Dispersion Relation Approach Research Articles

Dalitz plot studies of D0→KS0K+K− decays in a factorization approach

The $\textit{BABAR}$ Collaboration data of the $D^0 \to K^0_S K^+ K^-$ process are analyzed within a quasi two-body factorization framework. In earlier studies, assuming $D^0$ transitions to two kaons and the transitions between one kaon and two kaons to proceed through the dominant intermediate resonances, we approximated them as being proportional to the kaon form factors. To obtain good fits, one has to multiply the scalar-kaon form factors, derived from unitary relativistic coupled-channel models or in a dispersion relation approach, by phenomenological energy-dependent functions. The final state kaon-kaon interactions in the $S$-, $P$- and $D$- waves are taken into account. All $S$-wave channels are treated in a unitary way. The $K^+K^-$ and $\bar K^0 K^+$ $S$-wave effective mass squared distributions, corrected for phase space, are shown, in a model-independent manner, to be significantly different. Then the $f_0(980)$ resonance must be included at variance with the BABAR analysis. The best fit has 19 free parameters and indicates i) the dominance of annihilation amplitudes, ii) a large dominance of the $f_0(980)$ meson in the near threshold $K^+K^-$ invariant mass distribution and iii) a sizable branching fraction to the $K_S^0 \ [\rho(770)^+ + \rho(1450)^+ + \rho(1700)^+] $ final states. An appendix provides an update of the determination of the isoscalar-scalar meson-meson amplitudes based on an enlarged set of data. A second appendix proposes two alternative fits based on scalar-kaon form factors calculated from the Muskhelishvili-Omn\`es dispersion relation approach. These fits have $\chi^2$ quite close to that of the best fit but they show important contributions from both the $f_0$ and $a_0^0$ mesons and a weaker role of the $\rho^+$ mesons.

Muon g−2 and Δα connection

The discrepancy between the Standard Model theory and experimental measurement of the muon magnetic moment anomaly, $a_{\mu}=\left(g_{\mu}-2\right)/2$, is connected to precision electroweak (EW) predictions via their common dependence on hadronic vacuum polarization effects. The same data for the total $e^+e^- \rightarrow \text{hadrons}$ cross section, $\sigma_{\rm had}(s)$, are used as input into dispersion relations to estimate the hadronic vacuum polarization contributions, $a_{\mu}^{\rm had,\,VP}$, as well as the five-flavor hadronic contribution to the running QED coupling at the $Z$-pole, $\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2)$, which enters natural relations and global EW fits. The EW fit prediction of $\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02722(41)$ agrees well with $\Delta\alpha_{\rm had}^{(5)}(M_{Z}^2) = 0.02761(11)$ obtained from the dispersion relation approach, but exhibits a smaller central value suggestive of a larger discrepancy $\Delta a_{\mu}=a_{\mu}^{\rm exp} - a_{\mu}^{\rm SM}$ than currently expected. Postulating that the $\Delta a_{\mu}$ difference may be due to missing $\sigma_{\rm had}(s)$ contributions, implications for $M_W$, $\sin^2 \! \theta^{\rm lep}_{\rm eff}$ and $M_H$ obtained from global EW fits are investigated. Shifts in $\sigma_{\rm had}(s)$ needed to bridge $\Delta a_{\mu}$ are found to be excluded above $\sqrt{s} \gtrsim 0.7$ GeV at the 95\%CL. Moreover, prospects for $\Delta a_{\mu}$ originating below that energy are deemed improbable given the required increases in the cross section. Such hypothetical changes to the hadronic data are also found to affect other related observables, such as the electron anomaly, $a_e^{\rm SM}$, the ratio $R_{e/\mu} = (m_\mu/m_e)^2 (a_{e}^{\rm had,\,LO\,VP}/a_{\mu}^{\rm had,\,LO\,VP})$ and the running of the weak mixing angle at low energies, although the consequences of these are currently less constraining.

Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows

The path-following scheme in Loisel and Maxwell (SIAM J Matrix Anal Appl 39(4):1726–1749, 2018) is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current. This is equivalent to solving the Rayleigh stability equation with linearized free-surface boundary condition for each sought point on the curve. Taking advantage of the analyticity of the dispersion relation, a path-following or continuation approach is adopted. The problem is discretized using a collocation scheme, parametrized along either a radial or angular path in the wave vector plane, and differentiated to yield a system of ODEs. After an initial eigenproblem solve using QZ decomposition, numerical integration proceeds along the curve using linear solves as the Runge–Kutta F(cdot ) function; thus, many QZ decompositions on a size 2N companion matrix are exchanged for one QZ decomposition and a small number of linear solves on a size N matrix. A piecewise interpolant provides dense output. The integration represents a nominal setup cost whereafter very many points can be computed at negligible cost whilst preserving high accuracy. Furthermore, a two-dimensional interpolant suitable for scattered data query points in the wave vector plane is described. Finally, a comparison is made with existing numerical methods for this problem, revealing that the path-following scheme is the most competitive algorithm for this problem whenever calculating more than circa 1,000 data points or relative normwise accuracy better than 10^{-4} is sought.

Two-Photon Exchange Correction in Elastic Lepton–Proton Scattering

We present the dispersion relation approach based on unitarity and analyticity to evaluate the two-photon exchange contribution to elastic electron-proton scattering. The leading elastic and first inelastic $\pi N$ intermediate state contributions are accounted for in the region of small momentum transfer $Q^2 < 1~\mathrm{GeV}^2$ based on the available data input. The novel methods of analytical continuation allow us to exploit the MAMI form factor data and the MAID parameterization for the pion electroproduction amplitudes as input in the calculation. The results are compared to the recent CLAS, VEPP-3 and OLYMPUS data as well as to the full two-photon exchange correction in the near-forward approximation, which is based on the Christy and Bosted unpolarized structure functions fit. Additionally, predictions are given for a forthcoming muon-proton scattering experiment.

Linear stability analysis of collective neutrino oscillations without spurious modes

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or MSW oscillations. They obey integro-differential equations, numerical solutions of which are also very challenging. If one focuses on the onset of the collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes, unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We have found that the spurious modes originate from the artificial production of pole singularities instead of a branch cut in the Riemann surface by the discretizations. The branching point singularities in the Riemann surface for the original undiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription removes the spurious modes indeed. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multi-energy case as well as to the dispersion relation approach that was proposed very recently.

A coupled-channel formalism for three-body final state interaction

From dispersion relation approach, a formalism that describes final state interaction among three particles in a coupled-channel system is presented. Different representations of coupled-channel three-body formalism for spinless particles in both initial and final states are derived.

Review of Recent Calculations of the Hadronic Vacuum Polarisation Contribution

Recent calculations of the hadronic vacuum polarisation contribution are reviewed. The focus is put on the leading-order contribution to the muon magnetic anomaly involving $e^+e^-$ annihilation cross section data as input to a dispersion relation approach. Alternative calculation including tau data is also discussed. The $\tau$ data are corrected for various isospin-breaking sources which are explicitly shown source by source.

A high-resolution, hybrid compact-WENO scheme with minimized dispersion and controllable dissipation

In this paper, a high-resolution, hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique. Firstly, a sufficient condition for a family of tri-diagonal compact schemes to have independent dispersion and dissipation is derived. Then, a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed. Finally, the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme. Moreover, the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several test cases are carried out to verify the high-resolution as well as the robust shock-capturing capabilities of the proposed scheme.

Viscous corrections to the resistance of nanojunctions: A dispersion relation approach

It is well known that the viscosity of a homogeneous electron liquid diverges in the limits of zero frequency and zero temperature. A nanojunction breaks translational invariance and necessarily cuts off this divergence. However, the estimate of the ensuing viscosity is far from trivial. Here, we propose an approach based on a Kramers-Kr\"onig dispersion relation, which connects the zero-frequency viscosity, $\eta(0)$, to the high-frequency shear modulus, $\mu_{\infty}$, of the electron liquid via $\eta(0) =\mu_{\infty} \tau$, with $\tau$ the junction-specific momentum relaxation time. By making use of a simple formula derived from time-dependent current-density functional theory we then estimate the many-body contributions to the resistance for an integrable junction potential and find that these viscous effects may be much larger than previously suggested for junctions of low conductance.

Pion charge form factor and constraints from space-time translations

The role of Poincaré covariant space-time translations is investigated in the case of a relativistic quantum mechanics approach to the pion charge form factor. It is shown that the related constraints are generally inconsistent with the assumption of a single-particle current, which is most often referred to. The only exception is the front-form approach with q+ = 0. How accounting for the related constraints, as well as restoring the equivalence of different RQM approaches in estimating form factors, is discussed. Some extensions of this work and, in particular, the relationship with a dispersion-relation approach, are presented. Conclusions relative to the underlying dynamics are given.

Comment on “Polarizability of the pion: No conflict between dispersion theory and chiral perturbation theory”

The statement of the authors of the article Phys. Rev. C 77, 065211 (2008) that spurious singularities occur in the dispersion relation approach, where imaginary parts of the amplitudes of the process \gamma\gamma->\pi\pi are saturated by the contributions of meson resonances by using Breit-Wigner expressions, is analyzed. It is shown that there are no any additional singularities in this approach and the disagreement between the predictions of the dispersion relations and the chiral perturbation theory for (\alpha_1-\beta_1)_\pi^{\pm} remains.

Modeling electromagnetic form factors of light and heavy pseudoscalar mesons

The electromagnetic form factors of light and heavy pseudoscalar mesons are calculated within two covariant constituent-quark models, viz., a light-front and a dispersion relation approach. We investigate the details and physical origins of the model dependence of various hadronic observables: the weak decay constant, the charge radius and the elastic electromagnetic form factor.

Form factors in RQM approaches: Constraints from space-time translations

Different relativistic quantum-mechanic approaches have recently been used to calculate properties of various systems, form factors in particular. It is known that predictions, which most often rely on a single-particle current approximation, can lead to predictions with a very large range. It was shown that accounting for constraints related to space-time translations could considerably reduce this range. It is shown here that predictions can be made identical for a large range of cases. These ones include the following approaches: instant form, front form, and “point form” in arbitrary momentum configurations and a dispersion-relation approach which can be considered as the approach which the other ones should converge to. This important result supposes both an implementation of the above constraints and an appropriate single-particle-like current. The change of variables that allows one to establish the equivalence of the approaches is given. Some points are illustrated with numerical results for the ground state of a system consisting of scalar particles.

Relativistic three-quark equations and mass spectrum of the (70, 1−) baryon multiplet

A relativistic generalization of three-quark Faddeev equations for the particles of the (70, 1−) multiplet is constructed within the dispersion-relation approach. An approximate solution to the relativistic three-quark equations is obtained on the basis of the method of isolating leading singularities of scattering amplitudes. The mass spectrum of the (70, 1−) multiplet is calculated, and the result is in rather good agreement with corresponding experimental data.

FORM FACTORS IN B → f0(980) AND D → f0(980) TRANSITIONS FROM DISPERSION RELATIONS

Within the dispersion relation approach we give the double spectral representation for space-like and time-like B → f0(980) and D → f0(980) transition form factors in the whole q2 range. The spectral densities, being the input of the dispersion relations, are obtained from a triangle diagram of relativistic constituent quarks.

Heliopause stability in the presence of neutral atoms: Rayleigh‐Taylor dispersion analysis and axisymmetric MHD simulations

The heliopause, a surface separating the tenuous hot heliosheath flow and the dense, magnetized interstellar flow, is subject to instabilities of the Rayleigh‐Taylor and Kelvin‐Helmholtz types. The dynamic properties of this discontinuity is of considerable importance for understanding the neutral atom and cosmic ray filtration at the interface. Here we investigate the stability of the upwind segment of the heliopause in the presence of charge exchange collisions using both an analytic (dispersion relation) approach and a numerical model that includes the interstellar magnetic field. Linear analysis yields dispersion relations that admit imaginary solutions for a range of wave numbers, implying that the stagnation point on the heliopause is Rayleigh‐Taylor unstable to small perturbations propagating parallel to the discontinuity surface. Effects of interstellar and heliosheath atoms are analyzed separately. We confirm our analytic results by performing time‐dependent numerical simulations of the nonlinear development of this instability using a multifluid MHD‐neutral approach. For typical solar wind and LISM conditions we obtain cyclical evolution of the upwind heliopause with a period of about 100 years. The amplitude of these oscillations is found to be dependent on the presence of hot heliosheath neutrals. We discuss the effect of a strong LISM magnetic field on the heliopause stability and possible implications of the obtained instability on the X‐ray emission and cosmic ray transport in the outer heliosphere.

Heliopause stability revisited: dispersion analysis and numerical simulations

The heliopause, a surface separating the tenuous hot heliosheath flow and the dense, strongly magnetized interstellar flow, is subject to instabilities of the Rayleigh–Taylor and Kelvin–Helmholtz types. The dynamic evolution of this discontinuity is of considerable importance for understanding the neutral atom and cosmic-ray filtration at the interface. Here, we investigate the stability of the upwind heliopause in the presence of charge exchange collisions using both an analytic (dispersion relation) approach and a numerical model that includes the interstellar magnetic field. The linear analysis yields a cubic dispersion relation that admits imaginary solutions for the full range of wavenumbers, implying that the stagnation point on the heliopause is unconditionally Rayleigh–Taylor unstable to small perturbations propagating parallel to the discontinuity surface. We confirm this result by following the nonlinear development of the instability with a time-dependent simulation using a four fluid MHD-neutral numerical code. For the typical solar wind and LISM conditions, we obtain cyclical evolution of the upwind heliopause with a period of the order of 100 years. We also identify two areas of space physics where the instability may have important implications.

Cottingham formula and the pion electromagnetic mass difference at finite temperature

We generalize the Cottingham formula at finite ( T≠0) temperature by using the imaginary time formalism. The Cottingham formula gives the theoretical framework to compute the electromagnetic mass differences of the hadrons using a dispersion relation approach. It can be also used in other contexts, such as non leptonic weak decays, and its generalization to finite temperature might be useful in evaluating thermal effects in these processes. As an application we compute the π +− π 0 mass difference at T≠0; at small T we reproduce the behaviour found by other authors: δm 2(T)=δm 2(0)+ O ( αT 2), while for moderate T, near the deconfinement temperature, we observe deviations from this behaviour.

Delbrück and nuclear effects in photon scattering: recent developments

Delbrück scattering, which is the elastic scattering of a photon in a Coulomb field, is of interest both as an observable high-order non-linear process in quantum electrodynamics and as an interfering process in studies of nuclear structure. This process has been extensively investigated in the past from theoretical and computational points of view. In this paper, attention is mainly focused on the theoretical features and, in particular, the dispersion relation approach and a critical review of the numerical evaluations of the first-order Born approximation cross-section is presented. Applications and limits of these results are discussed in comparison with experimental data on elastic photon scattering by medium-and heavy-mass nuclei and theoretical models of nuclear structure. Copyright © 1999 John Wiley & Sons, Ltd.

Delbrück and nuclear effects in photon scattering: recent developments

Delbrück scattering, which is the elastic scattering of a photon in a Coulomb field, is of interest both as an observable high-order non-linear process in quantum electrodynamics and as an interfering process in studies of nuclear structure. This process has been extensively investigated in the past from theoretical and computational points of view. In this paper, attention is mainly focused on the theoretical features and, in particular, the dispersion relation approach and a critical review of the numerical evaluations of the first-order Born approximation cross-section is presented. Applications and limits of these results are discussed in comparison with experimental data on elastic photon scattering by medium-and heavy-mass nuclei and theoretical models of nuclear structure. Copyright © 1999 John Wiley & Sons, Ltd.