====== Minimal second order operator ======
----
====Code====
Let' s calculate one-loop counterterms of minimal second order operator using methods described in [[documentation:guide:calculating_one-loop_counterterms]]. The operator is:
 \[
D_i{}^j\,=\,\delta_i{}^j\,\Box + W_i{}^j
\]

The input quantities needed for the algorithm described in [[documentation:guide:calculating_one-loop_counterterms]] are:
\begin{eqnarray*}
&&K^{\mu\nu}{}_\alpha{}^\beta \,=\, g^{\mu\nu} \delta_\alpha^\beta,
\quad S^\mu{}_\alpha{}^\beta \,=\, 0\,,
\quad W_\alpha{}^\beta \,=\, W_\alpha{}^\beta\,,
\\&& (Kn)^{-1}{}_{\alpha}{}^\beta \,=\, \delta_{\alpha}^\beta\,,
\quad F_{\mu\nu}{}_\alpha{}^\beta \,=\, F_{\mu\nu}{}_\alpha{}^\beta\,.
\end{eqnarray*}

The following code produces the result:
<sxh groovy; gutter: true>
//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
</sxh>
<sxh plain; gutter: false>
   > 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}
</sxh>

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

\[
\Gamma^{(1)}_{\infty} = \frac{1}{16\pi(d-4)} \int d^4 x  \sqrt{-g} \left(  \frac{1}{30} R^2 +\frac{1}{12} F_{\nu\beta }{}^{\epsilon}{}_{\rho} F^{\nu\beta\rho}{}_{\epsilon}+\frac{1}{15} R_{\delta \nu } R^{\delta\nu }+\frac{1}{2} W^{\alpha }{}_{\rho} W^{\rho}_{\alpha}+\frac{1}{6} R W^{\beta }{}_{\beta}\right),
\]

where $F_{\mu\nu\alpha\beta}$ is a curvature tensor with respect to the principal bundle, $R_{\mu\nu}$ is a Ricci tensor and $R$ is a Riemann scalar curvature.
            
====See also====
  * Related guides: [[documentation:guide:calculating_one-loop_counterterms]]
  * Related tutorials: [[documentation:tutorials:minimal_fourth_order_operator]] 
  * Related reference material: [[documentation:ref:eliminateduesymmetries]]