Lecture IV: Re introduction to Fundamental Forces ala P Steinberg

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Lecture IV(Re) introduction to Fundamental
Forcesala P Steinberg
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Goal 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

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What 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

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Classical 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!

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Example 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!

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Basic Scattering Theory
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The 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!
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The 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
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Whats 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

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Why 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!
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Feynman 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
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Crossing 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
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Interactions

• 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

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Strength from Time

• Consider 3 decays

All of these have the same quarks (uds)
andsimilar Q values (liberated kinetic energy)
Why the drastically different lifetimes??
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How 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
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Electromagnetism

• 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

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Building 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
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EM Processes
Rutherford Scattering(easy to generalize
tonuclear scattering)
g

g
g
Bhabha scattering interference of exchange
annihilation channels
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EM Proceses in External Fields

Bremsstrahlung photon emission in nuclear EM field
Pair production in field of nucleus!
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Higher 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

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Self-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!

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Renormalization

• 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!

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Charge 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-
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Strong 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

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QCD 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
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Strong 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
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Confinement

• 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
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Running 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

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Particle 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

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Weak 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)
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Weak Force Diagrams
Charged Current
Neutral Current
Joins particledoublets
And all othercrossing diagrams!
time
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Crossing, 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
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Weak Decay examples
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A Hybrid Example
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Electroweak 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!