Forward-backward multiplicity fluctuations in ultrarelativistic nuclear collisions with wounded quarks and fluctuating strings

We analyze a generic model where wounded quarks are amended with strings in which both endpoint positions fluctuate in spatial rapidity. With the assumption that the strings emit particles independently of one another and with a uniform distribution in rapidity, we are able to analyze the model semianalytically, which allows for its detailed understanding. Using as a constraint the one-body string emission functions obtained from the experimental data for collisions at $\sqrt{s_{NN}}=200$ GeV, we explore the two-body correlations for various scenarios of string fluctuations. We find that the popular measures used to quantify the longitudinal fluctuations ($a_{nm}$ coefficients) are limited with upper and lower bounds. These measures can be significantly larger in the model where both endpoints are allowed to fluctuate, compared to the model with single endpoint fluctuations.

Figure 1

One-particle emission profiles obtained in the wounded quark model via Eqs. (2, 3, 4) from the PHOBOS rapidity spectra for $d$-Au collisions at $\sqrt{s_{NN}}=200$ GeV [54] in the indicated centrality classes (a), together with the corresponding symmetric (b) and antisymmetric (c) components. The shaded bands show the experimental uncertainties (propagated via the Gaussian method) for the $40–60$% and $60–80$% centrality classes, as well as for the PHOBOS minimum bias data [55].

Figure 2

One-particle emission profiles obtained in the wounded quark model from the PHOBOS rapidity spectra for Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV [56] in the indicated centrality classes. The shaded bands give the experimental uncertainties (propagated via the Gaussian method) for the most central and the most peripheral case.

Figure 3

Comparison of the wounded-quark model predictions (lines) with the experimental rapidity spectra (points) for $d$-Au [54] (a) and Au-Au [56] (b) collisions, with the experimental uncertainties shown as shaded bands. A universal profile discussed in the text is taken for the model calculations in all cases.

Figure 4

Distribution functions $g_1$ and $g_2$ (a) and cumulative distribution functions $G_1$ and $G_2$ (b) of the string endpoints for the cases of (i) $\omega =2f\left({\eta }_{\max }\right)$, (ii) $\omega =f\left({\eta }_{\max }\right)$, and (iii) $\alpha =-0.5, \beta =3$, as indicated in the legend. The light dot-dashed line corresponds to the model with one endpoint fixed and the other one fluctuating [1], which overlaps with case (ii) for $\eta \le {\eta }_{\max }$. The vertical line in panel (b) indicates $\eta ={\eta }_{\max }$. See the text for further details.

Figure 5

Covariance for the emission from a single string for cases (i) (a), (ii) (b), and (iii) (c).

Figure 6

Correlations $C_{AB}({\eta }_1,{\eta }_2)$ for the $6\%$ most central Au-Au collisions for model cases (i) (a) and (ii) (b), as well as $C_{AB}^{*}({\eta }_1,{\eta }_2)$ for case (i) (c).

Figure 7

$a_{11}$ (a) and $a_{11}^{*}$ (b) for Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV as a function of $\langle N_{+}\rangle$ (the selected values for $\langle N_{+}\rangle$ correspond to the 6 centrality classes $0–6\%, 6–15\%, 15–25\%, 25–35\%, 35–45\%$, and $45–55\%$) in cases (i) with $\omega =2f\left({\eta }_{\max }\right)$, (ii) with $\omega =f\left({\eta }_{\max }\right)$, (iii) with $\alpha =-0.5,\beta =3$, together with the model of [1] with one endpoint fixed, as indicated in the legend. Panel (c) displays the ratio $a_{11}^{*}/a_{11}$. To enhance visibility, the markers for the overlapping cases are slightly shifted to the left or right along the abscissa.

Figure 8

The product of $a_{11}$ and $\langle N_{+}\rangle$, showing the scaling discussed in the text.

Figure 9

Same as in Fig. 7 but for $d$-Au collisions at $\sqrt{s_{NN}}=200$ GeV, plotted as a function of the average number of wounded quarks in Au, $\langle N_B\rangle$ (selected values for $\langle N_B\rangle$ correspond to centrality classes $0–20\%, 20–40\%, 40–60\%, 60–80\%$).

Figure 10

Same as in Fig. 7 but for $a_{22}$ (a) and $-a_{13}$ (b).