The Three Faces of U ( 3 )

: U ( n ) is a semi-direct product group characterized by nontrivial homomorphisms mapping U ( 1 ) into the automorphism group of SU ( n ) . For U ( 3 ) , there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of U ( 3 ) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a mod3 permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U ( 3 ) quantum numbers. One can interpret this as three species of matter fields in the defining representation.


Introduction
Off hand it seems U(3) is a phenomenologically uninteresting symmetry group for constructing a gauge field theory. After all, standard gauge theory lore says the symmetry group can be expressed as the direct product of semi-simple and U(1) groups, so the group should really be SU(3) × U(1). Moreover, SU(3) C × U(1) B with SU(3) C being the color symmetry of QCD and U(1) B coupling to baryon number was already hypothesized as a gauge symmetry in [1], but it was effectively falsified by experiment [2]. So it appears SU(3) × U(1) gauge symmetry is out of the question -at least for normal matter. But, as we will discuss, the insistence on direct products of semi-simple and U(1) factor symmetry groups is too restrictive and U(3) = SU(3) × U (1).
From where comes the lore? For a physically realistic gauge field theory, one must impose a positive-definite Ad-invariant real inner product on the gauge symmetry Lie algebra. And it is well known that the Lie algebra g of a compact real group G decomposes into a direct sum of semisimple s i and u(1) j factors i,j s i ⊕ u(1) j if and only if the Killing inner product on g is non-degenerate and hence positive-definite (see e.g. [3]). So for example, given a starting Lie algebra basis and following the argument of Weinberg [4, ch. 15], it is possible to explicitly exhibit a basis with totally antisymmetric structure constants that respects this direct sum decomposition relative to the Killing inner product; which on Hermitian elements is positive-definite, Ad-invariant, and real and therefore renders viable gauge invariant Lagrangians.
However, g may admit additional suitable inner products, and the semisimple decomposition is not the most general compatible with a non-Killing inner product: The conditions on the inner product allow a real compact reductive Lie algebra g = i,j s i ⊕ a j where a j are abelian, s i ∩ a j = 0 relative to the non-Killing inner product, and Ad(A j )s i ⊆ s i . [5,6] (Here A j is a subgroup of G generated by a j and Ad(·) denotes the adjoint action of G on g.) If a suitable non-Killing inner product exists, the decomposition with antisymmetric structure constants is not the only valid option. And, in a gauge field theory context, the two decompositions along with their inner products describe different interactions and therefore different theories.
It happens that U(3) is a connected, compact reductive real group possessing a two-parameter class of positive-definite Ad-invariant real inner products. So it is possible to construct a consistent gauge theory with U(3) symmetry. But unlike SU(3) × U(1) where the gauge field associated with U(1) completely decouples, all U(3) gauge fields will mutually interact as a true U(3) symmetry dictates. To ensure this happens implies certain non-vanishing conditions on relevant structure constants that might otherwise vanish. As a consequence, the structure constants are not totally antisymmetric: the inner product is not Killing, u There is no reason to favor one particular representation over another, so when constructing a gauge field theory coupled to fermions the most general Lagrangian contains the standard Yang-Mills term − 1 2 F ·F and fermion terms rψ D (r) ψ summed over the three representations. Consider permuting some chosen basis of C 3 with some unitary permutation matrix in L B (C 3 ). There are two classes of permutations: one class induces "small" gauge transformations and the other induces "large" gauge transformations. Of course, the small gauge transformations represent a redundant state description in the quantum version. In contrast, the large gauge transformations represent the non-abelian analog of charge conjugation, and they effect a genuine matter field re-characterization. They essentially permute r. Accordingly, the U(3) gauge symmetry allows the fermion contribution rψ D (r) ψ to be rewritten in a single representation as rψ r Dψ r where ψ r are three different types of fermion matter fields -each type a U(3) triplet characterized by three quantum numbers coming from the action of the Cartan subalgebra. This is our main result: The most general U(3) gauge invariant Lagrangian for fermions in a chosen defining representation includes precisely three generations of matter fields relative to an imbedding U(1) ֒→ SU(3) ⋊ U(1).
is not a realistic theory, there are reasons to believe there might be some kind of charge-carrying abelian gauge field beyond the Standard Model. Along these lines, many models incorporate a "dark photon" that interacts with a hidden matter field sector but may or may not interact with the Standard Model sector. The dark photon literature is quite extensive: For a review see [10] and references therein. The idea of appending a hypercolor symmetry group SU(3) H ×U(1) H to the minimal supersymmetric SU(5) GU T was studied in [11][12][13][14]. The extra factor group 2 resolves some shortcomings of the model, and it can be viewed as a D3 − D7 brane system in type IIB supergravity. The semi-direct product group (SU(3) C × SU(2) L ) ⋊ U(1) Y and anomaly cancellation were used by [15] to put constraints on matter field hypercharge. As evidenced by the literature, it is possible to construct interesting models with extra U(1) factors.
There have been previous attempts to explain three generations using a variety of mechanisms. Most notable perhaps are preon models [16][17][18][19], and super string models [20,21]. But there are also models based on non-anomalous discrete R-symmetry [22], extra dimensions with anomaly cancellation [23], and the anthropic principle [24].  Proof : The Lie algebra brackets are [u ab , u cd ] = δ bc u ad − δ ad u cb where u ab ∈ u(3) are a chosen Hermitian basis with a, b, c, d ∈ {1, 2, 3}. From these brackets it follows that the adjoint map is given by ad X (u ab ) = c x ca u cb − x bc u ac with X = a,b x ab u ab . Hence, The center of u(3) is span R {1}, and it is easy to see that K(1, X) = 0 for all X ∈ u(3). Positivity follows from the hermiticity of X, Y.
This suggests to define an inner product on the Lie algebra u(3) in the defining representation ρ : where the basis elements {Λ α } = {ρ ′ (u ab )} are 3×3 Hermitian matrices with α ∈ {1, . . . , 9} and the parameters g 1 , g 2 ∈ R obey 0 < g 2 2 < g 2 1 . It is clearly positive-definite, Ad-invariant, and real. For a triangular decomposition of the basis {Λ α } denoted by {S ± a , H a } with a ∈ {1, 2, 3}, the structure constants associated with the brackets [S ± a , H a ] differ from those associated with the Killing form. These structure constants, which are functions of (g 1 , g 2 ), characterize quantum numbers of nonneutral gauge bosons, and eigenvalues of {H a } characterize quantum numbers of matter fields.

Semidirect structure of U (3)
Mathematically, it is fruitful to view U(3) as an extension of a group H ∼ = U(1) by a normal subgroup N ∼ = SU(3) ⊳ U(3). This is represented by the short exact sequence Since f is injective and im f = ker π, then s(h)f (n)s(h −1 ) = f (n ′ ) for some unique n ′ (n, h) ∈ N that depends on (n, h). It is convenient to write ϕ h (·) ≡ n ′ (·, h) so that ϕ h : N → N. Note that ϕ h (id N ) = id N for all h ∈ H since s is a homomorphism.

Lemma 2.3
The function ϕ h ∈ Aut N.
proof : First, f (ϕ id H (n)) = f (n) implies ϕ id H (n) = n for all n ∈ N. Next, for n 1 , n 2 ∈ N, where we used s is a homomorphism. On the other hand, from the definition of ϕ h , we have n 1 n 2 )). Injective f then implies ϕ h (n 1 n 2 ) = ϕ h (n 1 )ϕ h (n 2 ). ⊟ On the other hand, And, since the decomposition U(3) = NH is unique (which we won't bother to prove), the homomorphism θ −1 is bijective. One can go on to show that the semidirect product reduces to a direct product if and only if H ⊳ U(3); in which case N and H commute and ϕ is trivial.
Observe the homomorphismφ : s(H) ∼ = H ∈ N ⋊ ϕ H → Aut N induced by s is given bỹ In this sense,φ induced by the section s coincides with ϕ. It is important to note that there may be multiple homomorphisms ϕ and hence multiple sections s that render a semidirect product. Physically, a non-trivial ϕ corresponds to a direct interaction between the gauge fields of the respective subgroups.
In particular, for the matrix group U(3) as a semidirect product, there exist three such nontrivial sections; where ω ∈ R. Each section gives rise to a different conjugation of SU(3) by s(h), and each of these induces a different representation ̺ r : H → L B (C 3 ) where r ∈ {1, 2, 3}. These can then be extended to three defining representations ρ r : U(3) → L B (C 3 ).

Lagrangian matter field term
Given the existence of three representations, the remaining argument is rather elementary. The decisive step is to insist that all allowed defining representations be included in the Lagrangian; Proposition 2.5 The matter field portion of the Lagrangian of a gauge field theory must include all allowed defining representations.
Accordingly, in a toy model of Yang-Mills coupled to a massive matter field in the defining representations, the gauge field term uses the chosen inner product 1 2 F, F with F ∈ u(3) and the matter field term will be r iΨ / D give rise to a different propagator and hence a different renormalization of the r-dependent mass parameter, gauge fields, and matter fields. The renormalized matter field term is then r iΨ / D In effect, the quantum theory distinguishes the classically isomorphic vector spaces carrying the defining representations. Now, there exists a class of elements in U(3) of the form with θ 1 (x), θ 2 (x), θ 3 (x) ∈ R. The adjoint action of P (x) on the Lie algebra u(3) leaves the normal subalgebra su(3) invariant and cyclically permutes the generators of the s(H) matrices diag(iω, 0, 0) Similarly, P −1 (x) = P † (x) permutes in the reverse direction. Crucially, P 3 = e i(θ 1 +θ 2 +θ3) diag(1, 1, 1). We claim that θ 1 (x) + θ 2 (x) + θ 3 (x) = ±(2n)π with n ∈ N induces small gauge transformations while θ 1 (x) + θ 2 (x) + θ 3 (x) = ±(2n + 1)π induces large gauge transformations: The latter cannot be reached by a gauge transformation homotopic to the identity because det P = −1. 3 It then follows from tr log P = iπ(2k + 1) that log P in this case involves a combination of Cartan generators (not present in the small permutation case) that contributes a multivalued mod 3 phase to matter field configurations, and it transforms between three physically distinct classes of gauge field configurations that survive gauge fixing in the quantized theory. Given P we have / D

Summary
Our analysis started with the observation that U(1) gauge symmetry can be incorporated into gauge field theories via semi-direct products and not simply as direct products. In particular, for U(3) the construction of the semi-direct product is not unique; it comes in three versions. We argued these three versions can be interpreted as three generations of matter fields. The interpretation relies on including all three versions of the semi-direct product in the Lagrangian, the large-gaugetransformation status of certain permutation operators, and the identification Ψ r := P r−1 Ψ. This perspective can be turned around: One can view fermions as a single field, and different fermion masses are just a manifestation of the three faces of U(3) in the defining representation. We did not consider U(2) as a replacement for SU(2) × U(1), but off hand the same mechanism would appear to apply and it should be studied in the context of spontaneous symmetry breaking. Of course our toy model is far from being realistic, and further investigation is required to determine if U(3) is phenomenologically feasible. We conclude by noting once again that the gauge-field interactions for U(3) differ considerably from the SU(3) × U(1) case. All of the gauge fields associated with the Cartan subalgebra of U(3) take part in both gauge and matter field interactions. So if there is somehow any vestige of a long-range charge carrier coming from U(3), it will couple to both gauge and matter field mass-energy and therefore have a chance of being consistent with gravity. Less clear and perhaps more imperative is whether U(3) can agree with QCD. A companion paper begins to explore this question.