Towards realistic supersymmetric spectra and Yukawa textures from intersecting branes

We study the possible phenomenology of a three-family Pati-Salam model constructed from intersecting D6-branes in type IIA string theory on the ${\mathbf{T}}^{\mathbf{6}}/({\mathbb{Z}}_{\mathbf{2}}\times {\mathbb{Z}}_{\mathbf{2}})$ orientifold with some desirable semirealistic features. In the model, tree-level gauge-coupling unification is achieved automatically at the string scale, and the gauge symmetry may be broken to the standard model (SM) close to the string scale. The small number of extra chiral exotic states in the model may be decoupled via the Higgs mechanism and strong dynamics. We calculate the possible supersymmetry breaking soft terms and the corresponding low-energy supersymmetric particle spectra which may potentially be tested at the Large Hadron Collider (LHC). We find that for the viable regions of the parameter space the lightest $CP$-even Higgs boson mass usually satisfies $m_H\le 120\mathrm{GeV}$, and the observed dark matter density may be generated. Finally, we find that it is possible to obtain correct SM quark masses and mixings, and the tau lepton mass at the unification scale. Additionally, neutrino masses and mixings may be generated via the seesaw mechanism. Mechanisms to stabilize the open and closed-string moduli, which are necessary for the model to be truly viable and to make definite predictions are discussed.

Figure 1

Breaking of the effective gauge symmetry $SU(4)\times SU(2{)}_L\times SU(2{)}_R$ down to $SU(3{)}_C\times U(2{)}_L\times U(1{)}_{I3R}\times U(1{)}_{B-L}$ via brane splitting. This process corresponds to giving a VEV to adjoint scalars, which arise as open-string moduli associated with the positions of stacks $a$ and $c$ in the internal space.

Figure 2

RGE running of the gauge coupling for $USp(2{)}_1/USp(2{)}_2$, $USp(2{)}_3/USp(2{)}_4$ hidden sector gauge groups, which become confining at $\approx 3\times {10}^{13}\mathrm{GeV}$ and $\approx 6\times {10}^5$, respectively.

Figure 3

Contour map of the dark matter density for $\tan \beta =46$, ${\theta }_3>0$, ${\Theta }_s=0$, and $\mu >0$. The areas in black denote regions where $0.09\le \Omega h^2\le 0.12$ with a gravitino mass $m_{3/2}=1\mathrm{TeV}$. The light gray regions are excluded because they do not satisfy the constraints on the soft terms at high scale. The dark gray regions denote regions where either the neutralino is not the LSP, mass limits are not satisfied, or for which there is no RGE solution. Regions inside the white contour satisfy the LEP limit, $m_h>114\mathrm{GeV}$.

Figure 4

Contour map of the neutralino-stau mass difference and neutralino-chargino mass difference for $\tan \beta =46$, ${\theta }_3>0$, ${\Theta }_s=0$, and $\mu >0$. The areas in black denote regions where $0.095\le \Omega h^2\le 0.125$ with a gravitino mass $m_{3/2}=1\mathrm{TeV}$. The light gray regions are excluded because they do not satisfy the constraints on the soft terms at high scale. The dark gray regions denote regions where either the neutralino is not the LSP, mass limits are not satisfied, or for which there is no RGE solution. The dark matter density within the allowed range coincides with the ${\tilde{\chi }}_1^0\tilde{\tau }$ and ${\tilde{\chi }}_1^0{\tilde{\chi }}^{\pm }/{\tilde{\chi }}_1^0{\tilde{\chi }}_2^0$ coannihiliation regions where the stau and/or chargino/next-to-lightest neutralino mass is slightly bigger than the lightest neutralino mass.

Figure 5

Contour maps of the relic density for $\tan \beta =46$, ${\theta }_3>0$, ${\Theta }_s=0$, and $\mu >0$ with a gravitino mass $m_{3/2}=475\mathrm{GeV}$ and $m_{3/2}=450\mathrm{GeV}$, respectively. The areas in black denote regions where $0.09\le \Omega h^2\le 0.12$. The light gray regions are excluded due to tachyonic scalar masses at the high scale. The dark gray regions denote regions where LEP superpartner mass limits are not satisfied or there is no RGE solution. Regions inside the white contour satisfy the LEP limit, $m_h>114\mathrm{GeV}$. The viable region of the parameter space is much smaller for $m_{3/2}=450\mathrm{GeV}$ and disappears for $m_{3/2}=425\mathrm{GeV}$, which constrains the minimum gravitino mass to be in the range $425\mathrm{GeV}<m_{3/2}\le 450\mathrm{GeV}$.

Figure 6

Contour maps of the relic density for $\tan \beta =46$, ${\theta }_s=0.40\ne 0$, ${\theta }_3=0.75$, and $\mu >0$ with a gravitino mass $m_{3/2}=1\mathrm{TeV}$. The light gray regions are excluded because they do not satisfy the constraints on the soft terms at high scale. Regions inside the white contour satisfy the LEP limit, $m_h>114\mathrm{GeV}$.

Figure 7

The viable parameter space with $\Omega h^2\le 0.125$ for ${\Theta }_s\ge 0$.

Figure 8

Theta function as a function of the parameters $\delta$ and $\kappa$, with $\varphi =0$. The maximum values occur for $\delta =0$ while the minimum values occur when $\delta =1/2$.

Figure 9

Brane configuration for the three two-tori. The SM fermion mass hierarchies primarily result from the intersections on the first torus.

Figure 10

Ratio of theta functions $R=[e^{-\pi \kappa {\delta }^2}{\vartheta }_3(i\pi \kappa \delta ,e^{-\pi \kappa })]/[{\vartheta }_3(0,e^{-\pi \kappa })]$.