What is resonance?

Abstract

The concept of duality, discussed in the previous paper by Dr C. Schmid, refers to the notion that the analysis of an S matrix amplitude in terms of s channel resonances and its analysis in terms of t channel (and u channel) reggeon exchanges are not independent, but represent alternative descriptions, the former being the more appropriate in the low energy domain and the latter being the more appropriate in the low t (or low u ) domain for high energy interactions. Hence, at the time when this Discussion meeting was planned, it appeared necessary to attempt to clarify the notion of resonance , since there was then a good deal of discussion about how partial wave analyses of scattering data should be interpreted in terms of resonance states (Bransden, O’Donnell & Moorhouse 1965; Lovelace 1968; Donnachie 1968), and even whether such an interpretation was necessarily the case at all (Schmid 1968 a,b ; Collins, Johnson & Squires 1968; Alessandrini & Squires 1968; Alessandrini, Freund, Oehme & Squires 1968; Collins, Johnson & Ross 1968; Kreps & Logan 1969). However, in the actual use of duality in this meeting and in the model amplitudes which have been discussed thus far for the illustration of duality, there is really no controversy about the nature of resonance, since these models are still confined to the cases of resonances with zero width, characterized by poles on the real s axis, and there is essentially no ambiguity about the relation between resonances and poles in this limiting case. All the same, it still seems worthwhile to take a few moments to draw attention to a number of points about the description of resonance, which deserve to be more widely known.

Parameterization dependence ofT-matrix poles and eigenphases from a fit toπNelastic scattering data

“…However, a more robust measure (if a set of amplitudes is available) is given by the time-delay matrix [18], which is proportional to the sum of energy derivatives of all eigenphases. Other factors, such as threshold openings can also produce rapid energy dependence.…”

Extraction of resonances from meson-nucleon reactions

“…For the single-channel elastic scattering case, the time delay of the outgoing wave packet with respect to the non-interacting wave packet in a given partial wave is defined [6,16,17] as…”

“…The concept of time delay was originally introduced by Eisenbud [5] and discussed by Wigner [6], Dalitz and Moorehouse [16], and Nussenzveig [17]. It is defined as the difference between the time in which a wave packet passes through an interaction region and the time spent by a free wave packet passing though the same distance.…”

Time delay and time advance in resonance theory

“…[9,10,11,12,13,14,15] and the references therein). Among these many works, we limit ourselves to quote a quite recent approach which seems promising and consists in extract-ing the resonance parameters from the energy distribution of the time delay [16,17].…”