Lorentz Violation and Radiative Corrections in Gauge Theories

TL;DR

This paper reviews Lorentz-violating radiative corrections in gauge theories, clarifies conventions in the Standard-Model Extension, and improves bounds on Lorentz-violating coefficients through loop correction analysis.

Contribution

It clarifies the conventions used in Lorentz-violating models and provides an improved bound on a specific Lorentz-violating coefficient in QED.

Findings

Identified inconsistencies in previous literature conventions.

Demonstrated a loop correction to the $k_{F}$ coefficient.

Improved bounds on a Lorentz-violating fermion sector coefficient.

Abstract

Various studies have already considered radiative corrections in Lorentz-violating models unveiling many instances where a minimal or nonminimal operator generates, via loop corrections, a contribution to the photon sector of the Standard-Model Extension. However, an important fraction of this literature does not follow the widely accepted conventions and notations of the Standard-Model Extension, and this obscures the comparison between different calculations as well as possible phenomenological consequences. After reviewing some of these works, we uncover one example where a well defined loop correction to the $k_{F}$ coefficient, already presented in the literature, allows us to improve the bounds on one specific coefficient of the fermion sector of the Lorentz-violating QED extension.

Equations

$$b_{\mu }\bar{\psi }{\gamma }^{\mu }{\gamma }_5\psi ,$$

$$C{e}^2{ϵ}^{\mu \nu \lambda \rho }{b}_{\mu }{A}_{\nu }{\partial }_{\lambda }{A}_{\rho },$$

$${\mathcal{L}}_{eff}\supset \frac{e^2}{24{\pi }^2{m}^2}{ϵ}^{\beta \mu \nu \rho }{b}_{\beta }{A}_{\mu }\square F_{\nu \rho },$$

$$({\hat{k}}_{AF}^{\left(5\right)}{)}_{\kappa }^{\alpha \beta }=\frac{e^2}{48{\pi }^2{m}^2}{b}_{\kappa }{\eta }^{\alpha \beta }.$$

$${\mathcal{L}}_{eff}\supset -\frac{e^2}{6{\pi }^2{m}^2}{b}^{\mu }{b}_{\lambda }{F}_{\mu \nu }{F}^{\lambda \nu },$$

$$(k_F^{(4)}{)}^{\mu \nu \alpha \beta }=-\frac{e^2}{6{\pi }^2{m}^2}\left(b^{\mu }{b}^{\alpha }{\eta }^{\nu \beta }-b^{\nu }{b}^{\alpha }{\eta }^{\mu \beta }-b^{\mu }{b}^{\beta }{\eta }^{\nu \alpha }+b^{\nu }{b}^{\beta }{\eta }^{\mu \alpha }\right).$$

$$\left|b_T\right|<\pi m/e\sqrt{6\times 10^{-37}},$$

$$\left|(b_T{)}_p\right|<3\times 10^{-15}\text{GeV}$$

$$\left|(b_T{)}_e\right|<1.5\times 10^{-20}\text{GeV}$$

Full text

\bodymatter

The Standard-Model Extension (SME)[1, 2] is understood as an effective field theory based on the internal symmetries and field content of the Standard Model, but incorporating Lorentz violation (LV) in a very general way. A broad experimental program has used the SME framework to obtain stringent bounds on possible LV operators using different experiments and astrophysical observations.[3] The most studied and well constrained sector of the SME is the LV extension to the Maxwell theory, defined in its most general form in Ref. \refciteKostelMewes-EMHD. In particular, the minimal photon sector involves only two LV coefficients, $k_{AF}$ and $k_{F}$. Both terms in general induce birefringence in the vacuum leading to very strong constraints from astrophysics: of order $10^{-43}\,$GeV for $k_{AF}$ and $10^{-37}$ for the birefringent components of $k_{F}$.[3]

If taken as an effective parametrization for LV effects to be searched for in the laboratory, the SME might be understood as a strictly tree-level theory. From the theoretical viewpoint, however, it motivates many interesting studies concerning the consistency, as a full quantum field theory (QFT), of theories in which one of the central aspects of the QFT formalism—Lorentz symmetry—is in some sense violated. Different field-theoretical aspects, such as renormalization, the structure of asymptotic states, and others, have already been investigated.[5, 6, 7, 8, 9]

Radiative corrections, in particular, may lead to results of direct phenomenological interest: some set of LV coefficients may generate or contribute to other sets of LV coefficients via loop corrections. An early example was discussed in detail in Ref. \refciteJK: the $b$ term

[TABLE]

which is a LV correction to the fermion propagator, will induce a one-loop correction to the quadratic photon lagrangian of the form

[TABLE]

$e$ being the fermion charge and $C$ a constant. In the SME notation, this amounts to the generation of a minimal $k_{AF}$ term with $k_{AF}\sim b$ or to a correction to a $k_{AF}$ term already present at tree level. The phenomenological interest in such a result is that $k_{AF}$ can be strongly constrained by photon vacuum birefringence, and these constraints could be translated to $b^{\mu}$. However, it was readily recognized that the result presented in Eq. (2) is anomalous: the loop integral turns out to be finite but ambiguous, its result being dependent on the regularization scheme used to calculate it (for a recent discussion of this problem, see Ref. \refciteAltschul:2019eip). In this case, it is certainly not possible to use loop corrections to obtain sound phenomenological conclusions.

Many different instances of radiative generation of LV operators have been presented in the literature, and we might wonder whether finite and well-defined corrections can be calculated in some cases, and whether stringent bounds obtained in one sector of the SME might be transferred to other, perhaps not so well bounded sectors. This requires some work since many of the reported calculations do not follow the now standard SME notations.

As an example, higher-derivative corrections to the photon sector originating from (1) where presented in Ref. \refciteTMariz1 as

[TABLE]

corresponding in SME notation to the generation of the dimension five coefficient

[TABLE]

This is a finite result, free of ambiguities. Unfortunately, this form of $\hat{k}_{AF}^{\left(5\right)}$ does not modify free propagation of photons at leading order, so no interesting bounds can be obtained from this result at the moment.

Looking at one-loop corrections involving higher orders in $b$, we may also obtain finite and well-defined results, such as[13, 14]

[TABLE]

which, written in the SME notation, corresponds to

[TABLE]

At first sight, one might not expect to find competitive constraints from this expression, since it is of second order in $b$. However, birefringent components of $k_{F}$ are constrained at the order of $10^{-37}$, and so this result would provide a bound of the order $b^{2}<6\pi^{2}m^{2}/e^{2}\times 10^{-37}$ for the $b$ coefficient, corresponding to $\left|b_{p}^{\mu}\right|<3\times 10^{-15}\,$GeV for protons and $\left|b_{e}^{\mu}\right|<1.5\times 10^{-20}\,$GeV for electrons, for example. These are not better (but also not much worse) than the constraints already found for the space components of $b^{\mu}$, as measured in the Sun-centered reference frame.[3] On the other hand, the temporal component $b^{T}$ is not so well constrained: the best bounds are of order $\left|(b^{T})_{p}\right|<7\times 10^{-8}\,$GeV for protons and $\left|(b^{T})_{e}\right|<10^{-15}\,$GeV for electrons. Therefore, the radiative correction presented in Eq. (6) can translate the stringent constraints on birefringent components of $k_{F}$ to competitive bounds on the temporal components $b^{T}$. It remains to check that this particular form of $k_{F}^{(4)}$ does indeed induce birefringence. The easiest way to do this is by using the parametrization of birefringent components of $k_{F}$ in terms of the ten $k^{a}$ coefficients as given in Ref. \refciteKostelecky:2001mb: one may easily verify that Eq. (6) corresponds to non-vanishing coefficients $k^{a}=e^{2}b^{2}/6\pi^{2}m^{2}$ for $a=3,4$.

The end result is therefore the constraint

[TABLE]

for the temporal component, in the Sun-centered frame, of the $b$ coefficient for a given fermion, depending on its mass $m$. For example, we have

[TABLE]

for protons and

[TABLE]

for electrons.

This result was presented in Ref. \refciteFerrari:2018tps, together with an extensive study of other instances of radiative corrections in different sectors of the SME. It is an interesting example, where a weakly bounded coefficient for LV can be subjected to a stronger constraint, borrowed from the very well studied photon sector of the SME. We believe that, besides interesting questions regarding theoretical consistency and technical challenges of calculating ambiguous Feynman integrals, the study of radiative corrections might help to fill some of the gaps in the extensive set of searches for LV that have been developed in the last decades using the SME as the fundamental framework.

Acknowledgments

This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) via the following grants: CNPq 304134/2017-1 and FAPESP 2017/13767-9.