Baryon-antibaryon annihilation and reproduction in relativistic heavy-ion collisions

The quark rearrangement model for baryon-antibaryon annihilation and reproduction ($B\overline{B}\leftrightarrow 3M$)—incorporated in the Parton-Hadron-String Dynamics (PHSD) transport approach—is extended to the strangeness sector. A derivation of the transition probabilities for the three-body processes is presented and a strangeness suppression factor for the invariant matrix element squared is introduced to account for the higher mass of the strange quark compared to the light up and down quarks. In simulations of the baryon-antibaryon annihilation and reformation in a box with periodic boundary conditions, we demonstrate that our numerical implementation fulfills detailed balance on a channel-by-channel basis for more than 2000 individual $2\leftrightarrow 3$ channels. Furthermore, we study central Pb+Pb collisions within PHSD from $11.7A\mathrm{GeV}$ to $158A\mathrm{GeV}$ and investigate the impact of the additionally implemented reaction channels in the strangeness sector. We find that the new reaction channels have a visible impact essentially only on the rapidity spectra of antibaryons. The spectra with the additional channels in the strangeness sector are closer to the experimental data than without for all antihyperons. Due to the chemical redistribution between baryons-antibaryons and mesons we find a slightly larger production of antiprotons thus moderately overestimating the available experimental data. We additionally address the question if the antibaryon spectra (with strangeness) from central heavy-ion reactions at these energies provide further information on the issue of chiral symmetry restoration and deconfinement. However, by comparing transport results with and without partonic phase as well as including and excluding effects from chiral symmetry restoration we find no convincing signals in the strange antibaryon sector for either transition due to the strong final-state interactions.

Figure 1

Distribution in the final number of pions $P\left(N_{\pi }\right)$ for $p\overline{p}$ annihilation at invariant energies $2.3\mathrm{GeV}\le \sqrt{s}\le 4\mathrm{GeV}$ from the QRP (short lines). The solid line is a Gaussian parametrization fitted to the experimental data. The figure is taken from Ref. [20].

Figure 2

Illustration of the quark rearrangement model for a general baryon antibaryon pair $B\overline{B}$ annihilating into three mesons $M$ and vice versa. Here the meson $M_i$ may be any of the $0^{-}$ or $1^{-}$ nonets.

Figure 3

$p\overline{p}$ annihilation cross section as a function of momentum in the laboratory $P_{\text{lab}}$. The data points are taken from Ref. [51] and the solid line is a fit by the function $50{\mathrm{mb}/\mathrm{v}}_{\mathrm{rel}}$ with $v_{\mathrm{rel}}$ denoting the relative velocity in the laboratory system Eq. (5).

Figure 4

Particle densities of a $p\overline{p}$ initialized system as a function of time. The particle species correspond to the following lines: the red solid line corresponds to nucleons, the blue dashed line to antinucleons, the green short-dashed line to pions, the violet dotted line to $\rho$ mesons, the black dashed-dotted line to $\varphi$ mesons, the gray dashed-doubly-dotted line to $\mathrm{\Lambda }\mathrm{s}$, the brown doubly dashed line to kaons and the beige short-dashed-dotted line to the vector kaons $K^{*}$. The different charge states of the particles have been summed over and $K$ denotes the sum of $K^{+},K^{-},K^0$, and ${\overline{K}}^0$.

Figure 5

Total reaction rate (per volume $dV$) as a function of time for two different initializations. The solid (slightly transparent) red lines correspond to the baryon-antibaryon annihilation and the red dotted lines to the formation. The systems are initialized in (a) with only $p+\overline{p}$ and in (b) with only ${\mathrm{\Delta }}^0+\overline{\mathrm{\Lambda }}$.

Figure 6

Total reaction rate as a function of the invariant mass $\sqrt{s}$ in equilibrium for five different initializations. The solid (slightly transparent) red line corresponds to the baryon-antibaryon annihilation and the red dotted line to the formation. The systems are initialized in (a) with only $\mathrm{\Lambda }+{\overline{\mathrm{\Xi }}}^0$ and in (b) with only ${\mathrm{\Omega }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$.

Figure 7

The reaction rate of the $B\overline{B}\leftrightarrow 3M$ reactions (solid line) as a function of time in 5% central Pb+Pb collisions at $158A\mathrm{GeV}$ in comparison to the total three-meson fusion rate (dashed line).

Figure 8

Rapidity spectra of $\overline{p},\overline{\mathrm{\Lambda }}+{\overline{\mathrm{\Sigma }}}^0,{\mathrm{\Xi }}^{-},{\overline{\mathrm{\Xi }}}^{+},{\mathrm{\Omega }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$ in (12%) 7.2% central Pb+Pb collisions at 11.7, 20 and $30A\mathrm{GeV}$. The solid lines show the results when including all light and strange quark channels [denoted by SU(3)] while the dashed lines results from discarding strange or antistrange quarks in the reaction channels [denoted by SU(2)]. The error bands indicate the systematic uncertainty of the calculations due to a different ensemble size. The dotted lines show the results with $B\overline{B}\leftrightarrow 3M$ reactions switched off. The data points are taken from Refs. [54, 55, 56].

Figure 9

Rapidity spectra $\overline{p},\overline{\mathrm{\Lambda }}+{\overline{\mathrm{\Sigma }}}^0,{\mathrm{\Xi }}^{-},{\overline{\mathrm{\Xi }}}^{+},{\mathrm{\Omega }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$ in central Pb+Pb collisions at 40, 80 and $158A\mathrm{GeV}$ for $B\overline{B}\leftrightarrow 3M$ with only light quarks (dashed lines) and including strange quarks (solid lines) compared to experimental measurements. The error bands indicate the systematic uncertainty of the calculations due to a different ensemble size. The data points are taken from Refs. [55, 56, 57, 58].

Figure 10

Transverse mass spectra for central Pb+Pb collisions at midrapidity. The centrality selection for the particles at the different energies is the same as in Figs. 8 and 9. The particles in each panel are from top to bottom ${\mathrm{\Xi }}^{-},\overline{\mathrm{\Lambda }}+{\overline{\mathrm{\Sigma }}}^0,\overline{p},{\overline{\mathrm{\Xi }}}^{+}$ and ${\mathrm{\Omega }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$, only at $158A$ GeV the lowest lying line corresponds to ${\mathrm{\Omega }}^{-}$. The data points are taken from Refs. [56, 57, 58].

Figure 11

Rapidity spectra for a central Pb+Pb collision at 30 and $158A\mathrm{GeV}$; comparison between simulations with (PHSD) and without (HSD) the deconfinement transition and with activated and deactivated chiral symmetry restoration (CSR). The data points are taken from Refs. [55, 56, 57, 58].

Figure 12

Rapidity spectra for a central Pb+Pb collision at $40A\mathrm{GeV}$; comparison between PHSD results with the $B\overline{B}\leftrightarrow 3M$ reactions including strangeness (red solid line), UrQMD-2.3 [28] (violet short-dashed line) and 3FD with a two-phase equation of state [29] (blue dashed line). The experimental data are taken from Refs. [55, 56].

Figure 13

Two-body phase-space integral for particles with masses $m_1=1\mathrm{GeV}$ and $m_2=2\mathrm{GeV}$ as a function of the invariant energy above threshold.

Figure 14

Illustration of the subsequent decay of an initial state (black dot) into $n$ particles. The initial state may consist of $m$ particles as only the invariant mass is relevant for the phase-space integral due to Lorentz invariance.

Figure 15

Illustration of the three-, four-, and five-body phase-space integrals as a function of the invariant energy above threshold. The red solid line shows the three-body phase-space integral for $\pi \rho \rho$, the blue dashed line shows the four-body phase-space integral for $3\pi \rho$ and the green dashed line shows the five-body phase-space integral for five pions.

Figure 16

Consistency check for a change in the cell size $\mathrm{\Delta }V$ by 20%. (a) for ${\mathrm{\Delta }}^0+\overline{\mathrm{\Lambda }}$ and (b) for ${\mathrm{\Sigma }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$ initalizations. The red solid line shows the baryon-antibaryon annihilation for the cell volume $\mathrm{\Delta }V$, the green dashed line shows the baryon-antibaryon formation for $\mathrm{\Delta }V$, the blue short-dashed line shows the baryon-antibaryon annihilation for $1.2\mathrm{\Delta }V$ and the violet dotted line shows the baryon-antibaryon formation for $1.2\mathrm{\Delta }V$.

Figure 17

Comparison of the reaction rates between the cell algorithm (cell) and the next-neighbor (NN) realization of the in-cell method. The systems shown are in (a) the $p+\overline{p}$ and in (b) the $\mathrm{\Lambda }+{\overline{\mathrm{\Xi }}}^0$ initialization. The red solid line shows the baryon-antibaryon annihilation for the cell method, the green dashed line shows the baryon-antibaryon formation for the cell method, the blue short-dashed line shows the baryon-antibaryon annihilation for the NN method and the violet dotted line shows the baryon-antibaryon formation for the NN method.

Figure 18

Comparison of the reaction rate between the sum and the difference of the strange and antistrange quarks in the calculation of transition probabilities in $B\overline{B}\leftrightarrow 3M$ reactions (denoted by sum and diff). (a) shows the $p+\overline{p}$ and (b) the ${\mathrm{\Omega }}^{-}+{\overline{\mathrm{\Omega }}}^{+}$ initialization. The red solid line shows the baryon-antibaryon annihilation of the sum, the green dashed line shows the baryon-antibaryon formation of the sum, the blue short-dashed line shows the baryon-antibaryon annihilation of the difference and the violet dotted line shows the baryon-antibaryon formation of the difference.

Figure 19

Rapidity spectra for a central Pb+Pb collision at $30A\mathrm{GeV}$; comparison between simulations with a strangeness suppression factor $\lambda =0.5$ (dashed lines) and no strangeness suppression, i.e., $\lambda =1$ (solid lines).