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Complex bumblebee model

Willian Carvalho1,*, A. C. Lehum1,†, J. R. Nascimento2,‡, and A. Yu. Petrov2,§

  • *Contact author: willian.carvalho@icen.ufpa.br
  • †Contact author: lehum@ufpa.br
  • ‡Contact author: jroberto@fisica.ufpb.br
  • §Contact author: petrov@fisica.ufpb.br
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Phys. Rev. D 113, 125003 – Published 2 June, 2026

DOI: https://doi.org/10.1103/437b-8nlz

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 gl and a nonminimal magnetic-type coupling gm 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, gl, gm, and the bumblebee self-couplings λ and λ˜. 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.

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Phys. Rev. D 113, 125003– Published 2 June, 2026
  • Received 1 April 2026
  • Accepted 29 April 2026
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