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Vector field


Code

Let us consider one-loop counterterms of the the vector field operator, which appears in the theory of the massive vector field: Dαβ=δαβλαβ+Pαβ, where =gμνμν and λ=1+1/ξ. This is a second order operator, and in order to rewrite it in the symmetric form, it is necessary to symmetrize the second term by commutation of the covariant derivatives:

Dαβ=(gμνδβαλ2(gμβδνα+gνβδμα))μν+Pαβ+λ2Rαβ,

where Rαβ is the Ricci tensor.

It can be easily found that Kn and (Kn)1 are:

(Kn)αβ=δβαλnαnβ,(Kn)1αβ=δβα+λ1λnαnβ.

Hereby, at this point we have whole set of input tensors required by the algorithm: Kμναβ=gμνδβαλ2(gμβδνα+gνβδμα),Sμαβ=0,Wαβ=Pαβ+λ2Rαβ,(Kn)1αβ=δβα+λ1λnαnβ,Fμναβ=Rμναβ.

In the further calculations we shall use the definition λ=γ/(1+γ) for convenience (so γ=λ/(1λ)).

The following code calculates one-loop counterterms of the vector field theory in curved space-time (here g used for γ):

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//setup symmetries of Riemann tensor
addSymmetries 'R_abcd', -[[0, 1]].p, [[0, 2], [1, 3]].p
setSymmetric 'R_ab', 'P_ab'
 
//tensor (Kn)^{-1}
def iK = 'iK_a^b = d_a^b + g*n_a*n^b'.t
//tensor K
def K = '''K^{mn}_i^j =
           g^{mn}*d_{i}^{j}
           - g/(2*(1+g))*(g^{mj}*d_i^n + g^{nj}*d_i^m)'''.t
//tensor S
def S = 'S^p^l_m=0'.t
//tensor W
def W = 'W^{a}_{b}=P^{a}_{b} + g/(2*(1+g))*R^a_b'.t
//tensor F
def F = 'F_abcd = R_abcd'.t
//divergent part of one-loop effective action
def div = oneloopdiv2(iK, K, S, W, F)
//counterterms
def counterterms = div.counterterms[1];
//simplifying counterterms
counterterms = ('P^a_a = P'.t
        & Collect['R', 'P', Factor[[FactorScalars: false]]]
) >> counterterms
println counterterms
> (1/120)*(10*g-32+5*g**2)*R^{lm}*R_{lm}+(1/24)*P*R*(2*g+g**2+4)
       +(1/12)*g*(g+4)*P^{ml}*R_{lm}+(1/48)*g**2*P**2
       +(1/240)*(20*g+28+5*g**2)*R**2
       +(1/24)*(6*g+g**2+12)*P^{a_{5}}_{a}*P^{a}_{a_{5}}
In order to obtain one-loop counterterms in the dimensional regularization, one should multiply the result produced by Redberry by 1/16π(d4) and integrate it over the space-time volume:

Γ(1)=116π(d4)d4xg(1120(32+5γ2+10γ)RϵμRϵμ+148γ2P2++1240R2(28+5γ2+20γ)+124(γ2+12+6γ)PβαPαβ++112γ(4+γ)RνϵPνϵ+124R(γ2+4+2γ)P)

See also