7314-7360 Updates

Dec 12

For Yukawa terms, I found 12 ways to make SU(2) singlets, but only
three of these conserved hypercharge Y.  These are just enough to
give masses to e-, u, and d.  You need to use both methods for
making singlets.  Again, this is easiest if you use 2-component
Weyl notation for the fermions.

Note that if we included all three generations of quarks and
leptons, we'd have a copy of these three terms for each generation,
but also terms combining fermions from different generations.  The
collection of Yukawa couplings then forms the Cabibbo-Kobayashi-Maskawa
mixing matrix.


Dec 11

A couple comments about the last problem:

1.  You can assume that the neutrinos are massless and
purely left-handed, which is what we thought was true until
very recently.

2. Recall that when we discussed in lecture making singlets
(ie scalars or neutral combinations) in SU(N), there were two
ways to do it.  You can contract the components of a field
with an adjointed field, or you can use the N-dim antisymmetric
tensor contracted with either all fields, or all adjointed
fields.

3.  Focus on the Weyl (ie two-component) version of spinors,
(with sigma_mu instead of gamma_mu) which is more natural for both
the weak interactions, which treat left and right chirality differently,
and also for particles which would be massless except for the Higgs
mechanism.  After you've gotten things worked out, you can go to
Dirac (four-comp) notation by writing psi_L = (1-gamma5)/2 psi, etc.


I'll count the number of Yukawa terms I expect and post that
sometime soon, so you'll have an idea if you're done.  But you
know that all the massive fermions have to acquire their mass
by a Yukawa term, so that should give you a pretty good idea
of what's needed.

 

Nov 15

We haven't discussed crossing symmetry in lecture, and the
definition in the text seems at first a little involved.
However, if you apply it not to the matrix element but the
cross section, there are only a few ingredients left to
interchange, and it becomes very easy to apply.  In particular,
if you write d sigma/d Omega (for example) in terms of s, t and
u, the crossing symmetry just involves shuffling these.


Nov 13

As I look at 8.18, I realize you're being asked to consider a
limit that's a bit peculiar.  In particular, you're being asked
to neglect masses but work in the muon's rest frame, which isn't
possible.  What is meant (I think) is not to assume that M = 0,
which makes things awkward, but that M can be neglected relative
to momenta such as k and k'.  The easiest way I found to handle it
was to convert quantities like M k (which can't be neglected) to
invariants such as s before proceeding.


Nov 5

For AH 8.4, the prime on one of the fields is meant (I think) to mean
it can be any kind of fermion field; a muon field, for example, as
opposed to an electron field.  The two fields don't have to be for the
same type of particle.  (It doesn't matter, since all fermion
fields transform the same way.)  Don't confuse it with the primes AH
use to indicate a different Lorentz-transformed frame.  If you like,
you can take both to be the same field.  Also note that both fields are
 at the same x.  


Oct 29

For the helicity vector question, in the frame where k is along z,
you can answer it two ways.  First, just solve for k.eps = 0 in
that frame, and break the solutions into states with definite
helicity.  In that frame, helicity = J.P/|P|, is still just J3.
(It would be more complicated for a general k.)  You can use the
results from the previous question or from class to determine these.
The second method is to start in the rest frame with the three J3
eigenstates that satisfy k.eps =0, then boost them along z.  The
answers should be the same.

This is intended to be a simple problem.

Oct 15

For AH 10.5, you may assume that q^2 < 0 (that is, spacelike).  This
will generally be the case for the t or u channel diagrams AH discusses
in the text.  (The s channel case, where q^2 can be > 0, is a bit more
complicated, involving imaginary parts as we discussed in the optical
theorem.)


Oct 13

I've added one more section to the reading (AH 11.8).


Oct 9

I've added a couple missing solutions for the last set.


Oct 8 (still more)

Also, you may end up with an integral that oscillates wildly at the
endpoints.  You can make sense of these types of integrals (which show
up all over in fourier analysis, delta functions, field theory, ...)
by giving the integrand an exponential damping.  You can do this to
damp out a region of the integral that doesn't contribute anyway because
it oscillates to zero.  Ask me if this doesn't help.


Oct 8

For the problem where you extract the short-distance singular behavior
of the vev of two fields at nearly the same point, the dominant contribution
comes from large momentum.  So in estimating the leading behavior,
you may ignore the mass, and use E(p) approx= |p|.  This should
make the integrals much simpler.  Also, when I computed this, I treated
timelike and spacelike eps^2's separately, taking advantage of a
special frame in each case (since it's a scalar fn of eps).  This
may be useful, depending on how you set your calculation up.


Oct 4

For the problem on time order under Lorentz transformations, consider 
what the defining relation

 g_mu'_nu' = g_mu_nu Lambda^mu_mu' Lambda^nu_nu'

says about the components of Lambda by using the explicit form for
g_mu_nu.  It helps to then consider a generic frame as the transform
of the frame in which (x-y)^2 = tau^2; that is, the frame in which
the events occur at the same spatial point and tau is the proper time.
(This frame always exists for timelike separations.)


Sept 17

For the problem where you consider a non-parity invariant term for
the vector Lagrangian, recall that there is a tensor in addition to
g that you can use to make scalars, but which gives terms that are
only invariant under proper transformations, not parity or time
reversal.

Incidently, I've moved the due date to Thursday.


Sept 12

It might be helpful to know that, for an NxN matrix M

 eps_{i1 i2 .... iN} M^i1_j1  M^i2_j2 ... M^iN_jN = (Det[M]) eps_{j1 j2 ... jN}

(Here, ^ and _ are the usual tex notations for raised and lowered indices.  If
you find this unreadable, look here.

Just as we did for the metric, it's useful to turn this around and define a
proper Lorentz transformation as any transformation that leaves both g and epsilon
invariant.  (An improper transformation, such as parity and time reversal, leaves
only g invariant, but changes epsilon by a sign.)


Aug 31

I'm not sure why AH sometimes keep c's and hbars in formulae, but if
you would, please work consistently in natural units whenever possible.
In particular, set all those c's to 1 in the homework.


Aug 16

Class information will appear here.

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