•Renormalization

This is the sum of all unrenormalized amplitudes:

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

The infinities exactly cancel:

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

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

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

(-25 (m _ π^(ó  r ))^4 + 64 (5 (m _ K^(ó  r ))^2 - 3 (m _ η^(ó  r ))^2) (m _ π^(ó  r ))^2 - 5620 (m _ K^(ó  r ))^4 + 12 (m _ K^(ó  r ))^2 (89 (m _ η^(ó  r ))^2 + 270 (s + t)) - 27 (13 (m _ η^(ó  r ))^4 + 30 (s^2 + t s + t^2)))/(180 (f _ K^(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 ;

ampfinalren1 = FullSimplify /@ Collect[ampfinalren, {Pi, _DecayConstant, _LeutwylerJBar, _Log}] //. manrules // Simplify

1/(4608 (f _ K^(ó  r ))^4) (4608 (f _ K^(ó  r ))^2 (m _ K^(ó  r ))^2 + 1/(π^2 ((m _ π^(ó  r ))^2 - (m _ K^(ó  r ))^2)) (3 (16 (9 log((m _ π^(ó  r ))^2/μ^2) + 48 log((m _ K^(ó  r ))^2/μ^2) + 31 log((m _ η^(ó  r ))^2/μ^2) + 76) (m _ K^(ó  r ))^6 - 8 (4 (9 log((m _ π^(ó  r ))^2/μ^2) + 24 log((m _ K^(ó  r ))^2/μ^2) + 11 log((m _ η^(ó  r ))^2/μ^2) + 38) (m _ π^(ó  r ))^2 - 9 (log((m _ π^(ó  r ))^2/μ^2) + 6 log((m _ K^(ó  r ))^2/μ^2) + 5 log((m _ η^(ó  r ))^2/μ^2) + 10) (u - 4 (m _ K^(ó  r ))^2)) (m _ K^(ó  r ))^4 + 18 ((s^2 + t s + t^2) (log((m _ π^(ó  r ))^2/μ^2) + 6 log((m _ K^(ó  r ))^2/μ^2) + 5 (log((m _ η^(ó  r ))^2/μ^2) + 2)) - 2 (5 log((m _ π^(ó  r ))^2/μ^2) + 12 log((m _ K^(ó  r ))^2/μ^2) + 7 log((m _ η^(ó  r ))^2/μ^2) + 20) (m _ π^(ó  r ))^2 (u - 4 (m _ K^(ó  r ))^2)) (m _ K^(ó  r ))^2 - 9 (s^2 + t s + t^2) (5 log((m _ π^(ó  r ))^2/μ^2) + 12 log((m _ K^(ó  r ))^2/μ^2) + 7 log((m _ η^(ó  r ))^2/μ^2) + 20) (m _ π^(ó  r ))^2)) + 16 (16 (864 L _ 3^(r ) - 576 L _ 4^(r ) - 288 L _ 5^(r ) + 1728 L _ 6^(r ) + 864 L _ 8^(r ) + 18 Overscript[J, _] _ (m _ K^(ó  r ))^2(s) + 4 Overscript[J, _] _ (m _ η^(ó  r ))^2(s) + 18 Overscript[J, _] _ (m _ K^(ó  r ))^2(t) + 4 Overscript[J, _] _ (m _ η^(ó  r ))^2(t) + 27 Overscript[J, _] _ (m _ π^(ó  r ))^2(u) + 108 Overscript[J, _] _ (m _ K^(ó  r ))^2(u) + 49 Overscript[J, _] _ (m _ η^(ó  r ))^2(u) + 6 (Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(s) + Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(t) + 4 Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(u))) (m _ K^(ó  r ))^4 - 72 (4 s Overscript[J, _] _ (m _ K^(ó  r ))^2(s) + 2 s Overscript[J, _] _ (m _ η^(ó  r ))^2(s) + 4 t Overscript[J, _] _ (m _ K^(ó  r ))^2(t) + 2 t Overscript[J, _] _ (m _ η^(ó  r ))^2(t) + 2 s Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(s) + 2 t Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(t) - 128 L _ 3^(r ) (u - 4 (m _ K^(ó  r ))^2) - 3 Overscript[J, _] _ (m _ π^(ó  r ))^2(u) (u - 4 (m _ K^(ó  r ))^2) - 14 Overscript[J, _] _ (m _ K^(ó  r ))^2(u) (u - 4 (m _ K^(ó  r ))^2) - 7 Overscript[J, _] _ (m _ η^(ó  r ))^2(u) (u - 4 (m _ K^(ó  r ))^2) - 4 Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(u) (u - 4 (m _ K^(ó  r ))^2)) (m _ K^(ó  r ))^2 + 4608 L _ 1^(r ) (-10 (m _ K^(ó  r ))^4 + 4 u (m _ K^(ó  r ))^2 + s^2 + t^2 + s t) + 4608 L _ 2^(r ) (-10 (m _ K^(ó  r ))^4 + 4 u (m _ K^(ó  r ))^2 + s^2 + t^2 + s t) + 9 (256 (s^2 + t s + t^2) L _ 3^(r ) + 3 (Overscript[J, _] _ (m _ π^(ó  r ))^2(s) s^2 + 6 Overscript[J, _] _ (m _ K^(ó  r ))^2(s) s^2 + 3 Overscript[J, _] _ (m _ η^(ó  r ))^2(s) s^2 + Overscript[J, _] _ (m _ π^(ó  r ))^2(u) s^2 + 6 Overscript[J, _] _ (m _ K^(ó  r ))^2(u) s^2 + 3 Overscript[J, _] _ (m _ η^(ó  r ))^2(u) s^2 + 2 Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(s) s^2 + 2 t Overscript[J, _] _ (m _ π^(ó  r ))^2(u) s + 12 t Overscript[J, _] _ (m _ K^(ó  r ))^2(u) s + 6 t Overscript[J, _] _ (m _ η^(ó  r ))^2(u) s + 2 Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(u) (u - 4 (m _ K^(ó  r ))^2)^2 + t^2 (Overscript[J, _] _ (m _ π^(ó  r ))^2(t) + 6 Overscript[J, _] _ (m _ K^(ó  r ))^2(t) + 3 Overscript[J, _] _ (m _ η^(ó  r ))^2(t)) + t^2 Overscript[J, _] _ (m _ π^(ó  r ))^2(u) + 6 t^2 Overscript[J, _] _ (m _ K^(ó  r ))^2(u) + 3 t^2 Overscript[J, _] _ (m _ η^(ó  r ))^2(u) + 2 t^2 Overscript[J, _] _ ((m _ π^(ó  r ))^2 (m _ η^(ó  r ))^2)(t)))))

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

(8 (-2 (10 L _ 1^(r ) + 10 L _ 2^(r ) + 5 L _ 3^(r ) + 2 L _ 4^(r ) + L _ 5^(r ) - 6 L _ 6^(r ) - 3 L _ 8^(r )) (m _ K^(ó  r ))^4 + 4 u (2 (L _ 1^(r ) + L _ 2^(r )) + L _ 3^(r )) (m _ K^(ó  r ))^2 + (s^2 + t s + t^2) (2 (L _ 1^(r ) + L _ 2^(r )) + L _ 3^(r ))))/(f _ K^(ó  r ))^4

Converted by Mathematica  (July 10, 2003)