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 f1(k) is given by f1(k)=-1/[ik+1/(Vk2)+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 f1(k)=-1/[ik+1/(Veffk2)+1/(Seffk)+1/Reff] where Veff, Seff, and Reff 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 6Li and 40K when the scattering volume is enhanced by the resonance.