# Lorentzian Goldstone modes shared among photons and gravitons

## Abstract

It has long been known that photons and gravitons may appear as vector and tensor Goldstone modes caused by spontaneous Lorentz invariance violation (SLIV). Usually this approach is considered for photons and gravitons separately. We develop the emergent electrogravity theory consisting of the ordinary QED and the tensor-field gravity model which mimics the linearized general relativity in Minkowski spacetime. In this theory, Lorentz symmetry appears incorporated into higher global symmetries of the length-fixing constraints put on the vector and tensor fields involved, \(A_{\mu }^{2}=\pm M_{A}^{2}\) and \(H_{\mu \nu }^{2}=\pm M_{H}^{2}\) (\(M_{A}\) and \(M_{H}\) are the proposed symmetry breaking scales). We show that such a SLIV pattern being related to breaking of global symmetries underlying these constraints induces the massless Goldstone and pseudo-Goldstone modes shared by photon and graviton. While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry *SO*(1, 3) of the electrogravity Lagrangian, the tensor-field constraint itself possesses much higher global symmetry *SO*(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. Accordingly, while the photon may only contain true Goldstone modes, the graviton appears at least partially to be composed of pseudo-Goldstone modes rather than of pure Goldstone ones. When expressed in terms of these modes, the theory looks essentially nonlinear and contains a variety of Lorentz and CPT violating couplings. However, all SLIV effects turn out to be strictly cancelled in the lowest order processes considered in some detail. How this emergent electrogravity theory could be observationally different from conventional QED and GR theories is also briefly discussed.

## 1 Introduction

The extremely successful concept of the spontaneously broken internal symmetries in particle physics allows one to think that spontaneous violation of spacetime symmetries and, particularly, spontaneous Lorentz invariance violation (SLIV), could also provide some dynamical approach to quantum electrodynamics [1], gravity [2] and Yang–Mills theories [3] with photon, graviton and non-Abelian gauge fields appearing as massless Nambu–Goldstone (NG) bosons [4, 5] (for some later developments, see [6, 7, 8, 9, 10, 11, 12, 13, 14]). In this connection, we recently suggested [15, 16] an alternative approach to the emergent gravity theory in the framework of nonlinearly realized Lorentz symmetry for the underlying symmetric two-index tensor field in a theory, which mimics linearized general relativity in Minkowski space-time. It was shown that such a SLIV pattern, due to which a true vacuum in the theory is chosen, induces massless tensor Goldstone and pseudo-Goldstone modes, some of which can naturally be associated with the physical graviton.

*SO*(1, 3) formally breaks down to

*SO*(3) or

*SO*(1, 2) depending on the timelike (\(n^{2}>0\)) or spacelike (\(n^{2}<0\)) nature of SLIV. However, in sharp contrast to the nonlinear \(\sigma \) model for pions, the nonlinear QED theory, due to the starting gauge invariance involved, ensures that all the physical Lorentz violating effects turn out to be non-observable. It was shown [17], while only in the tree approximation and for the timelike SLIV (\(n^{2}>0\)), that the nonlinear constraint (1) implemented into the standard QED Lagrangian containing a charged fermion field \(\psi (x)\)

*S*-matrix remains unaltered under such a gauge convention. Really, this nonlinear QED contains a plethora of Lorentz and CPT violating couplings when it is expressed in terms of the pure emergent photon modes (\(a_{\mu }\)) according to the constraint condition (1)

Some time ago, this result was extended to the one-loop approximation and for both the timelike (\(n^{2}>0\)) and spacelike (\(n^{2}<0\)) Lorentz violation [19]. All the contributions to the photon–photon, photon–fermion and fermion–fermion interactions violating physical Lorentz invariance were shown to exactly cancel among themselves in the manner observed long ago by Nambu for the simplest tree-order diagrams. This means that the constraint (1), having been treated as a nonlinear gauge choice at the tree (classical) level, remains as a gauge condition when quantum effects are taken into account as well. So, in accordance with Nambu’s original conjecture, one can conclude that physical Lorentz invariance is left intact at least in the one-loop approximation, provided we consider the standard gauge invariant QED Lagrangian (3) taken in flat Minkowski space-time. Later this result was confirmed for the spontaneously broken massive QED [20], non-Abelian theories [21] and tensor-field gravity [15, 16]. The point is, however, that all these calculations represent somewhat “empirical” confirmation of gauge invariance of the nonlinear QED and other emergent theories rather than a theoretical one. Indeed, whether the constraint (1) amounts in general to a special gauge choice for a vector field is an open question unless the corresponding gauge function satisfying the constraint condition is explicitly constructed. We discuss this important issue in more detail in Sect. 4.

Usually, an emergent gauge field framework is considered either regarding emergent photons or regarding emergent gravitons. For the first time, we consider it regarding them both in the so-called electrogravity theory where together with the Nambu QED model [17] with its gauge invariant Lagrangian (3) we propose the linearized Einstein–Hilbert kinetic term for the tensor field preserving a diffeomorphism (diff) invariance. We show that such a combined SLIV pattern, conditioned by the constraints (1) and (7), induces the massless Goldstone modes which appear shared among photon and graviton. Note that one needs in common nine zero modes both for photon (three modes) and graviton (six modes) to provide all necessary (physical and auxiliary) degrees of freedom. They actually appear in our electrogravity theory due to spontaneous breaking of high symmetries of the constraints involved. While for the vector field case the symmetry of the constraint coincides with the Lorentz symmetry *SO*(1, 3), the tensor field constraint itself possesses a much higher global symmetry *SO*(7, 3) , whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. These modes are largely pseudo-Goldstone modes (PGMs) since *SO*(7, 3) is a symmetry of the constraint (7) rather than the electrogravity Lagrangian whose symmetry is only given by Lorentz invariance. The electrogravity theory we start with becomes essentially nonlinear, when expressed in terms of the Goldstone modes, and contains a variety of Lorentz (and CPT) violating couplings. However, as our calculations show, all SLIV effects turn out to be strictly cancelled in the low order physical processes involved once the tensor-field gravity part of the electrogravity theory is properly extended to general relativity (GR). This can be taken as an indication that in the electrogravity theory physical Lorentz invariance is preserved in this approximation. Thereby, the length-fixing constraints (1) and (7) put on the vector and tensor fields appear as the gauge fixing conditions rather than sources of the actual Lorentz violation just as it was in the pure nonlinear QED framework [17]. From this viewpoint, if this cancellation were to work in all orders, one could propose that emergent theories, like as the electrogravity theory, are not different from conventional gauge theories. We argue, however, that even in this case some observational difference between them could unavoidably appear, if gauge invariance were presumably broken by quantum gravity at the Planck scale order distances.

The paper is organized in the following way. In Sect. 2 we formulate the model for the tensor-field gravity and find corresponding massless Goldstone modes some of which are collected in the graviton. Then in Sect. 3 we consider in significant detail the combined electrogravity theory consisting of QED and tensor field gravity. In Sect. 4 we derive general Feynman rules for basic interactions in the emergent framework. The model appears to be in essence three-parametric containing the inverse Planck and SLIV scales, \(1/M_{P}\), \(1/M_{A}\) and \(1/M_{H},\) respectively, as the perturbation parameters, so that the SLIV interactions are always proportional to some powers of them. Further, some lowest order SLIV processes, such as an elastic photon–graviton scattering and photon–graviton conversion are considered in detail. We show that all these effects, taken in the tree approximation, appear in fact to be vanishing so that the physical Lorentz invariance is ultimately restored. Finally, in Sect. 5 we present our conclusion.

## 2 Tensor-field gravity

*H*-fields included.

^{1}Following the nonlinear \(\sigma \)-model for QED [17], we propose the SLIV condition (7) as some tensor field length-fixing constraint which is supposed to be substituted into the total Lagrangian \(\mathcal {L}(H,A)\) prior to the variation of the action. This eliminates, as was mentioned above, a massive Higgs mode in the final theory, thus leaving only massless Goldstone modes, some of which are then collected in a graviton.

*SO*(3) or

*SO*(1, 2), depending on the timelike (\(\mathfrak {n}^{2}>0\)) or spacelike (\( \mathfrak {n}^{2}<0\)) nature of SLIV. For the tensor-field constraint (7) the choice turns out to be wider. Indeed, this constraint can be written in the more explicit form

*SO*(1, 3) of the Lagrangian \(\mathcal {L}(H,A)\) given in (9) then formally breaks down at a scale \(M_{H}\) to one of its subgroups. If one assumes a “minimal” vacuum configuration in the

*SO*(1, 3) space with the VEV (15) developed on a single \(H_{\mu \nu }\) component, there are in fact the following three breaking channels:

^{2}Accordingly, there are only three Goldstone modes in the cases (

*a*,

*b*) and five modes in the cases (

*c*)–(

*d*). In order to associate at least one of the two transverse polarization states of the physical graviton with these modes, one could have any of the above-mentioned SLIV channels except for the case (

*a*) where the only nonzero Goldstone modes are given by the tensor components \(h_{0i}\) (\(i=1,2,3\)). Indeed, it is impossible for a graviton to have all vanishing spatial components, as in the case (

*a*). However, these components may be provided by some accompanying pseudo-Goldstone modes, as we argue below. Apart from the minimal VEV configuration, there are many others as well. A particular case of interest is that of the traceless VEV tensor \(\mathfrak {n}_{\mu \nu }\)

Aside from the pure Lorentz Goldstone modes, the question of the other components of the symmetric two-index tensor \(H_{\mu \nu }\) naturally arises. Remarkably, they turn out to be pseudo-Goldstone modes (PGMs) in the theory. Indeed, although we only propose Lorentz invariance of the Lagrangian \(\mathcal {L}(H,A)\), the SLIV constraint (7) formally possesses the much higher global accidental symmetry *SO*(7, 3) of the constrained bilinear form (14), which manifests itself when considering the \(H_{\mu \nu }\) components as the “vector” ones under *SO*(7, 3) . This symmetry is in fact spontaneously broken, side by side with Lorentz symmetry, at the scale \(M_{H}.\) Assuming again a minimal vacuum configuration in the *SO*(7, 3) space, with the VEV (15) developed on a single \(H_{\mu \nu }\) component, we have either timelike (\(SO(7,3) \rightarrow SO(6,3)\)) or spacelike (*SO*(7, 3) \(\rightarrow SO(7,2)\)) violations of the accidental symmetry depending on the sign of \(\mathfrak {n} ^{2}=\pm 1\) in (14). According to the number of broken *SO*(7, 3) generators, just nine massless Goldstone modes appear in both cases. Together with an effective Higgs component, on which the VEV is developed, they complete the whole ten-component symmetric tensor field \(H_{\mu \nu }\) of the basic Lorentz group as is presented in its parametrization (20). Some of them are true Goldstone modes of the spontaneous Lorentz violation, others are PGMs since, as was mentioned, an accidental *SO*(7, 3) symmetry is not shared by the whole Lagrangian \(\mathcal {L}(H,A)\) given in (9). Notably, in contrast to the scalar PGM case [18], they remain strictly massless being protected by the starting diff invariance which becomes exact when the tensor-field gravity Lagrangian (9) is properly extended to GR\(^{1}\). Owing to this invariance, some of the Lorentz Goldstone modes and PGMs can then be gauged away from the theory, as usual.

*SO*(7, 3) symmetry of the constraint (7), thus containing the Lorentz Goldstone modes and PGMs put together. If Lorentz symmetry is completely broken then the pure Goldstone modes appear enough to be solely collected in a physical graviton. On the other hand, when one has a partial Lorentz violation, some PGMs should be added.

*A*(and any other matter as well) thus leading to an unacceptably large Lorentz violation if the SLIV scale \( M_{H}\) were comparable with the Planck mass \(M_{P}\). However, this term can be gauged away [15, 16] by an appropriate redefinition of the vector field by going to the new coordinates

^{3}Usually, the spin 1 states (and one of the spin 0 states) are excluded by the conventional Hilbert–Lorentz condition,

*q*is an arbitrary constant, giving for \(q=-1/2\) the standard harmonic gauge condition). However, as we have already imposed the emergent constraint (20), we cannot use the full Hilbert–Lorentz condition (27) eliminating four more degrees of freedom in \(h_{\mu \nu }.\) Otherwise, we would have an “over-gauged” theory with a non-propagating graviton. In fact, the simplest set of conditions which conform with the emergent condition \(\mathfrak {n}\cdot h=0\) in (20) turns out to be

^{4}in \(h_{\mu \nu }\) and, besides, it automatically satisfies the Hilbert–Lorentz spin condition as well. So, with the Lagrangian (22) and the supplementary conditions (20) and (28) lumped together, one eventually arrives at a working model for the emergent tensor-field gravity [15, 16]. Generally, from ten components of the symmetric two-index tensor \(h_{\mu \nu }\) four components are excluded by the supplementary conditions (20) and (28). For a plane gravitational wave propagating in, say, the

*z*direction another four components are also eliminated, due to the fact that the above supplementary conditions still leave freedom in the choice of a coordinate system, \(x^{\mu }\rightarrow \, x^{\mu }+\xi ^{\mu }(t-z/c),\) much as in standard GR. Depending on the form of the VEV tensor \(\mathfrak {n}_{\mu \nu }\), caused by SLIV, the two remaining transverse modes of the physical graviton may consist solely of Lorentzian Goldstone modes or of pseudo-Goldstone modes, or include both of them. This theory, similar to the nonlinear QED [17], while suggesting an emergent description for the graviton, does not lead to physical Lorentz violation [15, 16].

## 3 Electrogravity theory

### 3.1 Emergent photons and gravitons together

### 3.2 Constraints and zero mode spectrum

*SO*(7, 3) symmetry, respectively, must be solely a function of \( A_{\mu }^{2}\equiv A_{\mu }A^{\mu }\) and \(H_{\mu \nu }^{2}\equiv H_{\mu \nu }H^{\mu \nu }\). Indeed, it cannot include any contracted and intersecting terms like \(H_\mathrm{tr}\), \(H^{\mu \nu }A_{\mu }A_{\nu }\) and others, which would immediately reduce the above symmetries to the common Lorentz one. So, one may only write

*SO*(7, 3) of the constraint (7), there are much more tensor zero modes than would appear from SLIV itself. In fact, they complete the whole tensor multiplet \(h_{\mu \nu }\) in the parametrization (20 ). However, as was discussed in the previous section, only a part of them are true Goldstone modes; the others are pseudo-Goldstone ones. In the minimal VEV configuration case, when these VEVs are developed only on the single \( A_{\mu }\) and \(H_{\mu \nu }\) components, one has several possibilities determined by the vacuum orientations \(n_{\mu }\) and \(\mathfrak {n}_{\mu \nu } \) in Eqs. (37), (16), (17) and (18), respectively. There appear 12 zero modes in total, three from Lorentz violation itself and nine from a violation of the

*SO*(7, 3) symmetry, which is more than enough to have the necessary three photon modes (two physical and one auxiliary ones) and six graviton modes (two physical and four auxiliary ones). We list below all interesting cases classifying them according to the corresponding \(n-\mathfrak {n}\) values.

(1) For the timelike–timelike SLIV, when both \(n_{0}\ne 0\) and \(\mathfrak {n} _{00}\ne 0\), the photon is determined by the space Goldstone components \(a_{i}\) (\(i=1,2,3\)) of the partially broken Lorentz symmetry \(SO(1,3)\rightarrow SO(3)\), while the space–space components \(h_{ij}\) needed for physical graviton and its auxiliary components can be only provided by the pseudo-Goldstone modes following from the timelike symmetry breaking *SO*(7, 3) \(\rightarrow SO(6,3)\) related to the tensor-field constraint (7).

(2) Another interesting case seems to be the timelike–spacelike SLIV, when \( n_{0}\ne 0\) and \(\mathfrak {n}_{i=j}\ne 0\) (one of the diagonal space components of the unit tensor \(\mathfrak {n}_{\mu \nu }\) is nonzero). Now, Lorentz symmetry is broken up to the plane rotations \(SO(1,3)\rightarrow SO(2),\) so that the five true Goldstone bosons appear shared among photon and graviton in the following way. The photon is given again by three space components \(a_{i}\), while the graviton is determined by two space–space components, \(h_{12}\) and \(h_{13}\) (if the VEV was developed along the direction \(\mathfrak {n}_{11}\)), as directly follows from the parametrization Eqs. (37) and (18). Thus, again one necessary component \( h_{23}\) for physical graviton, as well as its gauge degrees of freedom, should be provided by the proper pseudo-Goldstone modes following from the spacelike symmetry breaking *SO*(7, 3) \(\rightarrow SO(7,2)\) related to the tensor-field constraint (7).

(3) For the similar timelike–spacelike SLIV case, when \(n_{0}\ne 0\) and \( \mathfrak {n}_{i\ne j}\ne 0\) (one of the nondiagonal space components of the unit tensor \(\mathfrak {n}_{\mu \nu }\) is nonzero), the Lorentz symmetry appears to be fully broken so that the photon has the same three space components \( a_{i}\), while the graviton physical components are given by the tensor field space components \(h_{ij}\). This is the only case when all physical components of both photon and graviton are provided by the true SLIV Goldstone modes, whereas some gauge degrees of freedom for a graviton are given by the PGM states stemming from the spacelike symmetry breaking *SO*(7, 3) \(\rightarrow SO(7,1)\) related to the tensor field constraint (7).

(4) Using the parametrization equations (37) and (18) one can readily consider all other possibilities as well; particularly, the spacelike–timelike (nonzero \(n_{i}\) and \(\mathfrak {n}_{00}\)), spacelike–spacelike diagonal (nonzero \(n_{i}\) and \(\mathfrak {n}_{i=j}\)) and spacelike–spacelike nondiagonal (nonzero \(n_{i}\) and \(\mathfrak {n}_{i\ne j}\) ) cases. In all these cases, while the photon may only contain true Goldstone modes, some pseudo-Goldstone modes appear to be necessary so as to be collected in the graviton together with some true Goldstone modes.

### 3.3 Emergent electrogravity interactions

## 4 The lowest order SLIV processes

### 4.1 Preamble

The emergent gravity Lagrangian in (22) taken alone or considered together with the material vector and scalar fields presents in fact a highly nonlinear theory which contains lots of Lorentz and CPT violating couplings. Nevertheless, as shown in [15, 16] in the lowest order calculations, they all are cancelled and do not manifest themselves in physical processes. This may mean that the length-fixing constraints (7) put on the tensor fields appear as gauge fixing conditions rather than a source of an actual Lorentz violation.

*S*(

*x*) is an action of a system, while

*e*and

*m*stand for the particle charge and mass, respectively. Comparison of Eqs. (50 ) and (51) shows the correspondence \(\omega (x)=S(x)/e\) and \( n^{2}M_{A}^{2}=m^{2}/e^{2}\). Thus, the constraint equation (50) should have a solution inasmuch as there is a solution to the classical problem described by Eq. (51). This conclusion was actually confirmed by Nambu for the timelike SLIV (\(n^{2}=+1\)) in the lowest order calculation of the physical processes in [17] and then was extended to the one-loop approximation and for both the timelike (\(n^{2}>0\)) and spacelike (\(n^{2}<0\)) Lorentz violation in [19]. Thus, the status of the constraint (1) as a special gauge choice in QED is only partially confirmed by some low order calculations rather than having a serious theoretical reason.

The present electrogravity theory, in contrast to the pure QED and tensor-field gravity theories, contains both the photon and the graviton as the emergent gauge fields. This adds new variety of Lorentz and CPT violating couplings (47), being expressed in terms of tensor and vector Goldstone modes. In general, one cannot be sure that, even though both the emergent QED and the tensor-field gravity taken separately preserve Lorentz invariance (in the low order processes), the combined electrogravity theory does not lead to physical Lorentz violation as well. However, as shown by our calculations given below, just this appears to be the case. All Lorentz violation effects turn out again to be strictly cancelled among themselves at least in the lowest order SLIV processes in the electrogravity theory. Thus, similar to emergent vector field theories, both Abelian [17, 19, 20] and non-Abelian [21], as well as in the pure tensor-field gravity [15, 16], such a cancellation may only mean that at least in the lowest approximation the SLIV constraints (1, 7) amount to a special gauge choice in the otherwise diff and Lorentz invariant emergent electrogravity theory presented here.

We will consider the lowest order SLIV processes, once the corresponding Feynman rules are properly established. For simplicity, both in the above Lagrangians and in forthcoming calculations, we continue to use the tracelessness of the VEV tensor \(\mathfrak {n}_{\mu \nu }\) (19), while our results remain true for any type of vacuum configuration caused by SLIV.

### 4.2 Feynman rules

Though the Feynman rules and processes related to the nonlinear QED, as well as with emergent gravity with the matter scalar fields, are thoroughly discussed in our previous works [15, 20], there are many new Lorentz and CPT breaking interactions in the total interaction Lagrangian (47). We present below some basic Feynman rules which are needed for calculations of different SLIV processes just appearing in the emergent electrogravity.

**i**) The first and most important is the graviton propagator which only conforms with the emergent gravity Lagrangian (22) and the gauge conditions (20) and (28),

**ii**) Next is the three-graviton vertex

*hhh*, again from the Lagrangian (22), with graviton polarization tensors (and 4-momenta) given by \( \epsilon ^{\alpha \alpha ^{\prime }}(k_{1}),\) \(\epsilon ^{\beta \beta ^{\prime }}(k_{2})\) and \(\epsilon ^{\gamma \gamma ^{\prime }}(k_{3})\) we have

**(iii)**Next, we address the contact tensor–tensor–vector–vector interaction coupling

*hhaa*coming from the Lagrangian \(\mathcal {L}_{2}\) in (48). However, it would be useful to give first the standard tensor–vector–vector vertex

*haa*with tensor and vector field polarizations, \(\epsilon ^{\alpha \alpha ^{\prime }}\) and \(\xi ^{\mu ,\nu }\), respectively,

**iv**) We have also to derive the four-linear tensor–vector interaction vertex

*haaa*coming from the Lagrangian \(\mathcal {L}_{1}\) in (48). Note that the last term in it which is proportional to \(h_\mathrm{tr}\) will not contribute in the processes with graviton on external lines, since its polarization tensor is traceless. For the other terms one has the vertex

**v**) For the three-vector Goldstone mode interaction

*aaa*we have the well-known vertex [20] following for the pure vector field Lagrangian (32),

**vi**) And finally, let us give also the vector–scalar–scalar interaction \(a\varphi \varphi ^{*}\) stemming from the same Lagrangian \( \mathcal {L}_{3}\),

These are rules that are actually needed to calculate the lowest order SLIV processes mentioned above. Note also that some of these processes could in principle appear in the pure nonlinear QED [20] or in the nonlinear tensor-field gravity [15, 16] where, as is well known, all the physical Lorentz violation effects are eventually vanishing. Therefore, we consider the SLIV contributions which only appear in the combined nonlinear vector–tensor electrogravity theory presented here.

### 4.3 Elastic photon–graviton scattering

*hhaa*vertex (60),

*haa*(57) vertices,

*q*is the momentum of the propagating graviton).

### 4.4 Photon–graviton conversion

This SLIV process \(\gamma +g\rightarrow \gamma +\gamma \) appears in the order of \(1/M_{A}M_{P}\) (now, due to the emergent nature of photon). Again, this process in the tree approximation is basically related to the interplay between the contact and pole diagrams.

*haaa*diagram being determined by the interaction vertex (61) has a matrix element

*p*and \(k_{1}\) are incoming and \( k_{2}\) and \(k_{3}\) outgoing momenta).

*haa*(57) vertices consist in fact of three diagrams differing from each other by the interchangeable external photon legs. Their total matrix element is

*q*is the propagating momentum, while the momenta of the graviton and photons,

*p*and \(k_{1,2,3}\), refer to the polarizations \(\epsilon _{\alpha \alpha ^{\prime }}\) and \(\xi _{1,2,3}\).

Using again the orthogonality properties and mass shell conditions for polarizations of the photons and graviton one can split the contact amplitude (69) into three terms which exactly cancel the corresponding terms in the pole amplitude (70). So, we will not have any physical Lorentz violation in this process as well.^{5}

### 4.5 Elastic photon–scalar scattering

^{6}Again, there are contact and pole diagrams for this process which cancel each other. The contact diagram corresponds to the vertex \(a\varphi \varphi ^{*}\) (64) appearing from the Lagrangian \(\mathcal {L}_{3}\) in (48) and leads to the matrix element

*aaa*vertex (63) and the standard scalar field current (44) in \(\mathcal {L}_{3}\) one has, using the mass shell properties of the vector field polarization,

### 4.6 Other processes

Many other tree level Lorentz violating processes related to gravitons and vector fields (interacting with each other and the matter scalar field in the theory) appear in higher orders in the basic SLIV parameters \(1/M_{H}\) and \(1/M_{A}\), by iteration of the couplings presented in our basic Lagrangians (22) and (47) or from further expansions of the effective vector and tensor-field Higgs modes (5) and (21) inserted into the starting total Lagrangian (30). Again, their amplitudes are essentially determined by an interrelation between the longitudinal graviton and photon exchange diagrams and the corresponding contact interaction diagrams, which appear to cancel each other, thus eliminating physical Lorentz violation in the theory.

Most likely, the same conclusion could be expected for SLIV loop contributions as well. Actually, as in the massless QED case considered earlier [19], the corresponding one-loop matrix elements in our emergent electrogravity theory could either vanish by themselves or amount to the differences between pairs of similar integrals whose integration variables are shifted relative to each other by some constants (being in general arbitrary functions of the external 4-momenta of the particles involved) which, in the framework of dimensional regularization, could lead to their total cancellation.

So, the emergent electrogravity theory considered here is likely to eventually possess physical Lorentz invariance provided that the underlying gauge and diff invariance in the theory remains unbroken.

## 5 Conclusion

We have developed an emergent electrogravity theory consisting of the ordinary QED and the tensor-field gravity model (which mimics the linearized general relativity in Minkowski spacetime) where both photons and gravitons emerge as states solely consisting of massless Goldstone and pseudo-Goldstone modes. This appears due to spontaneous violation of Lorentz symmetry incorporated into global symmetries of the length-fixing constraints put on the starting vector and tensor fields, \(A_{\mu }^{2}=\pm M_{A}^{2}\) and \(H_{\mu \nu }^{2}=\pm M_{H}^{2}\) (\(M_{A}\) and \(M_{H}\) are the proposed symmetry breaking scales). While for the vector field case the symmetry of the constraint coincides with Lorentz symmetry *SO*(1, 3) of the electrogravity Lagrangian, the tensor field constraint itself possesses the much higher global symmetry *SO*(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. Accordingly, while the photon may only contain true Goldstone modes, the graviton appears at least partially composed from pseudo-Goldstone modes rather than from pure Goldstone ones. Thereby, the SLIV pattern related to breaking of the constraint symmetries, due to which the true vacuum in the theory is chosen, induces a variety of zero modes shared among photon and graviton.

This theory looks essentially nonlinear and contains a variety of Lorentz and CPT violating couplings, when expressed in terms of the pure tensor Goldstone modes. Nonetheless, all the SLIV effects turn out to be strictly cancelled in the lowest order processes considered. This can be taken as an indication that in the electrogravity theory physical Lorentz invariance is preserved in this approximation. Thereby, the length-fixing constraints (1) and (7) put on the vector and tensor fields appear as gauge fixing conditions rather than sources of the actual Lorentz violation in the gauge and diff invariant Lagrangian (30) we started with. In fact, some Lorentz violation through deformed dispersion relations for the material fields involved would appear in the interaction sector (34), which only possesses an approximate diff invariance. However, a proper extension of the tensor-field theory to GR, with its exact diff invariance, ultimately restores the normal dispersion relations and, therefore, the SLIV effects are cancelled at least in the lowest order considered. If this cancellation were to work in all orders, one could propose that emergent theories, like as the electrogravity theory, are not differed from conventional gauge theories. Accordingly, spontaneous Lorentz violation caused by the vector and tensor-field constraints (1) and (7 ) appear hidden in the gauge degrees of freedom, and only results in a noncovariant gauge choice in an otherwise gauge invariant emergent electrogravity theory.

From this standpoint, the only way for physical Lorentz violation to occur would be if the above gauge invariance were slightly broken at distances of the order of the Planck scale, which could be presumably caused by quantum gravity. This is in fact a place where the emergent vector- and tensor-field theories may drastically differ from conventional QED, Yang–Mills and GR theories where gauge symmetry breaking could hardly induce physical Lorentz violation. In contrast, in emergent electrogravity such breaking could readily lead to many violation effects including deformed dispersion relations for all matter fields involved. Another basic distinction of emergent theories with non-exact gauge invariance is a possible origin of a mass for graviton and other gauge fields (namely, for the non-Abelian ones, see [21]), if they, in contrast to the photon, are partially composed of pseudo-Goldstone modes rather than of pure Goldstone ones. Indeed, these PGMs are no longer protected by gauge invariance and may properly acquire tiny masses, which still do not contradict experiment. This may lead to a massive gravity theory where the graviton mass emerges dynamically, thus avoiding the notorious discontinuity problem [24].

So, while emergent theories with an exact local invariance are physically indistinguishable from conventional gauge theories, there are some principal distinctions when this local symmetry is slightly broken, which could eventually allow us to differentiate between the two types of theory in an observational way. We may return to a more detailed consideration of this interesting point elsewhere.

## Footnotes

- 1.Such an extension means that in all terms included in the GR action, particularly in the QED Lagrangian term, \((-g)^{1/2}g_{\mu \nu }g_{\lambda \rho }F^{\mu \lambda }F^{\nu \rho }\), one expands the metric tensorstaking into account the higher terms in$$\begin{aligned} g_{\mu \nu }=\eta _{\mu \nu }+H_{\mu \nu }/M_{P},\quad g^{\mu \nu }=\eta ^{\mu \nu }-H^{\mu \nu }/M_{P}+H^{\mu \lambda }H_{\lambda }^{\nu }/M_{P}^{2}+\cdot \cdot \cdot \end{aligned}$$
*H*-fields. - 2.One may alternatively argue starting from the vector representation of the higher
*SO*(7, 3)symmetry determined by the constraint equation (14) itself (see below). Thereby, one has a standard parametrizationwhere the “big” indices$$\begin{aligned} H_{A}=\left[ e^{i\mathfrak {n}_{M}\mathcal {J}^{MN}h_{N}/M_{H}}\right] _{A}^{B} \mathfrak {n}_{B}M_{H} \end{aligned}$$*A*,*B*,*M*,*N*correspond to the pairs of different values of the old indices (\(\mu \nu \)) appearing in (14). Consequently, one has the equality \(\mathfrak {n}_{M}\mathcal {J}^{MN}h_{N}=\mathfrak {n}_{\mu \nu }\mathcal {J}^{\mu \sigma }\eta ^{\nu \tau }h_{\sigma \tau }\) when going to the standard Lorentz indices so that antisymmetry in the indices (*M*,*N*) goes to antisymmetry in the index pairs (\(\mu \nu ,\sigma \tau \)). - 3.
Indeed, the spin-1 component must be necessarily excluded in the tensor \( h_{\mu \nu }\), since the sign of the energy for the spin-1 component is always opposite to that for the spin-2 and spin-0 ones.

- 4.
The solution for a gauge function \(\xi _{\mu }(x)\) satisfying the condition (28) can generally be chosen as \(\xi _{\mu }=\) \(\ \square ^{-1}(\partial ^{\rho }h_{\mu \rho })+\partial _{\mu }\theta \), where \( \theta (x)\) is an arbitrary scalar function, so that only three degrees of freedom in \(h_{\mu \nu }\) are actually eliminated.

- 5.
Note that together with the pure QED \(a^{3}\) vertex (62) we could also use the new \(a^{3}\) vertex (63) in the above pole diagrams. This would give some new contribution into this process with the lesser order \( M_{H}/M_{A}M_{P}^{2}\). One may expect, however, that such a contribution will be cancelled by the corresponding contact term appearing in the same order when going to GR (see the footnote\(^{1}\)).

- 6.

## Notes

### Acknowledgements

We would like to thank Colin Froggatt, Archil Kobakhidze, Rabi Mohapatra and Holger Nielsen for useful discussions and comments. Z.K. acknowledges financial support from Shota Rustaveli National Science Foundation (Grant # YS-2016-81).

## References

- 1.J.D. Bjorken, Ann. Phys. (N.Y.)
**24**, 174 (1963)Google Scholar - 2.P.R. Phillips, Phys. Rev.
**146**, 966 (1966)ADSCrossRefGoogle Scholar - 3.T. Eguchi, Phys. Rev. D
**14**, 2755 (1976)ADSCrossRefGoogle Scholar - 4.Y. Nambu, G. Jona-Lasinio, Phys. Rev.
**122**, 345 (1961)ADSCrossRefGoogle Scholar - 5.J. Goldstone, Nuovo Cimento
**19**, 154 (1961)MathSciNetCrossRefGoogle Scholar - 6.J.L. Chkareuli, C.D. Froggatt, H.B. Nielsen, Phys. Rev. Lett.
**87**, 091601 (2001)Google Scholar - 7.J.L. Chkareuli, C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B
**609**, 46 (2001)Google Scholar - 8.J.D. Bjorken. arXiv:hep-th/0111196
- 9.Per Kraus, E.T. Tomboulis. Phys. Rev. D
**66**, 045015 (2002)Google Scholar - 10.A. Jenkins, Phys. Rev. D
**69**, 105007 (2004)ADSCrossRefGoogle Scholar - 11.V.A. Kostelecky, Phys. Rev. D
**69**, 105009 (2004)ADSCrossRefGoogle Scholar - 12.Z. Berezhiani, O.V. Kancheli. arXiv:0808.3181
- 13.V.A. Kostelecky, R. Potting, Phys. Rev. D
**79**, 065018 (2009)ADSMathSciNetCrossRefGoogle Scholar - 14.S.M. Carroll, H. Tam, I.K. Wehus, Phys. Rev. D
**80**, 025020 (2009)ADSCrossRefGoogle Scholar - 15.J.L. Chkareuli, J.G. Jejelava, G. Tatishvili, Phys. Lett. B
**696**, 126 (2011)ADSCrossRefGoogle Scholar - 16.J.L. Chkareuli, C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B
**848**, 498 (2011)ADSCrossRefGoogle Scholar - 17.Y. Nambu, Progr. Theor. Phys. Suppl. Extra
**190**(1968)Google Scholar - 18.S. Weinberg,
*The Quantum Theory of Fields,*v.2, Cambridge University Press (2000)Google Scholar - 19.A.T. Azatov, J.L. Chkareuli, Phys. Rev. D
**73**, 065026 (2006)ADSCrossRefGoogle Scholar - 20.J.L. Chkareuli, Z.R. Kepuladze, Phys. Lett. B
**644**, 212 (2007)ADSCrossRefGoogle Scholar - 21.J.L. Chkareuli, J.G. Jejelava, Phys. Lett. B
**659**, 754 (2008)ADSCrossRefGoogle Scholar - 22.V.A. Kostelecky, S. Samuel, Phys. Rev. D
**40**, 1886 (1989)ADSCrossRefGoogle Scholar - 23.R. Bluhm, N.L. Cage, R. Potting, A. Vrublevskis, Phys. Rev. D
**77**, 125007 (2008)ADSCrossRefGoogle Scholar - 24.H. van Dam, M.J.G. Veltman, Nucl. Phys. B
**22**, 397 (1970)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}