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How can neutrinos have a magnetic moment in spite of being elementary (as opposed to a neutron) and electrically neutral (as opposed to a proton)? How can it even be defined, and measured?

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    $\begingroup$ Ask yourself this: what experimental obsevables would that imply and are they seen? $\endgroup$ – dmckee Apr 1 '18 at 21:38
  • $\begingroup$ Dipole moments are related to charge; neutrinos are neutral. $\endgroup$ – SRS Apr 2 '18 at 0:09
  • $\begingroup$ How can photons have intrinsic angular momentum in spite of being massless and elementary? $\endgroup$ – Ben Crowell Apr 2 '18 at 1:09
  • $\begingroup$ So the origin of the magnetic moment of the neutrino is its intrinsic spin? Why do photons don't have magnetoc moment then? Because it's massless? @BenCrowell $\endgroup$ – SRS Apr 2 '18 at 6:23
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The neutron has a magnetic moment despite being neutral. We usually attribute this to the internal structure of the neutron in terms of charged constituents, but this is not really necessary. Consider a theory (chiral perturbation theory), where elementary neutrons interact with protons and pions. Then there is a contribution to the magnetic moment of the neutron from intermediate states $n\to p\pi^-\to n$.

Similarly, neutrinos can have a magnetic moment because of intermediate states $\nu \to eW^+\to\nu$. This does not happen in the SM, because the magnetic moment operator is $$ \bar\psi\sigma_{\mu\nu}F^{\mu\nu}\psi=\bar\psi_L\sigma_{\mu\nu}F^{\mu\nu}\psi_R + (L\leftrightarrow R) $$ and requires a right handed neutrino. In extensions of the SM where the mass is Dirac, a neutrino magnetic moment is automatically generated (as pointed out by Ron below). The Majorana case is more complicated.

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  • $\begingroup$ Absence of which intrinsic property would have caused your magnetic moment operator to vanish? By your logic, why can't photons have any magnetic moment? @Thomas $\endgroup$ – SRS Apr 2 '18 at 6:42
  • $\begingroup$ You still have to be able to write down an appropriate operator (which in the case of the SM neutrino does not exist). By magnetic moment we mean something that looks like $\vec\Sigma\cdot\vec{B}$ in the rest frame. $\endgroup$ – Thomas Apr 2 '18 at 13:36
  • $\begingroup$ Not clear what this means for a massless particle (this is what happens for the neutrino), but the spatial part of the non-abelian three boson vertex $f^{abc}k_{[i}A^a_{j]}A^{bi}A^{cj}$ is usually called a "magnetic moment" coupling. But this only works for non-abelian fields (like the W) and not the photon. $\endgroup$ – Thomas Apr 2 '18 at 13:39
  • $\begingroup$ Note that the extending the Standard Model to describe massive neutrinos also gives you right-handed neutrinos (in the neutrino's rest frame). $\endgroup$ – rob Apr 2 '18 at 15:29
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As seen in the answer by Thomas, within the original standard model the neutrino has no magnetic moment, but also no mass, whereas neutrino oscillations have been measured, thus there should be an extension to the SM. Once one goes to extensions of the standard model the search starts. The standard model has already been extended with assuming that neutrinos do have mass, as seen in the table here.

Here one can find, in section IV, a review on the search for magnetic moment for the neutrino, with a list of experiments(page 29).

On how to measure: the recoil spectrum of neutrino interactions is measured and compared with theoretical predictions for a magnetic moment for the neutrino, in extensions of the standard model.

This experiment has put limits

A search for the solar neutrino effective magnetic moment has been performed using data from 1291.5 days exposure during the second phase of the Borexino experiment. No significant deviations from the expected shape of the electron recoil spectrum from solar neutrinos have been found, and a new upper limit on the effective neutrino magnetic moment of $\mu^\text{eff}_\nu < 2.8\times10^{-11}\mu_B$ at 90% confidence limits has been set using constraints on the sum of the solar neutrino fluxes implied by the radiochemical gallium experiments.Using the limit for the effective neutrino moment, new limits for the magnetic moments of the neutrino flavor states, and for the elements of the neutrino magnetic moments matrix for Dirac and Majorana neutrinos, are derived.

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