Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.
Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.
It is shown that the effective interaction strength of three bosons at small collision energies can be extracted from their wave function at zero energy. Asymptotic expansions of this wave function at large interparticle distances are derived, from which is defined a quantity D named three-body scattering hypervolume, which is an analog of the two-body scattering length. Given any finite-range interactions, one can thus predict the effective three-body force from a numerical solution of the Schr\"odinger equation. In this way, the constant D for hard-sphere bosons is computed, leading to the first complete result for the ground-state energy per particle of a dilute Bose-Einstein condensate (BEC) of hard spheres to order ${\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density. Effects of D are also demonstrated in the three-body energy in a finite box of size L, which is expanded to the order ${L}^{\ensuremath{-}7}$, and in the three-body scattering amplitude in vacuum. The three-body scattering amplitude calculated in this paper disagrees with an earlier calculation in the literature, because of the omission of the two-body effective range in that earlier work. Another key prediction is the condensate fractions of dilute BECs, which also disagree with an earlier work in the literature based on the effective field theory (EFT), as a result of short-range physics. An EFT prediction of the BEC ground-state energy, however, is corroborated.
Effective Field Theory (EFT) provides a powerful framework that exploits a separation of scales in physical systems to perform systematically improvable, model-independent calculations. Particularly interesting are few-body systems with short-range interactions and large two-body scattering length. Such systems display remarkable universal features. In systems with more than two particles, a three-body force with limit cycle behavior is required for consistent renormalization already at leading order. We will review this EFT and some of its applications in the physics of cold atoms and nuclear physics. In particular, we will discuss the possibility of an infrared limit cycle in QCD. Recent extensions of the EFT approach to the four-body system and N-boson droplets in two spatial dimensions will also be addressed.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.
We develop an effective field theory (EFT) for strongly interacting bosonic clusters, using 4He as a paradigmatic example of universality in systems with large scattering length. At leading order (LO), two- and three-body zero-range interactions are entirely determined by the dimer and trimer ground-state energies. We show that ground-state energies for up to N=15 particles converge to cutoff-independent limits with extrapolation coefficients of natural size. At next-to-leading order (NLO), corrections stemming from the two-body interaction range and a four-body force, calibrated to the tetramer ground-state energy, reduce cutoff sensitivity. Close agreement with results from a realistic potential is found at LO and improved at NLO, demonstrating systematic convergence with few parameters at each order. The resulting EFT is directly applicable to larger clusters and bulk helium.
We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for 4He atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.