D. Gutierrez-Reyes, I.Scimemi, A.Vladimirov
Matching coefficients for transversity TMD distributions at NNLO
based on arXiv:18??.????

Abstract:
This file contains the results of calculation of transversity and pretzelosity TMD PDF, and the small - b matching coefficients. The calculation for transversity TMD FF and the corresponding small - b matching coefficients are also included.
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.Unsubstracted TMD PDFs
	4.Unsubstracted TMD FFs
            5.Integrated PDFs
	6.Integrated FFs
	7. TMD PDFs
	8. TMD FFs
	9. Matching of TMD PDFs
	10. 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:
Through all the file the following convention for plus-distributions is used:
Only the singular at x->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;
Z2[2]=CF  (((CA+(CF/2))/(ϵ^2))+((-((17 CA)/4)+((3 CF)/4)+TrNf)/ϵ));

Zq[1]=-CF((2/(ϵ^2))+((4+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)))+(1/(108 ϵ))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]));

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])));

Clear[S2]

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 (((4 x)/((-1+x) ϵ))+((2 ((Lμ)^2) x ϵ)/(-1+x))+(((6+(-12+(π^2)) x+6 (x^2)) ϵ)/(3 (-1+x)))+((2 ((Lμ)^3) x (ϵ^2))/(3 (-1+x)))+Lμ (((4 x)/(-1+x))+(((6+(-12+(π^2)) x+6 (x^2)) (ϵ^2))/(3 (-1+x))))+(((ϵ^2) (6+6 (x^2)+4 x (-3+Zeta[3])))/(3 (-1+x)))+DELTA[1-x] (((4 Log[δp])/ϵ)+2 ((Lμ)^2) ϵ Log[δp]+(1/3) (π^2) ϵ Log[δp]+(2/3) ((Lμ)^3) (ϵ^2) Log[δp]+Lμ (4 Log[δp]+(1/3) (π^2) (ϵ^2) Log[δp])+(4/3) (ϵ^2) Log[δp] Zeta[3]));
Φ[q←q,2]=CF((1/(27 (-1+x)))4 TrNf (9+x (-74-6 Lμ (10+3 Lμ)+9 x)-3 x Log[x] (10+12 Lμ+3 Log[x])-2 (28+30 Lμ+9 ((Lμ)^2)) (-1+x) DELTA[1-x] Log[δp])+(1/(ϵ^2))(((CA (-44 x+(-1+x) DELTA[1-x] ((π^2)+Log[4096]-44 Log[δp])))/(6 (-1+x)))+(8/3) TrNf ((x/(-1+x))+DELTA[1-x] Log[δp])+((4 CF (6 x (-2 Log[1-x]+Log[x]+2 Log[δp])-(-1+x) DELTA[1-x] ((π^2)-6 ((Log[δp])^2))))/(3 (-1+x))))+(1/ϵ)((8/9) TrNf (-((x (5+3 Log[x]))/(-1+x))-5 DELTA[1-x] Log[δp])+(1/(3 (-1+x)))2 CF (3 ((((-1+x))^2)+(-3+8 Lμ) x Log[x]-4 x Log[1-x] (4 Lμ+Log[x])+16 Lμ x Log[δp])-4 Lμ (-1+x) DELTA[1-x] ((π^2)-6 ((Log[δp])^2)))+(1/(18 (-1+x)))CA (-2 (9+x (-152+6 (π^2)+9 x)-6 x Log[x] (11+3 Log[x]))+(-1+x) DELTA[1-x] (6 Lμ (π^2)+72 (1+Lμ) Log[2]+4 (67-3 (π^2)) Log[δp]-81 Zeta[3])))-(1/(9 (-1+x)))CF (2 (12 ((Lμ)^2)+(π^2)) (-1+x) DELTA[1-x] ((π^2)-6 ((Log[δp])^2))+3 (-12 (-1+Lμ) (((-1+x))^2)+8 x ((π^2)+6 x) ArcTanh[1-2 x]+36 x ((Log[1-x])^2) Log[x]+3 x Log[x] (4 (6+(3-4 Lμ) Lμ)+3 Log[x])+6 Log[1-x] (3-7 x+16 ((Lμ)^2) x+2 x Log[x] (4 Lμ+Log[x]))-8 (3+x (-6+12 ((Lμ)^2)+(π^2)+3 x)) Log[δp]+24 x ((Log[1-x]+3 Log[x]) PolyLog[2,x]+PolyLog[3,1-x]-5 PolyLog[3,x]+5 Zeta[3])))+(1/(216 (-1+x)))CA (16 (-63+99 ((Lμ)^2) x+(530-63 x) x-3 Lμ (9+x (-152+6 (π^2)+9 x)))+24 (-6 (3+(-3+2 (π^2)) x) Log[1-x]+72 x ((Log[1-x])^2) Log[x]+x Log[x] (4 (38+33 Lμ)+3 Log[x] (11+12 Lμ+2 Log[x]))+36 x (2 (Log[1-x]+Log[x]) PolyLog[2,x]+2 PolyLog[3,1-x]-4 PolyLog[3,x]-3 Zeta[3]))-(-1+x) DELTA[1-x] (-16 Log[δp] (404+3 Lμ (134+33 Lμ-6 (π^2))-378 Zeta[3])+9 (5 (π^4)+26 ((Log[2])^4)-8 (π^2) Log[2] (1+Log[4])-8 ((Lμ)^2) ((π^2)+Log[4096])-48 (Log[16]-8 PolyLog[4,(1/2)]+5 PolyLog[2,2,(1/2)])+306 Log[2] Zeta[3]+24 Lμ (-8 Log[2]+9 Zeta[3])))));

Q ← (OverBar[Q])

Φ[q←qbar,0]=0;
Φ[q←qbar,1]=0;
Φ[q←qbar,2]=(((CA-2 CF) CF (-3+2 (π^2) x+3 (x^2)-6 x ((Log[x])^2)+24 x Log[x] Log[1+x]+24 x PolyLog[2,-x]))/(3 (1+x) ϵ))+(1/(3 (1+x)))(CA-2 CF) CF (-3-6 Lμ+4 Lμ (π^2) x+3 (x^2)+6 Lμ (x^2)-4 (π^2) x Log[x]-12 Lμ x ((Log[x])^2)+24 x Log[1-x] ((Log[x])^2)-2 x ((Log[x])^3)-4 (π^2) x Log[1+x]+48 Lμ x Log[x] Log[1+x]+12 x ((Log[x])^2) Log[1+x]-24 x Log[x] ((Log[1+x])^2)+8 x ((Log[1+x])^3)+24 x Log[x] PolyLog[2,1-x]+24 x (2 Lμ-Log[x]) PolyLog[2,-x]+24 x PolyLog[3,-x]+48 x PolyLog[3,x]-48 x PolyLog[3,(x/(1+x))]+12 x Zeta[3]);

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(Lμ (((4 z)/(-1+z))+4 DELTA[1-z] Log[δp])+((((4 z)/(-1+z))+4 DELTA[1-z] Log[δp])/ϵ)+ϵ (((Lμ)^2) (((2 z)/(-1+z))+2 DELTA[1-z] Log[δp])+((6+(-12+(π^2)) z+6 (z^2)+(π^2) (-1+z) DELTA[1-z] Log[δp])/(3 (-1+z))))+(ϵ^2) (-2+(2/3) ((Lμ)^3) ((z/(-1+z))+DELTA[1-z] Log[δp])+(1/3) Lμ (((6+((π^2)+6 (-2+z)) z)/(-1+z))+(π^2) DELTA[1-z] Log[δp])+(4/3) DELTA[1-z] Log[δp] Zeta[3]+z (2+((4 Zeta[3])/(3 (-1+z))))));
Δ[q->q,2]=CF (((4 TrNf (9-2 (37+30 Lμ+9 ((Lμ)^2)) z+9 (z^2)+6 (5+6 Lμ) z Log[z]-9 z ((Log[z])^2)))/(27 (-1+z)))+((((22 CA z)/(3-3 z))+((8 TrNf z)/(3 (-1+z)))+((8 CF z (-2 Log[1-z]+Log[z]+2 Log[δp]))/(-1+z)))/(ϵ^2))+(1/ϵ)(((8 TrNf z (-5+3 Log[z]))/(9 (-1+z)))+(1/(3 (-1+z)))2 CF (3-6 z-2 (π^2) z+3 (z^2)-6 z ArcTanh[1-2 z]+3 z Log[1-z]-48 Lμ z Log[1-z]+6 z Log[z]+24 Lμ z Log[z]+24 z Log[1-z] Log[z]-12 z ((Log[z])^2)+48 Lμ z Log[δp]+12 z PolyLog[2,1-z]+12 z PolyLog[2,z])+(1/(9-9 z))CA (9-152 z+9 (z^2)+66 z Log[z]+36 z Log[1-z] Log[z]-18 z ((Log[z])^2)+36 z PolyLog[2,1-z]+36 z PolyLog[2,z]))+(1/(3 (-1+z)))CF (-12+12 Lμ+24 z-24 Lμ z-8 Lμ (π^2) z-12 (z^2)+12 Lμ (z^2)-12 z ArcTanh[1-2 z]-24 Lμ z ArcTanh[1-2 z]-12 z ((ArcTanh[1-2 z])^2)-24 Log[1-z]+48 z Log[1-z]+12 Lμ z Log[1-z]-96 ((Lμ)^2) z Log[1-z]-12 (π^2) z Log[1-z]-18 (z^2) Log[1-z]+3 z ((Log[1-z])^2)+24 z Log[z]+24 Lμ z Log[z]+48 ((Lμ)^2) z Log[z]-28 (π^2) z Log[z]+18 (z^2) Log[z]-6 z Log[1-z] Log[z]+96 Lμ z Log[1-z] Log[z]+36 z ((Log[1-z])^2) Log[z]-6 z ((Log[z])^2)-48 Lμ z ((Log[z])^2)-36 z Log[1-z] ((Log[z])^2)+16 z ((Log[z])^3)+24 Log[δp]-48 z Log[δp]+96 ((Lμ)^2) z Log[δp]+8 (π^2) z Log[δp]+24 (z^2) Log[δp]+24 z (2 Lμ+Log[z]) PolyLog[2,1-z]+24 z (2 Lμ+Log[1-z]-3 Log[z]) PolyLog[2,z]+24 z PolyLog[3,1-z]+144 z PolyLog[3,z]-144 z Zeta[3])-(1/(27 (-1+z)))CA (126+54 Lμ-1060 z-912 Lμ z-198 ((Lμ)^2) z+126 (z^2)+54 Lμ (z^2)-54 z Log[1-z]-36 (π^2) z Log[1-z]+54 (z^2) Log[1-z]+510 z Log[z]+396 Lμ z Log[z]-90 (π^2) z Log[z]-54 (z^2) Log[z]+216 Lμ z Log[1-z] Log[z]+216 z ((Log[1-z])^2) Log[z]-99 z ((Log[z])^2)-108 Lμ z ((Log[z])^2)-108 z Log[1-z] ((Log[z])^2)+54 z ((Log[z])^3)+108 z (2 Lμ+Log[z]) PolyLog[2,1-z]+108 z (2 Lμ+2 Log[1-z]-3 Log[z]) PolyLog[2,z]+216 z PolyLog[3,1-z]+648 z PolyLog[3,z]+108 z Zeta[3])+DELTA[1-z] (-(8/27) (28+30 Lμ+9 ((Lμ)^2)) TrNf Log[δp]+(((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)))/(ϵ^2))+(1/ϵ)(-(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)))+(1/36) CF (-96 ((Lμ)^2) (π^2)-7 (π^4)+30 ((Log[2])^4)+48 (12 ((Lμ)^2)+(π^2)) ((Log[δp])^2)-720 PolyLog[2,2,(1/2)]-90 Log[2] Zeta[3])+CA ((2/27) Log[δp] (404+99 ((Lμ)^2)-6 Lμ (-67+3 (π^2))-378 Zeta[3])+(1/24) (11 (π^4)+192 Log[2]-26 ((Log[2])^4)+8 (π^2) Log[2] (1+Log[4])+8 ((Lμ)^2) ((π^2)+Log[4096])-384 PolyLog[4,(1/2)]+240 PolyLog[2,2,(1/2)]+24 Lμ (Log[256]-9 Zeta[3])-306 Log[2] Zeta[3]))));

Q ->  (OverBar[Q])

Δ[q->qbar,0]=0;
Δ[q->qbar,1]=0;
Δ[q->qbar,2]=((2 (CA-2 CF) CF z ((Log[z])^3))/(1+z))+((4 (CA-2 CF) CF (π^2) z Log[1+z])/(3 (1+z)))-((8 (CA-2 CF) CF z ((Log[1+z])^3))/(3 (1+z)))+((Log[z])^2) (-((4 (CA-2 CF) CF Lμ z)/(1+z))+((4 (CA-2 CF) CF z Log[1+z])/(1+z)))+Log[z] (((4 (CA-2 CF) CF (π^2) z)/(3 (1+z)))+((16 (CA-2 CF) CF Lμ z Log[1+z])/(1+z)))+(((16 (CA-2 CF) CF Lμ z)/(1+z))+((8 (CA-2 CF) CF z Log[z])/(1+z))) PolyLog[2,-z]+(((((CA-2 CF) CF (-3+2 (π^2) z+3 (z^2)))/(3 (1+z)))-((2 (CA-2 CF) CF z ((Log[z])^2))/(1+z))+((8 (CA-2 CF) CF z Log[z] Log[1+z])/(1+z))+((8 (CA-2 CF) CF z PolyLog[2,-z])/(1+z)))/ϵ)+((8 (CA-2 CF) CF z Log[z] PolyLog[2,z])/(1+z))-((8 (CA-2 CF) CF z PolyLog[3,-z])/(1+z))-((16 (CA-2 CF) CF z PolyLog[3,z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[3,(1/(1+z))])/(1+z))+(((CA-2 CF) CF (-3+Lμ (-6+4 (π^2) z+6 (z^2))+3 z (z-4 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]=((-2CF)/ϵ)(((2x)/(1-x))+DELTA[1-x](3/2));
f[q←q,2]=(1/(ϵ^2))(CF (TrNf (((8 x)/(3 (-1+x)))-2 DELTA[1-x])+(11/2) CA (((4 x)/(3-3 x))+DELTA[1-x]))+(CF^2) (((9/2)-((4 (π^2))/3)) DELTA[1-x]-((4 x (3+4 Log[1-x]-2 Log[x]))/(-1+x))))+(1/ϵ)((CF^2) (2 (-1+x)-((2 x (3+4 Log[1-x]) Log[x])/(-1+x))+DELTA[1-x] (-(3/4)+(π^2)-12 Zeta[3]))+CF ((1/9) TrNf ((3+4 (π^2)) DELTA[1-x]-((8 x (5+3 Log[x]))/(-1+x)))+CA (((-9+(152-6 (π^2)-9 x) x+6 x Log[x] (11+3 Log[x]))/(9 (-1+x)))+DELTA[1-x] (-(17/12)-((11 (π^2))/9)+6 Zeta[3]))));

Q ← (OverBar[Q])

f[q←qbar,1]=0;
f[q←qbar,2]=-((2 CF (-(CA/2)+CF) (-1+x+((2 x ((π^2)-3 ((Log[x])^2)+12 Log[x] Log[1+x]+12 PolyLog[2,-x]))/(3 (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 (2z))/((1-z) ϵ))-((3 DELTA[1-z])/ϵ));
d[q->q,2]=((CF (((22 CA z)/(3-3 z))+((8 TrNf z)/(3 (-1+z))))+((CF^2) ((9/2)-((4 (π^2))/3))+CF (((11 CA)/2)-2 TrNf)) DELTA[1-z]-((4 (CF^2) z (3+4 Log[1-z]-2 Log[z]))/(-1+z)))/(ϵ^2))+(1/ϵ)(((2 (CF^2) ((((-1+z))^2)+z (3+4 Log[1-z]) Log[z]-4 z ((Log[z])^2)))/(-1+z))+CF (-((8 TrNf z (5+3 Log[z]))/(9 (-1+z)))+((CA (9-152 z+6 (π^2) z+9 (z^2)-66 z Log[z]-18 z ((Log[z])^2)))/(9-9 z)))+DELTA[1-z] ((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 -> (OverBar[Q])

d[q->qbar,1]=0;
d[q->qbar,2]=-((2 CF (-(CA/2)+CF) (-1+z+((2 z ((π^2)-3 ((Log[z])^2)+12 Log[z] Log[1+z]+12 PolyLog[2,-z]))/(3 (1+z)))))/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]=((4 CF Lμ x)/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x]+((((4 CF x)/(-1+x))-3 CF DELTA[1-x])/ϵ)+ϵ (((CF (6+(-12+6 ((Lμ)^2)+(π^2)) x+6 (x^2)))/(3 (-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) (((20 (CF^2) x)/(3-3 x))+(2/9) CF (-11 CA-9 CF (-1+lζ)+4 TrNf) DELTA[1-x])+((2 CF (3 CF (-3+7 x)+CA (-3+(3-2 (π^2)) x)) Log[1-x])/(3 (-1+x)))+(-((8 CF (-19 CA+27 CF+5 TrNf) x)/(9 (-1+x)))+((4 (2 CA-3 CF) CF x ((Log[1-x])^2))/(-1+x))) Log[x]+(-((CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((4 (CF^2) x Log[1-x])/(-1+x))) ((Log[x])^2)+((2 CA CF x ((Log[x])^3))/(3 (-1+x)))+((-((2 CF (11 CA+18 CF-4 TrNf) x)/(3 (-1+x)))+(1/6) CF (33 CA+CF (27-8 (π^2))-12 TrNf) DELTA[1-x]-((16 (CF^2) x Log[1-x])/(-1+x))+((8 (CF^2) x Log[x])/(-1+x)))/(ϵ^2))+((Lμ)^2) (((2 CF (11 CA+9 CF (-1+2 lζ)-4 TrNf) x)/(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]-((32 (CF^2) x Log[1-x])/(-1+x))+((16 (CF^2) x Log[x])/(-1+x)))+Lμ (-((2 CF (40 TrNf x+9 CF (-2+(4+(π^2)-2 x) x)+CA (9+x (-152+6 (π^2)+9 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]+(-((4 CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((16 (CF^2) x Log[1-x])/(-1+x))) Log[x]+((4 CA CF x ((Log[x])^2))/(-1+x)))+(((8 (CA-CF) CF x Log[1-x])/(-1+x))+((8 (CA-3 CF) CF x Log[x])/(-1+x))) PolyLog[2,x]+((8 (CA-CF) CF x PolyLog[3,1-x])/(-1+x))+((8 CF (-2 CA+5 CF) x PolyLog[3,x])/(-1+x))-(1/(27 (-1+x)))CF (9 CF (30+((3-2 lζ) (π^2)+30 (-2+x)) x)-4 TrNf (9+x (-74+9 x))+2 CA (63+x (-530+63 x))+72 CF x ((π^2)+6 x) ArcTanh[1-2 x]+36 (9 CA+34 CF) x 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])+CA (9712+9696 lζ-1206 (π^2)+27 (π^4)-270 ((Log[2])^4)+6480 PolyLog[2,2,(1/2)]-18 (308+504 lζ-45 Log[2]) Zeta[3]))+(1/ϵ)(((CF (-40 TrNf x+6 CF (3-(6+(π^2)) x+3 (x^2))-CA (9+x (-152+6 (π^2)+9 x))))/(9 (-1+x)))+((Lμ)^2) (-((4 (CF^2) x)/(-1+x))+3 (CF^2) DELTA[1-x])+(-((2 CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((8 (CF^2) x Log[1-x])/(-1+x))) Log[x]+((2 CA CF x ((Log[x])^2))/(-1+x))+Lμ (((4 (CF^2) (-3+2 lζ) x)/(-1+x))-(2/3) (CF^2) (9 lζ+4 (π^2)) DELTA[1-x]-((32 (CF^2) x Log[1-x])/(-1+x))+((16 (CF^2) x 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 ←  (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 (-3+2 (π^2) x+3 (x^2)-6 x ((Log[x])^2)+24 x Log[x] Log[1+x]+24 x PolyLog[2,-x]))/(3 (1+x) ϵ))+(1/(3 (1+x)))(CA-2 CF) CF (-3-6 Lμ+4 Lμ (π^2) x+3 (x^2)+6 Lμ (x^2)-4 (π^2) x Log[x]-12 Lμ x ((Log[x])^2)+24 x Log[1-x] ((Log[x])^2)-2 x ((Log[x])^3)-4 (π^2) x Log[1+x]+48 Lμ x Log[x] Log[1+x]+12 x ((Log[x])^2) Log[1+x]-24 x Log[x] ((Log[1+x])^2)+8 x ((Log[1+x])^3)+24 x Log[x] PolyLog[2,1-x]+24 x (2 Lμ-Log[x]) PolyLog[2,-x]+24 x PolyLog[3,-x]+48 x PolyLog[3,x]-48 x PolyLog[3,(x/(1+x))]+12 x Zeta[3]);

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]=((4 CF Lμ z)/(-1+z))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-z]+((((4 CF z)/(-1+z))-3 CF DELTA[1-z])/ϵ)+ϵ (((CF (6+(-12+6 ((Lμ)^2)+(π^2)) z+6 (z^2)))/(3 (-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) (((20 (CF^2) z)/(3-3 z))+(2/9) CF (-11 CA-9 CF (-1+lζ)+4 TrNf) DELTA[1-z])+((2 CF (CA (3+2 (π^2)-3 z) z-3 CF (4+z (-8+2 (π^2)+3 z))) Log[1-z])/(3 (-1+z)))+(((CF^2) z ((Log[1-z])^2))/(-1+z))+(((2 CF z (20 TrNf+CA (-85+15 (π^2)+9 z)+CF (36-42 (π^2)+27 z)))/(9 (-1+z)))-((2 (CF^2) z Log[1-z])/(-1+z))+((4 CF (-2 CA+3 CF) z ((Log[1-z])^2))/(-1+z))) Log[z]+(-((CF (-11 CA+6 CF+4 TrNf) z)/(3 (-1+z)))+((4 (CA-3 CF) CF z Log[1-z])/(-1+z))) ((Log[z])^2)+((2 CF (-3 CA+8 CF) z ((Log[z])^3))/(3 (-1+z)))+((-((2 CF (11 CA+18 CF-4 TrNf) z)/(3 (-1+z)))+(1/6) CF (33 CA+CF (27-8 (π^2))-12 TrNf) DELTA[1-z]-((16 (CF^2) z Log[1-z])/(-1+z))+((8 (CF^2) z Log[z])/(-1+z)))/(ϵ^2))+((Lμ)^2) (((2 CF (11 CA+9 CF (-1+2 lζ)-4 TrNf) z)/(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]-((32 (CF^2) z Log[1-z])/(-1+z))+((16 (CF^2) z Log[z])/(-1+z)))+((4 CF (-CA+2 CF) z Log[z] PolyLog[2,1-z])/(-1+z))+(((8 CF (-CA+CF) z Log[1-z])/(-1+z))+((12 (CA-2 CF) CF z Log[z])/(-1+z))) PolyLog[2,z]+Lμ (-((2 CF (40 TrNf z+3 CF (-6+(12+7 (π^2)-6 z) z)+CA (9+z (-152+9 z))+36 CF z ArcTanh[1-2 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]+((4 (CF^2) z Log[1-z])/(-1+z))+(((4 CF (-11 CA+6 CF+4 TrNf) z)/(3 (-1+z)))+((8 CF (-CA+4 CF) z Log[1-z])/(-1+z))) Log[z]+((4 (CA-4 CF) CF z ((Log[z])^2))/(-1+z))+((8 CF (-CA+2 CF) z PolyLog[2,1-z])/(-1+z))+((8 CF (-CA+2 CF) z PolyLog[2,z])/(-1+z)))+((8 CF (-CA+CF) z PolyLog[3,1-z])/(-1+z))+((24 CF (-CA+2 CF) z PolyLog[3,z])/(-1+z))-(1/(27 (-1+z)))CF (9 CF (30+((3-2 lζ) (π^2)+30 (-2+z)) z)-4 TrNf (9+z (-74+9 z))+2 CA (63+z (-530+63 z))+36 z (3 CF ArcTanh[1-2 z] (1+ArcTanh[1-2 z])+(3 CA+40 CF) Zeta[3]))+(1/648) CF DELTA[1-z] (8 TrNf (-328-336 lζ+45 (π^2)+252 Zeta[3])+CA (9712+9696 lζ-1206 (π^2)+459 (π^4)-270 ((Log[2])^4)+6480 PolyLog[2,2,(1/2)]-18 (308+504 lζ-45 Log[2]) Zeta[3])-9 CF (36 lζ (π^2)+13 (π^4)-60 ((Log[2])^4)+1440 PolyLog[2,2,(1/2)]+36 (-8+Log[32]) Zeta[3]))+(1/ϵ)(((CF (-40 TrNf z+CA (-9+(152-9 z) z)+18 CF (1-(2+(π^2)) z+(z^2))-36 CF z ArcTanh[1-2 z]))/(9 (-1+z)))+((Lμ)^2) (-((4 (CF^2) z)/(-1+z))+3 (CF^2) DELTA[1-z])+((2 (CF^2) z Log[1-z])/(-1+z))+(((2 CF (-11 CA+6 CF+4 TrNf) z)/(3 (-1+z)))+((4 CF (-CA+4 CF) z Log[1-z])/(-1+z))) Log[z]+((2 (CA-4 CF) CF z ((Log[z])^2))/(-1+z))+Lμ (((4 (CF^2) (-3+2 lζ) z)/(-1+z))-(2/3) (CF^2) (9 lζ+4 (π^2)) DELTA[1-z]-((32 (CF^2) z Log[1-z])/(-1+z))+((16 (CF^2) z Log[z])/(-1+z)))+((4 CF (-CA+2 CF) z PolyLog[2,1-z])/(-1+z))+((4 CF (-CA+2 CF) z PolyLog[2,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 -> (OverBar[Q])

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

DD[q->qbar,1]=0;
DD[q->qbar,2]=((2 (CA-2 CF) CF z ((Log[z])^3))/(1+z))+((4 (CA-2 CF) CF (π^2) z Log[1+z])/(3 (1+z)))-((8 (CA-2 CF) CF z ((Log[1+z])^3))/(3 (1+z)))+((Log[z])^2) (-((4 (CA-2 CF) CF Lμ z)/(1+z))+((4 (CA-2 CF) CF z Log[1+z])/(1+z)))+Log[z] (((4 (CA-2 CF) CF (π^2) z)/(3 (1+z)))+((16 (CA-2 CF) CF Lμ z Log[1+z])/(1+z)))+(((16 (CA-2 CF) CF Lμ z)/(1+z))+((8 (CA-2 CF) CF z Log[z])/(1+z))) PolyLog[2,-z]+(((((CA-2 CF) CF (-3+2 (π^2) z+3 (z^2)))/(3 (1+z)))-((2 (CA-2 CF) CF z ((Log[z])^2))/(1+z))+((8 (CA-2 CF) CF z Log[z] Log[1+z])/(1+z))+((8 (CA-2 CF) CF z PolyLog[2,-z])/(1+z)))/ϵ)+((8 (CA-2 CF) CF z Log[z] PolyLog[2,z])/(1+z))-((8 (CA-2 CF) CF z PolyLog[3,-z])/(1+z))-((16 (CA-2 CF) CF z PolyLog[3,z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[3,(1/(1+z))])/(1+z))+(((CA-2 CF) CF (-3+Lμ (-6+4 (π^2) z+6 (z^2))+3 z (z-4 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)((((1/18) (-108 lζ Lμ+54 ((Lμ)^2)+9 (π^2)-48 Lμ (π^2)) DELTA[1-x]-((2 (18 Lμ x-12 lζ Lμ x+6 ((Lμ)^2) x+(π^2) x+48 Lμ x Log[1-x]-24 Lμ x Log[x]))/(3 (-1+x))))/ϵ)+(1/18) DELTA[1-x] (-54 lζ ((Lμ)^2)+36 ((Lμ)^3)-9 lζ (π^2)+18 Lμ (π^2)-24 ((Lμ)^2) (π^2)-4 (π^4)+72 Zeta[3])+(1/(3 (-1+x)))(-18+36 x-18 ((Lμ)^2) x+12 lζ ((Lμ)^2) x-8 ((Lμ)^3) x-3 (π^2) x+2 lζ (π^2) x-4 Lμ (π^2) x-18 (x^2)-24 Log[1-x]+48 x Log[1-x]-48 ((Lμ)^2) x Log[1-x]-8 (π^2) x Log[1-x]-24 (x^2) Log[1-x]-24 x Log[x]+24 ((Lμ)^2) x Log[x]+4 (π^2) x Log[x]+24 (x^2) Log[x]-16 x 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]=((4 CF Lμ x)/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[q←q,2]=-((4 (CF^2) ((Lμ)^3) x)/(-1+x))+((2 CF (CA (-3+(3-2 (π^2)) x)+CF (3+x (-3+4 (π^2)+12 x))) Log[1-x])/(3 (-1+x)))+(-((4 CF x (-38 CA+10 TrNf+3 CF (12+(π^2)+6 x)))/(9 (-1+x)))+((4 (2 CA-3 CF) CF x ((Log[1-x])^2))/(-1+x))) Log[x]+(-((CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((4 (CF^2) x Log[1-x])/(-1+x))) ((Log[x])^2)+((2 CA CF x ((Log[x])^3))/(3 (-1+x)))+((Lμ)^2) (((8 (CF^2) lζ x)/(-1+x))+((2 CF (11 CA-4 TrNf) x)/(3 (-1+x)))-((16 (CF^2) x Log[1-x])/(-1+x))+((8 (CF^2) x Log[x])/(-1+x)))+Lμ (-((2 CF (40 TrNf x+3 CF (-6+(12+(π^2)-6 x) x)+CA (9+x (-152+6 (π^2)+9 x))))/(9 (-1+x)))+(-((4 CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((16 (CF^2) x Log[1-x])/(-1+x))) Log[x]+((4 CA CF x ((Log[x])^2))/(-1+x)))+(((8 (CA-CF) CF x Log[1-x])/(-1+x))+((8 (CA-3 CF) CF x Log[x])/(-1+x))) PolyLog[2,x]+((8 (CA-CF) CF x PolyLog[3,1-x])/(-1+x))+((8 CF (-2 CA+5 CF) x PolyLog[3,x])/(-1+x))-(1/(27 (-1+x)))2 CF (36 CF x ((π^2)+6 x) ArcTanh[1-2 x]+CA (63+x (-530+63 x+162 Zeta[3]))+2 (TrNf (-9+(74-9 x) x)+27 CF ((((-1+x))^2)+10 x Zeta[3])))+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)+8 TrNf (-328+45 (π^2)+252 Zeta[3])+CA (9712-1206 (π^2)+27 (π^4)-270 ((Log[2])^4)+6480 PolyLog[2,2,(1/2)]+18 (-308+45 Log[2]) Zeta[3])));

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]=-((2 (CA-2 CF) CF x ((Log[x])^3))/(3 (1+x)))+((Log[x])^2) (((8 (CA-2 CF) CF x Log[1-x])/(1+x))+((4 (CA-2 CF) CF x Log[1+x])/(1+x)))-((4 (CA-2 CF) CF x Log[x] ((π^2)+6 ((Log[1+x])^2)))/(3 (1+x)))+((8 (CA-2 CF) CF x Log[x] PolyLog[2,1-x])/(1+x))-((8 (CA-2 CF) CF x Log[x] PolyLog[2,-x])/(1+x))+Lμ (((2 (CA-2 CF) CF (-3+2 (π^2) x+3 (x^2)))/(3 (1+x)))-((4 (CA-2 CF) CF x ((Log[x])^2))/(1+x))+((16 (CA-2 CF) CF x Log[x] Log[1+x])/(1+x))+((16 (CA-2 CF) CF x PolyLog[2,-x])/(1+x)))+((8 (CA-2 CF) CF x PolyLog[3,-x])/(1+x))+((16 (CA-2 CF) CF x PolyLog[3,x])/(1+x))-((16 (CA-2 CF) CF x PolyLog[3,(x/(1+x))])/(1+x))+(((CA-2 CF) CF (-4 (π^2) x Log[1+x]+8 x ((Log[1+x])^3)+3 (-1+(x^2)+4 x 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=((1/ϵ)((((-12960 lζ Lμ+6480 ((Lμ)^2)+1080 (π^2)-5760 Lμ (π^2)) DELTA[1-z])/2160)-(1/(3 (-1+z)))2 (18 Lμ z-12 lζ Lμ z+6 ((Lμ)^2) z+(π^2) z+48 Lμ z Log[1-z]-36 z Log[z]-24 Lμ z Log[z]-48 z Log[1-z] Log[z]+24 z ((Log[z])^2)))+((DELTA[1-z] (-6480 lζ ((Lμ)^2)+4320 ((Lμ)^3)-1080 lζ (π^2)+2160 Lμ (π^2)-2880 ((Lμ)^2) (π^2)-480 (π^4)+8640 Zeta[3]))/2160)+(1/(180 (-1+z)))(-1080+2160 z-1080 ((Lμ)^2) z+720 lζ ((Lμ)^2) z-480 ((Lμ)^3) z-180 (π^2) z+120 lζ (π^2) z-240 Lμ (π^2) z-1080 (z^2)-1440 Log[1-z]+2880 z Log[1-z]-2880 ((Lμ)^2) z Log[1-z]-480 (π^2) z Log[1-z]-1440 (z^2) Log[1-z]-1440 z Log[z]+2880 lζ Lμ z Log[z]-1920 (π^2) z Log[z]+1440 (z^2) Log[z]-5760 Lμ z Log[1-z] Log[z]+4320 z ((Log[z])^2)+2880 Lμ z ((Log[z])^2)+5760 z Log[1-z] ((Log[z])^2)-3840 z ((Log[z])^3)-11520 z Log[z] PolyLog[2,z]+23040 z PolyLog[3,z]-24000 z Zeta[3]))(CF^2);

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}/.{ArcTanh[1-2 z]-> (1/2)(Log[1-z]-Log[z])}/.{PolyLog[2,1-z]-> -PolyLog[2,z]+((π^2)/6)-Log[z]Log[1-z]}/.{PolyLog[2,2,(1/2)]->(1/720) ((π^4)+30 ((Log[2])^4)-30 Log[8] Zeta[3])},{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]+((4 CF z (Lμ-2 Log[z]))/(-1+z));
Cff[q->q,2]=-((4 (CF^2) ((Lμ)^3) z)/(-1+z))+((2 (CA-CF) CF (3+2 (π^2)-3 z) z Log[1-z])/(3 (-1+z)))+(((2 CF (20 TrNf z+CF (-6+(39+4 (π^2)-9 z) z)+CA (3+z (-79+6 (π^2)+6 z))))/(3 (-1+z)))+((4 CF (-2 CA+3 CF) z ((Log[1-z])^2))/(-1+z))) Log[z]+(-((CF (-11 CA+45 CF+4 TrNf) z)/(3 (-1+z)))+((4 (2 CA-9 CF) CF z Log[1-z])/(-1+z))) ((Log[z])^2)+((2 CF (-9 CA+40 CF) z ((Log[z])^3))/(3 (-1+z)))+((Lμ)^2) (((8 (CF^2) lζ z)/(-1+z))+((2 CF (11 CA-4 TrNf) z)/(3 (-1+z)))-((16 (CF^2) z Log[1-z])/(-1+z))+((16 (CF^2) z Log[z])/(-1+z)))+Lμ (-((2 CF (40 TrNf z+3 CF (-6+(12+(π^2)-6 z) z)+CA (9+z (-152+6 (π^2)+9 z))))/(9 (-1+z)))-((16 (CF^2) lζ z Log[z])/(-1+z))+(((4 CF (-11 CA+9 CF+4 TrNf) z)/(3 (-1+z)))+((48 (CF^2) z Log[1-z])/(-1+z))) Log[z]+((4 (CA-8 CF) CF z ((Log[z])^2))/(-1+z)))+(((8 CF (-CA+CF) z Log[1-z])/(-1+z))+((16 CF (CA+2 CF) z Log[z])/(-1+z))) PolyLog[2,z]+((8 CF (-CA+CF) z PolyLog[3,1-z])/(-1+z))-((8 CF (3 CA+10 CF) z PolyLog[3,z])/(-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])))-(1/(27 (-1+z)))2 CF (CA (63+z (-530+63 z+54 Zeta[3]))+2 (TrNf (-9+(74-9 z) z)+27 CF ((((-1+z))^2)-20 z Zeta[3])));

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]=-2 (CA-2 CF) CF (-1+z) Log[z]+((6 (CA-2 CF) CF z ((Log[z])^3))/(1+z))+((12 CF (-CA+2 CF) z ((Log[z])^2) Log[1+z])/(1+z))+((8 CF (-CA+2 CF) z Log[z] PolyLog[2,-z])/(1+z))+Lμ (((2 (CA-2 CF) CF (-3+2 (π^2) z+3 (z^2)))/(3 (1+z)))+((4 CF (-CA+2 CF) z ((Log[z])^2))/(1+z))+((16 (CA-2 CF) CF z Log[z] Log[1+z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[2,-z])/(1+z)))+((8 (CA-2 CF) CF z Log[z] PolyLog[2,z])/(1+z))+((8 CF (-CA+2 CF) z PolyLog[3,-z])/(1+z))+((16 CF (-CA+2 CF) z PolyLog[3,z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[3,(1/(1+z))])/(1+z))+(((CA-2 CF) CF (4 (π^2) z Log[1+z]-8 z ((Log[1+z])^3)+3 (-1+(z^2)-4 z 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 FF 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]=((4 CF Lμ x)/(-1+x))+CF (2 lζ Lμ-((Lμ)^2)-((π^2)/6)) DELTA[1-x];
Cpdf[q←q,2]=-((4 (CF^2) ((Lμ)^3) x)/(-1+x))+((2 CF (CA (-3+(3-2 (π^2)) x)+CF (3+x (-3+4 (π^2)+12 x))) Log[1-x])/(3 (-1+x)))+(-((4 CF x (-38 CA+10 TrNf+3 CF (12+(π^2)+6 x)))/(9 (-1+x)))+((4 (2 CA-3 CF) CF x ((Log[1-x])^2))/(-1+x))) Log[x]+(-((CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((4 (CF^2) x Log[1-x])/(-1+x))) ((Log[x])^2)+((2 CA CF x ((Log[x])^3))/(3 (-1+x)))+((Lμ)^2) (((8 (CF^2) lζ x)/(-1+x))+((2 CF (11 CA-4 TrNf) x)/(3 (-1+x)))-((16 (CF^2) x Log[1-x])/(-1+x))+((8 (CF^2) x Log[x])/(-1+x)))+Lμ (-((2 CF (40 TrNf x+3 CF (-6+(12+(π^2)-6 x) x)+CA (9+x (-152+6 (π^2)+9 x))))/(9 (-1+x)))+(-((4 CF (-11 CA+9 CF+4 TrNf) x)/(3 (-1+x)))-((16 (CF^2) x Log[1-x])/(-1+x))) Log[x]+((4 CA CF x ((Log[x])^2))/(-1+x)))+(((8 (CA-CF) CF x Log[1-x])/(-1+x))+((8 (CA-3 CF) CF x Log[x])/(-1+x))) PolyLog[2,x]+((8 (CA-CF) CF x PolyLog[3,1-x])/(-1+x))+((8 CF (-2 CA+5 CF) x PolyLog[3,x])/(-1+x))-(1/(27 (-1+x)))2 CF (36 CF x ((π^2)+6 x) ArcTanh[1-2 x]+CA (63+x (-530+63 x+162 Zeta[3]))+2 (TrNf (-9+(74-9 x) x)+27 CF ((((-1+x))^2)+10 x Zeta[3])))+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)+8 TrNf (-328+45 (π^2)+252 Zeta[3])+CA (9712-1206 (π^2)+27 (π^4)-270 ((Log[2])^4)+6480 PolyLog[2,2,(1/2)]+18 (-308+45 Log[2]) Zeta[3])));

Q ← (OverBar[Q])

Cpdf[q←qbar,1]=0;
Cpdf[q←qbar,2]=-((2 (CA-2 CF) CF x ((Log[x])^3))/(3 (1+x)))+((Log[x])^2) (((8 (CA-2 CF) CF x Log[1-x])/(1+x))+((4 (CA-2 CF) CF x Log[1+x])/(1+x)))-((4 (CA-2 CF) CF x Log[x] ((π^2)+6 ((Log[1+x])^2)))/(3 (1+x)))+((8 (CA-2 CF) CF x Log[x] PolyLog[2,1-x])/(1+x))-((8 (CA-2 CF) CF x Log[x] PolyLog[2,-x])/(1+x))+Lμ (((2 (CA-2 CF) CF (-3+2 (π^2) x+3 (x^2)))/(3 (1+x)))-((4 (CA-2 CF) CF x ((Log[x])^2))/(1+x))+((16 (CA-2 CF) CF x Log[x] Log[1+x])/(1+x))+((16 (CA-2 CF) CF x PolyLog[2,-x])/(1+x)))+((8 (CA-2 CF) CF x PolyLog[3,-x])/(1+x))+((16 (CA-2 CF) CF x PolyLog[3,x])/(1+x))-((16 (CA-2 CF) CF x PolyLog[3,(x/(1+x))])/(1+x))+(((CA-2 CF) CF (-4 (π^2) x Log[1+x]+8 x ((Log[1+x])^3)+3 (-1+(x^2)+4 x 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 ← (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 ← (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]+((4 CF z (Lμ-2 Log[z]))/(-1+z));
Cff[q->q,2]=-((4 (CF^2) ((Lμ)^3) z)/(-1+z))+((2 (CA-CF) CF (3+2 (π^2)-3 z) z Log[1-z])/(3 (-1+z)))+(((2 CF (20 TrNf z+CF (-6+(39+4 (π^2)-9 z) z)+CA (3+z (-79+6 (π^2)+6 z))))/(3 (-1+z)))+((4 CF (-2 CA+3 CF) z ((Log[1-z])^2))/(-1+z))) Log[z]+(-((CF (-11 CA+45 CF+4 TrNf) z)/(3 (-1+z)))+((4 (2 CA-9 CF) CF z Log[1-z])/(-1+z))) ((Log[z])^2)+((2 CF (-9 CA+40 CF) z ((Log[z])^3))/(3 (-1+z)))+((Lμ)^2) (((8 (CF^2) lζ z)/(-1+z))+((2 CF (11 CA-4 TrNf) z)/(3 (-1+z)))-((16 (CF^2) z Log[1-z])/(-1+z))+((16 (CF^2) z Log[z])/(-1+z)))+Lμ (-((2 CF (40 TrNf z+3 CF (-6+(12+(π^2)-6 z) z)+CA (9+z (-152+6 (π^2)+9 z))))/(9 (-1+z)))-((16 (CF^2) lζ z Log[z])/(-1+z))+(((4 CF (-11 CA+9 CF+4 TrNf) z)/(3 (-1+z)))+((48 (CF^2) z Log[1-z])/(-1+z))) Log[z]+((4 (CA-8 CF) CF z ((Log[z])^2))/(-1+z)))+(((8 CF (-CA+CF) z Log[1-z])/(-1+z))+((16 CF (CA+2 CF) z Log[z])/(-1+z))) PolyLog[2,z]+((8 CF (-CA+CF) z PolyLog[3,1-z])/(-1+z))-((8 CF (3 CA+10 CF) z PolyLog[3,z])/(-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])))-(1/(27 (-1+z)))2 CF (CA (63+z (-530+63 z+54 Zeta[3]))+2 (TrNf (-9+(74-9 z) z)+27 CF ((((-1+z))^2)-20 z Zeta[3])));

Q -> (OverBar[Q])

Cff[q->qbar,1]=0;
Cff[q->qbar,2]=-2 (CA-2 CF) CF (-1+z) Log[z]+((6 (CA-2 CF) CF z ((Log[z])^3))/(1+z))+((12 CF (-CA+2 CF) z ((Log[z])^2) Log[1+z])/(1+z))+((8 CF (-CA+2 CF) z Log[z] PolyLog[2,-z])/(1+z))+Lμ (((2 (CA-2 CF) CF (-3+2 (π^2) z+3 (z^2)))/(3 (1+z)))+((4 CF (-CA+2 CF) z ((Log[z])^2))/(1+z))+((16 (CA-2 CF) CF z Log[z] Log[1+z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[2,-z])/(1+z)))+((8 (CA-2 CF) CF z Log[z] PolyLog[2,z])/(1+z))+((8 CF (-CA+2 CF) z PolyLog[3,-z])/(1+z))+((16 CF (-CA+2 CF) z PolyLog[3,z])/(1+z))+((16 (CA-2 CF) CF z PolyLog[3,(1/(1+z))])/(1+z))+(((CA-2 CF) CF (4 (π^2) z Log[1+z]-8 z ((Log[1+z])^3)+3 (-1+(z^2)-4 z 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 -> (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 -> (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]