Scaling relation between $pA$ and $AA$ collisions

We compare the flowlike correlations in high multiplicity proton-nucleus $\left(p+A\right)$ and nucleus-nucleus $\left(A+A\right)$ collisions. At fixed multiplicity, the correlations in these two colliding systems are strikingly similar, although the system size is smaller in $p+A$. Based on an independent cluster model and a simple conformal scaling argument, where the ratio of the mean free path to the system size stays constant at fixed multiplicity, we argue that flow in $p+A$ emerges as a collective response to the fluctuations in the position of clusters, just like in $A+A$ collisions. With several physically motivated and parameter free rescalings of the recent LHC data, we show that this simple model captures the essential physics of elliptic and triangular flow in $p+A$ collisions. We also explore the implications of the model for jet energy loss in $p+A$, and predict slightly larger transverse momentum broadening in $p+A$ than in $A+A$ at the same multiplicity.

Figure 1

The ratio of fluctuation-driven eccentricities $\delta {\epsilon }_2$ and $\delta {\epsilon }_3$, for a Gaussian profile compared to the phobos Glauber model [25] as a function of $N_{\mathrm{trk}}^{\mathrm{offline}}$. The precise definition of these quantities are given in Eqs. (3.6) and  (3.12).

Figure 2

The integrated $v_2\left\{2\right\}$ for PbPb and pPb vs multiplicity from [24]. (Left) Original values. (Right) The fluctuation-dependent elliptic flow, ${\left(v_2\left\{2\right\}\right)}_{\mathrm{PbPb},\mathrm{rscl}}=\sqrt{1-{\epsilon }_s^2/{\epsilon }_2{\left\{2\right\}}^2}{\left(v_2\left\{2\right\}\right)}_{\mathrm{PbPb}}$, compared to ${\left(v_2\left\{2\right\}\right)}_{\mathrm{pPb}}$. The scaling factor is extracted using the phobos Glauber model [25] in $A+A$ simulations, and is not a fit.

Figure 3

The integrated $v_3\left\{2\right\}$ for PbPb and pPb vs multiplicity from [24]. An approximately conformal response leads to ${\left(v_3\left\{2\right\}\right)}_{\mathrm{pPb}}\simeq {\left(v_3\left\{2\right\}\right)}_{\mathrm{PbPb}}$ at fixed multiplicity.

Figure 4

A comparison of the momentum dependent $v_2\left\{2\right\}$ in pPb and PbPb collisions. (Upper) Original data. (Lower) The PbPb data is rescaled to isolate the fluctuation-driven part of the elliptic flow as defined in Eq. (3.10). The momentum axis is also scaled by the conformal scaling factor $\kappa \approx 1.25$, Eq. (3.18). This is a parameter free rescaling. The agreement in the low $p_T$ region suggests that elliptic flow in $p+A$ results from a linear response to the fluctuations of the initial geometry which is conformally related to the $A+A$ response. The data are from Ref. [24].

Figure 5

The comparison of the momentum-dependent $v_3\left\{2\right\}$ in pPb and PbPb collisions. (Upper) Original data. (Lower) The momentum axis is scaled by the conformal scaling factor $\kappa \approx 1.25$, Eq. (3.18). This is a parameter free rescaling. The agreement in the low $p_T$ region suggests that the triangular flow in $p+A$ results from a linear response to the fluctuations in the initial geometry which is conformally related to the $A+A$ response. The data are from Ref. [24].