Symbolic Summation and Higher Orders in Perturbation TheoryHypergeometric summation Harmonic...

Symbolic Summation and Higher Orders inPerturbation Theory

Sven-Olaf [email protected]

DESY, Zeuthen

————————————————————————————————————–

– Advanced Computing and Analysis Techniques in Physics (ACAT 2005), DESY, Zeuthen, May 26, 2005 –

Sven-Olaf Moch Symbolic Summation and Higher Orders in Perturbation Theory – p.1

http://www-zeuthen.desy.de/~moch


Introduction

TaskGiven expression g(n) (depending on n) find expression f(n), suchthat

f(n) =

nX

i=0

g(i)

Summation algorithmsPolynomial summationHypergeometric summationHarmonic summationBeyond

Examples in particle physics

Summary



Polynomial summation

Examples

Polynomials

n−1X

i=0

i =1

2n(n− 1)

n−1X

i=0

i2 =

1

6n(n− 1)(2n− 1)

n−1X

i=0

i3 =

1

4n

2(n− 1)2

n−1X

i=0

i4 =

1

30n(n− 1)(2n− 1)(3n

2− 3n− 1)



Difference opDifference operator

Introduce operator ∆ with (∆f)(n) = f(n + 1)− f(n)

Symbolic summation problem (cf. symbolic integration)g = (∆f) with f = (

P

g), operator ∆ is left inverse ∆(P

f) = f

Differential operator D acts in continuum as D(xm) = mxm−1

Discrete analog ∆ on polynomials? E.g. ∆(n3) = 3n2 + 3n− 1

Rising and falling factorialsDefine rising factorials as fm = f(x)f(x + 1) . . . f(x + m− 1)

(also known as Pochhammer symbols (x)m )

Define falling factorials as fm = f(x)f(x− 1) . . . f(x−m + 1)



RiseFall FacThen, with falling factorials

∆(xm) = mxm−1

n−1X

i=0

im =

1

m + 1n

m+1

Conversion of polynomial powers xm

(decomposition with Stirling numbers of second kind

(

m

i

)

)

xm =

mX

i=0

(

m

i

)

xi

Stirling numbers of second kind denote # of ways to partition n

things in k non-empty sets

Examples

n−1X

i=0

i =

n−1X

i=0

i1 =

1

2n

2 =1

2n(n− 1)

n−1X

i=0

i2 =

n−1X

i=0

(i2 + i1) =

1

3n

3 +1

2n

2 =1

6n(n + 1)(2n + 1)



Hypergeometric summation

Definition

Hypergeometric function mFn

mFn

a1, . . . , am

b1, . . . , bn

˛

˛

˛

˛

z

!

=X

i≥0

ai1 . . . ai

m

bi1 . . . bi

n

zi

i!

Examples

0F0

„˛

˛

˛

˛

z

«

=X

i≥0

zi

i!= ez

2F1

a, 1

1

˛

˛

˛

˛

z

!

=X

i≥0

ai zi

i!=

1

(1− z)a

2F1

1, 1

2

˛

˛

˛

˛

z

!

= zX

i≥0

1i1i

2i

zi

i!= − ln(1− z)



RatiosRatios

A term gn is hypergeometric, if the ratio r(n) of two consecutive termsis a rational function of n.

r(n) =gn+1

gn

Example: binomial coefficient

m

n + 1

!

m

n

! =Γ(m + 1)Γ(n + 1)Γ(m− n + 1)

Γ(n + 2)Γ(m− n)Γ(m)=−n + m

n + 1

Given a hypergeometric term g, is there hypergeometric term f suchthat ∆f = g?

fn+1 − fn = gn



Gosper/WZGospers algorithm

Gospers algorithm for indefinite hypergeometric summationdetermines fn from a given recursion

fn = fn−1 + gn−1 = fn−2 + gn−1 + gn−2 = · · · = f0 +

n−1X

k=0

gk

Idea: recursive algorithm; telescoping

Wilf-Zeilberger algorithm

WZ algorithmdefinite hypergeometric summationIdea: telescoping

Definite vs. indefinite summation

ExamplesX

k

n

k

!

=X

k

n

k

!

nX

k=0

n

k

!

= 2n



Harmonic summation

Harmonic sums Sm1,...,mk(n)

Gonzalez-Arroyo, Lopez, Ynduráin ‘79; Vermaseren ‘98; Blümlein, Kurth ‘98;

S.M., Uwer, Weinzierl ‘01; Blümlein ‘03

recursive definition S±m1,...,mk(n) =

nX

i=1

(±1)i

im1Sm2,...,mk

(i)

Particle physicsdimensional regularization D = 4− 2ε requires expansion of theGamma-function around positive integers values (n ≥ 0)

Γ(n + 1 + ε)

Γ(1 + ε)= Γ(n + 1) exp

−

∞X

k=1

εk (−1)k)

kSk(n)

!



Algorithms 1Algorithms for harmonic sums

Multiplication (Hopf algebra)basic formula (recursion)

Sm1,...,mk(n)× Sm′

1,...,m′

l

(n) =

nX

j1=1

1

jm1

1

Sm2,...,mk(j1)Sm′

1,...,m′

l

(j1)

+nX

j2=1

1

jm′

1

2

Sm1,...,mk(j2)Sm′

2,...,m′

l

(j2)

−

nX

j=1

1

jm1+m′

1

Sm2,...,mk(j)Sm′

2,...,m′

l

(j)

Proof uses decomposition

nX

i=1

nX

j=1

aij =

nX

i=1

iX

j=1

aij +

nX

j=1

jX

i=1

aij −

nX

i=1

aii

-

6

j1

j2

=

-

6

j1

j2

+

-

6

j1

j2

−

-

6

j1

j2



Algorithms 2Algorithms for harmonic sums (cont’d)

Convolution (sum over n− j and j)

n−1X

j=1

1

jm1Sm2,...,mk

(j)1

(n− j)n1Sn2,...,nl

(n− j)

Conjugation

−

nX

j=1

n

j

!

(−1)j1

jm1Sm2,...,mk

(j)

Binomial convolution (sum over binomial, n− j and j)

−

n−1X

j=1

n

j

!

(−1)j1

jm1Sm2,...,mk

(j)1

(n− j)n1Sn2,...,nl

(n− j)



Beyond

Generalized sums S(n; m1, ..., mk; x1, ..., xk)

recursive definition

S(n; m1, ..., mk; x1, ..., xk) =

nX

i=1

xi1

im1S(i;m2, ..., mk; x2, ..., xk)

multiple scales x1, ..., xk

depth k, weight w = m1 + ... + mk

Example

Powers of logarithm ln(1− x)

∞X

j=1

xj

j!Γ(j − ε) =

∞X

j=1

xj

j− ε

∞X

j=1

xj

jS1(j − 1) + ε

2. . .

= − ln(1− x) − ε1

2ln(1− x)2 + ε

2. . .



Harmonic polylogarithmsHarmonic polylogarithms

Harmonic polylogarithms Hm1,...,mk(x)

Goncharov ‘98; Borwein, Bradley,Broadhurst, Lisonek ‘99; Remiddi, Vermaseren ‘99

physical quantities in momentum (x)-space (single scale)

basic functions of lowest weightH0(x) = ln x , H1(x) = − ln(1− x) , H−1(x) = ln(1 + x)

higher functions defined by recursion

Hm1,...,mw(x) =

Z x

0dz fm1(z) Hm2,...,mw

(z)

f0(x) =1

x, f1(x) =

1

1− x, f−1(x) =

1

1 + x

Harmonic polylogarithms related to generalized sumsS(∞; m1, ..., mk; x1, ..., xk)

(Inverse) Mellin transformation

f̃(N) =

Z 1

0dx x

Nf(x)

unique mappingHm1,...,mw

(x)

(1± x)←→ Sn1,...,nw+1(N)



Harmonic polylogarithmsHarmonic polylogarithms

Harmonic polylogarithms Hm1,...,mk(x)

Goncharov ‘98; Borwein, Bradley,Broadhurst, Lisonek ‘99; Remiddi, Vermaseren ‘99

physical quantities in momentum (x)-space (single scale)basic functions of lowest weightH0(x) = ln x , H1(x) = − ln(1− x) , H−1(x) = ln(1 + x)

higher functions defined by recursion

Hm1,...,mw(x) =

Z x

0dz fm1(z) Hm2,...,mw

(z)

f0(x) =1

x, f1(x) =

1

1− x, f−1(x) =

1

1 + x

Harmonic polylogarithms related to generalized sumsS(∞; m1, ..., mk; x1, ..., xk)

(Inverse) Mellin transformation

f̃(N) =

Z 1

0dx x

Nf(x)

unique mappingHm1,...,mw

(x)

(1± x)←→ Sn1,...,nw+1(N)



Algorithms beyondAlgorithms for nested sums

Same structures as for harmonic sums, in particularmultiplication

S(n; m1, ...; x1, ...) × S(n; m′1, ...; x′

1, ...)

convolutionconjugationbinomial convolution

Recursive algorithms analogous to harmonic sums solve multiplenested sums



Higher transcendental functionsHigher transcendental functions

Expansion of higher transcendental functions in small parameterexpansion parameter ε occurs in the rising factorials(Pochhammer symbols)

Hypergeometric function

2F1(a, b; c, x0) =

∞X

i=0

aibi

ci

xi0

i!

First Appell function

F1(a, b1, b2; c; x1, x2) =∞X

m1=0

∞X

m2=0

am1+m2bm1

1 bm2

2

cm1+m2

xm1

1

m1!

xm2

2

m2!

Second Appell function

F2(a, b1, b2; c1, c2; x1, x2) =∞X

m1=0

∞X

m2=0

am1+m2bm1

1 bm2

2

cm1

1 cm2

2

xm1

1

m1!

xm2

2

m2!



Examples in particle physics

Feynman integralsScalar diagram withexternal momenta P and Q;Four-point function withunderlying ladder topology

=

Z 3Y

n

dD

ln1

(P − l1)21

l21 . . . l28

N -th moment:coefficient of (2P ·Q)N

=(2P ·Q)N

(Q2)N+α

CN

Taylor expansion1

(P − l1)2=X

i

(2P · l1)i

(l21)i+1−→

(2P · l1)N

(l21)N



Difference eqDifference equations

Single-step difference equation in N

extremely simple example1

1

1 1

1

1

1

1

1 1

= −

N+3+3εN+2

2 p ·qq2

1

1

1 1

1

1

1

1

1 1

+2

N+2

1

1

1 1

1

1 2

1 1

Formal equation , formal solution , input to solution

I(N) = −N+3+3ε

N+2I(N− 1) +

2

N+2G(N)I(N) = (−1)N

QNj=1(j+3+3ε)QN

j=1(j+2)I(0) + (−1)N

NX

i=1

(−1)jQN

j=i+1(j+3+3ε)QN

j=i(j+2)G(i)

I(0) = −2

3

1

ε2+

23

3

1

ε− 42

G(i) =(−1)i

ε22

3

„

S1(i+2)

i+2−

S1,2(i)

2−

S2(i+1)

2(i+1)− S2(i)−

1

(i+1)2−

1

(i+2)2

«

+ . . .

Upshotautomatic build-up of nested sumsefficient implementation in FORM






1

1 1

1

1

1

1

1 1

= −

N+3+3εN+2

2 p ·qq2

1

1

1 1

1

1

1

1

1 1

+2

N+2

1

1

1 1

1

1 2

1 1

Formal equation

, formal solution , input to solution

I(N) = −N+3+3ε

N+2I(N− 1) +

2

N+2G(N)

I(N) = (−1)NQN

j=1(j+3+3ε)QN

j=1(j+2)I(0) + (−1)N

NX

i=1

(−1)jQN

j=i+1(j+3+3ε)QN

j=i(j+2)G(i)

I(0) = −2

3

1

ε2+

23

3

1

ε− 42

G(i) =(−1)i

ε22

3

„

S1(i+2)

i+2−

S1,2(i)

2−

S2(i+1)

2(i+1)− S2(i)−

1

(i+1)2−

1

(i+2)2

«

+ . . .







1

1 1

1

1

1

1

1 1

= −

N+3+3εN+2

2 p ·qq2

1

1

1 1

1

1

1

1

1 1

+2

N+2

1

1

1 1

1

1 2

1 1

Formal equation, formal solution

, input to solution

I(N) = −N+3+3ε

N+2I(N− 1) +

2

N+2G(N)

I(N) = (−1)NQN

j=1(j+3+3ε)QN

j=1(j+2)I(0) + (−1)N

NX

i=1

(−1)jQN

j=i+1(j+3+3ε)QN

j=i(j+2)G(i)

I(0) = −2

3

1

ε2+

23

3

1

ε− 42

G(i) =(−1)i

ε22

3

„

S1(i+2)

i+2−

S1,2(i)

2−

S2(i+1)

2(i+1)− S2(i)−

1

(i+1)2−

1

(i+2)2

«

+ . . .







1

1 1

1

1

1

1

1 1

= −

N+3+3εN+2

2 p ·qq2

1

1

1 1

1

1

1

1

1 1

+2

N+2

1

1

1 1

1

1 2

1 1

Formal equation, formal solution, input to solution

I(N) = −N+3+3ε

N+2I(N− 1) +

2

N+2G(N)

I(N) = (−1)NQN

j=1(j+3+3ε)QN

j=1(j+2)I(0) + (−1)N

NX

i=1

(−1)jQN

j=i+1(j+3+3ε)QN

j=i(j+2)G(i)

I(0) = −2

3

1

ε2+

23

3

1

ε− 42

G(i) =(−1)i

ε22

3

„

S1(i+2)

i+2−

S1,2(i)

2−

S2(i+1)

2(i+1)− S2(i)−

1

(i+1)2−

1

(i+2)2

«

+ . . .







1

1 1

1

1

1

1

1 1

= −

N+3+3εN+2

2 p ·qq2

1

1

1 1

1

1

1

1

1 1

+2

N+2

1

1

1 1

1

1 2

1 1

Formal equation, formal solution, input to solution

I(N) = −N+3+3ε

N+2I(N− 1) +

2

N+2G(N)

I(N) = (−1)NQN

j=1(j+3+3ε)QN

j=1(j+2)I(0) + (−1)N

NP

i=1

(−1)jQN

j=i+1(j+3+3ε)QN

j=i(j+2)G(i)

I(0) = −2

3

1

ε2+

23

3

1

ε− 42

G(i) =(−1)i

ε22

3

„

S1(i+2)

i+2−

S1,2(i)

2−

S2(i+1)

2(i+1)− S2(i)−

1

(i+1)2−

1

(i+2)2

«

+ . . .




LA13 exampleI(N) =

sign(N)*ep^-2 * ( 4/3*S(R(1),1 + N)*den(1 + N) + 8/3*S(R(1),1 + N)*den(1 + N)^2 + 4/3*S(R(1),2 + N)*den(2 + N) + 4/3*S(R(1),2 + N)*den(2 + N)^2 + 4/3*S(R(1),N) + 2/3*S(R(1,2),N) + 2/3*S(R(2),1 + N)*den(1 + N) + 2/3*S(R(2),2 + N)*den(2 + N) - 2*S(R(2),N) - 4/3*S(R(2),N)*N + 4*S(R(2,1),N) + 4/3*S(R(2,1),N)*N- 6*S(R(3),N) - 2*S(R(3),N)*N - 8/3*den(1 + N)^2 - 4*den(1 + N)^3 - 4/3*den(2+ N)^2 - 2*den(2 + N)^3 )

+ sign(N)*ep^-1 * ( - 16*S(R(1),1 + N)*den(1 + N) - 88/3*S(R(1),1 + N)*den(1 + N)^2 - 20/3*S(R(1),1 + N)*den(1 + N)^3 - 16*S(R(1),2 + N)*den(2 + N) -44/3*S(R(1),2 + N)*den(2 + N)^2 - 10/3*S(R(1),2 + N)*den(2 + N)^3 - 20*S(R(1),N)+ 8/3*S(R(1,1),1 + N)*den(1 + N) + 8/3*S(R(1,1),1 + N)*den(1 + N)^2 + 8/3*S(R(1

,1),2 + N)*den(2 + N) + 8/3*S(R(1,1),N) + 10/3*S(R(1,1,2),N) + 10/3*S(R(1,2),1+ N)*den(1 + N) + 10/3*S(R(1,2),2 + N)*den(2 + N) - 16*S(R(1,2),N) - 4*S(R(1,2)

,N)*N + 14*S(R(1,2,1),N) + 4*S(R(1,2,1),N)*N - 24*S(R(1,3),N) - 6*S(R(1,3),N)*N- 58/3*S(R(2),1 + N)*den(1 + N) - 40/3*S(R(2),1 + N)*den(1 + N)^2 - 46/3*S(R(2)

,2 + N)*den(2 + N) - 6*S(R(2),2 + N)*den(2 + N)^2 + 56/3*S(R(2),N) + 20*S(R(2),N)*N + 10*S(R(2,1),1 + N)*den(1 + N) + 6*S(R(2,1),2 + N)*den(2 + N) - 134/3*S(R(2,1),N) - 56/3*S(R(2,1),N)*N + 16/3*S(R(2,1,1),N) + 8/3*S(R(2,1,1),N)*N - 62/3*S(R(2,2),N) - 22/3*S(R(2,2),N)*N - 18*S(R(3),1 + N)*den(1 + N) - 12*S(R(3),2 + N)*den(2 + N) + 76*S(R(3),N) + 100/3*S(R(3),N)*N - 10*S(R(3,1),N) - 10/3*S(R(3,1),N)*N + 36*S(R(4),N) + 12*S(R(4),N)*N + 32*den(1 + N)^2 + 164/3*den(1 + N)^3 + 24*den(1 + N)^4 + 16*den(2 + N)^2 + 82/3*den(2 + N)^3 + 12*den(2 + N)^4 )

+ sign(N) * ( 100*S(R(1),1 + N)*den(1 + N) + 168*S(R(1),1 + N)*den(1 + N)^2 + 268/3*S(R(1),1 + N)*den(1 + N)^3 - 16/3*S(R(1),1 + N)*den(1 + N)^4 + 100*S(R(1),2 + N)*den(2 + N) + 84*S(R(1),2 + N)*den(2 + N)^2 + 134/3*S(R(1),2 + N)*den(2 + N)^3 - 8/3*S(R(1),2 + N)*den(2 + N)^4 + 160*S(R(1),N) - 32*S(R(1,1),1 +N)*den(1 + N) - 80/3*S(R(1,1),1 + N)*den(1 + N)^2 - 20/3*S(R(1,1),1 + N)*den(1+ N)^3 - 32*S(R(1,1),2 + N)*den(2 + N) - 4/3*S(R(1,1),2 + N)*den(2 + N)^2 - 10/

3*S(R(1,1),2 + N)*den(2 + N)^3 - 40*S(R(1,1),N) + 4/3*S(R(1,1,1),1 + N)*den(1 +N) - 40/3*S(R(1,1,1),1 + N)*den(1 + N)^2 + 4/3*S(R(1,1,1),2 + N)*den(2 + N) - 44/3*S(R(1,1,1),2 + N)*den(2 + N)^2 + 4/3*S(R(1,1,1),N) + 38/3*S(R(1,1,1,2),N) + 38/3*S(R(1,1,2),1 + N)*den(1 + N) + 38/3*S(R(1,1,2),2 + N)*den(2 + N) - 68*S(R(1,1,2),N) - 12*S(R(1,1,2),N)*N + 42*S(R(1,1,2,1),N) + 12*S(R(1,1,2,1),N)*N - 76*S(R(1,1,3),N) - 18*S(R(1,1,3),N)*N - 170/3*S(R(1,2),1 + N)*den(1 + N) + 40/3*S(R(1,2),1 + N)*den(1 + N)^2 - 134/3*S(R(1,2),2 + N)*den(2 + N) + 14*S(R(1,2),2 + N)*den(2 + N)^2 + 430/3*S(R(1,2),N) + 60*S(R(1,2),N)*N + 30*S(R(1,2,1),1 + N)*den(1+ N) + 18*S(R(1,2,1),2 + N)*den(2 + N) - 452/3*S(R(1,2,1),N) - 56*S(R(1,2,1),N)

*N + 74/3*S(R(1,2,1,1),N) + 8*S(R(1,2,1,1),N)*N - 248/3*S(R(1,2,2),N) - 22*S(R(1,2,2),N)*N - 58*S(R(1,3),1 + N)*den(1 + N) - 40*S(R(1,3),2 + N)*den(2 + N) + 886/

3*S(R(1,3),N) + 100*S(R(1,3),N)*N - 116/3*S(R(1,3,1),N) - 10*S(R(1,3,1),N)*N +410/3*S(R(1,4),N) + 36*S(R(1,4),N)*N + 186*S(R(2),1 + N)*den(1 + N) + 448/3*S(R(2),1 + N)*den(1 + N)^2 + 160/3*S(R(2),1 + N)*den(1 + N)^3 + 138*S(R(2),2 + N)*den(2 + N) + 206/3*S(R(2),2 + N)*den(2 + N)^2 + 80/3*S(R(2),2 + N)*den(2 + N)^3- 70*S(R(2),N) - 160*S(R(2),N)*N - 338/3*S(R(2,1),1 + N)*den(1 + N) - 64/3*S(R(

2,1),1 + N)*den(1 + N)^2 - 206/3*S(R(2,1),2 + N)*den(2 + N) - 10/3*S(R(2,1),2 +N)*den(2 + N)^2 + 760/3*S(R(2,1),N) + 140*S(R(2,1),N)*N + 50/3*S(R(2,1,1),1 + N)*den(1 + N) + 26/3*S(R(2,1,1),2 + N)*den(2 + N) - 170/3*S(R(2,1,1),N) - 100/3*S(R(2,1,1),N)*N - 12*S(R(2,1,1,1),N) + 4/3*S(R(2,1,1,1),N)*N + 38/3*S(R(2,1,2),N)- 2/3*S(R(2,1,2),N)*N - 182/3*S(R(2,2),1 + N)*den(1 + N) - 116/3*S(R(2,2),2 + N

)*den(2 + N) + 676/3*S(R(2,2),N) + 308/3*S(R(2,2),N)*N - 118/3*S(R(2,2,1),N) -18*S(R(2,2,1),N)*N + 296/3*S(R(2,3),N) + 36*S(R(2,3),N)*N + 694/3*S(R(3),1 + N)*den(1 + N) + 188/3*S(R(3),1 + N)*den(1 + N)^2 + 448/3*S(R(3),2 + N)*den(2 + N)+ 80/3*S(R(3),2 + N)*den(2 + N)^2 - 1454/3*S(R(3),N) - 290*S(R(3),N)*N - 86/3*

S(R(3,1),1 + N)*den(1 + N) - 56/3*S(R(3,1),2 + N)*den(2 + N) + 440/3*S(R(3,1),N)+ 164/3*S(R(3,1),N)*N - 10*S(R(3,1,1),N) - 10/3*S(R(3,1,1),N)*N + 80*S(R(3,2),N

) + 80/3*S(R(3,2),N)*N + 302/3*S(R(4),1 + N)*den(1 + N) + 194/3*S(R(4),2 + N)*den(2 + N) - 434*S(R(4),N) - 556/3*S(R(4),N)*N - 8*S(R(4,1),N) - 8/3*S(R(4,1),N)*N - 150*S(R(5),N) - 50*S(R(5),N)*N - 200*den(1 + N)^2 - 380*den(1 + N)^3 - 896/3*den(1 + N)^4 - 100*den(1 + N)^5 - 100*den(2 + N)^2 - 190*den(2 + N)^3 - 448/3*den(2 + N)^4 - 50*den(2 + N)^5 );

Result for I(N) in the G-scheme



Splitting functions (nut shell)Splitting functions in a nut shell

Calculate anomalous dimensions (Mellin moments of splittingfunctions) −→ divergence of Feynman diagrams in dimensionalregularization D = 4− 2ε

γ(n)ij (N) = −

Z 1

0dx x

N−1P

(n)ij (x)

One-loop Feynman diagrams

−→ in total 18 for γ(0)ij / P

(0)ij

(pencil + paper)

Two-loop Feynman diagrams


(1)ij

(simple computer algebra)

Three-loop Feynman diagrams


(2)ij

(cutting edge technology −→ computer al-gebra system FORM Vermaseren ‘89-‘04)



LO/NLO PLO and NLO splitting functionsP (0)

ns (x) = CF(2pqq(x)+3δ(1− x))

P (0)ps (x) = 0

P (0)qg (x) = 2nf pqg(x)

P (0)gq (x) = 2CF pgq(x)

P (0)gg (x) = CA

(

4pgg(x)+113

δ(1− x))

−

23

nf δ(1− x)

P(1)+ns (x) = 4CACF

(

pqq(x)[67

18−ζ2 +

116

H0 +H0,0

]

+ pqq(−x)[

ζ2 +2H−1,0 −H0,0

]

+143

(1− x)+δ(1− x)[17

24+

113

ζ2 −3ζ3

])

−4CFnf

(

pqq(x)[5

9+

13

H0

]

+23(1− x)

+δ(1− x)[ 1

12+

23

ζ2

])

+4CF2(

2pqq(x)[

H1,0 −34

H0 +H2

]

−2pqq(−x)[

ζ2 +2H−1,0

−H0,0

]

− (1− x)[

1−32

H0

]

−H0 − (1+ x)H0,0 +δ(1− x)[3

8−3ζ2 +6ζ3

])

P(1)−ns (x) = P(1)+

ns (x)+16CF

(

CF −

CA

2

)(

pqq(−x)[

ζ2 +2H−1,0 −H0,0

]

−2(1− x)

− (1+ x)H0

)

P(1)ps (x) = 4CFnf

(209

1x−2+6x−4H0 + x2

[83

H0 −569

]

+(1+ x)[

5H0 −2H0,0

])

P(1)qg (x) = 4CAnf

(209

1x−2+25x−2pqg(−x)H

−1,0 −2pqg(x)H1,1 + x2[44

3H0 −

2189

]

+4(1− x)[

H0,0 −2H0 + xH1

]

−4ζ2x−6H0,0 +9H0

)

+4CFnf

(

2pqg(x)[

H1,0 +H1,1 +H2

−ζ2

]

+4x2[

H0 +H0,0 +52

]

+2(1− x)[

H0 +H0,0 −2xH1 +294

]

−

152−H0,0 −

12

H0

)

P(1)gq (x) = 4CACF

(1x

+2pgq(x)[

H1,0 +H1,1 +H2 −116

H1

]

− x2[8

3H0 −

449

]

+4ζ2 −2

−7H0 +2H0,0 −2H1x+(1+ x)[

2H0,0 −5H0 +379

]

−2pgq(−x)H−1,0

)

−4CFnf

(23

x

−pgq(x)[2

3H1 −

109

])

+4CF2(

pgq(x)[

3H1 −2H1,1

]

+(1+ x)[

H0,0 −72

+72

H0

]

−3H0,0

+1−32

H0 +2H1x)

P(1)gg (x) = 4CAnf

(

1− x−109

pgg(x)−139

(1x− x2

)

−

23(1+ x)H0 −

23

δ(1− x))

+4CA2(

27

+(1+ x)[11

3H0 +8H0,0 −

272

]

+2pgg(−x)[

H0,0 −2H−1,0 −ζ2

]

−

679

(1x− x2

)

−12H0

−

443

x2H0 +2pgg(x)[67

18−ζ2 +H0,0 +2H1,0 +2H2

]

+δ(1− x)[8

3+3ζ3

])

+4CFnf

(

2H0

+23

1x

+103

x2−12+(1+ x)

[

4−5H0 −2H0,0

]

−

12

δ(1− x))

.



LO ns P@

NNLO non-singlet splitting functionsS.M., Vermaseren, Vogt ‘04

P(2)+ns (x) = 16CACFnf

(16

pqq(x)[10

3ζ2 −

20936

−9ζ3 −16718

H0 +2H0ζ2 −7H0,0 −2H0,0,0

+3H1,0,0 −H3

]

+13

pqq(−x)[3

2ζ3 −

53

ζ2 −H−2,0 −2H

−1ζ2 −103

H−1,0 −H

−1,0,0

+2H−1,2 +

12

H0ζ2 +53

H0,0 +H0,0,0 −H3

]

+(1− x)[1

6ζ2 −

25754

−

4318

H0 −16

H0,0 −H1

]

− (1+ x)[2

3H−1,0 +

12

H2

]

+13

ζ2 +H0 +16

H0,0 +δ(1− x)[5

4−

16754

ζ2 +1

20ζ2

2 +2518

ζ3

])

+16CACF2(

pqq(x)[5

6ζ3 −

6920

ζ22−H

−3,0 −3H−2ζ2 −14H

−2,−1,0 +3H−2,0 +5H

−2,0,0

−4H−2,2 −

15148

H0 +4112

H0ζ2 −172

H0ζ3 −134

H0,0 −4H0,0ζ2 −2312

H0,0,0 +5H0,0,0,0 +23

H3

−24H1ζ3 −16H1,−2,0 +679

H1,0 −2H1,0ζ2 +313

H1,0,0 +11H1,0,0,0 +8H1,1,0,0 −8H1,3 +H4

+679

H2 −2H2ζ2 +113

H2,0 +5H2,0,0 +H3,0

]

+ pqq(−x)[1

4ζ2

2−

679

ζ2 +314

ζ3 +5H−3,0

−32H−2ζ2 −4H

−2,−1,0 −316

H−2,0 +21H

−2,0,0 +30H−2,2 −

313

H−1ζ2 −42H

−1ζ3 +94

H0

−4H−1,−2,0 +56H

−1,−1ζ2 −36H−1,−1,0,0 −56H

−1,−1,2 −134

9H−1,0 −42H

−1,0ζ2 −H3,0

+32H−1,3 −

316

H−1,0,0 +17H

−1,0,0,0 +313

H−1,2 +2H

−1,2,0 +1312

H0ζ2 +292

H0ζ3 +679

H0,0

+13H0,0ζ2 +8912

H0,0,0 −5H0,0,0,0 −7H2ζ2 −316

H3 −10H4

]

+(1− x)[133

36+4H0,0,0,0

−

1674

ζ3 −2H0ζ3 −2H−3,0 +H

−2ζ2 +2H−2,−1,0 −3H

−2,0,0 +774

H0,0,0 −209

6H1 −7H1ζ2

+4H1,0,0 +143

H1,0

]

+(1+ x)[43

2ζ2 −3ζ2

2 +252

H−2,0 −31H

−1ζ2 −14H−1,−1,0 −

133

H−1,0

+24H−1,2 +23H

−1,0,0 +552

H0ζ2 +5H0,0ζ2 +1457

48H0 −

102536

H0,0 −1556

H2 +H2ζ2 −15H3

+2H2,0,0 −3H4

]

−5ζ2 −12

ζ22 +50ζ3 −2H

−3,0 −7H−2,0 −H0ζ3 −

372

H0ζ2 −2429

H0

−2H0,0ζ2 +185

6H0,0 −22H0,0,0 −4H0,0,0,0 +

283

H2 +6H3 +δ(1− x)[151

64+ζ2ζ3 −

20524

ζ2

−

24760

ζ22 +

21112

ζ3 +152

ζ5

])

+16CA2CF

(

pqq(x)[245

48−

6718

ζ2 +125

ζ22 +

12

ζ3 +1043216

H0

+H−3,0 +4H

−2,−1,0 −32

H−2,0 −H

−2,0,0 +2H−2,2 −

3112

H0ζ2 +4H0ζ3 +38972

H0,0 −2H2,0,0

−H0,0,0,0 +9H1ζ3 +6H1,−2,0 −H1,0ζ2 −114

H1,0,0 −3H1,0,0,0 −4H1,1,0,0 +4H1,3 +3112

H0,0,0

+1112

H3 +H4

]

+ pqq(−x)[67

18ζ2 −ζ2

2−

114

ζ3 −H−3,0 +8H

−2ζ2 +116

H−2,0 −4H

−2,0,0

−3H−1,0,0,0 +

113

H−1ζ2 +12H

−1ζ3 −16H−1,−1ζ2 +8H

−1,−1,0,0 +16H−1,−1,2 +

679

H−1,0

−8H−2,2 +11H

−1,0ζ2 +116

H−1,0,0 −

113

H−1,2 −8H

−1,3 −34

H0 −16

H0ζ2 −4H0ζ3 −6718

H0,0

−3H0,0ζ2 −3112

H0,0,0 +H0,0,0,0 +2H2ζ2 +116

H3 +2H4

]

+(1− x)[1883

108−

12

H0,0,0,0 +11H1

−H−2,−1,0 +

12

H−3,0 −

12

H−2ζ2 +

12

H−2,0,0 +

52336

H0 +H0ζ3 −133

H0,0 −52

H0,0,0 +2H1ζ2

−2H1,0,0

]

+(1+ x)[

8H−1ζ2 +4H

−1,−1,0 +83

H−1,0 −5H

−1,0,0 −6H−1,2 −

133

ζ2 +38

ζ22

−

434

ζ3 −52

H−2,0 −

112

H0ζ2 −12

H2ζ2 −54

H0,0ζ2 +7H2 −14

H2,0,0 +3H3 +34

H4

]

+12

H0,0ζ2

+14

ζ22−

83

ζ2 +172

ζ3 +H−2,0 −

192

H0 +52

H0ζ2 −H0ζ3 +133

H0,0 +52

H0,0,0 +12

H0,0,0,0

−δ(1− x)[1657

576−

28127

ζ2 +18

ζ22 +

979

ζ3 −52

ζ5

])

+16CFn2f

( 118

pqq(x)[

H0,0 −13

+53

H0

]

+(1− x)[13

54+

19

H0

]

−δ(1− x)[ 17

144−

527

ζ2 +19

ζ3

])

+16CF2nf

(13

pqq(x)[

5ζ3 −4H1,0,0

−

5516

+58

H0 +H0ζ2 +32

H0,0 −H0,0,0 −103

H1,0 −103

H2 −2H2,0 −2H3

]

+23

pqq(−x)[5

3ζ2

−

32

ζ3 +H−2,0 +2H

−1ζ2 +103

H−1,0 +H

−1,0,0 −2H−1,2 −

12

H0ζ2 −53

H0,0 −H0,0,0 +H3

]

− (1− x)[10

9+

1918

H0,0 −43

H1 +23

H1,0 +43

H2

]

+(1+ x)[4

3H−1,0 −

2524

H0 +12

H0,0,0

]

+29

H0

+79

H0,0 +43

H2 −δ(1− x)[23

16−

512

ζ2 −2930

ζ22 +

176

ζ3

])

+16CF3(

pqq(x)[ 9

10ζ2

2−2H

−3,0

+6H−2ζ2 +12H

−2,−1,0 −6H−2,0,0 −

316

H0 −32

H0ζ2 +H0ζ3 +138

H0,0 −2H0,0,0,0 +8H1,3

+12H1ζ3 +8H1,−2,0 −6H1,0,0 −4H1,0,0,0 +4H1,2,0 −3H2,0 +2H2,0,0 +4H2,1,0 +4H2,2

+4H3,0 +4H3,1 +2H4

]

+ pqq(−x)[7

2ζ2

2−

92

ζ3 −6H−3,0 +32H

−2ζ2 +8H−2,−1,0 +3H

−2,0

−26H−2,0,0 −28H

−2,2 +6H−1ζ2 +36H

−1ζ3 +8H−1,−2,0 −48H

−1,−1ζ2 +40H−1,−1,0,0

+48H−1,−1,2 +40H

−1,0ζ2 +3H−1,0,0 −22H

−1,0,0,0 −6H−1,2 −4H

−1,2,0 −32H−1,3 −

32

H0

−

32

H0ζ2 −13H0ζ3 −14H0,0ζ2 −92

H0,0,0 +6H0,0,0,0 +6H2ζ2 +3H3 +2H3,0 +12H4

]

+(1− x)[

2H−3,0 −

318

+4H−2,0,0 +H0,0ζ2 −3H0,0,0,0 +35H1 +6H1ζ2 −H1,0 +

52

H2,0

]

+(1+ x)[37

10ζ2

2−

934

ζ2 −812

ζ3 −15H−2,0 +30H

−1ζ2 +12H−1,−1,0 −2H

−1,0 −26H−1,0,0

−24H−1,2 −

53916

H0 −28H0ζ2 +1918

H0,0 +20H0,0,0 +854

H2 −3H2,0,0 −2H3,0 +13H3

−H4

]

+4ζ2 +33ζ3 +4H−3,0 +10H

−2,0 +672

H0 +6H0ζ3 +19H0ζ2 −25H0,0 −17H0,0,0

−2H2 −H2,0 −4H3 +δ(1− x)[29

32−2ζ2ζ3 +

98

ζ2 +185

ζ22 +

174

ζ3 −15ζ5

])

.

P(2)−ns (x) = P(2)+

ns (x)+16CACF

(

CF −

CA

2

)(

pqq(−x)[134

9ζ2 −4ζ2

2−11ζ3 −4H

−3,0

+32H−2ζ2 +

223

H−2,0 −16H

−2,0,0 −32H−2,2 +

443

H−1ζ2 +48H

−1ζ3 −64H−1,−1ζ2

+32H−1,−1,0,0 +64H

−1,−1,2 +268

9H−1,0 +44H

−1,0ζ2 +223

H−1,0,0 −12H

−1,0,0,0 −443

H−1,2

−32H−1,3 −3H0 −

23

H0ζ2 −16H0ζ3 −134

9H0,0 −12H0,0ζ2 −

313

H0,0,0 +4H0,0,0,0 +8H2ζ2

+223

H3 +8H4

]

+(1− x)[367

18+

12

ζ22 +2H

−3,0 −2H−2ζ2 −4H

−2,−1,0 −10H−2,0 −2H0,0

+2H−2,0,0 +2H0ζ3 +H0,0ζ2 −H0,0,0,0 +8H1ζ2 +

1403

H1

]

+(1+ x)[

32H−1ζ2 −18ζ2

−23ζ3 +263

H−1,0 −16H

−1,0,0 −32H−1,2 −

48118

H0 −29H0ζ2 +5H0,0,0 +24H3 +703

H2

]

−2ζ2 −2ζ3 +32H0 +14H0ζ2 +2H0,0,0 −16H3

)

+16CFnf

(

CF −

CA

2

)(

pqq(−x)[

2ζ3

−

209

ζ2 −43

H−2,0 −

83

H−1ζ2 −

409

H−1,0 −

43

H−1,0,0 +

83

H−1,2 +

23

H0ζ2 +209

H0,0 +43

H0,0,0

−

43

H3

]

+(1− x)[61

9−

83

H1

]

+(1+ x)[

2H0,0 −83

H−1,0 +

419

H0 −43

H2

])

+16CF2(

CF −

CA

2

)(

pqq(−x)[

9ζ3 −7ζ22 +12H

−3,0 −64H−2ζ2 −16H

−2,−1,0 −6H−2,0

+52H−2,0,0 +56H

−2,2 −12H−1ζ2 −72H

−1ζ3 −16H−1,−2,0 +96H

−1,−1ζ2 −80H−1,−1,0,0

−96H−1,−1,2 −80H

−1,0ζ2 −6H−1,0,0 +44H

−1,0,0,0 +12H−1,2 +8H

−1,2,0 +64H−1,3 +3H0

+3H0ζ2 +26H0ζ3 +28H0,0ζ2 +9H0,0,0 −12H0,0,0,0 −12H2ζ2 −6H3 −4H3,0 −24H4

]

− (1− x)[

15+8H−3,0 +8H

−2,0,0 +61H0 +6H0ζ3 +2H0,0ζ2 −6H0,0,0,0 +12H1ζ2 +60H1

+8H1,0

]

+(1+ x)[

24ζ2 +57ζ3 +10H−2,0 −48H

−1ζ2 −4H−1,0 +40H

−1,0,0 +48H−1,2

+59H0ζ2 −22H0,0 −35H0,0,0 −22H2 −4H2,0 −44H3

]

+8ζ2 −42ζ3 −4H−2,0 +42H0

−38H0ζ2 +14H0,0 −16H2 +26H0,0,0 +24H3

)

.

P (2)sns (x) = 16nf

dabcdabc

nc

(12(1− x)

[503

+4112

ζ2 −54

ζ22−H

−3,0 +H−2ζ2 −H

−2,0,0 +94

H3

+2H−2,−1,0 +

32

H0,0ζ2 −12

H1ζ2 −34

H1,0,0 +9112

H1

]

+12(1+ x)

[

H−1,−1,0 −

32

H−1ζ2 +

34

H0

−

136

H−1,0 +

12

H−1,0,0 +2H

−1,2 −32

H−2,0 +

94

H0ζ2 +2912

H0,0 +4112

H2 −H2ζ2 −12

H2,0,0

+32

H4

]

−

13

(1x

+ x2)[

3H−1ζ2 +2H

−1,−1,0 −2H−1,0,0 −2H

−1,2 +H1ζ2

]

+13

x2[

5ζ3 −2H3

+2H−2,0 +4H0ζ2 −2H0,0,0 +2H1ζ2

]

+9124

H0 +ζ3 −92

ζ2 +ζ22−H0ζ3 −H0ζ2 −2H0,0ζ2

+38

H0,0 −14

H0,0,0 +12

H0,0,0,0 +H−2,0 −H3

)

.



LO singlet P@

NNLO singlet splitting functionsS.M., Vermaseren, Vogt ‘04

P(2)ps (x) = 16CACFnf

(43(1x

+ x2)[13

3H−1,0 −

149

H0 +12

H−1ζ2 −H

−1,−1,0 −2H−1,0,0

−H−1,2

]

+23(1x− x2)

[163

ζ2 +H2,1 +9ζ3 +94

H1,0 −6761216

+57172

H1 +103

H2 +H1ζ2 −16

H1,1

−3H1,0,0 +2H1,1,0 +2H1,1,1

]

+(1− x)[182

9H1 +

1583

+39736

H0,0 −132

H−2,0 +3H0,0,0,0

+136

H1,0 +3xH1,0 +H−3,0 +H

−2ζ2 +2H−2,−1,0 +3H

−2,0,0 +12

H0,0ζ2 +12

H1ζ2 −94

H1,0,0

−

34

H1,1 +H1,1,0 +H1,1,1

]

+(1+ x)[ 7

12H0ζ2 +

316

ζ3 +9118

H2 +7112

H3 +11318

ζ2 −82627

H0

+52

H2,0 +163

H−1,0 +6xH

−1,0 +316

H0,0,0 −176

H2,1 +11720

ζ22 +9H0ζ3 +

52

H−1ζ2 +2H2,1,0

+12

H−1,0,0 −2H

−1,2 +H2ζ2 −72

H2,0,0 +H−1,−1,0 +2H2,1,1 +H3,1 −

12

H4

]

+5H−2,0 +H2,1

+H0,0,0,0 −12

ζ22 +4H

−3,0 +4H0ζ3 −329

H0,0 −2912

H0 −23512

ζ2 −51112

−

9712

H1 +334

H2 −H3

−

112

H0ζ2 −112

ζ3 −32

H2,0 −10H0,0,0 +23

x2[83

4H0,0 −

2434

H0 +10ζ2 +511

8+

978

H1 −43

H2

−4ζ3 −H0ζ2 +H3 +H2,0 −6H−2,0

])

+16CFnf2( 2

27H0 −2−H2 +ζ2 +

23

x2[

H2 −ζ2 +3

−

196

H0

]

+29(1x− x2)

[

H1,1 +53

H1 +23

]

+(1− x)[1

6H1,1 −

76

H1 + xH1 +3527

H0 +18554

]

+13(1+ x)

[43

H2 −43

ζ2 +ζ3 +H2,1 −2H3 +2H0ζ2 +296

H0,0 +H0,0,0

])

+16CF2nf

(8512

H1

−

254

H0,0 −H0,0,0 +58312

H0 −10154

+734

ζ2 −734

H2 +H3 −5H2,0 −H2,1 −H0ζ2 + x2[55

12

−

8512

H1 −223

H0,0 −109

6−

1354

H0 +289

ζ2 −289

H2 −163

H0ζ2 +163

H3 +4H2,0 +43

H2,1 −263

ζ3

+223

H0,0,0

]

+43(1x− x2)

[2312

H1,0 −523144

H1 −3ζ3 +5516

+12

H1,0,0 +H1,1 −H1,1,0 −H1,1,1

]

+(1− x)[1

2H1,0,0 +

712

H1,1 −2743

72H0 −

5312

H0,0 −25112

H1 −54

ζ2 +54

H2 −83

H1,0 +3xH1,0

+3H0ζ2 −3H3 −H1,1,0 −H1,1,1

]

+(1+ x)[1669

216+

52

H0,0,0 +4H2,1 +7H2,0 +10xζ3 −3710

ζ22

−7H0ζ3 +6H0,0ζ2 −4H0,0,0,0 +H2,0,0 −2H2,1,0 −2H2,1,1 −4H3,0 −H3,1 −6H4

])

.

P (2)qg (x) = 16CACF nf

(

pqg(x)[39

2H1ζ3 −4H1,1,1 +3H2,0,0 −

154

H1,2 +94

H1,1,0 +3H2,1,0

+H0ζ3 −2H2,1,1 +4H2ζ2 −17312

H0ζ2 −55172

H0,0 +643

ζ3 −ζ22−

494

H2 −32

H1,0,0,0 −13

H1,0,0

−

38572

H1,0 −312

H1,1 −11312

H1 +494

H2,0 +52

H1ζ2 +796

H0,0,0 +17312

H3 −1259

32+

2833216

H0

+6H2,1 +3H1,−2,0 +9H1,0ζ2 +6H1,1ζ2 +H1,1,0,0 +3H1,1,1,0 −4H1,1,1,1 −3H1,1,2 −6H1,2,1

−6H1,3 +494

ζ2

]

+ pqg(−x)[17

2H−1ζ3 −

52

H−1,−1,0 −

52

H−1,2 −

92

H−1,0 +

52

H−2,0 +

32

H−1,0,0

−2H3,1 −2H4 −6H−2,2 +6H

−2,−1,0 −6H−2,0,0 +2H0,0ζ2 +9H

−2ζ2 +3H−1,−2,0 −2H

−1,2,1

−6H−1,−1,−1,0 +6H

−1,−1,0,0 +6H−1,−1,2 +9H

−1,0ζ2 −9H−1,−1ζ2 −2H

−1,2,0 −112

H−1,0,0,0

−6H−1,3

]

+(1x− x2)

[5512

−4ζ3 +239

H1,0 −43

H1,1,0

]

+(1x

+ x2)[2

3H1,0,0 −

371108

H1 +239

H1,1

−

23

H1,1,1

]

+(1− x)[

6H2,1,0 +3H2,1,1 −56

H1,1,1 −7H2,0,0 −2H1,2 +39H0ζ3 −4H2ζ2 −163

ζ3

+H1,1,0 +154

3H0ζ2 +

89924

H0,0 +12110

ζ22 +

60736

H2 −52

H1ζ2 +656

H1,0,0 −2912

H1,0 −1318

H1,1

−

1189108

H1 −673

H2,1 −29H2,0 −94936

ζ2 −672

H0,0,0 −142

3H3 +

21532

−

398948

H0 +2H−3,0

]

+(1+ x)[

H−1,0,0 −10H

−2ζ2 +6H−2,0,0 +2H0,0ζ2 −9H

−1,−1,0 −7H−1,2 −9H

−2,0 −2H3,1

−4H−2,−1,0 −4H4 −4H3,0 −4H0,0,0,0 +

372

H−1,0 +

52(1+ x)H

−1ζ2

]

−4H−2,0,0 +2H0,0ζ2

+H2ζ2 −3H1,1,0 +2H0,0,0,0 +H−3,0 −9H2,1,0 −

92

H2,1,1 +113

H1,1,1 +192

H2,0,0 +92

H1,2

−

912

H0ζ3 +8H−2ζ2 +

52

H−1,−1,0 +

52

H−1,2 +

92

H−1,0 +

392

H−2,0 −

47312

H0ζ2 −1853

48H0,0

−

21712

ζ3 −594

ζ22−

16918

H2 −134

H1ζ2 −23

H1,0,0 +16724

H1,0 +19118

H1,1 +1283108

H1 +18512

H2,1

+754

H2,0 +170

9ζ2 +

854

H0,0,0 +42512

H3 +7693192

+3659

48H0 −2x

[

xH2,2 +4H3,0 −4H−2,2

])

+16CAnf2(1

6pqg(x)

[

H1,2 −H1ζ2 −H1,0,0 −H1,1,0 −H1,1,1 −22918

H0 +43

H0,0 +112

]

+ x[1

6H2

−

5318

H0 +176

H0,0 −ζ3 +1118

ζ2 −139108

]

+13

pqg(−x)H−1,0,0 −

53162

(1x− x2)−

29(1− x)

[

6H0,0,0

−

76

xH1 −H0,0 +72

xH1,1

]

+79

x(1+ x)H−1,0 +

74

H0 −1954

H1 +H0,0,0 +59

H1,1 +59

H−1,0

−

85216

)

+16CA2nf

(

pqg(x)[

3H1,3 +316

H1,0,0 −172

H2,1 +75

ζ22−

5512

H1,1,0 +3112

H3 −312

H1ζ3

−

512

H2,0 −638

H1,0 −2312

H1,2 −155

6ζ3 +

2524

H2 −2537

27H0 +

8678

−

232

H−1,0,0 +3H4 −H1,1,1

+38372

H1,1 −252

H−2,0 −

38

ζ2 −74

H1ζ2 −3H0,0ζ2 −3112

H0ζ2 +103216

H1 +52

H1,0,0,0 +2561

72H0,0

+H1,1,1 −2H2,0,0 −3H1,−2,0 −5H1,0ζ2 +3H0,0,0 −H1,1ζ2 −H1,1,0,0 −4H1,1,1,0 +2H1,1,1,1

−2H1,1,2 −2H1,2,0

]

+ pqg(−x)[

H−1,−1ζ2 −2H

−1,2 −6H−1,−1,0 +H1,1,1 +2H

−2ζ2 −H−2,0,0

+72736

H−1,0 −H

−1ζ2 −2H−2,2 −

52

H−1ζ3 −H

−1,−2,0 +2H−1,−1,0,0 +2H

−1,−1,2 −32

H−1,0,0,0

+6H−1,−1,−1,0 −2H

−1,3 +2H−1,2,1

]

+(1x− x2)

[23

H2,1 +329

ζ2 −2H1,0,0 +43

H1,1,0 −109

H1,1

−

83

H−1,0,0 +

32

H1,0 +6ζ3 +16136

H1 −2351108

]

+23(1x

+ x2)[26

3H−1,0 −

289

H0 −2H−1,−1,0

−2H−1,2 +H1ζ2 +H

−1ζ2 +103

H2 +H1,1,1

]

+(1− x)[

15H0,0,0,0 −5H2ζ2 −656

ζ3 +116

H1,1,1

−

32

H4 +52

H0,0ζ2 +H1,1,0 −316

H2,0 +1712

H1,0 −55120

ζ22−

294

H1,0,0 −113

4H2 +

1869172

H0

+2243108

+265

6H−1,0,0 +

332

H2,0,0 +19H2,1 +3112

H1,1 +232

H−2,0 −

49736

ζ2 +296

H1ζ2 −14312

H3

−

116

H1,1,1 −1912

H0ζ2 +1223

72H1 −

436

H0,0,0 −3011

36H0,0

]

+(1+ x)[

8H2,1,0 −4H−1,2

+7H−1,−1,0 −

356

H1,1,1 −5H−2ζ2 −11H

−2,0,0 +13

H−1,0 +

152

H−1ζ2 +8H3,1 −10H

−2,−1,0

+5H2ζ2 +4H2,1,1 −H−3,0 +36H0ζ3 −5H2ζ2

]

+2H−1,2 +6H

−1,−1,0 −6H2,1,0 −3H2,1,1

−11H0,0,0,0 −5H3,1 +254

H1,1,1 +132

H−2ζ2 +

272

H−2,0,0 +

112

H−3,0 +

132

H2ζ2 −174

H1,0,0

+13H−2,−1,0 −

1712

H1,1,1 −34

H4 −14

H0,0ζ2 +H1,2 +112

H1,1,0 +7912

H2,0 +678

H1,0 +263

8ζ2

2

+119

3ζ3 +

96724

H2 −30512

H−1,0 −24H0ζ3 +H

−1ζ2 −13375

72H0 −

188918

−38H−1,0,0 −

212

H2,1

−

794

H2,0,0 −21724

H1,1 −72

H−2,0 +

7972

ζ2 +43

H1ζ2 +1712

H1,1,1 +1712

H0ζ2 +3118

H1 +3H0,0,0

+14512

H3 +1553

24H0,0

)

+16CFnf2(7

6H0,0,0 +

1136

H1 −73996

+16324

H0 +7

24H0,0 +2H0,0,0,0

−

59

H1,1 −59

H2 −5

18H1,0 +

59

ζ2 +16

pqg(x)[

H2,1 +912−

353

H0 −223

H0,0 +H1,1,1 +6H0,0,0

−ζ3 −2H1,0,0 +79

H1

]

+7781

(1x− x2)+(1− x)

[ 112

H1 −6463432

−4H0,0,0,0 −163

H0,0,0 +79

xH1,1

+79

xH2 +89

xH1,0 −79

xζ2

]

− (1+ x)[3475

216H0 +

10312

H0,0

])

+16CF2nf

(

pqg(x)[

7H1,3 +7H4

−2H−3,0 −7H1ζ3 +5H2,2 +6H3,0 +6H3,1 +H2,1,0 +4H2,0,0 +3H2,1 +2H2,1,1 +

52

H2,0

+618

H2 −618

ζ2 +878

H1 +112

H1,2 +618

H1,1 +172

H1,0 −7H0,0ζ2 +52

H1,0,0 +52

H1,1,0 −192

ζ3

+8132

+112

H3 −112

H0ζ2 −72

H1ζ2 +152

H0,0,0 +878

H0 +115

ζ22 +3H1,1,1 −5H2ζ2 −7H0ζ3

+11H0,0 −2H1,−2,0 −7H1,0ζ2 +3H1,0,0,0 −5H1,1ζ2 +4H1,1,0,0 +H1,1,1,0 +2H1,1,1,1 +5H1,1,2

+6H1,2,0 +6H1,2,1

]

+4pqg(−x)[

H0,0,0,0 −H−2,0 +H

−1,−1,0 −H−2,0,0 +

12

H−1,−2,0 −

58

H−1,0

−

54

H−1,0,0 −

12

H−3,0 +

12

H−1ζ2 +H

−1,−1,0,0 −14

H−1,0,0,0

]

+2(1− x)[

H2,1,0 −H2,0,0 −H2,2

−H3,1 −2H3,0 −2H−1ζ2 +H1,2 −H1,0,0 −H1,1,0 +H2ζ2 −ζ2

2 +438

H2 +498

ζ2 +138

H1,1

−

3316

H1 +52

H1,0 +72

H0,0ζ2 +214

ζ3 +47964

−

12

H1,1,1 −12

H3 +14

H2,1 +12

H2,1,1 +32

H0ζ2

+12

H0ζ3 −72

H4 +H1ζ2 −192

H0,0,0 −23916

H0,0 −40532

H0

]

+8(1+ x)[

H−1,−1,0 −H

−1,0,0

−H0,0,0,0 +34

H−2,0 −

94

H−1,0

]

−4H−1,−1,0 +5H0,0,0,0 +5H

−1,0,0 −13H−2,0 +

12

H−1,0

+4xH−2,0,0 −

1138

H2 −718

ζ2 −354

H1 −112

H1,2 −338

H1,1 −72

H1,0 −72

H0,0ζ2 −52

H1,0,0

−

52

H1,1,0 −52

ζ3 −15764

−

94

H1,1,1 −94

H3 −54

H2,1 −12

H2,1,1 +14

H0ζ2 +H2ζ2 +52

H0ζ3

+95

ζ22 +

72

H4 +72

H1ζ2 +494

H0,0,0 +39116

H0,0 +40116

H0 −H2,0,0 −H2,1,0 +H2,2 +H3,1

+2H3,0 +6H−1ζ2 +

12

H2,0 +2H−2ζ2 +4H

−2,−1,0

)

P(2)gq (x) = 16CACFnf

(29

x2[25

6H1 −

1314

+3ζ2 −H−1,0 −3H2 +H1,1 +

1256

H0 −H0,0

]

+56

pgq(x)[

H1,2 +H2,1 +967120

+25190

H1 −3910

H1,1 −3ζ3 −25

H0ζ2 −15

H1ζ2 −43

H1,0 +H1,1,0

−

25

H1,0,0 +H1,1,1 +25

H2,0

]

+23

pgq(−x)[

2H−1ζ2 +

74

ζ2 +4112

H−1,0 −

15172

H0 +12

H−2,0

+53

H2 +2H−1,−1,0 −H

−1,0,0 −H−1,2

]

+23(1− x)

[

H−2,0 +2ζ3 −H3

]

+(1+ x)[179

108H1

+59

ζ2 +259

H−1,0 −

536

H1,1 −16736

H0,0 −13

H2,1 −43

H0ζ2

]

−

19372

+14

H1 +19

H−1,0 +4H2

−

14

H1,1 +22718

H0 −3512

H0,0 −H2,1 −23

H0ζ2 +103

H−2,0 +3ζ3 +2H3 +

23

H0,0,0 + x[11

4ζ2

−

523144

−

1936

H2 +271108

H0 −56

H1,0

])

+16CACF2(

x2[7

2+

17354

H1 −2ζ3 −23

H1,1,1 −269

H1,1

−6H2 +2H2,1 +6ζ2 +33554

H0 −289

H0,0 −83

H0,0,0

]

+ pgq(x)[3

2H1ζ3 +

16332

−5ζ2 +274

ζ3

+6503432

H1 +29

H1,1 +353

H1,1,1 +4H2 +92

H2,1 +4H1,0,0 +2H2,0,0 −H2ζ2 +4112

H1,2 +H2,2

+19124

H1,0 +3H2,0 −2H2,1,1 −32

H−1ζ2 −

5912

H1ζ2 +5H1,−2,0 +H1,0ζ2 +52

H1,0,0,0 −2H1,1ζ2

+1

12H1,1,0 +5H1,1,0,0 −3H1,1,1,0 −4H1,1,1,1 −H1,1,2 −2H1,2,1 +H2,1,0

]

+ pgq(−x)[

H−1,0

+H−1,0ζ2 +

32

H−1,0,0 +

2710

ζ22−3H

−1,−1,0 −112

H−1ζ3 −3H

−1,−2,0 −32

H−1,0,0,0 −3H

−1,2

+5H−1,−1ζ2 −4H

−1,−1,0,0 −2H−1,−1,2 +6H

−1,−1,−1,0 +2H−1,2,1

]

+(1− x)[

H2ζ2 −H2,2

+2312

H1,0 −7061432

H0 −4631144

H0,0 −383

H0,0,0 −H−3,0 −2H3,0 −

4433432

H1 −2H2,0,0 −212

H1,0,0

−

25

ζ22−

72

H1,2 +232

H1ζ2 −4H0ζ3

]

+(1+ x)[49

6H3 −H

−2,0 −556

H0ζ2 −12

H3,1 −1159

36ζ2

+655576

−

1516

ζ3 −18518

H1,1 +16

H1,1,1 −959

H2 +296

H2,1 −171

4H−1,0 −12H

−1,0,0 +7H−1ζ2

+16H−1,−1,0 +

53

H2,0 +32

H2,1,1 +4H0,0,0,0

]

−35H−2,0 −

17927

H0 +2041144

H0,0 −196

H0,0,0

−2H3,0 −132

H0ζ2 −13H−3,0 −

132

H3,1 +152

H3 −2005

64+

1574

ζ2 +8ζ3 +1291432

H1 +5512

H1,1

+32

H2 +12

H2,1 +274

H−1,0 −

112

H1,0,0 −8H2,0,0 −4ζ22 +

32

H1,2 −H2,2 +52

H1ζ2 +8H−1,−1,0

+4H2,0 +32

H2,1,1 −H−1ζ2 +7H2ζ2 +6H

−2ζ2 +12H−2,−1,0 −6H

−2,0,0 + x[

3H1,1,1 −H0,0ζ2

+92

H−1,0,0 −

358

H1,0 +2H4 +3H1,1,0 +H−1,2

])

+16CA2CF

(

x2[2

3H1ζ2 −

210581

−

7718

H0,0

−6H3 +163

ζ3 −10H−1,0 −

143

H2,0 −23

H−1ζ2 −

143

H0,0,0 +104

9H2 −

43

H1,0,0 +379

H1,1

+43

H−1,−1,0 −

1049

ζ2 −83

H2,1 +14518

H1,0 +43

H−1,2 +

23

H1,1,1 −10927

H1 +83

H−1,0,0 +6H0ζ2

+4H−2,0 +

58427

H0

]

+ pgq(x)[7

2H1ζ3 +

1383052592

−

13

H2,0 +134

H−1ζ2 +2H2,1,1 +

112

H1,0,0

+4H3,1 −436

H1,1,1 −10912

ζ2 −173

H2,1 −7124

H1,0 −116

H−2,0 −

212

ζ3 +32

H1,0,0,0 −H1,−2,0

+39554

H0 −2H1,0ζ2 −H1,1ζ2 −5512

H1,1,0 +2H1,1,0,0 +4H1,1,1,0 +2H1,1,1,1 +4H1,1,2 −5512

H1,2

+6H1,2,0 +4H1,2,1 +4H1,3 +3H2,1,0 +3H2,2

]

+ pgq(−x)[23

2H−1ζ3 +5H

−2ζ2 +2H−2,−1,0

+10912

H−1,0 +H0ζ3 +

175

ζ22 +

16

H1ζ2 +2H2ζ2 −6524

H1,1 −192

H−1,−1,0 −4H3,0 −3H2,0,0

−7H−2,0,0 −

32

H−1,2 +

3379216

H1 −4H−2,2 −

496

H−1,0,0 −

112

H−1,0,0,0 −13H

−1,−1ζ2 −8H−1,3

−6H−1,−1,−1,0 +12H

−1,−1,0,0 +10H−1,−1,2 +10H

−1,0ζ2 +5H−1,−2,0 −2H

−1,2,0 −2H−1,2,1

+116

H0ζ2

]

+(1− x)[41699

2592−3H

−2,−1,0 −32

H−2ζ2 −

1289

ζ2 −4H3,0 +263

ζ3 −52

H−2,0,0

−7H1ζ2 +9712

H1,0,0 +103

H−1,0,0 +

24512

H3 −8H0,0,0,0

]

+(1+ x)[

4H3,1 −H2,1,1 +296

H−1,2

+176

H−2,0 −12H2,0 −

3112

H2,1 +12

H2,0,0 −H2ζ2 +6136

H1,0 −4H0ζ3 −133

H−1ζ2 −

463

H−1,−1,0

+254

H4 +934

H0ζ2 −559

H1,1 −7118

H2 +4918

H0,0 −132

H0,0ζ2 −4740

ζ22]

+61312592

−

312

H−2ζ2

−15H−2,−1,0 +

92

H−1,0,0 −3H2,1,1 −

94

H2,1 +533

H−2,0 −

12

H−2,0,0 −5H2,0 −

76

H1,1,1 −8H0ζ3

−

6740

ζ22 +

296

H−1,2 −H

−1,0 +8H−2,2 +25H0ζ2 +

4129

H1 +928

9H0 +

14

H4 −65H3 −38H0,0

−9H−3,0 −

173

H0,0,0 + x[27

2H−1,0 −

12

H0,0,0,0 +34

H0,0ζ2 +12

H−3,0 −14H0,0,0 +

112

H1,1,1

−

4336

ζ2 −12

H2ζ2 +772

H0 +74954

H1 +135

4ζ3 +

9724

H1,0 +4312

H1ζ2 −8512

H−1ζ2 −

133

H1,0,0

+5312

H2 +394

H1,1 −2H3,1 +136

H−1,−1,0 +

74

H2,0,0 −4H1,1,0 −4H1,2

])

+16CFnf2(1

9−

19

1x

+29

x−16

xH1 +16

pgq(x)[

H1,1 −53

H1

])

+16CF2nf

(49

x2[

H0,0 −116

H0 −72

+H−1,0

]

+13

pgq(x)[

H1,2 −H1,0 −H1ζ2 +9ζ3 +8312

H1,1 +2H−2,0 −

736

H1 +2H0ζ2 −162548

+32

H1,0,0

+2H1,1,0 −52

H1,1,1

]

+3118

pgq(−x)[95

93H0 −ζ2 −H

−1,0

]

+13(2− x)

[

6H0,0,0,0 −H3 −13051

288

−

132

ζ3 −4H−2,0 −H2,0 −

12

H1,0 −12

H2,1 +2H0,0,0 −65324

H0,0

]

+(1+ x)[

H0ζ2 −1187216

H0

+89

H2 −8518

H−1,0 −

10118

ζ2

]

−

8027

H0 +2318

ζ2 −13

H1,1 +54

xH1,1 −19

H1 −3712

xH1 +2318

H−1,0

+1501

54+H0ζ2 −H0,0,0 +

1013

H0,0 −13

H1,0

)

+16CF3(

pgq(x)[

3H1,1ζ2 +3H1ζ2 +72

ζ2

−

238

H1,1 −8H1ζ3 −6H1,−2,0 −2H1,0ζ2 +3H1,1,0 −3H1,1,0,0 −H1,1,1,0 +2H1,1,1,1 −3H1,1,2

−2H1,2,0 −2H1,2,1 −92

H1,1,1 −32

H1,0,0 −4716

−

4716

H1 −152

ζ3

]

+ pgq(−x)[

2H−1,−2,0

+6H−1,−1,0 +3H

−1ζ2 +74

H1,0 −165

ζ22−6H

−1,0,0 −72

H−1,0 +4H

−1,−1,0,0 −2H−1,0ζ2

−H−1,0,0,0

]

+(1− x)[

9H1,0,0 +H1,1,1 −10H1ζ2 +3H0ζ3 +H2,2 −H2ζ2 +H0,0,0 +5H2,0,0

−4H3 +H2,1,1 +3H0,0ζ2 +3H3,1 −3H4 +21116

H1 +4920

ζ22]

+(1+ x)[

11ζ3 +14

H1,1 +14

H1,0

+9116

H0 +36H−1,0 +8H

−1,0,0 −14H−1,−1,0 −7H

−1ζ2 +2H1,2 +4H0ζ2 −H2,1 +2H−2,0,0

+5H−2,0 +

112

H2 −2H0,0,0,0

]

−2H−1,−1,0 −H

−1ζ2 −134

ζ2 +94

H1,0 +9

20ζ2

2 +28732

+1116

H1

+4H−1,0,0 +16H

−3,0 −4H−2ζ2 −8H

−2,−1,0 −5H2ζ2 +194

H2 +H2,2 −358

H0,0 +9H0ζ3

+25H−2,0 +6H

−2,0,0 +32

x[58

3ζ2 −

73

H1ζ2 +4H1,1 −32

H1,1,1 +52

H1,0,0 −17596

+H3,1 +193

ζ3

+2H2,0 −14H0 +H0,0ζ2 −H−1,0 −H4 −

32

H2,1 +13

H2,1,1 +3H2,0,0 −56

H3 −H1,2 −76

H0ζ2

+23

H1,1,0 −296

H0,0,0 −185

8H0,0

])

.

P (2)gg (x) = 16CACFnf

(

x2[4

9H2 +3H1,0 −

9712

H1 +83

H−2,0 −

23

H0ζ2 +10327

H0 −163

ζ2 +2H3

−6H−1,0 +2H2,0 +

12718

H0,0 −51112

]

+ pgg(x)[

2ζ3 −5524

]

+43(1x− x2)

[1724

H1,0 −4318

H0

−

521144

H1 −6923432

−

12

H2,1 +2H1ζ2 +H0ζ2 −2H1,0,0 +1

12H1,1 −H1,1,0 −H1,1,1

]

−

17512

H2

+6H−1,0 +8H0ζ3 −6H

−2,0 −536

H0ζ2 −492

H0 +1854

ζ2 +51112

−

12

H2,0 −3H1,0 −4H0,0,0,0

−

6712

H0,0 +432

ζ3 −H2,1 +9712

H1 −4ζ22−

92

H3 −8H−3,0 +

332

H0,0,0 +43(1x

+ x2)[1

2H2 −H2,0

+113

H−1,0 +H

−2,0 +196

ζ2 +2ζ3 −H−1ζ2 −4H

−1,−1,0 −12

H−1,0,0 −H

−1,2

]

+(1− x)[

9H1ζ2

+12H0,0,0,0 −293108

+616

H0ζ2 −73

H1,0 −85736

H1 −9H0ζ3 +16H−2,−1,0 −4H

−2,0,0 +8H−2ζ2

−

132

H1,0,0 +34

H1,1 −H1,1,0 −H1,1,1

]

+(1+ x)[1

6H2,0 −

953

H−1,0 −

14936

H2 +3451108

H0

−7H−2,0 +

3029

H0,0 +196

H3 −99136

ζ2 −163

6ζ3 −

353

H0,0,0 +176

H2,1 −4310

ζ22 +13H

−1ζ2

+18H−1,−1,0 −H3,1 −6H4 −4H

−1,2 +6H0,0ζ2 +8H2ζ2 −7H2,0,0 −2H2,1,0 −2H2,1,1 −4H3,0

−9H−1,0,0

]

−

241288

δ(1− x))

+16CAnf2(19

54H0 −

124

xH0 −1

27pgg(x)+

1354

(1x− x2)

[53−H1

]

+(1− x)[11

72H1 −

71216

]

+29(1+ x)

[

ζ2 +1312

xH0 −12

H0,0 −H2

]

+29

288δ(1− x)

)

+16CA2nf

(

x2[

ζ3 +119

ζ2 +119

H0,0 −23

H3 +23

H0ζ2 +1639108

H0 −2H−2,0

]

+13

pgg(x)[10

3ζ2

−

20936

−8ζ3 −2H−2,0 −

12

H0 −103

H0,0 −203

H1,0 −H1,0,0 −203

H2 −H3

]

+109

pgg(−x)[

ζ2

+2H−1,0 +

310

H0ζ2 −H0,0

]

+13(1x− x2)

[

H3 −H0ζ2 −133

H2 +5443108

−3H1ζ2 +20536

H1

−

133

H1,0 +H1,0,0

]

+(1x

+ x2)[151

54H0 −

83

ζ2 +13

H−1ζ2 −ζ3 +2H

−1,−1,0 −23

H−1,0,0

−

379

H−1,0 +

23

H−1,2

]

+(1− x)[5

6H−2,0 +H

−3,0 +2H0,0,0 −26936

ζ2 −4097216

−3H−2ζ2

−6H−2,−1,0 +3H

−2,0,0 −72

H1ζ2 +67772

H1 +H1,0 +74

H1,0,0

]

+(1+ x)[193

36H2 −

112

H−1ζ2

+3920

ζ22−

712

H3 −539

H0,0 +7

12H0ζ2 −

52

H0,0ζ2 +5ζ3 −7H−1,−1,0 +

776

H−1,0 +

92

H−1,0,0

+2H−1,2 −3H2ζ2 −

23

H2,0 +32

H2,0,0 +32

H4

]

+19

ζ2 +7H−2,0 +2H2 +

45827

H0 +H0,0ζ2

+32

ζ22 +4H

−3,0 − x[131

12H0,0 −

83

H0ζ2 +72

H3 −H0,0,0,0 +76

H0,0,0 +1943216

H0 +6H0ζ3

]

−δ(1− x)[233

288+

16

ζ2 +1

12ζ2

2 +53

ζ3

])

+16CA3(

x2[

33H−2,0 +33H0ζ2 −

124918

H0,0

−44H0,0,0 −110

3H3 −

443

H2,0 +856

ζ2 +6409108

H0

]

+ pgg(x)[245

24−

679

ζ2 −3

10ζ2

2 +113

ζ3

−4H−3,0 +6H

−2ζ2 +4H−2,−1,0 +

113

H−2,0 −4H

−2,0,0 −4H−2,2 +

16

H0 −7H0ζ3 +679

H0,0

−8H0,0ζ2 +4H0,0,0,0 −6H1ζ3 −4H1,−2,0 +10H2,0,0 −6H1,0ζ2 +8H1,0,0,0 +8H1,1,0,0 +8H4

+134

9H1,0 +

116

H1,0,0 +8H1,2,0 +8H1,3 +134

9H2 −4H2ζ2 +8H3,1 +8H2,2 +

116

H3 +10H3,0

+8H2,1,0

]

+ pgg(−x)[11

2ζ2

2−

116

H0ζ2 −4H−3,0 +16H

−2ζ2 −12H−2,2 −

1349

H−1,0 +2H2ζ2

+8H−2,−1,0 +12H

−1ζ3 −18H−2,0,0 +8H

−1,−2,0 −16H−1,−1ζ2 +24H

−1,−1,0,0 +16H−1,−1,2

+18H−1,0ζ2 −16H

−1,0,0,0 −4H−1,2,0 −16H

−1,3 −5H0ζ3 −8H0,0ζ2 +4H0,0,0,0 +2H3,0

−

679

ζ2 +679

H0,0 +8H4

]

+(1

x− x2

)[16619162

+223

H2,0 −552

ζ3 −112

H0ζ2 −679

H2 −679

H1,0

−

413108

H1 −112

H1ζ2 +332

H1,0,0

]

+11(1x

+ x2)[71

54H0 −

16

H3 −389198

ζ2 −23

H−2,0 −

12

H−1ζ2

+H−1,−1,0 −

523198

H−1,0 +

83

H−1,0,0 +H

−1,2

]

+(1− x)[31

36H1 +

272

H1,0 −252

H1,0,0 −4H−3,0

−

26312

H0,0 −293

H0,0,0 −193

H−2,0 −

11317108

−4H−2ζ2 −8H

−2,−1,0 −12H−2,0,0 −

32

H1ζ2

]

+(1+ x)[27

2H0ζ2 −

436

H3 +293

H2,0 +4651216

H0 −32918

ζ2 +112

(1+ x)ζ3 −435

ζ22−

2156

H−1,0

−22H0,0ζ2 −8H0ζ3 −3H−1,−1,0 +38H

−1,0,0 +25H−1,2 +10H2,0,0 −4H2ζ2 +16H3,0 +26H4

−

1589

H2 −532

H−1ζ2

]

−29H0,0 −403

H0,0,0 +27H−2,0 +

413

H0ζ2 −20H3 −24H2,0 +536

ζ2

+60112

H0 +24ζ3 +2ζ22 +27H2 −4H0,0ζ2 −16H0ζ3 −16H

−3,0 +28xH0,0,0,0 +δ(1− x)[79

32

−ζ2ζ3 +16

ζ2 +1124

ζ22 +

676

ζ3 −5ζ5

])

+16CFnf2(2

9x2

[116

H0 +H2 −ζ2 +2H0,0 −9]

+13

H2

−

13

ζ2 −103

H0 −13

H0,0 +2+29(1x− x2)

[83

H1 −2H1,0 −H1,1 −7718

]

− (1− x)[1

3H1,0 +

16

H1,1

+49

+136

H1 + xH1

]

+13(1+ x)

[689

H0 −43

H2 +43

ζ2 +296

H0,0 −ζ3 +2H0ζ2 −H0,0,0 −2H3

−H2,1 −2H2,0

]

+11

144δ(1− x)

)

+16CF2nf

(43

x2[163

16+

278

H0 +72

H0,0 −H2,0 −ζ2 +94

H1,0

−H2,1 +12

H0,0,0 +8516

H1 +H2 −2H−2,0 −

32

ζ3

]

+43(1x− x2)

[3116

H1 −1116

−

54

H1,0 +12

H1,0,0

−H1ζ2 −H1,1 +H1,1,0 +H1,1,1 +ζ3

]

+43(1x

+ x2)[

H−1ζ2 +2H

−1,−1,0 −H−1,0,0

]

+21512

H0,0

+203

H0 −131

6+3H2,0 +

20512

xζ2 −3H1,0 +H2,1 −8512

H1 +114

H2 +8H−2,0 +2ζ2

2−H0ζ2

+H3 +6H0ζ3 +8H−3,0 −4xH0,0,0 +(1− x)

[10712

H1 −56

H1,0 −4ζ2 +H0ζ3 −8H−2,−1,0

−4H−2ζ2 +4H

−2,0,0 −4H1ζ2 +72

H1,0,0 −7

12H1,1 +H1,1,0 +H1,1,1

]

+(1+ x)[5

4H2 +

338

−

994

H0,0 −8H2,0 −54124

H0 −4H2,1 −32

H0,0,0 −2xζ3 +92

ζ22 +5H0ζ2 −5H3 −4H

−1ζ2

−8H−1,−1,0 +

673

H−1,0 +4H

−1,0,0 +2H0,0ζ2 −2H0,0,0,0 −4H2ζ2 +3H2,0,0 +2H2,1,0

+2H2,1,1 +H3,1 −2H4

]

+1

16δ(1− x)

)

.



NLO C@

Coefficient functions in DIS at three loopsS.M., Vermaseren, Vogt ‘05

Exact (analytical) results for coefficient functions of F2 and FL

fill O(100) pages (normalsize fonts)



Summary

Symbolic summation

Polynomial summation −→ solved in Mathematica, Maple, ...

Hypergeometric summation −→ algorithms of Gosper, Wilf-Zeilberger

Harmonic summation −→ algebraic structures, recursion relations

Beyond −→ generalizations ...

Examples in particle physicsFeynman diagram calculations

QCD splitting functions

...



Symbolic Summation and Higher Orders in Perturbation TheoryHypergeometric summation Harmonic...

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Transcript of Symbolic Summation and Higher Orders in Perturbation TheoryHypergeometric summation Harmonic...