Baryons as Solitons

(information awaits update)

A pedagogical introduction to the concept of "Chiral Soliton Models for Baryons" is available as Springer Lecture Notes.

In the limit of infinitely many color degrees of freedom quantum-chromo-dynamics (QCD) is equivalent to an effective theory of weakly interacting mesons. In this effective meson theory baryons then emerge as soliton solitons, i.e. field configurations which solve the classical equations of motion and yield a localized energy density. Based of this feature of QCD, we attempt to obtain a comprehensive model for the physics of baryons.
Although the specific effective meson theory remains unknown many aspects of baryons physics can be illuminated from this perspective. Since at low energy only light mesons should be important the effective theory may well be approximated via the symmetries of QCD in that energy regime Besides Lorentz-covariance this is in particular chiral symmetry and its spontaneous breaking. In the simplest case one is lead to the Skyrme soliton model whose only degrees of freedom are the pions. The latter being the Goldstone bosons of spontaneous chiral symmetry breaking. Extensions of the Skyrme model can be obtained by applying the rules of chiral symmetry to the meson dynamics. In such an approach many static nucleon properties (magnetic moments, spin decomposition etc.) can be described reasonably well. In particular we are interested in the generalization to the three flavor case by incorporating strange degrees of freedom [1]. This not only allows us to discuss static properties of hyperons but also the effects of virtual strange quark-antiquark pairs in the nucleon.
A key issue of our research project is to consider the Nambu-Jona-Lasinio model as an approximation to the quark flavor dynamics. Using functional integral techniques the effective meson theory equivalent to the NJL model can be determined. Remarkably the resulting effective meson theory contains soliton solutions[2]. In contrast to Skyrme type models the connection to the quark degrees is known explicitly. Hence the NJL model soliton is well suited to study issues which are directly related to the quark substructure of the nucleon. Along this line we have recently succeeded to compute nucleon structure function in this chiral soliton model [3] [4] [5]. More recently very exhaustive numerical simulations on these ideas have been performed [6]. The ultimate goal of these investigations is to describe the connection between the soliton picture and the QCD parton model. More recently the soliton picture has acquired renewed interest in the context of pentaquark baryons. These are states that in (non-relativistic) quark model are composed of four quarks and an anti-quark. Consequently such models predict large pentaquark masses. The opposite is the case in the soliton model. Here pentaquarks arise as collective flavor excitations of the nucleon and hence should be comparable in mass. Mostly our interest has been in the mixing scenario between pentaquarks and radial excitations of the nucleon [7] as well as a proper description the pentaquark width (e.g. lifetime) [8]. This proper description can be verified in the limit of infinitely many color degrees of freedom, where the small and large amplitude fluctuations are equivalent. For the most prominent collective nucleon excitation, the Δ resonance, this test is not available because nucleon and Δ are degenerate and the Δ is hence stable in that limit. Thus, even though the pentaquarks may not be as fascinating as earlier assumed, they have been very fruitful in understanding the lifetimes of collective excitations in the continuum[9]. The chiral soliton model description has been extend to heavy baryons that, in a valence quark picture, contain strange and charm (or bottom) baryons[10]. Interestingly enough, any self-contained description of both exotic and and heavy baryons requires considerations beyond the mean field approach to chiral soliton models[11].

Some selected research publications:

[1] H. Weigel,
Baryons as three flavor solitons,
Int. J. Mod. Phys. A11 (1996) 2419.

[2] R. Alkofer, H. Reinhardt, H. Weigel,
Baryons as chiral solitons in the Nambu--Jona--Lasinio model,
Phys. Rep. 265 (1996) 139.

[3] H. Weigel, L. Gamberg, H. Reinhardt,
Nucleon structure functions from a chiral soliton,
Phys. Lett. B399 (1997) 287.

[4] O. Schröder, H. Reinhardt, H. Weigel,
Nucleon structure functions in the three flavor NJL soliton model,
Nucl. Phys. A651 (1999) 174.

[5] H. Weigel, E. Ruiz Arriola, L. Gamberg,
Hadron Structure Functions in a Chiral Quark Model: Regularization, Scaling and Sum Rules,
Nucl. Phys. B560 (1999) 383.

[6] I. Takyi, H. Weigel,
Nucleon structure functions from the NJL-model chiral soliton,
Eur. Phys. J. A55 (2019) 8
       H. Weigel I. Takyi,
Chiral Soliton Models and Nucleon Structure Functions,
Symmetry 2021 13(1), 108

[7] H. Weigel,
Radial excitations of low--lying baryons and the Z+ penta--quark,
Euro. Phys. J. A2 (1998) 391.
Exotic Baryons and Monopole Excitations in a Chiral Soliton Model,
Eur. Phys. J. A21 (2004) 133.

[8] H. Walliser, H. Weigel,
Bound State versus Collective Coordinate Approaches in Chiral Soliton Models and the Width of the Θ+ Pentaquark,
Eur. Phys. J. A26 (2005) 361.

[9] H. Weigel,
Collective Resonances in the Soliton Model Approach to Meson-Baryon Scattering,
Eur. Phys. J. A31 (2007) 495.

[10] J. P. Blanckenberg and H. Weigel,
Heavy Baryons with Strangeness in a Soliton Model,
Phys. Lett. B750 (2015) 230.

[11] H. Weigel,
Exotic Baryons in Chiral Soliton Models,
Universe 2018 4, 4120142.

Quantum Energies of Solitons

(information awaits update)

Energies that arise from quantum fluctations about a non-trivial background configuration can be computed with spectal methods that utilize scattering data associated with this background. A pedagogical introduction to "Spectral Methods in Quantum Field Theory" is available as Springer Lecture Notes. A more recent review has been published as a chapter in book on the future of the Casimir energy.

In renormalizable models one-loop quantum corrections are considered in the construction of the soliton configuration. In order to uniquely determine the finite parts of these corrections to the energy it is important to cancel the ultra-violet divergences by the same counterterms as in the perturbative sector. This is accomplished by identifying the Born approximants in the phase shift expression for the Casimir energy and perturbatively computed Feynman diagrams. Consequently also the scale dependence of the regularization procedure drops out. Finally the minimum of the energy functional is constructed by varying those parameters which characterize the localized field configuration. In some models those quantum corrections can actually stabilize a soliton which classically is unstable. [1]. Furthermore it is shown that the scattering phase shifts together with Levinson's theorem may be utilized to determine the charge of the vacuum that is polarized by the background soliton configuration. This is in particular the case for non-integer charges as for example carried by the chiral bag-model in 3+1 dimensions [2]. Furthermore this formalism can be generalized to geometric configurations with the background field dwelling only in a subset of the coordinates. In the other coordinates the background field equals the vacuum configuration. Renormalizability of the considered quantum field theory then yields novel sum rules between the phase shifts and the bound state energies. These sum rules can be interpreted as generalizations of Levinson's theorem [3].

These techniques are successfully applied to gain deeper understanding of the Casimir effect because it allows for the important distinction between divergences inherited from the quantum field theory and those that are induced by singular backgrounds. The former are curred by standard renormalization techniques but the latter can only be removed by proper consideration of material properties. Fortunately, these properties do not affect the standard results for the Casimir force between rigid bodies. Yet the similar approach to the Casimir self-tension of single objects must be questioned [4], [5].

Using the methods described in ref. [1] the vacuum polarization energies of magnetic vortices have been computed [6]. In particular, the obstacles that arise for regularization from the Feynman series not being gauge invariant term by term have been solved for string type configurations in the standard model [7]. We have found that the fermion contribution to the vacuum polarization energies of these configurations is much smaller than their classical energies unless the fermion mass is about ten times as large as that of the top quark [8]. This is a step towards investigating the existence of such structures that eventually have significant impact on our understanding of cosmology. On the technical side, the studies of ref. [7] provide further support for the use of spectral methods to compute the vacuum polarization energies as they verify the equality of the finite parts of the Born approximants and the Feynman diagrams. Recently we have shown [9] that by populating the bound fermion levels induced by a cosmic string dynamically stabilizes the string if the fermion mass parameter is only slightly larger than suggested by the top quark mass.

The calculation of the vacuum polarization energies for cosmic strings involves a number of complicated (numerical) computations that individually are not gauge invariant. It has been shown that, when collecting all pieces, gauge invariance is maintained[10]. The quantum stabilization of closed abelian strings has been discussed in [11].

Some recent work focuses on vacuum polarization energies of systems with mass gaps [12]. Here the conjecture emerges that quantum corrections destabilze a classically stable soliton when the field potential has degenerate minima with different curvatures.

Also some conceptual questions of the Casmir energy have been explored [13].

Abelian strings (vortices) are characterized by singular gauge potentials. The standard approach to find the vacuum polarization energy therefore requires modification that is derived and applied in [14]. The picture emerges that the quantum corrections favor a lump of vortices with the certain topological charge of isolated ones with the same total charge.

Some selected research publications:

[1] E. Farhi, N. Graham, R. L. Jaffe, H. Weigel,
Heavy Fermion Stabilization of Solitons in 1+1 Dimensions
Nucl. Phys. B585 (2000) 443.

[2] E. Farhi, N. Graham, R. L. Jaffe, H. Weigel,
Fractional and Integer Charges from Levinson's Theorem,
Nucl. Phys. B595 (2001) 536.

[3] N. Graham, R. L. Jaffe, M. Quandt, H. Weigel,
Quantum Energies of Interfaces,
Phys. Rev. Lett. 87 (2001) 31601.

[4] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, M. Scandurra, H. Weigel,
Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem,
Nucl. Phys. B645 (2002) 49.

[5] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, O. Schröder, H. Weigel,
The Dirichlet Casimir Problem,
Nucl. Phys. B677 (2004) 379.

[6] N. Graham, M. Quandt, O. Schröder, H. Weigel,
Quantum QED Flux Tubes in 2+1 and 3+1 Dimensions,
Nucl. Phys. B707 (2005) 233.

[7] N. Graham, M. Quandt, O. Schröder, H. Weigel,
Quantum Energies of Strings in a 2+1 Dimensional Gauge Theory,
Nucl. Phys. B707 (2005) 233
       H. Weigel, M. Quandt, N. Graham, O. Schröder,
Vacuum Energies of Non-Abelian String-Configurations in 3+1 Dimensions,
Nucl. Phys. B831 (2010) 306.

[8] H. Weigel, M. Quandt,
Gauge Invariance and Vacuum Energies of Non-Abelian String-Configurations,
Phys. Lett. B690 (2010) 514.

[9] H. Weigel, M. Quandt, N. Graham,
Stable Charged Cosmic Strings,
Phys. Rev. Lett. 106 (2011) 101601
       N. Graham, M. Quandt, H. Weigel,
Fermion Energies in the Background of a Cosmic String,
Phys. Rev. D84 (2011) 025017.

[10] H. Weigel, M. Quandt, N. Graham,
Isospin Invariance and the Vacuum Polarization Energy of Cosmic Strings
Phys. Rev. D94 (2016) 045015.

[11] M. Quandt, N. Graham, H. Weigel,
Quantum Stabilization of a Closed Nielsen-Olesen String,
Phys. Rev. D87 (2013) 085013.

[12] H. Weigel,
Vacuum Polarization Energy for General Backgrounds in One Space Dimension,
Phys. Lett. B766 (2017) 65
       H. Weigel, M. Quandt, N. Graham,
Spectral Methods for Coupled Channels with a Mass Gap,
Phys. Rev. D97 (2018) 036017
       H. Weigel, N. Graham,
Vacuum Polarization Energy of the Shifman-Voloshin Soliton,
Phys. Lett. B783 (2018) 434.

[13] N. Graham, M. Quandt, H. Weigel,
Attractive Electromagnetic Casimir Stress on a Spherical Dielectric Shell,
Phys. Lett. B726 (2013) 846
       N. Graham, M. Quandt, H. Weigel,
On the Casimir Energies of Frequency Dependent Interactions,
Phys. Rev. D90 (2014) 085004.

[14] N. Graham, H. Weigel,
Vacuum Polarization Energy of a Complex Scalar Field in a Vortex Background,
Phys. Rev. D101 (2020) 076006
       N. Graham, H. Weigel,
Winding Number Dependence of Quantum Vortex Energies at one-loop,
Phys. Rev. D104 (2021) L011901
       N. Graham, H. Weigel,
Quantum Energies of BPS Vortices in D=2+1 and D=3+1,
Phys. Rev. D106 (2022) 076013.

Kinks and Antikinks

(information awaits update)

More recent work focuses on collective corrdinates in kink-antikink scattering [1] in soliton models with one space dimension. In particular the question was considered under which circumstance bouncing configurations emerge and which mechanism(s) drives these bounces. The approach can be also applied to lattices by transforming the differential field equations to difference equation, i.e. avoiding the continuum limit [2].

Some selected research publications:

[1] I. Takyi, H. Weigel,
Collective Coordinates in One-Dimensional Soliton Models Revisited,
Phys. Rev. D94 (2016) 085005.
       H. Weigel,
Kink-Antikink Scattering in ϕ4 and φ6 Models,
J. Phys. Conf. Series, 482 (2014) 012045.

[2] D. Saadatmand, A. M. Marjaneh, A. Askari, H. Weigel,
Phonons Scattering off Discrete Asymmetric Solitons in the Absence of a Peierls-Nabarro Potential,
Chaos, Solitons and Fractals 180 (2024) 114550.