=> 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
n
pN
nucleons: doublet of SU(2)
=> SU(2) - transformations
Lawrence Berkeley Nat. Lab
delta: quadruplet ( 1230 MeV )
0
pions: triplet eta: singlet
MeV
MeV
548
1400
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 ,,
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
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
o
66
oK K
Ko
K
MeV892
MeV892
MeV783 MeV1020
MeV775
MeV
MeV
MeVo
1020:
783:
775:
ofmasstoequal
almostofmass
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
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