Effective Range Approximation Research Articles

General features and contact modeling of $N$-body isolated resonances near threshold

NN-body non efimovian bound or quasi-bound states for particles with short range interactions are considered in arbitrary dimensions. The different resonance regimes near the threshold are depicted by using a generalization of the effective range approximation. This two-parameter description can be used in various contexts from ultracold to hadronic physics. The universal character of these states makes it possible a formulation in terms of a contact model. The singularity at the contact imposes the introduction of a modified scalar product to solve the normalization catastrophe and to restore the self-adjoint character of the model. An equivalence with the standard scalar product used for realistic finite range models is derived.

New parametrization for the 3He(d, p) 4He fusion reaction rate and refinement of the Lawson criterion for d-3He thermonuclear reactors

We present a new parametrization of the d + 3He → p + 4He fusion reaction astrophysical factor based on the effective range approximation, which is an effective theoretical method for describing near-threshold, including resonance, nuclear reactions. In the framework of this approximation we describe experimental data on the energy dependence of the cross section and the astrophysical factor within the experimental uncertainties in the energy range of 0-800 keV. On this basis we calculate the temperature dependence of the Maxwellian-averaged reaction rate in the range of 0-400 keV. In conclusion, we discuss the effect of the calculated reaction rates on the Lawson criterion for thermonuclear reactors based on d-3He fuel.

Analytical version of the resonance coupled-channel model for D + T → 5He** → α + n reaction and its application for the description of low-energy D-T and D- 3He scattering

The purpose of the present paper is the formulation of the analytical version of the resonance coupled-channel model (RCCM) originally developed for D + T → 5He** → α + n nuclear fusion reaction. The integral in the denominator of the Breit-Wigner type is examined in the expression for S-matrix elements of binary processes in this model. Imaginary part of this integral determines the energy-dependent decay width for the near-threshold channel. It is demonstrated that this integral can be calculated explicitly with the Binet representation for the ψ-function (the logarithmic derivation of the gamma function). As the result the explicit expression for the S-matrix elements in the form of analytical functions of the channel momenta are obtained and the equivalence of the RCCM and the effective range approximation (Landau – Smorodinsky – Bethe approximation) is established on this basis. This allows expressing the parameters of the RCCM through the model independent system characteristics: the complex scattering length and the complex effective range. Several sets of model parameters of both approaches that provide a good description of the measured data on D + T → α + n reaction and D-T elastic scattering are derived. By this means we find the location of the S – matrix poles on different Riemann sheets which corresponds to Jπ = (3/2)+ state of 5He and 5Li nuclei. In particular, the location of the resonance (R) and shadow (S) poles is determined: 5He**: ZR = 46.9 – i37.2 (keV) ZS = 81.7 – i3.5 (keV) 5Li**: ZR = 205.7 – i146.8 (keV) ZS = 264.4 + i112.0 (keV). Our results agree well with previous findings. The possible generalizations of the results obtained are discussed.

Masses and decay constants of the Ds0*(2317) and Ds1(2460) from Nf=2 lattice QCD close to the physical point

We perform a high statistics study of the $J^{P}=0^{+}$ and $1^{+}$ charmed-strange mesons, $D_{s0}^*(2317)$ and $D_{s1}(2460)$, respectively. The effects of the nearby $DK$ and $D^{*}K$ thresholds are taken into account by employing the corresponding four quark operators. Six ensembles with $N_f=2$ non-perturbatively ${\cal O}(a)$ improved clover Wilson sea quarks at $a=0.07$ fm are employed, covering different spatial volumes and pion masses: linear lattice extents $L/a=24,32,40,64$, equivalent to 1.7 fm to 4.5 fm, are realised for $m_{\pi}=290$ MeV and $L/a=48,64$ or 3.4 fm and 4.5 fm for an almost physical pion mass of $150$ MeV. Through a phase shift analysis and the effective range approximation we determine the scattering lengths, couplings to the thresholds and the infinite volume masses. Differences relative to the experimental values are observed for these masses, however, this is likely to be due to discretisation effects as spin-averaged quantities and splittings are reasonably compatible with experiment. We also compute the weak decay constants of the scalar and axialvector and find $f_V^{0^+}=114(2)(0)(+5)(10)$ MeV and $f_A^{1^+}=194(3)(4)(+5)(10)$ MeV, where the errors are due to statistics, renormalisation, finite volume and lattice spacing effects.

Low-energy theorems for nucleon-nucleon scattering atMπ=450MeV

We apply the low-energy theorems to analyze the recent lattice QCD results for the two-nucleon system at a pion mass of $M_\pi\simeq 450$ MeV obtained by the NPLQCD collaboration. We find that the binding energies of the deuteron and dineutron are inconsistent with the low-energy behavior of the corresponding phase shifts within the quoted uncertainties and vice versa. Using the binding energies of the deuteron and dineutron as input, we employ the low-energy theorems to predict the phase shifts and extract the scattering length and the effective range in the $^3S_1$ and $^1S_0$ channels. Our results for these quantities are consistent with those obtained by the NPLQCD collaboration from effective field theory analyses but are in conflict with their determination based on the effective-range approximation.

Ultracold-atom collisions in atomic waveguides: A two-channel analysis

Low dimensional behavior of two ultra-cold atoms trapped in two-and one-dimensional waveguides is investigated in the vicinity of a magnetic Feshbach resonance. A quantitative two-channel model for the Feshbach mechanism is used allowing an exhaustive analysis of low-dimensional resonant scattering behavior and of the confinement induced bound states. The role of the different parameters of the resonance is depicted in this context. Results are compared with the ones of the zero-range approach. The relevance of the effective range approximation in low dimensions is studied. Examples of known resonances are used to illustrate the bound state properties.

Low-energy parameters of neutron-neutron interaction in the effective-range approximation

The effect of the mass difference between the charged and neutral pions on the low-energy parameters of nucleon-nucleon interaction in the 1 S 0 state is studied in the effective-range approximation. On the basis of experimental values of the singlet parameters of neutron-proton scattering and the experimental value of the virtual-state energy for the neutron-neutron systemin the 1 S 0 state, the following values were obtained for the neutron-neutron scattering length and effective range: a nn = −16.59(117) fm and r nn = 2.83(11) fm. The calculated values agree well with present-day experimental results.

Experimental Analysis and Modeling of the Crack Arrest Condition Under Severe Plastic Fretting Fatigue Conditions

The fretting-fatigue crack arrest condition is formalized as the condition for which a crack nucleates but stops propagating and so the sample never fails. Usually this domain is small as the fatigue loading greatly promotes the crack propagation. The aim of this research work is to investigate the crack arrest condition for severe plastic condition (maximal pressure is 2.5 times greater than the yield stress) and high fatigue stress ratio (R=0.94) thus for not much propagating conditions.A low alloyed industrial steel flat specimen (IS) fretted against a 35NCD16 steel cylinder pad was studied in order to identify the crack arrest condition for various fretting loadings (shear amplitude). An original experimental monitoring has been implemented on a fretting-fatigue test device to observe on-line the crack arrest using the potential drop technic. To rationalize these results, a fretting-fatigue map concept which consists to plot the crack arrest boundary as a function of the applied maximum fatigue stress and the fretting tangential force amplitude is applied [1].A 2D plain strain FEM modelling has been implemented to simulate the complex contact stressing including the contact plasticity. This plastic deformation promotes a significant extension of the contact and a proportional reduction of contact pressure and cyclic shear stresses. One consequence is that for a given contact and fatigue loading, this plastic accommodation of the interface tends to reduce the effective stress imposed to the material.The crack arrest boundary was formalized applying a pure mode I effective intensity stress range approximation [2]. It is shown that combining elastic-plastic FEM modelling, weight function and an adequate estimation of the effective intensity stress range, it is possible to predict with accuracy the fretting fatigue crack arrest boundary.

The characteristics of the 8Be ground state in the effective-range approximation

We find the vertex constant for the synthesis α + α → 8Be and the corresponding asymptotic normalization coefficient of the Gamov wave function for 8Be in the ground state. We use modern data on the position and width of the narrow resonance of this state as well as the energy dependence of the αα scattering phase shift in the s-wave known from the literature. The effective-range theory was applied with the Coulomb interaction taken into account. The parameters of the standard effective-range function expansion up to the member with k4 (k being the relative momentum) were found.

Decay of a negative molecular ion in a constant electric field

The problem of the shift and broadening of the electron energy levels in the field of two 3D short-range potentials (e.g., the model of a negative molecular ion) by a constant electric field F is considered. The interaction of an electron with attraction centers is taken into account in the effective range approximation. We analyze the cases when both centers maintain weakly bound s states and when the electron state in the field of one of the centers is a p state. Exact numerical results for the shift and the width of the energy levels of a quasi-molecule as functions of field F, distance R between atomic centers, and the orientation of the quasi-molecule axis relative to vector F are presented, as well as the results of analytic treatment for a number of limiting cases. The exact values of complex energies of the quasi-molecule are compared with analytic results for a weak field in the case of identical s centers [26], as well as nonequivalent s centers and s-p centers; the applicability boundaries of the weak field approximation are established. It is shown that for large values of R, the position and width of the levels in a strong field are correctly described in perturbation theory in the exchange interaction. We analyze the field-induced quasi-intersection of molecular energy levels of the system with nonequivalent atomic centers and peculiarities in the energy level widths associated with this intersection. The results make it possible to qualitatively interpret the results of numerical calculations of the probability of homo- and heteronuclear molecules being ionized by a low-frequency laser field.

THE VALIDATION AND CORRECTION OF THE APPROXIMATION EQUATION USED FOR THE DIRECT CALCULATION OF THE EFFECTIVE FREQUENCIES OF EMP WAVEFORMS

An effective frequency range approximation method was developed in this study. The method enables one to find the -3 dB effective cut off frequency ranges of EMP waveforms directly from the rise time and the FWHM. The conventional -3 dB as constant K was thought to be 0.35. However, this was found to be incorrect. In this paper, a correct formulation for K is presented. In addition, various effective frequency range approximations were acquired. The determined range was from -1 ~ 8 dB; a general equation for the total approximation of constant K was developed using a curve fitting tool. The K for the -X dB points was found to be in the range of 0.03 ~ 0.1075.

The unbound isotopes 9,10He

The unbound nuclei 9He and 10He have been produced in proton-knockout reactions from a 280 MeV/u 11Li beam impinging on a liquid hydrogen target at the ALADIN-LAND setup at GSI. Information on their nuclear structure has been obtained from the relative-energy spectra, He 8 + n and He 8 + 2 n , employing reaction models incorporating the structure of 11Li. The He 8 + n relative-energy spectrum is dominated by a strong peak-like structure at low energy, which may be interpreted within the effective-range approximation as the result of an s-wave interaction with a neutron scattering length of − 3.17 ( 66 ) fm . This spectrum also shows evidence for resonance states at 1.33 ( 8 ) MeV and at 2.4 MeV above the He 8 + n threshold. It is argued that the s-state might not be the 9He ground state. For He 8 + n + n , the analysis of the relative-energy spectrum gives two alternative possibilities for an interpretation: either a narrow resonance at 1.42 ( 10 ) MeV ( Γ = 1.11 ( 76 ) MeV ) superimposed on a correlated background, or two overlapping resonances, a ground state at 1.54(11) MeV ( Γ = 1.91 ( 41 ) MeV ) together with an excited state at 3.99(26) MeV ( Γ = 1.64 ( 89 ) MeV ). The two possible interpretations are discussed.

Extraction of scattering lengths from final-state interactions

A recently proposed method based on dispersion theory, that allows to extract the scattering length of a hadronic two-body system from corresponding final-state interactions, is generalized to the situation where the Coulomb interaction is present. The steps required in a concrete practical application are discussed in detail. In addition a thorough examination of the accuracy of the proposed method is presented and a comparison is made with results achieved with other methods like the Jost-function approach based on the effective-range approximation. Deficiencies of the latter method are pointed out. The reliability of the dispersion theory method for extracting also the effective range is investigated.

Analysis of the $\Lambda p$ final-state interaction in the reaction $p + p\to K^ + (\Lambda p)$

The missing mass spectrum measured in high resolution studies of the reaction pp -> K^+X is analyzed with respect to the strong final state interaction near the Lambda p production threshold. The observed spectrum can be described by factorizing the reaction amplitude in terms of a production amplitude and a final state scattering amplitude. Parametrizing the Lambda p final state interaction in terms of the inverse Jost function allows a direct extraction of the low-energy phase-equivalent potential parameters. Constraints on the singlet and triplet scattering lengths and effective ranges are deduced in a simultaneous fit of the Lambda p invariant mass spectrum and the total cross section data of the free Lambda p scattering using the effective range approximation.

Calculation of the binding energy and of the parameters of low-energy scattering in the Λnp system

The cluster-reduction method is used to solve the differential Faddeev equations for S=1/2, T=0 and S=3/2, T=0 spin-isospin states of the Λnp system in the s-wave approximation. The NN interaction is simulated on the basis of the MT I–III potential model, and the ΛN potential is set to VΛN=V NN /2. This simple option makes it possible to reproduce faithfully the binding energy of the hypertriton Λ 3 H. The doublet and quadruplet Λd scattering lengths and the low-energy phase shifts are calculated. It is shown that the effective-range approximation is applicable to the cases of doublet and quadruplet scattering.

A non-perturbative chiral approach for meson-meson interactions

A non-perturbative method which combines constraints from chiral symmetry breaking and coupled channel unitarity is used to describe meson-meson interactions up to \sqrt{s}\lesssim 1.2 GeV, extending in this way the range of applicability of the information contained in Chiral Perturbation Theory (\chi PT), since this perturbative series is typically restricted to \sqrt{s}\lesssim 500 MeV. The approach uses the O(p^2) and O(p^4) \chiPT Lagrangians. The seven free parameters resulting from the O(p^4) Lagrangian are fitted to the experimental data. The approach makes use of the expansion of T^{-1} instead of the amplitude itself as done in \chiPT. The former expansion is suggested by analogy with the effective range approximation in Quantum Mechanics and it appears to be very useful. The results, in fact, are in good agreement with a vast amount of experimental analyses.

Pp→ppωreaction near threshold

We analyze the total cross section data for $p\stackrel{\ensuremath{\rightarrow}}{p}\mathrm{pp}\ensuremath{\omega}$ near threshold measured recently at SATURNE. Using an effective range approximation for the on-shell $\mathrm{pp}$ S-wave final-state interaction, we extract from these data the modulus $|\ensuremath{\Omega}|=0.53$ ${\mathrm{fm}}^{4}$ of the threshold transition amplitude \ensuremath{\Omega}. We present a calculation of various (tree-level) meson exchange diagrams contributing to \ensuremath{\Omega}. It is essential that $\ensuremath{\omega}$ emission from the anomalous $\ensuremath{\omega}\ensuremath{\rho}\ensuremath{\pi}$ vertex interferes destructively with \ensuremath{\omega} emission from the proton lines. The contribution of scalar $\ensuremath{\sigma}$-meson exchange to $\ensuremath{\Omega}$ turns out to be negligibly small. Without introducing off-shell meson-nucleon form factors the experimental value $|\ensuremath{\Omega}|=0.53$ ${\mathrm{fm}}^{4}$ can be reproduced with an $\ensuremath{\omega}N$-coupling constant of ${g}_{\ensuremath{\omega}N}=10.7.$ The results of the present approach agree qualitatively with the J\"ulich model. We also perform a combined analysis of the reactions $p\stackrel{\ensuremath{\rightarrow}}{p}\mathrm{pp}{\ensuremath{\pi}}^{0},$ $\mathrm{pn}{\ensuremath{\pi}}^{+},$ $\mathrm{pp}\ensuremath{\eta},$ $\mathrm{pp}\ensuremath{\omega},$ and $p\stackrel{\ensuremath{\rightarrow}}{n}\mathrm{pn}\ensuremath{\eta}$ near threshold.

The reaction pp→pΛK + near threshold

We analyze the recent total cross section data for $pp \to p\Lambda K^+$ near threshold measured at COSY. Using an effective range approximation for the on-shell $p\Lambda$ S-wave final state interaction we extract from these data the combination ${\cal K} = \sqrt{2|K_s|^2+|K_t|^2} = 0.38$ fm$^4$ of the singlet ($K_s$) and triplet ($K_t$) threshold transition amplitudes. We present an exploratory calculation of various (tree-level) vector and pseudoscalar meson exchange diagrams. Pointlike $\omega$-exchange alone and the combined ($\rho^0,\omega,K^{*+} $)-exchange can explain the experimental value of $\cal K$. The pseudoscalar meson exchanges based on a SU(3) chiral Lagrangian turn out to be too large. However, when adding $\pi^0$-exchange in combination with the resonant $\pi N \to S_{11}(1650)\to K \Lambda$ transition and introducing monopole form factors with a cut-off $\Lambda_c= 1.5$ GeV one is again able to reproduce the experimental value of $\cal K$. More exclusive measurements are necessary to reveal the details of the $pp\to p\Lambda K^+$ production mechanism.

Asymptotic coefficients for atoms and ions

We derive convenient analytical formulas in the effective-range approximation for the asymptotic coefficient C κ of the radial wave function at infinity and for the average radius of the system. A comparison with the results of numerical calculations (done by the Hartree-Fock method) for multi-electron atoms and ions reveals that this approximation has good accuracy for valence s-electrons in all atoms from hydrogen to uranium. We calculate the values of the scattering lengths and the effective ranges for electron-atom and electron-ion scattering. We also examine the quasiclassical approximation for C κ. Finally, we discuss the logarithmic increase in the effective ranges of ns states as n→∞.

Novel approach to pion and eta production in proton-proton collisions near threshold

We evaluate the threshold matrix-element for the reaction $pp \to pp\pi^0$ in a fully relativistic Feynman diagrammatic approach. We employ a simple effective range approximation to take care of the S-wave $pp$ final-state interaction. The experimental value for the threshold amplitude ${\cal A} = (2.7 - i 0.3)$ fm$^4$ can be reproduced by contributions from tree level chiral (long-range) pion exchange and short-range effects related to vector meson exchanges, with $\omega$ exchange giving the largest individual contribution. Pion loop effects appear to be small. We stress that the commonly used heavy baryon formalism is not applicable in the NN-system above the pion production threshold due to the large external momentum, $|\vec p | \simeq \sqrt {Mm_\pi}$, with $M$ and $m_\pi$ the nucleon and the pion mass, respectively. We furthermore investigate the reaction $pp\to p n \pi^+$ near threshold within the same approach. We extract from the data the triplet threshold amplitude as ${\cal B}= (2.8 -i 1.5)$ fm$^4$. Its real part can be well understood from (relativistic) tree level meson-exchange diagrams. In addition, we investigate the process $pp \to pp \eta$ near threshold. We use a simple factorization ansatz for the $pp\eta$ final-state interaction and extract from the data the modulus of the threshold amplitude, $|{\cal C}| = 1.32 $fm$^4$. With $g_{\eta N}=5.3$, this value can be reproduced by (relativistic) tree level meson-exchange diagrams and $\eta$-rescattering, whose strength is fixed by the $\eta N$ scattering length. We also comment on the recent near threshold data for $\eta'$-production.