Let' s calculate one-loop counterterms of minimal second order operator using methods described in Calculating one-loop counterterms. The operator is: Dij=δij◻+Wij
The input quantities needed for the algorithm described in Calculating one-loop counterterms are: Kμναβ=gμνδβα,Sμαβ=0,Wαβ=Wαβ,(Kn)−1αβ=δβα,Fμναβ=Fμναβ.
The following code produces the result:
1 2 3 4 5 6 7 8 9 10 11 12 | //setup symmetries of Riemann tensor addSymmetries 'R_abcd' , - [[ 0 , 1 ]] .p , [[ 0 , 2 ], [ 1 , 3 ]] .p setSymmetric 'R_ab' //(Kn)^(-1) def iK = 'iK_a^b = d^b_a' .t def K = 'K^lm_a^b = d^b_a*g^{lm}' .t def S = 'S^lab = 0' .t def W = 'W_a^b = W_a^b' .t def F = 'F_lmab = F_lmab' .t def div = oneloopdiv2(iK, K, S, W, F) def counterterms = EliminateDueSymmetries > > div .counterterms println counterterms |
> counterterms = (1/30)*R**2+(1/12)*F_{bm}^{c}_{d}*F^{bmd}_{c} +(1/2)*W^{a_{5}}_{a}*W^{a}_{a_{5}} +(1/6)*R*W^{a_{5}}_{a_{5}}+(1/15)*R_{ad}*R^{ad} |
Multiplying the produced result by 1/16π(d−4) and integrating over the space-time volume gives:
Γ(1)∞=116π(d−4)∫d4x√−g(130R2+112FνβϵρFνβρϵ+115RδνRδν+12WαρWρα+16RWββ),
where Fμναβ is a curvature tensor with respect to the principal bundle, Rμν is a Ricci tensor and R is a Riemann scalar curvature.