Methods of Experimental Particle Physics Alexei Safonov Lecture

Methods of Experimental Particle Physics Alexei Safonov Lecture #3 1

Calculations in HEP • Last time we wrote down Feynman rules for QED • Today we will calculate the e+e- -> e+ecross 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 G encompasses the probability that particle will decay • d. F is the “phase space” for each final state particle • Lifetime t=1/G • Survival probability: • Can also calculate partial widths G=G 1+G 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 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 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: • 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/rpp 2012 -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/Pics. II/QFT 10 Mar 05 Bh 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