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

http://upload.wikimedia.org/wikipedia/commons/b/b8/Bubble-chamber.svg


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()(

http://demonstrations.wolfram.com/ExponentialDecay/


Cross Sections

• Effective area

• Inclusive vs. exclusive

• Elastic vs. inelastic (different reactions are called channels)

• Resonances


Cross Sections

• Effective area

• Differential cross section

dd


Cross Sections


Cross Sections• Some cross-sections diverge (e.g., for Rutherford scattering)

• Effective cut-off


Cross Sections


Mandelstam Variables

Universally used!


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


Cross Sections

• Luminosity: number of particles in a beam per unit area per unit time


Luminosity

• What about colliding beams?

• Luminosity = collision frequency n1 n2 / beam area


Luminosity


LHC luminosity: reality check

http://lhc.web.cern.ch/lhc/


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)



• Kinematic constraints:

All outgoing particles are on the mass shell

All outgoing particles have positive energy

Energy & momentum conservation



• 2π rules:

Every δ gets a 2π

Every d gets a 1/(2π)


The Fermi Golden Rule• With the kinematic constraints, the G.R. simplifies to:

• For two-body decay: 2

218M

cmpS


The Feynman-Diagram Rules

• Goal: figure out amplitude M

• Draw all possible diagrams for the

process

• The amplitudes from different

diagrams add


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


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….


Example/interlude: Diagrams in

http://www.oup.com/uk/catalogue/?ci=9780199532421


Example/interlude: Diagrams in

• Vanishes for

• Vanishes in the high-frequency limit

http://www.oup.com/uk/catalogue/?ci=9780199532421


Relativistic Equations

Nonrelativistic

Relativistic; spin zero


The Dirac Equation (relativistic, spin ½)• Introduce 44 Dirac Matrices:

Relativistic; spin 1/2


Solving the Dirac Equation• Assume wavefunction independent of position:


Solving the Dirac Equation• Four independent solutions:

• The Dirac Sea

• Plane wave solutions (Sec. 7.2)

Electron Electron

Positron Positron


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


Dirac Spinor Algebra

• Introduce another matrix:

0010000110000100

32105 i

arpseudoscal odd P55

0

What about 4 ?


Bilinear Covariants

Introduction to particle physics Part IV

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