Self-energy and critical temperature of weakly interacting bosons

Using the exact renormalization group we calculate the momentum-dependent self-energy $\Sigma \left(k\right)$ 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 $\Sigma \left(k\right)\propto k^{2-\eta }$, and the short-wavelength regime $k⪢k_c$, where $\Sigma \left(k\right)\propto k^{2(D-3)}$ in $D>3$ and $\Sigma \left(k\right)\propto \ln (k∕k_c)$ in $D=3$. Our approach yields the crossover scale $k_c$ on the same footing with a reasonable estimate for the critical exponent $\eta$ in $D=3$. From our $\Sigma \left(k\right)$ we find for the interaction-induced shift of $T_c$ in three dimensions $\Delta T_c∕T_c\approx 1.23a{n}^{1∕3}$, where $a$ is the $s$-wave scattering length and $n$ is the density.

Figure 1

Double logarithmic plot of the scaling function $\sigma \left(x\right)$ in $D=3$, see Eqs. (1, 16). Note that in our rescaled units we obtain a universal curve for $u_0\to 0$. The dashed lines indicate the asymptotic behavior for small and large $x$.