•Renormalization

This is the sum of all unrenormalized amplitudes:

ampfinal = amploopfull + ampl2mult + ampl4 ;

The infinities. Meissner, Mueller & Steininger seem to have a problem with their counterterms. They yield the amplitude finite only for π^0 π^0 -> π^0 π^0. See instead the calculations in "Pions+VirtualPhotons" following Urich & Knecht.

cc0 = Coefficient[Renormalize[ampl2mult], LeutwylerLambda[]] /. ParticleMass[Vector[1], RenormalizationState[0]] -> 0 // MandelstamReduce[#, Cancel -> MandelstamU] & // FullSimplify

-(16 (m _ π^+^(ó  ))^2 (m _ π^0^(ó  ))^2)/(3 (f _ π^(ó  ))^4)

cc1 = Coefficient[Renormalize[ampl4], LeutwylerLambda[]] /. ParticleMass[Vector[1], RenormalizationState[0]] -> 0 // MandelstamReduce[#, Cancel -> MandelstamU] & // FullSimplify

(4 ((7 (m _ π^0^(ó  ))^4 - 4 (s + t) (m _ π^0^(ó  ))^2 + s^2 + t^2 + s t) (f _ π^(ó  ))^2 + C^( ) (e^( ))^2 (m _ π^0^(ó  ))^2))/(f _ π^(ó  ))^6

cc2 = Coefficient[amploopfull, LeutwylerLambda[]] /. ParticleMass[Vector[1], RenormalizationState[0]] -> 0 // MandelstamReduce[#, Cancel -> MandelstamU] & // Simplify

-(2 (39 (m _ π^0^(ó  ))^4 - (5 (m _ π^+^(ó  ))^2 + 24 (s + t)) (m _ π^0^(ó  ))^2 + 6 (s^2 + t s + t^2)))/(3 (f _ π^(ó  ))^4)

cc0 + cc1 + cc2 // Expand // Simplify

(2 (m _ π^0^(ó  ))^2 (2 C^( ) (e^( ))^2 + (f _ π^(ó  ))^2 ((m _ π^0^(ó  ))^2 - (m _ π^+^(ó  ))^2)))/(f _ π^(ó  ))^6

% /. dmrules // Expand // FullSimplify

0

This is then the full renormalized amplitude:

ampfinalren = ((Renormalize[ampfinal /. dmrules] // Collect[#, LeutwylerLambda[]] & // Simplify // Collect[#, {_DecayConstant, _SU2Delta, _CouplingConstant}] & // Simplify) //. manrul // FullSimplify) //. manrul // FullSimplify

1/(288 π^2 C^( ) (f _ π^(ó  ))^4) (64 π^2 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) - 5 (k _ 7^(r ) - 2 k _ 10^(r ) + k _ 11^(r ))) (m _ π^0^(ó  ))^2 ((m _ π^0^(ó  ))^2 - (m _ π^+^(ó  ))^2) (f _ π^(ó  ))^4 + 3 C^( ) ((16 π^2 (576 L _ 1^(r ) + 576 L _ 2^(r ) + 288 L _ 3^(r ) - 384 L _ 4^(r ) - 192 L _ 5^(r ) + 768 L _ 6^(r ) + 384 L _ 8^(r ) - 2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(s) + 3 Overscript[J, _] _ (m _ π^0^(ó  ))^2(s) - 2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(t) + 3 Overscript[J, _] _ (m _ π^0^(ó  ))^2(t) - 2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(u) + 3 Overscript[J, _] _ (m _ π^0^(ó  ))^2(u)) - 2 log((m _ π^+^(ó  ))^2/μ^2) - 12 log((m _ π^0^(ó  ))^2/μ^2) - 11) (m _ π^0^(ó  ))^4 + 2 (9 log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^2 + 16 π^2 (3 (f _ π^(ó  ))^2 + (-s + t + u) Overscript[J, _] _ (m _ π^+^(ó  ))^2(s) + (s - t + u) Overscript[J, _] _ (m _ π^+^(ó  ))^2(t) + (s + t - u) Overscript[J, _] _ (m _ π^+^(ó  ))^2(u))) (m _ π^0^(ó  ))^2 + 4 (-s^2 + t s - t^2 - u^2 + (s + t) u - 384 (t u + s (t + u)) π^2 L _ 1^(r ) + 8 π^2 (2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(s) s^2 - t Overscript[J, _] _ (m _ π^+^(ó  ))^2(s) s - u Overscript[J, _] _ (m _ π^+^(ó  ))^2(s) s - t Overscript[J, _] _ (m _ π^+^(ó  ))^2(t) s - 48 (t u + s (t + u)) L _ 2^(r ) - 24 (s t + (s + t) u) L _ 3^(r ) + 2 t^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(t) - t u Overscript[J, _] _ (m _ π^+^(ó  ))^2(t) - (s + t - 2 u) u Overscript[J, _] _ (m _ π^+^(ó  ))^2(u)) - (s^2 - (t + u) s + t^2 + u^2 - t u) log((m _ π^+^(ó  ))^2/μ^2))))

Converted by Mathematica  (July 10, 2003)