Extension of the Bjorken energy density formula of the initial state for relativistic heavy ion collisions

For relativistic heavy ion collisions, the Bjorken formula is very useful for estimating the initial energy density once an initial time ${\tau }_0$ is specified. However, it cannot be trusted at low energies, e.g., well below $\sqrt{s_{{}_{\mathrm{NN}}}}\approx 50\mathrm{GeV}$ for central Au+Au collisions, when ${\tau }_0$ is smaller than the finite time it takes for the two nuclei to cross each other. Here I extend the Bjorken formula by including the finite time duration of the initial energy production. Analytical solutions for the formed energy density in the central spacetime-rapidity region are derived for several time profiles. Compared to the Bjorken formula at low energies, the maximum energy density reached is much lower, increases much faster with the collision energy, and is much less sensitive to the uncertainty of the formation time, while the energy density time evolution is much longer. Comparisons with results from a multiphase transport confirm the key features of these solutions. The effect of the finite longitudinal width of the initial energy production, which is neglected in the analytical results, is investigated with the transport model and shown to be small. This work thus provides a general model for the initial energy production of relativistic heavy ion collisions that is also valid at low energies.

Figure 1

(a) Schematic picture of the collision of two nuclei, where the initial energy production takes place over the shaded area of finite widths in $t$ and $z$. (b) This simplified picture is considered for analytical derivations of energy density at central spacetime rapidity ${\eta }_s\to 0$ (as $d\to 0$): particles could be initially produced at any time $x$ within $[0,d_t]$ at $z=0$ and then propagate to observation time $t$.

Figure 2

Time profiles for the initial energy production at central spacetime rapidity: a uniform profile (dashed curve), $\beta$ profiles with integer powers $n=1$ to 5 (solid curves), and a triangular profile (dot-dashed). Circles represent the time profile of partons within mid-spacetime-rapidity from the string melting AMPT model for central Au+Au collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}=11.5\mathrm{GeV}$.

Figure 3

Average formed energy densities at central spacetime rapidity as functions of time for central Au+Au collisions at (a) 4.84 GeV and (b) 200 GeV from the uniform time profile with the naive choice of $t_1=0$ and $t_2=d_t$ (dashed), the $\beta$ time profile for $n=4$ (solid), and the Bjorken formula (dotted). Three sets of curves of each type correspond to ${\tau }_{{}_{\mathrm{F}}}=0.1,0.3$ and $0.9\mathrm{fm}/\mathrm{c}$.

Figure 4

Average energy densities at central spacetime rapidity for the uniform (dashed curves), $\beta$ (solid curves), triangular (dot-dashed curves) time profiles and the Bjorken formula (dotted curves) for ${\tau }_{{}_{\mathrm{F}}}=0.1$ and $0.3\mathrm{fm}/\mathrm{c}$, in comparison with the corresponding AMPT results (circles), for central Au+Au collisions at (a) 4.84 GeV, (b) 11.5 GeV, (c) 27 GeV, and (d) 200 GeV. I have used $t_1=0.29d_t$ and $t_2=0.71d_t$ for the uniform profile and $t_1=0.20d_t$ and $t_2=0.80d_t$ for the triangular profile so that they both have the same mean and standard deviation as the $\beta$ profile.

Figure 5

Maximum energy density at central spacetime rapidity (${\eta }_s=0$) averaged over the transverse overlap area versus the collision energy for central Au+Au collisions from the Bjorken formula (dashed curves), the uniform profile (solid curves), and the triangular profile (dotted curves) for ${\tau }_{{}_{\mathrm{F}}}=0.1,0.3$ and $0.9\mathrm{fm}/\mathrm{c}$. I have used $t_1=0.29d_t$ and $t_2=0.71d_t$ for the uniform profile and $t_1=0.20d_t$ and $t_2=0.80d_t$ for the triangular profile.

Figure 6

AMPT results of average energy densities at central spacetime rapidity for central Au+Au collisions at (a) 4.84 GeV, (b) 11.5 GeV, and (c) 27 GeV when excluding the finite widths in $t$ and $z$ (open circles), including the finite widths in $t$ and $z$ (filled circles), and including the finite width in $t$ but not the finite width in $z$ (dashed curves).