Abstract
We demonstrate how an effective density of states can be derived from the S-matrix describing a coupled-channel system. Besides the locations of poles, the phase of the determinant of the S-matrix encodes essential details in characterizing the dynamics of resonant and non-resonant interactions. The density of states is computed for the two channel scattering problem ($\pi\pi, K \bar{K}$, S-wave), and the influences from the various dynamical structures: poles, roots, branch cuts, and Riemann sheets, are examined.
Highlights
Thermodynamics is essentially tied to the proper counting of states: the question is what states to count and how to count them
We find that the S-matrix density of states (DOS) is only influenced by those states which are directly connected to the physical sheet, suggesting not all states are counted in the partition sum
The smoothness of color in transiting the real line indicates the connectedness of the Riemann sheets. This gives an intuitive criterion for the relevance of poles in the complex plane when calculating the physical DOS
Summary
Thermodynamics is essentially tied to the proper counting of states: the question is what states to count and how to count them. A unique feature of this formulation, in contrast to the standard Matsubara approach, is the decoupling of zero temperature dynamics and statistics [1,3,4,5] This is what makes the scheme powerful: one can make progress in understanding the thermal medium by successively improving the S-matrix input, including relevant channels, extension to N > 2 scatterings, etc., working toward building an accurate virial/cluster expansion. For a single resonance with width, decaying into a single channel, similar principle applies: the DOS derived from the single-channel phase shift contains the contribution of the resonance, including its full width, and in addition a nonresonant scattering contribution from the asymptotic states [5,16,34,35] The latter tend to dominate at threshold and give a substantial contribution to the thermal pressure due to the smaller Boltzmann suppression. The influence from other dynamical features, e.g., roots and Riemann sheet structures, will be examined
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