Initial-state fluctuations in collisions between light and heavy ions

In high-energy collisions involving small nuclei ($p+p$ or $x+\text{Au}$ collisions where $x=p, d$, or ${}^3\mathrm{He}$) the fluctuating size, shape, and internal gluonic structure of the nucleon is shown to have a strong effect on the initial size and shape of the fireball of new matter created in the collision. A systematic study of the eccentricity coefficients describing this initial fireball state for several semirealistic models of nucleon substructure and for several practically relevant collision systems involving small nuclei is presented. The key importance of multiplicity fluctuations in such systems is pointed out. Our results show large differences from expectations based on conventional Glauber model simulations of the initial state created in such collisions.

Figure 1

Comparison of the normalized charged multiplicity distribution in $p+\text{Pb}$ collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}=5020\mathrm{GeV}$ measured by CMS [7, 26] (blue squares) with the Gaussian nucleon MC-Glauber model including $pp$ multiplicity fluctuations (green circles), as described in the text. 500 000 $p+\text{Pb}$ initial entropy density distributions were sampled for the model simulations; to increase statistics, for each event the Poisson distribution was oversampled five times, resulting in $2.5\times {10}^6$ theoretical data points. Vertical dashed lines labeled by red numbers define “centrality” classes as fractions of the total number of events.

Figure 2

Contour plots of the initial entropy density for five randomly selected $p+p$ collisions at $\sqrt{s}=200\mathrm{GeV}$ and impact parameter $b=1.3\mathrm{fm}$, computed with the MC-Glauber model using a smooth Gaussian nucleon density profile for collision detection and including multiplicity fluctuations in the deposited entropy.

Figure 3

Same as Fig. 1, but now the experimental data from CMS are compared with simulations in which the protons are subdivided into gluon clouds of width ${\sigma }_g$, with three choices for ${\sigma }_g$ as given in the legend, located around three valence quarks with fluctuating positions as described in the text. The inset is a linear plot of the same quantities to better see the sensitivity to ${\sigma }_g$ at low multiplicities.

Figure 4

Contour plots of the initial entropy density for five randomly selected $p+p$ collisions at $\sqrt{s}=200\mathrm{GeV}$ and impact parameter $b=1.3\mathrm{fm}$, computed with the MC-Glauber model using quark subdivision of the nucleon density profile for both collision detection and entropy deposition, including additional multiplicity fluctuations in the deposited entropy. See text for model description and discussion.

Figure 5

Elliptic and triangular rms eccentricities, ${\epsilon }_2\left\{2\right\}$ (a), (c) and ${\epsilon }_3\left\{2\right\}$ (b), (d) for $p+p$ collisions at $\sqrt{s}=200\mathrm{GeV}$ (a), (b) and 5020 GeV (c), (d). See text for detailed discussion.

Figure 6

Ensemble-averaged rms radii for $p+p$ collisions at $\sqrt{s}=200\mathrm{GeV}$ (a) and 5020 GeV (b) as a functions of collision centrality, using the same symbols and line styles as in Fig. 5. See text for discussion.

Figure 7

A representative sample of the $\mathrm{\Gamma }$-distributed random field (with transverse correlation length $\overline{a}=0.29\mathrm{fm}$) used for modulating the gluon-field strength inside a Gaussian nucleon, or within the Gaussian gluon cloud carried along by a quark when nucleons are subdivided into valence quarks.

Figure 8

Similar to Fig. 5, but including gluon-field fluctuations. The width of the gluon cloud carried by valence quarks for quark-subdivided nucleons has been set to ${\sigma }_g=0.3\mathrm{fm}$, and $\overline{a}=0.29\mathrm{fm}$ is taken for the transverse gluon-field correlation length in collisions at $\sqrt{s}=200\mathrm{GeV}$. See text for detailed discussion.

Figure 9

Contour plots of the initial entropy density for five randomly selected $p+p$ collisions at $\sqrt{s}=200\mathrm{GeV}$ and impact parameter $b=1.3\mathrm{fm}$, computed with the MC-Glauber model with Gaussian or quark-subdivided nucleons whose Gaussian gluon density distributions have been modulated by the $\mathrm{\Gamma }$-distributed random field shown in Fig. 7. The fluctuated gluon density distributions are used for both collision detection and entropy deposition. In the entropy deposition step multiplicity fluctuations are either included or not, as specified in the following: (a) Gaussian nucleons with gluon-field fluctuations (GFFs) of transverse correlation length $a=\overline{a}=0.29\mathrm{fm}$, without multiplicity fluctuations (MFs); (b) same as (a) except for $a=2\overline{a}$; (c) same as (a) except for $a=\overline{a}/2$; (d) same as (a) but including fluctuations in the multiplicity generated by each nucleon; (e) quark-subdivided nucleons with valence quark gluon clouds of width ${\sigma }_g=0.3\mathrm{fm}$ modulated by GFF with $a=\overline{a}$ without MF; (f) same as (e) but including independent fluctuations of the multiplicities created by each quark.

Figure 10

False-color scatter plot of multiplicity $dS/dy\propto d{N}_{\mathrm{ch}}/dy$ vs impact parameter $b$ (a) and vs the number of participant nucleons $N_{\mathrm{part}}$ (b) in $p+p, p+\text{Au}, d+\text{Au}$, and ${}^3\mathrm{He}+\text{Au}$ collisions at $\sqrt{s}=200A\mathrm{GeV}$. Quark-subdivided nucleons with width ${\sigma }_g=0.3\mathrm{fm}$ for the valence quark gluon clouds were used to produce this figure.

Figure 11

(Left) Probability distributions for the intrinsic eccentricities ${\epsilon }_2$ (left column) and ${\epsilon }_3$ (right column) for protons (red circles), deuterons (blue squares), and ${}^3\mathrm{He}$ nuclei (green triangles), calculated from the nuclear thickness functions for smooth Gaussian nucleons [row (a)], Gaussian nucleons modulated by gluon-field fluctuations [row (b)], and quark-subdivided nucleons modulated by gluon-field fluctuations [row (c)]. (Right) Probability distributions for the same eccentricities calculated from the deposited entropy density profile in central ($0\%–10\%$ centrality) $p+\text{Au}$ (red circles), $d+\text{Au}$ (blue squares), and ${}^3\mathrm{He}+\text{Au}$ collisions at $\sqrt{s}=200A\mathrm{GeV}$, including multiplicity fluctuations, again using smooth Gaussian nucleons, as well as Gaussian and quark-subdivided nucleons modulated by gluon-field fluctuations [rows (a)–(c), respectively]. Without multiplicity and subnucleonic density fluctuations, we observed a significant decrease of the mean ellipticity $\langle {\epsilon }_2\rangle$ for $p+\text{Au}$ collisions and of the mean triangularity $\langle {\epsilon }_3\rangle$ for $d+\text{Au}$ collisions (not shown).

Figure 12

Centrality dependence of ${\epsilon }_2\left\{2\right\}$ (solid lines, shifted up by 0.2 for better visibility) and ${\epsilon }_3\left\{2\right\}$ (dashed lines) for $p+p$ (a), $p+\mathrm{Au}$ (b), $d+\text{Au}$ (c), and ${}^3\mathrm{He}+\text{Au}$ (d) collisions at $\sqrt{s}=200A\mathrm{GeV}$. In the left (right) panels impact parameter (multiplicity) is used to characterize centrality. Purple triangles and black circles connected by solid lines show results from disklike and Gaussian nucleon density profiles used in the collision detection algorithm. Red, blue, and green squares connected by dotted, dash-dotted, and dashed lines, respectively, use quark-subdivided nucleon density profiles with widths ${\sigma }_g=0.25$, 0.3, and 0.4 fm, respectively. All results include multiplicity fluctuations in the entropy deposition process. See text for discussion.

Figure 13

Similar to Fig. 12, but using different panels for different models for the collision detection [disklike (a),(d), Gaussian (b),(e), and quark-subdivided nucleons with ${\sigma }_g=0.3\mathrm{fm}$ (c),(f)] and comparing in each panel different collision systems (black triangles, $p+p$; red circles, $p+\text{Au}$; blue upside-down triangles, $d+\text{Au}$; green squares, ${}^3\mathrm{He}+\mathrm{Au}$). Panels (a)–(c) use impact parameter; panels (d)–(f) use multiplicity to characterize collision centrality. See text for discussion.

Figure 14

Similar to Fig. 13, but for the mean rms radii of the initial entropy density distribution as functions of collision centrality, using different models for the collision detection: disklike (a), Gaussian (b), and quark-subdivided nucleons with ${\sigma }_g=0.3\mathrm{fm}$ (c). In each panel different collision systems (black triangles, $p+p$; red circles, $p+\text{Au}$; blue upside-down triangles, $d+\text{Au}$; green squares, ${}^3\mathrm{He}+\mathrm{Au}$) are compared. See discussion in text.

Figure 15

Eccentricity correlations between ${\epsilon }_3$ and ${\epsilon }_2$ in $p+p$ (a), $p+\text{Au}$ (b), $d+\text{Au}$ (c), and ${}^3\mathrm{He}+\text{Au}$ collisions (d) at $\sqrt{s}=200A\mathrm{GeV}$. The calculations assume quark-subdivided nucleons with gluon clouds of width ${\sigma }_g=0.3\mathrm{fm}$ and include multiplicity fluctuations in the entropy deposited by each struck valence quark. (In panel (a) the lower set of curves shows for comparison the corresponding results for nucleons with a smooth Gaussian density profile.) The gray line in the center of each panel [which degenerates to a point in panel (a) owing to the centrality independence of both ${\epsilon }_2$ and ${\epsilon }_3$ in $p+p$ collisions] connects the mean values of ${\epsilon }_3$ vs ${\epsilon }_2$ for the centrality bins shown in the legend. The colored points subdivide the events in each centrality bin into bins with different ${\epsilon }_2$ values and plot the mean triangularity $\langle {\epsilon }_3\rangle$ vs the mean ellipticity $\langle {\epsilon }_2\rangle$ in each such bin. See text for discussion.