•N _ (20 - 23)

tmp = ((Collect[Cancel[1/I * DecayConstant[PhiMeson]^4 * (end4 /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] /. CouplingConstant[ChPT3[4], ___] -> 0 /. Pair[_LorentzIndex, ___] -> Sequence[] /. _LeutwylerLambda -> 0 /. toEtaRules /. _RenormalizationState -> Sequence[])] // Expand, Prepend[{(MandelstamS | MandelstamT | MandelstamU | ParticleMass[a___]^2 | Pair[__]) (MandelstamS | MandelstamT | MandelstamU | ParticleMass[b___]^2 | Pair[__]) , (MandelstamS^2 | MandelstamT^2 | MandelstamU^2 | ParticleMass[a___]^4 | Pair[__]^4)}, CouplingConstant[ChPTW3[2], ___]]]) /. aa : Alternatives @@ ((# * r__) & /@ moms) -> Cancel[aa/r] * coll[Times[r], {_DecayConstant, Pi, _Log, _ParticleMass, _Pair, _CouplingConstant}] /. coll -> Collect)/(DecayConstant[PhiMeson]^4) * I

1/(f _ ϕ^(ó  ))^4 (i c _ 2^( ) ((-N _ 19^( ) + N _ 20^( ) - 8 N _ 21^( ) - 4 N _ 22^( ) - 8 N _ 23^( )) (m _ π^(ó  ))^4 + (N _ 19^( ) - N _ 20^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + t (-N _ 19^( ) + N _ 20^( ) + 4 N _ 21^( ) + 2 N _ 22^( ) + 4 N _ 23^( )) (m _ π^(ó  ))^2 + u (-N _ 19^( ) + N _ 20^( ) + 4 N _ 21^( ) + 2 N _ 22^( ) + 4 N _ 23^( )) (m _ π^(ó  ))^2 + (N _ 19^( ) - 3 N _ 20^( ) - (4 N _ 21^( ))/3 - 2 N _ 22^( ) - 4 N _ 23^( )) p _ 2^2 (m _ π^(ó  ))^2 - 2 N _ 20^( ) p _ 2^4 + t (-1/2 N _ 19^( ) + N _ 20^( )/2 - N _ 21^( ) - N _ 22^( )/2 - N _ 23^( )) (m _ K^(ó  ))^2 + u (-1/2 N _ 19^( ) + N _ 20^( )/2 - N _ 21^( ) - N _ 22^( )/2 - N _ 23^( )) (m _ K^(ó  ))^2 + (-N _ 19^( ) - 3 N _ 20^( ) - (2 N _ 21^( ))/3 + N _ 22^( ) + 2 N _ 23^( )) p _ 2^2 (m _ K^(ó  ))^2 + t u (2 N _ 19^( ) - 2 N _ 20^( )) + t^2 (N _ 19^( )/2 - N _ 20^( )/2) + u^2 (N _ 19^( )/2 - N _ 20^( )/2) + t ((7 N _ 20^( ))/2 - N _ 19^( )/2) p _ 2^2 + u ((7 N _ 20^( ))/2 - N _ 19^( )/2) p _ 2^2))

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 ;

trule = MandelstamT^2    ( CouplingConstant[ChPTW3[4], 19] - CouplingConstant[ChPTW3[4], 20]) + MandelstamU^2    ( CouplingConstant[ChPTW3[4], 19] - CouplingConstant[ChPTW3[4], 20]) -> (MandelstamT^2 + MandelstamU^2)    ( CouplingConstant[ChPTW3[4], 19] - CouplingConstant[ChPTW3[4], 20])

(N _ 19^( ) - N _ 20^( )) t^2 + u^2 (N _ 19^( ) - N _ 20^( )) -> (t^2 + u^2) (N _ 19^( ) - N _ 20^( ))

tmp1 = tmp //. tRules /. tuRule

1/(f _ ϕ^(ó  ))^4 (i c _ 2^( ) ((-N _ 19^( ) + N _ 20^( ) - 8 N _ 21^( ) - 4 N _ 22^( ) - 8 N _ 23^( )) (m _ π^(ó  ))^4 + (N _ 19^( ) - N _ 20^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (N _ 19^( ) - 3 N _ 20^( ) - (4 N _ 21^( ))/3 - 2 N _ 22^( ) - 4 N _ 23^( )) p _ 2^2 (m _ π^(ó  ))^2 + (-N _ 19^( ) + N _ 20^( ) + 4 N _ 21^( ) + 2 N _ 22^( ) + 4 N _ 23^( )) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) (m _ π^(ó  ))^2 - 2 N _ 20^( ) p _ 2^4 + (-N _ 19^( ) - 3 N _ 20^( ) - (2 N _ 21^( ))/3 + N _ 22^( ) + 2 N _ 23^( )) p _ 2^2 (m _ K^(ó  ))^2 + t u (2 N _ 19^( ) - 2 N _ 20^( )) + (t^2 + u^2) (N _ 19^( )/2 - N _ 20^( )/2) + (-1/2 N _ 19^( ) + N _ 20^( )/2 - N _ 21^( ) - N _ 22^( )/2 - N _ 23^( )) (m _ K^(ó  ))^2 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2) + ((7 N _ 20^( ))/2 - N _ 19^( )/2) p _ 2^2 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - s + p _ 2^2)))

ctss2 = (((Collect[Cancel[1/I * ParticleMass[Kaon]^2 * DecayConstant[PhiMeson]^4 * tmp1] // Expand, Prepend[moms, CouplingConstant[ChPTW3[2], ___]]]) /. aa : Alternatives @@ ((# * r_) & /@ moms) -> Cancel[aa/r] * coll[r, {_DecayConstant, Pi, _Log, _ParticleMass, _CouplingConstant}] /. coll -> Collect)/(ParticleMass[Kaon]^2 * DecayConstant[PhiMeson]^4) * I // FullSimplify) /. trule

1/(6 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (-3 (2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^4 + (3 (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + p _ 2^2) + 2 N _ 21^( ) (12 (m _ π^(ó  ))^2 + 3 s - 5 p _ 2^2)) (m _ K^(ó  ))^2 - 4 (3 s (N _ 22^( ) + 2 N _ 23^( )) + N _ 21^( ) (6 s - 4 p _ 2^2)) (m _ π^(ó  ))^2 + 3 N _ 19^( ) (-6 (m _ π^(ó  ))^4 + 2 s (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 + t^2 + u^2 - p _ 2^4 + (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 4 t u + p _ 2^2 (-2 (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2 + s)) - 3 N _ 20^( ) (-6 (m _ π^(ó  ))^4 + 2 s (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 + t^2 + u^2 - 3 p _ 2^4 + (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 4 t u + p _ 2^2 (-10 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 7 s))))

tucases = Cases[ctss2, (Plus[(___ * CouplingConstant[__]) .., ___]) * __, Infinity]

{3 s (N _ 22^( ) + 2 N _ 23^( )), -4 (3 s (N _ 22^( ) + 2 N _ 23^( )) + N _ 21^( ) (6 s - 4 p _ 2^2)) (m _ π^(ó  ))^2, 3 (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + p _ 2^2), (3 (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + p _ 2^2) + 2 N _ 21^( ) (12 (m _ π^(ó  ))^2 + 3 s - 5 p _ 2^2)) (m _ K^(ó  ))^2, -3 (2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^4}

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

cts2 = ctss2 //. turules // FullSimplify

1/(6 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (-3 (2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^4 + (3 (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + p _ 2^2) + 2 N _ 21^( ) (12 (m _ π^(ó  ))^2 + 3 s - 5 p _ 2^2)) (m _ K^(ó  ))^2 - 4 (3 s (N _ 22^( ) + 2 N _ 23^( )) + N _ 21^( ) (6 s - 4 p _ 2^2)) (m _ π^(ó  ))^2 + 3 N _ 19^( ) (-6 (m _ π^(ó  ))^4 + 2 s (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 + t^2 + u^2 - p _ 2^4 + (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 4 t u + p _ 2^2 (-2 (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2 + s)) - 3 N _ 20^( ) (-6 (m _ π^(ó  ))^4 + 2 s (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 + t^2 + u^2 - 3 p _ 2^4 + (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 4 t u + p _ 2^2 (-10 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 7 s))))

Converted by Mathematica  (July 10, 2003)