The effect of phase-shift ambiguities on finite-energy sum rules

Abstract

In the use of finite-energy sum rules for ampl i tudes at a fixed m o m e n t u m transfer some prel iminary work is a prerequisi te. Namely the ampl i tude in the h)wand intermediate-energy region mus t first of all be deduced via a phase-shift almlysis from the exper imental data. The quest ion of how well a phase-shift analysis can give a reliable value of the ampl i tude has no s traightforward answer aud we have recent ly found that some degree of ambigui ty cer ta inly exists (~). There is of course the obvious ambigu i ty tha t different phase-shift analyses may lead to quan t i t a t ive ly differing part ial waves (-~) al though the qual i ta t ive features may be very similar. I lowevcr in (x) we pointed out tha t an ((in principle )) ambigui ty may exist of the, form tha t (the mult ipl icat ion of) a given ampl i tude (or pair of amplitudes) by a factor exp [iqJ(E, 0)], where ~(~, 0) is real for physical E aud 0, would give rise to identical cross-section and polarizatiol~ data since these are ahvays given as a bil inear form, ampl i tude mult ipl ied by complex conjugate of ampl i tude. The form and s t rength of r 0) is of course restr icted by the requi remcnt tha t the resul t ing par t ia l waves satisfy uni ta ry bounds. We here invest igate what effect this type of ambigu i ty has on the use of F E S R for ~-.N' ampli tudes. We adopt the s tandard no ta t ion for the ~,-~\' ampl i tudes . t ' ' , B ~ (3) aud consider F E g R with a moment factor. A typical one is