Scattering amplitude of ultracold atoms near the p-wave magnetic Feshbach resonance

Most of the current theories on the p-wave superfluid in cold atomic gases are based on the effective-range theory for the two-body scattering, where the low-energy p-wave scattering amplitude f(1)(k) is given by f(1)(k) = -1/[ik + 1/(Vk(2)) + 1/R]. Here k is the incident momentum, V and R are the k-independent scattering volume and effective range, respectively. However, due to the long-range nature of the van der Waals interaction between two colliding ultracold atoms, the p-wave scattering amplitude of the two atoms is not described by the effective-range theory [J. Math. Phys. 4, 54 (1963); Phys. Rev. A 58, 4222 (1998)]. In this paper we provide an explicit calculation for the p-wave scattering of two ultracold atoms near the p-wave magnetic Feshbach resonance. We show that in this case the low-energy p-wave scattering amplitude f(1)(k) = -1/[ik + 1/(V(eff)k(2)) + 1/(S(eff)k) + 1/R-eff] where V-eff, S-eff, and R-eff are k-dependent parameters. Based on this result, we identify sufficient conditions for the effective-range theory to be a good approximation of the exact scattering amplitude. Using these conditions we show that the effective-range theory is a good approximation for the p-wave scattering in the ultracold gases of Li-6 and K-40 when the scattering volume is enhanced by the resonance.