Vortex solutions in gauge–Higgs models

Consider a static straight string in three spatial dimensions, invariant along its length. Its tension is an energy integral over the transverse plane. Let the complex scalar approach magnitude v at large transverse radius, with phase winding by an integer multiple of two pi around a circle. Choose covariant derivative D equal to partial minus i e A, with positive dimensionless gauge coupling e. Finite tension requires the covariant angular derivative to vanish asymptotically.

That condition makes the gauge circulation equal to the scalar phase change divided by e. Stokes’ theorem then fixes the magnetic flux through the plane. Regularity requires the Higgs magnitude to vanish in the core of the standard axisymmetric nonzero-winding ansatz; otherwise its phase would be ill-defined there. A gauge transformation can move apparent phase dependence between scalar and vector potential, but does not erase the quantized flux of a smooth finite-tension configuration with the stated boundaries.

Choose the potential coefficient so that the tension contains magnetic energy one half B squared, scalar gradient magnitude squared, and potential one half e squared times the squared departure of scalar magnitude squared from v squared. This is the critically coupled model under this normalization. Completing squares gives one covariant first-order scalar equation and one magnetic equation. The integrated remaining term is e v squared times flux, with sign chosen according to winding.

For positive winding the equations use D one plus i D two and magnetic field e times v squared minus scalar magnitude squared. Their signs reverse appropriately for negative winding. A solution saturating these equations has tension two pi v squared times absolute winding. Boundary terms discarded in the square completion must vanish under the finite-tension asymptotics. These first-order equations are special to the stated coupling; their existence and moduli properties are not automatic consequences of writing down a flux integer.

Without a gauge field, a global phase vortex in an infinite plane generally has a long-range gradient contribution and logarithmically divergent tension. The local gauge vortex avoids that particular divergence by canceling the asymptotic covariant gradient. Its flux and scalar core have characteristic scales set by the gauge and Higgs masses. Numerical factors in those scales depend on how scalar fields and couplings are normalized, so they should not be imported unchanged from another Lagrangian.

Away from critical coupling, vortices can still exist but their tension and interactions need not equal the Bogomolny result. Scalar and magnetic tails then compete, producing the familiar distinctions between attraction and repulsion in appropriate models. A classical vortex in a Higgs phase is also not by itself a derivation of color confinement in a non-Abelian theory. The useful lesson is a controlled connection among boundary winding, gauge compensation, flux, and a special energy bound.