•Check of scale independence

logRule = Log[a_] :> 0 /; FreeQ[{a}, ScaleMu] ;

logRule1 = Log[a_ * b_] -> Log[a] + Log[b] ;

scaleRule = CouplingConstant[l_[4], i_, r___] :> RenormalizationCoefficients[l[4]][[i]]/(32 Pi^2) (CouplingConstant[l[4], i, r] + Log[ParticleMass[Pion, RenormalizationState[1]]^2/ScaleMu^2]) /; RenormalizationCoefficients[l[4]][[i]] =!= 0 ;

rlogs = Simplify[logs /. {i1 -> 1, i2 -> 1, i3 -> 2, i4 -> 2} /. logRule1 /. logRule // MandelstamReduce[#, Cancel -> MandelstamU] & // Expand] // FullSimplify

-(log (1/μ^2) (9 (m _ π^(ó  r ))^4 - 4 (s + 4 t) (m _ π^(ó  r ))^2 + 4 (s^2 + t s + t^2)))/(96 π^2 (f _ π^(ó  r ))^4)

slogs = Simplify[(cts /. {i1 -> 1, i2 -> 1, i3 -> 2, i4 -> 2} /. scaleRule) - (cts /. {i1 -> 1, i2 -> 1, i3 -> 2, i4 -> 2} /. scaleRule /. _Log -> 0) /. logRule1 /. logRule // MandelstamReduce[#, Cancel -> MandelstamU] & // Expand] // FullSimplify

(log (1/μ^2) (9 (m _ π^(ó  r ))^4 - 4 (s + 4 t) (m _ π^(ó  r ))^2 + 4 (s^2 + t s + t^2)))/(96 π^2 (f _ π^(ó  r ))^4)

slogs + rlogs // Simplify

0

Converted by Mathematica  (July 10, 2003)