The $ n_f^ 2$ contributions to fermionic four-loop form factors

TTP17-026

arXiv:1705.06862v1 [hep-ph] 19 May 2017

The n2f contributions to fermionic four-loop form factors Roman N. Leea , Alexander V. Smirnovb, Vladimir A. Smirnovc, and Matthias Steinhauserd (a) Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia (b) Research Computing Center, Moscow State University 119991, Moscow, Russia (c) Skobeltsyn Institute of Nuclear Physics of Moscow State University 119991, Moscow, Russia (d) Institut f¨ ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT) 76128 Karlsruhe, Germany

Abstract We compute the four-loop contributions to the photon quark and Higgs quark form factors involving two closed fermion loops. We present analytical results for all non-planar master integrals of the two non-planar integral families which enter our calculation. PACS numbers: 11.15.Bt, 12.38.Bx, 12.38.Cy

1

Introduction

There are various aspects of form factors which promote them to important objects in any quantum field theory. In the framework of QCD form factors constitute building blocks for various production and decay processes, most prominently for Higgs boson production and the Drell-Yan process. Furthermore, form factors are the simplest Green’s functions with a non-trivial infrared structure which makes them ideal objects to investigate general infrared properties of gauge theories [1–7]. The main objects of this work are massless fermionic form factors where the fermions couple via a vector and scalar coupling to an external current. In the framework of the Standard Model they can be interpreted as photon quark and Higgs quark form factors. In the following we provide brief definitions of these objects. The quark-antiquark-photon form factor is conveniently obtained from the photon quark vertex function Γµq by applying an appropriate projector. In d = 4 − 2ǫ space-time dimensions we have Fq (q 2 ) = −


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

(1)

with q = q1 + q2 . q1 and q2 are the incoming quark and antiquark momenta and q is the momentum of the photon. In analogy, the Higgs quark form factor is constructed from the Higgs quark vertex function Γq . For definiteness we consider in the following the coupling to bottom quarks and write Fb (q 2 ) = −

1 Tr (q/2 Γb /q1 ) . 2q 2

(2)

Sample Feynman diagrams contributing to Fq and Fb are shown in Fig. 1. Two- and three-loop corrections to Fq have been computed in Refs. [8–15] and Fb has been considered in Refs. [16, 17]. Four-loop results for Fq in the planar limit have been obtained in Refs. [18, 19] and fermionic corrections with three closed quark loops have been computed in Ref. [20]. In this paper we consider fermionic contributions to Fq and Fb . More precisely, we compute four-loop corrections with two closed fermion loops (which in our notation are proportional to n2f ). This is the simplest well-defined gauge invariant subset which involves non-planar integral families. It is the main result of this work to study in detail these families and provide analytic results for all non-planar master integrals which are part of these families. Note, that all planar families including all master integrals have been considered in a systematic way in Refs. [18,19,21]. Non-planar integral families are already present in the n3f contribution to the Higgs gluon form factor which has been considered in Ref. [20]. Very recently the 1/ǫ2 pole of the four-loop form factor within N = 4 super Yang-Mills theory has been computed using numerical methods [22]. 2

If ǫ0 terms at four loops shall be computed the lower-order corrections need to be expanded to higher orders in ǫ. In particular, the one-, two- and three-loop results are needed to order ǫ6 , ǫ4 and ǫ2 , respectively. The three-loop ǫ2 terms for Fq have been computed in Ref. [13] and cross-checked in Ref. [19]. As a preparatory calculation for the results obtained in this work we could confirm the three-loop corrections to the gluonic from factor up to order ǫ2 as given in Ref. [13]. Furthermore, we provide the first independent check for the three-loop ǫ0 term of Fb [17] and extend it to order ǫ2 which is not yet available in the literature. It is convenient to parametrize the form factor in terms of the bare strong coupling constant and write X  α0 n  µ2 nǫ s Fq(n) , Fq = 1 + 2 − i0 4π −q n≥1 # " X  α0 n  µ2 nǫ (n) s 0 , (3) Fb = y 1 + Fb 4π −q 2 − i0 n≥1 where y 0 = m0b /v is the bare Yukawa coupling and m0b and v are the bare bottom quark mass and Higgs vacuum expectation value, respectively. The renormalized counterparts of the form factors are easily obtained by multiplicative renormalization of the parameters αs0 and y 0 using three- and four-loop renormalization constants, respectively, which can be obtained from Refs. [23–26] (see also Appendix F and the ancillary file of Ref. [27] for an explicit result of Zy up to four loops in terms of SU(Nc ) colour factors). The pole part of the logarithm of the renormalized form factor has a universal structure which is given by (see, e.g., Ref. [13, 28, 29]) log(Fx ) =

Figure 1: Sample Feynman diagrams contributing to Fq and Fb at the four-loop order and containing two closed fermion loops. The gray box indicates either a scalar or a vector coupling of the fermions to the external current. Solid and curly lines represent quarks and gluons, respectively. All particles are massless.

3

( " " #) # 1 1 1 0 − CF γcusp + γx0 ǫ2 2 ǫ ( " " #) # " #  α 2 1 3 1 1 1 1 γx1 s 0 0 1 + + − + β C γ β γ − C γ 0 F 0 F cusp x cusp 4π ǫ3 8 ǫ2 2 8 ǫ 2 ( " # "  #   α 3 1 11 2 1 1 5 2 s 0 0 1 − β0 CF γcusp + 3 CF + β02 γx0 β1 γcusp + β0 γcusp + 4π ǫ4 36 ǫ 9 36 3 " #) " # 1 1 1 γx2 1 1 2 + 2 − β1 γx0 − β0 γx1 − CF γcusp + ǫ 3 3 18 ǫ 3 "  ( " # #   α 4 1 25 13 1 5 1 s 0 1 0 + β 3 CF γcusp + 4 CF − β02 γcusp − β1 β0 γcusp − β03 γx0 4π ǫ5 96 0 ǫ 96 12 4 "  #  5 3 7 1 1 1 0 1 2 β2 γcusp + β1 γcusp + β0 γcusp + β02 γx1 + β1 β0 γx0 + 3 CF ǫ 32 32 96 4 2 " #) " # 1 1 1 1 1 1 γx3 3 + 2 − β2 γx0 − β1 γx1 − β0 γx2 − CF γcusp + ... , (4) + ǫ 4 4 4 32 ǫ 4

αs 4π

where the ellipse denote higher order terms in αs and ǫ. We have x ∈ {q, b}. CF and CA are the eigenvalues of the quadratic Casimir operators of the fundamental and adjoint representation for the SU(Nc ) colour group, respectively. In Eq. (4) µ2 = −q 2 has been chosen. The cusp and collinear anomalous dimensions are defined through X  αs (µ2 ) n γx = γxn , (5) 4π n≥0 with x ∈ {cusp, q, b}. Note that γq = γb which, together with the universality of γcusp , is an important check of our calculation. The coefficients of the β function in Eq. (4) are given by 11CA 2nf − , 3 3 34CA2 10CA nf + − 2CF nf , = − 3 3 1415 2 79 2857CA3 11 205 CA CF nf − CA nf + CA n2f + + CF n2f + CF2 nf . = − 18 54 54 54 9

β0 = β1 β2

(6)

where we have used T = 1/2 with T being the index of the fundamental representation. nf counts the number of massless active quarks. Note that the three leading pole terms of the four-loop contribution to log(Fx ) are determined from lower-order coefficients and serve as an important consistency check. The 1/ǫ2 3 and 1/ǫ1 terms have new ingredients, γcusp and γx3 , which we can extract by comparison of Eq. (4) with our explicit calculation of the form factor. 4

1 7 3

2

2

10

8 9

11

12

4

1 8

9 10

6

3 4

5

7

12 11 6 5

7

786

Figure 2: Graphs associated with the non-planar families 7 and 786 needed for the n2f contribution to Fq and Fb . The numbers n next to the lines correspond to the indices of the propagators, i.e. to the nth integer argument of the functions representing the integrals. In addition to the 12 propagators we have for each family six linear independent numerator factors. However, the corresponding indices are always zero for our master integrals.

The remainder of the paper is organized as follows: In the next section we provide details to the techniques which have been used to compute non-planar four-loop vertex integrals. Results for the n2f contributions are presented in Section 3. Fq and Fb are discussed in Sections 3.1 and 3.2, respectively. In particular, we provide results for the n2f terms of the four-loop cusp and collinear anomalous dimensions and the finite parts of the form factors. Section 4 contains our conclusions. In the Appendix we provide analytic results for all non-planar master integrals which are present in the non-planar integral families needed for our calculation. The analytic results presented in this paper are also provided in computer-readable form in an ancillary file [30].

2

Technical details

The amplitudes, which contribute to the form factors, are prepared with the help of a well-tested setup. In a first step they are generated with qgraf [31]. For the fermionic form factors we have in total 1, 15 and 337 diagrams at one, two and three loops. At four-loop order we have 77 diagrams proportional to n2f and one diagram proportional to n3f (cf. Fig. 1 for sample diagrams). Next, we transform the output to FORM [32] notation using q2e and exp [33,34]. The program exp furthermore maps each Feynman diagram to families for massless four-loop vertices with two different non-vanishing external momenta. Then we perform the Dirac algebra and obtain a set of input integrals for each family. It turns out that fourteen planar and two non-planar families are involved in the n2f contribution of the fermionic form factors. The graphs associated with the non-planar 5

families are shown in Fig. 2.1 For the reduction to master integrals we use FIRE [35–37] which we apply in combination with LiteRed [38, 39]. Using FIRE we reveal 26 and 40 one-scale master integrals for the two non-planar families 7 and 786, respectively. For the analytical computation of these master integrals we follow the same strategy as in our previous work [19] which we briefly summarize for convenience: 1. We introduce a second mass scale by removing one of the quark momenta, q2 , from the light cone, i.e., we have q22 6= 0. Furthermore we define q22 = xq 2 . With the help of FIRE we obtain 91 (101) two-scale master integrals for family 7 (786). The differential equations for these master integrals with respect to x are obtained with the help of LiteRed [38, 39]. 2. To solve our differential equations we turn from the primary basis to a canonical basis [40], where the corresponding master integrals satisfy a system of differential equations with the right-hand side proportional to ǫ and with only so-called Fuchsian singularities with respect to x. To construct our canonical basis we apply the private implementation of one of the authors (R.N.L.) of the algorithm2 discussed in Ref. [44]. 3. Since our equations are in a canonical (or ǫ) form, we write down a solution in a straightforward way order-by-order in ǫ in terms of harmonic polylogarithms (HPL) [47] with letters 0 and 1. 4. We determine the boundary conditions for the canonical master integrals, which are given as linear combinations of the primary master integrals, for x = 1. The primary master integrals are regular at this point and become of propagator type. Thus they are expressed as linear combinations of 28 master integrals. Their analytic ǫ-expansions are well-known [48, 49] up to weight 12. They have been cross-checked numerically in Ref. [50]. 5. We solve our differential equations asymptotically near the point x = 0 and fix these solutions by matching them to our solution at general x. Here we use the package HPL [51] to extract the leading order behaviour of the elements of the canonical basis in the limit x → 0. The asymptotic solutions are linear combinations of powers xkǫ with k = 0, 1, . . . , 8. We pick up asymptotic terms with k = 0 and obtain the so-called naive values of the canonical master integrals at x = 0. 6. From the analytic results for the naive part we obtain analytical results for the sought-after one-scale master integrals after changing back to the primary basis. 1

For convenience we use the internal numeration of the families also in the paper. Two public implementations, Fuchsia [41, 42] and epsilon [43], of the algorithm of Ref. [44] are available. See also [45, 46] where an algorithm for the case of two and more variables is described. 2

6

To make the transition from the point x = 1 to the point x = 0 (cf. items 4. and 5. in the above list) we could apply the prescriptions explained in our previous paper [19]. However, we prefer to use the following slightly modified approach which we find more effective. Let us assume that we have a differential equation in ǫ-form ∂x J(x) = ǫM(x)J(x) ,

M(x) =

X Ma . x − a a

(7)

In our case, the sum over a includes two terms, with a = 0 and a = 1. The formal solution of this equation is the path-ordered exponent   x Z U(x, x0 ) = P exp ǫ dξM(ξ) . (8) x0

This evolution operator can be readily expanded in ǫ with iterated integrals as coefficients. Since we want to put boundary conditions at x = 1, we need to consider the limit x0 → 1. Due to the presence of non-analytic terms of the form (1 − x0 )ǫ , this limit is not well defined when expanding in ǫ. This can be fixed by factoring out the non-analytic piece e (x, x0 )(1 − x0 )−ǫM1 , U(x, x0 ) = U

(9)

e x0 ) has a finite limit for x0 → 1. Therefore, we can write down the general where U(x, solution as J(x) = =

lim U(x, x0 )Cx0

x0 →1

lim U(x, x0 )(1 − x0 )−ǫM1 (1 − x0 )ǫM1 Cx0

x0 →1

e (x, 1)C , = U

(10)

where Cx0 and C are column vectors of constants (depending on ǫ). Cx0 depends in e (x, 1) can be addition on x0 whereas C does not. The “reduced” evolution operator U easily expanded in ǫ with the coefficients being harmonic polylogarithms of x. The column of constants can be fixed by considering the asymptotics of the canonical master integrals for x → 1. In this limit we have J(x) ∼ (1 − x)ǫM1 C .

(11)

Note, that we want to relate C to the coefficients of the asymptotic expansion of the primary master integrals, j(x), which are obtained from the canonical ones via j(x) = T (ǫ, x)J(x).3 This might seem nontrivial since T (ǫ, x) (and T −1 (ǫ, x)) have multiple poles 3

T (ǫ, x) is the transition matrix as introduced in Ref. [44] reducing the system to ǫ form.

7

for x → 1. Therefore, we need to know several terms of the asymptotic expansion of J(x). We identically rewrite Eq. (10) as J(x) = (1 − x)ǫM1 U (x, 1)C .

(12)

e (x, 1) can be expanded in 1 − x up to sufficiently The operator U (x, 1) = (1 − x)−ǫM1 U high power which makes it possible to connect the column of constants in C with the specific coefficients of the asymptotic expansion of the primary master integrals at x = 1. We stress that the described method is extremely economic in the sense that the overall number of asymptotic coefficients of the primary master integrals (each being a function of ǫ) to be fixed can be minimized and is equal to the number of constants in C. In addition, since the integrals are all analytic in the point x = 1, we set to zero all coefficients in front of non-integer powers of 1 − x in the generic solution. This reduces even further the number of coefficients which we need to calculate. We finally find that the boundary conditions are entirely fixed by those entries of j(1) which are present among the 28 integrals from Refs. [48, 49]. Note that within our present approach it is not necessary to calculate several expansion terms of the primary masters near x = 1, in contrast to Ref. [19]. The analysis at x = 0 is simplified in a similar way. Repeating similar considerations for the point x = 0, we finally connect the coefficients of the asymptotic expansion of the primary master integrals for x → 0 with those for x → 1. In particular, we extract the naive values of the primary master integrals at x = 0. Following the procedure outlined in this Section we could compute all master integrals contained in the families 7 and 786 up to weight 8. Analytic expressions for the 24 non-planar master integrals are given in the Appendix.

3

n2f results for form factors

In this Section we discuss the results which we have obtained for the various form factors using the techniques outlined in the previous Section and the analytic results for the master integrals given in the Appendix. We have used a general QCD gauge parameter ξ in the gluon propagator and have expanded each Feynman amplitude up to the linear term. We have checked that the coefficient of ξ vanishes for the bare form factors once all master integrals are mapped to a minimal set.

3.1

Photon quark form factor

For the n2f term of the photon quark form factor only the following three non-planar master integrals are needed: (7)

(7)

(786)

G110110100111 , G110110100112 , G111101101110 8

(13)

Analytic results are given in the Appendix. We insert these results together with the ones for the planar master integrals, expand in ǫ and renormalize αs . After taking the logarithm we can compare to Eq. (4) and extract the cusp and collinear anomalous dimension. We obtain 0 γcusp = 4,   268 4π 2 40nf 1 γcusp = CA − − , 9 3 9   88ζ3 44π 4 536π 2 490 2 2 + − + γcusp = CA 3 45 27 3      2 16n2f 110 836 112ζ3 80π + CF 32ζ3 − − + − , + nf CA − 3 27 27 3 27 3,n0

3,n1

3 γcusp = γcuspf + γcuspf nf "    # 4 2 4 2240ζ 640ζ 56π 304π 923 16π 2392 3 3 + n2f CA + CF − − − + + + 27 135 243 81 9 45 81   64ζ3 32 n3f , (14) − + 27 81

and γq0 = −3CF ,       11π 2 961 65 π 2 3 1 2 γq = 26ζ3 − CA CF + CF nf + −24ζ3 + 2π − CF2 , − + 6 54 27 3 2 "   964ζ3 11π 4 1297π 2 8659 256ζ3 14π 4 13π 2 γq2 = nf − CA CF + + + − − − 27 45 243 729 9 27 9  #   2953 8 844ζ3 247π 4 205π 2 151 + CF2 + − π 2 ζ3 − CA CF2 − 120ζ5 + + − 54 3 3 135 9 4   3526ζ3 83π 4 7163π 2 139345 44 CA2 CF − 136ζ5 − − − + − π 2 ζ3 + 9 9 90 486 2916    16π 2 ζ3 8π 4 8ζ3 10π 2 2417 2 CF nf + − + − 68ζ3 + 240ζ5 − − 3π 2 + − 27 27 729 3 5  29 CF3 , − 2 "  64 2 7436ζ3 592ζ5 19π 4 41579π 2 97189 3 2 − π ζ3 − γq = nf CA CF + − − + 27 243 9 135 8748 34992   # 56π 2 ζ3 2116ζ3 520ζ5 1004π 4 493π 2 9965 + CF2 + − + − − 27 81 9 1215 81 972 9

  712ζ3 16π 4 4π 2 18691 3,n0 3,n1 + − CF n3f + nf γq f + γq f , − − + 243 1215 81 6561

(15) 3,n0

3,n1

3,n0f

where ζn is Riemann’s zeta function evaluated at n. The coefficients γcuspf , γcuspf , γq 3,n1

and γq f are only known in the large-Nc limit [18, 19]. The one- to three-loop results for 3 γcusp and γq can be found in Refs. [12, 13, 29, 52–55] and the n3f terms of γcusp has been 3 2 obtained in Refs. [56, 57]. The nf term of γcusp agrees with [58]. The terms in γq3 which are beyond the large-Nc limit are new. For completeness we also present results for the finite part of Fq which is conveniently done for the bare form factors since at each loop order the µ dependence factorizes. In analogy to Eq. (3) we write X  α0 n  µ2 nǫ s log(Fq ) = log(Fq )|(n) . (16) 2 4π −q − i0 n≥1 The n3f and n2f terms of log(Fq )|(4) are given by (4)

log(Fq )|n2 ,n3 = " f f # "  #   2CF2 11 11 1 2 π 2 395 1 1 3 2 3 CA CF − + CF nf CF nf − CA CF nf + 4 nf − ǫ5 27 18 ǫ 18 54 9 27 "      1 2π 2 481 19ζ3 5π 2 75619 + 3 n2f CA CF + −6ζ3 + CF2 − − − ǫ 3 6 1296 3 108 # "     1 2 2170ζ3 13π 4 1022π 2 2953141 254 5π 2 3 CF nf + 2 nf CA CF + + − − + 81 81 ǫ 27 60 81 7776 #      4 2 2 7π 82ζ3 55π 197π 14309 29023 + −52ζ3 − CF2 + − CF n3f + − + + 90 27 324 81 81 1458 "   206 2 61459ζ3 784ζ5 503π 4 424399π 2 102630137 1 2 − CA CF π ζ3 + + + − − + nf ǫ 27 81 9 810 3888 46656    74π 2 ζ3 15121ζ3 31π 4 16993π 2 1308889 + CF2 − + 62ζ5 + + − 9 27 135 324 3888 #    1714ζ32 218π 2 ζ3 902ζ3 151π 4 1270π 2 331889 2 3 − CF nf + nf + + + − + − 81 2430 243 2916 3 9  2897315ζ3 150886ζ5 709π 6 1861π 4 5825827π 2 3325501813 CA CF + − − − − + 486 135 17010 2430 7776 279936  2702ζ32 1820π 2 ζ3 859249ζ3 1580ζ5 17609π 6 1141π 4 76673π 2 + + − + + + + 3 27 162 3 17010 1620 243    4 25891301 410 2 20828ζ3 2194ζ5 1661π 145115π 2 − CF2 + − π ζ3 − − + + 11664 243 243 135 2430 4374 10

 10739263 CF n3f . + 17496

(17)

The four-loop term in the large-Nc limit can be extracted from Ref. [19]; all other terms are new.

3.2

Higgs quark form factor

The calculation of Fb proceeds in close analogy to Fq . It is interesting to note that about 20% fewer integrals are needed in the case of Fb . However, the complexity of the most complicated integrals is the same and thus the CPU time needed for the reduction to master integrals is comparable for the two calculations. After renormalizing αs and y and taking the logarithm we again compare to Eq. (4) and extract the cusp and collinear anomalous dimension. We obtain the same results as in Eqs. (14) and (15) which constitutes a strong check for our calculation. If we define log(Fb ) expressed in terms of bare αs and y in analogy to Eq. (16) we get for the n3f and n2f terms of log(Fb )|(4) (4)

log(Fb )|n2 ,n3 = " f f # "  #   2 2 1 1 2C 5 π 197 11 1 F CA CF − + CF n3f CF n3f − CA CF n2f + 4 n2f − ǫ5 27 18 ǫ 18 54 9 27 "      1 2 2π 2 283 19ζ3 5π 2 19819 + 3 nf CA CF + −6ζ3 + CF2 − − − ǫ 3 6 1296 3 108 # "     1 1846ζ3 13π 4 491π 2 417697 65 5π 2 3 2 CF nf + 2 nf CA CF + + − − + 81 81 ǫ 27 60 81 7776 #      2 7π 4 119π 2 1244 82ζ 25π 2537 3 + −64ζ3 − CF2 + − CF n3f + − + + 90 27 81 81 81 729 "   1 2 206 2 35089ζ3 784ζ5 251π 4 121495π 2 7185425 + nf − CA CF π ζ3 + + + − − ǫ 27 81 9 810 3888 46656     410ζ3 38π 4 6871π 2 258229 74π 2 ζ3 12079ζ3 CF2 + − − + 62ζ5 − + − + 9 27 135 324 3888 81 # "   1714ζ32 14π 2 ζ3 1209401ζ3 151π 4 325π 2 11408 CF n3f + n2f − + + + + + 2430 243 729 3 9 486   144946ζ5 709π 6 2129π 4 1168375π 2 73152481 2702ζ32 + CA CF + − + − − 135 17010 2430 7776 279936 3   2 6 4 2 632π ζ3 489655ζ3 17609π 4693π 94031π 2614421 + CF2 − − 760ζ5 + − + − 27 162 17010 1620 972 11664 11

#   5330ζ3 2194ζ5 151π 4 12685π 2 159908 410 2 CF n3f . π ζ3 − − + + + + − 243 243 135 486 2187 2187 (18) To obtain this result the ǫ2 terms of the three-loop form factor are needed. We refrain from listing them explicitly but present the expressions, which are not yet available in the literature, in the ancillary file [30].

4

Conclusions

We have computed the complete n2f contributions for the massless four-loop fermionic form factors Fq and Fb and provide the corresponding cusp and collinear anomalous dimensions. This requires to consider two non-planar integral families. We systematically construct the solution applying algorithmic procedures and obtain analytic results for all master integrals contained in these families. Although only three non-planar master integrals are needed for the form factors we present analytic results for all 24 non-planar integrals, one of the main results of this paper. They constitute important ingredients for future calculations, e.g., the n1f and the nf -independent parts. Furthermore, we extend the three-loop result for the Higgs fermion form factor to order ǫ2 . All analytic results can be downloaded from [30].

Acknowledgments This work is supported by the Deutsche Forschungsgemeinschaft through the project “Infrared and threshold effects in QCD”. The work of A.S. and V.S. is supported by RFBR, grant 17-02-00175A.

Appendix: Explicit results for non-planar master integrals The 26 master integrals in family 7 (cf. Fig. 2) are (7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

(7)

G000001111001 , G000000111111 , G000001111110 , G000101111100 , G010000110111 , G110000011011 , G000001111111 , G000101111110 , G010000111111 , G010001111011 , G010001111012 , G010110110101 , G011010110101 , G110001101011 , G000110111111 , G001111111100 , G010010111111 , G010110110111 , G011010111101 , G011010111102 , G011011111001 , G110000111111 , G110001111011 , G110011101011 , G110110100111 , G110110100112 ,

(19) 12

and the 40 master integrals in family 786 are chosen as (786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

(786)

G001010011010 , G001001111010 , G011010001110 , G100010011110 , G100101001110 , G100101011010 , G110010010110 , G011001101110 , G011001110110 , G011010011110 , G100001111110 , G100101111010 , G100110011011 , G100110011110 , G101001011110 , G101001011120 , G110001110110 , G110110001110 , G111001001110 , G011001111110 , G100101111110 , G100111011110 , G101001111110 , G101011011110 , G101101111010 , G101110011011 , G101111011010 , G110101101110 , G111001110110 , G111010010111 , G111010010112 , G111010011110 , G111011010110 , G111110001110 , G101111011110 , G111001111110 , G111011011110 , G111011011120 , G111101101110 , G111111001110 .

(20)

The planar integrals have already been used in Ref. [19] to obtain Fq in the large-Nc limit. Their analytic results will be provided in Ref. [21]. Here, we concentrate on the non-planar integrals. Family 7 has twelve non-planar master integrals. Five of them can be mapped to family 786 using (7)

(786)

(7)

(786)

(7)

(786)

(7)

(786)

(7)

(786)

G010110110111 → G101001111110 , G011011111001 → G101110011011 , G110001111011 → G111010011110 , G110011101011 → G111110001110 , G110110100111 → G111010010111 .

(21)

The analytic results of the remaining seven integrals expanded up to weight eight are given by (7)

G000110111111 =

# " 6 4 2 6 π 67π ζ 47π ζ 591ζ 11π 5ζ5 3 5 7 + 9ζ32 + 55ζ5 + + ǫ 99ζ32 + − + 455ζ5 + + ǫ 30 90 3 2 30 " 76s8a 302ζ5ζ3 737π 4 ζ3 517π 2ζ5 6501ζ7 + ǫ2 − − 5π 2 ζ32 + 819ζ32 − + − + 3355ζ5 + 5 3 90 3 2 # 23689π 8 91π 6 , + + 378000 30

(7)

G001111111100 =

# " " # 1 1 107ζ3 π2 7 1 1498ζ3 119π 4 2 + − + + 63 − − 7π + 465 + − − − 63π 2 2ǫ4 ǫ3 ǫ2 2 ǫ 3 3 180 # " 4 833π 2437ζ 5 − − 465π 2 + 18873 + 3069 + ǫ 33π 2 ζ3 − 4506ζ3 − 5 90 13

"

34118ζ5 6721π 6 7519π 4 11593ζ32 + 462π 2 ζ3 − 33446ζ3 − − − − 3069π 2 9 5 11340 90 " # 162302ζ32 12757π 4ζ3 6871π 2 ζ5 + 110621 + ǫ3 + + 4170π 2 ζ3 − 222702ζ3 + 9 270 15 # 307822ζ5 46762ζ7 6721π 6 55841π 4 − − − − 18873π 2 + 626545 − 5 7 810 90 " 10169 2 2 528478ζ5ζ3 89299π 4 ζ3 + ǫ4 − 712s8a − π ζ3 + 163134ζ32 + + + 30966π 2ζ3 9 15 135 + ǫ2

− 1386486ζ3 +

43481π 8 60439π 6 41349π 4 96194π 2ζ5 − 456778ζ5 − 93524ζ7 − − − 15 # 680400 810 10

− 110621π 2 + 3459765 , (7)

G010010111111 = 71π 6 5ζ5 + 7ζ32 − 2π 2 ζ3 + 45ζ5 + ǫ " 2268 # 4 2 6 7π ζ 40π ζ 907ζ 3023π 3 5 7 + ǫ 47ζ32 − − 34π 2 ζ3 + + 285ζ5 + + 18 3 4 11340 " 582s8a 53 2 2 722ζ5ζ3 367π 4 ζ3 + ǫ2 − + π ζ3 + 127ζ32 − − − 386π 2 ζ3 + 128π 2 ζ5 5 3 3 90 # 9097ζ7 341261π 8 2473π 6 , + + + 1425ζ5 + 4 1701000 1620 (7)

G011010111101 = " # 5 2 351ζ3 3ζ5 143π 4 3ζ3 1 39ζ3 11π 4 + − + + π ζ + + + 3 2ǫ2 ǫ 2 360 2 2 2 360 " # 263ζ32 57π 2 ζ3 2715ζ3 79ζ5 2519π 6 143π 4 +ǫ − − + + − + 2 2 2 2 22680 40 " 3299ζ32 2491π 4ζ3 433π 2 ζ3 19383ζ3 269π 2 ζ5 1111ζ5 + ǫ2 − − − + − + 2 540 2 2 6 2 " # 755 2 2 28491ζ32 13531ζ5ζ3 2213ζ7 29219π 6 1991π 4 + ǫ3 316s8a + − + π ζ3 − − − 4 22680 72 6 2 5 −

6329π 4 ζ3 2677π 2 ζ3 131859ζ3 3263π 2 ζ5 11955ζ5 13445ζ7 13939π 8 − + − + − − 108 2 2 6 2 2 24300

14

# 2811π 6 71071π 4 , + − 280 360 (7)

G011010111102 = # # # # # " " " " " 1 1 1 89ζ3 2 1 89ζ3 7π 4 4 1 1 1 1 + 5 − + 4 − + 3 + 2 − − + − ǫ6 36 ǫ 18 ǫ 9 ǫ 54 9 ǫ 27 120 9 " # 1 7π 2 ζ3 178ζ3 2117ζ5 7π 4 8 + + + + − ǫ 9 27 90 60 9 " 2 6 4 2 14π ζ3 356ζ3 2117ζ5 2573π 7π 16 9271ζ32 9271ζ3 + + + + + − +ǫ − − 162 9 27 45 34020 30 9 81 4

2

6

4

#

667π ζ3 28π ζ3 712ζ3 7π 32 4234ζ5 21151ζ7 2573π + + + 9π 2 ζ5 + + + + − 180 9 27 45 252 17010 15 9 " 1528 2 2 18542ζ32 230803ζ5ζ3 667π 4ζ3 344π 2 ζ3 1424ζ3 2 848s8a − π ζ3 − − − + + +ǫ 5 27 81 135 90 9 27 # 15668ζ5 21151ζ7 500903π 8 2573π 6 14π 4 64 + 18π 2 ζ5 + , + − + + − 45 126 3402000 8505 15 9 −

(7)

G110000111111 = # " # " # " 1 1 1 1 37ζ3 π 2 93 π 2 ζ3 1 91 π 2 5ζ5 + 4 + 3+ 2 + − + − − + − 48ζ3 + ǫ 12 ǫ ǫ 12 36 ǫ 9 3 2 3 3 " 37π 4 95π 2 3025 5ζ32 115π 2 ζ3 3131ζ3 2038ζ5 88π 6 193π 4 − − + +ǫ + − − + − 540 36 12 3 27 9 45 8505 270 " # 2543ζ32 71π 4 ζ3 1039π 2ζ3 53390ζ3 π 2 ζ5 79163ζ5 955π 2 + 1262 + ǫ2 + + − − − − 54 27 270 27 27 9 135 " # 407ζ7 799π 6 7261π 4 35449π 2 71767 346s8a 94 2 2 + + ǫ3 − − − − + − π ζ3 12 14580 1620 324 12 15 3 25763ζ32 1694ζ5ζ3 1073π 4ζ3 26051π 2ζ3 761281ζ3 383π 2 ζ5 1657792ζ5 − + + − + − 27 9 270 81 81 135 405 # 8 6 4 2 278843ζ7 461π 14227π 24929π 157496π 54491 − , − − − − + 252 70875 15309 1215 243 2 +

(7)

G110110100112 = # # # " " " 1 1 π2 ζ3 π 2 2ζ3 π 4 π 2 1 + 3 − − + 2 − − − + 4 − ǫ 36 ǫ 3 18 ǫ 3 30 9

15

" # 361ζ32 74π 2 ζ3 8ζ3 π 4 2π 2 71π 6 1 37π 2 ζ3 4ζ3 + − + 6ζ5 − − + − + 12ζ5 + + ǫ 27 3 15 9 9 27 3 1890 " 2π 4 4π 2 3925ζ7 722ζ32 44π 4 ζ3 148π 2ζ3 16ζ3 886π 2ζ5 − − +ǫ + + − + + 24ζ5 + 15 9 9 15 27 3 45 8 " # 869s8a 2738 2 2 1444ζ32 11224ζ5ζ3 88π 4 ζ3 71π 6 4π 4 8π 2 + ǫ2 − − − − π ζ3 + + + + 945 15 9 5 81 9 15 15 296π 2ζ3 32ζ3 1772π 2ζ5 3925ζ7 365909π 8 142π 6 8π 4 − + + 48ζ5 + + + − 27# 3 45 4 972000 945 15 2 16π , − 9

+

(22)

where s8a = ζ8 + ζ5,3 ≈ 1.0417850291827918834 .

(23)

ζm1 ,...,mk are multiple zeta values given by ζm1 ,...,mk =

∞ iX 1 −1 X

···

i1 =1 i2 =1

ik−1 −1 k X Y ik =1 j=1

sgn(mj )ij |mj |

ij

.

(24)

Family 786 has 17 non-planar master integrals. Their analytic results read (786)

G101001111110 = # " # " # " 1 39ζ3 11π 4 1 2 1 3ζ3 351ζ3 23ζ5 143π 4 + + − π ζ3 + + 2 + + + ǫ 2 ǫ 2 360 2 2 2 360 " # 211ζ32 13π 2 ζ3 2715ζ3 299ζ5 121π 6 143π 4 +ǫ − − + + − + 2 2 2 2 3240 40 " 2743ζ32 2257π 4 ζ3 117π 2 ζ3 19383ζ3 28π 2 ζ5 2691ζ5 1925ζ7 2 +ǫ − − − + + + − 2 540 2 2 3 2 4 " # 98 2 2 24687ζ32 28928ζ5ζ3 29341π 4ζ3 1573π 6 1991π 4 3 1603s8a +ǫ + + π ζ3 − − − − 3240 72 5 3 2 15 540 905π 2ζ3 131859ζ3 364π 2 ζ5 20815ζ5 25025ζ7 907097π 8 1573π 6 + + + − − − 2 2 3 2 4 2268000 360 # 4 71071π + , 360 −

(786)

G101011011110 = 16

# " # " # " 1 33ζ3 π 4 5 2 3ζ3 201ζ3 71ζ5 91π 4 1 + − + − π ζ3 − − − − + 2 − ǫ 2 ǫ 2 40 6 2 2 360 " # 5ζ32 33π 2 ζ3 501ζ3 913ζ5 3673π 6 29π 4 +ǫ − − − − − 2 2 2 2 22680 24 " 793ζ32 283π 4 ζ3 1259π 2ζ3 5991ζ3 79π 2 ζ5 7529ζ5 6675ζ7 − − + + − − + ǫ2 − 2 180 6 2 2 2 4 " # 18701π 6 713π 4 4632s8a 2339 2 2 17473ζ32 26513ζ5ζ3 4421π 4ζ3 − + ǫ3 + + π ζ3 − + − 7560 360 5 18 2 15 108 4341π 2ζ3 118371ζ3 1107π 2 ζ5 48421ζ5 51301ζ7 1726121π 8 + + − − − 2 2 2 2 2 1701000 # 6 4 575791π 13823π − , + 22680 120 −

(786)

G101101111010 = # # " " " 1 1 1 112ζ3 19π 4 35π 2 5ζ3 5π 2 109 1 13 π 2 + + − − + − − + − − 4ǫ4 ǫ3 4 12 ǫ2 2 6 4 ǫ 3 144 6 # 157π 2ζ3 1024ζ3 125ζ5 1043π 4 71π 2 4677 753 + − − − − + + 4 36 3 4 720 2 4 " # 953ζ32 803π 2ζ3 7570ζ3 2471ζ5 893π 6 883π 4 403π 2 27225 +ǫ + − − − − − + 6 18 3 5 18144 80 2 4 " 16979ζ32 20909π 4ζ3 2896π 2ζ3 6563π 2 ζ5 46481ζ5 3805ζ7 + ǫ2 + + − 16670ζ3 + − + 9 2160 9 120 10 32 " # 3638 2 2 75931π 6 52279π 4 2199π 2 151933 + ǫ3 − 299s8a − − − + π ζ3 − 90720 720 2 4 27 6773π 2ζ5 138134ζ32 26623ζ5ζ3 22397π 4 ζ3 6100π 2ζ3 + + + − 103214ζ3 + 9 6 216 3 12 349893ζ5 108151ζ7 1637173π 8 241853π 6 319363π 4 35129π 2 − − + − − − 10 # 56 3628800 30240 720 6 823921 + , 4

+

(786)

G101110011011 =

# " # " 1 40ζ3 13π 2 753 538ζ3 53π 4 109π 2 13 1 π 2 109 1 + + − + − + + − 4− 3+ 2 4ǫ 4ǫ ǫ 12 4 ǫ 3 12 4 3 180 12

17

# " 46 2 4702ζ3 846ζ5 79π 4 251π 2 27225 4677 + ǫ − π ζ3 + + + + − − 4 9 3 5 20 4 4 " 4055ζ32 625π 2 ζ3 11428ζ5 1345π 6 413π 4 1559π 2 + ǫ2 − − + 11378ζ3 + + + + 9 9 5 4536 12 4 " # 56162ζ32 2107π 4 ζ3 5527π 2ζ3 706π 2 ζ5 151933 + ǫ3 − − − + 74906ζ3 − − 4 9 108 9 15 # 100384ζ5 89139ζ7 1963π 6 8963π 4 9075π 2 823921 + + + + + − 5 56 504 36 4 4 " 3308s8a 10597 2 2 507488ζ32 808π 4 ζ3 13553π 2ζ3 + ǫ4 + π ζ3 − − 11699ζ5ζ3 − − 5 54 9 3 3 9583π 2ζ5 732928ζ5 1154873ζ7 1191469π 8 50021π 6 + + − + 15 5 # 56 4536000 1512 4 2 97961π 151933π 4378933 + , + − 60 12 4

+ 466338ζ3 −

(786)

G101111011010 =

# " # " 1 293ζ3 23π 4 11π 2 1107 15 1 π 2 143 1 + − + + − 3ζ3 − + − − + 4ǫ4 4ǫ3 ǫ2 4 4 ǫ 6 360 3 4 " # " 17π 2 ζ3 2017ζ3 77ζ5 779π 4 821π 2 7599 + + ǫ 246ζ32 − − − − + 6 4 2 720 24 4 # 187π 2 ζ3 102037ζ3 12809ζ5 487π 6 16277π 4 4151π 2 48267 + − − + − − + 4 24 20 7560 1440 16 4 " 29987ζ32 290π 4ζ3 34609π 2 ζ3 513681ζ3 52π 2 ζ5 53407ζ5 + + − + − + ǫ2 9 27 72 16 3 8 " # 1771ζ7 7391π 6 54919π 4 55901π 2 290551 2014 2 2 + + ǫ3 − 1104s8a − + − − + π ζ3 2 15120 576 32 4 9 62409ζ32 96836ζ5ζ3 19859π 4ζ3 577855π 2ζ3 7274999ζ3 5638π 2 ζ5 + + + − + 2 15 135 144 32 15 4519813ζ5 231549ζ7 46621π 8 70573π 6 458701π 4 697507π 2 + + + − − − 80 # 28 37800 30240 640 64 1682259 + , 4 +

(786)

G111001110110 =

18

# " # " 1 1 28ζ3 13π 2 251 418ζ3 13 1 109 π 2 + − +− + + 2 − − + 4 3 12ǫ 12ǫ ǫ 12 36 ǫ 9 36 4 9 # " 29π 4 109π 2 1559 55π 2 ζ3 4078ζ3 16ζ5 41π 4 251π 2 9075 − − + +ǫ − − − − + 270 36 4 27 9 15 27 12 4 " 3593ζ32 742π 2 ζ3 11042ζ3 1873ζ5 293π 6 1894π 4 1559π 2 + − − + − − + ǫ2 27 27 3 15 7560 135 12 " # 151933 56888ζ32 2849π 4 ζ3 6508π 2ζ3 81290ζ3 1067π 2 ζ5 33379ζ5 + + ǫ3 + + − + − 12 27 324 27 3 90 15 " # 107831ζ7 2927π 6 1622π 4 3025π 2 823921 2556s8a 6131 2 2 + + ǫ4 − + − − + − π ζ3 84 7560 15 4 12 5 81 7408π 2 ζ5 585038ζ32 101287ζ5ζ3 51769π 4ζ3 15812π 2ζ3 + + + − 188038ζ3 + 27 45 405 9 45 127781ζ5 2737267ζ7 1794577π 8 15179π 6 34106π 4 151933π 2 + + + − − − 5 # 168 1944000 7560 45 36 4378933 + , 12

+

(786)

G111010011110 =

# " # " 1 1 23ζ3 7π 2 455 7 1 125 π 2 + − + + − − + 24ǫ4 12ǫ3 ǫ2 24 72 ǫ 9 36 12 " 50π 2 ζ3 3415ζ3 158ζ5 293π 4 349ζ3 79π 4 125π 2 1967 − − + +ǫ − − − − 9 1080 72 8 27 9 15 270 " # 455π 2 5929 3493ζ32 673π 2 ζ3 27545ζ3 3007ζ5 83π 6 − + ǫ2 + + − − + 36 4 27 27 9 15 2835 " # 2239π 4 1967π 2 204205 53789ζ32 10697π 4ζ3 5629π 2 ζ3 66661ζ3 − + ǫ3 − + + + − 216 24 24 27 1620 27 3 # 1538π 2 ζ5 7130ζ5 98507ζ7 205π 6 4403π 4 5929π 2 566335 − + + − − + + 45 3 168 648 54 12 12 " 1677s8a 9011 2 2 534284ζ32 112502ζ5ζ3 32201π 4ζ3 73315π 2ζ3 + ǫ4 − − π ζ3 + + + + 5 162 27 45 324 54 454363ζ3 84103π 2ζ5 67891ζ5 191359ζ7 1062491π 8 79579π 6 + − + + + 3 180 3 # 24 1701000 45360 2 4 204205π 6129101 69251π , − + − 120 72 24 −

19

(786)

G111011010110 =

# " # " 1 28ζ3 67π 2 319 1 71 5π 2 11 1 − 4− + − + − − − 12ǫ 12ǫ3 ǫ2 12 36 ǫ 9 36 12 " 463ζ3 31π 4 287π 2 269 731π 2 ζ3 1841ζ3 4339ζ5 1997π 4 + − − − +ǫ + + − 18 270 18 4 108 36 60 1080 " # 8047π 2 835 4609ζ32 20273π 2 ζ3 78925ζ3 88993ζ5 2459π 6 − + ǫ2 + + − + + 72 4 108 216 72 120 12960 # 41221π 4 33637π 2 54833 − + − 2160 48 12 " 2 34867π 4 ζ3 363883π 2ζ3 865033ζ3 1247π 2 ζ5 939131ζ5 3 287959ζ3 +ǫ + + − + + 216 3240 432 48 18 240 " # 9407s8a 1875815ζ7 404363π 6 697471π 4 393983π 2 516097 + ǫ4 − + − − + + 672 181440 4320 96 12 10 69871 2 2 8620349ζ32 6038ζ5ζ3 1137997π 4ζ3 5424973π 2ζ3 π ζ3 + + + + 324 432 45 6480 864 17813807ζ3 168103π 2ζ5 2973053ζ5 49508561ζ7 95485771π 8 − + + + + 96 180 480 1344 # 54432000 6 4 2 5863657π 3526939π 13246403π 3863977 + , − − + 362880 2880 576 12

−

(786)

G111110001110 =

# " " # 1 1 47ζ3 31π 2 29 5π 2 5 1 326ζ3 151π 4 247π 2 − 4− 3+ 2 − + − − − 37 + − − 6ǫ 3ǫ ǫ 3 18 ǫ 9 9 9 540 9 " # " 923ζ32 361π 2 ζ3 170ζ3 1187ζ5 110π 4 539π 2 + + − − + 723 + ǫ2 − 49 + ǫ 27 9 15 27 3 27 # 4462π 2 ζ3 5846ζ3 1642ζ5 223π 6 1031π 4 3167π 2 27997 + − + + − − + 27 3 3 2835 27 3 3 " 40046ζ32 8161π 4ζ3 35422π 2ζ3 76862ζ3 1616π 2 ζ5 3446ζ5 + ǫ3 + + − + + 27 405 27 3 9 15 " # 25913 2 2 231793 213683ζ7 191π 6 4408π 4 + ǫ4 − 560s8a − − − − 5809π 2 + π ζ3 + 84 1260 15 3 81 +

98404π 2 ζ5 597926ζ32 60682ζ5ζ3 119101π 4ζ3 76982π 2ζ3 + + + − 231202ζ3 + 27 45 405 9 45

20

152482ζ5 548876ζ7 2242787π 8 44237π 6 91172π 4 275869π 2 + + − − − 5 # 21 1360800 3780 45 9 1621429 , + 3 −

(786)

G111010010111 =

# " # " 1 1 73ζ3 65π 2 251 895ζ3 19π 4 13 1 109 5π 2 + − − + + + + + − 12ǫ4 12ǫ3 ǫ2 12 36 ǫ 9 36 4 9 540 # " 185 2 6931ζ3 2896ζ5 181π 4 1255π 2 9075 545π 2 1559 + +ǫ − π ζ3 − − − + + + 36 4 27 9 15 540 12 4 " 6419ζ32 2405π 2 ζ3 14309ζ3 36133ζ5 214π 6 817π 4 7795π 2 + ǫ2 − − − − − + 27 27 3 15 405 540 12 " # 73268ζ32 892π 4 ζ3 20165π 2ζ3 76169ζ3 1102π 2 ζ5 151933 + ǫ3 + − − − + 12 27 405 27 3 9 # 286879ζ5 679841ζ7 607π 6 137π 4 15125π 2 823921 − − − + + + 15 168 90 180 4 12 " 1237s8a 13690 2 2 506270ζ32 422812ζ5ζ3 5833π 4 ζ3 46435π 2 ζ3 + π ζ3 + + + − + ǫ4 5 81 27 45 405 9 14119π 2ζ5 614281ζ5 4406629ζ7 11991739π 8 14911π 6 − − − − 9 5 # 84 6804000 270 4 2 16381π 759665π 4378933 + , + + 180 36 12 − 117487ζ3 −

(786)

G111010010112 =

# # " " " 1 1 116ζ3 π 2 16 1 464ζ3 8π 4 π 2 2 π2 2 1 + 2 + − 5− 4+ 3 − − − − + − 18ǫ 9ǫ ǫ 3 36 ǫ 27 9 9 ǫ 27 135 3 # 40 37π 2ζ3 464ζ3 3557ζ5 32π 4 257π 2 32 − + + + + − − 9 27 9 45 135 54 3 " # 11323ζ32 148π 2 ζ3 2458ζ3 14228ζ5 691π 6 32π 4 1985π 2 224 +ǫ − + + + + + − − 81 27 27 45 3402 45 54 9 " 45292ζ32 319π 4 ζ3 148π 2ζ3 1910ζ3 1246π 2 ζ5 14228ζ5 675607ζ7 − + − + + + + ǫ2 − 81 81 9 27 45 15 504 " # 167s8a 2738 2 2 45292ζ32 1382π 6 371π 4 4705π 2 512 + ǫ3 − − − − π ζ3 − + 1701 135 18 9 5 81 27

21

614624ζ5ζ3 1276π 4 ζ3 19018π 2ζ3 20230ζ3 4984π 2ζ5 151444ζ5 − + − + + 135 81 81 9 45 45 # 8 6 4 2 1382π 991π 98941π 675607ζ7 2968111π + + − − − 128 , + 126 6804000 567 27 54 −

(786)

G101111011110 = # " # " " # 1 1 197 4 1 2 37ζ32 1793π 6 1 29π 2 ζ5 963ζ7 + 3 − π ζ3 − 5ζ5 + 2 − + − − π ζ3 + − ǫ 12 ǫ 2 45360 ǫ 360 6 2 + 380s8a +

1045 2 2 269ζ5 ζ3 1047407π 8 π ζ3 − − , 36 6 2721600

(786)

G111001111110 = " # 1 176π 6 + 2π 2 ζ3 + 10ζ5 + 10ζ32 + 20π 2 ζ3 + 100ζ5 + ǫ 2835 # " 2 6 4 3π ζ 1509ζ 352π 25π ζ 5 7 3 + 168π 2ζ3 − + 840ζ5 + + + ǫ 100ζ32 + 36 2 8 567 " 962s8a 547 2 2 4870ζ5ζ3 125π 4 ζ3 + ǫ2 − − π ζ3 + 840ζ32 − + + 1360π 2ζ3 − 15π 2 ζ5 5 3 3 18 # 7545ζ7 24163π 8 704π 6 , − + + 6800ζ5 + 4 567000 135 (786)

G111101101110 = # # " # " " 1 π2 1 7ζ3 π 2 1 17π 4 + 4 + 3 + 2 14ζ3 + + + π2 ǫ 12 ǫ 2 3 ǫ 72 " # 71 2 133ζ5 17π 4 8π 2 1 − π ζ3 + 42ζ3 + + + + ǫ 18 2 18 3 1429ζ32 142π 2 ζ3 1927π 6 17π 4 20π 2 − + 112ζ3 + 266ζ5 + + + 9 7560 6 3 "6 2 4 2 2 2858ζ3 8879π ζ3 142π ζ3 489π ζ5 395ζ7 1927π 6 +ǫ − − − + 280ζ3 − + 798ζ5 + + 3 540 3 10 4 1890 " # 68π 4 126577ζ5ζ3 8879π 4ζ3 2582s8a 6085 2 2 + + 16π 2 + ǫ2 + π ζ3 − 2858ζ32 − − 9 5 54 15 135 −

1136π 2ζ3 978π 2 ζ5 1170293π 8 1927π 6 170π 4 + 672ζ3 − + 2128ζ5 + 395ζ7 − + + 9 # 5 2268000 630 9 2 112π , + 3

−

22

(786)

G111111001110 = # " # " # " 1 4π 4 1 1 2π 4 2 + 3 6ζ3 + 2 60ζ3 + + − 6π ζ3 + 504ζ3 + 78ζ5 + ǫ ǫ 15 ǫ 3 " # " 38π 6 56π 4 1343π 4ζ3 + − 632ζ32 − 60π 2 ζ3 + 4080ζ3 + 780ζ5 − + ǫ − 6320ζ32 − + 135 5 45 # 76π 6 272π 4 − 504π 2ζ3 + 32736ζ3 + 2π 2 ζ5 + 6552ζ5 − 4476ζ7 − + 27 3 " 84184ζ5ζ3 2686π 4 ζ3 2 2 2 2 − − 4080π 2 ζ3 + 262080ζ3 + ǫ 5088s8a + 600π ζ3 − 53088ζ3 − 5 9 # 8 6 4 76469π 1064π 10912π + 20π 2 ζ5 + 53040ζ5 − 44760ζ7 − , − + 14175 45 15 (786)

G111011011110 = " # " # 1 7π 4 ζ3 5π 2 ζ5 441ζ7 87s8a 23 2 2 473ζ5ζ3 8069π 8 + + − , − − − π ζ3 − − ǫ 360 3 16 2 6 2 777600 (786)

G111011011120 = # # # # " " " " 1 7ζ3 1 463π 4 1 1247ζ5 127π 2 ζ3 π2 1 + 4 + 3 + 2 − + 5 − ǫ 96 ǫ 16 ǫ 8640 ǫ 48 144 " # 1 38761π 6 1079ζ32 78923π 4 ζ3 18301π 2ζ5 161ζ7 + − − + − ǫ 362880 12 12960 720 24 " # 3291s8a 55183 2 2 330689ζ5ζ3 122374187π 8 +ǫ . + π ζ3 − − 5 216 90 653184000

(25)

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23

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