Unified Pati-Salam from Noncommutative Geometry: Overview and Phenomenological Remarks (2511.07672v1)

Abstract: The lack of clear new-physics signals at the LHC searches motivates models that can guide current and future collider searches. The spectral action principle within the noncommutative geometry (NCG) framework yields such models with distinctive phenomenology. This formalism derives the actions of the Standard Model, General Relativity, and beyond from the underlying algebra, putting them on a common geometric footing. Certain versions of Pati-Salam (PS) models with gauge coupling unification and limited scalar content can be derived from an appropriate noncommutative algebra. In this paper, I review these gauge-coupling-unified Pati-Salam models and discuss their phenomenological aspects, focusing on the $S_1$ scalar leptoquark.

• The paper introduces unified Pati-Salam models using noncommutative geometry and the spectral action principle to merge gravity with gauge interactions.
• It demonstrates that geometric constraints yield a restricted scalar sector and precise gauge coupling unification while avoiding dangerous diquark couplings.
• The study explores phenomenological implications, notably resolving B-meson decay anomalies through the incorporation of the S1 leptoquark with left-handed couplings.

Unified Pati-Salam from Noncommutative Geometry: Formalism and Phenomenological Implications

Noncommutative Geometry and the Spectral Action Principle

The paper presents a comprehensive overview of the construction of Pati-Salam models with gauge coupling unification from noncommutative geometry (NCG) via the spectral action principle. NCG formulations replace the classical notion of geometry with the spectral data $(\mathcal{A}, \mathcal{H}, \mathcal{D})$, where $\mathcal{A}$ denotes an involutive algebra acting on the Hilbert space $\mathcal{H}$, and $\mathcal{D}$ is a generalized Dirac operator. This setup, enhanced by a $\mathbb{Z}_2$ grading and an antiunitary operator $\mathcal{J}$, encodes both the manifold and gauge content geometrically. The theory derives the Standard Model (SM), General Relativity (GR), and extended gauge models within the same geometric unification.

The spectral action,

$$\mathcal{S} = \left( J\psi, \mathcal{D}_A \psi \right) + \mathrm{Tr} \left[ \chi \left( \frac{\mathcal{D}_A}{\Lambda} \right) \right],$$

combines the fermionic and bosonic sectors. The operator $\mathcal{D}_A$ incorporates inner fluctuations, with the Higgs field arising naturally as the connection in the discrete dimension. The cutoff function $\chi$ restricts the spectrum to the scale $\Lambda$.

Implementing the SM within this framework requires choosing the finite algebra $$\mathcal{A}_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$$, directly reproducing the $U(1)$, $SU(2)$, and $SU(3)$ gauge group factors. The bosonic part of the action exhibits necessary gauge coupling unification conditions, $g_3^2 = g_2^2 = \frac{5}{3}g_1^2$, imposed at a unification scale $M_U$ much lower than $\Lambda$. This approach highlights constraints and limitations: e.g., the minimal construction yields the incorrect Higgs mass, which can be corrected by introducing an additional singlet field. Truncating the spectral action at order $O(1/\Lambda^2)$ implies the framework is viewed as an effective geometric theory, not UV complete. Notably, the formal QFT structure respecting NCG is not fully established, requiring pragmatic application of usual QFT methods at lower energies.

Construction of Pati-Salam Models From NCG

To extend beyond the minimal SM spectral geometry, the finite algebra is replaced by $$\mathcal{A}_F = \mathbb{H}_R \oplus \mathbb{H}_L \oplus M_4(\mathbb{C})$$, yielding the Pati-Salam gauge group $SU(4)_C \times SU(2)_L \times SU(2)_R$ (denoted $G_{422}$) or explicitly left-right symmetric $G_{422D}$, depending on the fulfillment of the order-one condition. The scalar sector becomes tightly restricted by the underlying algebra and geometry. Three versions of the model—A, B, and C—differ in scalar content and symmetry, summarized as follows:

Model	Symmetry	Scalar Content (main entries)
A	$G_{422}$	$\phi(1,2,2)$, $\Sigma(15,1,1)$, $\widetilde{\Delta}_R(4,1,2)$
B	$G_{422}$	$\phi(1,2,2)$, $\widetilde{\Sigma}(15,2,2)$, $\Delta_R(10,1,3)$, $H_R(6,1,1)$
C	$G_{422D}$	$B$ plus $\Delta_L(10,3,1)$, $H_L(6,1,1)$

The symmetry breaking proceeds through $\langle \Delta_R \rangle$, leading from the unified group $G_{422D}$ to the SM group $G_{321}$ at hierarchy scales $M_U \gg M_C$.

Fermionic and Scalar Sector: Predictivity and Restrictions

The fermion representations are dictated by the group structure, filling $(4,2,1)$ and $(4,1,2)$, matching the SM fermion content with inclusion of right-handed neutrinos. In the scalar sector, crucial fields include $\Sigma^{bJ}_{\dot{a}I}$, $H_{aIbJ}$, and $H_{\dot{a}I \dot{b}J}$, associated with bidoublets, triplets, and sextets. The NCG construction restricts possible interaction terms: several terms present in ordinary PS models due to larger scalar freedom are automatically eliminated by geometric constraints, yielding increased predictivity and profound implications for low-energy phenomenology.

Yukawa Lagrangian terms are generated via

$$\mathcal{L}_Y = \overline{\psi}^{\dot{a}I} \gamma_5 \Sigma^{bJ}_{\dot{a}I} \psi_{bJ} + \overline{\psi^C}_{aI} \gamma_5 H^{aIbJ} \psi_{bJ} + \overline{\psi^C}_{\dot{a}I} \gamma_5 H^{\dot{a}I \dot{b}J} \psi_{\dot{b}J} + \text{h.c.}$$

The grading ($\gamma_5$) arises from the product geometry $M\times F$. Symmetry breaking yields SM-like Yukawa terms with additional interactions of new scalars, with absence of certain couplings (e.g., diquark couplings) enforced by the NCG structure.

Phenomenology: Addressing $B$-Meson Decay Anomalies and Leptoquarks

A salient application is in the context of flavor anomalies in $B$-decays, such as $R_{D^{(*)}}$ measurements, which currently deviate from SM predictions by over $3\sigma$. Scalar leptoquarks—especially the $S_1 = (\overline{3},1,1/3)_{321}$—are able to resolve these anomalies at tree level. In the NCG-PS models, the specific leptoquark content and allowed couplings differ from the generic PS scenario, which often suffers from proton decay issues due to diquark couplings.

Figure 1: SM and $S_1$ leptoquark contributions to $$\overline{B}_0 \to D^{(*)+}\tau^-\overline{\nu_{\tau L}}$$: the $S_1$ mechanism yields tree-level new physics relevant to $R_{D^{(*)}}$.

The geometric constraints only allow the $S_1$ in $H_L(6,1,1)_{422}$ (Model C) to couple to left-handed fermions without diquark couplings, thus avoiding proton decay. The resulting interaction terms are:

$$\overline{d_L^C} \nu_L H_{3L}^* - \overline{u_L^C} e_L H_{3L}^* + \varepsilon^{ijk}\overline{u_{Li}^C} d_{Lj} H_{\overline{3}Lk}^*$$

where $H_{3L}^*$ is phenomenologically viable, whereas $H_{\overline{3}L}^*$ should be heavy to suppress dangerous modes. If needed, $H_{3R}^{*}$ from $H_R(6,1,1)_{422}$ can provide right-handed couplings.

Additionally, the absence of right-handed couplings reduces parameter space for $R_{D^{(*)}}$ fits compared to regular models, but the proton is automatically stable. In the context of $a_\mu$, contributions from $S_1$ with only left-handed couplings are suppressed and negative; given recent convergence between experimental and theoretical predictions, this feature is not problematic.

Figure 2: Leading-order $S_1$ leptoquark contribution to $a_\mu$ is suppressed and carries negative sign with left-handed couplings only.

The framework supports global phenomenological analyses for observables including $R_{K^{(*)}}$, $R_K^\nu$, rare tau decays, and $B_c \to \tau \nu$, as well as correlations among flavor anomalies and $g-2$, leveraging the highly predictive scalar and Yukawa sector. This restrictiveness sharpens constraints and enables model discrimination against broader new physics scenarios.

Implications and Outlook

The construction of Pati-Salam models within noncommutative geometry via the spectral action principle unifies gravity and particle physics on geometric grounds, enforces gauge coupling unification, and constrains the allowed scalar and Yukawa content. The removal of dangerous couplings (notably mediating proton decay) and the minimal scalar sector emerge naturally from the geometric formalism, enhancing theoretical consistency and phenomenological viability.

For future developments, establishing a QFT structure compatible with NCG, investigating the RG flow and possible UV completions, and exploring precision flavor physics in light of new data could further probe the framework’s predictions. The approach’s attention to flavor anomalies, restricted parameter space, and geometric control over couplings will continue to be relevant for collider and flavor experiments.

Conclusion

The reviewed work demonstrates that noncommutative geometry, implemented through the spectral action principle, provides a robust method of constructing Pati-Salam models with gauge coupling unification. Characteristic features—including constrained scalar sectors, absence of problematic couplings, and compatibility with flavor anomaly resolution—set these models apart from standard PS and GUT scenarios. The geometric foundation not only places gravity and particle physics on equal footing but also endows the low-energy phenomenology with distinctive, testable signatures, making the NCG-based Pati-Salam unification a prominent candidate for guiding future experimental and theoretical explorations.