Eventwise mean-$p_t$ fluctuations versus minimum-bias jets (minijets) at energies available at the CERN Large Hadron Collider

Fluctuation measurements of eventwise mean transverse momentum $\langle p_t\rangle$ for $p\text{−}p$ and Pb-Pb collisions at the CERN Large Hadron Collider (LHC) have been reported recently. In that study it was concluded that the strength of “nonstatistical” $\langle p_t\rangle$ fluctuations decreases with increasing particle multiplicity $n_{\text{ch}}$ (or $A\text{−}A$ centrality) and is nearly independent of collision energy over a large interval. Among several potential mechanisms for those trends the onset of thermalization and collectivity are mentioned. The LHC analysis employed one fluctuation measure selected from several possibilities. An alternative fluctuation measure reveals a strong increase of $p_t$ fluctuations with $n_{ch}$ (or $A\text{−}A$ centrality) and collision energy, consistent with previous measurements at the BNL Relativistic Heavy Ion Collider (RHIC). The $p_t$ fluctuation data for LHC $p\text{−}p$ collisions can be described accurately by a two-component $\left(\mathrm{soft}+\mathrm{hard}\right)$ model (TCM) in which the hard component represents dijet production. The data for Pb-Pb collisions are described accurately by a TCM reference for more-peripheral collisions (suggesting transparent collisions), but the data deviate quantitatively from the reference for more-central collisions, suggesting a modification of jet formation. Overall fluctuation data trends suggest that minimum-bias jets (minijets) dominate $p_t$ fluctuations at both the LHC and the RHIC.

Figure 1

(Left) Per-participant hadron production measured by $(2/N_{\text{part}})d{n}_{\text{ch}}/d\eta$ vs $\nu$ for 200-GeV Au-Au collisions (points) inferred from analysis of identified-hadron spectra [25]. The dash-dotted line is the conventional TCM with fixed $x=0.095$ [26]. The solid curve is described in the text. (Right) Hadron production vs $\nu$ for 2.76-TeV Pb-Pb collisions from Ref. [27] (solid triangles). The dash-dotted line and solid curve are the TCM for 200-GeV Au-Au collisions scaled up by factors 1.84 (soft component) and $1.{84}^2$ (hard component), respectively, reflecting soft multiplicity $n_s$ scaling as $ln(\sqrt{s}/\text{10}\text{GeV})$ [21, 28].

Figure 2

(Left) ${\overline{p}}_t$ vs $n_{\mathrm{ch}}$ for several collision energies (represented as $\langle p_t\rangle$ in Ref. [13]). The upper group of points is ${\overline{p}}_t^{\prime }$ from Ref. [14] biased by a low-$p_t$ acceptance cut near 0.15 GeV/$c$. (Right) LHC ${\overline{p}}_t^{\prime }$ data from the left panel multiplied by factor $n_{\text{ch}}^{\prime }/n_s$ that removes the bias from the low-$p_t$ cut and the jet contribution to the denominator of ${\overline{p}}_t^{\prime }$. The universal soft component ${\overline{p}}_{t,s}$ is then subtracted according to Eq. (25), isolating the ${\overline{P}}_t$ hard component as $x\left(n_s\right){\overline{p}}_{t,h}\left(\sqrt{s}\right)={\overline{P}}_{t,h}/n_s$.

Figure 3

(Left) Hard components ${\overline{p}}_{t,h}$ (represented here as ${\langle p_t\rangle }_h$ from Ref. [13]) isolated according to Eq. (26). Most data fall within narrow horizontal bands reflecting point-to-point data variation consistent with the $p\text{−}p$ TCM. The solid curve is a parametrization of 7-TeV data used in the present study. (Right) ${\overline{p}}_{t,h}$ mean values from the left panel plotted vs parameter $\mathrm{\Delta }{y}_{\text{max}}=ln(\sqrt{s}/\text{3}\text{GeV})$ describing variation of MB jet-spectrum widths with $p\text{−}p$ collision energy [21].

Figure 4

(Left) Representation of $\langle p_t\rangle$ fluctuation data from Ref. [1], Fig. 1 (points), with a measure that exhibits similar trends for three energies available at LHC. The curves are described in the text. ${\text{mpt}}^{\prime }$ stands for uncorrected ${\overline{p}}_t^{\prime }$ biased by a low-$p_t$ acceptance cut. (Right) Data from the left panel transformed according to the axis labels but employing corrected charge density $n_{\text{ch}}/\mathrm{\Delta }\eta$. The curves are described in the text.

Figure 5

(Left) $P_t|n$ variance difference $B$ derived from the data in Fig. 4 (left) according to the axis labels. Factor $n_{\text{ch}}^{\prime 2}$, biased by a low-$p_t$ acceptance cut, cancels the same bias in $C^{\prime }$. (Right) Ratio $B/n_s$ data (points) exhibiting the LS scaling (dashed line) expected for dijets. In both panels the dashed curve (line) represents the LS trend (participant-gluon-pair LS) in the $p\text{−}p$ TCM.

Figure 6

(Left) Representation of 2.76-TeV Pb-Pb data from Fig. 8 (solid points) and 7-TeV $p\text{−}p$ data from Fig. 1 (open circles) of Ref. [1] normalized to $A/\sqrt{n_{\text{ch}}}$ from a fit to hijing data. (Right) Transformation of the Pb-Pb data from the left panel according to the axis labels with $A=0.315$. The result is equivalent to $\mathrm{\Delta }{\sigma }_{P_t|n}$ from Ref. [3]. $n_{\text{ch}0}$ is an estimate of the charge multiplicity corresponding to $b=0$ central Pb-Pb collisions. The thin lines connecting solid points guide the eye. Other curves in both panels are described in the text.

Figure 7

(Left) $n_{\text{ch}}C\approx B/n_{\text{ch}}=\mathrm{\Delta }{\sigma }_{P_t|n}^2$ (points) transformed from the 2.76-TeV Pb-Pb data in Fig. 6 (right). (Right) Equivalent data for 200-GeV Au-Au collisions obtained from a study of $p_t$ fluctuation scale dependence in Ref. [4]. The solid lines guide the eye.

Figure 8

Per-particle data from Fig. 7 multiplied by factor $2n_{\text{ch}}/N_{\text{part}}$ to obtain per-participant trends for $A\text{−}A$ collisions equivalent to $B/n_s$ in Fig. 5 (right) for $p\text{−}p$ collisions. The dash-dotted lines represent TCM “first-hit” trends if $A\text{−}A{B}_h^{\prime }$ in Eq. (28) is replaced with in-vacuum $p\text{−}p{B}_h$. The solid lines guide the eye.

Figure 9

(Left) Representation of Pb-Pb data (solid points) from Fig. 8 of Ref. [1] including default ampt (open circles) and hijing (open triangles) MC results. (Right) The Pb-Pb data and ampt MC results transformed to per-participant format $(2/N_{\text{part}})B$. The solid lines guide the eye.

Figure 10

Data and trends from Fig. 7 compared to hijing MC results. The MC results in the left panel are transformed from Fig. 9 (left) assuming a fixed value ${\overline{p}}_t^{\prime }=0.5$ GeV/$c$. The MC results in the right panel are obtained from Ref. [6]. The solid lines guide the eye.

Figure 11

(Left) $\mathrm{\Delta }{\sigma }_{p_t:n}^2$ (GeV/${c)}^2$ distributions on scale $(\delta \eta ,\delta \varphi )$ for 45%–55% central 200-GeV Au-Au collisions. (Right) Corresponding autocorrelations on difference variables $\left({\eta }_{\mathrm{\Delta }},{\varphi }_{\mathrm{\Delta }}\right)$ inferred from data at left by fluctuation inversion.

Figure 12

(Left) Jet-related SS 2D peak amplitude (solid dots) vs path length $\nu$ from 2D fits to $p_t$ angular correlations from 200-GeV Au-Au collisions as in Fig. 11 (right). The dash-dotted and dotted lines represent corresponding hijing quench-off and quench-on results, respectively, from Ref. [6]. There is no soft component to those jet-related data. (Right) Per-particle $p_t$ fluctuation amplitude $B/n_{\text{ch}}$ as shown in Fig. 7 (right) for comparison with correlations, including the $p\text{−}p$ estimate $B_{pp}/n_s\approx 0.0112$ (GeV/${c)}^2$ from Sec. 6b. The solid lines guide the eye.

Figure 13

(Top) $p_t$ angular correlations for (a) 85%–95% and (b) 10%–20% central 200-GeV Au-Au collisions inferred by inverting $p_t$ fluctuation scale dependence [4]. AS dipole and nonjet quadrupole components of 2D model fits to the data are subtracted. (Bottom) Results for the same collision systems but $p_t$ correlations are obtained by direct pair counting rather than fluctuation inversion. Improved angular resolution and unfiltered statistical fluctuations are evident. Data for all panels include an additional acceptance factor $4\pi$.