Multiplicity dependence of inclusive $p_t$ spectra from $p\mathrm{\text{−}}p$ collisions at $\sqrt{s}=200\mathrm{GeV}$

We report measurements of transverse momentum $p_t$ spectra for ten event multiplicity classes of $p\mathrm{\text{−}}p$ collisions at $\sqrt{s}=200\mathrm{GeV}$. By analyzing the multiplicity dependence we find that the spectrum shape can be decomposed into a part with amplitude proportional to multiplicity and described by a Lévy distribution on transverse mass $m_t$, and a part with amplitude proportional to multiplicity squared and described by a Gaussian distribution on transverse rapidity $y_t$. The functional forms of the two parts are nearly independent of event multiplicity. The two parts can be identified with the soft and hard components of a two-component model of $p\mathrm{\text{−}}p$ collisions. This analysis then provides the first isolation of the hard component of the $p_t$ spectrum as a distribution of simple form on $y_t$.

Figure 1

Corrected and normalized charged-particle spectra on transverse momentum $p_t$ (left) and transverse rapidity $y_t$ (right) for 10 event multiplicity classes, displaced upward by successive factors 40 relative to ${\widehat{n}}_{\mathrm{ch}}=1$ at the bottom. Solid curves represent reference function $n_s/n_{\mathrm{ch}}·S_0(y_t)$ (cf. Sec. 4c). Dotted curves are spline fits to guide the eye.

Figure 2

Left: Relative residuals from power-law fits to $p_t$ spectra in Fig. 1. The hatched band represents the expected statistical errors for STAR data. Right: Exponents $n$ from power-law fits to data (solid points) and to corresponding two-component fixed-model functions (open circles, see Sec. 6) compared to the two-component fixed-model Lévy exponent $12.8\pm 0.15$ (hatched band).

Figure 3

Left: Spectra from Fig. 1 (right panel) replotted as spline curves and without offsets (solid curves) compared to reference $S_0$ (dash-dot curve). Right: Running integrals Eq. (1) of extrapolated $y_t$ spectra in Fig. 1 divided by $n_s/n_{\mathrm{ch}}$ (solid curves) compared to running integral $N_0(y_t)$ of reference $S_0(y_t)$ (dash-dot curve). The ten data curves from bottom to top correspond to increasing ${\widehat{n}}_{\mathrm{ch}}\in [1,11.5]$.

Figure 4

Left: Differences between integrals of extrapolated $y_t$ distributions in Fig. 1 according to Eq. (1), and integral $N_0(y_t)$ of soft reference $S_0(y_t)$. The ten curves correspond to $n_{\mathrm{ch}}\in [1,11.5]$ from bottom to top. Right: Distributions in the left panel divided by their end point values at $y_t=4.5$. The dashed curve is the running integral of $H_0$ (cf. Sec. 5). The dash-dot curve is $N_0$, the running integral of $S_0$.

Figure 5

Left panel: Distributions on $y_t$ in Fig. 1 (right panel) divided by $n_s/n_{\mathrm{ch}}$ minus reference $S_0(y_t)$. Dashed curves in the left panel and solid curves in the right panel are spline fits to guide the eye. The vertical dotted lines enclose region $B$ previously defined. Right panel: Distributions $H(n_{\mathrm{ch}},y_t)$ obtained by dividing the curves in the left panel by $n_h/n_s$. The dashed curve represents hard reference $H_0(y_t)$. The dash-dot curve represents soft reference $S_0(y_t)$. The solid curve underlying the dash-dot curve is an error function [19]. The hatched region estimates the systematic error from subtraction of $S_0$.

Figure 6

Left: Relative residuals between data $y_t$ spectra in Fig. 1 (right panel) and the two-component fixed parametrization, for all multiplicity classes. Right: Relative residuals between data $y_t$ spectra and two-component free ${\chi }^2$ fits, also for all multiplicity classes. The hatched bands represent the expected statistical errors for STAR data.

Figure 7

Left: Parameters $n$ and $n_h/{\tilde{n}}_s$ vs ${\widehat{n}}_{\mathrm{ch}}$ from free ${\chi }^2$ fits (solid symbols) in Table . The open symbols represent similar free fits but with ${\overline{y}}_t$ held fixed at 2.65 (see text for discussion). The bands represent the corresponding two-component fixed parametrizations with their stated errors also given in Table . Right: Ratio $n_h({\widehat{n}}_{\mathrm{ch}})/n_s({\widehat{n}}_{\mathrm{ch}})$ derived from integrals in Fig. 4 (left panel) (open squares) and from Gaussian amplitudes in Fig. 5 (left panel) (solid dots). The dashed and dotted lines have slopes 0.105 and 0.095, respectively. The solid curve is described in the text.

Figure 8

$⟨p_t⟩({\widehat{n}}_{\mathrm{ch}})$ derived from the two-component $H_0$ Gaussian amplitudes (solid dots), from the running integrals (open squares), and from power-law fits to STAR and UA1 data (open circles, triangles). The solid and dotted lines correspond to Eq. (6) with $\alpha =0.0095$ and 0.015, respectively.

Figure 9

Two-component analysis applied to Pythia Monte Carlo data with the same multiplicity classes as for STAR data. The left and right panels may be compared with Figs. 3 (left panel) and 5 (left panel), respectively. Dashed curves in the right panels represent the STAR data hard component for ${\widehat{n}}_{\mathrm{ch}}\sim 11$ ($H_0/10$). Dash-dot curves in the left panels represent soft component $S_0$ optimized for each Monte Carlo configuration: Pythia V6.222 default with parameters $T_0=0.147\mathrm{GeV}$ and $n=23$ (upper panel); Pythia tune $A$ with parameters $T_0=0.137\mathrm{GeV}$ and $n=14$ (lower panel).

Figure 10

Decomposition of inclusive $p_t$ spectra into a soft component represented by a Lévy distribution on $m_t$ and a hard component represented by a Gaussian on $y_t$. Dashed curves $H_0/9$ correspond to data for ${\widehat{n}}_{\mathrm{ch}}=11.5$, while the dotted curve $H_0/140$ corresponds to data for ${\widehat{n}}_{\mathrm{ch}}=1$ (cf. Fig. 1). The dash-dot curves are soft reference $S_0$, and the solid curves are the totals of soft and hard components for the model. The dotted curve in the right panel estimates the shape of the inclusive $y_t$ distribution for those $p\mathrm{\text{−}}p$ collisions containing at least one minimum-bias hard parton scatter (hard events).