Time delay in a multichannel formalism

Abstract

We reexamine the time-delay formalism of Wigner, Eisenbud and Smith, which was developed to analyze both elastic and inelastic resonances. An error in the paper of Smith has propagated through the literature. We correct this error and show how the results of Eisenbud and Smith are related. We also comment on some recent time-delay studies, based on Smith's erroneous interpretation of the Eisenbud result. I. THE RESULT OF WIGNER AND EISENBUDIn 1948, Eisenbud [1] used a simple wave-packet approach to show that, for an elastic resonance, the time-delay (∆t) in a collision process was related to the energy-derivative of the phase shiftthis result, apart from a factor of two [2], also appeared in a paper by Wigner [3], which used causality to place a limit on dδ/dE. Less well known is the Eisenbud result for scattering into several final states [1,4],In the case of elestic scattering, where S = e 2iδ , this gives the result in Eq.(1). Note that ∆t, in the multi-channel case, becomes a matrix. The matrix S being symmetric and unitary was diagonalized and one element was assumed to be resonant. The resulting matrix ∆t was again found to have the form of Eq. (1) for each entry, however, with the phase shift replaced by the resonant eigenphase. Unfortunately, the same notation was used for both the phase shift and the eigenphase, and (as we will see) this may have caused some confusion. II. THE RESULT OF SMITHSmith [5] derived the time-delay matrix based on the flux passing through some interaction region of radius R. His main result was for the average lifetime of a metastable state due to a collision beginning in the i th channel. The result wasSmith further claimed that his result and the result of Eisenbud could be connected, using the following representation for Eisenbud's resultand a relation for the average over all outgoing channelsgiving Eq. (3), as claimed. The problem with this argument lies in Eq. (4), which is not equivalent to Eisenbud's result given in Eq. (2) above.

Thermodynamics of the strange baryon system from a coupled-channels analysis and missing states

“…Consider a single relativistic Breit-Wigner (B-W) resonance of mass m res decaying via two channels. The S-matrix can be parametrized as [30][31][32]…”

Extraction of resonances from meson-nucleon reactions

“…We now consider the on-shell matrix element of the S-matrix Eq. (25). If the on-shell momentum is denoted as p i for channel i, we then have…”

“…(1) of the single-channel case. The trace of the life-time matrix can be expressed in terms of eigen phases δ i of the S-matrix [23,24,25] TrQ…”

“…Our findings could provide some information for investigating the differences between Refs. [25] and [34]. We also want to mention here that an improved SP method using higher order derivatives of the amplitudes was proposed in Ref.…”

“…(16), where the phase of the elastic channel is used instead of the eigen phases discussed in Refs. [23,24,25], gives both the resonance poles on the uu-sheet and the zeros(shadow poles). Our findings could provide some information for investigating the differences between Refs.…”