KSS2Amplitude.nb

•Calculation of the amplitude

Isospin and momenta reduction:

$res = {} ; Do[Clear[subres, subres1, summ, suminds, sub, sums, tmpi, tmpii, tmpsum, name, tmpsub, subsum, tmpres] ; summ = amplFC1[[rep]] //. {(SumOver[SUNIndex[ii_]] * SUNDelta[ExplicitSUNIndex[jj_Integer], SUNIndex[ii_]] * rest_) :> (rest /. ii -> jj), (SumOver[SUNIndex[ii_]] * (p : HoldPattern[Plus[(SUNDelta[ExplicitSUNIndex[jj_Integer], SUNIndex[ii_]] * _) ..]]) * rest_) :> (p * rest /. ii -> jj)} ; Print["Length of expression ", fcelements[[rep]], ": ", Length[summ]] ; subres = (Print["Finding summation indices"] ; suminds = (#[[1]]) & /@ Union[Cases[summ, _SumOver, Infinity]] ; sums = If[suminds === {}, {I1, 1}, Sequence @@ ((({#, If[FreeQ[summ, #], 1, 8]} & /@ suminds)))] ; Print["Sums: ", {sums}] ; tmpi = 0 ; tmpii = 0 ; Print["Summing..."] ; Sum[WriteString["stdout", (#[[1]]) & /@ {sums}] ; subsum = ( ((* WriteString["stdout", "."] ; *) SUNReduce[SUNReduce[SUNReduce[SUNReduce[#]]]]) & /@ (tmpres = WriteOutUMatrices[(* Print["Expanding matrices and momenta"] ; *) summ (* /. p1 -> -p3 - p4 *) /. subpar /. udrules // MomentumExpand // ExpandScalarProduct // MomentumCombine (* // Expand *)] (* ; Print["Length of expression: ", Length[tmpres], ". Reducing SU(3) structures and simplifying"] *) ; tmpres) ) /. {p2 + p3 -> -p1, -p2 - p3 -> p1} ; If[! FreeQ[subsum, (SU3F | SU3D)[___, _SUNIndex, ___], Infinity], WriteString["stdout", "Still contractions left. Summing explicitly "] ; subsum = (WriteString["stdout", "."] ; SUNReduce[#, Explicit -> True, HoldSums -> False]) & /@ Expand[subsum], subsum] ; Simplify[subsum], Evaluate[sums]] // MomentumExpand // ExpandScalarProduct // Simplify) ; res = Append[res, subres], {rep, 1, Length[amplFC1]}] ;$

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

$rep = 0 ; res3 = (++ rep ; WriteString["stdout", rep, " "] ; exp = Collect[#, {_FeynAmpDenominator}] ; If[Head[exp] === Plus, WriteString["stdout", Length, " ", Length[exp], " "] ; (WriteString["stdout", "."] ; OneLoop[q1, #, Dimension -> D]) & /@ exp, OneLoop[q1, exp, Dimension -> D]]) & /@ Take[res, {1, -1}] ;$

1 Length 3 ...2 Length 5 ...3 Length 3 ...4 Length 5 .....1 Length 3 ...2 Length 5 .....3 Length 3 ...4 Length 5 .....1 Length 3 ...2 Length 5 .....3 Length 3 ...4 Length 5 .....1 Length 3 ...2 Length 5 .....3 Length 3 ...4 Length 5 .....

Higher rank Passarino-Veltman symbols are reduced:

$res4 = CheckF[Simplify /@ ((Collect[If[FreeQ[#, PaVe, Infinity, Heads -> True], #, PaVeReduce[#]], {(* *) Pi, _DecayConstant, _B0, _A0 (* , _CouplingConstant, _Pair, _ParticleMass *)}] & /@ SUNReduce[res3, FullReduce -> True]) /. D -> Sequence[]), "KSS2res4"]$

${-((3 A _ 0 ( (m _ π^(ó ))^2 ) + 6 A _ 0 ( (m _ K^(ó ))^2 ) + A _ 0 ( (m _ η^(ó ))^2 )) c _ 5^( ) !, _ 0^( ) (δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)))/(24 π^2 (f _ ϕ^(ó ))^3), 1/(144 π^2 (f _ ϕ^(ó ))^3) (!, _ 0^( ) (-(12 B _ 0 (0, (m _ π^(ó ))^2, (m _ η^(ó ))^2) c _ 2^( ) p _ 1 · p _ 3 ((m _ π^(ó ))^2 - (m _ η^(ó ))^2) δ _ (3 i)^(3))/p _ 3^2 + (6 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) c _ 2^( ) (p _ 1 · p _ 3 (2 (m _ π^(ó ))^2 - 2 (m _ η^(ó ))^2 + p _ 3^2) + p _ 3^2 ((m _ π^(ó ))^2 + (m _ η^(ó ))^2 - p _ 3^2)) δ _ (3 i)^(3))/p _ 3^2 + B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) (4 c _ 5^( ) ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) + 3 c _ 2^( ) (2 (m _ η^(ó ))^2 + p _ 1 · p _ 3 - p _ 3^2)) (3 δ _ (0 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + 6 A _ 0 ( (m _ η^(ó ))^2 ) c _ 2^( ) (3 δ _ (0 i)^(3) + δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + 6 A _ 0 ( (m _ K^(ó ))^2 ) c _ 2^( ) (6 δ _ (0 i)^(3) + 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) - 3 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) (3 c _ 2^( ) (2 (m _ π^(ó ))^2 + p _ 1 · p _ 3 - p _ 3^2) + 4 c _ 5^( ) ((m _ K^(ó ))^2 - (m _ π^(ó ))^2)) (3 δ _ (0 i)^(3) + 3^(1/2) δ _ (8 i)^(3)) - 6 A _ 0 ( (m _ π^(ó ))^2 ) c _ 2^( ) (9 δ _ (0 i)^(3) - δ _ (3 i)^(3) + 3 3^(1/2) δ _ (8 i)^(3)) + 3 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) (4 c _ 5^( ) ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) (6 δ _ (0 i)^(3) - δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + c _ 2^( ) (2 (m _ K^(ó ))^2 + p _ 1 · p _ 3 - p _ 3^2) (6 δ _ (0 i)^(3) + 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3))))), 1/(144 π^2 (f _ ϕ^(ó ))^3 (p _ 2^2 - (m _ K^(ó ))^2)) (c _ 5^( ) ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) !, _ 0^( ) (A _ 0 ( (m _ η^(ó ))^2 ) (12 δ _ (0 i)^(3) - 3 δ _ (3 i)^(3) - 5 3^(1/2) δ _ (8 i)^(3)) + 12 A _ 0 ( (m _ K^(ó ))^2 ) (6 δ _ (0 i)^(3) - 2 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + 3 A _ 0 ( (m _ π^(ó ))^2 ) (12 δ _ (0 i)^(3) - 5 δ _ (3 i)^(3) + 3^(1/2) δ _ (8 i)^(3)))), -1/(144 π^2 (f _ ϕ^(ó ))^3 (p _ 2^2 - (m _ K^(ó ))^2)) (c _ 5^( ) ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) !, _ 0^( ) (-6 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) δ _ (0 i)^(3) (m _ π^(ó ))^2 + 2 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) δ _ (3 i)^(3) (m _ π^(ó ))^2 + 2 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) δ _ (8 i)^(3) (m _ π^(ó ))^2 + 18 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) (m _ K^(ó ))^2 δ _ (0 i)^(3) + 36 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) (m _ K^(ó ))^2 δ _ (0 i)^(3) + 18 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) (m _ K^(ó ))^2 δ _ (0 i)^(3) - 18 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) (m _ η^(ó ))^2 δ _ (0 i)^(3) - 18 A _ 0 ( (m _ η^(ó ))^2 ) δ _ (0 i)^(3) + 18 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 1 · p _ 2 δ _ (0 i)^(3) + 36 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 2 δ _ (0 i)^(3) + 18 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 2 δ _ (0 i)^(3) - 9 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 1 · p _ 3 δ _ (0 i)^(3) - 18 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 3 δ _ (0 i)^(3) - 9 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 3 δ _ (0 i)^(3) - 9 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 2 · p _ 3 δ _ (0 i)^(3) - 18 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 2 · p _ 3 δ _ (0 i)^(3) - 9 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 2 · p _ 3 δ _ (0 i)^(3) + 9 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 3^2 δ _ (0 i)^(3) + 18 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 3^2 δ _ (0 i)^(3) + 9 B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 3^2 δ _ (0 i)^(3) + 4 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) (m _ K^(ó ))^2 δ _ (3 i)^(3) - 6 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) (m _ K^(ó ))^2 δ _ (3 i)^(3) + 6 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) (m _ η^(ó ))^2 δ _ (3 i)^(3) + 6 A _ 0 ( (m _ η^(ó ))^2 ) δ _ (3 i)^(3) - 12 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 2 δ _ (3 i)^(3) - 6 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 2 δ _ (3 i)^(3) + 6 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 3 δ _ (3 i)^(3) + 3 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 3 δ _ (3 i)^(3) + 6 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) p _ 2 · p _ 3 δ _ (3 i)^(3) + 3 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 2 · p _ 3 δ _ (3 i)^(3) - 6 B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ η^(ó ))^2) p _ 3^2 δ _ (3 i)^(3) - 3 B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 3^2 δ _ (3 i)^(3) + 6 3^(1/2) B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) (m _ K^(ó ))^2 δ _ (8 i)^(3) - 6 3^(1/2) B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) (m _ K^(ó ))^2 δ _ (8 i)^(3) - 6 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) (m _ K^(ó ))^2 δ _ (8 i)^(3) + 6 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) (m _ η^(ó ))^2 δ _ (8 i)^(3) + 6 3^(1/2) A _ 0 ( (m _ η^(ó ))^2 ) δ _ (8 i)^(3) + 6 3^(1/2) B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 1 · p _ 2 δ _ (8 i)^(3) - 6 3^(1/2) B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 2 δ _ (8 i)^(3) - 6 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 2 δ _ (8 i)^(3) - 3 3^(1/2) B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 1 · p _ 3 δ _ (8 i)^(3) + 3 3^(1/2) B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 1 · p _ 3 δ _ (8 i)^(3) + 3 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 1 · p _ 3 δ _ (8 i)^(3) - 3 3^(1/2) B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 2 · p _ 3 δ _ (8 i)^(3) + 3 3^(1/2) B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 2 · p _ 3 δ _ (8 i)^(3) + 3 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 2 · p _ 3 δ _ (8 i)^(3) + 3 3^(1/2) B _ 0 (p _ 3^2, (m _ π^(ó ))^2, (m _ π^(ó ))^2) p _ 3^2 δ _ (8 i)^(3) - 3 3^(1/2) B _ 0 (p _ 3^2, (m _ K^(ó ))^2, (m _ K^(ó ))^2) p _ 3^2 δ _ (8 i)^(3) - 3 3^(1/2) B _ 0 (p _ 3^2, (m _ η^(ó ))^2, (m _ η^(ó ))^2) p _ 3^2 δ _ (8 i)^(3) - 6 A _ 0 ( (m _ π^(ó ))^2 ) (3 δ _ (0 i)^(3) - δ _ (3 i)^(3) + 3^(1/2) δ _ (8 i)^(3)) + 6 A _ 0 ( (m _ K^(ó ))^2 ) (-6 δ _ (0 i)^(3) + δ _ (3 i)^(3) + 3^(1/2) δ _ (8 i)^(3))))}$

Converted by Mathematica (July 10, 2003)