Complete CKM quark mixing via dimensional deconstruction

It is shown that the deconstruction of $[SU(2)\times U(1)]^N$ into $[SU(2)\times U(1)]$ is capable of providing all necessary ingredients to completely impliment the complex CKM mixing of quark flavors. The hierarchical structure of quark masses originates from the difference in the deconstructed chiral zero-mode distribution in theory space, while the CP-violating phase comes from the genuinely complex vacuum expectation value of link fields. The mixing is constructed in a specific model to satisfy experimental bounds on quarks' masses and CP violation.


I. INTRODUCTION
Dimensional deconstruction [1], [2] is a very interesting approach to dynamically generate the effects of extra dimensions departing from the four-dimensional (4D) renormalizable physics at ultraviolet scale. That is, apart from having the viability in the sense of renormalizability, whatever amusing mechanisms being dynamically raised by the virtue of extra dimensions (ED) now can also be easily arranged to rise dynamically in a pure 4D framework.
In this paper we look specifically into two such important mechanisms of extra dimension theories, namely the localization of matter fields in the bulk [3,4,5] and the dynamical breaking of CP symmetry by ED Wilson line [6,7,8,9]. Ultimately, the hybrid of these two mechanisms is just the well-known complex mixing of fermion flavors. And it is conceptually interesting to note that dimensional deconstruction (DD) nicely encompasses both of these issues. In other words, complete Cabibbo-Kobayashi-Maskawa (CKM) mixing can be generated naturally via dimensional deconstruction.
With the presence of extra dimensions, one has a new room to localize the matter fields differently along the transverse directions as in the so-called split fermion scenario. Various overlaps of fermions of different flavors then induce various fermion masses observed in nature (see e.g. [10,11,12]). Amazingly, the deconstruction interaction is also able to produce similar localization effects [13]. Indeed, after the spontaneous breaking of link fields, fermions get an extra contribution to their masses via the Higgs mechanism. Fermions then reorganize themselves into mass sequences and the lightest mass eigenstate of these towers exposes some interesting "localization" pattern in the theory space (also referred to as deconstruction group index space). We will first work out the analytical expressions and confirm the localization of these zero modes in a rather generic deconstruction set-up. The next question to raise is how to make these light modes chiral. Imposing some kind of chiral boundary conditions [2] is the answer again coming from the ED lessons. There is however one more subtle point to be mentioned here. If one truly wishes to relate the ED scenario to the dimensional deconstruction, one needs to latticize the extra dimensions to host the deconstruction group. There comes the lattice theory's issue of fermion doubling, and its standard remedy, such as adding to the Lagrangian a Wilson term [14] would remove half of original chiral degrees of freedom. This is the reason why most of previous works addressing the fermionic mixing in deconstructed picture (e.g. [2,13,15]) usually start out with only Weyl spinors. In the current work, we adopt a different and somewhat more general 4D deconstruction approach [16] where no extra dimension is actually invoked. As a result the fermions to begin with keep a standard 4-component Dirac spinor representation.
In any deconstruction set-up, the link fields transform non-trivially under at least two different gauge groups. This implies a complex vacuum expectation value (VEV) for these fields, whose phase would not be rotated away in general. After the deconstruction process, this phase is carried over into the complex value of wave functions and wave function overlaps of fermions. In turn, the induced complex-valued mass matrices can render a required CP-violating phase in the well-known KM mechanism. In contrast, we note that the generation of complex mass matrices within the split fermion scenario is a non-trivial problem and requires rather sophisticated techniques to solve [17,18]. Interestingly, the above CP violation induction via deconstruction can also be visualized in extra dimensional view point. Indeed, because of having the same symmetry transformation property, DD link field can be identified with the Wilson line pointing along a latticized transverse direction (Appendix B), and the latter then can naturally acquire a complex VEV in the generalized Hosotani's mechanism [6,7,8,9] of dynamical symmetry breaking. Apparently, the source of CP violation in this approach comes from the complex effective Yukawa couplings so it can be classified as hard CP violation. Nevertheless, those couplings acquire complex values after the spontaneous breaking of the DD link fields. In that sense this CP violation pattern could also be considered soft and dynamical. This paper is presented in the following order. In Sec. IIA we give the zero mass eigenstate of fermions obtained in the deconstructed picture, in Sec. IIB the resulting expression of mass matrix elements, and in Sec. IIC the symmetry breaking of [SU(2) × U (1) In Sec. III we present the numerical fit for quark mass spectrum and CKM matrix in a model where each "standard model" Higgs field is chosen to transform under only a single deconstruction subgroup. The conclusion and comments on numerical results is given in Sec. V. Appendix A provides a detailed derivation of zero mode wave functions in 4D deconstruction using combinatoric techniques. Appendix B outlines intuitive arguments on the complexity of link field inspired by lattice models. Appendix C presents analytical expressions for wave function overlaps used in the determination of mass matrix elements.
Finally, Appendix D gives referencing values of key physical quantities that have been used in the search algorithm (Table I), and numerical solution of our models' parameters (Table  II).

II. DECONSTRUCTION AND QUARK MASS MATRIX
In this section we describe how the mixing of quark flavors arises in the DD picture. But we first briefly recall the basic idea of the dimensional deconstruction applied to just a single quark generation. The family replication will be restored in the later sections.
These fields transform non-trivially only under their corresponding group [SU (2) (1)] n to "link" fermions of the same type.
Because of this, scalars φ's are also referred to as link fields hereafter. For the simplicity of the model, we assume a symmetry for the Lagrangian under the permutation of group index n.
We note that those conditions are in agreement with the gauge transformation property of fields, e.g. φ Q N −1,N Q N,L and Q N −1,L transform identically under the underlying gauge groups.
Essentially, these boundary conditions render one more left-handed degree of freedom over the right-handed for Q field, and the contrary holds for U and D fields. The actual calculation will show that the zero-mode of Q field indeed is left-handed while for U, D it is right-handed. When the link fields φ Q,U,D assume VEV proportional to V Q,U,D , above CBC become the very reminiscence of Neumann and Dirichlet boundary conditions.
In the deconstruction scenario, after the spontaneous symmetry breaking (SSB) the link fields acquire an uniform VEV V Q,U,D respectively, independent of site index n (in accordance with the assumed permutation symmetry), and the fermions obtain new mass structure.
Using the CBC (2), the fermion mass term can be written in the chiral basis as can be analogously found.
By coupling the following Dirac equations for chiral fermion sets and similar expressions hold . The quantitative derivation of the zeroeigenstates, which are identified with the SM chiral fermion, is presented in appendix A. In this section we just concentrate on some qualitative discussion. In general the diagonalization of matrices (6) leads to the transformation between gauge eigenstates Q nL and mass eigenstatesQ mL where the matrix The key observation, which will be analyzed in more details in appendix B, is that VEV can serve to generate the mass hierarchy among fermion families in a manner similar to that of ED split fermion scenario (see e.g. [12,18]). Thus we see that dimensional deconstruction indeed provides all necessary ingredients to construct a complete (complex) CKM structure of fermion family mixing.

B. Complex Mass Matrix
In order to give mass to the above chiral zero-mode of fermions, we introduce Higgs doublet fields just as in the SM. In the simplest and most evident scenario (see [15]), there is one doublet Higgs H n transforming as ( We also implement the replication of families by incorporating family indices i, j = 1, .., 3. Another scenario to generate the (vector-like) fermion mass hierarchy by assuming various link fields to connect arbitrary sites of the latticized fifth dimension has been proposed in [19].
The gauge-invariant Yukawa terms read In order to extract the terms involving zero modes, which are the only terms relevant at low energy limit, we rewrite (9) in the mass eigenbasis. However, this procedure depends explicitly on the specific CBCs being imposed on each of the fields Q, U, D. To be generic, let us consider the following configuration. We assume the "localization" of zero modesQ 0L , U 0R andD 0R to be at n = 1, n = 1 and n = N respectively. To achieve this localization pattern, we impose the following CBCs on these fields (see Eq. (2) and Appendix A, Eq. (A22)) Because of these boundary conditions, zero modesQ 0L ,Ũ 0R andD 0R would be localized at n = 1, n = 1 and n = N respectively, this also means that the first term of Eq. (9) would represent the overlap between 2 wave function localized at the same site n = 1, while the second term represents the overlap between wave functions localized at n = 1 and n = N.
Using (10) to eliminate the dependent components and after the SM spontaneous symmetry breaking H n = (0, v/ √ 2) T uniformly for all n's, we can rewrite the Yukawa term (9) as After going to the mass eigenbasis by the virtue of transformation of the type (7), keeping only zero-mode terms and together with the assumption of universality for the Yukawa couplings in the up and down sectors, we obtain the following effective mass terms with Because Thus in this simplest deconstruction approach, we might better understand the dynamical origin of CP-violation phase in the SM mass matrices. We also note that (13), (14) represent the specific case whereQ 0L ,Ũ 0R andD 0R are localized at n = 1, n = 1 and n = N respectively. All other localization configurations can be similarly found. Further, when we replace link fields φ's in (13), (14) by their VEVs following the deconstruction, these mass matrix elements will look much simpler (see (27), (28)).
Before moving on to give explicit expressions of these complex-valued mass matrices in term of zero mode wave functions (appendix A) and perform the numerical fit, let us briefly turn to the breaking pattern of product group N n=1 where B n is the gauge boson associated with U(1) n , while g ′ 0 is the common gauge coupling for all U(1)'s. For abelian groups, the opposite signs of the last two terms in (15) (and also in (16)) originate from the opposite charges of φ U n,n+1 (and φ D n,n+1 ) under U(1) n and U(1) n+1 (so that terms likeŪ n φ U n,n+1 U n are gauge-invariant). For non-abelian groups, the similar sign reversing will hold for terms in the expression of covariant derivatives (see Eq. (21) below), the nature of which also has its root in the gauge invariance of the theory. Indeed, under the Yang-Mills SU(2) n × SU(2) n+1 gauge the covariant derivative of φ Q n,n+1 must be formulated as follows (so that it transforms exactly like φ Q n,n+1 in (17)) where A n and T n are respectively the gauge bosons and some 2 × 2-special unitary matrix characterizing the SU(2) n transformation, while g 0 is the common gauge coupling for all SU(2)'s.
, the mass terms for gauge bosons are generated. Specifically, we obtain as parts of kinetic terms (D µ )] the following gauge bosons squared mass matrices where, after restoring the family replication index (i = 1, 2, 3), Both matrices in (22) In Eq. (23) it is also shown that the pattern of symmetry breaking is not spoiled by family replication as long as charges q U (and q D , q Q ) are independent of the site index n under a presumed permutation symmetry (just like V and similarly where x n 's are given in (A20) (corresponding to a zero mode localized at the end point n = 1) and y n 's in (A24) (corresponding to a zero mode localized at the end point n = N).
After the spontaneous symmetry breaking, the link fields acquire an uniform VEV V Q,U,D respectively (independent of site index n). In term of x n 's and y n 's, the SM mass matrices (13), (14) for up and down quark sectors become where x n 's, y n 's are given in (A20), (A24) respectively. The analytical forms of (27) Table II  The numerical approach to fit the parameters consists in minimizing a positive function which gets a zero value when all the predicted quantities are in the corresponding experimental ranges [18]. The minimization procedure is based on the simulated annealing method, which seems working better than other minimization approaches when the parameter space becomes larger [20], [21]. The input referencing physical quantities are given in Table I of Appendix D.
We consider eight different cases, which correspond to all the eight possible ways of localizing the left and right components. The eight different cases are the following: 8. U and D localized in n = 1, Q localized in n = N denoted as (UD1QN) We specially note that, due to the mirror complexity between CBCs (2) and (A22), the mass matrices obtained in the cases 1 and 5, cases 2 and 6, cases 3 and 7, cases 4 and 8, are complex conjugate pairwise. In the result, all eight cases are inequivalent.

B. Numerical results
In the following we present the characteristically important numerical results for the four cases out the eight mentioned above, for which we were able to find solutions. The cases are referred to in the above order. For each case we give one particular, but typical, numerical complete set of the 20 defining parameters (Table II), the quark mass matrices and quark mass spectra, the CKM matrix and the CP parameters. Complex phases are measured in radiant, and N = 10 for all cases. The masses are given in GeV and are evaluated at the M Z scale. For the sake of visualization, we also present graphically the comprehensive solutions of the quark wave function profiles in the theory space (Fig. 1), the mass spectrum (Fig.   2), the CKM matrix (Fig. 3) and theρ-η CP parameters (Fig. 4) for the case of all fields Q, U and D localized at the same site n = 1.
withρ andη defined asρ We are now ready for comments on the presented numerical solutions.

IV. CONCLUDING COMMENTS
In this paper we have reconstructed the observed complex mixing of quark flavors, starting with the product group N n=1 [SU(2) × U(1)] n at a higher energy scale. The deconstruction of this product group into the electroweak gauge group can indeed provide all necessary components to generate such mixing.
We have built a specific models with 20 parameters to fit the quark mass spectrum and the CP phase. However, the numerical fit is found only for the "preferred" configurations where fermion fields Q and U are localized at the same position in the theory space. Arguably, this is because the ratio κ U /κ D of Yukawa couplings can be responsible only for the difference in the overall scale of up and down-quark masses, while the more hierarchical internal mass spectrum of the up-quark sector (compared to that of the down-quark sector) would still require a higher degree of overlapping.
As far as the structure of mass matrices is concerned, the deviation from democracy is moderate. In all the cases, the mass matrices assume a hierarchy with two rows (or two columns) having similar absolute value matrix elements, with the third row (or third column) having different values, but still similar along that row (or that column). A quite close mass matrix structure was found in [18], but in a different approach.
We did not perform a study of the dependence on the number of deconstruction subgroups N. We expect anyway that the fitting would be more feasible for larger N as the wave functions and their overlaps then can be tuned more smoothly. In the other direction, the constraint from flavor changing neutral current that sets an upper limit on the length of extra dimension in the split fermion scenario (see e.g. [23]) is also expected to set an upper limit on the ratio N/V (between N and the VEV of link field) in the deconstruction theory.
We however leave a more careful analysis of these and other relevant phenomenological issues for future publications. In this appendix we will work out the general expression of zero eigenstate of the matrix of the type (6). This mode plays a special role because it will be identified with the SM chiral fermions. To simplify the writing, here we denote this zero eigenstate generally as {x 1 , x 2 , . . . , x N −1 } while in Section III we will restore all omitted scripts Q, U, D, i, j.

Zero-mode localization at the end-point n=1
The equation set determining the zero eigenstate (6) is . . .
After a bit of algebra, we can equivalently transform this equation set into . . .
where we have introduced new parameter and variables We note that V (and ρ) is a complex parameter in general (see appendix B). The new simple recursion relation allows us to analytically determine the set {X 1 , .., X N −1 } (and then the zero eigenstate {x 1 , .., x N −1 } ) for any ρ (i.e. for any real M and complex V ). After some combinatorics [25] we obtain for 1 ≤ n ≤ N − 3 and for n = N − 2 (see (A10)) Using the equality we can rewrite (A13) as (with 1 ≤ n ≤ N − 3) Again, using another equality [24] sinh px = sinh x we obtain for 1 ≤ n ≤ N − 3 and for |ρ| < 1 For |ρ| > 1 2 , the expression of X N is similar to (A18) but with hyperbolic functions (sinh and cosh) replaced respectively by trigonometric ones (sin and cos).
Finally, from (A12) we have altogether where C is the normalization constant determined by the normalization equation We note that this normalization is nothing other than the unitarity condition of the rotation matrix U (see (8) and (25), (26)).

Zero-mode localization at the end-point n=N
The chiral boundary conditions (CBC) and the value of parameter |ρ| ≡ |V |/|M| are two crucial factors that determine the localization pattern of the chiral zero-mode of fermion.
For e.g. in the previous subsection we have seen that, when |ρ| < 1/2, along with CBCs we can localize the left-handed zero mode of Q field around site n = 1 (A20).
On the intuitive ground, we expect that the "mirror image" of (2) (apart from the requirement |ρ| < 1/2) would produce a left-handed zero mode of Q field localized at n = N. A similar calculation indeed confirms this localization pattern. Specifically, if we denote y i (i = 2, .., N) the zeromode subject to CBCs (A22), and x j (j = 1, .., N − 1) subject to CBCs (2) as before, we find or even more explicitly (see (A20)) y n = Ce i(N +1−n)θ sinh (n − 1)α − e −iθ sinh (n − 2)α (2 ≤ n ≤ N) . Since the link fields φ n,n+1 transform non-trivially under two different groups, we may expect its VEV to be complex in general. It is because in this case the VEV's phase could not be rotated away in general. The standard and rigorous method to determine the VEV is to write down and then minimize the corresponding potential. It turns out [22] that there always exist ranges of potential parameters which generate complex VEV. In this appendix, however, we just recapitulate the complexity nature of link field VEV from the latticized extra dimension perception which is derived in [14] in details. Though the approach taken in this work does not strictly stem from latticizing the fifth dimension, this perception could still serve as the principle illustration.
To make the connection between DD theory and its latticized ED counterpart, we interpret the link field as a Wilson line connecting two neighboring branes where χ n essentially is the ED component of gauge field, g and a are gauge coupling and lattice spacing respectively.
where C 1 , C 2 are the normalization factors, which are determined also from the overlap of the respective wave function with itself.