Effective Range Corrections Research Articles

Applied nonrelativistic conformal field theory: Scattering-length and effective-range corrections to rate of production of three neutrons at low relative momenta

Due to an accidentally large s-wave scattering length, in a relatively wide range of energy, neutrons are approximately described by the nonrelativistic conformal field theory of unitarity fermions, perturbed by one relevant and an infinite number of irrelevant operators. We develop a formalism which provides a nonperturbative definition of local operators in that nonrelativistic conformal field theory. We compute the scattering-length and effective-range corrections to the two-point functions of primary charge-three operators using the technique of conformal perturbation theory. These calculations allow us to find the first corrections to the scale-invariant behavior of the rate of nuclear reactions with three neutrons in the final state in the regime when the neutrons have small relative momenta. Published by the American Physical Society 2024

An Alternative Scheme for Pionless EFT: Neutron-Deuteron Scattering in the Doublet S-Wave

Using the effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the breakdown of unitarity in the three-body amplitude. We discuss the extension of our alternative subtraction scheme for including effective range corrections in pionless effective field theory for spinless bosons to nucleons. In particular, we consider the neutron-deuteron system in the doublet S-wave channel explicitly.

Symmetries of the Nucleon–Nucleon S-Matrix and Effective Field Theory Expansions

The s-wave nucleon–nucleon scattering matrix (S-matrix) exhibits UV/IR symmetries which are hidden in the effective field theory (EFT) action and scattering amplitudes, and which explain some generic features of the phase shifts. These symmetries offer clarifying interpretations of existing pionless EFT expansions, and suggest starting points for novel expansions. The leading-order (LO) S-matrix obtained in the pionless EFT with scattering lengths treated exactly is shown to have a UV/IR symmetry which leaves the sum of s-wave phase shifts invariant. A new scheme, which treats effective range corrections exactly, and which possesses a distinct UV/IR symmetry at LO, is developed up to NLO (next-to-LO) and compared with data.

An alternative scheme for effective range corrections in pionless EFT

We discuss an alternative scheme for including effective range corrections in pionless effective field theory. The standard approach treats range terms as perturbative insertions in the T-matrix. In a finite volume this scheme can lead to singular behavior close to the unperturbed energies. We consider an alternative scheme that resums the effective range but expands the spurious pole of the T-matrix created by this resummation. We test this alternative expansion for several model potentials and observe good convergence.

Quantum Monte Carlo studies of a trimer scaling function with microscopic two- and three-body interactions

We present an energy scaling function to predict, in a specific range, the energy of bosonic trimers with large scattering lengths and finite range interactions, which is validated by quantum Monte Carlo calculations using microscopic Hamiltonians with two- and three-body potentials. The proposed scaling function depends on the scattering length, effective range, and a reference energy, which we chose as the trimer energy at unitarity. We obtained the scaling function as a limit cycle from the solution of the renormalized zero-range model with effective range corrections. We proposed a simple parametrization of the energy scaling function. Besides the intrinsic interest in theoretical and experimental investigations, this scaling function allows one to probe Efimov physics with only the trimer ground states, which may open opportunities to identify Efimov trimers whenever access to excited states is limited.

Effective-range corrections to the ground-state energy of the weakly-interacting Bose gas in two dimensions

Nonuniversal effects due to leading effective-range corrections are computed for the ground-state energy of the weakly-coupled repulsive Bose gas in two spatial dimensions. Using an effective field theory of contact interactions, these corrections are computed first by considering fluctuations around the mean-field free energy of a system of interacting bosons. This result is then confirmed by an exact calculation in which the energy of a finite number of bosons interacting in a square with period boundary conditions is computed and the thermodynamic limit is explicitly taken.

Effective Field Theory for Rydberg Polaritons.

We develop an effective field theory (EFT) to describe the few- and many-body propagation of one-dimensional Rydberg polaritons. We show that the photonic transmission through the Rydberg medium can be found by mapping the propagation problem to a nonequilibrium quench, where the role of time and space are reversed. We include effective range corrections in the EFT and show that they dominate the dynamics near scattering resonances in the presence of deep bound states. Finally, we show how the long-range nature of the Rydberg-Rydberg interactions induces strong effective N-body interactions between Rydberg polaritons. These results pave the way towards studying nonperturbative effects in quantum field theories using Rydberg polaritons.

Effective range from tetramer-dissociation data for cesium atoms

The shifts in the four-body recombination peaks, due to an effective range correction to the zero-range model close to the unitary limit, are obtained and used to extract the corresponding effective range of a given atomic system. The approach is applied to an ultracold gas of cesium atoms close to broad Feshbach resonances, where deviations of experimental values from universal model predictions are associated to effective range corrections. The effective range correction is extracted, with a weighted average given by 3.9$\pm 0.8 R_{vdW}$, where $R_{vdW}$ is the van der Waals length scale; which is consistent with the van der Waals potential tail for the $Cs_2$ system. The method can be generally applied to other cold atom experimental setups to determine the contribution of the effective range to the tetramer dissociation position.

Effective Range Corrections from Effective Field Theory with Dibaryon Fields and Perturbative Pions

We discuss a role of dibaryon fields in an effective field theory to study pion cloud effect around the unitary limit of two nucleon systems at low energies.

Effective-range dependence of resonantly interacting fermions

We extract the leading effective range corrections to the equation of state of the unitary Fermi gas from ab initio fixed-node quantum Monte Carlo (FNQMC) calculations in a periodic box using a density functional theory (DFT), and show them to be universal by considering several two-body interactions. Furthermore, we find that the DFT is consistent with the best available unbiased QMC calculations, analytic results, and experimental measurements of the equation of state. We also discuss the asymptotic effective-range corrections for trapped systems and present the first QMC results with the correct asymptotic scaling.

Effective range corrections from effective field theory with dibaryon fields and perturbative pions

Contributions of perturbative pions around a non-trivial fixed point are studied by utilizing di-baryon fields. We calculate ${}^1S_0$ and ${}^3S_1$ phase shifts for the $np$ scattering at low energies up to one-pion-exchange contributions. We also calculate effective range parameters, $v_2$, $v_3$ and $v_4$ in the higher order of the effective range expansion, and obtain corrections of the effective range to the expressions previously reported by Cohen and Hansen. After the scattering length and the effective range are renormalized, we study the role of renormailzation scale parameter $\mu$ and discuss the range of validity of the theory.

Radio-frequency spectra of Feshbach molecules in quasi-two-dimensional geometries

The line shape of radio frequency spectra of tightly bound Feshbach molecules in strong transverse confinement can be described by a simple analytic formula that includes final state interactions. By direct comparison to experimental data, we clarify the role of effective range corrections to two-body bound-state energies in lower dimensions.

Universal two-body spectra of ultracold harmonically trapped atoms in two and three dimensions

We consider the spectrum of two ultracold harmonically trapped atoms interacting via short-range interactions. The Green function approach is used to unify the two- and three-dimensional cases. We derive criteria for the universality of the spectrum, i.e. its independence of the details of the short-range interaction. The results in three dimensions are exemplified for narrow s-wave Feshbach resonances and we show how effective range corrections can modify the rearrangement of the level structure. However, this requires extremely narrow resonances or very tight traps that are not currently experimentally available. In the two-dimensional case, we discuss the p-wave channel in detail and demonstrate how the non-universality of the spectrum arises within the Green function approach. We then show that the spectrum is not particularly sensitive to the short-distance details in the case when the two-body interaction has a bound state.

On parity-violating three-nucleon interactions and the predictive power of few-nucleon EFT at very low energies

We address the typical strengths of hadronic parity-violating three-nucleon interactions in “pion-less” Effective Field Theory (EFT) in the nucleon-deuteron (iso-doublet) system. By analysing the superficial degree of divergence of loop diagrams, we conclude that no such interactions are needed at leading order, $\ensuremath {O}(\epsilon Q^{-1})$ . The only two distinct parity-violating three-nucleon structures with one derivative mix $\ensuremath ^2S_{\frac{1}{2}}$ and $\ensuremath ^2P_{\frac{1}{2}}$ waves with iso-spin transitions $ \Delta$ I = 0 or 1. Due to their structure, they cannot absorb any divergence ostensibly appearing at next-to-leading order, $\ensuremath {O}(\epsilon Q^0)$ . This observation is based on the approximate realisation of Wigner’s combined SU(4) spin-isospin symmetry in the two-nucleon system, even when effective-range corrections are included. Parity-violating three-nucleon interactions thus only appear beyond next-to-leading order. This guarantees renormalisability of the theory to that order without introducing new, unknown coupling constants and allows the direct extraction of parity-violating two-nucleon interactions from three-nucleon experiments.

Feshbach resonances and the unitary limit in a model of strongly correlated nucleons

A model of strongly interacting and correlated hadrons is developed. The interaction used contains a long range attraction and short range repulsive hard core. Using this interaction and various limiting situations of it, a study of the effect of bound states and Feshbach resonances is given. The limiting situations are a pure square well interaction, a delta-shell potential and a pure hard core potential. The limit of a pure hard core potential are compared with results for a spinless Bose and Fermi gas. The limit of many partial waves for a pure hard core interaction is also considered and result in expressions involving the hard core volume. This feature arises from a scaling relation similar to that for hard sphere scattering with diffractive corrections. The role of underlying isospin symmetries associated with the strong interaction of protons and neutrons in this two component model is investigated. Properties are studied with varying proton fraction. An analytic expression for the Beth Uhlenbeck continuum integral is developed which closely approximates exact results based on the potential model considered. An analysis of features associated with a unitary limit is given. In the unitary limit of very large scattering length, the ratio of effective range to thermal wavelength appears as a limiting scale. Thermodynamic quantities such as the entropy and compressibility are also developed. The effective range corrections to the entropy vary as the cube of this ratio for low temperatures and are therefore considerably reduced compared to the corrections to the interaction energy which varies linearly with this ratio. Effective range corrections to the compressibility are also linear in the ratio.

Renormalization-group study of the four-body problem

We perform a renormalization-group analysis of the nonrelativistic four-boson problem by means of a simple model with pointlike three- and four-body interactions. We investigate in particular the region where the scattering length is infinite and all energies are close to the atom threshold. We find that the four-body problem behaves truly universally, independent of any four-body parameter. Our findings confirm the recent conjectures of others that the four-body problem is universal, now also from a renormalization-group perspective. We calculate the corresponding relations between the four- and three-body bound states, as well as the full bound-state spectrum and comment on the influence of effective range corrections.

Range corrections for two-neutron halo nuclei in effective theory

The range corrections to the universal properties and structure of two-neutron halo nuclei are investigated within an effective quantum mechanics framework. Treating the nucleus as an effective three-body system, we make a systematic improvement upon previous calculations by calculating the linear range corrections at next-to-leading order. Since the effective ranges for the neutron–core interactions are not known, we estimate the effective range to be set by the inverse of the pion mass. We investigate the possibility of excited Efimov states in two-neutron halo nuclei and calculate their mean square radii to next-to-leading order. We find that the effective range corrections are generally small and the leading order predictions are very robust.

Thomas-Fermi approximation for a condensate with higher-order interactions

We consider the ground state of a harmonically trapped Bose-Einstein condensate within the Gross-Pitaevskii theory including the effective-range corrections for a two-body zero-range potential. The resulting nonlinear Schroedinger equation is solved analytically in the Thomas-Fermi approximation neglecting the kinetic-energy term. We present results for the chemical potential and the condensate profiles, discuss boundary conditions, and compare to the usual Thomas-Fermi approach. We discuss several ways to increase the influence of effective-range corrections in experiment with magnetically tunable interactions. The level of tuning required could be inside experimental reach in the near future.

Universality in few-body systems

Low-energy universality in atomic few-body systems as a result of a large two-body scattering length has gained a lot of attention recently. Here, I discuss recent progress in describing the three-body recombination of cold atoms in terms of a finite set of universal scaling functions and review results for the recombination length of 133Cs atoms obtained with these functions. Furthermore, I will consider the inclusion of effective range corrections and the relevance for further calculations in atomic and nuclear physics.

Universal properties of Efimov physics beyond the scattering length approximation

We calculate the low-lying energy spectrum for three trapped identical bosons interacting via a finite-range potential. The spectrum is compared to results from the zero-range model. The thresholds for trimer binding and atom-dimer binding are extracted and related to effective range corrections. Effective range corrections to Efimov physics and Borromean binding are two aspects of the same effect, and we connect these two regions qualitatively. The Borromean window becomes slightly narrower for substantial effective ranges. The structure at the atom-dimer threshold is an atom far away from the dimer and the major energy correction is due to the change of dimer energy with effective range. This structure becomes less pronounced when the effective range increases.