Title: Announcements
1Announcements
- Today 9.6, 9.8
- Friday 10A 10E
- Monday 10F 10H
9.6 Only do differential cross-section See
problem 7.7 to do most of the work for us
11/14
2Weak Interactions
Neutrinos
- First discovered in ?-decay
- Energy spectrum of electron
- Must be some particle carrying off the rest of
the energy - We now know it is neutron decay
- Probably an invisible, neutral particle
- Must be a fermion to conserve angular momentum
- Must be neutral
- Must be very light (lt 1 eV or so)
3Other Neutrino Interactions
- Muon like a heavy electron, but unstable
- Decay requires two invisible particles
- Pion strongly interacting particle
- Easily produced in proton-nucleus collisions
- Decays to muon plus neutrino
- Neutrinos can be converted back to the
corresponding charged particles
- Electron neutrinos make electrons, muon neutrinos
make muons
4The Leptons
- We have already discussed the quarks there are
six of them
- They each come in three colors, and have strong
interactions
- There are six other spin ½ fermions in the
standard model, called leptons
- There are anti-particles for each of these as
well - Neutrino physics is currently evolving better
names are ?1 ?2 ?3 - When weak interactions were first researched,
quarks werent known - We will focus on leptons first
5Fermi Theory of Muon Decay
- First attempt Fermi Theory assumed this was a
basic matrix element
- Original guess was something like this
- This is not the only form that respects Lorentz
invariance
- These five combinations (and linear combinations)
are only objects that respect Lorentz invariance
including P, T, and PT - No combination fit all the available date
6V A Theory of Muon Decay
- In 1956, it was proposed that parity might be
violated in weakinteractions - In early 1957, this was quickly experimentally
verified - Suddenly there were other possibilities
- A ?? coupling is called a vector coupling, and a
???5 is called axial vector - We will call this the V A theory of weak
interactions
- Only left-handed fields participate in these
interactions
7Muon decay rate calculation
- Treat electron as massless
- The ? terms are anti-symmetric under ????, the
other terms are symmetric - The cross terms will automatically vanish
8Muon decay rate calculation (2)
- This amplitude squared was solved in problem 4.11
- Note decay rate rises rapidly as mass increases
- Weak interactions get stronger as you go up in
energy - Eventually, get probabilities gt1 ? no good
9Announcements
- Today 10A 10E
- Monday 10F 10H
- Monday
11/16
10The W - particle
- This interaction is not renormalizable, since GF
GeV-2 - Maybe this is not really what is going on?
- To get this to work, we need W coupling something
like
- The factor of 2?2 is for convenience later
- The index ? implies the W particle must
havepolarization vector, like a photon - Spin 1, like a photon
- The W must be charged, unlike a photon
- The W must be massive, or it would have already
been discovered
11Dealing With Spin-1 Massive Particles
- Polarization vector satisfies same equations as
before
- But this time there are three such polarizations
- Then the three polarizations are
- We need to find
- For propagator
- For summing over initial/final states
12Questions from the Reading Quiz
I have no idea what's going on with the groups
and the electroweak coupling/interaction. I
understand that the SU(2) and U(1) aren't really
independent. But it's all confusing and it's
making my head hurt.
- Spin 1 particles run into trouble with
renormalization unless they are gauge-type
couplings
- What we think is going on so far is
13A Toy Model The Two Photon Model
- Surprisingly, it is sometimes ambiguous which are
the actual particles - Consider the following toy model The Carlson
two-photon model
- Classically, if you shake a particle with both
types of charge, you would make both types of
fields - Quantum mechanically, you would create states
that are superpositions of each type of field
- Unless there is something logically picking out
particular directions in A1A2-space, it is not
obvious which ones you want to think of as the
real fields.
14Rotating Fields Arbitrarily
- We can change the fields in any arbitrary way,
for example
- We can just as easily work with these fields
15Announcements
- Today 10F 10H
- Monday 10.1, 10.3
- Wednesday 10.4, 10.5, 10.8
11/16
16Weak Interactions with One Lepton Pair
- Naively, there is one charged lepton field and
one neutrino,
- The left- and right-handed pieces of the massive
electron have different weak interactions, and
should be divided
- Without mass, only the left-handed neutrino has
weak interactions - There is no reason to even believe there is a
right-handed neutrino
- Weak interactions connect the left-handed
neutrino and electron
- Masses connect the left- and right-handed
electrons
17Mass and Couplings with One Lepton
- The Feynman rule for W-coupling for one lepton
- There is no reason, in principle, that the mass
cant be apparently complex
- This can easily be fixed, for example, by
redefining the field eR by a phase
- Hence the phase is irrelevant
- We work with eL and eR, and drop the primes
18Weak Interactions with Multiple Leptons
- Naively, there are three charged lepton fields
and neutrinos,
- The left- and right-handed pieces of the massive
electron have different weak interactions, and
should be divided
- Without mass, only the left-handed neutrino has
weak interactions - There is no reason to even believe there is a
right-handed neutrino
- Weak interactions connect the left-handed
neutrino and lepton
- Masses connect the left- and right-handed leptons
19The Weak coupling with Many leptons
- The Feynman rule for W-coupling for many leptons
20Complicated mass?
- There is no reason, in principle, that the mass
cant be apparently complicated
- This matrix is completely arbitrary
- We can nonetheless always change basis to
straighten it out
- For example, suppose the mass matrix looked like
this
- The new mass matrix is then
21Complicated Couplings?
- But we now defined new states
- This makes our W-couplings complicated
- In the leptons, this can be fixed simply by
similarly redefining the neutrinos
- Drop the irrelevant primes
22Weak Interactions with One Quark Pair
- Naively, there is one up quark and one down quark
- The left- and right-handed pieces of the massive
quarks have different weak interactions, and
should be divided
- All four of these exist in the Standard Model
- Weak interactions connect the up and down quarks
- Masses connect the left- and right-handed quarks
23The Weak coupling with many quarks
- The Feynman rule for W-coupling for many quarks
- Warning This is actually incorrect!
- This is different from the leptons
- As I will explain soon (I hope)
24Complicated mass?
- There is no reason, in principle, that the masses
cant be apparently complicated
- These matrices are completely arbitrary
- We can nonetheless always change basis to
straighten them out
- For example, suppose the mass matrices looked
like this
- The new mass matrix is then
25Complicated Couplings?
- But we now defined new states
- This makes our W-couplings complicated
- In the quarkss, can this can fixed simply by
similarly redefining the up quarks?
- No! This messes up the mass matrix M
- The couplings really are complicated in the quark
sector
26The Weak coupling with many quarks
- By appropriate redefinition of the various
fields, the mass matrices for the up- and
down-type quarks can always be made diagonal and
real
- Such a redefinition will, however, introduce a
unitary matrix V into the charged current
interactions
- This matrix is called the Cabibbo-Kobayashi-Maskaw
a matrix, or CKM matrix
- Some, but not all, of the parameters of V can be
eliminated by appropriate redefinition of the
corresponding fields.