Abstract
HTML
Reference
Related
PDF
-
The investigation of the properties of isospin-asymmetric nuclear matter under extreme conditions is a timely issue in both nuclear physics and nuclear astrophysics [1−4]. It plays a crucial role in understanding the complex dynamic processes of heavy-ion collisions (HICs), nuclear structure, and formation and evolution of dense stars, such as neutron stars [5−7]. Over the past two decades, significant progress has been made in constraining the isospin-symmetric nuclear equation of state (EoS) at subnormal and normal densities through theoretical calculations and comparisons with experimental nuclear data. However, its density-dependent behavior, particularly in high-density regions, remains largely unclear, with uncertainty increasing rapidly as density rises [8−10]. Furthermore, the construction of advanced radioactive beam facilities and new HICs experiments in them, including the High Intensity heavy ion Accelerator Facility (HIAF) in China [11], the Facility for Antiproton and Ion Research (FAIR) in Germany [12], the Beam Energy Scan (BES) and fixed target (FXT) programs at the Relativistic Heavy Ion Collider (RHIC) in the United States [13], and the Nuclotron-based Ion Collider fAcility (NICA) in Russia [14], is expected to open up new opportunities for experimental and theoretical investigations in the higher energy and higher density EoS of isospin-asymmetric nuclear matter.
Charged-pion related observables are commonly used as sensitive probes for investigating the high-density asymmetric nuclear EoS in HICs at intermediate energies and have attracted considerable attention in recent years [15−22]. However, predictions from different hadronic transport models for charged-pion related observables differ, especially at high densities. For instance, pion yields and ratios, as well as rapidity and transverse momentum distributions predicted by these models, often lack consistency and fail to accurately reproduce experimental data [23−25]. In HICs, it is known that at intermediate energies, the pions are mostly produced from the decay of
$ \Delta $ (1232) particles. Therefore, the production, evolution, and decay of$ \Delta $ particles in the isospin-asymmetric nuclear medium are critical for accurately understanding and constraining the asymmetric nuclear EoS through experimental measurements and dynamic simulations [19,26−30].Regarding the cross sections of particle production, evolution, and decay used in the simulation of HICs, one can usually derive them from the Brueckner theory [31,32], Dirac-Brueckner theory [33,34], variational approach [35], and one-boson-exchange model [36−39]. They can also be parametrized from the comparison of theoretical calculations with experimental data [40−42]. With the help of the self-consistent relativistic BUU (RBUU) transport theory, isospin-dependent in-medium nucleon-nucleon (NN) elastic cross sections (
$ \sigma^{*}_{NN\rightarrow NN} $ ) have been systematically studied [43,44]. For$ \Delta $ -related cross sections, such asNNinelastic cross sections$ \sigma^{*}_{NN \rightarrow N\Delta} $ (hard-$ \Delta $ production), the soft-$ \Delta $ production$ \sigma^{*}_{N\pi \rightarrow \Delta} $ and$ \Delta $ absorption$ \sigma^{*}_{N\Delta \rightarrow NN} $ channels have been calculated within the framework of the RBUU approach in which the$ \sigma $ ,$ \omega $ ,$ \rho $ , and$ \delta $ meson fields are considered [19,26,27]. The calculated results not only confirm that these cross sections are energy-, density-, and isospin-dependent, but also indicate that the$ \delta $ meson field causes a splitting effect on the effective masses of nucleons and$ \Delta $ particles, leading to splitting in the cross sections of individual channels.Recently, charged-pion yields from Au+Au collisions at several GeV energies have been measured by the STAR and HADES Collaborations [20,24]. Although these beam energies are currently too high to accurately investigate nuclear symmetry energy using pion-related observables, they provide more precise experimental data that help improve the theoretical description of pion production in HICs, thereby enabling more accurate constraints on the EoS of high-density nuclear matter. However, a significant mismatch remains between the charged-pion yields calculated by various transport models and the experimentally measured values reported by the HADES collaboration [24]. By considering an isospin-dependent reduction factor on the
$ \Delta $ production, the charged-pion yields can be described properly [19,45]. To achieve a more accurate understanding of the dense nuclear EoS, it is essential to compare measured pion-related observables with transport model simulation results. In addition to the channels of single-$ \Delta $ production and absorption, other channels (e.g.,$ N\Delta $ elastic channels) should be self-consistently treated in the same transport model.In a previous study [46], the
$ N\Delta $ elastic cross section$ \sigma^{*}_{N\Delta\rightarrow N\Delta} $ was calculated within the RBUU approach; only the isoscalar$ \sigma $ and$ \omega $ meson exchanges were involved. Then, the isovector$ \rho $ meson exchange was further considered to investigate the contribution of the isovector field on$ \sigma^{*}_{N\Delta\rightarrow N\Delta} $ [47]. In the relativistic mean field theory, the bulk properties of nuclei, such as binding energy and charge radius, can be precisely predicted by introducing the isovector$ \rho $ meson field [48,49]. In addition, it has been pointed out that the$ \delta $ meson field plays a crucial role in accurately describing strongly isospin asymmetric matter at high densities in neutron stars, directly affecting the density dependence of the symmetry energy and causing a splitting of the Dirac mass for protons and neutrons in asymmetric matter [26,50−52]. For instance, Ref. [53] demonstrated that the inclusion of the$\delta$ meson field not only improves the accuracy of mass and radius predictions for finite nuclei but also influences the EoS at higher densities, resulting in much better agreement with heavy-ion collision data. Furthermore, the$\sigma^2\delta^2$ mixing terms in the nonlinear coupling of the effective Lagrangian significantly affect astrophysical observables, such as the radius and tidal deformability of neutron stars [54].In this study, based on the effective Lagrangian within the same framework of the RBUU microscopic transport theory, in which the scalar-isovector
$ \delta $ meson exchange is considered, we further studied the energy-, density-, and isospin-dependent$ N\Delta \rightarrow N\Delta $ cross sections more systematically.This paper is organized as follows. A brief review of the RBUU equation and analytic expressions of in-medium
$ N\Delta \rightarrow N\Delta $ cross sections are presented in Sec. II. Numerical results of total and individual$ \sigma^{*}_{N\Delta} $ , as well as the effective mass splitting effects on the cross section are presented in Sec. III. Conclusions and outlook are provided in Sec. IV. -
The same theoretical framework as that established in Refs. [27,43,44,46] was employed in this study. By using the closed time-path Green's function technique, which is extensively employed to process issues related to non-equilibrium systems [55], and incorporating the semi-classical and quasi-particle approximations, the RBUU equation for the
$ \Delta $ distribution function can be derived as [46]$ \begin{aligned}[b] &\left\{p_{\mu}\left[\partial_{x}^{\mu}-\partial_{x}^{\mu} \Sigma_{\Delta}^{v}(x) \partial_{v}^{p}+\partial_{x}^{v} \Sigma_{\Delta}^{\mu}(x) \partial_{v}^{p}\right]+m_{\Delta}^{*} \partial_{x}^{v} \Sigma_{\Delta}^{S}(x) \partial_{v}^{p}\right\} \\ & \times\frac{f_{\Delta}({\bf{x}}, {\bf{p}}, \tau)}{E_{\Delta}^{*}(p)}=C^{\Delta}(x, p). \end{aligned} $
(1) Here,
$ m_{\Delta}^{*} $ and$ f_{\Delta}({\bf{x}}, {\bf{p}}, \tau) $ represent the effective mass and distribution function of$ \Delta(1232) $ , respectively.$ \Sigma_{\Delta}^{S} $ and$ \Sigma_{\Delta}^{\mu,\nu} $ on the left side characterize the Hartree terms of the$ \Delta $ self-energies.$ C^{\Delta}(x, p) $ on the right side represents the collision term, which is determined by the collisional self-energy and is closely related to the in-medium elastic and inelastic cross sections.In the present study, we exploratorily introduced the scalar-isovector
$ \delta $ meson field in the effective Lagrangian, along with the scalar-isoscalar$ \sigma $ , vector-isoscalar$ \omega $ , and vector-isovector$ \rho $ meson fields, with the aim of understanding the impact of including the$ \delta $ meson field on the description of$ N \Delta \rightarrow N \Delta $ scattering. It should be noted that$ N\Delta $ elastic cross sections in free space can be understood primarily with the help of$ \pi $ meson exchanges. This is due to the long-range nuclear exchange characteristics of$ \pi $ mesons, which effectively provide cross section within free space, showing good agreement with Cugnon’s parametrization in the higher energy region. However, in the nuclear medium, other meson exchanges, such as$ \sigma $ ,$ \omega $ ,$ \rho $ , and$ \delta $ , become increasingly significant and dominant at higher densities.Thus, the effective Lagrangian can be written as
$ L=L_{F}+L_{I}, $
(2) where
$ L_{F} $ is the free Lagrangian density and$ L_{I} $ is for the interaction part,$ \begin{aligned}[b] L_{F}=\; & \bar{\Psi}\left[i \gamma_{\mu} \partial^{\mu}-m_{N}\right] \Psi+\bar{\Psi}_{\Delta \nu}\left[i \gamma_{\mu} \partial^{\mu}-m_{\Delta}\right] \Psi_{\Delta}^{\nu} \\ & +\frac{1}{2} \partial_{\mu} \sigma \partial^{\mu} \sigma+\frac{1}{2} \partial_{\mu} \vec{\delta} \partial^{\mu} \vec{\delta}-\frac{1}{4} F_{\mu \nu} \cdot F^{\mu v}-\frac{1}{4} \vec{L}_{\mu\nu} \cdot \vec{L}^{\mu \nu} \\ & -\frac{1}{2} m_{\sigma}^{2} \sigma^{2}-\frac{1}{2} m_{\delta}^{2} \vec{\delta}^{2} +\frac{1}{2} m_{\omega}^{2} \omega_{\mu} \omega^{\mu}+\frac{1}{2} m_{\rho}^{2} \vec{\rho}_{\mu} \vec{\rho}^{\mu}, \end{aligned} $
(3) $ \begin{aligned}[b] L_{I}=\; & g_{N N}^{\sigma} \bar{\Psi} \Psi \sigma+g_{N N}^{\delta} \bar{\Psi}\vec{\tau }\cdot \Psi \vec{\delta} -g_{N N}^{\omega} \bar{\Psi} \gamma_{\mu} \Psi \omega^{\mu}\\ & -g_{N N}^{\rho} \bar{\Psi} \gamma_{\mu} \vec{\tau } \cdot \Psi \vec{\rho }^{\mu} +g_{\Delta \Delta}^{\sigma} \bar{\Psi}_{\Delta} \Psi_{\Delta} \sigma+g_{\Delta \Delta}^{\delta} \bar{\Psi}_{\Delta} \vec{\tau } \cdot \Psi_{\Delta} \vec{\delta} \\ &-g_{\Delta \Delta}^{\omega} \bar{\Psi}_{\Delta} \gamma_{\mu} \Psi_{\Delta} \omega^{\mu}-g_{\Delta \Delta}^{\rho} \bar{\Psi}_{\Delta} \gamma_{\mu} \vec{\tau} \cdot \Psi_{\Delta} \vec{\rho }^{\mu}, \end{aligned} $
(4) where
$ F_{\mu \nu} \equiv \partial_{\mu} \omega_{v}-\partial_{v} \omega_{\mu} , L_{\mu \nu} \equiv \partial_{\mu} \vec{\rho}_{v}-\partial_{v} \vec{\rho}_{\mu} $ ,$ \psi $ is the Dirac spinor, and$ \psi_{\triangle} $ is the Rarita-Schwinger spinor.In this study, we adopted density-dependent coupling constants, which have been extensively applied in the calculation of both elastic and inelastic reaction channels. As a result, we provide a more accurate description of cross sections for
$ NN\rightarrow NN $ ,$ NN\rightarrow N\Delta $ , and$ N\pi \rightarrow \Delta $ [26,27,44,56]. Thus, it can be quantitatively parametrized as$ g_{q}(\rho_{b})=g_{q}(\rho_{0})f_{q}(u), \quad q=\sigma,\; \omega,\; \rho,\; \delta $
(5) where
$ u=\rho_{b}/\rho_{0}
Nucleon-Δ elastic cross section in isospin-asymmetric nuclear medium with inclusion of scalar-isovectorδmeson field
- Received Date:2025-03-09
- Available Online:2025-09-15
Abstract:The production, dynamic evolution, and decay of
