Moving rho-mesons at finite temperature

An exciting aspect of hadron physics is the question how a hadron changes its properties once it is put in a strongly interacting environment. Here, the vector mesons deserve special attention as they couple directly to (virtual) photons. The latter can decay into dileptons which leave the strongly interacting system untouched. Via that process information about possible in-medium modifications of the vector mesons can be carried to the detectors. In the following we shall describe how ρ-mesons change their properties at finite temperature. For not too high temperatures such a system consists mainly of pions as the by far lightest hadronic states. ρ-mesons are resonances, i.e. they can be observed in scattering events, e.g. in pionpion scattering. Consequently we will explore pion-pion scattering in the presence of a thermal gas of pions. At low energies pion-pion scattering is described by chiral perturbation theory (χPT). The scattering amplitude calculated in χPT becomes invalid in the resonance region. The resonances cause poles in the scattering amplitudes which terminates the region of convergence of χPT. In contrast, if one calculates the inverse scattering amplitude the poles turn to zeros. Here one can expect that the radius of convergence is larger. This approach — called inverse amplitude method (IAM) — has been shown to work rather well in the vacuum [1]. The ρ-meson could be reconstructed among several other resonances from the χPT expressions for the inverse scattering amplitude. Consequently this approach has been generalized to finite temperature. In [2] the χPT expressions for pion-pion scattering were developed. In [3] the in-medium properties of a ρ-meson which is at rest with respect to the pionic heat bath were determined within the IAM approach. Here we take up the formalism developed in [2, 3] and generalize it to the situation of a ρ-meson which moves with respect to the heat bath. Recall that the ρ-meson is reconstructed form pion-pion scattering. Hence we have to study scattering events were the center-of-mass frame of the reaction moves with respect to the heat bath frame. In general, such a situation is characterized by five independent variables (in vacuum there are only two): One can choose e.g. the Mandelstam variables s and t and in addition the relative velocity of the reaction plane with respect to the heat bath and the two relative angles between this velocity and the incoming and outgoing scattering lines, respectively. To extract the ρ-meson from a scattering amplitude one has to project on spin 1. This only makes sense if the angular momentum is conserved in the reaction. This is not fulfilled for the general kinematical situation described above. It is fulfilled, however, if the reaction plane of the scattering pions (which form the ρmeson) is perpendicular with respect to the boost vector which connects the heat bath frame with the center-ofmass frame of the reaction. In this case rotations within the reaction plane do not change the physical situation: The lines of incoming and outgoing pions, respectively, re600 620 640 660 680 700 720 740 760 780 800 820