Higher corrections to non-Abelian vortex low-energy dynamics

In the 1960s a new approach to quantum field theory developed becoming a new frontier of research. Physicists and mathematicians began to study the classical field equations in their fully non-linear form trying to interpret their solutions as new particle-like objects. Differently from the elementary particles that arise from the quantization of the wave-like excitations, these new particles, called solitons, appear as smooth, classical solutions, whose energy-density is concentrated in a localized area. Their stability is ensured by their topological structure that depends on the particular boundary condition of the field. Indeed, to such boundary map can be associated a topological classification that, once fixed, cannot be changed without breaking the finiteness of the energy or the continuity of the field. In this way, it represents an invariant of the solution during the time-evolution and prevents the soliton from decaying into the vacuum. In many cases, the topological information of the field is captured by an integer N, called topological charge, that is directly connected with the choice of the boundary conditions. Solitons arise in many different examples of field theories and their physical interpretation varies depending on the specific model considered. Prominent examples of solitons are istantons that arise in an Euclidean Yang-Mills theory. Being classical solutions of an Euclidean Action, their contribution becomes significant in the study of quantum processes in which the Euclidean functional integral is involved. Another example of soliton is given by the t'Hooft-Polyakov monopole that emerges in a non-Abelian gauge theory spontaneously broken. This theory allows for the existence of smooth soliton solutions that asymptotically behave as magnetic monopoles. Other important examples of solitons are the vortices, which are the main focus of this thesis, sigma model Lumps, Kinks, domain walls, skyrmions and so on. Vortices are one of the first examples of solitons that have been considered in Field Theory and their discovery was connected with the study of condensed matter phenomena, such as superconductors. Within the Ginzburg-Landau (GL) approach in describing superconductors, a complex field, that can be interpreted as the wave-function of the Cooper electron-pair, is coupled to the electromagnetic gauge potential. The superconducting phase is characterized by the spontaneous symmetry breaking of the Abelian gauge group U(1) that introduces a length scale to the electromagnetic field. Due to the existence of such a scale, the magnetic field cannot propagate freely into the superconducting medium but it decays exponentially on its surface. This physical phenomenon, which consists in the expulsion of the magnetic field from the superconductor, is known as Meissner effect. It was discovered by Abrikosov in 1957 that the GL energy function allows for the existence of topological vortices. Arising from the breaking of an Abelian gauge symmetry, these solutions are commonly known as Abelian vortices. In three dimensions a single Abelian vortex appears as an infinite magnetic flux tube, squeezed into the superconductor by the Meissner effect, whose total energy is proportional to its length. The magnetic flux of such tube is proportional to the topological charge N, that classifies the different solutions. Abelian vortices are experimentally observed in Type II superconductors under appropriate conditions of magnetic field. These features of the Abelian vortices remind us of the semi-classical picture of the QCD-string generated by a quark-antiquark pair. In such a picture, the chromo-electric field, generated by the quarks charges, is squeezed into a flux tube whose energy-density per unit length is constant. Following this idea, in the mid-1970s 't Hooft, Mandelstam and Nambu proposed a soliton model for the QCD confinement, known as dual-Meissner effect. In such a conjecture, the QCD vacuum might be a kind of dual-superconductor, in which a chromo-magnetic charge condenses and the chromo-electric field is expelled from the bulk. In such a medium, a quark-antiquark pair would be confined by a dual vortex. In spite of many investigations, the Nambu-'t Hooft-Mandelstam conjecture has not found convincing evidences. Anyway, a proposal for the realization of a non-Abelian superconductor, in which a chromo-magnetic charge pair would be confined, has been recently made. As in the Abelian case, the non-Abelian superconductor allows for the existence of topological vortices. A non-Abelian vortex consists in a magnetic flux tube, as in the Abrikosov theory, but with a further internal degree of freedom. Indeed, due to the presence of an exact non-Abelian symmetry, the static vortex solution can be rotated with respect to this group. Every solution is therefore characterized by a set of parameters, the orientation moduli, that represent the possible non-Abelian orientations of the vortex. Since the fluctuations of such moduli represent the zero modes of the system, in the limit of low-energy, they constitute the relevant degrees of freedom to describe the non-Abelian vortex dynamics. Therefore, as it happens in many examples of effective theories, a low-energy model for the non-Abelian vortex can be constructed by a derivative expansion of the Lagrangian with respect to these "light" degrees of freedom. The reason for using a derivative expansion depends on the fact that the value of a derivative is related to the scale of the kinetic energy and then, in the limit of low-energy, it can be used as parameter of a Taylor series. Such a procedure for the construction of an effective Lagrangian for the non-Abelian vortex has been proposed in ref[Eto et All. "Higher derivative corrections to Non-Abelian Vortex effective theory"]. In that work, the lower orders of the effective theory describe the free motion of the orientation moduli or, equivalently, the rigid evolution of the vortex along its zero modes. At this order of approximation, known as the moduli approximation, the soliton behaves like a rigid body that, as long as the internal excitations can be neglected, evolves without modifying its structure. In order to improve such an approximation, we need to take into account the small deformations of the soliton shape. To this end, in ref[Eto et All. "Higher derivative corrections to Non-Abelian Vortex effective theory"] a small fluctuation is added to the rigidly evolving non-Abelian vortex obtaining, in this way, a next order of the effective theory. The new effective Lagrangian contains now the higher derivative corrections for the orientation moduli that give a non-vanishing contribution to the physical observables. The presence of higher derivative corrections is a common feature of different effective Quantum Field Theories, such as the Chiral Perturbation Theory. In that model, the pions represent the zero-modes fluctuations, taking the role of the orientation moduli, and the low-energy Lagrangian is performed, in the same way, with a derivative expansion of the field. We thus deduce that the soliton effective theory constitutes an example of a more general framework of low-energy theories. In ref[Eto et All. "Higher derivative corrections to Non-Abelian Vortex effective theory"] it has been proved that the explicit form of the internal vortex fluctuations is not necessary in order to construct the effective Lagrangian with higher derivative corrections. This approach, however, does not give the opportunity of an appropriate analysis of such fluctuations, that can be useful for a future description of the exact vortex dynamics. With this aim, in this thesis, we solve the full equations for the vortex fluctuations and then we provide an appropriate discussion on the effects of the higher orders on the vortex dynamics. Afterwards, we repeat the same analysis for the case of a monopole in Higgs phase, a new soliton configuration that arises adding a new term to the vortex Lagrangian. The main original contribution of this thesis work is presented in Chapter 4 and Chapter 6 below. The present thesis is organized into six Chapters with the following structure. In Chapter 1 we review the general theory of solitons, giving the basic concepts of Topology. Here we analyse the case of the Abelian vortex. Chapter 2 is dedicated to reviewing different examples of low-energy effective theories. In particular, we analyse in detail an example of low-energy approximation in Classical Mechanics and we provide different examples of higher derivative corrections in field theories. This discussion shows how the derivative-expansion is a common feature of effective theories. In Chapter 3 we discuss the general theory of non-Abelian Vortex and the general procedure to construct a low-energy model for it. In particular, we analyse the lower orders of such effective theory that describe the rigid evolution of the vortex, i.e. the evolution along its zero modes. To this end, we reproduce an explicit calculation of the non-Abelian vortex zero modes using different gauge choices. In Chapter 4 we deal with a higher order of the low-energy approximation in which the internal fluctuations of the vortex are taken into account. Here, we discuss a strategy for the resolution of such fluctuations giving, at the end, an appropriate solution. Then, we calculate the higher derivative corrections to the non-Abelian vortex effective theory, proving how this result reproduces correctly the effective Lagrangian in ref[Eto et All. "Higher derivative corrections to Non-Abelian Vortex effective theory"]. Afterwards, we analyse the effective low-energy theory showing how the field-fluctuations deform the vortex profile. This discussion is proposed for different space-time d+1 dimensions with d=0,1,2. The Chapter 5 is dedicated to the case of a monopole in the Higgs phase, consisting in a deformed configuration of the non-Abelian vortex. This new configuration, obtained adding a new field to the Lagrangian, can be interpreted as a single Abelian monopole trapped inside the superconductor. For this model, we discuss the general theory and the low-energy dynamics. Here, we point out the similarities and the differences with respect to the vortex case. In Chapter 6 we take into consideration the higher corrections of the low-energy theory of the monopole and we resolve the problem of the soliton internal fluctuations. Then, we analyse the effective monopole theory with the higher derivative corrections, for which we provide a discussion in 0+1 dimensions