Correct Threshold Behavior Research Articles

A Comparison of Methods for Calculating the Basic Reproductive Number for Periodic Epidemic Systems.

When using mathematics to study epidemics, oftentimes the goal is to determine when an infection can invade and persist within a population. The most common way to do so uses threshold quantities called reproductive numbers. An infection's basic reproductive number (BRN), typically denoted [Formula: see text], measures the infection's initial ability to reproduce in a naive population and is tied mathematically to the stability of the disease-free equilibrium. Next-generation methods have long been used to derive [Formula: see text] for autonomous continuous-time systems; however, many diseases exhibit seasonal behavior. Incorporating seasonality into models may improve accuracy in important ways, but the resulting non-autonomous systems are much more difficult to analyze. In the literature, two principal methods have been used to derive BRNs for periodic epidemic models. One, based on time-averages, is simple to apply but does not always describe the correct threshold behavior. The other, based on linear operator theory, is more general but also more complicated, and no detailed explanations of the necessary computations have yet been laid out. This paper reconciles the two methods by laying out an explicit procedure for the second and then identifying conditions (and some important classes of models) under which the two methods agree. This allows the use of the simpler method, which yields interpretable closed-form expressions, when appropriate, and illustrates in detail the simplest possible case where they disagree. Results show that seasonality alone cannot affect disease persistence, but must act in conjunction with non-hierarchical heterogeneity in the infected population, in order to do so.

Finite-width Laplace sum rules for0−+pseudoscalar glueball in the instanton vacuum model

The correlation function of the ${0}^{\ensuremath{-}+}$ pseudoscalar glueball current is calculated based on the semiclassical expansion for quantum chromodynamics (QCD) in the instanton liquid background. Besides taking the pure classical contribution from instantons and the perturbative one into account, we calculate the contribution arising from the interaction (or the interference) between instantons and the quantum gluon fields, which is infrared free and more important than the pure perturbative one. Instead of the usual zero-width approximation for the resonances, the Breit-Wigner form with a correct threshold behavior for the spectral function of the finite-width resonance is adopted. The properties of the ${0}^{\ensuremath{-}+}$ pseudoscalar glueball are investigated via a family of the QCD Laplacian sum rules. A consistency between the subtracted and unsubtracted sum rules is very well justified. The values of the mass, decay width, and coupling constants for the ${0}^{\ensuremath{-}+}$ resonance in which the glueball fraction is dominant are obtained.

Variational calculations for the hydrogen-antihydrogen system with a mass-scaled Born-Oppenheimer potential

Abstract The problem of proton-antiproton motion in the H-$\bar H$ system is investigated by means of the variational method. We introduce a modified nuclear interaction through mass-scaling of the Born-Oppenheimer potential. This improved treatment of the interaction includes the nondivergent part of the otherwise divergent adiabatic correction and shows the correct threshold behaviour. Using this potential we calculate the vibrational energy levels with angular momentum 0 and 1 and the corresponding nuclear wave functions, as well as the S-wave scattering length. We obtain a full set of all bound states together with a large number of discretized continuum states that might be utilized in variational four-body calculations. The results of our calculations gives an indication of resonance states in the hydrogen-antihydrogen system.

Unusual potential behavior for the weakly bound nucleus 9Be in elastic scattering from 208Pb and 209Bi near the threshold

The elastic scattering angular distributions of the weakly bound 9Be projectile from 208Pb and 209Bi have been measured for 14 beam energies near the threshold from 37 to 50 MeV. The parameters of the optical potential are extracted by means of phenomenological optical model analysis with PTOLEMY. Both of the systems show unusual potential behavior in the vicinity of the Coulomb barrier that the strength of the imaginary (absorptive) part of the potential is increasing (rather than decreasing) with decreasing energy, which is quite different from the results of some previous reports. This unusual threshold phenomenon indicates that the breakup channel is strongly coupled with the elastic channel and has obvious effects on the optical potential. The analyses also show that high precision elastic scattering angular distributions, especially those below the Coulomb barrier, are very important for extracting correct threshold behavior of the optical potential.

Calculation of a hydrogen atom photoionization in a strong magnetic field by using the angular oblate spheroidal functions

A new efficient method for calculating the photoionization of a hydrogen atom in a strong magnetic field is developed based on the Kantorovich approach to the parametric boundary problems in spherical coordinates using the orthogonal basis set of angular oblate spheroidal functions. The progress as compared with our previous paper (Dimova M G, Kaschiev M S and Vinitsky S I 2005 J. Phys. B: At. Mol. Opt. Phys. 38 2337–52) consists of the development of the Kantorovich method for calculating the wavefunctions of a continuous spectrum, including the quasi-stationary states imbedded in the continuum. Resonance transmission and total reflection effects for scattering processes of electrons on protons in a homogenous magnetic field are manifested. The photoionization cross sections found for the ground and excited states are in good agreement with the calculations by other authors and demonstrate correct threshold behavior. The estimates using the calculated photoionization cross section show that due to the quasi-stationary states the laser-stimulated recombination may be enhanced by choosing the optimal laser frequency.

Structure functions at low [formula omitted]: higher twists and target mass effects

We review the physics of structure functions at low Q 2 , focusing on the phenomenon of quark-hadron duality and the resonance–scaling transition, both phenomenologically and in the context of quark models. We also present a new implementation of target mass corrections to nucleon structure functions which, unlike existing treatments, has the correct kinematic threshold behavior at finite Q 2 in the x → 1 limit.

Target mass corrections revisited

We propose a new implementation of target mass corrections to nucleon structure functions which, unlike existing treatments, has the correct kinematic threshold behavior at finite Q^2 in the x -> 1 limit. We illustrate the differences between the new approach and existing prescriptions by considering specific examples for the F_2 and F_L structure functions, and discuss the broader implications of our results, which call into question the notion of universal parton distribution at finite Q^2.

Nucleon structure functions, resonance form factors, and duality

The behavior of nucleon structure functions in the resonance region is explored. For form factors that describe resonance production, expressions are obtained that are dependent on the photon virtuality Q 2, which have a correct threshold behavior, and which take into account available experimental data on resonance decay. Resonance contributions to nucleon structure functions are calculated. The resulting expressions are used to investigate quark-hadron duality in electron-nucleon scattering by taking the example of the structure function F 2.

Cross Sections for (Multiple) Ionization of Ions in Collisions with Electrons

Recently we proposed a new semiempirical analytical parametrization for the shape of ionization cross sections resulting from collisions of ions with various projectiles [1]. It was shown that the incorporation of the correct threshold behaviour as given by the Wannier theory is the crucial point to obtain good agreement with experimental results from threshold to high energies and leads to a unified view of collision processes in very different systems. While the shape of the cross sections as given by this parametrization already compares favourably with the experiment, a simple extension of the formula leads to an even more accurate description for electron-impact ionization of ions [2]. Using recent results from an extended Wannier theory for multiple ionization [3,4], tests of similar parametrizations for multiple ionization processes become feasible.

Ordered analysis of heavy flavor production in deep-inelastic scattering

At low ${Q}^{2},$ charm production in deep-inelastic scattering is adequately described by assuming generation in electroweak boson-light parton scattering (dominantly boson-gluon fusion), which naturally incorporates the correct threshold behavior. At high ${Q}^{2}$ this description is inadequate, since it does not sum logs in ${Q}^{2}{/m}_{c}^{2},$ and is replaced by the treatment of the charm quark as a light parton. We show how the problem of going from one description to the other can be solved in a satisfactory manner to all orders. The key ingredient is the constraint of matching the evolution of the physical structure function ${F}_{2}$ order by order in ${\ensuremath{\alpha}}_{s}{(Q}^{2}),$ in addition, to the matching of the value of ${F}_{2}$ itself. This leads to new expressions for the coefficient functions associated with the charm parton, which are unique in incorporating both the correct threshold and asymptotic behaviors at each order in perturbation theory. The use of these improved coefficients leads to an improvement in global fits and an excellent description of the observed ${F}_{2,\mathrm{charm}}.$

A practical procedure for evolving heavy flavour structure functions

The transition from the low $Q^2$ region in deep inelastic scattering where charm production is described by photon-gluon fusion to a region where the charm structure function $F_2^c$ is largely generated by a charm quark density should be a smooth passage. The all orders prescription for matching between the two schemes does not uniquely determine all the relevant coefficient functions (CF's) or guarantee the correct threshold behaviour at fixed order in $\alpha_s$. The constraints of matching not only the value but also the evolution of $F_2^c$ order by order allows all CF's to be determined at any order, each of which reduces to the appropriate massless $\msb$ expression as $Q^2 \to \infty$, and leads to a smooth transition in the threshold region. This procedure gives an excellent description of the $F_2^c$ data for all $Q^2$.

Threshold law for the triplet state for electron-impact ionization in the Temkin-Poet model

We derive the analytical threshold behavior for the triplet cross section for electron-impact ionization in the Temkin-Poet model. The analytical results indicate that the most recent numerical calculations may fail to reproduce the correct threshold behavior in an energy regime below about E=0.1 a.u. We also present an analytical expression for the energy distribution of the two electrons near threshold.

Pion-nucleon scattering in the P 33 channel in the linear sigma model

A consistent study of the NN interaction requires the investigation of the Δ resonance, as the Δ can affect the properties of the effective light σ meson originating from correlated two-pion exchange and vice versa. We study how the Δ appears as an effective degree of freedom in the linear σ model. Since the field-theoretic scattering amplitudes respect certain analytic properties, it is appropriate to use partial-wave dispersion relations for πN scattering, which produce non-linear integral equations that are solved by the N/ D method. We choose an amplitude that guarantees the correct threshold behavior and that gives a P 33 resonance with a driving term from one-nucleon exchange. Then we investigate the effect of adding the one-sigma-exchange force. With either a very heavy or a light σ meson, we can find a P 33 resonance. For intermediate σ masses, we find a πN bound state. We confirm that the calculated resonance mass depends sensitively on the short-range πN attraction. Nevertheless, the existence of a P 33 resonance is compatible with the existence of an effective light σ meson in this model.

The quantum mechanical third cluster for a binary step potential

We calculate the quantum mechanical third cluster coefficient b 3 in two dimensions. The model of the interaction is a binary repulsive step potential with finite range. We expand the continuum three-body wave functions in hyperspherical harmonics and solve large finite sets of coupled equations in the hyper radius ρ, by using an adiabatic approximation. We obtain the eigenphase shifts of the three to three scattering and integrate a phase shift sum (weighted by a Boltzmann factor), over the energy, to give us the cluster as a function of temperature. We note that we do obtain the correct threshold behaviour for the crucial phase shift sum.

Higher-order WKB approximations in supersymmetric quantum mechanics.

In the framework of the recently proposed supersymmetric WKB (SWKB) approximation scheme, we obtain an explicit expression for the quantization condition which contains all terms up to order ${\ensuremath{\Elzxh}}^{6}$. For spherically symmetric potentials, we find that the SWKB approach automatically yields wave functions with the correct threshold behavior. This is in contrast to the usual WKB scheme, where proper r\ensuremath{\rightarrow}0 behavior necessitates the use of cumbersome ``Langer corrections.'' Previous authors have shown that the leading-order (${\ensuremath{\Elzxh}}^{0}$) SWKB quantization integral gives exact bound-state spectra for analytically solvable shape-invariant potentials. For these cases, we show that the higher-order correction terms vanish identically. Finally, for nonanalytically solvable potentials, a comparison of our results (comprising of higher-order corrections) with numerically determined eigenvalues reveals very good accuracy.

The rate constants for the H+H2 reaction and its isotopic analogs at low temperatures: Wigner threshold law behavior

Variational transition-state theory rate constants for the H+H2 reaction and its isotopic analogs at very low temperatures have been calculated using the LSTH potential surface. The Wigner threshold law is found to hold quantitatively below 1 K and to within 34% on average at 5 K. The transition-state theory rate constants are found to be consistent with the requirements of the Wigner threshold law provided that a tunneling probability having the correct threshold behavior is used in calculating the transmission coefficient. Using these calculated rate constants, decay rates of H and D atoms in solid H2, D2, and HD were estimated and compared with experiment. The good agreement found suggests that these decays occur through quantum mechanical tunneling exchange reactions.

Hadronic corrections to QCD sum rules and light quark masses

A parametrization of theJp=0− hadronic continuum, in the framework of Extended PCAC, is discussed with emphasis on finite-width effects and on the constraints imposed by the correct threshold behavior of the pion spectral function. As an application light quark masses are calculated using both Hilbert and Laplace transform QCD sum rules. The results for the runing quark masses are:\((\bar m_u + \bar m_d )|_{1 Gev} = 16 \pm 2 MeV,(\bar m_u + \bar m_s )|_{1 Gev} = 199 \pm 27 MeV\), and a ratio\(R \equiv 2(\bar m_u + \bar m_s )/(\bar m_u + \bar m_d )_{1 Gev} = 25 \pm 4\).

The long-range contribution to quasi-elastic collisions of atoms

A theoretical investigation of quasi-elastic atom-scattering is presented. The transition potential for the system, which consists of an excited atom in a neutral atom in the ground state, is derived and split into short- and long-range parts. A general expression for the leading long-range (two-photon exchange) part of the inelastic potential is obtained and the correct threshold behaviour of the partial-wave amplitude is discussed. The cross sections sigma short(nl to nl') and sigma long(nl to nl') of l-changing collisions of the Rydberg atoms are estimated for large values of n, and small as well as large angular momentum transitions. It is found that sigma short>> sigma long if n2qmin >l,l' when n to infinity .

Discrete two-variable expansions of scattering amplitudes for particles with spin

Scattering amplitudes for two-body reactions involving particles with arbitrary spins are expanded in terms of functions of the energy and scattering angle. The expansions are in terms of the representation matrices of the O(4) group and are a modification of previously obtained two-variable expansions in terms of the representations of the internal Lorentz group. The new expansions involve only sums, rather than integrals. They can be interpreted as partial-wave expansions, supplemented by a further expansion of each partial wave in terms of the O(4) $d$ functions. The partial waves automatically have the correct threshold behavior and the O(4) amplitudes (the expansion coefficients) have simple properties with respect to the usual discrete symmetries (parity, etc.). The expansions can be applied to analyze two-body scattering data simultaneously for all angles $0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}$, $0\ensuremath{\le}\ensuremath{\varphi}l2\ensuremath{\pi}$ and all energies from threshold to some a priori chosen maximal energy.

Comments on charmed baryon pair production in e +e −-annihilation

We show that the symmetric quark model calculation of charmed baryon pair creation in e +e −-annihilation has an incorrect threshold structure. The cc̄-pair creation model possesses the correct s-wave threshold behaviour, if the cc̄-pair and diquark pair are created in orbital s-waves. We show that the two models become identical for a specific orbital configuration and discuss implications for the relative production rates of Λ c Λ c :Σ c Σ c :Σ c Σ c ∗+Σ c ∗ Σ c :Σ c ∗ c ∗ .