Next Contents Previous

3.4. Macroscopic string description

Let us recapitulate briefly the microphysics setting before we see its connection with the macroscopic string description we will develop below. We consider a Witten-type bosonic superconductivity model in which the fundamental Lagrangian is invariant under the action of a U(1) × U(1) symmetry group. The first U(1) is spontaneously broken through the usual Higgs mechanism in which the Higgs field Phi acquires a non-vanishing vacuum expectation value. Hence, at an energy scale ms ~ lambdaphi1/2 eta (we will call ms = m hereafter) we are left with a network of ordinary cosmic strings with tension and energy per unit length T ~ U ~ m2, as dictated by the Kibble mechanism.

The Higgs field is coupled not only with its associated gauge vector but also with a second charged scalar boson sum, the current carrier field, which in turn obeys a quartic potential. A second phase transition breaks the second U(1) gauge (or global, in the case of neutral currents) group and, at an energy scale ~ m*, the generation of a current-carrying condensate in the vortex makes the tension no longer constant, but dependent on the magnitude of the current, with the general feature that T leq m2 leq U, breaking therefore the degeneracy of the Nambu-Goto strings (more below). The fact that |sum| neq 0 in the string results in that either electromagnetism (in the case that the associated gauge vector A(sigma)µ is the electromagnetic potential) or the global U(1) is spontaneously broken in the core, with the resulting Goldstone bosons carrying charge up and down the string.

Macroscopic quantities

So, let us define the relevant macroscopic quantities needed to find the string equation of state. For that, we have to first express the energy momentum tensor as follows

Equation 33 (33)

One then calculates the macroscopic quantities internal to the string worldsheet (recall `internal' means coordinates t,z)

Equation 34 (34)

The macroscopic charge density/current intensity is defined as

Equation 35 (35)

Now, the state parameter is nu ident sgn(w) |w|1/2. For vanishing coupling e we have w ~ k2 - omega2 and nu yields the energy of the carrier (in the case w < 0) or its momentum (w > 0).

We get the energy per unit length U and the tension of the string T by diagonalizing T barab

Equation 36 (36)

Figure 8a
Figure 8b

Figure 1.8. Variation of the relevant macroscopic quantities with the state parameter. In the left panel we show the variation of the amplitude of the macroscopic (integrated) charge density (for w < 0) and current intensity (for w > 0) along the string core versus the state parameter, as defined by nu ident sgn(w)|w|1/2. In the right panel one can see the corresponding variations of the integrated energy per unit length (upper set of curves) and tension (lower set of curves) for the string. Both the neutral (e = 0) and the charged cases are shown with, in the latter case, a rather exaggerated value of the coupling, in order to distinguish the curves in each set [Peter, 1992].

As shown in Figure (1.8) the general string dynamics in the neutral case does not get much modified when the electromagnetic e-coupling is included. Nevertheless, a couple of main features are worth to note:

Macroscopic description

Now, let us focus on the macroscopic string description. For a local U(1) we have

Equation 37 (37)

[to stick to usual notation in the literature, we are now changing erightarrow - e in our expressions of previous sections]. In this equation we have the conserved Noether current

Equation 38 (38)

Now, recall that Aµ(sigma) varies little inside the core, as the penetration depth was bigger than the string core radius. We can then integrate to find the macroscopic current

Equation 39 (39)

which is well-defined even for electromagnetic coupling e rightarrow 0.

The macroscopic dynamics is describable in terms of a Lagrangian function curlyL(w) depending only on the internal degrees of freedom of the string. Now it is phi's gradient that characterizes local state of string through

Equation 40 (40)

where gammaab is the induced metric on the worldsheet. The latter is given in terms of the background spacetime metric gµnu with respect to the 4-dimensional background coordinates xµ of the worldsheet. We use a comma to denote simple partial differentiation with respect to the worldsheet coordinates xia and using Latin indices for the worldsheet coordinates xi1 = sigma (spacelike), xi0 = tau (timelike). As we saw above, the gauge covariant derivative phi|a is expressible in the presence of a background electromagnetic field with Maxwellian gauge covector Aµ(sigma) (Aµ hereafter) by phi|a = phi, a - eAµ xµ,a. So, now a key rôle is played by the squared of the gradient of phi in characterizing the local state of the string through w.

The dynamics of the system is determined by the Lagrangian curlyL(w). Note there is no explicit appearance of phi in curlyL. From it we get the conserved particle current vector za, such that

Equation 41 (41)

Let's define -dcurlyL / dw = 1/2 curlyK-1. Matching Eqns. (41) and (39), viz. za (macro) dot eq Ia (micro) we find

Equation 42 (42)

which allows us to see the interpretation of the quantity curlyK-1. In fact, we have curlyK-1 propto tilde curlyK propto amplitude of sum-condensate. When w rightarrow 0 (null) we have curlyK rightarrow 1. (with kappa0 the zero current limit of tilde curlyK).

Next Contents Previous