Application of Group Theory in Particle Physics using the Young Tableaux Method 2006 PASSHE-MA CONFERENCE (March 31 – April 1) Akhtar Mahmood ( Assistant

Application of Group Theory in Application of Group Theory in Particle Physics using the Young Particle Physics using the Young

Tableaux MethodTableaux Method

2006 PASSHE-MA CONFERENCE (March 31 – April 1)2006 PASSHE-MA CONFERENCE (March 31 – April 1)

Akhtar Mahmood (Akhtar Mahmood (Assistant Professor of Physics)Assistant Professor of Physics)Jack Dougherty (Jack Dougherty (Undergraduate Research Assistant)Undergraduate Research Assistant)

Edinboro University of PennsylvaniaEdinboro University of Pennsylvania

Maury Gell-MannMaury Gell-Mann ((at CalTechat CalTech))Proposed the existence of Proposed the existence of Quarks as the fundamental Quarks as the fundamental building blocks of matter in building blocks of matter in the late 1960s.the late 1960s. Awarded Nobel Prize in Awarded Nobel Prize in Physics in 1969 for the Physics in 1969 for the development of the Quark development of the Quark model, and the classification model, and the classification of elementary particles.of elementary particles.

FlavorFlavor Q/eQ/e

uu +2/3+2/3

dd -1/3-1/3

ss -1/3-1/3

Three Families of QuarksThree Families of Quarks

GenerationsGenerations

II IIII IIIIII

Charge Charge ==

-1/3-1/3

dd(down)(down)

ss(strange(strange

))

b b (bottom(bottom

))

Charge Charge ==

+2/3+2/3

uu(up)(up)

c c (charm)(charm)

tt(top)(top)

Also, each quark has a corresponding antiquark.The antiquarks have opposite charge to the quarks

Increasing mass

STANDARD MODEL

QUARKS & ANTI-QUARKS

LEPTONS & ANTI-LEPTONS

VECTOR BOSONS

tb

cs

du

tb

cs

du

e e

e e

HH

ZW

Gg

,

,

,,

0

0

MESONS & ANTI-MESONS ( qq ) ( qq )

BARYONS & ANTI-BARYONS (qqq) ( qqq )

PENTAQUARKS & ANTI-PENTAQUARKS ( qqqqq ) ( qqqqq )

HEXAQUARKS & ANTI-HEXAQUARKS (qqqqqq) ( qqqqqq )

HYBRID MESONS & ANTI-HYBRID MESONS ( gqq ) ( qgq )

HYBRID BARYONS & ANTI-HYBRID BARYONS (qqqg ) ( gqqq )

BOUND STATE GLUONS (gg ) (ggg )

HadronsHadrons- the composites of - the composites of QuarksQuarks

BaryonsBaryons are a composites of three quarks are a composites of three quarks Mesons Mesons are a composites of a quark-antiquark pair are a composites of a quark-antiquark pair

Let’s make some more Let’s make some more baryons !baryons !

su

d

Lambda ()Q = 0

M=1116 MeV/c2

su

u

Sigma ()Q = +1

M=1189 MeV/c2

su

d

Sigma ()Q = 0

M=1192 MeV/c2

sd

d

Sigma ()Q = -1

M=1197 MeV/c2

u d s

Q

Mass

+2/3 -1/3 -1/3

Quark up down strange

u u d d ss

~5 [MeV/c2] ~10 [MeV/c2] ~200 [MeV/c2]

The Quark Configuration of The Quark Configuration of

the Charmed-Strange Baryonthe Charmed-Strange Baryon c

u s

c

d s

c

c 0

c

/(1/ 2) ,(1/ 2) ,(3/ 2)

Summary of Young Tableaux Summary of Young Tableaux Method – The SU(N) NotationMethod – The SU(N) Notation

Here N = 2 (for Spin up or down)Here N = 2 (for Spin up or down)

N = 1 to 5 (for quark flavors – N = 1 to 5 (for quark flavors – up, down, strange, charm, up, down, strange, charm,

beauty)beauty)

Young Tableaux Young Tableaux (Young Diagrams) (Young Diagrams)

In the SU(N) Notation, In the SU(N) Notation, N is denoted by a box.N is denoted by a box.

(N) - Total Number of (N) - Total Number of Spins or Quark Flavors Spins or Quark Flavors

The conjugate representation NThe conjugate representation Nfor Anti-quarks.for Anti-quarks. Denoted by a Denoted by a

column of N-1 boxes.column of N-1 boxes.If N = 4, then the conjugate If N = 4, then the conjugate representation of N is with 3 representation of N is with 3

boxes.boxes.

RULES FOR SU(N) RULES FOR SU(N)

REPRESENTATION REPRESENTATION

No row is longer than No row is longer than any above it. Descending rows any above it. Descending rows

are always shorter than the are always shorter than the ones above them.ones above them.

AllowedAllowed Not AllowedNot Allowed

Box values always increase Box values always increase from left to right in a rowfrom left to right in a row

3 544

NOT PERMITTEDNOT PERMITTED

3 142

Going from left to right, Going from left to right, no box- columns can be longer no box- columns can be longer

than the previous one.than the previous one.

AllowedAllowed Not AllowedNot Allowed AllowedAllowed

Box values always decrease Box values always decrease from top to bottom in a columnfrom top to bottom in a column

3

2

1


3

4

5

PERMITTEDPERMITTED

4 5

2

6

3

1

4

7

5

3


3 4

3

5

43

4

6

5

3

Fully Symmetric ConfigurationFully Symmetric Configuration

SS = =

Mixed Symmetric ConfigurationMixed Symmetric Configuration

MM = =

Fully Anti-symmetric ConfigurationFully Anti-symmetric Configuration

AA = =

WHAT IS THE NET VALUE OF A WHAT IS THE NET VALUE OF A VALID CONFIGURATION ?VALID CONFIGURATION ?

A RATIO OF…..A RATIO OF…..

Product of Box Values (n)Product of Box Values (n)

VV= -----------------------------------= ----------------------------------- Product of Possible “Hook” Values (h) Product of Possible “Hook” Values (h)

A numerator (n) A numerator (n) is defined as the product of the is defined as the product of the

actual value each consecutive box.actual value each consecutive box.

n = 3 n = 3 4 4 5 = 60 5 = 60

3 4 5

FOR N = 3FOR N = 3

A denominator is defined as A denominator is defined as the product of all of the the product of all of the

possible consecutive “hooks”possible consecutive “hooks”

HooksHooks

h = 3 h = 3 2 2 1 = 6 1 = 6

FOR N = 3, IN THIS FOR N = 3, IN THIS PARTICULAR CONFIGURATION PARTICULAR CONFIGURATION

3 3 4 4 5 5 V = ------------V = ------------ = 10= 10 3 3 2 2 1 1

LET’S CALCULATE THE LET’S CALCULATE THE VALUE OF THIS VALUE OF THIS

CONFIGURATION FOR ANY CONFIGURATION FOR ANY “N” !“N” !

N N+1

NN-1

““n” is defined as the product n” is defined as the product of the actual value of each of the actual value of each

consecutive box.consecutive box.

n = ( each box values ) (product)

n = ( N )( N + 1 )( N – 1 )( N )

FOR N = 3FOR N = 3

3 4

32

n = 3 n = 3 4 4 2 2 3 = 72 3 = 72

““h” is defined as the product h” is defined as the product of the possible “hook” values of of the possible “hook” values of

each consecutive box-each consecutive box-configurationconfiguration

n = (“hook” value of each box-configuration)

(product)

h = 3 h = 3 2 2 2 2 1 = 12 1 = 12

FOR N = 3, IN THIS FOR N = 3, IN THIS PARTICULAR CONFIGURATION PARTICULAR CONFIGURATION

3 3 4 4 2 2 3 3 V = ----------------V = ---------------- = 6= 6 3 3 2 2 2 2 1 1

Another Example (For N=3)Another Example (For N=3)

n = 3 n = 3 4 4 2 = 24 2 = 24

h = 3 h = 3 1 1 1 = 3 1 = 3

V = 24 V = 24 3 = 8 3 = 8MM

3 4

2

|Baryon(Qq|Baryon(Qq11qq22))AA=|Space=|SpaceSS|Color|ColorAASpinSpinS,MS,MAS,MS,MAFlavorFlavorS,MS,MAS,MS,MA

COMPLETE BARYON WAVE FUNCTIONCOMPLETE BARYON WAVE FUNCTION

q2

l´

l

q1

q3

SIZE = 1 fmSIZE = 1 fm

Space is described by the Space is described by the Parity (P) of the Baryon Parity (P) of the Baryon

which is defined as the state which is defined as the state of the particleof the particle

P ≡ (-1P ≡ (-1)(l + l´))(l + l´)

In the ground state l = 0 and l´=0In the ground state l = 0 and l´=0

P = (-1) P = (-1)0 0 = += +

(+) (+) Ground state (l = 0 and l’ = 0) Ground state (l = 0 and l’ = 0)

(-)(-) Excited state (l = 1 and Excited state (l = 1 and l’ = 0 or (l = 0 and l’ = 1)l’ = 0 or (l = 0 and l’ = 1)

ColorColorThe strong force that binds the quarks with The strong force that binds the quarks with

gluons carry “color charge” i.e. gluons carry “color charge” i.e. redred, , greengreen and and blueblue..

Since a baryon is a Fermion, it must obey Since a baryon is a Fermion, it must obey Pauli’s Exclusion Principle, and hence the color Pauli’s Exclusion Principle, and hence the color charge combination must be anti-symmetric charge combination must be anti-symmetric

for all baryons.for all baryons.

COLOR: From the SU(3)COLOR: From the SU(3)c c Color Color Symmetry Group Symmetry Group

3 3 3 10 8 8 1C C C S MS MA A

Only the color combination is validOnly the color combination is valid - Color Singlet (Colorless)- Color Singlet (Colorless)

1A

1A

The 3 quark color are Red, The 3 quark color are Red, Green and BlueGreen and Blue

3 3 3 3 5 3

2

4 4 3 4

2

3

2

1

3 4 5

3 2 1

3 4 2

3 1 1

3 4 2

3 1 1

3 2 1

3 2 1

10S 8MS 8MA 1A

1 2 3 1 23

1 32

1

2

3

11A A Color Singlet Color Singlet Colorless Colorless

R + G + B = White (Colorless) R + G + B = White (Colorless)

1166RGB-GRB+BRG-RBG+GBR-BGRRGB-GRB+BRG-RBG+GBR-BGRAA

Total Angular Momentum Total Angular Momentum ((JJPP))

J J L + S where L = l +l’ and L + S where L = l +l’ and P P (-1) (-1)(l +l´)(l +l´)

Each quark has a Spin (S) of ½ and Each quark has a Spin (S) of ½ and a Ja JPP of ½ of ½++..

WHAT ARE POSSIBLE WHAT ARE POSSIBLE JJPP VALUES OF VALUES OF A BARYON ?A BARYON ?

q2

l´

l

q1

q31 1

(0 1 )2 2

/1 1 1 1 1 3

(0 1 ) 0 12 2 2 2 2 2MA MS S

In Ground In Ground StateState

l = 0 and l = 0 and l’ =0l’ =0

q1(½)+ q2

(½)+

{q1q2}(1)+ q3(½)+

{q1q2q3}(3/2)+

If q1 q2 q3 (3 Distinct JP Values)

{q1q2}q3(1/2)+/

[q1q2]q3(1/2)+

q1 or q2 = u, d, s

q3 = c, b

(Symmetric)

(Mixed-Symmetric)

(Mixed-Antisymmetric)

[q1q2](0)+ q3(½)+

The Quark Configuration of the The Quark Configuration of the

Charmed-Strange Baryon in the Ground StateCharmed-Strange Baryon in the Ground State c

u s

c

d s

c

c 0

c

/(1/ 2) ,(1/ 2) ,(3/ 2)

q1(½)+ q2

(½)+

{q1q2}(1)+ q3(½)+

{q1q2q3}(3/2)+

If q1= q2 q3 (2 Distinct JP Values)

{q1q2}q3(1/2)+/

q1 or q2 = u, d, s

q3 = c, b

(Symmetric)

(Mixed-Symmetric)

q1(½)+ q2

(½)+

{q1q2}(1)+ q3(½)+

{q1q2q3}(3/2)+

If q1= q2= q3 (1 Distinct JP Value)

q1 or q2 = u, d, s

q3 = c, b

(Symmetric)

SPIN: From the SU(2)SPIN: From the SU(2)S S Spin Spin Symmetry Group Symmetry Group

2 2 2 4 2 2S MS MA

Three distinct spin states for each of the Three Three distinct spin states for each of the Three distinct Jdistinct Jpp values. values.

JJpp(3/2)(3/2)+ + 4 4S S

JJpp(1/2)(1/2)+/+/ 2 2MSMS

JJpp(1/2)(1/2)++ 2 2MAMA

SPIN CONFIGURATION USING SPIN CONFIGURATION USING YOUNG’S TABLEAUX YOUNG’S TABLEAUX

2 2 2 2 4 2

1

3 3 2 3

1

2

1

0

2 3 4

3 2 1

2 3 1

3 1 1

2 1 3

3 1 1

2 1 0

3 2 1

4S 2MS 2MA 0A

1 2 3 1 23

1 32

1

2

3

4 separate Spin orientations (S, S4 separate Spin orientations (S, Szz) for S = 3/2) for S = 3/2

Sz

+3/2

+1/2

-1/2

-3/2

(S, S(S, Szz) or (J, J) or (J, Jzz) for) for

SSzz = = 44S S and and J = J = (3/2)(3/2)++

(3/2, +3/2)(3/2, +3/2)(3/2, +1/2)(3/2, +1/2) (3/2, -1/2) (3/2, -1/2)(3/2, -3/2)(3/2, -3/2)

2 separate Spin orientations (S, S2 separate Spin orientations (S, Szz) for S = 1/2) for S = 1/2

Sz

+1/2

-1/2

(S, S(S, Szz) or (J, J) or (J, Jzz) for) for

SSzz = = 22MS MS OR OR SSzz = = 22MAMA

and and

J = J = (1/2)(1/2)+/ +/ OR OR J =J =(1/2)(1/2)+ +

(1/2, +1/2)(1/2, +1/2)(1/2, -1/2)(1/2, -1/2)

(1/2, +1/2)(1/2, +1/2)(1/2, -1/2)(1/2, -1/2)

|Spin|SpinS S FlavorFlavorSS

||SpinSpinMS MS FlavorFlavorMSMS

||SpinSpinMA MA FlavorFlavorMAMA

JJPP = (3/2) = (3/2)++

|Spin|SpinS S |Flavor |FlavorS S |4|4S S |(3/2) |(3/2)++S S

||SS | |{{qq33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}}}S S (3/2,+3/2)(3/2,+3/2)

||1/√3(1/√3( + + + + ))SS | |{{qq33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}}}S S (3/2,+1/2)(3/2,+1/2)

||1/√3(1/√3( + + + + ))SS | |{{qq33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}}}S S (3/2,-1/2)(3/2,-1/2)

||SS | |{{qq33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}}}S S (3/2,-3/2)(3/2,-3/2)

JJPP = (1/2) = (1/2)+/+/

|Spin|SpinMS MS |Flavor |FlavorMS MS |2|2MS MS |(1/2) |(1/2)+/+/MSMS

||1/√6(-1/√6(- - - + 2 + 2 ))MSMS |q |q33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}MS MS (1/2,+1/2)(1/2,+1/2)

||1/√6(+1/√6(+ + + - 2 - 2 ))MSMS |q |q33 1/√2 {1/√2 {qq11qq2 2 ++ qq22qq11}}MS MS (1/2,-1/2)(1/2,-1/2)

JJPP = (1/2) = (1/2)++

|Spin|SpinMA MA |Flavor |FlavorMA MA |2|2MA MA |(1/2) |(1/2)+/+/MAMA

||1/√2 1/√2 (( - - ))MAMA |q |q33 1/√2 {1/√2 {qq11qq2 2 -- qq22qq11}}MA MA (1/2,+1/2)(1/2,+1/2)

||1/√2 1/√2 (( - - ))MA MA |q |q33 1/√2 {1/√2 {qq11qq2 2 -- qq22qq11}}MA MA (1/2,-1/2)(1/2,-1/2)

Quark Mass and ChargeQuark Mass and Charge

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

QUANTUM PROPERTIES OF BARYONSQUANTUM PROPERTIES OF BARYONS

Baryon #:Baryon #: B = 1 (Each Quark has a Baryon # B = 1 (Each Quark has a Baryon # of 1/3)of 1/3)

Isospin:Isospin: II33 = Q -½(B+s+c+b+t) = Q -½(B+s+c+b+t)

Hypercharge:Hypercharge: Y = 2(Q - IY = 2(Q - I33) - (c ) - (c b b t) t)

Ordinary matter – SU(2)Ordinary matter – SU(2)FF Symmetry Symmetry Group - only up and down quarks Group - only up and down quarks

(N(NFF =2) =2)

2 2 2 2 4 2

1

3 3 2 3

1

2

1

0

2 3 4

3 2 1

2 3 1

3 1 1

2 1 3

3 1 1

2 1 0

3 2 1

4S 2MS 2MA 0A

1 2 3 1 23

1 32

1

2

3

For NF = 2 (u and d Quarks), We can have 4 Baryons with JP (3/2)+ and 2 Baryons with JP (1/2)+/

Particle quarks B Q s c b t Y I3 JP

p uud 1 0 0 0 0 0 1 -1/2 1/2+/

n udd 1 1 0 0 0 0 1 1/2 1/2+/

ddd 1 -1 0 0 0 0 1 -3/2 3/2+

udd 1 0 0 0 0 0 1 -1/2 3/2+

uud 1 1 0 0 0 0 1 1/2 3/2+

uuu 1 2 0 0 0 0 1 3/2 3/2+

u(½)+ u(½)+

{uu}(1)+ d(½)+

+ (3/2)+


p (1/2)+/

q1 or q2 = u, u

q3 = d

{uud}(Symmetric)

u{ud}(Mixed-Symmetric)

d(½)+ d(½)+

{dd}(1)+ u(½)+

0 (3/2)+


n (1/2)+/

q1 or q2 = d, d

q3 = d

{ddu}(Symmetric)

u{dd}(Mixed-Symmetric)

Light Baryons – SU(3)Light Baryons – SU(3)FF Symmetry Symmetry Group - up, down, and strange Group - up, down, and strange

quarks (Nquarks (NF F = 3)= 3)

3 3 3 3 5 3

2

4 4

3 4 5

3 2 1

3 4 2

3 1 1

10S 8M

1 2 3 1 23

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


dds 1 -1 -1 0 0 0 0 -1 1/2+/

uds 1 0 -1 0 0 0 0 0 1/2+/

uus 1 1 -1 0 0 0 0 1 1/2+/

dss 1 -1 -2 0 0 0 -1 -3/2 1/2+/

uss 1 0 -2 0 0 0 -1 1/2 1/2+/

uds 1 0 -3 0 0 0 0 0 1/2+

dds 1 -1 -1 0 0 0 0 -1 3/2+

uds 1 0 -1 0 0 0 0 0 3/2+

uus 1 1 -1 0 0 0 0 1 3/2+

dss 1 -1 -2 0 0 0 -1 -3/2 3/2+

uss 1 0 -2 0 0 0 -1 1/2 3/2+

sss 1 -1 -3 0 0 0 -2 0 3/2+


p uud 1 0 0 0 0 0 1 -1/2 1/2+/

n udd 1 1 0 0 0 0 1 1/2 1/2+/

ddd 1 -1 0 0 0 0 1 -3/2 3/2+

udd 1 0 0 0 0 0 1 -1/2 3/2+

uud 1 1 0 0 0 0 1 1/2 3/2+

uuu 1 2 0 0 0 0 1 3/2 3/2+


dds 1 -1 -1 0 0 0 0 -1 1/2+/

uds 1 0 -1 0 0 0 0 0 1/2+/

uus 1 1 -1 0 0 0 0 1 1/2+/

dss 1 -1 -2 0 0 0 -1 -3/2 1/2+/

uss 1 0 -2 0 0 0 -1 1/2 1/2+/

uds 1 0 -3 0 0 0 0 0 1/2+

dds 1 -1 -1 0 0 0 0 -1 3/2+

uds 1 0 -1 0 0 0 0 0 3/2+

uus 1 1 -1 0 0 0 0 1 3/2+

dss 1 -1 -2 0 0 0 -1 -3/2 3/2+

uss 1 0 -2 0 0 0 -1 1/2 3/2+

sss 1 -1 -3 0 0 0 -2 0 3/2+

u(½)+ d(½)+

{ud}(1)+ s(½)+


0 (1/2)+

q1 or q2 = u, d

q3 = s

[ud](0)+ s(½)+

[sud]

(Mixed-AntiSymmetric)

s{ud}

(Mixed-Symmetric)

0 (1/2)+/

{sud}

(Symmetric)

*0 (3/2)+

JP (1/2)+

8 Baryons

Y Axis – Y (Hypercharge)

X Axis – I3 or Iz

(z – Component of Isospin)

JP (3/2)+

10 Baryons


X Axis – I3 or Iz


1 m1 maa

JP (1/2)+

8 Baryons - 7 with JP (1/2)+/ and 1 with JP (1/2)+


X Axis – I3 or Iz



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)+

10 Baryons with JP (3/2)+


X Axis – I3 or Iz


Charmed Baryons – SU(4)Charmed Baryons – SU(4)FF Symmetry Symmetry Group - up, down, strange and charm Group - up, down, strange and charm

quarks (Nquarks (NF F = 4)= 4)

4 4 4 4 6 4

3

5 5

4 5 6

3 2 1

4 5 3

3 1 1

20/S 20M

1 2 3 1 23


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

Particle quarks B Q s c b t Y I3 Jp

c0

cdd 1 0 0 1 0 0 1 -1 1/2+/

c+

cud 1 1 0 1 0 0 1 0 1/2+

c+

cud 1 1 0 1 0 0 1 0 1/2+/

c++

cuu 1 2 0 1 0 0 1 1 1/2+/

c0

csd 1 0 -1 1 0 0 0 - 1/2 1/2+

c/0

csd 1 0 -1 1 0 0 0 - 1/2 1/2+/

c+

csu 1 1 -1 1 0 0 0 1/2 1/2+

c/+

csu 1 1 -1 1 0 0 0 1/2 1/2+/

c0

css 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+/

c*0

cdd 1 0 0 1 0 0 1 -1 3/2+

c*+

cud 1 1 0 1 0 0 1 0 3/2+

c*++

cuu 1 2 0 1 0 0 1 1 3/2+

c*0

csd 1 0 -1 1 0 0 0 - 1/2 3/2+

c*+

csu 1 1 -1 1 0 0 0 1/2 3/2+

c*0

css 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+

CC

II33

YY

CC

II33

YY

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)+ ??

CC

II33

YY

3 m3 maa

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

CC

II33

YY

The Quark Configuration of the The Quark Configuration of the

Charmed-Strange Baryon in the Ground StateCharmed-Strange Baryon in the Ground State c

u s

c

d s

c

c 0

c

/(1/ 2) ,(1/ 2) ,(3/ 2)

u(½)+ s(½)+

{us}(1)+ c(½)+


(1/2)+

q1 or q2 = u, s

q3 = c

[us](0)+ c(½)+

[csu]


c{su}

(Mixed-Symmetric)

(1/2)+/

{csu}

(Symmetric)

(3/2)+*+c

+/c

+c

JP (1/2)+

[csu]

(Mixed- AntiSymmetric)

Mass:

2465.7 1.3

MeV/c2

CLEO Experiment at CESR (Cornell Electron Storage Ring)

+c

JP (1/2)+/

c{su}

(Mixed-Symmetric)

Mass:

2575.0 2.0

MeV/c2


+/c

JP (3/2)+

{csu}

(Symmetric)

Mass:

2644.3 2.1

MeV/c2


*+c

d(½)+ s(½)+

{ds}(1)+ c(½)+


(1/2)+

q1 or q2 = d, s

q3 = c

[ds](0)+ c(½)+

[csd]


c{sd}

(Mixed-Symmetric)

(1/2)+/

{csd}

(Symmetric)

(3/2)+*0c

0/c

0c

JP (1/2)+

[csd]

(Mixed- AntiSymmetric)

Mass:

2468.8 1.2

MeV/c2

0c


0/c

JP (1/2)+/

c{sd}

(Mixed-Symmetric)

Mass:

2580.6 2.1

MeV/c2


JP (3/2)+

{csd}

(Symmetric)

Mass:

2644.5 1.7

MeV/c2


*0c

Beauty Baryons – SU(5)Beauty Baryons – SU(5)FF Symmetry Group - up, down, Symmetry Group - up, down,

strange, charm, and beauty quarks strange, charm, and beauty quarks (N(NFF = 5) = 5)

5 5 5 5 7 5

4

6 6

5 6 7

3 2 1

5 6 4

3 1 1

35S 40M

1 2 3 1 23

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 quarks Q s c b t Y I3 JpB

b- bdd -1 0 0 -1 0 -1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+ 1

b+ buu 1 0 0 -1 0 1 1 1/2+/ 1

b- bsd -1 -1 0 -1 0 0 - 1/2 1/2+ 1

b/- bsd -1 -1 0 -1 0 0 - 1/2 1/2+/ 1

b0 bsu 0 -1 0 -1 0 0 1/2 1/2+ 1

b/0 bsu 0 -1 0 -1 0 0 1/2 1/2+/ 1

b- bss -1 -2 0 -1 0 -1 0 1/2+/ 1

bc0 bcd 0 0 1 -1 0 1 - 1/2 1/2+ 1

bc/0 bcd 0 0 1 -1 0 1 - 1/2 1/2+/ 1

bc+ bcu 1 0 1 -1 0 1 1/2 1/2+ 1

bc/+ bcu 1 0 1 -1 0 1 1/2 1/2+/ 1

bc0 bcs 0 -1 1 -1 0 0 0 1/2+ 1

bc/0 bcs 0 -1 1 -1 0 0 0 1/2+/ 1

bcc+ bcc 1 0 2 -1 0 1 0 1/2+/ 1

bbd- bbd -1 0 0 -2 0 1 - 1/2 1/2+/ 1

bbu0 bbu 0 0 0 -2 0 1 1/2 1/2+/ 1

bbs- bbs -1 -1 0 -2 0 0 0 1/2+/ 1

bbc0 bbc 1 0 1 -2 0 3 0 1/2+/ 1

b*- bdd -1 0 0 -1 0 1 -1 3/2+ 1

b*0 bud 0 0 0 -1 0 1 0 3/2+ 1

b*+ buu 1 0 0 -1 0 1 1 3/2+ 1

b*- bsd -1 -1 0 -1 0 0 - 1/2 3/2+ 1

b*0 bsu 0 -1 0 -1 0 0 1/2 3/2+ 1

b*- bss -1 -2 0 -1 0 -1 0 3/2+ 1

bc*0 bcd 0 0 1 -1 0 1 - 1/2 3/2+ 1

bc*+ bcu 1 0 1 -1 0 1 1/2 3/2+ 1

bc*0 bcs 0 -1 1 -1 0 0 0 3/2+ 1

bcc*+ bcc 1 0 2 -1 0 1 0 3/2+ 1

bbd*- bbd -1 0 0 -2 0 1 - 1/2 3/2+ 1

bbu*0 bbu 0 0 0 -2 0 1 1/2 3/2+ 1

bbs*- bbs -1 -1 0 -2 0 0 0 3/2+ 1

bbc*0 bbc 1 0 1 -2 0 3 0 3/2+ 1

bbb- bbb -1 0 0 -3 0 1 0 3/2+ 1

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)Beauty Baryons – SU(5)FF Symmetry Symmetry Group - up, down, strange, charm, and Group - up, down, strange, charm, and

beauty quarks (Nbeauty quarks (NFF = 5) = 5)

AXIS PROBLEM !!AXIS PROBLEM !!

Need a 4Need a 4thth Axis ?? Axis ??

No Possible SU(5) No Possible SU(5) Representations Representations

can be added to the Existing can be added to the Existing Scheme.Scheme.

We can’t physically add a 4th We can’t physically add a 4th Beauty axis to the SU(4) Beauty axis to the SU(4) representation diagram!representation diagram!

CC

II33

YY

CC

II33

YY

AS OF TODAY NO SU(5)AS OF TODAY NO SU(5)FF QUARK REPRESENTATION QUARK REPRESENTATION

EXITS !!EXITS !!

My Solution !My Solution !

CONSIDER A NEW CONSIDER A NEW TYPE TYPE OF AXIS INSTEAD OF AXIS INSTEAD OF THE TRADITIONAL OF THE TRADITIONAL “CHARM”“CHARM” AXIS IN AXIS IN

THE Z-DIRECTION.THE Z-DIRECTION.

BUT HOW ??BUT HOW ??

FLAVOR AXIS !! FLAVOR AXIS !! (A NEW QUANTUM NUMBER (A NEW QUANTUM NUMBER !)!)

F = 1 FOR EACH HEAVY QUARKF = 1 FOR EACH HEAVY QUARK(CHARM AND BEAUTY)(CHARM AND BEAUTY)

F = 0 FOR EACH LIGHT QUARKF = 0 FOR EACH LIGHT QUARK(UP, DOWN AND STRANGE)(UP, DOWN AND STRANGE)

HOW DOES THIS FLAVOR AXIS HOW DOES THIS FLAVOR AXIS ACTUALLY WORK – ACTUALLY WORK – THE FLAVOR THE FLAVOR

QUANTUM #QUANTUM #

F = 3 (ccc) or (bbb) or (bcc) or (bbc) etc…F = 3 (ccc) or (bbb) or (bcc) or (bbc) etc…

F = 2 (ccu) or (ccs) or (ccd) or (bbu) or (bbs) or F = 2 (ccu) or (ccs) or (ccd) or (bbu) or (bbs) or (bbd) or (bcu) or (bcs) etc…(bbd) or (bcu) or (bcs) etc…

F = 1 (cuu) or (css) or (cdd) or (csu) or (bbu) or F = 1 (cuu) or (css) or (cdd) or (csu) or (bbu) or (bss) or (bsu) or (bdd) etc…(bss) or (bsu) or (bdd) etc…

F = 0 (uuu) or (sss) or (ddd) or (sud) or (uud) or F = 0 (uuu) or (sss) or (ddd) or (sud) or (uud) or (ddu) or (suu) or (ssu) etc…(ddu) or (suu) or (ssu) etc…

FF

YY

II33

SU(3) Light Baryons (u, d and SU(3) Light Baryons (u, d and s)s)

II33

YY

8 Baryons

SU(3) Light Baryons (u, d and SU(3) Light Baryons (u, d and s)s)

II33

YY

10 Baryons

3 m3 maa

CC

II33

YY

SU(4) Charmed SU(4) Charmed BaryonsBaryons

CC

II33

YY

SU(4) Charmed SU(4) Charmed BaryonsBaryons

Beauty Baryons – SU(5)Beauty Baryons – SU(5)FF Symmetry Group - up, down, Symmetry Group - up, down,

strange, charm, and beauty quarks strange, charm, and beauty quarks (N(NFF = 5) = 5)

5 5 5 5 7 5

4

6 6

5 6 7

3 2 1

5 6 4

3 1 1

35S 40M

1 2 3 1 23


b- bdd -1 0 0 -1 0 -1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+ 1

b+ buu 1 0 0 -1 0 1 1 1/2+/ 1

b- bsd -1 -1 0 -1 0 0 - 1/2 1/2+ 1

b/- bsd -1 -1 0 -1 0 0 - 1/2 1/2+/ 1

b0 bsu 0 -1 0 -1 0 0 1/2 1/2+ 1

b/0 bsu 0 -1 0 -1 0 0 1/2 1/2+/ 1

b- bss -1 -2 0 -1 0 -1 0 1/2+/ 1

bc0 bcd 0 0 1 -1 0 1 - 1/2 1/2+ 1

bc/0 bcd 0 0 1 -1 0 1 - 1/2 1/2+/ 1

bc+ bcu 1 0 1 -1 0 1 1/2 1/2+ 1

bc/+ bcu 1 0 1 -1 0 1 1/2 1/2+/ 1

bc0 bcs 0 -1 1 -1 0 0 0 1/2+ 1

bc/0 bcs 0 -1 1 -1 0 0 0 1/2+/ 1

bcc+ bcc 1 0 2 -1 0 1 0 1/2+/ 1

bbd- bbd -1 0 0 -2 0 1 - 1/2 1/2+/ 1

bbu0 bbu 0 0 0 -2 0 1 1/2 1/2+/ 1

bbs- bbs -1 -1 0 -2 0 0 0 1/2+/ 1

bbc0 bbc 1 0 1 -2 0 3 0 1/2+/ 1

b*- bdd -1 0 0 -1 0 1 -1 3/2+ 1

b*0 bud 0 0 0 -1 0 1 0 3/2+ 1

b*+ buu 1 0 0 -1 0 1 1 3/2+ 1

b*- bsd -1 -1 0 -1 0 0 - 1/2 3/2+ 1

b*0 bsu 0 -1 0 -1 0 0 1/2 3/2+ 1

b*- bss -1 -2 0 -1 0 -1 0 3/2+ 1

bc*0 bcd 0 0 1 -1 0 1 - 1/2 3/2+ 1

bc*+ bcu 1 0 1 -1 0 1 1/2 3/2+ 1

bc*0 bcs 0 -1 1 -1 0 0 0 3/2+ 1

bcc*+ bcc 1 0 2 -1 0 1 0 3/2+ 1

bbd*- bbd -1 0 0 -2 0 1 - 1/2 3/2+ 1

bbu*0 bbu 0 0 0 -2 0 1 1/2 3/2+ 1

bbs*- bbs -1 -1 0 -2 0 0 0 3/2+ 1

bbc*0 bbc 1 0 1 -2 0 3 0 3/2+ 1

bbb- bbb -1 0 0 -3 0 1 0 3/2+ 1

SU(5) Beauty BaryonsSU(5) Beauty Baryons



b- bdd -1 0 0 -1 0 -1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+/ 1

b0 bud 0 0 0 -1 0 1 0 1/2+ 1

b+ buu 1 0 0 -1 0 1 1 1/2+/ 1

b- bsd -1 -1 0 -1 0 0 - 1/2 1/2+ 1

b/- bsd -1 -1 0 -1 0 0 - 1/2 1/2+/ 1

b0 bsu 0 -1 0 -1 0 0 1/2 1/2+ 1

b/0 bsu 0 -1 0 -1 0 0 1/2 1/2+/ 1

b- bss -1 -2 0 -1 0 -1 0 1/2+/ 1

bc0 bcd 0 0 1 -1 0 1 - 1/2 1/2+ 1

bc/0 bcd 0 0 1 -1 0 1 - 1/2 1/2+/ 1

bc+ bcu 1 0 1 -1 0 1 1/2 1/2+ 1

bc/+ bcu 1 0 1 -1 0 1 1/2 1/2+/ 1

bc0 bcs 0 -1 1 -1 0 0 0 1/2+ 1

bc/0 bcs 0 -1 1 -1 0 0 0 1/2+/ 1

bcc+ bcc 1 0 2 -1 0 1 0 1/2+/ 1

bbd- bbd -1 0 0 -2 0 1 - 1/2 1/2+/ 1

bbu0 bbu 0 0 0 -2 0 1 1/2 1/2+/ 1

bbs- bbs -1 -1 0 -2 0 0 0 1/2+/ 1

bbc0 bbc 1 0 1 -2 0 3 0 1/2+/ 1

b*- bdd -1 0 0 -1 0 1 -1 3/2+ 1

b*0 bud 0 0 0 -1 0 1 0 3/2+ 1

b*+ buu 1 0 0 -1 0 1 1 3/2+ 1

b*- bsd -1 -1 0 -1 0 0 - 1/2 3/2+ 1

b*0 bsu 0 -1 0 -1 0 0 1/2 3/2+ 1

b*- bss -1 -2 0 -1 0 -1 0 3/2+ 1

bc*0 bcd 0 0 1 -1 0 1 - 1/2 3/2+ 1

bc*+ bcu 1 0 1 -1 0 1 1/2 3/2+ 1

bc*0 bcs 0 -1 1 -1 0 0 0 3/2+ 1

bcc*+ bcc 1 0 2 -1 0 1 0 3/2+ 1

bbd*- bbd -1 0 0 -2 0 1 - 1/2 3/2+ 1

bbu*0 bbu 0 0 0 -2 0 1 1/2 3/2+ 1

bbs*- bbs -1 -1 0 -2 0 0 0 3/2+ 1

bbc*0 bbc 1 0 1 -2 0 3 0 3/2+ 1

bbb- bbb -1 0 0 -3 0 1 0 3/2+ 1



Finally A Solution That Finally A Solution That Actually Works!Actually Works!

Paper in progress for Paper in progress for publication in publication in

PRL(Physical Review Letters)PRL(Physical Review Letters)

STANDARD MODEL

QUARKS & ANTI-QUARKS

LEPTONS & ANTI-LEPTONS

VECTOR BOSONS

tb

cs

du

tb

cs

du

e e

e e

HH

ZW

Gg

,

,

,,

0

0

MESONS & ANTI-MESONS ( qq ) ( qq )

BARYONS & ANTI-BARYONS (qqq) ( qqq )

PENTAQUARKS & ANTI-PENTAQUARKS ( qqqqq ) ( qqqqq )

HEXAQUARKS & ANTI-HEXAQUARKS (qqqqqq) ( qqqqqq )

HYBRID MESONS & ANTI-HYBRID MESONS ( gqq ) ( qgq )

HYBRID BARYONS & ANTI-HYBRID BARYONS (qqqg ) ( gqqq )

BOUND STATE GLUONS (gg ) (ggg )

Documents

Application of Group Theory in Particle Physics using the Young Tableaux Method 2006 PASSHE-MA CONFERENCE (March 31 – April 1) Akhtar Mahmood ( Assistant