(* Definition of the 5 tensor structures resulting from the symmetries of T[rho1,rho2;rho3,rho4](q):=i*int d^4x e^(iqx) T T[rho1,rho2](x)*T[rho3,rho4](0) t_1=(q[rho1]*q[rho2]*q[rho3]*q[rho4]) t_2=(q[rho1]*q[rho2]*g[rho3,rho4]+q[rho3]*q[rho4]*g[rho1,rho2])*q^2 t_3=(q[rho1]*q[rho3]*g[rho2,rho4]+q[rho1]*q[rho4]*g[rho2,rho3]+q[rho2]*q[rho3]*g[rho1,rho4]+q[rho2]*q[rho4]*g[rho1,rho3])*q^2 t_4=(g[rho1,rho2]*g[rho3,rho4])*(q^2)^2 t_5=(g[rho1,rho3]*g[rho2,rho4]+g[rho1,rho4]*g[rho2,rho3])*(q^2)^2 The Wilson coefficients can be expressed with 5 scalar pieces, e.g. C0[rho1,rho2;rho3,rho4](q)=C0_i(Q^2)*t_i The requirement q[rho1]*T[rho1,rho2;rho3,rho4](q) = local leads to the 3 conditions for every Wilson coefficient D0_1=C0_1+C0_2+2*C0_3 = local D0_2=C0_2+C0_4 = local D0_3=C0_3+C0_5 = local If these are fulfilled we can express the scale dependent (physical) part of the Wilson coefficient as C0[rho1,rho2;rho3,rho4](q)=C0_S(Q^2)*t_S + C0_T(Q^2)*t_T with t_S=eta[rho1,rho2]*eta[rho3,rho4] t_T=eta[rho1,rho3]*eta[rho2,rho4]+eta[rho1,rho4]*eta[rho2,rho3]-2/(D-1)*eta[rho1,rho2]*eta[rho3,rho4] where D=4-2*ep and eta[rho1,rho2]=q^2*g[rho1,rho2]-q[rho1]q[rho2] with the metric g as=alpha_s/(Pi) lmuq=Log(mu^2/Q^2) mu=MSbar renormalization scale *) (* Wilson coefficient C0 for the energy momentum tensor correlator, 3loop *) C0_1 = + ep^-2 * ( + 11/972*as^2*ca^2*ng - 109/7776*as^2*nl*ca*tr*ng + 7/1944*as^2*nl^2*tr^2*ng ) + ep^-1 * ( + 2/15*ng + 1/15*nl*dF - 1/27*as*ca*ng + 7/216*as*nl*tr*ng - 31/5832*as^2*ca^2*ng - 1/96*as^2*nl*tr*cf*ng + 1463/46656*as^2*nl*ca*tr*ng - 119/11664*as^2*nl^2*tr^2*ng ) + 17/225*ng + 31/225*nl*dF - 2/15*(-lmuq)*ng - 1/15*(-lmuq)*nl*dF + 181/1620*as*ca*ng + 1057/6480*as*nl*tr*ng + 2/27*as*(-lmuq)*ca*ng - 7/108*as*(-lmuq)*nl*tr*ng + 595559/699840*as^2*ca^2*ng - 431/1440*as^2*nl*tr*cf*ng + 607769/1399680*as^2*nl*ca*tr*ng - 93239/349920*as^2*nl^2*tr^2*ng - 1681/19440*as^2*(-lmuq)*ca^2*ng + 1/32*as^2*(-lmuq)*nl*tr*cf*ng - 8023/38880*as^2*(-lmuq)*nl*ca*tr*ng + 413/4860*as^2*(-lmuq)*nl^2*tr^2*ng - 11/324*as^2*(-lmuq)^2*ca^2*ng + 109/2592*as^2*(-lmuq)^2*nl*ca*tr*ng - 7/648*as^2*(-lmuq)^2*nl^2*tr^2*ng + 5/6*z5*as^2*ca^2*ng + 1/2*z5*as^2*nl*tr*cf*ng - 1/12*z5*as^2*nl*ca*tr*ng - 2/5*z3*as*ca*ng - 1/5*z3*as*nl*tr*ng - 563/360*z3*as^2*ca^2*ng - 37/120*z3*as^2*nl*tr*cf*ng + 29/360*z3*as^2*nl*ca*tr*ng + 19/90*z3*as^2*nl^2*tr^2*ng + 11/30*z3*as^2*(-lmuq)*ca^2*ng + 1/20*z3*as^2*(-lmuq)*nl*ca*tr*ng - 1/15*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_2 = + ep^-2 * ( + 11/1944*as^2*ca^2*ng - 109/15552*as^2*nl*ca*tr*ng + 7/3888*as^2*nl^2*tr^2*ng ) + ep^-1 * ( + 1/15*ng + 1/30*nl*dF - 1/54*as*ca*ng + 7/432*as*nl*tr*ng + 35/11664*as^2*ca^2*ng - 1/192*as^2*nl*tr*cf*ng + 809/93312*as^2*nl*ca*tr*ng - 77/23328*as^2*nl^2*tr^2*ng ) + 47/450*ng + 23/225*nl*dF - 1/15*(-lmuq)*ng - 1/30*(-lmuq)*nl*dF - 187/1620*as*ca*ng + 1987/12960*as*nl*tr*ng + 1/27*as*(-lmuq)*ca*ng - 7/216*as*(-lmuq)*nl*tr*ng - 160831/1399680*as^2*ca^2*ng - 61/480*as^2*nl*tr*cf*ng + 1733639/2799360*as^2*nl*ca*tr*ng - 140909/699840*as^2*nl^2*tr^2*ng + 941/9720*as^2*(-lmuq)*ca^2*ng + 1/64*as^2*(-lmuq)*nl*tr*cf*ng - 15943/77760*as^2*(-lmuq)*nl*ca*tr*ng + 593/9720*as^2*(-lmuq)*nl^2*tr^2*ng - 11/648*as^2*(-lmuq)^2*ca^2*ng + 109/5184*as^2*(-lmuq)^2*nl*ca*tr*ng - 7/1296*as^2*(-lmuq)^2*nl^2*tr^2*ng + 5/12*z5*as^2*ca^2*ng + 1/4*z5*as^2*nl*tr*cf*ng - 1/24*z5*as^2*nl*ca*tr*ng - 1/5*z3*as*ca*ng - 1/10*z3*as*nl*tr*ng - 563/720*z3*as^2*ca^2*ng - 37/240*z3*as^2*nl*tr*cf*ng + 29/720*z3*as^2*nl*ca*tr*ng + 19/180*z3*as^2*nl^2*tr^2*ng + 11/60*z3*as^2*(-lmuq)*ca^2*ng + 1/40*z3*as^2*(-lmuq)*nl*ca*tr*ng - 1/30*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_3 = + ep^-2 * ( - 11/1296*as^2*ca^2*ng + 109/10368*as^2*nl*ca*tr*ng - 7/2592*as^2*nl^2*tr^2*ng ) + ep^-1 * ( - 1/10*ng - 1/20*nl*dF + 1/36*as*ca*ng - 7/288*as*nl*tr*ng + 1/864*as^2*ca^2*ng + 1/128*as^2*nl*tr*cf*ng - 415/20736*as^2*nl*ca*tr*ng + 35/5184*as^2*nl^2*tr^2*ng ) - 9/100*ng - 3/25*nl*dF + 1/10*(-lmuq)*ng + 1/20*(-lmuq)*nl*dF + 1/540*as*ca*ng - 1367/8640*as*nl*tr*ng - 1/18*as*(-lmuq)*ca*ng + 7/144*as*(-lmuq)*nl*tr*ng - 343429/933120*as^2*ca^2*ng + 307/1440*as^2*nl*tr*cf*ng - 983059/1866240*as^2*nl*ca*tr*ng + 109129/466560*as^2*nl^2*tr^2*ng - 67/12960*as^2*(-lmuq)*ca^2*ng - 3/128*as^2*(-lmuq)*nl*tr*cf*ng + 10663/51840*as^2*(-lmuq)*nl*ca*tr*ng - 473/6480*as^2*(-lmuq)*nl^2*tr^2*ng + 11/432*as^2*(-lmuq)^2*ca^2*ng - 109/3456*as^2*(-lmuq)^2*nl*ca*tr*ng + 7/864*as^2*(-lmuq)^2*nl^2*tr^2*ng - 5/8*z5*as^2*ca^2*ng - 3/8*z5*as^2*nl*tr*cf*ng + 1/16*z5*as^2*nl*ca*tr*ng + 3/10*z3*as*ca*ng + 3/20*z3*as*nl*tr*ng + 563/480*z3*as^2*ca^2*ng + 37/160*z3*as^2*nl*tr*cf*ng - 29/480*z3*as^2*nl*ca*tr*ng - 19/120*z3*as^2*nl^2*tr^2*ng - 11/40*z3*as^2*(-lmuq)*ca^2*ng - 3/80*z3*as^2*(-lmuq)*nl*ca*tr*ng + 1/20*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_4 = + ep^-2 * ( - 11/1944*as^2*ca^2*ng + 109/15552*as^2*nl*ca*tr*ng - 7/3888*as^2*nl^2*tr^2*ng ) + ep^-1 * ( - 1/15*ng - 1/30*nl*dF + 1/54*as*ca*ng - 7/432*as*nl*tr*ng - 35/11664*as^2*ca^2*ng + 1/192*as^2*nl*tr*cf*ng - 809/93312*as^2*nl*ca*tr*ng + 77/23328*as^2*nl^2*tr^2*ng ) - 47/450*ng - 23/225*nl*dF + 1/15*(-lmuq)*ng + 1/30*(-lmuq)*nl*dF + 187/1620*as*ca*ng - 1987/12960*as*nl*tr*ng - 1/27*as*(-lmuq)*ca*ng + 7/216*as*(-lmuq)*nl*tr*ng + 160831/1399680*as^2*ca^2*ng + 61/480*as^2*nl*tr*cf*ng - 1733639/2799360*as^2*nl*ca*tr*ng + 140909/699840*as^2*nl^2*tr^2*ng - 941/9720*as^2*(-lmuq)*ca^2*ng - 1/64*as^2*(-lmuq)*nl*tr*cf*ng + 15943/77760*as^2*(-lmuq)*nl*ca*tr*ng - 593/9720*as^2*(-lmuq)*nl^2*tr^2*ng + 11/648*as^2*(-lmuq)^2*ca^2*ng - 109/5184*as^2*(-lmuq)^2*nl*ca*tr*ng + 7/1296*as^2*(-lmuq)^2*nl^2*tr^2*ng - 5/12*z5*as^2*ca^2*ng - 1/4*z5*as^2*nl*tr*cf*ng + 1/24*z5*as^2*nl*ca*tr*ng + 1/5*z3*as*ca*ng + 1/10*z3*as*nl*tr*ng + 563/720*z3*as^2*ca^2*ng + 37/240*z3*as^2*nl*tr*cf*ng - 29/720*z3*as^2*nl*ca*tr*ng - 19/180*z3*as^2*nl^2*tr^2*ng - 11/60*z3*as^2*(-lmuq)*ca^2*ng - 1/40*z3*as^2*(-lmuq)*nl*ca*tr*ng + 1/30*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_5 = + ep^-2 * ( + 11/1296*as^2*ca^2*ng - 109/10368*as^2*nl*ca*tr*ng + 7/2592*as^2*nl^2*tr^2*ng ) + ep^-1 * ( + 1/10*ng + 1/20*nl*dF - 1/36*as*ca*ng + 7/288*as*nl*tr*ng - 1/864*as^2*ca^2*ng - 1/128*as^2*nl*tr*cf*ng + 415/20736*as^2*nl*ca*tr*ng - 35/5184*as^2*nl^2*tr^2*ng ) + 9/100*ng + 3/25*nl*dF - 1/10*(-lmuq)*ng - 1/20*(-lmuq)*nl*dF - 1/540*as*ca*ng + 1367/8640*as*nl*tr*ng + 1/18*as*(-lmuq)*ca*ng - 7/144*as*(-lmuq)*nl*tr*ng + 343429/933120*as^2*ca^2*ng - 307/1440*as^2*nl*tr*cf*ng + 983059/1866240*as^2*nl*ca*tr*ng - 109129/466560*as^2*nl^2*tr^2*ng + 67/12960*as^2*(-lmuq)*ca^2*ng + 3/128*as^2*(-lmuq)*nl*tr*cf*ng - 10663/51840*as^2*(-lmuq)*nl*ca*tr*ng + 473/6480*as^2*(-lmuq)*nl^2*tr^2*ng - 11/432*as^2*(-lmuq)^2*ca^2*ng + 109/3456*as^2*(-lmuq)^2*nl*ca*tr*ng - 7/864*as^2*(-lmuq)^2*nl^2*tr^2*ng + 5/8*z5*as^2*ca^2*ng + 3/8*z5*as^2*nl*tr*cf*ng - 1/16*z5*as^2*nl*ca*tr*ng - 3/10*z3*as*ca*ng - 3/20*z3*as*nl*tr*ng - 563/480*z3*as^2*ca^2*ng - 37/160*z3*as^2*nl*tr*cf*ng + 29/480*z3*as^2*nl*ca*tr*ng + 19/120*z3*as^2*nl^2*tr^2*ng + 11/40*z3*as^2*(-lmuq)*ca^2*ng + 3/80*z3*as^2*(-lmuq)*nl*ca*tr*ng - 1/20*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_S = + 11/108*as*ca*ng - 1/27*as*nl*tr*ng + 313/864*as^2*ca^2*ng - 1/54*as^2*nl*tr*cf*ng - 85/324*as^2*nl*ca*tr*ng + 7/162*as^2*nl^2*tr^2*ng - 121/1296*as^2*(-lmuq)*ca^2*ng + 11/162*as^2*(-lmuq)*nl*ca*tr*ng - 1/81*as^2*(-lmuq)*nl^2*tr^2*ng ; C0_T = + ep^-2 * ( + 11/1296*as^2*ca^2*ng - 109/10368*as^2*nl*ca*tr*ng + 7/2592*as^2*nl^2*tr^2*ng ) + ep^-1 * ( + 1/10*ng + 1/20*nl*dF - 1/36*as*ca*ng + 7/288*as*nl*tr*ng - 1/864*as^2*ca^2*ng - 1/128*as^2*nl*tr*cf*ng + 415/20736*as^2*nl*ca*tr*ng - 35/5184*as^2*nl^2*tr^2*ng ) + 9/100*ng + 3/25*nl*dF - 1/10*(-lmuq)*ng - 1/20*(-lmuq)*nl*dF - 1/540*as*ca*ng + 1367/8640*as*nl*tr*ng + 1/18*as*(-lmuq)*ca*ng - 7/144*as*(-lmuq)*nl*tr*ng + 343429/933120*as^2*ca^2*ng - 307/1440*as^2*nl*tr*cf*ng + 983059/1866240*as^2*nl*ca*tr*ng - 109129/466560*as^2*nl^2*tr^2*ng + 67/12960*as^2*(-lmuq)*ca^2*ng + 3/128*as^2*(-lmuq)*nl*tr*cf*ng - 10663/51840*as^2*(-lmuq)*nl*ca*tr*ng + 473/6480*as^2*(-lmuq)*nl^2*tr^2*ng - 11/432*as^2*(-lmuq)^2*ca^2*ng + 109/3456*as^2*(-lmuq)^2*nl*ca*tr*ng - 7/864*as^2*(-lmuq)^2*nl^2*tr^2*ng + 5/8*z5*as^2*ca^2*ng + 3/8*z5*as^2*nl*tr*cf*ng - 1/16*z5*as^2*nl*ca*tr*ng - 3/10*z3*as*ca*ng - 3/20*z3*as*nl*tr*ng - 563/480*z3*as^2*ca^2*ng - 37/160*z3*as^2*nl*tr*cf*ng + 29/480*z3*as^2*nl*ca*tr*ng + 19/120*z3*as^2*nl^2*tr^2*ng + 11/40*z3*as^2*(-lmuq)*ca^2*ng + 3/80*z3*as^2*(-lmuq)*nl*ca*tr*ng - 1/20*z3*as^2*(-lmuq)*nl^2*tr^2*ng ; Adler functions of renormalized Correlator C0_t for 3 loop and t=1...5 C0_1 = + as * ( + 2/27*ca*ng - 7/108*nl*tr*ng ) + as^2 * ( - 1681/19440*ca^2*ng + 1/32*nl*tr*cf*ng - 8023/38880*nl*ca*tr*ng + 413/4860*nl^2*tr^2*ng - 11/162*(-lmuq)*ca^2*ng + 109/1296*(-lmuq)*nl*ca*tr*ng - 7/324*(-lmuq)*nl^2*tr^2*ng + 11/30*z3*ca^2*ng + 1/20*z3*nl*ca*tr*ng - 1/15*z3*nl^2*tr^2*ng ) - 2/15*ng - 1/15*nl*dF ; C0_2 = + as * ( + 1/27*ca*ng - 7/216*nl*tr*ng ) + as^2 * ( + 941/9720*ca^2*ng + 1/64*nl*tr*cf*ng - 15943/77760*nl*ca*tr*ng + 593/9720*nl^2*tr^2*ng - 11/324*(-lmuq)*ca^2*ng + 109/2592*(-lmuq)*nl*ca*tr*ng - 7/648*(-lmuq)*nl^2*tr^2*ng + 11/60*z3*ca^2*ng + 1/40*z3*nl*ca*tr*ng - 1/30*z3*nl^2*tr^2*ng ) - 1/15*ng - 1/30*nl*dF ; C0_3 = + as * ( - 1/18*ca*ng + 7/144*nl*tr*ng ) + as^2 * ( - 67/12960*ca^2*ng - 3/128*nl*tr*cf*ng + 10663/51840*nl*ca*tr*ng - 473/6480*nl^2*tr^2*ng + 11/216*(-lmuq)*ca^2*ng - 109/1728*(-lmuq)*nl*ca*tr*ng + 7/432*(-lmuq)*nl^2*tr^2*ng - 11/40*z3*ca^2*ng - 3/80*z3*nl*ca*tr*ng + 1/20*z3*nl^2*tr^2*ng ) + 1/10*ng + 1/20*nl*dF ; C0_4 = + as * ( - 1/27*ca*ng + 7/216*nl*tr*ng ) + as^2 * ( - 941/9720*ca^2*ng - 1/64*nl*tr*cf*ng + 15943/77760*nl*ca*tr*ng - 593/9720*nl^2*tr^2*ng + 11/324*(-lmuq)*ca^2*ng - 109/2592*(-lmuq)*nl*ca*tr*ng + 7/648*(-lmuq)*nl^2*tr^2*ng - 11/60*z3*ca^2*ng - 1/40*z3*nl*ca*tr*ng + 1/30*z3*nl^2*tr^2*ng ) + 1/15*ng + 1/30*nl*dF ; C0_5 = + as * ( + 1/18*ca*ng - 7/144*nl*tr*ng ) + as^2 * ( + 67/12960*ca^2*ng + 3/128*nl*tr*cf*ng - 10663/51840*nl*ca*tr*ng + 473/6480*nl^2*tr^2*ng - 11/216*(-lmuq)*ca^2*ng + 109/1728*(-lmuq)*nl*ca*tr*ng - 7/432*(-lmuq)*nl^2*tr^2*ng + 11/40*z3*ca^2*ng + 3/80*z3*nl*ca*tr*ng - 1/20*z3*nl^2*tr^2*ng ) - 1/10*ng - 1/20*nl*dF ; C0_S = + as^2 * ( - 121/1296*ca^2*ng + 11/162*nl*ca*tr*ng - 1/81*nl^2*tr^2*ng ); C0_T = + as * ( + 1/18*ca*ng - 7/144*nl*tr*ng ) + as^2 * ( + 67/12960*ca^2*ng + 3/128*nl*tr*cf*ng - 10663/51840*nl*ca*tr*ng + 473/6480*nl^2*tr^2*ng - 11/216*(-lmuq)*ca^2*ng + 109/1728*(-lmuq)*nl*ca*tr*ng - 7/432*(-lmuq)*nl^2*tr^2*ng + 11/40*z3*ca^2*ng + 3/80*z3*nl*ca*tr*ng - 1/20*z3*nl^2*tr^2*ng ) - 1/10*ng - 1/20*nl*dF ; (* Wilson coefficient C1 for the energy momentum tensor correlator, 2loop *) (* restrictions D1_1, D1_2, D1_3 *) D1_1 = 0; D1_2 = + ep^-1 * ( + 11/36*as*ca - 1/9*as*nl*tr + 17/72*as^2*ca^2 - 1/12*as^2*nl*tr*cf - 5/36*as^2*nl*ca*tr ) + 1/3 - 55/216*as*ca + 5/54*as*nl*tr - 85/432*as^2*ca^2 + 5/72*as^2*nl*tr*cf + 25/216*as^2*nl*ca*tr ; D1_3 = + ep^-1 * ( - 11/18*as*ca + 2/9*as*nl*tr - 17/36*as^2*ca^2 + 1/6*as^2*nl*tr*cf + 5/18*as^2*nl*ca*tr ) - 2/3 - 11/108*as*ca + 1/27*as*nl*tr - 17/216*as^2*ca^2 + 1/36*as^2*nl*tr*cf + 5/108*as^2*nl*ca*tr ; (* C1_t for t=1...5 *) C1_1 = + as * ( + 4/9*ca - 7/18*nl*tr ) + as^2 * ( + 3/8*nl*tr*cf + 1/36*nl*ca*tr - 1/18*nl^2*tr^2 ); C1_2 = + as * ( - ca + 1/4*nl*tr ) + as^2 * ( - 83/216*ca^2 + 25/48*nl*tr*cf + 35/216*nl*ca*tr + 5/108*nl^2*tr^2 ); C1_3 = + as * ( + 5/18*ca + 5/72*nl*tr ) + as^2 * ( + 83/432*ca^2 - 43/96*nl*tr*cf - 41/432*nl*ca*tr + 1/216*nl^2*tr^2 ); C1_4 = + as * ( + 11/36*ep^-1*ca - 1/9*ep^-1*nl*tr + 161/216*ca - 17/108*nl*tr ) + as^2 * ( + 17/72*ep^-1*ca^2 - 1/12*ep^-1*nl*tr*cf - 5/36*ep^-1*nl*ca*tr + 3/16*ca^2 - 65/144*nl*tr*cf - 5/108*nl*ca*tr - 5/108*nl^2*tr^2 ) + 1/3*Q.Q^-2 ; C1_5 = + as * ( - 11/18*ep^-1*ca + 2/9*ep^-1*nl*tr - 41/108*ca - 7/216*nl*tr ) + as^2 * ( - 17/36*ep^-1*ca^2 + 1/6*ep^-1*nl*tr*cf + 5/18*ep^-1*nl*ca*tr - 13/48*ca^2 + 137/288*nl*tr*cf + 61/432*nl*ca*tr - 1/216*nl^2*tr^2 ) - 2/3*Q.Q^-2 ; C1_S = + as * ( + 22/27*ca - 8/27*nl*tr ) + as^2 * ( + 83/324*ca^2 - 2/9*nl*tr*cf - 8/81*nl*ca*tr - 4/81*nl^2*tr^2 ); C1_T = + as * ( - 5/18*ca - 5/72*nl*tr ) + as^2 * ( - 83/432*ca^2 + 43/96*nl*tr*cf + 41/432*nl*ca*tr - 1/216*nl^2*tr^2 ); (* ================================================================================= *) (* Wilson coefficient in front of O_1 for the correlator O1(x)O1(0) (in momentum space)*) C1GG = + ep^-1*as^2 * ( - 17/24*ca^2 + 1/4*nl*tr*cf + 5/12*nl*ca*tr ) + as * ( - 49/36*ca + 5/9*nl*tr - 11/12*Lmuq*ca + 1/3*Lmuq*nl*tr ) + as^2 * ( - 11509/1296*ca^2 + 13/4*nl*tr*cf + 3095/648*nl*ca*tr - 25/81*nl^2*tr^2 - 1151/216*Lmuq*ca^2 + Lmuq*nl*tr*cf + 97/27*Lmuq*nl*ca*tr - 10/27*Lmuq*nl^2*tr^2 - 121/144*Lmuq^2*ca^2 + 11/18*Lmuq^2*nl*ca*tr - 1/9*Lmuq^2*nl^2*tr^2 + 33/8*z3*ca^2 - 3*z3*nl*tr*cf + 3/2*z3*nl*ca*tr ) - 1 ;