5.6. Hypermagnetic fields
At small temperatures and small densities of the different fermionic charges the SUL(2) UY(1) is broken down to the Uem(1) and the long range fields which can survive in the plasma are the ordinary magnetic fields. For sufficiently high temperatures (and for sufficiently high values of the various fermionic charges) the SUL(2) UY(1) is restored and non-screened vector modes correspond to hypermagnetic fields. In fact, Abelian electric fields decay within a typical time scale 1/ where is the conductivity. The long-ranged non-Abelian magnetic fields (corresponding, for instance, to the color SU(3) or to the weak SU(2)) cannot exist because at high temperatures the non-Abelian interactions induce a "magnetic" mass gap ~ g2T where g is the gauge coupling constant. Also the non-Abelian electric fields decay because of the finite value of the conductivity as it occurs for Abelian electric fields. Therefore, the only long scale field that can exist in the plasma for enough time must be associated with some Abelian U(1) group. This statement, valid to all orders in perturbation theory, has been confirmed non-perturbatively for the electroweak theory by recent lattice studies in . Under normal conditions (i.e. small temperatures and small densities of the different fermionic charges) the SU(2)×U(1)Y symmetry is "broken" down to U(1)EM, the massless field corresponding to U(1)EM is the ordinary photon and the only long-lived field in the plasma is the ordinary magnetic one. At sufficiently high temperatures, T > Tc, the SU(2)×U(1)Y symmetry is "restored", and non-screened vector modes Yµ correspond to the U(1)Y hypercharge group. Hence, if primordial fields existed at T > Tc, they did correspond to hypercharge rather than to U(1)EM.
At the electroweak epoch the typical size of the Hubble radius is of the order of 3 cm . The typical diffusion scale is of the order of 10-9 cm. Therefore, over roughly eight orders of magnitude hypermagnetic fields can be present in the plasma without being dissipated .
5.6.1. Anomalous MHD equations
The evolution of hypermagnetic fields can be obtained from the anomalous magnetohydrodynamical (AMHD) equations. The AMHD equations generalize the treatment of plasma effects involving hypermagnetic fields to the case of finite fermionic density .
There are essential differences between the interactions of magnetic fields and the ones of hypermagnetic fields with matter. The ordinary electromagnetic field has a vector-like coupling to the fermions, while the coupling of the hypercharge fields is chiral. Thus, if hyperelectric (Y) and hypermagnetic (Y) fields are present simultaneously, they cause a variation of the fermionic number according to the anomaly equation, µ jµ ~ g'2 / (4 2) Y . Y (here g' the hypercharge gauge coupling constant). Now, the presence of non-homogeneous hypermagnetic fields in the EW plasma with (hyper)conductivity c always implies the existence of a related electric field, Y ~ 1 / c × Y. Since for a general stochastic magnetic background <(Y . × Y)2> 0, the non-uniform hypermagnetic field may absorb or release fermions and produce, ultimately, baryon and lepton density perturbations because of the anomaly equation.
The behaviour of cold fermionic matter with non-zero anomalous Abelian charges was considered in  where it was pointed out that the anomalous fermionic matter is unstable against the creation of Abelian gauge field with non-zero Chern-Simons number, which eats up fermions because of the anomaly. The right electron number density may be converted to the hypercharge field because of a similar effect. Also the opposite effect is possible: hypercharge fields may be converted into fermions in a hot environment.
The electroweak plasma in complete thermal equilibrium at a temperature T can be characterized by nf chemical potentials µi, i = 1,..., nf corresponding to the exactly conserved global charges
(Li is the lepton number of the i-th generation, B is the baryon number, and nf is the number of fermionic generations). One should also introduce a chemical potential µY corresponding to weak hypercharge, but its value is fixed from the requirement of the hypercharge neutrality of the plasma, <Y> = 0.
It is interesting to study this plasma slightly out of thermal equilibrium, for instance in the situation where a non-uniform distribution of the hypermagnetic field is present. Because of the anomaly, this field is coupled to the fermionic number densities. In principle, different chemical potentials can be assigned to all the fermionic degrees of freedom of the electroweak theory (45 if nf = 3) and the coupled system of Boltzmann-type equations for these chemical potentials and the hypercharge fields may be written. Since we are interested in the slow processes in the plasma, this is not necessary. If the coupling, corresponding to some slow process, is switched off, then the electroweak theory acquires an extra conserved charge and a further chemical potential should be added to the system given in Eq. (5.105).
An interesting observation (which turns out to be quite important in our context) has been made in [167, 168, 169], where it was noticed that perturbative reactions with right-handed electron chirality flip are out of thermal equilibrium at temperatures higher than some temperature TR. (23) Thus, the number of right electrons is perturbatively conserved at temperatures T > TR and the chemical potential µR can be introduced for it. On the other hand, this charge is not conserved because of the Abelian anomaly,
and it is therefore coupled to the hypermagnetic field. Here and are, respectively, the UY(1) hypercharge field strengths and their duals, g' is the associated gauge coupling and yR = -2 is the hypercharge of the right electron.
Now we are ready to derive the anomalous MHD equations in flat space [163, 164]. The effective Lagrangian density describing the dynamics of the gauge fields at finite fermionic density is :
(g is the determinant of the metric defined in (5.1); Y = [ Y]; is the covariant derivative with respect to the metric (5.1) [notice that in the metric (5.1) [ Y] = [ Y]]; g' is the Abelian coupling constant). The first and the last terms in Eq. (5.107) are nothing but the curved space generalization of the flat-space effective Lagrangian for the hypercharge fields at finite fermion density [163, 164], J is the ohmic current. The equations of motion for the hyperelectric and hypermagnetic fields are then
with the same notations introduced in the case of the conventional MHD equations.
To Eqs. (5.108), the evolution equation of the right electron chemical potential, accounting for the anomalous and perturbative non-conservation of the right electron number density (nR), must be added:
where is the perturbative chirality-changing rate, = T TR / M0, nReq is the equilibrium value of the right electron number density, and the term proportional to Y . Y is the right electron anomaly contribution.
Finally, the relationship between the right electron number density and the chemical potential must be specified. This relation depends upon the particle content of the theory. In the case of the Minimal Standard Models (MSM) the evolution equation of the chemical potential becomes 
At finite hyperconductivity (in what we would call, in a MHD context, resistive approximation) we have that from Eq. (5.108) the induced hyperelectric field is not exactly orthogonal to the hypermagnetic one and, moreover, an extra "fermionic" current comes in the game thanks to the fact that we are working at finite chemical potential. Therefore in our context the resistive Ohm law can be written as
In the bracket appearing in Eq. (5.111) we can identify two different contributions. One is associated with the curl of the magnetic field. We will call this the MHD contribution, since it appears in the same way in ordinary plasmas. The other contribution contains the chemical potential and it is directly proportional to the magnetic field and to the chemical potential. This is a typical finite density effect. In fact the extra Ohmic current simply describes the possibility that the energy sitting in real fermionic degrees of freedom can be transferred to the hypermagnetic field. It may be of some interest to notice the analogy between the first term of Eq. (5.111) and the typical form of the ohmic current discussed in Eq. (4.17) appearing in the context of the dynamo mechanism. In the latter case the presence of a current (proportional to the vorticity through the dynamo term) was indicating that large length scales magnetic fields could grow by eating up fluid vortices. By inserting Y obtained from the generalized Ohm law (5.111) in the evolution equations (5.108) of the hypercharge fields, we obtain the generalized form of the magnetic diffusivity equation:
In order to be consistent with our resistive approach we neglected terms containing time derivatives of the electric field, which are sub-leading provided the conductivity is finite. In our considerations we will also make a further simplification, namely we will assume that the EW plasma is (globally) parity-invariant and that, therefore, no global vorticity is present. Therefore, since the length scale of variation of the bulk velocity field is much shorter than the correlation distance of the hypermagnetic field, the infrared modes of the hypercharge will be practically unaffected by the velocity of the plasma, which will be neglected when the large-scale part of the hypercharge is concerned. This corresponds to the usual MHD treatment of a mirror symmetric plasma (see, e.g. Eq. (4.32)-(4.34), when = 0).
Eqs. (5.112) and (5.110) form a set of AMHD equations for the hypercharge magnetic field and right electron chemical potential in the expanding Universe.
5.6.2. Hypermagnetic knots
The Abelian nature of the hypercharge field does not forbid that the hypermagnetic flux lines should have a trivial topological structure. This situation is similar to what already encountered in the case of conventional electromagnetic fields with the important difference that the evolution equations of hypermagnetic fields are different from the ones of ordinary magnetic fields. After a swift summary of the properties of hypermagnetic knots (based on [171, 172]), some interesting applications of these hypercharge profiles will be reviewed.
In the gauge Y0 = 0, . = 0, an example of topologically non-trivial configuration of the hypercharge field is the Chern-Simons wave [173, 174, 175]
This particular configuration is not homogeneous but it describes a hypermagnetic knot with homogeneous helicity and Chern-Simons number density
where Y = × , H(t) = k0 Y(t); g' is the U(1)Y coupling.
It is possible to construct hypermagnetic knot configurations with finite energy and helicity which are localized in space and within typical distance scale Ls. Let us consider in fact the following configuration in spherical coordinates 
where = r / Ls is the rescaled radius and B0 is some dimensionless amplitude and n is just an integer number whose physical interpretation will become clear in a moment. The hypermagnetic field can be easily computed from the previous expression and it is
The poloidal and toroidal components of can be usefully expressed as p = Hr r + H and t = . The Chern-Simons number is finite and it is given by
The total helicity of the configuration can also be computed
We can compute also the total energy of the field
and we discover that it is proportional to n2. This means that one way of increasing the total energy of the field is to increase the number of knots and twists in the flux lines.
This type of configurations can be also obtained by projecting a non-Abelian SU(2) (vacuum) gauge field on a fixed electromagnetic direction  (24). The resulting profile of the knot depends upon an arbitrary function of the radial distance.
These configurations have been also studied in [178, 179]. In particular, in , the relaxation of HK has been investigated with a technique different from the one employed in [171, 172] but with similar results. The problem of scattering of fermions in the background of hypermagnetic fields has been also studied in [180, 181].
Hypermagnetic knots with large correlation scale can be also generated dynamically provided an unknotted hypermagnetic background is already present.
Let us assume that dynamical pseudoscalar particles are evolving in the background geometry given by Eq. (5.1). The pseudoscalars are not a source of the background (i.e. they do not affect the time evolution of the scale factor) but, nonetheless, they evolve according to their specific dynamics and can excite other degrees of freedom.
The action describing the interaction of a dynamical pseudoscalar with hypercharge fields can be written as
This action is quite generic. In the case V() = (m2 / 2) 2 Eq. (5.120) is nothing but the curved space generalization of the model usually employed in direct searches of axionic particles [182, 183, 184]. The constant in front of the anomaly is a model-dependent factor. For example, in the case of axionic particles, for large temperatures T mW, the Abelian gauge fields present in Eq. (5.120) will be hypercharge fields and c = c Y ' / (2 ) where ' = g'2 / 4 and cY is a numerical factor of order 1 which can be computed (in a specific axion scenario) by knowing the Peccei-Quinn charges of all the fermions present in the model [185, 186]. For small temperatures T mW we have that the Abelian fields present in the action (5.120) will coincide with ordinary electromagnetic fields and c = c em / 2 where em is the fine structure constant and c is again a numerical factor.
The coupled system of equations describing the evolution of the pseudoscalars and of the Abelian gauge fields can be easily derived by varying the action with respect to and Yµ,
are the usual covariant derivatives defined from the background FRW metric Eqs. (5.121) can be written in terms of the physical gauge fields
We want now to study the amplification of gauge field fluctuations induced by the time evolution of . Then, the evolution equation for the hypermagnetic fluctuations Y can be obtained by linearizing Eqs. (5.123). We will assume that any background gauge field is absent. In the linearisation procedure we will also assume that the pseudoscalar field can be treated as completely homogeneous (i.e. | | << '). This seems to be natural if, prior to the radiation dominated epoch, an inflationary phase diluted the gradients of the pseudoscalar.
In this approximation, the result of the linearization can be simply written in terms of the vector potentials in the gauge Y0 = 0 and . = 0:
By combining the evolution equations for the gauge fields we can find a decoupled evolution equation for Y,
From this equation is already apparent that the pseudo-scalar vertex induces an interaction in the two physical polarizations of the hypermagnetic field. Giving initial conditions which are such that Y 0 with Y . × Y = 0 a profile with Y . × Y 0 can be generated provided ' 0.
23 This temperature depends on the particle physics model, see also the discussion reported in Section 5. In the MSM TR 80 TeV [167, 168, 169]. Back.
24 In order to interpret these solutions it is very interesting to make use of the Clebsh decomposition. The implications of this decomposition (beyond the hydrodynamical context, where it was originally discovered) have been recently discussed (see  and references therein). Back.