•The non-analytic parts

looplogs = ((Simplify /@ (ampinfinities /. Log -> log)) /. toEtaRules /. LeutwylerLambda[] -> 0 /. _LeutwylerJBar -> 0 // Simplify) ;

logpartloops = ((Plus @@ looplogs) // ExpandScalarProduct // FullSimplify) /. log -> Log

-1/(8640 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (5 c _ 2^( ) (-64 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 96 (m _ π^(ó  ))^4 + 230 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 24 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 288 s (m _ π^(ó  ))^2 + 36 s log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 32 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 36 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^4 + 81 s^2 + 135 s (m _ K^(ó  ))^2 - 216 s log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 12 log((m _ η^(ó  ))^2/μ^2) (-5 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 9 s) (m _ η^(ó  ))^2 - 9 log((m _ K^(ó  ))^2/μ^2) (3 s^2 + (m _ K^(ó  ))^2 (32 (m _ π^(ó  ))^2 - 16 (m _ K^(ó  ))^2 - 19 s)) + 18 log((m _ π^(ó  ))^2/μ^2) (16 (m _ π^(ó  ))^4 - (5 (m _ K^(ó  ))^2 + 26 s) (m _ π^(ó  ))^2 + 6 s ((m _ K^(ó  ))^2 + s))) (m _ K^(ó  ))^2 + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (45 log((m _ π^(ó  ))^2/μ^2) (2 (m _ π^(ó  ))^2 - 3 (m _ K^(ó  ))^2 + s) (m _ π^(ó  ))^2 + 90 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (s - 3 (m _ K^(ó  ))^2) - log((m _ η^(ó  ))^2/μ^2) (36 (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + (-10 (m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2 + 15 s) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)))))

logpartmult = multlogs /. D -> Sequence[] /. toEtaRules // FullSimplify

-1/(3456 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (9 c _ 2^( ) (-58 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + ((41 log((m _ π^(ó  ))^2/μ^2) - 20 log((m _ K^(ó  ))^2/μ^2) - 3 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 6 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 17 s log((m _ π^(ó  ))^2/μ^2) + 3 s log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (17 log((m _ K^(ó  ))^2/μ^2) + 6 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 - 2 s (7 log((m _ K^(ó  ))^2/μ^2) + 6 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 - 6 s log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 - 384 π^2 (f _ ϕ^(ó  ))^2 (-2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + s)) (m _ K^(ó  ))^2 + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-4 (9 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 + (33 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 36 s log((m _ π^(ó  ))^2/μ^2) - 9 s log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - 68 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 36 s (2 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 9 log((m _ η^(ó  ))^2/μ^2) (3 (m _ K^(ó  ))^2 + s) (m _ η^(ó  ))^2)))

The final loop function expression:

jpart = ((Plus @@ ampinfinities /. {CouplingConstant[_[4], ___] -> 0, _LeutwylerLambda -> 0, _Log -> 0} // Simplify) /. toEtaRules) // ScalarProductExpand // FullSimplify

-1/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2 (s - (m _ K^(ó  ))^2)) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (32 (2 π^2 (9 Overscript[J, _] _ (m _ π^(ó  ))^2(s) - Overscript[J, _] _ (m _ η^(ó  ))^2(s)) - 1) (m _ π^(ó  ))^6 + 8 ((m _ K^(ó  ))^2 + 12 s + 4 π^2 (-36 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ π^(ó  ))^2) (m _ K^(ó  ))^2 - 9 Overscript[J, _] _ (m _ π^(ó  ))^2(s) ((m _ K^(ó  ))^2 + 5 s) + Overscript[J, _] _ (m _ η^(ó  ))^2(s) (5 (m _ K^(ó  ))^2 - 3 s))) (m _ π^(ó  ))^4 + 3 (32 π^2 (15 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ π^(ó  ))^2) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ π^(ó  ))^2)) (m _ K^(ó  ))^4 - 15 s (m _ K^(ó  ))^2 - 9 s^2 + 48 s π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(s) ((m _ K^(ó  ))^2 - s) + 192 s π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) ((m _ K^(ó  ))^2 + s)) (m _ π^(ó  ))^2 - 96 π^2 (3 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ π^(ó  ))^2) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ π^(ó  ))^2)) (m _ K^(ó  ))^6))

Simplify[(jpart /. JBarToB /. _Log -> 0 /. fixdenominators /. gellmannOkubo // FullSimplify) /. toEtaRules]

1/(36 (f _ ϕ^(ó  ))^2 (m _ π^(ó  ))^2 (s - (m _ K^(ó  ))^2)) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (9 s B((m _ K^(ó  ))^2, (m _ K^(ó  ))^2, s) ((m _ K^(ó  ))^2 - s) (m _ π^(ó  ))^2 + 18 B((m _ π^(ó  ))^2, (m _ π^(ó  ))^2, s) (2 s - (m _ π^(ó  ))^2) (-2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + s) (m _ π^(ó  ))^2 - 2 (B((m _ η^(ó  ))^2, (m _ η^(ó  ))^2, s) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 3 s) (m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (B((m _ K^(ó  ))^2, (m _ η^(ó  ))^2, (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 3 B((m _ π^(ó  ))^2, (m _ K^(ó  ))^2, (m _ π^(ó  ))^2) ((m _ K^(ó  ))^2 - 4 (m _ π^(ó  ))^2)))))

logpartjbar = Collect[Simplify[(jpart /. JBarToB /. _B -> 0 /. fixdenominators /. gellmannOkubo // Simplify) /. toEtaRules // ScalarProductExpand], _Log]

-(i c _ 2^( ) log((m _ π^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (36 (m _ π^(ó  ))^6 - 90 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 90 s (m _ π^(ó  ))^4 + 18 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 36 s^2 (m _ π^(ó  ))^2 + 36 s (m _ K^(ó  ))^2 (m _ π^(ó  ))^2))/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2 (s - (m _ K^(ó  ))^2)) - (i c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (72 (m _ π^(ó  ))^2 (m _ K^(ó  ))^4 + 9 s (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 - 9 s^2 (m _ π^(ó  ))^2))/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2 (s - (m _ K^(ó  ))^2)) - (i c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-4 (m _ π^(ó  ))^6 + 10 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 6 s (m _ π^(ó  ))^4 + 6 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 - 24 (m _ K^(ó  ))^6))/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2 (s - (m _ K^(ó  ))^2))

(((Cancel[(logpartloops + logpartmult) * (MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] /. Log -> log /. gellmannOkubo // ScalarProductExpand)/I // Expand // Collect[#, {_CouplingConstant, _Pair, Pi, _DecayConstant, _log}] &) /. toEtaRules /. log -> Log)/(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)

1/(s - (m _ K^(ó  ))^2) (1/(π^2 (f _ ϕ^(ó  ))^4) (c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (log((m _ π^(ó  ))^2/μ^2) ((s (m _ π^(ó  ))^4)/(32 (m _ K^(ó  ))^2) - 1/32 (m _ π^(ó  ))^4 + 1/32 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 1/32 s (m _ π^(ó  ))^2) + log((m _ η^(ó  ))^2/μ^2) (-(s (m _ π^(ó  ))^4)/(96 (m _ K^(ó  ))^2) + 1/96 (m _ π^(ó  ))^4 - 5/96 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 5/96 s (m _ π^(ó  ))^2 + 1/24 (m _ K^(ó  ))^4 - 1/24 s (m _ K^(ó  ))^2) + log((m _ K^(ó  ))^2/μ^2) (1/16 (m _ K^(ó  ))^4 - 1/16 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 - 1/16 s (m _ K^(ó  ))^2 + 1/16 s (m _ π^(ó  ))^2))) + c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((2 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 - s)/(f _ ϕ^(ó  ))^2 + 1/(π^2 (f _ ϕ^(ó  ))^4) (1/18 (m _ π^(ó  ))^4 - 1/72 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 1/6 s (m _ π^(ó  ))^2 + (3 s^2)/64 + 5/64 s (m _ K^(ó  ))^2 + log((m _ π^(ó  ))^2/μ^2) (1/64 (m _ π^(ó  ))^4 + 7/128 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 29/128 s (m _ π^(ó  ))^2 + s^2/16 + 1/16 s (m _ K^(ó  ))^2) + log((m _ η^(ó  ))^2/μ^2) (-19/576 (m _ π^(ó  ))^4 + (131 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/1152 + 5/384 s (m _ π^(ó  ))^2 + 1/32 (m _ K^(ó  ))^4 - 3/32 s (m _ K^(ó  ))^2) + log((m _ K^(ó  ))^2/μ^2) (11/64 (m _ K^(ó  ))^4 - 7/32 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 1/16 s (m _ K^(ó  ))^2 - s^2/64))))

(Cancel[(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) * logpartjbar]/I // Expand // Collect[#, {_CouplingConstant, _Pair, Pi, _DecayConstant, _Log}] &)/(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)

1/(π^2 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (log((m _ K^(ó  ))^2/μ^2) (-1/8 (m _ K^(ó  ))^4 - 1/64 s (m _ K^(ó  ))^2 + s^2/64) + log((m _ π^(ó  ))^2/μ^2) (-1/16 (m _ π^(ó  ))^4 + 5/32 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 5/32 s (m _ π^(ó  ))^2 - 1/32 (m _ K^(ó  ))^4 - s^2/16 - 1/16 s (m _ K^(ó  ))^2) + log((m _ η^(ó  ))^2/μ^2) ((m _ K^(ó  ))^6/(24 (m _ π^(ó  ))^2) - 1/96 (m _ K^(ó  ))^4 - 5/288 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 1/144 (m _ π^(ó  ))^4 + 1/96 s (m _ π^(ó  ))^2)))

The final logarithmic expression:

endlogs = Collect[Simplify[Cancel[(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) (logpartjbar + logpartloops + logpartmult)] /. Log -> log /. gellmannOkubo // ScalarProductExpand] /. toEtaRules, {_DecayConstant, _CouplingConstant, _Pair, Pi, _log}]/(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) /. log -> Log

1/(s - (m _ K^(ó  ))^2) ((i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2304 (m _ π^(ó  ))^4 (m _ K^(ó  ))^2 - 1152 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 + s)))/(1152 (f _ ϕ^(ó  ))^2 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) + 1/(f _ ϕ^(ó  ))^4 (1/π^2 (c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((i (64 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 - 16 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4 - 192 s (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 90 s (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 54 s^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2))/(1152 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) + (i log((m _ K^(ó  ))^2/μ^2) (54 (m _ π^(ó  ))^2 (m _ K^(ó  ))^6 - 252 (m _ π^(ó  ))^4 (m _ K^(ó  ))^4 + 54 s (m _ π^(ó  ))^2 (m _ K^(ó  ))^4))/(1152 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) + 1/(1152 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2) (i log((m _ η^(ó  ))^2/μ^2) (48 (m _ K^(ó  ))^8 + 24 (m _ π^(ó  ))^2 (m _ K^(ó  ))^6 + 111 (m _ π^(ó  ))^4 (m _ K^(ó  ))^4 - 108 s (m _ π^(ó  ))^2 (m _ K^(ó  ))^4 - 30 (m _ π^(ó  ))^6 (m _ K^(ó  ))^2 + 27 s (m _ π^(ó  ))^4 (m _ K^(ó  ))^2)) + (i log((m _ π^(ó  ))^2/μ^2) (-54 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 - 81 (m _ K^(ó  ))^2 (s - 3 (m _ K^(ó  ))^2) (m _ π^(ó  ))^4 - 36 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2))/(1152 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2))) + 1/π^2 (c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((i log((m _ π^(ó  ))^2/μ^2) (s - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (m _ π^(ó  ))^2)/(32 (m _ K^(ó  ))^2) + (i log((m _ η^(ó  ))^2/μ^2) (s - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (m _ η^(ó  ))^2)/(32 (m _ K^(ó  ))^2) + 1/16 i log((m _ K^(ó  ))^2/μ^2) (s - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))))

To compare with Bijnens, Pallante and Prades, we go on the mass-shell and also multiply with the renormalization factor of f _ K:

Collect[Simplify[DiscardOrders[fkaonfac * Cancel[(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) endlogs] /. CouplingConstant[_[4, ___], ___] -> 0 /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2 /. Log -> log /. gellmannOkubo, PerturbationOrder -> 6]] /. toEtaRules, {_log, _DecayConstant, _CouplingConstant, _Pair, Pi}] /. log -> Log

(2 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(f _ ϕ^(ó  ))^2 + (i c _ 2^( ) log((m _ π^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-27 (m _ π^(ó  ))^6 + 54 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 9 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2))/(288 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2) + (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (16 (m _ π^(ó  ))^6 - 52 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 36 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2))/(288 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2) + (i c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (54 (m _ π^(ó  ))^2 (m _ K^(ó  ))^4 - 90 (m _ π^(ó  ))^4 (m _ K^(ó  ))^2))/(288 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2) + (i c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-3 (m _ π^(ó  ))^6 + 12 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 3 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^6))/(288 π^2 (f _ ϕ^(ó  ))^4 (m _ π^(ó  ))^2)

BijnensLogs = Collect[(mPi^2 (2 muK - muPi - mu8) + (mK^2 - mPi^2) (mK^2/(2 mPi^2) (muPi - mu8) - 5/2 muPi - 1/2 mu8 - 3 muK) /. {muPi -> μ _ Pion, muK -> μ _ Kaon, mu8 -> μ _ EtaMeson} /. {mPi -> ParticleMass[Pion, RenormalizationState[0]], mK -> ParticleMass[Kaon, RenormalizationState[0]]} /. {μ _ p_ -> ParticleMass[p, RenormalizationState[0]]^2 Log[ParticleMass[p, RenormalizationState[0]]^2/ScaleMu^2]/(32 π^2 DecayConstant[PseudoScalar[1], RenormalizationState[0]]^2)} /. Log -> log /. gellmannOkubo // Expand), {_DecayConstant, Pi, _log}] /. toEtaRules

1/(π^2 (f _ ϕ^(ó  ))^2) (log((m _ K^(ó  ))^2/μ^2) (5/32 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 - 3/32 (m _ K^(ó  ))^4) + log((m _ π^(ó  ))^2/μ^2) (3/64 (m _ π^(ó  ))^4 - 3/32 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1/64 (m _ K^(ó  ))^4) + log((m _ η^(ó  ))^2/μ^2) (-(m _ K^(ó  ))^6/(48 (m _ π^(ó  ))^2) + 1/192 (m _ K^(ó  ))^4 - 1/48 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + 1/192 (m _ π^(ó  ))^4))

Converted by Mathematica  (July 10, 2003)