On quantum aspects of the Myers-Pospelov extended QED

For the Myers-Pospelov extensions of QED, we study its tree-level dynamics, discuss the dispersion relation, and present one more scheme for its perturbative generation, including the finite temperature case. Also, we explicitly verify that in the absence of higher time derivatives, the unitarity is preserved at the one-loop level.


I. INTRODUCTION
Formulation of the Lorentz-breaking extension of the standard model called attention to studies of Lorentz-breaking extensions for different field theory models, and, first of all, for QED [1].
Conclusions obtained by treating different aspects of various extensions of QED in dozens of papers became paradigmatic results for the Lorentz-breaking theories in general.Among the most important results, one can emphasize studies of exact solutions, canonical quantization and different issues related to quantum corrections.Within this context, an important role is naturally played by higher-derivative Lorentz-breaking extensions of QED.Indeed, it is well known that the effective action is nonlocal and can be represented in the form of the derivative expansion.Moreover, the higher-derivative terms naturally emerge within the string context [2].Therefore, one naturally faces a problem of studying different issues related to higher-derivative Lorentz-breaking extensions of QED.The first step in such study has been performed in [3] where the so-called Myers-Pospelov (MP) term has been proposed, that is, the first higher-derivative Lorentz-breaking term in QED.This term attracted a great interest due to the fact that a special choice of the Lorentz-breaking vector allows to rule out the higher time derivatives from this term, thus avoiding the unitarity breaking which is known to be the main problem of higher-derivative theories.A number of studies of unitarity issues for QED with the MP term have been performed in [4].Some other tree-level results for this theory can be found in [5], and some phenomenological applications -in [6].Further, the higher-derivative terms were shown to arise as quantum corrections, first, for the case when the Lorentz symmetry breaking is introduced through the third-rank constant tensor [7] (which for a certain choice of this tensor yields the higher-derivative [8] CFJ-like term [9]), second, for the case where the Lorentz symmetry breaking is introduced through a constant vector, with the nonminimal coupling is present [10].It was shown that in these cases the resulting higher-derivative terms are finite.Therefore, one can naturally establish the questions, first, about other possible schemes allowing to generate the higher-derivative Lorentz-breaking terms, second, about the tree-level behavior of the QED with additive higher-derivative Lorentz-breaking terms, which clearly would modify propagators and ultraviolet behavior of the theory.In this paper, we address namely these questions.To be more precise, in this paper we induce the higher-derivative terms in the gauge sector and discuss the impacts of higher derivatives for the propagator and the unitarity.
The structure of this paper looks like follows.In the section 2, we introduce the classical action of the gauge sector of the Lorentz-breaking extended QED with higher derivatives.In the section 3, we carry out the one-loop calculation of the higher-derivative Lorentz-breaking terms in the gauge sector with use of the new coupling, both at zero and finite temperature.In the section 4 we discuss the related unitarity issues and explicitly demonstrate that in the absence of the higher time derivatives, the unitarity is preserved.Finally, in the section 5, we summarize our results.

II. CLASSICAL ACTION AND DISPERSION RELATIONS
Let us consider the higher-derivatives (HD) extension of QED looking like Here u µ is a dimensionless vector, M is a mass scale, which can be suggested to be of the order of the Planck mass [3], c 1 and c 2 are some dimensionless numbers.They accompany the Myers-Pospelov [3] and higher-derivative CFJ-like [8] terms respectively (we note that within many schemes these terms arise simultaneously, see f.e.[10]).
Since this theory is gauge invariant, in order to find the propagator we can carry out the change of variable A µ → A µ + ∂ µ Ξ, where we henceforth will suggest that A µ is transversal, and Ξ is some scalar.As a result, evidently, the Ξ goes away as usually occurs in gauge theories.
We use the notation Σ = c 2 u 2 − c 1 (u • ∂) 2 .The resulting quadratic Lagrangian for the essentially transversal A µ will be given by the expression with As a result, one will have just the propagator, whose explicit form is where one has Calculating as above, we get indicates that the UV dynamics will not everywhere be higher-derivative one.Hence, the UV behavior is the same as in usual theories without higher derivatives (for example, the term A 1 η νλ asymptotically behaves as 1 k 2 ), and renormalization properties will not be improved in comparison with the usual QED.One can remind that the similar situation occurs in three-dimensional QED with higher-derivative CFJ term m µνλ A µ ∂ ν A λ , where one has So, the term proportional to ∂ ν ∂ ρ asymptotically behaves as k −2 , thus the UV asymptotics is the same as in the usual case.
To find the dispersion relations, one should consider the denominators of ( 6) and carry out the Fourier transform, so, from the denominator Q one finds the unusual dispersion relation (where is the usual square of the vector n µ in Minkowski space), whose some aspects have been earlier studied also in [11]: This dispersion relation in general cannot be simplified since there is no fundamental reason to impose the relation c 1 = c 2 forever.Here we emphasize some typical situations (see also [11] for some discussions of this relation).
1.The vector u µ is light-like, u µ u µ = 0.In this case the CFJ-like term vanishes (the same situation is observed if c 2 = 0), and we have the simplified dispersion relation: 2. Let c 1 = c 2 (that is, if we have some kind of a finite renormalization so that the corresponding term is present from the very beginning).We have the simplified dispersion relation: 3. The vector u µ is space-like, with u 0 = 0 and c 2 = 0 (no CFJ-like term).In this case we can avoid the presence of higher time derivatives (so, the theory is unitary), and 4. If c 1 = 0 and u µ is time-like (with u i = 0), we also have the absence of higher time derivatives (so, the unitarity is again achieved).
5. If c 2 = 0 and u µ is space-like (with u 0 = 0), we avoid the higher time derivatives.
We see that one cannot eliminate higher time derivatives in both terms simultaneously through a simple choice of u µ .
To study the unitarity in our theory, it is useful to change the basis of the gauge field from the linear one e a µ , with a = 1, 2, to the circular one ε µ , with λ = ±1.For this construction, let us consider the projector P (λ) µν which projects any four vector v µ onto the hyperplane orthogonal to u ν and p λ vectors, with and In other words, these tensors are orthogonal to p and u, i.e.
They satisfy the properties Using these properties, one can show that and It is always possible to choose two linear polarization vectors e We can define the analogous of the circular polarization vectors such that The propagator in this new base is where We use this propagator in the Section IV.

III. PERTURBATIVE GENERATION
Let us consider the perturbative generation of the Myers-Pospelov term.One scheme, based on the magnetic coupling, has been developed in [10], where it was shown to yield the finite corrections.
Here we deal with another one.Let us consider the fermionic Lagrangian [3] where is the Planck mass (as above), and η 2 is some dimensionless number.So, we have the following explicit form of the Lagrangian: We note that the similar coupling, but including third derivatives, has been used in [12].
One can easily observe that the effective action of second order in A, in lower order in η 2 , is given by two contributions.The first of them involves only usual (minimal) vertices proportional to e which do not involve any v µ vector, and η 2 arises from the expansion of the propagator.It looks like

S
(2) where with Now, by applying the expansion with S(p) = ( / p − m) −1 , we can easily single out the terms of first order in η 2 , by writing where we have promoted the integral to the D-dimensional space-time, so that d 4 p/(2π) 4 goes to , with µ being an arbitrary parameter that identifies the mass scale.
In order to calculate the above integrals, we will use the Feynman parametrization, so that Eq. ( 34) is rewritten as where Then, after the calculating the trace, we obtain with In the following, after we integrate over d D p and expand the result around D = 4, we have where = 4 − D and µ 2 = 4πµ 2 e −γ .We note that, as Finally, after we integrate over the parameter x, we obtain Here, we can consider the limits k 2 m 2 (m = 0) and k 2 m 2 (m = 0), so that we get and respectively, where we have also defined Therefore, for the induced bosonic Myers-Pospelov term from the contribution (40), which corresponds to the non-zero mass, we have This enforces the fact that the above higher-derivative terms should be introduced from the very beginning (1), so that we have a consistent subtraction of the divergences.
Then, the quartic vertex evidently should give a zero contribution.Indeed, the quartic vertex can yield only the Proca-like term (v • A) 2 (with no derivatives, since there are no derivatives of A µ in the classical action, and the only relevant graph is a tadpole, so, the integration over the internal momentum cannot yield a contribution depending on the external momentum), and this term is inconsistent with the gauge symmetry (its vanishing can be shown explicitly, as well).
In order to find the remaining first-order contribution in η 2 , we should consider the contraction of two vertices: the first of them is the usual −e ψ / Aψ, and the second one is −ie , where the propagator is the free one (indeed, expanding (32), we will get only the next-orders contributions).Its explicit form is Here the factor 2k ν + p ν originates from the nonminimal vertex given by the expression )ψ (the moment k is for the spinor propagator, and the moment p is for external gauge field).It remains to expand this expression up to the third order in external p (actually, the first order in p disappears, so, it remains to deal only with the third order).Indeed, the trace in (43) can be calculated before of any expansion of the propagator in series in p: Now, this expression can be expanded into power series in external momenta, where only the third order should be taken into account (the first and second orders evidently vanish: for the first order, one evidently will have the contraction of the Levi-Civita symbol with two Lorentz-breaking vectors which immediately vanishes, and for the second order, the corresponding scalar simply does not exist).
However, to study it we can first present it as where with It is clear that, up to the second order in the external p (the highest relevant order), the tensor Indeed, there is no other possible tensor structures.Here Q 1 , Q 2 are two (divergent) constants.
Substituting this structure to the contribution (46), we find that it identically vanishes.Hence this "mixed" contribution is zero, and the only non-trivial result for the Myers-Pospelov term is given by ( 42).The divergent nature of this result immediately shows that for consistency of the theory, one should have the higher-derivative CFJ-like and Myers-Pospelov term presented in the classical action from the very beginning.Now, we can consider the renormalization.Since the theory is actually non-renormalizable (indeed, our coupling is α M , and it has a negative mass dimension; we note that the model considered in [10] for another scheme of generating the higher-derivative Lorentz-breaking terms is also nonrenormalizable, as well as that one used in [13]), we consider the fermionic determinant only.The superficial degree of divergence in the one-loop order, for purely external A µ , looks like where , and that by the gauge symmetry reasons there must be at least one derivative acting to gauge legs (to get the CFJ term) or two derivatives (to get the Maxwell or aether terms).Also, it is evident that the potentially divergent Feynman diagrams with V 2 = 1, 2 will be not gauge invariant since they will yield the contributions proportional to (v•A) 2 or (v•A) 4 , and they should vanish in some regularization.Hence in divergent diagrams one should have V 2 = 0.Then, the diagram with V 0a = 2 has been studied above (42), and the contribution with V 0a = 1 and V 0b + V 1 = 1 is just that one given by (44), and its contribution is zero.In principle we can also have divergences in contributions to the two-point function formed by only V 1 and V 0b vertices, however, they are strongly suppressed being proportional to 1 M 2 .We conclude our discussion with the statement that up to the order M −1 , our results are exact, and the only nontrivial divergent contribution is given by (42).
Therefore we can establish the action of the following higher-derivative extended QED given by the sum of ( 1) and (28).We note that, as usual, if we suggest the gauge field to be a purely external one, the one-loop result is exact.Now, let us introduce finite temperature.To do it, we apply the Matsubara formalism, i.e., in the integrals over momenta above, (34) and (46), we suggest the zero component of the internal momentum to be discrete, p 0 = 2πT (l + 1  2 ), with l being an integer, and, afterwards, we integrate over spatial components of the internal momentum and sum over l.As a result, at the finite temperature, our self-energy tensor, given by Π µν M P = Π µν 1M P + Π µν 2M P , turns out to look like where with ξ = m 2πT .Above, we have split the internal momentum as p µ = p µ + p 0 t µ , with p µ = (0, p) and t µ = (1, 0, 0, 0) being a constant vector along the time axis.We note that these functions of the temperature emerged as well in [14] where the finite-temperature extension of results obtained in [10] for another Lorentz-breaking extension of the QED, involving the magnetic coupling and the coupling of ψ to the constant axial vector b µ , was carried out.It was shown there that in the high temperature limit all these functions vanish.The result (50) is clearly gauge invariant.

IV. UNITARITY ASPECTS IN THE EXTENDED QED
It is well known that in the presence of higher time derivatives the quantum field theory can develop negative-metric states or ghosts.In the subclass of higher time derivative theories, called Lee-Wick theories, these additional degrees of freedom arise in complex conjugate poles [15].The structure of poles determines the discontinuities in phase space integrals and so, under some assumptions, both contributions of complex conjugate modes cancel each other, order by order in the perturbative series [16].The issue of analyticity in the complex energy plane and the resulting cutting equations has been intensively studied over the years (for the general discussion of ghosts states see f.e.[17]).The Lee-Wick prescription of removing the negative metric particles from the asymptotic space has been shown to be efficient in providing a unitary theory along with the good properties of convergence.
In general, in a case of presence higher time derivatives, one is confronted with the problem of analyticity of amplitude diagrams.The i prescription seems to be misleading in many cases, hence it is necessary to analyze the configurations of poles that allows to preserve unitarity, case by case.Together with this, the presence of Lorentz symmetry breaking makes study of the analyticity properties of the integrals to be more complicated, since in this case the extra poles related to the negative-metric states may come in odd numbers changing a property of a Lee-Wick theory.It is also difficult to deal with the perturbative solutions which may become complex for certain regions, with some of them becoming complex in disconnected regions [18].An early approach to deal with the analytic properties of phase space integrals in the presence of Lorentz violation, based on the Euclidean space or Wick rotation has been presented in [19].
Recently a new formulation for Lee-Wick theories as non-analytical Euclidean theories has been proposed in [20].We follow along similar lines doing some steps to deal with the unitarity in higher time derivative theories with Lorentz symmetry violation.The strategy we pursue to compute the relevant contributions to the cut integrals is to consider the Euclidean theory from the beginning and perform the Wick rotation including the rotation of the preferred four vector.In this way we arrive at a simplified integral with simplified poles.
The processes we study are the Bhabba scattering at tree level (we note that some earlier studies of Bhabha scattering in a Lorentz-breaking extension of QED were carried out in [21], where, however, no higher-derivative terms were studied) and Compton scattering at the oneloop level.In the first case, we select a general preferred four vector letting additional degrees of freedom associated to a negative metric to play a role, and in the second we choose a purely timelike preferred four vector without ghosts in the theory.For both cases, we consider the forward scattering of two particles with incoming momenta p = k and p related as p + p → p + p . (55) A. Bhabha scattering at tree level in the ghost sector We consider a generic preferred four vector u µ , so that, in general, ghosts can arise.We also consider the Bhabha scattering process at tree level given by the Fig. 1.
FIG. 1: The Bhabha scattering diagram at tree level.
The amplitude is given by with where µν and a, D are given by ( 11) and ( 27), respectively.The standard way to compute the imaginary part of the diagram in Eq. ( 56), is to fix the four-vector u µ , solve the dispersion relation, and, afterwards, to analyze discontinuities.However, in our model with modified photons, the dispersion relation is a very complicated expression and the solutions can be difficult to find.So, we introduce a novel method to deal with unitarity.
The strategy is to start with a theory in Minkowski space, which is defined as the one obtained from the Wick rotation in the Euclidean theory, perhaps non-analytically, as in the Ref. [20].This starting point ensures a well defined Wick rotation to the Euclidean space warranted by the position of positive and negative poles in the fourth and second quadrant of the energy plane respectively.
Hence, we perform the Wick rotation, changing variable q 4 = iq 0 , so that the dispersion relation decouples into usual and ghost solutions.This last step simplifies the calculation considerably.
The rotated energy integral will still depend on the i prescription which allows us to compute the discontinuity.Finally, we compute the integral using the residue theorem and we analyze the discontinuity of the phase space integral.Only in the final step we remove the i prescription going to the limit → 0.
From Eq. (56), using the expression (25), we can write where Then, by performing the Wick rotation, i.e., q 0 = −iq 4 , we arrive at where we have compensated with an extra i due to the delta function.The polarization vectors Eµ must be evaluated as ε µ (q 0 = −iq 4 , q).Now, we introduce the expressions in terms of the functions of the angle θ, given by We multiply both the numerator and denominator by −1 + iλβ γq 2 E , in order to have In Euclidean space we make explicit the dependence which for the calculation of integrals, however, is usually taken to zero.Hence, the pole structure of the propagator looks like From the dispersion relation one obtains the solutions Evaluating the q 4 integral with the delta function yields where s 4 = p 4 + p 4 refers to the external momenta.To compute the discontinuities in terms of s 4 , we focus on the element and we use the known formula where P.V denotes the principal value.Thus, we arrive at where in the last line we have evaluated → 0.
We restrict external momenta at effective energies at which they are always below the high energy scale defined by W .In this case we can set the two last delta functions to zero, obtaining Substituting this expression into Eq.(66), we find Now, we introduce an integral over q 0E together with a delta function, so that we write We can obtain this expression in an equivalent way by introducing a physical delta function δ defined to select only asymptotic degrees of freedom in Hilbert space.In [22] it has been used to test the unitarity in a higher-order Lorentz violating scalar theory.
The square parenthesis above can be written as which allows us to write Now, we carry out the inverse Wick rotation and use Disc where We see that it is equivalent of considering the denominators in the expression on-shell or replacing the propagator with the physical delta function.In this way the unitarity constraint is satisfied.

B. Compton scattering at one-loop level
Now, we consider the Compton scattering process at one-loop level.It is illustrated by Fig. 2.
The scattering amplitude, with the help of the propagator (26), is found to be FIG. 2: The Compton scattering process at one-loop level. with We focus on the integral In terms of the poles in Eq. ( 77) and the fermion one E q−p = ( q − p) 2 − m 2 , we write where We perform the q 0 integral by closing the contour downward and using the residue theorem.Taking into account the relevant poles in the fourth quadrant, we arrive at Using Eq. ( 68), the discontinuity of the integral turns out to be equal to where We have set = 0 in the numerators where the factors are not relevant.We can simplify further with the delta function, i.e., With the help of the identity d 3 q = d 3 kd 3 k δ (3) ( k + k − p), and introducing two additional integrals in k 0 and k 0 and with k = p − q, k = q, we can write Now, we use the fact that under the integral with the deltas, the two pieces behave as and together with the on-shell relation we can rewrite (87) as Hence we conclude that the optical theorem is satisfied both at the tree level and the one-loop level within this scattering process.Since it is natural to expect that the higher-loop results do not differ too much, we conclude that the unitarity is maintained in our theory.
V. SUMMARY We considered the higher-derivative Lorentz-breaking extension of QED which involves, first, additive terms, that is, Myers-Pospelov and higher-derivatives CFJ-like terms, in the purely gauge sector, second, a new, non-renormalizable spinor-vector coupling.For this model, we discussed the dispersion relations and found that, to achieve the unitarity, we can have only one higherderivative term, that is, either the MP term or the higher-derivative CFJ term, but not both terms at the same time.Besides of this, we carried out study of quantum corrections and showed that for a consistent subtraction of the divergences, the higher-derivative terms should be introduced from the very beginning.Nevertheless, it is very reasonable to treat this theory as an effective one.Indeed, all higher-order divergent terms will be very small carrying different degrees of 1 M , thus, they are strongly suppressed.The similar situation will occur in higher loops where all dangerous divergences will be suppressed by negative degrees of M .We carried a calculation of these corrections in the finite temperature case as well, and we see that our result tends to zero in the high temperature limit.
We verified the unitarity in our theory, both at the tree level and at the one-loop level.We checked directly that the optical theorem is satisfied in both cases, therefore, we conclude that, if higher time derivatives are absent, the unitarity in our theory is preserved.We conclude that this manner of introducing the higher derivatives is compatible with the unitarity as well as the Horava-Lifshitz methodology.However, the advantage of the approach presented in this paper is that, unlike the Horava-Lifshitz theories [23], in our case the Lorentz symmetry breaking continues to be small which is much more convenient from the viewpoint of achieving the consistency with experimental measurements which, as it is well known, impose very strong upper boundaries on Lorentz-breaking effects.