Relativistic Heavy Ion Physics-Opacity Effects In HBT Correlations at RHIC:Progress and Future Work

J.G. Cramer(UW), G.A. Miller, J. M. S. Wu(UW Grad Student), and J-H Yoon, UW

In relativistic heavy ion collisions it is usually assumed that the pion source resulting from the collision is cylindrically symmetric around the beam axis and is transparent to emitted pions. Thus the detector should receive particles from all kinematically-allowed regions of the source. This leads to a prediction that the radius parameter toward the detector, $R_O$, is larger than the corresponding radius perpendicular to the detector, $R_S$, for particles of zero rapidity: $R_O^2 \approx R_S^2+ \beta_0^2\delta\tau^2$, with $\delta\tau$ the duration of pion emission, and $\beta_0$ is the pion velocity in the direction of the detector.

Experimentally these HBT radius parameters are extracted from the two-particle momentum correlation functions, and it is found that for A+Au collisions at RHIC, $R_O$ is approximately equal to $R_S$[59]. Moreover, it is observed that at higher values of the average pion pair momentum, $R_O$ may actually be smaller than $R_S$[60]. These observations contradict the theoretical expectations above, and have been taken as an indication that the emission duration may be extremely short, i.e. that the source freezes out quite suddenly, even in the presence of pion-emitting resonances and transit-time effects.

To avoid this scenario, Heiselberg and Vischer[61] have proposed that the source should not be assumed to be transparent to pions, but rather should be treated as opaque. Their study, as well as a more detailed one by Tomasik and Heinz[62], showed that an opaque source would indeed lead naturally to $R_O$ smaller than $R_S$ for plausible emission durations. However, both of these studies used the high energy limit and the semi-classical (eikonal) approximation to treat the pions emerging from the source. This procedure cannot be justified for the pions of momenta of a few hundred MeV/c used in the experimental correlation studies. Furthermore, neither calculation produces $R_O = R_S$ at $K_T=0$, thus violating a basic symmetry: at zero momentum one cannot distinguish the momentum differences $q_O$ and $q_S$ or define any transverse momentum direction.

We are attempting to properly treat the effects of opacity by describing the system of two outgoing pions by solving quantum mechanical wave equations for the pions in an opaque medium, using a relativistic optical model. These calculations are presently in a preliminary form, but we can already state some of the results. We find that at $K_T=0$, $R_O = R_S$ as required. We find also that at larger momenta $R_O$ can be less than $R_S$, that there are saturation effects, and we predict some $K_T$ dependence for $R_O$. The latter results are seen in the data but cannot be accommodated by references [61,62]. The aim of future work would be to check, refine and apply the techniques to all of the available data. The ultimate aim is to remove the effects of opacity from the experimental data so that the correct underlying value of $\delta\tau$ could be determined.