**3.5. Neutrino spin-oscillations in the presence of a
magnetic field**

It is interesting to consider how the limit obtained in the
previous section changes if neutrinos carry non-standard
properties which may change the effective neutrino number during
BBN. We are especially interested here to the possibility that
neutrinos carry non-vanishing masses and magnetic moments. If this
is the case, the dipole interaction of the neutrinos with the magnetic field
may give rise to *spin-oscillations* of the neutrinos, i.e.
periodic conversion of a helicity state into another. In the case
of Dirac neutrinos, this phenomenon may have crucial consequences
for BBN. In fact, spin-oscillation can populate the right-handed
helicity state of the neutrino which, being practically sterile
(for *m*_{}
<< *T*) to weak interactions, would otherwise play no
effective role. By adding a new degree of freedom to the thermal
bath, such an effect may produce dangerous consequences for the
outcome of BBN. This problem was first pointed-out by
Shapiro and Wasserman
[112]
and, independently, by Lynn
[113]
who used the argument to put a constraint on the
product of the magnetic field with the neutrino magnetic moment.
In both works, however, the important role played by neutrino refractive
properties determined by the neutrino collective interaction with
the heat-bath, as well as that played by neutrino scattering
with leptons, were disregarded. A more complete treatment was developed by
Fukugita et al. [114].
They showed that the conditions under which
the neutrino wrong-helicity state can be effectively populated are the
following:

- the spin-oscillation frequency
*E*_{magn}= 2*µ*_{}*B*has to exceed the Universe expansion rate; - since neutrino scattering destroy the phase relationship
between the left-handed and right-handed helicity states,
*E*_{magn}has to be larger than the scattering rate; - since the refractive indices for left-handed and right-handed
states,
*n*_{L}and*n*_{R}, are not equal, oscillations can occur only if(3.35) where

*E*_{refr}(*n*_{L}-*n*_{R})*E*_{}.

The BBN is affected only if such conditions are simultaneously
satisfied at some temperature *T*_{osc} in the range
*T*_{dec}
*T*_{osc}
*T*_{QCD} where *T*_{dec}
1 MeV
is the neutrino decoupling temperature and *T*_{QCD}
200 MeV. Note that in
the case right-handed neutrinos
decouples before the QCD phase transition, the huge amount of entropy which
is expected to be released during this transition would dilute their
relative abundance so to prevent any effect on the BBN.
From the previous considerations the authors of Ref.
[114] derived the limit

(3.36) |

where *µ*_{B} is the Bohr magneton.
The work of Fukugita et al. has been reconsidered by several
authors. For example, Enqvist, Olesen and Semikoz
[115],
improved the previous analysis by considering the effect of the
neutrino refractive properties on the left-right transition probability.
Elmfors, Grasso and Raffelt
[116]
accounted for the effect of the magnetic field on the neutrino
refractive properties and used an
improved treatment of neutrino collisions. First of all, Elmfors et al.
noted that by affecting the thermodynamics properties of the
electromagnetic component of the heat-bath (see
Sec. 3.3)
a strong magnetic field changes also the neutrino potentials. This
may have relevant consequences both for neutrino spin-oscillations
and flavour oscillations in a magnetized medium
[117].
The interplay between spin-oscillations and collisions was then
accounted in Ref.
[116]
by means of the following evolution equation
[118]

(3.37) |

where **P** is the neutrino polarization vector and
**P**_{T} is its transverse component respect to the neutrino
direction of motion. **V** is a vector of effective magnetic
interaction energies which can be decomposed into its transverse
and longitudinal components

(3.38) | |

(3.39) |

where *µ*_{}
is the neutrino magnetic moment and
[116]

(3.40) |

is the left-right neutrino energy difference in the magnetized
medium ^{(11)}.
It is worthwhile to note that in Eq. (3.40)
the expression (3.27) has to be used for
_{e}(*B*).
As we wrote above, collisions destroy the phase coherence between the
left-handed and right-handed component of a neutrino state, which
amounts to a damping of the transverse part **P**_{T} of
the polarization vector. The main contribution to the damping rate
*D* comes from neutrino elastic and inelastic scattering with
leptons and equals half the total collision rate of the
left-handed component
[121].
In the early Universe at *T* ~ 1 MeV one finds:

(3.41) |

where *f*_{D} is a order one numerical factor.
Inserting the previous expressions in Eq. (3.37) it is
easy to derive the neutrino depolarization rate
_{depol}. In
the small mixing angle limit,

(3.42) |

one finds

(3.43) |

(3.44) |

which is not too different from the limit (3.36)
previously found by the authors of Ref.
[114].
Limits on
*µ*_{} were
also found by the authors of
Refs. [108,
122]
who considered the case of random magnetic fields.

In principle, right-handed neutrinos could also be populated by
direct spin-flip interactions mediated by virtual photons
produced by scattering on charged particles or by annihilation
processes [123],
as well as by the interaction with small
scale magnetic fields produced by thermal fluctuations
[124].
In practice, however, bounds on
*µ*_{} from
a possible large scale magnetic
field are found to be more stringent even for very weak magnetic fields.
The most stringent upper limit on Dirac type neutrino magnetic
moment with mass
*m*_{} < 1
MeV, comes from stellar evolution considerations.
It is *µ*_{}
3 ×
10^{-12}*µ*_{B}
[125,
126].
It is interesting
to observe that if one of the neutrinos saturate this limit,
Eq. (3.44) implies the following quite stringent
bound on the present time cosmic magnetic field, *B*_{0}
10^{-13} G.

In the particle physics standard model, neutrinos have no magnetic
dipole moment. However, if the neutrino has a Dirac mass
*m*_{},
radiative corrections automatically give rise to a finite dipole
moment [113]

(3.45) |

even without invoking any further extension of the standard model beside that required to account for the finite neutrino mass. On the basis of this consideration Enqvist et al. [127] derived the following upper limit for the present time local magnetic field

(3.46) |

Clearly, this limit cannot compete with the constraint derived in the previous section.

Spin oscillations in the presence of twisted primordial magnetic fields (i.e. magnetic field with a nonvanishing helicity, see Sec. 1.4) have been considered by Athar [128]. Athar showed that in such a situation the left-right conversion probabilities for neutrino and antineutrinos may be different. This result may open the interesting possibility that a neutrino-antineutrino asymmetry may be generated during the big-bang by a preexisting non trivial topology of a primeval magnetic field. As we shall see in Sec. 4.4, the production of a net magnetic helicity of the Universe is indeed predicted by some models.

It is also interesting to speculate on the effects when the number of dimensions change, and these are large [129]. In fact BBN is one of the most serious objections to this idea, together with the background diffuse gamma radiation [130]. Detailed studies of effects of magnetic fields in these scenarios are not available yet.

^{11} For a computation of the neutrino
refractive properties in a magnetized medium see also Refs.
[119,
120].
Back.

^{12} Note that in Ref.
[116]
*B*_{0} was defined as the magnetic field at BBN time.
Back.