Download The Quark model

Lecture
The Quark
model
WS2015/16: ‚Physics of Strongly Interacting Matter‘
1
Quark model
The quark model is a classification scheme for hadrons in
terms of their valence quarks — the quarks and antiquarks
which give rise to the quantum numbers of the hadrons.
The quark model in its modern form was developed by
Murray Gell-Mann - american physicist who received the
1969 Nobel Prize in physics for his work on the theory of
elementary particles.
* QM - independently proposed by George Zweig
1929
Hadrons are not ‚fundamental‘, but they are built from ‚valence quarks‘, i.e.
quarks and antiquarks, which give the quantum numbers of the hadrons
| Baryon 〉 = | qqq 〉
| Meson 〉 = | qq 〉
L=0
q= quarks, q – antiquarks
Meson (qq)
Baryon (qqq)
2
Quark quantum numbers
The quark quantum numbers:
flavor (6): u (up-), d (down-), s (strange-), c (charm-), t (top-), b(bottom-) quarks
q:
anti-flavor for anti-quarks
charge:
u , d , s , c , t, b
Q = -1/3, +2/3 (u: 2/3, d: -1/3, s: -1/3, c: 2/3, t: 2/3, b: -1/3 )
baryon number: B=1/3 - as baryons are made out of three quarks
spin: s=1/2 - quarks are the fermions!
strangeness: S s = −1,
charm: Cc = 1,
C c = −1,
bottomness: Βb = −1,
topness:
Tt = 1,
S s = 1,
q = u, d, c, t, b (and q )
Cq = 0 for q = u, d, s , t, b (and q )
Bb = 1,
Tt = −1,
S q = 0 for
Bq = 0 for
q = u, d, s, c, t (and q )
Tq = 0 for q = u, d, s , c, b (and q )
3
Quark quantum numbers
The quark quantum numbers:
hypercharge:
Y=B+S+C+B+T
(1)
(= baryon charge + strangeness + charm + bottomness +topness)
I3 (or Iz or T3) - 3‘d component of isospin
charge (Gell-Mann–Nishijima formula):
Q = I3 + Y/2
(2)
(= 3‘d component of isospin + hypercharge/2)
4
Quark quantum numbers
5
Quark quantum numbers
The quark model is the follow-up to the Eightfold Way classification scheme
(proposed by Murray Gell-Mann and Yuval Ne'eman )
The Eightfold Way may be understood as a consequence
of flavor symmetries between various kinds of quarks.
Since the strong nuclear force affects quarks the same
way regardless of their flavor, replacing one flavor
of a quark with another in a hadron should not change
its mass very much. Mathematically, this replacement
may be described by elements of the SU(3) group.
Consider u, d, s quarks :
then the quarks lie in the fundamental representation, 3
(called the triplet) of the flavour group SU(3) : [3]
The antiquarks lie in the complex conjugate representation 3 : [ 3 ]
6
Quark quantum numbers
triplet in SU(3)flavor group: [3]
anti-triplet in SU(3)flavor group: [ 3 ]
Y=2(Q-T3)
E.g. u-quark: Q=+2/3, T3=+1/2, Y=1/3
7
Quark quantum numbers
The quark quantum numbers:
Collor 3: red, green and blue
triplet in SU(3)collor group: [3]
Anticollor 3: antired, antigreen and antiblue
anti-triplet in SU(3)collor group [ 3 ]
• The quark colors (red, green, blue) combine to be colorless
•The quark anticolors (antired, antigreen, antiblue) also combine to be colorless
All hadrons
color neutral
= color singlet in the SU(3)collor group
History: The quantum number ‚color‘ has been introduced (idea from
Greenberg, 1964) to describe the state ∆++(uuu) (Q=+2, J=3/2) , discovered by
Fermi in 1951 as π+p resonance: ∆+ + ( uuu) → p( uud ) + π + (d u)
The state ∆+ + ( u ↑ u ↑ u ↑ ) with all parallel spins (to achieve J=3/2) is forbidden
according to the Fermi statistics (without color) !
8
Quark quantum numbers
The current quark masses:
masses of the quarks
mu = 1.7 - 3.3 MeV/c2
md = 4.1 – 5.8 MeV/c2
ms = 70 – 130 MeV/c2
mc = 1.1 –1.5 GeV/c2
mb = 4.1 - 4.4 GeV/c2
mb ~ 180 GeV/c2
The current quark mass is also called the mass of the 'naked‘ (‚bare‘) quark.
Note: the constituent quark mass is the mass of a 'dressed' current quark, i.e. for
quarks surrounded by a cloud of virtual quarks and gluons:
Mu(d)* ~ 350 MeV/c2
9
Building Blocks of Matter
mq,L
[MeV]
6
10
Periodensystem
Leptonen
Quarks
t
5
10
4
10
τ
b
c
3
10
µ
2
10
ντ
d
1
10
0
10
s
νµ
u
e
-1
10
-2
10
10
Hadrons in the Quark model
Gell-Mann (1964): Hadrons are not ‚fundamental‘, but they are built from
‚valence quarks‘,
| Baryon 〉 = | qqq 〉
Baryon charge:
BB = 1
| Meson 〉 = | qq 〉
(3)
Bm = 0
Constraints to build hadrons from quarks:
• strong color interaction (red, green, blue)
• confinement
• quarks must form color-neutral hadrons
State function for baryons – antisymmetric under interchange of two quarks
ΨA ≡| qqq 〉 A = [| color 〉 ⊗ | space〉 ⊗ | spin 〉 ⊗ | flavor 〉 ]A
(4)
Since all baryons are color neutral, the color part of ΨA must be antisymmetric,
i.e. a SU(3)color singlet
ΨA ≡| qqq 〉 A =| color 〉 A ⊗ [| space〉 ⊗ | spin 〉 ⊗ | flavor 〉 ]S
(5)
symmetric
11
Hadrons in Quark model
Possible states ΨA:
Ψ A =| color 〉 A ⊗ [| space〉 S ⊗ | spin 〉 A ⊗ | flavor 〉 A ]S
(6)
[| space〉 S ⊗ | spin 〉 S ⊗ | flavor 〉 S ]S
(7)
or a linear conbination of (6) and (7):
Ψ A = α | color 〉 A ⊗ [| space〉 S ⊗ | spin 〉 A ⊗ | flavor 〉 A ]S
+ β | color 〉 A ⊗ [| space〉 S ⊗ | spin 〉 S ⊗ | flavor 〉 S ]S
where
(8)
α2 + β2 =1
Consider flavor space (u,d,s quarks)
SU(3)flavor group
Possible states: |flavor> : (6) – antisymmetric
for baryons
(7) – symmetric
(8) – mixed symmetry
12
Mesons in the Quark model
| Meson 〉 = | qq 〉
Quark
triplet in SU(3)flavor group: [3]
Anti-quark
anti-triplet in SU(3)flavor group: [ 3 ]
From group theory: the nine states (nonet) made out of a pair can be
decomposed into the trivial representation, 1 (called the singlet), and the
adjoint representation, 8 (called the octet).
[3] ⊗ [ 3] = [8] ⊕ [1]
octet + singlet
13
Mesons in the Quark model
−
[3]
π (d u )
electric charge Q=+1
3 states: Y=0, I3=0
A,B,C:
Q= -1
in octet: A,B singlet state C
Q=0
linear combination of uu , dd , ss
C=
1
( uu + dd + ss )
3
A=
1
( uu − dd ),
2
B=
1
( uu + dd14− 2 ss )
6
Mesons in the Quark model
Classification of mesons:
Quantum numbers:
• spin S
• orbital angular momentumrL r r
• total angular momentum J = L + S
Properties with respect to Poincare transformation:
1) continuos transformation
UB ~ e
rr
iβα
Lorentz boost (3 parameters: β)
Casimir operator (invariant under transformation): M 2 = pµ p µ
2) rotations (3 parameters: Euler angle ϕ) :
UR ~ e
rr
iϕ J
Casimir operator: J2
3) space-time shifts (4 parameters: aµ)
U st ~ e
iα µ x µ
x ′µ → x µ + a µ
10 parameters
of Poincare group
15
Mesons in the Quark model
Classification of mesons:
Discrete operators:
4) parity transformation: flip in sign of the spacial coordinate
eigenvalue P = +1
P = (−1)L + 1
5) time reversal: t
eigenvalue T = +1
r
r
r = −r
-t
6) charge conjugation:
C = (−1)L + S
C = -C
C - parity: eigenvalue C = +1
P ⋅ C ⋅T = 1
General PCT –theorem:
due to the fact that discrete transformations correspond to the
U(1) group they are multiplicative
Properties of the distinguishable (not continuum!) particles are defined by
M 2 (or M ),
J 2 (or J ),
P, C
16
Mesons in the Quark model
Classification of mesons:
the mesons are classified in JPC multiplets
1) L=0 states:
J=0 or 1, i.e. J=S
P = (−1)L + 1 = -1
JPC =
C = (−1)L + S = (−1)S =
+1 for S=0
-1
for S=1
0-+ - pseudoscalar states
1-- - vector states
2) L=1 states - orbital exitations;
P = (−1)L + 1 = +1
S= -1 J=0
JPC= 0++ - scalar states
1++ - axial vectors
r r r S= 0 J=1
J = L+ S
1+- - axial vectors
S= 1 J=2
2++ - tensor
|L − S| ≤ J ≤ L + S
17
Mesons in the Quark model
isospin
L
S
J PC
L = 0 S = 0 0−+
− −
S =1 1
L = 1 S = 0 1+ −
S = 1 2+ +
1+ +
0++
I = 1 I = 1/ 2 I = 0
m [ MeV ]
π
η ,η ' 140 ( mπ ) − 500
K
ρ
B
A2
A1
K*
Q2
K '*
Q1
ω ,ϕ
H
f, f'
D
~ 800
1250
1400
1300
δ
κ
ε,S
ε, ∗
1150
18
Mesons in the Quark model
JPC = 0-+ - pseudoscalar nonet (L=0, S=0)
Strangeness
JPC = 1-- - vector nonet (L=0, S=1)
19
Baryons in the Quark model
| Baryon 〉 = | qqq 〉
Quark
triplet in SU(3)flavor group: [3]
Eqs. (4-8): state function for baryons – antisymmetric under interchange
of two quarks:
Ψ ≡| qqq 〉 = [| color 〉 ⊗ | space〉 ⊗ | spin 〉 ⊗ | flavor
A
A
〉 ]A
where |flavor> state can be symmetric (S), antisymmetric (A) or mixed symmetry (M)
From group theory: with three flavours, the decomposition in flavour is
[3] ⊗ [3] ⊗ [3] = ([6]S ⊕ [3 ]A ) ⊗ [3] =
= ([6]S ⊗ [3]) ⊕ ([ 3 ] ⊗ [3]) =
= [10]S ⊕ [8]M ⊕ [8]M ⊕ [1]A
The decuplet is symmetric in flavour, the singlet antisymmetric and the two octets
have mixed symmetry (they are connected by a unitary transformation and thus
describe the same states).
The space and spin parts of the states are then fixed once the orbital angular
20
momentum is given.
Baryons in the Quark model
1) Combine first 2 quark triplets:
[3] ⊗ [3] = [6]S ⊕ [ 3 ]A
2) Add a 3‘d quark:
[3] ⊗ [3] ⊗ [3] = ([6]S ⊕ [3 ]A ) ⊗ [3] =
= [10]S ⊕ [8]M ⊕ [8]M ⊕ [1]A
21
Baryons in the Quark model
Octet [8]
Spin:
J=S
L=0
1
JP =
2
Decuplet [10]
+
J=S+L
L=1
3
J =
2
+
P
22
Structure of known baryons
Ground states of Baryons
+ exitation spectra
23
Mesons in the SU(4) flavor Quark model
Now consider the basis states of meons in 4 flavour SU(4)flavor:
u, d, s, c quarks
[4] ⊗ [ 4 ] = [15] ⊕ [1]
SU(4) weight diagram showing the 16-plets for the pseudoscalar and
vector mesons as a function of isospin I, charm C and hypercharge Y. The
nonets of light mesons occupy the central planes to which the cc states
have been added.
24
Baryons in the SU(4) flavor Quark model
Now consider the basis states of baryons in 4 flavour SU(4)flavor: u, d, s, c quarks
SU(4) multiplets of baryons made of u, d, s, and c quarks:
the 20-plet with an SU(3) octet and the 20-plet with an SU(3) decuplet.
25
Exotic states
26
Exotic states
| Hybrid 〉 = | qqg 〉 + ...
| Baryonium 〉 = | qqqq 〉 + ...
| Glueball 〉 = | gg 〉 + ...
u
s
u
Experimental evidence:
π(1400)
σ(600)
fo(1500)
||
very broad width
(200-300 MeV) => short
lifetime < 1 fm/c
d
d
| Pentaquark 〉 = | qqqq q 〉 + ...
27
Pentaquarks
„Flavour“-exotic state, e.g.
+
| Θ 〉 = | uudd s〉
Decay:
| uudd s〉 → | udd 〉+ | us〉
| Θ + 〉 → | n〉+ | K + 〉
d
u
u
s
u
s
d
| uudd s〉 → | uud 〉+ | ds〉
| Θ + 〉 → | p〉+ | K 0 〉
u
d d
Very small life time (big width)?
28
(Quark)-Soliton-Model
Chiral Lagrangean:
invariant under SU(3)-flavor Rotation
[
(
L ef f = q i∂ − M exp iγ 5 π λ /f π
Pseudoscalar fields:
a
a
Diakonov, Petrov, Polyakov ('97)
)] q
π a = {π, K, η}
q
πa
q
q
Chiral Quark-Soliton Model:
•solution of the Euler-Lagrange equationof-motion => Solitons
• quantization of the soliton solutions
under SU(3)f
Predictions for the pentaquark state:
• Spin J=1/2, Parity P=positive: JP = 1/2+
• Width Γ < 30 MeV
• SU(3)f - Antidecuplet
29
Quark-Correlations (diquark) Modelle
Jaffe, Wilczek, Karliner, Lipkin ('03)
• [qq] correlations: antisymmetric in Color,
Flavor und Spin state = diquark
• Pentaquark: [ [q q ]
3C
1
2
⊗ [q 1q 2 ]
3C
]
3C
⊗q
J 1 2
3C
1 2
L
1
Predictions for the pentaquarks:
• Spin J=1/2, Parity P=positiv: JP = 1/2+
• Width Γ < 15 MeV
• SU(3)f - Antidecuplet + Oktet
Oktet: Partner with JP = 3/2+
30
Positive experimental signals of Θ+(1540)
Spring8
JLab
DIANA
ITEP
JLab
ELSA
SVD/IHEP
HERMES
ZEUS
COSY-TOF
NOMAD
pp
Σ+Θ+.
31
… but not seen by other experiments
BABAR
CDF
Delphi/LEP
ALEPH/LEP
E690/FermiLab
HyperCP
STAR/RHIC
HERA-B
PHENIX/RHIC
32
2004: PDG entry for pentaquark – NOT any more in 2014!
33