Possible octupole deformation of $^{208}$Pb and the ultracentral $v_2$ to $v_3$ puzzle

Recent measurements have established the sensitivity of ultracentral heavy-ion collisions to the deformation parameters of non-spherical nuclei. In the case of ${}^{ 129}$Xe collisions, a quadrupole deformation of the nuclear profile led to an enhancement of elliptic flow in the most central collisions. In ${}^{ 208}$Pb collisions a discrepancy exists in similar centralities, where either elliptic flow is over-predicted or triangular flow is under-predicted by hydrodynamic models; this is known as the $v_2$-to-$v_3$ puzzle in ultracentral collisions. Motivated by low-energy nuclear structure calculations, we consider the possibility that $^{208}$Pb nuclei could have a pear shape deformation (octupole), which has the effect of increasing triangular flow in central PbPb collisions. Using the recent data from ALICE and ATLAS, we revisit the $v_2$-to-$v_3$ puzzle in ultracentral collisions, including new constraints from recent measurements of the triangular cumulant ratio $v_3\left\{4\right\}/v_3\left\{2\right\}$ and comparing two different hydrodynamic models. We find that, while an octupole deformation would slightly improve the ratio between $v_2$ and $v_3$, it is at the expense of a significantly worse triangular flow cumulant ratio. In fact, the latter observable prefers no octupole deformation, with $\beta_3\lesssim 0.0375$ for ${}^{ 208}$Pb, and is therefore consistent with the expectation for a doubly-magic nucleus even at top collider energies. The $v_2$-to-$v_3$ puzzle remains a challenge for hydrodynamic models.