Neutrino masses weak eigenstates are NOT mass eigenstates i) construct with Then, the most general MAJORANA mass term upon diagonalization of the complex & symmetric matrix we get where are Majorana neutrinos ( )
ii) alternatively, introduce and write the most general DIRAC mass term diagonalization thereof leads to where are DIRAC neutrino fields
Neutrino oscillations A neutrino of definite flavor is produced at t=0 and has probability amplitude to be found in flavor state after time t with and Thus, the (vacuum) transition probability is P’s depend on 2 mass-squared differences, 3 mixing angles & 1 CP phase
The key parameter is the “oscillation length” Oscillations can be observed in an experiment if
Experiments fall roughly into 2 cathegories: i) atmospheric and long baseline accelerator experiments on the one hand & ii) solar and reactor long baseline experiments on the other. Analysis of data shows that and Then, neglecting small quantities and, oscillations in i) (for which holds) are mainly and thus
For experiments in cathegory ii), is relevant and the survival probabilities again can be given by standard 2-flavor formulas, i.e. for reactor (KamLAND) experiments, or for solar MSW matter oscillations. In short, oscillations in atmospheric-LBL and solar-KamLAND domains decouple.
A brief survey of experiments i) Super-Kamiokande atmospheric experiment if there were no oscillations, electron & muon events from up/down going should satisfy electron events satisfy this equality but muon events don’t best fit gave
Further support for disappearence is provided by K2K experiment MINOS experiment produced at KEK accelerator Fermilab-Soudan, 730km and detected at SKamiokande, 250km apart away 112 were observed and 204 events expected (if no oscillations) observed, expected (no oscillations) best fit
ii) Solar-KamLAND domain All solar neutrino experiments (Homestake, GALLEX-GNO, SAGE and Super-Kamiokande) show a factor 2-3 less rate than expected by the SSM. The SNO experiment gave model independent evidence for oscillations via reactions charged current neutral current elastic scattering From CC the flux of on Earth is obtained From NC the flux of on Earth is obtained SNO results:
Thus, about a factor 3 less solar electron neutrinos reach the Earth because they convert into other flavors on their way from the Sun to the Earth. The total flux measured is in agreement with the SSM prediction: Additional model independent evidence for oscillations comes from the reactor KamLAND experiment: In the Kamiokande mine the antineutrinos from 53 reactors in Japan are detected through. Their average distance to the detector is 170km. events were expected (no oscillations) 258 events were observed From a global analysis of solar & KamLAND data,
Static properties Majorana neutrinos have neither charge nor magnetic/electric dipole moments. This follows from and that implies However Dirac neutrinos (with mass) can have magnetic moment (and EDM, if CP is not conserved)
Indeed, the SM (with RH neutrino added) predicts a magnetic dipole moment: which is extremely small. The best empirical limits on MDM come from astrophysics: A larger MDM would cause excessive energy depletion from globular-cluster red giant cores via the plasma process (all flavors).
Dirac or Majorana? Best way to decide is neutrinoless double β decay: and other even-even nuclei Majorana ν The matrix element for the 0νββ-decay is proportional to an effective Majorana mass. Most competitive lower bounds on 0νββ-decay half-lives are,
New experiments in 0νββ-decay are presently being prepared (CUORE, GERDA,EXO, MAJORANA, …). Goal is to reach
What is the absolute scale of neutrino masses? From ν oscillations we only know mass squared differences. No absolute ν mass values are known. Upper bounds come from laboratory experiments and astrophysics/cosmology. Tritium β-decay experiments: Mainz and Troitsk experiments give The future Katrin experiment foresees a sensitivity
Astrophysics/Cosmology Bounds on neutrino masses have been derived from various astrophysical/cosmological settigs, e.g. Supernova 1987A, Lyman-α forests studies, Galaxy redshift surveys, CMB anisotropies, cosmic energy density, … The “classical” Gerstein-Zeldovich limit Light neutrinos (i.e. relativistic at neutrino decoupling, when T~1MeV) populate the Universe today (~100 per cubic cm). If they are nonrelativistic today, they contribute to the known matter density. Indeed, Observationally, and since
The most powerful constraint comes from the CMB radiation data in conjunction with the power spectrum obtained from Large Scale Structure (LSS) surveys. After their decoupling at T~1MeV, relativistic neutrinos free-stream at almost the speed of light and outflow from regions smaller than the horizon so that density perturbations at those scales are effectively erased. This comes to an end when neutrinos become non-relativistic and cluster with the cold components of dark matter. Hence, for all physical scales smaller than the size of the horizon at the time when neutrinos turn non-relativistic, the growth of perturbations is hindered. In Fourier space, scales are characterized by their “wavenumber” k and this limiting scale is given by Neutrino mass influences cosmic structure formation at small scales (i.e. ).
The power spectrum is defined as, i.e. the variance of the Fourier transformed density fluctuations. The power loss at small scales induced by a non-zero neutrino mass can be parameterized by W.Hu et al., PRL 80 (1998) 5255
From the large samples of data in galaxy redshift surveys, such as the 2dFGRS, the power spectrum of matter fluctuations can be analyzed and bounds on neutrino masses inferred. These bounds are particularly strong when the WMAP results on cosmological parameters (most importantly on and ) are incorporated in the analysis. In this way, cosmology sets the limit and cosmologists claim that future data will be sensitive to
SUMMARY 4 oscillation parameters are known (with approximate accuracies) The parameter is bounded The CP phase δ is unknown Cosmology sets a bound on absolute mass