Minimal conditions for collectivity in $e^{+}{e}^{-}$ and $p+p$ collisions

Signatures of collective behavior have been measured in highly relativistic $p+p$ collisions, as well as in $p+\mathrm{A}, d+\mathrm{A}$, and ${}^3\mathrm{He}+\mathrm{A}$ collisions. Numerous particle correlation measurements in these systems have been successfully described by calculations based on viscous hydrodynamic and transport models. These observations raise the question of the minimum necessary conditions for a system to exhibit collectivity. Recently, numerous scientists have raised the question of whether the quarks and gluons generated in $e^{+}{e}^{-}$ collisions may satisfy these minimum conditions. In this paper we explore possible signatures of collectivity, or lack thereof, in $e^{+}{e}^{-}$ and $p+p$ collisions utilizing A Multi-Phase Transport (ampt) framework which comprises melted color strings, parton scattering, hadronization via coalescence, and hadron rescattering.

Figure 1

Single-color-string event with partons in their initial positions shown as blue points, and initial parton momentum vectors shown as red arrows. The center-of-mass coordinate for the set of partons is shown as the black open circle and the spatial eccentricity shown as the ellipse.

Figure 2

ampt events with a single-color-string modeling $e^{+}{e}^{-} \to Z \to q\overline{q}$. (a) Long-range two-particle correlations $\left|\mathrm{\Delta }\eta \right|>2.0$ for hadrons with $p_T>0.5$ GeV/$c$, with and without final-state interactions. Fourier fits are also shown as lines and the $c_2$ coefficients displayed. (b) Azimuthal anisotropy ($v_2$) calculated with respect to the initial parton plane with and without final-state interactions, and then the net difference thus isolating the effects from final-state interactions (black curve).

Figure 3

Two-color-string event with initial parton positions shown as blue points, and initial parton momentum vectors shown as red arrows. The center-of-mass coordinate for the set of partons is shown as the black open point and the spatial eccentricity shown as the drawn ellipse.

Figure 4

ampt events with two color strings. (a) Long-range two-particle correlations $\left|\mathrm{\Delta }\eta \right|>2.0$ for hadrons with $p_T>0.5$ GeV/$c$, with and without final-state interactions. Fourier fits are also shown as lines and the $c_2$ coefficients displayed. (b) Azimuthal anisotropy ($v_2$) calculated with respect to the initial parton plane with and without final-state interactions, and then the net difference thus isolating the effects from final-state interactions (black curve).

Figure 5

ampt events with two color strings at total collision energies of 45, 30, and 10 GeV. Shown are the long-range two-particle correlations $\left|\mathrm{\Delta }\eta \right|>2.0$ for hadrons with $p_T>0.5$ GeV/$c$, with and without final-state interactions. Fourier fits are also shown as lines and the $c_2$ coefficients displayed.

Figure 6

ampt calculated $v_2$ at $p_T=0.75$ GeV/c with respect to the true geometry in the two-color-string configuration as a function of the total energy. Also shown superimposed is the fraction of partons that suffer at least one scattering.