Resonances extracted in truncated partial-wave analysis are effective mixtures of angular momenta (Possible implications for H\"ohler's clustering)

Abstract

In truncated partial-wave analysis one fits observables, not amplitudes, and the relevant observables are bilinear in the amplitudes. For angle-dependent observables from which partial-wave content is inferred, truncation therefore does more than simply discard higher partial waves. The extracted lower partial waves are determined by a coupled nonlinear fit and need not be direct projections of the corresponding quantities in the full non-truncated problem. Instead, truncation reshuffles pole-bearing content among partial waves, including the nominally retained lower ones, so that a resonance contribution associated with one exact angular-momentum sector can reappear in several extracted partial waves and lose a unique angular-momentum assignment. We demonstrate this explicitly in a minimal scalar toy model, where a Hermitian bilinear represented by a Legendre series truncated at order 2 is fitted by another series truncated at order 1. Even in this simplest case, the fitted low-order coefficients depend on bilinear combinations involving higher-order parts of the original amplitude. Resonance-related quantities extracted from such a truncated analysis should therefore not, in general, be interpreted as resonances with definite angular momentum. We then discuss a possible phenomenological consequence for H\"ohler's observation that resonance poles assigned to different partial waves in $\pi N$ scattering tend to cluster near a few common points in the complex energy plane. If the extracted pole-bearing quantities are effective mixtures of several angular-momentum sectors, the inferred spectrum can naturally exhibit cross-wave correlations. In this sense, truncation provides a plausible contribution to H\"ohler-type clustering.