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Minimal second order operator


Code

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:

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//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(d4) and integrating over the space-time volume gives:

Γ(1)=116π(d4)d4xg(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.

See also