Angular Momentum Plane Research Articles

Surface polaritons on metallic and semiconducting cylinders: A complex angular momentum analysis

We revisit scattering of electromagnetic waves from metallic and semiconducting cylinders in the framework of complex angular momentum techniques. We prove that "resonant surface polariton modes" are generated by a unique surface wave, i.e. a surface polariton, propagating close to the cylinder surface. This surface polariton corresponds to a particular Regge pole of the $S$-matrix of the cylinder. From the associated Regge trajectory we can construct semiclassically the spectrum of the complex frequencies of the resonant surface polariton modes which can be considered as Breit-Wigner-type resonances. Furthermore, by taking into account Stokes' phenomenon, we derive an asymptotic expression for the position in the complex angular momentum plane of the surface polariton Regge pole. We then describe semiclassically the surface polariton and provide analytical expressions for its the dispersion relation and its damping. All these features allow us to consider the photon-cylinder system as a kind of artificial atom where the photon plays the role of the electron. Finally, we briefly discuss the implication of our results for two-dimensional photonic crystals.

Study of resonances in the complex charge plane

Potential resonances are usually investigated either directly in the complex energy plane or indirectly in the complex angular momentum plane. Another formulation complementing these two is presented in this work. It is an indirect method which studies resonances in a complex charge plane (Z-plane). The complex scaling (rotation) method is employed in the development of this formulation. Bound states spectrum and resonance energies are mapped onto a discrete set of poles of a resolvent operator on the real line of the Z-plane. These poles move along trajectories as we vary the energy. A finite L2 basis is used in the numerical implementation of the method. Stability of poles and trajectories against variations in all computational parameters is demonstrated. Resonances for a given potential example are calculated and compared with those obtained elsewhere. In this presentation, modest effort will be devoted to mathematical rigor.

Time delay and time advance in resonance theory

We propose a theory of the resonance–antiresonance scattering process which differs considerably from the classical one (the Breit–Wigner theory), which is commonly used in the phenomenological analysis. Here both resonances and antiresonances are described in terms of poles of the scattering amplitude: the resonances by poles in the first quadrant while the antiresonances by poles in the fourth quadrant of the complex angular momentum plane. The latter poles are produced by non-local potentials, which derive from the Pauli exchange forces acting among the nucleons or the quarks composing the colliding particles.

Unified scheme for describing time delay and time advance in the interpolation of rotational bands of resonances

In this paper we show how rotational bands of resonances can be described by using trajectories of poles of the scattering amplitude in the complex angular momentum plane: each band of resonances is represented by the evolution of a single pole lying in the first quadrant of the plane. The main result of the paper consists in showing that also the antiresonances (or echoes) can be described by trajectories of the scattering amplitude poles, instead of using the hard-sphere potential scattering as prescribed by the classical Breit-Wigner theory. The antiresonance poles lie in the fourth quadrant of the complex angular momentum plane, and are associated with non-local potentials which take into account the exchange forces; it derives a clear-cut separation between resonance and antiresonance poles. The evolution of these latter poles describes the passage from quantum to semi-classical physics. The theory is tested on the rotational band produced by $\alpha$-$\alpha$ elastic scattering and on the hadronic rotational bands in $\pi^+$-p elastic scattering.

‘Quick and dirty’ methods for studying black-hole resonances

We discuss simple integration methods for the calculation of black-hole scattering resonances both in the complex frequency plane (quasinormal modes) and the complex angular momentum plane (Regge poles). Our numerical schemes are based on variations of ‘phase-amplitude’ methods. In particular, we discuss the Prüfer transformation, where the original (frequency domain) Teukolsky wave equation is replaced by a pair of first-order nonlinear equations governing the introduced phase functions. Numerical integration of these equations, performed along the real r∗-axis (where r∗ denotes the usual tortoise radial coordinate), or along rotated contours in the complex r∗-plane, provides the required \U0001d4ae-matrix element (the ratio of amplitudes of the outgoing and ingoing waves at infinity). Müller's algorithm is then employed to conduct searches in the complex plane for the poles of this quantity (which are, by definition, the desired resonances). We have tested this method by verifying known results for Schwarzschild quasinormal modes and Regge poles, and provide new results for the Kerr black-hole problem. Results produced by our scheme prove to be accurate as long as the imaginary part of the resonance is not much larger than the real part. We also describe a new method for estimating the ‘excitation coefficients’ for quasinormal modes. The method is applied to scalar waves moving in the Kerr geometry, and the obtained results shed light on the long-lived quasinormal modes that exist for black holes rotating near the extreme Kerr limit.

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.

Orbiting resonances and bound states in molecular scattering

A family of orbiting resonances in molecular scattering is globally described by using a single pole moving in the complex angular momentum plane. The extrapolation of this pole at negative energies gives the location of the bound states. Then a single pole trajectory, that connects a rotational band of bound states and orbiting resonances, is obtained. These complex angular momentum singularities are derived through a geometrical theory of the orbiting. The downward crossing of the phase-shifts through pi/2, due to the repulsive region of the molecular potential, is estimated by using a simple hard-core model. Some remarks about the difference between diffracted rays and orbiting are also given.

Predictions for high-energy real and virtual photon–photon scattering from color dipole BFKL–Regge factorization

High-energy virtual photon-virtual photon scattering can be viewed as interaction of small size color dipoles from the beam and target photons, which makes $\gamma^{*}\gamma^{*}, \gamma^{*}\gamma$ scattering at high energies (LEP, LEP200 & NLC) an indispensable probe of short distance properties of the QCD pomeron exchange. Based on the color dipole representation, we investigate consequences for the $\gamma^{*}\gamma^{*},\gamma^{*}\gamma$ scattering of the incorporation of asymptotic freedom into the BFKL equation which makes the QCD pomeron a series of isolated poles in the angular momentum plane. The emerging color dipole BFKL-Regge factorization allows us to relate in a model-independent way the contributions of each BFKL pole to $\gamma^{*}\gamma^{*},\gamma^{*} \gamma$ scattering and DIS off protons. Numerical predictions based on our early works on color dipole BFKL phenomenology of DIS on protons are in a good agreement with the experimental data on the photon structure function $F_{2\gamma}$ and most recent data on the $\gamma^*\gamma^*$ cross section $\sigma^{\gamma^*\gamma^*}(Y)$ from OPAL and L3 experiments at LEP200. We discuss the role of non-perturbative dynamics and predict pronounced effect of the Regge-factorization breaking due to large unfactorizable non-perturbative corrections to the perturbative vacuum exchange. We comment on the salient features of the BFKL-Regge expansion for $\gamma^{*}\gamma^{*},\gamma^{*}\gamma$ scattering including the issue of decoupling of subleading BFKL poles and the soft plus rightmost hard BFKL pole dominance .

Color dipole BFKL-Regge factorization and high-energy photon-photon scattering

Based on the color dipole representation, we investigate consequences for the γ*γ*, γ *γ scattering of the finding by Fadin, Kuraev, and Lipatov that the incorporation of asymptotic freedom into the BFKL equation makes the QCD pomeron a series of isolated poles in the angular momentum plane. The emerging color dipole BFKL-Regge factorization allows us to relate in a model-independent way the contributions of each BFKL pole to the γ *γ*, γ*γ scattteirng and the deep inelastic scattering on protons. Numerical predictions based on our early work on the color dipole BFKL phenomenology of the deep inelastic scattering on protons gives a good agreement with the recent experimental data from OPAL and L3 experiments at LEP200. We discuss the role of nonperturbative dynamics and predict a pronounced effect of the Regge-factorization breaking due to large unfactorizable nonperturbative corrections to the perturbative vacuum exchange. We comment on the salient features of the BFKL-Regge expansion for the γ*γ*, γ*γ scattering including the issue of the decoupling of subleading BFKL poles and the soft plus rightmost hard BFKL pole dominance.

Regge models of the proton structure function with and without hard pomeron: A comparative analysis

A comparative phenomenological analysis of Regge models with and without a hard Pomeron component is performed using a common set of recently updated data. It is shown that the data at small $x$ do not indicate explicitly the presence of the hard Pomeron. Moreover, the models with two soft-Pomeron components (simple and double poles in the angular momentum plane) with trajectories having intercept equal one lead to the best description of the data not only at $W>3$ GeV and at small $x$ but also at all $x\leq 0.75$ and $Q^{2}\leq 30000$ GeV$^{2}$.

Complex-angular-momentum analysis of atom-diatom angular scattering: Zeros and poles of theSmatrix

It is shown that in the complex plane of the total angular momentum, an atom-diatom S-matrix element may have complex zeros as well as resonance poles. The zeros close to the real axis correlate with interference minima of the partial reaction probability. With the help of the Pad\'e approximation we estimate positions of both poles and zeros, and construct a uniform semiclassical model for the angular distributions. Examples include $\mathrm{C}\mathrm{l}+\mathrm{H}\mathrm{C}\stackrel{\ensuremath{\rightarrow}}{l}\mathrm{C}\mathrm{l}\mathrm{H}+\mathrm{C}\mathrm{l}$ and $\mathrm{F}+\mathrm{H}\stackrel{\ensuremath{\rightarrow}}{D}\mathrm{F}\mathrm{H}+\mathrm{D}$ reactive scattering.

Padé reconstruction of Regge poles from scattering matrix data for chemical reactions

We describe a practical procedure for analytically continuing a scattering matrix element into the complex angular momentum plane, given its values at the physical angular momentum quantum numbers of 0, 1, 2, …. The procedure first applies a preconditioning function to the scattering matrix element; then the remaining residual function is interpolated using Padé approximations. The method is illustrated by extracting the leading Regge pole position and residue for the state selected reactions Cl+HCl( v=1, j=5) → ClH( v′=1, j′=5)+Cl and I+HI( v=0, j=0) → IH( v′=0, j′=2)+I, where v and j are vibrational and rotational quantum numbers respectively.

The color dipole BFKL-Regge expansion: from DIS on protons to pions to rise of hadronic cross sections

As noticed by Fadin, Kuraev and Lipatov in 1975 incorporation of asymptotic freedom into the BFKL equation splits the QCD pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier work on the proton structure function we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small- x structure function of the pion F 2 π ( x, Q 2). We calculate F 2 π in a small- x region and find good agreement with the recent H1 determination of F 2 π ( x, Q 2). We also present the BFKL-Regge factorization based evaluation of the contribution from hard scattering to the observed rise of the NN, πN and real photo-absorption γN and γπ total cross sections.

Semiclassical angular scattering in the F+H2→HF+H reaction: Regge pole analysis using the Padé approximation

We apply the complex angular momentum analysis to study state-selective forward scattering in the F+H2 reaction. The scattering matrix calculated by Castillo et al. is used. The S-matrix element is analytically continued into the complex plane of the total angular momentum with the help of the Padé approximation. Resonance (Regge) poles are identified and their residues evaluated. The semiclassical optical model is corrected to account for finite lifeangles of leading resonances. For chosen transitions, the forward scattering enhancement is shown to be a resonance effect due to the decay of a Regge state with angular life of about 32°.

Jost function for singular potentials

An exact method for direct calculation of the Jost function and Jost solutions for a repulsive singular potential is presented. Within this method the Schrodinger equation is replaced by an equivalent system of linear first-order differential equations, which after complex rotation, can easily be solved numerically. The Jost function can be obtained to any desired accuracy for all complex momenta of physical interest, including the spectral points corresponding to bound and resonant states. The method can also be used in the complex angular-momentum plane to calculate the Regge trajectories. The effectiveness of the method is demonstrated using the Lennard-Jones (12,6) potential. The spectral properties of the realistic inter-atomic He4-He4 potentials HFDHE2 and HFD-B of Aziz and collaborators are also investigated.

BFKL Pomeron in(2+1)-dimensional QCD

We investigate high-energy scattering in spontaneously broken Yang-Mills gauge theory in $2+1$ space-time dimensions and present the exact solution of the leading $\mathrm{ln}s$ BFKL equation. The solution is constructed in terms of special functions using the earlier results of two of us (L.N.L. and L.S.). The analytic properties of the $t$-channel partial wave as functions of the angular momentum and momentum transfer have been studied. We find in the angular momentum plane (i) a Regge pole whose trajectory has an intercept larger than 1 and (ii) a fixed cut with the rightmost singularity located at $j=1.$ The massive Yang-Mills theory can be considered as a theoretical model for the (nonperturbative) Pomeron. We study the main structure and property of the solution including the Pomeron trajectory at momentum transfer different from zero. The relation to the results of Li and Tan for the massless case is discussed.

The quark spin structure of the nucleon

Some non-perturbative aspects of the nucleon quark spin structure are reviewed. The first part is a brief summary of early theoretical developments in the field of polarized deep inelastic scattering of electrons on polarized nucleons and an illustration of the non-perturbative power of lattice QCD 25 years later. The second part is a short pedagogical introduction to the analysis of high energy scattering in the complex angular momentum plane, with particular emphasis on spin-dependent deep inelastic electron-nucleon scattering. The third part comprises a brief introduction to lattice QCD and its applications in the non-perturbative determination of the spin-dependent structure functions.

Rotational bands and surface waves in α−40Ca elastic scattering

The scattering of heavy ions gives clear evidence of resonances which may be grouped in families like the rotational bands. The classical Breit-Wigner theory makes use of fixed poles which describe locally the resonances, but the global character of the rotational sequences is completely lost. Furthermore, the phenomenology shows that the rotational sequences evolve into surface waves. Again the classical Breit-Wigner theory, in view of its local character, cannot describe this evolution. In this paper we describe the resonances by the use of poles of the scattering amplitude in the complex angular-momentum plane: moving poles. However, in order to interpolate a sequence of resonances belonging to the same family we must add to the poles a term which takes into account the repulsive forces due to the Pauli principle and to the hard core. This term describes the downward crossing, through {pi}/2, of the phase shifts after each resonance. At higher energies the effect of the exchange forces tends to vanish and simultaneously the resonances evolve towards diffractive effects: we have the surface waves creeping around the target. This phenomenon is described in our theory by the moving poles as the imaginary part of the angular momentum increases for increasingmore » energy. Besides a detailed study of this theory we present here an extensive analysis of the {alpha}-{sup 40}Ca elastic scattering which gives clear phenomenological support to the model. {copyright} {ital 1997} {ital The American Physical Society}« less

CHN, QCD, and $$\overline {SA} $$ (4,R)

“CHN≓ (1966)was an algebraic algorithm which reproduced and extended the predictions of the “non-interacting≓ quark model in the asymptotic high-energy region. It wus formulated within the conceptual framework of on- mass- shell physics and of the complex angular-momentum plane. Prior to the advent of the standard model, it was reinterpreted in terms of the Melosh transformation relating “current≓ to “constituent≓ quarks. It is now lied up to the QCD paradigm.

Bethe ansatz for QCD pomeron

The equivalence is found between high-energy QCD in the generalized leading logarithmic approximation and the one-dimensional Heisenberg magnet. According to Regge theory, the high-energy asymptotics of hadronic scattering amplitudes are related to singularities of partial waves in the complex angular momentum plane. In QCD, the partial waves are determined by nontrivial two-dimensional dynamics of the transverse gluonic degrees of freedom. The “bare” gluons interact with each other to form a collective excitation, the Reggeon. The partial waves of the scattering amplitude satisfy the Bethe-Salpeter equation whose solutions describe the color singlet compound states of Reggeons - Pomeron, Odderon and higher Reggeon states. We show that the QCD Hamiltonian for reggeized gluons coincides in the multi-color limit with the Hamiltonian of XXX Heisenberg magnet for spin s = 0 and spin operators being the generators of the conformal SL(2,C) group. As a result, the Schrödinger equation for the compound states of Reggeons has a sufficient number of conservation laws to be completely integrable. A generalized Bethe ansatz is developed for the diagonalization of the QCD Hamiltonian and for the calculation of hadron-hadron scattering. Using the Bethe Ansatz solution of high-energy QCD we investigate the properties of the Reggeon compound states which govern the Regge behavior of the total hadron-hadron cross sections and the small-x behavior of the structure functions deep inelastic scattering.