Uniﬁcation of elementary forces in gauge SL(2N,C) theories

We argue that the gauge SL (2 N, C ) theories may point to a possible way where the known elementary forces, including gravity, could be consistently uniﬁed. Remarkably, while all related gauge ﬁelds are presented in the same adjoint multiplet of the SL (2 N, C ) symmetry group, the tensor ﬁeld submultiplet providing gravity can be naturally suppressed in the weak-ﬁeld approach developed for accompanying tetrad ﬁelds. As a result, the whole theory turns out to eﬀectively possess the local SL (2 , C ) × SU ( N ) symmetry so as to naturally lead to the SL (2 , C ) gauge gravity, on the one hand, and the SU ( N ) grand uniﬁed theory, on the other. Since all states involved in the SL (2 N, C ) theories are addi-tionally classiﬁed according to their spin values, many possible SU ( N ) GUTs – including the conventional one-family SU (5) theory – appear not to be relevant for the standard 1 / 2 spin quarks and leptons. Meanwhile, the SU (8) grand uniﬁcation for all three families of composite quarks and leptons that stems from the SL (16 , C ) theory seems to be of special interest that is studied in some detail.


Introduction
It just so happens that there is a generic analogy between local frame spacetime in gauge gravity schemes [1,2,3] (see more references in [4]) and internal symmetry space in conventional quantum field theories. Indeed, the spin-connection fields gauging the local SL(2, C) symmetry group of gravity emerge much as photons and gluons appear in the Standard Model. In this connection, one may think that these spin-connection fields could be unified with the ordinary SM gauge fields in the framework of some non-compact symmetry group thus leading to the hyperunification of all known elementary gauge forces. We will refer below to such theories as the hyperunified theories, and specifically, hyperunified GUT (HUT) when speaking about an unification of the SL(2, C) gauge gravity with grand unified theory, respectively.
It may not be surprising that the similar ideas have been put forward for a long time. Indeed, there are many classes of models in the literature, where unification of gravity and other interactions goes through the presentation of the Lorentz and internal symmetry gauge fields as the unified spin-connection components acting in extended spacetime of some non-compact covering symmetry group [5,6,7]. While the spin-1 fields in the total gauge hypermultiplet of this group are proposed to mediate ordinary gauge interactions, the spin-2 fields in it should provide the tiny gravity type interactions in the theory. The point is, however, that generally these tensor fields turn out not only to cause interactions being sizeable with those stemming from vector fields but also lead to an existence of ghosts in the theory making it essentially instable that can only be cured by some enormously extended Higgs sector in it. In contrast, we show that the SL(2N, C) HUT being considered here suggests some new and crucial possibility that has not been yet actually explored. This is related to the generic option in the theory to selectively weaken tensor field multiplet as compared to the spin-1 field ones in the total SL(2N, C) gauge hypermultiplet. As a result, these tensor fields do not propagate, their interactions essentially decouple from other elementary forces and in the linear approximation their trace is only left in a form of Einstein-Cartan gravity in a final unified theory with the effective SL(2, C) × SU (N ) symmetry.
Note that some prototype for the SL(2N, C) hyperunification could be the well-known SL(6, C) symmetry which has been considered quite a long ago [8] as a possible relativistic version of the global SU (6) symmetry model describing the spin -unitary spin symmetry classification for mesons and baryons [9]. In this respect, one might expect that there would be a potential danger for our hyperunified theory due to the Coleman-Mandula theorem [10] on the impossibility of combining spacetime and internal symmetries that just first appeared in connection with the SL(6, C) symmetry mentioned above. Regarding to the hyperunified theories, however, this theorem seems not to be a serious obstacle as is usually claimed on the basis of the following two heuristic arguments. The first is that it only constrains symmetries of the S-matrix and, as such, places no constraints on spontaneously broken symmetries which do not show up directly at the S-matrix level. The second is that the theorem only works if there is a mass gap in the theory that certainly does not happen in the HUT where in the symmetry limit all fields, as gauge bosons so the matter fields, are massless. Apart from that, however, there is an argument being very specific for the HUT considered. The point is that, while all gauge fields in it are unified in a framework of SL(2N, C) symmetry group, its nondiagonal generators are solely related to the SU (N ) "flavored" tensor fields which appear to be naturally suppressed in the weakfield approach developed for accompanying tetrad fields. As a result, the whole theory turns out to effectively possess the local SL(2, C) × SU (N ) symmetry rather than the unified SL(2N, C), so as to naturally lead to the SL(2, C) gauge gravity, on the one hand, and the SU (N ) GUT, on the other, thus actually getting rid of the constraints of the Coleman-Mandula theorem.
The paper is organized in the following way. In Section 2 we give a standard presentation of the SL(2, C) gauge gravity which then it is discussed in the weak-field approach. In Section 3 the SL(2N, C) HUT is presented in detail and in Section 4 some particular HUT models are considered. Our summary is given in final Section 5.

Standard presentation
We first assume that a local frame at any spacetime point possesses the global SL(2, C) symmetry group. Accordingly, its transformations for the basic fermions in the theory are given by the matrix Ω which generally has the pseudounitary form, Ω −1 = γ 0 Ω + γ 0 . Furthermore, to provide invariance of their kinetic terms, iΨγ µ ∂ µ Ψ, one need to replace γ-matrices in them by a set of some tetrad matrices e µ which transform like Generally, the tetrad matrices e µ , as well as their conjugates e µ , contain the appropriate tetrad fields e µ a and e a µ , respectively, They, as usual, satisfy the orthogonality relations and determine the metric tensors in the theory Going now to the case when the SL(2, C) transformations (1) become local, one have to introduce the gauge field multiplet I µ transforming as usual thus providing the fermion field by covariant derivative The I µ multiplet gauging the SL(2, C) has by definition the form can be written in a conventional form µν , e ≡ − det T r(e µ e ν )/4 (10) once the commutator for tetrads and some of standard relations for γ-matrices have been used. This is in fact the simplest pure gravity Lagrangian being equivalent to the Palatini formulation of the standard Einstein-Cartan gravity Lagrangian. Remarkably, such a possibility appears due the starting SL(2, C) invariance involved. Indeed, when even being pure global it requires an introduction of tetrads which then after its localization provide the principally new invariant coupling being linear in the tensor field strength. Furthermore, this coupling happens to be invariant invariant not only under SL(2, C) but, as was claimed [3], under general four-coordinate transformations GL(4, R) as well. Meanwhile, the gauge invariant fermion matter coupling given by the covariant derivative (7,8) presents the spin-connection tensor field T µ[ab] interaction with the spin-density current This is a key feature of the Einstein-Cartan gravity which eventually results in, apart from a standard GR, the tiny four-fermion interaction once the constraint equation for the non-propagating tensor field is used.

SL(2, C) gravity in weak-field approach
Since we think that the SL(2, C) gauge gravity is a part of the unified set of all elementary forces assembled in the SL(2N, C) we have to be sure that its extraordinary smallness is compatible with pretty sizeable contributions of other interactions. In this connection we propose that this smallness related to the weak nature of the gravity tensor field itself rather than its vanishingly small coupling constant. This weakness supposedly manifests itself once spacetime described by the tetrad e a µ appears to be close to the flat Minkowski one in the weak-field approximation. Note that the process of decomposing the general spacetime described by the tetrad e a µ into the flat one δ a µ plus some perturbation term e a µ does not break general covariance. Actually, the perturbation tetrad e a µ which, we refer further as to the petrad, is defined in terms of the subset of a general set of diffeomorphisms on spacetime which leave e a µ sufficiently small that is required by the weak-field approximation e a µ = δ a µ + e a µ , e a µ ≪ 1 In a general case when no extra constraints on petrads are imposed the tetrads are no longer orthogonal but their orthogonality relations (4) include some vanishingly small deviations which can be directly calculated from the above tetrad parametrization. In this connection, the metric tensor will also include such a deviation which we define from a similar equation Multiplying the basic equations (13) by the proper tetrads and also require a general covariance for a shifted metric tensor (15), g µν e νb = e b µ , one can readily find relations between all deviations Using them the metric tensor itself can be directly calculated up to the second order terms in p. Now, instead of the "strong" gauge field I µ (8) we will construct new "weak" one as follows so that the SL(2, C) gauge theory can only be formulated in terms of the weak I µ field specially written with factor 1/4 to remain later a formal similarity with the starting I µ field (8). According to the equations (8, 18) this field -in weak tetrad approximationacquires the form We have only left here the first order terms in the petrad modes e c a and also used standard relations for products of γ-matrices, particularly, for Thus, the weak tensor field T µ[ab] appears to be vanishingly small being suppressed through the tetrad nonorthogonality parameter q c a (14) so that we will keep only the lowest-order terms in it for what follows.
Remarkably, in weak tetrad approximation the identity (20) triggers an important suppression mechanism in the SL(2, C) gravity theory according to which the starting tensor field multiplet T µ [ab] in (19) is largely extinguished and only some of its tiny part T µ[ab] (given in square brackets above) comes out. Another important thing is that, while the new tensor multiplet T µ[ab] globally transforms like the old T µ[ab] one, the right gauge transformation may alternatively appear for only one of them in the theory. Indeed, as follows from (19), their gauge functions could be related as which means that if one requires gauge invariance for the T µ[ab] field, then T µ[ab] field (or any superposition of them) is excluded as a gauge field candidate in a theory, and vice versa.
The basic Lagrangian terms for the SL(2, C) gravity with the new I field (18, 19) will also contain terms being linear in tensor field T µ[ab] that results in the appropriate analogs of the gravity and matter Lagrangian (10, 11), respectively. Indeed, the minimal gravity Lagrangian (10) with the proper replacements remains practically the same form In a linear weak-field T µ[ab] field approach taken the tensor field propagation is automatically neglected, while variation of this Lagrangian with respect to the tetrad e µ a (or petrad e µ a ) leads to the standard Einstein equation of motion for pure gravity.
3 Extension to the SL(2N, C) HUT

Some basic elements
Generally, the SL(2N, C) symmetry contains, as its main subgroups, the SL(2, C) which covers the orthochronous Lorentz group and the internal U (N ) symmetry, so that the SU (N ) appears as the maximal compact subgroup of the SL(2N, C)). Indeed, the 8N 2 −2 generators of SL(2N, C) are formed from the tensor products of the generators of SL(2, C) and generators of U (N ) so that the basic transformation applied to the fermions looks as while λ 0 is the unit matrix corresponding to U (1) generator (all θ parameters may be constant or, in general, depend on the spacetime coordinate). For further description of the fermion matter in the theory one needs again to introduce the generalized tetrad multiplet e µ = (e aK µ γ a + e aK µ5 γ a γ 5 )λ K which transforms, as before, according to (2). Despite its somewhat excessive extension form which generally appears in the SL(2N, C) framework, it would be natural for tetrad flat space components in (24) to have the same form as in the pure gravity case. This means that such an extension might not include the axial-vector part and SU (N ) symmetry components, that could be reached through some gauge invariant constraints put on tetrad. Indeed, one can introduce for that some special non-dynamical SL(2N, C) scalar multiplet in the theory which transforms like as S → ΩS. With this scalar multiplet one can form a new tetrad in terms of the gauge invariant construction, S −1 eS. So, choosing appropriately the flat space components in the S field one can turn the tetrad axial part to zero and establish symmetry between Greek and Latin spacetime indices [8]. In this way one may also exclude the tetrad SU (N ) symmetry components. This is especially apparent when the starting tetrad has a simple factorized form, e aK µ = e a µ ǫ K , where ǫ K (x) is some set the U (N ) symmetrical functions. As a result, the SU (N ) members ǫ k of this set can be then gauged away in S −1 eS by the proper choice of the s k field components (26). In the first order in them the corresponding conditions happen to be Now, the new tetrad (25), as well as the related metric tensor, will automatically satisfy all conditions discussed above in the pure gravity case including its the weak-field approximation.

Local SL(2N, C) symmetry and gauge hypermultiplet
Once the SL(2N, C) transformation (23) becomes local one also need, as ever, to introduce the gauge field multiplet I µ transforming as usual thus providing the fermion multiplet by covariant derivative The I µ hypermultiplet includes, as follows from its decomposition to the flat spacetime component fields the vector and axial-vector field multiplets, and tensor field multiplet as well. While the vector fields are supposedly mediating the electroweak and color forces, tensor fields provides the gravity interaction in the framework of the SL(2N, C) HUTs. In this connection, the crucial problem is how one could selectively suppress the tensor field when it is a member of the same gauge hypermultiplet as vector and axial-vector ones. Fortunately, the weak-field mechanism described above for the pure gravity case causes a radical distinction between them, thus allowing to naturally combine strong internal forces with tiny gravity.
For that purpose, we show again that the SL(2N, C) tetrad multiplets which is reduced to the simple pure gravity form (25) can project an essential part of tensor field components out of the SL(2N, C) gauge hypermultiplet. So, instead of the "strong" gauge fields contained in I µ we will construct new "weak" ones so that the SL(2N, C) gauge theory can only be formulated in terms of the weak I µ field specially written with the 1/4 factor to remain later a formal similarity with the starting I µ field (30). In contrast to standard tetrads, the e K µa field components (25) do not satisfy the orthogonality conditions like those in (4) due to which the tensor multiplet in the new field I µ would be completely excluded that were unacceptable. We will see, however, that some vanishingly small part of this multiplet may survive in the weak-field approximation. Indeed, leaving only the first order terms in the petrad components one goes to Using some standard algebra for γ-commutators [γ a , γ bc ] and [γ ab , γ cd ] one eventually has for a new gauge field hypermultiplet in the weak-field approximation As one can readily see, the vector and axial-vector submultiplets in the starting gauge multiplet (30) completely remain in the weak tetrad approximation Meanwhile, the tensor field components amount to are significantly weakened by the corresponding petrad components. Thus, the hyperunification of the basic elementary forces (appearing with some universal coupling constant for all the vector fields involved) does not prevent the tensor field submultiplet of having the vanishingly small couplings with each other, as well as with a matter. It is also worth noting that, though the new I µ field hypermultiplet globally transforms like the old I µ one (30), the right gauge transformation may alternatively appear for only one of them in the theory. Indeed, as follows from (33) and (35), while gauge functions for vector and axial-vector submultiplets in I µ and I µ can be practically the same, such a function for the modified tensor field multiplet should be quite different, just as we had it above in the pure gravity case (21). This means that if one chooses the I µ as the gauge hypermultiplet in a theory then the old I µ hypermultiplet or any superposition of I µ and I µ is excluded as a gauge field candidate.

Tensor fields
Let us now construct the field strength for the I µ field multiplet and then its total Lagrangian including the matter field part. This strength is given in main terms by for vector, axial-vector and tensor curls, (V + A) µν and T µν , respectively. One can see that kinetic terms of the tensor field multiplet which would provide its propagation can be neglected in the linear tensor field approximation taken. The total action will only contain terms being linear in T K µ[ab] field multiplet that amounts to the Palatini type Lagrangian (10) which for the SL(2N, C) case has a form Interestingly, there are no contributions from the vector and axial-vector multiplets collected in the curls (V + A) µν in (36). So, writing only tensor multiplet one comes -after using of standard trace relations for appearing products of γ-matrices T r(γ a γ b γ c γ d ) and T r(γ ab γ cd ) -to the general Lagrangian being practically a repetition of the pure gravity one The only difference are related to some extra SU (N ) flavored tensor self-interaction terms, while the corresponding flavor tensor field kinetic terms are absent. Let us turn now to the matter sector in the SL(2N, C) hyperunified theory proposed. The gauge invariant fermion matter couplings are given in terms of the modified gauge multiplet (33) by Leaving aside for the moment vector and axial-vector fields one has after using of standard trace relations for γ-and λ-matrices for the matter Lagrangian of the tensor multiplet Variation of the total Lagrangian (38, 40) with respect to e µ0 a leads to the GR type equation containing a standard stress-energy tensor ϑ a µ and a new spin source extension which also contains the flavored tensor fields given in (38). Meanwhile, variation under T 0 µ[ab] gives the constraint equation leading to the tiny 4-fermion spin density current interaction being solely proportional to the conventional GR coupling constant κ just as in the standard Einstein-Cartan gravity but also containing the extra SU (N ) symmetrical fourfermion interaction term.

Vector and axial-vector fields
Let us turn now to the spin-1 vector and axial-vector fields being basic carriers of the internal SU (N ) symmetry in this hyperunification scheme. Their gauge sector stemming from the common strength tensor (36) looks as where one can readily confirm that vector fields acquire a conventional gauge theory form, while axial-vector fields only contributed into the vector field interaction terms. Meanwhile, as follows from the matter sector of the theory (39) the vector fields interact with ordinary matter fermions L while axial-vector fields do not, thus being sterile to them. We could try to adapt such fields to reality though there seems no a direct indication that they might exist. Somewhat traditional way is to make these axial fields superheavy through the enormously extended Higgs sector in the theory so as to remain the Standard Model gauge bosons massless or enough light. This seems to be quite difficult since the axial-vector fields want to follow the same pattern of the mass formation as the vector ones.
Another and a rather interesting way could be if these axial-vector fields were condensed, thus providing a true vacuum in the theory. Remarkably, in this vacuum, as is shown below, the gauge invariance for vector field is completely restored, though a tiny spontaneous breaking of Lorentz symmetry at some Planck order scale M may also appear. To get that vacuum one can, instead of writing a conventional polynomial potential for the gauge hypermultiplet, put on it some nonlinear covariant constraint of the type 1 which in the dominating field components looks as The most appropriate solution to this constraint equation may be related to the special Goldstone type one This parametrization shows that, whereas the axial multiplet is condensed being provided by an effective Higgs mode (given by the second term), there are produced the zero mass excitations being orthogonal to the vacuum direction along the unit Lorentz vector n i µ , n µ i n i µ = 1. We further use its factorized form, n i µ ≡ n µ ǫ i , where n µ is the unit Lorentz vector (n 2 µ = ±1, while ǫ i is the internal SU (N ) symmetry one (ǫ i ǫ i = 1). Note that due to the constraint (46) one, apart from the Lorentz invariance violation, has a spontaneous violation of this internal symmetry as well, since the axial-vector field multiplet plays simultaneously the role of the scalar adjoint multiplet in a conventional SU (N ) GUT [14] (though the number k is not yet fixed). Now, turning back to the spin-1 field Lagrangian (43) and substituting the A i µ expression (47) one can confirm that the first order terms in the zero mode a i µ do not show up there provided that the orthogonality relations work. They can be treated as gauge conditions for zero modes a i µ and vector field V i µ , respectively, while the last relation means that zero modes are supposed not to depend on the x-coordinate component along the direction where Lorentz symmetry is broken. So, neglecting all the higher zero mode terms one eventually comes to the Lagrangian which presents the conventional vector field gauge invariant Lagrangian plus some small non-invariant and Lorentz violating terms stemming from the square root in (47) when the lowest order terms in (V i µ ) 2 /M 2 is taken. The point is, however, that, as follows from the total Lagrangian (43), the high-order terms in zero modes are also important. They contain the zero mode interaction terms with vector fields. Interestingly, some of them acquire the large masses since symmetry of the constraint (46) is much higher than symmetry of the Lagrangian (43) and part of zero modes are in fact pseudo Goldstone states. As follows from the proper mass term in (43) only modes related to the broken non-diagonal generators of the SU (N ) acquire superheavy masses, while modes corresponding to the diagonal ones (48) are left massless. They are the ones that will lead to various processes including those where the vector fields decay into these invisible massless modes being sterile to ordinary matter. Note that the present data allows in principle such a possibility for the Standard Model vector bosons whose total width fraction into invisible modes is still quite large [13].

Symmetry breaking
The entire symmetry breaking scenario will crucially depends on the way the starting SL(2N, C) symmetry breaks through the proper set of scalar fields into the SU (N ) × SL(2, C) symmetry and further to the Standard Model. Actually, one does not need to cause the first stage of symmetry breaking since all nondiagonal generators of SL(2N, C) are related to the SU (N ) "flavored" tensor fields which in the weak-field approach appear to be significantly suppressed and can be neglected beyond the first order terms. Due to this suppression tensor fields do not propagate and the whole theory appears to practically deal with gauge vector and axial-vector fields in a kind of the SU (N ) grand unification scheme. The only place where tensor fields can take part is the pure gravity sector with the Einstein-Cartan SL(2, C) theory provided by the "neutral" tensor field T . Remarkably, in the linear gravity Lagrangian (38) the "flavored" tensor fields T k[ab] µ contribute only to the tiny four-fermion spin current interactions.
As to the internal SU (N ) symmetry violation to the Standard model one actually need to have a number of scalars, its adjoint (Φ) and fundamental (H) multiplets which under SL(2N, C) transform as Φ → ΩΦΩ −1 , H → ΩH respectively. For simplicity, one can again use the tetrad projection mechanism which we used above for gauge hypermultiplet (30) to suppress the tensor field components in them. As a result, one comes to only scalar and pseudoscalar field components in the SU (N ) symmetry breaking multiplets, while their suppressed tensor components do not propagate and can be neglected beyond the the first order approximation. Particularly, for the "projected" adjoint and fundamental scalars one now has respectively.

Application to GUTs
Let us now consider more closely how the SL(2N, C) type model can be applied to the known GUTs starting from a conventional SU (5) [14] which would stem from the SL(10, C) HUT and then to the higher GUTs like as the SU (8) [15] or SU (11) [16] containing all three quark-lepton families which could emerge in turn from the SL(16, C) or SL(22, C) hyperunified theories, respectively. In the simplest SL(10, C) case some of its low-dimensional multiplets of the chiral (lefthanded for certainty) fermions can be given in terms of the SU (5) × SL(2, C) components as where we have used that that a common antisymmetry on two or more joint SL(10, C) indices means antisymmetry in internal indices (i, j, k = 1, ..., 5) and symmetry in the chiral spinor ones (a, b, c = 1, 2), and vice versa (dimension of representations are also indicated). One can see that, while the SU (5) antiquintet can easily be constructed (52), its decuplet is not contained in the pure antisymmetric SL(10, C) representations (53, 54). Moreover, the tensor (53) corresponds in fact to the collection of vector and scalar multiplets rather than the fermion ones, while the tensor (54) contains some unusual spin 3/2 and spin 1/2 fermion multiplets being out of use in the SU (5) GUT. Note in this connection, that all GUTs where fermions are assigned to the pure asymmetric representations seem to be also irrelevant since the spin values of appearing states are not in conformity with what we have in reality. The most known example of this kind is the SU (11) GUT [16] which certainly should be excluded in the framework of the considered SL(2N, C) theories. For the right 1/2 spin value of fermions involved the matter fermion sector in these theories should contain multiplets having in general the upper and lower indices.
In this regard, the most appealing case where all quarks and leptons are treated on an equal basis seems to be the SL(16, C) HUT with its subgroup SU (8) as the grand unified symmetry [15] containing all three families of them in the single representation. This is the 216 dimensional multiplet Ψ  where the first term corresponds the one-handed 1/2 spin fermion multiplet Ψ k , while all other multiplets containing the 3/2 spin states and extra 1/2 states should be made heavy. One way for a such arrangement could be introducing into theory the conjugated fermion multiplets Ψ [ia, jb] kc (L,R) with both of chiralities so that the left-right symmetry in the starting theory holds. Remarkably, such multiplets might appear in the unified preon model (of the type considered in [15]) with the lefthanded and righthanded preons transforming according to the fundamental multiplets P α Lia and P α ′ Ria of the SL(16, C) times some left-right metacolor symmetry M L × M ′ R . So the above chiral multiplets Ψ just containing the scalar and pseudoscalar components, respectively.

Summary
We have argued that the SL(2N, C) hyperunified theories may be a possible way where all elementary gauge forces could be unified. Remarkably, while all related gauge fields are unified in a framework of SL(2N, C) symmetry group, the tensor field submultiplet providing gravity appears to be naturally suppressed in the weak-field approach developed for accompanying tetrad fields. As a result, the whole theory turns out to effectively possess the local SL(2, C) × SU (N ) symmetry, so as to naturally lead to the SL(2, C) gauge gravity, on the one hand, and the SU (N ) GUT, on the other. An essential problem related to this type of HUT models is a possible presence of ghosts. Indeed, given that the unifying gauge group is noncompact, one may expect that some components of the connections will have wrong sign kinetic terms [5,6,7]. Remarkably, in our case this generically happens to the tensor field components in the SL(2N, C) gauge hypermultiplet rather than its vector and axial-vector components. But since just tensor fields appears to be naturally suppressed in the theory that allows to circumvent this problem in weak-field approximation.
Spontaneous breakdown of HUT down to the Standard Model will lead to the variety of new processes generalizing as the gravity sector through the torsion related phenomena, so the SM sector via new couplings and new particles that was discussed above. It is also worth noting somewhat principal differences of the SL(2N, C) HUTs from unification in the framework of the pseudo-orthogonal SO(1, N ) symmetries that also should be carefully explored. These and related questions are planned to be considered elsewhere [17].