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The Static Quark Model Particle Physics

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The Static Quark Model Particle Physics Particle Physics - Chapter 1 The static quark model Paolo Bagnaia last mod. 5-Mar-21 1 − The static quark model 1. Quantum numbers 2. Hadrons : elementary or composite ? 3. The eightfold way 4. The discovery of the Ω– 5. The static quark model 6. The mesons 7. Meson quantum numbers 8. Meson mixing 9. The baryons Caveats for this chapter: • arguments are presented in 10. SU(3) historical order; some of the results are incomplete, e.g. heavy flavors 11. Color are not mentioned here (wait a bit); 12. Symmetries and multiplets • large overlap with [FNSN, MQR, IE]. Paolo Bagnaia - PP - 01 2 short summary • in this chapter [see § 6 for QCD]: no dynamics, only static classification, The roadmap: i.e. algebraic regularities of the states; only hadrons, no photons / leptons; • operators associated with conserved quantum numbers; • modest program, but impressive results: • old attempts of classification; all hadrons are (may be classified as) composites of the same elementary • first successes (multiplets, Ω−); objects, called quarks; • modern classification: the quark dynamics, outside the scope quarks; of this chapter, follow simple group theory: flavor-SU(3); conservation rules; color: color-SU(3); QM and group theory are enough to symmetries; produce the hadron classification; • "construction" of a a a a a a a a a although quarks have not been mesons and baryons; observed, their static properties can be inferred from the particle spectra; → § 6, QCD. • does it mean that quarks are "real"? what really "real" means? [???] Paolo Bagnaia - PP - 01 3 1/4 quantum numbers : the Mendeleev way • Many hadrons exist, with different ( ) quantum numbers (qn). * even if D.I.M. lived long before • Some qn show regularities (spin, parity, …). the advent of QM. • Other qn are more intriguing (mass, ...). an example from ( ) • A natural approach (à la D.I.M. * ): many years ago investigate in detail the qn: [add antiparticles…] the associated operators; the qn conservation; the proliferation of hadrons started in look for regularities; Dmitri Ivanovich Mendeleev the '50s – now they (Дми́трий Ива́нович Менделе́ев) classify the states. are few hundreds … Name π± π0 K± K0 η p n Λ Σ±,0 ∆ Mass (MeV) 140 135 494 498 548 938 940 1116 1190 1232 Charge ±1 0 ±1 0 0 1 0 0 ±1,0 2,±1,0 many other − − − − − Parity + + + + + hadrons Baryon n. 0 0 0 0 0 1 1 1 1 1 3 Spin 0 0 0 0 0 ½ ½ ½ ½ /2 other qn … Paolo Bagnaia - PP - 01 4 2/5 quantum numbers : parity for complete definitions and discussion, [FNSN], [MQR], [BJ]. Definition* : |ψ(q,x,t)> = P|ψ(q,─x, t)> • Gauge theories → Pγ = Pg = −1. W± and Z do NOT conserveℙ parity in their • Particles atℙrest (= ⃗in their own ref⃗ .sys.) interactions, so their intrinsic parity is not defined. are parity eigenstates: |ψ(q,x=0,t)> = P|ψ(q,x=0,t)>. • For a many-body system, P is a multiplicative quantum number : • Eigenvalue P : intrinsic parity ℙ ⃗ ⃗ ψ(x1,x2…xn,t) = P1P2…Pnψ(-x1,-x2…-xn,t). 2 = , P real → P = (± 1). • Particlesℙ ⃗ ⃗ in⃗ a state of orbital⃗ ⃗ angular⃗ • ℙDiracequation → for spin ½ fermions, momentum are parity eigenstates : P(antiparticle) = -P(particle) k Ykm(θ,φ) = (−1) Ykm(θ ,φ+π) → k • Convention: P(quarks/leptons) = +1 → |ψkm (θ,φ)> = (−1) |ψkm(θ,φ)> +1 = P = P = P = P = P = P = …; e− u d s • Therefore, for a two- or a three-particle −1 = P = P = P = P = P = P = … ℙ e+ µ+ τ+ ū d̄ s̄ system: L • Field theory: for spin-0 bosons → Psys(12) = P1P2 (−1) ; P(antiparticle) = +P(particle) : L1+L2 L1 Psys(123) = P1P2P3(−1) . Pπ+ = P = P , … 1 2 * here and in the following slides : = +1 or −1 ? L2 • q : charges + additive qn; … be patient … L 3 • x,a : polar / axial vectors; 1 2 • t : time. ⃗ Paolo Bagnaia - PP - 01 5 3/5 quantum numbers : charge conjugation Definition : changes a particle p into its • Only particles (like π0, unlike K’s) which antiparticle p̄, leaving untouched the space are their own antiparticles,ℂ are and time variablesℂ : eigenstates of , with values C = (± 1) : |p,ψ(x,t)> = C |p̄, ψ(x,t)>. C = +1 for π0, η, η’; ℂ C = −1 for ρ0, ω, φ; • Therefore,ℂ ⃗ under : ⃗ C = −1 for γ. for Z, and are not defined charge q → −q; ℂ baryon n. B→ −B; • However, few particlesℂ areℙ an eigenstate flip lepton n. L → −L ; of ; e.g. |π+> = ─ |π−>. strangeness S → −S; ℂ position x → x ; • ℂWhy define ? E.g. use -conservation in e.-m. decays: momentum p → p; no-flip ⃗ ⃗ π0 → : +1ℂ→ (−1) (−1) ℂ ok; spin s → s. π0 → γ : +1→ (−1) (−1) (−1) no. • is hermitian; its eigenvalues are ±1; BR(π0 → γ ) measured to be ∼10−8. they are multiplicatively conserved in strongℂ and e.m. interactions. see later Paolo Bagnaia - PP - 01 6 4/5 quantum numbers : G-parity • charge conjugation is defined as proposed by Lee and Yang, 1956. |q, B, L, S > = ± |-q, -B, -L, -S >; • -parity is multiplicative : ℂ • therefore, only states q = B = L = S = 0 |nπ+ mπ− kπ0> = ℂ may be -eigenstates (e.g. π0, η, γ, = (−)n+m+k |nπ+ mπ− kπ0>; [π+π−]). |qq̄> = (−)L+S+I |qq̄> → G = (−)L+S+I; ℂ ≡ • is useful: Generalization [G-parity]: 2, where = rotation in the isospin space: -parity is conserved only in strong 2 ℂ ℝ ≡ exp (-i ); interactions ( and isospin are valid); 2ℝ 2 I-I3 it produces selection rules (e.g. a 2|I, I3> = (−) |I, −I3> ℝ ℂ π |q, x, t, I, I > = (−)I-I3|-q, x, t, I, -I >; decay in odd/even number of 's is ℝ2 3 3 allowed/forbidden). • hasℝ more⃗ eigenstates than ⃗ ; e.g.: • e.g. ω(782) is IG(JPC) = 0−(1− −) : π± − π | > = | >; BR (ω → π+π−π0) = (89.2±0.7)% 0 0∓ ℂ |π > = + |π >; + − ℂ BR (ω → π π ) = ( 1.5±0.1)% π± π 2| > = + | >; opposite to the obvious phase-space ℂ 0 ∓0 2|π > = − |π >. predictions (more room for 2π than 3π ℝ • therefore: decay). ℝ ±,0 ±,0 ±,0 |π > = 2 |π > = − |π >. • [see also J/ψ decay]. Paolo Bagnaia - PP -ℂ01 ℝ 7 5/5 quantum numbers : some proofs • See [FNSN, §7]. • For a qq̄ (or particle-antiparticle) state: |q,x,s -q,-x,-s> = • Be |q, x, a> an eigenstate of a generic = C |-q,x,s, q,-x,-s> = ℙ ℂ ⃗ ⃗ ⃗ ⃗ "flip" operator ( = , , , [spin- = C P |-q,-x,s, q,x,-s> = ⃗ ℙ ⃗ ⃗ ⃗ ⃗ flip], [time-reversal]). = C P |-q,-x,-s,q,x,s> = ℂ ℙ ⃗ ⃗ ⃗ ⃗ the spin s is an = C P |q,x,s, -q,-x,-s>. axial vector (a) ⃗ ⃗ ⃗ ⃗ • From their definition: → = CP = ± 1; ⃗ 2 → −1 ⃗ ⃗ ⃗ ⃗ = = . → = ± -1 -1 = ± . ℙ ℂ • an eigenstate of has eigenvalue K: • for suchℂ a state ℙ[BJ, 335 ℙ]: |q, x, a> = K |±q, ±x, ±a>; L+1 P = (−1) (see before); 2 |q, x, a> = K2 |q, x, a> = |q, x, a>; ⃗ ⃗ = (−1)S+1 (Pauli principle); K2 = 1 → K = real = ±1. ⃗ ⃗ ⃗ → C = P × = (−1)L+S; → G = (-1)L+S+I (see before). • P(γ) = −1 (vector potential [FNSN, 7.2.2]) : eigenvalue of ; S : value of the spin. Paolo Bagnaia - PP - 01 8 1/3 hadrons : "elementary" or composite ? Over time the very notion of "elementary (???) particle" 1890 1900 1910 1920 entered a deep crisis. e− p The existence of (too) many 1920 1930 1940 1950 hadrons was seen as a contradiction with the n e+ µ± π± K± elementary nature of the 1950 1960 fundamental component of matter. 0 0 ± 0 0 π Λ Σ p̄νe Σ Λ ρ νµ α2 and many more K0 ∆ Ξ n̄ Ξ0 ω φ η' It was natural to interpret the − � η f Ω hadrons as consecutive K* − resonances of elementary components. too many hadronic states: resonances ? The main problem was then to the figure shows the particle discoveries from 1898 to measure the properties of the the '60s; their abundance and regularity, as a function components and possibly to of quantum numbers like charge and strangeness, were observe them. suggesting a possible sequence, similar to the Mendeleev table [FNSN]. [and the leptons ? ... see later] Paolo Bagnaia - PP - 01 9 2/3 hadrons : "elementary" or composite ? 1949 : E.Fermi and C.N. Yang proposed that ALL the resonances were bound state p-n. Chen-Ning Yang Shoiki Sakata Enrico Fermi (杨振宁 - 楊振寧, (坂田 昌一, Yáng Zhènníng) Sakata Shōichi) 1956 : Sakata extended the Fermi- Yang model including the Λ, to account for strangeness : all hadronic states were then composed by (p, n, Λ) and their antiparticles. Paolo Bagnaia - PP - 01 10 3/3 hadrons : "elementary" or composite ? 1961 : M. Gell-Mann and Y. Ne’eman mention an internal structure. The (independently) proposed a new name was invented by Gell-Mann classification, the Eightfold Way, and comes from the "eight based on the symmetry group SU(3). commandments" of the Buddhism. The classification did NOT explicitly The Quark Idea bottom (up, down, strange) charm 1960 1970 1980 1990 Murray Yuval Ne'eman ± (יובל נאמן) Gell-Mann J/ψτ D ϒ Λcηc B W Z top ψ'χc ϒ' Σc Ds ψ" ϒ" Ξc 1990 2000 Bs t Many more hadrons Λb have been discovered. Warning : "t" is a quark, not a hadron (in modern language). Paolo Bagnaia - PP - 01 11 1/5 the Eightfold Way: 1961-64 All hadrons (known in the '60s) are classified in the plane (I3 − Y), (Y = strong hypercharge): Q=0 Q=+1 mesons Y JP = 0− K0 K+ I3 = Iz = third component of isospin; +1 Y = B + S [baryon number + strangeness].
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