Nucleon clustering at kinetic freezeout of heavy-ion collisions via path-integral Monte Carlo
Dallas DeMartini, Edward Shuryak
备注：
24 pages, 11 figures, 6 tables; version 3 of this paper; significantly reduced extraneous explanations and technical details and addressed some points raised by others; submitted to Nucl. Phys. A
Clustering of the four-nucleon system at kinetic freezeout conditions is studied using path-integral Monte Carlo techniques. This method seeks to improve upon previous calculations which relied on approximate semiclassical methods or few-body quantum mechanics. Estimates are given for the decay probabilities of the 4N system into various light nuclei decay channels and the strength of spatial correlations is characterized. Additionally, a simple model is presented to describe the impact of this clustering on nucleon multiplicity distributions. The effects of a possible modification of the inter-nucleon interaction due to the close critical line (and hypothetical QCD critical point) on the clustering are also studied.
We revisit the problem of the production of light atomic nuclei in ultrarelativistic heavy-ion collisions. While their production systematics is well produced by hadro-chemical freezeout at temperatures near the QCD pseudo-critical temperature, their small binding energies of a few MeV per nucleon suggest that they cannot survive as bound states under these conditions. Here, we adopt the concept of effective chemical potentials in the hadronic evolution from chemical to thermal freezeout (at typically $T_{\rm fo}$$\simeq$100\,MeV), which, despite frequent elastic rescatterings in hadronic matt…
Combining the covariant coalescence model and a blast-wave-like analytical parametrization for (anti-)nucleon phase-space freezeout configuration, we explore light (anti-)nucleus production in central Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV. Using the nucleon freezeout configuration (denoted by FO1) determined from the measured spectra of protons (p), deutrons (d) and $^{3}$He, we find the predicted yield of $^{4}$He is significantly smaller than the experimental data. We show this disagreement can be removed by using a nucleon freezeout configuration (denoted by FO2) in which the nucleo…
We study the instabilities of quark matter in the framework of a generalized Nambu--Jona-Lasinio model, in order to explore possible competition between three-quark clustering to form nucleons and diquark formation leading to color superconductivity. Nucleon and $Δ$ solutions are obtained for the relativistic Faddeev equation at finite density and their binding energies are compared with those for the scalar and axial-vector diquarks found from the Bethe-Salpeter equation. In a model with interactions in both scalar and axial diquark channels, bound nucleons exist up to nuclear matter density.…
We show that the measured yield ratio $^3_Λ$H/$^3$He ($^3_{\overlineΛ}\overline{\text{H}}$/$^3\overline{\text{He}}$) in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV and in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV can be understood within a covariant coalescence model if (anti-)$Λ$ particles freeze out earlier than (anti-)nucleons but their relative freezeout time is closer at $\sqrt{s_{NN}}=2.76$ TeV than at $\sqrt{s_{NN}}=200$ GeV. The earlier (anti-)$Λ$ freezeout can significantly enhance the yield of (anti)hypernucleus $^4_Λ$H ($^4_{\overlineΛ}\overline{\text{H}}$), leading to that $^4_{…
It has been proposed that one can look for the QCD critical point (CP) by the Beam Energy Scan (BES) accurately monitoring event-by-event fluctuations. This experimental program is under way at the BNL RHIC collider. Separately, it has been studied how clustering of nucleons at freezeout affects proton multiplicity distribution and light nuclei production. It was found that even a minor increase of the range of nuclear forces dramatically increases clustering, while large correlation length $ξ$ near CP makes attraction due to binary forces unrealistically large. In this paper we show that repu…
Nuclear mass data provide EMPIRICAL evidence of: 1. Clustering of nucleons; 2. Attractive n-p interactions; and 3. Repulsive but symmetric n-n and p-p interactions after correcting for the repulsive Coulomb interactions between positive nuclear charges. These findings suggest a possible source of energy in neutron stars, in stars which formed on them, and demonstrate the need for a Theoretical understanding of 1. Interactions between nucleons; 2. Clustering of nucleons; and 3. Neutron-emission by penetration of the gravitational barrier surrounding a neutron star.
Baryon number cumulants are invaluable tools to diagnose the primordial stage of heavy ion collisions if they can be measured. In experiments, however, proton number cumulants have been measured as substitutes. In fact, proton number fluctuations are further modified in the hadron phase and different from those of baryon number. We show that the isospin distribution of nucleons at kinetic freezeout is binomial and factorized. This leads to formulas that express the baryon number cumulants sorely in terms of proron number fluctuations, which are experimentally observable.
Starting with a quark model of nucleon structure in which the valence quarks are strongly correlated within a nucleon, the light nuclei are constructed by assuming similar correlations of the quarks of neighboring nucleons. Applying the model to larger collections of nucleons reveals the emergence of the face-centered cubic (FCC) symmetry at the nuclear level. Nuclei with closed shells possess octahedral symmetry. Binding of nucleons are provided by quark loops formed by three and four nucleon correlations. Quark loops are responsible for formation of exotic (borromean) nuclei, as well. The mo…
We study clustering of baryons at the freeze-out point of relativistic heavy-ion collisions. Using a Walecka-Serot model for the nucleon-nucleon (NN) interaction we analyze how the modified/critical $σ$ mode---responsible for the NN attraction---allows for clustering of nucleons when the system is close to a possible critical point of QCD. We investigate clusters of few nucleons, and also the internal cluster configuration when the system is long lived. For realistic heavy-ion collisions we study to how extend such clusters can be formed in a finite time, and perform the statistical analysis o…
It is shown that the instanton-induced interaction in qq pairs, iterated in t-channel, leads to a meson-exchange interactions between quarks. In this way one can achieve a simultaneous understanding of low-lying mesons, baryons and the nuclear force. The discussion is general and does not necessarily rely on the instanton-induced interaction. Any nonperturbative gluonic interaction between quarks, which is a source of the dynamical chiral symmetry breaking and explains the $π$ - $ρ$ mass splitting, will imply an effective meson exchange picture in baryons. Due to the (anti)screening there is a…
Using the framework of nuclear energy density functionals we examine the conditions for single-nucleon localization and formation of cluster structures in finite nuclei. We propose to characterize localization by the ratio of the dispersion of single-nucleon wave functions to the average inter-nucleon distance. This parameter generally increases with mass and describes the gradual transition from a hybrid phase in light nuclei, characterized by the spatial localization of individual nucleon states that leads to the formation of cluster structures, toward the Fermi liquid phase in heavier nucle…
Based on an improved multiphase transport (AMPT) model, which gives a good description of proton production with a smooth quark matter to hadronic matter transition in relativistic heavy ion collisions, we study deuteron and triton production from the coalescence of nucleons at the kinetic freezeout of these collisions. For Au+Au collisions at center-of-mass energies $\sqrt{s_\text{NN}}$ from 7.7 GeV to 200 GeV available at the Relativistic Heavy Ion Collider (RHIC), we find that the yield ratio $N_\text{t}N_\text{p}/N_\text{d}^2$ of proton, deuteron\com{,} and triton is essentially a constant…
Clustering plays an important role in the structure of nuclei, especially for light nuclei in the $p$-shell. In nuclear cluster models these degrees of freedom are introduced explicitly. In the Resonating Group Method or in the Generator Coordinate Method the clusters are built from individual nucleons interacting via an effective nucleon-nucleon interaction; the total wave function is antisymmetrized. Fermionic Molecular Dynamics (FMD) goes beyond pure cluster models. It is a microscopic many-body approach using a Gaussian wave packet basis that includes the harmonic oscillator shell model an…
Heavy ion collisions at RHIC are well described by the (nearly ideal) hydrodynamics. In the present paper we study propagation of perturbations induced by moving charges (jets) on top of the expanding fireball, using hydrodynamics and (dual) magnetohydrodynamics. Two experimentally observed structures, called a "cone" and a "hard ridge", have been discovered in dihadron correlation function with large-$p_t$ trigger, while "soft ridge" is a similar structure seen without hard trigger. All three can be viewed as traces left by a moving charge in matter, on top of overall expansion. A puzzle is w…
The scalar-isoscalar mode of QCD becomes lighter/nearly massless close to the chiral transition/second-order critical point. From nuclear physics we know that this mode is the main responsible for the attractive part of the nucleon-nucleon potential at inter-particle distances of 1-2 fm. Therefore one expects that close to the critical point there is a long-range strong attraction among nucleons. Using a Walecka-Serot model for the NN potential we study the effects of the critical point in a finite system of nucleons and mesons by solving classical Molecular Dynamics+Langevin equations for the…
The Borromean $^6$He nucleus is an exotic system characterized by two `halo' neutrons orbiting around a compact $^4$He (or $α$) core, in which the binary subsystems are unbound. The simultaneous reproduction of its small binding energy and extended matter and point-proton radii has been a challenge for {\em ab initio} theoretical calculations based on traditional bound-state methods. Using soft nucleon-nucleon interactions based on chiral effective field theory potentials, we show that supplementing the model space with $^4$He+$n$+$n$ cluster degrees of freedom largely solves this issue. We an…
We investigate the evolution of clustering structure through the momentum distributions in the $^{8-10}$Be isotopes. The nucleon dynamics within the inter-cluster antisymmetrization are discussed via the momentum distribution of a Brink type $α$-$α$ wave function. For the state with a small $α$-$α$ distance, we observe a significant depression with a dip structure at zero-momentum and an enhanced tail at relatively higher momentum region. In addition, we find the "cluster structure" in the intrinsic frame of momentum space, which is complementary to its significant $α$-cluster dissolution in t…
We explore possible observable signatures of $α$ clustering of light nuclei in ultra-relativistic nuclear collisions involving ${}^{7,9}$Be, ${}^{12}$C, and ${}^{16}$O. The clustering leads to specific spatial correlations of the nucleon distributions in the ground state, which are manifest in the earliest stage of the ultra-high energy reaction. The formed initial state of the fireball is sensitive to these correlations, and the effect influences, after the collective evolution of the system, the hadron production in the final stage. Specifically, we study effects on the harmonic flow in coll…
We present lattice calculations of the low-lying spectrum of $^{12}$C using a simple nucleon-nucleon interaction that is independent of spin and isospin and therefore invariant under Wigner's SU(4) symmetry. We find strong signals for all excited states up to $\sim 15$~MeV above the ground state, and explore the structure of each state using a large variety of $α$ cluster and harmonic oscillator trial states, projected onto given irreducible representations of the cubic group. We are able to verify earlier findings for the $α$ clustering in the Hoyle state and the second $2^+$ state of $^{12}$…
11Li is a Borromean nucleus, where two out of three objects as 9Li + n and two neutrons independently do not form bound systems. Two neutrons should form a di-neutron cluster in the nuclear field generated by the 9Li core nucleus. We treat di-neutron clustering by solving the two neutron relative wave function precisely by using the bare nucleon-nucleon interaction so that the spatial clustering structure is obtained quantitatively within the whole 11Li nucleus. This di-neutron clustering is an essential dynamics to form the halo structure by making a compact di-neutron cluster, which distribu…
Although SU(2) isospin symmetry is generally assumed in the basic theory of the strong interaction, a number of significant violations have been observed in scattering and bound states of nucleons. Many of these violations can be attributed to the electromagnetic interaction but the question of how much of the violation is due to it remains open. A schematic model based on clustering of quarks in the interior of the confinement region of the two-nucleon system is introduced and evaluated. In this model the Coulomb interaction is the source of all isospin breaking. It draws on a picture of the …
A Universal Four-Fermion Formation Framework and Odd-Even Staggering in $α$ Decay
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University（中山大学中法核工程与技术学院）；Department of Physics, Royal Institute of Technology（皇家理工学院物理系）
An abrupt change in $α$ decay systematics around the N=126 neutron shell closure is discussed. It is explained as a sudden hindrance of the clustering of the nucleons that eventually form the $α$ particle. This is because the clustering induced by the pairing mode acting upon the four nucleons is inhibited if the configuration space does not allow a proper manifestation of the pairing collectivity.
Collisions of light and heavy nuclei in relativistic heavy-ion collisions have been shown to be sensitive to nuclear structure. With a proposed $^{16}\mathrm{O}^{16}\mathrm{O}$ run at the LHC and RHIC we study the potential for finding $α$ clustering in $^{16}$O. Here we use the state-of-the-art iEBE-VISHNU package with $^{16}$O nucleonic configurations from {\rm ab initio} nuclear lattice simulations. This setup was tuned using a Bayesian analysis on pPb and PbPb systems. We find that the $^{16}\mathrm{O}^{16}\mathrm{O}$ system always begins far from equilibrium and that at LHC and RHIC it ap…
Following the idea of nucleon clustering and light-nuclei production in relativistic heavy-ion collisions close to the QCD critical-end point, we address the quantum effects affecting the interaction of several nucleons at finite temperature. For this aim we use the $K$-harmonics method to four-nucleon states ($α$ particle), and also develop a novel semiclassical "flucton" method at finite temperature, based on certain classical paths in Euclidean time, and apply it to two- and four-particle configurations. To study possible effects on the light-nuclei production close to the QCD critical poin…
The cumulants of baryon multiplicity distribution in relativistic heavy-ion collisions (HICs) have attracted considerable attention recently. It has been conjectured that they may serve as a promising observable to detect the critical end point in the QCD phase diagram, while the cumulants in HICs at intermediate energies have not been widely studied to date. How to interpret the cumulants data at intermediate energies and compare with the data at relativistic energies is now being actively discussed. Both meam-field potential and clustering are highly important to HICs at intermediate energie…
Fluctuations of the baryon number in relativistic heavy-ion collisions are a promising observable to explore the structure of the QCD phase diagram. The cumulant ratios in heavy ion collisions at intermediate energies ($\sqrt{s_{\textrm{NN}}} < 7$ GeV) have not been studied to date. We investigate the effects of mean field potential and clustering on the cumulant ratios of baryon and proton number distributions in Au+Au collisions at beam energy of 1.23 GeV$/$nucleon as measured by the HADES Collaboration at GSI. Ultrarelativistic Quantum Molecular Dynamics (UrQMD) and the JAM model are used t…
Nuclear clustering describes the appearance of structures resembling smaller nuclei such as alpha particles (4He nuclei) within the interior of a larger nucleus. While clustering is important for several well-known examples, much remains to be discovered about the general nature of clustering in nuclei. In this letter we present lattice Monte Carlo calculations based on chiral effective field theory for the ground states of helium, beryllium, carbon, and oxygen isotopes. By computing model-independent measures that probe three- and four-nucleon correlations at short distances, we determine the…
Subthreshold events (pion production, for instance) at energies $E< m_π$ are treated as rare events. The associated multiplicity distribution $P_c (n)$ and the unconstrained distribution P(n), when there is no rare event-trigger, are related by the model independent relation $P_c (n) = {n^2 \over \average{n^2}} P(n)$. This relation, and in particular its improved version inspired in clustering of nucleons, is in fair agreement with data. Moreover, it allows to extract from the multiplicity data information on the number of nucleons involved in meson production processes, occuring at energies b…
The clustering of nucleons in nuclei is a widespread but elusive phenomenon for study. Here, we wish to highlight the variety of theoretical approaches, and demonstrate how they are mutually supportive and complementary. On the experimental side, we describe recent advances in the study of the classic cluster nucleus 24Mg. Also, recent studies of clustering in nuclei approaching the neutron drip line are described. In the region near N/Z=2, both theory and experiment now suggest that multi-centre cluster structure is important, in particular for the very neutron rich beryllium isotopes.
The formation of deuterons in heavy-ion collisions at relativistic energies is investigated by employing two recently advanced models -- the Minimum Spanning Tree (MST) method and the coalescence model by embedding them in the PHQMD and the UrQMD transport approaches. While the coalescence mechanism combines nucleons into deuterons at the kinetic freeze-out hypersurface, the MST identifies the clusters during the different stages of time evolution. We find that both clustering procedures give very similar results for the deuteron observables in the UrQMD as well as in the PHQMD environment. Mo…
We discuss the role of clustering on monopole, dipole, and quadrupole excitations in nuclei in the framework of the ab initio symmetry-adapted no-core shell model (SA-NCSM). The SA-NCSM starts from nucleon-nucleon potentials and, by exploring symmetries known to dominate the nuclear dynamics, can reach nuclei up through the calcium region by accommodating ultra-large model spaces critical to descriptions of clustering and collectivity. The results are based on calculations of electromagnetic sum rules and discretized responses using the Lanczos algorithm, that can be used to determine response…
A collision system scan involving $α$-clustered $^{12}$C and $^{16}$O is studied by using a multiphase transport model for central collisions at $\sqrt{s_{NN}} = 6.37$ TeV. Background subtracted away-side dihadron azimuthal correlation is performed via the zero yield at minimum (ZYAM) method from raw signals, and the quantitative parameters, such as RMS width and Kurtosis, seem nicely follow the $A^{-1/3}$ law of the system size if the nucleus has the normal Woods-Saxon nucleon distribution. However, for $α$-clustering light nuclei, specifically for $^{12}$C and $^{16}$O, the RMS width and Kur…
Event-by-event (ebe) fluctuations in mean pseudorapidity values of relativistic charged particles in full phase space is studied by analysing experimental data on $^{16}O-AgBr$ collisions at 14.5A, 60A, and 200A GeV/c and $^{32}S-AgBr$ collisions at 200A GeV/c. The findings are compared with the prediction of A Multi-Phase Transport(\amm) model and those obtained from the analysis of correlation free Monte-Carlo events. Fluctuations in mean pseudorapidity distributions are noticed to be in excess to that expected from the statistically independent particle emission. The observed dependence of …
The excited states of Be-12, Be-14 and B-15 were studied by an antisymmetrized molecular dynamics method. The theoretical results reproduced the energy levels of recently measured excited states of Be-12, and also predicted rotational bands with innovative clustering structures in Be-12, Be-14 and B-15. Clustering states with new exotic clusters (He-6, He-8 and Li-9) were theoretically suggested. One new aspect in very neutron-rich nuclei is a 6-nucleon correlation among 4 neutrons and 2 protons, which plays an important role in the formation of He-6 clusters during clustering: 8He + 6He of Be…
The main purpose of the present paper is to discuss whether or not the collective flows in heavy-ion collision at Fermi energy can be taken as a tool to investigate the cluster configuration in light nuclei. In practice, within an Extended Quantum Molecular Dynamics model, four $α$-clustering (linear chain, kite, square, and tetrahedron) configurations of $^{16}$O are employed in the initialization, $^{16}$O+$^{16}$O around Fermi energy (40 - 60 MeV$/$nucleon) with impact parameter 1 - 3 fm are simulated, the directed and elliptic flows are analyzed. It is found that collective flows are influ…
Accurate quantum Monte Carlo calculations of ground and low-lying excited states of light p-shell nuclei are now possible for realistic nuclear Hamiltonians that fit nucleon-nucleon scattering data. At present, results for more than 30 different (J^pi;T) states, plus isobaric analogs, in A \leq 8 nuclei have been obtained with an excellent reproduction of the experimental energy spectrum. These microscopic calculations show that nuclear structure, including both single-particle and clustering aspects, can be explained starting from elementary two- and three-nucleon interactions. Various densit…
Through $^{12}$C + $^{197}$Au collisions at $\sqrt{s_{NN}} =$ 200 GeV using a multiphase transport (AMPT) model, the azimuthal angle dependences of the Hanbury Brown-Twiss (HBT) radii relative to the second- and third-order participant plane from $π$-$π$ correlations are discussed. Three initial geometric configurations of $^{12}$C, namely three-$α$-cluster triangle, three-$α$-cluster chain and Woods-Saxon distribution of nucleons, are taken into account, and their effects on the correlations are investigated. The ratio of the third- to the second-order HBT radii $R_{o(s),3}^2/R_{o(s),2}^2$ is…
We investigate the structures of $^{9,10}$Be and $^{10}$B with the tensor-optimized shell model (TOSM) using the effective interaction based on the bare nucleon-nucleon interaction AV8$^\prime$. The tensor correlation is treated in TOSM with the full optimization of 2p2h configurations including high-momentum components. The short-range correlation is described in the unitary correlation operator method (UCOM). It is found that the level orders of the low-lying states of $^{9,10}$Be and $^{10}$B are entirely reproduced. For $^9$Be, ground band states are located relatively in higher energy tha…
The isospin dependence of in-medium nuclear effective interactions is a fundamental issue in nuclear physics and has broad ramifications in astrophysics. Its uncertainties, especially the difference of neutron-proton interactions in the isosinglet and isotriplet channels, affect significantly the density and momentum dependence of the isovector single-nucleon potential and nucleon-nucleon short-range correlation in neutron-rich matter. Consequently, the neutron-proton effective mass splitting and the density dependence of nuclear symmetry energy are still rather uncertain. Heavy-ion reactions …
Electromagnetic fields produced in relativistic heavy-ion collisions are affected by the asymmetry of the projectile-target combination as well as the different initial configurations of the nucleus. In this study, the results of the electric and magnetic fields produced for different combinations of ions, namely $^{12}$C + $^{197}$Au, $^{24}$Mg + $^{197}$Au, $^{64}$Cu + $^{197}$Au, and $^{197}$Au + $^{197}$Au at $\sqrt{s_{NN}} = 200$ GeV are demonstrated with a multi-phase transport model (AMPT). The configuration of the distribution of nucleons of $^{12}$C is initialized by a Woods-Saxon sph…
Systematic study of flow of protons and light clusters in intermediate-energy heavy-ion collisions with momentum-dependent potentials
Joint Institute for Nuclear Research（俄罗斯杜布纳联合核子研究所）；SUBATECH, Nantes University, IMT Atlantique, IN2P3/CNRS（南特大学SUBATECH实验室）；Institut für Kernphysik（法兰克福大学核物理研究所）；Frankfurt Institute for Advanced Studies（法兰克福高等研究院）；GSI Helmholtzzentrum für Schwerionenforschung GmbH（德国重离子研究中心）；Helmholtz Research Academy Hessen for FAIR (HFHF)（黑森亥姆霍兹FAIR研究学院）；Institut für Theoretische Physik, Johann Wolfgang Goethe University（约翰·沃尔夫冈·歌德大学理论物理研究所）；Dipartimento di Fisica e Astronomia “E. Majorana”, Università degli Studi di Catania（卡塔尼亚大学物理与天文学系）
The richness of phenomena occurring in heavy-ion collisions calls for microscopic approaches where the motion of each nucleon is treated quantum mechanically. The most popular microscopic approach for low-energy collisions between atomic nuclei is the time-dependent Hartree-Fock (TDHF) theory, providing a quantum mean-field dynamics of the system. The TDHF approach and some of its extensions are used to predict the evolution of out-of-equilibrium nuclear systems. The formation of di-nuclear systems with a structure close to molecular states is investigated. In particular, lifetimes and exit ch…
Freeze-out model of light nuclei formation in heavy-ion collision transport
International Research Laboratory on Nuclear Physics and Astrophysics, Michigan State University and CNRS（核物理与天体物理学国际研究实验室，密歇根州立大学和法国国家科学研究中心）；Department of Physics and Astronomy and the Facility for Rare Isotope Beams, Michigan State University（物理与天文系及稀有同位素束设施，密歇根州立大学）
