General Procedure Draw Feynman diagrams Find intermediate momenta Write Feynman amplitude for each diagram Add or subtract them Simplify if you can Multiply by complex conjugate For unpolarized cross-sections: Sum over final spins Average over initial spins Rewrite as traces Recall, any number equals its trace You can then move back to front in a trace Use sum over spins rules Simplify using trace rules, etc. Finish the problem in the usual way
Plus or minus? For each pair of diagrams, should I add or subtract? Add if they differ only by switching external boson lines Subtract if they differ only by switching external fermion lines Possible answers: Add, Subtract, or Trick Question Subtract Trick Add Trick Only add diagrams with identical particles in initial and final states
A hard computation 6.8 Calculate the cross section for scattering. Treat all masses as zero. Plus or minus?
Multiply by complex conjugate
Sum/Average over spins Just as we sum over final momenta, we sum over final spins too Initial spin usually random, so average over it Combine them so they each start and end with the same Dirac spinor
Replace with traces
Use the sum rules
Announcements 10/5 Today: Problems 6.1, 6.2, 6.4 Monday: Read 6E - 6G Wednesday: Problems 6.6, 6.7 6.6 …For an added challenge, let m 0 but keep M =0. 6.7 Write the full Feynman amplitude for (k) (k) (p) (p) for pseudoscalar couplings. 6.8 Calculate the cross section for scattering. Treat all masses as zero.
Get rid of the 5 s
Simplify and take traces Note middle two terms are identical
Work on dot products What are the momenta? Recall all particles massless
Finish the problem … Find D: Recall we have identical final state particles:
A hard computation 6.5 Calculate the cross section for scattering.
Square and sum/average on spins
Do the traces Only even number of Dirac matrices contribute
Announcements 10/8 Today: Read 6E - 6G Wednesday: Problems 6.6, 6.7 6.6 …For an added challenge, let m 0 but keep M =0. 6.7 Write the full Feynman amplitude for (k) (k) (p) (p) for pseudoscalar couplings.
Write out the momenta explicitly In the cm frame, the initial particles must have equal and opposite momenta p But the initial energies will not match The final particles also have matching momenta p The final energies will be: But energy is conserved: To make this work, p = p
Replace all the dot products
How do you average over spins? When you have a spin ½ particle in the initial state, you sum over spins and divide by two. When you have two spin ½ particles in the initial state, you sum over spins and divide by four. What do you divide by if you have n spins in the initial state? Does your formula work for n = 0?
Questions from the Reading Quiz I'm still confused on the whole pseudoscalar vs scalar thing. How do we pick the "scalar theory" of the "pseudoscalar theory". Which one is right? Answer: Neither is right, because its not a real theory.
Questions from the Reading Quiz Could we please go over the B coupling on page 101?