Announcements 11/7 Today: 8.9, 8.11 Friday: 9.1, 9.2, 9.4 Monday: 9D-9F 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. 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

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

QCD – Quantum Chromodynanics 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)? QED and QCD e becomes g s, Q becomes T a A field becomes A a, one for each T a

In QED, we had fields: In QCD we will have fields: Chromoelectric and magnetic 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

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 Some Complications:

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

Feynman Rules for QCD:

Announcements 11/12 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

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(-  )

Questions from the Reading Quiz “Can you explain how the sum on equation 9.26 turns the T a '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:

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

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

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

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

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

Average and sum over colors Calculate the differential cross-section for qq*  g . Treat quarks as massless. The T a ’s are Hermitian matrices, so * and transpose are the same thing

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*:

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

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

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