Neutrino oscillation experiments have confirmed that neutrinos are massive particles and that lepton flavors are mixed. To explain the observed lepton mass spectra and flavor mixing patterns, flavor symmetry plays a crucial and unique role. In this paper, we propose a useful symmetry-breaking scheme by applying $S_{3L} \times S_{3R} \rightarrow $ $ S_{2L} \times S_{2R} \rightarrow \emptyset$ within both charged-lepton and neutrino sectors at the mass-matrix level. For the three distinct residual subgroups $S_{2L}^{(23)} \times S_{2R}^{(23)}$ , $S_{2L}^{(13)} \times S_{2R}^{(13)}$ , and $S_{2L}^{(12)} \times S_{2R}^{(12)}$ that are under consideration, we systematically analyze the various parameterizations of the lepton mass matrices. It is shown that all three scenarios are in good agreement with current neutrino oscillation data. Notably, within the latest best-fit values of neutrino oscillation parameters, the predicted values of the Dirac CP-violating phase $\textit{δ}$ are $294.6^\circ$ , $302.3^\circ$ , and $287.0^\circ$ , respectively. To further assess the viability of the model, a comprehensive numerical analysis is performed by utilizing neutrino oscillation parameters at the $3\sigma$ level. It is found that the allowed ranges of $\textit{δ}$ are $281.2^\circ \rightarrow 338.7^\circ$ , $287.0^\circ \rightarrow 342.2^\circ$ , and $282.7^\circ \rightarrow 297.0^\circ$ , which all fall within its $3\sigma$ range. These results indicate that the proposed symmetry-breaking scheme $S_{3L} \times S_{3R} \rightarrow S_{2L} \times S_{2R} \rightarrow \emptyset$ can naturally explain the realistic lepton mass hierarchy and mixing pattern, thereby providing valuable theoretical perspectives for future research.

China

Chang-Qi Hu

Chun-Yuan Li

Xiao Liang

Xing-Hua Yang

Dai-Xing Zhang

Further study on lepton mass spectra and flavor mixing with $\textit{S}$ 3$\textit{L}$ × $\textit{S}$ 3$\textit{R}$ flavor symmetry * Supported in part by the National Natural Science Foundation of China (12105162, 12305106), the Natural Science Foundation of Shandong Province (ZR2021QA058, ZR2021QA040) and the Youth Innovation Technology Project of Higher School in Shandong Province (2023KJ146)

Based on $ (10087\pm 44)\times10^{6} $ $ J/\psi $ events collected with the BESIII detector, we search for the lepton number violating decay $ J/\psi \to K^+K^+e^-e^- + c.c. $ for the first time. The upper limit on the branching fraction of this decay is set to $ 2.1 \times 10^{-9} $ at the 90% confidence level with a frequentist method. This is the first search for $ J/\psi $ decays with a lepton number change by two, offering valuable insights into the underlying physical processes.

M. Ablikim

Russia

M.N. Achasov

Sweden

P. Adlarson

X.C. Ai

Germany

R. Aliberti

Italy

A. Amoroso

Q. An

Y. Bai

O. Bakina

Y. Ban

H.R. Bao

Poland

V. Batozskaya

Mongolia

K. Begzsuren

N. Berger

M. Berlowski

M.B. Bertani

D. Bettoni

F. Bianchi

E. Bianco

A. Bortone

I. Boyko

United States

R.A. Briere

A. Brueggemann

H. Cai

M.H. Cai

X. Cai

A. Calcaterra

G.F. Cao

N. Cao

Türkiye

S.A. Cetin

X.Y. Chai

J.F. Chang

T.T. Chang

G.R. Che

Y.Z. Che

C.H. Chen

Chao Chen

G. Chen

H.S. Chen

H.Y. Chen

M.L. Chen

S.J. Chen

S.L. Chen

S.M. Chen

T. Chen

X.R. Chen

X.T. Chen

X.Y. Chen

Y.B. Chen

Y.Q. Chen

Z. Chen

Z.K. Chen

J.C. Cheng

L.N. Cheng

S.K. Choi

X. Chu

G. Cibinetto

F. Cossio

United Kingdom

J. Cottee-Meldrum

H.L. Dai

J.P. Dai

A. Dbeyssi

R.E. de Boer

D. Dedovich

C.Q. Deng

Z.Y. Deng

A. Denig

I. Denisenko

M. Destefanis

F. De Mori

B. Ding

X.X. Ding

Y. Ding

Y.X. Ding

J. Dong

L.Y. Dong

M.Y. Dong

X. Dong

M.C. Du

S.X. Du

X.L. Du

Y.Y. Duan

Z.H. Duan

P. Egorov

G.F. Fan

J.J. Fan

Y.H. Fan

J. Fang

S.S. Fang

W.X. Fang

Y.Q. Fang

R. Farinelli

L. Fava

F. Feldbauer

G. Felici

C.Q. Feng

J.H. Feng

Search for the lepton number violating process ${{\boldsymbol J}/{\boldsymbol\psi }{\bf\to}{\boldsymbol K}^{\bf +}{\boldsymbol K}^{\bf +}{\boldsymbol e}^{\bf -}{\boldsymbol e}^{\bf -}{\bf +}{\boldsymbol {c.c.}}}$ * This work is supported in part by National Key R&D Program of China under Contracts Nos. 2023YFA1606000, 2023YFA1606704; National Natural Science Foundation of China (NSFC) under Contracts Nos. 12035009, 11635010, 11935015, 11935016, 11935018, 12025502, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12361141819; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; CAS under Contract No. YSBR-101; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contract No. FOR5327; Istituto Nazionale di Fisica Nucleare, Italy; Knut and Alice Wallenberg Foundation under Contracts Nos. 2021.0174, 2021.0299; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology fund of Mongolia; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation of Thailand under Contract No. B50G670107; Polish National Science Centre under Contract No. 2024/53/B/ST2/00975; Swedish Research Council under Contract No. 2019.04595; U. S. Department of Energy under Contract No. DE-FG02-05ER41374

We investigate the valence quark distributions of pions at a low initial scale ( $Q^2_0$ ) using Tsallis entropy, a non-extensive measure that effectively captures long-range correlations among internal constituents. Utilizing the maximum entropy approach, we adopt two distinct functional forms and fit experimental data using the elegant GLR-MQ-ZRS evolution equation to derive the model parameters. Our findings indicate that the resulting valence quark distributions provide an optimal fit to the experimental data, with $\textit{q}$ values deviating from unity. This deviation indicates that correlations among valence quarks play a significant role in shaping understanding of the internal structures of pions. Additionally, our computations of the first three moments of the pion quark distributions at $ Q^2 = 4$ GeV$^{2}$ display consistency with other theoretical models, thereby reinforcing the importance of incorporating valence quark correlations within this analytical framework.

Jingxuan Chen

Xiaopeng Wang

Yanbing Cai

Xurong Chen

Qian Wang

Valence quark distributions of pions: insights from Tsallis entropy * Supported by National Key R&D Program of China (2024YFE0109800, 2024YFE0109802); National Natural Science Foundation of China (12375073) and Guizhou Provincial Basic Research Program (Natural Science) (QKHJC-ZK[2023]YB027)

The correct choice for the matter Lagrangian <math altimg='si17.svg'><mrow><mo>(</mo><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo>)</mo></mrow></math> in the framework of <italic>f</italic>( <italic>R, T</italic>) theory of gravity, has been a fundamental yet often overlooked ambiguity. It has been a long-standing issue, whether to choose <math altimg='si1.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mi>p</mi></mrow></math> or <math altimg='si18.svg'><mrow><mo>&#8722;</mo><mi>&#961;</mi></mrow></math> as the proper definition of matter sector. In this work, we show that both choices lead to the same energy-momentum tensor, from thermodynamic point of view. However, for these two choices, the structure of the TOV equations are different. We construct and solve TOV equations using MIT bag model equation of state for <math altimg='si1.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mi>p</mi></mrow></math> and <math altimg='si18.svg'><mrow><mo>&#8722;</mo><mi>&#961;</mi></mrow></math>, and study the impact of the choices for matter Lagrangian on the maximum mass limit as well as M-R plot of compact stars. It is interesting to note that allowed range of gravity-matter coupling constant ( <italic>&#945;<inf>c</inf></italic>) is also different for <math altimg='si1.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mi>p</mi></mrow></math> and <math altimg='si5.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mi>&#961;</mi></mrow></math>, i.e., <italic>&#945;<inf>c</inf></italic> can not be taken arbitrarily. Notably, through <math altimg='si1.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mi>p</mi></mrow></math>, we achieve a maximum mass of 2.78&#8201; <italic>M</italic> <inf>&#8857;</inf>, whereas for <math altimg='si5.svg'><mrow><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mi>&#961;</mi></mrow></math>, we obtain a maximum mass of 2.41&#8201; <italic>M</italic> <inf>&#8857;</inf>. So, despite the same energy-momentum tensor for different choices of <math altimg='si19.svg'><msub><mi mathvariant='script'>L</mi><mi mathvariant='script'>M</mi></msub></math>, the upper limit of maximum mass is significantly modified.

India

Debadri Bhattacharjee

Pradip Kumar Chattopadhyay

Thermodynamic invariance of the energy-momentum tensor under matter-Lagrangian choices and its astrophysical implications in <italic>f</italic>( <italic>R, T</italic>) gravity

This study investigates the dynamics of dissipative gravitational collapse within the framework of <italic>f</italic>( <italic>R</italic>) gravity, focusing on the exponential correction model <math altimg='si7.svg'><mrow><mi>f</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><mi>R</mi><mo linebreak='goodbreak'>&#8722;</mo><msub><mi>&#946;</mi><mn>1</mn></msub><msub><mi>R</mi><mi>s</mi></msub><mrow><mo stretchy='true'>(</mo><mn>1</mn><mo linebreak='badbreak'>&#8722;</mo><msup><mi>e</mi><mrow><mo>&#8722;</mo><mi>R</mi><mo>/</mo><msub><mi>R</mi><mi>s</mi></msub></mrow></msup><mo stretchy='true'>)</mo></mrow></mrow></math> and incorporating the concept of a horizon function. The model considers a radiating anisotropic stellar object with string density, matched to an exterior Vaidya spacetime. We thoroughly derive the junction conditions at the stellar boundary and emphasize the role of non-zero radial pressure induced by heat flux. By employing the horizon function alongside the exponential correction model and Finch-Skea metric assumptions, we examine the conditions under which black holes can form as a final state of collapse. A graphical analysis supports the theoretical model, demonstrating how the presence of string density influences the collapse process and can prevent total mass vanishing, thereby allowing black hole formation in <italic>f</italic>( <italic>R</italic>) gravity.

Azerbaijan

Pakistan

M. Ilyas

Zaib Un Nisa Shinwari

Fawad Khan

Sidra Marwat

Role of density inhomogeneity in dissipative collapse through Finch-Skea spacetime in <italic>f</italic>( <italic>R</italic>) gravity

In this work, we study the restricted phase space thermodynamics (RPST) of charged static and rotating black holes in $\textit{f}$($\textit{R}$) gravity. By keeping the cosmological constant fixed and introducing the central charge $\textit{C}$ with its conjugate chemical potential $$\mu $$ , RPST allows the black-hole mass to be interpreted as internal energy. We compute the corresponding thermodynamic quantities and analyse the $\textit{T}$–$\textit{S}$ and $\textit{F}$–$\textit{T}$ behaviours, which show the characteristic non-monotonicity and swallow-tail structures associated with first-order phase transitions, along with a second-order critical point. To confirm these results, we use the Legendre-invariant framework of geometrothermodynamics (GTD) and show that divergences of the GTD scalar curvature coincide with those of the specific heat, providing geometric confirmation of the phase structure in $\textit{f}$($\textit{R}$) gravity.

Amijit Bhattacharjee

Prabwal Phukon

Restricted phase space thermodynamics of charged static and charged rotating black holes in $\textit{f}$($\textit{R}$) gravity

This work examines the cosmological implications of two functional forms of <math altimg='si49.svg'><mrow><mi>f</mi><mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><mi>R</mi><mo linebreak='goodbreak'>+</mo><mi>&#945;</mi><msup><mi>T</mi><mi>n</mi></msup></mrow></math> gravity: for two different value of <italic>n</italic> where <math altimg='si2.svg'><mrow><mi>n</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn></mrow></math> and <italic>n</italic>&#8239;&#8800;&#8239;1, and <italic>&#945;</italic> and <italic>n</italic> are free parameters. The modified Friedmann equations are derived, and the cosmic evolution of the Hubble parameter <italic>H</italic>( <italic>z</italic>) is determined. Cosmological parameters are estimated through <italic>&#967;</italic> <sup>2</sup> minimization and MCMC analysis using the emcee algorithm, with model parameters constrained by various observational datasets. Both models reproduce late-time acceleration and remain observationally indistinguishable from &#923;CDM, while allowing small deviations parameterized by <italic>&#945;</italic> and <italic>n</italic>. The cosmological behavior of the deceleration parameter <italic>q</italic>( <italic>z</italic>), the jerk, the snap parameter <italic>s</italic>( <italic>z</italic>), and the effective equation of state parameter <italic>&#969;</italic> is also analyzed. The results indicate that the Universe transitioned from deceleration to late-time accelerated expansion, consistent with the &#923;CDM model. Analysis of the energy conditions reveals that the NEC and DEC are satisfied, while the SEC is violated, which explains the transition from a decelerating matter-dominated epoch to the present accelerated phase. These findings indicate that the proposed <italic>f</italic>( <italic>R, T</italic>) models are compatible with current observational data and provide a viable alternative to &#923;CDM in describing cosmic acceleration.

Morocco

Mustapha Lamaaoune

Exploring the universe expansion history with f(R,T) gravity: Constraints on cosmological parameters

We apply in a simple model derived from quadratic <math altimg='si2.svg'><msup><mi mathvariant='script'>R</mi><mn>2</mn></msup></math> gravity the technique of Dyson-Schwinger equations to solve for its corresponding quantum theory. Particularly, we solve the classical equations of motion to get a solution to the hierarchy of Dyson-Schwinger equations in the limit of large Ricci scalar, assumed to be constant and larger than the square of the Starobinsky mass. Moving to the Einstein frame, the model admits Higgs-like solutions with a single particle having a finite mass. We quantize the scalar field showing the appearing of a mass gap through a Higgs-like solution. The presence of the mass gap, that increases with the square root of the Ricci scalar, shows how the effect of the scalar sector at low-energy becomes ineffective, making it relevant only at short distances.

Sayantan Choudhury

Marco Frasca

Anish Ghoshal

Mass gap in non-perturbative quadratic <math altimg='si2.svg'><msup><mi mathvariant='script'>R</mi><mn>2</mn></msup></math> gravity via Dyson-Schwinger

This study introduces two new charged anisotropic interior configurations within the fluid-geometry associated <italic>f</italic>( <italic>R</italic>) gravity model. Our analysis begins with the modified Einstein-Maxwell field equations, which describe a static and charged interior spacetime structure. To solve these equations, two separate forms are used for the radial function and anisotropic pressure, satisfying established acceptability conditions. In both cases, differential equations arise, and integration constants appear in their solutions. These constants are determined by maintaining continuity between both the inner and outer Reissner-Nordstr&#246;m geometries at the border. Additionally, we assume the interface has a zero radial pressure. This study graphically explores both models employing observational data from the star LMC X-4. For certain parameter values, both solutions meet physical acceptance conditions in this non-conservative theoretical scenario, indicating its significance in constructing anisotropic fluid structures.

Tayyab Naseer

M. Sharif

Hammad Afzal

Saudi Arabia

Omer A. Magzoub

Mona Mahmoud

Role of stellar anisotropy and non-minimally coupled <italic>f</italic>( <italic>R</italic>) gravity theory in the construction of charged compact objects

The formation of the third <italic>r</italic>-process abundance peak near <italic>A</italic>&#8239;&#8764;&#8239;195 is highly sensitive to both nuclear structure far from stability and the astrophysical conditions that produce the heaviest elements. In particular, the <math altimg='si1.svg'><mrow><mi>N</mi><mo linebreak='goodbreak'>=</mo><mn>126</mn></mrow></math> shell closure plays a crucial role in shaping this peak. Experimental data hints that the shell weakens as proton number departs from <math altimg='si9.svg'><mrow><mi>Z</mi><mo linebreak='goodbreak'>=</mo><mn>82</mn></mrow></math>, a trend largely missed by global mass models. To investigate its impact on <italic>r</italic>-process nucleosynthesis, we employ both standard global models with strong closures and modified Duflo-Zuker (DZ) models that reproduce the weakening, combined with three sets of <math altimg='si5.svg'><msup><mi>&#946;</mi><mo>&#8722;</mo></msup></math>-decay rates. Strong shell closures generate sharply peaked abundances, whereas weakened closures consistent with the experimental trend produce broader, flatter patterns. Accurately reproducing the solar third peak under weakened shell strength requires sufficiently neutron-rich conditions that significant fission occurs, and slower decay rates. These results demonstrate that a weakening <math altimg='si1.svg'><mrow><mi>N</mi><mo linebreak='goodbreak'>=</mo><mn>126</mn></mrow></math> shell closure away from stability imposes significant constraints on the astrophysical environments of the <italic>r</italic>-process and underscores the need for precise mass measurements and improved characterization of <math altimg='si5.svg'><msup><mi>&#946;</mi><mo>&#8722;</mo></msup></math>-decay properties in this region.

Mengke Li

Gail C. McLaughlin

Rebecca Surman

Implications of a weakening <math altimg='si1.svg'><mrow><mi>N</mi><mo linebreak='goodbreak'>=</mo><mn>126</mn></mrow></math> shell closure away from stability for <italic>r</italic>-process astrophysical conditions