One-loop QCD self-energy diagrams using package X

T.Becher, Oct. 15, 2020

Needs["X`"]

Note: ξ = 0 corresponds to Feynman gauge.

Fermion self energy

Numerator of the diagram

diag=CF (g^(2 ))DiracMatrix[(Subscript[γ, μ]),γ.(p+k)+me 𝟙,(Subscript[γ, ν])](-(Subscript[𝕘, μ,ν])k.k +ξ (Subscript[k, μ]) (Subscript[k, ν]));

diag2=diag//Contract

diag3=diag2//FermionLineExpand

Add denominators and loop integration

diag4=LoopIntegrate[diag3,k,{k,0,2},{k+p,m}]

diag5=LoopRefine[diag4];

diag6=diag5//Simplify

Note that the package suppresses a factor

extra=((I (E^(-EulerGamma ϵ)))/(((4π))^(d/2)));

Result for the divergent term. 

The result corresponds to Σ

divergence=+I (extra/.d->4/.ϵ->0) Coefficient[diag6,ϵ,-1]/. (g^2)->4π αs//Simplify

((4π)/(αs)) divergence

Gluon self energy

Note: the results correspond to:

 -i (Subscript[Π, μν])

Fermion contribution to gluon self energy

vacuumPolFerm=(g^2)nf TF LoopIntegrate[Spur[(Subscript[γ, ν]),γ.k+m 𝟙,(Subscript[γ, μ]),γ.(k+q)+m 𝟙],k,{k,m},{k+q,m}]

vacuumPolFerm2=LoopRefine[vacuumPolFerm]

vacuumPolFerm3L=vacuumPolFerm2//Longitudinal//LoopRefine

vacuumPolFerm3=vacuumPolFerm2//Transverse//LoopRefine//DiscExpand

Note that the package suppresses a factor

extra=((I (E^(-EulerGamma ϵ)))/(((4π))^(d/2)));

vacuumPolFermD=+I (extra/.d->4/.ϵ->0) Coefficient[vacuumPolFerm2,ϵ,-1]/. (g^2)->4π αs//Simplify

((4π)/(αs))  vacuumPolFermD

Ghost contribution to gluon self energy

I’m suppressing the color conservation Kronecker (Subscript[δ, a,b])

vacuumGhost=(g^2)CA  LoopIntegrate[((Subscript[k, μ])+(Subscript[q, μ]))(Subscript[k, ν]),k,{k,0},{k+q,0}]

vacuumGhost2=LoopRefine[vacuumGhost]

vacuumGhost3L=vacuumGhost2//Longitudinal//LoopRefine

vacuumGhost3T=vacuumGhost2//Transverse//LoopRefine//DiscExpand

vacuumGhostD=+I (extra/.d->4/.ϵ->0) Coefficient[vacuumGhost2,ϵ,-1]/. (g^2)->4π αs//Simplify

((4π)/(αs))  vacuumGluonD

Gluon contribution to gluon self energy

feynman={VG3[k_,p_,q_,μ_,ν_,ρ_]-> I g ((Subscript[𝕘, μ,ν])(Subscript[(k-p), ρ])+(Subscript[𝕘, ν,ρ])(Subscript[(p-q), μ])+(Subscript[𝕘, ρ,μ]) (Subscript[(q-k), ν])),
(* Color structure: -I f[a,b,c]*)
Gα[k_,α_,β_]->((-I )/(k.k))((Subscript[𝕘, α,β])-ξ(((Subscript[k, α])(Subscript[k, β]))/(k.k)))
};

numer=(((k.k))^2) (((k.k+2 k.q+q.q))^2)(VG3[q,-q-k,k,μ,α1,β1]VG3[q+k,-q,-k,α2,ν,β2]Gα[k+q,α1,α2]Gα[k,β1,β2]/.feynman)//Simplify//Expand//Simplify//Contract//Simplify//Expand

vacuumGluon=-(CA/2) LoopIntegrate[numer ,k,{k,0,2},{k+q,0,2}];

vacuumGluon2=LoopRefine[vacuumGluon]

vacuumGluon3L=vacuumGluon2//Longitudinal//LoopRefine

vacuumGluon3T=vacuumGluon2//Transverse//LoopRefine//DiscExpand

vacuumGluonD=+I (extra/.d->4/.ϵ->0) Coefficient[vacuumGluon2,ϵ,-1]/. g->(Sqrt[4π αs])//Simplify

((4π)/(αs))  vacuumGluonD//Expand

Transversality

Note that the sum of the ghost and gluon loops are transverse

vacuumGluon3L+vacuumGhost3L

Explicitly:

vacuumGluonD+vacuumGhostD//Simplify

((4π)/(αs))(vacuumGluonD+vacuumGhostD)/.ξ->0//Simplify