Dolen-Horn-Schmid Duality and the Deck Effect

Abstract

A remark of profound import for strong-interaction theory has been made by Dolen, Horn, and Schmid 1 in connection with finite-energy sum rules. They have observed that high-energy Regge behavior is consistent with low-energy resonance behavior only if extrapolation of the smooth Regge representation down to low energy gives a certain semilocal average over the resonance peaks. In other words what is usually called the "peripheral" approximation to a reaction amplitude must, without containing energy poles, in a rough sense represent the resonances. (The converse presumably is also true.) We refer to this startling notion as "Dolen-Horn-Schmid duality." Its implication for bootstrap theory is being pursued vigorously by many authors 2 ; our object here is to suggest relevance to what has been called the "Deck effect." 3 We argue that the Deck peripheral model for a reaction such as irN^piiN, explaining a peak in the final irp mass spectrum without explicit insertion therein of a resonance, fails to imply the absence of a resonance. On the contrary, Dolen-Horn-Schmid duality means that when peripheral models of this kind predict large cross sections at low subenergies (the term "subenergy" is used to mean energy of a subsystem), there ). The curve of Fig. 2 is 7% lower than the approximate calculation of Ash et al. (Ref. 2).probably are resonances present. Such reasoning leads to enormous simplification of multiperipheral calculations.The step needed to relate Dolen-Horn-Schmid to Deck is the extension of single peripheralism to double peripheralism. Deck's model for the above reaction, for example, is depicted in Fig. 1, corresponding to a double Regge-pole representation, 4 a representation supposed to have validity when both the TTN and irp final subenergies are large. 5 The highest trajectory for the righthand momentum transfer is the Pomeranchuk; the highest for the left-hand momentum transfer is not the 77, but the small mass of the physical pion enhances the Regge residue so that this trap ir N FIG. 1. Diagram representing the Deck doubly peripheral model for the reaction irN^irpN.An extension to multiperipheralism is made of the Dolen-Horn-Schmid duality argument relating Regge poles to low-energy resonances. The Deck model is thereby interpreted as predicting the existence of the A it rather than as undermining experimental evidence for this resonance. It is shown in general that Dolen-Horn-Schmid duality permits a vast simplification in the calculation of multiple-production processes.1078

Review of Recent Work on Narrow Resonance Models

“…We would like to discuss the subject of "duality If. interms of an amplitude with-poles in two channels, sand t, which is symmetric under t -s. The term; duality, was first invented by Chew and Pignotti (1968) to describe the observation of Dolen,Horn, and Schmid (1968) that there exist intermediate energies where some FESR I S can be saturated on the left hand side by a few -14-dominant resonances,and on the right hand side by the leading Regge trajectory, as discussed above. Controversy has arisen regarding the definition and applicabilityof "dual" and liinterference" models.…”

Graphical form of Duality

“…However, it is not clear how to do this in a relativistically invariant way. 20 Our approach exhibits duality for production processes as well, 21 but the SU(3) couplings are harder to test. However, rule (3) above may provide useful estimates of production amplitude phases for use in multiperipheral bootstraps.…”

Coherent Production of theK−π+π−System inWerner1, Ammar2, Davis3 et al. 1969Phys. Rev.

“…In fact, recent interpretation of the duality hypothesis indicates that such a diagram could describe the Q region in some average sense and such a description is not inconsistent with an explanation involving KWT resonances. 22 With this in mind, it is noteworthy to consider the quantum numbers of the KTW system associated with the s-wave Kw system responsible for the fore-aft asymmetry. The dominant mode in the Q region is a 1+ R*ir system in a relative s wave.…”