Formalism used

The differential cross-section of Nπ electroproduction in the most general form can be presented as the following sum:

\begin{equation} \frac{\d\sigma}{\d\Omega} = \frac{\d\sigma_{T }}{\d\Omega} + \varepsilon \frac{\d\sigma_{L }}{\d\Omega} + \sqrt{2\varepsilon(1+\varepsilon)} \frac{\d\sigma_{TL }}{\d\Omega} \cos \varphi + \varepsilon \frac{\d\sigma_{TT }}{\d\Omega} \cos2\varphi + h\sqrt{2\varepsilon(1-\varepsilon)} \frac{\d\sigma_{TL'}}{\d\Omega} \sin \varphi \end{equation}

\( \varepsilon \) stands for virtual photon polarization parameter (please, note: here we do NOT use the longitudinal polarization \( \varepsilon_L \) so the longitudinal cross-sections differ from those in [1]), \( h \) describes the longitudinal polarization of the incident electron: \( h = +1(-1) \) if electrons are polarized parallel (anti-parallel) to the beam direction, \( \varphi \) stands for meson emission angle, \( \frac{\d\sigma_{i}}{\d\Omega} (i = T, L, TT, TL, TL′) \) are related to corresponding response functions \( R^{00}_{i} \) as follows:

\begin{align} \frac{\d\sigma_{i }}{\d\Omega} & = R^{00}_{i } \frac{p_m}{k_{\gamma}^{cm}} \end{align}

\( p_m \) is absolute value of meson three-momentum, \( k_{\gamma}^{cm} \) is the photon equivalent energy in the CM-frame:

\begin{align} k_{\gamma}^{cm} & = \frac{W^2 - M_B^2}{2W} \\ p_m & = \sqrt{E_m^2 - m_m^2} \\ E_m & = \frac{W^2 + m_m^2 - M_B^2}{2W} \end{align}

\( E_m \) stands for meson energy in the CM-frame, \( m_m \) and \( M_B \) stand for meson and baryon masses, \( Q^2 \) is photon virtuality, \( W \) is invariant mass of the final hadronic system.

\( R^{00}_i (i = T, L, TT, TL, TL′) \) are response functions:

\begin{align} \begin{split} R^{00}_{T } & = \frac{1}{2}(|H_1|^2 + |H_2|^2 + |H_3|^2 + |H_4|^2) \\ R^{00}_{L } & = |H_5|^2 + |H_6|^2 \\ R^{00}_{TL } & = \frac{1}{\sqrt{2}}(H_5^*H_1 - H_5^*H_4 + H_6^*H_2 + H_6^*H_3) \\ R^{00}_{TT } & = - H_1^*H_4 + H_2^*H_3 \\ R^{00}_{TL'} & = \frac{1}{\sqrt{2}}(-H_5^*H_1 + H_5^*H_4 - H_6^*H_2 - H_6^*H_3) \end{split} \end{align}

\( H_i \; (i = 1..6) \) stand for helicity amplitudes.