```Methods of Experimental
Particle Physics
Alexei Safonov
Lecture #2
1
Particle Physics and the Origin of Universe
• One example of
an open question
is the baryon
asymmetry
• Lots of protons,
very few
antiprotons
• Why? Shouldn’t
there be equal
numbers?
• Something must have happened in the very
early Universe at the level of basic
interactions that shifted things there
2
Standard Model of Particle Physics
• Physical content:
• 12 basic particles
• Each has an antiparticle
• Interact via force carriers
called gauge bosons
• Higgs boson giving mass to all
other particles
• Includes 2.5 forces:
• Electroweak=“electromagnetic
+ weak” combined force
• All basic particles participate,
transmitted by W/Z/g bosons
decays and electromagnetic
interactions
• Dark Matter is unexplained at all
• Discovery of neutrino
Only quarks participate,
oscillations makes Standard
transmitted by gluons
Holds proton and other composite Model at least “not quite right”
• Gravity is not included
• Strong force:
•
•
3
MATH IN PARTICLE PHYSICS
4
Special Relativity and QM
• Particle physics deals with very small
objects where quantum mechanics is the
only way to describe things
• These things tend to move very fast, so
special relativity is equally important
• Need to combine both
particle:

1 2

• i   H  
t
2m
• This is nothing but E=p2/2m
• The trouble is this equation is non-relativistic
5
Dirac’s Equation
p2
 E  p 2  m2
• Need to go from E 
2m
• So you write something like this:
 2
 ( )    2  m 2
t
• But the problem is that the density is not always
positively defined in this case
•
• Dirac took this equation:
•
• And tried to write the part under the square root as a
square of an operator:
• But you need to remove cross-terms, which is only
possible if AB+BA=0
• Too bad numbers don’t work that way
6
Dirac’s Equation
• But it works with matrices
• That’s great because electron has a spin, you actually want these wave
functions to be 2-component spinors like in QM
• The strange thing was that one needs 4x4 matrices to satisfy all the
requirements
• Write as
• Apply the same operator again (from left):
• That looks familiar if k=m and so
• Can write in an explicit 4-dimensional form:
• Where matrices
7
.
.
Solution of Dirac’s Equation
• Solutions:
• Fourier transformation for general solution:
• Where s is spin +/- ½ and a and b can be
thought of as operators creating/annihilating a
fermion/anti-fermion
• Lagrangian:
• This is because one should be able to get equations
of motion from the Lagrangian as in classic
mechanics:
• In field theory you do
 L  L

 
0
  (  )  



8
Propagator
• Take particle from point x to point y:
• Where T is “time ordering”:
• Plug in our solutions from before:
• where
9
Fermion Propagator
• We just learnt how to calculate the amplitude
(related to probability) for a particle to travel
from point x to point y
• In momentum space:
• This is something we will need to know how to
do to calculate probabilities for scattering
processes
• Can propagate other particles too
10
Anything Else?
• Yeah, we only know how to describe a free
electron and know how to propagate it from
point A to point B
interactions with electromagnetic field – need to
add photons that can interact with electrons
• Where D is the covariant derivative
• Aμ is the covariant four-potential of the electromagnetic field
generated by the electron itself;
• Bμ is the external field imposed by external source;
•
is electromagentic field tensor
11
QED Lagrangian and Feyman Rules
12
QED Lagrangian and Feyman Rules
13
Calculating Amplitudes
• This is what we need to do
at the end:
• Calculate quantum
mechanical probability of a
specific process happening
• First calculate amplitude,
amplitude squared is the
probability
• For scattering processes it is
called “cross-section”
14
Other Processes
• What else can happen?
• There could be more complex diagrams that
include more interconnections and
intermediate particles
15
Good vs “Not So Good” Theories
• Renormalizability
• Once you start
more complex
diagrams, all sorts of
ugly things start
happening
• Infinite amplitudes
and cross-sections
• Renormalization
takes care of it
• Or can be an
“effective” theory,
then no requirements
16
References
• Historical overview and derivation of Dirac’s equation:
• http://en.wikipedia.org/wiki/Quantum_electrodynamics
• A formal (and lengthy) introduction to QED:
• Peskin, Schroeder, “An Introduction to Quantum Field Theory”,
section 3
• Feynman rules for QED:
• Peskin, Schroeder, “An Introduction to Quantum Field Theory”,
section 4
17
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