Lecture The Quark model. WS2015/16: Physics of Strongly Interacting Matter

Transcription

1 Lecture The Quark model W2015/16: Physics of trongly Interacting Matter 1

2 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 L0 q quarks, q antiquarks Meson (qq) Baryon (qqq) 2

3 The quark quantum numbers: Quark quantum numbers flavor (6): u (up-), d (down-), s (strange-), c (charm-), t (top-), b(bottom-) quarks anti-flavor for anti-quarks q: u, d, s, c, t, b charge: 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: B1/3 - as baryons are made out of three quarks spin: s1/2 - quarks are the fermions! strangeness: s 1, s 1, q 0 for q u, d, c,t,b (and q) charm: C c 1, C c 1, C q 0 for q u, d, s,t,b (and q) bottomness: Β 1, B 1, Bq 0 for q u, d, s,c,t b b (and q) topness: T 1, T 1, Tq 0 for q u, d, s,c,b t t (and q) 3

4 The quark quantum numbers: Quark quantum numbers hypercharge: Y B + + C + B + T (1) ( baryon charge + strangeness + charm + bottomness +topness) I 3 (or I z or T 3 ) - 3 d component of isospin charge (Gell-Mann Nishijima formula): Q I 3 + Y/2 (2) ( 3 d component of isospin + hypercharge/2) 4

5 Quark quantum numbers 5

6 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. ince 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 U(3) group. Consider u, d, s quarks : then the quarks lie in the fundamental representation, 3 (called the triplet) of the flavour group U(3) : [3] The antiquarks lie in the complex conjugate representation 3 : [3] 6

7 Quark quantum numbers triplet in U(3) flavor group: [3] anti-triplet in U(3) flavor group: [3] Y2(Q-T 3 ) E.g. u-quark: Q+2/3, T 3 +1/2, Y1/3 7

8 Quark quantum numbers The quark quantum numbers: Collor 3: red, green and blue triplet in U(3) collor group: [3] nticollor 3: antired, antigreen and antiblue anti-triplet in U(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 ll hadrons color neutral color singlet in the U(3) collor group History: The quantum number color has been introduced (idea from Greenberg, 1964) to describe the state ++ (uuu) (Q+2, J3/2), discovered by Fermi in 1951 as π + p resonance: ++ + ( uuu) p( uud) + π ( du) + The state + ( u u u ) with all parallel spins (to achieve J3/2) is forbidden according to the Fermi statistics (without color)! 8

9 The current quark masses: masses of the quarks Quark quantum numbers m u MeV/c 2 m d MeV/c 2 m s MeV/c 2 m c GeV/c 2 m b GeV/c 2 m b ~ 180 GeV/c 2 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: M u(d) * ~ 350 MeV/c 2 9

10 Building Blocks of Matter m q,l [MeV] Periodensystem Leptonen Quarks t τ b c 10 2 µ s ν τ ν µ e d u

11 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: B B 1 B m 0 Constraints to build hadrons from quarks: strong color interaction (red, green, blue) confinement quarks must form color-neutral hadrons tate function for baryons antisymmetric under interchange of two quarks Ψ qqq [ color space spin flavor ] ince all baryons are color neutral, the color part of Ψ must be antisymmetric, i.e. a U(3) color singlet Ψ qqq color [ space spin flavor ] symmetric 11 (4) (5)

12 Hadrons in Quark model Possible states Ψ : Ψ color [ space spin flavor ] [ space spin flavor ] (6) (7) or a linear conbination of (6) and (7): Ψ α color + β color [ space [ space spin spin flavor flavor ] ] (8) α 2 2 where 1 + β Consider flavor space (u,d,s quarks) U(3) flavor group Possible states: flavor> : (6) antisymmetric for baryons (7) symmetric (8) mixed symmetry 12

13 Mesons in the Quark model Meson qq Quark nti-quark triplet in U(3) flavor group: [3] anti-triplet in U(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

14 Mesons in the Quark model π (du) [3] electric charge Q+1 3 states: Y0, I 3 0 Q -1,B,C: in octet:,b singlet state C C Q0 linear combination of 1 ( uu + dd + ss) 3 1 ( uu dd ), 2 B u u, dd, ss 1 ( uu + dd14 2ss) 6

15 Classification of mesons: Mesons in the Quark model Quantum numbers: spin orbital angular momentum rl total angular momentum J r L + r Properties with respect to Poincare transformation: 1) continuos transformationlorentz boost (3 parameters: β) B r iβ r α U ~ e 2 µ Casimir operator (invariant under transformation): 2) rotations (3 parameters: Euler angle ϕ) : U R ~ e rr iϕj Casimir operator: J 2 M p µ p 3) space-time shifts (4 parameters: a µ ) U ~ st e iα x µ µ x + µ xµ aµ 10 parameters of Poincare group 15

16 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 r r 5) time reversal: t -t eigenvalue T +1 6) charge conjugation: C -C C ( 1) L + 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 2 2 M ( or M ), J ( or J ), P, C 16

17 Classification of mesons: Mesons in the Quark model the mesons are classified in J PC multiplets 1) L0 states: J0 or 1, i.e. J P ( 1) L C ( 1) L + ( 1) J PC pseudoscalar states vector states +1 for 0-1 for 1 2) L1 states - orbital exitations; P ( 1) L r r r J L + -1 J0 J PC scalar states 0 J axial vectors axial vectors 1 J tensor L J L + 17

18 Mesons in the Quark model isospin L J PC I 1 I 1/ 2 I 0 m [ MeV ] L π ρ K K * η, η' ω, ϕ 140 ( m π ) ~ L B 2 1 Q 2 * K' Q 1 H f, f D 1250 ' δ κ ε,

19 Mesons in the Quark model J PC pseudoscalar nonet (L0, 0) trangeness J PC vector nonet (L0, 1) 19

20 Baryons in the Quark model Baryon qqq Quark triplet in U(3) flavor group: [3] Eqs. (4-8): state function for baryons antisymmetric under interchange of two quarks: Ψ qqq [ color space spin flavor ] where flavor> state can be symmetric (), antisymmetric () or mixed symmetry (M) From group theory: with three flavours, the decomposition in flavour is [3] [3] [3] ([6] [3] ) [3] ([6] [3]) ([3] [3]) [10] [8] M [8] M [1] 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

21 Baryons in the Quark model 1) Combine first 2 quark triplets: [3] [3] [6] [3 ] 2) dd a 3 d quark: [3] [3] [3] ([6] [3] ) [3] [10] [8] M [8] M [1] 21

22 Baryons in the Quark model Octet [8] Decuplet [10] pin: J L0 J P J+L L1 J P

23 tructure of known baryons Ground states of Baryons + exitation spectra 23

24 Mesons in the U(4) flavor Quark model Now consider the basis states of meons in 4 flavour U(4) flavor : u, d, s, c quarks [ 4] [4] [15] [1] U(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

25 Baryons in the U(4) flavor Quark model Now consider the basis states of baryons in 4 flavour U(4) flavor : u, d, s, c quarks U(4) multiplets of baryons made of u, d, s, and c quarks: the 20-plet with an U(3) octet and the 20-plet with an U(3) decuplet. 25

26 Exotic states 26

27 Exotic states u u s d Hybrid Baryonium Glueball d qqg gg... qqqq Pentaquark qqqq q + Experimental evidence: π(1400) σ(600) f o (1500) very broad width ( MeV) > short lifetime < 1 fm/c... 27

28 Pentaquarks Flavour -exotic state, e.g. Θ + uudd s Decay: uudd s udd + us u u s d d Θ + uudd s Θ + n + K + uud + ds 0 p + K d u d u s Very small life time (big width)? 28

29 (Quark)-oliton-Model Chiral Lagrangean: invariant under U(3)-flavor Rotation L eff q [ ( )] a a i M exp iγ π λ /f q Pseudoscalar fields: π a { π,k, η} 5 π Diakonov, Petrov, Polyakov ('97) q π a q q Chiral Quark-oliton Model: solution of the Euler-Lagrange equationof-motion > olitons quantization of the soliton solutions under U(3) f Predictions for the pentaquark state: pin J1/2, Parity Ppositive: J P 1/2 + Width Γ < 30 MeV U(3) f - ntidecuplet 29

30 Quark-Correlations (diquark) Modelle Jaffe, Wilczek, Karliner, Lipkin ('03) [qq] correlations: antisymmetric in Color, Flavor und pin state diquark Pentaquark: 3C [ q q ] [ q q ] [ ] 3 3 3C C C q J L 1 Predictions for the pentaquarks: pin J1/2, Parity Ppositiv: J P 1/2 + Width Γ < 15 MeV U(3) f - ntidecuplet + Oktet Oktet: Partner with J P 3/2 + 30

31 Positive experimental signals of Θ + (1540) pring8 DIN JLab EL JLab ITEP VD/IHEP HERME ZEU COY-TOF NOMD pp Σ + Θ +. 31

32 but not seen by other experiments BBR Delphi/LEP LEPH/LEP CDF E690/FermiLab HyperCP TR/RHIC HER-B PHENIX/RHIC 32

33 2004: PDG entry for pentaquark NOT any more in 2014! 33