### Femtoscopy Primer

We have asked Scott Pratt, one of the pioneers of the application of HBT to relativistic heavy-ion collisions and a leading expert on the topic to contribute this primer to the Virtual Journal. Scott’s early work related to HBT can be found in:

The goal of femtoscopy is to determine space-time characteristics of heavy-ion sources using two-particle correlation measurements. Femtoscopy also goes by the name of interferometry or Hanbury-Brown Twiss (HBT), referring to the use of correlated photons from uncorrelated sources to determine sizes. HBT measurements were originally used to determine sizes of very long-lived objects such as stars, which requires one to restrict the relative times of the arrival of the two photons:

- A New type of interferometer for use in radio astronomy
- Correlation between Photons in two Coherent Beams of Light

In contrast, heavy-ion sources are short-lived, and the correlation measurement can be confined to momentum space, which significantly simplifies the analysis. The first femtoscopic measurement for hadronic sources involved the correlation of two pions, and is sometimes referred to as GGLP:

Since that measurement, femtoscopic correlations have expanded to numerous other hadrons and light nuclei, and for beam energies of a few MeV per nucleon to the highest available collider energies. Of the numerous reviews, this recent review is fairly pedagogical and should serve as a decent primer for those entering the field:

Since measurements are confined to the asymptotic momenta of outgoing particles, the inference of space-time information is inherently indirect and depends on applying the Koonin equation:

Here, the source function S(P,r) is the probability of two particles with the same velocity, whose total momemtum is P, to be separated by a distance r in their asymptotic straight-line trajectories. The measurable correlation function C(P,q) and the source function are related through the outgoing wave function, which incorporates symmetrization and interactions between the two particles. The Koonin equation can be thought of as a convolution between a function with information in r, into one whose information is encoded in the relative momentum q, which is usually assumed to be small. The goal of analyzing the correlation function is to determine as many properties of the source as possible from measurement and to compare these properties with models. The source function is rather mis-named as it describes the shape of the outgoing phase-space distribution, and since it only describes the probability two particles are separated by a given amount, it does not provide the time or position of the centroid of the emitted packet. Details about the phenomenology and its theoretical justification are summarized in the previously mentioned review.

Six dimensions of information are encoded into the correlation function. The most common choice for expressing the information is to state three Gaussian source parameters, which are all functions of P. If symmetries do not determine the direction of the principal axes, there are also three Euler angles, or equivalently three cross terms, and if the particles are not identical one might also use three offsets to describe the centroid of the Gaussian. For central collisions at mid rapidity, the principal axes are known from symmetry and are referred to as the longitudinal (along the beam), outward (parallel to the total momentum) and sideward (perpendicular to the previous two). Additionally, due to the fact that some particles may come from very long distances and thus not be correlated, one also includes a normalization factor lambda, which represents the fraction of pairs where both particles are correlated. Most discussion focuses on the three dimensions R_{long}, R_{out} and R_{side}, but more sophisticated analyses have evaluated the orientation of the principal axes. This has led to a better understanding of the violation of boost-invariance away from mid-rapidity:

and has provided a picture of the transverse anisotropy due toe non-zero impact parameter:

- Azimuthal dependence of pion interferometry at the AGS
- Azimuthally sensitive HBT in Au + Au collisions at s(NN)**(1/2) = 200-GeV

The last two years has seen significant effort in evaluating non-Gaussian contributions to the source function:

- Obtaining femtoscopy results in models with resonances
- Three-Dimensional Two-Pion Emission Source at SPS: Extraction of Source Breakup Time and Emission Duration

This has involved applying both more sophisticated functional forms, such as Edgeworth expansions or two-Gaussian forms, and a process called “imaging”, where a spline-based description of the source function is adjusted to best fit the data. These analyses have quantitatively verified expectations of exponential tails to the source function, due to intermediate-lifetime resonances, such as the omega, or due to the exponential fall-off in the longitudinal direction expected from boost-invariant dynamics.

Collective expansion is a theme of all correlation analyses, especially those at high energy where collisions are extremely explosive. In the presence of velocity gradients the dimensions of a phase-space cloud are determined by the ratio of the thermal velocity to the velocity gradient, and since more massive particles have lower thermal velocities, their source dimensions tend to be smaller. Due to relativistic kinematics, the effective mass includes the energy due to motion transverse to the dimension in question. For example, Rlong falls with increasing mt due to longitudinal expansion. As a particle’s velocity exceeds that of the outer matter, it becomes increasingly correlated to the surface, which also contributes to the fall of the transverse dimensions with m_{t}. Finally, more energetic particles are emitted earlier, which also contributing to a fall-off with m_{t}. In contrast to these reasons for the radii to fall with m_{t}, a long-lived source has the opposite effect, in that it leads to elongated phase space clouds with R_{out} becoming larger than R_{side}. Experimentally, R_{out}/R_{side} is modestly greater than unity at AGS and falls to very near unity at RHIC, qualitatively suggesting increasingly explosive collisions. Analyses that combine femtoscopic observables and spectra, which then fit to parameteric forms of a thermal exploding source, e.g. a blast-wave model or the Buda-Lund model, make a compelling case that the mattter expands quickly for about 10 fm/c before dissolving within a few fm/c, with an exponential tail to the source due to intermediate-lifetime resonances:

- Observable implications of geometrical and dynamical aspects of freeze out in heavy ion collisions
- The Reconstructed final state of Pb + Pb 158-GeV/A reactions from spectra and correlation data of NA49, NA44 and WA98

Perhaps, the most striking verification of this picture is derived from the correlation of non-identical particles. Due to their slower thermal velocity, fast protons are more confined to the surface than are pions. This leads an offset of the Gaussian sources of pions and protons, with the protons and kaons being several fm ahead of the pions. This is quantitatively consistent with the thermal blast-wave picture:

Any dynamical model, be it based on hydrodynamics, the Boltzmann equation, or some combination, can predict the spatial shape of outgoint phase space clouds. Femtoscopy thus represents a stringent six-dimensional test of the dynamics. To the initial surprise of many, hydrodynamic models resulted in longer-lived, more gradually emitting sources than the extremely explosive sources suggested by blast-wave analyses of RHIC data. This failure was particularly acute in the R_{out} /R_{side} ratio, which experimentally was near unity, but was more than 50% higher in model calculations. This was referred to as the *HBT puzzle*:

- Pion interferometry at RHIC: Probing a thermalized quark gluon plasma?
- Two RHIC puzzles: Early thermalization and the HBT problem

In the last two years, it appears that a consensus is being reached with the cause(s) of the discrepancy. Transverse expansion during the pre-thermalized stage and shear viscosity, which increases the transverse pressure at early times, provide a boost to the expansion:

Combined with using a stiffer equation of state, one can readily construct a model which comes within 10% of the femtoscopic radii from RHIC. However, it should be emphasized that these explanations have not yet been consistently applied across the phenomenological triumvarate of spectra, flow observables and correlations. Only then can one claim success in finding a satisfactory explanation of the soft physics in high-energy heavy ion collisions.