•Counterterms

The counterterms of Bijnens, Pallante and Prades:

fpionfac = CheckF[dum, "ChPT3A00P20o2.Fac"]

(128 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 - 2 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 128 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))/(32 π^2 (f _ ϕ^(ó  ))^2) + 1

fkaonfac = CheckF[dum, "ChPT3A00P60o2.Fac"]

(-3 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 512 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 6 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 4 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 512 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))/(128 π^2 (f _ ϕ^(ó  ))^2) + 1

sqrtZPi0 = (1 + CheckF[dum, "ChPT3P02o2.Fac"])/2

1/2 ((144 π^2 (f _ ϕ^(ó  ))^2 + (-1152 π^2 (L _ 4^(r ) + L _ 5^(r ) - 4 (L _ 6^(r ) + L _ 8^(r ))) - 9 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - (2304 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) + 9 log((m _ K^(ó  ))^2/μ^2) + 4 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)/(144 π^2 (f _ ϕ^(ó  ))^2) + 1)

sqrtZPi0inv = (3 - CheckF[dum, "ChPT3P02o2.Fac"])/2

1/2 (3 - (144 π^2 (f _ ϕ^(ó  ))^2 + (-1152 π^2 (L _ 4^(r ) + L _ 5^(r ) - 4 (L _ 6^(r ) + L _ 8^(r ))) - 9 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - (2304 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) + 9 log((m _ K^(ó  ))^2/μ^2) + 4 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)/(144 π^2 (f _ ϕ^(ó  ))^2))

sqrtZK0 = (1 + CheckF[dum, "ChPT3P06o2.Fac"])/2

1/2 (1/(576 π^2 (f _ ϕ^(ó  ))^2) (576 π^2 (f _ ϕ^(ó  ))^2 + (-4608 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) - 27 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - 2 (2304 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 8 L _ 6^(r ) - 4 L _ 8^(r )) + 27 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2) + 1)

sqrtZK0inv = (3 - CheckF[dum, "ChPT3P06o2.Fac"])/2

1/2 (3 - 1/(576 π^2 (f _ ϕ^(ó  ))^2) (576 π^2 (f _ ϕ^(ó  ))^2 + (-4608 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) - 27 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - 2 (2304 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 8 L _ 6^(r ) - 4 L _ 8^(r )) + 27 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))

bres = Expand[sqrtZPi0 sqrtZK0 C/2^(1/2) 2 (2 - fpionfac) (2 - fkaonfac) (q^2 (-G8 + (fpionfac^2 + fkaonfac^2) GG8) - mPi^2 fpionfac^2 GG8 + 2/DecayConstant[PhiMeson]^2 (2 q^4 G8 Ε _ 3 + q^2 mK^2 (G8 (4 Ε _ 1 + 4 Ε _ 2 - 2 Ε _ 5 - 2 Ε _ 10) - 8 GG8 (4 L6 + L8)) + q^2 mPi^2 (G8 (2 Ε _ 2 + 2 Ε _ 5 - Ε _ 11) - 8 GG8 (2 L6 + L8)) - 2 mK^2 mPi^2 (G8 (Ε _ 1 + Ε _ 2) - 8 GG8 L6) - mPi^4 (G8 Ε _ 2 - 8 GG8 (L6 + L8)))) /. {G8 -> C2, GG8 -> C5, q^2 -> Pair[Momentum[p1], Momentum[p1]], q^4 -> Pair[Momentum[p1], Momentum[p1]]^2, mK -> ParticleMass[Kaon], mPi -> ParticleMass[Pion]}] ;

bbres = bres /. {CouplingConstant[_[4], ___]^_ -> 0, CouplingConstant[_[4], a___] CouplingConstant[_[4], b___] -> 0, CouplingConstant[_[4], a___] CouplingConstant[_[4], b___]^_ -> 0, Log[_]^_ -> 0, Log[a_] Log[b_] -> 0, Log[a_] Log[b_]^_ -> 0} /. _RenormalizationState -> Sequence[] /. BijnensToKambor ;

bbres // Length

466

BijnensRes = DiscardOrders[#, PerturbationOrder -> 4] & /@ bbres ;

Simplify[4 2^(1/2) QuarkCondensate[]^2 BijnensRes/((Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)) /. {_Log -> 0, C -> 1}]

-1/((f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (8 (c _ 2^( ) (p _ 1^2 (f _ ϕ^(ó  ))^2 + 2 (N _ 11^( ) (m _ π^(ó  ))^4 + 2 N _ 10^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 N _ 11^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 L _ 4^( ) p _ 1^2 (m _ π^(ó  ))^2 - 4 L _ 5^( ) p _ 1^2 (m _ π^(ó  ))^2 + 16 L _ 6^( ) p _ 1^2 (m _ π^(ó  ))^2 + 8 L _ 8^( ) p _ 1^2 (m _ π^(ó  ))^2 + N _ 8^( ) p _ 1^2 (m _ π^(ó  ))^2 - 2 N _ 11^( ) p _ 1^2 (m _ π^(ó  ))^2 + 2 N _ 12^( ) p _ 1^4 - 16 L _ 4^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 L _ 5^( ) p _ 1^2 (m _ K^(ó  ))^2 + 32 L _ 6^( ) p _ 1^2 (m _ K^(ó  ))^2 + 8 L _ 8^( ) p _ 1^2 (m _ K^(ó  ))^2 + 2 N _ 5^( ) p _ 1^2 (m _ K^(ó  ))^2 + 2 N _ 8^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 N _ 10^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 N _ 11^( ) p _ 1^2 (m _ K^(ó  ))^2 - 2 N _ 36^( ) (p _ 1^4 + ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) p _ 1^2 - (m _ π^(ó  ))^2 (m _ K^(ó  ))^2))) + c _ 5^( ) (((m _ π^(ó  ))^2 - 2 p _ 1^2) (f _ ϕ^(ó  ))^2 + 8 (L _ 4^( ) (2 p _ 1^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) + L _ 5^( ) (p _ 1^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) - (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) - 2 (L _ 6^( ) (2 p _ 1^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) + L _ 8^( ) (p _ 1^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) - (m _ π^(ó  ))^2 (m _ K^(ó  ))^2))))) (!, _ 0^( ))^2)

The counterterms of the present calculation:

Simplify[Plus @@ final2all + end4old /. _RenormalizationState -> Sequence[] /. {_Log -> 0, _LeutwylerJBar -> 0, p3 -> -p1, p2 -> 0}]

-1/((f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (8 (c _ 2^( ) (p _ 1^2 (f _ ϕ^(ó  ))^2 + 2 (N _ 11^( ) (m _ π^(ó  ))^4 + 2 N _ 10^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 N _ 11^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 L _ 4^( ) p _ 1^2 (m _ π^(ó  ))^2 - 4 L _ 5^( ) p _ 1^2 (m _ π^(ó  ))^2 + 16 L _ 6^( ) p _ 1^2 (m _ π^(ó  ))^2 + 8 L _ 8^( ) p _ 1^2 (m _ π^(ó  ))^2 + N _ 8^( ) p _ 1^2 (m _ π^(ó  ))^2 - 2 N _ 11^( ) p _ 1^2 (m _ π^(ó  ))^2 - 2 N _ 12^( ) p _ 1^4 - 16 L _ 4^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 L _ 5^( ) p _ 1^2 (m _ K^(ó  ))^2 + 32 L _ 6^( ) p _ 1^2 (m _ K^(ó  ))^2 + 8 L _ 8^( ) p _ 1^2 (m _ K^(ó  ))^2 + 2 N _ 5^( ) p _ 1^2 (m _ K^(ó  ))^2 + 2 N _ 8^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 N _ 10^( ) p _ 1^2 (m _ K^(ó  ))^2 - 4 N _ 11^( ) p _ 1^2 (m _ K^(ó  ))^2 + 2 N _ 36^( ) (p _ 1^4 - ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) p _ 1^2 + (m _ π^(ó  ))^2 (m _ K^(ó  ))^2))) + c _ 5^( ) (((m _ π^(ó  ))^2 - 2 p _ 1^2) (f _ ϕ^(ó  ))^2 + 8 (L _ 4^( ) (2 p _ 1^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) + L _ 5^( ) (p _ 1^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) - (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) - 2 (L _ 6^( ) (2 p _ 1^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) + L _ 8^( ) (p _ 1^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) - (m _ π^(ó  ))^2 (m _ K^(ó  ))^2))))) (!, _ 0^( ))^2)

As in the case of K->2π, there is a problem with N _ 12^( ) - N _ 36^( ), probably due to an error in the translation N's<-->E's.

% - %% // FullSimplify

(64 c _ 2^( ) (N _ 12^( ) - N _ 36^( )) p _ 1^4 (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))

Converted by Mathematica  (July 10, 2003)