Simple solutions of fireball hydrodynamics for rotating and expanding triaxial ellipsoids and final state observables

We present a class of analytic solutions of nonrelativistic fireball hydrodynamics for a fairly general class of equation of state. The presented solution describes the expansion of a triaxial ellipsoid that rotates around one of its principal axes. We calculate the hadronic final state observables such as single-particle spectra, directed, elliptic, and third flows, as well as two-particle Bose-Einstein (also named HBT) correlations and corresponding radius parameters, utilizing simple analytic formulas. The final tilt angle of the fireball, an important observable quantity, is shown to be not independent of its exact definition: one gets different tilt angles from the geometrical anisotropies, from the single-particle spectra, and from HBT measurements. Taken together, the tilt angle in the momentum space and in the relative momentum or HBT variable may be sufficient for the determination of the magnitude of the rotation of the fireball. We argue that observing this rotation and its dependence on collision energy could characterize the softest point of the equation of state. Thus determining the rotation may be a powerful tool for the experimental search for the critical point in the phase diagram of strongly interacting matter.

Figure 1

Illustration of the dependence of the time evolution of principal axes $X, Y, Z$ of the solution on the equation of state, for three different constant $\kappa$ values. Initial conditions and parameters are $m_0=938$ MeV, $T_0=300$ MeV, ${\omega }_0=0.15c/\mathrm{fm}, X_0=4$ fm, $Y_0=6$ fm, $Z_0=2$ fm, and ${\dot{X}}_0={\dot{Y}}_0={\dot{Z}}_0=0$. Markers denote the values at the respective freezeouts (when the temperature reaches $T_f=140$ MeV).

Figure 2

Illustration of the dependence of the time evolution oft the temperature $T$ (a) and the angular velocity $\omega$ (b) introduced in Eq. (28) on the equation of state, for three different constant $\kappa$ values. Initial conditions and parameters are as in the previous example: $m_0=938$ MeV, $T_0=300$ MeV, ${\omega }_0=0.15c/\mathrm{fm}, X_0=4$ fm, $Y_0=6$ fm, $Z_0=2$ fm, and ${\dot{X}}_0={\dot{Y}}_0={\dot{Z}}_0=0$. Markers denote the values at the respective freezeouts (when the temperature reaches $T_f=140$ MeV).

Figure 3

Time evolution of the various tilt angles in our new hydrodynamical solution, for three different EoSs. Parameters and initial conditions are the same as in Figs. 1 and 2, markers denote the values at the respective freezeouts. (a) Tilt angle of the coordinate-space ellipsoids in the $x-z$ plane ${\vartheta }_{\mathbf{r}}$, as introduced in the text. This is the tilt angle of the corotating $K^{\prime }$ frame. (b) ${\vartheta }_{\mathbf{p}}$ angle, the observable tilt angle of the eigenframe of the single-particle spectrum (${\vartheta }_{\mathbf{p}}=\vartheta +{\vartheta }_{\mathbf{p}}^{\prime }$). (c) ${\vartheta }_{\mathbf{q}}$, the tilt angle of the eigenframe of the HBT correlation function (${\vartheta }_{\mathbf{q}}=\vartheta +{\vartheta }_{\mathrm{HBT}}^{\prime }$). For plotting the ${\vartheta }_{\mathbf{q}}$ angle, the mass of the pion $m_{\pi }$ was used to evaluate ${\vartheta }_{\mathrm{HBT}}^{\prime }$. As mentioned after Eq. (90), in the $m\to 0$ limit, the coordinate-space tilt is recovered as ${\vartheta }_{\mathbf{q}}\to {\vartheta }_{\mathbf{r}}$.

Figure 4

The various tilt angles introduced in the text (${\vartheta }_{\mathbf{r}}\equiv \vartheta$, the coordinate-space tilt, ${\vartheta }_{\mathbf{p}}\equiv \vartheta +{\vartheta }_{\mathbf{p}}^{\prime }$, the tilt of the single-particle spectrum, and ${\vartheta }_{\mathbf{q}}\equiv \vartheta +{\vartheta }_{\mathrm{HBT}}^{\prime }$, the tilt of the HBT correlation function) at freezeout time. (a) Freeze-out time angles plotted as a function of initial angular velocity ${\omega }_0$ (for $\kappa =3/2$). (b) Freeze-out time angles plotted as a function of $\kappa$ (in this plot, the ${\omega }_0$ was taken to be $0.15c/\mathrm{fm}$). All other initial conditions are the same as in Fig. 1.

Figure 5

Bertsch-Pratt HBT radii vs $\phi$ azimuthal angle of the particle pair, calculated for a reasonable set of parameters: ${R^{\prime }}_{xx}^2=25{\mathrm{fm}}^2, {R^{\prime }}_{yy}^2=16{\mathrm{fm}}^2, {R^{\prime }}_{zz}^2=36{\mathrm{fm}}^2, {R^{\prime }}_{xz}^2=2{\mathrm{fm}}^2, {\vartheta }_f=\pi /8$. The oscillations in $R_{\mathrm{oo}}^2, R_{\mathrm{ss}}^2$, and $R_{\mathrm{os}}^2$ with $\pi$ periodicity are characteristic to an ellipsoidlike source. The oscillations in $R_{\mathrm{ol}}^2$ and $R_{\mathrm{sl}}^2$ with $2\pi$ periodicity are characteristic to a tilted or rotating ellipsoidlike source. A measurement of these Bertsch-Pratt radii enables one to deduce the $\mathbf{R}$ HBT radius matrix introduced in Eq. (90), and in turn the $\vartheta +{\vartheta }_{\mathrm{HBT}}^{\prime }$ angle introduced in Eq. (90), that characterizes the tilt of the eigenframe of the HBT correlation function. In this plot, the freezeout is assumed to be instantaneous ($\mathrm{\Delta }t=0$), although in real data analysis $\mathrm{\Delta }t$ plays an important role.

Figure 6

Illustration of the dependence of flow parameters $v_1, v_2, v_2$ on rapidity $y$. The parameters were set to ${T^{\prime }}_{xx}=300$ MeV, $T_{xz}^{\prime }=-20$ MeV, $T_{zz}^{\prime }=500$ MeV, $T_{yy}^{\prime }=200$ MeV, ${\vartheta }_f=\pi /8$. The $p_t$ of the particles are assumed to be $p_t=600\mathrm{MeV}/c$, while the particle mass $m$ was set to equal to the kaon mass, $m=494$ MeV. By measuring these flow coefficients and the angle-averaged spectrum, one can reconstruct the $\mathbf{T}$ slope matrix elements introduced in Eq. (66), and in turn the angle ${\vartheta }_{\mathbf{p}}^{\prime }$ that characterizes the tilt of the eigenframe of the momentum distribution, as introduced in Eq. (65).