![$Configuration = "ChPT2" ; $Lagrangians = {"ChPT2"[2], "ChPT2"[4]} ;](HTMLFiles/index_2.gif){kind=small}
![Needs["HighEnergyPhysics`FeynCalc`"] ;](HTMLFiles/index_3.gif){kind=small}
FeynCalc 4.2.0
For help, type ?FeynCalc,
use the built-in help system
or visit www.feyncalc.org
Loading PHI
Loading FeynArts
The standard lowest (energy) order ChPT lagrangian in raw form:
![Lagrangian[ChPT2[2]]](HTMLFiles/index_9.gif){kind=small}
All external sources are set to zero:
![IsoVector[QuantumField[Particle[AxialVector[0], ___], ___], ___][_] := 0 ; <br /> IsoVector[QuantumField[Particle[Vector[0], ___], ___], ___][_] := 0 ; <br /> IsoVector[QuantumField[Particle[Scalar[1 | 2], ___], ___], ___][_] := 0 ; <br /> QuantumField[Particle[Scalar[1 | 2], ___], ___][_] := 0 ; <br /> QuantumField[Particle[PseudoScalar[0], ___], ___][_] := 0 ;](HTMLFiles/index_11.gif)
![ll = ArgumentsSupply[Lagrangian[ChPT2[2]], x, RenormalizationState[0], GaugeGroup -> 2, ExpansionOrder -> 4, DropOrder -> 4] ;](HTMLFiles/index_12.gif){kind=small}
The lagrangian expanded to order 4 in the pion fields:
![lll = DiscardTerms[ll, Retain -> {Particle[Pion , RenormalizationState[0]] -> 4}, Method -> Expand] // Simplify](HTMLFiles/index_14.gif){kind=small}
DotExpand |
![1/(48 (f _ π^(ó ))^2) (< Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > (m _ π^(ó ))^2 - 2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + < Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + 3 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > - 2 < ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] >)](HTMLFiles/index_21.gif)
We could of course already here simplify things a lot. For the sake of illustration we keep it complicated.
![(< Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] > (m _ π^(ó ))^2 - 2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] > + 2 < Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] '6 Overscript[π^( ), ->] · Overscript[σ, ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[σ, ->] >)/(48 (f _ π^(ó ))^2)](HTMLFiles/index_23.gif)
The expansion of the Pauli matrices gives a lot of cross product terms:
![llle = ExpandU[lll, CommutatorReduce -> False] // Simplify](HTMLFiles/index_24.gif){kind=small}
![1/(24 (f _ π^(ó ))^2) (-(Overscript[π^( ), ->] × Overscript[π^( ), ->] × Overscript[π^( ), ->] · Overscript[π^( ), ->]) (m _ π^(ó ))^2 + (Overscript[π^( ), ->] · Overscript[π^( ), ->] '6 Overscript[π^( ), ->] · Overscript[π^( ), ->]) (m _ π^(ó ))^2 + 2 Overscript[π^( ), ->] × Overscript[π^( ), ->] × ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) - Overscript[π^( ), ->] × ∂ _ μ(Overscript[π^( ), ->]) × Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) - 3 Overscript[π^( ), ->] × ∂ _ μ(Overscript[π^( ), ->]) × ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->] + ∂ _ μ(Overscript[π^( ), ->]) × Overscript[π^( ), ->] × Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) - ∂ _ μ(Overscript[π^( ), ->]) × Overscript[π^( ), ->] × ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->] + 2 ∂ _ μ(Overscript[π^( ), ->]) × ∂ _ μ(Overscript[π^( ), ->]) × Overscript[π^( ), ->] · Overscript[π^( ), ->] - 2 (Overscript[π^( ), ->] · Overscript[π^( ), ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->])) + Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) '6 Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) + 3 (Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->]) - ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->] '6 Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]) + ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->] '6 ∂ _ μ(Overscript[π^( ), ->]) · Overscript[π^( ), ->] - 2 (∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]) '6 Overscript[π^( ), ->] · Overscript[π^( ), ->]))](HTMLFiles/index_25.gif)
When CommutatorReduce is invoked, the expression may not change, but often terms cancel:
![ExpandU[lll, CommutatorReduce -> True] // Simplify](HTMLFiles/index_26.gif){kind=small}
![(4 (Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]))^2 + (Overscript[π^( ), ->] · Overscript[π^( ), ->])^2 (m _ π^(ó ))^2 - 4 Overscript[π^( ), ->] · Overscript[π^( ), ->] ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]))/(24 (f _ π^(ó ))^2)](HTMLFiles/index_31.gif)
![(4 (Overscript[π^( ), ->] · ∂ _ μ(Overscript[π^( ), ->]))^2 + (Overscript[π^( ), ->] · Overscript[π^( ), ->])^2 (m _ π^(ó ))^2 - 4 Overscript[π^( ), ->] · Overscript[π^( ), ->] ∂ _ μ(Overscript[π^( ), ->]) · ∂ _ μ(Overscript[π^( ), ->]))/(24 (f _ π^(ó ))^2)](HTMLFiles/index_33.gif)
Converted by Mathematica (July 10, 2003)