Let us consider one-loop counterterms of the the vector field operator, which appears in the theory of the massive vector field: Dαβ=δαβ◻−λ∇α∇β+Pαβ, where ◻=gμν∇μ∇ν 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=(gμνδβα−λ2(gμβδνα+gνβδμα))∇μ∇ν+Pαβ+λ2Rαβ,
where Rαβ is the Ricci tensor.
It can be easily found that Kn and (Kn)−1 are:
(Kn)αβ=δβα−λnαnβ,(Kn)−1αβ=δβα+λ1−λnαnβ.
Hereby, at this point we have whole set of input tensors required by the algorithm: Kμναβ=gμνδβα−λ2(gμβδνα+gνβδμα),Sμαβ=0,Wαβ=Pαβ+λ2Rαβ,(Kn)−1αβ=δβα+λ1−λnαnβ,Fμναβ=Rμναβ.
In the further calculations we shall use the definition λ=γ/(1+γ) for convenience (so γ=λ/(1−λ)).
The following code calculates one-loop counterterms of the vector field theory in curved space-time (here g
used for γ):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | //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}} |
Γ(1)∞=116π(d−4)∫d4x√−g(1120(−32+5γ2+10γ)RϵμRϵμ+148γ2P2++1240R2(28+5γ2+20γ)+124(γ2+12+6γ)PβαPαβ++112γ(4+γ)RνϵPνϵ+124R(γ2+4+2γ)P)