**5.3. The effect of strong magnetic fields on the electroweak
vacuum**

It was pointed out by Ambj orn and Olesen
[223] (see also
Ref.[225])
that the Weinberg-Salam model of
electroweak interactions shows an instability at *B*
10^{24}
Gauss. The nature of such instability can be understood
by looking at the expression of the energy of a particle with
electric charge *e*, and spin **s**, moving in homogeneous
magnetic field **B** directed along the *z*-axis. As we already
discussed in Sec. 3.1, above a critical
field *B*_{c} = *m*^{2} / *e* particle
energy is discretized into Landau levels

(5.54) |

We observe that energy of scalar (*s* = 0) and spinor
(*s*_{z} = ± 1/2) is always positive, and indeed no
instability arise in QED (it is possible to verify that quantum one-loop
corrections do not spoil this conclusion). In the case of vector
particles (*s*_{z} = 0, ± 1), however, the lowest
energy level (*n* = 0, *k*_{z} = 0,
*s*_{z} = +1) becomes imaginary for *B* >
*B*_{c}, which could be the signal of vacuum
instability. The
persistence of imaginary values of the one loop corrected lowest
level energy [223]
seems to confirm the physical reality of the instability.

As it is well known the Weinberg-Salam model contains some charged
vector fields, namely the *W*^{±} gauge bosons. The
coupling of
the *W*_{µ} field to an external electromagnetic field
*A*^{ext}_{µ} is given by

(5.55) |

with

(5.56) |

The important term in the previous expression is the "anomalous" magnetic
moment term *ie*
*F*^{ext}_{µ}
*W*^{µ}
*W*^{}, which
arises because the
non-Abelian nature of the *SU*(2) component of the Weinberg-Salam
model gauge group structure. Due to this term the mass eigenvalues of
the *W* Lagrangian becomes

(5.57) |

As expected from the considerations in the above, a tachyonic mode appears
for *B* > *B*_{c}. The corresponding eigenvector
for zero kinetic energy is determined by solving the equation of motions

(5.58) |

where *W*_{1,2} = *W*_{x} ±
*iW*_{y}. Ambjorn and Olesen argued that a
suitable solution of this equation is

(5.59) |

corresponding to a vortex configuration where *W*-fields wind
around the *z*-axis. This configuration corresponds to the
Nielsen-Olesen vortex solution
[157]. A similar
phenomenon should also take place for *Z* bosons. Given the
linearity of the equations of motion it is natural to assume that
a superpositions of vortices is formed above the critical field.
This effect resemble the behaviour of a type-II superconductor in
the presence of a critical field magnetic field. In that case
*U*(1) symmetry is locally broken by the formations of a lattice
of Abrikosov vortices in the Cooper-pairs condensate through which
the magnetic field can flow. In the electroweak case this situation is
reversed, with the formation of a *W* condensate along the
vortices. Concerning the back-reaction of the *W* condensate on
the magnetic field, an interesting effect arises. By writing the
electric current induced by the *W* fields

(5.60) |

Ambj orn and Olesen noticed that its sign is opposite to the current
induced by the Cooper pairs in a type-II superconductor, which is
responsible
for the Meissner magnetic field screening effect. Therefore, they
concluded that the *W*-condensate induce *anti-screening* of
the external magnetic field.

Although the Higgs field
does not couple
directly to the electromagnetic field (this is different from the case of a
superconductor where the Cooper-pairs condensate couples directly
to *A*^{ext}_{µ}), it does through the
action of the *W*
condensate. This can be seen by considering the Higgs, *W*
potential in the presence of the magnetic fields:

(5.61) |

In the above _{0} and
_{+} are
respectively the Higgs field vev and charged component, *g* is the
SU(2) coupling
constant, and is the
Higgs the self-interaction coupling
constant. We see that the *W*-condensate influences the the Higgs
field at classical level due to the
^{2}
|*W* |^{2}
term. It is straightforward to verify that if *eB* <
*m*_{W}^{2} = 1/2 *g*^{2}
_{0}^{2} the
minimum of *V*(,
*W*) sits in the standard
field value =
_{0} with no
*W* condensate. Otherwise a
*W* condensate is energetically favoured with the minimum of the
potential sitting in

(5.62) |

where

(5.63) |

We see that the Higgs expectation value will vanish as the average
magnetic field strength approaches zero, provided the Higgs mass is
larger than the *W* mass. This seems to suggest that a
*W*-condensate should exist for

(5.64) |

and that the *SU*(2) × *U*_{Y}(1) symmetry is
restored above *H*^{(2)}_{c}
*m*_{H}^{2}/*e*. Thus, anti-screening should
produce restoration of the electroweak symmetry in the core of *W*
vortices. If *m*_{H} < *m*_{W} the
electroweak vacuum is expected to
behave like a type I superconductor with the formation of
homogeneous *W*-condensate above the critical magnetic field. The
previous qualitative conclusion have been confirmed by analytical
and numerical computations performed for *m*_{H} =
*m*_{W} in
Ref. [223],
and for arbitrary Higgs mass in
Refs. [226,
227].

A different scenario seems, however, to arise if thermal corrections are taken into account. Indeed, recent finite temperature lattice computations [187] showed no evidence of the Ambjorn and Olesen phase. According to Skalozub and Demchik [228] such a behaviour may be explained by properly accounting the contribution of Higgs and gauge bosons daisy diagrams to the effective finite temperature potential.

In conclusion, it is quite uncertain if the Ambj orn and Olesen phenomenon was really possible in the early Universe.