Physics 56400 Introduction to Elementary Particle Physics I Lecture 11 Fall 2019 Semester Prof. Matthew Jones Hadronic Isospin Multiplets 푝 • Nucleons: 푛 휋+ • Pions: 휋0 휋− ∆++ ∆+ • Delta resonances: ∆0 ∆− Baryon Antiparticles • Baryons have baryon number B=+1 • Anti-baryons will have baryon number B=-1 • Baryon number is conserved in strong interactions • Production of anti-protons must proceed as follow: 푝 + 푝 → 푝 + 푝 + 푝 + 푝ҧ or 푝 + 푛 → 푝 + 푛 + 푝 + 푝ҧ • What is the minimum beam energy needed to make anti-protons in a fixed target experiment? Baryon Antiparticles Beam proton: m 푝푏 = 퐸푏, 푝Ԧ푏 Target proton: 푝푡 = 푚푝, 0 Final state: 푝푓 = 4퐸푝, 4푝Ԧ푝 2 2 푝Ԧ 퐸푏 − 푚푝 푝Ԧ = 푏 = 푝 4 4 2 2 퐸푏 + 푚푝 2 퐸 − 푚푝 퐸 = = 푝Ԧ + 푚2 = 푏 + 푚2 푝 4 푝 푝 16 푝 2 2 2 2 퐸 + 2퐸푏푚푝 + 푚푝 퐸 + 15푚푝 푏 = 푏 16 16 퐸푏 = 7푚푝 = 6.57 GeV 푇푏 = 6푚푝 = 5.63 GeV (kinetic energy of beam proton) Anti-protons Brookhaven Cosmotron (1953) 퐸 = 3 GeV Berkeley Bevatron (1954) 퐸 = 6 GeV Anti-proton discovered in 1955 by Chamberlain and Segré. Nobel prize in 1959. Baryon Anti-particles • Using the Dirac sea idea, an anti-proton is equivalent to the absence of a negative-energy proton. • If a negative energy proton has isospin 퐼3 = +1/2 then its absence looks like 퐼3 = −1/2. 푛ത 1/2 • Anti-protons: = 푝ҧ −1/2 ∆ത+ 3/2 ത0 1/2 • Anti-deltas: ∆ = ∆ത− −1/2 ∆ത−− −3/2 • Anti-baryons also have odd-parity. • Pions: – 휋+ and 휋− are anti-particles, – 휋0 is its own anti-particle. Vector Mesons • 푒+푒− collisions can produce states of spin-1: • The 휌 770 multiplet contains three charge states: 휌+ 휌0 휌− • They form an isospin triplet (I=1). • They have odd parity. • They decay to pairs of pions: 휌+ → 휋+휋0 휌0 → 휋+휋− Strange Mesons (d) (e) (b) (a) (c) a) Incident proton b) Interaction with target nucleus c) Outgoing proton d) Outgoing charged particle e) Decay of a long-lived neutral particle, 푉0 → 휋+휋− Strange Mesons • First observed in high-altitude cloud chamber experiments • Strange mesons were easily produced in strong interactions, but only in pairs. • Strange mesons had long lifetimes so they must decay weakly even though there were no leptons in the final state. • Proposal: – These particles carry a new quantum number (strangeness) – Strangeness is conserved in strong interactions – Strange particles are produced in particle/anti-particle pairs to conserve strangeness – Strange particles cannot decay strongly – The lightest positively charged strange meson assigned S=+1 Strange Hadrons • Types of strange particles: – Neutral mesons: 푉0 → 휋+휋− – Charged mesons: 휏+ → 휋+휋−휋+ Parity -1 휃+ → 휋+휋0 Parity +1 – Neutral baryons: Λ → 푝휋− • The 휏-휃 puzzle: – The 휏+ and 휃+ had exactly the same charge, mass, lifetime (they seemed to be the same particle) – They decayed to final states with different parity (they seemed to have opposite quantum numbers) – Parity must not necessarily be conserved in weak decays • Both are now know as the charged kaon, 퐾+ Strange Hadrons + − • Charged kaons: 퐾 and 퐾 , 푚퐾+ = 493.7 MeV 0 ഥ0 • Neutral kaons: 퐾 and 퐾 , 푚퐾0 = 497.6 MeV • These seemed to form two isospin doublets: 퐾+ 퐾ഥ0 퐾0 퐾− • The neutral Λ is an isospin singlet, 푚Λ = 1116 MeV • A triplet of strange baryons was observed: + 0 − Σ , Σ , Σ with similar mass, 푚Σ~1193 MeV • Decays: – Σ+ → 푝휋0, 푛휋+ with lifetime 80 ns Not particle/anti-particle! – Σ− → 푛휋− with lifetime 148 ns – Σ0 → Λ훾 with lifetime 10-19 seconds (electromagnetic decay conserves strangeness) Doubly Strange Baryons 퐾−푝 → 퐾+휩− 휩− → Λ휋− Λ → 푝휋− The Ξ− has S=-2 and decays weakly. The Ξ is sometimes called a “cascade”. Strange Resonances • Strange vector mesons: 퐾∗(892) 퐾∗+ → 퐾+휋0, 퐾0휋+ 퐾∗0 → 퐾+휋−, 퐾0휋0 • They also (rarely) decay electromagnetically: 퐾∗ → 퐾훾 (퐵퐹 ~ 10−3) • Vector mesons with no strangeness: 휙 1020 → 퐾+퐾−, 퐾0퐾ഥ0 휔 782 → 휋+휋−휋0, 휋0훾 • Strange baryon resonances: Σ∗−, Σ∗0, Σ∗+ Meson Multiplets • Plot 퐼3 on the x-axis and strangeness on the y-axis: (charge) Baryon Multiplets • Do the same for the baryons: 푚 Σ∗ − 푚 Δ = 153 MeV 푚 Ξ∗ − 푚 Σ∗ = 145 MeV Therefore, the Ω− mass is estimated to be about 푚 Ω− = 푚 Σ∗ + 150 MeV = 1530 MeV + 150 MeV = 1680 MeV Predicted independently by Gell-Mann and Ne’eman in 1961. The Omega Baryon Discovered in 1964 using an 80-inch bubble chamber at the 33 GeV Brookhaven AGS. 풎 휴− = ퟏퟔퟕퟐ 퐌퐞퐕 S=-3 Interpreting Hadron Multiplets • Group theoretic approach (Gell-Mann): ퟑ ⊗ ퟑഥ = ퟖ ⊕ ퟏ ퟑ ⊗ ퟑ ⊗ ퟑ = ퟏퟎ ⊕ ퟖ ⊕ ퟖ ⊕ ퟏ • Constituent particle approach (Zweig): – Three types of quarks: 푢, 푑, 푠 – Mesons are made of quark/anti-quark pairs – Baryons are made of three quarks – Quarks have charge 푞푢 = +2/3, 푞푑 = 푞푠 = − 1Τ3. – Quarks have spin 1/2 Quark Composition of the Mesons • Pseudoscalar mesons are |푞 ↑ 푞′ഥ ↓ൿ states • Vector mesons are |푞 ↑ 푞′ഥ ↑ൿ states Meson Quarks Meson Quarks 휋+ (푢푑ҧ) 휌+ (푢푑ҧ) 휋0 푢푢ത or 푑푑ҧ 휌0 푢푢ത or 푑푑ҧ 휋− (푑푢ത) 휌− (푑푢ത) 퐾+ (푢푠ҧ) 퐾∗+ (푢푠ҧ) 퐾0 (푑푠ҧ) 퐾∗0 (푑푠ҧ) 퐾ഥ0 (푠푑ҧ) 퐾ഥ∗0 (푠푑ҧ) 퐾− (푠푢ത) 퐾∗− (푠푢ത) 휂 ? 휔 ? 휂′ ? 휙 ? Quark Composition of the Baryons 1 ۧ↓↑↑ | Spin Τ2 baryons are combinations with • 3 ۧ↑↑↑ | Spin Τ2 baryons have • J=3/2 Baryon Quarks J=1/2 Baryon Quarks Δ++ (푢푢푢) 푝 (푢푢푑) Δ+ (푢푢푑) 푛 (푢푑푑) Δ0 (푢푑푑) + Σ (푢푢푠) Δ− (푑푑푑) 0 Σ (푢푑푠) Σ∗+ (푢푢푠) − Σ (푑푑푠) Σ∗0 (푢푑푠) 0 Ξ (푢푠푠) Σ∗− (푑푑푠) − Ξ (푑푠푠) Ξ∗0 (푢푠푠) Λ ? Ξ∗− (푑푠푠) Ω− (푠푠푠) The Quark Model • We can start to think of mesons as bound states of quarks (like the Hydrogen atom) • Spins that are ↑↑ are excited states of ↑↓ • The potential energy function must be of the form 푠Ԧ ∙ 푠Ԧ 푉 = 푉 푟 + 휅 1 2 푚1푚2 • If the strong interaction doesn’t care about quark flavor, then all mesons have the same ground-state energy. • Empirical mass formula: 푠Ԧ1 ∙ 푠Ԧ2 푚 = 푚1 + 푚2 + 휅 푚1푚2 • What are the expectation values of the 푠Ԧ1 ∙ 푠Ԧ2 operator? The Quark Model 2 Recall that 푆Ԧ = 푠Ԧ1 + 푠Ԧ2 and that 푆Ԧ = 푆(푆 + 1) 2 푆Ԧ = 푠Ԧ1 + 푠Ԧ2 ∙ 푠Ԧ1 + 푠Ԧ2 2 2 = 푠Ԧ1 + 푠Ԧ2 + 2 푠Ԧ1 ∙ 푠Ԧ2 1 2 푠Ԧ ∙ 푠Ԧ = 푆Ԧ − 푠Ԧ 2 − 푠Ԧ 2 1 2 2 1 2 1 3 3 = 푆 푆 + 1 − − 2 4 4 − 3Τ4 when 푆 = 0 푠Ԧ ∙ 푠Ԧ = ቊ (pseudoscalar) 1 2 + 1Τ4 when 푆 = 1 (vector) Quark Model Wave Functions • The neutral mesons are orthogonal linear combinations of 푢푢ത , 푑푑ҧ and (푠푠ҧ) states. 1 휋0 = 푢푢ത − 푑푑ҧ 2 1 휂 ≈ 푢푢ത + 푑푑ҧ − 2푠푠ҧ 6 1 휂′ ≈ 푢푢ത + 푑푑ҧ + 푠푠ҧ 3 • Vector mesons: 1 휌0 = 푢푢ത − 푑푑ҧ 2 1 휔 ≈ 푢푢ത + 푑푑ҧ 2 휙 ≈ 푠푠ҧ Color Quantum Numbers • Pauli’s exclusion principle states that identical Fermions can’t occupy the same state. • There is apparently a problem with the spin 3/2 baryons: ∆++= 푢 ↑ 푢 ↑ 푢 ↑ ∆−= 푑 ↑ 푑 ↑ 푑 ↑ Ω− = 푠 ↑ 푠 ↑ 푠 ↑ • Also, the exchange of identical fermions must be anti- symmetric. • It was proposed that quarks must carry an additional quantum number that we now call “color”. • The “color” part of the wave function is completely anti- symmetric. • Physical hadrons are color singlets – they have no net color charge. Color Wave Functions • Mesons always have the color symmetric wave function: 1 휓푐 = (푅푅ത + 퐺퐺ҧ + 퐵퐵ത) 3 • Baryons always have the color anti-symmetric wave function: 1 휓푐 = (푅퐺퐵 − 푅퐵퐺 + 퐺퐵푅 − 퐺푅퐵 + 퐵푅퐺 − 퐵퐺푅) 6 • These factor from the rest of the hadron spin-flavor wave function. • The baryon spin-flavor wave function must be symmetric. • Color eventually became a central concept in the description of the strong interaction. Orbital Angular Momentum • Bound states of quarks behave sort of like hydrogen atoms except – The masses of the constituents are similar – The quarks are light enough that they are relativistic • Nevertheless, excited hadrons can be associated with the same quantum numbers used to describe the hydrogen atom • Spectroscopic notation: 2푠+1 푛 퐿푗 • Orbital quantum numbers are written S, P, D, F for L=0,1,2,3,… • Many excited mesons have been assigned such quantum numbers… Excited Meson States Strong Decays of Hadrons • The “full width”, Γ, is interpreted as a decay rate: 푑푁 = −Γ푁푑푡 푁 푡 = 푁 0 푒−Γ푡 • The total decay rate is the sum of the decay rates to exclusive final states: 푛 Γ푡표푡푎푙 = ෍ Γ푖 푖=1 • The branching fraction is the probability that a specific final state will be observed: Γ푖 퐵푖 = Γ푡표푡푎푙 Strong Decays of Hadrons • Not derived here: statistical factor 푛 푛 3 2 S 푑 푝Ԧ푘 4 4 푑Γ = ℳ푓푖 ෑ 3 × 2휋 훿 푃 − ෍ 푝푘 2M 2휋 2퐸푘 푘=1 푘=1 (reduced) matrix element squared phase space of 4-momentum particles in the conserving delta- final state function • The phase space is the density of allowed quantum mechanical states with a specific energy. Strong Decays of Hadrons • Two-body decays: 푆 푝Ԧ 2 Γ = ℳ 8휋푀 푓푖 • Strictly speaking though, because of Heisenberg’s uncertainty principle, the energy (ie, mass) of the state might not be precisely known… ℏ Δ퐸Δ푡 ~ 2 1 Δ푡 = Γ • In practice, we can replace the delta-function over energy with a Breit-Wigner function: Strong Decays of Hadrons 1 훿 퐸 − 푀푅 → 2 2 퐸 − 푀푅 − Γ /4 휞 푬 푴푹 • The “mass” and “width” are parameters, determined from analyses of the resonance shape. Strong Decays of Hadrons • Strong decays will dominate unless they are forbidden by conservation laws • Electromagnetic decays will also occur (unless they are forbidden) but with lower rates • Weak decays generally dominate only when other decays are forbidden Γ ≈ 46 MeV Quark Current Lines • We can describe decay mechanisms by drawing the flow of quark currents • These are not strictly speaking Feynman diagrams, but they borrow some of the concepts – Time flows from left to right – Anti-quark currents flow backwards 푑 ∗0 + − 휋− = ቊ • Example: 퐾 → 퐾 휋 푢ത 푑 푝 = 287 MeV 퐾∗0 = ቊ 푢 푠ҧ 퐾+ = ൜ 푠ҧ – Fractions of 퐾+휋− and 퐾0휋0 are determined from isospin analysis… Isospin Analysis of Strong Decays 퐾∗0 → 퐾+휋− and 퐾∗0 → 퐾0휋0 1 2 ۧ−퐾∗0ۧ = |퐾0휋0ۧ − |퐾+휋| 3 3 퐵푟 퐾∗0 → 퐾+휋− = 0.667 퐵푟 퐾∗0 → 퐾0휋0 = 0.333 Electromagnetic Decays • These are similar to transitions in hydrogen atoms • Dipole transitions: γ + ۧ↓↑ | → ۧ↑↑ | • The electromagnetic coupling constant (훼) must be about 30 times smaller than the strong coupling constant (훼푠).