New low-$Q^2$ measurements of the $\gamma^\ast N \to \Delta(1232)$ Coulomb quadrupole form factor, pion cloud parametrizations and Siegert's theorem

TL;DR

The paper presents new low-$Q^2$ measurements of the $ o ext{Delta}(1232)$ Coulomb quadrupole form factor, showing agreement with pion cloud models and Siegert's theorem, and refining parametrizations at low momentum transfer.

Contribution

It introduces new low-$Q^2$ data and improved parametrizations for the quadrupole form factors consistent with Siegert's theorem, enhancing understanding of nucleon-to-Delta transitions.

Findings

New measurements alter previous data trends.

Form factors agree with pion cloud contributions at low $Q^2$.

Parametrizations satisfy Siegert's theorem.

Abstract

The novel measurements of the $\gamma^\ast N \to \Delta(1232)$ Coulomb quadrupole form factor in the range $Q^2=0.04$--0.13 GeV$^2$ changed the trend of the previous data. With the new data the electric and Coulomb form factors are both in remarkable agreement with estimates of the pion cloud contributions to the quadrupole form factors at low $Q^2$. The pion cloud contributions to the electric and Coulomb form factors can be parametrized by the relations $G_E \propto G_{En}/ \left(1 + \frac{Q^2}{2M_\Delta(M_\Delta-M)}\right)$ and $G_C \propto G_{En}$, where $G_{En}$ is the neutron electric form factor, and $M$, $M_\Delta$ are the nucleon and $\Delta$ masses, respectively. Those parametrizations are in full agreement with Siegert's theorem, which states that $G_E= \frac{M_\Delta-M}{2M_\Delta} G_C$ at the pseudothreshold, when $Q^2=-(M_\Delta -M)^2$, and improve previous parametrizations. Also a small valence quark component estimated by a covariant quark model contributes to this agreement. The combination of the new data with the new parametrization for $G_E$ concludes an intense period of studying the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors at low $Q^2$, with the agreement between theory and data.