It is well known that nonrelativistic or semirelativistic constituent quark models can describe a large number of the meson and baryon properties with surprising accuracy. In this work, we study the Killingbeck potential using the WKB approximation in the constituent quark model. The implicit approximate solutions for the energy eigenvalues, together with analytical expressions for the root-mean-square radii and the probability densities at the origin, of the Schrödinger equation are obtained by using appropriate quantization conditions. Finally, we discuss applications to double-heavy-meson spectroscopy within the Killingbeck potential.

China

Cheng-Qun Pang

Lhamo Chosto

Ya-Rong Wang

Zhi-Bin Gao

Hao Chen

Yu-Long Kang

Wei Li

WKB approximation under a Killingbeck potential and its application to double-heavy-meson spectroscopy

The observed fluid behavior of the matter produced in heavy ion and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mi>p</mi></mrow></math> collisions implies that the strings stretched between the constituents of the projectile and target should have a hard core giving rise to a repulsion between them. It is shown that in these collisions there are two critical string densities, one corresponding to the well-studied percolation of strings and another to the percolation of the hard core of the strings. These two critical densities are related to two critical temperatures, one <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mspace width="4pt"></mspace><mo>≈</mo><mspace width="4pt"></mspace><mn>160</mn></mrow></math> MeV would correspond to the restoration of chiral symmetry and partial deconfinement and a second transition <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mspace width="4pt"></mspace><mo>≈</mo><mspace width="4pt"></mspace><mn>220</mn></mrow></math> MeV form the fluid behavior of quantum chromodynamics matter strongly interacting to a quasi free gas of quarks and gluons. This argument is supported by the recently published results on the sudden increase in the degrees of freedom above the deconfinement temperature in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mi>p</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mtext>−</mtext><mi>A</mi></mrow></math> collisions [Mishra et al., <a href="http://dx.doi.org/10.1016/j.nuclphysa.2024.122865">Nucl. Phys. A 1046, 122865 (2024)</a>].

Spain

C. Pajares

United States

R.P. Scharenberg

B.K. Srivastava

Existence of a second phase transition in <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mtext>−</mtext><mi>A</mi></mrow></math> collisions at energies available at the CERN Large Hadron Collider: A physical interpretation

We consider nonequilibrium evolution of non-Gaussian fluctuations in a hydrodynamic system undergoing a boost-invariant expansion described by Bjorken flow. We derive the evolution equations for two- and three-point velocity correlators using the effective field-theory framework and present analytical solutions for them. We show that the average Landau frame is better suited for studying non-Gaussian fluctuations of velocity when relativistic effects are important. In the Bjorken background, the average Landau frame corresponds to the density frame. We demonstrate that the three-point correlators depend nonlinearly on the nonequilibrium dynamics of the two-point functions and exhibit nontrivial effects such as memory. The importance of these effects in the context of the search for the QCD critical point via fluctuations is discussed.

Gökçe Başar

Shuo Song

Non-Gaussian hydrodynamic fluctuations in an expanding relativistic fluid

The energy-energy correlator (EEC) inside jets is a sensitive observable for studying jet modification in the quark-gluon plasma. However, its interpretation in heavy-ion collisions remains challenging, requiring a consistent understanding of jet evolution across multiple dynamical scales together with a proper treatment of the background subtraction. In this work, we employ an updated CoLBT-hydro framework in which a medium scale <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mi>M</mi></msub><mo>=</mo><mn>2.0</mn></mrow></math> GeV is introduced to separate the vacuum and in-medium stages of the parton shower, enabling a more self-consistent treatment of jet evolution. Using a theoretical background subtraction within the model, the resulting simulation reproduces the recent CMS measurement of the in-jet EEC and, through a decomposition of the different contributions, highlights the impact of medium modification on the observable. To further validate the experimental procedure, we also implement the CMS mixed-event background-subtraction method directly in the simulation and find that the results are consistent with those obtained with the theoretical background subtraction. Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>T</mi></msub></math>-ranked jets in each event, we further investigate the dependence of medium modification on the in-medium path length, reflected in the different EECs of leading and subleading jets. Finally, we explore the dependence of the leading-jet EEC on the dijet rapidity gap as a signal of the jet-induced diffusion wake.

Zhong Yang

Raghav Kunnawalkam Elayavalli

Xin-Nian Wang

Energy-energy correlators inside single inclusive jets in heavy-ion collisions with the CoLBT-hydro model

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>t</mi></msub></math>-differential quantity <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math> is meant to measure elliptic flow manifested by a dense QCD medium formed in high-energy nucleus-nucleus collisions. Elliptic flow may be referred to more neutrally as a cylindrical quadrupole component of the transverse motion of particle sources within a collision. As defined, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math> relies on an implicit assumption that almost all produced particles emerge from a single source. This article describes a detailed study of the algebraic structure of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math>. A procedure is developed to derive a common monopole boost (“radial flow”) value and quadrupole <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>t</mi></msub></math> spectra for several hadron species. The method is applied to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math> data for four hadron species from 2.76-TeV Pb-Pb collisions. According to available <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math> data the assumption of a single dominant particle source within <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi></mrow><mtext>−</mtext><mrow><mi>A</mi></mrow></math> collisions is unjustified. Combined with a previous study of quadrupole amplitude variation for 200-GeV <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mtext>−</mtext><mi>p</mi></mrow></math> collisions these results demonstrate that quadrupole structure is related to a novel QCD process separate from projectile-nucleon dissociation and jet production. Given quadrupole evolution it is unlikely that a hydrodynamic description is relevant to that process.

Thomas A. Trainor

Quadrupole spectra derived from 2.76-TeV Pb-Pb identified-hadron <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>p</mi><mi>t</mi></msub><mo>)</mo></mrow></mrow></math> data

Coherent hard exclusive reactions on light nuclei provide access to their quark and gluon structure and enable three-dimensional tomography of these complex systems. We study deeply virtual Compton scattering on a helium-4 target, including both kinematic twist-3 and twist-4 corrections, as well as next-to-leading-order corrections to the twist-2 amplitude in the strong coupling <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mi>s</mi></msub></math>. We show that these contributions are crucial for achieving a precise description of the data and, as a result, obtain the first tomographic image of the helium-4 nucleus at the quark-gluon level.

France

V. Martínez-Fernández

B. Pire

Poland

P. Sznajder

J. Wagner

Coherent deeply virtual Compton scattering on <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> beyond leading power

We present a new formulation for the mean-field propagation part of relativistic quantum molecular dynamics, simulating an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>-body system of interacting Gaussian wave packets via Lorentz scalar and vector potentials. Covariant equations of motion are derived based on the principle of least action with a weak form of mass-shell conditions and time-fixation constraints defined with respect to a chosen foliation. However, as is common with traditional relativistic quantum molecular dynamics, the dynamics exhibits a residual dependence on the chosen foliation, which is unavoidable because a finite number of interacting degrees of freedom is not strictly compatible with relativity. Nevertheless, we show that this dependence remains small for physically reasonable choices of the foliation in practical applications. By introducing a new approximation method to the spatial integral in the equations of motion, these covariant equations of motion can be solved with a computational cost comparable to that of conventional noncovariant quantum molecular dynamics. Furthermore, the new equations of motion accurately estimate the density-dependent potential, as demonstrated through comparison of the forces with the numerical integration. We apply them to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>-body systems interacting via the Skyrme-type potentials or the relativistic mean field to simulate heavy-ion collisions. Our results show that the derived equations of motion provide a robust approximation to the dynamics of the full numerical integrations.

Japan

Yasushi Nara

Asanosuke Jinno

Koichi Murase

Covariant formulation of relativistic quantum molecular dynamics for a system of interacting wave packets

The simultaneous observation of strong sequential bottomonium suppression and small azimuthal anisotropy constrains the time scale and geometry of quarkonium dissociation in relativistic heavy-ion collisions. This work investigates an early-time contribution in which strong pre-equilibrium color fields generate an effective proper-acceleration scale and an associated causal length. The maximal acceleration is fixed through the phenomenological anchoring assumption <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>T</mi><mi>U</mi><mo movablelimits="true" form="prefix">max</mo></msubsup><mo>≃</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></math>, without identifying the kinematic Unruh scale with an equilibrium temperature. The survival probability is described by an event-averaged one-scale exponential ansatz. With no state-by-state adjustment, the resulting horizon component gives a quantitatively reasonable description of the LHC centrality dependence. For the directly measured <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Υ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></mrow></math>-to-<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Υ</mi><mo>(</mo><mn>1</mn><mi>S</mi><mo>)</mo></mrow></math> double ratio, the fixed horizon term supplies a substantial part of the relative suppression, while the data require an additional state-dependent late-stage factor. A minimal one-parameter effective quark-gluon plasma (QGP) attenuation gives a quantitative multistage description. At RHIC, the absolute <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow><mi>A</mi><mi>A</mi></mrow></msub></math> central values lie below the isolated horizon component, while the relative <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>S</mi></mrow></math>-to-<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mi>S</mi></mrow></math> suppression remains compatible with the predicted state-size hierarchy within present uncertainties. Because the proposed early factor is local and scalar, it gives the exact leading-order null contribution <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>v</mi><mn>2</mn><mi>H</mi></msubsup><mo>=</mo><mn>0</mn></mrow></math> within this construction, consistent with current CMS measurements.

Taiwan

Yi Yang

Causal-horizon scaling of quarkonium suppression in strong QCD fields

Equations of a boost-invariant and cylindrically symmetric perfect hydrodynamics are solved numerically for initial conditions inspired by the wounded nucleon model. The energy-momentum and spin tensors are used in the form that describes a relativistic massive gas governed by Boltzmann statistics. In contrast to one-dimensional boost-invariant expansion, we find a coupling between the azimuthal and longitudinal components of the electric and magnetic components of the spin polarization tensor. This feature is similar to that found earlier for the Gubser symmetry; however, our treatment allows for a more general form of initial conditions and expansion geometry. Defining the freeze-out hypersurface by the constant temperature condition, we evaluate the Pauli-Lubański four-vector and find that for the assumed geometry the only nonzero total polarization may be induced by the longitudinal component of the magnetic part of the spin polarization tensor, coupled with the azimuthal electric component. The obtained results may serve as a reference point for more realistic models of hydrodynamic expansion.

Zbigniew Drogosz

Wojciech Florkowski

Jakub Witkowski

Boost-invariant and cylindrically symmetric perfect spin hydrodynamics

We study fluctuations and correlations of conserved charges in QCD using a string-based description of the hadronic mass spectrum. Mesons and baryons are modeled as open relativistic strings with quark-antiquark and quark-diquark endpoints, respectively, leading to an exponential Hagedorn growth of states with a limiting temperature fixed by the string tension. We find that continuous Hagedorn spectra constrained by experimentally established hadrons underestimate net-baryon number fluctuations obtained in lattice QCD (LQCD) calculations. By extracting the Hagedorn string spectrum directly from lattice QCD through a fit to the second-order net-baryon number susceptibility, we obtain a consistent description of a broad set of fluctuations of conserved charges from LQCD with the Hagedorn temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>T</mi><mi>H</mi></msub><mo>≃</mo><mn>323</mn><mspace width="0.16em"></mspace><mi>MeV</mi></mrow></math>, without introducing additional free parameters. Our results provide thermodynamic evidence in support of a string quark-diquark picture of baryons in the confined phase of QCD.

Michał Marczenko

Krzysztof Redlich

Evidence for quark-diquark structure of baryons from fluctuations of conserved charges