![Overscript[J, _]](../HTMLFiles/index_242.gif){kind=small}
terms vanish for 
->0.The terms vanish for ->0.
![limitjs = finalloops /. {p3 -> -p1, CouplingConstant[_[4], ___] -> 0} /. _RenormalizationState -> Sequence[] /. _Log -> 0 // FullSimplify](../HTMLFiles/index_244.gif){kind=small}
![1/(36 (f _ ϕ^(ó ))^2 p _ 2^2 (p _ 1^2 - (m _ π^(ó ))^2) (p _ 1^2 - (m _ K^(ó ))^2)) ((2 c _ 5^( ) (3 Overscript[J, _] _ ((m _ π^(ó ))^2 (m _ K^(ó ))^2)(p _ 2^2) (36 p _ 1 · p _ 2 ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) + p _ 2^2 (12 p _ 1^2 + 5 p _ 2^2 - 9 ((m _ π^(ó ))^2 + 3 (m _ K^(ó ))^2))) + Overscript[J, _] _ ((m _ K^(ó ))^2 (m _ η^(ó ))^2)(p _ 2^2) (p _ 2^2 (-8 (m _ π^(ó ))^2 + 11 (m _ K^(ó ))^2 - 15 (m _ η^(ó ))^2 - 12 p _ 1^2 + 3 p _ 2^2) + 36 p _ 1 · p _ 2 ((m _ η^(ó ))^2 - (m _ K^(ó ))^2))) (m _ K^(ó ))^2 + c _ 2^( ) (3 Overscript[J, _] _ ((m _ π^(ó ))^2 (m _ K^(ó ))^2)(p _ 2^2) (-(m _ π^(ó ))^2 - (m _ K^(ó ))^2 + p _ 2^2) (36 p _ 1 · p _ 2 ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) + p _ 2^2 (12 p _ 1^2 + 5 p _ 2^2 - 9 ((m _ π^(ó ))^2 + 3 (m _ K^(ó ))^2))) + Overscript[J, _] _ ((m _ K^(ó ))^2 (m _ η^(ó ))^2)(p _ 2^2) (-(m _ K^(ó ))^2 - (m _ η^(ó ))^2 + p _ 2^2) (p _ 2^2 (-8 (m _ π^(ó ))^2 + 11 (m _ K^(ó ))^2 - 15 (m _ η^(ó ))^2 - 12 p _ 1^2 + 3 p _ 2^2) + 36 p _ 1 · p _ 2 ((m _ η^(ó ))^2 - (m _ K^(ó ))^2)))) (!, _ 0^( ))^2)](../HTMLFiles/index_245.gif)
The logs of Bijnens, Pallante and Prades:
![PP[m2_] := mK^2/(16 π^2 DecayConstant[PhiMeson]^2) m2/(mK^2 - m2) Log[mK^2/m2] ;](../HTMLFiles/index_246.gif){kind=small}
![blogs = - C/2^(1/2) (G8 (q^2 (8/3 mu8 + 26/3 muK + 8 muPi + 1/2 PmPi - 1/6 Pm8) - mK^2 (-1/3 mu8 + 4/9 muK + 7/3 muPi + 7/6 PmPi + 1/18 Pm8) - mPi^2 (1/3 mu8 + 8/9 muK + 3 muPi + 1/3 PmPi + 1/9 Pm8)) - GG8 (q^2 (2/3 muK + 1/2 PmPi - 1/6 Pm8) + mK^2 (1/3 mu8 - 4/9 muK - 7/3 muPi - 7/6 PmPi - 1/18 Pm8) + mPi^2 (-mu8 - 26/9 muK + 3 muPi - 1/3 PmPi - 1/9 Pm8))) /. CouplingConstant[_[4], ___] -> 0 /. {PmPi -> PP[mPi^2], Pm8 -> PP[m8^2]} /. {G8 -> C2, GG8 -> C5, q^2 -> Pair[Momentum[p1], Momentum[p1]], q^4 -> Pair[Momentum[p1], Momentum[p1]]^2, m8 -> ParticleMass[EtaMeson], mK -> ParticleMass[Kaon], mPi -> ParticleMass[Pion]} /. {muPi -> μ _ Pion, muK -> μ _ Kaon, mu8 -> μ _ EtaMeson} /. {μ _ p_ -> ParticleMass[p]^2 Log[ParticleMass[p]^2/ScaleMu^2]/(32 π^2 DecayConstant[PhiMeson]^2)} /. _RenormalizationState -> Sequence[] // Expand ;](../HTMLFiles/index_247.gif)
![bblogs = Expand[blogs] ;](../HTMLFiles/index_248.gif){kind=small}
![btreelogs = Expand[C/2^(1/2) 2 (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]}] ;](../HTMLFiles/index_253.gif)
![bbtreelogs = btreelogs /. {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 ;](../HTMLFiles/index_254.gif)
![BijnensTreeLogs = DiscardOrders[#, PerturbationOrder -> 4] & /@ bbtreelogs ;](../HTMLFiles/index_255.gif){kind=small}
![bblogstree = BijnensTreeLogs /. {CouplingConstant[_[4], ___] -> 0, _LeutwylerLambda -> 0} ;](../HTMLFiles/index_256.gif){kind=small}
![BijnensLogs = CheckF[DiscardOrders[#, PerturbationOrder -> 4] & /@ blogs, "KPiBijensLogs"] ;](../HTMLFiles/index_257.gif){kind=small}
![BijnensLogs1 = Collect[(4 2^(1/2) QuarkCondensate[]^2 (ParticleMass[PseudoScalar[2]]^2 - ParticleMass[PseudoScalar[6]]^2) (BijnensLogs) /. gellmannOkubo /. C -> 1 /. Log -> log // Simplify) /. log[a_] :> log[a /. toEtaRules] /. log[ParticleMass[p_]^2/ParticleMass[q_]^2] -> log[ParticleMass[p]^2/ScaleMu^2] - log[ParticleMass[q]^2/ScaleMu^2] // Simplify, {_DecayConstant, Pi, _log}] /. log[a_] * b_ :> log[a] * Simplify[b]](../HTMLFiles/index_263.gif)
![BijnensLogs2 = Collect[(4 2^(1/2) QuarkCondensate[]^2 (ParticleMass[PseudoScalar[2]]^2 - ParticleMass[PseudoScalar[6]]^2) (bblogstree) /. gellmannOkubo /. C -> 1 /. Log -> log // Simplify) /. log[a_] :> log[a /. toEtaRules] /. log[ParticleMass[p_]^2/ParticleMass[q_]^2] -> log[ParticleMass[p]^2/ScaleMu^2] - log[ParticleMass[q]^2/ScaleMu^2] // Simplify, {_DecayConstant, Pi, _log}] /. log[a_] * b_ :> log[a] * Simplify[b]](../HTMLFiles/index_266.gif)
The logs of the present calculation:
![limitlogs = Collect[DiscardOrders[((ParticleMass[PseudoScalar[2]]^2 - ParticleMass[PseudoScalar[6]]^2) ((Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)) ((fpionfac - 1 + fkaonfac - 1 + sqrtZPi0inv + sqrtZK0inv - 1) (Plus @@ final2all) + finalloops) /. CouplingConstant[ChPT3[4], ___] -> 0 /. _RenormalizationState -> Sequence[] /. _LeutwylerJBar -> 0 /. gellmannOkubo // Simplify) /. toEtaRules /. MomentaRules /. p3 -> -p1 /. p2 -> 0, PerturbationOrder -> 6], {_DecayConstant, Pi, _Log}] /. Log[a_] * b_ :> Log[a] * Simplify[b]](../HTMLFiles/index_269.gif)
![looplogs = (((ParticleMass[PseudoScalar[2]]^2 - ParticleMass[PseudoScalar[6]]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)) * ampinfinities /. {_LeutwylerJBar -> 0, _LeutwylerLambda -> 0, p3 -> -p1, _RenormalizationState -> Sequence[]} /. gellmannOkubo // Simplify) /. toEtaRules /. p2 -> 0 // Simplify](../HTMLFiles/index_272.gif)
![treelogs = DiscardOrders[#, PerturbationOrder -> 6] & /@ (((ParticleMass[PseudoScalar[2]]^2 - ParticleMass[PseudoScalar[6]]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)) * ((fpionfac - 1 + fkaonfac - 1 + sqrtZPi0inv + sqrtZK0inv - 1) final2all) /. p3 -> -p1 /. p2 -> 0 /. {_LeutwylerJBar -> 0, CouplingConstant[_[4], ___] -> 0} /. _RenormalizationState -> Sequence[]) // Simplify](../HTMLFiles/index_275.gif)
![Collect[Plus @@ looplogs[[{1, 4}]] /. Log -> log // Expand, {_DecayConstant, Pi, _log}]](../HTMLFiles/index_278.gif){kind=small}
![Collect[Plus @@ treelogs[[{1, 2}]] /. Log -> log // Expand, {_DecayConstant, Pi, _log}]](../HTMLFiles/index_281.gif){kind=small}
We observe that Bijnens, Pallante and Prades seem to have an overall sign error on their ![Underscript[∏, K^0 π^0] (q^2) | _ Logs](../HTMLFiles/index_282.gif){kind=small}
.
![Collect[BijnensLogs1 - BijnensLogs2 + limitlogs /. Log -> log //. log[a_ b_] -> log[a] + log[b] // Simplify, {_DecayConstant, Pi, _log}]](../HTMLFiles/index_283.gif){kind=small}
Converted by Mathematica (July 10, 2003)