•Tree contribution of fourth order in the chiral expansion

lag = Lagrangian[ChPT3[4]] /. CouplingConstant[ChPT3[4], i_ ? (# < 4 || # > 8) &, ___] :> 0

L _ 7^( ) ((< χ '6 ÷„^† > - < ÷„ '6 χ^† >) '6 (< χ '6 ÷„^† > - < ÷„ '6 χ^† >)) + L _ 6^( ) ((< ÷„ '6 χ^† > + < χ '6 ÷„^† >) '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 4^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 (< ÷„ '6 χ^† > + < χ '6 ÷„^† >)) + L _ 1^( ) (< ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† > '6 < ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† >) + L _ 2^( ) (< ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > '6 < ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† >) + H _ 2^( ) < χ^† '6 χ > + H _ 1^( ) (< L _ (μ ν) '6 L _ (μ ν) > + < R _ (μ ν) '6 R _ (μ ν) >) + L _ 5^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 (÷„ '6 χ^† + χ '6 ÷„^†) > + i L _ 9^( ) (< L _ (μ ν) '6 ÷s _ μ(÷„) '6 ÷s _ ν(÷„)^† > + < R _ (μ ν) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) >) + L _ 8^( ) (< ÷„ '6 χ^† '6 ÷„ '6 χ^† > + < χ '6 ÷„^† '6 χ '6 ÷„^† >) + L _ 3^( ) < ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 ÷s _ ν(÷„) '6 ÷s _ ν(÷„)^† > + L _ 10^( ) < L _ (μ ν) '6 ÷„ '6 R _ (μ ν) '6 ÷„^† >

Expand[lag] // Length

15

lls = (WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 2]) & /@ Expand[lag] ;

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

lld = ArgumentsSupply[lls, x, RenormalizationState[0], ExpansionOrder -> 2, DropOrder -> 2] ;

ArgumentsSupply :: argxpr : Warning : The argument x is already in the expression.

Expand[lld] // Length

20

lle = (WriteString["stdout", "."] ; NMExpand[#]) & /@ Expand[lld] ;

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

Expand[lle] // Length

317

lll = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {ParticleField[PhiMeson , RenormalizationState[0]] -> 2}, CommutatorReduce -> False, Method -> Expand]) & /@ Expand[lle] ;

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

Expand[lll] // Length

294

llle = (WriteString["stdout", "."] ; ExpandU[#, CommutatorReduce -> True]) & /@ Expand[lll] ;

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

llle // Length

161

IsoIndicesCounter = 0 ;

llll = (WriteString["stdout", "."] ; # // IsoIndicesSupply // SUNReduce // SUNReduce // IndicesCleanup // CommutatorReduce[#, FullReduce -> True] & // Simplify) & /@ llle ;

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

field = FieldsSet[QuantumField[Particle[PhiMeson, RenormalizationState[0]]], ParticlesNumber -> 2]

{ϕ^( )^I _ 1, ϕ^( )^I _ 2}

llll // Expand // Length

143

amp4f[i1_, i2_] = ((WriteString["stdout", "."] ; IndicesCleanup[SUNReduce[FunctionalD[# // PhiToFC, fields]]]) & /@ Expand[llll]) /. {SUNDelta -> SU3Delta, I1 -> i1, I2 -> i2} // SUNReduce[#, Fullreduce -> True] & // Simplify ;

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

amp4Pion = amp4f[1, 1] /. udrules // Simplify // SUNReduce // SUNReduce // FullSimplify

(-8 (2 (L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^2 + 4 L _ 6^( ) (m _ K^(ó  ))^2 + L _ 5^( ) p _ 1 · p _ 2) (m _ π^(ó  ))^2 - 8 L _ 4^( ) p _ 1 · p _ 2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))/(f _ ϕ^(ó  ))^2

amp4Kaon = amp4f[4, 4] /. udrules // Simplify // SUNReduce // SUNReduce // FullSimplify

-(8 (2 (2 L _ 6^( ) + L _ 8^( )) (m _ K^(ó  ))^4 + (2 L _ 6^( ) (m _ π^(ó  ))^2 + (2 L _ 4^( ) + L _ 5^( )) p _ 1 · p _ 2) (m _ K^(ó  ))^2 + L _ 4^( ) p _ 1 · p _ 2 (m _ π^(ó  ))^2))/(f _ ϕ^(ó  ))^2

amp4Eta = amp4f[8, 8] /. FromK0Rules /. udrules // Simplify // SUNReduce // SUNReduce // FullSimplify

1/(3 (f _ ϕ^(ó  ))^2) (4 (-9 (2 L _ 6^( ) + 3 L _ 8^( )) (m _ η^(ó  ))^4 + 6 (-3 L _ 6^( ) (m _ π^(ó  ))^2 - L _ 5^( ) p _ 1 · p _ 2 + L _ 8^( ) ((m _ π^(ó  ))^2 + 4 (m _ K^(ó  ))^2)) (m _ η^(ó  ))^2 - 18 L _ 7^( ) ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^2)^2 - L _ 8^( ) (7 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^4) - 9 L _ 4^( ) p _ 1 · p _ 2 ((m _ π^(ó  ))^2 + (m _ η^(ó  ))^2)))

amp4 = AmplitudeProjection[amp4f, Channel -> {{PionZero} -> {PionZero}}] // SUNReduce // SUNReduce // FullSimplify

-1/(f _ ϕ^(ó  ))^2 (8 (L _ 5^( ) p _ 1 · p _ 2 (m _ π^(ó  ))^2 + L _ 4^( ) p _ 1 · p _ 2 ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) + 2 (L _ 6^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) (m _ π^(ó  ))^2 + 2 L _ 7^( ) ((m _ K^+^(ó  ))^2 - (m _ K^0^(ó  ))^2)^2 + L _ 8^( ) ((m _ π^(ó  ))^4 + ((m _ K^+^(ó  ))^2 - (m _ K^0^(ó  ))^2)^2))))

Converted by Mathematica  (July 10, 2003)