The nucleon and Delta-resonance masses in relativistic chiral effective-field theory

We study the chiral behavior of the nucleon and $\mathrm{\Delta }$-isobar masses within a manifestly covariant chiral effective-field theory, consistent with the analyticity principle. We compute the $\pi N$ and $\pi \mathrm{\Delta }$ one-loop contributions to the mass and field-renormalization constant, and find that they can be described in terms of universal relativistic loop functions, multiplied by appropriate spin, isospin and coupling constants. We show that these relativistic one-loop corrections, when properly renormalized, obey the chiral power-counting and vanish in the chiral limit. The results including only the $\pi N$-loop corrections compare favorably with the lattice QCD data for the pion-mass dependence of the nucleon and $\mathrm{\Delta }$ masses, while inclusion of the $\pi \mathrm{\Delta }$ loops tends to spoil this agreement.