Complex Angular Momentum Techniques Research Articles

Electromagnetic radiation generated by a charged particle falling radially into a Schwarzschild black hole: A complex angular momentum description

By using complex angular momentum techniques, we study the electromagnetic radiation generated by a charged particle falling radially from infinity into a Schwarzschild black hole. We consider both the case of a particle initially at rest and that of a particle projected with a relativistic velocity and we construct complex angular momentum representations and Regge pole approximations of the partial wave expansions defining the Maxwell scalar $\phi_2$ and the energy spectrum $dE/d\omega$ observed at spatial infinity. We show, in particular, that Regge pole approximations involving only one Regge pole provide effective resummations of these partial wave expansions permitting us (i) to reproduce with very good agreement the black hole ringdown without requiring a starting time, (ii) to describe with rather good agreement the tail of the signal and sometimes the pre-ringdown phase, and (iii) to explain the oscillations in the electromagnetic energy spectrum radiated by the charged particle. The present work as well as a previous one concerning the gravitational radiation generated by a massive particle falling into a Schwarzschild black hole [A. Folacci and M. Ould El Hadj, Phys. Rev. D$\textbf{98}$, 064052 (2018), arXiv:1807.09056 [gr-qc]] highlight the benefits of studying radiation from black holes in the complex angular momentum framework (they obviously appear when the approximations obtained involve a small number of Regge poles and have a clear physical interpretation) but also to exhibit the limits of this approach (this is the case when it is necessary to take into account background integral contributions).

Scattering from compact objects: Regge poles and the complex angular momentum method

We calculate the Regge poles of the scattering matrix for a gravitating compact body, for scalar fields and for gravitational waves in the axial sector. For a neutron-starlike body, the spectrum exhibits two distinct branches of poles, labeled surface waves and broad resonances; for ultracompact objects, the spectrum also includes a finite number of narrow resonances. We show, via a WKB analysis, that the discontinuity of the effective potential at the body's surface determines the imaginary component of the broad-resonance poles. Next, we examine the role of Regge poles in the time-independent scattering of monochromatic planar waves. We apply complex angular momentum techniques to re-sum the partial wave series for the scattering amplitude, expressing it as a residue series evaluated at poles in the first quadrant, accompanied by a background integral. We compute the scattering cross section at several frequencies, and show precise agreement with the partial-wave calculations. Finally, we show that compact bodies naturally give rise to orbiting, glory, and rainbow-scattering interference effects.

Regge pole description of scattering of gravitational waves by a Schwarzschild black hole

We revisit the problem of plane monochromatic gravitational waves impinging upon a Schwarzschild black hole using complex angular momentum techniques. By extending our previous study concerning scalar and electromagnetic waves [A. Folacci and M. Ould El Hadj, Phys.\, Rev.\, D {\bf 99}, 104079, (2019), arXiv:1901.03965], we provide complex angular momentum representations and Regge pole approximations of the helicity-preserving and helicity-reversing scattering amplitudes and of the total differential scattering cross section. We show, in particular, that for high frequencies (i.e., in the short-wavelength regime), a small number of Regge poles permits us to describe numerically with very good agreement the black hole glory and the orbiting oscillations and we then provide a semiclassical approximation that unifies these two phenomena.

Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole

We revisit the problem of scalar and electromagnetic waves impinging upon a Schwarzschild black hole from complex angular momentum techniques. We focus more particularly on the associated differential scattering cross sections. We derive an exact representation of the corresponding scattering amplitudes by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the complex angular momentum plane plus a sum over the Regge poles of the $S$-matrix involving the associated residues. We show that, surprisingly, the background integral is numerically negligible for intermediate and high frequencies and, as a consequence, that the cross sections can be reconstructed in terms of Regge poles with very good agreement. We show in particular that, for large values of the scattering angle, a small number of Regge poles permits us to describe the black hole glory and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the orbiting oscillations. In fact, in the short-wavelength regime, the sum over Regge poles allows us to extract by resummation the information encoded in the partial wave expansion defining a scattering amplitude and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the black hole. As a consequence, from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles -- ``surface waves'' propagating close to the photon sphere, we can provide an analytical approximation describing both the black hole glory and a large part of the orbiting oscillations. We finally discuss the role of the background integral for low frequencies.

Alternative description of gravitational radiation from black holes based on the Regge poles of the S -matrix and the associated residues

We advocate for an alternative description of gravitational radiation from black holes based on complex angular momentum techniques (analytic continuation of partial wave expansions, duality of the ${\cal S}$-matrix and effective resummations involving its Regge poles and the associated residues, Regge trajectories, semiclassical interpretations, etc.). Such techniques, which proved to be very helpful in various areas of physics to describe and analyze resonant scattering, were only marginally used in the context of black hole physics. Here, by considering the multipolar waveform generated by a massive particle falling radially from infinity into a Schwarzschild black hole, we show that they could play a fundamental role in gravitational-wave physics. More precisely, from the multipole expansion defining the Weyl scalar $\Psi_4$, we extract the Fourier transform of a sum over Regge poles and their residues which can be evaluated numerically from the associated Regge trajectories. This Regge pole approximation permits us to reconstruct, for an arbitrary direction of observation, a large part of the multipolar waveform $\Psi_4$. In particular, it can reproduce with very good agreement the quasinormal ringdown as well as with rather good agreement the tail of the signal. This is achieved even if we take into account only one Regge pole and, if a large number of modes are excited, the result can be improved by considering additional poles. Moreover, while quasinormal-mode contributions do not provide physically relevant results at "early times" due to their exponentially divergent behavior as time decreases, it is not necessary to determine from physical considerations a starting time for the Regge ringdown.

Strong gravitational lensing and black hole quasinormal modes: towards a semiclassical unified description

We examine in a semiclassical framework the deflection function of strong gravitational lensing, for static and spherically symmetric black holes, endowed with a photon sphere. From a first-order WKB analysis near the maximum of the Regge-Wheeler potential, we extract the real phase shifts from the S-matrix elements and then we derive the associated semiclassical deflection function, characterized by a logarithmic divergent behavior. More precisely, using the complex angular momentum techniques, we show that the Regge poles and the associated greybody factor residues, for a massless scalar field theory, from which one can recover the black hole quasinormal complex frequencies as well as the fluctuations of the high energy absorption cross section, play naturally the role of critical parameters in the divergent behavior of the semiclassical deflection function. For very high frequencies, we finally recover the logarithmic part of the classical strong deflection limit, which clarifies analytically the fundamental link between quasinormal modes and strong gravitational lensing, suggested in recent works.

Resonance and absorption spectra of the Schwarzschild black hole for massive scalar perturbations: A complex angular momentum analysis

We reexamine some aspects of scattering by a Schwarzschild black hole in the framework of complex angular momentum techniques. More precisely, we consider, for massive scalar perturbations, the high-energy behavior of the resonance spectrum and of the absorption cross section by emphasizing analytically the role of the mass. This is achieved (i) by deriving asymptotic expansions for the Regge poles of the $S$-matrix and then for the associated weakly damped quasinormal frequencies and (ii) by taking into account the analytic structure of the greybody factors which allows us to extract by resummation the physical information encoded in the absorption cross section.

Surface plasmon polaritons and surface phonon polaritons on metallic and semiconducting spheres: Exact and semiclassical descriptions

We study the interaction of an electromagnetic field with a non-absorbing or absorbing dispersive sphere in the framework of complex angular momentum techniques. We assume that the dielectric function of the sphere presents a Drude-like behavior or an ionic crystal behavior modelling metallic and semiconducting materials. We more particularly emphasize and interpret the modifications induced in the resonance spectrum by absorption. We prove that ``resonant surface polariton are generated by a unique surface wave, i.e., a surface (plasmon or phonon) polariton, propagating close to the sphere surface. This surface polariton corresponds to a particular Regge pole of the electric part (TM) of the $S$ matrix of the sphere. 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 the 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 dispersion relation and its damping in the non-absorbing and absorbing cases. In these analytic expressions, we more particularly exhibit well-isolated terms directly linked to absorption. Finally, we explain why the photon-sphere system can be considered as an artificial (a ``plasmonic atom or ``phononic atom) and we briefly discuss the implication of our results in the context of the Casimir effect.

Surface polaritons on left-handed spheres

We consider the interaction of an electromagnetic field with a sphere, i.e., with a sphere fabricated from a material, in the framework of complex angular momentum techniques. We emphasize more particularly, from a semiclassical point of view, the resonant aspects of the problem linked to the existence of surface polaritons. We prove that the long-lived resonant modes can be classified into distinct families, each family being generated by one surface polariton propagating close to the sphere surface and we physically describe all the surface polaritons by providing, for each one, its dispersion relation and its damping. This can be achieved by noting that each surface polariton corresponds to a particular Regge pole of the electric part (TM) or the magnetic part (TE) of the $S$ matrix of the sphere. Moreover, for both polarizations, we find that there exists a particular surface polariton which corresponds, in the large radius limit, to that supported by the plane interface. There also exists, for both polarizations, an infinite family of surface polaritons of whispering gallery type having no analogs in the plane interface case and specific to materials. They present a left-handed behavior (phase and group velocities are opposite) as well as a very weak damping. They could be very useful in the context of plasmonics or cavity quantum electrodynamics.

Surface polaritons on left-handed cylinders: A complex angular momentum analysis

We consider the scattering of electromagnetic waves by a left-handed cylinder -- i.e., by a cylinder fabricated from a left-handed material -- in the framework of complex angular momentum techniques. We discuss both the TE and TM theories. We emphasize more particularly the resonant aspects of the problem linked to the existence of surface polaritons. We prove that the long-lived resonant modes can be classified into distinct families, each family being generated by one surface polariton propagating close to the cylinder surface and we physically describe all the surface polaritons by providing, for each one, its dispersion relation and its damping. This can be realized by noting that each surface polariton corresponds to a particular Regge pole of the $S$ matrix of the cylinder. Moreover, for both polarizations, we find that there exists a particular surface polariton which corresponds, in the large-radius limit, to the surface polariton which is supported by the plane interface. There exists also an infinite family of surface polaritons of whispering-gallery type which have no analogs in the plane interface case and which are specific to left-handed materials.

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.

Complex angular momentum analysis of resonance scattering in the Cl+HCl→ClH+Cl reaction

Resonance effects in the differential cross sections of the Cl+HCl(v,j)→ClH(v′,j′)+Cl reaction are analyzed using Regge pole and complex angular momentum (CAM) techniques. This is the first detailed application of CAM theory to reactive molecular scattering. The rovibrational transitions studied are v=1, j=5→v′=0, j′=15, and v=1, j=5→v′=1, j′=5 at total energies E=0.66, 0.68, 0.70 eV. The CAM theory expresses the scattering amplitude as a background subamplitude plus a pole subamplitude. The uniform (and nonuniform) semiclassical evaluation of the background subamplitude is discussed. It is necessary to include explicitly the resonance Regge pole in the semiclassical theory because it has a small imaginary part. We derive a new generic semiclassical formula, involving the complementary error function for the resonance angular scattering. The position and residue of the resonance Regge pole at each E are extracted numerically from scattering matrix elements calculated by the centrifugal sudden hyperspherical (CSH) quantum scattering method. There is good agreement between the semiclassical CAM and CSH angular distributions. However, the latter involve summing a partial wave (PW) series with a large number of numerically significant terms—as a result the PW computations provide no physical insight. We also show that a simple semiclassical optical model becomes inaccurate when the rotational period of the ClHCl complex is comparable to the resonance lifetime. We derive a new ‘‘sticky’’ optical model which allows for rotation of the complex. All our calculations use the Bondi–Connor–Manz–Römelt semiempirical potential energy surface.

New uniform semiclassical theory of resonance angular scattering for reactive molecular collisions

The angular distribution of a reactive molecular collision is analysed when a resonance is present. A new generic semiclassical formula involving the complementary error function is derived for the scattering amplitude. Powerful complex angular momentum techniques are applied to the theory of chemical reactions for the first time. Results are reported for the Cl + HCl → ClH + Cl reaction.

Complex angular momentum theory of molecular collisions: New phase rules for rotationally inelastic diffraction scattering in atom–homonuclear diatomic molecule collisions

New phase rules have been derived for the rotationally inelastic diffraction scattering of an atom with a homonuclear diatomic molecule. The phase rules are obtained by applying semiclassical complex angular momentum techniques to the infinite order sudden approximation. The new phase rules are of three types, which are denoted weak coupling, intermediate coupling, and strong coupling. Higher-order terms have been derived for each type of phase rule. A systematic computational study of the accuracy of the phase rules has been carried out for a deformed inverse power potential energy of the form ε[r(γ)/r]n, n≥3, where r(γ)=r0+r2P2(cos γ). Additional calculations are reported for the Ne–D2 and He–N2 collision systems, using more realistic interaction potential energies.

Field correlations within a Bessel-correlated spherical source

The spatial field correlations within a Bessel-correlated spherical source with large size parameter (radius much greater than the wavelength) are investigated. Exact expressions for the field spectral density, in terms of a partial-wave series, as well as for the magnitude of the complex degree of spectral coherence (correlating the field at an arbitrary internal point with that at the center of the sphere), are obtained. Asymptotic approximations for a large size parameter are found by applying complex angular-momentum techniques. The results are discussed and plotted for a wide range of values of the ratio between the source spatial coherence length and the wavelength. The asymptotic approximations are found to be quite accurate. The field ranges from extreme incoherence to full spatial coherence. The latter is attained only when the source’s coherence length becomes much larger than its Fresnel radius. Results previously obtained in the limit of an unbounded source are verified. Features arising from finite size and boundary effects are discussed.

On the complex angular momentum theory of scattering

A contribution to the theory of complex angular momentum techniques in the field of atomic and molecular collisions is given. The author derives a new, flexible representation of the scattering amplitude on the basis of realistic assumptions for the behaviour of the S-matrix in the complex angular momentum plane. The representation has the form of a sum of steepest-descent integrals, S-matrix residue terms and a symmetry-type background integral. The flexibility is due to the presence of two integer parameters which may be chosen conveniently so as to make the residue sums sufficiently convergent and to minimise the total number of important terms.

Complex angular momentum analysis of diffraction scattering in atomic collisions

The application of complex angular momentum techniques to diffraction scattering in elastic atom–atom collisions is investigated. Two different semiclassical methods have been used. In the first method, complex saddle points associated with the semiclassical integral representation for the scattering amplitude are employed. The second method is the Regge pole approach to elastic scattering. The first calculations are reported in which the semiclassical uniform Airy approximation using complex angular momenta has been applied to the dark side of a rainbow. Good agreement with partial wave results is obtained for the uniform Airy and Regge pole theories, with the Regge pole approach the easier to apply. The accuracy of the transitional Airy and primitive semiclassical approximations has also been studied. The effect of neglecting the scattering from the repulsive core of the potential is investigated. This case arises in the elastic scattering of chemically reactive systems when strong absorption is present. In addition, the effect on diffraction scattering of damping out the attractive part of a Lennard-Jones potential has ben studied until just the repulsive core remains.

Calculation of angular distributions in complex angular momentum theories of elastic scattering

An important practical problem in the application of complex angular momentum techniques to atomic and molecular scattering is the evaluation of Legendre's function of the first kind of complex degree. An exact series representation, that can be summed numerically, has been derived using accelerated convergence techniques. The accuracy of various asymptotic representations for the Legendre function is also investigated. The use of these asymptotic formulae is shown to be valid for most applications of complex angular momentum techniques to atomic and molecular scattering. The asymptotic formulae have a physical interpretation involving backward glories and surface waves.

Regge poles and the Watson-Sommerfeld transformation in low-energy atomic collisions

The usefulness of complex angular momentum techniques is examined in the context of low-energy atomic collisions. For elastic e-Ar collisions at energies between 50 eV and 200 eV the observed structure in the differential cross sections is reproduced by the contribution from three Regge poles. For a Lennard-Jones potential representing p-Ar collisions the contributions from many poles are significant and the interference between these contributions leads to serious numerical difficulties in the application of the Watson-Sommerfeld transformation. The Remler model is shown to give a reliable phenomenological procedure for fitting measured differential cross sections for ion-atom collisions, even though the physical basis of the model is questionable.

Regge pole analysis of large angle elastic scattering in chemically reactive systems

Complex angular momentum techniques are applied to the theory of elastic scattering in chemically reactive systems. The Watson-Sommerfeld transformation allows the scattering amplitude to be written as a background integral and a sum over Regge Poles in the right-hand complex plane. It is shown the Regge Poles are restricted to the first quadrant for physically meaningful optical potentials. Neglect of the background integral and all poles except for the one closest to the real axis allows the differential cross section for large scattering angles to be written in terms of a Legendre function of complex degree. The theory agrees with experimental results for the elastic angular distributions of direct and complex reactions. It predicts oscillations and a glory in the backward direction of the angular distributions. The oscillations arise from interference between orbiting or surface waves that propagate around the region of chemical reaction. Measurements of angular distributions at different collision en...