•Seeley-DeWitt coefficients

SeeleyDeWitt[0][x_] = 1

1

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

1/8 (-2 < χ _ + '6 σ^k _ 1 '6 σ^k _ 1 > - < σ^k _ 1 '6 σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 > + 2 < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^k _ 1 '6 u _ ρ _ 1 > - < σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 '6 σ^k _ 1 >)

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]]

1/48 ((-< ∂ _ 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 (< ∂ _ 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 ((< χ _ + '6 σ^j '6 σ^k _ 1 > + < χ _ + '6 σ^k _ 1 '6 σ^j > + < σ^k _ 1 '6 σ^j '6 u _ ρ _ 1 '6 u _ ρ _ 1 > - 2 < σ^k _ 1 '6 u _ ρ _ 1 '6 σ^j '6 u _ ρ _ 1 > + < σ^k _ 1 '6 u _ ρ _ 1 '6 u _ ρ _ 1 '6 σ^j >) '6 (< 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 >))

A bit of reduction:

sdw2Res = SeeleyDeWitt[2][x] // NMExpand // Expand // SUNReduce // CycleUTraces ;

sdw2Res1 = sdw2Res // UGammaTrick // NMExpand // Expand // CycleUTraces // Simplify // IndicesCleanup ;

sdw2Res2 = sdw2Res1 /. $Substitutions // NMExpand // Expand // CycleUTraces // UReduce // IndicesCleanup ;

This is finally the divergent part of the one-loop functional (notice that  we do not use the inverse Bernese convention for the left/right handed fields  and the field strengths, but the one of Donoghue et al.):

endres = sdw2Res2 // NMExpand // Expand // UReduce[#, SMMToMM -> True] & // CayleyHamiltonTrick // IndicesCleanup // UReduce // Expand

3/16 < ÷s _ ρ _ 2(÷„)^† '6 ÷s _ ρ _ 1(÷„) >^2 + 11/144 < ÷„^† '6 χ >^2 + 11/144 < χ^† '6 ÷„ >^2 + 3/32 < ÷s _ ρ _ 1(÷„)^† '6 ÷s _ ρ _ 1(÷„) > < ÷s _ ρ _ 2(÷„)^† '6 ÷s _ ρ _ 2(÷„) > + 1/8 < ÷s _ ρ _ 1(÷„)^† '6 ÷s _ ρ _ 1(÷„) > < ÷„^† '6 χ > + 1/8 < ÷s _ ρ _ 1(÷„)^† '6 ÷s _ ρ _ 1(÷„) > < χ^† '6 ÷„ > + 11/72 < ÷„^† '6 χ > < χ^† '6 ÷„ > + 5/24 < χ^† '6 χ > - 1/8 < L^( ) _ (ρ _ 1 ρ _ 2) '6 L^( ) _ (ρ _ 1 ρ _ 2) > - 1/8 < R^( ) _ (ρ _ 1 ρ _ 2) '6 R^( ) _ (ρ _ 1 ρ _ 2) > + 1/4 i < ÷s _ ρ _ 1(÷„)^† '6 ÷s _ ρ _ 2(÷„) '6 R^( ) _ (ρ _ 1 ρ _ 2) > - 1/4 i < ÷s _ ρ _ 1(÷„)^† '6 L^( ) _ (ρ _ 1 ρ _ 2) '6 ÷s _ ρ _ 2(÷„) > - 3/8 < ÷„^† '6 ÷s _ ρ _ 1(÷„) '6 χ^† '6 ÷s _ ρ _ 1(÷„) > - 1/4 < ÷„^† '6 L^( ) _ (ρ _ 1 ρ _ 2) '6 ÷„ '6 R^( ) _ (ρ _ 1 ρ _ 2) > + 5/48 < ÷„^† '6 χ '6 ÷„^† '6 χ > + 5/48 < χ^† '6 ÷„ '6 χ^† '6 ÷„ > - 3/8 < ÷„ '6 ÷s _ ρ _ 1(÷„)^† '6 χ '6 ÷s _ ρ _ 1(÷„)^† >

Converted by Mathematica  (July 10, 2003)