Imaging the shape of atomic nuclei in high-energy nuclear collisions: a demonstration with Uranium-238

Updated on Tue, 2024-08-27 18:39. Originally created by chunjian on 2021-03-22 09:17.

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Abstract:
Atomic nuclei are ubiquitous self-organized many-body quantum systems, held together by strong nuclear forces within a femtometer-scale space. These complex systems manifest a diverse set of shapes across the nuclide chart, traditionally explored through non-invasive spectroscopic techniques at low energies $[1,2]$. Their instantaneous geometrical shapes are obscured by long-timescale quantum fluctuations involved in the measurement process, but can be deduced through model-dependent means [3, 4]. Here, we present a method to directly image the shapes of atomic nuclei by colliding them at ultrarelativistic speeds and analyzing the collective response of the resulting debris. Termed as "collective flow assisted nuclear shape imaging," our method captures an event-specific snapshot of the spatial matter distribution within the nuclei, which is preserved throughout the hydrodynamic expansion phase of the system, and is evident in the momentum distribution patterns of particles recorded in the detector. We benchmark the method in collisions of Uranium-238 nuclei, known for its elongated rugby-ball-like shape [5]. The shape parameters extracted from our analysis are remarkably consistent with previous extractions based on low-energy experiments. Our findings open new avenues for imaging the shapes of stable nuclei across the nuclide chart, in particular those with large uncertainties in their shape characteristics. By integrating this approach with established low-energy methods, we can better address the forefront questions in nuclear science across energy scales.

Figure1:

Figure 1: Methods for Determing the Nuclear Shape a, cartoon of the shape of a prolatedeformed nuclei. b, typical timescale associated with the rotational degrees-of-freedom for the nuclei. c, the quantum mechanical manifestation of the deformation in terms of the first rotational band of ${ }^{238} \mathrm{U}$. d, aligning the two nuclei with headon positions in the body-body configuration (top) and tip-tip configuration (bottom). e, appearance of two Lorentz-contracted nuclei in high-energy collisions and resulting 3D profile of the quark-gluon plasma. $\mathbf{f}$, the 3D profile of the quark-gluon plasma at the end of the hydrodynamic expansion prior to freezeout into particles, where the arrows indicate the velocities of fluid cells. g, the charged particle tracks measured in the detector. The timescales shown are in units of $\mathrm{fm} / c-$ the time for light to travel one femtometer.

Figure2:

Figure 2: Correlation between elliptic flow and radial flow. a, $v_2$ versus $\delta p_{\mathrm{T}} /\left\langle\left[p_{\mathrm{T}}\right]\right\rangle$ in $0-0.5 \%$ most central $\mathrm{Au}+\mathrm{Au}$ and $\mathrm{U}+\mathrm{U}$ collisions. $\mathbf{b}, \rho_2=\frac{\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle}{\left\langle v_2^2\right\rangle \sqrt{\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle}}$ as a function of collision centrality, quantifing the $v_2-\delta p_{\mathrm{T}}$ correlation. Representative elliptic-shaped overlap areas at different centrality are also displayed.

Figure3:

Figure 3: Constrain the Shape of ${ }^{238} \mathbf{U}$ The ratios of $\left\langle v_2^2\right\rangle(\mathbf{a}),\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ (b), and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ (c) as a function of centrality. The data are compared to the IP-Glasma+Music hydrodynamic model calculation assuming $\beta_{2 \mathrm{U}}=0.28$ (red) and $\beta_{2 \mathrm{U}}=0.25$ (blue), whose shaded bands denote the model uncertainties (Methods). The measurement are also shown for ratios of $\left\langle v_2^2\right\rangle$ (d) and $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ (d), and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ (e) in 0-5\% most central collisions. These ratios are compared to the hydrodynamic model as a function of $\beta_{2 \mathrm{U}}^2$ or $\beta_{2 \mathrm{U}}^3$ with four $\gamma_{\mathrm{U}}$ values. The colored quadrilaterals denote the allowed ranges of $\beta_{2 \mathrm{U}}^2$ or $\beta_{2 \mathrm{U}}^3$ from the intersection of data with calculations. $\mathbf{g}$ shows the constrained ranges of $\left(\beta_{2 \mathrm{U}}, \gamma_{\mathrm{U}}\right)$ from three observables separately, as well as the $68 \%$ and $95 \%$ confidence contours, obtained by either combining $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ (solid lines) or combining all three observables (dashed lines). The inset cartoon illustrates the three orthographic projections of the 3D shape of Uranium extracted from $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$.

Prospects and Conclusions:
Collective-flow-assisted imaging provides a new approach for probing the structure of atomic nuclei across the nuclide chart. The strength of the method lies in its ability to capture a rapid, almost instantaneous snapshot of nucleon spatial distribution, irrespective of the species involved in the collision. This stands in sharp contrast to traditional nuclear spectroscopy, where data complexity and interpretation depend on the specific prosition of a nucleus in the nuclide chart. Our method is particularly suited to precisely discerning shape differences between two species with similar mass numbers, ideally isobar pairs, which serve as additional constraints to nuclear structure models. A few potential applications are as follows:

- A promising direction is in the study of odd-mass nuclides where either $\mathrm{N}$ or $\mathrm{Z}$ is odd. While the shape of an odd-mass nucleus is presumed to resemble its adjacent even-even counterparts, its spectroscopic data tend to be more complex [11]. This complexity arises from the quantum mechanical coupling of the single unpaired nucleon's angular momentum with the nuclear core. In our highenergy approach, the shape of odd-mass nuclei can be extracted by comparing their flow observables to chosen neighboring even-even nuclei with wellestablished shapes.

- Additionally, our method offers a novel approach to investigate the elusive octupole and hexadecapole deformations in nuclei, which are less common and generally weaker than the quadrupole deformations [51]. The presence of these deformations should manifest in higher-order flow harmonics [25], specifically the triangular and quadrangular flows, which are routinely measured in high-energy experiments.

- Our method is particularly effective in studying dynamic deformations in certain nuclear species, where the shape fluctuates due to nucleon motion within the nucleus. Understanding these "soft" nuclei is at the forefront of nuclear physics research, requiring sophisticated experimental and theoretical efforts. By leveraging multi-particle correlations, routinely measured in high-energy collisions [52], our method could distinguish between average deformation and these dynamic shape fluctuations [40].

- Our method also has potential for advancing the experimental search for $0 \nu \beta \beta$ decay to reveal the properties of neutrinos. The decay rate of this process is influenced by the nuclear matrix elements (NME), describing the transition between the initial and final nuclear species, which are isobars [10]. Currently, large uncertainties in NME, partially attributable to nuclear shapes, pose a major challenge in experimental design. Our comparative method enables precise determination of the shape differences between the involved isobar species, potentially reducing the uncertainty in NME calculations.

Collective-flow-assisted imaging represents a discovery tool for exploring nuclear structure. Together with established low-energy methods, we can better address the forefront challenges in nuclear science. Future research should leverage collider facilities to conduct experiments with selected isobaric or isobar-like pairs. The combination of high and low-energy techniques promises to open new avenues in the study of nuclear matter, potentially leading to significant discoveries.

Methods:
Extended data Figure1:

The observables for $v_2-\left[p_{\mathrm{T}}\right]$ correlations. The centrality dependences of the components used to calculate the $\rho_2$ of Eq. (2) in $\mathrm{U}+\mathrm{U}$ and $\mathrm{Au}+\mathrm{Au}$ collisions: $\left\langle v_2^2\right\rangle^{1 / 2}$ (a), $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle^{1 / 2}(\mathrm{~b}),\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ (c) and $\rho_2$ (d). They are calculated using the two-subevent method.

Extended data Figure2:

Impact of "non-flow" background. The centrality dependence of $\rho_2$ in $\mathrm{Au}+\mathrm{Au}$ (left) and $\mathrm{U}+\mathrm{U}$ (right) collisions, compared between the standard, two-subevent and three-subevent correlation methods.

Extended data Figure3:

Model prediction of sensitivity on shear and bulk viscosities. IP-Glasma+Music model prediction of the $\left\langle v_2^2\right\rangle$ (left), $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ in Au+Au collisions and the corresponding ratios between $\mathrm{U}+\mathrm{U}$ and $\mathrm{Au}+\mathrm{Au}$ collisions, for different amount of shear and bulk viscosities.

Extended data Figur4:

Model prediction of sensitivity on nuclear structure parameters. IP-Glasma+Music model prediction of the ratios of $\left\langle v_2^2\right\rangle$ (left), $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ between $\mathrm{U}+\mathrm{U}$ and $\mathrm{Au}+\mathrm{Au}$ collisions, for different Glauber model paramters.

Extended data Figur5:

Comparison of prediction from two hydrodynamic models. The ratios of $\left\langle v_2^2\right\rangle$ (a), $\left\langle\left(\delta p_{\mathrm{T}}\right)^2\right\rangle$ (b) and $\left\langle v_2^2 \delta p_{\mathrm{T}}\right\rangle$ (c) as a function of centrality. The data are compared to IP-Glasma+Music (solid lines) and trajectum (dashed lines) hydrodynamic models calculations assuming $\beta_{2 \mathrm{U}}=0.28$ (red) and $\beta_{2 \mathrm{U}}=0.25$ (blue). In the right panel, different $\gamma_{\mathrm{U}}$ values are used.

Extended data Table 1:

The choices Woods-Saxon parameters, including deformations, in the IP-Glasma+Music model. The default values are denoted by bold font, while the rest are variations designed to constrain the values of $\left(\beta_{2 \mathrm{U}}, \gamma_{\mathrm{U}}\right)$ and derive theoretical uncertainties associated with other structure parameters.