Transverse-rapidity $y_t$ dependence of the nonjet azimuth quadrupole from 62- and 200-GeV Au-Au collisions

Previous measurements of a quadrupole component of azimuth correlations denoted by symbol $v_2$ have been interpreted to represent elliptic flow, a hydrodynamic phenomenon conjectured to play a major role in noncentral nucleus-nucleus collisions. $v_2$ measurements provide the main support for conclusions that a “perfect liquid” is formed in heavy-ion collisions at the Relativistic Heavy Ion Collider. However, conventional $v_2$ methods based on one-dimensional (1D) azimuth correlations give inconsistent results and may include a jet contribution. In some cases the data trends appear to be inconsistent with hydrodynamic interpretations. In this study we distinguish several components of 2D angular correlations and isolate a nonjet (NJ) azimuth quadrupole denoted by $v_2\left\{\text{2D}\right\}$. We establish systematic variations of the NJ quadrupole on $y_t$, centrality, and collision energy. We adopt transverse-rapidity $y_t$ as both a velocity measure and a logarithmic alternative to transverse momentum $p_t$. Based on NJ-quadrupole trends, we derive a completely factorized universal parametrization of quantity $v_2\left\{\text{2D}\right\}(y_t,b,\sqrt{s_{NN}})$ which describes the centrality, $y_t$, and energy dependence. From $y_t$-differential $v_2\left(y_t\right)$ data we isolate a quadrupole spectrum and infer a quadrupole source boost having unexpected properties. NJ quadrupole $v_2$ trends obtained with 2D model fits are remarkably simple. The centrality trend appears to be uncorrelated with a sharp transition in jet-related structure that may indicate rapid change of Au-Au medium properties. The lack of correspondence suggests that the NJ quadrupole may be insensitive to such a medium. Several quadrupole trends have interesting implications for hydro interpretations.

Figure 1

(Left) The relation between transverse momentum $p_t$ and transverse-rapidity $y_t\left(\pi \right)$ (assuming a pion mass). The grid shows the uniform $y_t$ bin system used for the present analysis. The dotted line provides a $log\left(p_t\right)$ reference for comparison. (Right) The symmetrized two-particle space $(y_{t1},y_{t2})$. An element of the joint distribution on $(y_{t1},y_{t2})$ is shown by the bold squares. An element of the marginal distribution on $y_t$ is shown by the bold rectangles.

Figure 2

(Left) Mean participant path length $\nu$ vs fractional cross-section measure $1-\sigma /{\sigma }_0$. The grid shows the 11 centrality bins for this analysis. The hatched region indicates the “sharp transition” in jet-related correlation properties reported in Ref. [16]. (Right) Path length $\nu$ vs fractional impact parameter $b/b_0=\sqrt{\sigma /{\sigma }_0}$. For Au-Au collisions $b_0\approx 14.7$ fm.

Figure 3

Centrality trends for optical and Monte Carlo Glauber IS azimuth multipoles with $m=2$, 3 on participant-projectile-nucleon number (left panel) and binary $N\text{−}N$ collisions $\nu$ per participant-nucleon pair (right panel).

Figure 4

2D data histograms from 40%–50% central 200-GeV Au-Au collisions for four $y_t$ bins: (a) $y_t<1.4$, (b) $1.8<y_t<2.2$, (c) $2.6<y_t<3.0$, (d) $3.4<y_t<3.8$. The narrow $\mathrm{BEC}+\mathrm{electron}$ peak at the origin decreases to zero amplitude for larger $y_t$.

Figure 5

Fits to 2D data histograms from 40%–50% central 200-GeV Au-Au collisions for two $y_t$ bins: (a), (b) $1.8<y_t<2.2$ and (c), (d) $3.4<y_t<3.8$. The model fits appear on the left in each case. The vertical scale is the same for fits and data. The upper limit has been adjusted to reveal the $\mathrm{BEC}+\mathrm{electron}$ contribution excluded from the fit (narrow peak in three central bins).

Figure 6

Per-pair fit parameters derived from 2D fits with Eq. (12) to $y_t$-differential data histograms from 200-GeV Au-Au collisions for eight centrality bins (0%–70% central). (a) SS 2D peak amplitude $B_{\text{2D}}(y_t,b)$; (b),(c) SS 2D peak rms widths ${\sigma }_{{\eta }_{\mathrm{\Delta }}}(y_t,b)$ and ${\sigma }_{{\varphi }_{\mathrm{\Delta }}}(y_t,b)$; (d) per-pair quadrupole amplitude $B_Q\left\{\text{2D}\right\}(y_t,b)$. The error bars indicate fit uncertainties. The $B_Q$ errors for $y_t=4.4$ are consistent with a 20% systematic uncertainty for that $y_t$ value, as noted in Sec. 10b.

Figure 7

(Left) Per-pair data $B_2\left\{\text{2D}\right\}(y_t,b)$ from the present analysis scaled as described in the text and compared with the minimum-bias pion data (solid curve) from Ref. [18] (left panel). The vertical dotted lines mark the TPC acceptance edge on $y_t$ for pions at $p_t=0.15$ GeV/$c$. (Right) The same procedure applied to data from 62-GeV Au-Au collisions.

Figure 8

Unit-normal quadrupole spectra for seven centralities of 200-GeV Au-Au collisions derived from Eq. (13). The spectrum shapes are independent of Au-Au centrality: All $Q(y_t,b)$ coincide within systematic uncertainties. The common shape denoted by $Q_0\left(y_t\right)$ is well described by a boosted Lévy distribution with parameters $T_2$ and $n_2$ [18].

Figure 9

$p_t$-differential $v_2\left\{\text{2D}\right\}(p_t,b)$ trends from 200-GeV Au-Au collisions (points) compared to a parametrization of $v_2\left\{\text{2D}\right\}(p_t,b)$ data given by Eq. (16) (solid curves) for (a) 40%–50%, (b) 30%–40%, (c) 20%–30%, (d) 10%–20%, (e) 5%–10%, (f) 0%–5% centralities. Solid points are for the AS 1D Gaussian peak model and open circles are for the AS dipole peak model in Eq. (12). Dotted curves are Eq. (16) without factor $f(y_t,b)$.

Figure 10

Residuals from 2D fits to $y_t$-differential histograms from 40%–50% central 200-GeV Au-Au collisions for (left) $y_t\in [1.8,2.2]$ ($p_t\in$ [0.4,0.65] GeV/$c$) and (right) $y_t\in [3.4,3.8]$ ($p_t\in$ [2,3.3] GeV/$c$). The vertical scales are half the range of the corresponding panels in Fig. 4.

Figure 11

Data-model deviations $Q(y_t,b)-Q_0\left(y_t\right)$ are shown relative to fixed spectrum model $Q_0\left(y_t\right)$. The upper hatched band indicates that the rms data deviation is less than 5% except near the acceptance end points. There are systematic deviations below 0.5 GeV/$c$ and larger fitting uncertainties for the largest $y_t$ bin. The lower hatched band provides an estimate of systematic uncertainties above $y_t=1.8$ and the range of variations corresponding to $v_2\left(b\right)$ vs $v_2^{\prime }\left(b\right)$ below $y_t=1.8$. The dashed curves are explained in the text.

Figure 12

(Left) Comparison of measured 2D angular correlations represented by $v_2^2\left\{\text{SS}\right\}(p_t,b)$ (dash-dotted curve) and $v_2^2\left\{\text{2D}\right\}(p_t,b)$ (hatched region, upper limit) with published $v_2^2\left\{\text{EP}\right\}(p_t,b)$ data (open circles) for 0%–5% central 200-GeV Au-Au collisions. (Right) Similar comparison for 5%–10% central Au-Au collisions showing the close correspondence between the sum $v_2^2\left\{\text{2D}\right\}(p_t,b)+v_2^2\left\{\text{SS}\right\}(p_t,b)$ (bold solid curve) and published $v_2^2\left\{\text{EP}\right\}(p_t,b)$ data (open circles) as in Eq. (26). The light solid curve through the $v_2^2\left\{\text{2D}\right\}$ data (solid dots) represents the parametrization in Eq. (16).

Figure 13

(Left) Comparison similar to Fig. 12 for 10%–20% central Au-Au collisions showing near-equal contributions from NJ quadrupole $v_2^2\left\{\text{2D}\right\}(p_t,b)$ and jet-related quadrupole $v_2^2\left\{\text{SS}\right\}(p_t,b)$. The bold solid curve is the sum of {2D} and {SS} data. The bold dotted curve is the parametrization of $v_2\left\{\text{2D}\right\}(p_t,b)$ from Eq. (16) combined with a $v_2\left\{\text{2D}\right\}\left(b\right)$ value derived from Refs. [14, 15] (Sec. 12b). (Right) The quadrupole amplitude of the SS 2D peak (dash-dotted curve) derived from Eq. (24) with data from Fig. 6 [same as Fig. 12 (left panel)] compared to a similar calculation using fixed SS 2D peak widths (dotted curve and points). The substantial differences illustrate the importance of accurately measured SS 2D peak properties for understanding jet biases in $v_2\left\{2\right\}\approx v_2\left\{\text{EP}\right\}$ data.

Figure 14

(Left) Collision energy dependence of $v_2$ data converted to per-particle measure $A_Q$. The solid points are from Refs. [14, 15] and Ref. [50]. They follow a logarithmic trend proportional to $R\left(\sqrt{s_{NN}}\right)$ (see text). The Bevalac-AGS trend below 13 GeV is proportional to $R^{\prime }\left(\sqrt{s_{NN}}\right)=ln(\sqrt{s_{NN}}/\text{3.2}\text{GeV})$. The open points are taken from Fig. 4 of Ref. [48]. (Right) The azimuth quadrupole amplitude $A_Q\left\{\text{2D}\right\}$ divided by optical ${\epsilon }_{\text{opt}}^2$ plotted vs energy-dependent factor $R\left(\sqrt{s_{NN}}\right)$ times 200-GeV Au-Au binary-collision number $N_{\text{bin}}\left(b\right)$. The $A_Q\left\{\text{2D}\right\}$ data are consistent with simple proportionality [Eq. (27)] over three decades.