•Loop results

finalloops = Pair[LorentzIndex[μ1], Momentum[p1]] Pair[LorentzIndex[μ1], Momentum[Polarization[p1, i]]]/DecayConstant[PhiMeson, RenormalizationState[0]]^4 * Collect[Together[DecayConstant[PhiMeson, RenormalizationState[0]]^4/Pair[LorentzIndex[μ1], Momentum[p1]] /Pair[LorentzIndex[μ1], Momentum[Polarization[p1, i]]] * endloops /. LeutwylerJBar[a__] :> LeutwylerJBar[seq[a] /. {MandelstamS -> s, MandelstamT -> t, MandelstamU -> u}] /. seq -> Sequence /. _LeutwylerLambda -> 0 (* /. symmetrize *) /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. toEtaRules /. _RenormalizationState -> Sequence[]] // Expand, {_Log, _LeutwylerJBar}] /. {Log[a_] * b_ :> Log[a] * Simplify[b], LeutwylerJBar[a__] * b_ :> LeutwylerJBar[a] * Simplify[b]} ;

The Overscript[J, _] and polynomial parts of the loop contribution in proportion to c _ 2:

loopJs2 = Collect[finalloops /. CouplingConstant[ChPTW3[2], 2] -> 0 /. _Log -> 0, {_DecayConstant, _Pair, _LeutwylerJBar}] /. (LeutwylerJBar[a__] * (b__ )) :> LeutwylerJBar[a] * Collect[Times[b], {MandelstamS, MandelstamT, MandelstamU, _ParticleMass}] ;

These are the logs originally present in the loop contribution in proportion to c _ 2:

looplogs2 = Collect[Simplify[(finalloops /. Pair[_LorentzIndex, ___] -> Sequence[] /. CouplingConstant[ChPTW3[2], 2] -> 0 /. _LeutwylerJBar -> 0) - (finalloops /. Pair[_LorentzIndex, ___] -> Sequence[] /. CouplingConstant[ChPTW3[2], 2] -> 0 /. _LeutwylerJBar -> 0 /. _Log -> 0)], {_DecayConstant, _Log, (ParticleMass[Kaon, RenormalizationState[1]]^2 - Pair[Momentum[p2], Momentum[p2]]), (Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Kaon, RenormalizationState[1]]^2)}] /. Log[a_] * b__ :> Log[a] * Simplify[Collect[Times[b], {_ParticleMass, _Pair}]]

1/(f _ ϕ^(ó  ))^4 (1/(27648 t u π^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) log((m _ η^(ó  ))^2/μ^2) (-4 (180 (t + u) (m _ K^(ó  ))^2 + 89 t u) (m _ π^(ó  ))^6 - 6 (-510 (t + u) (m _ K^(ó  ))^4 - (30 t^2 + 307 u t + 30 u^2 + 30 s (t + u)) (m _ K^(ó  ))^2 + t u (65 s - 4 (t + u)) + p _ 2^2 (31 t u - 90 (t + u) (m _ K^(ó  ))^2)) (m _ π^(ó  ))^4 + 3 (-240 (t + u) (m _ K^(ó  ))^6 - 10 (24 t^2 + 65 u t + 24 u^2 + 24 s (t + u)) (m _ K^(ó  ))^4 + t u (131 (t + u) - 214 s) (m _ K^(ó  ))^2 - 3 t u (15 t^2 + 14 u t + 15 u^2 - 13 s (t + u)) + p _ 2^4 (60 (t + u) (m _ K^(ó  ))^2 - 24 t u) + p _ 2^2 (-780 (t + u) (m _ K^(ó  ))^4 - 10 (6 t^2 - 31 u t + 6 u^2 + 6 s (t + u)) (m _ K^(ó  ))^2 + 3 t u (8 s + 11 (t + u)))) (m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2 (-36 (2 t u - 5 (t + u) (m _ K^(ó  ))^2) p _ 2^4 + 3 (-60 (t + u) (m _ K^(ó  ))^4 - 2 (30 t^2 + 41 u t + 30 u^2 + 30 s (t + u)) (m _ K^(ó  ))^2 + 3 t u (8 s + 11 (t + u))) p _ 2^2 + t u (154 (m _ K^(ó  ))^4 + (489 (t + u) - 906 s) (m _ K^(ó  ))^2 + 9 (-15 t^2 - 14 u t - 15 u^2 + 13 s (t + u)))))) + 1/(3072 t u π^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) log((m _ π^(ó  ))^2/μ^2) (4 (103 t u - 4 (t + u) (m _ K^(ó  ))^2) (m _ π^(ó  ))^6 + 2 (2 (t + u) (m _ K^(ó  ))^4 + (2 t^2 - 263 u t + 2 u^2 + 2 s (t + u)) (m _ K^(ó  ))^2 + t u (8 (t + u) - 475 s) + p _ 2^2 (6 (t + u) (m _ K^(ó  ))^2 + 59 t u)) (m _ π^(ó  ))^4 + (4 ((t + u) (m _ K^(ó  ))^2 + 2 t u) p _ 2^4 - (4 (t + u) (m _ K^(ó  ))^4 + 2 (2 t^2 + 59 u t + 2 u^2 + 2 s (t + u)) (m _ K^(ó  ))^2 + t u (104 s - 7 (t + u))) p _ 2^2 + t u (74 (m _ K^(ó  ))^4 + (998 s + 77 (t + u)) (m _ K^(ó  ))^2 + 288 s^2 - 75 t^2 - 75 u^2 + 26 t u + 17 s (t + u))) (m _ π^(ó  ))^2 - 96 s t u (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 + 3 s - p _ 2^2))) + 1/(1536 t u π^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) log((m _ K^(ó  ))^2/μ^2) (48 (t + u) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 - 4 (m _ K^(ó  ))^2 (3 t^2 + 3 s t + 37 u t + 3 u^2 + 43 (t + u) (m _ K^(ó  ))^2 + 3 s u + 9 (t + u) p _ 2^2) (m _ π^(ó  ))^4 + 2 (20 (t + u) (m _ K^(ó  ))^6 + (20 t^2 + 191 u t + 20 u^2 + 20 s (t + u)) (m _ K^(ó  ))^4 - 6 (t + u) p _ 2^4 (m _ K^(ó  ))^2 - 11 t u (4 (t + u) - 11 s) (m _ K^(ó  ))^2 - 18 s^2 t u + p _ 2^2 (66 (t + u) (m _ K^(ó  ))^4 + (6 t^2 - 3 u t + 6 u^2 + 6 s (t + u)) (m _ K^(ó  ))^2 + 6 s t u)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-8 (2 t u - 5 (t + u) (m _ K^(ó  ))^2) p _ 2^4 + (-40 (t + u) (m _ K^(ó  ))^4 - 10 (4 t^2 + 9 u t + 4 u^2 + 4 s (t + u)) (m _ K^(ó  ))^2 + t u (4 s + 13 (t + u))) p _ 2^2 + t u (-154 (m _ K^(ó  ))^4 + (127 (t + u) - 362 s) (m _ K^(ó  ))^2 + 36 s^2 + 15 t^2 + 15 u^2 - 34 t u + 11 s (t + u))))))

The Overscript[J, _] and polynomial parts of the loop contribution in proportion to c _ 5:

loopJs5 = Collect[finalloops /. CouplingConstant[ChPTW3[2], 1] -> 0 /. _Log -> 0, {_DecayConstant, _Pair, _LeutwylerJBar}] /. (LeutwylerJBar[a__] * (b__ )) :> LeutwylerJBar[a] * Collect[Times[b], {MandelstamS, MandelstamT, MandelstamU, _ParticleMass}] ;

These are the logs originally present in the loop contribution in proportion to c _ 5:

looplogs5 = Collect[Simplify[(finalloops /. Pair[_LorentzIndex, ___] -> Sequence[] /. CouplingConstant[ChPTW3[2], 1] -> 0 /. _LeutwylerJBar -> 0) - (finalloops /. Pair[_LorentzIndex, ___] -> Sequence[] /. CouplingConstant[ChPTW3[2], 1] -> 0 /. _LeutwylerJBar -> 0 /. _Log -> 0)], {_DecayConstant, _Log, (ParticleMass[Kaon, RenormalizationState[1]]^2 - Pair[Momentum[p2], Momentum[p2]]), (Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Kaon, RenormalizationState[1]]^2)}] /. Log[a_] * b__ :> Log[a] * Simplify[Collect[Times[b], {_ParticleMass, _Pair}]]

1/(f _ ϕ^(ó  ))^4 (1/(4608 t u π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) log((m _ π^(ó  ))^2/μ^2) (4 (47 t u - 36 (t + u) (m _ K^(ó  ))^2) (m _ π^(ó  ))^6 + 2 (18 (t + u) (m _ K^(ó  ))^4 + (18 t^2 + 421 u t + 18 u^2 + 18 s (t + u)) (m _ K^(ó  ))^2 - t u (71 s + 176 (t + u)) + p _ 2^2 (54 (t + u) (m _ K^(ó  ))^2 - t u)) (m _ π^(ó  ))^4 + (36 ((t + u) (m _ K^(ó  ))^2 - 2 t u) p _ 2^4 + (-36 (t + u) (m _ K^(ó  ))^4 - 2 (18 t^2 - 169 u t + 18 u^2 + 18 s (t + u)) (m _ K^(ó  ))^2 - 3 t u (11 (t + u) - 136 s)) p _ 2^2 + t u (-238 (m _ K^(ó  ))^4 - (146 s + 155 (t + u)) (m _ K^(ó  ))^2 + 3 (80 s^2 - 29 (t + u) s + 95 t^2 + 95 u^2 - 50 t u))) (m _ π^(ó  ))^2 - 48 s t u (m _ K^(ó  ))^2 (-5 (m _ K^(ó  ))^2 + 5 s - t - u + 7 p _ 2^2))) - 1/(11520 t u π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) log((m _ K^(ó  ))^2/μ^2) (-720 (t + u) (m _ K^(ó  ))^2 (m _ π^(ó  ))^6 - 4 (-405 (t + u) (m _ K^(ó  ))^4 - (45 t^2 - 43 u t + 45 u^2 + 45 s (t + u) + 135 (t + u) p _ 2^2) (m _ K^(ó  ))^2 + 45 s t u) (m _ π^(ó  ))^4 - 2 (180 (t + u) (m _ K^(ó  ))^6 + (180 t^2 - 253 u t + 180 u^2 + 180 s (t + u)) (m _ K^(ó  ))^4 - 90 (t + u) p _ 2^4 (m _ K^(ó  ))^2 - 2 t u (164 s + 59 (t + u)) (m _ K^(ó  ))^2 + 45 s t (5 s - t - u) u + p _ 2^2 (630 (t + u) (m _ K^(ó  ))^4 + (90 t^2 - 47 u t + 90 u^2 + 90 s (t + u)) (m _ K^(ó  ))^2 + 315 s t u)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-360 (t + u) p _ 2^4 (m _ K^(ó  ))^2 + t u (26 (m _ K^(ó  ))^4 + (484 s - 1301 (t + u)) (m _ K^(ó  ))^2 + 15 (30 s^2 - 17 (t + u) s + 25 t^2 + 25 u^2 + 2 t u)) + p _ 2^2 (360 (t + u) (m _ K^(ó  ))^4 + 2 (180 t^2 + 733 u t + 180 u^2 + 180 s (t + u)) (m _ K^(ó  ))^2 - 15 t u (13 (t + u) - 42 s))))) - 1/(69120 t u π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 5^( ) log((m _ η^(ó  ))^2/μ^2) (-4 (413 t u - 540 (t + u) (m _ K^(ó  ))^2) (m _ π^(ó  ))^6 - 2 (p _ 2^2 (810 (t + u) (m _ K^(ó  ))^2 - 67 t u) + 3 (1530 (t + u) (m _ K^(ó  ))^4 + (90 t^2 - 659 u t + 90 u^2 + 90 s (t + u)) (m _ K^(ó  ))^2 + 7 t u (47 s - 28 (t + u)))) (m _ π^(ó  ))^4 + (180 (2 t u - 3 (t + u) (m _ K^(ó  ))^2) p _ 2^4 - 5 (-1404 (t + u) (m _ K^(ó  ))^4 - 2 (54 t^2 - 163 u t + 54 u^2 + 54 s (t + u)) (m _ K^(ó  ))^2 + 9 t u (8 s + 5 (t + u))) p _ 2^2 - 3 (-720 (t + u) (m _ K^(ó  ))^6 - 6 (120 t^2 + 409 u t + 120 u^2 + 120 s (t + u)) (m _ K^(ó  ))^4 + 5 t u (269 (t + u) - 322 s) (m _ K^(ó  ))^2 + 45 t u (5 t^2 - 6 u t + 5 u^2 + s (t + u)))) (m _ π^(ó  ))^2 + 4 (m _ K^(ó  ))^2 (-180 (2 t u - 3 (t + u) (m _ K^(ó  ))^2) p _ 2^4 + (-540 (t + u) (m _ K^(ó  ))^4 - 2 (270 t^2 + 353 u t + 270 u^2 + 270 s (t + u)) (m _ K^(ó  ))^2 + 45 t u (8 s + 5 (t + u))) p _ 2^2 + t u (14 (m _ K^(ó  ))^4 + (411 (t + u) - 2334 s) (m _ K^(ó  ))^2 + 135 (5 t^2 - 6 u t + 5 u^2 + s (t + u)))))))

Converted by Mathematica  (July 10, 2003)