The Force that Holds the Nucleus Together

Before the discovery of quarks the force which held the nucleolus of atoms together was called the strong force or the strong nuclear force as opposed to the weak nuclear force responsible for the beta decay process. Today we know that this force is not fundamental but a residual force which results the fundamental color force of quarks. In this post I will expand on this strong force, showing how it emerges out of the color force. But first a brief look at the color force is in order. The color charge is based on the SU (3) symmetry group. The color charge structure can be expressed by a three vector. r  g 1  C   g b 2  br 

Based on SU (5) or SO (10) symmetry groups.

( r , g , b, p , y )

This gives us:

1

 1  2   r  0   1    2  1  2   1  g   2    0   

   0    1  b   2  1    2

The exchange bosons for these local charges are described by the SU (3) group. The generators for this group the Gell-Mann Matrixes. Given any special unitary group SU ( n) we get: n2  1

Generators of which n 1

are diagonal.

2

0 1 0    1 0 0  0 0 0    0 i 0  2    i 0 0  0 0 0   1 0 0 3    0 1 0  0 0 0   0 0 1 4    0 0 0  1 0 0    0 0 i  5    0 0 0  i 0 0    0 0 0 6    0 0 1  0 1 0   0 0 0  7    0 0 i  0 i 0    1 0 0  1   8   0 1 0  3   0 0 2  1

Here we see that  3 and  8 are the diagonal generators. This gives us a total of eight states given by”

ni 

1 2

c i c

3

Where c  r , g,b 

and r   c g b  

Giving us:

1 

1 2

2  i 3  4 

1 2 1 2

5  i 6 

1 2

7  i 8 

1 6

 rb  br  1 2

 rb  br 

 rr  bb   rg  gr  1 2

 rg  gr 

 bg  gb  1 2

 bg  gb 

 rr  bb  2 gg 

The six non diagonal gluon states are linear superpositions of two of the states above. We get;

4

rb  rb  gr  gr  gb  gb 

1 2 1 2 1

 1  i 2   1  i 2 

2 1 2 1 2 1 2

 4  i 5   4  i 5   6  i 7   6  i 7 

However, each color charge is associated with a color Isospin and color hypercharge based on SU (3) symmetry. We see this symmetry from the following diagram.

5

These color Isospin and color hypercharge results in a strong force charge carried by the gluons and the nucleons. We can define these charges in terms of the Gell-MannNishijima formula. Q  T3  Y

Which gives us; r0 g 1 b

1 2

The gluon states are given by:

gij 

1 2

u u i

j

 di d j 

So we get g 3  g8  0 g rg  1 g rg  1 1 2 1 g rb   2 3 g gb   2 3 g gb   2 g rb  

6

For any Nucleon we get;

r  g b 

1 2

So we see that while a nucleon has no net color charge it does retain a strong force charge. The meson which are combinations of color and anti color carry no strong charge, The Boson field that serves as the primary strong force carrier are the pion fields. These quanta are the Goldstone Bosons of the color force. We have two electrically charged bosons and one uncharged boson. This forms an Abelian SU  2  interaction with regard to color but also has two elements that couple to SU (2) Non Abelian Isospin. . 0 P   1 0 P   0

0  0 1  0 1 1 0  P0    2  0 1 

And we get flavor vectors F †   ui , d i  u  F  i  di 

Giving us”

   F † P F  0  F † P0 F

7

Since this coupling field is a scalar field ( even spin number) we get attraction for like charges and repulsion for opposite charges.

These interactions are illustrated by the tree level Feynman diagrams.

These pions which exchange the strong force carry no strong force charge. But these particles are massive causing the attractive force between nucleons to drop off faster than Inverse Square. The point at which the attractive force equals the electric force is given by;

8

 mcR   h 

Qs  e R2



Qem R2

Giving us:

R

Q  h h ln  em   mc  Qs  mc

For the   we get 8.9 1015 meters And for the  0 we get 9.2 1015 meters The approximate size of a nucleon (Diameter) is 2.5 1015 meters . This illustrates the very short range of the strong force. Bob Zannelli

9