**5.3.1. The electroweak phase transition in a magnetic field**

We shall now consider the possible effects of strong magnetic fields on the electroweak phase transition (EWPT). As it is well known, the properties of the EWPT are determined by the Higgs field effective potential. In the framework of the minimal standard model (MSM), taking into account radiative corrections from all the known particles and for finite temperature effects, one obtains that

(5.65) |

where is the radial
component of the Higgs field and *T* is the
temperature (for the definitions of the coefficients see *e.g.*
Ref. [186]).

A strong hypermagnetic field can produce corrections to the
effective potential as it affects the charge particles propagators
(see below). There is, however, a more direct and simpler effect of
magnetic and hypermagnetic fields on the EWPT which was recently
pointed-out by Giovannini and Shaposhnikov
[178] and by
Elmfors, Enqvist and Kainulainen
[186]. The authors of
Refs.
[178,
186]
noticed that hypermagnetic fields affect the Gibbs free energy
(in practice the pressure) difference between the broken and the
unbroken phase, hence the strength of the transition. The effect
can be understood by the analogy with the Meissner effect, *i.e.*
the expulsion of the magnetic field from superconductors as
consequence of photon getting an effective mass inside the
specimen. In our case, it is the *Z*-component of the hypercharge
*U*(1)_{Y} magnetic field which is expelled from the broken
phase.
This is just because *Z*-bosons are massive in that phase. Such a
process has a cost in terms of free energy. Since in the broken
phase the hypercharge field decomposes into

(5.66) |

we see that the Gibbs free energy in the broken and unbroken phases are

(5.67) | |

(5.68) |

where *B*^{ext}_{Y} is the external hypermagnetic
field. In other words, compared to the case in which no magnetic field is
present, the energy barrier between unbroken and broken phase,
hence the strength of the transition, is enhanced by the quantity
1/2 sin^{2}
_{w}
(*B*^{ext}_{Y})^{2}.
According to the authors of Refs.
[178,
186] this effect
can have important consequence for baryogenesis.

In any scenario of baryogenesis it is crucial to know at which
epoch do the sphaleronic transitions, which violate the sum (*B* +
*L*) of the baryon and lepton numbers, fall out of thermal
equilibrium. Generally this happens at temperatures below
such that

(5.69) |

where *E*(*T*) is the sphaleron energy at the temperature
*T* and *A*
35 - 45 , depending on the poorly known prefactor of the sphaleron rate.
In the case of baryogenesis at the electroweak scale one requires the
sphalerons to drop out of thermal equilibrium soon after the electroweak
phase transition. It follows that the requirement
= *T*_{c}, where
*T*_{c}
is the critical temperature, turns eq. (5.69) into a lower bound
on the Higgs vacuum expectation value (VEV),

(5.70) |

As we already discussed, it is by now agreed [143] that the standard model (SM) does not have a phase transition strong enough as to fulfill Eq. (5.70), whereas there is still some room left in the parameter space of the minimal supersymmetric standard model (MSSM) [144].

The interesting observation made in Refs.
[178,
186] is that
a magnetic field for the hypercharge *U*(1)_{Y} present for
*T* > *T*_{c}
may help to fulfill Eq. (5.70). In fact, it follows from the
Eqs. (5.67), that in presence of the magnetic field the critical
temperature is defined by the expression

(5.71) |

This expression implies a smaller value of *T*_{c} than that it
would take in the absence of the magnetic field, hence a larger
value of the ratio (5.70).

Two major problems, however, bar the way of this intriguing
scenario. The first problem is that by affecting fermion, Higgs
and gauge field propagators, the hypermagnetic field changes the
electroweak effective potential in a nontrivial way. Two different
approaches have been used to estimate the relevance of this kind of
effects based either on lattice simulations
[143] or
analytical computations
[228].
Both approaches agreed in
the conclusion that for a Higgs field mass compatible with the
experimental constraints (*m*_{H} > 75 GeV), and for field
strengths *B*, *B*_{Y}
10^{23}
G, the standard model EWPT is
second order or a cross-over. Although this negative result could,
perhaps, be overcome by adopting a supersymmetrical extension of
the standard model (see e.g. Ref.
[144]),
a second, and more
serious problem arises by considering the effect of the magnetic field on
the anomalous processes (sphalerons) which are responsible for
lepton and baryon violation at the weak scale. This effect will be
the subject of the next section.