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\begin{document}

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\title{Primordial Magnetic Fields and Triangle Anomaly}

%\vspace{1cm}

\tocauthor{M. E. Shaposhnikov}
\author{M.E. Shaposhnikov\\
\small\it Theory Division, CERN, CH-1211 Geneva 23, Switzerland\\
\small\it and\\
\small\it Institute for Nuclear Research, Russian Academy of Sciences,\\
\small\it Moscow 117312, Russia}

\date{}
\maketitle

\begin{abstract}
  A possible relationship between primordial hypermagnetic
fields, fermion number asymmetry and baryon number fluctuations is
discussed.
\end{abstract}

%\vspace{1cm}

The existence of large scale magnetic fields observed in our and other
galaxies is still a puzzle. If the galactic magnetic field is naively
blue-shifted to earlier epochs, one finds that the Universe was always
filled by a magnetic field of a considerable amplitude, with energy
density comparable with that of cosmic microwave background radiation.
Of course, this consideration does not take into account different
physical processes operating at the galactic scale, such as the dynamo
mechanism, which change considerably the naive scaling law
$|\vec{H}|\sim T^2$ (here $\vec{H}$ is the magnetic field, and $T$ is
the temperature). In any case, it is widely believed that some seed
fields of primordial origin are necessary for the successful generation
of the galactic magnetic fields.

There are neither compelling theoretical arguments nor motivated
phenomenological constraints that could exclude the existence of
magnetic fields at temperatures as high as that of the electroweak
phase transition, $T_c \sim 100$ GeV. At $T_c$ the SU(2)$\times$U(1)
group ``breaks" down to U(1)$_{EM}$, and a transition from
hypermagnetic $U(1)$ fields to ordinary magnetic fields takes place.
This talk addresses the question of a possible existence of any
cosmological implications associated with this transition. Since this
analysis has already been published in \cite{js,gs1,gs2}, I just make
some qualitative remarks and formulate the results.

The main observation is that there is an essential difference between
interaction of magnetic fields and that of hypermagnetic fields
with matter. An ordinary electromagnetic field has a vector-like
coupling to fermions, while the coupling of hypercharge fields
is chiral. Thus, if hyperelectric ($\vec{{\cal E}}_{Y}$) and
hypermagnetic ($\vec{{\cal H}}_{Y}$) fields are present simultaneously,
they cause a variation of the fermionic number according to the anomaly
equation,
\begin{equation}
\partial_\mu j_\mu \sim
\frac{g'^2}{4\pi^2} \vec{{\cal H}}_{Y}\cdot \vec{{\cal E}}_{Y}~,
\end{equation}
where $g'$ is the hypercharge gauge coupling constant. So, the
anomalous coupling of a hypercharge magnetic field to fermions
induces a transformation of the hypermagnetic energy density into
fermionic number and vice versa.

In \cite{gs1,gs2} we assumed that primordial hypermagnetic fields were
generated by some mechanism before the electroweak epoch and studied
their influence on the fermion content of the Universe. We showed that,
depending on the particle physics model and on the initial spectrum of
the primordial magnetic fields, quite large fluctuations of the baryon
and lepton numbers may be generated. These fluctuations can survive
until the onset of Big Bang Nucleosynthesis (BBN) and create unusual
initial conditions for the calculation of the light element abundances.
In particular, a natural outcome of our considerations is a non-uniform
distribution of baryon number, not necessarily positive-definite.
Matter--antimatter domains are then possible. The requirement that
these fluctuations be small enough not to conflict with the predictions
of the standard homogeneous BBN allowed us to put quite a strong
constraint on the spectrum of the primordial magnetic fields. Moreover,
we argued that the primordial magnetic field may change the nature of
the electroweak phase transition. Finally, the existence of primordial
fields with some specific topological structure may result in the
production of the net baryon number at the electroweak scale.

In ref. \cite{js} we studied a possible hypermagnetic field
generation from right-handed electron asymmetry, which may have existed
before the electroweak phase transition as a result of GUT
baryogenesis. We showed that quite large magnetic fields ($\sim
10^{22}$ gauss at $T\sim 100$ GeV with typical inhomogeneity scale $
\sim \frac{ 10^6}{T}$) can be generated. These fields may be of
cosmological interest, potentially acting as seeds for amplification to
larger scale magnetic fields through non-linear mechanisms \cite{kari}.

\begin{thebibliography}{99}
\bibitem{js} M. Joyce and M. Shaposhnikov,
\Journal{\PRL}{79}{1193}{1997}.

\bibitem{gs1} M. Giovannini and M. Shaposhnikov,
\Journal{\PRL}{79}{22}{1997}.

\bibitem{gs2} M. Giovannini and M. Shaposhnikov,
\Journal{\PRD}{57}{2186}{1998}.

\bibitem{kari} K. Enqvist, {\em Int. J. Mod. Phys.} D {\bf 7}, 331
(1998).

\end{thebibliography}

\end{document}



