Abstract
The one-loop divergences for the scalar field theory with Lorentz and/or CPT breaking terms are obtained in curved space-time. We analyze two separate cases: minimal coupled scalar field with gravity and nonminimal one. For the minimal case with real scalar field, the counterterms are evaluated in a nonperturbative form in the CPT-even parameter through a redefinition of space-time metric. In the most complicated case of complex scalar field nonminimally interacting with gravity, the solution for the divergences is obtained in the first order in the weak Lorentz violating parameter. The necessary form of the vacuum counterterms indicate the most important structures of Lorentz/CPT violations in the pure gravitational sector of theory. The conformal theory limit is also analyzed. It turns out that if we allow the violating fields to transform, the classical conformal invariance of massless scalar fields can be maintained in the $\,\xi = 1/6\,$ case. At quantum level the conformal symmetry is violated by trace anomaly. As a result, conformal anomaly and the anomaly-induced effective action are evaluated in the presence of extra Lorentz- and/or CPT-violating parameters. Such gravitational effective action is important for cosmological applications and can be used for searching of Lorentz violation in primordial universe in the cosmological perturbations, especially gravitational waves.
Highlights
In the last years, there was an intensive investigation of the theoretical and experimental aspects in theories where the Lorentz and/or CPT symmetries are violated
The symmetry-violating parameters were treated as fields, rather than constants
The solution for the one-loop counterterms was found in a closed form, while for the nonminimal complex scalar field, the solution has been obtained in the first order in the small symmetry-breaking parameters
Summary
There was an intensive investigation of the theoretical and experimental aspects in theories where the Lorentz and/or CPT symmetries are violated. One of the possibilities is to introduce only those terms which can emerge as vacuum divergences in a semiclassical theory of quantum matter fields These criteria concern the minimal necessary set of terms in the classical gravitational action which are required by renormalizability. The introduction of new terms in the matter fields sector requires, at the quantum level, the extension of a classical action of vacuum with a set of structures depending on the violating parameters. The form of those terms may be established on the basis of general covariance and power counting arguments, but only direct calculation of the counterterms can indicate which terms are truly necessary.
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