Three-Family SUSY Pati-Salam Models

• Three-family supersymmetric Pati-Salam models are N=1 extensions that embed exactly three generations of Standard Model fermions within the SU(4)_C × SU(2)_L × SU(2)_R gauge symmetry.
• They are constructed via intersecting D6-branes on orientifolded tori with rigid cycles, ensuring anomaly cancellation, moduli stabilization, and computable Yukawa couplings through worldsheet instanton effects.
• These models offer concrete mechanisms for gauge coupling unification, flavor hierarchy generation, and controlled supersymmetry breaking, with extensive classifications in both string and F-theory frameworks.

A three-family supersymmetric Pati-Salam (PS) model refers to a class of four-dimensional $\mathcal{N}=1$ supersymmetric extensions of the Pati-Salam gauge symmetry, $SU(4)_C \times SU(2)_L \times SU(2)_R$, embedding precisely three generations of Standard Model (SM) fermions. These constructions realize unification properties, family structure, and hierarchies naturally, often within string theory frameworks such as Type IIA orientifold compactifications with intersecting D6-branes, or in global F-theory models. Such models feature controlled anomaly cancellation, calculable Yukawa couplings, and symmetry-breaking chains down to the MSSM, and provide concrete mechanisms for moduli stabilization, supersymmetry breaking, and realistic flavor structures. Modern constructions have achieved a complete landscape classification, revealed rigid-cycle D-brane realizations that solve the adjoint moduli problem, and extended to generalized intersection patterns, symplectic gauge factors, and four-stack visible sectors.

1. Fundamental Frameworks and Geometric Construction

Three-family supersymmetric Pati-Salam models are realized on factorizable six-torus orientifolds, $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_2)$ or $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_2')$, with D6-branes wrapping three-cycles characterized by integer wrapping numbers $(n^i,l^i)$ per two-torus. The brane content specifies stacks $a$, $b$, $c$ (corresponding to $U(4)_C$, $U(2)_L$, $U(2)_R$), with possible additional stacks for extended visible or hidden sectors. The orientifold and orbifold actions create four types of O6-planes, each with their own RR charge and projection conditions. RR tadpole and K-theory constraints must be satisfied to ensure consistency and charge neutrality (Li et al., 2019, He et al., 2021, Mansha et al., 2023, Li et al., 2019).

A deterministic classification algorithm exploits the polynomial structure of RR- and SUSY-angle constraints, together with integer intersection number relations to enumerate all consistent three-family solutions. This yields $202{,}752$ distinct models, consolidating (after modding out physical equivalences such as torus permutations and T-duality) to $33$ genuinely inequivalent gauge-coupling classes at the string scale (He et al., 2021).

An essential extension is the use of rigid cycles (e.g., $T^6/(\mathbb{Z}_2\times\mathbb{Z}_2')$ with discrete torsion) to eliminate open-string moduli and adjoint fields. Each rigid fractional D6-brane wraps a combination of bulk and exceptional three-cycles fixed under the orbifold twists, with specific twisted charge assignments, such that all adjoint matter is absent and moduli stabilization is tractable (Mansha et al., 6 May 2025, Mansha et al., 15 Jan 2025).

2. Gauge Group Realizations and Three-Family Structure

The minimal stack configuration gives rise to $SU(4)_C \times SU(2)_L \times SU(2)_R$, with the chiral spectrum determined by intersection numbers between the a, b, and c stacks:

• Three families are enforced by

$I_{ab}+I_{ab'}=3$, $I_{ac}=-3$, $I_{ac'}=0$

or related constraints under orientifold images. The left-handed sector arises at $a\cap b$ and the right-handed at $a\cap c$, both yielding exactly three families of $(4,2,1)$ and $(\overline{4},1,2)$ multiplets, respectively.

Generalizations relax the requirement that $a$ and $c$ (or $c'$) be parallel along any torus, admitting intersection patterns such as $I_{ac}=-(3+h)$, $I_{ac'}=h$ with $h\in\mathbb{Z}^+$, which broadens the family of allowed models and leads to four new distinct PS model classes (Li et al., 20 Nov 2025). These can include extended visible sectors, for example with $$U(4)_C\times U(2)_L\times U(2)_{R_1}\times U(2)_{R_2}$$, realized via an extra d-stack, and with diagonal symmetry breaking implemented by bifundamental Higgs fields (Huangfu et al., 9 Dec 2025).

Replacing $U(2)$ with $USp(2)$ restricts the model-building landscape severely, but yields ten new classes (five with $USp(2)_L$ and their SU(2)$_L \leftrightarrow$ SU(2)$_R$ duals), each with unique string-scale gauge-coupling ratios and distinct physical implications (Mansha et al., 2022, Mansha et al., 2023).

3. Symmetry Breaking, The Higgs Sector, and Yukawa Couplings

The symmetry breaking chain from the PS gauge group down to the MSSM is achieved via D-brane splitting and the Higgs mechanism:

• Stack splitting (e.g., $a \to a_1 + a_2$, $c \to c_1 + c_2$) breaks $SU(4)_C \to SU(3)_C \times U(1)_{B-L}$ and $SU(2)_R \to U(1)_{I_{3R}}$, leaving $U(1)_Y = \tfrac12 U(1)_{B-L} + U(1)_{I_{3R}}$.
• The $bc$ and $bc'$ intersections (and, in four-stack/d-stack models, $bd$, $cd$ as well) provide Higgs bi-doublets $(1,2,2)$, which induce supersymmetric electroweak breaking via VEVs in D- and F-flat directions (Li et al., 2019, Huangfu et al., 9 Dec 2025).
• Models may use adjoint Higgs fields (adjoints of $SU(12)_C$, $SU(6)_{L,R}$) for higher-rank symmetry breaking (Li et al., 2019), or rely on fundamental representations and higher-dimensional effective operators to disentangle quark and lepton masses without adjoints (Leontaris et al., 21 Feb 2025).
• Yukawa couplings are geometrically determined as worldsheet instanton overlap integrals, explicitly: $Y_{ijk} \sim \exp(-A_{ijk}/2\pi\alpha')$, where $A_{ijk}$ is the area of the triangle stretched among the three relevant brane intersections. In rigid models, the calculation is particularly tractable due to the fixed geometric setup, and realistic mass hierarchies and mixing patterns are produced (Mansha et al., 6 May 2025, Sabir et al., 2024). Spot checks confirm that in the viable sector (17 out of 33 models), complete fitting of all quark/lepton masses and mixings with three- and four-point couplings is achieved (Sabir et al., 2024).

4. Hidden Sector, Supersymmetry Breaking, and Moduli Stabilization

Hidden sectors are engineered via additional "filler" D6-brane stacks (USp or $U(2)$ branes) wrapping O6-plane cycles. These sectors saturate RR-tadpoles and often possess negative one-loop $\beta$-functions, leading to infrared confinement and the possibility of gaugino condensation. The generated nonperturbative superpotential $$W_{\textrm{np}} \sim \exp(-8\pi^2/|\beta| f_{\rm USp})$$ enables dynamical SUSY breaking and moduli stabilization (Li et al., 2019, Mansha et al., 15 Jan 2025, Mansha et al., 6 May 2025).

Adjoint moduli from non-rigid branes create unwanted light states unless removed—this is solved in rigid-cycle constructions. Phenomenologically, exotic states (from intersections involving hidden sector) often decouple via confinement or vector-like mass terms, rendering the low-energy spectrum clean (Mansha et al., 6 May 2025, Mansha et al., 2022, Huangfu et al., 9 Dec 2025).

The landscape accommodates both confining and non-confining hidden sectors, with models lacking USp filler branes entirely achievable in certain four-stack (d-stack) configurations, minimizing the presence of exotics due to hidden sector dynamics (Huangfu et al., 9 Dec 2025).

5. Gauge Coupling Unification, Thresholds, and RGE Analysis

String-scale gauge coupling relations are model-dependent and determined by three-cycle volumes and dilaton/Kähler moduli: $$\frac{1}{g_a^2} \propto \prod_{i=1}^3 |n_a^i R_1^i| + |l_a^i R_2^i|$$ Tree-level unification ($g_4^2 = g_{2L}^2 = g_{2R}^2 = (5/3)g_Y^2$ at $M_s$) is realized in a unique class (He et al., 2021, Li et al., 2022), with most models showing non-unified gauge-coupling ratios at $M_s$ but precise unification restored via two-loop RGE running involving threshold corrections from vector-like exotics and adjoint chiral multiplets (e.g., $XG$, $XW$, $XQ$, $XD$, $XU$). Additions of these $\mathcal{N}=2$ hypermultiplet pairs at specific mass scales are guided by exact intersection numbers and the model's structure (He et al., 2021, Li et al., 2022).

Representative beta-function shifts: | Multiplet | Representation | $\Delta b$ | |------------|-------------------|-------------------| | XG | (8,1,0) | (0,0,3) | | XW | (1,3,0) | (0,2,0) | | XQ | (3,2,1/6) + c.c. | (1/5,3,2) | | XD | (3,1,-1/3) + c.c. | (2/5,0,1) |

The addition of an extra d-stack in four-stack models permits further control of unification properties, allows reduction (or elimination) of required USp filler branes, and facilitates the inclusion of additional vector-like matter to generate thresholds needed for two-loop corrections, supporting unification at $M_s \sim 5\times10^{17}$ GeV (Huangfu et al., 9 Dec 2025).

6. Flavor, Family Symmetries, and Phenomenology

Flavor structures are addressed via spontaneous breaking of continuous family symmetries (e.g., $SU(3)_F$ or $SU(3)_L \times SU(3)_R$), with matter and Higgs bi-doublets as flavor triplets (Hartmann et al., 2014, Varzielas, 2011). Flavon VEV patterns aligned by discrete symmetries (such as $Z_4$) or quartic soft terms generate hierarchical Yukawa textures consistent with observed quark and lepton spectra. Seesaw-induced neutrino mass matrices emerge from higher-dimensional operators, with type-I or hybrid seesaw structures and hierarchical right-handed neutrino Majorana masses.

In the string-derived models, the number and origin of Higgs doublets ($3$, $6$, $9$, or $12$ from $\mathcal{N}=2$ sectors) crucially determine which models can fit all SM masses and mixings (Sabir et al., 2024). The viable 12-Higgs models produce normal-ordered Dirac neutrino masses, consistent with both phenomenological and swampland constraints.

Low-energy predictions are model dependent: in unified GUT approaches, fitting to low-energy observables including $sin\,2\beta$, electric dipole moments, and rare decays is quantitatively possible, with representative gluino mass bounds $M_{\tilde g} \lesssim 1.9$–$2.7$ TeV and lightest neutralino in the $300$–$500$ GeV range (Poh et al., 2017). Decoupling of proton-decay operators and control of baryon-number violation are achieved via operator selection rules and suppression of dangerous operators by the PS gauge symmetry (Leontaris et al., 21 Feb 2025, Cvetic et al., 2015).

7. F-theory, Global Embedding, and Future Directions

Globally consistent F-theory compactifications with three-family PS structure are constructed using toric hypersurface Calabi–Yau fourfolds with fluxes, requiring the solution of $G_4$-flux quantization and D3-tadpole cancellation, and matching vertical cohomology to chiral matter and anomaly-free spectra. Explicit computation of the 3D Chern–Simons terms and induced chiral indices confirm three-family solutions, with Higgsing patterns and decoupling of vector-like exotics controlled by geometrically engineered couplings (Cvetic et al., 2015).

The full landscape of three-family supersymmetric PS vacua, as constructed in the intersecting D6-brane framework, is now established and classified both phenomenologically and in terms of their UV-consistent string embeddings, with ongoing work focused on complete moduli stabilization, precision calculation of soft SUSY-breaking terms, and searches for phenomenologically distinctive signatures related to the extended gauge and flavor sector, gravitational wave backgrounds from domain wall collapse, and flavor sum rules (Sabir et al., 2024, Leontaris et al., 21 Feb 2025, Mansha et al., 15 Jan 2025).