Backward Dispersion Relations Research Articles

The effective periodic homogeneous metamaterials for infinite complementary dielectric slab characteristics

A comparison of balanced and unbalanced backward wave dispersion relations and the constitutive electromagnetic effective parameters of dielectric constant, magnetic permeability are obtained. In the balanced case, the initial transmission line lies in left handed (LH) medium and the next to cut-off frequency of higher order regions transmission line becomes in right-handed (RH) medium. In the unbalanced case, there are a tiny reject band displays due to the attenuations. It is also desirable for metamaterial parameters extraction properties should behave in antiresonance behaviour in the band gap regions. The proposed extraction technique shows the suitable results for dual negative magnetic permeability and electric permittivity that match with the theory. To demonstrate the infinite wave investigations, there are four unit cell of complementary split- ring resonators (CSRRs) microstrip implementation is obtained and the equivalent lumped element circuit is analyzed. The theory and experiment result exhibits excellent agreement that ensures minimum loss and electrical small size due to less than wavelength dimensions.

Sum rule for the backward spin polarizability of the nucleon from a backward dispersion relation

A new sum rule for $\gamma_\pi$, the backward spin polarizability of the nucleon, is derived from a backward-angle dispersion relation. Taking into account single- and multi-pion photoproduction in the s-channel up to the energy 1.5 GeV and resonances in the t-channel with mass below 1.5 GeV, it is found for the proton and neutron that $[\gamma_\pi]_p$ = -39.5 +/- 2.4 and $[\gamma_\pi]_n$ = 52.5 +/- 2.4, respectively, in units of 10^{-4} fm^4.

Dispersion relations and the nucleon polarizability.

Recent experimental results on the proton and neutron polarizabilities are examined from the point of view of backward dispersion relations. Results are found to be in reasonable agreement with the measured values. A rigorous relationship between the nucleon and pion polarizabilities is derived and shown to be in excellent agreement with several models.

Backward dispersion relations and pion-4He phase-shift analyses

The usefulness of backward dispersion relations in nuclear physics is discussed. As an example, pion-4He elastic scattering is considered. It is shown how the use of backward dispersion relations can help in reducing ambiguities of phase-shift analyses. In particular, the solution with theS-wave changing sign twice between 150 and 260 MeV and with the real part of the backward scattering amplitude becoming positive at ∼140 MeV and exhibiting resonating behaviour has been suggested as the most probable one.

Nucleon polarizabilities andπ0→2γ,η0→2γdecay rates from Compton-scattering amplitudes

The difference of the dynamic electric and magnetic polarizabilities of the nucleon $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ is estimated on the basis of sum rules obtained from unsubtracted backward dispersion relations for the relevant nucleon Compton-scattering amplitudes in the framework of the two-particle unitarity approximation. The $s$- and $u$-channel contributions are then evaluated in terms of pion photoproduction data while the contribution of the two-pion state in the $t$ channel is shown to contain a large positive piece, explicitly computed, which could bring the theoretical prediction into qualitative agreement with the existing experimental results. It is consequently pointed out that if the ordering $\ensuremath{\alpha}>\ensuremath{\beta}$ is further confirmed experimentally, the annihilation-channel exchanges should be strongly responsible for this fact. The importance of the two-pion continuum for the dispersion determination of the ${\ensuremath{\eta}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decay rate from the superconvergence of certain nucleon Compton-scattering amplitudes for fixed $s$ and large $t$ is also particularly emphasized.

Dynamic electromagnetic polarizabilities of the photon

The dynamic electric and magnetic polarizabilities of the photon α and β are calculated by means of a backward dispersion relation for the relevant Compton amplitude. It is pointed out that the annihilation channel exchanges could be responsible for the ordering α > β given by the present experimental determinations.

Critical Comment on Resonance Saturation of Sum Rules

Using a generalization of the sum rules obtained by equating a fixed-$t$ and a backward dispersion relation, it is argued and illustrated for $\ensuremath{\pi}N$ elastic scattering that saturation with a few resonances can easily lead to inconsistent results.

Dispersion-relation analysis of K +p scattering below 2 GeV/ c

We have obtained K +p scattering amplitudes that fit a large body of data below 2 GeV/ c, satisfy phenomenological partial-wave backward dispersion relations, and are consistent with forward dispersion relations for the spin-non-flip amplitude, and forward dispersion-relation sum rules for the K +p P-wave scattering lengths. This has been achieved by firstly making an energy-dependent phase-shift analysis of 1 660 pieces of K +p data below 2 GeV/ c, using parameterised partial-wave dispersion relations, with the additional constraints of forward dispersion relations and P-wave scattering lengths obtained from forward dispersion-relation sum rules, and then checking the resulting solutions for consistency with backward K +p dispersion relations.

Consistency between the low-energy pion-pion interaction and pion-nucleon scattering

The ππ- N N helicity amplitudes f + 0 and f ± 1 are calculated assuming certain ππ phases δ 0 0 and δ 1 1 and taking careful account of errors. These values are then compared with πN data via partial wave and backward dispersion relations. Excellent agreement is obtained with a conventional ϱ-form for δ 1 1 but in the case of δ 0 0 we require the low-energy behaviour of the Morgan-Shaw favoured phase BD1 to be somewhat reduced to give a scattering length in the range −0.15 ⪅ a 0 0 ⪅ 0.10.

Electromagnetic couplings from the combined use of forward and backward dispersion relations

We derive sum rules for the isovector electromagnetic transition moments N ∗∗→ N ∗γ where N ∗∗ are arbitrary resonances. The derivation is based on the following assumptions: (i) the isospin odd amplitudes of the scattering process N ∗∗π← N ∗π obey unsubstracted dispersion relations in forward as well as in backward direction; (ii) the coupling constants of the s channel resonances are given, to a good approximation, by existing solutions of the algebra of axial charges; (iii) the t-channel contribution of these amplitudes is dominated by a single ϱ meson; and (iv) the N ∗∗N ∗ϱ coupling constant can be related to the electromagnetic couplings N ∗∗N ∗γ via vector meson dominance. Our equations are tested by a calculation of the known couplings of the nucleon and the ΔNγ transition. The results agree well with experiment. We then predict the isovector couplings R(1470) N γ ( K RN V ≈ 1.2), R(1470)R γ ( K RR V ≈ 1.3), where R (1470) is the 1 2 + Roper resonance. We conclude that R(1470) should be seen in photoproduction on deuteron.

K +p backward dispersion relations

We study various sets of K +p phase shifts for consistency with backward dispersion relations. The simplest interpretation of the backward invariant amplitudes is obtained for the Sens γ-solution with the phases smoothed below 700 MeV/ c. Using SU(3) to relate the couplings G ϱKK and G ωKK to G ϱππ and taking values for the ϱ-and ϵ-couplings from fits to πN backward dispersion relations, we obtain two relations between G ωNN (V), G ωNN (T)and G ϵKK / G ϵππ . Assuming G ωNN (T) ≈ 0we find − 1.9 ⩽ G ϵKK / G ϵππ ⩽ − 0.9 and − 14.7 ⩽ G ωNN (V) ⩽ − 5.0.

Boundary Dispersion Relations

We derive a dispersion relation for direct-channel reactions of the type $a+b\ensuremath{\rightarrow}c+d$ with ${m}_{b}={m}_{d}$, e.g., $\ensuremath{\gamma}N\ensuremath{\rightarrow}\ensuremath{\pi}N$, which involves contributions from both forward and backward scattering of the direct channel and backward scattering of the crossed channel but no contribution from unphysical regions of the direct channel. In particular, the integral involving the direct-channel contribution is evaluated along the boundary of the physical region. Our result, in the case of elastic scattering, i.e., ${m}_{a}={m}_{c}$, reduces to the familiar backward dispersion relation.

The Low Frequency Instability in a Plasma Produced by a Local Electron Cyclotron Resonance

A low frequency oscillation, having a backward wave-type dispersion relation, is spontaneously excited in a plasma produced by local electron cyclotron resonance. The fundamental frequency of the oscillation is smaller than ion cyclotron frequency and is nearly proportional to the magnetic field intensity, Radially, the wave amplitude has maxima near the regions of maximum density gradient. The wave is observed to propagate in the direction of the ion diamagnetic velocity. The phase angle between density and potential wave is about 90° and density wave is leading. These characteristics of the oscillation are compared with those of the collisional drift wave. Further, a nonlinear effect characterized by the self-excitation of the subharmonic oscillations of 1/2, 1/3 and 1/4, and their higher harmonics are also observed.

Backward dispersion relations for πN→ πΔ scattering and NΔγ couplings

Using unsubtracted dispersion relations for the invariant amplitudes of the process πN→ πΔ in forward and backward direction we derive sum rules relating t channel to s and u channel resonances. We saturate these sum rules by ϱ, N, and Δ and obtain for the ϱ coupling constants C 3m= 5 3 ; C 4m 2 =C 5m 2 = 5 4 3 where m is the average mass of N and Δ. Assuming vector meson dominance of the electromagnetic vertex NΔγ these become the values of the Gourdin-Salin coupling constants in reasonable agreement with experiment.

Meson-exchanges in the isospin zero t-channel of pion-nucleon scattering

The πN backward dispersion relations for even isospin ( I t = 0) are investigated with phase shifts and the baryon Regge-exchange model as input. The ππ N N cut contributions to the dispersion relations are then used to estimate in an ϵ(700) and f 0(1260) exchange model the meson-nucleon coupling constants.

Information on low-energy ππ interaction from elastic pion-nucleon phase shifts

The well-known methods of partial-wave dispersion relations and backward amplitude dispersion relations for analysing πN phase shifts with a view to obtain the low-energy ππ interaction have been modified by means of recent theories for performing analytic continuation of experimental data in order to obtain a unified model independent treatment. Good agreement between the two methods can be obtained only after a most important correction term has been taken into account, a fact which was overlooked in all previous applications of backward dispersion relations. First results of application of the FESR method of ref. [15] for calculating a partial-wave amplitude in an extended region on the left-hand cut are presented. Results for the T = J = 1 ππ → N N interaction are in accordance with results of Höhler et al. [12]. Results for the T = J = 0 case are against a phase shift that resonates clearly below 900 MeV and in favour of the up-down or down-down solution of ref. [13]. Some information of ππ d-waves is given.

N, Δ and ϱ-exchange in pion-nucleon scattering

The N, Δ and ϱ-exchange contributions to the πN charge exchange amplitude as derived from fixed- t and fixed angle θ = 180° dispersion relations are compared with the corresponding terms in a chiral Lagrangian. A new evaluation of the backward dispersion relation combined with the forward dispersion relation gives information on the physical origin and the magnitude of the corrections to the simple ϱ-dominance and N, Δ, ϱ-exchange models. The result for the πNΔ-coupling constant g ∗2/4π = 0.26 differs from that of other authors and the determination of the ϱNN coupling constants leads to difficulties which were ignored in earlier work. The equality between the forward and backward amplitudes at threshold gives a relation between coupling constants similar to a result which was recently derived from chiral Lagrangians.

Backward (θ=180°)πNDispersion Relations: Applications to the Interference Model,PandP′Trajectories, and to the Mechanical Form Factors of the Nucleon

On the basis of Mandelstam analyticity, crossing, and the observed drop of the backward (180\ifmmode^\circ\else\textdegree\fi{}) ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ differential cross sections with energy, a set of unsubtracted dispersion relations is written for the $\ensuremath{\pi}N$ amplitudes ${A}^{\ifmmode\pm\else\textpm\fi{}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}$ at fixed $\ensuremath{\theta}=\ensuremath{\pi}$. A further application of crossing allows the derivation of separate sum rules on ${A}^{\ensuremath{-}}$ and ${B}^{+}$, which are not of the superconvergent variety, and which provide us with information about $\overline{N}N\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ scattering. In particular, we are able to deduce a value of the spin-flip ${f}^{0}(1250)\ensuremath{-}N\ensuremath{-}N$ coupling constant, which is shown to permit the following observations: (1) The residue function of the $P$ (or ${P}^{\ensuremath{'}}$) trajectory in $\ensuremath{\pi}N$ scattering changes sign between $t=0$ and $t={{m}_{f}}^{2}=1.56$ ${\mathrm{GeV}}^{2}$, and (2) universal coupling of the ${f}^{0}(1250)$ meson to the gravitational stress-energy density and the knowledge of the aforementioned coupling constant fixes the zero-momentum-transfer values of two of the three mechanical form factors. The values are given in the text. Lastly, we present an extended discussion of the Barger-Cline model within the context of backward dispersion relations.