GUT Breaking on the Lattice

We construct a supersymmetric grand unified model in the framework of a latticized extra dimension. The SU(5) symmetries on the lattice are broken by the vacuum expectation values of the link fields connecting adjacent SU(5) sites, leaving just the MSSM at low energies. Below the SU(5) breaking scale, the theory gives rise to a similar spectrum as in orbifold breaking of SU(5) symmetry in 5 dimensions, and shares many features with the latter scenario. We discuss gauge coupling unification and proton decay emphasizing the differences with respect to the usual grand unified theories. Our model may be viewed as an effective four dimensional description of the orbifold symmetry breaking in higher dimensions.


Introduction
Recently, a new approach to gauge theories in extra dimensions has been introduced by considering extra dimensions on a transverse lattice [1,2]. This provides an "ultraviolet complete" gauge invariant description of the higher dimensional gauge theory. On the other hand, from a purely 4-dimensional point of view, the extra dimensions are "generated" through a series of gauge groups and link fields among them. This latticizing or desconstructing approach to the extra dimensions provides a great tool to understand higher dimensional gauge theories, and to obtain new models both in pure 4 dimensions and higher dimensions [3,4,5,6].
In this note, we examine the orbifold breaking of the grand unified (GUT) gauge symmetry [7,8,9,10,11] from the point of view of the deconstructed extra dimensions. In the case of SU(5) GUT breaking, a reflection around an orbifold fixed point y = 0 with the parity transformation A µ (−y) = P −1 A µ (y)P, P = diag(−1, −1, −1, +1, +1) projects out the zero modes of the X, Y gauge bosons and breaks SU(5) down to SU(3)×SU(2)×U (1). In field theory, the orbifold should be viewed as a theory defined on a finite interval with suitable boundary conditions. In this case, the boundary condition is such that the gauge symmetry at the boundary point y = 0 is only SU(3) × SU(2) × U(1), while the SU(5) gauge symmetry is preserved in the bulk. The usual gauge coupling unification can be preserved because the gauge couplings are dominated by the SU(5) symmetric bulk contributions which are enhanced by the volume factor relative to the contributions from the boundary. There are several nice features of this GUT breaking mechanism. It is easy to obtain doublet-triplet splitting in the Higgs sector, at the same time avoiding proton decay mediated by the colored triplet Higgs fields, which may already pose a problem with the current experimental bounds in the usual 4-dimensional GUT. The gaugino mediated supersymmetry (SUSY) breaking [12] can naturally be incorporated in this framework to solve the SUSY flavor problem.
In the following, we consider this orbifold GUT breaking on latticized extra dimensions. It becomes a 4-dimensional theory with a series of gauge groups, broken down to the diagonal subgroup by the link field vacuum expectation values (VEV's). The simplest realization is to have only SU(3) × SU(2) × U(1) gauge symmetry on one lattice point at the end, and SU(5)'s on all other lattice points. However, we prefer to start with SU(5)'s on all lattice points, and break them down to SU(3) × SU(2) × U(1) with the VEV's of the link fields. Since the link fields are identified with the A 5 component of the gauge field in the continuum limit, this is equivalent to the "Wilson line breaking," which has been shown to be equivalent to the orbifold breaking in the continuum theory [13,14,15]. In this model, the SU(3), SU (2), and U(1) gauge couplings are truly unified at some high scale. We will find that the spectrum of the continuum theory is reproduced in the limit of large number of lattice points. We will also discuss related issues such as doublet-triplet splitting and gauge coupling unification in this 4-dimensional picture. While finishing this work, we learned that a similar idea is being pursued by C. Csaki, G. D. Kribs, and J. Terning [16].

SUSY SU(5) on the orbifold lattice
We will begin with a supersymmetric SU(5) theory on a latticized extra dimention. We assume that we have N + 1 SU(5) gauge groups with common gauge coupling g, with N + 1 N = 1 vector multiplets V i (i = 0, · · · , N), one for each SU (5). There are also two sets of N chiral multiplets Φ i and Φ i , Φ i forms (5 i−1 , 5 i ) under the two nearest SU(5)'s, while Φ i has the opposite charges. The Lagrangian for the vector multiplets and the chiral fields is the following: As shown in [5], if the diagonal components of the link fields, Φ i and Φ i , acquire universal vacuum expectation values v/ √ 2 which preserve the N = 1 supersymmetry, then SU(5) N +1 is spontaneously broken down to a diagonal SU(5). 1 The vector multiplets have the mass spectrum M V,n = 2gv sin nπ 2(N +1) , n = 0 · · · N, while certain linear combinations of some components in the link fields Φ and Φ, which become part of the massive N = 1 vector multiplets, receive D term contributions and acquire the mass spectrum M Φ,Φ,n = 2gv sin nπ 2(N +1) , n = 1 · · · N. The other components of Φ and Φ acquire masses ∼ v or higher and thus decouple from the low energy effective theory. Therefore, one recovers an N = 1 SU(5) theory at the zero mode level.
In addition, we have four sets of chiral fields: where the subscripts show the charges of the fields under each SU (5). We assume that on the zeroth brane, one only has H 5,0 and H 5,0 , but not their conjugate partners. The superpotentital for these fields is the following, When Φ i and Φ i accquire VEV's, and assuming M H = λv/ √ 2, the mass spectra for the H fields arising from the superpotential are such that H 5 and H 5 have massless zero modes which preserve N = 1 SUSY, while all conjugate fields become massive. The massive H and H c fields have the spectra M H,H c ,n = 2M H sin nπ 2(N +1) , n = 1, · · · , N, which is the same as the massive vector multiplets and the massive chiral link fields, given the choice The results map onto a continuum five-dimensional theory with N = 1 supersymmetry compactified on a Z 2 orbifold of size L = (N +1)/gv. Orbifolding breaks the N = 1 SUSY in five dimensions (which is equivalent to N = 2 SUSY in four dimensions) down to N = 1 SUSY in four dimensions. The Higgs fields H 5 and H 5 are complete hypermultiplets in the 5D theory, while in 4D N = 1 language each of them includes two chiral multiplets that are conjugate of each other.

SU(5) breaking
To generate SU(5) breaking, we assume that the first set of link fields takes on a different form. We assume that there are four link fields, Φ 1 , Φ 1 , Φ ′ 1 , Φ ′ 1 that are charged under SU(5) 0 and SU(5) 1 . Φ 1 and Φ ′ 1 form (5 0 , 5 1 ) representation, and Φ 1 and Φ ′ 1 form (5 0 , 5 1 ). Their VEV's have the following structure, These VEV's can be obtained with suitable superpotential interaction [17,18]. All other link fields have the same structure and VEV's as previously discussed. The unbroken gauge group is then SU(3) × SU(2) × U(1), which is easily seen from the mass spectrum of the gauge bosons. For the SU(3) × SU(2) × U(1) gauge bosons, the mass matrix remains the same as in the case considered previously in Sec. 2.1, Hence, there is a zero mode for each of the gauge groups. The X, Y gauge bosons, however, accquire a different mass spectrum, due to the fact that the VEV's of Φ 1 and Φ ′ 1 do not generate off-diagonal mass terms between the gauge bosons of SU(5) 0 and SU (5) 1 , As a result, the X, Y gauge bosons on the 0th brane are decoupled from the rest of the lattice, and have masses M 0 = gv. The other X, Y bosons on branes 1..N accquire the mass spectrum M X,Y,n = 2gv sin (n−1/2)π (2N +1) , n = 1, · · · N. Since the model preserves N = 1 SUSY, we expect it to contain the full vector multiplets of SU(3) × SU(2) × U(1), and the X, Y vector multiplets to exhibit the same mass spectrum as their scalar components. The corresponding components in the Φ and Φ fields also split in a similar fashion.
In the Higgs sector, we modify the couplings between the Higgs fields on the 0th and 1st brane and the corresponding link fields, while keeping the couplings on all other branes the same. The superpotential takes the following form,  (3) × SU(2) × U(1) vector multiplets, as we previously discussed. Namely, M H U ,H D ,n = 2gv sin nπ 2(N +1) , n = 0 · · · N. However, the structure for the colored Higgs components is changed. Φ ′ 1 and Φ ′ 1 do not generate off-diagonal mass terms between the 0th and 1st colored Higgs field, hence, the colored triplets on the 0th brane H C,0 and H C,0 are decoupled from the rest of the lattice. The N × N mass matrix for the colored Higgses on the n = 1, · · · N branes takes the following form, (2.8) Therefore, there are N massive modes, with the spectrum M H C ,H C ,n = 2gv sin (n−1/2)π (2N +1) , n = 1, · · · N. Finally, the colored components in H 5,0 and H 5,0 accquire masses from the higher dimensional coupling that is localized on the first brane, as shown in eqn.(2.7). Their masses are λ ′ v 2 /2M. One can tune the parameter λ ′ , assuming that v is comparable to M, such that λ ′ v 2 /2M = gv. Hence, the complete colored Higgs spectrum matches onto that of the X, Y vector multiplets.
It is easy to verify that the H c fields also exhibit the same splitting between their colored components and their doublet components, due to the vacuum structure of the first set of link fields. The doublet components of the H c fields have the spectrum M H c U,D = 2gv sin nπ 2(N +1) , n = 1 · · · N, while the triplets have the spectrum M H c C = 2gv sin (n−1/2)π (2N +1) . In summary, the massless modes in our model include N = 1 SU(3) × SU(2) × U(1) vector multiplets and two Higgs chiral multiplets H U and H D . The massive modes fall into two types according to their spectrum.
• M 1n = 2gv sin( nπ 2(N +1) ), n = 1, · · · , N. The fields that have this type of mass spectrum are the KK modes of the SU(3) × SU(2) × U(1) gauge supermultiplets, which include components coming from the link fields Φ and Φ, and the KK towers of Higgs doublets including H U,D and H c U,D .
• M 0 = gv and M 2n = 2gv sin (n−1/2)π (2N +1) , n = 1, · · · N. This category includes the massive X, Y vector multiplets, which contain components from the link fields, and the KK towers of Higgs triplets which include H C,C . At the same time, the massive colored Higgs modes belonging to the H c 5,5 (there is a total of N of those) do not include M 0 .
There are other components of the link fields (and other possible fields required to generate the link VEV's) acquiring masses of order v or higher from minimizing the potential.
Hence, the masses of the two sets of KK modes have a relative shift of gvπ 2N . The low energy spectrum is the same as that of the KK modes in [10], in which a SUSY SU(5) model in five dimensions is compactified on a Z 2 × Z 2 orbifold.
One complete family of quarks and leptons comes from a 5 and a 10 of the SU(5). We can assume that these matter fields are localized on a single lattice point (i.e., transforming under a single gauge group). Having matter fields localized on the boundary which preserves (breaks) the SU (5)

Discussion
Given the spectrum presented in the previous section, the running of the gauge couplings at the 1-loop level including the threshold corrections from all massive modes can be easily calculated as follows, (3.9) Here α G = g 2 /(4π(N + 1)), and the numerical coefficients are determined only by the group structure of the fields. β a (a = 1, 2, 3 refers to U(1), SU(2) and SU(3)) includes the contribution from the zero modes, γ a includes the contribution from the modes which have a Type I mass spectrum, δ a accounts for the the modes with a Type II mass spectrum for n = 1, · · · N. These coefficients have been calculated in [10], where a model with a similar spectrum has been constructed from a Z 2 × Z 2 compactification of a supersymmetric 5D theory: β a = ( . ∆ a includes the threshold corrections from heavy link field components, which are near or above the SU(5) breaking scale.
As discussed in Refs. [10,19], gauge coupling unification is not ruined by the presence of the Kaluza-Klein spectrum. We now examine this in our model in more detail.
Let us define M G = 2 × 10 16 GeV as the scale where α 1 and α 2 meet in the MSSM. Previous studies [20,21] have shown that with the central values for the gauge couplings at the weak scale, and a SUSY spectrum which is not unnaturally heavy, the gauge couplings miss each other at the scale M G by (In what follows, we ignore the model-dependent effects from ∆ a .) Having determined M 0 = gv, there are no free parameters left, and for any given N we get a prediction for ε 3 at the unification scale M G : with, excluded by) proton decay, are denoted by circles (diamonds, crosses). We see that the predicted threshold correction ε 3 is negative, i.e. goes in the right direction. However, its magnitude is not large enough to completely fix gauge coupling unification. One might hope that the additional threshold effects ∆ a due to the heavy components of the link fields will ameliorate the situation. Alternatively, gauge coupling unification can be further improved by reducing λ ′ , hence lowering the mass of the colored triplet Higgs on lattice point 0, which results in an addtitional negative contribution to ε 3 .
From Fig. 1 we also see that the SU(5) breaking scale, defined as 2gv = 2M 0 , is a few times higher than the usual M G , and it grows for larger N. The mass of the lowest KK mode, namely, the effective compactification scale, is between 0.4 ∼ 0.8 × M G .
The colored triplet Higgs mediated proton decay is absent if the matter fields are localized away from the zeroth lattice point (i.e., do not transform under SU(5) 0 ), because the two sets of Higgs fields containing H U and H D do not couple to each other away from lattice point 0. Although the triplets on lattice point 0, H C,0 and H C,0 , couple through the non-renormalizable interaction λ ′ , they decouple from the triplets on the other lattice points. As a result, the proton decay process mediated by H C,0 and H C,0 can only take place if the quarks and the leptons are on the 0th brane. 2 If the matter fields are localized on branes away from the 0th brane, the dimension 6 proton decay operators from the X, Y gauge boson exchange will be enhanced compared to the usual SUSY GUT, because there are many X, Y gauge bosons contributing to the process and the lightest ones are lighter than those in the traditional 4D SUSY GUT. The experimental value of the proton lifetime thus imposes constraints on the scales in our construction. The decay mode p → e + π 0 through exchanging of X and Y gauge bosons requires that the lightest X and Y gauge bosons both should have mass gvπ/(2N + 1) ≥ 5 × 10 15 GeV. On the other hand, as we discussed ealier, the X, Y gauge bosons on lattice point 0 are decoupled from the other X, Y gauge bosons, and have mass gv, which is not supressed by the volume factor N and somewaht larger than the usual SUSY GUT scale. Therefore, if the matter fields are localized on the lattice 0, the dimension 6 proton decay operators will be suppressed compared to the case when matter is localized away from the 0th brane.
As mentioned in the Introduction, gaugino mediated SUSY breaking can be easily incoporated in the orbifold GUT breaking scenario. In our case, similar superpartner spectrum can be obtained if SUSY breaking only couples to the gauge group on the lattice point away from where matter fields are localized [5,6].
In summary, we have constructed a 4D SUSY GUT theory with many SU(5) gauge groups. The gauge symmetry breaking scale is somewhat higher than the GUT scale in the usual 4D theory. However, gauge coupling unification is achieved due to the threshold corrections from the "Kaluza-Klein" modes lighter than the symmetry breaking scale. It shares many features with the 5D orbifold GUT breaking models, and may be viewed as an effective 4D description of these higher dimensional mechanisms.