•The kaon

Cosmetics:

LoadLagrangian[ChPT3[4]]

ff2 = amp2 /. udrules

(4 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) δ _ (7 I _ 1)^(3))/f _ ϕ^(ó  )

amploop = ampinfinities /. i3 -> I1 /. udrules // SUNReduce // Simplify

1/(144 π^2 (f _ ϕ^(ó  ))^3) ((c _ 2^( ) ((832 π^2 λ + 27 log((m _ π^(ó  ))^2/μ^2) - log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/μ^2) + log (3)) (m _ π^(ó  ))^4 + 8 (32 π^2 λ + log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 (544 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 8 log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ K^(ó  ))^4) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((256 π^2 λ + 9 log((m _ π^(ó  ))^2/μ^2) - log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/μ^2) + log (3)) (m _ π^(ó  ))^2 + 2 (352 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 2 log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))

ampwf4 = amp4 /. p2 -> -p1 /. udrules // SUNReduce // Simplify

(8 c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 N _ 10^( ) (m _ K^(ó  ))^2 - N _ 21^( ) p _ 1^2 + N _ 11^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))/(f _ ϕ^(ó  ))^3

ff4 = (amploop + ampwf4 + ff2 /. gellmannOkubo // ExpandAll // Simplify) /. etalogs

1/(144 π^2 (f _ ϕ^(ó  ))^3) ((c _ 2^( ) (832 π^2 λ (m _ π^(ó  ))^4 + 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - (log((m _ η^(ó  ))^2/μ^2) + log (3)) (m _ π^(ó  ))^4 + log (3) (m _ π^(ó  ))^4 + 2304 π^2 N _ 10^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 256 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2304 π^2 N _ 10^( ) (m _ K^(ó  ))^4 - 1088 π^2 λ (m _ K^(ó  ))^4 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^( ) p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 1152 π^2 N _ 11^( ) ((m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^4)) - c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-576 π^2 (f _ ϕ^(ó  ))^2 + (512 π^2 λ + 18 log((m _ π^(ó  ))^2/μ^2) - 2 (log((m _ η^(ó  ))^2/μ^2) + log (3)) + log (9)) (m _ π^(ó  ))^2 + 4 (352 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))

ff0 = Renormalize[ff4 /. CouplingConstant[c_[4], i_] -> CouplingConstant[c[4], i, RenormalizationState[0]]] // Simplify

1/(144 π^2 (f _ ϕ^(ó  ))^3) ((c _ 2^( ) (1152 π^2 N _ 11^(r ) (m _ π^(ó  ))^4 + 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1152 π^2 N _ 11^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 960 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^4 - 2304 π^2 N _ 11^(r ) (m _ K^(ó  ))^4 - 960 π^2 λ (m _ K^(ó  ))^4 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^(r ) p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 960 π^2 λ p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-288 π^2 (f _ ϕ^(ó  ))^2 + (256 π^2 λ + 9 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (352 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))

The coefficient c is the renormalization factor relating the unrenormalized leading order amplitude to the renormalized next to leading order amplitude:

c = ff0/ff2 // Simplify

(c _ 2^( ) (1152 π^2 N _ 11^(r ) (m _ π^(ó  ))^4 + 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1152 π^2 N _ 11^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 960 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^4 - 2304 π^2 N _ 11^(r ) (m _ K^(ó  ))^4 - 960 π^2 λ (m _ K^(ó  ))^4 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^(r ) p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 960 π^2 λ p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-288 π^2 (f _ ϕ^(ó  ))^2 + (256 π^2 λ + 9 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (352 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))/(576 π^2 c _ 5^( ) (f _ ϕ^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))

Coefficient[ff4, LeutwylerLambda[]] // Simplify

(4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))/(9 (f _ ϕ^(ó  ))^3)

Coefficient[ff0, LeutwylerLambda[]] // Simplify

-(4 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (15 c _ 2^( ) (p _ 1^2 - (m _ K^(ó  ))^2) + 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)) δ _ (7 I _ 1)^(3))/(9 (f _ ϕ^(ó  ))^3)

Coefficient[c, LeutwylerLambda[]] // Simplify

-(15 c _ 2^( ) (p _ 1^2 - (m _ K^(ó  ))^2) + 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2))/(9 c _ 5^( ) (f _ ϕ^(ó  ))^2)

$VeryVerbose = 2 ;

CheckF[c, ToFileName[{$FeynCalcDirectory, "Phi", "Factors"}, "ChPTW3P70S10o2.Fac"]] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTW3P70S10o2.Fac

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)