1

lecture 4

=> unitary groupsU(n)

external symmetries=> orthogonal groups

O (n)

internal symmetries

Lorentz group:

O(3,1)

rotation group: O(3)

isospin: SU(2)

unitary symmetry: SU(3)

Internal symmetries isospin symmetry => nuclear physicsSU(3) – symmetry =>hadronschiral summetry => pionscolor symmetry =>quarks electroweak symmetry => SU(2)xU(1) model

>

Internal symmetries:broken by interaction ( electromagnetism breaks isospin )

broken by explicit symmetry breaking ( SU(3) – symmetry of hadrons )

unbroken ( color symmetry of quarks )

broken by spontaneous symmetry breaking ( chiral symmetry and electroweak symmetry)

Rutherford:He suggested in 1919 that there must exist a

neutral partner of the proton.

helium nucleus: charge: 2 x protonmass: 4 x proton

1932:discovery of the neutron

(J. Chadwick)

atomic nuclei are composed ofprotons and neutrons

9

n

pN

n

pN

nucleons: doublet of SU(2)

=> SU(2) - transformations

Lawrence Berkeley Nat. Lab

1953pion

nucleus

p

delta: quadruplet ( 1230 MeV )

0

pions: triplet eta: singlet

MeV

MeV

548

1400

16

17

SU(2)-representations:singlet

doublettriplet

quadruplet…

basics about unitary groups

U(n): group of complex unitary

n x n matrices

SU(n): n x n matrices with det U = 1

U = exp (iH)

H: Hermitean n x n matrix

matricesnnHermiteantindependenn 2

det U = exp i (trH)

SU(n): det U = 1tr H = 0

matricesnnSU )1(:)( 2

SU(n): (n x n - 1) generators

SU(2): 3 SU(3): 8 SU(4): 15 SU(5): 24

SU(2): 3 Hermitean matrices

3 Pauli matrices

10

01

0

0

01

10321

i

i

SU(3): 8 Hermitean

matrices 8 Gell-Mann

matrices

000

010

001

000

00

00

000

001

010

321 i

i

010

100

000

00

000

00

001

000

100

654 i

i

200

010

001

3

1

00

00

000

87 i

i

ijji

i

tr

tr

2

0

ijji

i

tr

tr

2

0

Commutation relations ofSU(2) and SU(3)

ii

kijkji

i

T

matricesPauli

TiTT

TgeneratorsSU

2

1

:

,

3:)2(

kijkji

i

TifTT

TgeneratorsSU

,

8:)3(

Algebra of SU(3)

f… : structure constants

iiT 2

1

2

3

2

12

11

678458

367156

345257246147123

ff

ff

fffff

kijkijji d 2

1

3

4

2

1,

2

1

3

1..

:

118 dge

symmetrictotallydijk

s

d

u

3 quarks

triplet fundamental representation

ud

s

isospin

spinU spinV

8

33

54

76

21

3

2

)3(arg:

FY

FT

iFFV

iFFU

iFFT

SUofescheightFi

hypercharge

two of the eight lambda matrices are diagonal:

T(3) and Y.

A state in a representation can be described by t(3) and y

ud

s

2

12

1

3

2

3t

yquark triplet

3

1

hypercharge:

3200

0310

0031

Y

irreducible representations

choose state with maximal value of t(3) –

proceed into the U, T and V directions to

the left, until it stops

p

q

steps p and q

External line of representation

2112

1 qpqpN

number of states in an irreducible representation

each state is described by 3 numbers:

ytt ,, 3

An irreducible representation

is described by (p,q)

(0,0): singlet(1,0): triplet

(0,1): anti -triplet

a representation is in general complex:

pqqp ,,

)0,1(3

)1,0(3

)0,2(6

)0,3(10

)1,1(8

46

=(2,2)

47(3,3) = 64 = 18 + 12 x 2 + 6 x 3 + 4

0 1 2 3

0 1 3 6 10

1 3* 8 15 24

2 6* 15* 27 42 3 10* 24* 42* 64

p

q

Lowest representations of SU(3)

direct product of representations

27011088188

10881333

3633

8133

Casimir operatorinvariant operatore.g. for angular momentum

2222)( zyx LLLL

8

1

22)( iFF

qpqpqpF

qptionrepresenta

222

3

1

:),(

1 0 3,3* 4/3 6,6* 10/3 8 3 10,10* 6 27 8

representation Casimir

),...,( 821 FFFF

22

21

221

21

2

1FFFFF

FFF

two representations

1961

Yuval Neeman

mesons

)(:1950

)(:1946

)(106::1936

:1935

0 Berkeleyfound

Bristolfoundand

fermionMeVmeson

mesonpredictsYukawa

Bevatron in Berkeley

K-mesons: 1947 =>Eta-meson: 1961

8 mesons

00

0,,

KKKK

00

hyperons

o

o

pn

64

K

o

oK

K oK

65

o

66

oK K

Ko

K

MeV892

MeV892

MeV783 MeV1020

MeV775

MeV

MeV

MeVo

1020:

783:

775:

ofmasstoequal

almostofmass

o

o

o?

68

SU(3)

breaking of SU(3): much larger than the breaking of isospin symmetry

o

o

pn

70

940 MeV

1190 MeV

1318 MeV

1116 MeV

o

o

o

71???

1232 MeV

1530 MeV

1385 MeV

Symmetry breaking

Wigner - Eckart

theorem

tripletarbitraryA

generatorsTSU

i

i

:)2(

tionrepresentaeirreducibltt 3

tti

t

t

tti

t

tTtaA

3´´33

´´3

)(

Physics given by a(t) - the various matrix elements => Clebsch-Gordan coefficients

kijkijji d 2

1

3

4

2

1,

2

1

3

1..

:

118 dge

symmetrictotallydijk

2228

322

3

,

4

1

3

2

9

2

3

23

2

3

1

,

YTFD

YTUVD

DifFD

FFdD

kijkji

kjkj

ijki

f - coupling

octetarbitraryASU i:)3(

),(),(

),(),(

),(),(

qpDqpa

qpFqpa

qpAqp

id

if

i

f - coupling

d - coupling

physics 2 numbers

Wigner-Eckart theorem -- SU(3)

Gell-Mann / Okubomass formula

octetanofcomponenteightsH

SUundersymmetricH

HHH

:

)3(:

8

0

80

Susumu Okubo (Rochester)

)4

1( 22

888

YTMYM

DmFmH

df

df

2228

322

3

4

1

3

2

9

2

3

23

2

3

1

,

YTFD

YTUVD

DifFD kijkji

MMMM

MMM

MMM

n

d

dn

3)(2

23

2)(2

Agreement better than 1 %

yMMM

yt

ttTdecuplet

baryon

0

2

12

1

)1(:

equal spacing rule

o

o

o

82

1236 MeV

1672 MeV ?

1232 MeV

1530 MeV

1385 MeV

83

84

K

o

oK

K oK

85

496 MeV

138 MeV958 MeV548 MeV

496 MeV

MeVM

MMM

MMMM

K

n

612

34

3)(2

experiment: 548 MeV

80

80

cossin

sincos

:

mixing

mixing changes the masses

lower state lower higher state higher

Experiment: mixing angle about 16 degrees

vector mesons

Gell-Mann / Okubo formula:

MeVMeriment

MeVM

MMKM

783)(:exp

931)(

)()(3)(4

8

8

mixing angle: ~ 54 degreespseudoscalar mesons:

~16 degrees

MeVM

MeVM

andofmixing

1020)(

783)(

:

Why pi mesons have a small mass?

Gell-Mann, Oakes, Renner(1968)

Chiral SymmetrySU(3) => SU(3,L) x SU(3,R)

Exact chiral symmetry:

3 pi mesons1 eta meson4 K mesons

mass zero

Goldstone bosons

Chiral symmetry breaking:

all eight mesons acquire masses

SU(3,L) x SU(3,R)

SU(2,L) x SU(2,R)

SU(2)

K-mesons and eta meson massive pions massless

pions massive

Why chiral symmetry?

QCD