Determination of the Neutron Skin of ${}^{208}\mathrm{Pb}$ from Ultrarelativistic Nuclear Collisions

Emergent bulk properties of matter governed by the strong nuclear force give rise to physical phenomena across vastly different scales, ranging from the shape of atomic nuclei to the masses and radii of neutron stars. They can be accessed on Earth by measuring the spatial extent of the outer skin made of neutrons that characterizes the surface of heavy nuclei. The isotope ${}^{208}\mathrm{Pb}$, owing to its simple structure and neutron excess, has been in this context the target of many dedicated efforts. Here, we determine the neutron skin from measurements of particle distributions and their collective flow in ${}^{208}\mathrm{Pb}+{}^{208}\mathrm{Pb}$ collisions at ultrarelativistic energy performed at the Large Hadron Collider, which are mediated by interactions of gluons and thus sensitive to the overall size of the colliding ${}^{208}\mathrm{Pb}$ ions. By means of state-of-the-art global analysis tools within the hydrodynamic model of heavy-ion collisions, we infer a neutron skin $\mathrm{\Delta }{r}_{np}=0.217\pm 0.058\mathrm{fm}$, consistent with nuclear theory predictions, and competitive in accuracy with a recent determination from parity-violating asymmetries in polarized electron scattering. We establish thus a new experimental method to systematically measure neutron distributions in the ground state of atomic nuclei.

Figure 1

Neutron skin and collective flow in relativistic nuclear collisions. (a) Two ions collide with impact parameter $b=8\mathrm{fm}$. Both ions are Lorentz contracted by a factor $\gamma \approx 2500$, and the relevant dynamics hence effectively takes place in the transverse plane, $x_{\perp }=(x,y)$. (b) The collision deposits energy in the interaction region depending on the extent of the neutron skin of the ${}^{208}\mathrm{Pb}$ nuclei. We consider $\mathrm{\Delta }{r}_{np}=0.086$ (top) and $\mathrm{\Delta }{r}_{np}=0.384\mathrm{fm}$ (bottom). The neutron skin is varied by keeping the half-width neutron radius $R_n$ constant while changing the neutron diffuseness, as displayed by the dotted lines [see also Eq. (2) below]. A larger neutron skin leads to a considerably larger total hadronic cross section, ${\sigma }_{\mathrm{tot}}$, and the resulting QGP is in addition more diffuse and less elliptical. (c) We show a single QGP evolving hydrodynamically and being converted into particles (marked in the figure with their respective symbols) as it cools, while expanding both in $z$ and in the transverse plane. The observation of millions of such events leads to characteristic azimuthal anisotropies in the momentum distribution of the produced particles, the most important of which is quantified by the rms value of its second Fourier component, the elliptic flow $v_2\{2\}$, which reflects the ellipticity of the QGP.

Figure 2

Signature of the neutron skin on bulk particle production in ultrarelativisitic ${}^{208}\mathrm{Pb}+{}^{208}\mathrm{Pb}$ collisions. Varying only the neutron skin size at our optimal parameter settings we show the charged particle multiplicity (left), the mean transverse momentum (middle), and the elliptic flow as measured by $v_2\{k\}$ (right) with a comparison to ALICE data [19, 20]. A larger neutron skin leads to more collisions, but per collision the multiplicity is lower at larger centralities. The larger size of the QGP leads to a reduced transverse momentum on average. Smearing of the elliptical shape leads to reduced elliptic flow as measured by $v_2\{2\}$ and $v_2\{4\}$. Theoretical error bars are statistical only, experimental uncertainties include systematics as well.

Figure 3

Inferred neutron skin and energy-deposition parameters. We show the posterior distribution of the neutron skin $\mathrm{\Delta }{r}_{np}$, the nucleon width $w$ and the energy deposition parameters $p$ and $q$, together with their expectation values (see top) and correlations. Uncertainties correspond to the standard deviations of the posterior distributions. Especially the $p$ parameter [see Eq. (1)] is highly anticorrelated with $\mathrm{\Delta }{r}_{np}$, as both have a strong effect on the centrality dependence of observables (see also Fig. 2).

Figure 4

State-of-the-art determinations of the neutron skin of ${}^{208}\mathrm{Pb}$. We show the final likelihood distribution of the neutron skin as determined from the LHC data as compared to the values obtained experimentally by the PREX Collaboration (including both experimental and theoretical uncertainties in the extraction) [6] and the estimate of ab initio nuclear theory (with an error bar corresponding to a 68% credibility interval) [7].