(* ::Package:: *) (* the integrals of the triple-collinear splitting functions for the fragmenting jet function as they enter eq. (45) in terms of master integrals *) (* the corresponding master integrals are given in insert_mi_fragmenting_jet.m *) (* z indicates which momentum fraction is measured, i.e. qzqqCF is the CF color structure of the integral where the momentum fraction of the quark is measured. *) (* the normalization is such that e.g. Integrate[qzqbarprimeqprime /. {(replacement in insert_mi_fragmenting_jet.m)},{z,0,1}] gives (2(1-eps)^2(2-4 eps+eps^2))/(eps^2(3-2eps))*Gamma[1-eps]^3/((1-2 eps)Gamma[3-3eps]) which is the right-hand side of the first line of (C1) except for the factor (alphaS^2) CF TF/(4 Pi)^2 mu^(4 eps)/s^(1+2 eps) *) qzggCF = ((2*(-2 + 8*eps - 9*eps^2 - 3*eps^3 + 10*eps^4 - 4*eps^5 - 2*z + 8*eps*z - 7*eps^2*z + 3*eps^3*z - 10*eps^4*z + 4*eps^5*z - 2*z^2 + 8*eps*z^2 - 8*eps^2*z^2)*YC[1, 1])/(eps^2*(-1 + z)) - 2*(1 - 2*eps^2 + eps^3 + z - 2*eps^2*z + eps^3*z - 2*z^2)* YC[s[1, 2]^(-1), 0] + (2*z*(-1 + eps - 2*eps*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[1, 3]*s[2, 4]), -1])/(-1 + z) + 4*z^3*YC[1/(s[1, 2]*s[2, 4]*s[2, 3, 4]), 0] + 4*z^3*YC[1/(s[1, 2]*s[3, 4]*s[2, 3, 4]), 0])/2 qzggCA = (-(((9 - 34*eps + 39*eps^2 - 6*eps^3 - 14*eps^4 + 6*eps^5 + 6*eps*z - 19*eps^2*z + 24*eps^4*z - 12*eps^5*z + 9*z^2 - 34*eps*z^2 + 42*eps^2*z^2 - 14*eps^3*z^2 - 10*eps^4*z^2 + 6*eps^5*z^2 - 6*z^3 + 28*eps*z^3 - 40*eps^2*z^3 + 16*eps^3*z^3)* YC[1, 1])/(eps^2*(-3 + 2*eps)*(-1 + z)^2)) + ((-eps + 3*eps^2 - 2*eps^3 + 2*eps*z + 2*z^2 - 2*eps*z^2 - eps^2*z^2 + 2*eps^3*z^2 - 2*z^3 + 4*eps*z^3)*YC[s[1, 2]^(-1), 0])/ ((-1 + 2*eps)*(-1 + z)) - (-1 + eps - 2*eps*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[2, 3]*s[3, 4]), -1] + 2*z^3*YC[1/(s[1, 2]*s[3, 4]*s[2, 3, 4]), 0] + ((-2*(-2 + 8*eps - 9*eps^2 - 3*eps^3 + 10*eps^4 - 4*eps^5 - 2*z + 8*eps*z - 7*eps^2*z + 3*eps^3*z - 10*eps^4*z + 4*eps^5*z - 2*z^2 + 8*eps*z^2 - 8*eps^2*z^2)*YC[1, 1])/(eps^2*(-1 + z)) + 2*(1 - 2*eps^2 + eps^3 + z - 2*eps^2*z + eps^3*z - 2*z^2)* YC[s[1, 2]^(-1), 0] - (2*z*(-1 + eps - 2*eps*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[1, 3]*s[2, 4]), -1])/(-1 + z) - 4*z^3*YC[1/(s[1, 2]*s[2, 4]*s[2, 3, 4]), 0] - 4*z^3*YC[1/(s[1, 2]*s[3, 4]*s[2, 3, 4]), 0])/2)/2 qgzgCF = -((-4 + 28*eps - 36*eps^2 - 8*eps^3 + 16*z - 80*eps*z + 101*eps^2*z - 8*eps^3*z + eps^4*z - 14*z^2 + 61*eps*z^2 - 73*eps^2*z^2 + 11*eps^3*z^2 + eps^4*z^2 + 4*z^3 - 16*eps*z^3 + 16*eps^2*z^3)*YC[1, 1])/(2*eps^2*(-1 + z)*z) + ((-2 + 12*eps - 16*eps^2 - 2*z + 16*eps*z - 30*eps^2*z + 24*eps^3*z + 8*eps*z^2 - 29*eps^2*z^2 + 20*eps^3*z^2 + 2*z^3 - 12*eps*z^3 + 16*eps^2*z^3)*YC[s[1, 2]^(-1), 0])/((-1 + 2*eps)*(-1 + 4*eps)*z) + ((-1 + z)*(-2 + 2*z - z^2 + eps*z^2)*YC[1/(s[1, 2]*s[2, 3]*s[3, 4]), -1])/z - (2*(1 - 4*eps - 2*z + 7*eps*z + 3*z^2 - 14*eps*z^2 - 2*z^3 + 7*eps*z^3)*YC[1/(s[1, 3]*s[1, 2, 4]), 0])/((-1 + 4*eps)*z) - 2*(-1 + z)^3*YC[1/(s[2, 3]*s[2, 4]*s[1, 2, 4]), 0] - 2*(-1 + z)^3*YC[1/(s[1, 2]*s[3, 4]*s[1, 3, 4]), 0] qgzgCA = (((-4 + 28*eps - 36*eps^2 - 8*eps^3 + 16*z - 80*eps*z + 101*eps^2*z - 8*eps^3*z + eps^4*z - 14*z^2 + 61*eps*z^2 - 73*eps^2*z^2 + 11*eps^3*z^2 + eps^4*z^2 + 4*z^3 - 16*eps*z^3 + 16*eps^2*z^3)*YC[1, 1])/(2*eps^2*(-1 + z)*z) - ((-6 - 4*eps + 162*eps^2 - 396*eps^3 + 246*eps^4 + 60*eps^5 - 24*eps^6 + 16*eps^7 - 12*z + 71*eps*z - 130*eps^2*z + 55*eps^3*z + 8*eps^4*z + 100*eps^5*z - 112*eps^6*z + 32*eps^7*z + 6*z^2 + 6*eps*z^2 - 168*eps^2*z^2 + 384*eps^3*z^2 - 192*eps^4*z^2 - 96*eps^5*z^2 + 6*z^3 - 29*eps*z^3 + 46*eps^2*z^3 + 65*eps^3*z^3 - 216*eps^4*z^3 - 28*eps^5*z^3 + 400*eps^6*z^3 - 256*eps^7*z^3 - 6*z^4 + 40*eps*z^4 - 126*eps^2*z^4 + 132*eps^3*z^4 + 58*eps^4*z^4 - 36*eps^5*z^4 - 264*eps^6*z^4 + 208*eps^7*z^4)*YC[1, 1])/ (3*eps^2*(-1 + 2*eps)*(-1 + z)*z*(1 + z)) - ((-12 - 8*eps + 252*eps^2 - 528*eps^3 + 300*eps^4 + 24*eps^5 - 48*eps^6 + 32*eps^7 - 14*eps*z + 37*eps^2*z + 56*eps^3*z - 287*eps^4*z + 374*eps^5*z - 224*eps^6*z + 64*eps^7*z + 6*z^2 + 33*eps*z^2 - 258*eps^2*z^2 + 423*eps^3*z^2 - 132*eps^4*z^2 - 84*eps^5*z^2 - 6*z^3 + 47*eps*z^3 - 55*eps^2*z^3 + 7*eps^3*z^3 - 69*eps^4*z^3 - 218*eps^5*z^3 + 800*eps^6*z^3 - 512*eps^7*z^3 + 8*eps*z^4 - 108*eps^2*z^4 + 168*eps^3*z^4 + 116*eps^4*z^4 - 72*eps^5*z^4 - 528*eps^6*z^4 + 416*eps^7*z^4)*YC[1, 1])/ (6*eps^2*(-1 + 2*eps)*(-1 + z)*z*(1 + z)) - (16*(-1 + eps)*(1 + 2*eps)^2*(-2 + 3*eps)*z^2*YC[1, 2])/(3*(1 + z)) - (16*(-1 + eps)*(-1 + 2*eps)*(1 + 2*eps)^2*z^2*YC[1, 3])/(3*(1 + z)) - ((-2 + 12*eps - 16*eps^2 - 2*z + 16*eps*z - 30*eps^2*z + 24*eps^3*z + 8*eps*z^2 - 29*eps^2*z^2 + 20*eps^3*z^2 + 2*z^3 - 12*eps*z^3 + 16*eps^2*z^3)*YC[s[1, 2]^(-1), 0])/((-1 + 2*eps)*(-1 + 4*eps)*z) - ((18 - 130*eps + 348*eps^2 - 408*eps^3 + 174*eps^4 + 12*eps^5 - 24*eps^6 + 16*eps^7 + 6*z - 66*eps*z + 216*eps^2*z - 228*eps^3*z - 78*eps^4*z + 276*eps^5*z - 168*eps^6*z + 48*eps^7*z + 15*z^2 - 96*eps*z^2 + 222*eps^2*z^2 - 192*eps^3*z^2 - 54*eps^4*z^2 + 228*eps^5*z^2 - 168*eps^6*z^2 + 48*eps^7*z^2 + 4*eps*z^3 + 30*eps^2*z^3 - 132*eps^3*z^3 + 134*eps^4*z^3 + 36*eps^5*z^3 - 120*eps^6*z^3 + 48*eps^7*z^3)*YC[s[1, 2]^(-1), 0])/ (3*(-1 + 2*eps)^3*z) - ((6 - 20*eps - 178*eps^2 + 1044*eps^3 - 1830*eps^4 + 924*eps^5 + 264*eps^6 - 112*eps^7 + 64*eps^8 - 24*z + 270*eps*z - 1206*eps^2*z + 2640*eps^3*z - 2682*eps^4*z + 516*eps^5*z + 1080*eps^6*z - 720*eps^7*z + 192*eps^8*z - 3*z^2 + 36*eps*z^2 - 219*eps^2*z^2 + 690*eps^3*z^2 - 990*eps^4*z^2 + 252*eps^5*z^2 + 792*eps^6*z^2 - 720*eps^7*z^2 + 192*eps^8*z^2 - 6*z^3 + 56*eps*z^3 - 230*eps^2*z^3 + 588*eps^3*z^3 - 854*eps^4*z^3 + 500*eps^5*z^3 + 264*eps^6*z^3 - 528*eps^7*z^3 + 192*eps^8*z^3)* YC[s[1, 2]^(-1), 0])/(3*(-1 + 2*eps)^3*(-1 + 4*eps)*z) - (16*(-1 + eps)^3*eps*(1 + 2*eps)^2*z^2*YC[s[1, 2]^(-1), 1])/ (3*(-1 + 2*eps)^2) - (16*(-1 + eps)^2*eps*(1 + 2*eps)^2*z^2* YC[s[1, 2]^(-1), 2])/(3*(-1 + 2*eps)) - (16*(-1 + eps)*eps*(1 + 2*eps)^2*z^2*YC[s[1, 2]^(-1), 3])/3 - (-2 + 2*z - z^2 + eps*z^2)*YC[1/(s[1, 2]*s[1, 3]*s[2, 4]), -1] - ((-1 + z)*(-2 + 2*z - z^2 + eps*z^2)*YC[1/(s[1, 2]*s[2, 3]*s[3, 4]), -1])/z + (16*(-1 + eps)*(-1 + 2*eps)*(1 + 2*eps)^2*z^2* YC[s[1, 2, 4]^(-1), 4])/3 + (16*(-1 + eps)*(1 + 2*eps)^2* (-5 + 21*eps - 31*eps^2 + 16*eps^3)*z^2*YC[s[1, 2]/s[1, 2, 4], 2])/ (3*(-1 + 2*eps)^2*(1 + z)) + (16*(-1 + eps)*(1 + 2*eps)^2* (4 - 11*eps + 8*eps^2)*z^2*YC[s[1, 2]/s[1, 2, 4], 3])/ (3*(-1 + 2*eps)*(1 + z)) + (16*(-1 + eps)*(1 + 2*eps)^2*(-3 + 4*eps)* z^2*YC[s[1, 2]/s[1, 2, 4], 4])/(3*(1 + z)) + ((8 - 30*eps - 3*z + 15*eps*z + 4*eps^2*z + 5*z^2 - 19*eps*z^2 + 4*eps^2*z^2)*YC[1/(s[1, 3]*s[1, 2, 4]), 0])/(-1 + 4*eps) + (2*(1 - 4*eps - 2*z + 7*eps*z + 3*z^2 - 14*eps*z^2 - 2*z^3 + 7*eps*z^3)* YC[1/(s[1, 3]*s[1, 2, 4]), 0])/((-1 + 4*eps)*z) - (-2 - 2*z - z^2 + eps*z^2)*YC[1/(s[1, 2]*s[1, 3]*s[1, 2, 4]), -1] - (16*(-1 + eps)^3*(1 + 2*eps)^2*z^2*YC[s[1, 3]/s[1, 2, 4], 2])/ (3*(-1 + 2*eps)^2*(1 + z)) - (16*(-1 + eps)^2*(1 + 2*eps)^2*z^2* YC[s[1, 3]/s[1, 2, 4], 3])/(3*(-1 + 2*eps)*(1 + z)) - (16*(-1 + eps)*(1 + 2*eps)^2*z^2*YC[s[1, 3]/s[1, 2, 4], 4])/ (3*(1 + z)) - (-2 + 2*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[2, 3]*s[1, 2, 4]), -1] + 2*(-1 + z)^3*YC[1/(s[2, 3]*s[2, 4]*s[1, 2, 4]), 0])/2 qzqbarprimeqprime = (-2*(-1 + eps)*(1 - 2*eps + eps^2 - 2*eps^2*z + z^2 - 2*eps*z^2 + eps^2*z^2)*YC[1, 1])/(eps*(-3 + 2*eps)*(-1 + z)^2) qqbarprimezqprime = -((-4 + 26*eps - 58*eps^2 + 44*eps^3 + 8*eps^4 - 16*eps^5 - 11*z + 54*eps*z - 77*eps^2*z - 6*eps^3*z + 88*eps^4*z - 48*eps^5*z - 11*z^2 + 54*eps*z^2 - 77*eps^2*z^2 - 6*eps^3*z^2 + 88*eps^4*z^2 - 48*eps^5*z^2 - 4*z^3 + 50*eps*z^3 - 34*eps^2*z^3 - 92*eps^3*z^3 - 56*eps^4*z^3 + 208*eps^5*z^3)*YC[1, 1])/ (3*eps*(-1 + 2*eps)*z*(1 + z)) - (8*(1 + 2*eps)^2*(-2 + 3*eps)*z^2* YC[1, 2])/(3*(1 + z)) - (8*(-1 + 2*eps)*(1 + 2*eps)^2*z^2*YC[1, 3])/ (3*(1 + z)) - (2*(-1 + eps)*(-2*eps + 11*eps^2 - 18*eps^3 + 4*eps^4 + 8*eps^5 + 3*z - 18*eps*z + 33*eps^2*z - 6*eps^3*z - 36*eps^4*z + 24*eps^5*z + 3*z^2 - 18*eps*z^2 + 33*eps^2*z^2 - 6*eps^3*z^2 - 36*eps^4*z^2 + 24*eps^5*z^2 + 2*eps*z^3 + 19*eps^2*z^3 - 30*eps^3*z^3 - 12*eps^4*z^3 + 24*eps^5*z^3)*YC[s[1, 2]^(-1), 0])/ (3*(-1 + 2*eps)^3*z) - (8*(-1 + eps)^2*eps*(1 + 2*eps)^2*z^2* YC[s[1, 2]^(-1), 1])/(3*(-1 + 2*eps)^2) - (8*(-1 + eps)*eps*(1 + 2*eps)^2*z^2*YC[s[1, 2]^(-1), 2])/ (3*(-1 + 2*eps)) - (8*eps*(1 + 2*eps)^2*z^2*YC[s[1, 2]^(-1), 3])/3 + (8*(-1 + 2*eps)*(1 + 2*eps)^2*z^2*YC[s[1, 2, 4]^(-1), 4])/3 + (8*(1 + 2*eps)^2*(-5 + 21*eps - 31*eps^2 + 16*eps^3)*z^2* YC[s[1, 2]/s[1, 2, 4], 2])/(3*(-1 + 2*eps)^2*(1 + z)) + (8*(1 + 2*eps)^2*(4 - 11*eps + 8*eps^2)*z^2*YC[s[1, 2]/s[1, 2, 4], 3])/ (3*(-1 + 2*eps)*(1 + z)) + (8*(1 + 2*eps)^2*(-3 + 4*eps)*z^2* YC[s[1, 2]/s[1, 2, 4], 4])/(3*(1 + z)) - (8*(-1 + eps)^2*(1 + 2*eps)^2*z^2*YC[s[1, 3]/s[1, 2, 4], 2])/ (3*(-1 + 2*eps)^2*(1 + z)) - (8*(-1 + eps)*(1 + 2*eps)^2*z^2* YC[s[1, 3]/s[1, 2, 4], 3])/(3*(-1 + 2*eps)*(1 + z)) - (8*(1 + 2*eps)^2*z^2*YC[s[1, 3]/s[1, 2, 4], 4])/(3*(1 + z)) qzqbarq = -(((1 - 5*eps + 15*eps^2 - 17*eps^3 + 6*eps^4 + 2*eps*z - 23*eps^2*z + 36*eps^3*z - 12*eps^4*z + z^2 - 5*eps*z^2 + 16*eps^2*z^2 - 19*eps^3*z^2 + 6*eps^4*z^2)*YC[1, 1])/ (eps^2*(-1 + z)^2)) + ((1 - 5*eps + 8*eps^2 - 4*eps^3 + 2*eps*z - 2*eps^2*z + 2*eps^3*z + z^2 + 2*eps*z^2 - 6*eps^2*z^2 + 2*eps^3*z^2)* YC[s[1, 2]^(-1), 0])/((-1 + 2*eps)*(-1 + z)) + (z*(-1 + eps - 2*eps*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[2, 3]*s[1, 2, 4]), -1])/(-1 + z) qqbarzq = (-2*(-1 + eps)*(4 - 11*eps + 4*eps^2)*YC[1, 1] + 2*(-2 + eps)*(-1 + eps)*eps*(1 + z)*YC[s[1, 2]^(-1), 0] - 2*(-1 + eps + 2*eps*z - z^2 + eps*z^2)*YC[1/(s[1, 3]*s[1, 2, 4]), 0] + (2*z*(-1 + eps + 2*eps*z - z^2 + eps*z^2)* YC[1/(s[1, 2]*s[1, 3]*s[1, 2, 4]), -1])/(1 + z))/2