•Preliminaries

cancelScales = Log[ParticleMass[PionPlus, r___]^2/ScaleMu^2] -> Log[ParticleMass[PionZero, r]^2/ScaleMu^2] + Log[ParticleMass[PionPlus, r]^2/ParticleMass[PionZero, r]^2] ;

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

scaleRuleinv = CouplingConstant[l_[4], i_, r___] :> (32 Pi^2)/RenormalizationCoefficients[l[4]][[i]] CouplingConstant[l[4], i, r] - Log[ParticleMass[PionPlus]^2/ScaleMu^2] /; RenormalizationCoefficients[l[4]][[i]] =!= 0 ;

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

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

manred[s_] := (MandelstamReduce[#, Cancel -> s, Masses -> ({ParticleMass[Pion, SUNIndex[i1], RenormalizationState[1]], ParticleMass[Pion, SUNIndex[i2], RenormalizationState[1]], ParticleMass[Pion, SUNIndex[i3], RenormalizationState[1]], ParticleMass[Pion, SUNIndex[i4], RenormalizationState[1]]} // IsoToChargedMasses)] &) ;

Numerics stuff below.

Rule for dropping higher orders in e , applied to numerical expressions:

eNumDrop = CouplingConstant[QED[1], ___]^i_ :> 0 /; i > 2 ;

(* eNumDrop = {} ; *)

po = (PaVeOrder[#, PaVeOrderList -> {{ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, MandelstamS, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, m^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2}, {ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, MandelstamT, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, m^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2}, {ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, MandelstamU, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2, m^2, ParticleMass[PseudoScalar[3], RenormalizationState[1]]^2}}]) &

PaVeOrder(#1, PaVeOrderList -> ( ó  r 2 ó  r 2 ó  r 2 ó  r 2 )) & (m ) (m ) (m ) (m ) + + + 2 + π π s π m π ó  r 2 ó  r 2 ó  r 2 ó  r 2 (m ) (m ) (m ) (m ) + + + 2 + π π t π m π ó  r 2 ó  r 2 ó  r 2 ó  r 2 (m ) (m ) (m ) (m ) + + + 2 + π π u π m π

po1 = (# /. C0[a__] -> VeltmanC0[a, SmallEpsilon -> 0.00000000001, C0Evaluation -> Infrared2, FCIntegrate -> NIntegrate, ExplicitLeutwylerSigma -> True] /. Log[m^2 mm_] :> Log[m^2] + Log[mm]) & ;

mPi = ParticleMass[Pion, RenormalizationState[1]] ;

mPip = ParticleMass[PionPlus, RenormalizationState[1]] ;

mPi0 = ParticleMass[PionZero, RenormalizationState[1]] ;

(* L_i ' s at scale m_rho *) numrules = {ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]], CouplingConstant[ChPTVirtualPhotons2[2], ___] -> DecayConstant[Pion, RenormalizationState[1]]^2/(2 CouplingConstant[QED[1], RenormalizationState[0]]^2) (ParticleMass[PionPlus, RenormalizationState[1]]^2 - ParticleMass[PionZero, RenormalizationState[1]]^2), CouplingConstant[QED[1], RenormalizationState[0]] -> (4 π 1/137)^(1/2), ParticleMass[PionPlus, RenormalizationState[1]] -> 139.570 (* Pi +, PDG *), ParticleMass[Pion, RenormalizationState[1]] -> 139.570 (* 134.98 *) (* Pi +, PDG *), ParticleMass[PionZero, RenormalizationState[1]] -> 134.976 (* Pi0, PDG *), DecayConstant[Pion, RenormalizationState[1]] -> 92.4 (* PDG - Knecht *), DecayConstant[Pion, RenormalizationState[0]] -> 1/(1.07) DecayConstant[Pion, RenormalizationState[1]], CouplingConstant[ChPTVirtualPhotons2[4], 1, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 1 ]]/(32 π^2) (-2.3 + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], 2, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 2 ]]/(32 π^2) (6.0 + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], 3, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 3 ]]/(32 π^2) (2.9 + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], 4, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 4 ]]/(32 π^2) (4.4 (* Knecht & Nehme *) (* 4.3 Knecht & Urech *) + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], 5, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 5 ]]/(32 π^2) (14 + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], 6, RenormalizationState[1]] :> RenormalizationCoefficients[ChPTVirtualPhotons2[4]] [[ 6 ]]/(32 π^2) (16.5 + Log[mPi^2/ScaleMu^2]), CouplingConstant[ChPTVirtualPhotons2[4], i_ ? ((# > 6) &), RenormalizationState[1]] :> 2/(32 π^2) (* Knecht & Urech ' s max *), ScaleMu -> 770 (* Gasser & Leutwyler 1984 *)} ;

zofst[s_, t_, m1_, m2_] := (m1^4 + m2^4 - 2 m2^2 s - 2 m1^2 (m2^2 + s) + s (s + 2 t))/(m1^4 + (m2^2 - s)^2 - 2 m1^2 (m2^2 + s)) ;

tofsz[s_, z_, m1_, m2_] = ((m1^4 + (m2^2 - s)^2 - 2 m1^2 (m2^2 + s)) (-1 + z))/(2 s) ;

Converted by Mathematica  (July 10, 2003)