====== Vector field ======
----
====Code====
Let us consider one-loop counterterms of the the vector field operator, which appears in the theory of the massive vector field:
\[
D_\alpha{}^\beta = 
\delta_\alpha{}^\beta \Box  - \lambda \, \nabla_\alpha \nabla^\beta +
P_\alpha{}^\beta,
\]
where \(\Box =g^{\mu\nu}\nabla_\mu\nabla_\nu\) and \(\lambda = 1 + 1/\xi\). 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:

\[
D_\alpha{}^\beta = \left( g^{\mu\nu} \delta_\alpha^\beta - \frac{\lambda}{2}
\left( g^{\mu\beta}\delta_\alpha^\nu + g^{\nu\beta}\delta_\alpha^\mu \right)
\right) \nabla_\mu\nabla_\nu + P_\alpha{}^\beta +
\frac{\lambda}{2}R_\alpha{}^\beta,
\]

where $R_{\alpha\beta}$ is the Ricci tensor.

It can be easily found that $Kn$ and $(Kn)^{-1}$ are:

\begin{equation*}
(Kn)_{\alpha}{}^\beta = \delta_{\alpha}^\beta - \lambda \, n_\alpha n^\beta,
\qquad
(Kn)^{-1}{}_{\alpha}{}^\beta = \delta_{\alpha}^\beta + \frac{\lambda}{1-
\lambda} n_\alpha n^\beta.
\end{equation*}

Hereby, at this point we have whole set of input tensors required by the algorithm:
\begin{eqnarray*}
&&K^{\mu\nu}{}_\alpha{}^\beta \,=\, g^{\mu\nu} \delta_\alpha^\beta -
\frac{\lambda}{2}\left( g^{\mu\beta}\delta_\alpha^\nu +
g^{\nu\beta}\delta_\alpha^\mu \right)\,,
\quad S^\mu{}_\alpha{}^\beta \,=\, 0\,,
\quad W_\alpha{}^\beta \,=\, P_\alpha{}^\beta + 
\frac{\lambda}{2}R_\alpha{}^\beta\,,
\\&& (Kn)^{-1}{}_{\alpha}{}^\beta \,=\, \delta_{\alpha}^\beta + 
\frac{\lambda}{1-
\lambda} n_\alpha n^\beta\,,
\quad F_{\mu\nu}{}_\alpha{}^\beta \,=\, R_{\mu\nu}{}_\alpha{}^\beta\,.
\end{eqnarray*}

In the further calculations we shall use the definition $\lambda =\gamma/(1+\gamma)$ for convenience (so $\gamma = \lambda/(1-\lambda)$).

The following code calculates one-loop counterterms of the vector field theory in curved space-time (here ''g'' used for $\gamma$):

<sxh groovy; gutter: true>
//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
</sxh>
<sxh plain; gutter: false>
   > (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}}
</sxh>
In order to obtain one-loop counterterms in the dimensional regularization, one should multiply the result produced by Redberry by $1/16\pi(d-4)$ and integrate it over the space-time volume:

\begin{multline*}
\Gamma^{(1)}_{\infty} = \frac{1}{16\pi(d-4)} \int d^4 x  \sqrt{-g} \left(  
\frac{1}{120}(-32+5 \gamma^2+10 \gamma) R_{\epsilon\mu} R^{\epsilon\mu}
+\frac{1}{48}\gamma^2 P^2 +\right.\\
+\frac{1}{240} R^2 (28+5 \gamma^2+20 \gamma)+\frac{1}{24} (\gamma^2+12+6 \gamma)
P_{\beta\alpha} P^{\alpha\beta} +\\
\left.+\frac{1}{12}\gamma (4+\gamma) R_{\nu\epsilon} P^{\nu\epsilon}
+\frac{1}{24} R (\gamma^2+4+2 \gamma) P\right)
\end{multline*}

====See also====
  * Related guides: [[documentation:guide:calculating_one-loop_counterterms]]
  * Related tutorials: [[documentation:tutorials:minimal_second_order_operator]], [[documentation:tutorials:minimal_fourth_order_operator]]
  * Related reference material: [[documentation:ref:collect]], [[documentation:ref:factor]]