Complex bumblebee model

Abstract

We formulate a renormalizable complex extension of the bumblebee theory in which the bumblebee field is promoted to a complex one and coupled to an Abelian gauge sector. Besides the minimal gauge covariant interaction, the model includes a longitudinal kinetic term controlled by a dimensionless parameter $g_l$ and a non-minimal magnetic-type coupling $g_m$ between the complex bumblebee and the photon. Using dimensional regularization and minimal subtraction, we determine the one-loop UV divergences of the two-, three-, and four-point functions relevant to the renormalization of the gauge, longitudinal, and quartic sectors. We obtain the corresponding counterterms and derive the one-loop renormalization-group functions for $e$, $g_l$, $g_m$, and the bumblebee self-couplings $λ$ and $\tildeλ$. Motivated by the known gauge- and field-reparametrization subtleties of the conventional Coleman--Weinberg analysis, we formulate an RG-covariant leading-logarithmic improvement scheme for the Vilkovisky--DeWitt effective potential in normal field coordinates, in which the RG operator is governed solely by the beta functions. We apply this framework to a real constant bumblebee background and obtain the leading-logarithmic one-loop effective potential, discussing the conditions under which a nontrivial vacuum is generated by dimensional transmutation and thereby provides a dynamical realization of Lorentz symmetry breaking in this class of models.

Figures (8)

• Figure 1: One-loop bumblebee contributions to the photon two-point function (vacuum polarization). Wavy and wiggly lines denote the photon and bumblebee propagators, respectively.
• Figure 2: Bumblebee self-energy (two-point 1PI function).
• Figure 3: The bumblebee--bumblebee--photon three-point function (1PI vertex).
• Figure 4: One-loop 1PI bumblebee four-point function. Its UV divergence determines the counterterms required to renormalize the quartic self-interactions governed by $\lambda$ and $\tilde{\lambda}$.
• Figure 5: One-loop 1PI contribution to the bumblebee--bumblebee--photon--photon four-point function.