Twisted Vortices in a Gauge Field Theory

We inspect a particular gauge field theory model that describes the properties of a variety of physical systems, including a charge neutral two-component plasma, a Gross-Pitaevskii functional of two charged Cooper pair condensates, and a limiting case of the bosonic sector in the Salam-Weinberg model. It has been argued that this field theory model also admits stable knot-like solitons. Here we produce numerical evidence in support for the existence of these solitons, by considering stable axis-symmetric solutions that can be thought of as straight twisted vortex lines clamped at the two ends. We compute the energy of these solutions as a function of the amount of twist per unit length. The result can be described in terms of a energy spectral function. We find that this spectral function acquires a minimum which corresponds to a nontrivial twist per unit length, strongly suggesting that the model indeed supports stable toroidal solitons. [email protected], [email protected], [email protected], [email protected] 1 supported by NFR Grant F-AA/FU 06821-308 2 supported by Göran Gustafssons Stiftelse UU/KTH Recently, a gauge field theory model with two charged bosons has been conceived, to describe a two-component plasma of negatively and positively charged particles [1]. But this model also appears to describe a large variety of other physical phenomena, including a Gross-Pitaevskii functional of two band superconductivity [2] and the bosonic sector in the Salam-Weinberg model in the limit where the Weinberg angle θW = 0 [3]. In [1] (see also [3]) it has been proposed that the model supports stable, self-confining and knot-like solitonic configurations. This would be somewhat remarkable, since it would contrast some of the widely held views in plasma physics that such configurations of plasma can not exist in general. This is due to a simple application of the Shafranov virial theorem which states that a static configuration of plasma in isolation is dissipative [4]. The proposed model, however, escapes this no-go theorem by incorporating non-linear interactions which are not accounted for by mean field variables such as the pressure [5]. The soliton solutions in the gauge model that we shall inspect can be viewed as a bundled filaments of twisted magnetic flux lines. The twisting is governed by a certain topological quantity, the Hopf invariant. Nontriviality of the Hopf invariant ensures that the flux lines are knotted, or linked. Numerical simulations, in the absence of effective analytical tools, seem so far to be the best way to help explore the nature of the soliton solutions. But even then their intricate topology makes full three dimensional simulations a daunting task. In this letter we present and analyse a tractable, yet challenging, simulation of the model, where the magnetic flux lines are twisted in an axis-symmetric manner. Such configuration of solutions can then be viewed as straight but twisted vortex tubes. As such, the stable vortices in the model we study can be applied to study a number of interesting physics. They may relate to the coronal loops on the solar photosphere [1], to Meissner effect in two-band superconductors [2] or to higher energy topological configurations in the weak sector of the standard model [3], [6]. Our study will then serve as a test bed to understand knot solitons in general. Serious, three dimensional searches for knotted structures in field theory models are attempted only recently. One prototype model, initiating these searches [7], [8] (see also [9, 10, 11]), is a Skyrme like O(3) non-linear sigma model. This model could be envisaged as describing the infrared phase of the pure SU(2) Yang-Mills theory, with glueballs represented by the knotted flux tubes of gluon [12]. We start from the classical kinetic theory model of a two-component plasma of electromagnetically interacting electrons and ions, given by the non-relativistic action [1], S = ∫ dx [ i~Ψ∗e ( ∂t + ieAt ~c ) Ψe + i~Ψ ∗ i ( ∂t − ieAt ~c ) Ψi (1) − ~ 2 2m ∣∣∣∣ ( ∂k + ieAk ~c ) Ψe ∣∣∣∣ 2 − ~ 2 2M ∣∣∣∣ ( ∂k − ieAk ~c ) Ψi ∣∣∣∣ 2 − 1 4 F 2 μν ]