D-state effects in the electromagnetic 𝑁𝛥 transition

We consider here a manifestly covariant quark model of the nucleon and the 𝛥, where one quark is off shell, and the other two quarks form an on-shell diquark pair. Using this model, we have shown previously that the nucleon form factors and the dominant form factor for the 𝛾⁢𝑁 →𝛥 transition (the magnetic dipole (M1) form factor) can be well described by nucleon and 𝛥 wave functions with S-state components only. In this paper, we show that nonvanishing results for the small electric (E2) and Coulomb (C2) quadrupole form factors can be obtained if D-state components are added to the 𝛥 valence quark wave function. We present a covariant definition of these components and compute their contributions to the form factors. We find that these components cannot, by themselves, describe the data. Explicit pion cloud contributions must also be added, and these contributions dominate both the E2 and the C2 form factors. By parametrizing the pion cloud contribution for the transition electric and Coulomb form factors in terms of the neutron electric form factor, we estimate that the contributions of the 𝛥 D-state coupled to quark core spin of 3/2 is of the order of 1%, and the contributions of the 𝛥 D-state coupled to quark core spin 1/2 is of the order of 4%.