International Journal of Modern Physics E

In this paper, we review recent developments that aim to achieve further understanding of the structure of atomic nuclei, by capitalizing on exact symmetries as well as approximate symmetries found to dominate low-lying nuclear states. The findings confirm the essential role played by the Sp(3, ℝ) symplectic symmetry to inform the interaction and the relevant model spaces in nuclear modeling. The significance of the Sp(3, ℝ) symmetry for a description of a quantum system of strongly interacting particles naturally emerges from the physical relevance of its generators, which directly relate to particle momentum and position coordinates, and represent important observables, such as, the many-particle kinetic energy, the monopole operator, the quadrupole moment and the angular momentum. We show that it is imperative that shell-model spaces be expanded well beyond the current limits to accommodate particle excitations that appear critical to enhanced collectivity in heavier systems and to highly-deformed spatial structures, exemplified by the second 0⁺ state in ¹²C (the challenging Hoyle state) and ⁸Be. While such states are presently inaccessible by large-scale no-core shell models, symmetry-based considerations are found to be essential.

In this paper, we have treated the light baryons as a relativistic three-body bound system. Inspired by lattice QCD calculations, we treated baryons as a spin-independent three-quark system within a relativistic three-quark model based on the three-particle Klein–Gordon equation. We presented the analytical solution of three-body Klein–Gordon equation with employing the constituent quark model based on a hypercentral approach through which two- and three-body forces are taken into account. Herewith the average energy values of the up, down and strange quarks containing multiplets are reproduced. To describe the hyperfine structure of the baryon, the splittings within the SU(6)-multiplets are produced by the generalized Gürsey Radicati mass formula. The considered SU(6)-invariant potential is popular "Coulomb-plus-linear" potential and the strange and non-strange baryons spectra are in general well reproduced.

The systematics of the Grodzins product rule (GPR) is studied from the perspective of the valence-proton and neutron product N_pN_n in the major shell space Z = 50–82, N = 82–126. The variation of nuclear structure from vibrator to deformed rotor is discussed. The Grodzins product shows more dependence on N_pN_n in the N ≤ 104 region, as it is a region of deformed nuclei. We present here for the first time the dependence of GPR on the N_pN_n product.

Transport properties of a charged droplet of weakly interacting particles in transverse electric field are investigated. Nonequilibrium, time-dependent distribution function which describes a process of the droplet transverse evolution with constant entropy in the field is calculated. With the help of this distribution function, shear viscosity coefficients in the transverse plane are calculated as well. They are found to be dependent on the ratio of the potential energy of the droplet in the electric field to the kinetic energy of the droplet; for weakly interacting particles, this parameter is small. Additionally, these coefficients are time-dependent and change during the hydrodynamical state of the droplet's expansion. Applicability of the results to the description of initial states of quark–gluon plasma (QGP) obtained in high-energy interactions of nuclei is also discussed.

The pion-nucleus microscopic optical potential (OP), defined by the pion-nucleon scattering amplitude and by the generalized density distribution of a target nucleus that incudes internal degrees of freedom, is applied to construct the pion-nucleus differential cross-sections of elastic and inelastic scattering on the nuclei ²⁸Si, ⁵⁸Ni, ²⁰⁸Pb at T_lab = 291 MeV. Calculations are based on the relativistic wave equation and thus relativistic effects and distortions on the relative motion wave functions are taken into account. The respective experimental data are analyzed and the in-medium parameters of the elementary πN-amplitude are established and compared with those from the pion scattering on free nucleons.

The dependences of the shapes of transverse momentum distributions of the negative pions, produced in minimum bias 4He+¹²C collisions at a momentum of 4.2 GeV/c per nucleon, on collision centrality and fitting range of p_t were analyzed systematically. To study the change in slopes (temperatures) of the p_t spectra of π^- with changing collision centrality and fitting p_t range, the p_t spectra, extracted from the experimental data and quark–gluon string model (QGSM) calculations, were fitted by the one- and two-temperature Hagedorn and Boltzmann functions. Fitting of the experimental p_t distributions of π^- in both the whole p_t and p_t = 0.1–1.2 GeV/c intervals required the two-temperature functions for adequate description of spectra, in agreement with the previous findings for different sets of colliding nuclei and various energies. On the whole, the absolute values of the extracted temperatures were lower in case of fitting range p_t = 0.1–0.7 GeV/c as compared to the fitting interval p_t = 0.1–1.2 GeV/c. The one-temperature functions were sufficient for fitting satisfactorily the experimental p_t distributions of the negative pions in range p_t = 0.1–0.7 GeV/c. In contrast to the experimental temperatures, the extracted QGSM temperatures were not sensitive to collision centrality and fitting range of p_t. The collision centrality dependences of the temperatures of soft (p_t = 0.1–0.5 GeV/c) and hard (p_t = 0.5–1.2 GeV/c) components of the experimental p_t distributions of the negative pions in 4He+¹²C collisions at 4.2A GeV/c were studied separately. The extracted temperatures of both soft and hard components of p_t distributions of π^- depended on geometry (size) and degree of overlap of colliding nuclei in peripheral, semicentral and central 4He+¹²C collisions. The temperature of soft p_t component of the negative pions was consistently larger in semicentral and central 4He+¹²C collisions than that in peripheral interactions. The temperature of hard p_t component of π^- in 4He+¹²C collisions decreased consistently with an increase in collision centrality. The physical interpretations of the observed centrality dependences of temperature (T) of soft and hard p_t components of the negative pions in 4He+¹²C collisions were given.

The quark exchange formalism is formulated to calculate the quark momentum distribution in the iso-scalar Lithium nucleus. Then by boosting the nucleus to an infinite momentum frame, the Lithium structure function is evaluated at different nucleon "sizes", i.e., b = 0.7, 0.8, 0.9 and 1 fm and the Bjorken scale (x) values. It is shown that the Lithium structure function becomes narrower, and it is pushed to the smaller x values, as the nucleon size is increased. Similar to our previous works for three nucleon systems, the Lithium nucleus European muon collaboration (EMC) ratio decreases, as we increase the x and b values and it shows larger effect, with respect to the free nucleon and three nucleons iso-scalar nucleus. On the other hand, present calculation of the EMC ratio for Lithium nucleus shows a good agreement with the corresponding NMC data, which is available for 1.4 × 10^-4 ≤ x ≤ 0.65. Since the atomic number is still small (A = 6), in this work as usual, we ignore the possibility of simultaneous exchange of quarks between more than two nucleons, which can be important as one moves to the heavy nuclei. Although, according to Hen et al., in the neutron rich nuclei the protons have a greater probability than neutrons to have momentum greater than the Fermi momentum, the three-body contribution may be suppressed.

This article is an extension of our recent studies of Ψ and ϒ production cross-sections in proton–proton collisions at the LHC with to E = 13 TeV.

The U(6)-phonon model of nuclear collective motion with the semi-direct product structure [HW(21)]U(6) is obtained as a hydrodynamic (macroscopic) limit of the fully microscopic proton–neutron symplectic model (PNSM) with Sp(12, R) dynamical group. The phonon structure of the [HW(21)]U(6) model enables it to simultaneously include the giant monopole and quadrupole, as well as dipole resonances and their coupling to the low-lying collective states. The U(6) intrinsic structure of the [HW(21)]U(6) model, from the other side, gives a framework for the simultaneous shell-model interpretation of the ground state band and the other excited low-lying collective bands. It follows then that the states of the whole nuclear Hilbert space which can be put into one-to-one correspondence with those of a 21-dimensional oscillator with an intrinsic (base) U(6) structure. The latter can be determined in such a way that it is compatible with the proton–neutron structure of the nucleus. The macroscopic limit of the Sp(12, R) algebra, therefore, provides a rigorous mechanism for implementing the unified model ideas of coupling the valence particles to the core collective degrees of freedom within a fully microscopic framework without introducing redundant variables or violating the Pauli principle.

We have studied the K* production within a multi-phase transport model (AMPT) for Au+Au collisions at to understand the hadronic re-scattering effect on the measured yields of the resonance. The hadronic re-scattering of the K* decay daughter particles (π and K) will alter their momentum distribution thereby making it difficult to reconstruct the K* signal through the invariant mass method. An increased hadronic re-scattering effect thus leads to a decrease in the reconstructed yield of K* in the heavy-ion collisions. Through this simulation study, we argue that a decrease in K*/K ratio with the increase in collision centrality necessarily reflects the hadronic re-scattering effect. Since the re-scattering occurs in the hadronic phase and K* has a lifetime of 4 fm/c, we present a toy model-based discussion on using measured K*/K to put a lower limit on the hadronic phase lifetime in high energy nuclear collisions.