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I discuss the problem of constructing an effective low-energy theory in the vicinity of a resonance or a bound state. The focus is on the example of the $\Delta(1232)$, the lightest resonance in the nucleon sector. Recent developments of the chiral effective-field theory in the $\Delta$-resonance region are briefly reviewed. I conclude with a comment on the merits of the manifestly covariant formulation of chiral EFT in the baryon sector.
We present a theoretical study of the radiative pion photoproduction on the nucleon ($\gamma N \rightarrow \pi N \gamma'$) in the $\De$-resonance region, with the aim to determine the magnetic dipole moment (MDM) of the $\Delta^+(1232)$. The study is done within the framework of chiral effective-field theory where the expansion is performed (to next-to-leading order) in the $\delta$ power-counting scheme which is an extension of chiral perturbation theory to the $\Delta$-resonance energy region. We present in detail the results for the absorptive part of the $\Delta$ MDM, as well as a sensitivity study for the radiative pion photoproduction observables on the real part of the $\Delta$ MDM. We find that an asymmetry for circular polarization of the photon beam may provide a model-independent way to measure the $\Delta$ MDM.
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