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Quarks in Hadrons

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Quarks in Hadrons Quarks in Hadrons • Charm and beauty • Quarkonium • Baryon decuplet • Quark spin and color • Baryon octet • Pseudoscalar mesons • Vector mesons • Other tests of the quark model • Mass relations and hyperfine splitting • EM mass differences and isospin symmetry • Baryon magnetic moments • Heavy quark mesons • The top quarks Physics 661, Chapter 4 1 Quark Model • Patterns of observed particles led to proposal in early 1960’s that hadrons were composed of quarks – u, d, and s (at that time) • Were quarks real? – exhaustive searches for free quarks were unsuccessful • With the discovery of “confined” quarks in the 1970’s it was realized that quarks truly exist, but cannot be freed Physics 661, Chapter 4 2 Eightfold Way - 1961 Murray Gell-Mann and Yuval Ne’eman Physics 661, Chapter 4 3 Internal Structure - 1964 • SU(3) • Murray Gell-Mann and George Zweig Physics 661, Chapter 4 4 Charm and Beauty • In 1970, three quarks explained all known hadrons. • During the 1970’s, two more quarks were discovered – charm, 1974 – bottom, 1977 • In 1995, the top quark was finally observed * Flavor I I3 S C B T Q/e u 1/2 1/2 0 0 0 0 +2/3 d 1/2 -1/2 0 0 0 0 -1/3 s 0 0 -1 0 0 0 -1/3 c 0 0 0 1 0 0 +2/3 b 0 0 0 0 -1 0 -1/3 t 0 0 0 0 0 1 +2/3 Physics 661, Chapter 4 5 Discovery of Charm • 1974 – two complementary experiments – SLAC: e+e- → ψ → hadrons → e+e-, µ+µ- – BNL: p + Be → J/ψ + anything → e+e- Physics 661, Chapter 4 6 Discovery of Charm • 1974 – SLAC: e+e- → ψ → hadrons hadrons → e+e-, µ+µ- width dominated by experimental resolution µ+µ- Also observed second peak at 3.7 GeV (ψʹ ) e+e- Physics 661, Chapter 4 7 Discovery of Charm • 1974 – BNL: p + Be → J/ψ + anything → e+e- Physics 661, Chapter 4 8 Discovery of Charm • Total width from integral of the cross section: i j ⌠σ(E) dE ⌡ e+e- Physics 661, Chapter 4 9 Discovery of Charm • Total width from integral of the cross section: i j ⌠σ(E) dE Γi = Γj = Γee ⌡ Let x = 2(E-ER)/Γ, dx = 2dE/Γ, dE=(Γ/2) dx 2J+1=3 and 2sa+1=2sb+1=2 2 2 ⌠σ(E) dE = 3π/2 λ (Γee/Γ) Γ ⌠ dx 2 ⌡ ⌡(1+x ) π Physics 661, Chapter 4 10 Discovery of Charm • Total width from integral of the cross section: i j 2 2 2 ⌠σ(E) dE = 3π /2 λ (Γee/Γ) Γ = 800 nb MeV ⌡ Γee/Γ = 0.06 λ = hc/pc = 197 MeV-fm/1550 MeV Γ = 0.093 MeV (very narrow) Ref: Particle Data Group Physics 661, Chapter 4 11 Discovery of Charm • Vector mesons: Γ . ρ(776 MeV) 150 MeV ω(784 MeV) 8.4 MeV J/ψ(3100 MeV) 0.093 MeV Γee ρ(776 MeV) 6.8 keV φ(1020 MeV) 1.4 keV ω(784 MeV) 0.6 keV J/ψ(3100 MeV) 5 keV Physics 661, Chapter 4 12 Discovery of Charm • Shape of the resonance was consistent with Jp = 1- – interference of the direct and virtual photon channels – ratio of decay modes (such as ρ0π0 and ρ-π+) shows J/ψ is I=0 state Physics 661, Chapter 4 13 Discovery of Charm • ψ(3700) → ψ(3100) π+ π- Physics 661, Chapter 4 14 Discovery of Charm Ref: Particle Data Group (e+ e- + -) R = σ (e+ e- → hadrons) / σ (e+ e- → µ+ µ-) σPT → µ µ PT = 4πα2 / 3s + - + - – consider e e → hadrons as e e → QQ, = 87nb / s(GeV2) summed over all quarks (point cross section) Physics 661, Chapter 4 15 Discovery of Charm • Narrowness of ψ(3100) and ψ(3700) explained by QCD, 3G exchange required Colorless, and C parity odd Physics 661, Chapter 4 16 Discovery of Charm • For heavier charmionium states, open charm decay modes open up -> broader resonances Physics 661, Chapter 4 17 Discovery of Charm ψ(3100) Γ = 0.093 MeV ψ(3700) Γ = 0.30 MeV Open charm threshold ψ(3770) Γ = 27 ± 1 MeV ψ(4040) Γ = 80 ± 10 MeV ψ(4160) Γ = 103 ± 8 MeV ψ(4415) Γ = 62 ± 20 MeV Ref: Particle Data Group Physics 661, Chapter 4 18 Discovery of Beauty • 1977 – history repeats itself – another narrow resonance – this time at Fermilab – p+Be -> µ+µ- + anything p+Cu -> µ+µ- + anything p+Pt -> µ+µ- + anything 400 GeV protons Physics 661, Chapter 4 19 Discovery of Beauty • Electron-positron colliders refine measurements: – DORIS at DESY, CLEO at Cornell Physics 661, Chapter 4 20 Discovery of Beauty Ref: Particle Data Group Physics 661, Chapter 4 21 Discovery of Beauty Mass Γee (keV) Γ (MeV) (MeV) ϒ(1S) 9460 1.34 0.054 ϒ(2S) 10023 0.61 0.032 ϒ(3S) 10355 0.020 ϒ(4S) 10579 0.27 20.5 ± 2.5 Ref: Particle Data Group Physics 661, Chapter 4 22 Quarkonium and Positronium Physics 661, Chapter 4 23 Positronium • e+e- -> γγ – τ = 1.25 x 10-10 sec – singlet state – even ang. momentum -> J=0 -> C = (-1)L+S =(-1)0 = +1 – C = (-1)nγ -> C = +1 • e+e- -> γγγ – τ = 1.4 x 10-7 sec – triplet state -> J=1 -> C = (-1)L+S = (-1)0+1 = -1 – C = (-1)nγ -> C = -1 Physics 661, Chapter 4 24 C & P of e+e- system • Interchange of particles – Spin symmetry (-1)S+1 eg. α(s=0, s3=0) = 1/√2 (éê−êé) – Spatial symmetry (-1)L+1 • Recall opposite intrinsic parities of e+ and e- – So – total symmetry is (-1)L+S • Interchange of space and spin is equivalent to Charge Conjugation Physics 661, Chapter 4 25 Positronium • Principal energy levels from non-relativistic Schroedinger equation in a Coulomb potential 2 2 2 En = - α µc / 2n , µ = m/2 • Relativistic corrections: – spin-orbit • S, P, D – spin-spin 3 1 • S1, S0, …… – these are about the same size in positronium: • ΔE ~ α4mc2 / n3 Physics 661, Chapter 4 26 Positronium • Lifetimes: – Two photon • two photons means rate is to order α2 • overlap of wavefunctions at origin to annihilate – |ψ(0)|2 = 1/ π a3 – a = 2 h/(2π mc α) • exact result: – Γ = α5 m/2 – Three photon • rate at higher order – Γ = 2(π2-9) α6 m/(9π) Physics 661, Chapter 4 27 Positronium • Excellent agreement on Theory and Experimental results for lifetimes and energy levels – (see textbook) Physics 661, Chapter 4 28 Quarkonium • Similar energy levels to positronium 2S+1 n L J Physics 661, Chapter 4 29 Quarkonium • Different potential: – positronium -> Coulomb – V = - α / r – quarkonium -> potential from QCD – expected potential of the form – V = -(4/3) αs / r + kr Physics 661, Chapter 4 30 Quarkonium • αs = 0.2 -1 • k ≈ 1 GeV fm Physics 661, Chapter 4 31 Quarkonium • For the heavy quarks, a non-relativistic approximation is valid: – p/m ~ 0.13 • Fine structure is of order αs, and therefore coarser than for positronium, as observed Physics 661, Chapter 4 32 Baryon Decuplet • Lightest spin 3/2 baryons isospin (3rd comp.) strangeness Before the Ω- was discovered, it (and its properties) were predicted by this pattern Physics 661, Chapter 4 33 Discovery of the Ω- Three, sequential decays of the strange quark Ω- = sss Ξ0 = ssu Λ = sud Physics 661, Chapter 4 34 Baryon Decuplet • Notice the masses M(Δ) = 1232 M(Σ) = 1384 = M(Δ) + 152 M(Ξ) = 1533 = M(Σ) + 149 M(Ω) = 1672 = M(Ξ) + 139 We see an orderly increase of mass with number of strange quarks Physics 661, Chapter 4 35 Group Theory • Quarks are fundamental representations of the group SU(3) 3 ⊗ 3 = 6 ⊕ 3 3 ⊗ 3 = 8 ⊕ 1 3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10 1 is anti-symmetric under interchange of two quarks 10 is symmetric under interchange of two quarks 8’s are mixed under interchange of two quarks Physics 661, Chapter 4 36 Group Theory – Combining Multiplets • Multiplet Labels – SU(n) -> (n-1) non negative integers (α, β, γ, …) so for SU(3) (α, β) (lengths of top, and bottom) (1,0) (1,1) (3,0) (0,1) Physics 661, Chapter 4 37 Group Theory – Combining Multiplets • Number of Particles in the Multiplet – For SU(3) N = (α+1) (β+1) (α+β+2)/2 (1,1) (3,0) N=8 N=10 N=3 N=3 (1,0) (0,1) Physics 661, Chapter 4 38 Group Theory – Combining Multiplets • Young’s diagrams – Top row is α boxes past end of 2nd row – 2nd row is β boxes past end of 3rd row (1,0) (1,1) (3,0) (0,1) Physics 661, Chapter 4 39 Group Theory – Combining Multiplets • Coupling multiplets – Sequence of letters okay if everywhere at least as many of an early letter (eg. a) has occurred as a later letter (eg. b) – In one diagram, replace boxes by a’s (1st row), b’s (2nd row), etc. – So becomes a becomes a a b becomes a a a becomes a b Physics 661, Chapter 4 40 Group Theory – Combining Multiplets 3 x 3 = x a = a + = (2,0) + (0,1) = 6 + 3 a 3 x 3 = 6 + 3 x = + Physics 661, Chapter 4 41 Group Theory – Combining Multiplets 3 x 3 = x a = ( a + ) b a b = a + = (1,1) + (0,0) = 8 + 1 b a b + = + Physics 661, Chapter 4 42 Group Theory – Combining Multiplets 3 x 3 x 3 = (6 + 3) x 3 = ( + ) x a = a + + a + a a = (3,0) + (1,1) + (1,1) + (0,0) = 10 + 8 + 8 + 1 x x = + + + Physics 661, Chapter 4 43 Quark Spin and Color • Consider the Δ++ – spin 3/2 (uuu) – therefore u↑u↑u↑ • now, this appears to violate the Pauli principle – two or more identical fermions cannot exist in the same quantum state • resolution, another quantum number (color) and each of the u quarks have a different value: u↑(red)u↑(green)u↑(blue) and we have to aniti-symmetrize the color u↑u↑u↑ (rgb-rbg+brg-bgr+gbr-grb) Physics 661, Chapter 4 44 Quark Spin and Color • We also know from the rate of decay of the π0 that there are three colors 0 2 Γ(π →γγ) = 7.73 eV (Nc/3) Γ(observed) = 7.76 ± 0.6 eV Nc = 2.99 ± 0.12 + - • Also the rate of e e → hadrons tells us Nc = 3 Physics 661, Chapter 4 45 π0 Lifetime 0 2 • Theory: Γ(π → γγ) = 7.73 eV (Nc/3) τ = h/Γ = 197 MeV-fm/(3 x 1023 fm/s Γ) -16 = 6.6 x 10 s / Γ(eV) -17 2 = 8.5 x 10 s (3/Nc) -2 2 d = γ β c τ = 2.6 x 10 µm (p/m) (3/Nc) 2 Suppose p = 5 GeV, then d ≈ 1 µm (3/Nc) Physics 661, Chapter 4 46 π0 Lifetime Experiment p + A → π0 + X γ γ γ Proton beam π0 γ d d ~ 1 µm for p = 5 GeV and Nc=3 Physics 661, Chapter 4 47 π0 Lifetime Experiment Technique – use thin foils for target, convert photons into e+e- pairs, and count as function of target thickness (t) γ Proton beam 0 π e- γ d e+ t Physics 661, Chapter 4 48 π0 Lifetime Experiment 1.
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