Generalized Three-Family Supersymmetric Pati-Salam Models from Type IIA Intersecting D6-Branes

TL;DR

This work broadens the landscape of ${ m N}=1$ supersymmetric Pati-Salam models from Type IIA intersecting D6-branes by generalizing the three-family condition to $I_{ac}=-(3+h)$, $I_{ac'}=h$ and identifying four new model classes that realize three generations without requiring parallel brane configurations. It introduces a systematic, Diophantine-based construction strategy, analyzes the massless spectra and hidden USp sectors of the four models, and demonstrates that distinct gauge-coupling relations emerge at the string scale. To achieve string-scale unification, the authors deploy two mechanisms: (i) symmetry breaking of ${ m SU}(2)_{L_1} imes{ m SU}(2)_{L_2}$ to a diagonal ${ m SU}(2)_{L'}$ which suppresses $g_{L'}$, and (ii) the inclusion of vector-like exotics from ${ m N}=2$ subsectors that modify two-loop RG running. Through detailed RG analyses, they show that these ingredients can yield precise gauge-coupling unification near $M_{ m string}\,(\sim 5 imes10^{17} ext{ GeV})$ for the extended Pati-Salam models. Overall, the paper expands the model-building landscape and provides viable pathways to reconcile low-energy phenomenology with a string-scale unification framework.

Abstract

Generalizing three-family chiral fermion conditions to $I_{ac}=-(3+h)$ and $I_{ac'}=h$, with positive integer $h$, we extend the landscape of three-family ${\cal N}=1$ supersymmetric Pati-Salam models in a broader region. Differing from the former investigation with $I_{ac}=-3$ and $I_{ac'}=0$, we do not restrict that the $a$ stack of D6-branes must be parallel to the orientifold image of the $c$-stack along one of the three two-tori. In this investigation, without the simple parallel construction, we find four new classes of supersymmetric Pati-Salam models that are allowed by the extended three generation condition with $I_{ac}=3, I_{ac'}=-6$ and $I_{ac}=-1, I_{ac'}=-2$ through the intersections of $a$- and $c/c'$-branes. Moreover, with the $SU(2)_{L'}$ gauge coupling realized from $SU(2)_{L_1}\times SU(2)_{L_2}$ symmetry breaking, the canonical normalization requirement of the gauge kinetic term provides an alternative approach that can be imposed before the renormalization group equation evolution for $SU(2)_{L'}$ gauge coupling. This turns out to be an effective mechanism to realize the string-scale gauge coupling relation, especially for the new supersymmetric Pati-Salam models with large $g_b/g_a$ ratio. We show that this symmetry-breaking modified renormalization group evolution can highly suppress $g_b/g_a$, and finally realizes string-scale gauge coupling relations for the extended supersymmetric Pati-Salam models as well.

Figures (3)

• Figure 1: The evolution of two-loop gauge couplings in the Model 1 with vector-like particles $(XU, \overline{XU})$ and $(XD, \overline{XD})$. The masses of these particles are set as $M_{XU}=1.0\times 10^{13}$ GeV (left panel) and $M_{XD}=5\times 10^{16}$ GeV, $M_{XU}=3.2\times 10^{13}$ GeV (right panel).
• Figure 2: The evolution of two-loop gauge couplings in the Model 2 with vector-like particles $(XU, \overline{XU})$ and $(XD, \overline{XD})$. The masses of these particles are set as $M_{XU}=2.0\times 10^{6}$ GeV (left panel) and $M_{XD}=2.24\times 10^{13}$ GeV, $M_{XU}=1.78\times 10^{10}$ GeV (right panel).
• Figure 3: The evolution of two-loop gauge couplings in the Model 3-m with vector-like particles $5(XQ, \overline{XQ})+3XG$. The masses of these particles are set as $M_{XQ}=M_{XG}=6.0\times 10^{15}$ GeV.