•Loop contribution of fourth order in the chiral expansion

tops = CreateTopologies[1, 1 -> 1, Adjacencies -> {3, 4}] ;

inserts = InsertFields[tops, {Photon[0]} -> {Photon[0]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

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

[Graphics:../HTMLFiles/index_77.gif]

amplFC = CreateFCAmp[inserts, AmplitudeLevel -> Classes, EqualMasses -> False, Sum -> True] // Simplify

{(i (e^( ))^2 g^(μ _ 1 μ _ 2) µ _ μ _ 1(p _ 1) µ _ μ _ 2^*(p _ 3) SumOver(I _ 1, 3) (δ _ (3 I _ 1)^2 - δ _ (I _ 1 I _ 1)))/(16 π^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2)), (i (e^( ))^2 (q _ 1^μ _ 1 + ((p _ 3 + q _ 1)^μ _ 1)) µ _ μ _ 1(p _ 1) (q _ 1^μ _ 2 + ((p _ 3 + q _ 1)^μ _ 2)) µ _ μ _ 2^*(p _ 3) (f _ (3 I _ 1 I _ 2)^(2))^2 SumOver(I _ 1, 3) SumOver(I _ 2, 3))/(32 π^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ π^(I _ 2 ))^2))}

$ConstantIsoIndices = Union[$ConstantIsoIndices, {i1, i2, I1, I2}]

{i _ 1, I _ 1, i _ 2, I _ 2, I _ 3, I _ 4, I _ 5, I _ 6}

aff = amplFC /. i2 -> i1 // SUNReduce[#, FullReduce -> True] & // Contract // Simplify

{(i (e^( ))^2 µ ( p _ 1 ) · µ^* ( p _ 3 ) (δ _ (3 I _ 1)^(2) - 1) SumOver(I _ 1, 3))/(16 π^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2)), (i (e^( ))^2 (p _ 3 · µ ( p _ 1 ) + 2 q _ 1 · µ ( p _ 1 )) q _ 1 · µ^* ( p _ 3 ) (f _ (3 I _ 1 I _ 2)^(2))^2 SumOver(I _ 1, 3) SumOver(I _ 2, 3))/(16 π^4 (q _ 1^2 - (m _ π^(I _ 1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ π^(I _ 2 ))^2))}

Cases[aff // SUNSimplify, _SumOver, Infinity] // Union // StandardForm

{SumOver[I1, 3], SumOver[I2, 3]}

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

ampsimple = ampreduced // SUNReduce[#, FullReduce -> True] & // Simplify

{-(A _ 0 ( (m _ π^(I _ 1 ))^2 ) (e^( ))^2 µ ( p _ 1 ) · µ^* ( p _ 3 ) (δ _ (3 I _ 1)^(2) - 1) SumOver(I _ 1, 3))/(16 π^2), -1/(288 π^2 p _ 3^2) ((e^( ))^2 µ ( p _ 1 ) · µ^* ( p _ 3 ) (-(3 B _ 0 (p _ 3^2, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) + 2) p _ 3^4 + 6 A _ 0 ( (m _ π^(I _ 2 ))^2 ) p _ 3^2 + 3 ((B _ 0 (0, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) + 2 B _ 0 (p _ 3^2, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) + 2) (m _ π^(I _ 1 ))^2 + (-B _ 0 (0, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) + 2 B _ 0 (p _ 3^2, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) + 2) (m _ π^(I _ 2 ))^2) p _ 3^2 + 3 (B _ 0 (0, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2) - B _ 0 (p _ 3^2, (m _ π^(I _ 1 ))^2, (m _ π^(I _ 2 ))^2)) ((m _ π^(I _ 1 ))^2 - (m _ π^(I _ 2 ))^2)^2) (f _ (3 I _ 1 I _ 2)^(2))^2 SumOver(I _ 1, 3) SumOver(I _ 2, 3))}

Converted by Mathematica  (July 10, 2003)