Application of Group Theory in
Particle Physics using the Young
Tableaux Method
2006 PASSHE-MA CONFERENCE (March 31 – April 1)
Akhtar Mahmood (Assistant Professor of Physics)
Jack Dougherty (Undergraduate Research Assistant)
Edinboro University of Pennsylvania
Maury Gell-Mann
(at CalTech)
Proposed the existence of
Quarks as the fundamental
building blocks of matter in
the late 1960s.
Awarded Nobel Prize in
Physics in 1969 for the
development of the Quark
model, and the classification
of elementary particles.
Flavor
Q/e
u
+2/3
d
s
-1/3
-1/3
Three Families of Quarks
Generations
Increasing mass
I
Charge =
-1/3
Charge =
+2/3
II
III
d
s
b
(down) (strange) (bottom)
u
(up)
c
(charm)
t
(top)
Also, each quark has a corresponding antiquark.
The antiquarks have opposite charge to the quarks
STANDARD MODEL
QUARKS &
ANTI-QUARKS
LEPTONS &
ANTI-LEPTONS
s c
e  e
 
W ,Z0
b
  
  
H 0, H 
u
s c
b t
d
t
MESONS & ANTI-MESONS
( qq )
BARYONS & ANTI-BARYONS
( qqq )
(qqq )
PENTAQUARKS & ANTI-PENTAQUARKS
( qqqqq )
( qqqqq )
HEXAQUARKS & ANTI-HEXAQUARKS
( qqqqqq )
 , g, G
e  e
 
u d
( qq )
VECTOR
BOSONS
( qqqqqq )
HYBRID MESONS & ANTI-HYBRID MESONS
( qq g )
( q qg )
HYBRID BARYONS & ANTI-HYBRID BARYONS
( qqqg )
( qqqg )
BOUND STATE GLUONS
( gg ) ( ggg )
Hadrons- the composites of Quarks


Baryons are a composites of three quarks
Mesons are a composites of a quark-antiquark pair
Let’s make some more
baryons !
Quark
Q
Mass
up
+2/3
~5 [MeV/c2]
u
u
u
u
s
down
-1/3
~10 [MeV/c2]
d
d
d
u
d
Lambda (L)
Q=0
M=1116 MeV/c2
s
strange
-1/3
~200 [MeV/c2]
s
s
u
u
Sigma (S)
Q = +1
M=1189 MeV/c2
s
s
d
d
Sigma (S0)
Q=0
M=1192 MeV/c2
s
d
Sigma (S)
Q = -1
M=1197 MeV/c2
The Quark Configuration of
the  c Charmed-Strange Baryon
u
s
d
c

s
c

c


/
0
c
(1/ 2) ,(1/ 2) ,(3/ 2)

Summary of Young Tableaux
Method – The SU(N) Notation
Here N = 2 (for Spin up or
down)
N = 1 to 5 (for quark flavors –
up, down, strange, charm,
beauty)
Young Tableaux
(Young Diagrams)
In the SU(N) Notation,
N is denoted by a box.
(N) - Total Number of
Spins or Quark Flavors
The conjugate representation N
for Anti-quarks. Denoted by a column
of N-1 boxes.
If N = 4, then the conjugate
representation of N is with 3 boxes.
RULES FOR SU(N)
REPRESENTATION
No row is longer than
any above it. Descending rows
are always shorter than the
ones above them.
Allowed
Not Allowed
Box values always increase
from left to right in a row
3
4
5
NOT PERMITTED
3
24
1
Going from left to right,
no box- columns can be longer
than the previous one.
Allowed
Not Allowed
Allowed
Box values always decrease
from top to bottom in a
column
3
2
1
NOT PERMITTED
3
4
5
PERMITTED
4
5
6
3
4
5
2
3
1
7
NOT PERMITTED
3
4
5
6
3
4
5
3
4
3
Fully Symmetric Configuration
YS =
Mixed Symmetric Configuration
YM =
Fully Anti-symmetric Configuration
YA =
WHAT IS THE NET VALUE OF A
VALID CONFIGURATION ?
A RATIO OF…..
Product of Box Values (n)
V= ----------------------------------Product of Possible “Hook” Values (h)
A numerator (n)
is defined as the product of the
actual value each consecutive box.
3
4
5
n = 3  4  5 = 60
FOR N = 3
A denominator is defined as
the product of all of the
possible consecutive “hooks”
h=321=6
Hooks
FOR N = 3, IN THIS
PARTICULAR CONFIGURATION
V =
345
------------ = 10
321
LET’S CALCULATE THE VALUE
OF THIS CONFIGURATION FOR
ANY “N” !
N
N+1
N-1
N
“n” is defined as the product
of the actual value of each
consecutive box.
n = P ( each box values )
(product)
n = ( N )( N + 1 )( N – 1 )( N )
FOR N = 3
3
4
2
3
n = 3  4  2  3 = 72
“h” is defined as the product
of the possible “hook” values of
each consecutive boxconfiguration
n = P (“hook” value of each box-configuration)
(product)
h = 3  2  2  1 = 12
FOR N = 3, IN THIS
PARTICULAR CONFIGURATION
3423
V = ---------------- = 6
3221
Another Example (For N=3)
n = 3  4  2 = 24
3
4
h=311=3
2
V = 24  3 = 8M
COMPLETE BARYON WAVE FUNCTION
|Baryon(Qq1q2)A=|SpaceS|ColorASpinS,MS,MAFlavorS,MS,MA
SIZE = 1 fm
l´
q1
l
q3
q2
Space is described by the
Parity (P) of the Baryon
which is defined as the state
of the particle
P ≡ (-1)(l + l´)
In the ground state l = 0 and l´=0
 P = (-1)0 = +
(+)  Ground state (l = 0 and l’ = 0)
(-) Excited state (l = 1 and
l’ = 0 or (l = 0 and l’ = 1)
Color
The strong force that binds the quarks with
gluons carry “color charge” i.e. red, green and
blue.
Since a baryon is a Fermion, it must obey
Pauli’s Exclusion Principle, and hence the color
charge combination must be anti-symmetric
for all baryons.
1A
COLOR: From the SU(3)c Color
Symmetry Group
3C  3C  3C  10S  8MS  8MA  1A
Only the 1A color combination is valid
- Color Singlet (Colorless)
The 3 quark color are Red,
Green and Blue
3

3

3

4
3
1

5
2
3

3
4
2
1 2
3


3
4
2
1 3
2


3
1
2
2
1
3

345
342
342 321
321
311
311 321
10S

8MS  8MA  1A
1A  Color Singlet  Colorless
R + G + B = White (Colorless)
16RGB-GRB+BRG-RBG+GBR-BGRA
Total Angular Momentum (JP)
J  L + S where L = l +l’ and
(l
+l´)
P  (-1)
Each quark has a Spin (S) of ½
and a JP of ½+.
WHAT ARE POSSIBLE JP VALUES
OF A BARYON ?
In Ground
State
l´
q1
l
q3
l = 0 and
l’ =0
q2


1
1


      (0  1 )
2
2



/





1
1
1
1
1
 3




(0  1 )     0      1             
2
 2   
 2    2  MA  2  MS  2  S


If q1 q2 q3 (3 Distinct JP Values)
+
(3/2)
{q1q2q3}
+
+
(1)
(½)
{q1q2}
q3
+
(½)
q1
q2(½)+
(Symmetric)
+/
(1/2)
{q1q2}q3
+
+
(0)
(½)
[q1q2]
q3
(Mixed-Symmetric)
+
(1/2)
[q1q2]q3
q1 or q2 = u, d, s
q3 = c, b
(Mixed-Antisymmetric)
The Quark Configuration of the  c
Charmed-Strange Baryon in the Ground State
u
s
d
c

s
c

c


/
0
c
(1/ 2) ,(1/ 2) ,(3/ 2)

If q1= q2 q3 (2 Distinct JP Values)
+
(3/2)
{q1q2q3}
+
+
(1)
(½)
{q1q2}
q3
+
(½)
q1
q2(½)+
(Symmetric)
+/
(1/2)
{q1q2}q3
(Mixed-Symmetric)
q1 or q2 = u, d, s
q3 = c, b
If q1= q2= q3 (1 Distinct JP Value)
+
(3/2)
{q1q2q3}
+
+
(1)
(½)
{q1q2}
q3
+  (½)+
(½)
q1
q2
q1 or q2 = u, d, s
q3 = c, b
(Symmetric)
SPIN: From the SU(2)S Spin
Symmetry Group
2  2  2  4S  2MS  2MA
Three distinct spin states for each of the Three
distinct Jp values.
Jp(3/2)+  4S
Jp(1/2)+/  2MS
Jp(1/2)+  2MA
SPIN CONFIGURATION USING
YOUNG’S TABLEAUX
2

2

2

3
2
1

4
2
3

2
3
1
1 2
3


2
3
1
1 3
2


2
1
1
2
0
3

234
231
213 210
321
311
311 321
4S

2MS  2MA  0A
4 separate Spin orientations (S, Sz) for S = 3/2
+3/2
(S, Sz) or (J, Jz) for
+1/2
Sz
-1/2
-3/2
Sz = 4S and J = (3/2)+
(3/2, +3/2)
(3/2, +1/2)
(3/2, -1/2)
(3/2, -3/2)
2 separate Spin orientations (S, Sz) for S = 1/2
(S, Sz) or (J, Jz) for
Sz = 2MS OR Sz = 2MA
+1/2
and
J = (1/2)+/ OR J =(1/2)+
Sz
-1/2
(1/2, +1/2)
(1/2, -1/2)
(1/2, +1/2)
(1/2, -1/2)
|SpinS  FlavorS
|SpinMS  FlavorMS
|SpinMA  FlavorMA
JP = (3/2)+
|SpinS  |FlavorS  |4S  |(3/2)+S
|S  |{q3 1/√2 {q1q2 + q2q1}}S (3/2,+3/2)
|1/√3( +  + )S  |{q3 1/√2 {q1q2 + q2q1}}S (3/2,+1/2)
|1/√3( +  + )S  |{q3 1/√2 {q1q2 + q2q1}}S (3/2,-1/2)
|S  |{q3 1/√2 {q1q2 + q2q1}}S (3/2,-3/2)
JP = (1/2)+/
|SpinMS  |FlavorMS  |2MS  |(1/2)+/MS
|1/√6(- -  + 2 )MS  |q3 1/√2 {q1q2 + q2q1}MS
(1/2,+1/2)
|1/√6(+ +  - 2 )MS  |q3 1/√2 {q1q2 + q2q1}MS (1/2,1/2)
JP = (1/2)+
|SpinMA  |FlavorMA  |2MA  |(1/2)+/MA
|1/√2 ( - )MA  |q3 1/√2 {q1q2 - q2q1}MA (1/2,+1/2)
|1/√2 ( - )MA  |q3 1/√2 {q1q2 - q2q1}MA (1/2,-1/2)
Quark Mass and Charge
FLAVOR






UP
DOWN
STRANGE
CHARM
BOTTOM
TOP
MASS (GeV)
0.003
0.006
0.1
1.3
4.3
175
ELECTRIC CHARGE
+ 2/3
- 1/3
- 1/3
+ 2/3
- 1/3
+ 2/3
QUANTUM PROPERTIES OF BARYONS
Baryon #: B = 1 (Each Quark has a Baryon
# of 1/3)
Isospin: I3 = Q -½(B+s+c+b+t)
Hypercharge: Y = 2(Q - I3) - (c  b  t)
Ordinary matter – SU(2)F Symmetry
Group - only up and down quarks
(NF =2)
2

2

2

3
2
1

4
2
3

2
3
1
1 2
3


2
3
1
1 3
2


2
1
1
2
0
3

234
231
213 210
321
311
311 321
4S

2MS  2MA  0A
For NF = 2 (u and d Quarks), We can have 4 Baryons
with JP (3/2)+ and 2 Baryons with JP (1/2)+/
Particle
p
n

0


 
quarks
uud
udd
ddd
udd
uud
uuu
B
1
1
1
1
1
1
Q
0
1
-1
0
1
2
s
0
0
0
0
0
0
c b
0 0
0 0
0 0
0 0
0 0
0 0
t
0
0
0
0
0
0
Y
1
1
1
1
1
1
I3
-1/2
1/2
-3/2
-1/2
1/2
3/2
JP
1/2+/
+/
1/2
3/2+
3/2+
+
3/2
3/2+
If q1= q2 q3 (2 Distinct JP Values)
+ (3/2)+
{uu}
u
(½)+
u
(½)+
(1)+ 
+
(½)
d
{uud}(Symmetric)
p (1/2)+/
u{ud}(Mixed-Symmetric)
q1 or q2 = u, u
q3 = d
If q1= q2 q3 (2 Distinct JP Values)
0 (3/2)+
+  (½)+
(1)
{dd}
u
+
(½)
d
d(½)+
{ddu}(Symmetric)
n
(1/2)+/
u{dd}(Mixed-Symmetric)
q1 or q2 = d, d
q3 = d
Light Baryons – SU(3)F Symmetry
Group - up, down, and strange
quarks (NF = 3)
3

3

3

4
3
1

5
2
3

3
4
2
1 2
3

345
342
321
311
10S

8M
NF = 3, in the SU(3) Symmetry Group
(u, d and s Quarks)
We can have 10 Baryons with JP (3/2)+
and
8 Baryons with JP (1/2)+ and (1/2)+/
TOTAL # OF BARYONS THAT CAN BE
CONSTRUCTED WITH u, d AND s QUARKS = 18
Particle
S
S0
S

0
L0
S 
S 0
S 
 
 0

quarks
dds
uds
uus
dss
uss
uds
dds
uds
uus
dss
uss
sss
B
1
1
1
1
1
1
1
1
1
1
1
1
Q
-1
0
1
-1
0
0
-1
0
1
-1
0
-1
s
-1
-1
-1
-2
-2
-3
-1
-1
-1
-2
-2
-3
c
0
0
0
0
0
0
0
0
0
0
0
0
b
0
0
0
0
0
0
0
0
0
0
0
0
t
0
0
0
0
0
0
0
0
0
0
0
0
Y
0
0
0
-1
-1
0
0
0
0
-1
-1
-2
I3
-1
0
1
-3/2
1/2
0
-1
0
1
-3/2
1/2
0
JP
1/2+/
1/2+/
1/2+/
1/2+/
1/2+/
1/2+
3/2+
3/2+
3/2+
3/2+
3/2+
3/2+
Particle
p
n

0


Particle
S
S0
S

0
L0
S 
S 0
S 
 
 0

quarks
uud
udd
ddd
udd
uud
uuu
B
1
1
1
1
1
1
Q
0
1
-1
0
1
2
s
0
0
0
0
0
0
c
0
0
0
0
0
0
b
0
0
0
0
0
0
t
0
0
0
0
0
0
Y
1
1
1
1
1
1
I3
-1/2
1/2
-3/2
-1/2
1/2
3/2
JP
1/2+/
1/2+/
3/2+
3/2+
3/2+
3/2+
quarks
dds
uds
uus
dss
uss
uds
dds
uds
uus
dss
uss
sss
B
1
1
1
1
1
1
1
1
1
1
1
1
Q
-1
0
1
-1
0
0
-1
0
1
-1
0
-1
s
-1
-1
-1
-2
-2
-3
-1
-1
-1
-2
-2
-3
c
0
0
0
0
0
0
0
0
0
0
0
0
b
0
0
0
0
0
0
0
0
0
0
0
0
t
0
0
0
0
0
0
0
0
0
0
0
0
Y
0
0
0
-1
-1
0
0
0
0
-1
-1
-2
I3
-1
0
1
-3/2
1/2
0
-1
0
1
-3/2
1/2
0
JP
1/2+/
1/2+/
1/2+/
1/2+/
1/2+/
1/2+
3/2+
3/2+
3/2+
3/2+
3/2+
3/2+
If q1 q2 q3 (3 Distinct JP Values)
S*0 (3/2)+
{ud}
(1)+ 
+
(½)
s
+  (½)+
(½)
u
d
{sud}
(Symmetric)
S0 (1/2)+/
+  (½)+
(0)
[ud]
s
s{ud}
(Mixed-Symmetric)
L0 (1/2)+
q1 or q2 = u, d
q3 = s
[sud]
(Mixed-AntiSymmetric)
JP (1/2)+
8 Baryons
Y Axis – Y (Hypercharge)
X Axis – I3 or Iz
(z – Component of Isospin)
JP (3/2)+
10 Baryons
Y Axis – Y (Hypercharge)
X Axis – I3 or Iz
(z – Component of Isospin)
1 ma
JP (1/2)+
Y Axis – Y (Hypercharge)
X Axis – I3 or Iz
(z – Component of Isospin)
8 Baryons - 7 with JP (1/2)+/ and
1 with JP (1/2)+
NF = 3, in the SU(3) Symmetry Group
(u, d and s Quarks)
From SU(3) we can have a total of 18
Baryons – 10 with JP (3/2)+ and 7 with
JP (1/2)+/ and 1 with JP (1/2)+
JP (3/2)+
Y Axis – Y (Hypercharge)
X Axis – I3 or Iz
(z – Component of Isospin)
10 Baryons with JP (3/2)+
Charmed Baryons – SU(4)F Symmetry
Group - up, down, strange and charm
quarks (NF = 4)
4

4

4

5
4
1

6
2
3

4
5
3
1 2
3

456
453
321
311
20/S

20M
NF = 4, in the SU(4) Symmetry Group
(u, d and s Quarks)
We can have 20 Baryons with JP (3/2)+
and
20 Baryons with JP (1/2)+ and (1/2)+/
TOTAL # OF BARYONS THAT CAN BE CONSTRUCTED
WITH u, d, s AND c QUARKS = 40
Q
s
c
b
t
Y
I3
Jp
cdd
B
1
0
0
1
0
0
1
-1
1/2+/
Lc+
cud
1
1
0
1
0
0
1
0
1/2+
S c+
cud
1
1
0
1
0
0
1
0
1/2+/
S c++
cuu
1
2
0
1
0
0
1
1
1/2+/
 c0
cs d
1
0
-1
1
0
0
0
- 1/2
1/2+
 c/0
cs d
1
0
-1
1
0
0
0
- 1/2
1/2+/
 c+
cs u
1
1
-1
1
0
0
0
1/2
1/2+
 c/+
cs u
1
1
-1
1
0
0
0
1/2
1/2+/
c0
cs s
1
0
-2
1
0
0
-1
0
1/2+/
 cc+
ccd
1
1
0
2
0
0
1
- 1/2
1/2+/
 cc++
ccu
1
2
0
2
0
0
1
1/2
1/2+/
cc+
ccs
1
2
-1
2
0
0
2
0
1/2+/
S c*0
cdd
1
0
0
1
0
0
1
-1
3/2+
S c*+
cud
1
1
0
1
0
0
1
0
3/2+
S c*++
cuu
1
2
0
1
0
0
1
1
3/2+
 c*0
cs d
1
0
-1
1
0
0
0
- 1/2
3/2+
 c*+
cs u
1
1
-1
1
0
0
0
1/2
3/2+
c*0
cs s
1
0
-2
1
0
0
-1
0
3/2+
 cc*+
ccd
1
1
0
2
0
0
1
- 1/2
3/2+
 cc*++
ccu
1
2
0
2
0
0
1
1/2
3/2+
cc*+
ccs
1
1
-1
2
0
0
0
0
3/2+
ccc++
ccc
1
2
0
3
0
0
1
0
3/2+
Particle
quark s
S c0
C
I3
Y
C
I3
Y
BUT HOW MANY BARYONS WAS CARRIED OVER
FROM THE SU(3) TO THE SU(4) SYMMETRY GROUP?
RECALL: From SU(3) we can have a total of 18
Baryons – 10 with JP (3/2)+ and 8 with
JP (1/2)+ [7 with JP (1/2)+/ and 1 with JP (1/2)+].
IN SU(4): A Total of 40 Baryons - 20 with JP (3/2)+
, and 20 with JP (1/2)+/ and 20 JP (1/2)+.
NEW – Actually, 10 Charmed Baryons with JP (3/2)+
and 12 Charmed Baryons with JP (1/2)+.
BUT How many Charmed Baryons with JP (1/2)+/
and with JP (1/2)+ ??
C
I3
Y
3 ma
How many Actual Charmed Baryons
with JP (1/2)+/ and with JP (1/2)+ ?
9 with JP (1/2)+/
and
3 with JP (1/2)+
TOTAL = 12
C
I3
Y
The Quark Configuration of the  c
Charmed-Strange Baryon in the Ground State
u
s
d
c

s
c

c


/
0
c
(1/ 2) ,(1/ 2) ,(3/ 2)

If q1 q2 q3 (3 Distinct JP Values)
 (3/2)+
*+
c
{us}
u
(½)+

(1)+ 
+
(½)
c
{csu}
(Symmetric)
 (1/2)+/
+/
c
+
(½)
s
+  (½)+
(0)
[us]
c
c{su}
(Mixed-Symmetric)
 (1/2)+
+
c
q1 or q2 = u, s
q3 = c
[csu]
(Mixed-AntiSymmetric)
JP (1/2)+
[csu]

+
c
(MixedAntiSymmetric)
Mass:
2465.7  1.3
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
JP (1/2)+/

+/
c
c{su}
(Mixed-Symmetric)
Mass:
2575.0  2.0
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
JP (3/2)+
{csu}

*+
c
(Symmetric)
Mass:
2644.3  2.1
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
If q1 q2 q3 (3 Distinct JP Values)
 (3/2)+
*0
c
+  (½)+
(1)
{ds}
c
+
(½)
d

{csd}
(Symmetric)
 (1/2)+/
0/
c
+
(½)
s
+  (½)+
(0)
[ds]
c
c{sd}
(Mixed-Symmetric)
 (1/2)+
0
c
q1 or q2 = d, s
q3 = c
[csd]
(Mixed-AntiSymmetric)
JP (1/2)+
[csd]

0
c
(MixedAntiSymmetric)
Mass:
2468.8  1.2
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
JP (1/2)+/
c{sd}

0/
c
(Mixed-Symmetric)
Mass:
2580.6  2.1
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
JP (3/2)+
{csd}

*0
c
(Symmetric)
Mass:
2644.5  1.7
MeV/c2
CLEO Experiment at CESR (Cornell Electron Storage Ring)
Beauty Baryons – SU(5)F Symmetry
Group - up, down, strange, charm, and
beauty quarks (NF = 5)
5

5

5

6
5
1

7
2
3

5
6
4
1 2
3

567
564
321
311
35S

40M
BUT HOW MANY BARYONS WAS CARRIED OVER
FROM THE SU(3) TO SU(4) TO THE SU(5)
SYMMETRY GROUP?
RECALL: From SU(3) we can have a total of 18
Baryons – 10 with JP (3/2)+ and 8 with
JP (1/2)+ [Actually, 7 with JP (1/2)+/ and 1 with JP
(1/2)+].
In SU(4): Total of 40 Baryons - 20 with JP (3/2)+ ,
and 20 with JP (1/2)+/ and 20 JP (1/2)+.
But, 10 Charmed Baryons with JP (3/2)+ and 12
Actual Charmed Baryons with JP (1/2)+ [Actually, 9
with JP (1/2)+/ and 3 with JP (1/2)+].
In SU(5): Total of 75 Baryons - 35 with JP (3/2)+ ,
and 40 with JP (1/2)+.
Particle
Sb
0
Sb
0
Lb
+
Sb
b
/b
0
b
/0
b
b
0
bc
/0
bc
+
bc
/+
bc
0
bc
/0
bc
+
bcc
bbd
0
bbu
bbs
0
bbc
*Sb
*0
Sb
*+
Sb
*b
*0
b
*b
*0
bc
*+
bc
*0
bc
*+
bcc
*bbd
*0
bbu
*bbs
*0
bbc
bbb
quarks
Q
s
c
b
t
Y
I3
J
p
B
1
bdd
-1
0
0
-1 0
-1
0
1/2
+/
bud
0
0
0
-1 0
1
0
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
bud
buu
bsd
bsd
bsu
bsu
bss
bcd
bcd
bcu
bcu
bcs
bcs
bcc
bbd
bbu
bbs
bbc
bdd
bud
buu
bsd
bsu
bss
bcd
bcu
bcs
bcc
bbd
bbu
bbs
bbc
bbb
0
1
-1
-1
0
0
-1
0
0
1
1
0
0
1
-1
0
-1
1
-1
0
1
-1
0
-1
0
1
0
1
-1
0
-1
1
-1
0
0
-1
-1
-1
-1
-2
0
0
0
0
-1
-1
0
0
0
-1
0
0
0
0
-1
-1
-2
0
0
-1
0
0
0
-1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
0
0
0
1
0
0
0
0
0
0
1
1
1
2
0
0
0
1
0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-3 0
1
1
0
0
0
0
-1
1
1
1
1
0
0
1
1
1
0
3
1
1
1
0
0
-1
1
1
0
1
1
1
0
3
1
0
1
- 1/2
- 1/2
1/2
1/2
0
- 1/2
- 1/2
1/2
1/2
0
0
0
- 1/2
1/2
0
0
-1
0
1
- 1/2
1/2
0
- 1/2
1/2
0
0
- 1/2
1/2
0
0
0
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
NEW: How many actual Beauty
Baryons - 15 with JP (3/2)+ and 20
with JP (1/2)+ ?
TOTAL # OF BEAUTY BARYONS = 35
How many Beauty Baryons with JP
(1/2)+/ and with JP (1/2)+ ??
How many actual Beauty Baryons with
JP (1/2)+/ and with JP (1/2)+ ?
14 with JP (1/2)+/
and
6 with JP (1/2)+
TOTAL = 20
Beauty Baryons – SU(5)F Symmetry
Group - up, down, strange, charm,
and beauty quarks (NF = 5)
AXIS PROBLEM !!
Need a
th
4
Axis ??
No Possible SU(5) Representations
can be added to the Existing
Scheme.
We can’t physically add a 4th
Beauty axis to the SU(4)
representation diagram!
C
I3
Y
C
I3
Y
AS OF TODAY NO SU(5)F
QUARK REPRESENTATION
EXITS !!
My Solution !
CONSIDER A NEW TYPE OF AXIS INSTEAD
OF THE TRADITIONAL “CHARM” AXIS IN
THE Z-DIRECTION.
BUT HOW ??
FLAVOR AXIS !! (A NEW QUANTUM
NUMBER !)
F = 1 FOR EACH HEAVY QUARK
(CHARM AND BEAUTY)
F = 0 FOR EACH LIGHT QUARK
(UP, DOWN AND STRANGE)
HOW DOES THIS FLAVOR AXIS
ACTUALLY WORK – THE FLAVOR
QUANTUM #
F = 3 (ccc) or (bbb) or (bcc) or (bbc) etc…
F = 2 (ccu) or (ccs) or (ccd) or (bbu) or (bbs) or
(bbd) or (bcu) or (bcs) etc…
F = 1 (cuu) or (css) or (cdd) or (csu) or (bbu) or
(bss) or (bsu) or (bdd) etc…
F = 0 (uuu) or (sss) or (ddd) or (sud) or (uud) or
(ddu) or (suu) or (ssu) etc…
F
I3
Y
SU(3) Light Baryons (u, d and s)
Y
I3
8 Baryons
SU(3) Light Baryons (u, d and s)
Y
I3
10 Baryons
3 ma
C
I3
Y
SU(4) Charmed Baryons
C
I3
Y
SU(4) Charmed Baryons
Beauty Baryons – SU(5)F Symmetry
Group - up, down, strange, charm, and
beauty quarks (NF = 5)
5

5

5

6
5
1

7
2
3

5
6
4
1 2
3

567
564
321
311
35S

40M
Particle
Sb
0
Sb
0
Lb
+
Sb
b
/b
0
b
/0
b
b
0
bc
/0
bc
+
bc
/+
bc
0
bc
/0
bc
+
bcc
bbd
0
bbu
bbs
0
bbc
*Sb
*0
Sb
*+
Sb
*b
*0
b
*b
*0
bc
*+
bc
*0
bc
*+
bcc
*bbd
*0
bbu
*bbs
*0
bbc
bbb
quarks
Q
s
c
b
t
Y
I3
J
p
B
1
bdd
-1
0
0
-1 0
-1
0
1/2
+/
bud
0
0
0
-1 0
1
0
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
bud
buu
bsd
bsd
bsu
bsu
bss
bcd
bcd
bcu
bcu
bcs
bcs
bcc
bbd
bbu
bbs
bbc
bdd
bud
buu
bsd
bsu
bss
bcd
bcu
bcs
bcc
bbd
bbu
bbs
bbc
bbb
0
1
-1
-1
0
0
-1
0
0
1
1
0
0
1
-1
0
-1
1
-1
0
1
-1
0
-1
0
1
0
1
-1
0
-1
1
-1
0
0
-1
-1
-1
-1
-2
0
0
0
0
-1
-1
0
0
0
-1
0
0
0
0
-1
-1
-2
0
0
-1
0
0
0
-1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
0
0
0
1
0
0
0
0
0
0
1
1
1
2
0
0
0
1
0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-3 0
1
1
0
0
0
0
-1
1
1
1
1
0
0
1
1
1
0
3
1
1
1
0
0
-1
1
1
0
1
1
1
0
3
1
0
1
- 1/2
- 1/2
1/2
1/2
0
- 1/2
- 1/2
1/2
1/2
0
0
0
- 1/2
1/2
0
0
-1
0
1
- 1/2
1/2
0
- 1/2
1/2
0
0
- 1/2
1/2
0
0
0
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
SU(5) Beauty Baryons
SU(5) Beauty Baryons
Particle
Sb
0
Sb
0
Lb
+
Sb
b
/b
0
b
/0
b
b
0
bc
/0
bc
+
bc
/+
bc
0
bc
/0
bc
+
bcc
bbd
0
bbu
bbs
0
bbc
*Sb
*0
Sb
*+
Sb
*b
*0
b
*b
*0
bc
*+
bc
*0
bc
*+
bcc
*bbd
*0
bbu
*bbs
*0
bbc
bbb
quarks
Q
s
c
b
t
Y
I3
J
p
B
1
bdd
-1
0
0
-1 0
-1
0
1/2
+/
bud
0
0
0
-1 0
1
0
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
+
1
+/
1
+
1
+/
1
+
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
1/2
+/
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
3/2
+
1
bud
buu
bsd
bsd
bsu
bsu
bss
bcd
bcd
bcu
bcu
bcs
bcs
bcc
bbd
bbu
bbs
bbc
bdd
bud
buu
bsd
bsu
bss
bcd
bcu
bcs
bcc
bbd
bbu
bbs
bbc
bbb
0
1
-1
-1
0
0
-1
0
0
1
1
0
0
1
-1
0
-1
1
-1
0
1
-1
0
-1
0
1
0
1
-1
0
-1
1
-1
0
0
-1
-1
-1
-1
-2
0
0
0
0
-1
-1
0
0
0
-1
0
0
0
0
-1
-1
-2
0
0
-1
0
0
0
-1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
0
0
0
1
0
0
0
0
0
0
1
1
1
2
0
0
0
1
0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-1 0
-2 0
-2 0
-2 0
-2 0
-3 0
1
1
0
0
0
0
-1
1
1
1
1
0
0
1
1
1
0
3
1
1
1
0
0
-1
1
1
0
1
1
1
0
3
1
0
1
- 1/2
- 1/2
1/2
1/2
0
- 1/2
- 1/2
1/2
1/2
0
0
0
- 1/2
1/2
0
0
-1
0
1
- 1/2
1/2
0
- 1/2
1/2
0
0
- 1/2
1/2
0
0
0
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
SU(5) Beauty Baryons
SU(5) Beauty Baryons
Finally A Solution That
Actually Works!
Paper in progress for
publication in
PRL(Physical Review Letters)
STANDARD MODEL
QUARKS &
ANTI-QUARKS
LEPTONS &
ANTI-LEPTONS
s c
e  e
 
W ,Z0
b
  
  
H 0, H 
u
s c
b t
d
t
MESONS & ANTI-MESONS
( qq )
BARYONS & ANTI-BARYONS
( qqq )
(qqq )
PENTAQUARKS & ANTI-PENTAQUARKS
( qqqqq )
( qqqqq )
HEXAQUARKS & ANTI-HEXAQUARKS
( qqqqqq )
 , g, G
e  e
 
u d
( qq )
VECTOR
BOSONS
( qqqqqq )
HYBRID MESONS & ANTI-HYBRID MESONS
( qq g )
( q qg )
HYBRID BARYONS & ANTI-HYBRID BARYONS
( qqqg )
( qqqg )
BOUND STATE GLUONS
( gg ) ( ggg )