Asymptotic Amplitude Research Articles

Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one-dimensional integrable lattice system of ODEs for complex functions Qn(�) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Qn(�) ∼ qexp(i(kn+!� + c)), in which q, k, and ! modulate on long time and long space scales t = and x = n. Such solutions arise from initial data of the form Qn(0) = q(n) exp(i�(n)/), the phase derivative � ′ giving the local value of the phase difference k. Formal asymptotic analysis as → 0 yields a first-order system of PDEs for q and � ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of t he solution in the case of initial data |Qn| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax-Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann-Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. c 1999 John Wiley & Sons, Inc.

Neutron reflectivity studies of critical adsorption: the correspondence between a critical adsorption profile and specular neutron reflection.

For interfacial order parameter profiles which decay as Az(-mu), such as the composition profile of the noncritical interface of a binary liquid mixture at a critical end point, there is a reported one-to-one correspondence between the profile and the reflectivity which can be described by an analytical theory [S. Dietrich and R. Schack, Phys. Rev. Lett. 58, 140 (1987)]. Evidence for mu=0.53+/-0.02 for adsorption at the hydrophilic silicon/liquid surface of the mixture (2-butoxyethanol+deuterium oxide) near its lower critical end point, determined from neutron reflectivity measurements, is presented. This value is in good agreement with the theoretical prediction of mu=0.516+/-0.004. Further examination of the data permits the determination of the asymptotic surface enrichment scaling factor amplitude P0 approximately 0.11, which is not in agreement with the theoretical value P(0) = 0.94+/-0.05 and values determined by other experimental methods.

Pion form factors with improved infrared factorization

We calculate electromagnetic pion form factors with an analytic model for α s( Q 2) which is infrared (IR) finite without invoking a “freezing” hypothesis. We show that for the asymptotic pion distribution amplitude, F π 0γ ∗γ agrees well with the data, whereas the IR-enhanced hard contribution to F π and the soft (nonfactorizing) part can jointly account for the data.

Coulomb effects on electromagnetic pair production in ultrarelativistic heavy-ion collisions

We calculate the asymptotic high-energy amplitude for electrons scattering at one ion, as well as at two colliding ions, by means of perturbation theory. We show that the interaction with one ion eikonalizes and that the interaction with two ions causally decouples. We are able to put previous results on perturbative grounds and propose further applications for the obtained rules for interactions on the light cone. We discuss the implications of the eikonal amplitude on the pair production probability in ultrarelativistic peripheral heavy-ion collisions. In this context the Weizs\"acker-Williams method is shown to be exact in the ultrarelativistic limit, irrespective of the produced particles' mass. A new equivalent single-photon distribution is derived, which correctly accounts for Coulomb distortions. The impact on single-photon induced processes is discussed.

Real-time relaxation and kinetics in hot scalar QED: Landau damping

The real time evolution of field condensates with soft length scales k^{-1}>(eT)^{-1} is solved in hot scalar electrodynamics, with a view towards understanding relaxational phenomena in the QGP and the electroweak plasma. We find that transverse gauge invariant non-equilibrium expectation values of fields relax via {\em power laws} to asymptotic amplitudes that are determined by the quasiparticle poles. The long time relaxational dynamics and relevant time scales are determined by the behaviour of the retarded self-energy not at the small frequencies, but at the Landau damping thresholds. This explains the presence of power laws and not of exponential decay. Furthermore, we derive the influence functional, the Langevin equation and the fluctuation-dissipation theorem for the soft modes, identifying the correlation functions that emerge in the classical limit. We show that a Markovian approximation fails to describe the dynamics {\em both} at short and long times. We also introduce a novel kinetic approach that goes beyond the standard Boltzmann equation and incorporates off-shell processes and find that the distribution function for soft quasiparticles relaxes with a power law through Landau damping. We also find an unusual dressing dynamics of bare particles and anomalous (logarithmic) relaxation of hard quasiparticles.

THE RUNNING COUPLING METHOD WITH NEXT-TO-LEADING ORDER ACCURACY AND PION, KAON ELM FORM FACTORS

The pion and kaon electromagnetic form factors FM(Q2) are calculated at the leading order of pQCD using the running coupling constant method. In calculations, the leading and next-to-leading order terms in αS((1-x)(1-y)Q2) expansion in terms of αS(Q2) are taken into account. The resummed expression for FM(Q2) is found. Results of numerical calculations for the pion (asymptotic distribution amplitude) are presented.

Lattice φ4 Theory of Finite-Size Effects Above the Upper Critical Dimension

We present a perturbative calculation of finite-size effects near Tc of the φ4 lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d&gt;4. The structural differences between the φ4 lattice theory and the φ4 field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters. One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite ξ/L where ξ is the bulk correlation length. At Tc, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to Tc of the lattice model, such as T max (L) of the maximum of the susceptibility χ, are found to scale asymptotically as Tc-T max (L) ~L-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ max ~Ld/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to Tc have not been observed in previous MC simulations at d=5 because of nonnegligible finite-size terms ~L(4-d)/2 caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the L(4-d)/2 and L4-d terms predicted by our theory.

Model of hydrodynamic perturbation growth in the start-up phase of laser implosion

A simple analytical model is presented to study hydrodynamic perturbation growths driven by nonuniform laser ablation in the start-up phase in laser fusion. Propagation of a rippled shock and deformation of an ablation surface are studied for cases of initial target roughness and nonuniform laser irradiation. The study of the perturbation growth in the start-up phase is very important because it seeds the Rayleigh-Taylor instability in the subsequent acceleration and stagnation phases. Analytical solutions are obtained for temporal evolutions of the shock front ripple and the ablation surface deformation. As a result, it is seen that the shock front ripples oscillate and decay in both cases. On the other hand, there is an asymptotic amplitude of the ablation surface deformation in the case of uniform laser irradiation on a target with a rippled surface, and an asymptotic growth rate of the ablation surface ripple in the case of nonuniform laser irradiation on a smooth target. In both cases, it can be shown that a high intensity of laser irradiation causes the ablation surface to distort, and a short wavelength laser inhibits its deformation. Approximate formulas expressing the temporal behaviors of the shock front and the ablation surface are obtained in the weak shock limit. Those formulas are also applicable to a relatively strong shock. Analytical results agree quite well with recent experimental data for the shock front ripple and the areal mass density perturbation in the initial target roughness case. The behaviors of the shock front ripple and the ablation surface deformation are also investigated in the case where the nonuniformity of the laser irradiation oscillates with time. It can be seen that the deformation of the ablation surface is inhibited for a high oscillation frequency of the laser nonuniformity.

Double-crossing mode conversion in nonuniform media

A new type of mode conversion between two linear waves in a nonuniform medium is investigated. Single-crossing conversion occurs when a ray of one wave crosses transversely the dispersion manifold of another wave. Double-crossing mode conversion theory describes when the ray punctures the dispersion manifold twice due to ray curvature. To study this new process, a one dimensionally nonuniform background medium is considered, which gives rise to four scenarios. These scenarios are distinguished on the basis of whether the two waves have equal or opposite energy signs, and whether they are copropagating or counterpropagating. Using modular-eikonal theory (suitable for multiple crossings), each scenario is first studied analytically by constructing an S-matrix relation between the outgoing and incoming asymptotic wave amplitudes. The analytical results are then compared with numerical results and excellent agreement is found.

Asymptotic flame shapes and speeds of hydrodynamically unstable laminar flames

The self-induced baroclinic instability of flames, the Landau-Darrieus instability, is studied numerically in the nonlinear range. A level set (G equation) based approach accounting for heat-release effects is used to follow the flame response to initial perturbation and shape evolution. It is shown that the instability leads to the development of product bubbles moving into the unburned mixture and cold mixture spikes penetrating into the burned gases similar to the bubble-spike configuration produced by the nonlinear Rayleigh-Taylor instability of interfaces in a gravitational field. Independently of the initial perturbation, the flame bubble approaches asymptotically a shape close to a paraboloid. The substantial growth of the flame-surface area due to the instability increases the flame propagation speed to the asymptotic value; + 0.29 α − 1 ) S l where α is the density ratio and Sl the laminar burning velocity. The asymptotic amplitude of the flame is approximately A = 0.37 d α − 1 , where d is the flame width. The burning velocity has a minor effect on the asymptotic shape of the flames. When the turbulence scale is much smaller than the size of the combustion apparatus, the results can be directly applied to turbulent flames by replacing S1 by the turbulent burning velocity.

Approximating infinite-k representations: surface relaxations and work functions of Al(001) and Be(0001)

Self-consistent techniques, such as the linearized augmented plane wave (LAPW) and surface embedding Green function (SEGF) methods, frequently yield calculated quantities which show damped oscillatory behaviour as a function of the number of special k-points. In the present work, the oscillatory dependence has been fitted to a damped harmonic oscillator in order to approximate the value corresponding to an infinite number of k-points in the Brillouin zone. The asymptotic value, amplitude, frequency, and phase are determined as functions of the damping constant by matching the value and derivative of the damped harmonic oscillator at consecutive turning points. It is shown that the asymptotic value of the damped oscillator may fall within the rms error of the asymptotic value of the fitting function. Results are reported for the surface relaxations and work functions of Al(001) and Be(0001).

Transverse momentum and Sudakov effects in exclusive QCD processes:γ*γπ0form factor

We analyze effects due to transverse degrees of freedom in QCD calculations of the fundamental hard exclusive amplitude of $\gamma^*\gamma \to \pi^0$ transition. A detailed discussion is given of the relation between the modified factorization approach (MFA) of Sterman et al. and standard factorization (SFA). Working in Feynman gauge, we construct basic building blocks of MFA from the one-loop coefficient function of the SFA, demonstrating that Sudakov effects are distinctly different from higher-twist corrections. We show also that the handbag-type diagram, contrary to naive expectations, does not contain an infinite chain of $(M^2/Q^2)^n$ corrections: they come only from diagrams with transverse gluons emitted from the hard propagator. A simpler picture emerges within the QCD sum rule approach: the sum over soft $\bar q G ... G q$ Fock components is dual to $\bar qq$ states generated by the local axial current. We combine the results based on QCD sum rules with pQCD radiative corrections and observe that the gap between our curves for the asymptotic and CZ distribution amplitudes is sufficiently large for an experimental discrimination between them.

Scaling behaviour of magnetization for temperatures in the vicinity of, and far from, the ferromagnetic - paramagnetic phase transition in amorphous and alloys

Asymptotic critical exponents and amplitudes as well as the leading `correction-to-scaling' (CTS) amplitudes have been accurately determined through an elaborate analysis of magnetization data taken on amorphous and alloys in the critical region. Consistent with the Harris criterion, asymptotic critical exponents and the universal amplitude ratio do not depend on composition and possess values the same as those predicted by theory for an ordered spin system with n = d = 3. The leading amplitude ratio , which is characteristic of ferromagnets with quenched random disorder and for which no theoretical estimate is presently available, is composition independent and probably universal. The fraction of spins actually participating in the ferromagnetic (FM) - paramagnetic (PM) transition occurring at is small and increases with Co substitution. While the magnetic equation of state (MES) in linear scaling variables and its counterpart in nonlinear scaling variables, valid for a second-order phase transition, form equivalent descriptions of magnetization, M(T,H), data in the asymptotic critical region (ACR), the latter version of MES alone reproduces closely the observed M(T,H) behaviour in a temperature range as wide as . Nonanalytic CTS terms dominate over analytic ones in the ACR but the reverse is true for temperatures outside the ACR. Initial susceptibility follows the generalized Curie - Weiss law from to and thereby permits an accurate determination of atomic moment in the PM state. The results of the present investigation provide strong experimental evidence for weak itinerant ferromagnetism in the glassy alloys in question.

On the Asymptotic Critical Behavior of Fe17Dy2, Fe17Ho2 and Fe17Er2 Single Crystals

The asymptotic critical behavior of Fe17Dy2, Fe17Ho2 and Fe17Er2 single crystals is investigated by deducing the asymptotic and correction-to-scaling critical exponents from detailed magnetic polarization measurements performed in the vicinity of the Curie temperature, where these systems undergo a magnetic phase transition from a ferrimagnetic to a paramagnetic state. These intermetallic compounds are characterized by a planar anisotropy for which, so far, measurements of critical exponents are not available. Well-defined phase transitions are found to exist in all these three systems with values of the asymptotic critical exponents very close to those predicted by the renormalization group calculations for a three-dimensional isotropic short-range Heisenberg ferromagnet. The nonanalytic corrections to the singular behavior at the Curie point, which originate from the nonlinear irrelevant scaling fields were found to have a significant role in deducing the true asymptotic critical amplitudes. Since the main interest of the present study lies only in the critical region, we have, for the time being, not investigated about these correction terms outside the critical region.

Eye-head coordination during head-unrestrained gaze shifts in rhesus monkeys.

We analyzed gaze shifts made by trained rhesus monkeys with completely unrestrained heads during performance of a delayed gaze shift task. Subjects made horizontal, vertical, and oblique gaze shifts to visual targets. We found that coordinated eye-head movements are characterized by a set of lawful relationships, and that the initial position of the eyes in the orbits and the direction of the gaze shift are two factors that influence these relationships. Head movements did not contribute to the change in gaze position during small gaze shifts (<20 degrees) directed along the horizontal meridian, when the eyes were initially centered in the orbits. For larger gaze shifts (25-90 degrees), the head contribution to the gaze shift increased linearly with increasing gaze shift amplitude, and eye movement amplitude saturated at an asymptotic amplitude of approximately 35 degrees. When the eyes began deviated in the orbits contralateral to the direction of the ensuing gaze shift, the head contributed less and the eyes more to amplitude-matched gaze shifts. The relative timing of eye and head movements was altered by initial eye position; head latency relative to gaze onset increased as the eyes began in more contralateral initial positions. The direction of the gaze shift also affected the relative amplitudes of eye and head movements; as gaze shifts were made in progressively more vertical directions, eye amplitude increased and head contribution declined systematically. Eye velocity was a saturating function of gaze amplitude for movements without a head contribution (gaze amplitude <20 degrees). As head contribution increased with increasing gaze amplitude (20-60 degrees), peak eye velocity declined by >200 degrees/s and head velocity increased by 100 degrees/s. For constant-amplitude eye movements (approximately 30 degrees), eye velocity declined as the velocity of the concurrent head movement increased. On the basis of these relationships, it is possible to accurately predict gaze amplitude, the amplitudes of the eye and head components of the gaze shift, and gaze, eye, and head velocities, durations and latencies if the two-dimensional displacement of the target and the initial position of the eyes in the orbits are known. These data indicate that signals related to the initial positions of the eyes in the orbits and the direction of the gaze shift influence separate eye and head movement commands. The hypothesis that this divergence of eye and head commands occurs downstream from the superior colliculus is supported by recent electrical stimulation and single-unit recording data.

A Delay Equation Representation of Pulse Circulation on a Ring in Excitable Media

This paper develops a theory for pulse circulation on a ring in a continuous excitable medium. Simulations of a partial differential equation (PDE) modeling propagation of electrical pulses on a one-dimensional ring of cardiac tissue are presented. The dynamics of the circulating pulse in this excitable medium are reduced to a single integral-delay equation. Stability conditions for steady circulation are obtained, and estimates are derived for the wavelength, growth rate, and asymptotic amplitude of oscillating solutions near the transition from steady rotation to oscillatory pulse dynamics. The analytical results agree with simulations of the delay equation and the PDE model and uncover previously uncharted solutions of the PDE equations.

Light-front heavy-quark effective theory and heavy-meson bound states.

The heavy quark effective theory is developed on the light-front. Based on this effective theory, a light-front heavy meson bound state with definite spin and parity is constructed. Within the effective theory, the Isgur-Wise function is derived in terms of the asymptotic light-front bound state amplitudes in the limit $m_Q\rightarrow\infty$; the result is a general expression for arbitrary recoil velocities. With the asymptotic form of the BSW amplitudes, the Isgur-Wise function is given by $\xi(v\cdot v')=1/v\cdot v'$. The slope at the zero-recoil point is $\rho^2=-\xi'(1)=1$, in excellent agreement with the recent CLEO result of $\rho^2=1.01\pm0.15\pm0.09$.

Determination of asymptotic critical exponents and amplitudes for amorphous 3D ferromagnets from bulk magnetization measurements

High-resolution bulk magnetization data have been taken in the critical region (−0.05 ≤ ϵ = ( T − T C)/ T C ≤ 0.05, where T C is the Curie temperature) on amorphous Fe x Ni 80- x P 14B 6 alloys with x = 20, 30 and 40. An elaborate data analysis, besides yielding accurate values for the spontaneous magnetization and initial susceptibility asymptotic (leading ‘correction-to-scaling’) critical exponents and amplitudes (amplitudes), reveals that the values of the asymptotic critical exponents and universal amplitude ratios m o/ M S( O) and Dm o δ / h o are composition-independent and exactly equal to those for the pure (ordered) spin system with space as well as spin dimensionality of three. However, the main difference between crystalline (ordered) and amorphous (quench-disordered) three-dimensional Heisenberg ferromagnets is that nearly all the spins participate in the ferromagnetic (FM)-paramagnetic (PM) phase transition in the former case while only a small fraction of spins is actually involved in such a transition in the latter case. This fraction reduces drastically as the percolation threshold for the long-range FM order is approached along the FMPM phase transition line in the magnetic phase diagram. Asymptotic critical exponents β, γ and δ obey Widom scaling relation, βδ = β + γ, and the magnetization data satisfy the scaling equation of state for second-order phase transition in the asymptotic critical region.

Bound-state effects in B-->K* gamma.

Applying perturbative QCD methods recently seen to give a good description of the two body hadronic decays of the B meson, we address the question of bound-state effects on the decay B into K* + gamma. Consistent with most analyses, we demonstrate that gluonic penguins, with photonic bremsstrahlung off a quark, change the decay rate by only a few percent. However, explicit off-shell b-quark effects normally discarded are found to be large in amplitude, although in the standard model accidents of phase minimize the effect on the rate. Using an asymptotic distribution amplitude for the K* and just the standard model, we can obtain a branching ratio of a few times 10^{-5}, consistent with the observed rate.

Complete unbinding of fluid membranes in the presence of short-ranged forces.

Lipid or surfactant bilayers which are bound by an external pressure and intereact via an additional short-ranged potential are studied theoretically. If the latter potential is not strong enough to bind the lamellae by itself, it has asymptotically no effect on the (complete) unbinding transition, which occurs in the limit of vanishing pressure; the separation and correlation lengths diverge as power laws as a function of the pressure, with the amplitudes being determined by characteristic amplitude ratios. If the potential strength exceeds the critical value, the bilayers are bound even for zero external pressure (incomplete unbinding). Exactly at the critical potential strength, all length scales again diverge as a function of the pressure. The critical exponents are found to be identical to those for a less attractive potential, but the asymptotic amplitude ratios have different values; also, the fluctuation amplitude, which measures the strength of the fluctuation-induced repulsion between the bilayers, is reduced by a factor of 12 as compared to the subcritical case. These results are obtained directly by Monte Carlo simulations of two fluid membranes and agree with exact calculations for the analogous system of two strings in 1+1 dimensions. Experimentally, the effects of short-ranged van der Waals attraction on the fluctuation amplitude ${\mathit{c}}_{\mathrm{fl}}$ should be observable for suitable systems by small-angle x-ray scattering on lamellar phases.