•Wavefuntion-renormalized lowest order graphs

final2all = end2all /. Pair[_LorentzIndex, ___] -> Sequence[] /. _LeutwylerLambda -> 0 /. toEtaRules /. _RenormalizationState -> Sequence[] ;

lows = Collect[final2all /. CouplingConstant[_[4], ___] -> 0, {_Log}] // Simplify ;

lows1 = lows /. _Log -> 0 // Simplify

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

lows2 = Collect[Simplify[lows - lows1], {_DecayConstant, _Log, (ParticleMass[Kaon, RenormalizationState[1]]^2 - Pair[Momentum[p2], Momentum[p2]]), (Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Kaon, RenormalizationState[1]]^2)}] /. Log[a_] * b__ :> Log[a] * Simplify[Collect[Times[b], {_ParticleMass, _Pair}]]

1/(f _ ϕ^(ó  ))^4 (1/(384 π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ π^(ó  ))^2/μ^2) (4 c _ 5^( ) (2 (m _ π^(ó  ))^4 + 2 (-7 (m _ K^(ó  ))^2 + 7 s + 6 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (9 (m _ K^(ó  ))^2 - 11 s - 9 p _ 2^2)) + c _ 2^( ) (-19 p _ 2^4 + (-70 (m _ π^(ó  ))^2 + 38 (m _ K^(ó  ))^2 + 57 s) p _ 2^2 + (m _ π^(ó  ))^2 (70 (m _ K^(ó  ))^2 - 18 s) - 19 ((m _ K^(ó  ))^4 + 3 s (m _ K^(ó  ))^2))) (m _ π^(ó  ))^2) + 1/(3456 π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ η^(ó  ))^2/μ^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (c _ 2^( ) ((m _ K^(ó  ))^4 + 3 s (m _ K^(ó  ))^2 + 9 p _ 2^4 + (m _ π^(ó  ))^2 (6 s - 34 (m _ K^(ó  ))^2) + p _ 2^2 (34 (m _ π^(ó  ))^2 - 10 (m _ K^(ó  ))^2 - 27 s)) + 4 c _ 5^( ) (2 (m _ π^(ó  ))^4 - 4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^4 + 9 s (m _ K^(ó  ))^2 + p _ 2^2 (11 (m _ K^(ó  ))^2 - 2 (m _ π^(ó  ))^2)))) + 1/(192 π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (4 c _ 5^( ) (-3 (m _ K^(ó  ))^2 + 5 s + 3 p _ 2^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (-7 p _ 2^4 + (-28 (m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^2 + 21 s) p _ 2^2 + (m _ K^(ó  ))^2 (28 (m _ π^(ó  ))^2 - 5 ((m _ K^(ó  ))^2 + 3 s))))))

lows3 = final2all - (final2all /. CouplingConstant[_[4], ___] -> 0) /. CouplingConstant[ChPT3[4], ___] -> 0 // Simplify

-(2 i c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 + 3 s - p _ 2^2) (2 N _ 10^( ) (m _ K^(ó  ))^2 - N _ 21^( ) p _ 2^2 + N _ 11^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(3 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

lows4 = final2all - (final2all /. CouplingConstant[_[4], ___] -> 0) /. CouplingConstant[ChPTW3[4], ___] -> 0 // FullSimplify

1/(3 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (8 i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-2 (2 L _ 4^( ) + L _ 5^( ) - 2 L _ 6^( ) - L _ 8^( )) (m _ K^(ó  ))^4 - ((-2 L _ 4^( ) + 3 L _ 5^( ) + 6 L _ 6^( ) - 6 L _ 8^( )) (m _ π^(ó  ))^2 - 4 (2 L _ 4^( ) + L _ 5^( )) p _ 2^2 + 6 (2 L _ 6^( ) + L _ 8^( )) (s + p _ 2^2)) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^2 (2 L _ 4^( ) ((m _ π^(ó  ))^2 + 2 p _ 2^2) + (L _ 5^( ) - 2 (L _ 6^( ) + L _ 8^( ))) (2 (m _ π^(ó  ))^2 + 3 (s + p _ 2^2)))))

Converted by Mathematica  (July 10, 2003)