1 Methods of Experimental Particle Physics Alexei Safonov Lecture #3

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 2

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 3

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 4

Particle Decays Simplest interaction is particle decay Width  encompasses the probability that particle will decay d  is the “phase space” for each final state particle Lifetime  =1/  Survival probability: Can also calculate partial widths  =  1+  2+… 5

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 N A and N B 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 cm 2 At colliders, often use “luminosity” L 6

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 7

Bhabha Scattering Lorentz-Invariant Mandelstam variables: 8

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

Amplitude Squared 10

Scattering Term 11

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

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

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 14

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

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 16

References S-matrix in scattering: 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: (The link is section 43 of the PDG book) Calculation of the e+e- scattering cross-section: abha.pdfhttp:// abha.pdf 17

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