Advances in High Energy Physics

Volume 2017 (2017), Article ID 1486912, 10 pages

https://doi.org/10.1155/2017/1486912

## Emerging Translational Variance: Vacuum Polarization Energy of the Kink

Institute for Theoretical Physics, Physics Department, Stellenbosch University, Matieland 7602, South Africa

Correspondence should be addressed to H. Weigel

Received 19 May 2017; Accepted 8 June 2017; Published 30 July 2017

Academic Editor: Ralf Hofmann

Copyright © 2017 H. Weigel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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 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.

#### 1. Motivation

It is of general interest to compute quantum corrections to classical field configurations like soliton solutions that are frequently interpreted as particles. On top of the wish list, we find the energies that predict particle masses. The quantum correction to the energy can be quite significant because the classical field acts as a background that strongly polarizes the spectrum of the quantum fluctuations about it. For that reason, the quantum correction to the classical energy is called vacuum polarization energy (VPE). Here, we will consider the leading (i.e., one-loop) contribution.

Field theories that have classical soliton solutions in various topological sectors deserve particular interest. Solitons from different sectors have unequal winding numbers and the fluctuation spectrum changes significantly from one sector to another. For example, the number of zero modes is linked to the number of (normalizable) zero modes that in turn arise from the symmetries that are spontaneously broken by the soliton. Of course, the pattern of spontaneous symmetry breaking is subject to the topological structure. On the other hand, the winding number is typically identified with the particle number. The prime example is the Skyrme model [1, 2] wherein the winding number determines the baryon number [3, 4]. Many properties of baryons have been studied in this soliton model and its generalization in the past [5]. More recently, configurations with very large winding numbers have been investigated [6] and these solutions were identified with nuclei. To obtain a sensible understanding of the predicted nuclear binding energies, it is, of course, important to consider the VPE, in particular when it is expected to strongly depend on the particle number. So far, this has not been attempted for the simple reason that the model is not renormalizable. A rough estimate [7] (see [8] for a general discussion of the quantum corrections of the Skyrmion and further references on the topic) in the context of the -dibaryon [9, 10] suggests that the VPE strongly reduces the binding energy of multibaryon states.

As already mentioned, one issue for the calculation of the VPE is renormalization. Another important one is, as will be discussed below, that the VPE is (numerically) extracted from the scattering data for the quantum fluctuations about the classical configuration [11]. Though this so-called* spectral method* allows for direct implementation of standard renormalization conditions, it has limitations as it requires sufficient symmetry for partial wave decomposition. This may not be possible for configurations with an intricate topological structure associated with large winding numbers.

The model in dimensions has soliton solutions with different topological structures [12, 13] and the fluctuations do not decouple into parity channels. The approach employed here is also based on scattering data but advances the spectral method such that no parity decomposition is required. We will also see that it is significantly more effective than previous computations [14–16] for the VPE of solitons in dimensions that are based on heat kernel expansions combined with -function regularization techniques [17–19].

Although the model is not fully renormalizable, at one-loop order, the ultraviolet divergences can be removed unambiguously. However, another very interesting phenomenon emerges. The distinct topological structures induce nonequivalent vacua that manifest themselves via different dispersion relations for the quantum fluctuations at positive and negative spatial infinity. At some intermediate position, the soliton mediates between these vacua. Since this position cannot be uniquely determined, the resulting VPE exhibits a translational variance. This is surprising since, after all, the model is defined through a local and translational invariant Lagrangian. In this paper, we will describe the emergence of this variance and link it to the different level densities that arise from the dispersion relations. To open these results for discussion (the present paper reflects the author’s invited presentation at the 5th* Winter Workshop on Non-Perturbative Quantum Field Theory* based on the methods derived in [20] making some overlap unavoidable), it is necessary to review in detail the methods developed in [20] to compute the VPE for backgrounds in one space dimension that are not (manifestly) invariant under spatial reflection.

Following this introductory motivation, we will describe the model and its kink solutions. In Section 3, we will review the spectral method that ultimately leads to a variant of the Krein–Friedel–Lloyd formula [21] for the VPE. The novel approach to obtain the relevant scattering data will be discussed in Section 4 and combined with the one-loop renormalization in Section 5. A comparison with known (exact) results will be given in Section 6 while Section 7 contains the predicted VPE for the solitons of the model. Translational variance of the VPE that emerges from the existence of nonequivalent vacua will be analyzed in Section 8. We conclude with a short summary in Section 9.

#### 2. Kinks in Models

In dimensions, thedynamics for the quantum field are governed solely by a field potential that is added to the kinetic termFor the model, we scale all coordinates, fields, and coupling constants such that the potential contains only a single dimensionless parameter :From Figure 1, we observe that there are three general cases. For , two degenerate minima at exist. For , an additional local minimum emerges at . Finally, for , the three minima at and are degenerate. Soliton solutions connect different vacua between negative and positive spatial infinity. For , the vacua are at and the corresponding soliton solution is [12]Its classical energy is The case is actually more interesting because two distinct soliton solutions do exist. The first one connects at to at :while the second one interpolates between and :These soliton configurations are shown in Figure 2. In either case, the classical mass is . This relation for the classical energies reflects the fact that as the solution disintegrates into two widely separated structures, one corresponding to and the other to .