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
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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+…
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 cm2
• At colliders, often use
“luminosity” L

number of scattering events
 A  B l Al B A
Number of events 
N A N B
A
Number of events  Lt
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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
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Summation over polarizations
• Tr stands for the regular matrix trace
• Next use completeness relations:
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Summation over polarizations
• Now use properties of traces for gamma
matrices:
•
•
• and trace of odd number =0
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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
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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/QFT10Mar05Bh
abha.pdf
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Near future
• Renormalization and running coupling
constants in QED
• Weal interactions and coming to the
Standard Model
• Standard Model Lagrangian
• Start talking about Higgs
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