Problem sets for Physics 230B fall 2012 ================================================== ================================================== Note: The most recent problem set appears first. ================================================== ================================================== Problem set 3 Due Thursday, Dec. 6 1) Consider the tree level contribution to gg->gg gluon scattering from the 4 gluon vertex, i.e. just one of the contributing graphs. We had this on the board in class a couple of times. The amplitude includes four gluon polarization vectors for physical initial and final state gluons. (Note: this is a bit of a fantasy that includes ignoring the fact that gluons are confined and never appear as physical external particles.) As we have already mentioned, there is some arbitrariness in the choice of those polarization vectors. They are determined only up to the addition of a multiple of the gluon momentum. I.e. \epsilon(q) and \epsilon(q) + aq are equivalent. Work in the CM frame. Show that for any given scattering angle and gluon helicities, the polarization vectors can be chosen so that the contribution from this diagram is zero. 2) Prove the third color algebra relation on page 299 of Gunion III. This is the relation with three external gauge vector bosons. 3) The context is the QCD sector of the standard model. Compute the CM differential cross section at O(g^4) (one gluon exchange) for u + dbar -> u + dbar separately for two helicity choices u_R + dbar_R -> u_R + dbar_R and u_R + dbar_L -> u_R + dbar_L Write your answers so that the dependence on the scattering angle theta is explicit. Neglect these light quark masses. Average on initial colors and sum on final colors. 4) Only a couple of people made progress on problem 3 in problem set 2. Now that you have new tools, you can approach the question again even more efficiently. ================================================== Problem set 2 Due Thursday, Nov. 15 1) Note that the Klein-Nishina formula for Compton scattering reproduces _classical_ Thompson scattering for low energy photons. This cross section vanishes as the target mass goes to infinity. What about e-P scattering with the proton treated as a structureless fermion? (See e.g. Zee p. 132-134). Look at it in the lab frame (initial proton at rest) and in the corresponding regime where the beam electron is highly relativistic and with the proton mass goes to infinity. I.e. m_P >> E_e >> m_e. Does this cross section vanish as m_P -> infinity? Is the behavior for these two processes the same or different? Can you give a simple physical explanation? The point here is that even though the tools of QFT and Feynman diagrams seem to be totally new, they can (better!) reproduce results from basic physics. 2) What is the difference between the physical processes described by the Compton scattering the cross sections Zee II.8.12 and either Gunion II eq. 195 or P&S eq.5.91? Can the correspondence Gunion II eg. 161 or P&S eg. 5.75 be used to go from the former to the latter? If not, do so by another direct method. 3) Now consider tree level Compton scattering off a massless electron. Find a choice of in-going and out-going photon _helicities_ for which the cross section is zero for all scattering angles and regardless of electron helicity. You will want to work in the center of mass frame. You will save a lot of algebra by making a good choice of photon polarization vectors. On way that helps a lot is to exploit gauge invariance to select polarization vectors with a definite helicity that have not only zero dot product the their own photon four momentum but also zero dot product with the incoming electron. Can you come up with a simple symmetry argument that explains why the cross section is zero for the choice of photon helicities that you have found? (I have not been able to do this.) 4) P&S 15.3 In part b, do not make your life overly difficult. If there are parts of the calculation that do not contribute to E( R ) as T -> infinity with R fixed, you need not calculate them. If there are divergent parts that do contribute, what is their physical origin, and what kind of contribution do they make to V( R )? 5) Prove the non-abelian Bianchi in at least one of the forms: Gunion III Eq. 379, P&S Eq. 15.89, or Zee problem IV.5.2. 6) Gunion problem 24 or Zee problem IV.5.4. If you do the Zee problem, you need to pay attention to a counterintuitive point from problem IV.5.3 about what FF = F^F (F wedge F) corresponds to. 7) In class, we did a toy example to illustrate the Fadeev-Popov method in a simple context. It was a two dimensional integral over a function f(x,y) for which f(x,y-alpha) = f(x,y). Make it it even more explicit with f(x,y) = exp(-i x^2/2) and gauge fixing function G_omega(x,y) = ax - (b^2 + c^2 x^2)y - omega. What is the F-P version of the integral of f over x and y? ================================================== Problem set 1 Due Thursday, Oct. 18 Gunion: 19, 20 P&S: 11.1a Zee: IV.3.1, IV.3.4 Gunion's problems are in the Resources section of our Smartsite. Gunion #19 is long and detailed. Do not wait until the night before it's due to start. ==================================================