•Final results

Leading order:

finallows1 = lows1 // FullSimplify

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

Logs:

finalLogs = (Collect[lows2 + finallooplogs2 + finallooplogs5, {_Log, _k}] // Simplify) /. b_Log * a___ :> b * FullSimplify[Times[a]]

1/(2304 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i (96 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) (c _ 2^( ) + (6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(p _ 2^2 - (m _ K^(ó  ))^2)) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2 + 96 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) (5 c _ 2^( ) + (6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(p _ 2^2 - (m _ K^(ó  ))^2)) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)^2 + 1/(π^2 ((m _ K^(ó  ))^2 - p _ 2^2)) (6 log((m _ K^(ó  ))^2/μ^2) (c _ 2^( ) (-23 (m _ K^(ó  ))^8 - (-158 (m _ π^(ó  ))^2 + 137 s - 57 p _ 2^2) (m _ K^(ó  ))^6 + (-156 (m _ π^(ó  ))^4 + 143 s (m _ π^(ó  ))^2 + 10 t^2 + 10 u^2 - 41 p _ 2^4 + 4 t u + 2 p _ 2^2 (81 s - 110 (m _ π^(ó  ))^2)) (m _ K^(ó  ))^4 - (-7 p _ 2^6 + (13 s - 62 (m _ π^(ó  ))^2) p _ 2^4 + 2 (-78 (m _ π^(ó  ))^4 + 76 s (m _ π^(ó  ))^2 + 5 t^2 + 5 u^2 + 2 t u) p _ 2^2 + 9 s^2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 3 s (3 s - p _ 2^2) p _ 2^2 (m _ π^(ó  ))^2) - 6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (15 (m _ K^(ó  ))^6 + (18 (m _ π^(ó  ))^2 - 23 s + 8 p _ 2^2) (m _ K^(ó  ))^4 - (-16 (m _ π^(ó  ))^4 + 9 (s - 2 p _ 2^2) (m _ π^(ó  ))^2 + 6 t^2 + 6 u^2 - 5 p _ 2^4 + 8 t u + 11 s p _ 2^2) (m _ K^(ó  ))^2 + 3 s (s + p _ 2^2) (m _ π^(ó  ))^2))) + 1/(π^2 ((m _ K^(ó  ))^2 - p _ 2^2)) (log((m _ η^(ó  ))^2/μ^2) (c _ 2^( ) (-256 (m _ K^(ó  ))^8 + 4 (-97 (m _ π^(ó  ))^2 + 124 s - 8 p _ 2^2) (m _ K^(ó  ))^6 + 4 (-10 (m _ π^(ó  ))^4 - 6 s (m _ π^(ó  ))^2 + 21 t^2 + 21 u^2 + 60 p _ 2^4 + 30 t u + p _ 2^2 (76 (m _ π^(ó  ))^2 - 96 s)) (m _ K^(ó  ))^4 - (-36 (m _ π^(ó  ))^6 + 4 (9 s - 22 p _ 2^2) (m _ π^(ó  ))^4 + 3 (7 t^2 + 10 u t + 7 u^2 + 4 (7 s - 8 p _ 2^2) p _ 2^2) (m _ π^(ó  ))^2 + 12 p _ 2^2 (7 t^2 + 10 u t + 7 u^2 + 4 (s - p _ 2^2) p _ 2^2)) (m _ K^(ó  ))^2 - 4 (s + 9 p _ 2^2) (m _ π^(ó  ))^6 - 24 p _ 2^2 (2 p _ 2^2 - 3 s) (m _ π^(ó  ))^4 + 3 p _ 2^2 (7 t^2 + 10 u t + 7 u^2 + 4 (s - p _ 2^2) p _ 2^2) (m _ π^(ó  ))^2) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (4 (2 p _ 2^2 + 3 ((m _ K^(ó  ))^2 - 4 s)) (m _ π^(ó  ))^4 + (20 (m _ K^(ó  ))^2 (3 (m _ K^(ó  ))^2 + 6 s - 4 p _ 2^2) - 9 (t - u)^2) (m _ π^(ó  ))^2 + 36 (m _ K^(ó  ))^2 ((t - u)^2 + 4 (p _ 2^2 - s) (m _ K^(ó  ))^2)))) - 1/(π^2 ((m _ K^(ó  ))^2 - p _ 2^2)) (3 log((m _ π^(ó  ))^2/μ^2) (6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-48 (m _ π^(ó  ))^6 + 4 (-13 (m _ K^(ó  ))^2 + 14 s - 8 p _ 2^2) (m _ π^(ó  ))^4 + (-8 (m _ K^(ó  ))^4 + 4 (s - 6 p _ 2^2) (m _ K^(ó  ))^2 + 17 t^2 + 17 u^2 + 14 t u + 4 (7 s - 3 p _ 2^2) p _ 2^2) (m _ π^(ó  ))^2 - 8 s (m _ K^(ó  ))^2 (-(m _ K^(ó  ))^2 + s + p _ 2^2)) + c _ 2^( ) (-36 s (m _ π^(ó  ))^6 - 64 p _ 2^6 (m _ π^(ó  ))^2 - (-300 (m _ π^(ó  ))^4 + 116 s (m _ π^(ó  ))^2 + 49 t^2 + 49 u^2 + 142 t u) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 4 (17 (m _ π^(ó  ))^2 + 6 s) (m _ K^(ó  ))^6 + 8 (9 s - 23 (m _ π^(ó  ))^2) (s - (m _ π^(ó  ))^2) (m _ K^(ó  ))^4 + 8 p _ 2^4 (-34 (m _ π^(ó  ))^4 + (14 s - 5 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 3 s (m _ K^(ó  ))^2) + p _ 2^2 (-300 (m _ π^(ó  ))^6 + 8 (11 (m _ K^(ó  ))^2 + 19 s) (m _ π^(ó  ))^4 + (49 t^2 + 142 u t + 49 u^2 + 36 ((m _ K^(ó  ))^4 + 4 s (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 - 24 s (m _ K^(ó  ))^2 (2 (m _ K^(ó  ))^2 + 3 s)))))))

finalLogsS = FullSimplify /@ (Collect[Expand[finalLogs] /. MandelstamT * MandelstamU -> ((ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)^2 - MandelstamT^2 - MandelstamU^2)/2, {_Log, _k}] /. MandelstamT | MandelstamU -> 0 /. b_Log * a___ :> b * FullSimplify[Times[a]]) // Simplify

1/(2304 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (96 k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + 6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + 96 k((m _ K^(ó  ))^2, (m _ η^(ó  ))^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (5 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + 6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) + 1/(π^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) (log((m _ η^(ó  ))^2/μ^2) (c _ 2^( ) (-196 (m _ K^(ó  ))^8 + (-163 (m _ π^(ó  ))^2 + 376 s + 28 p _ 2^2) (m _ K^(ó  ))^6 + (140 (m _ π^(ó  ))^4 - 234 s (m _ π^(ó  ))^2 + 60 s^2 + 180 p _ 2^4 + p _ 2^2 (289 (m _ π^(ó  ))^2 - 384 s)) (m _ K^(ó  ))^4 - (24 (m _ π^(ó  ))^6 + 8 (19 p _ 2^2 - 3 s) (m _ π^(ó  ))^4 + 3 (5 s^2 - 52 p _ 2^2 s + 43 p _ 2^4) (m _ π^(ó  ))^2 + 12 p _ 2^2 (5 s^2 - 6 p _ 2^2 s + p _ 2^4)) (m _ K^(ó  ))^2 - 4 (s - 6 p _ 2^2) (m _ π^(ó  ))^6 + 12 p _ 2^2 (s + p _ 2^2) (m _ π^(ó  ))^4 + 3 p _ 2^2 (5 s^2 - 6 p _ 2^2 s + p _ 2^4) (m _ π^(ó  ))^2) + 2 c _ 5^( ) (m _ π^(ó  ) - m _ K^(ó  )) (m _ π^(ó  ) + m _ K^(ó  )) (36 (m _ π^(ó  ))^6 + (-96 (m _ K^(ó  ))^2 - 84 s + 44 p _ 2^2) (m _ π^(ó  ))^4 + (-75 (m _ K^(ó  ))^4 + 2 (123 s - 103 p _ 2^2) (m _ K^(ó  ))^2 + 9 (s - p _ 2^2)^2) (m _ π^(ó  ))^2 - 36 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 + s - p _ 2^2)^2))) - 1/(π^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) (3 log((m _ π^(ó  ))^2/μ^2) (c _ 2^( ) (-4 (9 s + 4 p _ 2^2) (m _ π^(ó  ))^6 + 12 p _ 2^2 (p _ 2^2 - 11 s) (m _ π^(ó  ))^4 + p _ 2^2 (71 s^2 - 30 p _ 2^2 s + 7 p _ 2^4) (m _ π^(ó  ))^2 - 3 ((m _ π^(ó  ))^2 - 8 s) (m _ K^(ó  ))^6 - (100 (m _ π^(ó  ))^4 + 114 s (m _ π^(ó  ))^2 - 72 s^2 + p _ 2^2 (35 (m _ π^(ó  ))^2 + 48 s)) (m _ K^(ó  ))^4 + (16 (m _ π^(ó  ))^6 + 168 s (m _ π^(ó  ))^4 - 71 s^2 (m _ π^(ó  ))^2 + p _ 2^4 (31 (m _ π^(ó  ))^2 + 24 s) + p _ 2^2 (88 (m _ π^(ó  ))^4 + 144 s (m _ π^(ó  ))^2 - 72 s^2)) (m _ K^(ó  ))^2) - 6 c _ 5^( ) (m _ π^(ó  ) - m _ K^(ó  )) (m _ π^(ó  ) + m _ K^(ó  )) (20 (m _ π^(ó  ))^6 + 4 (6 (m _ K^(ó  ))^2 - 7 s + p _ 2^2) (m _ π^(ó  ))^4 + ((m _ K^(ó  ))^4 + 10 (s + p _ 2^2) (m _ K^(ó  ))^2 - 7 s^2 + 5 p _ 2^4 - 14 s p _ 2^2) (m _ π^(ó  ))^2 + 8 s (m _ K^(ó  ))^2 (-(m _ K^(ó  ))^2 + s + p _ 2^2)))) + 1/(π^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) (6 log((m _ K^(ó  ))^2/μ^2) (c _ 2^( ) (-21 (m _ K^(ó  ))^8 + (166 (m _ π^(ó  ))^2 - 141 s + 59 p _ 2^2) (m _ K^(ó  ))^6 + (-148 (m _ π^(ó  ))^4 + 5 (27 s - 44 p _ 2^2) (m _ π^(ó  ))^2 + 2 s^2 - 43 p _ 2^4 + 162 s p _ 2^2) (m _ K^(ó  ))^4 + (5 p _ 2^6 - 9 (s - 6 (m _ π^(ó  ))^2) p _ 2^4 - 2 (-74 (m _ π^(ó  ))^4 + 72 s (m _ π^(ó  ))^2 + s^2) p _ 2^2 - 9 s^2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 - 3 s p _ 2^2 (p _ 2^2 - 3 s) (m _ π^(ó  ))^2) - 6 c _ 5^( ) (m _ π^(ó  ) - m _ K^(ó  )) (m _ π^(ó  ) + m _ K^(ó  )) ((2 (m _ K^(ó  ))^4 + (7 s + 2 p _ 2^2) (m _ K^(ó  ))^2 + 3 s (s + p _ 2^2)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (11 (m _ K^(ó  ))^4 - 15 s (m _ K^(ó  ))^2 - 4 s^2 + p _ 2^4 - 3 s p _ 2^2))))))

finalLogsTU = Simplify /@ Collect[Expand[Collect[Expand[finalLogs] /. MandelstamT * MandelstamU -> ((ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)^2 - MandelstamT^2 - MandelstamU^2)/2, {_Log, _k}] - finalLogsS], {_Log, _k}]

(i (t^2 + u^2) log((m _ π^(ó  ))^2/μ^2) (11 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + 30 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) (m _ π^(ó  ))^2)/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - (i (t^2 + u^2) log((m _ η^(ó  ))^2/μ^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) ((m _ K^(ó  ))^2 - p _ 2^2)))/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - (i (t^2 + u^2) log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (2 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + 3 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)))/(96 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))

Loop functions:

finaljbars2KLMs = jbars2KLMs /. l : LeutwylerJBar[MandelstamS | s, __] * _ :> (l /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[]) // FullSimplify

1/(864 (f _ ϕ^(ó  ))^4) (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 + 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) + 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) + 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) + 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) - 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) - 108 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (3 (m _ K^(ó  ))^2 - 3 s + p _ 2^2) + 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)) + 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)) + 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)) + 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)) + 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 - 4 (t + 6 u) (m _ K^(ó  ))^2 + 2 u (t + 23 u) + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 u)) + 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 - 4 (6 t + u) (m _ K^(ó  ))^2 + 2 t (23 t + u) + p _ 2^2 (35 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^2 - 16 t)) + 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 - 60 (7 t + 8 u) (m _ K^(ó  ))^2 + 18 u (5 t + 7 u) - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 u)) + 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 - 60 (8 t + 7 u) (m _ K^(ó  ))^2 + 18 t (7 t + 5 u) - 3 p _ 2^2 (19 (m _ π^(ó  ))^2 - 91 (m _ K^(ó  ))^2 + 24 t))))

finaljbars5KLMs = jbars5KLMs /. l : LeutwylerJBar[MandelstamS | s, __] * _ :> (l /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[]) // FullSimplify

-1/(144 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-8 Overscript[J, _] _ (m _ η^(ó  ))^2(s) (-9 (m _ K^(ó  ))^2 + 9 s + p _ 2^2) (m _ π^(ó  ))^2 - 108 s Overscript[J, _] _ (m _ K^(ó  ))^2(s) (-(m _ K^(ó  ))^2 + s + p _ 2^2) - 72 Overscript[J, _] _ (m _ π^(ó  ))^2(s) (2 s - (m _ π^(ó  ))^2) (-(m _ K^(ó  ))^2 + s + p _ 2^2) + 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) + 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) + 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) + 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) - Overscript[J, _] _ ((m _ K^(ó  ))^2 (m _ η^(ó  ))^2)(t) (5 (m _ π^(ó  ))^4 - 3 (8 t + 4 u - 7 p _ 2^2) (m _ π^(ó  ))^2 - 46 (m _ K^(ó  ))^4 + (-31 (m _ π^(ó  ))^2 + 24 t + 84 u - 45 p _ 2^2) (m _ K^(ó  ))^2 + 18 t (t - u)) + 12 K[u, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (2 (m _ K^(ó  ))^4 - ((m _ π^(ó  ))^2 + t - 5 p _ 2^2) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^2 (-7 (m _ π^(ó  ))^2 + t + p _ 2^2)) + 12 K[t, (m _ K^(ó  ))^2, (m _ η^(ó  ))^2] (2 (m _ K^(ó  ))^4 - ((m _ π^(ó  ))^2 + u - 5 p _ 2^2) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^2 (-7 (m _ π^(ó  ))^2 + u + p _ 2^2)) + 36 K[u, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (2 (m _ K^(ó  ))^4 - ((m _ π^(ó  ))^2 + t - p _ 2^2) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^2 (-3 (m _ π^(ó  ))^2 + t + p _ 2^2)) + 36 K[t, (m _ π^(ó  ))^2, (m _ K^(ó  ))^2] (2 (m _ K^(ó  ))^4 - ((m _ π^(ó  ))^2 + u - p _ 2^2) (m _ K^(ó  ))^2 + (m _ π^(ó  ))^2 (-3 (m _ π^(ó  ))^2 + u + p _ 2^2)) + 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 + 6 (m _ K^(ó  ))^4 - 4 (t - 4 u) (m _ K^(ó  ))^2 + 2 (t - 9 u) u + p _ 2^2 (-13 (m _ π^(ó  ))^2 - 3 (m _ K^(ó  ))^2 + 8 u)) - 9 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ K^(ó  ))^2)(t) (-3 (m _ π^(ó  ))^4 + (25 (m _ K^(ó  ))^2 - 24 t + 4 u) (m _ π^(ó  ))^2 + 2 t (9 t - u) - 2 (m _ K^(ó  ))^2 (3 (m _ K^(ó  ))^2 + 8 t - 2 u) + p _ 2^2 (13 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 - 8 t)) + 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)))

Next to leading order polynomial:

finallooppolys = finallooppolys2 + finallooppolys5 /. MandelstamT * MandelstamU -> ((ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)^2 - MandelstamT^2 - MandelstamU^2)/2 // Simplify

1/(1152 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (78 (m _ π^(ó  ))^4 - (17 (m _ K^(ó  ))^2 + 237 s) (m _ π^(ó  ))^2 - 16 (m _ K^(ó  ))^4 + 87 s^2 - 3 t^2 - 3 u^2 + 3 p _ 2^4 + 111 s (m _ K^(ó  ))^2 + p _ 2^2 (37 (m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2 - 36 s)) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (3 p _ 2^4 + (-7 (m _ π^(ó  ))^2 - 9 (m _ K^(ó  ))^2 + 54 s) p _ 2^2 + 3 (2 (m _ π^(ó  ))^4 - (7 (m _ K^(ó  ))^2 + 3 s) (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^4 + 23 s^2 - t^2 - u^2 - 9 s (m _ K^(ó  ))^2))))

finallooppolysS = finallooppolys /. MandelstamT | MandelstamU -> 0 // Simplify

1/(1152 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (78 (m _ π^(ó  ))^4 - (17 (m _ K^(ó  ))^2 + 237 s) (m _ π^(ó  ))^2 - 16 (m _ K^(ó  ))^4 + 87 s^2 + 3 p _ 2^4 + 111 s (m _ K^(ó  ))^2 + p _ 2^2 (37 (m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2 - 36 s)) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (3 p _ 2^4 + (-7 (m _ π^(ó  ))^2 - 9 (m _ K^(ó  ))^2 + 54 s) p _ 2^2 + 3 (2 (m _ π^(ó  ))^4 - (7 (m _ K^(ó  ))^2 + 3 s) (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^4 + 23 s^2 - 9 s (m _ K^(ó  ))^2))))

finallooppolysTU = finallooppolys - finallooppolysS // Simplify

-(i (t^2 + u^2) (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/(384 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

Counterterms:

oldcts

-1/(3 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (-4 (6 N _ 5^( ) + 12 N _ 7^( ) + 3 N _ 8^( ) - 6 N _ 10^( ) - 14 N _ 11^( ) - 6 N _ 12^( )) (m _ π^(ó  ))^4 + 3 (s (4 N _ 5^( ) + 8 N _ 7^( ) + 3 N _ 8^( ) + 2 N _ 9^( )) - (N _ 8^( ) + 2 N _ 9^( )) p _ 2^2) (m _ π^(ó  ))^2 + 2 (3 N _ 5^( ) + 3 N _ 8^( ) - 8 (N _ 10^( ) + N _ 11^( ))) (m _ K^(ó  ))^4 + ((48 N _ 7^( ) - 21 N _ 8^( ) - 18 N _ 9^( ) - 8 (N _ 10^( ) + 5 N _ 11^( ) + 3 N _ 12^( ))) (m _ π^(ó  ))^2 + 6 s (N _ 5^( ) - 4 N _ 7^( ) + 3 N _ 8^( ) + 2 N _ 9^( )) - 6 (N _ 5^( ) + N _ 8^( )) p _ 2^2) (m _ K^(ó  ))^2 - (2 ((m _ K^(ó  ))^2 + 3 s - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (N _ 11^( ) (m _ π^(ó  ))^2 + 2 (N _ 10^( ) + N _ 11^( )) (m _ K^(ó  ))^2))/(p _ 2^2 - (m _ K^(ó  ))^2)))

newcts1 = newcts /. MandelstamT * MandelstamU -> ((ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 + Pair[Momentum[p2], Momentum[p2]] - MandelstamS)^2 - MandelstamT^2 - MandelstamU^2)/2 // Expand // FullSimplify

1/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) ((2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^6 - (2 N _ 21^( ) (4 (m _ π^(ó  ))^2 + s) + (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + 2 p _ 2^2)) (m _ K^(ó  ))^4 - (2 N _ 21^( ) (p _ 2^2 (s + p _ 2^2) - 2 (2 s + p _ 2^2) (m _ π^(ó  ))^2) - (N _ 22^( ) + 2 N _ 23^( )) (s + p _ 2^2) (4 (m _ π^(ó  ))^2 + p _ 2^2)) (m _ K^(ó  ))^2 - 4 (s (N _ 22^( ) + 2 N _ 23^( )) + N _ 21^( ) (s - p _ 2^2)) p _ 2^2 (m _ π^(ó  ))^2 + N _ 19^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (2 (m _ π^(ó  ))^4 - 6 s (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 + 2 s^2 - t^2 - u^2 + p _ 2^4 - 3 (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 - 3 p _ 2^2 (s - 2 (m _ π^(ó  ))^2)) + N _ 20^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (-2 (m _ π^(ó  ))^4 + 6 s (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4 - 2 s^2 + t^2 + u^2 + p _ 2^4 + 3 (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + p _ 2^2 (2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 3 s))))

newcts1S = newcts1 /. MandelstamT | MandelstamU -> 0 // FullSimplify

1/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) ((2 N _ 21^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ K^(ó  ))^6 - (2 N _ 21^( ) (4 (m _ π^(ó  ))^2 + s) + (N _ 22^( ) + 2 N _ 23^( )) (4 (m _ π^(ó  ))^2 + s + 2 p _ 2^2)) (m _ K^(ó  ))^4 - (2 N _ 21^( ) (p _ 2^2 (s + p _ 2^2) - 2 (2 s + p _ 2^2) (m _ π^(ó  ))^2) - (N _ 22^( ) + 2 N _ 23^( )) (s + p _ 2^2) (4 (m _ π^(ó  ))^2 + p _ 2^2)) (m _ K^(ó  ))^2 - 4 (s (N _ 22^( ) + 2 N _ 23^( )) + N _ 21^( ) (s - p _ 2^2)) p _ 2^2 (m _ π^(ó  ))^2 + N _ 19^( ) (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^4 - 3 (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + p _ 2^4 - 3 p _ 2^2 (s - 2 (m _ π^(ó  ))^2) + 2 ((m _ π^(ó  ))^4 - 3 s (m _ π^(ó  ))^2 + s^2)) + N _ 20^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (-(m _ K^(ó  ))^4 + 3 (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + p _ 2^4 - 2 ((m _ π^(ó  ))^4 - 3 s (m _ π^(ó  ))^2 + s^2) + p _ 2^2 (2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 3 s))))

newcts1TU = newcts1 - newcts1S // Simplify

-(i (t^2 + u^2) c _ 2^( ) (N _ 19^( ) - N _ 20^( )))/(2 (f _ ϕ^(ó  ))^4)

strongcts

-1/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (4 i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (10 L _ 3^( ) (m _ π^(ó  ))^4 - 6 s L _ 3^( ) (m _ π^(ó  ))^2 - 6 t L _ 3^( ) (m _ π^(ó  ))^2 - 6 u L _ 3^( ) (m _ π^(ó  ))^2 - 10 s L _ 4^( ) (m _ π^(ó  ))^2 - 4 s L _ 5^( ) (m _ π^(ó  ))^2 + 4 s L _ 6^( ) (m _ π^(ó  ))^2 + 4 s L _ 8^( ) (m _ π^(ó  ))^2 + 6 L _ 3^( ) p _ 2^2 (m _ π^(ó  ))^2 + 6 L _ 4^( ) p _ 2^2 (m _ π^(ó  ))^2 + 4 L _ 6^( ) p _ 2^2 (m _ π^(ó  ))^2 + 4 L _ 8^( ) p _ 2^2 (m _ π^(ó  ))^2 + 2 (2 L _ 4^( ) + L _ 5^( ) - 4 L _ 6^( ) - 2 L _ 8^( )) (m _ K^(ó  ))^4 + 3 L _ 3^( ) p _ 2^4 + (4 (2 L _ 5^( ) - 9 L _ 6^( ) - 5 L _ 8^( )) (m _ π^(ó  ))^2 + L _ 3^( ) p _ 2^2 - 2 (L _ 5^( ) - 4 L _ 6^( ) - 2 L _ 8^( )) (s + p _ 2^2) - 2 L _ 4^( ) (-13 (m _ π^(ó  ))^2 + 6 s + 2 p _ 2^2)) (m _ K^(ó  ))^2 + 3 s t L _ 3^( ) + 3 s u L _ 3^( ) + 2 t u L _ 3^( ) - 3 s L _ 3^( ) p _ 2^2 - 3 t L _ 3^( ) p _ 2^2 - 3 u L _ 3^( ) p _ 2^2 + 8 L _ 1^( ) (s - 2 (m _ π^(ó  ))^2) ((m _ K^(ó  ))^2 - s + p _ 2^2) + 4 L _ 2^( ) (-2 (m _ π^(ó  ))^4 + 2 s (m _ π^(ó  ))^2 - s^2 + (s - 2 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 + 2 t u + p _ 2^2 (-2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + s))))

finalall = finallows1 + finalLogs + finaljbars2KLMs + finaljbars5KLMs + finallooppolys + oldcts + newcts1 + strongcts (* /. MrToJBar /. KLToJBar /. k -> kk *) ;

Converted by Mathematica  (July 10, 2003)