6.3. SOLAR NEUTRINO PROBLEM 355
whereHˆ=−ihc ̄(~α·∇)+mc^2 α 0 , andVis as in (6.3.35).
The equations in (6.3.34) are also in the form
(6.3.37)
eiλ^1 t/h ̄
eiλ^2 t/h ̄
eiλ^3 t/ ̄h
A†Hˆ
νe
νμ
ντ
=
λ 1
λ 2
λ 3
φ 1
φ 2
φ 3
.
ReplacingHˆbyHˆin (6.3.37), we infer from (6.3.34) that
(6.3.38) A†HˆA
ν 1
ν 2
ν 3
=
β 1
β 2
β 3
ν 1
ν 2
ν 3
,
whereHˆis as in (6.3.36).
The equation (6.3.38) is the neutrino oscillation model with the MSW effect, where the
eigenvaluesβkand eigenstatesνk( 1 ≤k≤ 3 )are different from that of (6.3.34). In fact, the
MSW effect is just the weak interaction effect.
6.3.5 Massless neutrino oscillation model
There are several serious problems in the massive neutrino oscillation model (6.3.34), which
we briefly explain as follows.
1.Parity problem.It is known that all weak interaction decays and scatteringsinvolving
neutrinos violate the parity symmetry, discovered by Lee and Yang in 1956 and experimen-
tally verified by C. Wu (Lee and Yang, 1956 ;Wu, Ambler, Hayward, Hoppes and Hudson,
1957 ). It means that the neutrinos are parity non-conserved. Hence it requires that under the
space reflective transformation
(6.3.39) x−→ −x,
the equations governing neutrinos should violate the reflective invariance. Based on Remarks
6.7and6.8, the Dirac equations are invariant under the reflection (6.3.39), while the Weyl
equations are not invariant. Hence, the massive neutrino oscillation model is in conflict with
the violation of parity symmetry.
2.Handedness and speed of neutrinos. Experiments showed that all neutrinos possess
only the left-handed spinJ=−^12 , and anti-neutrinos possess the right-handed spinJ=^12. It
implies that the velocity of free neutrinos must be at the speed of light, which is a contradic-
tion with massive neutrino assumption.
In fact, the handedness is allowed only for massless particles. Otherwise, there exist two
coordinate systemsAandBsatisfying
vA<vp<vB,wherevA,vBandvpare the velocities ofA,Band the particle. When we look at the particle
νfromAandB, the spins would be reversed. Therefore, all massive particles must have both
left-handed and right-handed spins.