The Quark Parton Model
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1 The Quark Parton Model Quark Model Pseudoscalar J P = 0 Mesons Vector J P = 1 Mesons Meson Masses J P = 3 /2 + Baryons J P = ½ + Baryons Resonances Resonance Detection Discovery of the ω meson Dalitz Plots
2 Quark Model Beginning of 60s: Regularities in the hadron spectra indicated that they are formed by 3 (valence) quarks: u, d, s and their antiparticles u, d, s Quarks are not observed as free particles symmetry driven model in 1964; 10 more years were needed to develop a dynamical model Beginning of 70s: deep-inelastic scattering of leptons off hadrons (Rutherford-like experiment): result: inside the nucleons there are light, quasi-free, point-like fermions called partons (our quarks!) Their effective mass is: m ~1/3 of the baryon mass, with a momentum p: ~1/R (~1 fm) For heavy quarks the non-relativistic approximation holds: m>>1/r Regularities in the hadron spectra were accounted for by the quark model (multiplets)
3 Classification of Hadrons
4 Classification of Hadrons
5 πn Scattering
6 pn scattering Resonances
7 Strangeness
8 Quark Model s =0 d u s = 1 s Quarks Antiquarks s = -1 s Q = 2/3 s =0 u d Q = -1/3 Q = -2/3 Q = 1/3 Particles with same spin, parity and charge conjugation symmetry described as multiplet - Different I z and Y Raising and lowering operators to navigate around the multiplet Gell-Mann and Zweig: Patterns of multiplets explained if all hadrons were made of quarks Model originally developed using group theory alone - No need for physical quarks - Fact that quark charges non-integer suggested perhaps they were not real particles Q Q Q Q Q
9 Mesons Mesons are built with quark/anti-quark pairs. If we assume only u, d, s quarks (with their anti-quarks) we could have families with 3 2 =9 elements each (nonets). One has then: J=1 spin triplet states (, 1/ 2 ( + ), ) with J=1 and J z =1, 0, -1 J=0 spin singlet states (1/ 2 ( - )) with J=0 S e.g. the J P =1 - meson nonet K *0 +1 K *+ vector mesons: J PC =1 -- (as the photon) ρ - ρ 0 φ ρ /2 ω +1/2 I 3 +1 K * - -1 K *0
10 Mesons
11 Pseudoscalar J P = 0 Mesons Quark plus antiquark with spins ½ + -½ = 0 ( ) ( 9 combinations) 0 :(uū d d)/ 2 :(uū + d d 2s s)/ 6 :(uū + d d + s s)/ 3
12 Pseudoscalar J P = 0 Mesons
13 Pseudoscalar Mesons
14 Vector J P = 1 Mesons Quark plus antiquark with spins ½ + ½ = 1 ( ) ( 9 combinations) ρ 0 = 1 / 2 (dd uu) ω = 1 / 2 (dd + uu) ϕ = ss
15 Vector Mesons J = 1 P
16 Vector Mesons
17 Higher Masses States shown so far have no angular momentum between the qq or the qqq Higher mass states are obtained by having orbital angular momentum between the qq or the qqq J = 2, 3,... 5 /2, 7 /2...
18 Meson Masses
19 Vector meson decays into lepton pairs: one more proof of the hadron quark composition and charge assignment The leptonic partial width Γ(e + e - ) is proportional to the square of the quark charges (Rutherford): Γ(V l + l ) = 16πα 2 Q 2 with Q 2 = Σa i Q i 2 Vector Meson Decays M V 2 ψ(0) 2 Amplitude squared for the two quarks interacting with the photon in one point of the space-time =1/volume of the meson (square of the mean quark charge, with a i amplitude coefficients) The formula is derived taking into account a (1/q 2 ) 2 propagator term, the phase space for 2-body decay (q 2 ) and the coupling of the photon to the quarks in the meson: q l α Q + α e.g. for the ρ : ω : 1 q 1/q 2 2 (uu dd) one has: Q 2 = [ 1 2 (2 3 ( 1 3 )]2 = (uu + dd) Q 2 = [ 1 2 (2 3 + ( 1 3 )]2 = 1 18 l + φ : ss Q 2 = ( 1 3 )2 = 1 9
20 Vector Meson Decays Indeed, the measured leptonicwidths for the various vector mesons are quite different: Γ e + e (ρ) = 6.8 ± 0.3 Γ e + e (ω) = 0.6 ± 0.02 Γ e + e (φ) =1.37 ± 0.05 But their differences are completely understood within the quark meson charge assignment and composition: Γ e + e (ρ) Σa i Q i 2 =13.6 ± 0.6 Γ e (ω) + e 2 Σa i Q i Γ e (φ) + e 2 Σa i Q i =10.8 ± 0.4 =12.3± 0.5
21 The Quark Parton Model Experimentally hadron states classified by mass, spin and parity and associated into families Baryons with J P = 3 /2 + I 3 = 3 /2 1 1 / / /2 Mass (MeV/c 2 ) Strangeness I = 3 /2 Δ - Δ 0 Δ + Δ I = 1 Σ * Σ *0 Σ *+Ω I = ½ Ξ *- Ξ * I = 0 Ω
22 J P = 3 /2 + Baryons These can all be explained by a basic set of 3 different spin ½ 'quarks' u, d, s combined in sets of 3 i.e. qqq with their spins aligned to give: ½ + ½ + ½ = 3 /2 ( ) with m u m d and m s m u MeV/c 2 Quark B J I I 3 S Q u ⅓ ½ħ ½ +½ 0 ⅔e d ⅓ ½ħ ½ -½ 0 -⅓e s ⅓ ½ħ ⅓e
23 J P = 3 /2 + Baryons d u d s Not known at the time These are strongly decaying resonances
24 J P = 3 /2 + Baryons Regularities in the hadron spectra: baryon J P = 3/2 + decuplet I 3 = -3/2 I 3 = 0 I 3 = +3/2 S = 0 Λ - (ddd) S Λ 0 (ddu) Λ + (duu) Λ ++ (uuu) Q = +2 I = 3/2 S = -1 Σ - (dds) Σ 0 (dus) Σ + (uus) I 3 Q = +1 I = 1 S = -2 Ξ - (dss) Ξ 0 (uss) I = 1/2 Q = 0 S = -3 Ω - (sss) Q = - 1 I = 0 Q e = I + B + S 3 2 = I 3 + Y 2
25 Ground State J P = 1 /2 + Baryons 3 quarks with spins ½ + -½ + ½ = ½ ( ) Λ 0 = 1 / 2 (ud du) s Σ 0 = 1 / 2 (ud + du) s Lightest baryons Decay weakly (except proton which is stable)
26 J P = 1 /2 + Baryons Why are there no uuu, ddd or sss here? With u, d and s quarks there are = 27 combinations From symmetry arguments these can be grouped into 1 (Singlet) + 8 (Octet) + 8 (Octet) + 10 (Decuplet) The 10 are the J P = 3/2 + shown first and one of the 8 are these J P = 1/2 + Baryons are fermions (spin 1 /2, 3 /2,...) so the overall wavefunction must be antisymmetric (A) The space spin flavour part must be symmetric (S) With angular momentum l = 0 the space part ~ (-1) l = +1 so we want spin flavour to be symmetric
27 Resonances There are over 100 known hadrons most are 'Resonances' i.e. Excited States with the same quark content but higher internal angular momentum p, n N *, Δ Λ, Σ Λ *, Σ * K 0, K ± K * π 0, π ± ρ, ω Resonances are formed and decay via the Strong Interaction Lifetimes very short ~ to s By Uncertainty Principle: Width Γ~ 66 MeV At speed of light only travel ~ m leave no tracks in detectors
28 Resonance Detection Detect their presence via stable (or longer lived) particles into which they decay or are produced π + + p Δ ++ (1232) π + + p π + p Δ 0 (1232) π - + p Equivalent CM Energy Beam momentum
29 Discovery of the ω Meson Note: ω (omega) meson not Ω baryon Take bubble chamber pictures of p annihilation- look at events with exactly 5 pions produced p + p π + + π + + π + π + π 0 Plot invariant mass of all possible 3π combinations: π + + π + + π π + π + π + π + + π + + π 0 π + π + π 0 π + + π + π 0 All have net electric charge No structure seen The only neutral combination Sharp peak at M = 790 MeV/c 2, Γ = 12 MeV/c 2
30 Peak Discovery of the ω Meson
31 Dalitz Plots In π + p π + + π + n long lived there could be intermediate short lived states: Delta Δ excited nucleon (Nπ) rho ρ excited pion (ππ) (a) ρ 0 (= ππ resonance) is produced via π exchange (b) Δ (= nπ resonance) is produced via ρ exchange How to tell if either (or both) happen? Dalitz Plot Dick Dalitz
32 Dalitz Plots Measure many thousand 'events' (= collisions of this type) and plot result on 2-D histogram Intermediate states (r 0, D + ) cause clustering of points at specific values of M² giving 'bands' on Dalitz Plot Spread of M² (width of band) gives measure of width of resonance (sometimes increased or masked by experimental resolution)
33 Dalitz Plots Note: Statistical treatment cannot identify specific events So many hadrons discovered like this no longer considered 'fundamental'
34 Discovery of the Ω - Discovery of the Ω (after being postulated to complete the decuplet) 1964, Brookhaven Lab. Ω (sss)
35 uud Combinations For uud Can have flavour symmetric (uud + udu + duu) and spin symmetric ( + + ) or J = 3 /2 flavour antisymmetric (uud) and spin antisymmetric ( ) J = ½ Same quantum numbers so can form linear combinations It's symmetric if it stays the same if you swap any two flavours or spins So there is a uud state in both the J = 3 /2 decuplet (D + ) and the J = ½ octet (p)
36 uuu Combinations For uuu (or ddd or sss) Can have only have flavour symmetric (uuu) and spin symmetric ( ) J = 3 /2 No flavour antisymmetric and spin antisymmetric ( ) possible So there is only a uuu state in the J = 3 /2 decuplet (D ++ ) and not in the J = ½ octet
37 Constructing Baryon States
38 Baryon Masses
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