Fundamental principles of particle physics

1
Fundamental principles of particle physics
Source Ross (CERN-Summer lectures)
Our description of the fundamental interactions
and particles rests on two fundamental
structures
2
Fundamental principles of particle physics


3
Symmetries
Central to our description of the fundamental
forces
Relativity - translations and Lorentz
transformations
Lie symmetries -
Copernican principle Your system of
co-ordinates and units is nothing special
Physics independent of system choice
4
Basic Objects

• Relativistic 4-vectors
• Minkowski Metric
• Matrix notation!
• Relativistic dot product

5
Special relativity
Space time point
not invariant under translations
Space-time vector
Invariant under translations but not invariant
under rotations or boosts
Einstein postulate the real invariant distance
is
Physics invariant under all transformations that
leave all such distances invariant
Translations and Lorentz transformations
6
Lorentz Boosts

• In previous courses, you should have seen one
  dimensional Lorentz transformations (boosts)
  presented this way

b
r
7
Relativistic Effects I

• Time Dilation
• When viewed from a moving frame, a clock at rest
  seems to be ticking more slowly
• Proper time t is that measured by the clock at
  rest

Bouncing photon clock!
b
8
Relativistic Effects II

• Length Contraction
• Measure an object by the interval when the front
  and back pass an observer
• The observers clock is moving slower than the
  one in the objects rest frame
• Thus, the object passes in less time ? observer
  measures a shorter length

9
Matrix representation

• This is more easily remembered as a matrix
  operation (lets drop x,y coords)
• And lo it looks like a rotation
• But with a wrong sign
• Hyperbolic rotation!

where
10
Lorentz transformations
(Summation assumed)
Solutions
3 rotations R
11
Lorentz transformations
(Summation assumed)
Solutions
3 boosts B
3 rotations R
Space reflection parity P
Time reflection, time reversal T
12
The Lorentz transformations form a group, G
Rotations
Angular momentum operator
13
The Lorentz transformations form a group, G
Rotations
Angular momentum operator
The
are the generators of the group.
Their commutation relations define a Lie
algebra.
14
Demonstration that
15
Useful Lorentz invariants

• Mass
• Proper-time
• Phase of a wave function

16
Space-like Time-like

• The distance between two events is an interval
• Space-time intervals can be
• Time-like when a light signal could traverse Dz
• Space-like when a light signal could never
  traverse it
• Time-like separated events can never be made
  simultaneous, but space-like can be

t
Dr
z
17
Momentum space

• We can also think of intervals in momentum space
• Time-like when all of the energy can be
  considered to have been at a point
• Space-like when the information must travel
  between two points

18
Relativistic Kinematics

• Two main principles
• Conserve Energy-momentum
• Make sure Sp(in) Sp(out), E(in)E(out)
• Invariants stay invariant under boosts
• Eg. CMS Energy, momentum transfer
• Work in the frame that simplifies things
• Example, 2 body scattering

19
Example

• Momentum transfer in elastic scattering

p
q
p
q
lt 0 ? Spacelike!
20
Example 2

• Particle-antiparticle annihilation

M
gt0 Time-like
21
Relativistic Kinematics

• Many things you will see in this course requires
  you to have a good grasp of the concepts
  mentioned here
• Lorentz boosts
• Time dilation, Length contraction
• Lorentz invariants (mass, time, phase, momentum
  transfer)
• Worth the time to play around with it to get
  comfortable with the formalism and physical ideas

22
Relativistic quantum field theory
Fundamental division of physicists world
Action
speed
slow
fast
Ac t ion
Classical Newton
Classical relativity
large
Classical Quantum mechanics
Quantum Field theory
small
Principle of Least Action Feynman Lectures in
Physics Vol II Chapter 19
c
23
Relativistic quantum field theory
Fundamental division of physicists world
speed
slow
fast
Ac t ion
Classical Newton
Classical relativity
large
Classical Quantum mechanics
Quantum Field theory
small
c
24
Quantum Mechanics Quantization of dynamical
system of particles
Quantum Field Theory Application of QM to
dynamical system of fields
Why fields?
No right to assume that any relativistic process
can be explained by single particle since Emc2
allows pair creation
(Relativistic) QM has physical problems. For
example it violates causality
25
Quantum Mechanics
Classical non relativistic
Quantum Mechanical Schrodinger eq
26
Quantum Mechanics
Classical non relativistic
Quantum Mechanical Schrodinger eq
Classical relativistic
Quantum Mechanical - relativistic
27
Relativistic QM - The Klein Gordon equation (1926)
Scalar particle (field) (J0)
(natural units)
Energy eigenvalues
1927 Dirac tried to eliminate negative solutions
by writing a relativistic equation linear in E
(a theory of fermions)
1934 Pauli and Weisskopf revived KG equation with
Elt0 solutions as Egt0 solutions for particles of
opposite charge (antiparticles). Unlike Diracs
hole theory this interpretation is applicable
to bosons (integer spin) as well as to fermions
(half integer spin).
As we shall see the antiparticle states make the
field theory causal
28
Negative Energy States

• Wave function of a free particle
• If the physics is Lorentz invariant, wave
  function can only be a function of Lorentz
  scalars
• This wave function is the same for
• A free particle of energy E and momentum p
  travelling forward in time in the x-direction
• A free particle of energy -E and momentum -p
  travelling backward in time in the (-x)-direction

Egt0
x
t1
t2
Elt0
t2
t1
29
Anti-particles

• Fermions
• Dirac postulated that anti-particles can be
  liberated along with a particle from a sea of
  negative energy states
• Implies conservation of matter anti-matter,
  I.e. produced and destroyed in pairs
• Bosons
• No comparable sea of anti-bosons
• However, every particle can be annihilated by a
  negative energy partner
• In all cases, particle and anti-particle have
  opposite electric charge

m
DE2m
0
-m
30
Dirac Equation

• Dirac wanted an equation first-order in the
  derivatives
• Weyl equation worked for massless particles (and
  will be useful to us later!)
• To satisfy Klein-Gordon equation, square the
  operators in each equation and compare with

31
Dirac Equation II

• Comparing terms, we get the defining features of
  Pauli 2x2 spin matrices
• So we recognize the Weyl equation describing
  massless spin-1/2 particles!
• The wave function is a 2-component spinor
• The and versions of the Weyl equation
  represent 4 wave function solutions

32
Dirac Equation III

• To add the particle mass we require the full
  Dirac equation
• Where a,b are now 4x4 matrices, and y is a
  4-spinor

2 Particle spin states
2 Anti-particle spin states
33

• Each component has a simple interpretation for a
  particle at rest

Spin-up
Spin-down
Particle
Anti-Particle
34
Dirac Equation IV

• The usual form of the Dirac equation is
  explicitly Lorentz covariant
• With the gamma matrices

35
Why we ignore Dirac

• Normally, we should solve the Dirac equation to
  get our free-particle spin-1/2 states (electrons,
  quarks, etc.)
• However, in most cases we consider, particles are
  high energy
• Their mass can effectively be ignored
• Then Dirac?Weyl
• Two decoupled equations for particles and
  anti-particles

36
Helicity

• For massless particles, we find that
• Thus, free fermions have a special quantum number
  Helicity
• Helicity is dot product of
• Particle momentum
• Spin direction
• Handedness to particle
• RH spin and momentum are parallel
• LH spin and momentum are anti-parallel

RH
LH
spin
37
Helicity, cont.

• Helicity, or handedness is only well defined
  for massless particles, so the Weyl equations are
  valid
• For mgt0, can boost into a frame where particle is
  heading the other direction!
• But as we said, extreme relativistic particles
  (p/mgtgt1) are effectively m0
• In this relativistic limit, helicity is
  conserved for EM, Weak and Strong forces!
  (Vector Axial-vector)
• So LH particles remain LH, RH reman RH
• Well get back to this when discussing conserved
  quantities

38
Back to Leptons

• There are three lepton families
• Neutrinos are nearly massless (at least they were
  until 1998)
• All of them are left handed!
• Each lepton number is conserved separately
• Le, Lm, Lt must be the same coming into and
  leaving a reaction

L
1
39
Lepton Decay

• So we start with 1 tau lepton
• We end with 1 tau neutrino, 2 muon neutrinos
  (particle anti-particle), 1 electron, and one
  anti-electron neutrino!
• Nothing is violated!

40
Quarks
Charge
2/3
0 MeV
1600 MeV
180 GeV
-1/3
150 MeV
4.5 GeV
5 MeV

• Quarks are the building blocks of protons and
  neutrons, the stable non-leptonic matter in the
  universe.
• Although we assign them identities and charge
  states, no free quarks have ever been seen!

41
Hadrons

• In nature, quarks are hidden.
• Instead, they appear in pairs and triplets
• Mesons QQ (p, K, r, w)
• quark-antiquark pairs with integral spin (bosons)
• Baryons QQQ (p, n, L, D)
• 3 quarks with half-integral spin (fermions)
• Baryon number is conserved in nature
• Some baryons are stable (Nuclei!)
• No mesons are stable

u
d
u
d
u
42
Quark Quantum Numbers

• Quark numbers are conserved separately by the
  strong electromagnetic interactions
• Up-ness, Down-ness, Strange-ness, Charm, Bottom,
  Top
• Isospin (covered later) is the separate
  conservation of up and down, considered as a
  single spin quantum number
• Other flavor QNs have a simple rule
• If quark has q2/3 (e.g. c,t) then positive
• If quark has q-1/3 (e.g. s,b) then negative
• So s?S-1, c?C1, b?B-1, t?T1

43
Quark QNs in Action

• Strange particle production (strong)
• Charm particle production (EM)

44
Summary

• Quarks and leptons are the fundamental matter
  particles of nature
• They are spin ½ particles
• Dirac equation?Weyl solutions for massless
  particles
• Helicity is conserved in all the forces we know
• Particles have anti-particles of opposite charge
  and/or lepton number
• These are negative energy states travelling
  backwards in time (pass the pipe)
• Total fermion number is conserved since fermions
  can only be produced in pairs
• Quark and lepton number are conserved separately
  for each family

45
4 vector notation
contravariant
covariant
4 vectors
46
Field theory of
Scalar particle satisfies KG equation
Classical electrodynamics, motion of charge e in
EM potential
is obtained by the substitution
Quantum mechanics
The Klein Gordon equation becomes
, means that it is sensible to
The smallness of the EM coupling,
Make a perturbation expansion of V in powers of
47
Physical interpretation of Quantum Mechanics
Schrödinger equation (S.E.)
probability current
probability density
48
Physical interpretation of Quantum Mechanics
Schrödinger equation (S.E.)
probability current
probability density
Klein Gordon equation
Negative probability?
Pauli and Weisskopf
49
Want to solve
Solution
where
and
Feynman propagator
Dirac Delta function
50
Want to solve
Solution
where
and
Feynman propagator
Dirac Delta function
Simplest to solve for propagator in momentum
space by taking Fourier transform
51
The Born series
Since V(x) is small can solve this equation
iteratively
Interpretation
52
But energy eigenvalues
Feynman Stuckelberg interpretation
Two different time orderings giving same
observable event
time
space
53
(p0 integral most conveniently evaluated using
contour integration via Cauchys theorem )
54
where
are positive and negative energy solutions to
free KG equation
55
Theory confronts experiment - Cross sections and
decay rates
Scattering in Quantum Mechanics
Prepare state at
Observe resulting system in state
QM probability amplitude