Effective Range Formula Research Articles

Elastic scattering of positive muons from He3 and He4

The study of positive muon ${\ensuremath{\mu}}^{+}$-He scattering plays an important role in precision experiments involving positive muons. In this paper, we employed the confined variational method to investigate $S$-wave ${\ensuremath{\mu}}^{+}$-He scattering with scattering momenta below $0.1{a}_{0}^{\ensuremath{-}1}$, where ${a}_{0}$ denotes the Bohr radius. Our approach yielded accurate $S$-wave phase shifts and scattering lengths. By utilizing the modified effective range formula, we determined the $S$-wave scattering lengths to be $\ensuremath{-}12.3{a}_{0}$ and $\ensuremath{-}10.6{a}_{0}$ for ${\ensuremath{\mu}}^{+}\text{\ensuremath{-}}^{4}\mathrm{He}$ and ${\ensuremath{\mu}}^{+}\text{\ensuremath{-}}^{3}\mathrm{He}$ scattering, respectively. Furthermore, we examined the distortion effects on helium induced by ${\ensuremath{\mu}}^{+}$.

Non-Born-Oppenheimer confined variational calculation of low-energy Ps- H2 scattering

Using the confined variational method, the low-energy $S$-wave elastic scattering of positronium from molecular hydrogen is studied nonadiabatically at incident energy up to 0.13 eV. Accurate $S$-wave phase shifts, scattering lengths, and cross sections are calculated and compared with other theoretical and experimental values. Using the modified effective-range formula, the $S$-wave scattering length is determined to be 1.97 Bohr radii, which is about $2.5%$ smaller than that of the confined variational calculation under the fixed-nucleus approximation. The consistency of the present cross sections with those from the angular correlation of annihilation radiation experiment indicates that the results of Doppler broadening spectroscopy might be problematic. We also discuss the distortion effects of positronium and molecular hydrogen.

Studying theρresonance parameters with staggered fermions

We deliver a lattice study of $\rho$ resonance parameters with p-wave $\pi\pi$ scattering phases, which are extracted by finite-size methods at one center-of-mass frame and four moving frames for six MILC lattice ensembles with pion masses ranging from $346$ to $ 176$ MeV. The effective range formula is applied to describe the scattering phases as a function of the energy covering the resonance region, this allows us to extract $\rho$ resonance parameters and to investigate the quark-mass dependence. Lattice studies with three flavors of the Asqtad-improved staggered fermions enable us to use the moving-wall source technique on large lattice spatial dimensions ($L=64$) and small light $u/d$ quarks. Numerical computations are carried out at two lattice spacings, $a \approx 0.12$ and $0.09$ fm.

Hidden charm molecules in finite volume

In the present paper we address the interaction of pairs of charmed mesons with hidden charm in a finite box. We use the interaction from a recent model based on heavy quark spin symmetry that predicts molecules of hidden charm in the infinite volume. The energy levels in the box are generated within this model, and from them some synthetic data are generated. These data are then employed to study the inverse problem of getting the energies of the bound states and phase shifts for $D \bar D$ or $D^* {\bar D}^*$. Different strategies are investigated using the lowest two levels for different values of the box size, carrying a study of the errors produced. Starting from the upper level, fits to the synthetic data are carried out to determine the scattering length and effective range plus the binding energy of the ground state. A similar strategy using the effective range formula is considered with a simultaneous fit to the two levels, one above and the other one below threshold. This method turns out to be more efficient than the other one. Finally, a method based on the fit to the data by means of a potential and a loop function conveniently regularized, turns out to be very efficient and allows to produce accurate results in the infinite volume starting from levels of the box with errors far larger than the uncertainties obtained in the final results. A regularization method based on Gaussian wave functions turns out to be rather efficient in the analysis and as a byproduct a practical and fast method to calculate the L\"uscher function with high precision is presented.

Lattice QCD study of theK*(892)meson decay width

We deliver an exploratory lattice QCD examination of the $K^\ast(892)$ meson decay width with the help of the p-wave scattering phase $\delta_1$ of pion-kaon ($\pi K$) system in the isospin $I=1/2$ channel, which are extracted by the modified Rummukainen-Gottlieb formula for two-particle system with arbitrary mass, and it clearly reveals the entity of a resonance at a mass around $K^\ast(892)$ meson mass. The effective range formula is applied to describe the energy dependence of the scattering phase and we obtain the effective $K^\ast \to \pi K$ coupling constant as $g_{K^\ast \pi K} = 6.38(78)$, and subsequently achieve the decay width to be $64.9 \pm 8.0$ MeV, which is in reasonable accordance with the current experiment. Our lattice investigations are conducted on a $20^3\times48$ MILC full QCD gauge configuration at $(m_\pi + m_K) / m_{K^\ast} \approx 0.739$ and the lattice spacing $a \approx 0.15$ fm.

Pseudo-Static Approach to Single Rammer Energy Calculation in Dynamic Compaction of Saturated Soft Soil Foundations

The article puts forward a single-rammer energy calculation formula based on pseudo-static method which is to be applied in the dynamic compaction trials of saturated soft soil foundations. A comparison between the formula and the dynamic compaction effective reinforcement range formula is made. Then, data from actual engineering project are applied to the formula, and the calculated results are consistent with the testing results. That is to say, the formula can facilitate the forecast of single-rammer energy in dynamic foundation compaction.

Preliminary lattice QCD calculation of the σ meson decay width

We report an exploratory lattice QCD investigation of the σ meson decay width though the s-wave scattering phase shift for isospin I = 0 pion-pion (ππ) system. Rummukainen-Gottlieb formula is used to estimate the scattering phase for the two-pion system in the moving frame, which demonstrate the presence of a resonance around σ meson. Using the effective range formula we extract the effective σ → ππ coupling constant as g σππ = 2.69(44) GeV, which is consistent with the theoretical predictions. The estimated decay width is about 236 ± 49 MeV. These simulations are carried out on a 163 × 48 MILC gauge configuration with the N f = 2 + 1 flavor of the “Asqtad” improved staggered dynamical sea quarks at m π /m σ ≈ 0.414 and the lattice spacing a ≈ 0.15 fm.

The preliminary lattice QCD calculation of κ meson decay width

We present a direct lattice QCD calculation of the $\kappa$ meson decay width with the s-wave scattering phase shift for the isospin $I=1/2$ pion-kaon ($\pi K$) system. We employ a special finite size formula, which is the extension of the Rummukainen-Gottlieb formula for the $\pi K$ system in the moving frame, to calculate the scattering phase, which indicates a resonance around $\kappa$ meson mass. Through the effective range formula, we extract the effective $\kappa \to \pi K$ coupling constant $g_{\kappa \pi K} = 4.54(76)$ GeV and decay width $\Gamma = 293 \pm 101$ MeV. Our simulations are done with the MILC gauge configurations with $N_f=2+1$ flavors of the "Asqtad" improved staggered dynamical sea quarks on a $16^3\times48$ lattice at $(m_\pi + m_K) / m_\kappa \approx 0.8$ and lattice spacing $a \approx 0.15$ fm.

Coupled channel analysis of theρmeson decay in lattice QCD

We employ a variational basis with a number of $\bar{q}q$ and $\pi\pi$ lattice interpolating fields with quantum numbers of the $\rho$ resonance to extract the discrete energy spectrum in a finite volume. In the elastic region, this spectrum is related to the phase shift of the continuum scattering amplitude by L\"uscher's formula and the relation allows the extraction of resonance parameters from the spectrum calculation. The simulations are performed at three different total momenta of the coupled $\bar q q-\pi\pi$ system, which allows us to extract the p-wave scattering phase at five values of pion relative momenta near the resonance region. The effective range formula describes the phase-shift dependence nicely and we extract the resonance mass $m_\rho=792(7)(8)$ MeV and the coupling $g_{\rho\pi\pi}=5.13(20)$ at our $m_\pi\simeq 266 $MeV. The coupling $g_{\rho\pi\pi}$ is directly related to the width of the $\rho$ meson and our value is close to the value derived from the experimental width. The simulations are performed using dynamical gauge configurations with two mass-degenerate flavors of tree-level improved clover-Wilson fermions. Correlation functions are calculated using the recently proposed distillation method with Laplacian Heaviside (LapH) smearing of quarks, which enables flexible calculations, in many cases with unprecedented accuracy.

Resonance parameters of theρmeson from lattice QCD

We perform a nonperturbative lattice calculation of the $P$-wave pion-pion scattering phase in the $\ensuremath{\rho}$-meson decay channel using two flavors of maximally twisted mass fermions at pion masses ranging from 480 to 290 MeV. Making use of finite-size methods, we evaluate the pion-pion scattering phase in the center-of-mass frame and two moving frames. Applying an effective range formula, we find a good description of our results for the scattering phase as a function of the energy covering the resonance region. This allows us to extract the $\ensuremath{\rho}$-meson mass and decay width and to study their quark mass dependence.

Some remarks on effective range formulae in potential scattering

In this paper, we consider the radial Schrödinger equation with a real-valued potential having spherical symmetry, and we present alternative proofs of very recent results on the necessary, as well as sufficient, conditions on the decrease of the potential at infinity for the validity of effective range formulae in 3D in low energy potential scattering (Khuri et al 2008 arXiv:0812.4054v1. See theorem 1 below. This paper also contains a careful study of the 2D case, with some amazing new results). Our proofs are based on compact formulae for the phase shifts. The sufficiency conditions have been well known for a long time. But the necessity of the same conditions for potentials keeping a constant sign at large distances are new. All these conditions are established here for dimension 3 and for all angular momenta ℓ ⩾ 0.

Effective range theory

The effective range formula for scattering by a short range potential is derived for arbitrary angular momentum of the scattered particle. The derivation is a generalization of a method due to Bethe, and makes no reference to the analytical properties of Jost functions. The derivation is a useful supplement to the textbook coverage of the subject.

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.

Higher threshold parameters in ππ scattering

A family of threshold parameters which probe the stability of chiral predictions is considered. The relevant criteria for the choice of threshold parameters are discussed. Sum rules for these quantities are derived from dispersion relations and evaluated from effective range formulae. Good agreement with two-loop chiral estimates for many of these quantities is found and interesting discrepancies are discussed.

Paris NN-bar potential and recent proton-antiproton low energy data.

The inner part of the Paris nucleon-antinucleon optical potential has been reanalyzed in view of new experimental constraints. These are mainly provided by accurate measurements of the analyzing power in charge-exchange scattering, p\ifmmode\bar\else\textasciimacron\fi{}p\ensuremath{\rightarrow}n\ifmmode\bar\else\textasciimacron\fi{}n, together with elastic p\ifmmode\bar\else\textasciimacron\fi{}p\ensuremath{\rightarrow}p\ifmmode\bar\else\textasciimacron\fi{}p polarization data, allowing a better determination of the isospin dependence of the short-range forces. The fit is performed with a set of 3632 data. The good quality of the fit is illustrated over a large sample of observables for laboratory momenta from 180 to 910 MeV/c. Curves of the potentials and of the phase shifts are shown as well as the parameters of the potentials and effective range formula coefficients for the S and P waves. The spectrum of resonances and bound states predicted by our potential is also displayed. In particular, a bound state having the AX(1565) quantum numbers is found with the right mass.

Secular perturbation theory of long-range interactions.

Solutions of Schroedinger's equation for systems interacting with long-range power-law potentials, [ital V][sub int]([ital r])[proportional to][ital r][sup [minus][ital n]] with [ital n][ge]2, are cast in the form of a power series in [ital V][sub int]. These perturbations lead to secular divergences that are eliminated by renormalizing the angular-momentum quantum number. Long-known perturbation techniques in classical mechanics and quantum-field theory yield modified effective-range formulas, quantum-defect functions, and solutions of close-coupling equations for wave propagation in long-range fields. As an example, we extract in first-order perturbation theory a modified effective-range expansion for the phase shift of an electron interacting with an atomic 1/[ital r][sup 4] polarization potential. Near thresholds, the method is applicable to all power-law potentials with [ital n][ge]2, and to their combinations, as well as to multichannel problems involving anisotropic potentials.

Nucleon exchange effects at low bombarding energies for deuteron-neutron scattering

We show that for the quartet nd scattering at low bombarding energies the nucleon exchange mechanism plays an important role and we propose a new type of effective-range formula which incorporates this structural effect. Our approach can be generalized for many channel processes too.

Shape dependence in low-energy NN scattering

The connection between the Noyes-CFS modified effective-range formula for triplet S-wave n-p scattering and the N/D method is reviewed. Two- and three-pole approximations are compared with the exact numerical solution of the coupled integral equations of this method. Analytic iteration is performed to second order, yielding new approximations which retain much of the structure of the exact solution, and provide very accurate numerical results, as well as a new shape dependent version of the effective-range expansion. It is found that the one-pion-exchange cut plus two closely spaced effective interaction poles can reproduce the shape dependence given by realistic nucleon-nucleon potentials.

Rapidly convergent effective range formula

A new effective range formula is developed using conformal mapping techniques. In contrast to earlier attempts in this direction, our formula is valid for arbitrary high energies and for all processes. As an illustration we study the s wave in np scattering.

Improved effective range formula

A new effective range formula is derived using a conformal mapping variable that incorporates the knowledge of the left-hand cut. Our formula is valid up to higher energies and has good convergence and extrapolation properties. As an illustration, we use the formula for a study of the $S$ and $P$ waves in the $n\ensuremath{-}p$ scattering.NUCLEAR STRUCTURE New effective range formula derived from analyticity using conformal mappings.