•Projections

Before projecting out, we set Pion to PionPlus.

ampa[I1_, I2_, I3_, I4_] := {low, cts, poly, logs, jbars} /. {i1 -> I1, i2 -> I2, i3 -> I3, i4 -> I4, LeutwylerJBar[a__, b___Rule] :> LeutwylerJBar[a, LeutwylerJBarEvaluation -> "subthreshold", ExplicitLeutwylerSigma -> True, b]} ;

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

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

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

(* L_i ' s at scale m_rho *) numrules = {ParticleMass[PionPlus, RenormalizationState[1]] -> 139.57 (* Pi +, PDG *) (* 140.97 *), ParticleMass[Pion, RenormalizationState[1]] -> 139.57 (* 134.98 *) (* Pi +, PDG *), ParticleMass[PionZero, RenormalizationState[1]] -> 134.98 (* Pi +, PDG *), DecayConstant[Pion, RenormalizationState[1]] -> 93.3 (* Gasser & Leutwyler 1984 *) (* 92.4 *) (* Donoghue et al . *), CouplingConstant[ChPT2[4], 1, RenormalizationState[1]] -> 0.65 * 10^(-3), CouplingConstant[ChPT2[4], 2, RenormalizationState[1]] -> 1.89 * 10^(-3), CouplingConstant[ChPT2[4], 3, RenormalizationState[1]] -> -3.06 * 10^(-3), CouplingConstant[ChPT2[4], 4, RenormalizationState[1]] -> 0 * 10^(-3), CouplingConstant[ChPT2[4], 5, RenormalizationState[1]] -> 2.3 * 10^(-3), CouplingConstant[ChPT2[4], 6, RenormalizationState[1]] -> 0 * 10^(-3), CouplingConstant[ChPT2[4], 8, RenormalizationState[1]] -> 1.2 * 10^(-3) (* ? *), 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) ;

newScale = 770 ;

(* newScale = 1000 ; *)

scaleTrans := ({CouplingConstant[c_[4], i_, r___] :> CouplingConstant[c[4], i, r] + (RenormalizationCoefficients[c[4]] [[ i ]] Log[ScaleMu/s2])/(16 π^2), ScaleMu -> s2} /. s2 -> newScale) ;

ampaP[s_, t_, u_] := 1/(32 π) AmplitudeProjection[ampa, Channel -> chan, OnMassShell -> True, MassArguments -> {RenormalizationState[1]}] /. {MandelstamS -> s, MandelstamT -> t, MandelstamU -> u} ;

ampaPN[s_, t_, u_] := ampaP[s, t, u] /. scaleTrans /. numrules ;

ampaPNL[s_][l_] := 1/2 ∫ _ (-1)^1 ampaPN[s, tofsz[s, z, mPip, mPip] /. numrules, 4 mPip^2 - s - tofsz[s, z, mPip, mPip] /. numrules] LegendreP[l, z] d z ;

ampaPNL[s_][0] := Join[1/2 ∫ _ (-1)^1 Take[ampaPN[s, tofsz[s, z, mPip, mPip], 4 mPip^2 - s - tofsz[s, z, mPip, mPip]], {1, 3}] d z, N[1/2 ∫ _ (-1)^1 Take[ampaPN[s, tofsz[s, z, mPip, mPip] /. numrules, 4 mPip^2 - s - tofsz[s, z, mPip, mPip] /. numrules], {4, 5}] d z]] ;

chan = {{Pion, Pion} -> {Pion, Pion}, 0} (* Channel -> {{PionPlus, PionPlus} -> {PionPlus, PionPlus}} *) ;

1/2 ∫ _ (-1)^1 ampaP[s, tofsz[s, z, mPip, mPip], 4 mPip^2 - s - tofsz[s, z, mPip, mPip]][[1]] LegendreP[l, z] d z /. PionPlus -> Pion /. l -> 0 // Simplify

-((m _ π^(ó  r ))^2 - 2 s)/(32 π (f _ π^(ó  r ))^2)

ampaPNL[1.00000000000001 4 (mPip /. numrules)^2][0]

{0.15581851407953812`, 0.014664919847858096`, -0.009936483454835623`, 0.03393953475466716`, 0.015456752040809986` + 2.440416361843894`*^-9 i}

Plus @@ %

0.2099432372680377` + 2.440416361843894`*^-9 i

chan = {{Pion, Pion} -> {Pion, Pion}, 2} (* Channel -> {{PionPlus, PionPlus} -> {PionPlus, PionPlus}} *) ;

1/2 ∫ _ (-1)^1 ampaP[s, tofsz[s, z, mPip, mPip], 4 mPip^2 - s - tofsz[s, z, mPip, mPip]][[1]] LegendreP[l, z] d z /. PionPlus -> Pion /. l -> 0 // Simplify

-(s - 2 (m _ π^(ó  r ))^2)/(32 π (f _ π^(ó  r ))^2)

ampaPNL[1.00000000000001 4 (mPip /. numrules)^2][0] /. numrules

{-0.04451957545129699`, 0.0014664919847858762`, -0.000946331757603411`, 0.0032323366433017085`, 0.0012617756767049353` + 1.992176621913416`*^-10 i}

Plus @@ %

-0.039505302904107885` + 1.992176621913416`*^-10 i

Converted by Mathematica  (July 10, 2003)