Neutrino Mass and Spontaneous Parity Nonconservation

1. Neutrino Mass and Spontaneous Parity Nonconservation Rabindra N. Mohapatra (Department of Physics, City College of New York) GoranSenjanovic (Department of Physics and Astronomy, University of Maryland) Published on 10 December 1979 Presented by Stephen Bello, September 3 2013

2. Neutrino Background Info • Neutrinos are neutral ½ spin particles that very weakly interact with matter.. They are affected solely by the weak force, which means neutrinos can typically pass through something as thick as the Earth without interacting with any matter. • Comes in three ‘flavors.’ One for each type of lepton: • Electron: (0.510 MeV/c2) • Muon: (105.66 MeV/c2) • Tau: (1776.82 MeV/c2) • The Standard Model assumes that neutrinos were massless. However experimentally verified neutrino oscillation requires neutrinos to have masses. • The first person to conceive of a neutrino with non-zero mass was Bruno Pontecorvo in 1957. He hypothesized that neutrinos may convert to other flavors (electron, muon and tau) spontaneously between measurements.

3. What is Parity? Parity transformations change a right-handed system into a left-handed one. Taking a parity transformation of a parity restores the original system. Particles can also be left or right-handed. Neutrinos are left handed because their spins always point in the opposite direction of their velocity vector. Anti-neutrinos are right-handed because their spins and velocity point in the same direction. Image courtesy of http://hyperphysics.phy-astr.gsu.edu

4. At the time of publication the level of experimental accuracy did not allow for any distinguishing differences to be seen between the standard model gauge group SU(2)L U(1) and the hypothesized group SU(2)L SU(2)R U(1) when it came to the structure of the neutrino neutral-current interactions and the parity-nonconserving electron-hadron weak interactions. • A big difference between the two gauge groups is how they deal with the neutrino: • The Left-Handed model, SU(2)L U(1), says that the neutrino has no mass. • The Left-Right-Symmetric Model, SU(2)L SU(2)R U(1), suggests that the neutrino has an extremely small, but non-zero, mass. • This made it important to fully understand just how small the neutrino mass is in the Left-Right-Symmetric Model

5. Let us assume a model exists of spontaneous parity nonconservation based on the LRS gauge group (Left-Right-Symmetric), SU(2)L SU(2)R U(1). With this assumption we can calculate an estimate for the neutrino mass that directly relates it to the right-handed gauge boson mass as well as the mass of the lepton. This equation holds for all lepton flavors (electron, muon, tau). If we take the limit of the right-handed gauge boson mass to infinity the neutrino’s mass goes to zero and we receive a pure V-A theory of weak interactions.

6. How did we get this Equation? Let us derive this equation for just one generation… 1) Start with two Majorana neutrinos (ν and N) and take the left/right handed lepton multiplets prior to the break to be: 2) Using left-right discrete symmetry we find the domain of coupling parameters: 3) In the domain of ν2 >> κ2 , κ`2 the symmetry breaks down and the local group is reduced to SU(2)L U(1), which is the standard model. We can now go ahead and use these relations to calculate the neutrino masses.

7. How did we get this Equation? (2) The gauge-invariant Yukawa couplings (in the realm of κ>> κ`) is written as: C is the Dirac Charge-conjugation matrix & & If we take the coupling parameters and the Yukawa couplings we find the electron mass equation: me≅ h2κ

8. How did we get this Equation? (3) We started with two Majorana neutrinos (ν and N). The mass matrix for the ν-N sector: Can make some assumptions about the Yukawa coupling terms. Let us take them to be the same order of magnitude. This gives us back the equation: Which spits back out after some substitution:

9. For the muon and tau generations it was assumed that the heavy Majorana mass was independent of the generation and therefore the mixing between the generations was ignored. The three generation equations give a reasonable mass upper limit. In conjunction the currently available charged and neutral current phenomena gave a lower band on the right-handed W-boson mass: m(wR) ≥ 3m(wL) m(wR) ≥ 250-300 GeV Plugging this value for the right-handed W-boson into each of the three equations gives the upper-limit on the mass for the three neutrino flavors (in order) to be: m(νe) ≤ 1.5 eV, m(νμ) ≤ 56 keV, and m(ντ) ≤ 16 MeV.

10. What does this tell us?

11. At low energies this model is indistinguishable from the standard model. Since R0 keeps the symmetry of SU(2)L U(1) unbroken after the first symmetry breaking stage, the following mass relations between the neutral and charged gauge mesons appear ( in the limit of ν2 >> κ2 + κ`2)

12. To suppress the amount of lepton-number-changing processes (ie. -eγ and -3e), the mixing of the electron generation with  and  generations should be made as small as possible.The way to make this zero is by assuming the following symmetry on the earlier Lagrangian (i = 1,2,3 is the iteration for each lepton flavor) • An interesting thing to note is that this symmetry prevents e- and e-mixing as well as the two examples above due to the breaking of the parity for the 2nd and 3rd generations. Since this is still unbroken this applies to every order in perturbation theory

13. In Conclusion… • We have created a simple model that has spontaneous parity nonconservation for the second and third generations and where the suppression of the V+A currents is directly proportional to the mass of the neutrino. • Provides an understanding for the mass of a neutrino. • Just for comparison: The current predicted sum for ALL three flavors of neutrinos is 0.23 eV as reported by the Planck Collaboration in March 2013.