Effective Range Expansion Research Articles

Renormalization ofNNscattering: Contact potential

The renormalization of the T matrix for NN scattering with a contact potential is re-examined in a nonperturbative regime through rigorous nonperturbative solutions. Based on the underlying theory, it is shown that the ultraviolet divergences in the nonperturbative solutions of the T matrix should be subtracted through 'endogenous' counterterms, which in turn leads to a nontrivial prescription dependence. Moreover, employing the effective range expansion, the importance of imposing physical boundary conditions to remove the nontrivial prescription dependence, especially before making any physical claims, is discussed and highlighted. As by-products, some relations between the effective range expansion parameters are derived. We also discuss the power counting of the couplings for the nucleon-nucleon interactions and other subtle points related to the EFT framework beyond perturbative treatment.

Description of scattering and of a bound state in the two-nucleon system on the basis of the Bargmann representation of the S matrix

For the effective-range function $k\cot \delta $, a pole approximation that involves a small number of parameters is derived on the basis of the Bargmann representation of the $S$ matrix. The parameters of this representation, which have a clear physical meaning, are related to the parameters of the Bargmann $S$ matrix by simple equations. By using a polynomial least-squares fit to the function $k\cot \delta $ at low energies, the triplet low-energy parameters of neutron-proton scattering are obtained for the latest experimental data of Arndt et al. on phase shifts. The results are $a_{t}=5.4030 $fm, $r_{t}=1.7494 $fm, and $v_{2}=0.163 $fm$^{3}$. With allowance for the values found for the low-energy scattering parameters and for the pole parameter, the pole approximation of the function $k\cot \delta $ provides an excellent description of the triplet phase shift for neutron-proton scattering over a wide energy range ($T_{\text{lab}}\lesssim 1000 $MeV), substantially improving the description at low energies as well. For the experimental phase shifts of Arndt et al., the triplet shape parameters $v_{n}$ of the effective-range expansion are obtained by using the pole approximation. The description of the phase shift by means of the effective-range expansion featuring values found for the low-energy scattering parameters proves to be fairly accurate over a broad energy region extending to energy values approximately equal to the energy at which this phase shift changes sign, this being indicative of a high accuracy and a considerable value of the effective-range expansion in describing experimental data on nucleon-nucleon scattering. The properties of the deuteron that were calculated by using various approximations of the effective-range function comply well with their experimental values.

Renormalization ofNNscattering with one pion exchange and energy dependent boundary conditions

A nonperturbative renormalization scheme for nucleon-nucleon interaction based on boundary conditions at short distances is presented and applied to the one pion exchange potential. It is free of off-shell ambiguities and ultraviolet divergences, provides finite results at any step of the calculation, and allows us to remove the short distance cutoff in a suitable way. Low energy constants and their nonperturbative evolution can be directly obtained from experimental threshold parameters in a completely unique and model independent way when the long range explicit pion effects are eliminated. This allows us to compute scattering phase shifts which are, by construction, consistent with the effective range expansion to a given order in the center of mass (c.m.) momentum $p$. In the singlet $^{1}S_{0}$ and triplet $^{3}S_{1}\text{\ensuremath{-}}^{3}D_{1}$ channels ultraviolet fixed points and limit cycles are obtained, respectively, for the threshold parameters. Data are described satisfactorily up to c.m. momenta of about $p\ensuremath{\sim}{m}_{\ensuremath{\pi}}$.

Multiplet structure of Feshbach resonances in nonzero partial waves

We report a unique feature of magnetic field Feshbach resonances in which atoms collide with non-zero orbital angular momentum. P-wave ($l=1$) Feshbach resonances are split into two components depending on the magnitude of the resonant state's projection of orbital angular momentum onto the field axis. This splitting is due to the magnetic dipole-dipole interaction between the atoms and it offers a means to tune anisotropic interactions of an ultra-cold gas of atoms. A parameterization of the resonance in terms of an effective range expansion is given.

Renormalization of singlet NN-scattering with one pion exchange and boundary conditions

We present a simple and physically compelling boundary condition regularization scheme in the framework of effective field theory as applied to nucleon–nucleon interaction. It is free of off-shell ambiguities and ultraviolet divergences and provides finite results at any step of the calculation. Low-energy constants and their non-perturbative evolution can directly be obtained from experimental threshold parameters in a completely unique, one-valued and model independent way when the long range explicit pion effects are removed. This allows to compute scattering phase shifts which are, by construction consistent with effective range expansion to a given order in the CM momentum and are free from finite cut-off artifacts. We illustrate how the method works in the 1S0 channel for the one pion exchange potential.

Low-energy operators in effective theories

Modern effective-theory techniques are applied to the nuclear many-body problem. A novel approach is proposed for the renormalization of operators in a manner consistent with the construction of the effective potential. To test this approach, a one-dimensional, yet realistic, nucleon-nucleon potential is introduced. An effective potential is then constructed by tuning its parameters to reproduce the exact effective-range expansion and a variety of bare operators are renormalized in a fashion compatible with this construction. Predictions for the expectation values of these effective operators in the ground state reproduce the results of the exact theory with remarkable accuracy (at the 0.5% level). This represents a marked improvement over a widely practiced approach that uses effective interactions but retains bare operators. Further, it is shown that this improvement is more impressive as the operator becomes more sensitive to the short-range structure of the potential. We illustrate the main ideas of this work using the elastic form factor of the deuteron as an example.

ηdEffective-range formula

We obtain the effective-range expansion of the $\ensuremath{\eta}d$ system for seven different theoretical models of the two-body interactions. The expansion reproduces very well the theoretical scattering amplitudes for center-of-mass energies up to 100 MeV. Using this expansion we calculate the position of the $\ensuremath{\eta}\mathrm{NN}$ quasivirtual state in the complex energy plane as it is predicted by each theoretical model.

Renormalization group approach to two-body scattering in the presence of long-range forces

We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible fixed points of the short-range potential as this cut-off is lowered to zero. The expansions around these fixed points define the power countings for the corresponding effective field theories. Expansions around nontrivial fixed points are shown to correspond to distorted-wave versions of the effective-range expansion. These methods are applied to scattering in the presence of Coulomb, Yukawa and repulsive inverse-square potentials.

Phase shift effective range expansion from supersymmetric quantum mechanics

Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the potential at infinity which allows us to introduce adequate long-distance transformations. The approach is shown to be effective in getting the correct phase shift effective range expansion. Applications are considered for the $^1P_1$ and $^1D_2$ partial waves of the neutron-proton scattering.

Beyond the Fermi pseudopotential: A modified Gross-Pitaevskii equation

We present an effective potential and the corresponding modified Gross-Pitaevskii equation that account for the energy dependence of the two-body scattering amplitude through an effective-range expansion. For the ground state energy of a trapped condensate, the theory leads to what we call a shape-dependent confinement correction that improves agreements with diffusion Monte Carlo calculations. The theory illustrates, for relatively strong confinement and/or high density, how the shape dependence on atom-atom interaction can come into play in a many-atom quantum system.

Threshold behavior of the Jost function for the atomic polarization potential

We evaluate the / -wave Jost function for the polarization potential and calculate its effective range expansion for / = up to the order k 4 . We investigate not only the phase but also the modulus of the Jost function, generalizing previous works for the expansion up to the order k 2 of the s-wave phase shift. We discuss the validity domain of this generalized expansion for the interaction of an electron with a Helium atom in any of its three long-lived He states, i.e., 1 1 S, 2 1 S, and 2 3 S. Even though this expansion is shown to be appropriate for most of the applications in electron-atom collisions, we propose here a more precise and simpler approach that provides much better results in a broader energy range when comparing with elaborated ab initio calculations.

Electron scattering by nonspherically symmetric atoms: Zero-energy limit

A modified perturbation theory, previously introduced for the construction of asymptotic states accounting for a general class of long-range angle-dependent multipole potentials is developed in further detail. The threshold behavior of these states is presented in explicit form and these results are used, with the aid of a modified effective-range formulation, to determine the threshold energy dependence of the multichannel transition matrix. Unitarity properties are established formally and verified in the low-energy limit. A variational procedure for determining the parameters appearing in the modified effective-range expansion is described. The same procedure accounts for the presence of zero-energy bound states leading to resonant behavior that alters the threshold energy dependence.

Lagrange mesh calculation of the effective range expansion

The first terms of the effective range expansion are accurately determined with a small number of evaluations of the potential. The method is based on the R matrix theory with a Lagrange basis which leads to a simple meshlike approximation. It is valid for both neutral and charged collisions for arbitrary partial waves. The accuracy of the algorithm is illustrated with an analytically solvable example, with different cases of potential scattering and in particular with the Paris and Bonn nucleon-nucleon potentials.

Effective range function belowthreshold

We demonstrate thatthe kernel of the Lippmann-Schwinger equation, associated withinteractions consisting of a sum of the Coulomb plus ashort-rangenuclear potential, becomes degeneratebelow threshold. Taking advantageof this fact, we present a simple method of calculating the effectiverange function for negative energies. This may be useful inpractice since the effective range expansion extrapolated to thresholdallows us to extract the low-energy scattering parameters: the Coulombmodified scattering length and the effective range.

Quantum scattering in one dimension

A self-contained discussion of non-relativistic quantum scattering ispresented in the case of central potentials in one space dimension, which will facilitate the understanding of the more complex scattering theory in two and three dimensions. The present discussion illustrates in asimple way the concepts of partial-wave decomposition, phase shift, opticaltheorem and effective-range expansion.

Ultracold atom-ion collisions

Studies of charge transfer and total cross sections in elastic collisions of atom-ion alkali metals at ultralow temperatures are reported. Calculations for Na+Na{sup +} have been carried out with the best available {sup 2}{sigma}{sub g}{sup +} and {sup 2}{sigma}{sub u}{sup +} potential curves. As functions of energy, the cross sections show considerable structure and are large in the limit of low temperature. The scattering lengths were also computed, and the effective range expansion verified. For higher temperatures, we compare quantal and semiclassical results, and investigate the range of applicability of the Langevin formula. Even at temperatures of a few degrees kelvin, the charge-transfer cross sections are large, and could provide an efficient way to produce cold ions. (c) 2000 The American Physical Society.

Recent progress on the chiral unitary approach to meson meson and meson baryon interactions

We report on recent progress on the chiral unitary approach, analogous to the effective range expansion in Quantum Mechanics, which is shown to have a much larger convergence radius than ordinary chiral perturbation theory, allowing one to reproduce data for meson meson interaction up to 1.2 GeV. Applications to physical processes so far unsuited for a standard chiral perturbative approach are presented. Results for the extension of these ideas to the meson baryon sector are discussed, together with applications to kaons in a nuclear medium and K − atoms.

Zero-energy determination of the astrophysicalSfactor and effective-range expansions

The first three terms of a Taylor expansion of the astrophysical S factor are determined in the potential model of radiative-capture reactions. As input, the radial Schr\"odinger equation and its inhomogeneous energy derivatives are solved at zero energy. The radial wave function and its energy derivatives are obtained by matching the solutions of these equations with the corresponding exact asymptotic forms. Explicit expressions are derived for the S factor and for its first and second derivatives at zero energy. The same algorithm allows one to accurately determine the first terms of the effective-range expansion. In particular, the effective-range formula converges much faster than the Schwinger-Bethe formula. The method is illustrated with potential-model descriptions of the ${}^{3}\mathrm{He}(\ensuremath{\alpha},\ensuremath{\gamma}{)}^{7}\mathrm{Be},$ ${}^{6}\mathrm{Li}(p,\ensuremath{\gamma}{)}^{7}\mathrm{Be},$ ${}^{7}\mathrm{Be}(p,\ensuremath{\gamma}{)}^{8}\mathrm{B},$ and ${}^{16}\mathrm{O}(p,\ensuremath{\gamma}{)}^{17}\mathrm{F}$ reactions.

A renormalisation-group treatment of two-body scattering

Nonrelativistic two-body scattering by a short-ranged potential is studied using the renormalisation group. Two fixed points are identified: a trivial one and one describing systems with a bound state at zero energy. The eigenvalues of the linearised renormalisation group are used to assign a systematic power-counting to terms in the potential near each of these fixed points. The expansion around the nontrivial fixed point is shown to be equivalent to the effective-range expansion.

Testing low energy theorems in nucleon-nucleon scattering

Low energy theorems have been derived for the coefficients of the effective range expansion in s-wave nucleon-nucleon scattering valid to leading nontrivial order in an expansion based $Q$ counting, a scheme in which both $m_\pi$ and $1/a$ (where $a$ is the scattering length) are treated as small mass scales. Previous tests of these theorems based on coefficients extracted from scattering data indicate a pattern of gross violations which suggested serious problems for the perturbative treatment of pions implicit in $Q$ counting. We discuss the possibility that uncertainties associated with extracting the coefficients from the scattering data make such tests invalid. Here we show that errors in the s-wave phase shift extractions are sufficiently small to test direct test predictions from $Q$ counting at next to leading order. In particular we show that there exist low energy theorems for the sum of all terms in the effective range expansion beyond the first two which allow for precise tests. These low energy theorems fail badly which suggests that pionic aspects of $Q$ counting are not under control.