•Calculation and reduction of the amplitude

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert, AmplitudeLevel -> Classes, Sum -> False, EqualMasses -> True] ;

Isospin implification:

af = Collect[ScalarProductExpand /@ SUNReduce[#, FullReduce -> True] & /@ (amplFC /. {I1 -> j1, I2 -> j2, I3 -> j3, I4 -> j4}), {_SU2Delta, Pi, _DecayConstant, _ParticleMass, _Pair}] ;

The loop integrals are expressed in terms of Passarino-Veltman symbols:

ampreduced = OneLoop[q1, #] & /@ af ;

ampsimple = Simplify /@ (Collect[#, {Pi, _DecayConstant, _SU2Delta, _B0, _ParticleMass, (* MandelstamS, MandelstamT, MandelstamU *) _Pair}] & /@ ampreduced) ;

ampsimplest = (FullSimplify /@ ((FullSimplify /@ (MandelstamReduce[ampsimple, Cancel -> None] /. ParticleMass[Pion, RenormalizationState[1]] -> ParticleMass[Pion, RenormalizationState[0]])) /. manrules)) /. ParticleMass[Pion, RenormalizationState[0]] -> ParticleMass[Pion, RenormalizationState[1]]

{(A _ 0 ( (m _ π^(ó  r ))^2 ) (-155 (δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)) (m _ π^(ó  r ))^2 + (100 t - 155 (m _ π^(ó  r ))^2) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + 100 (u δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + s δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2))))/(1440 π^2 (f _ π^(ó  ))^4), 1/(288 π^2 (f _ π^(ó  ))^4) (((t - u) (s - 6 (m _ π^(ó  r ))^2) - 2 A _ 0 ( (m _ π^(ó  r ))^2 ) (-4 (m _ π^(ó  r ))^2 + 2 s + t - 2 u) + 3 B _ 0 (s, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) (2 (m _ π^(ó  r ))^4 - (3 s + t - 3 u) (m _ π^(ó  r ))^2 + s (s - u))) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + (-(t - u) (s - 6 (m _ π^(ó  r ))^2) - 2 A _ 0 ( (m _ π^(ó  r ))^2 ) (-4 (m _ π^(ó  r ))^2 + 2 s - 2 t + u) + 3 B _ 0 (s, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) (2 (m _ π^(ó  r ))^4 - (3 s - 3 t + u) (m _ π^(ó  r ))^2 + s (s - t))) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + (3 B _ 0 (s, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) (-3 (m _ π^(ó  r ))^4 + 4 s (m _ π^(ó  r ))^2 + s (3 s - 4 (m _ π^(ó  r ))^2)) - 10 s A _ 0 ( (m _ π^(ó  r ))^2 )) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)), 1/(288 π^2 (f _ π^(ó  ))^4) (-(A _ 0 ( (m _ π^(ó  r ))^2 ) (-8 (m _ π^(ó  r ))^2 - 4 s + 2 t + 4 u) + (s - t) (u - 6 (m _ π^(ó  r ))^2) + 3 B _ 0 (u, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) ((-2 (m _ π^(ó  r ))^2 - 3 s + t + 3 u) (m _ π^(ó  r ))^2 + (s - u) u)) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + (3 B _ 0 (u, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) (-3 (m _ π^(ó  r ))^4 + 4 u (m _ π^(ó  r ))^2 + u (3 u - 4 (m _ π^(ó  r ))^2)) - 10 u A _ 0 ( (m _ π^(ó  r ))^2 )) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + ((s - t) (u - 6 (m _ π^(ó  r ))^2) - 2 A _ 0 ( (m _ π^(ó  r ))^2 ) (-4 (m _ π^(ó  r ))^2 + s - 2 t + 2 u) - 3 B _ 0 (u, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) (-2 (m _ π^(ó  r ))^4 + (s - 3 t + 3 u) (m _ π^(ó  r ))^2 + (t - u) u)) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)), -1/(288 π^2 (f _ π^(ó  ))^4) ((s - u) (t - 6 (m _ π^(ó  r ))^2) (δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) - δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)) - 2 A _ 0 ( (m _ π^(ó  r ))^2 ) (-5 t δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + (4 (m _ π^(ó  r ))^2 + 2 s - 2 t - u) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) - (-4 (m _ π^(ó  r ))^2 + s + 2 t - 2 u) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)) + 3 B _ 0 (t, (m _ π^(ó  r ))^2, (m _ π^(ó  r ))^2) ((3 (m _ π^(ó  r ))^4 - 4 t (m _ π^(ó  r ))^2 + t (4 (m _ π^(ó  r ))^2 - 3 t)) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + (-2 (m _ π^(ó  r ))^4 + (-3 s + 3 t + u) (m _ π^(ó  r ))^2 + (s - t) t) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + (-2 (m _ π^(ó  r ))^4 + (s + 3 t - 3 u) (m _ π^(ó  r ))^2 + t (u - t)) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2)))}

ampinfinitiesfull = (VeltmanExpand[#, ExplicitLeutwylerJ0 -> True, B0Evaluation -> "jbar"] & /@ ampsimplest) // Simplify ;

amploopfull = Underoverscript[∑, j = 1, arg3] ampinfinitiesfull[[j]] // Simplify ;

Converted by Mathematica  (July 10, 2003)