Remarks on the renormalization properties of Lorentz- and CPT-violating quantum electrodynamics
In this work, we employ algebraic renormalization technique to show the renormalizability to all orders in perturbation theory of the Lorentz- and CPT-violating QED. Essentially, we control the breaking terms by using a suitable set of external sources. Thus, with the symmetries restored, a perturbative treatment can be consistently employed. After showing the renormalizability, the external sources attain certain physical values, which allow the recovering of the starting physical action. The main result is that the original QED action presents the three usual independent renormalization parameters. The Lorentz-violating sector can be renormalized by nineteen independent parameters. Moreover, vacuum divergences appear with extra independent renormalization. Remarkably, the bosonic odd sector (Chern-Simons-like term) does not renormalize and is not radiatively generated. One-loop computations are also presented and compared with the existing literature.
In the last few decades many efforts have been employed in order to understand models that present Lorentz- and CPT-symmetry breaking, see for instance [1, 2, 3, 4, 5, 6]. In particular, the main contributions are interested in how these models are situated under aspects of the usual quantum field theory. Due to the well known success of quantum field theory – specially, gauge field theory – in describing at least three of four fundamental interactions, any extension of the standard model respecting attributes as stability, renormalizability, unitarity and causality could be interesting. In fact, the Lorentz and gauge symmetry have a fundamental importance on the features mentioned before. For instance, the functional that describes the dynamics of the fields belonging to the standard model are built in a Lorentz covariant way and the classification of particles is performed by studying the Lorentz group representations [7, 8]. Moreover, besides restricting the coupling between fields, the gauge symmetry play an important role on unitarity and renormalizability of gauge theories [9, 10, 11].
The Abelian Lorentz- and CPT-violating minimal Standard Model Extension (mSME), i.e., Lorentz- and CPT-violating QED, is characterized by the presence of constant background tensorial (and pseudo-tensorial) fields coupled to the fundamental fields of the theory, and is power-counting renormalizable . These background fields are, in principle, natural consequences of more fundamental theories such as string theories , non-commutative field theories [13, 14, 15, 16, 17], supersymmetric field theories [18, 19, 20] and loop quantum gravity . For instance, there exists the possibility of spontaneous Lorentz symmetry breaking in string theory. This breaking manifests itself when tensorial fields acquire non-trivial vacuum expectation values. This feature implies on a preferred spacetime direction. Although many searches have been performed in order to detect signs of these background tensors [22, 23, 24, 25], nothing have been found so far. Nevertheless, these efforts have been useful to determine phenomenological and experimental upper bounds for the v.e.v. of these tensors .
In what concerns the theoretical consistency of these models, it has been verified that they can preserve causality and unitary [1, 2, 27, 28, 29, 30]. In this work, we confine ourselves to the formal analysis of renormalizability of the Lorentz- and CPT-violating QED. In fact, there are some works about the renormalizability of such models. For instance: in , the one-loop renormalization is discussed; the proof of renormalizability to all orders in perturbation theory, from algebraic renormalization technique point of view , was performed in . The latter makes use of the gauge symmetry and requires PT-invariance to prove that anomalies are not present. Essentially, they prove the renormalizability of the model with C and/or PT-invariance. Moreover, they find nine independent renormalization parameters; furthermore, they also show in  that no CPT-odd bosonic Lorentz violation is generated from the CPT-odd fermionic Lorentz violation sector; there also exist studies about the renormalization properties of the QED extension on curved manifolds . In this work the renormalization study was realized by assuming that Lorentz- and CPT-violating parameters are classical fields rather than constants. This last approach shares some resemblance with the present work.
It is worth mention that there also exists a class of Lorentz-violating quantum field theories which can also preserve unitarity and renormalizability, see for instance . These theories are characterized by the presence of higher order space derivatives while the time derivatives remain at the same order of the usual fermionic or bosonic models. The renormalizability is assured by modifying the usual power counting criterion by introducing the concept of “weighted power-counting” . Actually, they introduce a regulator to account for this discrepancy. This criterion can put on a renormalizable form vertexes that, in principle, are non renormalizable . For instance, a higher-energy Lorentz-violating QED shows itself to be super-renormalizable and its low-energy limit is recovered by choosing specific values for scale parameters . In this work, however, we deal with Carroll-Field-Jackiw theories , which constitute a whole different class of theories. For instance, space and time are treated on an equal footing, and the Lorentz violation manifests itself under particle Lorentz transformations. On the other hand, these theories are manifestly invariant under observer Lorentz transformations.
In this work, we employ the BRST quantization and algebraic renormalization theory to explore the renormalizability of Lorentz- and CPT-violating QED. In particular, we generalize the study made in  for all possible Lorentz breaking terms. Furthermore, in this work we add one more breaking term not considered in , which is a massive term coupled to a pseudo-scalar operator. The main idea can be summarized in the following way: The action which describes the bosonic Lorentz violation of CPT-odd is gauge invariant only because the Lorentz-violating coefficients are constant (and also neglecting surface terms). Thus, the gauge symmetry is ensured at the action level, but not at Lagrangian level.
The algebraic renormalization approach, which will be used here, relies on the quantum action principle (QAP) [32, 41, 42, 43, 44, 45]. Thus, in order to analyze the renormalizability of the Lorentz-violating QED we will employ here the Symanzik method  – vastly employed in non-Abelian gauge theories in order to control a soft BRST symmetry breaking, see [47, 48, 49, 50, 51, 52] – to treat the BRST quantization of the Lorentz-violating electrodynamics. We will follow here the procedure employed in the proof of the renormalizability to all orders in perturbation theory of pure Yang-Mills (YM) theory with Lorentz violation .
The main results obtained from our approach are: First, the model is renormalizable to all orders in perturbation theory; second, the usual QED sector has only three independent renormalization parameters (in accordance with the usual QED); third, the Chern-Simons-like violating term does not renormalize. Moreover, we attain extra important results: The Lorentz-violating sector has nineteen renormalization parameters; extra independent renormalization parameters are needed to account for extra vacuum divergences that do not appear from other approaches. However, these terms do not affect the dynamical content of the model; as pointed out in , the Abelian Chern-Simons-like term is not induced from the CPT-odd Lorentz-violating term of the fermionic sector, see also [3, 54, 55]; although one-loop computations have already been done in , we also perform these computations here in order to compare them with the algebraic results.
This work is organized as follows: In Sect. 2 we provide the definitions, conventions and some properties of the Lorentz-violating electrodynamics. In Sect. 3, the BRST quantization of the model with the extra set of auxiliary sources is provided: where we discuss the subtle quantization of the Lorentz-violating coefficients coupled with this respective composite operators within of this formalism. In Sect. 4, we study the renormalizability properties of the model, with a detailed study of the quantum stability of the model. Then, in Sect. 5, we present the one-loop explicit computations and its relation with our renormalization independent scheme is discussed. Our final considerations are displayed in Sect. 6.
2 Lorentz-violating electrodynamics
The QED extension, just like the standard QED, is a gauge theory for the group, where the electromagnetic field is coupled to the Dirac field through minimal coupling. However, this theory presents Lorentz violation in both, bosonic and fermionic, sectors. The breaking sectors are characterized by the presence of background fields. The model is described by following action [2, 4]
is the classical action of the usual QED. The covariant derivative is defined as , the field strength is written as and is the gauge potential. The parameter stands for the electron mass and for its electric charge. The matrices are in Dirac representation (see  for the full set of conventions111For self-consistency, we just define (2.3) ). The other term in (2.1) is the Lorentz-violating sector,
The violation of Lorentz symmetry in the fermionic sector is characterized by the following constant tensorial fields: , , , , , , , and . These tensors select privileged directions in spacetime, dooming it to anisotropy. Tensorial fields with even numbers of indexes preserve CPT while an odd number of indexes do not preserve CPT222For the explicit CPT features of the fields see Table 4 in terms of the sources at the App. B (See next section for the source-background correspondence).. The tensorial fields , , , and are dimensionless and , , and has mass dimension 1. The tensor field is anti-symmetric and is anti-symmetric only on its first two indexes. At the photonic sector, the Lorentz violation is characterized by the field , with mass dimension 1, and , which is dimensionless. The tensor obeys the same properties of the Riemann tensor, and is double traceless
As the reader can easily infer, the action (2.1) is a Lorentz scalar, being invariant under observers Lorentz transformations while, in contrast, presents violation with respect to particle Lorentz transformations 333This can be understood in the following way: let and be a generic field and a background vector field, respectively. Under observer Lorentz transformation, i.e., exchange of references systems, these fields behave as and . On the other hand, under particle Lorentz transformation the reference systems do not transform, but the fields transform as and . The generalization to (pseudo-)tensorial backgrounds are immediate. .
3 BRST quantization and restoration of Lorentz symmetry
In the process of quantization of the QED extension theory, as in the usual QED, gauge fixing is required. In the present work we employ the BRST quantization method and adopt, for simplicity, the Landau gauge condition . Thus, besides the photon and electron fields, we introduce the Lautrup-Nakanishi field and the Faddeev-Popov ghost and anti-ghost fields444Even though the ghost and anti-ghost fields are not required in the Abelian theory at Landau gauge, we opt by introduce them for following reasons: i) It is a direct way to keep the off-shell BRST symmetry of the action . ii) Due to the discrete Faddeev-Popov symmetry, the trivial and non-trivial sectors of the BRST cohomology becomes explicit (see table 1). iii) With the introduction of the field, it is easy to see that photon propagator keeps its transversality to all orders in perturbation theory. iv) As expected, the tree-level decoupling of the ghosts is kept at all orders in perturbation theory, see Sect. 4.1. This feature will bring important consequences for the renormalization properties of the CPT-odd bosonic violating sector of the action (2.1)., namely, and , respectively. The BRST transformations are
where is the nilpotent BRST operator. Thus, the Landau gauge fixed action is
Following the BRST quantization of the Lorentz-violating sector with the Symanzik prescription , we will have two distincts situations. The first situation concerns the CPT-even bosonic violating term and all fermionic breaking terms, all of them are BRST invariant. Then, they will couple to BRST invariant sources. Henceforth, we define the following set of invariant sources
On the second situation, the CPT-odd bosonic violating term, a BRST doublet is required because this term is not BRST invariant,
Thus, we replace the action (3.2) by
is the embedding555The embedding concept used here is discussed in detail in Ref. . of the Lorentz-violating bosonic sector while the embedding of the Lorentz-violating term for the fermionic sector is given by
It is easy to check that . The quantum number of the sources follow the quantum numbers of the background fields, as displayed in666The external sources and will be defined in Section 4 in order to control the nonlinear BRST transformations of the spinor fields. Table 3.
The action , at the physical value of the sources (3.6), reduces to
It is clear then that the kinematical content of the model does not change once the physical limit of the sources are taken. This is a peculiarity of the Abelian model, where the symmetries avoid many terms that are present at non-Abelian model . In fact, at the non-Abelian model with Lorentz violation the kinematics of the model is drastically changed when this approach is employed. See  for more details.
4 Algebraic proof of the renormalizability
Let us now face the issue of the renormalizability of the model. For that, we need one last set of external BRST invariant sources, namely, and , in order to control the non-linear BRST transformations of the original fields,
Thus, the complete action is given by
Indeed, it is easy to note that extra combinations among sources are possible, including the electron mass. However, these combinations do not interfere with the renormalization of the sources and they will be renormalizable as well. Moreover, extra dimensionless parameters will be needed in order to absorb vacuum divergences, see for instance . To avoid a cumbersome analysis, we omit these pure vacuum terms here. Nevertheless, for completeness, this issue is discussed in the Appendix A.
Explicitly, the action (4.2) has the form
As one can easily check that, at the physical values of the sources, this action is also contracted down to (LABEL:15aab).
4.1 Ward identities
The action (4.2) enjoys the following set of Ward identities
4.2 Most general counterterm
In order to obtain the most general counterterm which can be freely added to the classical action at any order in perturbation theory, we need a general local integrated polynomial with dimension bounded by four and vanishing ghost number. Thus, imposing the Ward identities (4.4)-(4.6) to the perturbed action , where is a small parameter, it is easy to find that the counterterm must obey the following constraints
where the operator is the nilpotent linearized Slavnov-Taylor operator,
The first constraint of (4.7) identifies the invariant counterterm as the solution of the cohomology problem for the operator in the space of the integrated local field polynomials of dimension four. From the general results of cohomology, it follows that can be written as 
where is the most general local polynomial counterterm with dimension bounded by four and ghost number , given by777Clearly, in contrast to the background fields, the external sources are not “frozen” with respect to CPT symmetries. Hence, they enjoy discrete mappings. Thus can be made CPT-invariant, avoiding a lot of counterterms and renormalization parameters. Furthermore, all of them, if included, are also avoided by the Ward identities of the model.
with being free coefficients, and
The contraction has the same symmetries as the source . Moreover, has the same discrete symmetries as . Indeed, this contraction extends the anti-symmetrization to all indexes of . From the second equation in (4.7), it follows that . Moreover, from the ghost equation, . Then, after the following redefinitions,
it is not difficult to verify that the form of the most general counterterm allowed by the Ward identities is given by
It remains to infer if the counterterm can be reabsorbed by the original action by means of the multiplicative redefinition of the fields, sources and parameters of the theory, according to
where the bare quantities are defined as
It is straightforward to check that this can be performed, proving the theory to be renormalizable to all orders in perturbation theory. Explicitly, the renormalization factors are listed below.
For the independent renormalization factors of the photon, electron and electron mass, we have
The renormalization factors of the ghosts, charge, Lautrup-Nakanishi field and sources are not independent, namely
Thus, the renormalization properties of the standard QED sector remain unchanged.
For the sector, due to the quantum numbers of and , there is a mixing between their respective operators, i.e., and . Thus, matricial renormalization is required, namely
where is a column matrix of sources that share the same quantum numbers. The quantity is a squared matrix with the associated renormalization factors. In this case,
where is a matrix depending on . Thus
As it is easy to infer from table 3, some external sources do not have exactly the same quantum numbers, specifically with respect to their mass dimensions. Then, in principle, they do not suffer quantum mixing. However, the model has a mass parameter, the electron mass . Thus, the mass parameter will enable extra mixing among sources . Firstly,