{'fraction_non_alphanumeric': 0.04082717707633759, 'fraction_numerical': 0.008954555630176853, 'mean_word_length': 4.371561701488051, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Family Unification from Universality

A direct consequence of the occurrence of fermion families is the invariance of currents under certain groups of (universality) transformations. We show how these universality groups can themselves be used to find and study grand family unification models. Identifying two independent - weak and strong - universality groups and assuming that the grand unification group is SU(8N), its subgroup respecting either weak or strong universality is shown to be G = SU(2)xU(1)xSU(3). The fundamental representation of SU(8N) decomposes as N families of leptons and quarks. In the G-invariant limit, all fermions are left-handed. A mechanism for generating the correct number of right-handed fermions with the correct couplings so as to give pure vector colour and electromagnetic currents is exhibited. Universality is shown to result most naturally from a preonic structure of fermions. In such a preonic picture there are no ultraheavy gauge bosons and no anomaly or hierarchy problem.


Introduction
The problem of accommodating the existence of several replicas of a family of leptons and quarks in unified theories of all their interactions (except, of course, gravitational) has preoccupied model makers for more than two decades now. Beginning with ad hoc impositions of invariance under (finite [1] or Lie [2]) groups of so called horizontal symmetries, this endeavour soon moved on to the recognition that certain orthogonal groups G have subgroups G and representations ρ such that the restriction of ρ to G is a direct sum of copies of a unique irreducible representation of G, making such groups plausible candidates for family unification [3]. Such models, however, have certain fundamental difficulties (primarily to do with unacceptably large right chiral interactions) which have resisted a satisfactory natural resolution. Subsequently, attention turned increasingly to supersymmetric models; in particular, very detailed work on superstring -inspired family unification models are going on apace now [4]. At the same time it has recently been shown that conventional nonsupersymmetric unification models, when combined with the topological properties of the Higgs phase of nonabelian gauge theories -i.e., going beyond the perturbative regimehave the richness to accommodate the family structure [5,6].
The approach of the present paper to dealing with families is, by and large, orthodox, at least in its fundamentals; supersymmetry is not invoked and topological aspects of the reduction of a unifying gauge group G to an observed or effective gauge group G are ignored in the Lie algebraic considerations below. It is also a pragmatic and phenomenological approach. Instead of looking for the source of the family structure, we shall focus primarily on its most characteristic empirical signature, namely the observed universality properties of the gauge interactions of all leptons and quarks, 8N in number where N is the number of families. We shall seek to determine a simple group G such that the subgroup of G whose currents respect the demand of universality is the (unbroken) standard model group G = SU(2) × U(1) × SU (3).
In the actual implementation, as described in the next section, we proceed as follows. Ignoring fermion masses and mixings, i.e., before G is further broken to U(1) em × SU (3), the G-gauge interactions of fermions are separately invariant under two types of -weak and strong -universality transformations forming two distinct subgroups S wk and S st of G. The G-gauge currents which are invariant under S wk (respectively S st ) are easily shown to couple to the gauge bosons of a subgroup G wk (G st ) of G which is the centraliser of S wk (S st ), namely the group of elements of G which commute with all elements of S wk (S st ). Moreover, the gauge group whose currents are invariant under either S wk or S st is G wk × G st . After identifying S wk and S st (taking account in particular of the fact that leptons have no strong interactions), it is established that if G is chosen to be SU(8N), then G wk = SU (2) and (3). Conversely if we want to have G = G wk × G st to be SU(2) × U(1) × SU (3), then G can only be SU(8N). The fundamental representation 8N decomposes under SU(2) × SU (3) as N copies of the representation (2, 1) ⊕ (2,3) i.e, N families of leptons and quarks. Thus, by embedding SU(2) × U(1) × SU (3) in SU(8N) via the imposition of weak and strong universalities, SU(8N) is made to play the role of a "grand" family unification group (no other simple group can serve this purpose) [7]. In particular, independently of the mechanism for the reduction of the gauge group from G to G (more fundamentally, even in ignorance of the true source of universality), the relative coupling constants of SU(2), SU (3) and U(1) get fixed.
It is noteworthy that the ("weak") hypercharge U(1) reflects the absence of strong interactions for the leptons (as it perhaps should since, empirically, it does distinguish between leptons and quarks). Other indications that the model is somewhat off the beaten track also become apparent at this point. For instance, the SU(2) coupling is smaller, in relation to SU (3) and U(1), by a factor √ 2 than in most conventional unification models, resulting in a (unification) value of sin 2 θ of 3/4. Moreover, since SU (2) currents must be left-chiral, all fermions are forced to be left-handed (on the scale of the G gauge theory), making the U(1) and SU(3) currents also left-chiral. Thus, in the G-gauge invariant limit, all fermions are massless and chiral symmetry is exact in the strongest possible form: there are no right-handed (R-) fermions to couple to L-fermions so as to generate masses.
This circumstance suggests strongly that the mechanisms responsible for the restoration of parity invariance of electromagnetic and colour interactions and for the appearance of fermion masses are one and the same, namely a suitable dynamical breaking of chiral symmetry, and that this mechanism is closely linked to the spontaneous breaking of SU (2) As a first step towards justifying such a linkage, we fall back on the standard Higgs mechanism for breaking SU(2) × U(1) to U(1) em . Accordingly, in the G theory, we take the Higgs to constitute one 8N representation, decomposing as N families of leptonic Higgs (∼ (2, 1) under SU(2)×SU (3)) and N of coloured Higgs (∼ (2,3) the U(1) em coupling strengths of R-fermions are also easily computed; they match precisely those of the corresponding L-fermions, making the electromagnetic current also pure V .
In the last section, we turn to the problem of the physics underlying the reduction of the primitive group G to the standard model group G. After exhibiting a simple Higgs representation that will achieve this, a more directly physical explanation for the manifestation of universality is sought in the very natural idea that the (L-) fermions are themselves composites of a set of preons or metafermions. The simplest explicit preonic model with built-in universality then requires 6 + N preons (2 for flavour, 4 for colour including leptonic colour and N for family) [9]. All of them have meta-interactions binding them into massless, L-chiral, leptons and quarks, but 6 of them have, additionally, SU(2) × U(1) × SU(3) gauge interactions -the photon, gluons, W and Z are thus elementary. In this perspective, our unifying SU(8N) is just 't Hooft's metaflavour group [10] unbroken as long as fermions are massless, but broken softly by the spontaneous generation of R-fermions and the concomitant masses -the gauging of only a subgroup of the metaflavour group is no more than a reflection of the fact that some of the preons already have such gauge interactions.
Obviously, these speculative ideas, touched upon only briefly in the concluding section, need to be worked out in detail. Another area for further work is the dynamics of chiral symmetry breaking. Particularly worthy of attention is the possibility that the Higgs fields themselves have a dynamical origin, i.e., that the spontaneous G-symmetry breaking and the chiral symmetry breaking both arise from some sort of Nambu-Jona-Lasinio mechanism involving the L-fermions. It is nevertheless encouraging that the purely group-theoretic and kinematic foundation of such an enterprise, as described in this paper, has proved to have no serious drawbacks.

2A. Implementing Universality
Considering all leptons and quarks together, a fermion is labelled by a family index i = 1, . . . , N, N assumed arbitrary, a flavour index f = 1, 2 ( 1 is "up" and 2 is "down", say) and a generalised colour index α = 0, 1, 2, 3, in fact it is a unique and nontrivial feature of the model that right-handed fermions cannot exist in the gauge -invariant limit of the model; they will be generated spontaneously (along with masses) in a very natural way.
The basic strategy pursued in this paper is to postulate first that underlying the standard G = SU(2) ×U(1) ×SU (3)) gauge model there is a unifying or embedding gauge theory with group G, treating all fermions on an equal footing and, then, to determine G as the group having the property that its subgroup respecting all observed universality properties is G. In the actual execution, it is simpler to proceed by choosing G as the smallest simple group that will suffice, namely G = SU(8N), and then to verify that its subgroup satisfying weak and strong universality is indeed G. Indeed, though the Higgs mechanism for reducing G to G certainly remains a viable option, we are now free to explore other, physically more compelling, reasons for the manifestation of universality and family structure. One such speculative possibility will be suggested at the end.

2B. Weak Universality
As already stated above, we follow conventional wisdom and begin by supposing that the weak interactions of leptons in the unbroken standard G gauge theory are unchanged by an arbitrary permutation π of the family index of the up leptons and the same permutation of the down family index.
However, this formulation of weak universality is incomplete. Firstly, if we ignore, once again, family mixings (i.e., at the G-invariant stage), the observed weak currents of a definite colour are of the general forms i u iα Γd iα (and its conjugate), u iα Γu iα and d iα Γd iα where Γ are (different) matrices which do not operate on i. So a unitary transformation of the family index i will commute with Γ; the currents are invariant not just under the discrete permutations π but the more general where U in both transformations is the same unitary matrix [11]. Next, as far as the weak currents are concerned, quark colour is no different from the family label; the coupling strengths of the currents do not depend on the quark colour and the matrices Γ do not operate on them. We may therefore immediately extend weak universality to encompass unitary transformations on the pair of indices (i, c = 1, 2, 3). Moreover the only empirical reason for not extending it to include leptons also would appear to be the lack of equality of the coupling strengths of leptons and quarks to the photon and the Z boson. The couplings of the physical γ and Z to fermions are however fixed only after SU(2) × U(1) ⊂ G invariance is broken down to U(1) em invariance and the neutral bosons mixed and hence this apparent reason is not so compelling. This leaves the theoretical objection that the so called weak hypercharges in the standard SU(2) × U(1) model does distinguish between quarks and leptons. The remarkable fact is, as will become clear in the next subsection, that in our unification scheme the hypercharge is not an attribute of weak but of strong universality and is a measure of the absence of strong interactions for the leptons.
In the light of the above considerations, we take the maximal weak uni-versality group S wk as the group consisting of unitary transformations with matrix elements U iα,jβ , i, j = 1, . . . , N, α, β = 0, 1, 2, 3, applied to the fermion basis |i, f, α leaving the flavour f unchanged: Abstractly, S wk is the group U(4N). If we write V as a tensor product of spaces spanned by family, flavour and colour basis vectors, In this abstract sense, weak universality is just the statement that, as far as weak interactions are concerned, family and colour directions in V can be chosen as an arbitrary orthonormal basis for V f am ⊗ V col .
In accordance with the general considerations of section 2A, the subgroup of G = SU(8N) whose currents respect weak universality is the centraliser C(S wk ). As a (8N ×8N) matrix group on V , S wk consists of pairs of identical U(4N) matrices s wk acting on V f am ⊗ V col : We write this as so that It is very easy to determine the group G wk . In the basis used in Eq. (4) (i.e., a fixed flavour labelling the first 4N rows and columns), write a general element g ∈ SU(8N) as the matrix with each g f f ′ a 4N ×4N matrix. For g to commute with s wk , each submatrix g f f ′ must commute with x wk and since x wk is an arbitrary matrix, each g f f ′ is trivial: where a f f ′ are complex numbers (the subscript on 1 indicating dimension will prove its usefulness soon). Hence a general element of G wk is of the form a f f ′ being arbitrary complex numbers such that g wk is unitary and has deter- The condition det g wk = (det a) 4N = 1 allows us to rewrite this as the diagonal group D 4N U(2) 4N , isomorphic to U(2) 4N , the group of unitary matrices having determinant equal to any 4Nth root of unity. This is obviously not a connected Lie group; its connected component is SU (2).
Since the currents in a gauge theory are completely specified in terms of the Lie algebra of the gauge group, possible lack of connectedness (and simple-connectedness) of the group is immaterial in the (perturbative) calculation of any process. Nonperturbative results may depend on the topology of the group, but such considerations are not pursued in this paper. With this qualification, we have thus shown that It is immediately evident that the 8N basis vectors {|i, f, α } of the fundamental fermion representation of SU(8N) form 4N copies of the fundamental representation of SU (2), the SU (2) acting on flavour and the copies labelled by family and colour. The relationship between weak universality and family structure has thus been made precise and concrete by the use of SU(8N) as an embedding or unifying group.

2C. Strong Universality
The formulation of a strong universality principle for quarks proceeds in a manner very similar to that of weak universality: the colour currents of quarks responsible for strong interactions are universal in the sense that they are sums over family and flavour, with a common coupling constant. This feature is usually recognised as being the reason for the phenomenon loosely called flavour invariance (approximate because of quark masses). Exactly as above, the strong universality group is then D 3 U(2N). Ignoring the existence of leptons for a moment and taking the primitive gauge group to be SU(6N), the resulting strong gauge group would then be S(D 2N U (3)).
As before, the strong gauge group G st is the centraliser of S st in SU(8N).
To compute it, write g ∈ SU(8N) as the matrix (in the basis used in Eg. (10)): each g αα ′ being a matrix on V f am ⊗ V f l . For g to commute with s st , we must have (2N) and c, c ′ = 1, 2, 3. These conditions are solved by where b (l) is an arbitrary complex number with |b (l) | = 1 and b (q) is a unitary 3×3 matrix, repeated 2N times along the diagonal. The matrix b operates on V col whose leptonic and quark subspaces are distinguished by the subscripts.
Thus, as a (8N × 8N) matrix group, the strong gauge group is The unit determinant condition says that b 2N (l) (det b (q) ) 2N = 1; this puts no restriction on det b (q) but merely says that its value is fixed in terms of the arbitrary phase b (l) . Hence G st is connected and is isomorphic to U(3). Restricting ourselves to the Lie algebra and ignoring topological niceties once again, we therefore write It is clear that the SU (3)  This will be ensured if we can conclude that the effective gauge group G is just the direct product of G wk and G st and that, in turn, requires that G wk and G st are disjoint as subgroups of G and that they are mutually commutative. These two properties are immediately obvious for the Lie algebras of G wk and G st (e.g., colour generators and flavour generators commute) and hence for the groups themselves if they are connected and simply connected.
Since we have already chosen to confine attention to the Lie algebra we may assert that the subgroup of G = SU(8N) describing the effective gauge theory which respects either weak or strong universality is indeed Under the SU(2) × SU (3) subgroup of G, leptons of each family transform as the representation (2, 1) and quarks of each family as the representation (2,3). (m and n in the notation (m, n) are the dimensions of SU(2) and SU (3) representations). The U(1) transformation properties of (i.e., the U(1) coupling constants to) the various fermions will be determined in the next subsection -it will turn out that U (1) (3) currents are pure V then poses a problem.
The usual ways of introducing R-fermions in a "grand" unified model broadly fall into two distinct strategies: i) They belong to a different representation of a simple unifying group G chosen carefully so that when G is broken to G, the parity properties of currents come out right. The prototype of this method is of course the SU(5) model [12] in which this is done for each family separately. ii) They transform trivially under G but nontrivially under The gauge bosons of the standard model then result from a mixing of the bosons of G and G ′ , induced by spontaneous symmetry breaking. The prototypes here are left-right symmetric models [13]. Such strategies are not natural for us. For the first option to work, the representation of SU(8N) to which R-fermions are to be assigned must evidently have dimension less than 8N and there are no such. The second option is excluded by universality itself. Observed universality does not discriminate between V − A (charged weak) and V (electromagnetic and colour) currents: if we apply universality to, say, G L × G R as the unifying group, we shall end up with G L × G R as the effective group, requiring further (Higgs?) gymnastics to end up with the correct V − A and V currents.
Thus, accepting universality as the sole guide in the choice of the unifying group forces us to look for novel ways of understanding the parity properties of currents. A possible solution of this problem is described in section 4B.

3B. Coupling Constants
The ratios of the squares of the coupling constants of SU(2) and SU (3) currents and of each individual fermion U(1) current are determined by the way G is embedded in G. (It is useful to remember that the physical mechanism of symmetry reduction is immaterial for this purpose). Fix a normal- for the generators of SU(8N). The generators of the SU(2) subgroup can then be written, as SU(8N) generators, as the matrices T a = 1 2 g diag (τ a , τ a , . . . , τ a ), a = 1, 2, 3, in the flavour basis, where τ a are the Pauli matrices (tr τ 2 a = 2), i.e., as the matrix having τ a repeated 4N times along the diagonal and all other entries zero. The normalisation (15) fixes the value of g 2 : i.e., Similarly we write the generators of the SU(3) subgroup as L n = 1 2 f diag (0, λ n , 0, λ n , . . . , 0, λ n ), n = 1, . . . , 8, in the colour basis, where λ n are the Gell-Mann matrices (tr λ 2 n = 2), with (0, λ n ) repeated 2N times along the diagonal. So The U(1) group commutes with SU(2) and SU (3) and hence the hypercharge Y ′ as a generator of SU(8N) commutes with {T a } and {L n }. Therefore (for real numbers g ′ and g ′′ ) in the flavour basis. Equivalently, in the colour basis, with 2N repetitions. Since Y ′ is traceless, and we write, to conform to usual practice, in the flavour basis. The correct ratio of the lepton and quark hypercharges follows strictly from the simplicity of SU(8N) and the fact that there are three colours of quarks and one of leptons (it is not directly related to baryon and lepton numbers). The common coupling constant g ′ is fixed by the normalisation δ = tr Y ′2 to be The ratios of the coupling constants are different from the almost canonical values expected [14] in a sequential embedding of the standard model in any simple unifying group with fermions in the (2, 1 The first of these numbers, in particular, shows that the unification regime of the model has to be at substantially higher energy then we have been used to (see later).

4A. The Choice of Higgs Fields
So far we have avoided specifying the mechanism that is responsible for reducing the gauge symmetry from the primitive ("would-be") gauge group G to the effective one of the standard model. In our model this mechanism is no different from that responsible for the universality properties of the G-gauge theory; hence, it is possible to think that looking for the underlying physical causes of universality might offer options other than the search for an astute Higgs representation of G. We postpone this question to the concluding section.
In contrast, the breaking of G to SU(3) × U(1) em is attributed in this paper to a conventional Higgs mechanism. The main reason is that our real concern is how to understand the (pre-Higgs) standard model and its observed systematics in terms of a unified model, rather than in the details of how it is further broken. In any case, the standard Higgs mechanism is extraordinarily successful compared to the many efforts to supplant it or to find a more fundamental explanation for it. At the same time, it is very simple -compare the complicated Higgs multiplets required in most (one-family) grand unified models. An additional reason is the central role the Higgs fields play in our approach to chiral symmetry breaking and in generating R-fermions. (See, however, the remark at the end of the Introduction).
Accordingly, we assume that the G gauge theory has one Higgs field Φ transforming as the defining (fundamental)  In addition, we have the colour triplets of quark-like Higgs (which may or may not be massive). They are presumably confined and cannot have SU (3) invariant couplings to quarks -indeed, Φ has no SU(8N) invariant coupling to the fermion 8N. They can interact with gluons however and so will form part of the "sea" in hadrons.

4B. Right-handed Fermions
We have already noted (Sec. 3A) the difficulty of introducing R-fermions in the G gauge model in such a way as to insure, in the effective G gauge theory, that SU (2)  Explicitly, our proposal is that R-fermions are composites of L-fermions and Higgs bosons in certain specific channels. First, the binding must be orbitally excited (to obtain the chirality flip) and must be, in a first approximation [15], within the same family (to preserve the universality properties of realistic Dirac fermions). Dropping the family index and writing ψ L for left handed Weyl spinor fields of a given SU(2) × SU(3) multiplet, the composite field ψ L ∂ µ φ transforms under the Lorentz group as 1 2 ) of which the right-handed (0, 1 2 ) component can be projected out by taking where σ 0 = 1 and σ i are the usual spin matrices. Evidently this equation can have, at this stage, only a schematic meaning, as a means of keeping track of quantum members. At the very least, we have ignored the need for proper regularisations of quantum composite fields. The dynamical problems which will have to be solved in implementing this idea cannot obviously be broached here. Nevertheless, Eq. (21) can already be seen to have an intriguing consequence. On writing it is clear that if both ψ L and ψ L φ are (L-projections of) massless Dirac fields, ψ R vanishes; right handed components can be generated dynamically in this way only if mass is also simultaneously generated. In the rest of this section, we shall disregard all dynamical problems concentrating only on the quantum number aspects.
In general ψ L φ composites can transform under SU(2) × SU(3) as any of the irreducible representations occurring in the decomposition of (2, 3 ⊕ 1) ⊗ (2, 3 ⊕ 1). To delimit the number of possible irreducible fields, let us assume that ψ L φ binding takes place for exactly the same irreducible representation (with respect to G) of φ which also condenses in the vacuum, namely the lepton-like (colourless) representation φ ∼ (2, 1), and, furthermore, in the antisymmetric tensor product ∧ of SU (2) representations. Then We now identify the two irreducible representations on the right as the down components of R-quarks and R-leptons respectively. Explicitly, in a general gauge, and The up R-fermions are correspondingly obtained by replacing φ by its conjugate, φ c : The subscripts u and d on R-fermions (as distinct from L-fermions for which they are written as arguments) are meant only to indicate in advance which composite will turn out to be the R-partner of which L-fermion in the currents coupling to U(1) × SU (3) -they are all invariant under SU (2). The totality of R-fermions falls into two sets of quarks ψ Ru (3) and ψ Rd (3) and two sets of leptons ψ Ru (1) and ψ Rd (1) in each family. This is exactly the pattern required by the standard model.
Finally, to fix the parity properties of the total fermionic currents of U(1) and SU (3), we need to know the corresponding coupling constants of the R-fermions. The transformation properties of ψ R under SU(2) × SU (3) immediately imply that no R-currents couple to SU (2) and no leptonic R currents couple to SU(3). The R-quark coupling to SU(3) is determined, just by SU (3) gauge invariance, to be the same as the L-quark coupling (= the intrinsic gauge coupling constant of SU (3)). Hence all SU (3) currents are pure V . The fermion couplings to abelian groups such as the hypercharge U(1) or (equivalently) the electromagnetic U(1) are not fixed in this way as there is no intrinsic gauge coupling defined by the gauge field; they have to be determined for each fermion individually. But being abelian charges, they are additive in each composite and we may compute them directly and easily.
For the electric charge, we get and, similarly, Hence the electromagnetic current is also pure V . It is pertinent to stress that the electric charges of ψ L and φ are themselves fixed completely by the embedding of the hypercharge in SU(8N) and by the Higgs structure of the model; they are not assigned a priori.
Thus the assumption that there exists an attractive force between Lfermions and Higgs in a specific "channel" corresponding to unique angular momentum, colour and flavour selection rules, strong enough to bind, leads to exactly the right global quantum numbers for the R-fermions. To make the picture complete, this is of course not enough. One needs to be able to construct local field operators for ψ R such that they transform correctly also under local gauge transformations, in other words, deal with the dynamical problems alluded to earlier. In conventional (unconstrained) quantum field theories, the way to construct local field operators for composites has been known since long [16], but its generalisation to gauge theories remains an open problem.

Open Questions
Of the questions we have so far left unaddressed, the most pressing is that of the physical mechanism which reduces the primitive gauge group SU(8N) to SU(2) × U(1) × SU (3) and, conversely, the significance and regime of validity of full SU(8N) gauge invariance. It may be reassuring to note at the outset that it is possible to find a Higgs representation which will serve the propose. It is shown in the Appendix that when a gauge group G is spontaneously broken to a subgroup G which is the centraliser of a group S ⊂ G, we can always find a set of Higgs fields belonging to the adjoint representation of S and a set of non-zero vacuum values for them such that their little group (stabiliser) is precisely G; it is also shown there that two copies of the adjoint representation of the family SU(N) group, suitably embedded in the adjoint representation of SU(8N), are sufficient to break If the Higgs option is chosen, then conventional wisdom dictates that the vacuum values of the adjoint Higgs will determine the energy scale at which unification will hold and all 64N 2 − 1 currents of SU(8N) will be operative. The relatively large values of sin 2 θ and of f 2 /g 2 that we have found in section 3B indicate that this unification energy (and hence the masses of the exotic (non-universal) gauge bosons of SU(8N)) will be many orders of magnitude larger than the 10 15 GeV or so that we are so used to and, even, the Planck mass. Conceptually, this is unknown teritory; in any case no reasonable physical sense can be attached to a procedure of evolving low energy parameters beyond the Planck mass while ignoring gravitational effects in the renormalisation group equations.
The other question touching on the magnitude of the unification energy scale is that of anomalies. Though the SU(8N) gauge theory is not anomalyfree, our low energy "universal" model, being indistinguishable from the standard model, is: the dynamically generated R-fermions cancel precisely the triangle anomalies arising from the L-fermions as long as no non-standard gauge boson internal lines occur in loops. Such loops will technically be non-renormalisable; however, their contribution will be negligibly small in any regularised computation, as long as the cut-off is much smaller than the unification scale. This is just a loose paraphrase of the set of results collectively known as decoupling theorems. Since gravitational effects effectively preclude a cut-off above the Planck mass, we need not take the demand of technical renormalisability too seriously at this stage. In any case, these points are only of academic interest if, as we suggest below, the reduction G −→ G is caused by a mechanism other than that of Higgs fields.
The option that we favour is to take leptons and quarks to be composites of a set of preons (or metaparticles) and to attribute the distinct gauge interactions of each fermion to the preons which themselves have the same gauge interactions. The model we are thus led to is one of the earliest (and conceptually simplest) preonic models proposed [9]: a fermion is assumed to be a composite of three types of spin 1 2 preons, |i, f, α = |i ⊗ |f ⊗ |α .
All of them are subject to a meta-interaction which is responsible for the binding (and about which it is futile to speculate at this stage). In addition, the flavour preons |f have an SU(2) gauge interaction and the quark-like colour preons |α = c , c = 1, 2, 3 have an SU(3) gauge interaction. If the preons are all L-chiral and the binding is assumed to be without "orbital excitation", the composites necessarily belong to one of the irreducibles in the representation of the Lorentz group. Since there are good reasons why massless particles of spin 3 2 are unlikely to exist [17], we end up with the right number of L-chiral spin 1 2 fermions if we assume that only one of the two ( 1 2 , 0) representations binds. In any case, chiral symmetry is valid strongly and the composite fermions are strictly massless. We have then a "metaflavour" group U (8N) which, when the effects of instantons are taken into account, leaves SU(8N) as the group of exact symmetries of all interactions of the composite fermions [10]. Thus a preonic picture provides a natural reason why the unifying group It is nevertheless important to highlight the one significant difference between the picture that emerges here and the one visualised by 't Hooft [10]. Once it is accepted that the universality properties of G gauge theory reflect the existence of flavour and colour preons |f and |α we are obliged to ascribe the (low energy) G-gauge interactions to the preons themselves.
They are not some residual interactions left over from the meta-interactions binding them, even less the manifestation of "spectator" gauge bosons. (An instructive down-to-earth parallel is provided by a set of nuclei having the same mass number but differing atomic numbers -their electromagnetic interactions at energies low enough for nuclear structure effects to be ignored arise from and are fully determined by the electromagnetic interactions of the proton and the neutron). In particular, the gauge bosons of G are themselves elementary.
We conclude by noting that there is no conceptual problem in having a "grand" unifying group G of which only a subgroup G is actually a gauge group with gauge bosons associated to its generators. The determination of the gauge coupling constants of G by embedding it in G is independent of whether all of G is gauged. The matrix elements of universality-violating currents are suppressed, not because they couple to superheavy gauge bosons, but because the states between which the matrix elements are taken respect universality. Some of the problems occurring in traditional "grand" unification models like the naturalness and hierarchy problems are then no longer relevant. Also, we need no longer worry about anomalies connected with G as along as the G theory is anomaly-free, which it is.
C(C(S)) has S as a subset. Moreover, since every x ∈ C(S) commutes with every g ∈ C(C(S)), C(S) centralises C(C(S)) and C(C(S)) is the maximal subgroup of G containing S and centralised by C(S). This observation is useful if we wish to find a subgroup S such that C(S) is a given subgroup, in our case G wk ×G st = C(S wk )×C(S st ). We have C(C(S)) = {g ∈ G | gx = xg, ∀ x ∈ G wk and gy = yg, ∀ y ∈ G st } = C(G wk ) ∩ C(G st ) = C(C(S wk )) ∩ C(C(S st )). It follows that S = S wk ∩ S st is the maximal subgroup centralised by G wk × G st .
Writing the defining representation space of SU(8N) as we may characterise S wk and S st as q (V f am ⊗ V f l ).
As the notation makes clear, S wk is the diagonal product with respect to flavour of the unitary group of V f am ⊗V cal and similarly for S st . The advantage of this explicit notation is that we can read off the intersection of S wk and S st as the group where, in the two factors on the right, the diagonal products are over V f l ⊗V q and V f l ⊗ V l respectively. Therefore S is isomorphic to U(N) × U(N).
In the argument given in section 2 for finding G given S wk and S st , these general considerations were unnecessary and were dispensed with. They become very useful however in finding a minimal Higgs scheme for reducing G to G. For this purpose, we begin by noting the standard identification of the Lie algebra of G with the vector space of the adjoint representation ρ ad of G.
If {t A } is a basis of matrices for Lie G and ξ ∈ V ad is a vector in the adjoint representation, with components {ξ A }, this identification is On the matrix X, the adjoint action of G corresponds to the usual one of conjugation by g ∈ G: ρ ad (g)ξ ←→ g X g −1 .
Under this identification, therefore, the centraliser of exp(iX) ∈ G corresponds to the set {g ∈ G | ρ ad (g)ξ = ξ}, in other words, the little group of ξ.
On the other hand, we have the general characterisation of the non-zero vacuum values of the Higgs fields as a numerical vector η of the representation (in general reducible) of G to which the Higgs belongs, such that the little group of η in G is the unbroken subgroup G. Hence when G is given as the centraliser of some subgroup S, we may conclude that i) the Higgs fields can be assigned to (sums of copies of) the adjoint representation of G, and ii) their