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IAIN STEWART: [INAUDIBLE]
theory graphs calculating
00:00:22.700 --> 00:00:24.075
the effective
theory graphs using
00:00:24.075 --> 00:00:27.290
the same infrared regulator,
renormalizing each of them
00:00:27.290 --> 00:00:29.930
separately, and then
subtracting them
00:00:29.930 --> 00:00:32.540
to figure out higher-order
corrections to the Wilson
00:00:32.540 --> 00:00:34.340
coefficients.
00:00:34.340 --> 00:00:40.618
And as part of that
discussion, we also
00:00:40.618 --> 00:00:42.410
talked about scheme
dependence because when
00:00:42.410 --> 00:00:44.270
you do that procedure
for normalization
00:00:44.270 --> 00:00:46.895
in the effective theory, you're
making a choice for the scheme.
00:00:46.895 --> 00:00:49.975
We picked m s bar, but you
could pick other choices.
00:00:49.975 --> 00:00:52.100
And at the end of lecture,
we talked about the fact
00:00:52.100 --> 00:00:55.010
that when you make
different scheme choices,
00:00:55.010 --> 00:00:56.610
it can affect things.
00:00:56.610 --> 00:00:59.180
So it can affect the
Wilson coefficients.
00:00:59.180 --> 00:01:01.340
It'll affect the matrix
elements of operators.
00:01:01.340 --> 00:01:03.380
It'll affect your
matching coefficients
00:01:03.380 --> 00:01:07.010
at one loop and your anomalous
dimensions at two loops,
00:01:07.010 --> 00:01:10.370
but those scheme dependencies
cancel out in observables.
00:01:10.370 --> 00:01:14.598
So it's important to take the
scheme into account and work
00:01:14.598 --> 00:01:15.390
in the same scheme.
00:01:15.390 --> 00:01:18.320
But if you consistently use
the same scheme for everything,
00:01:18.320 --> 00:01:20.550
then you will be OK.
00:01:20.550 --> 00:01:23.265
Now, if you're just doing
calculations on pen and paper,
00:01:23.265 --> 00:01:24.890
it's very easy that
you just-- well, we
00:01:24.890 --> 00:01:25.890
love the m s bar scheme.
00:01:25.890 --> 00:01:27.080
Let's use that.
00:01:27.080 --> 00:01:29.360
But remember that
sometimes, there
00:01:29.360 --> 00:01:31.288
will be things in
your result that you
00:01:31.288 --> 00:01:32.580
may want to get from elsewhere.
00:01:32.580 --> 00:01:34.350
For example, maybe there's
some matrix elements
00:01:34.350 --> 00:01:35.058
of the operators.
00:01:35.058 --> 00:01:36.760
You want to get them
from the lattice.
00:01:36.760 --> 00:01:38.510
Well, if you look up
results in a lattice,
00:01:38.510 --> 00:01:40.180
they're not using m s bar.
00:01:40.180 --> 00:01:43.130
There's no non-perturbative
definition of m s bar.
00:01:43.130 --> 00:01:45.440
So they'll be using
some other scheme.
00:01:45.440 --> 00:01:47.892
And those results will
have to be converted
00:01:47.892 --> 00:01:49.850
to m s bar if you're
going to put them together
00:01:49.850 --> 00:01:51.260
with your results.
00:01:51.260 --> 00:01:54.650
So that's just something
to be aware of.
00:01:54.650 --> 00:01:57.140
Before we leave this topic
and go on to something else,
00:01:57.140 --> 00:02:00.020
I thought I'd say a few
words about phenomenology.
00:02:00.020 --> 00:02:03.470
What is all this
technology good for?
00:02:03.470 --> 00:02:08.848
So a nice example of
that is beta s gamma.
00:02:08.848 --> 00:02:11.120
So beta s gamma--
00:02:11.120 --> 00:02:13.862
it's a neutral current process.
00:02:13.862 --> 00:02:16.070
It doesn't happen in the
Steiner model at tree level.
00:02:30.060 --> 00:02:32.010
And therefore, what that
means, if it doesn't
00:02:32.010 --> 00:02:33.427
happen at tree
level, is that it's
00:02:33.427 --> 00:02:36.780
sensitive to loop corrections.
00:02:36.780 --> 00:02:37.830
And you could go--
00:02:37.830 --> 00:02:43.050
since we talked about a
channel with b to c, you bar d,
00:02:43.050 --> 00:02:44.250
but this is very analogous.
00:02:44.250 --> 00:02:46.950
We have a b meson in
the initial-- b quark
00:02:46.950 --> 00:02:47.830
in the initial state.
00:02:47.830 --> 00:02:49.840
So it's the same
scales in the problem.
00:02:49.840 --> 00:02:52.740
Effectively, the b quark
scale is the light scale.
00:02:52.740 --> 00:02:54.720
We want to get rid
of the things that
00:02:54.720 --> 00:02:57.390
would mediate this decay that
are inside the loop, which
00:02:57.390 --> 00:03:02.530
will be w bosons and top
quarks in this example.
00:03:02.530 --> 00:03:05.670
And so you draw this
in the standard model.
00:03:05.670 --> 00:03:07.410
There's various diagrams
that contribute.
00:03:07.410 --> 00:03:09.960
One is this one.
00:03:15.270 --> 00:03:18.900
So here's a b quark changing
into a strange quark
00:03:18.900 --> 00:03:21.820
through a top quark and a w.
00:03:21.820 --> 00:03:25.560
And if we integrate
out the w in the top,
00:03:25.560 --> 00:03:27.600
then we get some local
operator, much as
00:03:27.600 --> 00:03:31.810
we were talking about
in our examples up here.
00:03:31.810 --> 00:03:34.740
It's just that we have
different diagrams.
00:03:34.740 --> 00:03:36.870
And we can also, from the
low-energy point of view,
00:03:36.870 --> 00:03:40.590
enumerate the operators.
00:03:40.590 --> 00:03:48.780
For some reason, I started
calling it Q instead of O.
00:03:48.780 --> 00:03:51.030
Let me give you an example
of some of these operators.
00:04:00.180 --> 00:04:03.960
So here's what you would
call a magnetic dipole
00:04:03.960 --> 00:04:07.020
operator, couples directly
to the photon, which
00:04:07.020 --> 00:04:09.540
is in the F mu nu.
00:04:09.540 --> 00:04:12.720
And it takes the b to an s bar.
00:04:12.720 --> 00:04:17.310
And that's an example of a
higher dimension operator.
00:04:17.310 --> 00:04:19.620
Remember, this is
dimension 2 and then plus 3
00:04:19.620 --> 00:04:21.300
there, so that's 5.
00:04:21.300 --> 00:04:23.940
There's a factor of the b
quark mass that just comes in
00:04:23.940 --> 00:04:26.670
because of the chiral structure
of the operator, which
00:04:26.670 --> 00:04:29.640
you can think of as there needs
to be a mass insertion here.
00:04:29.640 --> 00:04:32.280
You need one factor of that
mass in order for the diagram
00:04:32.280 --> 00:04:33.120
not to give 0.
00:04:37.950 --> 00:04:40.800
If you really start
to do beta s gamma,
00:04:40.800 --> 00:04:43.320
and you want to go
through the whole story,
00:04:43.320 --> 00:04:45.600
then you have to actually
think about other operators.
00:04:45.600 --> 00:04:48.305
And there's a whole
basis of them.
00:04:48.305 --> 00:04:49.680
And one thing you
could do is you
00:04:49.680 --> 00:04:52.290
could take the operator
up there and just
00:04:52.290 --> 00:04:53.760
replace the photon by a gluon.
00:04:57.450 --> 00:05:00.510
That won't give a tree-level
contribution to beta s gamma,
00:05:00.510 --> 00:05:03.043
but these guys are charged
under electromagnetism.
00:05:03.043 --> 00:05:04.710
So you could just
have a direct coupling
00:05:04.710 --> 00:05:06.930
to the b quark or
the strange quark,
00:05:06.930 --> 00:05:09.270
and then loop up the
gluon, and get a correction
00:05:09.270 --> 00:05:11.220
from an operator like this one.
00:05:11.220 --> 00:05:15.840
And then there's also
four quark operators,
00:05:15.840 --> 00:05:22.950
like the ones we
talked about before,
00:05:22.950 --> 00:05:24.480
but with different flavors.
00:05:32.550 --> 00:05:35.400
And if you enumerate
all of them,
00:05:35.400 --> 00:05:43.470
there's nine more different ways
of making four-quark operators.
00:05:43.470 --> 00:05:46.590
So you could build some basis
using the equation of motion
00:05:46.590 --> 00:05:48.540
to simplify the operators
as much as possible.
00:05:48.540 --> 00:05:52.190
And then you get
down to these ones.
00:05:52.190 --> 00:05:53.630
And you can go
through the program
00:05:53.630 --> 00:05:56.130
that we talked about, of doing
matching and renormalizations
00:05:56.130 --> 00:05:57.472
with evolution.
00:05:57.472 --> 00:05:59.180
And really what I want
to talk about here
00:05:59.180 --> 00:06:00.680
is a little bit
about phenomenology,
00:06:00.680 --> 00:06:03.110
because the interesting
thing about this loop
00:06:03.110 --> 00:06:06.470
is that if there was
nu physics-- since
00:06:06.470 --> 00:06:08.900
in the standard model at
tree level it doesn't happen,
00:06:08.900 --> 00:06:11.150
if there's nu physics, then
you can be sensitive to it
00:06:11.150 --> 00:06:13.340
here, because you're
sensitive to heavy particles
00:06:13.340 --> 00:06:14.460
in this loop.
00:06:14.460 --> 00:06:15.950
That's why beta
s gamma is always
00:06:15.950 --> 00:06:19.520
used to constrain nu physics.
00:06:19.520 --> 00:06:23.510
So in our effective theory, we
have just an operator that does
00:06:23.510 --> 00:06:30.800
this, which is this 07 gamma.
00:06:33.570 --> 00:06:38.670
And if you think about what
C 7 gamma is at lowest order,
00:06:38.670 --> 00:06:41.050
let's just calculate
that diagram over there.
00:06:41.050 --> 00:06:44.190
And the result from
doing that will
00:06:44.190 --> 00:06:47.010
be some function
of m w over m top,
00:06:47.010 --> 00:06:50.380
because those are the things
that are appearing in the loop.
00:06:50.380 --> 00:06:56.220
And if you do that calculation,
you get a number like that,
00:06:56.220 --> 00:07:00.180
if you stick values
for the top m w.
00:07:00.180 --> 00:07:03.390
And then you can start
doing loop corrections.
00:07:03.390 --> 00:07:05.350
The first thing you
might think about,
00:07:05.350 --> 00:07:07.680
which is actually
not even suppressed
00:07:07.680 --> 00:07:11.010
by any factors of the
coupling alpha s strong,
00:07:11.010 --> 00:07:19.280
would be to just loop up the Q1
operator, which I guess I don't
00:07:19.280 --> 00:07:22.610
know why I wrote it this way.
00:07:22.610 --> 00:07:25.280
This should be b.
00:07:25.280 --> 00:07:29.000
You could just loop the Q1
operator, contract the u u var,
00:07:29.000 --> 00:07:32.090
and then you can get a beta s
transition from this operator.
00:07:32.090 --> 00:07:36.770
So this is the u quark here,
and then attach a photon.
00:07:36.770 --> 00:07:39.940
And there's no factor in that
loop of the strong coupling,
00:07:39.940 --> 00:07:42.530
because this is an
electromagnetic coupling,
00:07:42.530 --> 00:07:45.920
and this guy here is
just this Q1 operator.
00:07:45.920 --> 00:07:48.470
So this doesn't look like
it's loop suppressed relative
00:07:48.470 --> 00:07:51.080
to that.
00:07:51.080 --> 00:07:53.610
And this is a little subtle,
but this guy actually is 0.
00:07:53.610 --> 00:08:01.860
So you have to use a
good scheme for gamma 5,
00:08:01.860 --> 00:08:04.460
but it turns out to be 0.
00:08:04.460 --> 00:08:07.110
So the first type of loop
corrections that you get,
00:08:07.110 --> 00:08:10.460
which are suppressed
by a factor of alpha s,
00:08:10.460 --> 00:08:12.170
come from diagrams like--
00:08:12.170 --> 00:08:14.600
well, there's various
diagrams, but one of them
00:08:14.600 --> 00:08:17.360
is like this, where I take
the same diagram there
00:08:17.360 --> 00:08:21.230
and I attach an extra
gluon on top of it.
00:08:21.230 --> 00:08:21.995
This guy diverges.
00:08:24.500 --> 00:08:33.080
And this two-loop calculation
is order alpha strong,
00:08:33.080 --> 00:08:36.530
and it gives the leading
order anomalous dimension,
00:08:36.530 --> 00:08:38.299
what we were calling gamma 0.
00:08:41.000 --> 00:08:42.230
It's not the only diagram.
00:08:42.230 --> 00:08:43.429
There's other diagrams, too.
00:08:45.950 --> 00:08:49.070
So we could go through that and
do the similar type of thing,
00:08:49.070 --> 00:08:52.290
just with more diagrams
than we had in our example.
00:08:52.290 --> 00:08:54.620
In particular, we
could construct
00:08:54.620 --> 00:08:56.990
from the tree level matching
in the one-loop anomalous
00:08:56.990 --> 00:09:03.350
dimension, the leading
log result. Let me
00:09:03.350 --> 00:09:04.610
just write that down for you.
00:09:17.810 --> 00:09:38.187
Putting in numbers for the
anomalous dimensions, at least
00:09:38.187 --> 00:09:39.770
sometimes putting
in numbers for them.
00:09:47.370 --> 00:09:52.460
So the eta factor here
is the ratio of alphas.
00:09:52.460 --> 00:09:56.420
So this is similar to what
we saw before in our example
00:09:56.420 --> 00:10:00.730
where we got a ratio of
alphas raised to a power.
00:10:00.730 --> 00:10:03.770
And if you want to pick a mu
for this process over here,
00:10:03.770 --> 00:10:08.150
beta s gamma, then the right
mu to think about is m b.
00:10:08.150 --> 00:10:13.010
So we want to take
mu to the m b.
00:10:24.650 --> 00:10:27.102
And so if I plug in numbers--
00:10:27.102 --> 00:10:29.060
and this is really what
I wanted to emphasize--
00:10:29.060 --> 00:10:32.480
if I plug in numbers here
for these various things,
00:10:32.480 --> 00:10:35.300
this guy here gives
a factor of 0.7.
00:10:35.300 --> 00:10:41.030
This guy here is this minus 0.2
that we talked about up there.
00:10:41.030 --> 00:10:47.090
This factor here is
a bit small, 0.085.
00:10:47.090 --> 00:10:50.690
0.96.
00:10:50.690 --> 00:10:52.360
That's right.
00:10:52.360 --> 00:10:59.130
And then this piece here is
a substantial correction.
00:10:59.130 --> 00:11:02.150
And if I wrote down all
these numbers correctly,
00:11:02.150 --> 00:11:06.440
then the final result
comes out to be,
00:11:06.440 --> 00:11:11.180
if I keep three digits,
minus 0.3, which you can see
00:11:11.180 --> 00:11:14.440
is a fairly substantial
change from minus 0.2.
00:11:14.440 --> 00:11:16.670
It's a 50% change.
00:11:16.670 --> 00:11:20.810
So just taking into
account the evolution
00:11:20.810 --> 00:11:23.215
gives a 50% correction.
00:11:23.215 --> 00:11:25.590
So if you didn't take it into
account, and you just said,
00:11:25.590 --> 00:11:27.090
well, forget about
effective theory.
00:11:27.090 --> 00:11:28.590
I just calculate this graph.
00:11:28.590 --> 00:11:32.940
That's the standard model, you'd
think that there is nu physics.
00:11:32.940 --> 00:11:36.710
We've certainly tested beta
s gamma at better than 50%,
00:11:36.710 --> 00:11:38.582
more like the 10% level.
00:11:38.582 --> 00:11:40.790
So you really have to take
into account these effects
00:11:40.790 --> 00:11:42.920
that we've been talking
about, like this leading log
00:11:42.920 --> 00:11:45.462
evolution, if you want to look
for nu physics, because you've
00:11:45.462 --> 00:11:52.745
got to get the right standard
model result. 50% larger.
00:11:56.860 --> 00:12:01.240
And actually, people go
two orders beyond what
00:12:01.240 --> 00:12:02.340
I'm talking about here.
00:12:02.340 --> 00:12:04.690
They go to the next, the
next leading log order,
00:12:04.690 --> 00:12:07.130
when they really do precision
beta s gamma physics.
00:12:07.130 --> 00:12:10.870
So some of the state of the
art calculations of multiloop
00:12:10.870 --> 00:12:13.150
diagrams have been done
exactly for beta s gamma
00:12:13.150 --> 00:12:15.250
because these effects are
so important for looking
00:12:15.250 --> 00:12:15.875
for nu physics.
00:12:42.850 --> 00:12:44.600
And it's even worse
when you get it put it
00:12:44.600 --> 00:12:47.895
in the branching ratio, because
a 50% enhancement in a coupling
00:12:47.895 --> 00:12:49.520
when you put it in
the branching ratio,
00:12:49.520 --> 00:12:50.790
you're squaring the amplitude.
00:12:50.790 --> 00:12:54.950
So that's a factor of 2.3.
00:12:54.950 --> 00:12:57.315
So these are really
crucial corrections
00:12:57.315 --> 00:12:58.190
to take into account.
00:13:00.490 --> 00:13:02.990
So that's what this electroweak
Hamiltonian is actually used
00:13:02.990 --> 00:13:06.620
for when you do phenomenology.
00:13:06.620 --> 00:13:10.370
So that's what I wanted to
say about the electroweak
00:13:10.370 --> 00:13:12.317
Hamiltonian, just to
give you a flavor for it.
00:13:12.317 --> 00:13:14.150
There's lots more that
you could do with it.
00:13:14.150 --> 00:13:16.550
We could talk about
more phenomenology,
00:13:16.550 --> 00:13:18.440
but let me stop there,
since the idea is
00:13:18.440 --> 00:13:21.080
to give you an introduction
to the concepts
00:13:21.080 --> 00:13:22.740
and we've done that.
00:13:22.740 --> 00:13:24.510
So now we'll move on
to something else,
00:13:24.510 --> 00:13:28.308
which is a different concept,
unless there's any questions.
00:13:36.930 --> 00:13:38.915
So all this business
of schemes and stuff
00:13:38.915 --> 00:13:41.040
comes in when people are
talking about beta s gamma
00:13:41.040 --> 00:13:44.738
and making this kind
of model prediction.
00:13:44.738 --> 00:13:47.280
And there's actually schemes
you can pick where you mess this
00:13:47.280 --> 00:13:49.760
up, but then, if
you're careful, you
00:13:49.760 --> 00:13:51.690
get the same answer
in the very end.
00:14:04.140 --> 00:14:07.020
So the next topic that
I want to talk about
00:14:07.020 --> 00:14:08.832
is an example of
something that's bottom up
00:14:08.832 --> 00:14:09.790
effective field theory.
00:14:09.790 --> 00:14:12.240
We've talked about top
down with this example
00:14:12.240 --> 00:14:14.340
of removing heavy particles.
00:14:14.340 --> 00:14:16.620
And the classic
example of bottom
00:14:16.620 --> 00:14:29.557
up is chiral perturbation
theory or chiral Lagrangians.
00:14:34.030 --> 00:14:36.370
So our purposes here are not--
00:14:36.370 --> 00:14:38.050
again, they're not
a full exploration
00:14:38.050 --> 00:14:41.380
of this topic, which
is a very large topic.
00:14:45.890 --> 00:14:48.260
So what are our goals?
00:14:48.260 --> 00:14:49.720
Bottom up effective
theory example.
00:15:00.540 --> 00:15:03.860
We will also see in
this example the utility
00:15:03.860 --> 00:15:06.260
of using something that's
called the nonlinear
00:15:06.260 --> 00:15:09.620
realization of symmetry,
non-linear symmetry
00:15:09.620 --> 00:15:14.780
representations, and
the kind of connection
00:15:14.780 --> 00:15:16.160
to field redefinitions.
00:15:27.907 --> 00:15:29.490
Since field redefinitions
is something
00:15:29.490 --> 00:15:31.490
we've been talking about,
we'll talk about that.
00:15:34.220 --> 00:15:36.120
Another thing that
this is an example of
00:15:36.120 --> 00:15:40.740
is an example where loops are
not suppressed by the coupling
00:15:40.740 --> 00:15:43.470
constant.
00:15:43.470 --> 00:15:47.040
Instead, they're actually
suppressed by powers
00:15:47.040 --> 00:15:48.744
in the power expansion.
00:16:02.300 --> 00:16:04.480
So that's kind of totally
different than what
00:16:04.480 --> 00:16:07.690
we saw when we were integrating
all the heavy particles,
00:16:07.690 --> 00:16:09.760
where you just put a loop.
00:16:09.760 --> 00:16:11.655
You're down by some
factor of alpha s,
00:16:11.655 --> 00:16:13.780
but you're at the same
order in the power expansion
00:16:13.780 --> 00:16:15.960
and 1 over large scale.
00:16:15.960 --> 00:16:18.810
This is going to be different.
00:16:18.810 --> 00:16:21.930
And therefore, in some ways
this has a non-trivial power
00:16:21.930 --> 00:16:24.870
accounting--
00:16:24.870 --> 00:16:27.630
more non-trivial,
anyway, than what
00:16:27.630 --> 00:16:28.810
we were just talking about.
00:16:28.810 --> 00:16:31.960
So we'd like to give this as
an example of non-trivial power
00:16:31.960 --> 00:16:34.680
accounting, and in fact,
prove something that's called
00:16:34.680 --> 00:16:36.905
a "power accounting theorem."
00:16:36.905 --> 00:16:38.280
What the power
accounting theorem
00:16:38.280 --> 00:16:40.230
means is that ahead
of time, you should
00:16:40.230 --> 00:16:42.690
be able to figure out from
your effective theory what
00:16:42.690 --> 00:16:44.070
order various things are.
00:16:44.070 --> 00:16:45.540
If you draw a
diagram, you should
00:16:45.540 --> 00:16:49.950
know even before you calculate
it how many powers in the power
00:16:49.950 --> 00:16:51.528
accounting expansion you have.
00:16:51.528 --> 00:16:53.070
If you didn't know
that, you wouldn't
00:16:53.070 --> 00:16:55.660
know which diagrams to compute
and which ones not to compute.
00:16:55.660 --> 00:16:57.160
You'd just have to
compute them all.
00:16:57.160 --> 00:16:58.930
That's, of course,
way too much work.
00:16:58.930 --> 00:17:01.900
So in order to formulate
the effective theory--
00:17:01.900 --> 00:17:04.773
especially since there's an
infinite number of diagrams.
00:17:04.773 --> 00:17:06.690
So in order to formulate
the theory, remember,
00:17:06.690 --> 00:17:09.630
you really have to have power
accounting under control.
00:17:09.630 --> 00:17:12.420
And that means you need
things like this in order
00:17:12.420 --> 00:17:15.210
to identify what's
leading order.
00:17:15.210 --> 00:17:16.950
So we'll talk about that.
00:17:16.950 --> 00:17:20.550
So I'm imagining that
maybe 50% of the class
00:17:20.550 --> 00:17:23.217
has seen chiral perturbation
theory before in some form.
00:17:23.217 --> 00:17:25.050
I know I teach it in
Quantum Field Theory 3,
00:17:25.050 --> 00:17:28.517
so if you took QFT 3 with me,
you saw a glimpse into it.
00:17:28.517 --> 00:17:30.600
The things we're going to
emphasize here are a bit
00:17:30.600 --> 00:17:34.230
different, but I am going to
assume that you have knowledge
00:17:34.230 --> 00:17:38.310
of it at some level, and I'll
assign you reading if you
00:17:38.310 --> 00:17:39.702
don't.
00:17:39.702 --> 00:17:41.160
The other thing
I'm going to assume
00:17:41.160 --> 00:17:43.452
that you have some knowledge
of is spontaneous symmetry
00:17:43.452 --> 00:17:45.125
breaking, because
that's not the topic.
00:17:45.125 --> 00:17:47.250
That's not one of the things
that I've listed here.
00:17:47.250 --> 00:17:50.130
That's not something that I
really want to delve into,
00:17:50.130 --> 00:17:55.110
but of course, if we talk about
the chiral Lagrangian, that's
00:17:55.110 --> 00:17:57.990
something that comes in,
in particular, when we're
00:17:57.990 --> 00:17:59.580
talking about it from QCD.
00:17:59.580 --> 00:18:07.830
So I'll remind you of some
things that are hopefully
00:18:07.830 --> 00:18:09.810
familiar, and anything
that's unfamiliar,
00:18:09.810 --> 00:18:11.310
you should do
additional reading on.
00:18:17.472 --> 00:18:19.180
So I know that there
are some people that
00:18:19.180 --> 00:18:21.990
are taking QFT 3 right now,
so this may be something
00:18:21.990 --> 00:18:22.990
they haven't got to yet.
00:18:22.990 --> 00:18:27.040
So I will do some review,
but for further reading,
00:18:27.040 --> 00:18:29.720
you should see QFT 3
or some other source.
00:18:29.720 --> 00:18:33.700
I've posted some
readings on the website.
00:18:40.590 --> 00:18:43.190
So if we start with
QCD, massless QCD,
00:18:43.190 --> 00:18:44.970
we can divide it into
a left-handed part
00:18:44.970 --> 00:18:46.040
and a right-handed part.
00:18:54.980 --> 00:18:57.200
And then we can talk
about the transformation
00:18:57.200 --> 00:18:59.780
where we take the
left-handed field
00:18:59.780 --> 00:19:02.750
and transform it by
a separate amount
00:19:02.750 --> 00:19:09.680
than the right-handed field
under a unitary transformation.
00:19:09.680 --> 00:19:13.440
And that's a symmetry
of the theory.
00:19:13.440 --> 00:19:15.620
Now, exactly what
symmetry you have
00:19:15.620 --> 00:19:18.498
depends on how many
components you're
00:19:18.498 --> 00:19:19.790
talking about in the psi field.
00:19:19.790 --> 00:19:22.610
If I say the psi field
is light, then you
00:19:22.610 --> 00:19:26.553
might think of up and down,
and that's one possibility.
00:19:26.553 --> 00:19:28.220
Or you could throw
the strange in there,
00:19:28.220 --> 00:19:29.570
and say, well, the
strange is light, too,
00:19:29.570 --> 00:19:31.237
and then you have up,
down, and strange.
00:19:33.480 --> 00:19:35.820
And that just the difference
between SU2 and SU3,
00:19:35.820 --> 00:19:41.590
and both of these
are viable choices.
00:19:41.590 --> 00:19:45.090
So let's make a little table.
00:19:45.090 --> 00:19:49.350
We have a group, which is
the transformations given
00:19:49.350 --> 00:19:51.510
by the left and the right.
00:19:51.510 --> 00:19:54.247
And it's going to be
broken spontaneously
00:19:54.247 --> 00:19:55.080
to some other group.
00:19:59.138 --> 00:19:59.930
So it could be SU3.
00:20:03.880 --> 00:20:06.040
So there's saying that
the left and the right
00:20:06.040 --> 00:20:08.470
are in SU3, each of them.
00:20:08.470 --> 00:20:12.235
And in QCD, that's broken
to the vector subgroup.
00:20:15.010 --> 00:20:17.623
So in this case, the
psi would be u d s.
00:20:20.800 --> 00:20:24.070
And you get Goldstone
bosons from that.
00:20:24.070 --> 00:20:26.700
And there's eight of them.
00:20:26.700 --> 00:20:28.796
And that's the pions,
the kaon, and the eta.
00:20:32.388 --> 00:20:34.180
And keeping thing from
our perspective what
00:20:34.180 --> 00:20:44.290
is the expansion parameter,
and it's not so great,
00:20:44.290 --> 00:20:47.260
because the strange quark
mass is not that light.
00:20:47.260 --> 00:20:50.740
So you can think that the
strange quark mass over lambda
00:20:50.740 --> 00:20:55.240
QCD is maybe something
like 1/3-ish,
00:20:55.240 --> 00:20:58.910
but you're not really
getting better than that.
00:20:58.910 --> 00:21:03.790
So there was 8 generations
here, 8 generators here,
00:21:03.790 --> 00:21:05.860
broken to 8 here.
00:21:05.860 --> 00:21:09.560
And then you have 8 Goldstones.
00:21:09.560 --> 00:21:14.250
You could do better, but
make less predictions,
00:21:14.250 --> 00:21:20.360
if you just considered SU2 and
just maybe up and down light.
00:21:20.360 --> 00:21:23.090
Then you just have the pions.
00:21:23.090 --> 00:21:28.350
And then you just have MU
and MD over lambda QCD,
00:21:28.350 --> 00:21:33.380
which is more like 1/50,
so a much better expansion.
00:21:33.380 --> 00:21:35.270
But then you can't
do kaon physics
00:21:35.270 --> 00:21:38.390
with this, because
the kaon is not
00:21:38.390 --> 00:21:43.020
part of the effective theory,
and that can be the case.
00:21:43.020 --> 00:21:46.663
So in order to construct a
theory for the Goldstones,
00:21:46.663 --> 00:21:48.330
which are the light
degrees of freedom--
00:21:48.330 --> 00:21:52.730
so this is we've identified
the light degrees of freedom--
00:21:52.730 --> 00:21:55.860
you'd like to construct an
effective theory for them.
00:21:55.860 --> 00:21:57.620
Those are bound states
of the particles
00:21:57.620 --> 00:22:01.590
in your original theory, and you
don't know, on pen and paper,
00:22:01.590 --> 00:22:05.120
how to calculate the matching.
00:22:05.120 --> 00:22:08.090
And that's characteristic of
a bottom up effective theory--
00:22:08.090 --> 00:22:12.230
that you don't do the
matching, that you just
00:22:12.230 --> 00:22:15.250
start from the bottom up.
00:22:15.250 --> 00:22:17.000
So in this case, you
don't do the matching
00:22:17.000 --> 00:22:21.339
because it's non-perturbative.
00:22:42.320 --> 00:22:44.150
So what you do is you
say, let's construct
00:22:44.150 --> 00:22:46.190
a type of field theory
based on the fact
00:22:46.190 --> 00:22:48.140
that I know what the
degrees of freedom are
00:22:48.140 --> 00:22:50.612
and I know something
about symmetry.
00:22:50.612 --> 00:22:52.070
And in particular
in this case, you
00:22:52.070 --> 00:22:53.695
know something about
symmetry breaking.
00:23:05.050 --> 00:23:08.080
So one kind of logic
is, then what you'll get
00:23:08.080 --> 00:23:10.330
is you'll get coefficients
times operators again.
00:23:12.880 --> 00:23:17.650
The operators will be built out
of your pion/kaon/eta fields.
00:23:17.650 --> 00:23:20.320
And you'll be able to calculate
matrix elements of these.
00:23:26.477 --> 00:23:29.060
But you'll get some coefficients
and you don't know the value.
00:23:29.060 --> 00:23:33.592
So these coefficients
you could fit to data,
00:23:33.592 --> 00:23:35.050
because you haven't
determined them
00:23:35.050 --> 00:23:36.910
from some high-level theory.
00:23:36.910 --> 00:23:41.290
You just could fit them to
the data and do phenomenology.
00:23:41.290 --> 00:23:42.820
So it's kind of
exactly the opposite
00:23:42.820 --> 00:23:44.290
from our high-energy
point of view,
00:23:44.290 --> 00:23:46.762
from the theory with the
electroweak Hamiltonian, where
00:23:46.762 --> 00:23:48.220
we thought the
matrix elements were
00:23:48.220 --> 00:23:49.532
going to be non-perturbative.
00:23:49.532 --> 00:23:50.740
Here are the matrix elements.
00:23:50.740 --> 00:23:51.760
They're something you calculate.
00:23:51.760 --> 00:23:53.177
And the coefficients
are including
00:23:53.177 --> 00:23:54.940
the non-perturbative physics.
00:23:57.810 --> 00:23:59.560
You could also get
these from the lattice.
00:24:03.560 --> 00:24:06.890
So a lattice QCD
calculation can tell you
00:24:06.890 --> 00:24:08.570
the values for the C's to use.
00:24:08.570 --> 00:24:10.612
And then you could just
use the chiral Lagrangian
00:24:10.612 --> 00:24:13.670
to do phenomenology.
00:24:13.670 --> 00:24:17.180
Now, because we have this
bottom-up point of view,
00:24:17.180 --> 00:24:20.400
there's something
that we don't know.
00:24:20.400 --> 00:24:24.320
And that is, we actually
don't know precisely what
00:24:24.320 --> 00:24:29.420
theory we started
with, because all we're
00:24:29.420 --> 00:24:34.220
encoding about that theory is
the symmetry breaking pattern.
00:24:34.220 --> 00:24:37.670
And so if there's a bunch
of upper-level theories
00:24:37.670 --> 00:24:41.364
that have the same
symmetry breaking pattern,
00:24:41.364 --> 00:24:42.810
they all look the same.
00:24:42.810 --> 00:24:45.337
They'll look like they have
the same chiral Lagrangian.
00:24:55.370 --> 00:24:58.782
And the thing that
distinguishes them
00:24:58.782 --> 00:25:00.740
is that they would have
different coefficients.
00:25:00.740 --> 00:25:02.782
And you wouldn't know that
unless you figured out
00:25:02.782 --> 00:25:06.080
what the coefficients are.
00:25:06.080 --> 00:25:07.640
So that's another
way of just saying
00:25:07.640 --> 00:25:09.440
that the high-energy
physics is being
00:25:09.440 --> 00:25:13.670
encoded in the coefficients.
00:25:13.670 --> 00:25:16.280
So the same chiral
Lagrangian would show up
00:25:16.280 --> 00:25:19.265
for different
high-level theories
00:25:19.265 --> 00:25:21.140
that have the same
symmetry breaking pattern.
00:25:30.580 --> 00:25:33.160
So this is just one example
of chiral Lagrangians.
00:25:33.160 --> 00:25:35.260
And we're going to use
it as an example, which
00:25:35.260 --> 00:25:38.920
is perhaps the most familiar,
to illustrate our bullets.
00:25:50.990 --> 00:25:53.210
So one of our
bullets was related
00:25:53.210 --> 00:25:55.632
to non-linear representations
in field redefinition,
00:25:55.632 --> 00:25:56.840
so let's start with that one.
00:26:12.960 --> 00:26:14.345
Problems set's due today.
00:26:14.345 --> 00:26:15.345
Problem set 2 is posted.
00:26:22.468 --> 00:26:24.760
So let me talk about something
called the "linear sigma
00:26:24.760 --> 00:26:26.650
model."
00:26:26.650 --> 00:26:28.690
And we'll use this as
an example of something
00:26:28.690 --> 00:26:30.482
that has a symmetry
breaking pattern that's
00:26:30.482 --> 00:26:33.610
similar to the one we want,
same as the one we want.
00:26:36.920 --> 00:26:40.450
And we'll construct, from
this, the chiral Lagrangian.
00:26:40.450 --> 00:26:43.040
And since it's the same
symmetry breaking pattern,
00:26:43.040 --> 00:26:49.390
it's also a viable chiral
Lagrangian directly for QCD.
00:26:49.390 --> 00:26:51.280
So what is the
linear sigma model?
00:26:51.280 --> 00:26:53.410
So I'll talk about
fields that I call pi
00:26:53.410 --> 00:26:55.540
without a vector symbol on top.
00:26:55.540 --> 00:26:56.840
They have two components.
00:26:56.840 --> 00:27:00.370
One component that's in the
diagonal, then one component--
00:27:00.370 --> 00:27:06.210
these are matrices
in the off diagonal,
00:27:06.210 --> 00:27:10.620
and different entries
in the sigma 3.
00:27:10.620 --> 00:27:13.740
So I want to think of this
is a kind of full theory.
00:27:18.452 --> 00:27:25.500
So the full theory for the
sigma model is the following--
00:27:25.500 --> 00:27:38.970
kinetic term, mass
term, interaction term.
00:27:45.040 --> 00:27:46.720
And I can couple it
to something else
00:27:46.720 --> 00:27:58.320
like a fermion to make
it more interesting,
00:27:58.320 --> 00:27:59.998
with Yukawa couplings.
00:28:11.460 --> 00:28:24.140
And this theory has an SU2 left
cross SU2 right symmetry, where
00:28:24.140 --> 00:28:32.090
I take psi left to L psi left,
psi right to R psi right.
00:28:32.090 --> 00:28:36.920
And I also transform
pi to L pi R dagger.
00:28:42.720 --> 00:28:45.120
And if you think of these
L's and these R's, you
00:28:45.120 --> 00:28:50.590
should think of them as
something with some generators.
00:28:50.590 --> 00:28:58.640
So L would have something like
that, with tau left generators,
00:28:58.640 --> 00:29:01.910
and then some parameters alpha
L. If I worked infinitesimally
00:29:01.910 --> 00:29:04.785
in those parameters, then
if I expanded this out,
00:29:04.785 --> 00:29:06.410
this would give a
linear transformation
00:29:06.410 --> 00:29:08.300
to the sigma and the pi vector.
00:29:10.905 --> 00:29:12.655
That's what it's meant,
that it's linear--
00:29:18.470 --> 00:29:22.830
so a linear infinitesimal
transformation to pi and sigma.
00:29:26.410 --> 00:29:29.050
So this theory has
spontaneous symmetry breaking.
00:29:39.880 --> 00:29:56.410
And if we write out the
potential by taking the traces,
00:29:56.410 --> 00:29:57.910
and we can write
in a way that makes
00:29:57.910 --> 00:30:01.810
that obvious by
completing the square.
00:30:09.220 --> 00:30:11.500
And we can always throw
away irrelevant constants
00:30:11.500 --> 00:30:13.420
when we do that.
00:30:13.420 --> 00:30:15.390
So we get something like that.
00:30:15.390 --> 00:30:16.600
So the minimum is shifted.
00:30:21.790 --> 00:30:24.340
And so if we want to
shift ourselves over
00:30:24.340 --> 00:30:26.830
to describe perturbations
around that vacuum,
00:30:26.830 --> 00:30:28.970
then we can do that.
00:30:28.970 --> 00:30:35.970
So let this guy have a
VEV, which I'll call curly
00:30:35.970 --> 00:30:39.705
V, square root mu
squared over lambda.
00:30:42.840 --> 00:30:45.420
Pi vector field has
no VEV, because we
00:30:45.420 --> 00:30:48.120
want to maintain
the vector symmetry.
00:30:48.120 --> 00:30:49.900
And we talk about
a shifted field,
00:30:49.900 --> 00:30:53.190
which describes
perturbations around the VEV
00:30:53.190 --> 00:30:57.450
when we do quantum field
theory in the sigma twiddle.
00:30:57.450 --> 00:31:00.750
And then we write our Lagrangian
in terms of the sigma twiddle,
00:31:00.750 --> 00:31:02.250
just by making a
change of variable.
00:31:12.380 --> 00:31:14.690
And that makes clear
who's getting masses
00:31:14.690 --> 00:31:17.240
when we expand
around that vacuum
00:31:17.240 --> 00:31:18.500
and who's remaining massless.
00:31:18.500 --> 00:31:20.708
And of course, the Goldstones
are remaining massless,
00:31:20.708 --> 00:31:23.060
and the Goldstones, by
clever choice of notation,
00:31:23.060 --> 00:31:23.810
are called Pi.
00:31:47.762 --> 00:31:50.800
Let's see if I can
squeeze it all in here.
00:31:53.570 --> 00:31:56.120
This is something they tell you
in Professor 101 never to do,
00:31:56.120 --> 00:31:57.370
but I'm going to do it anyway.
00:32:08.190 --> 00:32:10.617
Since if you can't
read something,
00:32:10.617 --> 00:32:12.450
you can always look at
my notes when I them.
00:32:15.480 --> 00:32:19.770
So I didn't write all the
fermion notes down again,
00:32:19.770 --> 00:32:23.550
but I could do the field
redefinition in those modes
00:32:23.550 --> 00:32:31.340
as well, the shift
in those terms.
00:32:36.360 --> 00:32:40.980
And the vector subgroup,
which remains unbroken,
00:32:40.980 --> 00:32:47.400
is no transformation for
sigma and the Pi field
00:32:47.400 --> 00:32:52.260
has left equal to right, so
we just get some matrix V
00:32:52.260 --> 00:32:54.940
on both sides.
00:32:54.940 --> 00:32:58.048
And if we look at the
Lagrangian that we have,
00:32:58.048 --> 00:33:00.090
then we can identify the
masses of various things
00:33:00.090 --> 00:33:00.720
at tree level.
00:33:09.430 --> 00:33:11.170
And I didn't write
down the fermion part,
00:33:11.170 --> 00:33:14.710
but the fermion gets mass 2
from the [INAUDIBLE] coupling,
00:33:14.710 --> 00:33:18.070
just like in the standard model.
00:33:18.070 --> 00:33:20.453
And the Pi, of course,
remain massless.
00:33:20.453 --> 00:33:22.870
And the idea of thinking about
this is an effective theory
00:33:22.870 --> 00:33:24.453
is that we can take
these to be large.
00:33:28.360 --> 00:33:29.920
And then there's
a clear separation
00:33:29.920 --> 00:33:33.010
between low-energy
degrees of freedom
00:33:33.010 --> 00:33:36.827
and heavy degrees of freedom.
00:33:36.827 --> 00:33:39.160
So if we want to describe
this with an effective theory,
00:33:39.160 --> 00:33:41.243
we would like to describe
the physics of the pions
00:33:41.243 --> 00:33:44.466
without worrying too much
about the sigma and the psi.
00:33:48.040 --> 00:33:50.830
So we could do that just by
thinking about this Lagrangian,
00:33:50.830 --> 00:33:53.590
but it turns out that you
can make field redefinitions
00:33:53.590 --> 00:33:55.930
and think about using
different formulations which
00:33:55.930 --> 00:33:57.700
are entirely equivalent.
00:33:57.700 --> 00:34:00.560
And some of those are more
useful than this linear sigma
00:34:00.560 --> 00:34:01.060
model.
00:34:03.690 --> 00:34:05.680
So we're thinking here
of field redefinitions
00:34:05.680 --> 00:34:07.170
as an organizational tool.
00:34:22.159 --> 00:34:24.960
So you'll see what I mean
when we go through this.
00:34:24.960 --> 00:34:30.239
So let's consider some
different choices.
00:34:30.239 --> 00:34:33.087
So there's something called the
"square root representation."
00:34:36.500 --> 00:34:53.989
So I just make a field
redefinition like that--
00:34:53.989 --> 00:34:56.280
involves a square root.
00:34:56.280 --> 00:34:59.645
And if I expand it, it
starts a sigma twiddle,
00:34:59.645 --> 00:35:01.933
and then it goes a
bunch of other terms.
00:35:01.933 --> 00:35:03.350
So it's certainly
within the realm
00:35:03.350 --> 00:35:06.590
of the things that are allowed
by our field redefinition
00:35:06.590 --> 00:35:08.060
theorem.
00:35:08.060 --> 00:35:14.270
And then I also talk about
making a field redefinition
00:35:14.270 --> 00:35:15.320
for the Pi as well.
00:35:30.870 --> 00:35:37.527
And again, this is
Pi plus other terms.
00:35:37.527 --> 00:35:39.110
So in this square
root representation,
00:35:39.110 --> 00:35:42.270
we're going over to
these fields S and psi.
00:35:42.270 --> 00:35:45.000
So these are our new fields.
00:35:45.000 --> 00:35:47.420
So we can make that
change of variable
00:35:47.420 --> 00:35:50.350
and just write out our
linear sigma model again.
00:35:55.790 --> 00:35:58.220
And it's the same theory,
it's just field redefined.
00:36:08.800 --> 00:36:13.257
And it looks kind of Godawful,
but it's more beautiful
00:36:13.257 --> 00:36:13.840
than it looks.
00:37:13.017 --> 00:37:15.350
So that's something you don't
want to do more than once.
00:37:19.520 --> 00:37:20.960
So that's one
possible way that we
00:37:20.960 --> 00:37:22.502
could deal with the
effective theory,
00:37:22.502 --> 00:37:25.570
is in terms of these
variables, S and phi.
00:37:25.570 --> 00:37:27.200
Let me write down a
couple other ways,
00:37:27.200 --> 00:37:31.010
and then we'll talk about
which we might pick.
00:37:31.010 --> 00:37:33.980
So there's something called the
"exponential representation,"
00:37:33.980 --> 00:37:35.550
also very common.
00:37:35.550 --> 00:37:37.820
These are very common
in the literature,
00:37:37.820 --> 00:37:42.380
where instead of using S
and phi, we use S and sigma.
00:37:42.380 --> 00:37:48.710
So we take our original
fields, and we rewrite it
00:37:48.710 --> 00:37:54.470
as v plus S times sigma.
00:37:54.470 --> 00:37:58.370
And then we think of sigma
as the exponential, and hence
00:37:58.370 --> 00:38:04.980
the name, of our
fundamental Goldstone field.
00:38:04.980 --> 00:38:06.140
So this is not the same Pi.
00:38:06.140 --> 00:38:08.170
It's some Pi prime, if you like.
00:38:08.170 --> 00:38:09.840
But I'm going to drop the prime.
00:38:12.720 --> 00:38:15.260
So you write everything
in terms of S and sigma,
00:38:15.260 --> 00:38:17.950
and you keep in mind that
inside the sigma is the Pi.
00:38:21.530 --> 00:38:24.200
So the S part at the
beginning is the same.
00:38:32.870 --> 00:38:34.870
And this guy's a little
bit nicer to write down.
00:38:43.400 --> 00:38:49.040
So the terms that are
pure S remain the same,
00:38:49.040 --> 00:38:50.000
so S cubed, S 4th.
00:39:06.560 --> 00:39:08.890
Since there's no funny
square roots in our field
00:39:08.890 --> 00:39:10.640
redefinition, things
are a little simpler.
00:39:14.120 --> 00:39:17.210
So that's, again, an equivalent
version of the sigma model,
00:39:17.210 --> 00:39:19.928
just using different fields.
00:39:19.928 --> 00:39:22.220
Everything I can calculate
in the original sigma model,
00:39:22.220 --> 00:39:25.690
I can calculate these
formulations as well.
00:39:41.740 --> 00:39:47.270
And the final one is
going to be different.
00:39:47.270 --> 00:39:53.860
And that's the non-linearity
chiral Lagrangian.
00:39:53.860 --> 00:39:55.780
And what I'm going to
do to get there is I'm
00:39:55.780 --> 00:39:58.000
just going to drop
the things that
00:39:58.000 --> 00:40:02.050
have mass from my previous
theory, which are S and psi.
00:40:02.050 --> 00:40:03.820
So I take this
exponential representation
00:40:03.820 --> 00:40:05.620
of the sigma model.
00:40:05.620 --> 00:40:08.800
These guys are massive.
00:40:08.800 --> 00:40:12.130
I think about-- well, I
integrate them out explicitly.
00:40:12.130 --> 00:40:15.850
In this case, what we're
doing here, which is not QCD,
00:40:15.850 --> 00:40:17.980
we can do that,
just remove them.
00:40:17.980 --> 00:40:21.010
That amounts to the lowest
order, just dropping them.
00:40:21.010 --> 00:40:31.000
And then we have just the
thing involving the pion,
00:40:31.000 --> 00:40:32.930
which is just very simple--
00:40:32.930 --> 00:40:33.430
that.
00:40:37.460 --> 00:40:40.870
So this is a viable
thing for low energies.
00:40:40.870 --> 00:40:48.560
The first three actions that
we wrote down are identical.
00:40:48.560 --> 00:40:51.937
They're just field redefined
versions of each other.
00:40:51.937 --> 00:40:53.395
They're really
equivalent theories.
00:40:59.860 --> 00:41:04.000
And this final one,
L chi, is equivalent
00:41:04.000 --> 00:41:10.750
for low-energy
phenomenology of the pions.
00:41:14.770 --> 00:41:18.040
So the first three are actually
equivalent to the last one
00:41:18.040 --> 00:41:21.540
if we restrict ourselves
to low-energy interactions.
00:41:25.290 --> 00:41:27.800
So in order to see that,
let's do an example
00:41:27.800 --> 00:41:30.080
where we calculate something.
00:41:30.080 --> 00:41:35.150
Let's think about calculating
Pi plus Pi 0 goes to Pi plus Pi
00:41:35.150 --> 00:41:39.680
0, so scattering of
Goldstone bosons.
00:41:39.680 --> 00:41:42.320
And q will be the
momentum transfer.
00:41:42.320 --> 00:41:43.910
So in kind of an
obvious notation,
00:41:43.910 --> 00:41:51.149
it's the difference between
the final and the initial,
00:41:51.149 --> 00:41:53.600
or the initial and the
final for the charge guy
00:41:53.600 --> 00:41:56.275
or the neutral guy.
00:41:56.275 --> 00:41:58.400
And there's two types of
diagrams that can come in.
00:41:58.400 --> 00:42:08.362
You could have a direct
scattering diagram
00:42:08.362 --> 00:42:09.945
or you could have
an exchange diagram.
00:42:14.950 --> 00:42:21.530
so this is Pi plus coupling
to either the sigma or the S,
00:42:21.530 --> 00:42:25.190
depending on which
representation we're using.
00:42:25.190 --> 00:42:29.000
And then the Pi 0 is down here.
00:42:29.000 --> 00:42:32.990
And I want to take these,
and I want to expand.
00:42:32.990 --> 00:42:36.710
And I'm going to expand in
q squared over v squared.
00:42:36.710 --> 00:42:38.727
That's what I'm going
to mean by "low energy".
00:42:38.727 --> 00:42:40.310
v was setting the
scale of the masses.
00:42:44.318 --> 00:42:46.610
So let's see how that works
out in the different cases.
00:42:46.610 --> 00:42:49.090
If I do it in the
linear case, and I
00:42:49.090 --> 00:42:50.840
have both of these
types of contributions,
00:42:50.840 --> 00:42:53.900
this guy just gives me
some symmetry factor,
00:42:53.900 --> 00:42:58.950
which is 2 minus 2i lambda.
00:42:58.950 --> 00:43:20.742
This guy as a propagator, square
root looks very different.
00:43:23.440 --> 00:43:24.940
And it turns out
in the square root,
00:43:24.940 --> 00:43:27.500
that you can find out that
this guy is order q to the 4th,
00:43:27.500 --> 00:43:29.810
so we won't even write it down.
00:43:36.180 --> 00:43:43.650
Exponential-- this guy
is again ditto and ditto,
00:43:43.650 --> 00:43:49.800
and then I don't have room,
but I'll put it up here.
00:43:49.800 --> 00:43:58.125
In the nonlinear, we don't
even have that diagram
00:43:58.125 --> 00:43:59.250
because we just dropped it.
00:44:02.110 --> 00:44:04.300
And if I take this line,
which is the only one that
00:44:04.300 --> 00:44:08.540
looks different, I can combine
these two terms together.
00:44:08.540 --> 00:44:11.020
And once I do that, then I
realize that this is also
00:44:11.020 --> 00:44:14.040
giving the same answer.
00:44:14.040 --> 00:44:20.380
So just rearrange it, put
in the relation between mass
00:44:20.380 --> 00:44:27.430
and coupling and VEV, which
looks like that, and expand.
00:44:27.430 --> 00:44:32.230
And you get i q squared
over v squared as well.
00:44:32.230 --> 00:44:34.880
So if I stop at order
q squared, all versions
00:44:34.880 --> 00:44:36.790
give the same thing.
00:44:36.790 --> 00:44:38.800
The linear and the square
root were equivalent.
00:44:38.800 --> 00:44:40.000
The linear, the square
root, and the exponential
00:44:40.000 --> 00:44:42.238
are just different
versions of the same thing.
00:44:42.238 --> 00:44:44.530
And all that's changed by
making the field redefinition
00:44:44.530 --> 00:44:46.390
is how I think about
these two diagrams.
00:44:46.390 --> 00:44:48.940
In the linear, the
leading order term is not
00:44:48.940 --> 00:44:50.590
coming from just
this diagram, it's
00:44:50.590 --> 00:44:52.237
also coming from that diagram.
00:44:52.237 --> 00:44:54.070
And it's actually coming
from a cancellation
00:44:54.070 --> 00:44:56.037
between those diagrams.
00:44:56.037 --> 00:44:58.120
In the square root and
exponential representation,
00:44:58.120 --> 00:45:00.862
this diagram alone gives
the leading order term
00:45:00.862 --> 00:45:02.320
and this diagram
gives higher order
00:45:02.320 --> 00:45:04.910
terms, which is kind
of what you want
00:45:04.910 --> 00:45:06.910
if you want to think about
these heavy particles
00:45:06.910 --> 00:45:10.490
as something that is giving
high-order corrections.
00:45:10.490 --> 00:45:15.500
So that can depend on
exactly how you formulate it.
00:45:15.500 --> 00:45:18.580
And from the point of
view of just keeping
00:45:18.580 --> 00:45:21.950
the low-energy
degree of freedom,
00:45:21.950 --> 00:45:24.280
the nonlinear just gives
you immediately the answer
00:45:24.280 --> 00:45:27.482
from very simple Lagrangian.
00:45:27.482 --> 00:45:29.690
Of course, it was just the
exponential with something
00:45:29.690 --> 00:45:32.530
dropped.
00:45:32.530 --> 00:45:35.170
So from the point of view
of doing calculations,
00:45:35.170 --> 00:45:39.220
the linear version here is the
one that you don't want to use,
00:45:39.220 --> 00:45:41.410
because there's cancellations
between diagrams
00:45:41.410 --> 00:45:43.360
that you have to figure out.
00:45:43.360 --> 00:45:46.180
And those cancellations
actually affect
00:45:46.180 --> 00:45:48.940
what you call "leading
order," because leading
00:45:48.940 --> 00:45:51.370
order for this calculation
is order q squared,
00:45:51.370 --> 00:45:53.620
and you don't see that until
there's that cancellation
00:45:53.620 --> 00:45:54.505
that's taken effect.
00:45:59.688 --> 00:46:01.730
So it's very hard to
formulate a power accounting
00:46:01.730 --> 00:46:05.767
for the linear theory
for that reason.
00:46:05.767 --> 00:46:07.350
But if any of these
nonlinear theories
00:46:07.350 --> 00:46:10.170
we could formulate
a power accounting,
00:46:10.170 --> 00:46:14.771
then everything is
nice and beautiful.
00:46:41.300 --> 00:46:44.150
So what happens is that
because of chiral symmetry,
00:46:44.150 --> 00:46:46.340
we have derivative couplings.
00:46:46.340 --> 00:46:48.350
And in the linear
version, we don't
00:46:48.350 --> 00:46:50.240
see that until we cancel terms.
00:46:54.600 --> 00:46:56.240
So if you like, if
you think about that
00:46:56.240 --> 00:46:59.270
as a property of the symmetry,
you'd like to make it explicit,
00:46:59.270 --> 00:47:01.910
and the other
representations do that.
00:47:17.323 --> 00:47:18.740
The version that's
most convenient
00:47:18.740 --> 00:47:21.380
is the non-linear guy,
since it's the simplest.
00:47:21.380 --> 00:47:24.650
And we just think
about it, forgetting
00:47:24.650 --> 00:47:27.170
about the heavy stuff.
00:47:27.170 --> 00:47:30.590
And it's what we really want for
our bottom-up effective theory
00:47:30.590 --> 00:47:36.630
because it only has
the low energy sigma
00:47:36.630 --> 00:47:41.790
field, which has a
pion in it, and it
00:47:41.790 --> 00:47:43.450
has the derivative couplings.
00:47:53.250 --> 00:47:56.228
If you look at how
the symmetry works,
00:47:56.228 --> 00:47:58.020
which I didn't talk
about when I wrote down
00:47:58.020 --> 00:48:04.290
all the different
versions, S is for singlet.
00:48:04.290 --> 00:48:08.370
That's what's behind the name
S, so it doesn't transform.
00:48:08.370 --> 00:48:12.540
Sigma transforms on
both sides with an L
00:48:12.540 --> 00:48:15.090
on the left and an R
dagger on the right.
00:48:15.090 --> 00:48:17.940
And that comes from
looking back, noting
00:48:17.940 --> 00:48:20.430
that S doesn't transform,
and looking back
00:48:20.430 --> 00:48:23.400
at how the Pi field transformed.
00:48:23.400 --> 00:48:24.590
It's just exactly that way.
00:48:28.480 --> 00:48:33.450
Now, if you write out the sigma,
which is transforming linearly,
00:48:33.450 --> 00:48:38.250
as the exponential
of i tau dot Pi,
00:48:38.250 --> 00:48:41.340
and if I put it in a
convenient normalization, which
00:48:41.340 --> 00:48:49.030
is the VEV, or the v,
then you can work out
00:48:49.030 --> 00:48:51.490
from this transformation
how the Pi field transforms.
00:48:51.490 --> 00:48:54.580
And it transforms
nonlinearly, and hence, that's
00:48:54.580 --> 00:48:55.270
the name for--
00:49:01.750 --> 00:49:05.140
that's where the
name comes from.
00:49:05.140 --> 00:49:11.350
If you do the infinitesimal
version of the transformation,
00:49:11.350 --> 00:49:13.630
and you work it out for
the Pi a field, which
00:49:13.630 --> 00:49:23.670
is a-th component of the
vector, a being 1, 2, or 3,
00:49:23.670 --> 00:49:26.760
part of the transformation
is simply a shift.
00:49:26.760 --> 00:49:29.790
And then there are
additional pieces,
00:49:29.790 --> 00:49:31.840
which are things that
you keep, that are
00:49:31.840 --> 00:49:33.510
order pi squared and higher.
00:49:33.510 --> 00:49:37.980
And that's those pi squared
and higher terms that they're
00:49:37.980 --> 00:49:39.297
saying it's nonlinear.
00:49:39.297 --> 00:49:41.880
This term here is what's telling
you it better be derivatively
00:49:41.880 --> 00:49:43.797
coupled, because if it's
derivatively coupled,
00:49:43.797 --> 00:49:47.960
if you have a derivative,
that kills this constant,
00:49:47.960 --> 00:49:50.460
and that was something that was
hidden in our linear version
00:49:50.460 --> 00:49:51.043
of the theory.
00:49:53.870 --> 00:49:57.370
Now, we've constructed it
here from this kind of point
00:49:57.370 --> 00:49:59.020
of view of integrating out.
00:49:59.020 --> 00:50:01.750
So I said QCD, we
can integrate it out,
00:50:01.750 --> 00:50:04.890
but all we care about is the
symmetry-breaking pattern.
00:50:04.890 --> 00:50:07.390
So we wrote down a theory, which
is this linear sigma model,
00:50:07.390 --> 00:50:09.460
had the same
symmetry-breaking pattern.
00:50:09.460 --> 00:50:11.800
And we could remove the
heavy fields in that,
00:50:11.800 --> 00:50:13.870
and then the low-energy
theory, I claim,
00:50:13.870 --> 00:50:16.270
is the same one that you
would use to describe QCD,
00:50:16.270 --> 00:50:19.330
because it's the same
symmetry-breaking pattern.
00:50:19.330 --> 00:50:22.360
But that's kind of a
roundabout way of getting
00:50:22.360 --> 00:50:24.160
to where we wanted to go.
00:50:24.160 --> 00:50:26.350
And you'd really
like to just get
00:50:26.350 --> 00:50:31.150
right to the chiral
Lagrangian from the start,
00:50:31.150 --> 00:50:32.170
and you can do that.
00:50:46.963 --> 00:50:49.130
So rather than going through
the linear sigma model,
00:50:49.130 --> 00:50:51.755
we could also just directly
get to where we want to go.
00:50:51.755 --> 00:50:52.880
And here's how you do that.
00:50:55.490 --> 00:51:02.250
So go back to what the
symmetry-breaking pattern was
00:51:02.250 --> 00:51:04.027
and figure out what
is it that we're
00:51:04.027 --> 00:51:06.110
trying to do with the
low-energy effective theory.
00:51:06.110 --> 00:51:14.250
So we have G that's broken
to H. The Goldstones are
00:51:14.250 --> 00:51:21.690
transformations in the
coset, which is G/H.
00:51:21.690 --> 00:51:23.900
And we'd like to
parameterize them.
00:51:23.900 --> 00:51:25.650
And what we're doing
with this sigma field
00:51:25.650 --> 00:51:27.255
is we're parameterizing
fluctuations
00:51:27.255 --> 00:51:29.790
to take us around the coset.
00:51:29.790 --> 00:51:34.080
AUDIENCE: I'm a little confused,
because if you don't expand
00:51:34.080 --> 00:51:37.920
sigma, that Lagrangian has
still the full symmetry as you
00:51:37.920 --> 00:51:40.740
do left, as you do right.
00:51:40.740 --> 00:51:43.480
But it contains
the v, [INAUDIBLE]
00:51:43.480 --> 00:51:45.105
parameter for the
rest of the symmetry.
00:51:47.790 --> 00:51:51.007
That Lagrangian is presenting
the unbroken phase or--
00:51:51.007 --> 00:51:52.590
IAIN STEWART: So
remember, I went over
00:51:52.590 --> 00:51:55.740
to the shifted fields,
the sigma twiddle,
00:51:55.740 --> 00:51:58.163
and then I was expanding
around the correct vacuum--
00:51:58.163 --> 00:52:00.330
I mean, the one that's the
lowest energy vacuum, not
00:52:00.330 --> 00:52:02.850
the unbroken.
00:52:02.850 --> 00:52:03.870
I went from sigma--
00:52:03.870 --> 00:52:05.950
I originally had sigma--
00:52:05.950 --> 00:52:07.980
sorry.
00:52:07.980 --> 00:52:10.930
Thank you for that.
00:52:10.930 --> 00:52:13.515
So really, when I talked
about the linear version,
00:52:13.515 --> 00:52:15.390
there was two versions
of the linear version.
00:52:15.390 --> 00:52:17.010
There was the original
version where I wrote it down
00:52:17.010 --> 00:52:19.218
in terms of sigma and Pi
vector, and then I went over
00:52:19.218 --> 00:52:21.337
to sigma twiddle and Pi vector.
00:52:21.337 --> 00:52:22.920
And when I went over
to sigma twiddle,
00:52:22.920 --> 00:52:25.830
it was just a shift
where I shifted myself
00:52:25.830 --> 00:52:27.128
to the proper vacuum.
00:52:27.128 --> 00:52:28.920
So I'm really doing
the perturbation theory
00:52:28.920 --> 00:52:32.100
in the linear here
about the proper vacuum.
00:52:32.100 --> 00:52:33.630
And all the other
versions, I'm also
00:52:33.630 --> 00:52:37.905
doing about that proper vacuum,
the lowest energy vacuum.
00:52:40.818 --> 00:52:42.610
So there was a step at
the beginning, which
00:52:42.610 --> 00:52:44.980
is the classic step for
spontaneous symmetry
00:52:44.980 --> 00:52:46.180
breaking, which is--
00:52:46.180 --> 00:52:47.980
in some sense, I had
to do that to get
00:52:47.980 --> 00:52:49.420
started when I
went from the sigma
00:52:49.420 --> 00:52:50.710
to the sigma twiddle field.
00:52:50.710 --> 00:52:52.210
But you should
really think about it
00:52:52.210 --> 00:52:53.720
as the sigma twiddle
field as being
00:52:53.720 --> 00:52:55.700
field in this linear case.
00:52:58.510 --> 00:53:01.870
That's a sigma twiddle there.
00:53:01.870 --> 00:53:02.800
Other questions?
00:53:06.210 --> 00:53:08.880
So how would we do this
construction of sigma
00:53:08.880 --> 00:53:11.935
if we didn't know about
this linear sigma model
00:53:11.935 --> 00:53:12.810
way of getting there?
00:53:15.990 --> 00:53:18.710
So we have a
generator, G, which is
00:53:18.710 --> 00:53:25.220
in L comma R. That's my notation
for the combined SU2 left
00:53:25.220 --> 00:53:27.552
cross SU2 right, for example.
00:53:27.552 --> 00:53:29.510
And that's going to be
broken down to something
00:53:29.510 --> 00:53:32.600
that's in the vector sub-group,
which I'll denote like this--
00:53:32.600 --> 00:53:38.090
V comma V. And then an
element of that we can call h.
00:53:41.390 --> 00:53:43.220
And if we want to
parameterize the coset,
00:53:43.220 --> 00:53:46.520
you can think that you
have a G left and a G right
00:53:46.520 --> 00:53:50.870
in a kind of obvious notation.
00:53:50.870 --> 00:53:55.400
And you want some kind of field
to parameterize fluctuations,
00:53:55.400 --> 00:53:57.140
where we pull out the h.
00:53:57.140 --> 00:53:58.640
So I'll call that "cascade."
00:54:19.170 --> 00:54:21.912
So let's just think
about an example.
00:54:21.912 --> 00:54:23.370
So say we had
something that looked
00:54:23.370 --> 00:54:30.700
like this, which is some
set of generators that
00:54:30.700 --> 00:54:35.380
is in the original
L comma R. And let
00:54:35.380 --> 00:54:38.830
me insert in here 1
in the form of gR,
00:54:38.830 --> 00:54:49.610
gR dagger, and then write it
as a product of generators
00:54:49.610 --> 00:54:52.640
that looks like this.
00:54:59.542 --> 00:55:01.250
So I just multiply
the various-- this guy
00:55:01.250 --> 00:55:04.610
multiplies that guy, this
guy multiplies that guy.
00:55:04.610 --> 00:55:06.980
So this guy here has the
same entries in both,
00:55:06.980 --> 00:55:10.150
so that's in h.
00:55:10.150 --> 00:55:16.520
This is in the v
comma v, which is h.
00:55:16.520 --> 00:55:19.970
And this guy here,
this gR dagger,
00:55:19.970 --> 00:55:22.430
is then the thing that's
parameterizing the cascade
00:55:22.430 --> 00:55:24.780
field.
00:55:24.780 --> 00:55:45.040
So this is a matrix that's
parameterizing the coset,
00:55:45.040 --> 00:55:48.160
and transforms in
the way that we said.
00:55:51.930 --> 00:55:55.170
So that's the idea behind
what the sigma field is doing.
00:55:55.170 --> 00:56:00.480
Now, you can ask, well, that
just seems like one choice.
00:56:00.480 --> 00:56:02.880
And I could think about
parameterizing the coset
00:56:02.880 --> 00:56:04.720
in many different ways.
00:56:04.720 --> 00:56:05.580
And that's true.
00:56:10.860 --> 00:56:12.480
And there's actually
a nice formalism
00:56:12.480 --> 00:56:15.790
that takes that into account.
00:56:15.790 --> 00:56:20.970
So if you just talk about broken
generators, which I can call X,
00:56:20.970 --> 00:56:26.460
then a very general definition
of what this cascade field is
00:56:26.460 --> 00:56:32.580
is just an exponential involving
those broken generators,
00:56:32.580 --> 00:56:35.850
and some fields to
describe the fluctuations,
00:56:35.850 --> 00:56:37.410
and normalized in some way.
00:56:41.950 --> 00:56:45.660
And this is due to Callan,
Coleman, Wess and Zamino.
00:56:49.740 --> 00:56:59.910
So it's the CCWZ prescription
for parameterizing a coset.
00:56:59.910 --> 00:57:01.530
And the point of
their prescription
00:57:01.530 --> 00:57:05.070
is that you have choices
here for how you represent
00:57:05.070 --> 00:57:06.300
the broken generators.
00:57:14.305 --> 00:57:15.680
We start out with
left generators
00:57:15.680 --> 00:57:18.200
and right generators,
and then you
00:57:18.200 --> 00:57:19.940
go over to the
vector generators,
00:57:19.940 --> 00:57:21.110
but what ones are broken?
00:57:21.110 --> 00:57:23.193
Well, you could take
different linear combinations
00:57:23.193 --> 00:57:24.080
that are broken.
00:57:24.080 --> 00:57:27.980
And those would be
equally viable choices.
00:57:27.980 --> 00:57:31.890
Doesn't have to be a group,
the broken generators,
00:57:31.890 --> 00:57:47.020
so you can pick different
choices of this X.
00:57:47.020 --> 00:57:54.520
And if we pick X, which
is the left generators,
00:57:54.520 --> 00:57:58.832
that's actually
what gives us what
00:57:58.832 --> 00:58:00.040
we were talking about before.
00:58:00.040 --> 00:58:11.200
Because then we end up
parameterizing the cosets
00:58:11.200 --> 00:58:15.425
by something as 1 in the
second entry, because we
00:58:15.425 --> 00:58:16.300
don't have the right.
00:58:16.300 --> 00:58:19.120
We're just saying pick the
left, the left are broken.
00:58:19.120 --> 00:58:23.217
One possible choice, left
plus right is unbroken.
00:58:23.217 --> 00:58:25.300
Then we end up with an
entry here, which is sigma.
00:58:28.040 --> 00:58:29.960
That's our sigma.
00:58:29.960 --> 00:58:33.220
And also, if you work
out the transformation,
00:58:33.220 --> 00:58:37.180
how the sigma transforms,
you could reproduce--
00:58:37.180 --> 00:58:45.820
you can also derive that sigma
goes to L, sigma R dagger.
00:58:45.820 --> 00:58:47.110
But that's just one choice.
00:58:47.110 --> 00:58:48.290
You could do other choices.
00:58:48.290 --> 00:58:52.030
So for example, you could
pick X a to be tau left
00:58:52.030 --> 00:58:54.640
a minus tau right a.
00:58:54.640 --> 00:58:56.500
That's another possible choice.
00:58:56.500 --> 00:59:09.520
And that would lead
to a different field,
00:59:09.520 --> 00:59:12.630
but an equally valid
parameterization of the coset.
00:59:12.630 --> 00:59:14.240
And this was actually
also popular.
00:59:14.240 --> 00:59:18.800
It's something that people
usually denote by C.
00:59:18.800 --> 00:59:22.280
I'm going to leave
further discussion of that
00:59:22.280 --> 00:59:23.540
to your reading.
00:59:23.540 --> 00:59:26.360
There's a very nice review
by Aneesh Manohar, where
00:59:26.360 --> 00:59:29.840
he goes through some of
these things for CCSW,
00:59:29.840 --> 00:59:32.600
and does a nice job of talking
about both of these in more
00:59:32.600 --> 00:59:35.850
detail than I've done here.
00:59:35.850 --> 00:59:39.753
So further discussion of this
point will be in your reading.
00:59:39.753 --> 00:59:42.170
You can also look back at the
original paper, if you like.
00:59:42.170 --> 00:59:44.360
But Aneesh has
distilled it nicely.
00:59:51.610 --> 00:59:53.130
So any questions about that?
00:59:53.130 --> 00:59:55.210
There's a way of thinking--
00:59:55.210 --> 00:59:57.335
I'll just keep talking
until someone raises their--
00:59:57.335 --> 00:59:59.502
there's a way of thinking
about field redefinitions,
00:59:59.502 --> 01:00:01.300
even from this
low-energy point of view.
01:00:01.300 --> 01:00:02.800
That's the key here.
01:00:02.800 --> 01:00:04.810
And it has to do with
the freedom that you have
01:00:04.810 --> 01:00:06.268
and how you
parameterize the coset.
01:00:15.910 --> 01:00:18.210
So even if we're constructing
it from the bottom up,
01:00:18.210 --> 01:00:20.380
there's different
representations.
01:00:20.380 --> 01:00:23.760
In fact, Weinberg likes
the square root version
01:00:23.760 --> 01:00:27.030
of parameterizing the coset.
01:00:27.030 --> 01:00:30.630
Maybe he's the only person.
01:00:30.630 --> 01:00:33.360
He has very good reasons
for being allowed
01:00:33.360 --> 01:00:37.830
to stick with his
original results,
01:00:37.830 --> 01:00:39.900
rather than the
exponential representation,
01:00:39.900 --> 01:00:42.850
which everyone else seems
to favor, including me.
01:00:46.630 --> 01:00:50.610
So if we go back to
talking about QCD,
01:00:50.610 --> 01:00:54.180
then it's common to call
this curly V f over root 2.
01:00:54.180 --> 01:00:55.290
1/2 is a decay constant.
01:00:58.715 --> 01:01:00.090
And so the
conventions, then, are
01:01:00.090 --> 01:01:03.322
the following-- at least one
possible choice of conventions.
01:01:08.050 --> 01:01:12.930
There's some choice about
where you put the 2's, and then
01:01:12.930 --> 01:01:16.000
everything else is fixed.
01:01:16.000 --> 01:01:17.880
So this is one
possible common choice
01:01:17.880 --> 01:01:20.350
for the exponential
representation.
01:01:20.350 --> 01:01:24.834
And then that gives you
the chiral Lagrangian--
01:01:30.270 --> 01:01:32.110
looks like that.
01:01:32.110 --> 01:01:35.550
And if you expand this out, you
get the standard kinetic term
01:01:35.550 --> 01:01:37.928
for the phi field.
01:01:37.928 --> 01:01:39.720
That's what this
normalization factor does.
01:01:51.500 --> 01:01:57.600
And then if you keep expanding,
you get some interaction terms
01:01:57.600 --> 01:02:02.945
like this one, et cetera.
01:02:02.945 --> 01:02:05.300
So there's four point
interactions in that Lagrangian
01:02:05.300 --> 01:02:05.800
as well.
01:02:09.170 --> 01:02:11.680
And that's because
we had a curved--
01:02:11.680 --> 01:02:13.870
our coset is nontrivial.
01:02:23.130 --> 01:02:26.760
So part of this story here
when you talk about symmetry
01:02:26.760 --> 01:02:28.230
breaking has to
be about the fact
01:02:28.230 --> 01:02:30.960
that symmetry is also
broken explicitly
01:02:30.960 --> 01:02:33.740
in the standard model.
01:02:33.740 --> 01:02:36.280
And so you have to add--
01:02:36.280 --> 01:02:38.480
the Goldstones are
only pseudogoldstones.
01:02:38.480 --> 01:02:40.310
You have to add a mass.
01:02:40.310 --> 01:02:44.210
And the way you do that is
by doing a spurion analysis.
01:02:49.480 --> 01:02:50.700
Remind you how that goes.
01:02:57.990 --> 01:03:00.900
You write a mass term like
this, and to left and right.
01:03:03.630 --> 01:03:06.420
The mass, you could say,
is just the diagonal matrix
01:03:06.420 --> 01:03:09.370
of up and down for SU2.
01:03:09.370 --> 01:03:12.330
And you could pretend, rather
than talking about things that
01:03:12.330 --> 01:03:14.970
are violating the
symmetry like these guys,
01:03:14.970 --> 01:03:19.590
you let the mass
transform and pretend
01:03:19.590 --> 01:03:25.080
that you actually have
something that's invariant.
01:03:25.080 --> 01:03:26.940
By pretending that
the mass transforms,
01:03:26.940 --> 01:03:28.950
you're able-- if you
pretend the mass transforms
01:03:28.950 --> 01:03:30.600
in the chiral theory--
of constructing
01:03:30.600 --> 01:03:33.300
things that violate the
symmetry in the same way.
01:03:33.300 --> 01:03:36.150
You construct invariance in
both theories with this kind
01:03:36.150 --> 01:03:37.740
of transformation law.
01:03:37.740 --> 01:03:40.650
And then when you go back
and fix them, and say no, no,
01:03:40.650 --> 01:03:43.080
it doesn't transform,
it's really a fixed thing,
01:03:43.080 --> 01:03:45.570
not a field that could
transform-- it's fixed,
01:03:45.570 --> 01:03:46.680
the number--
01:03:46.680 --> 01:03:49.600
then you violate the symmetry
explicitly in the same way
01:03:49.600 --> 01:03:50.100
as up here.
01:03:50.100 --> 01:03:54.210
It's a trick for how to
build things that are
01:03:54.210 --> 01:03:56.370
covariant in a certain form.
01:04:02.220 --> 01:04:05.550
So you could add the
variant operators
01:04:05.550 --> 01:04:07.078
of the chiral Lagrangian.
01:04:10.910 --> 01:04:14.490
This is a different
v, some parameter,
01:04:14.490 --> 01:04:16.650
different than the curly
one we were talking about
01:04:16.650 --> 01:04:22.140
earlier, which is now called f.
01:04:36.080 --> 01:04:38.790
So this is the spurion story--
01:04:38.790 --> 01:04:43.640
how to break symmetry explicitly
in a different theory,
01:04:43.640 --> 01:04:44.360
the same way.
01:04:49.420 --> 01:04:51.750
And if you expand
a quadratic order,
01:04:51.750 --> 01:04:53.325
you get mass terms
for the pions.
01:04:57.020 --> 01:04:58.020
I won't go through that.
01:05:11.420 --> 01:05:13.490
But one kind of an
important thing about it
01:05:13.490 --> 01:05:19.130
is that the mass is proportional
to v times the quark
01:05:19.130 --> 01:05:27.500
mass, so v0, so the parameter
in the Lagrangian, linear
01:05:27.500 --> 01:05:30.560
in the quark masses, quadratic
in the Goldstone mass.
01:05:40.830 --> 01:05:43.110
You can also a
couple of currents.
01:05:43.110 --> 01:05:45.210
And you can do a
similar type of analysis
01:05:45.210 --> 01:05:49.230
with spurion-type transformation
analysis for the currents.
01:05:52.170 --> 01:05:53.850
And I'm not going
to go through this
01:05:53.850 --> 01:05:56.430
in any gory detail,
but just enough
01:05:56.430 --> 01:05:58.780
to do what we need to do here.
01:05:58.780 --> 01:06:02.340
So here's a left-handed fermion
current, standard model.
01:06:02.340 --> 01:06:05.490
Could be coupling to the
W boson, for example.
01:06:05.490 --> 01:06:11.640
And you could think of getting
J by taking a functional
01:06:11.640 --> 01:06:15.270
derivative with respect to
a field that's left handed.
01:06:19.932 --> 01:06:21.640
And then I can do the
same kind of thing,
01:06:21.640 --> 01:06:25.300
where I think about this L and
let it transform, just like I
01:06:25.300 --> 01:06:27.070
was letting the M transform.
01:06:27.070 --> 01:06:34.220
So think about J as this
capital J times minus L.
01:06:34.220 --> 01:06:37.012
And if I then think about
this L mu transforming,
01:06:37.012 --> 01:06:38.720
I can build a chiral
Lagrangian, and then
01:06:38.720 --> 01:06:41.030
use this formula to
construct the current
01:06:41.030 --> 01:06:44.000
in the chiral Lagrangian.
01:06:44.000 --> 01:06:59.130
So you couple a spurion
current, L mu a tau a,
01:06:59.130 --> 01:07:03.370
and you, in this case,
in order to couple it,
01:07:03.370 --> 01:07:06.100
you can let it transform like
a left-handed gauge field.
01:07:13.450 --> 01:07:15.487
And then you get something
that's invariant.
01:07:21.700 --> 01:07:31.385
So the transformation that
will make it invariant
01:07:31.385 --> 01:07:33.260
is to think of it as a
left-hand gauge field.
01:07:33.260 --> 01:07:41.500
So L mu goes to
L, L mu, L dagger.
01:07:53.680 --> 01:08:02.900
And then that gives
you an invariant
01:08:02.900 --> 01:08:05.380
in the original theory.
01:08:05.380 --> 01:08:08.230
And then you build in a
variant of the x theory.
01:08:12.570 --> 01:08:14.500
And so if you do
this, and you're
01:08:14.500 --> 01:08:16.540
replacing, in your
chiral Lagrangian,
01:08:16.540 --> 01:08:19.223
your partial derivative
by a covariate one,
01:08:19.223 --> 01:08:20.890
it acts on the left
because that's where
01:08:20.890 --> 01:08:22.899
the sigma is transforming.
01:08:22.899 --> 01:08:25.450
And so it's an easy way
of building that in.
01:08:29.074 --> 01:08:31.399
And you just get an i L
times the sigma on the left.
01:08:31.399 --> 01:08:33.899
So you take everywhere you have
a derivative, you replace it
01:08:33.899 --> 01:08:35.585
by this combination.
01:08:35.585 --> 01:08:37.710
So we had two derivatives
in our chiral Lagrangian.
01:08:37.710 --> 01:08:40.770
Expand it out, and
we can figure out
01:08:40.770 --> 01:08:44.770
how to put it in the
left-handed spurion.
01:08:44.770 --> 01:08:47.082
So the spurion is just
kind of a general way
01:08:47.082 --> 01:08:48.540
of going from one
to the other just
01:08:48.540 --> 01:08:50.130
by tracking symmetry breaking.
01:09:07.515 --> 01:09:08.890
And you'll get
some more practice
01:09:08.890 --> 01:09:12.220
with some of this
on your problem set.
01:09:15.944 --> 01:09:17.319
So on your problem
set, what I've
01:09:17.319 --> 01:09:22.330
asked you to do on problem set 2
is to do a one-loop calculation
01:09:22.330 --> 01:09:24.640
in this chiral theory.
01:09:24.640 --> 01:09:28.652
So you can really see that
this is a quantum field
01:09:28.652 --> 01:09:30.069
theory, an effective
field theory,
01:09:30.069 --> 01:09:33.790
not only a tree-level
mnemonic theory, but really
01:09:33.790 --> 01:09:35.590
a field theory in
its own right where
01:09:35.590 --> 01:09:39.850
you have to consider loops,
contractions, Wick's theorem.
01:09:39.850 --> 01:09:43.210
Everything is the same as a real
quantum field theory should be.
01:09:43.210 --> 01:09:45.880
It's an effective field theory
of certain degrees of freedom.
01:09:53.180 --> 01:09:56.320
So you do a one-loop calculation
for the decay constant.
01:09:56.320 --> 01:09:59.710
And there's two problems
in the problem set.
01:09:59.710 --> 01:10:02.002
The first one is
fairly straightforward.
01:10:02.002 --> 01:10:03.460
And then that's
the second problem.
01:10:03.460 --> 01:10:06.002
That's all I'm asking you to do
because it's fairly involved.
01:10:09.190 --> 01:10:11.090
It's like maybe two
normal problems.
01:10:14.280 --> 01:10:17.785
So here's our chiral Lagrangian.
01:10:17.785 --> 01:10:19.910
And we're going to count
in this chiral Lagrangian,
01:10:19.910 --> 01:10:22.680
in terms of power counting,
partial squared as being order
01:10:22.680 --> 01:10:23.272
v0 ab q.
01:10:23.272 --> 01:10:24.980
That means we're going
to count p squared
01:10:24.980 --> 01:10:26.210
as being order M pi squared.
01:10:29.720 --> 01:10:32.400
That's going to
be our accounting.
01:10:32.400 --> 01:10:36.740
And so we're going to expand
in these things as small
01:10:36.740 --> 01:10:38.455
and something else
that is large.
01:10:38.455 --> 01:10:40.830
And we'll figure out what the
large thing is in a minute.
01:10:46.200 --> 01:10:48.410
So we do the expansion
like that, where something
01:10:48.410 --> 01:10:51.290
is downstairs that has mass
dimensions and the things that
01:10:51.290 --> 01:10:54.330
are upstairs are p squared
and then Pi squared.
01:10:54.330 --> 01:10:57.140
So the type of
expansion we have here
01:10:57.140 --> 01:11:02.360
is a combination of a derivative
expansion, although it's
01:11:02.360 --> 01:11:05.540
a bit of a different derivative
expansion than before,
01:11:05.540 --> 01:11:09.110
and an M q expansion,
simultaneously driven
01:11:09.110 --> 01:11:10.400
expansion in these two things.
01:11:15.090 --> 01:11:17.520
If we look at L0, then
it has things in it
01:11:17.520 --> 01:11:19.800
like a propagator--
01:11:19.800 --> 01:11:22.350
dashed line for the
scalar plan propagator.
01:11:26.580 --> 01:11:29.730
It has four point interactions.
01:11:29.730 --> 01:11:32.067
And if you look at
how those scale,
01:11:32.067 --> 01:11:33.900
we could write out what
the Feynman rule is,
01:11:33.900 --> 01:11:35.317
but if you look
at how they scale,
01:11:35.317 --> 01:11:41.700
they go like p squared over
f squared, or M Pi squared
01:11:41.700 --> 01:11:43.848
over f squared.
01:11:43.848 --> 01:11:45.390
The p squared you
could put on shell,
01:11:45.390 --> 01:11:47.430
and it would be M
Pi over f squared.
01:11:52.890 --> 01:11:56.510
And you will actually
need this Lagrangian
01:11:56.510 --> 01:11:58.500
and this Feynman rule
for your homework,
01:11:58.500 --> 01:11:59.750
so let me tell you what it is.
01:12:03.860 --> 01:12:06.800
And I'll leave it to you to
work out the Feynman rule.
01:12:22.490 --> 01:12:28.424
So commutators,
there's a factor of 6.
01:12:28.424 --> 01:12:40.550
And then there's
also a term that
01:12:40.550 --> 01:12:43.250
would give four-point
scattering from the M2 term
01:12:43.250 --> 01:12:44.810
in the Lagrangian.
01:12:44.810 --> 01:12:47.938
So that also gives an M Pi
squared over f squared term,
01:12:47.938 --> 01:12:48.980
and that looks like that.
01:12:53.190 --> 01:12:55.530
So you could build up a
higher point function,
01:12:55.530 --> 01:12:58.760
six-point function,
et cetera from
01:12:58.760 --> 01:13:01.290
the leading-order Lagrangian
that has all this in it.
01:13:01.290 --> 01:13:04.100
It has two parameters, f and v0.
01:13:07.130 --> 01:13:08.187
So what about loops?
01:13:08.187 --> 01:13:09.020
And what about this?
01:13:09.020 --> 01:13:11.647
What is this scale lambda chi?
01:13:11.647 --> 01:13:13.730
So the type of loops that
you can have if you have
01:13:13.730 --> 01:13:17.360
a four-point interaction, and
just draw a solid line since
01:13:17.360 --> 01:13:20.210
it's easier than
drawing dashed lines--
01:13:20.210 --> 01:13:21.440
think of all these as dashed.
01:13:31.170 --> 01:13:33.810
So there's cross diagrams as
well as the original diagram.
01:13:36.547 --> 01:13:38.130
This is not the
calculation I'm asking
01:13:38.130 --> 01:13:39.213
you to do in the homework.
01:13:42.810 --> 01:13:49.920
If we thought about our example
of Pi plus Pi 0 scattering,
01:13:49.920 --> 01:13:52.800
then we could contract up
Pi plus and Pi 0 fields,
01:13:52.800 --> 01:13:56.250
and we could have a loop
that looks like that.
01:13:56.250 --> 01:13:59.280
And in order for our
theory to be unitary,
01:13:59.280 --> 01:14:01.890
we have to think that
these loops make sense,
01:14:01.890 --> 01:14:04.380
because there's cutting
rules, and all that story
01:14:04.380 --> 01:14:06.330
should go over for this
quantum field theory.
01:14:06.330 --> 01:14:08.100
And it does.
01:14:08.100 --> 01:14:12.870
And that means that if you like
the imaginary part of this loop
01:14:12.870 --> 01:14:15.600
is needed, and you can't
get the imaginary part
01:14:15.600 --> 01:14:17.280
without the real
part, so we have
01:14:17.280 --> 01:14:19.200
to also think that
the real part is
01:14:19.200 --> 01:14:20.690
a physically meaningful thing.
01:14:23.137 --> 01:14:24.720
And so we can go
ahead and compute it,
01:14:24.720 --> 01:14:26.012
and figure out what's going on.
01:14:30.370 --> 01:14:35.160
So there's derivative couplings
that go like f squared, 1
01:14:35.160 --> 01:14:36.600
over p squared over f squared.
01:14:43.930 --> 01:14:46.700
L is the loop momenta.
01:14:46.700 --> 01:14:48.360
Primes are the final.
01:14:48.360 --> 01:14:49.745
Unprimes are the initial.
01:14:56.892 --> 01:14:59.970
You can parameterize
it however you like.
01:14:59.970 --> 01:15:02.070
I'm using here dimensional
regularization.
01:15:02.070 --> 01:15:05.700
I like to dimensional
regularization, and so do you.
01:15:05.700 --> 01:15:09.820
Dimensional regularization
here preserves chiral symmetry,
01:15:09.820 --> 01:15:11.860
which is something that
we want to preserve.
01:15:11.860 --> 01:15:14.070
And I'll tell you
a little later what
01:15:14.070 --> 01:15:16.960
would have happened if we'd
done this with a cut-off.
01:15:16.960 --> 01:15:20.670
So if you do this calculation,
just calculate those guys,
01:15:20.670 --> 01:15:23.460
you get terms that
go like p to the 4th.
01:15:23.460 --> 01:15:26.550
You get terms that go like
p squared, M Pi squared,
01:15:26.550 --> 01:15:28.200
terms that go like
M Pi to the 4th.
01:15:30.780 --> 01:15:33.490
You get a 4 Pi,
because it's a loop,
01:15:33.490 --> 01:15:34.920
and then you get
an f to the 4th,
01:15:34.920 --> 01:15:38.970
because there's
four factors of f.
01:15:38.970 --> 01:15:41.070
So what this behaves
like, it's p squared
01:15:41.070 --> 01:15:44.070
or M pi squared over
f squared, which
01:15:44.070 --> 01:15:51.930
was our tree-level, four-point
function, times something
01:15:51.930 --> 01:15:54.750
that's causing suppression,
which is p squared
01:15:54.750 --> 01:16:04.090
or M Pi squared
over 4 Pi f squared.
01:16:04.090 --> 01:16:07.350
So the loop is actually
down by a factor of p
01:16:07.350 --> 01:16:09.420
squared over 4 Pi
f, all squared.
01:16:13.200 --> 01:16:15.920
So just by explicit calculation
and not breaking the symmetry,
01:16:15.920 --> 01:16:22.100
we see that loops are suppressed
by p squared over lambda chi
01:16:22.100 --> 01:16:28.370
squared, where lambda chi
here is defined to be 4 Pi f.
01:16:31.200 --> 01:16:35.980
So the 4 Pi is important
because f is like 130 GeV,
01:16:35.980 --> 01:16:40.290
and you need this 4 Pi in
order to have some range
01:16:40.290 --> 01:16:42.370
for your chiral expansion.
01:16:42.370 --> 01:16:44.640
So if you plug in
numbers, depending
01:16:44.640 --> 01:16:46.738
on your conventions
for f-- and there's
01:16:46.738 --> 01:16:48.030
a couple of common conventions.
01:16:48.030 --> 01:16:53.010
You could get numbers that
are like 1.6 or 1.2 GeV, so
01:16:53.010 --> 01:16:55.523
some scale of order a GeV.
01:16:55.523 --> 01:16:56.940
On general grounds,
you could also
01:16:56.940 --> 01:16:58.953
ask the question, if
I just think about it
01:16:58.953 --> 01:17:00.370
from the effective
theory, what is
01:17:00.370 --> 01:17:03.480
a reasonable scale
for the denominator,
01:17:03.480 --> 01:17:05.227
for this lambda chi?
01:17:05.227 --> 01:17:07.560
Remember, you're going to
have to construct higher order
01:17:07.560 --> 01:17:09.130
operators as well.
01:17:09.130 --> 01:17:11.225
And so you need to
figure out what's
01:17:11.225 --> 01:17:13.350
the right scale for the
suppression of those higher
01:17:13.350 --> 01:17:14.820
order operators.
01:17:14.820 --> 01:17:16.470
And if you did it
on physical grounds,
01:17:16.470 --> 01:17:18.465
than what you'd expect
the lambda chi to be
01:17:18.465 --> 01:17:20.400
is about the mass of the rho.
01:17:20.400 --> 01:17:23.010
Because that's the lowest
energy hadron that you've
01:17:23.010 --> 01:17:24.360
left out of this chiral theory.
01:17:27.870 --> 01:17:31.830
You might expect
lambda chi's of order M
01:17:31.830 --> 01:17:38.130
rho, which is sort of 770 MeV.
01:17:38.130 --> 01:17:41.910
And so you can't really say
whether it's 770 or 1.2,
01:17:41.910 --> 01:17:44.370
because the difference
between those,
01:17:44.370 --> 01:17:46.500
although important
numerically, is not
01:17:46.500 --> 01:17:52.110
within the realm of the
kind of factor 2-ish meaning
01:17:52.110 --> 01:17:55.500
of these twiddle symbols.
01:17:55.500 --> 01:17:59.760
So these twiddle symbols mean
"parametrically of order."
01:17:59.760 --> 01:18:02.560
Then there's two
symbols-- there's
01:18:02.560 --> 01:18:03.930
much greater than or twiddle.
01:18:03.930 --> 01:18:05.762
"Twiddle" means factor of 2-ish.
01:18:05.762 --> 01:18:07.720
"Much greater" means
parametrically [INAUDIBLE]
01:18:07.720 --> 01:18:10.910
or much less than.
01:18:10.910 --> 01:18:15.200
And just with those two
symbols, there's some freedom.
01:18:15.200 --> 01:18:17.450
These are different ways of
doing an estimate for what
01:18:17.450 --> 01:18:20.480
lambda chi are, just saying
that there is that freedom,
01:18:20.480 --> 01:18:22.520
and you have to do
the full calculation
01:18:22.520 --> 01:18:24.890
in a particular channel
to really figure out--
01:18:24.890 --> 01:18:27.380
or a particular piece
of phenomenology--
01:18:27.380 --> 01:18:29.330
what the lambda is.
01:18:29.330 --> 01:18:31.040
Extract it from
data if you like.
01:18:37.040 --> 01:18:42.838
So that is likely a
good place to stop.
01:18:42.838 --> 01:18:44.630
And next time, we'll
talk a little bit more
01:18:44.630 --> 01:18:46.400
about these loops.
01:18:46.400 --> 01:18:48.560
And we'll talk
about what happens
01:18:48.560 --> 01:18:51.350
to these loops in the power
counting theorem that you have,
01:18:51.350 --> 01:18:53.450
that organizes all
diagrams in this theory,
01:18:53.450 --> 01:18:57.030
and tells you what you have
to do if you want to construct
01:18:57.030 --> 01:18:59.008
the Lagrangian order by order--
01:18:59.008 --> 01:19:00.800
how you deal with when
to put in the loops,
01:19:00.800 --> 01:19:02.960
when to put in
the counter terms.
01:19:02.960 --> 01:19:05.500
We'll talk about those
things next time.