Extrapolating parton fragmentation to low $Q^2$ in $e^{+}\mathrm{\text{−}}{e}^{-}$ collisions

We analyze the energy-scale dependence of fragmentation functions from $e^{+}\mathrm{\text{−}}{e}^{-}$ collisions using conventional momentum measures $x_p$ and ${\xi }_p$ and rapidity $y$. We find that replotting fragmentation functions on a normalized rapidity variable results in a compact form precisely represented by the beta distribution, its two parameters varying slowly and simply with parton energy scale $Q$. The resulting parameterization enables extrapolation of fragmentation functions to low $Q$ in order to describe fragment distributions at low transverse momentum $p_t$ in heavy ion collisions at RHIC.

Figure 1

Left panel: Schematic illustration of the double-log approximation (DLA) to parton fragmentation. Right panel: Self-similar variation with energy of the the corresponding fragment distribution on $y$.

Figure 2

Left panel: $e^{+}\mathrm{\text{−}}{e}^{-}$ fragmentation functions on fractional momentum $x_p=p_{\mathrm{fragment}}/p_{\mathrm{parton}}$ for three CM energies. The dashed line is an exponential reference. Right panel: The same fragmentation functions on logarithmic variable ${\xi }_p=\ln (1/x_p)$. The vertical dotted lines mark equivalent points on the two variables. The solid curves are determined by the parameterization from this analysis.

Figure 3

Fragmentation functions on rapidity $y$ for $e^{+}\mathrm{\text{−}}{e}^{-}$ collisions (left panel) and $\mathrm{p}\mathrm{\text{−}}\overline{\mathrm{p}}$ collisions (right panel).

Figure 4

Comparison of fragmentation functions for three energies on ${\xi }_p$ and $y_{\max }-y$. Differences are noticible only for small $y$ or $p$ (large ${\xi }_p$).

Figure 5

Fragmentation functions plotted on normalized rapidity $u$ in linear (left panel) and semilog (right panel) formats. The data distributions have been normalized by the corresponding dijet multiplicity at each energy (lower solid curves in Fig. 13) determined by parameters $(p,q)$. The data for three energies are plotted, but the curves for only 14 and 91 GeV are plotted to provide visible separation.

Figure 6

Fragmentation functions for two CM energies and for pion (left panel) and kaon (right panel) fragments plotted on normalized rapidity $u$.

Figure 7

Left panel: Fragmentation-function data for $\sqrt{s}=91\mathrm{GeV}$ and for three hadron species on fractional momentum $x_p$ and corresponding beta-distribution fits on normalized rapidity $u$ transformed to $x_p$ (dashed-dotted curves). Right panel: Dashed and dashed-dotted curves for identified hadrons with proper mass assignments transformed from the left panel. The solid curve is a sum on $x_p$ of the beta-distribution fits from identified fragments in the left panel transformed to rapidity $y$ with a pion mass assignment. The solid curve is close to the distribution obtained from inclusive hadrons (dotted curve).

Figure 8

Unit-normal fragment distributions on normalized rapidity $u$ for udsc quarks (upper left), b quarks (lower left) and gluons (upper right) fragmenting to inclusive hadrons. Lower-right panel: Normalized logarithmic variable $v(\beta )$ (see text) measures the FF shape for b-quark jets (solid) relative to those for gluon jets (dashed-dotted) and udsc-quark jets (dashed) as references.

Figure 9

Distributions on $u$ for several quark/meson flavor combinations, showing evolution of the $g(u,y_{\max })$ shape with quark/meson mass. The $c\to D$ data are from [50], and the $b\to B$ data are from 49. The energies are dijet energies.

Figure 10

A study of the systematic effect on fiducial FFs and their beta parameters $(p,q)$ of variations with $y_{\min }$. The low ends of the FFs are most sensitive, and therefore $p$ and FFs for low $y_{\max }$ are affected most. The trends in the lower-right panel may be compared with the $(p,q)$ entries in Table .

Figure 11

Details of beta distribution fits to fragmentation functions for identified partons fragmenting to unidentified hadrons. The solid curves are free fits to data. The dashed curves are determined by the $(p,q)$ systematics in Fig. 14 (left panel).

Figure 12

Unit-normal model functions $\beta (y\mathrm{\text{or}}u;p,q)$ from fits to FFs for identified fragments (pions, kaons and protons) and inclusive fragments, all from inclusive partons at $\sqrt{s}=91.2\mathrm{GeV}$, plotted on rapidity $y$ and normalized rapidity $u$.

Figure 13

Dijet charged-particle multiplicity vs energy scale $Q$ (dijet energy) plotted in a conventional format (left panel) and vs parton rapidity assuming the pion mass (right panel). The solid curves are quark and gluon dijet multiplicities $2n_{\mathrm{q}}$ and $2n_{\mathrm{g}}$ obtained from the $(p,q)$ parameterizations in Fig. 14 (left panel). The dashed-dotted curve in the left panel is from a 3NLO pQCD expression. The udsc quark-jet multiplicities for unidentified hadrons (solid dots) are taken from a survey in 54. The dotted curves in the right panel illustrate quadratic trends $A(y_{\max }-y_{\min }{)}^2$ (see text).

Figure 14

Left panel: Beta-distribution parameters $(p_{\mathrm{q}},q_{\mathrm{q}})$ and $(p_{\mathrm{g}},q_{\mathrm{g}})$ respectively for light-quark (solid) and gluon (dashed) jets and corresponding gluon-to-quark-jet multiplicity ratio $r$ vs parton rapidity $y_{\max }$. Right panel: Unit-normal FFs (beta distributions) obtained from parameters in the left panel plotted on normalized rapidity $u$ for quark (solid) and gluon (dashed) jets and for nine equal-spaced values of parton rapidity $y_{\max }$ illustrating peak shape evolution with energy scale.

Figure 15

Left panel: Joint fragment distribution $D(y,y_{\max })$ on fragment and parton rapidities for inclusive partons ($\sim \mathrm{\text{udsc quarks}}$) and inclusive hadrons. Fragmentation functions are vertical slices (conditional distributions) from the joint distribution. Right panel: The same distribution transformed to $(y-y_{\max },y_{\max })$, with $y_{\max }-y\sim {\xi }_p$, the logarithmic relative momentum. The vertical dashed-dotted lines define the interval determined by fiducial FFs plus dijet multiplicities. The intervals between vertical dashed-dotted and dotted lines are defined only by multiplicity trends. The upper-right region of the right panel illustrates scaling violations.

Figure 16

Left panel: Comparison of fragmentation-function modes vs parton rapidity from quark and gluon data (points) with “locus of modes” trends (solid and dashed curves) derived from $(p,q)$ energy systematics in Fig. 14 (left panel) and from the MLLA (dashed-dotted and dotted curves). Right panel: Comparison at two energies of FF data, beta distributions on $u$ and MLLA Gaussians suitably transformed to $u$.

Figure 17

Scaling violations in a conventional format. Each curve is a conditional slice at fixed $x_E$ from the distribution $2dn/d{x}_E(x_E,Q^2)$ suitably transformed from model function $2n(y_{\max })\beta (u(y);p,q)$ in Fig. 15 (right panel).

Figure 18

Left panel: Conventional presentation of ${\alpha }_s$ systematics on energy scale $Q$. Right panel: Systematics of $1/{\alpha }_s$ on rapidity $y_{\max }$. The dotted lines cross at ${\alpha }_s(M_{Z_0})$.

Figure 19

Left panel: Scaling violations in the form of logarithmic derivatives of the distributions in Fig. 17 (left panel). Solid curves represent udsc-quark jets, dashed curves represent gluon jets. The near uniformity to the right of the dotted line ($Q=20\mathrm{GeV}$) for larger $x_E$ is notable. Right panel: Logarithmic derivatives for quark and gluon jets at $y_{\max }=7.8$ vs energy fraction $x_E=\cosh (y)/\cosh (y_{\max })$.

Figure 20

Left panel: Beta distribution (solid) and KKP FF (dashed) curves compared to OPAL 91 GeV data points (open circles) on linear momentum variable $x_p$. Right panel: The same curves and data transformed to normalized rapidity $u$. The vertical dotted lines both correspond to $x_p=0.1$.