Vacuum Polarization Energy Research Articles

A Variational Approach to Excited Fermions on Kinks

We study the back-reaction of fermion fields on the kink solution in one space and one time dimension. We employ a variational procedure to determine an upper limit for the minimum of the total energy. This energy has three contributions: the classical kink energy, the energy of valence fermions and the fermion vacuum polarization energy. The latter arises from the interaction of the kink with the Dirac sea and is required for consistency of the semi-classical expansion for the fermions. Earlier studies only considered the valence part and observed a substantial back-reaction. This was reflected by a sizable distortion of the kink profile. We find that this distortion is strongly mitigated when the Dirac sea is properly accounted for. As a result, the back-reaction merely produces a slight squeeze or stretch of the kink profile.

Excited fermions on kinks and the Dirac sea

We study quantum effects of recently discovered kink solitons which are constructed self-consistently by coupling to a single, excited fermion bound state. Our studies are based on the observation that in a semi-classical expansion the energies of this single level and of the Dirac sea should be treated equally. For these kink solutions we compute the energy of the Dirac sea as the fermion vacuum polarization energy. We find it to be substantial and to typically outweigh the energy gain from binding the single level.

Quantum effects of solitons in the self-dual impurity model

We compute the vacuum polarization energies (VPE) of solitons in a self-dual impurity model in which the soliton profiles take the shape of a separated kink-antikink pair. Classically the soliton energies are invariant under the change of a continuous parameter that can be interpreted as the kink-antikink separation. This is not the case for the VPE so that quantum effects decide on the energetically most favorable separation. The considered configurations are classically stable so that its quantum fluctuations have only real frequency eigenvalues. Hence, in contrast to the kink-antikink configuration in the $\phi^4$ model, the VPE is well defined for any value of the separation and we gain insight into the quantum corrections to the kink-antikink potential

Vacuum polarization energy decline and spontaneous positron emission in QED under Coulomb supercriticality

The properties of QED-vacuum polarization, caused by the supercritical external Coulomb source with charge $Z$ and size $R$, are explored in an essentially nonperturbative approach with emphasis on the vacuum energy ${\mathcal{E}}_{\mathrm{VP}}$ (where VP stands for vacuum polarization.) It is shown that in the supercritical region ${\mathcal{E}}_{\mathrm{VP}}$ turns out to be a decreasing function of the Coulomb charge, resulting in decay into the negative range as $\ensuremath{\sim}\ensuremath{-}{Z}^{4}/R$. Moreover, it is indeed the decline of ${\mathcal{E}}_{\mathrm{VP}}$, which provides the required energy for (presumably possible) positron emission. Found this way, properties of the resonance decay are in agreement with those achieved by other methods. The additional problems of spontaneous emission, caused by lepton number, are also discussed.

Vacuum polarization energy of the kinks in the sinh-deformed models

We compute the one-loop quantum corrections to the kink energies of the sinh-deformed $\phi^{4}$ and $\varphi^{6}$ models in one space and one time dimensions. These models are constructed from the well-known polynomial $\phi^{4}$ and $\varphi^{6}$ models by a deformation procedure. We also compute the vacuum polarization energy to the non-polynomial function $U(\phi)=\frac{1}{4}(1-\sinh^{2}\phi)^{2}$. This potential approaches the $\phi^{4}$ model in the limit of small values of the scalar function. These energies are extracted from scattering data for fluctuations about the kink solutions. We show that for certain topological sectors with nonequivalent vacua the kink solutions of the sinh-deformed models are destabilized.

Winding number dependence of quantum vortex energies at one-loop

We compute the one-loop vacuum polarization energies of Abrikosov-Nielsen-Olesen vortices with topological charge $n$ in scalar electrodynamics, for the Bogomolny-Prasad-Sommerfeld case of equal gauge and scalar masses. This calculation allows us to investigate the relationship between the winding number and the quantum-corrected vortex energy, which in turn determines the stability of higher winding configurations against decay into configurations with unit winding. While the classical energy is proportional to $n$, we find that the vacuum polarization energy is negative and approximately proportional to $n\ensuremath{-}1$ with a small constant offset.

Quantum corrections to solitons in the ϕ8 model

We compute the vacuum polarization energy of kink solitons in the $\phi^{8}$ model in one space and one time dimensions. There are three possible field potentials that have eight powers of $\phi$ and that possess kink solitons. For these different field potentials we investigate whether the vacuum polarization destabilizes thesolitons. This may particularly be the case for those potentials that have degenerate ground states with different curvatures in field space yielding different thresholds for the quantum fluctuations about the solitons at negative and positive spatial infinity. We find that destabilization occurs in some cases, but this is not purely a matter of the field potential but also depends on the realized soliton solution for that potential. One of the possible field potentials has solitons with different topological charges. In that case the classical mass approximately scales like the topological charge. Even though destabilization precludes robust statements, there are indications that the vacuum polarization energy does not scale as the topological charge.

Vacuum polarization energy of a complex scalar field in a vortex background

Scattering methods make it possible to compute the effects of renormalized quantum fluctuations on classical field configurations. As a classic example of a topologically nontrivial classical solution, the Abrikosov-Nielsen-Olesen vortex in U(1) Higgs-gauge theory provides an ideal case in which to apply these methods. While physically measurable gauge-invariant quantities are always well-behaved, the topological properties of this solution give rise to singularities in gauge-variant quantities used in the scattering problem. In this paper we show how modifications of the standard scattering approach are necessary to maintain gauge invariance within a tractable calculation. We apply this technique to the vortex energy calculation in a simplified model, and show that to obtain accurate results requires an unexpectedly extensive numerical calculation, beyond what has been used in previous work.

Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac–Coulomb System with Z > Zcr: Vacuum Charge Density

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge Z > Zcr,1 and radius R0, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom,” we study the behavior of both the vacuum density ρVP(r) and the total induced charge and also the vacuum-polarization energy EVP. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for ρVP(r) and ɛVP, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρVP and ɛVP. The most important result is that for Z ≫ Zcr,1 in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z, which reaches large negative values and whose behavior is estimated from below (in absolute value) as ~ −|ηeffZ3|/R0. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with mj = ±1/2,±3/2. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge QVP in the overcritical region in detail.

Vacuum polarization energy of the Shifman–Voloshin soliton

We compute the vacuum polarization energy of soliton configurations in a model with two scalar fields in one space dimension using spectral methods. The second field represents an extension of the conventional ϕ4 kink soliton model. We find that the vacuum polarization energy destabilizes the soliton except when the fields have identical masses. In that case the model is equivalent to two independent ϕ4 models.

Spectral methods for coupled channels with a mass gap

We develop a method to compute the vacuum polarization energy for coupled scalar fields with different masses scattering off a background potential in one space dimension. As an example we consider the vacuum polarization energy of a kink-like soliton built from two real scalar fields with different mass parameters.

Nonperturbative effects of vacuum polarization for a quasi-one-dimensional Dirac–Coulomb system with Z > Z cr

Major nonperturbative properties of the vacuum charge density ρ VP (x) and vacuum-polarization energy ξ VP are considered for an overcritical Coulomb source with Z > Z cr in 1 + 1 D. We demonstrate that, for a broad range of external-field parameters, vacuum energy may significantly deviate from the perturbative quadratic increase and even decrease towards deeply negative values due to vacuum polarization in the overcritical region.

Emerging Translational Variance: Vacuum Polarization Energy of the ϕ6 Kink

We propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow decomposition into symmetric and antisymmetric channels in one space dimension. In particular, we compute the vacuum polarization energy of the kink soliton in the ϕ6 model. We link the dependence of this energy on the position of the center of the soliton to the different masses of the quantum fluctuations at negative and positive spatial infinity.

Vacuum polarization energy for general backgrounds in one space dimension

For field theories in one time and one space dimensions we propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow a decomposition into symmetric and anti--symmetric channels. The method also applies to scenarios in which the masses of the quantum fluctuations at positive and negative spatial infinity are different. As an example we compute the vacuum polarization energy of the kink soliton in the $\phi^6$ model. We link the dependence of this energy on the position of the soliton to the different masses.

Isospin invariance and the vacuum polarization energy of cosmic strings

We corroborate the previously applied spectral approach to compute the vacuum polarization energy of string configurations in models similar to the standard model of particle physics. The central observation underlying this corroboration is the existence of a particular global isospin transformation of the string configuration. Under this transformation the single particle energies of the quantum fluctuations are invariant, while the inevitable implementation of regularization and renormalization requires operations that are not invariant. We verify numerically that all such variances eventually cancel, and that the vacuum polarization energy obtained in the spectral approach is indeed gauge invariant.

QED radiative corrections and many-body effects in atoms: vacuum polarization and binding energy shifts in alkali metals

We calculate vacuum polarization corrections to the binding energies in neutral alkali atoms Na through to the superheavy element E119. We employ the relativistic Hartree–Fock method to demonstrate the importance of relaxation of the electronic core and the correlation potential method to study the effects of second and higher orders of perturbation theory. These many-body effects are sizeable for all orbitals, though particularly important for orbitals with angular momentum quantum number . The orders of magnitude enhancement for d waves produces shifts that, for Rb and the heavier elements, are larger than those for p waves and only an order of magnitude smaller than the s-wave shifts. The many-body enhancement mechanisms that operate for vacuum polarization apply also to the larger self-energy corrections.

Quantum stabilization of cosmic strings

In the standard model, stabilization of a classically unstable cosmic string may occur through the quantum fluctuations of a heavy fermion doublet. We review numerical results from a semiclassical expansion in a reduced version of the standard model. In this expansion, the leading quantum corrections emerge at one loop level for many internal degrees of freedom. The resulting vacuum polarization energy and the binding energies of occupied fermion energy levels are of the same order, and must therefore be treated on equal footing. Populating these bound states lowers the total energy compared to the same number of free fermions and assigns a charge to the string. Charged strings are already stabilized for a fermion mass only somewhat larger than the top quark mass. Though obtained in a reduced version, these results suggest that neither extraordinarily large fermion masses nor unrealistic couplings are required to bind a cosmic string in the standard model. Furthermore, we also review results for a quantum stabilization mechanism that prevents closed Nielsen–Olesen-type strings from collapsing.

Casimir energy of frequency dependent interactions

Vacuum polarization (or Casimir) energies can be straightforwardly computed from scattering data for static field configurations whose interactions with the fluctuating field are frequency independent. In effective theories, however,such interactions are typically frequency dependent. As a consequence, the relationship between scattering data and the Green's function is modified, which may or may not induce additional contributions to the vacuum polarization energy. We discuss several examples that naturally include frequency dependent interactions: (i) scalar electrodynamics with a static background potential, (ii) an effective theory that emerges from integrating out a heavy degree of freedom, and (iii) quantum electrodynamics coupled to a frequency dependent dielectric material. In the latter case, we argue that introducing dissipation as required by the Kramers-Kronig relations requires the consideration of the Casimir energy within a statistical mechanics formalism, while in the absence of dissipation we can work entirely within field theory, using an alternative formulation of the energy density.

Vacuum polarization and Casimir energy of a Dirac field induced by a scalar potential in one spatial dimension

We investigate the vacuum polarization and Casimir energy of a Dirac field coupled to a scalar potential in one spatial dimension. Both of these effects have a common cause, which is the distortion of the spectrum of the Dirac field due to its coupling with the background field. Choosing the potential to be a symmetrical square well renders the problem exactly solvable, and we can obtain the whole spectrum of the system analytically. We show that the total number of states and the total density remain unchanged as compared with the free case, as one expects. Furthermore, since there is a reflection symmetry between positive- and negative-energy eigenstates of the fermion, the total density and the total number of negative and positive states remain unchanged, separately. This, along with the fact that there is no zero mode, mandate that the vacuum polarization in this model is zero for any choice of the parameters of the potential. It is important to note that although the vacuum polarization is zero due to the symmetries of the model, the Casimir energy of the system is not zero in general. In the graph of the Casimir energy as a function of the depth of the well, there is a maximum approximately when the bound energy levels change direction and move back towards their continuum of origin. The Casimir energy for a fixed value of the depth is an almost linear increasing function of the width. Moreover, the Casimir energy density (the energy density of all the negative-energy states) and the energy density of all the positive-energy states are exactly the mirror images of each other. Finally, we compute the total energy of a valence fermion present in the lowest positive-energy fermionic bound state. We find that taking into account the Casimir energy does not result in the appearance of any local minima in the graphs of the total energy as a function of the parameters of the model, and this is in sharp contrast to the cases where there are levels crossing the line $E=0$.

COSMIC STRINGS STABILIZED BY FERMION FLUCTUATIONS

We provide a thorough exposition of recent results on the quantum stabilization of cosmic strings. Stabilization occurs through the coupling to a heavy fermion doublet in a reduced version of the standard model. The study combines the vacuum polarization energy of fermion zero-point fluctuations and the binding energy of occupied energy levels, which are of the same order in a semiclassical expansion. Populating these bound states assigns a charge to the string. Strings carrying fermion charge become stable if the Higgs and gauge fields are coupled to a fermion that is less than twice as heavy as the top quark. The vacuum remains stable in the model, because neutral strings are not energetically favored. These findings suggest that extraordinarily large fermion masses or unrealistic couplings are not required to bind a cosmic string in the standard model.