Gauge invariant nonlinear electrodynamics motivated by a spontaneous breaking of the Lorentz symmetry

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(1)PHYSICAL REVIEW D 81, 025007 (2010). Gauge invariant nonlinear electrodynamics motivated by a spontaneous breaking of the Lorentz symmetry Jorge Alfaro1 and Luis F. Urrutia2 1. 2. Facultad de Fı́sica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, 04510 México D.F. (Received 8 July 2009; revised manuscript received 12 November 2009; published 13 January 2010) We introduce a new version of nonlinear electrodynamics which is produced by a spontaneous symmetry breaking of Lorentz invariance induced by the nonzero vacuum expectation value of the gauge invariant electromagnetic field strength. The symmetry breaking potential is argued to effectively arise from the integration of massive gauge bosons and fermions in an underlying fundamental theory. All possible choices of the vacuum lead only to the remaining invariant subgroups Tð2Þ and HOM(2). We explore in detail the plane wave solutions of the linearized sector of the model for an arbitrary vacuum. They present two types of dispersion relations. One corresponds to the case of the usual Maxwell electrodynamics with the standard polarization properties of the fields. The other dispersion relation involves anisotropies determined by the structure of the vacuum. The corresponding fields reflect these anisotropies. The model is stable in the small Lorentz invariance violation (LIV) approximation. We have also embedded our model in the photon sector of the standard model extension, in order to translate the many bounds obtained in the latter into corresponding limits for our parameters. The one-way anisotropic speed of light is calculated for a general vacuum, and its isotropic component is strongly bounded by ~ c=c < 2 1032 . The anisotropic violation contribution is estimated by introducing an alternative definition for the difference of the two-way speed of light in perpendicular directions, c, that is relevant to Michelson-Morley type of experiments and which turns out to be also strongly bounded by c=c < 1032 . Finally, we speculate on the relation of the vacuum energy of the model with the cosmological constant and propose a connection between the vacuum fields and the intergalactic magnetic fields. DOI: 10.1103/PhysRevD.81.025007. PACS numbers: 11.30.Qc, 03.50.De, 11.10.Lm, 11.30.Cp. I. INTRODUCTION The possible violation of Lorentz invariance has recently received a lot of attention, both from the experimental and theoretical sides. In the latter case, mainly in connection with possible effects arising from drastic modifications of space-time at distances of the order of Planck length suggested by most of the current quantum gravity approaches. Experiments and astrophysical observations set stringent bounds upon the parameters describing such violations, which nevertheless are still been subject to improvement in their precision. Broadly speaking, there are two basic possibilities which produce such a breaking: (1) one introduces by hand a number of nondynamical tensor fields, whose fixed directions induce the corresponding breaking in a given reference frame. Examples of this approach are the phenomenological Myers-Pospelov model [1] together with QED in a constant axial vector background [2]. (2) a second possibility is to dwell upon spontaneous symmetry breaking. This is basically what is done in the standard model extension (SME) [3,4], where such nondynamical tensor fields are assumed to arise from vacuum expectation values of some basic fields belonging to a more fundamental model, like string theory, for example. Nevertheless, we are interested in studying a model of electrodynamics which incorporates spontaneous Lorentz. 1550-7998= 2010=81(2)=025007(19). symmetry breaking, without having to go to a string theory setting. In order to produce such a model we require the presence of the nonzero vacuum expectation value of a tensorial field of rank greater than zero. A first possibility arises from having a nonzero vacuum expectation value (VEV) of the potential field A . This case has been thoroughly studied and leads to the interesting idea that the photon arises as the corresponding Goldstone boson of the global spontaneous Lorentz symmetry breaking. Normally the masslessness of the photon is explained by invoking Abelian gauge invariance under the group Uð1Þ. This almost sacred principle has undoubtedly been fundamental in the development of physics, and it is very interesting to explore the possibility that it could have a dynamical origin [5]. This idea goes back to the works of Nambu [6] and Bjorken [7] together with many other contributions [8,9]. Recently it has been revived in Refs. [10–14]. One of the most explored approaches along these lines starts form a theory with a vector field B endowed with the standard electrodynamics kinetic term plus a potential designed to break Lorentz symmetry via a nonzero vacuum expectation value hB i, which defines a preferred direction in space-time. This potential also breaks gauge invariance. These are the so-called bumblebee models [15]. The subsequent symmetry breaking, obtained from the nonzero minimum of the potential, splits the original 4 degrees of freedom in B into three vectorial. 025007-1. Ó 2010 The American Physical Society.

(2) JORGE ALFARO AND LUIS F. URRUTIA . PHYSICAL REVIEW D 81, 025007 (2010). Nambu-Goldstone bosons A , satisfying the constraint A A ¼ M2 , to be identified with the photon, plus a massive scalar field , which is assumed to be excited at very high energies. Under this threshold one basically recovers the Lagrangian for electrodynamics in vacuum plus the above mentioned nonlinear constraint, which is interpreted as a gauge fixing condition. Calculations at the tree level [6] and to the one-loop level [12] have been carried on, showing that the possible Lorentz violating effects are not present in the physical observables and that the results are completely consistent with the standard gauge invariant electrodynamics. This is certainly a surprising result, which can be really appreciated from the complexity of the calculation in Ref. [12]. Nevertheless, alternative kinetic terms in the bumblebee models can lead to a theory differing substantially from electromagnetism [16]. The use of nonpolynomial interactions within this framework has been explored in [17]. The idea of a photon as a Goldstone boson has also been extended to gravitons in general relativity [15,18,19]. It is noteworthy to recall here that the constraint A A ¼ M2 was originally proposed by Dirac as a way to derive the electromagnetic current from the additional excitations of the photon field, which now ceased to be gauge degrees of freedom, avoiding in this way the problems arising from considering pointlike charges [20]. As reported in [21], this theory requires the existence of an ether emerging from quantum fluctuations, but not necessarily of the violation of Lorentz invariance, due to an averaging process of the quantum states producing such fluctuations. In this work we explore the possibility of having a nonzero VEV of the electromagnetic strength, so that we avoid the problem of disturbing gauge invariance from the very beginning. To this end we consider a nonlinear version of electrodynamics with a potential term having an stable minimum for a constant electromagnetic strength. The paper is organized as follows: In Sec. II, we make plausible the existence of such gauge invariant potential Veff ðF2 Þ, arising as an effective low energy contribution from the integration of degrees of freedom corresponding to massive gauge bosons and fermions in a more fundamental theory. This potential also exhibits the right behavior in the low-intensity field approximation. Section III contains the construction of the model starting from a nonlinear version of electrodynamics which is spontaneously broken by choosing a vacuum characterized by a constant electromagnetic tensor C . In Sec. IV, we discuss the possible realizations of such vacuum in terms of the associated electromagnetic fields e and b, identifying the Lorentz subgroups that remain invariant after the breaking. The equations of motion corresponding to the propagating sector, i.e. those arising from the quadratic terms in the Lagrangian, are examined in Sec. V using a covariant approach, as well as the standard 3 þ 1 decom-. position. Also, the modified Maxwell equations are presented. Section VI contains a summary of the modified photon dispersion relations ! ¼ !ðkÞ, together with the expressions for the electromagnetic fields E and B of plane waves propagating in the different vacua parameterized by e and b. The model is shown to be stable in the small Lorentz invariance violation approximation. In Sec. VII, we study the embedding of our model in the corresponding sector of the SME [22] and use the experimentally established bounds upon the LIV parameters to constrain the electromagnetic fields e and b. Finally, we conclude with a summary of the work together with some comments in Sec. VIII. The Appendix contains some details of the calculation leading to the motivation described in Sec. II. II. MOTIVATION One of the main challenges of the proposed approach is to make it plausible the existence of a gauge invariant potential VðF Þ depending upon the electromagnetic tensor, which is stable for large values of the electric and magnetic fields and which possess a minimum for some constant value of F . In particular, we will consider at this level only the case of magnetic fields in order to avoid further instabilities due to pair creation in strong electric fields. A first possibility that arises is to consider the effective photon interaction in QED after integrating the fermions. Unfortunately, a detailed calculation of this potential, in ð1Þ ð2Þ the one-loop (VEM ) and two-loop (VEM ) approximations [23], shows that it is given by 1 ð1Þ ð2Þ VEM ðBÞ ¼ B2 þ VEM þ VEM 2 1 e2 2 eB e4 eB 2 ¼ B2 B B ln ln ; 2 2 4 2 m m2 24 128 (1) in the limit when B B0 ¼ m2 =e, which clearly is unbounded from below. A new possibility arises in a further generalization of the mechanism proposed in Refs. [7,8], restricted to the gauge invariant case. We start from a conventional gauge theory including fermions, gauge fields, and Higgs fields, which provide masses, via spontaneous symmetry breaking, to the gauge fields, except for the photon potential A , which carries the electromagnetic Uð1Þ gauge invariance. For simplicity we consider only one fermion species, as in Ref. [7], but the idea can be generalized to many of them. In the latter case, the subsequent integration over the fermionic degrees of freedom can be made only for those having a mass larger than a given scale, thus leaving unintegrated the lightest ones. Integrating the massive gauge bosons of this theory leads to a power series in bilinears in the fermion fields, which are suppressed by powers of the energy scale that. 025007-2.

(3) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. characterizes the symmetry breaking mechanism providing the mass to the gauge vector bosons [8]. We focus upon the neutral sector of the model where the following contributions to the effective Lagrangian arise n X 1 a ðMa ÞðM Þ ; 6n4 a (2) X n 1 a D ÞðM a D Þ ; . . . ð M 8n4 a We are considering all possible contributions which are consistent with gauge invariance, the remaining symmetry of this sector of the theory. Notice also that terms containing the Uð1Þ covariant derivative D ¼ ð@ þ ieA~ Þ are further suppressed with respect to those without it. Also we consider only the n ¼ 1 case, which provides a gauge invariant generalization of the model in Ref. [7]. Thus, our starting point is ð@ þ ieA ~ Þ mÞ Leff ¼ ði X rM a ÞðM a Þ: ½ðM 2 M;a. 4 16 48 16 4. 4 0 16 0 4. (4) B ! 0: V ð2Þ ¼. . 1 162. 2 2 2 m eB 1 e2 B4 2 X þ rM 3m2 18 m4 M. M:. (5) Here, m is the mass of the integrated fermion which is such that m > . The main point of the estimation is that we obtain X ¼ rM M ¼ 4rS þ 16rV þ 48rT 16rPV 4rPS ; M. TABLE I. General notation and traces for the different contributions to lM in Eq. (A27). P P c c M Mc M ¼ 3M ¼ c trðM Mc Þ c trð3 M 3 Mc Þ I44 i 5 5. ent currents and only add them to the total effective potential at the end. In this way we are nor considering possible interference terms among the currents, which nevertheless could be calculated within the approximations performed. Calling V ð2Þ the finite part of the total effective potential arising from the currents jM we obtain 1 2 m 2 2 2 2eB 2 X e B rM M ; B ! 1: V ð2Þ ¼ ln 162 m2 M. (3). The index M: S, V, T, PV, PS labels the tensorial objects that constitute the Dirac basis: scalar, vector, tensor, pseudovector, and pseudoscalar, respectively. The generic index a labels the covariant (contravariant) components of each tensor class. Some details are given in Table I. We follow the Dirac algebra conventions of Ref. [24], with appropriate factors chosen in such a way that each current a ) is real. In this way, the proposed generalization (M includes all possible fermionic quartic interactions. Also, the coefficients rM are assumed to be of order one. The fermions can be integrated next by introducing the corresponding auxiliary fields ðCM Þa for each real current a , as suggested in Ref. [8], and following ðjM Þa ¼ M standard manipulations in the path integral formulation. Some details are given in the Appendix. Our main goal is to estimate the additional corrections provided by such currents to the purely electromagnetic effective potential (1) and to investigate whether or not it is possible to generate a net positive contribution in the highintensity field limit. To have a preliminary simpler expression for the resulting extension we choose to consider separately each of the contributions arising from the differ-. S V T PV PS. PHYSICAL REVIEW D 81, 025007 (2010). (6) which can be made positive from a judicious choice of the parameters rM . In the high-intensity field approximation, the dominant term in the electromagnetic contribution goes like eB VEM ðBÞ ’ e2 B2 ln 2 ; (7) m which can be overcame by the contribution from the additional currents 2 m 2 2 2eB 2 e B ln ; (8) V ð2Þ ’ m2 provided > 0. As we mention in the Appendix, this preliminary estimation has only considered an appropriate regularization of the divergent integrals, but is missing an adequate renormalization of the total effective Lagrangian Leff . Our renormalization conditions would be dLeff lim Leff ¼ 0; lim > 0; (9) B!0 B!0 dB2 in such a way that the effective potential is a decreasing function in the vicinity of B ¼ 0. The final normalization to the value of Leff ¼ b2 =2 will be imposed at the end of the procedure, after we have expanded the magnetic field around an stable minimum BMin of Veff , such that B ¼ BMin þ b. (10). and we are left with the physical component b. The renormalization procedure should be analogous to the one carried in Ref. [23] for the two-loop contribution Lð2Þ EM ðAÞ to the effective electromagnetic action and it is rather involved due to the complicated field dependence of the. 025007-3.

(4) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). regularized version of the divergent integrals. This calculation is beyond the scope of these preliminary estimations. Having an effective potential with the characteristics described above guarantees the existence of an absolute minimum, which we assume to arise at values lower that the critical magnetic and electric fields, in such a way to avoid fermion pair production in the later case. We also generalize this idea to the case of an arbitrary constant electromagnetic field F . The study of the stability of the model under radiative corrections is beyond the scope of the present work. III. THE MODEL In the previous section we have made it plausible the existence of a gauge invariant potential Veff ðF2 Þ having an stable minimum and arising as an effective low energy model from the integration of degrees of freedom corresponding to massive gauge bosons and fermions in a more fundamental theory. This potential also exhibits the right behavior in the low-intensity field approximation. To examine the dynamical consequences of the symmetry breaking we have to adopt a convenient parametrization of such potential. There are various choices in the literature which would translate into the following for our case [25]: VðFÞ ¼ ðF2 C2 Þ;. VðFÞ ¼. 4. ðF2 C2 Þ2 ;. (12). F ¼ F ;. F. i . @i F. i . . ¼ 0;. ¼ 0:. (16). (17). holds. In other words, to find the vacuum configuration we have to extremize the effective action, subjected to the condition that F and X are constant fields. Applying these requirements to (13) plus the choice (12) we obtain @V ¼ 0 ¼ ð þ F2 ÞF ; @F. (19). which is solved by a constant ðFExtr Þ C such that ðFExtr Þ2 ¼ . ¼ C2 0: . (20). The expansion around the minimum ðC ; C Þ is written as F ðxÞ ¼ C þ a ðxÞ;. X ¼ C þ X ðxÞ: (21). Introducing such expansion in the potential (12), we arrive to VðF Þ ¼ VðC Þ þ ðC a Þ2 þ ðC a Þ ða a Þ þ. (14). and the X are just Lagrange multipliers that will finally impose the condition that the field strength satisfies the Bianchi identity and can then be expressed in terms of the vector potential. The above Lagrangian is similar to that used in the standard formulation of nonlinear electrodynamics [26]. Our conventions are ¼ diagðþ;;;Þ, 0123 ¼ þ1, 123 ¼ þ1. The next step is to determine the vacuum configuration of the theory, corresponding to the minimum of the energy E. The energy density T 00 is calculated via the standard Noether theorem starting from the Lagrangian (13) and produces. (15). In order to maintain four dimensional translational invariance we look for extrema of the fields which are independent of the coordinates. From Eqs. (16) and (17) we verify that this is possible provided the condition @V ¼0 (18) @F Extr. where the dual field F is . d3 x½VðF Þ þ F i ð@i X Þ:. E 1 ¼ X 2. (13). F ¼ 12. Z. E @V ¼ þ ð@i X Þ F @F . Since the basic variable we want to deal with is F , we start from the effective Lagrangian LðF ; X Þ ¼ VðF Þ F @ X ;. d3 xT 00 ¼. The extremum conditions, obtained by varying the independent fields F and X , are. (11). where C is the constant value of the electromagnetic field at the minimum. Instead we will adopt the standard Ginzburg-Landau parametrization 1 VðF Þ ¼ F2 þ ðF2 Þ2 : 2 4. E¼. Z. ða a Þ2 ; 4 . Þ: VðC Þ þ Vða. (22) (23). We notice that (22) has no linear term in a , and also that the quadratic term ðC a Þ2 is positive provided > 0;. (24). thus verifying the expansion around a minimum. On the other hand, according to Eq. (20), the sign of is arbitrary, in such a way that it is opposite to the sign of C2 . In this way, the spontaneously symmetry broken action is. 025007-4.

(5) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. Sða ; X Þ ¼ . Z. 1 d4 x 2. SðA Þ ¼ . a @ X . Þ þ VðCÞ ; þ Vða. @V a : @ X 2 ¼ 0; @a . @ a ¼ 0:. (27). (28). (29). which is certainly harmonic because it is constant. Calling dA ¼ f we have a ðxÞ ¼ lC þ f ðxÞ:. (30). From (26) we obtain @. @V ¼ 0; @a. (31). which are the generalization of Maxwell equations for the potential A . From (26) we obtain @ X @ X yielding ð@ X @ X Þ ¼. @V R ðA Þ: @a. (32). We verify that R ðA Þ can in fact be written as the lefthand side of Eq. (32) by calculating @ R . The result is zero in virtue of the equations of motion for A . Fixing the corresponding gauge freedom by setting @ X ¼ 0, for example, we obtain @2 X ¼ @ R ðA Þ;. (34). (35). which coincide with the previous ones in Eq. (31). It is convenient to rename the dimensionless parameter l ¼ 1. Substituting (30) in (34) and defining C ¼. 1 D ; 2 . B¼. > 0; 4. (36). where. where A is a one-form, l is a constant and s is an harmonic two-form. Since in our case we naturally have one such form at our disposal, arising precisely from the chosen vacuum, we take s ¼ 12C dx ^ dx ;. Þ þ VðCÞÞ; d4 xðVða. The resulting equations of motion are @V A : @ ¼ 0; @a. (26). Equation (27) establishes that the two-form a is closed, i.e. da ¼ 0. The Hodge-De Rham theorem says that the most general solution to this is obtained by requiring a ¼ dA þ ls;. Z. a ðxÞ ¼ lC þ @ A @ A :. (25). where we have eliminated the total derivatives arising from the shifts of the fields (21) performed in the original Lagrangian (13). Next we consider the equations of motion derived from (25) and show that the Lagrange multiplier X is fully determined up to gauge transformation X ! X þ @ . The equations are. X :. PHYSICAL REVIEW D 81, 025007 (2010). ¼. 1 ; 1ÞðC C Þ. ð2 1ÞðC C Þ > 0; (37). we obtain our final effective action Z 1 1 SðA Þ ¼ d4 x ½1 D2 BD2 f f 4 4 B½ðD f Þ þ ðf f Þ2 ;. (38). where f ðxÞ ¼ @ A @ A . We recognize the standard Maxwell kinetic term in the right-hand side of Eq. (38). The only restriction now is B > 0, with D2 arbitrary. Let us emphasize that the case of purely spontaneously broken Lorentz invariance really corresponds to the singular choice ¼ 1, (l ¼ 0). In this case, the corresponding action would not possess the standard kinetic term 14 f f but will start with the quadratic term ðD f Þ2 . This very interesting case is not considered here, and its investigation is postponed for future work. IV. SYMMETRY ALGEBRAS ARISING FROM DIFFERENT CHOICES OF THE VACUUM In this section we study the possible vacua allowed by the tensor symmetry breaking and we also identify the corresponding subgroups of the Lorentz group which are left invariant after the breaking. In order to do this, it is convenient to parameterize the background field D in terms of three dimensional components Dij ¼ . ijm bm ;. 2. 0 6 e1 6 ½D ¼ 6 4 e2 e3. (33). which completely determines the Lagrange multiplier X . Going back to the action (25), integrating by parts and substituting the constraint (27) we are left with the reduced effective action. 2ð2. D0i ¼ ei ; e1 0 b3 b2. e2 b3 0 b1. 123. ¼ 1;. 3 e3 b2 7 7 7; b1 5 0. (39). (40). which will mix when going to another reference frame via a passive Lorentz transformation. Since we have two vec-. 025007-5.

(6) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). B. Case b ¼ 0, e3 0, e2 ¼ 0. tors that determine a plane we choose a coordinate frame where. Here, we have. e ¼ ð0; e2 ¼ jej sin c ; e3 ¼ jej cos c Þ:. b ¼ ð0; 0; bÞ;. (41) That is to say, we have chosen the plane of the two vectors as the (z-y) plane, with the vector b defining the z-direction and c being the angle between b and e. In this way the matrix representing the vacuum is 3 2 0 0 e2 e3 6 0 0 b 0 7 7 7: 6 (42) ½D ¼ 6 4 e2 b 0 05 e3 0 0 0 The most general infinitesimal generator G, including Lorentz transformations plus dilatations is 3 2 x2 x3 z x1 6 x1 z y3 y2 7 7 7: 6 G ¼ ½G ¼ i6 (43) 4 x2 y3 z y1 5 x3 y2 y1 z Motivated by the work in Ref. [27] we are including conformal dilatations D among our generators. Within this restricted Poincaré algebra, this generator commutes with the remaining ones corresponding to pure Lorentz transformations and can be realized as a multiple of the identity. We do this in order to explore the possibility of having the largest possible invariant subalgebra after the symmetry breaking. The condition for the vacuum to be invariant under the infinitesimal transformations is 0 ¼ G D þ G D ;. x1 e2 þ 2zb ¼ 0; x1 b þ y1 e3 ¼ 2ze2 ;. x2 ¼ y2 ¼ 0;. (50). so that we are left with a two-parameter ðx3 ; y3 Þ subalgebra with generator G ¼ x3 K3 y3 J 3 :. (51). C. Case b 0, e3 ¼ 0, e2 ¼ 0 This case is analogous to the previous one. We have x1 ¼ y1 ¼ 0;. z ¼ 0;. x2 ¼ y2 ¼ 0;. (52). with a two-parameter ðx3 ; y3 Þ subalgebra. D. Case b ¼ 0, e3 0, e2 0 In this case we have zðe22 þ e22 Þ ¼ 0 ! z ¼ 0; x1 ¼ 0;. y1 ¼ 0;. y2 ¼ y3. e2 ; e3. (53) x2 ¼ x3. e2 ; e3 (54). yielding a two-parameter ðx3 ; y3 Þ subalgebra with generator e2 2 e2 2 3 3 K þ K y3 J þ J : G ¼ x3 (55) e3 e3 Calling Gx3 ¼. (44). e Gy3 ¼ J 3 þ 2 J 2 ; e3. e2 2 K þ K3 ; e3. (56). we can show that. and the resulting equations are x2 b y2 e3 ¼ y3 e2 ;. x1 ¼ y1 ¼ 0;. z ¼ 0;. x2 e3 þ y2 b ¼ x3 e2 ;. (45). ½Gx3 ; Gy3 ¼ 0:. y1 e2 þ 2ze3 ¼ 0;. (46). E. Case b 0, e2 0, e3 ¼ 0. x1 e3 þ y1 b ¼ 0:. (47). Next we consider the solutions of the above system of equations corresponding to the seven nontrivial possibilities dictated by the choices in the arrangement ðb; e2 ; e3 Þ in the above coordinate system. In the sequel we use the notation of Ref. [28] for the Lorentz group generators.. (57). Here, we have y3 ¼ x2. b ; e2. x3 ¼ þy2. b ; e2. y1 ¼ 0. (58). and the consistency condition zðb2 e22 Þ ¼ 0;. (59). which leads to two possibilities A. Case b ¼ e3 ¼ 0, e2 0. 1. Subcase z ¼ 0. This leads to z ¼ 0;. x1 ¼ y1 ¼ 0;. x3 ¼ y3 ¼ 0:. (48). We have two free parameters, x2 , y2 and the generator is G ¼ x2 K 2 y2 J 2 :. (49). 025007-6. Here, we have y3 ¼ x2. b ; e2. y1 ¼ 0;. x3 ¼ þy2 x1 ¼ 0;. b ; e2. (60).

(7) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. and we are left with a two-parameter subgroup where b 3 b 3 2 2 K J : G ¼ x2 K þ J þ y2 (61) e2 e2. PHYSICAL REVIEW D 81, 025007 (2010). B ¼. sb > 0:. . b 3 2 K J ; Gy2 ¼ e2. (62). (63). 2. Subcase z 0. 1 pffiffiffi D ; 2. ¼ s ¼ 1;. (64). x1 ¼ 2sz;. y2 ¼ sx3 : (65). Here, we are left with a three-parameter Lie algebra ðz; x3 ; y3 Þ, and the generator is. z ¼ 0;. Gx ¼ K sJ ;. Gy ¼ ðJ þ sK 2 Þ;. y3 ¼ x2 (67). b e þ y2 3 ; e2 e2. x3 ¼ x2. ½Gz ; Gy ¼ 2iGy ;. (68). which is isomorphic to HOM(2). The condition (64) implies that D2 ¼ D D ¼ 0. Since B defined in Eq. (36) has to be finite and positive we have to be careful in the limiting procedure leading to this quantity. Also we need to have C C 0, in order that the original parameters and are well defined. Let us consider e3 ¼ 0;. ! 0;. e3 b þ y2 ; e2 e2. (77). zbe3 ¼ 0;. (78). which lead to z ¼ 0. We have two free parameters: x2 , y2 . The generators are e3 3 b 3 b 3 e3 3 2 2 G ¼ x2 K þ K þ J y2 J K þ J : e2 e2 e2 e2 (79) Defining e b G0x2 ¼ K2 þ 3 K3 þ J 3 ; e2 e2 b 3 e3 3 0 2 Gy2 ¼ J K þ J ; e2 e2. (69). such that. (76). together with the consistency conditions zðb2 e23 e22 Þ ¼ 0;. ½Gx ; Gy ¼ 0;. (80). we can verify that D2 ¼ 2ðb2 e22 Þ ¼ 4sb :. The choice. x2 ¼ y2 ¼ 0:. Here,. we obtain the algebra. e2 ¼ sb þ ;. x1 ¼ y1 ¼ 0;. G. Case b 0, e2 0, e3 0. 2. 3. ½Gz ; Gx ¼ 2iGx ;. (75). The free parameters here are x3 , y3 .. Defining Gz ¼ ð2sK þ iIÞ;. 1 ¼ ; 2. We have. G ¼ zð2sK 1 þ iIÞ þ x3 ðK3 sJ 2 Þ y3 ðsK 2 þ J 3 Þ: (66). 3. 2 > 0; 2sb. F. Case b 0, e3 0, e2 ¼ 0. x2 ¼ sy3 ;. 1. 4sb sb ¼ 2 < 0; (74) 2 4 . showing that the limiting procedure is well defined.. which corresponds to a plane wave VEV. Summarizing we have y1 ¼ 0;. C2 ¼ . which ensures that C2 is finite. The fact that C2 is negative is consistent with the second condition in (37). Summarizing, in this case we have. Here, we have b2 e22 ¼ 0 ! b ¼ se2 ;. (73). Next we consider C ¼. we can verify that ½Gx2 ; Gy2 ¼ 0:. (72). is finite. Besides we must demand that. Calling b 3 2 Gx2 ¼ K þ J ; e2. 2 8sb. pffiffiffi ¼ ;. with arbitrary, guarantees that. (70). ½G0x2 ; G0y2 ¼ 0:. (71). Summarizing, all the two-parameter subalgebras that leave the vacuum invariant are isomorphic to Tð2Þ, while the only three-parameter subalgebra, corresponding to the case IV E 2, is isomorphic to HOM(2).. 025007-7. (81).

(8) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). D0 @0 ¼ @l Dl :. H. The symmetries of the action To conclude this section, let us consider the full effective action 1 L ¼ f f B½ðD f Þ þ ðf f Þ2 4. (82). and verify the invariance under the Lorentz plus scale transformations that leave the vacuum invariant given by the generator (43) together with the condition (44). The term f f is invariant under the full Lorentz group plus scale transformations, which includes also the second term in the parenthesis proportional to B. Let us verify the invariance of the term (D f ) under the transformation (44). We have ðD f Þ ¼ ðD Þf þ ðD f Þ ¼ ðD f Þ; (83) because (44) is the subset of transformations that leave the vacuum D invariant. The remaining transformations are given by those of a contravariant tensor ðD f Þ ¼ D ðf G þ f G Þ ¼ f ðG. D. . þ G D Þ ¼ 0. (84). because the last equation in the right-hand side just reproduces the vacuum invariance condition (44). In these way, the symmetries of (82) are in fact those of the vacuum D . V. THE EQUATIONS OF MOTION: THE PROPAGATING SECTOR To study the propagation properties we consider only the quadratic terms in the effective Lagrangian 1 L0 ¼ f f Bðf D Þ2 ; 4. (85). A. Covariant formulation The modified dispersion relations are found very easily by manipulating Eqs. (86). In momentum space Z A ðxÞ ¼ d4 xA ðkÞeik x (92) and choosing the Lorentz gauge, these equations reduce to pffiffiffiffiffi p 2 BD k ; (93) k2 A þ 2p ðp A Þ ¼ 0; k A ¼ 0:. (86). We have verified the consistency of the above when taking @ . It is convenient to define X ¼ D @ A . (87). ðk2 þ 2p2 Þðp A Þ ¼ 0:. Dk. ¼ Dk ¼ ½D0k @0 Dlk @l ; D0 ¼ Di0 @i :. k2 þ 2p2 ¼ 0:. @2 ¼ 0;. (97). which leaves only 1 degree of freedom as it should be. Moreover from the first equation in (93), this remaining degree of freedom satisfies k2 A ¼ 0;. (98). so its dispersion relation is k2 ¼ 0:. (99). The general solution of (95) is p A ¼ 1 ðkÞðk2 þ 2p2 Þ:. (100). Putting it back into (93), we get A ¼ a ðkÞðk2 Þ þ 2 a. ðkÞk. ¼ 0;. a. 1 ðkÞ ðk2 k2. ðkÞp. þ 2p2 Þp ;. (101). ¼ 0:. From (101), we obtain the electromagnetic tensor. In this way we have X ¼ ðDj Aj þ D0 A0 Þ;. (96). If (k2 þ 2p2 ) is not zero in Eq. (95), it follows that p A ¼ 0. In four dimensions, this condition plus the Lorentz gauge leaves 2 degrees of freedom. But the Lorentz gauge permits a further gauge transformation with parameter such that. (88) (89). (95). Moreover, p A is gauge invariant. In fact, in coordinate space is proportional to D f . So this component is physical and has a dispersion relation given by. and to introduce the notation @. (94). The vector p is proportional to the momentum space version of the vector D introduced in Eqs. (88) and (89). Multiplying the first relation in (93) by p , it follows that. where we recall that f ¼ @ A @ A . The equations are ð@2 A @ @ A Þ ¼ 8BD @ ðD @ A Þ:. (91). (90). f ¼ ðk a ðkÞ k a ðkÞÞðk2 Þ þ2. together with. 025007-8. 1 ðkÞ ðk2 k2. þ 2p2 Þðk p k p Þ:. (102).

(9) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. It represents a plane wave with dispersion relation k ¼ 0 (the magnetic and electric fields are the standard ones, perpendicular to k), plus a plane wave propagating in the direction k with dispersion relation k2 þ 2p2 ¼ 0. The fields for the second wave are. PHYSICAL REVIEW D 81, 025007 (2010). 2. Ej ¼ f0j ¼ 2ðk0 pj kj p0 Þ Bj ¼ 2. jkl ðkk pl Þ. 1 ðkÞ ; k2. 1 ðkÞ : k2. B ¼ fBi g; 123. ¼ þ1;. Er ¼ i!Ar ikr A0 ; (103). (113). Bk ¼ i. klm kl Am. (114). in momentum space with the notation k ¼ fki g in the three dimensional subspace. C. Modified Maxwell equations They are r E þ 8Bðe rÞðB b E eÞ ¼ 4 ;. B. 3 þ 1 decomposition. (115). r ðB þ 8BbðB b E eÞÞ. An alternative way to determine the polarization of the plane wave solutions is by splitting Eq. (86) into the components 0, i, which yields. . 1 @ 4 ½E þ 8BeðB b E eÞ ¼ J; c @t c r B ¼ 0;. (104). ð@2 Ak @k @0 A0 þ @k @i Ai Þ ¼ 8BDk ðDj Aj þ D0 A0 Þ:. rEþ. (105). (116) (117). 1 @B ¼ 0: c @t. (118). In the notation of electrodynamics in a medium we can introduce. We choose the Coulomb gauge @r Ar ¼ 0:. 1 f ¼ klm @l Am ; 2 klm lm B ¼ r A;. which reduces to. Notice that E is perpendicular to B, B is perpendicular to k, but E is not necessarily orthogonal to k. Moreover, this wave exists only if k2 0.. ðr2 þ 2bD20 ÞA0 @0 @i Ai ¼ 8BD0 Dj Aj ;. Bk ¼ . (106). D ¼ E þ 8BeðB b E eÞ;. (119). H ¼ B þ 8BbðB b E eÞ;. The final equations are ðr2 þ 8BD20 ÞA0 ¼ 8BD0 Dj Aj ;. (107). ð@2 Ak @k @0 A0 Þ ¼ 8BDk ðDj Aj þ D0 A0 Þ:. (108). r D ¼ 4 ;. rH. 1 @D 4 ¼ J; c @t c. (120). As in the usual electrodynamics A0 can be expressed as an instantaneous function of Aj. r B ¼ 0;. rEþ. 1 @B ¼ 0: c @t. (121). Di ¼ ðij 8Bei ej ÞEj þ 8Bei bj Bj ;. (122). Hi ¼ ðij þ 8Bbi bj ÞBj 8Bbi ej Ej :. (123). A0 ¼ 8B. 1 D0 Dj Aj : ðr2 þ 8BD20 Þ. (109). such that. In components. The final vector equation leads to @2 Ak ¼ ðr2 kl þ @k @l Þ. 8B Dl Dj Aj ; ðr2 þ 8BD20 Þ (110). where it is a direct matter to verify the Coulomb gauge condition. The electromagnetic fields are given by A ¼ E ¼ fEi g;. fAi g;. Ei ¼ f0i ¼ @0 Ai @i A0 ; E ¼ @0 A rA0 ;. (111). Particular cases are bj ¼ 0 ! Di ¼ ðij 8Bei ej ÞEj ; ei ¼ 0 ! Di ¼ Ei ;. Hi ¼ Bi ; (124). Hi ¼ ðij þ 8Bbi bj ÞBj : (125). The first case induces a modification in Coulomb law B ¼ 0;. r ðE 8Beðe EÞÞ ¼ 4 ;. E ¼ r; (126). (112). 025007-9. ðr2 8Bðei @i Þ2 Þ ¼ 4 :. (127).

(10) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). VI. SUMMARY OF THE DISPERSION RELATIONS AND ELECTROMAGNETIC FIELDS Here, we use the previous decomposition of the background field D in terms of three dimensional components together with the 3 þ 1 formulation described in Sec. V B, to search for plane wave solutions of the Maxwell equations. We assume the space-time dependence of any field to be proportional to eiðkx!tÞ , where k is the momentum of the wave and we choose the potential A in the Coulomb gauge. The notation is b ¼ fbm g;. e ¼ fem g:. fDk g ¼ D ¼ i½!e k b;. (129). D0 ¼ ie k:. (130). There are two main cases (i) Dj Aj ¼ 0: In this case the dispersion relation is ! ¼ jkj. The fields are ^ k^ e þ b kg; ^ B ¼ fðe kÞ. (131). ^ k^ b e kg: ^ E ¼ fðb kÞ. (132). (128). The expressions of Eqs. (88) and (89) in momentum space are. (ii) Dj Aj 0: In this case the dispersion relation and the fields are. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 256B2 ðk^ ðb eÞÞ2 4ð1 8Be2 Þf8B½ðk^ eÞ2 ðk^ bÞ2 1g 16Bk^ ðb eÞ ! ¼ jkj ; 2ð1 8Be2 Þ. (133). to be very small we can rewrite (138) as B ¼ fk ðk bÞ !k eg;. (134). E ¼ f!k b !2 e þ kðe kÞg:. (135). !1 ¼ jkjð1 þ 4Be2 sin2 Þ:. (139). In both cases is an arbitrary constant. For small B, the dispersion relation (133) reduces to. The electromagnetic fields in momentum space are ðe kÞ E ¼ e k ; B¼ ½k e; (140) 2 !1 !1. ! ¼ jkj½1 þ Bð8k^ ðe bÞ þ 4ðk^ bÞ2. where is a dimensionless constant. The following orthogonality relations are satisfied:. þ 4ðk^ eÞ2 Þ:. (136). The above expression clearly shows that the model is stable in the small Lorentz invariance violation approximation, where the following quantities Be2 ; Bb2 ; Bjejjbj are very small compared to one. The anisotropic velocity of light arising from the above dispersion relation is. E B ¼ 0;. k^ E 0;. k^ B ¼ 0;. (141). together with the properties E B ½k 8Bðe kÞe m; m B ¼ 0;. m E ¼ 0:. (142) (143). The ratio between the amplitudes is. ^ rk ! ¼ cðkÞ. sin2 þ ð1 8Be2 Þ2 cos2 jEj2 ¼ : ð1 8Be2 cos2 Þð1 8Be2 Þ jBj2. ^ þ 8Bðb2 þ e2 Þ 4Bððk^ bÞ2 þ ðk^ eÞ2 ÞÞ ¼ kð1 ^ 8Bðe kÞe: ^ þ 8Bðe bÞ 8Bðb kÞb (137) Next we consider the separate cases where e 0, b ¼ 0, e ¼ 0, b 0, and e 0, b 0. The results are as follows:. 2. Case (I-2) e A ¼ 0, D A ¼ 0 The corresponding dispersion relation is !22 ¼ k2 ;. A. Case I: b ¼ 0, e 0 1. Case (I-1) e A 0, D A 0. !21 ¼. 1 8Be cos 1 8Be2. E ¼ ðk^ eÞ; 2. k2 ;. (145). in accordance with (99), with the electromagnetic fields. We find the dispersion relation 2. (144). (138). B ¼ k^ ðk^ eÞ;. (146). where is an arbitrary dimensionless constant. Also we have. which agrees with the one obtained from (96). Here, is the angle between k and e. Assuming the corrections terms. 025007-10. jEj ¼ jBj ¼ jej sin :. (147).

(11) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. PHYSICAL REVIEW D 81, 025007 (2010). We choose a reference frame such that. The orthogonality relations are k^ E ¼ 0;. k^ B ¼ 0;. E B ¼ 0;. n^ ¼ ð1; 0; 0Þ;. (148). b ¼ ð0; 0; bÞ;. e ¼ ð0; sb; 0Þ;. similarly to the standard case. Also jEj2 ¼ 1: jBj2. (158). s ¼ 1;. k A ¼ 0;. (159). Dj Aj ¼ ibððk1 !sÞA2 k2 A1 Þ:. (160). (149). B. Case II: e ¼ 0, b 0, D0 ¼ 0 1. Case Dk Ak ¼ 0.. 1. Case (II-1) b A 0 ) D A ¼ 0 We find the dispersion relation !23 ¼ k2 ;. (150). The dispersion relations are the usual ones. From the master formulae (131) and (132), the fields are. and we have E ¼ ½k^ ðk^ bÞ;. B ¼ ðk^ b; Þ. (151). ^ k^ sðb nÞ ^ ^ kÞ ^ þ b kg; B ¼ fðsðb nÞ. (161). ^ k^ b þ sk^ ðb nÞg: ^ E ¼ fðb kÞ. (162). with an arbitrary dimensionless constant. Also jEj2 ¼ 1; jBj2. k^ E ¼ k^ B ¼ 0;. 2. Case Dk Ak 0.. E B ¼ 0: (152). 2. Case (II-2) b A ¼ 0 ) D A 0 Here, the dispersion relation is !24 ¼ k2 ð1 þ 8Bb2 sin2 Þ;. (153). The exact modified dispersion relations are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 !5 ¼ ð ðk2 8Bb2 ðk2 k21 ÞÞ þ 8Bb2 sk1 Þ: ð1 8Bb2 Þ (163) In the small violation approximation Eq. (163) reduces to k 2 !5 ¼ jkj 1 þ 4Bb2 1 þ s 1 : (164) jkj. with being the angle between k and b. The above result agrees with that obtained from (96). The electromagnetic fields are determined, up to the arbitrary constant , by ðk bÞ; !4. (154). k ðk bÞ; !24. (155). E ¼ B ¼. k^ B ¼ 0;. E B ¼ 0;. !2 jEj2 ¼ 42 ¼ ð1 þ 8Bb2 sin2 Þ: 2 jkj jBj. ^ B ¼ fk^ ðk^ bÞ !5 sk^ ðb nÞg;. (165). ^ ^ ^ þ skððb ^ kÞg: nÞ E ¼ f!5 ðk^ bÞ !25 sðb nÞ (166) VII. THE MODEL AS A SECTOR OF THE SME. and they satisfy k^ E ¼ 0;. From the formulae (134) and (135), the fields are. (156) (157). Notice that half of the cases previously considered violate rotational invariance, with the exception of those having unmodified dispersion relations together with standard properties of the propagating electromagnetic fields.. In order to impose some bounds upon the parameters of the model it is convenient to recast the quadratic sector of the action (38) in the language of the SME and to make use of the numerous experimental constraints derived from it (see, for example, Ref. [22]) and summarized in Table II: Photon-sector summary, of Ref. [29]. To begin with, let us recall the dimension of the fields and parameters involved in the model ½A ¼ m;. ½e ¼ ½b ¼ m2 ;. ½B ¼. 1 ; m4. ½Be2 ¼ 0: (167). C. Case III:. b2. ¼. e2 ,. eb¼0. Next we make the identification. We consider this case corresponding to a plane wave vacuum because the final theory remains invariant under the three-parameter subgroup HOM(2).. Bðf D Þ2 ¼ 14ðkF Þ where the tensor ðkF Þ. 025007-11. ,. f. . f ;. (168). with 19 independent compo-.

(12) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). nents, has all the symmetries of the Riemann tensor and a vanishing double trace. In terms of the matrix elements D characterizing the vacuum expectation values of the electromagnetic tensor we obtain ðkF Þ ¼ 4B D D þ 12ðD D D D Þ 12BD2 ð. . . . . Þ;. (169). where D2 D D ¼ 2ðb2 e2 Þ:. (170). Next we have to identify the appropriate combinations of the components of ðkF Þ which are bounded. A first step in that direction constitutes the definitions of the components ðDE Þjk , ðHB Þjk and ðDB Þjk ¼ ðHE Þkj , which are written in terms of our parameters e, b and B as follows: ðDE Þjk 2ðkF Þ0j0k ¼ 12Bej ek BD2 jk ; (171) ðHB Þjk . 1 2. jpq krs ðkF Þ. pqrs. ¼ 12Bbj bk BD2 jk ; (172). ðDB Þjk ¼ ðHE Þkj ¼. 0jpq kpq ðkF Þ. ¼ 12Bðej bk 13ðe bÞjk Þ; tr½jk DB . (173). ¼ 0:. The final combinations in terms of which the bounds are presented turn out to be. only the light cone gets modified by the LIV parameters, while the propagation of the fermioms is the standard one [22,32]. All matrices from (174) to (177) are traceless, with ð 0þ Þjk being antisymmetric (3 independent components) while the three remaining ones are symmetric (5 independent components each). The above combinations (174) to (178) constitute a convenient alternative way to display the 19 independent components of the tensor ðkF Þ . The bounds are obtained in terms of such combinations referred to the standard inertial reference frame defined in Ref. [22] and are written at the end of each of the above equations. Notice that the bilinears which are odd under the duality transformations e ! b;. We consider only absolute values of the related quantities and we focus upon the stringent constraints in Eqs. (174) and (177). The nontrivial contributions are jð~ eþ Þ11 j ¼ jðx2 y2 Þj < 3;. ¼ 6B½bj bk ej ek 13ðb2 e2 Þjk < 1032 ; (174) ð e Þjk ¼ 12ðDE HB Þjk 13jk trðDE Þ. (182). jð~ eþ Þ33 j ¼ j2y2 þ ð1 3cos2 c Þx2 j < 3;. (183). (176) ð o. ¼. HE. jð~ eþ Þ23 j ¼ jx2 sin2 c j < 2;. (184). ð~ o Þ11 ¼ ð~ o Þ22 ¼ jxy cos c j < 32;. (185). ð~ o Þ33 ¼ jxy cos c j < 34;. (186). ð~ o Þ23 ¼ jxy sin c j < 1:. (187). together with. ð oþ Þjk ¼ 12ðDB þ HE Þjk ¼ 6Bðej bk ek bj Þ < 1012 ; 1 2ðDB. (181). jð~ eþ Þ22 j ¼ jð1 3sin2 c Þx2 y2 j < 3;. ¼ 6B½ðbj bk þ ej ek Þ 13jk ðb2 þ e2 Þ < 1016 ; (175). (179). are much more suppressed than those that are even. In this way, even though our model is not duality invariant, this transformation seems to explain the above-mentioned hierarchy in the LIV parameters exhibited in Eqs. (174)– (178). In order to obtain some specific consequences of the bounds (174) to (178) it is convenient to express them in the coordinate system defined by (41). Also we introduce the notation pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi x ¼ 6Be 1016 ; y ¼ 6Bb 1016 ; (180) x ¼ jxj; y ¼ jyj:. ð eþ Þjk ¼ 12ðDE þ HB Þjk. Þjk. b ! e;. Þjk. ¼ 6B½ðej bk þ ek bj Þ 23ðe bÞjk < 1032 ; (177) 2 2 15 : tr ¼ 13 tr½ij DE ¼ 2Bðe þ b Þ < 10. (178). The bounds for tr have recently been improved from 107 [29], to 1011 [30], and finally to 1015 [31]. In quoting our results arising from the bounds upon ð eþ Þjk , ð o Þjk and tr we have chosen our reference frame in such a way that. The allowed region in the (x-y) plane is shown in Fig. 1, where we plot the boundaries of the corresponding inequalities. In expressions (184)–(187) we consider the lower bound corresponding to the maximum value of the trigonometric function on the left-hand side of the inequality. This leads to pffiffiffi x < 2; jxyj < 34 < 1 < 32: (188). 025007-12.

(13) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. PHYSICAL REVIEW D 81, 025007 (2010). Eqs. (174)–(178). All the above relations are valid in the standard inertial reference frame centered in the Sun [22]. VIII. SUMMARY AND FINAL COMMENTS. FIG. 1. Boundaries of the allowed region obtained from the ~jk constraints in the parameters ~jk o . The allowed region is eþ and on the inside the dashed, dot-dashed, and dot-dot-dashed lines. An excellent approximation for it is the sector of the circle shown in solid line.. The boundary xy ¼ 3=4 is plotted in a dashed line. The most stringent bound from Eq. (183) results when c ¼ =2 and corresponds to sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 x2 y< : (189) 2 The boundary is shown in the dot-dot-dashed line. The most stringent bound from (182) arises again from c ¼ =2 and corresponds to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y < 3 2x2 ; (190) which boundary is plotted in the dot-dashed line. The expression (181) does not provide additional bounds and is plotted for completeness. The corresponding boundaries pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 are y ¼ x 3 shown in the dotted lines in Fig. 1. An upper bound including all previous ones is given by the interior of the circle sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 y¼ x2 ; (191) 2 shown in solid line in Fig. 1 and which translates into the restriction B ðe2 þ b2 Þ ¼. 1 ðe2 þ b2 Þ < 2:5 1033 : 4 MB. (192). This bound incorporates all the constraints established in. We study a novel way of implementing a model with spontaneously broken Lorentz symmetry by introducing a constant VEV of the field strength hF i ¼ C instead of imposing a VEV of the electromagnetic potential hA i as frequently investigated in the literature. In this way, our model preserves gauge invariance from the very beginning. We start from the effective Lagrangian (13) describing nonlinear electrodynamics and containing a potential Veff ðF Þ, which is argued to arise after integrating massive gauge bosons and fermions in an underlying conventional theory. This approach is an extension of the model in Refs. [7,8], which includes all quartic fermion interactions allowed by the Dirac algebra. For simplicity we have considered only one fermion species. The subsequent integration of them produces the effective gauge invariant potential which is bounded in the high-intensity field limit and has an stable minimum. In order to investigate the dynamical consequences of the electrodynamics constructed around such minimum we have chosen the standard Ginzburg-Landau parametrization for Veff ðF Þ. We have verified that the extremum conditions in the potential corresponds to the extremum conditions of the energy arising from (13), under the requirement that the involved fields are constant at the extreme points. Next we expand the fields around the minimum and arrive to the spontaneously broken action (25) from where we can further discuss the resulting nonlinear electrodynamics, which differs from the standard one described in Ref. [26], for example. The action (25) contains the field strength a ðE; BÞ, together with the auxiliary field X which equations of motion demand that the two-form a is such that da ¼ 0, thus introducing the electromagnetic potentials. The constitutive relations expressing the electromagnetic excitations D and H, are identified via the equations of motions (26) and (31) and provide a direct connection with the field strengths E and B. After eliminating the auxiliary field X and writing the most general solution to the condition da ¼ 0, we arrive at the action (38), which is the starting point of our subsequent analysis. Some redefinitions allow the symmetry breaking parameters to be characterized by the constant antisymmetrical tensor D (proportional to the VEV C ), plus an additional constant B. The search for the subgroups under which the model remains invariant after the symmetry breaking includes the generator of dilatations plus the standard Lorentz generators. In this restricted subalgebra the former commutes with the remaining ones, which allows its realization as a multiple of the identity. There is only one case, given in IV E 2, which incorporates this generator, resulting in the. 025007-13.

(14) JORGE ALFARO AND LUIS F. URRUTIA. PHYSICAL REVIEW D 81, 025007 (2010). breaking of the Lorentz group to the three generators subgroup HOM(2). All the other cases break to a subgroup isomorphic to Tð2Þ, with two generators. The modified photon dispersion relations together with the corresponding plane wave polarizations are classified according to the value of Di k Ai , where k is the plane wave four momentum and Ai is the corresponding vector potential in the Coulomb gauge. The amplitudes of the propagating fields are written in terms of those parameterizing D . The case Di k Ai ¼ 0 leads to dispersion relations ! ¼ jkj with the triad E, B, k having standard orthogonality properties. The situation Di k Ai 0 produces the following properties for the vectors involved: E is perpendicular to B, B is perpendicular to k, but E is not necessarily orthogonal to k. The corresponding dispersion relations are of the form ! ¼ jkjF ;. ^ 2 Þ: þ e2 ðe^ kÞ. ^ ¼ 1 þ 8Bðe2 þ b2 Þ 4Bððb kÞ ^ 2 cðkÞ. c ^ cTW ðqÞj; ^ jcTW ðkÞ c. From the last expression we obtain the bound c < 4Bsin2 ðb2 þ e2 Þ < 4Bðb2 þ e2 Þ < 1032 ; c (200) according to Eq. (192), where is the angle between the ^ The above anisotropy measures the difvectors n^ and k. ference in the two-way speed of light propagating in perpendicular directions in a given reference frame. The standard Michelson-Morley type of analysis can be made by measuring first the time difference along two perpen^ and L2 dicular trajectories L1 (propagating with cTW ðkÞ), ^ (propagating with cTW ðqÞ) ¼. L2 L1 : ^ ^ cTW ðqÞ cTW ðkÞ. (201). Subsequently, a similar measurement is made by rotating the apparatus by 90 degrees 0 ¼. L2 L1 : ^ ^ cTW ðkÞ cTW ðqÞ. (202). Finally, the difference in optical paths ¼ j 0 j ^ 2 þ e2 ðe^ kÞ ^ 2 Þ (203) ¼ 4BðL2 þ L1 Þsin2 ðb2 ðb^ kÞ. ^ ¼ 1ðcðkÞ ^ þ cðkÞÞ ^ cTW ðkÞ 2 ^ 2 þ ðe kÞ ^ 2 Þ: ¼ 1 þ 8Bðe2 þ b2 Þ 4Bððb kÞ (195) A standard measure for the anisotropy of the speed of light is the comparison of the two-way velocities in perpendicu^ ¼ n^ in our lar directions. We single out the vector ðe^ bÞ reference frame, which plays an analogous role to the relative velocity among the ether frame and the laboratory frame in the Robertson-Mansouri-Sexl (RMS) type of analysis [33,34]. The vectors k^ and n^ form a plane in which we define the vector q^ perpendicular to k^ given by ^ ¼ ðk^ nÞ ^ k^ n: ^ q^ ¼ k^ ðk^ nÞ. (198). c ^ 2 þ e2 ðe^ kÞ ^ 2 Þj: ^ 2 Þðb2 ðb^ kÞ ¼ j4Bð1 ðk^ nÞ c (199). (194). which we take as the one-way light speed in the direction ^ predicted by our model. The corresponding two-way k, light speed is then. (197). An appropriate definition for the anisotropy of the speed of light in this model is. (193). where F is independent of jkj, being only a function of the angles between k and e, k and b, k and e b. In this way our model predicts anisotropy in the speed of light. Also, as shown in Eq. (136), the model is stable in the small Lorentz invariance violation approximation where the quantities Be2 ; Bb2 ; Bjejjbj are very small compared to one. This is validated by the bound Bðe2 þ b2 Þ < 2:5 1033 derived from Fig. 1. In order to make a more quantitative statement about this anisotropy let us consider the different possibilities arising from the velocity (137). To the leading order Bjej2 , Bjbj2 , Bjejjbj the magnitude of such velocity is. ^ 2 2k^ ðe bÞÞ; þ ðe kÞ. ^ 2 ^ ¼ 1 þ 8Bðe2 þ b2 Þ 4Bðk^ nÞ ^ 2 ðb2 ðb^ kÞ cTW ðqÞ. measures the change in the interference pattern of the Michelson-Morley experiment, according to our model. It is important to emphasize that the bound (200) is not related to the anisotropy in the speed of light induced by the passage from a preferred frame (where propagation is isotropic) to another frame moving with relative speed v and subjected to alternative methods of time synchronization. As such, it cannot be directly compared with the RMS type bounds appearing in the literature. The difference in two perpendicular two-way speed of light in the RMS case is [34]. (196). Next we calculate the two-way light speed along this perpendicular direction obtaining. 025007-14. c ¼ jcTW ð þ =2Þ cTW ðÞj c 2 v 1 þ cos2 ¼ : c 2. (204).

(15) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. Here, is the angle between the photon direction and the relative speed v. Recent bounds upon the Mansouri-Sexl parameter ( 12 þ ) [35] are of the order of 1010 , implying a RMS bound c c. 1016 ;. (205). with the relative speed between the earth and the cosmic microwave background reference frame, where the cosmic background radiation looks isotropic, having the value v ¼ jvj ’ 300 km=s. Perhaps a more significant comparison of our results can be made with the work of Ref. [36], where the Euler model of nonlinear electrodynamics [37] is used to predict the following isotropic bound of the one-way speed of light ~ c c. 1:2 1023 :. (206). PHYSICAL REVIEW D 81, 025007 (2010). MB > 1:4 104 GeV:. Recalling that the lightest charged particle is the electron with me ¼ 5:1 104 GeV, we notice that the above bound is consistent with the QED Euler-Heisenberg type of nonlinear corrections to the photon interaction, which scale as 1=m4 , with m being the mass of the charged particle contributing to the loop in the effective action [39]. ACKNOWLEDGMENTS The work of J. A. was partially supported by Fondecyt Contract No. 1060646. L. F. U. acknowledges support from the DAGPA-UNAM-IN Grant No. 109107, CONACyT Grant No. 55310, and Fondecyt Contract Nos. 7080101 and 7060206. L. F. U. also acknowledges R. Lehnert for useful suggestions and comments.. In our case this would correspond to averaging over all angles in (194), leading to ~ c c. 8Bðe2 þ b2 Þ ’ 2 1032 :. (207). Let us emphasize that in all our estimations we are assuming that the involved references frames (cosmic microwave background, Sun based, earth based) are concordant, in such a way that first order quantities in the Lorentz invariance violation parameters remain of the same order in all of them. Assuming that our background fields e and b might represent some galactic or intergalactic fields in the actual era, we obtain a very reasonable bound for the magnetic intergalactic field by assuming that the constant appearing in the action (38) corresponds to an energy density ¼ 14½ð1 D2 BD2 ’ 12ðb2 e2 Þ;. < 1048 ðGeVÞ4 ;. (209). jbj < 5 105 Gauss;. (210). j. j. APPENDIX In this Appendix we extend the models in Refs [7,8] to include all quartic gauge invariant fermionic interactions allowed by the Dirac algebra and look for their contribution to the effective gauge invariant electromagnetic action by integrating the fermion field. We heavily rely on the discussion and results of Ref. [23], which presents a complete and detailed description of the standard electromagnetic calculation. We do not attempt to cite the original papers, which references can be found in Ref. [23]. After integrating the massive gauge bosons in an underlying conventional theory [8], let us consider the effective Lagrangian. (208). which we can associate with the cosmological constant, since it would represent a global property of the universe. The fact that this constant is positive favors ðb2 e2 Þ > 0, so that one can perform a passive Lorentz transformation to a reference frame where e ¼ 0. Supposing further that this frame, which describes the intergalactic fields, is concordant with the standard inertial reference frame and taking the upper limit [38]. we obtain the bound. which is consistent with observations of intergalactic magnetic fields. Let us recall that 1 Gauss ¼ 1:95 1020 ðGeVÞ2 . It is amusing to observe that this bound on jbj translates into the following condition for the mass scale MB introduced in Eq. (192):. (211). ð@ þ ieA Þ mÞ Leff ¼ ði X rM a ÞðM a Þ: ½ðM 2 2 M;a. (A1). The index M: S, V, T, PV, PS labels the tensorial objects that constitute the Dirac basis: scalar, vector, tensor, pseudovector, and pseudoscalar, respectively. The generic index a labels the covariant (contravariant) components of each tensor class. More details are given in Table I. We follow the conventions of Ref. [24], with appropriate fac a ) is tors chosen in such a way that each current (M real. Our goal is to estimate the contributions W ð2Þ of the additional quartic fermionic couplings to the standard efð1Þ fective electromagnetic action WEM . The latter is obtained by integrating the quadratic contribution of the fermions in Eq. (A1) and it is given by. 025007-15.

(16) JORGE ALFARO AND LUIS F. URRUTIA. h0þ j0 iAð1Þ ¼. PHYSICAL REVIEW D 81, 025007 (2010). Z. ½D½D Z 4 exp i d xði ð@ þ ieA Þ mÞ ;. ¼ detðði ð@ þ ieA Þ mÞÞ Z ð1Þ exp½iWEM ¼ exp i d4 xLð1Þ EM :. (A2). To this end we introduce the gauge invariant auxiliary fields ðCM Þa , via the couplings Leff ¼ ði ð@ þ ieA Þ mÞ X pM a Þ : þ ðCM Þa ðCM Þa qM ðCM Þa ðM M;a 2. (A3) The equations of motion for the auxiliary fields are (in the Appendix we do not use the Einstein convention for repeated indices unless explicitly stated) q ðCM Þa ¼ M ðM a Þ; pM. (A4). which reproduce (A1) when rM q2 ¼ M: 2 pM. (A5). Next we integrate over the auxiliary fields following the standard steps which begin with the introduction of the currents ðjM Þa together with the replacements : ðCM Þa ! iðjM Þa. ðjM. Þa ¼0. (A7). where ðx x0 Þ is proportional to ðx x0 Þ in each case, we can rewrite the above equation as Z X 1 h0þ j0 iA ¼ exp i d4 x a M;a 2pM ðXM Þ ðXM Þa Z ½D½D Z ði ð@ þ ieA Þ exp i d4 x X ; mÞ qM Ma ðXM Þa . . M;a. ¼ exp i. Z. XM ¼0. X 1 a M;a 2pM ðXM Þ ðXM Þa. d4 x. det ði ð@ þ ieA Þ mÞ X a : qM M ðXM Þa XM ¼0. M;a. (A11). where we have already taken the limit ðjM Þa ¼ 0. The next step is to rewrite X ði ð@ þ ieA Þ mÞ qM Ma ðXM Þa M;a. with Leff. Z exp i d4 x i ð@ þ ieA Þ ½D½D X m qM M a iðjM Þa M;a Z X i ðjM Þa ðjM Þa ; (A9) exp d4 x a pM. where the limit ðjM Þa ! 0 is implicit. Notice that in order for the functional integral of the fields ðCM Þa to be well defined in the usual Euclidean analytic continuation we must require that pa > 0. Using the standard formula (see, for example, Eq. (1.53) in Ref. [23]) Z Z i i i F exp jj ¼ exp jj exp ij 2 2 2 Z FðXÞ ; X X X¼j (A10). (A6). The total vacuum transition amplitude can then be written as Z Z 4 h0þ j0 iA ¼ ½D½D½DðC Þ exp i d xL M a eff XZ exp i d4 xðjM Þa ðCM Þa M;a. h0þ j0 iA ¼. Z. X ¼ i ð@ þ ieA Þ m qM Ma þ. X pM M;a. 2. ðCM Þa ðCM. Þa. M;a. :. X ¼ ði ð@ þ ieA Þ mÞ 1 GA qM Ma ðXM Þa ;. iðjM Þa (A8). Now we can perform the Gaussian integral over the fields ðCM Þa , which includes the linear contribution of the currents, to obtain. M;a. (A12) where GA is the fermion Green function in the presence of the external electromagnetic field A , satisfying ði ð@ þ ieA Þ mÞGA ¼ 1: Also we recall the expression. 025007-16. (A13).

(17) GAUGE INVARIANT NONLINEAR ELECTRODYNAMICS . . .. detP ¼ exp½Tr lnP;. ðBM Þb c ðx; x0 Þ ¼ iq2M GA ðx; x0 ÞMb GA ðx0 ; xÞMc ;. (A14). where the trace is understood both in coordinate as well as in internal spaces. In this way we have ð1Þ h0þ j0 iA ¼ exp½iWEM Z X 1 exp i d4 x a M;a 2pa ðXM Þ ðXM Þa X exp Tr ln 1 GA qM Ma ðXM Þa :. To make an estimation analogous to the two-loop correcð2Þ ð1Þ tions WEM ðAÞ to the effective action WEM ðAÞ, which is already calculated in Ref. [23], we expand the third exponent up to terms quadratic in ðXM Þa . Calling the operator X X GA qM Ma ðXM Þa ¼ G ¼ GM (A16). Ma ! ; (A21). In the sequel we also restrict ourselves to the case of a constant (or slowly varying) external electromagnetic field. In this case GA ðx; x0 Þ is independent of position so that the transformation properties under the Lorenz group imply that FMa ¼ 0. (See, for example, Eq. (7.8) of Ref. [23]). This further simplifies Eq. (A17) to ð1Þ exp½þ12 Tr lnð1 BM AM Þ1 h0þ j0 iAM ¼ exp½iWEM ð1Þ ’ exp½iWEM exp½þ12 TrðBM AM Þ:. M. (A22). This defines the sought correction. we are left with ð1Þ h0þ j0 iA ¼ exp½iWEM Z X 1 exp i d4 x a M;a 2pM ðXM Þ ðXM Þa i exp½i TrðiGÞ exp TrðiG2 Þ : 2 XM ¼0. (A17) With the purpose of having a preliminary order of magnitude estimation of the corrections we next consider separately each of the terms GM in the summation appearing in Eq. (A16). In this way we are neglecting the contribution of the interference terms among different currents jM , even though their contribution could also be calculated along the lines presented here. In order to proceed we make use of another well-known identity Z Z iZ i exp A exp XBX þ i FX 2 X X 2 1 iZ 1 1 ¼ exp þ Tr lnð1 BAÞ XBð1 ABÞ X exp 2 2 Z exp þi Xð1 BAÞ1 F iZ 1 FAð1 BAÞ F ; (A18) exp þ 2. ð2Þ exp½iWM. 1 b 4 ðx x0 Þ; pM c. FMa ¼ iqM GA ðx; x0 ÞMa ;. (A19). . 1 ¼ exp þ TrðBM AM Þ ; 2. M fixed; (A23). where Tr ðBM AM Þ ¼ i. rM X Z 4 d x trðGA ðx; xÞMc GA ðx; xÞMc Þ; 2 c (A24). where tr is the trace in the Dirac-matrices space. In other words, we have 2rM X Z 4 d x trðGA ðx; xÞMc GA ðx; xÞMc Þ 2 c r Z M2 d4 xlM : (A25) . ð2Þ ¼ WM. In order to estimate each contribution we further consider the case where the external field is a constant magnetic B field in the z direction. We avoid the constant electric field case because of the inherent instability due to pair creation in the strong field regime. For the case under consideration we have [40] m Z 1 ds z i3 z e ; exp½ism2 sinz 162 0 s2 (A26) s ¼ eBs; 3 ¼ i1 2 ;. GA ðx; xÞ ¼. for a fixed M. Next we identify the corresponding operators ðAM Þb c ðx x0 Þ ¼. b c ! ;. 1 b c ! ð Þ: 2. (A15). (A20). where bc denotes the identity in the corresponding case. For example, Ma ! ;. M;a. M;a. PHYSICAL REVIEW D 81, 025007 (2010). where m is the mass of the integrated fermion. In this way, the general expression for the quantity lM defined in Eq. (A25) is. 025007-17.

(18) JORGE ALFARO AND LUIS F. URRUTIA. lM ¼. . PHYSICAL REVIEW D 81, 025007 (2010). 2 Z 1. m ds ds0 exp½ism2 exp½is0 m2 2 2 02 16 0 s s z z0 X ½ðcosz cosz0 trðMc Mc Þ sinz sinz0 c. Our renormalization conditions will be dLeff lim Leff ¼ 0; lim > 0; B!0 B!0 dB2. sinz sinz0 trð3 Mc 3 Mc ÞÞ þ sinðz z0 Þtrð3 Mc Mc Þ:. (A27) P. The last term in Eq. (A27) vanishes because c Mc Mc is always a multiple of the identity and trð3 Þ ¼ 0. Let us introduce the notation X X cM ; c M ; ¼ tr M ¼ tr M M c 3M 3 3 c c. c. (A28) which values are given in Table I. Also it is convenient to define Z 1 ds z cosz ; exp½ism2 I1 ¼ 2 sinz 0 s Z 1 ds Z 1 ds 2 z ¼ eB exp½ism2 : I2 ¼ exp½ism 2 0 s 0 s (A29) In terms of the above quantities we have m 2 lM ¼ ½ M I12 3M I22 : 162. (A30). 2. The second contribution behaves as B so that we consider it as part of the normalization condition that we demand of the low energy limit of the effective potential lim Veff ðBÞ ¼ B2 ;. B!0. > 0:. (A31). In this way we consider the full expression for the contribution due to the additional currents to be X m 2 rM ð2Þ 2 ; L ¼ (A32) Lð2Þ ¼ Lð2Þ M I1 : M M 162 2 M The effective action (A32) leads to the effective potential m 2 rM ð2Þ 2 VM ¼ Lð2Þ ¼ (A33) M I1 : M 162 2 The main point to be made is that each contribution to the effective potential arising from the subspace M contributes with an specific sign, as can be seen from Table I. In these way, as it will be argued in the sequel, it is possible to have a net positive contribution for the total effective potential in the limit B=B0 ! 1, where B0 ¼ m2 =e is the critical value ð2Þ of the magnetic field. Let us emphasize that VM is still not well defined because the integral I1 diverges. This means that we should regularize it and further implement a renormalization prescription for the full effective Lagrangian ð2Þ ð2Þ Leff ¼ B2 þ Lð1Þ EM ðBÞ þ LEM ðBÞ þ L ðBÞ:. (A35). in such a way that the effective potential is a decreasing function in the vicinity of B ¼ 0. The final normalization to the value of Leff ¼ b2 =2; b ! 0; will be imposed at the end of the procedure, after we expand the magnetic field around the expected stable minimum BMin of Veff , such that B ¼ BMin þ b. (A36). and we are left with the physical component b. The renormalization procedure should be analogous to the one carð2Þ ried out in Ref. [23] for the contribution WEM ðAÞ and it is rather involved due to the complicated field dependence of the regularized version of the divergent integrals. This calculation is out of the scope of these preliminary estimations. Thus, we will only consider a particular regularization of I1 to proceed with the comparison of the different highintensity field contributions to the effective potential. We take Z 1 ds 2 z cosz 1 ; (A37) I1 ! I1 ¼ exp½ism 2 sinz 0 s which produces a zero contribution when B ! 0. This integral can be readily calculated from the first order electromagnetic correction of the effective action given by [23] 1 Z 1 ds z cosz 1 ð1Þ 2 LEM ¼ 2 þ z1 : exp½ism sinz 3 8 0 s3 (A38) The result is @Lð1Þ eB2 I1 ¼ i 82 EM þ ; @m2 3m2. (A39). which yields our final result ð2Þ VM ¼. . m 162. 2. rM 2. M. 82. @Lð1Þ eB2 2 EM þ : @m2 3m2. (A40). In order to produce the required results in the low (high)intensity field approximations, we list here the behavior of the relevant terms obtained from Ref. [23]. For B ! 1 we have e2 2 eB ð1Þ VEM ! B ln ; (A41) 2 m2 24. (A34). 025007-18. ð2Þ ! VEM. e4 eB 2 B ln ; m2 1284. (A42).