Self-energy and critical temperature of weakly interacting bosons

Using the exact renormalization group we calculate the momentum-dependent self-energy Σ(k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3⩽D<4 . We obtain the complete crossover function interpolating between the critical regime k≪ k_c , where Σ(k)∝ k^2-η , and the short-wavelength regime k≫ k_c , where Σ(k)∝ k^2(D-3) in D>3 and Σ(k)∝ln(k/ k_c ) in D=3 . Our approach yields the crossover scale k_c on the same footing with a reasonable estimate for the critical exponent η in D=3 . From our Σ(k) we find for the interaction-induced shift of T_c in three dimensions Δ T_c / T_c ≈1.23a n^1/3 , where a is the s -wave scattering length and n is the density.