Gauge Unification and Quark Masses in a Pati-Salam Model from Branes

We investigate the phase space of parameters in the Pati-Salam model derived in the context of D-branes scenarios, requiring low energy string scale. We find that a non-supersymmetric version complies with a string scale as low as 10 TeV, while in the supersymmetric version the string scale raises up to ~2 x 10^7 TeV. The limited energy region for RGE running demands a large tan(beta) in order to have experimentally acceptable masses for the top and bottom quarks.


Introduction
The last few years, there has been considerable work in trying to derive a low energy theory of fundamental interactions through a D-brane construction [1,2,3,4,5,6,7,8,9,10,11,12,13] Recent investigations have shown that there is a variety of possibilities, concerning the group structure of the theory as well as the magnitude of the string scale and the nature of the particle spectrum.
A particularly interesting possibility in this context, is the case of models with low scale unification of gauge and gravitational interactions. This is indeed a very appealing framework for solving the hierarchy problem as one dispenses with the use of supersymmetry. There are a number of phenomenological questions however that should be answered in this case, including the smallness of neutrino mass 1 .
Another interesting possibility which could solve a number of puzzles (as the neutrino mass problem mentioned previously), is the intermediate scale scenario. A variety of models admit an intermediate unification scale, however supersymmetry is needed in this case to solve the hierarchy problem.
In this letter we concentrate on phenomenogical issues of the Pati-Salam [14] gauge symmetry proposed as a D-brane alternative [11] to the traditional grand unified version. In particular we investigate the gauge coupling relations in two cases: for a non-supersymmetric version and for a supersymmetric one. In both cases, in order to achieve a low string scale, we relax the idea of strict gauge coupling unification. However, this should not be considered as a drawback. Indeed, the various gauge group factors are associated with different stacks of branes and therefore it is natural that gauge couplings may differ at the string scale. In the non-supersymmetric case the string scale could be as small as a few TeVs. On the other hand, the absence of a large mass scale puts the see-saw type mechanism (usually responsible for giving neutrino masses in the experimentally acceptable region) in trouble. In the supersymmetric case, the string scale is of the order of 10 3 TeV and a sufficiently suppressed neutrino mass may be obtained.

The Model
We assume here a class of models which incorporate the Pati-Salam symmetry [14], having representations that can be derived within a D-brane construction. In these models, gauge interactions are described by open strings with ends attached on various stacks of D-brane configurations and therefore fermions are constrained to be in representations smaller than the adjoint. A novelty of these constructions is the appearance of additional anomalous U(1) factors. At most, one linear combination of these U(1)'s is anomaly free and may remain unbroken in low energies. As we will see, the role of this extra U(1) is important since when it is included in the hypercharge definition allows the possibility of a low string scale.
We start with a brief review of the model [11]. The embedding of the Pati-Salam (PS) model in the brane context leads to a SU(4) C × SU(2) L × SU(2) R × U(1) C × U(1) L × U(1) R gauge symmetry. Open strings with ends on two different branes carry quantum numbers of the corresponding groups. The Standard Model particles appear under the following multiplets of the PS group: where we have also shown the quantum numbers under the three U(1)'s 2 and the breaking to the SM group. The Higgs which breaks the PS down to the SM is: The U(1)-charge parameter δ can take two values δ = ±1. Each one of them is associated with a different symmetry breaking pattern. The down-quark like triplets are the only remnants after the PS breaking while one HiggsH (and its complex conjugate) is enough to achieve this breaking. Additional states, such as: can arise which could provide masses to the PS breaking remnants (colored triplets with down-type quark charges d H , d c H ) or break an additional abelian symmetry (by a non-vanishing vev of η and/or h R ).
While all three of the U(1)'s that come with the PS group are anomalous, there exists only one combination which is anomaly free (even from gravitational anomalies): where Y X , X = C, L, R corresponds to the quantum number under the U(1) X . None of the SM fermions and Higgs bidoublet (providing the SM higgses) are charged under this U(1) H . To this end, we assume that all anomalous abelian combinations break and we are left with a gauge symmetry SU(4 The SM hypercharge is given by the usual PS generators plus a contribution from the U(1) H : The interesting case is when c differs from zero. Indeed, there exists a breaking pattern where c = 1/2 and the parameter δ determining theH charge under the U(1) H (namely δ − 1), takes the value δ = −1 [11]. We are interested in that case and we shall develop the RGE for gauge and Yukawa couplings running.

Setting the RGE's
Three different scales appear in our approach: the string scale M U , the Pati-Salam breaking scale M R and the low energy scale M Z . In principle, since the various groups leave in different stacks of branes, the corresponding gauge couplings may differ as well. However, in order not to loose predictability at the unification scale M U , we require a"petit" unification, namely α 4 = α R = α L (see [11] for discussion).
For further convenience we introduce the parameter ξ = α L (M U )/α 4 (M U ) (α 4 , α L and α R correspond to the three groups of the model: SU(4), SU(2) L and SU(2) R ). The value of α H at M U is given by the following relation: At M R we have the following relations due to the Pati-Salam group breaking: where α 3 , α 2 and α Y correspond to the three groups of the SM. As has been mentioned above, the parameter c can take two acceptable values. The value c = 0 corresponds to the standard definition of the hypercharge. Assuming petit unification, we find [11] M U ≥ 10 10 GeV. The c = 1/2 introduces a component of the extra U(1) H in Y without affecting the SM charge assignment. This case allows the possibility of low unification in the TeV range. For the rest of the paper we will work with c = 1/2. Now for completeness we give the β-functions for all groups: where n g is the number of families (n g = 3) while all other notation is in accordance with that of Eqs. (1, 2, 3). First we would like to set the range for the parameter ξ = α L /α 2 in order to achieve a low energy M U , while keeping M R < M U as an upper limit and M R > 1TeV as a lower limit. We use the following low energy (M Z ) experimental values: sin 2 θ W = .23151, α em = 1/128.9 and α 3 = 0.119 ± 0.003. Our particle content is the following: The charge γ is not fully constrained (as opposed to the case ofH) and, in principle, can take two values γ = ±1. However, if supersymmetry is assumed, as the corresponding charge of the fieldH has been determined to δ = −1, the value of γ should be fixed to γ = 1. Further, the following exotic representations could appear (1, 1, 2; 0, 0, −1)  Fig.(1). The particle content is the minimum one (see text).
Keeping the same conditions as in the non-supersymmetric case, Eqs(6,7) and fixing again the value of c to 1/2, we plot M U and M R vs ξ in Fig.(2). The content is the minimum possible, i.e.
We observe that, in contrast to the non-supersymmetric case examined in the previous section, here the limiting case M R = M U is realized at the highest ξ value, while the lower ξ is correlated to the lower acceptable M R value (∼ 1 TeV). The energy scale of M U and M R now is three orders of magnitude higher than the corresponding non-supersymmetric case. In Fig.(3) we show the same graph for the minimal and a non minimal content for the supersymmetric case (γ = 1 and δ = −1). The non minimal content drives the ξ parameter to lower values but expands the acceptable region of the scales by almost one order of magnitude.

Yukawa Coupling Running for Top and Bottom
In the PS model with the minimal Higgs content, the Yukawa couplings for the top and the bottom quarks are equal at M R , i.e. h t = h b . In this section we check whether such a constraint is compatible with the bottom and top quark masses as they are measured by the experiments. If v 1 and v 2 are the two v.e.v.'s that correspond to H d and H u , we have of course: where the factor η = 1.4 takes care for the QCD renormalisation effects from the scale m t down to the mass of the bottom quark. Since we have two v.e.v.'s (although we do not have supersymmetry), the relation with M Z is: while we insert, as usual, the parameter tan β = v 2 /v 1 . The RGE for the two couplings are: where we have ignored all other Yukawa couplings. We run the equations from M R down to scale M where h t (M)v 2 = M, which is the top mass m t . In Fig(4a) we plot m t vs tan β in order to have m b in the acceptable experimental region (4.0-4.4)GeV. The choice of ξ (in the acceptable region defined above) makes a very small effect which shows itself in the thickness of the lines. Since we require unification of the two Yukawa couplings at M R , the large difference in the mass of the two quarks can only be provided by a large angle, therefore the large values of tan β were expected. Moreover, being in the large tan β regime, m t changes by a The form of the Eq.(11) also shows that the two couplings run almost "parallel" to each other and actually the main contribution to the running comes from the gauge couplings (as we can see in the next figure, the value of the Yukawas at M R are small). The corresponding figure with n d c H = 2 does not show any significant difference.
In Fig.(4b)  Finally, in figure (5 we plot the unified value of the Yukawa coupling h u versus the ξ parameter. We note that, -in contrast to the non-supersymmetric case which is exhibited in figure (5)-here higher h u values are obtained for lower ratios ξ.

Conclusions
In the present work, we have examined the gauge and bottom − top Yukawa coupling evolution in models with Pati-Salam symmetry obtained in the context of brane scenarios. In the case of 'petit' unification of gauge couplings, i.e., a 4 = a R = a L , it turns out that in the non-supersymmetric version of the above model one may have a string scale at a few TeV. Further, assuming h b − h t Yukawa unification at the string scale, one finds that the correct m b,t quark masses are obtained for a a 4 approximately twice as big as a L . A similar analysis for the supersymmetric case shows that the string scale raises up to 10 7 T eV while h b − h t unification reproduces also the right mass relations m b , m t .