Amplitude reconstruction from complete electroproduction experiments and truncated partial-wave expansions

We compare the methods of amplitude reconstruction, for a complete experiment and a truncated partial-wave analysis, applied to the electroproduction of pseudoscalar mesons off nucleon targets. We give examples which show, in detail, how the amplitude reconstruction (observables measured at a single energy and angle) is related to a truncated partial-wave analysis (observables measured at a single energy and a number of angles). A connection is made to existing data.

Figure 1

Kinematics of an electroproduction experiment. The scattering plane {1,3} is defined by the respective incoming and outgoing electron momenta ${\vec{k}}_i,{\vec{k}}_f$ with the electron-scattering angle ${\mathrm{\Theta }}_e$. The reaction plane is spanned by the virtual photon $\vec{q}$ and the outgoing meson $\vec{k}$, scattered by the angle $\theta$. The reaction plane is tilted vs the scattering plane by the azimuthal angle $\varphi$.

Figure 2

Frames for polarization vectors. Target and recoil polarization are commonly defined as $\{x,y,z\}$ and $\{x^{\prime },y^{\prime },z^{\prime }\}$ in the c.m. frame, with the $z^{\prime }$ direction along the outgoing meson $\pi \left(k\right)$. The virtual photon $\gamma \left(q\right)$ can carry different types of polarization, including the linear and circular polarizations, $P_T$ in the $\{x,y\}$ plane and $P_{\odot }$ along the $z$ axis, as in photoproduction. In addition, the longitudinal photon carries a polarization, ${\varepsilon }_L$, with further polarization types appearing in the $LT$ interferences of Eq. (3).