**4.3. Magnetic fields from the electroweak phase transition**

**4.3.1. Magnetic fields from a turbulent charge flow**

Some of the ingredients which may give rise to magnetogenesis at the QCDPT may also be found at the electroweak phase transition (EWPT). As for the case of the QCDPT, magnetogenesis at the weak scale seems to require a first order transition. Although recent lattice computations performed in the framework of the standard electroweak theory [143] give a strong evidence against a first order transition, this remains a viable possibility if supersymmetric extension of the standard model are considered [144]. It is noticeable that a first order EWPT is also required for the successful realization of the electroweak baryogenesis scenario [145]. Indeed, as we shall see in the rest of this review, this is only one among several common aspects of baryogenesis and magnetogenesis.

According to Baym, Bödeker and McLerran
[146] strong
magnetic fields can be generated by a first order EWPT
^{(13)}
via a dynamo mechanism. In this
scenario seed fields are provided by random magnetic field fluctuations
which are always present on a scale of the order of a thermal wavelength
^{(14)}.
The amplification of such seed
fields proceeds as follows. When the Universe supercooled below
the critical temperature (*T*_{c} ~ 100 GeV) the Higgs field
locally tunneled from the unbroken SU(2) × U(1)_{Y}
phase to the broken *U*(1)_{em} phase. The tunneling
gave rise to the
formation of broken phase bubbles which then expanded by
converting the false vacuum energy into kinetic energy. Although
the bubble wall velocity is model dependent, one can find that for
a wide range of the standard model parameters the expansion is
subsonic (deflagration) which give rise to a supersonic shock wave
ahead of the burning front. As the shock fronts collided
turbulence should have formed in the cone associated with the
bubble intersection. The Reynold number for the collision of two
bubbles is

(4.8) |

where *v*_{fluid}~ *v*_{wall} ~
10^{-1} is the typical fluid
velocity, *R*_{bubble} is the size of a bubble at the
collision time and is
the scattering length of excitations in the electroweak plasma.
The typical size of a bubble after the phase transition is completed is in
the range

(4.9) |

where

(4.10) |

is the size of the event horizon at the electroweak scale,
*m*_{Pl}
is the Planck mass, *g*_{*}~ 10^{2} is the
number of massless degrees of
freedom in the matter, and the fractional size *f*_{b} is ~
10^{-2}-10^{-3}. The typical
scattering length of
excitations in the plasma is of order

(4.11) |

where _{w} is
the fine structure constant at the
electroweak scale, and *g*_{ew} ~
*g*_{*} is the number of degrees of
freedom that scatter by electroweak processes.
By substituting these expressions Baym et al. found

(4.12) |

Such a huge Reynolds number means that turbulence fully develops
at all scales smaller than *R*_{bubble}. Since conductivity
is expected to be quite large at that time
[49], magnetic fields
followed the fluid motion so that a strong magnetic turbulence
should also have been produced. In such a situation it is known that
the kinetic energy of the turbulent flow is equipartitioned with
the magnetic field energy. Therefore

(4.13) |

where (*T*_{c}) ~ *g*_{*}
*T*_{c}^{4} is the energy density
of the electroweak plasma.

In order to estimate the magnetic field strength on scale larger than
*R*_{bubble}, Baym et al. treated the large scale field as
a superposition of the field of dipoles with size *R*_{bubble}.
This is similar to what done by other authors
[105,
140]
(see what we wrote above for the QCDPT)
apart for the fact that Baym et al. used a continuum approximation for
the distribution of dipoles rather than to assume a random walk of
the field lines. The density
^{i}(**r**) of
dipoles pointing
in the *i*-th direction was assumed to be Gaussianly distributed.
This implies the following correlation functions for the density
of dipoles

(4.14) |

and for the magnetic field

(4.15) |

The logarithmic divergence of the integral in these regions
is cut off by the size of the typical dipole, *f*_{b}
*H*^{-1}_{ew}, so
that for *r* >> *f*_{b}
*H*^{-1}_{ew},

(4.16) |

By using this expression and the equipartition relation
(4.13)
one finds that the strength of *B*^{2} measured by
averaging on a size scale *R* is

(4.17) |

This result can be better expressed in terms of the ratio *r* of
<*B*^{2}> to the energy
_{} in
photons which is a constant during Universe expansion in the absence of
flux diffusion. From Eqs. (4.17) one gets

(4.18) |

where _{ew} is
the Hubble radius at the electroweak
phase transition (~ 1 cm) times the scale factor, *T*_{c} /
*T*_{},
where *T*_{} is the photon temperature.

From the previous results the authors of Ref. [146] estimated the average magnetic field strength at the present time. This is

(4.19) |

where *l*_{diff} ~ 10 AU is present time diffusion length,
and

(4.20) |

on the galactic scale *l*_{gal} ~ 10^{9} AU.

^{13} At the time the paper by Baym et
al. was written, a
first order EWPT was thought to be compatible with the standard
model. Therefore all computation in
[146] were done in
the framework of that model.
Back.

^{14} It is worthwhile to observe here that
thermal fluctuation in a dissipative plasma can actually produce
stochastic magnetic fields on a scale larger than the thermal
wavelength [147,
148].
Back.