•Reduction of the loop amplitude

Changing to Mandelstam variables:

res5 = CheckF[((WriteString["stdout", "."] ; ((# /. RenormalizationState[1] -> RenormalizationState[0]) // MomentumExpand // ExpandScalarProduct) /. MandelstamRules) // MomentumCombine // Simplify) & /@ (res4 * Pair[LorentzIndex[μ1], Momentum[Polarization[p1, i]]]), "KSPiPiAmpsSS.m"] ;

................

LeafCount /@ res5

{67, 184, 6022, 5411, 271, 156, 269, 1539, 13798, 13798, 6146, 5510, 261, 13391, 13391, 1546}

Some of the diagrams are related by crossing:

Collect[res5[[14]] - (res5[[15]] /. {p3 -> p4, p4 -> p3, MandelstamT -> MandelstamU, MandelstamU -> MandelstamT}), _B0] // Cancel

0

Changing from B0s to Overscript[J, _]'s and λ's:

ampinfinities = ((WriteString["stdout", "."] ; VeltmanExpand[Collect[# /. D -> Sequence[], {_B0, _A0}], ExplicitLeutwylerJ0 -> True]) & /@ res5) ;

................

If[# === 0, Null, Length[Cases[#, _LeutwylerJBar | _LeutwylerLambda, Infinity, Heads -> True]]] & /@ ampinfinities

{3, 8, 9, 9, 3, 3, 3, 9, 9, 9, 9, 9, 8, 9, 9, 9}

LeafCount /@ ampinfinities

{244, 440, 5216, 5200, 471, 388, 545, 1319, 10756, 10756, 4780, 4770, 620, 8610, 8610, 1388}

We divide off the kaon propagator and put the kaon on-mass-shell:

ampinfonshell = CheckF[(WriteString["stdout", "."] ; Simplify[Cancel[# * (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon, RenormalizationState[0]]^2)] /. {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[0]]^2} /. gellmannOkubo]) & /@ ampinfinities, "KSPiPiLoopsOnShellS.m"] ;

................

LeafCount /@ ampinfonshell

{1, 1, 1, 1, 629, 1, 822, 1680, 5319, 5319, 1, 1, 1, 4953, 4951, 2064}

ampinfonshell // Length

16

endloops = CheckF[Plus @@ ampinfonshell // Simplify, "KSPiPiLoopsOnShellfinS.m"] ;

The coefficient on the λ's:

looplambdas = Simplify[Coefficient[#, LeutwylerLambda[]] /. cancelU /. RenormalizationState[1] -> RenormalizationState[0]] & /@ ampinfonshell

{0, 0, 0, 0, 1/(270 (f _ ϕ^(ó  ))^4) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (8 c _ 5^( ) (43 (m _ π^(ó  ))^4 + 28 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 71 (m _ K^(ó  ))^4) + 5 c _ 2^( ) (-232 (m _ π^(ó  ))^4 + (-430 (m _ K^(ó  ))^2 + 63 s - 9 p _ 1^2 - 9 p _ 2^2) (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 (196 (m _ K^(ó  ))^2 + 171 s - 63 p _ 1^2 - 63 p _ 2^2)))), 0, (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (150 (m _ π^(ó  ))^4 + ((m _ K^(ó  ))^2 - 60 s + 3 p _ 1^2 + 6 p _ 2^2) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (77 (m _ K^(ó  ))^2 - 165 s + 36 p _ 1^2 + 87 p _ 2^2)))/(135 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)), 1/(36 (f _ ϕ^(ó  ))^4) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (24 c _ 5^( ) (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 7 s) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (224 (m _ π^(ó  ))^4 + 16 ((m _ K^(ó  ))^2 - 24 s) (m _ π^(ó  ))^2 - 3 p _ 2^2 (-16 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 9 s) + 3 (-6 (m _ K^(ó  ))^4 + 35 s (m _ K^(ó  ))^2 + 27 s^2)))), -1/(432 (f _ ϕ^(ó  ))^4) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (592 (m _ π^(ó  ))^4 - 1493 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 1404 s (m _ π^(ó  ))^2 - 1152 t (m _ π^(ó  ))^2 + 613 (m _ K^(ó  ))^4 + 540 s^2 + 432 t^2 + 405 p _ 1^4 + 324 p _ 2^4 + 1269 s (m _ K^(ó  ))^2 + 585 t (m _ K^(ó  ))^2 + 972 s t - 9 p _ 2^2 (-132 (m _ π^(ó  ))^2 + 101 (m _ K^(ó  ))^2 + 96 s + 84 t) - 3 p _ 1^2 (-427 (m _ π^(ó  ))^2 + 238 (m _ K^(ó  ))^2 + 315 s + 279 t - 243 p _ 2^2)) - 24 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2 + 6 s + 6 t - 5 p _ 1^2 - 6 p _ 2^2))), 1/(432 (f _ ϕ^(ó  ))^4) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-16 (m _ π^(ó  ))^4 + 323 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 s (m _ π^(ó  ))^2 + 576 t (m _ π^(ó  ))^2 - 613 (m _ K^(ó  ))^4 - 432 t^2 - 684 s (m _ K^(ó  ))^2 + 585 t (m _ K^(ó  ))^2 + 108 s t + 36 p _ 2^2 (-7 (m _ π^(ó  ))^2 + 9 (m _ K^(ó  ))^2 + 3 t) + 3 p _ 1^2 (-61 (m _ π^(ó  ))^2 + 43 (m _ K^(ó  ))^2 + 9 t)) - 24 c _ 5^( ) (-14 (m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2 + 6 t - p _ 1^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))), 0, 0, 0, 1/(216 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (712 (m _ π^(ó  ))^4 - 923 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 1188 s (m _ π^(ó  ))^2 - 1296 t (m _ π^(ó  ))^2 + 427 (m _ K^(ó  ))^4 + 540 s^2 + 648 t^2 + 495 p _ 1^4 + 396 p _ 2^4 + 687 s (m _ K^(ó  ))^2 + 1227 t (m _ K^(ó  ))^2 + 1188 s t - 3 p _ 2^2 (-348 (m _ π^(ó  ))^2 + 265 (m _ K^(ó  ))^2 + 312 s + 348 t) - 3 p _ 1^2 (-381 (m _ π^(ó  ))^2 + 302 (m _ K^(ó  ))^2 + 345 s + 381 t - 317 p _ 2^2))), 1/(216 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (712 (m _ π^(ó  ))^4 + 1531 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 108 s (m _ π^(ó  ))^2 - 1296 t (m _ π^(ó  ))^2 + 427 (m _ K^(ó  ))^4 + 648 t^2 - 540 s (m _ K^(ó  ))^2 - 1227 t (m _ K^(ó  ))^2 + 108 s t - 36 p _ 2^2 (-7 (m _ π^(ó  ))^2 - 12 (m _ K^(ó  ))^2 + 7 t) + p _ 1^2 (153 (m _ π^(ó  ))^2 + 321 (m _ K^(ó  ))^2 - 153 t + 60 p _ 2^2))), 1/(36 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (32 (m _ π^(ó  ))^4 - 88 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 80 s (m _ π^(ó  ))^2 - 18 (m _ K^(ó  ))^4 + 126 s^2 + 3 s (m _ K^(ó  ))^2 + p _ 1^2 (8 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 - 21 s) + p _ 2^2 (16 (m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^2 - 42 s)))}

looplambdas /. CouplingConstant[ChPTW3[2], 2] -> 0 // Simplify

{0, 0, 0, 0, -(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (232 (m _ π^(ó  ))^4 + (430 (m _ K^(ó  ))^2 - 63 s + 9 p _ 1^2 + 9 p _ 2^2) (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 (-196 (m _ K^(ó  ))^2 - 171 s + 63 p _ 1^2 + 63 p _ 2^2)))/(54 (f _ ϕ^(ó  ))^4), 0, 0, (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (224 (m _ π^(ó  ))^4 + 16 ((m _ K^(ó  ))^2 - 24 s) (m _ π^(ó  ))^2 - 3 p _ 2^2 (-16 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 9 s) + 3 (-6 (m _ K^(ó  ))^4 + 35 s (m _ K^(ó  ))^2 + 27 s^2)))/(36 (f _ ϕ^(ó  ))^4), -1/(432 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (592 (m _ π^(ó  ))^4 - 1493 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 1404 s (m _ π^(ó  ))^2 - 1152 t (m _ π^(ó  ))^2 + 613 (m _ K^(ó  ))^4 + 540 s^2 + 432 t^2 + 405 p _ 1^4 + 324 p _ 2^4 + 1269 s (m _ K^(ó  ))^2 + 585 t (m _ K^(ó  ))^2 + 972 s t - 9 p _ 2^2 (-132 (m _ π^(ó  ))^2 + 101 (m _ K^(ó  ))^2 + 96 s + 84 t) - 3 p _ 1^2 (-427 (m _ π^(ó  ))^2 + 238 (m _ K^(ó  ))^2 + 315 s + 279 t - 243 p _ 2^2))), 1/(432 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-16 (m _ π^(ó  ))^4 + 323 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 s (m _ π^(ó  ))^2 + 576 t (m _ π^(ó  ))^2 - 613 (m _ K^(ó  ))^4 - 432 t^2 - 684 s (m _ K^(ó  ))^2 + 585 t (m _ K^(ó  ))^2 + 108 s t + 36 p _ 2^2 (-7 (m _ π^(ó  ))^2 + 9 (m _ K^(ó  ))^2 + 3 t) + 3 p _ 1^2 (-61 (m _ π^(ó  ))^2 + 43 (m _ K^(ó  ))^2 + 9 t))), 0, 0, 0, 0, 0, 0}

(looplambdas[[3]]) - (looplambdas[[4]] /. MandelstamT -> MandelstamU /. cancelU /. RenormalizationState[1] -> RenormalizationState[0]) // Simplify

0

(looplambdas[[12]]) - (looplambdas[[13]] /. MandelstamT -> MandelstamU /. cancelU /. RenormalizationState[1] -> RenormalizationState[0]) // Simplify

0

LOOPlambdaCoeff = Plus @@ looplambdas /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[0]]^2 // FullSimplify

1/(108 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (56 (m _ π^(ó  ))^4 + (-742 (m _ K^(ó  ))^2 - 666 s + 432 t) (m _ π^(ó  ))^2 + 281 (m _ K^(ó  ))^4 - 81 p _ 2^4 + 9 (83 s + 24 t) (m _ K^(ó  ))^2 + 108 (s^2 - 2 t s - 2 t^2) + 9 p _ 2^2 (-26 (m _ π^(ó  ))^2 - 16 (m _ K^(ó  ))^2 + 15 s + 24 t)) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (284 (m _ π^(ó  ))^4 - 4 (-95 (m _ K^(ó  ))^2 + 99 s + 162 t) (m _ π^(ó  ))^2 - 25 (m _ K^(ó  ))^4 + 81 p _ 2^4 - 9 (43 s + 36 t) (m _ K^(ó  ))^2 + 324 (s^2 + t s + t^2) + p _ 2^2 (316 (m _ π^(ó  ))^2 + 236 (m _ K^(ó  ))^2 - 27 (s + 12 t)))))

Converted by Mathematica  (July 10, 2003)