Unitarity in Maxwell-Carroll-Field-Jackiw electrodynamics

In this work we focus on the Carroll-Field-Jackiw (CFJ) modified electrodynamics in combination with a CPT-even Lorentzviolating contribution. We add a photon mass term to the Lagrange density and study the question whether this contribution can render the theory unitary. The analysis is based on the pole structure of the modified photon propagator as well as the validity of the optical theorem. We find, indeed, that the massive CFJ-type modification is unitary at tree-level. This result provides a further example for how a photon mass can mitigate malign behaviors.


Introduction
Lorentz symmetry violation has been of permanent interest in the past two decades whereby the major part of the investigations have been carried out within the minimal Standard-Model Extension (SME) [1,2]. The minimal SME incorporates powercounting renormalizable contributions for Lorentz violation in all particle sectors and has been subject to various studies.
The CPT-even photon sector of SME has been investigated thoroughly with the main objective to obtain stringent bounds on its 19 coefficients [3]. The absence of vacuum birefringence has led to bounds at the level of 10 −32 to 10 −37 [4] for the 10 birefringent coefficients. Investigating vacuum Cherenkov radiation [5] for ultra-high energy cosmic rays [6,7] has provided a set of tight constraints on the remaining coefficients [8].
Furthermore, there has been a vast interest in understanding the properties of the CPT-odd photon sector of the SME that is represented by the Carroll-Field-Jackiw (CFJ) electrodynamics [9]. This theory has been extensively examined in the literature with respect to its consistency [10], modifications that it induces in quantum electrodynamics (QED) [11,12], its radiative generation [13], and many other aspects. As vacuum birefringence has not been observed, the related coefficients are strongly bounded at the level of 10 −43 GeV [8].
It is known that the timelike sector of CFJ electrodynamics is plagued by several problems such as negative contributions to the energy density [1] and dispersion relations that become imaginary in the infrared regime. In particular, it was shown that violations of unitarity are present, at least for small mo-Email addresses: manojr.ufma@gmail.com (Manoel M. Ferreira, Jr.), helayel@cbpf.br (Jose A. Helayël-Neto), creyes@biobio.cl (Carlos M. Reyes), marco.schreck@ufma.br (Marco Schreck), pdiego.10@hotmail.com (Pedro D.S. Silva) menta [11]. In the current work, our objective is to study unitarity of the timelike sector of CFJ electrodynamics.
We would like to find out whether the inclusion of a photon mass can, indeed, solve these issues. This idea is not unreasonable, as it is well-known that the introduction of a mass for an otherwise massless particle helps to get rid of certain problems. For example, a photon mass can act as a regulator for infrared divergences. Furthermore, adding a mass to the graviton renders gravity renormalizable (despite being plagued by the Boulware-Deser ghost [14] that is removed by the construction of de Rham, Gabadadze, and Tolley [15]). It was indicated in [1] and demonstrated in [16] that a photon mass is capable of mitigating the malign behavior in CFJ electrodynamics.
The paper is organized as follows. In Sec. 2 we introduce the theory to be considered and discuss some of its properties. We determine the photon polarization vectors as well as the modified photon propagator in Sec. 3. Subsequently, we express the propagator in terms of the polarization vectors, which is a procedure that was introduced in [16]. The resulting object turns out to be powerful to obtain quite general statements on perturbative unitarity in Sec. 4. In Sec. 5 we briefly argue how the previous results can be incorporated into the electroweak sector of the SME. Finally, we conclude on our findings in Sec. 6. Natural units with = c = 1 are used unless otherwise stated.

Theoretical setting
We start from a Lagrange density that is a Lorentz-violating modification of the Stückelberg theory: with the U(1) gauge field A µ , the associated field strength tensor F µν = ∂ µ A ν − ∂ ν A µ , a photon mass m γ , and a real parameter ξ. Lorentz violation is encoded in the CPT-even and CPTodd background fields (k F ) κλµν and (k AF ) κ , respectively. All fields are defined on Minkowski spacetime with metric signature (+, −, −, −). Furthermore, ε µν σ is the Levi-Civita symbol in four spacetime dimensions where we use the convention 0123 = 1. Restricting Eq. (1) to the first and third term only, corresponds to the theory originally investigated by Carroll, Field, and Jackiw in [9]. For a timelike choice of k AF , the latter theory is known to have stability problems, which explicitly show up, e.g., in the corresponding energy-momentum tensor [1] or the dispersion relations [11]. In analyses carried out in the past, the introduction of a photon mass as a regulator [16] turned out to resolve these issues. It prevents the dispersion relation from becoming imaginary in the infrared region, i.e., for low momenta. Note that the current upper limit for a photon mass is 10 −27 GeV [17]. Although this constraint on a violation of U(1) gauge invariance is very strict, it still lies many orders of magnitude above the constraints on the coefficients of CFJ theory. Finally, as the propagator of Proca theory is known to have a singularity for m γ → 0, we also include the last term in Eq. (1), which was introduced by Stückelberg. Now, we employ a specific parameterization of the background fields as follows: with a symmetric and traceless (4 × 4) matrixk µν and a fourvector b µ that gives rise to a preferred direction in spacetime. If Lorentz violation arises from a vector-valued background field, a reasonable assumption could be that the latter is responsible for both CPT-even and CPT-odd contributions. Furthermore, ζ 1 and ζ 2 are Lorentz-violating coefficients of mass dimension 0 and 1, respectively, that are introduced to control the strength of CPT-even and CPT-odd Lorentz violation independently from each other. The parameterization of (k F ) µν σ stated in Eq. (2a) is sometimes called the nonbirefringent Ansatz [18], as it contains the 9 coefficients of (k F ) µν σ that do not provide vacuum birefringence at leading order in Lorentz violation. Let us rewrite the Lagrange density of Eq. (1) in terms of the preferred four-vector b µ introduced before: with a = 1 − ζ 1 b 2 . Theories with a similar structure can be generated by radiative corrections and were, for example, studied in [19]. Performing suitable integrations by parts, the latter Lagrange density is expressed as a tensor-valued opera-torM νµ sandwiched in between two gauge fields according to This form directly leads us to the equation of motion for the massive gauge field:

Polarization vectors
Analyzing the properties of the polarization vectors for modified photons will allow us to find a relation between the sum over polarization tensors and the propagator [16]. In turn, this relation will be useful to compute imaginary parts of forward scattering amplitudes and to reexpress these in terms of amplitudes associated with cut Feynman diagrams. The latter is required by the optical theorem to test perturbative unitarity. Note that studies of Lorentz-violating modifications based on the optical theorem have already been performed in several papers for field operators of mass dimension 4 [20] as well as higher-dimensional ones [21].
The operator of Eq. (4) transformed to momentum space (with a global sign dropped) is given by We consider the eigenvalue problem for a basis {v (λ) } of polarization vectors which diagonalize the equation of motion. We use λ = {0, +, −, 3} as labels for these vectors. The eigenvalue Λ λ (p) = Λ λ corresponds to the dispersion equation of the mode λ. To find a real basis, we choose the temporal polarization vector as and the longitudinal one as with D = (b · p) 2 − p 2 b 2 , which is the Gramian of the two fourvectors p µ and b µ . It is not difficult to check that p·v ( Proceeding with the evaluation of Eq. (7) for λ = 0, 3 we obtain The theory is gauge-invariant for vanishing photon mass. In this case, ξ can be interpreted as a gauge fixing parameter. The dependence of Λ 0 on ξ tells us that this associated degree of freedom is nonphysical. We will come back to this point later. Now, we find the remaining two polarization states, which we label as λ = ±. First, let us define the two real four-vectors with an auxiliary four-vector n µ . We normalize these vectors as which fixes the normalization constants: Both vectors of Eqs. (12) are orthogonal to p µ and b µ . Besides, they satisfy We now introduce the linear combinations that obey the properties Constructing suitable linear combinations of Eqs. (15) results in Hence, the dispersion equation for the transverse modes reads In analyses performed in the past, the sum over two-tensors formed from the polarization vectors and weighted by the dispersion equations turned out to be extremely valuable [16]. In particular, with the dispersion equations Λ 0,3 of Eq. (11a) and Λ ± of Eq. (18b). Inserting the explicit expressions for the basis {v (λ) } results in By computation, we showed that P µν is equal to the negative of the inverse of the operator M µν in Eq. (6): M µν P ν = −δ µ . Therefore, we define iP µν as the modified photon propagator of the theory given by Eq. (1). For ζ 1 = ζ 2 = 0 and m γ → 0 we observe that which is the standard result for the propagator in Fermi's theory, as expected.

Perturbative unitarity at tree level
Now were are interested in studying probability conservation for the theory given by Eq. (1). It is known that the timelike sector of CFJ theory has unitarity issues [11] while it was also demonstrated that a photon mass helps to perform a consistent quantization [16]. Therefore, we would like to investigate the question whether the presence of a photon mass can render the theory unitary. For brevity, we choose a purely timelike background field: (b µ ) = (1, 0, 0, 0). Note that in this case, the theory of Eq. (1) is isotropic and we can identify ζ 1 with the isotropic CPT-even coefficientκ tr [22]. For completeness, we keep the CPT-even contributions, although they are not expected to cause unitarity issues for small enough ζ 1 .
In the classical regime, a four-derivative applied to the field equations (5) provides where = ∂ µ ∂ µ is the d'Alembertian. Even if A µ couples to a conserved current, ∂ · A behaves as a free field. It can be interpreted as a nonphysical scalar mode that exhibits the dispersion relation Therefore, in the classical approach, the Stückelberg term proportional to 1/ξ can actually be removed from the field equations. In this case, we automatically get the subsidiary requirement ∂ · A = 0 that corresponds to the Lorenz gauge fixing condition. Furthermore, the modified Gauss and Ampère law for the electric field E and magnetic field B are obtained directly from Eq. (5) and read as follows: where φ is the scalar and A the vector potential, respectively. By expressing the physical fields in terms of the potentials and using the Lorenz gauge fixing condition, the modified Gauss law can be brought into the form with the Laplacian = ∇ 2 . This massive wave equation leads to the dispersion relation The associated mode is interpreted as longitudinal. Furthermore, the modified Ampère law yields The latter provides the modified transverse dispersion relations Note that the dispersion relations found above correspond to the poles of the propagator that we have calculated in Eq. (21). These results will be valuable in the quantum treatment to be performed as follows.
First, we would like to construct the amplitude of a scattering process involving two external, conserved four-currents J µ , J * ν without specifying them explicitly: The latter object is sometimes called the saturated propagator in the literature [23,24] (cf. [25] for an application of this concept in the context of massive gravity in (1 + 2) dimensions). Inserting the decomposition (20) of the propagator in terms of polarization vectors, we obtain: where the mode labeled with λ = 0 is eliminated because of current conservation: p · J = 0. To guarantee the validity of unitarity, the imaginary part of the residues of S evaluated at the positive poles in p 0 should be nonnegative. The numerator of the latter expression is manifestly nonnegative. Hence, the outcome only depends on the pole structure of S. For the case of a purely timelike b µ , which is the interesting one to study, we obtain for each one of the physical modes λ = ±, 3 with dispersion relations (27), (29). For perturbative ζ 1 , ζ 2 , and m γ > |ζ 2 |/ 1 − ζ 1 it holds that ω (λ) k > 0 for λ = ±, 3. In this case, the right-hand side of Eq. (32) is nonnegative. This result is already an indication for the validity of unitarity. We see how a decomposition of the form of Eq. (20) leads to a very elegant study of the saturated propagator without the need of considering explicit configurations of the external currents such as done in the past [24].
To test unitarity more rigorously, we couple Eq. (1) to standard Dirac fermions and consider the theory Here, e is the elementary charge, m ψ the fermion mass, γ µ are the standard Dirac matrices satisfying the Clifford algebra {γ µ , γ ν } = 2η µν , ψ is a Dirac spinor, and ψ = ψ † γ 0 its Dirac conjugate. We intend to check the validity of the optical theorem for the tree-level forward scattering amplitude of the particular electron-positron scattering process in Fig. 1. The amplitude M F associated with the corresponding Feynman graph is with particle spinors u (s) and antiparticle spinors v (s) of spin projection s. The momentum of the internal photon line is k = p 1 + p 2 . Furthermore, P F µν (k) is the Feynman propagator obtained from Eq. (20) by employing the usual prescription k 2 → k 2 + i . Let us define the four-currents J * µ ≡ū (s) (p 1 )γ µ v (r) (p 2 ) .
Due to current conservation at the ingoing and outgoing vertices, p · J = p · J * = 0. Introducing an integral and a δ function for momentum conservation, we can write the forward scattering amplitude as Now we insert Eq. (20) and decompose the denominator in terms of the poles as follows: with the dispersion relation (24) for the unphysical scalar mode, Eq. (27) for the massive mode, and those of Eq. (29) for the transverse modes. Since the zeroth mode points along the direction of the four-momentum, we are left with the sum over λ = ±, 3. We then employ Taking the imaginary part of Eq. (37) by using the general relation with the principal value P, results in In the final step we exploited that for each λ. The negative-energy counterparts k 0 = −ω (λ) k do not contribute due to energy-momentum conservation. The righthand side of Eq. (40) corresponds to the total cross section of e + e − → γ with both the transverse photon modes and the massive mode contributing. Also, it is nonnegative under the conditions stated below Eq. (32). Therefore, we conclude that the optical theorem at tree-level and, therefore, unitarity are valid for the theory defined by Eq. (1) as long as the photon mass is large enough. Note that the latter computation is generalized to an arbitrary timelike frame by computing the imaginary part of Eq. (36) directly with employed. This result means that the decay rate or cross section of a particular process at tree level can be safely obtained in the context of massive CFJ theory where problems are not expected to occur (cf. [26] for the particular example of Cherenkov-like radiation in vacuo).

Application to electroweak sector
Several CFJ-like terms are included in the electroweak sector of the SME before spontaneous symmetry breaking. We consider the Abelian contribution that is given by [1] with the U Y (1) gauge field B µ , the associated field strength tensor B µν , and the controlling coefficients (k 1 ) κ . After spontaneous symmetry breaking SU L (2) ⊗ U Y (1) → U em (1), the field B µ is interpreted as a linear combination of the photon field A µ and the Z boson field Z µ . The corresponding field strength tensor reads where θ w is the Weinberg angle and Z µν the field strength tensor associated with the Z boson. Hence, the Lorentz structure of the field operator in L B of Eq. (43) after spontaneous symmetry breaking has the form Therefore, CFJ-like terms are induced for the massive Z boson as well as for the photon. Our analysis shows that unitarity issues are prevented in the Z sector due to the mass of this boson that emerges via the coupling of Z µ to the vacuum expectation value of the Higgs field. In the photon sector, a mass has to be added by hand, though.

Conclusions
In this work, we considered both CPT-even and CPT-odd Lorentz-violating modifications for photons that were constructed from a single preferred spacetime direction. The timelike sector of the CPT-odd CFJ electrodynamics is known to exhibit issues with unitarity in the infrared regime. Therefore, our intention was to find out whether the inclusion of a photon mass can mitigate these effects.
To perform the analysis, we derived the modified propagator of the theory and decomposed it into a sum of polarization tensors weighted by the dispersion equations for each photon mode. To get a preliminary idea on the validity of unitarity, we contracted this propagator with general conserved currents. The imaginary part of the residue evaluated at the positive poles was found to be nonnegative as long as the modified dispersion relations for each mode stays real. This property is guaranteed by the presence of a sufficiently large photon mass. A second more thorough check involved the evaluation of the optical theorem for a particular tree-level process. The optical theorem was found to be valid for the same conditions encountered previously. It is clear how unitarity issues arise in the limit of a vanishing photon mass when the dispersion relations of certain modes can take imaginary values.
In general, it is challenging to decide whether the imaginary part of the residue of the propagator contracted with conserved currents is positive for arbitrary preferred directions and currents. We emphasize that the decomposition of the propagator into polarization tensors allows for a quite elegant proof of this property independently of particular choices for background fields and currents. Similar relations are expected to be valuable for showing unitarity in alternative frameworks.
Hence, we conclude that CFJ electrodynamics is, indeed, unitary when a photon mass is included into the theory. This finding clearly demonstrates how a mild violation of gauge invariance is capable of solving certain theoretical issues. Note that unitarity issues are prevented automatically for a CFJ-like term in the Z-boson sector where the Z boson mass is generated via the Higgs mechanism.