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

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,ε {4}, in relativistic 238 U+ 238 U collisions. We demonstrate that both | c 3,ε {4}| and the ratio | c 3 , ε { 4 } / c 3 , ε 2 { 2 } | scale linearly with the fourth-order moment of octupole deformation, 〈 β 3 , U 4 〉 . Combined with the known linear relation of c 3,ε {2} to 〈 β 3 , U 2 〉 , 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.