(* ::Package:: *)

(* Dominik Kara, "Corrections of Order \alpha \alpha_s to W Boson Decays *)
(* TTP13-026, arxiv:1307.7190 *)

(* This file is designed to present the main formulas of the mentioned publication. *)

(* Definitions
alpha = fine-structure constant
alphas = strong coupling constant
Mw = W boson mass
nc = number of colors in QCD
sw^2 = 1 - cw^2 (sine squared of the weak mixing angle)
CKM = CKM matrix element Vqq' corresponding to the decay W \[Rule] qq'
ep = parameter of dimensional regularization in D = 4 - 2*ep dimensions
mu = Renormalization scale
x = q^2/Mw^2, where q is the external momentum of the W boson two-point function
gq = (Iq3 - sw^2*Qq)/(sw*cw) and gq' = (Iq'3 - sw^2*Qq')/(sw*cw) are the couplings of the Z boson to the quarks q and q',
	where Iq3 = 1/2 (Iq'3 = -1/2) is the third component of the quarks' weak isospin and Qq = 2/3 (Qq' = -1/3) is their charge.
*)


(* Born decay width (Eqs. (6,7)) *)

Gamma0 = Mw*alpha*nc/12/sw^2*CKM^2;

Gamma0ep = Gamma0*(
			1 + ep*(5/3 + Log[mu^2/Mw^2] - Log[x])
			);


(* NLO QCD Corrections (Eqs. (8,9)) *)

GammaQCD = Gamma0*alphas/Pi;

GammaQCDep = Gamma0*alphas/Pi*(
			1 + ep*(55/6 - 8*Zeta[3] + 2*Log[mu^2/Mw^2] - 2*Log[x])
			);


(* NLO Electroweak Corrections (Eqs. (12,22-24,26-31)) *)

s1a = - 7/6 - 1/4*1/ep - 1/2*Log[mu^2/Mw^2] + 1/2*Log[x];

s1b = cw/sw* (gq - gq')*(
	+ 9/4 + 3/2*1/ep + 3*Log[mu^2/Mw^2] - 3/2*Log[x]
	- 5/18*x
	- 5/144*x^2
	- 23/4200*x^3
	- 73/75600*x^4
	- 53/291060*x^5
	- 145/4036032*x^6
	- 19/2594592*x^7
	- 241/157528800*x^8
	- 149/457271100*x^9
	- 19/269549280*x^{10}
	);

s1c = 1/sw^2*(
	- 7/24 - 1/4*1/ep - 1/2*Log[mu^2/Mw^2] + 1/4*Log[x]
	);

s1d = (gq^2 + gq'^2)*sw^2*s1c;

s1e = gq*gq'*(
	+ 7/12 + 1/2*1/ep + Log[mu^2/Mw^2] - 1/2*Log[x]
	+ (11/18 - 1/3*Log[x])*x
	+ (-13/144 + 1/12*Log[x])*x^2
	+ (47/1800 - 1/30*Log[x])*x^3
	+ (-37/3600 + 1/60*Log[x])*x^4
	+ (107/22050 - 1/105*Log[x])*x^5
	+ (-73/28224 + 1/168*Log[x])*x^6
	+ (191/127008 - 1/252*Log[x])*x^7
	+ (-121/129600 + 1/360*Log[x])*x^8
	+ (299/490050 - 1/495*Log[x])*x^9
	+ (-181/435600 + 1/660*Log[x])*x^{10}
	);

s1f = (
	+ 15/4 + 3/2*1/ep + 3*Log[mu^2/Mw^2] - 3/2*Log[x]
	+ (-109/96 + 1/2*Log[x])*x
	+ (-647/1440 + 1/3*Log[x])*x^2
	+ (-349/1440 + 1/4*Log[x])*x^3
	+ (-2549/16800 + 1/5*Log[x])*x^4
	+ (-419/16800 + 1/6*Log[x])*x^5
	+ (-6403/84672 + 1/7*Log[x])*x^6
	+ (-1159/20160 + 1/8*Log[x])*x^7
	+ (-1171/25920 + 1/9*Log[x])*x^8
	+ (-787/21600 + 1/10*Log[x])*x^9
	+ (-4531/151008 + 1/11*Log[x])*x^{10}
	);

s1g = (
	- 1/16*x^2
	- 19/180*x^3
	- 37/336*x^4
	- 59/560*x^5
	- 85/864*x^6
	- 23/252*x^7
	- 149/1760*x^8
	- 17/216*x^9
	- 229/3120*x^{10}
	);

Gamma1f = Gamma0*alpha/Pi*(
			- 2*Zeta[2] + 317/72 +3/2*1/ep + 3*Log[mu^2/Mw^2]
			);

P1f[5,5] = (- 0.047*x^5 + 1.046*x^4 - 6.245*x^3 + 15.004*x^2 - 15.674*x + 5.939)/
			(- 3.04*10^{-3}*x^5 + 0.092*x^4 - 0.680*x^3 + 1.939*x^2 - 2.336*x +1);

t1f = (
	- 44/9 + 1/6*1/ep + 1/3*Log[mu^2/Mw^2] - 17/6*Log[x]
	+ (43/24 + 7/4*1/ep + 7/2*Log[mu^2/Mw^2] - 51/8*Log[x])*1/x
	+ (161/72 - 5/12*1/ep - 5/6*Log[mu^2/Mw^2] + 7/24*Log[x])*1/x^2
	+ (101/288 - 1/3*Log[x])*1/x^3
	+ (49/160 - 1/4*Log[x])*1/x^4
	+ (1559/7200 - 1/5*Log[x])*1/x^5
	+ (317/2016 - 1/6*Log[x])*1/x^6
	+ (1859/15680 - 1/7*Log[x])*1/x^7
	+ (1117/12096 - 1/8*Log[x])*1/x^8
	+ (13403/181440 - 1/9*Log[x])*1/x^9
	+ (1063/17600 - 1/10*Log[x])*1/x^{10}
	);

P1gh[5,5] = (- 0.010*x^5 - 0.085*x^4 + 0.504*x^3 + 0.092*x^2 - 2.001*x + 1.573)/
			(1.04*10^{-3}*x^5 + 0.036*x^4 - 0.459*x^3 + 1.611*x^2 - 2.174*x +1);

Gamma1g = Gamma0*alpha/Pi*(
			29/9 + 1/2*1/ep + Log[mu^2/Mw^2]
			);

t1g = (
	+ 689/72 + 11/12*1/ep + 11/6*Log[mu^2/Mw^2] - 11/6*Log[x]
	+ (-169/24 - 11/4*1/ep - 11/2*Log[mu^2/Mw^2] + 11/2*Log[x])*1/x
	+ (-361/72 + 5/6*1/ep + 5/3*Log[mu^2/Mw^2] - 5/3*Log[x])*1/x^2
	- 15/16*1/x^3
	- 19/20*1/x^4
	- 37/48*1/x^5
	- 177/288*1/x^6
	- 17/32*1/x^7
	- 115/252*1/x^8
	- 447/1220*1/x^9
	- 17/48*1/x^{10}
	);

Gamma1h = Gamma0ep*alpha/Pi*(
		+ 131/72 + 19/24*1/ep + 19/24*Log[mu^2/Mw^2] - 1 - 1/2*1/ep - 1/2*Log[mu^2/Mw^2]
		);
	
S1h = Gamma0ep*alpha/Pi*(
	+ 143/144 + 19/24*1/ep
	+ 43/48*x
	+ 15/32*x^2
	+ 23/72*x^3
	- 163/672*x^4
	+ 219/1120*x^5
	+ 283/1728*x^6
	+ 71/504*x^7
	+ 87/704*x^8
	+ 523/4752*x^9
	+ 619/6240*x^{10}
	);
	
T1h = Gamma0ep*alpha/Pi*(
	+ 43/36 + 19/24*1/ep + 19/24*Log[mu^2/Mw^2] - 19/24*Log[x]
	+ (13/16 -5/8*Log[x]) *1/x
	+ (8/9 +5/12*Log[x])*1/x^2
	+ 3/32*1/x^3
	+ 19/120*1/x^4
	+ 43/288*1/x^5
	+ 15/112*1/x^6
	+ 23/192*1/x^7
	+ 163/1512*1/x^8
	+ 219/2240*1/x^9
	+ 283/3168*1/x^{10}
	);


(* NNLO Mixed QCD/Electroweak Corrections (Eqs. (45,52-54,56-59,62-63)) *)

s2a = (
	- 433/144 + 2*Zeta[3] - 1/8*1/ep - 3/8*Log[mu^2/Mw^2] + 3/8*Log[x]
	);
   
s2b = cw/sw*(gq - gq')*(
	+ 103/8 - 12*Zeta[3] + 5/4*1/ep + 15/4*Log[mu^2/Mw^2] - 11/4*Log[x]
	- 7/18*x
	- 89/2160*x^2
	- 29/4725*x^3
	- 53/50400*x^4
	- 851/4365900*x^5
	- 1381/36324288*x^6
	- 349/45405360*x^7
	- 3017/1890345600*x^8
	- 4177/12346319700*x^9
	);
   
s2c = 1/sw^2*(
	- 89/48 + 2*Zeta[3] - 1/8*1/ep - 3/8*Log[mu^2/Mw^2] + 3/8*Log[x]
	+ (- 121/243 + 22/81*Log[x] - 1/27*Log[x]^2)*x
	+ (- 169/15552 + 13/1296*Log[x] - 1/432*Log[x]^2)*x^2
	+ (- 2209/2430000 + 47/40500*Log[x] - 1/2700*Log[x]^2)*x^3  
	+ (- 1369/9720000 + 37/162000*Log[x] - 1/10800*Log[x]^2)*x^4
	+ (- 11449/364651875 + 214/3472875*Log[x] - 1/33075*Log[x]^2)*x^5
	+ (- 5329/597445632 + 73/3556224*Log[x] - 1/84672*Log[x]^2)*x^6
	+ (- 36481/12098274048 + 191/24004512*Log[x] - 1/190512*Log[x]^2)*x^7
	+ (- 14641/12597120000 + 121/34992000*Log[x] - 1/388800*Log[x]^2)*x^8
	+ (- 89401/180111751875 + 598/363862125*Log[x] - 1/735075*Log[x]^2)*x^9
	);

s2d = (gq^2 + gq'^2)*sw^2*s2c;

s2e = gq*gq'*(
	+ 89/24 - 4*Zeta[3] + 1/4*1/ep + 3/4*Log[mu^2/Mw^2] - 3/4*Log[x]
	+ (71/27 - 4/3*Zeta[3] - 2/3*Log[x] - 1/3*Log[x]^2)*x
	+ (- 1159/2592 + 1/3*Zeta[3] + 5/6*Log[x] - 7/36*Log[x]^2)*x^2
	+ (- 1853/10800 - 2/15*Zeta[3] - 13/90*Log[x])*x^3
	+ (146179/972000 + 1/15*Zeta[3] + 83/810*Log[x] - 11/180*Log[x]^2)*x^4
	+ (- 78941/595350) - 4/105*Zeta[3] - 37/2835*x^5
	+ (514328497/5334336000 + 1/42*Zeta[3] + 221/12960*Log[x] - 143/5040*Log[x]^2)*x^6
	+ (- 262758413/3429216000 - 1/63*Zeta[3] + 799/136080*Log[x] - 1/1080*Log[x]^2)*x^7
	+ (13907067061/240045120000 + 1/90*Zeta[3] + 853/6804000*Log[x] - 403/25200*Log[x]^2)*x^8
	+ (- 42384988309/907670610000 - 4/495*Zeta[3] + 19058/2338875*Log[x] - 11/9450*Log[x]^2)*x^9
	);

s2f = (
	+ 111/8 + 5/4*1/ep - 12*Zeta[3] + 15/4*Log[mu^2/Mw^2] - 11/4*Log[x]
	+ (- 1157/864 + 5/9*Log[x])*x
	+ (- 2071/4320 + 25/72*Log[x])*x^2
	+ (- 5423/21600 + 23/90*Log[x])*x^3
	+ (- 23459/151200 + 73/360*Log[x])*x^4
	+ (- 223399/2116800 + 53/315*Log[x])*x^5
	+ (- 1619/21168 + 145/1008*Log[x])*x^6
	+ (- 220961/3810240 + 95/756*Log[x])*x^7
	+ (- 8843/194400 + 241/2160*Log[x])*x^8
	+ (- 861827/23522400 + 149/1485*Log[x])*x^9
	);

s2g = (
	- 1/16*x^2
	- 19/180*x^3
	- 37/336*x^4
	- 59/560*x^5
	- 85/864*x^6
	- 23/252*x^7
	- 149/1760*x^8
	- 17/216*x^9
	);

P2f[5,4] = (- 1.37*10^{-4}*x^5 + 9.11*10^{-3}*x^4 - 0.056*x^3 + 0.090*x^2 - 0.014*x - 0.033)/
			(0.026*x^4 - 0.349*x^3 + 1.371*x^2 - 2.025*x +1);

t2f = (
	+ 77/24 - 20/3*Zeta[3] + 7/12*1/ep + 7/4*Log[mu^2/Mw^2] - 17/4*Log[x]
	+ (325/48 - 7*Zeta[3] + 7/8*1/ep + 21/8*Log[mu^2/Mw^2] - 5*Log[x])*1/x
	+ (- 61/432 + 5/3*Zeta[3] - 5/24*1/ep - 5/8*Log[mu^2/Mw^2] + 1/36*Log[x])*1/x^2
	+ (151/864 - 25/72*Log[x])*1/x^3
	+ (3817/21600 - 23/90*Log[x])*1/x^4
	+ (2773/21600 - 73/360*Log[x])*1/x^5
	+ (99823/1058400 - 53/315*Log[x])*1/x^6
	+ (15149/211680 - 145/1008*Log[x])*1/x^7
	+ (42647/762048 - 95/756*Log[x])*1/x^8
	+ (15271/340200 - 241/2160*Log[x])*1/x^9
	);

t2g = (
	+ 159/16 - 11/3*Zeta[3] + 11/24*1/ep + 11/8*Log[mu^2/Mw^2] - 11/8*Log[x]
	+ (- 217/16 + 11*Zeta[3] - 11/8*1/ep - 33/8*Log[mu^2/Mw^2] + 33/8*Log[x])*1/x
	+ (- 17/24 - 10/3*Zeta[3] + 5/12*1/ep + 5/4*Log[mu^2/Mw^2] - 5/4*Log[x])*1/x^2
	- 5/8*1/x^3
	- 19/30*1/x^4
	- 37/72*1/x^5
	- 59/140*1/x^6
	- 17/48*1/x^7
	- 115/378*1/x^8
	- 149/560*1/x^9
	);

Gamma2h = GammaQCDep*alpha/Pi*(
		+ 131/72 + 19/24*1/ep + 19/24*Log[mu^2/Mw^2] - 1 - 1/2*1/ep - 1/2*Log[mu^2/Mw^2]
		);

S2h = GammaQCDep*alpha/Pi*(
	+ 143/144 + 19/24*1/ep
	+ 43/48*x
	+ 15/32*x^2
	+ 23/72*x^3
	- 163/672*x^4
	+ 219/1120*x^5
	+ 283/1728*x^6
	+ 71/504*x^7
	+ 87/704*x^8
	+ 523/4752*x^9
	+ 619/6240*x^{10}
	);

T2h = GammaQCDep*alpha/Pi*(
	+ 43/36 + 19/24*1/ep + 19/24*Log[mu^2/Mw^2] - 19/24*Log[x]
	+ (13/16 -5/8*Log[x]) *1/x
	+ (8/9 +5/12*Log[x])*1/x^2
	+ 3/32*1/x^3
	+ 19/120*1/x^4
	+ 43/288*1/x^5
	+ 15/112*1/x^6
	+ 23/192*1/x^7
	+ 163/1512*1/x^8
	+ 219/2240*1/x^9
	+ 283/3168*1/x^{10}
	);

P2gh[5,4] = (- 2.38*10^{-4}*x^5 - 2.57*10^{-3}*x^4 + 0.024*x^3 - 0.032*x^2 - 0.024*x +0.039)/
			(0.027*x^4 - 0.357*x^3 + 1.385*x^2 - 2.033*x +1);

s2d2e = gq*gq'*(
		+ (397/243 - 4/3*Zeta[3] - 10/81*Log[x] - 11/27*Log[x]^2)*x
		+ (- 1823/3888 + 1/3*Zeta[3] + 553/648*Log[x] - 43/216*Log[x]^2)*x^2
		+ (- 421343/2430000 - 2/15*Zeta[3] - 1439/10125*Log[x] - 1/1350*Log[x]^2)*x^3
		+ (364763/2430000 + 1/15*Zeta[3] + 2779/27000*Log[x] - 331/5400*Log[x]^2)*x^4
		+ (- 32249507/243101250 - 4/105*Zeta[3] - 44897/3472875*Log[x] - 2/33075*Log[x]^2)*x^5
		+ (1799816677/18670176000 + 1/42*Zeta[3] + 50657/2963520*Log[x] - 6011/211680*Log[x]^2)*x^6
		+ (- 115885580383/1512284256000 - 1/63*Zeta[3] + 176657/30005640*Log[x] - 223/238140*Log[x]^2)*x^7
		+ (62579290843/1080203040000 + 1/90*Zeta[3] + 16201/122472000*Log[x] - 21769/1360800*Log[x]^2)*x^8
		+ (- 46158233533877/988453294290000 - 4/495*Zeta[3] + 20762534/2547034875*Log[x] - 12007/10291050*Log[x]^2)*x^9
		);

t2d2eCK = gq*gq'*(
		- 1/4
		+ 1/4*1/x^2
		+ (397/243 - 4/3*Zeta[3] + 10/81*Log[x] - 11/27*Log[x]^2)*1/x^3
		+ (- 1823/3888 + 1/3*Zeta[3] - 553/648*Log[x] - 43/216*Log[x]^2)*1/x^4
		+ (- 421343/2430000 - 2/15*Zeta[3] + 1439/10125*Log[x] - 1/1350*Log[x]^2)*1/x^5
		+ (364763/2430000 + 1/15*Zeta[3] - 2779/27000*Log[x] - 331/5400*Log[x]^2)*1/x^6
		+ (- 32249507/243101250 - 4/105*Zeta[3] + 44897/3472875*Log[x] - 2/33075*Log[x]^2)*1/x^7
		+ (1799816677/18670176000 + 1/42*Zeta[3] - 50657/2963520*Log[x] - 6011/211680*Log[x]^2)*1/x^8
		+ (- 115885580383/1512284256000 - 1/63*Zeta[3] - 176657/30005640*Log[x] - 223/238140*Log[x]^2)*1/x^9
		+ (62579290843/1080203040000 + 1/90*Zeta[3] - 16201/122472000*Log[x] - 21769/1360800*Log[x]^2)*1/x^{10}
		);


(* General Results: Coefficients for Mw \[NotEqual] Mz (Eqs. (A.2-A.7)) *)
(* These coefficients are connected to the s-series (Eqs. (12,45)) as described in Eqs. (15,16,47,A.1). *)

(* Coefficients of s1b *)
Ab[0,1] = 3/4;
Ab[0,2] = 1/4;
Ab[0,3] = 1/8;
Ab[0,4] = 3/40;
Ab[0,5] = 1/20;
Ab[0,6] = 1/28;
Ab[0,7] = 3/112;
Ab[0,8] = 1/48;
Ab[0,9] = 1/60;
Ab[0,10] = 3/220;
Ab[1,1] = - 5/36;
Ab[1,2] = - 1/10;
Ab[1,3] = - 29/360;
Ab[1,4] = - 43/630;
Ab[1,5] = - 5/84;
Ab[1,6] = - 10/189;
Ab[1,7] = - 103/2160;
Ab[1,8] = - 43/990;
Ab[1,9] = - 79/1980;
Ab[2,1] = - 5/144;
Ab[2,2] = - 9/280;
Ab[2,3] = - 149/5040;
Ab[2,4] = - 55/2016;
Ab[2,5] = - 17/672;
Ab[2,6] = - 7/297;
Ab[2,7] = - 131/5940;
Ab[2,8] = - 237/11440;
Ab[3,1] = - 23/2800;
Ab[3,2] = - 1/105;
Ab[3,3] = - 17/1680;
Ab[3,4] = - 191/18480;
Ab[3,5] = - 91/880;
Ab[3,6] = - 329/32175;
Ab[3,7] = - 201/20020;
Ab[4,1] = - 73/37800;
Ab[4,2] = - 5/1848;
Ab[4,3] = - 547/166320;
Ab[4,4] = - 287/77220;
Ab[4,5] = - 259/64350;
Ab[4,6] = - 7/1650;
Ab[5,1] = - 53/116424;
Ab[5,2] = - 3/4004;
Ab[5,3] = - 445/432432;
Ab[5,4] = - 173/135135;
Ab[5,5] = - 3/2002;
Ab[6,1] = - 145/1345344;
Ab[6,2] = - 7/34320;
Ab[6,3] = - 41/131040;
Ab[6,4] = - 2319/5445440;
Ab[7,1] = - 19/741312;
Ab[7,2] = - 2/36465;
Ab[7,3] = - 977/10501920;
Ab[8,1] = - 241/39382200;
Ab[8,2] = - 27/1847560;
Ab[9,1] = - 149/101615800;

(* Coefficients of s1d *)
Ad[0,1] = - 1/4;
Ad[0,2] = - 1/8;
Ad[0,3] = - 1/12;
Ad[0,4] = - 1/16;
Ad[0,5] = - 1/20;
Ad[0,6] = - 1/24;
Ad[0,7] = - 1/28;
Ad[0,8] = - 1/32;
Ad[0,9] = - 1/36;
Ad[0,10] = - 1/40;
Ad[1,m] = 0;
Ad[2,m] = 0;
Ad[3,m] = 0;
Ad[4,m] = 0;
Ad[5,m] = 0;
Ad[6,m] = 0;
Ad[7,m] = 0;
Ad[8,m] = 0;
Ad[9,m] = 0;

(* Coefficients of s1e *)
Ae[0,1] = 1/2;
Ae[0,2] = 1/4;
Ae[0,3] = 1/6;
Ae[0,4] = 1/8;
Ae[0,5] = 1/10;
Ae[0,6] = 1/12;
Ae[0,7] = 1/14;
Ae[0,8] = 1/16;
Ae[0,9] = 1/18;
Ae[0,10] = 1/20;
Ae[1,1] = 5/18 - 1/3*Log[x];
Ae[1,2] = 1/9 - 1/3*Log[x];
Ae[1,3] = - 1/3*Log[x];
Ae[1,4] = - 1/12 - 1/3*Log[x];
Ae[1,5] = - 3/20 - 1/3*Log[x];
Ae[1,6] = - 37/180 - 1/3*Log[x];
Ae[1,7] = - 319/1260 - 1/3*Log[x];
Ae[1,8] = - 743/2520 - 1/3*Log[x];
Ae[1,9] = - 2509/7560 - 1/3*Log[x];
Ae[2,1] = - 7/72 + 1/6*Log[x];
Ae[2,2] = - 1/16 + 1/4*Log[x];
Ae[2,3] = 1/3*Log[x];
Ae[2,4] = 1/12 + 5/12*Log[x];
Ae[2,5] = 11/60 + 1/2*Log[x];
Ae[2,6] = 107/360 + 7/12*Log[x];
Ae[2,7] = 533/1260 + 2/3*Log[x];
Ae[2,8] = 1879/3360 + 3/4*Log[x];
Ae[3,1] = 9/200 - 1/10*Log[x];
Ae[3,2] = 1/25 - 1/5*Log[x];
Ae[3,3] = - 1/3*Log[x];
Ae[3,4] = - 1/12 - 1/2*Log[x];
Ae[3,5] = - 13/60 - 7/10*Log[x];
Ae[3,6] = - 73/180 - 14/15*Log[x];
Ae[3,7] = - 55/84 - 6/5*Log[x];
Ae[4,1] = - 11/450 + 1/15*Log[x];
Ae[4,2] = - 1/36 + 1/6*Log[x];
Ae[4,3] = 1/3*Log[x];
Ae[4,4] = 1/12 + 7/12*Log[x];
Ae[4,5] = 1/4 + 14/15*Log[x];
Ae[4,6] = 191/360 + 7/5*Log[x];
Ae[5,1] = 13/882 - 1/21*Log[x];
Ae[5,2] = 1/49 - 1/7*Log[x];
Ae[5,3] = - 1/3*Log[x];
Ae[5,4] = - 1/12 - 2/3*Log[x];
Ae[5,5] = - 17/60 - 6/5*Log[x];
Ae[6,1] = - 15/1568 + 1/28*Log[x];
Ae[6,2] = - 1/64 + 1/8*Log[x];
Ae[6,3] = 1/3*Log[x];
Ae[6,4] = 1/12 + 3/4*Log[x];
Ae[7,1] = 17/2592 - 1/36*Log[x];
Ae[7,2] = 1/81 - 1/9*Log[x];
Ae[7,3] = - 1/3*Log[x];
Ae[8,1] = - 19/4050 + 1/45*Log[x];
Ae[8,2] = - 1/100 + 1/10*Log[x];
Ae[9,1] = 21/6050 - 1/55*Log[x];

(* Coefficients of s2b *)
Bb[0,1] = 1/2;
Bb[0,2] = 1/6;
Bb[0,3] = 1/12;
Bb[0,4] = 1/20;
Bb[0,5] = 1/30;
Bb[1,1] = - 7/36;
Bb[1,2] = - 73/540;
Bb[1,3] = - 19/180;
Bb[1,4] = - 11/126;
Bb[1,5] = - 113/1512;
Bb[2,1] = - 89/2160;
Bb[2,2] = - 127/3360;
Bb[2,3] = - 13/378;
Bb[2,4] = - 95/3024;
Bb[2,5] = - 97/3360;
Bb[3,1] = - 29/3150;
Bb[3,2] = - 67/6300;
Bb[3,3] = - 17/1512;
Bb[3,4] = - 95/8316;
Bb[3,5] = - 1351/118800;
Bb[4,1] = - 53/25200;
Bb[4,2] = - 587/199584;
Bb[4,3] = - 593/166320;
Bb[4,4] = - 4963/1235520;
Bb[4,5] = - 5023/1158300;
Bb[5,1] = - 851/1746360;
Bb[5,2] = - 2021/2522520;
Bb[5,3] = - 509/463320;
Bb[5,4] = - 79/57915;
Bb[6,1] = - 1381/12108096;
Bb[6,2] = - 41/190080;
Bb[6,3] = - 134/405405;
Bb[7,1] = - 349/12972960;
Bb[7,2] = - 4759/82702620;
Bb[8,1] = - 3017/472586400;

(* Coefficients of s2d *)
Bd[0,1] = 0;
Bd[0,2] = 0;
Bd[0,3] = 0;
Bd[0,4] = 0;
Bd[0,5] = 0;
Bd[1,1] = - 55/243 + 16/81*Log[x] - 1/27*Log[x]^2;
Bd[1,2] = - 31/243 + 13/81*Log[x] - 1/27*Log[x]^2;
Bd[1,3] = - 2/27 + 11/81*Log[x] - 1/27*Log[x]^2;
Bd[1,4] = - 13/324 + 19/162*Log[x] - 1/27*Log[x]^2;
Bd[1,5] = - 1/60 + 83/810*Log[x] - 1/27*Log[x]^2;
Bd[2,1] = - 91/7776 + 5/324*Log[x] - 1/216*Log[x]^2;
Bd[2,2] = - 17/1728 + 1/54*Log[x] - 1/144*Log[x]^2;
Bd[2,3] = - 1/144 + 13/648*Log[x] - 1/108*Log[x]^2;
Bd[2,4] = - 19/5184 + 53/2592*Log[x] - 5/432*Log[x]^2;
Bd[2,5] = - 1/3240 + 43/2160*Log[x] - 1/72*Log[x]^2;
Bd[3,1] = - 47/30000 + 37/13500*Log[x] - 1/900*Log[x]^2;
Bd[3,2] = - 119/67500 + 59/13500*Log[x] - 1/450*Log[x]^2;
Bd[3,3] = - 1/675 + 47/8100*Log[x] - 1/270*Log[x]^2;
Bd[3,4] = - 1/1296 + 37/5400*Log[x] - 1/180*Log[x]^2;
Bd[3,5] = 47/162000 + 199/27000*Log[x] - 7/900*Log[x]^2;
Bd[4,1] = - 407/1215000 + 59/81000*Log[x] - 1/2700*Log[x]^2;
Bd[4,2] = - 23/48600 + 47/32400*Log[x] - 1/1080*Log[x]^2;
Bd[4,3] = - 1/2160 + 37/16200*Log[x] - 1/540*Log[x]^2;
Bd[4,4] = - 31/129600 + 199/64800*Log[x] - 7/2160*Log[x]^2;
Bd[4,5] = 1/4320 + 293/81000*Log[x] - 7/1350*Log[x]^2;
Bd[5,1] = - 1391/14586075 + 172/694575*Log[x] - 1/6615*Log[x]^2;
Bd[5,2] = - 263/1620675 + 137/231525*Log[x] - 1/2205*Log[x]^2;
Bd[5,3] = - 2/11025 + 107/99225*Log[x] - 1/945*Log[x]^2;
Bd[5,4] = - 37/396900 + 323/198450*Log[x] - 2/945*Log[x]^2;
Bd[6,1] = - 365/11063808 + 59/592704*Log[x] - 1/14112*Log[x]^2;
Bd[6,2] = - 89/1354752 + 47/169344*Log[x] - 1/4032*Log[x]^2;
Bd[6,3] = - 1/12096 + 73/127008*Log[x] - 1/1512*Log[x]^2;
Bd[7,1] = - 3247/246903552 + 155/3429216*Log[x] - 1/27216*Log[x]^2;
Bd[7,2] = - 463/15431472 + 247/1714608*Log[x] - 1/6804*Log[x]^2;
Bd[8,1] = - 2299/393660000 + 197/8748000*Log[x] - 1/48600*Log[x]^2;

(* Coefficients of s2e *)
Be[0,1] = 0;
Be[0,2] = 0;
Be[0,3] = 0;
Be[0,4] = 0;
Be[0,5] = 0;
Be[1,1] = 53/27 - 4/3*Zeta[3] - 4/3*Log[x] - 1/3*Log[x]^2;
Be[1,2] = 35/27 - 4/3*Zeta[3] - 5/3*Log[x] - 1/3*Log[x]^2;
Be[1,3] = 20/27 - 4/3*Zeta[3] - 17/9*Log[x] - 1/3*Log[x]^2;
Be[1,4] = 29/108 - 4/3*Zeta[3] - 37/18*Log[x] - 1/3*Log[x]^2;
Be[1,5] = - 77/540 - 4/3*Zeta[3] - 197/90*Log[x] - 1/3*Log[x]^2;
Be[2,1] = - 79/1296 + 2/3*Zeta[3] + 23/18*Log[x] - 7/18*Log[x]^2;
Be[2,2] = 473/864 + Zeta[3] + 55/36*Log[x] - 7/12*Log[x]^2;
Be[2,3] = 803/648 + 4/3*Zeta[3] + 89/54*Log[x] - 7/9*Log[x]^2;
Be[2,4] = 5083/2592 + 5/3*Zeta[3] + 361/216*Log[x] - 35/36*Log[x]^2;
Be[2,5] = 43/16 + 2*Zeta[3] + 97/60*Log[x] - 7/6*Log[x]^2;
Be[3,1] = - 791/1200 - 2/5*Zeta[3] - 13/30*Log[x];
Be[3,2] = - 307/200 - 4/5*Zeta[3] - 13/15*Log[x];
Be[3,3] = - 205/72 - 4/3*Zeta[3] - 13/9*Log[x];
Be[3,4] = - 667/144 - 2*Zeta[3] - 13/6*Log[x];
Be[3,5] = - 4981/720 - 14/5*Zeta[3] - 91/30*Log[x];
Be[4,1] = 171079/243000 + 4/15*Zeta[3] + 233/810*Log[x] - 11/45*Log[x]^2;
Be[4,2] = 185059/97200 + 2/3*Zeta[3] + 769/1620*Log[x] - 11/18*Log[x]^2;
Be[4,3] = 192749/48600 + 4/3*Zeta[3] + 439/810*Log[x] - 11/9*Log[x]^2;
Be[4,4] = 1375583/194400 + 7/3*Zeta[3] + 1093/3240*Log[x] - 77/36*Log[x]^2;
Be[4,5] = 2767561/243000 + 56/15*Zeta[3] - 1279/4050*Log[x] - 154/45*Log[x]^2;
Be[5,1] = - 16099/23814 - 4/21*Zeta[3] - 37/567*Log[x];
Be[5,2] = - 8179/3969 - 4/7*Zeta[3] - 37/189*Log[x];
Be[5,3] = - 8290/1701 - 4/3*Zeta[3] - 37/81*Log[x];
Be[5,4] = - 67097/6804 - 8/3*Zeta[3] - 74/81*Log[x];
Be[6,1] = 529489097/889056000 + 1/7*Zeta[3] + 689/15120*Log[x] - 143/840*Log[x]^2;
Be[6,2] = 535276697/254016000 + 1/2*Zeta[3] - 65/6048*Log[x] - 143/240*Log[x]^2;
Be[6,3] = 534935447/95256000 + 4/3*Zeta[3] - 9659/22680*Log[x] - 143/90*Log[x]^2;
Be[7,1] = - 259882013/489888000 - 1/9*Zeta[3] + 763/19440*Log[x] - 7/1080*Log[x]^2;
Be[7,2] = - 257478563/122472000 - 4/9*Zeta[3] + 1463/9720*Log[x] - 7/270*Log[x]^2;
Be[8,1] = 13910828791/30005640000 + 4/45*Zeta[3] - 52699/1701000*Log[x] - 403/3150*Log[x]^2;

