Off-shell characteristics of pion generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs) are examined within the framework of the Nambu–Jona-Lasinio model. In our previous studies, we separately investigated the properties of on-shell pion GPDs and light-front wave functions. Comparing the differences between on-shell and off-shell pion GPDs is particularly intriguing becuase it enables us to explore the effects associated with off-shellness. The absence of crossing symmetry causes the moments of GPDs to incorporate odd powers of the skewness parameter, resulting in new off-shell form factors. Through our calculations, we derived correction functions that account for modifications in pion GPDs attributed to off-shell effects. Unlike their on-shell counterparts, certain properties break down in the off-shell scenario; for example, symmetry properties and polynomiality conditions may no longer hold. In addition, we evaluate off-shell TMDs and compare them with their on-shell equivalents while also investigating their dependence on $ {\boldsymbol{k}}_{\perp} $ .

China

Jin-Li Zhang

Off-shell modifications of pion generalized parton distributions and transverse momentum dependent parton distributions * Supported by the Scientific Research Foundation of Nanjing Institute of Technology (YKJ202352) and the General Program of the Natural Science Foundation of Jiangsu Higher Education Institutions (25KJD140001)

Inspired by the observation of the ${\eta}_{t}$ meson at the LHC and the promising prospect of the ${\eta}_{t}$ meson available at the approaching HL-LHC, branching ratios for the ${\eta}_{t}$ ${\to}$ $f\bar{f}$ , $gg$ , ${\gamma}{\gamma}$ , $W^{+}W^{-}$ , $Z^{0}Z^{0}$ , $Z^{0}{\gamma}$ , and $Z^{0}H$ decays are roughly estimated. It is found that tens of opposite-charge dilepton events from the ${\eta}_{t}$ ${\to}$ $W^{+}W^{-}$ decay and hundreds of events from the ${\eta}_{t}$ ${\to}$ $Z^{0}H$ ${\to}$ ${\ell}^{+}{\ell}^{-}H$ decay using the single $Z^{0}$ boson tagging method are expected to be accessible. This estimation provides a reference for future experimental study on the ${\eta}_{t}$ meson.

Yueling Yang

Bingbing Yang

Jiazhi Li

Zhaojie Lü

Junfeng Sun

Searching for the toponium ${{\boldsymbol\eta}_{\boldsymbol t}}$ with the ${{\boldsymbol\eta}_{\boldsymbol t}}$ ${\boldsymbol\to}$ ${{\boldsymbol W}^{\bf +}{\boldsymbol W}^{\bf -}}$ decay * Supported by the National Natural Science Foundation of China (12275068, 12275067), the National Key R&D Program of China (2023YFA1606000), the Natural Science Foundation of Henan Province (252300421491, 242300420250), and the Science and Technology R&D Program Joint Fund Project of Henan Province (225200810030), as well as the Science and Technology Innovation Leading Talent Support Program of Henan Province (254200510039)

We investigate the discovery prospects of a leptophilic gauge boson $ Z_\ell $ at future $ e^+e^- $ colliders, focusing on a comparative study of the Circular Electron–Positron Collider (CEPC) and the International Linear Collider (ILC). Such a state naturally arises from an additional $ U(1)'_\ell $ gauge symmetry, under which quarks remain neutral while all leptons carry a universal charge, motivated by neutrino oscillations and scenarios of physics beyond the Standard Model. As a clean benchmark, we study the process $ e^+e^- \to \mu^+\mu^- $ , including realistic effects of initial-state radiation and beamstrahlung. Our results indicate that the CEPC, with its very high luminosity at $ \sqrt{s}=240 $ GeV, can probe couplings down to $ g_\ell \approx 10^{-3} $ for $ Z_\ell $ masses up to about 220 GeV. The ILC extends the sensitivity to heavier states in the multi-hundred GeV range through its higher $ \sqrt{s} $ stages. These findings demonstrate the strong complementarity of circular and linear colliders in exploring purely leptophilic interactions.

S.O. Kara

Discovery prospects for a leptophilic gauge boson Z$_{ℓ}$ at CEPC and ILC

Heavy Majorana neutrinos ($\textit{N}$) are predicted in many models of physics beyond the Standard Model. In this work, we explore the production and detection prospects of TeV-scale heavy neutrinos ( $ m_N \gtrsim 1 $ TeV) at a future 3 TeV Compact Linear Collider (CLIC). We focus on two distinct decay topologies: (i) $ N \to \ell^\pm W^\mp $ with hadronic $\textit{W}$ boson decay, leading to a final state with one charged lepton and a hadronic fat jet $ J_W $ ; and (ii) $ N \to \nu h $ with subsequent Higgs decay $ h \to b\bar{b} $ , yielding a Higgs-tagged fat jet $ J_h $ and $\not {E}_T$ . Based on comprehensive detector-level simulations and background analysis, we present both $ 2\sigma $ exclusion limits and $ 5\sigma $ discovery reaches in the $ m_N $ – $ |V_{\ell N}|^2 $ plane. We further extract 95% confidence level upper limits on the mixing parameter $ |V_{\ell N}|^2 $ and perform a detailed comparison with existing constraints from direct searches at future colliders and indirect global fits. Our findings demonstrate that a 3 TeV CLIC can improve the sensitivity to $ |V_{\ell N}|^2 $ by about two orders of magnitude compared with the projected reaches of future hadron colliders while remaining competitive with other CLIC search channels.

Yao-Bei Liu

Jing-Wei Lian

Searches for heavy neutrinos at a 3 TeV CLIC in fat jet final states * Yao-Bei Liu is supported by the Natural Science Foundation of Henan Province, China (252300421988), Jing-Wei Lian is supported by the National Natural Science Foundation of China (12447140) and the Natural Science Foundation of Henan Province, China (252300421771)

One-loop contributions for the decay process $ H^{\pm} \rightarrow W^{\pm}Z $ within the Two-Higgs-Doublet Model (THDM) are computed in the general $ {\cal{R}}_{\xi} $ gauge, and its phenomenological applications at future muon–TeV colliders are investigated. Analytic results are confirmed by several consistency tests, including those of $\textit{ξ}$-independence, renormalization-scale stability, and the ultraviolet finiteness of the one-loop amplitude. We first perform an updated parameter scan of the Type-X THDM in phenomenological studies. The production of charged Higgs boson pairs at future muon–TeV colliders is then investigated through two processes: $ \mu^+\mu^- \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh $ and $ \mu^+\mu^- \rightarrow \gamma\gamma \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh $ . Both signal events and their significance are evaluated considering the corresponding Standard Model backgrounds. We find that the signal significances can exceed $ 5\sigma $ at several benchmark points in the viable parameter space of the Type-X THDM.

Vietnam

Dzung Tri Tran

Quang Hoang-Minh Pham

Khoa Ngo-Thanh Ho

Khiem Hong Phan

One-loop expressions for ${H^\pm \rightarrow W^\pm Z} $ and their implications at muon-TeV colliders * This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) (103.01-2023.16)

Quantum chromodynamics (QCD) is a fundamental theory describing quark interactions. Thus far, various quark models based on QCD have been widely used to study the properties of hadrons, including their structures and mass spectra. However, unlike quantum electrodynamics and Bohr's model of the hydrogen atom, a direct classical analogy is lacking for hadronic structures. This paper presents a classical interpretation of the nonrelativistic quark potential model, providing a more intuitive and visualizable description of strong interactions through the quantitative formulation of color charge and color flux. In addition, we establish the relationship between meson mass and its structural radius in the nonrelativistic framework and estimate key parameters of our model using available data from $ \eta_b(1S) $ and $ \Upsilon(1S) $ . Subsequently, we extend this relationship to a broader range of excited meson states and obtain their structural radii, which show good agreement with the root mean square radius or charge radius predicted by QCD calculations.

Zhiguang Tan

Youneng Guo

Shengjie Wang

Hua Zheng

Classical interpretation of nonrelativistic quark potential model: Color charge definition and meson mass-radius relationship * Supported in part by the Department of Education (21A0541) and Natural Science Foundation (2025JJ50382) of Hunan province, China

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.

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)

We investigate how the positive geometry framework for loop integrands in $$ \mathcal{N}=4 $$ super Yang-Mills theory constrains the structure of the integrated answers. This is done in the context of a geometric expansion of Wilson loops with a Lagrangian insertion, called negative geometries, extending ideas previously used for scattering amplitudes related to the Amplituhedron. The procedure we adopt combines the knowledge of all maximal codimension boundaries of the geometry, which characterize all possible leading singularities of the integral, with a geometrically informed Landau analysis. The interplay between geometry and Landau analysis arises from associating Landau diagrams to geometric boundaries. The boundary structure of the geometry then determines which solutions to the Landau equations are spurious and which ones are physical, that is, which singularities are actually present in the integral. This method allows us to efficiently determine the symbol alphabet of the associated integral, and serves as a starting point for the symbol bootstrap. We successfully implement this procedure and compute the six-point two-loop and five-point three-loop ladder negative geometries at the symbol level. We also present the conjectural alphabet for ladder negative geometries at two loops for all multiplicities. These are finite integrals that serve as building blocks for the Wilson loop with Lagrangian insertion, and therefore provide insights into the function space of the latter.

France

Dmitry Chicherin

Germany

Johannes Henn

Elia Mazzucchelli

United States

Jaroslav Trnka

Qinglin Yang

Shun-Qing Zhang

Geometric Landau analysis and symbol bootstrap

<p>The first observation of the decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">→</mo><mi>p</mi><mover accent="true"><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> is presented using proton-proton collision data collected by the LHCb experiment between 2016 and 2018 at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>5.4</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fb</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>. The signal significance exceeds seven standard deviations. Using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msubsup><mi>K</mi><mi mathvariant="normal">S</mi><mn>0</mn></msubsup><msup><mi>π</mi><mo>+</mo></msup></math> decay as a normalization channel, the branching fraction is measured and combined with previous LHCb results based on data collected at 7 and 8 TeV in 2011 and 2012, yielding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">→</mo><mi>p</mi><mover accent="true"><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1.24</mn><mo>±</mo><mn>0.17</mn><mo>±</mo><mn>0.05</mn><mo>±</mo><mn>0.03</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow></msup><mo>,</mo></mrow></math> where the first uncertainty is statistical, the second is systematic, and the third comes from the uncertainty on the branching fraction of the normalization channel. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">→</mo><mi>p</mi><mover accent="true"><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> weak decay parameter is measured to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>=</mo><mn>0.8</mn><msubsup><mrow><mn>7</mn></mrow><mrow><mo>−</mo><mn>0.29</mn></mrow><mrow><mo>+</mo><mn>0.26</mn></mrow></msubsup><mo>±</mo><mn>0.09</mn></mrow></math>, indicating the presence of comparable S-wave and P-wave decay amplitudes.</p>

Netherlands

R. Aaij

United Kingdom

A.S.W. Abdelmotteleb

Switzerland

C. Abellan Beteta

F. Abudinén

T. Ackernley

A.A. Adefisoye

Spain

B. Adeva

M. Adinolfi

P. Adlarson

C. Agapopoulou

C.A. Aidala

Z. Ajaltouni

S. Akar

K. Akiba

Italy

P. Albicocco

J. Albrecht

R. Aleksiejunas

CERN

F. Alessio

P. Alvarez Cartelle

R. Amalric

Brazil

S. Amato

J.L. Amey

Y. Amhis

L. An

L. Anderlini

M. Andersson

P. Andreola

M. Andreotti

S. Andres Estrada

A. Anelli

D. Ao

C. Arata

F. Archilli

Z. Areg

M. Argenton

Costa Rica

S. Arguedas Cuendis

L. Arnone

A. Artamonov

M. Artuso

E. Aslanides

R. Ataíde Da Silva

M. Atzeni

B. Audurier

J.A. Authier

D. Bacher

I. Bachiller Perea

S. Bachmann

M. Bachmayer

J.J. Back

P. Baladron Rodriguez

V. Balagura

A. Balboni

W. Baldini

Z. Baldwin

L. Balzani

H. Bao

J. Baptista de Souza Leite

C. Barbero Pretel

M. Barbetti

I.R. Barbosa

R.J. Barlow

M. Barnyakov

S. Barsuk

W. Barter

J. Bartz

Poland

S. Bashir

B. Batsukh

P.B. Battista

A. Bay

A. Beck

M. Becker

F. Bedeschi

I.B. Bediaga

N.A. Behling

S. Belin

A. Bellavista

K. Belous

I. Belov

I. Belyaev

G. Benane

G. Bencivenni

E. Ben-Haim

A. Berezhnoy

R. Bernet

S. Bernet Andres

A. Bertolin

F. Betti

J. Bex

O. Bezshyyko

S. Bhattacharya

J. Bhom

M.S. Bieker

N.V. Biesuz

A. Biolchini

M. Birch

F.C.R. Bishop

A. Bitadze

A. Bizzeti

T. Blake

F. Blanc

Observation of the Rare Baryonic Decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">→</mo><mi>p</mi><mover accent="true"><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> and Measurement of its Weak Decay Parameter

<p>We propose a stringy description of a system composed of two heavy quarks and two heavy antiquarks, mimicking that in pure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> gauge theory. We present both analytical and numerical studies of the string configurations for rectangular geometries. As an application, we analyze the two lowest Born-Oppenheimer potentials. Our results suggest that, depending on the geometry and quark ordering, the ground state of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>Q</mi><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover></math> system may be a hadronic molecule, a tetraquark state, or a superposition of the two. For general geometries, we derive the asymptotic expression for the energy of the tetraquark configuration in the infrared limit and extend this result to multiquark configurations. Here we also demonstrate the universality of the string tension.</p>

Oleg Andreev

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>Q</mi><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover></math> quark system and gauge/string duality