  A systematic investigation of the effects of disorder on the BCS-BEC
crossover at the lowest order in the impurity potential is presented for the
normal phase above the critical temperature Tc. Starting with the t-matrix
approach for the clean system, by which pairing correlations between
opposite-spin fermions evolve from the weak-coupling (BCS) to the
strong-coupling (BEC) limits by increasing the strength of the attractive
inter-particle interaction, all possible diagrammatic processes are considered
where the effects of a disordered potential are retained in the self-energy at
the lowest order. An accurate numerical investigation is carried out for all
these diagrammatic terms, to determine which of them are mostly important
throughout the BCS-BEC crossover. Explicit calculations for the values of Tc,
the chemical potential, and the Tan's contact are carried out. In addition, the
effect of disorder on the single-particle spectral function is analyzed, and a
correlation is found between an increase of Tc and a widening of the pseudo-gap
energy at Tc on the BCS side of unitarity in the presence of disorder, while on
the BEC side of unitarity the presence of disorder favors the collapse of the
underlying Fermi surface. The present investigation is meant to orient future
studies when the effects of disorder will be considered at higher orders, with
the purpose of limiting the proliferation of diagrammatic terms in which
interaction and disorder are considered simultaneously.
