•Pϕ amplitude to fourth order in the chiral expansion

The lagrangian in raw form (non-relevant terms have been set to zero):

lag = Lagrangian[ChPT3[4]] /. UCouplingConstant[ChPT3[4], 1 | 2 | 3 | 7 | 9 | 10 | 11 | 12, ___][___] :> 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 ÷„^† >

llu = (WriteString["stdout", "."] ; UNMSplit[#, x, DropOrder -> 1]) & /@ lag ;

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

Expand[llu] // Length

23

Remaining 'raw' quantites are put on arguments:

ll = ArgumentsSupply[llu, x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 1, DiagonalToU -> True] // Simplify ;

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

Matrices are traced:

Expand[ll] // Length

9

llld = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PseudoScalar[0] , RenormalizationState[0]] -> 1, Particle[PhiMeson , RenormalizationState[0]] -> 1}, CommutatorReduce -> False, Method -> Expand]) & /@ Expand[ll] ;

.........

llle = ExpandU[llld, CommutatorReduce -> True] ;

Indices are supplied:

$IsoIndicesCounter = 0 ;

tmp = Expand[llle] ; tmp // Length

36

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

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

fields = {QuantumField[Particle[PseudoScalar[0], RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I2]][p2]}

{p^( )^I _ 1, ϕ^( )^I _ 2}

amp4 = FeynRule[llll, fields] // Simplify // SUNReduce[#, Fullreduce -> True] & // IndicesCleanup // CommutatorReduce // FullSimplify

1/(3 f _ ϕ^(ó  )) (32 i !, _ 0^( ) (3 L _ 6^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) δ _ (I _ 1 I _ 2)^(3) + L _ 8^( ) ((2 3^(1/2) d _ (8 I _ 1 I _ 2)^(3) + δ _ (I _ 1 I _ 2)^(3)) (m _ π^(ó  ))^2 + 3 ((m _ K^+^(ó  ))^2 - (m _ K^0^(ó  ))^2) d _ (3 I _ 1 I _ 2)^(3) - ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) (3^(1/2) d _ (8 I _ 1 I _ 2)^(3) - δ _ (I _ 1 I _ 2)^(3)))))

Converted by Mathematica  (July 10, 2003)