Vacuum Polarization Tensor Research Articles

Analytic properties of the vacuum polarization tensor in the temporal gauge. Trouble with unitarity

The UV-finite part of the vacuum polarization tensor Π μβ ab in the temporalplanar gauge is computed using the Leibbrandt-Vienna (or generalized Leibbrandt-Mandelstam) prescription for the gauge dependent poles of the gauge-field propagator. It is shown that Π μν ab has analytic properties that differ dramatically from those familiar in covariant gauges. By explicit calculation it is also shown that graph-by-graph unitarity is not satisfied. Comments on the absence of non-local counterterms in Burnel and Caprasse's formulation of the Leibbrandt-Vienna prescriptions are given.

VACUUM POLARIZATION TENSOR IN THREE-DIMENSIONAL QUANTUM ELECTRODYNAMICS

We evaluate the one-loop vacuum polarization tensor for three-dimensional quantum electrodynamics (QED), using an analytic regularization technique, implemented in a gauge-invariant way. We show thus that a gauge boson mass is generated at this level of radiative correction to the photon propagator. We also point out in our conclusions that the generalization for the non Abelian case is straightforward.

Perturbative Chern-Simons theory in the light-cone gauge. The one-loop vacuum polarization tensor in a gauge-invariant formalism

Perturbative Chern-Simons (CS) theory, with gauge group SU( N), is studied in the physical light-cone gauge n σ A σ a = 0, n 2 = 0. Using a gauge-invariant regularization procedure, we compute the vacuum polarization tensor to one-loop order. The regularized theory is consistent with BRS invariance. It is shown that the vacuum polarization tensor is UV finite and transverse, but contains one non-local term which is strictly gauge-dependent. A finite, non-multiplicative (gauge-dependent) wave function renormalization enables us to express the one-loop effective action as Γ one-loop( A μ (ren), k) = (( k + sign( k) c v)/4 π) S CS( A μ (ren), where S CS is the classical action and k is the bare Chern-Simons parameter.

Vacuum polarization at finite temperature and density in QED

We calculate to first order in α, finite temperature and density (FTD) corrections to vacuum polarization in the framework of QED. The limiting values of the vacuum polarization tensor when the temperature of the heat bath T is sufficiently high are subject to physical interpretation and compared to the previously existing works. The FTD corrections to the] electric permittivity, the magnetic permeability of the medium, the anomalous magnetic moment of the electron, the charge renormalization constant Z 3, and the running coupling constant are also calculated and comparison with the previous work furnished wherever possible.

Polarization of the magnetized scalar and spinor vacua

The vacuum polarization tensor of the magnetized boson system, taken within the random phase or one-loop approximation, is presented in three representation forms: the Landau, the proper-time, and the dispersion sum representation. Detailed numerical investigations into many aspects of the physical properties of both the bosonic and fermionic vacua are presented on the basis of these analytical results. From the static and uniform limit of the polarization scalars it is found that the boson vacuum state enhances small electrostatic fields parallel to the external field as well in the transverse direction contrary to the screening behaviour of the fermion vacuum, and all scalars have a weak logarithmic growth in strong fields. Also contrary to the fermionic behaviour, for wavevectors parallel to the field, the longitudinal dielectric function does not exhibit any singularities at the pair production thresholds. Furthermore the 'massive longitudinal photon' mode found for the magnetized fermion vacuum does not exist in the bosonic equivalent. Like the spin-1/2 case the dispersion solutions show that the purely transverse 'photon' mode (3) acquires mass and is channelled along the field lines, thus manifesting the mixed state of a photon and a boson-antiboson quasibound state. However, the other mode (2), which is a combined longitudinal-transverse mode does not deviate significantly from the free-space dispersion law and its inverse lifetime has no singularities at the pair threshold.

The three-gluon vertex in the temporal-planar gauge with the Leibbrandt-Vienna prescription: The one-loop effective action

The UV-divergent part of the one-loop contribution to the three-gluon vertex, Γ μνϱ abc , is computed in the temporal-planar gauge. The generalized Leibbrandt-Mandelstam (or Leibbrandt-Vienna) prescription - which is free from the inconsistencies that are known to affect the principal value prescription - is used to define the unphysical poles 1/( k· n) j . We compute the (one-loop) UV-divergent part of the gauge-field effective action up to order g 3. We show that all the awkward non-local divergences can be written as the BRS variation of some functional X whereas the β-function coincides with the usual (covariant) result. In the homogeneous case we show that a non-local renormalization of the gauge field A μ a of the form A 0μ a = Z g − 1 2 A μ a − h ̷ ƒ μ a(A), A 0μ a and A μ a having identical gauge transformations, and the usual coupling constant renormalization, g 0 = Z g μ 2 ϵ g, suffice to get rid of all the UV-divergences in Γ μνϱ abc and the vacuum polarization tensor gP μν ab .

QCD corrections to vector boson self-energies in the standard model

The O (α s) corrections to the vacuum polarization tensor of the gauge bosons are presented for both light and heavy quark flavours in the standard model of electroweak interactions on the basis of a running coupling constant α s. The influence of ϒ, T b and θ resonance as well as the proper threshold behaviour of the imaginary part of the vacuum polarization function are incorporated in the analysis. The real part is calculated numerically through the properly subtracted dispersion relation. A set of FORTRAN routines that implements the scheme is described.

THE DEPENDENCE OF THE TEMPORAL GAUGE GREEN FUNCTIONS ON THE GAUGE-FIXING VECTORS THROUGH BRS SYMMETRY

The UV-divergent part of the one-loop contribution to the vacuum polarization tensor [Formula: see text] is shown to satisfy the BRS identities governing its dependence on the time-like vector n and the auxiliary space-like vector n′ introduced by the generalized Leibbrandt-Mandelstam (LM) prescription. As far as we know the BRS identity for the n′ dependence cannot be obtained within the pragmatic approach to the LM prescription. In order to circumvent the problem the LM prescription is incorporated into a BRS-invariant action. The BRS identity for n′ provides an example of a situation where terms of order η combine with loop integrals that are singular as η→0 yielding a finite contribution.

Lorentz non-covariance of lattice chiral gauge theories

We show that, in the continuum limit, the four-dimensional chiral gauge theory built from Wilson fermions with radially frozen Higgs fields leads to Lorentz non-covariant terms in the vacuum polarization tensor, despite the gauge invariance of the regularization. The computation is done when n f, the number of flavors, is large.

UV-divergent part of the vacuum polarization tensor in the temporal gauge with the LM prescription. BRS identities

The one-loop UV-divergent contribution to the vacuum polarization tensor ( Π μν ab ) is computed in a temporal-planar gauge using the LM prescription that gives the correct exponential behaviour for the Wilson loop. Its dependence on the gauge parameter and on the magnitude of the unphysical vectors n and n′ - introduced by the LM prescription - is derived from the BRS invariance of the theory and shown to be consistent with our result for Π ab μν .

P- and CP-odd terms in the photon self-energy within a medium.

For the propagation of electromagnetic fields within a medium (e.g., a plasma or a heat bath at finite temperature), we write down the most general expression for the vacuum-polarization tensor that is consistent with gauge and Lorentz invariance in four dimensions. The expression contains a term which signals parity and CP violation either in the Lagrangian, or in the background, or both. The effect of this term is to split the otherwise degenerate transverse degrees of polarization of the photon. The physical implications of this term are discussed.

Schwinger model on the random block lattice

The fermion propagator and the vacuum polarization tensor of the Schwinger model are studied on the random block lattice, in the free field limit and weak coupling perturbation theory, respectively. The ensemble averaged fermion propagator and the vacuum polarization tensor agree with their respective values in the continuum theory for distances greater than three average lattice spacings. In momentum space, the ensemble averaged fermion propagator also agrees with the continuum propagator. These results support the possibility of having a chiral fermion interacting with the dynamical gauge field, on the random block lattice, with the correct continuum limit.

Stochastic regularization and the axial gauge

We study the compatibility of Breit, Gupta and Zaks' stochastic regularization and the axial gauge in the principal value prescription, by analysing the transversality of the vacuum polarization tensor. A one-loop computation shows the breakdown of gauge invariance by the above-mentioned regularization method.

Background-field quantization and the light-cone planar gauge: An exception to Kallosh's theorem.

The one-loop vacuum-polarization tensor for pure Yang-Mills theory is examined using background-field quantization in the light-cone planar gauge, i.e. , with the gauge-fixing Lagrangian Wgf — ( 1 /2a )n Q'D '( A )n Q, where n =0. The divergent part of the vacuum polarization, [H„,(p)]d;„ is found to depend on both a and n„, and hence is gauge dependent This result d.oes not comply with Kallosh's theorem, according to which the counterterms should be independent of gauge choice. We argue that the occurrence of nonlocal counterterms in the light-cone-type gauges violates one of the implicit assumptions of Kallosh's theorem. We also point out that a similar violation of Kallosh's theorem occurs also in the ordinary light-cone gauge, i.e., using the gaugefixing Lagrangian Wst= — (1/2a)(n. g), where n =0. The role of the light-cone gauge' in supersymmetric and superstring models has recently stimulated an examination of radiative processes in Yang-Mills theories quantized in this gauge. One definite advantage of this gauge choice is that it allows one to eliminate unphysical degrees of freedom from the analysis. In earlier studies of this gauge singularities of the form (k n) ' pres.ented a problem in Feynman integrals when n =0. The principal value prescription, first suggested for handling such singularities in the context of dimensional regularization, proved to be deficient as it led to integrals which do not obey naive power counting. Explicit calculations using this prescription also led to results inconsistent with the usual axial anomaly, renormalizability, ' ' and the vanishing of the )33 function in the N=4 supersymmetric theory. Subsequently Mandelstam and Leibbrandt found a prescription that overcomes all the above-mentioned deficiencies of the principal-value prescription, is consistent with canonical quantization, ' and preserves gauge invariance.

Calculation of nonperturbative QCD parameters.

To calculate the values of quark and gluon condensates, the nonperturbative part of the vacuum-polarization tensor is extrapolated in an optimal manner from low to fairly high values of negative s. The extrapolation is made using a variable obtained by conformally mapping the cut plane of analyticity of a polarization function into the inside of a unifocal ellipse. Taking the canonical value of the quark condensate, the gluon condensate is found to have the value G\ensuremath{\sim}(0.0115${\ifmmode\pm\else\textpm\fi{}}_{0.0001}^{0.000}$ $^{15}$) ${\mathrm{GeV}}^{4}$.

Non-Abelian Debye screening: The color-averaged potential.

The non-Abelian analogue of the Debye screening effect is investigated by means of perturbative calculations in high-temperature Yang-Mills theory. It is recalled that the gluon vacuum-polarization tensor, which is gauge dependent, does not directly yield the Debye screening mass beyond leading order. Instead, the correlation function of two Polyakov loops, representing the color-averaged potential energy of a static quark-antiquark pair, is examined as a possible means of defining a gauge-invariant Debye mass. Though apparently satisfactory to leading order, an examination of higher orders reveals two problems: (i) an unexpected breakdown of perturbation theory in the electrostatic sector and (ii) the emergence of the magnetostatic mass gap as the dominant decay mass of this correlation. These results might be the signal for a breakdown of gauge symmetry in the plasma phase and indicate the need for examining a gauge-invariant electrostatic correlation function that is free from color averaging.

Parity anomalies in gauge theories in 2 + 1 dimensions

Abstract It is shown that the introduction of massless fermions in an abelian gauge theory in 2 + 1 dimensions does not lead to any parity anomaly despite a non-commutativity of limits in the structure function of the odd part of the vacuum polarisation tensor. However, a parity anomaly does exist in non-abelian theories due to a conflict between gauge invariance under large gauge transformations and the parity symmetry.

Stochastic regularization of fermions.

Stochastic quantization is applied to the U(1)-gauge theory including fermions (QED). The early formulation of stochastically quantized QED used non-gauge-invariant Langevin equations, whereas nowadays the explicitly gauge-invariant equations of Ishikawa are preferred. We will comment on the older representation, where gauge invariance can be violated by the regularization because gauge invariance is already broken by the Langevin equations from the beginning. This is explicitly demonstrated in a few examples, including anomalous Ward-Takahashi identities and the vacuum polarization tensor ${\ensuremath{\Pi}}_{\ensuremath{\mu}\ensuremath{\nu}}$. Deeper insight into the breakdown of gauge invariance is gained by the use of the Fokker-Planck equation and the construction of equilibrium probability distributions in the case of a classical electromagnetic background field. Our investigations result in the conclusion that only Ishikawa's explicitly gauge-invariant Langevin equations can be applied without contradictions.

Relativistic quantum kinetic analysis of a pion-nucleon system

A relativistic plasma of nucleons interacting through pions via the usual isospin-invariant Yukawa coupling is analyzed in the framework of the covariant Wigner function technique. The method is manifestly covariant and the temperature effects are considered. The relativistic quantum BBGKY hierarchy for the pion-nucleon system is derived. By generalizing the Bogolioubov analysis of the classical BBGKY hierarchy a non-perturbative renormalizable method is elaborated which allows the solution of the kinetic problem in form of power series of two cluster parameters which measure the importance of correlations. In the lowest order of the cluster expansion (Hartree approximation or zero-order approximation) the quasi-nucleon Fock space is introduced, the fermion Wigner function in the thermodynamic equilibrium is obtained and the vacuum effects are renormalized. In this approximation the plasma behaves as a perfect Fermi gas of nucleons and antinucleons, but there exists an abnormal configuration with a uniform pion condensate which is unstable. In the next approximation (quadratic in the small parameters) the quasi-pion dispersion relation is obtained and the vacuum polarization tensor is renormalized. The quasi-pion rest-mass spectra (“plasma frequency”) and the effective-coupling behaviour as functions of the thermodynamic state are given. By estimating the size of the cluster parameters the self-consistency of the approximation scheme is proved. The quasi-pion Fock space is introduced and the quasi-pion equilibrium Wigner function is obtained. From these results the problem of the higher-order corrections to the Hartree thermodynamics is outlined.

Quantum-electrodynamics without renormalization: III. Vacuum polarization and the mass of the photon

The vacuum-polarization tensor is calculated to second order. The fourth-order photon mass is calculated. The photon mass is shown to be determine to all orders. Putting it equal to zero may fix the value of the fine-structure constant.