Quark Modelmodel

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QuarkQuark ModelModel

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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 IsospinIsospin

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 = 1 (n − n + n − n ) 3 2 u d d u ni number of i quarks Isospin works well Masses of u and d quark are almost equal Nuclear and Particle Physics Franz Muheim 2 IsospinIsospin ConservationConservation

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

Production Mass 1232 MeV π + p → ∆++ → π + p Width 120 MeV π − p → ∆0 → π − p π − p → ∆0 → π 0n Isospin addition

+ 1 1 3 3 π p : 1,1 2 , 2 = 2 , 2

− 1 1 1 3 1 2 1 1 π p : 1,−1 2 , 2 = 3 2 ,− 2 − 3 2 ,− 2

0 1 1 2 3 1 1 1 1 π n : 1,0 2 ,− 2 = 3 2 ,− 2 + 3 2 ,− 2

3 3 Matrix element M 3 = 2 H 3 2 1 1 depends on I, not I3 M1 = 2 H1 2 + ++ + M(π p → ∆ → π p) = M 3 − 0 − 1 2 M ()π p → ∆ → π p = 3 M 3 + 3 M1 Cross sections − 0 0 2 2 M()π p → ∆ → π n = 3 M 3 − 3 M1 2 σ ∝ M σ (π + p → ∆++ → π + p)≈ 200 mb ≈ 9x In agreement with σ ()π − p → ∆0 → all ≈ 70 mb ≈ 3x I=3/2 Isospin prediction σ ()π − p → ∆0 → π − p ≈ 23 mb ≈ 1x

Nuclear and Particle Physics Franz Muheim 3 StrangenessStrangeness

Strange Particles Discovered in 1947 Rochester and Butler V, “fork”, and K, “kink”

Production of V(K0, Λ) and K± π − p → K 0Λ τ = O(10−23 s) via strong interaction, 0 + − −10 K → π π τ 0 = 0.89×10 s weak decay K Λ → π − p τ = 2.63×10−10 s Associated Production Λ Strange particles produced in pairs Pais Strangeness S Additive quantum number Gell-Mann Nishijima Conserved in strong and electromagnetic interactions Violated in weak decays Non-zero for Kaons S = 0 : π , p, n, ∆, ... S = 1: K + , K 0 and hyperons S = −1: K − , K 0 , Λ, Σ, ... S = −2 : Ξ

Naturally explained in quark model S = ns − ns Nuclear and Particle Physics Franz Muheim 4 QuarkQuark ModelModel

33 QuarkQuark FlavoursFlavours u,u, d,d, ss 1964 - introduced by Gell-Mann & Zweig Quark Charge Isospin Strange- Q[e] |I, I3 > ness S up (u) +2/3 |½, +½ › 0 Gell-Mann down (d) -1/3 |½, -½ › 0 strange (s) -1/3 |0,0› -1

Zweig

Charge, Isospin and Strangeness Additive quark quantum numbers are related

Q = I3 + ½(S + B) not all independent 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 Nuclearfor and ParticleIsospin Physicsand Strange Franzness Muheim 5 MesonsMesons

Bound q q States Zero net colour charge Zero net baryon number B = +1/3 +(-1/3) = 0 Angular Momentum L 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 L P()qq = Pq Pq ()− 1 = ()+ 1 (− 1)(− 1)L = −1 for L = 0 Total Angular Momentum J r r J = L + S include quark spins S = 0 qq spins anti-aligned ↑↓ or ↓↑ Î JP = 0- Pseudo-scalar mesons S = 1 qq spins aligned ↑↑ or ↓↓ Î JP = 1- Vector mesons Quark flavours ud , us, du, ds, su, sd non-zero flavour states uu, dd , ss zero net flavour states have identical additive quantum numbers Physical states are mixtures

Nuclear and Particle Physics Franz Muheim 6 MesonsMesons

Pseudoscalar Mesons JP = 0-

Kaons: K+, K0, anti-K0, K- Pions: π+, π0, π- Etas: η, η’ Strangeness S

Isospin I3 Vector Mesons JP = 1-

Kstar: K*+, K*0, anti-K*0, K*- rho: ρ+, ρ0, ρ- omega/phi: ω, φ Strangeness S

Isospin I3

Nuclear and Particle Physics Franz Muheim 7 BaryonBaryon DecupletDecuplet

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+ Delta 1232 MeV uuu

Sigma* 1385 MeV

Cascade* 1533 MeV Strangeness S

Omega- 1672 MeV

Isospin

Quark model predicted unobserved state Ω- (sss)

Nuclear and Particle Physics Franz Muheim 8 BaryonBaryon OctetOctet

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+ p,n 938.9 MeV

Sigma 1193 MeV Lambda 1116 MeV Strangeness S Cascade 1318 MeV (Xi)

Isospin

Lightest baryons stable or long-lived Antibaryons ()p, n, ... also form Octet and Decuplet

Nuclear and Particle Physics Franz Muheim 9 DiscoveryDiscovery ofof ΩΩ-

Ω- (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 0 + a Ξ +π 0 0 a Λ + π a γ + γ + − a e e + − a e e − a π p Nuclear and Particle Physics Franz Muheim 10 HadronHadron MassesMasses

Quark Masses u, d & s quark masses light at short distance 2 2 q > 1 GeV mu < md ~ 5 MeV ms ~ 100 MeV Constituent mass is relevant for quark model 2 2 q < 1 GeV 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 S ⋅ S S ⋅ S1 2 ∆E ∝ α 1 2 mm()qq ==mm+1 +mm+2A+ A1 2 S () 1 2 m m m1m2 m1m2 1 2

r r 1 r 2 r 2 r 2 1 S1 ⋅ S2 = ()S − S1 − S2 = ()S(S + 1) − S1 (S1 + 1) − S2 (S2 + 1) 2 2 ⎧ 3 1 ⎪ 1− = S = 1 = 4 4 ⎨ 3 3 Mass [MeV] ⎪ 0 − = − S = 0 ⎩ 4 4 Meson Prediction Experiment Meson Masses π 140 138 mu = md = 310 MeV K 484 496 ms = 483 MeV 2 ρ 780 770 A = (2mu) · 160 MeV Excellent agreement ω 780 782 What about eta(‘)? K* 896 894

Nuclear and Particle Physicsφ Franz Muheim1032 1019 11 HeavyHeavy QuarksQuarks

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

P − 0 + + J = 0 D = cu, D = cd , Ds = cs, P − *0 *+ *+ J = 1 D = cu, D = cd , Ds = cs, 1 − J P = Λ+ = cud 2 c Bottom-quark hadrons

P − + 0 0 J = 0 B = ub, B = db, Bs = sb, P − *+ *0 *0 J = 1 B = ub, B = db, Bs = sb, 1 − J P = Λ0 = bud 2 b Top quark Decays before forming bound states

Nuclearm andt ~ Particle 174 PhysicsGeV discovered Franz Muheim in 1995 at Fermilab 12