Lecture 5: Feb 7th 2011
Reading: Griffiths Chapter 2, Perkins Chapter 2 handed out in class.
Homework: Griffiths: 2.2, 2.4, 2.5, Perkins 2.4, Hints for homework See end
1) Details of the Weak interaction
Examples:
Same types of diagrams as for the other two forces but now the W (not the Z) changes
particle types.
The weak force involves a large number in interactions with all the quark and leptons
since they all have flavor. When determining what sort of interactions are possible or
probable the guiding rules are that the interactions should conserve momentum and
energy, electric charge and various types of weak flavor when appropriate.
Some example interactions are
d-> uW- -> ue- anti e: seen for neutrons, n(ddu) -> p(duu) e- antie
- -> W-  -> e- anite 
d anti u -> W- -> e- anti e: seen for charged pions - -> e- anti e
s -> uW- -> ue- anti e: seen for charged kaons, for K-(s anti u) -> 0(u anit u) e- antie
changes generations.
Flavor changing between the quark generations can be explained by the idea that the
three down type particles d, s, and b and the three up type particles u, c, and t are
eigenstates of the strong force, but that the eigenstates of the weak force are slightly
different for the down type particles d’, s’ and b’ which can be expressed as a linear
combination of the d, s and b strong eigenstates. These coefficients can be summed up in
the 3x3 CKM rotation matrix.
Consider the b quark. When the b quark is in a meson or baryon coupled to other particles
by gluons so it is definitely in a strong eigenstate. Therefore, it is in some superposition
of d’, s’, and b’. When it interacts with the W+ it might interact as any of the three with
a probability governed by the components of the linier combination it is in (CKM matrix)
The W particle would convert b’ to t’ type eigenstates(particles) b(-1/3) -> W- t(2/3) with
highest probability, but this decay is forbidden since the t quark is much more massive
than the b quark If it interacts as a d’ what you will see is -> W- u or if a s’ what you
will see is -> W- c. Therefore there is a finite, but reduced, probability of decaying to a c
or a u quark. t is forbidden by energy conservation. According to the CKM matrix
constants c is much more likely.
This is the only way a particle from second and third generation can decay to the lowest
energy generation. These particles all have long lifetime and in fact hadrons with b
quarks have very long lifetimes since the b transitions to the other generations have small
probabilities. The superposition of b’, s’, d’ for the b quark is mostly b’.
The neutral Z particles have scattering and annihilation or pair production interactions
much like the photons. Though these interactions can also involve neutrinos. Flavor
changing between the generations like B->Zd is not possible for the Z particle. This can
be understood mathematically. The W an interaction can be written as <u|WUCKM|s>,
where a s strong eigenstate has some probability of being in a d’ weak eigenstate and
being converted to a u quark. The matrix U is unitary because it conserves probability.
For the Z, <d|U+ZU|s>, where U+ is the complex transform of U. However, U doesn’t act
on the Z since it isn’t a quark and U+ZU = Z, and <d|Z|s> has zero probability since d and
s are orthogonal eigenstates.
The idea of strong vs. weak eigenstates and the CMK matrix is designed to not allow
flavor changing by the neutral Z interaction at tree level. These types of interactions
were not observed and idea of the strong and weak eigenstates was a simple idea based
on the well established concept of having multiple basis’s of eigenstates that would
explain why these processes were not observed.
Note FCNC, flavor changing neutral currents does happen at higher order at very low
probability through higher order diagrams.
Note that the weak boson couple to each other also but the interactions are very low
probability so you don’t see them very often. They don’t lead to interesting new effects
like in QCD.
2) Review Particles and Interactions
Particles and charges
electron: electric charge -1, weak lepton charge +1e
anti electron neutrino: weak lepton charge(flavor) -1e, near 0 mass
anti-electron: electron charge +1, weak lepton charge -1e
electron neutrino: weak lepton charge(flavor) +1e, near 0 mass
3 generations of leptons: electrons, muons, and taus. 2nd and 3rd generations have
substantially more mass
charged leptons can’t convert between generation. neutrinos can convert between
generations via the non SM process of neutrino mixing.
up quark: electric charge +2/3, weak flavor, color r g or b
down quark: electric charge -1/3, weak flavor, color r g or b
anti up quark: electric charge -2/3, weak flavor, color anti r g or b
anti down quark: electric charge +1/3, weak flavor, color anti r g or b
3 generations of quarks c,s and t,b. 2nd and 3rd generations have substantially more mass.
Can convert between generations via the weak force with low probability.
EM interaction
Mediated by massless, electric charge 0 photon. Couples to all particles with electron
charge. Infinite range. Can form bound states.
Scattering and annihilation/pair production diagrams possible.
Strong interaction
Mediated by mass less, electric charge 0, bicolored gluon. Gluon can self interact which
results in short range nature of force, confinement of quarks, and formation of colorless
bound states.
Scattering and annihilation/pair production diagrams possible.
Weak interaction
Mediated by massive W+, W- and Z bosons. Interacts with objects with weak flavor
charge and self interacts. No bound states possible.
For scattering and annihilation/pair production diagrams possible.
For W a large number of diagrams are possible including ones that cross generations.
Violates many discreet symmetries that the other forces respect.
Coupling constants: Strong s = 1.3 (1.3 at 0.217GeV low energy, 0.3 at 1 GeV mid
energy, 0.1 at 100 GeV high energy)
EM:  = 1/137
Weak W = 10-5
Cross section: Strong 10-30 m2
EM 10-33
Weak 10-44 m2
Lifetime: Strong 10-23 sec
EM 10-20 sec
Weak 10-8 sec or longer
3) Unification
A milestone toward the standard model was the Z particle, the third quanta of the weak
force. This particle was neutral and had similar interactions to the electromagnetic force
such as e+e- -> Z -> e+e-. However it took a long time to find this particle since no one
expected it! Any type of interaction involving the Z had an equivalent higher probability
EM interaction, except for the neutrino interactions, which are very difficult to observe.
Later it was seen that as you observed interactions such as particle-particle annihilation at
higher and higher energies the constants that governed the electromagnetic and weak
interactions changed and eventually unified at a high value. At energies well above the Z
mass(91 times the proton mass) the mass of the carrier becomes irrelevant to the
interaction. This was the first strong hint that there had to be some unifying theory
behind all the diverse quantum field theories that had existed.
Another way to put this is the both interactions have the same coupling constant and it’s
just the extra factor of the mass of the gauge boson in the denominator that makes that
gives the difference between the strengths of the interaction. Doing that calculation using
the measured strength of the weak and electromagnetic interactions in similar
configurations involving photons and Zs gives mZ=91GeV. Exactly what was later
measured.
Also a milestone for theory since it was predicted before it was found!
In addition the strong force is seen to weaken at higher energies so it may also unify.
One mystery in this picture is still present. Why do the W and Z have large mass. To
explain this you need to introduce an additional field and particle that by coupling with
particles can give them mass. This is the Higgs particle. The Higgs may also gives mass
to all the other massive particles.
In fact the b, s and d states can be considered eigenstates of the Higgs interaction and the
strong interaction. As before the b’, s’ and d’ states are still the Weak eigenstates. The
Higgs particle interacts with any particle that has mass. Since these particles have mass
they are in eigenstates of the Higgs particle (for instance when in a configuration where
they have definite mass, such as a hadron). The neutrinos, if they had no mass would be
pure eigenstates of the Weak force. But since they do have some mass you can have
interactions change lepton flavor generation just like with the quarks.
These quantum field theories including the unified electroweak theory with the Higgs
particle to explain the mass of the W and Z make up the standard model (SM).
4) New physics
The above set of quantum field theories account for every process we have observed and
measured in collider experiments. So why would be we interested in new theories of
particle physics. The answer is that we have observed a number of phenomena that can’t
be explained by the SM. Also the Higgs piece of the SM is undiscovered
0)The Higgs particle - This is the subject of my current research.
1) Neutrinos have mass. Not in original SM. Also why is the scale for neutrino
masses so different from the other fermions.
2) Dark matter. Observations of galactic rotation curves, the expansion of the
Universe and gravitational lensing indicate that there is some sort of particle out
there that doesn’t interact via any of the forces except gravity. This matter would
not radiate so we call it dark matter.
3) Dark energy. The universal expansion rate is bigger than expected given the
amount of matter and dark matter. Something is pushing it apart. We call this
dark energy
4) Why is everything in the Universe made of matter? CP violating processes can
result in antimatter decaying slightly more often than matter, but nothing like the
observed rate.(A part of my research is investigating CP violating and matter
antimatter conversion processes)
There are a number of other problems as well that involve the theory, results of
calculations, and complexity of the standard model.
1) We would like the forces to unify at high energy. If you try to calculate the
strengths of the interaction at very high energies, like those at the big bang, you
find that they get very close to each other but don’t match up.
2) Some diagrams in the SM have divergent contributions at high energy.
Probabilities greater than 1 or 100%. WW scattering is the main example.
3) Why three generations, why all the masses we observe including such a large
range, why the coupling constants we measure and why so complex?
4) Why are some of the coupling constants as large or small as they are. To be
precisely the values we observe there has to be a near complete cancelation
between some contributions to the observed processes from loop diagrams. This
is known as the Hierarchy and/or fine tuning problem.
However, even though we know the SM has to be wrong it’s very important to
understand it in detail so we can understand how our particle detectors work and when
we have seen something new that we need to pursue it to start understanding the new
physics. Like Newton’s laws compared to relativity the SM should be considered a valid
low energy effective theory consistent with the full theory in the correct energy range.
Our particle detectors generally function in this lower energy range.
0) Hints for homework
Griffiths 2.4: A key elements in many high energy physics relativity problems is energy
and momentum conservation. You should choose a frame, rest frame of a particle or lab
frame as appropriate and then equate energy and momentum before and after the process.
Also remember that m2 = E2 +p2 and since E = mc2 and p = mv then v = pc2/E = p/E.
In almost all cases you can avoid velocity transformations when considering different
frames, which can make the problem quite complex.
Griffiths 2.5: How likely a process is will be determined by how many vertices there are
in the process, what coupling constants are present at each vertex, and whether there are
any additional multiplicative factors such as CKM matrix factors which make it less
likely to change flavor between generations. The best way to approach a problem like
this is to write down the starting and ending quarks on either end of your diagram and
then try to figure out what goes on in-between. An interesting example related to the
problem is D0 -> pi+pi compared to D0 -> K+K-. We will go over this on Thursday.