Universality, that is the possibility of describing all the interaction physics with a single parameter, the s-wave scattering length, is one of the most outstanding properties of ultracold quantum gases. In the universal regime, the interaction potential is modeled through a contact (zero-range) effective potential, called Fermi pseudopotential. The validity of the universal description of the physics of the gas is very wide, but in some regimes the predictions have been shown to depart from numerical simulations, for example in the case of Bose-Bose mixtures, as shown by recent Monte Carlo calculations [1]. It has been shown that using an improved model for the interaction potential, one can carry out calculations of the equation of state of the gas using finite-temperature functional integration [2], and also compute a modification to the Gross-Pitaevskii equation [3], thus going beyond universality. In a recently published article [4] we investigate the two-body scattering theory in a generic D-dimensional space. By using the T-matrix formalism, and setting an on-shell approximation on the Lippman-Schwinger equation, we systematically obtain explicit expressions of the low-momentum expansion coefficients of the interaction potential, in terms of the low-energy scattering parameters, i.e. the s-wave scattering length and effective range. In the calculations, we use Dimensional Regularization (DR) [5] with Minimal Subtraction (MS). We compare our approach to previous formulas obtained using Effective Field Theory (EFT) [6], and another one based on Born approximation [7]. The former investigation stemmed from the context of the nucleon-nucleon scattering problem [8], and our computation represents an alternative method for obtaining an equivalent potential expansion. As an application, we determine the 2D equation of state with the effective range correction.
[1] V. Cikojevi ́c et al., Universality in ultradilute liquid Bose-Bose mixtures, Phys. Rev. A 99, 023618 (2019).
[2] A. Cappellaro and L. Salasnich, Thermal field theory of bosonic gases with finite-range effective interaction, Phys. Rev. A 95, 033627 (2017).
[3] J. J. Garcìa-Ripoll et al., A quasi-local Gross–Pitaevskii equation for attractive Bose–Einstein condensates, Math. Comput. Simul. 62, 21 (2003).
[4] F. Lorenzi, A. Bardin and L. Salasnich, On-shell approximation for the s-wave scattering theory, Phys. Rev. A 107, 033325 (2023).
[5] L. Salasnich and F. Toigo, Zero-point energy of ultracold atoms, Phys. Rep. 640, 1 (2016).
[6] E. Braaten et al., Nonuniversal effects in the homogeneous Bose gas, Phys. Rev. A 63, 063609 (2001).
[7] A. Collin et al., Energy dependent effective interactions for dilute many-body systems, Phys. Rev. A 75, 013615 (2007).
[8] S. R. Beane and R. C. Farrell, Symmetries of the Nucleon–Nucleon S-Matrix and Effective Field Theory Expansions, Few-Body Syst. 63, 45 (2022).
