• quantum field theory
• electroweak interaction
• non-topological soliton

In quantum field theory, elementary particles arise from the quantization of wave-like excitations of the fields around a stable Poincaré-invariant equilibrium configuration. Canonical quantization in interaction picture allows to write the fields as linear combinations of creation and annihilation operators that, acting on the vacuum state, generate a Fock space. This picture is inevitably incomplete, because wave-like field configurations constitute only a small part of the phase space of the classical theory.

In order to explore the regions of the Hilbert space that are invisible to perturbation theory, we must first identify and classify the static and stable solutions of the classical equationsof motion. These objects are commonly called lumps, or solitons, and are local minima of the static energy functional of the theory. The relevance of these field configurations at the quantum level can be intuitively grasped in the Schrödinger picture: given a local minimum of the static energy functional, we expect that the Hilbert space contains at least one metastable state whose wavefunction is sharply peaked around it. The expectation value of an operator on such a state is, by construction, equal to its classical counterpart up to some quantum corrections that are due to the tails of the wavefunction. These corrections can be determined by studying the small fluctuations of the fields around the equilibrium point. In this way, we find a sort of correspondence between a class of classical field configurations and a class of quantum states that cannot be reached perturbatively.

Solitons are not individual objects, but rather classes of objects that are connected by symmetry transformations. Indeed, by acting on a solution of some equations of motion with the symmetry group of the theory, it is possible to obtain other legitimate solutions with the same energy. Hence, each individual soliton of a given class is labelled by a set of coordinates that describe its collective degrees of freedom e.g. its position and orientation in space. Contrarily to the other fluctuations, these degrees of freedom have no mass gap, and therefore are expected to be dominant at low energies. When they are quantized, they generate a Hilbert space that can be decomposed into irreducible representations of the Galileo group and of the internal symmetry group of the theory. Hence, in the low-energy limit, soliton states behave as one-particle states. Furthermore, when fermions are coupled to the constituent fields of a soliton, the soliton acquires a generally non-integer fermion number, whose fractional part can be computed perturbatively.

Several solitonic field configurations have been studied in literature so far: kinks, magnetic
monopoles, dyons, Q-balls, vortices. Some of them are protected by a topological conservation law, that forbids their decay into the perturbative vacuum both at the classical and at the quantum level. One of the field configurations that enjoy topological protection is the skyrmion, a soliton of the chiral theory that was found by T.H.R. Skyrme in 1961 in an attempt to formulate a unified theory of mesons and baryons before the quark model was available. After being neglected for twenty years, the Skyrme model was revived by G.S. Adkins, C.R. Nappi and E. Witten in 1983, with a work aimed to study their exquisitely quantum properties. It was found that the predictions of the Skyrme model agree with a 30% accuracy with the current experimental data on baryons, and that this discrepancy disappears in the ’t Hooft limit.

Skyrmionic field configuration have been found to exist also in the theory of the electroweak interactions, with some relevant differences: they are metastable and lack topological protections. This soliton is often called techniskyrmion or, more commonly, electroweak skyrmion. Several authors have studied this subject from a purely classical point of view, under different assumptions.

The electroweak skyrmion appeared in literature for the first time in 1984, in an article of E. D’Hoker and E. Farhi that studied the phenomenology of the electroweak field configuration with non-trivial topology. The authors assumed the presence of a new non-renormalizable term, that must be added to the electroweak lagrangian in order to stabilize the skyrmion. Two years later, two articles, one by G. Eliam, D. Klabucar and A. Stern, and one by J. Ambjørn and V.A. Rubakov, studied numerically this possibility, finding that the addition of this term was a necessity. The existence of multi-skyrmion solutions was studied in a 1988 paper by Y. Brihaye and J. Kunz, and the classical process of production and destruction of electroweak skyrmions was studied numerically in a 1996 article by E. Farhi, J. Goldstone, A. Lue and K. Rajagopal. All these works made the simplifying assumptions of a non-dynamical Higgs scalar and of a decoupled hypercharge field.

The topic was revived in 2012 by J.C. Criado, V.V. Khoze and M. Spannowsky, who performed a numerical study relaxing the assumption of frozen Higgs. They found the region of the parameter space that allows the existence of the electroweak skyrmion, deriving an upper bound of approximately 8 TeV for its mass, and noting how the production and destruction of these states are B+L- violating processes. Other papers on the subject appeared subsequently, discussing in particular the possibility of the electroweak skyrmion as a dark matter constituent.

The present thesis has the aim to progress along these lines by studying the quantum properties of the electroweak skyrmion, following mainly an approach à la Adkins-Nappi-Witten. It is organized into five Chapters with the following structure

• In Chapter 1 we review the general theory of solitons, presenting a fully quantum-mechanical treatment of the subject. A brief discussion on the role and meaning of Euclidean solutions is also present.

• In Chapter 2 we present the methods and techniques adopted to study the effects of the coupling of fermions to the soliton fields.

• In Chapter 3 we introduce the Skyrme model, reviewing the main results obtained by various authors on the subject and discussing its virtues and limitations.

• In Chapter 4 we review the existing literature on the electroweak skyrmion and present some new results, discussing the assumptions and limitations of the model. In particular, we construct an Hilbert space for the collective degrees of freedom of the skyrmion, and find its spin, statistics and other quantum numbers. Afterwards, we compute perturbatively the lepton number of the soliton and find an expression for the long-distance interaction between two skyrmions in semiclassical approximation. Then, we propose a spherically symmetric Euclidean solution that describes its decay, and discuss the possibility of observing electroweak skyrmions in present or future collider experiments.

• Chapter 5 is an appendix that contains some derivations and technicalities omitted in the rest of the thesis.