Integrals In Momentum Space Research Articles

Renormalization group equations and the Lifshitz point in noncommutative Landau–Ginsburg theory

A one-loop renormalization group (RG) analysis is performed for noncommutative Landau–Ginsburg theory in an arbitrary dimension. We adopt a modern version of the Wilsonian RG approach, in which a shell integration in momentum space bypasses the potential IR singularities due to UV-IR mixing. The momentum-dependent trigonometric factors in interaction vertices, characteristic of noncommutative geometry, are marginal under RG transformations, and their marginality is preserved at one loop. A negative Θ-dependent anomalous dimension is discovered as a novel effect of the UV-IR mixing. We also found a noncommutative Wilson–Fisher (NCWF) fixed point in less than four dimensions. At large noncommutativity, a momentum space instability is induced by quantum fluctuations, and a consequential first-order phase transition is identified together with a Lifshitz point in the phase diagram. In the vicinity of the Lifshitz point, we introduce two critical exponents ν m and β k , whose values are determined to be 1/4 and 1/2, respectively, at mean-field level.

Liftshitz-point critical behaviour to O(ϵ2)

We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $\epsilon=4-d+\frac{m}{2}$ were presented for the critical exponents $\nu_{{\mathrm{L}}2}$, $\eta_{{\mathrm{L}}2}$ and $\gamma_{{\mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects.

Formulisms of path integrals in momentum space and in mixed spaces

We define the Lagrangians in the momentum space and in mixed position-momentum spaces. In terms of the so-defined Lagrangians, the actions and the propagators are directly defined in the spaces, not as the Fourier transforms of their counterparts in the configuration space as previously known in the literature.

Light-front-quantized QCD in a covariant gauge

The light-front (LF) canonical quantization of quantum chromodynamics in covariant gauges is discussed. The Dirac procedure is used to eliminate the constraints in the gauge-fixed front form theory quantum action and to construct the LF Hamiltonian formulation. The physical degrees of freedom emerge naturally. The propagator of the dynamical ${\ensuremath{\psi}}_{+}$ part of the free fermionic propagator in the LF quantized field theory is shown to be causal and not to contain instantaneous terms. Since the relevant propagators in the covariant gauge formulation are causal, rotational invariance---including the Coulomb potential in the static limit---can be recovered, avoiding the difficulties encountered in light-cone gauge. Wick rotation may also be performed allowing the conversion of momentum space integrals into Euclidean space forms. Some explicit computations are done in quantum electrodynamics to illustrate the equivalence of front form theory with the conventional covariant formulation. LF quantization thus provides a consistent formulation of gauge theory, despite the fact that the hyperplanes ${x}^{\ifmmode\pm\else\textpm\fi{}}=0$ used to impose boundary conditions constitute characteristic surfaces of a hyperbolic partial differential equation.

Variational solution of the single-particle Dirac equation in the field of two nuclei using relativistically adapted Slater basis functions

A variational method for solving the time-independent single-particle Dirac equation in the Coulomb field of two nuclei is described. A minimax variational principle and basis functions that have the proper analytic behavior, i.e. behave like rγ,γ non-integer, in the neighborhood of a nucleus, are used. A momentum space integration scheme for computing the necessary two-center integrals is described. Results are given for a standard test problem on two nuclei with Z=90 with an internuclear separation of 2.0/Z. The results confirm those of a previous calculation [F.A. Parpia and A.K. Mohanty, Chem Phys Lett 238: 209 (1995)].

Structure of the Pauli and Correlation-Kinetic Components of the Kohn–Sham Exchange Potential at a Metal Surface

According to the rigorous physical interpretation of Kohn–Sham (KS) density-functional theory in terms of the components of the true wavefunction, the KS exchange potentialνKSx(r)=δEKSx[ρ]/δρ(r), whereEKSx[ρ] is the exchange energy functional, is the work done to move an electron in a conservative fieldR(r). This field comprises a component EKSx(r) representative of Pauli correlations and anotherZ(1)tc(r) that constitutespartof the correlation contribution to the kinetic energy. The field EKSx(r) is derived via Coulomb's law from the KS Fermi hole charge, and the fieldZ(1)tc(r) from the kinetic-energy-density tensor involving the first-order correction to the KS single-particle density matrix. For systems in which the curls of these component fields separately vanish, the potentialvKSx(r) is the sum of the work doneWKSx(r) andW(1)tc(r) in the fields EKSx(r) andZ(1)tc(r), respectively. In this paper we study the structure of the workWKSx(r) andW(1)tc(r) at a simple-metal surface as represented by the jellium and structureless-pseudopotential models for which the workWKSx(r) andW(1)tc(r) are separately path-independent. A general expression for the field EKSx(r) is derived in terms of momentum-space integrals of the electron orbitals. This enables its easy determination, and thereby determination of the potentialWKSx(r). The field expression further allows for the derivation of theexact analyticalasymptotic structure of the potentialWKSx(r) in the vacuum region, a result valid for thefully self-consistentlydetermined orbitals of both models. With the exact analytical asymptotic structure ofvKSx(r) in the vacuum known, that of the potentialW(1)tc(r) in this region is then determinedanalytically. As is the case forvKSx(r) which decays asymptotically in the vacuum as −αKS/x, the potentialsWKSx(r) andW(1)tc(r) also decay as −αW/xandα(1)tc/x, respectively, the decay coefficients depending upon the metal Fermi energy and barrier height. It is further shown that for metallic densities,WKSx(r) does not approach thevKSx(r) asymptotic metal-bulk value of (−kF/π), so thatWtc(1)(r) is also finite in this region. Thus, at a metal surface, the KS exchange potentialvKSx(r) comprises principally its Pauli componentWKSx(r), with the correlation-kinetic partW(1)tc(r) beingfiniteandlong-rangedin both the vacuum and metal-bulk regions, its contribution diminishing with increasing metal density.

Macroscopic Einstein equations to second order in the interaction constant

The present paper is a direct continuation of an earlier paper [JETP 83, 1 (1996)] devoted to the derivation of the macroscopic Einstein equations to within terms of second order in the interaction constant. Ensemble averaging of the microscopic Einstein equations and the Liouville equation for the random functions leads to a closed system of macroscopic Einstein equations and kinetic equations for one-particle distribution functions. The macroscopic Einstein equations differ from the classical equations in that their left-hand side contains additional terms due to particle interaction. The terms are traceless tensors with zero divergence. An explicit covariant expression for these terms is given in the form of momentum-space integrals of expressions dependent on one-particle distribution functions of the interacting particles of the medium. The given expressions are proportional to the cube of the Einstein constant and the square of the particle number density. The latter relationship implies that interaction effects manifest themselves in systems of very high density (the universe in the early stages of its evolution, dense objects close to gravitational collapse, etc.)

One-loop tensor integrals in dimensional regularisation

We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of n- and ( n − 1)-point scalar integrals that are finite in the limit of vanishing Gram determinant. These non-trivial combinations of dilogarithms, logarithms and constants are systematically obtained by either differentiating with respect to the external parameters - essentially yielding scalar integrals with Feynman parameters in the numerator - or by developing the scalar integral in D = 6 − 2 ϵ or higher dimensions. An additional advantage is that other spurious kinematic singularities are also controlled. As an explicit example, we develop the tensor integrals and associated scalar integral combinations for processes where the internal particles are massless and where up to five (four massless and one massive) external particles are involved. For more general processes, we present the equations needed for deriving the relevant combinations of scalar integrals.

Non-relativistic non-dipole two-photon bremsstrahlung in a Coulomb field

We present analytical and numerical results on the cross section of the two-photon bremsstrahlung (2BrS) of a non-relativistic projectile scattered in a point Coulomb field. The effect of the radiation retardation is taken into account. The exact analytic evaluation of the point Coulomb two-photon non-dipole free - free matrix element, which characterizes the amplitude of the 2BrS process, is carried out following a method of momentum space integration developed by Gavrila. The obtained formulae are the generalization of the expressions derived earlier by several workers within the framework of the non-relativistic dipole approximation. Results of numerical calculations are compared with predictions of other theoretical approaches and with the experimental data on 2BrS.

Momentum-space diffusion due to resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma

The momentum-space diffusion equation and the kinetic wave equation for resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma are derived from the relativistic Vlasov–Maxwell equations by perturbation theory. The p-dependent diffusion coefficient and the nonlinear wave—wave coupling coefficient are given in terms of third-order tensors which are amenable to analysis. The transport equations describing energy and momentum transfer between waves and particles are obtained by momentum-space integration of the momentum-space diffusion equation, and are expressed in terms of the nonlinear wave—wave coupling coefficient in the kinetic wave equation. The conservation laws for the total energy and momentum densities of waves and particles are verified from the kinetic wave equation and the transport equations. These equations are very useful for the theoretical analysis of transport phenomena or the acceleration and generation of high-energy or relativistic particles caused by quasi-linear and resonant wave—wave scattering processes.

Excitation of electron Bernstein and ion Bernstein waves by extraordinary electromagnetic pump: Kinetic theory

In the present paper, parametric decay instability of an extraordinary electromagnetic (EM) wave into a high-frequency electron Bernstein wave and a low-frequency ion Bernstein wave is studied. The guiding center formalism is followed to obtain the coupling coefficients including kinetic effects. The momentum space integrals in the coupling coefficients are solved in the dipole approximation of the pump wave and in the limits, k⊥ρe<1 for electron Bernstein wave and k⊥ρi<1 for ion Bernstein wave. Analytical expressions for homogeneous growth rate and threshold are given explicitly. Applications of the present investigation are pointed out to ionospheric modification experiment and fusion plasmas.

A Gaussian quadrature for the optimal evaluation of integrals involving Lorentzians over a semi-infinite interval

A discretized Stieltjes procedure is used to construct orthogonal polynomials over the interval [0, ∞], based on weight functions of the form (1 + x 2) − n , where n is a positive integer. Gaussian quadrature formulae subsequently derived from these polynomials are used to numerically integrate functions with a Lorentzian decaying aspect. As a test case, we use these quadratures to evaluate the momentum expectation values of the helium atom and show that similar accuracy can be obtained with considerably less number of functional evaluations, as compared to other quadrature rules. The results obtained indicate that quadratures of this form should be well recommended for the numerical evaluation of momentum space integrals over [0, ∞] or more generally, for the integration of any function possessing a Lorentzian type behavior.

A new derivation of the photoelectric effect

A first order time dependent electronic wave function is obtained by summing contributing terms through complex integration in momentum space. The angular dependence of the current found from this wave function agrees exactly with known results for the surface effect, whereas the volume effect is found to be proportional to the second power of the penetration depth l.

Analytical expressions for the internal and kinetic energy distributions of sputtered clusters and molecules

Analytical expressions are presented for the internal and kinetic energy distributions of sputtered molecules and clusters. The molecules and clusters are treated as systems built up of two identical sub-units which are activated independently in the sputter process. First general expressions are obtained from approximate momentum space integrations. Then explicit functions are given for collision cascade and thermal sputtering. The quality of the analytical expressions for the collision cascade model is evaluated by comparison with a numerical calculation of the momentum space integrals.

Calculation of the energy-momentum tensor of the vacuum in an anisotropic universe

Subtractive methods (N-wave and adiabatic) which are applicable to the calculation of the energy-momentum tensor of quantum fields in curved space-time are in need of a foundation in terms of renormalizations. In the example of a scalar field in an anisotropic universe of Bianchi type I it is shown that the Pauli-Villars scheme, in which the renormalization is in fact realized separately in each mode, provides such a foundation. The technical difficulty obstructing the explicit regularization of divergent integrals in momentum space is shown. We calculate the polarization of the vacuum of a scalar field with arbitrary coupling to the curvature in a weakly anisotropic universe.

BPHZL-subtraction scheme and axial gauges

The application of the BPHZL subtraction scheme to Yang-Mills theories in axial gauges is presented. In the auxiliary mass formulation we show the validity of the convergence theorems for subtracted momentum space integrals, and we give the integral formulae necessary for one-loop calculations.

One-loop integrals in axial-type gauges

A general formula for momentum-space integrals containing one noncovariant denominator is derived. These denominators arise in ghost-free gauges such as the axial gauge. The method works in dimensionally regularized Minkowski space and employs Feynman parameters as well as the Wick rotation. The pole in the noncovariant denominator is circumvented by a general iepsilon prescription. With the final formula the principal value can also be evaluated. Several explicit examples are considered.

Dimensional regularization for zero-mass particles in quantum field theory

We show by a suitable redefinition of momentum-space integrals that the technique of dimensional regularization can be extended consistently to include nonnormally ordered theories describing zero-mass particles.

Electron Scattering by Pair Production in Silicon

The energy-dependent rate for inelastic scattering of electrons by production of electron-hole pairs is computed by first-order perturbation theory for silicon using a screened Coulomb interaction with a frequency- and momentum-dependent dielectric function calculated for silicon in the random-phase approximation. The threshold for momentum-conserving pair creation is found to be very close to that determined by energy conservation alone. This follows from the silicon band structure. The absolute scattering rate is close to that obtained experimentally by Bartelink, Moll, and Meyer. Pair production dominates the scattering rate for electrons of energy greater than 6.5 eV above the valence-band maximum. For electrons between 4 and 6.5 eV the scattering rate is dominated by phonon scattering, but energy loss is dominated by pair scattering. Below 4 eV phonon processes dominate inelastic processes as well. The momentum space integrals for pair-creation scattering are performed by a Monte Carlo method. It is found that the calculations with momentum conservation can be reproduced surprisingly well by a simple "random-k" approximation, which effectively ignores momentum conservation. This leads to a very simple expression for pair scattering in the form of a two-dimensional energy fold over one-electron state-density functions, a result first obtained by Berglund and Spicer. This simple form facilitates the calculation tremendously and should make calculations for other materials very simple if the state-density function is known. Using this "random-k" method, the scattering rate for primary holes is obtained and found to be almost identical with that for primary electrons of comparable energy. The secondary-particle energy distribution functions are also determined for primary holes and electrons. One-electron state-density structure is prominent in these distributions.