1. Announcements Homework returned now 9/19 Switching to more lecture-style class starting today Good luck on me getting powerpoint lectures ready every day Schedule slowing down a bit Monday – reading assignment only Wednesday – homework only

2. Questions from the Reading Quiz “Can you explain how to obtain and the meaning of the Feynman invariant amplitude? ” How to obtain it – depends on the theory For  *  theory – read chapter 5 What it means I don’t know It’s an amplitude – the amplitude that you go into a particular state But with some factors taken out to make it simpler

3. Questions from the Reading Quiz “I'm also kind of confused on the concept of D, mainly whether or not is has any physical significance” Definition: What is its physical significance? It is most of the factors in the probability for a process to occur It allows for any number of particles in the final state Works for 2 or 3 (or more) particle final states It ignores details of the incoming state Can work for decay rates or cross-sections Mostly, it’s just a calculational tool

4. Questions from the Reading Quiz “How is it that we just assume using first order perturbation theory will work?” It is an approximation In general, for perturbation theory to work, we need a small parameter The small parameter in this theory is g If g is not small, then this approximation is poor In QED, for example, the parameter is e e = 0.3, which is kind of small As you go to higher order, you get factors of about e 2 /16  2 In QCD, the coupling is significantly larger than one Perturbation theory fails

5. How to calculate everything Fermi’s Golden Rule The general formula for probability is: For 1-2 particles in the initial state, and 2-3 particles in the final state, these become:

6. Comments on D,  and  : The quantity D is Lorentz invariant Usually calculated in the c.m. frame The quantity p is the momentum of either particle in this frame The quantity E cm in any frame is just  s Decay rate is not Lorentz invariant Listed value is in comoving (c.m.) frame In arbitrary frame, formula becomes D/2E Cross section is Lorentz invariant in any frame where particles are colinear Head-on collisions or stationary target Then magnitude symbols not really necessary

7. Differential cross-sections and decays: Combining these: Sometimes you are asked for the “differential decay rate” That means: don’t do the integral Similarly for cross-sections

8. The general procedure Find the Feynman invariant amplitude Multiply it by its complex conjugate Later, this will be harder than it looks now Rewrite this quantity in terms of the given or final energies or angles Calculate D using one of the two formulas we have Find the decay rate or cross-section

9. An easy problem: Calculate the rate for  decay in the  *  theory. This is a decay, so we use the decay formula Two particles in final state, so we need that formula too Center of mass energy is the mass Final particles have equal and opposite 3-momentum Final particles have identical energies

10. A hard problem: Calculate the rate for muon decay. Treat all final state particles as massless. This is a decay, so we use the decay formula Three particles in final state, so we need that formula too We need to: Write all quantities in terms of the final energies or angles Determine limits of integration Perform all integrals

11. Rewriting the Amplitude Conservation of momentum Square this quantity, remembering everything except the muon is massless Working in the rest frame of the muon, so

12. Limits on integration Particle 1 can go any direction we want The azimuthal angle runs from 0 to 2  The energy integrals are tricky: The total momentum is zero: they form a triangle The total energy is m  : this is the perimeter No side can have more than half the total perimeter p1p1 p2p2 p3p3

13. Announcements Homework returned in boxes 9/21 Small error in problem 4.9 Note my solutions are online for past homework problems (a formula for momentum of the final particles was found in problem 2.8b)

14. Finishing the Problem Calculate the rate for muon decay. Treat all final state particles as massless.

15. Turning it Into Numbers Calculate the rate for muon decay. Treat all final state particles as massless.

16. The general procedure 1.Find the Feynman invariant amplitude 2.Multiply it by its complex conjugate Later, this will be harder than it looks now 3.Rewrite this quantity in terms of the given or final energies or angles 4.Calculate D using one of the two formulas we have 5. Find the decay rate or cross-section

17. Sample Calculation – Step 3 The amplitude for the scattering process e - (p 1 ) +  - (p 2 )  e - (p 3 ) +  - (p 4 ) is given at right, in the limit where all masses are negligible. Find the differential cross-section if they are colliding head-on, each with energy E. 3. Rewrite this quantity in terms of the given or final energies or angles p1p1 p2p2 p4p4 p3p3

18. The dot products

19. Sample Calculation – Steps 4 and 5 …Find the differential cross section… 4. Calculate D using one of the two formulas we have 5. Find the decay rate or cross-section

20. Comments on this problem Total cross section is infinity Caused by  = 0 Classically, because particles always scatter a little Experimentally, small angles cannot be detected This expression assumes you are in the c.m. frame Since cross-section is invariant, good idea to write in terms of s e - (p 1 ) +  - (p 2 )  e - (p 3 ) +  - (p 4 ) Unfortunately,  is not Lorentz invariant If we had already found total cross-section, this problem would not occur

21. Sample Calculation – Step 3 The amplitude for H  e + e - is given by the formula at right, where p and p’ are the momenta of the final state particles, m is the mass of the electron, and v is a constant. What is the rate for this decay? Let M be the mass of the Higgs. 3. Rewrite this quantity in terms of the given or final energies or angles

22. Sample Calculation – steps 4 and 5 4. Calculate D using one of the two formulas we have 5. Find the decay rate or cross-section

23. Announcements Homework not yet graded 9/24 Wednesday: Problems 4.9 & 4.10 Friday: Reading quiz I and problems 5.1, 5.2, 5.3, 5.5, and 5.6

24. Feynman Diagrams Where they fit: 1.Find the Feynman invariant amplitude 2.Multiply it by its complex conjugate Later, this will be harder than it looks now 3.Rewrite this quantity in terms of the given or final energies or angles 4.Calculate D using one of the two formulas we have 5. Find the decay rate or cross-section

25. How to draw Feynman Diagrams Make a list of initial particles on the left and final particles on the right Label them by their four-momenta (and spin) Start drawing the right type of line for each initial and final particle Line for  particle Arrow right for  particle Arrow left for  * particle Find every possible way to connect everything together using the allowed couplings To tree level only You now have a series of Feynman diagrams You need to calculate each one You need to add their contribution

26. How to calculate Feynman Diagrams FOR EACH DIAGRAM Conserve four-momentum at each vertex Multiply the following factors: Include one factor of – ig for each vertex Include one factor of i/(k 2 -M 2 ) for each  propagator Include one factor of i/(p 2 -m 2 ) for each  propagator Add all the diagrams together