•The fourth order loop amplitude

mesonstop = CreateTopologies[1, 1 -> 2, Adjacencies -> {3, 4, 5}, ExcludeTopologies -> {SelfEnergies, WFCorrections, Tadpoles}] ;

loopinsert = InsertFields[mesonstop, {Scalar[2][0, {i1}]} -> {Pion[0, {i2}], Pion[0, {i3}]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

Paint[loopinsert, PaintLevel -> {Classes}, AutoEdit -> False, SheetHeader -> False, Numbering -> False, ColumnsXRows -> {2, 1}] ;

[Graphics:../HTMLFiles/index_12.gif]

$ConstantIsoIndices = {} ;

ampFC = CreateFCAmp[loopinsert] /. _SumOver -> 1 // SUNReduce // SUNReduce // Simplify

{(5 i !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(48 π^4 (f _ π^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2)), (i (5 (m _ π^(ó  ))^2 + 4 p _ 3 · p _ 4 - 2 p _ 3 · q _ 1 + 2 (p _ 3 + p _ 4 + q _ 1) . (p _ 3) - 2 p _ 4 · q _ 1 + 2 (p _ 3 + p _ 4 + q _ 1) . (p _ 4) - 4 (p _ 3 + p _ 4 + q _ 1) . (q _ 1)) !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(48 π^4 (f _ π^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2))}

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

ampsimple = Simplify /@ ampreduced // ExpandScalarProduct // Simplify

{-(5 A _ 0 ( (m _ π^(ó  ))^2 ) !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(48 π^2 (f _ π^(ó  ))^2), ((4 A _ 0 ( (m _ π^(ó  ))^2 ) - B _ 0 (p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 4 p _ 3^2 + 12 p _ 3 · p _ 4 + 4 p _ 4^2)) !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(48 π^2 (f _ π^(ó  ))^2)}

ampinfinitiesfull = VeltmanExpand[#, ExplicitLeutwylerJ0 -> True, ExplicitLeutwylerSigma -> True, B0Evaluation -> "jbar"] & /@ ampsimple // Simplify

{(5 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(48 π^2 (f _ π^(ó  ))^2), 1/(12 π^2 (f _ π^(ó  ))^2) ((-(32 π^2 λ + log((m _ π^(ó  ))^2/μ^2) + 1) (m _ π^(ó  ))^2 + (m _ π^(ó  ))^2 + 1/4 (-16 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) + 32 π^2 λ + log((m _ π^(ó  ))^2/μ^2) + 1) ((m _ π^(ó  ))^2 + 4 p _ 3^2 + 12 p _ 3 · p _ 4 + 4 p _ 4^2)) !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))}

Converted by Mathematica  (July 10, 2003)