•Loop contribution of fourth order in the chiral expansion

Construction of topologies:

mesonstop = CreateTopologies[1, 1 -> 1, Adjacencies -> {4}] ;

It is important to avoid that the internal summation index I1 be summed over automatically, like in SU3D[SUNIndex[k1],SUNIndex[I1],SUNIndex[I1] , which would be set to zero were it not for the definition below.

$ConstantIsoIndices = {I1, i1, i2} ;

Field insertion. Takes a while because of the large (unused here) Feynman rules.

mesontreeinsert = InsertFields[mesonstop, {PhiMeson[0, {i1}]} -> {PhiMeson[0, {i2}]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

Graphical representation of the process:

Paint[mesontreeinsert, PaintLevel -> {Classes}, AutoEdit -> False, ColumnsXRows -> 1, Numbering -> False, SheetHeader -> False] ;

[Graphics:../HTMLFiles/index_47.gif]

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert] ;

The one-loop integrals are simplified:

aff[j1_, j2_][k1_] := amplFC[[1]] /. _SumOver -> 1 /. {i1 -> j1, i2 -> j2, I1 -> k1} ;

aff1[j1_, j2_] := Table[SUNReduce[SUNReduce[SUNReduce[#]], Explicit -> True, HoldSums -> False] & /@ Expand[aff[j1, j2][kk]] /. p3 -> -p1, {kk, 8}] /. subpar /. {PionPlus -> Pion, PionMinus -> Pion, PionZero -> Pion} // SUNReduce // Simplify ;

af[j1_, j2_] := OneLoopSimplify[Plus @@ aff1[j1, j2], q1] // Collect[#, FeynAmpDenominator[__]] & ;

The loop integrals are expressed in terms of Passarino-Veltman symbols.

ampreduced[j1_, j2_] := OneLoop[q1, af[j1, j2]] // Simplify ;

ampsimple[j1_, j2_] := Collect[ampreduced[j1, j2], {_B0, _DecayConstant, Pi, _ParticleMass, _Pair}] ;

The divergences are singled out:

ampinfinitiesf[j1_, j2_] := Collect[VeltmanExpand[ampsimple[j1, j2], ExplicitLeutwylerJ0 -> True], {_Log, _DecayConstant, Pi, _ParticleMass, _Pair}] ;

ampinfinitiesPion = ampinfinitiesf[1, 1] /. udrules // Simplify

1/(96 π^2 (f _ ϕ^(ó  ))^2) ((m _ π^(ó  ))^2 ((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2) - 2 p _ 1^2 (2 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))

ampinfinitiesKaon = ampinfinitiesf[4, 4] /. udrules // Simplify

-1/(192 π^2 (f _ ϕ^(ó  ))^2) (((32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2 - 3 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 3 p _ 1^2 ((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2) - 3 (2 (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2 (m _ K^(ó  ))^2 - (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^4))

ampinfinitiesEta = ampinfinitiesf[8, 8] /. FromK0Rules /. udrules /. Log -> log // Simplify

1/(384 π^2 (f _ ϕ^(ó  ))^2) (3 (96 π^2 λ + 4 log((m _ π^(ó  ))^2/μ^2) - log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^4 - 2 ((-128 π^2 λ + log((m _ K^(ó  ))^2/μ^2) - 5 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ K^(ó  ))^2 + 2 (128 π^2 λ + log((m _ η^(ó  ))^2/μ^2) + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ η^(ó  ))^2) (m _ π^(ó  ))^2 - 512 π^2 λ (m _ K^(ó  ))^4 - 8 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 8 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^4 + 224 π^2 λ (m _ η^(ó  ))^4 + 16 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^4 - 9 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ η^(ó  ))^4 + 768 π^2 λ (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + 6 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + 18 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 - 6 p _ 1^2 ((32 π^2 λ + log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 2 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^2 + 96 π^2 λ (m _ η^(ó  ))^2 + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ η^(ó  ))^2))

ampinfinities = Collect[AmplitudeProjection[ampinfinitiesf, Channel -> {{PionZero} -> {PionZero}}], {_Log, _DecayConstant, Pi, _ParticleMass, _Pair}]

(log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 (m _ η^(ó  ))^2)/(96 π^2 (f _ ϕ^(ó  ))^2) + (log((m _ π^(ó  ))^2/μ^2) (1/96 (m _ π^(ó  ))^4 - 1/24 p _ 1^2 (m _ π^(ó  ))^2))/(π^2 (f _ ϕ^(ó  ))^2) + (log((m _ K^0^(ó  ))^2/μ^2) (1/96 (m _ K^0^(ó  ))^4 + 1/96 (m _ π^(ó  ))^2 (m _ K^0^(ó  ))^2 - 1/96 (m _ K^+^(ó  ))^2 (m _ K^0^(ó  ))^2 - 1/96 p _ 1^2 (m _ K^0^(ó  ))^2))/(π^2 (f _ ϕ^(ó  ))^2) + (log((m _ K^+^(ó  ))^2/μ^2) (1/96 (m _ K^+^(ó  ))^4 + 1/96 (m _ π^(ó  ))^2 (m _ K^+^(ó  ))^2 + (-1/96 (m _ K^0^(ó  ))^2 - p _ 1^2/96) (m _ K^+^(ó  ))^2))/(π^2 (f _ ϕ^(ó  ))^2) + 1/(f _ ϕ^(ó  ))^2 (1/3 λ (m _ π^(ó  ))^4 + (1/3 λ (m _ K^+^(ó  ))^2 + 1/3 λ (m _ K^0^(ó  ))^2 + 1/3 λ (m _ η^(ó  ))^2 - (4 λ p _ 1^2)/3) (m _ π^(ó  ))^2 + 1/3 λ (m _ K^+^(ó  ))^4 + 1/3 λ (m _ K^0^(ó  ))^4 - 1/3 λ p _ 1^2 (m _ K^0^(ó  ))^2 + (m _ K^+^(ó  ))^2 (-2/3 λ (m _ K^0^(ó  ))^2 - (λ p _ 1^2)/3))

Coefficient[ampinfinities, LeutwylerLambda[]] // Simplify

((m _ π^(ó  ))^4 + ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2 + (m _ η^(ó  ))^2) (m _ π^(ó  ))^2 + ((m _ K^+^(ó  ))^2 - (m _ K^0^(ó  ))^2)^2 - p _ 1^2 (4 (m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)