•Renormalization

The coefficients of the infinities:

amploopfull

1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (12 (16 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + 32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (-48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + 96 π^2 λ + 3 log((m _ π^(ó  ))^2/μ^2) + 1) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))

Coefficient[amploopfull, LeutwylerLambda[]] // Simplify

(i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (4 (m _ π^(ó  ))^2 + p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))/(3 (f _ π^(ó  ))^2)

ampl2mult

i e^( ) ( p _ 3 - p _ 4 ) · µ ( p _ 1 ) (1 - ((384 π^2 L _ 4^( ) + 192 π^2 L _ 5^( ) - 32 π^2 λ - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2)) f _ (3 i _ 1 i _ 2)^(2)

Coefficient[Renormalize[ampl2mult], LeutwylerLambda[]]

-(8 i e^( ) ( p _ 3 - p _ 4 ) · µ ( p _ 1 ) (m _ π^(ó  ))^2 f _ (3 i _ 1 i _ 2)^(2))/(3 (f _ π^(ó  ))^2)

ampl4

-(2 i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (4 (2 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^2 + L _ 9^( ) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))/(f _ π^(ó  ))^2

Coefficient[Renormalize[ampl4], LeutwylerLambda[]] // Simplify

-(i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (12 (m _ π^(ó  ))^2 + p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))/(3 (f _ π^(ó  ))^2)

Coefficient[amploopfull, LeutwylerLambda[]] + Coefficient[Renormalize[ampl2mult], LeutwylerLambda[]] + Coefficient[Renormalize[ampl4], LeutwylerLambda[]] // ScalarProductExpand // Simplify

0

This is the full unrenormalized amplitude:

ampfinal = amploopfull + ampl2mult + ampl4 /. D -> Sequence[] // ExpandScalarProduct // Simplify

1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) (p _ 3 · µ ( p _ 1 ) - p _ 4 · µ ( p _ 1 )) (288 π^2 (f _ π^(ó  ))^2 - 192 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) (m _ π^(ó  ))^2 + (576 π^2 L _ 9^( ) + 48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) - 96 π^2 λ - 3 log((m _ π^(ó  ))^2/μ^2) - 1) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))

This is the full renormalized amplitude:

ampfinalren = Renormalize[ampfinal, InfinityFactor -> LeutwylerLambda[]] // MomentumCombine // Simplify

1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) ( p _ 3 - p _ 4 ) · µ ( p _ 1 ) (288 π^2 (f _ π^(ó  ))^2 - 192 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) (m _ π^(ó  ))^2 + (576 π^2 L _ 9^(r ) + 48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) - 3 log((m _ π^(ó  ))^2/μ^2) - 1) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))

We may project out in specific particle channels:

ampp[I1_, I2_] = ampfinalren /. {i1 -> I1, i2 -> I2, Pion -> PionPlus} ;

AmplitudeProjection[ampp, Channel -> {{PionPlus} -> {PionPlus}}, OnMassShell -> False] // Simplify

(e^( ) ( p _ 3 - p _ 4 ) · µ ( p _ 1 ) (288 π^2 (f _ π^+^(ó  ))^2 - 192 π^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2 + (576 π^2 L _ 9^(r ) + 48 π^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) - 3 log((m _ π^+^(ó  ))^2/μ^2) - 1) p _ 1^2))/(288 π^2 (f _ π^+^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)