We present a renormalizable SO(10) grand unified theory with a $\textit{minimal}$ Yukawa sector consisting of a 126 , a real 10 and a real 120 , where $\textit{CP}$ violation has a spontaneous origin. We show that the Yukawa sector in this setup, which consists of only 19 real parameters, is capable of simultaneously reproducing the observed fermion masses and mixings, including neutrino oscillations, as well as the baryon asymmetry of the Universe via thermal leptogenesis. In this framework, $\textit{CP}$ is spontaneously broken when a $\textit{CP}$-odd Higgs field 54 , used for GUT symmetry breaking, acquires a non-zero vacuum expectation value. The Yukawa sector of the model contains four independent phases which determine the Dirac phases $\textit{δ}$ and $\textit{δ}$ , the neutrino Majorana phases, as well as those responsible for leptogenesis. We show that these four phases can arise from a single complex parameter in the Higgs potential involving the $\textit{CP}$-odd Higgs field 54 . The proposed minimal setup predicts a $\textit{normal}$ ordering of neutrino masses, with the atmospheric mixing angle $\textit{θ}$ preferred in the first octant and the $\textit{δ}$ lying in the range (–37°, +31°). The fermion fits in our scenario further yield a strongly hierarchical mass spectrum for the three right-handed neutrinos, ($\textit{M}$ , $\textit{M}$ , $\textit{M}$ ) ∼ (10 , 10 , 5 · 10 ) GeV, which is shown to result in successful $\textit{N}$ $\textit{-dominated}$ leptogenesis, consistent with current cosmological data.

United States

K. Babu

Brazil

Chee Fong

Slovenia

Shaikh Saad

Spontaneous CP violation and leptogenesis in SO(10)

We investigate $\textit{Higgs Parity unification}$ — a realization of SO(10) grand unification based on the Higgs Parity mechanism in which the Standard Model (SM) Higgs resides in a spinor representation. The theory has an intermediate left-right symmetric stage where the SU(2) symmetry breaking scale is fixed by the vanishing of the SM Higgs quartic coupling. The strong $\textit{CP}$ problem is solved by parity. Gauge coupling unification successfully predicts $\textit{α}$ ($\textit{M}$ ) to within 1%. The spinor Higgs naturally leads to a seesaw origin for SM flavor observables. We identify a novel mechanism where large mixing of third generation fermions with additional heavy vector-like fermions accounts for the anarchical nature of the PMNS matrix and the lack of hierarchy in the neutrino mass spectrum, relative to the up-quarks. A fit to quark and lepton masses and mixings, with a minimal parameter set, predicts 1) A testable relation between the top quark mass and $\textit{α}$ ($\textit{M}$ ) which is about (1 – 2)$\textit{σ}$ from current best fit values, 2) The order of magnitude of the baryon asymmetry of the universe, via leptogenesis from second-generation right-handed neutrino decays. 3) The proton decay and the neutron EDM are likely observable in next generation experiments, and 4) A normal ordered neutrino mass spectrum where 0$\textit{νββ}$ decay and the mass of the lightest neutrino are out of reach of next generation experiments.

Juanca Carrasco-Martinez

Lawrence Hall

Japan

Keisuke Harigaya

Kevin Langhoff

A flavor of SO(10) unification with a spinor Higgs

<p>We have studied an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> Skyrmion on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi mathvariant="normal">I</mi><mspace depth="0.0ex" height="0.0ex" width="-0.2em"></mspace><mi mathvariant="normal">R</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math>, which can be seen as a static soliton in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mo>+</mo><mn>1</mn></math> dimensions. This is a sequel of the known <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></math> gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mi>D</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math> Skyrmions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi mathvariant="normal">I</mi><mspace depth="0.0ex" height="0.0ex" width="-0.2em"></mspace><mi mathvariant="normal">R</mi></mrow><mrow><mi>D</mi></mrow></msup></mrow></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo>=</mo><mn>2</mn></math> and in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo>=</mo><mn>3</mn></math>, which are likewise localized to an absolute scale and topologically stable, their energies being bounded below by the winding number. In the absence of an analytic proof of existence, some such solutions are constructed numerically. Two families of solutions are found, of which only one possesses a gauge decoupling limit. The curvatures of both of these solutions decay as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, a property they share with the Yang-Mills instantons on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi mathvariant="normal">I</mi><mspace depth="0.0ex" height="0.0ex" width="-0.2em"></mspace><mi mathvariant="normal">R</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math>.</p>

Spain

Francisco Navarro-Lérida

Ireland

D.H. Tchrakian

SO(4) gauged O(5) Skyrmion on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">R</mi><mn>4</mn></msup></math>

We propose a gauge-invariant formulation of magnetic translation (GMT) operators, eliminating the gauge dependence that conventional definitions suffer from due to specific gauge choices. We extend this framework by incorporating <math altimg='si1.svg'><mrow><mi mathvariant='fraktur'>so</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math> conformal symmetry, demonstrating how GMT operators can be naturally embedded within this algebraic structure. We then show the fundamental role of <math altimg='si2.svg'><mrow><mi mathvariant='fraktur'>sim</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math> duality between pseudo-dilatation and pseudo-angular momentum operators in constructing GMT operators. A key result of this approach is the derivation of the Curtright-Zachos (CZ) generators from gauge-invariant <math altimg='si1.svg'><mrow><mi mathvariant='fraktur'>so</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math> operators, providing a novel perspective on their algebraic properties such as the relationship between FFZ internal symmetry and <math altimg='si2.svg'><mrow><mi mathvariant='fraktur'>sim</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math> duality.

Haru-Tada Sato

Gauge invariant and generic formulation of magnetic translations and <math altimg='si1.svg'><mrow><mi mathvariant='fraktur'>so</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math> Curtright-zachos generators

We consider the dimensional reduction on a torus of the family of 6d (1$\textit{,}$ 0) SCFTs UV completing an $\textit{so}$($\textit{N}$) gauge theory with $\textit{N}$ − 8 vector hypermultiplets. These SCFTs are known to possess a rich structure of discrete symmetries, notably 0-form and 1-form symmetries, which often merge to form a higher group structure, both split and non-split. We investigate what happens to this symmetry structure once the theory is reduced on a circle to 5d and on a torus to 4d, especially when a non-trivial Stiefel-Whitney class for the flavor symmetry is turned on. Unlike in Lagrangian theories, here the 1-form symmetries of the 6d theory reduce to non-trivially acting 1-form and 0-form symmetries, and the original higher group structure leads to an extension of the 0-form symmetries.

Israel

Gabi Zafrir

Generalized symmetries and the dimensional reduction of 6d $\textit{so}$ SCFTs

We discuss the charged and neutral fermion masses and mixings in a supersymmetric SO(10) model with a minimal Higgs sector consisting of the multiplets 10 , 45 , and $${16}_{H}+{\overline{16} }_{H}$$ . In addition to the renormalizable Yukawa couplings involving the Higgs 10-plet, we include non-renormalizable Yukawa couplings, which are important for reproducing with good accuracy the observed masses and mixings in the quark and lepton sectors. We identify a preferred solution which is compatible with third family quasi-Yukawa unification, namely $\textit{y}$ ≈ $\textit{y}$ ≈ 0$\textit{.}$73$\textit{y}$ at the unification scale, with the MSSM parameter tan $\textit{β}$ ~ 58. Acceptable solutions with lower tan $\textit{β}$ values are also realized in our framework, and we provide an example with tan $\textit{β}$ = 10. Based on our fits, the masses for the three right-handed neutrinos turn out to be ($\textit{M}$ $\textit{, M}$ $\textit{, M}$ ) ~ (10 $\textit{,}$ 8∙10 $\textit{,}$ 9∙10 ) GeV. We briefly discuss the metastable and quasistable string scenarios that can be realized in this class of supersymmetric SO(10) models.

Qaisar Shafi

Fermion masses and mixings in supersymmetric SO(10) with third-generation quasi-Yukawa unification

We apply a universal two-zero texture (UTZT) to all mass matrices for matter in its flavor space within the $\textit{SO}$(10) GUT framework. This texture can be realized by assigning different charges to each family in a $ Z_6 $ symmetry. By fixing the charged fermion masses at their best-fit values, we fit the remaining nine precisely measured observables (three angles and one CP-violating phase in quark mixing, three angles in lepton mixing, and two neutrino mass-squared differences) with seven model parameters. The model fits all data on fermion masses and mixing accurately, and the leptonic CP-violating phase is predicted to lie in the range $ (90^\circ, 230^\circ) $ . The model further predicts the right-handed neutrino masses, with the lightest and heaviest ones being on orders of $ 10^9 $ and $ 10^{12} $ GeV, respectively. Gauge unification and proton decay were checked under the assumption of a breaking chain with two intermediate symmetries above the electroweak scale. The result indicates that $ \alpha_{\rm{GUT}} $ varies in the range of (0.022,0.032) as long as the assumption of an economical choice of Higgs contents is fulfilled, and $ M_{\rm{GUT}} $ should be bigger than $ 4.5\times 10^{15} $ GeV to meet the Super-K bound. We report on the effective mass $ m_{ee} $ for neutrinoless double beta decay, which provides a possibility to test grand unification with neutrinoless double beta decay experiments.

China

Gao-Xiang Fang

Ye-Ling Zhou

Fermion masses and mixing in $\textit{SO}$(10) GUT with a universal two-zero texture * Supported by the National Natural Science Foundation of China (NSFC) (12205064, 12347103) and Zhejiang Provincial Natural Science Foundation of China (LDQ24A050002)

We show that electroweak triplet scalar can significantly impact baryogenesis via leptogenesis in concrete and predictive SO(10) GUTs, even when light neutrino masses arise predominantly from the type-I seesaw mechanism. This is illustrated within a minimal renormalisable SO(10) model with and $$ \overline{\textbf{126}} $$ scalars in the Yukawa sector and a global Peccei-Quinn-like symmetry. The quark-lepton unification and the flavour structure of the fundamental Yukawa couplings enforce type-I dominance in the light neutrino masses, also suppressing the triplet-induced CP asymmetries in the right-handed neutrino decays. However, the triplet’s own decays introduce a new CP-violating source, which can enhance or suppress the total baryon asymmetry. For triplet mass near the right-handed neutrino mass scale, this contribution can dominate, making it essential in assessing the viability of SO(10) leptogenesis scenarios.

India

Ketan Patel

Electroweak triplet scalar contribution to SO(10) leptogenesis

<p>Non-Abelian toroidal heterotic orbifolds have received comparatively little attention, mainly because of the significant computational challenges they pose, even at the level of computing their matter spectrum. Similarly, the SO(32) heterotic string remains relatively unexplored. In this paper, we provide some useful tools to handle this situation. We find that certain non-Abelian orbifolds can be studied using the techniques that are common to Abelian compactifications. In such cases, we show how to compute the gauge groups and massless matter spectrum for non-Abelian orbifolds of the SO(32) heterotic string with standard embedding. A general feature of these constructions is the reduction of the rank of the gauge group. Our findings motivate further research on non-Abelian orbifolds with nonstandard embedding, where realistic, rank-reduced models are expected to emerge.</p>

Mexico

Miguel Hernández-Segura

Saúl Ramos-Sánchez

Non-Abelian orbifolds of the SO(32) heterotic string

Asymptotic grand unification provides an alternative approach to gradually unify gauge couplings in the UV limit, where they reach a non-trivial UV fixed point. Using an economical and realistic particle content setup, we demonstrate that asymptotic grand unification can be achieved in a 5D SO(10) model with one extra dimension. The top, bottom and tau masses are split, and the smallness of the neutrino mass is explained via inverse seesaw. One intermediate scale, the Pati-Salam symmetry breaking scale, is included below the compactification scale. Due to the absence of large-dimensional Higgs representations, gauge couplings exhibit asymptotic safety and are thus asymptotically unified, regardless of their initial values. In contrast, Yukawa couplings can achieve asymptotic freedom if the negative gauge contributions dominate over the positive Yukawa terms, requiring exact unification at the compactification scale. The widely-used 126-dimensional Higgs is not recommended in this 5D asymptotic SO(10) GUT, as it tends to drive the gauge beta function positive, compromising asymptotic safety.