Scaling approach to rigid and soft nuclear deformation through flow fluctuations in high-energy nuclear collisions

Abstract: The nature of octupole deformation, whether static or vibrational, remains an open question in nuclear physics. Here, we propose a scaling approach to probe this ambiguity by triangular flow fluctuations using multi-particle cumulants, $c_{3,\varepsilon}{4}$, in relativistic $^{238}$U+$^{238}$U collisions. We demonstrate that both $|c_{3,\varepsilon}{4}|$ and the ratio $|c_{3,\varepsilon}{4}/c^2_{3,\varepsilon}{2}|$ scale linearly with the fourth-order moment of octupole deformation, $\langle \beta^4_{3,\mathrm{U}} \rangle$. Combined with the known linear relation of $c_{3,\varepsilon}{2}$ to $\langle \beta^2_{3,\mathrm{U}} \rangle$, this new relation provides a direct extraction of both the mean and variance of the octupole deformation fluctuations, finally discriminating between static and dynamic origins. This work establishes a new tool to probe the static and dynamic collective modes in high-energy nuclear collisions, advancing a significant step toward refining the initial conditions of quark-gluon plasma.