Quarks

1. Experience the strong, weak, and EM interactions There are anti-quarks as well Quark masses are not well-defined Quarks carry color (RGB) Color is the charge of the strong interaction (SI) Free quarks do not exist? Quarks form bound states through the SI to produce the hadron spectrum of several hundred observed particles These bound states are colorless Structureless and pointlike Quarks

2. Experience the strong, weak, and EM interactions There are anti-quarks as well Quark masses are not well-defined Quarks carry color (RGB) Color is the charge of the strong interaction (SI) Free quarks do not exist? Quarks form bound states through the SI to produce the hadron spectrum of several hundred observed particles These bound states are colorless Structureless and pointlike Quarks

3. Quark Content • Here are some particles for which you should know the quark content • p = uud, n = udd • Δ’s = uuu, uud, udd, ddd • π= ud, (uu + dd)/√2, du • K0 = ds, K0 = sd, there are also K+, K- • Λ = uds, Ω- = sss • J/ψ = cc, Υ = bb (the “oops Leon”) • D0 = cu, D0 = uc, there are also D+, D- • B0 = db, B0 = bd, there are also B+, B- • Note there are no bound states of the top quark • This is because the top quark decays before it hadronizes

4. Hadrons • Hadrons == particles that have strong interactions • Baryons (fermions) • Mesons (bosons) • Baryons == 3 quarks (or antiquarks) • p = uud, n = ddu, Λ = uds, Ω- = sss • Mesons == quark plus antiquark • π+ = u(d-bar), π- = d(u-bar), • π0 = (u(u-bar)+d(d-bar))/√2) • Hadrons can decay via the strong, weak, or electromagnetic interaction

5. Quark Model • By the 1960’s scores of “elementary particles” had been discovered suggesting a periodic table • “The discoverer of a new particles used to be awarded the Nobel Prize; now, he should be fined $10000” – Lamb • Underlying structure to this spectrum was suggested by Gell-Mann in the 1960’s • First through the “Eightfold Way” and later through the quark model • It took approximately a decade for physicists to accept quarks as being “real” • Discovery of J/ψ and deep inelastic scattering experiments gave evidence that partons = quarks

6. Quark Model • One of the early successes of the quark model (Eightfold Way) was the prediction of the existence of the Ω- before its discovery

7. A Little More (review) on Spin • Physics should be unchanged under symmetry operations • Rotations form a symmetry group • So do infinitesimal rotations • The angular momentum operators are the generators of the infinitesimal rotation group • An infinitesimal rotation ε about z is • U ψ(x,y,z) = ψ (R-1r) ~ ψ (x+εy,y-εx,z) • = ψ(x,y,z) + ε(y∂ψ/∂x - x∂ψ/∂y) • = (1 - iε(xpy – ypx))ψ = (1 – iεJ3)ψ • And the generators (angular momentum operators) satisfy commutation relations and have eigenvalues shown on the previous page

8. SU(2) Group (Jargon) • SU(2) group is the set of all traceless unitary 2x2 matrices (detU = 1) • U(2) group is the set of all unitary 2x2 matrices • U†U = 1 • U(θi) = exp(-iθiσi/2) • σi are the Pauli matrices and Ji = σi/2 • The generators of this group are the Ji • The SU(2) algebra is just the algebra of the generators Ji • The lowest, nontrivial representation of the group are the Pauli matrices • The basis for this representation are the column vectors

9. SU(2) Group Representations • Higher order representations (higher order spin states) can be built from the fundamental representation (by adding spin states via the CG coefficients) • A composite system is described in terms of the basis |jAjBJM> == |jAmA>|jBmB> • The J’s and M’s follow the normal rules for addition of angular momentum • |jAjBJM> = ∑ CG(mAmB;JM>|jAjBmAmB> where the CG are the Clebsch-Gordon coefficients we talked about earlier in the course

10. SU(2) Representations • The product of 2 irreducible representations of dimension 2jA+1and 2jB+1 may be decomposed into the sum of irreducible representations of dimension 2J+1 where J = jA+jB, …, |jA+jB| • Irreducible means … • What is he talking about???

11. SU(3) Group (Jargon) • SU(3) group is the set of all traceless unitary 3x3 matrices (detU = 1) • The generators of this group are the Fa • There are 32-1 = 8 generators Fa • They satisfy the algebra [Fa,Fb] = ifabcFc • fabc== structure constants • The generators Fa = 1/2λa where λa are the Gell-Mann matrices (see next page) • The basis for this representation are the column vectors

12. SU(3) Group • Note F3 and F8 are diagonal • F3 == Isospin operator • F8 == Hypercharge operator • Later we’ll define Y = B+S and • Experimentally we find Q = I3 + Y/2

13. SU(3) Represenations • Combining 2 SU(3) objects • 3 x 3 = 6 + 3 • It’s a 3 because in Y, I3 space the u, d, s triangle looks like the ud, us, ds triangle

14. SU(3) Representations • Combining 3 SU(3) objects • 3 x 3 x 3 = 3 x (6 + 3) = 10 + 8 + 8 + 1 • Note the 8’s! • Note the symmetry is S, MS, MA, A • The mixed symmetry representations are given on the next page

15. Quark Model • Hopefully you’ve caught on to what we’ve done • Let u, d, s be the SU(3) basis states • Define isospin Ii = λi/2 • Define hypercharge Y = λ8/√3 = B+S • Since λ3 and λ8 are diagonal, I3 and Y are conserved and represent additive quantum numbers • Note I2, S, Q = I3 + Y/2 are also diagonal and hence are conserved and represent additive quantum numbers

16. Quark Model

17. Quark Model • A convenient way to display the multiplet is to show its elements on a weight diagram in Y-I3 space • Note that the combinations ud, us, ds would appear in the same triangle as s, d, u

18. Mesons • 3 x 3 = 8 + 1 • One can determine the multiplet by explicit calculation of the representation or by the following trick