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Lecture The model

WS2018/19: ‚ of Strongly Interacting ‘ 1

The quark model is a classification scheme for in terms of their valence — 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 for his on the theory of elementary . * QM - independently proposed by 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

|   | qqq  |   | qq 

q= quarks, q – antiquarks

2 Quark quantum numbers

The quark quantum numbers:

. flavor (6): u (up-), d (down-), s (strange-), c (-), t (top-), b(bottom-) quarks

anti-flavor for anti-quarks q: 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 )

. : B=1/3 - as are made out of three quarks

. : s=1/2 - quarks are the !

. : Ss  1, Ss  1, Sq  0 for q  u, d, c,t,b (and q)

charm: . Cc  1, Cc  1, Cq  0 for q  u, d, s,t,b (and q)

. : Βb  1, Bb  1, Bq  0 for q  u, d, s,c,t (and q)

. : Tt  1, Tt  1, Tq  0 for q  u, d, s,c,b (and q) 3 Quark quantum numbers

The quark quantum numbers: : Y = B + S + C + B + T (1)

(= baryon charge + strangeness + charm + bottomness +topness)

. I3 (or Iz or T3) - 3‘d component of

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 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 affects quarks the same way regardless of their flavor, replacing one flavor of a quark with another in a should not change its very much. Mathematically, this replacement may be described by elements of the SU(3) .

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 ‚color‘ has been introduced (idea from Greenberg, 1964) to describe the state D++(uuu) (Q=+2, J=3/2) , discovered by Fermi in 1951 as p+p resonance: D (uuu)  p(uud)  p  (du) The state D   ( 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 : . masses of the quarks 2 mu = 1.7 - 3.3 MeV/c 2 md = 4.1 – 5.8 MeV/c 2 ms = 70 – 130 MeV/c 2 mc = 1.1 –1.5 GeV/c 2 mb = 4.1 - 4.4 GeV/c 2 mt ~ 180 GeV/c

The current quark mass is also called the mass of the 'naked‘ (‚bare‘) quark. Note: the mass is the mass of a 'dressed' current quark, i.e. for quarks surrounded by a cloud of virtual quarks and :

* 2 Mu(d) ~ 350 MeV/c 9 Building Blocks of Matter

Periodensystem mq,L [MeV] 106 Leptonen Quarks t 105

104 b  c 103  s 102

 1 10 d u

100 e  10-1

10-2

10 Hadrons in the Quark model

Gell-Mann (1964): Hadrons are not ‚fundamental‘, but they are built from ‚valence quarks‘, | Baryon   | qqq  | Meson   | qq  (3)

Baryon charge: BB = 1 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 (4) A | qqq  A  [| color| space| spin| flavor ]A

Since all baryons are color neutral, the color part of A must be antisymmetric, i.e. a SU(3)color singlet (5) A | qqq  A | color A [| space | spin | flavor ]S symmetric 11 Hadrons in Quark model

 Possible states A:

Ψ A | color A [| space S  | spin A  | flavor  A ]S (6) (7) [| space S  | spin S  | flavor  S ]S

or a linear conbination of (6) and (7):

Ψ A   | color A [| space S  | spin A  | flavor  A ]S (8)   | color A [| space S  | spin S  | flavor  S ]S

where  2   2  1

. Consider flavor space (u,d,s quarks)  SU(3)flavor group Possible states: |flavor> : (6) – antisymmetric for baryons (7) – symmetric (8) – mixed

12 in the Quark model

| Meson   | qq 

Quark Anti-quark triplet in SU(3) group: [3] flavor 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 , 8 (called the octet). [3]  [3]  [8]  [1] octet + singlet

13 Mesons in the Quark model

[3] π (du)

Q=+1

Q= -1 Q=0 3 states: Y=0, I3=0 linear combination of uu, dd , ss 1 1 A  (uu  dd )  p 0 , B  (uu  dd  2ss)  2 6 1 A,B,C: in octet: A,B C C  (uu  dd  ss)  14 3 Mesons in the Quark model

Classification of mesons:

Quantum numbers: • spin S • orbital angular L    • total J  L  S

 Properties with respect to Poincare transformation:

1) continuos transformation  Lorentz boost (3 parameters: )  i U ~ e 2  B Casimir ( under transformation): M  p p

2) (3 parameters: Euler angle j) :

 ijJ 2 U R ~ e Casimir operator: J

3) space- shifts (4 parameters: a)  10 parameters i  x  x  a U st ~ e of Poincare group 15 Mesons in the Quark model

Classification of mesons:

 Discrete operators: 4) transformation: flip in sign of the spacial coordinate r  r eigenvalue P = +1 P = (−1)L + 1

5) time reversal: t  -t eigenvalue T = +1 6) charge conjugation: C = -C C = (−1)L + S C - parity: eigenvalue C = +1

General PCT –theorem: P C T  1 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

1) L=0 states: J=0 or 1, i.e. J=S +1 for S=0 P = (−1)L + 1 = -1 C = (−1)L + S = (−1)S = -1 for S=1 0-+ - states PC J = 1-- - vector states

2) L=1 states - orbital exitations; P = (−1)L + 1 = +1 S= -1 J=0 JPC= 0++ - states ++    S= 0 J=1 1 - axial vectors J  L  S 1+- - axial vectors S= 1 J=2 2++ -

|L − S| ≤ J ≤ L + S 17 Mesons in the Quark model

isospin L S J PC I  1 I  1/ 2 I  0 m [MeV ]  L  0 S  0 0 p K ,' 140 (mp )  500 S  1 1   K * ,j ~ 800  L  1 S  0 1 B Q2 H 1250  * S  1 2 A2 K' f , f ' 1400  1 A1 Q1 D 1300 0++ d k e,S* 1150

18 Mesons in the Quark model

PC -+ J = 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: A | qqq  A  [| color| space| spin| flavor ]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 momentum is given. 20 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] Decuplet [10]

  Spin: P 1 J=S+L 3 J=S J  J P  2 L=1 2 L=0 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 ccbar 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

Experimental evidence: | Hybrid   | qqg   ... p(1400) s(600) | Baryonium   | qqqq   ... fo(1500) || |   | gg   ... very broad width (200-300 MeV) => short lifetime < 1 fm/c u d s u d | Pentaquark   | qqqq q   ... 27

„Flavour“-exotic state, e.g. | Θ   | uudd s

Decay: | uudd s  | udd  | us | Θ   | n | K   s u u d s | uudd s  | uud  | ds u d u | Θ   | p | K 0  d d

Very small life time (big width)?

28 (Quark)-Soliton-Model

Chiral Lagrangean: Diakonov, Petrov, Polyakov ('97) invariant under SU(3)-flavor q a a L eff  q i  M expiγ 5 π λ /f π  q pa a Pseudoscalar fields: π  π,K, η q q

Chiral Quark-Soliton Model: •solution of the Euler-Lagrange equation- of-motion => Solitons • of the soliton solutions under SU(3)f Predictions for the state: • Spin J=1/2, Parity P=positive: JP = 1/2+ • Width G < 30 MeV • SU(3) - Antidecuplet f 29 Quark-Correlations () Model

Jaffe, Wilczek, Karliner, Lipkin ('03)

[qq] correlations: antisymmetric in Color, • J 1 2 Flavor und Spin state = diquark 3 C 3 • Pentaquark: 3C 3C C 1 2  q1q 2   q1q 2    q

L 1

Predictions for the pentaquarks: • Spin J=1/2, Parity P=positiv: JP = 1/2+ • Width G < 15 MeV

• SU(3)f - Antidecuplet + Oktet Oktet: Partner with JP = 3/2+

30 Other models of pentaquarks

A five-quark "bag" A "meson-baryon "

31 Positive experimental signals of Θ+(1540)

Spring8 DIANA JLab ELSA

ITEP SVD/IHEP JLab

HERMES

COSY-TOF ZEUS NOMAD

pp  ++.

32 … but not seen by other experiments

BABAR Delphi/LEP ALEPH/LEP

E690/FermiLab CDF

HyperCP

STAR/RHIC HERA-B

PHENIX/RHIC

33 2004: PDG entry for pentaquark – NOT any more in 2016!

34 2015 LHCb results

In July 2015, the LHCb collaboration at CERN identified 0 − pentaquarks in the Λ b→J/ψK p channel

+ +  Two states, named P c(4380) and P c(4450)

35