Gauge-invariant coupling to spin-(3)/(2) resonances

A gauge invariant interaction of the Δ-resonance to the final πN and γN is discussed. It is shown that the gauge-invariant coupling invented by Pascalutsa eliminates 'off-shell' degrees of freedom only for the Rarita-Schwinger propagator but fails in the general case. The origin of the problem is discussed and a generalized gauge-invariant coupling is deduced. Finally, the full Lagrangian depends on one free parameter which reflects a freedom in choosing an off-shell content of the theory. The calculated πN scattering amplitude shows no dependency on the specific choice of the parameter. This reflect the fact that physical observables do not depend on any off-shell contributions and do not depend on the free parameter of Lagrangian.