We calculate the vacuum polarization energy (VPE) of a scalar field in the background of a nonsingular cosmic string in 2+1 spacetime dimensions. Our calculation expresses the VPE as a renormalized sum and integral over scattering data, which can be expressed in terms of Legendre and Bessel functions for the "ballpoint pen" model, analogous to a square well in curvature, and which can be obtained numerically for a generic string profile. We show how relationships between the local density of states, expressed in terms of the Green's function, and the global density of states, expressed in terms of the Jost function, extend to this curved spacetime background and allow for precise implementation of perturbative renormalization conditions.

Generalizing quantum chromo dynamics (QCD) from three to arbitrarily many color degrees of freedom suggests that baryons can be described as solitons in an effective meson theory whose interaction strength decreases with the number of colors. The exact form of that theory is unknown, but at low energies chiral symmetry and its breaking are considered as the construction recipes for modeling the theory. The Skyrmion is a static, localized solution in a non-linear field theory for pions and it is the most prominent version of a soliton in a chirally symmetric meson theory. Upon quantization it reproduces the spectrum of the low-lying baryons and their static properties reasonably well. Extending that theory by vector mesons improves on the agreement between predicted and empirical data. Chiral solitons within models for the quark flavor dynamics facilitate the investigation of nucleon structure functions. Here we provide a pedagogical overview of these facets.

We study an extended Proca model with one scalar field and one massive vector field in one space dimension and one time dimension. We construct the soliton solution and subsequently compute the vacuum polarization energy (VPE), which is the leading quantum correction to the classical energy of the soliton. For this calculation, we adopt the spectral methods approach, which heavily relies on the analytic properties of the Jost function. This function is extracted from the interaction of the quantum fluctuations with a background potential generated by the soliton. Particularly, we explore eventual non-analytical components that may be induced by mass gaps and the unconventional normalization for the longitudinal component of the vector field fluctuations. By numerical simulation, we verify that these obstacles do not actually arise and that the real and imaginary momentum formulations of the VPE yield equal results. The Born approximation to the The Jost function is crucial when implementing standard renormalization conditions. In this context, we solve problems arising from the Born approximation being imaginary for real momenta associated with energies in the mass gap.

We investigate serval models for two scalar fields in one space dimension with topologically stable solitons that are constructed from BPS equations. The asymptotic behavior of these solitons fully determines their classical energies. A particular feature of the considered models is that there are several translationally invariant ground states that we call primary and secondary vacua. The former are those that are asymptotically assumed by the solitons. Solitons that occupy a secondary vacuum in finite but eventually large portions of space are classically degenerate. Thus the quantum contributions to the energies are decisive for the energetically favored soliton. While some of these solitons were constructed previously, we, for the first time, compute the leading (one-loop) quantum contribution their energies. In all cases considered we find that this contribution is not bounded from below and that it is the more negative the larger the region is in which the soliton approaches a secondary vacuum. This corroborates the conjecture, earlier inferred from the Shifman-Voloshin soliton, that the availability of secondary vacua destabilizes these solitons on the quantum level.

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.

We compute the renormalized one-loop quantum corrections to the energy density T00(x) and pressure T11(x) for solitons in the 1+1 dimensional scalar sine-Gordon and kink models. We show how precise implementation of counterterms in dimensional regularization resolves previously identified discrepancies between the integral of T00(x) and the known correction to the total energy.

Nonlinear field theories often possess the so-called soliton solutions that have localized energy densities and that are characterized by topological charges n. To explore the stability of solitons with n>1, we consider vortices in scalar electrodynamics which is one of the rare renormalizable models that contain soliton type solutions with different topological charges. We focus on the BPS case which is particularly interesting because the quantum corrections decide on the stability of vortices with n>1. We also explain how to overcome technical obstacles that are encountered when computing these corrections and that arise from the singular structure of the vortex profiles.

We analyze the interaction of lattice vibrations (phonon wave-packets) with an asymmetric kink soliton initially at rest. We employ the $\phi^6$ model in one space and one time dimensions for various lattice spacings and consider two different discretization prescriptions for the field potential that do not generate Peierls-Nabarro potentials, i.e. the kink can be placed anywhere along the lattice beyond discrete translational invariance. Since the $\phi^6$ model kink is neither symmetric nor anti-symmetric under spatial reflections we simulate the cases where the wave-packet approaches the kink from negative or positive spatial infinity. We extract the energy transmission and reflection coefficients as functions of the central frequency of the phonon wave-packet for the different lattice spacings. For large lattice spacings the wave-packet is always fully reflected while for smaller spacing the amount of reflection and transmission depends on the central frequency. We also identify scenarios in which the target kink acquires a non-zero velocity from its interaction with the wave-packet.

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 semiclassical 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.

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

We consider vortices in scalar electrodynamics and compute the leading quantum correction to their energies for the BPS case of identical classical masses of the Higgs and gauge fields. In particular, we focus on the winding number $n$ dependence of these corrections, from which we can extract the binding energies of configurations with larger $n$. For both dimensionalities, $D=2+1$ and $D=3+1$, we find that quantum corrections are negative and scale approximately linearly with $n$, so that combined vortices are favored over isolated ones.

We review recent progress in the computation of leading quantum corrections to the energies of classical solitons with topological structure, including multi-soliton models in one space dimension and string configurations in three space dimensions. Taking advantage of analytic continuation techniques to efficiently organize the calculations, we show how quantum corrections affect the stability of solitons in the Shifman-Voloshin model, stabilize charged electroweak strings coupled to a heavy fermion doublet, and bind Nielsen-Olesen vortices at the classical transition between type I and type II superconductors.

We outline and review the computations of polarized and unpolarized nucleon structure functions within the bosonized Nambu-Jona-Lasinio chiral soliton model. We focus on a consistent regularization prescription for the Dirac sea contribution and present numerical results from that formulation. We also reflect on previous calculations on quark distributions in chiral quark soliton models and attempt to put them into perspective.

Baryons containing heavy quarks have drawn renewed attention since the potential discovery of pentaquark states with hidden charm. Here we present a model calculation for the spectrum of baryons with a single heavy quark. We start from a chiral soliton model with pseudoscalar and vector meson fields that reasonably well describes the spectrum, static properties and meson nucleon scattering. These studies are reviewed in Ref.2. We then augment this model by coupling mesons with a single heavy quark (charm or bottom). This coupling is constructed to reproduce the heavy spin–flavor symmetry as the mass of the heavy meson is sent to infinity. The soliton then produces an attractive potential for the heavy mesons and the resulting bound states are central to the investigation of heavy baryons. In this presentation, which is mainly based on Ref.5, we particularly discuss how flavor symmetry breaking between the light non–strange and strange quarks is included when coupling the chiral soliton and the heavy meson bound state to form a baryon with a heavy quark.

We compute the one-loop vacuum polarization energies of Abrikosov-Nielsen-Olesen (ANO) vortices with topological charge $n$ in scalar electrodynamics, for the BPS 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 $n-1$ with a small constant offset.

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 the solitons. 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.

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.

. We present numerical simulations for unpolarized and polarized structure functions in a chiral soliton model. The soliton is constructed self-consistently from quark fields from which the structure functions are extracted. Central to the project is regularizing the Dirac sea (or vacuum) contribution to structure functions directly from the regularized action functional that defines the model. In turn, the structure functions are obtained from matrix elements of symmetry currents without assumptions on the nature of quark bilocal and bilinear operators. We discuss in detail how sum rules are realized at the level of the quark wave-functions in momentum space. The comparison with experimental data is convincing for the polarized structure functions but exhibits some discrepancies in the unpolarized case. The vacuum contribution to the polarized structure functions is particularly small.

We compute the vacuum polarization energies for a couple of soliton models in one space and one time dimensions. These solitons are mappings that connect different degenerate vacua. From the considered sample solitons we conjecture that the vacuum polarization contribution to the total energy leads to instabilities whenever degenerate vacua with different curvatures in field space are accessible to the soliton.

Collective coordinate methods are frequently applied to study dynamical properties of solitons. These methods simplify the field equations - typically partial differential equations - to ordinary differential equations for selected excitations. More importantly though, collective coordinates provide a practical means to focus on particular modes of otherwise complicated dynamical processes. We review the application of collective coordinate methods in the analysis of the kink-antikink interaction within the φ^4 soliton model and illuminate discrepancies between these methods and the exact results from the field equations.