Accuracy of traditional Legendre estimators of quadrupole ratios for the $N\to \Delta$ transition

We evaluate the accuracy of traditional estimators often used to extract $N\to \Delta$ quadrupole ratios from cross-section angular distributions for pion electroproduction. We find that neither $M_{1+}$ dominance nor $\ell \le 1$ truncation is sufficiently accurate for this purpose. Truncation errors are especially large for $R_{\mathrm{EM}}$, for which it is also essential to perform Rosenbluth separation. The accuracy of similar truncated Legendre analyses for $E_{0+},S_{0+}$, and especially $M_{1-}$ is even worse.

Figure 1

W dependence of quadrupole estimators for $p{\pi }^0$ at $Q^2=1.0$ (GeV/c)${}^2$ using MAID2003 multipoles. Solid curves show $R_{\mathrm{EM}}$ and $R_{\mathrm{SM}}$ while dashed and dash-dotted curves show ${\tilde{R}}_{\mathrm{EM}}$ and ${\tilde{R}}_{\mathrm{SM}}$ for $\ell \le 5$ using $\epsilon =0$ and $\epsilon =0.9$, respectively. The vertical dashed line denotes $W=1.232$ GeV.

Figure 2

Accuracy of traditional quadrupole estimators for $p{\pi }^0$ at $W=1.232$ GeV using MAID2003 multipoles. Solid curves use $\epsilon =0$; dashed curves, $\epsilon =0.9$.