•Calculation and reduction of the amplitude

Calculation of the amplitude:

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

Isospin implification:

LeafCount /@ amplFC

{101552, 10677, 10677, 10677}

Below we don't use DoSumOver because speed is gained by operating on each term in the sum "on the fly". Some such functionality should perhaps be built into DoSumOver.

af = CheckF[(WriteString["stdout", "\n\n+"] ; Sum[tmp = Expand[# /. channel] ; WriteString["stdout", " {I1, I2} = ", {I1, I2}, " Length: ", Length[tmp], ". "] ; ii = 0 ; (++ ii ; ScalarProductExpand[SUNReduce[WriteOutUMatrices[If[IntegerQ[ii/100], WriteString["stdout", ii, " "]] ; #], FullReduce -> True]]) & /@ tmp, Evaluate[Sequence @@ ({#[[1]], 1, 8} & /@ (Union[Cases[#, _SumOver, Infinity]]))]]) & /@ Take[amplFC, {1, -1}], "MesonMesonAmpaf"] ;

aaf = Simplify /@ (af /. subpar /. udrules /. gellmannOkubo) /. toEtaRules

The one-loop integrals are simplified:

aff = OneLoopSimplify[#, q1] & /@ aaf ;

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

ampreduced = CheckF[OneLoop[q1, #] & /@ aff, "MesonMesonampreduced"] ;

ampsimple = Simplify /@ (Collect[#, {Pi, _DecayConstant, _SU3Delta, _B0, _ParticleMass}] & /@ ampreduced) ;

The momentum variables are substituted with Mandelstam variables:

ampsimplest = FullSimplify /@ MandelstamReduce[ampsimple]

{((-13 A _ 0 ( (m _ π^(ó  ))^2 ) + 4 A _ 0 ( (m _ K^(ó  ))^2 ) + 3 A _ 0 ( (m _ η^(ó  ))^2 )) (m _ π^(ó  ))^2)/(288 π^2 (f _ π^(ó  ))^4), 1/(1152 π^2 (f _ π^(ó  ))^4) (4 (27 B _ 0 (s, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (s, (m _ η^(ó  ))^2, (m _ η^(ó  ))^2)) (m _ π^(ó  ))^4 + 48 (2 A _ 0 ( (m _ π^(ó  ))^2 ) - 3 s B _ 0 (s, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2)) (m _ π^(ó  ))^2 + s (-80 A _ 0 ( (m _ π^(ó  ))^2 ) - 10 A _ 0 ( (m _ K^(ó  ))^2 ) + 9 s (8 B _ 0 (s, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (s, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2)))), 1/(1152 π^2 (f _ π^(ó  ))^4) (4 (171 B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 36 B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2) + B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ η^(ó  ))^2, (m _ η^(ó  ))^2)) (m _ π^(ó  ))^4 - 8 (28 A _ 0 ( (m _ π^(ó  ))^2 ) + 5 A _ 0 ( (m _ K^(ó  ))^2 ) + 9 (s + t) (6 B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2))) (m _ π^(ó  ))^2 + (s + t) (80 A _ 0 ( (m _ π^(ó  ))^2 ) + 10 A _ 0 ( (m _ K^(ó  ))^2 ) + 9 (s + t) (8 B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (4 (m _ π^(ó  ))^2 - s - t, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2)))), 1/(1152 π^2 (f _ π^(ó  ))^4) (4 (27 B _ 0 (t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (t, (m _ η^(ó  ))^2, (m _ η^(ó  ))^2)) (m _ π^(ó  ))^4 + 48 (2 A _ 0 ( (m _ π^(ó  ))^2 ) - 3 t B _ 0 (t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2)) (m _ π^(ó  ))^2 + t (-80 A _ 0 ( (m _ π^(ó  ))^2 ) - 10 A _ 0 ( (m _ K^(ó  ))^2 ) + 9 t (8 B _ 0 (t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + B _ 0 (t, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2))))}

The divergences are singled out:

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

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

Converted by Mathematica  (July 10, 2003)