Self-energy of an impurity in an ideal Fermi gas to second order in the interaction strength

We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass m perturbed by the presence of a single distinguishable impurity of mass M. The interaction between the impurity and the fermions involves only the partial s wave through the scattering length a and has negligible range b compared to the inverse Fermi wave number 1/k(F) of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector K not equal 0: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around K = 0. We rely thus upon perturbation theory, where the small and negative parameter k(F)a -> 0(-) excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy Sigma((2))(K, omega) up to second order included in a. Remarkably, we obtain an analytical explicit expression for Sigma((2))(K, omega), allowing us to study its derivatives in the plane (K, omega). These present interesting singularities, which in general appear in the third-order derivatives. partial derivative(3)Sigma((2))(K, omega). In the special case of equal masses, M = m, singularities appear already in the physically more accessible second-order derivatives partial derivative(2)Sigma((2))(K, omega); using a self-consistent heuristic approach based on Sigma((2)) we then regularize the divergence of the second-order derivative partial derivative(2)(K) Delta E(K) of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at K = k(F), and we predict an interesting scaling law in the neighborhood of K = k(F). As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.