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Asymptotic states and renormalization in Lorentz-violating quantum field theory

Asymptotic single-particle states in quantum field theories with small departures from Lorentz symmetry are investigated perturbatively with focus on potential phenomenological ramifications. To this end, one-loop radiative corrections for a sample Lorentz-violating Lagrangian contained in the Standard-Model Extension (SME) are studied at linear order in Lorentz breakdown. It is found that the spinor kinetic operator, and thus the free-particle physics, is modified by Lorentz-violating operators absent from the original Lagrangian. As a consequence of this result, both the standard renormalization procedure as well as the Lehmann-Symanzik-Zimmermann reduction formalism need to be adapted. The necessary adaptations are worked out explicitly at first order in Lorentz-breaking coefficients.


I. INTRODUCTION
Current understanding of physics at the fundamental level is based on two distinct theories: general relativity (GR) and the Standard Model (SM) of particle physics. It is commonly believed that these two theories arise as the low-energy limit of an underlying Planck-scale framework that consistently merges gravity and quantum mechanics. Since direct measurements at this scale are presently impractical, experimental research in this field relies largely on ultrahigh-precision searches for Plancksuppressed effects at attainable energies.
One possible effect in this context is a minute breakdown of Lorentz invariance. Lorentz symmetry is a fundamental feature of both GR and the SM, so that any observed deviation from this symmetry would imply new physics. A number of theoretical approaches to physics beyond the SM, such as strings [1], non-commutative field theories [2], cosmologically varying scalar fields [3], quantum gravity [4], random-dynamics models [5], multiverses [6], brane-world scenarios [7], and massive gravity [8], are believed to allow for small violations of Lorentz invariance at low energies. Searches for such violations are also motivated by the apparent fundamental character of Lorentz symmetry. Consequently, Lorentz invariance ought to be supported as firmly as possible by experimental evidence.
It is natural to expect that Lorentz-violating effects can be described within effective field theory, at least at currently attainable energies [9]. The framework generally adopted in this context is the Standard-Model Extension (SME) [10,11], which contains both GR and the SM as limiting cases. The additional Lagrangian terms present in the SME include all operators for Lorentz violation that are scalars under coordinate changes. The SME has constituted the basis for the analysis of numerous exper-imental searches for Lorentz breakdown [12].
Paralleling the conventional Lorentz-symmetric case, perturbative quantum-field analyses within the SME also rely on a few key theoretical concepts. Some of these, such as canonical quantization [13,14] and renormalization [16][17][18], have previously been studied and generalized to the SME. Another such core concept concerns the treatment of external states. They span the asymptotic Hilbert space, so their determination is of fundamental importance for perturbation theory. For example, explicit S-matrix calculations require a separate, independent determination of the external legs up to the desired order. This special status of externalleg physics is highlighted by the usual Feynman rules: external-leg corrections cannot be incorporated into the diagram for a scattering process; the rules for S-matrix calculations specifically call for "amputed" diagrams. The usual treatment of radiative corrections to external legs involves sophisticated theoretical concepts like the Källén-Lehmann representation [19] and the Lehmann-Symanzik-Zimmermann (LSZ) reduction formalism [20]. Although a number of prior investigations have considered radiative corrections from various other perspectives [21,22], we are unaware of any dedicated study to generalize the Lorentz-invariant external-leg treatment to Lorentz-violating field theories.
A second need for a proper understanding of the asymptotic Hilbert space in the presence of Lorentz breakdown derives from its phenomenological importance: external-state effects govern the physics of free particles and are therefore also crucial for numerous Lorentz tests. Examples include various kinematical threshold effects in cosmic rays [23], photon birefringence and dispersion [24,25], collider kinematics and interferometry [26][27][28][29], and neutrino propagation [30]. Paralleling the conventional Lorentz-symmetric case, all previous analyses have been performed under the tacit assumption that the physics of free particles is determined by the quadratic pieces of the corresponding Lagrangian [31]. However, this approach disregards the selfinteractions of the particle, although such effects are always present, even for asymptotic states. Consequently, they need to be considered, e.g., in any scattering process beyond tree level. In a conventional renormalizable quantum field theory (QFT), Lorentz symmetry implies that the quadratic Lagrangian can only acquire a mass shift and a field-strength factor, both of which can be treated by renormalization of existing quantities. The external QFT legs are then identical in structure to the quadratic-Lagrangian solutions, which therefore indeed describe the propagation of free particles correctly. A non-perturbative rigorous justification for this feature is given by the aforementioned LSZ reduction formalism [20]. However, in the presence of Lorentz violation a similar line of reasoning fails, and the question regarding the determination of free-particle properties arises.
The present work is intended to initiate a theoretical investigation of these issues. In particular, we demonstrate that in the absence of Lorentz symmetry the external legs in perturbative quantum field theory exhibit a different structure than the plane-wave solutions arising from the quadratic Lagrangian. This result is in accordance with a recent work [32] in which a generalization of the Källén-Lehmann representation for the propagator was derived for a field-theoretic model with fermions that are coupled to the same Lorentz-violating SME coefficients as the ones we are considering in this work. In fact, we will use the results obtained in Ref. [32] to extract consistently the one-particle poles defining the external states of scattering amplitudes. To this end, we generalize the conventional LSZ reduction formula to the Lorentz-violating SME-type case. For this analysis, we restrict ourselves to a subset of the minimal SME's electrodynamics sector for simplicity. Moreover, since potential Lorentz violations must be feeble on observational grounds, we will consider only leading-order effects in SME coefficients. Throughout, we adopt natural units c = = 1, and our convention for the metric signature is timelike η µν = diag(+, −, −, −).
The outline of this work is as follows. In Sec. II, our Lorentz-violating model Lagrangian is introduced and some of its properties are reviewed. Section III contains a discussion of the fermion two-point function within this model, the correct way to extract the one-particle pole in the Lorentz-violating case, and the derivation of a general formula for the corresponding spinor wave-function renormalization factor. In Sec. IV, the one-loop radiative corrections to the fermion propagator are evaluated, the one-particle pole is extracted, and the dispersion relation as well as the spinor wave-function renormalization factor are obtained. Section V extends the LSZ formalism to the Lorentz-violating case and establishes the associated Feynman expansion of the scattering matrix. In Sec. VI, the formalism developed in this paper is applied to the example of Coulomb scattering. Our summary and an outlook are contained in Sec. VII. Supplementary material is collected in various appendices.

II. MODEL BASICS AND SCOPE
Our model is based on the bare gauge-invariant flatspacetime Lagrange density for single-flavor quantum electrodynamics (QED) within the minimal SME: The label B denotes bare quantities, ψ B is a Dirac spinor and We have also implemented the conventional notation for the U(1)covariant derivative D B µ = ∂ µ + ie B A B µ and for the dual field-strength tensorF µν B = 1 2 ε µνρσ F B ρσ . The Lorentzviolating effects are contained in the quantities (k B F ) µνρσ and (k B AF ) µ as well as in the the generalized gamma matrices Γ µ B and the generalized mass matrix M B . The latter are given by the explicit expressions The nondynamical spacetime constant quantities and H µν B control the type and extent of Lorentz and CPT breakdown. It has been shown that this flat-spacetime Lagrangian is multiplicatively renormalizable at one-loop order [16], and that this renormalizability property is maintained in curved spacetimes [18].
The complete one-loop structure of Lagrangian (1) would be of interest, but lies beyond the scope of this work. Our present goal is rather to initiate the study of finite radiative corrections in the presence of Lorentz violation by highlighting several theoretical issues that can arise within this context. For such illustrative purposes, it seems appropriate to simplify the model (1) such that tractability, phenomenological importance, and theoretical relevance are optimized. Considerations along these lines are presented next.
A key simplification is setting to zero all Lorentzviolating coefficients, with the exception of c µν B and (k B F ) µνρσ . We may also take the c B µν coefficient to be symmetric because its antisymmetric piece can be removed from the Lagrangian by a field redefinition at leading order [10]. Moreover, we will choose k B F to be of the form wherek µν B is taken as symmetric, traceless, and given bỹ We remark that the above choice of Lorentz-violating couplings is compatible with the structure of the renormalization constants, as will become apparent below. In particular, no additional operators are needed to absorb ultraviolet divergences in the perturbative quantum-field expansion of the model. The above choice of SME coefficients also requires a number of additional considerations, which we present next. First, we note that Eqs. (3) and (4) are incompatible in spacetime dimensions d = 4. When dimensional regularization is employed, it might then appear that this could lead to interpretational difficulties with previously determined (k B F ) µνρσ and c B µν counterterms [16] in the context of minimal subtraction, affect finite radiative corrections, or may even be associated with trace anomalies. However, it turns out that such spurious issues can be avoided altogether by considering a model withk µν B in its own right rather than as the limit of the full (k B F ) µνρσ Lagrangian. Throughout this work, we follow this latter, independent interpretation of our model.
Second, the identification of observables in Lorentzviolating field theories requires special care due various types field redefinitions and reinterpretions [33]. In the present case, it turns out that the c µν andk µν coefficients are observationally indistinguishable at leading order in any fermion-photon system; only their difference 2c µν −k µν can be measured within the context of Lagrangian (1). This feature has previously been discussed from various perspectives [24,27,34]. For example, suitable coordinate rescalings can eliminate the c µν coefficient in favor ofk µν , or vice versa. Such coordinate redefinitions can be exploited to simplify calculations. In what follows, however, we will for the most part avoid the choice of a particular coordinate scaling by keeping both c µν andk µν nonzero. This will provide an independent partial test of our results, since coordinate-scalar expressions for physically observable radiative effects should only depend on 2c µν −k µν .
Third, we also note that various experimental investigations have sought to constrain 2c µν −k µν . In particular, measurements have been performed in the context of resonance cavities [35], kinematical threshold studies at colliders [27], synchrotron radiation [28], Compton-edge investigations in electron-photon scattering [29], and astrophysical observations [36]. Through these investigations, all components of 2c µν −k µν are currently obeying bounds at the levels of 10 −13 . . . 10 −17 . At present, 2c µν −k µν nevertheless remains the parameter combination in Lagrangian (1) with the weakest experimental limits providing additional phenomenological justification for dropping the other SME coefficients from our analysis. We finally mention that further constraints on 2c µν −k µν may, for example, also be determined with spectroscopic studies of hydrogen [37] and ultrahigh-energy photonshower measurements [38].
Paralleling the conventional case, perturbative calculations within the present model are conveniently performed by fixing a gauge and allowing for the need to reg-ularize infrared divergences. For the general description of massive Lorentz-violating photons, a modified Stueckelberg procedure has recently been developed [39]. It essentially consists of amending any Lorentz-violating QED Lagrange density by where ξ denotes a gauge parameter and m γ parametrizes the gap in the photon dispersion relation. The tensorial structureη µν B ≡ η µν + δη µν B can involve small, but otherwise arbitrary Lorentz-breaking contributions δη µν . Note that both the ξ and the m γ term need to contain the same tensorη µν B [39]. The choice of δη µν B in the present context needs to be compatible with the specific purpose of the m γ term as a regulator: no Lorentz violation in addition to c andk should be introduced. One obvious possibility would be the Lorentz-symmetric choice δη µν B = 0. Another possibility is to match the Lorentz-violating structure of the effective kinetic term of the photon. At leading order in c andk, this kinetic term depends on the combination η µν +k µν B . If radiative corrections are included, the c µν B coefficient may also appear, but only in the combination (2c µν B −k µν B ), as discussed above. These considerations suggest the following choice for δη µν : Here, f is a free multiplicative coefficient that may for example be chosen to match the radiative corrections to free-photon propagation. In the present work, it will be convenient to select the choice (6). Since our primary focus is the fermion two-point function, we will set f = 0. Altogether, the above considerations lead to the following bare Lagrange density [54]: The next step is to define finite fields and couplings. To this end, we employ the usual multiplicative renormalization procedure with its Lorentz-violating generalization established in Ref. [16]: Adopting Feynman gauge (ξ = 1), working in 4 − ǫ dimensions, and employing minimal subtraction, we have at one-loop order [16]: We remark that these expressions are compatible with the usual QED Ward-Takahashi identity Z e √ Z A = 1. In terms of the above physical couplings and fields, our model Lagrange density reads

III. THE FERMION TWO-POINT FUNCTION
Our main objective being the treatment of external fermion states in the context of Lorentz-violating field theory, we will turn in this section to the general procedure of determining the on-shell limit of the two-point function in our model (13). We are interested in particular in the Lorentz-breaking radiative corrections to this limit. In Lorentz-symmetric field theory, one extracts from the general off-shell two-point function the one-particle pole and its residue, which respectively determine the asymptotic single-particle solutions and the wave-function renormalization coefficient. Lorentz invariance strongly restricts the form of the different types of terms that can occur in the fermion two-point function. In the presence of the Lorentz-violating parameters c µν andk µν , this procedure has to be generalized because more general terms can (and do) occur, the only fundamental restriction being that they are observer Lorentz scalars. What makes this generalization particularly nontrivial is the fact that some of the terms involving gamma matrices become noncommuting, an effect that does not occur in the usual Lorentz-symmetric case.
In a recent study [32], a generalization of the Källén-Lehmann spectral representation was derived for scalar and fermion field theories in the presence of a Lorentzviolating background of the type considered in this work. We will use the form derived for the one-particle pole in that work as a guiding principle for extracting the oneparticle fermion pole in the present analysis. We begin our general discussion of the two-point function with a few general remarks about the choice of perturbation scheme. (An actual perturbative calculation of this function will have to wait until the next section.) To set up perturbation theory for calculating the radiative corrections to the two-point function we have to make a suitable choice of a zeroth-order system with known solutions, such that the remaining piece can be considered a small perturbation relative to this zeroth-order system. While usually one takes as the zeroth-order system the full quadratic part of the action, in the case at hand there are at least two reasonable choices one might consider.
In the first scheme, one defines as a basis the renormalized quadratic Lagrangian of the conventional Lorentzsymmetric case. All Lorentz-violating contributions to the Lagrangian are then taken as perturbations. One can, for example, use the minimal-subtraction scheme to define the counterterms (either Lorentz-symmetric or Lorentz-violating).
In the second scheme, one defines as a basis the full renormalized quadratic Lagrangian, including the Lorentz-violating part. The perturbations are then just the non-quadratic contributions to the Lagrangian. For the latter, it is convenient to join the corresponding Lorentz-symmetric and Lorentz-violating vertices in the same diagram. One can do this also for the counterterms.
A key difference between the two schemes concerns their kinematical features, which is best explained by an example. Consider the one-loop fermion self energy. In the conventional Lorentz-symmetric case, energymomentum kinematics prohibits the internal photon and fermion lines from going on-shell simultaneously for physical incoming momenta. Let us next look at the leadingorder Lorentz-violating generalization of this process in the above two schemes with focus on the special case with a photonk coefficient only.
In the first scheme, this process involves the conventional diagram plus another version of the diagram with ak insertion on the photon line represented by a box in Fig. 1. Both of these diagrams exhibit the same Lorentzsymmetric propagators and the same Lorentz-symmetric dispersion relation for external momenta. The energymomentum kinematics is therefore unchanged relative to the conventional case. In particular, the internal photon and fermion propagators cannot go on-shell simultaneously for physical incoming momenta at this order in perturbation theory. In the second scheme, there is a single diagram analogous to the conventional one, but with the usual photon propagator replaced by a Lorentz-violating propagator containingk, represented by a double photon line in Fig. 2. At this order, the dispersion relation of the incoming momentum remains Lorentz symmetric, but thek propagator modifies the internal kinematics of the self-energy diagram. In particular, the internal photon and fermion lines can now go on-shell simultaneously for physical incoming momenta in certain regimes. This is evident from the well-established result [44] that photons that are slowed down by certain values ofk lead to vacuum Cherenkov radiation at ultrahigh energies: a real free fermion can now emit a real photon. The nonzero cross section for this effect is then directly related to imaginary contributions to the fermion self-energy via the optical theorem. Note that imaginary contributions are absent in the first scheme.
Such Cherenkov instabilities are relatively rare: they do not occur for all Lorentz-violating coefficients and are in any case only present at ultrahigh energies in our model. They correspond exactly to the two-particle regime discussed in Ref. [32], where the Källén-Lehmann representation of the fermion two-point function was analyzed in the presence of a c µν -type Lorentz-violating perturbation. As was shown there, for ultrahigh momenta the two-point function can pass from a stable one-particle regime into an unstable two-particle regime and provoke a Cherenkov-type decay. In the case at hand, the latter consists of the fermion plus a photon.
In this work, we are primarily interested in the properties of external states before such rare instabilities lead to their decay. We therefore omit imaginary contributions to the fermion self-energy. If needed, the physics of such instabilities may still be included subsequently, for example via cut diagrams using Cutkosky's rules [45]. For the above purpose, which disregards instabilities, the two schemes become equivalent-a fact we have verified explicitly at one-loop order. For definiteness, we present our analysis in the second scheme. It has the advantage of being more economical, as the number of diagrams is considerably reduced. In particular, the diagrams in this scheme are in one-to-one correspondence with the diagrams in the Lorentz-symmetric case. Moreover, the in the present context this scheme is free of momentumrouting ambiguities. Although the prescription for calcu-lating the corresponding amplitude is more involved due to the Lorentz-violating coefficients, we have found that the full calculation is easier to carry out in the second scheme and is also less error-prone.
Explicitly, the second scheme implies that our model Lagrange density (13) is split into the following three pieces: where and The corresponding Feynman rules, which are collected in Appendix A, now facilitate an order-by-order calculation of our model's two-point function Paralleling the conventional perturbative determination of this function, Σ denotes the contribution of the oneparticle irreducible Feynman diagrams. Before proceeding with an actual one-loop calculation, it is instructive to determine the general structure of Σ. We may decompose this quantity as where we have defined c p γ ≡ c µν γ µ p ν andk p γ ≡k µν γ µ p ν . In this decomposition, Σ LI ( / p) denotes the Lorentzsymmetric contributions equivalent to the conventional diagrams; it can thus only be a function of / p, as usual: where both f 0 (p 2 ) and f 1 (p 2 ) are understood to depend on the fine-structure constant α = e 2 /4π and the square of the 4-momentum p 2 . The remaining two terms involve deviations from Lorentz symmetry, so we will describe them in more detail. The second term Σ LV (p 2 , c p γ ,k p γ ) contains all those Lorentz-violating terms with a gamma-matrix structure that is already present in the fermion Lagrange density (i.e., a Lorentz-breaking symmetric traceless 2-tensor contracted with a gamma matrix and a single momentum factor). For example, Σ LV includes the counterterms for c µν andk µν together with the corresponding (regulated) infinities they cancel to yield an ultraviolet finite expression: where f c 2 (p 2 ) and fk 2 (p 2 ) depend only on α and p 2 since we are working at leading order in Lorentz violation. Explicit expressions for f c 2 (p 2 ) and fk 2 (p 2 ) can in principle be determined within perturbation theory to any given order in α. Below we will calculate f c 2 (p 2 ) and fk 2 (p 2 ) at one loop, i.e., at O(α). Initially, f c 2 (p 2 ) and fk 2 (p 2 ) may also depend on an infrared regulator and an arbitrary mass scale introduced by the chosen ultraviolet regularization procedure. But a consistent treatment of infrared effects and the renormalization conditions should remove free parameters from Σ LV (p 2 , c p γ ,k p γ ). The remaining term δΣ(p µ , c µν ,k µν ) contains novel Lorentz-breaking structures that are not already present in the original Lagrange density (13). Like the second term, it must involve combinations of p µ factors, γ matrices, and-since we are working at linear order in Lorentz violation-a single c µν ork µν coefficient. Up to factors consisting of powers of p 2 , a multitude of terms can be constructed that satisfy these requirements. They are as well as an additional seven terms with c µν replaced bỹ k µν [41]. The above list (22) can be constrained further by noting that electromagnetic interactions preserve C, P, and T. Quantum corrections linear in c µν andk µν must exhibit the same discrete symmetries as the original Lorentz-violating operators. This fact together with our scope set out earlier (i.e., omitting instabilities and thus nonhermitian expressions) leaves only the first two terms in the list (22) and theirk µν analogues [43]. For this reason, δΣ(p µ , c µν ,k µν ) can only depend on c p p ≡ c µν p µ p ν andk p p ≡k µν p µ p ν , and we may write Here, we have introduced the dimensionless functions f c 3 (p 2 ), f c 4 (p 2 ), fk 3 (p 2 ), and fk 4 (p 2 ), which can in principle be calculated within perturbation theory to any given order in α. These functions may initially still contain infrared regulators, which can presumably be removed by a soft-photon treatment. Disregarding the aforementioned possibility of high-energy nonhermitian contributions, Eqs. (19), (20), (21), and (23) determine the full off-shell structure of the fermion two-point function in our ck model (13) at all orders in α and at linear order in Lorentz violation.
Before deriving explicit expressions for the scalar functions appearing in the corrections (20), (21), and (23)-a task to which we will turn in Sec. IV-it is instructive to construct a general procedure for extracting the onshell external-leg physics determined by the structure of these corrections. The standard method in the Lorentzinvariant case is to consider Γ (2) (p) as a function of / p, and to determine the value of / p = m phys for which the two-point function vanishes. Laurent-expanding the inverse of the latter around / p = m phys yields the coefficient of the simple pole (residue), which determines the fieldstrength renormalization.
Unfortunately, for the Lorentz-violating expression (19) the situation is more complicated: Γ (2) (p) depends not only on / p, but also on c p γ ,k p γ , as well as on the Lorentz-violating coordinate scalars c p p andk p p . Consequently, the propagator can be expected to have a pole in terms of a value of some functional combination of all those quantities, rather than / p only. Of course, this is already the case before introducing loop effects, as Γ µ p µ depends on c p γ as well as / p. Let us briefly review the way the propagator pole is extracted in the Lorentz-invariant case. Displaying the 4 × 4 identity matrix 1 1 for clarity, one can then write Γ (2) (p) as a function of / p only Γ (2) (24) and determine the physical mass m phys by requiring One can then expand where Z R is the wave-function renormalization and Σ 2 ( / p) a (finite) function. How can this procedure be generalized to the Lorentzviolating case? The obstacle arises due to the form of the last term in Eq. (26): the inclusion of structures that involve c p γ andk p γ becomes ambiguous, as these expressions do not commute with / p. Nevertheless, it is possible to make a meaningful generalization. First of all, we write the momentum-space two-point function in the most gen-eral possible form (27) Here, the expressions A, C, K, and M are functions of p 2 and (c, k) p p , where the latter are quantities of the form with t µν i equaling either c µν ork µν . Following Ref. [32], it should be possible to extract from the full fermion propagator a pole of the form where the ellipsis stands for terms proportional to with t µν i again equal to either c µν ork µν (cf. definition (28) above). Note that the coefficient functionsx (c,k) andm only depend on the quantities (28), not on p 2 . In this paper, we will be working only to first order in Lorentz violation. Consequently, (29) reduces tō where we can takē andm We should expect to be able to write down a generalization of relation (26) with Γ (2) (p) being equal to a sum of a term linear inP (p) and a rest term of quadratic order. This task is complicated by the presence of the quantities c p γ andk p γ , which do not commute with / p. But it turns out that a self-consistent generalization of Eq. (26) to the Lorentz-violating case is given by where is some Lorentzviolating expression involving the momentum. Note the order of the noncommuting functionsP (p) and Σ 2 (p) in the second term. The function Z R (c, k) p p is the generalization of the wave-function renormalization constant in the Lorentz-violating case. Its form is in accordance with the results in Ref. [32]. Indeed,Eq. (34) implies that so thatP (p) −1 and Γ (2) (p) −1 have the same pole structure.
To determine the coefficient functionsx,ȳ, andm in Eq. (31), we write which leads to wherē We can now rewrite (note that the coefficient functions A, C, and K commute with the gamma matrices) and evaluate the anticommutators. In the above relation, we consider A ≡ A PP + 2mP +β, (c, k) p p with analogous definitions for C, K, and M . Using that the anticommutators in Eq. (39) can be worked out. It then follows that where the first argument of the coefficient functions (originally p 2 ) should be taken equal toβ. We see from Eq. (37) that this amounts to taking p 2 on the mass shell defined byP = 0. The symbol ∂ 1 indicates a partial derivative with respect to the first argument. It is apparent that Γ (2) (p) in Eq. (42) takes the expected form of Eq. (34), provided the relations are satisfied. Moreover, in Eq. (42) we can identify the expression in square brackets multiplyingP with the inverse of the wave-function renormalization Z R (c, k) p p . Equations (43)-(45) determine the three coefficient functionsx,ȳ, andm. Note that these are implicit relations, asβ, which is an argument of the functions A, M , C, and K, depends itself onx,ȳ, andm.

IV. ONE-LOOP CALCULATION OF THE MODIFIED PROPAGATOR
An interesting question concerns the determination of the functions , and fk 4 (p 2 ) perturbatively at leading order in the fine-structure constant α. To this end, we will adopt the perturbative scheme based on the expressions (14)- (17), in which the propagators are built from the full quadratic Lagrange density, including the Lorentzviolating parts. The corresponding Feynman rules are presented in Appendix A, and the only loop diagram involved in the fermion self-energy is: This diagram together with the corresponding counterterm contributions gives the conventional Lorentz- Here, γ E = 0.57721 . . . denotes the Euler-Mascheroni constant, m γ a ficticious photon mass introduced as an infrared regulator, and the arbitrary mass scale µ is a remnant from dimensional regularization. The integrations over y can be performed requiring p 2 to be close to the mass shell: (m − m γ ) 2 < p 2 < (m + m γ ) 2 . They yield the infrared finite limits on the (conventional) mass shell For the on-shell values of their derivatives we find An explicit calculation also yields ultraviolet-finite expressions for the remaining, Lorentz-violating contributions to Σ. For the functions f c i (p 2 ) we obtain: While f c 2 (p 2 ) is infrared finite, this is not the case for f c 3 (p 2 ) and f c 4 (p 2 ): the latter both diverge for p 2 = m 2 when taking m γ → 0: Below, we will also need the derivatives of these functions at their on-shell value p 2 = m 2 . To this effect, we use that Applying these relations to (54)-(56), and evaluating the integrals at their on-shell value one finds Note that f c 3 ′ (p 2 ) and f c 4 ′ (p 2 ) have linear, rather than logarithmic, infrared divergences.
For the coefficients fk i (p 2 ) one finds: All three coefficients fk i (p 2 ) are infrared finite on the conventional mass shell: Applying the relations (60) to Eqs. (64)-(66) and evaluating the integrals at p 2 = m 2 one finds the infrareddivergent expressions Let us now see how the formalism developed in Sec. III can be used to incorporate the radiative corrections derived above. First, let us express the quantities A, C, K, and M in Eq. (27) in terms of the coefficients f c i and fk i calculated above. Explicitly, we have Using the expansions (32) and (33) it follows, by taking Eq. (43) to zeroth order in Lorentz violation, that As the functions f i are O(α), we obtain Taking relations (44) and (45) to zeroth order in Lorentz violation, it follows that We continue by extracting m 10 and m 01 from Eq. (44), taken to first order in Lorentz violation. We now have to expand so that and . .) and taking the coefficients of c p p andk p p it follows that These results allow us to evaluate the explicit form of the wave-function renormalization Z R (c, k) p p , which is the inverse of the coefficient ofP in Eq. (42): In order to evaluate the expressions ∂ 1 A(β) and ∂ 1 C(β) in Eq. (87) to the required order it is necessary to Taylorexpand the functions f ′ 0 (β) and f ′ 1 (β) just like in Eq. (81). The Lorentz-symmetric part of Z R (c, k) p p is identical to the usual (one-loop) wave-function renormalization constant for the fermion in QED. In particular, it has a logarithmic infrared divergence. The linear and logarithmic infrared divergences that are present in various terms proportional to c p p andk p p in Eq. (87) cancel in the final expression Eq. (88). As is well known, the Lorentz-symmetric infrared divergences cancel in scattering cross sections when we take into account the contributions of soft-photon emission from the corresponding external legs. In Sec. VI, we will check that these contributions do not introduce additional infrared divergences proportional to c p p ork p p . We are now in a position to determine the radiative corrections to the fermion dispersion relation. It is easy to check that the dispersion relation that follows from P (p) = 0 is (to first order in the Lorentz-violating coefficients and α) It is gratifying to note that the O(α) Lorentz-violating radiative corrections to the dispersion relation turn out to be proportional to the combination (2c p p −k p p ), in accordance with the general expectation (see Sec. II).
At this point it is useful to give a physical interpretion of the various quantities that were calculated in this section. From Eqs. (31)- (33), (78)-(80), (84), and (85), we see that the loop-corrected (Dirac-)operatorP (p) can be written, to first order in c µν ,k µν , and in the finestructure constant α as where is the usual loop-corrected mass in the minimalsubtraction scheme, and Equation (92) expresses the radiatively corrected, physical value of the Lorentz-violating parameter c µν in terms of its tree-level value. Note that both c µν and the mass scale µ are unphysical, renormalization-schemedependent quantities, unlike (c phys ) µν , which is in principle measurable. This situation is completely analogous to relation (91) expressing m phys in terms of m and µ.
In Eq. (90), we have anticipated the loop-corrected version (k phys ) µν of the parameterk µν . While we do not compute this expression in this work, it will of course differ from the tree-level value by an expression of order α, so that it can be substituted for the latter in Eq. (90), as we disregard terms of order α 2 .
The last term in Eq. (90) is a radiative correction in the form of an operator that is of a different form than any of the terms present in the tree-level Dirac-operator This might seem surprising, as such a situation never arises in the Lorentz-symmetric case. However, it is fully compatible with the general form of the one-particle fermion pole of the Källén-Lehmann representation derived in Ref. [32]. This (µ-independent) correction constitutes one of the central results of this work. It is pleasing to note that the procedure outlined in this section deals nicely with the potential infrared divergences that are present in some of the coefficients f i and their derivatives. We see that the dispersion relation (89) is infrared finite. One can also readily verify that, at least up to the order considered (linear in c,k and in α), this is in fact the case for all coefficients A β , (c, k) p p , M β , (c, k) p p , C β , (c, k) p p , and K β , (c, k) p p , and hence for the coefficientsx,ȳ, andm. Note the cancellation of infrared divergences in the expression multiplying c p p , both in Eq. (82) and Eq. (83). We conjecture that infrared divergences will continue to cancel at any order in the perturbative expansion.
Passing the momenta to derivatives, it follows that the Lagrangian describing the on-shell free fermion field acquires higher spacetime derivatives. This is the case for either the Lagrangian derived from the original proper two-point function or for the one derived from the operator P (p). Often it might be possible to avoid working directly with the physical field and employ the bare field instead treating the higher-dimensional Lorentz-violating terms perturbatively. However, this becomes problematic or impossible if we consider on-shell external states, as in the derivation of the Lehmann-Symanzik-Zimmermann reduction formula, which computes scattering amplitudes for on-shell external physical states. In the next section, we will analyze how this situation generalizes to loopcorrected Lorentz-violating Lagrangians.

V. EXTERNAL STATES IN FEYNMAN DIAGRAMS AND THE LSZ FORMULA
How do the Lorentz-violating radiative corrections parametrized by the coefficient functionsx,ȳ, andm in Eq. (31) contribute to S-matrix elements? Let us reflect a moment on how we can determine the latter.
In the quantum-field description of scattering experiments, it is presupposed that the Fock space of physical states is generated from a unique vacuum by free fields ψ in (x) andψ in (x) (here we will only consider fermions in the asymptotic states and ignore the possibility of photons). One assumes that the coupling terms in the equations of motion are affected by some adiabatic cutoff function equal to unity at finite times and vanishing smoothly as |t| → ∞, and the particles in the initial and final states have become well separated. Then, according to the usual adiabatic hypothesis the interacting fields ψ(x) andψ(x) are presumed to satisfy, in a weak sense, (and similarly forψ(x)) for some normalization constant Z that should be smaller than one, in order to account for the fact that the content of the state ψ(x)|0 is not exhausted by the matrix elements with one-particle states, while ψ in |0 is. In Sec. IV, we saw that for the Lorentz-violating model we are considering the normalization constant analogous to the constant Z in Eq. (94) is not only Lorentz violating, but becomes dependent on the momentum of the external particle: Z → Z R (p), see Eqs. (87) and (88). To see how this will affect the usual treatment of external states in scattering amplitudes, let us begin by looking at the free field ψ in (x) (the out-field ψ out (x) will be analogous). Consider the spinor wave functions u s in ( p) and v s in ( p) of the physical field ψ in (x). They are modified with respect to the Lorentz-invariant situation. While in the latter case we have ( / p − m)u s in ( p) = 0 (with p 0 = ω p > 0) and ( / p−m)v s in ( p) = 0 (with p 0 = −ω p < 0), the spinors now satisfȳ for the positive-energy solutions and for the negative-energy solutions corresponding to a given 3-momentum. Thus, we conclude that our model's external spinors, unlike in the Lorentz-invariant case, are modified by the one-loop radiative-correction terms calculated in the previous section. Note also that we get the Lorentz-violating multiplicative contribution Z R (c, k) p p (wave-function renormalization) to the S-matrix for every external fermion that factors out of the fermion propagator pole (see Eq. (35)).
Let us analyze in some more detail the new equations of motion for the spinors, Eqs. (95) and (96). To first order in radiative corrections and in Lorentz-violating parameters, we can use the approximations of Eqs. (32) and (33), which yield For fixed 3-momentum p, the value of p 0 in Eq. (97) is determined as the positive root of the dispersion relation (89). Every term in the dispersion relation is of even order in the 4-momentum, so that when (p 0 , p ), where p 0 > 0, satisfies the dispersion relation (89), so does (−p 0 , − p ). The latter solution is taken to correspond to v s in ( p), an anti-fermion with momentum p and energy p 0 . Thus, where p 0 takes the same value as in Eq. (97). The fact that a fermion and an anti-fermion with the same momentum have equal energy is a consequence of CPT invariance, which is unbroken by the c µν (and by thek µν ) coefficients.
On the other hand, in general there are Lorentzviolating quadratic terms in the dispersion relation (89) that mix p 0 and p. As a consequence, the fact that (p 0 , p ) (with p 0 > 0) satisfies Eq. (89) does not imply the same for (−p 0 , p ). This expresses the fact that the c 0i (and k 0i ) violate parity. Another useful observation is that the dispersion relation (89) is not sensitive to the spin label s. Note that spin-dependence does play a role for some of the other types of SME coefficients, but we will not consider them in this work.
We see from Eqs. (97) and (98) that the equation of motion has terms quadratic in the momentum due to the presence of c p p andk p p . These terms make a rigorous analysis of the equation of motion for the external fermion field and a quantization of the latter along the lines of Appendix B problematic. For instance, they likely introduce spurious non-physical solutions. For this reason, we use the zeroth-order dispersion relation p 0 = p 2 + m 2 ≡ ω p to substitute for c p p : and similarly fork p p . For simplicity, we will suppress thẽ k terms in the following. Equation (97) becomes The spinor u s in,1 ( p) satisfying Eq. (100) differs from the original one u s in ( p) by terms of second order (and higher) in the Lorentz-violating coefficients. We remark that this higher-order difference between the spinors permits a selfconsistent treatment of the asymptotic Hilbert space in terms of u s in,1 ( p), while also allowing us to switch back to the original spinors u s in ( p) at a later point in the calculation.
With these considerations, we can proceed as in Appendix B. The equation of motion can be written in the form of an eigenvalue equation: The operator acting on the left-hand side of Eq. (102) on the spinor is hermitian with respect to the inner product which is different from that in Eq. (B13). Consequently, it has real eigenvalues, with the corresponding eigenspinors forming an orthonormal basis in spinor space as well as in analogy to Eqs. (B14) and (B15). Incidentally, note the hermiticity relationΓ 0 ( p ) † = γ 0Γ0 ( p )γ 0 forΓ 0 ( p ).
The free field has the Fourier decomposition The creation and annihilation operators can be expressed by the following projections and their hermitian conjugates. We remind the reader that the zeroth components of the momentum in the plane-wave exponentials in Eqs. (109) and (110) depend on the corresponding mode: p 0 u,s and p 0 v,s . The results derived in Appendix B for the free-field quantization in the presence of Lorentz violation hold analogously for the field ψ in (x). Note in particular the (Feynman) propagator In the last step, we have retained only the leading-order Lorentz-violating corrections to the denominator of the integrand. This approximation holds provided one ignores any possible unphysical poles far from the mass shell that might appear in the integration over p 0 when takingP (p) as the momentum-space two-point function.
From the discussion at the beginning of this section, we can now give a more precise formulation of the adiabatic hypothesis for the interacting field ψ(x) and the free field ψ in (x). Comparing Eq. (111) with the on-shell limit of Eq. (35) it follows that instead of relation (94) we now have, in the limit x 0 → −∞: For large negative times, Eq. (109) can thus be written in terms of the interacting field: and similarly for d † in . In the same way we can express the out-oscillators in terms of the interacting field for large positive times.
We will now use the above results to derive the LSZ reduction formula for the Lorentz-violating case. For this expression to exist, it is important that the Cauchy initial-value problem of the field theory be well defined.
Fortunately, the above procedure leading to a spinor equation of motion linear in the zeroth component of the 4-momentum is exactly what is needed for a consistent derivation of the LSZ formula, as we will see below.
We begin by defining a smearing procedure for the definite-momentum creation and annihilation operators, so that the states created by the smeared operators can be localized in position space. For example, with analogous definitions for the various other creation and annihilation operators. Here, we have abbreviated describing the creation of a particle localized in 3momentum space near q and localized in 3-position space near the origin. In the Schrödinger picture, a state created by this operator evolves in time. Applying this smearing to b in s † given in the form of Eq. (113) We can now use the equation of motion (102) for u s in,1 ( p) to expressΓ 0 p 0 in the last equation in terms of p. We then trade the p i components for partial derivatives acting to the right on the exponential. By performing partial integrations they can be converted to partial derivatives acting to the left, which yields The last identity is valid to first order in Lorentz violation, on the physical mass shell (i.e., any spurious, unphysical solutions ofP (p) = 0 far from the mass shell should be disregarded). Similarly, if we start with an antifermion in the initial state: can be expressed, using Eqs. (116) and (117) and their hermitian conjugates, as the LSZ reduction formula As in the derivation for the Lorentz-invariant case (see, e.g., Ref. [46]), the introduction of the time-ordered product in Eq. (119) is necessary so that the field operators are in the convenient order with respect to the in-and out-vacua. In deriving Eq. (119), we have taken the momentum distributions f q ( p ) to the delta-function limit In practical calculations it is most useful to express the scattering amplitude in terms of truncated Green's functions. Using the definitioñ in Eq. (119), the connected scattering amplitude can be expressed as If we now introduce the truncated Green's functions in which all external legs are multiplied by the inverses of the corresponding complete propagators, it follows from Eqs. (42), (122), and (121) that Note that G (2n) trunc carries 2n Dirac indices (that are contracted with the spinors), which are suppressed here for readability.
Formula (124) embodies the Feynman rules for the scattering amplitude, incorporating: • a momentum-conserving delta-function; • the amputated Green's function; • a momentum-dependent wave-function renormalization factor ±iZ 1 2 R (c,k) p p for every external leg; • a Dirac spinor for every external leg: u s in ( p) for an incoming fermion; -ū s in ( p) for an outgoing fermion; v s in ( p) for an outgoing anti-fermion; -v s in ( p) for an incoming anti-fermion.
We will end this section with a derivation of some explicit formulas for the spinors u s in (p) and v s in (p) satisfying Eqs. (95) and (96). The most convenient way to achieve this is to take them proportional to the usual Lorentzinvariant spinor functions u s LI (p) and v s LI (p), but then not calculated for the real, physical momentum p µ , but for a redefined momentum valuep µ satisfying Thus where C u ( p ) and C v ( p ) are normalization constants to be determined below. One easily checks that satisfies Eq. (125) to first order in the Lorentz-violating parameters and obeys the dispersion relationp 2 = m 2 0 . Let us work out the normalization constants C u ( p ) and C v ( p ), in accordance with Eq. (105). Consider the case of C u ( p ) first. For u s LI ( p) we have the usual relations u r LI (p)u s LI (p) = δ rs .
Demanding now that u s in (p) satisfies the normalization condition (105) it follows that Using Eqs. (129) and (130) and working to first order in Lorentz violation one obtains the following expression for the normalization constant: In the last equation, we defined ωp ≡p 0 = p ipi + m 2 0 . Note that the same analysis can be done for the v s in (p) spinors. The normalization constant turns out to be the same as for the u spinors, so that we can safely suppress the u and v indices: As an additional simplification, the normalization constants are also independent of the spin index s. This allows us to determine spin-sum formulas. They follow directly from the usual expressions for the Lorentzinvariant case: In Eqs. (134) and (135), it is understood that p 0 ≡ p 0 u = p 0 v (see Eqs. (97) and (98)).

VI. SAMPLE CALCULATION: INFRARED DIVERGENCES IN COULOMB SCATTERING
It is instructive to apply the techniques described above to a particular case. We will do this for the Coulomb (or rather Mott) scattering of a fermion off a stationary charge. For simplicity, we will assume that only the Lorentz-violating parameterk µν is nonzero.
Let us review quickly the Lorentz-invariant case. We have for the scattering amplitude at tree level (136) Here, we have normalized the states in a finite volume V . For the Coulomb problem, we can take A = 0 and A 0 = Ze/4π| x|, so We can now pass to the cross section by squaring the absolute value of Eq. (137), multiplying by the number of possible final states V d 3 p f /(2π) 3 and dividing by the incident flux | v i |/V and the time interval T . Note that for large time intervals T , one can take |2πδ( Using now that it follows that If we do not observe the final polarization, we must sum over s, while for an unpolarized incident wave we average over the initial polarizations r. With the usual formulas for the spin sums one obtains When turning on Lorentz violation, various adaptations have to be made to the formulas (137)-(141) at tree level: 1. The Maxwell equations become Lorentz violating: whereη µν = η µν +k µν and˜ = ∂ αη αβ ∂ β . This means that the Fourier transform of the Coulomb potential becomeŝ 2. The incident velocity is now given by the group velocity v g i = ∂E/∂p i , which is fixed by the dispersion relation (89). However, note that, as in this example we choose c µν = 0, there is no Lorentz-violating effect at tree level.

The dispersion relation (89) also implies Lorentz-
violating modifications to the integration-variable transformation from dp f to dE f implied by Eq. (139). However, also here there is no effect at tree level because we take c µν = 0. Incidentally, note that the factors √ E i and E f in the denominator of Eq. (137) remain equal to their Lorentzinvariant form √ ω i,f (ω ≡ p 2 + m 2 ).
4. The spinors are modified according to the relations (126)- (128) and (132). In the unpolarized cross section (141), we have to use the modified spin sum (134) or (135), as appropriate. Again, there is no effect at tree level as we take c µν = 0.
It is straightforward but tedious to adapt the formulas for the tree-level cross sections (140) and (141) to the Lorentz-violating case accordingly. Rather than doing this explicitly, we will move our attention to radiative corrections. The diagrams shown in Fig. 3 contribute to one-loop order. Note that the fermion self-energy diagrams are taken into account implicitly in the order α corrections to the external spinors in formulas (126)-(132), as well as by the inclusion of the wave-function renormalization Z 1/2 R for each external fermion leg.
Instead of carrying out the full calculation of the oneloop diagrams, we will just concentrate on the infrareddivergent contributions to the scattering amplitude. We will then show that they indeed cancel in the experimental cross section, just as in the Lorentz-invariant case.
Paralleling the usual Lorentz-invariant case, the vacuum-polarization diagram is infrared finite, so that we only have to consider the vertex-correction diagram. With the modified photon propagator (recall that we have chosen c µν = 0) the amplitude for the vertex correction is given by The the integral on the right-hand-side of Eq. (144) is infrared divergent. This divergence arises from the residue of the photon propagator. In order to isolate it, we can use the fermion on-shell relations by using the (approximate) equations of motion for the spinors. Moreover, we can drop terms linear in k in the numerator and quadratic in k in the fermion pole factors in the denominator. It then follows that We can evaluate the integral (146) by using the identity where P denotes the principle-value part. Omitting the infrared-finite contribution from the principal value we find It follows that the one-loop contributions to the elastic Coulomb scattering amplitude are obtained by substituting where the ellipsis indicates infrared-finite contributions. For the elastic cross section, this implies The term proportional to α as well as the factors Z R in expression (150) are infrared-divergent in the limit m γ → 0.
In the Lorentz-invariant case, infrared-divergences are canceled if one incorporates the fact that some final states that include soft photons are experimentally indistinguishable. We will now proceed to check that this continues to hold true in the case at hand.
Following the usual procedure, we consider final states that include one soft photon with energy smaller than the detector resolution ∆E. The amplitude for this process is where M 0 is the amplitude for elastic scattering without photon emission. To get the cross section for this process, we have to include also the final-state volume element for the photon. As shown in Appendix C, a consistent way to do this is to take rather than the conventional factor d 3 k/(2ω k (2π) 3 ).
Here, k 0± ( k) are the absolute values of the solutions for k 0 to the modified dispersion relation. Combining Eqs. (151) and (152) and employing formula (C25) one then finds for the cross section for this (soft-bremsstrahlung) process Using the polarization-sum formula (C24), Eq. (153) reduces to Comparing the third term inside brackets in Eq. (154) with Eq. (150) we see that both contain integrals over the same p, p ′ term, with opposite sign, so that the corresponding infrared divergences cancel.
To verify that the same happens for the p, p and the p ′ , p ′ terms in Eq. (154) and the Z R (p ′ )Z R (p) factors in Eq. (150), we evaluate the k integral over, say, the p, p term in Eq. (154). To this end, we perform a change of variables k µ →k µ such that To first order, this means thatk µ = k µ + 1 2k µν k ν . To lowest order, the measure d 4 k is invariant under this transformation, as where we have definedp µ = (η µν − 1 2k µν )p ν . Note that strictly speaking, the upper limit of the transformed energyω is modified by the transformation, but the result is an effect of higher order ink and can be ignored. The resulting integration is a standard one with result [46] 4π ln ∆E where the ellipsis indicates terms that are finite as m γ → 0. Substitution in Eq. (154) gives (for the second term) It follows that the infrared divergence in Eq. (159) indeed cancels the corresponding one in the elastic cross section (150) arising from the multiplicative renormalization function Z R (p) that was evaluated in Eq. (88):

VII. SUMMARY AND OUTLOOK
Perturbative Lorentz-invariant quantum field theory rests on a few core field-theoretic techniques. One of these concerns the order-by-order determination of the asymptotic Hilbert space, and thus the calculation of quantum corrections to the external states. Such effects govern the propagation of free particles and are indispensable for scattering amplitudes. The present work for the first time has addressed the issue how to generalize this core technique to Lorentz-violating quantum-field theories. To illustrate the salient features of this generalization, we have focused on a particular sector of the SME. We expect, though, that our reasoning can also be applied to other Lorentz-violating field theories with a more complex structure and a wider variety of coefficients.
Specifically, we considered the SME's single-flavor QED sector in the presence of the c µν and the nonbirefringent piece of the k µναβ F coefficients. We found that the presence of these Lorentz-breaking terms in the Lagrangian has some profound consequences for the radiative corrections to the pole structure of the external states. In particular, the Dirac equation satisfied by the external-state spinors turns out to be modified by Lorentz-violating operators not present in the Lagrangian, a feature that is unknown in usual Lorentzsymmetric field theories. Our analysis also shows that the wave-function renormalization will typically contain Lorentz-breaking coefficients contracted with momenta. We note that this is in contrast to the usual one-loop QED result, where Z ψ is a momentum-independent constant. Momentum dependence of the wave-function renormalization is known to occur in certain other contexts [48].
We have limited our present study primarily to theoretical techniques for determining quantum corrections to external states in Lorentz-violating backgrounds. However, our results indicate that such corrections may have profound phenomenological implications for Lorentz tests, which can be seen as follows. The new, radiatively induced term exhibits two powers of the momentum, whereas the existing terms contain only up to a single power. The correction term should therefore grow faster with the momentum than the existing terms. This opens the possibility-at least in principle-that the Lorentzbreaking radiative corrections become larger than the original tree-level Lorentz violation. Note that this does not necessarily signal a breakdown of perturbation theory because the perturbation Hamiltonian also includes the conventional electromagnetic interaction, which is comparatively much larger. In our simple model, the radiative-correction term of size ∼ α 3πm (2c p p −k p p ) can reach the size of the tree-level contribution c p γ when α p ∼ m. It follows that for Lorentz tests involving free electrons with energies ∼ > 100 MeV, radiative Lorentz-breaking corrections may not always be negligible relative to the tree-level Lorentz violation. Note that electrons in such an energy range are routinely employed in various Lorentz tests. We remark, however, that in our particular model the resulting fermion eigenenergies are free of this effect. This may be a special property of our model as both the tree-level Lorentz violation and the induced correction have the same C, P, and T properties. Nevertheless, the model discussed in this work could still exhibit other observables, such as ones involving the one-loop eigenspinors, in which the Lorentz-breaking correction dominates the tree-level effects. In the context of more general models involving the parity-odd weak interaction, the radiatively induced terms are likely to display a greater variety of structures since they do not necessarily have to share the same C, P, and T properties of the tree-level Lorentz violation. Then, even the eigenenergies may show the effects mentioned above.
Another immediate consequence of our result concerns multimetric theories [49], such as recently proposed bimetric models [50,51]. The basic idea in models of this type is that different fields experience different effective metrics. But our analysis shows that the concept of two metrics is difficult to maintain in a quantum theory: beyond tree level, radiative corrections to particle propagation typically induce higher-order terms incompatible with an effective-metric interpretation. This difficulty by itself does not affect the consistency of such models; it rather illustrates, for example, that the trajectory of the particle is not a geodesic with respect to some metric.
To see this more explicitly, consider first the free electromagnetic field in our model. Inspection of Eqs. (6) and (7) establishes thatη µν = η µν +k µν can be interpreted as the effective (inverse) metric that governs photon propagation at tree level. Similarly, comparison of our fermion kinetic term 1 2 iψ(δ µ a + c µ a )γ a ↔ ∂µ ψ with that in general coordinates 1 2 ieψe µ a γ a ↔ ∂µ ψ reveals that we may interpret δ µ a + c µ a as the vierbein e µ a [55]. It is then apparent that the fermion propagation is controlled by the (inverse) effective metric (g f ) µν = e µ a e ν b η ab = η µν + 2c µν + O(c 2 ) at tree level. We see that in the absence of quantum corrections our Lorentz-violating QED extension can indeed be interpreted as a bimetric model in the flat-spacetime limit. We remark in passing that this is consistent with our earlier discussion that only 2c µν −k µν is observable. On the other hand, our analysis has shown that the leading radiative corrections to the free-fermion propagation-displayed in Eq. (90)-are determined by a term of the form ψ(2c µν −k µν )∂ µ ∂ ν ψ. But such a term precludes an interpretation of the fermion's propagation as being governed by an effective metric.
On a more practical level, we applied our formalism to Coulomb scattering for the case c µν = 0,k µν = 0. We showed that, just like in the usual Lorentz-symmetric case, infrared divergences cancel when soft-photon emission is taken into account in the final fermion states. It should be stressed that this result involves a non-trivial cancellation between various infrared-divergent Lorentzviolating terms.
Our study also demonstrates how to extract the Smatrix of a process with external fermion states in the presence of Lorentz-breaking coefficients, generalizing the usual LSZ reduction formula to the Lorentz-violating case.
In this work, we have only considered Lorentz-violating radiative effects on fermion external states. In particular, we have left unaddressed the effects of the Lorentz breakdown on vacuum polarization. The main reason for this omission is that a Källén-Lehmann representation for the photon propagator in the presence of the k µναβ F coefficient (and in particular its birefringent part) has not yet been developed, which makes a proper extraction of the photon pole(s) unclear. We expect to come back to this issue in the future.
In this appendix, we carry out the explicit quantization of the Dirac field in the presence of Lorentz violation. One possible approach is to use a field redefinition ψ = Aχ that transforms the terms with time derivatives to the standard Dirac form [13,47]. Alternatively, one may use an unconventional scalar product in spinor space to bypass the hermiticity issues associated with unconventional time derivatives [52]. The method presented below is based on this latter approach; it is more more direct; it maintains spinor coordinate covariance, and it introduces explicit creation and annihilation operators for the particle modes corresponding to the physical field ψ as well as an expression for the Hamiltonian in terms of these. These features are particularly suitable for the quantization of the spinor equation of motion (108).
We start with the free-fermion Lagrange density with The Lorentz-violating background is taken as real-valued, so that Γ µ † = γ 0 Γ µ γ 0 and M † = γ 0 M γ 0 . The canonical momentum is given by and the canonical Hamiltonian becomes The matrix Γ 0 is indeed invertible for perturbatively small Lorentz violation [13]. From this Hamiltonian, one can recover the equation of motioṅ Next, we expand the solutions of the equation of motion for ψ in Fourier modes: where ω p ≡ +(m 2 + p 2 ) 1/2 denotes the conventional fermion energies, and p 0 takes the absolute value of the respective (four) eigenvalues of (Γ 0 ) −1 (Γ i p i + M ). The latter are, in general, all different. But for small enough Lorentz-violating parameters, two of them (which we will denote p 0 u,s ) are positive, and two (−p 0 v,s ) are negative [13]. Below, we will show that these eigenvalues are, in fact, real. The corresponding eigenvectors are u s ( p) and v s ( p), which satisfy Here, Latin superscripts r, s, . . . from the middle of the alphabet label the spin-type state. To quantize, we replace the Poisson brackets with i×anti-commutators: [ψ a ( x, t),ψ b ( y, t)] + = [ψ a ( x, t), −iπ c ( y, t)] and [ψ a ( x, t), ψ b ( y, t)] + = [ψ a ( x, t),ψ b ( y, t)] + = 0 , (B10) where Latin subscripts a, b, c, . . . from the beginning of the alphabet denote spinor components. We now proceed to check that the anticommutation relations (B9) and (B10) correspond to taking for the oscillators the following nonzero anticommutators: To see that the matrix Γ −1 0 ( Γ· p + M ) has real eigenvalues, we note that u|v ≡ūΓ 0 v (B13) represents an (unconventional) inner product on spinor space. In this expression, u and v are arbitrary spinors with the usual notationū ≡ u † γ 0 . The definition (B13) clearly satisfies the appropriate linearity conditions of an inner product. To establish positive definiteness, note first thatūΓ 0 u is always real since (Γ 0 ) † = γ 0 Γ 0 γ 0 . Moreover, γ 0 Γ 0 differs from 1 1 only by SME corrections, which implies positive definiteness for sufficiently small Lorentz violation [13]. The matrix Γ −1 0 ( Γ· p + M ) turns out to be hermitian with respect to the modified inner product (B13), which shows that this matrix indeed possesses real eigenvalues, as claimed. Moreover, the presumed small size of the SME corrections implies that these eigenvalues are perturbations around the conventional ones, so two eigenenergies are positive and two negative [13]. The four corresponding eigenspinors are orthogonal and can be normalized in the usual way: u r ( p)Γ 0 u s ( p) =v r (− p)Γ 0 v s (− p) = ω p m δ rs .
As {u 1 ( p), u 2 ( p), v 1 (− p), v 2 (− p)} forms a complete set of eigenspinors of the operator (Γ 0 ) −1 ( Γ· p + M ), we have the completeness relation We can now use this completeness relation in Eq. (B12), recovering the result of Eq. (B9). As a further application, let us derive an expression for the Hamiltonian in terms of the oscillators. The quantum-field version of Eq. (B5) is given by With the Fourier decomposition (B7) we find, using the equations of motion (B8) for u s ( p) and v s ( p), an explicitly positive-definite expression for the Hamiltonian: where E s u and E s v are the (positive) energies corresponding to the solutions of Eq. (B8). We also note the useful expressions and their hermitian conjugates. The time-ordered product is defined in the usual way [53] and satisfies T ψ a (x)ψ b (y) = 0|T ψ a (x)ψ b (y)|0 + : ψ a (x)ψ b (y) : , (B20) where 0|T ψ a (x)ψ b (y)|0 ≡ iS(x − y) ab (B21) is the modified Feynman propagator: Indeed, one can verify that Here, we used that ψ(x) satisfies the modified Dirac equation (B6) and the anticommutator relation (B9).
Appendix C: Canonical quantization of the radiation field with Lorentz-violating parameterk µν Our starting point is the following Lorentz-violating Stueckelberg Lagrange density in ξ = 1 Feynman gauge: whereη µν = η µν +k µν , withk µν =k νµ andk µ µ = 0. The quantization of such photon models has recently been studied by various authors [15]. Here, we summarize the main results and tailor the presentation to the case at hand. We begin by finding the canonical momenta and impose the fundamental equal-time commutation relations [A µ (t, x), Π ν (t, y)] = iδ ν µ δ 3 ( x − y) .
From Eq. (C3) it follows that the spatial derivatives of A µ commute at equal times. Using Eqs. (C4) and (C2), one then deduces that whereη µν is defined as the inverse ofη µν : The equation of motion following from (C1) is which implies that the dispersion relation is the same for all four modes. In other words, our model is strictly free of any birefringence. Consider now the vacuum expectation value of the time-ordered product 0|T A µ (x)A ν (y)|0 .
Acting on it with the kinetic operator we find where we have used relation (C5). We infer that 0|T A µ (x)A ν (y)|0 must indeed be equal to the (modified) Feynman propagator: paralleling the Lorentz-invariant case.
It is useful to cast Eq. (C10) in an alternative form. To this end, we write with respect to the effective metricη µν . This choice of the transverse modes corresponds to defining the physical states satisfying phys|∂ µη µν A ν |phys = 0 .
It follows from Eqs. (C20) and (C22) that the physicalpolarization sum becomes Finally, we note that with the choice (C19) the 3momentum measure appearing in Eq. (C15) satisfies the property where f (k µ ) is an arbitrary even function of k µ .