Imprints of octupole collectivity in uranium-238 on relativistic heavy-ion flow observables

Technical Summary: Imprints of Octupole Collectivity in Uranium-238 on Relativistic Heavy-Ion Flow Observables

Problem Statement
While low-energy spectroscopic studies have established finite octupole strength in the $^{238}\text{U}$ nucleus, the nature of this collectivity—whether it represents a rigid static pear-shaped ground state or soft/vibrational correlations—remains a subject of debate. Traditional low-energy measurements (typically $< 30$ MeV/nucleon) infer nuclear shapes from transition probabilities but cannot directly probe the equal-time coordinate-space correlations relevant to high-energy collisions. The central question addressed is whether these octupole correlations can be encoded in the initial geometry of relativistic heavy-ion collisions and subsequently mapped to final-state flow observables, providing a complementary probe to low-energy studies.

Methodology
The authors employ a state-of-the-art hydrodynamic framework, IP-Glasma + MUSIC + UrQMD, to simulate event-by-event $(2+1)$D boost-invariant collisions at $\sqrt{s_{NN}} = 193$ GeV for $^{238}\text{U}+^{238}\text{U}$ and $\sqrt{s_{NN}} = 200$ GeV for $^{197}\text{Au}+^{197}\text{Au}$.

1. Initial State Modeling: Nuclear density is parameterized using a deformed Woods-Saxon (WS) profile. The nuclear surface is expanded in spherical harmonics, incorporating quadrupole ($\beta_2$), octupole ($\beta_3$), and hexadecapole ($\beta_4$) deformations. Crucially, $\beta_3$ is treated not as a static deformation of the ground state, but as an effective amplitude controlling reflection-asymmetric octupole correlations in the event-by-event coordinate-space sampling.
2. Parameter Variation: To isolate octupole effects, the study varies the effective octupole amplitude $\beta_{3,U}$ from 0 to 0.2 while fixing $\beta_{2,U} = 0.28$ and triaxiality $\gamma_U = 0^\circ$. Hexadecapole effects ($\beta_{4,U}$) are also scanned between 0 and 0.09. For $^{197}\text{Au}$, parameters are fixed based on low-energy constraints ($\beta_{2,\text{Au}}=0.14, \gamma_{\text{Au}}=45^\circ$), assuming negligible higher-order deformations.
3. Observables: The study focuses on two primary observables:
   • Triangular Flow ($\langle v_3^2 \rangle$): Sensitive to initial triangularity ($\varepsilon_3$).
   • Correlation with Transverse Momentum ($\langle v_3^2 \delta p_T \rangle$): A three-particle correlator linking triangular flow to event-wise deviations in mean transverse momentum.
4. Analysis Strategy: To suppress system-dependent effects (such as final-state interactions and odd-$A$ specific effects), the authors construct ratios of these observables between $^{238}\text{U}+^{238}\text{U}$ and $^{197}\text{Au}+^{197}\text{Au}$ collisions ($R_{v_3^2}$ and $R_{v_3^2 \delta p_T}$).
5. Validation: Results are cross-checked using constrained Skyrme Hartree-Fock-Bogoliubov (SHFB) densities as alternative initial conditions to ensure the findings are not artifacts of the Woods-Saxon parameterization.

Key Contributions and Results
The paper identifies three distinct features characterizing the response to octupole collectivity:

1. Inversion of Hierarchy in Ultra-Central Collisions (UCC): For $\beta_{3,U} \gtrsim 0.05$, the ordering of $\langle v_3^2 \rangle$ reverses in the 0–2% centrality range. While the smaller $^{197}\text{Au}$ system typically exhibits larger fluctuations than $^{238}\text{U}$ (yielding $R_{v_3^2} < 1$), the presence of finite octupole correlations in $^{238}\text{U}$ enhances its triangularity sufficiently to make $R_{v_3^2} > 1$. This inversion serves as a unique signature for $\beta_3$.
2. Linear Dependence on $\beta_3^2$: In the 0–2% centrality range, the ratio $R_{v_3^2}$ exhibits a linear dependence on $\beta_{3,U}^2$. Assuming low-energy derived values of $\beta_{3,U} \approx 0.08 - 0.10$, the model predicts $R_{v_3^2} \approx 1.05 - 1.12$, which is consistent with recent high-precision STAR measurements.
3. Suppression in $\langle v_3^2 \delta p_T \rangle$: The ratio $R_{v_3^2 \delta p_T}$ shows a near-linear dependence on $\beta_{2,U}^2 \beta_{3,U}^2$ in central collisions (0–5%). The presence of $\beta_3$ introduces a negative contribution, driving the ratio further below unity and inducing anti-correlation behavior at larger $\beta_3$ magnitudes. This observable is also sensitive to $\beta_4$, suggesting that combined analysis of $R_{v_3^2}$ and $R_{v_3^2 \delta p_T}$ is necessary to disentangle octupole and hexadecapole effects.

The study confirms that these trends are robust against variations in transverse momentum ($p_T$) intervals and are consistent between Woods-Saxon and SHFB density inputs.

Significance
The paper claims that these findings provide a complementary probe of odd-order nuclear collectivity in heavy nuclei, distinct from traditional low-energy spectroscopy. By demonstrating quantitative sensitivity to octupole-induced features in the initial-state geometry, the work helps constrain the initial conditions of the Quark-Gluon Plasma (QGP). Specifically, the results support the existence of a finite effective octupole-correlation strength in $^{238}\text{U}$ ($\beta_{3,U} \sim 0.08 - 0.1$) without requiring a rigid static pear-shaped ground state. The "Imaging-by-Smashing" approach presented here offers a method to reduce uncertainties in QGP initial-state modeling and provides a new avenue to investigate nuclear structure dynamics at the yoctosecond scale.