Introduction to particle physics Part IV
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Transcript of Introduction to particle physics Part IV
INTRODUCTION TO PARTICLE PHYSICS
PART IV
Physics 129, Fall 2010; Prof. D. Budker
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Bubble chamber
Great topics for oral presenantion!
The Gargamelle at CERN: discovered weak neutral currents in 1973
Professor Donald A. Glaser
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
How particles decay
• Decay probability goes as dt :
• Particles do not age!
• Board work: Mean Lifetime = 1/
• Branching Ratios
• Partial decay rates add
teNtNdtNdN )0()(
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections
• Effective area
• Inclusive vs. exclusive
• Elastic vs. inelastic (different reactions are called channels)
• Resonances
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections
• Effective area
• Differential cross section
dd
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections• Some cross-sections diverge (e.g., for Rutherford scattering)
• Effective cut-off
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Mandelstam Variables
Universally used!
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Units of cross section
Origin: Uranium nucleus 10-24 cm2 --- as "big as a barn"
Unit Symbol m2 cm2
megabarn Mb 10−22 10−18
barn b 10−28 10−24
millibarn mb 10−31 10−27
microbarn (or "outhouse"[3]) μb 10−34 10−30
nanobarn nb 10−37 10−33
picobarn pb 10−40 10−36
femtobarn fb 10−43 10−39
attobarn ab 10−46 10−42
shed[4][5] (10−24 barn) [none] 10−52 10−48
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Cross Sections
• Luminosity: number of particles in a beam per unit area per unit time
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Luminosity
• What about colliding beams?
• Luminosity = collision frequency n1 n2 / beam area
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Luminosity
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
LHC luminosity: reality check
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Fermi Golden Rule
• mi – mass of ith particle
• pi – 4-momentum of ith particle
• S – statistical factor accounting for identical particles
• M – amplitude (p1, …. , pn)
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Fermi Golden Rule
• Kinematic constraints:
All outgoing particles are on the mass shell
All outgoing particles have positive energy
Energy & momentum conservation
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Fermi Golden Rule
• 2π rules:
Every δ gets a 2π
Every d gets a 1/(2π)
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Fermi Golden Rule• With the kinematic constraints, the G.R. simplifies to:
• For two-body decay: 2
218M
cmpS
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Feynman-Diagram Rules
• Goal: figure out amplitude M
• Draw all possible diagrams for the
process
• The amplitudes from different
diagrams add
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Feynman-Diagram Rules• For each diagram:
1. Label external momenta pi , label internal momenta qi, draw arrows (arbitrary for internal lines)
2. For each vertex, write coupling constant 3. Each internal line propagator:
4. For each vertex: energy/momentum conservation: (minus for outgoing lines)
5. Add for each internal line; integrate
6. Erase the resulting ; multiply by
• The result is M ; examples in Ch. 6 of Griffiths
ig
222 cmqi
jj
321442 kkk
qd 4
421
nppp ...2 2144 i
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Higher-order diagrams
• Problem: loop integrals (logarithmically) diverge at large q
• This is not because the diagrams are bad!
• Regularization: introduce a heavy particle cut-off (p. 219)
• Renormalization; running coupling constants….
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Example/interlude: Diagrams in
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Example/interlude: Diagrams in
• Vanishes for
• Vanishes in the high-frequency limit
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Relativistic Equations
Nonrelativistic
Relativistic; spin zero
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
The Dirac Equation (relativistic, spin ½)• Introduce 44 Dirac Matrices:
Relativistic; spin 1/2
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Solving the Dirac Equation• Assume wavefunction independent of position:
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Solving the Dirac Equation• Four independent solutions:
• The Dirac Sea
• Plane wave solutions (Sec. 7.2)
Electron Electron
Positron Positron
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Dirac Spinor Algebra
• Some useful facts about spinors:
• How do Dirac spinors transform under P?
invariant! icRelativist
SpinorAdjoint 2
42
32
22
10
43210
scalar trueinvariant P
0
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Dirac Spinor Algebra
• Introduce another matrix:
0010000110000100
32105 i
arpseudoscal odd P55
0
What about 4 ?
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html
Bilinear Covariants
Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html