Bled Workshops in Physics Vol. 18, No. 2 JLV Proceedings to the 20th Workshop What Comes Beyond ... (p. 25) Bled, Slovenia, July 9-20, 2017 3 Eightfold Way for Composite Quarks and Leptons J.L. Chkareuli * Center for Elementary Particle Physics, ITP, Ilia State University 0162 Tbilisi, Georgia To memory of V.N. Gribov (1930-1997) Abstract. The L-R symmetric composite model for quarks and leptons where constituent preons are bound by the SO(n)L x SO(n)R gauge forces is reconsidered. We find that just the eight left-handed and right-handed preons, their local metaflavor symmetry SU(8)mf and accompanying global chiral symmetry SU(8)L x SU(8)R may determine physical world at small distances. This result for an admissible number of preons filling the fundamental multiplet of some SU(N)mf symmetry group appears as a solution to the't Hooft's anomaly matching condition precisely for N = 8, provided that this condition is satisfied separately for the L-preon and R-preon composites which fill individually a single multiplet of the SU(N) rather than a set of its multiplets. We next show that an appropriate L-R symmetry violation reduces an initially emerged vectorlike SU(8) theory down to the conventional SU(5) GUT with an extra local family symmetry SU(3)F and three standard generations of quarks and leptons. Though the tiny radius of compositeness for universal preons composing both quarks and leptons makes it impossible to immediately confirm their composite nature, theory predicts the several extra heavy SU(5) x SU(3)F multiplets located at the scales from O(1) TeV up to the Planck mass scale that may appear of actual experimental interest. Povzetek. Avtor obravnava model sestavljenih kvarkov in leptonov s simetrijo L-R, v katerem sestavne delce (preone) veže sila umeritve SO(n)L x SO(n)R. Fiziko na majhnih razdaljah v tem primeru doloca samo osem levorocnih in desnorocnih preonov z njihovo lokalno meta-okusno simetrijo SU(N)mf in ustrezno globalno kiralno simetrijo SU(N)L x SU(N)R. Ta rezultat za N = 8 zadosti 't Hooftovemu pogoju za ujemanje anomalij, ce velja posebej za gruce levorocnih in posebej za gruce desnorocnih preonov, ce vsako od gruc doloca bodisi levorocni bodisi desnrocni multiplet. Avtor pokaze, da priblizna zlomitev simetrije L — R vodi do GUT SU(5), ki pa ima dodatno lokalno simetrijo SU(3)F, s katero opise tri druzine druzine kvarkov in leptonov. Teorija napove vec zelo tezkih multipletov SU(5) x SU(3)F, ki imajo mase od O(1) TeV do Planckove skale, kar bi lahko opazili pri poskusih. Keywords: Composite, Preon, Family, Grand unified theory * E-mail: j.chkareuli@iliauni.edu.ge 26 J.L. Chkareuli 3.1 Preamble It has long recognized that there is no meaningful internal symmetry scheme beyond the known Grand Unified Theories like as the SU(5), S0(10), or E(6) GUTs which could be well suited for classification of all observed quarks and lep-tons. Any attempt to describe all three quark-lepton families in the grand unified framework leads to higher symmetries with enormously extended representations which also contain lots of exotic states that never been detected in an experiment. This may motivate us to continue seeking a solution in some subparticle or preon models for quark and leptons just like as in the nineteen-sixties the spectroscopy of hadrons had required to seek a solution in the quark model for hadrons in the framework of the so-called Eightfold Way. This term was coined by Murray Gell-Mann in 1961 to describe a classification scheme for hadrons, that he had devised, according to which the known baryons and mesons are grouped into the eight-member families of some global hadron flavor symmetry SU(3) [1]. This concept had finally led to the hypothesis of quarks locating in the fundamental triplet of this symmetry, and consequently to a compositeness of baryons and mesons observed. We try to show now that the Eightfold Way idea looks much more adequate when it is applied to a next level of the matter elementarity, namely, to elementary preons and composite quarks and leptons. Remarkably, just the eight preons and their generic SU(8) symmetry seem to determine in a somewhat special way the fundamental entities of the physical world and its total internal symmetry. Interestingly, not only the number "eight" for preons but also its breakdown into some special subdivisions corresponds to the spirit of the Eightfold Way that will be seen from a brief sketch given below. In more detail, the Eightfold Way or Noble Eightfold Path [2] is a summary of the path of Buddhist practices leading, as supposed, to a true liberation. Keeping in mind the particle physics we propose that the eight spoke Dharma wheel which symbolizes the Noble Eightfold Path could be associated with eight preon fields (or superfields, in general) Pi (i = 1,..., 8) being the fundamental octet of the basic flavor symmetry SU(8). They may carry out the eight fundamental quantum numbers which has been detected so far. These numbers are related to the weak isospin, color and families of quarks and leptons. Accordingly, we will refer to these preons as a collection of "isons" Pw (w = 1,2), "chromons" Pc (c = 1,2,3) and "famons" Pf (f = 1,2,3). Surprisingly, the Noble Eightfold Path is also originally divided into three similar basic divisions. They are: (1) The Insight consisting of the Right view and the Right resolve, (2) The Moral virtue consisting of the Right speech, the Right action and the Right livelihood, (3) The Meditation consisting of the Right effort, the Right mindfulness and the Right Concentration. This analogy with a similar decomposition of the sacred number eight, 8 = 2 + 3 + 3, which appears in the expected breakdown of the generic preon SU(8) symmetry SU(8) -> SU(2)w x SU(3)c x SU(3)f , (3.1) looks indeed rather impressive. 3 Eightfold Way for Composite Quarks and Leptons 27 In principle, it is not necessary to generically relate the Eightfold Way concept to preons and composite quarks and leptons. First of all, it is related to the eight fundamental quantum charges of particle physics presently observed. They correspond in fact to the two weak isospin orientations, the three types of colors and the three species of quark-lepton families, all of which may be accommodated in the unified SU(8) theory. Their carriers could be or could not be the elementary preons, though the preon model composing the observed quark and leptons at appropriate distances seems to reflect this concept in the most transparent way. We find, resurrecting to an extent the old Eightfold Way idea in an initially L-R symmetric and SU(N) invariant physical world, that just the eight left-handed and right-handed preons and their basic flavor symmetry SU(8) appear as a solution to the't Hooft's anomaly matching condition [3] providing the chiral symmetry preservation at all distances involved and, therefore, masslessness of emerged composite fermions. We show that this happens if (1) this condition is satisfied separately for the L-preon and R-preon composites and (2) each of these two series of composites fill only one irreducible representation of the starting SU(N) symmetry group rather than a set of its representations. We next show that an appropriate L-R symmetry violation reduces an emerged vectorlike SU(8) theory down to one of its chiral remnants being of significant physical interest. Particularly, this violation implies that, while there still remains the starting chiral symmetry for the left-handed preons and their composites, for the right-handed states we only have the broken chiral symmetry [SU(5) x SU(3)]R. Therefore, whereas nothing changes for the left-handed preon composites still filling the total multiplet of the SU(8), the right-handed preon composites will fill only some particular submultiplets in it. As a result, we eventually come to the conventional SU(5) GUT with an extra local family symmetry SU(3)F and three standard generations of quarks and leptons. Moreover, the theory has the universal gauge coupling constant running down from the SU(8) unification scale, and also predicts some extra heavy SU(5) x SU(3)f multiplets located at the scales from O(1) TeV up to the Planck mass that may appear of actual experimental interest. For simplicity, we largely work in an ordinary spacetime framework, though extension to the conventional N = 1 supersymmetry with preons and composites treated as standard scalar superfields could generally be made. All these issues are successively considered in the subsequent sections 3.2-3.7, and in the final section 3.8 we present our conclusion. Some attempt to classify quark-lepton families in the framework of the SU(8) GUT with composite quarks and leptons had been made quite a long ago [4], though with some special requirements which presently seem not necessary or could be in principle derived rather than postulated. Since then also many other things became better understood, especially the fact that the chiral family symmetry subgroup SU(3)F of the SU(8), taken by its own, was turned out to be rather successful in description of quark-lepton generations. At the same time, there have not yet appeared, as mentioned above, any other meaningful internal symmetry for an appropriate classification of all the observed quarks and leptons. All that motivates us to address this essential problem once again. 28 J.L. Chkareuli 3.2 Preons - metaflavors and metacolors We start formulating a few key elements of preon models (for some significant references, see [5,6]), partially refining some issues given in our old paper [4]. • We propose that there is an exact L-R symmetry at small distances where N elementary massless left-handed and right-handed preons, PiL and PiR (i = 1,..., N), possess a local metaflavor symmetry SU(N)MF including the known physical charges, such as weak isospin, color, and family number. The preons, both PiL and PiR, are located in its fundamental representation. • The preons also possess a local metacolor symmetry GMC = GMC x GjMc with n metacolors (n is odd) which bind preons into composites - quarks, leptons and other states. In contrast to their common metaflavors, preons have different metacolors for the left-right and left-handed components, P?L and P[R, where a and a' are indices of the corresponding metacolor subgroups GMC (a = 1,...,n) and GjMc (a' = 1,...,n), respectively. As a consequence, there are two types of composites at large distances being composed from them separately with a radius of compositeness, RMC ~ 1/AMC, where AMC corresponds to the scale of the preon confinement for the asymptotically free (or infrared divergent) GMc symmetries. Obviously, the preon condensate (PlPr) which could cause the AMC order masses for composites is principally impossible. This is in sharp contrast to an ordinary QCD case where the left-handed and right-handed quarks forms the (qTqR) condensate thus leading to the AC order masses (AC ~ (0.1 ^ 1 )GeV) for mesons and baryons. Due to the L-R symmetry, the metacolor symmetry groups GMC and GMC are taken identical with a similar confinement for both of sets of preons. If one also proposes that the preon metacolor symmetry GMC is generically anomaly-free for any matter content involved, then for independent left-handed and right-handed preons one comes to an input chiral orthogonal symmetry of the type Gmc = SOnmc x SO(U)mc, n = 3,5,... (3.2) for the n-preon configurations of composites. For reasons of economy, it is usually proposed that fermion composites have the minimal 3-preon configuration. • Apart from the local symmetries, metacolors and metaflavors, the preons P?L and P-R possess an accompanying chiral global symmetry K(N) = SU(N)l x SU(N)r (3.3) being unbroken at the small distances. We omitted above the Abelian chiral U(1)l,r symmetries in K(N) since the corresponding currents have Adler-Bell-Jackiw anomalies in the triangle graph where they couple to two metagluons [7]. In fact, their divergences for massless preons are given by jl^ d T^ — n gL,R Gp 2 + 2N/11, due to which the composite quarks and leptons have to appear at least as five-preon states. Checking generally all possible n-index representations of SU(N) we find that the AM condition only works for some combination of its "traced" tensors ^|jk]L R and YiL,R obtained after taking traces out of the proper n-index tensors Yjk' ]LR. This eventually leads to the equation generalizing the above anomaly matching condition (3.9) N2/2 — 7N/2 — 1 + p = n (3.11) where p is a number of the traced fundamental multiplets YiL,R for composites. One can see that there appear some reasonable solutions only for n — p = 3 and, therefore, one has again solutions for the "eightfold" metaflavor symmetry SU(8). Apart from the AM condition (3.5) there would be in general another kind of constraint on composite models which has been also proposed in [3]. This constraint requires the anomaly matching for preons and composites, even if some of introduced N preons become successively heavier than the scale of compositeness and consequently decouple from the entire theory. As a result, the AM condition should work for any number of preons remained massless (thus basically being independent of N) that could make generally classification of composite fermions quite arbitrary. Fortunately, such an extra constraint is not applicable to our L-R symmetric model where the Dirac masses for preons are not possible by definition, whereas the Majorana masses would mean breaking of the input local metaflavor SU(N)MF symmetry1. Most importantly, the orthogonal symmetry for metacolor (3.2) allows to consider more possible composite configurations than it is in the case of an unitary metacolor symmetry, as in the conventional SU(3) color for QCD. The above strengthening of the AM condition, according to which composites only fill a 1 Apart from that, it has been generally argued [5] that the nonperturbative effects may not be analytic in the preon mass so that for the large and small preon masses the theories may be quite different, thus avoiding this additional constraint. 32 J.L. Chkareuli single multiplet of the metaflavor SU(N)MF symmetry group, has unambiguously led us to the composite multiplets R having the same classical U(1)L,R fermion numbers or hypercharges YL,R as the preons themselves. We argued in the previous section that these hypercharges may be considered in the valent preon approximation as the almost conserved classical charges according to which the preon and composite states could be classified. With all that in mind, one could assume that there may work some extra selection rule according to which only composites satisfying the condition Yl,r (preons) = YL,R (composites) (3.12) appear in physical spectrum in the orthogonal left-right metacolor case. We can directly see that the condition (3.12) trivially works for the simplest composite states which could be constructed out of a single preon PaL or P?R, whose metacolor charge is screened by the metagluon fields A*^ and Aj^ of SO(n)MC and SO( n)Mc, respectively. These composites will also satisfy the general AM condition (3.5) provided that one admits the n left-handed and right-handed fundamental multiplets of the SU(N)MF to participate (iN = n). In our L-R symmetric model, however, such massless composites will necessarily pair up, thus becoming very massive and decoupling from the low-lying particle spectrum, no matter the starting L-R symmetry becomes later broken or not. This in sharp contrast to the models [8] with the orthogonal metacolor group SO(n) for the single chirality preons, where such massless composite generally appear to be in contradiction with observations. Moreover, in this case the composite multi-preon states for quarks and leptons seem hardly to be stable, since they could freely dissociate into three screened preon states. One could wonder why the condition (3.12) does not work in the familiar QCD case with elementary quarks and composite baryons. The point is that, despite some conceptual similarity, QCD is the principally different theory. The first and immediate is that the unitary color SU(3)C, in contrast to the orthogonal ones, allows by definition no other quark number for baryons but YB = 3Yq. The most important aspect of this difference is, however, that the color symmetry SU(3) c is vectorlike due to which chiral symmetry in QCD is broken by quark-antiquark condensates with the corresponding zero-mass Goldstone bosons (pions, kaons etc.) providing the singularity of the three-point function. As a consequence, the AM condition implies in this case that dynamics requires spontaneous breakdown of chiral symmetry rather than an existence of massless composite fermions, as happens in the orthogonal metacolor case discussed above. We find below in section 3.5 that, though the proposed condition (3.12) looks rather trivial in the L-R symmetry phase of the theory, it may become rather significant when this symmetry becomes spontaneously broken. 3.4 Composites - the L-R symmetry phase So, we have at small distances the preons given by the Weil fields POL , P-ir (i = 1,...,8; a = 1,2,3; a' = 1,2,3) (3.13) 3 Eightfold Way for Composite Quarks and Leptons 33 belonging to the fundamental octet of the local metaflavor symmetry SU(8)MF and to triplets of the metacolor symmetry SO(3)MC x SO(3)MC which are local, and there is also the accompanying global chiral symmetry K(8) = SU(8)L x SU(8)R (3.14) of the eight preon species (3.13). At large distances, on the other hand, we have composites located, respectively, in the left-handed and right-handed multiplets of the SU(8)mf j(216),¥|jk]R(216) , (3.15) where their dimensions are explicitly indicated. The chiral symmetry (3.14), according to the AM condition taken, remains at large distances. Due to a total L-R symmetry of preons and composites the triangle anomalies both at small and large distances appears automatically compensated. Decomposing the SU(8)MF composite multiplets (3.15) into the SU(5) x SU(3)F components one has 216l,r = [(5 + 10, 3) + (45,1)+ (5,8)+ (24,3) +(1,3) +(1,6)] L,R (3.16) where the first term for the left-handed composites, (5+10,3)L, could be associated with the standard SU(5) GUT assignment for quarks and leptons [5] extended by some family symmetry SU(3)F, while other multiplets are somewhat exotic and, hopefully, could be made heavy to decouple them from an observed low-lying particle spectrum. The determination of the explicit form of the wave function for composite states (3.15) is a complicated dynamical problem related to the yet unknown dynamics of the preon confinement. We propose that some basic feature of these composites are simply given by an expression ¥|jk]L(x) SU(8)l x [SU(5) x SU(3)]r (3.20) For convenience, we consider a supersymmetric model where this breaking may be caused presumably due to the asymmetric preon condensation e«pT (PiLPfLPYL) = 0 , £a'P'y (PfRPfRPyr) = SaS^abcAMc (3.21) emerging for preon superfields with their fermion and scalar field components involved. Here antisymmetric third-rank tensors eapY and ea/p/Y/ belong to the metacolor symmetries SO(3)MC and SO(3)MC, respectively, while eabc (a, b, c = 1,2,3) to the symmetry SU(3)R. Remarkably, the breaking (3.20) is only possible when the number of metacolors n is equal 3 or 5, as is actually implied in our model. For the minimal case, n = 3, the vacuum configurations (3.21) could spontaneously appear in some L-R symmetric model with the properly arranged high-dimensional preon interactions2 {GnL [(PlPlPl) (PlPlPl)]n + G"R [(PrPrPr) (PrPrPr)]n n=1 +G"r [(PlPlPl) (PrPrPr)]" + G"l[(PrPrPl) (PlPlPl)]"} (3.22) with coupling constants satisfying the conditions G"L = GnR and G"R = G"L. This model is evidently non-renormalizable and can be only considered as an effective theory valid at sufficiently low energies. The dimensionful couplings Gn 2 This L-R symmetry breaking model looks somewhat similar to the well-known multi-fermion interaction schemes used in the other contexts for chiral symmetry breaking [10] or spontaneous Lorentz violation [11]. 3 Eightfold Way for Composite Quarks and Leptons 35 are proportional to appropriate powers of some UV cutoff A which in our case can be ultimately related to the preon confinement energy scale AMC, Gn ~ AMCn. For some natural choice of these coupling constants one may come to the asymmetric solution (3.21). A more conventional way of getting the L-R asymmetry may follow from the symmetric scalar field potential [5] U = M2(®L + ®r) + h(®L + ®R)2 + h'®L®R + P(®l, ®R ) (3.23) containing two elementary third-rank antisymmetric scalar fields, ®Lijk] and ® Rjk], interacting with L- and R-preons, respectively. For some natural area of the parameters in the potential, M2 < 0 and h, h' > 0, and properly chosen couplings for scalars ®Ljk] and ®Rjk] in the polynomial P(®L, ®R) they may readily develop the totally asymmetric VEV configuration (®Lijk]) = 0 , (®Rijk]) = 6aa5{,5keabcMLR (a,b,c = 1,2,3) (3.24) where the mass MLR corresponds to the L-R symmetry breaking scale and indices a, b, c belong to the SU(3)R. Due to these VEVs, the higher dimension terms in the effective superpotential induced generally by gravity MG^ (PiLPjLPkL) U(1)R5) x Z(3)R3) (3.27) while the U(1 )L symmetry is left intact. Here, U(1 )R5) and Z(3)R3) stand for the survived continuous and discrete symmetries of quintet preons PsR (s = 1,..., 5) of SU(5)R and triplet preons PaR (a = 1,2,3) of SU(3)R, respectively, which are thereby separated. Namely, the R-preon hypercharge group in the broken L-R symmetry phase is given by the product (3.27) rather than the universal U(1 )R for all eight preons, as was in its unbroken phase. Now, if we require the preon number matching for preons and composites the states collected in (5 + 10, 3)R will never appear in physical spectrum. Indeed, as one can easily check, both the U(1)R5) hypercharge and discrete Z(3)R3) symmetry values for these states are quite different from those for the preons PsR and PaR, respectively. At the same time, all other composite submultiplets in 216R (3.16) readily match the both symmetry values for preons. One way or another, the simplest combination of the 216R submultiplets which may simultaneously satisfy the AM conditions for the [SU(5) x SU(3)]R symmetry, as well as the above preon number matching condition is in fact given by the collection (45,1 )r + (5,8 + 1)r + 3(1, 3)r (3.28) where the submultiplet (1,3)R has to appear three times in (3.28) in order to appropriately restore the anomaly coefficient balance for the R-preon composites. Of course, this collection of states can also appear by its own without any reference to the preon number matching condition that we have used above as some merely heuristic argument. 3.6 Physical sector - quarks and leptons We can see that after chiral symmetry breaking in the sector of the right-handed preon composites the starting metaflavor symmetry SU (8) at large distances is reduced to the product of the standard SU (5) GUT and chiral family symmetry SU(3)f SU(8)mf -> SU(5) x SU(3)f (3.29) presumably with the equal gauge coupling constants g5 and g3F at the grand unification scale. This is in essence the chiral remnant of the initially emerged vectorlike SU(8)MF symmetry. The massless composite fermions, due to pairing up of the similar L-preon and R-preon composites and decoupling them from a low-energy spectrum, are given now by the collection of the SU(5) x SU(3)F multiplets (5 + 10,3) l + (24,3)l + 2(1,3)r + (1,6)r (3.30) which automatically appear free from both the SU(5) and SU(3)F anomalies. They contain just three conventional families of quarks and leptons plus massive multiplets located on the family symmetry scale MF. In order to sufficiently suppress 3 Eightfold Way for Composite Quarks and Leptons 37 all flavor-changing transitions, which would induce the family gauge boson exchanges, this scale should be at least of the order 105^6 GeV, though in principle it could be as large as the SU(5) GUT scale. In the latter case, some of the heavy states in (3.30) could be considered as candidates for the superheavy right-handed neutrinos. One can argue that the physical composite multiplets (3.30) appear not only for the triple metacolor, n = 3, but in general case as well. Indeed, using the remark concerning the generalized AM condition (3.11) and properly extending the left-handed multiplets in (3.16) and the right-handed multiplets in (3.28) by the new n — 3 fundamental composite octets [(5,1) + (1,3)]L,R to have anomaly matching for any number n of metacolors, one comes after pairing of the identical multiplets to the same physical remnant (3.30) as in the triple metacolor case. It is important to note that the tiny radius of compositeness for universal preons composing both quarks and leptons makes it impossible to directly observe their composite nature [12]. Indeed, one can readily see that the quark pair u + d contains the same preons as the antiquark-antilepton pair u + e+ that will lead to the process u + d->u + e+ (3.31) and consequently to the proton decay p —» n0 + e+ just due to a simple rearrangement of preons in a proton. To prevent this one should take the compos-iteness scale AMC of the order of the scale of the SU(5) GUT or even larger, Amc > Mgut ~ 2 • 1016 GeV, and, respectively, Rmc < 5 • 10-31 sm. This limit on the radius of compositeness may in turn cause limits on the composite fermions masses appearing as a result of the quantum gravitational transitions of the identical states in the left-handed multiplets (3.15) and right-handed multiplets (3.28), (45,1 )l,r + (5,8 + 1 )l,r + (1,3)l,r , (3.32) into each other. From dimensional arguments related to a general structure of the composites proposed above (3.17), these masses could be of the order (Amc/Mpi)5Amc (that corresponds in fact to the 6-fermion interaction of the left-handed and right-handed preons) and, in fact, are very sensitive to the confinement scale AMC. Actually, for the metacolor scales, MGUT < AMC < MPl, the heavy fermion masses may be located at the scales from O(1) TeV up to the Planck mass scale. Therefore, the heavy composite states may be of direct observation interest if they are located near the low limit, or otherwise they will populate the SU(5) GUT desert. Interestingly, the screened preon states (3.19) (5,1 )l,r + (1,3)l,r (3.33) acquire much heavier masses when being pairing with each other. Again, from the dimensional arguments one may conclude that these masses has a natural order (Amc/Mpi) Amc that is significantly larger than masses of the 3-preon states (3.17, 3.18). 38 J.L. Chkareuli Note that some of the heavy states (3.32) can mix with ordinary quarks and leptons given by the multiplet (5 + 10,3) L in (3.30). Particularly, there could be the large mixing term of the part (5,3)L containing the lepton doublet and down antiquarks with the multiplet (5,8 + 1 )R in (3.32). This term has a form (5,3)l(5,8 + 1) r (1,3) (3.34) where (1,3) stands for some pure "horizontal" scalar field being a triplet of the family symmetry SU(3)F. Actually, this mixing is related again to the 6-fermion gravitational interaction of the left-handed and right-handed preons, thus leading to the nondiagonal masses of the order (Amc/Mpi)5Mf. Thereby, in order not to significantly disturb the masses of quarks and leptons in (3.30) one has to generally propose Mf C Amc. This in fact is readily satisfied even for high family scales, namely, in the case when the scale MF is taken near the grand unification scale Mgut, while the scale AMC near the Planck scale MPl. The more liberal limitations appears when that part (5,3)L mixes with the screen preon states (5,1) R in (3.33) due to the same scalar triplet (1,3) of the SU(3)F. Now, this mixing caused by the 4-fermion interaction leads to the nondiagonal mass of the order (Amc/Mpi)2Mf that may be naturally much lesser than diagonal mass (AMC/MPl) AMC derived above for the screened preon state. Nevertheless, depending on real values of the scales Amc and MF there could be expected some violation of unitarity in the conventional 3 x 3 mass matrices of leptons and down quarks which may be of a special interest for observations. Other mixings of quarks and leptons with heavy states (3.32) and (3.33) will necessarily include an ordinary Higgs quintet of the grand unified SU(5) (or a doublet of the SM) and, therefore, are negligibly small. To conclude, our preon model predicts three types of states which are (1) three families of ordinary quarks and leptons (5 + 10,3)L in (3.30) with masses at the electroweak scale, (2) the heavy chiral multiplets (24,3)L + 2(1,3)R + (1,6)R (3.30) with the Majorana type masses at the family scale MF = 106^16 GeV and (3) the heavy paired multiplets (3.32) with masses in the interval 103^19 GeV which are related to the gravitational transition amplitudes of the L-preon composites into the R-preon ones. However, the most important prediction of the left-right preon model considered here is, indeed, an existence of the local chiral family (or horizontal) symmetry SU(3)F for quark-lepton generations which is briefly presented below. 3.7 The chiral family symmetry SU(3)F The flavor mixing of quarks and leptons is certainly one of the major problems that presently confront particle physics. Many attempts have been made to interpret the pattern of this mixing in terms of various family symmetries - discrete or continuous, global or local. Among them, the chiral family symmetry SU(3)F derived first in the similar preon framework [4] and developed then by its own by many authors [13-22] seems most promising. As was shown, the spontaneous breaking of this symmetry gives some guidance to the observed hierarchy between elements of the quark-lepton mass matrices, on the one hand, and to presence 3 Eightfold Way for Composite Quarks and Leptons 39 of texture zeros in them, on the other, that leads to relationships between the mass and mixing parameters. In the framework of the supersymmetric Standard Model, it leads, at the same time, to an almost uniform mass spectrum for the superpartners, with a high degree of flavor conservation, that makes its existence even more significant in the SUSY case. Generically, the chiral family symmetry SU(3)F possesses four basically attractive features: (i) It provides a natural explanation of the number three of observed quark-lepton families, correlated with three species of massless or light (mv < MZ/2) neutrinos contributing to the invisible Z boson partial decay width; (ii) Its local nature conforms with the other local symmetries of the Standard Model, such as the weak isospin symmetry SU(2)w or color symmetry SU(3)c, thus leading to the family-unified SM with a total symmetry SM x SU(3)F; (iii) Its chiral nature, according to which both left-handed and right-handed fermions are proposed to be fundamental triplets of the SU(3)F, provides the hierarchical mass spectrum of quark-lepton families as a result of a spontaneous symmetry breaking at some high scale MF which could in principle located in the area from 105^6 GeV (to properly suppress the flavor-changing processes) up to the grand unification scale Mqut and even higher. Actually, any family symmetry should be completely broken in order to conform with reality at lower energies. This symmetry should be chiral, rather than a vectorlike, since a vectorlike symmetry would not forbid the invariant mass, thus leading in general to degenerate rather than hierarchical mass spectra. Interestingly, both known examples of local vectorlike symmetries, electromagnetic U(1 and color SU(3)C, appear to be exact symmetries, while all chiral symmetries including conventional grand unifications [5] SU(5), SO(10) and E(6) (where fermions and antifermions lie in the same irreducible representations) appear broken; (iv) Thereby, due to its chiral structure, the SU(3)F admits a natural unification with all known GUTs in a direct product form, both in an ordinary and super-symmetric framework, thus leading to the family-unified GUTs, GUT x SU(3)F, beyond the Standard Model. So, if one takes these naturality criteria seriously, all the candidates for flavor symmetry can be excluded except for local chiral SU(3)F symmetry. Indeed, the U(1) family symmetry does not satisfy the criterion (i) and is in fact applicable to any number of quark-lepton families. Also, the SU(2) family symmetry can contain, besides two light families treated as its doublets, any number of additional (singlets or new doublets of SU(2)) families. All the global non-Abelian symmetries are excluded by criterion (ii), while the vectorlike symmetries are excluded by the last criteria (iii) and (iv). Among applications of the SU(3)F symmetry, the most interesting ones are the description of the quark and lepton masses and mixings in the Standard Model and GUTs [13], neutrino masses and oscillations [15] and rare processes [16] including their astrophysical consequences [22]. Remarkably, the SU(3)F invariant Yukawa coupling are always accompanied by an accidental global chiral U(1) symmetry, which can be identified with the Peccei-Quinn symmetry [18] provided it is not explicitly broken in the Higgs sector, thus giving a solution to 40 J.L. Chkareuli the strong CP problem [17]. In the SUSY context [19], the SU(3)F model leads to a special relation between (s)fermion masses and the soft SUSY breaking terms at the GUT scale in a way that all the dangerous flavor-changing processes are naturally suppressed. The special sector of applications is related to a new type of topological defects - flavored cosmic strings and monopoles appearing during the spontaneous violation of the SU(3)F which may be considered as possible candidates for the cold dark matter in the Universe [20]. Let us note in conclusion that if the family symmetry SU(3)F arises from the preon model proposed above one can expect that in the emerged SU(5) x SU(3)F GUT the gauge coupling constants g5 and g3F should be equal at the SU(8)MF unification scale. The study of flavor changing processes ^ —» e + y, D0 — D , B0 — B° and others caused by the SU(3)F gauge boson exchanges could in principle show whether the family symmetry has an origin in the preon model or it is, rather, independently postulated. However, the most crucial difference between these two cases is related to the existence in the preon model of some heavy SU(5) x SU(3)f multiplets located at scales from O(1) TeV up to the Planck mass. If they are relatively light, they may be of direct observational interest by them own. If they are heavy, they still strongly affect the quark-lepton mass matrices due to their large mixings with the down quarks and leptons, as was shown in (3.34). Remarkably, even if the family symmetry SU(3)F is taken at the GUT scale the difference between these cases is still left. Indeed, now all flavor-changing transitions due to the family gauge boson exchange will be extremely suppressed, while for the independently introduced family symmetry these transitions may significantly contribute into rates of the nondiagonal processes. Moreover, for the high scale family symmetry one has some natural candidates for massive right-handed neutrinos in terms of the extra heavy states given in (3.30). 3.8 Conclusion and outlook We have shown that, apart from somewhat inspirational religious and philosophical aspects ensured by the Eightfold Way, the SU(8) symmetry as a basic internal symmetry of the physical world is indeed advocated by preon model for composite quarks and leptons. In fact, many preon models have been discussed and considered in the past (some significant references can be found in [5,6]), though they were not turned out to be too successful and attractive, especially compared with other theory developments, like as supersymmetry and supergravity, appeared at almost the same time. However, there is still left a serious problem in particle physics with classification of all observed quark-lepton families. As in the hadron spectroscopy case, this may motivate us to continue seeking a solution in some subparticle or preon models for quarks and leptons, rather than in the less definitive extra dimension or superstring theories. Let us briefly outline the main results presented here. We have started with the L-R symmetric preon model and found that an admissible metaflavor symmetry SU(8)mf appears as a solution to the't Hooft's anomaly matching condition 3 Eightfold Way for Composite Quarks and Leptons 41 providing preservation of the accompanying chiral symmetry SU(8)L x SU(8)R at all scales involved. In contrast to a common point of view, we require that states composed from the left-handed and right-handed preons with their own metacolors, SO(3)MC x SO(3)MC, have to satisfy AM condition separately, though their triangle anomalies may compensate each other. The point is, however, this SU(8)mf theory emerges as the vectorlike theory with respect both to preons and composites. As a consequence, while preons are left massless being protected by their metacolors, all L-preon and R-preon composites being metacolor singlets will pair up and, therefore, acquire superheavy Dirac masses. It is rather clear that such a theory is meaningless unless the L-R symmetry is partially broken that seems to be a crucial point in our model. In this connection, the natural mechanisms for spontaneous L-R symmetry breaking have been proposed according to which some R-preons, in contrast to L-preons, may be condensed or such asymmetry may be caused by the properly arranged scalar field potential. As result, an initially emerged vectorlike SU(8) theory reduces down to the conventional SU(5) GUT with an extra local family symmetry SU(3)F and three standard generations of quarks and leptons. Though the tiny radius of compositeness for universal preons composing both quarks and leptons makes it impossible to immediately confirm their composite nature, the theory necessarily predicts a few special SU(5) x SU(3)F multiplets of composite fermions located at the scales from O(1) TeV up to the Planck mass scale that may appear of actual experimental interest. Some of them may be directly observed, the others populate the SU(5) GUT desert. Due to their mixing with ordinary quark-lepton families there may be expected some violation of unitarity in the mass matrices for leptons and down quarks depending on the interplay between the compositeness scale AMC and scale of the family symmetry SU(3)f. For the reasons of simplicity, we have not considered here boson composites which could appear as the effective scalar or vector fields in the theory. Generally, they will become very heavy (with masses of the order of the compositeness scale Amc) unless their masses are specially protected by the low-scale supersymmetry. The point is, however, that some massless composite vector fields could nonetheless appear in a theory as the Goldstone bosons related to spontaneous violation of Lorentz invariance through the multi-preon interactions similar to those given in the section 3.5 (3.22). In principle, one could start with a global metaflavor symmetry SU(N)MF which is then converted into the local one through the contact multi-preon interactions [11] or some nonlinear constraint put on the preon currents (see in this connection [23] and the later works [24]). If so, the quarks and leptons, on the one hand, and the gauge fields (photons, weak bosons, gluons etc.), on the other, could be composed at the same order distances determined by the preon confinement scale AMC. In other words, there may be a lower limit to the division of matter beyond which one can not go. Indeed, a conventional division of matter from atoms to quarks is naturally related to the fact that matter is successively divided, whereas the mediator gauge fields (photons, gluons, gravitons etc.) are left intact. However, situation may be drastically changed if these spontaneously emerging gauge fields become composite as well. We will address this and other related questions elsewhere. 42 J.L. Chkareuli Acknowledgments I am grateful to many people who had significantly contributed to the ideas presented here during the years when they were developed, especially to A.A. Anselm, V.N. Gribov, O.V. Kancheli, S.G. Matinyan and V.I. Ogievetsky. I would also like to thank the organizers and participants of the 20th International Workshop "What Comes Beyond the Standard Model?" (9-17 July 2017, Bled, Slovenia) M.Yu. Khlopov, N.S. Mankoc Borstnik, H.B.F. Nielsen and K.V. Stepanyantz for a warm hospitality and interesting discussions. References 1. M. Gell-Mann and Y. Ne'eman, The Eightfold Way (W.A. Benjamin, New York, 1964). 2. https://en.wikipedia.org/wiki/Noble_Eightfold_Path. 3. G. 't Hooft, in Recent Developments in Gauge Theories, edited by G.'t Hooft et al (Plenum, New-York, 1980). 4. J.L. Chkareuli, JETP Lett. 32 (1980) 671; J.L. Chkareuli, Composite Quarks And Leptons: From Su(5) To Su(8) Symmetry, In Proc. of the Int. Seminar QUARKS-82, pp 149-156, edited by A.L. Kataev (INR, Moscow, 1982). 5. R.N. Mohapatra, Unification and Supersymmetry (Springer-Verlag, New-York, 2003). 6. I.A. D'Souza and C.S. Kalman, Preons: Models of Leptons, Quarks and Gauge Bosons as Composite Objects (World Scientific, Singapore, 1992); H. Terazawa, Y. Chikashige and K. Akama, Phys. Rev. D 15 (1977) 480 ; H. Terazawa, ibid. 22 (1980) 184. 7. G. 't Hooft, Phys. Rev. Letters 37 (1976) 8. 8. R. Barbieri, L. Maiani and R. Petronzio, Phys. Lett. B 96 (1980) 63. 9. K.G. Wilson, Phys. Rev. 10 (1974) 2245; M. Creutz, Phys. Rev. Lett. 43 (1979) 553. 10. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. 11. J.D. Bjorken, Ann. Phys. (N.Y.) 24 (1963) 174; Per Kraus and E.T. Tomboulis, Phys. Rev. D 66 (2002) 045015. 12. A.A. Anselm, Sov. Phys. JETP 53 (1981) 23. 13. Z.G. Berezhiani and J.L. Chkareuli, Sov. J. Nucl. Phys. 37 (1983) 618; Z.G. Berezhiani, Phys. Lett. B129 (1983) 99; ibid B150 (1985) 177; Z. Berezhiani and M. Yu.Khlopov, Sov. J. Nucl. Phys. 51(1990) 935; T. Appelquist, Y. Bai and M. Piai, Phys. Lett. B 637 (2006) 245 . 14. F. Wilczek, AIP Conf. Proc. 96 (1983) 313. 15. Z. G. Berezhiani, J. L. Chkareuli, JETP Lett. 37 (1983) 338; Z.G. Berezhiani and J.L. Chkareuli, Sov. Phys. Usp. 28 (1985) 104; T. Appelquist, Y. Bai and M. Piai, Phys. Rev. D 74 (2006) 076001 . 16. J.L. Chkareuli, Phys. Lett. B246 (1990) 498; ibid B 300 (1993) 361. 17. Z.G. Berezhiani and J.L. Chkareuli, Horizontal Unification Of Quarks And Leptons, In Proc. of the Int. Seminar QUARKS-84, vol. 1, pp 110-121, edited by A.N. Tavkhelidze et al (INR, Moscow, 1984); Z. Berezhiani and M. Yu.Khlopov, Sov. J. Nucl. Phys. 51(1990) 739; T. Appelquist, Y. Bai and M. Piai, Rev. D 75(2007) 073005. 18. R.D. Peccei and H.R. Quinn, Phys. Rev. D16 (1977) 1891. 3 Eightfold Way for Composite Quarks and Leptons 43 19. Z.G. Berezhiani, J.L. Chkareuli, G.R. Dvali and M.R. Jibuti, Supersymmetry and generations of quarks and leptons, In Proc. of the Int. Seminar QUARKS-86, pp 209-223, edited by A.N. Tavkhelidze et al (INR, Moscow, 1986); Z. Berezhiani, Phys. Lett. B 417 (1998) 287; S.F. King and G.G. Ross, Phys. Lett. B 520 (2001) 243; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B 626 (2002) 307. 20. J.L. Chkareuli, Phys. Lett. B272 (1991) 207; G. Dvali and G. Senjanovic, Phys. Rev. Lett. 72 (1994) 9; D. Spergel and U.-Li Pen, Astrophys. J. 491 (1997) L67; S.M. Carroll and M. Trodden , Phys. Rev. D57 (1998) 5189. 21. P. Ramond, hep-ph/9809459. 22. M. Yu. Khlopov, Cosmoparticle Physics (World Scientific, Singapore, 1999). 23. Y. Nambu, Progr. Theor. Phys. Suppl. E 68 (1968) 190. 24. J.L. Chkareuli, C.D. Froggatt, J.G. Jejelava and H.B. Nielsen, Nucl. Phys. B 796 (2008) 211; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B 848 (2011) 498.