358 CHAPTER 6. QUANTUM PHYSICS
which are generated by the weak interaction attracting force, as demonstrated in the weak
charge potentials
(6.3.52) Φi=gwe−r/r^0
[
1
r−
Bi
ρν( 1 +
2 r
r 0)e−r/r^0]
for 1≤i≤ 3 ,wherer 0 = 10 −^16 cm,B 1 ,B 2 ,B 3 >0 are the weak interaction constants forνe,νμ,ντrespec-
tively, andρνis the neutrino radius.
The formula (6.3.52) defines attractive radiiRifor the neutrinos and antineutrinos of the
same flavors. Namely, whenνiandνiare in the radiusRi, the reaction (6.3.51) may occur:
(6.3.53) νi+νi−→ν(νiνi) if dist(νi,νi)<Ri for 1≤i≤ 3 ,
whereν 1 =νe,ν 2 =νμ,ν 3 =ντ, and dist(νi,νi)is the distance betweenνiandνi. The
condition (6.3.53) implies that the transition probabilityΓidepends onRi:
(6.3.54) Γi=Γi(Ri) for 1≤i≤ 3.
The attracting radiusRisatisfies that
(6.3.55)
d
dr
Φi(Ri) = 0.Thus we can give a non-oscillation mechanism of neutrinos toexplain the solar neutrino
problem. Namely, due to theβ-decay, there are large amounts of electronic anti-neutrinosνe
around the earth, which generate the reaction (6.3.51) withνe, leading to the discrepancy of
the solar neutrino.
In addition, there are three axis eigenvalues of the Weyl equations given by (6.3.48) and
(6.3.49). We believe that they are the three flavors of neutrinosνe,νμ,ντ. Namely, the fol-
lowing three wave functions
(6.3.56)
ψ 1 =c 1(
sinky
−cosky)
+c 2(
cosky
sinky)
,
ψ 2 =(
e−ikx
e−ikx)
,
ψ 3 =c 3(
e−ikz
0)
+c 4(
0
eikz)
represent the three flavors of neutrinos. In fact, massive particles in field equations are distin-
guished by different masses, and flavors of neutrinos by different axis eigenstates.
Also, neutrinos have significant scattering effect of the weak interaction. In fact, they
can have weak interactions with all subatomic particles. This can certainly cause deficits of
neutrinos, and may be one of the main reasons for the loss of solar neutrinos.
6.4 Energy Levels of Subatomic Particles
6.4.1 Preliminaries
For subatomic particles we can establish energy levels, andthis theory is based on two basic
ingredients: