Gauge Fields as Constrained Composite Bosons

We reconsider a scenario in which photons and other gauge fields appear as the composite vector bosons made of the fermion pairs that may happen with or without spontaneous violation of Lorentz invariance. The class of composite models for emergent gauge fields is proposed, where these fields are required to be restricted by some nonlinear covariant constraint, $ A_{\mu }^{2}=M^{2}$. Such a constraint may only appear if the corresponding fermion currents in the prototype model, being invariant under some global internal symmetry $G$, are properly constrained as well. In contrast to the conventional approach, the composite bosons emerged in this way appear naturally massless, the global symmetry $G$ in the model turns into the local symmetry $G_{loc}$, while the vector field constraint reveals itself as the gauge fixing condition. Finally, we consider the case when the constituent fermions generating emergent gauge bosons could be at the same time the preons composing the known quark-lepton species in the Standard Model and Grand Unified Theories.


Introduction
One can think that local gauge invariance, contrary to a global symmetry case, may look like a cumbersome geometrical input rather than a true physical principle, especially in the framework of an effective quantum field theory becoming, presumably, irrelevant at very high energies. In this connection, one could wonder whether there is any basic dynamical reason that necessitates gauge invariance and an associated masslessness of gauge fields as some emergent phenomenon arising from a more profound level of dynamics related to their truly elementary constituents. By analogy with a dynamical origin of massless scalar particle excitations, which is very well understood in terms of spontaneously broken global internal symmetries [1], one could think that the origin of composite massless gauge fields as the vector Nambu-Goldstone (NG) bosons are presumably related to spontaneous violation of Lorentz invariance which is in fact the minimal spacetime global symmetry underlying particle physics. This well-known approach which might in principle provide a viable alternative to quantum electrodynamics [2], gravity [3] and Yang-Mills theories [4,5] has a long history started over fifty years ago.
This scenario proposes that a given current-current interaction of basic constituent fermions induces some almost gauge invariant effective theory which, apart from the invariant kinetic terms, contain the nontrivial vector-field potential terms. These terms, no matter whether they exist at the tree level or appear through radiative corrections, may lead in principle to spontaneous Lorentz invariance violation (SLIV) according to which some of components of the emergent composite vector bosons can be viewed as the corresponding NG zero modes. It is worth pointing out that one can generally talk about SLIV regardless of whether it is observable or not. In both of cases, the corresponding zero modes associated with gauge fields are necessarily generated. Indeed, in the gauge invariance limit this violation may normally be hidden in gauge degrees of freedom of emergent vector fields. However, when these superfluous degrees are eliminated by gauge symmetry breaking, SLIV becomes observable. In this connection, we will use later on the terms "inactive SLIV" and "active SLIV", respectively, in order to distinguish these two cases.
The important point, however, is that the potential-based vector field models which can only lead to the active or physical SLIV appear to be generically unstable and contradictory and, therefore, seems to be hardly acceptable in their present form, as we will argue below. On the other hand, why should one necessarily insist on physical Lorentz violation, if emergent gauge fields are anyway generated through the "safe" inactive SLIV models which recover a conventional Lorentz invariance?
In the light of this reasoning, we propose here an alternative approach for composite gauge bosons replacing the potential-based vector field models by the constraint-based ones which we briefly sketch below in Sections 2 and 3, respectively. Later in Section 4, we show how such constrained gauge bosons are induced by the properly constrained currents of the constituent fermions involved. In contrast to the potential-based approach, the composite gauge bosons appear now to be naturally massless, though being restricted by some nonlinear constraint which is then treated as their nonlinear gauge. So, the starting global symmetry of constituent fermions G turns into the local internal symmetry G loc of the gauge and matter fields involved. Further, in Section 5, we discuss a possible scenario when the constituent fermions generating emergent gauge bosons could be at the same time the preons composing the known quark-lepton species in the Standard Model and Grand Unified Theories. And, finally, our conclusion is provided in Section 6, where we discuss why basically the proposed constraint pattern for vector fields could make the whole physical field system involved to adjust itself in a gauge invariant way. Otherwise, as is argued, it could loose too many degrees of freedom, thus getting unphysical.

Models with vector field potential
For an emergence of the potential-based vector field models one can start with the generalized prototype Lagrangian with all possible multi-fermi current-current interactions [6] rather than with the original four-fermi one [2] where the fermion set includes N constituent fermion species ψ s (hereafter, summation over all repeated indices is implied). The Lagrangian L(ψ) possesses the U (N ) global flavor symmetry under which the above all-fermion current j µ is invariant in itself. This model is evidently non-renormalizable and can only be considered as an effective theory valid at sufficiently low energies. The dimensionful couplings G p are proportional to appropriate powers of the UV cutoff Λ being ultimately related to some energy scale up to which this effective theory is valid, G p ∼ Λ 4−6p . Factors of N in (1) are chosen in such a way to provide a well defined large N limit. The Lagrangian (1) can be re-written using the standard trick of introducing an aux- The potential V is a power series in A µ A µ that can generally be written in the form with the coefficients λ p and mass parameter M chosen such that by solving the algebraic equations of motion for A µ and substituting back into (3) one recovers the starting Lagrangian (1). If instead one integrates out the fermions ψ s , one gets an effective action in terms of the properly renormalized composite A µ field which acquires its own dynamics Since the fermions ψ s are minimally coupled to the vector field A µ in (3), its kinetic term generated in this way appears gauge invariant provided that a gauge invariant cutoff is chosen. Furthermore, since there are N species of fermions ψ s the effective action (5) has an overall factor of N . And the last point is that one has to introduce in the basic Lagrangian (1), apart from the pure constituent fermions ψ s , some actual matter fields Ψ t interacting through their own conserved current J µ This will generate in turn the standard gauge invariant matter coupling given in the action (5). Indeed, in contrast to the heavy fermions ψ s whose masses are normally proposed to be of the cutoff order, the matter fermions Ψ t can be taken to be light and even massless. This allows to keep them in the low energy effective action, whereas the heavy fermions ψ s are integrated out. They are in some sense hidden ones which are solely needed to properly induce the composite vector bosons as appropriate gauge fields. Note that, while their number N is in fact somewhat technical one allowing to properly suppress the high-order dangerous terms, N Ψ is a number of the actually observed similar matter fermion species, say, those being among known quarks and leptons. At the first glance, everything looks good in the effective action (5). However, some generic problems related to SLIV and stability of the emerged theory may necessarily appear. Let us consider at the beginning only the first non-trivial term in the multifermi interactions (1) and, respectively, in the potential V (5) that just corresponds to the original Bjorken model [2]. As matter of fact, this model naturally leads to the massive QED theory rather than to the conventional massless QED one. Indeed, any conclusion that such a massive vector boson might be condensed due to its radiatively produced quartic term [2] or even be massless in itself through a strict cancellation of its tree-level and radiative masses [4], seems to be rather problematic being related to somewhat formal manipulations with divergent integrals. The same could be said about the non-Abelian symmetry case as well 1 .
It might seem that the above problem would be over when the higher-order terms beyond the four-fermi interaction are activated in the basic fermion Lagrangian (1) and, respectively, in the potential (4). As is readily seen from (4), the next term in it really gives the quartic A µ field term in the effective action S ef f (5). This is enough to generate the familiar Mexican hat structure of the potential V(A µ A µ ), thus coming to spontaneous Lorentz violation at a scale determined by the mass parameter |M |. Rewriting the action (5) in terms of the renormalized A µ field and leaving in the potential only bilinear and quartic vector field terms one gets the effective theory being sometimes referred to as the "bumblebee" model [7]. This partially gauge invariant model means in fact that the vector field A µ develops a constant background value and Lorentz symmetry SO (1,3) breaks down to SO(3) or SO(1, 2) depending on whether the M 2 is positive or negative, respectively. In both of cases, there are three zero massless modes and a heavy Higgs mode in the symmetry broken phase.
The point is, however, that not only this bumblebee-like model but all possible potentialbased models appear generally unstable. Their Hamiltonians (as was argued specifically for the bumblebee case [8] but those arguments are applicable to a general V potential as well) are not bounded from below beyond the constrained phase space determined by the nonlinear condition put on the vector field, as can be readily shown using Dirac constraint analysis [9]. With this condition imposed, the massive Higgs mode never appears, the Hamiltonian turns to be positive, and the model is physically equivalent to the nonlinear constraint-based QED, which we consider in the next section. Note also that, apart from the instability, these models, in contrast to some viable effective field theories, do not posses a consistent ultraviolet completion [10]. This makes their 2 application rather problematic in a sense that one can not draw relevant conclusions about the strength of physical effects in such an effective theory.
And one more argument against the potential-based models may follow from their supersymmetric extension. As one can readily confirm, SUSY may only admit the bilinear mass term in the vector field potential energy (see [11] for more details). As a result, without the stabilizing quartic (and higher order) vector field terms, this type of spontaneous Lorentz violation can in no way be realized in the SUSY context. The same could be said about the prototype Lagrangian with the multi-fermi current-current interactions (3) which can not be constructed from any matter chiral superfields.
All that means that the above-mentioned composite models leading eventually to the vector field potential V(A µ A µ ), whether it contains only their mass terms or the higher order terms as well, seems to be hardly acceptable in the present form. One might only expect that, since a mass scale |M | of SLIV is normally presumed to be near the Planck scale, the quantum gravity theory would make the ultimate conclusion on physical viability of such models.

Models with vector field constraint
We give here some brief sketch of the constraint-based models for vector fields considering first the constraint-based QED model. This model starts directly with the nonlinearly realized Lorentz symmetry for underlying vector field A µ (x) through the "length-fixing" constraint (7) implemented into a conventional QED. The constraint-based models were first studied by Nambu a long ago [12], and in more detail in recent years [13,15,16,14,17,18].
The constraint (7) is in fact very similar to the constraint appearing in the nonlinear σ-model for pions [19], σ 2 + π 2 = f 2 π , where f π is the pion decay constant. Rather than impose by postulate, the constraint (7) may be implemented into the standard QED Lagrangian (for some charged matter fields Ψ t ) through the invariant Lagrange multiplier term according to which the variation under the multiplier field λ(x) leads to the vector field constraint (7). On the other hand, the λ field is acting as an extra source of charge density. Actually, the Lagrange multiplier term in (8) corresponds to interaction of vector field with the conserved extra source current once proposed by Dirac [20] for an alternative introduction of classical electric charge. This theory goes to the conventional electrodynamics in the vanishing λ limit that one takes it having first used the constraint itself 3 . One way or another, the constraint (7) means in essence that the vector field A µ develops the VEV along the direction given by the unit Lorentz vector n µ causing again an appropriate time-like (M 2 > 0) or space-like (M 2 < 0) Lorentz violation 4 at a scale |M |. The point is, however, that in sharp contrast to the nonlinear σ model for pions, the constrained QED theory (8) ensures that all the physical Lorentz violating effects turn out to be non-observable. Actually, as was shown in the tree [12] and oneloop approximations [13], the nonlinear constraint (7) implemented as a supplementary condition appears in essence as a possible gauge choice for the vector field A µ , while the S-matrix remains unaltered under such a gauge convention. Indeed, the SLIV inspired by the nonlinear constraint (7) produces an ordinary photon as a true Goldstone vector boson (a µ ) being orthogonal to the vacuum direction given by n µ However, as mentioned, it leaves physical Lorentz invariance intact. Similar result was also confirmed for spontaneously broken non-Abelian theories [15,16] and tensor field gravity [17]. Let us describe in some detail just the non-Abelian symmetry case since we are going to consider later the Yang-Mills theories with composite gauge fields. Let us assume there 3 Note that an arbitrary λ field may generally cause instability of the theory making its Hamiltonian negative [8]. For this reason, we propose that initial values for all fields (and their momenta) involved are chosen so as to restrict their phase space to values providing the infinitely small or vanishing multiplier function λ(x). Fortunately, due to an automatic conservation of the matter and extra source currents, Jµ (6) and J ext µ (9), such a type of λ field will then remain for all time. 4 Let us note that there could also be the light-like Lorentz violation [21] appearing through the zero "length-fixing" constraint put on a gauge vector field, AµA µ = 0. In that case the Lorentz invariance breaks at some arbitrary scale |M ′ | to the so-called Euclidean E(2) symmetry group that leads again to an emergence of the zero NG modes collected in physical photon.
is such a theory possessing some internal symmetry G having D generators so that the nonlinear constraint for the vector field multiplet has now the form Remarkably, this time not only the pure Lorentz symmetry SO(1, 3), but the much larger accidental symmetry SO(D, 3D) of the SLIV constraint (12) also happens to be spontaneously broken. As a result, although the pure Lorentz violation still generates only one true Goldstone vector boson, the accompanying pseudo-Goldstone vector bosons related to the SO(D, 3D) breaking also come into play properly completing the whole gauge multiplet of the internal symmetry group G taken. In contrast to the known scalar pseudo-Goldstone modes, they remain strictly massless, being protected by the simultaneously generated non-Abelian gauge invariance [15,16].
To conclude, the constraint-based theories are in fact emergent gauge theories appearing due to spontaneous Lorentz violation which we refer to as inactive SLIV. These theories, both Abelian and non-Abelian, when being expressed in terms of the pure Goldstone vector modes (a µ and a i µ , respectively) look essentially nonlinear and contain plethora of Lorentz and CP T violating couplings. However, due to total cancellations of their contributions among themselves, they appear to be physically indistinguishable from the conventional QED and Yang-Mills theories. Their emergent nature could only be seen when taking the covariant gauge condition (7) into account. Any other gauge, e.g. Coulomb gauge, is not in line with emergent picture, since it "breaks" Lorentz invariance in an explicit rather than spontaneous way. As to an observational evidence in favor of emergent theories, the only way for SLIV to become active, thus causing physical Lorentz violation, would only appear if gauge invariance in these theories were really broken [22] rather than merely constrained by the gauge-fixing term. In substance, the above SLIV ansatz, due to which the vector field A µ (x) develops the VEV (10), may itself be treated as a pure gauge transformation with a gauge function linear in coordinates, ω(x) = n µ x µ |M |. From this viewpoint, gauge invariance in QED or Yang-Mills theory leads to the conversion of SLIV into gauge degrees of freedom of the massless gauge fields emerged. This is what one could refer to as the generic non-observability of SLIV in gauge invariant theories. Moreover, as was shown some time ago [23], gauge theories, both Abelian and non-Abelian, can be obtained by themselves from the requirement of the physical non-observability of SLIV induced by condensation of vector fields rather than from the standard gauge principle.

Constrained composite bosons
We are coming now to the most interesting question: how can the constrained QED Lagrangian (8) be generated from some underlying dynamics of the elementary constituent fermions? One can readily confirm following the standard procedure that such basic Lagrangian for both hidden constituent and real matter fermions, ψ s and Ψ t , has to have the form where j µ and J µ stand for their Noether currents (2) and (6), respectively. Indeed, employing the standard trick of introducing an auxiliary field A µ (x) one has instead with L 0 (ψ) and L 0 (Ψ) being the free Lagrangians for fermions ψ s and Ψ t . Now, solving the algebraic equations of motion for A µ and substituting back into the Lagrangian (14) one recovers the starting fermion Lagrangian (13). Remarkably, as one can readily see, the equation (15) connects the Dirac extra source current (9) with the standard fermion currents in the theory Now, variation of the prototype vector field Lagrangian (14) under the multiplier function λ leads to the vector field constraint (7), mentioned above, while a similar variation of the fermion Lagrangian (13) gives in turn the constraint for the total fermion current in itself And integrating then all constituent fermions ψ s out from the prototype Lagrangian (14) we come to the effective Lagrangian (8) expressed in terms of the properly renormalized composite A µ field interacting with matter fermions Ψ t in gauge invariant way. Indeed, since the constituent fermions ψ s were minimally coupled to A µ in the prototype Lagrangian, all the generated terms in the effective Lagrangian (including any high-order ones) appear gauge invariant provided a gauge invariant cutoff is chosen. The vector field kinetic term appears in the form where the renormalization constant Z 3 is given by the usual vacuum polarization integral with some momentum-space cutoff Λ. The Z 3 is then absorbed into the wave-function renormalization of A µ field which in turn renormalizes the gauge coupling, multiplier field and SLIV scale as well that leads eventually to the gauge invariant effective Lagrangian (8) (with all the former notations remained). The nonlinear vector field constraint term in it, though being noninvariant, appears in fact as gauge condition in an otherwise gauge invariant and Lorentz invariant theory. So, in sharp contrast to the potential-based models considered above, all radiative corrections which may appear in the effective Lagrangian (8) have to necessarily be both Lorentz invariant and gauge invariant. In fact, in one-loop approximation it was explicitly demonstrated some time ago [13].
It is worth pointing out that the constraint-based models possess one more remarkable advantage as compared to the potential-based ones -they do not need in general to have the large number N of the hidden constituent fermion species. Actually such a number N was mainly imposed to properly suppress the large physical Lorentz violation at low energies which could otherwise appear through the uncontrollably large radiative corrections to the effective Lagrangians. That is not expected at all in the constraintbased models and, therefore, the number N of the constituent fermions ψ sa may be solely determined now by their own dynamics.
While the nonlinear vector field constraint (7) turns into the gauge condition, the constraint (17) put on the currents could mean some relation between currents of the hidden constituents and matter fermions. Actually, this relation also includes an extra source of charge density given by the λ field. However, when going to the conventional QED limit with the vanishing λ field one has instead (j µ + J µ ) 2 = 0 (20) that means that the hidden and the actual matter currents may appear to be opposite to each other, j µ = −J µ . Let us finally briefly describe a possible extension of this approach to the non-Abelian G symmetry case employing the non-Abelian model with constrained vector field multiplet that we discussed in the previous section. Now, all the N and N Ψ species of hidden constituent and matter fermions, ψ s and Ψ t belong to the fundamental representations of G, ψ sa and Ψ ta (a = 1, 2, ..., n), while the emerging composite vector fields will complete its adjoint multiplet. This extension can readily be made just by the corresponding replacements in the above equations (13-17) (D stands for the G symmetry group dimension). One can then proceed in a conventional way [4] to generate the vector field kinetic energy term together with the three-and four-vector self-couplings from the relevant loop diagrams. We eventually arrive at the totally gauge invariant theory with an obvious replacement in the QED Lagrangian (8), where F i µν is the standard strength-tensor for the constrained vector field multiplet A i µ . The modified Lagrange multiplier term providing the constraint (12) appears again as the pure gauge-fixing condition.

Composite quarks, leptons and gauge bosons
We present here some scenario how one could in principle combine this type models of composite gauge bosons with models for quarks and leptons composed from preons (some significant references can be found in [24]). The preons are usually viewed as the truly elementary carriers of all known physical charges, such as weak isospin, color, family number, etc. (which we refer to as "metaflavors"). Apart from metaflavors, they normally possess some metacolor forces that bind them inside quarks and leptons. And last but not least, they should be massless (or having some tiny masses) that is required to have at large distances the observed quarks and leptons with masses which much less than their composition scale. For that, as is well known, the chiral symmetry of the preons should be remained so that the anomaly matching condition of preons at small distances and their composites at the large ones has to be satisfied [25].
In the light of this, the above massive constituent fermions ψ s composing gauge bosons can not be used for composition of quarks and leptons. The most direct way would be to treat the matter fields as the massless preon candidates supplying them with both metacolor and metaflavor symmetries G M C and G M F Ψ ak , a = 1, 2, ..., n M C , k = 1, 2, ..., n M F (22) where indices a and k belong to their fundamental representations, respectively. They are still global symmetries which then will be converted into the local ones by the proper current-current interaction of the hidden constituent fermions, such as it was described above. We propose that they are assigned to the symmetry groups G M C and G M F in the same way ψ ak , a = 1, 2, ..., n M C , k = 1, 2, ..., n M F (23) Note that in the both cases (22) and (23), we identify the number of fermions, the hidden constituent and matter ones, with number of metaflavors in the symmetry group G M F One might, of course, introduce some independent large N number of the hidden constituent fermions (23). However, as was mentioned above, in the constraint-based models, one can have it as low as it is required by the composite dynamics itself. So, technically, we can start with the pure fermion Lagrangian like that of (13) but being extended to the non-Abelian symmetry corresponding to the still global metacolor and metaflavor symmetries G M C and G M F , respectively. One can then readily use the above procedure (14,15)

of introducing the auxiliary vector fields
in the Lagrangian through the interrelated Noether currents of the above fermions (22) and (23) Here T i and T r stand for generators of the symmetry groups G M C and G M F having dimensions D M C and D M F , respectively, while λ M C (x) and λ M F (x) are the corresponding Lagrange multiplier fields. Note also that the metacolor currents j i µ and J i µ are invariant under the metaflavor symmetry G M F , as well as the metaflavor currents j r µ and J r µ are invariant under the metacolor symmetry G M C (the corresponding fermion indices are properly summed up in them). Further, a subsequent integration of the massive constituent fermion multiplets ψ ak out will convert the global symmetries G M C and G M F into the local ones with the metacolor and metaflavor gauge fields A i µ and A r µ interacting with the presumably massless preon multiplets Ψ ak . After the appropriate renormalization of all the composite vector fields involved one arrives at the final preon Lagrangian with the corresponding vector field constraints included. The equations (25) show that both metacolor and metaflavor gauge field multiplets, A i µ and A r µ , apart from the hidden constituent fermions ψ ak (23), are also consisted of the preons Ψ ak (22) which at the same time compose the observed quarks and lepton through the metacolor forces. It goes without saying that their composition scale Λ M C is always less the cutoff Λ of the prototype fermion theory.
After all that, some whole scenario with composite quarks and leptons interacting with composite gauge bosons may be properly developed depending on the particular symmetry groups G M C and G M F taken. One possible scenario could be realized in the model of composite quarks and leptons that was recently presented in [26]. We give below some of its key elements: (1) It is proposed that there are 2K elementary massless left-handed and right-handed preons at small distances, P kL and Q kR (k = 1, . . . , K), which possess a common local symmetry SU (K) M F unifying all known metaflavors, such as weak isospin, color, family number, etc. The preons, both P kL and Q kR , transform under fundamental representation of SU (K) M F and their metaflavor theory has presumably an exact L-R symmetry. Actually, the SU (K) M F appears at the outset as some vectorlike symmetry which then breaks down at large distances to some of its chiral subgroup.
(2) In contrast to their common metaflavors, the left-handed and right-handed preon multiplets are taken to be chiral under the local metacolor symmetry G M C = SO(3) L × SO(3) R . They appear with different metacolors, P a kL and Q a ′ kR , where a and a ′ are indices of the corresponding metacolor subgroups SO(3) L (a = 1, 2, 3) and SO(3) R (a ′ = 1, 2, 3), respectively. They are generically anomaly-free and provide the minimal threepreon configurations for composite quarks and leptons. Due to the chiral metacolor, there are two types of composites at large distances being composed individually from the leftright and left-handed preons, respectively.
(3) Obviously, the preon condensate P L Q R which could cause the metacolor scale Λ M C order masses for composites is principally impossible in the left-right metacolor model taken. This may be generally considered as a necessary but not yet a sufficient condition for masslessness of composites. The genuine massless fermion composites are presumably only those which preserve the chiral SU (K) L × SU (K) R symmetry of preons at large distances that is controlled by the 't Hooft's anomaly matching condition [25].
(4) The strengthening of this condition in a way that the massless fermion composites, both left-handed and right-handed, are required to complete a single representation of the SU (K) M F rather than some set of its representations, allows to fix the number of basic metaflavors K. We can see now that in order to adapt the constrained gauge bosons to this scenario one has to generate the local vectorlike metaflavor symmetry SU (8) M F together with the local chiral metacolor symmetry SO(3) L × SO(3) R binding separately the left-handed and right-handed preons. This means that, while the metaflavor part in the above preon Lagrangian (26) is left intact, its metacolor part has to be properly changed. Namely, it should now include separately both left-handed and right-handed preons, P a kL and Q a ′ kR , being assigned to the different metacolor groups SO(3) L and SO(3) R though to the same metaflavor symmetry SU (8) M F . Also, there are the two metacolor currents, J a µ and J a ′ µ of SO(3) L and SO(3) R with the corresponding gauge bosons A a µ and A a ′ µ , as well as the two constraining Lagrange multiplier terms with the λ M C and λ ′ M C fields in the modified Lagrangian. It is clear that this Lagrangian is emerged in turn after integration out of the corresponding massive constituent fermion fields ψ a k and ψ a ′ k which, in contrast to chiral preons, are vectorlike (ψ a k ≡ (ψ L,R ) a k , ψ a ′ k ≡ (ψ L,R ) a ′ k ) with respect to their own symmetry group, SO(3) L or SO(3) R , respectively. It is worth noting that these constituents will produce themselves the composite states due metacolor forces they possess. However, they will definitely appear very heavy (of the order of the metacolor scale taken) and might only influence physics at high energies comparable with the grand unification scale or so.
We have thus briefly demonstrated one particular way of unification of the constrained composite gauge bosons with composite quarks and leptons that could appear due to the effective preon Lagrangian (26) and its extensions. There are presumably some other ways as well.

Conclusion
We have considered the class of composite models for gauge fields, which are emerged from the prototype fermion model taken in the current-current interaction form. In contrast to the conventional approach where such interactions leads generally to the unstable vector field potential, in our model with the properly constrained fermion currents the compos-ite gauge bosons come out to be only restricted by the nonlinear covariant constraint, A 2 µ = M 2 . They appear naturally massless as the NG modes of SLIV, so that the global internal symmetry G in the model turns into the local symmetry G loc, while the vector field constraint reveals itself as the gauge fixing condition 5 .
Whereas the field-current identity for the massive vector fields, underlying the standard argumentation, has a long story dating back to the paper of Kroll, Lee and Zumino [29], such an identity for massless vector fields seems hardly possible unless they are properly constrained. The crucial point, as we could see, happens to be the relation (16) between the standard matter currents and extra classical current introduced by Dirac. Actually, one now has in a sense the current-current identity rather than field-current one.
We can go a bit further and wonder why basically the proposed constraint pattern for vector fields could make the whole physical field system involved to adjust itself in a gauge invariant way. The possible answer seems to be that the only theories compatible with the nonlinear vector field constraints taken are the gauge invariant ones, as has been generally argued in [16] (see also [27,28]). Indeed, once the SLIV constraint (7) or 12) is imposed, it is therefore not possible to satisfy another supplementary condition since this would superfluously restrict the number of degrees of freedom for the vector field. To avoid this, its equation of motion should be automatically divergenceless that is only possible in the gauge invariant theory. Actually, gauge invariance in such theories appears in essence as a response of an interacting field system to putting the covariant constraint (7, 12) on its dynamics, provided that we allow parameters in the corresponding Lagrangian density to be adjusted so as to ensure self-consistency without losing too many degrees of freedom. Otherwise, a given field system could get unphysical in a sense that a superfluous reduction in the number of degrees of freedom would make it impossible to set the required initial conditions in the appropriate Cauchy problem. Namely, it would be impossible to specify arbitrarily the initial values of the vector and other field components involved, as well as the initial values of the momenta conjugated to them. Furthermore, in quantum theory, to choose self-consistent equal-time commutation relations would also become impossible [30]. So, the nonlinear SLIV condition (7,12), due to which true vacuum in the theory is chosen and massless gauge fields are generated, may provide a dynamical setting for all underlying internal symmetries involved through such an emergence conjecture.
One can see that the gauge theory framework, be it taken from the outset or emerged, makes in turn SLIV to be physically unobservable both in Abelian and non-Abelian symmetry case that is also favored by the supersymmetric SLIV version. We referred to it above as the inactive SLIV in contrast to the active SLIV case where physical Lorentz invariance could effectively occur. From the present standpoint, the only way for an inactive SLIV to be activated would be if emergent gauge symmetries presented above were 5 Generally, apart from the current-current interaction terms, there are many other fermion bilinears and their interactions that could be included in our prototype Lagrangian (13). The above "bosonization" procedure could then be generalized so as to introduce a new auxiliary field (scalar, vector or tensor one) for each bilinear that eventually would lead to an effective action for a set of interacting auxiliary fields [6]. In contrast to the above massless Aµ fields induced by the current bilinears, they are not constrained and, presumably, acquire large masses. On general grounds, these masses have to be of the cutoff order and, therefore, all such states can be neglected at low energies.
slightly broken at small distances being presumably controlled by quantum gravity. One might even think that quantum gravity could in principle hinder the setting of the required initial conditions in the appropriate Cauchy problem, thus admitting a superfluous restriction of vector fields in terms of some high-order operators which occur at the Planck scale order distances. This is just the range of distances where the composite gauge fields, as well as the composite quarks and leptons, should presumably emerge in order to be properly adapted to the grand unification landscape. So, some trace of the gauge symmetry breaking and therefore physical Lorentz violation might accompany their composition processes. We may return to this special issue elsewhere.