•Calculation and reduction of the amplitude

Calculation of the amplitude:

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

amplFC // Length

67

Count[amplFC, 0]

59

LeafCount /@ amplFC

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 710, 61, 1, 1, 1, 1, 899, 87, 899, 87, 899, 87, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

Isospin implification:

af = (WriteString["stdout", "."] ; Simplify[Collect[Expand[Contract[ScalarProductExpand[MomentumCombine[Simplify[SUNReduce[DoSumOver[#], FullReduce -> True]]] /. {p2 + p3 + p4 -> -p1, -p2 - p3 - p4 -> p1}]]], {_SumOver, _DecayConstant, Pi, _CouplingConstant, _SU2Delta, _FeynAmpDenominator, _Pair, _ParticleMass}] //. a_Pair * b_ -> a * Simplify[b]]) & /@ amplFC ;

...................................................................

LeafCount /@ af

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 258, 1, 1, 1, 1, 1, 657, 1, 653, 1, 653, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

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

cou = 0 ; ampreduced = (++ cou ; WriteString["stdout", cou, " "] ; OneLoop[q1, #, OneLoopSimplify -> True]) & /@ af ;

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

LeafCount /@ ampreduced

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 625, 1, 1, 1, 1, 1, 5306, 1, 5306, 1, 5306, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

ampsimple = (WriteString["stdout", "."] ; Collect[# // PaVeReduce, {_DecayConstant, Pi, _CouplingConstant, _SU2Delta, _SU2F, _A0, _B0, _C0}] /. (a : ((A0 | B0 | C0)[__])) * b_ -> a * tt[b] /. tt -> Together /. (a : ((A0 | B0 | C0)[__])) * b_ -> a * Simplify[b]) & /@ ampreduced ;

...................................................................

LeafCount /@ ampsimple

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 240, 1, 1, 1, 1, 1, 1745, 1, 1745, 1, 1745, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

count = 0 ; ampsimplest = (++ count ; WriteString["stdout", count, " "] ; Collect[MandelstamReduce[(WriteString["stdout", "."] ; Simplify[#]) & /@ Collect[Expand[#] /. pirule /. subpar /. dmrules, {_A0, _B0, _C0, _D0}], Cancel -> None], {_A0, _B0, _C0, _D0}]) & /@ ampsimple ;

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ..26 27 28 29 30 31 ...32 33 ...34 35 ...36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

LeafCount /@ ampsimplest

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 173, 1, 1, 1, 1, 1, 532, 1, 532, 1, 263, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

count = 0 ; ampsimplestus = (++ count ; WriteString["stdout", count, " "] ; Simplify /@ Collect[#, {_A0, _B0, _C0, _D0}]) & /@ ampsimplest ;

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

LeafCount /@ ampsimplestus

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 68, 1, 1, 1, 1, 1, 133, 1, 133, 1, 133, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

Count[ampsimplestus, 0]

63

ampsimplestu = (Simplify /@ Collect[Plus @@ ampsimplestus, {_A0, _B0, _C0, _D0}]) // Simplify ;

ampsimplestu // LeafCount

271

count = 0 ; ampsimplestusim = ((++ count ; WriteString["stdout", count, " "] ; Simplify[Together[#]]) & /@ (ampsimplestu - (ampsimplestu /. (_A0 | _B0 | _C0 | _D0) -> 0))) + Simplify[ampsimplestu /. (_A0 | _B0 | _C0 | _D0) -> 0] /. manrul ;

1 2 3 4

ampsimplestusim // LeafCount

381

amploopfull = VeltmanExpand[ampsimplestusim, ExplicitLeutwylerJ0 -> True] ;

Converted by Mathematica  (July 10, 2003)