Constraining the Nucleon Size with Relativistic Nuclear Collisions

The notion of the “size” of nucleons and their constituents plays a pivotal role in the current paradigm of the formation and the fluctuations of the quark-gluon plasma produced in high-energy nuclear collision experiments. We report on state-of-the-art hydrodynamic results showing that the correlation between anisotropic flow $v_n^2$ and the mean transverse momentum of hadrons $[p_t]$ possesses a unique sensitivity to the nucleon size in off-central heavy-ion collisions. We argue that existing experimental measurements of this observable support a picture where the relevant length scale characterizing the colliding nucleons is of order 0.5 fm or smaller, and we discuss the broad implications of this finding for future global Bayesian analyses aimed at extracting initial-state and medium properties from nucleus-nucleus collision data, including $v_n^2\text{−}[p_t]$ correlations. Determinations of the nucleon size in heavy-ion collisions will provide a solid independent constraint on the initial state of small system collisions and will establish a deep connection between collective flow data in nucleus-nucleus experiments and data on deep inelastic scattering on protons and nuclei.

Figure 1

Estimated $\rho (v_n^2,[p_t])$ in the TRENTO model. Left: results for smooth nucleons. Right: results for smooth nucleons and composite nucleons with three hot spots with $w_q=0.27\mathrm{fm}$. Different line styles correspond to different parametrizations of nucleon structure. Top: $n=2$. Bottom: $n=3$.

Figure 2

Results of the $\mathrm{IP}\text{−}\mathrm{Glasma}+\mathrm{MUSIC}+\mathrm{UrQMD}$ framework for $\rho (v_2^2,[p_t])$ (left) and $\rho (v_3^2,[p_t])$ (right). Different shaded bands correspond to different values of the nucleon width. The orange band with starry hatches corresponds to a calculation with reduced viscosities. The dashed lines are estimators using JETSCAPE initial conditions. Symbols are preliminary ATLAS data [26] (diamonds) for charged particles with $0.5<p_t<2\mathrm{GeV}$, $|\eta |<2.5$ and the centrality defined via $\sum E_T$, and ALICE data [27] (circles) for charged particles with $0.2<p_t<3\mathrm{GeV}$, $|\eta |<0.8$, and the centrality defined via V0M amplitude. We note that the theoretical calculations implement the same $p_t$ cut used by the ALICE Collaboration and that the larger upper $p_t$ cut in the ALICE analysis explains partly the larger ${\rho }_n$ compared to ATLAS data for noncentral collisions.

Figure 3

$\mathrm{IP}\text{−}\mathrm{Glasma}+\text{MUSIC}+\mathrm{UrQMD}$ results for the components of $\rho (v_n^2,[p_t])$. We show results for composite nucleons with $w=0.4\mathrm{fm}$ plus three hot spots with $w_q=0.11\mathrm{fm}$ (blue shaded band), and smooth nucleons with $w=1.2\mathrm{fm}$ (gray hatched band or orange starry-hatched band for low viscosities). Top left: standard deviation of $[p_t]$ fluctuations. Top right: standard deviation of $v_n^2$ fluctuations. Bottom left: covariance of $v_2^2$ and $[p_t]$. Bottom right: covariance of $v_3^2$ and $[p_t]$.