•The limit K->2π

Off-shell:

final2all /. {s -> MandelstamS, t -> MandelstamT, u -> MandelstamU} /. {(* MandelstamS -> ParticleMass[Pion]^2, *) MandelstamT -> ParticleMass[Pion]^2, CouplingConstant[_[4], ___] -> 0} /. _RenormalizationState -> Sequence[] /. _Log -> 0 /. p2 -> 0 // FullSimplify

-(i (c _ 2^( ) (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s) (m _ K^(ó  ))^2 + 2 c _ 5^( ) (s - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)))/(2 (f _ ϕ^(ó  ))^2 (m _ K^(ó  ))^2)

This expression agrees with the previously calculated expression for the Overscript[J, _]'s of K->2π:

limitjs = final2all + finalloops /. Pair[___, _LorentzIndex] -> 1 /. {s -> MandelstamS, t -> MandelstamT, u -> MandelstamU} /. cancelU /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. {MandelstamS -> ParticleMass[Kaon]^2, MandelstamT -> ParticleMass[Pion]^2, CouplingConstant[_[4], ___] -> 0} /. _RenormalizationState -> Sequence[] /. _Log -> 0 /. p2 -> 0 // Expand // FullSimplify

1/(72 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2) (i c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((8 π^2 (Overscript[J, _] _ (m _ η^(ó  ))^2((m _ K^(ó  ))^2) - 9 Overscript[J, _] _ (m _ π^(ó  ))^2((m _ K^(ó  ))^2)) + 4) (m _ π^(ó  ))^4 + 144 π^2 (f _ ϕ^(ó  ))^2 (m _ π^(ó  ))^2 + 9 (16 π^2 (Overscript[J, _] _ (m _ π^(ó  ))^2((m _ K^(ó  ))^2) + Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ π^(ó  ))^2)) - 1) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 12 π^2 (3 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ π^(ó  ))^2) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ π^(ó  ))^2)) (m _ K^(ó  ))^4))

This expression agrees with the previously calculated expression for the logs of K->2π:

limitlogs = (final2all + finalloops /. Pair[___, _LorentzIndex] -> 1 /. cancelU /. p2 -> 0 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. {MandelstamS -> ParticleMass[Kaon]^2, MandelstamT -> ParticleMass[Pion]^2, CouplingConstant[_[4], ___] -> 0} /. _RenormalizationState -> Sequence[] /. _LeutwylerJBar -> 0 // Simplify // Collect[#, {_CouplingConstant, Pi, _DecayConstant, _Log}] &)

c _ 2^( ) ((2 i ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(f _ ϕ^(ó  ))^2 + 1/(π^2 (f _ ϕ^(ó  ))^4) (1/288 i log((m _ η^(ó  ))^2/μ^2) (14 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 - 5 (m _ π^(ó  ))^4) + 1/288 i (16 (m _ π^(ó  ))^4 - 52 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 (m _ K^(ó  ))^4) + 1/288 i log((m _ π^(ó  ))^2/μ^2) (-9 (m _ π^(ó  ))^4 - 36 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 (m _ K^(ó  ))^4) + 1/288 i log((m _ K^(ó  ))^2/μ^2) (90 (m _ K^(ó  ))^4 - 90 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)))

Converted by Mathematica  (July 10, 2003)