We systematically investigate the scaling properties of the transverse momentum spectra for pions, kaons, and protons in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=7.7$, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV, as well as in $\mathrm{U}+\mathrm{U}$ collisions at $\sqrt{s_{NN}}=193\mathrm{GeV}$, across different centrality classes, using experimental data from the collaborations at the Relativistic Heavy Ion Collider (RHIC). Universal scaling emerges when the particle transverse momentum spectra are scaled by global physical quantities, i.e., the average total particle multiplicity and mean transverse momentum, confirming recent scaling findings from the data at the Large Hadron Collider (LHC) by the ExTrEMe Collaboration. The scaling behavior breaks down in the high $p_{\mathrm{T}}$ region and in peripheral collisions. We provide a natural explanation for these observations in the framework of a blast-wave-type model built upon the Cooper-Frye formula. Furthermore, we demonstrate the equivalence between the scaling found by the ExTrEMe Collaboration and the Hwa-Yang scaling, which was proposed two decades ago.

We develop the theoretical framework for semi-inclusive deep-inelastic scattering on a polarized spin-1 target and apply it to scattering on the polarized deuteron with spectator nucleon tagging. In Part I (this article) we present the general form of the semi-inclusive cross section and polarization observables for the spin-1 target. A relativistically covariant formulation in terms of four-vectors and invariant polarization parameters is employed. The target polarization is described by a spin density matrix with vector and tensor polarization. The spin and azimuthal angle dependence of the semi-inclusive cross section is derived and parametrized in terms of invariant structure functions. To validate the result, the structure functions are expressed as photon-target helicity amplitudes with known symmetry properties. The expressions presented here are kinematic (no assumptions about particle production dynamics) and valid in all regions of the deep-inelastic final state (current and target fragmentation regions). In Part II (following article), we consider deep-inelastic scattering on the polarized deuteron with spectator nucleon tagging as a special case of target fragmentation. The semi-inclusive structure functions are computed by separating nuclear and hadronic structure, and the polarization observables are explored as functions of the tagged nucleon momentum.

We develop the theoretical framework for semi-inclusive deep-inelastic scattering on a polarized spin-1 target and apply it to scattering on the polarized deuteron with spectator nucleon tagging. In Part I (previous article), we present the general form of the semi-inclusive cross section and polarization observables for the spin-1 target. In Part II (this article), we consider deep-inelastic scattering on the polarized deuteron with spectator nucleon tagging as a special case of target fragmentation. Methods of light-front quantization are employed to separate nuclear and hadronic structure in the high-energy process and achieve a composite description. The light-front wave function of the polarized deuteron is obtained from a rotationally covariant three-dimensional wave function in the center-of-mass frame of the proton-neutron system. The tagged structure functions are computed in the impulse approximation. The momentum and spin distribution of the active nucleon are controlled by the deuteron polarization and the detected spectator momentum ($D/S$ wave ratio). The cross section and spin asymmetries are evaluated for general deuteron polarization (vector and tensor, longitudinal and transverse) as functions of the spectator momentum. Tensor-polarized spin asymmetries of order unity are achieved for spectator momenta of approximately 300 MeV, which select configurations with a large $D$ wave. Sum rules for the tagged spin structure functions are derived. The results can be used for simulations of spectator tagging in future polarized fixed-target experiments (Jefferson Lab) or at the Electron-Ion Collider.

An effective operator is exactly equivalent to the long-wavelength form of the $M1$ operator in transition matrix elements. It allows us to analytically and numerically analyze the $M1$ contribution to the ${}^4\mathrm{He}{(d,\gamma )}^6\mathrm{Li}$ reaction. Isoscalar $M1$ transitions from an initial $S$ wave are shown to be forbidden in radiative capture reactions when distortion is neglected in the initial state. An illustrative calculation in a three-body model with proton, neutron, and a structureless $\alpha$ interacting through effective forces leads to a negligible $M1 S$ factor at small energies. The dominant $M1$ contribution comes from transitions from an initial $S$ wave to isospin 1 components of the ${}^6\mathrm{Li}$ ground state. The importance of the orthogonality between initial and final states is emphasized. It is suggested that using this effective $M1$ operator in other models should clarify the origin of large discrepancies between $M1 S$ factors appearing in the literature.

We revisit the neutron-deuteron scattering using the wave-packet continuum discretization (WPCD) method with the Epelbaum-Krebs-Meißner (EKM) chiral nuclear force at various chiral orders. We rederive the permutation operator and solve the Faddeev-AGS equations directly, without rewriting the initial Faddeev kernel $t{G}_0$ or introducing pseudostates, thereby rendering the approach easily extendable to a relativistic framework. This is the first application of the EKM interaction within the WPCD approach. We find that, up to the next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$), the differential cross sections are well described, while the long-standing $A_y$ puzzle remains unresolved, consistent with previous studies. The overall agreement provides a useful benchmark for the present framework and supports its application to future, more comprehensive calculations.

The symmetric double-$\alpha$ decay of ${}^{216,218}\mathrm{Rn}$ and ${}^{220,222}\mathrm{Ra}$ has been investigated at the ISOLDE facility. Radioactive molecular ${}^{220,222}\mathrm{RaF}$ ion beams were produced and delivered at an energy of 30 keV with intensities of approximately ${10}^3$ and ${10}^5\mathrm{pps}$, respectively. The ions were implanted into a thin self-supported carbon foil surrounded by four double-sided silicon strip detectors. Three ${}^{14}\mathrm{C}$ cluster emission events from ${}^{222}\mathrm{Ra}$ were observed. The measured branching ratio, $B=[10.1_{-6.4}^{+7.8}\left(\mathrm{stat}\right)\pm 1.1\left(\mathrm{syst}\right)]\times {10}^{-10}$, is consistent with the adopted value. For the simultaneous symmetric double-$\alpha$ decay ($E_{{\alpha }_1}=E_{{\alpha }_2}$ and ${\theta }_{{\alpha }_1{\alpha }_2}={180}^{\circ }$), no excess events were observed in the region of interest. Consequently, experimental upper limits on the branching ratios were derived at the level of ${10}^{-7}$ and ${10}^{-9}$ for the ${}^{220}\mathrm{Ra}$ and ${}^{222}\mathrm{Ra}$ decay chains, respectively.

The ${}^3\mathrm{He}$ puzzle, namely, that ${}^3\mathrm{He}$ exhibits anomalously higher kinetic energies per nucleon than tritons ($t$) and $\alpha$ particles in the multifragmentation processes, remains a prominent open issue in the field of heavy-ion collision, as existing interpretations rely primarily on the measured energy spectra and lack constraints from multiple experimental observables. In this work, focusing on the ${}^3\mathrm{He}$ puzzle in comparison to $t$, the differences between $t$ and ${}^3\mathrm{He}$ in in-plane/out-of-plane emission dynamics and emission chronology from the necklike region in the midcentral ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ collisions at 35 MeV/u are experimentally investigated using the two-particle correlation functions. It is found that, compared to $t, {}^3\mathrm{He}$ shows stronger in-plane collective motion, weaker out-of-plane collective motion, and delayed emission time, collectively indicating that, compared to $t, {}^3\mathrm{He}$ tends to be emitted in a less violent collision process or the ${}^3\mathrm{He}$ production is suppressed in a more violent process from an inverse perspective. This work offers novel insights into the ${}^3\mathrm{He}$ puzzle by combining dynamical and chronological observables.

We investigate microscopically systematic behavior of shape coexistence in even-even ${}^{106–120}\mathrm{Sn}$ isotopes in the framework of the shell model. The two-particle and two-hole (2p-2h) proton excitations across the $Z=50$ closed shell gap are essential to describe the deformed intruder states built on the first excited $0_2^{+}$ state, which intrudes into the spherical normal states, indicating the shape coexistence. In addition, the neutron excitations to the $1h_{11/2}$ orbit are important for a parabolic behavior of the intruder band with a minimum in the excitation energy near the neutron midshell $N=66$ point. The calculations predict that the crossing between the intruder and normal bands occurs between spin $I=4$ and $I=6$. From the potential energy surface (PES) with particle-number and angular-momentum projections, collective wave function, and quadrupole moments, it is shown that the calculations describe shape coexistence structure for the midshell Sn nuclei.

Sub-barrier photo-induced fission in ${}^{236}\mathrm{U}(\gamma ,f)$ is investigated within the nonequilibrium Green's function method. A model space for the fission process is constructed by superposing Skyrme-Hartree-Fock wave functions along the fission path allowing the particle-hole excitation. Then, the transition from the photoabsorption channel to the fission channel is described by the nonequilibrium Green's function formalism. The calculated fission cross section in the incident gamma-ray energy range $5\mathrm{MeV}\le E_{\gamma }\le 6\mathrm{MeV}$ reproduces the overall behavior of the experimental data, including the suppression below the fission barrier. An eigenchannel analysis of the wave propagation in the present fission model space is also performed, and the first eigenchannel is found to dominate the fission probability. This result supports the Bohr-Wheeler transition-state picture from a microscopic viewpoint.

The deformation properties within the ${}^{98–106}\mathrm{Ru}$ even-even isotopic chain, are investigated by means of the covariant density functional theory with a density-dependent point-coupling x parametrization. The considered nuclei are found to exhibit very shallow prolate and triaxial ground-state deformation. This information is used to ascertain their dynamical behavior within prolate $\gamma$-stable and $\gamma$-unstable instances of a phenomenological Bohr-Mottelson Hamiltonian with an octic potential in the axial deformation variable. The comparative study of the low-lying collective states, revealed the presence of a shape phase transition from low to high deformation, as well as evidence of shape coexistence and mixing between spherical vibrator, $\gamma$-unstable or prolate configurations in ground and excited states. It is also shown that the effect of shape coexistence and mixing on the $\gamma$-band states can account to some extent for the typical $\gamma$-unstable staggering even in prolate $\gamma$-stable conditions.

Based on the octupole core-quasiparticle coupling model, the effect of different octupole-correlated configurations—$\pi (2p_{3/2},1g_{9/2}), \pi (2d_{5/2},1h_{11/2})$, and $\pi (2f_{7/2},1i_{13/2})$—on wobbling motion in octupole deformed odd-$A$ nuclei is systematically investigated. Spin coherent state plots are constructed to visualize the angular momentum geometry. The results demonstrate that wobbling motion can occur in both positive- and negative-parity bands built on these octupole-correlated orbitals. As the spin increases, a transition from transverse wobbling to longitudinal wobbling is revealed, with the critical spin increasing with the particle angular momentum. This shows the crucial role of high-$j$ particles in stabilizing the transverse wobbling mode.

The Ichimura-Austern-Vincent (IAV) sum-rule formalism for inclusive breakup reactions $a+A\to b+\mathrm{anything}$ treats the detected fragment $b$ as a spectator by replacing its interaction with the target by an optical potential. This assumption becomes questionable when $b$ is a loosely bound composite particle such as a deuteron. I derive a generalization that removes the spectator approximation and retains $b\text{"s}$ internal degrees of freedom, providing state-resolved inclusive cross sections. Within the distorted-wave Born approximation, all nonspectator effects enter through the source function via the operator $V_{bA}-U_{bA}$. The exact sum rule involves the full $x+A$ resolvent ${(E_{x,0}^{+}-H_{xA})}^{-1}$, while a single-channel IAV-like expression is recovered only when the explicit target dependence of $V_{bA}$ is neglected; post-prior equivalence is preserved in both cases, since $V_{bA}$ cancels in the difference between the post- and prior-form residual interactions, and is verified explicitly in the single-channel limit. A key conceptual finding is that the standard IAV result for structureless $b$ corresponds, under closure and a reduction of $b\text{"s}$ internal space, to the total inclusive cross section summed over all of $b\text{"s}$ internal states, rather than the cross section for $b$ in a specific state. An operator-level estimate for $b=d$ on ${}^{208}\mathrm{Pb}$ shows that the nonspectator correction is not a small perturbation at the nuclear surface. The present work is purely formal: it establishes the theoretical framework and identifies the relevant operators, while quantitative assessment of the cross-section impact awaits a full numerical evaluation of the source integrals.

A systematic Bayesian analysis of experimentally measured nuclear level densities for ${}^{204,206,207,208}\mathrm{Bi}$ and ${}^{209}\mathrm{Po}$ nuclei has been performed to investigate the damping of shell effects with increasing excitation energy. The analysis indicates high shell-damping rates (${\gamma }_0$ = $0.{59}_{-0.10}^{+0.22}, 0.{53}_{-0.06}^{+0.16}$, and $0.{49}_{-0.07}^{+0.20} {\mathrm{MeV}}^{-1}$) in neutron-deficient bismuth nuclei (${}^{204}\mathrm{Bi},{}^{206}\mathrm{Bi}$, and ${}^{207}\mathrm{Bi}$), implying a rapid attenuation of shell effects as the excitation energy increases. Systematic investigation incorporating different treatments of pairing correlations and spin cutoff prescriptions demonstrate that the deduced shell-damping rates have minimal model dependence. These findings provide valuable insight into the excitation energy dependence of shell structure near the closed-shell nuclei.

The proton evaporation spectrum has been measured in an exclusive experiment for ${}^{60}\mathrm{Co}$, which is populated in the ${}^{57}\mathrm{Fe}$(${}^7\mathrm{Li},\alpha$) reaction and decays into ${}^{59}\mathrm{Fe}+\mathrm{p}$. A statistical-model analysis of the coincident proton spectra enabled the extraction of the excitation-energy-dependent nuclear level density (NLD) of ${}^{59}\mathrm{Fe}$. The extracted NLD shows good consistency with available microscopic predictions and the RIPL-3 evaluated data, thereby providing an important experimental benchmark for this short-lived nucleus. It was further used to estimate the Maxwellian-averaged neutron-capture cross section relevant for astrophysical reaction rates, particularly for the ${}^{58}\mathrm{Fe}$ nucleus.

The thermal modification of the exclusive decay $K_1^{+}\left(1270\right)\to {\pi }^{+}{\pi }^{-}{K}^{+}$ in a hot hadronic medium is studied. The decay amplitude is constructed from effective hadronic interactions dominated by the $\rho$- and $K^{*}$-pole contributions, thereby enabling a Dalitz-level analysis of the three-body decay in the medium. Thermal effects associated with partial chiral-symmetry restoration are incorporated through a phenomenological interpolation toward vector–axial-vector degeneracy near the chiral crossover. As the temperature increases, the reduction of the parent $K_1$ mass strongly compresses the available three-body phase space, leading to substantial deformation of the Dalitz distribution and invariant-mass spectra, as well as to a pronounced suppression of ${\mathrm{\Gamma }}_{K_1\to {\pi }^{+}{\pi }^{-}{K}^{+}}\left(T\right)$. As a further step toward future transport-based phenomenological comparisons, Normalized shape observables are introduced to quantify the thermal evolution of the $K^{*}$-dominated $\pi K$ region, the upper-edge weight of the $\pi \pi$ spectrum, and the compactification of the Dalitz population. The dominant effect identified in the present framework is therefore kinematic: thermal phase-space reduction in the strange axial-vector channel. These results suggest that the exclusive $K_1\left(1270\right)$ channel may provide a useful qualitative probe of in-medium strange axial-vector dynamics near the pseudocritical region.

A Bayesian analysis of the astrophysical $S$ factor for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction is presented using available low-energy information at carbon-carbon relative energies $E<3.5\mathrm{MeV}$, including direct measurements and recent inverse-kinematics data. The goal of the global Bayesian fit is not to reproduce the local resonance-by-resonance structure of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system, but rather to constrain the smooth global component of $S^{*}\left(E\right)$ using the direct and inverse-kinematics constraints. A conservative uncertainty model is adopted in order to combine heterogeneous datasets with different systematic uncertainties and normalizations. The inference is performed for $y\left(E\right)={log}_{10}{S}^{*}\left(E\right)$, which reduces the dynamic range of the data and leads to approximately Gaussian uncertainties. The function $y\left(E\right)$ is represented by a quadratic polynomial, and the posterior distribution of its coefficients is determined. Propagation of this posterior gives the median smooth curve and the corresponding $68\%$ posterior interval for $S^{*}\left(E\right)$. The resulting posterior $S^{*}\left(E\right)$ lies systematically below the traditional FCZ75 reference normalization over the energy interval considered. For example, at $E=1.5\mathrm{MeV}$ the median posterior value is $S^{*}\left(1.5\mathrm{MeV}\right)=1.20\times {10}^{16}\mathrm{keV}\mathrm{b}$, whereas the Fowler–Caughlan–Zimmerman reference value is $S_{\mathrm{FCZ}75}^{*}=3.0\times {10}^{16}\mathrm{keV}\mathrm{b}$ [W. A. Fowler, G. R. Caughlan, and B. A. Zimmerman, Annu. Rev. Astron. Astrophys. 13, 69 (1975)]. Thus, the updated posterior favors a lower smooth low-energy trend and does not support a sharp increase of $S^{*}\left(E\right)$ as the energy decreases. The astrophysical consequence of the analysis is assessed through the thermonuclear reaction rate $N_A\langle \sigma v\rangle$. The resulting median rate is lower than the CF88 analytic rate [G. R. Caughlan and W. A. Fowler, At. Data Nucl. Data Tables 40, 283 (1988)] over the temperature interval considered, with $R_{\mathrm{present}}/R_{\mathrm{CF}88}\simeq 0.33–0.46$ for $0.3\le T_9\le 1.2$. At higher temperatures, where the Gamow window extends above the upper end of the adopted integration interval, the calculated rate should be regarded as a truncated rate rather than a complete stellar rate.

Understanding the emergence of complex structures of nuclei from chiral effective field theory (EFT) is a central challenge. The large number of low-energy couplings (LECs) in the EFT expansion and the significant cost of ab initio many-body calculations render large-scale sensitivity studies of many-body observables computationally prohibitive, necessitating the use of emulators as low-cost surrogates. In this work, we study a Hartree-Fock emulator based on eigenvector continuation to investigate trends in nuclear charge radii of neutron-rich calcium isotopes. We systematically vary the five LECs entering the leading three-nucleon ($3N$) interactions, and demonstrate the precision of the emulator through cross validation over a wide parameter space. Our findings indicate that the large charge-radius increase from ${}^{48}\mathrm{Ca}$ to ${}^{52}\mathrm{Ca}$ is likely not explained by variations of the leading $3N$ couplings. This suggests that other effects, such as sensitivities to chiral two-nucleon interactions or neglected many-body effects, e.g., associated with nuclear collectivity, play an important role.

The impact of hexadecapole correlations on the low-energy spectroscopic properties of Th, U, and Pu nuclei, within the mass range $232\le A\le 240$, is studied systematically using the mapped $sdg$-IBM model. Fermionic input is obtained via the quadrupole-hexadecapole constrained Hartree-Fock-Bogoliubov approximation, based on the parametrization D1S of the Gogny energy density functional. The $sdg$-IBM Hamiltonian parameters are determined by mapping the quadrupole-hexadecapole fermionic mean-field potential-energy surfaces onto the corresponding bosonic surfaces. The low-energy spectra and transition strengths, obtained via the diagonalization of the $sdg$-IBM Hamiltonian, compare well with the available experimental data. It is shown that the effects of hexadecapole collectivity can be observed in high-spin yrast states with spins $J^{\pi }⩾{10}^{+}$. The mapped $sdg$-IBM improves the excitation energies of those states, as compared with the simpler $sd$-IBM model. The $sdg$-IBM also improves the description of the $E2$ transition strengths between high-spin yrast states and predicts strong $E4$ transitions from nonyrast $4^{+}$ states to the $0^{+}$ ground state.

Background: double-$\alpha$ decay is an exotic nuclear decay mode that offers a sensitive probe of quantum tunneling, nuclear clustering, and barrier penetration phenomena in medium-heavy and heavy nuclei. Despite being energetically allowed for a significant fraction of nuclei, this decay mode remains experimentally unexplored. An experimental proposal to search for double-$\alpha$ decay in ${}^{224}\mathrm{Ra}$ has been advanced at the ISOLDE facility of CERN, highlighting ongoing efforts to probe this rare decay channel.

Purpose: This work presents a systematic investigation of double-$\alpha$-decay half-lives and branching ratios across the nuclear chart, to identify energetically allowed and experimentally promising candidates, as well as evaluating the predictive performance of machine-learning approaches for such rare nuclear-decay modes.

Methods: A machine-learning framework based on the extreme gradient boosting regression algorithm is employed to predict double-$\alpha$-decay half-lives for nuclei with positive double-$\alpha$-decay energy in the atomic number range $Z=60–101$. The atomic mass table AME2016 is used to determine decay energetics, and the machine-learning predictions are compared with results from the modified generalized liquid drop model [Phys. Rev. C 104, 064604 (2021)] and the semiempirical formulation of Tretyak [Nucl. Phys. At. Energy 22, 121 (2021)]. In addition, a Geiger-Nuttall-type linear relation for double-α decay is extracted from the global systematics.

Results: The predicted double-$\alpha$-decay half-lives span many orders of magnitude and show good overall agreement with established theoretical models. The machine-learning approach successfully captures the dominant dependence of the decay probability on decay energy and Coulomb barrier penetration. At the same time, deviations are attributed to shell effects, deformation, and pairing correlations in specific nuclei. A simple empirical relation linking ${log}_{10}{T}_{1/2}^{2\alpha }$ to $Z_d/\sqrt{Q_{2\alpha }}$ is proposed for double-$\alpha$ decay. The calculated branching ratios confirm the rare nature of this decay mode and identify a subset of nuclei with enhanced probabilities that may be accessible to future experiments.

Conclusion: The present study demonstrates that machine-learning methods provide a reliable and efficient framework for exploring double-$\alpha$ decay across a wide region of the nuclear chart. The results offer quantitative guidance for experimental searches and establish machine learning as a powerful complementary tool for investigating rare and exotic nuclear-decay processes.

The ratio of the nuclear binding energy to the charge radius is systematically analyzed based on the experimental data of both. The linear relationship is observed between the ratios for nuclei $(Z,N)$ and those for $(Z,N-2)$. Together with this linear relationship, the nuclear charge radius can be extracted based on the radial basis function approach. The root-mean-square (rms) deviation with respect to 1063 measured nuclear charge radii reaches 0.007 fm, and the rms deviation of the test set for the neutron-rich region is 0.014 fm. The charge radii for 1333 nuclei are extracted, including 270 unmeasured nuclei. The new probe indicates that large deformations occur in atomic nuclei starting from neutron number $N=89$ in the isotope chains with proton numbers $Z=63$ and 65, based on the analysis of nuclear binding energy and charge radius. The proposed method enables charge-radius extraction from binding-energy data alone, providing a new approach for nuclei with unmeasurable charge radii.