1. Phys 450 Spring 2003 Quarks  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

2. Phys 450 Spring 2003 Quarks  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

3. Phys 450 Spring 2003 Quark Content  Here are some particles for which you should know the quark content u p = uud, n = udd u Δ’s = uuu, uud, udd, ddd  π = ud, (uu + dd)/√2, du u K 0 = ds, K 0 = sd, there are also K +, K - u Λ = uds, Ω - = sss u J/ψ = cc, Υ = bb (the “oops Leon”) u D 0 = cu, D 0 = uc, there are also D +, D - u B 0 = db, B 0 = bd, there are also B +, B - u Note there are no bound states of the top quark  This is because the top quark decays before it hadronizes

4. Phys 450 Spring 2003 Hadrons  Hadrons == particles that have strong interactions u Baryons (fermions) u Mesons (bosons)  Baryons == 3 quarks (or antiquarks) u p = uud, n = ddu, Λ = uds, Ω - = sss  Mesons == quark plus antiquark u π + = u(d-bar), π - = d(u-bar), u π 0 = (u(u-bar)+d(d-bar))/√2)  Hadrons can decay via the strong, weak, or electromagnetic interaction

5. Phys 450 Spring 2003 Quark Model  By the 1960’s scores of “elementary particles” had been discovered suggesting a periodic table u “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 u First through the “Eightfold Way” and later through the quark model  It took approximately a decade for physicists to accept quarks as being “real” u Discovery of J/ψ and deep inelastic scattering experiments gave evidence that partons = quarks

6. Phys 450 Spring 2003 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. Phys 450 Spring 2003 A Little More (review) on Spin  Physics should be unchanged under symmetry operations u Rotations form a symmetry group u So do infinitesimal rotations  The angular momentum operators are the generators of the infinitesimal rotation group u An infinitesimal rotation ε about z is  U ψ(x,y,z) = ψ (R -1 r) ~ ψ (x+εy,y-εx,z)  = ψ(x,y,z) + ε(y∂ψ/∂x - x∂ψ/∂y)  = (1 - iε(xp y – yp x ))ψ = (1 – iεJ 3 )ψ u And the generators (angular momentum operators) satisfy commutation relations and have eigenvalues shown on the previous page

8. Phys 450 Spring 2003 SU(2) Group (Jargon)  SU(2) group is the set of all traceless unitary 2x2 matrices (detU = 1) u U(2) group is the set of all unitary 2x2 matrices u U † U = 1 u U(θ i ) = exp(-iθ i σ i /2) u σ i are the Pauli matrices and J i = σ i /2  The generators of this group are the J i  The SU(2) algebra is just the algebra of the generators J i  The lowest, nontrivial representation of the group are the Pauli matrices  The basis for this representation are the column vectors

9. Phys 450 Spring 2003 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) u A composite system is described in terms of the basis |j A j B JM> == |j A m A >|j B m B > u The J’s and M’s follow the normal rules for addition of angular momentum u |j A j B JM> = ∑ CG(m A m B ;JM>|j A j B m A m B > where the CG are the Clebsch-Gordon coefficients we talked about earlier in the course

10. Phys 450 Spring 2003 SU(2) Representations  The product of 2 irreducible representations of dimension 2j A +1 and 2j B +1 may be decomposed into the sum of irreducible representations of dimension 2J+1 where J = j A +j B, …, |j A +j B | u Irreducible means …  What is he talking about???

11. Phys 450 Spring 2003 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 F a u There are 3 2 -1 = 8 generators F a u They satisfy the algebra [F a,F b ] = if abc F c u f abc == structure constants u The generators F a = 1/2λ a where λ a are the Gell-Mann matrices (see next page)  The basis for this representation are the column vectors

12. Phys 450 Spring 2003 SU(3) Group  Note F 3 and F 8 are diagonal u F 3 == Isospin operator u F 8 == Hypercharge operator u Later we’ll define Y = B+S and u Experimentally we find Q = I 3 + Y/2

13. Phys 450 Spring 2003 SU(3) Represenations  Combining 2 SU(3) objects  3 x 3 = 6 + 3 u It’s a 3 because in Y, I 3 space the u, d, s triangle looks like the ud, us, ds triangle

14. Phys 450 Spring 2003 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. Phys 450 Spring 2003 Quark Model  Hopefully you’ve caught on to what we’ve done  Let u, d, s be the SU(3) basis states  Define isospin I i = λ i /2  Define hypercharge Y = λ 8 /√3 = B+S u Since λ 3 and λ 8 are diagonal, I 3 and Y are conserved and represent additive quantum numbers u Note I 2, S, Q = I 3 + Y/2 are also diagonal and hence are conserved and represent additive quantum numbers

16. Phys 450 Spring 2003 Quark Model uds I1/2 0 I3I3 -1/20 Y1/3 -2/3 Q2/3-1/3 B1/3 S00 Spin1/2 P+++ uds I½ 0 I3I3 -1/2½0 Y-1/3 2/3 Q-2/31/3 B-1/3 S001 Spin1/2 P---

17. Phys 450 Spring 2003 Quark Model  A convenient way to display the multiplet is to show its elements on a weight diagram in Y-I 3 space  Note that the combinations ud, us, ds would appear in the same triangle as s, d, u

18. Phys 450 Spring 2003 Mesons  3 x 3 = 8 + 1  One can determine the multiplet by explicit calculation of the representation or by the following trick