•Loop log and polynomial pieces

finallooplogs2 = ((((((looplogs2coll + jbars2KLMLogs) /. gellmannOkubo // Expand) /. _k -> 0 /. toEtaRules /. cancelLogs /. Log -> log // FullSimplify) /. tuRule) // FullSimplify) /. MandelstamT^2 * a_ + MandelstamU^2 * a_ -> (MandelstamT^2 + MandelstamU^2) * a) + Simplify[Plus @@ (# * Coefficient[jbars2KLMLogs, #] & /@ Union[Cases[jbars2KLMLogs, _k, Infinity]])] /. log -> Log

(i c _ 2^( ) (k((m _ π^(ó  ))^2, (m _ K^(ó  ))^2) + 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))/(24 (f _ ϕ^(ó  ))^4) + 1/(6912 π^2 (f _ ϕ^(ó  ))^4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) (-18 log((m _ K^(ó  ))^2/μ^2) (13 (m _ K^(ó  ))^6 + (-92 (m _ π^(ó  ))^2 + 107 s - 20 p _ 2^2) (m _ K^(ó  ))^4 - (-100 (m _ π^(ó  ))^4 + 113 s (m _ π^(ó  ))^2 - 7 p _ 2^4 + 4 t u + 10 (t^2 + u^2) + p _ 2^2 (13 s - 48 (m _ π^(ó  ))^2)) (m _ K^(ó  ))^2 + 3 s (3 s - p _ 2^2) (m _ π^(ó  ))^2) - log((m _ η^(ó  ))^2/μ^2) (-40 (m _ π^(ó  ))^6 + 18 (9 s - 7 p _ 2^2) (m _ π^(ó  ))^4 + 9 (10 t u + 7 (t^2 + u^2) + 4 (s - p _ 2^2) p _ 2^2) (m _ π^(ó  ))^2 + 760 (m _ K^(ó  ))^6 + 6 (241 (m _ π^(ó  ))^2 - 252 s + 156 p _ 2^2) (m _ K^(ó  ))^4 - 6 (37 (m _ π^(ó  ))^4 - 9 (s + 3 p _ 2^2) (m _ π^(ó  ))^2 + 6 (10 t u + 7 (t^2 + u^2) + 4 (s - p _ 2^2) p _ 2^2)) (m _ K^(ó  ))^2) + 9 log((m _ π^(ó  ))^2/μ^2) (-160 (m _ π^(ó  ))^6 + (-362 (m _ K^(ó  ))^2 + 38 s - 234 p _ 2^2) (m _ π^(ó  ))^4 + (-30 (m _ K^(ó  ))^4 + 370 s (m _ K^(ó  ))^2 - 64 p _ 2^4 + 142 t u + 49 (t^2 + u^2) + 2 p _ 2^2 (56 s - 71 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 - 24 s (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 + 3 s - p _ 2^2))))

finallooplogs5 = ((((((looplogs5coll + jbars5KLMLogs) /. gellmannOkubo // Expand) /. _k -> 0 /. toEtaRules /. cancelLogs /. Log -> log // FullSimplify) /. tuRule) // FullSimplify) /. MandelstamT^2 * a_ + MandelstamU^2 * a_ -> (MandelstamT^2 + MandelstamU^2) * a) + Simplify[Plus @@ (# * Coefficient[jbars5KLMLogs, #] & /@ Union[Cases[jbars5KLMLogs, _k, Infinity]])] /. log -> Log

(i c _ 5^( ) (k((m _ π^(ó  ))^2, (m _ K^(ó  ))^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)^2)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + 1/(3456 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) (18 log((m _ K^(ó  ))^2/μ^2) (33 (m _ K^(ó  ))^6 + (66 (m _ π^(ó  ))^2 - 49 s + 36 p _ 2^2) (m _ K^(ó  ))^4 - (-48 (m _ π^(ó  ))^4 + (47 s - 42 p _ 2^2) (m _ π^(ó  ))^2 + 3 (-5 p _ 2^4 + 11 s p _ 2^2 + 8 t u + 6 (t^2 + u^2))) (m _ K^(ó  ))^2 + 9 s (s + p _ 2^2) (m _ π^(ó  ))^2) + 9 log((m _ π^(ó  ))^2/μ^2) (-152 (m _ π^(ó  ))^6 + 4 (-25 (m _ K^(ó  ))^2 + 28 s - 36 p _ 2^2) (m _ π^(ó  ))^4 + (-60 (m _ K^(ó  ))^4 + 56 s (m _ K^(ó  ))^2 - 36 p _ 2^4 + 42 t u + 51 (t^2 + u^2) + 12 p _ 2^2 (7 s - 3 (m _ K^(ó  ))^2)) (m _ π^(ó  ))^2 - 24 s (m _ K^(ó  ))^2 (-(m _ K^(ó  ))^2 + s + p _ 2^2)) + log((m _ η^(ó  ))^2/μ^2) (-8 (m _ π^(ó  ))^6 + 4 (3 (m _ K^(ó  ))^2 + 36 s - 4 p _ 2^2) (m _ π^(ó  ))^4 + (27 (t - u)^2 - 4 (m _ K^(ó  ))^2 (54 (m _ K^(ó  ))^2 + 99 s - 41 p _ 2^2)) (m _ π^(ó  ))^2 - 112 (m _ K^(ó  ))^6 + 64 (9 s - 4 p _ 2^2) (m _ K^(ó  ))^4 - 108 (t - u)^2 (m _ K^(ó  ))^2)))

finallooppolys2 = ((polys2 + jbars2KLMPolys // FullSimplify) /. tuRule /. MandelstamT * a_ + MandelstamU * a_ -> (MandelstamT + MandelstamU /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[]) * a // FullSimplify) /. MandelstamT^2 * a_ + MandelstamU^2 * a_ -> (MandelstamT^2 + MandelstamU^2) * a

1/(1152 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (-103 (m _ K^(ó  ))^4 + 5 (57 s - 73 (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 - 84 p _ 2^4 + 3 (-90 (m _ π^(ó  ))^4 + 37 s (m _ π^(ó  ))^2 + 58 t u + 28 (t^2 + u^2)) + p _ 2^2 (-311 (m _ π^(ó  ))^2 - 187 (m _ K^(ó  ))^2 + 138 s)))

finallooppolys5 = (((polys5 + jbars5KLMPolys // FullSimplify) /. tuRule // Simplify) //. {MandelstamT * a_ + MandelstamU * a_ -> (MandelstamT + MandelstamU) * a, MandelstamT^2 * a_ + MandelstamU^2 * a_ -> (MandelstamT^2 + MandelstamU^2) * a} /. tuRule // FullSimplify) /. MandelstamT^2 * a_ + MandelstamU^2 * a_ -> (MandelstamT^2 + MandelstamU^2) * a

1/(576 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (66 p _ 2^4 + (283 (m _ π^(ó  ))^2 + 147 (m _ K^(ó  ))^2 - 192 s) p _ 2^2 - 3 (-90 (m _ π^(ó  ))^4 + (89 s - 99 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 27 (m _ K^(ó  ))^4 + 37 s (m _ K^(ó  ))^2 + 46 t u + 22 (t^2 + u^2))))

Converted by Mathematica  (July 10, 2003)