•Renormalization

This is the sum of all unrenormalized amplitudes:

ampfinal = amploopfull + ampl2mult + ampl4 /. gellmannOkubo /. dcren // Simplify ;

The infinities exactly cancel:

cc1 = Coefficient[Renormalize[ampl2mult + ampl4 /. dcren], LeutwylerLambda[]] // Simplify

(484 (m _ π^(ó  r ))^4 - 4 ((m _ K^(ó  r ))^2 + 81 (s + t)) (m _ π^(ó  r ))^2 + 81 (s^2 + t s + t^2))/(18 (f _ π^(ó  r ))^4)

cc = Coefficient[amploopfull /. dcren, LeutwylerLambda[]] // Simplify

(-162 (m _ π^(ó  r ))^4 + 2 (2 (m _ K^(ó  r ))^2 - (m _ η^(ó  r ))^2 + 54 (s + t)) (m _ π^(ó  r ))^2 - 27 (s^2 + t s + t^2))/(6 (f _ π^(ó  r ))^4)

cc + cc1 /. gellmannOkubo // Simplify

0

This is then the full renormalized amplitude:

ampfinalren = (Renormalize[ampfinal] // Collect[#, LeutwylerLambda[]] & // Simplify // Collect[#, {_LeutwylerJBar, _DecayConstant, _CouplingConstant}] & // Simplify) /. toEtaRules // Simplify

1/(72 (f _ π^(ó  r ))^4) (4 Overscript[J, _] _ (m _ η^(ó  r ))^2(s) (m _ π^(ó  r ))^4 + 4 Overscript[J, _] _ (m _ η^(ó  r ))^2(t) (m _ π^(ó  r ))^4 + 4 Overscript[J, _] _ (m _ η^(ó  r ))^2(4 (m _ π^(ó  r ))^2 - s - t) (m _ π^(ó  r ))^4 + 72 (f _ π^(ó  r ))^2 (m _ π^(ó  r ))^2 + 9 Overscript[J, _] _ (m _ K^(ó  r ))^2(4 (m _ π^(ó  r ))^2 - s - t) (-4 (m _ π^(ó  r ))^2 + s + t)^2 + 9 s^2 Overscript[J, _] _ (m _ K^(ó  r ))^2(s) + 9 t^2 Overscript[J, _] _ (m _ K^(ó  r ))^2(t) + 36 Overscript[J, _] _ (m _ π^(ó  r ))^2(s) (3 (m _ π^(ó  r ))^4 - 4 s (m _ π^(ó  r ))^2 + 2 s^2) + 36 Overscript[J, _] _ (m _ π^(ó  r ))^2(t) (3 (m _ π^(ó  r ))^4 - 4 t (m _ π^(ó  r ))^2 + 2 t^2) + 36 Overscript[J, _] _ (m _ π^(ó  r ))^2(4 (m _ π^(ó  r ))^2 - s - t) (19 (m _ π^(ó  r ))^4 - 12 (s + t) (m _ π^(ó  r ))^2 + 2 (s + t)^2) + 72 (-32 L _ 4^(r ) (m _ π^(ó  r ))^4 - 16 L _ 5^(r ) (m _ π^(ó  r ))^4 + 96 L _ 6^(r ) (m _ π^(ó  r ))^4 + 48 L _ 8^(r ) (m _ π^(ó  r ))^4 + 16 L _ 1^(r ) (6 (m _ π^(ó  r ))^4 - 4 (s + t) (m _ π^(ó  r ))^2 + s^2 + t^2 + s t) + 16 L _ 2^(r ) (6 (m _ π^(ó  r ))^4 - 4 (s + t) (m _ π^(ó  r ))^2 + s^2 + t^2 + s t) + 8 L _ 3^(r ) (6 (m _ π^(ó  r ))^4 - 4 (s + t) (m _ π^(ó  r ))^2 + s^2 + t^2 + s t) + 1/(192 π^2) (-2 (75 log((m _ π^(ó  r ))^2/μ^2) + 12 log((m _ K^(ó  r ))^2/μ^2) + log((m _ η^(ó  r ))^2/μ^2) + 88) (m _ π^(ó  r ))^4 + 12 (s + t) (8 log((m _ π^(ó  r ))^2/μ^2) + log((m _ K^(ó  r ))^2/μ^2) + 9) (m _ π^(ó  r ))^2 - 3 (s^2 + t s + t^2) (8 log((m _ π^(ó  r ))^2/μ^2) + log((m _ K^(ó  r ))^2/μ^2) + 9))))

ampfinalren - (ampfinalren /. CouplingConstant[ChPT3[4], __] -> 0) // FullSimplify

(8 (2 (6 L _ 1^(r ) + 6 L _ 2^(r ) + 3 L _ 3^(r ) - 2 L _ 4^(r ) - L _ 5^(r ) + 6 L _ 6^(r ) + 3 L _ 8^(r )) (m _ π^(ó  r ))^4 - 4 (s + t) (2 (L _ 1^(r ) + L _ 2^(r )) + L _ 3^(r )) (m _ π^(ó  r ))^2 + (s^2 + t s + t^2) (2 (L _ 1^(r ) + L _ 2^(r )) + L _ 3^(r ))))/(f _ π^(ó  r ))^4

Converted by Mathematica  (July 10, 2003)