当前位置：文档之家› Some higher moments of deep inelastic structure functions at next-to-next-to leading order

Some higher moments of deep inelastic structure functions at next-to-next-to leading order

a r X i v :h e p -p h /0007294v 2 13 M a r 2001

TTP00-13

NIKHEF-2000-018

Some higher moments of deep inelastic structure functions at next-to-next-to leading order of perturbative QCD

A.R′e tey a and J.A.M.Vermaseren b

Institut f¨u r Theoretische Teilchenphysik,

Universit¨a t Karlsruhe,D-76128Karlsruhe,Germany

NIKHEF Theory Group

Kruislaan 409,1098SJ Amsterdam,The Netherlands

Abstract

We present the analytic next-to-next-to-leading QCD calculation of some higher moments of deep inelastic structure functions in the leading twist approximation.We give results for the moments N =1,3,5,7,9,11,13of the structure function F 3.Similarly we present the moments N =10,12for the ?avour singlet and N =12,14for the non-singlet structure functions F 2and F L .We have calculated both the three-loop anomalous dimensions of the corresponding operators and the three-loop coe?cient functions of the moments of these structure functions.

1Introduction

The determination of the next-to-next-to-leading(NNL)order QCD approximation for the struc-ture functions of deep inelastic scattering has become important for the understanding of pertur-bative QCD and necessary for an accurate comparison of perturbative QCD with the increasing precision of experiments.Such calculations however are rather complicated and hence a complete NNL result does not exist as of yet.The one-loop anomalous dimensions were calculated in[1,2]. In[3](see also the references therein)the complete one-loop coe?cient functions were obtained. Anomalous dimensions at2-loop order were obtained in[4,5,6,7,8,9,10,11,12].The2-loop coe?cient functions were calculated in[13,14,15,16,17,18,19,20],[21].

Analytical results of the3-loop anomalous dimensions and coe?cient functions of the moments of F2and F L are only known for the moments N=2,4,6,8in both the singlet and non-singlet case and additionally for N=10in the non-singlet case from[22]and[23].In addition the Gross-

Llevellyn Smith sum rule,which corresponds to the?rst moment of Fνp+ˉνp

3has been calculated

at this order[24].

For a complete reconstruction of the x-dependence of the structure functions via an inverse Mellin-transformation one would need the moments for all N(that is either all even or all odd integer values).Additionally one needs them both for F2and F3in order to untangle the various quark and gluon contributions.The determination of the NNL approximation for generic N is work in progress[25],but probably will not be?nished in the near future.

The available moments of F2have been used by a number of authors to make a reconstruction of the complete structure functions at NNL by a variety of means[23,26,27,28].Additionally they can be used to obtain a better value ofαS[29]It should be clear that it is important to have as large a number of moments as possible.First,these results can be immediately used to increase the precision of phenomenological investigations of deep inelastic scattering[30].Second, the moments will be a very important check for the new methods and programs needed for the determination of the3-loop results for arbitrary N.Unfortunately it is not very easy to increase the number of moments,because each new moment requires roughly?ve times the computer resources that its predecessor needs.With the advent of better computers this means that by now it has been possible to obtain two more moments for the singlet case and one additional moment for the non singlet F2case.This should allow for instance a somewhat better determination of

αS.More important however is the determination of the?rst seven odd moments of Fνp+ˉνp

3to

three loops.To this end we used the the same programs as in[23],state of the art computers and a new version of the symbolic manipulation program FORM[31]that supports now64-bit architectures and to some extend parallel computers(see also[32]).We could push the limit in these calculations to include two new moments(N=10,12)in the calculation of the?avour singlet structure functions F L and F2,and two new moments(N=12,14)for the?avour nonsinglet structure functions F L and F2.Additionally we have computed the moments N=3,5,7,9,11,13 of the structure function F3.We do not expect more moments to become available before the complete results for all N will be presented.

2The formalism

This calculation follows the one presented in[23](see also[21])in every detail,so we only will give a very short review on the methods used.

We need to calculate the hadronic part of the amplitude for unpolarized deep inelastic scattering

which is given by the hadronic tensor

W μν(x,Q 2)=

2x F L (x,Q 2)

?g μν?p μp ν

4x 2

q 2

p ·q

F 3(x,Q 2)

(1)

where the J μare either electromagnetic or weak hadronic currents and x =Q 2/(2p ·q )is the Bjorken scaling variable with 0

Using the dispersion relation technique one can relate the hadronic tensor to the following 4-point Green functions:

W μν(p,q )=

g ν1ν2?q ν1q ν2μ2

,a s )

? g ν1μ1g μ2ν2q 2

?g ν1μ1q ν2q μ2?g ν2μ2q ν1q μ1+g ν1ν2q μ1q μ2 C j 2,N (

Q 2μ

2,a s )

×q μ3...q μN O j,{μ1,...μN }(0)+higher twists ,j =α,ψ,G

(2)

Here we have introduced the notation a s =αs /(4π)=g 2/(4π)2and everything is assumed to be

renormalized.The sum over N runs over the standard set of the spin-N twist-2irreducible ?avour non-singlet quark operators and the singlet quark and gluon operators:

O α,{μ1,...,μN }=ˉψλ

αγ{μ1D μ2···D μN }ψ,α=1,2,...,(n 2f ?1)

O Ψ,{μ1,...,μN }=ˉψγ

{μ1D μ2···D μN }ψO G,{μ1,...,μN }=G {μμ1D μ2···D μN ?1G μN μ}

Application of this OPE to Eq.(1)leads to an expansion for the unphysical values x →∞.From

the proper analytical continuation to the physical region 0

M k,N ?2=

d xx

N ?2

F k (x,Q 2

i =α,ψ,G

C i

k,N (

Q 2

μ2

,a s )A i nucl ,N

(4)

with the spin averaged matrix elements

p,nucl |O

j,{μ1···μN }

|p,nucl =p

{μ1

···p

μN }

A j nucl ,N (

?μ2

+β(a s )

?μ2,a s

=0,i =2,3,L (6)

k =ψ,G

μ2

?a s

δjk ?

γjk N

C k i,N

2N !

(p ·q )2

p μp ν

P 2=?

3?2?

(p ·q )

2p μp ν+1

μ2,a s ,?)= C ψk,N (Q 2?)+C G

k,N (Q 2

?) A ψ,tree

quark ,N (?)

T gγgγ,s k,N (Q 2

μ2,a s ,?)Z ψG N (a s ,1μ2,a s ,?)Z GG

N (a s ,1μ2,a s ,?)=C ns

k,N (Q 2?

)A ns ,tree quark ,N (?)k =2,L

(11)

From these equations the coe?cient functions and from the Z ij

N the anomalous dimensions can be

calculated in the usual way.

It should be mentioned that in the Eqs.(11)on the left hand side after applying the projec-tors(10)we are left with only diagrams of the massless propagator type,a problem solved at 3-loop order long ago[35]and implemented in an e?cient way in the FORM package MINCER [36].On the right hand side only the tree level diagrams contributing to the Matrix elements survive.

It turns out that much computing time can be saved when calculating additionally Green functions with external ghosts to get rid of the unphysical polarization states of the external gluons instead of using the very complicated projection onto physical states.Also,from Eqs.(11)

one can determine the Z GG

N and Z Gψ

only to orderα2s.To obtain theα3s-contributions one can

calculate additionally Greens functions with external scalar?eldsφthat couple to gluons only at tree level.Altogether,to obtain the coe?cient functions and anomalous dimensions for the even moments with N=10,12of F2and F L the following diagrams had to be calculated(q=quark, g=gluon,γ=photon,h=ghost,φ=scalar?eld):

1-loop3-loops

1272

qφqφ1697

202

hγhγ53

12411

hφhφ111266

Total2310846

?μνρσγμγνγργσ

Projecting out the?avour non-singlet part and the corresponding Lorentz structure with:

P3=?i 1

p·q

one?nds products of metric tensors which have to be considered as D-dimensional objects.Since this de?nition ofγ5in D dimensions violates the axial Ward identity one needs to renormalize Aμwith a renormalization constant Z A and additionally apply a?nite renormalization with Z5,both of these constants are given to3-loop order in[24].Combining all this?nally leads to

Z A(a s,1

μ2

,a s,?)=C3,N(

)A ns N,tree(?)

Due to theγ5insertion at one of the vertices,some of the symmetries that were used to minimize the number of diagrams could not be applied in this case and to determine the T ns3,N1076(=1 +4+55+1016)diagrams had to be evaluated,which took about6weeks for the moments N=1,3,5,7,9,11,13.

5Results

Using the strategies sketched in the previous section we?nd the following results for the coe?cient functions and anomalous dimensions.Again following Ref.[23],we present the combined singlet and non-singlet results for F2and F L in terms of?avour factors which are de?ned in the following table for n f number of?avours:

fl11fl g2

non-singlet30-

singlet1 n f f=1e2f1n f( n f f=1e f)2

γψG

=?0.6666666667a s n f?7.543209877a2s n f

+a3s ?37.62337275n f+12.11248285n2f

γψG

=?0.3666666667a s n f+1.290703704a2s n f

+a3s 33.58149273n f+6.06027262n2f

γψG

=?0.2619047619a s n f+2.761104812a2s n f

+a3s 33.41602135n f+3.537682102n2f

γψG

=?0.2055555556a s n f+3.243957223a2s n f

+a3s 28.7612615n f+2.225433112n2f

=?0.1696969697a s n f+3.407168695a2s n f

γψG

+a3s 23.93704198n f+1.449828678n2f

=?0.1446886447a s n f+3.438705999a2s n f

γψG

+a3s 19.63230379n f+0.9524545446n2f

=?3.555555556a s+a2s(?48.32921811+5.135802469n f)

γGψ

+a3s ?859.4478372+175.649986n f+4.674897119n2f

=?0.9777777778a s+a2s(?16.1752428+0.6182716049n f)

γGψ

+a3s ?315.276255+39.82571027n f+1.801843621n2f

=?0.5587301587a s+a2s(?9.496317796+0.0888*******n f)

γGψ

+a3s ?188.9088124+19.67944546n f+1.087843741n2f =?0.3915343915a s+a2s(?6.757603506?0.07061952353n f)

γGψ

+a3s ?134.7055042+12.3754454n f+0.7536013741n2f =?0.3016835017a s+a2s(?5.297576945?0.1348941718n f)

γGψ

+a3s ?104.911278+8.796702078n f+0.5579674847n2f =?0.2455322455a s+a2s(?4.398625917?0.1639529655n f)

γGψ

+a3s ?86.18107998+6.735609285n f+0.4293379075n2f =0.6666666667a s n f+7.543209877a2s n f

γGG

+a3s 37.62337275n f?12.11248285n2f

=a2s(128.178?13.64948148n f)+a s(12.6+0.6666666667n f)

γGG

+a3s 2066.19278?401.3127939n f?10.43150645n2f

γGG

=a2s(183.0538144?20.46668466n f)+a s(17.78571429+0.6666666667n f) 6

+a3s 2987.042058?566.6373298n f?10.78060861n2f

=a2s(219.6240988?24.69926432n f)+a s(21.26666667+0.6666666667n f)γGG

+a3s 3609.35419?673.9430658n f?11.20133837n2f

=a2s(247.6655484?27.82178573n f)+a s(23.92337662+0.6666666667n f)γGG

+a3s 4089.236943?755.1340541n f?11.57068198n2f

=a2s(270.6428892?30.31377688n f)+a s(26.08168498+0.6666666667n f)γGG

+a3s 4483.563048?821.1236576n f?11.88665683n2f The corresponding coe?cient functions read:

Cψ2,2=1+0.4444444444a s+a2s(17.69376589?5.333333333n f?2.189300412?02n f) +a3s 442.7409693?165.1971095n f?24.09201335?11n f+6.030272415n2f

+?02 ?79.04486142n f+3.325504478n2f

Cψ2,4=1+6.066666667a s+a2s(142.3434719?16.98791358n f+0.4858308642?02n f) +a3s 4169.267888?901.2351626n f?18.21884618?11n f+23.35503924n2f

+?02 16.64834849n f?2.208630689n2f

Cψ2,6=1+11.17671958a s+a2s(302.398735?28.0130504n f+0.4868787285?02n f) +a3s 10069.63085?1816.322929n f?16.14271761?11n f+42.66273116n2f

+?02 24.11778813n f?1.525489143n2f

Cψ2,8=1+15.52989418a s+a2s(470.807419?37.9248228n f+0.3859585393?02n f) +a3s 17162.37245?2787.297692n f?15.09203827?11n f+61.91177997n2f

+?02 22.33201938n f?1.036308122n2f

Cψ2,10=1+19.30061568a s+a2s(639.210663?46.86131842n f+0.3045901308?02n f) +a3s 24953.13497?3770.10212n f?14.45874451?11n f+80.52097973n2f

+?02 19.53359559n f?0.7372434464n2f

Cψ2,12=1+22.62841097a s+a2s(804.5854321?54.99446579n f+0.2451231747?02n f) +a3s 33171.45501?4746.440949n f?14.03541028?11n f+98.3483124n2f

+?02 16.98652635n f?0.5471547625n2f

=1+2.561093284a s+a2s(965.8132564?62.46549093n f)

Cψ,NS

2,14

+a3s 41657.11568?5708.215623n f?13.73240102?11n f+115.3919490n2f

C G2,2=?0.5a s n f?8.918338961a2s n f

+a3s ?130.7340963n f+29.37933515n2f?0.9007972776?g11n2f

C G2,4=?0.7388888889a s n f?14.27158692a2s n f

+a3s ?346.4612756n f+46.52017564n2f?1.611816512?g11n2f

C G2,6=?0.7051587302a s n f?20.06849828a2s n f

+a3s ?715.0372438n f+61.28545096n2f?1.496036938?g11n2f

C G2,8=?0.6440873016a s n f?23.17873524a2s n f

+a3s ?996.5038709n f+68.66467304n2f?1.286400915?g11n2f

C G2,10=?0.5861279461a s n f?24.76678064a2s n f

+a3s ?1201.206903n f+72.23614791n2f?1.094394334?g11n2f

C G2,12=?0.5358430591a s n f?25.51669345a2s n f

+a3s ?1351.047836n f+73.7936445n2f?0.9344248731?g11n2f

Cψ

=1.777777778a s+a2s(56.75530152?4.543209877n f?3.950617284?02n f) L,2

+a3s 2544.598087?421.6908885n f?7.736698288?11n f+11.8957476n2f

+?02 ?213.9253076n f+17.91326528n2f

CψL,4=1.066666667a s+a2s(47.99398931?3.413333333n f?0.6945185185?02n f) +a3s 2523.73902?383.0520013n f?5.058869512?11n f+10.88895473n2f

+?02 ?55.5530456n f+2.348005487n2f

CψL,6=0.7619047619a s+a2s(40.9961976?2.69569161n f?0.2524824533?02n f) +a3s 2368.193775?340.0691069n f?3.705612526?11n f+9.472190428n2f

+?02 ?24.01322539n f+0.7652692585n2f

=0.5925925926a s+a2s(35.87664406?2.231471683n f?0.1217397796?02n f)

Cψ

L,8

+a3s 2215.210875?305.4730329n f?2.913702563?11n f+8.337149534n2f

+?02 ?12.97185267n f+0.344362391n2f

CψL,10=0.4848484848a s+a2s(32.01765947?1.908598248n f?0.06856377261?02n f) +a3s 2081.213222?278.0172177n f?2.397641695?11n f+7.452505612n2f

+?02 ?7.947555677n f+0.1841598535n2f

CψL,12=0.4102564103a s+a2s(29.0058065?1.671007435n f?0.04265241396?02n f) +a3s 1965.791047?255.8431044n f?2.035689631?11n f+6.751061503n2f

+?02 ?5.284573837n f+0.1098463658n2f

=0.3555555556a s+a2s(26.5848844?1.488624298n f)

Cψ,NS

L,14

+a3s 1866.009187?237.5642566n f?1.768138102?11n f+6.182499654n2f

C G L,2=0.6666666667a s n f+12.94776709a2s n f

+a3s 407.280632n f?20.23959748n2f?0.388939664?g11n2f

C G L,4=0.2666666667a s n f+13.81659259a2s n f

+a3s 767.7125421n f?36.78419232n2f?0.3984298404?g11n2f

C G L,6=0.1428571429a s n f+10.26095364a2s n f

+a3s 694.5092121n f?27.9895081n2f?0.3055276256?g11n2f

C G L,8=0.0888*******a s n f+7.733545104a2s n f

+a3s 592.3307972n f?21.30333681n2f?0.2322211886?g11n2f

C G L,10=0.06060606061a s n f+6.023053074a2s n f

+a3s 504.5424832n f?16.70544537n2f?0.1805482229?g11n2f

C G L,12=0.0439*******a s n f+4.830676204a2s n f

+a3s 433.9050534n f?13.47418408n2f?0.1439099345?g11n2f The numerical values of the anomalous dimensions for the odd moments of F3read:

γns1=0

γns3=5.555555556a s+a2s(70.88477366?5.12345679n f)

+a3s 1244.913602?196.4738081n f?1.762002743n2f

γns5=8.088888889a s+a2s(98.19940741?7.68691358n f)

+a3s 1720.942172?278.1581739n f?2.366211248n2f

γns7=9.780952381a s+a2s(116.4158903?9.437457798n f)

+a3s 2036.492478?330.8816595n f?2.739358023n2f

γns9=11.05820106a s+a2s(130.3414045?10.77682428n f)

+a3s 2277.19805?370.5905277n f?3.006446616n2f

γns11=12.08581049a s+a2s(141.6907901?11.86441897n f)

+a3s 2473.311857?402.691565n f?3.212753995n2f

γns13=12.94606135a s+a2s(151.2989044?12.78102552n f)

+a3s 2639.409887?429.7314605n f?3.37996738n2f

and the coe?cient functions are:

C ns3,1=1?4a s+a2s(?73.33333333+5.333333333n f)

+a3s ?2652.154437+513.3100408n f?11.35802469n2f

C ns3,3=1+1.666666667a s+a2s(14.25404015?6.742283951n f)

+a3s ?839.7638717?45.09953407n f+1.747689309n2f

C ns3,5=1+7.748148148a s+a2s(173.000629?19.39801646n f)

+a3s 4341.081057?961.2756356n f+22.24125078n2f

C ns3,7=1+12.72248677a s+a2s(345.9910777?30.52332666n f)

+a3s 11119.00053?1960.237096n f+43.10377964n2f

C ns3,9=1+16.9152381a s+a2s(520.0059615?40.35464229n f)

+a3s 18771.99642?2975.924131n f+63.17127673n2f

C ns3,11=1+20.54831329a s+a2s(690.8719666?49.17096968n f)

+a3s 26941.47987?3984.411605n f+82.24581704n2f

C ns3,13=1+23.76237745a s+a2s(857.1778817?57.18099124n f)

+a3s 35426.82868?4976.080869n f+100.3509187n2f

6Acknowledgements

A.R.would like to thank Sven Moch,Timo van Ritbergen and Thomas Gehrmann for many useful discussions and Denny Fliegner for technical support.J.V.would like to thank the University of Karlsruhe and the DFG for repeated hospitality.

This work was supported by the DFG under Contract Ku502/8-1(DFG-Forschergruppe “Quantenfeldtheorie,Computeralgebra und Monte-Carlo-Simulationen”)and the Graduiertenkol-leg“Elementarteilchenphysik an Beschleunigern”at the Univerit¨a t Karlsruhe.

References

[1]D.J.Gross and F.Wilczek,Phys.Rev.D8,3633(1973).

[2]D.J.Gross and F.Wilczek,Phys.Rev.D9,980(1974).

[3]W.A.Bardeen,A.J.Buras,D.W.Duke,and T.Muta,Phys.Rev.D18,3998(1978).

[4]E.G.Floratos,D.A.Ross,and C.T.Sachrajda,Nucl.Phys.B129,66(1977).

[5]Nucl.Phys.B139,545(1978).

[6]E.G.Floratos,D.A.Ross,and C.T.Sachrajda,Nucl.Phys.B152,493(1979).

[7]A.Gonzalez-Arroyo,C.Lopez,and F.J.Yndurain,Nucl.Phys.B153,161(1979).

[8]A.Gonzalez-Arroyo and C.Lopez,Nucl.Phys.B166,429(1980).

[9]C.Lopez and F.J.Yndurain,Nucl.Phys.B183,157(1981).

[10]W.Furmanski and R.Petronzio,Phys.Lett.B97,437(1980).

[11]G.Curci,W.Furmanski,and R.Petronzio,Nucl.Phys.B175,27(1980).

[12]R.Hamberg and W.L.van Neerven,Nucl.Phys.B379,143(1992).

[13]D.W.Duke,J.D.Kimel,and G.A.Sowell,Phys.Rev.D25,71(1982).

[14]A.Devoto,D.W.Duke,J.D.Kimel,and G.A.Sowell,Phys.Rev.D30,541(1984).

[15]D.I.Kazakov and A.V.Kotikov,Nucl.Phys.B307,721(1988).

[16]D.I.Kazakov and A.V.Kotikov,Phys.Lett.B291,171(1992).

[17]D.I.Kazakov,A.V.Kotikov,G.Parente,O.A.Sampayo,and J.S.Guillen,Santiago de

Compostela Univ.-US-FT-90-06(90,rec.Jul.)6p.

[18]W.L.van Neerven and E.B.Zijlstra,Phys.Lett.B272,127(1991).

[19]E.B.Zijlstra and W.L.van Neerven,Phys.Lett.B273,476(1991).

[20]E.B.Zijlstra and W.L.van Neerven,Nucl.Phys.B383,525(1992).

[21]S.Moch and J.A.M.Vermaseren,Nucl.Phys.B573,853(2000),hep-ph/9912355.

[22]https://www.doczj.com/doc/3f9122738.html,rin,T.van Ritbergen,and J.A.M.Vermaseren,Nucl.Phys.B427,41(1994).

[23]https://www.doczj.com/doc/3f9122738.html,rin,P.Nogueira,T.van Ritbergen,and J.A.M.Vermaseren,Nucl.Phys.B492,

338(1997),hep-ph/9605317.

[24]https://www.doczj.com/doc/3f9122738.html,rin and J.A.M.Vermaseren,Phys.Lett.B259,345(1991).

[25]J.A.M.Vermaseren and S.Moch,(2000),hep-ph/0004235.

[26]G.Parente,A.V.Kotikov,and V.G.Krivokhizhin,Phys.Lett.B333,190(1994),hep-

ph/9405290.

[27]A.Kataev,A.V.Kotikov,G.Parente and A.V.Sidorov,Phys.Lett.B388,179(1996),

hep-ph/9605367.

[28]W.L.van Neerven and A.Vogt,Nucl.Phys.B568,263(2000),hep-ph/9907472.

[29]J.Santiago and F.J.Yndurain,Nucl.Phys.B563,45(1999).

[30]W.L.van Neerven and A.Vogt(2000),hep-ph/0006154.

[31]J.A.M.Vermaseren,Symbolic Manipulation with Form version2,Tutorial and Reference

Manual(Computer Algebra Nederland,Amsterdam,1991).

[32]D.Fliegner,A.Retey,and J.A.M.Vermaseren,(1999),hep-ph/9906426.

[33]A.J.Buras,Rev.Mod.Phys.52,199(1980).

[34]S.G.Gorishnii and https://www.doczj.com/doc/3f9122738.html,rin,Nucl.Phys.B283,452(1987).

[35]K.G.Chetyrkin and https://www.doczj.com/doc/3f9122738.html,achev,Nucl.Phys.B192,159(1981).

[36]https://www.doczj.com/doc/3f9122738.html,rin,https://www.doczj.com/doc/3f9122738.html,achev,and J.A.M.Vermaseren,NIKHEF-H-91-18.

[37]https://www.doczj.com/doc/3f9122738.html,rin,T.van Ritbergen,and J.A.M.Vermaseren,Prepared for3rd International

Workshop on Software Engineering,Arti?cial Intelligence and Expert systems for High-energy and Nuclear Physics,Oberammergau,Germany,4-8Oct1993.

[38]J.A.Gracey,Phys.Lett.B322,141(1994),hep-ph/9401214.

A Conventions

Here we give the complete expressions for the newly computed moments and coe?cient functions.

The notation of the color factors is as usual:The Casimir operators of the fundamental and adjoint representation are denoted by C F and C A and their values for the color group SU(3) are4

3.For the trace normalization of the fundamental representation we have

inserted T F=1

3ζ5).The numerical values

given in this reference are correct.

It should be noted that the terms in d abc d abc enter for the?rst time at the three loop level and help in the determination of P S qq?P S q

2310 +a s n f +2

12326391000 +a2s C F n f +92841821715175

+a3s C3A +239083526238286750523

36243287457300000+

17831164

3883209370425000?

1344

2196562876200000?

17746492

420260754375 +a3s C A n2f

?2752314359

γGG 12

=a s C A

71203

+a 2s C 2

+165198392441572108106

+a 2s C F n f

+7482350342112541869725600000

+a 3s C F C A n f

?49693541388602890695713585585ζ3 +a 3s C 2

F n f

?4699124115250144376149507

ζ3 +a 3s C 2

A n f

+74338654569222233539585585

ζ3 +a 3s C F n 2

?50627726543561953759677078160

γGψ10

=a s C F

112

80858250

+a 2s C 2

88631998

735075

+a 3s C F C 2

?16589443127472580381675

ζ3

+a 3s C 2F C A

422442179103147495127225

ζ3

+a 3s C 3

+645789745908437189381675

ζ3

+a 3s C F C A n f

?18846629176433495

ζ3 +a 3s C 2

F n f

+529979902254031495

ζ3

+a 3s C F n 2

+152267426

γGψ12

=a s C F

50645138544

+a 2s C 2

9387059226553

115945830

+a 3s C F C 2

?247519699097672401191536441435

ζ3

+a 3s C 2F C A

1159865991831228099740232147145

ζ3

+a 3s C 3

+21659273057249922158947036441435

ζ3 +a 3s C F C A n f

?64190493078139789429

ζ3

+a 3s C 2

F n f

+1401404001326440151429

ζ3 +a 3s C F n 2

13454024393417165 +a 2s C F n f

?379479917

150935400

+a 3s C F C A n f

+926990216580622991

27225

ζ3 +a 3s C 2

F n f

?10919800485362138333025

ζ3 +a 3s C 2

A n f

?2102543085765897127225

ζ3 +a 3s C F n 2

+15847133257543697906140000

γψG 12

=a s n f

3197294100

+a 2s C A n f

653436358741

104630014026728910720000?

171207527

143866269286752252240000+

2563

543532540398591744000

+129763817

249119081016021216000

+a 3s C A n 2

226617401255197

γψψ10

=a s C F +12055

523908000 +a2s C2F ?9579051036701

288149400 +a2s?02C F n f ?27284

435681892800000+

151796299

465937579800000?

151796299

127866318149354400000+

151796299

389001690000?

48220

1230075210672000+

48220

11981252052000

+a3s?02C F C A n f ?102876641210704327225ζ3 +a3s?02C2F n f +20996606374679827225ζ3 +a3s?02C F n2f ?33230913134

γψψ12

=a s C F +423424

486972486000 +a2s C2F ?5507868301548461

9739449720 +a2s?02C F n f ?249775

6016642459027200000+

25648239313

2634913356900224400000?

25648239313

4747586886462824323920000+

25648239313

1052912430329760000?

3387392

526982671380044880000+

3387392

26322810758244000

+a3s?02C F C A n f ?6969748954384649469139039ζ3 +a3s?02C2F n f +8603325540244325619739039ζ3 +a3s?02C F n2f ?2566080055386457

18018

+a2s C F C A +288858136265399

48697248600

+a2s C2F ?22819142381313407

3685193506154160000?

720484

515927090861582400000+

3663695353

26322810758244000

+a3s C2F C A ?9600133371690362148838764414350ζ3 +a3s C2F n f ?379085447979756145127339009ζ3 +a3s C3F ?4055239574606487124221170996621525ζ3

C ψ2,10

=1+a s C F

200629929045459520000

104674

15975002736000

+a 2s C 2

+5587087999873240131155ζ3

+a 2s ?02C F n f

+358420378849123550349033411200000+10519793104

693

ζ4

+a 3s C F C 2

+7092211199654579390952376243237000

ζ3+151796299231

ζ5

+a 3s C F n 2

+5708442804785155191118711

ζ3 +a 3s C 2

F C A

+163500093049269333896088296163195500

ζ3

151796299

ζ5

+a 3s C 2

n f

?152138746003699406101004942567525

ζ3+2411092155812816602703168000000

21822088252452828004150

ζ4?

75212861140509985448000000?7952746327225

ζ4

+a 3s ?02C F n 2

?148475806971656561735075

ζ3 +a 3s ?02C 2

F n f

+42257725087595445377161711037151125

ζ3+

6272n

+3753913187503606375

ζ3?

448

C ψ2,12

=1+a s C F

183473419

87742702527480000

?1477711

131745667845011220000

1261726

1403883240439680

+a 2s ?02C F n f

+10355592866326975885504678726462720000+

3684228204145045

ζ4 +a 3s C F C 2

+163181620367687907864404054279243486243000ζ3+256482393133003ζ5

+a 3s C F n 2

f +20037551000996574384238871216215

ζ3

+a 3s C 2

F C A

+484319198652684915730033380410997491891697200

ζ3?256482393131287

ζ5

+a 3s C 2

F n f

?84838981067097560083179855712174312150

ζ3+

16936963421680820811486746735622400000

42634681331415644

1352701350

ζ4?8800426931765610480021619328000000

310947622280339039ζ4 +a 3s ?02C F n 2

?196086760373306062485195823ζ3 +a 3s ?02C 2

n f

+33528586559068805780843402423791330289750

ζ3+

6241n

+809917806143013559851350500

ζ3?

200

C ns 2,14

=1+a s C F

90849502

4913591341538880000

?315626

35097081010992000

+a 2s C 2

31002322638187643268973

3003

ζ3

+a 3s

C F C A n f

?28812973254576289068812626927103481653275

ζ3

?360242

1896378761921374303372800000

83168919211026563

386486100

ζ4+

273814628457064254522423520000+

72048417091312791266167566112000000

13682992796062061

128828700

ζ4?

865562377340672014967335875200000

12775152582499

9009

ζ4

+a 3s C 3

129244077799850312684282800666009211695029480644500

ζ3

+36636953539009ζ5

+a 3s fl 11n f

d abc d abc

49587712574400000

78376866703

ζ5

C ψL,10

=a s C F

8731800

?48

114345

+a 2s C 2

199951060711

ζ3

+a 2s ?02C F n f

415796

1699159381920000

+55485434

6796637527680000

95022195887

11ζ5

+a 3s C F n 2

+632726393354750574560000+22904191

11ζ5

+a 3s C 2

F n f +904887432630763715015

ζ3

+a 3s C 3

?887562386698645967383468242775ζ3+1344043491944948760000

36224156897348300

+a 3s ?02C 2

F n f

319520059852805113

1911195

ζ3

+a 3s ?11n f

d abc d abc

528099264000

?1820773

ζ5