Quantum gravity and matter fields in a general background gauge

Abstract

We analyse the gauge-dependence of the effective action in an interacting quantum theory of gravitational and matter fields. An explicit off-shell result is obtained in a general background gauge at one-loop order, which reduces in a particular gauge to the effective action found by 't Hooft-Veltman. We confirm the validity of DeWitt-Kallosh theorem, which implies that the on-shell effective action should be independent of the gauge-fixing parameter. We employ this theorem to expose the non-renormalizability of the theory in a general background gauge.

Figures (4)

• Figure 1: One-loop contributions to $\left \langle \mathfrak{g} ^{\mu \nu} \mathfrak{g} ^{\alpha \beta}\right\rangle$ [(a), (b), (c)] and to $\left \langle \phi \phi\right\rangle$ [(d)]. The wavy, curly, dotted, dashed and solid lines denote the quantum graviton, background graviton, ghost, quantum scalar and the background scalar, respectively. The momenta satisfies the relation: $q =p+k$.
• Figure 2: One-loop contributions (permutations have been omitted) to the background $3$-point function $\langle \mathfrak{g}^{\mu \nu} (k) \phi (k_1) \phi (k_2) \rangle$. We have that $q = p+k$, $r =p+k-k_1$ and $k = k_{1} + k_{2}$.
• Figure 3: One-loop contributions (permutations have been omitted) to the background $4$-point function $\left \langle \phi (k_{1} )\phi (k_{2} )\phi (k_{3} )\phi (k_{4} )\right\rangle$. Here, $q = p+k_{1} + k_{2}$, $r = p + k_{4}$ and $s = p +k_{2}$.
• Figure 4: One-loop contributions to the ghost-background graviton-ghost vertex.