STABILITY OF Z-STRINGS IN STRONG MAGNETIC FIELDS

We show that the Z-strings of the standard electroweak theory can be stabilized by strong external magnetic fields, provided that $\beta^{1/2} \equiv M_H/M_Z\leq 1$, where $M_H$ and $M_Z$ are the Higgs and Z masses. The magnetic fields needed are larger than $\beta^{1/2} B_c$ and smaller than $B_c$, where $B_c\equiv M^2_W/e$ is the critical magnetic field which causes W-condensation in the usual broken phase vacuum. If such magnetic fields were present after the electroweak transition, they would stabilize strings for a period comparable to the inverse Hubble rate at that time. Pair creation of monopoles and antimonopoles linked by segments of string is briefly considered.

They can be described as flux tubes of the Z gauge boson trapped in a thin core where the Higgs field vanishes. The Higgs develops an expectation value outside the core, where its phase has a net winding around the string. Like the sphaleron [2], Z-strings can carry baryon number and may have played a role in the generation of baryon asymetry [3]. Nambu's early work [4] showed that a finite segment of string would be bounded by a magnetic monopole on one end and an antimonopole at the other. Detailed studies of the vortex, by Vachaspati and collaborators [1,5] stirred a new wave of attention. Unlike the vortices in the Abelian Higgs model, which are topologically stable, Z-strings are embedded in a larger theory and their stability depends on the dynamical details. Unfortunately, the analysis showed that, for sin 2 θ W ≈ .23, Z-strings are unstable [5,17,16]. Then, even if they formed at the phase transition, their lifetime would probably be too short for them to play a cosmological role.
The above studies, however, ignored the effect of a possible background magnetic field.
Right after the electroweak phase transition one may expect strong magnetic fields on small scales [6], and it is natural to ask whether these would affect the stability of vortices.The aim of this letter is to show that there is a range for the magnetic field for which Z-strings would be stable. Intuitively, this result can be understood as follows. Recall that the instability of strings is related to the phenomenon of W-condensation [7,8]. The strong Z-flux in the core of the string is coupled to the anomalous magnetic moment of the W-bosons, and this causes the W's to condensate. We shall see that by opposing the Z-flux with an external magnetic field, the condensation can be avoided.
To be definite, let us consider the bosonic part of the electroweak energy functional. For static configurations Here As usual, g = α cos θ W , e = g sin θ W and γ = g cos θ W , and we follow the conventions of Ref. [9].
Ambjorn and Olesen have shown [7] that the anomalous mass term acts like a negative contribution to the W mass squared. An instability develops when the combined field strength gB T = |eB A +γB Z | exceeds a certain threshold. For instance, in the usual vacuum (without strings), this happens when the magnetic field contribution exceeds the usual positive W mass term M W = gη/2, Perkins, on the other hand, has pointed out [8] that in the core of a string the field strength B Z causes W-condensation for the physical value of cos θ W .
It is clear that a constant external magnetic field B A antiparallel to B Z would reduce the anomalous term (2) in the core of the string, and hence strings might be stabilized in the physical region of parameter space. Of course, detailed analysis is required before we jump to conclusions. If we need magnetic fields larger than B c to counteract B Z in the core of the string, these would cause W -condensation outside the core. Also we have to consider all possible modes of fluctuation, not just the formation of W -condensates.

Stability analysis
The Z-string solution is given by with all the other fields set to zero. The profiles f (r) and v(r) satisfy the Nielsen-Olesen differential equations [10]. We can always add to the solution a constant magnetic field B A , with potential

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This satisfies the field equations without altering f (r) and v(r), since Φ and Z are not magnetically charged. On the other hand, the fluctuations of W and of the upper component of the Higgs field are charged and will feel the presence of B A .
The stability analysis can be carried out as in the case with no magnetic field [5]. Full details will be reported elsewhere [11]. To check whether the string is a local minimum of (1), one must consider generic perturbations to (3). It can be shown that the energy is minimized when the perturbations to the Z, A, and to the lower component of the Higgs field are set to zero. The reason can be traced to the fact that the abelian Nielsen-Olesen string is stable. The perturbed configuration is then written as with the Z and A fields as given in (3) and (4). The perturbations can be taken to be independent of the z coordinate, and the z components of the gauge fields can be set to zero, since relaxing these conditions always results in an increase of energy [5]. For the same reasons we have set the temporal components of the gauge fields to zero in (1). Generic perturbations can then be expanded in Fourier series where χ, Θ and R are arbitrary complex functions of r. Substituting into (1), expanding to quadratic order in the fluctuations and integrating the angular dependence we find that the energy fluctuation is δE = l E l . Here E l is a quadratic functional depending on χ n−l , Θ l and R l . Actually R l appears only as a Lagrange multiplier. The energy is minimized by imposing δE l /δR l = 0, from which R l can be eliminated. The functional E l depends now on two functions χ n−l and Θ l , but by diagonalizing the kinetic term we readily find that there is only one dynamical degree of freedom. Defining the new fields This simplification is due to gauge invariance: it can be shown that ζ (g) l corresponds to a pure gauge mode [11,5]. Finally, after lengthy algebra [11] and a prime indicates d/dr.
Just as in the case with no magnetic fields, the problem reduces to finding whether the equation Oζ l = wζ l has any negative eigenvalues w. The boundary conditions of the eigenvalue equation are ζ(r → 0) → r |n−l| if l = 0 and ζ(r → ∞) → 0 for all l. For n = 1 and 2, the l = 0 mode is never unstable [11]. For n ≥ 3 and l = 0 the boundary condition at the origin is ζ → r n .
Results for n=1 For B A = 0 the result coincides with the critical line found in Ref. [5]. As the magnetic field B A is increased, the critical line moves to the left, so that this mode is stable for hep-ph/9505424 smaller values of sin 2 θ W . The interpretation is that the magnetic field opposes B Z and hence W -condensation is inhibited. In particular, when K ≈ 1 the l = 1 mode is stable for sin 2 θ W ≈ .23 . The vertical lines correspond to B A = B c . To the right of the vertical lines we have W -condensation in the trivial vacuum, outside the core of the string [7]. Note that for β = 1 the critical lines l = 1 meet with the vertical lines. This is consistent with Ref. [12], where the Bogomol'ny limit (β = 1) was considered.
However, this is not the whole story, since we have to consider all values of l. When The experimental lower bound on the Higgs mass gives β 1/2 > ∼. 6. For .6 < ∼β 1/2 ≤ 1 the n = 1 strings are stable over a range of magnetic fields of relative width If the magnetic field is frozen in with the plasma, as it is usually assumed [6], we have where a(t) ∝ t 1/2 is the cosmological scale factor. Given a sufficiently large initial magnetic field, this means that strings can be metastable for a time of cosmological order: ∆t ≈ (∆B A /B A )t ≈ .2(β −1/2 − 1)t. This may be sufficient for strings to fall out of thermal equilibrium, which in turn may have consequences for baryogenesis [3].
For β 1/2 > 1 and B A < B c there are no stable Z-strings. When B A > B c the vacuum decays into a lattice of W-condensate vortices [7]. Since the W bosons couple to the Z flux,

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we expect that the simple Z-string solution will not exist in this background. The stable regions for n = 1 and n = 2 have some overlap, but the case n = 2 extends to somewhat lower values of the applied magnetic field. This "overlap" pattern continues for higher n. The loci where the anomalous term vanishes at r = 0 are represented in Fig. 3 for different values of n. As n increases, the dashed line moves to lower values of the magnetic field. The stability regions for each n are in the vicinity of these lines. We find [11] that the width of these regions decreases with n, but the stable patch for winding n always has some overlap with the stable patch for winding n + 1, which in turn extends to somewhat smaller values of the magnetic field. For n → ∞ the dashed line tends to the straight line B A = β 1/2 B c , and the width of the stable region tends to zero. Hence B A = β 1/2 B c is the lowest value of the magnetic field for which we can find metastable strings of any winding.

Results for higher winding
As we cross the critical lines l = ±1 into the unstable region, the strings may unwind and release their energy in the form of linear excitations on the broken phase vaccuum, or settle down into a possible "dressed string" configuration [18,17]. The pattern of stability described above may also suggest that, as magnetic field decreases, the n = 1 string becomes unstable and decays into a metastable n = 2 string. By decreasing B A further, this decays into a n = 3 string, and so forth. In principle, it is not clear whether this cascade is energetically allowed; n = 2 strings are heavier than n = 1 strings, but we have to take into account that these strings end in monopoles of twice the charge (in other words, energy can be extracted from the magnetic field). We shall return to this question below.

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Pair creation The magnetic fields needed above have an energy density comparable to the electroweak scale, and one should worry about instabilities of the broken phase in the presence of such fields. As we shall see, for β 1/2 B c < B A < B c the broken phase is only metastable.
It is well known that a magnetic field can pair-produce monopoles and antimonopoles [13]. Electroweak monopoles are linked by Z-strings, so pair production can only happen if the force exerted by the magnetic field on the monopoles exceeds the string tension [14], Here Q = (4π/α) tan θ W is the n = 1 monopole charge [4], and µ n is the string tension. For large n the tension is dominated by the constant energy density in the core of the string, and we can neglect gradient terms Here S n is the area of a cross section of the core of the string, and B Z = φ n /S n , where φ n = 4πn/α is the quantized flux. Minimizing the tension with respect to S n we find µ n = β 1/2 πη 2 n. Comparing to the magnetic force, we conclude that in the large n limit the strings will grow provided that B A > β 1/2 B c (larger values of B A are needed for lower n).
The decay of the metastable phase can proceed through pair production by thermal activation or by quantum tunneling [13,14]. However, the actual decay rate and the "fate" of this "false vacuum" are difficult to assess. The problem is that condition (7) is only satisfied in the regions where the Z-strings of winding n are unstable. As a result, the corresponding instantons will have too many negative modes, and will probably not represent a semiclassical channel of decay [15]. Further work will be needed to clarify this issue.
Finally, we may ask whether a n = 1 string can decay into a n = 2 string. This process can be thought of as the nucleation of a monopole antimonopole pair on top of an existing n = 1 string. For this to be energetically possible, we need QB A > µ 2 − µ 1 . The transition line QB A = µ 2 − µ 1 is depicted in Fig. 2b  in the intersection of the regions of metastability for n = 1 and n = 2. Since the strings involved are both metastable, this process may be mediated by and instanton with just one negative mode, as is required by the general theory of semiclassical tunneling [15].
We are grateful to Ana Achúcarro and Tanmay Vachaspati for stimulating discussions.