Hanbury-Brown–Twiss correlation functions and radii from event-by-event hydrodynamics

Hanbury-Brown–Twiss correlation functions and radii from event-by-event hydrodynamics (HoTCoffeeh) is a new computational tool which determines Hanbury-Brown–Twiss (HBT) charged-pion (${\pi }^{+}$) correlation functions and radii for event-by-event hydrodynamics with fluctuating initial conditions in terms of Cooper–Frye integrals, including resonance-decay contributions. In this paper, we review the basic formalism for computing the HBT correlation functions and radii with resonance-decay contributions included and discuss our implementation of this formalism in the form of HoTCoffeeh. This tool may easily be integrated with other numerical packages [see, e.g., [Comput. Phys. Commun. 199, 61 (2016)]] for the purpose of simulating the evolution of heavy-ion collisions and thereby extracting predictions for heavy-ion observables.

Figure 1

The out-side-long ($osl$) coordinate system used for defining the HBT radii. Here, $\vec{r}$, ${\vec{p}}_{1,2}$, $\vec{q}$, and $\vec{K}$ are seen projected onto the transverse plane, so that the transverse component of $\vec{K}$ makes an angle ${\mathrm{\Phi }}_K$ with the $x$ axis, defined to point in the direction of the impact parameter $\vec{b}$ (or a proxy for it, such as the elliptic flow angle ${\mathrm{\Psi }}_2$). The longitudinal direction (i.e., the $z$ direction) is defined to point out of the page.

Figure 2

Slices of the full ensemble-averaged correlation function $C_{\mathrm{avg}}$ (2) including all resonance decays (solid lines) along the $q_o$, $q_s$, and $q_l$ axes (from left to right), compared with the analogous results for directly emitted (“thermal”) pions only (dashed lines), for three choices of the pair momentum, $K_T=0$, 0.4, and 1.0 GeV (from top to bottom). The pair momentum ${\vec{K}}_T$ was chosen to point in x direction (${\mathrm{\Phi }}_K=0$) such that $q_x=q_o$ and $q_y=q_s$. The ensemble consists of $N_{\mathrm{ev}}=1000$ hydrodynamically evolved central (0%–10% centrality) Au-Au collisions with $\eta /s=0.08$ at $\sqrt{s}=200A\mathrm{GeV}$.

Figure 3

The intercept parameter $\lambda \left(\vec{K}\right)$ [defined in Eq. (4)] as a function of $\vec{K}$, for a three-dimensional Gaussian fit to the full correlation function including resonances shown in Fig. 2.

Figure 4

Event-by-event distributions of the azimuthally averaged SV $R_{s,0}^2$ [43] (denoted simply as $R_s^2$ in the figure), (a) with and (b) without the $\delta f$ correction.

Figure 5

Same as Fig. 4, but for the outward radius parameter SV $R_{o,0}^2$.

Figure 6

Same as Fig. 4, but for the longitudinal radius parameter SV $R_{l,0}^2$.

Figure 7

(a) The azimuthally and ensemble-averaged SV HBT radii and (b) their normalized variances as a function of pair momentum $K_T$, including all resonance decays, for the central Au-Au collision events studied in this paper. In panel (a) solid lines show results that include all resonance-decay contributions while dashed lines show the HBT radii for only the directly emitted pions.

Figure 8

Event-by-event distributions of GF HBT radii, for several different values of $K_T$.

Figure 9

Same as Fig. 7, but for the mean radii and normalized variances of the GF radii. Solid lines correspond to the radii with all resonances included, while the dashed lines represent the GF radii using thermal ${\pi }^{+}$ only.

Figure 10

Slices of the ensemble-averaged ($N_{\mathrm{ev}}=1000$ events) correlation function $C_{\mathrm{ev}}$ (solid lines), including all resonance decays, for (from top to bottom) $K_T=0$, 0.4, and 1 GeV, compared with the same slices of the best-fit three-dimensional Gaussian correlation function (dashed lines). These best-fit curves clearly reproduce the shape of the true correlation function better at large $q$ than at small $q$.

Figure 11

One-dimensional fits to $q_x$ slice of the correlation function shown in the uppermost, left-hand panel of Fig. 10, for different ranges of $q_x$: $|q_x|\le 20$ MeV (red, dashed curve); $|q_x|\ge 20$ MeV (green, dash-dotted curve); all $q_x$ points in the range shown (blue, dotted curve). We see that the Gaussian best fits depend strongly on the precise distribution of points used.

Figure 12

Three different slices ($q_x=0$, $q_y=0$, and $q_z=0$, respectively) of the correlation function at fixed $K_T$ and ${\mathrm{\Phi }}_K$, comparing the truncated resonance calculation at 60%, with (solid) and without (dashed) extrapolation over the remaining resonances, compared with the full 100% calculation (solid black). The correlation function was computed and extrapolated at seven equally spaced nodes along each $q$ axis (cf. Sec. 3c) and then interpolated by using a quadratic spline for aesthetic purposes.

Figure 13

Three different slices ($q_x=0$, $q_y=0$, and $q_z=0$, respectively) of the correlation function at fixed $K_T$ and ${\mathrm{\Phi }}_K$, illustrating our correlation function constructed by fleshing out a sparse grid (solid), compared with the same correlation function computed on a dense grid over a similar range of $q$ points (dashed). For the first term in the numerator of Eq. (56), the algorithm interpolates the logarithm of the thermal contribution linearly in $q^2$, which is an excellent approximation, since the thermal contribution is nearly Gaussian. For the remaining two terms in the numerator of (56), the algorithm uses cubic interpolation. This approach clearly works well at all $q$, although small discrepancies emerge at large $q$ in the longitudinal ($q_z$) direction.