•L _ i

tmp = ((Collect[Cancel[1/I * (ParticleMass[Kaon]^2 - Pair[Momentum[p2], Momentum[p2]]) * DecayConstant[PhiMeson]^4 * (end4 /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] /. CouplingConstant[ChPTW3[4], ___] -> 0 /. Pair[_LorentzIndex, ___] -> Sequence[] /. _LeutwylerLambda -> 0 /. toEtaRules /. _RenormalizationState -> Sequence[])] // Expand, Prepend[moms, CouplingConstant[ChPTW3[2], ___]]]) /. (aa : (Alternatives @@ ((# * r__) & /@ moms))) -> Cancel[Times[aa]/Times[r]] * coll[Times[r], {_DecayConstant, Pi, _Log, _CouplingConstant}] /. coll -> Collect)/((ParticleMass[Kaon]^2 - Pair[Momentum[p2], Momentum[p2]]) * DecayConstant[PhiMeson]^4) * I

1/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (i c _ 5^( ) ((32 L _ 2^( ) + 8 L _ 3^( ) + (224 L _ 4^( ))/3 + (32 L _ 5^( ))/3 + (32 L _ 6^( ))/3 + (32 L _ 8^( ))/3) (m _ π^(ó  ))^6 + (64 L _ 1^( ) + 16 L _ 3^( ) - 48 L _ 4^( ) - (32 L _ 5^( ))/3 + (448 L _ 6^( ))/3 + (160 L _ 8^( ))/3) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + t (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) - 40 L _ 4^( ) - 8 L _ 5^( )) (m _ π^(ó  ))^4 + u (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) - 40 L _ 4^( ) - 8 L _ 5^( )) (m _ π^(ó  ))^4 + (64 L _ 1^( ) + 32 L _ 2^( ) + 24 L _ 3^( ) + (16 L _ 4^( ))/3) p _ 2^2 (m _ π^(ó  ))^4 + (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) + 16 L _ 4^( ) + (16 L _ 5^( ))/3 - (416 L _ 6^( ))/3 - (160 L _ 8^( ))/3) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + t (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) - 8 L _ 4^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + u (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) - 8 L _ 4^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (-64 L _ 1^( ) - 16 L _ 3^( ) + (112 L _ 4^( ))/3 + (16 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + t^2 (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) (m _ π^(ó  ))^2 + u^2 (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) (m _ π^(ó  ))^2 + t u (64 L _ 1^( ) + 16 L _ 3^( )) (m _ π^(ó  ))^2 + t (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) p _ 2^2 (m _ π^(ó  ))^2 + u (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) p _ 2^2 (m _ π^(ó  ))^2 + (-128/3 L _ 4^( ) - (16 L _ 5^( ))/3 - (64 L _ 6^( ))/3 - (32 L _ 8^( ))/3) (m _ K^(ó  ))^6 + t (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + 48 L _ 4^( ) + 8 L _ 5^( )) (m _ K^(ó  ))^4 + u (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + 48 L _ 4^( ) + 8 L _ 5^( )) (m _ K^(ó  ))^4 + (-32 L _ 2^( ) - 8 L _ 3^( ) - (128 L _ 4^( ))/3 - (16 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^4 + t u (-64 L _ 1^( ) - 16 L _ 3^( )) (m _ K^(ó  ))^2 + t^2 (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) (m _ K^(ó  ))^2 + u^2 (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) (m _ K^(ó  ))^2 + t (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) p _ 2^2 (m _ K^(ó  ))^2 + u (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) p _ 2^2 (m _ K^(ó  ))^2))

tCases = Select[Cases[tmp, (Plus[(__ * CouplingConstant[__]) .., __]) * __, Infinity], (FreeQ[#, MandelstamU] && ! FreeQ[#, MandelstamT]) &] ;

tRules = (# + (# /. MandelstamT -> MandelstamU) -> ((Select[#, ! FreeQ[#, MandelstamT] &] + (Select[#, ! FreeQ[#, MandelstamT] &] /. MandelstamT -> MandelstamU)) * (# /. MandelstamT -> 1))) & /@ tCases ;

tmp1 = tmp //. tRules /. tuRule

1/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (i c _ 5^( ) ((32 L _ 2^( ) + 8 L _ 3^( ) + (224 L _ 4^( ))/3 + (32 L _ 5^( ))/3 + (32 L _ 6^( ))/3 + (32 L _ 8^( ))/3) (m _ π^(ó  ))^6 + (64 L _ 1^( ) + 16 L _ 3^( ) - 48 L _ 4^( ) - (32 L _ 5^( ))/3 + (448 L _ 6^( ))/3 + (160 L _ 8^( ))/3) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + (64 L _ 1^( ) + 32 L _ 2^( ) + 24 L _ 3^( ) + (16 L _ 4^( ))/3) p _ 2^2 (m _ π^(ó  ))^4 + (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) - 40 L _ 4^( ) - 8 L _ 5^( )) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) (m _ π^(ó  ))^4 + (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) + 16 L _ 4^( ) + (16 L _ 5^( ))/3 - (416 L _ 6^( ))/3 - (160 L _ 8^( ))/3) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + (-64 L _ 1^( ) - 16 L _ 3^( ) + (112 L _ 4^( ))/3 + (16 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (t^2 + u^2) (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) (m _ π^(ó  ))^2 + t u (64 L _ 1^( ) + 16 L _ 3^( )) (m _ π^(ó  ))^2 + (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) - 8 L _ 4^( )) (m _ K^(ó  ))^2 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) (m _ π^(ó  ))^2 + (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) p _ 2^2 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) (m _ π^(ó  ))^2 + (-128/3 L _ 4^( ) - (16 L _ 5^( ))/3 - (64 L _ 6^( ))/3 - (32 L _ 8^( ))/3) (m _ K^(ó  ))^6 + (-32 L _ 2^( ) - 8 L _ 3^( ) - (128 L _ 4^( ))/3 - (16 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^4 + t u (-64 L _ 1^( ) - 16 L _ 3^( )) (m _ K^(ó  ))^2 + (t^2 + u^2) (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) (m _ K^(ó  ))^2 + (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + 48 L _ 4^( ) + 8 L _ 5^( )) (m _ K^(ó  ))^4 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) + (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) p _ 2^2 (m _ K^(ó  ))^2 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2)))

ctss3 = (((Collect[Cancel[1/I * (ParticleMass[Kaon]^2 - Pair[Momentum[p2], Momentum[p2]]) * DecayConstant[PhiMeson]^4 * tmp1], Prepend[moms, CouplingConstant[ChPTW3[2], ___]]]) /. aa : (Alternatives @@ ((# * r_) & /@ moms)) -> Cancel[aa/r] * coll[r, {_DecayConstant, Pi, _Log, _ParticleMass, _Pair, _CouplingConstant}] /. coll -> Collect)/((ParticleMass[Kaon]^2 - Pair[Momentum[p2], Momentum[p2]]) * DecayConstant[PhiMeson]^4) * I) //. tRules

1/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (i c _ 5^( ) ((-128 L _ 1^( ) - 32 L _ 2^( ) - 40 L _ 3^( ) - (16 L _ 4^( ))/3 - (16 L _ 5^( ))/3 + (32 L _ 6^( ))/3 + (32 L _ 8^( ))/3) (m _ π^(ó  ))^6 + (64 L _ 1^( ) + 16 L _ 3^( ) - 104 L _ 4^( ) - (56 L _ 5^( ))/3 + (448 L _ 6^( ))/3 + (160 L _ 8^( ))/3) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + s (64 L _ 1^( ) + 32 L _ 2^( ) + 24 L _ 3^( ) + 40 L _ 4^( ) + 8 L _ 5^( )) (m _ π^(ó  ))^4 + (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) - (104 L _ 4^( ))/3 - 8 L _ 5^( )) p _ 2^2 (m _ π^(ó  ))^4 + (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + 104 L _ 4^( ) + (64 L _ 5^( ))/3 - (416 L _ 6^( ))/3 - (160 L _ 8^( ))/3) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( ) + 8 L _ 4^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (32 L _ 2^( ) + 8 L _ 3^( ) + (88 L _ 4^( ))/3 + (16 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (t^2 + u^2) (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) (m _ π^(ó  ))^2 + t u (64 L _ 1^( ) + 16 L _ 3^( )) (m _ π^(ó  ))^2 + s (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) p _ 2^2 (m _ π^(ó  ))^2 + 4/3 (-24 L _ 1^( ) p _ 2^4 - 12 L _ 2^( ) p _ 2^4 - 9 L _ 3^( ) p _ 2^4) (m _ π^(ó  ))^2 + (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + (16 L _ 4^( ))/3 + (8 L _ 5^( ))/3 - (64 L _ 6^( ))/3 - (32 L _ 8^( ))/3) (m _ K^(ó  ))^6 + s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( ) - 48 L _ 4^( ) - 8 L _ 5^( )) (m _ K^(ó  ))^4 + (64 L _ 1^( ) + 16 L _ 3^( ) + (16 L _ 4^( ))/3 + (8 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^4 + t u (-64 L _ 1^( ) - 16 L _ 3^( )) (m _ K^(ó  ))^2 + (t^2 + u^2) (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) (m _ K^(ó  ))^2 + s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) p _ 2^2 (m _ K^(ó  ))^2 + 4/3 (24 L _ 1^( ) p _ 2^4 + 12 L _ 2^( ) p _ 2^4 + 9 L _ 3^( ) p _ 2^4) (m _ K^(ó  ))^2))

tucases = Cases[ctss3, (Plus[(___ * CouplingConstant[__]) .., ___]) * __, Infinity]

{(t^2 + u^2) (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) (m _ π^(ó  ))^2, t u (64 L _ 1^( ) + 16 L _ 3^( )) (m _ π^(ó  ))^2, s (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( )) p _ 2^2 (m _ π^(ó  ))^2, 4/3 (-24 L _ 1^( ) p _ 2^4 - 12 L _ 2^( ) p _ 2^4 - 9 L _ 3^( ) p _ 2^4) (m _ π^(ó  ))^2, s (64 L _ 1^( ) + 32 L _ 2^( ) + 24 L _ 3^( ) + 40 L _ 4^( ) + 8 L _ 5^( )) (m _ π^(ó  ))^4, (-64 L _ 1^( ) - 32 L _ 2^( ) - 24 L _ 3^( ) - (104 L _ 4^( ))/3 - 8 L _ 5^( )) p _ 2^2 (m _ π^(ó  ))^4, (-128 L _ 1^( ) - 32 L _ 2^( ) - 40 L _ 3^( ) - (16 L _ 4^( ))/3 - (16 L _ 5^( ))/3 + (32 L _ 6^( ))/3 + (32 L _ 8^( ))/3) (m _ π^(ó  ))^6, t u (-64 L _ 1^( ) - 16 L _ 3^( )) (m _ K^(ó  ))^2, (t^2 + u^2) (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) (m _ K^(ó  ))^2, s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( )) p _ 2^2 (m _ K^(ó  ))^2, 4/3 (24 L _ 1^( ) p _ 2^4 + 12 L _ 2^( ) p _ 2^4 + 9 L _ 3^( ) p _ 2^4) (m _ K^(ó  ))^2, s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( ) + 8 L _ 4^( )) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2, (32 L _ 2^( ) + 8 L _ 3^( ) + (88 L _ 4^( ))/3 + (16 L _ 5^( ))/3) p _ 2^2 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2, (64 L _ 1^( ) + 16 L _ 3^( ) - 104 L _ 4^( ) - (56 L _ 5^( ))/3 + (448 L _ 6^( ))/3 + (160 L _ 8^( ))/3) (m _ π^(ó  ))^4 (m _ K^(ó  ))^2, s (-32 L _ 1^( ) - 16 L _ 2^( ) - 12 L _ 3^( ) - 48 L _ 4^( ) - 8 L _ 5^( )) (m _ K^(ó  ))^4, (64 L _ 1^( ) + 16 L _ 3^( ) + (16 L _ 4^( ))/3 + (8 L _ 5^( ))/3) p _ 2^2 (m _ K^(ó  ))^4, (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + 104 L _ 4^( ) + (64 L _ 5^( ))/3 - (416 L _ 6^( ))/3 - (160 L _ 8^( ))/3) (m _ π^(ó  ))^2 (m _ K^(ó  ))^4, (32 L _ 1^( ) + 16 L _ 2^( ) + 12 L _ 3^( ) + (16 L _ 4^( ))/3 + (8 L _ 5^( ))/3 - (64 L _ 6^( ))/3 - (32 L _ 8^( ))/3) (m _ K^(ó  ))^6}

turules = (pt = Select[#, ! FreeQ[#, CouplingConstant[__]] &] ; mpt = -pt ; {a_ * pt + b_ * pt -> (a + b) * pt, a_ * pt + b_ * mpt -> (a - b) * pt}) & /@ tucases // Flatten ;

cts3s = ctss3 //. turules //. Plus[a__] :> ss /@ Plus[a] /; FreeQ[a, ss, Infinity] /. ss -> Simplify

1/(3 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (4 i c _ 5^( ) (-2 (48 L _ 1^( ) + 12 L _ 2^( ) + 15 L _ 3^( ) + 2 L _ 4^( ) + 2 L _ 5^( ) - 4 L _ 6^( ) - 4 L _ 8^( )) (m _ π^(ó  ))^6 + 2 (24 L _ 1^( ) + 6 L _ 3^( ) - 39 L _ 4^( ) - 7 L _ 5^( ) + 56 L _ 6^( ) + 20 L _ 8^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 6 s (8 L _ 1^( ) + 4 L _ 2^( ) + 3 L _ 3^( ) + 5 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^4 - 2 (24 L _ 1^( ) + 12 L _ 2^( ) + 9 L _ 3^( ) + 13 L _ 4^( ) + 3 L _ 5^( )) p _ 2^2 (m _ π^(ó  ))^4 + (24 L _ 1^( ) + 12 L _ 2^( ) + 9 L _ 3^( ) + 78 L _ 4^( ) + 16 L _ 5^( ) - 104 L _ 6^( ) - 40 L _ 8^( )) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 - 3 s (8 L _ 1^( ) + 4 L _ 2^( ) + 3 L _ 3^( ) - 2 L _ 4^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 (12 L _ 2^( ) + 3 L _ 3^( ) + 11 L _ 4^( ) + 2 L _ 5^( )) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (24 L _ 1^( ) + 12 L _ 2^( ) + 9 L _ 3^( ) + 4 L _ 4^( ) + 2 L _ 5^( ) - 16 L _ 6^( ) - 8 L _ 8^( )) (m _ K^(ó  ))^6 - 3 s (8 L _ 1^( ) + 4 L _ 2^( ) + 3 L _ 3^( ) + 12 L _ 4^( ) + 2 L _ 5^( )) (m _ K^(ó  ))^4 + 2 (24 L _ 1^( ) + 6 L _ 3^( ) + 2 L _ 4^( ) + L _ 5^( )) p _ 2^2 (m _ K^(ó  ))^4 - 3 (8 L _ 1^( ) + 4 L _ 2^( ) + 3 L _ 3^( )) p _ 2^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 12 t u (4 L _ 1^( ) + L _ 3^( )) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 3 (8 L _ 1^( ) + 4 L _ 2^( ) + 3 L _ 3^( )) (t^2 + u^2 + s p _ 2^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))

cts3 = ((cts3s /. {MandelstamT^2 -> (2 ParticleMass[Pion]^2 + ParticleMass[Kaon]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS - MandelstamU) MandelstamT, MandelstamU^2 -> (2 ParticleMass[Pion]^2 + ParticleMass[Kaon]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS - MandelstamT) MandelstamU} // FullSimplify) /. tuRule // FullSimplify) //. MandelstamS * a_ + MandelstamS * b_ -> MandelstamS * (Collectt[Expandd[a + b], ParticleMass[__]]) /. Expandd -> Expand /. Collectt -> Collect

1/(3 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (4 i c _ 5^( ) (m _ π^(ó  ) - m _ K^(ó  )) (m _ π^(ó  ) + m _ K^(ó  )) (-30 L _ 3^( ) (m _ π^(ó  ))^4 - 4 L _ 4^( ) (m _ π^(ó  ))^4 - 4 L _ 5^( ) (m _ π^(ó  ))^4 + 8 L _ 6^( ) (m _ π^(ó  ))^4 + 8 L _ 8^( ) (m _ π^(ó  ))^4 + 18 t L _ 3^( ) (m _ π^(ó  ))^2 + 18 u L _ 3^( ) (m _ π^(ó  ))^2 - 18 L _ 3^( ) p _ 2^2 (m _ π^(ó  ))^2 - 26 L _ 4^( ) p _ 2^2 (m _ π^(ó  ))^2 - 6 L _ 5^( ) p _ 2^2 (m _ π^(ó  ))^2 - 2 (2 L _ 4^( ) + L _ 5^( ) - 4 (2 L _ 6^( ) + L _ 8^( ))) (m _ K^(ó  ))^4 - 9 L _ 3^( ) p _ 2^4 + (6 (-3 L _ 5^( ) + 20 L _ 6^( ) + 8 L _ 8^( )) (m _ π^(ó  ))^2 + 2 L _ 5^( ) (3 s - p _ 2^2) - 3 L _ 3^( ) p _ 2^2 + L _ 4^( ) (-82 (m _ π^(ó  ))^2 + 36 s - 4 p _ 2^2)) (m _ K^(ó  ))^2 - 6 t u L _ 3^( ) + 9 t L _ 3^( ) p _ 2^2 + 9 u L _ 3^( ) p _ 2^2 + s ((18 L _ 3^( ) + 30 L _ 4^( ) + 6 L _ 5^( )) (m _ π^(ó  ))^2 - 9 t L _ 3^( ) - 9 u L _ 3^( ) + 9 L _ 3^( ) p _ 2^2) + 24 L _ 1^( ) (s - 2 (m _ π^(ó  ))^2) (-(m _ K^(ó  ))^2 + s - p _ 2^2) + 12 L _ 2^( ) (2 (m _ π^(ó  ))^4 - 2 (s - (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + s^2 - s (m _ K^(ó  ))^2 - 2 t u + p _ 2^2 (2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 - s))))

Converted by Mathematica  (July 10, 2003)