Application of Group Theory in Particle Physics using the Young Tableaux Method 2006 PASSHEMA CONFER

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Application of Group Theory in Particle Physics
using the Young Tableaux Method2006 PASSHE-MA
CONFERENCE (March 31 April 1)

• Akhtar Mahmood (Assistant Professor of Physics)
• Jack Dougherty (Undergraduate Research
  Assistant)
• Edinboro University of Pennsylvania

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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.
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Three Families of Quarks
Increasing mass
Also, each quark has a corresponding
antiquark.The antiquarks have opposite charge to
the quarks
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Hadrons- the composites of Quarks

• Baryons are a composites of three quarks
• Mesons are a composites of a quark-antiquark
  pair

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Lets make some more baryons !

u
u
u
d
d
u
d
d
s
s
s
s
Q 0M1116 MeV/c2
Q 1M1189 MeV/c2
Q 0M1192 MeV/c2
Q -1M1197 MeV/c2
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The Quark Configuration of the
Charmed-Strange Baryon
u
s
d
s
c
c
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Summary of Young Tableaux Method The SU(N)
NotationHere N 2 (for Spin up or down)N 1
to 5 (for quark flavors up, down, strange,
charm, beauty)
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Young Tableaux (Young Diagrams) In the SU(N)
Notation, N is denoted by a box.
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(N) - Total Number of Spins or Quark
Flavors
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The conjugate representation Nfor Anti-quarks.
Denoted by a column of N-1 boxes.If N 4, then
the conjugate representation of N is with 3
boxes.
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RULES FOR SU(N) REPRESENTATION
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No row is longer than any above it. Descending
rows are always shorter than the ones above them.

• Not Allowed

• Allowed

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Box values always increase from left to right in
a row
3
5
4
4
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NOT PERMITTED
3
1
4
2
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Going from left to right, no box- columns can be
longer than the previous one.

• Allowed

• Not Allowed

Allowed
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Box values always decrease from top to bottom in
a column
3
2
1
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NOT PERMITTED
3
4
5
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PERMITTED
4
5
6
7
3
4
5
2
3
1
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NOT PERMITTED
3
4
5
6
4
3
5
3
4
3
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Fully Symmetric Configuration

• YS

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Mixed Symmetric Configuration

• YM

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Fully Anti-symmetric Configuration

• YA

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WHAT IS THE NET VALUE OF A VALID CONFIGURATION
?A RATIO OF.. Product of Box Values
(n)V -----------------------------------
Product of Possible Hook Values (h)
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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
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A denominator is defined as the product of all
of the possible consecutive hooks

• h 3 ? 2 ? 1 6

• Hooks

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FOR N 3, IN THIS PARTICULAR CONFIGURATION
3 ? 4 ? 5 V ------------
10 3 ? 2 ? 1
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LETS CALCULATE THE VALUE OF THIS CONFIGURATION
FOR ANY N !
N
N1
N
N-1
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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 )
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FOR N 3
3
4
3
2
n 3 ? 4 ? 2 ? 3 72
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h is defined as the product of the possible
hook values of each consecutive
box-configuration
n P (hook value of each box-configuration)
(product)  
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h 3 ? 2 ? 2 ? 1 12
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FOR N 3, IN THIS PARTICULAR CONFIGURATION
3 ? 4 ? 2 ? 3 V
---------------- 6 3 ? 2 ? 2 ?
1
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Another Example (For N3)

• n 3 ? 4 ? 2 24
• h 3 ? 1 ? 1 3
• V 24 ? 3 8M

3
4
2
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Baryon(Qq1q2)?ASpace?S?Color?A??Spin?S,MS,MA??
Flavor?S,MS,MA
COMPLETE BARYON WAVE FUNCTION

l
q1
SIZE 1 fm
q2
l
q3
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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 l0 ? P (-1)0
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• () ? Ground state (l 0 and l 0)

• (-)? Excited state (l 1 and
• l 0 or (l 0 and l 1)

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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 Paulis
  Exclusion Principle, and hence the color charge
  combination must be anti-symmetric for all
  baryons.

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COLOR From the SU(3)c Color Symmetry Group
Only the color combination is valid - Color
Singlet (Colorless)
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The 3 quark color are Red, Green and Blue
3
?
?
3
?
3
5
?
3
4
4
?
3
4
?
3
3
1
2
2
2
1
2
3
1
2
1
3
2
3
2
1
3
?
?
?
?
3 ? 4 ? 5 3 ? 2 ? 1
3 ? 4 ? 2 3 ? 1 ? 1
3 ? 4 ? 2 3 ? 1 ? 1
3 ? 2 ? 1 3 ? 2 ? 1
8MS
10S
8MA
1A
?
?
?
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1A ? Color Singlet ? Colorless R G B
White (Colorless)1??6?RGB-GRBBRG-RBGGBR-BGR?A
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Total Angular Momentum (JP)
J ? L S where L l l and
P ? (-1)(l l) Each quark has a Spin (S) of
½ and a JP of ½. WHAT ARE POSSIBLE JP VALUES OF
A BARYON ?
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In Ground Statel 0 and l 0

l
q1
q2
l
q3
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If q1? q2? q3 (3 Distinct JP Values)
q1q2q3(3/2)
(Symmetric)
q1q2(1) ? q3(½)
q1q2q3(1/2)/
q1(½) ?q2(½)
(Mixed-Symmetric)
q1q2(0) ? q3(½)
q1q2q3(1/2)
q1 or q2 u, d, s
(Mixed-Antisymmetric)
q3 c, b
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The Quark Configuration of the
Charmed-Strange Baryon in the Ground State
u
s
d
s
c
c
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If q1 q2? q3 (2 Distinct JP Values)
q1q2q3(3/2)
(Symmetric)
q1q2(1) ? q3(½)
q1q2q3(1/2)/
q1(½) ?q2(½)
(Mixed-Symmetric)
q1 or q2 u, d, s
q3 c, b
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If q1 q2 q3 (1 Distinct JP Value)
q1q2q3(3/2)
(Symmetric)
q1q2(1) ? q3(½)
q1(½) ?q2(½)
q1 or q2 u, d, s
q3 c, b
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SPIN From the SU(2)S Spin Symmetry Group
Three distinct spin states for each of the Three
distinct Jp values. Jp(3/2) ? 4S Jp(1/2)/ ?
2MS Jp(1/2) ? 2MA
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SPIN CONFIGURATION USING YOUNGS TABLEAUX
2
?
?
2
?
2
4
?
2
3
3
?
2
3
?
2
2
1
1
1
1
1
2
3
1
2
1
3
2
3
2
0
3
?
?
?
?
2 ? 3 ? 4 3 ? 2 ? 1
2 ? 3 ? 1 3 ? 1 ? 1
2 ? 1 ? 3 3 ? 1 ? 1
2 ? 1 ? 0 3 ? 2 ? 1
2MS
4S
2MA
0A
?
?
?
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4 separate Spin orientations (S, Sz) for S 3/2
(S, Sz) or (J, Jz) for Sz 4S and J
(3/2)(3/2, 3/2)(3/2, 1/2) (3/2,
-1/2)(3/2, -3/2)
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2 separate Spin orientations (S, Sz) for S 1/2
(S, Sz) or (J, Jz) for Sz 2MS OR Sz 2MA
and J (1/2)/ OR J (1/2) (1/2,
1/2)(1/2, -1/2)(1/2, 1/2)(1/2, -1/2)
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Spin?S ? ?Flavor?S
Spin?MS ? ?Flavor?MS
Spin?MA ? ?Flavor?MA
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JP (3/2)Spin?S ?
Flavor?S ? 4?S ? (3/2)?S ????S ? q3
1/v2 q1q2 q2q1?S (3/2,3/2)1/v3(???
??? ???)?S ? q3 1/v2 q1q2 q2q1?S
(3/2,1/2)1/v3(??? ??? ???)?S ? q3 1/v2
q1q2 q2q1?S (3/2,-1/2)????S ? q3 1/v2
q1q2 q2q1?S (3/2,-3/2)
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JP (1/2)/Spin?MS ?
Flavor?MS ? 2?MS ? (1/2)/?MS 1/v6(-??? -
??? 2 ???)?MS ? q3 1/v2 q1q2 q2q1?MS
(1/2,1/2)1/v6(??? ??? - 2 ???)?MS ? q3
1/v2 q1q2 q2q1?MS (1/2,-1/2)
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JP (1/2)Spin?MA ?
Flavor?MA ? 2?MA ? (1/2)/?MA 1/v2 ?(?? -
??)?MA ? q3 1/v2 q1q2 - q2q1?MA
(1/2,1/2)1/v2 ?(?? - ??)?MA ? q3 1/v2 q1q2
- q2q1?MA (1/2,-1/2)
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Quark Mass and Charge

• FLAVOR MASS (GeV)
  ELECTRIC CHARGE
• UP   0.003  
   2/3
• DOWN   0.006  
  - 1/3
• STRANGE   0.1
    - 1/3
• CHARM   1.3  
   2/3
• BOTTOM   4.3  
  - 1/3
• TOP   175  
   2/3

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QUANTUM PROPERTIES OF BARYONS Baryon B 1
(Each Quark has a Baryon of 1/3)Isospin I3
Q -½(Bscbt) Hypercharge Y 2(Q - I3) -
(c ? b ? t)
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Ordinary matter SU(2)F Symmetry Group - only up
and down quarks (NF 2)
2
?
?
2
?
2
4
?
2
3
3
?
2
3
?
2
2
1
1
1
1
1
2
3
1
2
1
3
2
3
2
0
3
?
?
?
?
2 ? 3 ? 4 3 ? 2 ? 1
2 ? 3 ? 1 3 ? 1 ? 1
2 ? 1 ? 3 3 ? 1 ? 1
2 ? 1 ? 0 3 ? 2 ? 1
2MS
4S
2MA
0A
?
?
?
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For NF 2 (u and d Quarks), We can have 4
Baryons with JP (3/2) and 2 Baryons with JP
(1/2)/
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If q1 q2? q3 (2 Distinct JP Values)
? (3/2)
uud(Symmetric)
uu(1) ? d(½)
p (1/2)/
u(½) ?u(½)
uud(Mixed-Symmetric)
q1 or q2 u, u
q3 d
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If q1 q2? q3 (2 Distinct JP Values)
?0 (3/2)
ddu(Symmetric)
dd(1) ? u(½)
n (1/2)/
d(½) ?d(½)
udd(Mixed-Symmetric)
q1 or q2 d, d
q3 d
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Light Baryons SU(3)F Symmetry Group - up, down,
and strange quarks (NF 3)
3
?
?
3
?
3
5
?
3
4
4
3
2
1
2
3
1
2
3
?
?
3 ? 4 ? 5 3 ? 2 ? 1
3 ? 4 ? 2 3 ? 1 ? 1
8M
10S
?
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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
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If q1? q2? q3 (3 Distinct JP Values)
?0 (3/2)
sud (Symmetric)
ud(1) ? s(½)
u(½) ? d(½)
?0 (1/2)/
sud (Mixed-Symmetric)
ud(0) ? s(½)
?0 (1/2)
q1 or q2 u, d
sud (Mixed-AntiSymmetric)
q3 s
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JP (1/2) 8 Baryons
Y Axis Y (Hypercharge) X Axis I3 or Iz
(z Component of Isospin)
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JP (3/2) 10 Baryons
Y Axis Y (Hypercharge) X Axis I3 or Iz
(z Component of Isospin)
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1 ma
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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)
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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)
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JP (3/2)
Y Axis Y (Hypercharge) X Axis I3 or Iz
(z Component of Isospin)
10 Baryons with JP (3/2)
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Charmed Baryons SU(4)F Symmetry Group - up,
down, strange and charm quarks (NF 4)
4
?
?
4
?
4
6
?
4
5
5
4
3
1
2
3
1
2
3
?
?
4 ? 5 ? 6 3 ? 2 ? 1
4 ? 5 ? 3 3 ? 1 ? 1
20M
20/S
?
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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
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C
I3
Y
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C
I3
Y
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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) ??
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C
I3
Y
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3 ma
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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
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C
I3
Y
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The Quark Configuration of the
Charmed-Strange Baryon in the Ground State
u
s
d
s
c
c
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If q1? q2? q3 (3 Distinct JP Values)
(3/2)
csu (Symmetric)
us(1) ? c(½)
u(½) ? s(½)
(1/2)/
csu (Mixed-Symmetric)
us(0) ? c(½)
(1/2)
q1 or q2 u, s
csu (Mixed-AntiSymmetric)
q3 c
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JP (1/2) csu (Mixed- AntiSymmetric)
Mass 2465.7 ? 1.3 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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JP (1/2)/ csu (Mixed-Symmetric)
Mass 2575.0 ? 2.0 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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JP (3/2) csu (Symmetric)
Mass 2644.3 ? 2.1 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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If q1? q2? q3 (3 Distinct JP Values)
(3/2)
csd (Symmetric)
ds(1) ? c(½)
d(½) ? s(½)
(1/2)/
csd (Mixed-Symmetric)
ds(0) ? c(½)
(1/2)
q1 or q2 d, s
csd (Mixed-AntiSymmetric)
q3 c
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JP (1/2) csd (Mixed- AntiSymmetric)
Mass 2468.8 ? 1.2 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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JP (1/2)/ csd (Mixed-Symmetric)
Mass 2580.6 ? 2.1 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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JP (3/2) csd (Symmetric)
Mass 2644.5 ? 1.7 MeV/c2
CLEO Experiment at CESR (Cornell Electron
Storage Ring)
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Beauty Baryons SU(5)F Symmetry Group - up,
down, strange, charm, and beauty quarks (NF 5)
5
?
?
5
?
5
7
?
5
6
6
5
4
1
2
3
1
2
3
?
?
5 ? 6 ? 7 3 ? 2 ? 1
5 ? 6 ? 4 3 ? 1 ? 1
40M
35S
?
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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).
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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) ??
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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
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Beauty Baryons SU(5)F Symmetry Group - up,
down, strange, charm, and beauty quarks (NF
5) AXIS PROBLEM !! Need a 4th Axis ??
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No Possible SU(5) Representations can be
added to the Existing Scheme.We cant
physically add a 4th Beauty axis to the SU(4)
representation diagram!
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C
I3
Y
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C
I3
Y
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AS OF TODAY NO SU(5)F QUARK REPRESENTATION
EXITS !!
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My Solution !
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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)
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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
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F
I3
Y
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SU(3) Light Baryons (u, d and s)
Y
I3
8 Baryons
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SU(3) Light Baryons (u, d and s)
Y
I3
10 Baryons
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3 ma
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C
I3
Y
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SU(4) Charmed Baryons
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C
I3
Y
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SU(4) Charmed Baryons
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Beauty Baryons SU(5)F Symmetry Group - up,
down, strange, charm, and beauty quarks (NF 5)
5
?
?
5
?
5
7
?
5
6
6
5
4
1
2
3
1
2
3
?
?
5 ? 6 ? 7 3 ? 2 ? 1
5 ? 6 ? 4 3 ? 1 ? 1
40M
35S
?
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SU(5) Beauty Baryons
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SU(5) Beauty Baryons
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SU(5) Beauty Baryons
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SU(5) Beauty Baryons
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Finally A Solution That Actually
Works! Paper in progress for publication in
PRL(Physical Review Letters)
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