Azimuth quadrupole component spectra on transverse rapidity $y_t$ for identified hadrons from Au-Au collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV

I present the first isolation of azimuth quadrupole components from published $v_2(p_t)$ data (called elliptic flow) as spectra on transverse rapidity $y_t$ for identified pions, kaons, and Λ's/protons from minimum-bias Au-Au collisions at 200 GeV. The form of the spectra on $y_t$ indicates that the three hadron species are emitted from a common boosted source with boost $\Delta y_{t0}~0.6$. The quadrupole spectra have a Lévy form similar to the soft component of the single-particle spectrum, but with significantly reduced ($~0.7$ times) slope parameters $T$. Comparison of quadrupole spectra with single-particle spectra suggests that the quadrupole component comprises a small fraction (<5%) of the total hadron yield, contradicting the hydrodynamic picture of a thermalized, flowing bulk medium. The form of $v_2(p_t)$ is, within a constant factor, the product of $p_t^{\text{'}}$ ($p_t$ in the boost frame) times the ratio of quadrupole spectrum to single-particle spectrum. That ratio in turn implies that above $0.5\hspace{0.5em}\mathrm{GeV}/c$, the form of $v_2(p_t)$ is dominated by the hard component of the single-particle spectrum (interpreted as being due to minijets). It is therefore unlikely that the so-called constituent-quark scaling attributed to $v_2$ is relevant to soft hadron production mechanisms (e.g., chemical freeze-out).

Figure 1

Left panel: $v_2(p_t)$ data for three hadron species plotted in the usual format [24, 25]. Right panel: The same $v_2(p_t)$ data divided by $p_t$ in the laboratory frame suggest a universality on proper transverse rapidity for each hadron species—in particular the correspondence of data near $y_t=1$. Dotted curves A and B in each panel are viscous hydro predictions from Refs. [1] and [27], respectively. The three curves through data are derived from this analysis.

Figure 2

Summary of pion (left) and proton (right) per-participant-pair single-particle spectra from Au-Au collisions at 200 GeV and five centralities [14]. $H_{\mathit{NN}}$ is the hard component (minimum-bias transverse parton fragmentation), and $S_{\mathit{NN}}$ is the soft component (longitudinal nucleon fragmentation), both inferred for $N\text{−}N$ collisions. The solid points in the left panel represent the NSD $p$-$p$ spectrum [28].

Figure 3

Relative residuals (data–model)/model for pions (left) and protons (right) from five centralities of Au-Au collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. The two-component spectrum model with modification factors $r_{\mathit{AA}}$ inferred in Ref. [14] describes data to the statistical limits from 0.5 to 10 GeV/$c$ ($y_t\in [2,5]$).

Figure 4

Left panel: $p_t^{\text{'}}$ ($p_t$ in the boost frame) vs $p_t$ in the laboratory frame. Factor ${\gamma }_t(1-{\beta }_t)$ in the denominator ensures that the combination $\to p_t$ for large $p_t$. Right panel: The same ratio is plotted on proper $y_t$ for each hadron species, demonstrating a fundamental relationship applicable to any hadron species.

Figure 5

Left panel: Quadrupole component modeled by a thermal spectrum boosted by $\Delta y_t(\varphi )$ containing monopole and quadrupole terms: monopole boost $\Delta y_{t0}=1$ and quadrupole boost amplitude $\Delta y_{t2}=0.5$. Right panel: Projection of the left panel onto $y_t$ revealing the edge structure.

Figure 6

Left panels: Quadrupole component modeled as in Fig. 5 for $\Delta y_{t0}=0.5$ and two values of quadrupole boost amplitude $\Delta y_{t2}$. Right panels: Corresponding Fourier amplitudes $V_2(y_t)$ from Eq. (19) normalized by quadrupole boost amplitudes $\Delta y_{t2}$. Widths of the negative regions are $2\Delta y_{t2}$. The dash-dot curve is the quadrupole spectrum ${\rho }_2(y_t;\Delta y_{t0})$.

Figure 7

Left panel: Structure of factor $F_2$ in Eq. (21), dominated by $\Delta y_{t2}$ at smaller $y_t$. Right panel: Structure of $O(1)$ factor $f(y_t;\Delta y_{t0},\Delta y_{t2})$ defined in Eq. (22).

Figure 8

Formation of quadrupole spectra from $v_2(p_t)$ data and single-particle spectra—measured quantities combined with full-spectrum two-component parametrizations. The open symbols are the values of ${\rho }_0(y_t)$ used for the conversion. The solid symbols are the resulting approximations to quadrupole spectra. The dashed curves are from the present analysis. The solid curves result from removing a $p_t^{\text{'}}/p_t$ factor. The dash-dotted curves are hard-component models from Ref. [14].

Figure 9

Spectra from Fig. 8 transformed to proper $y_t$ for each hadron species. The dotted curves are soft components from respective single-particle spectra for comparison. The prominent feature is the common edge at $y_t~0.6$, implying that the three hadron species originate from a common boosted source. The hadron abundances and spectrum shapes are the same as the single-particle spectrum soft components.

Figure 10

Left panel: $v_2(p_t)$ data from Fig. 1 (left panel) are repeated for comparison. The curves through data are from the present analysis. Right panel: Relation between $p_t$ in the boost frame and laboratory frame for three hadron masses. Intercepts on the abscissa are at $p_{t0}$ values defined in the text.

Figure 11

Ratio of quadrupole (boosted soft component) spectra to single-particle spectra plotted on $p_t$ (left panel) and proper $y_t$ for each hadron species (right panel; solid, dashed, dash-dot) for minimum-bias Au-Au collisions ($\nu ~3.5$). Spectrum ratios with no hard component (dotted, $\nu =0$) are relevant to a hydro description. Ratios for no monopole boost (dashed, $\Delta y_{t0}=0$) are also plotted. The dominant role of the hard component in $v_2(p_t)$ is most evident in the right panel.

Figure 12

Left panel: Ratio of quadrupole component (boosted soft component) to spectrum soft component (both Lévy distributions) for three hadron species illustrating typical hydro mass dependence of the ratio. The dotted curve includes the factor $p_t^{\text{'}}/p_t{\gamma }_t(1-{\beta }_t)$ for comparison with the right panel. Right panel: $v_2/p_t$ data compared with hydro theory curves as described in the text. Different boost distributions are apparent at lower left.

Figure 13

Lower (solid) and upper (dotted) limits on quadrupole spectra, the latter obtained by direct comparison to single-particle spectra (dash-dot curves).

Figure 14

Left panel: Minimum values of quadrupole spectra compared with minimum-bias spectrum residuals relative to the two-component model [14]. Right panel: Expected centrality dependence of the quadrupole integral $(2/n_{\mathrm{part}})\Delta y_{t2}\hspace{0.5em}{n}_{\mathrm{ch}2}$ compared with that for the spectrum hard component.

Figure 15

Lower limits on quadrupole spectra in the form $(2/n_{\mathrm{part}}){\rho }_2(y_t)$ (solid points and curves) compared with single-particle spectrum residuals (open points and thin curves of various styles) for five Au-Au centralities relative to a two-component reference [14]. Soft components $S_{\mathit{NN}}(y_t)$ provide a reference. The dashed curves corresponding to hadrons from a boosted source with $\Delta y_t~1.1$ and substantially smaller slope parameters $T$ than the quadrupole components suggest a possible mechanism for the residual spectrum structure.

Figure 16

Comparison of the quadrupole component with soft and hard spectrum components for pions and protons.

Figure 17

Left panel: $p_t$ in the boost frame compared with transverse kinetic energy $m_t-m_0$ in the laboratory frame for monopole boost value $\Delta y_{t0}=0.6$. The curve intercepts are at $m_{t0}-m_0$ as defined in the text. Right panel: The same relations rescaled by constituent quark number as $2/n_q$ so that meson trends are unchanged. The shift for baryons is indicated.

Figure 18

Left panel: $v_2(p_t)$ vs kinetic energy $m_t-m_0$. The intercept spacing at small $p_t$ is reduced by a factor of 3. The lower dotted curve is the zero-viscosity hydro curve [1]. The upper dotted curve is the hydro curve $\times \hspace{0.5em}2.7$. Right panel: Same as left panel, except with constituent quark scaling in the form $2/n_q$ so that meson trends remain unchanged. The shift for baryons is indicated by the arrows.