We investigate the temperature dependence of the shear viscosity (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math>) and bulk viscosity (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi></math>) of the gluon plasma using lattice QCD over the range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0.76</mn><mo>−</mo><mn>2.25</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>, extending from below the transition temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math> across the transition region and into the deconfined phase. At each temperature, we employ three large, fine lattices, which enables controlled continuum extrapolations of the energy-momentum tensor correlators. Using gradient flow together with a recently developed blocking technique, we achieve percent-level precision for these correlators, providing strong constraints for a model-based spectral analysis. Since the inversion to real-time information is intrinsically ill posed, we extract viscosities by fitting spectral functions whose ultraviolet behavior is matched to the best available perturbative result, while the infrared region is described by a Lorentzian transport peak. The dominant modeling uncertainty associated with the transport peak width is bracketed by varying it over a physically motivated range set by thermal scales. We find that the shear-viscosity-to-entropy-density ratio, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi><mo>/</mo><mi>s</mi></math>, exhibits a minimum near the transition temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mi>c</mi></msub></math> and increases for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mo>&gt;</mo><msub><mi>T</mi><mi>c</mi></msub></math>, whereas the bulk-viscosity-to-entropy-density ratio, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi><mo>/</mo><mi>s</mi></math>, decreases monotonically over the entire temperature range studied.

Heng-Tong Ding

Hai-Tao Shu

Cheng Zhang

Shear and bulk viscosities of the gluon plasma across the transition temperature from lattice QCD

We present a unified free field realization of the Ding-Iohara algebra at arbitrary levels, which satisfies a generalized form of the Serre relations. This realization, constructed using six free boson fields, arises from a specialized factorization of the structure function in the defining relations of the algebra. Based on this construction, we then develop intertwining operators for the Ding-Iohara algebra.

China

Zitao Chen

Xiang-Mao Ding

Free field realization of the Ding-Iohara algebra at general levels

We investigated the impact of a constant external magnetic field on the dressed propagators of up-, down-, and strange quarks. In the weak field limit, we derive a general momentum-space representation for the propagator and numerically solve the corresponding gap equation. Our analysis reveals that the vector term of the propagator can be decomposed into components parallel and perpendicular to the magnetic field, resulting in anisotropic effective masses, with the transverse mass consistently exceeding the longitudinal mass. This mass disparity exhibits a power law dependence on the magnetic field strength and is less pronounced for the strange quark compared to up and down quarks. Additionally, the magnetic field induces axial-vector and tensor terms, highlighting the Zeeman effect, resulting from quark interactions with the magnetic field. These findings have important implications for (inverse) magnetic catalysis and phenomena such as vector meson and pion condensations.

Germany

Minghui Ding

Fei Gao

Sebastian M. Schmidt

Anisotropic quark propagation and Zeeman effect in an external magnetic field * Fei Gao is supported by the National Science Foundation of China (12305134). Minghui Ding appreciates the support received from the Helmholtz-Zentrum Dresden-Rossendorf High Potential Programme

We present continuum-estimated (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD results for the leading-order Taylor expansion coefficients of the equation of state in strong magnetic fields and at nonzero baryon chemical potential. Simulations employ the highly improved staggered quark action with physical pion masses on lattices of temporal extent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>8</mn></math>, 12, covering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>145</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>≲</mo><mi>T</mi><mo>≲</mo><mn>165</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi><mo>≲</mo><mn>0.8</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>, imposing strangeness neutrality with baseline results at electric charge to baryon number ratio <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi><mo>=</mo><mn>0.4</mn></math>. We determine the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mi>–</mi><mi>e</mi><mi>B</mi></math> dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>q</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>s</mi><mn>1</mn></msub></math> (electric charge and strangeness chemical potential ratios), pressure coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mn>2</mn></msub></math>, baryon number density coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>N</mi><mn>1</mn><mi mathvariant="normal">B</mi></msubsup></math>, and energy-related coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Θ</mi><mn>2</mn></msub></math> (trace anomaly), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ε</mi><mn>2</mn></msub></math> (energy density), and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>σ</mi><mn>2</mn></msub></math> (entropy density). Magnetic fields induce temperature-band crossings for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>q</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mn>2</mn></msub></math> and nonmonotonic structures in the energy-related coefficients, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Θ</mi><mn>2</mn></msub></math> at strong fields possibly vanishing or turning negative at higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math>, indicating dominance of the pressure term over the energy contribution. We also examine the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi></math>-dependence, finding that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi><mo>=</mo><mn>0</mn></math> (charge-neutral matter) shows the most muted magnetic-field enhancement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mn>2</mn></msub></math> despite larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>q</mi><mn>1</mn></msub><mo stretchy="false">|</mo></math>, providing a useful reference for neutron-star–relevant conditions. Comparisons with the hadron resonance gas model show qualitative agreement at low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math> and weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math>, with clear deviations near the crossover and at strong fields. These results provide useful input for constraining models and effective theories of QCD matter in strong magnetic fields at finite baryon density.

Jin-Biao Gu

Arpith Kumar

Sheng-Tai Li

Leading-order QCD equation of state in strong magnetic fields at nonzero baryon chemical potential

We discuss the landscape of flavor physics at the Circular Electron-Positron Collider (CEPC), based on the nominal luminosity outlined in its Technical Design Report. The CEPC is designed to operate in multiple modes to address a variety of tasks. At the $\textit{Z}$ pole, the expected production of 4 Tera $\textit{Z}$ bosons will provide unique and highly precise measurements of $\textit{Z}$ boson couplings, while the substantial number of boosted heavy-flavored quarks and leptons produced in clean $\textit{Z}$ decays will facilitate investigations into their flavor physics with unprecedented precision. We investigate the prospects of measuring various physics benchmarks and discuss their implications for particle theories and phenomenological models. Our studies indicate that, with its highlighted advantages and anticipated excellent detector performance, the CEPC can explore beauty and $\textit{τ}$ physics in ways that are superior to or complementary with the Belle II and Large-Hadron-Collider-beauty experiments, potentially enabling the detection of new physics at energy scales of 10 TeV and above. This potential also extends to the observation of yet-to-be-discovered rare and exotic processes, as well as testing fundamental principles such as lepton flavor universality, lepton and baryon number conservation, etc., making the CEPC a vibrant platform for flavor physics research. The $\textit{WW}$ threshold scan, Higgs-factory operation and top-pair productions of the CEPC further enhance its merits in this regard, especially for measuring the Cabibbo-Kobayashi-Maskawa matrix elements, and Flavor-Changing-Neutral-Current physics of Higgs boson and top quarks. We outline the requirements for detector performance and considerations for future development to achieve the anticipated scientific goals. The role of machine learning for innovative detector design and advanced reconstruction algorithms is also stressed. The CEPC flavor physics program not only develops new capabilities for exploring flavor physics beyond existing projects but also enriches the physics opportunities of this machine. It should be remarked that, given the richness of the CEPC flavor physics, this manuscript is not meant to be a comprehensive survey, but rather an investigation of representative cases. Uncovering the full potential of flavor physics at the CEPC will require further dedicated explorations in the future.

Xiaocong Ai

United States

Wolfgang Altmannshofer

Peter Athron

Xiaozhi Bai

Lorenzo Calibbi

Lu Cao

Yuzhi Che

Chunhui Chen

Ji-Yuan Chen

Long Chen

Mingshui Chen

Shanzhen Chen

Xuan Chen

Shan Cheng

Cheng-Wei Chiang

Switzerland

Andreas Crivellin

Hanhua Cui

France

Olivier Deschamps

Sébastien Descotes-Genon

Xiaokang Du

Shuangshi Fang

Yu Gao

Yuanning Gao

Li-Sheng Geng

Pablo Goldenzweig

Jiayin Gu

Feng-Kun Guo

Yuchen Guo

Zhi-Hui Guo

Tao Han

Hong-Jian He

Jibo He

Miao He

Xiaogang He

Yanping Huang

Gino Isidori

Quan Ji

Jianfeng Jiang

Xu-Hui Jiang

Slovenia

Jernej F. Kamenik

Tsz Hong Kwok

Gang Li

Geng Li

Haibo Li

Haitao Li

Hengne Li

Honglei Li

Liang Li

Lingfeng Li

Qiang Li

Shu Li

Xiaomei Li

Xin-Qiang Li

Yiming Li

Yubo Li

Yuji Li

Zhao Li

Hao Liang

Zhijun Liang

Libo Liao

Zoltan Ligeti

Jia Liu

Jianbei Liu

Tao Liu

Yi Liu

Yong Liu

Zhen Liu

Xinchou Lou

Peng-Cheng Lu

Italy

Alberto Lusiani

Hong-Hao Ma

Kai Ma

Farvah Mahmoudi

Yajun Mao

Yaxian Mao

David Marzocca

Juan-Juan Niu

Soeren Prell

Huirong Qi

Sen Qian

Zhuoni Qian

Qin Qin

Ariel Rock

Jonathan L. Rosner

Manqi Ruan

Dingyu Shao

Chengping Shen

Xiaoyan Shen

Haoyu Shi

United Kingdom

Liaoshan Shi

Zong-Guo Si

Cristian Sierra

Huayang Song

Shufang Su

Wei Su

Zhijia Sun

Michele Tammaro

Dayong Wang

En Wang

Flavor physics at the CEPC: a general perspective * We acknowledge financial support from the National Natural Science Foundation of China (NSFC) (12125507, 12047503, 12035008, 2211530479, 12475094, 12135006, 12075097, 12375086, 2022YFA1601903, 12061141007, 12375091, 12342502, 12235018, 12335003, 12105100, 12475106, 11961141015, 12188102, 12175245, 12205171, 12321005, tsqn202312052, 2024HWYQ-005, 12405121, 12447167, 12061141006, 12405102, 12125503,12305115, 12075213, 12335005, 12235008); the Chinese Academy of Sciences (YSBR-101, XDB34030000); the National Key R&D Program of China (2022YFE0116900, 2023YFA1606703, 2022YFA1601901); the National Key Research and Development Program of China (2023YFA1606300); the Excellent Postdoctoral Program of Jiangsu Province (2023ZB891); the Shenzhen Science and Technology Program (202206193000001, 20220816094256002); the Natural Science Foundation for Distinguished Young Scholars of Henan Province (242300421046); the Beijing Municipal Natural Science Foundation (JQ22002); the Area of Excellence (AoE/P-404/18-3) and the General Research Fund (16304321) (both grants are issued by the Research Grants Council of Hong Kong S.A.R); the MOST National Key R&D Program (2023YFA1606303); the Shanghai Key Laboratory for Particle Physics and Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (Ministry of Education), Shanghai Jiao Tong University

Lepton flavor violation (LFV) offers a powerful probe of physics beyond the Standard Model, particularly in models addressing neutrino masses and the baryon asymmetry of the universe. In this study, we investigate LFV processes within the framework of type II seesaw leptogenesis, where the Standard Model is extended by an $S U(2)_L$ triplet Higgs field. We focus on key LFV processes, including $\mu^+\to e^+\gamma$ , $\mu^+ \to e^+e^-e^+$ , and $\mu \rightarrow e$ conversion in nuclei, deriving stringent constraints on the parameter space from current experimental data. We scan the 3$\textit{σ}$ range of neutrino oscillation parameters and identify the most conservative bounds consistent with existing measurements. Our results reveal that the MEG experiment currently provides the most stringent constraints in the normal ordering (NO) scenario, while the SINDRUM experiment offers comparable sensitivity in the inverted ordering (IO) case. Future experiments, such as MEG II, Mu3e, Mu2e, and COMET, are predicted to significantly improve the sensitivity, testing larger regions of the parameter space.

Chengcheng Han

Yijun Han

Sihui Huang

Zhanhong Lei

Probing type II seesaw leptogenesis through lepton flavor violation * Chengcheng Han acknowledges supports from the National Key R&D Program of China (2023YFA1606100), the Sun Yat-Sen University Science Foundation, the Fundamental Research Funds for the Central Universities at Sun Yat-sen University (24qnpy117) and the Key Laboratory of Particle Astrophysics and Cosmology (MOE) of Shanghai Jiao Tong University

We present a first-principles lattice QCD investigation of second-order fluctuations of and correlations among conserved charges—baryon number (B), electric charge (Q), and strangeness (S)—in the presence of external magnetic fields. Our study employs lattice simulations of (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD with physical pion masses using highly improved staggered fermions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>48</mn><mn>3</mn></msup><mo>×</mo><mn>12</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math> lattices, covering a wide range of magnetic field strengths up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi><mo>≃</mo><mn>0.8</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. We identify clear signals of magnetic field-induced modifications to these fluctuations and correlations, with the baryon-electric charge correlation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>BQ</mi></mrow></msubsup></math>, exhibiting particularly strong sensitivity to the magnetic field. To bridge theoretical predictions with experimental observables, we implement systematic kinematic cuts that emulate detector acceptances of the STAR and ALICE experiments within the hadron resonance gas (HRG) model and construct proxy observables for fluctuations measurable in heavy-ion collision experiments. Our findings highlight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>BQ</mi></mrow></msubsup></math> as a promising “magnetometer” for probing the presence of magnetic fields in QCD matter. Furthermore, we explore experimentally relevant ratios involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mn>11</mn><mrow><mi>BQ</mi></mrow></msubsup></math>, demonstrating their potential in mitigating volume effects and enhancing sensitivity to magnetic fields in collision environments. Additionally, we assess the limitations of the HRG model at strong magnetic fields, revealing deviations that indicate nontrivial modifications to hadronic degrees of freedom. These results offer new insights into the interplay between thermal and magnetic effects in the QCD phase diagram and provide experimentally relevant guidance for the detection of magnetic fields in heavy-ion collisions.

Second order fluctuations of conserved charges in external magnetic fields

We calculate the temperature dependence of bottomonium correlators in (2+1)-flavor lattice QCD with the aim to constrain in-medium properties of bottomonia at high temperature. The lattice calculations are performed using HISQ action with physical strange quark mass and light quark masses twenty times smaller than the strange quark mass at two lattice spacings $\textit{a}$ = 0$\textit{.}$0493 fm and 0$\textit{.}$0602 fm, and temporal extents $\textit{N}$ = 16 − 30, corresponding to the temperatures $\textit{T}$ = 133 − 250 MeV. We use a tadpole-improved NRQCD action including spin-dependent $\textit{v}$ corrections for the heavy quarks and extended meson operators in order to be sensitive to in-medium properties of the bottomonium states of interest. We find that within estimated errors the bottomonium masses do not change compared to their vacuum values for all temperatures under our consideration; however, we find different nonzero widths for the various bottomonium states.

Wei-Ping Huang

Rasmus Larsen

Stefan Meinel

Swagato Mukherjee

Peter Petreczky

Zhanduo Tang

In-medium bottomonium properties from lattice NRQCD calculations with extended meson operators

The ultralight dark photon is an intriguing dark matter candidate. The interaction between visible and dark photons is introduced by the gauge kinetic mixing between the field strength tensors of the Abelian gauge groups in the Standard Model and dark sector. Relativistic electrodynamics was generalized to quantum electromagnetodynamics (QEMD) in the presence of both electric and magnetic charges. The photon is described by two four-potentials corresponding to two $ U(1) $ gauge groups and satisfying non-trivial commutation relations. In this work, we construct low-energy dark photon-photon interactions in the QEMD framework and obtain new dark photon-photon kinetic mixings. Then, we derive the consequent field and Maxwell's equations. We also investigate the detection strategies of dark photons as light dark matter and generic kinetic mixings at haloscope experiments.

Chang-Jie Dai

Tong Li

Rui-Jia Zhang

Dark photon dark matter in quantum electromagnetodynamics and detection at haloscope experiments * Li Tong is supported by the National Natural Science Foundation of China (12375096, 12035008, 11975129)

We investigate the effects of temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math> and external magnetic fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math> on the chiral condensates and screening masses of neutral pseudoscalar mesons, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mn>0</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>η</mi></mrow><mrow><mi>s</mi><mover accent="true"><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></math>, in (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD with physical quark masses. The chiral condensates are intrinsically connected to the screening masses via Ward-Takahashi identities, with the latter characterizing the inverse of the spatial correlation length in the pseudoscalar channel. Using highly improved staggered quark (HISQ), we perform simulations on lattices with temporal extents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>8</mn></math>, 12, 16 and an aspect ratio of 4, covering five temperatures from 145 MeV to 166 MeV. For each temperature, eight magnetic field strengths are simulated, reaching up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>e</mi><mi>B</mi><mo>∼</mo><mn>0.8</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. These simulations allow us to provide continuum estimates for the chiral condensates and screening masses. We observe intricate behavior in the light (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi><mi>d</mi></math>), strange-light (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mi>s</mi></math>), and strange (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi></math>) quark condensates as functions of the magnetic field and temperature, reflecting the competition between magnetic catalysis and inverse magnetic catalysis effects. This complex behavior is also mirrored in the screening masses of the neutral pseudoscalar mesons. Notably, the screening masses of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mn>0</mn></msup></math> exhibit a nonmonotonic dependence on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math>, closely following the variations in their corresponding chiral condensates. Meanwhile, the screening mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>η</mi><mrow><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><mn>0</mn></msubsup></math> decreases monotonically with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math>. These findings provide valuable insights for understanding the behavior of QCD in a thermomagnetic medium and can serve as benchmarks for low-energy QCD models and effective theories.

Rishabh Thakkar

Chiral condensates and screening masses of neutral pseudoscalar mesons from lattice QCD at physical quark masses