Photon Polarization Tensor Research Articles

Photon propagation in slowly varying inhomogeneous electromagnetic fields

Starting from the Heisenberg-Euler effective Lagrangian, we determine the photon current and photon polarization tensor in inhomogeneous, slowly varying electromagnetic fields. To this end, we consider background field configurations varying in both space and time, paying special attention to the tensor structure. As a main result, we obtain compact analytical expressions for the photon polarization tensor in realistic Gaussian laser pulses, as generated in the focal spots of high-intensity lasers. These expressions are of utmost importance for the investigation of quantum vacuum nonlinearities in realistic high-intensity laser experiments.

Publisher’s Note: Zeta function regularization of the photon polarization tensor for a magnetized vacuum [Phys. Rev. D90, 045033 (2014)

Received 6 January 2015DOI:https://doi.org/10.1103/PhysRevD.91.029903© 2015 American Physical Society

Zeta function regularization of the photon polarization tensor for a magnetized vacuum

In this paper, we have developed a systematic technique to regularize double summations of Landau levels and analytically evaluated the photon vacuum polarization at an external magnetic field. The final results are described by Lerch transcendent $\Phi(z,s,v)$ or its $z$-derivation. We have found that the tensor of vacuum polarization is split into not only longitudinal and transverse parts but also another mixture component. We have obtained a complete expression of the magnetized photon vacuum polarization at any kinematic regime and any strength of magnetic field for the first time. In the weak $B$-fields, after canceling out a logarithmic counter term, all three scalar functions are limited to the usual photon polarization tensor without turning on magnetic field. In the strong $B$-fields, the calculations under Lowest Landau Level approximation are only valid at the region $M^2\gg q_{\shortparallel}^2$, but not correct while $q_{\shortparallel}^2\gg M^2$, where, an imaginary part has been missed. It reminds us, a recalculation of the gap equation under a full consideration of all Landau Levels is necessary in the next future.

Impossibility of the strong magnetic fields generation in an electron-positron plasma

We examine the issue whether a magnetic field can be amplified in a background matter consisting of electrons and positrons self-interacting within the Fermi model. For this purpose we compute the antisymmetric contribution to the photon polarization tensor in this matter having nonzero temperature and chemical potential. It is shown that this contribution is vanishing in the static limit. Then we study a particular case of a degenerate relativistic electron gas present in a magnetar. We demonstrate that a seed magnetic field is attenuated in this case. Thus, contrary to the recent claim, we show that there is no magnetic field instability in such a system, which can lead to the magnetic field growth. Therefore recently proposed mechanism cannot be used for the explanation of strong magnetic fields of magnetars.

Laser photon merging in an electromagnetic field inhomogeneity

We study the effect of laser photon merging, or equivalently high harmonic generation, in the quantum vacuum subject to inhomogeneous electromagnetic fields. Such a process is facilitated by the effective nonlinear couplings arising from charged particle-antiparticle fluctuations in the quantum vacuum subject to strong electromagnetic fields. We derive explicit results for general kinematic and polarization configurations involving optical photons. Concentrating on merged photons in reflected channels which are preferable in experiments for reasons of noise suppression, we demonstrate that photon merging is typically dominated by the competing nonlinear process of quantum reflection, though appropriate polarization and signal filtering could specifically search for the merging process. As a byproduct, we devise a novel systematic expansion of the photon polarization tensor in plane wave fields.

Instability of magnetic fields in electroweak plasma driven by neutrino asymmetries

The magnetohydrodynamics (MHD) is modified to incorporate the parity violation in the Standard Modelleading to a new instability of magnetic fields in the electroweak plasma in the presence ofnonzero neutrino asymmetries. The main ingredient for such a modified MHD is the antisymmetric part ofthe photon polarization tensor in plasma, where the parity violating neutrino interaction with charged leptonsis present. We calculate this contribution to the polarizationtensor connected with the Chern-Simons term in effective Lagrangian ofthe electromagnetic field. The general expression for such a contribution which depends onthe temperature and the chemical potential of plasma as well as on the photon's momentum is derived.The instability of a magnetic field driven by the electron neutrino asymmetry for the ν-burst during the first second of a supernova explosion can amplify a seed magnetic field of a protostar, and, perhaps, can explain the generation of strongest magnetic fields in magnetars.The growth of a cosmological magnetic field driven by the neutrino asymmetry density Δnν = nν−nν̄≠0 is provided by a lower bound on |ξνe| = |μνe|/T which is consistent with the well-known Big Bang nucleosynthesis (upper) bound on neutrino asymmetries in a hot universe plasma.

Photon polarization tensor in a homogeneous magnetic or electric field

We revisit the photon polarization tensor in a homogeneous external magnetic or electric field. The starting point of our considerations is the momentum space representation of the one-loop photon polarization tensor in the presence of a homogeneous electromagnetic field, known in terms of a double parameter integral. Our focus is on explicit analytical insights for both on- and off-the-light-cone dynamics in a wide range of well-specified physical parameter regimes, ranging from the perturbative to the manifestly nonperturbative strong field regime. The basic ideas underlying well-established approximations to the photon polarization tensor are carefully examined and critically reviewed. In particular, we systematically keep track of all contributions, both the ones to be neglected and those to be taken into account explicitly, to all orders. This allows us to study their ranges of applicability in a much more systematic and rigorous way. We point out the limitations of such approximations and manage to go beyond at several instances.

Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels

Photons propagating in strong magnetic fields are subject to a phenomenon called the “vacuum birefringence” where refractive indices of two physical modes both deviate from unity and are different from each other. We compute the vacuum polarization tensor of a photon in a static and homogeneous magnetic field by utilizing Schwinger’s proper-time method, and obtain a series representation as a result of double integrals analytically performed with respect to proper-time variables. The outcome is expressed in terms of an infinite sum of known functions which is plausibly interpreted as summation over all the Landau levels of fermions. Each contribution from infinitely many Landau levels yields a kinematical condition above which the contribution has an imaginary part. This indicates decay of a sufficiently energetic photon into a fermion–antifermion pair with corresponding Landau level indices. Since we do not resort to any approximation, our result is applicable to the calculation of refractive indices in the whole kinematical region of a photon momentum and in any magnitude of the external magnetic field.

Photon polarization tensor in the light front field theory at zero and finite temperatures

In this work, we consider the light front quantum electrodynamics in ($3+1$) dimensions and evaluate the photon polarization tensor at one loop for both zero and finite temperatures. In the first case, we apply the dimensional regularization method to extract the finite contribution and find the transverse structure for the amplitude in terms of the light front coordinates. The result agrees with one-loop covariant calculation. For the thermal corrections, we generalize the hard thermal loop approximation to the light front and calculate the dominant temperature contribution to the polarization tensor, consistent with the Ward identity. In both zero as well as finite temperature calculations, we use the oblique light front coordinates.

OPTICAL PROBES OF THE QUANTUM VACUUM: THE PHOTON POLARIZATION TENSOR IN EXTERNAL FIELDS

The photon polarization tensor is the central building block of an effective theory description of photon propagation in the quantum vacuum. It accounts for the vacuum fluctuations of the underlying theory, and in the presence of external electromagnetic fields, gives rise to such striking phenomena as vacuum birefringence and dichroism. Standard approximations of the polarization tensor are often restricted to on-the-light-cone dynamics in homogeneous electromagnetic fields, and are limited to certain momentum regimes only. We devise two different strategies to go beyond these limitations: First, we aim at obtaining novel analytical insights into the photon polarization tensor for homogeneous fields, while retaining its full momentum dependence. Second, we employ wordline numerical methods to surpass the constant-field limit.

Heavy quark medium polarization at next-to-leading order

We compute the imaginary part of the heavy quark contribution to the photon polarization tensor, i.e. the quarkonium spectral function in the vector channel, at next-to-leading order in thermal QCD. Matching our result, which is valid sufficiently far away from the two-quark threshold, with a previously determined resummed expression, which is valid close to the threshold, we obtain a phenomenological estimate for the spectral function valid for all non-zero energies. In particular, the new expression allows to fix the overall normalization of the previous resummed one. Our result may be helpful for lattice reconstructions of the spectral function (near the continuum limit), which necessitate its high energy behaviour as input, and can in principle also be compared with the dilepton production rate measured in heavy ion collision experiments. In an appendix analogous results are given for the scalar channel.

Virtual photon emission from a quark-gluon plasma

We present phenomenological formulas for virtual photon emission rates from a thermalized quark-gluon plasma (QGP) that include bremsstrahlung and annihilation with scattering (AWS) mechanisms along with the Landau-Pomeranchuk-Migdal (LPM) effects. For this purpose we follow the approach of generalized emission functions (GEF) for virtual photon emission, we showed earlier for a fixed temperature and strong coupling constant. In the present work, we extend the LPM calculations for several temperatures and strong coupling strengths, photon energies (q{sub 0}), photon mass (Q{sup 2}), and quark energies (p{sub 0}). We generalize the dynamical scaling variables, x{sub T},x{sub L}, for bremsstrahlung and AWS processes that are now functions of variables p{sub 0},q{sub 0},Q{sup 2},T,{alpha}{sub s}. The GEF introduced earlier, g{sub T}{sup b},g{sub T}{sup a},g{sub L}{sup b},g{sub L}{sup a}, are also generalized for any temperatures and coupling strengths. From this, the imaginary part of the photon polarization tensor as a function of photon mass and energy has been calculated as a one-dimensional integral over these GEF and parton distribution functions in the plasma. By fitting these polarization tensors obtained from GEF method, we obtained a phenomenological formula for virtual photon emission rates as a function of (q{sub 0},Q{sup 2},T,{alpha}{sub s}) that includes bremsstrahlung and AWSmore » mechanisms with LPM effects.« less

Generalized emission functions for photon emission from quark-gluon plasma

The Landau-Pomeranchuk-Migdal effects on photon emission from the quark-gluon plasma have been studied as a function of photon mass, at a fixed temperature of the plasma. The integral equations for the transverse vector function [$\stackrel{\texttildelow{}}{\mathbf{f}}({\stackrel{\texttildelow{}}{\mathbf{p}}}_{\ensuremath{\perp}})$] and the longitudinal function [$\stackrel{\texttildelow{}}{g}({\stackrel{\texttildelow{}}{\mathbf{p}}}_{\ensuremath{\perp}})$] consisting of multiple scattering effects are solved by the self-consistent iterations method and also by the variational method for the variable set ${{p}_{0},{q}_{0},{Q}^{2}}$. We considered the bremsstrahlung and the off shell annihilation ($\mathbf{aws}$) processes. We define two new dynamical scaling variables, ${x}_{T},{x}_{L}$, for bremsstrahlung and $\mathbf{aws}$ processes which are functions of variables ${p}_{0},{q}_{0},{Q}^{2}$. We define four new emission functions for massive photon emission represented by ${g}_{T}^{b},{g}_{T}^{a},{g}_{L}^{b},{g}_{L}^{a}$ and we constructed these using the exact numerical solutions of the integral equations. These four emission functions have been parametrized by suitable simple empirical fits. Using the empirical emission functions, we calculated the imaginary part of the photon polarization tensor as a function of photon mass and energy.

Out-of-equilibrium electromagnetic radiation

We derive general formulas for photon and dilepton production rates from an arbitrary non-equilibrated medium from first principles in quantum field theory. At lowest order in the electromagnetic coupling constant, these relate the rates to the unequal-time in-medium photon polarization tensor and generalize the corresponding expressions for a system in thermodynamic equilibrium. We formulate the question of electromagnetic radiation in real time as an initial value problem and consistently describe the virtual electromagnetic dressing of the initial state. In the limit of slowly evolving systems, we recover known expressions for the emission rates and work out the first correction to the static formulas in a systematic gradient expansion. Finally, we discuss the possible application of recently developed techniques in non-equilibrium quantum field theory to the problem of electromagnetic radiation. We argue, in particular, that the two-particle-irreducible (2PI) effective action formalism provides a powerful resummation scheme for the description of multiple scattering effects, such as the Landau-Pomeranchuk-Migdal suppression recently discussed in the context of equilibrium QCD.

Photon self-energy in a color superconductor

In a color superconductor the diquark condensates break spontaneously both the color and ordinary electromagnetism, leaving a remaining rotated $U(1)$ symmetry unbroken. The gauge interactions associated with this rotated symmetry may be considered as the in-medium electromagnetism. We compute the in-medium photon self-energy in the presence of diquark condensates at high baryonic density and weak coupling. This is done to one-loop order for the cases of two and three quark flavors. For vanishing temperature, a detailed discussion is given of the low momentum behavior of the photon polarization tensor. A simple physical picture for the propagation of light in color superconducting media is obtained. The main new effect is due to the diquark condensates, which lead to a strong dielectric constant of the medium. The magnetic permeability remains unchanged, because the primary condensates have vanishing spin and angular momentum. In the two flavor case, an additional contribution arises due to gapless quarks and electrons, which is responsible for Debye screening effects. We also discuss the low energy effective theory for the three flavor case in the presence of electromagnetic interactions.

Electromagnetic properties of a neutrino stream

In a medium that contains a neutrino background in addition to the matter particles, the neutrinos contribute to the photon self-energy as a result of the effective electromagnetic vertex that they acquire in the presence of matter. We calculate the contribution to the photon self-energy in a dense plasma, due to the presence of a gas of charged particles, or neutrinos, that moves as a whole relative to the plasma. General formulas for the transverse and longitudinal components of the photon polarization tensor are obtained in terms of the momentum distribution functions of the particles in the medium, and explicit results are given for various limiting cases of practical interest. The formulas are used to study the electromagnetic properties of a plasma that contains a beam of neutrinos. The transverse and longitudinal photon dispersion relations are studied in some detail. Our results do not support the idea that neutrino streaming instabilities can develop in such a system. We also indicate how the phenomenon of optical activity of the neutrino gas is modified due to the velocity of the neutrino background relative to the plasma. The general approach and results can be adapted to similar problems involving relativistic plasmas and high-temperature gauge theories in other environments.

Invariant representations of finite rotation matrices and some applications

Standard representations of finite rotation matrices (FRM) are defined by expressions that are wedded to two particular coordinate frames such as, e.g., are involved in the definition of Euler angles. We present here representations for the FRM in invariant tensor forms comprised of vectors defined in a space-fixed coordinate frame $K.$ Three explicit expressions for a FRM are presented: First, in terms of tensor products of the spherical or Cartesian basis vectors of frame $K,$ second in a differential form containing the tensor products of gradient operators, and, third, as the superposition of so-called ``minimal'' bipolar harmonics depending on any pair of unit vectors a, b connected with a frame $K.$ Based on these results for the FRM, the transformation rule for an irreducible tensor set under space rotations may be written in terms of bipolar harmonics. Our results are especially useful for analyzing angular distributions in atomic processes involving a precise accounting of all effects of photon and target polarizations. Four examples are considered as illustrations of the techniques presented. First, an invariant representation of the photon polarization tensor is found in terms of linear and circular polarization degrees of the photon beam. Second, an invariant decomposition of tripolar harmonics of second rank in terms of very simple, rank 2 tensors is presented. Third, a convenient parametrization is proposed for the polarization state multipoles of a polarized atomic target. Fourth, a simple invariant formula is derived for the angular distribution of polarized photons resulting from electric dipole photon emission by an arbitrary polarized atom.

Self-consistent resummation scheme in scalar QED.

In this paper we derive a resummation scheme that may be useful in the calculation of finite temperature processes that involve infrared-divergent diagrams. We discuss the inclusion of self-consistent vertices in calculations of diagrams with very soft external momenta. We work with scalar QED and show that the use of self-consistent vertices in the infrared limit of the retarded photon polarization tensor is equivalent to the resummation of dominant diagrams. To lowest order in an expansion about the parameter that is to be determined self-consistently, we find that the result is independent of this parameter and equal to the expression obtained with uncorrected lines and vertices. The motivation for this work is the hope that it will be possible to use this technique to perform self-consistent calculations beyond leading order.

Photon polarization tensor and gauge dependence in three-dimensional quantum electrodynamics.

We calculate the three-dimensional QED (${\mathrm{QED}}_{3}$) photon polarization tensor using dressed fermion propagators and a fermion-photon vertex that satisfies the Ward-Takahashi identity. Irrespective of the structural details of the transverse part of the fermion-photon vertex the photon remains massless; i.e., there is no photon mass generation in the manner of the Schwinger mechanism. Our calculation suggests that ${\mathrm{QED}}_{3}$ is confining when the polarization tensor is calculated using a dressed fermion propagator and fermion-photon vertex. The gauge parameter dependence of the fermion propagator, fermionphoton vertex, and photon polarization tensor is discussed in connection with the Landau-Khalatnikov gauge transformation laws.

Nonlinear electrodynamics in a superstrong magnetic field

Using a Feynman technique in a two-dimensional space we have calculated an electron propagator and a photon polarization tensor in a superstrong magnetic field B ⪢ B 0 = m 2 e = 4.41 × 10 13 G. Some applications are briefly discussed.