•The log part of the loops

The logs are seen to cancel exactly the above logs coming from the expansion of the Overscript[J, _]'s:

tmp2 = (((MandelstamT^2 * MandelstamU^2 * looplogs2 /. gellmannOkubo /. MandelstamS -> -MandelstamU - MandelstamT + ParticleMass[Kaon]^2 + Pair[Momentum[p2], Momentum[p2]] + 2 ParticleMass[Pion]^2 // Expand) /. (MandelstamT)^(_ ? ((# > 1) &)) (MandelstamU)^(_ ? ((# > 1) &)) -> 0 // Simplify) // Simplify)

1/(384 π^2 (f _ ϕ^(ó  ))^4) (i t u (t + u) c _ 2^( ) (p _ 2^2 - (m _ π^(ó  ))^2) (m _ K^(ó  ))^2 ((log((m _ π^(ó  ))^2/μ^2) - 6 log((m _ K^(ó  ))^2/μ^2) + 5 log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ π^(ó  ))^2 + 20 (log((3 (m _ K^(ó  ))^2)/μ^2) - log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/μ^2)) (m _ K^(ó  ))^2))

(tmp1 + tmp2 /. cancelLogs /. _RenormalizationState -> Sequence[] /. {s -> MandelstamS, t -> MandelstamT, u -> MandelstamU} // Expand // Simplify) //. {Log[a_] + Log[b_] :> Log[a b], Log[a_] - Log[b_] :> Log[a /b]} // Simplify

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Converted by Mathematica  (July 10, 2003)