354 CHAPTER 6. QUANTUM PHYSICS
It is very difficult to computeEe,Eμ,Eτby (6.3.31). However, sinceA∈SU( 3 )is norm-
preserving:
Ee^2 +Eμ^2 +Eτ^2 =E 12 +E^22 +E 32 ,
by (6.3.32) andEk^2 =p^2 c^2 +m^2 kc^4 , we deduce that
m^2 e+m^2 μ+m^2 τ=m^21 +m^22 +m^23 ,which leads to
(6.3.33) m^2 e+m^2 μ+m^2 τ=∆ 32 + 2 ∆ 21 + 3 m^21 ,
where∆ 32 and∆ 21 are as in (6.3.29).
If neutrinos have masses, then only the mass square differences∆ijin (6.3.29) can be
measured by current experimental methods. Hence, the only mass information ofνe,νμ,ντ
is given by the relation (6.3.33).
6.3.4 MSW effect
In 1978, L. Wolfenstein (Wolfenstein, 1978 ) first noted that as neutrinos pass through matter
there are additional effects due to elastic scattering
νe+e−→νe+e.This phenomenonwas also observed and expanded by S. Mikheyev and A. Smirnov (Mikheev and Smirnov,
1986 ), and is now called the MSW effect.
The MSW effect can be reflected in the neutrino oscillation model. We recall the oscilla-
tion model without MSW effect expressed as
(6.3.34)
νk=φk(x)e−iλkt/h ̄,
[−i ̄hc(~α·∇)+mc^2 α 0 ]φk=λkφk k= 1 , 2 , 3 ,
νe
νμ
ντ
=A
ν 1
ν 2
ν 3
, Ais as in (6.3.26).To consider the MSW effect, we have to add weak interaction potentials in the Hamilto-
nian operatorHˆfor neutrinosνe,νμ,ντ. The weak potential energy is as given in (4.6.32):
(6.3.35) Vν=gs(ρν)ρs(ρe)Nee−kr
[
1
r−
B
ρ
( 1 + 2 kr)e−kr]
,
whereρν,ρeare the radii of neutrinos and electron,gsis the weak charge, andNwis the weak
charge density. Namely the Hamiltonian with MSW effect for(νe,νμ,ντ)is
(6.3.36) Hˆ
νe
νμ
ντ
=
Hˆ+Ve 0 0
0 Hˆ+Vμ 0
0 0 Hˆ+Vτ
νe
νμ
ντ