Causal electric charge diffusion and balance functions in relativistic heavy-ion collisions

We study the propagation and diffusion of electric charge fluctuations in high-energy heavy-ion collisions using the Cattaneo form for the dissipative part of the electric current. As opposed to the ordinary diffusion equation this form limits the speed at which charge can propagate. Including the noise term in the current, which arises uniquely from the fluctuation-dissipation theorem, we calculate the balance functions for charged hadrons in a simple 1+1-dimensional Bjorken hydrodynamical model. Limiting the speed of propagation of charge fluctuations increases the height and reduces the width of these balance functions when plotted versus rapidity. We also estimate the numerical value of the associated diffusion time constant from anti–de Sitter-space/conformal-field theory.

Figure 1

The density-density self-correlation function versus $\mathrm{\Delta }\xi$ for various values of $v_Q^2$ evaluated at the final time ${\tau }_f$. For large $v_Q^2$ it approaches the exponential form of Eq. (57) and eventually a Dirac $\delta$ function.

Figure 2

Time evolution of the density-density correlation function (with self-correlations subtracted) versus $k$. Top panel: $v_Q^2=1/3$. Middle panel: $v_Q^2=1$. Bottom panel: ordinary diffusion with $v_Q^2\to \infty$. The different curves in each panel correspond to elapsed times of 5, 10, 15, 25, and 100% of the system's lifetime ${\tau }_f-{\tau }_0=5.852$ fm/c, starting at the top and working down at $k=0$.

Figure 3

The regular part of the density-density correlator in $\xi$ space after 5% of the total expansion time of ${\tau }_f-{\tau }_0=5.852$ fm/c has elapsed. The smooth broad curve corresponds to ordinary diffusion with white noise.

Figure 4

Schematic depiction of fluctuations and their horizons.

Figure 5

Balance functions for pions, protons, and kaons. Looking at $\mathrm{\Delta }y=0$ the curves correspond to $v_Q^2$ = 1/3 (top), 1 (middle), and 10 (bottom), with 10 being indistinguishable from the case of white noise in the ordinary diffusion equation.