The one-loop generating functional of χPT with virtual photons

Configuration:  "ChPTVirtualPhotons3"
Lagrangians:  ChPTVirtualPhotons3[2]

$LoadPhi = True ; $LoadFeynArts = True ;

$Configuration = "ChPTVirtualPhotons3" ; $Lagrangians = {"ChPTVirtualPhotons3"[2], "ChPTVirtualPhotons3"[4]} ;

Needs["HighEnergyPhysics`FeynCalc`"] ;

FeynCalc 4.1.1     Evaluate ?FeynCalc for help or visit www.feyncalc.org

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FeynArts 3

by Hagen Eck, Sepp Kueblbeck, and Thomas Hahn

last revised 7 May 01

patched for use with FeynCalc by Frederik Orellana

Comparison is made with the thesis of Res Urech (not the paper)

•Preliminaries

•Expansion of the lagrangian

•Splitting in components

•Seeley-DeWitt coefficients

SeeleyDeWitt[0][x_] = 1

1

SeeleyDeWitt[1][x_] = -Y[k1, k1]

-1/(8 n (f _ ϕ^(ó  ))^2) (δ _ (μ _ 1, k _ 1) (f _ ϕ^(ó  ) δ _ (μ _ 1, k _ 1) (< H _ L >^2 + 3 n < H _ L '6 H _ L >) + 2 n (< ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > + i (< H _ R '6 σ^k _ 1 '6 u _ μ _ 1 > - < H _ R '6 u _ μ _ 1 '6 σ^k _ 1 >))) (f _ ϕ^(ó  ))^3 + n ((-2 (f _ ϕ^(ó  ))^2 < H _ L '6 σ^k _ 1 >^2 + 2 f _ ϕ^(ó  ) δ _ (μ _ 1, k _ 1) (< ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > + i (< H _ R '6 σ^k _ 1 '6 u _ μ _ 1 > - < H _ R '6 u _ μ _ 1 '6 σ^k _ 1 >)) + 2 < χ _ + '6 σ^k _ 1 '6 σ^k _ 1 > - < σ^k _ 1 '6 σ^k _ 1 '6 u _ μ _ 1 '6 u _ μ _ 1 > + 2 < σ^k _ 1 '6 u _ μ _ 1 '6 σ^k _ 1 '6 u _ μ _ 1 > - < σ^k _ 1 '6 u _ μ _ 1 '6 u _ μ _ 1 '6 σ^k _ 1 >) (f _ ϕ^(ó  ))^2 + 2 C^( ) (-< σ^k _ 1 '6 H _ L '6 H _ L '6 σ^k _ 1 > + 2 < σ^k _ 1 '6 H _ L '6 σ^k _ 1 '6 H _ L > + < σ^k _ 1 '6 H _ R '6 H _ R '6 σ^k _ 1 > - 2 < σ^k _ 1 '6 H _ R '6 σ^k _ 1 '6 H _ R > - < σ^k _ 1 '6 σ^k _ 1 '6 H _ L '6 H _ L > + < σ^k _ 1 '6 σ^k _ 1 '6 H _ R '6 H _ R >)))

XFST[k1, j, μ1, μ2] // NMExpand // Expand

(δ _ (μ _ 1, k _ 1) δ _ (μ _ 2, j) < H _ L >^2 (f _ ϕ^(ó  ))^2)/(8 n) - (δ _ (μ _ 1, j) δ _ (μ _ 2, k _ 1) < H _ L >^2 (f _ ϕ^(ó  ))^2)/(8 n) - 1/8 δ _ (μ _ 1, k _ 1) δ _ (μ _ 2, j) < H _ L '6 H _ L > (f _ ϕ^(ó  ))^2 + 1/8 δ _ (μ _ 1, j) δ _ (μ _ 2, k _ 1) < H _ L '6 H _ L > (f _ ϕ^(ó  ))^2 + 1/4 δ _ (μ _ 2, k _ 1) < ∇ _ μ _ 1(H _ L) '6 σ^j > f _ ϕ^(ó  ) - 1/4 δ _ (μ _ 2, j) < ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > f _ ϕ^(ó  ) - 1/4 δ _ (μ _ 1, k _ 1) < ∇ _ μ _ 2(H _ L) '6 σ^j > f _ ϕ^(ó  ) + 1/4 δ _ (μ _ 1, j) < ∇ _ μ _ 2(H _ L) '6 σ^k _ 1 > f _ ϕ^(ó  ) - 1/2 < ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^j '6 σ^k _ 1 > + 1/2 < ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^k _ 1 '6 σ^j > + 1/2 < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^j '6 σ^k _ 1 > - 1/2 < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^k _ 1 '6 σ^j > - 1/2 < σ^k _ 1 '6 σ^j '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + 1/2 < σ^k _ 1 '6 σ^j '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 > + 1/2 < σ^k _ 1 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 σ^j > - 1/2 < σ^k _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 '6 σ^j >

Y[k1, j] // NMExpand // Expand

(δ _ (μ _ 1, j) δ _ (μ _ 1, k _ 1) < H _ L >^2 (f _ ϕ^(ó  ))^2)/(8 n) + 3/8 δ _ (μ _ 1, j) δ _ (μ _ 1, k _ 1) < H _ L '6 H _ L > (f _ ϕ^(ó  ))^2 - 1/4 < H _ L '6 σ^j > < H _ L '6 σ^k _ 1 > (f _ ϕ^(ó  ))^2 + 1/4 δ _ (μ _ 1, k _ 1) < ∇ _ μ _ 1(H _ L) '6 σ^j > f _ ϕ^(ó  ) + 1/4 δ _ (μ _ 1, j) < ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > f _ ϕ^(ó  ) + 1/4 i δ _ (μ _ 1, k _ 1) < H _ R '6 σ^j '6 u _ μ _ 1 > f _ ϕ^(ó  ) + 1/4 i δ _ (μ _ 1, j) < H _ R '6 σ^k _ 1 '6 u _ μ _ 1 > f _ ϕ^(ó  ) - 1/4 i δ _ (μ _ 1, k _ 1) < H _ R '6 u _ μ _ 1 '6 σ^j > f _ ϕ^(ó  ) - 1/4 i δ _ (μ _ 1, j) < H _ R '6 u _ μ _ 1 '6 σ^k _ 1 > f _ ϕ^(ó  ) + 1/8 < χ _ + '6 σ^j '6 σ^k _ 1 > + 1/8 < χ _ + '6 σ^k _ 1 '6 σ^j > - 1/8 < σ^k _ 1 '6 σ^j '6 u _ μ _ 1 '6 u _ μ _ 1 > + 1/4 < σ^k _ 1 '6 u _ μ _ 1 '6 σ^j '6 u _ μ _ 1 > - 1/8 < σ^k _ 1 '6 u _ μ _ 1 '6 u _ μ _ 1 '6 σ^j > - (C^( ) < σ^k _ 1 '6 H _ L '6 H _ L '6 σ^j >)/(4 (f _ ϕ^(ó  ))^2) + (C^( ) < σ^k _ 1 '6 H _ L '6 σ^j '6 H _ L >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < σ^k _ 1 '6 H _ R '6 H _ R '6 σ^j >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < σ^k _ 1 '6 H _ R '6 σ^j '6 H _ R >)/(2 (f _ ϕ^(ó  ))^2) - (C^( ) < σ^k _ 1 '6 σ^j '6 H _ L '6 H _ L >)/(4 (f _ ϕ^(ó  ))^2) + (C^( ) < σ^k _ 1 '6 σ^j '6 H _ R '6 H _ R >)/(4 (f _ ϕ^(ó  ))^2)

SeeleyDeWitt[2][x_] = 1/2 CNM[Y[k1, j], Y[j, k1]] + 1/12 NM[XFST[k1, j, μ1, μ2], XFST[j, k1, μ1, μ2]]

1/(768 n^2) ((δ _ (μ _ 1, j) δ _ (μ _ 2, k _ 1) - δ _ (μ _ 1, k _ 1) δ _ (μ _ 2, j)) (< H _ L >^2 - n < H _ L '6 H _ L >) (f _ ϕ^(ó  ))^2 + 2 n (-δ _ (μ _ 2, k _ 1) < ∇ _ μ _ 1(H _ L) '6 σ^j > + δ _ (μ _ 2, j) < ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > + δ _ (μ _ 1, k _ 1) < ∇ _ μ _ 2(H _ L) '6 σ^j > - δ _ (μ _ 1, j) < ∇ _ μ _ 2(H _ L) '6 σ^k _ 1 >) f _ ϕ^(ó  ) - 4 n (-< ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^j '6 σ^k _ 1 > + < ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^k _ 1 '6 σ^j > + < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^j '6 σ^k _ 1 > - < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^k _ 1 '6 σ^j > - < σ^k _ 1 '6 σ^j '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + < σ^k _ 1 '6 σ^j '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 > + < σ^k _ 1 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 σ^j > - < σ^k _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 '6 σ^j >)) '6 ((δ _ (μ _ 1, k _ 1) δ _ (μ _ 2, j) - δ _ (μ _ 1, j) δ _ (μ _ 2, k _ 1)) (< H _ L >^2 - n < H _ L '6 H _ L >) (f _ ϕ^(ó  ))^2 + 2 n (δ _ (μ _ 2, k _ 1) < ∇ _ μ _ 1(H _ L) '6 σ^j > - δ _ (μ _ 2, j) < ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > - δ _ (μ _ 1, k _ 1) < ∇ _ μ _ 2(H _ L) '6 σ^j > + δ _ (μ _ 1, j) < ∇ _ μ _ 2(H _ L) '6 σ^k _ 1 >) f _ ϕ^(ó  ) - 4 n (< ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^j '6 σ^k _ 1 > - < ∂ _ μ _ 2(Γ _ μ _ 1) '6 σ^k _ 1 '6 σ^j > - < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^j '6 σ^k _ 1 > + < ∂ _ μ _ 1(Γ _ μ _ 2) '6 σ^k _ 1 '6 σ^j > - < σ^j '6 σ^k _ 1 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + < σ^j '6 σ^k _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 > + < σ^j '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 σ^k _ 1 > - < σ^j '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 '6 σ^k _ 1 >)) + 1/(128 n^2 (f _ ϕ^(ó  ))^4) (δ _ (μ _ 1, k _ 1) (f _ ϕ^(ó  ) δ _ (μ _ 1, j) (< H _ L >^2 + 3 n < H _ L '6 H _ L >) + 2 n (< ∇ _ μ _ 1(H _ L) '6 σ^j > + i (< H _ R '6 σ^j '6 u _ μ _ 1 > - < H _ R '6 u _ μ _ 1 '6 σ^j >))) (f _ ϕ^(ó  ))^3 + n ((-2 < H _ L '6 σ^j > < H _ L '6 σ^k _ 1 > (f _ ϕ^(ó  ))^2 + 2 δ _ (μ _ 1, j) (< ∇ _ μ _ 1(H _ L) '6 σ^k _ 1 > + i (< H _ R '6 σ^k _ 1 '6 u _ μ _ 1 > - < H _ R '6 u _ μ _ 1 '6 σ^k _ 1 >)) f _ ϕ^(ó  ) + < χ _ + '6 σ^j '6 σ^k _ 1 > + < χ _ + '6 σ^k _ 1 '6 σ^j > - < σ^k _ 1 '6 σ^j '6 u _ μ _ 1 '6 u _ μ _ 1 > + 2 < σ^k _ 1 '6 u _ μ _ 1 '6 σ^j '6 u _ μ _ 1 > - < σ^k _ 1 '6 u _ μ _ 1 '6 u _ μ _ 1 '6 σ^j >) (f _ ϕ^(ó  ))^2 + 2 C^( ) (-< σ^k _ 1 '6 H _ L '6 H _ L '6 σ^j > + 2 < σ^k _ 1 '6 H _ L '6 σ^j '6 H _ L > + < σ^k _ 1 '6 H _ R '6 H _ R '6 σ^j > - 2 < σ^k _ 1 '6 H _ R '6 σ^j '6 H _ R > - < σ^k _ 1 '6 σ^j '6 H _ L '6 H _ L > + < σ^k _ 1 '6 σ^j '6 H _ R '6 H _ R >))) '6 (δ _ (ζ, j) (f _ ϕ^(ó  ) δ _ (ζ, k _ 1) (< H _ L >^2 + 3 n < H _ L '6 H _ L >) + 2 n (< ∇ _ ζ(H _ L) '6 σ^k _ 1 > + i (< H _ R '6 σ^k _ 1 '6 u _ ζ > - < H _ R '6 u _ ζ '6 σ^k _ 1 >))) (f _ ϕ^(ó  ))^3 + n ((-2 < H _ L '6 σ^j > < H _ L '6 σ^k _ 1 > (f _ ϕ^(ó  ))^2 + 2 δ _ (ζ, k _ 1) (< ∇ _ ζ(H _ L) '6 σ^j > + i (< H _ R '6 σ^j '6 u _ ζ > - < H _ R '6 u _ ζ '6 σ^j >)) f _ ϕ^(ó  ) + < χ _ + '6 σ^j '6 σ^k _ 1 > + < χ _ + '6 σ^k _ 1 '6 σ^j > - < σ^j '6 σ^k _ 1 '6 u _ ζ '6 u _ ζ > + 2 < σ^j '6 u _ ζ '6 σ^k _ 1 '6 u _ ζ > - < σ^j '6 u _ ζ '6 u _ ζ '6 σ^k _ 1 >) (f _ ϕ^(ó  ))^2 + 2 C^( ) (-< σ^j '6 H _ L '6 H _ L '6 σ^k _ 1 > + 2 < σ^j '6 H _ L '6 σ^k _ 1 '6 H _ L > + < σ^j '6 H _ R '6 H _ R '6 σ^k _ 1 > - 2 < σ^j '6 H _ R '6 σ^k _ 1 '6 H _ R > - < σ^j '6 σ^k _ 1 '6 H _ L '6 H _ L > + < σ^j '6 σ^k _ 1 '6 H _ R '6 H _ R >)))

A bit of reduction:

sdw2Res = (SeeleyDeWitt[2][x] // NMExpand // Expand // CommutatorReduce) ;

sdw2Res // LeafCount

24228

sdw2Res1 = sdw2Res // applyProjectionRules // CommutatorReduce // SUNReduce // CycleUTraces // CommutatorReduce // SUNReduce // CommutatorReduce ;

sdw2Res1 // LeafCount

1823

sdw2Res1 // IndicesCleanup

(< H _ L >^4 (f _ ϕ^(ó  ))^4)/(8 n^2) + 3/8 < H _ L '6 H _ L >^2 (f _ ϕ^(ó  ))^4 + (< H _ L > < H _ L '6 χ _ + > (f _ ϕ^(ó  ))^2)/(2 n) - 1/4 i < ∇ _ μ _ 1(H _ L) '6 H _ R '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 + 1/4 i < ∇ _ μ _ 1(H _ L) '6 u _ μ _ 1 '6 H _ R > (f _ ϕ^(ó  ))^2 - 1/4 < χ _ + '6 H _ L '6 H _ L > (f _ ϕ^(ó  ))^2 + 1/4 < H _ L '6 H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 - 1/4 < H _ L '6 u _ μ _ 1 '6 H _ L '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 + 1/4 < H _ R '6 H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 - 1/4 < H _ R '6 u _ μ _ 1 '6 H _ R '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 - (< H _ L >^2 < χ _ + > (f _ ϕ^(ó  ))^2)/(4 n^2) - 1/3 < ∂ _ μ _ 2(Γ _ μ _ 1) >^2 + 1/8 < u _ μ _ 1 '6 u _ μ _ 2 >^2 + < χ _ + >^2/(8 n^2) + < χ _ + >^2/16 + 1/3 < ∂ _ μ _ 2(Γ _ μ _ 1) > < ∂ _ μ _ 1(Γ _ μ _ 2) > + 1/3 n < ∂ _ μ _ 2(Γ _ μ _ 1) '6 ∂ _ μ _ 2(Γ _ μ _ 1) > - 1/3 n < ∂ _ μ _ 2(Γ _ μ _ 1) '6 ∂ _ μ _ 1(Γ _ μ _ 2) > + 1/16 n < χ _ + '6 χ _ + > - < χ _ + '6 χ _ + >/(4 n) + 1/16 < u _ μ _ 1 '6 u _ μ _ 1 > < u _ μ _ 2 '6 u _ μ _ 2 > - 2/3 n < ∂ _ μ _ 2(Γ _ μ _ 1) '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + 2/3 n < ∂ _ μ _ 2(Γ _ μ _ 1) '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 > - 1/8 n < χ _ + '6 u _ μ _ 1 '6 u _ μ _ 1 > - 1/2 C^( ) < H _ L '6 H _ L '6 H _ R '6 H _ R > + 1/2 C^( ) < H _ L '6 H _ R '6 H _ L '6 H _ R > - 1/3 n < Γ _ μ _ 1 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 2 > + 1/3 n < Γ _ μ _ 1 '6 Γ _ μ _ 2 '6 Γ _ μ _ 1 '6 Γ _ μ _ 2 > + 1/16 n < u _ μ _ 1 '6 u _ μ _ 1 '6 u _ μ _ 2 '6 u _ μ _ 2 > - 1/8 < u _ μ _ 1 '6 u _ μ _ 1 > < χ _ + > + 1/4 < χ _ + '6 u _ μ _ 1 > < u _ μ _ 1 > - 1/8 < u _ μ _ 1 '6 u _ μ _ 2 '6 u _ μ _ 2 > < u _ μ _ 1 > - 1/8 < u _ μ _ 2 '6 u _ μ _ 1 '6 u _ μ _ 1 > < u _ μ _ 2 > + (C^( ) < H _ L '6 u _ μ _ 1 >^2)/(2 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R '6 u _ μ _ 1 >^2)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ L > < H _ L '6 χ _ + >)/(2 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R > < H _ R '6 χ _ + >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ L '6 H _ L > < u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R '6 H _ R > < u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ L > < H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ R > < H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) - (n C^( ) < χ _ + '6 H _ L '6 H _ L >)/(4 (f _ ϕ^(ó  ))^2) + (n C^( ) < χ _ + '6 H _ R '6 H _ R >)/(4 (f _ ϕ^(ó  ))^2) + (n C^( ) < H _ L '6 H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (n C^( ) < H _ R '6 H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ L '6 H _ L > < χ _ + >)/(4 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ R '6 H _ R > < χ _ + >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < u _ μ _ 1 '6 H _ L '6 H _ L > < u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < u _ μ _ 1 '6 H _ R '6 H _ R > < u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (3 (C^( ))^2 < H _ L '6 H _ L >^2)/(4 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L '6 H _ R >^2)/(f _ ϕ^(ó  ))^4 + (3 (C^( ))^2 < H _ R '6 H _ R >^2)/(4 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L '6 H _ L > < H _ R '6 H _ R >)/(2 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L > < H _ L '6 H _ L '6 H _ L >)/(f _ ϕ^(ó  ))^4 + ((C^( ))^2 < H _ L > < H _ L '6 H _ R '6 H _ R >)/(f _ ϕ^(ó  ))^4 + ((C^( ))^2 < H _ R > < H _ R '6 H _ L '6 H _ L >)/(f _ ϕ^(ó  ))^4 - ((C^( ))^2 < H _ R > < H _ R '6 H _ R '6 H _ R >)/(f _ ϕ^(ó  ))^4 + (n (C^( ))^2 < H _ L '6 H _ L '6 H _ L '6 H _ L >)/(4 (f _ ϕ^(ó  ))^4) - (n (C^( ))^2 < H _ L '6 H _ L '6 H _ R '6 H _ R >)/(2 (f _ ϕ^(ó  ))^4) + (n (C^( ))^2 < H _ R '6 H _ R '6 H _ R '6 H _ R >)/(4 (f _ ϕ^(ó  ))^4)

sdw2Res2 = sdw2Res1 // (* UGammaTrick must be applied before IndicesCleanup !! *) UGammaTrick // NMExpand // Expand // UReduce // IndicesCleanup // HLeftRightTrick // NMExpand // Expand // CommutatorReduce // CycleUTraces // UReduce ;

sdw2Res2 // LeafCount

1425

sdw2Res2

(< H _ L >^4 (f _ ϕ^(ó  ))^4)/(8 n^2) + 3/8 < H _ L '6 H _ L >^2 (f _ ϕ^(ó  ))^4 + (< H _ L > < H _ L '6 χ _ + > (f _ ϕ^(ó  ))^2)/(2 n) - 1/4 < χ _ + '6 H _ L '6 H _ L > (f _ ϕ^(ó  ))^2 + 1/4 < H _ L '6 H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 - 1/4 < H _ L '6 u _ μ _ 1 '6 H _ L '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 + 1/2 < H _ R '6 H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 - 1/2 < H _ R '6 u _ μ _ 1 '6 H _ R '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 + 1/4 i < öÆ^† '6 H _ R '6 u _ μ _ 1 '6 öÆ '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 i < öÆ^† '6 c _ μ _ 1^R Q _ R '6 öÆ '6 H _ R '6 u _ μ _ 1 > (f _ ϕ^(ó  ))^2 + 1/4 i < öÆ^† '6 c _ μ _ 1^R Q _ R '6 öÆ '6 u _ μ _ 1 '6 H _ R > (f _ ϕ^(ó  ))^2 - 1/4 i < öÆ^† '6 u _ μ _ 1 '6 H _ R '6 öÆ '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - (< H _ L >^2 < χ _ + > (f _ ϕ^(ó  ))^2)/(4 n^2) + 1/8 < u _ μ _ 1 '6 u _ μ _ 2 >^2 + < χ _ + >^2/(8 n^2) + < χ _ + >^2/16 + 1/24 < G _ (μ _ 1 μ _ 2)^L >^2 + 1/24 < G _ (μ _ 1 μ _ 2)^R >^2 + 1/16 n < χ _ + '6 χ _ + > - < χ _ + '6 χ _ + >/(4 n) - 1/24 n < G _ (μ _ 1 μ _ 2)^L '6 G _ (μ _ 1 μ _ 2)^L > - 1/24 n < G _ (μ _ 1 μ _ 2)^R '6 G _ (μ _ 1 μ _ 2)^R > + 1/16 < u _ μ _ 1 '6 u _ μ _ 1 > < u _ μ _ 2 '6 u _ μ _ 2 > - 1/8 n < χ _ + '6 u _ μ _ 1 '6 u _ μ _ 1 > - 1/2 C^( ) < H _ L '6 H _ L '6 H _ R '6 H _ R > + 1/2 C^( ) < H _ L '6 H _ R '6 H _ L '6 H _ R > + 1/24 n < u _ μ _ 1 '6 u _ μ _ 1 '6 u _ μ _ 2 '6 u _ μ _ 2 > + 1/48 n < u _ μ _ 1 '6 u _ μ _ 2 '6 u _ μ _ 1 '6 u _ μ _ 2 > - 1/24 i n < öÆ^† '6 G _ (μ _ 1 μ _ 2)^R '6 öÆ '6 u _ μ _ 1 '6 u _ μ _ 2 > + 1/24 i n < öÆ^† '6 G _ (μ _ 1 μ _ 2)^R '6 öÆ '6 u _ μ _ 2 '6 u _ μ _ 1 > - 1/24 i n < öÆ^† '6 u _ μ _ 1 '6 u _ μ _ 2 '6 öÆ '6 G _ (μ _ 1 μ _ 2)^L > + 1/24 i n < öÆ^† '6 u _ μ _ 1 '6 u _ μ _ 2 '6 öÆ '6 G _ (μ _ 2 μ _ 1)^L > - 1/12 n < G _ (μ _ 1 μ _ 2)^L '6 öÆ^† '6 öÆ^† '6 G _ (μ _ 1 μ _ 2)^R '6 öÆ '6 öÆ > - 1/8 < u _ μ _ 1 '6 u _ μ _ 1 > < χ _ + > + 1/12 < G _ (μ _ 1 μ _ 2)^L > < G _ (μ _ 1 μ _ 2)^R > + 1/4 < χ _ + '6 u _ μ _ 1 > < u _ μ _ 1 > - 1/8 < u _ μ _ 1 '6 u _ μ _ 2 '6 u _ μ _ 2 > < u _ μ _ 1 > - 1/8 < u _ μ _ 2 '6 u _ μ _ 1 '6 u _ μ _ 1 > < u _ μ _ 2 > + (C^( ) < H _ L '6 u _ μ _ 1 >^2)/(2 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R '6 u _ μ _ 1 >^2)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ L > < H _ L '6 χ _ + >)/(2 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R > < H _ R '6 χ _ + >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ L '6 H _ L > < u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ R '6 H _ R > < u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ L > < H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ R > < H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) - (n C^( ) < χ _ + '6 H _ L '6 H _ L >)/(4 (f _ ϕ^(ó  ))^2) + (n C^( ) < χ _ + '6 H _ R '6 H _ R >)/(4 (f _ ϕ^(ó  ))^2) + (n C^( ) < H _ L '6 H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (n C^( ) < H _ R '6 H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < H _ L '6 H _ L > < χ _ + >)/(4 (f _ ϕ^(ó  ))^2) + (C^( ) < H _ R '6 H _ R > < χ _ + >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < u _ μ _ 1 '6 H _ L '6 H _ L > < u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (C^( ) < u _ μ _ 1 '6 H _ R '6 H _ R > < u _ μ _ 1 >)/(2 (f _ ϕ^(ó  ))^2) + (3 (C^( ))^2 < H _ L '6 H _ L >^2)/(4 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L '6 H _ R >^2)/(f _ ϕ^(ó  ))^4 + (3 (C^( ))^2 < H _ R '6 H _ R >^2)/(4 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L '6 H _ L > < H _ R '6 H _ R >)/(2 (f _ ϕ^(ó  ))^4) - ((C^( ))^2 < H _ L > < H _ L '6 H _ L '6 H _ L >)/(f _ ϕ^(ó  ))^4 + ((C^( ))^2 < H _ L > < H _ L '6 H _ R '6 H _ R >)/(f _ ϕ^(ó  ))^4 + ((C^( ))^2 < H _ R > < H _ R '6 H _ L '6 H _ L >)/(f _ ϕ^(ó  ))^4 - ((C^( ))^2 < H _ R > < H _ R '6 H _ R '6 H _ R >)/(f _ ϕ^(ó  ))^4 + (n (C^( ))^2 < H _ L '6 H _ L '6 H _ L '6 H _ L >)/(4 (f _ ϕ^(ó  ))^4) - (n (C^( ))^2 < H _ L '6 H _ L '6 H _ R '6 H _ R >)/(2 (f _ ϕ^(ó  ))^4) + (n (C^( ))^2 < H _ R '6 H _ R '6 H _ R '6 H _ R >)/(4 (f _ ϕ^(ó  ))^4)

sdw2Res3 = (sdw2Res2 /. {CQLeft -> UMatrix[cql], CQRight -> UMatrix[cqr], GLeft -> UMatrix[gl], GRight -> UMatrix[gr], UTrace1[HLeft[___]] :> 0} /. $Substitutions // UReduce[#, SMMToMM -> True] & // NMExpand // Expand // IndicesCleanup // Simplify) /. {UMatrix[cql] -> CQLeft, UMatrix[cqr] -> CQRight, UMatrix[gl] -> GLeft, UMatrix[gr] -> GRight} ;

sdw2Res3 // LeafCount

1866

sdw2Res3

1/(48 n^2 (f _ ϕ^(ó  ))^4) (18 n^2 (< Q _ L '6 Q _ L > + < Q _ R '6 Q _ R > - 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)^2 (f _ ϕ^(ó  ))^8 + 12 n^2 (3 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 Q _ L > - < ÷s _ μ _ 1(÷„)^† '6 c _ μ _ 1^R Q _ R '6 ÷„ '6 Q _ L > - 2 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > - < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ '6 c _ μ _ 1^L Q _ L > + 3 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ _ 1(÷„) > - < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L '6 Q _ L > + < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 c _ μ _ 1^L Q _ L > - < ÷„^† '6 c _ μ _ 1^R Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > + < ÷„^† '6 c _ μ _ 1^R Q _ R '6 Q _ R '6 ÷s _ μ _ 1(÷„) > - < ÷„^† '6 χ '6 Q _ L '6 Q _ L > - < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L > - < ÷„^† '6 Q _ R '6 c _ μ _ 1^R Q _ R '6 ÷s _ μ _ 1(÷„) > + < ÷„^† '6 Q _ R '6 χ '6 Q _ L > - < ÷„^† '6 Q _ R '6 Q _ R '6 χ > - < χ^† '6 ÷„ '6 Q _ L '6 Q _ L > + < χ^† '6 Q _ R '6 ÷„ '6 Q _ L > - < χ^† '6 Q _ R '6 Q _ R '6 ÷„ > + 3 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > + 3 < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > + < ÷„^† '6 Q _ R '6 ÷„ '6 χ^† '6 ÷„ '6 Q _ L > + < ÷„ '6 Q _ L '6 ÷„^† '6 χ '6 ÷„^† '6 Q _ R > - < Q _ L '6 ÷s _ μ _ 1(÷„)^† '6 ÷„ '6 ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ > - < Q _ L '6 ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 ÷s _ μ _ 1(÷„) >) (f _ ϕ^(ó  ))^6 + (6 < χ^† '6 χ > n^3 - 2 < G _ (μ _ 1 μ _ 2)^L '6 G _ (μ _ 1 μ _ 2)^L > n^3 - 2 < G _ (μ _ 1 μ _ 2)^R '6 G _ (μ _ 1 μ _ 2)^R > n^3 - 2 i < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 G _ (μ _ 1 μ _ 2)^L > n^3 + 2 i < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 G _ (μ _ 2 μ _ 1)^L > n^3 + 2 i < ÷s _ μ _ 1(÷„)^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷s _ μ _ 2(÷„) > n^3 - 2 i < ÷s _ μ _ 2(÷„)^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷s _ μ _ 1(÷„) > n^3 + < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) > n^3 + 2 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 1(÷„) > n^3 + 6 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 χ^† '6 ÷s _ μ _ 1(÷„) > n^3 - 4 < ÷„^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷„ '6 G _ (μ _ 1 μ _ 2)^L > n^3 + 3 < ÷„^† '6 χ '6 ÷„^† '6 χ > n^3 + 3 < χ^† '6 ÷„ '6 χ^† '6 ÷„ > n^3 + 6 < ÷„ '6 ÷s _ μ _ 1(÷„)^† '6 χ '6 ÷s _ μ _ 1(÷„)^† > n^3 + 6 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) >^2 n^2 + 3 < ÷„^† '6 χ >^2 n^2 + 3 < χ^† '6 ÷„ >^2 n^2 + 2 < G _ (μ _ 1 μ _ 2)^L >^2 n^2 + 2 < G _ (μ _ 1 μ _ 2)^R >^2 n^2 + 6 < ÷„^† '6 χ > < χ^† '6 ÷„ > n^2 + 3 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) > (< ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 2(÷„) > - 2 (< ÷„^† '6 χ > + < χ^† '6 ÷„ >)) n^2 - 96 C^( ) < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > n^2 + 96 C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > n^2 + 4 < G _ (μ _ 1 μ _ 2)^L > < G _ (μ _ 1 μ _ 2)^R > n^2 - 24 < χ^† '6 χ > n - 12 < ÷„^† '6 χ '6 ÷„^† '6 χ > n - 12 < χ^† '6 ÷„ '6 χ^† '6 ÷„ > n + 6 < ÷„^† '6 χ >^2 + 6 < χ^† '6 ÷„ >^2 + 12 < ÷„^† '6 χ > < χ^† '6 ÷„ >) (f _ ϕ^(ó  ))^4 + 24 n^2 C^( ) (4 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > - 2 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > + 2 < ÷„^† '6 χ > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > + 2 < χ^† '6 ÷„ > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > + n < ÷„^† '6 Q _ R '6 χ '6 Q _ L > + n < χ^† '6 Q _ R '6 ÷„ '6 Q _ L > + n < ÷„^† '6 Q _ R '6 ÷„ '6 χ^† '6 ÷„ '6 Q _ L > + n < ÷„ '6 Q _ L '6 ÷„^† '6 χ '6 ÷„^† '6 Q _ R > + n < Q _ L '6 ÷s _ μ _ 1(÷„)^† '6 ÷„ '6 ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ > + n < Q _ L '6 ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 ÷s _ μ _ 1(÷„) > + < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > < Q _ L > + < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > < Q _ L > - < ÷„^† '6 χ '6 Q _ L > < Q _ L > - < ÷„^† '6 Q _ R '6 χ > < Q _ L > - < χ^† '6 ÷„ '6 Q _ L > < Q _ L > - < χ^† '6 Q _ R '6 ÷„ > < Q _ L > + < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > < Q _ R > + < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > < Q _ R > - < ÷„^† '6 χ '6 Q _ L > < Q _ R > - < ÷„^† '6 Q _ R '6 χ > < Q _ R > - < χ^† '6 ÷„ '6 Q _ L > < Q _ R > - < χ^† '6 Q _ R '6 ÷„ > < Q _ R >) (f _ ϕ^(ó  ))^2 + 96 n^2 (C^( ))^2 (4 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + 2 < Q _ L '6 Q _ L > < Q _ R '6 Q _ R > + n < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > + n < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > - 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > < Q _ L > - 2 < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L > < Q _ L > - 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > < Q _ R > - 2 < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L > < Q _ R >))

This is finally the divergent part of the one-loop functional:

endres = (sdw2Res3 // NMExpand // Expand // UReduce[#, SMMToMM -> True] & // IndicesCleanup // NMExpand // Expand) /. UTrace1[UMatrix[(UChiralSpurionLeft | UChiralSpurionRight)[a___]][x_]] :> UTrace1[UMatrix[UChiralSpurion[a]][x]] // Expand ;

endres // LeafCount

2183

endres1 = Simplify /@ Collect[endres /. NM -> nm /. (* Transform back to Minkowski space *) nm[a__] :> (-1)^(Count[{a}, LorentzIndex[__], Infinity, Heads -> True]/2) * NM[a], {_UTrace1}]

3/8 < Q _ L '6 Q _ L >^2 (f _ ϕ^(ó  ))^4 + 3/8 < Q _ R '6 Q _ R >^2 (f _ ϕ^(ó  ))^4 - 3/2 < Q _ R '6 Q _ R > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > (f _ ϕ^(ó  ))^4 - 3/4 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 + 1/4 < ÷s _ μ _ 1(÷„)^† '6 c _ μ _ 1^R Q _ R '6 ÷„ '6 Q _ L > (f _ ϕ^(ó  ))^2 + 1/2 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > (f _ ϕ^(ó  ))^2 + 1/4 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 - 3/4 < ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 + 1/4 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 + 1/4 < ÷„^† '6 c _ μ _ 1^R Q _ R '6 ÷s _ μ _ 1(÷„) '6 Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 c _ μ _ 1^R Q _ R '6 Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 χ '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 + 1/4 < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 c _ μ _ 1^L Q _ L > (f _ ϕ^(ó  ))^2 + 1/4 < ÷„^† '6 Q _ R '6 c _ μ _ 1^R Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 - 1/4 < ÷„^† '6 Q _ R '6 Q _ R '6 χ > (f _ ϕ^(ó  ))^2 - 1/4 < χ^† '6 ÷„ '6 Q _ L '6 Q _ L > (f _ ϕ^(ó  ))^2 - 1/4 < χ^† '6 Q _ R '6 Q _ R '6 ÷„ > (f _ ϕ^(ó  ))^2 - 3/4 < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > (f _ ϕ^(ó  ))^2 - 3/4 < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 + 1/8 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) >^2 + ((n^2 + 2) < ÷„^† '6 χ >^2)/(16 n^2) + ((n^2 + 2) < χ^† '6 ÷„ >^2)/(16 n^2) + (3/2 (f _ ϕ^(ó  ))^4 + (8 (C^( ))^2)/(f _ ϕ^(ó  ))^4) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + 1/24 < G _ (μ _ 1 μ _ 2)^L >^2 + 1/24 < G _ (μ _ 1 μ _ 2)^R >^2 + ((n^2 - 4) < χ^† '6 χ >)/(8 n) - 1/24 n < G _ (μ _ 1 μ _ 2)^L '6 G _ (μ _ 1 μ _ 2)^L > - 1/24 n < G _ (μ _ 1 μ _ 2)^R '6 G _ (μ _ 1 μ _ 2)^R > - 1/24 i n < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 G _ (μ _ 1 μ _ 2)^L > + 1/24 i n < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 G _ (μ _ 2 μ _ 1)^L > + 1/24 i n < ÷s _ μ _ 1(÷„)^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷s _ μ _ 2(÷„) > - 1/24 i n < ÷s _ μ _ 2(÷„)^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷s _ μ _ 1(÷„) > + 1/48 n < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) > + 1/24 n < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 2(÷„) '6 ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 1(÷„) > - 1/8 n < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 χ^† '6 ÷s _ μ _ 1(÷„) > - 1/12 n < ÷„^† '6 G _ (μ _ 1 μ _ 2)^R '6 ÷„ '6 G _ (μ _ 1 μ _ 2)^L > + ((n^2 - 4) < ÷„^† '6 χ '6 ÷„^† '6 χ >)/(16 n) + < ÷„^† '6 χ > (((n^2 + 2) < χ^† '6 ÷„ >)/(8 n^2) + (C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2) + < Q _ L '6 Q _ L > ((3/4 (f _ ϕ^(ó  ))^4 + (4 (C^( ))^2)/(f _ ϕ^(ó  ))^4) < Q _ R '6 Q _ R > - 3/2 (f _ ϕ^(ó  ))^4 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >) + 1/16 < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) > (< ÷s _ μ _ 2(÷„)^† '6 ÷s _ μ _ 2(÷„) > + 2 (< ÷„^† '6 χ > + < χ^† '6 ÷„ > + (8 C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2)) + ((n^2 - 4) < χ^† '6 ÷„ '6 χ^† '6 ÷„ >)/(16 n) - 1/8 n < ÷„ '6 ÷s _ μ _ 1(÷„)^† '6 χ '6 ÷s _ μ _ 1(÷„)^† > + 2 C^( ) ((n C^( ))/(f _ ϕ^(ó  ))^4 - 1) < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > + 2 C^( ) ((n C^( ))/(f _ ϕ^(ó  ))^4 + 1) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > + 1/12 < G _ (μ _ 1 μ _ 2)^L > < G _ (μ _ 1 μ _ 2)^R > - (2 C^( ) < ÷„^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > < ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) >)/(f _ ϕ^(ó  ))^2 + (C^( ) < χ^† '6 ÷„ > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(f _ ϕ^(ó  ))^2 + (((f _ ϕ^(ó  ))^4 + 2 n C^( )) < ÷„^† '6 Q _ R '6 χ '6 Q _ L >)/(4 (f _ ϕ^(ó  ))^2) + (((f _ ϕ^(ó  ))^4 + 2 n C^( )) < χ^† '6 Q _ R '6 ÷„ '6 Q _ L >)/(4 (f _ ϕ^(ó  ))^2) + (((f _ ϕ^(ó  ))^4 + 2 n C^( )) < ÷„^† '6 Q _ R '6 ÷„ '6 χ^† '6 ÷„ '6 Q _ L >)/(4 (f _ ϕ^(ó  ))^2) + (((f _ ϕ^(ó  ))^4 + 2 n C^( )) < ÷„ '6 Q _ L '6 ÷„^† '6 χ '6 ÷„^† '6 Q _ R >)/(4 (f _ ϕ^(ó  ))^2) + (((f _ ϕ^(ó  ))^4 - 2 n C^( )) < Q _ L '6 ÷s _ μ _ 1(÷„)^† '6 ÷„ '6 ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷„ >)/(4 (f _ ϕ^(ó  ))^2) + (((f _ ϕ^(ó  ))^4 - 2 n C^( )) < Q _ L '6 ÷„^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) '6 ÷„^† '6 ÷s _ μ _ 1(÷„) >)/(4 (f _ ϕ^(ó  ))^2) - (C^( ) < ÷s _ μ _ 1(÷„)^† '6 ÷s _ μ _ 1(÷„) '6 Q _ L > < Q >)/(f _ ϕ^(ó  ))^2 - 1/(f _ ϕ^(ó  ))^4 (C^( ) (< ÷s _ μ _ 1(÷„)^† '6 Q _ R '6 ÷s _ μ _ 1(÷„) > (f _ ϕ^(ó  ))^2 + < ÷„^† '6 χ '6 Q _ L > (f _ ϕ^(ó  ))^2 + < ÷„^† '6 Q _ R '6 χ > (f _ ϕ^(ó  ))^2 + < χ^† '6 ÷„ '6 Q _ L > (f _ ϕ^(ó  ))^2 + < χ^† '6 Q _ R '6 ÷„ > (f _ ϕ^(ó  ))^2 + 8 C^( ) < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > + 8 C^( ) < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L >) < Q >)

•SU(3) Cayley-Hamilton rules

su3calhamruleH = CayleyHamiltonRules[{{HLeft[x], HRight[x], HLeft[x], HRight[x]}}, UDimension -> 3] /. UTrace1[HLeft[_]] -> 0 // CycleUTraces

{< H _ L '6 H _ R '6 H _ L '6 H _ R > -> -1/2 < H _ L '6 H _ L > < H _ R >^2 + 2 < H _ R '6 H _ L '6 H _ L > < H _ R > + < H _ L '6 H _ R >^2 + 1/2 < H _ L '6 H _ L > < H _ R '6 H _ R > - 2 < H _ L '6 H _ L '6 H _ R '6 H _ R >}

su3calhamruleUHL = CayleyHamiltonRules[{{HLeft[x], USmall[LorentzIndex[μ1]][x], HLeft[x], USmall[LorentzIndex[μ1]][x]}}, UDimension -> 3] /. UTrace1[(HLeft | USmall[_])[_]] -> 0 // CycleUTraces

{< H _ L '6 u _ μ _ 1 '6 H _ L '6 u _ μ _ 1 > -> < H _ L '6 u _ μ _ 1 >^2 + 1/2 < H _ L '6 H _ L > < u _ μ _ 1 '6 u _ μ _ 1 > - 2 < H _ L '6 H _ L '6 u _ μ _ 1 '6 u _ μ _ 1 >}

su3calhamruleUHR = CayleyHamiltonRules[{{HRight[x], USmall[LorentzIndex[μ1]][x], HRight[x], USmall[LorentzIndex[μ1]][x]}}, UDimension -> 3] /. UTrace1[(HLeft | USmall[_])[_]] -> 0 // CycleUTraces

{< H _ R '6 u _ μ _ 1 '6 H _ R '6 u _ μ _ 1 > -> -1/2 < u _ μ _ 1 '6 u _ μ _ 1 > < H _ R >^2 + 2 < H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 > < H _ R > + < H _ R '6 u _ μ _ 1 >^2 + 1/2 < H _ R '6 H _ R > < u _ μ _ 1 '6 u _ μ _ 1 > - 2 < H _ R '6 H _ R '6 u _ μ _ 1 '6 u _ μ _ 1 >}

su3calhamruleQl = CayleyHamiltonRules[{{NM[Adjoint[MM[x]], UMatrix[UChiralSpurionLeft[]][x], MM[x]], UMatrix[UChiralSpurionLeft[]][x], NM[Adjoint[MM[x]], UMatrix[UChiralSpurionLeft[]][x], MM[x]], UMatrix[UChiralSpurionLeft[]][x]}}, UDimension -> 3] /. UTrace1[UMatrix[UChiralSpurionLeft[]][_]] -> 0 // CycleUTraces

{< ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L '6 Q _ L '6 ÷„^† '6 Q _ L '6 ÷„ > -> -1/2 < Q _ L '6 Q _ L > < ÷„^† '6 Q _ L '6 ÷„ >^2 + 2 < ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L '6 Q _ L > < ÷„^† '6 Q _ L '6 ÷„ > + < ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L >^2 + 1/2 < Q _ L '6 Q _ L > < ÷„^† '6 Q _ L '6 ÷„ '6 ÷„^† '6 Q _ L '6 ÷„ > - < ÷„^† '6 Q _ L '6 ÷„ '6 ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L '6 Q _ L > - < ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ L '6 ÷„ '6 Q _ L >}

su3calhamruleQlr = CayleyHamiltonRules[{{NM[Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]], UMatrix[UChiralSpurionLeft[]][x], NM[Adjoint[MM[x]], UMatrix[UChiralSpurionRight[]][x], MM[x]], UMatrix[UChiralSpurionLeft[]][x]}}, UDimension -> 3] /. UTrace1[UMatrix[UChiralSpurionLeft[]][_]] -> 0 // CycleUTraces

{< ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ > -> -1/2 < Q _ L '6 Q _ L > < ÷„^† '6 Q _ R '6 ÷„ >^2 + 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > < ÷„^† '6 Q _ R '6 ÷„ > + < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + 1/2 < Q _ L '6 Q _ L > < ÷„^† '6 Q _ R '6 ÷„ '6 ÷„^† '6 Q _ R '6 ÷„ > - < ÷„^† '6 Q _ R '6 ÷„ '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > - < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >}

clr1 = CayleyHamiltonRules[{{HLeft[x], HRight[x], HLeft[x], HRight[x]}}, UDimension -> 3] /. UTrace1[(HLeft | USmall[_])[_]] -> 0

{< H _ L '6 H _ R '6 H _ L '6 H _ R > -> -1/2 < H _ L '6 H _ L > < H _ R >^2 + 2 < H _ R '6 H _ L '6 H _ L > < H _ R > + < H _ L '6 H _ R >^2 + 1/2 < H _ L '6 H _ L > < H _ R '6 H _ R > - 2 < H _ L '6 H _ L '6 H _ R '6 H _ R >}

su3calham = (clr1[[1, 1]] - (clr1[[1, 2]]) /. $Substitutions // UReduce[#, SMMToMM -> True] &) /. su3calhamrule4 /. UTrace1[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][x]] -> 0 // NMExpand // Expand // UOrder

2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + < Q _ L '6 Q _ L > < Q _ R '6 Q _ R > - 4 < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L > - 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >

List @@ su3calham

{< Q _ L '6 Q _ L > < Q _ R '6 Q _ R >, 2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2, -4 < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L >, -2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >}

su3calhamruleQlQrQlQr = (-1/2 su3calham[[-1]] /. {μ -> μ_, x -> x_}) -> 1/2 Plus @@ Drop[su3calham, {-1}]

< ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > -> 1/2 (2 < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >^2 + < Q _ L '6 Q _ L > < Q _ R '6 Q _ R > - 4 < ÷„^† '6 Q _ R '6 Q _ R '6 ÷„ '6 Q _ L '6 Q _ L >)

clr1 = CayleyHamiltonRules[{{HLeft[x], USmall[LorentzIndex[μ]][x], HLeft[x], USmall[LorentzIndex[μ]][x]}}, UDimension -> 3] /. UTrace1[(HLeft | USmall[_])[_]] -> 0

{< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 + 1/2 < H _ L '6 H _ L > < u _ μ '6 u _ μ > - 2 < H _ L '6 H _ L '6 u _ μ '6 u _ μ >}

su3calham = (clr1[[1, 1]] - (clr1[[1, 2]]) /. $Substitutions // UReduce[#, SMMToMM -> True] &) /. su3calhamrule4 /. UTrace1[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][x]] -> 0 // NMExpand // Expand // UOrder

< ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >^2 + 2 < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L > < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > + < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2 - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L > - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R > + 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L > - 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷s _ μ(÷„) '6 Q _ L > + 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) > + < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > - 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ > - < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > - < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L > - 2 < Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >

List @@ su3calham

{-1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L >, -1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R >, < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >^2, 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >, < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2, 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L >, -2 < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷s _ μ(÷„) '6 Q _ L >, 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) >, < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L >, -2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ >, -< ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >, -< ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >, -2 < Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >}

su3calhamruleQlQl = (-su3calham[[-2]] /. {μ -> μ_, x -> x_}) -> Plus @@ Drop[su3calham, {-2}] // UOrder

< ÷„^† '6 ÷s _ μ_(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ_(÷„) '6 Q _ L > -> < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >^2 + 2 < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L > < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > + < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2 - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L > - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R > + 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L > - 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷s _ μ(÷„) '6 Q _ L > + 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) > + < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < ÷„^† '6 Q _ R '6 ÷„ '6 Q _ L > - 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 Q _ R '6 ÷„ > - < ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ > - 2 < Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 ÷s _ μ(÷„)^† '6 Q _ R '6 ÷„ >

clr1 = CayleyHamiltonRules[{{UMatrix[UChiralSpurionLeft[]][x], NM[Adjoint[MM[x]], CovariantFieldDerivative[MM[x], x, {μ}]], UMatrix[UChiralSpurionLeft[]][x], NM[Adjoint[MM[x]], CovariantFieldDerivative[MM[x], x, {μ}]]}}, UDimension -> 3] /. UTrace1[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][_] | NM[Adjoint[MM[x]], CovariantFieldDerivative[MM[x], x, {μ}]]] -> 0

{< ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L > -> < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2 + 1/2 < Q _ L '6 Q _ L > < ÷„^† '6 ÷s _ μ(÷„) '6 ÷„^† '6 ÷s _ μ(÷„) > - 2 < ÷„^† '6 ÷s _ μ(÷„) '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L >}

su3calham = clr1[[1, 1]] - (clr1[[1, 2]]) // UReduce // NMExpand // Expand // UOrder

-< ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2 + 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L > - 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L > + < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >

List @@ su3calham

{1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L >, -< ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2, -2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L >, < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >}

su3calhamruleQlQl = (su3calham[[-1]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su3calham, {-1}]

< ÷„^† '6 ÷s _ μ_(÷„) '6 Q _ L '6 ÷„^† '6 ÷s _ μ_(÷„) '6 Q _ L > -> < ÷„^† '6 ÷s _ μ(÷„) '6 Q _ L >^2 - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ L '6 Q _ L > + 2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) '6 Q _ L '6 Q _ L >

clr1 = CayleyHamiltonRules[{{UMatrix[UChiralSpurionRight[]][x], NM[CovariantFieldDerivative[MM[x], x, {μ}], Adjoint[MM[x]]], UMatrix[UChiralSpurionRight[]][x], NM[CovariantFieldDerivative[MM[x], x, {μ}], Adjoint[MM[x]]]}}, UDimension -> 3] /. UTrace1[UMatrix[(UChiralSpurionRight | UChiralSpurionLeft)[]][_] | NM[Adjoint[MM[x]], CovariantFieldDerivative[MM[x], x, {μ}]]] -> 0

{< ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) > -> < ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >^2 + 1/2 < Q _ R '6 Q _ R > < ÷„^† '6 ÷s _ μ(÷„) '6 ÷„^† '6 ÷s _ μ(÷„) > - 2 < ÷„^† '6 ÷s _ μ(÷„) '6 ÷„^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) >}

su3calham = clr1[[1, 1]] - (clr1[[1, 2]]) // UReduce // NMExpand // Expand

-< ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >^2 + 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R > - 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) > + < ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >

List @@ su3calham

{1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R >, -< ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >^2, -2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) >, < ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >}

su3calhamruleQrQr = (su3calham[[-1]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su3calham, {-1}]

< ÷„^† '6 Q _ R '6 ÷s _ μ_(÷„) '6 ÷„^† '6 Q _ R '6 ÷s _ μ_(÷„) > -> < ÷„^† '6 Q _ R '6 ÷s _ μ(÷„) >^2 - 1/2 < ÷s _ μ(÷„)^† '6 ÷s _ μ(÷„) > < Q _ R '6 Q _ R > + 2 < ÷s _ μ(÷„)^† '6 Q _ R '6 Q _ R '6 ÷s _ μ(÷„) >

su3calha = CayleyHamiltonRules[{{UMatrix[a], UMatrix[a], UMatrix[a], UMatrix[a]}}, UDimension -> 3] // ExpandAll

{< a '6 a '6 a '6 a > -> < a >^4/6 - < a '6 a > < a >^2 + 4/3 < a '6 a '6 a > < a > + 1/2 < a '6 a >^2}

su3calhamrule4 = (su3calha[[1, 1]] /. UMatrix[a] -> a_) -> (su3calha[[1, 2]] /. UMatrix[a] -> a)

< a_ '6 a_ '6 a_ '6 a_ > -> < a >^4/6 - < a '6 a > < a >^2 + 4/3 < a '6 a '6 a > < a > + 1/2 < a '6 a >^2

•SU(3) equations of motion

•The identity ∇ _ μ(H _ (R, L)) = H _ (μ ±)+i/2[u _ μ,H _ (L, R)],              H _ (μ ±):= u^†c _ R^μQ _ Ru ± u c _ L^μQ _ Lu^†

•Equations of motion

•SU(3) intermediate results

•SU(3) final results

Converted by Mathematica  (July 10, 2003)