Microscopic study of deuteron production in PbPb collisions at $\sqrt{s}=2.76\mathrm{TeV}$ via hydrodynamics and a hadronic afterburner

The deuteron yield in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76\mathrm{TeV}$ is consistent with thermal production at a freeze out temperature of $T=155\mathrm{MeV}$. The existence of deuterons with binding energy of 2.2 MeV at this temperature was described as “snowballs in hell” [P. Braun-Münzinger, B. Dönigus, and N. Löher, CERN Courier, August 2015]. We provide a microscopic explanation of this phenomenon, utilizing relativistic hydrodynamics and switching to a hadronic afterburner at the above-mentioned temperature of $T=155\mathrm{MeV}$. The measured deuteron $p_T$ spectra and coalescence parameter $B_2\left(p_T\right)$ are reproduced without free parameters, only by implementing experimentally known cross sections of deuteron reactions with hadrons, most importantly $\pi d\leftrightarrow \pi np$.

Figure 1

Deuteron-pion interaction cross sections from SAID database [40] and partial wave analysis [41] are compared to our parametrizations (Tables 2 and 3 in the Appendix). Inelastic $d\pi \leftrightarrow np\pi$ reactions are the most important for deuteron production and disintegration at high energies. The large total $\pi d$ cross section is responsible for the late deuteron freeze out. Also, the inelastic $\pi d$ cross section is larger than elastic. As a consequence, for deuteron chemical and kinetic freeze out coincide, see Fig. 6.

Figure 2

Deuteron disintegration cross sections by nucleons (a) and antinucleons (b) as given by Tables 2 and 3, compared to data [42, 46]. Although the cross sections are large, these processes are of minor importance compared to $\pi d\leftrightarrow \pi np$, because pions are much more abundant in Pb+Pb collisions at $\sqrt{s}=2.76\mathrm{TeV}$.

Figure 3

Identified particle transverse momentum spectra compared to experimental measurements [1, 47].

Figure 4

Deuteron coalescence parameter $B_2$, extracted from the hydro + SMASH simulation (which does not involve any coalescence, only collisions with experimentally known deuteron cross sections) is compared to ALICE measurement [48].

Figure 5

Reaction rates of the most important $\pi d\leftrightarrow \pi pn$ reaction in forward and reverse direction.

Figure 6

Distribution of deuteron last collision time in case last collision was inelastic (circles) or elastic (triangles). Kinetic freeze out for deuterons coincides with chemical freeze out, which follows from inelastic $\pi d$ cross section being larger than elastic, see Fig. 1.

Figure 7

Top: deuteron yield (both in hydrodynamics and afterburner) versus time for the scenarios, described in the text. Bottom: relative amount of energy in the afterburner. This is to indicate, how much of the system is already treated by the afterburner.

Figure 8

Evolution of yields (top) and thermodynamic variables (bottom) in our toy model without annihilations for $T_0=155\mathrm{MeV}$. The deuteron yield grows, which is similar to our simulation within the fourth scenario in Fig. 7.

Figure 9

Equilibration in the box, where interactions are artificially limited to those involving deuterons. Nucleon and deuteron numbers equilibrate for 10–15 fm/c via $\pi d\leftrightarrow \pi np$ and $Nd\leftrightarrow Nnp$ reactions with large cross sections. Pions equilibrate via $\pi d\leftrightarrow NN$, which has a small cross section and takes around 500 fm/$c$ to equilibrate.

Figure 10

Demonstration, that detailed balance is obeyed with 1% precision. Differential reaction rates against the center of mass energy of the reaction (a) and integrated numbers of reactions, scaled over isospin average (b) are shown. Forward reactions are marked by triangles pointing right, reverse reactions—by triangles pointing left. In case of a perfect detailed balance both should point at the same value.