Questioning tests of hadron production models based on minimum-bias distributions on transverse energy

A recent study reports measurements of transverse-energy $E_t$ distributions at midrapidity for several high-energy nuclear collision systems. The $E_t$ data are analyzed in the context of constituent-quark (CQ) participants estimated with a Glauber-model simulation. The study concludes that systematic variations of hadron and $E_t$ yields previously interpreted in terms of a two-component $\mathrm{soft}+\mathrm{hard}$ model (TCM) of hadron production including a dijet (hard) contribution are actually the result of CQ participant trends with only soft production. It is claimed that deviations from linear scaling with the number of nucleon participants of hadron yields vs $A-A$ centrality do not actually arise from dijet production, as previously assumed. In the present study I examine the new $E_t$ data in the context of the TCM and compare those results with previous differential spectrum and minimum-bias correlation analysis. I present substantial evidence supporting a significant dijet contribution to all high-energy nuclear collisions consistent with the TCM and conclude that the $E_t$ data, given their systematic uncertainties, fail to support claimed CQ model interpretations.

Figure 1

(a) Ratio $N_{\mathrm{qp}}/N_{\mathrm{part}}$ derived from a Glauber Monte Carlo simulation of 200-GeV Au-Au collisions from Ref. [1], with $N_{\mathrm{part}}$ values from that reference (open circles) and from Eq. (1) (solid points) with a parametrization used in the present study (curve). The two values for 200-GeV $p-p$ collisions (triangles) are discussed in Secs. 4c and 7b. (b) The ratio parametrization from panel (a) vs participant path length $\nu$ (solid curve). Also plotted are the TCM for 200-GeV Au-Au with fixed $x=0.1$ (upper dashed curve) and $(2/N_{\mathrm{part}})d{n}_{\mathrm{ch}}/d\eta$ trends with variable $x$ for three energies (lower dashed curve and dash-dotted and dotted curves). Those curves are scaled down relative to data by factor 1.35.

Figure 2

(a) Per-participant transverse-energy production measured by $(2/N_{\mathrm{part}})d{E}_t/d\eta$ vs centrality for 200-, 130-, and 62.4-GeV Au-Au collisions. The data and uncertainties ($\mathrm{stat}+\mathrm{sys}$) are from Tables V–VII of Ref. [1]. The $N_{\mathrm{part}}$ and $\nu$ values are from Eq. (1). The solid triangle is the corrected $p-p$ value from Table X of Ref. [1]. (b) Data from panel (a) divided by ratio $N_{\mathrm{qp}}/N_{\mathrm{part}}$ from Fig. 1. The $p-p$ points are obtained from the point in panel (a) with $N_{\mathrm{qp}}/N_{\mathrm{part}}$ ratio values 1.95 (solid point, extrapolation of the Fig. 1 solid curve) and 1.4 (open point, separate $p-p$ Glauber CQ simulation reported in Ref. [1]).

Figure 3

(a) The relation of measured charged-hadron production to Glauber-simulated number of CQs in Au-Au collisions for three energies derived from the curves in Fig. 1. The points are for 200-GeV $p-p$ collisions. (b) Per-hadron mean $E_t$ defined as $\langle e_t\rangle =d{E}_t/d\eta /d{n}_{tot}/d\eta$, with $n_{\mathrm{tot}}$ (all hadrons contributing to $E_t$) approximated by $1.5n_{\mathrm{ch}}$ assuming that pions dominate (approximately 85% of all hadrons). The $d{E}_t/d\eta$ data and uncertainties are taken from Tables V–VII of Ref. [1]. The $d{n}_{\mathrm{tot}}/d\eta$ trends are derived from STAR data represented by TCM parametrizations with varying $x$ in Fig. 1. The solid curve is a prediction from the TCM for 200-GeV Au-Au collisions [11]. Other curves are described in the text.

Figure 4

(a) Hard/soft multiplicity ratio $n_h/n_s$ (points) vs SC $n_s$ consistent with a linear trend (line). Those data are the basis for Eq. (8). Assuming that $n_s$ represents the density of small-$x$ participant partons (gluons) and $n_h$ represents dijet production by parton scattering, an eikonal model of $p-p$ collision geometry (analogous to the Glauber model of $A-A$ collisions) would predict the $n_s^{1/3}$ trend (dashed curve). (b) The $\langle p_t\rangle$ trend predicted by the $p-p$ spectrum TCM (line) and as determined by direct spectrum integration (points). The hatched band represents the uncertainty in extrapolating spectra to zero momentum.

Figure 5

(a) Minimum-bias correlated-pair density on 2D transverse-rapidity space $y_t\times y_t$ from 200 GeV $p-p$ collisions showing soft (smaller $y_t$) and hard (larger $y_t$) components as peak structures. (b) Correlated-pair density on 2D angular difference space $({\eta }_{\Delta },{\varphi }_{\Delta })$. Hadrons are selected with $p_t\approx 0.6$ GeV/$c$ ($y_t\approx 2$). Nevertheless, features expected from dijets are observed: (a) SS 2D peak representing intrajet correlations and (b) AS 1D peak on azimuth representing interjet (back-to-back jet) correlations.

Figure 6

(a) Hard components from 0–12% central 200-GeV Au-Au collisions for identified pions (solid points [13]) and from 200-GeV $p-p$ collisions (open points [3]). The dashed and dash-dotted curves show the GLS references for $p-p$ and Au-Au collisions, respectively. The solid curve is a pQCD description of Au-Au HC data assuming a slight modification of $e^{+}-e^{-}$ FFs [11]. (b) Spectra for identified protons (green solid curves) from several centralities of Au-Au collisions. The red dotted curve is the TCM SC. The dashed curve is the TCM proton HC inferred from Au-Au spectrum systematics. The dash-dotted curves show GLS references for peripheral and central Au-Au collisions. The data HCs for more-central collisions show an increase relative to GLS above the HC peak mode at $y_t=2.7$ (1 GeV/$c$), in contrast to the identified-pion result.

Figure 7

(a) Angular correlations from 85–95% central 200-GeV Au-Au collisions equivalent to NSD $N-N$ collisions. The SC is a 1D peak on ${\eta }_{\Delta }$. The HCs are a SS 2D peak and AS 1D peak on azimuth. Conversion electrons and Bose-Einstein correlations contribute a narrow exponential peak at the origin confined to $p_t<0.5$ GeV/$c$. (b) The same for 0–5% central Au-Au collisions. The SS 2D peak is strongly elongated on ${\eta }_{\Delta }$. The SC falls to zero by midcentrality. The nonjet quadrupole ($v_2$) amplitude is negligible [18, 19].

Figure 8

(a) The SS 2D peak amplitude vs centrality measured by path length $\nu$ for two collision energies [15]. The 62.4-GeV data coincide with 200-GeV data when multiplied by factor 1.6, as discussed in the text. The data follow the GLS reference (dashed curve) up to $\nu =3$ (ST), after which the slope changes by factor 3 or more. (b) Equivalent results for the AS 1D peak amplitude. The 62- and 200-GeV data correspond for more-peripheral collisions without additional factor. That relation is expected for dijets.

Figure 9

(Top) $p_t$ angular correlations for (a) 85–95% and (b) 10–20% central 200-GeV Au-Au collisions inferred by inverting $\langle p_t\rangle$ fluctuation scale dependence [21]. AS dipole and nonjet quadrupole components of 2D model fits to the data have been subtracted. (Bottom) Results for the same collision systems but the $p_t$ correlations are obtained by direct pair counting rather than fluctuation inversion. Improved angular resolution is evident, as well as unfiltered statistical fluctuations.

Figure 10

(a) Dijet number within angular acceptance $\Delta \eta =2$ at midrapidity for 200-GeV Au-Au collisions scaling with $N_{\mathrm{bin}}$ from Eq. (1) (GLS reference, solid curve). The horizontal hatched band indicates the measured dijet rate for NSD $p-p$ collisions in the same acceptance. The upper-right hatched band indicates the ST near $\nu =3$ for jet correlations in Au-Au collisions. The dashed curve and leftmost hatched band are discussed in Appendix B 3. (b) Per-participant hadron production measured by $(2/N_{\mathrm{part}})d{n}_{\mathrm{ch}}/d\eta$ vs $\nu$ for 200-GeV Au-Au collisions (solid points) inferred from analysis of identified-hadron spectra [13]. The dash-dotted line is the conventional TCM with fixed $x=0.095$ [12]. The solid curve is a prediction obtained by analysis of jet-related angular correlations based on the $n_j$ solid curve in panel (a) [30]. The dashed curve is a prediction derived from color-glass condensate theory [12, 34].

Figure 11

(a) $\langle p_t\rangle$ vs $n_{\mathrm{ch}}$ data for several energies from the SPS, RHIC, and LHC [3, 6, 36]. The curves through data are defined by Eq. (13). (b) $\langle p_t\rangle$ HCs inferred from the data in panel (a).

Figure 12

(a) $n_{\mathrm{ch}}$-averaged HC ${\langle p_t\rangle }_h\left(\sqrt{s}\right)$ values vs jet spectrum width measure $\Delta y_{\max }$ demonstrating the linear correlation that confirms a dijet origin for the $p-p$ spectrum HC, as established in Ref. [11]. (b) $p$-Pb and Pb-Pb $\langle p_t\rangle$ data from the LHC [36] (points) compared to the $\langle p_t\rangle$ TCM (curves). The $p$-Pb data for larger $n_{\mathrm{ch}}$ are well described by a Glauber linear superposition (GLS) form of the TCM consistent with transparent $p$-Pb collisions (no rescattering). For smaller $n_{\mathrm{ch}}$ the $p$-Pb data correspond quantitatively to $p-p$ collisions at the same energy.

Figure 13

(a) MB distributions on $n_{\mathrm{ch}}^{1/4}$ derived from Glauber Monte Carlo simulations [23] assuming participant scaling (dotted lines) and the conventional TCM with fixed $x=0.1$ (dashed lines) according to Eq. (A2). (b) The same trends plotted in the conventional MB semilog format on $n_{\mathrm{ch}}$.

Figure 14

(a) MB distribution of negative hadrons $h^{-}$ from 130-GeV Au-Au collisions (solid curve [39]) compared to power-law participant scaling (dotted curve) and the TCM with fixed $x=0.08$ (dashed curve). The dash-dotted curve is discussed in the text. (b) The same distributions plotted in power-law format. Information in the data is only the more-central mean slope (reflecting the TCM $x$ value for more-central collisions) and the slope at the terminus reflecting hadron production fluctuations in Au-Au collisions with $N_{\mathrm{part},\max }=382$. The step in the dashed curve at midcentrality is a schematic model of the ST (see the text).

Figure 15

(a) Models of MB frequency distributions on $n_{\mathrm{ch}}$ in the power-law format including no fluctuations (dash-dotted curve), fluctuations for the STAR TPC acceptance (dotted curve) and for the fiducial PHENIX EMCal acceptance (solid curve), both corrected to the reference acceptance. (b) Measured production centrality trends for STAR (solid dots) [13] and PHENIX (open squares) and “RHIC average” (open triangles) [40] compared to integrated model MB distributions from panel (a) (same line styles). The large effect of angular acceptance on fluctuation distortions for more-central collisions is demonstrated.

Figure 16

(a) Hadron production vs centrality for 2.76-TeV Pb-Pb collisions (inverted solid triangles [36]) compared to TCM trends (dash-dotted and dashed lines) extrapolated from the 200-GeV TCM based on measured RHIC energy trends as described in the text. The upright solid triangle denotes $p-p$ data. (b) Hadron production vs centrality for 200-GeV Au-Au collisions (open triangles [40]) compared to a TCM prediction from analysis of jet-related angular correlations (dotted curve) [30]. The dash-dotted line indicates the conventional 200-GeV Au-Au TCM with fixed $x=0.095$ [12]. The dashed line indicates a GLS extrapolation from $p-p$ collisions with $x=0.015$. The solid triangles and solid line are the solid triangles and dash-dotted line in panel (a) scaled down by factor 2.1. The two line slopes differ by factor 1.55.