The heavy fermion contributions to the massive three loop form factors

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Nuclear Physics B ••• (••••) ••••••

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The heavy fermion contributions to the massive three loop form factors

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J. Blümlein , P. Marquard , N. Rana , C. Schneider

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a Deutsches Elektronen–Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany b INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

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c Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, A–4040,

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Linz, Austria

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Received 2 August 2019; received in revised form 28 August 2019; accepted 28 August 2019

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Editor: Tommy Ohlsson

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Abstract

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We compute the non-singlet nh terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals have to be dealt with only in their moment representation, allowing to also consider quantities which obey differential equations, which are not first order factorizable (elliptic and higher), already at this level. To obtain all the associated recursions, up to 8000 moments had to be calculated. A new technique has been applied to solve the associated differential equation systems. Here the decoupling is performed such, that only minimal depth ε–expansions had to be performed for non–firstorder factorizing systems, minimizing the calculation of initial values. The pole terms in the dimensional parameter ε can be completely predicted using renormalization group methods, as confirmed by the present results. A series of contributions at O(ε0 ) have first order factorizable representations. For a smaller number of color–zeta projections this is not the case. All first order factorizing terms can be represented by harmonic polylogarithms. We also obtain analytic results for the non–first-order factorizing terms by Taylor series in a variable x, for which we have calculated at least 2000 expansion coefficients, in an approximation. Based on this representation the form factors can be given in the Euclidean region and in the region q 2 ≈ 0. Numerical results are presented. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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E-mail address: [email protected] (P. Marquard). https://doi.org/10.1016/j.nuclphysb.2019.114751 0550-3213/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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1. Introduction

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The knowledge of the massive three–loop form factor is essential ingredient to the calculation for a series of massive processes at e+ e− and hadron colliders, determined by vector, axialvector, scalar and pseudoscalar currents. It has been calculated to two–loop order in Refs. [1–6]. At three–loop order the color planar contributions have been computed in Refs. [7–12] and its asymptotic behavior has been studied in [13,14], including partial results at four–loop order. In the present paper, we compute the non-singlet nh contributions of the massive three–loop form factor for vector, axialvector, scalar and pseudoscalar currents. As the basic computational method we use the method of arbitrarily large moments [15]. Here the differential equations given by the integration by parts (IBP) relations [16–23] are transformed into recursions, through which a large number of moments for the master integrals and the form factors are generated using the package SolveCoupledSystems [15]. These moments are sequences in Q parameterized by multiple zeta values (MZVs) [25] and color factors. Using the method of guessing [26]1 we determine the associated difference equations, which are finally solved using Sigma [24,28]. In the expansion of the form factors to master integrals usually higher order terms in ε = 2 − D/2 are contributing. These are containing, however, elliptic and more involved contributions. Although being present, these terms cannot be distinguished from the simpler contributions considering moments since they appear only encoded as rational numbers. The advantage of the present method is that it always allows to obtain difference equations for all MZV and color projections. In the case of the pole terms and a large number of contributions at O(ε 0 ) the corresponding difference equations are first order factorizable and can therefore be solved by Sigma. In case of the remaining terms we are able to factorize the first order factors. The remainder terms need other methods to be solved. The first order factorizable contributions are given by iterative integrals, cf. [29]. In the present case these iterative integrals are harmonic polylogarithms (HPLs) [30]. The paper is organized as follows. After some basic definitions given in Section 2 main steps of the calculation are described in Section 3. In Section 4 the universal infrared structure of the form factors is presented which is later compared with the unrenormalized three–loop form factors providing a check of the calculation. In Section 5 we describe a new decoupling strategy, which allows to work with a minimal–depth expansion concerning the initial values. A brief summary on the recurrences, which are not factorizing to first order, is given in Section 6. In Section 7 we present the analytic results for the non-singlet nh –contributions to the different form factors and also give numerical illustrations. Section 8 contains the conclusions. In the appendix we present a series of deeper ε–expansions for some integrals defining the initial conditions.

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The basic structure of the massive form factors has been described in Ref. [6] before. We consider vector, axialvector, scalar and pseudoscalar currents coupling to a heavy quark pair of mass m

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uc (q1 )Xcd vd (q2 ), with q = q1 + q2 . The main variable considered is x given by

(2.1)

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2. The form factors

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1 For an early application of this method in perturbative calculations in Quantum Field Theory, cf. [27].

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q2 (1 − x)2 =z=− . (2.2) 2 x m We work in D = 4 − 2ε dimensions. In the axialvector and pseudoscalar case we can use an anticommuting γ5 since we only consider the non-singlet contributions. We consider the decay amplitude (μ ) of the Z-boson into a pair of heavy quarks. The general structure of μ consists of six form factors, two of which are CP odd. As we consider only higher order QCD effects and Standard Model (SM) neutral current interactions to lowest order, the CP invariance holds. This implies that μ has four form factors FV ,i (s), FA,i (s) i = 1, 2 comprising the following general form μ

μ

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= constants as defined by where σ μν

i μ ν 2 [γ , γ ], q

Q

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gV2 ,1 =

(3 − 2ε)x 2

(1 − ε)(1 + x)4 2x 2 [−1 + ε(1 − x)2 + (4 − x)x] gV2 ,2 = (1 − ε)(1 − x)2 (1 + x)4 x2 2 gA,1 = (1 − ε)(1 − x 2 )2 2x 2 [1 − ε(1 + x)2 + x(4 + x)] 2 gA,2 = . (1 − ε)(1 − x)4 (1 + x)2

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(2.4)

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(2.5)

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(2.6)

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(2.7)

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(2.8)

We will use this decomposition throughout the present calculation. Furthermore, we consider a general neutral particle h that couples to heavy quarks through the following Yukawa interaction  m ¯ ¯ 5 Q h, Lint = − sQ QQ (2.9) + ipQ Qγ v √ where m denotes the heavy quark mass, v = ( 2GF )−1/2 is the SM Higgs vacuum expectation value, with GF being the Fermi constant, sQ and pQ are the scalar/pseudo-scalar coupling, respectively, and Q and h are the heavy quark and scalar and pseudo-scalar field, respectively. The decay amplitude of h → Q¯ + Q, Xcd ≡ cd , consists of two form factors with the following general structure

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Q

x 4(1 − ε)(1 + x)2 x2 gV1 ,2 = (1 − ε)(1 − x 2 )2 x 1 gA,1 = 4(1 − ε)(1 + x)2 x2 1 gA,2 = (1 − ε)(1 − x 2 )2

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(2.3)

e is the charge of positron, θw is the weak mixing angle, T3 is the third component of the weak isospin, and QQ is the charge of the heavy quark. In case of the vector and axialvector form factors it is convenient to use their decomposition into two parts, respectively, which are labeled by the functions (cf. [6]2) gV1 ,1 =

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T3 e . aQ = − sin θw cos θw 2

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= q1 +q2 , and vQ and aQ are the SM vector and axial-vector coupling

T  e 3 vQ = − sin2 θw QQ , sin θw cos θw 2

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cd = V ,cd + A,cd      i μν 1 μ = −iδcd vQ γ μ FV ,1 + σ qν FV ,2 + aQ γ μ γ5 FA,1 + q γ5 FA,2 2m 2m

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2 Note, that due to different conventions g presented here slightly differ from [6]. i

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cd = S,cd + P ,cd  m  = − δij sQ FS + ipQ γ5 FP , (2.10) v where FS and FP denote the renormalized scalar and pseudo-scalar form factors, respectively. The form factors obey the expansion ∞  (k) Fi,l (x, as ) = δl,1 + ask Fi (x), (2.11) k=1

with i = V , A, S, P and l = 1, 2 for i = A, V and as = αs /(4π) denotes the strong coupling constant.

(3.1)

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(3.2)

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The systems of differential equations are thus mapped into associated systems of difference equations. To work with moments for the master integrals has the advantage that also general non–first order factorizing cases can be treated. The difference equations are now solved consecutively starting from the highest pole in ε working through to the required power in the dimensional parameter ε. The way of decoupling is here very important, cf. [37] and Section 5 for details, since the required depth in expanding in ε for the associated initial values, may easily go beyond the level, which is currently known. We have performed those extensions in a series of cases, see Appendix A. In general those extensions are costly and time consuming. In the present case they could be avoided and the initial values are either known from the literature [38] or having been calculated for other previous applications, cf. [39–42], turned out to be sufficient. We could work with a minimal number of additional terms in the ε-expansion, here in the non–first order factorizing case. For each color–zeta projection of the given massive form factor one obtains a series of rational numbers mk,l,n , where −n labels the power in ε. We seek now a minimal recurrence determined by the set {mk,l,n |l = 0..Nmax }.

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The diagrams of the nh –contributions of the different massive three loop form factors are generated using QGRAF [31] and the color structures are evaluated using Color [32]. Furthermore, we used q2e/exp [33,34] and perform the Dirac-algebra using Form [35,36]. In the present paper we perform the calculation of the form factor for QCD, setting CA = Nc = 3, CF = (Nc2 − 1)/(2Nc ) = 4/3 and TF = 1/2, to reduce the complexity of the problem. The only free parameters are the numbers of equal mass heavy flavors nh and massless flavors nl , which partly occur together with zeta-values. In the expansions, in which we mainly work, this decomposition is unique. By transforming to x–space additional ζ –terms may occur e.g. due to regularizations. The IBP reduction is performed using the package Crusher [23] and systems of linear ordinary differential equations are obtained for the master integrals. To the nh case 14 families with a total of 103 master integrals contribute. We map to the new variable y in which the master integrals obey the Taylor expansions ∞  Mk (ε, x) = m ˜ k,l (ε)y l .

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(3.3)

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For each family we obtain a system of differential equations, the largest of which has a coefficient matrix of 7 × 7. The form factor can then be rewritten as 0  1 (3) F (3) (x) = F (y), ε k −k k=3

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∞ 

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a−k,l y l .

(3.4)

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Here the expansion coefficients a−k,l obey difference equations in l for k ∈ {3, 2, 1, 0}, which are parameterized by polynomials of color factors and multiple zeta values (MZVs) [25] over Q, irrespective of the fact whether in the general l representation elliptic or higher structures contribute or not, see also [43,44]. Also the master integrals are rewritten in moment–form (3.4). Their recurrences are finally used to calculate a large set of moments, assembled to ak,l , using the method of large moments of Ref. [15], which are projected to the different contributing monomials in MZVs and nl , nh . The number of moments, now given by sequences of rational numbers, needs to be large enough to allow the determination of their recurrence by the method of guessing [26], implemented in Sage [45], which has been successfully applied in different calculations in Refs. [15,27] before. In the case of the pole terms of the three–loop form factors all the corresponding recurrences have to be solvable in difference rings [24,28,46–58], since they are expected to factorize at first order. The solution can therefore be found using the package Sigma [24,28]. This may be as well the case for some of the MZV-factors contributing to the constant term, as in the case for (3) the massive operator matrix element AQg , cf. Ref. [15]. As result one obtains representations of ak,l in terms of harmonic sums [59,60] and generalized harmonic sums [61,70] in l. The latter ones occur because of the necessary transformation x → (1 − y) to also deal with the logarithmic contributions ∝ lnm (x). Cyclotomic or finite binomial sums [62,63] do not contribute. The infinite sums appearing in (3.4) can now be performed using a series of procedures of the package HarmonicSums [59–67] and one obtains harmonic polylogarithms (HPLs) [30] or Kummer–Poincaré iterated integrals [61,68–70] in the variable y. The transformation y = 1 − x then yields representations in terms of HPLs in x, which are further reduced to suitable bases, to reduce the numbers of contributing functions as much as possible. We remark, that the different master integrals for the present representation partly need deep expansion in the dimensional variable ε. If we needed to write them in real terms, also elliptic and even higher integral representations would be needed in explicit form. It is an advantage of the present method that these contributions do earliest show up in the factorization of some of the recursions to be solved for the physical quantity under consideration, but cancel otherwise. The automated solution of differential equations over general bases, presented in Ref. [12], working in the case of first order factorization, can therefore not be applied here. In the following we discuss a sample calculation to illustrate the general method. For all computation we used the qftquad–cluster equipped with Xeon Gold 6128 and Xeon 6C E5-2643v4 processors. For this we choose all contributions to the massive three–loop vector form factors ∝ nl . They have been calculated up to O(ε 0 ) by using different methods in Refs. [10,12]. By setting Nc = 3, 60 different recurrences have to be found and solved. A linear combination of these quantities yields the vector current form factors FV ,1 and FV ,2 . The reduction to master integrals led to maximally 5 × 5 systems, which were decoupled by using the Gauss approach implemented in the package Oresys [108] with a decoupling time of 3.1 min. 28 integrals contribute. The largest depth of the initial values to be provided were 10 moments with an ε-expansion up to ε 3 .

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The most demanding terms were g1 nl and g2 nl . All other contributions are significantly simpler. For g1 nl 3000 generated moments were not enough to determine the associated recurrence. Therefore we generated 4000 moments, which took 12.88 days instead of 9.96 days. The guessing time amounted to 16.1 min and 23.2 min and led to recurrences of degree and order (d = 474, o = 22), (d = 537, o = 35) for the two largest cases, respectively. Their solution using Sigma and representation in terms of HPLs in the variable x took 3.28 days. In the representation in the variable y 13 harmonic sums and 185 generalized harmonic sums, transcendent to each other, contributed. In x–space we obtain representations in terms of 55 HPLs. The results agree with those given in Refs. [10,12]. The needed computational times of the present example let it appear feasible to compute the pole terms and some of the terms of ε 0 of the massive three–loop form factor. Here a wider range of initial values, to high order in ε, is needed. We therefore had to extend the results given in [38] significantly.

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4. The universal infrared structure of QCD amplitudes

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Scattering amplitudes in perturbative QCD contain infrared singularities arising from soft gluon contributions and collinear parton divergences. In this respect much work has been performed for massless scattering amplitudes [71–76]. Especially in the case of massless QCD amplitudes with two partons, i.e. the form factors, the infrared (IR) structure becomes interesting due to its prominent form in terms of anomalous dimensions. The interplay of the collinear and soft anomalous dimensions to shape the singular structure of the massless form factors was noticed in [76] at two–loop order and later established at three–loop order in [77]. In the case of massive form factors, the finding of a Sudakov type integro-differential equation was a challenge as the massive form factors do not exponentiate. However, in the asymptotic limit i.e. in the limit where the quark mass is small compared to the center of mass energy, the massless QCD corrections to the massive form factor do exponentiate. In [78], the first step was taken by obtaining the singular behavior of massive QCD amplitudes in the asymptotic limit. Meanwhile, a factorization theorem was also proposed in [79,80], also in the asymptotic limit. Recently, following the method proposed for massless form factors in [81,82], a rigorous study has been performed in [14] in the asymptotic limit to obtain all the poles and also all logarithmic contributions to finite pieces of the three loop heavy quark form factors for vector, axial-vector, scalar and pseudo-scalar currents. In this scenario, one can relate the massless form factors to the massive ones and hence use the massless results [83,84] to obtain these predictions. A general IR structure is needed for the exact computation, which was obtained in [85], following a soft–collinear effective theory (SCET) approach at two-loop. However, the argument can be extended to three–loop appropriately. The IR singularities of the massive form factors can be factorized as a multiplicative renormalization factor, whose structure is constrained by the renormalization group equation (RGE), as follows, FI (αs , x, ε) = Z(αs , x, ε, μ)FIfin (αs , x, ε, μ) ,

(4.1)

where FIfin is finite as ε → 0. We note that Z does not carry any process dependent information (I ). Here μ is the scale introduced corresponding to this particular factorization. Now one can write down the renormalization group equation (RGE) for Z which is characterized by the massive cusp anomalous dimension, . However, before we proceed, we note that  does not contain any contributions from internal heavy quark loops, i.e. like the QCD β function, or light quark

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mass anomalous dimension, the massive cusp anomalous dimension also has been computed considering the massless QCD corrections with nl light quark flavors. On the other hand, the form factors are defined for (nl + 1) flavors. Hence,  cannot describe the singularities arising in case of a massive quark loop contributions to the heavy quark form factors. To overcome the hurdle, the immediate solution is to use the decoupling relations [86–91]. To obtain these decoupling relations, one constructs an effective theory with nl light quark flavors and then demands consistency with the full theory of nl + nh flavors by relating the couplings and light quark masses in the two cases. For our case, we need the decoupling relation for the strong coupling constant i.e. the relation between α¯ s and αs , where α¯ s is defined for an effective theory with nl light quark only and αs is defined for the full theory with nl + nh quark flavors. ¯ the equivalent of Z in the effective Keeping this in mind, we now write down the RGE for Z, theory with nl light quark, which reads

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d ¯ s , x, ε, μ) = −(αs , x, μ) . ln Z(α d ln μ

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(4.2)

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Z¯ =

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 is by now available up to the three–loop level [92–95]. Both Z¯ and  can be expanded in a perturbative series in αs as follows

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(4.3)

n .

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Next, we solve the RGE, Eq. (4.2), in massless QCD, i.e. considering only nl light quarks.

   2  2 0 β¯0 0  1  α ¯ 1 α ¯ s 0 s + − + Z¯ = 1 + 4π 2ε 4π ε2 8 4 4ε

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+ O(α¯ s4 ) .

(4.4)

¯ We use the following β¯ is the QCD β function for nl light quark flavors. Now to obtain Z from Z. decoupling relation obtained using the background field method [91,96,97] to obtain the relation between α¯ s and αs    α 2 s α¯ s = αs 1 − ζ2 nh TF ε + O(ε2 ) 4π 3  α 2  32   s + (4.5) CA TF nh − 15CF TF nh + O(ε) + O(αs3 ) 4π 9

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and obtain  Z = Z¯ α¯

s →αs

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α    α 2 1  2 β¯    s 0 s 0 0 1 0 =1+ + − + 4π 2ε 4π ε2 8 4 4ε

   α 3  1 3 β¯ 2 β¯ 2  1 0 1 β¯1 0 + β¯0 1 0 0 s 0 0 0 − + + − + 4π 8 6 8 6 ε 3 48 ε2

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1 + ε



 2 β¯   2    32   2 0 0 0 0 +2 − − ζ2 nh TF + CA TF nh − 15CF TF nh 6 8 4 3 2 9

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+ O(α¯ s4 ) .

(4.6)

The above representation leads to the prediction of the pole terms for all the massive form factors to three–loop order, which has been an open problem in Ref. [78]. 5. Refined versions of the large moment method

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We will now describe a general toolbox that enables one to calculate large numbers of moments in the integer variable, say n = 0, 1, 2, . . . , μ, for a finite number of Feynman integrals Fi (n, ε) with 1 ≤ i ≤ λ. Here the moments Fi (n, ε) depend also on the dimensional parameter ε and the corresponding ε-expansion

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Fi (n, ε) =

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ri 

Fi,k (n) ε + O(ε

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∞ 

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ri

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Gi (n, ε) =

fλ (x, ε)

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(5.3)

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gλ (x, ε)

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Gi (x, ε)x n ,

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can be determined by • other coupled systems to which the large moment method under consideration is applied recursively;

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with A(x, ε) being an invertible λ × λ matrix with entries from the polynomial ring3 K[x, ε] and where the inhomogeneous parts gi (x, ε) are given in terms of linear combinations of simpler master integrals. Here we assume that their moments Gi,k (n), n = 1, . . . , μ, with

n=0

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fλ (x, ε)

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(5.2)

are a solution of a given coupled system ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ f1 (x, ε) f1 (x, ε) g1 (x, ε) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ .. .. .. Dx ⎝ ⎠ = A(x, ε) ⎝ ⎠+⎝ ⎠, . . .

gi (x, ε) =

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Fi (n, ε)x n

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is calculated for each moment n = 0, . . . , μ up to the order ri . Standard procedures like Mincer [98] or MATAD [99] allow the calculation of a comparable small number of moments, e.g., μ  50. The idea to calculate expansions from differential equations has already frequently been used, see e.g. [100–106]. Only recently, we obtained a new method in [15] that can compute thousands of such moments. In general, this method assumes that their (formal) power series representations ∞ 

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k

k=l

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• symbolic summation or integration methods [24,28,107] that yield representations in terms of indefinite nested sums or integrals from which one can produce a large number of moments; • by standard procedures like Mincer [98] or MATAD [99] if only a small number of moments contributes.

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Summarizing, it is assumed that already μ + 1 moments for the inhomogeneous parts gi (x, ε) in (5.3) are computed. Then given such an input, we propose the following strategy to compute the first μ + 1 moments for f1 (x, ε), . . . , fλ (x, ε). Strategy 1:

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b1,i (x, ε)Dxi f1 (x, ε) =

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ok 

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bk,i (x, ε)Dxi fk (x, ε) =

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15

21



dk,i,j (x, ε)Dxi gj (x, ε) +

k−1  

22

ek,i,j (x, ε)Dxi fj (x, ε)

j =1 i

oλ 

bλ,i (x, ε)Dxi fλ (x, ε) =

i=0

24

26



dλ,i,j (x, ε)Dxi gj (x, ε) +

i,j

λ−1  

27

eλ,i,j (x, ε)Dxi fj (x, ε)

j =1 i

for explicitly given polynomials bk,i (x, ε) ∈ K[x, ε] and rational functions dk,i,j (x, ε), ek,i,j (x, ε) ∈ K(x, ε). Here the kth equation, 1 ≤ k ≤ λ, is considered as a linear differential equation of order ok in fk (x, ε) where the right hand side is given in terms of the inhomogeneous parts gj (x, ε), the functions f1 (x, ε), . . . , fk−1 (x, ε), that will be treated already within our iterative method, and their derivatives. In general it can happen that the orders oi might be larger than λ. However, in all our calculations we ended up at surprisingly nice orders; see also Remark 5.1 below. 2. We suppose that the greatest common divisor of the coefficients b1,0 (x, ε), . . . , b1,o1 (x, ε) in the first equation (5.5) is 1, i.e., there is no common polynomial factor that depends on x or ε. This implies that there is at least one i such that b1,i (x, 0) = 0. In addition, let p1 (x) ∈ K[x] be the greatest common divisor of b1,0 (x, 0), . . . , b1,o1 (x, 0), i.e., p1 (x) contains all common polynomial factors in x of the b1,i (x, 0). Dividing the first equation of (5.5) by p1 (x) yields

i=0

 d1,i,j (x, ε) b1,i (x, ε) i Dx f1 (x, ε) = Dxi gj (x, ε). p1 (x) p1 (x) i,j

28 29

(5.5)

o1 

23

25

31

33

14

20

30

32

13

18

.. .

26

28

9

19

i,j

i=0

25

27

8

17

d1,i,j (x, ε)Dxi gj (x, ε)

.. .

21

24

 i,j

20

22

7

16

o1 

18 19

5

12

1. Uncouple the system (5.3). Experiments showed that the implementation of Gauss’s elimination method in OreSys [108] is an excellent choice. In general one obtains k linear differential equations of the form

16 17

4

11

12 13

3

10

10 11

2

6

6 7

1

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

(5.6)

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

10

Plugging (5.4) into the right-hand side of (5.6) (with r1 sufficiently large) and performing its ε-expansion up to ri and its x-expansion up to μ produces4 o1  b1,i (x, ε)

4 5

i=0

6

8

p1 (x)

Dxi f1 (x, ε) =

μ r1  

˜ k (n)ε k x n + O(ε r1 +1 x μ+1 ). G

4

(5.7)

k=l n=0

Consequently, by coefficient comparison in (5.7) w.r.t. equation

we obtain the linear differential

11

i=0

12

15 16 17 18 19 20

8

23 24 25 26

b˜i (x)Dxi

∞ 

F1,l (n)x n =

n=0

∞ 

10

˜ l (n) x n G

11

n=0

12

with

13 14

bi (x, 0) ∈ K[x], b˜i (x) = pi (x)

15 16

not all 0. Finally, by coefficient comparison w.r.t. recurrence relation ν1 

21 22

xn

in the last equation we get a linear

29 30 31 32 33 34 35 36 37 38 39 40 41 42

17 18 19

˜ l+l1 (n) βi (n)F1,l (n + i) = G

(5.8)

i=0

20 21 22

of order ν1 for some polynomials βi (n) ∈ K[n] with βν1 (n) = 0 and an integer l1 ∈ Z. In particular, the following bound on the order of the recurrence holds: ν1 ≤ o1 + max degx (b˜i (x)) = o1 + max degx (b1,i (x, 0)) − degx (p1 (x)); i

i

23 24 25

(5.9)

26 27

27 28

7

9

o1 

10

14

5 6

εl

9

13

2 3

3

7

1

in the generic case we have equality – in any case the above bound is a good indication which order we can expect. Using this recurrence plus the first ν1 initial values of F1,l , one can now calculate5 in linear time the moments for F1,l (n) for n = 0, . . . , μ. 3. Next, we plug in these moments F1,l (n) with n = 0, . . . , μ into (5.7) and update its righthand side. In this way, l is replaced by l + 1 and F1,l+1 takes over the role of F1,l . Now we repeat this method for k = l + 1, . . . , r1 to get the moments of the remaining ε-contributions; we remark that the coefficients on the left-hand side of the recurrence (5.8) remain unchanged - only the inhomogeneous part on the right-hand side has to be updated. 4. Finally, we repeat the steps (2)–(3) for the second equation (k = 2) in (5.5) using the moments of F1,k (n). Together with sufficiently many initial values, say ν2 , we can calculate the moments F2 (n, ε) for n = 0 . . . , μ. Similarly, we calculate iteratively the moments for Fk (n, ε) with k = 3, . . . , λ. Note that the orders νi (for i = 1 see (5.8)) are usually small (in many cases ≤ 5 and in harder cases ∼ 25), but μ can be arbitrarily large (e.g., μ = 2000 or μ = 8000).

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

43 44

4 Efficient (and parallelized) methods have been implemented in the package SolveCoupledSystem.m in order to

44

45

˜ k (n) efficiently from the given moments Gi,k (n) in (5.4). calculate the moments G 5 Special care has to be taken if β (n) has integer zeroes: this might require extra initial values. Furthermore, it might n1 happen that |l1 | extra moments for G1,l (n) are needed.

45

46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

11

Remark 5.1. In the original version of the large moment method [15] we followed a slightly different approach. (1) For the uncoupling task we used Zürcher’s algorithm [109] from OreSys [108]. This method provides a system (5.3) where usually only the first equation is of higher-order (namely of order λ). All other equations are of order 0, i.e., o2 = · · · = oλ = 0. In general, such an uncoupled system leads to a simpler method to calculate the desired moments. On the other side, in all our calculations it turned out that the Gaussian elimination method delivered much smaller orders than λ. As a consequence, switching to the Gaussian elimination tactic, the recurrence orders can be reduced considerably (compare the order bound (5.9)). In addition, using the Gaussian uncoupling method, the coefficients b1,i (x, ε) are less complicated and factors of the form ε1q arise with smaller q ∈ N. As a consequence, using the Gaussian uncoupling strategy we could decrease notably the necessary orders ri of the ε-expansions. (2) Furthermore, our original method from [15] differs as follows: the first differential equation in (5.5) (and usually the only differential equation) is directly transformed to a linear recurrence in n whose coefficients also depend on ε. This allows one to utilize a rather efficient machinery [15,110] to compute the moments F1,k (n). However, in order to compute such a recurrence as a preprocessing step, operations have to be carried out in K(x, ε) which are rather costly. In addition, this approach misses the opportunity to cancel the polynomial p1(x) (which in applications is usually large) and thus to reduce the recurrence order substantially (compare again the order bound (5.9)). Summarizing, with our improved Strategy 1, the recurrence orders could be reduced significantly and the required ε-orders ri can be kept rather small.

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Strategy 1 has been implemented in the package SolveCoupledSystem.m and works for the calculations of the nh contributions in many cases highly efficient. Example. For a system (5.3) with λ = 4 we needed the moments (5.1) for n = 0, . . . , μ up to the orders r1 = r3 = r4 = 4 and r2 = 3. The uncoupled system has the form (5.5) with o1 = 4, o2 = 2, o3 = 0 and o4 = 0. Using this output, we obtain a recurrence of the form (5.8) for F1,l (n) of order ν1 = 4. We note that this small recurrence order was possible by sneaking in the polynomial p1 (x) ∈ Q[x] of degree 13 within the linear differential equation (5.7); setting p1 (x) = 1 would have delivered a recurrence of order 4 + 13 = 17. Similarly, we obtain recurrences for F2,l (n), F3,l (n), F4,l (n) of orders 2, 0, 0, respectively (again we reduced the recurrence order from 9 to 2 for F2 (n, ε) by factoring out a polynomial p2 (x) of degree 7). Finally, with the corresponding initial values, we calculated the moments (5.1) for n = 0, . . . , μ. E.g., for μ = 4000 we needed 14000 seconds (257723 CPU seconds6 ), for μ = 6000 we needed 25598 seconds (257723 CPU seconds) and for μ = 8000 we needed 62063 seconds (678321 CPU seconds) in order to get the moments of the rational contribution (ignoring the moments that depend on ζ2, ζ3 etc.). In general, Strategy 1 turned out to be optimal for systems with the dimension λ ≤ 4. However, if λ ≥ 5, the uncoupling step (in particular, using Gaussian elimination or Zürcher’s algorithm) failed by space-time resources or produced a not digestible output: the degrees of the polynomials bi,j (x, ε) in (5.5) were very high yielding linear recurrences with orders close to 1000. For these more complicated systems λ ≥ 5 we developed another variant of our large moment method, that is also implemented in our package SolveCoupledSystem.m.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

45

46

46 47

2

23

23 24

1

6 Adding up the calculation time of the 15 CPUs that we used in our parallelized implementation.

47

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1 2

Strategy 2. Here we assume that not only the matrix A(x, ε) in (5.3) but also A(x, 0) is invertible. Consider the Laurent series expansions

3 4

fi (x, ε) =

5 6

gi (x, ε) =

7 8 9 10 11 12 13

16 17 18 19 20 21 22 23 24 25 26

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

k=l ∞ 

47

2

4

fi,k (x)ε ,

5 6

gi,k (x)ε

k

7 8

fλ,l (x)

fλ,1 (x)

gλ,l (x)

which is free of ε; see also [12]. Now we carry out steps7 (1)–(2) of Strategy 1 to this simpler system. This leads to two central improvements: In step (1) the uncoupling method only deals with univariate rational functions in K(x) and not in the very expensive multivariate case K(x, ε). Furthermore, the rational functions in (5.5) (now free of ε!) are much smaller and usually have lower degree. As a consequence, this leads in step (2) to linear recurrences with rather small orders. Furthermore, no factors of the form ε1q can occur, i.e., the ε–order cannot be increased by carrying out steps (1) and (2). After calculating the moments Fi,l (n) for n = 0, . . . , μ and 1 ≤ i ≤ λ, one plugs them into (5.3) and obtains a new system (similar to (5.10)) where fi,l+1 (x) takes over the role fi,l (x) and the right hand side is adapted accordingly by taking into account the computed moments of Fi,l (n). Now we repeat this calculation iteratively looping through k = l + 1, . . . , ρ with ρ = max(r1 , . . . , rλ )

9 10 11 12 13 14

(5.11)

in order to get the moments for Fi,k (n), n = 0, . . . , μ. Example. For a system (5.3) with λ = 6 we needed the moments (5.1) for n = 0, . . . , μ up to the orders ri = 4 for 1 ≤ i ≤ 6. The uncoupled system of (5.10) (free of ε) has the form (5.5) with o1 = o2 = 4, o3 = 2 and o4 = o5 = o6 = 0. Using this information, we obtain recurrences for Fi,l (n) of orders ν1 = 28, ν2 = 11, ν3 = 10 and ν4 = ν5 = ν6 = 0 with l = −3. Finally, with the corresponding initial values, we can calculate the moments Fi,l (n) for 1 ≤ i ≤ 6. Plugging them into (5.3) one obtains a new system of the form (5.3) and repeats this process for k = l + 1, . . . , 4. E.g., for μ = 4000 we needed 115209 seconds (637678 CPU seconds), for μ = 6000 we needed 262156 seconds (1.764 · 106 CPU seconds), and for μ = 8000 we needed 3.724 · 106 seconds (5.161 · 106 CPU seconds) in order to get the moments of the rational contribution (ignoring the moments that depend on ζ2 , ζ3 , etc.). Remark 5.2. For all systems (5.3) with λ ≥ 5 the following modification was sufficient to obtain a matrix A(x, ε) such that A(x, 0) was invertible. We simply carried out a substitution fi (x, ε) → ε1λi f¯i (x, ε) for some suitable λi ∈ N with f¯i (x, ε) := x λi fi (x, ε), cleared the arising denominators and canceled common factors. As a side-effect, this modification increased the

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

45 46

1

3

k

for 1 ≤ i ≤ λ with l ∈ Z. Then by coefficient comparison w.r.t. ε l in (5.3) we get the system ⎛ ⎛ ⎞ ⎞ ⎛ ⎞ f1,l (x) g1,l (x) f1,l (x) ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ (5.10) Dx ⎝ ... ⎠ = A(x, 0) ⎝ ... ⎠ + ⎝ ... ⎠

27 28

∞ 

k=l

14 15

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

12

7 Since A(x, 0) is invertible, the available uncoupling methods from OreSys, in particular Gauss’ method, are appli-

cable.

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[m1+; v1.304; Prn:6/09/2019; 12:57] P.13 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3

13

necessary ε-orders ri for f¯i (x, ε) to obtain in the end the required ε-order for fi (x, ε). However, this phenomenon arises only in this initialization phase when the matrix A is usually simple. In our cases the λi turned out to be rather small (λi ≤ 2).

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

If Strategy 1 was applicable (which was the case for λ ≤ 4), it was superior to Strategy 2: One has to loop up simultaneously for all f1 (x, ε), . . . , fλ (x, ε) to ε ρ with (5.11), while in Strategy 1 one computes the ε-expansions for smaller recurrences taking care individually to which order ri the fi (x, ε) have to be expanded. The latter approach with individual ri reduces substantially the calculation cost. Furthermore, taking ρ with (5.11) (instead of considering the individual orders ri ) often implied that the right hand sides gi (x, ε) in (5.3), and thus the simpler master integrals arising in the gi (x, ε), have to be calculated to a higher ε-expansion. As it turns out, this enlargement of the required ε-orders even raised to a higher power when one applies the large moment method iteratively to the recursively defined systems coming from IBP relations. In order to tackle the master integrals coming from the nh -contributions, we applied the two strategies (Strategy 1 if the dimension of the system is ≤ 4 and Strategy 2 if the dimension is ≥ 5) to 41 systems, more precisely, we tackled 16 systems with λ = 1, 15 with λ = 2, 2 with λ = 3, 3 with λ = 4, 3 with λ = 5, 1 with λ = 6, and 1 with λ = 7. Here the systems depend recursively on each other: the inhomogeneous parts gi (x, ε) in (5.3) depend on master integrals that are solutions of simpler systems. The calculation time of the moments up to the required ε-order for these 92 master integrals can be summarized as follows. For μ = 2000 moments we needed in total 438432 second (2.630 · 106 CPU seconds). From these 2000 moments we produced the corresponding number of moments of the color-factors and succeeded in guessing recurrences for almost all cases: only the recurrences for the ζ3 -contributions and the constant free contributions could not be guessed with the given number of moments. Thus we restarted our large moment method to produce μ = 4000 moments in 720874 seconds (6.751 · 106 CPU seconds) ignoring all constants that have been tackled already. This time we succeeded in computing the recurrences for the ζ3 -contribution but not yet for the constant-free terms. Therefore we restarted our method for μ = 6000 moments ignoring also the ζ3 contributions and obtained them in 1.735 · 106 seconds (2.091 · 106 CPU seconds). However, we failed to derive further recurrences. Finally, we produced 8000 moments in 3.724 · 106 seconds (5.161 · 106 CPU seconds) and succeeded in guessing the remaining recurrences. 6. Non-first order factorizing contributions

36 37 38 39 40 41 42 43 44 45 46 47

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

33 34

34 35

3

32

32 33

2

4

4 5

1

Using the method of arbitrarily high moments [15] and guessing [26], the recurrences for all pole terms and of a series of color-zeta contributions at the O(ε0 ) can be obtained using 2000 moments, while for other color-zeta projections the corresponding recurrences can be obtained from 4000 moments. For the purely rational contributions we tried the guessing method first with 6000 moments, which were not sufficient. As the next step, we generated 8000 moments by which we obtained the recurrences for the purely rational terms. While for many of these projections the corresponding recurrences are first order factorizable and can thus be solved using the difference ring methods encoded in the package Sigma, for a smaller number non–first order factorizing terms contribute. In Table 1 we characterize those recurrences. Separating the first order factorizing parts, we find remaining non–first order factorizing contributions of order o = 6, 10 and 15. This is uniformly the case for all currents. The remaining recurrences have to be studied with other techniques. One may translate these remaining equations into systems of ordinary differential equations again. In other cases it has been

35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2

Table 1 Structure of the recurrences for the non–first order contributions.

3 4

FV

5 6 7 8 9 10

FA

11 12 13 14 15

FS

16 17 18

[m1+; v1.304; Prn:6/09/2019; 12:57] P.14 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

14

FP

19 20

Degree

Order

Remaining order

3

g1 nh g1 nh ζ3 g1 nh ζ2 g2 nh g2 nh ζ3 g2 nh ζ2

1288 409 295 1324 430 273

54 29 24 55 30 23

15 10 6 15 10 6

4

g1 nh g1 nh ζ3 g1 nh ζ2 g2 nh g2 nh ζ3 g2 nh ζ2

1314 419 280 1130 352 232

54 29 23 52 28 23

15 10 6 15 10 6

nh nh ζ3 nh ζ2

1114 350 230

50 27 22

15 10 6

15

nh nh ζ3 nh ζ2

1130 352 232

52 28 23

15 10 6

18

FS

Color

Degree

Order

Remaining order

Nc2 nh Nc2 nh ζ3

901 257

46 23

5 4

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

8

10 11 12 13 14

16 17

19 20

23 24 25 26 27 28 29

29 30

7

22

Table 2 Structure of the recurrences for the non-first order contributions in leading color approximation for the scalar three loop massive form factor.

26

28

6

21

25

27

5

9

22

24

2

Color

21

23

1

observed [44,111] that systems of this kind may decompose into series of smaller systems. This has to be investigated in further studies. We also analyzed the leading color case for the scalar current. Here 10 color-ζ structures contribute, out of which 8 have a representation, which results from a difference equation which factorizes at first order. The largest recurrences could be found using 6000 moments. In Table 2 we summarize the characteristics for the recurrences for the constant term, which contain non– first order factorizing parts of order o = 5 and o = 4, respectively. The solution for the ζ2 –term at leading color, unlike in the full color case, can be represented in terms of nested sums. The corresponding recurrence is of degree d = 150 and order o = 17 and one obtains  NC2 nh ζ2 R27 8R28 nh Nc2 ζ2 FS =− ln(2) + 2 4 2(1 + x) 9x(1 + x) 54(1 − x)2 (1 + x)4 8R23 R29 + − ln(2) − 3(1 − x)(1 + x)3 18(1 − x)3 (1 + x)4

R24 16R2 8R8 ln(2) − H2 + H−1 + (1 − x 2 ) 3(1 − x)(1 + x)3 3(1 − x 2 ) −1

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

15

4R17 48x 2 ζ2 + ζ3 H 0 − 3(1 − x)2 (1 + x) (1 − x 2 )  24x 2 R26 8R15 ζ2 + ln(2) + + 2 2 3 (1 − x ) 36(1 − x) (1 + x) (1 − x)2 (1 + x)

8R14 4R16 2x 2 2 3 − H H H4 + − H −1 0 0 3(1 − x)2 (1 + x) 9(1 − x)2 (1 + x) (1 − x 2 ) 0

  80(1 + x) 1 + x 2 16R5 8R22 + − − ln(2) − H−1 H0 (1 − x) 9(1 − x)(1 + x)3 3(1 − x 2 )

  8 5 − 13x + 54x 2 − x 3 − x 4 2 16R6 + H0 − ζ2 H 1 3(1 − x 2 ) 3(1 − x 2 )

  32(1 + x) 1 − 3x + x 2 8R28 16R7 2 − − H0 H1 + − ζ2 H−1 3(1 − x) 9x(1 + x)4 3(1 − x 2 )   80(1 + x) 1 + x 2 8R19 8R22 + − ln(2) + H0 (1 − x) 9(1 − x)(1 + x)3 3(1 − x)2 (1 + x)

  32x 2 H02 64(1 + x) 1 − 3x + x 2 128R1 + H−1 H0,1 − H1 + 3(1 − x) (1 − x 2 ) 3(1 − x 2 ) R21 8R11 16R2 64x 2 ln(2) − H2 + H − + 0 3(1 − x)2 (1 − x 2 ) 3(1 − x)(1 + x)3 (1 − x 2 ) 0

32R4 32R9 16R2 H1 − H−1 H0,−1 − H0 H−1,1 + 3(1 − x 2 ) 3(1 − x 2 ) (1 − x 2 )

16xR13 192x 2 + + H0 H0,0,1 3(1 − x)2 (1 + x) (1 − x 2 )

16R20 288x 2 − + H0 H0,0,−1 3(1 − x)2 (1 + x) (1 − x 2 )   64(1 + x) 1 − 3x + x 2 16R10 − H0,1,1 − H0,1,−1 3(1 − x) 3(1 − x 2 ) 32R4 16R12 384x 2 − + − H H H0,0,0,1 0,−1,1 0,−1,−1 3(1 − x 2 ) 3(1 − x 2 ) (1 − x 2 )

480x 2 16R3 R25 + H0,0,0,−1 + ln(2) − ζ2 (1 − x 2 ) (1 − x 2 ) 18(1 − x)(1 + x)3  72x 2 2R18 2 − ζ + ζ3 , (6.1) 5(1 − x 2 ) 2 3(1 − x)2 (1 + x)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

with the polynomials

46

46 47

1

R1 = 7x 4 + 3x 3 + 28x 2 + 3x + 7,

(6.2)

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

16

R2 = 11x 4 + 4x 3 + 34x 2 + 4x + 11,

(6.3)

R3 = 17x − 2x + 58x − 2x + 17,

(6.4)

R4 = 19x 4 + 21x 3 + 76x 2 + 21x + 19,

(6.5)

4

5

R5 = 23x + 12x + 122x + 12x + 23,

(6.6)

5

6 7

R6 = 23x 4 + 17x 3 + 60x 2 + 17x + 23,

(6.7)

8

R7 = 24x 4 + 5x 3 + 130x 2 + 5x + 24,

(6.8)

8

9

R8 = 31x + 26x + 86x + 26x + 31,

(6.9)

9

10 11

R9 = 32x 4 + 19x 3 + 94x 2 + 19x + 32,

(6.10)

12

R10 = 71x 4 + 54x 3 + 254x 2 + 54x + 71,

(6.11)

12

R11 = 83x − 96x + 214x − 96x + 83,

(6.12)

13

R12 = 97x 4 + 50x 3 + 290x 2 + 50x + 97,

(6.13)

15

R13 = 100x 4 − 19x 3 + 197x 2 − 105x + 11,

(6.14)

16

R14 = 3x + 23x − 22x + 148x + 7x + 57,

(6.15)

R15 = 15x 5 − 3x 4 + 14x 3 − 16x 2 − x − 1,

(6.16)

19

20

R16 = 19x − 5x + 212x − 112x + 43x + 19,

(6.17)

20

21 22

R17 = 31x 5 + 39x 4 + 258x 3 − 70x 2 + 91x + 27,

(6.18)

23

R18 = 89x 5 − 289x 4 + 22x 3 − 134x 2 − 55x − 1,

(6.19)

23

24

R19 = 101x − 19x + 142x − 38x − 7x + 5,

(6.20)

24

26

R20 = 125x 5 − 45x 4 + 182x 3 − 78x 2 − 23x + 23,

(6.21)

27

R21 = −617x 6 − 690x 5 + 2033x 4 + 3572x 3 + 2033x 2 − 690x − 617,

(6.22)

27

R22 = 45x + 398x − 481x − 2148x − 481x + 398x + 45,

(6.23)

28

R23 = 48x 6 + 131x 5 − 28x 4 + 18x 3 − 28x 2 + 131x + 48,

(6.24)

30

R24 = 233x 6 − 358x 5 − 1809x 4 − 3716x 3 − 1809x 2 − 358x + 233,

(6.25)

31

R25 = 2571x + 8438x − 13795x − 45084x − 13795x + 8438x + 2571,

(6.26)

1 2 3 4

13 14 15 16 17 18 19

25

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

4

4

47

2

3

4

2

3

4

2

3

5

2

4

5

3

4

2

4

6

3

5

6

2

4

3

5

2

4

3

2

3

6 7

10 11

(6.27)

R27 = −100331x − 173294x − 40776x + 237806x + 226918x + 237806x 6

5

17 18

21 22

25 26

4

32 33 34

− 325x − 709, 7

2

29

R26 = 917x 7 + 12853x 6 − 3767x 5 − 24311x 4 − 11289x 3 + 9223x 2 8

1

14

2

3

5

35 36

3

− 40776x 2 − 173294x − 100331,

37

(6.28)

38

R28 = 124x − 5x − 609x − 2635x − 3950x − 2635x − 609x − 5x + 124, (6.29)

39

8

7

6

5

4

3

2

R29 = 1984x 10 − 5471x 9 − 11702x 8 − 28981x 7 + 9401x 6 + 43521x 5 + 12817x 4 − 24727x 3 − 13829x 2 − 8150x + 561.

44 45

dyfb (y)Ha (y), H∅ = 1, ai , b ∈ {0, 1, −1} 0

42 43

x Hb,a (x) =

40 41

(6.30)

Here the functions Ha (x) denote the harmonic polylogarithms [30], which are defined by

45 46

3

(6.31)

46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3

with

1

f0 (x) =

1 , x

f1 (x) =

4 5 6 7 8

17

1 , 1−x

f−1 (x) =

1 . 1+x

2

(6.32)

In the case of the color–ζ terms the recurrences of which do not factorize to first order we can use the finite analytic representation given by the number of analytic moments Nmax = 2000 for numerical results and graphical illustrations given below, which already gives a good representation. In some cases they have a logarithmic divergence near x = 0, however.

11 12 13 14 15 16 17 18 19 20 21 22 23

7. The results In the following we present the results for the non-singlet contribution ∝ nkh , k = 1, 2, of the massive three–loop form factors in the vector-, axialvector-, scalar- and pseudoscalar case. The pole terms are given in terms of HPLs of the variable x. For the constant contributions, for a series of color–zeta terms also first order factorizing solutions are obtained, which are expressed using HPLs. Other contributions contain non–first order factorizing parts. For these we have derived the corresponding recurrences of their Taylor coefficients also using the method of arbitrarily high moments [15]. In the present paper we present these contributions in terms of analytic series expansions, for which we derived at least 2000 terms and in some cases 4000 or 8000 terms, cf. Section 6. (0) The renormalized form factors, which are lengthy expressions, except the functions FC,i ,i = 1...3, C = V1 , V2 , A1 , A2 , P , S defined below, are given in an attachment to this paper, where we also present the other results in computer-readable form. 7.1. The vector form factors

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

6 7 8

10 11 12 13 14 15 16 17 18 19 20 21 22 23

25 26

26 27

5

24

24 25

4

9

9 10

3

The vector form factor FV ,1 is given by    

 2 H 1 + x 32 2 589 + 602x + 589x 2 1 16 0 2 FV ,1 = 3 nh − − nh ε 27 27(1 − x 2 ) 27(1 + x)2

    16 41 + 190x + 154x 2 + 190x 3 + 41x 4 H0 16 32 1 + x 2 H0 − − nl − 27 27(1 − x 2 ) 27(1 − x)(1 + x)3 

   2  128 1 + x 2 16 3 + 2x + 3x 2 1 2 2 16 + H − + 2 nh 27 27(1 − x 2 )2 0 ε 27(1 − x 2 )

       

64 1 + x 2 16 1 + x 2 2 64 1 + x 2 32 1 + x 2 − H−1 H0 − H − H0,−1 + ζ2 27(1 − x 2 ) 27(1 − x 2 ) 0 27(1 − x 2 ) 27(1 − x 2 )     64 7 + 3x + 7x 2 4 263 − 66x + 263x 2 128 + nh − + nl + − 81 27(1 + x)2 81(1 − x 2 )

      128 1 + x 2 32 1 + x 2 2 128 1 + x 2 + H−1 H0 − H − H0,−1 27(1 − x 2 ) 27(1 − x 2 ) 0 27(1 − x 2 )     64 1 + x 2 8 371 + 1622x + 1334x 2 + 1622x 3 + 371x 4 ζ2 + + 27(1 − x 2 ) 27(1 − x)(1 + x)3

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

    64 1 + x 2 128 23 + 73x + 64x 2 + 73x 3 + 23x 4 + H−1 H1 − 3(1 − x 2 ) 27(1 − x)(1 + x)3

  2 2 64 1 + x 2 1088 1 + x 2 + H − H0,−1 H0 0,1 3(1 − x 2 )2 27(1 − x 2 )2   16 13 + 51x − 46x 2 − 10x 3 − 167x 4 − 97x 5 + 27(1 − x)2 (1 + x)3

2       512 1 + x 2 64 − 2 + x 2 1 + x 2 3 64 1 + x 2 2 + H−1 H0 + H0 − H0,1 27(1 − x)2 (1 + x)2 27(1 − x)2 (1 + x)2 3(1 − x 2 )   128 23 + 73x + 64x 2 + 73x 3 + 23x 4 H0,−1 + 3 27(1 − x)(1 + x)   2 2 128 1 + x 2 128 1 + x 2 − H0,0,1 + H0,0,−1 3(1 − x)2 (1 + x)2 3(1 − x)2 (1 + x)2   64 23 + 73x + 46x 2 + 37x 3 + 5x 4 + − 27(1 − x)(1 + x)3      2  32 1 + x 2 − 1 + 35x 2 32 1 + x 2 + H0 ζ2 + ζ3 27(1 − x)2 (1 + x)2 3(1 − x)2 (1 + x)2  (1) (1) 8H02 P9 16H03 P13 16H0 P4(1) 1 2 + − + nh − ε 243(1 − x)(1 + x)3 81(1 − x)(1 + x)4 27(1 − x)(1 + x)5       64 13 − 268x + 13x 2 64 1 + x 2 2 32 3 + 2x + 3x 2 − − H H0 + H0 243(1 + x)2 27(1 − x 2 ) −1 27(1 − x 2 )

      32 1 + x 2 2 128 1 + x 2 32 3 + 2x + 3x 2 + H + H0,−1 H−1 − H0,−1 27(1 − x 2 ) 0 27(1 − x 2 ) 27(1 − x 2 )     (1) 64 1 + x 2 128 1 + x 2 8P11 − − + H H 0,0,−1 0,−1,−1 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x)(1 + x)4

   

(1) 64 1 + x 2 64 1 + x 2 16H0 P16 − − H−1 ζ2 + ζ3 27(1 − x 2 ) 27(1 − x)(1 + x)5 27(1 − x 2 ) (1) (1) 4P6(1) 128H0,−1,−1 P3 64H0,1 P1 + nh − + − 27(1 − x)(1 + x)3 27(1 − x)(1 + x)3 243(1 + x)4

40 41

+

(1) 128H0,0,1 P18 27(1 − x)2 (1 + x)4

−

(1) 128H0,0,0,1 P30 27(1 − x)2 (1 + x)5

42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.18 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

18



−

(1) 64H0,0,−1 P21 27(1 − x)2 (1 + x)4

−

(1) 16ζ22 P37 135(1 − x)2 (1 + x)5

+

(1) 128H0,0,0,−1 P27 9(1 − x)2 (1 + x)5

−

(1) 8H04 P39 81(1 − x)2 (1 + x)6

  (1) (1) 32 1 − 194x + x 2 16H02 P7 32H0 P2 + nl − − − 81(1 + x)2 81(1 − x)(1 + x)3 81(1 − x)(1 + x)4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.19 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

19

    (1) 256 1 + x 2 2 256 7 + 3x + 7x 2 32H03 P14 − − H H0 + H0 27(1 − x 2 ) −1 81(1 − x 2 ) 27(1 − x)(1 + x)5

      128 1 + x 2 2 512 1 + x 2 256 7 + 3x + 7x 2 + H + H0,−1 H−1 − H0,−1 27(1 − x 2 ) 0 27(1 − x 2 ) 81(1 − x 2 )     (1) 256 1 + x 2 512 1 + x 2 8P12 − H0,0,−1 + H0,−1,−1 + 27(1 − x 2 ) 27(1 − x 2 ) 81(1 − x)(1 + x)4

    (1) 256 1 + x 2 256 1 + x 2 16H0 P17 H−1 ζ2 + ζ3 − − 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x)(1 + x)5 (1) (1) (1) 128H0,1 P15 64H0,−1 P19 4P23 + − − 243(1 − x)(1 + x)5 27(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)4 (1) (1) (1) 128H0,0,−1 P29 512H0,0,1 P24 64P1 − + + 27(1 − x)(1 + x)3 27(1 − x)2 (1 + x)5 27(1 − x)2 (1 + x)5

   2  2 256 1 + x 2 64 1 + x 2 2 640 1 + x 2 + [H0,1 − H0,−1 ] H1 + H0,1,1 H + 27(1 − x 2 )2 3(1 − x 2 ) 1 27(1 − x 2 )2

   2 2 2 896 1 + x 2 896 1 + x 2 128 1 + x 2 − H0,1,−1 − H0,−1,1 + H0,−1,−1 H0 27(1 − x 2 )2 27(1 − x 2 )2 27(1 − x 2 )2 (1) (1) (1) 64H1 P8 64H0,1 P31 64H0,−1 P33 + + − 27(1 − x)(1 + x)4 27(1 − x)2 (1 + x)5 27(1 − x)2 (1 + x)5

(1) (1) (1) 16P35 64H1 P28 4P41 2 − − H0 + − 243(1 − x)2 (1 + x)6 81(1 − x)2 (1 + x)5 81(1 − x)2 (1 + x)5   2    128 1 + x 2 128 1 + x 2 512 1 + x 2 3 H0,0,1 H0,1 + H0,−1 − × H0 + − 3(1 − x 2 ) (1 − x 2 ) 27(1 − x 2 )2

2  (1) 512 1 + x 2 128H0,−1 P3(1) 32H02 P22 + H + − + H 0,0,−1 1 27(1 − x 2 )2 27(1 − x)(1 + x)3 27(1 − x)2 (1 + x)4    2 (1) 128 1 + x 2 256 1 + x 2 16P5 − + H0,1 H1 + 27(1 − x)(1 + x)3 (1 − x 2 ) 3(1 − x 2 )2

   2    4352 1 + x 2 64 1 + x 2 − 1 + 5x 2 3 128 1 + x 2 − H0,−1 H0 + H0 + H0,1 27(1 − x 2 )2 27(1 − x 2 )2 (1 − x 2 )

  2 2 (1) 512 1 + x 2 512 1 + x 2 64H0 P3 + H − H + H 0,0,1 0,0,−1 −1 3(1 − x 2 )2 3(1 − x 2 )2 27(1 − x)(1 + x)3

  2 2 1024 1 + x 2 448 1 + x 2 16P5(1) 2 2 2 − H − H + H −1 27(1 − x 2 )2 0 27(1 − x 2 )2 0,1 27(1 − x)(1 + x)3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1

+

2 3 4

−

5 6

−

7 8 9

−

10 11

   2 2  3200 1 + x 2 2240 1 + x 2 128 1 + x 2 2 H0,1 H0,−1 − H + H0,1,1 27(1 − x 2 )2 27(1 − x 2 )2 0,−1 3(1 − x 2 )      2 128 1 + x 2 128 1 + x 2 512 1 + x 2 H0,0,1,1 H0,1,−1 − H0,−1,1 + (1 − x 2 ) (1 − x 2 ) 27(1 − x 2 )2    2 2 2 512 1 + x 2 512 1 + x 2 512 1 + x 2 H0,0,1,−1 − H0,0,−1,1 + H0,0,−1,−1 3(1 − x 2 )2 3(1 − x 2 )2 3(1 − x 2 )2 2    (1) 2048 1 + x 2 128 ln(2) 1 + x 2 64H0,−1 P25 H + − − 0,−1,0,1 27(1 − x 2 )2 3(1 + x)2 27(1 − x)2 (1 + x)5 (1)

(1)

(1)

64H0,1 P32 P38 32H02 P40 + − − 27(1 − x)2 (1 + x)5 27(1 − x)(1 + x)6 27(1 − x)2 (1 + x)6

  (1) (1) 64 1 + x 2 256H1 P26 8P36 − − H1 H0 − 3(1 − x)(1 + x) 27(1 − x)2 (1 + x)5 27(1 − x)2 (1 + x)5

   (1)  256 1 + x 2 13 + 4x 2 64P10 + − H0 H−1 ζ2 27(1 − x)(1 + x)4 27(1 − x)2 (1 + x)2  2 (1) (1) 128 1 + x 2 32P20 32H0 P34 + + + H1 27(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)5 27(1 − x 2 )2

  2 128 1 + x 2 − H−1 ζ3 (7.1) + FV(0) ,1 , 3(1 − x 2 )2

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

[m1+; v1.304; Prn:6/09/2019; 12:57] P.20 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

20

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

and the polynomials

27

27 28

1

(1)

P1 = 37x 4 + 104x 3 + 86x 2 + 104x + 37

(7.2)

28

(1) P2 (1) P3 (1) P4 P5(1) (1) P6 (1) P7 (1) P8 (1) P9 (1) P10 (1) P11 (1) P12 (1) P13 (1) P14 (1) P15

= 103x + 98x + 702x + 98x + 103

(7.3)

29

= 115x 4 + 284x 3 + 266x 2 + 284x + 115

(7.4)

31

= 337x 4 + 20x 3 + 3638x 2 + 20x + 337

(7.5)

32

2

= 913x + 1462x + 6506x + 1462x + 913

(7.6)

33

= 14921x 4 + 130220x 3 + 280134x 2 + 130220x + 14921

(7.7)

35

= 9x 5 + 117x 4 − 60x 3 + 356x 2 + 75x + 47

(7.8)

36

= 14x − 89x + 761x − 617x + 197x + 22

(7.9)

4

3

4

2

3

5

4

3

2

30

34

37 38

= 29x 5 − 75x 4 + 362x 3 − 470x 2 + 9x − 47

(7.10)

39

= 55x + 264x + 277x + 391x + 210x + 67

(7.11)

40

= 87x 5 + x 4 + 1502x 3 − 1430x 2 + 43x − 75

(7.12)

= 331x 5 + 249x 4 + 4942x 3 − 3758x 2 + 519x − 107

(7.13)

43

= x + 4x + 3x + 48x + 3x + 4x + 1

(7.14)

44

= x 6 + 4x 5 + 5x 4 + 28x 3 + 5x 2 + 4x + 1

(7.15)

46

= 2x 6 + 123x 5 − 870x 4 + 1234x 3 − 870x 2 + 123x + 2

(7.16)

47

5

6

4

5

3

4

3

2

2

41 42

45

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J. Blümlein et al. / Nuclear Physics B ••• (••••) •••••• (1)

1 2 3 4 5 6 7 8 9 10 11

P16 = 5x 6 + 20x 5 + 11x 4 + 280x 3 + 11x 2 + 20x + 5

(7.17)

1

(1) P17 (1) P18 (1) P19 (1) P20 (1) P21 (1) P22 (1) P23

= 7x + 28x + 25x + 296x + 25x + 28x + 7

(7.18)

2

= 18x 6 + 143x 5 − 890x 4 + 1090x 3 − 926x 2 + 71x − 18

(7.19)

4

= 43x 6 − 685x 5 + 5403x 4 − 8498x 3 + 5403x 2 − 685x + 43

(7.20)

5

= 137x + 1490x − 6779x + 12512x − 7257x + 606x − 197

(7.21)

= 142x 6 + 1359x 5 − 6912x 4 + 11114x 3 − 7268x 2 + 683x − 142

(7.22)

8

= 228x − 11x + 3894x − 5882x + 3538x − 687x − 56

(7.23)

9

(1) P24 (1) P25 (1) P26 (1) P27 (1) P28 (1) P29 (1) P30 (1) P31 (1) P32 (1) P33 (1) P34 (1) P35 (1) P36 (1) P37

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

(1) P38

33 34

36 37 38 39

45 46 47

6

3

5

6

2

4

5

3

4

2

3

2

5

4

3

and

3

6 7

10

= 46991x + 217814x + 668705x + 1193908x + 668705x + 217814x 6

2

11

(7.24)

12

= 5x − 37x + 421x − 1247x + 1268x − 406x + 46x − 2

(7.25)

13

= 7x 7 + 15x 6 + 17x 5 − 239x 4 + x 3 − 187x 2 − 117x − 41

(7.26)

= 13x 7 − 14x 6 + 449x 5 − 1326x 4 + 1319x 3 − 454x 2 + 11x − 14

(7.27)

16

= 15x + 201x − 1135x + 3933x − 3709x + 1295x − 105x + 17

(7.28)

17

= 25x 7 − 31x 6 + 893x 5 − 2659x 4 + 2631x 3 − 913x 2 + 19x − 29

(7.29)

= 30x 7 − 171x 6 + 2127x 5 − 6392x 4 + 6448x 3 − 2087x 2 + 195x − 22

(7.30)

20

= 41x + 382x − 1814x + 6397x − 5949x + 2134x − 190x + 23

(7.31)

21

= 42x 7 − 31x 6 + 1377x 5 − 3742x 4 + 3966x 3 − 1217x 2 + 127x − 10

(7.32)

= 43x 7 − 83x 6 + 1751x 5 − 5367x 4 + 5213x 3 − 1861x 2 + 17x − 65

(7.33)

24

= 50x + 45x + 1011x − 2442x + 2750x − 791x + 87x − 6

(7.34)

25

= 71x 7 − 207x 6 + 3511x 5 − 10527x 4 + 10653x 3 − 3421x 2 + 261x − 53

(7.35)

= 229x 7 − 245x 6 + 6189x 5 − 4655x 4 − 731x 3 + 399x 2 − 167x + 5

(7.36)

28

= 281x + 4495x − 12289x + 14449x − 953x + 2801x − 671x + 79

(7.37)

29

7

6

5

7

4

6

7

6

7

3

4

5

6

2

4

5

6

7

3

5

5

2

3

4

2

3

2

4

3

2

6

5

4

3

15

18 19

22 23

26 27

31

(7.38)

= 18177x + 16453x + 426009x + 662997x − 333541x − 273065x 7

14

30

= 809x 7 + 655x 6 + 16429x 5 − 41801x 4 + 43971x 3 − 14879x 2 2

32 33

(7.39)

34 35

(1) = 8x 8 + 31x 7 + 42x 6 + 693x 5 + 212x 4 + 795x 3 + 214x 2 + 145x + 36 P39

(7.40)

36

(1) P40 (1) P41

(7.41)

37

= 71x + 232x + 832x − 74x + 498x + 1490x − 256x + 80x + 7 8

7

6

5

4

3

2

38

= 13421x 8 + 46436x 7 + 167926x 6 + 195276x 5 + 214672x 4 − 46932x 3 − 22982x 2 − 25564x − 5677,

40

44

4

+ 23243x − 8305

35

43

5

+ 275x − 499

32

42

6

+46991

12

41

21

39

(7.42)

40 41



   1 + x2 x 1 FV ,2 = 3 nh 128 − + H0 ε 3(1 + x)2 3(−1 + x)(1 + x)3    64H0 x 64 2 + nh + 2 −nh −73(1 − x 2 ) + 18(1 + x 2 )ζ2 2 3 ε 27(1 − x ) 27(1 − x)(1 + x)

42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.22 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••



    128 18 − 73x + 18x 2 256 1 + x 2 128H0 − − nl − H−1 H0 27(1 − x 2 ) 27(1 − x)(1 + x)3 3(1 − x)(1 + x)3

     64(−17 + x) 1 + x 2 2 256 1 + x 2 + H − H0,−1 27(1 − x)2 (1 + x)3 0 3(1 − x)(1 + x)3 

  64 37 − 298x + 37x 2 512x 2 ζ2 x 2 2176 + − + n − + H0 ε h 27(1 + x)2 81(1 − x)(1 + x)3 3(1 − x)(1 + x)5   256x 2 H03 128H−1 H0 32 − 3 + 39x − 45x 2 + x 3 2 + + H + 0 27(1 − x 2 ) 27(1 − x)(1 + x)4 9(1 − x)(1 + x)5

  64 − 7 − 213x + 207x 2 + 5x 3 128H0,−1 − ζ2 − 27(1 − x 2 ) 27(1 − x)(1 + x)4 (2) (2) (2) 256ζ3 P5 128H0,0,−1 P9 512H0,0,1 P4 − + + nh − 27(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)4   (2) 160 635 + 1654x + 635x 2 64ζ2 P18 − + 81(1 + x)4 27(1 − x)2 (1 + x)6

  128 37 − 112x + 37x 2 512x 2 ζ2 2176 + nl − + − + H0 27(1 + x)2 81(1 − x)(1 + x)3 3(1 − x)(1 + x)5   256x 2 H03 512H−1 H0 64 3 − 15x + 27x 2 + x 3 2 + H + − 0 27(1 − x 2 ) 27(1 − x)(1 + x)4 9(1 − x)(1 + x)5

  (2) 128 − 5 − 111x + 99x 2 + x 3 128H0,−1 P6 512H0,−1 − ζ − + − 2 27(1 − x 2 ) 27(1 − x)(1 + x)4 9(1 − x)2 (1 + x)4   (2) (2) 256 5 + 22x + 5x 2 64ζ2 P16 16P12 − + + 27(1 − x)(1 + x)3 81(1 − x)(1 + x)5 27(1 − x)3 (1 + x)5

    5120x 7 − 16x + 7x 2 512 6 − 55x + 6x 2 − ζ2 H1 + H0,1 9(1 + x)4 9(1 − x)(1 + x)5     26624x 8 − 17x + 8x 2 512x 521 − 1142x + 521x 2 − H0,0,1 + H0,0,−1 27(1 − x)(1 + x)5 27(1 − x)(1 + x)5     512x 209 − 479x + 209x 2 256 1 + x 2 + ζ3 H0 − H 2 H0 3(1 − x)(1 + x)3 −1 27(1 − x)(1 + x)5 (2) 256H1 P3(2) 64P17 + − − 27(1 − x)2 (1 + x)4 81(1 − x)2 (1 + x)6   (2) 256x 313 − 688x + 313x 2 256xP8 + H − H0,−1 0,1 27(1 − x)(1 + x)5 27(1 − x)3 (1 + x)5

(2) (2) 64P15 128xP14 2 − ζ + H 2 0 27(1 − x)3 (1 + x)6 81(1 − x)3 (1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.23 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

23

  (2) 2560x 7 − 16x + 7x 2 32xP13 3 − H − H4 H 1 0 27(1 − x)(1 + x)5 81(1 − x)3 (1 + x)6 0   (2) (2) 128 143 − 318x + 143x 2 64H02 P7 128ζ2 P2 + + − H0 9(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)4 27(1 − x)(1 + x)3

    512 1 + x 2 256 5 + 22x + 5x 2 + H0,−1 H−1 + − 3(1 − x)(1 + x)3 27(1 − x)(1 + x)3

    128 143 − 318x + 143x 2 5120x 7 − 16x + 7x 2 + ζ2 H0,1 + 27(1 − x)(1 + x)3 9(1 − x)(1 + x)5

  (2) 512 1 + x 2 512xP1 − ζ2 H0,−1 − H0,−1,−1 3(1 − x)(1 + x)3 9(1 − x)3 (1 + x)5   (2) 512x 173 − 356x + 173x 2 512xP10 + H − H0,0,0,−1 0,0,0,1 9(1 − x)(1 + x)5 9(1 − x)3 (1 + x)5

 (2) 128xP11 (0) − ζ2 (7.43) + FV ,2 , 45(1 − x)3 (1 + x)5 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 21

1

with

21

(2) P1 P2(2) (2) P3 P4(2) (2) P5 P6(2) (2) P7 P8(2) (2) P9 (2) P10 (2) P11 (2) P12 (2) P13 (2) P14 (2) P15 (2) P16 (2) P17 (2) P18

22

= x − 9x + 10x − 9x + 1

(7.44)

= 11x 4 − 12x 3 + 26x 2 − 12x − 1

(7.45)

24

= 14x − 215x + 370x − 215x + 14

(7.46)

25

= 22x 4 − 187x 3 + 362x 2 − 187x + 22

(7.47)

= 58x 4 − 937x 3 + 1616x 2 − 937x + 40

(7.48)

28

= 65x − 876x + 1578x − 876x + 65

(7.49)

29

= 125x 4 − 1772x 3 + 3262x 2 − 1772x + 161

(7.50)

31

= 209x 4 − 887x 3 + 1354x 2 − 887x + 209

(7.51)

32

= 265x − 3484x + 6206x − 3484x + 229

(7.52)

33

= 312x 4 − 1297x 3 + 1972x 2 − 1297x + 312

(7.53)

35

= 590x 4 − 2447x 3 + 3760x 2 − 2447x + 590

(7.54)

36

= 609x + 21136x + 56414x + 21136x + 609

(7.55)

= x 5 + 148x 4 − 143x 3 + 145x 2 − 140x + 1

(7.56)

39

= 104x 5 − 263x 4 + 173x 3 + 317x 2 − 407x + 104

(7.57)

40

= 86x 6 − 1756x 5 + 2685x 4 − 1008x 3 − 470x 2 + 76x − 45

(7.58)

= 157x 6 − 2540x 5 + 3393x 4 − 1728x 3 + 299x 2 − 532x + 87

(7.59)

43

= 671x + 284x − 322x − 2888x + 863x + 188x + 212

(7.60)

44

4

3

4

2

3

4

2

3

2

4

3

4

6

2

3

5

2

4

3

2

26 27

30

34

37 38

41 42

45

= 24 ln(2)(x − 1)2 (x + 1)4 − 61x 6 + 7514x 5 + 2979x 4 − 17684x 3 +1601x 2 + 7194x + 185.

23

46

(7.61)

47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11

(0)

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(0),r

The constant parts FV ,1(2) are given by the genuine terms and more simple terms, FV ,1(2) , resulting from the pole-terms in the ε-expansion. The latter terms are not displayed explicitly, because of space reasons. In an attachment we will present the complete renormalized form factors. Because of contributions due to non first order factorizing terms (elliptic and higher) in some color–zeta combinations higher functions contribute, for which we have calculated analytically at least 2000 moments, and in some cases 4000, 6000 and 8000 to determine their recurrence relations. They are used to define the corresponding numeric representations. The corresponding recurrences, which do not factorize in first order completely will be studied elsewhere.  (3) (3) (3) 256P5 128H0,0,1 P8 64H0,−1,−1 P3 (0) 2 FV ,1 = nh − + ln(2)ζ − 2 27(1 − x)(1 + x)3 27(1 + x)4 81(1 + x)4

12 13

[m1+; v1.304; Prn:6/09/2019; 12:57] P.24 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

24

−

(3) 64H0,−1 P12

243(1 − x)(1 + x)3

+

(3) 8P14

729(1 + x)4

+ (3)

(3) 32H0,0,−1 P17

81(1 − x)(1 + x)4

(3) (3) 64H0,0,0,−1 P25 4H04 P31 64ζ22 P22 + − − 9(1 − x)(1 + x)5 27(1 − x)(1 + x)5 27(1 − x)(1 + x)5 (3) (3) (3) 8H03 P52 16P46 128H0,1 P8 + + − 243(1 − x)(1 + x)7 81(1 + x)4 729(1 − x)(1 + x)5

  (3) 512 1 + 6x + x 2 P1 128(1 + x 2 ) H0,0,−1 H0 + H0,0,1 − 27(1 − x 2 ) 27(1 − x)(1 + x)5   (3) 128 1 + x 2 3 64H1 P8(3) 32P47 + H + − + H 0 81(1 − x 2 ) −1 81(1 + x)4 243(1 − x)(1 + x)6

   (3)  128 1 + 6x + x 2 P1 64 1 + x 2 − H0,1 + H0,−1 H02 27(1 − x 2 ) 27(1 − x)(1 + x)5 (3) (3) (3) 64H0,−1 P3 16H02 P17 64H0 P12 + + − 27(1 − x)(1 + x)3 243(1 − x)(1 + x)3 81(1 − x)(1 + x)4

  (3) 128 1 + x 2 32H03 P25 + + [H0,0,−1 + 2H0,−1,−1 ] H−1 27(1 − x 2 ) 81(1 − x)(1 + x)5

    (3) 32 1 + x 2 2 128 1 + x 2 32H0 P3 2 H − H0,−1 H−1 + − − 27(1 − x)(1 + x)3 27(1 − x 2 ) 0 27(1 − x 2 )    (3)  256 1 + 6x + x 2 P1 128 1 + x 2 − H0,0,0,1 − H0,0,−1,−1 27(1 − x 2 ) 9(1 − x)(1 + x)5   (3) (3) 256 1 + x 2 8H02 P26 32H0,−1 P32 − + − + H 0,−1,−1,−1 27(1 − x 2 ) 3(1 − x)(1 + x)5 9(1 − x)(1 + x)5 (3) (3) (3) 32P50 8H0 P53 32P18 + + − 27(1 − x)(1 + x)4 1215(1 − x)(1 + x)6 81(1 − x)(1 + x)7

  (3) (3) 64 1 + x 2 2 16P20 32H0 P24 H − + + H ζ −1 2 27(1 − x 2 ) −1 81(1 − x)(1 + x)4 27(1 − x)(1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.25 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

   (3) 128 1 + x 2 32H0 P34 + − H−1 ζ3 27(1 − x 2 ) 27(1 − x)(1 + x)5    (3) (3)   1024x 2 5 − 2x + 5x 2 128P9 2048P30 + nh H0 + − − H 1 27(1 + x)4 27(1 − x)(1 + x)5 (1 − x)(1 + x)5

      256x 2 5 − 2x + 5x 2 2 1024x 2 5 − 2x + 5x 2 1 − H0 + H0,1 Li4 5 5 2 (1 − x)(1 + x) (1 − x)(1 + x)

  (3) 128x 2 5 − 2x + 5x 2 16P9(3) 256P30 + + − − H1 H0 81(1 + x)4 81(1 − x)(1 + x)5 3(1 − x)(1 + x)5     32x 2 5 − 2x + 5x 2 2 128x 2 5 − 2x + 5x 2 − H0 + H0,1 3(1 − x)(1 + x)5 3(1 − x)(1 + x)5   (3) (3) 128x 2 5 − 2x + 5x 2 128H0,1 P36 64ζ22 P41 4 − ζ (2) + + − ln 2 3(1 − x)(1 + x)5 9(1 − x)(1 + x)5 9(1 − x)(1 + x)5 (3)

39 40 41 42 43 44 45 46 47

(3)

(3) 64H02 P51 16P54 128H0,−1 P40 − + + 9(1 − x)(1 + x)5 9(1 − x)2 (1 + x)6 81x(1 + x)6

  (3) (3) 64x 2 7 − 22x + 7x 2 128H1 P36 32P43 + − H0 H0 − 9(1 − x)(1 + x)5 27(1 − x)(1 + x)5 (1 − x)(1 + x)5



(3) (3) 128H0 P39 128P4 3 + − + H−1 ζ2 ln(2) 3(1 + x)4 9(1 − x)(1 + x)5   (3) (3) 256x 2 5 − 2x + 5x 2 128P35 32P11 + + H1 H0 + 27(1 + x)4 27(1 − x)(1 + x)5 (1 − x)(1 + x)5

    64x 2 5 − 2x + 5x 2 2 256x 2 5 − 2x + 5x 2 + H0 − H0,1 ζ2 (1 − x)(1 + x)5 (1 − x)(1 + x)5  

(3) 256x 2 5 − 2x + 5x 2 2 256P5 2 + ζ (2) + n ln(2)ζ2 ln l 2 27(1 + x)4 (1 − x)(1 + x)5

−

(3) 128H0,0,−1 P16

81(1 − x)(1 + x)4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

(3)

(3) (3) (3) 512H0,−1,−1 P6 128H0,0,1 P8 128H0,−1 P10 64P13 − − − + 81(1 − x)(1 + x)3 81(1 + x)4 81(1 − x)(1 + x)3 729(1 + x)4

37 38

25

−

(3) 128H0,0,0,−1 P28 9(1 − x)(1 + x)5

−

(3) 8H04 P33

27(1 − x)(1 + x)5 (3) (3) (3) 64ζ22 P37 16H03 P38 128H0,1 P8 + + + 81(1 + x)4 135(1 − x)(1 + x)5 81(1 − x)(1 + x)5   (3)   (3) 1 + 6x + x 2 P1 64P45 128 512 − + H0,0,1 − H0,0,−1 H0 27 9 729(1 − x)(1 + x)5 (1 − x)(1 + x)5   (3) (3) 1024 1 + x 2 3 64H1 P8 32P48 H + H − + 0 81(1 − x 2 ) −1 81(1 + x)4 81(1 − x)(1 + x)6

35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

   (3)  (3) 128 1 + 6x + x 2 P1 64 1 + 6x + x 2 P1 + H0,1 − H0,−1 H02 27(1 − x)(1 + x)5 9(1 − x)(1 + x)5 (3) (3) (3) 512H0,−1 P6 64H02 P16 128H0 P10 + + + 81(1 − x)(1 + x)3 81(1 − x)(1 + x)3 81(1 − x)(1 + x)4

    (3) 1024 1 + x 2 2048 1 + x 2 64H03 P28 + + H0,0,−1 + H0,−1,−1 H−1 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x)(1 + x)5

    (3) 256 1 + x 2 2 1024 1 + x 2 256H0 P6 2 − + H + H0,−1 H−1 81(1 − x)(1 + x)3 27(1 − x 2 ) 0 27(1 − x 2 )     (3)   256 1 + 6x + x 2 P1 1024 1 + x 2 2048 1 + x 2 H0,0,−1,−1 − − H0,0,0,1 − 27(1 − x 2 ) 27(1 − x 2 ) 9(1 − x)(1 + x)5 (3) (3) (3) 64H0,−1 P23 16H02 P27 16H0 P42 × H0,−1,−1,−1 + − + 9(1 − x)(1 + x)5 3(1 − x)(1 + x)5 81(1 − x)(1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

(3)

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

(0) (0) (0) (0),r + nh FV ,1,1 (x) + nh ζ2 FV ,1,2 (x) + nh ζ3 FV ,1,3 (x) + FV ,1 (x),

37 38

40 41 42 43 44 45 46 47

(3)

(3)

  8P49 128P19 64H0 P29 H−1 + + − + 4 6 5 81(1 − x)(1 + x) 81(1 − x)(1 + x) 27(1 − x)(1 + x)

    (3) 512 1 + x 2 2 32 1 + x 2 16P21 + + H H0 ζ2 + 27(1 − x 2 ) −1 81(1 − x)(1 + x)4 3(1 − x 2 )

      1024 1 + x 2 1024x 2 5 − 2x + 5x 2 1 H − Li ζ + − −1 3 4 2 27(1 − x 2 ) (1 − x)(1 + x)5

  (3) (3) 32x 2 5 − 32x + 5x 2 8P15 8P7 + − H − + − ζ ζ 0 3 2 (1 + x)4 135(1 + x)4 (1 − x)(1 + x)5

  (3) 64x 2 43 + 98x + 43x 2 32P44 + H − H0 1 5(1 − x)(1 + x)5 135(1 − x)(1 + x)5

    16x 2 43 + 98x + 43x 2 2 64x 2 43 + 98x + 43x 2 + H0 − H0,1 ζ22 5(1 − x)(1 + x)5 5(1 − x)(1 + x)5

     (3) 64x 2 43 + 98x + 43x 2 3 80x 2 1 − 16x + x 2 20P2 + ζ2 + H0 + ζ5 (1 + x)4 5(1 − x)(1 + x)5 (1 − x)(1 + x)5

18

39

[m1+; v1.304; Prn:6/09/2019; 12:57] P.26 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

26

(7.62)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

with the polynomials (3) P1 (3) P2 (3) P3 (3) P4 P5(3) (3) P6

1

39 40

= x − 2x + 12x − 2x + 1,

(7.63)

= x 4 + 8x 3 + 50x 2 + 8x + 1,

(7.64)

42

= x 4 + 12x 3 + 6x 2 + 12x + 1,

(7.65)

43

= 2x − 7x + 12x − 7x + 2,

(7.66)

44

= 11x 4 + 14x 3 + 238x 2 + 14x + 11,

(7.67)

46

= 11x 4 + 40x 3 + 34x 2 + 40x + 11,

(7.68)

47

4

3

4

2

3

2

41

45

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.27 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(3)

27

P7 = 13x 4 + 56x 3 + 170x 2 + 56x + 13,

(7.69)

1

(3) P8 (3) P9 (3) P10 (3) P11 (3) P12 (3) P13 (3) P14 (3) P15 (3) P16 (3) P17 (3) P18 (3) P19 (3) P20 (3) P21 (3) P22 (3) P23 (3) P24 (3) P25 (3) P26 (3) P27 (3) P28 (3) P29 (3) P30 (3) P31 (3) P32 (3) P33 (3) P34 (3) P35 (3) P36 (3) P37 (3) P38 (3) P39 (3) P40 (3) P41

= 19x − 2x + 206x − 2x + 19,

(7.70)

2

= 53x 4 + 184x 3 + 502x 2 + 184x + 53,

(7.71)

4

= 89x + 108x + 710x + 108x + 89,

(7.72)

5

= 91x 4 − 94x 3 − 70x 2 − 94x + 91,

(7.73)

= 155x 4 + 10x 3 + 1846x 2 + 10x + 155,

(7.74)

8

= 377x + 3452x + 6006x + 3452x + 377,

(7.75)

9

= 1367x 4 + 26372x 3 + 46362x 2 + 26372x + 1367,

(7.76)

= 5363x 4 + 15190x 3 + 15322x 2 + 15190x + 5363,

(7.77)

12

= 3x + 123x − 60x + 356x + 81x + 41,

(7.78)

13

= 35x 5 − 81x 4 + 362x 3 − 470x 2 + 3x − 41,

(7.79)

= 39x 5 − 29x 4 + 434x 3 − 398x 2 + 55x − 37,

(7.80)

16

= 79x + 39x + 772x − 476x + 165x − 35,

(7.81)

17

= 253x 5 + 435x 4 + 7450x 3 − 7234x 2 − 279x − 241,

(7.82)

19

= 463x 5 + 837x 4 + 7414x 3 − 5046x 2 + 795x − 111,

(7.83)

20

= x + 4x + 78x + 4x + 1,

(7.84)

21

= x 6 + 4x 5 − 21x 4 + 288x 3 − 21x 2 + 4x + 1,

(7.85)

23

= x 6 + 4x 5 − 17x 4 + 248x 3 − 17x 2 + 4x + 1,

(7.86)

24

= x + 4x − 5x + 128x − 5x + 4x + 1,

(7.87)

25

= x 6 + 4x 5 − x 4 + 88x 3 − x 2 + 4x + 1,

(7.88)

27

= x 6 + 4x 5 + 3x 4 + 48x 3 + 3x 2 + 4x + 1,

(7.89)

28

= x + 4x + 9x − 12x + 9x + 4x + 1,

(7.90)

= x 6 + 4x 5 + 19x 4 − 112x 3 + 19x 2 + 4x + 1,

(7.91)

31

= x 6 + 10x 5 + 36x 4 + 84x 3 + 36x 2 + 10x + 1,

(7.92)

32

= 3x + 12x + x + 224x + x + 12x + 3,

(7.93)

= 3x 6 + 12x 5 + 5x 4 + 184x 3 + 5x 2 + 12x + 3,

(7.94)

35

= 3x + 12x + 11x + 124x + 11x + 12x + 3,

(7.95)

36

= 11x 6 + 44x 5 + 5x 4 + 808x 3 + 5x 2 + 44x + 11,

(7.96)

= 13x 6 + 67x 5 + 243x 4 + 228x 3 + 243x 2 + 67x + 13,

(7.97)

39

= 21x − 76x − 193x − 36x − 193x − 76x + 21,

(7.98)

40

= 31x 6 + 124x 5 + 304x 4 − 622x 3 + 304x 2 + 124x + 31,

(7.99)

42

= 35x 6 − 118x 5 + 325x 4 − 404x 3 − 423x 2 − 110x − 41,

(7.100)

43

= 99x + 70x + 61x + 24x + 61x + 70x + 99,

(7.101)

44

= 105x 6 + 88x 5 + 127x 4 − 48x 3 + 127x 2 + 88x + 105,

(7.102)

46

= 147x 6 − 64x 5 − 259x 4 − 120x 3 − 259x 2 − 64x + 147,

(7.103)

47

4

3

2

4

3

4

2

3

5

2

4

5

6

3

4

3

5

6

4

5

6

6

2

3

4

2

3

5

5

2

3

4

5

6

3

4

5

6

2

3

5

6

2

4

4

2

3

3

2

2

3

6 7

10 11

14 15

18

22

26

29 30

33 34

37 38

41

45

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.28 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

28 (3)

1 2 3 4 5 6 7 8 9 10 11

P42 = 189x 6 − 390x 5 + 1827x 4 − 1684x 3 − 1165x 2 − 358x − 115,

(7.104)

1

(3) P43 (3) P44 (3) P45 (3) P46 (3) P47 (3) P48 (3) P49

= 381x + 316x − 2779x + 13392x − 3139x + 568x + 309,

(7.105)

2

= 806x 6 + 2501x 5 + 5574x 4 + 9924x 3 + 5574x 2 + 2501x + 806,

(7.106)

4

= 1471x + 5920x + 19873x + 30272x + 19873x + 5920x + 1471,

(7.107)

5

= 2945x 6 + 9044x 5 + 56687x 4 + 91384x 3 + 56687x 2 + 9044x + 2945,

(7.108)

= 19x 7 − 43x 6 + 476x 5 − 2038x 4 − 3695x 3 − 2817x 2 − 432x − 174,

(7.109)

8

= 62x + 269x + 965x + 2426x + 2876x + 1637x + 481x + 116,

(7.110)

9

6

5

6

7

4

5

3

4

6

2

3

5

2

4

3

(3) P50

13 14

= 4452x + 17001x + 81141x + 78032x − 20702x − 57731x 6

5

4

(7.112)

(3)

(3) P52

17 18

= 181x + 240x + 1444x + 1952x − 3522x − 4000x − 1788x 8

7

6

5

4

− 336x − 91,

22

(7.115)

− 222044x 2 − 60837x + 2768. (0)

In the variable y = 1 − x the first expansion coefficients of the functions FV ,1,i (x), i = 1..3 are given by

28

2222242 1047067y 2 1047067y 3 3436873681y 4 + + + 243 729 729 3499200 923912881y 5 (7.117) + O(y 6 ) + 1749600 2390434 8763197y 2 8763197y 3 4103868673y 4 (0) − − − FV ,1,2 (x) = 243 4860 4860 3402000 345583241y 5 (7.118) + O(y 6 ) − 567000 311488 259276y 2 259276y 3 571282067y 4 156440467y 5 (0) − − − − FV ,1,3 (x) = 81 243 243 777600 388800 6 + O(y ) (7.119) (0)

FV ,1,1 (x) = −

29 30 31 32 33 34 35 36 37 38 39 40

44 45 46 47

21

23

(7.116)

25

19

22

(3)

24

16

20

P54 = 2768x 8 − 60837x 7 − 222044x 6 − 979155x 5 − 1790040x 4 − 979155x 3

23

15

18

(7.114)

(3)

21

14

17

P53 = 213x 8 + 240x 7 + 1452x 6 + 1264x 5 − 6642x 4 − 4688x 3 − 1780x 2

20

43

(7.113)

2

− 336x − 123,

19

42

3

12 13

P51 = 70x 8 + 173x 7 − 342x 6 + 245x 5 + 860x 4 + 241x 3 − 458x 2 + 45x − 2,

16

41

2

− 12251x − 2902,

15

27

3

7

11

(7.111)

7

6

10

= 1969x 7 + 7833x 6 + 31477x 5 + 37933x 4 + 4483x 3 − 10661x 2 − 1833x − 545,

12

26

2

3

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

and



(0) FV ,2





  (4) 128 451 + 830x + 451x 2 128P11 2 = x nh − + 243(1 + x)4 243(x − 1)(1 + x)5

  (4) 2 128(x − 1)H−1 512 7 − 79x + 7x 2 128P14 − H − + − H −1 0 81(x − 1)(1 + x)3 27(1 + x)3 81(x − 1)(1 + x)6

42 43 44 45 46 47

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

29



  (4) 64 1 − 40x + 3x 2 32P18 512x 2 H−1 2 + H−1 H0 + − + 27(1 + x)4 81(x − 1)(1 + x)7 9(x − 1)(1 + x)5   256 1 − 20x + x 2 H02 H1 320x 2 H04 3 × H0 − − 27(1 + x)4 9(x − 1)(1 + x)5

    512 1 − 20x + x 2 H0 512 7 − 79x + 7x 2 1024x 2 H02 + − H0,1 + 27(1 + x)4 81(x − 1)(1 + x)3 3(x − 1)(1 + x)5

  512 1 − 20x + x 2 256(x − 1)H−1 4096x 2 H0 + + H0,−1 + − H0,0,1 27(1 + x)3 27(1 + x)4 3(x − 1)(1 + x)5   128 1 − 40x + 3x 2 256(x − 1)H0,−1,−1 2048x 2 H0,0,0,1 − H − − 0,0,−1 27(1 + x)4 27(1 + x)3 (x − 1)(1 + x)5   (4) 1024 1 + 22x + x 2 32P16 1024x 2 H0,0,0,−1 − + − ln(2) − 9(1 + x)4 3(x − 1)(1 + x)5 405(x − 1)(1 + x)6

(4) 32P19 256x 2 H02 1024x 2 H−1 + − + − H 0 27(x − 1)(1 + x)7 3(x − 1)(1 + x)5 (x − 1)(1 + x)5

  128 7 − 120x + 5x 2 1792x 2 ζ22 2048x 2 H0,−1 + H + + ζ −1 2 27(1 + x)4 3(x − 1)(1 + x)5 3(x − 1)(1 + x)5

    (4) 128 27 + 364x + 29x 2 160ζ5 P1 1024x 2 H0 + + ζ + n − 3 h 27(1 + x)4 (x − 1)2 (1 + x)4 (x − 1)(1 + x)5   (4) 128x 1 − x + x 2 2 128P8 128xH0 P7(4) 4 + ln (2) − − H 27(x − 1)3 (1 + x)5 81(x − 1)2 (1 + x)4 3(x − 1)(1 + x)5 0

    512x 1 − x + x 2 512x 1 − x + x 2 − H0 H1 + H0,1 3(x − 1)(1 + x)5 3(x − 1)(1 + x)5   1024P8(4) 1024xH0 P7(4) 1 + Li4 − 2 9(x − 1)3 (1 + x)5 27(x − 1)2 (1 + x)4     1024x 1 − x + x 2 2 4096x 1 − x + x 2 − H0 − H0 H1 (x − 1)(1 + x)5 (x − 1)(1 + x)5

  (4) 4096x 1 − x + x 2 256P9 + H + n 0,1 l (x − 1)(1 + x)5 81(x − 1)(1 + x)5

  2 2048xH−1 512 23 − 130x + 23x 2 − H−1 + H0 81(x − 1)(1 + x)3 27(x − 1)(1 + x)3   (4) 1024 14 + 27x + 14x 2 128P15 − + 81(1 + x)4 81(x − 1)(1 + x)6

  (4) 256 − 1 + 17x − 25x 2 + x 3 64P4 2 + H + − H −1 0 27(x − 1)(1 + x)4 27(x − 1)(1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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  256 1 − 20x + x 2 H02 H1 320x 2 H04 512x 2 H−1 3 − − H0 − 27(1 + x)4 9(x − 1)(1 + x)5 9(x − 1)(1 + x)5

  512 1 − 20x + x 2 H0 1024x 2 H02 + − H0,1 4 27(1 + x) 3(x − 1)(1 + x)5

  512 23 − 130x + 23x 2 512x 2 H02 4096xH−1 + + − H0,−1 81(x − 1)(1 + x)3 (x − 1)(1 + x)5 27(x − 1)(1 + x)3

  512 1 − 20x + x 2 4096x 2 H0 1024x 2 H0 + − + H0,0,1 − 4 5 27(1 + x) 3(x − 1)(1 + x) (x − 1)(1 + x)5   512 − 1 + 17x − 25x 2 + x 3 4096xH0,−1,−1 + H0,0,−1 + 4 27(x − 1)(1 + x) 27(x − 1)(1 + x)3   1024 1 + 22x + x 2 2048x 2 H0,0,0,1 1024x 2 H0,0,0,−1 − + + − ln(2) 9(1 + x)4 (x − 1)(1 + x)5 3(x − 1)(1 + x)5

(4) (4) 128P6 32P17 1024x 2 H−1 + − − H0 − 81(x − 1)(1 + x)6 27(x − 1)(1 + x)5 3(x − 1)(1 + x)5   512 − 3 + 67x − 59x 2 + 3x 3 256x 2 H02 − + H−1 27(x − 1)(1 + x)4 (x − 1)(1 + x)5

7424x 2 ζ22 7168x 2 H0,−1 + − ζ 2 3(x − 1)(1 + x)5 15(x − 1)(1 + x)5       256 − 7 − 155x + 139x 2 + 7x 3 4096x 1 − x + x 2 1 + ζ Li + − 3 4 27(x − 1)(1 + x)4 2 (x − 1)(1 + x)5   (4) (4) 512x 1 − x + x 2 4 256H−1 P5 1024H0,−1 P20 − ln (2) + ln(2) − + 3(x − 1)2 (1 + x)4 9(x − 1)3 (1 + x)5 3(x − 1)(1 + x)5 (4)

(4)

(4)

512H0 H1 P23 512H0,1 P23 256xH02 P24 − + + 3 5 3 5 9(x − 1) (1 + x) 9(x − 1) (1 + x) 9(x − 1)3 (1 + x)6

(4) (4) (4) 64P26 256P25 1024H−1 P22 + + − + H0 81(x − 1)2 x(1 + x)6 9(x − 1)3 (1 + x)5 27(x − 1)3 (1 + x)5   (4) (4) 256x 1 − 7x + x 2 3 256xH0 P3 128P10 2 − H (2) − + + ln 0 (x − 1)(1 + x)5 9(x − 1)3 (1 + x)5 27(x − 1)2 (1 + x)4

      256x 1 − x + x 2 2 1024x 1 − x + x 2 1024x 1 − x + x 2 + H + H0 H1 − H0,1 (x − 1)(1 + x)5 0 (x − 1)(1 + x)5 (x − 1)(1 + x)5   (4) 768x 2 2 − 3x + 2x 2 64P2 + + H0 ζ3 ζ2 (x − 1)2 (1 + x)4 (x − 1)3 (1 + x)5 (4) (4) (4) 32P13 64xH0 P12 1024P21 + − − − ln(2) 45(x − 1)3 (1 + x)5 135(x − 1)2 (1 + x)4 9(x − 1)3 (1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

    1024x 1 − x + x 2 2 64x 11 + 25x + 11x 2 2 + ln (2) + H0 (x − 1)(1 + x)5 5(x − 1)(1 + x)5

    256x 11 + 25x + 11x 2 256x 11 + 25x + 11x 2 + H0 H1 − H0,1 ζ22 5(x − 1)(1 + x)5 5(x − 1)(1 + x)5    256x 11 + 25x + 11x 2 3 640(−2 + x)x 2 (−1 + 2x) + ζ2 − H0 ζ5 5(x − 1)(1 + x)5 (x − 1)3 (1 + x)5

1 2 3 4 5 6 7 8

(0) (0) (0),r + nh FV(0) ,2,1 (x) + nh FV ,2,2 (x)ζ2 + nh FV ,2,3 (x)ζ3 + FV ,2 (x),

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

31

1 2 3 4 5 6 7 8

(7.120)

10

with the polynomials P1(4) (4) P2 (4) P3 (4) P4 (4) P5 P6(4) (4) P7 (4) P8 (4) P9 (4) P10 (4) P11 (4) P12 (4) P13 (4) P14 (4) P15 (4) P16 (4) P17 (4) P18 (4) P19 (4) P20 (4) P21 (4) P22 (4) P23 (4) P24 (4) P25 (4) P26

9

11

= x + 5x − 10x + 5x + 1,

(7.121)

= x 4 + 9x 3 − 26x 2 + 9x + 1,

(7.122)

= x 4 + 82x 3 + 62x 2 + 82x + 1,

(7.123)

15

= 3x − 64x − 6x + 16x − 1,

(7.124)

16

= 5x 4 − 12x 3 + 26x 2 − 12x + 5,

(7.125)

= 7x 4 − 147x 3 − 8x 2 + 13x − 1,

(7.126)

19

= 7x + 28x + 122x + 28x + 7,

(7.127)

20

= 17x 4 + 102x 3 + 50x 2 + 102x + 17,

(7.128)

= 67x 4 − 10x 3 − 138x 2 − 10x + 67,

(7.129)

23

= 115x + 96x + 262x + 96x + 115,

(7.130)

24

= 187x 4 − 1300x 3 − 2590x 2 − 1300x + 187,

(7.131)

= 671x 4 + 1042x 3 − 702x 2 + 1042x + 671,

(7.132)

27

= 1145x − 960x − 8542x − 960x + 1145,

(7.133)

28

= 2x 5 − 61x 4 + 137x 3 + 281x 2 + 177x − 16,

(7.134)

30

= 14x 5 − 35x 4 − 271x 3 − 325x 2 − 87x + 32,

(7.135)

31

= 77x − 19649x − 22606x + 5886x + 15009x + 483,

(7.136)

32

= 361x 5 − 3677x 4 − 5558x 3 + 790x 2 + 2701x + 7,

(7.137)

34

= 13x 6 − 174x 5 − 381x 4 + 60x 3 + 243x 2 + 114x − 3,

(7.138)

35

= 17x − 206x − 449x + 124x + 175x + 82x + 1,

(7.139)

36

= 27x 6 − 31x 5 + 21x 4 − 10x 3 + 21x 2 − 31x + 27,

(7.140)

38

= 27x 6 − 28x 5 − 24x 4 + 56x 3 − 24x 2 − 28x + 27,

(7.141)

39

= 27x − 28x − 6x + 20x − 6x − 28x + 27,

(7.142)

40

= 27x 6 − 25x 5 − 69x 4 + 122x 3 − 69x 2 − 25x + 27,

(7.143)

42

= 27x 6 + 136x 5 − 245x 4 − 221x 3 + 202x 2 + 223x − 134,

(7.144)

43

= 165x − 101x − 2215x + 5984x − 2404x + 7x + 120,

(7.145)

44

4

3

4

2

3

4

2

3

2

4

3

2

4

3

5

2

4

6

3

5

6

4

5

6

4

5

3

4

2

2

3

14

17 18

21 22

25 26

2

33

37

41

= 1280x 8 − 7735x 7 + 155816x 6 + 33391x 5 − 401120x 4 + 33391x 3 + 155816x 2 − 7735x + 1280,

13

29

2

3

12

(7.146)

45 46 47

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32

and

1

1083812 32229191y 2 32229191y 3 488527686149y 4 − − − 243 18225 18225 428652000 5 109512399989y − + O(y 6 ) 214326000 1126457 150475907y 2 150475907y 3 462427840529y 4 (0) + + + FV ,2,2 (x) = − 243 72900 72900 357210000 93761868379y 5 + O(y 6 ) + 178605000 334736 8944046y 2 8944046y 3 92909575471y 4 (0) + + + FV ,2,3 (x) = − 81 6075 6075 95256000 22788254831y 5 + O(y 6 ). + 47628000

2

(0) FV ,2,1 (x) =

3 4

(7.147)

8 9

(7.148)

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

10 11 12 13

(7.149)

14 15 16

7.2. The scalar form factor

17 18

18 19

6 7

16 17

5

The scalar form factor is given by   

64(1 + x) 1 + x 2 1 1 64 2 2 FS = − 3 H0 n − (1 + x) + ε 2(1 + x)2 h 27 27(1 − x)  32H0 P8(5) 4 32 2 997 + 1418x + 997x − − nl (1 + x)2 + nh 2 27 27(1 − x ) 9

 

2   64(1 + x) 1 + x 2 256 1 + x 2 − H2 H0 + 27(1 − x) 27(1 − x)2 0  1 1 256x(1 + x)H0 832 − 2 (1 + x)2 − n2h − 2 ε 2(1 + x) 81 27(1 − x)     32(1 + x) 1 + x 2 2 128(1 + x) 1 + x 2 H−1 H0 + H0 − 27(1 − x) 27(1 − x)  

  128(1 + x) 1 + x 2 64(1 + x) 1 + x 2 + H0,−1 − ζ2 27(1 − x) 27(1 − x)  16  64 + nh 897 + 1786x + 897x 2 + nl − (1 + x)2 27 3     64(1 + x) 5 − 24x + 5x 2 256(1 + x) 1 + x 2 + H0 − H−1 H0 81(1 − x) 27(1 − x)      

64(1 + x) 1 + x 2 2 256(1 + x) 1 + x 2 128(1 + x) 1 + x 2 + H0 + H0,−1 − ζ2 27(1 − x) 27(1 − x) 27(1 − x)

(5) (5) 64P26 16P13 128H−1 P7(5) + − H + 0 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x)2 (1 + x)

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

33

  2   1024 1 + x 2 128 − 2 + x 2 1 + x 2 3 2 − H−1 H0 − H0 27(1 − x)2 27(1 − x)2      2 128(1 + x) 1 + x 2 128(1 + x) 1 + x 2 128 1 + x 2 − H0 H0,1 H0 H1 + − 3(1 − x) 3(1 − x) 3(1 − x)2 2 2   256 1 + x 2 2176 1 + x 2 128P7(5) − H0 H0,−1 + H0,0,1 − 27(1 − x 2 ) 27(1 − x)2 3(1 − x)2   2   (5) 256 1 + x 2 64 1 + x 2 − 1 + 35x 2 64P5 − H0,0,−1 + H0 ζ2 − 3(1 − x)2 27(1 − x 2 ) 27(1 − x)2  2    64 1 + x 2 1 1 32  − ζ3 107 + 294x + 107x 2 − n2h − 2 2 ε 2(1 + x) 81 3(1 − x)

  (5) 64P10 512x(1 + x)H−1 128(1 + x) 1 + x 2 2 + + H−1 H0 + 243(1 − x 2 ) 27(1 − x) 27(1 − x)

    (5) 64(1 + x) 1 + x 2 32(1 + x) 1 + x 2 3 64P17 2 − + H−1 H0 + H0 81(1 − x)(1 + x)2 27(1 − x) 27(1 − x)

    128(1 + x) 1 + x 2 512x(1 + x) 256(1 + x) 1 + x 2 − + H−1 H0,−1 + H0,0,−1 27(1 − x) 27(1 − x) 27(1 − x)     (5) 256(1 + x) 1 + x 2 160(1 + x) 1 + x 2 32P25 + + H0,−1,−1 + − H0 27(1 − x) 27(1 − x)(1 + x)2 27(1 − x)

   

(5) 128(1 + x) 1 + x 2 128(1 + x) 1 + x 2 64P15 + H−1 ζ2 − ζ3 + nh 27(1 − x) 27(1 − x) 243(1 + x)2   (5)  512(1 + x) 1 + x 2 2 128P4 640  2 + nl − 37 + 86x + 37x + H−1 + 243 27(1 − x) 81(1 − x 2 )

  (5) 256(1 + x) 5 − 24x + 5x 2 128P2 H−1 H0 + − − 81(1 − x) 81(1 − x)(1 + x)2

    256(1 + x) 1 + x 2 64(1 + x) 1 + x 2 3 2 − H−1 H0 + H0 27(1 − x) 27(1 − x)

    256(1 + x) 5 − 24x + 5x 2 1024(1 + x) 1 + x 2 + − H−1 H0,−1 81(1 − x) 27(1 − x)   (5) 512(1 + x) 1 + x 2 16P32 + [H0,0,−1 + 2H0,−1,−1 ] + − 27(1 − x) 81(1 − x)(1 + x)2

    224(1 + x) 1 + x 2 512(1 + x) 1 + x 2 + H0 + H−1 ζ2 27(1 − x) 27(1 − x)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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  (5) 2 P (5) 256H−1 512(1 + x) 1 + x 2 32H−1 P14 9 − ζ3 + − + 27(1 − x) 27(1 − x 2 ) 27(1 − x 2 )

(5) (5) (5) 32P34 32P37 8P41 + − H−1 − H 0 243(1 − x)(1 + x)3 243(1 − x)2 (1 + x)4 27(1 − x)2 (1 + x)

 2 (5) 2048 1 + x 2 32P39 2 2 + H + H −1 0 27(1 − x)2 81(1 − x)2 (1 + x)3

   (5) (5) 128 1 + x 2 − 1 + 5x 2 16P19 256H02 P1 3 4 H − H + + H −1 0 27(1 − x)2 81(1 − x)2 (1 + x) 0 27(1 − x)2

    256(1 + x) 7 − 32x + 7x 2 256(1 + x) 1 + x 2 − − H−1 H0 27(1 − x) 1−x

  (5) 128(1 + x) 1 + x 2 128P24 3 + H H1 − H0 H12 3(1 − x) 81(1 − x)2 (1 + x) 0

   2 (5) 256(1 + x) 7 − 32x + 7x 2 512 1 + x 2 512P3 + + H−1 H0 + 27(1 − x) 27(1 − x)2 3(1 − x)2    2 (5) 256(1 + x) 1 + x 2 512 1 + x 2 256P23 H02 − H0 H1 + − 27(1 − x)2 (1 + x) 3(1 − x) 27(1 − x)2     2 2  256(1 + x) 1 + x 2 6400 1 + x 2 896 1 + x 2 − H0,−1 H0,1 + H2 H−1 − 1−x 27(1 − x)2 27(1 − x)2 0,1 (5) (5) 512H−1 P9 32P14 + − + 27(−1 + x)(1 + x) 27(−1 + x)(1 + x)

 2 (5) (5) 8704 1 + x 2 128P29 64P12 + H + H2 H − −1 0 27(1 − x)2 27(1 − x)2 27(1 − x)2 (1 + x) 0  2   256(1 + x) 1 + x 2 512 1 + x 2 + − H0 H1 H0,−1 + 1−x 27(1 − x)2  2 (5) (5) 4480 1 + x 2 512P6 1024P16 2 H0 + H + − + 0,−1 27(1 − x)2 27(1 − x)2 27(1 − x)2 (1 + x)

  2 2 (5) 1024 1 + x 2 1024 1 + x 2 64P35 + H − H + H 1 −1 0,0,1 27(1 − x)2 3(1 − x)2 27(1 − x)2 (1 + x)

2 2   (5) 1024 1 + x 2 1024 1 + x 2 512P21 H1 + H−1 H0,0,−1 H0 − − 27(1 − x)2 (1 + x) 27(1 − x)2 3(1 − x)2     2  256(1 + x) 1 + x 2 256(1 + x) 1 + x 2 1280 1 + x 2 + − H0 H0,1,1 + − 3(1 − x) 1−x 27(1 − x)2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

 2 (5) 1792 1 + x 2 512P9 + H + H ] + [H 0 0,1,−1 0,−1,1 27(1 − x)2 27(−1 + x)(1 + x)

  2 (5) 256 1 + x 2 256P27 H + H0,0,0,1 − H 0 0,−1,−1 27(1 − x)2 27(1 − x)2 (1 + x)  2  (5) 1024 1 + x 2 256P22 1 H0,0,0,−1 + − − H0,0,1,1 + H0,0,1,−1 9 9(1 − x)2 (1 + x) 3(1 − x)2  (5)  64H−1 P11  4 256 + H0,0,−1,1 − H0,0,−1,−1 + H0,−1,0,1 + ln(2) 1 + x 2 − 9 3 27(1 − x 2 )

   (5) (5) 512 1 + x 2 13 + 4x 2 2P40 16P38 + + − + H−1 27(1 − x)(1 + x)4 27(1 − x)2 (1 + x)3 27(1 − x)2

  (5) (5) 128(1 + x) 1 + x 2 64P31 512P20 2 H + + H0 × H0 + 27(1 − x)2 (1 + x) 0 3(1 − x) 27(1 − x)2 (1 + x)

(5) (5) 128P28 128P18 H0,1 + H0,−1 ζ2 × H1 − 27(1 − x)2 (1 + x) 27(1 − x)2 (1 + x) 2  (5) (5) (5) 32P36 64(P33 + P30 H0 ) 256 1 + x 2 2 + − H1 ζ + 135(1 − x)2 (1 + x) 2 27(1 − x)2 (1 + x) 27(1 − x)2

  2 256 1 + x 2 (0) − H−1 ζ3 (7.150) + FS , 3(1 − x)2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

35

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

with the polynomials (5) P1 (5) P2 P3(5) (5) P4 (5) P5 (5) P6 (5) P7 (5) P8 (5) P9 (5) P10 (5) P11 (5) P12 (5) P13 (5) P14 (5) P15

1

27

= x − 32x + 48x − 32x − 17,

(7.151)

= 3x 4 + 42x 3 + 10x 2 + 6x − 5,

(7.152)

= 13x 4 + 47x 3 − 56x 2 + 47x + 13,

(7.153)

31

= 17x − 40x − 34x − 40x + 17,

(7.154)

32

= 19x 4 + 92x 3 + 110x 2 + 164x + 55,

(7.155)

34

= 27x 4 + 62x 3 − 64x 2 + 62x + 9,

(7.156)

35

= 55x + 164x + 146x + 164x + 55,

(7.157)

36

= 59x 4 + 226x 3 + 190x 2 + 226x + 59,

(7.158)

38

= 62x 4 + 151x 3 + 142x 2 + 151x + 62,

(7.159)

39

= 65x − 68x + 214x − 68x + 65,

(7.160)

40

= 137x 4 + 436x 3 + 352x 2 + 340x + 143,

(7.161)

42

= 163x 4 + 1050x 3 − 1146x 2 + 1050x + 163,

(7.162)

43

= 255x − 214x − 1530x − 214x + 255,

(7.163)

44

= 499x 4 − 730x 3 − 3050x 2 − 730x + 499,

(7.164)

46

= 11134x 4 + 45469x 3 + 67950x 2 + 45469x + 11134,

(7.165)

47

4

3

4

2

3

4

2

3

4

2

3

4

2

3

2

28 29 30

33

37

41

45

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

(5)

P16 = 5x 5 − 11x 4 + 89x 3 − 83x 2 + 14x − 2,

(7.166)

(5) P17 (5) P18 (5) P19 (5) P20 (5) P21 (5) P22 (5) P23 (5) P24 (5) P25 (5) P26 (5) P27 (5) P28 (5) P29 (5) P30 (5) P31 (5) P32 (5) P33 (5) P34 (5) P35 (5) P36 (5) P37 (5) P38 (5) P39 (5) P40

= 5x 5 + 3x 4 + 50x 3 − 14x 2 + 9x − 5,

(7.167)

= 7x 5 + 7x 4 − 52x 3 − 16x 2 − 41x − 41,

(7.168)

= 8x + 8x + 35x + 53x + 36x + 36,

(7.169)

= 13x 5 − 3x 4 + 79x 3 − 81x 2 + 2x − 14,

(7.170)

7

= 15x − 25x + 216x − 208x + 29x − 11,

(7.171)

8

= 15x 5 + 63x 4 − 225x 3 + 289x 2 − 31x + 17,

(7.172)

= 21x 5 − 3x 4 + 142x 3 − 110x 2 + 19x − 5,

(7.173)

11

= 25x − 7x + 156x − 164x + 3x − 29,

(7.174)

12

= 33x 5 + 19x 4 + 130x 3 − 178x 2 − 35x − 33,

(7.175)

14

= 41x + 36x − 59x − 41x − 82x − 23,

(7.176)

15

= 41x 5 + 121x 4 − 372x 3 + 500x 2 − 57x + 23,

(7.177)

= 43x 5 − 21x 4 + 298x 3 − 342x 2 − x − 65,

(7.178)

18

= 50x + 18x + 211x − 123x + 26x − 6,

(7.179)

19

= 71x 5 − 57x 4 + 682x 3 − 646x 2 + 75x − 53,

(7.180)

21

= 71x + 55x + 157x − x + 23x + 7,

(7.181)

22

= 133x 5 − 225x 4 + 298x 3 − 1130x 2 + 81x − 53,

(7.182)

= 221x 5 + 799x 4 − 194x 3 − 122x 2 + 213x − 149,

(7.183)

= 273x − 425x − 196x − 124x − 1245x − 331,

(7.184)

= 599x 5 + 2001x 4 − 388x 3 − 316x 2 + 1181x − 5,

(7.185)

28

= 809x + 265x + 3072x − 2452x + 45x − 499,

(7.186)

29

= 5411x 6 + 5015x 5 − 11707x 4 − 28382x 3 − 11707x 2 + 5015x + 5411,

(7.187)

31

= 155x 7 + 1277x 6 + 1073x 5 − 2993x 4 − 1563x 3 − 1661x 2 − 433x + 49,

(7.188)

32

= 181x + 191x − 288x − 210x − 1103x − 1097x − 646x − 100,

(7.189)

38 39

5

4

3

5

2

4

5

3

4

5

4

5

4

5

2

3

2

3

4

7

6

2

3

5

2

4

3

2

42 43 44 45 46 47

3 4 5

9 10

17

20

23 24 25 26 27

33 34 35

(7.190)

36 37

(5)

P41 = 5069x 8 − 892x 7 − 75926x 6 − 69132x 5 + 50716x 4 + 90780x 3 + 17638x 2 − 5140x − 4921.

16

30

= 13209x 7 + 22973x 6 + 44113x 5 − 24083x 4 − 106013x 3 − 82753x 2

38

(7.191)

39 40

40 41

2

13

2

3

4

5

2

3

4

5

2

3

1

6

− 11469x − 7017,

36 37

[m1+; v1.304; Prn:6/09/2019; 12:57] P.36 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

36

(0)

The constant part, FS , reads  (6) (6) 2 (6) 32P26 H H 256P 16P 1 1 (0) 0 2 2 14 FS = − − + n 2(1 + x)2 h 81(1 + x)2 243(1 + x)2 729(1 − x)(1 + x)3

  (6) 256(1 + x) 1 + x 2 3 128P12 H−1 512x(1 + x) 2 − − H − H−1 H0 27(1 − x) −1 81(1 − x) 243(1 − x 2 )

41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.37 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

37



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

  (6) (6) 64(1 + x) 1 + x 2 2 128P16 64P27 + + H−1 + H−1 H02 27(1 − x) 243(1 − x)(1 + x)4 81(1 − x)(1 + x)2

    (6) 64(1 + x) 1 + x 2 8(1 + x) 1 + x 2 4 64P30 3 + − − + H H0 H −1 0 81(1 − x) 9(1 − x) 243(1 − x)(1 + x)5

  (6) 256(1 + x) 1 + x 2 2 512H0 P2(6) 128P12 + + + H0 − H 0,1 27(1 − x) 81(1 + x)2 243(1 − x)(1 + x)

    128(1 + x) 1 + x 2 2 1024x(1 + x)H−1 256(1 + x) 1 + x 2 2 − H0 + + H−1 27(1 − x) 27(1 − x) 27(1 − x)   (6) 1024(1 + x) 1 + x 2 512P2 − × H0,−1 + H0 H0,0,1 27(1 − x) 81(1 + x)2

    (6) 256(1 + x) 1 + x 2 256(1 + x) 1 + x 2 256P16 + − + H0 − H−1 27(1 − x) 27(1 − x) 81(1 − x)(1 + x)2

  1024x(1 + x) 512(1 + x) 1 + x 2 − H−1 H0,−1,−1 × H0,0,−1 + − 27(1 − x) 27(1 − x)     512(1 + x) 1 + x 2 128(1 + x) 1 + x 2 + H0,0,0,1 + H0,0,0,−1 9(1 − x) 27(1 − x)     256(1 + x) 1 + x 2 512(1 + x) 1 + x 2 + H0,0,−1,−1 + H0,−1,−1,−1 27(1 − x) 27(1 − x) (6) (6) (6) 512P5 128P29 32P28 + − ln(2) − + − 27(1 + x)2 1215(1 − x)(1 + x)4 81(1 − x)(1 + x)5

    (6) 64(1 + x) 1 + x 2 16(1 + x) 1 + x 2 2 256P15 − H−1 H−1 H0 + H0 + 27(1 − x) 3(1 − x) 27(1 − x)(1 + x)2

    128(1 + x) 1 + x 2 2 64(1 + x) 1 + x 2 − H−1 − H0,−1 ζ2 27(1 − x) 3(1 − x)     (6) 128(1 + x) 1 + x 2 2 704(1 + x) 1 + x 2 256P20 − − ζ2 + − H0 9(1 − x) 81(1 − x)(1 + x)2 27(1 − x)



   256(1 + x) 1 + x 2 32  + H−1 ζ3 + nh ln4 (2) − 35 + 106x + 35x 2 27(1 − x) 81

  (6)  64P4 256  1 + H0 + Li4 − 35 + 106x + 35x 2 2 81(1 − x ) 2 27

(6)  256H02 H1 P2(6) 512P4 1664  + H0 + nl − 77 + 142x + 77x 2 + 27(1 − x)(1 + x) 243 81(1 + x)2   (6) 512(1 + x) 5 − 24x + 5x 2 2048P6(6) 512P7 + − H−1 + 729(1 − x)(1 + x) 81(1 − x)(1 + x) 81(1 − x)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.38 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

38



  (6) 2048(1 + x) 1 + x 2 3 128P19 − H−1 H0 + 81(1 − x) 81(1 − x)(1 + x)2

  (6) 512(1 + x) 1 + x 2 2 512P1 + H−1 + H−1 H02 27(1 − x) 81(1 − x)(1 + x)2

  (6) 128(1 + x) 1 + x 2 128P17 + − − H−1 H03 27(1 − x) 81(1 − x)(1 + x)2     (6) 16(1 + x) 1 + x 2 4 256(1 + x) 1 + x 2 2 512H0 P2 + + H0 + − H0 H0,1 9(1 − x) 27(1 − x) 81(1 + x)2     (6) 128(1 + x) 1 + x 2 2 1024(1 + x) 5 − 24x + 5x 2 512P7 + − H0 − 81(1 − x)(1 + x) 9(1 − x) 81(1 − x)

  (6) 2048(1 + x) 1 + x 2 2 512P2 × H−1 + H−1 H0,−1 + 27(1 − x) 81(1 + x)2

  (6) 1024(1 + x) 1 + x 2 1024P1 − H0 H0,0,1 + − 27(1 − x) 81(1 − x)(1 + x)2

    256(1 + x) 1 + x 2 2048(1 + x) 1 + x 2 + H0 − H−1 H0,0,−1 9(1 − x) 27(1 − x)

    1024(1 + x) 5 − 24x + 5x 2 4096(1 + x) 1 + x 2 + − H−1 H0,−1,−1 81(1 − x) 27(1 − x)     512(1 + x) 1 + x 2 256(1 + x) 1 + x 2 + H0,0,0,1 + H0,0,0,−1 9(1 − x) 9(1 − x)     2048(1 + x) 1 + x 2 4096(1 + x) 1 + x 2 + H0,0,−1,−1 + H0,−1,−1,−1 27(1 − x) 27(1 − x) (6) (6) (6) 512P5 32P24 32P22 + − ln(2) − − 27(1 + x)2 81(1 − x)(1 + x)2 81(1 − x)(1 + x)2

    128(1 + x) 1 + x 2 32(1 + x) 1 + x 2 2 + H−1 H0 + H0 27(1 − x) 3(1 − x)   (6) 1024(1 + x) 1 + x 2 2 1024P18 + H−1 − H−1 81(1 − x)(1 + x)2 27(1 − x)

    128(1 + x) 1 + x 2 3968(1 + x) 1 + x 2 2 − H0,−1 ζ2 − ζ2 9(1 − x) 135(1 − x)   (6) 64(1 + x) 1 + x 2 32P23 + − − H0 81(1 − x)(1 + x)2 3(1 − x)

   2048(1 + x) 1 + x 2 128  2 + H−1 ζ3 + ln (2) − 41 − 125x + 41x 2 27(1 − x) 27 2 × H−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.39 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

(6) (6) (6) 64P25 128P8 128P31 − + H0 H0 + ln(2) − 27(1 − x)(1 + x) 81x(1 + x)4 27(1 − x)(1 + x)3 (6) (6)  256P3 128P21 128  2 2 − − H + H H 1 − 16x + x 0 1 0 9(1 − x)2 (1 + x) 3(1 − x)(1 + x) 3

(6) (6) (6) 256P9 256P3 256P10 H0 H−1 + H0,1 + H0,−1 − 3(1 − x)(1 + x) 3(1 − x)(1 + x) 3(1 − x)(1 + x)

2   192x + 16 13 + 20x + 13x 2 + H0 ζ3 ζ2 (1 − x)(1 + x)

1 2 3 4 5 6 7 8 9 10 11 12

(6) 128P11

 32  2146 + 3383x + 2146x 2 − ln(2) 135 3(1 − x)(1 + x)

 (6) 32P13 160x 2 2 2 H0 ζ2 − 40 1 + 4x + x ζ5 + H0 ζ5 + 135(1 − x)(1 + x) (1 − x)(1 + x)

+

13 14 15 16 17

(0)

20

(0)

(0)

+ nh FS,1 (x) + nh ζ2 FS,2 (x) + nh ζ3 FS,3 (x),

18 19

39

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

(7.192)

18 19

with

20

(6)

21

P1 = 3x 4 + 42x 3 + 10x 2 + 6x − 5,

(7.193)

22

(6) P2 (6) P3 (6) P4 (6) P5 (6) P6 (6) P7 (6) P8 (6) P9 (6) P10 (6) P11 (6) P12 (6) P13 (6) P14 (6) P15 (6) P16 (6) P17 (6) P18 (6) P19 (6) P20

= 5x 4 + 2x 3 + 34x 2 + 2x + 5,

(7.194)

= 7x 4 − 13x 3 + 34x 2 − 13x + 7,

(7.195)

24

= 8x − 11x − 2x − 11x + 8,

(7.196)

25

= 11x 4 + 26x 3 + 70x 2 + 26x + 11,

(7.197)

= 16x 4 − 193x 3 − 445x 2 − 193x + 16,

(7.198)

28

= 17x − 40x − 34x − 40x + 17,

(7.199)

29

= 26x 4 + 25x 3 − 20x 2 + 25x + 26,

(7.200)

31

= 33x 4 + 12x 3 + 100x 2 + 12x + 33,

(7.201)

32

= 35x + 16x + 98x + 16x + 35,

(7.202)

= 49x 4 − 10x 3 + 166x 2 − 10x + 49,

(7.203)

35

= 65x 4 − 68x 3 + 214x 2 − 68x + 65,

(7.204)

36

= 1612x + 3737x + 6068x + 3737x + 1612,

(7.205)

= 3859x 4 + 15236x 3 + 22882x 2 + 15236x + 3859,

(7.206)

39

= 5x − 5x + 26x − 38x + x − 5,

(7.207)

40

= 5x 5 + 3x 4 + 50x 3 − 14x 2 + 9x − 5,

(7.208)

= 5x 5 + 3x 4 + 86x 3 + 22x 2 + 9x − 5,

(7.209)

43

= 10x − 9x + 22x − 74x − 5,

(7.210)

44

= 16x 5 + 13x 4 − 25x 3 − 123x 2 − 59x + 18,

(7.211)

46

= 35x 5 + 54x 4 + 107x 3 − 143x 2 − 66x − 35,

(7.212)

47

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

4

3

4

3

4

2

3

4

5

2

3

4

5

2

2

3

4

2

3

2

21 22 23

26 27

30

33 34

37 38

41 42

45

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11

(6)

P21 = 70x 5 − 11x 4 + 71x 3 − 70x 2 − 2x − 2,

(7.213)

1

(6) P22 (6) P23 (6) P24 (6) P25 (6) P26 (6) P27 (6) P28

= 105x + 51x + 1548x + 524x + 147x − 55,

(7.214)

2

= 325x 5 + 87x 4 + 274x 3 − 1938x 2 − 375x − 165,

(7.215)

4

= 391x 5 − 979x 4 − 1930x 3 + 746x 2 + 611x − 119,

(7.216)

5

= 261x + 790x − 1127x + 1600x − 569x + 1042x + 243,

(7.217)

= 599x 6 − 4456x 5 − 20887x 4 − 30512x 3 − 20887x 2 − 4456x + 599,

(7.218)

8

= 11x 7 + 202x 6 + 445x 5 + 304x 4 + 131x 3 − 240x 2 − 75x + 54,

(7.219)

9

12 13 14 15 16

19

5

6

28

3

2

6

5

4

3

2

35 36 37 38 39 40 41 42 43 44 45 46 47

10 11 12

(7.222)

14

= 25x + 84x + 358x + 944x + 1122x + 464x + 54x − 12x − 15, 7

6

5

4

3

2

= 536x 8 − 15135x 7 − 63923x 6 − 147609x 5 − 194394x 4 − 147609x 3 − 63923x 2 − 15135x + 536.

(7.223)

(0)

FS,1 (x) = −

(0) FS,2 (x) =

(0)

3932123y 5 16041283y 4 2421832y 3 2421832y 2 343864 + + + + + O(y 6 ) 18225 36450 3645 3645 81 (7.225)

FS,3 (x) = −

48600

−

21262303y 4 97200

−

22516y 3 81

−

16 17

19

96756433y 5 316061833y 4 731018y 3 731018y 2 874750 − − − − + O(y 6 ) 218700 437400 729 729 243 (7.224)

7752703y 5

15

18

(0)

The first expansion coefficients of FS,i , i = 1..3 are given by

22516y 2 81

+

62968 + O(y 6 ). 27 (7.226)

20 21 22 23 24 25 26 27 28 29 30

7.3. The pseudoscalar form factor

31 32

32

34

7

13

8

30

33

6

(6) P30 (6) P31

29

31

3

(7.221)

26 27

4

(6)

23

25

5

2

= 7239x + 16287x + 11607x + 7399x + 1301x − 7507x − 13747x − 5939, (7.220)

22

24

3

7

20 21

4

P29 = 15x 8 + 57x 7 + 264x 6 + 779x 5 + 1062x 4 + 539x 3 + 112x 2 + 9x − 5,

17 18

[m1+; v1.304; Prn:6/09/2019; 12:57] P.40 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

40

The unrenormalized pseudoscalar form factor reads   

64(1 − x) 1 + x 2 H0 1 1 3988 2 64 2 FP = 3 (1 − x) − + nh − (1 − x)2 n 27(1 + x) 27 ε 2(1 − x)2 h 27     64(1 − x) 1 + x 2 H0 32(1 − x) 59 + 36x + 59x 2 32 2 + nl (1 − x) − + H0 9 27(1 + x) 27(1 + x)

   2 256 1 + x 2 H02 1 1 832 − (1 − x)2 + 2 n2 27(1 + x)2 ε 2(1 − x)2 h 81     128(1 − x) 1 + x 2 H−1 H0 32(1 − x) 1 + x 2 H02 + − 27(1 + x) 27(1 + x)    

2 128(1 − x) 1 + x H0,−1 64(1 − x) 1 + x 2 ζ2 − + 27(1 + x) 27(1 + x)

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.41 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

  16(1 − x)2 299 + 694x + 299x 2 64 + nh − + nl (1 − x)2 2 3 9(1 + x)     320(1 − x) 1 + x 2 H0 256(1 − x) 1 + x 2 H−1 H0 − + 81(1 + x) 27(1 + x)     2  

2 64(1 − x) 1 + x H0 256(1 − x) 1 + x 2 H0,−1 128(1 − x) 1 + x 2 ζ2 − − + 27(1 + x) 27(1 + x) 27(1 + x)

  (7) 128(1 − x) 55 + 18x + 55x 2 16(1 − x)P12 − H−1 H0 + 27(1 + x)3 27(1 + x)

   2 64(1 − x) 23 + 9x + 41x 2 1024 1 + x 2 H−1 + + H02 27(1 + x) 27(1 + x)2      128 2 − x 2 1 + x 2 3 128(1 − x) 1 + x 2 H0 H1 − H0 + 3(1 + x) 27(1 + x)2   2  2 2 128(1 − x) 1 + x 128 1 + x H0 − + H0,1 3(1 + x) 3(1 + x)2    2 128(1 − x) 55 + 18x + 55x 2 2176 1 + x 2 H0 + − H0,−1 27(1 + x) 27(1 + x)2   2 2 256 1 + x 2 H0,0,1 256 1 + x 2 H0,0,−1 − + 3(1 + x)2 3(1 + x)2      64(1 − x) 55 + 18x + 19x 2 64 1 + x 2 1 − 35x 2 − H0 ζ2 + 27(1 + x) 27(1 + x)2  2     2 2 64 1 + x 2 ζ3  1 1 2 32(1 − x) 107 + 198x + 107x + + n 3(1 + x)2 ε 2(1 − x)2 h 81(1 + x)2  2  (7) 128(1 − x) 1 + x 2 H−1 64(1 − x)P8 − + H0 243(1 + x)3 27(1 + x)      64(1 − x) 1 + x 2 H−1 64(1 − x)2 1 + x 2 5 + 14x + 5x 2 + − H02 27(1 + x) 81(1 + x)4     32(1 − x) 1 + x 2 H03 256(1 − x) 1 + x 2 H−1 H0,−1 − + 27(1 + x) 27(1 + x)     2 128(1 − x) 1 + x H0,0,−1 256(1 − x) 1 + x 2 − − H0,−1,−1 27(1 + x) 27(1 + x)

    (7) 160(1 − x) 1 + x 2 H0 128(1 − x) 1 + x 2 H−1 32(1 − x)2 P6 − + + ζ2 27(1 + x) 27(1 + x) 27(1 + x)4  

(7) 128(1 − x) 1 + x 2 ζ3 128(1 − x)2 P16 + + nh − 27(1 + x) 243(1 + x)4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.42 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

42

  (7) 128(1 − x)2 185 + 358x + 185x 2 128(1 − x)P3 + nl − 243(1 + x)2 81(1 + x)3  2    1280(1 − x) 1 + x 2 H−1 512(1 − x) 1 + x 2 H−1 − + H0 81(1 + x) 27(1 + x)

     128(1 − x) 1 + x 2 5 + 12x + 3x 2 256(1 − x) 1 + x 2 H−1 + − + H02 27(1 + x) 81(1 + x)4

      1280(1 − x) 1 + x 2 64(1 − x) 1 + x 2 H03 1024(1 − x) 1 + x 2 H−1 − − − 27(1 + x) 81(1 + x) 27(1 + x)     512(1 − x) 1 + x 2 H0,0,−1 1024(1 − x) 1 + x 2 H0,−1,−1 × H0,−1 − − 27(1 + x) 27(1 + x)

    (7) 224(1 − x) 1 + x 2 H0 512(1 − x) 1 + x 2 H−1 16(1 − x)P27 − − ζ2 + 81(1 + x)4 27(1 + x) 27(1 + x)   (7) (7) 512(1 − x) 1 + x 2 32(1 − x)P29 32(1 − x)H−1 P14 + − ζ3 − 27(1 + x) 27(1 + x)3 243(1 + x)5

  (7) (7) 256(1 − x) 62 + 9x + 62x 2 2 32H−1 P13 8P32 − − H−1 H0 + 27(1 + x) 27(1 + x)2 243(1 + x)6

2 2  (7) 2048 1 + x 2 H−1 32P31 2 − + H 0 27(1 + x)2 81(1 − x)(1 + x)5

   (7) 128 1 + x 2 − 1 + 5x 2 16P19 3 + H + H4 H −1 0 27(1 + x)2 81(1 − x)(1 + x)2 0   (7) 1792(1 − x) 1 + x 2 128H03 P4 + − + 81(1 + x)2 27(1 + x)

    256 − 17 − 6x − 6x 3 + x 4 2 256(1 − x) 1 + x 2 H−1 H0 H1 H0 − − 1+x 27(1 + x)2     128(1 − x) 1 + x 2 H0 H12 1792(1 − x) 1 + x 2 + + − 3(1 + x) 27(1 + x)

 2 (7) (7) 512 1 + x 2 H−1 512P2 256P22 − + − H2 H 0 27(1 + x)2 3(1 + x)2 27(1 − x)(1 + x)2 0     2  256(1 − x) 1 + x 2 256(1 − x) 1 + x 2 H−1 512 1 + x 2 H0 − − H1 + 3(1 + x) 27(1 + x)2 1+x

    2 2 2 (7) 896 1 + x 2 H0,1 6400 1 + x 2 H0,−1 32(1 − x)P14 + − + H 0,1 27(1 + x)2 27(1 + x)2 27(1 + x)3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.43 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

 2 (7) (7) 8704 1 + x 2 H−1 64P10 128P24 + + + H2 H 0 27(1 + x)2 27(1 + x)2 27(1 − x)(1 + x)2 0    2 256(1 − x) 1 + x 2 512 1 + x 2 H0 + − H1 1+x 27(1 + x)2

 2 2   4480 1 + x 2 H0,−1 512(1 − x) 62 + 9x + 62x 2 − H−1 H0,−1 − 27(1 + x) 27(1 + x)2 2  (7) (7) 1024 1 + x 2 H1 512P5 1024P17 + + H − 0 27(1 + x)2 27(1 − x)(1 + x)2 27(1 + x)2

 2 (7) (7) 1024 1 + x 2 H−1 64P15 512P20 + + H0 H0,0,1 − 3(1 + x)2 27(1 + x)2 27(1 − x)(1 + x)2

   2 2  1024 1 + x 2 H1 1024 1 + x 2 H−1 256(1 − x) 1 + x 2 − + H0,0,−1 + 3(1 + x) 27(1 + x)2 3(1 + x)2    2  2 1280 1 + x 2 H0 256(1 − x) 1 + x 2 1792 1 + x 2 H0 + + H0,1,1 − 27(1 + x)2 1+x 27(1 + x)2    2 256(1 − x) 1 + x 2 1792 1 + x 2 H0 × H0,1,−1 + − − H0,−1,1 1+x 27(1 + x)2    2 512(1 − x) 62 + 9x + 62x 2 256 1 + x 2 H0 + + H0,−1,−1 27(1 + x) 27(1 + x)2  2 (7) (7) 1024 1 + x 2 H0,0,1,1 256P23 H0,0,0,1 256P21 + − H0,0,0,−1 + 27(1 − x)(1 + x)2 9(1 − x)(1 + x)2 27(1 + x)2      2 2 2 1024 1 + x 2 H0,0,1,−1 1024 1 + x 2 H0,0,−1,1 1024 1 + x 2 H0,0,−1,−1 − − + 3(1 + x)2 3(1 + x)2 3(1 + x)2  2 (7) (7) (7) 4096 1 + x 2 H0,−1,0,1 64H−1 P9 2P33 128H0,1 P7 − + − + 27(1 + x)2 27(1 + x)2 27(1 + x)2 27(1 + x)6   (7) 256(1 − x)2 3 + 4x + 3x 2 16P30 − ln(2) + − 9(1 + x)2 27(1 − x)(1 + x)5

   (7) 512 1 + x 2 13 + 4x 2 64P26 − H + H2 H −1 0 27(1 + x)2 27(1 − x)(1 + x)2 0

  (7) (7) 128(1 − x) 1 + x 2 128P18 512H0 P1 + − H0,−1 ζ2 − H1 + 3(1 + x) 27(1 + x)2 27(1 − x)(1 + x)2 (7) (7) (7) 64P25 32P28 64P11 2 + ζ + − H0 135(1 − x)(1 + x)2 2 27(1 + x)2 27(1 − x)(1 + x)2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.44 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

44

   2 2 256 1 + x 2 H1 256 1 + x 2 H−1 (0) + − ζ3 + FP , 27(1 + x)2 3(1 + x)2

1 2

(7.227)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

2 3

3 4

1

with

4

P1(7) = 13x 4 − 16x 3 − x 2 + 16x − 14,

(7.228)

(7)

5 6

P2 = 13x 4 + 5x 3 − 8x 2 + 5x + 13,

(7.229)

7

P3(7) (7) P4 P5(7) (7) P6 (7) P7 (7) P8 (7) P9 (7) P10 (7) P11 (7) P12 (7) P13 (7) P14 (7) P15 (7) P16 (7) P17 (7) P18 (7) P19 (7) P20 (7) P21 (7) P22 (7) P23 (7) P24 (7) P25 (7) P26 (7) P27 (7) P28 (7) P29 (7) P30

= 17x + 48x + 46x + 48x + 17,

(7.230)

8

= 25x 4 − 32x 3 − 4x 2 + 32x − 29,

(7.231)

10

= 27x + 4x − 16x + 4x + 9,

(7.232)

11

= 33x 4 + 108x 3 + 118x 2 + 108x + 33,

(7.233)

= 43x 4 − 64x 3 − 22x 2 + 64x − 65,

(7.234)

14

= 65x + 196x + 166x + 196x + 65,

(7.235)

15

= 137x 4 + 54x 3 − 54x 2 − 18x − 143,

(7.236)

17

= 163x 4 + 174x 3 + 54x 2 + 174x + 163,

(7.237)

18

= 221x + 166x + 164x + 94x − 149,

(7.238)

= 255x 4 + 370x 3 + 806x 2 + 370x + 255,

(7.239)

21

= 273x − 94x − 170x − 166x − 331,

(7.240)

22

= 499x 4 + 998x 3 + 1574x 2 + 998x + 499,

(7.241)

= 599x 4 + 254x 3 − 62x 2 + 182x − 5,

(7.242)

25

= 5567x + 21986x + 33054x + 21986x + 5567,

(7.243)

26

= 5x 5 − 21x 4 + 21x 3 + 15x 2 − 18x + 2,

(7.244)

28

= 7x 5 − 7x 4 − 28x 3 + 40x 2 − 41x + 41,

(7.245)

29

= 8x − 8x + 39x − 49x + 36x − 36,

(7.246)

= 15x 5 − 55x 4 + 48x 3 + 40x 2 − 51x + 11,

(7.247)

32

= 15x + 33x − 21x − 85x + 65x − 17,

(7.248)

33

= 21x 5 − 45x 4 + 42x 3 + 10x 2 − 29x + 5,

(7.249)

= 41x 5 + 39x 4 − 28x 3 − 156x 2 + 103x − 23,

(7.250)

36

= 50x − 82x + 79x − 9x − 38x + 6,

(7.251)

37

= 71x 5 − 199x 4 + 154x 3 + 118x 2 − 181x + 53,

(7.252)

39

= 71x 5 − 87x 4 + 89x 3 − 67x 2 − 9x − 7,

(7.253)

40

= 133x + 255x + 10x + 310x − 15x − 53,

(7.254)

= 809x 5 − 1353x 4 + 776x 3 + 156x 2 − 1043x + 499,

(7.255)

43

= 5411x + 25439x + 53201x + 69802x + 53201x + 25439x + 5411, (7.256)

44

4

3

4

2

3

2

4

3

4

2

3

4

2

4

3

4

5

4

5

4

6

2

3

2

3

5

2

4

3

2

= 155x 8 + 794x 7 + 852x 6 − 778x 5 − 2634x 4 − 346x 3 + 396x 2 + 74x − 49,

13

19 20

23 24

27

2

3

4

5

2

3

12

16

2

3

5

9

30 31

34 35

38

41 42

45 46

(7.257)

47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.45 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) •••••• (7)

P31 = 181x 8 + 266x 7 − 359x 6 − 726x 5 + 123x 4 + 278x 3 + 275x 2 + 246x + 100, (7.258)

1 2 3

(7)

P32 = 5069x 8 + 25148x 7 + 56746x 6 + 50652x 5 + 21436x 4 − 10044x 3 − 13562x 2

4

− 14812x − 4921,

5 6

(7)

+ 19428x + 7017,

8

10

13

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

1 2 3 4 5 6 7 8 9



FP(0) =

1 n2 4(1 − x)2 h

14 15

(7.260)

and

11 12

(7.259)

P33 = 13209x 8 + 44196x 7 + 34324x 6 + 5148x 5 − 48650x 4 − 38052x 3 − 12044x 2

7

9

45

−



(8) 64P21

729(1 + x)5

(8) 32(1 − x)2 P14 243(1 + x)4



H0 +

+ (1 − x)

(8) 128(1 − x)P22 − 243(1 + x)6

256H−1 P9(8) 243(1 + x)3

+

 3  512 1 + x 2 H−1 81(1 + x)

 2  128(1 − x) 1 + x 2 H−1 5 + 14x + H−1 − H02 81(1 + x)4 27(1 + x)

   (8)  128(1 − x) 1 + x 2 P20 128(1 − x) 1 + x 2 H−1 + + H03 243(1 + x)7 81(1 + x)      16(1 − x) 1 + x 2 H04 512(1 − x)2 1 + x 2 5 + 14x + 5x 2 2 − H0 H1 − 9(1 + x) 81(1 + x)4      1024(1 − x)2 1 + x 2 5 + 14x + 5x 2 512(1 − x) 1 + x 2 H02 + H0 − H0,1 81(1 + x)4 27(1 + x)  2    256(1 − x)P9(8) 256(1 − x) 1 + x 2 H02 512(1 − x) 1 + x 2 H−1 − − + H0,−1 243(1 + x)3 27(1 + x) 27(1 + x)      1024(1 − x)2 1 + x 2 5 + 14x + 5x 2 2048(1 − x) 1 + x 2 H0 − − H0,0,1 27(1 + x) 81(1 + x)4      512(1 − x)2 1 + x 2 5 + 14x + 5x 2 512(1 − x) 1 + x 2 H0 + − − 81(1 + x)4 27(1 + x)

    512(1 − x) 1 + x 2 H−1 1024(1 − x) 1 + x 2 + H0,0,−1 + H−1 H0,−1,−1 27(1 + x) 27(1 + x)  1 − 3H0,0,0,1 − (H0,0,0,−1 + 2H0,0,−1,−1 ) − H0,−1,−1,−1 4   (8) (8) (8) 256(1 − x) 1 + x 2 P19 64(1 − x)P23 1024(1 − x)2 P3 ln(2) + + + 27(1 + x)4 81(1 + x)7 1215(1 + x)6

    128(1 − x) 1 + x 2 H−1 32(1 − x) 1 + x 2 H02 + H0 − 27(1 + x) 3(1 + x) 256(1 − x)2



1 + x2



+ 5x 2



10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.46 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

 2     256(1 − x) 1 + x 2 H−1 512(1 − x)2 1 + x 2 5 + 14x + 5x 2 + H−1 + 27(1 + x) 27(1 + x)4

    2 2 128(1 − x) 1 + x H0,−1 256(1 − x) 1 + x + ζ2 + ζ22 3(1 + x) 9(1 + x)     1408(1 − x) 1 + x 2 H0 512(1 − x) 1 + x 2 H−1 + − 27(1 + x) 27(1 + x)   

2560(1 − x)2 1 + x + x 2 7 + 18x + 7x 2 ζ3 − 81(1 + x)4

 128P2(8) 64  4 2 + nh ln (2) 35 − 22x + 35x − H0 81 81(1 − x)(1 + x)

  (8)  1024P2 512  1 2 + Li4 35 − 22x + 35x − H0 2 27 27(1 − x)(1 + x)   256(1 − x)2 1001 + 1966x + 1001x 2 512(1 − x)(1 + x 2 ) + nl − H0 + 9(1 + x) 243(1 + x)2  2  (8) (8) 5120(1 − x) 1 + x 2 H−1 1024(1 − x)P8 1024(1 − x)H−1 P4 + − − 81(1 + x)3 729(1 + x)3 81(1 + x)

     3 4096(1 − x) 1 + x 2 H−1 1024(1 − x) 1 + x 2 5 + 12x + 3x 2 + H−1 H0 − − 81(1 + x) 81(1 + x)4  2  (8) 1024(1 − x) 1 + x 2 H−1 256(1 − x)P15 + + H02 81(1 + x)4 27(1 + x)      256(1 − x) 1 + x 2 H−1 256(1 − x)2 1 + x 2 5 + 14x + 5x 2 + + − H03 27(1 + x) 81(1 + x)4      32(1 − x) 1 + x 2 H04 512(1 − x)2 1 + x 2 5 + 14x + 5x 2 2 − H0 H1 − 9(1 + x) 81(1 + x)4      1024(1 − x)2 1 + x 2 5 + 14x + 5x 2 512(−1 + x) 1 + x 2 H02 + H0 + H0,1 81(1 + x)4 27(1 + x)     (8) 256(1 − x) 1 + x 2 H02 10240(1 − x) 1 + x 2 H−1 1024(1 − x)P4 + − + + 81(1 + x)3 9(1 + x) 81(1 + x)

    2 4096(1 − x) 1 + x 2 H−1 2048(1 − x) 1 + x 2 − H0,−1 + H0 27(1 + x) 27(1 + x)      1024(1 − x)2 1 + x 2 5 + 14x + 5x 2 512(1 − x) 1 + x 2 H0 − H0,0,1 + 81(1 + x)4 9(1 + x)

     2 2 4096(1 − x) 1 + x 2 H−1 2048(1 − x) 1 + x 5 + 12x + 3x − H0,0,−1 − 27(1 + x) 81(1 + x)4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.47 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

47



    10240(1 − x) 1 + x 2 8192(1 − x) 1 + x 2 H−1 + − + H0,−1,−1 81(1 + x) 27(1 + x)

  512(1 − x) 1 + x 2 8 16 − 2H0,0,0,1 + H0,0,0,−1 + H0,0,−1,−1 + H0,−1,−1,−1 9(1 + x) 3 3   (8) (8) 256(1 − x) 1 + x 2 H−1 64(1 − x)P18 1024(1 − x)2 P3 ln(2) + + + + 27(1 + x)4 81(1 + x)4 27(1 + x)

   64(1 − x) 1 + x 2 − 55 − 69x + 219x 2 + 105x 3 + H0 81(1 + x)4      64(1 − x) 1 + x 2 H02 2048(1 − x) 1 + x 2 − 5 − 6x + 21x 2 + 10x 3 − H−1 − 3(1 + x) 81(1 + x)4

 2    2048(1 − x) 1 + x 2 H−1 256(1 − x) 1 + x 2 H0,−1 + + ζ2 27(1 + x) 9(1 + x)     (8) 7936(1 − x) 1 + x 2 2 128(1 − x) 1 + x 2 H0 64(1 − x)P17 + + ζ2 + 135(1 + x) 3(1 + x) 81(1 + x)4

  4096(1 − x) 1 + x 2 H−1 256P7(8) − ζ3 + ln2 (2) 27(1 + x) 27(1 + x)2

(8) (8) (8) 256P6 256P25 128P24 + − H0 H0 + ln(2) 27(1 − x)(1 + x) 81x(1 + x)6 27(1 − x)(1 + x)5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

−

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

(0)

H2 2 0

(0)

(0)

+ nh FP ,1 (x) + nh ζ2 FP ,2 (x) + nh ζ3 FP ,3 (x),

44

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

(7.261)

44 45

with

46

46 47

2

26

(8) 512P5

+ H0 H1 9(1 − x)(1 + x) 9(1 − x)(1 + x)

(8) (8) (8) 512P5 512P10 256P1 − − − H H0,1 H 0 −1 3(1 + x)2 9(1 − x)(1 + x) 9(1 − x)(1 + x)

(8)   512P11 − H0,−1 + −32 13 − 12x + 13x 2 9(1 − x)(1 + x)

 896x 2 64  − H0 ζ3 ζ2 − 2146 − 3893x + 2146x 2 (1 − x)(1 + x) 135

(8) (8)   64P13 256P12 − ln(2) + H0 ζ22 + 80 1 + 4x + x 2 ζ5 9(1 − x)(1 + x) 135(1 − x)(1 + x)

 960x 2 512(1 − x)(1 + x 2 ) + H 0 ζ5 − H0 H0,0,−1 (1 − x)(1 + x) 9(1 + x)

27

45

(8) 256P16

1

(8)

P1 = x 4 − 6x 3 + 18x 2 − 6x + 1,

(7.262)

47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

(8)

P2 = 8x 4 − 43x 3 + 22x 2 − 43x + 8,

(7.263)

1

(8) P3 (8) P4 (8) P5 (8) P6 (8) P7 (8) P8 (8) P9 (8) P10 (8) P11 (8) P12 (8) P13 (8) P14 (8) P15 (8) P16 (8) P17 (8) P18 (8) P19 (8) P20 (8) P21 (8) P22 (8) P23

= 11x + 38x + 46x + 38x + 11,

(7.264)

2

= 17x 4 + 48x 3 + 46x 2 + 48x + 17,

(7.265)

4

= 21x 4 − 69x 3 + 94x 2 − 69x + 21,

(7.266)

5

= 26x − 79x + 76x − 79x + 26,

(7.267)

6

= 41x 4 − 87x 3 + 32x 2 − 87x + 41,

(7.268)

8

= 64x 4 + 263x 3 + 218x 2 + 263x + 64,

(7.269)

9

= 65x + 196x + 166x + 196x + 65,

(7.270)

10

= 99x 4 − 252x 3 + 308x 2 − 252x + 99,

(7.271)

12

= 105x 4 − 264x 3 + 326x 2 − 264x + 105,

(7.272)

13

= 147x − 402x + 514x − 402x + 147,

(7.273)

14

= 1612x 4 − 2711x 3 + 2996x 2 − 2711x + 1612,

(7.274)

16

= 3859x 4 + 15092x 3 + 22402x 2 + 15092x + 3859,

(7.275)

17

= 16x + 57x + 107x + 81x + 73x + 18,

(7.276)

18

= 70x 5 − 97x 4 + 13x 3 + 12x 2 − 2x + 2,

(7.277)

20

= 325x 5 + 807x 4 + 418x 3 + 222x 2 − 327x − 165,

(7.278)

21

= 391x + 1357x + 854x + 650x − 317x − 119,

(7.279)

22

= 15x 6 + 78x 5 + 159x 4 + 152x 3 + 27x 2 − 18x − 5,

(7.280)

24

= 25x 6 + 126x 5 + 225x 4 + 152x 3 − 39x 2 − 66x − 15,

(7.281)

25

= 599x + 3824x + 7193x + 7360x + 7193x + 3824x + 599,

(7.282)

26

= 11x 7 + 70x 6 + 361x 5 + 712x 4 + 635x 3 + 588x 2 + 321x + 54,

(7.283)

30 31 32 33 34 35

4

3

4

2

3

4

2

3

4

2

3

5

2

4

5

3

4

6

3

5

40

(8) P25

= 7239x 7 + 36399x 6 + 60903x 5 + 38023x 4 − 11083x 3 − 41923x 2

= 848x

10

− 14847x − 55535x − 37906x + 58575x + 123650x + 58575x 9

8

7

6

5

− 37906x − 55535x − 14847x + 848. 2

The first expansion coefficients of

(0) FP ,i , i

4

(7.286)

= 1...3, are given by

1488375 6 + O(y )

46

(0)

FP ,3 (x) = −

804767441y 4 381536y 3 381536y 2 4990072 − − − + 2976750 2025 2025 729 (7.288)

4050340711y 5 19051200

−

622908463y 4 4233600

11

15

19

23

27 28

30 31 32 33 34 35 36

38

19068183229y 5 22625094013y 4 756146y 3 756146y 2 5529994 + + + − 85730400 171460800 18225 18225 729 6 + O(y ) (7.287) 5

7

37

FP(0) ,1 (x) =

524338481y (0) FP ,2 (x) = −

3

29

(7.284)

3

45

47

2

(8)

42

44

3

P24 = 261x 8 + 1096x 7 − 702x 6 − 3888x 5 − 1232x 4 − 4392x 3 − 810x 2 + 1168x + 243, (7.285)

41

43

2

4

38 39

2

− 28579x − 5939,

36 37

[m1+; v1.304; Prn:6/09/2019; 12:57] P.48 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

48

−

496121y 3 6075

−

496121y 2 6075

+

426952 243

39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.49 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1

+ O(y 6 ).

49

(7.289)

3

7.4. The axialvector form factors

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The axialvector form factors are given by 

    2 32 1 + x 2 589 + 602x + 589x 2 1 16 2 FA,1 = 3 nh − H 0 + nh − ε 27 27(1 − x)(1 + x) 27(1 + x)2

   2  (9) 128 1 + x 2 32 1 + x 2 16P8 16 + nl H0 − H2 − H0 + 27 27(1 − x)(1 + x) 27(1 − x)(1 + x)3 27(1 − x 2 )2 0

     2 16 3 − 2x + 3x 2 64 1 + x 2 H−1 1 2 16(1 + x ) + − + H0 + 2 nh ε 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x 2 )

    16 1 + x 2 H02 64 1 + x 2 H0,−1 32(1 + x 2 ) − + ζ2 − 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x 2 ) 988 4 1 1052 988 − x + 680x 2 + 680x 3 − x − + nh 2 3 9(1 − x) (1 + x) 3 3 3

    64 7 − 3x + 7x 2 128 1 + x 2 H−1 1052 5 128(1 + x 2 ) − x + nl + − + H0 3 81(1 − x 2 ) 81(1 − x 2 ) 27(1 − x 2 )

    32 1 + x 2 H02 128 1 + x 2 H0,−1 64(1 + x 2 ) − + ζ2 − 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x 2 )

26 27

+

28 29 30

−

31 32 33

+

34 35 36

+

37 38

+

39 40 41

+

42 43 44

+

45 46 47

3 4

4 5

1 2

2

−

2968 (1 + 2x − x 2 + x 3 − 2x 4 − x 5 )H0 27(1 − x)2 (1 + x)3 (9) (9) 16P22 128P7 H H + − −1 0 27(1 − x)(1 + x)3 27(1 − x)2 (1 + x)3

  2   512 1 + x 2 64 − 2 + x 2 1 + x 2 3 2 H−1 H0 + H0 27(1 − x)2 (1 + x)2 27(1 − x 2 )2      2 64 1 + x 2 H0 H1 64 1 + x 2 H0,1 64 1 + x 2 − + H0 H0,1 3(1 − x 2 ) 3(1 − x 2 ) 3(1 − x 2 )2   2 2 (9) 1088 1 + x 2 128 1 + x 2 128P7 H − H H − H0,0,1 0,−1 0 0,−1 27(1 − x)(1 + x)3 27(1 − x 2 )2 3(1 − x 2 )2  2 128 1 + x 2 1 64 1472 3200 H0,0,−1 + − x + 64x 2 + x 3 − 2 2 2 3 3(1 − x ) (1 − x) (1 + x) 27 27 3

2048 4 320 5 32(1 + x + x 2 + x 3 )H0 x + x − 27 27 27(1 − x)2 (1 + x)3

1120 2 2 3 x (1 + x + x + x )H0 ζ2 27(1 − x)2 (1 + x)3

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

 1 32 2 3 4 5 + (1 + x + 2x + 2x + x + x )ζ3 3(1 − x 2 ) (1 + x)3    (9) 64 13 − 34x + 13x 2 16P12 1 2 + − H0 nh − ε 243(1 + x)2 243(1 − x)(1 + x)3  2    (9) H0 64 1 + x 2 H−1 32 3 − 2x + 3x 2 8P18 + H − H2 H + −1 0 27(1 − x 2 ) 27(1 − x 2 ) 81(1 − x)(1 + x)4 0     (9) 32 1 + x 2 H−1 H02 32 3 − 2x + 3x 2 16P1 3 + H − − H0,−1 27(1 − x 2 ) 27(1 − x)(1 + x)3 0 27(1 − x 2 )       128 1 + x 2 H−1 H0,−1 64 1 + x 2 H0,0,−1 128 1 + x 2 H0,−1,−1 + − − 27(1 − x 2 ) 27(1 − x 2 ) 27(1 − x 2 )

  (9) (9) 64 1 + x 2 H−1 16P3 8P21 ζ2 − H0 − + 27(1 − x)(1 + x)4 27(1 − x)(1 + x)3 27(1 − x 2 ) 

 (9) (9) (9) 64 1 + x 2 64H02 H1 P5 128H0,0,1 P6 128H0 H0,1 P2 + − + ζ + n − 3 h 27(1 − x 2 ) 27(1 − x 2 )2 27(1 − x 2 )2 27(1 − x 2 )2 (9)

(9) (9) 32H−1 H02 P25 64H0 H0,−1 P9(9) 4P14 64H0,0,−1 P24 − − − + 2 2 4 2 3 27(1 − x ) 243(1 + x) 27(1 − x) (1 + x) 27(1 − x)2 (1 + x)3

23 24

+

(9) 512H0 H0,0,1 P27 27(1 − x 2 )3

+

(9) 128H0,0,0,1 P31 27(1 − x 2 )3

25 26 27 28 29 30

+

31 32 33

−

34 35

−

36 37 38

−

39 40 41

−

42 43 44

−

45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.50 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

50

−

−

−

(9) 128H0,0,0,−1 P29 9(1 − x 2 )3

(9) 64H02 H0,1 P32 27(1 − x 2 )3

+

−

+

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

(9) 128H0 H0,0,−1 P30 27(1 − x 2 )3

(9) 64H02 H0,−1 P33 27(1 − x 2 )3

1

24 25

(9) 16P168

ζ2 135(1 − x 2 )3 2   (9) (9) 32 1 − 38x + x 2 8H04 P37 4H02 P41 − + nl − 81(1 − x)3 (1 + x)4 243(1 − x)2 (1 + x)6 81(1 + x)2  2    (9) H0 256 1 + x 2 H−1 256 7 − 3x + 7x 2 32P10 H−1 H0 − H0 + 81(1 − x)(1 + x)3 81(1 − x 2 ) 27(1 − x 2 )     (9) 128 1 + x 2 H−1 H02 32 1 + 2x + 2x 3 + x 4 3 16P15 2 − H + H0 81(1 − x)(1 + x)4 0 27(1 − x 2 ) 27(1 − x)(1 + x)3       256 7 − 3x + 7x 2 512 1 + x 2 H−1 H0,−1 256 1 + x 2 H0,0,−1 + H − 0,−1 81(1 − x 2 ) 27(1 − x 2 ) 27(1 − x 2 )   (9) (9) 512 1 + x 2 H0,−1,−1 8P26 16P4 − H0 + 27(1 − x 2 ) 81(1 − x)(1 + x)4 27(1 − x)(1 + x)3

   

(9) 256 1 + x 2 H−1 256 1 + x 2 ζ3 4P36 ζ + + H0 2 27(1 − x 2 ) 27(1 − x 2 ) 243(1 − x)(1 + x)5

(9) (9) (9) 16P39 16P13 64H1 P17 H−1 H0 + − + 27(1 − x)(1 + x)3 81(1 − x)2 (1 + x)3 81(1 − x)3 (1 + x)5

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.51 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

51

     64 1 + x 2 − 1 + 5x 2 64 37 − 14x + 37x 2 3 + H−1 H0 + H0 H1 27(1 − x 2 )2 27(1 − x 2 )     (9) 128 1 + x 2 H−1 H0 H1 64 1 + x 2 H0 H12 64P11 − H0 + + (1 − x 2 ) 3(1 − x 2 ) 27(1 − x)(1 + x)3

  2  1024 1 + x 2 64 37 − 14x + 37x 2 2 2 H0,1 − H H−1 − 27(1 − x 2 )2 0 27(1 − x 2 )    2  2 256 1 + x 2 128 1 + x 2 H1 H0,1 256 1 + x 2 − H−1 H0 H0,1 − H0 H1 H0,1 + 3(1 − x 2 )2 3(1 − x 2 ) 27(1 − x 2 )2     2 2 128 1 + x 2 H−1 H0,1 3200 1 + x 2 448 1 + x 2 + H0,−1 H0,1 − H2 + (1 − x 2 ) 27(1 − x 2 )2 27(1 − x 2 )2 0,1  2 (9) 4352 1 + x 2 16P13 + H0,−1 + H−1 H0 H0,−1 27(1 − x)(1 + x)3 27(1 − x 2 )2    2 128 1 + x 2 H1 H0,−1 256 1 + x 2 + − H0 H1 H0,−1 1 − x2 27(1 − x 2 )2 2  (9) 2240 1 + x 2 128P11 − H−1 H0,−1 − H2 27(1 − x)(1 + x)3 27(1 − x 2 )2 0,−1    2 2 2 512 1 + x 2 512 1 + x 2 512 1 + x 2 − H H + H H + H1 H0,0,−1 1 0,0,1 −1 0,0,1 27(1 − x 2 )2 3(1 − x 2 )2 27(1 − x 2 )2    2  2 512 1 + x 2 128 1 + x 2 H0,1,1 640 1 + x 2 − H−1 H0,0,−1 + H0 H0,1,1 + 3(1 − x 2 )2 3(1 − x 2 ) 27(1 − x 2 )2     2  128 1 + x 2 H0,1,−1 896 1 + x 2 128 1 + x 2 H0,−1,1 − − H0 H0,1,−1 − (1 − x 2 ) 27(1 − x 2 )2 1 − x2   2 (9) 896 1 + x 2 128P11 − H H + H0,−1,−1 0 0,−1,1 27(1 − x 2 )2 27(1 − x)(1 + x)3 2 2 2    128 1 + x 2 512 1 + x 2 512 1 + x 2 + H0 H0,−1,−1 + H0,0,1,1 − H0,0,1,−1 27(1 − x 2 )2 27(1 − x 2 )2 3(1 − x 2 )2    2 2 2 512 1 + x 2 512 1 + x 2 2048 1 + x 2 − H0,0,−1,1 + H0,0,−1,−1 − H0,−1,0,1 3(1 − x 2 )2 3(1 − x 2 )2 27(1 − x 2 )2 (9) (9) (9) 256H0 H1 P16 64H0,1 P19 64H−1 P20 + − + − 27(1 − x)2 (1 + x)3 27(1 − x)2 (1 + x)3 27(1 − x)2 (1 + x)3 × H03

(9)

(9)

(9)

64H0,−1 P28 32H02 P38 8H0 P40 + + − 2 3 3 4 27(1 − x ) 27(1 − x) (1 + x) 27(1 − x)3 (1 + x)5     

(9) 256 1 + x 2 13 + 4x 2 64 1 + x 2 H1 P42 ζ2 + − H−1 H0 − 27(1 − x 2 )2 3(1 − x 2 ) 27(1 − x)2 (1 + x)6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.52 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

52



 2 (9) (9) 128 1 + x 2 32P23 32H0 P34 + − + H1 27(1 − x)2 (1 + x)3 27(1 − x 2 )3 27(1 − x 2 )2

  2 128 1 + x 2 (0) − H−1 ζ3 + FA,1 , 3(1 − x 2 )2

1 2 3 4 5

1 2 3

(7.290)

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

5 6

6 7

4

with

7

P1(9) (9) P2 (9) P3 P4(9) (9) P5 (9) P6 (9) P7 (9) P8 (9) P9 (9) P10 (9) P11 (9) P12 (9) P13 (9) P14 (9) P15 (9) P16 (9) P17 (9) P18 (9) P19 (9) P20 (9) P21 (9) P22 (9) P23 (9) P24 (9) P25 (9) P26 (9) P27 (9) P28 (9) P29 (9) P30

= x + 2x − 2x + 2x + 1,

(7.291)

= 2x 4 + 125x 3 − 346x 2 + 125x + 2,

(7.292)

10

= 5x 4 + 10x 3 − 14x 2 + 10x + 5,

(7.293)

11

= 7x + 14x − 10x + 14x + 7,

(7.294)

12

= 14x 4 − 121x 3 + 310x 2 − 121x − 22,

(7.295)

14

= 18x 4 + 129x 3 − 382x 2 + 129x − 18,

(7.296)

15

= 23x + 73x + 64x + 73x + 23,

(7.297)

16

= 41x 4 + 190x 3 + 154x 2 + 190x + 41,

(7.298)

18

= 43x 4 − 777x 3 + 1970x 2 − 777x + 43,

(7.299)

19

= 103x + 70x + 94x + 70x + 103,

(7.300)

20

= 115x 4 + 284x 3 + 266x 2 + 284x + 115,

(7.301)

22

= 337x 4 − 64x 3 + 158x 2 − 64x + 337,

(7.302)

23

2

= 913x + 1014x − 390x + 1014x + 913,

(7.303)

24

= 14921x 4 + 59084x 3 + 82566x 2 + 59084x + 14921,

(7.304)

26

= 9x 5 + 105x 4 + 48x 3 + 104x 2 + 39x + 47,

(7.305)

27

= 13x − 38x + 226x − 228x + 37x − 14,

(7.306)

28

= 25x 5 − 77x 4 + 450x 3 − 458x 2 + 73x − 29,

(7.307)

30

= 29x 5 − 87x 4 + 38x 3 − 74x 2 + 45x − 47,

(7.308)

31

= 43x − 161x + 886x − 930x + 139x − 65,

(7.309)

32

= 55x 5 + 136x 4 − 27x 3 + 9x 2 − 94x − 67,

(7.310)

34

= 87x 5 − 151x 4 + 86x 3 − 62x 2 + 179x − 75,

(7.311)

35

= 97x + 135x − 22x + 14x − 83x − 13,

(7.312)

36

= 137x 5 + 1241x 4 − 1412x 3 − 1340x 2 + 691x − 197,

(7.313)

38

= 142x 5 + 1207x 4 − 1655x 3 − 1619x 2 + 815x − 142,

(7.314)

39

= 228x − 261x + 731x + 767x − 653x − 56,

(7.315)

40

= 331x 5 − 279x 4 + 478x 3 + 130x 2 + 855x − 107,

(7.316)

42

= 5x 6 − 52x 5 + 277x 4 − 458x 3 + 274x 2 − 52x + 2,

(7.317)

43

= 7x + 6x − 65x + 24x − 31x + 6x + 41,

(7.318)

44

= 15x 6 + 156x 5 − 823x 4 + 1378x 3 − 855x 2 + 156x − 17,

(7.319)

46

= 30x 6 − 259x 5 + 1378x 4 − 2298x 3 + 1370x 2 − 259x + 22,

(7.320)

47

4

3

2

4

3

4

2

3

4

2

3

2

4

3

5

4

5

4

5

2

3

4

5

2

3

4

5

6

3

2

3

4

2

3

2

8 9

13

17

21

25

29

33

37

41

45

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.53 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) •••••• (9)

1 2 3 4 5 6 7 8

P31 = 41x 6 + 261x 5 − 1367x 4 + 2282x 3 − 1431x 2 + 261x − 23,

(7.321)

(9) P32 (9) P33 (9) P34 (9) P35 (9) P36

= 42x − 155x + 828x − 1374x + 796x − 155x + 10,

(7.322)

= 50x 6 − 103x 5 + 554x 4 − 920x 3 + 510x 2 − 103x + 6,

(7.323)

4

= 71x − 412x + 2199x − 3696x + 2181x − 412x + 53,

(7.324)

5

= 809x 6 − 1764x 5 + 8941x 4 − 15196x 3 + 8631x 2 − 1764x + 499,

(7.325)

(9) P37 (9) P38 (9) P39

= 8x + 9x − 21x + 102x − 38x + 31x − 35x − 36,

(7.327)

= 71x 7 + 19x 6 + 211x 5 − 135x 4 − 261x 3 + 181x 2 − 59x − 7,

(7.328)

9 10 11 12 13 14 15

(9) P41

18 19

(9) P42

21 22

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

5

3

4

2

3

2

= 46991x 6 + 160202x 5 + 253361x 4 + 257260x 3 + 253361x 2 + 160202x + 46991, (7.326) 7

6

5

4

3

2

= 229x − 318x + 98x + 1828x − 1004x − 1266x − 54x + 300x − 5, (7.329) 8

7

6

5

4

3

2

= 13421x + 46988x + 29734x + 3876x + 15184x + 26916x − 11702x 8

7

6

5

4

3

  = −1152 ln(2)(1 − x) (x + 1) x 2 + 1 + 18177x 8 − 28796x 7 − 86188x 6 2

2

(7.331)

4

1 2 3

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

+ 6780x 5 + 175430x 4 + 42236x 3 − 91148x 2 − 57084x + 8305,

23

27

6

4

− 13780x − 5677,

20

26

5

(9)

17

25

6

P40 = 281x 8 + 3062x 7 + 48x 6 − 6750x 5 + 2006x 4 + 1130x 3 + 688x 2 + 382x − 79, (7.330)

16

24

53

(7.332)

23 24

and

 (0)

FA,1 = n2h

25

(10)

(10) (10) 64ζ22 P3 256H0,0,0,1 P4 64H0,0,0,−1 P2 − + 27(x − 1)(1 + x)3 9(x − 1)(1 + x)3 9(x − 1)(1 + x)3 (10)

(10) (10) 4H04 P9 64H02 H1 P14 8P24 + − + 27(x − 1)(1 + x)3 81(1 + x)4 729(1 + x)4 (10) (10) 2 32(x − 1)H−1 320H−1 P15 16P40 + − + + 27(1 + x) 243(x − 1)(1 + x)3 729(x − 1)(1 + x)5

  (10) (10) 128 1 + x 2 16H−1 P17 32P41 3 − − H−1 H0 + 81(x − 1)(1 + x) 81(1 + x)4 243(x − 1)(1 + x)6

  (10) (10) 32 1 + x 2 32H−1 P2 8P44 2 2 + − H H0 + − 27(x − 1)(1 + x) −1 81(x − 1)(1 + x)3 243(x − 1)(1 + x)7

(10) (10) (10) 320P15 128H02 P4 128H0 P14 3 × H0 + + + H 0,1 27(x − 1)(1 + x)3 81(1 + x)4 243(x − 1)(1 + x)3

    2 64 1 + x 2 128 1 + x 64(x − 1)H −1 − H2 − + H 2 H0,−1 27(x − 1)(1 + x) 0 27(1 + x) 27(x − 1)(1 + x) −1

(10) (10) (10) 32P17 512H0 P4 128P14 + − − + − H 0,0,1 27(x − 1)(1 + x)3 81(1 + x)4 81(1 + x)4

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

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[m1+; v1.304; Prn:6/09/2019; 12:57] P.54 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

54



    128 1 + x 2 128 1 + x 2 64(x − 1) + H0 − H−1 H0,0,−1 + 27(x − 1)(1 + x) 27(x − 1)(1 + x) 27(1 + x)

    2 2 256 1 + x 128 1 + x − H−1 H0,−1,−1 + H0,0,−1,−1 27(x − 1)(1 + x) 27(x − 1)(1 + x)     256 1 + x 2 256(x − 1)2 11 + 12x + 11x 2 + ln(2)ζ2 H0,−1,−1,−1 + 27(x − 1)(1 + x) 27(1 + x)4 (10) (10) (10) 32H0,−1 P10 32H−1 P18 64P43 + − + − 9(x − 1)(1 + x)3 27(1 + x)4 1215(x − 1)(1 + x)6

   (10) 32 1 − 4x + x 2 1 + 6x + x 2 8P45 + − − H−1 H0 81(x − 1)(1 + x)7 27(x − 1)(1 + x)3

    8(x − 1) 1 + 4x + x 2 2 64 1 + x 2 2 + H0 − H ζ2 27(x − 1)(1 + x) −1 3(1 + x)3

   (10) (10) 128 1 + x 2 16P23 32H0 P12 + − − + H−1 ζ3 27(x − 1)(1 + x)3 81(1 + x)4 27(x − 1)(1 + x)    (10) (10) 128P19 1024H0 P34 256x 2 H02 1 + nh Li4 + + 2 27(x − 1)2 (1 + x)2 27(x − 1)3 (1 + x)3 (x − 1)(1 + x)3

(10) 16P19 1024x 2 H0 H1 1024x 2 H0,1 4 + − (2) + ln (x − 1)(1 + x)3 (x − 1)(1 + x)3 81(x − 1)2 (1 + x)2 +

(10) 128H0 P34 81(x − 1)3 (1 + x)3

+

32x 2 H02 3(x

− 1)(1 + x)3

+

128x 2 H0 H1 3(x

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

− 1)(1 + x)3

1

(10)

64ζ22 P38 128x 2 H0,1 128x 2 ζ2 − + + ln(2) − 3(x − 1)(1 + x)3 3(x − 1)(1 + x)3 9(x − 1)3 (1 + x)3 (10) (10) (10) 128H0 H1 P36 128H0,1 P36 128H−1 P8 + − − + 3(x − 1)2 (1 + x)2 9(x − 1)3 (1 + x)3 9(x − 1)3 (1 + x)3 (10) (10) (10) 64H02 P42 128H−1 P37 16P47 + + + − 9(x − 1)3 (1 + x)4 81(x − 1)2 x(1 + x)6 9(x − 1)3 (1 + x)3

  (10) (10) 128 1 + x 2 P22 32P46 192x 2 H03 + + H0,−1 ζ2 H0 + (x − 1)(1 + x)3 9(x − 1)3 (1 + x)3 27(x − 1)3 (1 + x)5 (10) (10) 128H0 P35 64x 2 H02 32P21 2 + ln (2) − − 27(x − 1)2 (1 + x)2 27(x − 1)3 (1 + x)3 (x − 1)(1 + x)3

256x 2 ζ22 256x 2 H0 H1 256x 2 H0,1 − + ζ2 − (x − 1)(1 + x)3 (x − 1)(1 + x)3 (x − 1)(1 + x)3 (10) (10) 128H0,0,0,−1 P6(10) 8H04 P11 256H0,0,0,1 P4 + nl + + 9(x − 1)(1 + x)3 9(x − 1)(1 + x)3 27(x − 1)(1 + x)3

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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[m1+; v1.304; Prn:6/09/2019; 12:57] P.55 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

55

  (10) (10) 64 377 + 106x + 377x 2 64ζ22 P16 64H02 H1 P14 − − + 81(1 + x)4 135(x − 1)(1 + x)3 729(1 + x)2   (10) (10) 256 11 − 6x + 11x 2 2 64P25 128H−1 P20 + − + + H−1 81(x − 1)(1 + x) 81(x − 1)(1 + x)3 729(x − 1)(1 + x)3

  (10) (10) 1024 1 + x 2 32P29 64H−1 P27 3 − + H−1 H0 + − 81(x − 1)(1 + x) 81(x − 1)(1 + x)4 81(x − 1)(1 + x)4

  (10) (10) 256 1 + x 2 64H−1 P6 16P28 2 2 + − H H + − H03 27(x − 1)(1 + x) −1 0 27(x − 1)(1 + x)3 81(x − 1)(1 + x)4

(10) (10) (10) 128H02 P4 64H02 P4 128H0 P14 + + + − H 0,1 27(x − 1)(1 + x)3 81(1 + x)4 9(x − 1)(1 + x)3

    (10) 512 11 − 6x + 11x 2 1024 1 + x 2 128P20 2 + − H−1 + H 81(x − 1)(1 + x) 27(x − 1)(1 + x) −1 81(x − 1)(1 + x)3

(10) (10) (10) 512H0 P4 128P14 128H0 P4 × H0,−1 + − − + H 0,0,1 27(x − 1)(1 + x)3 81(1 + x)4 9(x − 1)(1 + x)3

  (10) 1024 1 + x 2 128P27 + − H−1 H0,0,−1 81(x − 1)(1 + x)4 27(x − 1)(1 + x)

    512 11 − 6x + 11x 2 2048 1 + x 2 + − H−1 H0,−1,−1 81(x − 1)(1 + x) 27(x − 1)(1 + x)     1024 1 + x 2 2048 1 + x 2 + H0,0,−1,−1 + H0,−1,−1,−1 27(x − 1)(1 + x) 27(x − 1)(1 + x)   (10) 256(x − 1)2 11 + 12x + 11x 2 64H0,−1 P1 + ln(2)ζ + − 2 27(1 + x)4 9(x − 1)(1 + x)3 (10)

(10) (10) 16H02 P5 128H−1 P30 8P33 + + − 3(x − 1)(1 + x)3 81(x − 1)(1 + x)4 81(x − 1)(1 + x)4

  (10) (10)   512 1 + x 2 16P31 64H−1 P7 H0 − + − − H 2 ζ2 27(x − 1)(1 + x) −1 27(x − 1)(1 + x)3 81(x − 1)(1 + x)4



    (10) 32 1 + x 2 1024 1 + x 2 16P32 + − − H0 + H−1 ζ3 81(x − 1)(1 + x)4 3(x − 1)(1 + x) 27(x − 1)(1 + x)   (10) 8P13 1024x 2 1 + Li + − 4 (x − 1)(1 + x)3 2 (x − 1)2 (1 + x)2

  (10) 32x 2 5 − 24x + 5x 2 8P26 + H0 ζ3 ζ2 + − (x − 1)3 (1 + x)3 135(x − 1)2 (1 + x)2 (10)

32H0 P39 16x 2 H02 64x 2 H0 H1 + + + 3 3 3 135(x − 1) (1 + x) 5(x − 1)(1 + x) 5(x − 1)(1 + x)3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA



  20 1 + x 2 1 − 8x + x 2 64x 2 ζ23 64x 2 H0,1 2 − + ζ + 5(x − 1)(1 + x)3 2 5(x − 1)(1 + x)3 (x − 1)2 (1 + x)2

   80x 2 1 − 8x + x 2 (0) (0) (0) (0),r − H0 ζ5 + FA,1,1 + FA,1,2 ζ2 + FA,1,3 ζ3 + FA,1 , (7.333) (x − 1)3 (1 + x)3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.56 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

56

2 3 4 5 6

with the polynomials (10) P1 (10) P2 (10) P3 P4(10) (10) P5 (10) P6 (10) P7 (10) P8 P9(10) (10) P10 (10) P11 (10) P12 (10) P13 (10) P14 (10) P15 (10) P16 (10) P17 (10) P18 (10) P19 (10) P20 (10) P21 (10) P22 (10) P23 (10) P24 (10) P25 (10) P26 (10) P27 (10) P28 (10) P29 (10) P30

1

7

= x + 2x − 26x + 2x + 1,

(7.334)

= x 4 + 2x 3 − 10x 2 + 2x + 1,

(7.335)

= x 4 + 2x 3 − 5x 2 + 2x + 1,

(7.336)

11

= x + 2x − 4x + 2x + 1,

(7.337)

12

= x 4 + 2x 3 − 2x 2 + 2x + 1,

(7.338)

= x 4 + 2x 3 + 4x 2 + 2x + 1,

(7.339)

15

= x + 2x + 14x + 2x + 1,

(7.340)

16

= 2x 4 − 9x 3 + 12x 2 − 9x + 2,

(7.341)

= 3x 4 + 6x 3 − 14x 2 + 6x + 3,

(7.342)

19

= 3x + 6x − 10x + 6x + 3,

(7.343)

20

= 3x 4 + 6x 3 − 4x 2 + 6x + 3,

(7.344)

= 11x 4 + 22x 3 − 50x 2 + 22x + 11,

(7.345)

23

= 13x − 24x − 6x − 24x + 13,

(7.346)

24

= 19x 4 − 14x 3 + 14x 2 − 14x + 19,

(7.347)

26

= 31x 4 − 10x 3 + 14x 2 − 10x + 31,

(7.348)

27

= 31x + 62x + 149x + 62x + 31,

(7.349)

28

= 35x 4 − 34x 3 + 28x 2 − 22x + 41,

(7.350)

30

= 39x 4 − 26x 3 + 28x 2 − 30x + 37,

(7.351)

31

= 53x − 188x + 174x − 188x + 53,

(7.352)

32

= 89x 4 + 48x 3 + 78x 2 + 48x + 89,

(7.353)

34

= 91x 4 − 190x 3 + 258x 2 − 190x + 91,

(7.354)

35

= 105x − 208x + 198x − 208x + 105,

(7.355)

36

= 253x 4 − 296x 3 − 310x 2 − 320x + 241,

(7.356)

38

= 1367x 4 + 3428x 3 + 4506x 2 + 3428x + 1367,

(7.357)

39

= 1471x + 2042x − 10x + 2042x + 1471,

(7.358)

40

= 5363x 4 − 4754x 3 − 2814x 2 − 4754x + 5363,

(7.359)

42

= 3x 5 + 87x 4 + 24x 3 + 80x 2 + 21x + 41,

(7.360)

43

= 35x − 141x + 34x − 78x − 9x − 41,

(7.361)

44

= 62x 5 + 125x 4 + 205x 3 + 47x 2 + 149x + 116,

(7.362)

46

= 79x 5 − 45x 4 + 136x 3 − 32x 2 + 153x − 35,

(7.363)

47

4

3

4

2

3

4

2

3

4

2

3

4

2

3

4

2

3

4

2

3

4

3

4

5

2

2

3

4

2

3

2

8 9 10

13 14

17 18

21 22

25

29

33

37

41

45

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.57 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

(10)

(7.364)

P32 = 463x 5 − 219x 4 + 598x 3 + 234x 2 + 1083x − 111,

(10)

(7.365)

(10) P33 = 1969x 5 + 1147x 4 − 1734x 3 + 3750x 2 + 1045x − 545,

(7.366)

4

(10) P34 (10) P35 (10) P36 (10) P37 (10) P38 (10) P39 (10) P40 (10) P41 (10) P42 (10) P43

= 2x + 15x − 37x + 64x − 37x + 15x + 2,

(7.367)

5

= 13x 6 + 39x 5 − 119x 4 + 164x 3 − 119x 2 + 39x + 13,

(7.368)

= 21x − 110x + 195x − 208x + 195x − 110x + 21,

(7.369)

= 99x 6 − 214x 5 + 333x 4 − 440x 3 + 333x 2 − 214x + 99,

(7.370)

= 147x − 428x + 693x − 832x + 693x − 428x + 147,

(7.371)

= 806x 6 − 951x 5 + 1172x 4 − 2852x 3 + 1172x 2 − 951x + 806,

(7.372)

13

= 2945x 6 + 6836x 5 + 287x 4 − 6056x 3 + 287x 2 + 6836x + 2945,

(7.373)

14

= 19x 7 − 133x 6 − 466x 5 − 238x 4 + 73x 3 + 81x 2 − 282x − 174,

(7.374)

= 70x 7 + 15x 6 − 69x 5 − 94x 4 + 6x 3 + 127x 2 − 53x + 2,

(7.375)

17

= 2226x + 4218x − 1317x − 2729x + 3554x + 3242x − 2143x − 1451, (7.376)

18

P31 = 189x 5 − 579x 4 + 222x 3 − 226x 2 − 51x − 115,

20 21

57

6

5

6

4

5

6

3

2

4

5

7

3

4

2

3

6

4

24 25 26

3

2

(10)

P44 = 181x 8 + 228x 7 − 212x 6 + 44x 5 + 942x 4 + 428x 3 + 12x 2 − 156x − 123,

27

(10)

P45 = 213x 8 + 276x 7 − 300x 6 + 268x 5 + 1854x 4 + 652x 3 − 76x 2 − 108x − 91, (7.378)

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

7

6

5

4

3

2

8

10 11

15 16

19 20

22 23 24 25 26

(7.379)

(10)

P47 = 2768x 10 − 63429x 9 − 44774x 8 + 54896x 7 + 46326x 6 − 43414x 5 + 46326x 4 + 54896x 3 − 44774x 2 − 63429x + 2768. The first expansion coefficients of

(0) FA,1,i , i

(7.380)

19976198y 2

19976198y 3

28 29 30 31 32

= 1...3 are given by

33 34

647051207603y 4

1105690 − − − 729 18225 18225 857304000 5 177211030643y + O(y 6 ) − 428652000 1979131 17033692y 2 17033692y 3 27237088943y 4 (0) + + + FA,1,2 (x) = 729 18225 18225 44651250 5 6370816243y + O(y 6 ) + 22325625 24544 1061573y 2 1061573y 3 255928217y 4 (0) + + + FA,1,3 (x) = 243 6075 6075 2352000 5 4084720937y + O(y 6 ). + 95256000 (0) FA,1,1 (x) = −

7

27

+ 394x + 309,

28 29

= 381x + 214x − 7742x + 5550x + 7394x + 5154x − 7526x 8

6

21

(7.377)

(10) P46

3

12

22 23

2

9

2

5

1

35 36

(7.381)

37 38 39 40 41

(7.382)

42 43 44 45 46

(7.383)

47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.58 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

58

The form factor FA,2 is given by       1 + x2 128nh 1 1 128 FA,2 = x − 3 + H0 + 2 n2h − 2 3 ε 3(1 + x) 3(1 − x)(1 + x) ε 27(1 − x)2

  64 3 − 2x + 3x 2 23488x(1 + x) 4672 − + H0 + nh 27(1 − x)3 (1 + x) 27(1 − x)2 (1 + x)3 27(1 − x)2 (1 + x)3 (11) 64H02 P24 4672x 3 256 + − − nl 2 3 4 3 27(1 − x) (1 + x) 27(1 − x) (1 + x) 27(1 − x)2



    (11) 128 3 − 2x + 3x 2 256 1 + x 2 128P4 + + H−1 H0 H0 + 27(1 − x)3 (1 + x) 27(1 − x)3 (1 + x)3 3(1 − x)(1 + x)3     (11) 256 1 + x 2 64H0 P7 128(1 + x 2 ) 1 2 − H + ζ n + − 0,−1 2 ε h 3(1 − x)(1 + x)3 3(1 − x)(1 + x)3 81(1 − x 2 )3     (11) 256 1 + 26x + x 2 128 3 − 2x + 3x 2 32H02 P20 + + + H−1 H0 27(1 − x)3 (1 + x)4 81(1 − x)2 (1 + x)2 27(1 − x)3 (1 + x)   (11) 128 3 − 2x + 3x 2 64P26 256x 2 H03 H + − + 0,−1 27(1 − x 2 )3 27(1 − x)3 (1 + x) 27(1 − x)3 (1 + x)4



(11) (11) 512H0 H0,1 P3 256H02 H1 P2 512x 2 H0 + + ζ2 + nh − 9(1 − x 2 )3 27(1 − x)4 (1 + x)2 27(1 − x)4 (1 + x)2 (11)

(11)

(11) 512H0,0,1 P6 128H−1 H0 P16 128P13 − + + 4 2 2 4 27(1 − x) (1 + x) 81(10x) (1 + x) 27(1 − x)3 (1 + x)3

+



+

−

(11)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

28 29

 2

(11) 128 23 − 2x + 23x 64H02 P22 128H0 P8(11) + nl − − − 2 3 3 4 81(1 − x ) 27(1 − x) (1 + x) 81(1 − x 2 )2     512 3 − 2x + 3x 2 512 3 − 2x + 3x 2 256x 2 H03 H−1 H0 + H0,−1 + − 27(1 − x)3 (1 + x) 27(1 − x 2 )3 27(1 − x)3 (1 + x)



  (11) 256 1 + x 2 128P25 512x 2 H0 + + H 2 H0 ζ2 − 27(1 − x)3 (1 + x)4 9(1 − x 2 )3 3(1 − x)(1 + x)3 −1   (11) (11) 512x 35 − 208x + 35x 2 32xP19 64P31 3 + + − H + H4 H 1 0 81(1 − x 2 )3 81(1 − x 2 )5 81(1 − x)5 (1 + x)4 0     256 23 − 14x + 23x 2 256 23 − 14x + 23x 2 + H0 H1 − H0,1 27(1 − x)3 (1 + x) 27(1 − x)3 (1 + x) (11) 256xP10 256xP11 + H02 H0,1 − H 2 H0,−1 5 3 27(1 − x) (1 + x) 27(1 − x)5 (1 + x)3 0

4

27

(11) 64H02 P33 81(1 − x)4 (1 + x)6



3

26

(11)

(11) 16H0 P30 81(1 − x)3 (1 + x)5

2

25

(11) (11) 128H0,−1 P16 64H−1 H02 P28 128H0 H0,−1 P17 + − + 27(1 − x 2 )3 27(1 − x)4 (1 + x)2 27(1 − x)4 (1 + x)3 (11) 128H0,0,−1 P29 27(1 − x)4 (1 + x)3

1

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.59 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

  (11) 512 1 + x 2 2048xP5 + H H − H0 H0,0,1 −1 0,−1 3(1 − x)(1 + x)3 27(1 − x)5 (1 + x)3   (11) 512 1 + x 2 512xP14 + H0 H0,0,−1 − H0,−1,−1 3(1 − x)(1 + x)3 27(1 − x)5 (1 + x)3

1 2 3 4 5

1 2 3 4 5

(11) (11) 512xP15 512x 2 P12 + H − H0,0,0,−1 0,0,0,1 27(1 − x)5 (1 + x)3 9(1 − x)5 (1 + x)3 (11) (11) (11) 64H0 P32 64P34 128H−1 P21 + − − 3(1 − x)4 (1 + x)3 27(1 − x)5 (1 + x)5 27(1 − x)4 (1 + x)6   (11) 1024x 35 − 208x + 35x 2 128xP23 H02 − − H0 H1 27(1 − x 2 )3 9(1 − x)5 (1 + x)4

  (11) 1024x 35 − 208x + 35x 2 512xP1 + H0,1 + H0,−1 ζ2 27(1 − x 2 )3 9(1 − x)5 (1 + x)3

6 7 8 9 10 11 12 13 14 15 16

(11) 128xP18 − ζ2 135(1 − x)5 (1 + x)3 2

 (11) (11) 512xP9 256P27 (0) + − + H0 ζ3 + FA,2 27(1 − x)4 (1 + x)3 27(1 − x)5 (1 + x)3

17 18 19 20 21 22

59

6 7 8 9 10 11 12 13 14 15 16 17 18 19

(7.384)

20 21 22

with the polynomials

23

23

(11)

24

P1

= x 4 − 5x 3 + 2x 2 − 5x + 1,

(7.385)

25

(11) P2 (11) P3 (11) P4 P5(11) (11) P6 (11) P7 P8(11) (11) P9 (11) P10 (11) P11 (11) P12 (11) P13 (11) P14 (11) P15 (11) P16 (11) P17 (11) P18

= 12x − 109x + 310x − 109x + 12,

(7.386)

= 24x 4 − 115x 3 + 330x 2 − 115x + 24,

(7.387)

27

= 29x 4 + 167x 3 − 16x 2 + 167x + 29,

(7.388)

28

= 36x − 277x + 494x − 277x + 36,

(7.389)

= 36x 4 − 121x 3 + 350x 2 − 121x + 36,

(7.390)

31

= 69x 4 − 152x 3 − 58x 2 − 152x + 69,

(7.391)

32

= 69x − 26x + 2x − 26x + 69,

(7.392)

= 71x 4 − 561x 3 + 992x 2 − 561x + 71,

(7.393)

35

= 71x − 557x + 990x − 557x + 71,

(7.394)

36

= 107x 4 − 834x 3 + 1478x 2 − 834x + 107,

(7.395)

= 108x 4 − 835x 3 + 1484x 2 − 835x + 108,

(7.396)

39

= 158x + 469x + 1006x + 469x + 158,

(7.397)

40

= 179x 4 − 1392x 3 + 2474x 2 − 1392x + 179,

(7.398)

42

= 181x 4 − 1378x 3 + 2466x 2 − 1378x + 181,

(7.399)

43

= 207x + 8x − 982x + 8x + 207,

(7.400)

44

= 249x 4 − 1380x 3 + 3994x 2 − 1380x + 249,

(7.401)

46

= 610x 4 − 4759x 3 + 8064x 2 − 4759x + 610,

(7.402)

47

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

4

3

4

2

3

4

2

3

4

2

3

4

4

2

3

3

2

2

24 25 26

29 30

33 34

37 38

41

45

JID:NUPHB AID:114751 /FLA

(11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

P19 = x 5 − 52x 4 + 81x 3 − 15x 2 + 44x + 1,

(7.403)

(11) P20 (11) P21 (11) P22 (11) P23 (11) P24 (11) P25 (11) P26 (11) P27 (11) P28 (11) P29 (11) P30 (11) P31

= 3x 5 − 41x 4 + 18x 3 − 30x 2 + 27x − 9,

(7.404)

= 3x 5 − 9x 4 + 6x 3 − 10x 2 + 3x − 1,

(7.405)

4

= 3x + 31x + 24x − 3x + 9,

(7.406)

5

= 12x 5 − 75x 4 + 53x 3 + 69x 2 − 91x + 12,

(7.407)

= 15x + 35x − 4x + 68x − 19x + 33,

(7.408)

= 21x 5 − 31x 4 + 48x 3 − 24x 2 + 59x − 9,

(7.409)

= 33x 5 − 83x 4 + 78x 3 − 66x 2 + 97x − 27,

(7.410)

= 48x − 421x + 859x + 787x − 367x + 30,

(7.411)

= 183x 5 − 701x 4 + 1726x 3 + 1870x 2 − 809x + 219,

(7.412)

14

= 315x − 1561x + 3502x + 3358x − 1453x + 279,

(7.413)

15

19 20

25 26 27

34 35 36 37 38 39 40 41 42 43 44 45 46 47

4

5

3

4

5

2

3

4

2

3

2

= 8449x 6 + 19482x 5 + 13519x 4 − 7316x 3 + 13519x 2 + 19482x + 8449, (7.414) = 48x 8 − 584x 7 + 457x 6 + 2554x 5 − 1127x 4 − 1544x 3 − 329x 2 + 54x − 105, (7.415)

8

7

6

5

1 2 3

6

4

3

2

7 8 9 10 11 12 13

16 17 18 19 20 21 22 23 24 25 26 27

+ 2814x 2 + 2652x − 613

28

33

2

= 696x + 1820x − 2251x − 3898x − 521x + 3080x + 155x + 86x − 63, (7.417)  4 (11) P34 = 24 ln(2) x 2 − 1 − 843x 8 + 2432x 7 + 5402x 6 + 2028x 5 − 8680x 4 − 2888x 3

24

32

5

(11) P33

23

31

4

(11)

22

30

5

P32 = 63x 8 + 1134x 7 − 402x 6 − 2986x 5 + 2320x 4 + 738x 3 + 146x 2 + 154x − 15, (7.416)

21

29

[m1+; v1.304; Prn:6/09/2019; 12:57] P.60 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

60

(7.418)

28 29

and

 (0)

FA,2 = x

256H02 H1 P2(12) n2h − 27(x − 1)2 (1 + x)4



30



−

128H0,0,−1 P3(12) 27(x − 1)2 (1 + x)4

+

(12) 512P5 243(x − 1)2 (1 + x)4

(12) (12) 2 128H−1 512H−1 P6 128P28 + − + + H0 81(x − 1)3 (1 + x)3 243(x − 1)3 (1 + x)5 27(x − 1)(1 + x)

(12) (12) (12) 128P29 64H−1 P3 32P31 2 + − + − H 0 27(x − 1)2 (1 + x)4 81(x − 1)3 (1 + x)6 81(x − 1)3 (1 + x)7

(12) 320x 2 H04 512H0 P2 512x 2 H−1 3 + − + H 0 27(x − 1)3 (1 + x)3 27(x − 1)3 (1 + x)3 27(x − 1)2 (1 + x)4

(12) 512P6 1024x 2 H02 256H−1 − − H0,−1 H0,1 + 9(x − 1)3 (1 + x)3 81(x − 1)3 (1 + x)3 27(x − 1)(1 + x)

(12) 512P2 4096x 2 H0 256H0,−1,−1 + − + H0,0,1 + 27(x − 1)(1 + x) 27(x − 1)2 (1 + x)4 9(x − 1)3 (1 + x)3

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.61 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

61

(12) 128H−1 P8 2048x 2 H0,0,0,1 1024x 2 H0,0,0,−1 − − + 3(x − 1)3 (1 + x)3 9(x − 1)3 (1 + x)3 27(x − 1)2 (1 + x)4   (12) (12) 1024 3 + 2x + 3x 2 32P30 32P32 − + ln(2) + − 9(1 + x)4 27(x − 1)3 (1 + x)7 405(x − 1)3 (1 + x)6

256x 2 H02 1024x 2 H−1 2048x 2 H0,−1 + + H0 − ζ2 9(x − 1)3 (1 + x)3 3(x − 1)3 (1 + x)3 9(x − 1)3 (1 + x)3

 (12) 1792x 2 ζ22 128P11 1024x 2 H0 + + − + ζ3 9(x − 1)3 (1 + x)3 27(x − 1)2 (1 + x)4 3(x − 1)3 (1 + x)3    (12) (12) 1024xH0 P17 160ζ5 P1 1024P7(12) 1 + Li + + nh 4 (x − 1)4 (1 + x)2 2 27(x − 1)4 (1 + x)2 27(x − 1)5 (1 + x)3

1024x 2 H02 4096x 2 H0 H1 4096x 2 H0,1 + + − (x − 1)3 (1 + x)3 (x − 1)3 (1 + x)3 (x − 1)3 (1 + x)3 (12) 128x 2 H02 128xH0 P17 128P7(12) 4 + + + ln (2) 81(x − 1)4 (1 + x)2 81(x − 1)5 (1 + x)3 3(x − 1)3 (1 + x)3

256H02 H1 P2(12) 512x 2 H0 H1 512x 2 H0,1 + − + n − l 3(x − 1)3 (1 + x)3 3(x − 1)3 (1 + x)3 27(x − 1)2 (1 + x)4   (12) (12) 512 11 + 32x + 11x 2 256P19 256ζ3 P27 − − + 27(x − 1)3 (1 + x)4 27(x − 1)2 (1 + x)2 81(x − 1)3 (1 + x)3

    512 55 + 62x + 55x 2 2048 1 − x + x 2 2 − H−1 + H H0 81(x − 1)(1 + x)3 27(x − 1)3 (1 + x) −1

(12) (12) (12) 64P25 128P10 256H−1 P22 2 + − + − H 0 81(x − 1)2 (1 + x)4 27(x − 1)3 (1 + x)4 81(x − 1)3 (1 + x)4

(12) 320x 2 H04 512H0 P2 512x 2 H−1 3 − − + H 0 27(x − 1)3 (1 + x)3 27(x − 1)3 (1 + x)3 27(x − 1)2 (1 + x)4

  512 55 + 62x + 55x 2 1024x 2 H02 512x 2 H02 − + H0,1 + 3 3 3 9(x − 1) (1 + x) 81(x − 1)(1 + x) 3(x − 1)3 (1 + x)3

  (12) 4096 1 − x + x 2 512P2 H − + − H −1 0,−1 27(x − 1)3 (1 + x) 27(x − 1)2 (1 + x)4

(12) 512P22 4096x 2 H0 1024x 2 H0 + − H0,0,1 + 9(x − 1)3 (1 + x)3 27(x − 1)3 (1 + x)4 3(x − 1)3 (1 + x)3   4096 1 − x + x 2 2048x 2 H0,0,0,1 H0,−1,−1 − × H0,0,−1 + 3 27(x − 1) (1 + x) 3(x − 1)3 (1 + x)3

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(12) (12) 32P20 512H−1 P23 1024x 2 H0,0,0,−1 + + − + 9(x − 1)3 (1 + x)3 81(x − 1)2 (1 + x)4 27(x − 1)3 (1 + x)4

  (12) 1024 3 + 2x + 3x 2 128P24 1024x 2 H−1 + ln(2) + − − H0 9(1 + x)4 27(x − 1)3 (1 + x)4 9(x − 1)3 (1 + x)3

256x 2 H02 7424x 2 ζ22 7168x 2 H0,−1 − + ζ2 − 3(x − 1)3 (1 + x)3 9(x − 1)3 (1 + x)3 45(x − 1)3 (1 + x)3   4096x 2 512x 2 1 + Li ln4 (2) + 4 (x − 1)3 (1 + x)3 2 3(x − 1)3 (1 + x)3 (12) (12) (12) 512xH0 H1 P16 512xH0,1 P16 1024xH0,−1 P13 + ln(2) − + − 9(x − 1)5 (1 + x)3 9(x − 1)5 (1 + x)3 9(x − 1)5 (1 + x)3

(12) (12) (12) 64P34 256P33 1024xH−1 P14 − + + H0 27(x − 1)4 x(1 + x)6 9(x − 1)5 (1 + x)3 27(x − 1)5 (1 + x)5

   (12) 256 1 + x 2 1 − 4x + x 2 256xH02 P26 768x 2 H03 − + − H−1 (x − 1)4 (1 + x)2 9(x − 1)5 (1 + x)4 (x − 1)3 (1 + x)3 (12) 128P9(12) 256xH0 P18 256x 2 H02 2 + ln (2) − − 4 2 5 3 27(x − 1) (1 + x) (x − 1)3 (1 + x)3 27(x − 1) (1 + x)

(12) 64P4 1024x 2 H0 H1 1024x 2 H0,1 − + + − (x − 1)3 (1 + x)3 (x − 1)3 (1 + x)3 (x − 1)4 (1 + x)2

  (12) 256x 2 1 + 19x + x 2 32P12 − H + ζ ζ 0 3 2 135(x − 1)4 (1 + x)2 (x − 1)5 (1 + x)3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

(12)

(12) 1024xP15 64xH0 P21 1024x 2 − + ln(2) − ln2 (2) 5 3 5 3 (x − 1)3 (1 + x)3 135(x − 1) (1 + x) 9(x − 1) (1 + x)

64x 2 H02 256x 2 H0 H1 256x 2 H0,1 + − ζ2 + 5(x − 1)3 (1 + x)3 5(x − 1)3 (1 + x)3 5(x − 1)3 (1 + x)3 2    640x 2 1 + 7x + x 2 256x 2 ζ23 + + H0 ζ5 5(x − 1)3 (1 + x)3 (x − 1)5 (1 + x)3

30 31 32 33 34 35 36 37

(0)

39

41 42 43 44 45 46 47

(0)

(0)

(0),r

+ FA,2,1 + FA,2,2 ζ2 + FA,2,3 ζ3 + FA,2 ,

38

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62

(7.419)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

with the polynomials (12) P1 (12) P2 (12) P3 (12) P4 (12) P5

1

40 41

= 2x − 7x − 8x − 7x + 2,

(7.420)

= 3x 4 − 14x 3 − 2x 2 − 14x + 3,

(7.421)

43

= 5x − 30x − 4x − 26x + 7,

(7.422)

44

= 6x 4 − 19x 3 − 16x 2 − 19x + 6,

(7.423)

46

= 14x 4 + 35x 3 + 54x 2 + 35x + 14,

(7.424)

47

4

4

3

2

3

2

42

45

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63

1

P6

= 15x 4 − 41x 3 − 16x 2 − 41x + 15,

(7.425)

2

(12) P7 (12) P8 (12) P9 (12) P10 (12) P11 (12) P12 (12) P13 (12) P14 (12) P15 (12) P16 (12) P17 (12) P18 (12) P19 (12) P20 (12) P21 (12) P22 (12) P23 (12) P24 (12) P25 (12) P26 (12) P27 (12) P28 (12) P29 (12) P30

= 17x 4 − 118x 3 + 58x 2 − 118x + 17,

(7.426)

= 19x 4 − 82x 3 − 12x 2 − 86x + 17,

(7.427)

4

= 29x − 16x + 154x − 16x + 29,

(7.428)

5

= 30x 4 + 41x 3 + 32x 2 − 83x − 80,

(7.429)

7

2

= 49x − 2x + 56x − 6x + 47,

(7.430)

8

= 65x 4 + 5552x 3 − 6446x 2 + 5552x + 65,

(7.431)

= 77x − 200x + 270x − 200x + 77,

(7.432)

= 80x 4 − 215x 3 + 276x 2 − 215x + 80,

(7.433)

= 80x 4 − 209x 3 + 264x 2 − 209x + 80,

(7.434)

= 83x − 218x + 258x − 218x + 83,

(7.435)

= 87x 4 − 92x 3 + 298x 2 − 92x + 87,

(7.436)

17

= 105x − 182x + 334x − 182x + 105,

(7.437)

18

= 135x 4 + 110x 3 − 98x 2 + 110x + 135,

(7.438)

20

= 1709x + 1708x − 690x + 716x + 829,

(7.439)

21

= 2415x 4 − 3158x 3 + 6274x 2 − 3158x + 2415,

(7.440)

23

= x 5 + 25x 4 − 8x 3 + 16x 2 − 9x + 7,

(7.441)

24

= 13x 5 − 43x 4 + 40x 3 − 32x 2 + 59x − 5,

(7.442)

= 14x 5 − 133x 4 + 31x 3 − 65x 2 + 3x − 10,

(7.443)

= 15x − 177x + 50x − 94x + 27x − 21,

(7.444)

= 28x 5 − 42x 4 + 35x 3 + 29x 2 − 117x + 55,

(7.445)

30

= 29x − 33x + 50x − 34x + 65x − 13,

(7.446)

31

= 391x 6 + 502x 5 − 1151x 4 − 2236x 3 − 1151x 2 + 502x + 391,

(7.447)

33

= 10x − 21x − 35x + 158x + 236x + 223x + 13x − 40,

(7.448)

34

(12) P31 (12) P32 (12) P33

= 27x − 92x − 404x − 356x + 66x + 28x + 76x + 36x − 21,

(7.450)

= 31x 8 − 116x 7 − 564x 6 − 524x 5 + 122x 4 − 140x 3 − 84x 2 + 12x − 17,

(7.451)

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

4

3

4

2

3

4

3

4

4

4

5

5

7

2

3

4

6

3

2

5

4

7

6

3

5

4

2

3

2

13 14 15 16

(12)

P34 = 208x 10 + 10715x 9 − 21650x 8 − 31368x 7 + 25186x 6 + 85658x 5 + 25186x 4 − 31368x 3 − 21650x 2 + 10715x + 208.

25 26 27 28 29

35 36 37 38 39

= 60x 8 − 441x 7 − 3520x 6 + 4719x 5 + 4313x 4 + 4773x 3 − 3358x 2 − 387x + 33, (7.452)

(7.453)

40 41 42 43 44 45 46

46 47

12

32

= 4647x 7 + 4667x 6 − 12281x 5 − 14941x 4 − 819x 3 + 4761x 2 − 4347x − 3447, (7.449) 8

11

22

2

3

10

19

2

4

3

9

2

3

2

6

2

3

1

(0)

The first expansion coefficients of FA,2,i , i = 1...3 are given by

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82929376 264976 264976 53768196023y 2 − + + 18225 81y 53581500 81y 2 3 4 53768196023y 42082197145871y 14983026350279y 5 + + + + O(y 6 ) 53581500 54010152000 27005076000 (7.454) 2 81870064 11132 11132 7224542428y (0) FA,2,2 (x) = − + − 18225 27y 7441875 27y 2 3 7224542428y 164328135367y 4 581273616754y 5 − − − + O(y 6 ) (7.455) 7441875 218791125 1093955625 6230776 1024 1024 22054357y 2 (0) + − − FA,2,3 (x) = 6075 27y 11907000 27y 2 3 22054357y 12191383321y 4 19907290871y 5 − (7.456) − − + O(y 6 ). 11907000 266716800 222264000 (0)

FA,2,1 (x) = −

The 1/y k behavior of these expressions is canceled by corresponding terms of other contributions. The renormalization of the three–loop massive form factors is performed in the same way as in earlier calculations, cf. [9,12]. We use a mixed scheme. The heavy quark mass and wave function have been renormalized in the on-shell (OS) renormalization scheme, while the strong coupling constant is renormalized in the MS scheme, where we set the universal factor Sε = exp(−ε(γE − ln(4π)) for each loop order to one at the end of the calculation. The required renormalization constants are available and are denoted by Zm,OS [38,41,42,112,113], Z2,OS [38, 42,112,114] and Zas [115–121] for the heavy quark mass, wave function and strong coupling constant, respectively. The renormalization of the heavy-quark wave function and the strong coupling constant are multiplicative, while the renormalization of massive fermion lines has been taken care of by properly considering the counter terms. Pseudoscalar and axialvector form factors are related by a Ward identity. We explicitly verified our results fulfill this relation. 7.5. Numerical results

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

30 31

31 32

2

29

29 30

1

We present now numerical results for the ε0 parts of the different unrenormalized form factors. (0) For comparison the functions FC,i , i = 1...3 are shown using their first 20, 50, 100. 200 and 500 expansion coefficients. In Figs. 1–7 we show the results in the Euclidean region 0 < x < 1. In Figs. 8–13 the results in the region below threshold (0 < z < 4) are shown, expanding around z = 0. At small values of x the form factor F1 has logarithmic singularities both in its nh (Fig. 1, left panel) and n2h parts (Fig. 1, right panel). Here and in the following we also illustrate taking into account a rising number of terms n = 20 to 500 from the non–first order factorizing contributions to illustrate the degree of convergence. The vector form factor F2 , cf. Fig. 2 is proportional to x, damping out further lnk (x) contributions. Despite taking only 500 expansion coefficients, one obtains the correct representation in the whole x range. In Fig. 3 we show the behavior of the axialvector form factor FA,1 under the same conditions as in Fig. 1, and for the form factor FA,2 in Fig. 4 similar to those in Fig. 2. In Fig. 5 we illustrate for the ratio of the vector form factor F1 evaluated of n terms of the non–first order factorizing contributions for n = 20, 50, 100, 200 and 500 to the case of n = 2000 to see the relative convergence both for the n2h and nh contribution, which approves towards x → 0. However, the complete logarithmic behavior cannot be resembled by this representation.

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1

2

2

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3

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5

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6

7

7

8

8

9

9 10

10 11 12 13

Fig. 1. Vector form factor FV ,1 : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

11 12 13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24 25

25 26 27

Fig. 2. Vector form factor FV ,2 : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

26 27

28

28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36

37

37

38

38

39 40

Fig. 3. Axial vector form factor FA,1 : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

42

42

44 45 46 47

40 41

41

43

39

The contributions to the scalar form Factor FS are shown in Fig. 6. Its x–dependence is similar to the one of the vector form factor. The pseudoscalar form factor is illustrated in Fig. 7. It has similar behavior as the axialvector form factor. Now we turn to the illustration of the threshold expansion of the form factors. All form factors are multiplied by the factor (4 − z)3/2 for the nh terms and by (4 − z) for the n2h terms for con-

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2

2

3

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5

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7

7

8

8

9

9 10

10 11 12

Fig. 4. Axial vector form factor FA,2 : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

11 12

13

13

14

14

15

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16

16

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17

18

18

19

19

20

20

21

21

22

22

23

23

24

24 25

25 26 27 28

Fig. 5. Ratios of the approximations with 20, 50, 100, 200, 500 terms and our best approximation using 2000 terms for the vector form factor FV ,1 . Left ε0 n1h , right ε0 n2h . The ratio using 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

26 27 28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36

37

37

38

38

39

39 40

40 41 42

Fig. 6. Scalar form factor FS : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

44

44

46 47

42 43

43

45

41

venience, with z = q 2 /m2 for the vector- (Fig. 7, 8), axialvector- (Figs. 9, 10), scalar- (Fig. 11) and pseudoscalar form factor (Fig. 12). In large z region differences due to the number of terms used in the expansion of the non–first order factorizing contributions are seen.

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7

7

8

8

9

9 10

10 11 12

Fig. 7. The pseudoscalar form factor FP : left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

11 12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22 23

23 24 25

Fig. 8. The vector form factor FV ,1 in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

24 25

26

26

27

27

28

28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36 37

37 38 39

Fig. 9. The vector form factor FV ,2 in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

8. Conclusions

44 45 46 47

41 42

42 43

39 40

40 41

38

We have calculated the non-singlet nh contributions to the massive three–loop form factors for vector-, axialvector-, scalar- and pseudoscalar currents. The contributing Feynman integrals were reduced to master integrals using the package Crusher [23]. The calculation of the analytic expansion coefficients of the master integrals in x required a new way of decoupling of the differential equations provided by the IBP–relations w.r.t. their optimal expansion in the dimen-

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1

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2

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3

4

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6

7

7

8

8

9

9 10

10 11 12

Fig. 10. The axialvector form factor FA,1 in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

11 12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24 25

Fig. 11. The axialvector form factor FA,2 in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

24 25

26

26

27

27

28

28

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29

30

30

31

31

32

32

33

33

34

34

35

35

36

36 37

37 38 39

Fig. 12. The scalar form factor FS in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

42 43 44 45 46 47

39 40

40 41

38

sional parameter ε. Otherwise, it would have been very demanding to provide the required initial values. Here we used the method of arbitrary high moments [15], partly requiring to calculate 8000 moments. The recursions were derived using the guessing method [26]. The recursions for the pole terms and a part of the contributions of O(ε 0 ) are first order factorizing and one obtains representations in terms of harmonic polylogarithms in the variable x. The recursions were solved using the packages Sigma [24,28] and EvaluateMultisums and SumProduction [125,126]. The resulting infinite sums were then converted into harmonic polylogarithms using

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6

7

7

8

8

9

9 10

10 11 12

Fig. 13. The pseudoscalar form factor FP in the threshold region as a function of z: left ε0 n1h , right ε0 n2h , the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

15 16 17 18 19 20 21 22 23 24 25 26

O(ε 0 )

the package HarmonicSums [59–67]. For some color-ζ contributions at non–first order factorizing parts are contained in the recurrences. Here the largest of the remaining recurrences are of order o = 15, resulting from of original recurrences of order up to o = 55 and degree d = 1324. For those we have obtained analytic polynomial expansions in x of degree d = 2000, which can be extended in case needed. They already allow for precise numerical representations in a wide range of x. However, in the case of logarithmic divergences for x → 0 these representations diverge. The analytic solution of the non–first order factorizing recurrences in terms of special higher functions still needs to be performed in the future. We also considered the leading color contributions for the scalar form factor as an example to see whether here simplifications can be obtained. This is indeed the case since the most involved of the remaining non–first order factorizing recurrences is only of order o = 5, stemming from an original recurrence of order o = 46 and degree d = 901. Complete first order factorization can, however, not be obtained in this case. Acknowledgement

31 32 33 34

We would like to thank J. Ablinger, A. Behring, A. De Freitas, K. Schönwald and J. Vermaseren for discussions. This work was supported in part by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15), by the EU TMR network SAGEX Marie Skłodowska-Curie grant agreement No. 764850 and COST action CA16201: Unraveling new physics at the LHC through the precision frontier. Appendix A. Initial values to high order in ε

39 40 41 42 43 44 45 46 47

17 18 19 20 21 22 23 24 25 26

28

30 31 32 33 34

36 37

37 38

16

35

35 36

15

29

29 30

14

27

27 28

12 13

13 14

11

Boundary values for the master integrals appearing in the present calculation can be obtained using three loop propagator integrals as have been dealt with in Ref. [38] to orders in the dimensional parameter up to O(ε3 ) and lower. Depending on the method of uncoupling of the associated differential equations terms up to O(ε 9 ), i.e. in total 13 orders in ε, may be necessary. For some of the integrals all-order in ε representations exist, [25,112,122–124]. For the integrals I9 , I11 , I12 , I15 and I16 of [38,123,124] we have derived the corresponding representations. They are thoroughly given in terms of multiple zeta values, cf. [25]. The calculation of ε-expansions at such depth is very time consuming, despite the fact that the techniques to be used are rather standard ones available in the packages Sigma [24,28], EvaluateMultisums and SumProduction [125,126] together with the asymptotic expansions of harmonic sums of the package

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Table 3 Number of basis elements by weight.

1

2 3

weight

# basis elements

3

4

1 2 3 4 5 6 7 8 9 10 11

1 1 1 1 2 2 3 5 8 11 18

4

1

5 6 7 8 9 10 11 12 13

2

5 6 7 8 9 10 11 12 13 14

14 15 16 17 18 19 20 21

HarmonicSums. In the final results all constants of higher transcendentality will be canceled. Given the associated calculational effort, it is of advantage having decoupling methods, in which only initial values requiring an expansion in ε to a far lower order in ε is needed. This has been possible by the algorithm described in Section 5. It is interesting to see that all constants spanning the MZVs [25] are contributing to this integral. (See Table 3.) These are (together with their approximate numerical values)

23

ln(2) ≈ 0.69314718055994530942

(A.1)

24

ζ2 ≈ 1.64493406684822643647

(A.2)

ζ3 ≈ 1.20205690315959428540   1 Li4 ≈ 0.51747906167389938633 2 ζ5 ≈ 1.03692775514336992633   1 ≈ 0.50840057924226870746 Li5 2 s6 = S−5,−1 (∞) ≈ 0.98744142640329971377   1 Li6 ≈ 0.50409539780398855069 2 s7a = S−5,1,1 (∞) ≈ −0.95296007575629860341

(A.3)

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

16 17 18 19 20 21 22

22

25

15

23 24 25 26 27

(A.4) (A.5) (A.6)

28 29 30 31 32

(A.7)

33 34

(A.8) (A.9)

35 36 37

s7b = S5,−1,−1 (∞) ≈ 1.02912126296432453422

(A.10)

38

ζ7 ≈ 1.00834927738192282684   1 ≈ 0.50201456332470849457 Li7 2 s8a = S5,3 (∞) ≈ 1.04178502918279188339

(A.11)

39

(A.12) (A.13)

43

s8b = S−7,−1 (∞) ≈ 0.99644774839783766598

(A.14)

44

s8c = S−5,−1,−1,−1 (∞) ≈ 0.98396667382173367092

(A.15)

46

s8d = S−5,−1,1,1 (∞) ≈ 0.99996261346268344770

(A.16)

47

40 41 42

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1

  1 ≈ 2 ζ9 ≈   1 Li9 ≈ 2 s9a = S7,−1,−1 (∞) ≈ Li8

71

1

0.50099665909705191056

(A.17)

1.00200839282608221442

(A.18)

0.50049488810595361004

(A.19)

1.00640196269235635901

(A.20)

s9b = S−7,−1,1 (∞) ≈ 0.99842952512288855440

(A.21)

8

s9c = S−6,−2,−1 (∞) ≈ −0.98747515763691535588

(A.22)

9

s9d = S−5,−1,1,1,1 (∞) ≈ 1.00219817413397743630

(A.23)

s9e = S−5,−1,−1,−1,1 (∞) ≈ 0.98591171955244547262

(A.24)

12

(A.25)

13

17

s9f = S−5,−1,−1,1,−1 (∞) ≈ 0.97848117128116624248   1 ≈ 0.50024632060600677501 Li10 2 h313111 ≈ 0.00000059060152818192

(A.27)

17

18

h331111 ≈ 0.00000040503762995428

(A.28)

18

h3331 ≈ 0.00003761590257651408

(A.29)

19

h511111 ≈ 0.00000030113797559813

(A.30)

21

h5113 ≈ 0.00008812461415599555

(A.31)

22

h5131 ≈ 0.00002330610214579640

(A.32)

23

h5311 ≈ 0.00001097851878750415

(A.33)

25

26

h7111 ≈ 0.00000654709718938270

(A.34)

26

27

h73 ≈ 0.00735713658369574885

(A.35)

h91 ≈ 0.00188302419181393899

(A.36)

29

(A.37)

30

(A.38)

34

ζ11 ≈ 1.00049418860411946456   1 Li11 ≈ 0.50012279152986795519 2 h3113111 ≈ 0.00000004259335538990

35

h3131111 ≈ 0.00000002903877500717

(A.40)

35

36

h3311111 ≈ 0.00000002146685479915

(A.41)

36

37 38

h33131 ≈ 0.00000203422625738433

(A.42)

39

h33311 ≈ 0.00000095250833508071

(A.43)

39

40

h5111111 ≈ 0.00000001677568330189

(A.44)

40

42

h51113 ≈ 0.00000535991478634246

(A.45)

42

43

h51131 ≈ 0.00000141456111518871

(A.46)

43

h51311 ≈ 0.00000066489238255553

(A.47)

44

h53111 ≈ 0.00000039564332931785

(A.48)

46

h533 ≈ 0.00039368649887395471

(A.49)

47

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

19 20 21 22 23 24 25

28 29 30 31 32 33

41

44 45 46 47

2 3 4 5 6 7

10 11

14

(A.26)

15 16

20

24

27 28

31

(A.39)

32 33 34

37 38

41

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h551 ≈ 0.00010106766377056452

(A.50)

1

h71111 ≈ 0.00000026780063665860

(A.51)

2

h731 ≈ 0.00004821281485407120

(A.52)

4

h713 ≈ 0.00018649660146306540

(A.53)

5

h911 ≈ 0.00002219036425770103,

(A.54)

with

6 7 8

habcd ≡ H−a,−b,−c,−d (1)

9

(A.55)

in the (collected) notation as e.g. H−3,−1 (1) ≡ H0,0,−1,−1, (1). Up to weight w = 9 we follow the representation of summer.h. Earlier a similar representation has been available for the weights w = 10–12. These files do not exist anymore [127]. Therefore we change to the basis which is used in the package HarmonicSums.m, where also the respective constant files are now available8 The MZV data mine used another basis for the  HPLs at argument one. This basis is of course equivalent. We used the values ζ2k+1 and Lik 12 as basis elements, through which one of the high weight HPLs is replaced.     1 1 17 1 1 1 Li10 = h31111111 − h511111 + h7111 − h91 − Li9 ln(2) 2 2 4 2 8 2       1 1 1 1 1 1 ln2 (2) − Li7 ln3 (2) − Li6 ln4 (2) − Li8 2 2 6 2 24 2     1 1 1 1 1 5 Li5 ln (2) − Li4 ln6 (2) − ln10 (2) − 120 2 120 2 43200 1 789 5 1 27 2 2 3 + ln8 (2)ζ2 − ζ − ln7 (2)ζ3 + ζ ζ + ζ 2 ζ3 ζ5 11520 8800 2 5760 160 2 3 8 3 3 + ζ52 + ζ3 ζ7 (A.56) 16 8     1 1 17 1 1 1 Li11 = h311111111 − h5111111 + h71111 − h911 − Li1 ln(2) 2 2 4 2 8 2       1 1 1 1 1 1 ln2 (2) − Li8 ln3 (2) − Li7 ln4 (2) − Li9 2 2 6 2 24 2       1 1 1 1 1 1 Li6 ln5 (2) − Li5 ln6 (2) − Li4 ln7 (2) − 120 2 720 2 5040 2 1 1 2533 4 1 ln11 (2) + ln9 (2)ζ2 − ln8 (2)ζ3 − ζ ζ3 − 332640 90720 46080 5600 2 5 99 3 3 57 5 30945 + ζ2 ζ33 − (A.57) ζ ζ5 + ζ32 ζ5 − ζ22 ζ7 − ζ2 ζ9 + ζ11 48 112 2 16 40 3 2048 The integrals I9 and I11 are given by   2 10 26 398 52 2 248 16 1 I9 = − 3 − 2 + − − 2ζ2 − 2 − ζ3 − 22ζ2 + ε − ζ2 − ζ3 3 ε 3 3 5 3 3ε 3ε

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

45

46

46 47

3

8 We thank J. Ablinger for making these files available.

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  64 ln4 (2) 1 + 96 ln(2)ζ2 − 146ζ2 + ε 1038 − 96ζ5 − 16ζ2 ζ3 − 512Li4 − 2 3

108 2 1888 − 256 ln2 (2)ζ2 + ζ − ζ3 + 960 ln(2)ζ2 − 774ζ2 5 2 3   2944 3 64 2 512 ln5 (2) 1 3 17470 +ε − ζ2 − ζ3 + 2496ζ5 − 4096Li5 + 3 35 3 2 15   2048 3 2944 1 + ln (2)ζ2 − ln(2)ζ22 + 16ζ2 ζ3 − 5120Li4 − 2560 ln2 (2)ζ2 3 5 2

640 ln4 (2) 3004 2 − + ζ − 3600ζ3 + 6144 ln(2)ζ2 − 3634ζ2 3 5 2   416 2 1 4 85562 +ε − 1328ζ7 − 288ζ2 ζ5 − ζ2 ζ3 − 12288s6 − 32768Li6 3 5 2   2048 ln6 (2) 4288 4 13696 2 1 − 1536Li4 ζ2 − − ln (2)ζ2 + ln (2)ζ22 2 45 3 5   19728 3 14680 2 1024 ln5 (2) 1 − 2112 ln(2)ζ2 ζ3 + ζ2 + ζ3 − 40960Li5 + 5 3 2 3   1 20480 3 ln (2)ζ2 − 5888 ln(2)ζ22 + 28512ζ5 + 752ζ2 ζ3 − 32768Li4 + 3 2 2

4096 ln4 (2) 23396 2 53264 − − 16384 ln2 (2)ζ2 + ζ2 − ζ3 + 32256 ln(2)ζ2 3 5 3

325632s7a 389806 133968 4 − 15894ζ2 + ε 5 − ζ2 − 64ζ2 ζ32 − 768ζ3 ζ5 − 3 175 7       1 1 1 362496s7b − 262144Li7 − 12288Li5 ζ2 − 4096Li4 ζ3 + 7 2 2 2 325632 ln(2)s6 16384 ln7 (2) 34304 5 512 4 + + ln (2)ζ2 − ln (2)ζ3 7 315 15 3 109568 3 905344 − ln (2)ζ22 + 9472 ln2 (2)ζ2 ζ3 − 95232 ln2 (2)ζ5 − ln(2)ζ23 15 35 407040 3311468 421296 2 1016280 − ln(2)ζ32 + ζ7 − ζ 2 ζ3 − ζ2 ζ5 − 122880s6 7 7 35 7     4096 ln6 (2) 42880 4 1 1 − 327680Li6 − 15360Li4 ζ2 − − ln (2)ζ2 2 2 9 3 1489888 3 149168 2 + 27392 ln2 (2)ζ22 − 21120 ln(2)ζ2 ζ3 + ζ2 + ζ3 35 3   188416 32768 ln5 (2) 131072 3 1 − 262144Li5 + + ln (2)ζ2 − ln(2)ζ22 2 15 3 5   1 + 190176ζ5 + 6096ζ2 ζ3 − 172032Li4 − 7168 ln4 (2) − 86016 ln2 (2)ζ2 2 +

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133996 2 160960 6 + ζ2 − 80400ζ3 + 152064 ln(2)ζ2 − 66626ζ2 + ε 566958 − ζ9 5 9 512 3 23552 3 7488 2 21695376s8a 28157952s8b − 3984ζ2 ζ7 − ζ − ζ ζ3 − ζ ζ5 − − 9 3 35 2 5 2 35 7 2 1449984 ln (2)s6 7704576 2899968s8c 2605056s8d − + + s6 ζ2 − 7 7 7 7       4980736 1 1 1 − 2097152Li8 − 98304Li6 ζ2 + Li5 ζ3 2 2 7 2   2884608 622592 5 1 2 16384 ln8 (2) 137216 6 − Li4 ζ2 − − ln (2)ζ2 − ln (2)ζ3 35 2 315 45 105 1173376 4 772096 3 468736 2 407808 2 + ln (2)ζ22 + ln (2)ζ2 ζ3 + ln (2)ζ23 + ln (2)ζ32 105 21 35 7 942502872ζ24 3316480 16493824 8558784 + ln(2)ζ22 ζ3 + ln(2)ζ2 ζ5 + + ζ2 ζ32 35 7 6125 7 3194496 3256320s7a 3624960s7b 3256320 ln(2)s6 − ζ 3 ζ5 − + + 7 7 7 7       32768 ln7 (2) 1 1 1 − 2621440Li7 − 122880Li5 ζ2 − 40960Li4 ζ3 + 2 2 2 63 5120 4 219136 3 68608 5 ln (2)ζ2 − ln (2)ζ3 − ln (2)ζ22 + 94720 ln2 (2)ζ2 ζ3 + 3 3 3 1810688 4070400 33458632 − 952320 ln2 (2)ζ5 − ln(2)ζ23 − ln(2)ζ32 + ζ7 7 7 7   10088208 4105216 2 1 − ζ2 ζ5 − ζ2 ζ3 − 786432s6 − 2097152Li6 7 35 2   6 131072 ln (2) 274432 4 876544 2 1 − 98304Li4 ζ2 − − ln (2)ζ2 + ln (2)ζ22 2 45 3 5   9771392 3 57344 ln5 (2) 1 − 135168 ln(2)ζ2 ζ3 + ζ2 + 319936ζ32 − 1376256Li5 + 35 2 5   989184 1 3 2 + 229376 ln (2)ζ2 − ln(2)ζ2 + 35440ζ2 ζ3 + 1019040ζ5 − 811008Li4 5 2 663876 2 1037584 ζ2 − ζ3 + 672768 ln(2)ζ2 − 33792 ln4 (2) − 405504 ln2 (2)ζ2 + 5 3

1956032 5 1664 2 2 7221278 − 272326ζ2 + ε 7 − 6912ζ52 − 2304ζ2 ζ3 ζ5 − ζ2 − ζ ζ 3 275 5 2 3

43 44 45 46 47

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

41 42

1

114294784s9a 90902528s9c 9240576s9d − 10624ζ3 ζ7 − + 6750208s9b + − 7 21 7 11599872s9e 143271936 206985216 144875904 ln(2)s8a − + s7a ζ2 − s7b ζ2 + 7 49 49 35 3 181805056 ln(2)s8b 1933312 ln (2)s6 183871488 − − ln(2)s6 ζ2 + 7 7 49

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75

    1 1 + 227696640s6 ζ3 − 16777216Li9 − 786432Li7 ζ2 2 2       39845888 81076224 1 1 2 283149312 1 + Li6 ζ3 − Li5 ζ2 − Li4 ζ5 7 2 35 2 7 2   46116864 131072 ln9 (2) 1097728 7 1 + Li4 ζ2 ζ3 + + ln (2)ζ2 7 2 2835 315 2490368 6 2137088 5 4220416 4 + ln (2)ζ3 − ln (2)ζ22 + ln (2)ζ2 ζ3 315 525 21 11797888 4 3332096 3 543744 3 − ln (2)ζ5 − ln (2)ζ23 − ln (2)ζ32 7 15 7 93715456 2 102425088 2 − ln (2)ζ22 ζ3 + ln (2)ζ2 ζ5 + 606208 ln2 (2)ζ2 ζ3 35 7 70934369088 ln(2)ζ24 168809664 113401856 2 + ln (2)ζ7 − + ln(2)ζ2 ζ32 7 6125 49 224415616 7007214274 1046207136 3 48494416 3 − ln(2)ζ3 ζ5 − ζ9 + ζ2 ζ3 − ζ3 7 63 245 9 2042564800 2 1637594676 43390752s8a 281579520s8b + ζ 2 ζ5 + ζ 2 ζ7 − − 49 49 7 7 2 14499840 ln(2) s6 77045760 28999680s8c 26050560s8d − + + s6 ζ2 − 7 7 7 7       49807360 1 1 1 − 20971520Li8 − 983040Li6 ζ2 + Li5 ζ3 2 2 7 2   5769216 1245184 5 1 2 32768 ln8 (2) 274432 6 − Li4 ζ − − ln (2)ζ2 − ln (2)ζ3 7 2 2 63 9 21 2346752 4 7720960 3 937472 2 + ln (2)ζ22 + ln (2)ζ2 ζ3 + ln (2)ζ23 21 21 7 1919703456ζ24 33181376 4078080 2 85587840 + ln (2)ζ32 + ln(2)ζ2 ζ5 + + ζ2 ζ32 7 7 1225 7 31746048 20840448s7a 23199744s7b 20840448 ln(2)s6 − ζ 3 ζ5 − + + 7 7 7 7       1048576 ln7 (2) 1 1 1 − 786432Li5 ζ2 − 262144Li4 ζ3 + − 16777216Li7 2 2 2 315 32768 4 7012352 3 214880784 2195456 5 ln (2)ζ2 − ln (2)ζ3 − ln (2)ζ22 + ζ7 + 15 3 15 7 5207968 2 57942016 64402848 − ζ 2 ζ3 − ln(2)ζ23 − ζ2 ζ5 − 4128768s6 7 35 7 32987648 26050560 − 6094848 ln2 (2)ζ5 + ln(2)ζ22 ζ3 − ln(2)ζ32 7 7     229376 ln6 (2) 1 1 − 11010048Li6 − 516096Li4 ζ2 − − 480256 ln4 (2)ζ2 2 2 15 4601856 2 51932032 3 5052736 2 + ln (2)ζ22 − 709632 ln(2)ζ2 ζ3 + ζ2 + ζ3 5 35 3

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  4663296 270336 ln5 (2) 1 + + 1081344 ln3 (2)ζ2 − ln(2)ζ22 2 5 5   1 + 4863456ζ5 + 176976ζ2 ζ3 − 3588096Li4 − 149504 ln4 (2) 2

1

− 6488064Li5

3033164 2 4325936 − 1794048 ln2 (2)ζ2 + ζ2 − ζ3 + 2863104 ln(2)ζ2 − 1097042ζ2 5 3   160960 1 2261065728 8 Li4 ln(2)ζ5 − 243216ζ11 − ζ2 ζ 9 + ε 10037278 + 7 2 3 103584 2 423936 3 512 1071744 4 − ζ2 ζ7 − 3072ζ32 ζ5 − ζ 2 ζ5 − ζ2 ζ33 − ζ2 ζ3 5 35 3 175 1202564511072256h91 19687710418944h73 5326405632h7111 + + − 780419 780419 29 257949696h511111 24978358272h5113 15639478272h5131 − − + 17 493 493 8004796416h5311 2606579712h3331 185597952h331111 + − + 493 493 17 914358272 ln(2)s9a 727220224 ln(2)s9c + − 54001664 ln(2)s9b − 7 21 92798976 ln(2)s9e 73924608 ln(2)s9d + − 13278720 ln2 (2)s8a + 7 7 553385984 ln2 (2)s8b 92798976 ln2 (2)s8c 83361792 ln2 (2)s8d − + + 7 7 7 673097882736 84473856 8699904 7815168 − s8a ζ2 − s8b ζ2 − s8c ζ2 − s8d ζ2 17255 7 7 7 27787264 ln3 (2)s7a 30932992 ln3 (2)s7b 1200881664 + − − ln(2)s7a ζ2 7 7 49 1716781056 132420673536s7a ζ3 2899968s7b ζ3 + ln(2)s7b ζ2 − + 49 3451 7 164856262656s6 ζ22 1278001152 ln(2)s6 ζ3 + − 6291456 ln4 (2)s6 − 7 17255     227500032 2 1 1 − ln (2)s6 ζ2 − 134217728Li10 − 6291456Li8 ζ2 49 2 2       333447168 2555904 1 1 1 2 − 2097152Li7 ζ3 − Li6 ln(2)ζ3 − Li6 ζ 2 7 2 5 2 2     166723584 630718464 1 1 − Li5 ln2 (2)ζ3 + Li5 ln(2)ζ22 7 2 35 2       14942208 83361792 1 1 1 − 589824Li5 ζ5 + Li5 ζ2 ζ3 + Li4 ln(2)2 ζ22 2 7 2 35 2       369623040 2015232 1 1 3 1 2 − Li4 ln(2)ζ2 ζ3 − Li4 ζ2 − 16384Li4 ζ 7 2 7 2 2 3

2 3 4

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524288 ln10 (2) 1097728 8 41811968 7 − ln (2)ζ2 + ln (2)ζ3 14175 315 315 32247808 6 727211008 5 345571328 5 − ln (2)ζ22 + ln (2)ζ5 − ln (2)ζ2 ζ3 1575 35 105 297220096 4 8366080 4 1193759744 3 + ln (2)ζ23 + ln (2)ζ32 − ln (2)ζ7 105 3 7 22380544 3 − 145342464 ln3 (2)ζ2 ζ5 + ln (2)ζ22 ζ3 + 270697472 ln2 (2)ζ3 ζ5 3 21425512704 2 2129968640 2 12544430784 + ln (2)ζ24 − ln (2)ζ2 ζ32 − ln(2)ζ2 ζ7 245 49 49 41723390470912 ln(2)ζ22 ζ5 300811136 56012645392 + ln(2)ζ9 − + ln(2)ζ33 63 120785 7 8089162496 2721028640820472ζ3 ζ7 11685311414928ζ52 − ln(2)ζ23 ζ3 − − 245 5462933 26911 1088749068720ζ2 ζ3 ζ5 877301874261743216ζ25 1430424876432ζ22 ζ32 + − − 3451 4780066375 17255 1142947840s9a 909025280s9c 92405760s9d − + 67502080s9b + − 7 21 7 115998720s9e 289751808 ln(2)s8a 1818050560 ln(2)s8b + + − 7 7 7   19333120 ln3 (2)s6 1838714880 1 − ln(2)s6 ζ2 − 167772160Li9 − 7 49 2       398458880 162152448 1 1 1 2 − 7864320Li7 ζ2 + Li6 ζ3 − Li5 ζ 2 7 2 7 2 2     2831493120 461168640 262144 9 1 1 − Li4 ζ5 + Li4 ζ2 ζ3 + ln (2) 7 2 7 2 567 2195456 7 4980736 6 4274176 5 + ln (2)ζ2 + ln (2)ζ3 − ln (2)ζ22 63 63 105 117978880 4 42204160 4 6664192 3 − ln (2)ζ5 + ln (2)ζ2 ζ3 − ln (2)ζ23 7 21 3 5437440 3 1134018560 2 1024250880 2 − ln (2)ζ32 + ln (2)ζ7 + ln (2)ζ2 ζ5 7 7 7 141868738176 ln(2)ζ24 187430912 2 2244156160 − ln (2)ζ22 ζ3 − ln(2)ζ3 ζ5 − 7 7 1225 1688096640 23343484700 16383169752 2 + ln(2)ζ2 ζ3 − ζ9 + ζ 2 ζ7 49 21 49 102141815744 2 484925216 3 10468171328 3 1388504064s8a + ζ 2 ζ5 − ζ3 + ζ 2 ζ3 − 245 9 245 35 1432719360 1802108928s8b 185597952s8c 166723584s8d − − + s7a ζ2 − 7 7 7 49 2069852160 92798976 ln2 (2)s6 493092864 − s7b ζ2 + + s6 ζ2 49 7 7 −

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      318767104 1 1 1 − 6291456Li6 ζ2 + Li5 ζ3 2 2 7 2   184614912 1 2 1048576 ln8 (2) 8781824 6 − Li4 ζ − − ln (2)ζ2 35 2 2 315 45 39845888 5 75096064 4 49414144 3 − ln (2)ζ3 + ln (2)ζ22 + ln (2)ζ2 ζ3 105 105 21 29999104 2 26099712 2 547762176 + ln (2)ζ23 + ln (2)ζ32 + ln(2)ζ2 ζ5 35 7 7 61806558768ζ24 212396736 1055604736 202743552 + ln(2)ζ22 ζ3 − ζ3 ζ5 + + ζ2 ζ32 35 7 6125 7   1 − 15630336s7a + 17399808s7b + 15630336 ln(2)s6 − 88080384Li7 2     7 262144 ln (2) 3842048 5 1 1 − 4128768Li5 ζ2 − 1376256Li4 ζ3 + + ln (2)ζ2 2 2 15 5 12271616 3 − 57344 ln4 (2)ζ3 − ln (2)ζ22 − 31997952 ln2 (2)ζ5 + 3182592 ln2 (2)ζ2 ζ3 5 43456512 − ln(2)ζ23 − 19537920 ln(2)ζ32 + 161445776ζ7 − 48240288ζ2 ζ5 5     19440544 2 1 1 − ζ2 ζ3 − 19464192s6 − 51904512Li6 − 2433024Li4 ζ2 5 2 2 − 134217728Li8

360448 ln (2) 21694464 2 − 2264064 ln4 (2)ζ2 + ln (2)ζ22 5 5   22610657280 246644352 3 1 + Li4 ζ5 ln(2) − 3345408 ln(2)ζ2 ζ3 + ζ2 7 2 35   1196032 ln5 (2) 1 2 + 7953216ζ3 − 28704768Li5 + + 4784128 ln(2)3 ζ2 2 5   20631552 1 2 − ln(2)ζ2 + 21694368ζ5 + 812528ζ2 ζ3 − 15269888Li4 5 2 1908736 ln4 (2) 13199844 2 17697424 − − 7634944 ln2 (2)ζ2 + ζ2 − ζ3 3 5 3

+ 11894784 ln(2)ζ2 − 4384758ζ2 + 227696640s6 ζ3

39

41 42 43 44 45 46 47

6

−

38

40

[m1+; v1.304; Prn:6/09/2019; 12:57] P.78 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

78

+ ε9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

124045582 1287680 − ζ3 ζ9 − 191232ζ5 ζ7 − 31872ζ2 ζ3 ζ7 − 20736ζ2 ζ52 3 9

59904 2 1024 4 94208 3 2 2321418752 6 − ζ2 ζ3 ζ5 − ζ − ζ ζ − ζ2 5 9 3 35 2 3 35035 1130454496528082960384h911 3788115950202254344192h731 − + 394189048955 8277970028055 1193251487848937005056h713 6057099264h71111 − + 2759323342685 731

39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.79 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

226872518626348089344h551 3233842372976071696384h533 − 1655594005611 74501730252495 23750770688h53111 2464531349504h51113 72970403840h51311 + − + 731 47515 731 2063597568h5111111 1484783616h3311111 1201267933184h33311 + − − 17 17 47515 9620516088578048h91 ln(2) 157501683351552h73 ln(2) − − 780419 780419 42611245056h7111 ln(2) 199826866176h5113 ln(2) + + 29 493 125115826176h5131 ln(2) 64038371328h5311 ln(2) − − 493 493 2063597568h511111 ln(2) 20852637696h3331 ln(2) + + 17 493 1484783616h331111 ln(2) 3657433088 ln2 (2)s9a − + 216006656 ln2 (2)s9b − 17 7 2908880896 ln2 (2)s9c 295698432 ln2 (2)s9d 371195904 ln2 (2)s9e + − − 21 7 7 20605079846912 346075561984 90902528 − s9a ζ2 + s9b ζ2 + s9c ζ2 332605 47515 7 27721728 34799616 933351424 ln3 (2)s8a − s9d ζ2 − s9e ζ2 + 7 7 35 3 3 1012137984 ln (2)s8b 494927872 ln (2)s8c 444596224 ln3 (2)s8d − − + 7 7 7 346806855835578375136s8a ζ3 4200071353238881144832s8b ζ3 + − 1970945244775 8277970028055 23199744 13929023471616 5342354835072 − s8c ζ3 − s8d ζ3 + ln(2)s8a ζ2 7 332605 17255 545415168 34803127545856 3021041664 + ln(2)s8b ζ2 + s7a ζ22 − s7b ζ22 7 2328235 245 184549376 ln5 (2)s6 250412578603008 ln2 (2)s6 ζ3 166723584 ln4 (2)s7a + + − 5 332605 7 4 185597952 ln (2)s7b 365283511369728 ln(2)s7a ζ3 + + 7 1377935 4803526656 2 6867124224 2 + ln (2)s7a ζ2 − ln (2)s7b ζ2 49 49 7442343481344 2356066299904s6 ζ5 3369369600 3 − ln(2)s6 ζ22 − + ln (2)s6 ζ2 120785 47515 49     61328336277504s6 ζ2 ζ3 1 1 + − 1073741824Li11 − 50331648Li9 ζ2 332605 2 2       20447232 1 1 2 1 − 16777216Li8 ζ3 − Li7 ζ2 − 4718592Li6 ζ5 2 5 2 2 −

79

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.80 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

80

    119537664 1333788672 1 1 Li6 ζ2 ζ3 + Li6 ln2 (2)ζ3 7 2 7 2     27814451740672 1 2 2522873856 1 + Li5 ζ3 − Li5 ln2 (2)ζ22 332605 2 35 2     1 3 889192448 1 − 7274496Li5 ζ + Li5 ln3 (2)ζ3 2 2 7 2     444596224 31735658955776 1 1 3 2 − Li4 ln (2)ζ2 − Li4 ζ2 ζ5 35 2 332605 2     1478492160 1 1 + Li4 ln2 (2)ζ2 ζ3 − 1019904Li4 ζ7 7 2 2     9044262912 18894660870144 1 1 2 2 − Li4 ln (2)ζ5 + Li4 ζ ζ3 7 2 1663025 2 2 +

11

9

4194304 ln (2) 8781824 ln (2)ζ2 250216448 8 + − ln (2)ζ3 155925 2835 315 57638912 7 10204172288 6 699662336 6 + ln (2)ζ22 − ln (2)ζ5 + ln (2)ζ2 ζ3 1575 105 45 85119722061824 ln5 (2)ζ32 201367552 5 5347831296 4 − − ln (2)ζ23 + ln (2)ζ7 4989075 15 7 602089461569408 ln4 (2)ζ2 ζ5 53577818647552 ln4 (2)ζ22 ζ3 + − 997815 4989075 3 376156375212032 ln (2)ζ3 ζ5 1403329698221056 ln3 (2)ζ2 ζ32 + − 332605 6984705 224050581568 2 301110728704 3 − ln (2)ζ9 − ln (2)ζ24 63 875 2527896877478656 ln2 (2)ζ2 ζ7 1433322075232256 ln2 (2)ζ23 ζ3 + + 2328235 11641175 100081795637701376 ln2 (2)ζ22 ζ5 + 2706974720 ln2 (2)ζ3 ζ5 + 67518815 2 3 13144557640192 ln (2)ζ3 250218587846220419425554176 ln(2)ζ3 ζ7 + − 66521 54288110011131555 2 268488970046321792 ln(2)ζ5 1370557129830144 ln(2)ζ2 ζ3 ζ5 + + 75216245 567385 2847832763676416 ln(2)ζ22 ζ32 16178324992 + − ln(2)ζ23 ζ3 3971695 49 44020721797951110997888 ln(2)ζ25 276298183385563985523133ζ11 − + 29392628139875 24833910084165 2551544671187332995658ζ2 ζ9 6411687453232793159672ζ22 ζ7 − + 14900346050499 17042879469525 2 3958383139188142278976ζ3 ζ5 42386525725006839853952ζ23 ζ5 + − 2759323342685 51128638408575 +

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.81 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

81

1629519807808ζ2 ζ33 6271064327915000770722032ζ24 ζ3 − 66521 7243223774548125 12025645110722560h91 196877104189440h73 53264056320h7111 + + − 780419 780419 29 80047964160h5311 249783582720h5113 156394782720h5131 − + + 493 493 493 2579496960h511111 1855979520h331111 26065797120h3331 − + − 17 17 493 9143582720 ln(2)s9a 7272202240 ln(2)s9c + − 540016640 ln(2)s9b − 7 21 739246080 ln(2)s9d 927989760 ln(2)s9e + + − 132787200 ln2 (2)s8a 7 7 5533859840 ln2 (2)s8b 927989760 ln2 (2)s8c 833617920 ln2 (2)s8d − + + 7 7 7 86999040 78151680 1346195765472s8a ζ2 844738560 − s8b ζ2 − s8c ζ2 − s8d ζ2 − 3451 7 7 7 277872640 ln3 (2)s7a 309329920 ln3 (2)s7b 1324206735360s7a ζ3 + − − 7 7 3451 28999680 12008816640 17167810560 + s7b ζ3 − ln(2)s7a ζ2 + ln(2)s7b ζ2 7 49 49 329712525312 2 2275000320 2 + s6 ζ2 − 62914560 ln4 (2)s6 − ln (2)s6 ζ2 3451 49     12780011520 1 1 − ln(2)s6 ζ3 − 1342177280Li10 − 62914560Li8 ζ2 7 2 2       3334471680 1 1 1 2 − 20971520Li7 ζ3 − Li6 ln(2)ζ3 − 5111808Li6 ζ 2 7 2 2 2       1261436928 149422080 1 1 1 + Li5 ln(2)ζ22 + Li5 ζ2 ζ3 − 5898240Li5 ζ5 7 2 7 2 2     1667235840 166723584 1 1 2 − Li5 ln (2)ζ3 + Li4 ln2 (2)ζ22 7 2 7 2       1 2 20152320 1 3 3696230400 1 − 163840Li4 ζ3 − Li4 ζ2 − Li4 ln(2)ζ2 ζ3 2 7 2 7 2 −

1048576 ln10 (2) 2195456 8 83623936 7 − ln (2)ζ2 + ln (2)ζ3 2835 63 63 64495616 6 1454422016 5 691142656 5 − ln (2)ζ22 + ln (2)ζ5 − ln (2)ζ2 ζ3 315 7 21 83660800 4 594440192 4 11937597440 3 + ln (2)ζ32 + ln (2)ζ23 − ln (2)ζ7 3 21 7 223805440 3 21299686400 2 − 1453424640 ln3 (2)ζ2 ζ5 + ln (2)ζ22 ζ3 − ln (2)ζ2 ζ32 3 49 42851025408 2 560126453920 125444307840 + ln (2)ζ24 + ln(2)ζ9 − ln(2)ζ2 ζ7 49 63 49 −

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.82 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

82

83446780941824 ln(2)ζ22 ζ5 3008111360 10803818496 + ln(2)ζ33 + ln(2)ζ2 ζ32 24157 7 49 27208138994797616ζ3 ζ7 116846231822496ζ52 10887196496352ζ2 ζ3 ζ5 − − − 5462933 26911 3451 2 2 5 14304036293152ζ2 ζ3 19303408822744671456ζ2 7314866176s9a − + − 17255 10516146025 7 5817761792s9c 591396864s9d 742391808s9e − − + 432013312s9b + 21 7 7 9272057856 ln(2)s8a 11635523584 ln(2)s8b 9169403904 + + s7a ζ2 + 35 7 49 13247053824 11767775232 123731968 ln3 (2)s6 − s7b ζ2 − ln(2)s6 ζ2 − 49 49 7     1 1 − 50331648Li7 ζ2 + 1457258496s6 ζ3 − 1073741824Li9 2 2       2550136832 5188878336 1 1 2 18121555968 1 + Li6 ζ3 − Li5 ζ2 − Li4 ζ5 7 2 35 2 7 2   2951479296 8388608 ln9 (2) 70254592 7 1 + Li4 ζ2 ζ3 + + ln (2)ζ2 7 2 2835 315 159383552 6 136773632 5 755064832 4 + ln (2)ζ3 − ln (2)ζ22 − ln (2)ζ5 315 525 7 270106624 4 34799616 3 213254144 3 + ln (2)ζ2 ζ3 − ln (2)ζ32 − ln (2)ζ23 21 7 15 7257718784 2 6555205632 2 5997789184 2 + ln (2)ζ7 + ln (2)ζ2 ζ5 − ln (2)ζ22 ζ3 7 7 35 4539799621632 ln(2)ζ24 14362599424 448104543296 − − ln(2)ζ3 ζ5 − ζ9 6125 7 63 104867942736 653737047104 2 1034493440 3 + ζ2 ζ7 + ζ 2 ζ5 − ζ3 49 245 3 67009518592 3 1041378048s8a + ζ2 ζ3 − − 1351581696s8b − 139198464s8c 245 5   1 2 − 125042688s8d + 69599232 ln (2)s6 + 369819648s6 ζ2 − 704643072Li8 2       138461184 1 1 1 2 − 33030144Li6 ζ2 + 239075328Li5 ζ3 − Li4 ζ 2 2 5 2 2 −

262144 ln8 (2) 15368192 6 9961472 5 18774016 4 − − ln (2)ζ2 − ln (2)ζ3 + ln (2)ζ22 15 15 5 5 22499328 2 + 12353536 ln3 (2)ζ2 ζ3 + 19574784 ln2 (2)ζ32 + ln (2)ζ23 5 791703552 + 410821632 ln(2)ζ2 ζ5 + ln(2)ζ22 ζ3 − 151892736ζ3 ζ5 5 46498767216 4 515801088s7a 574193664s7b + 159311296ζ2 ζ32 + ζ2 − + 875 7 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.83 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

    515801088 ln(2)s6 1 1 − 415236096Li7 − 19464192Li5 ζ2 7 2 2   2883584 ln7 (2) 18112512 5 1 − 6488064Li4 ζ3 + + ln (2)ζ2 − 270336 ln4 (2)ζ3 2 35 5 57851904 3 − ln (2)ζ22 − 150847488 ln(2)2 ζ5 + 15003648 ln2 (2)ζ2 ζ3 5 644751360 1434064896 5333463504 1590681888 − ln(2)ζ32 − ln(2)ζ23 + ζ7 − ζ2 ζ5 7 35 7 7     127947168 2 1 1 − ζ2 ζ3 − 86114304s6 − 229638144Li6 − 10764288Li4 ζ2 7 2 2 +

4784128 ln6 (2) 61079552 3 95981568 2 − − 10016768 ln4 (2)ζ2 + ln (2)ζ2 + ln (2)ζ22 15 3 5   105679040 2 1096650368 3 1 − 14800896 ln(2)ζ2 ζ3 + ζ3 + ζ2 − 122159104Li5 3 35 2 5

15269888 ln (2) 87801856 − ln(2)ζ22 + 92863200ζ5 + 3547600ζ2 ζ3 15 5   55731628 2 7929856 ln4 (2) 1 − 63438848Li4 − − 31719424 ln2 (2)ζ2 + ζ2 2 3 5 − 23863952ζ3 + 48685056 ln(2)ζ2 − 17459170ζ2

  22610657280 1 + (A.58) Li4 ζ5 ln(2) 7 2

1 7 253 2501 59437 128ζ2 2831381 1792 I11 = 3 + 2 + + +ε − + ε2 − ζ3 ε 2ε 36ε 216 1296 3 7776 9

  4544 8192 1024 4 1 3 117529021 + 512 ln(2)ζ2 − ζ2 + ε − Li4 − ln (2) 9 46656 3 2 9

11008 2 63616 18176 99680 7168 2 − ln (2)ζ2 + ζ − ζ3 + ln(2)ζ2 − ζ2 3 15 2 27 3 27   4096 5 28672 3 87296 1 4 4081770917 − 32768Li5 + ζ5 + ln (2) + ln (2)ζ2 +ε 279936 2 3 15 3   44032 3584 290816 36352 4 1 − ln(2)ζ22 + ζ 2 ζ3 − Li4 − ln (2) 5 3 9 2 27

254464 2 390784 2 1395520 398720 1750448 − ln (2)ζ2 + ζ2 − ζ3 + ln(2)ζ2 − ζ2 9 45 81 9 81   633344 2 8192 6 1 5 125873914573 − 180224s6 − 393216Li6 − ln (2) + ζ3 +ε 1679616 2 15 9 +

46 47

83

− 28672 ln4 (2)ζ2 +

264192 2 745472 3 ln (2)ζ22 − 14336 ln(2)ζ2 ζ3 + ζ2 5 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.84 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

84

  1163264 145408 5 1017856 3 3099008 1 Li5 + ζ5 + ln (2) + ln (2)ζ2 3 2 9 45 9   1563136 127232 6379520 797440 4 1 − ln(2)ζ22 + ζ2 ζ 3 − Li4 − ln (2) 15 9 27 2 81 5582080 2 1714496 2 24506272 7001792 − ln (2)ζ2 + ζ2 − ζ3 + ln(2)ζ2 27 243 27

27 27091736 25493504 3593750577461 19922944 − ζ2 + ε 6 − s7a + s7b 243 10077696 21 21   72259840 32768 7 19922944 1 − 4718592Li7 + ln(2)s6 + ζ7 + ln (2) 2 21 7 35 344064 5 1056768 3 + ln (2)ζ2 − ln (2)ζ22 + 86016 ln2 (2)ζ2 ζ3 − 2095104 ln2 (2)ζ5 5 5 5758976 24903680 7158784 2 76080640 − ln(2)ζ23 − ln(2)ζ32 − ζ 2 ζ3 − ζ2 ζ5 15 21  21 21 6397952 1017856 4 290816 6 1 − s6 − 4653056Li6 − ln (2) − ln (2)ζ2 3 2 45 3 3126272 2 508928 26464256 3 22483712 2 + ln (2)ζ22 − ln(2)ζ2 ζ3 + ζ2 + ζ3 5 3 45 27   25518080 637952 5 22328320 3 67981760 1 − Li5 + ζ5 + ln (2) + ln (2)ζ2 9 2 27 27 27   6857984 2791040 112028672 14003584 4 1 − ln(2)ζ22 + ζ 2 ζ3 − Li4 − ln (2) 9 27 81 2 243 98025088 2 150538528 2 108366944 379284304 − ln (2)ζ2 + ζ2 + ln(2)ζ2 − ζ3 81 405 81 729

387541868 897384448 97480790072029 2183312384 − ζ2 + ε 7 − s8a − s8b 729 60466176 105 7 −

101974016 79691776 50987008 2 250216448 − s8c − s8d + ln (2)s6 + ζ2 s6 7 7  7     7 159383552 79691776 2 1 1 1 − 56623104Li8 + ζ3 Li5 − ζ2 Li4 2 7 2 35 2 49152 8 688128 6 19922944 5 56614912 4 − ln (2) − ln (2)ζ2 − ln (2)ζ3 + ln (2)ζ22 35 5 105 105 32620544 3 4751360 2 14340096 2 + ln (2)ζ2 ζ3 − ln (2)ζ23 + ln (2)ζ32 21 7 7 572674048 305018880 378638336 + ln(2)ζ22 ζ3 + ln(2)ζ2 ζ5 − ζ3 ζ5 35 7 21 22032280576 4 331879424 707264512 905019392 + ζ2 + ζ2 ζ32 − s7a + s7b 6125 21 63   63 707264512 1163264 7 2565224320 1 + ln(2)s6 − 55836672Li7 + ζ7 + ln (2) 63 2 21 105 4071424 5 12505088 3 + ln (2)ζ2 − ln (2)ζ22 − 24792064 ln2 (2)ζ5 5 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

[m1+; v1.304; Prn:6/09/2019; 12:57] P.85 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1

3 4 5 6

− −

7 8

−

9 10 11 12 13

+ +

14 15

−

16 17 18

+

19 20

−

21 22

−

23 24 25 26 27

+ −

28 29

+

30 31 32 33 34

+ +

35 36

+

37 38

−

39 40

+

41 42 43 44 45 46 47

204443648 884080640 ln(2)ζ23 − ln(2)ζ32 45 63   2700862720 254136832 2 140349440 102072320 1 ζ 2 ζ5 − ζ 2 ζ3 − s6 − Li6 63 63 9 3 2 1275904 6 22328320 4 13715968 2 ln (2) − ln (2)ζ2 + ln (2)ζ22 27 9 3   11164160 493216640 2 116107264 3 448114688 1 ln(2)ζ2 ζ3 + ζ3 + ζ2 − Li5 9 81 27 27 2 56014336 5 392100352 3 602154112 1193805536 ln (2) + ln (2)ζ2 − ln(2)ζ22 + ζ5 405 81 135 81   49012544 1733871104 216733888 4 1 ζ2 ζ3 − Li4 − ln (2) 81 243 2 729 1517137216 2 2329889296 2 5425586152 ln (2)ζ2 + ζ2 − ζ3 243 1215 2187

1550167472 5263826150 2553476823634373 ln(2)ζ2 − ζ2 + ε 8 243 2187 362797056

+ 1017856 ln2 (2)ζ2 ζ3 −

2

− + −

85

16011624448 1157496832 13187219456 344457216 s9a + s9b + s9c − s9d 21 3 63 7 611844096 7005710336 26374438912 s9e + ln(2)s8a + ln(2)s8b 7 35 21 5416157184 11319115776 s7a ζ2 − s7b ζ2 + 1117716480s6 ζ3 49 49   101974016 7557611520 1 3 s6 ln (2) − s6 ln(2)ζ2 − 679477248Li9 7 49 2     4015521792 1912602624 1 1 2 Li6 ζ3 − Li5 ζ 7 2 35 2 2     1722286080 13673512960 380773366144 1 1 Li4 ζ2 ζ3 − Li4 ζ5 − ζ9 7 2 7 2 63 65536 9 1179648 7 39845888 6 98992128 5 ln (2) + ln (2)ζ2 + ln (2)ζ3 − ln (2)ζ22 35 5 105 175 39141376 4 1709189120 4 249135104 3 ln (2)ζ2 ζ3 − ln (2)ζ5 − ln (2)ζ23 7 21 35 28680192 3 4011491328 2 4822073344 2 ln (2)ζ32 − ln (2)ζ22 ζ3 + ln (2)ζ2 ζ5 7 35 7 15886533632 2 3435955044352 6177746944 ln (2)ζ7 − ln(2)ζ24 + ln(2)ζ2 ζ32 21 6125 49 32555948032 61344280576 3 7109786624 3 ln(2)ζ3 ζ5 + ζ2 ζ3 − ζ3 21 245 27 1581235817984 2 78334057728 77507589632 ζ 2 ζ5 + ζ2 ζ7 − s8a 735 49 315   31857147904 3620077568 2829058048 1 s8b − s8c − s8d − 670040064Li8 21 21 21 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:6/09/2019; 12:57] P.86 (1-97)

J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

86

  1810038784 8882683904 5658116096 1 s6 ln2 (2) + s6 ζ2 + Li5 ζ3 21 21 21 2   2829058048 392100352 4 1 2 581632 8 − Li4 ζ2 − ln (2) − ln (2)ζ2 105 2 35 27 8142848 6 707264512 5 2009829376 4 − ln (2)ζ2 − ln (2)ζ3 + ln (2)ζ22 5 315 315 1158029312 3 168673280 2 169691136 2 + ln (2)ζ2 ζ3 − ln (2)ζ23 + ln (2)ζ32 63 21 7 20329928704 3609390080 782145960448 4 + ln(2)ζ22 ζ3 + ln(2)ζ2 ζ5 + ζ2 105 7 18375 11781719552 13441660928 2216427520 2836152320 + ζ2 ζ32 − ζ3 ζ5 − s7a + s7b 63 63 27 27   8038907200 729088 7 2216427520 1 − 408289280Li7 + ln(2)s6 + ζ7 + ln (2) 2 27 9 9 54863872 3 22328320 2 17862656 5 ln (2)ζ2 − ln (2)ζ22 + ln (2)ζ2 ζ3 + 3 3 3 543854080 2 896960512 2770534400 − ln (2)ζ5 − ln(2)ζ23 − ln(2)ζ32 3 27 27   796414720 2 8463971200 1792458752 2464630784 1 − ζ 2 ζ3 − ζ 2 ζ5 − Li6 − s6 27 27 9 2 27 112028672 6 1204308224 2 196050176 − ln (2) + ln (2)ζ22 − ln(2)ζ2 ζ3 405 45 27   10194609152 3 8661216704 2 6935484416 866935552 5 1 + ζ2 + ζ3 − Li5 + ln (2) 405 243 81 2 1215 9319557184 758568608 6068548864 3 ln (2)ζ2 − ln(2)ζ22 + ζ 2 ζ3 + 243 405 243   18476563952 24802679552 3100334944 4 1 + ζ5 − Li4 − ln (2) 243 729 2 2187 33328600648 2 73693566100 21702344608 2 ln (2)ζ2 + ζ2 − ζ3 − 729 3645 6561

21055304600 69020223371 65282718863433709 + ln(2)ζ2 − ζ2 + ε 9 729 6561 2176782336 +

14684258304 19013284397056 24574427136 h331111 − h3331 − h511111 17 54723 17 407096785895424 38026568794112 128886518054912 − h5113 + h5131 + h5311 127687 18241 127687 77744179249152 982846081352335360 498389163900928000 − h7111 + h73 + h91 7511 606385563 5462933       1 1 1 − 8153726976Li10 − 8040480768Li9 − 4899471360Li8 2 2 2       7169835008 27741937664 99210718208 1 1 1 − Li7 − Li6 − Li5 3 2 27 2 243 2

+

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1

−

2 3 4 5 6 7 8

+ − −

9 10

−

11 12

−

13 14

−

15 16 17 18 19 20 21 22 23

− − + +

24 25

+

26 27

−

28 29

+

30 31 32 33 34 35 36 37 38

− + + +

39 40

−

41 42

+

43 44 45

+

46 47

−

87

  336884873600 2326528 9 364544 8 393216 10 1 Li4 − ln (2) + ln (2) − ln (2) 2187 2 175 105 3 1733871104 6 12401339776 5 64016384 7 ln (2) − ln (2) + ln (2) 135 1215 3645 1912602624 42110609200 4 ln (2) − 467140608s6 ln4 (2) + s7a ln3 (2) 6561 7 2447376384 49804746752 38921961472 s7a − s7b ln3 (2) + s7b 81 7 81 3620077568 5672304640 38921961472 s6 ln3 (2) + s6 ln2 (2) + s6 ln(2) 21 9 81 4524146688 248702716928 38145164288 s6 − s8a ln2 (2) + s8a ln(2) 81 5 105 99834019840 936292581376 48578700544 s8a − s8b + s8b ln(2) 27 9 63 11344609280 7342129152 40886075392 s8b ln2 (2) − s8c + s8c ln2 (2) 7 9 7 5737807872 568412667904 8865710080 s8d + s8d ln2 (2) − s9a 9 7 63 64046497792 41091137536 s9a ln(2) + s9b − 4629987328s9b ln(2) 7 9 52748877824 4076077056 468146290688 s9c − s9c ln(2) − s9d 189 21 7 4133486592 7240155136 7342129152 s9d ln(2) − s9e + s9e ln(2) 7 7 7 276080893484 294774264400 2 1766128887523 ζ2 + ln(2)ζ2 − ln (2)ζ2 39366 2187 2187 86809378432 3 6068548864 4 1568401408 5 ln (2)ζ2 − ln (2)ζ2 + ln (2)ζ2 729 81 45 35725312 6 13959168 7 1769472 8 ln (2)ζ2 + ln (2)ζ2 − ln (2)ζ2 3 5 5 27836579840 89431736320 35417751552 s6 ζ2 − s6 ln(2)ζ2 − s6 ln2 (2)ζ2 9 49 49 64091193344 64993886208 133942870016 s7a ζ2 − s7a ln(2)ζ2 − s7b ζ2 49 49 49 135829389312 42561416368128 90537809780 2 ln(2)s7b ζ2 − s8a ζ2 + ζ2 49 18241 2187     47517007872 1 2 1773142016 1 2 133314402592 Li5 ζ − Li4 ζ − ln(2)ζ22 35 2 2 9 2 2 1215   48186261504 18639114368 2 1 Li5 ln(2)ζ22 + ln (2)ζ22 35 2 135   5737807872 4817232896 3 1259681792 4 1 Li4 ln2 (2)ζ22 − ln (2)ζ22 + ln (2)ζ22 35 2 45 27 1171406848 5 279773184 6 50887098236928 2 ln (2)ζ22 − ln (2)ζ22 + s6 ζ2 175 175 127687

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

88

157782270464 3 78756156416 528588800 2 ζ2 − ln(2)ζ23 − ln (2)ζ23 1215 405 9 8844296192 3 7067271168 4 490218242816 4 − ln (2)ζ23 + ln (2)ζ23 + ζ2 105 35 1575 121976404074496 39645136330752 2 − ln(2)ζ24 + ln (2)ζ24 18375 6125   262144173134771720192 5 966283127194 17731420160 1 + ζ2 − ζ3 + Li5 ζ3 23822289975 19683 9 2     22632464384 22951231488 1 1 + Li6 ζ3 − Li6 ln(2)ζ3 7 2 7 2   11475615744 443285504 5 1414529024 6 1 − Li5 ln2 (2)ζ3 − ln (2)ζ3 + ln (2)ζ3 7 2 27 315 318767104 7 + ln (2)ζ3 + 13226311680s6 ζ3 − 13412597760s6 ln(2)ζ3 35   305322589421568 10851172304 20380385280 1 − s7a ζ3 + ζ2 ζ3 + Li4 ζ2 ζ3 127687 729 7 2   3034274432 20667432960 1 − ln(2)ζ2 ζ3 − Li4 ln(2)ζ2 ζ3 81 7 2 392100352 2 3629035520 3 1389518848 4 + ln (2)ζ2 ζ3 + ln (2)ζ2 ζ3 + ln (2)ζ2 ζ3 9 27 21 6783172608 5 13985579392 2 12741997568 − ln (2)ζ2 ζ3 − ζ2 ζ3 + ln(2)ζ22 ζ3 35 81 9 47469314048 2 2068316160 3 2177721960448 3 − ln (2)ζ22 ζ3 + ln (2)ζ22 ζ3 + ζ 2 ζ3 35 7 735 736131366912 134049909728 2 48652451840 − ln(2)ζ23 ζ3 + ζ3 − ln(2)ζ32 245 729 81 339382272 3 + 177259520 ln2 (2)ζ32 − ln (2)ζ32 + 182403072 ln4 (2)ζ32 7 36921585920 219310016512 129888706560 2 + ζ2 ζ32 + ln(2)ζ2 ζ32 − ln (2)ζ2 ζ32 27 147 49 9358852028348416 2 2 252397425152 3 28439146496 − ζ 2 ζ3 − ζ + ln(2)ζ33 1915305 81  3 9 264303553976 485409710080 1 + ζ5 − Li4 ζ5 729 21 2   164082155520 9550444288 2 1 + Li4 ln(2)ζ5 − ln (2)ζ5 7 2 9 60676213760 4 53190565888 5 148633040320 − ln (2)ζ5 + ln (2)ζ5 − ζ2 ζ5 63 35 81 11311116800 171183603712 2 + ln(2)ζ2 ζ5 + ln (2)ζ2 ζ5 − 10360356864 ln3 (2)ζ2 ζ5 3 21 56133871538432 2 118824071572576256 42123514880 + ζ 2 ζ5 − ln(2)ζ22 ζ5 − ζ3 ζ5 2205 4469045 27 1155736155136 − ln(2)ζ3 ζ5 + 19901587456 ln2 (2)ζ3 ζ5 63 +

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14

89

2399840466176000 186051331138705408 2 141168629920 ζ 2 ζ3 ζ5 − ζ5 + ζ7 127687 6969949 27 563971943936 2 84356968448 3 926953016448 + ln (2)ζ7 − ln (2)ζ7 + ζ2 ζ7 63 7 49 940008692736 18265279696831990784 − ln(2)ζ2 ζ7 − ζ3 ζ7 49 606385563

13517454498112 1523093464576 − ζ9 + ln(2)ζ9 (A.59) 189 21  2 23 35 275 7  14917 136π 4 I12 = 3 + 2 + + + ε − 81 + 128ζ3 + ε 2 − − 2ε 12 24 48 45 ε 3ε

  32 ln4 (2) 1 32 48005 68π 4 + 256Li4 + 280ζ3 + ε 3 − − − π 2 ln2 (2) + 3 3 2 32 3 −

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

64 64 ln5 (2) 272π 4 ln(2) − 80π 2 ln2 (2) + π 2 ln3 (2) + 80 ln4 (2) − + 15 3 5

    4060ζ3 1 1 1108525 986π 4 + 1920Li4 + 1536Li5 + − 1240ζ5 + ε 4 − − 2 2 3 192 9 1160 2 2 272 4 2 32π 6 + 136π 4 ln(2) − π ln (2) − π ln (2) + 160π 2 ln3 (2) − 5 3 5   1160 ln4 (2) 64 ln6 (2) 1 2 4 5 + − 32π ln (2) − 96 ln (2) + + 9280Li4 3 5 2

    2 4880ζ3 1 1 + 9216Li6 + 3840s6 + 5390ζ3 − − 9300ζ5 + 11520Li5 2 2 3 1972π 4 ln(2) 3824π 6 ln(2) 2570029 1309π 4 5 − − 48π 6 + + +ε − 128 3 3 135 2320 2 3 544 4 3 π ln (2) + π ln (2) + 1540 ln4 (2) 3 5 192 2 5 384 ln7 (2) π ln (2) + 96 ln6 (2) − − 240π 2 ln4 (2) − 464 ln5 (2) + 5 35         1 1 1 1 + 36960Li4 + 55680Li5 + 69120Li6 + 55296Li7 + 28800s6 2 2 2 2

− 1540π 2 ln2 (2) − 408π 4 ln2 (2) +

38 39 40 41 42 43 44 45

−

74240 74240s7a 87040s7b 57967ζ3 ln(2)s6 + − + + 7 7 7 3

720π 4 ζ

3

− 12200ζ32 7

92800 130360π 2 ζ5 772868ζ7 2 2 + ln(2)ζ3 − 44950ζ5 + + 22320 ln (2)ζ5 − 7 21 7 593716π 8 50743957 140777π 4 − − 232π 6 − + 2618π 4 ln(2) + ε6 − 768 90 33075

46 47

2 3 4 5 6 7 8 9 10 11 12 13 14 15

15 16

1

+

1912π 6 ln(2) 9

−

3328 6 2 16562 2 2 π ln (2) − 1972π 4 ln2 (2) − π ln (2) 3 315

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

90

16562 ln4 (2) 46768 4 4 − 1160π 2 ln4 (2) − π ln (2) 3 315 192 2 6 576 ln7 (2) − 1848 ln5 (2) + 288π 2 ln5 (2) + 464 ln6 (2) − π ln (2) − 5 7 1     4 8 7424π Li4 2 1 1 288 ln (2) + 132496Li4 + + 221760Li5 + 35 2 21 2       229120 2 1 1 1 + 334080Li6 + 414720Li7 + 331776Li8 + 139200s6 − π s6 2 2 2 7 556800 261120 2 556800s7a 652800s7b 767504s8a − ln(2)s6 − ln (2)s6 + − + 7 7 7 7 7 4862976s8b 522240s8c 445440s8d 130095ζ3 5400π 4 ζ3 + + + + + 7 7 7 2 7   51952 4 37120 2 3 7424 5 890880 1 − π ln(2)ζ3 − π ln (2)ζ3 + ln (2)ζ3 − Li5 ζ3 21 21 7 7 2 + 3080π 2 ln3 (2) + 816π 4 ln3 (2) +

176900ζ32 96080 2 2 696000 73440 2 − − π ζ3 + ln(2)ζ32 − ln (2)ζ32 − 179025ζ5 3 7 7 7 325900π 2 ζ5 261120 2 636208ζ3 ζ5 + − π ln(2)ζ5 + 167400 ln2 (2)ζ5 + 7 7 7

4 5796510ζ7 296858π 8 107716245 21063π − + ε7 − − − 924π 6 − 7 512 4 2205 +

140777π 4 ln(2) 15

+

27724π 6 ln(2) 27

+

38646953π 8 ln(2) 33075

− 18585π 2 ln2 (2)

1664 6 2 33124 2 3 π ln (2) + π ln (2) + 3944π 4 ln3 (2) 21 3 23384 4 4 43376 6 3 π ln (2) + 18585 ln4 (2) − 4620π 2 ln4 (2) − π ln (2) + 315 21 22112 4 5 33124 ln5 (2) + 1392π 2 ln5 (2) + π ln (2) + 1848 ln6 (2) − 288π 2 ln6 (2) − 5 175   2784 ln7 (2) 1152 2 7 432 ln8 (2) 192 ln9 (2) 1 − + π ln (2) + − + 446040Li4 7 35 7 35 2 1       4 4 18560π Li4 2 58368π 1 1 1 + 794976Li5 + Li5 + 1330560Li6 + 7 2 7 2 2       1 1 1 + 2004480Li7 + 2488320Li8 + 1990656Li9 + 554400s6 2 2 2 1718400 2 2691200 3878400 2 1958400 2 − π s6 − ln(2)s6 + π ln(2)s6 − ln (2)s6 7 7 49 7 261120 3 2691200s7a 2964480 2 3155200s7b 4830720π 2 s7b + ln (2)s6 + − π s7a − + 7 7 49 7 49 36472320s8b 23764480 5756280s8a 3787464 − ln(2)s8a + − ln(2)s8b + 7 7 7 7 − 7854π 4 ln2 (2) −

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••

1

+

2 3 4 5 6

+ −

7 8

+

9 10 11

−

12 13

+

14 15 16 17 18

+ −

19 20

+

21 22 23

−

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

I15 =

91

3916800s8c 3340800s8d 14821888s9a 11882240s9c + + − 951808s9b − 7 7 7 21 1105920s9d 1566720s9e 2526055ζ3 26100π 4 ζ3 18482398π 6 ζ3 + + + − 7 7 12 7 6615 67912 4 2 92800 2 3 19840 2 4 129880 4 π ln(2)ζ3 + π ln (2)ζ3 − π ln (2)ζ3 − π ln (2)ζ3 7 7 7 7     55680 5 7424 6 921600 2 6681600 1 1 ln (2)ζ3 − ln (2)ζ3 − π Li4 ζ3 − Li5 ζ3 7 7 7 2 7 2   5345280 720600 2 2 1 Li6 ζ3 − 3011328s6 ζ3 − 234850ζ32 − π ζ3 7 2 7 3364000 3448560 2 550800 2 73440 3 ln(2)ζ32 − π ln(2)ζ32 − ln (2)ζ32 + ln (2)ζ32 7 49 7 7 6408524ζ33 1283555ζ5 4725550π 2 ζ5 690281971π 4 ζ5 − + − 9 2 21 4410 2232728 1545608 4 1958400 2 π ln(2)ζ5 + 809100 ln2 (2)ζ5 − π 2 ln2 (2)ζ5 + ln (2)ζ5 7 7 7 1 37094592Li4 2 ζ5 4771560ζ3 ζ5 29334280 28016465ζ7 + + ln(2)ζ3 ζ5 − 7 7 7 7

70452569 2 14706092 2 216126121ζ9 π ζ7 − ln (2)ζ7 + (A.60) 98 7 14

1 7 1 959 25π 2 7π 2 5 2 + 2+ 75 + 8π − + + 4ζ3 + ε − + 3 2ε 4ε 24ε 16 6 32 12



16π 4 8ζ3 10493 56π 4 5 2 2 + + 14ζ3 + ε − + +π − + + 25ζ3 + 72ζ5 45 64 45 24 3



5 85175 20π 4 458π 6 959 28ζ3 3 2 2 +ε − + + +π − + − ζ3 + 16ζ3 + 252ζ5 128 9 945 48 3 2



610085 229π 6 10493 50ζ3 2 128ζ3 4 2 4 +ε − + +π − + + 48ζ5 + π − + 256 135 96 3 9 45

959 4087919 1145π 6 3337π 8 − ζ3 + 56ζ32 + 450ζ5 + 996ζ7 + ε 5 − + + 4 512 378 4725



32 85175 5 959 448ζ3 + π2 − − ζ3 + ζ32 + 168ζ5 + π 4 − + 192 3 3 45 45

10493 26332493 + − + 576ζ5 ζ3 + 100ζ32 − 45ζ5 + 3486ζ7 + ε 6 − 8 1024

3337π 8 10493 160ζ3 256ζ5 610085 959 + + π4 − + + + π2 − − ζ3 1350 90 9 5 384 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:114751 /FLA

92

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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J. Blümlein et al. / Nuclear Physics B ••• (••••) ••••••





112 2 85175 229 3664ζ3 6 + ζ + 300ζ5 + 664ζ7 + π − + − − 2016ζ5 ζ3 3 3 756 945 16

128 8631 40240 165503975 3337π 8 − 10ζ32 + ζ33 − ζ5 + 6225ζ7 + ζ9 + ε 7 − + 3 2 3 2048 756

512 2 896 17035 16 70336π 10 + π4 − − ζ3 + ζ + ζ5 + 66825 36 9 45 3 5

200 10493 4087919 + − + 384ζ5 ζ3 + ζ 2 − 30ζ5 + 2324ζ7 + π2 − 768 12 3 3



610085 31373 1832ζ3 6 + + − + 3600ζ5 + 7968ζ7 ζ3 − 959ζ32 +π − 1080 135 32

448 3 94437 1245 140840 2 + ζ − ζ5 + 5184ζ5 − ζ7 + ζ9 3 3 4 2 3

1023933365 246176π 10 343271 4580ζ3 7328ζ5 8 6 + +π − + + +ε − 4096 66825 2160 189 105

1792 2 10624 122017 7672 + π4 − − ζ3 + ζ + 320ζ5 + ζ7 72 45 45 3 15

20 256 3 85175 26332493 2 + − + 1344ζ5 ζ3 − ζ32 + ζ − 2877ζ5 +π − 1536 24 3 9 3



80480 4087919 3337 26696ζ3 8 + 4150ζ7 + ζ9 + π − + + − − 360ζ5 9 7560 4725 64

10493 800 3 766575 + 27888ζ7 ζ3 + − + 2304ζ5 ζ32 + ζ − ζ5 + 18144ζ52 2 3 3 8

238791 251500 − ζ7 + ζ9 + 182412ζ11 (A.61) 4 3

1 35 559 π 2 2737 35π 2 16 I16 = − 2 − − − +ε − − ζ3 36ε 216 3 1296 18 3 6ε

552041 559π 2 37π 4 280 25027345 259π 4 − − − ζ3 + ε 3 − + ε2 7776 108 45 9 46656 54



2236 2737 32ζ3 855963737 20683π 4 − − ζ3 − 208ζ5 + ε 4 − + π2 648 3 27 279936 1620



256 2 3640 560ζ3 5474 2318π 6 2 552041 +π − + ζ3 − ζ − ζ5 − 945 3888 9 81 3 3 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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93



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42



4472ζ3 25728647329 1159π 6 2 25027345 +ε − +π − − 416ζ5 1679616 81 23328 27



4480 2 29068 1184ζ3 552041 4 101269 +π − + ζ3 − ζ − ζ5 − 6168ζ7 9720 45 243 9 3 9 647881π 6 37189π 8 10948 6 718599153833 2 855963737 − − +π + ζ3 +ε 10077696 17010 4725 139968 81





512 2 7280 4144ζ3 25027345 4 20425517 − ζ − ζ5 + π − + − 6656ζ5 ζ3 3 3 3 58320 27 1458

35776 2 71162 37189π 8 7 19166358676465 − ζ3 + ζ5 − 35980ζ7 + ε − 27 27 60466176 810

185202353 165464ζ3 15392ζ5 25728647329 1104082 − − + π2 + ζ3 + π4 69984 405 15 839808 243

8960 2 58136 74176ζ3 6 453169 − ζ − ζ5 − 12336ζ7 + π − 9 3 9 14580 945

855963737 116480ζ5 87584 2 8192 3 7176533 + − ζ3 + ζ − ζ + ζ5 8748 3 81 3 9 3 81

287326 1629280 20788651π 8 8 495815946702713 − ζ7 − ζ9 + ε − 3 9 362797056 170100

405076ζ3 18944 2 53872 1738426π 10 4 31670658269 +π + − ζ − ζ5 − 66825 2099520 1215 45 3 9

71552 2 25027345 2 718599153833 +π + − 13312ζ5 ζ3 − ζ 5038848 729 27 3



142324 37088ζ3 25728647329 6 91402217 + ζ5 − 71960ζ7 + π − + 27 87480 81 52488

8832656 2 143360 3 325355485 930176ζ5 − 197376ζ7 ζ3 + ζ3 − ζ3 + ζ5 − 9 243 27 486

703409 28512400 2 − 129792ζ5 + ζ7 − ζ9 (A.62) 9 27 5

Appendix B. Supplementary material

47

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

44 45

45 46

2

43

43 44

1

Supplementary material related to this article can be found online at https://doi.org/10.1016/ j.nuclphysb.2019.114751.

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References

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