Renormalization in the three-body problem with resonant $p$-wave interactions

Resonant $p$-wave interactions can be described by a minimal zero-range model defined by a truncated effective range expansion so that the only two-body interaction parameters are the inverse scattering volume $1/a_p$ and the $p$-wave effective range $r_p$. This minimal model can be formulated as a local quantum field theory with a $p$-wave interaction between atom fields and a molecular field. In the two-atom sector, the model is renormalizable, but it has unphysical behavior at high energies because there are negative-probability states with momentum scale $r_p$. In the sector with three atoms, two of which are identical, renormalization in some parity and angular momentum channels involves an ultraviolet limit cycle, indicating asymptotic discrete scale invariance. The Efimov effect occurs in the unitary limit $a_p^{-1/3},r_p\to 0$, but this limit is unphysical because there are low-energy states with negative probability. The minimal model can be of physical relevance only at energies small compared to the energy scale set by $r_p$ where the effects of negative-probability states are suppressed.

Figure 1

Discrete scaling factor ${\lambda }_0=\exp (\pi /s_0)$ as a function of the mass ratio $r=m_1/m_2$. If the identical particles are bosons, the Efimov effect occurs only in the $J^P=1^{+}$ sector. If the identical particles are fermions, the Efimov effect occurs in the $0^{+},1^{+},1^{-}$, and $2^{+}$ sectors.

Figure 2

Efimov plot showing the energy variable $K$ as a function of ${(1/a_p)}^{1/3}$ with $r_p=0$ for the case of identical fermions, equal masses, and $J^P=0^{+}$. The scaling factor ${\lambda }_0\approx 34.2$ was divided by 17 to fit more states in the plot.

Figure 3

The dimensionless binding momentum $\sqrt{8\mu \left|E\right|}/\left|r_p\right|$ as a function of the dimensionless cutoff $2\Lambda /|r_p|$ for fermions with $J^P=0^{+},1/a_p=0,r_p<0$, and $m_1/m_2=1/8$.