Mathematical Principles of Theoretical Physics

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 371

By (6.4.7), the strong interaction potential for quarks is taken as

(6.4.50) S 0 =

(

ρw

ρ 0

) 3

gs

[

1

r

−

Aq

ρ 0

( 1 +

r

r 0

)e−r/r^0

]

,

wherer 0 = 10 −^16 cm,ρ 0 is the quark radius,Aqthe constant of quarks.
Similar to the relation (6.4.43), from (6.4.48) and (6.4.50) we can obtain the parameter
relation of nucleons as follows

(6.4.51) λn=

1

2

mqv^2 + 2

(

ρw

ρ 0

) 6

g^2 s

(

1

rn

−

κn

ρ 0

)

whereλnis the bound energy of nucleons,mqthe quark masses,vthe average velocity of
quarks,rnthe nucleon radius,κnthe bound parameter of nucleons, these parameters are
expressed as

rn=

∫ρ 1

ρ 0

rφn^2 (r)dr,

κn=Aq

∫ρ 1

ρ 0

r^2 ( 1 +

r

r 0

(6.4.52) e−r/r^0 φn^2 (r)dr,

(m

qv

h ̄

) 2

=

∫ρ 1

ρ 0

r^2

(

dφn

dr

) 2

dr,

whereλnandφnare the first eigenvalue and eigenfucntion of the following equation

(6.4.53)

− ̄h

2

2 mq

1

r^2

d

dr(r

2 d

dr)φn+^2 gsS^0 φn=λ φn,

φn(ρ 0 ) =φn(ρ 1 ) = 0.

Remark 6.19.The two sets of parameters (6.4.47) and (6.4.52) can characterize the physical
properties of naked electrons and naked nucleons respectively. For the real electrons and
nucleons, we have to consider their mediator clouds.

6.4.5 Energy spectrum of mediators

Mediators are massless bosons, which consists of a weakton and its anti-particle. From the
viewpoint of bound energy, gluons are generated by both weakand strong interactions, and
others are bound only by weak interaction. Hence, the spectral equations of gluons are dif-
ferent from those the other mediators.

1.Gluons.The weakton constituent of gluons is given by

(6.4.54) g=w∗w∗.

Based onthe weakton model,w∗andw∗contain a weak chargegwand a strong chargegs. By
(6.4.6) and (6.4.9),
g^2 w
g^2 s

= 0. 25 ,