Convergence of a renormalization-group approach to dimer-dimer scattering

We study the convergence of a functional renormalization group technique by looking at the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for a system of nonrelativistic fermions. We find that in a systematic expansion in powers of the fields there is a rapid convergence of the result that agrees with known exact results.

Figure 1

Skeletons of the diagrams that contribute to the evolution of the couplings. Dashed lines represent bosons, solid lines fermions. The dots represent the interaction vertices. In (a) we show the diagram for the evolution of $u_1$, $Z_m$, and $Z_{\varphi }$, and in (b) for $u_2$, (c) for $\lambda$, (d) for $g^{\prime }$, and (e) for $\nu$.

Figure 2

Ratio of dimer-dimer to fermion-fermion scattering lengths as a function of the relative scale parameter $c_B$. The black (dash-dotted) curve shows results for the minimal action; the red (dashed) curve shows the effect of adding the local three-body term; the green (solid) curve displays the full local four-body calculation, and is enlarged in the inset. The purple (dotted) line shows the effect of using a smooth cutoff ($\sigma =0.5$) on the full results.

Figure 3

Ratio of dimer-dimer to fermion-fermion scattering lengths as a function of the relative scale parameter $c_B$. The black (right-slanted hash) curve shows results for the minimal action; the red (vertical hash) curve shows the effect of adding the local three-body term; the green (left-leaning hash) curve displays the full local four-body calculation. The width of each curve denotes the sensitivity of the result to changes in $a_F$ at $a_F=\mathit{gM}/\hslash {}^2$.

Figure 4

Ratio of dimer-dimer to fermion-fermion scattering lengths as a function of the relative scale parameter $\log {}_{10}{c}_B$, and the base 10 logarithm of the scattering length $a_F$ (divided by $\mathit{gM}/\hslash {}^2$). Notice the insensitivity to $a_F$ for $c_B<1$, a clear indication of convergence.