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Announcements

Announcements. Today: 8.9, 8.11 Friday: 9.1, 9.2, 9.4 Monday: 9D-9F. For 9.2 We have moved into quark model. Therefore our operators are no longer called T 1 2 but now T ud 1 = u, 2 = d, 3 = s.

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Announcements

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  1. Announcements • Today: 8.9, 8.11 • Friday: 9.1, 9.2, 9.4 • Monday: 9D-9F For 9.2 We have moved into quark model. Therefore our operators are no longer called T12but now Tud 1 = u, 2 = d, 3 = s 9.4 Draw at least thirteen tree-level Feynman diagrams for the process qq*gg, where q is any quark, and g is a gluon. You don’t have to do anything with the diagrams. For extra credit, find them all. 11/7

  2. Announcements • Today: 9.1, 9.2, 9.4 • Monday: 9D-9F • Wednesday: 9.6, 9.8 9.6 Only do differential cross-section See problem 7.7 to do most of the work for us 11/9

  3. QCD – Quantum Chromodynanics QED and QCD • Dirac equation: • Make it gauge invariant: • Gauge invariance: • The exponent can be thought of as an arbitrary U(1) matrix • QED is considered a U(1) gauge theory • Can we do the same thing with SU(3)? • e becomes gs, Q becomes Ta • Afield becomes Aa, one for each Ta

  4. Chromoelectric and magnetic fields • In QED, we had fields: • In QCD we will have fields: • This is non-linear • Qualitatively – Fields have charges and produce more fields • This will lead to additional Feynman rules, involving self-couplings • The fields that result are called gluon fields • The true strong force is this force • The strong force between composite colorless objects is a side-effect • Much like chemical interactions between neutral atoms

  5. Some Complications: • We are now dealing with multiple types of particles • To keep them straight, generally need to label the lines • In principle, with flavor and color • Color i takes on three values • Gluons a take on eight values • When doing old computations, such as QED, only identical particles (flavor and color) can contribute

  6. Sample QED computation: What is the cross section as a function of s for e+e- uu*? For uu*  e+e-? • This is correct for cross-section to particular color • If we want cross-section to any color, we get • For the reverse process, you get the same cross section • This time we average over colors

  7. Feynman Rules for QCD:

  8. Announcements • Today: 9D-9F • Wednesday: 9.6, 9.8 • Friday: 10A – 10C 9.6 Only do differential cross-section See problem 7.7 to do most of the work for us 11/12

  9. Questions from the Reading Quiz “Why can't quarks quantum tunnel out of a baryon?” • Force is constant, independent of distance • Therefore potential is linear • Probability of quantum tunneling is exp(-)

  10. Questions from the Reading Quiz “Can you explain how the sum on equation 9.26 turns the Ta's into traces?” • A (square) matrix is an NxN arrangement of numbers • Individual elements of a matrix A are denoted by a row i and column j • Two matrices are multiplied by attaching the second (column) index of the first matrix to the first (row) index of the second matrix and summing • The trace of a matrix is obtained by matching the row and column matrices and summing • Therefore:

  11. Questions from the Reading Quiz “I think walking through the analogy of calculations would be helpful”

  12. Strategy for solving problems in QCD • Label all particles with their momentum, spin and color • Quarks get a color i = 1,2,3 • Gluons get a color a = 1, …, 8 • Write down the Feynman amplitude • When possible, find a similar process with gluons replaced by photons • Steal formulas from QED computation • Modify formulas appropriately • Average over initial colors; sum over final colors • Use identities when needed • Replace, as appropriate, strong fine structure constant

  13. Sample Problem Calculate the differential cross-section for qq*  g. Treat quarks as massless. • Label all initial and final particles by type, momentum, spin/polarization and color • Also label the intermediate states • Now write the Feynman amplitude for each

  14. Simplify the color stuff Calculate the differential cross-section for qq*  g. Treat quarks as massless.

  15. Find a similar problem to one before Calculate the differential cross-section for qq*  g. Treat quarks as massless. Calculate the differential cross-section for e+e- 

  16. Average and sum over colors Calculate the differential cross-section for qq*  g. Treat quarks as massless. • The Ta’s are Hermitian matrices, so * and transpose are the same thing

  17. Sample Problem Calculate the total cross-section for uu*  cc*. Treat quarks as massless. • Let the intermediate gluon color be a • There is an implied sum on a, since this index is repeated • Compare with e+e- ff*:

  18. Sum and average over colors Calculate the total cross-section for uu*  cc*. Treat quarks as massless. • Danger! There is a double sum on a. For safety, change second a to b • Summation symbol no longer needed • * is the same as transpose

  19. Sample Problem - Revised Calculate the total cross-section for uu*  cc*. Include both QCD and QED contributions. Treat quarks as massless.

  20. Sum and average over colors Calculate the total cross-section for uu*  cc*. Include both QCD and QED contributions. Treat quarks as massless.

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