(* Load definitions from Kpinunu.m *)

Definitions

Definition of the coefficients of the beta function

be0=(-((11 nc)/3)+((2 nf)/3));
be1=(-((34 (nc^2))/3)-(nf/nc)+((13 nc nf)/3));
be2=(-((2857 (nc^3))/54)-((187 nf)/36)-(nf/(4 (nc^2)))+((1709 (nc^2) nf)/54)+((11 (nf^2))/(18 nc))-((56 nc (nf^2))/27));

nc=3;

Definition of the functions (Subscript[U, x])(μ,(Subscript[μ, 0])) and (Subscript[U, x,ν])(μ,(Subscript[μ, 0])) as U[a(μ), a((Subscript[μ, 0])),(Subscript[γ, x]^0),(Subscript[γ, x]^1),(Subscript[γ, x]^2)] and Und[a(μ), a((Subscript[μ, 0])),(Subscript[γ, x]^0),(Subscript[γ, x]^1),(Subscript[γ, x]^2),(Subscript[γ, x,ν]^0),(Subscript[γ, x,ν]^1),(Subscript[γ, x,ν]^2)], respectively

U[aMu_,aMu0_,g0_,g1_,g2_]=((((aMu/aMu0)))^(g0/be0)) (1+(-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)) (((aMu0 be1 g0)/(be0^2))-((aMu0 g1)/be0))+((-aMu be1 g0+aMu0 be1 g0+aMu be0 g1-aMu0 be0 g1)/(be0^2))+(((aMu0^2) (-be0 (be1^2) g0+(be0^2) be2 g0+(be1^2) (g0^2)+(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)-(be0^3) g2))/(2 (be0^4)))+(((aMu0^2) (-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)) (-be0 (be1^2) g0+(be0^2) be2 g0+(be1^2) (g0^2)+(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)-(be0^3) g2))/(2 (be0^4)))+(((aMu^2) (be0 (be1^2) g0-(be0^2) be2 g0+(be1^2) (g0^2)-(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)+(be0^3) g2))/(2 (be0^4)))+(((aMu^2) (((aMu0 be1 g0)/(be0^2))-((aMu0 g1)/be0)) (be0 (be1^2) g0-(be0^2) be2 g0+(be1^2) (g0^2)-(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)+(be0^3) g2))/(2 (be0^4)))+(1/(6 (be0^6)))(aMu^3) (-2 (be0^2) (be1^3) g0+4 (be0^3) be1 be2 g0-2 (be0^4) be3 g0-3 be0 (be1^3) (g0^2)+3 (be0^2) be1 be2 (g0^2)-(be1^3) (g0^3)+2 (be0^3) (be1^2) g1-2 (be0^4) be2 g1+6 (be0^2) (be1^2) g0 g1-3 (be0^3) be2 g0 g1+3 be0 (be1^2) (g0^2) g1-3 (be0^3) be1 (g1^2)-3 (be0^2) be1 g0 (g1^2)+(be0^3) (g1^3)-2 (be0^4) be1 g2-3 (be0^3) be1 g0 g2+3 (be0^4) g1 g2)+(1/(6 (be0^6)))(aMu0^3) (2 (be0^2) (be1^3) g0-4 (be0^3) be1 be2 g0+2 (be0^4) be3 g0-3 be0 (be1^3) (g0^2)+3 (be0^2) be1 be2 (g0^2)+(be1^3) (g0^3)-2 (be0^3) (be1^2) g1+2 (be0^4) be2 g1+6 (be0^2) (be1^2) g0 g1-3 (be0^3) be2 g0 g1-3 be0 (be1^2) (g0^2) g1-3 (be0^3) be1 (g1^2)+3 (be0^2) be1 g0 (g1^2)-(be0^3) (g1^3)+2 (be0^4) be1 g2-3 (be0^3) be1 g0 g2+3 (be0^4) g1 g2));
Und[aMu_,aMu0_,g0_,g1_,g2_,gnd0_,gnd1_,gnd2_]=((((-((((aMu/aMu0)))^(g0/be0))+((((aMu/aMu0)))^(gnu0/be0))) gnd0)/(-g0+gnu0))+(((-((((aMu/aMu0)))^(g0/be0))+((((aMu/aMu0)))^(gnu0/be0))) gnd0 (-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)+((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0)))/(-g0+gnu0))+(((aMu ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) aMu0) gnd0 (-(be1/be0)+((be1 g0-be0 g1)/(be0^2))+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))))/(be0-g0+gnu0))+(((aMu ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) aMu0) gnd0 (-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)+((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0)) (-(be1/be0)+((be1 g0-be0 g1)/(be0^2))+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))))/(be0-g0+gnu0))+(1/(-g0+gnu0))(-((((aMu/aMu0)))^(g0/be0))+((((aMu/aMu0)))^(gnu0/be0))) gnd0 ((((aMu^2) (be0 (be1^2) g0-(be0^2) be2 g0+(be1^2) (g0^2)-(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)+(be0^3) g2))/(2 (be0^4)))+(-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)) (((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0))+(((aMu0^2) (-be0 (be1^2) gnu0+(be0^2) be2 gnu0+(be1^2) (gnu0^2)+(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)-(be0^3) gnu2))/(2 (be0^4))))+(1/(be0-g0+gnu0))(aMu ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) aMu0) gnd0 (-(be1/be0)+((be1 g0-be0 g1)/(be0^2))+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))) ((((aMu^2) (be0 (be1^2) g0-(be0^2) be2 g0+(be1^2) (g0^2)-(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)+(be0^3) g2))/(2 (be0^4)))+(-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)) (((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0))+(((aMu0^2) (-be0 (be1^2) gnu0+(be0^2) be2 gnu0+(be1^2) (gnu0^2)+(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)-(be0^3) gnu2))/(2 (be0^4))))+(1/(2 be0-g0+gnu0))((aMu^2) ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) (aMu0^2)) gnd0 (((-be0 (be1^2) g0+(be0^2) be2 g0+(be1^2) (g0^2)+(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)-(be0^3) g2)/(2 (be0^4)))+(((be1^2) gnd0-be0 be2 gnd0-be0 be1 gnd1+(be0^2) gnd2)/((be0^2) gnd0))+(((-(be1/be0)+(gnd1/gnd0)) (-be1 gnu0+be0 gnu1))/(be0^2))+(((be1 g0-be0 g1) (-(be1/be0)+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))))/(be0^2))+((be0 (be1^2) gnu0-(be0^2) be2 gnu0+(be1^2) (gnu0^2)-(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)+(be0^3) gnu2)/(2 (be0^4))))+(1/(2 be0-g0+gnu0))((aMu^2) ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) (aMu0^2)) gnd0 (-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)+((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0)) (((-be0 (be1^2) g0+(be0^2) be2 g0+(be1^2) (g0^2)+(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)-(be0^3) g2)/(2 (be0^4)))+(((be1^2) gnd0-be0 be2 gnd0-be0 be1 gnd1+(be0^2) gnd2)/((be0^2) gnd0))+(((-(be1/be0)+(gnd1/gnd0)) (-be1 gnu0+be0 gnu1))/(be0^2))+(((be1 g0-be0 g1) (-(be1/be0)+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))))/(be0^2))+((be0 (be1^2) gnu0-(be0^2) be2 gnu0+(be1^2) (gnu0^2)-(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)+(be0^3) gnu2)/(2 (be0^4))))+(1/(-g0+gnu0))(-((((aMu/aMu0)))^(g0/be0))+((((aMu/aMu0)))^(gnu0/be0))) gnd0 ((1/(6 (be0^6)))(aMu^3) (-2 (be0^2) (be1^3) g0+4 (be0^3) be1 be2 g0-2 (be0^4) be3 g0-3 be0 (be1^3) (g0^2)+3 (be0^2) be1 be2 (g0^2)-(be1^3) (g0^3)+2 (be0^3) (be1^2) g1-2 (be0^4) be2 g1+6 (be0^2) (be1^2) g0 g1-3 (be0^3) be2 g0 g1+3 be0 (be1^2) (g0^2) g1-3 (be0^3) be1 (g1^2)-3 (be0^2) be1 g0 (g1^2)+(be0^3) (g1^3)-2 (be0^4) be1 g2-3 (be0^3) be1 g0 g2+3 (be0^4) g1 g2)+(((aMu^2) (be0 (be1^2) g0-(be0^2) be2 g0+(be1^2) (g0^2)-(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)+(be0^3) g2) (((aMu0 be1 gnu0)/(be0^2))-((aMu0 gnu1)/be0)))/(2 (be0^4)))+(1/(2 (be0^4)))(aMu0^2) (-((aMu be1 g0)/(be0^2))+((aMu g1)/be0)) (-be0 (be1^2) gnu0+(be0^2) be2 gnu0+(be1^2) (gnu0^2)+(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)-(be0^3) gnu2)+(1/(6 (be0^6)))(aMu0^3) (2 (be0^2) (be1^3) gnu0-4 (be0^3) be1 be2 gnu0+2 (be0^4) be3 gnu0-3 be0 (be1^3) (gnu0^2)+3 (be0^2) be1 be2 (gnu0^2)+(be1^3) (gnu0^3)-2 (be0^3) (be1^2) gnu1+2 (be0^4) be2 gnu1+6 (be0^2) (be1^2) gnu0 gnu1-3 (be0^3) be2 gnu0 gnu1-3 be0 (be1^2) (gnu0^2) gnu1-3 (be0^3) be1 (gnu1^2)+3 (be0^2) be1 gnu0 (gnu1^2)-(be0^3) (gnu1^3)+2 (be0^4) be1 gnu2-3 (be0^3) be1 gnu0 gnu2+3 (be0^4) gnu1 gnu2-2 (be0^5) gnu3))+(1/(3 be0-g0+gnu0))((aMu^3) ((((aMu/aMu0)))^(gnu0/be0))-((((aMu/aMu0)))^(g0/be0)) (aMu0^3)) gnd0 ((1/(6 (be0^6)))(2 (be0^2) (be1^3) g0-4 (be0^3) be1 be2 g0+2 (be0^4) be3 g0-3 be0 (be1^3) (g0^2)+3 (be0^2) be1 be2 (g0^2)+(be1^3) (g0^3)-2 (be0^3) (be1^2) g1+2 (be0^4) be2 g1+6 (be0^2) (be1^2) g0 g1-3 (be0^3) be2 g0 g1-3 be0 (be1^2) (g0^2) g1-3 (be0^3) be1 (g1^2)+3 (be0^2) be1 g0 (g1^2)-(be0^3) (g1^3)+2 (be0^4) be1 g2-3 (be0^3) be1 g0 g2+3 (be0^4) g1 g2)+((-(be1^3) gnd0+2 be0 be1 be2 gnd0-(be0^2) be3 gnd0+be0 (be1^2) gnd1-(be0^2) be2 gnd1-(be0^2) be1 gnd2)/((be0^3) gnd0))+((((be1^2) gnd0-be0 be2 gnd0-be0 be1 gnd1+(be0^2) gnd2) (-be1 gnu0+be0 gnu1))/((be0^4) gnd0))+(((-be0 (be1^2) g0+(be0^2) be2 g0+(be1^2) (g0^2)+(be0^2) be1 g1-2 be0 be1 g0 g1+(be0^2) (g1^2)-(be0^3) g2) (-(be1/be0)+(gnd1/gnd0)+((-be1 gnu0+be0 gnu1)/(be0^2))))/(2 (be0^4)))+(((-(be1/be0)+(gnd1/gnd0)) (be0 (be1^2) gnu0-(be0^2) be2 gnu0+(be1^2) (gnu0^2)-(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)+(be0^3) gnu2))/(2 (be0^4)))+(1/(be0^2))(be1 g0-be0 g1) ((((be1^2) gnd0-be0 be2 gnd0-be0 be1 gnd1+(be0^2) gnd2)/((be0^2) gnd0))+(((-(be1/be0)+(gnd1/gnd0)) (-be1 gnu0+be0 gnu1))/(be0^2))+((be0 (be1^2) gnu0-(be0^2) be2 gnu0+(be1^2) (gnu0^2)-(be0^2) be1 gnu1-2 be0 be1 gnu0 gnu1+(be0^2) (gnu1^2)+(be0^3) gnu2)/(2 (be0^4))))+(1/(6 (be0^6)))(-2 (be0^2) (be1^3) gnu0+4 (be0^3) be1 be2 gnu0-2 (be0^4) be3 gnu0-3 be0 (be1^3) (gnu0^2)+3 (be0^2) be1 be2 (gnu0^2)-(be1^3) (gnu0^3)+2 (be0^3) (be1^2) gnu1-2 (be0^4) be2 gnu1+6 (be0^2) (be1^2) gnu0 gnu1-3 (be0^3) be2 gnu0 gnu1+3 be0 (be1^2) (gnu0^2) gnu1-3 (be0^3) be1 (gnu1^2)-3 (be0^2) be1 gnu0 (gnu1^2)+(be0^3) (gnu1^3)-2 (be0^4) be1 gnu2-3 (be0^3) be1 gnu0 gnu2+3 (be0^4) gnu1 gnu2+2 (be0^5) gnu3)));

Decoupling at (Subscript[μ, b]) and evolution to (Subscript[μ, c]) for (Subscript[Q, ν]) and (Subscript[A, ±]^(NS,S))

(*  (([(Subscript[Q, ν])((Subscript[μ, b]))])^((5))) in terms ((of [(Subscript[Q, ν])((Subscript[μ, c]))])^((4))) *)

Qnub=(Bnu/.a-> a5mub)*(U[a4mub,a4muc,gnu0,gnu1,gnu2]/.nf->4)*Qnu;


(* (([(Subscript[A, ±]^NS)((Subscript[μ, b]))])^((5))) in terms ((of [(Subscript[Q, ν])((Subscript[μ, c]))])^((4))) ((and [(Subscript[A, ±]^S)((Subscript[μ, b]))])^((4))) *)

ANSpb=(Bp/.a->a5mub)*(4/5*(Cb/.a->a5mub)-1/5)ASp4+(4/5*Bbpnu-1/5*Bcupnu/.a->a5mub)*(U[a4mub,a4muc,gnu0,gnu1,gnu2]/.nf->4)*Qnu;
ANSmb=(Bm/.a->a5mub)*(4/5*(Cb/.a->a5mub)-1/5)ASm4+(4/5*Bbmnu-1/5*Bcumnu/.a->a5mub)*(U[a4mub,a4muc,gnu0,gnu1,gnu2]/.nf->4)*Qnu;


(* (([(Subscript[A, ±]^S)((Subscript[μ, b]))])^((5))) in terms ((of [(Subscript[Q, ν])((Subscript[μ, c]))])^((4))) ((and [(Subscript[A, ±]^S)((Subscript[μ, b]))])^((4))) *)

ASpb=(Bp/.a->a5mub)*(Cb+1/.a->a5mub)ASp4+(Bbpnu+Bcupnu/.a->a5mub)*(U[a4mub,a4muc,gnu0,gnu1,gnu2]/.nf->4)Qnu;
ASmb=(Bm/.a->a5mub)*(Cb+1/.a->a5mub)ASm4+(Bbmnu+Bcumnu/.a->a5mub)*(U[a4mub,a4muc,gnu0,gnu1,gnu2]/.nf-> 4)Qnu;


(* (([(Subscript[A, ±]^S)((Subscript[μ, b]))])^((4))) in terms ((of [(Subscript[Q, ν])((Subscript[μ, c]))])^((4))) ((and [(Subscript[A, ±]^S)((Subscript[μ, c]))])^((4))) *)

ASp4=(U[a4mub,a4muc,gp0+gS0,gp1+gS1,gp2+gS2]/.nf->4)(CSpnu/.a->a4muc) Qnu + (Und[a4mub,a4muc,gp0+gS0,gp1+gS1,gp2+gS2,gSpnd0,gSpnd1,gSpnd2]/.nf->4)Qnu;
ASm4=(U[a4mub,a4muc,gm0+gS0,gm1+gS1,gm2+gS2]/.nf->4)(CSmnu/.a->a4muc) Qnu + (Und[a4mub,a4muc,gm0+gS0,gm1+gS1,gm2+gS2,gSmnd0,gSmnd1,gSmnd2]/.nf->4)Qnu;

Expressions for the resummed anomalous diagrams

(* Top quark anomalous diagrams *)

At=-(Ct/.a->a6mut)((CSnu/.a->a5mu)*(U[a5mu,a5mub,gnu0,gnu1,gnu2]/.nf->5)Qnub + (Cp/.a->a5mu/.nf->5)*((U[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2]/.nf->5)ASpb+(Und[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2,gSpnd0,gSpnd1,gSpnd2]/.nf->5)Qnub) + (Cm/.a->a5mu/.nf->5)*((U[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2]/.nf->5)ASmb+(Und[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2,gSmnd0,gSmnd1,gSmnd2]/.nf->5)Qnub));

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^b)((Subscript[μ, W]))])^((5))) *)

Ab=((Cbnu/.a->a5mu)(U[a5mu,a5mub,gnu0,gnu1,gnu2]/.nf->5)Qnub
+(Cp/.a->a5mu/.nf->5)(((U[a5mu,a5mub,gp0,gp1,gp2]/.nf->5)ANSpb+1/5*(Und[a5mu,a5mub,gp0,gp1,gp2,gNSpnd0,gNSpnd1,gNSpnd2]/.nf->5)Qnub) + 1/5*((U[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2]/.nf->5)ASpb+(Und[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2,gSpnd0,gSpnd1,gSpnd2]/.nf->5)Qnub))
+(Cm/.a->a5mu/.nf->5)(((U[a5mu,a5mub,gm0,gm1,gm2]/.nf->5)ANSmb+1/5*(Und[a5mu,a5mub,gm0,gm1,gm2,gNSmnd0,gNSmnd1,gNSmnd2]/.nf->5)Qnub) + 1/5*((U[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2]/.nf->5)ASmb+(Und[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2,gSmnd0,gSmnd1,gSmnd2]/.nf->5)Qnub)));

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^S)((Subscript[μ, W]))])^((5))) *)

AS=1/5*((U[a5mut,a5mu,gS0,gS1,gS2]/.nf->5)-1)*((CSnu/.a->a5mu)*(U[a5mu,a5mub,gnu0,gnu1,gnu2]/.nf->5)Qnub + (Cp/.a->a5mu/.nf->5)*((U[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2]/.nf->5)ASpb+(Und[a5mu,a5mub,gp0+gS0,gp1+gS1,gp2+gS2,gSpnd0,gSpnd1,gSpnd2]/.nf->5)Qnub) + (Cm/.a->a5mu/.nf->5)*((U[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2]/.nf->5)ASmb+(Und[a5mu,a5mub,gm0+gS0,gm1+gS1,gm2+gS2,gSmnd0,gSmnd1,gSmnd2]/.nf->5)Qnub));

Analytic results

Results in terms of (Subscript[m, c])((Subscript[μ, c])) and (Subscript[m, b])((Subscript[μ, b]))

(* Top quark anomalous diagrams *)

At1=Collect[Normal[((Coefficient[At,Qnu]/.x->1)/.a6mut->x a6mut/.a5mut->x a5mut/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^b)((Subscript[μ, W]))])^((5))) *)

Ab1=Collect[Normal[((Coefficient[Ab,Qnu]/.x->1)/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^S)((Subscript[μ, W]))])^((5))) *)

AS1=Collect[Normal[((Coefficient[AS,Qnu]/.x->1)/.a5mut->x a5mut/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

Results in terms of (Subscript[m, c])((Subscript[m, c]))and (Subscript[m, b])((Subscript[m, b]))

(* Anomalous dimensions for (Subscript[m, c])(μ) *)

gM0=-4;
gM1=(-(202/3)+((20 nf)/9));
gM2=(1/81) (-101169+6648 nf+140 (nf^2)+4320 nf z3);

(* Conversion factor between (Subscript[m, c]^2)((Subscript[μ, c])) and (Subscript[m, c]^2)((Subscript[m, c])) *)
mamuctomc=Collect[Normal[((((U[a4muc,a4mc,gM0,gM1,gM2]))^2)/.a4mc-> x a4mc/.a4muc-> x a4muc/.nf->nf)+O[x]^4],{x},Simplify]/.x-> 1;

(* Conversion factor between (Subscript[m, b]^2)((Subscript[μ, b])) and (Subscript[m, b]^2)((Subscript[m, b])) *)mamubtomb=Collect[Normal[((((U[a5mub,a5mb,gM0,gM1,gM2]))^2)/.a5mb-> x a5mb/.a5mub-> x a5mub/.nf->nf/.nc->3)+O[x]^4],{x},Simplify]/.x-> 1;

(* Conversion factors for the masses inside logs *)

Lcorr=Collect[Normal[(Log[mamuctomc]/.a4mc-> x a4mc/.a4muc-> x a4muc)+O[x]^4],{x},Simplify]/.x-> 1;

Lcorr2=Collect[Normal[(Log[mamubtomb]/.a5mb-> x a5mb/.a5mub-> x a5mub/.nf-> 5)+O[x]^4],{x},Simplify]/.x-> 1;

At2=Collect[Normal[((Coefficient[At,Qnu]/.x->1)*mamuctomc/.Lmucmc-> Lmucmc - Lcorr/.Lmubmb-> Lmubmb - Lcorr2/.nf-> 4/.a6mut->x a6mut/.a5mut->x a5mut/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

Ab2=Collect[Normal[((Coefficient[Ab,Qnu]/.x->1)*mamuctomc/.Lmucmc-> Lmucmc - Lcorr/.Lmubmb-> Lmubmb - Lcorr2/.nf-> 4/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

AS2=Collect[Normal[((Coefficient[AS,Qnu]/.x->1)*mamuctomc/.Lmucmc-> Lmucmc - Lcorr/.Lmubmb-> Lmubmb - Lcorr2/.nf-> 4/.a5mut->x a5mut/.a5mu->x a5mu/.a5mub-> x a5mub/.a5mb-> x a5mb/.a4mub-> x a4mub/.a4muc-> x a4muc/.a4mc-> x a4mc)+O[x]^3],{x},Simplify];

(* Top quark anomalous diagrams *)

ResultAt=Collect[Normal[(Collect[Normal[(Expand[(At2 )]/.((((a4muc/a4mc)))^(24/25)) ->(K1^(24/25))/.((((a4mub/a4muc)))^(24/25))-> (K2^(-24/25))/.((((a4mub/a4muc)))^(13/25))-> (K2^(-13/25))/.((((a4mub/a4muc)))^(6/25))-> (K2^(-6/25))/.((((a5mu/a5mub)))^(1/23))-> (K3^(-1/23))/.((((a5mu/a5mub)))^(6/23))-> (K3^(-6/23))/.((((a5mu/a5mub)))^(11/23))-> (K3^(-11/23)) /.(a4muc/a4mub)-> K2/.((a4muc^2)/a4mub)-> K2 a4muc/.(a5mub/a5mu)-> K3/.((a5mub^2)/a5mu)-> K3 a5mub)+O[x]^3],{x,a6mt,a5mu,a4mub,a5mub,a4mc,a4muc,Lmucmc,Log,K1,K2,K3,K4,K5},Simplify]/.(1/(K3^(6/23)))->  (iK3^(6/23))/.(1/(K3^(1/23)))->  (iK3^(1/23))/.(1/(K3^(11/23)))->  (iK3^(11/23))/.(1/(K2^(6/25)))->  (iK2^(6/25))/.(1/(K2^(24/25)))->  (iK2^(24/25))/.(1/(K2^(13/25)))->  (iK2^(13/25))/. Log[((((a5mub/a5mb)))^(24/23))]-> (24/23) Log[((a5mub/a5mb))]/. (iK2^(24/25))-> (a4mub/a4muc)(K2^(1/25))/.(iK3^(11/23))-> (a5mu/a5mub)M12/.  (iK2^(13/25))-> (a4mub/a4muc)L12/. (iK2^(6/25))-> iL6 /.(K2^(1/25))->L1/.(K2^(12/25))-> L12/.(K3^(1/23))-> M1/.(K3^(12/23))-> M12/.(iK3^(6/23))->iM6/.(iK3^(1/23))->iM1 /.K1-> Kc/.iK1-> iKc/.a4mc-> a4muc(1+ x  a4muc b0 Lmucmc+ (x^2) YYY (a4muc^2) /.nf->4)/.a5mb-> a5mub(1+ x  a5mub b0 Lmubmb+ (x^2) XXX (a5mub^2)/.nf->5)/.a4mub-> a5mub(1+ 4x a5mub(-(1/6)Lmubmb)))+O[x]^3]/.(-1+4 Lmutmt)->Z,{Kc,x,a6mut,Z},Expand]/.Log[(a4muc/a4mc)]-> Log[Kc]/.Log[(1/iK1)]-> Log[Kc]/.Z-> (-1+4 Lmutmt)

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^b)((Subscript[μ, W]))])^((5))) *)

ResultAb=Collect[Normal[(Collect[Normal[(Expand[(Ab2 )]/.((((a4muc/a4mc)))^(24/25)) ->(K1^(24/25))/.((((a4mub/a4muc)))^(24/25))-> (K2^(-24/25))/.((((a4mub/a4muc)))^(13/25))-> (K2^(-13/25))/.((((a4mub/a4muc)))^(6/25))-> (K2^(-6/25))/.((((a5mu/a5mub)))^(1/23))-> (K3^(-1/23))/.((((a5mu/a5mub)))^(6/23))-> (K3^(-6/23))/.((((a5mu/a5mub)))^(11/23))-> (K3^(-11/23)) /.(a4muc/a4mub)-> K2/.((a4muc^2)/a4mub)-> K2 a4muc/.(a5mub/a5mu)-> K3/.((a5mub^2)/a5mu)-> K3 a5mub)+O[x]^3],{x,a6mt,a5mu,a4mub,a5mub,a4mc,a4muc,Lmucmc,Log,K1,K2,K3,K4,K5},Simplify]/.(1/(K3^(6/23)))->  (iK3^(6/23))/.(1/(K3^(1/23)))->  (iK3^(1/23))/.(1/(K3^(11/23)))->  (iK3^(11/23))/.(1/(K2^(6/25)))->  (iK2^(6/25))/.(1/(K2^(24/25)))->  (iK2^(24/25))/.(1/(K2^(13/25)))->  (iK2^(13/25))/. Log[((((a5mub/a5mb)))^(24/23))]-> (24/23) Log[((a5mub/a5mb))]/. (iK2^(24/25))-> (a4mub/a4muc)(K2^(1/25))/.(iK3^(11/23))-> (a5mu/a5mub)M12/.  (iK2^(13/25))-> (a4mub/a4muc)L12/. (iK2^(6/25))-> iL6 /.(K2^(1/25))->L1/.(K2^(12/25))-> L12/.(K3^(1/23))-> M1/.(K3^(12/23))-> M12/.(iK3^(6/23))->iM6/.(iK3^(1/23))->iM1 /.K1-> Kc/.iK1-> iKc/.a4mc-> a4muc(1+ x  a4muc b0 Lmucmc+ (x^2) YYY (a4muc^2) /.nf->4)/.a5mb-> a5mub(1+ x  a5mub b0 Lmubmb+ (x^2) XXX (a5mub^2)/.nf->5)/.a4mub-> a5mub(1+ 4x a5mub(-(1/6)Lmubmb)))+O[x]^3],{Qnu,Kc,x,a6mt,a5mu,a4mub,a5mub,a4mc,a4muc,L1,Lmucmc,LnuMW,Lmubmb,},Expand]/.Log[(a4muc/a4mc)]-> Log[Kc]/.Log[(1/iK1)]-> Log[Kc]

(* Bottom quark anomalous diagrams: Contribution ((from [(Subscript[A, ±]^S)((Subscript[μ, W]))])^((5))) *)

ResultAS=Collect[Collect[Normal[(Collect[Normal[(Expand[(AS2 )]/.((((a4muc/a4mc)))^(24/25)) ->(K1^(24/25))/.((((a4mub/a4muc)))^(24/25))-> (K2^(-24/25))/.((((a4mub/a4muc)))^(13/25))-> (K2^(-13/25))/.((((a4mub/a4muc)))^(6/25))-> (K2^(-6/25))/.((((a5mu/a5mub)))^(1/23))-> (K3^(-1/23))/.((((a5mu/a5mub)))^(6/23))-> (K3^(-6/23))/.((((a5mu/a5mub)))^(11/23))-> (K3^(-11/23)) /.(a4muc/a4mub)-> K2/.((a4muc^2)/a4mub)-> K2 a4muc/.(a5mub/a5mu)-> K3/.((a5mub^2)/a5mu)-> K3 a5mub)+O[x]^3],{x,,a4mub,a5mub,a4mc,a4muc,Lmucmc,Log,K1,K2,K3,K4,K5},Simplify]/.(1/(K3^(6/23)))->  (iK3^(6/23))/.(1/(K3^(1/23)))->  (iK3^(1/23))/.(1/(K3^(11/23)))->  (iK3^(11/23))/.(1/(K2^(6/25)))->  (iK2^(6/25))/.(1/(K2^(24/25)))->  (iK2^(24/25))/.(1/(K2^(13/25)))->  (iK2^(13/25))/. Log[((((a5mub/a5mb)))^(24/23))]-> (24/23) Log[((a5mub/a5mb))]/. (iK2^(24/25))-> (a4mub/a4muc)(K2^(1/25))/.(iK3^(11/23))-> (a5mu/a5mub)M12/.  (iK2^(13/25))-> (a4mub/a4muc)L12/. (iK2^(6/25))-> iL6 /.(K2^(1/25))->L1/.(K2^(12/25))-> L12/.(K3^(1/23))-> M1/.(K3^(12/23))-> M12/.(iK3^(6/23))->iM6/.(iK3^(1/23))->iM1 /.K1-> Kc/.iK1-> iKc/.a4mc-> a4muc(1+ x  a4muc b0 Lmucmc+ (x^2) YYY (a4muc^2) /.nf->4)/.a5mb-> a5mub(1+ x  a5mub b0 Lmubmb+ (x^2) XXX (a5mub^2)/.nf->5)/.a4mub-> a5mub(1+ 4x a5mub(-(1/6)Lmubmb)))+O[x]^3],{Kc,x,a6mt,a4mub,a5mub,a4mc,a4muc,L1,Lmucmc,LnuMW,Lmubmb,Log},Simplify]/.Log[(a4muc/a4mc)]-> Log[Kc]/.Log[(1/iK1)]-> Log[Kc]/.(a5mu-a5mut)->Z,{Kc,x,Z,a5mu,a4mub,a5mub,a4mc,a4muc},Expand]/.Z-> (a5mu-a5mut)

Reexpansion of the results

(* Connection between alpha at nf+1 and alpha at nf, where mb is the large scale being decoupled *)

zdecamb=1+ 4a(-(1/6)Lnumb)+16(a^2)((11/72)-(11/24)Lnumb+(1/36)(Lnumb^2))+64(a^3)((564731/124416)-(82043/27648)z3-(955/576)Lnumb+(53/576)(Lnumb^2)-(1/216)(Lnumb^3)+nf(-(2633/31104)+(67/576)Lnumb-(1/36)(Lnumb^2)));

(* Connection between M4 at nf+1 and M4 at nf, where mb is the large scale being decoupled *)

zdecMmb=1+16(a^2)((89/432)-(5/36)Lnumb+(1/12)(Lnumb^2))+64(a^3)((2951/2916)-(407/864)z3+(5/4)z4-(1/36)B4+(-(311/2592)-(5/6)z3)Lnumb+(175/432)(Lnumb^2)+(29/216)(Lnumb^3)+nf((1327/11664)-(2/27)z3-(53/432)Lnumb-(1/108)(Lnumb^3)));

(* Connection between xil at nf+1 and xil at nf, where mb is the large scale being decoupled *)

zdecxilmb=1+4a((1/6)Lnumb)+16(a^2)((91/1152)+(29/96)Lnumb+(3/32)(Lnumb^2))+64(a^3)(-(284023/62208)+(86183/27648)z3+(99/128)z4-(1/32)B4+((52433/27648)-(33/64)z3)Lnumb+(383/2304)(Lnumb^2)+(119/768)(Lnumb^3)+nf((3307/15552)-(1/12)z3-(293/1152)Lnumb+(1/36)(Lnumb^2)-(1/96)(Lnumb^3))+(xil+1)(-(677/1536)+(3/32)z3+(233/1024)Lnumb-(3/32)(Lnumb^2)+(3/256)(Lnumb^3)));

(* (m^2)((Subscript[μ, 0])) in terms of (m^2)(μ) *)

mamuctoNu=Collect[Normal[((((U[a5muc,a5mu,gM0,gM1,gM2]))^2)/.a5mu-> x a5mu/.a5muc-> x a5muc/.nf->nf/.nc->3)+O[x]^4],{x},Simplify]/.x-> 1;

(* (Subscript[(m^2), c])((Subscript[μ, 0])) in terms of (Subscript[(m^2), c])(μ) inside logs *)

Lcorr3=Collect[Normal[(Log[mamuctoNu]/.a5mu-> x a5mu/.a5muc-> x a5muc)+O[x]^4],{x},Simplify]/.x-> 1;

(* nf = number of massless flavors *)

Collect[Normal[((1/32)(((zdecMmb^2)/.a->a6mut/.Lnumb-> Lmutmt/.nf->5)(mamuctoNu/.a5mu-> a5mut/.a5muc->a5mu/.nf-> 5)(mamuctoNu/.a5muc-> a5mub/.nf-> 5)((zdecMmb^2)/.a->a5mub/.Lnumb-> Lmubmb/.nf->4)(mamuctoNu/.a5mu-> a4mub/.a5muc->a4muc/.nf-> 4)( Ab1+AS1+At1)((a6mut/a4muc)))/.Lmucmc-> Lmucmc -(Lcorr3/.a5mu-> a4mub/.a5muc->a4muc/.nf->4)/.Lmucmc-> Lmucmc-( Log[(zdecMmb^2)]/.a->a5mub/.Lnumb-> Lmubmb/.nf->4 )/.Lmucmc-> Lmucmc -(Lcorr3/.a5mu->a5mut/.a5muc->a5mub/.nf->5)/.Lmucmc-> Lmucmc-( Log[(zdecMmb^2)]/.a->a6mut/.Lnumb-> Lmutmt/.nf->5 )/.a4muc-> a4mub(1- a4mub be0 Lmubmuc+a4mub^2(-be1 Lmubmuc+be0^2Lmubmuc^2)/.nf->4)/.a4mub-> (zdecamb*a/.a-> a5mub/.Lnumb-> Lmubmb/.nf->4)/.Lmubmb-> Lmubmb -(Lcorr3/.a5mu->a5mut/.a5muc->a5mub/.nf->5)/.Lmubmb-> Lmubmb-( Log[(zdecMmb^2)]/.a->a6mut/.Lnumb-> Lmutmt/.nf->5 )/.a5mub-> a5mu(1- a5mu be0 Lmumub+a5mu^2(-be1 Lmumub+be0^2Lmumub^2)/.nf->5)/.a5mu-> a5mut(1+ a5mut be0 Lmumut+a5mut^2(be1 Lmumut+be0^2Lmumut^2)/.nf->5)/.a5mut-> (zdecamb*a/.a->  a6mut/.Lnumb-> Lmutmt/.nf->5)/.a6mut-> a/.nc-> 3)+O[a]^4]/.x->1,{a,x,Lmubmuc,Lmumub},Simplify]

Numerical results

(* Rundec.m obtained from hep-ph/0004189 *)

<< "~/Desktop/Work/RunDec/RunDec.m"

Parameter values

(* mb(mb) and mc(mc) taken from hep-ph/0702103 *)

MW=80.4;
mt=162;
mb=4.16;
mc=1.29;
mtau=1.777;

Definition of α

a6[μ_]:=Module[{a1,adec,a2},
a1=AlphasExact[0.118,91.18,mt,5,2];
adec=DecAsUpMS[a1,mt,mt,5,1];
a2=AlphasExact[adec,mt,μ,6,2];
Return[a2];];

a5[μ_]:=Module[{a1,adec,a2},
a1=AlphasExact[0.118,91.18,μ,5,2];
Return[a1];];

a4[μ_]:=Module[{a1,adec,a2},
a1=AlphasExact[0.118,91.18,10,5,2];
adec=DecAsDownOS[a1,4.75,10,4,1];
a2=AlphasExact[adec,10,μ,4,2];
Return[a2];];

Numerical results, top diagrams

AtN[μ_,mub_,nu_,nut_]:=(1/(4π a4[μ]))(At2/.x->1/.a6mut-> ((a6[nut])/(4π))/.a5mu-> ((a5[nu])/(4π))/.a5mub-> ((a5[mub])/(4π))/.a5mb-> ((a5[mb])/(4π))/.a4mub-> ((a4[mub])/(4π))/.a4muc-> ((a4[μ])/(4π))/.a4mc-> ((a4[mc])/(4π))/.Lmucmc-> Log[((μ^2)/(mc^2))]/.Lmubmb-> Log[((mub^2)/(mb^2))]/.LmuMW-> Log[((nu^2)/(MW^2))]/.Lmutmt-> Log[((nut^2)/(mt^2))]/.z2->Zeta[2]/.z3->Zeta[3]);

(* Central value *)

XAt=N[((π^2)/2)((mc^2)/(MW^2))(AtN[3,10,MW,mt])]

(* Uncertainties *)

errmucAt=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[1.5,10,MW,mt])-(AtN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,10,MW,mt])-(AtN[4.5,10,MW,mt]))/.c4->1]]]

errmubAt=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,5,MW,mt])-(AtN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,15,MW,mt])-(AtN[3,10,MW,mt]))/.c4->1]]]

errmuWAt=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,10,40,mt])-(AtN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,10,160,mt])-(AtN[3,10,MW,mt]))/.c4->1]]]

errmutAt=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,10,MW,(mt/2)])-(AtN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AtN[3,10,MW,2mt])-(AtN[3,10,MW,mt]))/.c4->1]]]

errXAt=(Abs[errmucAt]+Abs[errmubAt]+Abs[errmuWAt])

(* Total *)

XAt + err*errXAt

Numerical results, bottom diagrams

AbSN[μ_,mub_,nu_,nut_]:=(1/(4π a4[μ]))(Ab2+ AS2/.x->1/.a5mut-> ((a5[nut])/(4π))/.a5mu-> ((a5[nu])/(4π))/.a5mub-> ((a5[mub])/(4π))/.a5mb-> ((a5[mb])/(4π))/.a4mub-> ((a4[mub])/(4π))/.a4muc-> ((a4[μ])/(4π))/.a4mc-> ((a4[mc])/(4π))/.Lmucmc-> Log[((μ^2)/(mc^2))]/.Lmubmb-> Log[((mub^2)/(mb^2))]/.LmuMW-> Log[((nu^2)/(MW^2))]/.z2->Zeta[2]/.z3->Zeta[3]);

(* Central value *)

XAb=N[((π^2)/2)((mc^2)/(MW^2))(AbSN[3,10,MW,mt])]

(* Uncertainties *)

errmucAb=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[1.5,10,MW,mt])-(AbSN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,10,MW,mt])-(AbSN[4.5,10,MW,mt]))/.c4->1]]]

errmubAb=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,5,MW,mt])-(AbSN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,15,MW,mt])-(AbSN[3,10,MW,mt]))/.c4->1]]]

errmuWAb=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,10,40,mt])-(AbSN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,10,160,mt])-(AbSN[3,10,MW,mt]))/.c4->1]]]

errmutAb=Max[Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,10,MW,(mt/2)])-(AbSN[3,10,MW,mt]))/.c4->1]],Abs[N[((π^2)/2)((mc^2)/(MW^2))((AbSN[3,10,MW,2mt])-(AbSN[3,10,MW,mt]))/.c4->1]]]

errXAb=(Abs[errmucAb]+Abs[errmubAb]+Abs[errmuWAb])

(* Total *)

XAb + err*errXAb