It is intrinsic for human mind to search for principles organizing the world around us. Among different ways of building causal chains out of events we witness, the scientific method proved in centuries to be the most useful in providing us with the means of knowledge about our universe. Here, physics and, in particular, particle physics and cosmology are intended to uncover the rules governing Na-ture at very small as well as very large length scales. It is very interesting to note in this regard that the laws of microscopic physics have direct imprint on the structure of the world at cosmological distances, and by observing remote patches of the universe, one can judge about elementary particles, their properties and interactions.
There has to be an irresistible attraction in the idea of a final theory unifying all known forms of matter and interaction. The 20th century saw the impressive progress towards implementing this idea. The Standard Model (SM) of particle physics, which unifies three out of four fundamental forces, shows a great success of theoretical physics in explaining phenomena that occur at distances as small as 10−16 cm (or, equivalently, at energies as large as∼100 GeV). However, despite of its great predictive power, we cannot treat the SM as the final theory. The list of experimental and observational data that require a step out of the Model includes neutrino oscillations, dark matter, dark energy, baryon asymmetry of the universe. This is why a significant amount of efforts today are focused on searches for new physics. The latter is often placed at energy scales at the frontier of what we can access with the current experimental facilities, or much above that frontier.
A reasonable strategy to probe new physics at high energy scales, to which we have no direct access, is to ask what kind of high-energy behaviour can possibly explain a given fact about low-energy observables, which otherwise lacks a complete explanation. One way to address such questions is to do it with non-perturbative tools. The latter imply the use of classical aspects of a theory at hand and, in particular, of solutions of classical equations of motion — solitons and instantons. This thesis is dedicated to several problems in particle physics and cosmology, which seem to be well suited for being treated in this way.
Although the current understanding of fundamental laws of Nature is based on principles of quantum theory, it is true that a lot of information can be obtained by studying classical configurations arising in a theory and, in particular, solu-tions of classical equasolu-tions of motion [1]. Often these solusolu-tions represent localized lumps of fields that exhibit a particle-like behavior, although the standard def-inition of a particle in quantum field theory refers to second quantization [2].
Existing thanks to nonlinear effects, such lumps are of non-perturbative nature.
Hence, they do not show up within the perturbation theory built on top of a ho-mogeneous classical vacuum. In turn, perturbation theory must be built above a classical solution, and the latter is a valid leading-order approximation in the semiclassical expansion provided that the corresponding semiclassical parameter is small (see, e.g., [3,4]).
Localized particle-like or extended configurations of finite energy or energy den-sity, which live in a spacetime with lorentzian signature, are referred to as soli-tons [5] (see also [6]). Often they are absolutely stable due to their topological properties or conservation laws and can propagate with maintaining their shape [7, 8].1 The notion of soliton is crucial in many branches of modern physics.
Their significance is due to numerous applications in nonlinear optics [9], con-densed matter physics [10], particle physics and cosmology [11,12]. Solitons are predicted in theories beyond the SM [13] and in gravity [14], hence they are of large phenomenological interest.
Another class of nonlinear classical objects are finite-action configurations arising in a theory continued to euclidean space. These objects are similar to solitons in their mathematical structure; however, unlike solitons, they are localized in time, albeit euclidean time [3]. Because of this, they are referred to as instantons or
1One also speaks of these objects as solitary waves, while reserving the name soliton for configurations that maintain their form when interacting with each other.
pseudoparticles [15].2 At the classical level, one often has a correspondence be-tween static soliton solutions of a theory inddimensions and instanton solutions of the same theory ind+1 dimensions [1]. However, the effects engendered by in-stantons in quantum field theory are very different from those caused by solitons.
Instantons are responsible for many non-perturbative phenomena which are im-portant for the current understanding of Nature. Among them are violation of quantum numbers, false vacuum decay and phase transitions [17]. Instantons are ample in modern theories including the SM and its extensions and gravity [13,18]. The non-perturbative nature of instantons allows them to link physics at different energy domains of a theory.
In this thesis,3we use semiclassical method to obtain and study euclidean classi-cal configurations of certain types arising in theories of sclassi-calar fields and gravity.
Despite the fact that the large part of the thesis is devoted to theoretical dis-course, our goal is to address actual problems standing in particle physics and cosmology. It is these problems that help us navigate through the vastness of opportunities and select particulars for close analysis. They also limit the depth of the analysis, although many features of the objects we will study are of in-terest on their own. This particularly refers to instantons in theories of gravity, which are allowable for analytical treatment.
One type of classical configurations we are interested in is a regular bounce deter-mining the semiclassical decay of false vacuum [16,23]. In euclidean space, this solution describes a motion from a false vacuum to a region of true vacuum and back, hence the name. Our purpose here is to clarify some questions regarding metastability of the electroweak (EW) vacuum in different environments. This is the topic of the first part of the thesis, which comprises chapters2 and 3.
The second part of the thesis includes chapters 4—9. There we study specific singular configurations in theories of one or two scalar fields and dynamical gravity. Our focus is on scale-invariant models and on the models where global scale symmetry is broken explicitly in the gravitational sector. Our goal is to use singular solutions to address the so-called hierarchy problem — a long-standing challenge of particle physics that concerns with the smallness of the ratio of the observed values of the EW and the Planck energy scales.4
2A regular solution of euclidean equations of motion with exactly one negative mode is also called a bounce [16].
3The content of the thesis is based on [19–22].
4Speaking more strictly, the problem is seen whenever an energy scale much above the EW scale appears, associated with particles that are coupled to the Higgs field; see chapter4and references therein.
Below we outline more specifically the questions addressed in the thesis. For convenience, the rest of this chapter is divided in two sections corresponding to the two parts of the thesis.