Mathematical Principles of Theoretical Physics

370 CHAPTER 6. QUANTUM PHYSICS

6.4.4 Baryons and mesons

Hadrons include baryons and mesons, their spectral equations are given respectively in the
following.

1.Spectral equations of baryons.Baryons consist of three quarks:B=qqq, and each
quarkqconsists of threew-weaktons

q=w∗ww and q=w∗ww.

Hence, each quark possesses one strong chargegsand three weak charges 3gw. It looks as
if the bound energy of baryons is provided by both weak and strong interactions. However,
since the weak interaction is short-ranged, i.e.

range of weak force≤ 10 −^16 cm,

and the radii of baryons are
r> 10 −^16 cm.

Hence the main interaction to hold three quarks together is the strong force. Letm 1 ,m 2 ,m 3 be
the masses of three quarks in a baryon, andψk= (ψ 1 k,ψ 2 k)T( 1 ≤k≤ 3 )be the wave functions.
Then the spectral equations (6.4.21)-(6.4.23) for baryons are in the form

−

̄h^2

2 mk

(∇+i

2 gs

hc ̄

~s)^2 ψk+ 2 gsS 0 ψk+ 2 ~μk·curl~Sψk

(6.4.48) =λ ψk inρ 0 <|x|<ρ 1 , for 1≤k≤ 3 ,

ψ= (ψ^1 ,ψ^2 ,ψ^3 ) =0 at|x|=ρ 0 ,ρ 1 ,

whereρ 0 is the quark radius,ρ 1 the strong attracting radius,Sμ= (S 0 ,~S)as in (6.4.1) is 4-
dimensional strong potential, and~μk=hg ̄ s~σ/ 2 mkthe strong magnetic moment.ψksatisfy
the nomalization
∫

Ω

|ψk|^2 dx= 1 , inΩ={x∈R^3 |ρ 0 <|x|<ρ 1 }.

2.Spectral equations of mesons. Mesons consist of a quark and a antiquark:M=qq.
Hence, the bound energy of mesons is mainly provided by strong interaction potentialSμ=

(S 0 ,~S).Letm 1 ,m 2 be the masses of quark and antiquark,ψ^1 andψ^2 are their wave functions.
Then the spectral equations for mesons take the form

−

h ̄^2

2 mk

(∇+i

gs

̄hc

~S)^2 ψk+gsS 0 ψk+~μk·curl~Sψk

(6.4.49) =λ ψk inΩ={x∈R^3 |ρ 0 <|x|<ρ 1 }, k= 1 , 2 ,

(ψ^1 ,ψ^2 ) =0 on∂Ω,

andψk(k= 1 , 2 )satisfy the normalization.

3.Physical parameters of nucleons.In hadrons, only nucleons (protons and neutrons) are
long life-time. Therefore the parameters of nucleons are very important.