•The c _ 5 pieces

The Overscript[J, _] parts of the loop contribution:

jbars5 = Collect[loopJs5 - (loopJs5 /. _LeutwylerJBar -> 0) /. Pair[_LorentzIndex, ___] -> Sequence[] // Simplify, {_LeutwylerJBar}] /. l : (LeutwylerJBar[t | u, __] * __) :> (l /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] // FullSimplify)

-(i c _ 5^( ) Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (-2 (m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2 - 5 s + t + u - 7 p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(12 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + 1/(288 t^2 u^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) Overscript[J, _] _ (m _ η^(ó  ))^2(s) (48 t^2 u^2 (m _ π^(ó  ))^4 - 24 t^2 u^3 (m _ π^(ó  ))^2 - 24 t^3 u^2 (m _ π^(ó  ))^2 + 120 s t^2 u^2 (m _ π^(ó  ))^2 - 120 t^2 u^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 40 t^2 u^2 p _ 2^2 (m _ π^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + 1/(72 t^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)(t) (2 (-(m _ π^(ó  ))^2 + t + p _ 2^2) (m _ K^(ó  ))^6 - (-6 (m _ π^(ó  ))^4 + 6 (p _ 2^2 - 3 t) (m _ π^(ó  ))^2 + t (23 t + u) + 16 t p _ 2^2) (m _ K^(ó  ))^4 + 2 (-3 (m _ π^(ó  ))^6 + 3 (p _ 2^2 - 5 t) (m _ π^(ó  ))^4 + t (-10 t + u + 10 p _ 2^2) (m _ π^(ó  ))^2 + 3 t^2 (2 t + 7 u) - 9 t^2 p _ 2^2) (m _ K^(ó  ))^2 + ((m _ π^(ó  ))^2 + 3 t) (2 (m _ π^(ó  ))^6 + (4 t - 2 p _ 2^2) (m _ π^(ó  ))^4 - t (5 t + u - 2 p _ 2^2) (m _ π^(ó  ))^2 + 3 t^2 (t - u))) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + 1/(8 t^2 (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) (2 (-(m _ π^(ó  ))^2 + t + p _ 2^2) (m _ K^(ó  ))^6 - (-6 (m _ π^(ó  ))^4 + 2 (t + 3 p _ 2^2) (m _ π^(ó  ))^2 + t (3 t + u)) (m _ K^(ó  ))^4 - 2 (3 (m _ π^(ó  ))^6 + (t - 3 p _ 2^2) (m _ π^(ó  ))^4 - t (6 t + u) (m _ π^(ó  ))^2 + t^2 (4 t - u - p _ 2^2)) (m _ K^(ó  ))^2 + (t - (m _ π^(ó  ))^2)^2 (2 (m _ π^(ó  ))^4 + 6 t (m _ π^(ó  ))^2 + t (9 t - u) - 2 p _ 2^2 ((m _ π^(ó  ))^2 + 2 t))) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + 1/(72 u^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)(u) (2 (-(m _ π^(ó  ))^2 + u + p _ 2^2) (m _ K^(ó  ))^6 - (-6 (m _ π^(ó  ))^4 + 6 (p _ 2^2 - 3 u) (m _ π^(ó  ))^2 + u (t + 23 u) + 16 u p _ 2^2) (m _ K^(ó  ))^4 + 2 (-3 (m _ π^(ó  ))^6 - 3 (5 u - p _ 2^2) (m _ π^(ó  ))^4 + u (t - 10 u + 10 p _ 2^2) (m _ π^(ó  ))^2 + 3 u^2 (7 t + 2 u - 3 p _ 2^2)) (m _ K^(ó  ))^2 + ((m _ π^(ó  ))^2 + 3 u) (2 (m _ π^(ó  ))^6 + (4 u - 2 p _ 2^2) (m _ π^(ó  ))^4 - u (t + 5 u - 2 p _ 2^2) (m _ π^(ó  ))^2 + 3 u^2 (u - t))) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + 1/(8 u^2 (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) (2 (-(m _ π^(ó  ))^2 + u + p _ 2^2) (m _ K^(ó  ))^6 - (-6 (m _ π^(ó  ))^4 + 2 (u + 3 p _ 2^2) (m _ π^(ó  ))^2 + u (t + 3 u)) (m _ K^(ó  ))^4 + 2 (-3 (m _ π^(ó  ))^6 - (u - 3 p _ 2^2) (m _ π^(ó  ))^4 + u (t + 6 u) (m _ π^(ó  ))^2 + u^2 (t - 4 u + p _ 2^2)) (m _ K^(ó  ))^2 + (u - (m _ π^(ó  ))^2)^2 (2 (m _ π^(ó  ))^4 + 6 u (m _ π^(ó  ))^2 + u (9 u - t) - 2 p _ 2^2 ((m _ π^(ó  ))^2 + 2 u))) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) + (i s c _ 5^( ) Overscript[J, _] _ (m _ K^(ó  ))^2(s) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(8 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

Crossing symmetry:

jbars5 - (jbars5 /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Simplify

0

We change to some different loop functions:

jbars5KLM = Simplify /@ Collect[Expand[Expand[jbars5] /. JBarToMr] /. JBarToKL /. gellmannOkubo /. {s -> MandelstamS, t -> MandelstamT, u -> MandelstamU} // Simplify, {_CouplingConstant, _K, _L, _LeutwylerJBar, _Log}] /. toEtaRules

1/(576 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) ((9 (t + u) log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 2^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(t u π^2 (f _ ϕ^(ó  ))^4) + (16 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (6 (m _ π^(ó  ))^2 - 15 (m _ K^(ó  ))^2 + 15 s - 3 t - 3 u + 5 p _ 2^2) (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^4 - (27 (t + u) log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2)/(t u π^2 (f _ ϕ^(ó  ))^4) + (72 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2))/(f _ ϕ^(ó  ))^4 + (48 Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2))/(f _ ϕ^(ó  ))^4 - 1/(t u π^2 (f _ ϕ^(ó  ))^4) (3 (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (t (m _ π^(ó  ))^2 + u (m _ π^(ó  ))^2 + 5 t (m _ K^(ó  ))^2 + 5 u (m _ K^(ó  ))^2 - 2 t u + 48 t u π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 48 t u π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 144 t u π^2 Mr(t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 144 t u π^2 Mr(t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2) + 144 t u π^2 Mr(u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 144 t u π^2 Mr(u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2))) - (144 K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (-3 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + t + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - t + p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (144 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (-3 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - u + p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (48 K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-7 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + t + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - t + 5 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (48 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-7 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - u + 5 p _ 2^2)))/(f _ ϕ^(ó  ))^4 + (4 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (5 (m _ π^(ó  ))^4 - (31 (m _ K^(ó  ))^2 + 24 t + 12 u - 21 p _ 2^2) (m _ π^(ó  ))^2 - 46 (m _ K^(ó  ))^4 + 3 (8 t + 28 u - 15 p _ 2^2) (m _ K^(ó  ))^2 + 18 t (t - u)))/(f _ ϕ^(ó  ))^4 - (4 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-5 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 12 t + 24 u - 21 p _ 2^2) (m _ π^(ó  ))^2 + 46 (m _ K^(ó  ))^4 - 3 (28 t + 8 u - 15 p _ 2^2) (m _ K^(ó  ))^2 + 18 (t - u) u))/(f _ ϕ^(ó  ))^4 + 1/(f _ ϕ^(ó  ))^4 (36 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-3 (m _ π^(ó  ))^4 + (25 (m _ K^(ó  ))^2 + 4 (u - 6 t)) (m _ π^(ó  ))^2 + p _ 2^2 (13 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 - 8 t) + 2 (-3 (m _ K^(ó  ))^4 + (2 u - 8 t) (m _ K^(ó  ))^2 + t (9 t - u)))) - 1/(f _ ϕ^(ó  ))^4 (36 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (3 (m _ π^(ó  ))^4 + (-25 (m _ K^(ó  ))^2 - 4 t + 24 u) (m _ π^(ó  ))^2 + p _ 2^2 (-13 (m _ π^(ó  ))^2 - 3 (m _ K^(ó  ))^2 + 8 u) + 2 (3 (m _ K^(ó  ))^4 - 2 (t - 4 u) (m _ K^(ó  ))^2 + (t - 9 u) u)))))

Limit[MandelstamT * jbars5KLM /. _Log -> 0, MandelstamT -> 0] // Simplify

(i c _ 5^( ) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2 ((m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2))/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

jbars5KLMs = Plus @@ ((Coefficient[jbars5KLM, #] * #) & /@ Union[Cases[jbars2KLM, _LeutwylerJBar | _K | _Mr, Infinity]]) // Simplify

1/(144 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) ((4 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (6 (m _ π^(ó  ))^2 - 15 (m _ K^(ó  ))^2 + 15 s - 3 t - 3 u + 5 p _ 2^2) (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^4 + (18 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2))/(f _ ϕ^(ó  ))^4 + (12 Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (2 (m _ π^(ó  ))^2 - 5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2))/(f _ ϕ^(ó  ))^4 - (108 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))/(f _ ϕ^(ó  ))^4 - (108 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))/(f _ ϕ^(ó  ))^4 - (108 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))/(f _ ϕ^(ó  ))^4 - (108 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))/(f _ ϕ^(ó  ))^4 - (36 K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (-3 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + t + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - t + p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (36 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (-3 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - u + p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (12 K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-7 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + t + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - t + 5 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (12 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-7 (m _ π^(ó  ))^4 + (-(m _ K^(ó  ))^2 + u + p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 - u + 5 p _ 2^2)))/(f _ ϕ^(ó  ))^4 + (Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (5 (m _ π^(ó  ))^4 - (31 (m _ K^(ó  ))^2 + 24 t + 12 u - 21 p _ 2^2) (m _ π^(ó  ))^2 - 46 (m _ K^(ó  ))^4 + 3 (8 t + 28 u - 15 p _ 2^2) (m _ K^(ó  ))^2 + 18 t (t - u)))/(f _ ϕ^(ó  ))^4 - (Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-5 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 12 t + 24 u - 21 p _ 2^2) (m _ π^(ó  ))^2 + 46 (m _ K^(ó  ))^4 - 3 (28 t + 8 u - 15 p _ 2^2) (m _ K^(ó  ))^2 + 18 (t - u) u))/(f _ ϕ^(ó  ))^4 + 1/(f _ ϕ^(ó  ))^4 (9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-3 (m _ π^(ó  ))^4 + (25 (m _ K^(ó  ))^2 + 4 (u - 6 t)) (m _ π^(ó  ))^2 + p _ 2^2 (13 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 - 8 t) + 2 (-3 (m _ K^(ó  ))^4 + (2 u - 8 t) (m _ K^(ó  ))^2 + t (9 t - u)))) - 1/(f _ ϕ^(ó  ))^4 (9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (3 (m _ π^(ó  ))^4 + (-25 (m _ K^(ó  ))^2 - 4 t + 24 u) (m _ π^(ó  ))^2 + p _ 2^2 (-13 (m _ π^(ó  ))^2 - 3 (m _ K^(ó  ))^2 + 8 u) + 2 (3 (m _ K^(ó  ))^4 - 2 (t - 4 u) (m _ K^(ó  ))^2 + (t - 9 u) u)))))

jbars5KLMPolys = jbars5KLM - jbars5KLMs /. {_Log -> 0, _k -> 0} // Simplify

(i c _ 5^( ) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2 ((t + u) (m _ π^(ó  ))^2 + 5 (t + u) (m _ K^(ó  ))^2 - 2 t u))/(192 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

jbars5KLMLogs = jbars5KLM - jbars5KLMs - jbars5KLMPolys // Simplify

1/(64 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((t + u) (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ π^(ó  ))^2 + 3 log((m _ η^(ó  ))^2/(m _ K^(ó  ))^2) (m _ η^(ó  ))^2) (m _ K^(ó  ))^2 + 16 t u π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 16 t u π^2 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))

The log parts of the loop contribution:

looplogs5coll = Collect[looplogs5 /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] // Expand, Log[__]] /. Log[a_] * b__ :> Log[a] * Together[Times[b]] /. a_Plus :> Collect[a, {ParticleMass[__], CouplingConstant[__]}] /; FreeQ[a, _Log]

1/(384 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ π^(ó  ))^2/μ^2) (((-6 t - 6 u) c _ 5^( ) (m _ K^(ó  ))^2 + 72 t u c _ 5^( )) (m _ π^(ó  ))^6 + ((6 t + 6 u) c _ 5^( ) (m _ K^(ó  ))^4 + c _ 5^( ) (28 t u + 6 p _ 2^2 u + 6 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 5^( ) (-112 u t^2 - 112 u^2 t + 136 u p _ 2^2 t)) (m _ π^(ó  ))^4 + (c _ 5^( ) (-52 t u - 6 p _ 2^2 u - 6 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 5^( ) (40 u t^2 + 40 u^2 t - 40 u p _ 2^2 t) (m _ K^(ó  ))^2 + c _ 5^( ) (51 u t^3 + 42 u^2 t^2 - 84 u p _ 2^2 t^2 + 51 u^3 t + 48 u p _ 2^4 t - 84 u^2 p _ 2^2 t)) (m _ π^(ó  ))^2 + c _ 5^( ) (24 u t^2 + 24 u^2 t - 48 u p _ 2^2 t) (m _ K^(ó  ))^4 + c _ 5^( ) (-24 u t^3 - 48 u^2 t^2 + 72 u p _ 2^2 t^2 - 24 u^3 t - 48 u p _ 2^4 t + 72 u^2 p _ 2^2 t) (m _ K^(ó  ))^2)) + 1/(3456 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ η^(ó  ))^2/μ^2) (((-54 t - 54 u) c _ 5^( ) (m _ K^(ó  ))^2 + 280 t u c _ 5^( )) (m _ π^(ó  ))^6 + ((270 t + 270 u) c _ 5^( ) (m _ K^(ó  ))^4 + c _ 5^( ) (-636 t u + 54 p _ 2^2 u + 54 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 5^( ) (-144 u t^2 - 144 u^2 t + 128 u p _ 2^2 t)) (m _ π^(ó  ))^4 + ((-216 t - 216 u) c _ 5^( ) (m _ K^(ó  ))^6 + c _ 5^( ) (540 t u - 270 p _ 2^2 u - 270 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 5^( ) (396 u t^2 + 396 u^2 t - 232 u p _ 2^2 t) (m _ K^(ó  ))^2 + (27 u t^3 - 54 u^2 t^2 + 27 u^3 t) c _ 5^( )) (m _ π^(ó  ))^2 + c _ 5^( ) (464 t u + 216 p _ 2^2 u + 216 t p _ 2^2) (m _ K^(ó  ))^6 + c _ 5^( ) (-576 u t^2 - 576 u^2 t + 320 u p _ 2^2 t) (m _ K^(ó  ))^4 + (-108 u t^3 + 216 u^2 t^2 - 108 u^3 t) c _ 5^( ) (m _ K^(ó  ))^2)) + 1/(192 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i log((m _ K^(ó  ))^2/μ^2) (((6 t + 6 u) c _ 5^( ) (m _ K^(ó  ))^2 + 36 t u c _ 5^( )) (m _ π^(ó  ))^6 + ((-18 t - 18 u) c _ 5^( ) (m _ K^(ó  ))^4 + c _ 5^( ) (-10 t u - 6 p _ 2^2 u - 6 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 5^( ) (-36 u t^2 - 36 u^2 t + 54 u p _ 2^2 t)) (m _ π^(ó  ))^4 + ((12 t + 12 u) c _ 5^( ) (m _ K^(ó  ))^6 + c _ 5^( ) (-70 t u + 18 p _ 2^2 u + 18 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 5^( ) (29 u t^2 + 29 u^2 t - 44 u p _ 2^2 t) (m _ K^(ó  ))^2 + c _ 5^( ) (9 u t^3 + 18 u^2 t^2 - 27 u p _ 2^2 t^2 + 9 u^3 t + 18 u p _ 2^4 t - 27 u^2 p _ 2^2 t)) (m _ π^(ó  ))^2 + c _ 5^( ) (-16 t u - 12 p _ 2^2 u - 12 t p _ 2^2) (m _ K^(ó  ))^6 + c _ 5^( ) (49 u t^2 + 49 u^2 t - 46 u p _ 2^2 t) (m _ K^(ó  ))^4 + c _ 5^( ) (-18 u t^3 - 24 u^2 t^2 + 33 u p _ 2^2 t^2 - 18 u^3 t - 18 u p _ 2^4 t + 33 u^2 p _ 2^2 t) (m _ K^(ó  ))^2))

looplogs5coll - (looplogs5coll /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Simplify

0

The polynomial parts of the loop contribution:

polys5 = Collect[loopJs5 /. Pair[_LorentzIndex, ___] -> Sequence[] /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] /. _LeutwylerJBar -> 0 // FullSimplify, {_DecayConstant, _CouplingConstant, _ParticleMass}] // Simplify

1/(576 t u π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (126 t u p _ 2^4 + (-3 (t + u) (m _ π^(ó  ))^4 + 374 t u (m _ π^(ó  ))^2 + 15 (t + u) (m _ K^(ó  ))^4 + 6 (25 t u - 2 (t + u) (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 - 192 t u (t + u)) p _ 2^2 + 3 ((t + u) (m _ π^(ó  ))^6 + (4 (t + u) (m _ K^(ó  ))^2 + 86 t u) (m _ π^(ó  ))^4 - (5 (t + u) (m _ K^(ó  ))^4 - 66 t u (m _ K^(ó  ))^2 + 89 t u (t + u)) (m _ π^(ó  ))^2 + t u (10 (m _ K^(ó  ))^4 - 37 (t + u) (m _ K^(ó  ))^2 + 22 t^2 + 22 u^2 + 46 t u))))

Limit[MandelstamT * polys5, MandelstamT -> 0]

-1/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (-c _ 5^( ) (m _ π^(ó  ))^8 - 3 c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 + c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^6 + 9 c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4 + 3 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 5 c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2 - 9 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 5 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6))

Limit[MandelstamU * polys5, MandelstamU -> 0]

-1/(192 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (-c _ 5^( ) (m _ π^(ó  ))^8 - 3 c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 + c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^6 + 9 c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^4 + 3 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 - 5 c _ 5^( ) (m _ K^(ó  ))^6 (m _ π^(ó  ))^2 - 9 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 5 c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^6))

Converted by Mathematica  (July 10, 2003)