# Three Family Neutrino Oscillations

This applet provides a tool for investigations of the effect of neutrino oscillations. It includes three family oscillations formalism, but ignores possible CP violation and matter effects. For the moment, it assumes that the primary neutrino beam is the µ beam.

If you have comments or suggestions, or if you find errors, please send mail to Adam Para.

## Mass eigenstates and flavor eigenstates

There are three mass eigenstates:
• 1 with mass m1
• 2 with mass m2
• 3 with mass m3
If masses of the mass eigenstates are different, the physical neutrinos will, in general, oscillate with the frequencies related to the differences of squares of masses, dmij=mi2-mj2. Solar neutrino deficit is usually explained by postulating dm12 to be in the range of 10-10 - 10-5 eV2. Atmospheric neutrinos anomaly (SuperKamiokande) is likely to be related to dm23 of the order of 3.5x10-3 eV2.

This program takes dm12 and dm23 as parameters, and computes masses to be:

• m2= sqrt(m12+dm12)
• m3= sqrt(m22+dm23)
• mass m1 provides an overall shift of the masses. It has no effect on neutrino oscillations, and it is assumed that m1=0

Physical states e, µ and  are combinations of the mass eigenstates. Mixing matrix is conveniently parametrized in terms of three mixing angles 12, 13 23. The resulting mixing matrix is shown at the bottom of the applet window. We use the PDG convention for the CKM matrix:

```

c12*c13                    s12*c13                s13

-s12*c23 - c12*s23*s13     c12*c23 - s12*s23*s13       s23*c13

s12*s23 - c12*c23*s13     -c12*s23 - s12*c23*s13       c23*c13

```
where cij and sij denote cosine and sine of the corresponding mixing angle.

Squares of the matrix elements in a given row give the relative contribution of different mass eigenstates to a given physical states, whereas squares of elements in a given column give decomposition of a mass eigenstates in terms of the physical states.

The bar chart in the left lower corner shows a composition of the physical state µ in terms of the mass eigenstates.

## Propagation of µ beam

Muon neutrino beam is produced by -> µ + µ decays. The produced µ state is a mixture of different mass eigenstates, and they evolve in time with their proper frequencies. As time evolution of mass eigenstates is different, the propagating beam will have different composition at different distances. The effect will depend on the beam energy.

The top plot illustrates the change of the beam composition with the distance from the source. The distance scale is from 0 to 10exp km, where exp can be selected from the 'Configuration' line. [Distance from Fermilab to Soudan is 735 km, from the Earth to the Sun is 1.6x109 km.]

The blue/green/red curves show probabilities of observation of e, µ and  , respectively, as a function of a distance from the neutrino source.

Energy of the propagating beam can be selected on the 'configuration' line too.

A detector located at the distance dist (selected in the 'Configuration' line) will detect neutrinos of different flavor. The relative probabilities of detecting different flavors is illustrated by a bar chart , bottom right.

## Modification of the µ energy spectrum as a result of neutrino oscillations

Oscillation frequency of mass eigenstates depends on the beam energy, therefore the neutrino oscillations depend on the energy too. If a beam of µ is produced with energies between elow and ehigh (in GeV!, Energy Spectrum line) and the neutrino detector is placed at the distance dist (in km, Configuration' line), than the observed energy spectrum of detected µ will exhibit oscillatory pattern shown in the middle graph.