H. Weigel

Calculating vacuum energies in renormalizable quantum field theories:: A new approach to the Casimir problem

Abstract The Casimir problem is usually posed as the response of a fluctuating quantum field to externally imposed boundary conditions. In reality, however, no interaction is strong enough to enforce a boundary condition on all frequencies of a fluctuating field. We construct a more physical model of the situation by coupling the fluctuating field to a smooth background potential that implements the boundary condition in a certain limit. To study this problem, we develop general new methods to compute renormalized one-loop quantum energies and energy densities. We use analytic properties of scattering data to compute Greens functions in time-independent background fields at imaginary momenta. Our calculational method is particularly useful for numerical studies of singular limits because it avoids terms that oscillate or require cancellation of exponentially growing and decaying factors. To renormalize, we identify potentially divergent contributions to the Casimir energy with low orders in the Born series to the Greens function. We subtract these contributions and add back the corresponding Feynman diagrams, which we combine with counterterms fixed by imposing standard renormalization conditions on low-order Greens functions. The resulting Casimir energy and energy density are finite functionals for smooth background potentials. In general, however, the Casimir energy diverges in the boundary condition limit. This divergence is real and reflects the infinite energy needed to constrain a fluctuating field on all energy scales; renormalizable quantum field theories have no place for ad hoc surface counterterms. We apply our methods to simple examples to illustrate cases where these subtleties invalidate the conclusions of the boundary condition approach.

Baryons as chiral solitons in the Nambu-Jona-Lasinio model

Abstract The description of baryons as chiral solitons of the Nambu-Jona-Lasinio (NJL) model is reviewed. A motivation for the soliton description of baryons is provided from large NC QCD. Rigorous results on the spontaneous breaking of chiral symmetry in QCD are discussed. It is then argued that the NJL model provides a fair description of low-energy hadron physics. The NJL model is therefore employed to mimic the low-energy chiral flavor dynamics of QCD. The model is bosonized by functional integral techniques and the physical content of the emerging effective meson theory is discussed. In particular, its relation to the Skyrme model is established. The static soliton solutions of the bosonized NJL model are found, their properties discussed, and the influence of various meson fields studied. These considerations provide strong support of Wittens conjecture that baryons can be understood as soliton solutions of effective meson theories. The chiral soliton of the NJL model is then quantized in a semiclassical fashion and various static properties of the nucleon are studied. The dominating 1 N C corrections to the semiclassically quantized soliton are investigated. Time-dependent meson fluctuations off the chiral soliton are explored and employed to estimate the quantum corrections to the soliton mass. Finally, hyperons are described as chiral solitons of the NJL model. This is done in both, the collective rotational approach of Yabu and Ando as well as in the bound state approach of Callan and Klebanov.

The Dirichlet Casimir Problem

Casimir forces are conventionally computed by analyzing the effects of boundary conditions on a fluctuating quantum field. Although this analysis provides a clean and calculationally tractable idealization, it does not always accurately capture the characteristics of real materials, which cannot constrain the modes of the fluctuating field at all energies. We study the vacuum polarization energy of renormalizable, continuum quantum field theory in the presence of a background field, designed to impose a Dirichlet boundary condition in a particular limit. We show that in two and three space dimensions, as a background field becomes concentrated on the surface on which the Dirichlet boundary condition would eventually hold, the Casimir energy diverges. This result implies that the energy depends in detail on the properties of the material, which are not captured by the idealized boundary conditions. This divergence does not affect the force between rigid bodies, but it does invalidate calculations of Casimir stresses based on idealized boundary conditions.

Mesons in a Poincaré covariant Bethe-Salpeter approach

We develop a covariant approach to describe the low{lying scalar, pseudoscalar, vector and axialvector mesons as quark{antiquark bound states. This approach is based on an eective interaction modeling of the non{ perturbative structure of the gluon propagator that enters the quark Schwinger{Dyson and meson Bethe{Salpeter equations. We consistently treat these integral equations by precisely implementing the quark propagator functions that solve the Schwinger{Dyson equations into the Bethe{Salpeter equations in the relevant kinematical region. We extract the meson masses and compute the pion and kaon decay constants. We obtain a quantitatively correct description for pions, kaons and vector mesons while the calculated spectra of scalar and axialvector mesons indicate that their structure is more complex than being quark{antiquark bound states.

The skyrme soliton in pion, vector- and scalar-meson fields: πN-scattering and photoproduction☆

Abstract The non-linear σ-model for pions is extended to incorporate ϱ and ω-mesons as massive gauge bosons in a non-linear realization of chiral symmetry. The chiral anomalies are included in gauge-invariant form. An additional scalar-isoscalar meson reproduces the QCD trace anomaly. Baryons are obtained as hedgehog solitons in the meson fields, and the S-matrix for meson-baryon scattering is derived by solving the coupled channels equations for the mesonic fluctuations around the static soliton. Coupling to the electromagnetic sector is established by gauging the meson lagrangian. Results for elastic πN-scattering and photoproduction of pions are discussed in detail. For a description of the Δ33 resonance dynamic al effects of the soliton isorotation are considered.

Radial excitations of low{lying baryons and the Z + penta{quark

Abstact: Within an extended Skyrme soliton model for baryons the interplay between the collective radial motion and the SU(3)–flavor–rotations is investigated. The coupling between these modes is mediated by flavor symmetry breaking. Collective coordinates which describe the corresponding large amplitude fluctuations are introduced and treated canonically. When diagonalizing the resulting Hamiltonian flavor symmetry breaking is fully taken into consideration. As eigenstates not only the low–lying (1/2)+ and (3/2)+ baryons but also their radial excitations are obtained and compared to the empirical data. In particular the relevance of radial excitations for the penta–quark baryon Z+ (Y=2, I=0, Jπ=(1/2)+) is discussed. In this approach its mass is predicted to be 1.58 GeV. Furthermore the widths for various hadronic decays are estimated which, for example, yields Γ(Z+→NK) ∼ 100 MeV for the only permissible decay process of the Z+.

Casimir energies in light of quantum field theory

We study the Casimir problem as the limit of a conventional quantum field theory coupled to a smooth background. The Casimir energy diverges in the limit that the background forces the field to vanish on a surface. We show that this divergence cannot be absorbed into a renormalization of the parameters of the theory. As a result, the Casimir energy of the surface and other quantities like the surface tension, which are obtained by deforming the surface, cannot be defined independently of the details of the coupling between the field and the matter on the surface. In contrast, the energy density away from the surface and the force between rigid surfaces are finite and independent of these complications.

CASIMIR EFFECTS IN RENORMALIZABLE QUANTUM FIELD THEORIES

We present a framework for the study of one–loop quantum corrections to extended field configurations in renormalizable quantum field theories. We work in the continuum, transforming the standard Casimir sum over modes into a sum over bound states and an integral over scattering states weighted by the density of states. We express the density of states in terms of phase shifts, allowing us to extract divergences by identifying Born approximations to the phase shifts with low order Feynman diagrams. Once isolated in Feynman diagrams, the divergences are canceled against standard counterterms. Thus regulated, the Casimir sum is highly convergent and amenable to numerical computation. Our methods have numerous applications to the theory of solitons, membranes, and quantum field theories in strong external fields or subject to boundary conditions.

BARYONS AS THREE-FLAVOR SOLITONS

The description of baryons as soliton solutions of effective meson theories for three flavor (up, down, strange) degrees of freedom is reviewed and the phenomenological implications are illuminated. In the collective approach the soliton configuration is equipped with baryon quantum numbers by canonical quantization of the coordinates describing the flavor orientation. The baryon spectrum resulting from exact diagonalization of the collective Hamiltonian is discussed. The prediction of static properties such as the baryon magnetic moments and the Cabibbo matrix elements for semi--leptonic hyperon decays are explored with regard to the influence of flavor symmetry breaking. In particular, the role of strange degrees of freedom in the nucleon is investigated for both the vector and axial--vector current matrix elements. The latter are discussed extensively within in the context of the {\it proton spin puzzle}. The influence of flavor symmetry breaking on the shape of the soliton is examined and observed to cause significant deviations from flavor covariant predictions on the baryon magnetic moments. Short range effects are incorporated by a chiral invariant inclusion of vector meson fields. These extensions are necessary to properly describe the singlet axial--vector current and the neutron proton mass difference. The effects of the vector meson excitations on baryon properties are also considered. The bound state description of hyperons and its generalization to baryons containing a heavy quark are illustrated. In the case of the Skyrme model a comparison is performed between the collective quantization scheme and bound state approach. Finally, the Nambu--Jona--Lasinio model is employed to demonstrate that hyperons can be described as solitons in a microscopic theory of the quark

Quantum energies of interfaces.

We present a method for computing the one-loop, renormalized quantum energies of symmetrical interfaces of arbitrary dimension and codimension using elementary scattering data. Internal consistency requires finite-energy sum rules relating phase shifts to bound state energies.