Bogomol’nyi equations of classical solutions

Abstract:We use a recent on-shell method, developed in [1], to construct Bogomol'nyi equations of the three-dimensional generalized Maxwell-Higgs model [2]. The resulting Bogomol'nyi equations are parametrized by a constant C 0 and they can be classified into two types determined by the value of C 0 = 0 and C 0 = 0. We identify that the Bogomol'nyi equations obtained by Bazeia et al. [2] are of the (C 0 = 0)-type Bogomol'nyi equations. We show that the Bogomol'nyi equations of this type do not admit the Prasad-Sommerfield limit in its spectrum. As a resolution, the vacuum energy must be lifted up by adding some constant to the potential. Some possible solutions whose energy equal to the vacuum are discussed briefly. The on-shell method also reveals a new (C 0 = 0)-type Bogomol'nyi equations. This non-zero C 0 is related to a non-trivial function f C 0 defined as a difference between energy density of the scalar potential term and of the gauge kinetic term. It turns out that these Bogomol'nyi equations correspond to vortices with locally non-zero pressures, while their average pressure P remain zero globally by the finite energy constraint.

“…The similar study was also done on generalized BPS monopoles [12][13][14]. 1 Here in this paper we look for something more modest by following a different route. Our aim is twofold.…”

Section: Jhep02(2016)117mentioning

confidence: 99%

Section: Jhep02(2016)117mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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More on Bogomol’nyi equations of three-dimensional generalized Maxwell-Higgs model using on-shell method

Atmaja

Ramadhan

Hora

2016

“…This is the simplest way we found to make sure that the Abelian Higgs-Galileon system has an energy density bounded from below -other possibility might exist though. Given these preliminary results, it would be interesting to study more in general the stability of our configurations under small fluctuations, and to explore alternative methods to obtain the BPS equations for Galileon vortices, such as methods based on the energy-momentum tensor [35] or on the Lagrangian of the system rather than on the Hamiltonian [36]. We end this section showing that the static configurations discussed in section 3.4 have positive definite energy for the values of β we considered.…”

Section: The Energy Functional For the Galileon Vortexmentioning

confidence: 99%

Galileon Higgs vortices

Chagoya

Tasinato

2016

Vortex solutions are topologically stable field configurations that can play an important role in condensed matter, field theory, and cosmology. We investigate vortex configuration in a 2+1 dimensional Abelian Higgs theory supplemented by higher order derivative self-interactions, related with Galileons. Our vortex solutions have features that make them qualitatively different from well-known Abrikosov-Nielsen-Olesen configurations, since the derivative interactions turn on gauge invariant field profiles that break axial symmetry. By promoting the system to a 3+1 dimensional string configuration, we study its gravitational backreaction. Our results are all derived within a specific, analytically manageable system, and might offer indications for understanding Galileonic interactions and screening mechanisms around configurations that are not spherically symmetric, but only at most cylindrically symmetric.

The first-order Euler-Lagrange equations and some of their uses

Adam

Santamaría

2016

In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.PACS numbers: I. INTRODUCTIONNonlinear field theories are ubiquitous in the description of physical systems from particle physics [1] -[4] to condensed matter systems [5] -[7] and cosmology [8], where any genuine interaction is generally related to the nonlinearity of the underlying field theory. In these theories, one powerful strategy to obtain solutions of physical importance is to reduce the order of the original field equations (the Euler-Lagrange (EL) equations) of the system. The resulting equations of lower order -Bogomolnyi equations, self-duality equations, Bäcklund transformations, etc. -are easier to solve and allow to obtain a large number of relevant solutions with particular characteristics, like solitons, nonlinear waves, vortices, monopoles, instantons, etc. There exist several known methods to achieve this reduction of order, where the best-known one is probably the Bogomolnyi trick [9], [10], [11] of completing a square. To consider an example, let us assume that we have the energy functional of a field theory which for static fields may be expressed as a sum of two terms, E = d d x(A 2 + B 2 ) (typically, A depends on first derivatives, whereas B only depends on the fields). This may trivially be rewritten asIf, in addition, Q is a homotopy invariant (i.e., AB is locally a total derivative), then it does not contribute to the EL equations, and its value only depends on the boundary conditions imposed on the fields. As a consequence, E andĒ lead to the same EL equations. Further,Ē is non-negative, so E obeys the inequality E ≥ |Q| (Bogomolnyi bound) which is saturated by solutions to the reduced-order (usually, first-order) equation A = ±B (Bogomolnyi equation or BPS equation).Recently it has been observed [12] that it can be useful to partly invert the logic of this construction. That is to say, let us assume that we have two functionals (functions of the fields, their derivatives, and possibly also of the coordinates x µ ) A, B which are in some sense "duals" of each other, and which are such that the product AB is locally a total derivative (the integral Q = 2 d d xAB is a homotopy invariant). This automatically implies that the "energy functional" E = d d x(A 2 + B 2 ) is a BPS action, and the "self-duality equations" (BPS equations) A = ±B