Ensemble mean $p_t$ versus charged-hadron multiplicities in high energy nuclear collisions

Measurements of event-ensemble mean transverse momentum $\langle p_t\rangle$ vs charged-hadron multiplicity $n_{\text{ch}}$ for $p_t$ spectra from 5-TeV $p$-Pb and 2.76-TeV Pb-Pb collisions and from $p\text{−}p$ collisions for several energies have been reported recently. While in all cases $\langle p_t\rangle$ increases monotonically with $n_{\mathrm{ch}}$ the rate of increase is very different from system to system. Comparisons with several theory Monte Carlos reveal substantial disagreements and lead to considerable uncertainty on how to interpret the $\langle p_t\rangle$ data. In the present study we develop a two-component ($\mathrm{soft}+\mathrm{hard})$ model (TCM) of $p_t$ production in high energy nuclear collisions and apply it to the $\langle p_t\rangle$ data. The soft component is assumed to be a universal feature of high energy collisions independent of the A-B system or energy. The hard-component model is based on the observation that dijet production in $p-p$ collisions does not satisfy the eikonal approximation but does so in $A\text{−}A$ collisions. Hard-component properties are determined from previous measurements of hadron spectrum hard components, jet spectra, and fragmentation functions. The TCM describes the $p-p$ and Pb-Pb $\langle p_t\rangle$ data accurately, within data uncertainties, and the $p$-Pb data appear to transition smoothly from $p-p$ to $A\text{−}A$ $n_{\mathrm{ch}}$ trends.

Figure 1

(Left) Hard components $H(y_t,n_{\text{ch}})$ for transverse-rapidity $y_t$ spectra from 11 multiplicity classes of 200-GeV NSD $p-p$ collisions normalized by hard multiplicity $n_h\approx {\rho }_h$ (within $\Delta \eta =1)$ [1]. The dashed curve is unit-normal hard-component Gaussian reference $H_0$. (Right) The ratio of hard to soft multiplicity (solid dots) plotted vs the soft multiplicity density. The trend indicates that dijets scale as $n_h=\alpha n_s^2$ within $\Delta \eta =1$ (solid line) inconsistent with the trend $n_s^{4/3}$ (dashed curve) implied by the eikonal approximation.

Figure 2

(Left) The $p-p$ spectrum soft-component model $S_0\left(y_t\right)=(1/n_s)d{n}_s/y_{t}d{y}_t$ (solid curve) and product $y_t{S}_0\left(y_t\right)$ (dashed curve). The nominal $p_t$ spectrum cutoff for Ref. [17] at 0.15 GeV/$c$ is denoted by the hatched band. (Right) The integral of $p_t{S}_0\left(p_t\right)$ above $p_{t,\text{cut}}$ vs the cut value (solid curve). The nominal cut position is denoted by the vertical hatched band. The horizontal hatched band indicates the range of $n_s^{\prime }/n_s$ ratios inferred from $\langle p_t\rangle$ data. The effect of the cut on $\langle p_t\rangle {}_s$ is denoted by ${\langle p_t\rangle }_s^{\prime }$ (dashed curve). The dash-dotted curve labeled $P_{t,s}^{\prime }=n_s^{\prime }{\langle p_t\rangle }_s^{\prime }$ indicates that the $P_{t,s}$ product is much less sensitive to the $p_t$ cut for smaller cut values.

Figure 3

(Left) $\langle p_t\rangle$ vs $n_{\mathrm{ch}}$ for several collision energies. The upper group of points is from Ref. [17]. The lower 900-GeV data from UA1 derived from a “power-law” spectrum model [12] fall substantially above the TCM for that energy (solid curve) but are consistent with the TCM form with amplitude adjusted (dashed curve). (Right) Data from the left panel multiplied by factor $n_{\text{ch}}^{\prime }/n_s$ that removes the jet contribution and the effect of the $p_t$ cut on the soft component from the denominator of $\langle p_t\rangle$. The universal soft component ${\langle p_t\rangle }_s$ is then subtracted according to Eq. (6), isolating product $x\left(n_s\right){\langle p_t\rangle }_h\left(\sqrt{s}\right)$.

Figure 4

(Left) $\langle p_t\rangle$ hard components isolated according to Eq. (7). The data fall within narrow horizontal bands reflecting the point-to-point data consistency with the TCM. The decreases for smaller $n_s$ reflect a real change in the hard component as seen in Fig. 1 (left panel) for 200-GeV spectra, here shown to be common to a range of collision energies. (Right) The ${\langle p_t\rangle }_h$ mean values from the left panel plotted vs parameter $\Delta y_{\text{max}}=ln(\sqrt{s}/\text{3}\text{GeV})$ that describes variation of the MB jet spectrum width with $p-p$ collision energy [23].

Figure 5

(Left) $\langle p_t\rangle {}^{\prime }$ data for 2.76-TeV Pb-Pb collisions from Ref. [17] (points). The solid curve is the Pb-Pb TCM. The dashed curve is a Glauber linear superposition (GLS) reference assuming that all $N\text{−}N$ collisions are equivalent to $p-p$ collisions and scale according to the Glauber model of Pb-Pb collisions following the eikonal approximation. ${\langle p_t\rangle }_s^{\prime }=P_{t,s}/n_{\text{ch}}^{\prime }$. (Right) The product $x_h\left(\nu \right){\langle p_t\rangle }_h\left(\nu \right)=P_{t,h}\left(\nu \right)/n_{pp}$ extracted from data in the left panel according to Eq. (8) with $x_{pp}{\langle p_t\rangle }_{h,pp}\approx 0.05$ GeV/$c$. The solid curve is part of the TCM.

Figure 6

(Left) Hadron production data vs mean participant path length $\nu$ for 2.76-TeV Pb-Pb collisions from Ref. [30] (solid triangles). The lines and solid curve are the TCM for 200-GeV Au-Au collisions scaled up by factors 1.85 (soft component) and $1.{85}^2$ (hard component) reflecting soft multiplicity $n_s$ scaling as $ln(\sqrt{s}/\text{10}\text{GeV})$, as noted in Refs. [23, 31]. (Right) The product data from Fig. 5 (right panel) divided by ratio model $x\left(\nu \right)$ defined by Eq. (10) and shown as the dashed curve in this panel. The solid curve is the ${\langle p_t\rangle }_h\left(\nu \right)$ model defined by Eq. (11). The combination is the basis for the Pb-Pb TCM.

Figure 7

(Left) The model functions $x\left(\nu \right)$ (dashed curve) and ${\langle p_t\rangle }_h\left(\nu \right)$ (solid curve) plotted vs path length $\nu$ over a larger centrality interval ($\nu \approx 6$ for central $A\text{−}A$ collisions). The $x\left(\nu \right)$ trend for 200-GeV Au-Au collisions (dash-dotted curve) is included for comparison. Corresponding 200-GeV ${\langle p_t\rangle }_h\left(\nu \right)$ data are not currently available. (Right) $\langle p_t\rangle$ data for 5-TeV $p$-Pb collisions from Ref. [17] (open circles). The dash-dotted curve is the TCM for 5-TeV $p-p$ collisions interpolated from the general $p-p$ TCM developed in this study. The dotted curve is the GLS for Pb-Pb collisions (dashed curve) with ratio $x$ increased by factor 2.5 and all else the same. The Pb-Pb data and TCM from Fig. 5 (left panel) are included for comparison.

Figure 8

(Left) Spectrum HC from 200-GeV NSD $p-p$ collisions (points) [3]. The solid curve is a pQCD prediction for the corresponding fragment distribution derived from measured FFs and a dijet total cross section of 2.5 mb. (Right) Spectrum HC for 0%–12% central 200-GeV Au-Au collisions (points) [16]. The solid curve is a pQCD prediction based on a simple modification of FFs [3]. The dotted curve is a GLS prediction of the TCM extrapolated from $p-p$ collisions.