•The soft pion limit

The Overscript[J, _] content:

((((# * Coefficient[((jbars2KLMs + (jbars2KLMs /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}))), #]) & /@ Union[Cases[((jbars2KLMs + (jbars2KLMs /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}))), _LeutwylerJBar, Infinity]]) /. softPionLimit)) // Simplify

{(i c _ 2^( ) Overscript[J, _] _ (m _ π^(ó  ))^2((m _ π^(ó  ))^2) (m _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 + p _ 2^2))/(2 (f _ ϕ^(ó  ))^4), -(i c _ 2^( ) Overscript[J, _] _ (m _ K^(ó  ))^2((m _ π^(ó  ))^2) (m _ π^(ó  ))^2 (-3 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 + p _ 2^2))/(4 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ (m _ η^(ó  ))^2((m _ π^(ó  ))^2) (m _ π^(ó  ))^2 (13 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 3 p _ 2^2))/(18 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) (-13 (m _ π^(ó  ))^4 + 39 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 20 (m _ K^(ó  ))^4 + p _ 2^2 (31 (m _ π^(ó  ))^2 - 37 (m _ K^(ó  ))^2)))/(48 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) (-97 (m _ π^(ó  ))^4 + 419 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 52 (m _ K^(ó  ))^4 + 3 p _ 2^2 ((m _ π^(ó  ))^2 - 43 (m _ K^(ó  ))^2)))/(432 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) (-13 (m _ π^(ó  ))^4 + 99 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 46 (m _ K^(ó  ))^4 + 30 p _ 2^4 - p _ 2^2 (29 (m _ π^(ó  ))^2 + 41 (m _ K^(ó  ))^2)))/(48 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (-97 (m _ π^(ó  ))^4 + 503 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 118 (m _ K^(ó  ))^4 + 54 p _ 2^4 - 9 p _ 2^2 (9 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2)))/(432 (f _ ϕ^(ó  ))^4)}

((((((# * Coefficient[((jbars2KLMs + (jbars2KLMs /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}))), #]) & /@ Union[Cases[((jbars2KLMs + (jbars2KLMs /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}))), _LeutwylerJBar, Infinity]]) /. softPionLimit)) /. Sign[ParticleMass[PseudoScalar[6]]] -> ParticleMass[PseudoScalar[6]] /. DirectedInfinity -> Identity /. Sqrt[a_^2] -> a // Simplify) (* /. gellmannOkubo // Simplify *)) /. Log[l_] :> Log[l /. toEtaRules] /. ParticleMass[Pion] -> 0

{0, 0, 0, (i c _ 2^( ) Overscript[J, _] _ (0 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) (-20 (m _ K^(ó  ))^4 - 37 p _ 2^2 (m _ K^(ó  ))^2))/(48 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) (-52 (m _ K^(ó  ))^4 - 129 p _ 2^2 (m _ K^(ó  ))^2))/(432 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) (-46 (m _ K^(ó  ))^4 - 41 p _ 2^2 (m _ K^(ó  ))^2 + 30 p _ 2^4))/(48 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (-118 (m _ K^(ó  ))^4 - 117 p _ 2^2 (m _ K^(ó  ))^2 + 54 p _ 2^4))/(432 (f _ ϕ^(ó  ))^4)}

softjbars2 = ((jbars2KLMs) /. softPionLimit /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> LeutwylerJBar[a] /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], Mr[l__] :> Mr[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]} /. cancelLogs /. Log -> log (* /. ParticleMass[Pion] -> 0 *) // Simplify) /. log -> Log

-1/(864 (f _ ϕ^(ó  ))^4 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ K^(ó  ))^2) (i c _ 2^( ) ((f _ ϕ^(ó  ))^4 ((-108 (Mr(p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 5 Mr(p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + Mr((m _ K^(ó  ))^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 5 Mr((m _ K^(ó  ))^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2)) p _ 2^2 (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) (34 (m _ π^(ó  ))^6 + 14 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 70 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 22 (m _ K^(ó  ))^6 + 30 p _ 2^6 - p _ 2^4 (29 (m _ π^(ó  ))^2 + 41 (m _ K^(ó  ))^2) + p _ 2^2 (-35 (m _ π^(ó  ))^4 + 83 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4)) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (-54 p _ 2^6 + 9 (9 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2) p _ 2^4 + 9 (7 (m _ π^(ó  ))^4 - 55 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^4) p _ 2^2 + 2 (35 (m _ π^(ó  ))^6 - 71 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 67 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 - 31 (m _ K^(ó  ))^6))) (m _ K^(ó  ))^2 + 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) p _ 2^2 (-34 (m _ π^(ó  ))^6 + (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 27 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + p _ 2^2 (20 (m _ π^(ó  ))^4 - 11 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 3 (m _ K^(ó  ))^4)) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) p _ 2^2 (5 (14 (m _ π^(ó  ))^4 - 7 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 61 (m _ K^(ó  ))^4) (m _ π^(ó  ))^2 + p _ 2^2 (-44 (m _ π^(ó  ))^4 + 25 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 145 (m _ K^(ó  ))^4))) - 12 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 (18 Overscript[J, _] _ (m _ π^(ó  ))^2((m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 + p _ 2^2) - 9 Overscript[J, _] _ (m _ K^(ó  ))^2((m _ π^(ó  ))^2) (-3 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 + p _ 2^2) - 2 Overscript[J, _] _ (m _ η^(ó  ))^2((m _ π^(ó  ))^2) (-13 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2 + 3 p _ 2^2))))

softjbars5 = ((jbars5KLMs) /. softPionLimit /. KLToJBar /. LeutwylerJBar[a__, ___Rule] -> LeutwylerJBar[a] /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], Mr[l__] :> Mr[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]} /. cancelLogs /. Log -> log (* /. ParticleMass[Pion] -> 0 *) // Simplify) /. log -> Log

(i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (2 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 (2 Overscript[J, _] _ (m _ η^(ó  ))^2((m _ π^(ó  ))^2) (21 (m _ π^(ó  ))^2 - 18 (m _ K^(ó  ))^2 + 2 p _ 2^2) + 6 Overscript[J, _] _ (m _ π^(ó  ))^2((m _ π^(ó  ))^2) (7 (m _ π^(ó  ))^2 - 6 (m _ K^(ó  ))^2 + 6 p _ 2^2) + 9 Overscript[J, _] _ (m _ K^(ó  ))^2((m _ π^(ó  ))^2) (7 (m _ π^(ó  ))^2 - 6 (m _ K^(ó  ))^2 + 6 p _ 2^2)) - (f _ ϕ^(ó  ))^4 ((108 (Mr(p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + Mr(p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + Mr((m _ K^(ó  ))^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + Mr((m _ K^(ó  ))^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2)) p _ 2^2 (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (-18 p _ 2^6 + 3 ((m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2) p _ 2^4 - 3 ((m _ π^(ó  ))^4 - 17 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^4) p _ 2^2 - 2 (7 (m _ π^(ó  ))^6 - 7 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^6)) - 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) (10 p _ 2^6 - (11 (m _ π^(ó  ))^2 + 15 (m _ K^(ó  ))^2) p _ 2^4 + (29 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 - 5 (m _ π^(ó  ))^4) p _ 2^2 + 2 (3 (m _ π^(ó  ))^6 - 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^6))) (m _ K^(ó  ))^2 + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) p _ 2^2 (-14 (m _ π^(ó  ))^6 + 7 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 61 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + p _ 2^2 (4 (m _ π^(ó  ))^4 - 5 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^4)) + 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) p _ 2^2 (-6 (m _ π^(ó  ))^6 + 7 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 5 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + p _ 2^2 (4 (m _ π^(ó  ))^4 - 21 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^4)))))/(144 (f _ ϕ^(ó  ))^4 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ K^(ó  ))^2 (p _ 2^2 - (m _ K^(ó  ))^2))

The original loop polynomial is of order (m _ π)^2 in the soft pion limit. The polynomial introduced be the change to the K and Mr loop functions are irrelevant, since we change back in the comparison below.

polys2 /. softPionLimit /. kaonOnShell /. _RenormalizationState -> Sequence[] // Simplify

-(i c _ 2^( ) (m _ π^(ó  ))^2 (-((m _ π^(ó  ))^2 - 49 (m _ K^(ó  ))^2) p _ 2^4 + ((m _ π^(ó  ))^4 + 33 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 172 (m _ K^(ó  ))^4) p _ 2^2 + (m _ K^(ó  ))^2 ((m _ π^(ó  ))^4 - 20 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 19 (m _ K^(ó  ))^4)))/(1152 π^2 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ K^(ó  ))^2)

polys5 /. softPionLimit /. kaonOnShell /. _RenormalizationState -> Sequence[] // Simplify

(i c _ 5^( ) (m _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((95 (m _ K^(ó  ))^2 - 3 (m _ π^(ó  ))^2) p _ 2^4 + 3 ((m _ π^(ó  ))^4 + 89 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 32 (m _ K^(ó  ))^4) p _ 2^2 + 3 (m _ K^(ó  ))^2 ((m _ π^(ó  ))^4 + 4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^4)))/(576 π^2 (f _ ϕ^(ó  ))^4 p _ 2^2 (m _ K^(ó  ))^2 (p _ 2^2 - (m _ K^(ó  ))^2))

This is the final soft pion result (neglecting terms of order (m _ π)^2). It agrees perfectly with the KS->π result apart from the pieces involving the kaon tadpole.

endlim = (Limit[# /. k -> kk /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], k[l__] :> k[Sequence @@ ({l} /. toEtaRules)], Mr[l__] :> Mr[Sequence @@ ({l} /. toEtaRules)], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]} /. softPionLimit, ParticleMass[Pion] -> 0] /. gellmannOkubo /. toEtaRules // Simplify) & /@ {lows1, lows2, oldcts, newcts, strongcts, finallooplogs2, finallooplogs5, softjbars2, softjbars5, finallooppolys2, finallooppolys5}

{(i (2 c _ 5^( ) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2)))/(2 (f _ ϕ^(ó  ))^2), 1/(1728 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) (i (m _ K^(ó  ))^2 (4 c _ 5^( ) ((27 log((m _ K^(ó  ))^2/μ^2) + 22 log((m _ η^(ó  ))^2/μ^2)) p _ 2^2 - (27 log((m _ K^(ó  ))^2/μ^2) + 14 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (9 (7 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) p _ 2^2 - (45 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))), (2 i c _ 2^( ) (m _ K^(ó  ))^2 (2 (N _ 10^( ) + N _ 11^( )) (m _ K^(ó  ))^2 + N _ 5^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + N _ 8^( ) (p _ 2^2 - (m _ K^(ó  ))^2)))/(f _ ϕ^(ó  ))^4, (i c _ 2^( ) (2 N _ 20^( ) p _ 2^2 (p _ 2^2 - (m _ K^(ó  ))^2) - (m _ K^(ó  ))^2 (2 N _ 21^( ) ((m _ K^(ó  ))^2 + p _ 2^2) - (N _ 22^( ) + 2 N _ 23^( )) (p _ 2^2 - (m _ K^(ó  ))^2))))/(2 (f _ ϕ^(ó  ))^4), (8 i (2 L _ 4^( ) + L _ 5^( ) - 2 (2 L _ 6^( ) + L _ 8^( ))) c _ 5^( ) (m _ K^(ó  ))^4)/(f _ ϕ^(ó  ))^4, -(i c _ 2^( ) (-(27 log((m _ K^(ó  ))^2/μ^2) + 254 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + 9 (17 log((m _ K^(ó  ))^2/μ^2) - 22 log((m _ η^(ó  ))^2/μ^2)) p _ 2^2 (m _ K^(ó  ))^2 + 27 (log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) p _ 2^4))/(3456 π^2 (f _ ϕ^(ó  ))^4), -(i c _ 5^( ) (m _ K^(ó  ))^2 (5 (27 log((m _ K^(ó  ))^2/μ^2) - 22 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + (81 log((m _ K^(ó  ))^2/μ^2) + 34 log((m _ η^(ó  ))^2/μ^2)) p _ 2^2 (m _ K^(ó  ))^2 - 27 (log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) p _ 2^4))/(1728 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)), 1/(864 (f _ ϕ^(ó  ))^4 p _ 2^2) (i c _ 2^( ) ((27 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) - 145 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) - 108 (Mr(p _ 2^2, 0, (m _ K^(ó  ))^2) + 5 Mr(p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + Mr((m _ K^(ó  ))^2, 0, (m _ K^(ó  ))^2) + 5 Mr((m _ K^(ó  ))^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2))) p _ 2^4 (m _ K^(ó  ))^2 + 9 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) (22 (m _ K^(ó  ))^6 - 8 p _ 2^2 (m _ K^(ó  ))^4 - 41 p _ 2^4 (m _ K^(ó  ))^2 + 30 p _ 2^6) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (62 (m _ K^(ó  ))^6 - 144 p _ 2^2 (m _ K^(ó  ))^4 - 117 p _ 2^4 (m _ K^(ó  ))^2 + 54 p _ 2^6))), 1/(144 (f _ ϕ^(ó  ))^4 p _ 2^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) (m _ K^(ó  ))^2 ((-27 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)((m _ K^(ó  ))^2) + 29 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)((m _ K^(ó  ))^2) + 108 (Mr(p _ 2^2, 0, (m _ K^(ó  ))^2) + Mr(p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + Mr((m _ K^(ó  ))^2, 0, (m _ K^(ó  ))^2) + Mr((m _ K^(ó  ))^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2))) p _ 2^4 (m _ K^(ó  ))^2 + 9 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) (2 (m _ K^(ó  ))^6 - 15 p _ 2^4 (m _ K^(ó  ))^2 + 10 p _ 2^6) + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (2 (m _ K^(ó  ))^6 + 48 p _ 2^2 (m _ K^(ó  ))^4 - 39 p _ 2^4 (m _ K^(ó  ))^2 + 18 p _ 2^6))), -(i c _ 2^( ) (m _ K^(ó  ))^2 (19 (m _ K^(ó  ))^2 + 13 p _ 2^2))/(1152 π^2 (f _ ϕ^(ó  ))^4), (i c _ 5^( ) (m _ K^(ó  ))^4 (5 (m _ K^(ó  ))^2 + 3 p _ 2^2))/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))}

endlimna = (Simplify /@ Collect[Limit[Expand[# /. fixzeros /. MrToJBar /. k -> kk /. gellmannOkubo] /. JBarToKL1 /. {Log[l_] :> Log[l /. toEtaRules], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]}, ParticleMass[Pion] -> 0] /. fixzeros, {_LeutwylerJBar, _K, _Log}]) & /@ endlim[[{8, 9}]]

{-(3 i c _ 2^( ) K[p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^4)/(8 (f _ ϕ^(ó  ))^4) - (7 i c _ 2^( ) K[p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^4)/(24 (f _ ϕ^(ó  ))^4) - (7 i c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2)/(384 π^2 (f _ ϕ^(ó  ))^4) + (5 i c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2)/(192 π^2 (f _ ϕ^(ó  ))^4) - (5 i c _ 2^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 + p _ 2^2) (m _ K^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) (19 (m _ K^(ó  ))^2 + 13 p _ 2^2) (m _ K^(ó  ))^2)/(1152 π^2 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) (-(m _ K^(ó  ))^4 - 7 p _ 2^2 (m _ K^(ó  ))^2 + 5 p _ 2^4))/(16 (f _ ϕ^(ó  ))^4) + (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (11 (m _ K^(ó  ))^4 - 27 p _ 2^2 (m _ K^(ó  ))^2 + 9 p _ 2^4))/(144 (f _ ϕ^(ó  ))^4), (3 i c _ 5^( ) K[p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (m _ K^(ó  ))^6)/(4 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) K[p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (m _ K^(ó  ))^6)/(4 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (i c _ 5^( ) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 + p _ 2^2) (m _ K^(ó  ))^4)/(16 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) (5 (m _ K^(ó  ))^2 + 3 p _ 2^2) (m _ K^(ó  ))^4)/(192 π^2 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) log((m _ K^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^4)/(64 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i c _ 5^( ) log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^4)/(32 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i c _ 5^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) ((m _ K^(ó  ))^4 - 5 p _ 2^2 (m _ K^(ó  ))^2 + 3 p _ 2^4) (m _ K^(ó  ))^2)/(24 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (i c _ 5^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) ((m _ K^(ó  ))^4 + 7 p _ 2^2 (m _ K^(ó  ))^2 - 5 p _ 2^4) (m _ K^(ó  ))^2)/(8 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2))}

softKMs = Simplify /@ Collect[(Plus @@ endlimna) - (Plus @@ endlimna /. {_K -> 0, _LeutwylerJBar -> 0}) // Expand, {_LeutwylerJBar, _K}]

-(3 i K[p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] ((2 c _ 5^( ) (m _ K^(ó  ))^2)/(p _ 2^2 - (m _ K^(ó  ))^2) + c _ 2^( )) (m _ K^(ó  ))^4)/(8 (f _ ϕ^(ó  ))^4) - (i K[p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] ((6 c _ 5^( ) (m _ K^(ó  ))^2)/(p _ 2^2 - (m _ K^(ó  ))^2) + 7 c _ 2^( )) (m _ K^(ó  ))^4)/(24 (f _ ϕ^(ó  ))^4) + (i Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) (-(m _ K^(ó  ))^4 - 7 p _ 2^2 (m _ K^(ó  ))^2 + 5 p _ 2^4) (2 c _ 5^( ) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2)))/(16 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (6 c _ 5^( ) ((m _ K^(ó  ))^4 - 5 p _ 2^2 (m _ K^(ó  ))^2 + 3 p _ 2^4) (m _ K^(ó  ))^2 + c _ 2^( ) (-11 (m _ K^(ó  ))^6 + 38 p _ 2^2 (m _ K^(ó  ))^4 - 36 p _ 2^4 (m _ K^(ó  ))^2 + 9 p _ 2^6)))/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

softKMs /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 // Simplify

-1/(144 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (6 (9 K[p _ 2^2, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] + 7 K[p _ 2^2, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2]) (m _ K^(ó  ))^4 + Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (-11 (m _ K^(ó  ))^4 + 27 p _ 2^2 (m _ K^(ó  ))^2 - 9 p _ 2^4) + 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(p _ 2^2) ((m _ K^(ó  ))^4 + 7 p _ 2^2 (m _ K^(ó  ))^2 - 5 p _ 2^4)))

softLogs = Simplify /@ Collect[Limit[((Plus @@ endlimna) - (Plus @@ endlimna /. _Log -> 0) + (Plus @@ endlim[[{6, 7}]]) + endlim[[2]]) /. cancelLogs /. gellmannOkubo /. Log[l_] :> Log[l /. toEtaRules // Simplify], ParticleMass[Pion] -> 0] // Expand, _Log]

-(i log((m _ η^(ó  ))^2/μ^2) (18 c _ 5^( ) ((m _ K^(ó  ))^2 + p _ 2^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-13 (m _ K^(ó  ))^4 - 12 p _ 2^2 (m _ K^(ó  ))^2 + 9 p _ 2^4)))/(576 π^2 (f _ ϕ^(ó  ))^4) - (i log((m _ K^(ó  ))^2/μ^2) (2 c _ 5^( ) (5 (m _ K^(ó  ))^2 + p _ 2^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-11 (m _ K^(ó  ))^4 + 6 p _ 2^2 (m _ K^(ó  ))^2 + p _ 2^4)))/(128 π^2 (f _ ϕ^(ó  ))^4)

Simplify /@ Collect[Expand[softLogs /. CouplingConstant[ChPTW3[2], 2, ___] -> 0], _Log]

-(i c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) (-13 (m _ K^(ó  ))^4 - 12 p _ 2^2 (m _ K^(ó  ))^2 + 9 p _ 2^4))/(576 π^2 (f _ ϕ^(ó  ))^4) - (i c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) (-11 (m _ K^(ó  ))^4 + 6 p _ 2^2 (m _ K^(ó  ))^2 + p _ 2^4))/(128 π^2 (f _ ϕ^(ó  ))^4)

softcts = endlim[[{3, 4, 5}]]

{(2 i c _ 2^( ) (m _ K^(ó  ))^2 (2 (N _ 10^( ) + N _ 11^( )) (m _ K^(ó  ))^2 + N _ 5^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + N _ 8^( ) (p _ 2^2 - (m _ K^(ó  ))^2)))/(f _ ϕ^(ó  ))^4, (i c _ 2^( ) (2 N _ 20^( ) p _ 2^2 (p _ 2^2 - (m _ K^(ó  ))^2) - (m _ K^(ó  ))^2 (2 N _ 21^( ) ((m _ K^(ó  ))^2 + p _ 2^2) - (N _ 22^( ) + 2 N _ 23^( )) (p _ 2^2 - (m _ K^(ó  ))^2))))/(2 (f _ ϕ^(ó  ))^4), (8 i (2 L _ 4^( ) + L _ 5^( ) - 2 (2 L _ 6^( ) + L _ 8^( ))) c _ 5^( ) (m _ K^(ó  ))^4)/(f _ ϕ^(ó  ))^4}

softlow = endlim[[1]]

(i (2 c _ 5^( ) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2)))/(2 (f _ ϕ^(ó  ))^2)

Limit[(lows1 + lows2 + oldcts + cts2 + strongcts + finallooplogs2 + finallooplogs5 + endlimna[[1]] + endlimna[[2]] + finallooppolys2 + finallooppolys5) /. k -> kk /. softPionLimit /. KLToJBar /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], k[l__] :> k[Sequence @@ ({l} /. toEtaRules)], Mr[l__] :> Mr[Sequence @@ ({l} /. toEtaRules)], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]} /. cancelLogs, ParticleMass[Pion] -> 0] // Expand // Collect[#, {_Log}] & // Simplify

1/(1152 π^2 (f _ ϕ^(ó  ))^4 p _ 2^2) (i (c _ 2^( ) (216 π^2 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) (m _ K^(ó  ))^6 + 56 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (m _ K^(ó  ))^6 - 2304 π^2 N _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 - 2304 π^2 N _ 8^( ) p _ 2^2 (m _ K^(ó  ))^4 + 4608 π^2 N _ 10^( ) p _ 2^2 (m _ K^(ó  ))^4 + 4608 π^2 N _ 11^( ) p _ 2^2 (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^( ) p _ 2^2 (m _ K^(ó  ))^4 - 576 π^2 N _ 22^( ) p _ 2^2 (m _ K^(ó  ))^4 - 1152 π^2 N _ 23^( ) p _ 2^2 (m _ K^(ó  ))^4 - 72 π^2 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) p _ 2^2 (m _ K^(ó  ))^4 + 88 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) p _ 2^2 (m _ K^(ó  ))^4 + 99 log((m _ K^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^4 + 26 log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^4 + 2304 π^2 N _ 5^( ) p _ 2^4 (m _ K^(ó  ))^2 + 2304 π^2 N _ 8^( ) p _ 2^4 (m _ K^(ó  ))^2 - 1920 π^2 N _ 21^( ) p _ 2^4 (m _ K^(ó  ))^2 + 576 π^2 N _ 22^( ) p _ 2^4 (m _ K^(ó  ))^2 + 1152 π^2 N _ 23^( ) p _ 2^4 (m _ K^(ó  ))^2 - 504 π^2 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) p _ 2^4 (m _ K^(ó  ))^2 - 216 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) p _ 2^4 (m _ K^(ó  ))^2 - 54 log((m _ K^(ó  ))^2/μ^2) p _ 2^4 (m _ K^(ó  ))^2 + 24 log((m _ η^(ó  ))^2/μ^2) p _ 2^4 (m _ K^(ó  ))^2 + 360 π^2 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) p _ 2^6 + 72 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) p _ 2^6 - 9 log((m _ K^(ó  ))^2/μ^2) p _ 2^6 - 18 log((m _ η^(ó  ))^2/μ^2) p _ 2^6 + 1152 π^2 N _ 20^( ) p _ 2^4 (p _ 2^2 - (m _ K^(ó  ))^2) + 576 π^2 (f _ ϕ^(ó  ))^2 p _ 2^2 (p _ 2^2 - (m _ K^(ó  ))^2)) - 6 c _ 5^( ) (m _ K^(ó  ))^2 (8 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) (m _ K^(ó  ))^4 - 3072 π^2 L _ 4^( ) p _ 2^2 (m _ K^(ó  ))^2 - 1536 π^2 L _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 + 6144 π^2 L _ 6^( ) p _ 2^2 (m _ K^(ó  ))^2 + 3072 π^2 L _ 8^( ) p _ 2^2 (m _ K^(ó  ))^2 + 16 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) p _ 2^2 (m _ K^(ó  ))^2 + 15 log((m _ K^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2 + 6 log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2 - 24 π^2 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 2^2) p _ 2^4 + 3 log((m _ K^(ó  ))^2/μ^2) p _ 2^4 + 6 log((m _ η^(ó  ))^2/μ^2) p _ 2^4 - 192 π^2 (f _ ϕ^(ó  ))^2 p _ 2^2 - 24 π^2 Overscript[J, _] _ (0 (m _ K^(ó  ))^2)(p _ 2^2) (-3 (m _ K^(ó  ))^4 - 2 p _ 2^2 (m _ K^(ó  ))^2 + 5 p _ 2^4))))

Limit[(lows1 + lows2 + oldcts + cts2 + strongcts + finallooplogs2 + finallooplogs5 + endlimna[[1]] + endlimna[[2]] + finallooppolys2 + finallooppolys5) /. k -> kk /. softPionLimit /. p2 -> 0 /. KLToJBar /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], k[l__] :> k[Sequence @@ ({l} /. toEtaRules)], Mr[l__] :> Mr[Sequence @@ ({l} /. toEtaRules)], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]} /. cancelLogs, ParticleMass[Pion] -> 0] // Expand // Collect[#, {_Log}] & // FullSimplify

-1/(2304 π^2 (f _ ϕ^(ó  ))^4) (i (m _ K^(ó  ))^2 (1152 π^2 (c _ 2^( ) - 2 c _ 5^( )) (f _ ϕ^(ó  ))^2 + (18 c _ 5^( ) (-1024 π^2 (2 L _ 4^( ) + L _ 5^( ) - 4 L _ 6^( ) - 2 L _ 8^( )) + 22 log((m _ K^(ó  ))^2/μ^2) - 8 log((m _ η^(ó  ))^2/μ^2) + 5) + c _ 2^( ) (1152 π^2 (4 N _ 5^( ) + 4 N _ 8^( ) - 8 N _ 10^( ) - 8 N _ 11^( ) + 2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) - 450 log((m _ K^(ó  ))^2/μ^2) + 200 log((m _ η^(ó  ))^2/μ^2) - 87)) (m _ K^(ó  ))^2))

Converted by Mathematica  (July 10, 2003)