Lattice vortices in the two-dimensional Abelian Higgs model

We generate and identify multi-vortices of the 2D Abelian Higgs model on a finite lattice by relaxation of Monte Carlo equilibrium configurations. The lattice vortices have action and a uniquely defined topological charge corresponding to the continuum ones. They exhibit the expected exponential decay behaviour and satisfy approximately the classical equations of motion. Vortex-antivortex superpositions are seen as well, supporting the dilute gas picture. Single vortices finally relax into “dislocations” and disappear. A background charge construction turns out nearly insensitive with respect to dislocations.     

Abelian vortices from sinh-Gordon and Tzitzeica equations

It is shown that both the sinh-Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in R^2,1${\mathbb{R}}^{2,1}$, or on a surface conformally related to a hyperbolic affine sphere in R³${\mathbb{R}}^3$. In both cases the Higgs field and the U(1)$U(1)$ vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh-Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlevé transcendents are used to compute explicit values for the strength of the vortices.     

Dynamics of Abelian Higgs vortices in the near Bogomolny regime

The aim of this paper is to give an analytical discussion of the dynamics of the Abelian Higgs multi-vortices whose existence was proved by Taubes ([JT82]). For a particular value of a parameter of the theory, , called the Higgs self-coupling constant, there is no force between two vortices and there exist static configurations corresponding to vortices centred at any set of points in the plane. This is known as the Bogomolny regime. We will develop some formal asymptotic expansions to describe the dynamics of these multi-vortices for close, but not equal to, this critical value. We shall then prove the validity of these asymptotic expansions. These expansions allow us to give a finite dimensional Hamiltonian system which describes the vortex dynamics. The configuration space of this system is the moduli space—the space of solutions of the static equations modulo gauge equivalence. The kinetic energy term in the Hamiltonian is obtained from the natural metric on the moduli space given by theL ² inner product of the tangent vectors. The potential energy gives the intervortex potential which is non-zero when is not given by its critical value. Thus the reduced equations for the evolution of the vortex parameters take the form of geodesics, with force terms to express the departure from the Bogomolny regime. The geodesics are geodesics on the moduli space with respect to the metric defined by theL ² inner product of the tangent vectors, in accordance with Manton's suggestion ([Man82]). This allows an understanding of the two main phenomenological issues—first of all there is the right angle scattering phenomenon, according to which two vortices passing through one another scatter through ninety degrees. Secondly there is the conjecture from numerical calculations that vortices repel for greater than the critical value, and attract for less than this value. The results of this paper allow a rigorous understanding of the right angle scattering phenomenon ([Sam92, Hit88]) and reduce the question of attraction or repulsion in the near Bogomolny regime to an understanding of the potential energy term in the Hamiltonian ([JR79]).     

Fun with the Abelian Higgs model

In calculations of the elementary scalar spectra of spontaneously broken gauge theories there are a number of subtleties which, though it is often unnecessary to deal with them in the order-of-magnitude type of calculations, have to be taken into account if fully consistent results are sought for. Within the “canonical” effective-potential approach these are, for instance: the need to handle infinite series of nested commutators of derivatives of field-dependent mass matrices, the need to cope with spurious IR divergences emerging in the consistent leading-order approximation and, in particular, the need to account for the fine interplay between the renormalization effects in the one- and two-point Green functions which, indeed, is essential for the proper stable vacuum identification and, thus, for the correct interpretation of the results. In this note we illustrate some of these issues in the realm of the minimal Abelian Higgs model and two of its simplest extensions including extra heavy scalars in the spectrum in attempt to exemplify the key aspects of the usual “hierarchy problem” lore in a very specific and simple setting. We emphasize that, regardless of the omnipresent polynomial cut-off dependence in the one-loop corrections to the scalar two-point function, the physical Higgs boson mass is always governed by the associated symmetry-breaking VEV and, as such, it is generally as UV-robust as all other VEV-driven masses in the theory.     

Dimensional reduction of the Abelian Higgs Carroll-Field-Jackiw model

Taking as a starting point a Lorentz non-invariant abelian Higgs model defined in 1 + 3 dimensions, we carry out its dimensional reduction to D = 1 + 2, obtaining a new planar model composed by a Maxwell-Chern-Simons-Proca gauge sector, a massive scalar sector, and a mixing term (involving the fixed background ) that imposes the Lorentz violation to the reduced model. The propagators of the scalar and massive gauge field are evaluated and the corresponding dispersion relations determined. Based on the poles of the propagators, a causality and unitarity analysis is carried out at tree level. We then show that the model is totally causal, stable and unitary.Received: 2 July 2004, Revised: 21 September 2004, Published online: 1 December 2004     

Fermionic string from Abelian Higgs model with monopoles and Θ-term

The four-dimensional Abelian Higgs model with monopoles and Θ-term is considered in the limit of large mass of the Higgs boson. We show that for Θ=2π the theory is equivalent, at large distances, to summation over all possible world-sheets of fermionic strings with Dirichlet-type boundary conditions on the string coordinates. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 76–80 (25 July 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

String tension in the three-dimensional Abelian Higgs model

We measure the expectation values of the Wilson loops for the radially active Abelian Higgs Model in three dimensions with Higgs chargeq=1 andq=2. We observe a drastic fall-off of the area term as we pass to the Higgs phase, as well as a peak of the perimetric term at the phase transition. Implications of our results for other Higgs models are also discussed.     

Abelian Higgs model with a Chern-Simons term in 3-dimensional gravitation

We consider the Abelian Higgs model with a Chern-Simons term coupled to the Einstein theory of gravitation in 3-dimensional space-time. We seek a finite solution, regular everywhere, having a stationary, cylindrically symmetric metric. We analyze these field equations and we suggest that such a solution exists. We find that the asymptotic metric of this solution corresponds to that which describes gravitationally a massive particle with spin. We obtain explicitly the expression of the spin. We give only the expression of the mass in the first order with respect to the gravitational coupling constant.     

Prescribing topological defects for the coupled Einstein and Abelian Higgs equations

We construct multi-string solutions of the coupled Einstein and Abelian Higgs equations so that the spacetime is uniform along the time axis and a vertical direction and nontrivial geometry is coded on a Riemann surfaceM. We concentrate on the critical Bogomol'nyi phase. WhenM is compact, the Abelian Higgs model is defined by a complex line bundleL overM. We prove that, due to the coupling of the Einstein equations, the Euler characteristic ofM and the first Chern number of the line bundleL identified as the total string number impose an exact obstruction to the existence of a string solution. Such an obstruction leads to some interesting implications. We then study the existence of multi-string solutions which can realize a prescribed string distribution. We show that there are such solutions when the local string winding numbers do not exceed half of the total string number. WhenM is noncompact and globally conformal to a plane, we show that the energy scale of symmetry breaking plays a crucial role and there are finite-energy radially symmetric string solutions realizing a given string number if and only if the symmetry breaking scale is sufficiently small but nonvanishing. Finally, we obtain finite-energy multistring solutions with an arbitrary string distribution and associated local winding numbers. These solutions are not radially symmetric and are regular everywhere and topologically nontrivial so that both the energy of the matter-gauge sector and the energy of the gravitational sector viewed as the total Gauss curvature ofM are quantized.     

A generalized Slavnov-invariant Abelian Higgs-Kibble model

The supersymmetric structure of the Becchi-Rouet-Stora invariance is used for investigating the Abelian Higgs-Kibble model. It allows for a symmetric photon mass M, in which case a massless scalar state corresponding to the Goldstone model belongs to the physical sector. If M = 0, i.e. the photon mass is generated by the Higgs mechanism, this state is decoupled from the physical sector, as usual. The BPHZ renormalization procedure is used.