We consider a class of spontaneously broken <inline-formula><tex-math id='TM0001' notation='LaTeX'>$SU(2)$</tex-math></inline-formula> gauge theories with an adjoint scalar and look for exact magnetic monopole solutions in the Bogomol&#8217;nyi&#8211;Prasad&#8211;Sommerfield (BPS) limit. We find that some of the resulting solutions exhibit a new internal degree of freedom (a moduli space parameter) that controls the energy density profile of the monopole while keeping the total energy (mass) constant.

Czechia

Petr Beneš

Filip Blaschke

Magnetic Monopoles with an Internal Degree of Freedom

Abstract We explore the relevance of the central hill for a symmetric double-well potential and its impact on the scattering of kinks in a scalar field theory in (1+1) dimensions. This region controls the inner core structure of the kink. We study how the disappearance of analyticity in this region of the potential affects the resonant features in  scattering.

Ondřej Nicolas Karpíšek

Lukáš Rafaj

Scattering of Kinks in Coreless Potentials

We present a systematic exploration of a general family of effective <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> models with an adjoint scalar. First, we discuss a redundancy in this class of models and use it to identify seemingly different, yet physically equivalent models. Next, we construct the Bogomol’nyi-Prasad-Sommerfield limit and derive analytic monopole solutions. In contrast to the ’t Hooft-Polyakov monopole, included here as a special case, these solutions tend to exhibit more complex energy density profiles. Typically, we obtain monopoles with a hollow cavity at their core where virtually no energy is concentrated; accordingly, most of the monopole’s energy is stored in a spherical shell around its core. Moreover, the shell itself can be structured, with several “subshells.” A recipe for the construction of these analytic solutions is presented.

Shapes of magnetic monopoles in effective <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> models

We present a “primitive” way of realizing finite-mass Dirac monopoles in <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge theories involving a single nonminimally interacting scalar field. Typically, the energy density of this type of monopole is not concentrated at its core, but it is distributed in a spherical shell, as we illustrate on several exact solutions in the Bogomol’nyi-Prasad-Sommerfield limit. We show that our construction can be interpreted as a limit of infinitely massive <math><mi>W</mi></math> bosons coupled to electromagnetic field strength via a dipole moment. Combining our approach with ideas of Weinberg and Lee, we present a general landscape of <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge models that support a finite-mass Dirac monopole. In fact, all classical monopoles, i.e., Wu-Yang, ’t Hooft-Polyakov, Cho-Maison, etc., are special points on this landscape.

All finite-mass Dirac monopoles

Abstract “Mechanization” is a procedure of replacing a scalar field in 1+1 dimensions with a piecewise linear function, i.e. a finite graph consisting of N joints (vertices) and straight segments (edges). As a result, the field theory is approximated by a sequence of algebraically tractable, general-purpose collective coordinate mechanical models. We observe the step-by-step emergence of dynamical objects and associated phenomena as N increases. “Mech-kinks” and “mech-oscillons”—mechanical analogs of kinks and oscillons (bions)—appear in the simplest models, while more intricate dynamical patterns, such as the bouncing phenomenon and bion pair-production, emerge gradually as decay states of high-N mech-oscillons.

Mechanization of scalar field theory in 1+1 dimensions

Abstract A brane-world SU(5) grand unified theory model with global non-Abelian vortices is constructed in six-dimensional spacetime. We find a solution with a vortex associated to SU(3) separated from another vortex associated to SU(2). This 3–2 split configuration achieves a geometric Higgs mechanism for SU(5) → SU(3) × SU(2) × U(1) symmetry breaking. A simple deformation potential induces a domain wall between non-Abelian vortices, leading to a linear confining potential. The confinement stabilizes the vortex separation moduli, and ensures the vorticities of the SU(3) and SU(2) groups are identical. This dictates the equality of the numbers of fermion zero modes in the fundamental representation of SU(3) (quarks) and SU(2) (leptons), leading to quark/lepton generations. The standard model massless gauge fields are localized on the non-Abelian vortices thanks to a field-dependent gauge kinetic function. We perform fluctuation analysis with an appropriate gauge fixing and obtain a four-dimensional effective Lagrangian of unbroken and broken gauge fields at quadratic order. We find that SU(3) × SU(2) × U(1) gauge fields are localized on the vortices and exactly massless. Complications in analyzing the spectra of gauge fields with the nontrivial gauge kinetic function are neatly worked out by a vector-analysis-like method.

Japan

Masato Arai

Minoru Eto

Masaki Kawaguchi

Norisuke Sakai

Standard model gauge fields localized on non-Abelian vortices in six dimensions

It is important to obtain (nearly) massless localized modes for the low-energy four-dimensional effective field theory in the brane-world scenario. We propose a mechanism for bosonic zero modes using the field-dependent kinetic function in the classical field theory setup. As a particularly simple case, we consider a domain wall in five dimensions and show that massless states for scalar (0-form), vector (1-form), and tensor (2-form) fields appear on a domain wall, which may be called topological because of the robustness of their existence (insensitive to continuous deformations of parameters). The spin of localized massless bosons is selected by the shape of the nonlinear kinetic function, analogously to the chirality selection of the fermion by the well-known Jackiw-Rebbi mechanism. Several explicitly solvable examples are given. We consider not only (anti) Bogomol’nyi-Prasad-Sommerfield (BPS) domain walls in a noncompact extra dimension but also non-BPS domain walls in a compact extra dimension.

Massless bosons on domain walls: Jackiw-Rebbi-like mechanism for bosonic fields

Abstract We propose a minimal and self-contained model in non-compact flat five dimensions that localizes the Standard Model (SM) on a domain wall. Localization of gauge fields is achieved by condensation of the Higgs field via a Higgs-dependent gauge kinetic term in the 5D Lagrangian. The domain wall connecting vacua with unbroken gauge symmetry drives the Higgs condensation, which provides both electroweak symmetry breaking and gauge field localization at the same time. Our model predicts higher-dimensional interactions  in the low-energy effective theory. This leads to two expectations: One is a new tree-level contribution to  () decay whose signature will be testable in future LHC experiments. The other is a finite electroweak monopole that may be accessible to the MoEDAL experiment. Interactions of the translational Nambu–Goldstone boson are shown to satisfy a low-energy theorem.

Localization of the Standard Model via the Higgs mechanism and a finite electroweak monopole from non-compact five dimensions

Abstract We present exact solutions to the Cho–Maison magnetic monopole in a family of effective electroweak models that have a Bogomol’nyi–Prasad–Sommerfield limit. We find that the lower bound to the mass of the magnetic monopole is . We argue that this bound holds universally, not just in theories with a BPS limit.

BPS Cho–Maison monopole

Abstract The dynamical localization of non-Abelian gauge fields in non-compact flat  dimensions is worked out. The localization takes place via a field-dependent gauge kinetic term when a field condenses in a finite region of spacetime. Such a situation typically arises in the presence of topological solitons. We construct a 4D low-energy effective Lagrangian up to the quadratic order in a universal manner applicable to any spacetime dimensions. We devise an extension of the  gauge to separate physical and unphysical modes clearly. Out of the -dimensional non-Abelian gauge fields, the physical massless modes reside only in the 4D components, whereas they are absent in the extra-dimensional components. The universality of non-Abelian gauge charges holds due to the unbroken 4D gauge invariance. We illustrate our methods with models in  (domain walls),  (vortices), and .