Neutrino electromagnetic properties: a window to new physics - II
Alexander Studenikin
hep-ph



§ ELECTROMAGNETIC PROPERTIES OF NEUTRINO

1. After the Nobel Prize of 2015 in physics was awarded to Arthur McDonald and Takaaki Kajita for
the discovery of neutrino oscillations, there can be no doubt that neutrinos are massive particles.
It has been know since quite many years <cit.> that massive neutrinos should have nonzero
magnetic moments. Although up to now there are no indications in favour of nonzero neutrino electromagnetic properties, neither from terrestrial experiments nor from astrophysical observations, the electromagnetic properties is one of popular issues related to neutrinos and this
problem has been discussed many times in recent literature (see, for instance,
<cit.>-<cit.>


). A complete review on neutrino electromagnetic properties and neutrino electormagnetic interactions is given in <cit.>.
In <cit.> an addendum to previous reviews are provided and the most recent new aspects and prospects related to neutrino electromagnetic interactions that have appeared after publication of the review paper <cit.> are discussed.

It has been often claimed (see, for instance, <cit.>) that neutrino electromagnetic properties open a window to new physics. In this short note we would like to justify this statement and to focus a discussion on to what extend and what kind of new physics neutrino electromagnetic properties do communicate with.




§ NEUTRINO MAGNETIC MOMENT


 Consider the magnetic moment as the most well theoretically appreciated and experimentally studied (constrained) electromagnetic characteristic of neutrinos.  Within the initial formulation of the Standard Model neutrinos are massless particles with zero magnetic moment. Thus, the would be nonzero neutrino magnetic moment regardless of its value should indicate the existence of new physics beyond the Standard Model. Indeed, as it has been shown in <cit.> a minimal extension of the Standard Model with right-handed neutrinos yields for the diagonal magnetic  moment of a Dirac neutrino to be proportional to the neutrino mass m_i,

    μ^D_ii
      = 3e G_F m_i/8√(2)π ^2≈ 3.2× 10^-19(m_i/1  eV) μ_B,

where μ_B is the Bohr magneton. For Majorana neutrinos the diagonal
magnetic moments are zero in the neutrino mass basis and only transition (off-diagonal) magnetic
moments  μ^M_ij (i≠ j) can be nonzero in this case.

The value of neutrino magnetic moment (<ref>) is several orders of magnitude smaller than the present
experimental limits if to account for the existed constraints on
neutrino masses. Note that in general transition magnetic moments are even smaller due to the GIM cancelation mechanism.

The best laboratory upper limit on neutrino magnetic moment has been obtained by the GEMMA collaboration that investigates the reactor antineutrino-electron scattering at the Kalinin Nuclear Power Plant (Russia) <cit.>. Within the presently reached electron recoil energy threshold of T ∼ 2.8 keV the neutrino magnetic moment is bounded from above by the value

    μ_ν < 2.9 × 10^-11μ_B  (90% C.L.).

This limit, obtained from unobservant distortions in the recoil electron energy
spectra, is valid for both Dirac and Majorana neutrinos and for both diagonal
and transition moments.
The most recent stringent constraint on the electron effective magnetic
moment

    μ_ν_e≤ 2.8 ×
    10^-11μ _B

has been reported by the Borexino Collaboration <cit.>.

A strict astrophysical bound on the neutrino magnetic moment is provided by the observed properties of globular cluster stars and amounts to <cit.> (see also <cit.>)

    ( ∑ _i,j| μ_ij| ^2) ^1/2≤ (2.2-2.6) ×
    10^-11μ _B.

This stringent astrophysical constraint on neutrino magnetic moments is applicable to both Dirac and Majorana neutrinos.

There is a huge gap of many orders of magnitude
between the present experimental limits on
neutrino magnetic moments and the prediction of a minimal
extension of the Standard Model. Therefore, if any direct
experimental confirmation of nonzero neutrino magnetic moment were
obtained in a reasonable future, it would open a window to new physics beyond a minimal
extension of the Standard Model.

 Much larger values for a neutrino magnetic moments are predicted in
different other extensions of the Standard Model. However, there is a general
problem for a theoretical model of how to get large
magnetic moment for a neutrino and simultaneously to avoid an
unacceptable large contribution to the neutrino mass (see the corresponding discussion in <cit.> and references therein).
If a contribution to the neutrino magnetic moment of an order μ_ν∼eG/Λ is generated by physics beyond a minimal
extension of the Standard Model at an energy scale characterized by
Λ, then the corresponding contribution to the neutrino mass
is

δ m_ν∼Λ ^2/2m_eμ_ν/μ_B=
μ_ν/10^-18μ_B(Λ/1  Tev)^2eV.

Therefore, a particular fine tuning is needed to get large value
for a neutrino magnetic moment while keeping the neutrino mass
within experimental bounds. Different possibilities to have large magnetic moment for a
neutrino were considered  in the literature (see in <cit.>).

A general and termed model-independent upper bound on the Dirac neutrino
magnetic moment, that can be generated by an effective theory beyond
a minimal extension of the Standard Model, has been derived in
<cit.>: μ_ν≤
10^-14μ_B. Note that the corresponding limit for transition moments of Majorana neutrinos is much weaker <cit.>. Thus, the value of a neutrino magnetic moment once observed
experimentally at the level not less than μ_ν∼10^-14μ_B would provide
information on the nature of neutrinos. This can be also considered as a view
on the realm of new physics.



§ NEUTRINO ELECTRIC MOMENT


  From the most general form of the neutrino
electromagnetic vertex function Λ _ij(q^2)
(see for detailed discussion <cit.>)
there are  three other sets (in addition to the magnetic moments μ_ij) of electromagnetic
characteristics that determine a neutrino coupling with real photons (q^2=0). They are namely the dipole electric moments ϵ_ij, anapole moments a_ij
and millicharges q_ij. In the theoretical framework with CP violation a neutrino
can have nonzero electric moments ϵ_ij. In the laboratory neutrino
scattering experiments for searching the neutrino magnetic moment (like, for instance,
the mentioned above GEMMA experiment) the electric moment contributions interfere with
those due to magnetic moments. Thus, these kind of experiments also provide constraints
on ϵ_ij. The astrophysical bounds (<ref>)

are also applicable for constraining ϵ_ij <cit.>-
<cit.>.



§ NEUTRINO ELECTRIC MILLICHARGE


There are extensions of the Standard Model that allow for nonzero
neutrino electric millicharges. This option can be provided by
not excluded experimentally possibilities for hypercharhge dequantization or
another new physics related with an additional U(1) symmetry
peculiar for extended theoretical frameworks. Neutrino millicharges
are strongly constrained on the level q_ν∼ 10^-21 e_0
(e_0 is the value of an electron charge) from neutrality of the hydrogen atom.

 A nonzero neutrino millicharge q_ν would contribute to the neutrino electron scattering in the terrestrial experiments. Therefore, it is possible to get bounds on q_ν in the reactor antineutrino experiments GEMMA. The most stringent constraint using the GEMMA data  is q_ν≤ 1.5 × 10^-11 e_0 <cit.> (see also <cit.>).

A neutrino millicharge might have specific phenomenological consequences
in astrophysics because of new electromagnetic processes are opened
due to a nonzero charge. Following this line, the most stringent astrophysical constraint on neutrino millicharges
q_ν≤ 1.3 × 10^-19 e_0 was obtained in <cit.>. This bound
follows from the impact of the neutrino star turning mechanism (STν) <cit.> that can be charged as a new physics phenomenon end up with a pulsar rotation frequency shift engendered by the motion of escaping from the
star neutrinos on curved trajectories due to millicharge interaction with a constant
magnetic field.



§ NEUTRINO CHARGE RADIUS


Even if a neutrino millicharge is vanishing, the electric form factor
f_ij^q(q^2) can still contain nontrivial information about
neutrino electromagnetic properties. The corresponding electromagnetic characteristics is
determined by the derivative of f_ij^q(q^2) over q^2  at
q^2=0 and is termed neutrino charge radius,


r_ij^2
=-
6

df_ij^q(q^2)/dq^2_| _ q^2=0.

A neutrino charge radius (that is indeed the charges radius squared) contributes to the neutrino scattering cross section on electrons and thus
can be constrained by the corresponding laboratory experiments <cit.>.
In all (see, for instance, <cit.>) but one previous studies it was claimed
 that the effect of the neutrino
charge radius can be included just as a shift of the vector coupling constant g_V
in the weak
contribution to the cross section.
However, as it has been recently illustrated, in <cit.> within the direct calculations of
the cross section accounting for all possible neutrino electromagnetic characteristics
and neutrino mixing, this is not the fact. The neutrino charge radius dependence of the cross section
indeed is more complicated and there is, in particular, the dependence on the interference terms of the type
g_Vr_ij^2 that can't be obtained just only by the corresponding shift of the constant g_V.



§ NEUTRINO SPIN PRECESSION IN MAGNETIC FIELD


One of an important phenomenon among several processes of neutrino electromagnetic interacts is neutrino spin and spin-flavour precession in magnetic fields. The origin of these effects is the neutrino magnetic moment interaction with a transversal magnetic field determined by μ_νB_⊥. The neutrino spin precession
in a transverse magnetic field can result in the neutrino helicity flip that can have important phenomenological consequences because an active neutrino ν_L can be converted to a sterile one ν_R in invironments with a magnetic field.  The precession ν_eL→ν_eR in the magnetic field of the Sun was first considered in <cit.>, a similar effect in magnetic fields of supernovae and neutron stars came into sight for the first time  in <cit.>.



§ NEUTRINO SPIN PRECESSION IN TRANSVERSAL MATTER CURRENTS


There is a phenomenon of new physics related to the neutrino spin precession in magnetic fields. For many years, until  2004, it was believed that a neutrino helicity precession
and the corresponding spin oscillations can be induced by the neutrino magnetic
interactions with the transversal magnetic field. A new and very interesting possibility for neutrino spin
(and spin-flavour) oscillations engendered by the neutrino interaction with matter background was
proposed and investigated in <cit.>. It
was shown that neutrino spin oscillations
can be induced not only by the neutrino interaction with a  magnetic field but also by neutrino interactions with matter in the case when there is a transversal matter current (or a transversal matter matter polarization). The is no need for neutrino magnetic moment interaction in this case. The origin of the oscillations ν_L⇔ν_R in the transversal matter currents j_⊥ is the neutrino weak interactions with moving matter and the corresponding mixing between neutrino states ν_L and ν_R  is determined by G_F j_⊥. This new effect has been explicitly highlighted in <cit.>, recently the existence of this effect was confirmed in <cit.>. For historical notes reviewing studies and the detailed derivation of the discussed effect see <cit.> and <cit.>.



§ CONCLUSIONS AND FUTURE PROSPECTS


 The foreseen progress in constraining neutrino electromagnetic characteristics is related, first of all, with the expected new results from the GEMMA experiment measurements of the reactor antineutrino cross section on electrons at Kalinin Power Plant. The new set of data is expected to arrive next year. The electron energy threshold will be as low as 350  eV ( or even lower, ∼ 200  eV). This will provide possibility to test the neutrino magnetic moment on the level of μ_ν∼ 0.9 × 10^-12μ_B and also to test the millicharge on the level of q_ν∼ 1.8 × 10^-13 e_0 <cit.>. For the next future, presently it seems unclear whether further progress in constraining the neutrino electromagnetic characteristics would be achievable with this type of the reactor antineutrino  experiment. In this concern, a rather promising claim was made in  <cit.>. It was shown that even much smaller values of the Majorana neutrino transition moments
would probably be tested in future high-precision experiments with the astrophysical neutrinos.  In particular,
observations of supernova fluxes  in the JUNO  experiment (see <cit.>-

<cit.>)  may reveal the effect of  collective  spin-flavour oscillations  due to the Majorana neutrino transition moment μ^M_ν∼ 10^-21μ_B.

To conclude, the existing current constraints on the flavour neutrino charge radius r_e,μ,τ^2≤ 10^-32 - 10^-31 cm ^2  from the scattering experiments differ only by 1 to 2
orders of magnitude from the values r_e,μ,τ^2≤ 10^-33 cm ^2 calculated within the minimally extended Standard Model with right-handed neutrinos <cit.>. This indicates that the minimally extended Standard Model neutrino charge radii could be experimentally tested in the near future. Note that there is a need to re-estimate experimental constraints on r_e,μ,τ^2  from the scattering experiments following new derivation of the cross section <cit.> that properly accounts for the interference of the weak and charge radius electromagnetic interactions and also for
the neutrino mixing.





§ ACKNOWLEDGEMENTS

This work was supported by the Russian Foundation for Basic Research under grants
No. 16-02-01023 A and No. 17-52-53133 GFEN_a.


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