•Loop contribution of fourth order in the chiral expansion

mesonstop = (CreateCTTopologies[2, 1 -> 2, Adjacencies -> {3, 4, 5}, ExcludeTopologies -> {SelfEnergyCTs, WFCorrectionCTs, TadpoleCTs}] // DiscardCT // Union[#, AddExternalLegs[#, ExternalPropagators -> 1, SelfEnergies -> False], AddExternalLegs[#, ExternalPropagators -> 2, SelfEnergies -> False]] & // Flatten) ;

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

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

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ampFC = (CreateFCAmp[loopinsert] // DoSumOver // SUNReduce) /. deltaReduce // Simplify

{(5 i (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (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^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2))}

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

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

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

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^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^2 ((m _ π^(ó  ))^2 - p _ 3^2) ((m _ π^(ó  ))^2 - p _ 4^2)), -(20 (Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) - 2 λ - (log((m _ π^(ó  ))^2/μ^2) + 1)/(16 π^2)) (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(3 (p _ 3^2 - (m _ π^(ó  ))^2)), (5 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^2 ((m _ π^(ó  ))^2 - p _ 3^2) ((m _ π^(ó  ))^2 - p _ 4^2)), -(20 (Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) - 2 λ - (log((m _ π^(ó  ))^2/μ^2) + 1)/(16 π^2)) (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(3 (p _ 4^2 - (m _ π^(ó  ))^2)), (5 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^2 ((m _ π^(ó  ))^2 - p _ 3^2) ((m _ π^(ó  ))^2 - p _ 4^2)), ((4 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (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^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(12 π^2 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))}

Converted by Mathematica  (July 10, 2003)