Limit cycles of effective theories

A simple example is used to show that renormalization group limit cycles of effective quantum theories can be studied in a new way. The method is based on the similarity renormalization group procedure for Hamiltonians. The example contains a logarithmic ultraviolet divergence that is generated by both real and imaginary parts of the Hamiltonian matrix elements. Discussion of the example includes a connection between asymptotic freedom with one scale of bound states and the limit cycle with an entire hierarchy of bound states.

Figure 1

The solid line displays the cycle. The broken line is an approximate result explained in Sec. 3.

Figure 2

Full lines show the $\lambda$-dependence of lowest energy components, i.e., ${\mathtt{a}}_{\lambda }={\mathtt{a}}_k(\lambda )/{\mathtt{a}}_k(\infty )$, for 5 eigenvectors belonging to one cycle with $k=-6$, $-5$, $-4$, $-3$, and $-2$, from left to right, respectively, and the broken lines display results of the approximate equation (10).

Figure 3

Behavior of $g_{\lambda }$ near the scale of binding in two cases: limit cycle and asymptotic freedom. The solid curve is obtained for the cycle with $p=50$ and the dashed curve for asymptotic freedom with $g_{\Delta }\sim 0.04$ (see text for details).