Low-$Q^2$ empirical parametrizations of the $N^\ast$ helicity amplitudes

The data associated with the electromagnetic excitations of the nucleon ($\gamma^\ast N \to N^\ast$) are usually parametrized by helicity amplitudes at the resonance $N^\ast$ rest frame. The properties of the $\gamma^\ast N \to N^\ast$ transition current at low $Q^2$ can be, however, better understood when expressed in terms of structure form factors, particularly near the pseudothreshold, when the magnitude of the photon three-momentum vanishes ($|{\bf q}| =0$). At the pseudothreshold the invariant four-momentum square became $q^2= (M_R-M_N)^2$, well in the timelike region $Q^2 =- q^2 < 0$ [$M_N$ and $M_R$ are the mass of the nucleon and of the resonance, respectively]. In the helicity amplitude representation, the amplitudes have well-defined dependences on $|{\bf q}|$, near the pseudothreshold, and there are correlations between different amplitudes. Those constraints are often ignored in the empirical parametrizations of the helicity amplitudes. In the present work, we show that the structure of the transition current near the pseudothreshold has an impact on the parametrizations of the data. We present a method which modifies analytic parametrizations of the data at low $Q^2$, in order to take into account the constraints of the transition amplitudes near the pseudothreshold. The model dependence of the parametrizations on the low-$Q^2$ data is studied in detail.