Methods of Experimental Particle Physics

1. Methods of Experimental Particle Physics Alexei Safonov Lecture #3

2. Calculations in HEP • Last time we wrote down Feynman rules for QED • Today we will calculate the e+e- -> e+e- cross section • Your homework would be to calculate eeee scattering cross section • The calculation requires some mathematical manipulations, which you should do at least once • It’s okay to use literature and help, but I want you to get through the entire calculation

3. QED Lagrangian and Feyman Rules • Needed to calculate the amplitude M, which tells you what is the probability of the interaction you wrote with the diagrams

4. Scattering Matrix • S is essentially probability amplitude for states on the left to transition to states on the right • Includes two options: nothing happens (they fly by) or they interact • In our calculations, we usually want to know the probability of something specific happening so we calculate M

5. Particle Decays • Simplest interaction is particle decay • Width G encompasses the probability that particle will decay • dF is the “phase space” for each final state particle • Lifetime t=1/G • Survival probability: • Can also calculate partial widths G=G1+G2+…

6. Particle Scattering Cross-Section • Also very important in high energy physics • Imagine you are colliding two beams of particles A and B, each beam has: • Number of particles NA and NB • lengths l(A) and l(B) • cross-section area A • Density r(A) and r(B) • Cross-section is used to calculate the probability of scattering • Units are cm2 • At colliders, often use “luminosity” L

7. Particle Scattering Cross-Section • Cross-section can be differential: • You may want to know not just the probability of any scattering, you want to know how often particles fly in a particular direction • Can calculate if you know M

8. Bhabha Scattering • Lorentz-Invariant Mandelstam variables:

9. Amplitude Calculation • Scattering and annihilation diagrams: • Note that we need to average over electron/positron polarizations

10. Amplitude Squared

11. Scattering Term

12. Summation over polarizations • Tr stands for the regular matrix trace • Next use completeness relations:

13. Summation over polarizations • Now use properties of traces for gamma matrices: • and trace of odd number =0

14. Summation over polarizations • Assuming we deal with a high energy scattering, drop m terms: • But this is only the scattering term, need to calculate three other terms

15. Annihilation term: • Adding the interference term to scattering and annihilation terms:

16. Bhabha Differential Cross-Section • Tells you the distribution of probabilities for different scatter directions of the particle: • We assumed the incoming come along z direction

17. References • S-matrix in scattering: • http://en.wikipedia.org/wiki/S-matrix • Feynman rules and calculations in QED: • Peskin, Schroeder, “An Introduction to Quantum Field Theory”, sections 3, 4, 5 • Brief review of cross-section and decay width calculations: • http://pdg.lbl.gov/2012/reviews/rpp2012-rev-kinematics.pdf • (The link is section 43 of the PDG book) • Calculation of the e+e- scattering cross-section: • http://en.wikipedia.org/wiki/Bhabha_scattering • http://www.physics.usu.edu/Wheeler/QFT/PicsII/QFT10Mar05Bhabha.pdf

18. Near future • Renormalization and running coupling constants in QED • Weal interactions and coming to the Standard Model • Standard Model Lagrangian • Start talking about Higgs