M.Echevarria, I.Scimemi, A.Vladimirov
“Unpolarized Transverse Momentum Dependent Parton Distributions and Fragmentation Functions at next-to-next-to-leading order”
arXiv:16??.????

Abstract:
This file contains the results of calculation of unpolarized TMD PDF and TMD FF, and the small - bT matching coefficients.
This is full version, that includes intermidiate expressions, and details.

VERSION:1.0

Table of content :

  Intermidiate results : (expressions for unsubtracted TMDs, soft factor, renormalization constants, as well as, their combination into TMD)
	1.Renormalization constants
	2.Soft factor
	3.unsubtracted TMD PDFs
	4.unsubtracted TMD FFs
	5.integrated PDFs
	6.integrated FFs
	5. TMD PDFs
	6. TMD FFs
	7. matching of TMD PDFs
	8. matching of TMD FFs
	
Final expressions : (expressions for matching coefficeints, including Logarithmical part)
	1.Matching coefficients for TMD PDF (complete expression)
	2.Matching coefficients for TMD PDF (logarithmical part)
	3.Matching coefficients for TMD PDF (finite part)
	4.Matching coefficients for TMD FF (complete expression)
	5.Matching coefficients for TMD FF (logarithmical part)
	6.Matching coefficients for TMD FF (finite part)

List of variables:

coupling constant:
as=((g^2)/(((4π))^2))

Dimension regularization:
ϵ=((4-d)/2),  d=4-2ϵ

color factors:
CF=(Subscript[C, F])=((((Subscript[N, c])^2)-1)/(2(Subscript[N, c])))
CA=(Subscript[C, A])=(Subscript[N, c])
TrNf=(Subscript[T, r])(Subscript[N, f])
TR=(Subscript[T, r])

Logarithms:
Lμ = ln(((((Subscript[b, T])^2)(μ^2))/(4 (e^(-2(Subscript[γ, E]))))))
lζ=ln(((μ^2)/ζ))

Special function:
DELTA[x] is Dirac delta-function

PLUS Distributions:
Though all the file the following convention for plus-distributions is used:
Only the singular at x,z->1 part of expression should be “plus”-regulated.
I.e.
f[x]=fr[x]+(SubPlus[(fs[x])]) 
where
fs[x]=Series[f[x],{x,1,-1}]
fr[x]=f[x]-fs[x]

Intermidiate results

Renomalization constants

Z2[n]=((Subscript[Z, 2])^([n])) quark field renormalization constant
Z3[n]=((Subscript[Z, 3])^([n]))gluon field renormalization constant
Zq[n]=((Subscript[Z, q])^([n]))quark TMD operator renormalization constant
Zg[n]=((Subscript[Z, g])^([n]))gluon TMD operator renormalization constant

Clear[Z2,Z3]
Z2[1]=((-1)/ϵ)CF;
Z3[1]=-(1/ϵ)(((-5)/3)CA+(4/3)TrNf);
Z2[2]=CF  (((CA+(CF/2))/(ϵ^2))+((-((17 CA)/4)+((3 CF)/4)+TrNf)/ϵ));
Z3[2]=((-((25 (CA^2))/12)+((5 CA TrNf)/3))/(ϵ^2))-((-((23 (CA^2))/8)+((5 CA TrNf)/2)+2 CF TrNf)/ϵ);

Zq[1]=-CF((2/(ϵ^2))+((4+2 lζ)/ϵ));
Zg[1]=-CA((2/(ϵ^2))+((2+2 lζ)/ϵ));
Zq[2]=((2 (CF^2))/(ϵ^4))+((CF (11 CA+8 CF (2+lζ)-4 TrNf))/(2 (ϵ^3)))+((CF (CA (50+66 lζ+3 (π^2))+4 (9 CF (((2+lζ))^2)-2 (2+3 lζ) TrNf)))/(18 (ϵ^2)))+((CF (4 ((92+60 lζ+9 (π^2)) TrNf+27 CF ((π^2)-12 Zeta[3]))+CA (-1420-99 (π^2)+12 lζ (-67+3 (π^2))+1404 Zeta[3])))/(108 ϵ));
Zg[2]=((2 (CA^2))/(ϵ^4))+((CA (CA (19+8 lζ)-4 TrNf))/(2 (ϵ^3)))+((CA (CA (55+276 lζ+72 ((lζ)^2)+6 (π^2))+(4-48 lζ) TrNf))/(36 (ϵ^2)))+((CA (4 (121+120 lζ-6 (π^2)) TrNf+CA (-2147+66 (π^2)+24 lζ (-67+3 (π^2))+216 Zeta[3])))/(216 ϵ));

Soft factor

The soft factor is universal for all TMDs. 
Soft factor is defined as 
S=exp(as S1+(as^2)S2)
The flavour dependance is given by CK, that is CA for adjoint Wilson lines, and CF for fundamental Wilson lines.

S1=CK (-4 lζ Lμ+(1/3) (6 ((Lμ)^2)+(π^2))-(4/(ϵ^2))+8 Lμ Log[δp]+((-4 lζ+8 Log[δp])/ϵ)+ϵ (lζ (-2 ((Lμ)^2)-((π^2)/3))+(2/3) (6 ((Lμ)^2)+(π^2)) Log[δp]+(2/3) (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3]))+(ϵ^2) ((((Lμ)^4)/2)+((((Lμ)^2) (π^2))/2)+((3 (π^4))/40)+(1/3) lζ (-2 ((Lμ)^3)-Lμ (π^2)-4 Zeta[3])+4 Lμ Zeta[3]+(2/3) Log[δp] (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));

S2=CK (((11 CA-4 TrNf)/(ϵ^3))+(((2/3) lζ (11 CA-4 TrNf)+(1/9) (20 TrNf+CA (-67+6 (π^2)+36 Log[2]))-(4/3) (11 CA-4 TrNf) Log[δp])/(ϵ^2))+(2/27) lζ (4 (28+30 Lμ+9 ((Lμ)^2)) TrNf+CA (-404-99 ((Lμ)^2)+6 Lμ (-67+3 (π^2))+378 Zeta[3]))-(4/27) Log[δp] (4 (28+30 Lμ+9 ((Lμ)^2)) TrNf+CA (-404-99 ((Lμ)^2)+6 Lμ (-67+3 (π^2))+378 Zeta[3]))+(1/ϵ)((2/9) lζ (CA (-67+3 (π^2))+20 TrNf)-(4/9) (CA (-67+3 (π^2))+20 TrNf) Log[δp]+(1/54) (4 (56-3 (π^2)) TrNf+CA (-808+(33+36 Lμ) (π^2)+54 Log[256]+36 Lμ Log[4096]+270 Zeta[3])))+(1/162) (-4 TrNf (-328+180 ((Lμ)^2)+72 ((Lμ)^3)+45 (π^2)+36 Lμ (π^2)+252 Zeta[3])+CA (792 ((Lμ)^3)-81 (π^4)+36 ((Lμ)^2) (67+18 Log[4])+9 (π^2) (67+24 ((Log[2])^2)+Log[4096])+36 Lμ (11 (π^2)+6 Log[4096]-81 Zeta[3])-4 (1214+54 ((Log[2])^4)-324 Log[4]+1296 PolyLog[4,(1/2)]-693 Zeta[3]+1134 Log[2] Zeta[3]))));

unsubtracted TMD PDFs

Φ[f←r,n]=((Subscript[Φ, f←r])^([n]))
NLO expressions are given up to (ϵ^2) order
Singularities at x->1 understood as "plus"-distribution

Q ← Q

Φ[q←q,0]=DELTA[1-x];
Φ[q←q,1]=CF (((2 (-(((-1+x))^2)+Lμ (1+(x^2))))/(-1+x))+((2 (1+(x^2)))/((-1+x) ϵ))+(((-12 Lμ (((-1+x))^2)+6 ((Lμ)^2) (1+(x^2))+(π^2) (1+(x^2))) ϵ)/(6 (-1+x)))+(((ϵ^2) (-6 ((Lμ)^2) (((-1+x))^2)-(π^2) (((-1+x))^2)+2 ((Lμ)^3) (1+(x^2))+Lμ (π^2) (1+(x^2))+4 (1+(x^2)) Zeta[3]))/(6 (-1+x)))+DELTA[1-x] (4 Lμ Log[δp]+((4 Log[δp])/ϵ)+(1/3) (6 ((Lμ)^2)+(π^2)) ϵ Log[δp]+(1/3) (ϵ^2) Log[δp] (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));
Φ[q←q,2]=CF (-((2 TrNf (2 (19+18 x+19 (x^2)+9 ((Lμ)^2) (1+(x^2))+12 Lμ (4-3 x+4 (x^2)))+6 (5+6 Lμ) (1+(x^2)) Log[x]+9 (1+(x^2)) ((Log[x])^2)))/(27 (-1+x)))+(1/(ϵ^2))(-((11 CA (1+(x^2)))/(3 (-1+x)))+((4 TrNf (1+(x^2)))/(3 (-1+x)))+((2 CF (2-4 x+2 (x^2)-4 (1+(x^2)) Log[1-x]+Log[x]+3 (x^2) Log[x]+4 Log[δp]+4 (x^2) Log[δp]))/(-1+x))+DELTA[1-x] ((1/6) CA ((π^2)+6 Log[4]-44 Log[δp])+(8/3) TrNf Log[δp]+CF (-((4 (π^2))/3)+8 ((Log[δp])^2))))+(1/ϵ)(-((4 TrNf (11-12 x+11 (x^2)+3 (1+(x^2)) Log[x]))/(9 (-1+x)))+((CA (187-240 x+187 (x^2)-3 (π^2) (1+(x^2))+3 (17+5 (x^2)) Log[x]+9 (1+(x^2)) ((Log[x])^2)))/(9 (-1+x)))+1/(-1+x)CF (2 (-5+2 x+Lμ (2+6 (x^2))) Log[x]+(-1+(x^2)) ((Log[x])^2)-4 Log[1-x] (-2 (((-1+x))^2)+4 Lμ (1+(x^2))+(1+(x^2)) Log[x])+2 ((-9+4 Lμ) (((-1+x))^2)+(-4 (((-1+x))^2)+8 Lμ (1+(x^2))) Log[δp]))+DELTA[1-x] (-(40/9) TrNf Log[δp]-(8/3) CF Lμ ((π^2)-6 ((Log[δp])^2))+CA (Log[16]+Lμ (((π^2)/3)+Log[16])-(2/9) (-67+3 (π^2)) Log[δp]-((9 Zeta[3])/2))))+(1/(54 (-1+x)))CA (16+1848 Lμ+198 ((Lμ)^2)+54 (π^2)-36 Lμ (π^2)+1584 x-2088 Lμ x-108 (π^2) x+16 (x^2)+1848 Lμ (x^2)+198 ((Lμ)^2) (x^2)+54 (π^2) (x^2)-36 Lμ (π^2) (x^2)-36 (π^2) Log[1-x]-108 x Log[1-x]+108 (x^2) Log[1-x]-36 (π^2) (x^2) Log[1-x]+348 Log[x]+612 Lμ Log[x]-432 x Log[x]+996 (x^2) Log[x]+180 Lμ (x^2) Log[x]-216 Log[1-x] Log[x]+432 x Log[1-x] Log[x]-216 (x^2) Log[1-x] Log[x]+216 ((Log[1-x])^2) Log[x]+216 (x^2) ((Log[1-x])^2) Log[x]+99 ((Log[x])^2)+108 Lμ ((Log[x])^2)+108 x ((Log[x])^2)-9 (x^2) ((Log[x])^2)+108 Lμ (x^2) ((Log[x])^2)+18 ((Log[x])^3)+18 (x^2) ((Log[x])^3)+216 (-(((-1+x))^2)+(1+(x^2)) Log[1-x]+(1+(x^2)) Log[x]) PolyLog[2,x]+216 (1+(x^2)) PolyLog[3,1-x]-432 PolyLog[3,x]-432 (x^2) PolyLog[3,x]-324 Zeta[3]-324 (x^2) Zeta[3])-(1/(3 (-1+x)))CF (-66+108 Lμ-24 ((Lμ)^2)+2 (π^2)+132 x-216 Lμ x+48 ((Lμ)^2) x-4 (π^2) x-66 (x^2)+108 Lμ (x^2)-24 ((Lμ)^2) (x^2)+2 (π^2) (x^2)-48 Lμ Log[1-x]+48 ((Lμ)^2) Log[1-x]+2 (π^2) Log[1-x]-6 x Log[1-x]+96 Lμ x Log[1-x]+6 (x^2) Log[1-x]-48 Lμ (x^2) Log[1-x]+48 ((Lμ)^2) (x^2) Log[1-x]+2 (π^2) (x^2) Log[1-x]+30 Log[x]+60 Lμ Log[x]-12 ((Lμ)^2) Log[x]-(π^2) Log[x]-78 x Log[x]-24 Lμ x Log[x]+96 (x^2) Log[x]-36 ((Lμ)^2) (x^2) Log[x]-3 (π^2) (x^2) Log[x]-36 Log[1-x] Log[x]+24 Lμ Log[1-x] Log[x]+72 x Log[1-x] Log[x]-36 (x^2) Log[1-x] Log[x]+24 Lμ (x^2) Log[1-x] Log[x]+18 ((Log[1-x])^2) Log[x]+18 (x^2) ((Log[1-x])^2) Log[x]+9 ((Log[x])^2)+6 Lμ ((Log[x])^2)+6 x ((Log[x])^2)-6 (x^2) ((Log[x])^2)-6 Lμ (x^2) ((Log[x])^2)+6 Log[1-x] ((Log[x])^2)+6 (x^2) Log[1-x] ((Log[x])^2)+((Log[x])^3)-(x^2) ((Log[x])^3)+48 Lμ Log[δp]-48 ((Lμ)^2) Log[δp]-4 (π^2) Log[δp]-96 Lμ x Log[δp]+48 Lμ (x^2) Log[δp]-48 ((Lμ)^2) (x^2) Log[δp]-4 (π^2) (x^2) Log[δp]+12 (-2 (((-1+x))^2)+(1+(x^2)) Log[1-x]+3 (1+(x^2)) Log[x]) PolyLog[2,x]+12 (1+(x^2)) PolyLog[3,1-x]-60 PolyLog[3,x]-60 (x^2) PolyLog[3,x]+60 Zeta[3]+60 (x^2) Zeta[3])+DELTA[1-x] (-(8/27) (28+30 Lμ+9 ((Lμ)^2)) TrNf Log[δp]-(2/9) CF (12 ((Lμ)^2)+(π^2)) ((π^2)-6 ((Log[δp])^2))+CA ((1/3) ((Lμ)^2) ((π^2)+Log[4096])+(2/27) Log[δp] (404+99 ((Lμ)^2)-6 Lμ (-67+3 (π^2))-378 Zeta[3])+Lμ (Log[256]-9 Zeta[3])+(1/36) (-7 (π^4)+12 (π^2) Log[2] (1+Log[4])-3 (8 ((Log[2])^4)+192 PolyLog[4,(1/2)]+5 Log[8] Zeta[3]+3 Log[2] (-32+51 Zeta[3]))))));

Q ← G

Φ[q←g,0]=0;
Φ[q←g,1]=TR (-2 (2 (-1+x) x+Lμ (1-2 x+2 (x^2)))+((-2+4 x-4 (x^2))/ϵ)+(-4 Lμ (-1+x) x+((Lμ)^2) (-1+2 x-2 (x^2))+(1/6) (-24 (-1+x) x+(π^2) (-1+2 x-2 (x^2)))) ϵ+(1/6) (ϵ^2) (-12 ((Lμ)^2) (-1+x) x+((Lμ)^3) (-2+4 x-4 (x^2))-Lμ (24 (-1+x) x+(π^2) (1-2 x+2 (x^2)))-4 Zeta[3]+2 x (12+(π^2)+4 Zeta[3])-2 (x^2) (12+(π^2)+4 Zeta[3])));
Φ[q←g,2]=(1/(ϵ^2))(CA TR (6+(8/(3 x))+8 x-((38 (x^2))/3)+(4-8 x+8 (x^2)) Log[1-x]+4 (1+4 x) Log[x])+CF TR (-9+20 x-16 (x^2)+(4-8 x+8 (x^2)) Log[1-x]+(-2+4 x-8 (x^2)) Log[x]-8 Log[δp]+16 x Log[δp]-16 (x^2) Log[δp]))+(1/ϵ)((1/3) CF TR (-18-48 Lμ+2 (π^2)+123 x+108 Lμ x-4 (π^2) x-120 (x^2)-84 Lμ (x^2)+4 (π^2) (x^2)-6 (1-2 x+2 (x^2)) ((Log[1-x])^2)-3 (-1-12 x+8 (x^2)+4 Lμ (1-2 x+4 (x^2))) Log[x]-3 (1-2 x+4 (x^2)) ((Log[x])^2)+12 Log[1-x] (2 (Lμ-x-2 Lμ x+(x^2)+2 Lμ (x^2))+(1-2 x+2 (x^2)) Log[x])-48 Lμ Log[δp]+48 x Log[δp]+96 Lμ x Log[δp]-48 (x^2) Log[δp]-96 Lμ (x^2) Log[δp])+1/(9 x)2 CA TR (-32+24 Lμ+54 x+36 Lμ x-81 (x^2)+108 Lμ (x^2)+6 (π^2) (x^2)+50 (x^3)-150 Lμ (x^3)+36 x (Lμ-x-2 Lμ x+(x^2)+2 Lμ (x^2)) Log[1-x]+9 x (1-2 x+2 (x^2)) ((Log[1-x])^2)-9 x Log[x]+36 Lμ x Log[x]+72 (x^2) Log[x]+144 Lμ (x^2) Log[x]-132 (x^3) Log[x]+9 x ((Log[x])^2)+18 (x^2) ((Log[x])^2)+18 x Log[x] Log[1+x]+36 (x^2) Log[x] Log[1+x]+36 (x^3) Log[x] Log[1+x]+18 x (1+2 x+2 (x^2)) PolyLog[2,-x]))+(1/6) CF TR (-78-72 Lμ-90 ((Lμ)^2)-8 (π^2)+8 Lμ (π^2)+546 x+468 Lμ x+204 ((Lμ)^2) x+10 (π^2) x-16 Lμ (π^2) x-528 (x^2)-456 Lμ (x^2)-156 ((Lμ)^2) (x^2)-6 (π^2) (x^2)+16 Lμ (π^2) (x^2)+48 ((Lμ)^2) Log[1-x]+4 (π^2) Log[1-x]-36 x Log[1-x]-96 Lμ x Log[1-x]-96 ((Lμ)^2) x Log[1-x]-8 (π^2) x Log[1-x]+48 (x^2) Log[1-x]+96 Lμ (x^2) Log[1-x]+96 ((Lμ)^2) (x^2) Log[1-x]+8 (π^2) (x^2) Log[1-x]-24 Lμ ((Log[1-x])^2)+24 x ((Log[1-x])^2)+48 Lμ x ((Log[1-x])^2)-24 (x^2) ((Log[1-x])^2)-48 Lμ (x^2) ((Log[1-x])^2)-4 ((Log[1-x])^3)+8 x ((Log[1-x])^3)-8 (x^2) ((Log[1-x])^3)+48 Log[x]+12 Lμ Log[x]-24 ((Lμ)^2) Log[x]+2 (π^2) Log[x]+90 x Log[x]+144 Lμ x Log[x]+48 ((Lμ)^2) x Log[x]-4 (π^2) x Log[x]-48 (x^2) Log[x]-96 Lμ (x^2) Log[x]-96 ((Lμ)^2) (x^2) Log[x]+48 Lμ Log[1-x] Log[x]-48 x Log[1-x] Log[x]-96 Lμ x Log[1-x] Log[x]+48 (x^2) Log[1-x] Log[x]+96 Lμ (x^2) Log[1-x] Log[x]+12 ((Log[1-x])^2) Log[x]-24 x ((Log[1-x])^2) Log[x]+24 (x^2) ((Log[1-x])^2) Log[x]+3 ((Log[x])^2)-12 Lμ ((Log[x])^2)+36 x ((Log[x])^2)+24 Lμ x ((Log[x])^2)-24 (x^2) ((Log[x])^2)-48 Lμ (x^2) ((Log[x])^2)-12 Log[1-x] ((Log[x])^2)+24 x Log[1-x] ((Log[x])^2)-24 (x^2) Log[1-x] ((Log[x])^2)-2 ((Log[x])^3)+4 x ((Log[x])^3)-8 (x^2) ((Log[x])^3)-96 ((Lμ)^2) Log[δp]-8 (π^2) Log[δp]+96 x Log[δp]+192 Lμ x Log[δp]+192 ((Lμ)^2) x Log[δp]+16 (π^2) x Log[δp]-96 (x^2) Log[δp]-192 Lμ (x^2) Log[δp]-192 ((Lμ)^2) (x^2) Log[δp]-16 (π^2) (x^2) Log[δp]+24 (1-2 x+2 (x^2)) (Log[1-x]-Log[x]) PolyLog[2,1-x]-24 (1-2 x+2 (x^2)) PolyLog[3,1-x]-24 PolyLog[3,x]+48 x PolyLog[3,x]-48 (x^2) PolyLog[3,x]+192 Zeta[3]-384 x Zeta[3]+384 (x^2) Zeta[3])+(1/(27 x))CA TR (272-384 Lμ+144 ((Lμ)^2)+12 (π^2)-414 x+648 Lμ x+162 ((Lμ)^2) x+18 (π^2) x+1638 (x^2)-756 Lμ (x^2)+756 ((Lμ)^2) (x^2)+90 (π^2) (x^2)+72 Lμ (π^2) (x^2)-1604 (x^3)+384 Lμ (x^3)-1008 ((Lμ)^2) (x^3)-75 (π^2) (x^3)+216 ((Lμ)^2) x Log[1-x]+18 (π^2) x Log[1-x]-270 (x^2) Log[1-x]-432 Lμ (x^2) Log[1-x]-432 ((Lμ)^2) (x^2) Log[1-x]-36 (π^2) (x^2) Log[1-x]+216 (x^3) Log[1-x]+432 Lμ (x^3) Log[1-x]+432 ((Lμ)^2) (x^3) Log[1-x]+36 (π^2) (x^3) Log[1-x]+108 Lμ x ((Log[1-x])^2)-108 (x^2) ((Log[1-x])^2)-216 Lμ (x^2) ((Log[1-x])^2)+108 (x^3) ((Log[1-x])^2)+216 Lμ (x^3) ((Log[1-x])^2)+18 x ((Log[1-x])^3)-36 (x^2) ((Log[1-x])^3)+36 (x^3) ((Log[1-x])^3)+252 x Log[x]-108 Lμ x Log[x]+216 ((Lμ)^2) x Log[x]+18 (π^2) x Log[x]+504 (x^2) Log[x]+864 Lμ (x^2) Log[x]+864 ((Lμ)^2) (x^2) Log[x]+816 (x^3) Log[x]-1584 Lμ (x^3) Log[x]-27 x ((Log[x])^2)+108 Lμ x ((Log[x])^2)+108 (x^2) ((Log[x])^2)+216 Lμ (x^2) ((Log[x])^2)-396 (x^3) ((Log[x])^2)+432 (x^2) Log[1-x] ((Log[x])^2)+18 x ((Log[x])^3)+36 (x^2) ((Log[x])^3)-18 (π^2) x Log[1+x]-36 (π^2) (x^2) Log[1+x]-36 (π^2) (x^3) Log[1+x]+216 Lμ x Log[x] Log[1+x]+216 (x^2) Log[x] Log[1+x]+432 Lμ (x^2) Log[x] Log[1+x]+216 (x^3) Log[x] Log[1+x]+432 Lμ (x^3) Log[x] Log[1+x]+54 x ((Log[x])^2) Log[1+x]+108 (x^2) ((Log[x])^2) Log[1+x]+108 (x^3) ((Log[x])^2) Log[1+x]-108 x Log[x] ((Log[1+x])^2)-216 (x^2) Log[x] ((Log[1+x])^2)-216 (x^3) Log[x] ((Log[1+x])^2)+36 x ((Log[1+x])^3)+72 (x^2) ((Log[1+x])^3)+72 (x^3) ((Log[1+x])^3)-36 (4-6 x+24 (x^2)-22 (x^3)+3 x (1-2 x+2 (x^2)) Log[1-x]-12 (x^2) Log[x]) PolyLog[2,1-x]+108 x (2 (Lμ+x+2 Lμ x+(x^2)+2 Lμ (x^2))-(1+2 x+2 (x^2)) Log[x]) PolyLog[2,-x]+108 x PolyLog[3,1-x]-216 (x^2) PolyLog[3,1-x]+216 (x^3) PolyLog[3,1-x]+108 x PolyLog[3,-x]+216 (x^2) PolyLog[3,-x]+216 (x^3) PolyLog[3,-x]+864 (x^2) PolyLog[3,x]-216 x PolyLog[3,(x/(1+x))]-432 (x^2) PolyLog[3,(x/(1+x))]-432 (x^3) PolyLog[3,(x/(1+x))]-108 x Zeta[3]+432 (x^2) Zeta[3]-216 (x^3) Zeta[3]);

G ← Q

Φ[g←q,0]=0;
Φ[g←q,1]=CF(Lμ (4-(4/x)-2 x)+2 x+((4-(4/x)-2 x)/ϵ)-(((-12 Lμ (x^2)+6 ((Lμ)^2) (2-2 x+(x^2))+(π^2) (2-2 x+(x^2))) ϵ)/(6 x))+(((ϵ^2) (6 ((Lμ)^2) (x^2)+(π^2) (x^2)-2 ((Lμ)^3) (2-2 x+(x^2))-Lμ (π^2) (2-2 x+(x^2))-4 (2-2 x+(x^2)) Zeta[3]))/(6 x)));
Φ[g←q,2]=(1/(3 x (ϵ^2)))CF (-42 CA+12 CF-16 TrNf+28 CA x+16 TrNf x+16 CA (x^2)+3 CF (x^2)-8 TrNf (x^2)+8 CA (x^3)+12 (CA+CF) (2-2 x+(x^2)) Log[1-x]-6 (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x]-24 CA (2-2 x+(x^2)) Log[δp])+(1/(9 x ϵ))CF (-18 CA-444 CA Lμ+36 CF Lμ+80 TrNf-38 CA x+9 CF x+360 CA Lμ x+36 CF Lμ x-12 CA (π^2) x-80 TrNf x-38 CA (x^2)+27 CF (x^2)+64 TrNf (x^2)-88 CA (x^3)+48 CA Lμ (x^3)-18 (CA-CF) (2-2 x+(x^2)) ((Log[1-x])^2)-144 CA Lμ Log[x]+216 CA x Log[x]-36 CF x Log[x]-144 CA Lμ x Log[x]+72 CF Lμ x Log[x]+90 CA (x^2) Log[x]-27 CF (x^2) Log[x]-144 CA Lμ (x^2) Log[x]-36 CF Lμ (x^2) Log[x]+48 CA (x^3) Log[x]-36 CA x ((Log[x])^2)+18 CF x ((Log[x])^2)-18 CA (x^2) ((Log[x])^2)-9 CF (x^2) ((Log[x])^2)+6 Log[1-x] (18 CF+24 CF Lμ+8 TrNf-18 CF x-24 CF Lμ x-8 TrNf x+3 CF (x^2)+12 CF Lμ (x^2)+4 TrNf (x^2)+CA (-22+22 x-17 (x^2)+12 Lμ (2-2 x+(x^2)))+6 CA (2-2 x+(x^2)) Log[x])-72 CA Log[x] Log[1+x]-72 CA x Log[x] Log[1+x]-36 CA (x^2) Log[x] Log[1+x]-288 CA Lμ Log[δp]+288 CA Lμ x Log[δp]+72 CA (x^2) Log[δp]-144 CA Lμ (x^2) Log[δp]-36 CA (2+2 x+(x^2)) PolyLog[2,-x])+(1/(54 x))CF (-6320 CA+42 CA (π^2)+18 CF (π^2)+896 TrNf+6328 CA x+540 CF x-108 CA (π^2) x+18 CF (π^2) x-896 TrNf x-2144 CA (x^2)-54 CF (x^2)+72 CA (π^2) (x^2)+208 TrNf (x^2)+1216 CA (x^3)-24 CA (π^2) (x^3)-1824 CA Log[1-x]+1728 CF Log[1-x]+72 CA (π^2) Log[1-x]+72 CF (π^2) Log[1-x]+480 TrNf Log[1-x]+1824 CA x Log[1-x]-1728 CF x Log[1-x]-72 CA (π^2) x Log[1-x]-72 CF (π^2) x Log[1-x]-480 TrNf x Log[1-x]-516 CA (x^2) Log[1-x]+540 CF (x^2) Log[1-x]+36 CA (π^2) (x^2) Log[1-x]+36 CF (π^2) (x^2) Log[1-x]+96 TrNf (x^2) Log[1-x]-396 CA ((Log[1-x])^2)+324 CF ((Log[1-x])^2)+144 TrNf ((Log[1-x])^2)+396 CA x ((Log[1-x])^2)-324 CF x ((Log[1-x])^2)-144 TrNf x ((Log[1-x])^2)-90 CA (x^2) ((Log[1-x])^2)+54 CF (x^2) ((Log[1-x])^2)+72 TrNf (x^2) ((Log[1-x])^2)-72 CA ((Log[1-x])^3)+72 CF ((Log[1-x])^3)+72 CA x ((Log[1-x])^3)-72 CF x ((Log[1-x])^3)-36 CA (x^2) ((Log[1-x])^3)+36 CF (x^2) ((Log[1-x])^3)-72 CA (π^2) Log[x]-2988 CA x Log[x]-810 CF x Log[x]-72 CA (π^2) x Log[x]+36 CF (π^2) x Log[x]+72 CA (x^2) Log[x]+270 CF (x^2) Log[x]-72 CA (π^2) (x^2) Log[x]-18 CF (π^2) (x^2) Log[x]-1056 CA (x^3) Log[x]-1584 CA Log[1-x] Log[x]+1728 CA x Log[1-x] Log[x]-648 CA (x^2) Log[1-x] Log[x]+288 CA (x^3) Log[1-x] Log[x]+216 CA ((Log[1-x])^2) Log[x]-216 CA x ((Log[1-x])^2) Log[x]+108 CA (x^2) ((Log[1-x])^2) Log[x]+648 CA x ((Log[x])^2)-108 CF x ((Log[x])^2)+162 CA (x^2) ((Log[x])^2)-81 CF (x^2) ((Log[x])^2)+144 CA (x^3) ((Log[x])^2)+216 CA Log[1-x] ((Log[x])^2)-216 CA x Log[1-x] ((Log[x])^2)+108 CA (x^2) Log[1-x] ((Log[x])^2)-72 CA x ((Log[x])^3)+36 CF x ((Log[x])^3)-36 CA (x^2) ((Log[x])^3)-18 CF (x^2) ((Log[x])^3)+72 CA (π^2) Log[1+x]+72 CA (π^2) x Log[1+x]+36 CA (π^2) (x^2) Log[1+x]+216 CA (x^2) Log[x] Log[1+x]-216 CA ((Log[x])^2) Log[1+x]-216 CA x ((Log[x])^2) Log[1+x]-108 CA (x^2) ((Log[x])^2) Log[1+x]+432 CA Log[x] ((Log[1+x])^2)+432 CA x Log[x] ((Log[1+x])^2)+216 CA (x^2) Log[x] ((Log[1+x])^2)-144 CA ((Log[1+x])^3)-144 CA x ((Log[1+x])^3)-72 CA (x^2) ((Log[1+x])^3)-72 CA (π^2) (2-2 x+(x^2)) Log[δp]+18 ((Lμ)^2) (-180 CA+6 CF+16 TrNf+152 CA x+18 CF x-16 TrNf x-16 CA (x^2)-3 CF (x^2)+8 TrNf (x^2)+16 CA (x^3)+24 (CA+CF) (2-2 x+(x^2)) Log[1-x]-12 (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x]-48 CA (2-2 x+(x^2)) Log[δp])+216 CA ((x^2)+(2+2 x+(x^2)) Log[x]) PolyLog[2,-x]+12 Lμ (-18 CA+80 TrNf-38 CA x+9 CF x-12 CA (π^2) x-80 TrNf x+10 CA (x^2)+36 CF (x^2)+40 TrNf (x^2)-88 CA (x^3)-18 (CA-CF) (2-2 x+(x^2)) ((Log[1-x])^2)+216 CA x Log[x]-36 CF x Log[x]+90 CA (x^2) Log[x]-27 CF (x^2) Log[x]+48 CA (x^3) Log[x]-36 CA x ((Log[x])^2)+18 CF x ((Log[x])^2)-18 CA (x^2) ((Log[x])^2)-9 CF (x^2) ((Log[x])^2)+6 Log[1-x] (18 CF+8 TrNf-18 CF x-8 TrNf x+3 CF (x^2)+4 TrNf (x^2)+CA (-22+22 x-17 (x^2))+6 CA (2-2 x+(x^2)) Log[x])-72 CA Log[x] Log[1+x]-72 CA x Log[x] Log[1+x]-36 CA (x^2) Log[x] Log[1+x]+72 CA (x^2) Log[δp]-36 CA (2+2 x+(x^2)) PolyLog[2,-x])+72 CA (-22+24 x-6 (x^2)+4 (x^3)+3 (6-2 x+3 (x^2)) Log[x]) PolyLog[2,x]-432 CA PolyLog[3,-x]-432 CA x PolyLog[3,-x]-216 CA (x^2) PolyLog[3,-x]-2160 CA PolyLog[3,x]+432 CA x PolyLog[3,x]-1080 CA (x^2) PolyLog[3,x]+864 CA PolyLog[3,(x/(1+x))]+864 CA x PolyLog[3,(x/(1+x))]+432 CA (x^2) PolyLog[3,(x/(1+x))]+2592 CA Zeta[3]-3024 CA x Zeta[3]+1296 CA (x^2) Zeta[3]);

G ← G

Φ[g←g,0]=DELTA[1-x];
Φ[g←g,1]=CA (((4 Lμ (((1-x+(x^2)))^2))/((-1+x) x))+((4 (((1-x+(x^2)))^2))/((-1+x) x ϵ))+(((6 ((Lμ)^2)+(π^2)) (((1-x+(x^2)))^2) ϵ)/(3 (-1+x) x))+(((((1-x+(x^2)))^2) (ϵ^2) (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3]))/(3 (-1+x) x))+DELTA[1-x] (4 Lμ Log[δp]+((4 Log[δp])/ϵ)+(1/3) (6 ((Lμ)^2)+(π^2)) ϵ Log[δp]+(1/3) (ϵ^2) Log[δp] (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));
Φ[g←g,2]=CF TrNf (((2 (-1+x) (12 (1+(-23+x) x)+24 Lμ (1+(-8+x) x)-12 ((Lμ)^2) (4+x (7+4 x))-(π^2) (4+x (7+4 x))))/(9 x))+(4/3) (12 ((Lμ)^2) (1+x)+(18+(π^2)) (1+x)+6 Lμ (3+x)) Log[x]+2 (3+x+4 Lμ (1+x)) ((Log[x])^2)+(4/3) (1+x) ((Log[x])^3))+CA TrNf (((4 (-121-18 ((Lμ)^2) (((1+(-1+x) x))^2)+x (231+x (-276+(249-139 x) x))-6 Lμ (23+x (-42+x (48+x (-42+23 x))))))/(27 (-1+x) x))-(4/3) x Log[1-x]+(4/9) (13+10 x+12 Lμ (1+x)) Log[x]+(4/3) (1+x) ((Log[x])^2)-(8/27) (28+30 Lμ+9 ((Lμ)^2)) DELTA[1-x] Log[δp])+(1/(ϵ^2))(CF TrNf (-((4 (-1+x) (4+x (7+4 x)))/(3 x))+8 (1+x) Log[x])+CA TrNf (((8 (((1+(-1+x) x))^2))/(3 (-1+x) x))+(8/3) DELTA[1-x] Log[δp])+(CA^2) (((2 (33+x (-58+x (39+x (-58+33 x)))))/(3 (-1+x) x))-((16 (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((8 (1+(-3+x) (-1+x) (x^2)) Log[x])/((-1+x) x))+((16 (((1+(-1+x) x))^2) Log[δp])/((-1+x) x))+DELTA[1-x] (-((7 (π^2))/6)+Log[4]-((22 Log[δp])/3)+8 ((Log[δp])^2))))+(1/ϵ)(CF TrNf (-((8 (-1+x) (-1-(-8+x) x+Lμ (4+x (7+4 x))))/(3 x))+4 (3+x+4 Lμ (1+x)) Log[x]+4 (1+x) ((Log[x])^2))+CA TrNf (-((4 (23+x (-42+x (48+x (-42+23 x)))))/(9 (-1+x) x))+(8/3) (1+x) Log[x]-(40/9) DELTA[1-x] Log[δp])+(CA^2) (((48 Lμ (((-1+x))^2) (1+x) (11+x (2+11 x))+x (109+109 (x^3)+25 x (1+x)-12 (π^2) (-1+(x^2)+(x^4))))/(9 x (-1+(x^2))))+((4 (((1+x-(x^2)))^2) ((Log[x])^2))/(-1+(x^2)))+Log[1-x] (-((32 Lμ (((1+(-1+x) x))^2))/((-1+x) x))-((8 (((1+(-1+x) x))^2) Log[x])/((-1+x) x)))+Log[x] ((2/3) (25+11 x (-1+4 x)+((24 Lμ (1+(-3+x) (-1+x) (x^2)))/((-1+x) x)))-((8 (((1+x+(x^2)))^2) Log[1+x])/(x (1+x))))+((32 Lμ (((1+(-1+x) x))^2) Log[δp])/((-1+x) x))-((8 (((1+x+(x^2)))^2) PolyLog[2,-x])/(x (1+x)))+DELTA[1-x] (-((7 Lμ (π^2))/3)+(1+Lμ) Log[16]+(2/9) (67-3 (π^2)) Log[δp]+16 Lμ ((Log[δp])^2)-((9 Zeta[3])/2))))+(CA^2) (-((4 (((1+(-1+x) x))^2) ((Log[1-x])^2) Log[x])/((-1+x) x))+((4 (((1+x-(x^2)))^2) ((Log[x])^3))/(3 (-1+(x^2))))+Log[1-x] (((2 (-1+x) (x^2)-96 ((Lμ)^2) (((1+(-1+x) x))^2)-8 (π^2) (((1+(-1+x) x))^2))/(3 (-1+x) x))-((8 (6 Lμ (((1+(-1+x) x))^2)-(((-1+x))^2) (11+x (-1+11 x))) Log[x])/(3 (-1+x) x))-((4 (((1+(-1+x) x))^2) ((Log[x])^2))/((-1+x) x)))+((4 (π^2) (((1+x+(x^2)))^2) Log[1+x])/(3 x (1+x)))-((8 (((1+x+(x^2)))^2) ((Log[1+x])^3))/(3 x (1+x)))+((Log[x])^2) (((24 Lμ (((1+x-(x^2)))^2)+(-1+x) (1+x) (25+11 x (-1+4 x)))/(3 (-1+(x^2))))-((4 (((1+x+(x^2)))^2) Log[1+x])/(x (1+x))))+Log[x] (((144 ((Lμ)^2) (1+(-3+x) (-1+x) (x^2))+12 (π^2) (1+(-3+x) (-1+x) (x^2))+12 Lμ (-1+x) x (25+11 x (-1+4 x))-(-1+x) x (701+x (149+536 x)))/(9 (-1+x) x))-((16 Lμ (((1+x+(x^2)))^2) Log[1+x])/(x (1+x)))+((8 (((1+x+(x^2)))^2) ((Log[1+x])^2))/(x (1+x))))+((8 (12 ((Lμ)^2)+(π^2)) (((1+(-1+x) x))^2) Log[δp])/(3 (-1+x) x))+(-((16 Lμ (((1+x+(x^2)))^2))/(x (1+x)))+((8 (((1+x+(x^2)))^2) Log[x])/(x (1+x)))) PolyLog[2,-x]+(((8 (-1+x) (11+x (-1+11 x)))/(3 x))-((8 (3+x (-1+x (3+x-3 (x^2)+(x^3)))) Log[x])/(x (-1+(x^2))))) PolyLog[2,x]-((8 (((1+x+(x^2)))^2) PolyLog[3,-x])/(x (1+x)))+((8 (5+x (-1+x (5+x-5 (x^2)+(x^3)))) PolyLog[3,x])/(x (-1+(x^2))))+((16 (((1+x+(x^2)))^2) PolyLog[3,(x/(1+x))])/(x (1+x)))+DELTA[1-x] (-(7/3) ((Lμ)^2) (π^2)-((31 (π^4))/72)+(2/3) (π^2) ((Log[2])^2)-((13 ((Log[2])^4))/12)+Log[256]+(4/3) (12 ((Lμ)^2)+(π^2)) ((Log[δp])^2)-16 PolyLog[4,(1/2)]+10 PolyLog[2,2,(1/2)]+(1/27) Log[δp] (808+6 Lμ (134+33 Lμ-6 (π^2))-756 Zeta[3])+Log[2] (4 Lμ (2+Lμ)+((π^2)/3)-((51 Zeta[3])/4))-9 Lμ Zeta[3])+(1/(27 x (-1+(x^2))))2 (3 Lμ x (109+109 (x^3)+25 x (1+x)-12 (π^2) (-1+(x^2)+(x^4)))+2 (1+x) (790+x (-1514+9 (π^2) (((-1+x))^2)+x (1686+5 x (-319+167 x))))+9 ((Lμ)^2) (99+x (-83+x (-5+x (-5+x (-83+99 x)))))-108 (6+x (-7+x (6+x (7+x (-6+7 x))))) Zeta[3]));

Q ← Q’

Φ[q←qp,0]=0;
Φ[q←qp,1]=0;
Φ[q←qp,2]=-((CF TR (-1+x) (272+2 x (-71+208 x)+36 ((Lμ)^2) (4+x (7+4 x))+3 (π^2) (4+x (7+4 x))-24 Lμ (16+x (-11+16 x))))/(27 x))+(2/9) CF TR (42+36 ((Lμ)^2) (1+x)+3 (π^2) (1+x)+8 x (-3+8 x)-6 Lμ (3+x (3+8 x))) Log[x]+(1/3) CF TR (-3+12 Lμ (1+x)-x (3+8 x)) ((Log[x])^2)+(2/3) CF TR (1+x) ((Log[x])^3)+((-((2 CF TR (-1+x) (4+x (7+4 x)))/(3 x))+4 CF TR (1+x) Log[x])/(ϵ^2))+((-((4 CF TR (-1+x) (-16+(11-16 x) x+3 Lμ (4+x (7+4 x))))/(9 x))+(2/3) CF TR (-3+12 Lμ (1+x)-x (3+8 x)) Log[x]+2 CF TR (1+x) ((Log[x])^2))/ϵ)+((8 CF TR (-1+x) (2+x (-1+2 x)) PolyLog[2,1-x])/(3 x));

Q ← (OverBar[Q])

Φ[q←qbar,0]=0;
Φ[q←qbar,1]=0;
Φ[q←qbar,2]=(((CA-2 CF) CF (1+(x^2)) ((Log[x])^3))/(3 (1+x)))+((2 (CA-2 CF) CF (1+(x^2)) ((Log[x])^2) (Lμ-2 Log[1-x]-Log[1+x]))/(1+x))+(((CA-2 CF) CF Log[x] (12 Lμ (((1+x))^2)-3 (1+x) (3+11 x)+2 (π^2) (1+(x^2))+12 Log[1+x] ((((1+x))^2)-2 Lμ (1+(x^2))+(1+(x^2)) Log[1+x])))/(3 (1+x)))+(-4 (CA-2 CF) CF (-1+x)-((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,1-x]+(((4 (CA-2 CF) CF ((((1+x))^2)-2 Lμ (1+(x^2))))/(1+x))+((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,-x]+((-(((CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (x^2)))/(3 (1+x)))+(((CA-2 CF) CF (1+(x^2)) ((Log[x])^2))/(1+x))+((2 (CA-2 CF) CF Log[x] ((((1+x))^2)-2 (1+(x^2)) Log[1+x]))/(1+x))-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[2,-x])/(1+x)))/ϵ)-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[3,-x])/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,x])/(1+x))+((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,(x/(1+x))])/(1+x))-(((CA-2 CF) CF (45-45 (x^2)-(π^2) (((1+x))^2)+2 Lμ (-12+(π^2)+(12+(π^2)) (x^2))-2 (1+(x^2)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+6 (1+(x^2)) Zeta[3]))/(3 (1+x)));

unsubtracted TMD FFs

Δ[f←r,n]=(z^2)((Subscript[Δ, f←r])^([n]))
NLO expressions are given up to (ϵ^2) order
Singularities at z->1 understood as "plus"-distribution

Q -> Q

Δ[q->q,0]=DELTA[1-z];
Δ[q->q,1]=CF (2-2 z+Lμ (2+(4/(-1+z))+2 z+4 DELTA[1-z] Log[δp])+((((2 (1+(z^2)))/(-1+z))+4 DELTA[1-z] Log[δp])/ϵ)+ϵ (Lμ (2-2 z)+((Lμ)^2) (((1+(z^2))/(-1+z))+2 DELTA[1-z] Log[δp])+(((π^2) (1+(z^2)+2 (-1+z) DELTA[1-z] Log[δp]))/(6 (-1+z))))+(ϵ^2) (((Lμ)^2) (1-z)+(1/3) ((Lμ)^3) (((1+(z^2))/(-1+z))+2 DELTA[1-z] Log[δp])+((Lμ (π^2) (1+(z^2)+2 (-1+z) DELTA[1-z] Log[δp]))/(6 (-1+z)))+(1/6) (-(π^2) (-1+z)+((4 (1+(z^2)) Zeta[3])/(-1+z))+8 DELTA[1-z] Log[δp] Zeta[3])));
Δ[q->q,2]=CF (-((2 TrNf (2 (19+18 z+19 (z^2)+9 ((Lμ)^2) (1+(z^2))+12 Lμ (4-3 z+4 (z^2)))-6 (5+6 Lμ) (1+(z^2)) Log[z]+9 (1+(z^2)) ((Log[z])^2)))/(27 (-1+z)))+(1/(ϵ^2))(-((11 CA (1+(z^2)))/(3 (-1+z)))+((4 TrNf (1+(z^2)))/(3 (-1+z)))+((2 CF (2-4 z+2 (z^2)-4 (1+(z^2)) Log[1-z]+Log[z]+3 (z^2) Log[z]+4 Log[δp]+4 (z^2) Log[δp]))/(-1+z))+DELTA[1-z] ((1/6) CA ((π^2)+6 Log[4]-44 Log[δp])+(8/3) TrNf Log[δp]+CF (-((4 (π^2))/3)+8 ((Log[δp])^2))))+(1/ϵ)(((4 TrNf (-11+12 z-11 (z^2)+3 (1+(z^2)) Log[z]))/(9 (-1+z)))+((CA (187-240 z+187 (z^2)-3 (π^2) (1+(z^2))-3 (5+17 (z^2)) Log[z]+9 (1+(z^2)) ((Log[z])^2)))/(9 (-1+z)))+1/(-1+z)CF (2 (-4+6 z+(z^2)+Lμ (2+6 (z^2))) Log[z]-(3+5 (z^2)) ((Log[z])^2)-4 Log[1-z] (-2 (((-1+z))^2)+4 Lμ (1+(z^2))-(1+(z^2)) Log[z])+2 ((-9+4 Lμ) (((-1+z))^2)+(-4 (((-1+z))^2)+8 Lμ (1+(z^2))) Log[δp]))+DELTA[1-z] (-(40/9) TrNf Log[δp]-(8/3) CF Lμ ((π^2)-6 ((Log[δp])^2))+CA (Log[16]+Lμ (((π^2)/3)+Log[16])-(2/9) (-67+3 (π^2)) Log[δp]-((9 Zeta[3])/2))))+(1/(3 (-1+z)))CF (66-108 Lμ+24 ((Lμ)^2)+6 (π^2)-132 z+216 Lμ z-48 ((Lμ)^2) z-12 (π^2) z+66 (z^2)-108 Lμ (z^2)+24 ((Lμ)^2) (z^2)+6 (π^2) (z^2)-6 Log[1-z]+48 Lμ Log[1-z]-48 ((Lμ)^2) Log[1-z]-6 (π^2) Log[1-z]+6 z Log[1-z]-96 Lμ z Log[1-z]+48 Lμ (z^2) Log[1-z]-48 ((Lμ)^2) (z^2) Log[1-z]-6 (π^2) (z^2) Log[1-z]+102 Log[z]-48 Lμ Log[z]+12 ((Lμ)^2) Log[z]-13 (π^2) Log[z]-84 z Log[z]+72 Lμ z Log[z]+30 (z^2) Log[z]+12 Lμ (z^2) Log[z]+36 ((Lμ)^2) (z^2) Log[z]-11 (π^2) (z^2) Log[z]-36 Log[1-z] Log[z]+24 Lμ Log[1-z] Log[z]+72 z Log[1-z] Log[z]-36 (z^2) Log[1-z] Log[z]+24 Lμ (z^2) Log[1-z] Log[z]+18 ((Log[1-z])^2) Log[z]+18 (z^2) ((Log[1-z])^2) Log[z]+30 ((Log[z])^2)-18 Lμ ((Log[z])^2)-54 z ((Log[z])^2)+15 (z^2) ((Log[z])^2)-30 Lμ (z^2) ((Log[z])^2)-30 Log[1-z] ((Log[z])^2)-30 (z^2) Log[1-z] ((Log[z])^2)+7 ((Log[z])^3)+9 (z^2) ((Log[z])^3)-48 Lμ Log[δp]+48 ((Lμ)^2) Log[δp]+4 (π^2) Log[δp]+96 Lμ z Log[δp]-48 Lμ (z^2) Log[δp]+48 ((Lμ)^2) (z^2) Log[δp]+4 (π^2) (z^2) Log[δp]+12 ((1+(z^2)) Log[1-z]-2 ((((-1+z))^2)+2 (1+(z^2)) Log[z])) PolyLog[2,z]+12 (1+(z^2)) PolyLog[3,1-z]+72 PolyLog[3,z]+72 (z^2) PolyLog[3,z]-72 Zeta[3]-72 (z^2) Zeta[3])-(1/(54 (-1+z)))CA (-16-1848 Lμ-198 ((Lμ)^2)+18 (π^2)+36 Lμ (π^2)-1584 z+2088 Lμ z-36 (π^2) z-16 (z^2)-1848 Lμ (z^2)-198 ((Lμ)^2) (z^2)+18 (π^2) (z^2)+36 Lμ (π^2) (z^2)-108 Log[1-z]-36 (π^2) Log[1-z]+108 z Log[1-z]-36 (π^2) (z^2) Log[1-z]+1104 Log[z]+180 Lμ Log[z]-72 (π^2) Log[z]-540 z Log[z]+348 (z^2) Log[z]+612 Lμ (z^2) Log[z]-72 (π^2) (z^2) Log[z]-216 Log[1-z] Log[z]+432 z Log[1-z] Log[z]-216 (z^2) Log[1-z] Log[z]+216 ((Log[1-z])^2) Log[z]+216 (z^2) ((Log[1-z])^2) Log[z]+117 ((Log[z])^2)-108 Lμ ((Log[z])^2)-324 z ((Log[z])^2)+9 (z^2) ((Log[z])^2)-108 Lμ (z^2) ((Log[z])^2)-216 Log[1-z] ((Log[z])^2)-216 (z^2) Log[1-z] ((Log[z])^2)+54 ((Log[z])^3)+54 (z^2) ((Log[z])^3)+216 (-(((-1+z))^2)+(1+(z^2)) Log[1-z]-2 (1+(z^2)) Log[z]) PolyLog[2,z]+216 (1+(z^2)) PolyLog[3,1-z]+648 PolyLog[3,z]+648 (z^2) PolyLog[3,z]+108 Zeta[3]+108 (z^2) Zeta[3])+DELTA[1-z] (-(8/27) (28+30 Lμ+9 ((Lμ)^2)) TrNf Log[δp]-(2/9) CF (12 ((Lμ)^2)+(π^2)) ((π^2)-6 ((Log[δp])^2))+CA ((1/3) ((Lμ)^2) ((π^2)+Log[4096])+(2/27) Log[δp] (404+99 ((Lμ)^2)-6 Lμ (-67+3 (π^2))-378 Zeta[3])+Lμ (Log[256]-9 Zeta[3])+(1/36) (17 (π^4)+12 (π^2) Log[2] (1+Log[4])-3 (8 ((Log[2])^4)+192 PolyLog[4,(1/2)]+5 Log[8] Zeta[3]+3 Log[2] (-32+51 Zeta[3]))))));

Q -> G

Δ[q->g,0]=0;
Δ[q->g,1]=CF (Lμ (4-(4/z)-2 z)+2 z+((4-(4/z)-2 z)/ϵ)-(((-12 Lμ (z^2)+6 ((Lμ)^2) (2+(-2+z) z)+(π^2) (2+(-2+z) z)) ϵ)/(6 z))+(((ϵ^2) (6 ((Lμ)^2) (z^2)+(π^2) (z^2)-2 ((Lμ)^3) (2+(-2+z) z)-Lμ (π^2) (2+(-2+z) z)-4 (2+(-2+z) z) Zeta[3]))/(6 z)));
Δ[q->g,2]=CF(((2 (CA-CF) (2+(-2+z) z) ((Log[1-z])^3))/(3 z))-(((CF (-2+z) z+2 CA (8+z (2+5 z))) ((Log[z])^3))/(3 z))+((Log[1-z])^2) (((2 (CA-CF) (-(z^2)+2 Lμ (2+(-2+z) z)))/z)-((2 (3 CA-2 CF) (2+(-2+z) z) Log[z])/z))+Log[1-z] (((2 CA (-3 z-12 Lμ (z^2)+12 ((Lμ)^2) (2+(-2+z) z)-(π^2) (2+(-2+z) z))+6 CF (z-4 Lμ (z^2)+4 ((Lμ)^2) (2+(-2+z) z)+(π^2) (2+(-2+z) z)))/(3 z))-((4 CA (-22+6 Lμ (2+(-2+z) z)+z (24+z (-9+4 z))) Log[z])/(3 z))+((6 CA (2+(-2+z) z) ((Log[z])^2))/z))-((2 CA (π^2) (2+z (2+z)) Log[1+z])/(3 z))+((4 CA (2+z (2+z)) ((Log[1+z])^3))/(3 z))+((Log[z])^2) (((3 CF (8+4 Lμ (-2+z)-z) z+2 CA (-88+12 Lμ (4+z (2+3 z))+z (48+z (-33+8 z))))/(6 z))-((2 CA (2+z (2+z)) Log[1+z])/z))+Log[z] (-(1/(9 z))(3 CF z (-6 Lμ (-8+z)+12 ((Lμ)^2) (-2+z)+(π^2) (-2+z)+3 (5+7 z))+2 CA (268+3 z+51 (z^2)-6 (π^2) (3+(-7+z) z)+72 ((Lμ)^2) (1+z+(z^2))-6 Lμ (44+15 (z^2))))+4 CA (z-((2 Lμ (2+z (2+z)))/z)) Log[1+z])-((4 CF (-12 Lμ (z^2)+12 ((Lμ)^2) (2+(-2+z) z)+(π^2) (2+(-2+z) z)) Log[δp])/(3 z))+((CF (20-(16/z)-9 z)+CA (8-(38/(3 z))+6 z+((8 (z^2))/3))+((4 (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((2 (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])/z)-((8 CF (2+(-2+z) z) Log[δp])/z))/(ϵ^2))+(4 CA (z-((2 Lμ (2+z (2+z)))/z))-((4 CA (2+z (2+z)) Log[z])/z)) PolyLog[2,-z]+(1/ϵ)(1/(9 z)(3 CF (9 (-1+z) (8+z)+8 (π^2) (2+(-2+z) z)-12 Lμ (7+z (-9+4 z)))+2 CA (107+z (-117+(27-44 z) z)-6 (π^2) (4+z (-3+2 z))+6 Lμ (-25+2 z (9+z (3+2 z)))))+((2 (CA-CF) (2+(-2+z) z) ((Log[1-z])^2))/z)+(CF (-2+z)+CA (4+(8/z)+6 z)) ((Log[z])^2)+Log[1-z] (((4 (CA+CF) (-(z^2)+2 Lμ (2+(-2+z) z)))/z)-((4 CA (2+(-2+z) z) Log[z])/z))+Log[z] (((3 CF z (-8-4 Lμ (-2+z)+z)+CA (88+30 (z^2)-48 Lμ (1+z+(z^2))))/(3 z))-((4 CA (2+z (2+z)) Log[1+z])/z))+8 CF (Lμ (4-(4/z)-2 z)+z) Log[δp]-((4 CA (2+z (2+z)) PolyLog[2,-z])/z))+(-((8 CA (-1+z) (11+z (-1+2 z)))/(3 z))-((4 (CA-CF) (2+(-2+z) z) Log[1-z])/z)+((4 CA (2+(-6+z) z) Log[z])/z)) PolyLog[2,z]-((4 (CA-CF) (2+(-2+z) z) PolyLog[3,1-z])/z)+4 CA (2+(2/z)+z) PolyLog[3,-z]+4 CA (10-(2/z)-z) PolyLog[3,z]-((8 CA (2+z (2+z)) PolyLog[3,(1/(1+z))])/z)+(1/(27 z))(CA (-1904+9 (π^2) (-23+2 z (11+z (3+2 z)))+18 ((Lμ)^2) (-56+z (42+z (9+8 z)))-12 Lμ (-107+6 (π^2) (4+z (-3+2 z))+z (117+4 z (-9+11 z)))+2 z (1062+z (-369+232 z)+108 Zeta[3]))+9 CF ((π^2) (-7+3 (3-4 z) z)-3 ((Lμ)^2) (26+z (-34+15 z))+2 Lμ (-72+63 z+6 (z^2)+8 (π^2) (2+(-2+z) z))+3 (-48+50 z-7 (z^2)+28 (2+(-2+z) z) Zeta[3]))));

G -> Q

Δ[g->q,0]=0;
Δ[g->q,1]=TR(-2 Lμ+4 z+4 Lμ z-4 (z^2)-4 Lμ (z^2)+((-2+4 z-4 (z^2))/ϵ)+(-((Lμ)^2)-((π^2)/6)+4 z+4 Lμ z+2 ((Lμ)^2) z+(((π^2) z)/3)-4 (z^2)-4 Lμ (z^2)-2 ((Lμ)^2) (z^2)-(((π^2) (z^2))/3)) ϵ+(ϵ^2) (-(((Lμ)^3)/3)-((Lμ (π^2))/6)+4 z+4 Lμ z+2 ((Lμ)^2) z+((2 ((Lμ)^3) z)/3)+(((π^2) z)/3)+(1/3) Lμ (π^2) z-4 (z^2)-4 Lμ (z^2)-2 ((Lμ)^2) (z^2)-((2 ((Lμ)^3) (z^2))/3)-(((π^2) (z^2))/3)-(1/3) Lμ (π^2) (z^2)-((2 Zeta[3])/3)+(4/3) z Zeta[3]-(4/3) (z^2) Zeta[3]));
Δ[g->q,2]=-(2/3) (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^3)+(1/3) TR (CA (2+20 z)+CF (-1+2 z-4 (z^2))) ((Log[z])^3)+((Log[1-z])^2) ((1/3) TR (4 TrNf+8 TrNf (-1+z) z-CA (11+34 (-1+z) z+12 Lμ (1+2 (-1+z) z))+3 CF (3+10 (-1+z) z+Lμ (4+8 (-1+z) z)))-2 CF TR (1+2 (-1+z) z) Log[z])+Log[1-z] ((4/9) TR (2 TrNf (5+16 (-1+z) z+6 Lμ (1+2 (-1+z) z))+CA (-38-91 (-1+z) z+Lμ (-33-30 (-1+z) z)+18 ((Lμ)^2) (1+2 (-1+z) z)+6 (π^2) (1+2 (-1+z) z))+3 CF (12+39 (-1+z) z+(π^2) (-1-2 (-1+z) z)+6 ((Lμ)^2) (1+2 (-1+z) z)+Lμ (9+30 (-1+z) z)))+((2 TR (CA (-8+z (23-70 z+66 (z^2)))+z (-4 TrNf (1+2 (-1+z) z)-3 CF (3+10 (-1+z) z+Lμ (4+8 (-1+z) z)))) Log[z])/(3 z))+2 (2 CA+CF) TR (1+2 (-1+z) z) ((Log[z])^2))+(2/3) CA (π^2) TR (1+2 z (1+z)) Log[1+z]-(4/3) CA TR (1+2 z (1+z)) ((Log[1+z])^3)+((Log[z])^2) (((TR (z (8 TrNf (1+2 (-1+z) z)+3 CF (1+4 Lμ-4 (5+2 Lμ) z+4 (5+4 Lμ) (z^2)))-4 CA (-8+z (10+6 Lμ (1+6 z)+z (-35+33 z)))))/(6 z))+2 CA TR (1+2 z (1+z)) Log[1+z])+Log[z] ((1/(9 z))TR (z (3 CF (-48-6 Lμ-12 ((Lμ)^2)+5 (π^2)+(201+24 Lμ (5+Lμ)-10 (π^2)) z-4 (39+6 Lμ (5+2 Lμ)-2 (π^2)) (z^2))-8 TrNf (5+16 (-1+z) z+6 Lμ (1+2 (-1+z) z)))+2 CA (64+36 ((Lμ)^2) z (1+4 z)+z (61+z (-5+218 z)+6 (π^2) (2+z+3 (z^2)))+12 Lμ (-4+z (4+z (-17+11 z)))))+8 CA TR (z+(z^2)+Lμ (1+2 z (1+z))) Log[1+z])+(4/3) CA TR (-12 ((Lμ)^2)-(π^2)+2 (6+12 Lμ (1+Lμ)+(π^2)) z-2 (6+12 Lμ (1+Lμ)+(π^2)) (z^2)) Log[δp]+(1/(ϵ^2))(((TR (CA (8+2 z (8+7 (2-3 z) z))+z (-8 TrNf (1+2 (-1+z) z)+3 CF (1+4 (z^2)))))/(3 z))+4 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+2 TR (CA (2+8 z)+CF (-1+2 z-4 (z^2))) Log[z]-8 CA TR (1+2 (-1+z) z) Log[δp])+(8 CA TR (z+(z^2)+Lμ (1+2 z (1+z)))+4 CA TR (1+2 z (1+z)) Log[z]) PolyLog[2,-z]+(1/ϵ)(1/(9 z)TR (z (8 TrNf (5+4 (-1+z) z)+9 CF (4+z+4 Lμ z+4 (1+Lμ) (z^2)+(π^2) (-2-4 (-1+z) z)))+2 CA (-32+6 Lμ (4+(30-37 z) (z^2))+z (5+(23-72 z) z+6 (π^2) (2+z (-3+4 z)))))-2 (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)+TR (CF-2 (CA+6 CA z+CF (1-2 z) z)) ((Log[z])^2)+Log[1-z] ((2/3) TR (4 TrNf+8 TrNf (-1+z) z+CA (-11-10 (-1+z) z+12 Lμ (1+2 (-1+z) z))+3 CF (3+10 (-1+z) z+Lμ (4+8 (-1+z) z)))-4 CF TR (1+2 (-1+z) z) Log[z])+Log[z] (((TR (z (-8 TrNf (1+2 (-1+z) z)-3 CF (1+4 Lμ-4 (5+2 Lμ) z+4 (5+4 Lμ) (z^2)))+4 CA (-4+z (4+6 Lμ (1+4 z)+z (-17+11 z)))))/(3 z))+4 CA TR (1+2 z (1+z)) Log[1+z])-16 CA TR ((-1+z) z+Lμ (1+2 (-1+z) z)) Log[δp]+4 CA TR (1+2 z (1+z)) PolyLog[2,-z])+(((8 CA TR (-1+z) (2+z (-1+11 z)))/(3 z))+4 CA TR (3-2 z+6 (z^2)) Log[z]) PolyLog[2,z]-4 CA TR (1+2 z (1+z)) PolyLog[3,-z]-4 CA TR (5+2 z (-1+5 z)) PolyLog[3,z]+8 CA TR (1+2 z (1+z)) PolyLog[3,(1/(1+z))]+(1/(27 z))TR (z (4 TrNf (60 Lμ (1+2 (-1+z) z)+18 ((Lμ)^2) (1+2 (-1+z) z)-9 (π^2) (1+2 (-1+z) z)+8 (7+17 (-1+z) z))+9 CF (9+3 z (11+4 z)+(π^2) (-9+(31-29 z) z)+3 ((Lμ)^2) (-1+2 z (3+z))-6 Lμ (-4-z (3+2 z)+(π^2) (2+4 (-1+z) z))))+CA (9 (π^2) (4+z (7+z (-8+11 z)))-36 ((Lμ)^2) (-4+z (4+z (-38+45 z)))+12 Lμ (-32+z (5+(71-120 z) z+6 (π^2) (2+z (-3+4 z))))+2 (208+z (-521+1651 z-1796 (z^2)+108 (3+z (-7+6 z)) Zeta[3]))));

G -> G

Δ[g->g,0]=DELTA[1-z];
Δ[g->g,1]=((4 CA Lμ (((1+(-1+z) z))^2))/((-1+z) z))+4 CA Lμ DELTA[1-z] Log[δp]+((((4 CA (((1+(-1+z) z))^2))/((-1+z) z))+4 CA DELTA[1-z] Log[δp])/ϵ)+ϵ (((CA (6 ((Lμ)^2)+(π^2)) (((1+(-1+z) z))^2))/(3 (-1+z) z))+(1/3) CA (6 ((Lμ)^2)+(π^2)) DELTA[1-z] Log[δp])+(ϵ^2) (((CA (((1+(-1+z) z))^2) (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3]))/(3 (-1+z) z))+(1/3) CA DELTA[1-z] Log[δp] (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3]));
Δ[g->g,2]=(1/(9 z (-1+(z^2))))(4 CA TrNf z (-1+(z^2)) (7+13 z+12 Lμ (1+z))+12 CF TrNf z (-1+(z^2)) (12 ((Lμ)^2) (1+z)+(18+(π^2)) (1+z)+6 Lμ (1+3 z))+(CA^2) (144 ((Lμ)^2) (1+z) (1+(-3+z) (-1+z) (z^2))+12 Lμ (-1+z) (1+z) (44+z (-11+25 z))-(-1+z) (1+z) (536+z (143+701 z))-12 (π^2) (3+z (-7+z (1+z (7+z (-1+5 z))))))) Log[z]+((4 (CA^2) (((1+(-1+z) z))^2) ((Log[1-z])^2) Log[z])/((-1+z) z))-(1/(3 z (-1+(z^2))))(4 CA TrNf (-1+z) z (((1+z))^2)+6 CF TrNf z (-1+(z^2)) (1+3 z+4 Lμ (1+z))+(CA^2) (24 Lμ (2+z (1+(-4+z) (-1+z) z (1+z)))-(-1+z) (1+z) (-88+z (59+z (-73+44 z))))) ((Log[z])^2)+((4 (CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (4+z (1+(-6+z) (-1+z) z (1+z)))) ((Log[z])^3))/(3 z (-1+(z^2))))+Log[1-z] (((2 CA (2 TrNf (-1+z) z+CA (z-(z^2)-48 ((Lμ)^2) (((1+(-1+z) z))^2)-4 (π^2) (((1+(-1+z) z))^2))))/(3 (-1+z) z))+((8 (CA^2) (6 Lμ (((1+(-1+z) z))^2)-(((-1+z))^2) (11+z (-1+11 z))) Log[z])/(3 (-1+z) z))-((12 (CA^2) (((1+(-1+z) z))^2) ((Log[z])^2))/((-1+z) z)))+(-((4 (CA^2) (π^2) (((1+z+(z^2)))^2))/(3 z (1+z)))-((16 (CA^2) Lμ (((1+z+(z^2)))^2) Log[z])/(z (1+z)))-((4 (CA^2) (((1+z+(z^2)))^2) ((Log[z])^2))/(z (1+z)))) Log[1+z]+((8 (CA^2) (((1+z+(z^2)))^2) ((Log[1+z])^3))/(3 z (1+z)))+(1/(ϵ^2))(-((16 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+((8 (CF TrNf z (-1+(z^2))+(CA^2) (1+(-3+z) (-1+z) (z^2))) Log[z])/((-1+z) z))+1/(3 (-1+z) z)2 (4 CA TrNf (((1+(-1+z) z))^2)+2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (33+z (-58+z (39+z (-58+33 z))))+24 (CA^2) (((1+(-1+z) z))^2) Log[δp])+(1/6) CA DELTA[1-z] (CA (-7 (π^2)+6 Log[4])+4 Log[δp] (-11 CA+4 TrNf+12 CA Log[δp])))+(-((16 (CA^2) Lμ (((1+z+(z^2)))^2))/(z (1+z)))-((8 (CA^2) (((1+z+(z^2)))^2) Log[z])/(z (1+z)))) PolyLog[2,-z]+(-((8 (CA^2) (-1+z) (11+z (-1+11 z)))/(3 z))-((8 (CA^2) (1+z (-3+z (1+z (3+z (-1+3 z))))) Log[z])/(z (-1+(z^2))))) PolyLog[2,z]+((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,-z])/(z (1+z)))+((8 (CA^2) (1+z (-5+z (1+z (5+z (-1+5 z))))) PolyLog[3,z])/(z (-1+(z^2))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,(1/(1+z))])/(z (1+z)))+(1/ϵ)(1/(3 (-1+z) z)(8 CA TrNf z (-1+(z^2))+12 CF TrNf (-1+z) z (1+3 z+4 Lμ (1+z))+2 (CA^2) (24 Lμ (1+(-3+z) (-1+z) (z^2))+(-1+z) (44+z (-11+25 z)))) Log[z]-((4 (CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (2+z (1+(-4+z) (-1+z) z (1+z)))) ((Log[z])^2))/(z (-1+(z^2))))+Log[1-z] (-((32 (CA^2) Lμ (((1+(-1+z) z))^2))/((-1+z) z))+((8 (CA^2) (((1+(-1+z) z))^2) Log[z])/((-1+z) z)))-((8 (CA^2) (((1+z+(z^2)))^2) Log[z] Log[1+z])/(z (1+z)))+1/(9 z (-1+(z^2)))(-24 CF TrNf (((-1+z))^2) (1+z) (-1-(-8+z) z+Lμ (4+z (7+4 z)))+(CA^2) (48 Lμ (((-1+z))^2) (1+z) (11+z (2+11 z))+z (109+109 (z^3)+25 z (1+z)-12 (π^2) (-1+(z^2)+(z^4))))-4 CA TrNf (1+z) (23+z (-42+z (48+z (-42+23 z))))+288 (CA^2) Lμ (1+z) (((1+(-1+z) z))^2) Log[δp])-((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[2,-z])/(z (1+z)))+(1/18) CA DELTA[1-z] (6 CA (-7 Lμ (π^2)+3 (1+Lμ) Log[16])+4 Log[δp] (CA (67-3 (π^2))-20 TrNf+72 CA Lμ Log[δp])-81 CA Zeta[3]))+(1/(27 z (-1+(z^2))))2 (-3 CF TrNf (((-1+z))^2) (1+z) (-12 (1+(-23+z) z)-24 Lμ (1+(-8+z) z)+12 ((Lμ)^2) (4+z (7+4 z))+(π^2) (4+z (7+4 z)))+2 CA TrNf (1+z) (-139-18 ((Lμ)^2) (((1+(-1+z) z))^2)+z (249+z (-276+11 (21-11 z) z))-6 Lμ (23+z (-42+z (48+z (-42+23 z)))))+36 (CA^2) (12 ((Lμ)^2)+(π^2)) (1+z) (((1+(-1+z) z))^2) Log[δp]+(CA^2) (3 Lμ z (109+109 (z^3)+25 z (1+z)-12 (π^2) (-1+(z^2)+(z^4)))+2 (1+z) (835+3 (π^2) (((-1+z))^2) (22+z+22 (z^2))+z (-1595+2 z (843+z (-757+395 z))))+9 ((Lμ)^2) (99+z (-83+z (-5+z (-5+z (-83+99 z)))))-108 (7+z (-6+z (7+z (6+z (-7+6 z))))) Zeta[3]))+DELTA[1-z] (-16 (CA^2) PolyLog[4,(1/2)]+(1/108) CA (9 CA (3 (π^4)+96 Log[2]+96 Lμ Log[2]-8 ((Log[2])^4)+((Lμ)^2) (-28 (π^2)+48 Log[2])+4 (π^2) Log[2] (1+Log[4]))+8 Log[δp] (CA (404+3 Lμ (134+33 Lμ-6 (π^2)))-4 (28+30 Lμ+9 ((Lμ)^2)) TrNf+18 CA (12 ((Lμ)^2)+(π^2)) Log[δp])-108 CA (9 Lμ+14 Log[2]+28 Log[δp]) Zeta[3]));

Q -> Q’

Δ[q->qp,0]=0;
Δ[q->qp,1]=0;
Δ[q->qp,2]=((-((2 CF TR (-1+z) (4+z (7+4 z)))/(3 z))+4 CF TR (1+z) Log[z])/(ϵ^2))+((2 CF TR (-2 (-1+z) (-16+(11-16 z) z+3 Lμ (4+z (7+4 z)))-3 Log[z] (8-3 (-1+4 Lμ) z (1+z)+3 z (1+z) Log[z])))/(9 z ϵ))+(1/(27 z))CF TR ((1-z) (416+2 z (-71+136 z)+36 ((Lμ)^2) (4+z (7+4 z))+3 (π^2) (4+z (7+4 z))-24 Lμ (16+z (-11+16 z)))+3 Log[z] (128+72 ((Lμ)^2) z (1+z)+6 z (-8+14 z+(π^2) (1+z))-12 Lμ (8+3 z (1+z))+3 Log[z] (16-z (9+12 Lμ (1+z)+z (-15+8 z))+2 z (1+z) Log[z])))-((8 CF TR (-1+z) (2+z (-1+2 z)) PolyLog[2,1-z])/(3 z));

Q ->  (OverBar[Q])

Δ[q->qbar,0]=0;
Δ[q->qbar,1]=0;
Δ[q->qbar,2]=-(((CA-2 CF) CF (1+(z^2)) ((Log[z])^3))/(1+z))-((2 (CA-2 CF) CF (π^2) (1+(z^2)) Log[1+z])/(3 (1+z)))+((4 (CA-2 CF) CF (1+(z^2)) ((Log[1+z])^3))/(3 (1+z)))+((Log[z])^2) (((2 (CA-2 CF) CF (-2+Lμ-2 z+Lμ (z^2)))/(1+z))-((2 (CA-2 CF) CF (1+(z^2)) Log[1+z])/(1+z)))+Log[z] ((((CA-2 CF) CF (12 Lμ (((1+z))^2)-3 (1+z) (11+3 z)-2 (π^2) (1+(z^2))))/(3 (1+z)))-4 (CA-2 CF) CF (-1+z) Log[1-z]-((4 (CA-2 CF) CF (-(((1+z))^2)+2 Lμ (1+(z^2))) Log[1+z])/(1+z)))+(-((4 (CA-2 CF) CF (-(((1+z))^2)+2 Lμ (1+(z^2))))/(1+z))-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,-z]+(1/ϵ)(-(((CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (z^2)))/(3 (1+z)))+(((CA-2 CF) CF (1+(z^2)) ((Log[z])^2))/(1+z))+Log[z] (2 (CA-2 CF) CF (1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[1+z])/(1+z)))-((4 (CA-2 CF) CF (1+(z^2)) PolyLog[2,-z])/(1+z)))+(-4 (CA-2 CF) CF (-1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,z]+((4 (CA-2 CF) CF (1+(z^2)) PolyLog[3,-z])/(1+z))+((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,z])/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,(1/(1+z))])/(1+z))-(((CA-2 CF) CF (45+(π^2)+2 Lμ (-12+(π^2)+(12+(π^2)) (z^2))-6 Zeta[3]-z (45 z+(π^2) (2+3 z)+6 z Zeta[3])))/(3 (1+z)));

integrated PDFs

f[f←r,n]=((Subscript[f, f←r])^([n]))
Singularities at x->1 understood as "plus"-distribution

Q ← Q

f[q←q,1]=CF(((2 (1+(x^2)))/((-1+x) ϵ))-((3 DELTA[1-x])/ϵ));
f[q←q,2]=((CF (-((11 CA (1+(x^2)))/(3 (-1+x)))+((4 TrNf (1+(x^2)))/(3 (-1+x))))+((CF^2) ((9/2)-((4 (π^2))/3))+CF (((11 CA)/2)-2 TrNf)) DELTA[1-x]-((2 (CF^2) (1+4 x+(x^2)+4 (1+(x^2)) Log[1-x]-(1+3 (x^2)) Log[x]))/(-1+x)))/(ϵ^2))+(1/ϵ)((((CF^2) (-10 (((-1+x))^2)-2 (3+2 x-2 (x^2)+2 (1+(x^2)) Log[1-x]) Log[x]+(-1+(x^2)) ((Log[x])^2)))/(-1+x))+CF (-((4 TrNf (11-12 x+11 (x^2)+3 (1+(x^2)) Log[x]))/(9 (-1+x)))+((CA (187-240 x+187 (x^2)-3 (π^2) (1+(x^2))+3 (17+5 (x^2)) Log[x]+9 (1+(x^2)) ((Log[x])^2)))/(9 (-1+x))))+DELTA[1-x] ((CF^2) (-(3/4)+(π^2)-12 Zeta[3])+CF ((1/9) (3+4 (π^2)) TrNf+CA (-(17/12)-((11 (π^2))/9)+6 Zeta[3]))));

Q ← G

f[q←g,1]=TR(((-2)/ϵ)(1-2x(1-x)));
f[q←g,2]=(1/(ϵ^2))(-(8/3) TR TrNf (1-2 x+2 (x^2))+((2 CA TR (4+14 x+2 (x^2)-9 (x^3)+6 x (1-2 x+2 (x^2)) Log[1-x]+6 x (1+4 x) Log[x]))/(3 x))+CF TR (-1+4 x+(4-8 x+8 (x^2)) Log[1-x]+(-2+4 x-8 (x^2)) Log[x]))+(1/ϵ)((1/2) CF TR (-8+18 x-8 Log[1-x]-2 (-1+4 x) Log[x]-2 (-1+2 x) ((Log[x])^2)+(4/3) (1-2 x+2 (x^2)) (-15+(π^2)-3 ((Log[1-x])^2)-6 Log[x]-3 ((Log[x])^2)+6 Log[1-x] (1+Log[x])))-(2/9) CA TR (91+(20/x)+7 x-6 (π^2) x-18 Log[1-x]-(1-2 x+2 (x^2)) (109-18 Log[1-x]+9 ((Log[1-x])^2)-66 Log[x])+3 (-19+68 x) Log[x]-9 (1+2 x) ((Log[x])^2)-18 ((x^2)+(((1+x))^2)) (Log[x] Log[1+x]+PolyLog[2,-x])));

G ← Q

f[g←q,1]=((-2CF)/ϵ)((2/x)-2+x);
f[g←q,2]=(((((CF^2) (4 (2-2 x+(x^2)) Log[1-x]-x (-4+x+2 (-2+x) Log[x])))/x)+CF (-((8 TrNf (2-2 x+(x^2)))/(3 x))+((2 CA (-9+2 x+14 (x^2)+4 (x^3)+6 (2-2 x+(x^2)) Log[1-x]-12 (1+x+(x^2)) Log[x]))/(3 x))))/(ϵ^2))+(1/ϵ)((((CF^2) (2 (6-6 x+5 (x^2)) Log[1-x]+2 (2-2 x+(x^2)) ((Log[1-x])^2)+x (5+7 x-(4+7 x) Log[x]-(-2+x) ((Log[x])^2))))/x)+CF ((8/3) TrNf (x+(-2+(2/x)+x) ((5/3)+Log[1-x]))+(1/9) CA (-56-12 (π^2)-65 x-88 (x^2)-36 x Log[1-x]-((3 (2-2 x+(x^2)) (3+6 ((Log[1-x])^2)+Log[1-x] (22-12 Log[x])))/x)+6 (36+15 x+8 (x^2)) Log[x]-18 (2+x) ((Log[x])^2)-((36 (2+2 x+(x^2)) (Log[x] Log[1+x]+PolyLog[2,-x]))/x))));

G ← G

f[g←g,1]=((-1)/ϵ)(4 CA (-2+(1/(1-x))+(1/x)+x-(x^2))+(((11 CA)/3)-((4 TrNf)/3)) DELTA[1-x]);
f[g←g,2]=(1/(ϵ^2))(((8 CA TrNf (((1-x+(x^2)))^2))/((-1+x) x))+((CA^2) ((121/9)-((4 (π^2))/3))-((88 CA TrNf)/9)+((16 (TrNf^2))/9)) DELTA[1-x]+(4/3) CF TrNf (3+(4/x)-3 x-4 (x^2)+6 (1+x) Log[x])+(((CA^2) (22-28 x-54 (x^2)-28 (x^3)+22 (x^4)-48 (((1-x+(x^2)))^2) Log[1-x]+24 (1+3 (x^2)-4 (x^3)+(x^4)) Log[x]))/(3 (-1+x) x)))+(1/ϵ)(-((4 CA TrNf (23-42 x+48 (x^2)-42 (x^3)+23 (x^4)-6 x (-1+(x^2)) Log[x]))/(9 (-1+x) x))+((4 CF TrNf (-2 (1-12 x+6 (x^2)+5 (x^3))+3 x (3+5 x) Log[x]+3 x (1+x) ((Log[x])^2)))/(3 x))+(1/9) (CA^2) (243 (-1+x)-((134 (-1+(x^3)))/x)+6 (25-11 x+44 (x^2)) Log[x]-72 (1+x) ((Log[x])^2)-((2 (((1-x+(x^2)))^2) (-67+3 (π^2)+36 Log[1-x] Log[x]-9 ((Log[x])^2)))/((-1+x) x))-((6 (((1+x+(x^2)))^2) ((π^2)-3 ((Log[x])^2)+12 Log[x] Log[1+x]+12 PolyLog[2,-x]))/(x (1+x))))+DELTA[1-x] (((8 CA TrNf)/3)+2 CF TrNf-(2/3) (CA^2) (8+9 Zeta[3])));

Q ← Q’

f[q←qp,1]=0;
f[q←qp,2]=-((2 CF TR (-4-3 x+3 (x^2)+4 (x^3)-6 x (1+x) Log[x]))/(3 x (ϵ^2)))-((CF TR (-4+(40/(9 x))+12 x-((112 (x^2))/9)+2 (1+5 x+((8 (x^2))/3)) Log[x]-2 (1+x) ((Log[x])^2)))/ϵ);

Q ← (OverBar[Q])

f[q←qbar,1]=0;
f[q←qbar,2]=-((4 CF (-(CA/2)+CF) (2-2 x+(1+x) Log[x]-(((1+(x^2)) ((π^2)-3 ((Log[x])^2)+12 Log[x] Log[1+x]+12 PolyLog[2,-x]))/(6 (1+x)))))/ϵ);

integrated FFs

d[f←r,n]=((Subscript[d, f←r])^([n]))
Singularities at z->1 understood as "plus"-distribution

Q -> Q

d[q->q,1]=CF(-((2 (1+(z^2)))/((1-z) ϵ))-((3 DELTA[1-z])/ϵ));
d[q->q,2]=(((1/2) CF (((11 CA)/3)-((4 TrNf)/3)) (2 (-1+(2/(1-z))-z)+3 DELTA[1-z])+(1/2) (CF^2) ((9-((8 (π^2))/3)) DELTA[1-z]-((4 (1+4 z+(z^2)+4 (1+(z^2)) Log[1-z]-(1+3 (z^2)) Log[z]))/(-1+z))))/(ϵ^2))-(1/(36 ϵ))CF (((4 TrNf (-(3+4 (π^2)) (-1+z) DELTA[1-z]+4 (11-12 z+11 (z^2)+3 (1+(z^2)) Log[z])))/(-1+z))+((CA (4 (-187+240 z-187 (z^2)+3 (π^2) (1+(z^2))-3 (17+5 (z^2)) Log[z]-9 (1+(z^2)) ((Log[z])^2))+(-1+z) DELTA[1-z] (51+44 (π^2)-216 Zeta[3])))/(-1+z))+(1/(-1+z))9 CF (-4 (-10 (((-1+z))^2)+(-4+4 z+6 (z^2)+4 (1+(z^2)) Log[1-z]) Log[z]-(3+5 (z^2)) ((Log[z])^2))-(-1+z) DELTA[1-z] (-3+4 (π^2)-48 Zeta[3])));

Q -> G

d[q->g,1]=((-CF)/ϵ)(2 (-2+(2/z)+z));
d[q->g,2]=CF((1/(3 z (ϵ^2)))(-18 CA-16 TrNf+4 CA z+12 CF z+16 TrNf z+28 CA (z^2)-3 CF (z^2)-8 TrNf (z^2)+8 CA (z^3)+12 (CA+CF) (2-2 z+(z^2)) Log[1-z]-6 (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])+(1/(2 ϵ))(-(4/9) CA (45+6 (π^2)+(17/z)-9 z-44 (z^2)-18 z Log[1-z]+((3 (-18+24 z+27 (z^2)+8 (z^3)) Log[z])/z)-((9 (4+2 z+3 (z^2)) ((Log[z])^2))/z)+((18 (2+2 z+(z^2)) (Log[z] Log[1+z]+PolyLog[2,-z]))/z)-((9 (2-2 z+(z^2)) (Log[1-z] (Log[1-z]-6 Log[z])-8 PolyLog[2,z]))/z))-CF (-2+18 z+8 z Log[1-z]+2 (-16+z) Log[z]-2 (-2+z) ((Log[z])^2)+((4 (2-2 z+(z^2)) (Log[1-z] (Log[1-z]-4 Log[z])-8 PolyLog[2,z]))/z))));

G -> Q

d[g->q,1]=-((2 TR (1-2 z+2 (z^2)))/ϵ);
d[g->q,2]=(1/(3 z (ϵ^2)))TR (8 CA+28 CA z-3 CF z-8 TrNf z+4 CA (z^2)+12 CF (z^2)+16 TrNf (z^2)-18 CA (z^3)-16 TrNf (z^3)+12 (CA+CF) z (1-2 z+2 (z^2)) Log[1-z]-6 z (-2 CA+CF-8 CA z-2 CF z+4 CF (z^2)) Log[z])-(1/(2 ϵ))TR (-(16/9) TrNf (5-4 z+4 (z^2)+(3-6 z+6 (z^2)) Log[1-z]+(3-6 z+6 (z^2)) Log[z])+CF (-24+46 z-40 (z^2)-4 (3-2 z+2 (z^2)) Log[1-z]+2 (-5+4 z+4 (z^2)) Log[z]-2 (1-2 z+4 (z^2)) ((Log[z])^2)-(4/3) (1-2 z+2 (z^2)) ((π^2)+3 ((Log[1-z])^2)-18 Log[1-z] Log[z]-24 PolyLog[2,z]))+(4/9) CA (13-(20/z)-95 z-6 (π^2) z+178 (z^2)+3 (11-10 z+10 (z^2)) Log[1-z]-6 (2+17 z+2 (z^2)) Log[z]+9 (1+6 z) ((Log[z])^2)-18 (1+2 z+2 (z^2)) (Log[z] Log[1+z]+PolyLog[2,-z])+9 (1-2 z+2 (z^2)) (Log[1-z] (Log[1-z]-4 Log[z])-8 PolyLog[2,z])));

G -> G

d[g->g,1]=((-1)/ϵ)(4 CA (-2+(1/(1-z))+(1/z)+z-(z^2))+(((11 CA)/3)-((4 TrNf)/3)) DELTA[1-z]);
d[g->g,2]=(1/(ϵ^2))((1/9) ((CA^2) (121-12 (π^2))-88 CA TrNf+16 (TrNf^2)) DELTA[1-z]+1/(3 (-1+z) z)2 (11 (CA^2)+12 CA TrNf-8 CF TrNf-14 (CA^2) z-24 CA TrNf z+2 CF TrNf z-27 (CA^2) (z^2)+36 CA TrNf (z^2)+12 CF TrNf (z^2)-14 (CA^2) (z^3)-24 CA TrNf (z^3)+2 CF TrNf (z^3)+11 (CA^2) (z^4)+12 CA TrNf (z^4)-8 CF TrNf (z^4)-24 (CA^2) (((1-z+(z^2)))^2) Log[1-z]+12 (CF TrNf z (-1+(z^2))+(CA^2) (1+3 (z^2)-4 (z^3)+(z^4))) Log[z]))+(1/ϵ)((1/9) (-243 (CA^2)-36 CA TrNf+((134 (CA^2))/z)+((52 CA TrNf)/z)+243 (CA^2) z+36 CA TrNf z-134 (CA^2) (z^2)-52 CA TrNf (z^2)+24 CA TrNf (1+z) Log[z]+((6 (CA^2) (44-11 z+25 (z^2)) Log[z])/z)+72 (CA^2) (1+z) ((Log[z])^2)-((8 CA TrNf (((1-z+(z^2)))^2) (5+6 Log[z]))/((-1+z) z))-((2 (CA^2) (((1-z+(z^2)))^2) (-67+3 (π^2)-6 (11+6 Log[1-z]) Log[z]+27 ((Log[z])^2)))/((-1+z) z))-((4 CF TrNf (46-18 z+54 (z^2)-82 (z^3)+3 (8+15 z+21 (z^2)+8 (z^3)) Log[z]+9 z (1+z) ((Log[z])^2)))/z)-((6 (CA^2) (((1+z+(z^2)))^2) ((π^2)-3 ((Log[z])^2)+12 Log[z] Log[1+z]+12 PolyLog[2,-z]))/(z (1+z))))+DELTA[1-z] (((8 CA TrNf)/3)+2 CF TrNf-(2/3) (CA^2) (8+9 Zeta[3])));

Q -> Q’

d[q->qp,1]=0;
d[q->qp,2]=-((2 CF TR (-4-3 z+3 (z^2)+4 (z^3)-6 z (1+z) Log[z]))/(3 z (ϵ^2)))-((2 CF TR (-72-(20/z)+36 z+56 (z^2)-3 (15+27 z+8 (z^2)) Log[z]+9 (1+z) ((Log[z])^2)))/(9 ϵ));

Q -> (OverBar[Q])

d[q->qbar,1]=0;
d[q->qbar,2]=-((CF (-(CA/2)+CF) (32-32 z+16 (1+z) Log[z]-((8 (1+(z^2)) ((π^2)-3 ((Log[z])^2)+12 Log[z] Log[1+z]+12 PolyLog[2,-z]))/(3 (1+z)))))/(4 ϵ));

TMD PDFs

In this section we recombine NLO and NNLO expressions for TMD PDF.
They are given by
F[f ← f', 1] = ((Subscript[F, f←f'])^([1]))=(Φ^([1]))-(((S^([1]))(Φ^([0])))/2)+(((Subscript[Z, f])^([1]))-((Subscript[Z, f'])^([1])))(Φ^([0])),
F[f ← f', 2] = ((Subscript[F, f←f'])^([2]))=(Φ^([2]))-(((S^([1]))(Φ^([1])))/2)+(3/8)(S^([1]))(S^([1]))(Φ^([0]))-((((S^([1]))(S^([1])))/2)+(S^([2])))((Φ^([0]))/2)+(((Subscript[Z, f])^([1]))-((Subscript[Z, f'])^([1])))((Φ^([1]))-(((S^([1]))(Φ^([0])))/2))+(((Subscript[Z, f])^([2]))-((Subscript[Z, f'])^([2]))-((Subscript[Z, f])^([1]))((Subscript[Z, f'])^([1]))+((Subscript[Z, f'])^([1]))((Subscript[Z, f'])^([1])))(Φ^([0])),
where (Subscript[Z, f'])=(Subscript[Z, 2,3])
In the presented results one can observe the cancelation of rapidity-divegences (ln(δp))

Q ← Q

F[q←q,1]=Collect[Normal@Series[Φ[q←q,1]-((S1 Φ[q←q,0])/2)+(Zq[1]-Z2[1])Φ[q←q,0]/.CK->CF,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

F[q←q,2]=Collect[Normal@Series[Φ[q←q,2]-((S1 Φ[q←q,1])/2)+((3/4)(S1^2)-((S1^2)/2)-S2)((Φ[q←q,0])/2)+(Zq[1]-Z2[1])(Φ[q←q,1]-(S1/2) Φ[q←q,0])+(Zq[2]-Z2[2]-Zq[1]Z2[1]+((Z2[1])^2))Φ[q←q,0]/.CK->CF,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

F[q←q,1]=((2 CF (-(((-1+x))^2)+Lμ (1+(x^2))))/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x]+((((2 CF (1+(x^2)))/(-1+x))-3 CF DELTA[1-x])/ϵ)+ϵ (((CF (-12 Lμ (((-1+x))^2)+6 ((Lμ)^2) (1+(x^2))+(π^2) (1+(x^2))))/(6 (-1+x)))+(1/6) CF DELTA[1-x] (lζ (6 ((Lμ)^2)+(π^2))-2 (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));
F[q←q,2]=(1/2) (CF^2) ((Lμ)^4) DELTA[1-x]+((Lμ)^3) (-((10 (CF^2) (1+(x^2)))/(3 (-1+x)))+(2/9) CF (-11 CA-9 CF (-1+lζ)+4 TrNf) DELTA[1-x])-((2 CF ((CA+CF) (π^2)+3 (CA-CF) x+(-3 CA+3 CF+(CA+CF) (π^2)) (x^2)) Log[1-x])/(3 (-1+x)))+(((CF (-20 TrNf (1+(x^2))+3 CF (-30+(π^2)+78 x+3 (-32+(π^2)) (x^2))+2 CA (29+x (-36+83 x))))/(9 (-1+x)))-4 (CA-3 CF) CF (-1+x) Log[1-x]+((2 (2 CA-3 CF) CF (1+(x^2)) ((Log[1-x])^2))/(-1+x))) Log[x]+(((CF (CA (11-(-12+x) x)+2 (CF (-9+6 (-1+x) x)-2 TrNf (1+(x^2)))))/(6 (-1+x)))-((2 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) ((Log[x])^2)+((CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^3))/(3 (-1+x)))+((-((CF (11 CA (1+(x^2))-4 TrNf (1+(x^2))+6 CF (1+x (4+x))))/(3 (-1+x)))+(1/6) CF (33 CA+CF (27-8 (π^2))-12 TrNf) DELTA[1-x]-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))+((2 (CF^2) (1+3 (x^2)) Log[x])/(-1+x)))/(ϵ^2))+((Lμ)^2) (((CF (11 CA (1+(x^2))-4 TrNf (1+(x^2))+3 CF (7+6 lζ-20 x+(7+6 lζ) (x^2))))/(3 (-1+x)))+(1/18) CF (9 CF (-6 lζ+4 ((lζ)^2)-5 (π^2))+2 CA (-67+33 lζ+3 (π^2))+8 (5-3 lζ) TrNf) DELTA[1-x]-((16 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))+((4 (CF^2) (1+3 (x^2)) Log[x])/(-1+x)))+Lμ (-((CF (16 TrNf (4+x (-3+4 x))+9 CF (30 (((-1+x))^2)+4 lζ (((-1+x))^2)+(π^2) (1+(x^2)))+CA (6 (π^2) (1+(x^2))-4 (77+x (-87+77 x)))))/(9 (-1+x)))+(1/9) CF (134 CA lζ-3 CF (-3+lζ) (π^2)-CA (11+6 lζ) (π^2)+4 (-10 lζ+(π^2)) TrNf) DELTA[1-x]+16 (CF^2) (-1+x) Log[1-x]+(((2 CF (6 CF (-5+2 x)-4 TrNf (1+(x^2))+CA (17+5 (x^2))))/(3 (-1+x)))-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) Log[x]+((2 CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^2))/(-1+x)))+(-4 (CA-2 CF) CF (-1+x)+((4 (CA-CF) CF (1+(x^2)) Log[1-x])/(-1+x))+((4 (CA-3 CF) CF (1+(x^2)) Log[x])/(-1+x))) PolyLog[2,x]+((4 (CA-CF) CF (1+(x^2)) PolyLog[3,1-x])/(-1+x))-((4 (2 CA-5 CF) CF (1+(x^2)) PolyLog[3,x])/(-1+x))+(1/(54 (-1+x)))CF (1188 CF (((-1+x))^2)+54 CA (π^2) (((-1+x))^2)+16 CA (1+x (99+x))-8 TrNf (19+x (18+19 x))+9 CF (π^2) (-5+(4-5 x) x+2 lζ (1+(x^2)))-36 (9 CA+34 CF) (1+(x^2)) Zeta[3])+(1/648) CF DELTA[1-x] (-27 CF (12 lζ (π^2)+5 (π^4)-96 Zeta[3])+8 TrNf (-328-336 lζ+45 (π^2)+252 Zeta[3])+2 CA (4856+4848 lζ-603 (π^2)+18 (π^4)-252 (11+18 lζ) Zeta[3]))+(1/ϵ)(((CF (-4 TrNf (11+x (-12+11 x))+CA (187+x (-240+187 x)-3 (π^2) (1+(x^2)))-3 CF (36 (((-1+x))^2)+(π^2) (1+(x^2)))))/(9 (-1+x)))+((Lμ)^2) (-((2 (CF^2) (1+(x^2)))/(-1+x))+3 (CF^2) DELTA[1-x])+8 (CF^2) (-1+x) Log[1-x]+(((CF (6 CF (-5+2 x)-4 TrNf (1+(x^2))+CA (17+5 (x^2))))/(3 (-1+x)))-((4 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) Log[x]+((CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^2))/(-1+x))+Lμ (((2 (CF^2) (1+(-8+x) x+2 lζ (1+(x^2))))/(-1+x))-(2/3) (CF^2) (9 lζ+4 (π^2)) DELTA[1-x]-((16 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))+((4 (CF^2) (1+3 (x^2)) Log[x])/(-1+x)))+(1/36) CF DELTA[1-x] (4 (3+4 (π^2)) TrNf+27 CF (-1+2 (π^2)-16 Zeta[3])+CA (-51-44 (π^2)+216 Zeta[3])));

Q ← G

F[q←g,1]=Collect[Normal@Series[Φ[q←g,1]-((S1 Φ[q←g,0])/2)+(Zq[1]-Z3[1])Φ[q←g,0]/.CK->CF,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

F[q←g,2]=Collect[Normal@Series[Φ[q←g,2]-((S1 Φ[q←g,1])/2)+((3/4)(S1^2)-((S1^2)/2)-S2)((Φ[q←g,0])/2)+(Zq[1]-Z3[1])(Φ[q←g,1]-(S1/2) Φ[q←g,0])+(Zq[2]-Z2[2]-Zq[1]Z2[1]+((Z2[1])^2))Φ[q←g,0]/.CK->CF,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

F[q←g,1]=-2 TR (2 (-1+x) x+Lμ (1-2 x+2 (x^2)))-((2 TR (1-2 x+2 (x^2)))/ϵ)-(1/6) TR (24 (-1+x) x+24 Lμ (-1+x) x+6 ((Lμ)^2) (1-2 x+2 (x^2))+(π^2) (1-2 x+2 (x^2))) ϵ;
F[q←g,2]=(10/3) CF ((Lμ)^3) TR (1+2 (-1+x) x)+(2/3) TR ((CA+CF) (π^2)-(15 CA+9 CF+2 (CA+CF) (π^2)) x+2 (CA+CF) (6+(π^2)) (x^2)) Log[1-x]+4 (CA-CF) TR (-1+x) x ((Log[1-x])^2)+(2/3) (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^3)+(1/3) TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^3)+((((TR (CA (8+2 x (14+(2-9 x) x))+x (3 CF (-1+4 x)-8 TrNf (1+2 (-1+x) x))))/(3 x))+4 (CA+CF) TR (1+2 (-1+x) x) Log[1-x]+2 TR (CA (2+8 x)+CF (-1+2 x-4 (x^2))) Log[x])/(ϵ^2))+((Lμ)^2) (((TR (CA (16+x (23+2 (37-51 x) x))+x (-4 TrNf (1+2 (-1+x) x)+3 CF (-11+2 (11-7 x) x-6 lζ (1+2 (-1+x) x)))))/(3 x))+8 (CA+CF) TR (1+2 (-1+x) x) Log[1-x]+4 TR (CA (2+8 x)+CF (-1+2 x-4 (x^2))) Log[x])+((Log[x])^2) (-2 TR (-8 CA x+CF (1+2 (-1+x) x)) Log[1-x]+(1/6) TR (CA (-6+8 (3-11 x) x)+3 CF (1+4 (3-2 x) x)+12 CA (1+2 x (1+x)) Log[1+x]))+Log[x] (8 CF TR (-1+x) x Log[1-x]+2 CF TR (1+2 (-1+x) x) ((Log[1-x])^2)+(1/9) TR (3 CF (24+(π^2) (1-2 x)+3 (15-8 x) x)+2 CA (42+3 (π^2)+4 x (21+34 x))+36 CA Log[1+x] (2 x (1+x)-(1+2 x (1+x)) Log[1+x])))+(((8 CA TR (-1+x) (2+x (-1+11 x)))/(3 x))+4 (-CA+CF) TR (1+2 (-1+x) x) Log[1-x]+4 TR (4 CA x+CF (-1-2 (-1+x) x)) Log[x]) PolyLog[2,1-x]+(8 CA TR x (1+x)-4 CA TR (1+2 x (1+x)) Log[x]) PolyLog[2,-x]+(1/ϵ)(2 CF ((Lμ)^2) TR (1+2 (-1+x) x)+((TR (2 CA (-32+x (54+x (-111+6 (π^2)+80 x)))+3 x (-16 TrNf (-1+x) x+3 CF (-6+(25-24 x) x+(π^2) (1+2 (-1+x) x)))))/(9 x))+8 (CA+CF) TR (-1+x) x Log[1-x]+2 (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^2)+TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^2)+Lμ (((2 TR (CA (8+x (17+26 x-40 (x^2)))+2 x (-2 TrNf (1+2 (-1+x) x)-3 CF (2+lζ-(5+2 lζ) x+(3+2 lζ) (x^2)))))/(3 x))+8 (CA+CF) TR (1+2 (-1+x) x) Log[1-x]+4 TR (CA (2+8 x)+CF (-1+2 x-4 (x^2))) Log[x])+Log[x] (4 CF TR (1+2 (-1+x) x) Log[1-x]+(1/3) TR (CA (-6+8 (6-11 x) x)+3 CF (1+4 (3-2 x) x)+12 CA (1+2 x (1+x)) Log[1+x]))+4 CA TR (1+2 x (1+x)) PolyLog[2,-x])+Lμ (((TR (4 CA (-32+x (54+x (-78+6 (π^2)+47 x)))+3 x (-16 TrNf (-1+x) x+CF (7 (π^2) (1+2 (-1+x) x)+6 (-6+x (31+4 lζ-30 x-4 lζ x))))))/(9 x))+16 (CA+CF) TR (-1+x) x Log[1-x]+4 (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^2)+2 TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^2)+Log[x] (8 CF TR (1+2 (-1+x) x) Log[1-x]+(2/3) TR (CA (-6+8 (6-11 x) x)+3 CF (1+4 (3-2 x) x)+12 CA (1+2 x (1+x)) Log[1+x]))+8 CA TR (1+2 x (1+x)) PolyLog[2,-x])+4 (CA-CF) TR (1+2 (-1+x) x) PolyLog[3,1-x]+4 CA TR (1+2 x (1+x)) PolyLog[3,-x]-4 TR (-8 CA x+CF (1+2 (-1+x) x)) PolyLog[3,x]-8 CA TR (1+2 x (1+x)) PolyLog[3,(x/(1+x))]-(1/(54 x))TR (6 x (2 TrNf (24 (-1+x) x+(π^2) (1+2 (-1+x) x))+3 CF (39-225 x+216 (x^2)+(π^2) (2+lζ+x-2 lζ x+(-3+2 lζ) (x^2))))+CA (-544+828 x+4 (x^2) (-729+712 x)+3 (π^2) (-8+x (-17+10 x (-5+4 x))))+36 CA x (1+2 x (1+x)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+72 x (3 CA (1+2 (-2+x) x)-26 CF (1+2 (-1+x) x)) Zeta[3]);

G ← Q

F[g←q,1]=Collect[Normal@Series[Φ[g←q,1]-(S1 Φ[g←q,0])/2+(Zg[1]-Z2[1])Φ[g←q,0]/.CK->CA,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

F[g←q,2]=Collect[Normal@Series[Φ[g←q,2]-(S1 Φ[g←q,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Φ[g←q,0]/2+(Zg[1]-Z2[1])(Φ[g←q,1]-S1/2 Φ[g←q,0])+(Zg[2]-Z2[2]-Zg[1]Z2[1]+((Z2[1])^2))Φ[g←q,0]/.CK->CA,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

F[g←q,1]=2 CF (Lμ (2-(2/x)-x)+x)+((2 CF (2-(2/x)-x))/ϵ)-((CF (-12 Lμ (x^2)+6 ((Lμ)^2) (2-2 x+(x^2))+(π^2) (2-2 x+(x^2))) ϵ)/(6 x));
F[g←q,2]=((10 CA CF ((Lμ)^3) (2+(-2+x) x))/(3 x))+((2 CF (8 TrNf (5+(-5+x) x)+3 CF (48+3 x (-16+5 x)+(π^2) (2+(-2+x) x))+CA (-152+(152-43 x) x+3 (π^2) (2+(-2+x) x))) Log[1-x])/(9 x))+((CF (CA (-22+(22-5 x) x)+3 CF (6+(-6+x) x)+4 TrNf (2+(-2+x) x)) ((Log[1-x])^2))/(3 x))-((2 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^3))/(3 x))-(1/3) CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^3)+((Lμ)^2) (((2 CF (-3 CF (-4+x) x+4 TrNf (2+(-2+x) x)+CA (-84-9 lζ (2+(-2+x) x)+2 x (35+4 (-1+x) x))))/(3 x))+((8 CF (CA+CF) (2+(-2+x) x) Log[1-x])/x)-((4 CF (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x])/x))+((((CF (-3 CF (-4+x) x-8 TrNf (2+(-2+x) x)+2 CA (-9+2 x (1+x (7+2 x)))))/(3 x))+((4 CF (CA+CF) (2+(-2+x) x) Log[1-x])/x)-((2 CF (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x])/x))/(ϵ^2))+((Log[x])^2) (((2 CA CF (2+(-2+x) x) Log[1-x])/x)+((CF x (-3 CF (4+3 x)+2 CA (36+x (9+8 x)))-12 CA CF (2+x (2+x)) Log[1+x])/(6 x)))+Log[x] (((4 CA CF (-22+x (24+x (-9+4 x))) Log[1-x])/(3 x))+((2 CA CF (2+(-2+x) x) ((Log[1-x])^2))/x)+((CF (-3 CF (-15 (-3+x)+(π^2) (-2+x)) x-2 CA (6 (π^2) (1+x+(x^2))+x (249-6 x+88 (x^2)))+36 CA Log[1+x] ((x^2)+(2+x (2+x)) Log[1+x])))/(9 x)))+(4 CA CF x+((4 CA CF (2+x (2+x)) Log[x])/x)) PolyLog[2,-x]+Lμ (((CF (2 (9 CF x (1+5 x)+40 TrNf (2+(-2+x) x))+CA (3 (π^2) (6+x (-14+3 x))-4 (9+x (19+x (4-9 lζ+44 x))))))/(9 x))+((4 CF (CA (-22+(22-17 x) x)+3 CF (6+(-6+x) x)+4 TrNf (2+(-2+x) x)) Log[1-x])/(3 x))-((4 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^2))/x)-2 CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^2)+Log[x] (((8 CA CF (2+(-2+x) x) Log[1-x])/x)-((2 CF (x (3 CF (4+3 x)-2 CA (36+x (15+8 x)))+12 CA (2+x (2+x)) Log[1+x]))/(3 x)))-((8 CA CF (2+x (2+x)) PolyLog[2,-x])/x))+(1/ϵ)(((2 CA CF ((Lμ)^2) (2+(-2+x) x))/x)+((CF (9 CF x (1+5 x)+16 TrNf (5+x (-5+4 x))+CA (3 (π^2) (2+(-6+x) x)-2 (9+x (19+x (37+44 x))))))/(9 x))+((2 CF (CA (-22+(22-17 x) x)+3 CF (6+(-6+x) x)+4 TrNf (2+(-2+x) x)) Log[1-x])/(3 x))-((2 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^2))/x)-CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^2)+Lμ (((2 CF (-3 CF (-4+x) x+2 CA (-3 lζ (2+(-2+x) x)+(-1+x) (31+x (7+4 x)))))/(3 x))+((8 CF (CA+CF) (2+(-2+x) x) Log[1-x])/x)-((4 CF (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x])/x))+Log[x] (((4 CA CF (2+(-2+x) x) Log[1-x])/x)+((CF x (-3 CF (4+3 x)+2 CA (36+x (15+8 x)))-12 CA CF (2+x (2+x)) Log[1+x])/(3 x)))-((4 CA CF (2+x (2+x)) PolyLog[2,-x])/x))+(((8 CA CF (-1+x) (11+x (-1+2 x)))/(3 x))+((4 CA CF (6+x (-2+3 x)) Log[x])/x)) PolyLog[2,x]-((4 CA CF (2+x (2+x)) PolyLog[3,-x])/x)-((4 CA CF (10+x (-2+5 x)) PolyLog[3,x])/x)+((8 CA CF (2+x (2+x)) PolyLog[3,(x/(1+x))])/x)+(1/(54 x))CF (9 CF (60-(π^2) (-4+x)-6 x) x+16 TrNf (56+x (-56+13 x))+2 CA (-3 (π^2) (-13+4 x (6+(-3+x) x)+3 lζ (2+(-2+x) x))+4 (-790+x (791+4 x (-67+38 x))))+36 CA ((2+x (2+x)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+4 (20+x (-23+10 x)) Zeta[3]));

G ← G

F[g←g,1]=Collect[Normal@Series[Φ[g←g,1]-(S1 Φ[g←g,0])/2+(Zg[1]-Z3[1])Φ[g←g,0]/.CK->CA,{ϵ,0,1}],{ϵ,Lμ,DELTA[_]},Simplify]

F[g←g,2]=Collect[Normal@Series[Φ[g←g,2]-(S1 Φ[g←g,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Φ[g←g,0]/2+(Zg[1]-Z3[1])(Φ[g←g,1]-S1/2 Φ[g←g,0])+(Zg[2]-Z3[2]-Zg[1]Z3[1]+((Z3[1])^2))Φ[g←g,0]/.CK->CA,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

F[g←g,1]=-CA ((Lμ)^2) DELTA[1-x]-(1/6) CA (π^2) DELTA[1-x]+Lμ (((4 CA (((1-x+(x^2)))^2))/((-1+x) x))+2 CA lζ DELTA[1-x])+((((4 CA (((1-x+(x^2)))^2))/((-1+x) x))+(1/3) (-11 CA+4 TrNf) DELTA[1-x])/ϵ)+ϵ (((CA (π^2) (((1-x+(x^2)))^2))/(3 (-1+x) x))-(2/3) CA ((Lμ)^3) DELTA[1-x]-(1/3) CA Lμ (π^2) DELTA[1-x]+((Lμ)^2) (((2 CA (((1-x+(x^2)))^2))/((-1+x) x))+CA lζ DELTA[1-x])+(1/6) CA DELTA[1-x] (lζ (π^2)-8 Zeta[3]));
F[g←g,2]=(1/2) (CA^2) ((Lμ)^4) DELTA[1-x]+((Lμ)^3) (-((20 (CA^2) (((1+(-1+x) x))^2))/(3 (-1+x) x))-2 (CA^2) lζ DELTA[1-x])+((2 CA (CA (x^2)-2 TrNf (x^2)-((4 CA (π^2) (((1+(-1+x) x))^2))/(-1+x))) Log[1-x])/(3 x))+(4/3) ((CA^2) (-2+x) x+CF TrNf (1+x)+((CA^2)/(-1+(x^2)))) ((Log[x])^3)+((Lμ)^2) (((4 (2 CF TrNf (-4+x+6 (x^2)+(x^3)-4 (x^4))+(CA^2) (9 lζ (((1+(-1+x) x))^2)+4 (((-1+x))^2) (11+x (2+11 x)))))/(3 (-1+x) x))+(1/18) CA (CA (-134+36 ((lζ)^2)-39 (π^2))+40 TrNf) DELTA[1-x]-((32 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+16 (((CA^2)/((-1+x) x))+(CA^2) (-3+x) x+CF TrNf (1+x)) Log[x])+(1/(ϵ^2))(((2 (12 CA TrNf (((1+(-1+x) x))^2)+2 CF TrNf (-4+x+6 (x^2)+(x^3)-4 (x^4))+(CA^2) (11+x (-14+x (-27+x (-14+11 x))))))/(3 (-1+x) x))+(1/9) ((CA^2) (121-12 (π^2))-88 CA TrNf+16 (TrNf^2)) DELTA[1-x]-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((8 (CF TrNf x (-1+(x^2))+(CA^2) (1+(-3+x) (-1+x) (x^2))) Log[x])/((-1+x) x)))+((Log[x])^2) (-((4 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((x (1+x) (4 CA TrNf (1+x)+6 CF TrNf (3+x)+(CA^2) (25+11 x (-1+4 x)))-12 (CA^2) (((1+x+(x^2)))^2) Log[1+x])/(3 x (1+x))))+Log[x] (((8 (CA^2) (-1+x) (11+x (-1+11 x)) Log[1-x])/(3 x))-((4 (CA^2) (((1+(-1+x) x))^2) ((Log[1-x])^2))/((-1+x) x))+(1/9) (52 CA TrNf+40 CA TrNf x+12 CF (18+(π^2)) TrNf (1+x)+((12 (CA^2) (π^2) (1+(-3+x) (-1+x) (x^2)))/((-1+x) x))-(CA^2) (701+x (149+536 x))+((72 (CA^2) (((1+x+(x^2)))^2) ((Log[1+x])^2))/(x (1+x)))))+((8 (CA^2) (((1+x+(x^2)))^2) Log[x] PolyLog[2,-x])/(x (1+x)))+Lμ ((1/(9 x (-1+(x^2))))2 (24 CF TrNf (((-1+x))^2) (1+x) (1+(-8+x) x)-4 CA TrNf (1+x) (23+x (-42+x (48+x (-42+23 x))))+(CA^2) (x (1+x) (109+x (-84+109 x))-3 (π^2) (3+x (-7+x (3+x (7+x (-3+7 x)))))))-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf) DELTA[1-x]+(8 CF TrNf (1+x)+((8 (CA^2) (((1+x-(x^2)))^2))/(-1+(x^2)))) ((Log[x])^2)+Log[x] (-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+(4/3) (4 CA TrNf (1+x)+6 CF TrNf (3+x)+(CA^2) (25+11 x (-1+4 x))-((12 (CA^2) (((1+x+(x^2)))^2) Log[1+x])/(x (1+x)))))-((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[2,-x])/(x (1+x))))+(((8 (CA^2) (-1+x) (11+x (-1+11 x)))/(3 x))-((8 (CA^2) (3+x (-1+x (3+x-3 (x^2)+(x^3)))) Log[x])/(x (-1+(x^2))))) PolyLog[2,x]-((8 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,-x])/(x (1+x)))+((8 (CA^2) (5+x (-1+x (5+x-5 (x^2)+(x^3)))) PolyLog[3,x])/(x (-1+(x^2))))+((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,(x/(1+x))])/(x (1+x)))+(1/(27 x (-1+(x^2))))((1+x) (-6 CF TrNf (((-1+x))^2) (-12 (1+(-23+x) x)+(π^2) (4+x (7+4 x)))+4 CA TrNf (-121+3 (π^2) (((1+(-1+x) x))^2)+x (231+x (-276+(249-139 x) x)))+(CA^2) (3 (π^2) (-11+6 lζ (((1+(-1+x) x))^2)-(-1+x) x (34+x (-23+11 x)))+4 (790+x (-1514+x (1686+5 x (-319+167 x))))))+36 (CA^2) ((-1+x) (((1+x+(x^2)))^2) Log[1+x] ((π^2)-2 ((Log[1+x])^2))-2 (20+x (-23+x (20+x (23+x (-20+23 x))))) Zeta[3]))+(1/324) CA DELTA[1-x] (4 TrNf (-328+45 (π^2)+6 lζ (-56+3 (π^2))+108 Zeta[3])+CA (4856-9 (π^2) (67+6 (π^2))-135 ((Log[2])^4)+3240 PolyLog[2,2,(1/2)]+27 (-44+15 Log[2]) Zeta[3]-6 lζ (-808+33 (π^2)+756 Zeta[3])))+(1/ϵ)(1/(9 x (-1+(x^2)))(24 CF TrNf (((-1+x))^2) (1+x) (1+(-8+x) x)-4 CA TrNf (1+x) (23+x (-42+x (48+x (-42+23 x))))+(CA^2) (x (1+x) (109+x (-84+109 x))-6 (π^2) (1+x (-3+x (1+x (3+x (-1+3 x)))))))+((Lμ)^2) (-((4 (CA^2) (((1+(-1+x) x))^2))/((-1+x) x))+(1/3) CA (11 CA-4 TrNf) DELTA[1-x])+(4 CF TrNf (1+x)+((4 (CA^2) (((1+x-(x^2)))^2))/(-1+(x^2)))) ((Log[x])^2)+Lμ (((4 (4 CA TrNf (((1+(-1+x) x))^2)+2 CF TrNf (-4+x+6 (x^2)+(x^3)-4 (x^4))+(CA^2) (33+6 lζ (((1+(-1+x) x))^2)+x (-58+x (39+x (-58+33 x))))))/(3 (-1+x) x))-(2/3) CA (11 CA lζ+4 CA (π^2)-4 lζ TrNf) DELTA[1-x]-((32 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+16 (((CA^2)/((-1+x) x))+(CA^2) (-3+x) x+CF TrNf (1+x)) Log[x])+Log[x] (-((8 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+(2/3) (4 CA TrNf (1+x)+6 CF TrNf (3+x)+(CA^2) (25+11 x (-1+4 x))-((12 (CA^2) (((1+x+(x^2)))^2) Log[1+x])/(x (1+x)))))-((8 (CA^2) (((1+x+(x^2)))^2) PolyLog[2,-x])/(x (1+x)))+(1/18) DELTA[1-x] (36 CF TrNf-4 CA (-12+(π^2)) TrNf+(CA^2) (11 (π^2)-12 (8+9 Zeta[3]))));

Q ← Q’

Since (Subscript[((Φ^([0,1]))), q←q'])=0, TMD concides with unsubtracted TMD.

F[q←qp,1]=0;
F[q←qp,2]=-((CF TR (-1+x) (272+2 x (-71+208 x)+36 ((Lμ)^2) (4+x (7+4 x))+3 (π^2) (4+x (7+4 x))-24 Lμ (16+x (-11+16 x))))/(27 x))+(2/9) CF TR (42+36 ((Lμ)^2) (1+x)+3 (π^2) (1+x)+8 x (-3+8 x)-6 Lμ (3+x (3+8 x))) Log[x]+(1/3) CF TR (-3+12 Lμ (1+x)-x (3+8 x)) ((Log[x])^2)+(2/3) CF TR (1+x) ((Log[x])^3)+((-((2 CF TR (-1+x) (4+x (7+4 x)))/(3 x))+4 CF TR (1+x) Log[x])/(ϵ^2))+((-((4 CF TR (-1+x) (-16+(11-16 x) x+3 Lμ (4+x (7+4 x))))/(9 x))+(2/3) CF TR (-3+12 Lμ (1+x)-x (3+8 x)) Log[x]+2 CF TR (1+x) ((Log[x])^2))/ϵ)+((8 CF TR (-1+x) (2+x (-1+2 x)) PolyLog[2,1-x])/(3 x));

Q ←  (OverBar[Q])

Since (Subscript[((Φ^([0,1]))), q←qbar])=0, TMD concides with unsubtracted TMD.

F[q←qbar,1]=0;
F[q←qbar,2]=(((CA-2 CF) CF (1+(x^2)) ((Log[x])^3))/(3 (1+x)))+((2 (CA-2 CF) CF (1+(x^2)) ((Log[x])^2) (Lμ-2 Log[1-x]-Log[1+x]))/(1+x))+(((CA-2 CF) CF Log[x] (12 Lμ (((1+x))^2)-3 (1+x) (3+11 x)+2 (π^2) (1+(x^2))+12 Log[1+x] ((((1+x))^2)-2 Lμ (1+(x^2))+(1+(x^2)) Log[1+x])))/(3 (1+x)))+(-4 (CA-2 CF) CF (-1+x)-((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,1-x]+(((4 (CA-2 CF) CF ((((1+x))^2)-2 Lμ (1+(x^2))))/(1+x))+((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,-x]+((-(((CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (x^2)))/(3 (1+x)))+(((CA-2 CF) CF (1+(x^2)) ((Log[x])^2))/(1+x))+((2 (CA-2 CF) CF Log[x] ((((1+x))^2)-2 (1+(x^2)) Log[1+x]))/(1+x))-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[2,-x])/(1+x)))/ϵ)-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[3,-x])/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,x])/(1+x))+((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,(x/(1+x))])/(1+x))-(((CA-2 CF) CF (45-45 (x^2)-(π^2) (((1+x))^2)+2 Lμ (-12+(π^2)+(12+(π^2)) (x^2))-2 (1+(x^2)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+6 (1+(x^2)) Zeta[3]))/(3 (1+x)));

TMD FFs

In this section we recombine NLO and NNLO expressions for TMD PDF.
They are given by
DD[f ← f', 1] = (z^2)((Subscript[D, f←f'])^([1]))=(Δ^([1]))-((S^([1]))(Δ^([0])))/2+(((Subscript[Z, f])^([1]))-((Subscript[Z, f'])^([1])))(Δ^([0])),
DD[f ← f', 2] = (z^2)((Subscript[D, f←f'])^([2]))=(Δ^([2]))-((S^([1]))(Δ^([1])))/2+3/8(S^([1]))(S^([1]))(Δ^([0]))-(((S^([1]))(S^([1])))/2+(S^([2])))(Δ^([0]))/2+(((Subscript[Z, f])^([1]))-((Subscript[Z, f'])^([1])))((Δ^([1]))-((S^([1]))(Δ^([0])))/2)+(((Subscript[Z, f])^([2]))-((Subscript[Z, f'])^([2]))-((Subscript[Z, f])^([1]))((Subscript[Z, f'])^([1]))+((Subscript[Z, f'])^([1]))((Subscript[Z, f'])^([1])))(Δ^([0])),
where (Subscript[Z, f'])=(Subscript[Z, 2,3])
In the presented results one can observe the cancelation of rapidity-divegences (ln(δp))

Q -> Q

DD[q->q,1]=Collect[Normal@Series[Δ[q->q,1]-(S1 Δ[q->q,0])/2+(Zq[1]-Z2[1])Δ[q->q,0]/.CK->CF,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

DD[q->q,2]=Collect[Normal@Series[Δ[q->q,2]-(S1 Δ[q->q,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Δ[q->q,0]/2+(Zq[1]-Z2[1])(Δ[q->q,1]-S1/2 Δ[q->q,0])+(Zq[2]-Z2[2]-Zq[1]Z2[1]+((Z2[1])^2))Δ[q->q,0]/.CK->CF,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

DD[q->q,1]=((2 CF (-(((-1+z))^2)+Lμ (1+(z^2))))/(-1+z))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((((2 CF (1+(z^2)))/(-1+z))-3 CF DELTA[1-z])/ϵ)+ϵ (((CF (-12 Lμ (((-1+z))^2)+6 ((Lμ)^2) (1+(z^2))+(π^2) (1+(z^2))))/(6 (-1+z)))+(1/6) CF DELTA[1-z] (lζ (6 ((Lμ)^2)+(π^2))-2 (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));
DD[q->q,2]=(1/2) (CF^2) ((Lμ)^4) DELTA[1-z]+((Lμ)^3) (-((10 (CF^2) (1+(z^2)))/(3 (-1+z)))+(2/9) CF (-11 CA-9 CF (-1+lζ)+4 TrNf) DELTA[1-z])+((2 CF (CA (3-3 z+(π^2) (1+(z^2)))-3 CF (1-z+(π^2) (1+(z^2)))) Log[1-z])/(3 (-1+z)))+(((CF (20 TrNf (1+(z^2))-3 CF (-102+13 (π^2)+84 z+(-30+11 (π^2)) (z^2))+2 CA (-92+(45-29 z) z+6 (π^2) (1+(z^2)))))/(9 (-1+z)))+4 (CA-3 CF) CF (-1+z) Log[1-z]-((2 (2 CA-3 CF) CF (1+(z^2)) ((Log[1-z])^2))/(-1+z))) Log[z]+(((CF (-CA (13+(-36+z) z)-4 TrNf (1+(z^2))+6 CF (10+z (-18+5 z))))/(6 (-1+z)))+((2 (2 CA-5 CF) CF (1+(z^2)) Log[1-z])/(-1+z))) ((Log[z])^2)+((CF (-3 CA (1+(z^2))+CF (7+9 (z^2))) ((Log[z])^3))/(3 (-1+z)))+((-((CF (11 CA (1+(z^2))-4 TrNf (1+(z^2))+6 CF (1+z (4+z))))/(3 (-1+z)))+(1/6) CF (33 CA+CF (27-8 (π^2))-12 TrNf) DELTA[1-z]-((8 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))+((2 (CF^2) (1+3 (z^2)) Log[z])/(-1+z)))/(ϵ^2))+((Lμ)^2) (((CF (11 CA (1+(z^2))-4 TrNf (1+(z^2))+3 CF (7+6 lζ-20 z+(7+6 lζ) (z^2))))/(3 (-1+z)))+(1/18) CF (9 CF (-6 lζ+4 ((lζ)^2)-5 (π^2))+2 CA (-67+33 lζ+3 (π^2))+8 (5-3 lζ) TrNf) DELTA[1-z]-((16 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))+((4 (CF^2) (1+3 (z^2)) Log[z])/(-1+z)))+Lμ (-((CF (16 TrNf (4+z (-3+4 z))+9 CF (30 (((-1+z))^2)+4 lζ (((-1+z))^2)+(π^2) (1+(z^2)))+CA (6 (π^2) (1+(z^2))-4 (77+z (-87+77 z)))))/(9 (-1+z)))+(1/9) CF (134 CA lζ-3 CF (-3+lζ) (π^2)-CA (11+6 lζ) (π^2)+4 (-10 lζ+(π^2)) TrNf) DELTA[1-z]+16 (CF^2) (-1+z) Log[1-z]+(((2 CF (4 TrNf (1+(z^2))-CA (5+17 (z^2))+6 CF (-4+z (6+z))))/(3 (-1+z)))+((8 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))) Log[z]+((2 CF (CA-3 CF+(CA-5 CF) (z^2)) ((Log[z])^2))/(-1+z)))+(4 (CA-2 CF) CF (-1+z)-((4 (CA-CF) CF (1+(z^2)) Log[1-z])/(-1+z))+((8 (CA-2 CF) CF (1+(z^2)) Log[z])/(-1+z))) PolyLog[2,z]-((4 (CA-CF) CF (1+(z^2)) PolyLog[3,1-z])/(-1+z))-((12 (CA-2 CF) CF (1+(z^2)) PolyLog[3,z])/(-1+z))+(1/(54 (-1+z)))CF (1188 CF (((-1+z))^2)-18 CA (π^2) (((-1+z))^2)+16 CA (1+z (99+z))-8 TrNf (19+z (18+19 z))+9 CF (π^2) (11+z (-28+11 z)+2 lζ (1+(z^2)))-36 (3 CA+40 CF) (1+(z^2)) Zeta[3])+(1/648) CF DELTA[1-z] (-27 CF (12 lζ (π^2)+5 (π^4)-96 Zeta[3])+8 TrNf (-328-336 lζ+45 (π^2)+252 Zeta[3])+2 CA (4856+4848 lζ-603 (π^2)+234 (π^4)-252 (11+18 lζ) Zeta[3]))+(1/ϵ)(((CF (-4 TrNf (11+z (-12+11 z))+CA (187+z (-240+187 z)-3 (π^2) (1+(z^2)))-3 CF (36 (((-1+z))^2)+(π^2) (1+(z^2)))))/(9 (-1+z)))+((Lμ)^2) (-((2 (CF^2) (1+(z^2)))/(-1+z))+3 (CF^2) DELTA[1-z])+8 (CF^2) (-1+z) Log[1-z]+(((CF (4 TrNf (1+(z^2))-CA (5+17 (z^2))+6 CF (-4+z (6+z))))/(3 (-1+z)))+((4 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))) Log[z]+((CF (CA-3 CF+(CA-5 CF) (z^2)) ((Log[z])^2))/(-1+z))+Lμ (((2 (CF^2) (1+(-8+z) z+2 lζ (1+(z^2))))/(-1+z))-(2/3) (CF^2) (9 lζ+4 (π^2)) DELTA[1-z]-((16 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))+((4 (CF^2) (1+3 (z^2)) Log[z])/(-1+z)))+(1/36) CF DELTA[1-z] (4 (3+4 (π^2)) TrNf+27 CF (-1+2 (π^2)-16 Zeta[3])+CA (-51-44 (π^2)+216 Zeta[3])));

Q -> G

DD[q->g,1]=Collect[Normal@Series[Δ[q->g,1]-(S1 Δ[q->g,0])/2+(Zq[1]-Z3[1])Δ[q->g,0]/.CK->CF,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

DD[q->g,2]=Collect[Normal@Series[Δ[q->g,2]-(S1 Δ[q->g,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Δ[q->g,0]/2+(Zq[1]-Z3[1])(Δ[q->g,1]-S1/2 Δ[q->g,0])+(Zq[2]-Z3[2]-Zq[1]Z3[1]+((Z3[1])^2))Δ[q->g,0]/.CK->CF,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

DD[q->g,1]=2 CF (Lμ (2-(2/z)-z)+z)+((2 CF (2-(2/z)-z))/ϵ)-((CF (-12 Lμ (z^2)+6 ((Lμ)^2) (2-2 z+(z^2))+(π^2) (2-2 z+(z^2))) ϵ)/(6 z));
DD[q->g,2]=((10 (CF^2) ((Lμ)^3) (2+(-2+z) z))/(3 z))+((2 CF (3 CF (z+(π^2) (2+(-2+z) z))-CA (3 z+(π^2) (2+(-2+z) z))) Log[1-z])/(3 z))+2 CF (-CA+CF) z ((Log[1-z])^2)+((2 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^3))/(3 z))-((CF (CF (-2+z) z+2 CA (8+z (2+5 z))) ((Log[z])^3))/(3 z))+((Lμ)^2) (((CF (-18 CF (3+2 lζ)-8 TrNf-(3 CF (13+6 lζ)+4 TrNf) (-2+z) z+CA (-102+z (74+z (23+16 z)))))/(3 z))+((8 CF (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((4 CF (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])/z))+((((CF (-3 CF (-4+z) z-8 TrNf (2+(-2+z) z)+2 CA (-9+2 z (1+z (7+2 z)))))/(3 z))+((4 CF (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((2 CF (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])/z))/(ϵ^2))+Log[z] (((4 CA CF (22+z (-24+(9-4 z) z)) Log[1-z])/(3 z))-((2 (3 CA-2 CF) CF (2+(-2+z) z) ((Log[1-z])^2))/z)-((CF (3 CF z ((π^2) (-2+z)+3 (5+7 z))+2 CA (268+3 z+51 (z^2)-6 (π^2) (3+(-7+z) z))-36 CA (z^2) Log[1+z]))/(9 z)))+((Log[z])^2) (((6 CA CF (2+(-2+z) z) Log[1-z])/z)+((CF (-3 CF (-8+z) z+2 CA (-88+z (48+z (-33+8 z)))-12 CA (2+z (2+z)) Log[1+z]))/(6 z)))+(4 CA CF z-((4 CA CF (2+z (2+z)) Log[z])/z)) PolyLog[2,-z]+Lμ ((1/(9 z))CF (24 TrNf (z^2)-2 CA (-214+12 (π^2) (4+z (-3+2 z))+z (234+z (-57+88 z)))+3 CF (19 (π^2) (2+(-2+z) z)+6 (-24+z (21+2 (-1+lζ) z))))-8 CF (CA+CF) z Log[1-z]+((4 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^2))/z)+2 CF (CF (-2+z)+CA (4+(8/z)+6 z)) ((Log[z])^2)+Log[z] (-((8 CA CF (2+(-2+z) z) Log[1-z])/z)+((2 CF (3 CF (-8+z) z+CA (88+30 (z^2))-12 CA (2+z (2+z)) Log[1+z]))/(3 z)))-((8 CA CF (2+z (2+z)) PolyLog[2,-z])/z))+(1/ϵ)(((2 (CF^2) ((Lμ)^2) (2+(-2+z) z))/z)+((CF (24 TrNf (z^2)+9 CF (-24+(21-5 z) z+3 (π^2) (2+(-2+z) z))+CA (214-12 (π^2) (4+z (-3+2 z))-2 z (117+4 z (-3+11 z)))))/(9 z))-4 CF (CA+CF) z Log[1-z]+((2 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^2))/z)+CF (CF (-2+z)+CA (4+(8/z)+6 z)) ((Log[z])^2)+Lμ (((2 CF (-4 TrNf (2+(-2+z) z)-6 CF (3+2 lζ-(5+2 lζ) z+(2+lζ) (z^2))+CA (-40+z (26+z (17+8 z)))))/(3 z))+((8 CF (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((4 CF (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])/z))+Log[z] (-((4 CA CF (2+(-2+z) z) Log[1-z])/z)+((CF (3 CF (-8+z) z+CA (88+30 (z^2))-12 CA (2+z (2+z)) Log[1+z]))/(3 z)))-((4 CA CF (2+z (2+z)) PolyLog[2,-z])/z))+(-((8 CA CF (-1+z) (11+z (-1+2 z)))/(3 z))-((4 (CA-CF) CF (2+(-2+z) z) Log[1-z])/z)+((4 CA CF (2+(-6+z) z) Log[z])/z)) PolyLog[2,z]-((4 (CA-CF) CF (2+(-2+z) z) PolyLog[3,1-z])/z)+4 CA CF (2+(2/z)+z) PolyLog[3,-z]+4 CA CF (10-(2/z)-z) PolyLog[3,z]-((8 CA CF (2+z (2+z)) PolyLog[3,(1/(1+z))])/z)+(1/(54 z))CF (CA (3 (π^2) (-128+z (122+z (41+24 z)))+4 (-952+z (1062+z (-369+232 z))))-36 CA (2+z (2+z)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))-6 (2 (π^2) TrNf (2+(-2+z) z)+3 CF (144+3 z (-50+7 z)+(π^2) (3+z (-5+11 z)+lζ (2+(-2+z) z))-184 Zeta[3]))+72 (6 CA+23 CF (-2+z)) z Zeta[3]);

G -> Q

DD[g->q,1]=Collect[Normal@Series[Δ[g->q,1]-(S1 Δ[g->q,0])/2+(Zg[1]-Z2[1])Δ[g->q,0]/.CK->CA,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

DD[g->q,2]=Collect[Normal@Series[Δ[g->q,2]-(S1 Δ[g->q,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Δ[g->q,0]/2+(Zg[1]-Z2[1])(Δ[g->q,1]-S1/2 Δ[g->q,0])+(Zg[2]-Z2[2]-Zg[1]Z2[1]+((Z2[1])^2))Δ[g->q,0]/.CK->CA,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

DD[g->q,1]=-2 TR (2 (-1+z) z+Lμ (1-2 z+2 (z^2)))-((2 TR (1-2 z+2 (z^2)))/ϵ)-(1/6) TR (24 (-1+z) z+24 Lμ (-1+z) z+6 ((Lμ)^2) (1-2 z+2 (z^2))+(π^2) (1-2 z+2 (z^2))) ϵ;
DD[g->q,2]=(10/3) CA ((Lμ)^3) TR (1+2 (-1+z) z)+(4/9) TR (2 TrNf (5+16 (-1+z) z)-3 CF (-12-39 (-1+z) z+(π^2) (1+2 (-1+z) z))+CA (-38-91 (-1+z) z+6 (π^2) (1+2 (-1+z) z))) Log[1-z]+(1/3) TR (9 CF+4 TrNf+2 (15 CF+4 TrNf) (-1+z) z+CA (-11-34 (-1+z) z)) ((Log[1-z])^2)-(2/3) (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^3)+(1/3) TR (CA (2+20 z)+CF (-1+2 z-4 (z^2))) ((Log[z])^3)+(1/(ϵ^2))(((TR (CA (8+2 z (14+(2-9 z) z))+z (3 CF (-1+4 z)-8 TrNf (1+2 (-1+z) z))))/(3 z))+4 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+2 TR (CA (2+8 z)+CF (-1+2 z-4 (z^2))) Log[z])+((Lμ)^2) (((2 TR (z (4 TrNf+8 TrNf (-1+z) z+3 CF (-1+4 z))+CA (8-z (5-64 z+78 (z^2)+9 lζ (1+2 (-1+z) z)))))/(3 z))+8 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+4 TR (CA (2+8 z)+CF (-1+2 z-4 (z^2))) Log[z])+Log[z] ((2/3) TR (CF (-9-30 (-1+z) z)-4 TrNf (1+2 (-1+z) z)+CA (23-(8/z)-70 z+66 (z^2))) Log[1-z]-2 CF TR (1+2 (-1+z) z) ((Log[1-z])^2)+(1/(9 z))TR (2 CA (64+z (61+z (-5+218 z)+6 (π^2) (2+z+3 (z^2))))+z (-8 TrNf (5+16 (-1+z) z)+3 CF ((π^2) (5+2 z (-5+4 z))-3 (16+z (-67+52 z))))+72 CA (z^2) (1+z) Log[1+z]))+((Log[z])^2) (2 (2 CA+CF) TR (1+2 (-1+z) z) Log[1-z]+((TR (-4 CA (-1+z) (8+z (-2+33 z))+z (8 TrNf (1+2 (-1+z) z)+CF (3+60 (-1+z) z))+12 CA z (1+2 z (1+z)) Log[1+z]))/(6 z)))+(8 CA TR z (1+z)+4 CA TR (1+2 z (1+z)) Log[z]) PolyLog[2,-z]+(1/ϵ)(2 CA ((Lμ)^2) TR (1+2 (-1+z) z)+((TR (z (8 TrNf (5+4 (-1+z) z)-9 CF (-4-5 z+(π^2) (2+4 (-1+z) z)))+CA (-64+z (10-2 z (13+36 z)+3 (π^2) (9+2 z (-7+9 z))))))/(9 z))+(2/3) TR (9 CF+4 TrNf+2 (15 CF+4 TrNf) (-1+z) z+CA (-11-10 (-1+z) z)) Log[1-z]-2 (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)+TR (CF-2 (CA+6 CA z+CF (1-2 z) z)) ((Log[z])^2)+Lμ (((2 TR (3 CF z (-1+4 z)+2 CA (4+z (3-3 lζ+6 (4+lζ) z-(31+6 lζ) (z^2)))))/(3 z))+8 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+4 TR (CA (2+8 z)+CF (-1+2 z-4 (z^2))) Log[z])+Log[z] (-4 CF TR (1+2 (-1+z) z) Log[1-z]+((TR (-z (8 TrNf (1+2 (-1+z) z)+CF (3+60 (-1+z) z))+4 CA (-4+z (4+z (-17+11 z)))+12 CA z (1+2 z (1+z)) Log[1+z]))/(3 z)))+4 CA TR (1+2 z (1+z)) PolyLog[2,-z])+Lμ ((1/(9 z))TR (2 z (40 TrNf (1+2 (-1+z) z)-9 CF (-4-5 z+(π^2) (2+4 (-1+z) z)))+CA (-128+z (3 (π^2) (19-30 z+38 (z^2))+4 (5+z (53+18 lζ-6 (17+3 lζ) z)))))+(4/3) TR (9 CF+4 TrNf+2 (15 CF+4 TrNf) (-1+z) z+CA (-11-10 (-1+z) z)) Log[1-z]-4 (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)+2 TR (CF-2 (CA+6 CA z+CF (1-2 z) z)) ((Log[z])^2)+Log[z] (-2 CF TR (4+8 (-1+z) z) Log[1-z]+(2/3) TR (-3 CF+60 CF z-60 CF (z^2)-8 TrNf (1+2 (-1+z) z)+((4 CA (-4+z (4+z (-17+11 z))))/z)+12 CA (1+2 z (1+z)) Log[1+z]))+8 CA TR (1+2 z (1+z)) PolyLog[2,-z])+(((8 CA TR (-1+z) (2+z (-1+11 z)))/(3 z))+4 CA TR (3-2 z+6 (z^2)) Log[z]) PolyLog[2,z]-4 CA TR (1+2 z (1+z)) PolyLog[3,-z]-4 CA TR (5+2 z (-1+5 z)) PolyLog[3,z]+8 CA TR (1+2 z (1+z)) PolyLog[3,(1/(1+z))]+(1/(54 z))TR (z (8 TrNf (-9 (π^2) (1+2 (-1+z) z)+8 (7+17 (-1+z) z))-9 CF (-18 (1+5 z)+(π^2) (19-64 z+60 (z^2))))+2 CA (416-2 z (521+z (-1543+1688 z))-9 (π^2) (-4+z (-8+3 (4-5 z) z+lζ (1+2 (-1+z) z))))+36 CA z ((1+2 z (1+z)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))+(40-92 z+80 (z^2)) Zeta[3]));

G -> G

DD[g->g,1]=Collect[Normal@Series[Δ[g->g,1]-(S1 Δ[g->g,0])/2+(Zg[1]-Z3[1])Δ[g->g,0]/.CK->CA,{ϵ,0,1}],{ϵ,Lb,DELTA[_]},Simplify]

DD[g->g,2]=Collect[Normal@Series[Δ[g->g,2]-(S1 Δ[g->g,1])/2+(3/4(S1^2)-(S1^2)/2-S2)Δ[g->g,0]/2+(Zg[1]-Z3[1])(Δ[g->g,1]-S1/2 Δ[g->g,0])+(Zg[2]-Z3[2]-Zg[1]Z3[1]+((Z3[1])^2))Δ[g->g,0]/.CK->CA,{ϵ,0,0}],{ϵ,Lμ,DELTA[_],PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

DD[g->g,1]=((4 CA Lμ (((1+(-1+z) z))^2))/((-1+z) z))+CA (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((((4 CA (((1+(-1+z) z))^2))/((-1+z) z))+(1/3) (-11 CA+4 TrNf) DELTA[1-z])/ϵ)+ϵ (((CA (6 ((Lμ)^2)+(π^2)) (((1+(-1+z) z))^2))/(3 (-1+z) z))+(1/6) CA DELTA[1-z] (lζ (6 ((Lμ)^2)+(π^2))-2 (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));
DD[g->g,2]=(1/2) (CA^2) ((Lμ)^4) DELTA[1-z]+((Lμ)^3) (-((20 (CA^2) (((1+(-1+z) z))^2))/(3 (-1+z) z))-2 (CA^2) lζ DELTA[1-z])-((2 CA (-2 TrNf (-1+z) z+CA ((-1+z) z+4 (π^2) (((1+(-1+z) z))^2))) Log[1-z])/(3 (-1+z) z))+((1/9) (28 CA TrNf+52 CA TrNf z+12 CF (18+(π^2)) TrNf (1+z)-(((CA^2) (536+z (143+701 z)))/z)-((12 (CA^2) (π^2) (3+z (-7+z (1+z (7+z (-1+5 z))))))/(z (-1+(z^2)))))-((8 (CA^2) (-1+z) (11+z (-1+11 z)) Log[1-z])/(3 z))+((4 (CA^2) (((1+(-1+z) z))^2) ((Log[1-z])^2))/((-1+z) z))) Log[z]+((4 (CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (4+z (1+(-6+z) (-1+z) z (1+z)))) ((Log[z])^3))/(3 z (-1+(z^2))))+(1/(ϵ^2))(((2 (12 CA TrNf (((1+(-1+z) z))^2)+2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (11+z (-14+z (-27+z (-14+11 z))))))/(3 (-1+z) z))+(1/9) ((CA^2) (121-12 (π^2))-88 CA TrNf+16 (TrNf^2)) DELTA[1-z]-((16 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+((8 (CF TrNf z (-1+(z^2))+(CA^2) (1+(-3+z) (-1+z) (z^2))) Log[z])/((-1+z) z)))+((Lμ)^2) (((4 (2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (9 lζ (((1+(-1+z) z))^2)+4 (((-1+z))^2) (11+z (2+11 z)))))/(3 (-1+z) z))+(1/18) CA (CA (-134+36 ((lζ)^2)-39 (π^2))+40 TrNf) DELTA[1-z]-((32 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+16 (1+z) (CF TrNf+(((CA^2) (1+(-3+z) (-1+z) (z^2)))/(-z+(z^3)))) Log[z])+((Log[z])^2) (-((12 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+(((1+z) (-4 CA TrNf z (1+z)-6 CF TrNf z (1+3 z)+(CA^2) (-88+z (59+z (-73+44 z))))-12 (CA^2) (((1+z+(z^2)))^2) Log[1+z])/(3 z (1+z))))-((8 (CA^2) (((1+z+(z^2)))^2) Log[z] PolyLog[2,-z])/(z (1+z)))+Lμ ((1/(9 z (-1+(z^2))))2 (24 CF TrNf (((-1+z))^2) (1+z) (1+(-8+z) z)-4 CA TrNf (1+z) (23+z (-42+z (48+z (-42+23 z))))+(CA^2) (z (1+z) (109+z (-84+109 z))-3 (π^2) (3+z (-7+z (3+z (7+z (-3+7 z)))))))-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf) DELTA[1-z]+(-8 CF TrNf (1+z)-((8 (CA^2) (2+z (1+(-4+z) (-1+z) z (1+z))))/(z (-1+(z^2))))) ((Log[z])^2)+Log[z] (((16 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))-((4 (-(1+z) (4 CA TrNf z (1+z)+6 CF TrNf z (1+3 z)+(CA^2) (44+z (-11+25 z)))+12 (CA^2) (((1+z+(z^2)))^2) Log[1+z]))/(3 z (1+z))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[2,-z])/(z (1+z))))+(-((8 (CA^2) (-1+z) (11+z (-1+11 z)))/(3 z))-((8 (CA^2) (1+z (-3+z (1+z (3+z (-1+3 z))))) Log[z])/(z (-1+(z^2))))) PolyLog[2,z]+((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,-z])/(z (1+z)))+((8 (CA^2) (1+z (-5+z (1+z (5+z (-1+5 z))))) PolyLog[3,z])/(z (-1+(z^2))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,(1/(1+z))])/(z (1+z)))+(1/648) CA DELTA[1-z] (8 TrNf (-328+45 (π^2)+6 lζ (-56+3 (π^2))+108 Zeta[3])+CA (9712+9696 lζ-1206 (π^2)-396 lζ (π^2)+333 (π^4)-216 (11+42 lζ) Zeta[3]))+(1/(27 z (-1+(z^2))))((1+z) (-6 CF TrNf (((-1+z))^2) (-12 (1+(-23+z) z)+(π^2) (4+z (7+4 z)))+4 CA TrNf (-139+3 (π^2) (((1+(-1+z) z))^2)+z (249+z (-276+11 (21-11 z) z)))+(CA^2) (3340-6380 z+8 (z^2) (843+z (-757+395 z))+3 (π^2) (77+6 lζ (((1+(-1+z) z))^2)+z (-150+z (135+z (-150+77 z))))))+36 (CA^2) ((-1+z) (((1+z+(z^2)))^2) Log[1+z] (-(π^2)+2 ((Log[1+z])^2))-2 (23+z (-20+z (23+z (20+z (-23+20 z))))) Zeta[3]))+(1/ϵ)(1/(9 z (-1+(z^2)))(24 CF TrNf (((-1+z))^2) (1+z) (1+(-8+z) z)-4 CA TrNf (1+z) (23+z (-42+z (48+z (-42+23 z))))+(CA^2) (z (1+z) (109+z (-84+109 z))-6 (π^2) (1+z (-3+z (1+z (3+z (-1+3 z)))))))+((Lμ)^2) (-((4 (CA^2) (((1+(-1+z) z))^2))/((-1+z) z))+(1/3) CA (11 CA-4 TrNf) DELTA[1-z])-((4 (CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (2+z (1+(-4+z) (-1+z) z (1+z)))) ((Log[z])^2))/(z (-1+(z^2))))+Lμ (((4 (4 CA TrNf (((1+(-1+z) z))^2)+2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (33+6 lζ (((1+(-1+z) z))^2)+z (-58+z (39+z (-58+33 z))))))/(3 (-1+z) z))-(2/3) CA (11 CA lζ+4 CA (π^2)-4 lζ TrNf) DELTA[1-z]-((32 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+16 (((CA^2)/((-1+z) z))+(CA^2) (-3+z) z+CF TrNf (1+z)) Log[z])+Log[z] (((8 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+((2 ((1+z) (4 CA TrNf z (1+z)+6 CF TrNf z (1+3 z)+(CA^2) (44+z (-11+25 z)))-12 (CA^2) (((1+z+(z^2)))^2) Log[1+z]))/(3 z (1+z))))-((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[2,-z])/(z (1+z)))+(1/18) DELTA[1-z] (36 CF TrNf-4 CA (-12+(π^2)) TrNf+(CA^2) (11 (π^2)-12 (8+9 Zeta[3]))));

Q -> Q’

Since (Subscript[((Δ^([0,1]))), q←q'])=0, TMD concides with unsubtracted TMD.

DD[q->qp,1]=0;
DD[q->qp,2]=((-((2 CF TR (-1+z) (4+z (7+4 z)))/(3 z))+4 CF TR (1+z) Log[z])/(ϵ^2))+((2 CF TR (-2 (-1+z) (-16+(11-16 z) z+3 Lμ (4+z (7+4 z)))-3 Log[z] (8-3 (-1+4 Lμ) z (1+z)+3 z (1+z) Log[z])))/(9 z ϵ))+(1/(27 z))CF TR ((1-z) (416+2 z (-71+136 z)+36 ((Lμ)^2) (4+z (7+4 z))+3 (π^2) (4+z (7+4 z))-24 Lμ (16+z (-11+16 z)))+3 Log[z] (128+72 ((Lμ)^2) z (1+z)+6 z (-8+14 z+(π^2) (1+z))-12 Lμ (8+3 z (1+z))+3 Log[z] (16-z (9+12 Lμ (1+z)+z (-15+8 z))+2 z (1+z) Log[z])))-((8 CF TR (-1+z) (2+z (-1+2 z)) PolyLog[2,1-z])/(3 z));

Q -> (OverBar[Q])

Since (Subscript[((Δ^([0,1]))), q←qbar])=0, TMD concides with unsubtracted TMD.

DD[q->qbar,1]=0;
DD[q->qbar,2]=-(((CA-2 CF) CF (1+(z^2)) ((Log[z])^3))/(1+z))-((2 (CA-2 CF) CF (π^2) (1+(z^2)) Log[1+z])/(3 (1+z)))+((4 (CA-2 CF) CF (1+(z^2)) ((Log[1+z])^3))/(3 (1+z)))+((Log[z])^2) (((2 (CA-2 CF) CF (-2+Lμ-2 z+Lμ (z^2)))/(1+z))-((2 (CA-2 CF) CF (1+(z^2)) Log[1+z])/(1+z)))+Log[z] ((((CA-2 CF) CF (12 Lμ (((1+z))^2)-3 (1+z) (11+3 z)-2 (π^2) (1+(z^2))))/(3 (1+z)))-4 (CA-2 CF) CF (-1+z) Log[1-z]-((4 (CA-2 CF) CF (-(((1+z))^2)+2 Lμ (1+(z^2))) Log[1+z])/(1+z)))+(-((4 (CA-2 CF) CF (-(((1+z))^2)+2 Lμ (1+(z^2))))/(1+z))-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,-z]+(1/ϵ)(-(((CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (z^2)))/(3 (1+z)))+(((CA-2 CF) CF (1+(z^2)) ((Log[z])^2))/(1+z))+Log[z] (2 (CA-2 CF) CF (1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[1+z])/(1+z)))-((4 (CA-2 CF) CF (1+(z^2)) PolyLog[2,-z])/(1+z)))+(-4 (CA-2 CF) CF (-1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,z]+((4 (CA-2 CF) CF (1+(z^2)) PolyLog[3,-z])/(1+z))+((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,z])/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,(1/(1+z))])/(1+z))-(((CA-2 CF) CF (45+(π^2)+2 Lμ (-12+(π^2)+(12+(π^2)) (z^2))-6 Zeta[3]-z (45 z+(π^2) (2+3 z)+6 z Zeta[3])))/(3 (1+z)));

matching of TMD PDFs

In this section perform the small-(Subscript[b, T]) matching NLO and NNLO expressions for TMD PDF.
They are given by
Cpdf[f ← f', 1] = ((Subscript[C, f←f'])^([1]))=(F^([1]))-(f^([1])),
Cpdf[f ← f', 2] = ((Subscript[C, f←f'])^([2]))=(F^([2]))-(∑_r)(C^([1]))⊗(f^([1]))-(f^([2])),
In the presented results one can observe the cancelation all divergent parts.
The convolutions (C^([1]))⊗(f^([1])) for each case presented by variable CTIMESf

Q ← Q

Cpdf[q←q,1]=Collect[Normal@Series[F[q←q,1]- f[q←q,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=(CF^2)((1/ϵ)((-6 lζ Lμ+3 ((Lμ)^2)+((π^2)/2)-((8 Lμ (π^2))/3)) DELTA[1-x]-1/(3 (-1+x))(6-6 Lμ-12 lζ Lμ+6 ((Lμ)^2)+(π^2)-12 x+48 Lμ x+6 (x^2)-6 Lμ (x^2)-12 lζ Lμ (x^2)+6 ((Lμ)^2) (x^2)+(π^2) (x^2)+24 (-(((-1+x))^2)+2 Lμ (1+(x^2))) Log[1-x]-12 (Lμ-(((-1+x))^2)+3 Lμ (x^2)) Log[x]))+DELTA[1-x] (2 ((Lμ)^3)-((lζ (π^2))/2)+Lμ (π^2)-((2 (π^4))/9)-(1/3) ((Lμ)^2) (9 lζ+4 (π^2))+4 Zeta[3])+(1/(6 (-1+x)))(-12 Lμ+6 ((Lμ)^2)+12 lζ ((Lμ)^2)-8 ((Lμ)^3)+(π^2)+2 lζ (π^2)-4 Lμ (π^2)+24 Lμ x-48 ((Lμ)^2) x-8 (π^2) x-12 Lμ (x^2)+6 ((Lμ)^2) (x^2)+12 lζ ((Lμ)^2) (x^2)-8 ((Lμ)^3) (x^2)+(π^2) (x^2)+2 lζ (π^2) (x^2)-4 Lμ (π^2) (x^2)-8 (-6 Lμ (((-1+x))^2)+6 ((Lμ)^2) (1+(x^2))+(π^2) (1+(x^2))) Log[1-x]+2 (-12 Lμ (((-1+x))^2)+6 ((Lμ)^2) (1+3 (x^2))+(π^2) (1+3 (x^2))) Log[x]-16 Zeta[3]-16 (x^2) Zeta[3]));

Cpdf[q←q,2]=Collect[F[q←q,2]-CTIMESf-f[q←q,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[q←q,1]=((2 CF (-(((-1+x))^2)+Lμ (1+(x^2))))/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[q←q,2]=-((2 (CF^2) ((Lμ)^3) (1+(x^2)))/(-1+x))-((2 (CA-CF) CF ((π^2)+3 x+(-3+(π^2)) (x^2)) Log[1-x])/(3 (-1+x)))+(-((2 CF (CA (-29+(36-83 x) x)+10 TrNf (1+(x^2))+9 CF (5+x (-13+16 x))))/(9 (-1+x)))-4 (CA-3 CF) CF (-1+x) Log[1-x]+((2 (2 CA-3 CF) CF (1+(x^2)) ((Log[1-x])^2))/(-1+x))) Log[x]+(((CF (CA (11-(-12+x) x)+2 (CF (-9+6 (-1+x) x)-2 TrNf (1+(x^2)))))/(6 (-1+x)))-((2 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) ((Log[x])^2)+((CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^3))/(3 (-1+x)))+((Lμ)^2) (((4 (CF^2) lζ (1+(x^2)))/(-1+x))+((CF (11 CA (1+(x^2))+2 (9 CF (((-1+x))^2)-2 TrNf (1+(x^2)))))/(3 (-1+x)))-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))+((2 (CF^2) (1+3 (x^2)) Log[x])/(-1+x)))+Lμ (-4 (CF^2) lζ (-1+x)+((CF (-16 TrNf (4+x (-3+4 x))-3 CF (84 (((-1+x))^2)+(π^2) (1+(x^2)))+CA (-6 (π^2) (1+(x^2))+4 (77+x (-87+77 x)))))/(9 (-1+x)))+8 (CF^2) (-1+x) Log[1-x]+(((2 CF (17 CA-4 (6 CF+TrNf)+(5 CA+6 CF-4 TrNf) (x^2)))/(3 (-1+x)))-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) Log[x]+((2 CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^2))/(-1+x)))+(-4 (CA-2 CF) CF (-1+x)+((4 (CA-CF) CF (1+(x^2)) Log[1-x])/(-1+x))+((4 (CA-3 CF) CF (1+(x^2)) Log[x])/(-1+x))) PolyLog[2,x]+((4 (CA-CF) CF (1+(x^2)) PolyLog[3,1-x])/(-1+x))-((4 (2 CA-5 CF) CF (1+(x^2)) PolyLog[3,x])/(-1+x))+((CF (27 CA (π^2) (((-1+x))^2)-27 CF (-22+(π^2)) (((-1+x))^2)+8 CA (1+x (99+x))-4 TrNf (19+x (18+19 x))-54 (3 CA+10 CF) (1+(x^2)) Zeta[3]))/(27 (-1+x)))+DELTA[1-x] ((((CF^2) ((Lμ)^4))/2)+((Lμ)^3) (-2 (CF^2) lζ+(2/9) CF (-11 CA+4 TrNf))+((Lμ)^2) (2 (CF^2) ((lζ)^2)+(1/3) CF lζ (11 CA-4 TrNf)+(1/18) CF (-134 CA+3 (2 CA-7 CF) (π^2)+40 TrNf))+Lμ ((1/9) CF (π^2) (-11 CA+4 TrNf)-(1/9) CF lζ (-134 CA+3 (2 CA+CF) (π^2)+40 TrNf))-(2/27) CF lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CF (9 CF (π^4)+2 CA (4856-603 (π^2)+18 (π^4)-2772 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q ← G

Cpdf[q←g,1]=Collect[Normal@Series[F[q←g,1]- f[q←g,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=TR(-(2/9) TrNf (24 (-1+x) x+24 Lμ (-1+x) x+6 ((Lμ)^2) (1-2 x+2 (x^2))+(π^2) (1-2 x+2 (x^2)))+(1/(18 x))CA (-48-48 Lμ+48 ((Lμ)^2)+8 (π^2)+144 x+144 Lμ x+102 ((Lμ)^2) x+17 (π^2) x+456 (x^2)+456 Lμ (x^2)+156 ((Lμ)^2) (x^2)+26 (π^2) (x^2)-552 (x^3)-552 Lμ (x^3)-240 ((Lμ)^2) (x^3)-40 (π^2) (x^3)+12 x (24 (-1+x) x+24 Lμ (-1+x) x+6 ((Lμ)^2) (1-2 x+2 (x^2))+(π^2) (1-2 x+2 (x^2))) Log[1-x]+12 x (48 x+48 Lμ x+6 ((Lμ)^2) (1+4 x)+(π^2) (1+4 x)) Log[x])+(1/ϵ)(-(8/3) TrNf (2 (-1+x) x+Lμ (1-2 x+2 (x^2)))+1/(3 x)CA (-8+16 Lμ+24 x+34 Lμ x+76 (x^2)+52 Lμ (x^2)-92 (x^3)-80 Lμ (x^3)+24 x (2 (-1+x) x+Lμ (1-2 x+2 (x^2))) Log[1-x]+24 x (Lμ+4 x+4 Lμ x) Log[x])+(1/3) CF (24-24 Lμ-12 lζ Lμ+6 ((Lμ)^2)+(π^2)-12 x+60 Lμ x+24 lζ Lμ x-12 ((Lμ)^2) x-2 (π^2) x-12 (x^2)-36 Lμ (x^2)-24 lζ Lμ (x^2)+12 ((Lμ)^2) (x^2)+2 (π^2) (x^2)+24 Lμ (1-2 x+2 (x^2)) Log[1-x]-12 (-1+Lμ-2 x-2 Lμ x+4 Lμ (x^2)) Log[x]))+(1/3) CF (24 Lμ-12 ((Lμ)^2)-6 lζ ((Lμ)^2)+4 ((Lμ)^3)-2 (π^2)-lζ (π^2)+2 Lμ (π^2)-12 Lμ x+30 ((Lμ)^2) x+12 lζ ((Lμ)^2) x-8 ((Lμ)^3) x+5 (π^2) x+2 lζ (π^2) x-4 Lμ (π^2) x-12 Lμ (x^2)-18 ((Lμ)^2) (x^2)-12 lζ ((Lμ)^2) (x^2)+8 ((Lμ)^3) (x^2)-3 (π^2) (x^2)-2 lζ (π^2) (x^2)+4 Lμ (π^2) (x^2)+2 (6 ((Lμ)^2)+(π^2)) (1-2 x+2 (x^2)) Log[1-x]-(-12 Lμ (1+2 x)+6 ((Lμ)^2) (1-2 x+4 (x^2))+(π^2) (1-2 x+4 (x^2))) Log[x]+8 Zeta[3]-16 x Zeta[3]+16 (x^2) Zeta[3]));

Cpdf[q←g,2]=Collect[F[q←g,2]-CTIMESf-f[q←g,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[q←g,1]=-2 TR (2 (-1+x) x+Lμ (1-2 x+2 (x^2)));
Cpdf[q←g,2]=2 CF ((Lμ)^3) TR (1+2 (-1+x) x)-2 (CA-CF) TR x (-3+4 x) Log[1-x]+4 (CA-CF) TR (-1+x) x ((Log[1-x])^2)+(2/3) (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^3)+(1/3) TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^3)+((Lμ)^2) (-4 CF lζ TR (1+2 (-1+x) x)+((TR (-3 CF x (7+4 x (-3+2 x))-2 CA (-1+x) (4+x (7+31 x))))/(3 x))+4 (CA+CF) TR (1+2 (-1+x) x) Log[1-x]+2 TR (CA (2+8 x)+CF (-1+2 x-4 (x^2))) Log[x])+((Log[x])^2) (-2 TR (-8 CA x+CF (1+2 (-1+x) x)) Log[1-x]+(1/6) TR (CA (-6+8 (3-11 x) x)+3 CF (1+4 (3-2 x) x)+12 CA (1+2 x (1+x)) Log[1+x]))+Log[x] (8 CF TR (-1+x) x Log[1-x]+2 CF TR (1+2 (-1+x) x) ((Log[1-x])^2)+(1/9) TR (4 CA (21-6 (5+(π^2)) x+68 (x^2))+3 CF (24+3 (15-8 x) x+(π^2) (2+4 (-1+x) x))+36 CA Log[1+x] (2 x (1+x)-(1+2 x (1+x)) Log[1+x])))+(((8 CA TR (-1+x) (2+x (-1+11 x)))/(3 x))+4 (-CA+CF) TR (1+2 (-1+x) x) Log[1-x]+4 TR (4 CA x+CF (-1-2 (-1+x) x)) Log[x]) PolyLog[2,1-x]+(8 CA TR x (1+x)-4 CA TR (1+2 x (1+x)) Log[x]) PolyLog[2,-x]+Lμ (-8 CF lζ TR (-1+x) x+((TR (3 CF x (5 (π^2) (1+2 (-1+x) x)-6 (10+x (-33+28 x)))+4 CA (-26+x (36+x (-135+6 (π^2)+116 x)))))/(9 x))+16 CF TR (-1+x) x Log[1-x]+4 (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^2)+2 TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^2)+Log[x] (8 CF TR (1+2 (-1+x) x) Log[1-x]+(1/3) (-2 TR (3 CF (1+8 (-1+x) x)+CA (6+88 (x^2)))+24 CA TR (1+2 x (1+x)) Log[1+x]))+8 CA TR (1+2 x (1+x)) PolyLog[2,-x])+4 (CA-CF) TR (1+2 (-1+x) x) PolyLog[3,1-x]+4 CA TR (1+2 x (1+x)) PolyLog[3,-x]-4 TR (-8 CA x+CF (1+2 (-1+x) x)) PolyLog[3,x]-8 CA TR (1+2 x (1+x)) PolyLog[3,(x/(1+x))]+(1/(27 x))TR (27 CF x (-13+(75+2 (π^2) (-1+x)-72 x) x)+CA (344+2 x (-315+(387+18 (π^2)-298 x) x))+18 x (CA (1+2 x (1+x)) Log[1+x] (-(π^2)+2 ((Log[1+x])^2))-6 (CA (1+2 (-2+x) x)-8 CF (1+2 (-1+x) x)) Zeta[3]));

G ← Q

Cpdf[g←q,1]=Collect[Normal@Series[F[g←q,1]- f[g←q,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=(1/(3 x ϵ))CF (-124 CA Lμ-24 CA lζ Lμ+12 CA ((Lμ)^2)+2 CA (π^2)-12 CF x+96 CA Lμ x+24 CF Lμ x+24 CA lζ Lμ x-12 CA ((Lμ)^2) x-2 CA (π^2) x-6 CF (x^2)+12 CA Lμ (x^2)-6 CF Lμ (x^2)-12 CA lζ Lμ (x^2)+6 CA ((Lμ)^2) (x^2)+CA (π^2) (x^2)+16 CA Lμ (x^3)+24 (CA Lμ (2-2 x+(x^2))+CF (-(x^2)+Lμ (2-2 x+(x^2)))) Log[1-x]-12 (CF (Lμ (-2+x)-x) x+4 CA Lμ (1+x+(x^2))) Log[x])+(1/(18 x))CF (-372 CA ((Lμ)^2)-72 CA lζ ((Lμ)^2)+48 CA ((Lμ)^3)-62 CA (π^2)-12 CA lζ (π^2)+24 CA Lμ (π^2)-72 CF Lμ x+288 CA ((Lμ)^2) x+72 CF ((Lμ)^2) x+72 CA lζ ((Lμ)^2) x-48 CA ((Lμ)^3) x+48 CA (π^2) x+12 CF (π^2) x+12 CA lζ (π^2) x-24 CA Lμ (π^2) x-36 CF Lμ (x^2)+36 CA ((Lμ)^2) (x^2)-18 CF ((Lμ)^2) (x^2)-36 CA lζ ((Lμ)^2) (x^2)+24 CA ((Lμ)^3) (x^2)+6 CA (π^2) (x^2)-3 CF (π^2) (x^2)-6 CA lζ (π^2) (x^2)+12 CA Lμ (π^2) (x^2)+48 CA ((Lμ)^2) (x^3)+8 CA (π^2) (x^3)+12 (CA (6 ((Lμ)^2)+(π^2)) (2-2 x+(x^2))+CF (-12 Lμ (x^2)+6 ((Lμ)^2) (2-2 x+(x^2))+(π^2) (2-2 x+(x^2)))) Log[1-x]-6 (CF x (6 ((Lμ)^2) (-2+x)+(π^2) (-2+x)-12 Lμ x)+4 CA (6 ((Lμ)^2)+(π^2)) (1+x+(x^2))) Log[x]+96 CA Zeta[3]-96 CA x Zeta[3]+48 CA (x^2) Zeta[3]);

Cpdf[g←q,2]=Collect[F[g←q,2]-CTIMESf-f[g←q,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[g←q,1]=2 CF (Lμ (2-(2/x)-x)+x);
Cpdf[g←q,2]=((2 CA CF ((Lμ)^3) (2+(-2+x) x))/x)+((2 CF (CA (-152+(152-43 x) x)+8 TrNf (5+(-5+x) x)+9 CF (16+x (-16+5 x))) Log[1-x])/(9 x))+((CF (CA (-22+(22-5 x) x)+3 CF (6+(-6+x) x)+4 TrNf (2+(-2+x) x)) ((Log[1-x])^2))/(3 x))-((2 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^3))/(3 x))-(1/3) CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^3)+((Lμ)^2) (-((4 CA CF lζ (2+(-2+x) x))/x)+((CF (-3 CF (-4+x) x+8 TrNf (2+(-2+x) x)+2 CA (-53+x (46+x (-11+4 x)))))/(3 x))+((4 CF (CA+CF) (2+(-2+x) x) Log[1-x])/x)-((2 CF (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x])/x))+((Log[x])^2) (((2 CA CF (2+(-2+x) x) Log[1-x])/x)+((CF x (-3 CF (4+3 x)+2 CA (36+x (9+8 x)))-12 CA CF (2+x (2+x)) Log[1+x])/(6 x)))+Log[x] (((4 CA CF (-22+x (24+x (-9+4 x))) Log[1-x])/(3 x))+((2 CA CF (2+(-2+x) x) ((Log[1-x])^2))/x)+((CF (x (45 CF (-3+x)-2 CA (249-6 x+88 (x^2)))+36 CA Log[1+x] ((x^2)+(2+x (2+x)) Log[1+x])))/(9 x)))+(4 CA CF x+((4 CA CF (2+x (2+x)) Log[x])/x)) PolyLog[2,-x]+Lμ (4 CA CF lζ x+((CF (2 (27 CF x (1+2 x)+40 TrNf (2+(-2+x) x))+CA (3 (π^2) (2+(-10+x) x)-4 (9+x (19+4 x+44 (x^2))))))/(9 x))+((4 CF (CA (-22+(22-17 x) x)+(9 CF+4 TrNf) (2+(-2+x) x)) Log[1-x])/(3 x))-((4 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^2))/x)-2 CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^2)+Log[x] (((8 CA CF (2+(-2+x) x) Log[1-x])/x)-((2 CF (x (3 CF (4+5 x)-2 CA (36+x (15+8 x)))+12 CA (2+x (2+x)) Log[1+x]))/(3 x)))-((8 CA CF (2+x (2+x)) PolyLog[2,-x])/x))+(((8 CA CF (-1+x) (11+x (-1+2 x)))/(3 x))+((4 CA CF (6+x (-2+3 x)) Log[x])/x)) PolyLog[2,x]-((4 CA CF (2+x (2+x)) PolyLog[3,-x])/x)-((4 CA CF (10+x (-2+5 x)) PolyLog[3,x])/x)+((8 CA CF (2+x (2+x)) PolyLog[3,(x/(1+x))])/x)+(1/(27 x))CF (-27 CF (-10+x) x+8 TrNf (56+x (-56+13 x))+CA (-3 (π^2) (-44+x (48+x (-9+8 x)))+4 (-790+x (791+4 x (-67+38 x))))+18 CA (2+x (2+x)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+216 CA (6+x (-7+3 x)) Zeta[3]);

G ← G

Cff[g←g,1]=Collect[Normal@Series[F[g←g,1]- f[g←g,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=((2 CF TrNf (-(-1+x) (-36 Lμ x (1+2 x)+6 ((Lμ)^2) (4+7 x+4 (x^2))+(π^2) (4+7 x+4 (x^2)))+6 x (-12 Lμ x+6 ((Lμ)^2) (1+x)+(π^2) (1+x)) Log[x]))/(9 x))+(1/ϵ)(CA (((16 Lμ TrNf (((1-x+(x^2)))^2))/(3 (-1+x) x))-(2/9) (-12 lζ Lμ+6 ((Lμ)^2)+(π^2)) TrNf DELTA[1-x])-((8 CF TrNf ((-1+x) (-3 x (1+2 x)+Lμ (4+7 x+4 (x^2)))-6 x (Lμ-x+Lμ x) Log[x]))/(3 x))+(CA^2) ((1/18) (-132 lζ Lμ+66 ((Lμ)^2)+11 (π^2)-48 Lμ (π^2)) DELTA[1-x]-1/(3 (-1+x) x)2 (-66 Lμ-12 lζ Lμ+6 ((Lμ)^2)+(π^2)+116 Lμ x+24 lζ Lμ x-12 ((Lμ)^2) x-2 (π^2) x-78 Lμ (x^2)-36 lζ Lμ (x^2)+18 ((Lμ)^2) (x^2)+3 (π^2) (x^2)+116 Lμ (x^3)+24 lζ Lμ (x^3)-12 ((Lμ)^2) (x^3)-2 (π^2) (x^3)-66 Lμ (x^4)-12 lζ Lμ (x^4)+6 ((Lμ)^2) (x^4)+(π^2) (x^4)+48 Lμ (((1-x+(x^2)))^2) Log[1-x]-24 Lμ (1+3 (x^2)-4 (x^3)+(x^4)) Log[x])))+(CA^2) ((1/18) DELTA[1-x] (44 ((Lμ)^3)+22 Lμ (π^2)-24 ((Lμ)^2) (π^2)-4 (π^4)-11 lζ (6 ((Lμ)^2)+(π^2))+88 Zeta[3])+(1/(9 (-1+x) x))(198 ((Lμ)^2)+36 lζ ((Lμ)^2)-24 ((Lμ)^3)+33 (π^2)+6 lζ (π^2)-12 Lμ (π^2)-348 ((Lμ)^2) x-72 lζ ((Lμ)^2) x+48 ((Lμ)^3) x-58 (π^2) x-12 lζ (π^2) x+24 Lμ (π^2) x+234 ((Lμ)^2) (x^2)+108 lζ ((Lμ)^2) (x^2)-72 ((Lμ)^3) (x^2)+39 (π^2) (x^2)+18 lζ (π^2) (x^2)-36 Lμ (π^2) (x^2)-348 ((Lμ)^2) (x^3)-72 lζ ((Lμ)^2) (x^3)+48 ((Lμ)^3) (x^3)-58 (π^2) (x^3)-12 lζ (π^2) (x^3)+24 Lμ (π^2) (x^3)+198 ((Lμ)^2) (x^4)+36 lζ ((Lμ)^2) (x^4)-24 ((Lμ)^3) (x^4)+33 (π^2) (x^4)+6 lζ (π^2) (x^4)-12 Lμ (π^2) (x^4)-24 (6 ((Lμ)^2)+(π^2)) (((1-x+(x^2)))^2) Log[1-x]+12 (6 ((Lμ)^2)+(π^2)) (1+3 (x^2)-4 (x^3)+(x^4)) Log[x]-48 Zeta[3]+96 x Zeta[3]-144 (x^2) Zeta[3]+96 (x^3) Zeta[3]-48 (x^4) Zeta[3]))+CA (((4 (6 ((Lμ)^2)+(π^2)) TrNf (((1-x+(x^2)))^2))/(9 (-1+x) x))+(2/9) TrNf DELTA[1-x] (lζ (6 ((Lμ)^2)+(π^2))-2 (2 ((Lμ)^3)+Lμ (π^2)+4 Zeta[3])));

Cpdf[g←g,2]=Collect[F[g←g,2]-CTIMESf-f[g←g,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[g←g,1]=((4 CA Lμ (((1-x+(x^2)))^2))/((-1+x) x))+CA (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[g←g,2]=-((4 (CA^2) ((Lμ)^3) (((1+(-1+x) x))^2))/((-1+x) x))+(2/3) CA (CA-2 TrNf) x Log[1-x]+(4/3) ((CA^2) (-2+x) x+CF TrNf (1+x)+((CA^2)/(-1+(x^2)))) ((Log[x])^3)+((Lμ)^2) (((8 (CA^2) lζ (((1+(-1+x) x))^2))/((-1+x) x))+((2 (-4 CA TrNf (((1+(-1+x) x))^2)+2 CF TrNf (-4+x+6 (x^2)+(x^3)-4 (x^4))+(CA^2) (55+x (-102+x (105+x (-102+55 x))))))/(3 (-1+x) x))-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((8 (CF TrNf x (-1+(x^2))+(CA^2) (1+(-3+x) (-1+x) (x^2))) Log[x])/((-1+x) x)))+((Log[x])^2) (-((4 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((x (1+x) (4 CA TrNf (1+x)+6 CF TrNf (3+x)+(CA^2) (25+11 x (-1+4 x)))-12 (CA^2) (((1+x+(x^2)))^2) Log[1+x])/(3 x (1+x))))+Log[x] (((8 (CA^2) (-1+x) (11+x (-1+11 x)) Log[1-x])/(3 x))-((4 (CA^2) (((1+(-1+x) x))^2) ((Log[1-x])^2))/((-1+x) x))+((-x (1+x) (-216 CF TrNf (1+x)-4 CA TrNf (13+10 x)+(CA^2) (701+x (149+536 x)))+72 (CA^2) (((1+x+(x^2)))^2) ((Log[1+x])^2))/(9 x (1+x))))+((8 (CA^2) (((1+x+(x^2)))^2) Log[x] PolyLog[2,-x])/(x (1+x)))+Lμ ((1/(9 x (-1+(x^2))))2 (-12 CF TrNf (((-1+x))^2) (1+x) (-2+x (19+4 x))-4 CA TrNf (1+x) (23+x (-42+x (48+x (-42+23 x))))+(CA^2) (x (1+x) (109+x (-84+109 x))-3 (π^2) (1+x (-5+x (1+x (5+x (-1+5 x)))))))+(8 CF TrNf (1+x)+((8 (CA^2) (((1+x-(x^2)))^2))/(-1+(x^2)))) ((Log[x])^2)+Log[x] (-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))-((4 (-x (1+x) (4 CA TrNf (1+x)+18 CF TrNf (1+x)+(CA^2) (25+11 x (-1+4 x)))+12 (CA^2) (((1+x+(x^2)))^2) Log[1+x]))/(3 x (1+x))))-((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[2,-x])/(x (1+x))))+(((8 (CA^2) (-1+x) (11+x (-1+11 x)))/(3 x))-((8 (CA^2) (3+x (-1+x (3+x-3 (x^2)+(x^3)))) Log[x])/(x (-1+(x^2))))) PolyLog[2,x]-((8 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,-x])/(x (1+x)))+((8 (CA^2) (5+x (-1+x (5+x-5 (x^2)+(x^3)))) PolyLog[3,x])/(x (-1+(x^2))))+((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,(x/(1+x))])/(x (1+x)))+(1/(27 x (-1+(x^2))))4 (-(1+x) (-18 CF TrNf (((-1+x))^2) (1+(-23+x) x)+(CA^2) (-790+3 (π^2) (((-1+x))^2) (11+x (-1+11 x))+x (1514+x (-1686+5 (319-167 x) x)))+CA TrNf (121+x (-231+x (276+x (-249+139 x)))))+9 (CA^2) ((-1+x) (((1+x+(x^2)))^2) Log[1+x] ((π^2)-2 ((Log[1+x])^2))-6 (6+x (-7+x (6+x (7+x (-6+7 x))))) Zeta[3]))+DELTA[1-x] ((((CA^2) ((Lμ)^4))/2)+((Lμ)^2) (2 (CA^2) ((lζ)^2)-(1/18) CA (CA (134+15 (π^2))-40 TrNf)+(1/3) CA lζ (11 CA-4 TrNf))+((Lμ)^3) (-2 (CA^2) lζ+(2/9) CA (-11 CA+4 TrNf))+Lμ ((1/9) CA (π^2) (-11 CA+4 TrNf)-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf))-(2/27) CA lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/324) CA (4 TrNf (-328+45 (π^2)+252 Zeta[3])+CA (4856-603 (π^2)+18 (π^4)-135 ((Log[2])^4)+3240 PolyLog[2,2,(1/2)]+9 (-308+45 Log[2]) Zeta[3])));

Q ← Q’

Cpdf[q←qp,1]=Collect[Normal@Series[F[q←qp,1]- f[q←qp,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=((CF TR (-24+48 Lμ+72 x+36 Lμ x-36 Lμ (x^2)-48 (x^3)-48 Lμ (x^3)+72 Lμ x Log[x]+72 (x^2) Log[x]+72 Lμ (x^2) Log[x]))/(9 x ϵ))+(1/(9 x))CF TR (-24-24 Lμ+24 ((Lμ)^2)+4 (π^2)+72 x+72 Lμ x+18 ((Lμ)^2) x+3 (π^2) x-18 ((Lμ)^2) (x^2)-3 (π^2) (x^2)-48 (x^3)-48 Lμ (x^3)-24 ((Lμ)^2) (x^3)-4 (π^2) (x^3)+36 ((Lμ)^2) x Log[x]+6 (π^2) x Log[x]+72 (x^2) Log[x]+72 Lμ (x^2) Log[x]+36 ((Lμ)^2) (x^2) Log[x]+6 (π^2) (x^2) Log[x]);

Cpdf[q←qp,2]=Collect[F[q←qp,2]-CTIMESf-f[q←qp,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[q←qp,1]=0;
Cpdf[q←qp,2]=-((2 CF TR (-1+x) (172+x (-143+136 x)))/(27 x))+(4/9) CF TR (21-30 x+32 (x^2)) Log[x]-(1/3) CF TR (3+x (3+8 x)) ((Log[x])^2)+(2/3) CF TR (1+x) ((Log[x])^3)+((Lμ)^2) (-((2 CF TR (-1+x) (4+x (7+4 x)))/(3 x))+4 CF TR (1+x) Log[x])+Lμ (((8 CF TR (-1+x) (13+x (-5+22 x)))/(9 x))-(4/3) CF TR (3+x (9+8 x)) Log[x]+4 CF TR (1+x) ((Log[x])^2))+((8 CF TR (-1+x) (2+x (-1+2 x)) PolyLog[2,1-x])/(3 x));

Q ← (OverBar[Q])

Cpdf[q←qbar,1]=Collect[Normal@Series[F[q←qbar,1]- f[q←qbar,1],{ϵ,0,0}],{ϵ,Lb,DELTA[_]},Simplify]

CTIMESf=0;

Cpdf[q←qbar,2]=Collect[F[q←qbar,2]-CTIMESf-f[q←qbar,2],{DELTA[_],Lμ,lζ,PolyLog[_,_],Log[x],Log[1-x]},FullSimplify]

Cpdf[q←qbar,1]=0;
Cpdf[q←qbar,2]=(((CA-2 CF) CF (1+(x^2)) ((Log[x])^3))/(3 (1+x)))+((Log[x])^2) (-((4 (CA-2 CF) CF (1+(x^2)) Log[1-x])/(1+x))-((2 (CA-2 CF) CF (1+(x^2)) Log[1+x])/(1+x)))+(((CA-2 CF) CF Log[x] (-3 (1+x) (3+11 x)+2 (π^2) (1+(x^2))+12 Log[1+x] ((((1+x))^2)+(1+(x^2)) Log[1+x])))/(3 (1+x)))+(-4 (CA-2 CF) CF (-1+x)-((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,1-x]+(4 (CA-2 CF) CF (1+x)+((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,-x]+Lμ (-((2 (CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (x^2)))/(3 (1+x)))+((2 (CA-2 CF) CF (1+(x^2)) ((Log[x])^2))/(1+x))+((4 (CA-2 CF) CF Log[x] ((((1+x))^2)-2 (1+(x^2)) Log[1+x]))/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[2,-x])/(1+x)))-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[3,-x])/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,x])/(1+x))+((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,(x/(1+x))])/(1+x))+(((CA-2 CF) CF (-45+45 (x^2)+(π^2) (((1+x))^2)+2 (1+(x^2)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))-6 (1+(x^2)) Zeta[3]))/(3 (1+x)));

matching of TMD FFs

In this section perform the small-(Subscript[b, T]) matching NLO and NNLO expressions for TMD PDF.
They are given by
Cff[f ← f', 1] = (z^2)((Subscript[C, f←f'])^([1]))=(D^([1]))-(d^([1])),
Cff[f ← f', 2] = (z^2)((Subscript[C, f←f'])^([2]))=(D^([2]))-(∑_r)(C^([1]))⊗((d^([1]))/(z^(-2ϵ)))-((d^([2]))/(z^(-2ϵ))),
In the presented results one can observe the cancelation all divergent parts.
The convolutions (C^([1]))⊗((d^([1]))/(z^(-2ϵ))) for each case presented by variable CTIMESd

Q -> Q

Cff[q->q,1]=Collect[Expand@Normal@Series[DD[q->q,1]- d[q->q,1](z^(2ϵ)),{ϵ,0,0}]/.DELTA[1-z]Log[z]->0,{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=-((4 (CF^2) ((Lμ)^3) (1+(z^2)))/(3 (-1+z)))-(((CF^2) ((Lμ)^2) (-1+8 z-(z^2)+8 (1+(z^2)) Log[1-z]-2 (-1+(z^2)) Log[z]))/(-1+z))+lζ (((2 (CF^2) ((Lμ)^2) (1+(z^2)))/(-1+z))+(((CF^2) (π^2) (1+(z^2)))/(3 (-1+z)))+((8 (CF^2) Lμ (1+(z^2)) Log[z])/(-1+z)))-((2 (CF^2) Lμ (3+(π^2)-6 z+3 (z^2)+(π^2) (z^2)-6 (((-1+z))^2) Log[z]-6 (1+3 (z^2)) ((Log[z])^2)+12 Log[1-z] (-(((-1+z))^2)+2 (1+(z^2)) Log[z])))/(3 (-1+z)))+(1/ϵ)(((4 (CF^2) lζ Lμ (1+(z^2)))/(-1+z))-((2 (CF^2) ((Lμ)^2) (1+(z^2)))/(-1+z))+(-6 (CF^2) lζ Lμ+3 (CF^2) ((Lμ)^2)+(((CF^2) (π^2))/2)-(8/3) (CF^2) Lμ (π^2)) DELTA[1-z]-((2 (CF^2) Lμ (-1+8 z-(z^2)+8 (1+(z^2)) Log[1-z]-2 (1+3 (z^2)) Log[z]))/(-1+z))-(((CF^2) (6+(π^2)-12 z+6 (z^2)+(π^2) (z^2)-72 z Log[z]+12 (1+3 (z^2)) ((Log[z])^2)-24 Log[1-z] ((((-1+z))^2)+2 (1+(z^2)) Log[z])))/(3 (-1+z))))-(1/(6 (-1+z)))(CF^2) (-192+15 (π^2)+384 z-24 (π^2) z-192 (z^2)+15 (π^2) (z^2)+8 (π^2) Log[1-z]+8 (π^2) (z^2) Log[1-z]-48 Log[z]+34 (π^2) Log[z]-96 z Log[z]+144 (z^2) Log[z]+30 (π^2) (z^2) Log[z]-96 Log[1-z] Log[z]+192 z Log[1-z] Log[z]-96 (z^2) Log[1-z] Log[z]-144 z ((Log[z])^2)-96 Log[1-z] ((Log[z])^2)-96 (z^2) Log[1-z] ((Log[z])^2)+32 ((Log[z])^3)+96 (z^2) ((Log[z])^3)+96 (-(((-1+z))^2)+2 (1+(z^2)) Log[z]) PolyLog[2,z]-384 (1+(z^2)) PolyLog[3,z]+400 Zeta[3]+400 (z^2) Zeta[3])+DELTA[1-z] (2 (CF^2) ((Lμ)^3)+(CF^2) Lμ (π^2)-(4/3) (CF^2) ((Lμ)^2) (π^2)+lζ (-3 (CF^2) ((Lμ)^2)-(((CF^2) (π^2))/2))+(CF^2) (-((2 (π^4))/9)+4 Zeta[3]));

Cff[q->q,2]=Collect[Expand[DD[q->q,2]-CTIMESd-Normal@Series[d[q->q,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

Cff[q->q,1]=CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((2 CF (-(((-1+z))^2)+Lμ (1+(z^2))-2 (1+(z^2)) Log[z]))/(-1+z));
Cff[q->q,2]=-((2 (CF^2) ((Lμ)^3) (1+(z^2)))/(-1+z))+((2 (CA-CF) CF (3-3 z+(π^2) (1+(z^2))) Log[1-z])/(3 (-1+z)))+(((2 CF (2 TrNf (9+z (-8+9 z))+CF (69+9 z (-14+9 z)+2 (π^2) (1+(z^2)))+CA (-93+(95-72 z) z+3 (π^2) (1+(z^2)))))/(3 (-1+z)))+4 (CA-7 CF) CF (-1+z) Log[1-z]-((2 (2 CA-3 CF) CF (1+(z^2)) ((Log[1-z])^2))/(-1+z))) Log[z]+(-((CF (CA (37-z (36+23 z))+2 (2 TrNf (1+(z^2))+3 CF (-22+z (34+3 z)))))/(6 (-1+z)))+((2 (2 CA-9 CF) CF (1+(z^2)) Log[1-z])/(-1+z))) ((Log[z])^2)+((CF (-9 CA (1+(z^2))+CF (29+51 (z^2))) ((Log[z])^3))/(3 (-1+z)))+((Lμ)^2) (((4 (CF^2) lζ (1+(z^2)))/(-1+z))+((CF (11 CA (1+(z^2))+2 (9 CF (((-1+z))^2)-2 TrNf (1+(z^2)))))/(3 (-1+z)))-((8 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))+((2 (CF^2) (3+5 (z^2)) Log[z])/(-1+z)))+Lμ (((CF (-16 TrNf (4+z (-3+4 z))-3 CF (84 (((-1+z))^2)+(π^2) (1+(z^2)))+CA (-6 (π^2) (1+(z^2))+4 (77+z (-87+77 z)))))/(9 (-1+z)))+8 (CF^2) (-1+z) Log[1-z]+(((2 CF (6 CF (-5+8 z)+4 TrNf (1+(z^2))-CA (5+17 (z^2))))/(3 (-1+z)))+((24 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))) Log[z]+((2 CF (CA-5 CF+(CA-11 CF) (z^2)) ((Log[z])^2))/(-1+z))+lζ (-4 (CF^2) (-1+z)-((8 (CF^2) (1+(z^2)) Log[z])/(-1+z))))+(4 (CA-6 CF) CF (-1+z)-((4 (CA-CF) CF (1+(z^2)) Log[1-z])/(-1+z))+((8 CF (CA+2 CF) (1+(z^2)) Log[z])/(-1+z))) PolyLog[2,z]-((4 (CA-CF) CF (1+(z^2)) PolyLog[3,1-z])/(-1+z))-((4 CF (3 CA+10 CF) (1+(z^2)) PolyLog[3,z])/(-1+z))+((CF (-9 CA (π^2) (((-1+z))^2)+9 CF (-30+13 (π^2)) (((-1+z))^2)+8 CA (1+z (99+z))-4 TrNf (19+z (18+19 z))-54 (CA-20 CF) (1+(z^2)) Zeta[3]))/(27 (-1+z)))+DELTA[1-z] ((((CF^2) ((Lμ)^4))/2)+((Lμ)^3) (-2 (CF^2) lζ+(2/9) CF (-11 CA+4 TrNf))+((Lμ)^2) (2 (CF^2) ((lζ)^2)+(1/3) CF lζ (11 CA-4 TrNf)+(1/18) CF (-134 CA+3 (2 CA-7 CF) (π^2)+40 TrNf))+Lμ ((1/9) CF (π^2) (-11 CA+4 TrNf)-(1/9) CF lζ (-134 CA+3 (2 CA+CF) (π^2)+40 TrNf))-(2/27) CF lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CF (9 CF (π^4)+2 CA (4856-603 (π^2)+234 (π^4)-2772 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q -> G

Cff[q->g,1]=Collect[Expand@Normal@Series[DD[q->g,1]- d[q->g,1](z^(2ϵ)),{ϵ,0,1}]/.DELTA[1-z]Log[z]->0,{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=((4 (CF^2) ((Lμ)^3) (2-2 z+(z^2)))/(3 z))-(1/(3 z))CF ((Lμ)^2) (40 CA+18 CF+12 CF lζ+8 TrNf-26 CA z-30 CF z-12 CF lζ z-8 TrNf z-17 CA (z^2)+12 CF (z^2)+6 CF lζ (z^2)+4 TrNf (z^2)-8 CA (z^3)-12 (CA+CF) (2-2 z+(z^2)) Log[1-z]+6 (-CF (4-2 z+(z^2))+4 CA (1+z+(z^2))) Log[z])+(1/(9 z))2 CF Lμ (124 CA-108 CF-24 CA (π^2)+30 CF (π^2)-72 CA z+90 CF z+24 CA (π^2) z-30 CF (π^2) z+3 CA (z^2)+18 CF (z^2)-12 CA (π^2) (z^2)+15 CF (π^2) (z^2)+12 TrNf (z^2)-88 CA (z^3)-108 CA Log[z]-108 CF Log[z]-72 CF lζ Log[z]+216 CA z Log[z]+36 CF z Log[z]+72 CF lζ z Log[z]+144 CA (z^2) Log[z]-54 CF (z^2) Log[z]-36 CF lζ (z^2) Log[z]+48 CA (z^3) Log[z]-72 CA ((Log[z])^2)-72 CA z ((Log[z])^2)+36 CF z ((Log[z])^2)-72 CA (z^2) ((Log[z])^2)-18 CF (z^2) ((Log[z])^2)+36 CA Log[1-z] (-(z^2)+2 (2-2 z+(z^2)) Log[z])+72 (CA-CF) (2-2 z+(z^2)) PolyLog[2,z])+(1/ϵ)(((2 (CF^2) ((Lμ)^2) (2-2 z+(z^2)))/z)-1/(3 z)2 CF Lμ (40 CA+18 CF+12 CF lζ+8 TrNf-26 CA z-30 CF z-12 CF lζ z-8 TrNf z-17 CA (z^2)+12 CF (z^2)+6 CF lζ (z^2)+4 TrNf (z^2)-8 CA (z^3)-12 (CA+CF) (2-2 z+(z^2)) Log[1-z]+6 (CF (-2+z) z+4 CA (1+z+(z^2))) Log[z])+1/(9 z)CF (248 CA-216 CF-48 CA (π^2)+54 CF (π^2)-144 CA z+180 CF z+48 CA (π^2) z-54 CF (π^2) z+6 CA (z^2)+36 CF (z^2)-24 CA (π^2) (z^2)+27 CF (π^2) (z^2)+24 TrNf (z^2)-176 CA (z^3)+264 CA Log[z]+96 TrNf Log[z]+120 CA z Log[z]-288 CF z Log[z]-96 TrNf z Log[z]+84 CA (z^2) Log[z]+36 CF (z^2) Log[z]+48 TrNf (z^2) Log[z]+144 CA ((Log[z])^2)+144 CA z ((Log[z])^2)-72 CF z ((Log[z])^2)+144 CA (z^2) ((Log[z])^2)+36 CF (z^2) ((Log[z])^2)-72 Log[1-z] (CA (z^2)+2 CF (2-2 z+(z^2)) Log[z])+144 (CA-CF) (2-2 z+(z^2)) PolyLog[2,z]))-(1/(54 z))CF (10064 CA+2592 CF+120 CA (π^2)+54 CF (π^2)+36 CF lζ (π^2)+24 (π^2) TrNf-6480 CA z-4752 CF z-78 CA (π^2) z-90 CF (π^2) z-36 CF lζ (π^2) z-24 (π^2) TrNf z-1296 CA (z^2)+2160 CF (z^2)-195 CA (π^2) (z^2)+36 CF (π^2) (z^2)+18 CF lζ (π^2) (z^2)+12 (π^2) TrNf (z^2)-2288 CA (z^3)-24 CA (π^2) (z^3)-72 CA (π^2) Log[1-z]-72 CF (π^2) Log[1-z]+72 CA (π^2) z Log[1-z]+72 CF (π^2) z Log[1-z]-36 CA (π^2) (z^2) Log[1-z]-36 CF (π^2) (z^2) Log[1-z]+3456 CA Log[z]+2592 CF Log[z]+72 CA (π^2) Log[z]-648 CF (π^2) Log[z]+7776 CA z Log[z]-864 CF z Log[z]+72 CA (π^2) z Log[z]+612 CF (π^2) z Log[z]+2592 CA (z^2) Log[z]-1296 CF (z^2) Log[z]+72 CA (π^2) (z^2) Log[z]-306 CF (π^2) (z^2) Log[z]+2112 CA (z^3) Log[z]+864 CA (z^2) Log[1-z] Log[z]-1584 CA ((Log[z])^2)-576 TrNf ((Log[z])^2)-720 CA z ((Log[z])^2)+1728 CF z ((Log[z])^2)+576 TrNf z ((Log[z])^2)-504 CA (z^2) ((Log[z])^2)-216 CF (z^2) ((Log[z])^2)-288 TrNf (z^2) ((Log[z])^2)+1728 CF Log[1-z] ((Log[z])^2)-1728 CF z Log[1-z] ((Log[z])^2)+864 CF (z^2) Log[1-z] ((Log[z])^2)-1152 CA ((Log[z])^3)-1152 CA z ((Log[z])^3)+576 CF z ((Log[z])^3)-1152 CA (z^2) ((Log[z])^3)-288 CF (z^2) ((Log[z])^3)-864 CA (-(z^2)+2 (2-2 z+(z^2)) Log[z]) PolyLog[2,z]+1728 (CA+CF) (2-2 z+(z^2)) PolyLog[3,z]-3456 CA Zeta[3]-3744 CF Zeta[3]+3456 CA z Zeta[3]+3744 CF z Zeta[3]-1728 CA (z^2) Zeta[3]-1872 CF (z^2) Zeta[3]);

Cff[q->g,2]=Collect[Expand[DD[q->g,2]-CTIMESd-Normal@Series[d[q->g,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^(a_)):>0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[_]},FullSimplify]

Cff[q->g,1]=((2 CF ((z^2)-Lμ (2-2 z+(z^2))+2 (2-2 z+(z^2)) Log[z]))/z);
Cff[q->g,2]=((2 (CF^2) ((Lμ)^3) (2+(-2+z) z))/z)+((2 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^3))/(3 z))-((CF (11 CF (-2+z) z+CA (80+44 z+62 (z^2))) ((Log[z])^3))/(3 z))+((Log[1-z])^2) (2 CF (-CA+CF) z-((2 (5 CA-4 CF) CF (2+(-2+z) z) Log[z])/z))+((Lμ)^2) (-((4 (CF^2) lζ (2+(-2+z) z))/z)+((CF (2 CA (-1+z) (31+z (7+4 z))-3 CF (12+z (-16+9 z))))/(3 z))+((4 CF (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((2 CF (CF (4+3 (-2+z) z)+4 CA (1+z+(z^2))) Log[z])/z))+Log[1-z] (-((2 (CA-CF) CF (3 z+2 (π^2) (2+(-2+z) z)))/(3 z))+((4 CF (6 CF (z^2)+CA (22+z (-24+(15-4 z) z))) Log[z])/(3 z))+((2 (11 CA-4 CF) CF (2+(-2+z) z) ((Log[z])^2))/z))-((2 CA CF (π^2) (2+z (2+z)) Log[1+z])/(3 z))+((4 CA CF (2+z (2+z)) ((Log[1+z])^3))/(3 z))+Log[z] (((CF (-9 CF ((3+z) (-16+13 z)+6 (π^2) (2+(-2+z) z))+2 CA (54+12 (π^2) (2+(-2+z) z)+z (735+z (147+88 z)))))/(9 z))+4 CA CF z Log[1+z])+((Log[z])^2) (((CF (-3 CF z (8+z)+2 CA (-212+3 z (32+z (-3+8 z)))))/(6 z))+((6 CA CF (2+z (2+z)) Log[1+z])/z))+(4 CA CF z+((4 CA CF (2+z (2+z)) Log[z])/z)) PolyLog[2,-z]+(((8 CA CF (11+z (-12+(9-2 z) z)))/(3 z))-((4 (CA-CF) CF (2+(-2+z) z) Log[1-z])/z)+((4 CF (CA (2+(-6+z) z)-8 CF (2+(-2+z) z)) Log[z])/z)) PolyLog[2,z]+Lμ (((CF (3 CF (-24+22 z-8 (z^2)+3 (π^2) (2+(-2+z) z))+4 CA (3 (5+3 (-3+z) z)-2 (π^2) (2+(-1+z) z))))/(3 z))+((4 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^2))/z)+((2 CF (3 CF (-2+z) z+2 CA (8+z (6+7 z))) ((Log[z])^2))/z)+Log[1-z] (-8 (CF^2) z-((24 CA CF (2+(-2+z) z) Log[z])/z))+lζ (4 (CF^2) z+((8 (CF^2) (2+(-2+z) z) Log[z])/z))+Log[z] (((2 CF (3 CF (12+z (-12+7 z))-2 CA (-62+z (36+z (9+8 z)))))/(3 z))-((8 CA CF (2+z (2+z)) Log[1+z])/z))-((8 CA CF (2+z (2+z)) PolyLog[2,-z])/z)-((16 (CA-CF) CF (2+(-2+z) z) PolyLog[2,z])/z))-((4 (CA-CF) CF (2+(-2+z) z) PolyLog[3,1-z])/z)+((4 CA CF (2+z (2+z)) PolyLog[3,-z])/z)+((4 CF (8 CF (2+(-2+z) z)+CA (14+z (-6+7 z))) PolyLog[3,z])/z)-((8 CA CF (2+z (2+z)) PolyLog[3,(1/(1+z))])/z)+(1/(27 z))CF (27 CF (z (-38-3 (-11+(π^2)) z)-4 (2+(-2+z) z) Zeta[3])+2 CA (1564-558 z-693 (z^2)-340 (z^3)+6 (π^2) (-1+z) (11+z (-1+2 z))-108 (8+z (-9+4 z)) Zeta[3]));

G -> Q

Cff[g->q,1]=Collect[Expand@Normal@Series[DD[g->q,1]- d[g->q,1](z^(2ϵ)),{ϵ,0,1}]/.DELTA[1-z]Log[z]->0,{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=(1/(9 z ϵ))TR (-104 CA+48 CA Lμ+36 CA z-72 CF z+36 CA Lμ z-18 CF Lμ z-36 CA lζ Lμ z+18 CA ((Lμ)^2) z+27 CA (π^2) z-24 CF (π^2) z-216 CA (z^2)+252 CF (z^2)+288 CA Lμ (z^2)+72 CF Lμ (z^2)+72 CA lζ Lμ (z^2)-36 CA ((Lμ)^2) (z^2)-54 CA (π^2) (z^2)+48 CF (π^2) (z^2)+284 CA (z^3)-180 CF (z^3)-372 CA Lμ (z^3)-72 CA lζ Lμ (z^3)+36 CA ((Lμ)^2) (z^3)+54 CA (π^2) (z^3)-48 CF (π^2) (z^3)-96 CA Log[z]-144 CA z Log[z]-36 CF z Log[z]+72 CA Lμ z Log[z]-36 CF Lμ z Log[z]-432 CA (z^2) Log[z]+144 CF (z^2) Log[z]+288 CA Lμ (z^2) Log[z]+72 CF Lμ (z^2) Log[z]+216 CA (z^3) Log[z]-144 CF (z^3) Log[z]-144 CF Lμ (z^3) Log[z]-72 CA z ((Log[z])^2)+36 CF z ((Log[z])^2)-288 CA (z^2) ((Log[z])^2)-72 CF (z^2) ((Log[z])^2)+144 CF (z^3) ((Log[z])^2)+72 z Log[1-z] (CA Lμ (1-2 z+2 (z^2))+CF (2 (-1+z) z+Lμ (1-2 z+2 (z^2)))-2 CA (1-2 z+2 (z^2)) Log[z])-144 (CA-CF) z (1-2 z+2 (z^2)) PolyLog[2,z])+(1/54) TR (-9 CF (360+48 Lμ+6 ((Lμ)^2)+(π^2)+16 Lμ (π^2)-696 z-168 Lμ z-24 ((Lμ)^2) z-36 (π^2) z-32 Lμ (π^2) z+336 (z^2)+120 Lμ (z^2)+32 (π^2) (z^2)+32 Lμ (π^2) (z^2)-24 ((Lμ)^2) Log[1-z]-4 (π^2) Log[1-z]+96 z Log[1-z]+96 Lμ z Log[1-z]+48 ((Lμ)^2) z Log[1-z]+8 (π^2) z Log[1-z]-96 (z^2) Log[1-z]-96 Lμ (z^2) Log[1-z]-48 ((Lμ)^2) (z^2) Log[1-z]-8 (π^2) (z^2) Log[1-z]+192 Log[z]+48 Lμ Log[z]+12 ((Lμ)^2) Log[z]+2 (π^2) Log[z]-240 z Log[z]-192 Lμ z Log[z]-24 ((Lμ)^2) z Log[z]-4 (π^2) z Log[z]+96 (z^2) Log[z]+96 Lμ (z^2) Log[z]+48 ((Lμ)^2) (z^2) Log[z]+8 (π^2) (z^2) Log[z]-96 Lμ Log[1-z] Log[z]+192 z Log[1-z] Log[z]+192 Lμ z Log[1-z] Log[z]-192 (z^2) Log[1-z] Log[z]-192 Lμ (z^2) Log[1-z] Log[z]+24 ((Log[z])^2)+24 Lμ ((Log[z])^2)-96 z ((Log[z])^2)-48 Lμ z ((Log[z])^2)+96 (z^2) ((Log[z])^2)+96 Lμ (z^2) ((Log[z])^2)-32 ((Log[z])^3)+64 z ((Log[z])^3)-128 (z^2) ((Log[z])^3)-96 (2 (-1+z) z+Lμ (1-2 z+2 (z^2))+(2-4 z+4 (z^2)) Log[z]) PolyLog[2,z]+192 (1-2 z+2 (z^2)) PolyLog[3,z]-192 Zeta[3]+384 z Zeta[3]-384 (z^2) Zeta[3])+(1/z)2 CA (712-312 Lμ+72 ((Lμ)^2)+12 (π^2)+1404 z+108 Lμ z+54 ((Lμ)^2) z-54 lζ ((Lμ)^2) z+36 ((Lμ)^3) z+9 (π^2) z-9 lζ (π^2) z+90 Lμ (π^2) z+2808 (z^2)-648 Lμ (z^2)+432 ((Lμ)^2) (z^2)+108 lζ ((Lμ)^2) (z^2)-72 ((Lμ)^3) (z^2)+72 (π^2) (z^2)+18 lζ (π^2) (z^2)-180 Lμ (π^2) (z^2)-4924 (z^3)+852 Lμ (z^3)-558 ((Lμ)^2) (z^3)-108 lζ ((Lμ)^2) (z^3)+72 ((Lμ)^3) (z^3)-93 (π^2) (z^3)-18 lζ (π^2) (z^3)+180 Lμ (π^2) (z^3)+108 ((Lμ)^2) z Log[1-z]+18 (π^2) z Log[1-z]-216 ((Lμ)^2) (z^2) Log[1-z]-36 (π^2) (z^2) Log[1-z]+216 ((Lμ)^2) (z^3) Log[1-z]+36 (π^2) (z^3) Log[1-z]+1512 z Log[z]-216 Lμ z Log[z]-216 lζ Lμ z Log[z]+216 ((Lμ)^2) z Log[z]+180 (π^2) z Log[z]+3024 (z^2) Log[z]+432 Lμ (z^2) Log[z]+432 lζ Lμ (z^2) Log[z]+216 ((Lμ)^2) (z^2) Log[z]-252 (π^2) (z^2) Log[z]+3216 (z^3) Log[z]-1584 Lμ (z^3) Log[z]-432 lζ Lμ (z^3) Log[z]+216 ((Lμ)^2) (z^3) Log[z]+324 (π^2) (z^3) Log[z]-288 ((Log[z])^2)-432 z ((Log[z])^2)+216 Lμ z ((Log[z])^2)-1296 (z^2) ((Log[z])^2)+864 Lμ (z^2) ((Log[z])^2)+648 (z^3) ((Log[z])^2)-432 z Log[1-z] ((Log[z])^2)+864 (z^2) Log[1-z] ((Log[z])^2)-864 (z^3) Log[1-z] ((Log[z])^2)-288 z ((Log[z])^3)-1152 (z^2) ((Log[z])^3)-432 Lμ z (1-2 z+2 (z^2)) PolyLog[2,z]-864 z (1-2 z+2 (z^2)) PolyLog[3,z]+936 z Zeta[3]-1872 (z^2) Zeta[3]+1872 (z^3) Zeta[3]));

Cff[g->q,2]=Collect[Expand[DD[g->q,2]-CTIMESd-Normal@Series[d[g->q,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

Cff[g->q,1]=-2 TR (2 (-1+z) z+Lμ (1-2 z+2 (z^2))+(-2+4 z-4 (z^2)) Log[z]);
Cff[g->q,2]=2 CA ((Lμ)^3) TR (1+2 (-1+z) z)+(2/9) TR (4 TrNf (5+16 (-1+z) z)-9 CF ((π^2) (1+2 (-1+z) z)-2 (4+9 (-1+z) z))+CA (9 (π^2) (1+2 (-1+z) z)-2 (38+91 (-1+z) z))) Log[1-z]+(1/3) TR (9 CF+4 TrNf+2 (15 CF+4 TrNf) (-1+z) z+CA (-11-34 (-1+z) z)) ((Log[1-z])^2)-(2/3) (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^3)+(1/3) TR (2 CA (11+62 z)-11 CF (1-2 z+4 (z^2))) ((Log[z])^3)+((Lμ)^2) (-4 CA lζ TR (1+2 (-1+z) z)+((TR (z (3 CF (-1+4 z)+8 TrNf (1+2 (-1+z) z))+CA (8-2 z (8+z (-40+47 z)))))/(3 z))+4 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+(-2 CF TR+4 (6 CA+CF) TR z-8 (CA+CF) TR (z^2)) Log[z])+Log[z] (((2 TR (-3 z (4 TrNf+8 TrNf (-1+z) z+CF (9+30 (-1+z) z))+CA (-8+z (45-90 z+86 (z^2)))) Log[1-z])/(3 z))+2 (2 CA-3 CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)+(1/(9 z))TR (2 CA (24-z (165+(699-38 z) z+18 (π^2) (1+2 (-1+z) z)))+3 z (-8 TrNf (5+8 (-1+z) z)+CF ((π^2) (2+4 (-1+z) z)-3 (8+z (-73+76 z))))+72 CA (z^2) (1+z) Log[1+z]))+((Log[z])^2) (-2 (2 CA-9 CF) TR (1+2 (-1+z) z) Log[1-z]+((TR (4 CA (16+z (-22+(35-59 z) z))+z (8 TrNf (1+2 (-1+z) z)+3 CF (-7-52 z+68 (z^2)))-36 CA z (1+2 z (1+z)) Log[1+z]))/(6 z)))+(8 CA TR z (1+z)-4 CA TR (1+2 z (1+z)) Log[z]) PolyLog[2,-z]+(((8 TR (-1+z) (-12 CF (z^2)+CA (2+z (-1+11 z))))/(3 z))-4 CA TR (5+2 z (-7+5 z)) Log[z]) PolyLog[2,z]+Lμ ((1/(9 z))TR (CA (-24+z (-16+428 z-692 (z^2)+3 (π^2) (9+2 z (-5+9 z))))-2 z (-40 TrNf (1+2 (-1+z) z)+3 CF ((π^2) (2+4 (-1+z) z)-3 (8+z (-9+10 z)))))+(4/3) TR (CA (-11-10 (-1+z) z)+(9 CF+4 TrNf) (1+2 (-1+z) z)) Log[1-z]-4 (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)-2 TR (-3 CF+6 CF (1-2 z) z+CA (6+28 z)) ((Log[z])^2)+lζ (-8 CA TR (-1+z) z+8 CA TR (1+2 (-1+z) z) Log[z])+Log[z] (-24 CF TR (1+2 (-1+z) z) Log[1-z]+((2 TR (z (CF (9+12 (1-3 z) z)-8 TrNf (1+2 (-1+z) z))+4 CA (-4+z (7+z (-23+33 z)))+12 CA z (1+2 z (1+z)) Log[1+z]))/(3 z)))+8 CA TR (1+2 z (1+z)) PolyLog[2,-z]+16 (CA-CF) TR (1+2 (-1+z) z) PolyLog[2,z])-4 CA TR (1+2 z (1+z)) PolyLog[3,-z]+4 TR (8 CF (1+2 (-1+z) z)+CA (3+2 z (-7+3 z))) PolyLog[3,z]+8 CA TR (1+2 z (1+z)) PolyLog[3,(1/(1+z))]+(1/(27 z))TR (CA (-296+2 z (-1223+z (139+774 z))+3 (π^2) (8+z (21-60 z+76 (z^2))))+z (4 TrNf (-9 (π^2) (1+2 (-1+z) z)+8 (7+17 (-1+z) z))-9 CF ((π^2) (9+14 (-1+z) z)-3 (63+z (-101+56 z))))+18 CA z (1+2 z (1+z)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))-216 z (CA+4 CF-(CA+8 CF) z+2 (CA+4 CF) (z^2)) Zeta[3]);

G -> G

Cff[g->g,1]=Collect[Expand@Normal@Series[DD[g->g,1]- d[g->g,1](z^(2ϵ)),{ϵ,0,1}]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=-((2 CA ((Lμ)^3) (12 CA (((1-z+(z^2)))^2)-(11 CA-4 TrNf) (-1+z) z DELTA[1-z]))/(9 (-1+z) z))+(1/(3 (-1+z) z))((Lμ)^2) (-CA (11 CA lζ+4 CA (π^2)-4 lζ TrNf) (-1+z) z DELTA[1-z]+2 (33 (CA^2)+6 (CA^2) lζ+4 CA TrNf-8 CF TrNf-58 (CA^2) z-12 (CA^2) lζ z-8 CA TrNf z+2 CF TrNf z+39 (CA^2) (z^2)+18 (CA^2) lζ (z^2)+12 CA TrNf (z^2)+12 CF TrNf (z^2)-58 (CA^2) (z^3)-12 (CA^2) lζ (z^3)-8 CA TrNf (z^3)+2 CF TrNf (z^3)+33 (CA^2) (z^4)+6 (CA^2) lζ (z^4)+4 CA TrNf (z^4)-8 CF TrNf (z^4)-24 (CA^2) (((1-z+(z^2)))^2) Log[1-z]-12 (2 (CA^2)-CF TrNf) z (-1+(z^2)) Log[z]))+(1/(9 (-1+z) z))Lμ (CA (π^2) (11 CA-4 TrNf) (-1+z) z DELTA[1-z]+4 (-3 (CA^2) (π^2)-40 CF TrNf+6 (CA^2) (π^2) z+4 CF TrNf z-9 (CA^2) (π^2) (z^2)+36 CF TrNf (z^2)+6 (CA^2) (π^2) (z^3)+76 CF TrNf (z^3)-3 (CA^2) (π^2) (z^4)-76 CF TrNf (z^4)+12 (2 CF TrNf (-2-z+3 (z^3))+(CA^2) (3 lζ (((1-z+(z^2)))^2)+(((-1+z))^2) (11+2 z+11 (z^2)))-6 (CA^2) (((1-z+(z^2)))^2) Log[1-z]) Log[z]+36 (CF TrNf z (-1+(z^2))+(CA^2) (1+3 (z^2)-4 (z^3)+(z^4))) ((Log[z])^2)))+(1/ϵ)(((CA ((Lμ)^2) (-12 CA (((1-z+(z^2)))^2)+(11 CA-4 TrNf) (-1+z) z DELTA[1-z]))/(3 (-1+z) z))+1/(3 (-1+z) z)2 Lμ (-CA (11 CA lζ+4 CA (π^2)-4 lζ TrNf) (-1+z) z DELTA[1-z]+2 (33 (CA^2)+6 (CA^2) lζ+4 CA TrNf-8 CF TrNf-58 (CA^2) z-12 (CA^2) lζ z-8 CA TrNf z+2 CF TrNf z+39 (CA^2) (z^2)+18 (CA^2) lζ (z^2)+12 CA TrNf (z^2)+12 CF TrNf (z^2)-58 (CA^2) (z^3)-12 (CA^2) lζ (z^3)-8 CA TrNf (z^3)+2 CF TrNf (z^3)+33 (CA^2) (z^4)+6 (CA^2) lζ (z^4)+4 CA TrNf (z^4)-8 CF TrNf (z^4)-24 (CA^2) (((1-z+(z^2)))^2) Log[1-z]+12 (CF TrNf z (-1+(z^2))+(CA^2) (1+3 (z^2)-4 (z^3)+(z^4))) Log[z]))+1/(18 (-1+z) z)(CA (π^2) (11 CA-4 TrNf) (-1+z) z DELTA[1-z]-4 (3 (CA^2) (π^2) (((1-z+(z^2)))^2)+8 CF TrNf (((-1+z))^2) (10+19 z+19 (z^2))-12 (-4 CA TrNf (((1-z+(z^2)))^2)+2 CF TrNf z (-3-6 z+5 (z^2)+4 (z^3))+(CA^2) (-11+18 z-3 (z^2)+18 (z^3)-11 (z^4))+12 (CA^2) (((1-z+(z^2)))^2) Log[1-z]) Log[z]+72 (CF TrNf z (-1+(z^2))+(CA^2) (1+3 (z^2)-4 (z^3)+(z^4))) ((Log[z])^2))))+(1/(54 (-1+z) z))(-3 CA (-1+z) z DELTA[1-z] (11 CA lζ (π^2)+4 CA (π^4)-4 lζ (π^2) TrNf-88 CA Zeta[3]+32 TrNf Zeta[3])+2 (6608 (CA^2)+99 (CA^2) (π^2)+18 (CA^2) lζ (π^2)-1040 CF TrNf+12 CA (π^2) TrNf-24 CF (π^2) TrNf-10496 (CA^2) z-174 (CA^2) (π^2) z-36 (CA^2) lζ (π^2) z-2848 CF TrNf z-24 CA (π^2) TrNf z+6 CF (π^2) TrNf z+7776 (CA^2) (z^2)+117 (CA^2) (π^2) (z^2)+54 (CA^2) lζ (π^2) (z^2)+6480 CF TrNf (z^2)+36 CA (π^2) TrNf (z^2)+36 CF (π^2) TrNf (z^2)-10496 (CA^2) (z^3)-174 (CA^2) (π^2) (z^3)-36 (CA^2) lζ (π^2) (z^3)-256 CF TrNf (z^3)-24 CA (π^2) TrNf (z^3)+6 CF (π^2) TrNf (z^3)+6608 (CA^2) (z^4)+99 (CA^2) (π^2) (z^4)+18 (CA^2) lζ (π^2) (z^4)-2336 CF TrNf (z^4)+12 CA (π^2) TrNf (z^4)-24 CF (π^2) TrNf (z^4)-72 (CA^2) (π^2) Log[1-z]+144 (CA^2) (π^2) z Log[1-z]-216 (CA^2) (π^2) (z^2) Log[1-z]+144 (CA^2) (π^2) (z^3) Log[1-z]-72 (CA^2) (π^2) (z^4) Log[1-z]+3216 (CA^2) Log[z]-288 (CA^2) (π^2) Log[z]-960 CF TrNf Log[z]+1536 (CA^2) z Log[z]+648 (CA^2) (π^2) z Log[z]-2496 CF TrNf z Log[z]-36 CF (π^2) TrNf z Log[z]-864 (CA^2) (π^2) (z^2) Log[z]-432 CF TrNf (z^2) Log[z]-1536 (CA^2) (z^3) Log[z]+504 (CA^2) (π^2) (z^3) Log[z]+3888 CF TrNf (z^3) Log[z]+36 CF (π^2) TrNf (z^3) Log[z]-3216 (CA^2) (z^4) Log[z]-288 (CA^2) (π^2) (z^4) Log[z]-792 (CA^2) ((Log[z])^2)-288 CA TrNf ((Log[z])^2)+1296 (CA^2) z ((Log[z])^2)+576 CA TrNf z ((Log[z])^2)-432 CF TrNf z ((Log[z])^2)-216 (CA^2) (z^2) ((Log[z])^2)-864 CA TrNf (z^2) ((Log[z])^2)-864 CF TrNf (z^2) ((Log[z])^2)+1296 (CA^2) (z^3) ((Log[z])^2)+576 CA TrNf (z^3) ((Log[z])^2)+720 CF TrNf (z^3) ((Log[z])^2)-792 (CA^2) (z^4) ((Log[z])^2)-288 CA TrNf (z^4) ((Log[z])^2)+576 CF TrNf (z^4) ((Log[z])^2)+864 (CA^2) Log[1-z] ((Log[z])^2)-1728 (CA^2) z Log[1-z] ((Log[z])^2)+2592 (CA^2) (z^2) Log[1-z] ((Log[z])^2)-1728 (CA^2) (z^3) Log[1-z] ((Log[z])^2)+864 (CA^2) (z^4) Log[1-z] ((Log[z])^2)-576 (CA^2) ((Log[z])^3)+576 CF TrNf z ((Log[z])^3)-1728 (CA^2) (z^2) ((Log[z])^3)+2304 (CA^2) (z^3) ((Log[z])^3)-576 CF TrNf (z^3) ((Log[z])^3)-576 (CA^2) (z^4) ((Log[z])^3)-1728 (CA^2) (((1-z+(z^2)))^2) Log[z] PolyLog[2,z]+3456 (CA^2) (((1-z+(z^2)))^2) PolyLog[3,z]-3600 (CA^2) Zeta[3]+7200 (CA^2) z Zeta[3]-10800 (CA^2) (z^2) Zeta[3]+7200 (CA^2) (z^3) Zeta[3]-3600 (CA^2) (z^4) Zeta[3]));

Cff[g->g,2]=Collect[Expand[DD[g->g,2]-CTIMESd-Normal@Series[d[g->g,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

Cff[g->g,1]=CA (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((4 CA (((1-z+(z^2)))^2) (Lμ-2 Log[z]))/((-1+z) z));
Cff[g->g,2]=-((4 (CA^2) ((Lμ)^3) (((1+(-1+z) z))^2))/((-1+z) z))-(2/3) CA (CA-2 TrNf) Log[1-z]+((1/(9 (-1+z) z))(-8 CF TrNf (-1+z) (-6+z (135+z (81+82 z)))+4 CA TrNf (46+z (-91+z (90+z (-71+46 z))))+(CA^2) (-536+60 (π^2) (((1+(-1+z) z))^2)+z (-1123+z (726+z (-407+1072 z)))))-((8 (CA^2) (-1+z) (11+z (-1+11 z)) Log[1-z])/(3 z))+((4 (CA^2) (((1+(-1+z) z))^2) ((Log[1-z])^2))/((-1+z) z))) Log[z]+(((44 CF TrNf (-1+z) z (((1+z))^2)+4 (CA^2) (20+z (11+(-1+z) z (1+z) (-42+11 z)))) ((Log[z])^3))/(3 z (-1+(z^2))))+((Lμ)^2) (((8 (CA^2) lζ (((1+(-1+z) z))^2))/((-1+z) z))+((2 (-4 CA TrNf (((1+(-1+z) z))^2)+2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (55+z (-102+z (105+z (-102+55 z))))))/(3 (-1+z) z))-((16 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+((8 (2 (CA^2) (1+(((-1+z))^3) z)+CF TrNf z (-1+(z^2))) Log[z])/((-1+z) z)))+((Log[z])^2) (-((28 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+(1/(3 z (-1+(z^2))))((1+z) (4 CA TrNf (4+(-1+z) z (3+z (-9+4 z)))+2 CF TrNf (-1+z) (16+z (21+z (15+16 z)))+(CA^2) (220-3 z (93+z (-48+43 z))))+36 (CA^2) (-1+z) (((1+z+(z^2)))^2) Log[1+z]))+((8 (CA^2) (((1+z+(z^2)))^2) Log[z] PolyLog[2,-z])/(z (1+z)))+Lμ ((1/(9 z (-1+(z^2))))2 (8 CF TrNf (((-1+z))^2) (1+z) (13+z (-5+22 z))-4 CA TrNf (1+z) (23+z (-42+z (48+z (-42+23 z))))+(CA^2) (z (1+z) (109+z (-84+109 z))-3 (π^2) (1+z (-5+z (1+z (5+z (-1+5 z)))))))-((16 (CA^2) lζ (((1+(-1+z) z))^2) Log[z])/((-1+z) z))-((8 (3 CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (4+z (-1+(-3+z) z) (-3+z (-1+3 z)))) ((Log[z])^2))/(z (-1+(z^2))))+Log[z] (((48 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))-((4 (1+z) (-4 CA TrNf z (1+z)+2 CF TrNf (8+3 z (3+z))+(CA^2) (-88+z (47+z (-61+44 z))))+48 (CA^2) (((1+z+(z^2)))^2) Log[1+z])/(3 z (1+z))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[2,-z])/(z (1+z))))+(-((8 (CA^2) (-1+z) (11+z (-1+11 z)))/(3 z))+((8 (CA^2) (7+z (-5+z (7+z (5+z (-7+5 z))))) Log[z])/(z (-1+(z^2))))) PolyLog[2,z]+((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,-z])/(z (1+z)))-((8 (CA^2) (15+z (-11+z (15+z (11+z (-15+11 z))))) PolyLog[3,z])/(z (-1+(z^2))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,(1/(1+z))])/(z (1+z)))+(1/(27 z (-1+(z^2))))4 ((1+z) (2 CF TrNf (((-1+z))^2) (139+z (409+301 z))+CA TrNf (-139+z (249+z (-276+11 (21-11 z) z)))+(CA^2) (-817+3 (π^2) (((-1+z))^2) (11+z (-1+11 z))+z (1029-2 z (129+z (-555+431 z)))))+9 (CA^2) ((-1+z) (((1+z+(z^2)))^2) Log[1+z] (-(π^2)+2 ((Log[1+z])^2))+6 (9+z (-10+z (9+z (10+z (-9+10 z))))) Zeta[3]))+DELTA[1-z] ((((CA^2) ((Lμ)^4))/2)+((Lμ)^2) (2 (CA^2) ((lζ)^2)-(1/18) CA (CA (134+15 (π^2))-40 TrNf)+(1/3) CA lζ (11 CA-4 TrNf))+((Lμ)^3) (-2 (CA^2) lζ+(2/9) CA (-11 CA+4 TrNf))+Lμ ((1/9) CA (π^2) (-11 CA+4 TrNf)-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf))-(2/27) CA lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CA (CA (9712-1206 (π^2)+477 (π^4)-5544 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q -> Q’

Cff[q->qp,1]=Collect[Expand@Normal@Series[DD[q->qp,1]- d[q->qp,1](z^(2ϵ)),{ϵ,0,1}]/.DELTA[1-z]Log[z]->0,{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=(1/(27 z ϵ))CF TR (-312+144 Lμ-108 z+108 Lμ z-108 (z^2)-108 Lμ (z^2)+528 (z^3)-144 Lμ (z^3)-288 Log[z]-432 z Log[z]+216 Lμ z Log[z]-432 (z^2) Log[z]+216 Lμ (z^2) Log[z]-216 z ((Log[z])^2)-216 (z^2) ((Log[z])^2))+(1/(27 z))CF TR (712-312 Lμ+72 ((Lμ)^2)+12 (π^2)+864 z-108 Lμ z+54 ((Lμ)^2) z+9 (π^2) z-432 (z^2)-108 Lμ (z^2)-54 ((Lμ)^2) (z^2)-9 (π^2) (z^2)-1144 (z^3)+528 Lμ (z^3)-72 ((Lμ)^2) (z^3)-12 (π^2) (z^3)+1080 z Log[z]-216 Lμ z Log[z]+108 ((Lμ)^2) z Log[z]+18 (π^2) z Log[z]+1296 (z^2) Log[z]-648 Lμ (z^2) Log[z]+108 ((Lμ)^2) (z^2) Log[z]+18 (π^2) (z^2) Log[z]+1056 (z^3) Log[z]-288 Lμ (z^3) Log[z]-288 ((Log[z])^2)-432 z ((Log[z])^2)+216 Lμ z ((Log[z])^2)-432 (z^2) ((Log[z])^2)+216 Lμ (z^2) ((Log[z])^2)-288 z ((Log[z])^3)-288 (z^2) ((Log[z])^3));

Cff[q->qp,2]=Collect[Expand[DD[q->qp,2]-CTIMESd-Normal@Series[d[q->qp,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

Cff[q->qp,1]=0;
Cff[q->qp,2]=((2 CF TR (-1+z) (148+z (859+436 z)))/(27 z))-((4 CF TR (-12+z (174+z (51+32 z))) Log[z])/(9 z))-((CF TR (-32+3 z (11+z (11+8 z))) ((Log[z])^2))/(3 z))+(22/3) CF TR (1+z) ((Log[z])^3)+((Lμ)^2) (-((2 CF TR (-1+z) (4+z (7+4 z)))/(3 z))+4 CF TR (1+z) Log[z])+Lμ (-((4 CF TR (-1+z) (-2+z (19+4 z)))/(3 z))+((4 CF TR (-8+z (3+z (15+8 z))) Log[z])/(3 z))-12 CF TR (1+z) ((Log[z])^2))-((8 CF TR (-1+z) (2+z (-1+2 z)) PolyLog[2,1-z])/(3 z));

Q -> (OverBar[Q])

Cff[q->qbar,1]=Collect[Expand@Normal@Series[DD[q->qbar,1]- d[q->qbar,1](z^(2ϵ)),{ϵ,0,0}]/.DELTA[1-z]Log[z]->0,{ϵ,Lb,DELTA[_]},Simplify]

CTIMESd=0;

Cff[q->qbar,2]=Collect[Expand[DD[q->qbar,2]-CTIMESd-Normal@Series[d[q->qbar,2](z^(2ϵ)),{ϵ,0,0}]]/.{DELTA[1-z]Log[z]->0,DELTA[1-z]((Log[z])^2)->0},{ϵ,DELTA[_],Lμ,lζ,PolyLog[_,_],Log[z],Log[1-z]},FullSimplify]

Cff[q->qbar,1]=0;
Cff[q->qbar,2]=-((3 (CA-2 CF) CF (1+(z^2)) ((Log[z])^3))/(1+z))+((2 (CA-2 CF) CF ((Log[z])^2) (-2 (1+z) (2+z)+3 (1+(z^2)) Log[1+z]))/(1+z))+Log[z] (-4 (CA-2 CF) CF (-1+z) Log[1-z]+(CA-2 CF) CF (-19+5 z+4 (1+z) Log[1+z]))+(4 (CA-2 CF) CF (1+z)+((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,-z]+Lμ (-((2 (CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (z^2)))/(3 (1+z)))+((2 (CA-2 CF) CF (1+(z^2)) ((Log[z])^2))/(1+z))+((4 (CA-2 CF) CF Log[z] ((((1+z))^2)-2 (1+(z^2)) Log[1+z]))/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[2,-z])/(1+z)))+(-4 (CA-2 CF) CF (-1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,z]+((4 (CA-2 CF) CF (1+(z^2)) PolyLog[3,-z])/(1+z))+((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,z])/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,(1/(1+z))])/(1+z))-(((CA-2 CF) CF (45-45 (z^2)-(π^2) (1+z) (-1+3 z)+2 (1+(z^2)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))-6 (1+(z^2)) Zeta[3]))/(3 (1+z)));

Final expressions

The coefficient functions for TMD PDF given by variables Cpdf[f ← f, n]
Cpdf[f ← f, n]=(∑_(m,l)) Cpdf[f ← f, n,m,l] ((Subscript[L, μ])^(m-l))((Subscript[l, ζ])^l)
		= Exp[-D Lζ](∑_m)Cpdf[f ← f, n,m] ((Subscript[L, μ])^m)
		
The coefficient functions for TMD F given by variables Cff[f ← f, n]
Cff[f ← f, n]=(∑_(m,l)) Cff[f ← f, n,m,l] ((Subscript[L, μ])^(m-l))((Subscript[l, ζ])^l)
		= Exp[-D Lζ](∑_m)Cff[f ← f, n,m] ((Subscript[L, μ])^m)
Note: expression for finite part presented in the article are simplfied by means of various relations for (Subscript[Li, {2,3}])

Matching coefficients for TMD PDF

Q ← Q

Cpdf[q←q,1]=((2 CF (-(((-1+x))^2)+Lμ (1+(x^2))))/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[q←q,2]=-((2 (CF^2) ((Lμ)^3) (1+(x^2)))/(-1+x))-((2 (CA-CF) CF ((π^2)+3 x+(-3+(π^2)) (x^2)) Log[1-x])/(3 (-1+x)))+(-((2 CF (CA (-29+(36-83 x) x)+10 TrNf (1+(x^2))+9 CF (5+x (-13+16 x))))/(9 (-1+x)))-4 (CA-3 CF) CF (-1+x) Log[1-x]+((2 (2 CA-3 CF) CF (1+(x^2)) ((Log[1-x])^2))/(-1+x))) Log[x]+(((CF (CA (11-(-12+x) x)+2 (CF (-9+6 (-1+x) x)-2 TrNf (1+(x^2)))))/(6 (-1+x)))-((2 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) ((Log[x])^2)+((CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^3))/(3 (-1+x)))+((Lμ)^2) (((4 (CF^2) lζ (1+(x^2)))/(-1+x))+((CF (11 CA (1+(x^2))+2 (9 CF (((-1+x))^2)-2 TrNf (1+(x^2)))))/(3 (-1+x)))-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))+((2 (CF^2) (1+3 (x^2)) Log[x])/(-1+x)))+Lμ (-4 (CF^2) lζ (-1+x)+((CF (-16 TrNf (4+x (-3+4 x))-3 CF (84 (((-1+x))^2)+(π^2) (1+(x^2)))+CA (-6 (π^2) (1+(x^2))+4 (77+x (-87+77 x)))))/(9 (-1+x)))+8 (CF^2) (-1+x) Log[1-x]+(((2 CF (17 CA-4 (6 CF+TrNf)+(5 CA+6 CF-4 TrNf) (x^2)))/(3 (-1+x)))-((8 (CF^2) (1+(x^2)) Log[1-x])/(-1+x))) Log[x]+((2 CF (CA-CF+(CA+CF) (x^2)) ((Log[x])^2))/(-1+x)))+(-4 (CA-2 CF) CF (-1+x)+((4 (CA-CF) CF (1+(x^2)) Log[1-x])/(-1+x))+((4 (CA-3 CF) CF (1+(x^2)) Log[x])/(-1+x))) PolyLog[2,x]+((4 (CA-CF) CF (1+(x^2)) PolyLog[3,1-x])/(-1+x))-((4 (2 CA-5 CF) CF (1+(x^2)) PolyLog[3,x])/(-1+x))+((CF (27 CA (π^2) (((-1+x))^2)-27 CF (-22+(π^2)) (((-1+x))^2)+8 CA (1+x (99+x))-4 TrNf (19+x (18+19 x))-54 (3 CA+10 CF) (1+(x^2)) Zeta[3]))/(27 (-1+x)))+DELTA[1-x] ((((CF^2) ((Lμ)^4))/2)+((Lμ)^3) (-2 (CF^2) lζ+(2/9) CF (-11 CA+4 TrNf))+((Lμ)^2) (2 (CF^2) ((lζ)^2)+(1/3) CF lζ (11 CA-4 TrNf)+(1/18) CF (-134 CA+3 (2 CA-7 CF) (π^2)+40 TrNf))+Lμ ((1/9) CF (π^2) (-11 CA+4 TrNf)-(1/9) CF lζ (-134 CA+3 (2 CA+CF) (π^2)+40 TrNf))-(2/27) CF lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CF (9 CF (π^4)+2 CA (4856-603 (π^2)+18 (π^4)-2772 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q ← G

Cpdf[q←g,1]=-2 TR (2 (-1+x) x+Lμ (1-2 x+2 (x^2)));
Cpdf[q←g,2]=2 CF ((Lμ)^3) TR (1+2 (-1+x) x)-2 (CA-CF) TR x (-3+4 x) Log[1-x]+4 (CA-CF) TR (-1+x) x ((Log[1-x])^2)+(2/3) (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^3)+(1/3) TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^3)+((Lμ)^2) (-4 CF lζ TR (1+2 (-1+x) x)+((TR (-3 CF x (7+4 x (-3+2 x))-2 CA (-1+x) (4+x (7+31 x))))/(3 x))+4 (CA+CF) TR (1+2 (-1+x) x) Log[1-x]+2 TR (CA (2+8 x)+CF (-1+2 x-4 (x^2))) Log[x])+((Log[x])^2) (-2 TR (-8 CA x+CF (1+2 (-1+x) x)) Log[1-x]+(1/6) TR (CA (-6+8 (3-11 x) x)+3 CF (1+4 (3-2 x) x)+12 CA (1+2 x (1+x)) Log[1+x]))+Log[x] (8 CF TR (-1+x) x Log[1-x]+2 CF TR (1+2 (-1+x) x) ((Log[1-x])^2)+(1/9) TR (4 CA (21-6 (5+(π^2)) x+68 (x^2))+3 CF (24+3 (15-8 x) x+(π^2) (2+4 (-1+x) x))+36 CA Log[1+x] (2 x (1+x)-(1+2 x (1+x)) Log[1+x])))+(((8 CA TR (-1+x) (2+x (-1+11 x)))/(3 x))+4 (-CA+CF) TR (1+2 (-1+x) x) Log[1-x]+4 TR (4 CA x+CF (-1-2 (-1+x) x)) Log[x]) PolyLog[2,1-x]+(8 CA TR x (1+x)-4 CA TR (1+2 x (1+x)) Log[x]) PolyLog[2,-x]+Lμ (-8 CF lζ TR (-1+x) x+((TR (3 CF x (5 (π^2) (1+2 (-1+x) x)-6 (10+x (-33+28 x)))+4 CA (-26+x (36+x (-135+6 (π^2)+116 x)))))/(9 x))+16 CF TR (-1+x) x Log[1-x]+4 (CA-CF) TR (1+2 (-1+x) x) ((Log[1-x])^2)+2 TR (CA (2+4 x)+CF (-1+2 x-4 (x^2))) ((Log[x])^2)+Log[x] (8 CF TR (1+2 (-1+x) x) Log[1-x]+(1/3) (-2 TR (3 CF (1+8 (-1+x) x)+CA (6+88 (x^2)))+24 CA TR (1+2 x (1+x)) Log[1+x]))+8 CA TR (1+2 x (1+x)) PolyLog[2,-x])+4 (CA-CF) TR (1+2 (-1+x) x) PolyLog[3,1-x]+4 CA TR (1+2 x (1+x)) PolyLog[3,-x]-4 TR (-8 CA x+CF (1+2 (-1+x) x)) PolyLog[3,x]-8 CA TR (1+2 x (1+x)) PolyLog[3,(x/(1+x))]+(1/(27 x))TR (27 CF x (-13+(75+2 (π^2) (-1+x)-72 x) x)+CA (344+2 x (-315+(387+18 (π^2)-298 x) x))+18 x (CA (1+2 x (1+x)) Log[1+x] (-(π^2)+2 ((Log[1+x])^2))-6 (CA (1+2 (-2+x) x)-8 CF (1+2 (-1+x) x)) Zeta[3]));

G ← Q

Cpdf[g←q,1]=2 CF (Lμ (2-(2/x)-x)+x);
Cpdf[g←q,2]=((2 CA CF ((Lμ)^3) (2+(-2+x) x))/x)+((2 CF (CA (-152+(152-43 x) x)+8 TrNf (5+(-5+x) x)+9 CF (16+x (-16+5 x))) Log[1-x])/(9 x))+((CF (CA (-22+(22-5 x) x)+3 CF (6+(-6+x) x)+4 TrNf (2+(-2+x) x)) ((Log[1-x])^2))/(3 x))-((2 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^3))/(3 x))-(1/3) CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^3)+((Lμ)^2) (-((4 CA CF lζ (2+(-2+x) x))/x)+((CF (-3 CF (-4+x) x+8 TrNf (2+(-2+x) x)+2 CA (-53+x (46+x (-11+4 x)))))/(3 x))+((4 CF (CA+CF) (2+(-2+x) x) Log[1-x])/x)-((2 CF (CF (-2+x) x+4 CA (1+x+(x^2))) Log[x])/x))+((Log[x])^2) (((2 CA CF (2+(-2+x) x) Log[1-x])/x)+((CF x (-3 CF (4+3 x)+2 CA (36+x (9+8 x)))-12 CA CF (2+x (2+x)) Log[1+x])/(6 x)))+Log[x] (((4 CA CF (-22+x (24+x (-9+4 x))) Log[1-x])/(3 x))+((2 CA CF (2+(-2+x) x) ((Log[1-x])^2))/x)+((CF (x (45 CF (-3+x)-2 CA (249-6 x+88 (x^2)))+36 CA Log[1+x] ((x^2)+(2+x (2+x)) Log[1+x])))/(9 x)))+(4 CA CF x+((4 CA CF (2+x (2+x)) Log[x])/x)) PolyLog[2,-x]+Lμ (4 CA CF lζ x+((CF (2 (27 CF x (1+2 x)+40 TrNf (2+(-2+x) x))+CA (3 (π^2) (2+(-10+x) x)-4 (9+x (19+4 x+44 (x^2))))))/(9 x))+((4 CF (CA (-22+(22-17 x) x)+(9 CF+4 TrNf) (2+(-2+x) x)) Log[1-x])/(3 x))-((4 (CA-CF) CF (2+(-2+x) x) ((Log[1-x])^2))/x)-2 CF (CF (-2+x)+2 CA (2+x)) ((Log[x])^2)+Log[x] (((8 CA CF (2+(-2+x) x) Log[1-x])/x)-((2 CF (x (3 CF (4+5 x)-2 CA (36+x (15+8 x)))+12 CA (2+x (2+x)) Log[1+x]))/(3 x)))-((8 CA CF (2+x (2+x)) PolyLog[2,-x])/x))+(((8 CA CF (-1+x) (11+x (-1+2 x)))/(3 x))+((4 CA CF (6+x (-2+3 x)) Log[x])/x)) PolyLog[2,x]-((4 CA CF (2+x (2+x)) PolyLog[3,-x])/x)-((4 CA CF (10+x (-2+5 x)) PolyLog[3,x])/x)+((8 CA CF (2+x (2+x)) PolyLog[3,(x/(1+x))])/x)+(1/(27 x))CF (-27 CF (-10+x) x+8 TrNf (56+x (-56+13 x))+CA (-3 (π^2) (-44+x (48+x (-9+8 x)))+4 (-790+x (791+4 x (-67+38 x))))+18 CA (2+x (2+x)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))+216 CA (6+x (-7+3 x)) Zeta[3]);

G ← G

Cpdf[g←g,1]=((4 CA Lμ (((1-x+(x^2)))^2))/((-1+x) x))+CA (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[g←g,2]=-((4 (CA^2) ((Lμ)^3) (((1+(-1+x) x))^2))/((-1+x) x))+(2/3) CA (CA-2 TrNf) x Log[1-x]+(4/3) ((CA^2) (-2+x) x+CF TrNf (1+x)+((CA^2)/(-1+(x^2)))) ((Log[x])^3)+((Lμ)^2) (((8 (CA^2) lζ (((1+(-1+x) x))^2))/((-1+x) x))+((2 (-4 CA TrNf (((1+(-1+x) x))^2)+2 CF TrNf (-4+x+6 (x^2)+(x^3)-4 (x^4))+(CA^2) (55+x (-102+x (105+x (-102+55 x))))))/(3 (-1+x) x))-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((8 (CF TrNf x (-1+(x^2))+(CA^2) (1+(-3+x) (-1+x) (x^2))) Log[x])/((-1+x) x)))+((Log[x])^2) (-((4 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))+((x (1+x) (4 CA TrNf (1+x)+6 CF TrNf (3+x)+(CA^2) (25+11 x (-1+4 x)))-12 (CA^2) (((1+x+(x^2)))^2) Log[1+x])/(3 x (1+x))))+Log[x] (((8 (CA^2) (-1+x) (11+x (-1+11 x)) Log[1-x])/(3 x))-((4 (CA^2) (((1+(-1+x) x))^2) ((Log[1-x])^2))/((-1+x) x))+((-x (1+x) (-216 CF TrNf (1+x)-4 CA TrNf (13+10 x)+(CA^2) (701+x (149+536 x)))+72 (CA^2) (((1+x+(x^2)))^2) ((Log[1+x])^2))/(9 x (1+x))))+((8 (CA^2) (((1+x+(x^2)))^2) Log[x] PolyLog[2,-x])/(x (1+x)))+Lμ ((1/(9 x (-1+(x^2))))2 (-12 CF TrNf (((-1+x))^2) (1+x) (-2+x (19+4 x))-4 CA TrNf (1+x) (23+x (-42+x (48+x (-42+23 x))))+(CA^2) (x (1+x) (109+x (-84+109 x))-3 (π^2) (1+x (-5+x (1+x (5+x (-1+5 x)))))))+(8 CF TrNf (1+x)+((8 (CA^2) (((1+x-(x^2)))^2))/(-1+(x^2)))) ((Log[x])^2)+Log[x] (-((16 (CA^2) (((1+(-1+x) x))^2) Log[1-x])/((-1+x) x))-((4 (-x (1+x) (4 CA TrNf (1+x)+18 CF TrNf (1+x)+(CA^2) (25+11 x (-1+4 x)))+12 (CA^2) (((1+x+(x^2)))^2) Log[1+x]))/(3 x (1+x))))-((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[2,-x])/(x (1+x))))+(((8 (CA^2) (-1+x) (11+x (-1+11 x)))/(3 x))-((8 (CA^2) (3+x (-1+x (3+x-3 (x^2)+(x^3)))) Log[x])/(x (-1+(x^2))))) PolyLog[2,x]-((8 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,-x])/(x (1+x)))+((8 (CA^2) (5+x (-1+x (5+x-5 (x^2)+(x^3)))) PolyLog[3,x])/(x (-1+(x^2))))+((16 (CA^2) (((1+x+(x^2)))^2) PolyLog[3,(x/(1+x))])/(x (1+x)))+(1/(27 x (-1+(x^2))))4 (-(1+x) (-18 CF TrNf (((-1+x))^2) (1+(-23+x) x)+(CA^2) (-790+3 (π^2) (((-1+x))^2) (11+x (-1+11 x))+x (1514+x (-1686+5 (319-167 x) x)))+CA TrNf (121+x (-231+x (276+x (-249+139 x)))))+9 (CA^2) ((-1+x) (((1+x+(x^2)))^2) Log[1+x] ((π^2)-2 ((Log[1+x])^2))-6 (6+x (-7+x (6+x (7+x (-6+7 x))))) Zeta[3]))+DELTA[1-x] ((((CA^2) ((Lμ)^4))/2)+((Lμ)^2) (2 (CA^2) ((lζ)^2)-(1/18) CA (CA (134+15 (π^2))-40 TrNf)+(1/3) CA lζ (11 CA-4 TrNf))+((Lμ)^3) (-2 (CA^2) lζ+(2/9) CA (-11 CA+4 TrNf))+Lμ ((1/9) CA (π^2) (-11 CA+4 TrNf)-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf))-(2/27) CA lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/324) CA (4 TrNf (-328+45 (π^2)+252 Zeta[3])+CA (4856-603 (π^2)+18 (π^4)-135 ((Log[2])^4)+3240 PolyLog[2,2,(1/2)]+9 (-308+45 Log[2]) Zeta[3])));

Q ← Q’

Cpdf[q←qp,1]=0;
Cpdf[q←qp,2]=-((2 CF TR (-1+x) (172+x (-143+136 x)))/(27 x))+(4/9) CF TR (21-30 x+32 (x^2)) Log[x]-(1/3) CF TR (3+x (3+8 x)) ((Log[x])^2)+(2/3) CF TR (1+x) ((Log[x])^3)+((Lμ)^2) (-((2 CF TR (-1+x) (4+x (7+4 x)))/(3 x))+4 CF TR (1+x) Log[x])+Lμ (((8 CF TR (-1+x) (13+x (-5+22 x)))/(9 x))-(4/3) CF TR (3+x (9+8 x)) Log[x]+4 CF TR (1+x) ((Log[x])^2))+((8 CF TR (-1+x) (2+x (-1+2 x)) PolyLog[2,1-x])/(3 x));

Q ← (OverBar[Q])

Cpdf[q←qbar,1]=0;
Cpdf[q←qbar,2]=(((CA-2 CF) CF (1+(x^2)) ((Log[x])^3))/(3 (1+x)))+((Log[x])^2) (-((4 (CA-2 CF) CF (1+(x^2)) Log[1-x])/(1+x))-((2 (CA-2 CF) CF (1+(x^2)) Log[1+x])/(1+x)))+(((CA-2 CF) CF Log[x] (-3 (1+x) (3+11 x)+2 (π^2) (1+(x^2))+12 Log[1+x] ((((1+x))^2)+(1+(x^2)) Log[1+x])))/(3 (1+x)))+(-4 (CA-2 CF) CF (-1+x)-((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,1-x]+(4 (CA-2 CF) CF (1+x)+((4 (CA-2 CF) CF (1+(x^2)) Log[x])/(1+x))) PolyLog[2,-x]+Lμ (-((2 (CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (x^2)))/(3 (1+x)))+((2 (CA-2 CF) CF (1+(x^2)) ((Log[x])^2))/(1+x))+((4 (CA-2 CF) CF Log[x] ((((1+x))^2)-2 (1+(x^2)) Log[1+x]))/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[2,-x])/(1+x)))-((4 (CA-2 CF) CF (1+(x^2)) PolyLog[3,-x])/(1+x))-((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,x])/(1+x))+((8 (CA-2 CF) CF (1+(x^2)) PolyLog[3,(x/(1+x))])/(1+x))+(((CA-2 CF) CF (-45+45 (x^2)+(π^2) (((1+x))^2)+2 (1+(x^2)) Log[1+x] ((π^2)-2 ((Log[1+x])^2))-6 (1+(x^2)) Zeta[3]))/(3 (1+x)));

Matching coefficients for TMD PDF (logarithmical part)

Q ← Q

Cpdf[q←q,1,2,0]=Coefficient[Cpdf[q←q,1]/.lζ->0,((Lμ)^2)]
Cpdf[q←q,1,1,1]=Coefficient[Cpdf[q←q,1],Lμ lζ]
Cpdf[q←q,1,1,0]=Simplify@Coefficient[Cpdf[q←q,1]/.lζ->0,Lμ]

Cpdf[q←q,1,2]=Coefficient[Cpdf[q←q,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[q←q,1,1]=Simplify@Coefficient[Cpdf[q←q,1]/.lζ->Lμ,Lμ]

Cpdf[q←q,2,4,0]=Coefficient[Cpdf[q←q,2]/.lζ->0,((Lμ)^4)]
Cpdf[q←q,2,4,1]=Coefficient[Cpdf[q←q,2],((Lμ)^3)lζ]
Cpdf[q←q,2,4,2]=Coefficient[Cpdf[q←q,2],((Lμ)^2) ((lζ)^2)]
Cpdf[q←q,2,3,0]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,3,1]=Collect[Coefficient[Cpdf[q←q,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,3,2]=Collect[Coefficient[Cpdf[q←q,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,2,0]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,2,1]=Collect[Coefficient[Cpdf[q←q,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,2,2]=Collect[Coefficient[Cpdf[q←q,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,1,0]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,1,1]=Collect[Coefficient[Cpdf[q←q,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[q←q,2,4]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,3]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,2]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←q,2,1]=Collect[Coefficient[Cpdf[q←q,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q ← G

Cpdf[q←g,1,2,0]=Coefficient[Cpdf[q←g,1]/.lζ->0,((Lμ)^2)]
Cpdf[q←g,1,1,1]=Coefficient[Cpdf[q←g,1],Lμ lζ]
Cpdf[q←g,1,1,0]=Simplify@Coefficient[Cpdf[q←g,1]/.lζ->0,Lμ]

Cpdf[q←g,1,2]=Coefficient[Cpdf[q←g,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[q←g,1,1]=Simplify@Coefficient[Cpdf[q←g,1]/.lζ->Lμ,Lμ]

Cpdf[q←g,2,4,0]=Coefficient[Cpdf[q←g,2]/.lζ->0,((Lμ)^4)]
Cpdf[q←g,2,4,1]=Coefficient[Cpdf[q←g,2],((Lμ)^3)lζ]
Cpdf[q←g,2,4,2]=Coefficient[Cpdf[q←g,2],((Lμ)^2) ((lζ)^2)]
Cpdf[q←g,2,3,0]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,3,1]=Collect[Coefficient[Cpdf[q←g,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,3,2]=Collect[Coefficient[Cpdf[q←g,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,2,0]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,2,1]=Collect[Coefficient[Cpdf[q←g,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,2,2]=Collect[Coefficient[Cpdf[q←g,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,1,0]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,1,1]=Collect[Coefficient[Cpdf[q←g,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[q←g,2,4]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,3]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,2]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←g,2,1]=Collect[Coefficient[Cpdf[q←g,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

G ← Q

Cpdf[g←q,1,2,0]=Coefficient[Cpdf[g←q,1]/.lζ->0,((Lμ)^2)]
Cpdf[g←q,1,1,1]=Coefficient[Cpdf[g←q,1],Lμ lζ]
Cpdf[g←q,1,1,0]=Simplify@Coefficient[Cpdf[g←q,1]/.lζ->0,Lμ]

Cpdf[g←q,1,2]=Coefficient[Cpdf[g←q,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[g←q,1,1]=Simplify@Coefficient[Cpdf[g←q,1]/.lζ->Lμ,Lμ]

Cpdf[g←q,2,4,0]=Coefficient[Cpdf[g←q,2]/.lζ->0,((Lμ)^4)]
Cpdf[g←q,2,4,1]=Coefficient[Cpdf[g←q,2],((Lμ)^3)lζ]
Cpdf[g←q,2,4,2]=Coefficient[Cpdf[g←q,2],((Lμ)^2) ((lζ)^2)]
Cpdf[g←q,2,3,0]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,3,1]=Collect[Coefficient[Cpdf[g←q,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,3,2]=Collect[Coefficient[Cpdf[g←q,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,2,0]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,2,1]=Collect[Coefficient[Cpdf[g←q,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,2,2]=Collect[Coefficient[Cpdf[g←q,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,1,0]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,1,1]=Collect[Coefficient[Cpdf[g←q,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[g←q,2,4]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,3]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,2]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←q,2,1]=Collect[Coefficient[Cpdf[g←q,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

G ← G

Cpdf[g←g,1,2,0]=Coefficient[Cpdf[g←g,1]/.lζ->0,((Lμ)^2)]
Cpdf[g←g,1,1,1]=Coefficient[Cpdf[g←g,1],Lμ lζ]
Cpdf[g←g,1,1,0]=Simplify@Coefficient[Cpdf[g←g,1]/.lζ->0,Lμ]

Cpdf[g←g,1,2]=Coefficient[Cpdf[g←g,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[g←g,1,1]=Simplify@Coefficient[Cpdf[g←g,1]/.lζ->Lμ,Lμ]

Cpdf[g←g,2,4,0]=Coefficient[Cpdf[g←g,2]/.lζ->0,((Lμ)^4)]
Cpdf[g←g,2,4,1]=Coefficient[Cpdf[g←g,2],((Lμ)^3)lζ]
Cpdf[g←g,2,4,2]=Coefficient[Cpdf[g←g,2],((Lμ)^2) ((lζ)^2)]
Cpdf[g←g,2,3,0]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,3,1]=Collect[Coefficient[Cpdf[g←g,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,3,2]=Collect[Coefficient[Cpdf[g←g,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,2,0]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,2,1]=Collect[Coefficient[Cpdf[g←g,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,2,2]=Collect[Coefficient[Cpdf[g←g,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,1,0]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,1,1]=Collect[Coefficient[Cpdf[g←g,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[g←g,2,4]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,3]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,2]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[g←g,2,1]=Collect[Coefficient[Cpdf[g←g,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q ← Q’

Cpdf[q←qp,1,2,0]=Coefficient[Cpdf[q←qp,1]/.lζ->0,((Lμ)^2)]
Cpdf[q←qp,1,1,1]=Coefficient[Cpdf[q←qp,1],Lμ lζ]
Cpdf[q←qp,1,1,0]=Simplify@Coefficient[Cpdf[q←qp,1]/.lζ->0,Lμ]

Cpdf[q←qp,1,2]=Coefficient[Cpdf[q←qp,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[q←qp,1,1]=Simplify@Coefficient[Cpdf[q←qp,1]/.lζ->Lμ,Lμ]

Cpdf[q←qp,2,4,0]=Coefficient[Cpdf[q←qp,2]/.lζ->0,((Lμ)^4)]
Cpdf[q←qp,2,4,1]=Coefficient[Cpdf[q←qp,2],((Lμ)^3)lζ]
Cpdf[q←qp,2,4,2]=Coefficient[Cpdf[q←qp,2],((Lμ)^2) ((lζ)^2)]
Cpdf[q←qp,2,3,0]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,3,1]=Collect[Coefficient[Cpdf[q←qp,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,3,2]=Collect[Coefficient[Cpdf[q←qp,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,2,0]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,2,1]=Collect[Coefficient[Cpdf[q←qp,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,2,2]=Collect[Coefficient[Cpdf[q←qp,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,1,0]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,1,1]=Collect[Coefficient[Cpdf[q←qp,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[q←qp,2,4]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,3]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,2]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qp,2,1]=Collect[Coefficient[Cpdf[q←qp,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q ← (OverBar[Q])

Cpdf[q←qbar,1,2,0]=Coefficient[Cpdf[q←qbar,1]/.lζ->0,((Lμ)^2)]
Cpdf[q←qbar,1,1,1]=Coefficient[Cpdf[q←qbar,1],Lμ lζ]
Cpdf[q←qbar,1,1,0]=Simplify@Coefficient[Cpdf[q←qbar,1]/.lζ->0,Lμ]

Cpdf[q←qbar,1,2]=Coefficient[Cpdf[q←qbar,1]/.lζ->Lμ,((Lμ)^2)]
Cpdf[q←qbar,1,1]=Simplify@Coefficient[Cpdf[q←qbar,1]/.lζ->Lμ,Lμ]

Cpdf[q←qbar,2,4,0]=Coefficient[Cpdf[q←qbar,2]/.lζ->0,((Lμ)^4)]
Cpdf[q←qbar,2,4,1]=Coefficient[Cpdf[q←qbar,2],((Lμ)^3)lζ]
Cpdf[q←qbar,2,4,2]=Coefficient[Cpdf[q←qbar,2],((Lμ)^2) ((lζ)^2)]
Cpdf[q←qbar,2,3,0]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,3,1]=Collect[Coefficient[Cpdf[q←qbar,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,3,2]=Collect[Coefficient[Cpdf[q←qbar,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,2,0]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,2,1]=Collect[Coefficient[Cpdf[q←qbar,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,2,2]=Collect[Coefficient[Cpdf[q←qbar,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,1,0]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,1,1]=Collect[Coefficient[Cpdf[q←qbar,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cpdf[q←qbar,2,4]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,3]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,2]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cpdf[q←qbar,2,1]=Collect[Coefficient[Cpdf[q←qbar,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Matching coefficients for TMD PDF (finite part)

Q ← Q

Cpdf[q←q,1,0]=Collect[Cpdf[q←q,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[q←q,2,0]=Collect[Cpdf[q←q,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

Q ← G

Cpdf[q←g,1,0]=Collect[Cpdf[q←g,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[q←g,2,0]=Collect[Cpdf[q←g,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

G ← Q

Cpdf[g←q,1,0]=Collect[Cpdf[g←q,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[g←q,2,0]=Collect[Cpdf[g←q,2]/.{Lμ->0,lζ->0,PolyLog[2,2,1/2]->1/720 ((π^4)+30 ((Log[2])^4)-30 Log[8] Zeta[3])},{DELTA[_],CF,CA,TrNf},Simplify]

G ← G

Cpdf[g←g,1,0]=Collect[Cpdf[g←g,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[g←g,2,0]=Collect[Cpdf[g←g,2]/.{Lμ->0,lζ->0,PolyLog[2,2,(1/2)]->(1/720) ((π^4)+30 ((Log[2])^4)-30 Log[8] Zeta[3])},{DELTA[_],CF,CA,TrNf},Simplify]

Q ← Q’

Cpdf[q←qp,1,0]=Collect[Cpdf[q←qp,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[q←qp,2,0]=Collect[Cpdf[q←qp,2]/.{Lμ->0,lζ->0},{TR CF,PolyLog[2,1-x],Log[x]},Simplify]

Q ← (OverBar[Q])

Cpdf[q←qbar,1,0]=Collect[Cpdf[q←qbar,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cpdf[q←qbar,2,0]=Collect[Cpdf[q←qbar,2]/.{Lμ->0,lζ->0},{PolyLog[_,_],Log[x]},Simplify]

Matching coefficients for TMD FF

Q -> Q

Cff[q->q,1]=CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((2 CF (-(((-1+z))^2)+Lμ (1+(z^2))-2 (1+(z^2)) Log[z]))/(-1+z));
Cff[q->q,2]=-((2 (CF^2) ((Lμ)^3) (1+(z^2)))/(-1+z))+((2 (CA-CF) CF (3-3 z+(π^2) (1+(z^2))) Log[1-z])/(3 (-1+z)))+(((2 CF (2 TrNf (9+z (-8+9 z))+CF (69+9 z (-14+9 z)+2 (π^2) (1+(z^2)))+CA (-93+(95-72 z) z+3 (π^2) (1+(z^2)))))/(3 (-1+z)))+4 (CA-7 CF) CF (-1+z) Log[1-z]-((2 (2 CA-3 CF) CF (1+(z^2)) ((Log[1-z])^2))/(-1+z))) Log[z]+(-((CF (CA (37-z (36+23 z))+2 (2 TrNf (1+(z^2))+3 CF (-22+z (34+3 z)))))/(6 (-1+z)))+((2 (2 CA-9 CF) CF (1+(z^2)) Log[1-z])/(-1+z))) ((Log[z])^2)+((CF (-9 CA (1+(z^2))+CF (29+51 (z^2))) ((Log[z])^3))/(3 (-1+z)))+((Lμ)^2) (((4 (CF^2) lζ (1+(z^2)))/(-1+z))+((CF (11 CA (1+(z^2))+2 (9 CF (((-1+z))^2)-2 TrNf (1+(z^2)))))/(3 (-1+z)))-((8 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))+((2 (CF^2) (3+5 (z^2)) Log[z])/(-1+z)))+Lμ (((CF (-16 TrNf (4+z (-3+4 z))-3 CF (84 (((-1+z))^2)+(π^2) (1+(z^2)))+CA (-6 (π^2) (1+(z^2))+4 (77+z (-87+77 z)))))/(9 (-1+z)))+8 (CF^2) (-1+z) Log[1-z]+(((2 CF (6 CF (-5+8 z)+4 TrNf (1+(z^2))-CA (5+17 (z^2))))/(3 (-1+z)))+((24 (CF^2) (1+(z^2)) Log[1-z])/(-1+z))) Log[z]+((2 CF (CA-5 CF+(CA-11 CF) (z^2)) ((Log[z])^2))/(-1+z))+lζ (-4 (CF^2) (-1+z)-((8 (CF^2) (1+(z^2)) Log[z])/(-1+z))))+(4 (CA-6 CF) CF (-1+z)-((4 (CA-CF) CF (1+(z^2)) Log[1-z])/(-1+z))+((8 CF (CA+2 CF) (1+(z^2)) Log[z])/(-1+z))) PolyLog[2,z]-((4 (CA-CF) CF (1+(z^2)) PolyLog[3,1-z])/(-1+z))-((4 CF (3 CA+10 CF) (1+(z^2)) PolyLog[3,z])/(-1+z))+((CF (-9 CA (π^2) (((-1+z))^2)+9 CF (-30+13 (π^2)) (((-1+z))^2)+8 CA (1+z (99+z))-4 TrNf (19+z (18+19 z))-54 (CA-20 CF) (1+(z^2)) Zeta[3]))/(27 (-1+z)))+DELTA[1-z] ((((CF^2) ((Lμ)^4))/2)+((Lμ)^3) (-2 (CF^2) lζ+(2/9) CF (-11 CA+4 TrNf))+((Lμ)^2) (2 (CF^2) ((lζ)^2)+(1/3) CF lζ (11 CA-4 TrNf)+(1/18) CF (-134 CA+3 (2 CA-7 CF) (π^2)+40 TrNf))+Lμ ((1/9) CF (π^2) (-11 CA+4 TrNf)-(1/9) CF lζ (-134 CA+3 (2 CA+CF) (π^2)+40 TrNf))-(2/27) CF lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CF (9 CF (π^4)+2 CA (4856-603 (π^2)+234 (π^4)-2772 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q -> G

Cff[q->g,1]=((2 CF ((z^2)-Lμ (2-2 z+(z^2))+2 (2-2 z+(z^2)) Log[z]))/z);
Cff[q->g,2]=((2 (CF^2) ((Lμ)^3) (2+(-2+z) z))/z)+((2 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^3))/(3 z))-((CF (11 CF (-2+z) z+CA (80+44 z+62 (z^2))) ((Log[z])^3))/(3 z))+((Log[1-z])^2) (2 CF (-CA+CF) z-((2 (5 CA-4 CF) CF (2+(-2+z) z) Log[z])/z))+((Lμ)^2) (-((4 (CF^2) lζ (2+(-2+z) z))/z)+((CF (2 CA (-1+z) (31+z (7+4 z))-3 CF (12+z (-16+9 z))))/(3 z))+((4 CF (CA+CF) (2+(-2+z) z) Log[1-z])/z)-((2 CF (CF (4+3 (-2+z) z)+4 CA (1+z+(z^2))) Log[z])/z))+Log[1-z] (-((2 (CA-CF) CF (3 z+2 (π^2) (2+(-2+z) z)))/(3 z))+((4 CF (6 CF (z^2)+CA (22+z (-24+(15-4 z) z))) Log[z])/(3 z))+((2 (11 CA-4 CF) CF (2+(-2+z) z) ((Log[z])^2))/z))-((2 CA CF (π^2) (2+z (2+z)) Log[1+z])/(3 z))+((4 CA CF (2+z (2+z)) ((Log[1+z])^3))/(3 z))+Log[z] (((CF (-9 CF ((3+z) (-16+13 z)+6 (π^2) (2+(-2+z) z))+2 CA (54+12 (π^2) (2+(-2+z) z)+z (735+z (147+88 z)))))/(9 z))+4 CA CF z Log[1+z])+((Log[z])^2) (((CF (-3 CF z (8+z)+2 CA (-212+3 z (32+z (-3+8 z)))))/(6 z))+((6 CA CF (2+z (2+z)) Log[1+z])/z))+(4 CA CF z+((4 CA CF (2+z (2+z)) Log[z])/z)) PolyLog[2,-z]+(((8 CA CF (11+z (-12+(9-2 z) z)))/(3 z))-((4 (CA-CF) CF (2+(-2+z) z) Log[1-z])/z)+((4 CF (CA (2+(-6+z) z)-8 CF (2+(-2+z) z)) Log[z])/z)) PolyLog[2,z]+Lμ (((CF (3 CF (-24+22 z-8 (z^2)+3 (π^2) (2+(-2+z) z))+4 CA (3 (5+3 (-3+z) z)-2 (π^2) (2+(-1+z) z))))/(3 z))+((4 (CA-CF) CF (2+(-2+z) z) ((Log[1-z])^2))/z)+((2 CF (3 CF (-2+z) z+2 CA (8+z (6+7 z))) ((Log[z])^2))/z)+Log[1-z] (-8 (CF^2) z-((24 CA CF (2+(-2+z) z) Log[z])/z))+lζ (4 (CF^2) z+((8 (CF^2) (2+(-2+z) z) Log[z])/z))+Log[z] (((2 CF (3 CF (12+z (-12+7 z))-2 CA (-62+z (36+z (9+8 z)))))/(3 z))-((8 CA CF (2+z (2+z)) Log[1+z])/z))-((8 CA CF (2+z (2+z)) PolyLog[2,-z])/z)-((16 (CA-CF) CF (2+(-2+z) z) PolyLog[2,z])/z))-((4 (CA-CF) CF (2+(-2+z) z) PolyLog[3,1-z])/z)+((4 CA CF (2+z (2+z)) PolyLog[3,-z])/z)+((4 CF (8 CF (2+(-2+z) z)+CA (14+z (-6+7 z))) PolyLog[3,z])/z)-((8 CA CF (2+z (2+z)) PolyLog[3,(1/(1+z))])/z)+(1/(27 z))CF (27 CF (z (-38-3 (-11+(π^2)) z)-4 (2+(-2+z) z) Zeta[3])+2 CA (1564-558 z-693 (z^2)-340 (z^3)+6 (π^2) (-1+z) (11+z (-1+2 z))-108 (8+z (-9+4 z)) Zeta[3]));

G -> Q

Cff[g->q,1]=-2 TR (2 (-1+z) z+Lμ (1-2 z+2 (z^2))+(-2+4 z-4 (z^2)) Log[z]);
Cff[g->q,2]=2 CA ((Lμ)^3) TR (1+2 (-1+z) z)+(2/9) TR (4 TrNf (5+16 (-1+z) z)-9 CF ((π^2) (1+2 (-1+z) z)-2 (4+9 (-1+z) z))+CA (9 (π^2) (1+2 (-1+z) z)-2 (38+91 (-1+z) z))) Log[1-z]+(1/3) TR (9 CF+4 TrNf+2 (15 CF+4 TrNf) (-1+z) z+CA (-11-34 (-1+z) z)) ((Log[1-z])^2)-(2/3) (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^3)+(1/3) TR (2 CA (11+62 z)-11 CF (1-2 z+4 (z^2))) ((Log[z])^3)+((Lμ)^2) (-4 CA lζ TR (1+2 (-1+z) z)+((TR (z (3 CF (-1+4 z)+8 TrNf (1+2 (-1+z) z))+CA (8-2 z (8+z (-40+47 z)))))/(3 z))+4 (CA+CF) TR (1+2 (-1+z) z) Log[1-z]+(-2 CF TR+4 (6 CA+CF) TR z-8 (CA+CF) TR (z^2)) Log[z])+Log[z] (((2 TR (-3 z (4 TrNf+8 TrNf (-1+z) z+CF (9+30 (-1+z) z))+CA (-8+z (45-90 z+86 (z^2)))) Log[1-z])/(3 z))+2 (2 CA-3 CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)+(1/(9 z))TR (2 CA (24-z (165+(699-38 z) z+18 (π^2) (1+2 (-1+z) z)))+3 z (-8 TrNf (5+8 (-1+z) z)+CF ((π^2) (2+4 (-1+z) z)-3 (8+z (-73+76 z))))+72 CA (z^2) (1+z) Log[1+z]))+((Log[z])^2) (-2 (2 CA-9 CF) TR (1+2 (-1+z) z) Log[1-z]+((TR (4 CA (16+z (-22+(35-59 z) z))+z (8 TrNf (1+2 (-1+z) z)+3 CF (-7-52 z+68 (z^2)))-36 CA z (1+2 z (1+z)) Log[1+z]))/(6 z)))+(8 CA TR z (1+z)-4 CA TR (1+2 z (1+z)) Log[z]) PolyLog[2,-z]+(((8 TR (-1+z) (-12 CF (z^2)+CA (2+z (-1+11 z))))/(3 z))-4 CA TR (5+2 z (-7+5 z)) Log[z]) PolyLog[2,z]+Lμ ((1/(9 z))TR (CA (-24+z (-16+428 z-692 (z^2)+3 (π^2) (9+2 z (-5+9 z))))-2 z (-40 TrNf (1+2 (-1+z) z)+3 CF ((π^2) (2+4 (-1+z) z)-3 (8+z (-9+10 z)))))+(4/3) TR (CA (-11-10 (-1+z) z)+(9 CF+4 TrNf) (1+2 (-1+z) z)) Log[1-z]-4 (CA-CF) TR (1+2 (-1+z) z) ((Log[1-z])^2)-2 TR (-3 CF+6 CF (1-2 z) z+CA (6+28 z)) ((Log[z])^2)+lζ (-8 CA TR (-1+z) z+8 CA TR (1+2 (-1+z) z) Log[z])+Log[z] (-24 CF TR (1+2 (-1+z) z) Log[1-z]+((2 TR (z (CF (9+12 (1-3 z) z)-8 TrNf (1+2 (-1+z) z))+4 CA (-4+z (7+z (-23+33 z)))+12 CA z (1+2 z (1+z)) Log[1+z]))/(3 z)))+8 CA TR (1+2 z (1+z)) PolyLog[2,-z]+16 (CA-CF) TR (1+2 (-1+z) z) PolyLog[2,z])-4 CA TR (1+2 z (1+z)) PolyLog[3,-z]+4 TR (8 CF (1+2 (-1+z) z)+CA (3+2 z (-7+3 z))) PolyLog[3,z]+8 CA TR (1+2 z (1+z)) PolyLog[3,(1/(1+z))]+(1/(27 z))TR (CA (-296+2 z (-1223+z (139+774 z))+3 (π^2) (8+z (21-60 z+76 (z^2))))+z (4 TrNf (-9 (π^2) (1+2 (-1+z) z)+8 (7+17 (-1+z) z))-9 CF ((π^2) (9+14 (-1+z) z)-3 (63+z (-101+56 z))))+18 CA z (1+2 z (1+z)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))-216 z (CA+4 CF-(CA+8 CF) z+2 (CA+4 CF) (z^2)) Zeta[3]);

G -> G

Cff[g->g,1]=CA (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((4 CA (((1-z+(z^2)))^2) (Lμ-2 Log[z]))/((-1+z) z));
Cff[g->g,2]=-((4 (CA^2) ((Lμ)^3) (((1+(-1+z) z))^2))/((-1+z) z))-(2/3) CA (CA-2 TrNf) Log[1-z]+((1/(9 (-1+z) z))(-8 CF TrNf (-1+z) (-6+z (135+z (81+82 z)))+4 CA TrNf (46+z (-91+z (90+z (-71+46 z))))+(CA^2) (-536+60 (π^2) (((1+(-1+z) z))^2)+z (-1123+z (726+z (-407+1072 z)))))-((8 (CA^2) (-1+z) (11+z (-1+11 z)) Log[1-z])/(3 z))+((4 (CA^2) (((1+(-1+z) z))^2) ((Log[1-z])^2))/((-1+z) z))) Log[z]+(((44 CF TrNf (-1+z) z (((1+z))^2)+4 (CA^2) (20+z (11+(-1+z) z (1+z) (-42+11 z)))) ((Log[z])^3))/(3 z (-1+(z^2))))+((Lμ)^2) (((8 (CA^2) lζ (((1+(-1+z) z))^2))/((-1+z) z))+((2 (-4 CA TrNf (((1+(-1+z) z))^2)+2 CF TrNf (-4+z+6 (z^2)+(z^3)-4 (z^4))+(CA^2) (55+z (-102+z (105+z (-102+55 z))))))/(3 (-1+z) z))-((16 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+((8 (2 (CA^2) (1+(((-1+z))^3) z)+CF TrNf z (-1+(z^2))) Log[z])/((-1+z) z)))+((Log[z])^2) (-((28 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))+(1/(3 z (-1+(z^2))))((1+z) (4 CA TrNf (4+(-1+z) z (3+z (-9+4 z)))+2 CF TrNf (-1+z) (16+z (21+z (15+16 z)))+(CA^2) (220-3 z (93+z (-48+43 z))))+36 (CA^2) (-1+z) (((1+z+(z^2)))^2) Log[1+z]))+((8 (CA^2) (((1+z+(z^2)))^2) Log[z] PolyLog[2,-z])/(z (1+z)))+Lμ ((1/(9 z (-1+(z^2))))2 (8 CF TrNf (((-1+z))^2) (1+z) (13+z (-5+22 z))-4 CA TrNf (1+z) (23+z (-42+z (48+z (-42+23 z))))+(CA^2) (z (1+z) (109+z (-84+109 z))-3 (π^2) (1+z (-5+z (1+z (5+z (-1+5 z)))))))-((16 (CA^2) lζ (((1+(-1+z) z))^2) Log[z])/((-1+z) z))-((8 (3 CF TrNf (-1+z) z (((1+z))^2)+(CA^2) (4+z (-1+(-3+z) z) (-3+z (-1+3 z)))) ((Log[z])^2))/(z (-1+(z^2))))+Log[z] (((48 (CA^2) (((1+(-1+z) z))^2) Log[1-z])/((-1+z) z))-((4 (1+z) (-4 CA TrNf z (1+z)+2 CF TrNf (8+3 z (3+z))+(CA^2) (-88+z (47+z (-61+44 z))))+48 (CA^2) (((1+z+(z^2)))^2) Log[1+z])/(3 z (1+z))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[2,-z])/(z (1+z))))+(-((8 (CA^2) (-1+z) (11+z (-1+11 z)))/(3 z))+((8 (CA^2) (7+z (-5+z (7+z (5+z (-7+5 z))))) Log[z])/(z (-1+(z^2))))) PolyLog[2,z]+((8 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,-z])/(z (1+z)))-((8 (CA^2) (15+z (-11+z (15+z (11+z (-15+11 z))))) PolyLog[3,z])/(z (-1+(z^2))))-((16 (CA^2) (((1+z+(z^2)))^2) PolyLog[3,(1/(1+z))])/(z (1+z)))+(1/(27 z (-1+(z^2))))4 ((1+z) (2 CF TrNf (((-1+z))^2) (139+z (409+301 z))+CA TrNf (-139+z (249+z (-276+11 (21-11 z) z)))+(CA^2) (-817+3 (π^2) (((-1+z))^2) (11+z (-1+11 z))+z (1029-2 z (129+z (-555+431 z)))))+9 (CA^2) ((-1+z) (((1+z+(z^2)))^2) Log[1+z] (-(π^2)+2 ((Log[1+z])^2))+6 (9+z (-10+z (9+z (10+z (-9+10 z))))) Zeta[3]))+DELTA[1-z] ((((CA^2) ((Lμ)^4))/2)+((Lμ)^2) (2 (CA^2) ((lζ)^2)-(1/18) CA (CA (134+15 (π^2))-40 TrNf)+(1/3) CA lζ (11 CA-4 TrNf))+((Lμ)^3) (-2 (CA^2) lζ+(2/9) CA (-11 CA+4 TrNf))+Lμ ((1/9) CA (π^2) (-11 CA+4 TrNf)-(1/9) CA lζ (CA (-134+9 (π^2))+40 TrNf))-(2/27) CA lζ (56 TrNf+CA (-202+189 Zeta[3]))+(1/648) CA (CA (9712-1206 (π^2)+477 (π^4)-5544 Zeta[3])+8 TrNf (-328+45 (π^2)+252 Zeta[3])));

Q -> Q’

Cff[q->qp,1]=0;
Cff[q->qp,2]=((2 CF TR (-1+z) (148+z (859+436 z)))/(27 z))-((4 CF TR (-12+z (174+z (51+32 z))) Log[z])/(9 z))-((CF TR (-32+3 z (11+z (11+8 z))) ((Log[z])^2))/(3 z))+(22/3) CF TR (1+z) ((Log[z])^3)+((Lμ)^2) (-((2 CF TR (-1+z) (4+z (7+4 z)))/(3 z))+4 CF TR (1+z) Log[z])+Lμ (-((4 CF TR (-1+z) (-2+z (19+4 z)))/(3 z))+((4 CF TR (-8+z (3+z (15+8 z))) Log[z])/(3 z))-12 CF TR (1+z) ((Log[z])^2))-((8 CF TR (-1+z) (2+z (-1+2 z)) PolyLog[2,1-z])/(3 z));

Q -> (OverBar[Q])

Cff[q->qbar,1]=0;
Cff[q->qbar,2]=-((3 (CA-2 CF) CF (1+(z^2)) ((Log[z])^3))/(1+z))+((2 (CA-2 CF) CF ((Log[z])^2) (-2 (1+z) (2+z)+3 (1+(z^2)) Log[1+z]))/(1+z))+Log[z] (-4 (CA-2 CF) CF (-1+z) Log[1-z]+(CA-2 CF) CF (-19+5 z+4 (1+z) Log[1+z]))+(4 (CA-2 CF) CF (1+z)+((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,-z]+Lμ (-((2 (CA-2 CF) CF (-12+(π^2)+(12+(π^2)) (z^2)))/(3 (1+z)))+((2 (CA-2 CF) CF (1+(z^2)) ((Log[z])^2))/(1+z))+((4 (CA-2 CF) CF Log[z] ((((1+z))^2)-2 (1+(z^2)) Log[1+z]))/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[2,-z])/(1+z)))+(-4 (CA-2 CF) CF (-1+z)-((4 (CA-2 CF) CF (1+(z^2)) Log[z])/(1+z))) PolyLog[2,z]+((4 (CA-2 CF) CF (1+(z^2)) PolyLog[3,-z])/(1+z))+((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,z])/(1+z))-((8 (CA-2 CF) CF (1+(z^2)) PolyLog[3,(1/(1+z))])/(1+z))-(((CA-2 CF) CF (45-45 (z^2)-(π^2) (1+z) (-1+3 z)+2 (1+(z^2)) Log[1+z] ((π^2)-2 ((Log[1+z])^2))-6 (1+(z^2)) Zeta[3]))/(3 (1+z)));

Matching coefficients for TMD FF (logarithmical part)

Q -> Q

Cff[q->q,1,2,0]=Coefficient[Cff[q->q,1]/.lζ->0,((Lμ)^2)]
Cff[q->q,1,1,1]=Coefficient[Cff[q->q,1],Lμ lζ]
Cff[q->q,1,1,0]=Simplify@Coefficient[Cff[q->q,1]/.lζ->0,Lμ]

Cff[q->q,1,2]=Coefficient[Cff[q->q,1]/.lζ->Lμ,((Lμ)^2)]
Cff[q->q,1,1]=Simplify@Coefficient[Cff[q->q,1]/.lζ->Lμ,Lμ]

Cff[q->q,2,4,0]=Coefficient[Cff[q->q,2]/.lζ->0,((Lμ)^4)]
Cff[q->q,2,4,1]=Coefficient[Cff[q->q,2],((Lμ)^3)lζ]
Cff[q->q,2,4,2]=Coefficient[Cff[q->q,2],((Lμ)^2) ((lζ)^2)]
Cff[q->q,2,3,0]=Collect[Coefficient[Cff[q->q,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,3,1]=Collect[Coefficient[Cff[q->q,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,3,2]=Collect[Coefficient[Cff[q->q,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,2,0]=Collect[Coefficient[Cff[q->q,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,2,1]=Collect[Coefficient[Cff[q->q,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,2,2]=Collect[Coefficient[Cff[q->q,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,1,0]=Collect[Coefficient[Cff[q->q,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,1,1]=Collect[Coefficient[Cff[q->q,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[q->q,2,4]=Collect[Coefficient[Cff[q->q,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,3]=Collect[Coefficient[Cff[q->q,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,2]=Collect[Coefficient[Cff[q->q,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->q,2,1]=Collect[Coefficient[Cff[q->q,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q -> G

Cff[q->g,1,2,0]=Coefficient[Cff[q->g,1]/.lζ->0,((Lμ)^2)]
Cff[q->g,1,1,1]=Coefficient[Cff[q->g,1],Lμ lζ]
Cff[q->g,1,1,0]=Simplify@Coefficient[Cff[q->g,1]/.lζ->0,Lμ]

Cff[q->g,1,2]=Coefficient[Cff[q->g,1]/.lζ->Lμ,((Lμ)^2)]
Cff[q->g,1,1]=Simplify@Coefficient[Cff[q->g,1]/.lζ->Lμ,Lμ]

Cff[q->g,2,4,0]=Coefficient[Cff[q->g,2]/.lζ->0,((Lμ)^4)]
Cff[q->g,2,4,1]=Coefficient[Cff[q->g,2],((Lμ)^3)lζ]
Cff[q->g,2,4,2]=Coefficient[Cff[q->g,2],((Lμ)^2) ((lζ)^2)]
Cff[q->g,2,3,0]=Collect[Coefficient[Cff[q->g,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,3,1]=Collect[Coefficient[Cff[q->g,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,3,2]=Collect[Coefficient[Cff[q->g,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,2,0]=Collect[Coefficient[Cff[q->g,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,2,1]=Collect[Coefficient[Cff[q->g,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,2,2]=Collect[Coefficient[Cff[q->g,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,1,0]=Collect[Coefficient[Cff[q->g,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,1,1]=Collect[Coefficient[Cff[q->g,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[q->g,2,4]=Collect[Coefficient[Cff[q->g,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,3]=Collect[Coefficient[Cff[q->g,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,2]=Collect[Coefficient[Cff[q->g,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->g,2,1]=Collect[Coefficient[Cff[q->g,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

G -> Q

Cff[g->q,1,2,0]=Coefficient[Cff[g->q,1]/.lζ->0,((Lμ)^2)]
Cff[g->q,1,1,1]=Coefficient[Cff[g->q,1],Lμ lζ]
Cff[g->q,1,1,0]=Simplify@Coefficient[Cff[g->q,1]/.lζ->0,Lμ]

Cff[g->q,1,2]=Coefficient[Cff[g->q,1]/.lζ->Lμ,((Lμ)^2)]
Cff[g->q,1,1]=Simplify@Coefficient[Cff[g->q,1]/.lζ->Lμ,Lμ]

Cff[g->q,2,4,0]=Coefficient[Cff[g->q,2]/.lζ->0,((Lμ)^4)]
Cff[g->q,2,4,1]=Coefficient[Cff[g->q,2],((Lμ)^3)lζ]
Cff[g->q,2,4,2]=Coefficient[Cff[g->q,2],((Lμ)^2) ((lζ)^2)]
Cff[g->q,2,3,0]=Collect[Coefficient[Cff[g->q,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,3,1]=Collect[Coefficient[Cff[g->q,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,3,2]=Collect[Coefficient[Cff[g->q,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,2,0]=Collect[Coefficient[Cff[g->q,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,2,1]=Collect[Coefficient[Cff[g->q,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,2,2]=Collect[Coefficient[Cff[g->q,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,1,0]=Collect[Coefficient[Cff[g->q,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,1,1]=Collect[Coefficient[Cff[g->q,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[g->q,2,4]=Collect[Coefficient[Cff[g->q,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,3]=Collect[Coefficient[Cff[g->q,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,2]=Collect[Coefficient[Cff[g->q,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->q,2,1]=Collect[Coefficient[Cff[g->q,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

G -> G

Cff[g->g,1,2,0]=Coefficient[Cff[g->g,1]/.lζ->0,((Lμ)^2)]
Cff[g->g,1,1,1]=Coefficient[Cff[g->g,1],Lμ lζ]
Cff[g->g,1,1,0]=Simplify@Coefficient[Cff[g->g,1]/.lζ->0,Lμ]

Cff[g->g,1,2]=Coefficient[Cff[g->g,1]/.lζ->Lμ,((Lμ)^2)]
Cff[g->g,1,1]=Simplify@Coefficient[Cff[g->g,1]/.lζ->Lμ,Lμ]

Cff[g->g,2,4,0]=Coefficient[Cff[g->g,2]/.lζ->0,((Lμ)^4)]
Cff[g->g,2,4,1]=Coefficient[Cff[g->g,2],((Lμ)^3)lζ]
Cff[g->g,2,4,2]=Coefficient[Cff[g->g,2],((Lμ)^2) ((lζ)^2)]
Cff[g->g,2,3,0]=Collect[Coefficient[Cff[g->g,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,3,1]=Collect[Coefficient[Cff[g->g,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,3,2]=Collect[Coefficient[Cff[g->g,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,2,0]=Collect[Coefficient[Cff[g->g,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,2,1]=Collect[Coefficient[Cff[g->g,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,2,2]=Collect[Coefficient[Cff[g->g,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,1,0]=Collect[Coefficient[Cff[g->g,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,1,1]=Collect[Coefficient[Cff[g->g,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[g->g,2,4]=Collect[Coefficient[Cff[g->g,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,3]=Collect[Coefficient[Cff[g->g,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,2]=Collect[Coefficient[Cff[g->g,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[g->g,2,1]=Collect[Coefficient[Cff[g->g,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q -> Q’

Cff[q->qp,1,2,0]=Coefficient[Cff[q->qp,1]/.lζ->0,((Lμ)^2)]
Cff[q->qp,1,1,1]=Coefficient[Cff[q->qp,1],Lμ lζ]
Cff[q->qp,1,1,0]=Simplify@Coefficient[Cff[q->qp,1]/.lζ->0,Lμ]

Cff[q->qp,1,2]=Coefficient[Cff[q->qp,1]/.lζ->Lμ,((Lμ)^2)]
Cff[q->qp,1,1]=Simplify@Coefficient[Cff[q->qp,1]/.lζ->Lμ,Lμ]

Cff[q->qp,2,4,0]=Coefficient[Cff[q->qp,2]/.lζ->0,((Lμ)^4)]
Cff[q->qp,2,4,1]=Coefficient[Cff[q->qp,2],((Lμ)^3)lζ]
Cff[q->qp,2,4,2]=Coefficient[Cff[q->qp,2],((Lμ)^2) ((lζ)^2)]
Cff[q->qp,2,3,0]=Collect[Coefficient[Cff[q->qp,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,3,1]=Collect[Coefficient[Cff[q->qp,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,3,2]=Collect[Coefficient[Cff[q->qp,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,2,0]=Collect[Coefficient[Cff[q->qp,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,2,1]=Collect[Coefficient[Cff[q->qp,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,2,2]=Collect[Coefficient[Cff[q->qp,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,1,0]=Collect[Coefficient[Cff[q->qp,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,1,1]=Collect[Coefficient[Cff[q->qp,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[q->qp,2,4]=Collect[Coefficient[Cff[q->qp,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,3]=Collect[Coefficient[Cff[q->qp,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,2]=Collect[Coefficient[Cff[q->qp,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qp,2,1]=Collect[Coefficient[Cff[q->qp,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Q -> (OverBar[Q])

Cff[q->qbar,1,2,0]=Coefficient[Cff[q->qbar,1]/.lζ->0,((Lμ)^2)]
Cff[q->qbar,1,1,1]=Coefficient[Cff[q->qbar,1],Lμ lζ]
Cff[q->qbar,1,1,0]=Simplify@Coefficient[Cff[q->qbar,1]/.lζ->0,Lμ]

Cff[q->qbar,1,2]=Coefficient[Cff[q->qbar,1]/.lζ->Lμ,((Lμ)^2)]
Cff[q->qbar,1,1]=Simplify@Coefficient[Cff[q->qbar,1]/.lζ->Lμ,Lμ]

Cff[q->qbar,2,4,0]=Coefficient[Cff[q->qbar,2]/.lζ->0,((Lμ)^4)]
Cff[q->qbar,2,4,1]=Coefficient[Cff[q->qbar,2],((Lμ)^3)lζ]
Cff[q->qbar,2,4,2]=Coefficient[Cff[q->qbar,2],((Lμ)^2) ((lζ)^2)]
Cff[q->qbar,2,3,0]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->0,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,3,1]=Collect[Coefficient[Cff[q->qbar,2],((Lμ)^2)lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,3,2]=Collect[Coefficient[Cff[q->qbar,2],Lμ ((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,2,0]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->0,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,2,1]=Collect[Coefficient[Cff[q->qbar,2],Lμ lζ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,2,2]=Collect[Coefficient[Cff[q->qbar,2]/.Lμ->0,((lζ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,1,0]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->0,Lμ ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,1,1]=Collect[Coefficient[Cff[q->qbar,2]/.Lμ->0,lζ ],{DELTA[_],CF,CA,TrNf},Simplify]

Cff[q->qbar,2,4]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->Lμ,((Lμ)^4) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,3]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->Lμ,((Lμ)^3) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,2]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->Lμ,((Lμ)^2) ],{DELTA[_],CF,CA,TrNf},Simplify]
Cff[q->qbar,2,1]=Collect[Coefficient[Cff[q->qbar,2]/.lζ->Lμ,((Lμ)^1) ],{DELTA[_],CF,CA,TrNf},Simplify]

Matching coefficients for TMD FF (finite part)

Q -> Q

Cff[q->q,1,0]=Collect[Cff[q->q,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[q->q,2,0]=Collect[Cff[q->q,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

Q -> G

Cff[q->g,1,0]=Collect[Cff[q->g,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[q->g,2,0]=Collect[Cff[q->g,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

G -> Q

Cff[g->q,1,0]=Collect[Cff[g->q,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[g->q,2,0]=Collect[Cff[g->q,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

G -> G

Cff[g->g,1,0]=Collect[Cff[g->g,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[g->g,2,0]=Collect[Cff[g->g,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

Q -> Q’

Cff[q->qp,1,0]=Collect[Cff[q->qp,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[q->qp,2,0]=Collect[Cff[q->qp,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]

Q -> (OverBar[Q])

Cff[q->qbar,1,0]=Collect[Cff[q->qbar,1]/.{lζ->0,Lμ->0},DELTA[_],Simplify]

Cff[q->qbar,2,0]=Collect[Cff[q->qbar,2]/.{Lμ->0,lζ->0},{DELTA[_],CF,CA,TrNf},Simplify]