Title: Lecture IV: Re introduction to Fundamental Forces ala P Steinberg
1Lecture IV(Re) introduction to Fundamental
Forcesala P Steinberg
2Goal of Lecture
- (Re) introduce the forces
- Describe basic features of the forces
- Force carriers
- Propagators
- A bit about renormalization
- High energy behavior
- Describe how we use this knowledge to calculate
cross sections lifetimes - Matrix elements
- Phase space
3What does each force do?
- Gravity
- Attractive acts on all particles
- Electromagnetic
- Attracts or repels electric charges
- Bends charges in magnetic fields
- Weak
- Responsible for transmuting particles
- Up??Down, Electron/Muon??Neutrino
- Strong
- Holds hadrons together by gluing quarks
- Exchanges information between hadrons, e.g. to
hold nuclei together
4Classical Quantum Fields
- Classically fields are defined throughout space
and act on particles - Quantum mechanically, fields are force particles
exchanged between matter particles - Heisenbergs Uncertainty Principle sets the scale
for the space-time extent of a force - If a field quantum is massless, it can travel
long distances - If it is massive, then cannot live longer than
?/m - Mass restricts the range of the field!
5Example Klein-Gordon
- Yukawa suggested that strong force carried by
massive field ? Nuclear size suggested 10-15m - We can interpret y as a potential, or as a wave
function of the force particle ? Either way,
lets solve this! - Damped potential, reduces to 1/r if m?0, Coulomb
force!
6Basic Scattering Theory
7The Yukawa Potential
- In quantum mechanics, the scattering amplitude
for a free particle off of a weak potential is
the Fourier transform of the potential (Born
Approximation) - The charge is called the coupling strength
Assume central force
Plug in Yukawa potential
Our final result!
8The propagator
- This was a non-relativistic calculation
- Relativity simply requires replacing q2 with q2.
- Then the matrix element is a fully covariant
object
Charge ofscatterer
Strength of potential
Mass of the exchange particle
Transfer of 4-momentum
9Whats it for?
- Why do we call it a propagator?
- It expresses how much momentum is propagated
between particles when they interact - Combined with the coupling strengths we have a
matrix element - This is just quantum mechanics lingo expressing
the overlap of incoming and final state wave
functions interacting through a potential
10Why is it called a propagator?
- Can be thought of in terms of higher-order terms
of the Born expansion
Spherical wave
g
g
g
Plane wave
g
g
V
g
V
V
yo
yo
yo
yo
V
V
V
Now iterate!
11Feynman Diagrams
- Feynman Diagrams are a way of writing down and
organizing matrix elements - Any quantum field theory specifies
- Particles ? Propagators to transport them
- Interactions ? Where particles meet at a vertex
- Fermions also get a time direction
- Particles that run backwards are anti-particles
- Electromagnetism
e-
e-
e-
Time
12Crossing Symmetry
u
n
- Feynman diagrams are also agnostic with respect
to how the external lines come and go - Matrix elements are the same, if an incoming
particle becomes an outgoing anti-particle - Only difference is the kinematics
(energy-momentum conservation)
W
e-
d
W-production
e
n
e-
W
b-decay
n
W
u
d
charged currentinteraction
u
d
13Interactions
- Now that we have more of a language to describe
particles and forces, lets discuss the familiar
forces in more detail than before! - Basic idea is to consider two factors
- The matrix element the precise form of the
force (couplings, exchange particles) - Phase space how much energy is left over for
the final state particles
14Strength from Time
All of these have the same quarks (uds)
andsimilar Q values (liberated kinetic energy)
Why the drastically different lifetimes??
15How to think about couplings
- The matrix elements relate to lifetime via the
width - The available energy Q indicates how easy it is
to decay - In this case, similar to within a factor of 2-3
- Thus, the ratio of the strengths is
Stronger coupling, shorter life
16Electromagnetism
- The simplest gauge theory QED
- Only one type of charge (electric)
- Weak coupling of charge to massless exchange
boson - Perturbative series converges
- First few terms sufficient for predicting many
phenomena - Many useful processes have been calculated in
leading order QED
17Building Blocks of QED
- For our purposes, we can understand QED as a
fundamental vertex diagram - And a propagator for the photon
- We build up the amplitude matrix element
for a process by summing together all possible
diagrams - We get the probability, or rate, of a reaction by
squaring this - Thus, each vertex comes in with a factor of e in
the amplitude ? factor of a in the probability - So each propagator will contribute factor of a2
in the end - The number of a factors order of a process
e-
e-
e
g
18EM Processes
Rutherford Scattering(easy to generalize
tonuclear scattering)
g
g
g
Bhabha scattering interference of exchange
annihilation channels
19EM Proceses in External Fields
Bremsstrahlung photon emission in nuclear EM field
Pair production in field of nucleus!
20Higher Orders
- Nothing keeps us from exchanging more than one
photon! - However the higher order diagrams are suppressed
by factors of (1/137)! - This is why we typically consider lowest order
diagrams in QED
21Self-Energy
- We can also get some pesky self-energy diagrams
at O(a2) - This modifies the electron propagator (trickier
than the photon) - Modifies both charge mass
- Also gets higher-order terms
- These loops are divergent!
22Renormalization
- We can remove these divergent diagrams by
absorbing the infinite integrals into the
electron charge and mass! - This means that the real charge and real mass
are unmeasurable in principle - We have to measure these paramters
- It is not trivial to absorb these infinities
- Only certain theories (including the standard
model QED, QCD, electroweak) can do this - Related to gauge invariance
- Discussed in a later lecture!
23Charge screening
- The picture that emerges is that the vacuum is
quite complicated! - A charge is surrounded by lots of virtual
electron-positron pairs which flicker in and out
of existence, but are polarized by the bare
charge - Reduces effective charge seen by a probe from the
outside - However, if we probe deeply, we see a stronger
charge! - Coupling constant a?1/128 at high energies!
e-
24Strong Force
- Quantum Electrodynamics ? Quantum Chromodynamics
- We call it QCD
- Similar in some ways to electromagnetism
- Force carriers (gluons) are massless vector
bosons - Charge structure is completely different
- Each quark carries color (red, blue, green)
- Anti-quark carries anti-color (anti-red, etc.)
- Each gluon carries color-anti-color pairs
- Red-antiblue, Red-antigreen, Blue-antired, etc.
u
u
u
25QCD Feynman Rules
- QCD feynman diagrams are similar to QED
- However, there are also vertex diagrams where
gluons interact among themselves
A gluon changes ared quark into a blue quark
3-gluonvertex
4-gluonvertex
26Strong Decays
- The basic idea is that
- Initial and final states are color neutral
- Only QQ mesons, and QQQ baryons
- Strong interaction conserves flavor
- Quark lines can emit quark-antiquark pairs by
radiating a gluon that fragments ? qq
u
u
u
u
d
d
s
s
27Confinement
- Part of QCD looks like QED at short range
- At long range, self coupling of gluons leads to a
linearly rising potential
Colour factor 8 gluon states3 colors(factor of
2)
Confining potentialDoesnt emerge
fromdiagrams. Only seenon in numerical
calculations
28Running Coupling
- Renormalizing the strong coupling is similar as
for EM - However, the non-abelian nature of the force
leads to anti-screening - The force gets smaller, the harder you probe it
- This has measureable consequences, e.g. at LEP
LEPII
29Particle Production
- At long distances and soft momentum scales the
QCD force becomes enormous - Consider a quark separating from an anti-quark
- Field lines are confined to a 1-D configuration
- When the energy density is large enough, pop
out qq pairs - These make pions, kaons, etc.
- String model
30Weak Force
- The weak force was so named since it was seen in
the slow radioactive beta-decays of nuclei - Similarly slow rates are seen in the decays of
strange particles, which lose a unit of
strangeness - How do we get such slow rates?
- What if weak decays are mediated by very heavy
bosons? - Short range, weak force
(measured)
31Weak Force Diagrams
Charged Current
Neutral Current
Joins particledoublets
And all othercrossing diagrams!
time
32Crossing, again
A
C
- As well as for
- A?BCD
- B?ACD
- C?ABD
- D?ABC
- Matrix element
- AB?CD
- Is the same for
- AC?BD
- AD?BC
- BC?AD
- BD?AC
- CD?BA
D
B
33Weak Decay examples
34A Hybrid Example
35Electroweak Unification
- In principle, weak force is just another force
- However, the major victory of theoretical physics
was finding a structure which could include
electromagnetism and weak forces in a single set
of fields - If we assume ge, we can extract M
- The W Z bosons were discovered in 1981, exactly
where they were predicted to be!