Quark Model Outline Outline Hadrons Hadrons known in 1960 Isospin, Strangeness Quark Model 3 Flavours u, d, s Mesons Pseudoscalar and vector mesons Baryons Decuplet, octet Hadron Masses Spin-spin coupling Heavy Quarks Charm, bottom, Heavy quark Mesons Top quark Motivation for Quark Model Particle “Zoo” proliferates “ … the finder of a new particle used to be rewarded by a Nobel prize, but such a discovery ought to be punished by a $10000 fine” Lamb, 1955 Nuclear and Particle Physics Franz Muheim 1 Isospin Nucleons Proton and neutron have almost equal mass Strong nuclear force is charge independent Vpp≈ Vpn ≈ Vnn Isospin p and n form part of single entity with isospin ½ analogous to ↑ and ↓ of spin ½ Isospin I is conserved in strong interactions Addition by rules of angular momentum Isospin Multiplets Useful for classification of hadrons, see slide 1 2I+1 states in a isospin muliplet |I, I3 > Quark Model Gives natural explanation for Isospin I 3 = 12 (nu − nd + nd − nu ) ni number of i quarks Isospin works well Masses of u and d quark are almost equal Nuclear and Particle Physics Franz Muheim 2 Isospin Conservation Conservation Law Isospin I is conserved in strong interactions Allows to calculate ratios of cross sections and branching fractions in strong interactions Delta(1232) Resonance Mass 1232 MeV Width 120 MeV Production π + p → ∆+ + → π + p π − p → ∆0 → π − p π − p → ∆0 → π 0 n Isospin addition π+ p: 1,1 π−p: 1,−1 π 0n : 1,0 1 2 , 12 = , 32 , 12 = 1 3 3 2 ,− 12 − 2 1 3 2 ,− 12 ,− 12 = 2 3 3 2 ,− 12 + 1 1 3 2 ,− 12 1 2 1 2 3 2 Matrix element M3 = 3 2 H3 3 2 depends on I, not I3 M1 = 1 2 H1 1 2 2 3 1 ( ) M (π p → ∆ → π p ) = M + M Cross sections M (π p → ∆ → π n ) = M − M 2 σ∝ M σ (π p → ∆ → π p ) ≈ 200 mb ≈ 9x In agreement with σ (π p → ∆ → all ) ≈ 70 mb ≈ 3x I=3/2 Isospin prediction σ (π p → ∆ → π p ) ≈ 23 mb ≈ 1x M π + p → ∆+ + → π + p = M 3 − − 0 + 0 ++ − − Nuclear and Particle Physics − 0 Franz Muheim 1 3 2 3 3 2 3 3 + 0 0 − 3 1 Strangeness Strange Particles Discovered in 1947 V, “fork”, and K, “kink” Rochester and Butler Production of V(K0, Λ) and K± π − p → K 0 Λ τ = O (10 −23 s ) via strong interaction, K 0 → π +π − τ K = 0.89 × 10 −10 s weak decay 0 Associated Production Λ → π − p τ Λ = 2.63 × 10 −10 s Strange particles produced in pairs Pais Strangeness S Additive quantum number Gell-Mann Nishijima Conserved in strong and electromagnetic interactions Violated in weak decays S = 1: K +, K0 Non-zero for Kaons S = 0 : π , p, n, ∆ , ... S = −1 : K − , K 0 , Λ , Σ , ... S = −2 : Ξ and hyperons S = ns − ns Naturally explained in quark model Nuclear and Particle Physics Franz Muheim 4 Quark Model 33 Quark Quark Flavours Flavours u, u, d, d, ss 1964 - introduced by Gell-Mann & Zweig Quark Charge Q [e] Isospin |I, I3 > Strangeness S up (u) +2/3 |½, +½ › 0 down (d) -1/3 |½, -½ › 0 strange (s) -1/3 |0,0› -1 Gell-Mann Zweig Charge, Isospin and Strangeness Additive quark quantum numbers are related not all independent Q = I3 + ½(S + B) Gell-Mann Nishijima predates quark model valid also for hadrons Baryon number B quarks B = +1/3 anti-quarks B = -1/3 Hypercharge Y = S + B is useful quantum number Quark model gives natural explanation Isospin and Strangeness Nuclearfor and Particle Physics Franz Muheim 5 Mesons Bound qq States Zero net colour charge Zero net baryon number Angular Momentum L B = +1/3 +(-1/3) = 0 For lightest mesons Ground state L = 0 between quarks Parity P Intrinsic quantum number of quarks and leptons P=+1 for fermions P=-1 for anti-fermions P (qq ) = Pq Pq (− 1) L = (+ 1)(− 1)(− 1) = −1 for L = 0 L Total Angular Momentum J r r J = L+ S S=0 Î J P = 0S=1 Î J P = 1- include quark spins qq spins anti-aligned ↑↓ or ↓↑ Pseudo-scalar mesons qq spins aligned ↑↑ or ↓↓ Vector mesons Quark flavours non-zero flavour states zero net flavour states uu , dd , ss have identical additive quantum numbers Physical states are mixtures ud , us , du , ds , su , sd Nuclear and Particle Physics Franz Muheim 6 Mesons Pseudoscalar Mesons JP = 0- Strangeness S Kaons: K+, K0, anti-K0, KPions: π+, π0, πEtas: η, η’ Isospin I3 Strangeness S Vector Mesons JP = 1Kstar: K*+, K*0, anti-K*0, K*rho: ρ+, ρ0, ρomega/phi: ω, φ Isospin I3 Nuclear and Particle Physics Franz Muheim 7 Baryon Decuplet Baryon Wavefunction Ψ(total) = Ψ(space) Ψ(spin) Ψ(flavour) Ψ(colour) Space symmetric - L = 0 Flavour symmetric, e.g. uuu, (udu+duu+uud)/√3 Spin symmetric all 3 quarks aligned → S = 3/2 Colour antisymmetric Total antisymmetric - obeys Pauli Exclusion Principle Baryon Decuplet JP = 3/2+ <Mass> Delta Strangeness S uuu 1232 MeV Sigma* 1385 MeV Cascade* 1533 MeV Omega- 1672 MeV Isospin Quark model predicted unobserved state Ω- (sss) Nuclear and Particle Physics Franz Muheim 8 Baryon Octet Baryon Wavefunction Ψ(space) symmetric (L = 0) Ψ(colour) antisymmetric Mixed symmetric Ψ(spin, flavour) Flavour mixed symmetric: e.g. (ud - du) u/√2 Spin as flavour: e.g. (↑↓ - ↑↓) ↑/√2 Spin-flavour e.g. (u↑d↓ - d↑u↓ - u↓d↑ + d↓u↑) u↑/√6 Symmetrisation by cyclic permutations Ψ(proton, s=+½) = ( 2u↑u↑d↓ - u↑u↓d↑- u↓u↑d↑ +2d↓u↑u↑ - d↑u↑u↓- d↑u↓u↑ +2u↑d↑u↓ - u↑d↓u↑- u↓d↑u↑) /√18 Baryon Octet JP = ½+ Strangeness S <Mass> 938.9 MeV p,n Sigma 1193 MeV Lambda 1116 MeV Cascade 1318 MeV (Xi) Isospin Lightest baryons Antibaryons ( p, n , ...) Nuclear and Particle Physics stable or long-lived also form Octet and Decuplet Franz Muheim 9 Discovery of ΩΩ- (sss) Hyperon Hyperon - baryon with at least one s quark Quark model predicted existence and mass Missing member of baryon decuplet JP = 3/2+ discovered 1964 at Brookhaven K- beam onto hydrogen target Bubble Chamber detector K − + p → .Ω − + K − + K 0 a Ξ 0 +π + a Λ0 + π 0 aγ +γ a e+e− a e+e− Nuclear and Particle Physics aπ−p Franz Muheim 10 Hadron Masses Quark Masses u, d & s quark masses light at short distance mu < md ~ 5 MeV ms ~ 100 MeV q2 > 1 GeV2 Constituent mass is relevant for quark model q2 < 1 GeV2 mu = md ~ 300 MeV ms ~ 500 MeV Meson Masses m(K) > m(π) due to ms > mu, md m(ρ) > m(π) same quark content e.g. ρ+, π+: (u-dbar) Mass difference is due to quark spins Chromomagnetic Mass Splitting Spin-spin coupling of quarks S1 = S2 = 1/2 analogous to hyperfine splitting in el. mag. interaction r r r r r Sr ⋅ S 1 2 S1 ⋅ S 2 S m q q m m = + + 1 ⋅ S2 1 m +2 A A ∆E ∝ α S m (qq ) = m1 + 2 m2 m1 m 2 m1mm 21 r r r2 r2 1 1 r S1 ⋅ S 2 = S 2 − S1 − S 2 = ( S ( S + 1) − S1 ( S1 + 1) − S 2 ( S 2 + 1)) 2 2 3 1 ⎧ S =1 ⎪ 1− 4 = 4 =⎨ Mass 3 3 ⎪ 0− = − S=0 4 4 ⎩ ( ) ( ) Meson Masses mu = md = 310 MeV ms = 483 MeV A = (2mu)2 · 160 MeV Excellent agreement What about eta(‘)? Nuclear and Particle Physics [MeV] Meson Prediction Experiment π 140 138 K 484 496 ρ 780 770 ω 780 782 K* 896 894 φFranz Muheim 1032 1019 11 Heavy Quarks Charm and bottom quarks Charmonium (c-cbar) --- see QCD lecture 1977 Discovery of Upsilon States Interpretation is Bottomonium (b-bar) Spectroscopy Charmonium and Upsilon mc ~ 1.1 … 1.4 GeV mb ~ 4.1 … 4.5 GeV Heavy-light Mesons and Baryons Charmed (c-quark) hadrons J P = 0− D 0 = cu , D + = cd , Ds+ = cs , J P = 1− D *0 = cu , D *+ = cd , Ds*+ = cs , 1 J = 2 − Λ+c = cud P Bottom-quark hadrons J P = 0− B + = ub , B 0 = db , Bs0 = sb , J P = 1− B * + = ub , B *0 = db , Bs*0 = sb , 1 J = 2 P − Λ0b = bud Top quark Decays before forming bound states 174 Physics GeV discovered in 1995 at Fermilab mt ~Particle Nuclear and Franz Muheim 12
