•Decomposition in functions of one variable

finjbar2s = finaljbars2KLMs /. (Mr | K | LeutwylerJBar)[MandelstamU | u, ___] -> 0 /. MandelstamU -> 1/2 (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT) - ν // Simplify ;

finjbar5s = finaljbars5KLMs /. (Mr | K | LeutwylerJBar)[MandelstamU | u, ___] -> 0 /. MandelstamU -> 1/2 (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT) - ν // Simplify ;

jbarApprox[s_, m12_, m22_] = Normal[(Series[LeutwylerJBar[s, m12, m22, LeutwylerJBarEvaluation -> "subthreshold"], {s, 0, 3}] /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y} // Simplify) /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y, 1/Sqrt[x_^2] -> 1/x, 1/Sqrt[x_^2 * y_^2] -> 1/(x * y)} // Simplify]

((m12^4 + 28 m22 m12^3 - 28 m22^3 m12 + 12 m22 (m12^2 + 3 m22 m12 + m22^2) log(m22/m12) m12 - m22^4) s^3)/(192 (m12 - m22)^7 π^2) + ((m12^3 + 9 m22 m12^2 - 9 m22^2 m12 + 6 m22 (m12 + m22) log(m22/m12) m12 - m22^3) s^2)/(96 (m12 - m22)^5 π^2) + ((m12^2 + 2 m22 log(m22/m12) m12 - m22^2) s)/(32 (m12 - m22)^3 π^2)

jbarApprox[s_, m12_] = Limit[jbarApprox[s, m12, m22], m22 -> m12]

(s (70 m12^2 + 7 s m12 + s^2))/(6720 m12^3 π^2)

N1c2tmp = Coefficient[finjbar2s, ν] // Simplify

-1/(144 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (5 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (m _ π^(ó  ))^2 - 14 (m _ K^(ó  ))^2 + 3 t) + 3 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + t) + 2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + 5 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))

N1c2res = (Limit[N1c2tmp /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a], MandelstamT -> 0] /. gellmannOkubo // Simplify) /. toEtaRules

(i c _ 2^( ) ((m _ π^(ó  ))^4 + (-3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 15 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 20) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (19 - 60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) (m _ K^(ó  ))^4))/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))

The c _ 2 part of the final N _ 1(t):

N1c2 = Collect[(N1c2tmp - N1c2res /. KLToJBar // Expand) /. (JBarToKL /. t -> MandelstamT), {_Log, _K, _LeutwylerJBar}] // FullSimplify

-1/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) ((32 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + 5 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) + 1) (m _ π^(ó  ))^4 - (64 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + 5 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) + 3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 15 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 20) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (32 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + 5 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) - 60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 19) (m _ K^(ó  ))^4 + 80 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (m _ π^(ó  ))^2 - 14 (m _ K^(ó  ))^2 + 3 t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 48 π^2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^4 + t (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4 - t (m _ K^(ó  ))^2)))

Limit[N1c2 /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. gellmannOkubo, MandelstamT -> 0] // Simplify

0

N1c5tmp = Coefficient[finjbar5s, ν] // Simplify

1/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (6 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 - 3 t) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^2 + 14 (m _ K^(ó  ))^2 - 3 t)))

N1c5res = (Limit[N1c5tmp /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a], MandelstamT -> 0] /. gellmannOkubo // Simplify) /. toEtaRules

-(i c _ 5^( ) ((m _ π^(ó  ))^4 + (3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 3 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 4) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 5) (m _ K^(ó  ))^4))/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

The c _ 5 part of the final N _ 1(t):

N1c5 = Collect[(N1c5tmp - N1c5res /. KLToJBar // Expand) /. (JBarToKL /. t -> MandelstamT), {_Log, _K, _LeutwylerJBar}] // FullSimplify

-1/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((32 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) - 1) (m _ π^(ó  ))^4 - (64 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) + 3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 3 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 4) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (32 π^2 (3 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) - 12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 5) (m _ K^(ó  ))^4 + 16 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (m _ π^(ó  ))^2 - 14 (m _ K^(ó  ))^2 + 3 t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 48 π^2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^4 + t (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4 - t (m _ K^(ó  ))^2)))

qqc2 = Collect[-ParticleMass[Pion]^2 (ParticleMass[Kaon]^2 - ParticleMass[Pion]^2)/(2 MandelstamT) N1c2, {_LeutwylerJBar, _K, _k, _Mr, _Log}] // Simplify

-1/(4608 t π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (m _ π^(ó  ))^2 (96 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ π^(ó  ))^4 + 160 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ π^(ó  ))^4 + (m _ π^(ó  ))^4 - 192 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 320 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 15 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 20 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 96 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^4 + 160 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^4 - 60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^4 + 19 (m _ K^(ó  ))^4 + 80 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (m _ π^(ó  ))^2 - 14 (m _ K^(ó  ))^2 + 3 t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 48 π^2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^4 + t (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4 - t (m _ K^(ó  ))^2)))

Limit[qqc2 /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. gellmannOkubo, MandelstamT -> 0] /. toEtaRules // Simplify

1/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) (i c _ 2^( ) (m _ π^(ó  ))^2 (5 (m _ π^(ó  ))^6 + (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 15 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 43) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 165 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 163) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 5 (84 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 25) (m _ K^(ó  ))^6))

qqc5 = Collect[-ParticleMass[Pion]^2 (ParticleMass[Kaon]^2 - ParticleMass[Pion]^2)/(2 MandelstamT) N1c5, {_LeutwylerJBar, _K, _k, _Mr, _Log}] // Simplify

1/(768 t π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (i c _ 5^( ) (m _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (96 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ π^(ó  ))^4 + 32 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ π^(ó  ))^4 - (m _ π^(ó  ))^4 - 192 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 64 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 96 π^2 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^4 + 32 π^2 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^4 - 12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^4 + 5 (m _ K^(ó  ))^4 + 16 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (m _ π^(ó  ))^2 - 14 (m _ K^(ó  ))^2 + 3 t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 48 π^2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-2 (m _ π^(ó  ))^4 + t (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4 - t (m _ K^(ó  ))^2)))

Limit[qqc5 /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. gellmannOkubo, MandelstamT -> 0] /. toEtaRules // Simplify

-(i c _ 5^( ) (m _ π^(ó  ))^2 (5 (m _ π^(ó  ))^6 + 3 (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 3 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 9) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 3 (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 33 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 33) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 7 (36 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 11) (m _ K^(ó  ))^6))/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2))

MNrestc2 = Simplify /@ Collect[finaljbars2KLMs - (ν N1c2 + q2t /. ν -> (MandelstamS - (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT - MandelstamS))/2) - (ν N1c2 + q2u /. ν -> (MandelstamS - (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT - MandelstamS))/2 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Expand, {_Log, _K, _LeutwylerJBar, _Mr}]

-(i c _ 2^( ) (m _ π^(ó  ))^6)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (13 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i s c _ 2^( ) (m _ π^(ó  ))^4)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i t c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i u c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i c _ 2^( ) Overscript[J, _] _ (m _ η^(ó  ))^2(s) (4 (m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2 + 9 s - 3 p _ 2^2) (m _ π^(ó  ))^2)/(36 (f _ ϕ^(ó  ))^4) - (i c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(128 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (5 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(576 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i s c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(288 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i t c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i u c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (i c _ 2^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 4 s + t + u - 2 p _ 2^2) (m _ π^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - q2t - q2u + (i s c _ 2^( ) Overscript[J, _] _ (m _ K^(ó  ))^2(s) (-3 (m _ K^(ó  ))^2 + 3 s - p _ 2^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) - (i c _ 2^( ) Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s - p _ 2^2))/(4 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + 2 t + u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + 2 t + u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + t + 2 u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + t + 2 u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s + 60 t - 84 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 u^2 - 420 s (m _ K^(ó  ))^2 - 420 t (m _ K^(ó  ))^2 - 735 u (m _ K^(ó  ))^2 + 90 s u + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 u))) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s - 84 t + 60 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 t^2 - 420 s (m _ K^(ó  ))^2 - 735 t (m _ K^(ó  ))^2 - 420 u (m _ K^(ó  ))^2 + 90 s t + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 t))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-9 (m _ π^(ó  ))^4 - (4 (s + 17 t + u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 t^2 - 4 s (m _ K^(ó  ))^2 - 27 t (m _ K^(ó  ))^2 - 4 u (m _ K^(ó  ))^2 + 2 s t + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + t)))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (-9 (m _ π^(ó  ))^4 - (4 (s + t + 17 u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 u^2 - 4 s (m _ K^(ó  ))^2 - 4 t (m _ K^(ó  ))^2 - 27 u (m _ K^(ó  ))^2 + 2 s u + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + u)))) + (19 i s c _ 2^( ) (m _ K^(ó  ))^4)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i t c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i u c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (5 i c _ 2^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 4 s + t + u - 2 p _ 2^2))/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i c _ 2^( ) (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (19 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))

MNrestc5 = Simplify /@ Collect[finaljbars5KLMs - (ν N1c5 + q5t /. ν -> (MandelstamS - (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT - MandelstamS))/2) - (ν N1c5 + q5u /. ν -> (MandelstamS - (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamT - MandelstamS))/2 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Expand, {_Log, _K, _LeutwylerJBar, _Mr}]

(i c _ 5^( ) (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i s c _ 5^( ) (m _ π^(ó  ))^4)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i t c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i u c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) Overscript[J, _] _ (m _ η^(ó  ))^2(s) (-9 (m _ K^(ó  ))^2 + 9 s + p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (m _ π^(ó  ))^2)/(18 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i c _ 5^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 4 s + t + u - 2 p _ 2^2) (m _ π^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i u c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (3 i c _ 5^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - q5t - q5u + (3 i s c _ 5^( ) Overscript[J, _] _ (m _ K^(ó  ))^2(s) (-(m _ K^(ó  ))^2 + s + p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s + t - 5 u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 u^2 + 4 s (m _ K^(ó  ))^2 + 4 t (m _ K^(ó  ))^2 - 13 u (m _ K^(ó  ))^2 - 2 s u - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 u))) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s - 5 t + u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 t^2 + 4 s (m _ K^(ó  ))^2 - 13 t (m _ K^(ó  ))^2 + 4 u (m _ K^(ó  ))^2 - 2 s t - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 t^2 - 84 s (m _ K^(ó  ))^2 - 75 t (m _ K^(ó  ))^2 - 84 u (m _ K^(ó  ))^2 + 18 s t + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 u^2 - 84 s (m _ K^(ó  ))^2 - 84 t (m _ K^(ó  ))^2 - 75 u (m _ K^(ó  ))^2 + 18 s u + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 u))) - (5 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i s c _ 5^( ) (m _ K^(ó  ))^4)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i t c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i u c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 4 s + t + u - 2 p _ 2^2))/(256 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i c _ 5^( ) (m _ K^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (-(m _ K^(ó  ))^2 + s + p _ 2^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2))

M0c2tmp = MNrestc2 /. {q2t -> 0, q2u -> 0} /. (K | LeutwylerJBar | Mr)[___, MandelstamT | MandelstamU, ___] -> 0 /. MandelstamT | MandelstamU -> 0 // FullSimplify

-1/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) (-(60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 19) (m _ K^(ó  ))^6 + ((60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 19) (2 s - p _ 2^2) - (832 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) + 3 (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 35 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 6)) (m _ π^(ó  ))^2) (m _ K^(ó  ))^4 + (m _ π^(ó  ))^2 (-39 (m _ π^(ó  ))^2 + 40 s - 20 p _ 2^2 + 3 (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 5 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) (-2 (m _ π^(ó  ))^2 + 2 s - p _ 2^2) + 64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (17 (m _ π^(ó  ))^2 + 9 s - 3 p _ 2^2)) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^4 (2 (m _ π^(ó  ))^2 - 2 s + p _ 2^2 - 64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (4 (m _ π^(ó  ))^2 + 9 s - 3 p _ 2^2)) + 576 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s - p _ 2^2) + 288 s π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(s) (-3 (m _ K^(ó  ))^2 + 3 s - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)))

M0c5tmp = MNrestc5 /. {q5t -> 0, q5u -> 0} /. (K | LeutwylerJBar | Mr)[___, MandelstamT | MandelstamU, ___] -> 0 /. MandelstamT | MandelstamU -> 0 // FullSimplify

-1/(1152 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) (-3 (12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 5) (m _ K^(ó  ))^6 + 3 ((12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 5) (2 s - p _ 2^2) - 3 (-64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) + log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 7 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 2) (m _ π^(ó  ))^2) (m _ K^(ó  ))^4 - (m _ π^(ó  ))^2 (9 (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) (2 (m _ π^(ó  ))^2 - 2 s + p _ 2^2) + 3 (9 (m _ π^(ó  ))^2 - 8 s + 4 p _ 2^2) + 64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (p _ 2^2 + 9 ((m _ π^(ó  ))^2 + s))) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^4 (-6 (m _ π^(ó  ))^2 + 6 s - 3 p _ 2^2 + 64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (9 s + p _ 2^2)) + 864 s π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(s) (-(m _ K^(ó  ))^2 + s + p _ 2^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 576 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (-(m _ K^(ó  ))^2 + s + p _ 2^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))

N0c2tmp = Simplify /@ Collect[MNrestc2 - M0c2tmp // Expand, {_Log, _K, _LeutwylerJBar, _Mr}]

(19 i c _ 2^( ) (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i c _ 2^( ) (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (19 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i t c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i u c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - (i (t + u) c _ 2^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i t c _ 2^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i u c _ 2^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (5 i (t + u) c _ 2^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (m _ K^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - q2t - q2u + (i c _ 2^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + 2 t + u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + 2 t + u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + t + 2 u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + t + 2 u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s + 60 t - 84 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 u^2 - 420 s (m _ K^(ó  ))^2 - 420 t (m _ K^(ó  ))^2 - 735 u (m _ K^(ó  ))^2 + 90 s u + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 u))) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s - 84 t + 60 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 t^2 - 420 s (m _ K^(ó  ))^2 - 735 t (m _ K^(ó  ))^2 - 420 u (m _ K^(ó  ))^2 + 90 s t + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 t))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-9 (m _ π^(ó  ))^4 - (4 (s + 17 t + u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 t^2 - 4 s (m _ K^(ó  ))^2 - 27 t (m _ K^(ó  ))^2 - 4 u (m _ K^(ó  ))^2 + 2 s t + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + t)))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (-9 (m _ π^(ó  ))^4 - (4 (s + t + 17 u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 u^2 - 4 s (m _ K^(ó  ))^2 - 4 t (m _ K^(ó  ))^2 - 27 u (m _ K^(ó  ))^2 + 2 s u + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + u)))) + (i t c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i u c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))

N0c2res = Limit[N0c2tmp /. {q2t -> qqc2, q2u -> (qqc2 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU})} /. gellmannOkubo /. KLToJBar /. MrToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. MandelstamU -> MandelstamT, MandelstamT -> 0] /. toEtaRules // Simplify

1/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) (i c _ 2^( ) ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 67) (m _ π^(ó  ))^8 - ((288 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 1440 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - 87 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 105 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 32) (m _ K^(ó  ))^2 + 2 s + (96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 43) p _ 2^2) (m _ π^(ó  ))^6 + 3 (m _ K^(ó  ))^2 ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - 21 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 47) p _ 2^2 + 2 (2 (24 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 120 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 11 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 27 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 75) (m _ K^(ó  ))^2 + s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 5 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 7))) (m _ π^(ó  ))^4 - (m _ K^(ó  ))^4 ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 57 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 2835 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 608) (m _ K^(ó  ))^2 + 6 s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 25 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 13) + 3 (96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 35 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 245 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 321) p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^6 ((193 - 564 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) (m _ K^(ó  ))^2 + 2 s (19 - 60 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) + (96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 480 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 2892 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 779) p _ 2^2)))

N0c2res1 = 1/2 * Limit[(N0c2tmp - N0c2res)/MandelstamT /. {q2t -> qqc2, q2u -> (qqc2 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU})} /. gellmannOkubo /. KLToJBar /. MrToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. MandelstamU -> MandelstamT, MandelstamT -> 0] /. toEtaRules // Simplify

1/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) (i c _ 2^( ) (33 (m _ π^(ó  ))^10 + (2 (22 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 42 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 95) (m _ K^(ó  ))^2 - 20 s + 54 p _ 2^2) (m _ π^(ó  ))^8 + 2 (m _ K^(ó  ))^2 ((17 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 39 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 112) p _ 2^2 - 2 ((-91 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 771 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 1354) (m _ K^(ó  ))^2 + s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 15 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 48))) (m _ π^(ó  ))^6 + 4 (m _ K^(ó  ))^4 ((-59 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 6237 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 5259) (m _ K^(ó  ))^2 + 2 s (90 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 103) + (-54 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 324 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 913) p _ 2^2) (m _ π^(ó  ))^4 + (m _ K^(ó  ))^6 ((38 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 52806 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 12319) (m _ K^(ó  ))^2 + 4 s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 585 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 288) - 2 (17 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 9999 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 9712) p _ 2^2) (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^8 ((1729 - 6036 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) (m _ K^(ó  ))^2 + 10 s (84 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 25) + (27132 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 7801) p _ 2^2)))

The c _ 2 part of the final 1/3 (N _ 0(t) + N _ 0(u)) +2/3 (R _ 0(t) + R _ 0(u)):

N0c2 = Simplify /@ Collect[(newcts1TU + finalLogsTU + finallooppolysTU /. C5 -> 0) + N0c2tmp - N0c2res - N0c2res1 * MandelstamT - N0c2res1 * MandelstamU // Expand, {_Log, _k, _K, _LeutwylerJBar, _Mr}]

-(11 i t c _ 2^( ) (m _ π^(ó  ))^10)/(1024 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (11 i u c _ 2^( ) (m _ π^(ó  ))^10)/(1024 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (67 i c _ 2^( ) (m _ π^(ó  ))^8)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (95 i t c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (95 i u c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (9 i t c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^8)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (9 i u c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^8)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (5 i s t c _ 2^( ) (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (5 i s u c _ 2^( ) (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(72 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (43 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (i s c _ 2^( ) (m _ π^(ó  ))^6)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (677 i t c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (677 i u c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (7 i t c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (7 i u c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (i s t c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(16 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (i s u c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(16 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (i t c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i u c _ 2^( ) (m _ π^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (25 i c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (47 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (7 i s c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (1753 i t c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (1753 i u c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (913 i t c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (913 i u c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (103 i s t c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (103 i s u c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + 1/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) (i c _ 2^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (-58 (m _ π^(ó  ))^8 + (28 (m _ K^(ó  ))^2 - 4 s - 23 t - 23 u + 42 p _ 2^2) (m _ π^(ó  ))^6 + (156 (m _ K^(ó  ))^4 + (12 s - 179 (t + u)) (m _ K^(ó  ))^2 + 2 s (t + u) - p _ 2^2 (14 (m _ K^(ó  ))^2 + 17 (t + u))) (m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 ((-164 (m _ K^(ó  ))^2 - 12 s + 115 (t + u)) (m _ K^(ó  ))^2 + 2 p _ 2^2 (54 (t + u) - 49 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (p _ 2^2 (70 (m _ K^(ó  ))^2 + 17 (t + u)) - 2 (-19 (m _ K^(ó  ))^4 + (9 (t + u) - 2 s) (m _ K^(ó  ))^2 + s (t + u)))) (m _ π^(ó  ))^2) - (5 i t c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (5 i u c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (11 i (t^2 + u^2) c _ 2^( ) log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (19 i c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(72 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (107 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (13 i s c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (12319 i t c _ 2^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (12319 i u c _ 2^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (607 i t c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (607 i u c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (3 i s t c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(8 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (3 i s u c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(8 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - q2t - q2u + (i c _ 2^( ) k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(24 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(24 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + 2 t + u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + 2 t + u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (32 (m _ π^(ó  ))^4 + (47 (m _ K^(ó  ))^2 + 2 s + t + 2 u - 23 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 2 s + t + 2 u + 37 p _ 2^2)))/(48 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + 2 t + u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-80 (m _ π^(ó  ))^4 + (67 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) + 29 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (47 (m _ K^(ó  ))^2 + 5 (2 s + t + 2 u) - 31 p _ 2^2)))/(144 (f _ ϕ^(ó  ))^4) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s + 60 t - 84 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 u^2 - 420 s (m _ K^(ó  ))^2 - 420 t (m _ K^(ó  ))^2 - 735 u (m _ K^(ó  ))^2 + 90 s u + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 u))) + 1/(864 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-157 (m _ π^(ó  ))^4 + (833 (m _ K^(ó  ))^2 + 60 s - 84 t + 60 u) (m _ π^(ó  ))^2 + 512 (m _ K^(ó  ))^4 + 171 t^2 - 420 s (m _ K^(ó  ))^2 - 735 t (m _ K^(ó  ))^2 - 420 u (m _ K^(ó  ))^2 + 90 s t + 90 t u - 3 p _ 2^2 (29 (m _ π^(ó  ))^2 - 161 (m _ K^(ó  ))^2 + 39 t))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-9 (m _ π^(ó  ))^4 - (4 (s + 17 t + u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 t^2 - 4 s (m _ K^(ó  ))^2 - 27 t (m _ K^(ó  ))^2 - 4 u (m _ K^(ó  ))^2 + 2 s t + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + t)))) + 1/(96 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (-9 (m _ π^(ó  ))^4 - (4 (s + t + 17 u) - 109 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 40 (m _ K^(ó  ))^4 + 47 u^2 - 4 s (m _ K^(ó  ))^2 - 4 t (m _ K^(ó  ))^2 - 27 u (m _ K^(ó  ))^2 + 2 s u + 2 t u + p _ 2^2 (37 (m _ π^(ó  ))^2 - 17 ((m _ K^(ó  ))^2 + u)))) - 1/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) (i c _ 2^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (70 (m _ π^(ó  ))^8 + (356 (m _ K^(ó  ))^2 - 20 s + 37 t + 37 u + 2 p _ 2^2) (m _ π^(ó  ))^6 - 3 (276 (m _ K^(ó  ))^4 + (453 (t + u) - 20 s) (m _ K^(ó  ))^2 + 10 s (t + u) + p _ 2^2 (162 (m _ K^(ó  ))^2 - 13 (t + u))) (m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 (308 (m _ K^(ó  ))^4 + (6963 (t + u) - 60 s) (m _ K^(ó  ))^2 + 240 s (t + u) + p _ 2^2 (966 (m _ K^(ó  ))^2 + 804 (t + u))) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (94 (m _ K^(ó  ))^4 + 2 (10 s + 757 (t + u)) (m _ K^(ó  ))^2 - 210 s (t + u) - p _ 2^2 (482 (m _ K^(ó  ))^2 + 6783 (t + u))))) + (19 i c _ 2^( ) (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i t c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i u c _ 2^( ) (m _ K^(ó  ))^4)/(4608 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i (t^2 + u^2) c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (i (t^2 + u^2) c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2))/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (19 i c _ 2^( ) (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (19 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - (193 i c _ 2^( ) (m _ K^(ó  ))^8)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (779 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (19 i s c _ 2^( ) (m _ K^(ó  ))^6)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (i t^2 c _ 2^( ))/(384 π^2 (f _ ϕ^(ó  ))^4) - (i u^2 c _ 2^( ))/(384 π^2 (f _ ϕ^(ó  ))^4) - (i t^2 c _ 2^( ) N _ 19^( ))/(2 (f _ ϕ^(ó  ))^4) - (i u^2 c _ 2^( ) N _ 19^( ))/(2 (f _ ϕ^(ó  ))^4) + (i t^2 c _ 2^( ) N _ 20^( ))/(2 (f _ ϕ^(ó  ))^4) + (i u^2 c _ 2^( ) N _ 20^( ))/(2 (f _ ϕ^(ó  ))^4) - (1729 i t c _ 2^( ) (m _ K^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (1729 i u c _ 2^( ) (m _ K^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (7801 i t c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (7801 i u c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (125 i s t c _ 2^( ) (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (125 i s u c _ 2^( ) (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4)

N0c5tmp = Simplify /@ Collect[MNrestc5 - M0c5tmp // Expand, {_Log, _K, _LeutwylerJBar, _Mr}]

(5 i c _ 5^( ) (m _ K^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i c _ 5^( ) (m _ K^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i t c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i u c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i (t + u) c _ 5^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (m _ K^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i (t + u) c _ 5^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i t c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i u c _ 5^( ) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (3 i c _ 5^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - q5t - q5u + (i c _ 5^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s + t - 5 u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 u^2 + 4 s (m _ K^(ó  ))^2 + 4 t (m _ K^(ó  ))^2 - 13 u (m _ K^(ó  ))^2 - 2 s u - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 u))) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s - 5 t + u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 t^2 + 4 s (m _ K^(ó  ))^2 - 13 t (m _ K^(ó  ))^2 + 4 u (m _ K^(ó  ))^2 - 2 s t - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 t^2 - 84 s (m _ K^(ó  ))^2 - 75 t (m _ K^(ó  ))^2 - 84 u (m _ K^(ó  ))^2 + 18 s t + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 u^2 - 84 s (m _ K^(ó  ))^2 - 84 t (m _ K^(ó  ))^2 - 75 u (m _ K^(ó  ))^2 + 18 s u + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 u))) - (i t c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i u c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

N0c5res = Limit[N0c5tmp /. q5t -> qqc5 /. q5u -> (qqc5 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) /. gellmannOkubo /. KLToJBar /. MrToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. MandelstamU -> MandelstamT, MandelstamT -> 0] /. toEtaRules // Simplify

-(i c _ 5^( ) ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 96 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - 11) (m _ π^(ó  ))^8 + (-(288 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 288 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 33 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 21 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 8) (m _ K^(ó  ))^2 + 2 s + (-96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) - 96 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 3) p _ 2^2) (m _ π^(ó  ))^6 + 3 (m _ K^(ó  ))^2 ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 96 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 5 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 9) p _ 2^2 + 2 (2 (24 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 24 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 9 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 17) (m _ K^(ó  ))^2 + s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 1))) (m _ π^(ó  ))^4 - (m _ K^(ó  ))^4 ((96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 96 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - 15 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 783 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 248) (m _ K^(ó  ))^2 + 6 s (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 5 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 3) + 3 (96 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 96 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) - 5 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 73 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 71) p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^6 (3 (32 π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 32 π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 212 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 63) p _ 2^2 - (12 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 5) (2 s - 5 (m _ K^(ó  ))^2))))/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))

N0c5res1 = 1/2 * (Limit[(N0c5tmp - N0c5res)/MandelstamT /. q5t -> qqc5 /. q5u -> (qqc5 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) /. gellmannOkubo /. KLToJBar /. MrToJBar /. LeutwylerJBar[a__, ___Rule] -> jbarApprox[a] /. MandelstamU -> MandelstamT, MandelstamT -> 0] /. toEtaRules // Simplify) // Simplify

(i c _ 5^( ) (-21 (m _ π^(ó  ))^10 + 2 ((6 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 18 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 139) (m _ K^(ó  ))^2 + 10 s + 5 p _ 2^2) (m _ π^(ó  ))^8 + 2 (m _ K^(ó  ))^2 ((21 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 9 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 64) p _ 2^2 + 2 ((75 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 333 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 826) (m _ K^(ó  ))^2 + s (3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 9 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 32))) (m _ π^(ó  ))^6 - 12 (m _ K^(ó  ))^4 ((17 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 1341 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 1221) (m _ K^(ó  ))^2 + 6 s (6 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 7) + (10 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 132 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 259) p _ 2^2) (m _ π^(ó  ))^4 + (m _ K^(ó  ))^6 ((30 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) + 40842 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 11675) (m _ K^(ó  ))^2 - 4 s (3 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 351 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 176) - 2 (21 log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) - 7713 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) + 6560) p _ 2^2) (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^8 (-5 (36 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) - 7) (m _ K^(ó  ))^2 + 14 s (11 - 36 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) + (5065 - 17604 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2)) p _ 2^2)))/(1536 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3)

The c _ 5 part of the final 1/3 (N _ 0(t) + N _ 0(u)) +2/3 (R _ 0(t) + R _ 0(u)):

N0c5 = Simplify /@ Collect[(newcts1TU + finalLogsTU + finallooppolysTU /. C2 -> 0) + N0c5tmp - N0c5res - N0c5res1 * MandelstamT - N0c5res1 * MandelstamU // Expand, {_Log, _k, _K, _LeutwylerJBar, _Mr}]

(7 i t c _ 5^( ) (m _ π^(ó  ))^10)/(512 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (7 i u c _ 5^( ) (m _ π^(ó  ))^10)/(512 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (11 i c _ 5^( ) (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (139 i t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (139 i u c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (5 i t c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (5 i u c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (5 i s t c _ 5^( ) (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (5 i s u c _ 5^( ) (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) - (i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) + (413 i t c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (413 i u c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s u c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^6)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (i s c _ 5^( ) (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (i u c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (i t c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i u c _ 5^( ) (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (9 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (17 i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(32 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) - (1221 i t c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (1221 i u c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (21 i s t c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (21 i s u c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (259 i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (259 i u c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) - (i c _ 5^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (22 (m _ π^(ó  ))^8 + (-36 (m _ K^(ó  ))^2 - 4 s + t + u - 6 p _ 2^2) (m _ π^(ó  ))^6 + (-4 (m _ K^(ó  ))^4 + (12 s + 53 (t + u)) (m _ K^(ó  ))^2 + 2 s (t + u) + p _ 2^2 (2 (m _ K^(ó  ))^2 + 7 (t + u))) (m _ π^(ó  ))^4 - (m _ K^(ó  ))^2 ((-28 (m _ K^(ó  ))^2 + 12 s + 37 (t + u)) (m _ K^(ó  ))^2 + 2 p _ 2^2 (10 (t + u) - 7 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 (p _ 2^2 (10 (m _ K^(ó  ))^2 + 7 (t + u)) + 2 (5 (m _ K^(ó  ))^4 - (2 s + 3 (t + u)) (m _ K^(ó  ))^2 + s (t + u)))) (m _ π^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i (t^2 + u^2) c _ 5^( ) log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i t^2 c _ 5^( ) (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i u^2 c _ 5^( ) (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i u c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (31 i c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) - (3 i s c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) + (11675 i t c _ 5^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11675 i u c _ 5^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11 i s t c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11 i s u c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (71 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (205 i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (205 i u c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (i c _ 5^( ) k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - q5t - q5u + (i c _ 5^( ) K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-8 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 3 p _ 2^2)))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + t + 2 u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - t - 2 u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-16 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + 2 s + 2 t + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (5 (m _ K^(ó  ))^2 - 2 s - 2 t - u + 11 p _ 2^2)))/(24 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s + t - 5 u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 u^2 + 4 s (m _ K^(ó  ))^2 + 4 t (m _ K^(ó  ))^2 - 13 u (m _ K^(ó  ))^2 - 2 s u - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 u))) + 1/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-7 (m _ π^(ó  ))^4 + (19 (m _ K^(ó  ))^2 + 4 (s - 5 t + u)) (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4 + 17 t^2 + 4 s (m _ K^(ó  ))^2 - 13 t (m _ K^(ó  ))^2 + 4 u (m _ K^(ó  ))^2 - 2 s t - 2 t u + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 - 7 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 t^2 - 84 s (m _ K^(ó  ))^2 - 75 t (m _ K^(ó  ))^2 - 84 u (m _ K^(ó  ))^2 + 18 s t + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 t))) + 1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-17 (m _ π^(ó  ))^4 + (109 (m _ K^(ó  ))^2 + 12 (s + t + u)) (m _ π^(ó  ))^2 + 88 (m _ K^(ó  ))^4 - 9 u^2 - 84 s (m _ K^(ó  ))^2 - 84 t (m _ K^(ó  ))^2 - 75 u (m _ K^(ó  ))^2 + 18 s u + 18 t u - 3 p _ 2^2 (9 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^2 + 3 u))) - (i c _ 5^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (-14 (m _ π^(ó  ))^8 + (-100 (m _ K^(ó  ))^2 + 4 s + 7 t + 7 u - 10 p _ 2^2) (m _ π^(ó  ))^6 + 3 (84 (m _ K^(ó  ))^4 + (81 (t + u) - 4 s) (m _ K^(ó  ))^2 + 2 s (t + u) - p _ 2^2 (-42 (m _ K^(ó  ))^2 + t + u)) (m _ π^(ó  ))^4 - (m _ K^(ó  ))^2 (148 (m _ K^(ó  ))^4 - 3 (4 s - 565 (t + u)) (m _ K^(ó  ))^2 + 48 s (t + u) + 6 p _ 2^2 (37 (m _ K^(ó  ))^2 + 46 (t + u))) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (p _ 2^2 (106 (m _ K^(ó  ))^2 + 1467 (t + u)) + 2 (5 (m _ K^(ó  ))^4 + (7 (t + u) - 2 s) (m _ K^(ó  ))^2 + 21 s (t + u)))))/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (5 i c _ 5^( ) (m _ K^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i t c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i u c _ 5^( ) (m _ K^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i t^2 c _ 5^( ) (m _ K^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i u^2 c _ 5^( ) (m _ K^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i (t^2 + u^2) c _ 5^( ) log((m _ η^(ó  ))^2/μ^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2))/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i c _ 5^( ) (m _ K^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i (t^2 + u^2) c _ 5^( ) log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2)/(32 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (63 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (5 i s c _ 5^( ) (m _ K^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (25 i c _ 5^( ) (m _ K^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - (5065 i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (5065 i u c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (35 i t c _ 5^( ) (m _ K^(ó  ))^10)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (35 i u c _ 5^( ) (m _ K^(ó  ))^10)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (77 i s t c _ 5^( ) (m _ K^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (77 i s u c _ 5^( ) (m _ K^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3)

M0fromN02 = Simplify /@ Collect[(N0c2res + N0c2res1 * (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)) // Expand, {_Log, _k, _K, _LeutwylerJBar, _Mr}]

(11 i c _ 2^( ) (m _ π^(ó  ))^12)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (347 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^10)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (47 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^10)/(1024 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (73 i s c _ 2^( ) (m _ π^(ó  ))^10)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (67 i c _ 2^( ) (m _ π^(ó  ))^8)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (5321 i c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (9 i c _ 2^( ) p _ 2^4 (m _ π^(ó  ))^8)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (73 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (277 i s c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (37 i s c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (5 i s^2 c _ 2^( ) (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(72 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (43 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (i s c _ 2^( ) (m _ π^(ó  ))^6)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (2291 i c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (11 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(16 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (859 i s c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (7 i c _ 2^( ) p _ 2^4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (13 i s c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (i s^2 c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(16 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (25 i c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (47 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (7 i s c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (1801 i c _ 2^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^4)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (295 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (5629 i s c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (913 i c _ 2^( ) p _ 2^4 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (707 i s c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (103 i s^2 c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + 1/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) (i c _ 2^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (102 (m _ π^(ó  ))^8 + (358 (m _ K^(ó  ))^2 - 22 s + 14 p _ 2^2) (m _ π^(ó  ))^6 + (17 p _ 2^4 - (3 (m _ K^(ó  ))^2 + 19 s) p _ 2^2 + 2 (-105 (m _ K^(ó  ))^4 - 98 s (m _ K^(ó  ))^2 + s^2)) (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (42 (m _ K^(ó  ))^4 + 67 s (m _ K^(ó  ))^2 - 54 p _ 2^4 + 27 p _ 2^2 (2 s - 3 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 (19 (m _ K^(ó  ))^4 + 21 s (m _ K^(ó  ))^2 + 2 s^2 + 17 p _ 2^4 + p _ 2^2 (68 (m _ K^(ó  ))^2 - 19 s))) (m _ π^(ó  ))^2) + (19 i c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(72 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (107 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (13 i s c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (19235 i c _ 2^( ) (m _ K^(ó  ))^10 (m _ π^(ó  ))^2)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (539 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (12167 i s c _ 2^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(3072 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (607 i c _ 2^( ) p _ 2^4 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (571 i s c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (3 i s^2 c _ 2^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(8 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (i c _ 2^( ) k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(24 (f _ ϕ^(ó  ))^4) - (5 i c _ 2^( ) k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(24 (f _ ϕ^(ó  ))^4) + 1/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) (i c _ 2^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (154 (m _ π^(ó  ))^8 - 2 (1175 (m _ K^(ó  ))^2 + 61 s - 61 p _ 2^2) (m _ π^(ó  ))^6 + 3 (3918 (m _ K^(ó  ))^4 + 628 s (m _ K^(ó  ))^2 + 10 s^2 + 13 p _ 2^4 - p _ 2^2 (71 (m _ K^(ó  ))^2 + 23 s)) (m _ π^(ó  ))^4 - 2 (m _ K^(ó  ))^2 (-5152 (m _ K^(ó  ))^4 + 3609 s (m _ K^(ó  ))^2 + 120 s^2 - 402 p _ 2^4 + 3 p _ 2^2 (803 (m _ K^(ó  ))^2 + 94 s)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (1603 (m _ K^(ó  ))^4 - 1699 s (m _ K^(ó  ))^2 + 210 s^2 - 6783 p _ 2^4 + p _ 2^2 (6573 s - 5756 (m _ K^(ó  ))^2)))) + (193 i c _ 2^( ) (m _ K^(ó  ))^8)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) - (779 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^6)/(2304 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (19 i s c _ 2^( ) (m _ K^(ó  ))^6)/(1152 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2) + (1729 i c _ 2^( ) (m _ K^(ó  ))^12)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (253 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^10)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (1979 i s c _ 2^( ) (m _ K^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) - (7801 i c _ 2^( ) p _ 2^4 (m _ K^(ó  ))^8)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (2517 i s c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^8)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4) + (125 i s^2 c _ 2^( ) (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^4)

M0fromN05 = Simplify /@ Collect[(N0c5res + N0c5res1 * (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)) // Expand, {_Log, _k, _K, _LeutwylerJBar, _Mr}]

(7 i c _ 5^( ) (m _ π^(ó  ))^12)/(256 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (535 i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (61 i s c _ 5^( ) (m _ π^(ó  ))^10)/(1536 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (1055 i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^8)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (257 i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11 i c _ 5^( ) (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (5 i s^2 c _ 5^( ) (m _ π^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (5 i c _ 5^( ) p _ 2^4 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^8)/(48 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (5 i s c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^6)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i s c _ 5^( ) (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) + (1625 i c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^6)/(96 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (523 i s c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i s^2 c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) p _ 2^4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^6)/(12 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (29 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^6)/(16 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (17 i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(32 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) - (i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) - (4349 i c _ 5^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^4)/(768 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (3889 i s c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) - (21 i s^2 c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (9 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (265 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^4)/(48 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (217 i s c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (259 i c _ 5^( ) p _ 2^4 (m _ K^(ó  ))^4 (m _ π^(ó  ))^4)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (i c _ 5^( ) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (-26 (m _ π^(ó  ))^8 + 2 (-33 (m _ K^(ó  ))^2 + s - 5 p _ 2^2) (m _ π^(ó  ))^6 + (-7 p _ 2^4 + (5 s - 19 (m _ K^(ó  ))^2) p _ 2^2 + 2 (11 (m _ K^(ó  ))^4 + 18 s (m _ K^(ó  ))^2 + s^2)) (m _ π^(ó  ))^4 - 2 (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^4 + 9 s (m _ K^(ó  ))^2 - 10 p _ 2^4 + p _ 2^2 (10 s - 27 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (5 (m _ K^(ó  ))^4 + 3 s (m _ K^(ó  ))^2 - 2 s^2 + 7 p _ 2^4 + p _ 2^2 (12 (m _ K^(ó  ))^2 - 5 s))) (m _ π^(ó  ))^2)/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (71 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - (31 i c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) + (3 i s c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2)) - (3845 i c _ 5^( ) (m _ K^(ó  ))^10 (m _ π^(ó  ))^2)/(512 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11587 i s c _ 5^( ) (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (11 i s^2 c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3 ((m _ K^(ó  ))^2 - p _ 2^2)) + (4535 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^8 (m _ π^(ó  ))^2)/(1536 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (205 i c _ 5^( ) p _ 2^4 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(24 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^3) + (97 i s c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6 (m _ π^(ó  ))^2)/(12 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (i c _ 5^( ) k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i c _ 5^( ) k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i c _ 5^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ K^(ó  ))^2 (4 (m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-2 (m _ π^(ó  ))^8 + 2 (199 (m _ K^(ó  ))^2 + 5 s - 5 p _ 2^2) (m _ π^(ó  ))^6 - 3 (p _ 2^4 + (61 (m _ K^(ó  ))^2 - 3 s) p _ 2^2 + 2 (483 (m _ K^(ó  ))^4 + 58 s (m _ K^(ó  ))^2 + s^2)) (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (-908 (m _ K^(ó  ))^4 + 873 s (m _ K^(ó  ))^2 + 24 s^2 - 138 p _ 2^4 + 3 p _ 2^2 (123 (m _ K^(ó  ))^2 + 38 s)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 (25 (m _ K^(ó  ))^4 + 23 s (m _ K^(ó  ))^2 - 42 s^2 + 1467 p _ 2^4 + p _ 2^2 (1588 (m _ K^(ó  ))^2 - 1425 s))))/(256 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (25 i c _ 5^( ) (m _ K^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (63 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6)/(128 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (5 i s c _ 5^( ) (m _ K^(ó  ))^6)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (35 i c _ 5^( ) (m _ K^(ó  ))^12)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (425 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^10)/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (119 i s c _ 5^( ) (m _ K^(ó  ))^10)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (5065 i c _ 5^( ) p _ 2^4 (m _ K^(ó  ))^8)/(768 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) - (1637 i s c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^8)/(256 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3) + (77 i s^2 c _ 5^( ) (m _ K^(ó  ))^8)/(384 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)^3)

The c _ 2 part of the final M _ 0:

finalM0c2 = finallows1 + finalLogsS + M0c2tmp + M0fromN02 + finallooppolysS + oldcts + newcts1S + strongcts /. _Pair -> 0 /. k -> kk /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 ;

The c _ 5 part of the final M _ 0:

finalM0c5 = finallows1 + finalLogsS + M0c5tmp + M0fromN05 + finallooppolysS + oldcts + newcts1S + strongcts /. _Pair -> 0 /. k -> kk /. CouplingConstant[ChPTW3[2], 1, ___] -> 0 ;

A few checks:

jbarApprox1[s_, m12_, m22_] = Normal[(Series[LeutwylerJBar[s, m12, m22, LeutwylerJBarEvaluation -> "subthreshold"], {s, mpi2, 1}] /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y, mpi2 -> ParticleMass[Pion]^2} // Simplify) /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y, 1/Sqrt[x_^2] -> 1/x, 1/Sqrt[x_^2 * y_^2] -> 1/(x * y)} // Simplify]

1/(32 (m12 - m22) π^2 (m _ π^(ó  ))^2) ((m12 - m22) ((m12 - m22) log(m22/m12) - log((-(m _ π^(ó  ))^2 + m12 + m22 - ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))/(-(m _ π^(ó  ))^2 + m12 + m22 + ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))) ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2)) - (-2 m12 + 2 m22 + (m12 + m22) log(m22/m12)) (m _ π^(ó  ))^2) + ((s - (m _ π^(ó  ))^2) (-2 (m _ π^(ó  ))^6 + (4 (m12 + m22) + (m22 - m12) log(m22/m12)) (m _ π^(ó  ))^4 + (-2 (m12 - m22)^2 + 2 (m12^2 - m22^2) log(m22/m12) - (m12 + m22) log((-(m _ π^(ó  ))^2 + m12 + m22 - ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))/(-(m _ π^(ó  ))^2 + m12 + m22 + ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))) ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2)) (m _ π^(ó  ))^2 - (m12 - m22)^2 ((m12 - m22) log(m22/m12) - log((-(m _ π^(ó  ))^2 + m12 + m22 - ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))/(-(m _ π^(ó  ))^2 + m12 + m22 + ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))) ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2)^(1/2))))/(32 π^2 (m _ π^(ó  ))^4 ((m _ π^(ó  ))^4 - 2 (m12 + m22) (m _ π^(ó  ))^2 + (m12 - m22)^2))

jbarApprox1[s_, m12_] = Limit[jbarApprox1[s, m12, m22], m22 -> m12] /. (-4 m12 ParticleMass[PseudoScalar[2]]^2 + ParticleMass[PseudoScalar[2]]^4)^(1/2) :> ParticleMass[PseudoScalar[2]] (-4 m12 + ParticleMass[PseudoScalar[2]]^2)^(1/2) // Simplify

1/(32 π^2 (m _ π^(ó  ))^3 ((m _ π^(ó  ))^2 - 4 m12)) (6 (m _ π^(ó  ))^5 - log(-((m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) - 2 m12)/(-(m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) + 2 m12)) ((m _ π^(ó  ))^2 - 4 m12)^(1/2) (m _ π^(ó  ))^4 - 2 (12 m12 + s) (m _ π^(ó  ))^3 + 6 m12 log(-((m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) - 2 m12)/(-(m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) + 2 m12)) ((m _ π^(ó  ))^2 - 4 m12)^(1/2) (m _ π^(ó  ))^2 + 8 m12 s m _ π^(ó  ) - 2 m12 s log(-((m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) - 2 m12)/(-(m _ π^(ó  ))^2 + ((m _ π^(ó  ))^2 - 4 m12)^(1/2) m _ π^(ó  ) + 2 m12)) ((m _ π^(ó  ))^2 - 4 m12)^(1/2))

Series[jbarApprox1[s, m12] /. ParticleMass[Pion] -> mpi2^(1/2), {mpi2, 0, 2}] /. mpi2 -> ParticleMass[Pion]^2 // Normal // Simplify

((3 s - 7 m12) (m _ π^(ó  ))^4 + 14 m12 s (m _ π^(ó  ))^2 + 70 m12^2 s)/(6720 m12^3 π^2)

finalM0c2series = Series[finalM0c2 /. {q2t -> qqc2, q2u -> (qqc2 /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU})} /. {LeutwylerJBar[a__, ___Rule] -> jbarApprox1[a]}, {MandelstamS, ParticleMass[Pion]^2, 1}] // Normal ;

finalM0c2trunc = ((Series[Expand[finalM0c2series /. gellmannOkubo] /. ParticleMass[Pion, ___] -> Sqrt[m2], {m2, 0, 2}] // Simplify) // Normal) /. m2 -> ParticleMass[Pion]^2 ;

$ExpansionQuantities = {ParticleMass[Pion, a___]} ;

finalM0c2exp = Collect[DiscardOrders[finalM0c2trunc, PerturbationOrder -> 0, DiscardMomenta -> False], MandelstamS] // FullSimplify

-1/(4608 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (2304 π^2 ((m _ K^(ó  ))^2 + 3 s) (f _ ϕ^(ó  ))^2 + (m _ K^(ó  ))^2 (4 (576 π^2 (4 N _ 5^( ) + 4 N _ 8^( ) - 8 N _ 10^( ) - 8 N _ 11^( ) - N _ 19^( ) + N _ 20^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) - 161 log(-(i m _ K^(ó  ))/μ) - 161 log((i m _ K^(ó  ))/μ) + 4741 log(4/3)) (m _ K^(ó  ))^2 - 5547 (m _ K^(ó  ))^2 - 4 s (576 π^2 (-4 N _ 5^( ) + 16 N _ 7^( ) - 12 N _ 8^( ) - 8 N _ 9^( ) - 8 N _ 10^( ) - 8 N _ 11^( ) - 3 N _ 19^( ) + 3 N _ 20^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) + 36 log(-i/μ) + 36 log(i/μ) + 36 log((m _ π^(ó  ))^2) + 235 log(-(i m _ K^(ó  ))/μ) + 235 log((i m _ K^(ó  ))/μ) + 4969 log(4/3)) + s (5709 - 32 3^(1/2) π))))

Notice that we have dropped the O(m _ π^2) contribution above.

testa = finalM0c2exp /. MandelstamS -> 0 // FullSimplify

-1/(4608 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (m _ K^(ó  ))^2 (2304 π^2 (f _ ϕ^(ó  ))^2 + (2304 π^2 (4 N _ 5^( ) + 4 N _ 8^( ) - 8 N _ 10^( ) - 8 N _ 11^( ) - N _ 19^( ) + N _ 20^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) - 644 log(-(i m _ K^(ó  ))/μ) - 644 log((i m _ K^(ó  ))/μ) + 18964 log(4/3) - 5547) (m _ K^(ó  ))^2))

testb = (finalM0c2exp - testa /. MandelstamS -> 1 /. gellmannOkubo // FullSimplify) /. toEtaRules

-1/(4608 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (6912 π^2 (f _ ϕ^(ó  ))^2 + (2304 π^2 (4 N _ 5^( ) - 16 N _ 7^( ) + 12 N _ 8^( ) + 8 N _ 9^( ) + 8 N _ 10^( ) + 8 N _ 11^( ) + 3 N _ 19^( ) - 3 N _ 20^( ) - 2 N _ 21^( ) - N _ 22^( ) - 2 N _ 23^( )) - 144 log(-i/μ) - 144 log(i/μ) - 144 log((m _ π^(ó  ))^2) - 940 log(-(i m _ K^(ó  ))/μ) - 940 log((i m _ K^(ó  ))/μ) - 19876 log(4/3) - 32 3^(1/2) π + 5709) (m _ K^(ó  ))^2))

This should agree with Gilberto's expression:

gilbertoX = (Series[(testb/(3 testa/ParticleMass[Kaon]^2) /. _DecayConstant -> 1/f2^(1/2)), {f2, 0, 1}] // Normal) /. gellmannOkubo //. cancelLogs1 // FullSimplify

1/(3456 π^2) (f2 (-2304 π^2 (4 N _ 5^( ) + 8 N _ 7^( ) - 4 N _ 9^( ) - 16 N _ 10^( ) - 16 N _ 11^( ) - 3 N _ 19^( ) + 3 N _ 20^( ) + 4 N _ 21^( ) + 2 N _ 22^( ) + 4 N _ 23^( )) - 848 log(μ) - 144 log(m _ π^(ó  )) + 992 log(m _ K^(ó  )) - 38384 log(4/3) - 16 3^(1/2) π + 11175) (m _ K^(ó  ))^2 + 3456 π^2)

gilbertoX /. scaleRule //. cancelLogs1 // FullSimplify

1/(3456 π^2) (f2 (-2304 π^2 (4 N _ 5^( ) + 8 N _ 7^( ) - 4 N _ 9^( ) - 16 N _ 10^( ) - 16 N _ 11^( ) - 3 N _ 19^( ) + 3 N _ 20^( ) + 4 N _ 21^( ) + 2 N _ 22^( ) + 4 N _ 23^( )) - 144 log(m _ π^(ó  )) + 144 log(m _ K^(ó  )) - 38384 log(4/3) - 16 3^(1/2) π + 11175) (m _ K^(ó  ))^2 + 3456 π^2)

The counterterms agree, the difference of the logs and constants must come from different treatment of the expansion of Overscript[J, _]'s and Gellmann-Okubo and should be of order O(m _ π^2).

Converted by Mathematica  (July 10, 2003)