Gauge-Coupling Unification in Pati-Salam Models

• The paper demonstrates that gauge-coupling unification is achieved by geometrically, algebraically, or dynamically aligning the Pati-Salam groups at a high scale.
• It details how intersecting D-brane constructions and spectral actions in noncommutative geometry enforce unification through precise moduli tuning and matching conditions.
• The study emphasizes practical phenomenological implications including proton decay suppression, leptoquark signatures, and controlled renormalization group evolution.

Gauge-coupling-unified Pati-Salam models are extensions of the Standard Model that realize the Pati-Salam gauge symmetry $SU(4)_C \times SU(2)_L \times SU(2)_R$ and feature precise or highly constrained unification of the corresponding gauge couplings at a high scale. These constructions are realized in diverse frameworks, including intersecting D-brane Type IIA string models, spectral action formulations in noncommutative geometry (NCG), and field-theoretic embeddings allowing for intermediate scales, extended matter content, or UV fixed points. The unification of gauge couplings is enforced geometrically, algebraically, or dynamically, depending on the underlying construction, and is closely tied to the possible breaking chains, matter content, and moduli stabilization mechanisms available in the given context.

1. Theoretical Structure: Pati-Salam Gauge Symmetry and Unified Coupling

The Pati-Salam group $SU(4)_C \times SU(2)_L \times SU(2)_R$ naturally accommodates the Standard Model matter content and motivates the possibility of gauge coupling unification. In geometric string constructions (e.g., Type IIA on $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_2)$), the gauge group is engineered by arrangements of intersecting D6-branes, where each stack wraps a specific three-cycle defined by integer wrapping numbers $(n^i_x, l^i_x)$, leading to gauge couplings

$$g_x^{-2} = \Re f_x = \frac{M_s^3}{(2\pi)^3 g_s} \prod_{i=1}^3 \left(n_x^i R_1^i + 2^{-\beta_i} l_x^i R_2^i\right)$$

with $\beta_i$ encoding whether the $i$-th two-torus is tilted.

In the noncommutative geometry (NCG) and spectral action approach, the unification of couplings at a scale $M_U$ is a consequence of the normalization of kinetic terms arising from the underlying finite algebra $$\mathcal{A}_F = \mathbb{H}_R \oplus \mathbb{H}_L \oplus M_4(\mathbb{C})$$ (Aydemir, 10 Nov 2025, Chamseddine et al., 2015), resulting in

$g_4(M_U) = g_L(M_U) = g_R(M_U) \equiv g_U$

at the emergence or unification scale.

2. Mechanisms for Gauge Coupling Unification

Unification occurs via one of several precise mechanisms:

• Geometric Unification: In Type IIA intersecting D-brane models, unification is achieved by tuning moduli (ratios of internal torus radii) so that the three-cycle volumes of the stacks corresponding to $SU(4)_C$, $SU(2)_L$, and $SU(2)_R$ are equal, enforcing $g_4 = g_{2L} = g_{2R}$ at the string scale. Exact tree-level unification is realized in specific models through such tuning, supported by particular choices of complex structure moduli and wrapping numbers (He et al., 2021, Mansha et al., 2023, Li et al., 2022).
• Field-Theoretic Matching and Symmetry Breaking Chains: Unification can also arise across a cascade of breakings, with each breaking stage defined by characteristic matching conditions and threshold effects. For instance, the matching at the Pati-Salam breaking scale $M_{PS}$ involves

$$\frac{1}{\alpha_1(M_{PS})} = \frac{3}{5}\frac{1}{\alpha_R(M_{PS})} + \frac{2}{5}\frac{1}{\alpha_4(M_{PS})}$$

and similar relations for $\alpha_2$, $\alpha_3$ (Chamseddine et al., 2015, Hartmann et al., 2014).

• Spectral Action in NCG: The spectral action yields, upon Seeley–DeWitt expansion, bosonic actions with identically normalized kinetic terms for all PS factors, enforcing algebraic unification at $M_U$ (Aydemir, 10 Nov 2025, Chamseddine et al., 2015).

3. Model Building: Spectrum, Matter Content, and Thresholds

The construction of gauge-coupling-unified models is sensitive to details of spectrum engineering:

• D6-Brane Models (Type IIA): The gauge sector arises from open strings at D-brane intersections; three chiral families are achieved by precise intersection numbers (e.g., $I_{ab} + I_{ab'} = 3$, $I_{ac} = -3$, $I_{ac'} = 0$) (He et al., 2021, Li et al., 2022). The requirement of gauge coupling unification selects unique wrapping configurations, usually with no exotics or exotics arranged in complete multiplets that do not spoil unification at the one-loop level.
• NCG/Spectral Models: The finite algebra dictates the number and type of scalar multiplets (e.g., bidoublet $(1,2,2)$, adjoint $(15,1,1)$, multiplets responsible for PS breaking such as $(4,1,2)$), with unification preserved for a variety of Higgs sectors. The minimal set suffices for both breaking and maintaining the coupling identity at $M_U$ (Aydemir, 10 Nov 2025, Chamseddine et al., 2015).
• Intermediate Exotics and Moduli Stabilization: In intersecting-brane and orbifold realizations, unification at the string or compactification scale sometimes requires introducing vector-like matter from $\mathcal{N}=2$ subsectors (e.g., seven pairs of vector-like $(3,1,-1/3)$ from parallel branes) or adjoint multiplets at intermediate scales. The counting and mass scales of such exotics are determined by brane intersection numbers or geometric data (He et al., 2021, Li et al., 2022).

4. Renormalization Group Evolution and Loop Effects

Rigorous treatment of RGEs and threshold effects is key for precise unification:

• One-Loop vs. Two-Loop Running: While many analyses establish unification at one-loop accuracy (Chamseddine et al., 2015, Aydemir, 10 Nov 2025), precision demands two-loop analysis, particularly when exotics are numerous or when mass hierarchies are introduced. The standard two-loop formula

$$\frac{d g_i}{dt} = \frac{b_i}{16\pi^2}g_i^3 + \frac{1}{(16\pi^2)^2}\sum_j b_{ij} g_i^3 g_j^2$$

is supplemented by beta function and matching coefficient shifts from extra matter (He et al., 2021, Li et al., 2022).

• Threshold Corrections: At each mass threshold (e.g., SUSY scale, exotic mass, string scale), gauge couplings are matched according to

$$\frac{1}{\alpha_i(\mu^+)} = \frac{1}{\alpha_i(\mu^-)} - \frac{\Delta b_i}{2\pi}\ln\left(\frac{M_V}{\mu}\right)$$

with the spectrum of states at each interval driving the running (Li et al., 2022).

• Examples: Precise two-loop running with exotics from $\mathcal{N}=2$ sectors yielded unification at $M_U \simeq 7.2 \times 10^{17}$ GeV for benchmark models, with differences in $\alpha_i^{-1}(M_U)$ at the $<1\%$ level, even for non-unified tree-level models, once exotics are included according to topology (He et al., 2021, Li et al., 2022).

5. Distinct Approaches: String, NCG, and Safe Unification

The gauge-coupling-unified paradigm in PS models appears in several distinctive approaches:

• String-Derived Models: In intersecting brane vacua, unification naturally emerges in the unique model with appropriate wrapping data and moduli tuning (He et al., 2021). In other models, unification is achieved only after incorporating vector-like matter or adjoint fields with multiplicities determined by geometry.
• Noncommutative Geometry/Spectral Action: Unified couplings are imposed at the "emergence" scale via spectral normalization, realizing unification robustly against threshold corrections and admitting variations with minimal or extended scalar content (Aydemir, 10 Nov 2025, Chamseddine et al., 2015, Aydemir et al., 2018). The NCG framework restricts the particle spectrum and relates algebraic properties of the finite Dirac operator to the running and matching of gauge couplings.
• Asymptotically Safe Pati-Salam Models: Here the unification of couplings arises dynamically from the existence of a common nontrivial UV fixed point for all gauge, Yukawa, and quartic couplings, realized via the inclusion of $\mathcal{O}(10^2)$ vector-like fermions in appropriate representations and large-$N_f$ resummations (Molinaro et al., 2018). The RG flow drives all couplings to finite interacting values below the Planck scale, a dynamical unification principle distinct from traditional GUT matching.

6. Phenomenological Implications and Model Discrimination

The high-scale unification in these models has several key phenomenological consequences:

• Proton Decay and Exotics: Scalar and fermionic exotics can be arranged to avoid rapid proton decay, either by symmetry (e.g., absence of diquark couplings enforced by NCG or selection rules) or by heavy masses (Aydemir, 10 Nov 2025, Aydemir et al., 2018). Near-unification at or above $10^{16}$ GeV generally suppresses proton decay below experimental bounds.
• Leptoquark Phenomenology: Certain models predict light scalar leptoquarks (notably the $S_1 \equiv (\bar{3},1,+1/3)$ from the $(6,1,1)$), with couplings constrained by both unification and low-energy flavor observables. These states are motivated in the context of recent $B$-physics anomalies and are naturally protected from mediating proton decay (Aydemir, 10 Nov 2025, Aydemir et al., 2018).
• Neutrino Masses and SUSY Breaking: In intersecting-brane models with confining hidden sector $USp(2)$ factors, moduli stabilization, supersymmetry breaking via gaugino condensation, and natural see-saw mechanisms for neutrino masses are implemented alongside unification (Li et al., 2022, Mansha et al., 2023).
• Intermediate Scales and Collider Phenomenology: Multi-step breaking chains allow for intermediate Pati-Salam or left-right symmetric stages, potentially predicting new exotics at accessible energies if breaking scales are appropriately arranged (Hartmann et al., 2014).

7. Obstacles, Tuning, and Future Directions

Attaining exact gauge-coupling unification in Pati-Salam models requires addressing several nontrivial constraints:

• Fine Tuning via Moduli or Matter: Unification at the string or Planck scale depends on subtle balancing of moduli, spectrum, and threshold corrections, especially in scenarios with tilted or partially factorized internal tori. Hidden sector effects and exotics must be included carefully and in accordance with global geometric or algebraic consistency conditions (tadpole cancellation, anomaly cancellation) (Li et al., 2022, He et al., 2021, Mansha et al., 2022).
• Rigidity of NCG Models: The unification condition $g_4 = g_L = g_R$ at $M_U$ in NCG-based models is inflexible; lowering certain symmetry breaking scales (e.g., for a TeV-scale $W_R$) is extremely challenging without either relaxing the algebraic unification constraint or introducing well-chosen extra matter (Aydemir et al., 2015).
• Dynamically Controlled Phenomena: Asymptotically safe models provide a novel paradigm wherein all couplings unify at an interacting fixed point, sidestepping some of the tuning required in other frameworks but requiring extensive vector-like field content (Molinaro et al., 2018).
• Experimental Probes: The presence of TeV-scale exotics, leptoquarks, or left-right symmetry can provide direct collider signatures or indirect precision constraints. Consistency with flavor physics, proton decay, and neutrino mass generation restricts the allowed parameter space further (Aydemir, 10 Nov 2025, Aydemir et al., 2018, Hartmann et al., 2014).

In sum, gauge-coupling-unified Pati-Salam models constitute a class of high-scale completions of the Standard Model where unification is achieved via precise geometric, spectral, or dynamical mechanisms, with each framework presenting highly constrained scenarios for beyond-the-Standard-Model phenomenology and testable implications arising from their unification structure.