IntroIntro toto NuclearNuclear andand ParticleParticle PhysicsPhysics (5110)(5110)
Apr 17, 2009
4/17/2009 1 PathsPaths toto thethe StandardStandard ModelModel • We’ve reached the point of understanding elementary particles roughly comparable to the world of the early 1960’s – Four fundamental forces. Three of these we (somewhat) understand: symmetries, conservation laws and either powerful (EM) or rudimentary (strong, weak) quantitative tools – Many, many particles: hadrons (mesons and baryons), leptons, the photon. Many more (esp. hadrons) being discovered – A few really powerful clues to the details: patterns in masses and decay properties among particles, CP violation, absence of “strangeness- changing neutral currents” • Some particle physicists were content with the situation, continually adding to the complexity with more particles, and more details of their properties and behavior • Others found it intolerable and were already looking for ways to bring order to the confusion. Still others were methodically pushing farther into the structure of nuclear matter with “deep inelastic scattering” of electron beams • Both paths would converge by the early 1970’s on the Standard Model of quarks and leptons
4/17/2009 2 ManyMany particles!particles! Patterns!Patterns!
Nucleons: p (938 MeV) n (940 MeV) Pions: π+ (140 MeV) π0 (135 MeV) π− (140 MeV) Kaons: K+ (494 MeV) K0 (498 MeV) K− (494 MeV)
• Isospin Members of multiplets are different states of the same particle: nucleon doublet, pion triplet, kaon and anti-kaon doublets But…the original premise of isospin was that only EM broke an otherwise perfect symmetry. If this is valid, how do we explain… MM>> and M M, but M>M? np KK00±±ππ
Perhaps…observed elementary particles are different assemblies of a small set of basic building blocks. First suggested by Fermi and Yang in 1949!
4/17/2009 3 Fermi-Yang “Pre-Quark Model” π + = pn 1 π 0 =−()pp nn SU(2) 2 π − = np Problems: No experimental or theoretical basis for baryons as building blocks With more particles, some repetition of patterns for excited versions of known particles, but no place for strangeness!
4/17/2009 4 MoreMore Patterns:Patterns: Strangeness,Strangeness, Hypercharge,…Hypercharge,… Gell-Mann, Nishijima and others noted that electric charge (Q) of all particles could be related to isospin (I3), baryon number (B) and strangeness (S): ()SB+ Y QI=+ =+I YS= + B Y: Hypercharge 3322 Interesting patterns became evident when Y was plotted vs. I3: Y 0 Model of Sakata (1956): K K + → +1 Known mesons (7) and baryons (8) can
− 0 + be built from three fundamental baryons π π π (p, n, Λ) and their antiparticles: -1 0+1I 1 3 π + = pn π 0 =+()pp nn π − = np − 2 K K 0 K + = pΛ K 0 = nΛ -1 K 0 = nΛ K − = pΛ (Baryons : Σ=+ Λpn, etc.) Good step! Explained known particles, got statistics right. But why should p, n, Λ be fundamental? Lots of tinkering, incl. weakening of connection between the building blocks and the physical baryons. Sakata model could not accommodate some decays and new particle discoveries 4/17/2009 5 Next step: SU(3) and the “Eightfold Way” Gell-Mann, Ne’eman (1961) Isospin SU(2) + Strangeness → SU(3)
Pseudoscalar Vector Spin-1/2 Spin-3/2 Mesons Mesons Baryons Baryons K 0 K + K ∗0 K ∗+ n p ∆− ∆0 ∆+ ∆++
− 0 + − 0 + − 0 + ∗− ∗0 ∗+ π π η π ρ ρ ω ρ Σ Σ Λ Σ Σ Σ Σ
K − K 0 K ∗− K ∗0 Ξ− Ξ0 Ξ∗− Ξ∗0
Octet of SU(3), consisting of an isotriplet, a singlet and two ? isodoublets is asserted to be the basic unit of organization: 8⊗8=18⊕⊕⊕810⊕10∗ ⊕27 Spin-3/2 baryons pose a puzzle – not an octet. 10? 27?
4/17/2009 6 • Many competing predictions were made of particle masses and decays expected under the Sakata model and the “10” and “27” interpretations of the “eightfold way” • Gell-Mann predicted that the “10” is correct and that the “?” in the spin- 3/2 decuplet would be the Ω−, a baryon with S = −3 at ~ 1700 MeV
Samios, et al. BNL - 1964
• Last step! Gell-Mann and Zweig in 1964 independently postulated that the constitutents of hadrons were not known particles, but as-yet- unseen building blocks in the fundamental (“3”) representation of SU(3). Zweig thought they were observable, Gell-Mann just “bookkeeping” 4/17/2009tools 7 TheThe QuarkQuark ModelModel • Initially these hypothetical particles were named p, n and λ, but within a decade the modern names u (“up”), d (“down”), and s (“sideways”→”strange”) had been adopted Observed hadrons constructed from quarks: + 1 − π = ud π 0 =−()uu dd π = du Mesons 2 quark/antiquark pairs Ku+ = s Kd0 = s Ks0 = d Ks− = u
Baryons pu= u d nu= d d Λ = ud s three quarks (antibaryons are p= uud three antiquarks) nu= dd Λ=uds +− P 11 11 Fundamental, pointlike; J = ; baryon number − ; Quark 22 33 2121 (antiquark) charges +−ee and −e and +e; nonzero masses T.B.D.; properties: 3333 isodoublet (ud, ) and isosinglet ()s; "feel" all four interactions
4/17/2009 8 3 3
Quark Model Successes Quark Model Problems − Classified known particles (early 60s), − Free quarks? Many looked, no one predicted others found later found − Accommodated many not-yet imagined − Quantum statistics problems, e.g. ∆++ − Explained absence of some particles, is a fermion, but constructed wave like S = +1, Q = 0 baryon function is totally symmetric − Explained mass splittings, isospin, − No satisfactory understanding of magnetic moments, cross-section quark binding in hadrons
ratios (σπp/σpp = 2/3) − “Family problem”
4/17/2009 9 SU(2)SU(2) →→ SU(3)SU(3) Isospin Quarks • n = 2, n2 –1 = 3generators (2×2 • n = 3, n2 –1 = 8generators (3×3 linearly independent, traceless linearly independent, traceless Hermitian matrices) Hermitian matrices)
Pauli Matrices Gell-Mann Matrices
One is diagonal ⇒ one quantum number:
I3
Two are diagonal ⇒ two quantum numbers:
I3, Y = B + S More quarks? SU(3) → SU(4) → SU(5) → …
4/17/2009 10 GroupGroup Theory:Theory: thethe MathMath ofof SymmetrySymmetry
We say that the set of transformations G G = (gg12, ,..., gn ) constitutes a group if the following four conditions hold:
1. Closure: If gi and gj are in G, then the product gigj is also in G.
2. Existence of Identity: ∃ I in G such that Igi = giI = gi ∀ gi -1 -1 -1 3. Existence of Inverse: ∀ gi ∃ gi such that gi gi = gi gi = I ∀ gi
4. Associativity: gi (gj gk) = (gigj )gk More terminology: – A representation of a group is the action of the transformations in the group on a specific set of objects – Generating set of a group – subset of the group such that every element of the group can be generated as a product of elements of the generating set and their inverses – Types of groups: finite, infinite, continuous, discrete – correspond to different transformations/symmetries – Abelian group ⇒ elements commute (e.g. translation in space) – Non-Abelian group ⇒ elements do not commute (e.g. rotation) 4/17/2009 11 GroupGroup TheoryTheory (continued)(continued) • Consider a transformation U that acts on a wave function ψ: ψ' = U(ψ) • Let U be a continuous transformation and it can be expressed U = eiθ where θ is an operator • If θ = θ† (Hermitian), then U is a unitary transformation:
∗T ∗T Ue† ==( iiθ ) e− θθ=e−i UU † ==eiiθθe− I (Note: U ≠ U† ⇒ U is not Hermitian)
• Important groups in physics – Members are matrices, operation is matrix multiplication U(n) n × n unitary group SU(n) n × n special unitary group (det = 1) SO(3) (rotations O(n) n × n orthogonal group (UTU = UUT = I) in 3D) is almost identical to SU(2) SO(n) n × n special orthogonal group (det = 1)
4/17/2009 12 SUSU(2)(2) isis especiallyespecially importantimportant becausebecause itit describesdescribes angularangular momentummomentum andand givesgives aa modelmodel forfor internalinternal symmetriessymmetries ofof elementaryelementary particlesparticles • SU(2) – the set of all 2 × 2 unitary matrices with unit determinant - forms a group under matrix multiplication
Elements of SU ()2 can be represented as r Ue()αr = iT ()α ααrr is a vector parameter that gives the phase and T ( ) is a 22× matrix generated by the Pauli matrices :
3 iTα r ∑ j j r iT ()α j=1 Ue()α ==e 01 0−i 1 0 σσ12== σ3= 1 10 i 0 0−1 T = σ jj2 (This is spin! We now apply it to something new.) 4/17/2009 13 Charge independence! IsospinIsospin Heisenberg’s assertion: Turn off EM and n and p would be the same
• Determine experimentally that isospin is conserved by the strong interaction, not by the electromagnetic or weak
[HIweak ,0]≠ ⇒= HIstrong ,0 []HIEM , ≠ 0
• Isospin conservation means strong interactions are invariant under rotations of states in “I space,” which means that the interactions for the particles with the same total I but different projections I3 are identical. This is the case for the proton and neutron, for the three pions, and for the other isomultiplets • Isospin is an internal quantum number that must be incorporated into the description of a state, along with all of the other internal quantum numbers to come • How do we use it?
4/17/2009 14 FundamentalFundamental RepresentationRepresentation ofof SU(2)SU(2)
(That representation from which all others can be constructed.) • Two-component spinor ( j = ½ ) – the “2” of SU(2) • These are symmetries – we should be able to construct them by manipulating physical models (graphically):
m m m m 1 + +1 1 + 2 0 = 0 + 0 2 1 − 2 −1 −1 22× = 3 + 1
Isospin is all about counting Triplet (“3”) Singlet (“1”) carefully. Next job is to see of SU(2) of SU(2) how isospin symmetry Trivial quantitatively constrains the Representation rates of related processes 4/17/2009 15 Determine relative rates for processes ApplicationsApplications ofof IsospinIsospin involving different charge states Modeled after the treatment of angular momentum: r rr JJ= 12+ J m m m m +1 1 +1 + 2 0 0 0 1 = + − 2 −1 −1 11 1 1 22 1 0 0 0 11 − 22 1 −1
jj12+ jm j m= Cjj,,12j j m mm= + m 112 2 ∑ mm,12,m 12 jj=−j 12 Clebsch-Gordan Coefficients
4/17/2009 16