  We study the chiral behavior of the nucleon and $\Delta$-isobar masses within
a manifestly covariant chiral effective-field theory, consistent with the
analyticity principle. We compute the $\pi N$ and $\pi\Delta$ 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 $\Delta$ masses, while
inclusion of the $\pi \Delta$ loops tends to spoil this agreement.
