The masses of exotic quantum-number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> compact tetraquark states are calculated in a constituent quark model, where a Cornell-like potential is employed as the central potential, spin-spin and spin-orbit coupling derived from the Breit-Fermi interaction are treated as hyperfine corrections, and model parameters are taken from previous works. The ground state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> P-wave tetraquarks are predicted at 1.9, 4.2, and 6.6 GeV for the light, charmoniumlike, and fully charm sectors, respectively. The decay width ratios of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> tetraquark states are calculated for two-body strong decay channels within the rearrangement mechanism, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi><msub><mi>h</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi><msub><mi>f</mi><mn>1</mn></msub></math> for isospin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> light tetraquarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><msub><mi>h</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><msub><mi>f</mi><mn>1</mn></msub></math> for isospin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math> light tetraquarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo>/</mo><mi>η</mi><mo>+</mo><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo>/</mo><mi>ω</mi><mo>+</mo><msub><mi>h</mi><mi>c</mi></msub></math> for charmoniumlike tetraquarks, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><msub><mi>h</mi><mi>c</mi></msub></math> for fully charm tetraquarks. The theoretical results are compared with the observed exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> states, and promising search channels for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> tetraquarks are discussed. The work suggests that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1855</mn><mo stretchy="false">)</mo></math> is unlikely to be a compact tetraquark state.

Kai Xu

Zheng Zhao

Nattapat Tagsinsit

Attaphon Kaewsnod

Ayut Limphirat

Christoph Herold

Yupeng Yan

Systematic study of exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math> tetraquark spectroscopy

We present a comprehensive study of the axial-vector resonance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></math> within the unitarized chiral perturbation theory, focusing on its two-pole structure and manifestation in femtoscopic observables. By considering the dominant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mi>K</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><mi>π</mi></math> coupled channels, we reproduce the well-established double-pole structure and trace the chiral evolution of both poles as functions of the pion mass, using the vector-meson mass trajectories fitted to lattice-QCD data and experimental values. The lower pole, dominantly coupled to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><mi>π</mi></math>, evolves from an above-threshold resonance to a virtual or bound state with increasing pion mass. In comparison, the higher pole, dominantly coupled to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mi>K</mi></math>, moves downward in energy, reflecting the strengthening of the chiral attraction. The influence of the finite vector-meson widths is systematically examined, showing that their inclusion smooths the pole trajectories without altering their qualitative behavior. Furthermore, femtoscopic correlation functions are calculated for all relevant vector-pseudoscalar channels in both charged sectors. The results exhibit distinct resonance and bound-state features consistent with the two-pole dynamics. The weak impact of higher channels, such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mi>η</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math>, confirms that the simplified two-channel treatment captures the essential dynamics of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></math> resonance. This study demonstrates that combining chiral extrapolation and femtoscopic correlation analyses provides a powerful and complementary framework for connecting lattice-QCD calculations, chiral effective theory, and experimental measurements, offering new insights into the molecular nature and chiral origin of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></math> resonance.

Jia-Ming Xie

Zhi-Wei Liu

Jun-Xu Lu

Haozhao Liang

Raquel Molina

Li-Sheng Geng

Chiral evolution and femtoscopic signatures of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></mrow></math> resonance

Global analyses of particle physics data are integral for validating and scrutinizing published results of experiments. Global fits of anomalous oscillation data which search for one or more eV-scale sterile neutrinos are particularly challenging both to evaluate and to reconcile in the global picture. Fits (especially joint ones) to oscillation data suffer from significant computational burdens, such as likelihood intractability, making traditional Markov Chain-Monte Carlo all but impossible. Given evidence both supporting and challenging beyond Standard Model physics across neutrino experiments of various baselines, energies, and detection techniques, the global search for sterile neutrinos requires additional tools in order to determine whether sterile neutrinos remain a viable solution to unexplained anomalies. Furthermore, both a Bayesian and frequentist interpretation of sterile neutrino data is needed for a complete assessment of longstanding tensions in the field. Techniques from the machine learning subfield of simulation-based inference have a natural application to such a problem. In this contribution, we illustrate some of the outstanding questions of the global picture of light sterile neutrinos by focusing on experiments searching with the disappearance of electron (anti)neutrinos, and look to posterior density estimation strategies to craft answers, including comparisons to a machine-learning-based frequentist approach.

United States

Joshua Villarreal

Julia Woodward

John Hardin

Janet Conrad

Machine learning-informed 3+1 sterile neutrino global fits using posterior density estimation of electron disappearance data

We investigate the collider signatures of a maximally flavor-violating $ U(1)_{L_\mu-L_\tau} $ model, where a new gauge boson $ Z^\prime $ and scalar triplets induce lepton flavor-changing interactions in the $\textit{μ}$-$\textit{τ}$ sector. Focusing on four-lepton final states at multi-TeV lepton colliders, we conduct a detailed analysis of cross sections, asymmetries, and polarization effects. We show that the signal cross section is highly sensitive to $ m_{Z^\prime} $ and the effective parameters $ \tilde{g}/m_{Z^\prime} $ , whereas it remains largely insensitive to the triplet Yukawa couplings within the phenomenologically allowed region. The forward-backward asymmetry exhibits a characteristic monotonic dependence on $ m_{Z^\prime} $ , and beam polarization can significantly suppress the Standard Model backgrounds while enhancing new physics contributions. We find that, in the phenomenologically allowed parameter space, the predicted observables remain highly sensitive to the underlying model parameters. These results show that multi-lepton final states are powerful probes of the $ U(1)_{L\mu - L_\tau} $ framework and can offer valuable guidance for future searches at muon and electron-positron colliders.

China

Jianing Qin

Fei Huang

Honglei Li

Zhi-Long Han

Jin Sun

Collider probes of four-lepton final states in maximally flavor-violating ${U(1)_{L_\mu-L_\tau}} $ model * Supported by National Natural Science Foundation of China (12505112, 11805081), Shandong Province Natural Science Foundation (ZR2024QA138, ZR2022MA056) and State Key Laboratory of Dark Matter Physics, University of Jinan Disciplinary Cross-Convergence Construction Project 2024 (XKJC- 202404) and Jin Sun is supported by IBS under the project code, (IBS-R018-D1)

We investigate thermal photon production in the quark-gluon plasma (QGP) under strong magnetic fields using a magnetohydrodynamic (MHD) framework. We employ relativistic ideal fluid dynamics under the non-resistive approximation by adopting the Bjorken flow model with power-law decaying magnetic fields $ \mathbf{B}(\tau) = \mathbf{B}_0 (\tau_0/\tau)^a $ , span> $\textit{a}$ controls the decay rate, $ B_0 = \sqrt{\sigma} T_0^2 $ , and $\textit{σ}$ characterizes the initial field strength. The resulting QGP temperature evolution exhibits distinct $\textit{a}$- and $\textit{σ}$-dependent behaviors. Thermal photon production rates are calculated for three dominant processes: Compton scattering with $ q\bar{q} $ annihilation (C+A), bremsstrahlung (Brems), and $ q\bar{q} $ annihilation with additional scattering (A+S). These rates are integrated over the space-time volume to obtain the photon transverse momentum $ (p_T) $ spectrum. Our results demonstrate that increasing $\textit{a}$ enhances photon yields across all $ p_T $ , with $ a \to \infty $ (super-fast decay) providing an upper bound. For $ a = 2/3 $ , a larger $\textit{σ}$ suppresses yields through accelerated cooling, whereas for $ a \to \infty $ , a larger $\textit{σ}$ enhances yields via prolonged thermal emission. Low- $ p_T $ photons receive significant contributions from all QGP evolution stages, while high- $ p_T $ photons originate predominantly from early times. The central rapidity region $ (y=0) $ dominates the total yield. This work extends the photon yield studies to the MHD regime under strong magnetic fields, clarifying the magnetic field effects on QGP electromagnetic signatures and establishing foundations for future investigations of magnetization and dissipative phenomena.

Jie Xiong

Xiang Fan

Jing Jing

Weishan Yang

Duan She

Ze-Fang Jiang

Thermal photon emission from quark-gluon plasma: 1+1D magnetohydrodynamics results * Supported by the National Natural Science Foundation of China (NSFC) (12305138). Duan She's research is funded by the Startup Research Fund of Henan Academy of Sciences (231820058) and the 2024 Henan Province International Science and Technology Cooperation Projects (242102521068)

The leading-order low-energy constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Δ</mi><mrow><mi>mix</mi></mrow></msub></math> in mixed-action chiral perturbation theory is calculated using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math>-flavor gauge ensembles with HISQ fermions and a tadpole-improved Symanzik gauge action at four lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0.048</mn><mo>,</mo><mn>0.111</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. By comparing our results to those from different actions and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor case, We find that the fermion action has the dominant impact, the gauge action has a secondary but measurable effect, and the contribution from charm quark loops is negligible within our current uncertainties.

Zun-Xian Zhang

Mengchu Cai

Bolun Hu

Xiangyu Jiang

Xiao-Lan Meng

Yi-Bo Yang

Dian-Jun Zhao

Comparison of the mixed-fermion-action effects using different fermion and gauge actions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> flavors

In bound systems whose constituent particles are heavier than the electron, the dominant radiative correction to energy levels is given by light-fermion (electronic) vacuum polarization. In consequence, relativistic and recoil corrections to the one-loop vacuum-polarization correction are phenomenologically relevant. Here, we generalize the treatment, previously accomplished for systems with orbiting muons, to bound systems of constituents with more general spins: spin-0, spin-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>, and spin-1. We discuss the application of our more general expressions to various systems of interest, including spinless systems (pionium), muonic hydrogen and deuterium, and devote special attention to the excited non-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math> states of deuteronium, the bound system of a deuteron and its antiparticle. The obtained energy corrections are of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>α</mi><mn>5</mn></msup><msub><mi>m</mi><mi>r</mi></msub></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math> is the fine-structure constant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>r</mi></msub></math> is the reduced mass.

Gregory S. Adkins

Ulrich D. Jentschura

Relativistic and recoil corrections to light-fermion vacuum polarization for bound systems of spin-0, spin-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>, and spin-1 particles

We use Soft Collinear Effective Theory (SCET) to factorize the polarized Deep Inelastic Scattering (DIS) structure functions $\textit{g}$ ($\textit{x}$) and $\textit{g}$ ($\textit{x}$), and to sum Sudakov double logarithms of 1 − $\textit{x}$. The analysis is done both in terms of lightcone parton distributions and their moments. Computing $\textit{g}$ requires subleading SCET operators which contain gluons. We calculate the one-loop matching coefficients from QCD onto these subleading SCET operators, and the one-loop matching from SCET onto the parton distribution function (PDF). The PDF in SCET is given by a bilocal operator, rather than the trilocal operator used in the QCD analysis of $\textit{g}$ for generic $\textit{x}$. We compute the one-loop anomalous dimension of the PDF operator for any $\textit{x}$, and show that as $\textit{x}$ → 1, it factors into a single-variable evolution.

Jaipratap Grewal

Aneesh Manohar

Jyotirmoy Roy

Polarized deep inelastic scattering as $\textit{x}$ → 1 using soft collinear effective theory

Results are presented on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>ω</mi><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>1</mn><mi>S</mi><mo stretchy="false">)</mo></mrow></math> transitions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">→</mo><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>3</mn><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>γ</mi><msub><mrow><mi>χ</mi></mrow><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></mrow></math> decays. The data were collected with the BABAR detector at the PEP-II asymmetric-energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> collider at SLAC. The integrated luminosity of the data sample is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>28.0</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>121.3</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>3</mn><mi>S</mi><mo stretchy="false">)</mo></math> decays. Signals of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></math> are observed over a negligible background. Improved precision measurements of branching fractions are obtained. First measurements of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></math> angular distributions are performed. No evidence is found for the presence of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mi>χ</mi><mrow><mi>b</mi><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">→</mo><mi>ω</mi><mrow><mi mathvariant="normal">ϒ</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mi>S</mi><mo stretchy="false">)</mo></mrow></mrow></math> decay mode.

J.P. Lees

V. Poireau

V. Tisserand

E. Grauges

A. Palano

G. Eigen

D.N. Brown

Yu. G. Kolomensky

M. Fritsch

H. Koch

R. Cheaib

C. Hearty

T.S. Mattison

J.A. McKenna

R.Y. So

V.E. Blinov

A.R. Buzykaev

V.P. Druzhinin

E.A. Kozyrev

E.A. Kravchenko

S.I. Serednyakov

Yu. I. Skovpen

E.P. Solodov

K. Yu. Todyshev

A.J. Lankford

B. Dey

J.W. Gary

O. Long

A.M. Eisner

W.S. Lockman

W. Panduro Vazquez

D.S. Chao

C.H. Cheng

B. Echenard

K.T. Flood

D.G. Hitlin

Y. Li

D.X. Lin

S. Middleton

T.S. Miyashita

P. Ongmongkolkul

J. Oyang

F.C. Porter

M. Röhrken

B.T. Meadows

M.D. Sokoloff

J.G. Smith

S.R. Wagner

D. Bernard

M. Verderi

D. Bettoni

C. Bozzi

R. Calabrese

G. Cibinetto

E. Fioravanti

I. Garzia

E. Luppi

V. Santoro

A. Calcaterra

R. de Sangro

G. Finocchiaro

S. Martellotti

P. Patteri

I.M. Peruzzi

M. Piccolo

M. Rotondo

A. Zallo

S. Passaggio

C. Patrignani

B.J. Shuve

H.M. Lacker

B. Bhuyan

U. Mallik

C. Chen

J. Cochran

S. Prell

A.V. Gritsan

N. Arnaud

M. Davier

F. Le Diberder

A.M. Lutz

G. Wormser

D.J. Lange

D.M. Wright

J.P. Coleman

D.E. Hutchcroft

D.J. Payne

C. Touramanis

A.J. Bevan

M. Bona

F. Di Lodovico

G. Cowan

Sw. Banerjee

C.L. Davis

A.G. Denig

W. Gradl

K. Griessinger

A. Hafner

K.R. Schubert

Study of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>ω</mi><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>1</mn><mi>S</mi><mo stretchy="false">)</mo></mrow></math> transitions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>3</mn><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>γ</mi><msub><mrow><mi>χ</mi></mrow><mrow><mi>b</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></mrow></math> decays at BABAR

We study the nonmaximal symmetry breaking patterns of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">u</mi></mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="fraktur">g</mi></mrow><mrow><mn>531</mn></mrow></msub><mo>/</mo><msub><mrow><mi mathvariant="fraktur">g</mi></mrow><mrow><mn>351</mn></mrow></msub></mrow></math> allowed in a supersymmetric theory with the affine Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">u</mi></mrow><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><mn>8</mn><msub><mo stretchy="false">)</mo><mrow><msub><mi>k</mi><mi>U</mi></msub><mo>=</mo><mn>1</mn></mrow></msub></math>, which was recently proposed to explain the three-generational standard model quark/lepton mass hierarchies as well as the Cabbibo-Kobayashi-Maskawa mixing pattern. Three gauge couplings are found to achieve the unification below the Planck scales, according to the unification relations determined by the affine Lie algebra at a level of 1. We also prove that a separate nonmaximal symmetry breaking pattern of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">u</mi></mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mi mathvariant="fraktur">g</mi><mn>621</mn></msub></math> is unrealistic due to the additional massless vectorlike quarks in the spectrum.

Ning Chen

Jianan Tian

Bin Wang

Nonmaximal symmetry breaking patterns in the supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">u</mi></mrow><mrow><mo stretchy="true">^</mo></mrow></mover><mo stretchy="false">(</mo><mn>8</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></msub></mrow></math> theory