Proper-time Integral Research Articles

Low-energy limit of N-photon amplitudes in a constant field: Part II

We employ the worldline formalism to derive a series representation of the low-energy limit of the N-photon amplitude in a constant background field for both scalar and spinor QED. The amplitudes are then written in terms of a single proper-time integral. The above-mentioned series representation terminates when considering a constant crossed field. This allows us to obtain even more compact expressions for these particular amplitudes for which the result of the proper-time integral, for fixed parameters, takes the form of a factorial function. In addition, we derive all helicity components of these amplitudes and express them explicitly in terms of Bernoulli numbers and spinor products.

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

Revisiting regularization with Kaluza-Klein states and Casimir vacuum energy from extra dimensional spaces

In the present paper, we investigate regularization of the one-loop quantum corrections with infinite Kaluza-Klein (KK) states and evaluate Casimir vacuum energy from extra dimensions. The extra dimensional models always involve the infinite massless or massive Kaluza-Klein states, and therefore, the regularization of the infinite KK corrections is highly problematic. In order to avoid the ambiguity, we adopt the proper time integrals and the Riemann zeta function regularization in evaluating the summations of infinite KK states. In the calculation, we utilize the KK regularization method with exchanging the infinite summations and the infinite loop integrals. At the same time, we also evaluate the correction by the the dimensional regularization method without exchanging the summations and the loop integrals. Then, we clearly show that the regularized Casimir corrections from the KK states have the form of $\propto 1/R^2$ for the Higgs mass and $\propto 1/R^4$ for the cosmological constant, where $R$ is the compactification radius. We also evaluate the Casimir energy in supersymmetric extra-dimensional models. The contributions from bulk fermions and bulk bosons are not offset because we choose SUSY breaking boundary conditions. The non-zero supersymmetric Casimir corrections from extra dimensions undoubtedly contribute to the Higgs mass and the cosmological constant. We conclude the coefficients of such corrections are enhanced compared to the case without bulk supersymmetry.

On Vacuum Polarization and Schwinger Pair Production in Intense Lasers

We review and elaborate the complex effective action at one-loop and at zero or finite temperature in the in-out formalism for scalar QED and the vacuum persistence in time-dependent electric fields. Using the gamma-function regularization, we find the effective action in the proper-time integral and the pair-production rate in an exponentially increasing electric field. We apply the quantum invariant theory to the scalar field in time-dependent electric fields and clarify the meaning of multi-pair states. And pair-production rates are derived when the initial state is the Minkowski vacuum or the adiabatic vacuum and are compared with two representations for the Vlasov equation. Finally, the contour integral method gives the pair-production rate in the exponentially increasing electric field.

One-loop effective action and Schwinger effect in (anti-) de Sitter space

We study the Schwinger mechanism by a uniform electric field in ${\rm dS}_2$ and ${\rm AdS}_2$ and the curvature effect on the Schwinger effect, and further propose a thermal interpretation of the Schwinger formula in terms of the Gibbons-Hawking temperature and the Unruh temperature for an accelerating charge in ${\rm dS}_2$ and an analogous expression in ${\rm AdS}_2$. The exact one-loop effective action is found in the proper-time integral in each space, which is determined by the effective mass, the Maxwell scalar, and the scalar curvature, and whose pole structure gives the imaginary part of the effective action and the exact pair-production rate. The exact pair-production rate is also given the thermal interpretation.

A Novel Approach to Model Moving Heat Sources

Abstract The solution of quasistationary heat-conduction problems with moving heat sources on the surface of a body is a demanding task and still a topic of ongoing research. The authors have shown in several contributions that an exact analytical solution exists for a constant velocity of the heat source and constant thermal properties of the body. Since also convection at the surface can be treated properly, cooling and quenching problems can be solved, accordingly. The main characteristic of the governing equation is that the partial derivative of the temperature with respect to time must be replaced by the convective term being the velocity of the heat source times the gradient of the temperature. In this contribution it is shown that the quasistationary problem can be transformed into a stationary one, if a modified heat-conduction coefficient is introduced. The application of this concept and its limits are demonstrated in comparison with existing analytical solutions. For this purpose a model is set up which consists of two parts. In the first part the concept of transforming to a stationary problem is employed, whereas in the second part fictitious heat sources are defined. The methodology is shown to be very efficient as it avoids difficulties arising from the necessity to use very dense meshes and proper time integration routines necessary when applying standard fine element codes for a spatially fixed configuration of a body with moving heat sources on its surface.

Coupled Problems of Dynamics in Materially Incompressible Saturated Porous Media

AbstractThe mechanical behavior of saturated porous materials is largely governed by the interaction between the solid skeleton and the pore fluid. This interaction is particularly strong in dynamic problems and leads to numerical challenges especially in the case of incompressible constituents. In fact, the permeability plays a significant role in this coupling and influences the choice of a proper time integration scheme. Proceeding from the macroscopic Theory of Porous Media (TPM) within the isothermal and geometrical linear regime, the governing balance equations of the dynamic binary solid–fluid model are the solid and fluid momentum balances, and the overall volume balance of the biphasic mixture. This set of coupled partial differential equations (PDEs) is solved within the framework of the mixed Finite Element Method (FEM) applying two different time solution methods, viz., a monolithic implicit and a splitted implicit–explicit scheme. The time stepping algorithms are implemented into the FE program PANDAS and a Scilab FE routine and compared on a one–dimensional wave propagation example. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Semiclassical treatment of induced schwinger processes at a finite temperature

The induced pair production in an external field at a finite temperature is considered. An one-loop correction to the Green’s function of a meson is calculated semiclassically within the framework of a saddle-point analysis of the Schwinger proper time integrals. This correction appears to be exponentially small in terms of an inverse temperature dependence. The low-temperature limit is shown to be in full agreement with previously obtained zero-temperature results. The corrections in the low-temperature limits are estimated up to the leading exponential and preexponential terms. A comparison is made to earlier calculations of the vacuum decay.

Hypermultiplet dependence of one-loop effective action in the Script N = 2 superconformal theories

We study one-loop low-energy effective action in the hypermultiplet sector for ${\cal N}=2$ superconformal models. Any such a model contains ${\cal N}=2$ vector multiplet and some number of hypermultiplets. Gauge group $G$ is assumed to be broken down to $\tilde{G}\times K$ where $K$ is an Abelian subgroup and a background vector multiplet belongs to the Cartan subalgebra corresponding to $K$. We find a general expression for the low-energy effective action in a form of a proper-time integral. The leading space-time dependent contributions to the effective action are derived and their bosonic component structure is analyzed. The component action contains the terms with three and four space-time derivatives of component fields and has the Chern-Simons form.

PROPER-TIME FORMALISM IN A CONSTANT MAGNETIC FIELD AT FINITE TEMPERATURE AND CHEMICAL POTENTIAL

We investigate scalar and spinor field theories in a constant magnetic field at finite temperature and chemical potential. In an external constant magnetic field the exact solution of the two-point Green functions are obtained by using the Fock–Schwinger proper-time formalism. We extend it to the thermal field theory and find the expressions of the Green functions exactly for the temperature, the chemical potential and the magnetic field. For practical calculations the contour of the proper-time integral is carefully selected. The physical contour is discussed in a constant magnetic field at finite temperature and chemical potential. As an example, behavior of the vacuum self-energy is numerically evaluated for the free scalar and spinor fields.

Spontaneous symmetry breaking and proper-time flow equations

We discuss the phenomenon of spontaneous symmetry breaking by means of a class of non-perturbative renormalization group flow equations which employ a regulating smearing function in the proper-time integration. We show, both analytically and numerically, that the convexity property of the renormalized local potential is obtained by means of the integration of arbitrarily low momenta in the flow equation. Hybrid Monte Carlo simulations are performed to compare the lattice effective potential with the numerical solution of the renormalization group flow equation. We find very good agreement both in the strong and in the weak coupling regime.

Green functions of the Dirac equation with magnetic-solenoid field

Various Green functions of the Dirac equation with a magnetic-solenoid field (the superposition of the Aharonov–Bohm field and a collinear uniform magnetic field) are constructed and studied. The problem is considered in 2+1 and 3+1 dimensions for the natural extension of the Dirac operator (the extension obtained from the solenoid regularization). Representations of the Green functions as proper time integrals are derived. The nonrelativistic limit is considered. For the sake of completeness the Green functions of the Klein–Gordon particles are constructed as well.

Low-energy dynamics inN= 2 super QED: two-loop approximation

The two-loop (Euler-Heisenberg-type) effective action for N = 2 supersymmetric QED is computed using the N = 1 superspace formulation. The effective action is expressed as a series in supersymmetric extensions of F^{2n}, where n=2,3,..., with F the field strength. The corresponding coefficients are given by triple proper-time integrals which are evaluated exactly. As a by-product, we demonstrate the appearance of a non-vanishing F^4 quantum correction at the two-loop order. The latter result is in conflict with the conclusion of hep-th/9710142 that no such quantum corrections are generated at two loops in generic N = 2 SYM theories on the Coulomb branch. We explain a subtle loophole in the relevant consideration of hep-th/9710142 and re-derive the F^4 term from harmonic supergraphs.

A proper time integration with split stepping for the explicit free-surface modeling

Errors due to split time stepping are discussed for an explicit free–surface ocean model. In commonly used split time stepping, the way of time integration for the barotropic momentum equation is not compatible with that of the baroclinic one. The baroclinic equation has three–time–level structure because of leapfrog scheme. The barotropic one, however, has two–time–level structure when represented in terms of the baroclinic time level, on which the baroclinic one is integrated. This incompatibility results in the splitting errors as shown in this paper.

QUANTUM SPINOR FIELD IN THE FRW UNIVERSE WITH A CONSTANT ELECTROMAGNETIC BACKGROUND

This paper is a natural continuation of our paper Quantum Scalar Field in FRW Universe with Constant Electromagnetic Background, Int. J. Mod. Phys.A12, 4837 (1997). We generalize our previous work to the case of a massive spinor field, which is placed in a FRW universe of a special type with a constant electromagnetic field. So as to achieve this, special sets of exact solutions of the Dirac equation in the background under consideration are constructed and classified. Using these solutions representations for out–in, in–in, and out–out spinor Green functions are explicitly constructed as proper-time integrals over the corresponding contours in the complex proper-time plane. The vacuum-to-vacuum transition amplitude and the number of created particles are found and vacuum instability is discussed. The mean values of the current and of the energy–momentum tensor are evaluated, and different approximations for them are presented. The back reaction related to particle creation and to unstable vacuum polarization is estimated in different regimes.

Quantum Scalar Field in the FRW Universe with a Constant Electromagnetic Background

We discuss a massive scalar field with conformal coupling in the Friedmann–Robertson–Walker (FRW) Universe of a special type with a constant electromagnetic field. Treating an external gravitational–electromagnetic background exactly, at the first time the proper-time representations for out–in, in–in and out–out scalar Green functions are explicitly constructed as proper-time integrals over the corresponding (complex) contours. The vacuum-to-vacuum transition amplitudes and the number of created particles are found and vacuum instability is discussed. The mean values of the current and the energy–momentum tensor are evaluated, and different approximations for them are investigated. The back reaction of the particles created to the electromagnetic field is estimated in different regimes. The connection between the proper-time method and the effective action is outlined. The effective action in scalar QED in the weakly curved FRW Universe (de Sitter space) with a weak constant electromagnetic field is found as a derivative expansion over curvature and electromagnetic field strength. Possible further applications of the results are mentioned.

Connection between momentum cutoff and operator cutoff regularizations.

Operator cutoff regularization based on the original Schwinger's proper-time formalism is examined. By constructing a regulating smearing function for the proper-time integration, we show how this regularization scheme simulates the usual momentum cutoff prescription yet preserves gauge symmetry even in the presence of the cutoff scales. Similarity between the operator cutoff regularization and the method of higher (covariant) derivatives is also observed. The invariant nature of the operator cutoff regularization makes it a promising tool for exploring the renormalization group flow of gauge theories in the spirit of Wilson-Kadanoff blocking transformation.

The use of a regularized heat-kernel expansion on supersymmetric and/or gauge invariant field theories

We regularize (superfield) Green functions using a cut-off in the proper time integration of the heat-kernel expansion. Counterterms are easily computed using the background field method, up to two-loop order. Supersymmetry is explicitly maintained through use of supergraphs. The supermultiplet structure of the superconformal anomaly is easily obtained.

StrongCPviolation and proper-time integration

It is demonstrated that the proper-time integration approach to background-field problems may be usefully applied to the evaluation of radiative corrections to the strong CP-violating parameter q-bar. Analysis of the one- and two-loop contributions reveals a class of models consistent with the phenomenological constraint q-bar<10/sup -8/. Estimated contributions in the range q-barapprox.10/sup -11/ to 10/sup -30/ are obtained for various models.

Evaluation of the effective potential in quantum electrodynamics

Starting with Schwinger's proper-time integral in QED, we here present the analytical and numerical evaluation of the one-loop effective potential in the presence of a constant prescribed magnetic (electric) field.