Consistence between $\pi N\Delta$ spin-3/2 gauge couplings and electromagnetic gauge invariance

We review the consistency between the recently proposed spin-3/2 gauge interaction for Δ resonance with nucleons (N) and pions (π), and the fundamental electromagnetic gauge invariance in any radiative amplitude. We show that: (1) the electromagnetic interaction introduced through minimal substitution in all derivatives breaks the spin-3/2 gauge invariance; (2) radiative corrections (in general not zero for any effective Lagrangian theory) of the spin-3/2 gauge strong vertexes at one loop reintroduce the conventional π derivative interaction. In fact, analyzing elastic and radiative $\pi N$ scattering amplitude, we show that chiral symmetric π-derivative $\pi N\Delta$ couplings can be substituted through a linear transformation to get Δ-derivative ones, which have the property of decoupling the 1/2 field components of the Δ propagator, plus contact terms where the Δ-field is absent. One could intend to apply new transformations (now linear and nonlinear) to restore this property in the radiative case, but the electromagnetic gauge invariance does not survive and nonlinear transformations generate an infinite number of Δ-field terms in the Lagrangian. We conclude that we can only content ourselves to fulfil approximately the electromagnetic gauge invariance at a given order n without destroying the spin-3/2 one, by dropping n + 1 order terms within an effective field theory framework, where the 'order n' is defined as a power counting expansion applicable only in the resonance region. In addition, we show that the Ward identity for the $\Delta \gamma \Delta$ vertex cannot be fulfilled with a trimmed 3/2 propagator—added ad hoc by some authors with the intention of avoiding 1/2 propagation in the radiative case for the $\Delta \gamma \Delta$ amplitude. (paper)