•The c _ 2 pieces

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

jbars2 = Collect[loopJs2 - (loopJs2 /. _LeutwylerJBar -> 0) /. Pair[_LorentzIndex, ___] -> Sequence[] // Simplify, {_LeutwylerJBar}] /. l : (LeutwylerJBar[t | u, __] * __) :> (l /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] // FullSimplify)

(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^( ) 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^( ) Overscript[J, _] _ (m _ η^(ó  ))^2(s) (192 t^2 u^2 (m _ π^(ó  ))^4 + 432 s t^2 u^2 (m _ π^(ó  ))^2 - 624 t^2 u^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 144 t^2 u^2 p _ 2^2 (m _ π^(ó  ))^2))/(1728 t^2 u^2 (f _ ϕ^(ó  ))^4) + 1/(48 t^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (2 ((m _ π^(ó  ))^2 + 5 t - p _ 2^2) (m _ K^(ó  ))^6 - (6 (m _ π^(ó  ))^4 + 34 t (m _ π^(ó  ))^2 + t (21 t - u) - 2 p _ 2^2 (3 (m _ π^(ó  ))^2 + 10 t)) (m _ K^(ó  ))^4 - 2 (-3 (m _ π^(ó  ))^6 + 3 (p _ 2^2 - t) (m _ π^(ó  ))^4 + t (-26 t + u + 4 p _ 2^2) (m _ π^(ó  ))^2 + t^2 (6 t + u) + 5 t^2 p _ 2^2) (m _ K^(ó  ))^2 + (t - (m _ π^(ó  ))^2)^2 (-2 (m _ π^(ó  ))^4 + 2 (7 t + p _ 2^2) (m _ π^(ó  ))^2 + t (23 t + u - 8 p _ 2^2)))) + 1/(48 u^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (2 ((m _ π^(ó  ))^2 + 5 u - p _ 2^2) (m _ K^(ó  ))^6 + (-6 (m _ π^(ó  ))^4 - 34 u (m _ π^(ó  ))^2 + (t - 21 u) u + p _ 2^2 (6 (m _ π^(ó  ))^2 + 20 u)) (m _ K^(ó  ))^4 - 2 (-3 (m _ π^(ó  ))^6 - 3 (u - p _ 2^2) (m _ π^(ó  ))^4 + u (t - 26 u + 4 p _ 2^2) (m _ π^(ó  ))^2 + u^2 (t + 6 u + 5 p _ 2^2)) (m _ K^(ó  ))^2 + (u - (m _ π^(ó  ))^2)^2 (-2 (m _ π^(ó  ))^4 + 2 (7 u + p _ 2^2) (m _ π^(ó  ))^2 + u (t + 23 u - 8 p _ 2^2)))) + 1/(432 t^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (2 (5 (m _ π^(ó  ))^2 + 13 t - 5 p _ 2^2) (m _ K^(ó  ))^6 + (-30 (m _ π^(ó  ))^4 - 6 (27 t - 5 p _ 2^2) (m _ π^(ó  ))^2 + t (151 t + 5 u + 92 p _ 2^2)) (m _ K^(ó  ))^4 - 2 (-15 (m _ π^(ó  ))^6 + 3 (5 p _ 2^2 - 31 t) (m _ π^(ó  ))^4 + t (-122 t + 5 u + 62 p _ 2^2) (m _ π^(ó  ))^2 + 15 t^2 (8 t + 7 u) - 57 t^2 p _ 2^2) (m _ K^(ó  ))^2 + ((m _ π^(ó  ))^2 + 3 t) (-10 (m _ π^(ó  ))^6 + 10 (p _ 2^2 - 2 t) (m _ π^(ó  ))^4 + t (-11 t + 5 u + 2 p _ 2^2) (m _ π^(ó  ))^2 + 3 t^2 (7 t + 5 u) - 12 t^2 p _ 2^2))) + 1/(432 u^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (2 (5 (m _ π^(ó  ))^2 + 13 u - 5 p _ 2^2) (m _ K^(ó  ))^6 + (-30 (m _ π^(ó  ))^4 - 6 (27 u - 5 p _ 2^2) (m _ π^(ó  ))^2 + u (5 t + 151 u + 92 p _ 2^2)) (m _ K^(ó  ))^4 - 2 (-15 (m _ π^(ó  ))^6 - 3 (31 u - 5 p _ 2^2) (m _ π^(ó  ))^4 + u (5 t - 122 u + 62 p _ 2^2) (m _ π^(ó  ))^2 + 3 u^2 (35 t + 40 u - 19 p _ 2^2)) (m _ K^(ó  ))^2 + ((m _ π^(ó  ))^2 + 3 u) (-10 (m _ π^(ó  ))^6 - 10 (2 u - p _ 2^2) (m _ π^(ó  ))^4 + u (5 t - 11 u + 2 p _ 2^2) (m _ π^(ó  ))^2 + 3 u^2 (5 t + 7 u - 4 p _ 2^2))))

Crossing symmetry:

jbars2 - (jbars2 /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Simplify

0

We change to some different loop functions:

jbars2KLM = Simplify /@ Collect[Expand[Expand[jbars2] /. JBarToMr] /. JBarToKL /. gellmannOkubo /. {s -> MandelstamS, t -> MandelstamT, u -> MandelstamU} // Simplify, {_CouplingConstant, _K, _L, _LeutwylerJBar, _Log}] /. toEtaRules

-1/3456 (i c _ 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) + (96 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)/(f _ ϕ^(ó  ))^4 - (135 (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) + (864 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))/(f _ ϕ^(ó  ))^4 + (432 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (3 (m _ K^(ó  ))^2 - 3 s + 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 + 19 t (m _ K^(ó  ))^2 + 19 u (m _ K^(ó  ))^2 - 6 t u + 48 t u π^2 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 240 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) + 720 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) + 720 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] (17 (m _ π^(ó  ))^4 + (23 (m _ K^(ó  ))^2 + t - 11 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (10 (m _ K^(ó  ))^2 + t + 19 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (144 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (17 (m _ π^(ó  ))^4 + (23 (m _ K^(ó  ))^2 + u - 11 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (10 (m _ K^(ó  ))^2 + u + 19 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (48 K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-35 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 5 t + 17 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-26 (m _ K^(ó  ))^2 - 5 t + 13 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (48 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-35 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 5 u + 17 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-26 (m _ K^(ó  ))^2 - 5 u + 13 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - 1/(f _ ϕ^(ó  ))^4 (36 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (-13 (m _ π^(ó  ))^4 + (103 (m _ K^(ó  ))^2 - 4 t - 64 u) (m _ π^(ó  ))^2 - 42 (m _ K^(ó  ))^4 + 46 u^2 - 4 t (m _ K^(ó  ))^2 - 24 u (m _ K^(ó  ))^2 + 2 t u + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 u))) - 1/(f _ ϕ^(ó  ))^4 (36 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-13 (m _ π^(ó  ))^4 + (103 (m _ K^(ó  ))^2 - 64 t - 4 u) (m _ π^(ó  ))^2 - 42 (m _ K^(ó  ))^4 + 46 t^2 - 24 t (m _ K^(ó  ))^2 - 4 u (m _ K^(ó  ))^2 + 2 t u + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 t))) - 1/(f _ ϕ^(ó  ))^4 (4 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-97 (m _ π^(ó  ))^4 + (443 (m _ K^(ó  ))^2 + 60 t - 24 u) (m _ π^(ó  ))^2 + 302 (m _ K^(ó  ))^4 + 126 u^2 - 420 t (m _ K^(ó  ))^2 - 480 u (m _ K^(ó  ))^2 + 90 t u - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 u))) - 1/(f _ ϕ^(ó  ))^4 (4 Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-97 (m _ π^(ó  ))^4 + (443 (m _ K^(ó  ))^2 - 24 t + 60 u) (m _ π^(ó  ))^2 + 302 (m _ K^(ó  ))^4 + 126 t^2 - 480 t (m _ K^(ó  ))^2 - 420 u (m _ K^(ó  ))^2 + 90 t u - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 t)))))

Limit[MandelstamT * jbars2KLM /. _Log -> 0, MandelstamT -> 0] // Simplify

-(i c _ 2^( ) (p _ 2^2 - (m _ π^(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)

jbars2KLMs = Plus @@ ((Coefficient[jbars2KLM, #] * #) & /@ Union[Cases[jbars2KLM, _LeutwylerJBar | _K | _Mr, Infinity]]) // Simplify

-1/864 i c _ 2^( ) ((24 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)/(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 - (540 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 - (540 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 + (216 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))/(f _ ϕ^(ó  ))^4 + (108 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (3 (m _ K^(ó  ))^2 - 3 s + p _ 2^2))/(f _ ϕ^(ó  ))^4 - (36 K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (17 (m _ π^(ó  ))^4 + (23 (m _ K^(ó  ))^2 + t - 11 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (10 (m _ K^(ó  ))^2 + t + 19 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (36 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (17 (m _ π^(ó  ))^4 + (23 (m _ K^(ó  ))^2 + u - 11 p _ 2^2) (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 (10 (m _ K^(ó  ))^2 + u + 19 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (12 K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-35 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 5 t + 17 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-26 (m _ K^(ó  ))^2 - 5 t + 13 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - (12 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (-35 (m _ π^(ó  ))^4 + (31 (m _ K^(ó  ))^2 + 5 u + 17 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-26 (m _ K^(ó  ))^2 - 5 u + 13 p _ 2^2)))/(f _ ϕ^(ó  ))^4 - 1/(f _ ϕ^(ó  ))^4 (9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(u) (-13 (m _ π^(ó  ))^4 + (103 (m _ K^(ó  ))^2 - 4 t - 64 u) (m _ π^(ó  ))^2 - 42 (m _ K^(ó  ))^4 + 46 u^2 - 4 t (m _ K^(ó  ))^2 - 24 u (m _ K^(ó  ))^2 + 2 t u + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 u))) - 1/(f _ ϕ^(ó  ))^4 (9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-13 (m _ π^(ó  ))^4 + (103 (m _ K^(ó  ))^2 - 64 t - 4 u) (m _ π^(ó  ))^2 - 42 (m _ K^(ó  ))^4 + 46 t^2 - 24 t (m _ K^(ó  ))^2 - 4 u (m _ K^(ó  ))^2 + 2 t u + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 t))) - 1/(f _ ϕ^(ó  ))^4 (Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(u) (-97 (m _ π^(ó  ))^4 + (443 (m _ K^(ó  ))^2 + 60 t - 24 u) (m _ π^(ó  ))^2 + 302 (m _ K^(ó  ))^4 + 126 u^2 - 420 t (m _ K^(ó  ))^2 - 480 u (m _ K^(ó  ))^2 + 90 t u - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 u))) - 1/(f _ ϕ^(ó  ))^4 (Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (-97 (m _ π^(ó  ))^4 + (443 (m _ K^(ó  ))^2 - 24 t + 60 u) (m _ π^(ó  ))^2 + 302 (m _ K^(ó  ))^4 + 126 t^2 - 480 t (m _ K^(ó  ))^2 - 420 u (m _ K^(ó  ))^2 + 90 t u - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 t))))

jbars2KLMPolys = jbars2KLM - jbars2KLMs /. {_Log -> 0, _k -> 0} // Simplify

-(i c _ 2^( ) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((t + u) (m _ π^(ó  ))^2 - 19 (t + u) (m _ K^(ó  ))^2 + 6 t u))/(1152 t u π^2 (f _ ϕ^(ó  ))^4)

jbars2KLMLogs = jbars2KLM - jbars2KLMs - jbars2KLMPolys // Simplify

1/(384 t u π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (p _ 2^2 - (m _ π^(ó  ))^2) ((t + u) (log((m _ K^(ó  ))^2/(m _ π^(ó  ))^2) (m _ π^(ó  ))^2 + 15 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) + 80 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:

looplogs2coll = Collect[looplogs2 /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] // Expand, Log[__]] /. Log[a_] * b__ :> Log[a] * Together[Times[b]] /. a_Plus :> Collect[a, {ParticleMass[__], CouplingConstant[__]}] /; FreeQ[a, _Log]

1/(768 t u π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i log((m _ π^(ó  ))^2/μ^2) (((-2 t - 2 u) c _ 2^( ) (m _ K^(ó  ))^2 - 84 t u c _ 2^( )) (m _ π^(ó  ))^6 + ((2 t + 2 u) c _ 2^( ) (m _ K^(ó  ))^4 + c _ 2^( ) (128 t u + 2 p _ 2^2 u + 2 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-38 u t^2 - 38 u^2 t + 28 u p _ 2^2 t)) (m _ π^(ó  ))^4 + (c _ 2^( ) (4 t u - 2 p _ 2^2 u - 2 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 2^( ) (-82 u t^2 - 82 u^2 t + 100 u p _ 2^2 t) (m _ K^(ó  ))^2 + c _ 2^( ) (49 u t^3 + 142 u^2 t^2 - 112 u p _ 2^2 t^2 + 49 u^3 t + 48 u p _ 2^4 t - 112 u^2 p _ 2^2 t)) (m _ π^(ó  ))^2 - 96 t u c _ 2^( ) (m _ K^(ó  ))^6 + c _ 2^( ) (168 u t^2 + 168 u^2 t - 144 u p _ 2^2 t) (m _ K^(ó  ))^4 + c _ 2^( ) (-72 u t^3 - 144 u^2 t^2 + 120 u p _ 2^2 t^2 - 72 u^3 t - 48 u p _ 2^4 t + 120 u^2 p _ 2^2 t) (m _ K^(ó  ))^2)) - 1/(384 t u π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i log((m _ K^(ó  ))^2/μ^2) (((-6 t - 6 u) c _ 2^( ) (m _ K^(ó  ))^2 + 36 t u c _ 2^( )) (m _ π^(ó  ))^6 + ((26 t + 26 u) c _ 2^( ) (m _ K^(ó  ))^4 + c _ 2^( ) (-90 t u + 6 p _ 2^2 u + 6 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-36 u t^2 - 36 u^2 t + 30 u p _ 2^2 t)) (m _ π^(ó  ))^4 + ((-20 t - 20 u) c _ 2^( ) (m _ K^(ó  ))^6 + c _ 2^( ) (18 t u - 26 p _ 2^2 u - 26 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 2^( ) (95 u t^2 + 95 u^2 t - 76 u p _ 2^2 t) (m _ K^(ó  ))^2 + c _ 2^( ) (9 u t^3 + 18 u^2 t^2 - 15 u p _ 2^2 t^2 + 9 u^3 t + 6 u p _ 2^4 t - 15 u^2 p _ 2^2 t)) (m _ π^(ó  ))^2 + c _ 2^( ) (120 t u + 20 p _ 2^2 u + 20 t p _ 2^2) (m _ K^(ó  ))^6 + c _ 2^( ) (-107 u t^2 - 107 u^2 t + 74 u p _ 2^2 t) (m _ K^(ó  ))^4 + c _ 2^( ) (-10 u t^3 - 4 u^2 t^2 + 13 u p _ 2^2 t^2 - 10 u^3 t - 6 u p _ 2^4 t + 13 u^2 p _ 2^2 t) (m _ K^(ó  ))^2)) - 1/(6912 t u π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i log((m _ η^(ó  ))^2/μ^2) (((90 t + 90 u) c _ 2^( ) (m _ K^(ó  ))^2 + 284 t u c _ 2^( )) (m _ π^(ó  ))^6 + ((-450 t - 450 u) c _ 2^( ) (m _ K^(ó  ))^4 + c _ 2^( ) (48 t u - 90 p _ 2^2 u - 90 t p _ 2^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-162 u t^2 - 162 u^2 t + 108 u p _ 2^2 t)) (m _ π^(ó  ))^4 + ((360 t + 360 u) c _ 2^( ) (m _ K^(ó  ))^6 + c _ 2^( ) (-1524 t u + 450 p _ 2^2 u + 450 t p _ 2^2) (m _ K^(ó  ))^4 + c _ 2^( ) (-54 u t^2 - 54 u^2 t - 36 u p _ 2^2 t) (m _ K^(ó  ))^2 + c _ 2^( ) (63 u t^3 + 90 u^2 t^2 - 36 u p _ 2^2 t^2 + 63 u^3 t - 36 u^2 p _ 2^2 t)) (m _ π^(ó  ))^2 + c _ 2^( ) (-752 t u - 360 p _ 2^2 u - 360 t p _ 2^2) (m _ K^(ó  ))^6 + c _ 2^( ) (1512 u t^2 + 1512 u^2 t - 720 u p _ 2^2 t) (m _ K^(ó  ))^4 + c _ 2^( ) (-252 u t^3 - 360 u^2 t^2 + 144 u p _ 2^2 t^2 - 252 u^3 t + 144 u^2 p _ 2^2 t) (m _ K^(ó  ))^2))

looplogs2coll - (looplogs2coll /. {t -> u, u -> t, MandelstamU -> MandelstamT, MandelstamT -> MandelstamU}) // Simplify

0

The polynomial parts of the loop contribution:

polys2 = Collect[loopJs2 /. Pair[_LorentzIndex, ___] -> Sequence[] /. cancelS /. kaonOnShell /. _RenormalizationState -> Sequence[] /. _LeutwylerJBar -> 0 // FullSimplify, {_DecayConstant, _CouplingConstant, _ParticleMass}] // Simplify

1/(1152 t u π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (-(t + u) (m _ π^(ó  ))^6 + (20 (t + u) (m _ K^(ó  ))^2 - 54 t u) (m _ π^(ó  ))^4 - (19 (t + u) (m _ K^(ó  ))^4 - 322 t u (m _ K^(ó  ))^2 + 111 t u (t + u)) (m _ π^(ó  ))^2 + 54 t u p _ 2^4 + t u (182 (m _ K^(ó  ))^4 - 285 (t + u) (m _ K^(ó  ))^2 + 84 t^2 + 84 u^2 + 174 t u) + p _ 2^2 ((t + u) (m _ π^(ó  ))^4 + (82 t u - 20 (t + u) (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 19 (t + u) (m _ K^(ó  ))^4 + 230 t u (m _ K^(ó  ))^2 - 138 t u (t + u))))

Limit[MandelstamT * polys2, MandelstamT -> 0]

(i (-c _ 2^( ) (m _ π^(ó  ))^6 + 20 c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^4 - 19 c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 - 20 c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 19 c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4))/(1152 π^2 (f _ ϕ^(ó  ))^4)

Limit[MandelstamU * polys2, MandelstamU -> 0]

(i (-c _ 2^( ) (m _ π^(ó  ))^6 + 20 c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^4 - 19 c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 - 20 c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 19 c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4))/(1152 π^2 (f _ ϕ^(ó  ))^4)

Converted by Mathematica  (July 10, 2003)