**4.4. Magnetic helicity and electroweak baryogenesis**

As we discussed in the introduction of our report, the cosmological magnetic
flux is a nearly conserved quantity due to the high conductivity of the
Universe. In this section we will focus on another quantity which, for the
same reason, is approximatively conserved during most of the Universe
evolution. This is the so called *magnetic helicity* defined by

(4.60) |

where **A** is the electromagnetic vector potential and **B**
is the magnetic field.
In the presence of a small value of the electric conductivity
the
time evolution of *H* is given by

(4.61) |

Besides for the fact to be a nearly conserved quantity, the magnetic helicity is a very interesting quantity for a number of different reasons. The main among these reasons are:

- In a field theory language
*H*coincides with the Chern-Simon number which is know to be related to the topological properties of the gauge fields. - Since
*H*is a P (parity) and CP-odd function, the observation of a nonvanishing net value of this quantity would be a manifestation of a macroscopic violation of both these symmetries. - It is know from magneto-hydro-dynamics that the presence of magnetic helicity can lead to the amplification of magnetic fields and contribute to their self-organization into a large scale ordered configuration (see Sec. 1.4). The same phenomenon could take place at a cosmological level.

In the last few years, several authors proposed mechanisms for the
production of magnetic helicity in the early Universe starting from particle
physics processes. Cornwall
[174]
suggested that magnetic helicity
was initially stored in the Universe under the form of baryons (B) and
leptons (L) numbers possibly generated by some GUT scale baryogenesis
mechanism.
He assumed that an order one fraction of the total classically conserved
*B*+*L*
charge was dissipated by anomalous processes at the EW scale and showed
that a small fraction of this dissipated charge, of the order of
*n*_{B+L} *T*^{-3}, may have been converted
into a magnetic helicity of the order of

(4.62) |

Another possibility is that before symmetry breaking of a non-Abelian gauge symmetry vacuum configurations existed which carried nonvanishing winding number. It was shown by Jackiw and Pi [175] that after symmetry breaking, one direction in isospin space is identified with electromagnetism, and the projection of the vacuum configuration becomes a magnetic field with non vanishing helicity.

A different mechanism was proposed by Joyce and Shaposhnikov
[176].
In this case it was assumed that some excess of
right-handed electrons over left-handed positrons was produced by
some means (*e.g.* from some GUT scale leptogenesis) above a
temperature *T*_{R}. At temperatures higher than
*T*_{R} perturbative
processes which changes electron chirality are out of thermal
equilibrium (*T*_{R} ~ 3 TeV in the SM
[177]). Therefore,
a chemical potential for right electrons *µ*_{R} can
be introduced
above *T*_{R}. On the other hand, the corresponding charge
is not conserved because of the Abelian anomaly, which gives

(4.63) |

In the above,
*f*_{µ}
and ^{µ} are,
respectively, the *U*(1)_{Y} hypercharge field strength and its
dual, *g*' is the associated gauge coupling constant, and
*y*_{R} = - 2 is the hypercharge of the right electron. As
it is well known, Eq. (4.63) relates the variation in the number of
the right handed electrons *N*_{R} to the variation of the
topological properties (Chern-Simon number) of the hypercharge
field configuration. By rewriting this expression in terms of the
hypermagnetic
_{Y} and of
the hyperelectric
_{Y} fields,

(4.64) |

it is evident the relation of
*j*_{R}^{µ} with the hypermagnetic
helicity. It is worthwhile to observe that only the hypermagnetic
helicity is coupled to the fermion number by the chiral anomaly
whereas such a coupling is absent for the Maxwell magnetic
helicity because of the vector-like coupling of the electromagnetic
field to fermions. From Eq. (4.63) it follows that the
variation in *N*_{R}, is related to the variation in the
Chern-Simon number,

(4.65) |

by *N*_{R}
= 1/2 *y*_{R}^{2}
*N*_{CS}.
In the above
*b*_{k} represents the hypercharge field potential. The energy
density sitting in right electrons is of order
*µ*_{R}^{2} *T*^{2} and
their number density of order *µ*_{R}
*T*^{2}. Such fermionic number
can be reprocessed into hypermagnetic helicity of order
*g*'^{2} *k**b*^{2}, with energy of order
*k*^{2} *b*^{2}, where *k* is the
momentum of
the classical hypercharge field and *b* is its amplitude.
Therefore, at *b* > *T* / *g*'^{2} it is
energetically convenient for the
system to produce hypermagnetic helicity by "eating-up"
fermions. It was showed by Joyce and Shaposhnikov
[176],
and in more detail by Giovannini and Shaposhnikov
[178],
that such a phenomenon corresponds to a *magnetic dynamo
instability*. In fact, by adding the anomaly term to the Maxwell
like equations for the hyperelectric and hypermagnetic fields
these authors were able to write the anomalous
magneto-hydrodynamical (AMHD) equations for the electroweak plasma
in the expanding Universe, including the following generalized
hypermagnetic diffusivity equation

(4.66) |

In the above *a* is the Universe scale factor,
'
*g*'^{2} /
4,
=
*a*^{-1}(*t*) *dt* is the conformal time, and
the electric
conductivity of the electroweak plasma
[49].
By comparing Eq. (4.6615])
one sees that the first term on the r.h.s. of Eq. (4.66)
corresponds to the so-called dynamo term which is know to be
related to the vorticity in the plasma flux
[11]. We see
that the fermion asymmetry, by providing a macroscopic parity
violation, plays a similar role of that played by the fluid
vorticity in the dynamo amplification of magnetic fields. In the
scenario advocated in Ref.
[176,
178],
however, it is not
clear what are the seed fields from which the dynamo amplification
start from. Clearly, no amplification takes place if the initial
value of the hypermagnetic field vanishes. Perhaps, seeds field
may have been provided by thermal fluctuations or from a previous
phase transition although this is a matter of speculation.
According to Joyce and Shaposhnikov
[176],
assuming that a large right electron asymmetry *µ*_{R}
/ *T* ~ 10^{-2} was
present when *T* = *T*_{R}, magnetic field of strength
*B* ~ 10^{22} G may have survived until the EWPT time with
typical inhomogeneity scales ~ 10^{6} / *T*.

Another interesting point raised by Giovannini and Shaposhnikov [178] is that the Abelian anomaly may also process a preexisting hypermagnetic helicity into fermions. In this sense the presence of tangled magnetic fields in the early Universe may provide a new leptogenesis scenario.

Indeed, assuming that a primordial hypermagnetic field
_{Y} was
present before the EWPT with some non-trivial topology
(i.e. <**B**_{Y} ·
×
**B**_{Y}>
0) the kinetic equation of
right electrons for *T* > *T*_{c} is

(4.67) |

where

(4.68) |

is the rate of the *N*_{R} nonconserving anomalous
processes whereas
is the
rate of the perturbative ones. In the case
_{np} >
, as a consequence
of Eq. (4.67), one finds

(4.69) |

Below the critical temperature the hypermagnetic fields are converted into ordinary Maxwell magnetic fields. Similarly to the usual EW baryogenesis scenario, the fermion number asymmetry produced by the Abelian anomaly may survive the sphaleron wash-out only if the EWPT is strongly first order, which we know to be incompatible with the standard model in the absence of primordial magnetic fields. However, Giovannini and Shaposhinikov argued that this argument might not apply in the presence of strong magnetic fields (we shall discuss this issue in Chap. 5). If this is the case a baryon asymmetry compatible with the observations might have been generated at the EW scale. Another prediction of this scenario is the production of strong density fluctuations at the BBN time which may affect the primordial synthesis of light elements [179].

Primordial magnetic fields and the primordial magnetic helicity may also have been produced by the interaction of the hypercharge component of the electromagnetic field with a cosmic pseudoscalar field condensate which provides the required macroscopic parity violation. This idea was first sketched by Turner and Widrow [45] in the framework of an inflationary model of the Universe which we shall discuss in more details in the next section. Turner and Widrow assumed the pseudoscalar field to be the axion, a particle which existence is invoked for the solution of the strong CP problem (for a review see Ref. [180]). Although the axion is supposed to be electrically neutral it couples to electromagnetic field by means of the anomaly. Indeed, the effective Lagrangian for axion electrodynamics is

(4.70) |

where *g*_{a} is a coupling constant of order
, the vacuum
angle
= _{a}
/ *f*_{a},
_{a} is the axion
field, *f*_{a}
is the Peccei-Quinn symmetry breaking scale (see
Ref. [180]),
*F*_{µ}
is the electromagnetic field strength and
^{µ} is its dual. Since the axion
field, as any other scalar field, is not conformally invariant
(see the next section), it will be amplified during the
inflationary expansion of the Universe starting from quantum
fluctuations, giving rise to
<^{2}>
~ (*H*_{0} / *f*_{a})^{2}, which can
act as a source term for the electromagnetic field
^{(17)}.

Carroll and Field
[182]
reconsider in more details
the idea of Turner and Widrow and found that the evolution of a Fourier
mode of the magnetic field with wave number *k* is governed by the
equation

(4.71) |

where *F*_{ ± } =
*a*^{2}(*B*_{x} ± *iB*_{y})
are the Fourier modes
corresponding to different circular polarizations, and
is
the conformal time. One or both polarization modes will be
unstable for *k* < *g*_{a}
|*d*_{a} /
*d*|, whereas both
polarization modes can becomes unstable to exponential growth if
is oscillating. In this
case it seems as if a quite strong magnetic field
could be produced during inflation. However, such a conclusion was
recently criticized by Giovannini
[183]
who noted that above the EWPT temperature QCD sphalerons
[184] are
in thermal equilibrium which can effectively damp axion
oscillations. In fact, because of the presence of QCD sphaleron
the axion equation of motion becomes

(4.72) |

where [184]

(4.73) |

(where _{s} =
*g*_{s}^{2} /
4). Giovannini founds that
sphaleron
induced damping dominates over damping produced by the expansion
of the Universe if *f*_{a} > 10^{9} GeV. Since
astrophysical and cosmological bounds
[125]
leave open a window
10^{10} GeV < *f*_{a} < 10^{12} GeV,
it follows that no
magnetic fields amplification was possible until QCD sphaleron
went out of thermal equilibrium. A very tiny magnetic helicity
production from axion oscillations may occur at lower
temperatures. In fact, Giovannini
[183]
estimated that in the temperature range 1 GeV
> *T* > 10MeV a magnetic helicity of the order of

(4.74) |

may be generated, which is probably too small to have any phenomenological relevance.

Generation of magnetic fields from coherent oscillations or roll-down of
pseudoscalar fields different from the axion has been also
considered in the literature. It is interesting that pseudoscalar
fields with an axion-like anomalous coupling to the
electromagnetic field appear in several possible extension of the
SM. Typically these fields have only perturbative derivative
interactions and therefore vanishing potential at high
temperatures, and acquire a potential at lower temperatures
through non-perturbative interactions. The generic form of this
potential is *V*()
= *V*_{0}^{4}
*A*(/*f*), where
*A* is a bounded
function of the "Peccei-Quinn" scale *f* which in this case can
be as large as the Planck scale. The pseudoscalar mass *m* ~
*V*_{0}^{2} / *f* may range from a few eV to
10^{12} GeV.
The amplification of magnetic fields proceeds in a way
quite similar to that discussed for axions. The evolution equation
of the electromagnetic Fourier modes were derived by Brustein and
Oaknin [185]
who added to Eq. (4.71) the effect of a finite electric conductivity,
finding

(4.75) |

If the pseudoscalar field is oscillating, the field velocity
*d* /
*d* changes sign
periodically and both polarization
modes are amplified, each during a different semi-cycle. Each mode
is amplified during one part of the cycle and damped during the
other part of the cycle. Net amplification results when *g*
*k**d* /
*d* >
^{2} and the
total amplification is exponential
in the number of cycles. Taking into account that
~ 10*T* before the
EWPT [49],
Brustein and Oaknin found
that a huge amplification can be obtained from pseudoscalar field
oscillations at *T* ~ 1 TeV, for a scalar mass *m* of few TeV
and *g* *f* ~ 10. This is a particularly interesting range of
parameters as in this case the pseudoscalar field dynamics could
be associated with breaking of supersymmetry at the TeV scale.
Since only a limited range of Fourier modes are amplified, with
*k* not too different from *T*, and modes with *k* /
*T* <
_{EW}
<< 1 are rapidly
dissipated, amplified fields may survive
until the EWPT only if amplification occurred just before the
transition. If this is a natural assumption may be a matter of
discussion. However, it was pointed-out by Brustein and Oaknin
[185]
that if, depending on the form of the pseudoscalar
field potential, this field rolls instead of oscillating, then the
hypermagnetic fields would survive until EWPT even if the
amplification takes place at higher temperatures before the
transition. In this case there could be interesting consequences
for the EW baryogenesis. This is so, because the pseudoscalar
field may carry a considerable amount of helicity. According to what
discussed in the above, this
number will be released in the form of fermions and baryons if the
EWPT is strongly first order. Brustein and Oaknin argued that this
mechanism could naturally generate the observed BAU.

Clearly, a more serious problem of this mechanism is the same of other EW baryogenesis scenarios, namely to have a strongly first order EWPT. An interesting possibility which was proposed by two different groups [178, 186] is that strong magnetic fields may enhance the strength of the EWPT (see Chap. 5). Unfortunately, detailed lattice computations [187] showed that this is not the case. Furthermore, in a recent work by Comelli et al. [188] it was shown that strong magnetic fields also increase the rate of EW sphalerons so that the preservation of the baryon asymmetry calls for a much stronger phase transition than required in the absence of a magnetic field. The authors of Ref. [188] showed that this effect overwhelms the gain in the phase transition strength (see Sec. 5.3.2). Therefore, the only way for the kind of EW baryogenesis mechanism discussed in the above to work is to invoke for extensions of the standard model which allow for a strong first order EWPT [144].

**Electroweak strings**, which we introduced in the
previous section, may also carry a net hypermagnetic helicity and
act as a source of the observed BAU. One of the interesting
features of this objects is that they should have been formed
during the EWPT even if this transition is second order or just a
cross over [173].
It is known that electroweak strings
can carry magnetic helicity, hence a baryon number, which is
related to the twisting and linking of the string gauge fields
[159,
189].
Several authors (see *e.g.*
Ref. [190])
tried to construct a viable EW baryogenesis
scenario based on these embedded defects. Many of these models,
however, run into the same problems of more conventional EW
baryogenesis scenarios. An interesting attempt was done by
Barriola [191]
who invoked the presence of a primordial
magnetic field at the EWPT time. Barriola observed that the production and
the following decay of electroweak strings give rise, in the
presence of an external magnetic field, to a variation in the baryon number
*B* in the time
interval *dt*, given by

(4.76) |

where _{Z} and _{Z} are the Z-electric and
Z-magnetic fields of the strings whereas
and
are the
corresponding standard Maxwell fields. The first term
in the r.h.s. of (4.76) represents the change in the
helicity of the string network, and the second and third terms
come form the coupling of the string fields with the external
Maxwell fields. Whereas, the first term averages to zero over a
large number of strings, the other terms may not. Clearly, some
bias is required to select a direction in the baryon number
violation. In the specific model considered by Barriola this is
obtained by a CP violation coming from the extension of the Higgs
sector of the SM, and from the dynamics of strings which are
supposed to collapse along their axis. It was concluded by
Barriola that such a mechanism could account for the BAU.
Unfortunately, it is quite unclear if the invoked out-of-equilibrium
mechanism based on the string collapse could indeed be effective
enough to avoid the sphaleron wash-out. We think, however, that
this possibility deserves further study.

^{17} The careful reader may wonder
what is the fate of axions in the presence of cosmic magnetic fields.
Interestingly, it was showed by Ahonen, Enqvist and Raffelt
[181]
that although oscillating cosmic axions drives an oscillating electric
field, the ensuing dissipation of axions is found to be inversely
proportional to the plasma conductivity and is, therefore, negligible.
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