Glauber modeling of high-energy nuclear collisions at the subnucleon level

Glauber models based on nucleon-nucleon interactions are commonly used to characterize the initial state in high-energy nuclear collisions and the dependence of its properties on impact parameter or number of participating nucleons. In this paper, an extension to the Glauber model is presented, which accounts for an arbitrary number of effective subnucleon degrees of freedom, or active constituents, in the nucleons. Properties of the initial state, such as the number of constituent participants and collisions, as well as eccentricity and triangularity, are calculated and systematically compared for different assumptions of how to distribute the subnuclear degrees of freedom and for various collision systems. It is demonstrated that at high collision energy the number of produced particles scales with an average number of subnucleon degrees of freedom of between 3 and 5. The source codes for the constituent Monte Carlo Glauber extension, as well as for the calculation of the overlap area and participant density in a standard Glauber model, are made publicly available.

Figure 1

Available data of total, elastic, and inelastic cross sections measured in pp and $\mathrm{p}\overline{\mathrm{p}}$ collisions [18, 19, 20, 21, 22]. The data [24, 25] at 13 TeV are preliminary. The curves are fits performed by the COMPETE Collaboration [23].

Figure 2

Calculated total cross sections for PbPb and pPb collisions as a function of ${\sigma }_{\mathrm{NN}}$.

Figure 3

Geometric properties ($2N_{\mathrm{coll}}{N}_{\mathrm{part}},S,{\varepsilon }_2$, and ${\varepsilon }_3$ from top left to bottom right panels) computed with Glauber MC for AuAu collisions at $\sqrt{s_{\mathrm{NN}}}=19$ GeV and PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV.

Figure 4

Radial distribution of constituents after recentering when constructed from the standard parametrization [Eq. (4)] for different $N_{\mathrm{c}}$.

Figure 5

Radial distribution of constituents after recentering when constructed from the standard [Eq. (4)] or modified [Eq. (5)] parametrizations for $N_{\mathrm{c}}=3$.

Figure 6

Impact parameter distribution for pp (left) and PbPb (right) collisions. In case of pp, ${\sigma }_{\mathrm{cc}}=3.6$ and 25.2 mb with $N_{\mathrm{c}}=3$ are used for pp collisions at $\sqrt{s}=0.019$ and 13 TeV, respectively. For PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, the standard nucleon-based approach is compared to bound and freely distributed cases using ${\sigma }_{\mathrm{cc}}=3$ mb and $N_{\mathrm{c}}=10$.

Figure 7

Calculated ${\sigma }_{\mathrm{NN}}$ for various choices of $N_{\mathrm{c}}$ vs ${\sigma }_{\mathrm{cc}}$. Parameters for commonly used ${\sigma }_{\mathrm{NN}}$ are listed in Table 2.

Figure 8

Calculated total cross section for PbPb (top) and pPb (bottom) collisions for various choices of $N_{\mathrm{c}}$ vs ${\sigma }_{\mathrm{cc}}$ for the bound and free cases.

Figure 9

Average values of $N_{\mathrm{ccoll}}$ (top) and $N_{\mathrm{cpart}}/2$ (middle), as well as of the ratio $N_{\mathrm{ccoll}}/N_{\mathrm{cpart}}$ (bottom) vs ${\sigma }_{\mathrm{cc}}$ for various $N_{\mathrm{c}}$ in pp collisions.

Figure 10

Average eccentricity (top) and triangularity (bottom) for $b<0.5$ fm vs ${\sigma }_{\mathrm{cc}}$ for various $N_{\mathrm{c}}$ in pp collisions.

Figure 11

Eccentricity (top) and triangularity (bottom) vs $b$ for various $N_{\mathrm{c}}$ in pp collisions at 13 TeV. The calculation for ${\sigma }_{\mathrm{cc}}=21.60$ mb corresponds to the modified case.

Figure 12

Eccentricity (left) and triangularity (right) vs $b$ for $N_{\mathrm{c}}=20$ and ${\sigma }_{\mathrm{cc}}=0.8$ mb corresponding to pp collisions at 13 TeV using exponential, single, and double Gaussian density profiles (see text).

Figure 13

$N_{\mathrm{cpart}}/\mu$ (top) and $N_{\mathrm{ccoll}}/\nu$ (bottom panels) for AuAu (left) and PbPb (right panels) collisions. The parameters for the calculations are summarized in Tables 3 and 4.

Figure 14

Ratio $N_{\mathrm{cpart}}/N_{\mathrm{part}}$ normalized to $\mu /2$ for AuAu (top) and PbPb (bottom panel) collisions. The parameters for the calculations are summarized in Tables 3 and 4.

Figure 15

Values of $dN/d\eta$ in PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=17.2$ GeV and $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV scaled by $N_{\mathrm{cpart}}/\mu$ for subnucleon ($N_{\mathrm{c}}>1$) and $N_{\mathrm{part}}/2$ for nucleon ($N_{\mathrm{c}}=1$) participants. The data are from [41, 42], drawn with only point-to-point uncorrelated systematic uncertainties. The 17.2-GeV data are scaled using $N_{\mathrm{c}}=3$ (${\sigma }_{\mathrm{cc}}=5.5$ mb, modified case). The 5.02-TeV data are scaled using $N_{\mathrm{c}}=5$ (${\sigma }_{\mathrm{cc}}=10.3$ mb). The lines show the central points if the data were scaled by $N_{\mathrm{c}}=3$ (${\sigma }_{\mathrm{cc}}=17.9$ mb, modified case) and $N_{\mathrm{c}}=7$ (${\sigma }_{\mathrm{cc}}=5.7$ mb), respectively.

Figure 16

Ratio of fits to central AA (scaled by 160 to approximately account for $N_{\mathrm{part}}/2$) and inelastic pp collisions compared to $N_{\mathrm{cpart}}/\mu$ from constituent Glauber calculations for $b<3.5$ fm. The values for the power-law fits are taken from [42].

Figure 17

Power-law fit of $dN/d\eta$ from inelastic pp collisions compared to scaled central AA data. The AA curves are obtained from a power-law fit to $2dN/d\eta /N_{\mathrm{part}}$ scaled by $\mu /N_{\mathrm{cpart}}$ (multiplied by 160 to approximately account for $N_{\mathrm{part}}/2$) for constituent Glauber calculations with $b<3.5$ fm. In the case of $N_{\mathrm{c}}=1,N_{\mathrm{cpart}}=N_{\mathrm{part}}$, and $\mu =2$, the shown curve essentially represents the original fit to $2dN/d\eta /N_{\mathrm{part}}$. The values for the power-law fits are taken from [42].

Figure 18

Eccentricity (left) and triangularity (right panels) for AuAu (top) and PbPb (bottom panels) collisions. The parameters for the calculations are summarized in Tables 3 and 4.

Figure 19

Area calculated directly by counting of the overlap area and scaled using the participant widths for AuAu collisions at $\sqrt{s_{\mathrm{NN}}}=19$ GeV and PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV.

Figure 20

Participant transverse area density in an area given by radius $R=1,3$, and 5 fm for AuAu collisions at $\sqrt{s_{\mathrm{NN}}}=19$ GeV and PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV.