Minimal fourth order operator

Code

Let' s calculate one-loop counterterms of minimal fourth order operator using methods described in Calculating one-loop counterterms. The operator is: $D_i{}^j = \delta_i{}^j \Box^2 + W^{\mu\nu}{}_{i}{}^j \nabla_\mu\nabla_\nu + M_i{}^{j}$

The input quantities needed for the algorithm described in Calculating one-loop counterterms are: $\begin{eqnarray*} &&K^{\lambda\mu\gamma\delta}{}_{\alpha}{}^\beta = \delta_\alpha^\beta \,\frac{1}{3}\,(g^{\lambda\mu} g^{\gamma\delta} + g^{\lambda\gamma} g^{\mu\delta} + g^{\lambda \delta} g^{\mu \gamma} )\,,\\ && S^{\mu\nu\gamma}{}_\alpha{}^\beta \,=\, 0\,,\\ && W^{\mu\nu}{}_\alpha{}^\beta \,=\, W^{\mu\nu}{}_\alpha{}^\beta\,,\\ && N^{\mu}{}_\alpha{}^\beta \,=\, 0\,,\\ && M_\alpha{}^\beta \,=\, M_\alpha{}^\beta\,,\\ && (Kn)^{-1}{}_{\alpha}{}^\beta \,=\, \delta_{\alpha}^\beta\,,\\ && F_{\mu\nu}{}_\alpha{}^\beta \,=\, F_{\mu\nu}{}_\alpha{}^\beta\,. \end{eqnarray*}$

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'
addSymmetry 'W_lmab', [[0, 1]].p
 
def iK = 'iK_a^b = d_a^b'.t
def K = ('K^lmcd_a^b = d_a^b*1/3*(g^lm*g^cd + g^lc*g^md + g^ld*g^mc)').t
def S = 'S^lmpab = 0'.t
def W = 'W^lm_a^b = W^lm_a^b'.t
def N = 'N^pab=0'.t
def M = 'M_a^b = M_a^b'.t
def F = 'F_lmab = F_lmab'.t
 
def div = oneloopdiv4(iK, K, S, W, N, M, F)
def counterterms = EliminateDueSymmetries >> div.counterterms
counterterms = 'M^l_l = M'.t >> counterterms
counterterms = 'W_lm^a_a = W_lm'.t >> counterterms
counterterms = 'W^a_a = W'.t >> counterterms
println counterterms

> counterterms = (2/3)*F_{mb}^{e}_{a_{5}}*F^{mba_{5}}_{e}-(1/9)*W^{lm}*R_{lm}
          +(1/9)*R*W-M+(1/24)*W^{bda}_{a_{5}}*W_{bd}^{a_{5}}_{a}
          +(1/48)*W_{d}^{da}_{a_{5}}*W_{b}^{ba_{5}}_{a}
          -(32/135)*R^{al}*R_{al}+(44/135)*R**2

Multiplying the produced result by $1\left/16\pi(d-4)\right.$ and integrating over the space-time volume gives:

$\begin{multline*} \Gamma^{(1)}_{\infty} = \frac{1}{16\pi(d-4)} \int d^4 x \sqrt{-g} \left( \,-M\,+\,\frac{2}{3} F_{\nu\beta}{}^{\epsilon}{}_{\rho} F^{\nu\beta\rho}{}_{\epsilon} \right.\\ -\frac{32}{135} R_{\mu\nu} R^{\mu\nu} + \frac{44}{135} R^2+\frac{1}{9} R W - \frac{1}{9} R_{\mu\nu}W^{\mu\nu}\\ \left.+\frac{1}{24} W^{\epsilon\delta\alpha}{}_{\rho} W_{\epsilon\delta}{}^{\rho}{}_{\alpha}+\frac{1}{48} W_{\delta}{}^{\delta\alpha}{}_{\rho} W_{\beta}{}^{\beta\rho}{}_{\alpha} \right), \end{multline*}$

where $F_{\mu\nu\alpha\beta}$ is a curvature tensor with respect to the principal bundle, $R_{\mu\nu}$ is a Ricci tensor, $R$ is a Riemann scalar curvature, $M_{\mu}{}^{\mu} = M$ , $W_{\mu\nu}{}^\alpha{}_\alpha = W_{\mu\nu}$ and $W^\alpha{}_\alpha = W$ .

Minimal fourth order operator

Code

See also