Legendre analysis of differential distributions in hadronic reactions

Modern experimental facilities have provided a tremendous volume of reaction data, often with wide energy and angular coverage, and with increasing precision. For reactions with two hadrons in the final state, these data are often presented as multiple sets of panels, with angular distributions at numerous specific energies. Such presentations have limited visual appeal, and their physical content is typically extracted through some model-dependent treatment. Instead, we explore the use of a Legendre series expansion with a relatively small number of essential coefficients. This approach has been applied in several recent experimental investigations. We present some general properties of the Legendre coefficients in the helicity framework and consider what physical information can be extracted without any model-dependent assumptions.

Figure 1

Top panel: Samples of the $\gamma p\to {\pi }^{0}p$ differential cross sections, $d\sigma /d\mathrm{\Omega }$, from A2 Collaboration at MAMI measurements (blue filled circles) [2] with the best fit results using Legendre polynomials (red dashed lines). The error bars on all data points represent statistical uncertainties only. Values of $E$ in each plot indicate the laboratory photon energies. Bottom panel: Coefficients of Legendre polynomials (blue filled circles). The error bars of all values represent ${\mathrm{A}}_J^{\left(\sigma \right)}$ uncertainties from the fits in which only the statistical uncertainties were used. Solid lines are plotted to help guide the eye. Red vertical arrows indicate masses of the four-star resonances (BW masses) known in this energy range [10]. The upper row of arrows corresponds to $N^{*}$ states with isospin $I=1/2$ and the lower row corresponds to ${\mathrm{\Delta }}^{*}$ with $I=3/2$.

Figure 2

Zoom for A2 $\gamma p\to {\pi }^{0}p$ data below $W=1400$ MeV to cover the $\mathrm{\Delta }$-isobar region [2] as shown in Fig. 1 (bottom panel).

Figure 3

Same as Fig. 1, but for $\gamma p\to \eta p$ data from [1].