Assessing saturation physics explanations of collectivity in small collision systems with the ip-jazma model

Experimental measurements in relativistic collisions of small systems from $p+p$ to $p/d/{}^3\mathrm{He}+\mathrm{A}$ at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) reveal particle emission patterns that are strikingly similar to those observed in $A+A$ collisions of large nuclei. One explanation of these patterns is the formation of small droplets of quark-gluon plasma (QGP) followed by hydrodynamic evolution. A geometry engineering program was proposed to further investigate these emission patterns, and the experimental data from that program in $p+\mathrm{Au}, d+\mathrm{Au}, {}^3\mathrm{He}+\mathrm{Au}$ collisions for elliptic and triangular anisotropy coefficients $v_2$ and $v_3$ follow the pattern predicted by hydrodynamic calculations [C. Aidala et al. (PHENIX Collaboration), Nat. Phys. 15, 214 (2019)]. One alternative approach, referred to as initial-state correlations, suggests that for small systems the patterns observed in the final-state hadrons are encoded at the earliest moments of the collision and therefore require no final-state parton scattering or hydrodynamic evolution. Recently, new calculations using only initial-state correlations, in the dilute-dense approximation of gluon saturation physics, reported striking agreement with the $v_2$ patterns observed in $p/d/{}^3\mathrm{He}+\mathrm{Au}$ data at RHIC [M. Mace, V. V. Skokov, P. Tribedy, and R. Venugopalan, Phys. Rev. Lett. 121, 052301 (2018)]. The results reported by Mace, Skokov, Tribedy and Venugopalan (MSTV) are counterintuitive and thus we aim here to reproduce some of the basic features of these calculations. In this first investigation, we provide a description of our publicly available model, ip-jazma, and investigate its implications for saturation scales, multiplicity distributions, and eccentricities, reserving for later work the analysis of momentum spectra and azimuthal anisotropies. We find that our implementation of the saturation physics model reproduces the results of the MSTV calculation of the multiplicity distribution in $d+\mathrm{Au}$ collisions at RHIC. However, additional aspects of studies, together with existing data, call into question some of the essential elements reported by MSTV. Resolution of these issues will require further developments of ip-jazma, in order to determine if it can replicate the qualitative agreement with the $v_2$ reported by MSTV. Both the work reported here and future studies will establish which features in the experimental data are uniquely attributable to the color glass condensate description.

Figure 1

Two-dimensional separation distance $r_T$ distribution between the proton and neutron constituents of the deuteron. Shown are results from the deuteron Hulthén wave function as well as for events selected within the 5% highest multiplicity from the ip-jazma dilute-dense calculation. In addition, we calculate the overlap fraction defined in Sec. 3d as a function of $r_T$ between the two nucleons following two-dimensional Gaussian distributions of width $\sigma =0.40$ and $0.56\mathrm{fm}$, and quote the values integrating over the ip-jazma distribution.

Figure 2

Functional form for fluctuations in $Q_s/\langle Q_s\rangle$ with standard deviation of 0.5 on $log\left[{(Q_s/\langle Q_s\rangle )}^2\right]$.

Figure 3

ip-jazma single $d+\mathrm{Au}$ interaction event display. Left and middle panels show the $Q_s^2(x,y)$ distribution for the projectile and target nuclei respectively. The right panel shows the calculated energy density from the resulting collision in arbitrary units.

Figure 4

Calculations of eccentricity moments ${\varepsilon }_2-{\varepsilon }_6$ for Au+Au collisions at 200 GeV. Shown are results from the full ip-glasma calculation [19] compared with calculations from the ip-jazma code. Also shown are calculations from Ref. [3] in the $N_{\mathrm{coll}}$ and $N_{\mathrm{part}}$ cases, noting that the $N_{\mathrm{coll}}$ case is the same algorithm as in ip-jazma modulo issues such as the $r_{\max }$ cutoff.

Figure 5

Functional form $F\left(x\right)$ where $x=Q_s\left(\text{targ}\right)/m$.

Figure 6

ip-jazma result for the distribution of $N_g/\langle N_g\rangle$ in the dilute-dense case and with no $Q_{s,0}^2$ fluctuations. The left (right) panel has the $y$ axis on a linear (log) scale.

Figure 7

ip-jazma results in $p+\mathrm{Au}$ collisions for the distribution of $N_g/\langle N_g\rangle$ in the dilute-dense case and with the inclusion of $Q_{s,0}^2$ fluctuations. The vertical dashed line indicates the cutoff for the 5% highest multiplicity events. The right panel shows the average projectile proton $Q_{s,0}^2$ as a function of event selected $N_g/\langle N_g\rangle$.

Figure 8

ip-jazma $d+\mathrm{Au}$ minimum bias results for the distribution of $N_g/\langle N_g\rangle$ in the dilute-dense case and with no $Q_{s,0}^2$ fluctuations. The left (right) panel has the $y$ axis on a linear (log) scale.

Figure 9

ip-jazma results in $d+\mathrm{Au}$ collisions for the distribution of $N_g/\langle N_g\rangle$ in the dilute-dense case and with the inclusion of $Q_{s,0}^2$ fluctuations. The vertical dashed line indicates the cutoff for the 5% highest multiplicity events. Also shown are STAR experimental data as a function of $d{N}_{ch}/d\eta /\langle d{N}_{ch}/d\eta \rangle$. The right panel shows the average projectile proton $Q_{s,0}^2$ as a function of event selected $N_g/\langle N_g\rangle$.

Figure 10

ip-jazma calculations of the interaction area (right) and the average ${\left(Q_s^p\right)}^2$ for the projectile over that area (left) as a function of the number of produced gluons $N_g$ in arbitrary units. While arbitrary, the $x$-axis units are such that one can make a direct comparison between the $p+\mathrm{Au}$ and $d+\mathrm{Au}$ results. The shaded circles indicate the approximate location for the 5% highest multiplicity selection in both cases, demonstrating that high-multiplicity $d+\text{Au}$ events result from a larger net interaction area than $p+\text{Au}$ events, but with similar saturation scales. See text for further discussion.

Figure 11

PHENIX published data for $v_2$ in $p+\mathrm{Au}$ and $d+\mathrm{Au}$ collisions in 0–5% and 20–40% multiplicity selections, which have comparable average $d{N}_{ch}/d\eta$ values as shown. Also shown is the MSTV calculation for $p+\mathrm{Au}$ at this multiplicity, which should be identical to their result for $d+\mathrm{Au}$ 20–40% multiplicity since the $d{N}_{ch}/d\eta$ is essentially the same.

Figure 12

ip-jazma $d+\mathrm{Au}$ highest multiplicity 5% distribution for $Q_s^2$(proj) /$Q_s^2$(targ) weighted by the contribution to the gluon density.

Figure 13

ip-jazma $d+\mathrm{Au}$ minimum bias results for the distribution of $N_g/\langle N_g\rangle$ in the dense-dense case and with no $Q_{s,0}^2$ fluctuations. Also shown are the STAR experimental data as a function of $d{N}_{ch}/d\eta /\langle d{N}_{ch}/d\eta \rangle$.

Figure 14

ip-jazma $\mathrm{Au}+\mathrm{Au} b=9$ fm events at 200 GeV and their energy density distribution calculated in the dense-dense limit. Note that no $Q_{s,0}^2$ fluctuations are included. The Gaussian fit to the distribution does not describe the tails of the distribution, while the $\mathrm{\Gamma }$ functional fit does. Also shown is a comparison with the full IP-Glasma result which is essentially identical to the ip-jazma result.