We present a comprehensive bottom-up analysis of lepton mass and mixing based on the non-holomorphic $A_{4}$ modular symmetry. Neutrinos are assumed to be Majorana particles, and the light neutrino masses are generated through the Weinberg operator. In this framework, we construct all phenomenologically viable models with minimal number of free parameters, with the Yukawa couplings expressed in terms of polyharmonic Maaß forms of weights $\pm4$ , $\pm2$ , and 0 at level $N=3$ . Without imposing generalized CP (gCP) symmetry, we identify 147 (six) viable models with eight real free parameters that successfully reproduce the current experimental data of the lepton sector for the normal (inverted) mass ordering. When gCP symmetry consistent with $A_{4}$ modular symmetry is included, the number of free parameters is reduced by one, yielding 47 (five) phenomenologically viable models in the normal (inverted) mass ordering. Finally, we present detailed numerical analyses of a representative model for both mass orderings to illustrate these results.

China

Xiang-Yan Gao

Cai-Chang Li

Minimal lepton models with non-holomorphic modular $\textit{A}$ $_{4}$ symmetry * Supported by Natural Science Basic Research Program of Shaanxi (2024JC-YBQN-0004) and the National Natural Science Foundation of China (12547106, 12247103)

The Schwinger effect, a non-perturbative mechanism for particle production in strong fields, plays a crucial role in understanding quantum vacuum decay and high-energy phenomena, including heavy-ion collisions (HIC). Although holographic quantum chromodynamics (QCD) models have been widely used to study this effect, most treatments assume isotropy or consider only a single type of anisotropy, neglecting the interplay between spatial and magnetic anisotropies that arise in realistic HIC scenarios. A unified framework accounting for both anisotropies is needed to accurately model particle production. We investigate the Schwinger effect in a twice anisotropic holographic QCD model incorporating both spatial and magnetic anisotropies. Using the anti-de Sitter/conformal field theory correspondence, we compute the total potential of a particle-antiparticle pair to quantify how these anisotropies influence pair production. Our results show that the magnetic field (parameterized by $c_B$ and $q_3$ ) enhances the Schwinger effect by lowering and narrowing the potential barrier, while increasing spatial anisotropy (controlled by $\nu$ ) suppresses the process by raising and widening the barrier. These findings demonstrate that magnetic and spatial anisotropies exert competing effects on particle production, emphasizing the necessity of treating both concurrently in holographic models. This work advances the theoretical description of the Schwinger effect in anisotropic environments, with implications for understanding non-equilibrium dynamics in HIC and other strongly coupled systems.

Wen-Bin Chang

Defu Hou

Schwinger effect in a twice anisotropic holographic model * Defu Hou's research is supported in part by the National Key Research and Development Program of China (2022YFA1604900). Additionally, Defu Hou receives partial support from the National Natural Science Foundation of China (NSFC) (12435009, 12275104). Wen-Bin Chang is supported by the Ph.D. Research Startup Project at Hubei Minzu University (RZ2500000857)

The real-time correlators of quantum field theories can be directly probed through new approaches to simulation, such as quantum computing and tensor networks. This provides a new framework for computing scattering observables in lattice formulations of strongly interacting theories, such as lattice quantum chromodynamics. In this paper, we prove that the proposal of real-time estimators of scattering observables is universally applicable to all scattering observables of gapped quantum field theories. All finite-volume errors are exponentially suppressed, and the rate of this suppression is controlled by the regulator considered, namely, a displacement of the spectrum of the theory into the complex plane. A partial restoration of Lorentz symmetry by averaging over different boosts gives an additional suppression of finite volume errors. Our results also apply to the simulation of wave packet scattering, where a similar averaging is performed to construct the wave packets that regulate the finite volume effects. This result represents a necessary key step toward determining a broad class of scattering observables via quantum computing that are currently inaccessible via classical computing. Such observables are relevant for various applications, including hadron spectroscopy, hadron structure, and precision tests of the Standard Model. We also comment on potential applications of our results to traditional computational schemes.

Ivan M. Burbano

Marco A. Carrillo

Rana Urek

Anthony N. Ciavarella

Raúl A. Briceño

Real-time estimators for scattering observables: A full account of finite-volume errors for quantum simulation

We present a systematic analysis of Bell nonlocality and entanglement in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo stretchy="false">(</mo><mi>J</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></math> decays into a baryon pair (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover></math>), with particular emphasis on their production via the process <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><mo stretchy="false">→</mo><mi>ψ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>γ</mi><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> at BESIII. From the baryon-antibaryon spin density matrix, we construct measurable Bell observables and concurrence, revealing a striking hierarchy of quantum correlations: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub></math> decays exhibit maximal violation and entanglement; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> decays violate Bell inequalities for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>θ</mi><mn>1</mn></msub><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></math> with angle-modulated strength; we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover></math> pair in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></math> decays is in a separable state, and no indication of Bell inequality violation is observed. We provide complete analytical results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>=</mo><mn>0</mn></math>, 1 and quantitative, uncertainty-aware estimations for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>=</mo><mn>2</mn></math> based on experimental inputs from BESIII. These results establish the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> system produced via this radiative transition as a novel and promising platform for testing quantum entanglement and Bell nonlocality in high-energy collisions.

Peng-Cheng Hong

Rong-Gang Ping

Wei-Min Song

Bell nonlocality and entanglement in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> decays into a baryon pair

Guided by the flipping principle, we propose a novel extension of the Standard Model based on a double right-handed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> gauge symmetry. In this framework, all left-handed fermions are neutral, while right-handed fermions of the third generation carry charges distinct from those of the first two generations. This structure provides an explanation for the observed Standard Model fermion mass hierarchy; the heavy masses of the third generation are generated at tree level, while the lighter masses of the first and second generations arise radiatively at the one-loop level. For the active neutrino sector, the tiny masses are generated through a combination of tree-level and two-loop seesaw mechanisms. Crucially, this approach successfully reproduces the observed neutrino mass hierarchy, with the atmospheric mass-squared difference generated at tree level and the solar neutrino mass squared difference emerging at the two-loop level. These hierarchical patterns stem from the interplay between gauge invariance and a residual parity symmetry that survives the spontaneous breaking of the extended gauge group. The same residual symmetry stabilizes a viable scalar singlet dark matter candidate, which we show can reproduce the observed relic abundance while remaining consistent with current direct detection bounds. After addressing constraints from electroweak precision tests and flavor-changing neutral currents, we explore the discovery prospects for the new neutral bosons at existing and future colliders, including the LEP, LHC, and a future ILC.

Duong Van Loi

A.E. Cárcamo Hernández

N.T. Duy

D.T. Binh

Cao H. Nam

Physical implications of a double right-handed gauge symmetry

Jet quenching—the modification of high-energy jets in the quark-gluon plasma—has been extensively studied through weakly coupled scattering amplitudes embedded in parton-shower frameworks. These models, often combined with bulk hydrodynamic evolution, successfully describe a wide range of observables, though they typically rely on assumptions of rapid thermalization and simplified treatments of medium response. Parallel to these developments, jet thermalization has been investigated within the finite-temperature QCD effective kinetic theory, which provides our best microscopic understanding of equilibration in heavy-ion collisions. Early studies of linearized perturbations have highlighted both the promise and the limitations of current approaches, as existing MC implementations face challenges—particularly in the treatment of recoils and particle merging. Building on this foundation, we introduce a new parton-shower algorithm that exactly reproduces the dynamics of the linearized effective kinetic theory, enabling a first-principles description of jet thermalization with proper inclusion of recoils, holes, quantum statistics, and merging processes.

Ismail Soudi

Adam Takacs

Deriving a parton shower for jet thermalization in QCD plasmas

We provide an elementary pedagogical introduction to some basic concepts and techniques of soft (or Sudakov) resummation, specifically in QCD, paying particular attention to simple but useful tricks of the trade. We briefly review collinear (Altarelli–Parisi) and infrared (eikonal) factorization, cancellation of infrared singularities, and factorization of mass singularities. We recall basic concepts on renormalization group invariance and the solution of renormalization group equations. We then show how threshold resummation can be derived from a renormalization group argument following from the cancellation of infrared singularities. We discuss various equivalent forms of the resummed result, and we briefly present transverse momentum resummation. Abstract Published by the Jagiellonian University 2026 authors

Italy

S. Forte

G. Ridolfi

A Simple Introduction to Soft Resummation

In the presence of a strong electric field, the vacuum is unstable to the production of pairs of charged particles—the Schwinger effect. The created pairs extract energy from the electric field, resulting in nontrivial backreaction. In this paper, we study <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">D</mi></mrow></math> massive QED subject to strong external electric fields in a self-consistent and fully quantum manner. We use the bosonized version of the theory, which attains a cosine interaction term in the presence of nonzero fermion mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math>. However, the assumption of a strong electric field justifies a perturbative treatment of the cosine interaction, i.e., an expansion in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math>. We calculate the vacuum expectation value of the electric field to first order in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math> and show that—surprisingly—it satisfies a classical nonlinear partial differential equation (related to the sine-Gordon equation). We show that the electric field exhibits dissipation-free oscillations (analogous to ordinary plasma oscillations) and calculate the plasma frequency analytically. We also compare to the semiclassical approximation commonly used to study backreaction, showing that it fails to capture the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></math> shift in the plasma frequency.

Samuel E. Gralla

Morifumi Mizuno

Schwinger effect with backreaction in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">D</mi></mrow></math> massive QED with a strong external field

Based on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>2712.4</mn><mo>±</mo><mn>14.4</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup><mtext> </mtext><mi>ψ</mi><mo stretchy="false">(</mo><mn>3686</mn><mo stretchy="false">)</mo></mrow></math> events collected with the BESIII detector, we present a high-precision study of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> mass spectrum in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3686</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mi>J</mi><mo>/</mo><mi>ψ</mi></math> decays. A clear resonancelike structure is observed near the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> mass threshold for the first time. A fit with a Breit-Wigner function yields a mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>285.6</mn><mo>±</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> and a width of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>16.3</mn><mo>±</mo><mn>0.9</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> with a statistical significance exceeding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>10</mn><mi>σ</mi></mrow></math>. To interpret the data, we incorporate final-state interactions (FSI) within two theoretical frameworks: chiral perturbation theory (ChPT) and QCD multipole expansion (QCDME). ChPT describes the spectrum above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0.3</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></math> but fails to reproduce the threshold enhancement. In contrast, the QCDME model—assuming the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3686</mn><mo stretchy="false">)</mo></math> is an admixture of S- and D-wave charmonium—reproduces the data well. The pronounced dip near <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0.3</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></math> offers new insight into the interplay between chiral dynamics and low-energy QCD.

M. Ablikim

M.N. Achasov

P. Adlarson

X.C. Ai

R. Aliberti

A. Amoroso

Q. An

Y. Bai

O. Bakina

Y. Ban

H.R. Bao

V. Batozskaya

K. Begzsuren

N. Berger

M. Berlowski

M.B. Bertani

D. Bettoni

F. Bianchi

E. Bianco

A. Bortone

I. Boyko

R.A. Briere

A. Brueggemann

H. Cai

M.H. Cai

X. Cai

A. Calcaterra

G.F. Cao

N. Cao

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.M. Chen

T. Chen

X.R. Chen

X.T. Chen

X.Y. Chen

Y.B. Chen

Y.Q. Chen

Z.K. Chen

J.C. Cheng

L.N. Cheng

S.K. Choi

X. Chu

G. Cibinetto

F. Cossio

J. Cottee-Meldrum

H.L. Dai

J.P. Dai

X.C. Dai

A. Dbeyssi

R.E. de Boer

D. Dedovich

C.Q. Deng

Z.Y. Deng

A. Denig

I. Denisenko

M. Destefanis

F. De Mori

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

L. Fava

F. Feldbauer

G. Felici

C.Q. Feng

J.H. Feng

L. Feng

Q.X. Feng

Y.T. Feng

M. Fritsch

Observation of a Threshold Enhancement in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> Spectrum in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mn>3686</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> Decays

We investigate the possibility of $\textit{freeze-in}$ dark matter production and baryogenesis via leptogenesis in a $$\psi '$$ SM model, which is an $$E_6$$ extension of the Standard Model, featuring a residual $$U(1)_{\psi '}$$ gauge symmetry. This symmetry arises from a linear combination of $$U(1)_\chi $$ and $$U(1)_{\psi }$$ , both of which are subgroups of the $$E_6$$ . The spontaneous breaking of $$U(1)_{\psi '}$$ symmetry governs the dynamics of a singlet fermion, which serves as a freeze-in dark matter candidate. The dark matter mass arises from dimension-five operators, and a discrete symmetry ensures its stability. We show that freeze-in production from scalar decay can yield the correct relic abundance for dark matter masses between a few MeV to a few hundred GeV. Simultaneously, heavy right-handed neutrinos generate light neutrino masses via the type-I seesaw and produce the observed baryon asymmetry via leptogenesis.

Pakistan

Russia

Adeela Afzal

United Kingdom

Rishav Roshan

Freeze-in dark matter and leptogenesis: a $$\psi '$$ SM route