q 2 + ζ5 + ζ3 − ζ2 + ζ2 ζ3 − ζ F3 |NF = +CF NF ǫ − 26244 135 243 81 3 81 2

– 20 –

+CF CA NF ǫ

1214351 3988 2 4933141 1552436 11596 − ζ5 − ζ3 + ζ + ζ2 729 45 729 27 3 4374 ! 1966 4579 2 2762 3 + ζ2 ζ3 + ζ + ζ 27 405 2 945 2

15199979 10769 553882 6881 2 961699 − ζ5 + ζ3 − ζ − ζ2 8748 135 243 27 3 972 ! 4627 94747 2 425813 3 − ζ2 ζ3 + ζ − ζ 54 3240 2 7560 2  2  N −4 94 344 2 260 170 752 + ζ5 + ζ3 − ζ + ζ2 ǫ N 3 9 9 3 3 3 ! 196 2 9728 3 +30ζ2 ζ3 − , (5.10) ζ − ζ 15 2 315 2

+CF2 NF ǫ −

+CF NF,V

and for the gluon form factor F3g |NF

8396 25315 172 2453 2 6599393 1844 + ζ5 + ζ3 − ζ2 − ζ2 ζ3 + ζ = +CA NF2 ǫ 26244 135 81 1458 27 405 2

!

18825781 1682 270232 6251 2 867919 + ζ5 + ζ3 − ζ + ζ2 8748 45 729 27 3 4374 ! 881 33403 2 133627 3 − ζ2 ζ3 + ζ + ζ 9 405 2 7560 2 ! 360181 224 1960 277 32 208 2 2 +CF NF ǫ − ζ5 − ζ3 − ζ2 + ζ2 ζ3 − ζ 972 9 9 9 3 15 2 2 +CA NF ǫ −

+CF CA NF ǫ −

+CF2 NF ǫ

7017335 7588 92894 4064 2 986 + ζ5 − ζ3 + ζ + ζ2 5832 27 243 9 3 54 ! 59987 2 23624 3 1960 ζ2 ζ3 − ζ + ζ + 9 540 2 315 2

18613 3080 10552 74 − ζ5 + ζ3 − 272ζ32 − ζ2 54 3 9 3 ! 328 2 35648 3 −16ζ2 ζ3 + ζ − ζ . 5 2 315 2

(5.11)

6. Infrared pole structure According to ref. [14, 23], the general infrared pole structure of a renormalised QCD amplitude is related to the ultraviolet behaviour of an effective operator in soft-collinear effective

– 21 –

theory. These poles can therefore be subtracted by means of a multiplicative renormalization factor Z. This means that the finite remainders of a scattering amplitude MF is obtained from the full amplitude M via the relation, MF = Z−1 M.

(6.1)

In general, the scattering amplitude M and Z are matrices in colour space. However, in the context of the quark and gluon form factors, the colour matrix is trivial. The UV renormalised amplitudes M and MF have perturbative expansions, X  αs (µ2 ) i M = 1+ Mi , (6.2) 4π i=1 X  αs (µ2 ) i F M = 1+ MiF , (6.3) 4π i=1

while log(Z) =

X  αs (µ2 ) i i=1

4π

Zi .

(6.4)

We can now solve eq. (6.1) order by order in the strong coupling, Poles(M1 ) = Z1 , Poles(M2 ) = Z2 +

(6.5) M12

, 2 M3 Poles(M3 ) = Z3 − 1 + M2 M1 , 3 M2 M14 − M12 M2 + M1 M3 + 2 , Poles(M4 ) = Z4 + 4 2 M15 Poles(M5 ) = Z5 − + M13 M2 − M12 M3 − M1 M22 + M1 M4 + M2 M3 . 5

(6.6) (6.7) (6.8) (6.9)

The deepest infrared pole for the i-loop amplitude is ǫ−2i . However, the deepest pole in the Zi -factor is ǫ−i−1 . All of the deepest poles are obtained directly from the lower loop amplitudes - which must be known to an appropriately high order in ǫ. For example, to obtain the correct pole structure for Mi , one needs knowledge of M1 through to O(ǫ2i−3 ). We find that the infrared pole structure of the renormalised form factors is given by (i = q, g and Cq = CF , Cg = CA for the cusp anomalous dimension): Ci γ0cusp γ0i + , 2ǫ2 ǫ
2   F1i 1 Ci γ1cusp β0 γ0i γ1i 3Ci γ0cusp β0 i + 2 − − + , + Poles(F2 ) = 8ǫ3 ǫ 2 8 2ǫ 2   11β02 Ci γ0cusp 1 5β0 Ci γ1cusp β02 γ0i 2Ci γ0cusp β1 i Poles(F3 ) = − + 3 + + 36ǫ4 ǫ 36 3 9
3   cusp i i i i F β1 γ0 β0 γ1 Ci γ2 γ 1 − − + 2 − 1 + F2q F1q . + 2 − ǫ 3 18 3 3ǫ 3

Poles(F1i ) = −

– 22 –

(6.10) (6.11)

(6.12)

Note that the full (all-orders) expressions for Fiq are recycled on the right-hand-side. The coefficients of the cusp soft anomalous dimension γicusp are known to three-loop order [17] and are given by: γ0cusp = 4 ,  2

268 4π 40NF − , (6.14) − 9 3 9     536π 2 44π 4 88ζ3 112ζ3 836 80π 2 2 490 = CA − + + + − + C A NF − 3 27 45 3 27 27 3   2 16NF 110 +CF NF − + 32ζ3 − . (6.15) 3 27

γ1cusp = CA γ2cusp

(6.13) 

while the quark and gluon collinear anomalous dimensions γiq and γig in the conventional dimensional regularisation scheme are also known to three-loop order [22, 23] and are given by: γ0q = −3CF , (6.16)     2 961 11π 3 γ1q = CF2 − + 2π 2 − 24ζ3 + CF CA − − + 26ζ3 2 54 6   65 π 2 + , (6.17) +CF NF 27 3     2953 13π 2 14π 4 256ζ3 8ζ3 2417 10π 2 γ2q = CF2 NF − − + − − + CF NF2 54 9 27 9 729 27 27   2 4 11π 964ζ3 8659 1297π + + − +CF CA NF − 729 243 45 27   4 2 8π 16π ζ3 29 3 2 +CF − − 3π − − 68ζ3 + + 240ζ5 2 5 3   151 205π 2 247π 4 844ζ3 8π 2 ζ3 +CA CF2 − + + − − − 120ζ5 4 9 135 3 3   2 4 83π 3526ζ3 44π 2 ζ3 139345 7163π 2 − − + − − 136ζ5 , (6.18) +CA CF − 2916 486 90 9 9 11CA 2NF γ0g = − + , (6.19) 3 3     128 π 2 692 11π 2 2 + + 2ζ3 + CA NF − + 2CF NF (6.20) γ1g = CA − 27 18 27 9   97186 6109π 2 319π 4 122ζ3 20π 2 ζ3 g 3 γ2 = CA − + − + − − 16ζ5 , 729 486 270 3 9   41π 4 356ζ3 30715 599π 2 2 − + + +CA NF 1458 243 135 27   2 4 1217 π 4π 152ζ3 +CF CA NF − − − − CF2 NF 27 3 45 9   11CF NF2 10π 2 56ζ3 269 2 + − . (6.21) − +CA NF − 1458 81 27 9

– 23 –

Taking this one step further, we find that the pole structure of the renormalised fourloop quark form factor is given by Poles(F4i ) =

25β03 Ci γ0cusp β0 (24β02 γ0i + 13β0 Ci γ1cusp + 40Ci γ0cusp β1 ) − 5 96ǫ 96ǫ4   cusp cusp 3β1 Ci γ1 β2γi β1 β0 γ0i 5Ci γ0cusp β2 1 7β0 Ci γ2 + + 0 1+ + + 3 ǫ 96 32 4 2 32   cusp i i i i 1 β0 γ2 β2 γ0 β1 γ1 Ci γ3 γ + 2 − − − − + 3 ǫ 4 32 4 4 4ǫ

2 4
2 Fi Fi (6.22) + 1 + F1i F2i − 2 − F1i F3i . 4 2

In this expression, we assume Casimir scaling of the cusp anomalous dimension to hold at four loops [23, 24], such that only a universal γ3cusp appears. If, contrary to expectations, Casimir scaling should be violated at this order, different γ3cusp would appear in the double pole terms of the quark and gluon form factors at four loops. Eq. (6.22) shows that in order to make use of a calculation of the pole parts of the four-loop form factors to extract the cusp and collinear anomalous dimensions, one requires the finite parts of the three-loop form factor for γ3cusp , and of the subleading O(ǫ) parts for γ3q,g . For all colour-factor contributions proportional to NF , these are provided in the previous section. The required subleading terms in higher orders in ǫ from the one-loop and two-loop form factors were summarized in Section 2 above.

7. Effective Theory Matching Coefficients It is well known that fixed-order perturbation theory is not necessarily reliable for physical quantities involving several disparate scales. In such cases, higher-order corrections are enhanced by large logarithms of scale ratios. Experimentally relevant examples are the Drell-Yan and Higgs production processes in hadron-hadron colliders. When the phase space for soft gluon emission is constrained, large logarithmic threshold corrections appear of the form   m−1 ln (1 − z) , (m ≤ 2k), (7.1) αks (1 − z) + where (1 − z) is the fraction of centre-of-mass energy of the initial partons available for soft gluon radiation. These can spoil the convergence of the perturbative series. The resummation of these so-called Sudakov-logarithms has been accomplished to fourth logarithmic order [59], using the exponentiation properties of the coefficient functions in moment space [60]. An alternative resummation framework is provided by soft-collinear effective field theory (SCET), which is based on the idea to split the calculation into a series of single-scale problems by successively integrating out the physics associated with the largest remaining scale. The SCET framework [10] originated in the study of heavy quarks, and has been subsequently generalized to massless collider processes [61]. The infrared poles in the high

– 24 –

energy theory (QCD) get transformed into ultraviolet poles in the effective theory [9, 14] and can then be resummed by renormalization-group (RG) evolution from the larger scales to the smaller ones. Of course the SCET must match precisely onto the high energy theory, and this is achieved by computing matrix elements in both the SCET and QCD and adjusting the Wilson coefficients so that they agree. If the matching is performed onshell, then the matching coefficients relevant for Drell Yan and Higgs production can be obtained from the quark and gluon form factors respectively. Therefore, we can utilise the results presented in the previous sections to compute the matching conditions through to three-loops. Results up to two loops were obtained previously in [11–13]. The renormalised form-factors are infrared divergent. In the effective field theory, these infrared divergences are transformed into ultraviolet poles. The matching coefficient C i (i = q, g) is obtained by extracting the poles using a renormalisation factor such that, C i (αs (µ2 ), s12 , µ2 ) = lim Zi−1 (ǫ, s12 , µ)F i (ǫ, s12 , µ2 ). ǫ→0

(7.2)

The matching coefficients have the perturbative expansion, i

2

2

C (αs (µ ), s12 , µ ) = 1 +

n ∞  X αs (µ2 )

n=1

4π

Cni (s12 , µ2 ).

(7.3)

They are are known to two loop order for Drell-Yan [11, 12] and Higgs [13] production,   q 2 C1 = CF − L + 3L − 8 + ζ2 , (7.4)      1 4 25 45 C2q = CF2 L − 3L3 + − ζ 2 L2 + − + 24ζ3 − 9ζ2 L 2 2 2  255 83 2 + − 30ζ3 + 21ζ2 − ζ2 8 10      2545 233 22 11 3 L + − + 2ζ2 L2 + − 26ζ3 + ζ2 L +CF CA 9 18 54 3  51157 313 337 44 − + ζ3 − ζ2 + ζ22 648 9 18 5     2 19 209 4 23 4085 2 +CF NF − L3 + L2 + − − ζ2 L + + ζ3 + ζ2 , (7.5) 9 9 27 3 324 9 9   (7.6) C1g = CA − L2 + ζ2 ,      11 3 67 22 80 g 2 2 1 4 L + L + − + ζ2 L + − 2ζ3 − ζ2 L C2 = CA 2 9 9 27 3  5105 143 67 1 + − ζ3 + ζ2 + ζ22 162 9 6 2     52 4 5 916 46 2 3 10 2 + ζ2 L − − ζ3 − ζ2 +CA NF − L + L + 9 9 27 3 81 9 3   67 +CF NF 2L − + 8ζ3 (7.7) 6

– 25 –

where L = log(−s12 /µ2 ). Exploiting the expressions for the renormalised quark and gluon form factors given in eqs. (5.8) and (5.9) respectively, we find that the three-loop matching coefficients are

C3q



    1 6 3 5 17 1 4 = − L + L + − + ζ2 L + 9ζ2 + 27 − 24ζ3 L3 6 2 2 2   507 105 83 2 2 + 102ζ3 − − ζ2 + ζ2 L 8 2 10   357 207 2 785 + − 214ζ3 − 240ζ5 − 8ζ2 ζ3 + ζ2 + ζ + L 2 10 2 8  6451 37729 3 2539 413 2 2 ζ + 664ζ5 − ζ2 + ζ − 470ζ3 + 250ζ2 ζ3 − + 16ζ3 − 5 2 24 630 2 12      11 5 299 1 2585 2 4 +CF CA − L + − 2ζ2 L + − + 26ζ3 − ζ2 L3 9 18 27 9   206317 1807 502 34 2 2 − ζ3 + ζ2 − ζ2 L + 648 9 9 5   13805 2441 11260 162 2 + − + 120ζ5 + ζ3 − ζ2 − 10ζ2 ζ3 + ζ L 24 3 27 5 2 18770 296 2 538835 3751 415025 2756 − ζ5 − ζ3 + ζ3 + ζ2 − ζ2 ζ3 + 648 9 3 648 9  27 4943 2 12676 3 − ζ − ζ 270 2 315 2      2 5 25 4 410 10 112 12815 70 +CF2 NF L − L + + ζ 2 L3 + − + ζ3 − ζ 2 L2 9 9 27 9 324 9 9   3121 610 1618 28 − ζ3 + ζ2 + ζ22 L + 108 9 27 5  13184 31729 38 331 2 41077 416 − ζ5 + ζ3 − ζ2 − ζ2 ζ3 − ζ2 + 972 9 81 324 9 27      2869 44 121 4 18682 26 44 2 2 2 3 +CF CA − L + − ζ2 L + − + 88ζ3 + ζ2 − ζ2 L 54 81 9 81 9 5   1045955 17464 17366 88 94 + + 136ζ5 − ζ3 + ζ2 + ζ2 ζ3 − ζ22 L 1458 27 81 3 3 505087 1136 2 412315 416 51082685 434 − ζ5 + ζ3 − ζ3 − ζ2 + ζ2 ζ3 − 52488 9 9 729 3  486 22157 2 6152 3 ζ − ζ + 270 2 189 2      22 4 974 8 16 5876 3 +CF CA NF L + − + ζ2 L + − 8ζ3 + ζ2 L2 27 81 9 81 3   5864 44 154919 724 + ζ3 − ζ2 + ζ22 L + − 729 9 81 15  1700171 4 4288 115555 4 2 2 + − ζ5 − ζ3 + ζ2 + ζ2 ζ3 + ζ2 6561 3 27 729 3 27 CF3

– 26 –

+CF NF2



+CF NF,V and, C3g

    2 4 76 3 406 8 152 9838 16 L + L + − − ζ 2 L2 + + ζ3 + ζ2 L 27 81 81 9 729 27 27  190931 416 824 188 2 − − ζ3 − ζ2 − ζ 13122 243 81 135 2    2 14 2 2 80 N −4 4 − ζ5 + ζ3 + 10ζ2 − ζ2 N 3 3 5 −

(7.8)



    1 6 11 5 281 3 1540 11 4 = − L − L + − ζ2 L + ζ2 + + 2ζ3 L3 6 9 54 2 3 81   6740 685 73 2 2 143 ζ3 − + ζ2 − ζ2 L + 9 81 18 10   2048 34 13420 176 2 373975 + ζ3 + 16ζ5 + ζ2 ζ3 − ζ2 + ζ − L 27 3 81 5 2 1458 105617 24389 3 152716 605 1939 2 2222 ζ + ζ5 + ζ2 − ζ2 − ζ3 − ζ2 ζ3 − 270 2 9 729 1890 243 9  29639273 104 2 + − ζ 26244 9 3      8 4 734 2 103 377 118 2 5 3 2 L − L + − − ζ2 L + + ζ3 − ζ 2 L2 +CA NF 9 27 81 3 27 9 9   133036 28 3820 48 + ζ3 + ζ2 − ζ22 L + 729 9 81 5  3765007 428 460 14189 82 73 2 − + ζ5 − ζ3 − ζ2 − ζ2 ζ3 + ζ2 6561 9 81 729 9 45      116 8 14057 128 40 80 2 + ζ 2 L2 + − − ζ3 − ζ2 L +CA NF2 − L4 + L3 + 27 81 81 9 729 27 27  611401 4576 4 4 + + ζ3 + ζ2 + ζ22 13122 243 9 27     4 2 52 32 20 16 4481 112 +CF NF2 L + − + ζ3 L + − ζ3 − ζ2 − ζ22 3 3 3 81 3 9 45      16 2 8 3 3833 376 2 +CF CA NF − L + 13 − 16ζ3 L + − ζ3 + 6ζ2 + ζ2 L 3 54 9 5  14564 68 64 64 341219 608 + ζ5 + ζ3 − ζ2 + ζ2 ζ3 − ζ22 − 972 9 81 9 3 45   304 296 2 +CF NF − 2L + (7.9) − 160ζ5 + ζ3 . 9 3 3 CA

These matching coefficients allow to perform the three-loop matching of the SCET-based resummation onto the full QCD calculation.

8. Conclusions In this paper, we described the calculation of the three-loop quark and gluon form factors in detail. Our results confirm earlier expressions obtained by Baikov et al. [42], which we extended by subleading terms in the fermionic corrections.

– 27 –

The form factors are the simplest QCD objects with non-trivial infrared structure. Recent findings on the relation between massless on-shell QCD amplitudes and operators in soft-collinear effective theory [27], combined with constraints from factorization, has led to the conjecture [23] that their pole terms at a given loop level contain all information needed to predict the pole structure of massless on-shell multi-leg amplitudes at the same loop order. In particular, the cusp anomalous dimension can be extracted from the double pole, and the collinear anomalous dimension from the single pole. At a given loop order, finite and subleading terms from lower loop orders are also required. In this respect, the finite terms presented here will be instrumental for the extraction of the four-loop cusp anomalous dimension, while the subleading terms contribute to the four-loop quark and gluon collinear anomalous dimension. The three-loop form factors are key ingredients for the fourth order (N3 LO) corrections to the inclusive Drell-Yan and Higgs boson production cross sections. The calculation of these, at least in an improvement to the soft approximation [8, 12], could be envisaged in future work. In view of this application, we derived the hard matching coefficients of the SCET operators to this order. Inclusion of these corrections will lead to a further stabilization of the perturbative prediction under scale variations, and are thus important for precision physics at hadron colliders.

Acknowledgements This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200020-126691, by the Forschungskredit der Universit¨ at Z¨ urich, the UK Science and Technology Facilities Council, by the European Commission’s Marie-Curie Research Training Network under contract MRTN-CT-2006-035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’, and by the Helmholtz Alliance “Physics at the Terascale”. EWNG gratefully acknowledges the support of the Wolfson Foundation and the Royal Society.

A. Master Integrals for three-loop form factors In this appendix, we summarize the ǫ-expansions of all master integrals needed for the three-loop form factors. Our notation for the integrals follows [38], using a Minkowskian loop integration measure dd k/(2π)d . All master integrals are defined in Section 4 above. With the normalization SΓ defined in (2.19), all M -loop integrals have an overall factor of
M sn12 iSΓ (−s12 − i0)−ǫ (A.1) where n is fixed by dimensional arguments. Unlike Refs. [38–40], there is no (−1)n factor. We expand to the required order for the three-loop form factors (which is typically the order where transcendentality 6 first appears). The one-loop master integral is:   2 1 2 3 6ζ2 + 2 + 4ǫ − ǫ (2ζ3 − 8) − ǫ + 4ζ3 − 16 B2,1 = ǫ 5

– 28 –

 12ζ22 −ǫ + 8ζ3 + 6ζ5 − 32 5   16ζ23 24ζ22 + − 2ζ32 + 16ζ3 + 12ζ5 − 64 −ǫ5 7 5   48 2 12 2 32 3 6 2 +ǫ 128 − 18ζ7 − 24ζ5 − 32ζ3 + 4ζ3 − ζ2 + ζ2 ζ3 − ζ2 + O(ǫ7 ) . 5 5 7 (A.2) 4



The two-loop two-point and three-point master integrals are:     2 1 13 115ǫ 865 65ζ3 5971 2 5ζ3 3 3ζ2 B3,1 = − − − +ǫ − + − +ǫ 4ǫ 8 16 2 32 2 4 64   2 575ζ3 27ζ5 39193 39ζ2 + + − +ǫ4 4 8 2 128   3 2 2 345ζ2 25ζ3 4325ζ3 351ζ5 249355 5 44ζ2 +ǫ + − + + − 7 8 2 16 4 256   1555105 165 3105 29855 325 2 2595 2 286 3 6 2 +ǫ − + ζ7 + ζ5 + ζ3 − ζ + ζ − 15ζ2 ζ3 + ζ 512 2 8 32 4 3 16 2 7 2 B4,2

C4,1

C6,2

+O(ǫ7 ) ,   2 4 1 2 12ζ2 + 16ζ3 − 80 = + 2 + + 12 − ǫ (4ζ3 − 32) − ǫ ǫ ǫ 5   2 3 48ζ2 + 48ζ3 + 12ζ5 − 192 −ǫ 5   3 144ζ22 4 32ζ2 2 −ǫ + − 8ζ3 + 128ζ3 + 48ζ5 − 448 7 5   384 2 48 2 128 3 5 2 +ǫ 1024 − 36ζ7 − 144ζ5 − 320ζ3 + 32ζ3 − ζ + ζ2 ζ3 − ζ 5 2 5 7 2

+O(ǫ6 ) ,     19 5 65 1 + ζ2 + + ǫ 5ζ2 − 4ζ3 + = + 2+ 2ǫ 2ǫ 2 2  2  6ζ2 211 −ǫ2 − 19ζ2 + 20ζ3 − 5 2   665 3 2 −ǫ 6ζ2 + 8ζ2 ζ3 − 65ζ2 + 76ζ3 + 24ζ5 − 2   2 3 114ζ2 2059 528ζ2 + + 40ζ2 ζ3 − 16ζ32 − 211ζ2 + 260ζ3 + 120ζ5 − −ǫ4 35 5 2  6305 +ǫ5 − 156ζ7 − 456ζ5 − 844ζ3 + 80ζ32 + 665ζ2 − 48ζ2 ζ5 − 152ζ2 ζ3 − 78ζ22 2  48 528 3 + ζ22 ζ3 − ζ2 + O(ǫ6 ) , 5 7   3 5ζ2 27ζ3 1 2 2 456ζ2 2 − 23ζ2 + ǫ (48ζ2 ζ3 − 117ζ5 ) − ǫ − 267ζ3 = + 4− 2 − ǫ ǫ ǫ 35

– 29 –

(A.3)

(A.4)

(A.5)

  1962 2 +ǫ3 6ζ7 + 240ζ2 ζ5 + ζ2 ζ3 + O(ǫ4 ) . 5

(A.6)

The Bt,i -type and Ct,i -type master integrals read at three loops: B4,1

    71 3115ǫ 7ζ3 109403 497ζ3 7π 4 3386467 1 2 3 + + +ǫ − + − + +ǫ − = 36ǫ 216 1296 9 7776 54 540 46656   4 21805ζ3 497π 96885467 +ǫ4 − − 7ζ5 − + 324 3240 279936   4 765821ζ3 497ζ5 4361π 4π 6 98ζ32 2631913075 5 +ǫ − − − − + + 1944 6 3888 243 9 1679616

B5,1 =

B5,2 =

B6,1 =

B6,2 =

B8,1

+O(ǫ6 ) ,     17 183 1597 π 4 12359 1 2 51ζ3 − + ǫ 3ζ3 − + − +ǫ − 2− 4ǫ 8ǫ 16 32 2 20 64   4 549ζ3 17π 88629 +ǫ3 + 15ζ5 + − 4 40 128   4 255ζ5 183π 2π 6 603871 4791ζ3 + + + − 18ζ32 − + O(ǫ5 ) , +ǫ4 8 2 80 63 256     1 22ζ3 11π 4 10 64 220ζ3 2 − 2− − + ǫ −112 + + + ǫ −528 + 3ǫ 3ǫ 3 3 3 90   4 11π 1408ζ3 + 70ζ5 + +ǫ3 −2336 + 3 9   4 29824 94π 6 242ζ32 4 352π +ǫ + 2464ζ3 + 700ζ5 − + − + O(ǫ5 ) , 45 3 567 3     1 6 24 π4 + + + 80 − 6ζ3 + ǫ 240 − 36ζ3 − ǫ3 ǫ2 ǫ 10   4 3π +ǫ2 672 − 144ζ3 − 18ζ5 − 5   2π 6 12π 4 2 3 − + 18ζ3 + O(ǫ4 ) , +ǫ 1792 − 480ζ3 − 108ζ5 − 5 63     1 8ζ3 103 7 31 235 56ζ3 2π 4 + + + + + + +ǫ 3ǫ3 3ǫ2 3ǫ 3 3 3 3 45   4 320ζ3 14π 19 + 120ζ5 + + +ǫ2 3 3 45   3953 1832ζ3 16π 4 176π 6 292ζ32 3 +ǫ − + 840ζ5 + + + − + O(ǫ4 ) , 3 3 9 567 3   10π 6 2 = 20ζ5 + ǫ 68ζ3 + 40ζ5 + 189   4 20π 6 34π ζ3 2 2 + 80ζ5 + + 450ζ7 +ǫ 136ζ3 + 15 189 +O(ǫ3 ) ,

– 30 –

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

C6,1

C8,1

   2  1 7π 7 1 π 2 33 131 = 3+ 2+ + − 5ζ3 + + 2ǫ 2ǫ ǫ 6 2 6 2   4 2 π 473 11π − 35ζ3 − + +ǫ 2 20 2   2 2 5π ζ3 7π 4 1611 2 131π +ǫ − − 165ζ3 − 27ζ5 − + 6 3 20 2   2 2 4 35π ζ3 33π 61π 6 5281 3 473π 2 +ǫ − − 655ζ3 − 189ζ5 − − + 25ζ3 + 6 3 20 756 2

+O(ǫ4 ) , (A.13)     2 2 2 1 1 5π 5π 1 +4 + 2 8− − 29ζ3 = 5+ 4+ 3 − ǫ ǫ ǫ 6 ǫ 3   1 121π 4 10π 2 + 16 − − 58ζ3 − ǫ 3 180   2 2 29π ζ3 121π 4 20π + − 116ζ3 − 123ζ5 − + 32 − 3 3 90   2 2 58π ζ3 40π 121π 4 163π 6 2 +ǫ − 232ζ3 − 246ζ5 − + 323ζ3 − + 64 − 3 3 45 3780 +O(ǫ2 ) .

(A.14)

The genuine three-loop vertex integrals are:     233 π 2 2363 19π 2 19 11ζ3 1 + + + + − +ǫ A5,1 = 24ǫ2 48ǫ 96 24 192 48 12   2 4 7227 233π π 209ζ3 +ǫ2 + + − 128 96 80 24   2 4 2363π 19π 2563ζ3 11π 2 ζ3 35ζ5 3 62641 + + − − − +ǫ 256 192 160 48 12 4  2 4 6 1575481 7227π 233π 919π 25993ζ3 +ǫ4 + + − − 1536 128 320 45360 96  2 2 209π ζ3 121ζ3 665ζ5 − + − 24 12 8 A5,2

A6,1

+O(ǫ5 ) , (A.15)     2 2 5 32 π 11ζ3 5π 1 + − − + + ǫ −56 − =− 2− 6ǫ 3ǫ 3 12 6 3   2 4 16π 19π 110ζ3 +ǫ2 −264 − + + 3 720 3   4 704ζ3 11π 2 ζ3 19π 3 2 + + + 35ζ5 +ǫ −1168 − 28π + 72 3 6   76π 4 9011π 6 55π 2 ζ3 121ζ32 14912 +ǫ4 − − 132π 2 + + + 1232ζ3 + − + 350ζ5 3 45 90720 3 3 +O(ǫ5 ) ,    2  208 16ζ3 8 1 44 π 2 8π 1 + + − + = 3+ 2+ 3ǫ 3ǫ ǫ 3 3 3 3 3

– 31 –

(A.16)

  44π 2 2π 4 128ζ3 +ǫ 304 + + − 3 15 3   4 2 16π 704ζ3 16π 2 ζ3 208π + − − − 56ζ5 +ǫ2 1280 + 3 15 3 3   4 6 88π 55π 3328ζ3 128π 2 ζ3 128ζ32 3 15808 2 +ǫ + 304π + − − − + − 448ζ5 3 15 567 3 3 3

A6,2

A6,3

A7,1

A7,2

A7,3

+O(ǫ4 ) , (A.17)  4  2ζ3 7π = + + 18ζ3 ǫ 180   4 2π 2 ζ3 7π + 122ζ3 − + 10ζ5 +ǫ 20 3   4 163π 6 2 427π 2 2 +ǫ (A.18) − + 738ζ3 − 6π ζ3 − 76ζ3 + 90ζ5 + O(ǫ3 ) , 180 7560    2  95 17ζ3 3 1 55 π 2 3π 1 + + − + = 3+ 2+ 6ǫ 2ǫ ǫ 6 6 2 2 3   2 4 1351 55π π +ǫ + + − 51ζ3 6 6 90   2023 95π 2 π 4 935ζ3 10π 2 ζ3 +ǫ2 + + − − − 65ζ5 2 2 10 3 3   2 4 6 1351π 11π 7π 268ζ32 2 3 26335 + + − − 1615ζ3 − 30π ζ3 + − 585ζ5 +ǫ 6 6 18 54 3 +O(ǫ4 ) ,     1 1 1 π2 π2 1 = 5 + 4 + 3 1− − 10ζ3 + 2 2− 4ǫ 2ǫ ǫ 6 ǫ 3   2π 2 11π 4 1 − − 20ζ3 + 4− ǫ 3 45   22π 4 4π 2 14π 2 ζ3 + − − + + 8 − 40ζ3 − 88ζ5 45 3 3   8π 2 44π 4 943π 6 28π 2 ζ3 2 +ǫ 16 − − − − 80ζ3 + + 196ζ3 − 176ζ5 3 45 7560 3

+O(ǫ2 ) ,     1 π2 1 π 2 83π 4 π2 + + 2ζ3 + + + 4ζ3 = 12ǫ3 ǫ2 6 ǫ 3 720  2  2π 83π 4 5π 2 ζ3 + + + 8ζ3 − + 15ζ5 3 360 3   2 83π 4 2741π 6 10π 2 ζ3 4π 2 + + + 16ζ3 − − 73ζ3 + 30ζ5 +ǫ 3 180 90720 3 +O(ǫ2 ) ,  2    1 31ζ32 π ζ3 119π 6 =+ − − 10ζ5 + − − + O(ǫ) , ǫ 6 2160 2

– 32 –

(A.19)

(A.20)

(A.21) (A.22)

A7,4

A7,5 A8,1

    6ζ3 1 11π 4 11π 4 2 = 2 + + 36ζ3 + 216ζ3 − 2π ζ3 + + 46ζ5 ǫ ǫ 90 15   19π 6 22π 4 − + 1296ζ3 − 12π 2 ζ3 − 282ζ32 + 276ζ5 + O(ǫ2 ) , +ǫ 5 270     11π 6 2 2 2 = + 2π ζ3 + 10ζ5 + ǫ + 12π ζ3 + 18ζ3 + 60ζ5 + O(ǫ2 ) , 162   4   8ζ3 1 5π 52π 2 ζ3 352ζ5 5π 4 =− 2+ + 8ζ3 + − 24ζ3 + − − 3ǫ ǫ 27 9 9 3   6 2 2 4 1709π 52π ζ3 332ζ3 5π − + 72ζ3 − + + 352ζ5 + O(ǫ2 ) . +ǫ − 3 8505 3 3

(A.23) (A.24)

(A.25)

The most complicated three-loop vertex integrals are the nine-line master integrals [41]:   1 1 53 4π 2 1 + + − − A9,1 = − 18ǫ5 2ǫ4 ǫ3 18 27   2 1 29 22π + 2 + − 2ζ3 ǫ 2 27   1 129 8π 2 158ζ3 20π 4 + − + − − ǫ 3 9 81 2   2 4 14π ζ3 537 578ζ3 238ζ5 322π 2 + 6π − + − − + 405 3 2 9 3   4 6 2133 302π 2398π 26π 2 ζ3 466ζ32 826ζ5 2 +ǫ − − 4π − − + 158ζ3 − − + 2 135 5103 3 3 3 A9,2

A9,4

+O(ǫ2 ) , (A.26)   2 5 1 20 7π 2 = 6+ 5+ 4 − − 9ǫ 6ǫ ǫ 9 27   2 1 50 17π 91ζ3 + 3 − − ǫ 9 27 9  2  1 4π 166ζ3 373π 4 110 + 2 − − − ǫ 3 9 1080 9   2 16π 2 187π 4 170 1 494ζ3 179π ζ3 + − 167ζ5 − − + + ǫ 9 27 9 540 9   2 4 2 130 32π 1466ζ3 679π 682π ζ3 1390ζ5 59797π 6 169ζ32 + − − + + − − + 9 9 9 540 27 3 136080 9 +O(ǫ) , (A.27)   2 1 8 1 10π = 6 + 5 + 4 −1 − 9ǫ 9ǫ ǫ 27   2 14 47π 1 − 12ζ3 + 3 − − ǫ 9 27   1 47π 4 71π 2 200ζ3 + 2 17 + − − ǫ 27 3 810   671π 4 652π 2 ζ3 692ζ5 940ζ3 1 2 − + − + −84 − π + ǫ 9 540 27 9

– 33 –

  448ζ3 2689π 4 2188π 2 ζ3 53563π 6 4564ζ32 2 + 339 − 15π − + + − 524ζ5 + + 9 1620 27 102060 9 +O(ǫ) , (A.28) where the analytic expressions for A9,1 and for the pole parts of A9,2 and A9,4 were obtained independently in [40]. For the corresponding integrals with homogeneous transcendentality, one finds: π2 14ζ3 47π 4 1 + + + 6 4 3 2 36ǫ 18ǫ  9ǫ 405ǫ 85 2 137 2 1 1160π 6 + π ζ3 + 20ζ5 + + ζ + O(ǫ) (A.29) 27 ǫ 5103 3 3 2 7π 2 91ζ3 373π 4 = 6− − − 9ǫ 27ǫ4  9ǫ3 1080ǫ2  59797 6 169 2 1 179 2 π ζ3 − 167ζ5 − π + ζ + O(ǫ) (A.30) + 27 ǫ 136080 9 3 1 10π 2 12ζ3 47π 4 =− 6+ + + 9ǫ 27ǫ4 ǫ3 810ǫ2  1 692 53563 6 4564 2 652 2 π ζ3 + ζ5 − π − ζ + O(ǫ) (A.31) + − 27 9 ǫ 102060 9 3

A9,1n =

A9,2n

A9,4n

B. Form factors in terms of master integrals The unrenormalised three-loop form factors can be expressed as a linear combination of master integrals. In the colour factor decomposition as defined in (5.1) and (5.2), these coefficients read: XCq 3 = F

43304589D 2 615952127D 34015 109222498 489406D 3 − + + − 625 3125 7500 4(2D − 7) 75(2D − 9) 6816654 89728 12581 6489724 50720 + + − + + 9(3D − 10) 11(3D − 14) 25(D − 2) 12(D − 3) 15(D − 4) 19326056092 7186019918 643118017984 1024 779 + − + − − 2 7734375(5D − 16) 78125(5D − 18) 703125(5D − 22) 3(D − 2) 12(D − 3)2  1187096 745376 91648 53258146831 884312 + + + − + 2 3 4 5 5(D − 4) 15(D − 4) 15(D − 4) 5(D − 4) 562500  2 3 16060301D 135964099D 7315 59657807 54568D −A5,1 + − + − − 625 9375 11250 2(2D − 7) 1300(2D − 9) 142784 106 770008 3481535536 36208 + − + + − 27(3D − 10) 75(D − 2) 3(D − 3) 15(D − 4) 3046875(5D − 16) 2887120096 32265012416 83104 12800 26112 + − + + + 2 3 78125(5D − 18) 234375(5D − 22) 3(D − 4) (D − 4) 5(D − 4)4  35332079719 − 1687500 

−B4,1 +

– 34 –

87316D 3 8532244D 2 15436454D 418 3273151 − + + + 1875 9375 3125 (2D − 7) 325(2D − 9) 2913100 400 152056 70952 9920 + + + − + 27(3D − 10) 99(3D − 14) 27(3D − 8) 225(D − 2) 15(D − 4) 368980436 37325556352 512 97088 9365062376 − − − − − 2 33515625(5D − 16) 78125(5D − 18) 703125(5D − 22) 3(D − 2) 15(D − 4)2  22016 3578943149 19392 − − − 5(D − 4)3 15(D − 4)4 421875  3 2 46827D 7169631D 221676243D 177975 2274503 −B5,1 + − + + − 5000 50000 100000 64(2D − 7) 130(2D − 9) 151 44476 50689072 648026848 9728 − + − + + 15(D − 2) 2(D − 3) 5(D − 4) 3046875(5D − 16) 78125(5D − 18)  53792 33824 19072 3774996391 1055401984 + + + − + 78125(5D − 22) 5(D − 4)2 5(D − 4)3 5(D − 4)4 1000000  3 2 105167D 33225224D 792891607D 1654184 4912 +B5,2 + − + − − 1875 28125 84375 65(2D − 9) 81(3D − 10) 29696 150868 500415488 470084528 3580901184 + + + + + 15(D − 2) 15(D − 4) 3046875(5D − 16) 78125(5D − 18) 78125(5D − 22)  29936 22112 46592 21148347004 + + + − 3(D − 4)2 3(D − 4)3 15(D − 4)4 1265625  2 3 7837D 62616143D 291310 2252 2207D − + + + −A6,1 + 375 50 56250 297(3D − 14) 9(D − 2) 3136 1553908 5323758 74762464 23 + + − − + (D − 3) 15(D − 4) 61875(5D − 16) 15625(5D − 18) 46875(5D − 22)  192 183504334 496 + − + 5(D − 4)2 5(D − 4)3 84375  3 2 39857D 4628009D 22107268D 627 235409 +A6,2 + − + − + 3000 15000 9375 16(2D − 7) 600(2D − 9) 4993 1024 36333448 446887648 98768 − + + + + 225(D − 2) 280(D − 3) 15(D − 4) 140625(5D − 16) 984375(5D − 22)  58 256 817543919 1024 + + − + 3(D − 2)2 3(D − 3)2 15(D − 4)2 150000  3 2 3422D 2017249D 835107683D 1045 4880379 −A6,3 + − + + + 125 3750 225000 8(2D − 7) 5200(2D − 9) 80 28736 161 4336 30753004 + + − + + 27(3D − 8) 25(D − 2) 4(D − 3) 5(D − 4) 203125(5D − 16)  1526704 85442816 576 3392 10891722217 − − + + − 3125(5D − 18) 15625(5D − 22) (D − 4)2 15(D − 4)3 1350000 2 3 (D − 7D + 16) +B6,1 (D − 4)3  7D 3 1109D 2 29395D 8475 200 −B6,2 + − + + + 8 48 288 64(2D − 7) 27(3D − 8) 

+A5,2 +

– 35 –

 264 152 160 374753 − − − − (D − 4) (D − 4)2 (D − 4)3 1728  2 3 1109D 29395D 8475 200 7D − + + + −C6,1 + 8 48 288 64(2D − 7) 27(3D − 8)  264 152 160 374753 − − − − (D − 4) (D − 4)2 (D − 4)3 1728  3 2 21D 5907D 523857D 1213 29539 −A7,1 + − + − + 50 500 5000 12(2D − 7) 624(2D − 9)  64 80 655856 388064 2151447 + + − − − 3(D − 2) (D − 4) 121875(5D − 16) 9375(5D − 18) 10000  2 3 733D 228267D 42745 232399 15D − + + + +A7,2 + 16 32 1600 288(2D − 7) 8320(2D − 9)  488 128 2821088 47928 4633049 + − − + − 45(D − 2) (D − 4) 14625(5D − 16) 125(5D − 18) 16000  2 3 60199D 760189D 1213 29539 601D +A7,3 + − + − + 1250 6250 6250 12(2D − 7) 2340(2D − 9) 8329 909167104 101225984 15661504 496 + − + − + (D − 2) 210(D − 3) 3046875(5D − 16) 703125(5D − 18) 546875(5D − 22)  8803773 21 − + (D − 3)2 15625  3 686707D 2 7042751D 865 235409 2489D − + + + −A7,4 + 5000 50000 60000 72(2D − 7) 2880(2D − 9) 556 4397 16 93131696 9430916 + − − + − 45(D − 2) 420(D − 3) 5(D − 4) 703125(5D − 16) 703125(5D − 18)  365471216 1 277480707 − + − 4921875(5D − 22) 3(D − 3)2 1000000  3 2 2489D 686707D 7042751D 865 235409 −A7,5 + − + + + 5000 50000 60000 72(2D − 7) 2880(2D − 9) 4397 16 93131696 9430916 556 − − + − + 45(D − 2) 420(D − 3) 5(D − 4) 703125(5D − 16) 703125(5D − 18)  1 277480707 365471216 + − − 4921875(5D − 22) 3(D − 3)2 1000000  3 2 3411D 758793D 3243781D 33573 32 +A8,1 + − + + + 80000 800000 320000 1024(2D − 7) (D − 2) 4 3411716 8269536 1425936 3015 + + − − − 448(D − 3) 5(D − 4) 78125(5D − 16) 78125(5D − 18) 546875(5D − 22)  663954073 4389 − + 256(2D − 7)2 16000000 3 2 (D − 20D + 104D − 176)(D 2 − 7D + 16) −B8,1 8(2D − 7)(D − 4) 3 2 (D − 20D + 104D − 176)(D 2 − 7D + 16) −C8,1 8(2D − 7)(D − 4)

– 36 –

243D 3 14661D 2 257769D 256 225 − + + − 1250 3125 6250 5(D − 2) 14(D − 3)  4083992 6463488 2127008 4086513 48 + + + − − 5(D − 4) 78125(5D − 16) 78125(5D − 18) 546875(5D − 22) 31250 6 5 4 3 2 3(3D − 14)(D − 41D + 661D − 4992D + 19276D − 37104D + 28288) −A9,2 10(5D − 16)(5D − 18)(5D − 22)(D − 3)  3 125091D 2 1808937D 4067 232399 567D − + + + −A9,4 + 80000 800000 1600000 2304(2D − 7) 998400(2D − 9) 16 225 8388688 574016 7557808 − − + − − 75(D − 2) 448(D − 3) 3046875(5D − 16) 234375(5D − 18) 4921875(5D − 22)  38866491 − 16000000 

+A9,1 +

XCq 2 C = F A  225717D 3 9995657D 2 893831341D 45685 990312631 +B4,1 + − + + − 250 625 9375 24(2D − 7) 975(2D − 9) 116080 707967 371482 125032 875 + + − + − 21(3D − 10) (3D − 14) 6237(D − 1) 25(D − 2) 4(D − 3) 5622111978 345702357 5032168544 1280 172908907 − − − − + 405(D − 4) 2234375(5D − 16) 3125(5D − 18) 46875(5D − 22) 3(D − 2)2 22959056 3875224 2622656 340096 9 + + + + + 4(D − 3)2 135(D − 4)2 45(D − 4)3 45(D − 4)4 15(D − 4)5  2728978211 − 25000  12785517D 2 156484099D 17815 97578701 71304D 3 +A5,1 + − + − − 625 6250 12500 3(2D − 7) 2600(2D − 9) 2467480 108736 1 6422264 73232 + + − + − 63(3D − 10) 27027(D − 1) 75(D − 2) (D − 3) 135(D − 4) 9290021216 26729196704 10782693376 179840 130816 + + − + + 2 11171875(5D − 16) 1015625(5D − 18) 78125(5D − 22) 9(D − 4) 15(D − 4)3  23118311953 50176 − + 4 15(D − 4) 1125000  3 24183202D 2 437583827D 2036 7969573 25782D −A5,2 + − + + + 625 28125 84375 3(2D − 7) 650(2D − 9) 100850 16 2178524 71828 190048 + + − + + 567(3D − 10) 3(3D − 14) (3D − 8) 81081(D − 1) 75(D − 2) 1092208 9583273136 4168502038 14910736064 640 − − − − − 405(D − 4) 33515625(5D − 16) 1015625(5D − 18) 234375(5D − 22) 3(D − 2)2  90176 35072 21872512759 523712 − − − − 135(D − 4)2 45(D − 4)3 45(D − 4)4 2531250

– 37 –

114279D 3 22958287D 2 448177891D 177975 326249 − + + − 10000 100000 200000 128(2D − 7) 26(2D − 9) 2 35294 25589872 831628448 6128 − + − + + 15(D − 2) (D − 3) 5(D − 4) 3046875(5D − 16) 78125(5D − 18)  7272 17696 10624 7313613927 130364416 + + + − − 78125(5D − 22) (D − 4)2 5(D − 4)3 5(D − 4)4 2000000  2 3 4405736D 287753909D 237272 16 44339D − + − + −B5,2 + 625 3125 28125 13(2D − 9) 27(3D − 10) 25456 154624 76140512 650239528 1665432384 + + − + + 15(D − 2) 15(D − 4) 3046875(5D − 16) 78125(5D − 18) 78125(5D − 22)  47624 32416 41984 7361965928 + + + − 5(D − 4)2 5(D − 4)3 15(D − 4)4 421875  2 3 609667D 25888633D 10085 545 4678D − + + + +A6,1 + 625 3125 18750 9(3D − 14) 54(D − 1) 18 31492 5042024 23129849 5366 + + + − + 15(D − 2) (D − 3) 135(D − 4) 78125(5D − 16) 78125(5D − 18)  1264 512 777739429 142238992 + + − − 78125(5D − 22) 9(D − 4)2 15(D − 4)3 281250  3 2 9671D 1150061D 27382277D 509 5481191 −A6,2 + − + − + 1200 6000 18000 8(2D − 7) 15600(2D − 9) 728 257261 3164 92725544 20493136 + − + + + 25(D − 2) 1680(D − 3) 15(D − 4) 121875(5D − 16) 65625(5D − 22)  1 3952 379243601 1280 + + − + 3(D − 2)2 2(D − 3)2 45(D − 4)2 112500  2 3 33322501D 436696447D 2545 14641137 60679D − + + + +A6,3 + 2500 75000 150000 12(2D − 7) 10400(2D − 9) 16 250462 37352 1 106784 + − + − + 5(3D − 8) 19305(D − 1) 75(D − 2) 4(D − 3) 135(D − 4) 433927224 487676544 27568 640 3533209912 − − + + + 33515625(5D − 16) 1015625(5D − 18) 78125(5D − 22) 45(D − 4)2 3(D − 4)3  2875843347 − 500000  D 3 71D 2 283D 8475 8 −B6,2 + + − − − 16 32 64 128(2D − 7) (3D − 8)  1000 80 64 1587 46 + + + + − 9(D − 1) 9(D − 4) 3(D − 4)2 (D − 4)3 128  3 2 D 71D 283D 8475 8 −C6,1 + + − − − 16 32 64 128(2D − 7) (3D − 8)  46 1000 80 64 1587 − + + + + 9(D − 1) 9(D − 4) 3(D − 4)2 (D − 4)3 128  2 3 7009D 584559D 301 21185 27D − + − + +A7,1 + 50 500 5000 6(2D − 7) 624(2D − 9) 

+B5,1 +

– 38 –

 32 80 6069872 827368 2492249 + + − − − (D − 2) (D − 4) 121875(5D − 16) 9375(5D − 18) 10000  2 3 142459D 9023129D 42745 697197 517D − + + + −A7,2 + 800 8000 80000 192(2D − 7) 16640(2D − 9) 1124 76 112 106444064 9644612 − − − − + 143(D − 1) 15(D − 2) (D − 4) 446875(5D − 16) 40625(5D − 18)  33226167 − 160000  152417D 301 4237 601D 3 29899D 2 − + − + −A7,3 + 1250 6250 3125 6(2D − 7) 468(2D − 9) 3683 10424832 54045184 12403904 544 + + + − + 3(D − 2) 210(D − 3) 1015625(5D − 16) 703125(5D − 18) 546875(5D − 22)  15077947 3 − − (D − 3)2 62500  3 19D 458683D 2 40603349D 235 5481191 +A7,4 + − + − + 80 60000 600000 32(2D − 7) 74880(2D − 9) 118 26393 24 46026288 501158 + − − + − 15(D − 2) 840(D − 3) 5(D − 4) 203125(5D − 16) 140625(5D − 18)  21760904 62067409 − − 328125(5D − 22) 400000  2 3 458683D 40603349D 235 5481191 19D − + − + +A7,5 + 80 60000 600000 32(2D − 7) 74880(2D − 9) 118 26393 24 46026288 501158 + − − + − 15(D − 2) 840(D − 3) 5(D − 4) 203125(5D − 16) 140625(5D − 18)  62067409 21760904 − − 328125(5D − 22) 400000  3 979611D 2 8338443D 29027 160 1197D − + + + −A8,1 + 160000 1600000 640000 2048(2D − 7) 3(D − 2) 34 23109548 3147936 1018736 13665 − + − − − 896(D − 3) 5(D − 4) 234375(5D − 16) 78125(5D − 18) 546875(5D − 22)  2065843091 3563 − + 128(2D − 7)2 32000000 3 2 (D − 20D + 104D − 176)(D 2 − 7D + 16) +B8,1 16(2D − 7)(D − 4) 3 2 (D − 20D + 104D − 176)(D 2 − 7D + 16) +C8,1 16(2D − 7)(D − 4)  3 14661D 2 257769D 256 225 243D −A9,1 + − + + − 1250 3125 6250 5(D − 2) 14(D − 3)  48 4083992 6463488 2127008 4086513 − + + + − 5(D − 4) 78125(5D − 16) 78125(5D − 18) 546875(5D − 22) 31250 6 5 4 3 2 (3D − 14)(3D − 108D + 1586D − 11304D + 41928D − 78208D + 57984) +A9,2 10(5D − 16)(5D − 18)(5D − 22)(D − 3)

– 39 –

1701D 3 375273D 2 5426811D 4067 232399 − + + + 160000 1600000 3200000 1536(2D − 7) 665600(2D − 9) 675 4194344 287008 3778904 8 − + − − − 25(D − 2) 896(D − 3) 1015625(5D − 16) 78125(5D − 18) 1640625(5D − 22)  116599473 − 32000000 

+A9,4 +

XCq

2 F CA

= 

45111262D 2 3307905503D 7045 140183197 153701D 3 − + − − 625 9375 112500 6(2D − 7) 975(2D − 9) 2060 9383166 165455 44164 659 − + − + + 27(3D − 10) 55(3D − 14) 2673(D − 1) 25(D − 2) 12(D − 3) 138263099401 5093619454 144904142656 128 416301857 − − − − + 4860(D − 4) 402187500(5D − 16) 234375(5D − 18) 703125(5D − 22) (D − 2)2 8793673 915068 345128 20864 143 + + + + + 12(D − 3)2 405(D − 4)2 135(D − 4)3 45(D − 4)4 5(D − 4)5  31855488829 − 843750  4277D 3 243647D 2 106547887D 49315 18960447 −A5,1 + − + − − 125 375 28125 24(2D − 7) 2600(2D − 9) 357584 7618840 10064 10 3703898 + + + + + 1323(3D − 10) 567567(D − 1) 25(D − 2) 3(D − 3) 405(D − 4) 563608248 1415791832 3488 81212 332977894 + − + + + 2 1340625(5D − 16) 203125(5D − 18) 46875(5D − 22) 189(D − 1) 45(D − 4)2  256 452116231 11584 + − + 45(D − 4)3 3(D − 4)4 67500  3 2 20594D 6630308D 25700042D 1409 2348211 +A5,2 + − + + + 1875 28125 16875 6(2D − 7) 650(2D − 9) 11944 801980 10360820 75128 559813 − + − + − 567(3D − 10) 99(3D − 14) 243243(D − 1) 225(D − 2) 1215(D − 4) 1052157372 11705152928 64 306428 1022764469 − − − − − 33515625(5D − 16) 1015625(5D − 18) 703125(5D − 22) (D − 2)2 405(D − 4)2  1408 3344023858 43696 − − − 135(D − 4)3 15(D − 4)4 1265625  3 2 7497D 394839D 10983001D 75999 1424 −B5,1 + − + − + 1250 3125 12500 40(2D − 9) 15(D − 2) 1 1055 712201 200430972 146129984 − + − + − (D − 3) (D − 4) 234375(5D − 16) 78125(5D − 18) 78125(5D − 22)  4176 296 1152 189456643 + + + − 5(D − 4)2 (D − 4)3 5(D − 4)4 125000  2 3 4406452D 92206756D 13818 320 13666D − + − + +B5,2 + 625 9375 28125 5(2D − 9) 27(3D − 10) −B4,1 +

– 40 –

10325 3748279 169244232 122152296 2536 + − + + 5(D − 2) 6(D − 4) 156250(5D − 16) 78125(5D − 18) 78125(5D − 22)  15736 7552 2504444962 22774 + + − + 15(D − 4)2 15(D − 4)3 15(D − 4)4 421875  3 2 17033D 1493603D 95481487D 80198 1711 −A6,1 + − + + + 7500 25000 225000 297(3D − 14) 288(D − 1) 1090 113 479 294544271 3011532 + + + + − 9(D − 2) 32(D − 3) 18(D − 4) 15468750(5D − 16) 78125(5D − 18)  104077288 1853 136 64 1505097247 − + + − − 234375(5D − 22) 144(D − 1)2 9(D − 4)2 15(D − 4)3 1687500  3 2 4249D 322949D 61126331D 1409 403479 +A6,2 + − + − + 6000 15000 300000 64(2D − 7) 5200(2D − 9) 5584 117431 9704 5192329489 872 + − + + − 297(D − 1) 225(D − 2) 1680(D − 3) 135(D − 4) 20109375(5D − 16)  128 25 176 846754451 41976608 + − + − + 984375(5D − 22) (D − 2)2 12(D − 3)2 5(D − 4)2 1800000  2 3 831416D 41933917D 7045 4880379 3073D − + + + −A6,3 + 625 9375 75000 96(2D − 7) 10400(2D − 9) 9004 13 293 1849361647 13867 + + + + − 429(D − 1) 75(D − 2) 4(D − 3) 30(D − 4) 67031250(5D − 16)  118945472 236 96 51691069 49983292 − + + − − 1015625(5D − 18) 78125(5D − 22) 5(D − 4)2 5(D − 4)3 46875  3 2 33D 8111D 645261D 3 329 −A7,1 + − + + + 200 2000 20000 16(2D − 7) 64(2D − 9)  32 20 220844 105556 2833051 + + − − − 3(D − 2) (D − 4) 9375(5D − 16) 3125(5D − 18) 40000  2 3 25417D 1658227D 42745 232399 71D − + + + +A7,2 + 800 8000 80000 576(2D − 7) 16640(2D − 9) 236 24 285048488 928156 562 − − − + − 143(D − 1) 45(D − 2) (D − 4) 4021875(5D − 16) 40625(5D − 18)  5030461 − 160000  3D 3 2023D 2 41819D 3 329 −A7,3 + − − − − 625 12500 12500 16(2D − 7) 240(2D − 9) 44 263 2204608 33404 2035336 − − − − + 3(D − 2) 105(D − 3) 234375(5D − 16) 78125(5D − 18) 546875(5D − 22)  7 421802 + + 4(D − 3)2 15625  3 57D 116647D 2 2694797D 3845 134493 +A7,4 + + − + − 10000 300000 600000 576(2D − 7) 8320(2D − 9) 1833 8 732913468 368278 38 + + − − − 45(D − 2) 140(D − 3) 5(D − 4) 9140625(5D − 16) 234375(5D − 18) +

– 41 –

 71838976 1 16428169 + + + 4921875(5D − 22) 12(D − 3)2 2000000  2 3 116647D 2694797D 3845 134493 57D + − + − +A7,5 + 10000 300000 600000 576(2D − 7) 8320(2D − 9) 38 1833 8 732913468 368278 − + + − − 45(D − 2) 140(D − 3) 5(D − 4) 9140625(5D − 16) 234375(5D − 18)  1 16428169 71838976 + + + 4921875(5D − 22) 12(D − 3)2 2000000  2 3 110409D 2547331D 2273 56 1107D + − + − −A8,1 + 160000 1600000 640000 2048(2D − 7) 3(D − 2) 18 8995987 493416 152884 5325 + − − + + 896(D − 3) 5(D − 4) 234375(5D − 16) 78125(5D − 18) 546875(5D − 22)  700944509 9863 + − 1024(2D − 7)2 32000000  3 243D 62289D 2 283527D 88 285 +A9,1 + − + + − 5000 50000 25000 5(D − 2) 56(D − 3)  2 1549163 1250592 406132 4692843 − + + + − (D − 4) 78125(5D − 16) 78125(5D − 18) 546875(5D − 22) 125000 6 5 4 3 2 (3D − 14)(3D − 93D + 1189D − 7632D + 26028D − 45104D + 31104) −A9,2 40(5D − 16)(5D − 18)(5D − 22)(D − 3)  3 125091D 2 1808937D 4067 232399 567D −A9,4 + − + + + 160000 1600000 3200000 4608(2D − 7) 1996800(2D − 9) 225 4194344 287008 3778904 8 − + − − − 75(D − 2) 896(D − 3) 3046875(5D − 16) 234375(5D − 18) 4921875(5D − 22)  38866491 − 32000000

XCq 2 N = F F  37120 742964 4 72D 2 11524D − − − + +B4,1 + 5 75 63(3D − 10) 6237(D − 1) (D − 3) 10576 319872 3088 1024 256 + + − + + 2 3 81(D − 4) 1375(5D − 16) 27(D − 4) 9(D − 4) 3(D − 4)4  412948 + 1125  512 2293120 19648 1216D 2 83936D − − − − −A5,1 + 75 375 3(3D − 10) 11583(D − 1) 81(D − 4)  297024 3698688 6272 1024 983968 + + − − + 6875(5D − 16) 8125(5D − 18) 27(D − 4)2 9(D − 4)3 1875  2 78712D 6656 32 4357048 4D − − − − −A5,2 + 75 1125 189(3D − 10) 9(3D − 8) 81081(D − 1)

– 42 –

131376 1585152 5312 1024 20464 + + − − 2 81(D − 4) 6875(5D − 16) 8125(5D − 18) 27(D − 4) 9(D − 4)3  2009608 + 16875 (D 2 − 7D + 16)(6D 3 − 65D 2 + 238D − 288)(D − 2) −A6,1 2(D − 3)(D − 1)(D − 4)2  32 500924 400 28D 2 2868D +A6,3 + − − − − 25 125 45(3D − 8) 19305(D − 1) 27(D − 4)  39984 176128 128 334516 − − − + 6875(5D − 16) 8125(5D − 18) 9(D − 4)2 5625 2 3 2 (D − 7D + 16)(3D − 31D + 110D − 128)(D − 2) −B6,2 (D − 1)(3D − 8)(D − 4)2 (D 2 − 7D + 16)(3D 3 − 31D 2 + 110D − 128)(D − 2) −C6,1 (D − 1)(3D − 8)(D − 4)2 8(D − 2)(D 4 − 28D 3 + 220D 2 − 696D + 784) −A7,2 (5D − 18)(D − 1)(5D − 16) −

XCq F CA NF =  250 330910 170 24D 2 396D +B4,1 + − − + − 5 25 3(3D − 10) 2673(D − 1) 3(D − 2) 92064 32 14624 3904 34408 + + + + + 2 2 243(D − 4) 1375(5D − 16) 3(D − 2) 81(D − 4) 27(D − 4)3  4364 256 + + 4 9(D − 4) 125  2 3656D 25352 4990592 184 208D − + − + +A5,1 + 75 125 441(3D − 10) 63063(D − 1) 3(D − 2) 992 148512 1849344 6976 16 − + + − − 2 9(D − 4) 6875(5D − 16) 8125(5D − 18) 189(D − 1) (D − 2)2  512 205952 256 − + + 27(D − 4)2 9(D − 4)3 5625  2 84D 6984D 460 20721640 20 +A5,2 + − + − + 25 125 63(3D − 10) 243243(D − 1) (D − 2) 24312 792576 16 5024 24568 − + − − − 243(D − 4) 6875(5D − 16) 8125(5D − 18) 3(D − 2)2 81(D − 4)2  700768 1024 + − 27(D − 4)3 5625  2 47D 3073 86 11 3D − − + + +A6,1 + 2 4 72(D − 1) 3(D − 2) 8(D − 3)  8 109 8 16 133 + − − + + 9(D − 4) 4(D − 1)2 (D − 2)2 3(D − 4)2 4

– 43 –

90D 7 − 1803D 6 + 15301D 5 − 70848D 4 + 191676D 3 − 299024D 2 + 242976D − 74880 9(5D − 16)(D − 3)(D − 2)2 (D − 1)(D − 4) 42D 7 − 656D 6 + 3854D 5 − 11430D 4 + 24896D 3 − 65144D 2 + 134560D − 113856 −A6,3 3(D − 1)(D − 2)2 (5D − 18)(5D − 16) 4(D − 2)(D 4 − 28D 3 + 220D 2 − 696D + 784) +A7,2 (5D − 18)(D − 1)(5D − 16)

+A6,2

XCq

2 F NF

+A6,1

= (6D 3 − 65D 2 + 238D − 288)(D − 2)2 6(D − 4)(D − 3)(D − 1)2

XCq F NF,V =  380 50245888 280 9088443 119132D +B4,1 + + − + + 625 9(2D − 7) 975(2D − 9) 3(3D − 10) 242(3D − 14) 4169543 182 1194157 57818921783 9269061200 − + + − + 722007(D − 1) 675(D − 2) 3(D − 3) 54(D − 4) 265443750(5D − 16) 221659776 15748 35 750554 808885693 + + + + + 2 2 243750(5D − 18) 53125(5D − 22) 9(D − 2) 3(D − 3) 45(D − 4)2  25148 2048 12730856 608 + + + − 3 3 4 3(D − 2) 3(D − 4) (D − 4) 9375  665 448624 56 92219200 12243D +A5,1 + + − − + 625 9(2D − 7) 325(2D − 9) 3(3D − 10) 65637(D − 1) 83479 10 148888 14475728 29074423 − + + − − 225(D − 2) 3(D − 3) 135(D − 4) 446875(5D − 16) 40625(5D − 18) 1496 14608 256 2048 292095986 − − + − − 53125(5D − 22) 3(D − 2)2 45(D − 4)2 (D − 2)3 15(D − 4)3  2436926 − 9375  14644D 76 40784 56 215775 −A5,2 + − + + + 1875 9(2D − 7) 325(2D − 9) 9(3D − 10) 121(3D − 14) 1082473 34666 176454986 831538 4075274240 − + − − + 2166021(D − 1) 675(D − 2) 405(D − 4) 132721875(5D − 16) 40625(5D − 18) 8024 17912 640 2176 99059576 + + − + − 2 2 3 53125(5D − 22) 9(D − 2) 135(D − 4) 3(D − 2) 45(D − 4)3  2370412 + 28125  196273 25123840 144542 2 3474D +B5,1 + − − + − 625 325(2D − 9) 65637(D − 1) 225(D − 2) (D − 3) 1437614 9370564 39796848 1984 93338 + − − − + 135(D − 4) 4021875(5D − 16) 40625(5D − 18) 53125(5D − 22) 3(D − 2)2

– 44 –

 2900 224 1664 190269 + + + + 9(D − 4)2 (D − 2)3 15(D − 4)3 3125  285488 80 28564480 163559 37136D −B5,2 + − + + + 1875 325(2D − 9) 9(3D − 10) 65637(D − 1) 450(D − 2) 150403 25678283 13477879 10007466 2972 + − − + − 135(D − 4) 4021875(5D − 16) 81250(5D − 18) 53125(5D − 22) 3(D − 2)2  480 3328 4577488 32612 + + + + 45(D − 4)2 (D − 2)3 15(D − 4)3 28125  14385 58221088 931 1 49D −A6,1 + − − + + 625 242(3D − 14) 722007(D − 1) 18(D − 2) 2(D − 3) 3759722 432134 3537842 16 418 − − + − + 135(D − 4) 3403125(5D − 16) 40625(5D − 18) 53125(5D − 22) (D − 2)2  100399 64 + − 45(D − 4)2 18750  19 10196 1382570 411274 17533D + + + − −A6,2 + 3000 24(2D − 7) 975(2D − 9) 1683(D − 1) 675(D − 2) 1088 18324568 1024224 3680 1623 + + + + − 56(D − 3) 45(D − 4) 2413125(5D − 16) 74375(5D − 22) 9(D − 2)2  256 42883 11 − + − 6(D − 3)2 3(D − 2)3 1875  26253D 95 41793584 13967 3 +A6,3 + − + − + 2500 36(2D − 7) 21879(D − 1) 9(D − 2) (D − 3) 359662 883413 12129744 2048 3548 − + − + + 45(D − 4) 103125(5D − 16) 40625(5D − 18) 53125(5D − 22) 3(D − 2)2  64 243387 352 − − + 15(D − 4)2 (D − 2)3 12500  9D 37 2549 280 37 +A7,1 + + + + + 100 18(2D − 7) 1560(2D − 9) 143(D − 1) 90(D − 2)  144076 6313 4 3927 − + + + 160875(5D − 16) 9750(5D − 18) (D − 2)2 1000 5D 6 − 93D 5 + 892D 4 − 4656D 3 + 12528D 2 − 15472D + 6080 −A7,2 (D − 1)(D − 2)2 (5D − 18)(5D − 16)  37 2549 5424 2566 211D + + − + −A7,3 + 625 18(2D − 7) 5850(2D − 9) 187(D − 1) 75(D − 2) 26995456 79688 56582 328 2 + − − − + 3(D − 3) 4021875(5D − 16) 28125(5D − 18) 53125(5D − 22) 3(D − 2)2  128 4576 2 + + − (D − 3)2 (D − 2)3 3125  37 2549 22736 754 162D +A7,4 + − + − + 625 72(2D − 7) 1170(2D − 9) 2431(D − 1) 45(D − 2) 2696828 97442 1877744 8 323 − + − − − 84(D − 3) 4021875(5D − 16) 365625(5D − 18) 1115625(5D − 22) (D − 2)2

– 45 –

 1 273763 − + 12(D − 3)2 75000  37 2549 22736 754 162D +A7,5 + − + − + 625 72(2D − 7) 1170(2D − 9) 2431(D − 1) 45(D − 2) 323 2696828 97442 1877744 8 − − + − − 84(D − 3) 4021875(5D − 16) 365625(5D − 18) 1115625(5D − 22) (D − 2)2  273763 1 + − 12(D − 3)2 75000  1643 8085 349 1305 1107D +A8,1 + + − + + 80000 768(2D − 7) 884(D − 1) 45(D − 2) 448(D − 3) 77484 85352 133 8 22363 − + + − + 28125(5D − 16) 40625(5D − 18) 1115625(5D − 22) 384(2D − 7)2 (D − 2)2  1511991 − 800000  24255 24 75 10076 243D + − − − −A9,1 + 5000 884(D − 1) (D − 2) 56(D − 3) 9375(5D − 16)  83136 252703 16 5103 − + + + 40625(5D − 18) 1115625(5D − 22) (D − 2)2 25000 3(3D − 14)(D − 4)(D 5 + 25D 4 − 290D 3 + 1036D 2 − 1560D + 928) +A9,2 20(D − 1)(D − 2)(D − 3)(5D − 22)(5D − 18)(5D − 16)

XCg 3 = A

1120 223915 11621367296 141 D 2 2592693 D − − − + 2 500 D 126 (2D − 7) 16575 (2D − 9) 26512 17781309 32 9391405657 2004296 + − + − + 21 (3D − 10) 110 (3D − 14) 5 (3D − 8) 1216215 (D − 1) 225 (D − 2) 158773009 360 7497336208 1340410849 280 − + − + + 3 (D − 3) 1215 (D − 4) (D − 6) 30121875 (5D − 14) 4021875 (5D − 16) 2529983456 55039 25736 1787 3469841417 − − − − − 243750 (5D − 18) 3125 (5D − 22) 432 (D − 1)2 5 (D − 2)2 16 (D − 3)2 32740532 832 5426656 89824 512 − + − − + 2 3 3 4 405 (D − 4) (D − 2) 135 (D − 4) 9 (D − 4) 3 (D − 4)5  300276143 + 18750  152 3380 103762208 64048 528664 D −A5,1 + + + − − 375 5 (2D − 5) 9 (2D − 7) 5525 (2D − 9) 2205 (3D − 10) 2189086 8 1487996 1440 1621951952 + + + + − 567567 (D − 1) 225 (D − 2) (D − 3) 81 (D − 4) 7 (D − 6) 1985701772 194176682 140342384 5504 65552704 + + − − + 4303125 (5D − 14) 28153125 (5D − 16) 40625 (5D − 18) 3125 (5D − 22) 189 (D − 1)2 23952 40016 768 315872 5888 − + + + + 2 2 3 3 5 (D − 2) 3 (D − 4) (D − 2) 45 (D − 4) 3 (D − 4)4 

+B4,1 +

– 46 –

 248426212 − 28125  144 9424 2704 9432928 797251 D −A5,2 + − − + − 2250 D 1485 (2D − 5) 63 (2D − 7) 5525 (2D − 9) 34088 253295 112 734044771 1047656 − − + − + 945 (3D − 10) 33 (3D − 14) 15 (3D − 8) 810810 (D − 1) 225 (D − 2) 576 857312704 1909981852 402543587 2009948 + + + + + 1215 (D − 4) 7 (D − 6) 90365625 (5D − 14) 28153125 (5D − 16) 446875 (5D − 18) 497 3184 1267888 768 35345856 − − + + + 3125 (5D − 22) 54 (D − 1)2 (D − 2)2 405 (D − 4)2 (D − 2)3  7040 225667646 333248 + − + 135 (D − 4)3 9 (D − 4)4 84375  5681 13125 45395966 495373 D −B5,1 +51 D 2 − + + − 1000 288 (2D − 5) 32 (2D − 7) 5525 (2D − 9) 317395 235954 5 859702 51714784 − + + + + 162 (D − 1) 75 (D − 2) 2 (D − 3) 135 (D − 4) 4303125 (5D − 14) 103936 3138226 101873408 9056 47116 − − + − + 2 40625 (5D − 16) 3125 (5D − 18) 9375 (5D − 22) 5 (D − 2) 5 (D − 4)2  20464 1920 14613393 320 + + + + (D − 2)3 3 (D − 4)3 (D − 4)4 12500  769576 66030496 4784 32 218919 D + − + + +B5,2 + 250 10395 (2D − 5) 5525 (2D − 9) 105 (3D − 10) 5 (3D − 8) 582929 565904 72 24448544 1921435783 + + + − − 810810 (D − 1) 75 (D − 2) 135 (D − 4) 7 (D − 6) 4303125 (5D − 14) 415352077 81364304 4267 23272 25102328 − + + − + 2 1340625 (5D − 16) 446875 (5D − 18) 3125 (5D − 22) 54 (D − 1) 5 (D − 2)2  791264 960 33184 2432 45656086 + + + + − 2 3 3 4 135 (D − 4) (D − 2) 9 (D − 4) 3 (D − 4) 9375  2 1581 D 50659 593 179 D + + + + −A6,1 + 4 125 198 (3D − 14) 162 (D − 1) 3 (D − 2) 1676696 155971 19461 1262352 199 + − − − + 3 (D − 4) 253125 (5D − 14) 34375 (5D − 16) 6250 (5D − 18) 3125 (5D − 22)  96 132 5861159 787 − + − − 36 (D − 1)2 5 (D − 2)2 5 (D − 4)2 112500  28093 D 160 551 169 2358232 +A6,2 + − − + + 225 D 42 (2D − 5) 42 (2D − 7) 16575 (2D − 9) 25856 91264 4136 5894 16 − + + − − 297 (D − 1) 75 (D − 2) 105 (D − 3) 135 (D − 4) 7 (D − 6) 1316274 960816 832 10 265141864 − + − − − 2 2008125 (5D − 14) 625625 (5D − 16) 4375 (5D − 22) (D − 2) (D − 3)2  320 3682658 − − 9 (D − 4)2 5625

– 47 –

 33372 D 845 304 4688143 −A6,3 +D 2 + − + + 125 63 (2D − 7) 15 (3D − 8) 38610 (D − 1) 25 24319 29688032 2863027 131356 + + − + + 45 (D − 2) 2 (D − 3) 45 (D − 4) 590625 (5D − 14) 309375 (5D − 16) 4328064 10688 3124 384 19412 − − + + + 2 2 40625 (5D − 18) 3125 (5D − 22) 5 (D − 2) 15 (D − 4) (D − 2)3  18671174 32 − + 3 (D − 4) 9375  192 528 −B6,1 +D 3 − 30 D 2 + 300 D + − (D − 2) (D − 4)  288 192 64 64 − − + − − 1024 (D − 2)2 (D − 4)2 (D − 2)3 (D − 4)3  2261 625 350 27 D 2 1689 D − + + − −B6,2 + 2 8 288 (2D − 5) 32 (2D − 7) 9 (D − 1) 788 610 512 700 128 − − + − − 2 2 (D − 2) 3 (D − 4) (D − 2) 3 (D − 4) (D − 2)3  1765 96 + − 3 (D − 4) 2  2 1689 D 2261 625 350 27 D − + + − −C6,1 + 2 8 288 (2D − 5) 32 (2D − 7) 9 (D − 1) 610 512 700 128 788 − + − − − 2 2 (D − 2) 3 (D − 4) (D − 2) 3 (D − 4) (D − 2)3  1765 96 + − 3 (D − 4) 2  76 1198 294779 2268 27 D −A7,1 + + − + − 10 77 (2D − 5) 63 (2D − 7) 13260 (2D − 9) 715 (D − 1) 20 12 72 56684 349 + + + − − 45 (D − 2) (D − 4) 35 (D − 6) 119 (5D − 14) 32175 (5D − 16)  8 873 139433 + + + 10725 (5D − 18) (D − 2)2 100  196 D 3098 26 40 48 +A7,2 + − + − − 25 715 (D − 1) (D − 2) (D − 4) 35 (D − 6)  5304 122054 16 4618 176 + + − − − 125 (5D − 14) 1925 (5D − 16) 1625 (5D − 18) (D − 2)2 125  57 1198 294779 502 6516 D +A7,3 + + − + − 125 (2D − 5) 63 (2D − 7) 49725 (2D − 9) (D − 1) 4258 14232368 5003392 176798 361018 + − − + + 225 (D − 2) 35 (D − 3) 74375 (5D − 14) 73125 (5D − 16) 5625 (5D − 18)  1840 10 512 328828 164176 − − + − − 13125 (5D − 22) (D − 2)2 (D − 3)2 (D − 2)3 625  19 599 294779 43 8401 D − + + + −A7,4 + 1000 16 (2D − 5) 126 (2D − 7) 9945 (2D − 9) 60 (D − 1)

– 48 –

717 4 11198396 1234996 1234 − − + − 45 (D − 2) 140 (D − 3) 5 (D − 6) 1115625 (5D − 14) 365625 (5D − 16)  1761496 16 1 1219407 119686 − − + − − 28125 (5D − 18) 65625 (5D − 22) (D − 2)2 4 (D − 3)2 50000  8401 D 19 599 294779 43 −A7,5 + − + + + 1000 16 (2D − 5) 126 (2D − 7) 9945 (2D − 9) 60 (D − 1) 717 4 11198396 1234996 1234 − − + − + 45 (D − 2) 140 (D − 3) 5 (D − 6) 1115625 (5D − 14) 365625 (5D − 16)  1761496 16 1 1219407 119686 − − + − − 28125 (5D − 18) 65625 (5D − 22) (D − 2)2 4 (D − 3)2 50000  837 D 463 844 15 4 −A8,1 + − + + − 1000 84 (2D − 7) 45 (D − 2) 14 (D − 3) (D − 4) 112651 95172 80068 169 79508 + + + + − 21875 (5D − 14) 28125 (5D − 16) 3125 (5D − 18) 65625 (5D − 22) 96 (2D − 7)2  16 807543 − − (D − 2)2 100000 3 (D − 3) (3D − 8) (D 3 − 16 D 2 + 68 D − 88) −B8,1 4 (2D − 5) (D − 2) (2D − 7) (D − 4) 3 (D − 3) (3D − 8) (D 3 − 16 D 2 + 68 D − 88) −C8,1 4 (2D − 5) (D − 2) (2D − 7) (D − 4)  132 15 2 16464 729 D + + − − +A9,1 + 500 5 (D − 2) 28 (D − 3) (D − 4) 3125 (5D − 14)  13794 1872 30056 96 156321 + + + − − 3125 (5D − 16) 3125 (5D − 18) 21875 (5D − 22) 5 (D − 2)2 12500 6 5 4 3 2 3 (3D − 14) (75 D − 1048 D + 5956 D − 17776 D + 30208 D − 29440 D + 13952) −A9,2 10 (D − 2) (D − 3) (5D − 22) (5D − 14) (5D − 16) (5D − 18) +

XCg 2 N = A F  5809119 D 2240 34925 104123558 37168 D 2 − + + + −B4,1 + 125 1250 3D 126 (2D − 7) 3315 (2D − 9) 2318004 64 77206856351 657533 11162 − + + − + 21 (3D − 10) 385 (3D − 14) 45 (3D − 8) 4864860 (D − 1) 90 (D − 2) 41 8729134 180 563997052 538277353 − − + − + 12 (D − 3) 1215 (D − 4) (D − 6) 6024375 (5D − 14) 4021875 (5D − 16) 1207808264 7637 6740 151 16666937 − + + − + 2 2 81250 (5D − 18) 28125 (5D − 22) 36 (D − 1) 3 (D − 2) 4 (D − 3)2  896 313448 29312 27982342 783268 − + + + + 405 (D − 4)2 3 (D − 2)3 135 (D − 4)3 45 (D − 4)4 5625  2 7596253 D 608 1075 27315013 6881 D − + + + −A5,1 + 125 11250 5 (2D − 5) 18 (2D − 7) 22100 (2D − 9)

– 49 –

3290734688 239912 8 3016 212536 − + + − 6615 (3D − 10) 567567 (D − 1) 75 (D − 2) (D − 3) 3 (D − 4) 720 19049176 160235632 67577636 10234588 − − − + + 7 (D − 6) 4303125 (5D − 14) 5630625 (5D − 16) 40625 (5D − 18) 9375 (5D − 22) 4928 62704 256 19264 11008 − − + − − 189 (D − 1)2 3 (D − 2)2 135 (D − 4)2 (D − 2)3 45 (D − 4)3  183292489 + 67500  7688 D 2 822412 D 96 37696 430 +A5,2 + − + + − 375 1875 D 1485 (2D − 5) 63 (2D − 7) 20648 66040 224 247020002 2489693 − − + + − 5525 (2D − 9) 945 (3D − 10) 231 (3D − 14) 135 (3D − 8) 135135 (D − 1) 377408 288 67345168 73779424 111043 + + + + − 75 (D − 2) 1215 (D − 4) 7 (D − 6) 3614625 (5D − 14) 9384375 (5D − 16) 19692512 994 504 125248 137313893 + + + + − 2 2 446875 (5D − 18) 28125 (5D − 22) 27 (D − 1) (D − 2) 405 (D − 4)2  4480 3858677 256 + + − 3 3 3 (D − 2) 27 (D − 4) 3375  2 1397 D 646933 D 5681 6125 2169937 +B5,1 + + − − − 1000 10000 72 (2D − 5) 96 (2D − 7) 5525 (2D − 9) 541766 1 1862 49642912 317395 − − − + + 81 (D − 1) 225 (D − 2) 2 (D − 3) 45 (D − 4) 4303125 (5D − 14) 1246 14295008 3136 3136 488704 + + + + − 2 365625 (5D − 16) 125 (5D − 18) 9375 (5D − 22) 3 (D − 2) 45 (D − 4)2  1312 11790061 128 + − − 3 3 (D − 2) 15 (D − 4) 20000  2 871 D 1716962 D 3078304 3156272 256 +B5,2 + − + + − 75 5625 10395 (2D − 5) 5525 (2D − 9) 945 (3D − 10) 1836706258 1384157 15388 36 64 − + − − − 45 (3D − 8) 405405 (D − 1) 450 (D − 2) 405 (D − 4) 7 (D − 6) 137885032 88774237 6784148 502 1683416 + + − + + 53125 (5D − 14) 28153125 (5D − 16) 893750 (5D − 18) 3125 (5D − 22) 3 (D − 1)2  21104 224 4672 18770368 5348 − + − + − 3 (D − 2)2 135 (D − 4)2 (D − 2)3 45 (D − 4)3 16875  2 197 D 2993 D 6604 4333 159 −A6,1 + − − − − 50 50 693 (3D − 14) 81 (D − 1) 5 (D − 2) 3850072 4564 11563 17696 44 − + − + + 45 (D − 4) 354375 (5D − 14) 4125 (5D − 16) 625 (5D − 18) 1875 (5D − 22)  8 1203449 443 − + − 9 (D − 1)2 (D − 4)2 5625  2 84531 D 320 1102 215 119 D − − − + +A6,2 + 50 1000 3 D 21 (2D − 5) 336 (2D − 7) −

– 50 –

51712 19264 4183 1084 93023 − − − − 7800 (2D − 9) 297 (D − 1) 75 (D − 2) 210 (D − 3) 135 (D − 4) 8 38037128 161431412 331172 272 + + − − + 7 (D − 6) 118125 (5D − 14) 5630625 (5D − 16) 39375 (5D − 22) 3 (D − 2)2  64 40336667 4 − + + (D − 3)2 3 (D − 2)3 90000  2 1571239 D 1075 4757193 608 921 D −A6,3 + − − − − 100 9000 504 (2D − 7) 88400 (2D − 9) 135 (3D − 8) 116434 3 112 139041824 4688143 − − + + + 19305 (D − 1) 225 (D − 2) (D − 3) 45 (D − 4) 2008125 (5D − 14) 1488052 2138 20224 32 24 − − + + − 2 804375 (5D − 16) 1625 (5D − 18) 625 (5D − 22) (D − 2) 5 (D − 4)2  128 182032831 + + (D − 2)3 270000  2 185 D 2261 875 700 7D − + + − −B6,2 + 8 16 72 (2D − 5) 288 (2D − 7) 9 (D − 1)  592 62 160 8 793 + + − + + 9 (D − 2) 9 (D − 4) 3 (D − 2)2 (D − 4)2 32  2 7D 185 D 2261 875 700 −C6,1 + − + + − 8 16 72 (2D − 5) 288 (2D − 7) 9 (D − 1)  62 160 8 793 592 + − + + + 9 (D − 2) 9 (D − 4) 3 (D − 2)2 (D − 4)2 32  2 4D 1437 D 304 451 28181 −A7,1 + − + − − 25 500 77 (2D − 5) 126 (2D − 7) 26520 (2D − 9) 9248 263 6 2253164 38046376 − − − + + 2145 (D − 1) 30 (D − 2) 35 (D − 6) 223125 (5D − 14) 5630625 (5D − 16)  7073303 20 10761 − − + 536250 (5D − 18) 3 (D − 2)2 1000  2 17 D 2237 D 71585 226533 6196 +A7,2 + − − − − 80 400 8064 (2D − 7) 141440 (2D − 9) 715 (D − 1) 24 63088 105104 55686 142 + + + + + 45 (D − 2) 35 (D − 6) 14875 (5D − 14) 375375 (5D − 16) 1625 (5D − 18)  35181 32 + − 3 (D − 2)2 1600  2 4966 D 228 451 28181 11 D − + − − +A7,3 + 125 625 (2D − 5) 126 (2D − 7) 99450 (2D − 9) 141554 202 63075448 1714432 1004 + − + + − (D − 1) 225 (D − 2) 5 (D − 3) 371875 (5D − 14) 73125 (5D − 16) 129298 2948 480 2 128 + + − + + 2 2 28125 (5D − 18) 9375 (5D − 22) (D − 2) (D − 3) (D − 2)3  52438 + 625 −

– 51 –

21 D 2 15043 D 19 6805 93023 − − + − 1000 5000 4 (2D − 5) 4032 (2D − 7) 37440 (2D − 9) 779 467 2 1100828 43 − + + + + 30 (D − 1) 45 (D − 2) 210 (D − 3) 5 (D − 6) 65625 (5D − 14)  4403 393514 1 916069 2621912 − + + + − 365625 (5D − 16) 5625 (5D − 18) 196875 (5D − 22) 4 (D − 3)2 60000  2 15043 D 19 6805 93023 21 D − − + − −A7,5 + 1000 5000 4 (2D − 5) 4032 (2D − 7) 37440 (2D − 9) 43 779 467 2 1100828 + − + + + 30 (D − 1) 45 (D − 2) 210 (D − 3) 5 (D − 6) 65625 (5D − 14)  2621912 4403 393514 1 916069 − − + + + 365625 (5D − 16) 5625 (5D − 18) 196875 (5D − 22) 4 (D − 3)2 60000  2377 712 45 169858 113787 D − + − − −A8,1 + 80000 1792 (2D − 7) 45 (D − 2) 28 (D − 3) 21875 (5D − 14)  105754 12804 3356 215 920019 + + − + − 28125 (5D − 16) 3125 (5D − 18) 65625 (5D − 22) 768 (2D − 7)2 80000 3 2 3 2 (2 D − 25 D + 94 D − 112) (D − 16 D + 68 D − 88) −B8,1 8 (2D − 7) (D − 2)2 (2D − 5) 3 2 (2 D − 25 D + 94 D − 112) (D 3 − 16 D 2 + 68 D − 88) −C8,1 8 (2D − 7) (D − 2)2 (2D − 5)  44 45 8232 81 D 2 9369 D − − + + +A9,1 + 1000 5000 3 (D − 2) 56 (D − 3) 625 (5D − 14)  40172 2928 3106 54801 − + + + 9375 (5D − 16) 3125 (5D − 18) 65625 (5D − 22) 5000 5 4 3 (3D − 14) (D − 4) (135 D − 2200 D + 15156 D − 54336 D 2 + 99776 D − 74112) +A9,2 20 (D − 3) (5D − 22) (5D − 14) (5D − 16) (5D − 18) (D − 2)  2 513 D 973 75511 4 81 D − − − − −A9,4 + 16000 10000 9216 (2D − 7) 5657600 (2D − 9) 75 (D − 2)  14036 1168 2234 36117 756 − + + + + 10625 (5D − 14) 121875 (5D − 16) 3125 (5D − 18) 28125 (5D − 22) 320000 

−A7,4 +

XCg A CF NF =  169208 D 2 28533686 D 1120 4465 1070886586 +B4,1 + − − + + 125 1875 3D 7 (2D − 7) 16575 (2D − 9) 18168111 1664 19856072 3342712 56648 + − + − − 63 (3D − 10) 770 (3D − 14) 45 (3D − 8) 405405 (D − 1) 675 (D − 2) 480653 2161801556 868101691 669068729 1138 + + + + − 3 (D − 3) 135 (D − 4) 30121875 (5D − 14) 2413125 (5D − 16) 48750 (5D − 18) 1607924552 13376 19 554888 471544 − + − + + 2 2 2 9375 (5D − 22) 9 (D − 2) 3 (D − 3) 15 (D − 4) 15 (D − 4)3

– 52 –

 46784 876803866 + + 5 (D − 4)4 28125  2 21754033 D 1216 2390 64410511 25803 D − + + + +A5,1 + 125 11250 5 (2D − 5) 9 (2D − 7) 22100 (2D − 9) 63976 13760 295132 40 711848 − − + + − 945 (3D − 10) 567 (D − 1) 225 (D − 2) 3 (D − 3) 405 (D − 4) 6633272 9607724 7186816 3712 11375744 + + − − − 478125 (5D − 14) 73125 (5D − 16) 3125 (5D − 18) 3125 (5D − 22) 3 (D − 2)2  19328 1255139429 5776 − + − 135 (D − 4)2 45 (D − 4)3 337500  2 27472 D 6814294 D 48 75392 1912 −A5,2 + − − + − 375 5625 D 1485 (2D − 5) 63 (2D − 7) 27436 258805 992 2189468 7070567 − + − − − 5525 (2D − 9) 945 (3D − 10) 231 (3D − 14) 27 (3D − 8) 81081 (D − 1) 1712 1146502544 482461664 464255741 1407278 + + − − − 675 (D − 2) (D − 4) 30121875 (5D − 14) 12065625 (5D − 16) 446875 (5D − 18)  10192 6544 71776 10496 124446769 − + + + + 375 (5D − 22) 9 (D − 2)2 45 (D − 4)2 15 (D − 4)3 28125  2 7331 D 205803 D 5681 6125 270116 +B5,1 + + + + + 500 5000 36 (2D − 5) 48 (2D − 7) 325 (2D − 9) 19 10708 31328 10336 40704 + + − − + 25 (D − 2) (D − 3) 15 (D − 4) 5625 (5D − 14) 40625 (5D − 16) 11927552 928 224 192 3005968 − − − − − 2 2 3125 (5D − 18) 1875 (5D − 22) (D − 2) 3 (D − 4) 5 (D − 4)3  52885663 − 50000  7409 D 2 5308534 D 6156608 392896 28528 −B5,2 + − + + + 75 5625 10395 (2D − 5) 325 (2D − 9) 945 (3D − 10) 1921904 423956 129704 13379128 16 + + + + + 9 (3D − 8) 27027 (D − 1) 225 (D − 2) 81 (D − 4) 253125 (5D − 14) 184123036 4877264 5344 118976 1042664096 − − − + + 2 28153125 (5D − 16) 446875 (5D − 18) 625 (5D − 22) 3 (D − 2) 135 (D − 4)2  112377904 10624 + + 3 45 (D − 4) 84375  2 5321 D 340129 D 51761 430 2273 +A6,1 + − + − − 375 1875 1386 (3D − 14) 27 (D − 1) 15 (D − 2) 1645052 823378 121841 148988 178 − − − − + 3 (D − 4) 590625 (5D − 14) 103125 (5D − 16) 6250 (5D − 18) 3125 (5D − 22)  272 17076368 + + 15 (D − 4)2 28125  2 334171 D 160 2204 239 1489 D − + − + −A6,2 + 100 1500 3 D 21 (2D − 5) 84 (2D − 7)

– 53 –

461624 8131 304 169118132 3947149 − + + + 132600 (2D − 9) 675 (D − 2) 210 (D − 3) 9 (D − 4) 669375 (5D − 14)  232964 544 5 14824063 123983528 − + + + − 1535625 (5D − 16) 13125 (5D − 22) 9 (D − 2)2 12 (D − 3)2 15000  2 24791 D 26566007 D 1195 14271579 1520 +A6,3 + − − − + 500 45000 126 (2D − 7) 88400 (2D − 9) 27 (3D − 8) 91192 2 13466 146291696 4052666 − − + − + 225 (D − 2) (D − 3) 45 (D − 4) 3346875 (5D − 14) 365625 (5D − 16)  200184 510816 1552 2491573703 + − + + 3125 (5D − 18) 3125 (5D − 22) 15 (D − 4)2 1350000  2 23 D 729 D 2261 875 1264 +B6,2 + − + + − 4 8 36 (2D − 5) 144 (2D − 7) 9 (D − 2)  32 80 5401 380 − + + + 9 (D − 4) 3 (D − 2)2 3 (D − 4)2 16  2 23 D 729 D 2261 875 1264 +C6,1 + − + + − 4 8 36 (2D − 5) 144 (2D − 7) 9 (D − 2)  380 32 80 5401 + − + + 9 (D − 4) 3 (D − 2)2 3 (D − 4)2 16  2 771 D 608 139 877 16 D − + − − +A7,1 + 25 125 77 (2D − 5) 21 (2D − 7) 390 (2D − 9) 1192 10112 72675488 9210328 1744 − + + − + 429 (D − 1) 45 (D − 2) 525 (5D − 14) 5630625 (5D − 16) 268125 (5D − 18)  21831 32 + − 3 (D − 2)2 1250  2 83 D 1307 D 71585 679599 246 −A7,2 + − − − − 80 80 2688 (2D − 7) 141440 (2D − 9) 5 (D − 2)  226392 7712 12886 645603 + + + + 14875 (5D − 14) 4875 (5D − 16) 125 (5D − 18) 8000  2 24 D 456 139 1754 169 D − + − − −A7,3 + 125 5 (2D − 5) 21 (2D − 7) 2925 (2D − 9) 15704 472 1432656 7709696 47912 + + + − + 225 (D − 2) 15 (D − 3) 21875 (5D − 14) 365625 (5D − 16) 5625 (5D − 18)  1792 1 238436 176 − + − + 1875 (5D − 22) 3 (D − 2)2 (D − 3)2 3125  2 313 D 10751 D 19 1807 3947149 +A7,4 + − − + − 1000 3000 2 (2D − 5) 448 (2D − 7) 636480 (2D − 9) 1817 12096842 6605536 57247 1681 + + − − − 45 (D − 2) 420 (D − 3) 371875 (5D − 14) 365625 (5D − 16) 28125 (5D − 18)  111166 32 5 5654087 + + + + 21875 (5D − 22) 3 (D − 2)2 12 (D − 3)2 300000  2 10751 D 19 1807 3947149 313 D − − + − +A7,5 + 1000 3000 2 (2D − 5) 448 (2D − 7) 636480 (2D − 9) −

– 54 –

1817 12096842 6605536 57247 1681 + + − − 45 (D − 2) 420 (D − 3) 371875 (5D − 14) 365625 (5D − 16) 28125 (5D − 18)  32 5 5654087 111166 + + + + 21875 (5D − 22) 3 (D − 2)2 12 (D − 3)2 300000  37071 D 1853 1132 15 368628 +A8,1 + − + − − 20000 448 (2D − 7) 45 (D − 2) 7 (D − 3) 21875 (5D − 14)  202312 31416 1468 239 1622997 + + − + − 28125 (5D − 16) 3125 (5D − 18) 13125 (5D − 22) 192 (2D − 7)2 100000 3 2 3 2 (2 D − 25 D + 94 D − 112) (D − 16 D + 68 D − 88) +B8,1 4 (2D − 7) (D − 2)2 (2D − 5) 3 2 (2 D − 25 D + 94 D − 112) (D 3 − 16 D 2 + 68 D − 88) +C8,1 4 (2D − 7) (D − 2)2 (2D − 5)  81 D 2 6939 D 144 15 49392 −A9,1 + − − + + 250 1250 5 (D − 2) 14 (D − 3) 3125 (5D − 14)  5808 12864 1752 175239 − − + + 3125 (5D − 16) 3125 (5D − 18) 4375 (5D − 22) 6250 5 4 3 2 3 (D − 4) (3D − 14) (15 D − 266 D + 1708 D − 5016 D + 6624 D − 2944) +A9,2 5 (D − 3) (5D − 22) (5D − 14) (5D − 16) (5D − 18) (D − 2)  1539 D 973 226533 4 243 D 2 − − − − +A9,4 + 16000 10000 3072 (2D − 7) 5657600 (2D − 9) 25 (D − 2)  2268 14036 3504 2234 108351 + − + + + 10625 (5D − 14) 40625 (5D − 16) 3125 (5D − 18) 9375 (5D − 22) 320000 −

XCg 2 N = F F  188119 D 2 61815901 D 23257388 280 13405743 −B4,1 + − − − + 125 3750 5525 (2D − 9) 9 (3D − 10) 385 (3D − 14) 160 23276 499 773374 1533843736 − − − + − 9 (3D − 8) 25 (D − 2) 6 (D − 3) 45 (D − 4) 3346875 (5D − 14) 79707397 2286575824 64 53 4855107866 + − + − + 4021875 (5D − 16) 9375 (5D − 18) 28125 (5D − 22) (D − 2)2 6 (D − 3)2  135408 36096 296368802 573968 + + + + 15 (D − 4)2 5 (D − 4)3 5 (D − 4)4 9375  2 27082 D 1697483 D 7475589 56 89768 −A5,1 + − + + − 125 625 11050 (2D − 9) 9 (3D − 10) 75 (D − 2) 32 282832 16480952 1500528 2180808 − + − + + 3 (D − 3) 135 (D − 4) 286875 (5D − 14) 40625 (5D − 16) 3125 (5D − 18)  48975472 94688 8704 87298177 − + + + 9375 (5D − 22) 45 (D − 4)2 15 (D − 4)3 11250  2 1283138 D 4077594 344 127310 21317 D − − − + +A5,2 + 375 1875 5525 (2D − 9) 27 (3D − 10) 77 (3D − 14)

– 55 –

46972 29872 141597536 73031344 400 − + + + 27 (3D − 8) 225 (D − 2) 45 (D − 4) 3346875 (5D − 14) 1340625 (5D − 16) 1607982 6282976 32 8576 256 − − + + + 2 2 3125 (5D − 18) 5625 (5D − 22) (D − 2) 15 (D − 4) (D − 4)3  152810662 + 84375  446671 D 1784 11 184976 7239 D 2 −B5,1 + − − − + 125 625 5 (D − 2) (D − 3) 135 (D − 4) 2576 1879752 2550912 37696 2031568 + − − + − 84375 (5D − 14) 3125 (5D − 16) 3125 (5D − 18) 625 (5D − 22) 45 (D − 4)2  6406811 4736 + + 15 (D − 4)3 3125  2 11609 D 726482 D 728 400 4948 +B5,2 + − − − − 75 375 27 (3D − 10) 27 (3D − 8) 5 (D − 2) 55288 117104 12584 133092 515088 + + + − − 27 (D − 4) 3375 (5D − 14) 625 (5D − 16) 625 (5D − 18) 125 (5D − 22)  13280 6272 3837572 + + + 9 (D − 4)2 15 (D − 4)3 675  2 4732 D 300428 D 12731 5458 2524 −A6,1 + − + − + 375 1875 231 (3D − 14) 45 (D − 2) 45 (D − 4) 289948 733 665896 64 1669664 − − − + − 196875 (5D − 14) 61875 (5D − 16) 125 (5D − 18) 9375 (5D − 22) 3 (D − 4)2  4892596 + 9375  215473 D 226533 154016 98 1013 D 2 − − − + +A6,2 + 50 750 22100 (2D − 9) 225 (D − 2) 3 (D − 3) 1662152 5141248 8936 64 208 + − + − + 15 (D − 4) 6375 (5D − 14) 73125 (5D − 16) 5625 (5D − 22) (D − 2)2  8774231 5 + − 6 (D − 3)2 7500  2 14331 D 1856633 D 4757193 60536 9 −A6,3 + − − − + 250 2500 44200 (2D − 9) 75 (D − 2) (D − 3) 4396 2026304 56236 33752 761024 + + + + − 15 (D − 4) 159375 (5D − 14) 24375 (5D − 16) 625 (5D − 18) 3125 (5D − 22)  512 64132247 + + 5 (D − 4)2 25000 2 16 (D − 3) (D − 7 D + 16) (5 D 3 − 62 D 2 + 236 D − 288) −A7,1 (D − 2) (5D − 14) (5D − 16) (5D − 18)  2 3181 D 71585 226533 1556 49 D − − − − +A7,2 + 40 200 4032 (2D − 7) 70720 (2D − 9) 45 (D − 2)  2752 6548 50413 215664 + + + + 14875 (5D − 14) 2925 (5D − 16) 125 (5D − 18) 800 −

– 56 –

419 D 2 33136 D 328 1126 1169616 − − + + 125 625 (D − 2) 105 (D − 3) 3125 (5D − 14)  16616 3504 7 787461 60416 − − − + + 3125 (5D − 16) 9375 (5D − 18) 4375 (5D − 22) (D − 3)2 3125  2 271 D 48887 D 3197 75511 622 −A7,4 + − + − − 500 7500 2016 (2D − 7) 35360 (2D − 9) 45 (D − 2) 694636 2925056 1314 49148 3 − − − + + (D − 3) 74375 (5D − 14) 365625 (5D − 16) 3125 (5D − 18) 28125 (5D − 22)  47263 1 + − 3 (D − 3)2 2000  2 271 D 48887 D 3197 75511 622 −A7,5 + − + − − 500 7500 2016 (2D − 7) 35360 (2D − 9) 45 (D − 2) 3 694636 2925056 1314 49148 + − − − + (D − 3) 74375 (5D − 14) 365625 (5D − 16) 3125 (5D − 18) 28125 (5D − 22)  47263 1 + − 3 (D − 3)2 2000  2 81 D 6939 D 144 15 49392 +A9,1 + − − + + 250 1250 5 (D − 2) 14 (D − 3) 3125 (5D − 14)  12864 1752 175239 5808 − + + − 3125 (5D − 16) 3125 (5D − 18) 4375 (5D − 22) 6250  2 81 D 513 D 973 75511 8 −A9,4 + − − − − 8000 5000 4608 (2D − 7) 2828800 (2D − 9) 75 (D − 2)  1512 28072 2336 4468 36117 + − + + + 10625 (5D − 14) 121875 (5D − 16) 3125 (5D − 18) 28125 (5D − 22) 160000 

+A7,3 +

XCg

=  715 1693529 55 37 968 D + − − − +B4,1 + 25 21 (3D − 10) 162162 (D − 1) 3 (D − 2) 6 (D − 3) 12056 20832 299376 9217 12496 + + + + − 81 (D − 4) 3375 (5D − 14) 1375 (5D − 16) 1625 (5D − 18) 108 (D − 1)2  1 1504 896 46756 4 + − − − + (D − 2)2 4 (D − 3)2 9 (D − 4)2 9 (D − 4)3 375  68 2806094 80 256 238 D −A5,2 + − − + − 75 63 (3D − 10) 81081 (D − 1) 3 (D − 2) 81 (D − 4) 2496 23328 994 8 21296 − − − − + 2 10125 (5D − 14) 1375 (5D − 16) 1625 (5D − 18) 27 (D − 1) (D − 2)2  4096 256 − − 2 27 (D − 4) 1125  32 278326 128 17072 154 D +B5,2 + − − − − 75 63 (3D − 10) 81081 (D − 1) 81 (D − 4) 10125 (5D − 14) 2 A NF

– 57 –

2208 11664 502 128 10408 − + − − 1375 (5D − 16) 1625 (5D − 18) 27 (D − 1)2 27 (D − 4)2 1125 (5D − 16) (D − 4) +A6,1 3 (D − 1)2 (D − 2)2 8 (D − 3) (D − 4) (5 D 3 − 62 D 2 + 236 D − 288) −A7,1 (D − 1) (5D − 14) (5D − 16) (5D − 18) −



XCg

=  2336 D 1360 64 20262752 4256 −B4,1 + − − + − 25 21 (3D − 10) 15 (3D − 8) 135135 (D − 1) 9 (D − 4) 24112 41664 598752 576 256 + + + − − 3375 (5D − 14) 1375 (5D − 16) 1625 (5D − 18) (D − 4)2 (D − 4)3  158512 − 375  160 32 4378936 1024 1976 D +A5,2 + + − + − 75 63 (3D − 10) 9 (3D − 8) 81081 (D − 1) 27 (D − 4)  4992 46656 256 49064 42592 − − − − + 10125 (5D − 14) 1375 (5D − 16) 1625 (5D − 18) 9 (D − 4)2 375  160 32 1481384 512 136 D −B5,2 + + + + − 25 63 (3D − 10) 9 (3D − 8) 81081 (D − 1) 27 (D − 4)  4416 23328 128 35816 34144 − − − − − 10125 (5D − 14) 1375 (5D − 16) 1625 (5D − 18) 9 (D − 4)2 1125 3 2 16 (D − 3) (D − 4) (5 D − 62 D + 236 D − 288) +A7,1 (D − 1) (5D − 14) (5D − 16) (5D − 18) 2 F NF

References [1] G. Altarelli, R. K. Ellis and G. Martinelli, Nucl. Phys. B 157 (1979) 461. [2] R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B 359 (1991) 343; B 644 (2002) 403(E). [3] R.V. Harlander and W. B. Kilgore, Phys. Rev. Lett. 88 (2002) 201801 [hepph/0201206]. [4] S. Dawson, Nucl. Phys. B 359 (1991) 283. [5] D. Graudenz, M. Spira and P.M. Zerwas, Phys. Rev. Lett. 70 (1993) 1372; M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, Nucl. Phys. B 453 (1995) 17

– 58 –

[hep-ph/9504378]; A. Djouadi, M. Spira and P.M. Zerwas, Z. Phys. C 70 (1996) 427 [hep-ph/9511344]; M. Spira, Fortsch. Phys. 46 (1998) 203 [hep-ph/9705337]; R. Harlander and P. Kant, JHEP 0512 (2005) 015 [hep-ph/0509189]. [6] S. Catani, D. de Florian and M. Grazzini, JHEP 0105 (2001) 025 [hep-ph/0102227]; C. Anastasiou and K. Melnikov, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004]; V. Ravindran, J. Smith and W. L. van Neerven, Nucl. Phys. B 665 (2003) 325 [hepph/0302135], 704 (2005) 332 [hep-ph/0408315]. [7] L. Magnea and G. Sterman, Phys. Rev. D 42 (1990) 4222. [8] S. Moch and A. Vogt, Phys. Lett. B 631 (2005) 48 [hep-ph/0508265]; E. Laenen and L. Magnea, Phys. Lett. B 632 (2006) 270 [hep-ph/0508284]; V. Ravindran, Nucl. Phys. B 746 (2006) 58 [hep-ph/0512249]. [9] G. P. Korchemsky, Phys. Lett. B 217 (1989) 330. [10] C.W. Bauer, S. Fleming and M.E. Luke, Phys. Rev. D 63 (2000) 014006 [hepph/0005275]; C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336]; C.W. Bauer and I.W. Stewart, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001]; C.W. Bauer, D. Pirjol and I.W. Stewart, Phys. Rev. D 65 (2002) 054022 [hepph/0109045]; M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152]; M. Beneke and T. Feldmann, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358]. [11] A. Idilbi and X. Ji, Phys. Rev. D 72 (2005) 054016 [hep-ph/0501006]. [12] A. Idilbi, X. Ji and F. Yuan, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068]. [13] A. Idilbi, X. Ji, J.P. Ma and F. Yuan, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294]; V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Eur. Phys. J. C 62 (2009) 333 [arXiv:0809.4283]. [14] G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B 171 (1986) 459; Nucl. Phys. B 283 (1987) 342. [15] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 688 (2004) 101 [hepph/0403192]. [16] A. Vogt, S. Moch and J. A. M. Vermaseren, Nucl. Phys. B 691 (2004) 129 [hepph/0404111]. [17] S. Moch, J. A. M. Vermaseren and A. Vogt, JHEP 0508 (2005) 049 [hep-ph/0507039].

– 59 –

[18] S. Moch, J. A. M. Vermaseren and A. Vogt, Phys. Lett. B 625, 245 (2005) [hepph/0508055]. [19] G. Kramer and B. Lampe, Z. Phys. C 34 (1987) 497; 42 (1989) 504(E); T. Matsuura and W.L. van Neerven, Z. Phys. C 38 (1988) 623; T. Matsuura, S.C. van der Maarck and W.L. van Neerven, Nucl. Phys. B 319 (1989) 570. [20] R.V. Harlander, Phys. Lett. B 492 (2000) 74 [hep-ph/0007289]. [21] T. Gehrmann, T. Huber and D. Maˆıtre, Phys. Lett. B 622 (2005) 295 [hepph/0507061]. [22] T. Becher, M. Neubert and B. D. Pecjak, JHEP 0701 (2007) 076 [hep-ph/0607228]. [23] T. Becher and M. Neubert, JHEP 0906 (2009) 081 [arXiv:0903.1126]. [24] G. P. Korchemsky, Mod. Phys. Lett. A 4 (1989) 1257. [25] L. F. Alday and J. M. Maldacena, JHEP 0711 (2007) 019 [arXiv:0708.0672]. [26] J. Frenkel and J. C. Taylor, Nucl. Phys. B 246 (1984) 231. [27] T. Becher and M. Neubert, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722]. [28] E. Gardi and L. Magnea, JHEP 0903 (2009) 079 [arXiv:0901.1091]. [29] S. Catani, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439]. [30] G. Sterman and M. E. Tejeda-Yeomans, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130]. [31] S. M. Aybat, L. J. Dixon and G. Sterman, Phys. Rev. D 74 (2006) 074004 [hepph/0607309]. [32] S. M. Aybat, L. J. Dixon and G. Sterman, Phys. Rev. Lett. 97 (2006) 072001 [hepph/0606254]. [33] L. J. Dixon, E. Gardi and L. Magnea, JHEP 1002 (2010) 081 [arXiv:0910.3653]. [34] S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033]. [35] C. Anastasiou and A. Lazopoulos, JHEP 0407 (2004) 046 [hep-ph/0404258]. [36] A. V. Smirnov, JHEP 0810 (2008) 107 [arXiv:0807.3243]. [37] C. Studerus, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546]. [38] T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Phys. Lett. B 640 (2006) 252 [hep-ph/0607185]. [39] G. Heinrich, T. Huber and D. Maˆıtre, Phys. Lett. B 662 (2008) 344 [arXiv:0711.3590].

– 60 –

[40] G. Heinrich, T. Huber, D. A. Kosower and V. A. Smirnov, Phys. Lett. B 678 (2009) 359 [arXiv:0902.3512]. [41] R.N. Lee, A.V. Smirnov and V.A. Smirnov, JHEP 1004 (2010) 020 [arXiv:1001.2887]. [42] P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519]. [43] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Phys. Lett. B 78 (1978) 443; J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos and C.T. Sachrajda, Phys. Lett. B 83 (1979) 339; T. Inami, T. Kubota and Y. Okada, Z. Phys. C 18 (1983) 69. [44] K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Phys. Rev. Lett. 79 (1997) 353 [hep-ph/9705240]. [45] B.A. Kniehl and M. Spira, Z. Phys. C 69 (1995) 77 [hep-ph/9505225]; K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Nucl. Phys. B 510 (1998) 61 [hepph/9708255]. [46] W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D 18 (1978) 3998. [47] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H. D. Politzer, Phys. Rev. Lett. 30 (1973) 1346. [48] W. E. Caswell, Phys. Rev. Lett. 33 (1974) 244; D. R. T. Jones, Nucl. Phys. B 75 (1974) 531; E. Egorian and O. V. Tarasov, Teor. Mat. Fiz. 41 (1979) 26 [Theor. Math. Phys. 41 (1979) 863]. [49] O. V. Tarasov, A. A. Vladimirov and A. Y. Zharkov, Phys. Lett. B 93 (1980) 429; S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208]. [50] P. Nogueira, J. Comput. Phys. 105 (1993) 279. [51] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159. [52] T. Gehrmann and E. Remiddi, Nucl. Phys. B580 (2000) 485 [hep-ph/9912329]. [53] T. Huber and D. Maˆıtre, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094]. [54] S.G. Gorishnii, S.A. Larin, L.R. Surguladze and F.V. Tkachov, Comput. Phys. Comm. 55 (1989) 381; S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, NIKHEF-H-91-18. [55] S. Bekavac, Comput. Phys. Commun. 175 (2006) 180 [hep-ph/0505174]. [56] M. Czakon, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200].

– 61 –

[57] H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, An Integer Relation Finding Algorithm, Math. Comput. 68 (1999) 351 [58] J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B 724 (2005) 3 [hepph/0504242]. [59] A. Vogt, Phys. Lett. B 497 (2001) 228 [hep-ph/0010146]; S. Catani, D. de Florian, M. Grazzini and P. Nason, JHEP 0307 (2003) 028 [hepph/0306211]; S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 726 (2005) 317 [hepph/0506288]. [60] G. Sterman, Nucl. Phys. B 281 (1987) 310; S. Catani and L. Trentadue, Nucl. Phys. B 327 (1989) 323; Nucl. Phys. B 353 (1991) 183; S. Catani, M. L. Mangano, P. Nason and L. Trentadue, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351]; H. Contopanagos, E. Laenen and G. Sterman, Nucl. Phys. B 484 (1997) 303 [hepph/9604313]. [61] C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088].

– 62 –