•Tree contribution of fourth order in the chiral expansion

$ConstantIsoIndices = {I1, I2} ;

The N _ (1 - 13) part of the lagrangian does not have the weak scalar source, so we multiply it on:

lag = QuantumField[Particle[Scalar[1], RenormalizationState[0]]] * (Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], i_ ? ((# > 13) &)] -> 0) + (Lagrangian[ChPTW3[4]] /. CouplingConstant[ChPTW3[4], i_ ? ((# < 14) &)] -> 0) // Expand ;

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

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

lluu = NMExpand[llu] ;

lux = Expand[lluu] ;

'raw' quantites are given arguments:

lla = ArgumentsSupply[lux, x, RenormalizationState[0], ExpansionOrder -> 1, DropOrder -> 2, DiagonalToU -> True] ;

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

lla // Length

10

lll = (WriteString["stdout", "."] ; DiscardTerms[#, Retain -> {Particle[PhiMeson , RenormalizationState[0]] -> 1, Particle[Scalar[1] , RenormalizationState[0]] -> 1}, CommutatorReduce -> True, Method -> Coefficient]) & /@ lla // Simplify ;

..........

The traces over generator matrices are expanded:

llle = ExpandU[lll, CommutatorReduce -> True] // Simplify ;

$IsoIndicesCounter = 0 ;

llll = llle // IsoIndicesSupply // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // Simplify ;

fields = {QuantumField[Particle[PhiMeson, RenormalizationState[0]], SUNIndex[I1]][p1], QuantumField[Particle[Scalar[1], RenormalizationState[0]]][p2]}

{ϕ^( )^I _ 1, s^( )}

melsimplified = FeynRule[llll, fields] // Simplify // SUNReduce[#, FullReduce -> True] & // IndicesCleanup // Simplify

(8 i c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) (2 N _ 10^( ) (m _ K^0^(ó  ))^2 + N _ 21^( ) p _ 1 · p _ 2 + N _ 11^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2)) δ _ (7 I _ 1)^(3))/(f _ ϕ^(ó  ))^3

amp4 = -I melsimplified // Simplify

(8 c _ 2^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) (2 N _ 10^( ) (m _ K^0^(ó  ))^2 + N _ 21^( ) p _ 1 · p _ 2 + N _ 11^( ) ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2)) δ _ (7 I _ 1)^(3))/(f _ ϕ^(ó  ))^3

Converted by Mathematica  (July 10, 2003)