Mathematical Principles of Theoretical Physics

240 CHAPTER 4. UNIFIED FIELD THEORY

The weak force between the two particles is given by

(4.6.33) F=−

d

dr

V=gw(ρ 1 )gw(ρ 2 )e−kr

[

1

r^2

+

k

r

−

4 B

ρ

k^2 re−kr

]

.

3.Repulsive condition. For the two particles as above, if their weak interaction constant
Bsatisfies the inequality

1

r^2

+

k

r

≥

4 B

ρ

k^2 re−kr, ∀ 0 <r≤

1

k

,

or equivalentlyBsatisfies

(4.6.34) B≤

e

2

ρk (e= 2. 718 ,k= 1016 cm−^1 ),

then the weak force between these two particles is always repulsive.
It follows from this conclusion that for the neutrinos:

(ν 1 ,ν 2 ,ν 3 ) = (νe,νμ,ντ),

(ν 1 ,ν 2 ,ν 3 ) = (νe,νμ,ντ),

the weak interaction constants

Bijforνiandνj ∀ 1 ≤i,j≤ 3 ,

Bijforνiand anti-neutrinosνj ∀i 6 =j,

satisfy the exclusion condition (4.6.34).

4.Value of weak charge gw. Based on the Standard Model, the coupling constantGwof
theβ-decay of nucleons and the Fermi constantGfhave the following relation

(4.6.35) G^2 w=

8

√

2

(m

Wc

h ̄

) 2

Gf,

andGfis given by

(4.6.36) Gf= 10 −^5 ̄hc/

(mpc

h ̄

) 2

,

wheremWandmpare masses ofW±bosons and protons. By the gauge theory,Gwis also the
coupling constant ofSU( 2 )gauge fields. Therefore we can regardGwas the weak charge of
nucleons, i.e.
Gw=gw(ρn), ρn the nucleon radius.

In addition, it is known that

gw(ρn) = 9

(

ρw

ρn

) 3

gs.