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PROFESSOR: Question.
00:00:25.145 --> 00:00:26.770
We know how to do
matching and running.
00:00:26.770 --> 00:00:28.560
We've seen an example of that.
00:00:28.560 --> 00:00:31.620
In this area, there's this
label v on the fields,
00:00:31.620 --> 00:00:33.900
and we want to figure
out how that affects
00:00:33.900 --> 00:00:35.428
doing matching and running.
00:00:35.428 --> 00:00:37.470
So we started out talking
about the wave function
00:00:37.470 --> 00:00:41.550
for normalization calculation.
00:00:41.550 --> 00:00:43.560
So we have these
fields, which we
00:00:43.560 --> 00:00:46.410
can have a bare version of
and a renormalized version
00:00:46.410 --> 00:00:49.510
of in the usual way.
00:00:49.510 --> 00:00:52.560
So this is renormalized,
and this is bare.
00:00:52.560 --> 00:00:57.120
Some Z factor between them
that I'll call Zh for Z heavy.
00:00:57.120 --> 00:01:05.570
And so we have to calculate at
the lowest order, this diagram,
00:01:05.570 --> 00:01:12.200
send in a momentum
P. This is Q plus p.?
00:01:12.200 --> 00:01:13.220
Use the Feynman rules.
00:01:19.330 --> 00:01:21.356
Use dimensional regularization.
00:01:30.600 --> 00:01:34.066
Use dimensional
regularization with MS bar,
00:01:34.066 --> 00:01:38.115
so there's some extra factors.
00:01:41.220 --> 00:01:44.700
And use Feynman gauge, which
is usually the simplest gauge
00:01:44.700 --> 00:01:45.720
choice.
00:01:45.720 --> 00:01:48.300
So each of these
vertices here gets
00:01:48.300 --> 00:01:53.790
a v. There's a v mu
from this vertex.
00:01:53.790 --> 00:01:56.490
A v mu from that vertex.
00:01:56.490 --> 00:01:59.440
And that's where this
v squared comes from.
00:01:59.440 --> 00:02:03.090
And then there's
one propagator here.
00:02:03.090 --> 00:02:05.490
It's very traditional
to denote a heavy quark
00:02:05.490 --> 00:02:07.020
by two lines rather
than one line,
00:02:07.020 --> 00:02:08.395
so you know which
lines are heavy
00:02:08.395 --> 00:02:09.660
and which lines are light.
00:02:09.660 --> 00:02:11.970
So these two lines
is a heavy quark,
00:02:11.970 --> 00:02:16.350
and that gives this propagator
here, this v dot q plus P.
00:02:16.350 --> 00:02:18.900
And then there's a relativistic
propagator for the gluon,
00:02:18.900 --> 00:02:20.970
and that's the q squared.
00:02:20.970 --> 00:02:23.890
I've taken into account all the
i's when I put this minus sign,
00:02:23.890 --> 00:02:26.040
and this is just the
fundamental Casimir
00:02:26.040 --> 00:02:28.815
from dotting two TAs together.
00:02:31.620 --> 00:02:32.970
OK.
00:02:32.970 --> 00:02:35.530
So when you have an
integral like this--
00:02:35.530 --> 00:02:36.490
so v squared is 1.
00:02:36.490 --> 00:02:39.372
That's one simplification.
00:02:39.372 --> 00:02:40.830
When you have an
integral like that
00:02:40.830 --> 00:02:43.920
where you have a linear
momentum in one propagator
00:02:43.920 --> 00:02:47.160
and a quadratic momentum
in the other propagator,
00:02:47.160 --> 00:02:49.290
you don't want to use the
standard Feynman trick.
00:02:52.050 --> 00:02:53.690
I call this trick
the George I trick.
00:02:56.520 --> 00:03:00.030
So it's very similar
to the Feynman trick
00:03:00.030 --> 00:03:03.880
but slightly different.
00:03:03.880 --> 00:03:09.720
So you use an integral that
goes from 0 to infinity,
00:03:09.720 --> 00:03:15.180
and you can convince
yourself that this is true.
00:03:15.180 --> 00:03:21.900
And so then set a equal the q
squared and b equal to v dot
00:03:21.900 --> 00:03:22.770
q plus p.
00:03:25.668 --> 00:03:27.960
And the reason that you'd
want to use this trick rather
00:03:27.960 --> 00:03:30.865
than the usual one is that
if you use the usual one,
00:03:30.865 --> 00:03:32.490
you'd get an x
multiplying this and a 1
00:03:32.490 --> 00:03:34.320
minus x multiplying that.
00:03:34.320 --> 00:03:37.300
But then you would have an
x multiplying the q squared.
00:03:37.300 --> 00:03:39.687
So when you complete
this, it would
00:03:39.687 --> 00:03:41.520
be-- when you'd want
to complete the square,
00:03:41.520 --> 00:03:43.240
you'd like nothing to
multiply the q squared.
00:03:43.240 --> 00:03:45.323
You'd like the q squared
just to be bare by itself
00:03:45.323 --> 00:03:47.380
with no Feynman
parameter multiplying it.
00:03:47.380 --> 00:03:49.110
And this trick does
that, because a
00:03:49.110 --> 00:03:52.650
has no Feynman parameter
multiplying it,
00:03:52.650 --> 00:03:55.920
or George I parameter
in this case.
00:03:55.920 --> 00:04:05.070
OK, so we combine
denominators in the usual way,
00:04:05.070 --> 00:04:09.130
and the denominator
would become this.
00:04:09.130 --> 00:04:13.260
And if I kept the
i epsilon it would
00:04:13.260 --> 00:04:17.100
be that if we combine these
two denominators here.
00:04:17.100 --> 00:04:19.920
So this factor would be that.
00:04:23.650 --> 00:04:26.310
And we can then complete
the square, right.
00:04:26.310 --> 00:04:28.020
This is some momentum
squared minus
00:04:28.020 --> 00:04:37.800
whatever is left over
where t is q plus lambda v,
00:04:37.800 --> 00:04:44.790
and then A is the rest of
the stuff, which is this.
00:04:49.905 --> 00:04:59.040
OK, so then this
guy, now we just
00:04:59.040 --> 00:05:02.310
have our usual
quadratic integral
00:05:02.310 --> 00:05:05.880
that we can use the
standard rules to do.
00:05:16.945 --> 00:05:18.570
And then instead of
an integral over x,
00:05:18.570 --> 00:05:19.987
we have this
integral over lambda.
00:05:33.280 --> 00:05:34.905
There's some factors
that I'm dropping.
00:05:40.380 --> 00:05:44.100
So there's that e to the Euler
gamma times epsilon and the 4
00:05:44.100 --> 00:05:47.020
pi minus epsilon.
00:05:47.020 --> 00:05:50.850
So write in everything with
d equals 4 minus 3 epsilon.
00:05:56.100 --> 00:06:01.950
Do that integral, which is just
giving some gamma functions.
00:06:17.990 --> 00:06:20.290
And if you think
about the dimensions
00:06:20.290 --> 00:06:23.092
here, so we end up with
something that's d minus 3.
00:06:23.092 --> 00:06:25.300
If you think-- if you want
to look at the dimensions,
00:06:25.300 --> 00:06:29.380
d over 2 minus 2 is
dimensionless, OK.
00:06:29.380 --> 00:06:30.970
The lambda had dimensions.
00:06:30.970 --> 00:06:32.620
If we go back here,
that's obvious.
00:06:32.620 --> 00:06:34.960
q is dimensionful,
v is dimensionless,
00:06:34.960 --> 00:06:37.150
so lambda has dimension 1.
00:06:37.150 --> 00:06:38.925
So the dimensions of the--
00:06:38.925 --> 00:06:43.390
actually, the dimensions
of the mu to the 2 epsilon
00:06:43.390 --> 00:06:45.950
are compensating
the dimensions here,
00:06:45.950 --> 00:06:48.130
and then there's one
power of dimension left.
00:06:48.130 --> 00:06:49.960
And that's why if
I take d equals 4,
00:06:49.960 --> 00:06:53.560
I'm getting one power of
momentum upstairs, which
00:06:53.560 --> 00:06:55.630
is what we would expect
for an inverse propagator
00:06:55.630 --> 00:06:59.770
for a heavy quark, is
1 factor of v dot P.
00:06:59.770 --> 00:07:01.030
So expand this.
00:07:12.438 --> 00:07:17.350
And we get a divergence.
00:07:17.350 --> 00:07:25.288
So add a counter term for
wave function renormalization.
00:07:39.940 --> 00:07:43.873
So if we did that in MS bar,
then the Zh would be just this.
00:07:43.873 --> 00:07:45.790
And that would be the
appropriate counter term
00:07:45.790 --> 00:07:48.947
to kill off the 1 over
epsilon divergence.
00:07:48.947 --> 00:07:50.530
I'm going to carry
out the calculation
00:07:50.530 --> 00:07:53.800
today, or this discussion
of matching in MS bar
00:07:53.800 --> 00:07:54.760
for everything.
00:07:54.760 --> 00:07:57.010
And I'll show you some of
the slight complication that
00:07:57.010 --> 00:07:59.020
shows up in that case.
00:07:59.020 --> 00:08:03.920
But we said that we
could do matching--
00:08:03.920 --> 00:08:06.350
basically, you could
have two choices here.
00:08:06.350 --> 00:08:08.440
You could either use
on-shell renormalization,
00:08:08.440 --> 00:08:10.000
or you could use MS bar.
00:08:10.000 --> 00:08:11.590
What's the difference?
00:08:11.590 --> 00:08:14.260
In MS bar, you just keep
the divergence in the z.
00:08:14.260 --> 00:08:17.950
In on-shell renormalization,
you keep some extra terms here,
00:08:17.950 --> 00:08:19.270
all right.
00:08:19.270 --> 00:08:21.098
And either way that we
choose to do things,
00:08:21.098 --> 00:08:23.140
we should actually end up
with the same matching,
00:08:23.140 --> 00:08:26.210
and I want to show
you why that's true.
00:08:26.210 --> 00:08:29.050
So in order to show
you why that's true,
00:08:29.050 --> 00:08:31.360
I'll pick to use MS
bar at this point,
00:08:31.360 --> 00:08:35.700
and we'll see what complication
that leads to later on.
00:08:35.700 --> 00:08:38.620
OK, so that's one thing that
needs to be renormalized.
00:08:38.620 --> 00:08:43.870
And if we look at this guy
and we compare it to z psi,
00:08:43.870 --> 00:08:47.980
this is not the same
as z psi in QCD.
00:08:47.980 --> 00:08:50.138
So this is something
different, and the reason
00:08:50.138 --> 00:08:52.180
it's different is because
we did a different loop
00:08:52.180 --> 00:08:55.150
integral that had this
heavy quark propagator, not
00:08:55.150 --> 00:08:56.701
the light quark propagator.
00:09:02.830 --> 00:09:06.040
So we have to renormalize
also local operators.
00:09:06.040 --> 00:09:08.200
And so let's think about
something that would make
00:09:08.200 --> 00:09:09.790
a heavy to light transition.
00:09:09.790 --> 00:09:14.430
So for example, if you
looked at b goes to u,
00:09:14.430 --> 00:09:20.680
electron neutrino,
then that would
00:09:20.680 --> 00:09:22.010
be a heavy to light transition.
00:09:22.010 --> 00:09:22.870
To b quark is heavy.
00:09:22.870 --> 00:09:24.520
The b quark is light.
00:09:24.520 --> 00:09:26.050
So to describe
that, you would use
00:09:26.050 --> 00:09:29.830
some operator that has one
heavy quark and one light quark.
00:09:29.830 --> 00:09:34.690
So let me call the light quark
small q and the heavy quark
00:09:34.690 --> 00:09:40.780
big Q. And so you'd have some
operator looks like that.
00:09:40.780 --> 00:09:52.830
And we could write down
a renormalized operator
00:09:52.830 --> 00:09:57.370
with renormalized fields and
then group all the z factors
00:09:57.370 --> 00:09:59.040
into a counter term.
00:10:06.506 --> 00:10:09.140
So let's think about doing the
perturbation theory that way.
00:10:09.140 --> 00:10:10.700
So this is the
renormalized operator.
00:10:10.700 --> 00:10:11.742
This is the counter term.
00:10:16.580 --> 00:10:18.955
So there's a wave function
factor Zq for the light quark
00:10:18.955 --> 00:10:21.070
and Zh for the heavy
quark, and then there's
00:10:21.070 --> 00:10:24.381
some Zo for the operator
renormalization.
00:10:34.210 --> 00:10:38.150
OK, so to renormalize
this operator at one loop,
00:10:38.150 --> 00:10:42.440
we insert it, and we
do a one-loop diagram.
00:10:46.370 --> 00:10:49.160
I'm just going to
tell you the answers,
00:10:49.160 --> 00:10:51.526
but let's draw the diagram.
00:10:51.526 --> 00:10:53.080
So here's the operator inserted.
00:10:53.080 --> 00:10:54.080
Here's your heavy quark.
00:10:54.080 --> 00:10:56.000
Here's your light quark.
00:10:56.000 --> 00:10:57.230
You have a diagram like that.
00:11:03.880 --> 00:11:10.030
And then you combine this
calculation with the wave
00:11:10.030 --> 00:11:16.150
function renormalize Zh and Zq.
00:11:16.150 --> 00:11:18.530
Zq is the same as Z psi.
00:11:18.530 --> 00:11:24.620
We should've called this
Zq for the light quark.
00:11:24.620 --> 00:11:26.908
Combine these things
together, and then,
00:11:26.908 --> 00:11:29.450
because that graph is telling
you to count this counter term,
00:11:29.450 --> 00:11:31.950
you need Zo.
00:11:31.950 --> 00:11:37.820
And that calculation, which
you can look at in your reading
00:11:37.820 --> 00:11:40.700
if you want to look
at more details,
00:11:40.700 --> 00:11:43.145
just gives you something
about what you'd expect.
00:11:43.145 --> 00:11:48.390
It gives you a 1 over epsilon
divergence factor of g squared.
00:11:48.390 --> 00:11:53.150
So the operator here has
renormalization that's
00:11:53.150 --> 00:11:59.700
just minus alpha s over pi.
00:11:59.700 --> 00:12:01.385
That's the anomalous dimension.
00:12:04.000 --> 00:12:07.335
So what is this anomalous
dimension doing?
00:12:07.335 --> 00:12:11.850
If you were to consider this
kind of process in full QCD,
00:12:11.850 --> 00:12:15.000
then you'd have here
gamma mu 1 minus 5,
00:12:15.000 --> 00:12:17.550
and that's a partially
conserved current.
00:12:17.550 --> 00:12:19.830
So there would be actually
no anomalous dimension
00:12:19.830 --> 00:12:20.970
to this operator.
00:12:20.970 --> 00:12:23.880
The vertex graph that
we just drew over
00:12:23.880 --> 00:12:26.250
there would cancel the wave
function graphs exactly.
00:12:26.250 --> 00:12:28.290
There'd be no
anomalous dimension.
00:12:28.290 --> 00:12:30.420
But here we have one.
00:12:30.420 --> 00:12:33.580
And that's because these guys
are not equivalent anymore.
00:12:33.580 --> 00:12:36.810
You saw that the Z for
the heavy quark changed,
00:12:36.810 --> 00:12:39.240
and the vertex
graph also changes.
00:12:39.240 --> 00:12:41.520
And we're left with something.
00:12:41.520 --> 00:12:45.150
And this remainder has to do
with renormalization group
00:12:45.150 --> 00:12:47.933
evolution below the
mass of the heavy quark.
00:12:47.933 --> 00:12:49.350
Above the mass of
the heavy quark,
00:12:49.350 --> 00:12:52.140
there's no renormalize group
evolution of this current.
00:12:52.140 --> 00:12:54.000
Below the mass of the
heavy quark, there is.
00:13:22.577 --> 00:13:24.160
So there's additional
logs, and that's
00:13:24.160 --> 00:13:27.580
because MQ is now being
treated as infinite.
00:13:27.580 --> 00:13:31.690
So things that, from the point
of view of QCD, were logs of MQ
00:13:31.690 --> 00:13:34.000
have now become UV
divergences, and that gives
00:13:34.000 --> 00:13:35.691
an extra anomalous dimension.
00:13:39.380 --> 00:13:42.320
One thing you can note about
this anomalous dimension
00:13:42.320 --> 00:13:47.150
is that I didn't really specify
for you what the gamma was.
00:13:47.150 --> 00:13:49.790
I told you for this process it
would be gamma mu 1 minus gamma
00:13:49.790 --> 00:13:50.720
5.
00:13:50.720 --> 00:13:56.250
And the results here,
actually, if you carry out
00:13:56.250 --> 00:13:58.380
this calculation
with arbitrary gamma,
00:13:58.380 --> 00:14:04.680
you find that it's
independent of gamma.
00:14:04.680 --> 00:14:08.310
So you get the same universal
anomalous convention
00:14:08.310 --> 00:14:11.520
for any spin structure.
00:14:11.520 --> 00:14:13.800
And that's partly
related to the things
00:14:13.800 --> 00:14:17.670
that we talked about last time
with the spin symmetry of HQET,
00:14:17.670 --> 00:14:21.360
which is telling you that
certain couplings are not
00:14:21.360 --> 00:14:22.890
sensitive to the spin.
00:14:22.890 --> 00:14:26.030
And effectively,
in this diagram,
00:14:26.030 --> 00:14:27.740
you're getting a v slash here.
00:14:27.740 --> 00:14:30.000
Well, let me not
try to go through it
00:14:30.000 --> 00:14:32.427
but leave it for looking
at in your reading,
00:14:32.427 --> 00:14:34.760
but we'll talk a little bit
more about this in a minute.
00:14:38.970 --> 00:14:41.160
I won't try to
explain where that
00:14:41.160 --> 00:14:43.960
comes from from the diagram.
00:14:43.960 --> 00:14:47.350
So let's look at another case.
00:14:47.350 --> 00:14:49.470
The only real interesting
thing that happened here
00:14:49.470 --> 00:14:51.750
was that we got-- well, the
fact that the answer was
00:14:51.750 --> 00:14:54.060
non-0 was interesting,
and the fact that it
00:14:54.060 --> 00:14:57.060
was independent of gamma.
00:14:57.060 --> 00:14:58.610
But the v didn't show up.
00:14:58.610 --> 00:15:01.110
And the reason that the v didn't
show up in this calculation
00:15:01.110 --> 00:15:03.870
here is because
there was only one v,
00:15:03.870 --> 00:15:06.120
and v squared was equal to 1.
00:15:06.120 --> 00:15:07.830
So v couldn't really
show up because we
00:15:07.830 --> 00:15:12.740
had to get a scalar answer, and
since v squared is equal to 1,
00:15:12.740 --> 00:15:15.160
just, it's not showing up.
00:15:15.160 --> 00:15:18.360
So something more interesting
is to look at, instead
00:15:18.360 --> 00:15:21.230
of a heavy to light
transition like that,
00:15:21.230 --> 00:15:22.980
a heavy to heavy transition.
00:15:28.326 --> 00:15:32.450
So we'll spend a little
bit more time on this one.
00:15:32.450 --> 00:15:36.560
So let's have two heavy fields.
00:15:36.560 --> 00:15:38.540
And I'm going to take
them in a current
00:15:38.540 --> 00:15:43.168
where they have different
velocities, v and v prime.
00:15:43.168 --> 00:15:45.710
So let me imagine I went through
this procedure of separating
00:15:45.710 --> 00:15:55.370
out counter term and
renormalized the operator just
00:15:55.370 --> 00:15:56.300
like I did over there.
00:15:59.650 --> 00:16:02.000
So I have these two terms,
two types of structures.
00:16:02.000 --> 00:16:04.650
Now I don't have a Zq, but
I have two heavy quarks.
00:16:04.650 --> 00:16:09.740
So I have Zh, root Zh
root Zh, which is just Zh.
00:16:09.740 --> 00:16:12.620
So an example of this
would be something
00:16:12.620 --> 00:16:19.160
like B meson changing to,
say, a D star meson electron
00:16:19.160 --> 00:16:22.890
and a neutrino, so having a
charm quark replace our up
00:16:22.890 --> 00:16:23.390
quark.
00:16:28.790 --> 00:16:30.590
OK, so now the charm
quark and the B quark
00:16:30.590 --> 00:16:32.270
could both be
thought of as heavy.
00:16:32.270 --> 00:16:34.880
They have different masses, but
we take both of those masses
00:16:34.880 --> 00:16:37.310
to infinity, so we can
use HQET for both of them.
00:16:42.210 --> 00:16:45.320
So Mb and Mc are
going to infinity.
00:16:45.320 --> 00:16:49.190
And there's no reason
why, in this process,
00:16:49.190 --> 00:16:51.290
that the b quark
and the charm quarks
00:16:51.290 --> 00:16:54.143
should have the same
velocity, and so we'll
00:16:54.143 --> 00:16:56.060
give them different
velocities, v and v prime.
00:16:59.230 --> 00:17:02.800
OK, so we can go
through the same thing.
00:17:02.800 --> 00:17:04.630
We already calculated
Zh, so we just
00:17:04.630 --> 00:17:11.230
have to calculate a
graph like this or two,
00:17:11.230 --> 00:17:14.710
where you have two heavy quarks.
00:17:14.710 --> 00:17:17.109
But the heavy quarks have
different velocities.
00:17:21.222 --> 00:17:22.930
So what would that
calculation look like?
00:17:27.619 --> 00:17:34.970
Again in Feynman gauge,
and let me just take
00:17:34.970 --> 00:17:39.290
the external quarks to have
zero momentum for simplicity,
00:17:39.290 --> 00:17:42.510
zero residual momentum.
00:17:42.510 --> 00:17:47.900
So this guy has k
equals 0 and k equals 0.
00:17:47.900 --> 00:17:50.480
So the HQET Feynman
rule for this guy
00:17:50.480 --> 00:17:54.380
has a v dot q if q
is the loop momenta,
00:17:54.380 --> 00:17:57.890
and this guy has a
v prime dot q, OK.
00:17:57.890 --> 00:18:01.550
So the integral that have
to do is that integral.
00:18:01.550 --> 00:18:04.880
And you can do this again
with one of these tricks where
00:18:04.880 --> 00:18:07.100
you use lambda parameters.
00:18:07.100 --> 00:18:10.220
And then one of the
handouts on the web page,
00:18:10.220 --> 00:18:12.080
I've given you the
appropriate trick
00:18:12.080 --> 00:18:14.090
that's two lambda parameters
for this integral.
00:18:19.430 --> 00:18:21.710
This integral actually
has both ultraviolet
00:18:21.710 --> 00:18:27.340
and infrared divergences.
00:18:27.340 --> 00:18:28.780
We're here in our
discussion only
00:18:28.780 --> 00:18:33.235
interested in the
ultraviolet ones,
00:18:33.235 --> 00:18:35.860
because we're worrying about the
anomalous dimension right now.
00:18:39.760 --> 00:18:41.260
And again, I'm not
going to drag you
00:18:41.260 --> 00:18:43.440
through the details
of this calculation.
00:18:46.300 --> 00:18:48.910
It's essentially just,
do the Feynman parameter,
00:18:48.910 --> 00:18:52.840
combine denominators with
two Feynman parameters,
00:18:52.840 --> 00:18:55.240
complete the square,
do the loop integral,
00:18:55.240 --> 00:18:59.290
do the Feynman parameter
integrals, get an answer.
00:18:59.290 --> 00:19:01.840
Combine it together with the
counter terms of the wave
00:19:01.840 --> 00:19:05.470
function renormalize, and
see what type of counter term
00:19:05.470 --> 00:19:07.750
you get left over.
00:19:07.750 --> 00:19:14.560
And it's more interesting
than the heavy to light case.
00:19:14.560 --> 00:19:29.400
So here's what it looks like,
where w is v dot v prime,
00:19:29.400 --> 00:19:33.660
and this function, r
of w, is the following.
00:19:33.660 --> 00:19:43.840
It's log w plus square
root w squared minus 1
00:19:43.840 --> 00:19:45.940
over the square root
of w squared minus 1.
00:19:52.080 --> 00:19:57.660
OK, so it's a
non-trivial structure.
00:19:57.660 --> 00:20:00.870
So that counter term would
lead to an anomalous dimension,
00:20:00.870 --> 00:20:02.670
which depends on this r of w.
00:20:13.320 --> 00:20:15.090
So the reason that
that can happen
00:20:15.090 --> 00:20:18.960
is because v squared is
1, v prime squared is 1,
00:20:18.960 --> 00:20:20.880
but v dot v prime need not be 1.
00:20:24.220 --> 00:20:29.350
So v squared was 1, v
prime squared was 1,
00:20:29.350 --> 00:20:32.755
but v dot v prime is not fixed.
00:20:36.890 --> 00:20:40.121
Well, it's not fixed simply by--
00:20:40.121 --> 00:20:41.620
that's a poor choice of words.
00:20:45.763 --> 00:20:47.180
This is a parameter
that can vary.
00:20:50.402 --> 00:20:52.340
So it will be fixed
by kinematics,
00:20:52.340 --> 00:20:59.570
but it can depend
on the kinematics.
00:20:59.570 --> 00:21:02.720
So let me go through
this and organize it
00:21:02.720 --> 00:21:04.430
as a bunch of notes, comments.
00:21:07.140 --> 00:21:13.220
So this is what I just said.
00:21:13.220 --> 00:21:14.837
The answer depends
on v dot v prime.
00:21:14.837 --> 00:21:16.670
And the way that you
should think about this
00:21:16.670 --> 00:21:20.590
is that you have a current
in the effective theory.
00:21:20.590 --> 00:21:23.870
So this is in the HQET current.
00:21:28.670 --> 00:21:33.320
And just like we label it
by its Lorentz index mu,
00:21:33.320 --> 00:21:35.570
we should label it also
by the v and v prime,
00:21:35.570 --> 00:21:38.940
because the fields
involve v and v prime.
00:21:38.940 --> 00:21:43.610
So if we thought of this as some
vector current or axial vector
00:21:43.610 --> 00:21:48.650
current labeled by mu, we
would also label the current
00:21:48.650 --> 00:21:49.640
by v and v prime.
00:21:52.500 --> 00:21:55.082
So these are indices that are
just indices on the current,
00:21:55.082 --> 00:21:56.540
and when you think
about the Wilson
00:21:56.540 --> 00:21:57.980
coefficient or the
anomalous dimension,
00:21:57.980 --> 00:21:59.272
it can depend on those indices.
00:22:06.770 --> 00:22:12.050
Now because the Wilson
coefficient is a scalar,
00:22:12.050 --> 00:22:15.470
it really only depends on w.
00:22:15.470 --> 00:22:18.560
So if you think about
Wilson coefficients,
00:22:18.560 --> 00:22:19.700
they depend on alpha s.
00:22:19.700 --> 00:22:22.730
They depend on scale mu.
00:22:22.730 --> 00:22:26.540
They depend on the hard
scales in your problem,
00:22:26.540 --> 00:22:28.010
and the hard scales
in the problem
00:22:28.010 --> 00:22:32.960
here for us are Mb times
v and Mc times v prime,
00:22:32.960 --> 00:22:35.750
because those are the heavy
scales of the heavy charm
00:22:35.750 --> 00:22:38.810
quark and the heavy b
quark that we're removing.
00:22:38.810 --> 00:22:40.940
But since we know that
this thing is a scalar,
00:22:40.940 --> 00:22:43.760
we can just square these,
dot them into each other.
00:22:43.760 --> 00:22:50.990
So it's really a function of
mu of alpha s mu Mb squared,
00:22:50.990 --> 00:22:54.890
from squaring the Mb term,
M charm squared, and then
00:22:54.890 --> 00:22:55.865
this w factor.
00:22:59.088 --> 00:23:01.130
Those are the scalar
quantities it can depend on.
00:23:11.580 --> 00:23:15.180
So what should I take for the w?
00:23:15.180 --> 00:23:17.340
Well, we could work
that out for an example
00:23:17.340 --> 00:23:24.058
like the one I was saying that
we're doing, B to D star L nu.
00:23:24.058 --> 00:23:25.600
So I want to show
you how that works.
00:23:39.710 --> 00:23:43.720
So for the B meson,
for momentum,
00:23:43.720 --> 00:23:48.280
we can take it to be M
capital B meson times v mu.
00:23:48.280 --> 00:23:50.770
And by momentum
conservation, that's
00:23:50.770 --> 00:23:53.680
going to be the D
star momentum, which
00:23:53.680 --> 00:23:58.480
we can take to be MD
Star times v prime mu
00:23:58.480 --> 00:24:02.260
then some momentum
transfer, which I call q.
00:24:02.260 --> 00:24:05.170
This is a different q
than my loop momentum q.
00:24:05.170 --> 00:24:06.040
Sorry about that.
00:24:10.998 --> 00:24:13.540
So then we can just take this
relation, and we can square it,
00:24:13.540 --> 00:24:14.950
and we can get v dot v primes.
00:24:14.950 --> 00:24:36.660
So if you look at 2 squared,
we can solve this for v dot v
00:24:36.660 --> 00:24:38.220
prime, and it's just fixed.
00:24:38.220 --> 00:24:40.770
You see that v dot v prime is
fixed in terms of the meson
00:24:40.770 --> 00:24:43.380
masses and the
momentum transfer,
00:24:43.380 --> 00:24:46.710
and that's the momentum
transfer to the leptons here.
00:24:46.710 --> 00:24:48.665
How much momentum
do they carry away?
00:24:48.665 --> 00:24:50.790
So you could think of all
those things as external.
00:24:50.790 --> 00:24:53.940
You fixed how much momentum
the leptons carry away.
00:24:53.940 --> 00:24:54.990
These are fixed numbers.
00:24:54.990 --> 00:24:57.960
You look them up in the PDG,
and then what value of v
00:24:57.960 --> 00:25:00.390
dot v prime to plug-in here.
00:25:00.390 --> 00:25:01.770
Now it's a function
of q squared.
00:25:01.770 --> 00:25:04.570
Q squared can vary
in the process.
00:25:04.570 --> 00:25:08.430
So in that sense, it's a
non-trivial function, not just
00:25:08.430 --> 00:25:10.230
a fixed number.
00:25:10.230 --> 00:25:12.750
But for any fixed
kinematic configuration,
00:25:12.750 --> 00:25:17.380
any fixed q, then it
would just be a number.
00:25:17.380 --> 00:25:19.230
So if you look at
this in practice,
00:25:19.230 --> 00:25:23.070
you find that this guy
for this particular case
00:25:23.070 --> 00:25:24.585
goes from 1 to 1.5.
00:25:24.585 --> 00:25:26.460
So that's the kinematic
range that's allowed.
00:25:47.370 --> 00:25:52.020
So it is fixed by external
kinematics, kinematics
00:25:52.020 --> 00:25:54.780
that is external to the
dynamics inside the loop.
00:25:59.072 --> 00:26:00.780
And then that way,
the Wilson coefficient
00:26:00.780 --> 00:26:04.150
here is more non-trivial
than the ones we saw earlier,
00:26:04.150 --> 00:26:05.442
which just depended on masses.
00:26:05.442 --> 00:26:07.650
Now it's depending on masses
as well as this function
00:26:07.650 --> 00:26:10.590
of v dot v prime, OK?
00:26:17.170 --> 00:26:21.220
Again, one finds that
gamma T comes out
00:26:21.220 --> 00:26:24.580
to be exactly the same,
independent of the choice
00:26:24.580 --> 00:26:27.850
of the spin structure.
00:26:27.850 --> 00:26:30.520
So we could do this calculation
with any spin structure
00:26:30.520 --> 00:26:33.880
we like, and heavy quark
symmetry in this case
00:26:33.880 --> 00:26:40.030
is all it takes to show that
this gamma T is independent
00:26:40.030 --> 00:26:42.872
of the spin structure.
00:26:42.872 --> 00:26:44.830
So if you think about
that from the loop graph,
00:26:44.830 --> 00:26:48.190
actually in this case,
it's pretty easy to see,
00:26:48.190 --> 00:26:51.910
because remember that this
vertex didn't have any spin
00:26:51.910 --> 00:26:52.420
structure.
00:26:52.420 --> 00:26:53.980
The propagator had
no spin structure.
00:26:53.980 --> 00:26:56.478
So nothing in the calculation
had spin structure.
00:26:56.478 --> 00:26:58.270
So the only thing that's
had spin structure
00:26:58.270 --> 00:27:00.500
is the gamma you stuck in there.
00:27:00.500 --> 00:27:02.500
So it's just a
scalar times gamma,
00:27:02.500 --> 00:27:04.660
so it doesn't care
about the gamma.
00:27:04.660 --> 00:27:07.270
On whatever the tree level
gamma is, it just goes through.
00:27:07.270 --> 00:27:10.960
It's not touched by heavy
quark, by the HQET Lagrangian.
00:27:15.100 --> 00:27:17.530
So what is physically
going on here,
00:27:17.530 --> 00:27:20.260
and what is HQET
doing for you is
00:27:20.260 --> 00:27:25.420
that there's logs like this,
MQ over lambda QCD and QCD.
00:27:25.420 --> 00:27:32.470
And by going over to
HQET, this becomes a log
00:27:32.470 --> 00:27:37.810
of mu over lambda
QCD, which is encoded
00:27:37.810 --> 00:27:43.990
in HQET operators
like this current,
00:27:43.990 --> 00:27:49.750
as well as a log
of mu over MQ or MQ
00:27:49.750 --> 00:27:56.410
over mu, which is in the HQET
coefficient functions, HQET
00:27:56.410 --> 00:27:58.220
Wilson coefficients.
00:27:58.220 --> 00:28:00.400
So just how we--
much exactly the same
00:28:00.400 --> 00:28:02.650
as how we talked about
it for integrating out
00:28:02.650 --> 00:28:06.520
a heavy particle, the logs
get split up into pieces,
00:28:06.520 --> 00:28:08.950
and the Wilson coefficient
into pieces, and the matrix
00:28:08.950 --> 00:28:11.800
elements, operators.
00:28:11.800 --> 00:28:15.220
Here we're separating
MQ heavy quark mass,
00:28:15.220 --> 00:28:19.540
and it's both the charm and
the bottom in the case of what
00:28:19.540 --> 00:28:22.370
we're talking about.
00:28:22.370 --> 00:28:25.370
And the anomalous dimension
that we calculated sums up those
00:28:25.370 --> 00:28:30.645
logs, and summing up those logs
involves a non-trivial function
00:28:30.645 --> 00:28:31.145
of this w.
00:28:44.840 --> 00:28:47.240
But we actually know exactly
the non-trivial function,
00:28:47.240 --> 00:28:50.650
because we can calculate it.
00:28:50.650 --> 00:28:53.600
And it's just this guy here.
00:29:00.450 --> 00:29:03.540
OK, so the new wrinkle
that can come in HQET
00:29:03.540 --> 00:29:06.210
is that the anomalous
dimension can become a more
00:29:06.210 --> 00:29:07.040
non-trivial thing.
00:29:14.000 --> 00:29:17.260
So if you look at it
at leading log order,
00:29:17.260 --> 00:29:19.250
the rest is pretty
straightforward.
00:29:19.250 --> 00:29:23.698
So if you go through
the leading log result,
00:29:23.698 --> 00:29:25.990
you would do the same type
of thing that we did before.
00:29:25.990 --> 00:29:29.980
You would match it, some scale.
00:29:29.980 --> 00:29:32.230
And at that scale, you
could normalize things
00:29:32.230 --> 00:29:37.420
so that the Wilson coefficient
at mu equals MQ is just 1
00:29:37.420 --> 00:29:39.640
at tree level.
00:29:39.640 --> 00:29:43.600
And then the leading log
result is the function
00:29:43.600 --> 00:29:47.980
of these various things,
which in general is C of MQ
00:29:47.980 --> 00:29:52.240
times some evolution
from MQ to mu,
00:29:52.240 --> 00:29:55.540
suppressing some of
the dependencies.
00:29:55.540 --> 00:29:59.140
The leading log
result is 1 for this.
00:29:59.140 --> 00:30:00.820
And then this guy
at the lowest order
00:30:00.820 --> 00:30:02.500
is just a ratio of alphas again.
00:30:07.730 --> 00:30:10.810
And then there's the gamma.
00:30:10.810 --> 00:30:12.900
For the purpose of
solving the RGE is just,
00:30:12.900 --> 00:30:14.650
the only thing that
matters is it's alpha.
00:30:21.270 --> 00:30:26.450
So a gamma's a constant for
heavy to light, just a number.
00:30:30.120 --> 00:30:33.300
So for that current,
it was a constant.
00:30:33.300 --> 00:30:36.760
This solution is actually
valid for both of them.
00:30:36.760 --> 00:30:47.190
And then it's a function of
this w for the current, where
00:30:47.190 --> 00:30:48.720
we have two heavy quarks.
00:30:48.720 --> 00:30:50.640
So the w dependence
just goes along
00:30:50.640 --> 00:30:52.973
for the ride when you're
solving the anomalous dimension
00:30:52.973 --> 00:30:54.130
equation.
00:30:54.130 --> 00:30:54.630
OK?
00:30:54.630 --> 00:30:58.920
So that's what re-summing
the logs would look like.
00:30:58.920 --> 00:31:01.890
Essentially, each log
is getting extra powers
00:31:01.890 --> 00:31:03.330
of this factor of gamma.
00:31:06.697 --> 00:31:11.360
So with number four, is there
any questions about that?
00:31:14.776 --> 00:31:17.009
OK, pretty straightforward.
00:31:21.320 --> 00:31:23.690
So much of this, essentially
all of the story,
00:31:23.690 --> 00:31:26.590
except for this one wrinkle, is
very similar to integrating out
00:31:26.590 --> 00:31:28.970
a massive particle.
00:31:28.970 --> 00:31:33.130
And the other part of
the story that's similar
00:31:33.130 --> 00:31:37.060
is that the HQET matrix
elements depend on mu as well.
00:31:46.090 --> 00:31:49.050
And so in our example that
we talked about last time,
00:31:49.050 --> 00:31:50.350
we had a matrix element.
00:31:50.350 --> 00:31:55.810
So let me give it to you in
the context of that example.
00:31:55.810 --> 00:31:57.750
So last time we were
talking about something
00:31:57.750 --> 00:32:02.708
which was a decay constant,
and that's one example
00:32:02.708 --> 00:32:04.000
of this heavy to light current.
00:32:07.970 --> 00:32:10.310
So we had our current which
had one heavy quark, one
00:32:10.310 --> 00:32:13.280
light quark, and
then a heavy meson.
00:32:13.280 --> 00:32:18.920
And we figured out that this
was giving some a times v mu,
00:32:18.920 --> 00:32:21.380
and now I'm telling you that
you should think of the a
00:32:21.380 --> 00:32:23.330
as a function of mu.
00:32:23.330 --> 00:32:25.190
The matrix element here
is a function of mu.
00:32:30.415 --> 00:32:33.440
OK, so that's just a
slight modification to what
00:32:33.440 --> 00:32:36.870
we talked about last time.
00:32:36.870 --> 00:32:42.470
And again, if you want,
for this matrix element,
00:32:42.470 --> 00:32:47.690
you'd want mu to be, say,
a GeV or some scale that's
00:32:47.690 --> 00:32:51.788
greater than lambda QCD,
and so what you would do
00:32:51.788 --> 00:32:53.330
is you would evaluate
matrix elements
00:32:53.330 --> 00:32:56.180
and define that parameter
at some scale like a GeV
00:32:56.180 --> 00:32:59.360
and do renormalization group
evolution from the heavy quark
00:32:59.360 --> 00:33:02.480
mass down to a GeV.
00:33:02.480 --> 00:33:03.020
OK?
00:33:03.020 --> 00:33:12.250
So that's how the
renormalization group evolution
00:33:12.250 --> 00:33:13.030
story would be.
00:33:16.292 --> 00:33:18.500
You don't want to run all
the way down to lambda QCD,
00:33:18.500 --> 00:33:21.500
because the anomalous dimension
has to remain perturbative.
00:33:21.500 --> 00:33:24.715
So you would decide
what your cutoff
00:33:24.715 --> 00:33:26.840
is for where you think
perturbation theory is still
00:33:26.840 --> 00:33:27.950
valid.
00:33:27.950 --> 00:33:32.270
Often people pick something like
1 GeV or 1 1/2 GeV for these
00:33:32.270 --> 00:33:33.050
types of problems.
00:33:44.910 --> 00:33:47.810
And again, this is just
separating out all the MQs,
00:33:47.810 --> 00:33:51.080
making sure that your matrix
element here has no MQs,
00:33:51.080 --> 00:33:54.230
it does have an extra cutoff mu.
00:33:54.230 --> 00:33:55.850
OK?
00:33:55.850 --> 00:33:58.580
So that's the RGE story.
00:33:58.580 --> 00:34:01.010
Let's also talk a little bit
about the matching story.
00:34:05.210 --> 00:34:13.000
So these are the perturbative
corrections at the scale MQ,
00:34:13.000 --> 00:34:16.239
or alpha s at MQ.
00:34:16.239 --> 00:34:18.172
We had perturbative
corrections at the w scale
00:34:18.172 --> 00:34:19.630
when we integrated
out the w boson.
00:34:19.630 --> 00:34:21.310
Now we have another set of
perturbative corrections
00:34:21.310 --> 00:34:23.409
at the heavy quark mass
scale when we integrated
00:34:23.409 --> 00:34:25.000
out the heavy quark mass.
00:34:25.000 --> 00:34:27.670
It's different because we're
now passing from something
00:34:27.670 --> 00:34:29.679
that looked like
a full QCD theory
00:34:29.679 --> 00:34:31.330
with some external operators.
00:34:31.330 --> 00:34:34.120
We're now passing to this HQET
theory for the heavy quark.
00:34:40.159 --> 00:34:43.940
So if you like, we
previously had Mw.
00:34:43.940 --> 00:34:45.710
We knew how to do
renormalization group
00:34:45.710 --> 00:34:50.929
evolution there, and we
had a Hw theory here.
00:34:50.929 --> 00:34:53.630
Now we integrate out
the heavy quark mass,
00:34:53.630 --> 00:34:57.620
and we pass to an HQET
theory below that scale.
00:35:03.065 --> 00:35:09.870
So if you go back to the Hw
theory, if we can call it that,
00:35:09.870 --> 00:35:13.140
and we want to match
that onto HQET,
00:35:13.140 --> 00:35:17.230
then we do it with a
calculation like this.
00:35:17.230 --> 00:35:20.400
And I'll just use this
heavy, light example still.
00:35:25.482 --> 00:35:27.190
So here's a matrix
element that you could
00:35:27.190 --> 00:35:31.110
consider for the matching.
00:35:31.110 --> 00:35:33.328
And let me write it with a
bunch of schematic objects
00:35:33.328 --> 00:35:34.620
and then explain what they are.
00:35:55.460 --> 00:35:57.480
So we use our spinners.
00:35:57.480 --> 00:36:01.980
I'm taking zero momentum
here, just for simplicity.
00:36:05.010 --> 00:36:08.430
These Rs are residue
factors that come in.
00:36:08.430 --> 00:36:14.550
So so far when we calculated the
wave function renormalization
00:36:14.550 --> 00:36:19.050
graphs, we just took
the divergent piece.
00:36:19.050 --> 00:36:21.960
And if you do that when you
do the matching computation,
00:36:21.960 --> 00:36:23.580
you have to use LSZ.
00:36:23.580 --> 00:36:25.770
And so the finite
pieces come back in,
00:36:25.770 --> 00:36:28.020
and you call them residues.
00:36:28.020 --> 00:36:37.460
So these are finite residues
that you have to take account
00:36:37.460 --> 00:36:40.670
of, finite in the UV sense.
00:36:40.670 --> 00:36:45.780
So UV finite residues you have
to take into account if you're
00:36:45.780 --> 00:36:48.390
just using MS bar,
and this here would
00:36:48.390 --> 00:36:51.150
be the vertex
renormalization graph.
00:36:54.782 --> 00:36:57.340
OK, both diagrams
look like this.
00:36:57.340 --> 00:37:03.590
In QCD, they're both the
same type of structure.
00:37:03.590 --> 00:37:07.390
And then in HQET,
it's a similar thing.
00:37:07.390 --> 00:37:09.310
We can write down a
formula for the s matrix
00:37:09.310 --> 00:37:14.600
element, the same states,
now with our effective theory
00:37:14.600 --> 00:37:15.100
current.
00:37:17.830 --> 00:37:22.000
And we know how to transition
from effective theory
00:37:22.000 --> 00:37:26.350
and full theory states.
00:37:26.350 --> 00:37:29.130
We talked about that last time.
00:37:29.130 --> 00:37:32.240
And so there again would
be some residue factors.
00:37:32.240 --> 00:37:33.682
And if the residue
factors are not
00:37:33.682 --> 00:37:36.140
the same in the two theories,
you have to account for that.
00:37:36.140 --> 00:37:38.890
And they won't be, because
one of the heavy quark residue
00:37:38.890 --> 00:37:41.200
will be different than the
heavy quark residue here.
00:38:02.260 --> 00:38:07.720
So this guy here is this
finite piece of this graph.
00:38:07.720 --> 00:38:10.460
This guy here will
be the same as above.
00:38:10.460 --> 00:38:19.240
And this guy here is
the heavy quark vertex
00:38:19.240 --> 00:38:22.270
graph, which is independent
of the spin structure.
00:38:22.270 --> 00:38:24.640
It's not independent of the
spin structure up there.
00:38:34.713 --> 00:38:36.880
So we could carry out the
calculations of those loop
00:38:36.880 --> 00:38:39.403
diagrams, and then
we could subtract,
00:38:39.403 --> 00:38:40.820
and we could see
what's left over.
00:38:40.820 --> 00:38:44.290
And whatever's left over
is the Wilson coefficient.
00:38:44.290 --> 00:38:47.320
OK, so very similar
to what we did before.
00:38:50.400 --> 00:38:53.430
Calculate, subtract.
00:38:59.410 --> 00:39:02.140
What you find when you do
that is that there's actually
00:39:02.140 --> 00:39:02.950
two currents.
00:39:02.950 --> 00:39:11.430
If you consider a vector
current where gamma is gamma mu,
00:39:11.430 --> 00:39:15.600
then the effective field
theory, which is HQET,
00:39:15.600 --> 00:39:21.450
has two effective theory
currents that are vector.
00:39:24.140 --> 00:39:27.145
So you have C1 and C2.
00:39:32.278 --> 00:39:34.820
The reason that there's two is
because we have another vector
00:39:34.820 --> 00:39:37.670
to play with, which is v mu.
00:39:37.670 --> 00:39:43.140
So that can have q bar v
mu replacing the gamma mu.
00:39:46.420 --> 00:39:48.880
So v mu wasn't an
external thing in QCD.
00:39:48.880 --> 00:39:50.920
It was part of the dynamics.
00:39:50.920 --> 00:39:52.420
Here it's an external
thing, so it's
00:39:52.420 --> 00:39:55.900
allowed to replace gamma mu
as one of the structures.
00:39:58.430 --> 00:40:00.253
And if you go through
the calculation,
00:40:00.253 --> 00:40:02.920
this is the result, just to show
you what the result looks like.
00:40:14.915 --> 00:40:16.290
Remember, the
heavy to light case
00:40:16.290 --> 00:40:18.520
is the case where you're not
getting a non-trivial function
00:40:18.520 --> 00:40:19.478
on the right hand side.
00:40:24.370 --> 00:40:26.660
So you'd get a non-zero
result of order alpha
00:40:26.660 --> 00:40:32.490
s for both of those
coefficients, OK?
00:40:32.490 --> 00:40:35.640
So the reading also goes
through this whole thing
00:40:35.640 --> 00:40:38.040
for the heavy to heavy case,
which is more interesting.
00:40:38.040 --> 00:40:40.890
But it's not really more
non-trivial than what
00:40:40.890 --> 00:40:42.368
we've talked about
so far already
00:40:42.368 --> 00:40:43.410
with anomalous dimension.
00:40:43.410 --> 00:40:45.830
You get results that
are functions of w.
00:40:45.830 --> 00:40:50.300
Wilson coefficient would have
functions of w showing up here,
00:40:50.300 --> 00:40:51.590
OK.
00:40:51.590 --> 00:40:55.360
So I won't go through that.
00:40:55.360 --> 00:40:57.970
Now if you wanted to carry
out this calculation,
00:40:57.970 --> 00:41:00.560
it looks like it's
kind of involved,
00:41:00.560 --> 00:41:02.470
this graph, this
graph, this graph,
00:41:02.470 --> 00:41:05.510
all these diagrams to consider.
00:41:05.510 --> 00:41:07.578
You'd like to make your
life as easy as possible,
00:41:07.578 --> 00:41:09.370
and there's actually
a very nice trick here
00:41:09.370 --> 00:41:11.650
for doing that I have
to mention to you,
00:41:11.650 --> 00:41:14.887
because it's kind of magical.
00:41:14.887 --> 00:41:17.220
So what is the fastest way
that I could get this result?
00:41:28.442 --> 00:41:30.400
So this is a nice trick
to remember if you ever
00:41:30.400 --> 00:41:33.580
have to do a
calculation like that,
00:41:33.580 --> 00:41:35.470
because it's not
specific to HQET.
00:41:38.060 --> 00:41:43.070
So let's pick our
infrared regulator
00:41:43.070 --> 00:41:45.960
to make the effective theory
as simple as possible.
00:41:45.960 --> 00:41:48.860
We have some choice in how to
pick the infrared regulator.
00:41:48.860 --> 00:41:50.480
The result for
Wilson coefficients
00:41:50.480 --> 00:41:54.120
and anomalous dimensions will
not depend on that choice.
00:41:54.120 --> 00:41:56.180
So let's use that
freedom and make
00:41:56.180 --> 00:41:58.540
things as simple as we can.
00:42:03.010 --> 00:42:05.770
And the choice
that does that here
00:42:05.770 --> 00:42:13.540
is to use dimensional
regularization for the UV,
00:42:13.540 --> 00:42:19.010
as we've been discussing,
but also for the infrared.
00:42:19.010 --> 00:42:23.350
So let's use dimensional
regularization for both.
00:42:23.350 --> 00:42:28.690
If you do that, you can convince
yourself that all heavy quark
00:42:28.690 --> 00:42:32.575
effective theory graphs with
on-shell external momentum--
00:42:32.575 --> 00:42:34.450
so I can take the external
momentum on-shell.
00:42:34.450 --> 00:42:36.280
I don't need it to
regulate divergences,
00:42:36.280 --> 00:42:40.428
because I'm going to
use div reg to do that.
00:42:40.428 --> 00:43:00.880
So all the integrals
are scaleless,
00:43:00.880 --> 00:43:06.520
and that means that they come
out to be something that,
00:43:06.520 --> 00:43:10.990
if you think about it, is either
zero or zero in a way where you
00:43:10.990 --> 00:43:13.240
have the ultraviolet divergence
cancelling an infrared
00:43:13.240 --> 00:43:16.720
divergence, which is still zero.
00:43:16.720 --> 00:43:18.640
Now you have to
think about the fact
00:43:18.640 --> 00:43:20.140
that there's both
of these going on,
00:43:20.140 --> 00:43:21.973
because you still have
to think about adding
00:43:21.973 --> 00:43:24.520
counter terms to HQET to
cancel the UV divergences.
00:43:31.890 --> 00:43:34.758
But the answers are very simple,
because you can throw away
00:43:34.758 --> 00:43:35.675
all the finite pieces.
00:43:35.675 --> 00:43:37.545
If you have 1 over
epsilon minus 1
00:43:37.545 --> 00:43:42.930
over epsilon, if you multiply by
epsilon, that term's not there.
00:43:42.930 --> 00:43:50.160
So 1 over epsilon gets
removed by counter terms,
00:43:50.160 --> 00:43:54.100
and there's no finite
pieces left over.
00:43:54.100 --> 00:44:00.095
So just use MS bar, so you
just strip off that exactly,
00:44:00.095 --> 00:44:01.720
and you're left with
1 over epsilon IR.
00:44:06.480 --> 00:44:13.250
Now if you-- so the effective
theory diagrams are just
00:44:13.250 --> 00:44:16.310
simply all 1 over epsilon IR.
00:44:16.310 --> 00:44:18.920
Now the reason that this
is making things simple
00:44:18.920 --> 00:44:21.050
is because you also
know a fact, which
00:44:21.050 --> 00:44:23.090
is that the IR divergences
in the full theory
00:44:23.090 --> 00:44:25.700
and the effective
theory have to match up.
00:44:25.700 --> 00:44:28.160
So these 1 over epsilon
IRs have to match up
00:44:28.160 --> 00:44:29.610
with your full
theory calculation.
00:44:32.430 --> 00:44:35.510
So if you renormalize the
QCD calculation, which
00:44:35.510 --> 00:44:37.817
you can't really
get around doing,
00:44:37.817 --> 00:44:39.150
you have to do that calculation.
00:44:39.150 --> 00:44:47.540
So you do that calculation
in pure dim reg, same IR
00:44:47.540 --> 00:44:52.340
regulator, which is a nice
regulator to use for QCD.
00:44:52.340 --> 00:44:54.860
You do the UV renormalization
using the standard counter
00:44:54.860 --> 00:44:58.405
terms, and what will you get?
00:44:58.405 --> 00:44:59.780
You will get
something that looks
00:44:59.780 --> 00:45:03.410
like a number over epsilon
IR, and then you'll
00:45:03.410 --> 00:45:12.890
get numbers times logs mu
over MQ plus other things.
00:45:12.890 --> 00:45:17.600
This thing here just cancels
with the-- if I subtract HQET,
00:45:17.600 --> 00:45:21.200
this is just canceling this.
00:45:21.200 --> 00:45:30.275
So this guy here cancels
when we subtract HQET.
00:45:34.190 --> 00:45:36.010
And so the matching
is then just this.
00:45:40.790 --> 00:45:43.750
So I don't actually even have to
consider calculating the heavy.
00:45:43.750 --> 00:45:46.910
If I use all these facts
that I know, if I trust them,
00:45:46.910 --> 00:45:50.630
then I don't even have to
calculate the HQET graphs.
00:45:50.630 --> 00:45:52.983
I just say, let me imagine
that I calculate them.
00:45:52.983 --> 00:45:53.900
They're all scaleless.
00:45:53.900 --> 00:45:55.277
They look like that.
00:45:55.277 --> 00:45:57.110
Let me imagine that I
renormalized them all.
00:45:57.110 --> 00:45:59.120
The 1 over epsilon UVs are gone.
00:45:59.120 --> 00:46:01.700
I'm left with 1
over epsilon IRs.
00:46:01.700 --> 00:46:02.790
I do this calculation.
00:46:02.790 --> 00:46:06.060
I say, let me imagine that
these cancel each other.
00:46:06.060 --> 00:46:08.090
And then I have the matching.
00:46:08.090 --> 00:46:09.980
So that's exploiting
all the facts
00:46:09.980 --> 00:46:12.740
that we know about effective
theories and full theories
00:46:12.740 --> 00:46:15.235
to get the matching as
quickly as possible by just
00:46:15.235 --> 00:46:16.610
doing the full
theory calculation
00:46:16.610 --> 00:46:19.130
with a particular regulator.
00:46:19.130 --> 00:46:20.300
It's not checking anything.
00:46:20.300 --> 00:46:22.925
It's not checking that the full
theory and the effective theory
00:46:22.925 --> 00:46:27.930
have the IR divergences matching
up, et cetera, et cetera.
00:46:27.930 --> 00:46:29.840
But if you know
that that's true,
00:46:29.840 --> 00:46:31.520
if you trust that
it's true, then this
00:46:31.520 --> 00:46:33.145
is the fastest way
to get the matching.
00:46:39.470 --> 00:46:40.490
Seems like magic, right?
00:46:45.580 --> 00:46:47.560
OK, so sometimes
you can exploit what
00:46:47.560 --> 00:46:50.080
you know about the
effective theory
00:46:50.080 --> 00:46:52.840
to get things more quickly.
00:46:52.840 --> 00:46:54.070
So questions about that?
00:46:58.830 --> 00:47:00.734
AUDIENCE: What is [INAUDIBLE]?
00:47:03.600 --> 00:47:05.110
PROFESSOR: Yeah.
00:47:05.110 --> 00:47:09.990
So if you think about
the loop integral,
00:47:09.990 --> 00:47:12.240
then the d here,
right, and 4 minus 2
00:47:12.240 --> 00:47:15.630
epsilon have an
epsilon greater than 0,
00:47:15.630 --> 00:47:19.050
decreasing the powers of q in
the numerator is making it more
00:47:19.050 --> 00:47:21.720
UV convergent.
00:47:21.720 --> 00:47:23.490
So to regulate the
IR, you want to have
00:47:23.490 --> 00:47:26.140
epsilon on the other side.
00:47:26.140 --> 00:47:28.500
So this is what you
need for regulating IR,
00:47:28.500 --> 00:47:30.910
and this is what you
need for regulating UV.
00:47:30.910 --> 00:47:32.910
So it may seem contradictory
that you could even
00:47:32.910 --> 00:47:34.618
do both of these things
at the same time,
00:47:34.618 --> 00:47:38.160
because greater than
zero and less than zero.
00:47:38.160 --> 00:47:41.010
But you could always think
of splitting up this integral
00:47:41.010 --> 00:47:43.770
with a hard cutoff somewhere
in between and then
00:47:43.770 --> 00:47:46.053
just using this above and
this below that cutoff.
00:47:46.053 --> 00:47:47.970
And the cutoff dependence
will cancel when you
00:47:47.970 --> 00:47:49.810
put the pieces back together.
00:47:49.810 --> 00:47:53.190
So it's actually valid
to just do calculations.
00:47:53.190 --> 00:47:56.280
And for the most part, you
can just close your eyes,
00:47:56.280 --> 00:47:58.140
and you'll get some
gamma of epsilons
00:47:58.140 --> 00:47:59.700
and some gamma of
minus epsilons,
00:47:59.700 --> 00:48:02.448
and those will be separately
regulating the divergences.
00:48:02.448 --> 00:48:04.740
And if you ever worry about
it, you can do what I said.
00:48:04.740 --> 00:48:06.900
You could put a cutoff in
and check that you're not
00:48:06.900 --> 00:48:09.780
making mistakes, but
for the most part,
00:48:09.780 --> 00:48:14.120
it just works automatically.
00:48:14.120 --> 00:48:14.740
Any other--
00:48:14.740 --> 00:48:17.710
AUDIENCE: [INAUDIBLE] all of the
epsilon UV [INAUDIBLE] epsilon
00:48:17.710 --> 00:48:18.210
IR.
00:48:18.210 --> 00:48:18.877
PROFESSOR: Yeah.
00:48:18.877 --> 00:48:21.280
AUDIENCE: [INAUDIBLE] minus
epsilon UV [INAUDIBLE] zero
00:48:21.280 --> 00:48:21.780
[INAUDIBLE].
00:48:21.780 --> 00:48:22.080
PROFESSOR: Yeah.
00:48:22.080 --> 00:48:23.033
AUDIENCE: --negative.
00:48:23.033 --> 00:48:23.700
PROFESSOR: Yeah.
00:48:23.700 --> 00:48:25.658
AUDIENCE: But formally,
you can set the epsilon
00:48:25.658 --> 00:48:26.840
UV plus an epsilon IR.
00:48:26.840 --> 00:48:29.030
PROFESSOR: That's right, yeah.
00:48:29.030 --> 00:48:32.490
So formally, this is zero.
00:48:32.490 --> 00:48:34.490
And the reason that you
have to worry about zero
00:48:34.490 --> 00:48:38.510
is because you have to add a
counter term to cancel this.
00:48:38.510 --> 00:48:40.640
Your UV counter term,
you have to still add it.
00:48:40.640 --> 00:48:42.223
And then it cancels
this, and then you
00:48:42.223 --> 00:48:43.610
get something non-zero.
00:48:43.610 --> 00:48:46.280
So the bare graph is zero.
00:48:46.280 --> 00:48:48.080
The counter term's
non-zero, and there
00:48:48.080 --> 00:48:51.310
renormalized graph is non-zero.
00:48:51.310 --> 00:48:57.890
This is a subtlety that's worth
remembering if you ever want
00:48:57.890 --> 00:48:59.170
to do calculations this way.
00:49:07.296 --> 00:49:10.900
OK, so that's some
of the complications
00:49:10.900 --> 00:49:15.535
and fascinating facts about
HQET in the perturbative sector.
00:49:19.360 --> 00:49:30.030
Let's come back and talk about
power corrections, which are--
00:49:30.030 --> 00:49:32.050
I'll go under the
title of-- well,
00:49:32.050 --> 00:49:35.150
maybe I should just call
them power corrections.
00:49:35.150 --> 00:49:35.880
Better title.
00:49:41.640 --> 00:49:44.670
So we have an effective theory.
00:49:44.670 --> 00:49:46.830
We've so far talked
about it at lowest order.
00:49:46.830 --> 00:49:48.120
We stopped at lowest order.
00:49:48.120 --> 00:49:50.512
We had this HQET
Lagrangian, and we
00:49:50.512 --> 00:49:52.470
talked about using that
Lagrangian to carry out
00:49:52.470 --> 00:49:53.827
some perturbative calculations.
00:49:53.827 --> 00:49:56.160
What if we went to higher
order in the power [INAUDIBLE]
00:49:56.160 --> 00:49:59.580
expansion, which is 1 over MQ?
00:49:59.580 --> 00:50:02.338
OK, so power corrections
here means higher order
00:50:02.338 --> 00:50:02.880
in 1 over MQ.
00:50:07.419 --> 00:50:14.500
So let me show you how you
can construct those terms.
00:50:14.500 --> 00:50:17.850
So let me go back to a
representation of the full QCD
00:50:17.850 --> 00:50:23.110
Lagrangian, which we had in
terms of this B field and the Q
00:50:23.110 --> 00:50:23.610
field.
00:50:26.340 --> 00:50:27.930
And when we first
talked about this,
00:50:27.930 --> 00:50:30.382
we just dropped all
the terms of the B,
00:50:30.382 --> 00:50:32.340
but now I'm going to do
something a little more
00:50:32.340 --> 00:50:38.100
sophisticated with them and
really just integrate them out.
00:50:41.740 --> 00:50:43.460
So we had this,
and this was really
00:50:43.460 --> 00:50:47.030
just us writing QCD in
a fancy way that was
00:50:47.030 --> 00:50:48.560
convenient for this discussion.
00:50:51.710 --> 00:50:53.960
So this is really just QCD
written in a fancy way.
00:51:00.080 --> 00:51:02.590
So if we want to take this
Lagrangian at tree level,
00:51:02.590 --> 00:51:06.040
we can just integrate out Bv.
00:51:06.040 --> 00:51:10.720
This is a Lagrangian that has
quadratic dependence on Bv.
00:51:10.720 --> 00:51:13.790
So you could think that the
path interval in this formula
00:51:13.790 --> 00:51:15.790
here would be quadratic
path interval, and those
00:51:15.790 --> 00:51:16.832
we can always just solve.
00:51:19.500 --> 00:51:23.760
And what effectively
integrating out Bv amounts
00:51:23.760 --> 00:51:27.270
to is solving for the
equations of motion of Bv
00:51:27.270 --> 00:51:28.560
and plugging that back in.
00:51:57.038 --> 00:51:58.580
So the type of
diagrams I was drawing
00:51:58.580 --> 00:52:00.038
before where I had
this wiggly line
00:52:00.038 --> 00:52:02.720
and it was a Bv propagator,
we can integrate out that
00:52:02.720 --> 00:52:05.320
by solving for the
equation of motion.
00:52:05.320 --> 00:52:08.810
So we look for
variation with respect
00:52:08.810 --> 00:52:12.890
to Bv bar and set it to 0.
00:52:12.890 --> 00:52:25.370
And that gives this
equation, and then
00:52:25.370 --> 00:52:44.075
we solve this formally for Bv
by just inverting this operator
00:52:44.075 --> 00:52:46.915
to get that equation.
00:52:46.915 --> 00:52:48.540
And then we can plug
that equation back
00:52:48.540 --> 00:53:15.120
into this equation,
which is still a QCD
00:53:15.120 --> 00:53:17.250
Lagrangian, actually, but--
00:53:17.250 --> 00:53:18.360
and then we expand.
00:53:18.360 --> 00:53:21.810
And once we expand, we match
onto the HQET Lagrangian order
00:53:21.810 --> 00:53:26.110
by order, tree level.
00:53:26.110 --> 00:53:29.670
So the first term is the
term we've been discussing.
00:53:29.670 --> 00:53:34.770
The next term, we drop the v dot
D here, because that's small.
00:53:34.770 --> 00:53:43.852
We just have 1 over 2 Q.
There's two D transverses,
00:53:43.852 --> 00:53:46.060
and they'd be higher
order terms as well,
00:53:46.060 --> 00:53:49.130
but we'll stop in that order.
00:53:49.130 --> 00:53:56.107
So this is L0, first order
term, and this would be L1.
00:53:56.107 --> 00:53:57.604
They'd be higher order terms.
00:54:01.600 --> 00:54:04.373
So what is this guy?
00:54:04.373 --> 00:54:06.040
It's useful to write
that guy, actually,
00:54:06.040 --> 00:54:07.415
in terms of two
different things,
00:54:07.415 --> 00:54:10.348
and you'll see why momentarily.
00:54:10.348 --> 00:54:12.640
So it's got two covariant
derivatives, and both of them
00:54:12.640 --> 00:54:14.680
are dotted into gamma matrices.
00:54:14.680 --> 00:54:18.550
It involves this thing, D
transverse, which, remember,
00:54:18.550 --> 00:54:22.180
is the full D minus
a projection onto
00:54:22.180 --> 00:54:26.200
v. So it's something
that's transverse to v. So
00:54:26.200 --> 00:54:28.780
what I want to do to
simplify this guy here
00:54:28.780 --> 00:54:31.040
is I want to do the following.
00:54:31.040 --> 00:54:32.240
I'm going to use the fact--
00:54:32.240 --> 00:54:34.407
I'm going to write it in
terms of the field strength
00:54:34.407 --> 00:54:38.230
by using the fact that
the commutator of two Ds
00:54:38.230 --> 00:54:45.440
gives me a G. And the commutator
of two sigmas, I'll write--
00:54:45.440 --> 00:54:45.940
sorry.
00:54:45.940 --> 00:54:47.800
The commutator of
two gammas gives me
00:54:47.800 --> 00:54:48.925
something I can call sigma.
00:54:56.000 --> 00:54:58.600
So let me write this as the
symmetric piece and then
00:54:58.600 --> 00:54:59.800
the anti-symmetric piece.
00:55:03.660 --> 00:55:07.530
And I do a symmetrization
in both the fields
00:55:07.530 --> 00:55:10.167
and the Dirac structures.
00:55:14.937 --> 00:55:23.700
Doing DT slash DT
slash anti-commutator.
00:55:38.170 --> 00:55:40.020
So then since it's
anti-symmetric,
00:55:40.020 --> 00:55:42.990
it automatically forces
that anti-symmetric.
00:55:42.990 --> 00:55:48.970
This guy is a GB nu, so for this
piece, we just get DT squared.
00:55:48.970 --> 00:55:55.350
And this piece, once
we track all the 2s,
00:55:55.350 --> 00:56:02.180
the i's give sigma dot G.
00:56:02.180 --> 00:56:06.650
And so the usual way of
writing L1 is as follows.
00:56:06.650 --> 00:56:12.250
You say L1, using this formula,
plugging it in, has two terms.
00:56:15.200 --> 00:56:18.367
And you'll see why
when I write them down
00:56:18.367 --> 00:56:19.450
that we wanted to do this.
00:56:32.836 --> 00:56:36.560
OK, so that's L1 is
after plugging that in.
00:56:36.560 --> 00:56:39.590
Now the reason to do this is
that if we ask about symmetry
00:56:39.590 --> 00:56:41.120
breaking, that's
something that can
00:56:41.120 --> 00:56:42.940
happen from sub-leading terms.
00:56:42.940 --> 00:56:46.070
Lowest order, we had a
symmetry, heavy quark symmetry,
00:56:46.070 --> 00:56:48.730
and that is broken by
these interactions.
00:56:48.730 --> 00:56:50.480
But it's actually
broken in different ways
00:56:50.480 --> 00:56:52.400
by these two terms.
00:56:52.400 --> 00:56:54.620
This term here doesn't
have any spin structure,
00:56:54.620 --> 00:56:56.690
so it doesn't break
the spin symmetry.
00:56:56.690 --> 00:56:59.030
It does have flavor
structure because of the MQ,
00:56:59.030 --> 00:57:02.370
so it breaks the
flavor symmetry.
00:57:02.370 --> 00:57:08.690
So this is a kinetic
energy type correction,
00:57:08.690 --> 00:57:12.050
and it breaks flavor symmetry
because of the dependence on Q
00:57:12.050 --> 00:57:12.950
in the MQ.
00:57:18.980 --> 00:57:21.050
And this guy breaks
both, because it
00:57:21.050 --> 00:57:24.180
has a spin structure now, and
it's a magnetic moment type
00:57:24.180 --> 00:57:24.680
term.
00:57:39.680 --> 00:57:49.186
So it's got the sigma dot
B field type interaction.
00:57:49.186 --> 00:57:51.760
This is what I mean by the
magnetic moment type term.
00:57:54.260 --> 00:57:54.760
OK?
00:57:54.760 --> 00:57:57.218
So that's what the sub-leading
power corrections look like,
00:57:57.218 --> 00:57:59.560
and if we wanted to use the
effective theory to talk
00:57:59.560 --> 00:58:02.440
about power corrections,
we could do that.
00:58:02.440 --> 00:58:05.050
We're constructing them here
by knowing the full theory,
00:58:05.050 --> 00:58:09.870
just integrating out
explicitly the fields, OK,
00:58:09.870 --> 00:58:11.810
which is a very nice
thing if you can do it.
00:58:14.770 --> 00:58:16.420
Now you could do
it the other way,
00:58:16.420 --> 00:58:19.750
which would be to think
about just writing them
00:58:19.750 --> 00:58:22.420
from the bottom up.
00:58:22.420 --> 00:58:23.920
And there is one
way in which that's
00:58:23.920 --> 00:58:26.710
more general than what
we've talked about,
00:58:26.710 --> 00:58:29.080
and that's because what we
talked about was tree level.
00:58:34.480 --> 00:58:37.150
And if you wanted to
include loop corrections,
00:58:37.150 --> 00:58:39.580
how do we know that there's
not some other operator here
00:58:39.580 --> 00:58:42.310
that we missed because it
just vanished at tree level,
00:58:42.310 --> 00:58:44.230
for example?
00:58:44.230 --> 00:58:46.042
We've seen examples
where that happens.
00:58:46.042 --> 00:58:48.250
There's an operator that
only shows up at loop level.
00:58:51.193 --> 00:58:53.110
So we could think about
it from the bottom up,
00:58:53.110 --> 00:58:55.600
even though this is a top-down
effective theory, in order
00:58:55.600 --> 00:58:57.730
to make sure we're
not missing anything.
00:58:57.730 --> 00:58:59.320
And if we wanted to
do that, we should
00:58:59.320 --> 00:59:02.650
enumerate all the possible
things, the symmetries and all
00:59:02.650 --> 00:59:06.470
that we can use to constrain
the form of the operators.
00:59:06.470 --> 00:59:09.042
So let's enumerate.
00:59:09.042 --> 00:59:10.750
So there's the power
counting, of course.
00:59:14.735 --> 00:59:15.610
That's pretty simple.
00:59:15.610 --> 00:59:19.280
Here all the powers of 1 over
MQ are being made explicit,
00:59:19.280 --> 00:59:21.280
and they just tell us
what dimension of operator
00:59:21.280 --> 00:59:30.200
to look for, just as
in our integrating
00:59:30.200 --> 00:59:33.488
out heavy particles.
00:59:33.488 --> 00:59:35.280
So we just know the
dimension of the fields
00:59:35.280 --> 00:59:36.780
that we have to put
in the numerator
00:59:36.780 --> 00:59:38.580
from how many MQs
we're talking about.
00:59:45.180 --> 00:59:46.770
There's gait
symmetry, of course.
00:59:46.770 --> 00:59:48.330
So use covariant derivatives.
00:59:51.930 --> 00:59:53.490
Very easy to take into account.
00:59:59.620 --> 01:00:02.710
There's discreet symmetries,
charge conjugation,
01:00:02.710 --> 01:00:05.240
parity at time reversal,
which are symmetries of QCD
01:00:05.240 --> 01:00:07.270
if we drop the theta term.
01:00:07.270 --> 01:00:12.628
And we can impose them as
well, and again, that's easy.
01:00:15.227 --> 01:00:17.810
I wouldn't be making a list if
there wasn't at least one thing
01:00:17.810 --> 01:00:25.280
that was hard and non-trivial.
01:00:25.280 --> 01:00:26.860
But discreet
symmetries are easy.
01:00:33.570 --> 01:00:36.540
The thing that actually is the
hardest is Lorentz symmetry.
01:00:41.000 --> 01:00:44.590
Oh, you say, just dot Lorentz
indices into Lorentz indices.
01:00:47.303 --> 01:00:49.970
But you have to ask the question
whether we even have Lorentzian
01:00:49.970 --> 01:00:53.060
variance in this theory.
01:00:53.060 --> 01:00:55.505
And it turns out that
part of the Lorentz group
01:00:55.505 --> 01:00:58.430
was actually broken by
having this heavy quark
01:00:58.430 --> 01:01:03.518
and doing this type of expansion
that we've been talking about.
01:01:03.518 --> 01:01:05.810
So if you think about the
six generators of the Lorentz
01:01:05.810 --> 01:01:08.690
group, the boosts
and the rotations,
01:01:08.690 --> 01:01:11.600
there's a part that I could
call the transverse part, which
01:01:11.600 --> 01:01:13.890
is transverse to the velocity.
01:01:13.890 --> 01:01:18.620
So in the rest frame, that
would be M12, M23, and M13.
01:01:21.490 --> 01:01:24.820
And those are the rotations.
01:01:24.820 --> 01:01:28.600
So this i-- v is like this.
01:01:28.600 --> 01:01:30.340
So no matter what
v you pick, there's
01:01:30.340 --> 01:01:34.570
always three generators
that are rotations.
01:01:34.570 --> 01:01:37.098
And then there's the boosts.
01:01:37.098 --> 01:01:38.515
And you should
think of the boosts
01:01:38.515 --> 01:01:43.090
as taking v mu and
then M dotted into v
01:01:43.090 --> 01:01:45.070
and then making the
other guy transverse,
01:01:45.070 --> 01:01:47.260
so when we denote it like that.
01:01:47.260 --> 01:01:50.020
So the new index is transverse,
and in the rest from,
01:01:50.020 --> 01:01:54.008
that's M01, M02, and M03.
01:01:59.490 --> 01:02:07.320
And so introducing v mu actually
breaks the boosts' symmetry.
01:02:26.970 --> 01:02:29.370
And if you like, you
could think the reason
01:02:29.370 --> 01:02:31.290
it breaks the symmetry
is because it gives
01:02:31.290 --> 01:02:34.340
a preferred frame, which is the
rest frame of the heavy quark.
01:02:34.340 --> 01:02:36.090
If you have a preferred
frame, then you've
01:02:36.090 --> 01:02:37.250
broken Lorentzian frames.
01:02:41.300 --> 01:02:43.945
So that's bad.
01:02:43.945 --> 01:02:45.320
And it turns out
there's actually
01:02:45.320 --> 01:02:47.695
a hidden symmetry of this
effective theory that partially
01:02:47.695 --> 01:02:48.695
restores this breaking.
01:02:51.830 --> 01:02:53.930
And it restores it
in exactly the amount
01:02:53.930 --> 01:02:55.120
that it needs to restore it.
01:03:04.060 --> 01:03:07.170
That is, it restores
it at low energies.
01:03:10.410 --> 01:03:16.170
And that's called
reparameterization invariance,
01:03:16.170 --> 01:03:18.420
which I will write once,
and then forever more,
01:03:18.420 --> 01:03:23.790
we talk about it as RPI,
Reparameterization Invariance.
01:03:23.790 --> 01:03:27.465
And it's an additional symmetry
that we have on v mu itself.
01:03:38.082 --> 01:03:40.165
So let's go back and think
about how we introduced
01:03:40.165 --> 01:03:41.890
v mu in the first place.
01:03:41.890 --> 01:03:44.140
So we're saying that v mu
breaks part of the cemetery,
01:03:44.140 --> 01:03:48.183
but how did we decide
on what v mu was?
01:03:48.183 --> 01:03:49.600
How much freedom
was there when we
01:03:49.600 --> 01:03:51.422
defined v mu at the beginning?
01:03:58.815 --> 01:04:00.440
If we're saying that
it breaks, then we
01:04:00.440 --> 01:04:03.200
should know how
much freedom there
01:04:03.200 --> 01:04:05.990
was, because a freedom
to define different vs
01:04:05.990 --> 01:04:09.670
could restore symmetry, just
realized in a different way.
01:04:09.670 --> 01:04:10.670
And that's what happens.
01:04:14.870 --> 01:04:16.860
So where did it come from?
01:04:16.860 --> 01:04:19.540
We had P, have a heavy
quark, and we split that
01:04:19.540 --> 01:04:25.330
into two pieces, MQv plus k.
01:04:25.330 --> 01:04:29.800
But this split into two pieces
is arbitrary by some amount.
01:04:29.800 --> 01:04:33.160
We could move pieces back and
forth between here and here,
01:04:33.160 --> 01:04:35.170
and we would still
have the same theory.
01:04:41.017 --> 01:04:43.600
We have to be careful that we're
moving back pieces that don't
01:04:43.600 --> 01:04:45.610
violate the power counting.
01:04:45.610 --> 01:04:48.850
And that's what I mean by
somewhat arbitrary here,
01:04:48.850 --> 01:04:50.530
not completely arbitrary.
01:04:50.530 --> 01:04:52.360
There was a point to
doing this, because we
01:04:52.360 --> 01:04:54.820
wanted to separate out the
big piece and the small piece.
01:04:54.820 --> 01:04:57.070
But we could always move a
small piece back over here,
01:04:57.070 --> 01:05:01.450
and that wouldn't change
this decomposition.
01:05:01.450 --> 01:05:06.680
So the invariance that
you have is the following.
01:05:06.680 --> 01:05:10.390
You can take v mu
and send it to v mu
01:05:10.390 --> 01:05:14.680
plus some epsilon mu over MQ.
01:05:14.680 --> 01:05:22.330
And k mu comes to k
mu minus epsilon mu.
01:05:22.330 --> 01:05:24.500
That moves the piece back
and forth between them,
01:05:24.500 --> 01:05:30.150
and as long as I think that
epsilon mu is some parameter
01:05:30.150 --> 01:05:32.220
that doesn't have a power
counting in it, i.e.
01:05:32.220 --> 01:05:35.490
it's order doesn't
have any MQs in it--
01:05:35.490 --> 01:05:39.810
it's just something of
order lambda QCD, say--
01:05:42.900 --> 01:05:45.720
then that makes the power
counting still true.
01:05:45.720 --> 01:05:48.150
That was the point of
this decomposition.
01:05:48.150 --> 01:05:50.400
And it allows us to move a
small piece back and forth.
01:05:50.400 --> 01:05:52.950
The small piece is this epsilon.
01:05:52.950 --> 01:05:54.090
So that's a symmetry.
01:05:54.090 --> 01:05:56.465
That's called
reparameterization invariance.
01:05:56.465 --> 01:05:57.840
So we have to make
sure that when
01:05:57.840 --> 01:05:59.730
we construct our
effective theory that it
01:05:59.730 --> 01:06:01.140
satisfies the
symmetry if we want
01:06:01.140 --> 01:06:03.150
it to be a boost invariant--
01:06:03.150 --> 01:06:07.360
if we want to restore boost
invariance to the theory.
01:06:07.360 --> 01:06:10.140
So this parameter epsilon
you can think of as--
01:06:10.140 --> 01:06:13.110
you could consider it to be
a finite reparameterization
01:06:13.110 --> 01:06:14.580
symmetry, but you
don't really have
01:06:14.580 --> 01:06:16.163
to worry about finite
transformations.
01:06:16.163 --> 01:06:18.090
You can just do
the infinitesimal.
01:06:18.090 --> 01:06:21.237
So we'll think about epsilon
mu as an infinitesimal.
01:06:29.840 --> 01:06:34.160
And it has that counting
that I put over there.
01:06:34.160 --> 01:06:37.490
Now v squared was equal to
1, and that's also something
01:06:37.490 --> 01:06:39.120
we don't want to spoil.
01:06:39.120 --> 01:06:41.070
But that's easy.
01:06:41.070 --> 01:06:44.790
We just say that epsilon
dot v is equal to 0.
01:06:44.790 --> 01:06:46.712
That maintains this condition.
01:06:46.712 --> 01:06:48.920
So that means that there's
three different components
01:06:48.920 --> 01:06:57.410
of epsilon, non-trivial
components to epsilon.
01:06:57.410 --> 01:06:59.120
And those three
components of epsilon
01:06:59.120 --> 01:07:02.144
are exactly related to
the three boosts here.
01:07:06.884 --> 01:07:08.350
OK, we have a three family--
01:07:08.350 --> 01:07:16.632
three-parameter family
of transformations,
01:07:16.632 --> 01:07:18.840
which are the three components
of the epsilon, which,
01:07:18.840 --> 01:07:21.382
in the rest frame, would just
be the one, the two, and three.
01:07:23.460 --> 01:07:25.260
What did we do-- what
about the fields?
01:07:25.260 --> 01:07:27.810
How does the field, the
Qv change under this type
01:07:27.810 --> 01:07:29.114
of transformation?
01:07:33.760 --> 01:07:36.640
Let me take the field
that x equals 0 for now.
01:07:40.850 --> 01:07:44.530
So v slash on Qv,
it was equal to 0.
01:07:44.530 --> 01:07:48.140
And if I do the transformation,
then this v slash changes.
01:07:48.140 --> 01:07:53.020
It becomes v slash plus
epsilon slash over MQ.
01:07:53.020 --> 01:07:54.325
Let me imagine Qv changes.
01:07:57.820 --> 01:08:00.785
It goes to Qv plus
delta Qv, and I
01:08:00.785 --> 01:08:02.035
have to do that on both sides.
01:08:06.740 --> 01:08:08.530
Then I can take this--
so this thing here
01:08:08.530 --> 01:08:11.380
is some order epsilon change.
01:08:11.380 --> 01:08:14.770
Then I can take this equation,
and I can just solve.
01:08:14.770 --> 01:08:16.359
So the piece that's
order epsilon
01:08:16.359 --> 01:08:18.670
to the 0, just satisfied.
01:08:18.670 --> 01:08:20.920
Solve for the piece
that's order epsilon,
01:08:20.920 --> 01:08:24.750
and that gives me an
equation for delta Qv.
01:08:24.750 --> 01:08:28.750
So rearranging this
equation, I find
01:08:28.750 --> 01:08:39.250
that 1 minus v slash delta Qv is
epsilon slash over MQ times Qv.
01:08:39.250 --> 01:08:46.729
And this equation
has the solution,
01:08:46.729 --> 01:08:53.410
but delta QV is
epsilon slash over 2MQ.
01:08:53.410 --> 01:08:57.470
Remember that epsilon
is transverse to v,
01:08:57.470 --> 01:09:04.040
so if I push the v slash-- so
if I plug that solution in here
01:09:04.040 --> 01:09:07.010
and I push a v slash
through the epsilon,
01:09:07.010 --> 01:09:10.279
well, then I can push the v
slash through the epsilon,
01:09:10.279 --> 01:09:11.359
let it hit the QV.
01:09:11.359 --> 01:09:14.180
That's giving a factor of 2,
because it's anti-commutes.
01:09:14.180 --> 01:09:16.279
That's this 2 here.
01:09:16.279 --> 01:09:20.130
And then I would get what I
wrote on the right hand side.
01:09:20.130 --> 01:09:23.420
So if it's not obvious,
check for yourself
01:09:23.420 --> 01:09:26.550
that that's a solution.
01:09:26.550 --> 01:09:28.729
So that's how you derive
the change to the field
01:09:28.729 --> 01:09:33.649
under this reparameterization.
01:09:33.649 --> 01:09:35.240
And so when we talk
about operators
01:09:35.240 --> 01:09:37.520
and the effective
theory, we have
01:09:37.520 --> 01:09:39.700
to worry, how does the
symmetry act on them?
01:09:47.279 --> 01:09:50.362
And it's a kind of a
non-trivial symmetry.
01:09:50.362 --> 01:09:52.279
Was not apparent to us
when we started, right.
01:09:52.279 --> 01:09:55.790
[CHUCKLES]
01:09:55.790 --> 01:09:58.400
OK.
01:09:58.400 --> 01:10:13.060
So the full
reparameterization is v mu
01:10:13.060 --> 01:10:18.250
goes to v plus epsilon over MQ.
01:10:18.250 --> 01:10:26.662
And then if I take Qv of x,
then that is what I said.
01:10:26.662 --> 01:10:31.395
There's this epsilon
slash over 2MQ piece.
01:10:31.395 --> 01:10:32.770
This is the
transformation, so it
01:10:32.770 --> 01:10:35.290
goes to itself plus
this extra piece.
01:10:35.290 --> 01:10:38.980
And the fact that I take it at x
adds one little slight wrinkle,
01:10:38.980 --> 01:10:41.260
and it just gives this
extra phase factor.
01:10:41.260 --> 01:10:43.360
And that extra phase
factor is exactly what
01:10:43.360 --> 01:10:45.820
encodes the change of k.
01:10:45.820 --> 01:10:49.390
So this, if you like,
encodes that derivatives.
01:10:49.390 --> 01:10:53.260
Should go to derivatives minus
epsilon or in momentum space,
01:10:53.260 --> 01:10:57.420
that k should go
to k minus epsilon.
01:10:57.420 --> 01:10:59.273
OK, so previously
we had a rule for k,
01:10:59.273 --> 01:11:00.940
but now I've encoded
that in this phase.
01:11:05.220 --> 01:11:05.720
OK?
01:11:05.720 --> 01:11:10.220
So that's the symmetry
that we should look into.
01:11:15.740 --> 01:11:18.540
So what does it do?
01:11:18.540 --> 01:11:21.460
So what this does is it restores
invariance under boosts,
01:11:21.460 --> 01:11:22.445
but only small boosts.
01:11:34.897 --> 01:11:36.605
The reason that I call
them a small boost
01:11:36.605 --> 01:11:39.453
is because epsilon here had
to be of order lambda QCD.
01:11:39.453 --> 01:11:40.745
It couldn't have been order MQ.
01:11:47.440 --> 01:11:49.410
That's what I mean by small.
01:11:49.410 --> 01:11:51.160
And from the point of
view of this theory,
01:11:51.160 --> 01:11:52.243
this is all we care about.
01:12:00.460 --> 01:12:02.440
Because we want to
remain within the region
01:12:02.440 --> 01:12:04.260
where the effective
theory was valid,
01:12:04.260 --> 01:12:06.340
the whole setup of
the effective theory
01:12:06.340 --> 01:12:08.800
involved dividing out a large
piece and a small piece.
01:12:08.800 --> 01:12:12.490
If we allow back large
pieces, then the game is over,
01:12:12.490 --> 01:12:15.070
and you wouldn't be
formulating correctly
01:12:15.070 --> 01:12:19.660
the effective theory, because
you'd spoil the power counting.
01:12:19.660 --> 01:12:21.940
OK, so this is this
hidden symmetry
01:12:21.940 --> 01:12:23.690
we're calling
reparameterization variance.
01:12:23.690 --> 01:12:25.270
And it's not special to HQET.
01:12:25.270 --> 01:12:28.060
Any time you have fields that
are labeled by something,
01:12:28.060 --> 01:12:30.560
you should think about whether
there's a symmetry like this.
01:12:32.650 --> 01:12:35.230
OK, so that's the entire
list of the symmetries
01:12:35.230 --> 01:12:37.360
that you should
consider in order
01:12:37.360 --> 01:12:40.630
to think about doing a
bottom-up approach to HQET.
01:12:40.630 --> 01:12:42.670
Simple ones, and then
there's this one that's
01:12:42.670 --> 01:12:44.069
a little more complicated.
01:12:48.290 --> 01:12:51.670
So let's go back and
now consider the 1
01:12:51.670 --> 01:12:58.895
over MQ operators in general.
01:13:03.900 --> 01:13:07.380
And it turns out that there's
not any missing operators,
01:13:07.380 --> 01:13:11.220
that the two operators
we have are actually
01:13:11.220 --> 01:13:14.760
the complete set that
you can write down
01:13:14.760 --> 01:13:20.345
at this dimension using all
the properties of the field.
01:13:20.345 --> 01:13:22.470
So we didn't miss anything
from that point of view.
01:13:26.630 --> 01:13:29.800
So let me write them down
again, and let me write them
01:13:29.800 --> 01:13:33.010
down in a way where I imagine
that radiative corrections have
01:13:33.010 --> 01:13:34.387
come in as well.
01:13:34.387 --> 01:13:36.220
And I'll give them some
Wilson coefficients,
01:13:36.220 --> 01:13:38.620
which are generically
called Ck and Cf.
01:13:38.620 --> 01:13:41.210
That's the standard notation.
01:13:41.210 --> 01:13:42.936
So this is a Wilson coefficient.
01:13:46.810 --> 01:13:48.280
That's a Wilson coefficient.
01:13:48.280 --> 01:13:49.330
This is not 4/3.
01:13:49.330 --> 01:13:52.710
it's Wilson coefficient.
01:13:52.710 --> 01:13:55.490
The name is the same
as the Cf that is 4/3,
01:13:55.490 --> 01:13:58.240
but this is a little
cf, not a big CF.
01:14:09.820 --> 01:14:11.230
So if we want to--
01:14:11.230 --> 01:14:12.540
so it's gauge invariant.
01:14:12.540 --> 01:14:15.250
It has the right parity,
et cetera, et cetera.
01:14:15.250 --> 01:14:17.746
We should worry about the
reparameterization invariance.
01:14:23.580 --> 01:14:25.320
So let's do that.
01:14:25.320 --> 01:14:33.810
So at lowest order, the
phase is what changes.
01:14:33.810 --> 01:14:36.090
And the leading order
Lagrangian is invariant,
01:14:36.090 --> 01:14:37.630
because v dot epsilon is 0.
01:14:44.550 --> 01:14:49.920
So at order MQ to the 0,
since v dot epsilon is 0,
01:14:49.920 --> 01:14:51.480
you don't get a
leading order change.
01:14:51.480 --> 01:14:54.240
So our Lagrangian was variant.
01:14:54.240 --> 01:14:56.490
This invariance, this
reparameterization invariance
01:14:56.490 --> 01:14:58.830
mixes orders.
01:14:58.830 --> 01:15:03.370
It connects orders
in the expansion.
01:15:03.370 --> 01:15:06.220
There was a term that was
order 1, which is this piece,
01:15:06.220 --> 01:15:08.020
and there's a term
that's order 1 over MQ.
01:15:08.020 --> 01:15:09.520
So the symmetry
is actually making
01:15:09.520 --> 01:15:11.590
a connection between leading
order and sub-leading order
01:15:11.590 --> 01:15:12.250
operators.
01:15:22.560 --> 01:15:24.770
So we could ask
about this delta L 0,
01:15:24.770 --> 01:15:27.980
and there will be a
piece at order 1 over MQ.
01:15:27.980 --> 01:15:30.350
So let's just write
out all these things.
01:15:42.170 --> 01:15:43.810
Transforming everything.
01:15:49.310 --> 01:15:53.210
This is our leading
order Lagrangian.
01:15:53.210 --> 01:15:57.620
After imposing the field
change as well as v change,
01:15:57.620 --> 01:16:00.530
the v dot D becomes this,
and the field becomes that.
01:16:04.140 --> 01:16:08.990
So there's three things
here that are being changed.
01:16:08.990 --> 01:16:11.010
Expand this out.
01:16:11.010 --> 01:16:14.780
Use things like 1
plus v slash over 2,
01:16:14.780 --> 01:16:19.700
epsilon slash 1
plus v slash over 2.
01:16:19.700 --> 01:16:22.325
C equal to epsilon
dot v is equal to 0.
01:16:25.130 --> 01:16:29.810
Simplify, do some Dirac
algebra, and you can boil this
01:16:29.810 --> 01:16:32.390
down to something
simple, which is
01:16:32.390 --> 01:16:35.870
that the entire change
is just an epsilon dot
01:16:35.870 --> 01:16:41.540
D over MQ times QV.
01:16:41.540 --> 01:16:44.420
And if you look at this, it
has to cancel against something
01:16:44.420 --> 01:16:46.856
that's order 1 over MQ.
01:16:46.856 --> 01:16:50.270
And if you look at the terms
that we had at order 1,
01:16:50.270 --> 01:16:53.120
which are 1 over MQ,
there was a kinetic piece.
01:16:53.120 --> 01:16:56.670
We called it kinetic
energy piece.
01:16:56.670 --> 01:16:58.430
And if we do the
change there, there
01:16:58.430 --> 01:17:01.880
is a contribution from
the phase in this case,
01:17:01.880 --> 01:17:06.230
because we had
transverse derivatives.
01:17:06.230 --> 01:17:07.963
So we can add epsilon
dot D transverse.
01:17:07.963 --> 01:17:08.630
That's non-zero.
01:17:15.005 --> 01:17:16.630
And if you go through
the leading order
01:17:16.630 --> 01:17:18.940
change to this guy, as
well as the guy that's
01:17:18.940 --> 01:17:24.580
the magnetic guy, you
find that the magnetic guy
01:17:24.580 --> 01:17:28.756
is 0 at this order.
01:17:28.756 --> 01:17:32.320
It's non-zero at higher orders,
but at this order, it's zero.
01:17:32.320 --> 01:17:35.170
And the kinetic guy does
have a transformation.
01:17:35.170 --> 01:17:39.040
It has exactly the same
form as this guy here.
01:17:39.040 --> 01:17:43.340
And if epsilon's dotted into
the D, then it's a D transverse.
01:17:43.340 --> 01:17:45.080
But this guy has a
Wilson coefficient.
01:17:45.080 --> 01:17:46.715
This guy doesn't.
01:17:46.715 --> 01:17:48.590
So in order for these
to cancel, you actually
01:17:48.590 --> 01:17:51.060
learn something non-trivial.
01:17:51.060 --> 01:17:53.060
The symmetry teaches you
something non-trivial
01:17:53.060 --> 01:17:54.950
about the sub-leading
Lagrangian.
01:17:54.950 --> 01:17:56.330
That Wilson
coefficient has to be
01:17:56.330 --> 01:18:02.040
1 to [INAUDIBLE]
perturbation theory in order
01:18:02.040 --> 01:18:03.540
for the symmetry
not to be violated.
01:18:09.700 --> 01:18:12.300
AUDIENCE: But you are
enforcing the symmetry?
01:18:12.300 --> 01:18:13.350
PROFESSOR: Yeah.
01:18:13.350 --> 01:18:15.150
So the symmetry is
boost invariance,
01:18:15.150 --> 01:18:17.850
and it seems like a
reasonable symmetry to impose.
01:18:17.850 --> 01:18:18.350
Yeah.
01:18:25.550 --> 01:18:26.850
It'd be small boosts.
01:18:43.470 --> 01:18:46.170
So as long as your
scheme and your regulator
01:18:46.170 --> 01:18:49.866
don't break the symmetry,
which is always something
01:18:49.866 --> 01:18:54.900
that you have to worry
about in general,
01:18:54.900 --> 01:18:58.640
then this guy is
1 to all orders.
01:18:58.640 --> 01:18:59.140
OK?
01:18:59.140 --> 01:19:00.960
So if you did something like
dimensional regularization
01:19:00.960 --> 01:19:02.490
and you thought you
should calculate this guy,
01:19:02.490 --> 01:19:04.050
you'd just find
that it would be 1.
01:19:04.050 --> 01:19:06.290
And you'd wonder,
well, why is that?
01:19:06.290 --> 01:19:09.694
It's the symmetry
that tells you it's 1.
01:19:09.694 --> 01:19:12.270
OK, so you don't have
to figure out that guy.
01:19:12.270 --> 01:19:14.760
The other guy, which
we've called Cf,
01:19:14.760 --> 01:19:18.000
you do have to figure out,
because it wasn't constrained.
01:19:18.000 --> 01:19:27.060
And at lowest order,
the other coefficient
01:19:27.060 --> 01:19:29.920
is not constrained in this way.
01:19:29.920 --> 01:19:33.030
And so it does get an
anomalous dimension.
01:19:33.030 --> 01:19:36.420
And so we could calculate it.
01:19:36.420 --> 01:19:38.280
It's a good homework problem.
01:19:38.280 --> 01:19:40.140
You may see it on a
future homework set.
01:19:49.860 --> 01:19:51.360
And there is an
anomalous dimension,
01:19:51.360 --> 01:19:52.920
and when you solve that
anomalous dimension,
01:19:52.920 --> 01:19:54.462
you're again getting
something that's
01:19:54.462 --> 01:19:56.190
the ratio of alphas
to some power.
01:19:56.190 --> 01:19:59.475
In this case, it's a non Abelian
power, so the adjoint Casimir.
01:20:06.522 --> 01:20:10.140
OK, so that guy does have
an anomalous dimension.
01:20:10.140 --> 01:20:12.080
I think we'll stop there today.
01:20:12.080 --> 01:20:15.260
And we'll talk more
about power corrections
01:20:15.260 --> 01:20:17.960
and the phenomenology
of them, how
01:20:17.960 --> 01:20:20.990
we can make non-trivial
predictions of from them
01:20:20.990 --> 01:20:22.840
next time.