•Loop contribution of fourth order in the chiral expansion

Construction of topologies:

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

Field insertion:

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

Graphical representation of the process:

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

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

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert] ;

The one-loop integral is simplified:

aff = (amplFC // DoSumOver) /. {i1 -> 1, i3 -> 1} // SUNReduce // Simplify

{(i (5 (m _ π^(ó  ))^2 + 4 p _ 1 · p _ 3 - 4 q _ 1^2))/(96 π^4 (f _ π^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2))}

The loop integral is expressed in terms of Passarino-Veltman symbols:

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

{-(A _ 0 ( (m _ π^(ó  ))^2 ) (m _ π^(ó  ))^2 + 4 A _ 0 ( (m _ π^(ó  ))^2 ) p _ 1 · p _ 3)/(96 π^2 (f _ π^(ó  ))^2)}

ampsimple = FullSimplify /@ ampreduced

{-(A _ 0 ( (m _ π^(ó  ))^2 ) ((m _ π^(ó  ))^2 + 4 p _ 1 · p _ 3))/(96 π^2 (f _ π^(ó  ))^2)}

The divergence is singled out:

ampinfinities = VeltmanExpand[ampsimple[[1]], ExplicitLeutwylerJ0 -> True] // Simplify

((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 ((m _ π^(ó  ))^2 + 4 p _ 1 · p _ 3))/(96 π^2 (f _ π^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)