•Derivation of the equations

We'll work with unexpanded operators:

SetOptions[#, Explicit -> False] & /@ {MM, SMM, UChi, CovariantFieldDerivative} ;

Notice that because det(U)=1, U=e^(i Overscript[ϕ, ->] · Overscript[λ, ->]), that is, there is no traceful part of Overscript[ϕ, ->].

Abbreviation for Overscript[U, _] e^(i Overscript[ξ, ->] · Overscript[λ, ->]). Overscript[U, _]=UMatrix[U] is the solution to the eqation of motion.

up = NM[UMatrix[U][x], UFieldMatrix[QuantumField[Particle[UPerturbation]][x], ExpansionOrder -> 1, DropOrder -> 1, Constant -> DecayConstant[PhiMeson]]] // NMExpand

(i ℵ (U '6 Overscript[ξ^( ), ->] · Overscript[σ, ->]))/f _ ϕ^(ó  ) + U

The lagrangian:

lag = ArgumentsSupply[Lagrangian[ChPT3[2]], x] /. MM[x, ___] -> up /. CovariantFieldDerivative -> FieldDerivative ;

Higher orders are dropped:

s = DiscardTerms[lag, Retain -> {Particle[UPerturbation] -> 1}] // Simplify ;

Vanishing of total derivatives is used (surfaceRules is defined in the "Preliminaries" section):

s1 = s // SurfaceReduce[#, UFields -> UPerturbation] & // Simplify ;

Rewrite using indices and simplify a bit:

s2 = s1 // IsoIndicesSupply // IndicesCleanup // CycleUTraces // Simplify

-(i (f _ ϕ^(ó  ))^2 (< ∂ _ ρ1(∂ _ ρ1(U))^† '6 U '6 σ^k1 > - < U^† '6 ∂ _ ρ1(∂ _ ρ1(U)) '6 σ^k1 > + < U^† '6 χ '6 σ^k1 > - < χ^† '6 U '6 σ^k1 >) ξ^( )^k1)/(4 f _ ϕ^(ó  ))

The functional derivative:

dsdpi = FunctionalDerivative[s2, {QuantumField[Particle[UPerturbation], SUNIndex[i1]][p1]}] // SUNReduce // Simplify

-(i (f _ ϕ^(ó  ))^2 (< ∂ _ ρ1(∂ _ ρ1(U))^† '6 U '6 σ^i _ 1 > - < U^† '6 ∂ _ ρ1(∂ _ ρ1(U)) '6 σ^i _ 1 > + < U^† '6 χ '6 σ^i _ 1 > - < χ^† '6 U '6 σ^i _ 1 >))/(4 f _ ϕ^(ó  ))

With  A ≡ a + b ÷¬öé, the condition < A σ^i > = 0 is equivalent to < a σ^i > = 0, where a = A-1/n< a^ >. This stems from the fact that there is no traceful part of Overscript[ϕ, ->].

Factor:

fac = UTrace[dsdpi /. UTrace1 -> Identity // SUNReduce // NMFactor // Simplify]

-(i (f _ ϕ^(ó  ))^2 < (∂ _ ρ1(∂ _ ρ1(U))^† '6 U - U^† '6 ∂ _ ρ1(∂ _ ρ1(U)) + U^† '6 χ - χ^† '6 U) '6 σ^i _ 1 >)/(4 f _ ϕ^(ó  ))

fac // StandardForm

-(i DecayConstant[PseudoScalar[1], RenormalizationState[0]]^2 UTrace1[NM[NM[Adjoint[FieldDerivative[FieldDerivative[UMatrix[U][x], x, LorentzIndex[ρ1]], x, LorentzIndex[ρ1]]], UMatrix[U][x]] - NM[Adjoint[UMatrix[U][x]], FieldDerivative[FieldDerivative[UMatrix[U][x], x, LorentzIndex[ρ1]], x, LorentzIndex[ρ1]]] + NM[Adjoint[UMatrix[U][x]], UMatrix[UChi[]][x]] - NM[Adjoint[UMatrix[UChi[]][x]], UMatrix[U][x]], UMatrix[UGenerator[SUNIndex[i1]]]]])/(4 DecayConstant[PseudoScalar[1]])


Finally, the equations of motion are found to read (well, the left-hand side,
to be set equal to 0)

eqsU = NM[Adjoint[FieldDerivative[FieldDerivative[UMatrix[U][x], x, LorentzIndex[μ]], x, LorentzIndex[μ]]], UMatrix[U][x]] - NM[Adjoint[UMatrix[U][x]], FieldDerivative[FieldDerivative[UMatrix[U][x], x, LorentzIndex[μ]], x, LorentzIndex[μ]]] + NM[Adjoint[UMatrix[U][x]], UMatrix[χ][x]] - NM[Adjoint[UMatrix[χ][x]], UMatrix[U][x]] - 1/3 UTrace[NM[Adjoint[UMatrix[U][x]], UMatrix[χ][x]] - NM[Adjoint[UMatrix[χ][x]], UMatrix[U][x]]] UIdentityMatrix[]

∂ _ μ(∂ _ μ(U))^† '6 U - U^† '6 ∂ _ μ(∂ _ μ(U)) + U^† '6 χ - χ^† '6 U - 1/3 ÷¬öé (< U^† '6 χ > - < χ^† '6 U >)

Converted by Mathematica  (July 10, 2003)