We study the hot medium effects in high-multiplicity proton–proton (pp) collisions at  $$\sqrt{s_{NN}}=13$$  <math> <mrow> <msqrt> <msub> <mi>s</mi> <mrow> <mi>NN</mi> </mrow> </msub> </msqrt> <mo>=</mo> <mn>13</mn> </mrow> </math>   TeV via the charmonium probes. The hot medium is described with the hydrodynamic model, while charmonium evolutions in the medium are studied with a time-dependent Schrödinger equation. The hot medium dissociation on charmonium is considered with the temperature-dependent complex potential parametrized with the results from lattice QCD calculations. The ratio  $$\sigma _{\psi (2S)}/\sigma _{J/\psi }$$  <math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </msub> <mo>/</mo> <msub> <mi>σ</mi> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </msub> </mrow> </math>   of  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   and  $$\psi (2S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   production cross sections are calculated and compared with the LHCb experimental data in pp collisions. Our calculations explain the charmonium relative suppression in different transverse momentum and multiplicity bins. The suppression of this ratio is mainly affected by the effects of the deconfined medium. It is less affected by the initial effects before the generation of the heavy quark pair. We suggest this to be a clear signal of the small QGP droplets generated in high multiplicity pp collisions.

United Kingdom

Yunfan Bai

China

Baoyi Chen

Probing QGP droplets with charmonium in high-multiplicity proton–proton collisions

Understanding the polarization property of  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   is critical to constrain its production mechanism. In addition, the polarization of  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   can reveal the impact of strong electromagnetic and vorticity fields in relativistic heavy ion collisions. In this study, we analyzed the yield and polarization of  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   in relativistic heavy ion collisions at different centrality and transverse momentum regions, using three different reference frames: the Collins–Soper frame, the helicity frame, and the event plane frame. The polarization of initially produced  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   is determined by the NRQCD calculation and is similar to that of pp collisions. However, both unpolarization and transverse polarization are considered for the regenerated  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>  . Our results indicate that the polarization at high  $$p_T$$  <math> <msub> <mi>p</mi> <mi>T</mi> </msub> </math>   is similar to that observed in pp collisions. However, at low  $$p_T$$  <math> <msub> <mi>p</mi> <mi>T</mi> </msub> </math>  , where regenerated  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   dominates, it is likely that the polarized charm quarks in the rotational QGP medium are responsible for this phenomenon. Our study supplies a baseline for future research on the effects of strong electromagnetic and vorticity fields on  $$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   polarization.

France

Jiaxing Zhao

$$J/\psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math>   polarization in relativistic heavy ion collisions

The static properties and dynamic evolution of bottomonium states in a hot QCD medium are investigated through the Schrödinger equation with complex heavy quark potentials, which have been presented recently in a lattice QCD study and with three different extractions. This approach builds a direct connection between the in-medium heavy quark potentials from the lattice QCD to the experimental observables. The yields and nuclear modification factors  $$R_{AA}$$  <math> <msub> <mi>R</mi> <mrow> <mi>AA</mi> </mrow> </msub> </math>   of bottomonium in Pb–Pb collisions at  $$\sqrt{s_\textrm{NN}}=5.02$$  <math> <mrow> <msqrt> <msub> <mi>s</mi> <mtext>NN</mtext> </msub> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </math>   TeV are calculated in this work. Our results show a large suppression of the bottomonium yield in heavy ion collisions due to the large imaginary potential. To understand bottomonium  $$R_{AA}$$  <math> <msub> <mi>R</mi> <mrow> <mi>AA</mi> </mrow> </msub> </math>   based on lattice QCD potentials, we propose a formation time for bottomonium states and find that experimental data can be well explained with the heavy quark potential extracted by the Padé fit, which shows no color screening in the real part potential.

Ge Chen

Bottomonium evolution with in-medium heavy quark potential from lattice QCD

We study the <math altimg='si1.svg'><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msubsup></math> production in Pb-Pb collisions at <math altimg='si2.svg'><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>5.02</mn></math> TeV. In the quark-gluon plasma (QGP) produced in heavy-ion collisions, heavy quarks make random motions with the energy loss. We employ the Langevin equations to study the non-equilibrium distributions of heavy quarks and the Instantaneous Coalescence Model (ICM) to study the hadronization process. Due to abundant charm and bottom quarks in the QGP, their coalescence probability is significantly enhanced compared with the situations in proton-proton collisions. We find that the final production of <math altimg='si1.svg'><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msubsup></math> is increased by the coalescence process, which makes the nuclear modification factor ( <math altimg='si3.svg'><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>A</mi></mrow></msub></math>) of <math altimg='si1.svg'><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msubsup></math> larger than unit. Our model explains the experimental data well at semi-central and central collisions. The observation of <math altimg='si4.svg'><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>A</mi></mrow></msub><mo stretchy='false'>(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msubsup><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>&gt;</mo><mn>1</mn></math> is regarded as an evident and strong signal of the existence of the deconfined medium generated in heavy-ion collisions.

Liuyuan Wen

Yunpeng Liu

<math altimg='si1.svg'><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msubsup></math> formation from random charm and anti-bottom quarks in the quark-gluon plasma

Abstract The open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW$_{4}$) model. The equation of motion of the WZW$_{4}$ model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW$_{4}$ model in the split signature by calculating the action density of the WZW$_{4}$ model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n-soliton solutions is a “non-linear superposition” of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

Japan

Masashi Hamanaka

Shan-Chi Huang

Hiroaki Kanno

Solitons in open = 2 String Theory Kahler-Chen-Simons theory A tour through N=2 strings On soliton solutions of the anti-self-dual Yang-Mills equations from the perspective of integrable systems Soliton solutions in an integrable chiral model in 2+1 dimensions On the space-time monopole equation A review on tachyon condensation in open string field theories Soliton equations as dynamical systems on infinite dimensional Grassmann manifold

In this work, we propose to search for the exotic doubly charmed meson  $$T_{cc}^+$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> </math>   and its analog  $$T_{cc\bar{s}}^+$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>+</mo> </msubsup> </math>   in  $$B_c^+$$  <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>+</mo> </msubsup> </math>   decays, which provide a good environment for the formation of the exotic state containing double charm quarks. Within the molecular scheme, the production of  $$T_{cc}^+$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> </math>   and  $$T_{cc\bar{s}}^+$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>+</mo> </msubsup> </math>   through various rescattering processes with different intermediate states are investigated. For the moderate values of model parameters, the branching ratios of  $$B_c^+$$  <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>+</mo> </msubsup> </math>   decaying into  $$T_{cc}^+ \bar{D}^{0}$$  <math> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mn>0</mn> </msup> </mrow> </math>  ,  $$T_{cc}^+ \bar{D}^{*0}$$  <math> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> <mn>0</mn> </mrow> </msup> </mrow> </math>  ,  $$T_{cc\bar{s}}^+ \bar{D}^{0}$$  <math> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>+</mo> </msubsup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mn>0</mn> </msup> </mrow> </math>   and  $$T_{cc\bar{s}}^+ \bar{D}^{*0}$$  <math> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>+</mo> </msubsup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> <mn>0</mn> </mrow> </msup> </mrow> </math>   are estimated to be of the order of  $$10^{-7}$$  <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </msup> </math>  ,  $$10^{-5}$$  <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </math>  ,  $$10^{-6}$$  <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </math>   and  $$10^{-4}$$  <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </math>  , respectively, which may be tested by future experiments.

Yuan Li

Ying-Bo He

Xiao-Hai Liu

Hong-Wei Ke

Searching for doubly charmed tetraquark candidates  $$T_{cc}$$  <math> <msub> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> </msub> </math>   and  $$T_{cc\bar{s}}$$  <math> <msub> <mi>T</mi> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </msub> </math>   in  $$B_c$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   decays

We employ the time-dependent Schrödinger equation with different complex potentials to study the bottomonium sequential suppression in Pb-Pb collisions at <math><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.76</mn></math> TeV and 5.02 TeV and Au-Au collisions at <math><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn></math> GeV. Both color screening effect and the random scatterings with thermal partons are considered in the real and imaginary parts of the heavy-quark potentials. As the real part of the heavy-quark potential is between the free energy <math><mi>F</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math> and the internal energy <math><mi>U</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math> of heavy quarkonium, we parametrize different potentials with a function of F and U to evolve the bottomonium wave packages in the medium. We find that when the real part of the potential is close to <math><mi>U</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>, it can explain well the pattern of bottomonium sequential suppression where their nuclear modification factors satisfy the relation <math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>A</mi></mrow></msub><mo>(</mo><mn>1</mn><mi>s</mi><mo>)</mo><mo>&gt;</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>A</mi></mrow></msub><mo>(</mo><mn>2</mn><mi>s</mi><mo>)</mo><mo>&gt;</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mi>A</mi></mrow></msub><mo>(</mo><mn>3</mn><mi>s</mi><mo>)</mo></math> observed in experiments. In the other limit of <math><mi>F</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>r</mi><mo>)</mo></math>, bottomonium wave packages tend to expand due to weak attracted force, which results in evident transitions from <math><mi>ϒ</mi><mo>(</mo><mn>2</mn><mi>s</mi><mo>)</mo></math> to <math><mi>ϒ</mi><mo>(</mo><mn>3</mn><mi>s</mi><mo>)</mo></math> components and does not satisfy the sequential suppression pattern. We suggest that the bottomonium sequential suppression can be a probe of strong heavy-quark potential in the medium.

Bottomonium sequential suppression and strong heavy-quark potential in heavy-ion collisions

As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables’ estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As a demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman path integral solving.

Shile Chen

Germany

Ukraine

Oleh Savchuk

United States

Shiqi Zheng

Horst Stoecker

Lingxiao Wang

Kai Zhou

Fourier-flow model generating Feynman paths

Heavy-ion collisions provide a unique opportunity to study the nature of X(3872) compared with electron-positron and proton-proton (antiproton) collisions. We investigate the centrality and momentum dependence of <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> in heavy-ion collisions via the Langevin equation and instant coalescence model (LICM). When <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> is treated as a compact tetraquark state, the tetraquarks are produced via the coalescence of heavy and light quarks near the quantum chromodynamic phase transition due to the restoration of the heavy quark potential at <math><mrow><mi>T</mi><mo>→</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></math>. In the molecular scenario, loosely bound <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> is produced via the coalescence of <math><mrow><msup><mi>D</mi><mn>0</mn></msup><mtext>−</mtext><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></mrow></math> mesons in a hadronic medium after kinetic freeze-out. We employ the LICM to explain both <math><msup><mi>D</mi><mn>0</mn></msup></math> and <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> production as a benchmark. Then we give predictions regarding <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> production and the nuclear modification factor <math><msubsup><mi>R</mi><mrow><mi>A</mi><mi>A</mi></mrow><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></msubsup></math>. We find that the total yield of tetraquark is several times larger than the molecular production in Pb-Pb collisions. Although the geometric size of the hadronic molecule is huge, the coalescence probability is small due to strict constraints on the relative momentum between <math><msup><mi>D</mi><mn>0</mn></msup></math> and <math><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></math> in the molecular Wigner function, which significantly suppresses the molecular yield.

Liu Jiang

<math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> production in relativistic heavy-ion collisions

We calculate <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> triangular flow in high energy nuclear collisions. The charmonium motion in medium is controlled by a transport equation with loss and gain terms, and the evolution of the hot medium is governed by a single-shot hydrodynamic equation with a triangularly deformed initial condition. In comparison with the elliptic flow <math><msub><mi>v</mi><mn>2</mn></msub></math>, the triangular flow <math><msub><mi>v</mi><mn>3</mn></msub></math> comes almost completely from the charmonium regeneration in the quark gluon plasma and therefore is more sensitive to the medium properties.