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IAN STEWART: So last time, we
were midway through this proof
00:00:24.880 --> 00:00:28.930
that the equations of motion,
i.e., field redefinitions,
00:00:28.930 --> 00:00:32.470
can be used to
simplify the theory.
00:00:32.470 --> 00:00:35.320
And I stated last
time that really,
00:00:35.320 --> 00:00:38.530
all we have to worry about is
the change to the Lagrangian.
00:00:38.530 --> 00:00:40.270
But when we looked
at the path integral
00:00:40.270 --> 00:00:42.640
to do things properly and make
a change variable in the path
00:00:42.640 --> 00:00:44.550
integral, it wasn't just
the Lagrangian that changed.
00:00:44.550 --> 00:00:47.050
The Lagrangian changed, and
we could set up our field
00:00:47.050 --> 00:00:50.200
redefinition to do what we want,
but there were also changes
00:00:50.200 --> 00:00:51.970
to the Jacobian and
the source, and that's
00:00:51.970 --> 00:00:56.120
what we had started to
talk about last time.
00:00:56.120 --> 00:01:00.460
So when I look at the change the
Jacobian, which is this thing,
00:01:00.460 --> 00:01:03.750
we could write that
as a ghost Lagrangian.
00:01:03.750 --> 00:01:07.030
And the argument for why we
don't need to worry about this
00:01:07.030 --> 00:01:07.795
is as follows.
00:01:13.880 --> 00:01:15.940
So the effective
field theory is going
00:01:15.940 --> 00:01:19.180
to be valid for small momentum.
00:01:19.180 --> 00:01:24.280
There's an expansion and
there's some scale introduced
00:01:24.280 --> 00:01:27.310
by the higher dimensional
operators, lamda new,
00:01:27.310 --> 00:01:31.120
and we're looking at low
momentum relative to that.
00:01:31.120 --> 00:01:34.150
In the particular case
that we're dealing with,
00:01:34.150 --> 00:01:37.100
this parameter eta
had dimensions,
00:01:37.100 --> 00:01:40.570
and so lambda nu
was 1 over root eta.
00:01:40.570 --> 00:01:43.312
And we put factors of eta in
front of our higher dimension
00:01:43.312 --> 00:01:44.770
operators, so that's
like putting 1
00:01:44.770 --> 00:01:47.230
over lambda nus in front
of those operators.
00:01:49.695 --> 00:01:51.070
And the reason
that we don't have
00:01:51.070 --> 00:01:52.960
to worry so much about
this ghost Lagrangian
00:01:52.960 --> 00:01:55.240
is that these ghosts
are going to get mass
00:01:55.240 --> 00:01:57.220
that's of the size lambda nu.
00:02:20.200 --> 00:02:23.577
So what that means is that
the ghost, along with perhaps
00:02:23.577 --> 00:02:25.660
other particles that are
up at the scale lamda new
00:02:25.660 --> 00:02:27.860
are things that we don't
have to worry about,
00:02:27.860 --> 00:02:30.010
and we can effectively
integrate out
00:02:30.010 --> 00:02:32.440
the ghost from the
theory in the same way
00:02:32.440 --> 00:02:34.720
that we would think about
removing some particles that
00:02:34.720 --> 00:02:37.120
had masses of order lambda nu.
00:02:37.120 --> 00:02:39.100
So I'll show you
why they get masses
00:02:39.100 --> 00:02:40.510
of order lambda nu in a second.
00:03:09.340 --> 00:03:12.310
So let's do that by way of
picking a particular example.
00:03:12.310 --> 00:03:14.880
So far, we've kept things fairly
general without specifying
00:03:14.880 --> 00:03:18.060
what this set of fields
it's in this thing
00:03:18.060 --> 00:03:19.980
that we called T was.
00:03:19.980 --> 00:03:22.740
We just left it as
an arbitrary thing.
00:03:22.740 --> 00:03:36.640
Let's pick a particular T.
The argument here is actually
00:03:36.640 --> 00:03:39.340
fairly general, but I think
it'd be easier to see it
00:03:39.340 --> 00:03:41.390
for particular example.
00:03:41.390 --> 00:03:47.240
So if we pick this
to be T, we have
00:03:47.240 --> 00:03:51.146
this term which has no relation
to T. That term is important.
00:03:54.428 --> 00:03:55.970
And then we have
the terms from this.
00:04:04.512 --> 00:04:05.970
And so we'd have
a ghost Lagrangian
00:04:05.970 --> 00:04:08.830
that would look like that.
00:04:08.830 --> 00:04:11.220
So this here is going
to be the mass term,
00:04:11.220 --> 00:04:14.520
and so far, it doesn't look like
it has the right dimensions.
00:04:14.520 --> 00:04:18.930
And that's because we've got
our kinetic term, if you like,
00:04:18.930 --> 00:04:21.310
with the wrong dimensions.
00:04:21.310 --> 00:04:26.550
So if we want ghost fields
of standard dimensions,
00:04:26.550 --> 00:04:29.950
or canonically normalized
kinetic term in this case,
00:04:29.950 --> 00:04:34.230
we would take c and rescale
it to c over root eta.
00:04:34.230 --> 00:04:38.850
And if we do that, then this
does become a lambda nu.
00:04:49.080 --> 00:04:51.480
So that removes the eta in
the interaction term here.
00:04:51.480 --> 00:04:54.340
It removes it from this kinetic
term that happened to be there,
00:04:54.340 --> 00:04:56.970
and then the only place that
the eta's showing up now
00:04:56.970 --> 00:04:59.520
in this case is in this term.
00:04:59.520 --> 00:05:02.550
But now you see explicitly in
terms of these ghost fields
00:05:02.550 --> 00:05:04.688
that they have mass lambda nu.
00:05:14.936 --> 00:05:17.340
OK, and that's just
what I was claiming.
00:05:20.360 --> 00:05:23.313
So the important point here
was that you did need this 1.
00:05:23.313 --> 00:05:24.980
This 1 was playing a
very important role
00:05:24.980 --> 00:05:25.688
in this argument.
00:05:25.688 --> 00:05:27.890
If the 1 wasn't there,
this wouldn't work.
00:05:33.140 --> 00:05:34.850
And that 1 was
related to something
00:05:34.850 --> 00:05:38.850
that was part of one of
our starting assumptions,
00:05:38.850 --> 00:05:40.680
that when we made the
field redefinition,
00:05:40.680 --> 00:05:44.230
that the one-particle
states would stay the same.
00:05:44.230 --> 00:05:52.310
So if you look back at what our
field redefinition was and you
00:05:52.310 --> 00:06:00.517
trace back where
that 1 came from,
00:06:00.517 --> 00:06:02.600
we had a field redefinition
that was of this form,
00:06:02.600 --> 00:06:04.910
and this guy was a
function of other fields.
00:06:04.910 --> 00:06:07.400
But we always had this
term, and that term
00:06:07.400 --> 00:06:09.290
was related to the
presence of this 1,
00:06:09.290 --> 00:06:12.650
and that was important
for this argument, OK.
00:06:34.350 --> 00:06:39.080
So there has to be a term
that's linear in the new field.
00:06:39.080 --> 00:06:43.730
You want the same
one-particle states.
00:06:43.730 --> 00:06:46.580
So these ghosts, as
you know about ghosts,
00:06:46.580 --> 00:06:47.780
always appear in loops.
00:06:51.490 --> 00:06:53.240
And so if you want to
think about removing
00:06:53.240 --> 00:06:55.460
this massive particle, it's
like a massive particle
00:06:55.460 --> 00:06:56.600
that occurs in some loops.
00:07:00.200 --> 00:07:03.620
It's really just like that.
00:07:03.620 --> 00:07:05.440
So it's just like
a heavy particle
00:07:05.440 --> 00:07:07.190
that you would remove
from the theory that
00:07:07.190 --> 00:07:08.315
would only appear in loops.
00:07:12.110 --> 00:07:14.900
And we're going to discuss
exactly how that works
00:07:14.900 --> 00:07:18.320
and how to remove heavy
particles that appear in loops
00:07:18.320 --> 00:07:22.280
in detail a little
bit later, so for now,
00:07:22.280 --> 00:07:26.007
that's enough detail
for number two.
00:07:26.007 --> 00:07:27.590
So are there any
questions about that?
00:07:32.570 --> 00:07:35.740
OK, so the change
to the Jacobian
00:07:35.740 --> 00:07:38.170
is effectively a massive ghost
that we could integrate out
00:07:38.170 --> 00:07:38.753
of the theory.
00:07:38.753 --> 00:07:40.750
When we integrate it
out of the theory,
00:07:40.750 --> 00:07:45.010
it would shift the values of
the couplings in the Lagrangian,
00:07:45.010 --> 00:07:48.190
but it doesn't have
any impact that
00:07:48.190 --> 00:07:49.920
can't be absorbed
in other operators
00:07:49.920 --> 00:07:50.920
in the effective theory.
00:07:57.633 --> 00:08:00.175
So then the final thing we have
to worry about is the source.
00:08:05.900 --> 00:08:10.510
So there is this term with j
phi dagger, and when we take--
00:08:10.510 --> 00:08:12.280
there was an extra
term with j phi dagger
00:08:12.280 --> 00:08:13.687
that we had last time.
00:08:13.687 --> 00:08:15.520
I'm going to remind you
what it looked like.
00:08:18.790 --> 00:08:28.520
So in our path integral,
we had an exponential
00:08:28.520 --> 00:08:30.610
of a bunch of stuff,
but then there
00:08:30.610 --> 00:08:37.820
was one final term that was
induced that involved j phi
00:08:37.820 --> 00:08:41.200
dagger from the
field redefinition.
00:08:41.200 --> 00:08:43.840
So when you take derivatives
with respect to j phi dagger,
00:08:43.840 --> 00:08:46.270
there's sort of the usual
term that we want to source,
00:08:46.270 --> 00:08:47.833
and then there's this new term.
00:08:47.833 --> 00:08:49.000
So what about that new term?
00:08:52.000 --> 00:08:54.950
That's the final thing
we have to worry about.
00:08:54.950 --> 00:08:57.220
And this is where it's
important that we're actually
00:08:57.220 --> 00:09:00.070
considering observables.
00:09:00.070 --> 00:09:03.850
So we need to
consider observables.
00:09:03.850 --> 00:09:08.328
So let's start by considering
some Green's functions
00:09:08.328 --> 00:09:09.370
or time-ordered products.
00:09:23.210 --> 00:09:25.150
So I'm just taking
a string of fields.
00:09:25.150 --> 00:09:28.120
I could have some other
fields here as well.
00:09:28.120 --> 00:09:32.560
I'm going to make my life a
little easier by taking the phi
00:09:32.560 --> 00:09:37.980
to be real, and that's just
for notational simplicity.
00:09:41.490 --> 00:09:43.920
I could write phi
dagger and phis, but.
00:09:56.010 --> 00:09:59.767
So when we make the field
redefinition, what happens
00:09:59.767 --> 00:10:00.975
to that time-ordered product?
00:10:08.020 --> 00:10:10.697
So this is one way of thinking
about these extra terms
00:10:10.697 --> 00:10:11.280
in the source.
00:10:16.672 --> 00:10:18.630
So you take functional
derivatives with respect
00:10:18.630 --> 00:10:21.457
to the other piece that
involves j phi, you get phi.
00:10:21.457 --> 00:10:23.040
So if you take
[INAUDIBLE] that piece,
00:10:23.040 --> 00:10:24.082
you get the extra eta Ts.
00:10:26.760 --> 00:10:29.880
Or you can just think about
making the field redefinition
00:10:29.880 --> 00:10:34.350
directly in this matrix
element, and you'd
00:10:34.350 --> 00:10:37.488
have a matrix element
that looks like that.
00:10:37.488 --> 00:10:39.780
So it's not at all obvious
that these extra pieces that
00:10:39.780 --> 00:10:42.900
involve fields can't change the
value of this Green's function,
00:10:42.900 --> 00:10:45.880
and in fact, they will change
the value of that Green's
00:10:45.880 --> 00:10:47.753
function.
00:10:47.753 --> 00:10:49.170
But the important
thing is that we
00:10:49.170 --> 00:10:54.790
have to consider observables
and not just Green's functions.
00:10:54.790 --> 00:10:56.790
So when we're going to
think about observables,
00:10:56.790 --> 00:11:02.940
we should think about the LSZ
formula, which connects Green's
00:11:02.940 --> 00:11:05.010
functions to S-matrix elements.
00:11:07.750 --> 00:11:09.180
So let me remind you about that.
00:11:18.300 --> 00:11:20.850
And I'll remind you
what it looks like.
00:11:20.850 --> 00:11:22.560
First one is fixed
set of fields,
00:11:22.560 --> 00:11:25.870
which are scalar fields, but
we could put fermions in it
00:11:25.870 --> 00:11:28.800
as well.
00:11:28.800 --> 00:11:32.180
So what this says is,
if I look at this--
00:11:32.180 --> 00:11:34.160
so it's an integral
over the spacetime.
00:11:34.160 --> 00:11:35.930
I'm sticking in
particular momentum
00:11:35.930 --> 00:11:37.870
for the particular fields.
00:11:37.870 --> 00:11:41.570
And if I look at it, if I
look at the leading term,
00:11:41.570 --> 00:11:51.620
the leading pole term as the
particles are taken on shell,
00:11:51.620 --> 00:11:54.680
then that's an observable
known as the S-matrix.
00:12:15.040 --> 00:12:17.200
So some number of these
particles are incoming.
00:12:17.200 --> 00:12:19.150
Some number of
them are outgoing.
00:12:19.150 --> 00:12:22.360
That affects the sign that
I put in this plus minus.
00:12:22.360 --> 00:12:24.955
I'm not trying to be too
detailed about which ones are
00:12:24.955 --> 00:12:27.080
outgoing, which ones are
incoming, but some of them
00:12:27.080 --> 00:12:30.280
are incoming, some
of them are outgoing.
00:12:30.280 --> 00:12:37.143
And for each one, I want to
strip off the leading pole.
00:12:37.143 --> 00:12:38.560
I want to look at
the coefficient,
00:12:38.560 --> 00:12:40.390
and it has to have
a pole, so this is
00:12:40.390 --> 00:12:42.280
a product over all particles.
00:12:42.280 --> 00:12:46.390
We really need to pull in all
the different cases and all
00:12:46.390 --> 00:12:51.460
for all the external
particles, OK.
00:12:51.460 --> 00:12:54.100
So if I wanted to
make this an equality,
00:12:54.100 --> 00:12:56.380
I'd say plus dot dot dot.
00:12:56.380 --> 00:13:00.310
There's other terms, and
the thing that's observable
00:13:00.310 --> 00:13:03.130
is the coefficient
of these poles, not
00:13:03.130 --> 00:13:05.115
the Z but this thing.
00:13:09.480 --> 00:13:12.870
So if I make changes to my
theory and they effect this
00:13:12.870 --> 00:13:15.607
thing but get cancelled
by this thing,
00:13:15.607 --> 00:13:17.690
as long as they don't
affect this thing, we're OK.
00:13:21.610 --> 00:13:24.850
And the claim is that we make
this change to the source
00:13:24.850 --> 00:13:28.467
and it won't affect the
matrix on the S, the S-matrix.
00:13:58.810 --> 00:14:01.440
So again, we can
do some examples,
00:14:01.440 --> 00:14:05.520
and I think the examples are
fairly quickly convincing
00:14:05.520 --> 00:14:07.950
that this is the case.
00:14:07.950 --> 00:14:10.980
So we'll do three
different examples,
00:14:10.980 --> 00:14:14.160
first a rather trivial one.
00:14:14.160 --> 00:14:15.630
What if T was just by itself?
00:14:18.250 --> 00:14:23.910
So T is just phi, so this
is just 1 plus eta phi.
00:14:34.760 --> 00:14:37.535
So if you think
back, when we have T,
00:14:37.535 --> 00:14:39.410
this was the form of
the terminal Lagrangian,
00:14:39.410 --> 00:14:43.640
so this is just eta
phi del squared phi,
00:14:43.640 --> 00:14:47.900
so it's just changing
the kinetic term for phi.
00:14:47.900 --> 00:14:54.590
If we look at our
matrix on it, we're
00:14:54.590 --> 00:15:01.050
just getting a factor of 1 plus
eta for each of the fields.
00:15:01.050 --> 00:15:04.160
So let's say we had 4 of them
just so we don't have to--
00:15:07.760 --> 00:15:11.380
so say I had 4 of them.
00:15:11.380 --> 00:15:13.642
If I had n of them, it would
just be to the nth power,
00:15:13.642 --> 00:15:16.100
but if I have power of them,
they'd be to the fourth power,
00:15:16.100 --> 00:15:18.170
and I'd just get
this extra prefactor.
00:15:18.170 --> 00:15:20.870
And it looks like
it's changed G.
00:15:20.870 --> 00:15:23.390
And indeed, it has changed
the left-hand side of this
00:15:23.390 --> 00:15:27.920
equation, but when we calculate
the z factor and we canonically
00:15:27.920 --> 00:15:29.700
normalize--
00:15:29.700 --> 00:15:31.650
if you want to think
about it that way--
00:15:31.650 --> 00:15:35.190
then we would exactly cancel
off these 1 plus etas.
00:15:35.190 --> 00:15:38.420
So the root Z here is going
to be 1 plus eta as well.
00:15:48.210 --> 00:15:49.700
So when we look at the--
00:15:49.700 --> 00:15:52.010
this is just the residue
of the free propagator.
00:15:52.010 --> 00:15:57.255
The residue gets changed when
you change the kinetic term,
00:15:57.255 --> 00:16:00.597
so that doesn't do
anything to the S-matrix.
00:16:04.107 --> 00:16:05.690
So that's one way
we can be protected.
00:16:05.690 --> 00:16:06.898
We change the left-hand side.
00:16:06.898 --> 00:16:10.730
It's compensated by a change to
the Z and leaves S invariant.
00:16:10.730 --> 00:16:14.390
Let's do another example,
a little more non-trivial.
00:16:21.080 --> 00:16:29.890
So let's just take some cubic
term in our field redefinition.
00:16:36.950 --> 00:16:39.500
So this is like having a 5
cubed del squared phi term.
00:16:46.400 --> 00:16:49.960
So this extra term
will give rise
00:16:49.960 --> 00:16:52.540
to extra terms in
the primary product.
00:16:52.540 --> 00:16:54.700
And let me just
write one of them,
00:16:54.700 --> 00:16:59.390
again thinking of it
as a 4-point function.
00:16:59.390 --> 00:17:01.750
So let's just imagine I look
at the change that comes
00:17:01.750 --> 00:17:04.510
from changing the 4th guy.
00:17:07.839 --> 00:17:12.530
So there's other terms from
changing 5x1, 5x2 and 5x3,
00:17:12.530 --> 00:17:16.119
but if I work to order eta,
then I'd change one of them
00:17:16.119 --> 00:17:21.339
at a time, and we're only
working to order eta here.
00:17:21.339 --> 00:17:25.180
So the claim is actually
that this matrix element has
00:17:25.180 --> 00:17:28.150
no effect on this structure.
00:17:28.150 --> 00:17:31.335
It can affect the
dots, but it's not
00:17:31.335 --> 00:17:32.710
going to affect
the leading term,
00:17:32.710 --> 00:17:35.590
and that's because
having a 5 cubed
00:17:35.590 --> 00:17:38.120
means that you're not
having a one-particle state.
00:17:38.120 --> 00:17:43.000
So if you try to draw it
as a [INAUDIBLE] diagram,
00:17:43.000 --> 00:17:46.810
in the position space, you'd
label these external points
00:17:46.810 --> 00:17:49.300
and you inject momenta there.
00:17:49.300 --> 00:17:51.920
And then we label point 4, but
there's three fields there,
00:17:51.920 --> 00:17:55.830
so maybe we have to tie it
up like this or something.
00:17:55.830 --> 00:17:58.810
And when you look at
asymptotically what's
00:17:58.810 --> 00:18:01.000
going to happen from
[INAUDIBLE] a 5 cubed,
00:18:01.000 --> 00:18:03.640
you do not get a single-particle
pole from a 5 cubed.
00:18:07.800 --> 00:18:11.415
So this guy is less singular.
00:18:19.750 --> 00:18:27.420
There's no single-particle
pole, and hence,
00:18:27.420 --> 00:18:28.980
gives no contribution
to scatter.
00:18:37.720 --> 00:18:38.563
Yeah.
00:18:38.563 --> 00:18:40.105
AUDIENCE: As far as
the theorem goes,
00:18:40.105 --> 00:18:45.010
this step that you're showing,
it proves my hypothesis, right?
00:18:45.010 --> 00:18:47.315
IAN STEWART: In what sense?
00:18:47.315 --> 00:18:48.940
AUDIENCE: The hypothesis
of the theorem
00:18:48.940 --> 00:18:51.315
is that [INAUDIBLE] does not
change a one-particle state.
00:18:51.315 --> 00:18:52.398
IAN STEWART: That's right.
00:18:52.398 --> 00:18:53.650
That was an assumption.
00:18:53.650 --> 00:18:57.220
AUDIENCE: I mean, is that
the same as what you're--
00:18:57.220 --> 00:18:59.297
IAN STEWART: It's
the same, yeah.
00:18:59.297 --> 00:19:03.083
It's effectively the
same, though it's
00:19:03.083 --> 00:19:05.500
being more careful about what
the statement of that means,
00:19:05.500 --> 00:19:09.400
right, because I mean,
you're changing the residue
00:19:09.400 --> 00:19:13.360
but you're not
changing the S-matrix.
00:19:13.360 --> 00:19:14.162
AUDIENCE: Right.
00:19:14.162 --> 00:19:16.120
This is what I was thinking
when you said that.
00:19:16.120 --> 00:19:18.070
IAN STEWART: Yeah, all right.
00:19:25.470 --> 00:19:27.390
So really, this statement of--
00:19:27.390 --> 00:19:30.708
yeah, I could have been
a little more careful.
00:19:30.708 --> 00:19:33.000
So instead of saying that
the hypothesis of the theorem
00:19:33.000 --> 00:19:35.580
is that it wouldn't change
one-particle states, what
00:19:35.580 --> 00:19:38.550
I should have really said is
that there's a linear term
00:19:38.550 --> 00:19:40.880
in the field redefinition.
00:19:40.880 --> 00:19:46.160
Yeah, which now I'm showing you
is equivalent to not changing
00:19:46.160 --> 00:19:48.740
the one-particle states.
00:19:48.740 --> 00:19:54.560
And let's do one
final example just
00:19:54.560 --> 00:19:56.510
to rule out all
possible things we could
00:19:56.510 --> 00:19:58.530
think of that are different.
00:19:58.530 --> 00:20:03.860
So what if we had a derivative
[INAUDIBLE] squared phi prime?
00:20:03.860 --> 00:20:05.660
So then we would
do the following.
00:20:16.230 --> 00:20:19.337
We could add and
subtract a mass term.
00:20:19.337 --> 00:20:21.920
The reason we would want to do
that is, if we looked at a term
00:20:21.920 --> 00:20:25.643
like this, it's, again, less
singular than this term.
00:20:25.643 --> 00:20:27.560
If this term is giving
the one particle state,
00:20:27.560 --> 00:20:30.860
this term has got a factor
of the propagator upstairs.
00:20:30.860 --> 00:20:33.800
So if you look for a pole that
would come from that term,
00:20:33.800 --> 00:20:34.700
it's canceling out.
00:20:34.700 --> 00:20:36.260
You get p squared
minus m squared
00:20:36.260 --> 00:20:37.760
over p squared minus m squared.
00:20:41.130 --> 00:20:43.980
So there's no pole
from this term.
00:20:43.980 --> 00:20:47.030
And this term is,
again, just of the type
00:20:47.030 --> 00:20:48.840
from our first example.
00:20:48.840 --> 00:20:51.520
It's just shifting
5 by some constant.
00:20:51.520 --> 00:20:54.650
So probably to get
dimensions right,
00:20:54.650 --> 00:20:56.760
I should put some
factors of eta in here.
00:21:03.450 --> 00:21:06.268
So again, something like
that doesn't change anything,
00:21:06.268 --> 00:21:08.060
because it can be
decomposed into something
00:21:08.060 --> 00:21:10.070
that has no pole
and then something
00:21:10.070 --> 00:21:14.020
that's just a shift, OK?
00:21:14.020 --> 00:21:16.540
So as long as you have this
linear term in your field
00:21:16.540 --> 00:21:21.608
redefinition, and it doesn't
have to even have a trivial--
00:21:21.608 --> 00:21:23.650
it doesn't have to even
have a trivial prefactor.
00:21:23.650 --> 00:21:27.310
You could have 2 times
phi or whatever you like,
00:21:27.310 --> 00:21:30.400
because that'll cancel out when
you normalize things correctly.
00:21:30.400 --> 00:21:33.550
As long as you have that
linear term, you're fine.
00:21:33.550 --> 00:21:37.570
And you don't have to worry
about changes to the Jacobian
00:21:37.570 --> 00:21:41.287
or changes from the source term.
00:21:41.287 --> 00:21:43.870
So you just have the changes to
the Lagrangian to worry about.
00:21:43.870 --> 00:21:45.870
You don't have to think
about the path integral.
00:21:45.870 --> 00:21:48.893
Just make field redefinitions
on the Lagrangian.
00:21:48.893 --> 00:21:50.560
And that's what you'll
get some practice
00:21:50.560 --> 00:21:54.170
with on the problem set.
00:21:54.170 --> 00:21:57.400
So as I said at the beginning
before everybody was here,
00:21:57.400 --> 00:21:59.362
there's a problem set
number 1 that's posted.
00:21:59.362 --> 00:22:01.820
Everyone should make sure that
they can actually access it.
00:22:01.820 --> 00:22:04.480
And if they can't access it,
they should let me know--
00:22:04.480 --> 00:22:08.110
if you can't get
access to the web page.
00:22:08.110 --> 00:22:11.600
Any final questions about
this before we move on?
00:22:11.600 --> 00:22:13.660
Yeah?
00:22:13.660 --> 00:22:15.160
AUDIENCE: I get
nervous about trying
00:22:15.160 --> 00:22:18.490
to cancel a pole by adding an
m squared term, because then, I
00:22:18.490 --> 00:22:20.220
guess, at some point,
you start to worry
00:22:20.220 --> 00:22:22.210
that masses get shifted when you
start [? normalizing these. ?]
00:22:22.210 --> 00:22:23.127
IAN STEWART: Oh, yeah.
00:22:23.127 --> 00:22:24.640
Yeah.
00:22:24.640 --> 00:22:27.790
Yeah, you should be
careful about that, too.
00:22:27.790 --> 00:22:29.830
But you don't have
to worry about it.
00:22:36.210 --> 00:22:36.710
Yeah.
00:22:43.300 --> 00:22:46.630
You could think of it as doing
the mass renormalization first
00:22:46.630 --> 00:22:48.940
and then worry about this, or--
00:22:58.373 --> 00:23:00.290
so that you're using a
renormalized mass here.
00:23:06.310 --> 00:23:06.810
All right.
00:23:12.720 --> 00:23:14.160
So we'll start a new section.
00:23:35.052 --> 00:23:36.760
And the goal of this
section is basically
00:23:36.760 --> 00:23:40.240
to be more careful
about loop diagrams
00:23:40.240 --> 00:23:41.740
and, in particular,
to show you what
00:23:41.740 --> 00:23:44.590
matching is between two
effective field theories.
00:23:44.590 --> 00:23:48.190
We'll still be thinking in the
context of mass of particles.
00:23:48.190 --> 00:23:51.130
That's the simplest
place to describe this
00:23:51.130 --> 00:23:55.510
that we'll probably do an
example later on of a case that
00:23:55.510 --> 00:23:57.310
doesn't just involve
mass of particles.
00:24:00.590 --> 00:24:02.800
So let's start out with a
very trivial example just
00:24:02.800 --> 00:24:05.140
to see what we're talking about.
00:24:07.970 --> 00:24:09.155
So we'll take two particles.
00:24:09.155 --> 00:24:10.405
One of them is a heavy scalar.
00:24:17.660 --> 00:24:20.865
It has mass capital
M. Actually, I
00:24:20.865 --> 00:24:22.240
put a little
underline so you can
00:24:22.240 --> 00:24:26.230
tell my capitals
from my lowercase,
00:24:26.230 --> 00:24:28.150
because the light
fermion that we're also
00:24:28.150 --> 00:24:37.560
going to have has a
lowercase mass, little m.
00:24:37.560 --> 00:24:38.060
OK.
00:24:38.060 --> 00:24:41.450
So heavy scalar, light fermion.
00:24:41.450 --> 00:24:43.450
And so what we want
to talk about here
00:24:43.450 --> 00:24:47.920
is really this picture that
we described to you earlier
00:24:47.920 --> 00:24:51.700
where we've got two
theories, 1 and 2,
00:24:51.700 --> 00:24:53.920
and we want to pass
from theory 1 to 2
00:24:53.920 --> 00:24:56.800
by removing something
from theory 1.
00:24:56.800 --> 00:25:00.590
So theory 1 here will be just
the theory of these things.
00:25:00.590 --> 00:25:03.670
And we'll make it a
renormalizable theory,
00:25:03.670 --> 00:25:07.150
in the traditional sense,
just so we know where to stop.
00:25:18.320 --> 00:25:19.700
And this guy should be capital.
00:25:23.825 --> 00:25:25.700
And then there's some
[INAUDIBLE] interaction
00:25:25.700 --> 00:25:27.910
between them.
00:25:27.910 --> 00:25:32.390
So we'll think of this
as being the theory 1,
00:25:32.390 --> 00:25:36.110
where we're in a situation
where capital M is
00:25:36.110 --> 00:25:37.413
much bigger than little m.
00:25:41.577 --> 00:25:44.160
So then we want to think about
describing psi at low energies.
00:25:47.220 --> 00:25:49.140
And that means we can
get rid of the scalar
00:25:49.140 --> 00:25:50.820
as an explicit
degree of freedom.
00:25:58.900 --> 00:26:01.510
So low energy is relative
to this mass scale,
00:26:01.510 --> 00:26:03.495
which is heavy--
00:26:03.495 --> 00:26:17.370
M, capital M. We're
going to remove
00:26:17.370 --> 00:26:19.440
the scalar from the theory.
00:26:19.440 --> 00:26:21.375
So you could just
say remove it, or you
00:26:21.375 --> 00:26:22.500
could say integrate it out.
00:26:22.500 --> 00:26:25.920
When you say integrate
out, the words you're using
00:26:25.920 --> 00:26:27.630
have a path-integral
connotation,
00:26:27.630 --> 00:26:30.120
where you had this [INAUDIBLE]
in the path integral,
00:26:30.120 --> 00:26:32.820
and you think about
just doing the path
00:26:32.820 --> 00:26:35.565
integral over that
field and removing it.
00:26:35.565 --> 00:26:37.815
But the words of "removing
it" or "integrating it out"
00:26:37.815 --> 00:26:40.230
are synonymous.
00:26:40.230 --> 00:26:42.000
So what is theory 2
going to look like?
00:26:45.007 --> 00:26:47.340
Well, we'll still have the
kinetic term for our fermion.
00:26:52.800 --> 00:26:56.370
And then removing
the scalar will
00:26:56.370 --> 00:26:57.840
generate some new operators.
00:26:57.840 --> 00:27:01.260
In particular, there'll be
a dimension-six operator
00:27:01.260 --> 00:27:03.220
like this.
00:27:03.220 --> 00:27:07.295
And then there could
well be other terms.
00:27:07.295 --> 00:27:08.670
And so if we wanted
to figure out
00:27:08.670 --> 00:27:11.520
what this dimension-six
operator is at tree level,
00:27:11.520 --> 00:27:14.580
that's a pretty
straightforward exercise.
00:27:14.580 --> 00:27:21.000
We would simply think about the
Feynman diagram in theory 1.
00:27:21.000 --> 00:27:25.780
So here's a Feynman diagram
in theory 1 with [INAUDIBLE]
00:27:25.780 --> 00:27:28.933
couplings here.
00:27:28.933 --> 00:27:30.600
We would calculate
this Feynman diagram.
00:27:40.202 --> 00:27:41.910
And we would assume
that all the momentum
00:27:41.910 --> 00:27:44.520
of the external
particles are small.
00:27:44.520 --> 00:27:46.628
And that means
that this is small.
00:27:46.628 --> 00:27:47.670
And we would just expand.
00:28:02.130 --> 00:28:05.040
And then we would just ask
that, when I take the Feynman
00:28:05.040 --> 00:28:06.150
rule from this--
00:28:06.150 --> 00:28:08.940
whoops, should have
been a square there--
00:28:08.940 --> 00:28:10.830
when I take the
Feynman rule from that,
00:28:10.830 --> 00:28:14.970
then I should get the same thing
as the Feynman rule from that.
00:28:14.970 --> 00:28:16.980
And that would fix
this coefficient a,
00:28:16.980 --> 00:28:18.990
which is, so far, arbitrary.
00:28:18.990 --> 00:28:21.450
But we can determine
it by using theory 1.
00:28:21.450 --> 00:28:24.720
And that's the idea of doing a
matching calculation that you
00:28:24.720 --> 00:28:26.550
have introduced the theory 2.
00:28:26.550 --> 00:28:28.650
It has some parameters
in front of operators.
00:28:28.650 --> 00:28:31.420
In this case, I called it a.
00:28:31.420 --> 00:28:33.730
And we want to determine
those parameters by doing
00:28:33.730 --> 00:28:35.890
calculations in theory 1.
00:28:35.890 --> 00:28:37.330
And in particular,
what you do is
00:28:37.330 --> 00:28:40.690
you make sure that S-matrix
elements in the two theories
00:28:40.690 --> 00:28:42.430
agree.
00:28:42.430 --> 00:28:44.605
But at tree level, that's
just matching up diagrams.
00:29:16.840 --> 00:29:18.460
So a is simply g squared.
00:29:29.310 --> 00:29:31.470
So we just have to match
up this guy with that guy.
00:29:31.470 --> 00:29:36.770
And so taking the leading-order
term, a is just g squared.
00:29:36.770 --> 00:29:39.392
Very simple.
00:29:39.392 --> 00:29:41.600
So the place where this gets
more involved and more--
00:29:41.600 --> 00:29:43.790
where you have to think
a little bit more-- this
00:29:43.790 --> 00:29:45.650
seems almost automatic,
that you just do
00:29:45.650 --> 00:29:48.740
calculations in here and
here, and just match them up.
00:29:48.740 --> 00:29:49.982
And that's the basic idea.
00:29:49.982 --> 00:29:51.440
It's only a little
more complicated
00:29:51.440 --> 00:29:53.510
when you have to take
into account loops.
00:29:53.510 --> 00:29:56.600
So that's what we'll spend most
of this section discussing,
00:29:56.600 --> 00:29:59.790
since this part is easier.
00:29:59.790 --> 00:30:00.635
So what about loops?
00:30:03.612 --> 00:30:05.570
What are some of the
issues that come up there?
00:30:09.260 --> 00:30:12.222
Well, the first one is that
the Feynman diagrams diverge,
00:30:12.222 --> 00:30:13.430
so you have to regulate them.
00:30:24.810 --> 00:30:27.520
And the thing that can be
confusing about thinking
00:30:27.520 --> 00:30:30.020
about effective field theories
and thinking about divergence
00:30:30.020 --> 00:30:33.660
diagrams is separating the
ideas of regularization
00:30:33.660 --> 00:30:38.280
and the mass scale lambda nu
that we've been talking about.
00:30:38.280 --> 00:30:41.660
So that's something
I want to talk
00:30:41.660 --> 00:30:49.910
about in some detail, because
they're not the same thing.
00:30:52.675 --> 00:30:54.800
So we would need to cut
off ultraviolet divergences
00:30:54.800 --> 00:30:57.020
to obtain finite results.
00:30:57.020 --> 00:31:01.970
And that means we
introduce cutoff parameters
00:31:01.970 --> 00:31:03.185
into our results.
00:31:12.480 --> 00:31:20.580
So examples would be taking
the Minkowski momentum,
00:31:20.580 --> 00:31:23.070
continuing it to be Euclidean,
and then just putting
00:31:23.070 --> 00:31:26.970
a hard cutoff on it--
that's one example.
00:31:26.970 --> 00:31:29.520
Or you could use dim reg.
00:31:29.520 --> 00:31:30.870
That's another example.
00:31:30.870 --> 00:31:32.340
Or you could use
a lattice spacing.
00:31:32.340 --> 00:31:34.540
That's another example.
00:31:34.540 --> 00:31:36.390
So these are all
examples of how you
00:31:36.390 --> 00:31:39.537
might cut off the theory to
remove ultraviolet divergences.
00:31:43.120 --> 00:31:51.340
And then there's a second
step, renormalization,
00:31:51.340 --> 00:31:53.110
distinct from regularization.
00:31:56.140 --> 00:31:58.030
So some of the things
I'm teaching you here
00:31:58.030 --> 00:31:59.650
should be familiar.
00:31:59.650 --> 00:32:02.170
And the only real generalization
that we're going to do
00:32:02.170 --> 00:32:04.390
is we're going to be able
to apply some of the things
00:32:04.390 --> 00:32:05.973
that you learned
about renormalization
00:32:05.973 --> 00:32:07.600
and regularization
to any operators
00:32:07.600 --> 00:32:09.017
that you might
have in the theory,
00:32:09.017 --> 00:32:13.030
whether they're renormalizable
in the dimension-four
00:32:13.030 --> 00:32:16.850
traditional sense or in a
higher-dimensional sense.
00:32:16.850 --> 00:32:19.660
So here, what we're doing when
we talk about renormalization
00:32:19.660 --> 00:32:22.930
is we're picking a
scheme that gives
00:32:22.930 --> 00:32:27.870
precise or definite
meaning to the parameters
00:32:27.870 --> 00:32:28.870
in the effective theory.
00:32:33.220 --> 00:32:38.530
Or, even more explicitly, each
coefficient in the Lagrangian,
00:32:38.530 --> 00:32:44.230
as well as the operators
in the Lagrangian,
00:32:44.230 --> 00:32:49.450
are given a meaning
by this procedure.
00:32:49.450 --> 00:32:51.850
If we didn't have a
renormalization procedure,
00:32:51.850 --> 00:32:54.640
then, because of the
ultraviolet divergences,
00:32:54.640 --> 00:32:57.190
there'd be ambiguities in how
we define the coefficients
00:32:57.190 --> 00:32:59.410
and the operators.
00:32:59.410 --> 00:33:01.720
Or, even if you don't have
ultraviolet divergences,
00:33:01.720 --> 00:33:05.290
you still have freedom
to pick different schemes
00:33:05.290 --> 00:33:09.760
for the definitions
of the coefficients.
00:33:09.760 --> 00:33:14.080
And when you do this, you
also can introduce parameters.
00:33:18.933 --> 00:33:20.850
So you can get parameters
from regularization.
00:33:20.850 --> 00:33:24.130
You can also get
parameters from this.
00:33:24.130 --> 00:33:26.670
So some examples
that are familiar--
00:33:26.670 --> 00:33:29.205
there's this scale mu that
shows up when you do MS bar.
00:33:32.485 --> 00:33:34.110
If you do something
that's a little bit
00:33:34.110 --> 00:33:36.870
different, if you take
Green's functions,
00:33:36.870 --> 00:33:40.290
and you go to some
offshell point,
00:33:40.290 --> 00:33:45.300
that's another renormalization
scheme called offshell momentum
00:33:45.300 --> 00:33:47.460
subtraction.
00:33:47.460 --> 00:33:54.100
And that scheme also has
a parameter that shows up.
00:33:54.100 --> 00:33:58.820
And if you did a
Wilsonian renormalization,
00:33:58.820 --> 00:34:02.580
there would also be a
cutoff associated to that.
00:34:02.580 --> 00:34:05.115
And this Wilsonian
cutoff here doesn't
00:34:05.115 --> 00:34:06.990
have to be the same as
this lambda uv cutoff.
00:34:14.040 --> 00:34:14.540
OK.
00:34:14.540 --> 00:34:16.130
So every coefficient,
every operator,
00:34:16.130 --> 00:34:18.409
no matter the
dimension, no matter
00:34:18.409 --> 00:34:21.650
where it turns up
in the series, we're
00:34:21.650 --> 00:34:23.900
going to have to think
about whether there's
00:34:23.900 --> 00:34:26.750
diagrams that generate
ultraviolet divergences.
00:34:26.750 --> 00:34:31.363
And if those diagrams
look like these operators,
00:34:31.363 --> 00:34:33.030
then you're going to
get renormalization
00:34:33.030 --> 00:34:33.822
of those operators.
00:34:33.822 --> 00:34:37.350
And you're going to have to
be careful about taking care
00:34:37.350 --> 00:34:39.725
of those divergences
and also the scheme
00:34:39.725 --> 00:34:41.100
that you define
the operators in.
00:34:46.020 --> 00:34:49.409
So you can think about
this, just as far
00:34:49.409 --> 00:34:59.061
as the coefficients
are concerned,
00:34:59.061 --> 00:35:04.710
as starting out with some
bare coefficients that
00:35:04.710 --> 00:35:07.860
depend on your
ultraviolet regulator,
00:35:07.860 --> 00:35:11.282
and switching to some
renormalized coefficients which
00:35:11.282 --> 00:35:12.990
don't depend on the
ultraviolet regulator
00:35:12.990 --> 00:35:15.810
but do depend on the scheme.
00:35:15.810 --> 00:35:18.855
And then, also, you
have some counterterms
00:35:18.855 --> 00:35:19.980
that depend on both things.
00:35:23.230 --> 00:35:24.960
So that's how it
would look if you
00:35:24.960 --> 00:35:30.090
wrote that down for a
cutoff, in a Wilsonian sense.
00:35:30.090 --> 00:35:34.140
If you wrote it down in
dimensional regularization,
00:35:34.140 --> 00:35:35.190
it would look like this.
00:35:39.630 --> 00:35:43.102
Same idea, different
names for the parameters.
00:35:46.760 --> 00:35:47.260
OK.
00:35:47.260 --> 00:35:48.635
So in dimensional
regularization,
00:35:48.635 --> 00:35:50.620
epsilon is the
ultraviolet regulator.
00:35:50.620 --> 00:35:53.320
It shows up in the
bare coefficients.
00:35:53.320 --> 00:35:57.400
The renormalized ones don't
depend on epsilon anymore.
00:35:57.400 --> 00:35:59.140
And these are your
1-over-epsilon poles.
00:36:05.340 --> 00:36:06.900
All right.
00:36:06.900 --> 00:36:09.480
So one of the things
that makes this tricky
00:36:09.480 --> 00:36:11.970
is when you start thinking
about your power counting.
00:36:11.970 --> 00:36:17.700
And so I want to spend a few
minutes talking about that.
00:36:27.850 --> 00:36:30.240
So let's consider this
example that I wrote down
00:36:30.240 --> 00:36:33.370
with the four-fermion operator.
00:36:33.370 --> 00:36:37.290
And if we just loop up
the four-fermion operator,
00:36:37.290 --> 00:36:39.552
then we can get mass
renormalization.
00:36:42.510 --> 00:36:45.600
So this is psi
dagger psi squared.
00:36:45.600 --> 00:36:53.920
And I'm sticking it
in and looping it up.
00:36:59.470 --> 00:37:02.265
So we start out with a
tree-level mass, little m.
00:37:02.265 --> 00:37:03.640
But there's a
one-loop correction
00:37:03.640 --> 00:37:05.807
that involves one of our
higher-dimension operators.
00:37:08.343 --> 00:37:10.010
And there's some
connection to the mass,
00:37:10.010 --> 00:37:12.240
which I'll call delta m.
00:37:12.240 --> 00:37:14.740
I'm not going to worry too much
about the overall prefactor.
00:37:17.980 --> 00:37:21.260
This operator came with
an a over M squared.
00:37:21.260 --> 00:37:24.370
And then there's
a loop integral.
00:37:24.370 --> 00:37:30.014
There's a fermionic propagator,
so that's a k slash plus m
00:37:30.014 --> 00:37:31.690
over k squared minus m squared.
00:37:35.290 --> 00:37:38.980
This guy here drops away.
00:37:38.980 --> 00:37:41.230
And really, we just have a
correction that's a scalar.
00:37:41.230 --> 00:37:47.060
And that's a correction to the
mass scalar in the spin space.
00:37:47.060 --> 00:37:51.100
So it's a, little m,
over capital M, and then
00:37:51.100 --> 00:37:53.840
just this integral.
00:37:53.840 --> 00:37:59.950
And if we continue
it to be Euclidean,
00:37:59.950 --> 00:38:04.510
then it looks like that if
I continue from Minkowski
00:38:04.510 --> 00:38:05.860
to Euclidean.
00:38:05.860 --> 00:38:09.440
That's why I put the i there.
00:38:09.440 --> 00:38:12.340
So if we just assume
that this integral that's
00:38:12.340 --> 00:38:15.730
sitting there-- there's only
one mass scale that seems
00:38:15.730 --> 00:38:17.740
explicit there-- the little m.
00:38:17.740 --> 00:38:20.110
So if we just assume
that that integral is
00:38:20.110 --> 00:38:21.460
dominated by the scale m--
00:38:30.040 --> 00:38:38.360
little m-- then you
could do a power counting
00:38:38.360 --> 00:38:39.860
for this loop integral.
00:38:39.860 --> 00:38:41.930
You just assume that
all the factors of k
00:38:41.930 --> 00:38:42.970
are of order little m.
00:38:42.970 --> 00:38:46.140
You've got an explicit little
m, four powers in the numerator,
00:38:46.140 --> 00:38:48.740
and two powers downstairs.
00:38:48.740 --> 00:38:55.610
So you would say that this
integral scales like m squared.
00:38:58.310 --> 00:39:02.240
And then you would put that
together with your delta m
00:39:02.240 --> 00:39:08.090
and say that delta m scales like
a, little m cubed, capital M
00:39:08.090 --> 00:39:09.330
squared.
00:39:09.330 --> 00:39:11.780
And so this is something that
would then be suppressed.
00:39:11.780 --> 00:39:14.030
Relative to the original
tree-level contribution,
00:39:14.030 --> 00:39:16.028
it would be suppressed
by, perhaps,
00:39:16.028 --> 00:39:17.570
some [? 4 pi ?]
[? is in ?] the loop,
00:39:17.570 --> 00:39:20.570
but also little m squared
over big M squared.
00:39:24.690 --> 00:39:29.490
So it would be a small
correction, which is really
00:39:29.490 --> 00:39:31.952
what we would like to be
the case when we're talking
00:39:31.952 --> 00:39:33.660
about some higher-dimension
operator that
00:39:33.660 --> 00:39:34.910
was supposed to be suppressed.
00:39:34.910 --> 00:39:39.120
We'd like it to be
a small correction.
00:39:39.120 --> 00:39:40.500
All right.
00:39:40.500 --> 00:39:43.128
So does anyone have any
idea what can go wrong
00:39:43.128 --> 00:39:43.920
with this argument?
00:39:47.970 --> 00:39:49.180
I've been glib about it.
00:39:49.180 --> 00:39:52.200
I just said, if the integral
is dominated by that.
00:39:56.270 --> 00:39:58.020
So the thing is that
we have to be careful
00:39:58.020 --> 00:40:00.970
about our choice of regulator,
because regulators also
00:40:00.970 --> 00:40:01.845
can introduce scales.
00:40:13.910 --> 00:40:15.759
So let's do this more carefully.
00:40:33.564 --> 00:40:35.490
Well, let's consider
two different choices
00:40:35.490 --> 00:40:37.110
for the regulator.
00:40:37.110 --> 00:40:40.967
So we'll start out with
just a cutoff, which
00:40:40.967 --> 00:40:43.050
seems very natural, from
an effective field theory
00:40:43.050 --> 00:40:46.200
point of view-- that your theory
is supposed to be only valid up
00:40:46.200 --> 00:40:48.030
to some scale.
00:40:48.030 --> 00:40:52.080
So why not just take and cut
off the momentum explicitly
00:40:52.080 --> 00:40:53.100
above some scale?
00:40:59.060 --> 00:41:02.210
And you can think of that
scale as being of order big M.
00:41:02.210 --> 00:41:04.520
So we just don't
include any momentum
00:41:04.520 --> 00:41:07.050
that are higher than that
scale in our loop integrals.
00:41:07.050 --> 00:41:09.772
So we're only integrating over
the region in momentum space
00:41:09.772 --> 00:41:11.480
where the theory is
supposed to be valid.
00:41:11.480 --> 00:41:13.820
It seems like a perfectly
reasonable thing to do.
00:41:27.260 --> 00:41:30.070
So in this case, we can
take this integral that
00:41:30.070 --> 00:41:33.250
has some angular parts to
it as well as a radial part,
00:41:33.250 --> 00:41:36.610
decompose it into the radial
piece and the angular piece.
00:41:40.480 --> 00:41:44.830
I'll give you a handout, or I'll
post a page of lecture notes
00:41:44.830 --> 00:41:49.090
that have all those fun formulas
that are useful to remember
00:41:49.090 --> 00:41:54.040
but not fun to talk about
for decomposing integrals
00:41:54.040 --> 00:42:00.302
in arbitrary dimensions into
radial and angular pieces.
00:42:00.302 --> 00:42:02.260
So if we do that, in this
case, then the radial
00:42:02.260 --> 00:42:04.390
integral-- we can cut
off the radial integral
00:42:04.390 --> 00:42:07.210
with a hard cutoff lambda uv.
00:42:07.210 --> 00:42:12.520
And just by dimensions, the d
4 k becomes kE times kE cubed.
00:42:12.520 --> 00:42:15.673
So this is a radial k now.
00:42:15.673 --> 00:42:17.215
And this integral,
we can do exactly.
00:42:24.340 --> 00:42:27.460
There's a loop
factor, 4 pi squared.
00:42:27.460 --> 00:42:30.340
There's a lambda uv squared.
00:42:30.340 --> 00:42:33.091
And then there's a logarithm.
00:42:33.091 --> 00:42:36.910
It goes like, lambda uv
squared over little m squared.
00:42:42.443 --> 00:42:44.110
There's two scales
that are showing up--
00:42:44.110 --> 00:42:45.760
lambda uv and little m.
00:42:45.760 --> 00:42:47.708
The answer depends
on both of them.
00:42:47.708 --> 00:42:49.750
And then we can start
expanding, because little m
00:42:49.750 --> 00:42:51.667
is supposed to be much
smaller than lambda uv.
00:43:00.890 --> 00:43:04.970
Let me pull out an a
m over 4 pi squared.
00:43:15.270 --> 00:43:18.530
So there would be
a logarithmic term.
00:43:18.530 --> 00:43:20.420
There's this lambda uv
squared over capital
00:43:20.420 --> 00:43:24.401
M squared if I put
that factor in.
00:43:24.401 --> 00:43:25.707
And then there's some--
00:43:31.920 --> 00:43:33.920
so everything I'm writing
in the square brackets
00:43:33.920 --> 00:43:36.200
here is dimensionless.
00:43:36.200 --> 00:43:37.880
This has the right
dimensions of a mass.
00:43:40.950 --> 00:43:41.880
OK.
00:43:41.880 --> 00:43:45.090
So you can see
that, if I do this,
00:43:45.090 --> 00:43:47.460
it's not satisfying
what I said over here.
00:43:47.460 --> 00:43:50.910
I'd like the correction to
be m cubed over M squared.
00:43:50.910 --> 00:43:54.360
There are corrections that go
like m cubed over M squared,
00:43:54.360 --> 00:43:55.600
like this one.
00:43:55.600 --> 00:43:57.130
These ones are even higher.
00:43:57.130 --> 00:43:59.370
But there's this term.
00:43:59.370 --> 00:44:01.392
And that's not a
small correction.
00:44:13.200 --> 00:44:17.460
So for that particular
piece, what you're finding
00:44:17.460 --> 00:44:19.410
is that your power
counting-- your naive power
00:44:19.410 --> 00:44:20.785
counting-- of the
loop was wrong,
00:44:20.785 --> 00:44:23.020
because there's a
piece from k of order--
00:44:23.020 --> 00:44:28.760
the cutoff that's contributing
for that part of it.
00:44:31.330 --> 00:44:35.100
And that's why your naive
scaling argument didn't work.
00:44:35.100 --> 00:44:38.400
Now, this is the
bare result. And we
00:44:38.400 --> 00:44:40.690
have to go through the
renormalization procedure.
00:44:45.760 --> 00:44:48.030
And so if we go through the
renormalization procedure,
00:44:48.030 --> 00:44:50.857
you can think of that as taking
a piece of the integral--
00:44:55.250 --> 00:44:58.770
so in a Wilsonian
sense, you would
00:44:58.770 --> 00:45:02.440
take a piece of the integral and
absorb it into the counterterm.
00:45:07.840 --> 00:45:10.020
So as promised, the
counterterm depends
00:45:10.020 --> 00:45:11.430
on both lambda uv and lambda.
00:45:11.430 --> 00:45:14.432
And those are just explicitly--
00:45:14.432 --> 00:45:16.140
I'm cutting off the
radial integral here.
00:45:21.210 --> 00:45:23.670
And that does improve things,
because then, I can lower
00:45:23.670 --> 00:45:27.610
this cutoff, make it smaller.
00:45:27.610 --> 00:45:31.920
And what is left would just be
lambda squared over capital M
00:45:31.920 --> 00:45:39.880
squared and log of m squared
over capital lambda squared.
00:45:39.880 --> 00:45:42.780
So I change the lambda
uv's up here into lambdas
00:45:42.780 --> 00:45:44.231
by that procedure.
00:45:51.780 --> 00:45:53.580
So when I renormalize,
the psi bar psi
00:45:53.580 --> 00:45:56.570
squared matrix element--
there's a correction from that
00:45:56.570 --> 00:45:57.260
to the mass.
00:45:57.260 --> 00:46:01.520
And the renormalized thing would
depend on this Wilsonian cutoff
00:46:01.520 --> 00:46:04.040
lambda, OK?
00:46:04.040 --> 00:46:06.230
But I'm not fully
getting around the issue
00:46:06.230 --> 00:46:07.880
that I'm generating
this type of term.
00:46:15.020 --> 00:46:21.770
So let's do it with a
different regulator, which
00:46:21.770 --> 00:46:23.420
is dimensional regularization.
00:46:23.420 --> 00:46:41.840
Let's see what happens
using the MS-bar scheme.
00:46:41.840 --> 00:46:42.920
Same calculation.
00:46:57.170 --> 00:47:04.100
So again, split it into
radial and angular parts.
00:47:04.100 --> 00:47:06.150
And again, I'll
give you formulas
00:47:06.150 --> 00:47:07.400
for doing something like that.
00:47:09.992 --> 00:47:11.690
But I won't write
them down in lecture.
00:47:18.160 --> 00:47:21.280
Actually, I think I will
write one down a little later.
00:47:21.280 --> 00:47:24.690
But I'll give you a more
complete set as a handout.
00:47:34.532 --> 00:47:35.990
So leaving over
the radial integral
00:47:35.990 --> 00:47:37.520
but regulating it
dimensionally just
00:47:37.520 --> 00:47:40.290
means that, instead of having
this guy to the cubed power,
00:47:40.290 --> 00:47:41.930
it's to the d minus 1.
00:47:41.930 --> 00:47:45.290
And there's some pieces
here that depend on epsilon.
00:47:45.290 --> 00:47:47.400
And my convention
for the entire course
00:47:47.400 --> 00:47:50.850
is that d is 4 minus 2
epsilon, not d minus epsilon.
00:47:50.850 --> 00:47:55.390
So again, this is an
integral we can do exactly.
00:48:19.540 --> 00:48:21.942
We get something like that.
00:48:21.942 --> 00:48:22.900
And then we can expand.
00:48:37.176 --> 00:48:40.330
We get a 1-over-epsilon pole.
00:48:40.330 --> 00:48:41.340
We get a logarithm.
00:48:43.930 --> 00:48:46.850
There is some constant as well.
00:48:46.850 --> 00:48:49.180
And then there's
order-epsilon pieces.
00:48:53.240 --> 00:48:57.435
So you should contrast
this result here
00:48:57.435 --> 00:48:59.730
with the result
we had over here.
00:48:59.730 --> 00:49:03.017
The order-epsilon pieces
are like these terms.
00:49:03.017 --> 00:49:05.600
They're the terms that would go
away if I took the cutoff here
00:49:05.600 --> 00:49:09.350
to infinity-- lambda uv to
infinity or epsilon to 0.
00:49:09.350 --> 00:49:12.660
There's terms here that go
like m cubed over M squared.
00:49:12.660 --> 00:49:14.485
That's like this term.
00:49:14.485 --> 00:49:15.610
And this term is not there.
00:49:22.750 --> 00:49:26.200
This 1 over epsilon is
related to the log divergence.
00:49:26.200 --> 00:49:28.960
And this thing here
is a power divergence.
00:49:28.960 --> 00:49:32.080
So the 1 over epsilon is related
to this log of lambda uv.
00:49:32.080 --> 00:49:33.760
You should think
of log of lambda uv
00:49:33.760 --> 00:49:38.288
as going to 1 over epsilon
plus log of mu squared.
00:49:38.288 --> 00:49:39.830
But we don't see
the power divergence
00:49:39.830 --> 00:49:46.140
in dimensional
regularization in MS bar.
00:49:46.140 --> 00:49:46.640
OK.
00:49:46.640 --> 00:49:50.750
So what's happening
is, from this result,
00:49:50.750 --> 00:50:02.450
we are getting something that's
the size that we expected
00:50:02.450 --> 00:50:04.070
by our power-counting argument.
00:50:04.070 --> 00:50:09.560
The regularized result is the
right size by power counting.
00:50:09.560 --> 00:50:21.300
The logarithm here is the same
log as in a with, if you like,
00:50:21.300 --> 00:50:24.660
a correspondence
between mu and lambda.
00:50:24.660 --> 00:50:26.927
So we're seeing that
logarithm there.
00:50:26.927 --> 00:50:28.260
We're just not seeing this term.
00:50:37.070 --> 00:50:41.440
And if we wanted to write
down the MS-bar counterterm.
00:50:41.440 --> 00:50:47.935
Some correction to the mass,
it would go like, a, m squared.
00:50:53.610 --> 00:50:56.775
The diagram would look like
just the 1-over-epsilon pole
00:50:56.775 --> 00:50:59.710
in MS bar.
00:50:59.710 --> 00:51:00.210
OK.
00:51:00.210 --> 00:51:04.290
So the two regulators seem to
give us similar results but not
00:51:04.290 --> 00:51:06.420
exactly-identical results.
00:51:06.420 --> 00:51:09.340
And the question is, how
should I think about this?
00:51:09.340 --> 00:51:11.670
What's the right way of
thinking about this extra term?
00:51:14.990 --> 00:51:16.540
So there's a
language that people
00:51:16.540 --> 00:51:19.510
use in effective field
theory and dimensional
00:51:19.510 --> 00:51:21.670
counting for the
difference between thinking
00:51:21.670 --> 00:51:28.110
about doing a regularization
of type A and type B.
00:51:28.110 --> 00:51:31.290
And what you say
is that, in type B,
00:51:31.290 --> 00:51:35.670
your regularization
of the problem
00:51:35.670 --> 00:51:52.330
does not break the
power counting, which
00:51:52.330 --> 00:51:57.400
means that you can do power
counting for regulated graphs
00:51:57.400 --> 00:51:58.750
prior to renormalization them.
00:52:08.250 --> 00:52:13.978
But you can see that, in case
A, that would be problematic,
00:52:13.978 --> 00:52:16.270
because our naive power
counting wouldn't have given us
00:52:16.270 --> 00:52:20.770
the right result.
00:52:20.770 --> 00:52:22.424
So what do we say about case A?
00:52:28.635 --> 00:52:30.690
Well, if case B didn't
break the power counting,
00:52:30.690 --> 00:52:31.590
then case A does.
00:52:35.970 --> 00:52:38.820
But you can say something
a little bit more positive
00:52:38.820 --> 00:52:45.600
about case A. And that is that,
when you think about case A,
00:52:45.600 --> 00:52:51.390
you can set up the theory with
renormalization conditions
00:52:51.390 --> 00:52:57.390
such that you can
restore power counting
00:52:57.390 --> 00:53:00.720
in the renormalized graphs
and renormalized couplings
00:53:00.720 --> 00:53:02.697
and renormalized operators.
00:53:02.697 --> 00:53:04.780
But you don't have power
counting-- explicit power
00:53:04.780 --> 00:53:07.715
counting-- in just the
regularized results.
00:53:07.715 --> 00:53:09.090
So you can think
that I just take
00:53:09.090 --> 00:53:10.890
the Wilsonian cutoff
small enough that I
00:53:10.890 --> 00:53:15.990
make that annoying term small.
00:53:15.990 --> 00:53:18.630
And order by order,
I can do that--
00:53:21.180 --> 00:53:22.860
order by order in
my loop expansion,
00:53:22.860 --> 00:53:24.480
order by order in
my calculations.
00:53:31.730 --> 00:53:32.340
OK.
00:53:32.340 --> 00:53:34.130
So in some sense,
there's nothing wrong
00:53:34.130 --> 00:53:35.960
with doing A. It's
just that you have
00:53:35.960 --> 00:53:38.060
to work a little harder
to think about it.
00:53:38.060 --> 00:53:40.213
And when you think about
the power counting,
00:53:40.213 --> 00:53:42.380
you have to do that for the
renormalized quantities,
00:53:42.380 --> 00:53:45.707
not for the bare quantities.
00:53:45.707 --> 00:53:47.290
So if you like, what
you're doing here
00:53:47.290 --> 00:53:49.540
is you're adding counterterms.
00:53:49.540 --> 00:53:55.670
Another way you can say this is
that you're adding counterterms
00:53:55.670 --> 00:53:59.060
to restore the power
counting that you want.
00:54:02.940 --> 00:54:05.142
And that's not too
different than--
00:54:05.142 --> 00:54:06.600
you could always
do that, actually.
00:54:06.600 --> 00:54:08.340
I made this analogy
that you should
00:54:08.340 --> 00:54:14.310
think about power counting
as being like gauge symmetry.
00:54:14.310 --> 00:54:17.550
So let's say you had a theory,
and it had nice gauge symmetry,
00:54:17.550 --> 00:54:19.560
but you picked some
crazy regulator
00:54:19.560 --> 00:54:21.805
that broke gauge symmetry.
00:54:21.805 --> 00:54:23.430
Well, you could always
put counterterms
00:54:23.430 --> 00:54:24.847
in that would break
gauge symmetry
00:54:24.847 --> 00:54:26.925
and restore it in the
renormalized quantity.
00:54:26.925 --> 00:54:28.440
That would be OK.
00:54:28.440 --> 00:54:29.850
It would be more work.
00:54:29.850 --> 00:54:31.590
We don't like to do that.
00:54:31.590 --> 00:54:33.810
We avoid it at all costs.
00:54:33.810 --> 00:54:36.600
But if we had to,
we could do it.
00:54:36.600 --> 00:54:38.748
If you do
supersymmetry, you might
00:54:38.748 --> 00:54:40.290
try to use dimensional
regularization
00:54:40.290 --> 00:54:42.540
and supersymmetry, standard
dimensional regularization
00:54:42.540 --> 00:54:44.880
and supersymmetry,
break supersymmetry.
00:54:44.880 --> 00:54:47.340
If you do that, you have
to introduce counterterms
00:54:47.340 --> 00:54:49.140
that restore supersymmetry.
00:54:49.140 --> 00:54:52.240
So the same language of
symmetry is being applied here,
00:54:52.240 --> 00:54:54.720
except now to power
counting, where
00:54:54.720 --> 00:54:57.780
we say that if your regulator
messes up your power counting,
00:54:57.780 --> 00:55:00.420
you can restore it in the
renormalized couplings.
00:55:00.420 --> 00:55:02.790
But you may be smart enough
to think up a regulator--
00:55:02.790 --> 00:55:05.130
in this case, dimensional
regularization--
00:55:05.130 --> 00:55:08.210
where you don't have to
deal with that complication.
00:55:08.210 --> 00:55:08.710
OK.
00:55:20.656 --> 00:55:21.820
So let me write that.
00:56:15.070 --> 00:56:16.740
So in an effective
theory, you should
00:56:16.740 --> 00:56:19.860
think about regulating
to preserve symmetries
00:56:19.860 --> 00:56:23.130
as well as to preserve
power counting, if you can.
00:56:36.660 --> 00:56:40.620
And one way, in a
more formal language,
00:56:40.620 --> 00:56:43.350
that you could say what
happens with the generation
00:56:43.350 --> 00:56:45.960
of that term that we talked
about with the cutoff
00:56:45.960 --> 00:56:50.310
is that you'd mix up different
orders in the expansion,
00:56:50.310 --> 00:56:54.840
and it looks like your
naively higher-order term
00:56:54.840 --> 00:56:58.325
is mixing back to a
term of lower dimension.
00:57:02.880 --> 00:57:06.060
And so if you can get away with
taking a regulator that doesn't
00:57:06.060 --> 00:57:09.510
have that mixing back to
more relevant operators,
00:57:09.510 --> 00:57:12.720
then you could preserve
your power counting,
00:57:12.720 --> 00:57:15.730
make it simpler.
00:57:15.730 --> 00:57:20.950
This is true irrespective of
what the power counting is in.
00:57:20.950 --> 00:57:23.800
This is a general statement.
00:57:23.800 --> 00:57:25.940
In the context of what
we've been talking about,
00:57:25.940 --> 00:57:29.230
which is dimensional
power counting,
00:57:29.230 --> 00:57:31.480
there's a particular phrase
that goes along with this.
00:57:39.980 --> 00:57:42.820
And that is that we talk
about using a mass independent
00:57:42.820 --> 00:57:50.415
regulator, like dim reg.
00:57:56.845 --> 00:58:00.070
If you like, it has
a mass scale, mu,
00:58:00.070 --> 00:58:02.260
but it's put in
softly in a way that
00:58:02.260 --> 00:58:04.360
doesn't mess up power counting.
00:58:04.360 --> 00:58:07.780
We call that using a mass
independent regulator.
00:58:07.780 --> 00:58:11.110
So we want to avoid
having different orders
00:58:11.110 --> 00:58:13.780
in the expansion mix
up with each other.
00:58:13.780 --> 00:58:16.810
In general, I should
comment, then,
00:58:16.810 --> 00:58:18.700
that terms that
are the same order
00:58:18.700 --> 00:58:23.530
will definitely typically
mix up with each other
00:58:23.530 --> 00:58:25.990
under renormalization.
00:58:25.990 --> 00:58:27.760
So even if you thought
you were smart,
00:58:27.760 --> 00:58:30.895
and you enumerated all the
operators, but you missed one,
00:58:30.895 --> 00:58:33.520
and then you
started calculating,
00:58:33.520 --> 00:58:35.740
that operator might
just pop out at you,
00:58:35.740 --> 00:58:37.900
because you could
have some calculation
00:58:37.900 --> 00:58:41.260
with another operator
mix into that operator.
00:58:41.260 --> 00:58:44.680
Or you could have an operator
that you did a matching at tree
00:58:44.680 --> 00:58:47.080
level, and you didn't
generate, but then you
00:58:47.080 --> 00:58:50.060
start renormalizing that
tree-level operator,
00:58:50.060 --> 00:58:52.925
and another operator
pops out at you.
00:58:52.925 --> 00:58:55.300
So it's important that you
include all the operators that
00:58:55.300 --> 00:58:58.893
have the same dimension and
the same quantum numbers,
00:58:58.893 --> 00:59:00.310
because if you
don't include them,
00:59:00.310 --> 00:59:04.963
you're bound to get
them from loops anyway.
00:59:04.963 --> 00:59:06.130
And you want to be complete.
00:59:11.140 --> 00:59:18.690
So if you like, in a matrix
[INAUDIBLE] notation,
00:59:18.690 --> 00:59:22.530
you could say that the bare
operators mix up with the set
00:59:22.530 --> 00:59:24.420
of possible
renormalized operators,
00:59:24.420 --> 00:59:27.420
and there's some matrix of
counterterms that would correct
00:59:27.420 --> 00:59:28.020
them--
00:59:28.020 --> 00:59:31.030
connect them.
00:59:31.030 --> 00:59:31.570
OK.
00:59:31.570 --> 00:59:33.120
So any questions so far?
00:59:37.030 --> 00:59:40.480
So sometimes, in the literature,
you'll see that these kinds
00:59:40.480 --> 00:59:44.050
of things-- regulator
discussions in effective field
00:59:44.050 --> 00:59:44.710
theory--
00:59:44.710 --> 00:59:47.860
generate all sorts of papers.
00:59:47.860 --> 00:59:50.410
Keep this in mind if
you ever run into that.
01:00:01.045 --> 01:00:02.920
You should be able to
think about the physics
01:00:02.920 --> 01:00:04.712
that you're after with
different regulators
01:00:04.712 --> 01:00:07.500
and come to the same conclusion.
01:00:07.500 --> 01:00:12.140
And it just may be easier with
one regulator versus another.
01:00:12.140 --> 01:00:14.112
So we've said, in
these kind of theories
01:00:14.112 --> 01:00:15.820
where we have dimensional
power counting,
01:00:15.820 --> 01:00:21.040
that dimensional
regularization is special.
01:00:21.040 --> 01:00:23.020
So I want to talk
a bit more about
01:00:23.020 --> 01:00:24.739
dimensional regularization.
01:00:45.123 --> 01:00:47.540
So sometimes you'll hear people
say that you should always
01:00:47.540 --> 01:00:51.130
use dimensional regularization
for doing the power counting.
01:00:51.130 --> 01:00:55.250
But that's not quite true in
the way that I've told you.
01:00:55.250 --> 01:00:56.750
It's not that you
have to, it's just
01:00:56.750 --> 01:00:58.940
that it makes things simpler.
01:00:58.940 --> 01:01:01.650
But given that we want to make
things as simple as possible,
01:01:01.650 --> 01:01:06.170
let's take dimensional
regularization seriously.
01:01:06.170 --> 01:01:08.540
So you can actually derive
dimensional regularization
01:01:08.540 --> 01:01:11.420
by just imposing axioms.
01:01:11.420 --> 01:01:16.445
If you say that you want a loop
integration that's linear--
01:01:22.260 --> 01:01:24.240
I should have said this earlier.
01:01:24.240 --> 01:01:26.190
So my notation with
dimensional regularization
01:01:26.190 --> 01:01:27.607
is, I put a little
cross on the d.
01:01:27.607 --> 01:01:30.640
And that means
dividing by the 2 pi.
01:01:30.640 --> 01:01:36.720
So that means d d p
over 2 pi to the d.
01:01:36.720 --> 01:01:39.720
So linearity means that if I'm
integrating some function that
01:01:39.720 --> 01:01:42.360
can be decomposed into
a sum of two pieces, a
01:01:42.360 --> 01:01:48.680
and b being constants,
f and g being functions,
01:01:48.680 --> 01:01:55.030
then I can write that
out as an integral over f
01:01:55.030 --> 01:01:59.590
plus an integral over g,
which really is something
01:01:59.590 --> 01:02:05.948
that almost every reasonable
definition of the integration
01:02:05.948 --> 01:02:06.490
will satisfy.
01:02:12.090 --> 01:02:18.660
The second one is translations,
which is more restricting.
01:02:18.660 --> 01:02:21.240
So that says, if you have
some integral over f,
01:02:21.240 --> 01:02:24.330
but it's a function of p
plus q-- q is some external--
01:02:24.330 --> 01:02:30.345
I can always shift away the q.
01:02:30.345 --> 01:02:32.205
It just goes-- p
goes to p minus q.
01:02:32.205 --> 01:02:34.013
And then I just have
an integral over p.
01:02:37.120 --> 01:02:38.800
And along with
translations, you can
01:02:38.800 --> 01:02:40.990
think about having rotations.
01:02:40.990 --> 01:02:44.560
My whole notation is covariance,
so we won't worry so much
01:02:44.560 --> 01:02:46.210
about rotations.
01:02:46.210 --> 01:02:47.340
and Lorentz group.
01:02:51.880 --> 01:02:56.560
And then the final
one that's obviously
01:02:56.560 --> 01:03:00.370
a little bit special to
dim reg is a scaling.
01:03:00.370 --> 01:03:04.150
So let's say we have a scalar
s multiplying our momentum p.
01:03:07.610 --> 01:03:12.290
Then we can rescale the
momentum p and get rid of--
01:03:12.290 --> 01:03:18.410
pull the s outside by just
taking p goes to p over s.
01:03:18.410 --> 01:03:20.000
So that changes this to a p.
01:03:20.000 --> 01:03:22.336
We get an s to the minus d.
01:03:22.336 --> 01:03:25.140
It pulls out front.
01:03:25.140 --> 01:03:29.210
And if we demand
that, then that's
01:03:29.210 --> 01:03:30.830
special to dimensional
regularization,
01:03:30.830 --> 01:03:34.760
because you can see
that this depends on d.
01:03:34.760 --> 01:03:37.340
Even if I call this measure
some abstract thing,
01:03:37.340 --> 01:03:38.910
now there's a d showing up.
01:03:38.910 --> 01:03:40.130
It's outside the measure.
01:03:45.600 --> 01:03:48.480
And these three
together actually give
01:03:48.480 --> 01:03:52.410
a unique definition
to the integration up
01:03:52.410 --> 01:03:54.285
to the overall normalization.
01:04:07.160 --> 01:04:09.263
And that unique
thing is dim reg.
01:04:12.890 --> 01:04:17.150
So I'm going to
refer you to reading,
01:04:17.150 --> 01:04:19.640
I have posted a chapter
from Collins' book
01:04:19.640 --> 01:04:21.410
on regularization.
01:04:21.410 --> 01:04:24.860
And around page 65, he talks
about how you prove that.
01:04:24.860 --> 01:04:25.730
It's not too hard.
01:04:34.310 --> 01:04:36.970
The standard definition
of the normalization,
01:04:36.970 --> 01:04:41.184
which is something
you have to specify,
01:04:41.184 --> 01:04:44.520
is that you let,
say, this Gaussian
01:04:44.520 --> 01:04:46.140
integral be pi to the d over 2.
01:04:50.580 --> 01:04:55.770
So then, from that, you have
some measure in some space
01:04:55.770 --> 01:04:56.895
that you can then just use.
01:05:01.220 --> 01:05:03.350
So one formula that
I used earlier on
01:05:03.350 --> 01:05:08.720
was the ability to
split that into pieces
01:05:08.720 --> 01:05:13.160
which were a radial piece
and then an angular piece.
01:05:13.160 --> 01:05:15.310
And in general,
this is a property
01:05:15.310 --> 01:05:17.060
that this integration
[? measure obeys, ?]
01:05:17.060 --> 01:05:19.393
that you could split it, and
you could split it further.
01:05:19.393 --> 01:05:28.670
You could pull out another
angle, for example,
01:05:28.670 --> 01:05:34.810
and get one less dimension
in the angular parts.
01:05:34.810 --> 01:05:38.110
And the uv divergences, if we're
talking about us divergences--
01:05:38.110 --> 01:05:40.500
they're occurring in
this radial part--
01:05:40.500 --> 01:05:41.920
Euclidean radial part.
01:05:48.070 --> 01:05:50.950
So by thinking about this
kind of decomposition,
01:05:50.950 --> 01:05:53.620
you're moving the
uv divergences to
01:05:53.620 --> 01:05:57.280
a one-dimensional integration,
at least at one loop.
01:05:57.280 --> 01:05:59.635
And you can always do that.
01:05:59.635 --> 01:06:01.510
So in general, in
dimensional regularization,
01:06:01.510 --> 01:06:03.220
there's many ways you could
evaluate the integral.
01:06:03.220 --> 01:06:04.750
You're not used
to using this one.
01:06:04.750 --> 01:06:07.030
You're used to keeping
things covariant,
01:06:07.030 --> 01:06:11.770
using some Feynman parameters,
combining propagators together,
01:06:11.770 --> 01:06:14.330
and then doing the integral.
01:06:14.330 --> 01:06:16.450
But you could also
do it this way,
01:06:16.450 --> 01:06:19.070
and you get the same answer.
01:06:19.070 --> 01:06:22.355
So it's really a well-defined
measure in the sense
01:06:22.355 --> 01:06:24.230
that you can manipulate
it in different ways.
01:06:24.230 --> 01:06:26.860
And they should all lead to
the same answer for your loop
01:06:26.860 --> 01:06:27.769
integrals.
01:06:31.763 --> 01:06:33.180
And that's part
of what I'm trying
01:06:33.180 --> 01:06:37.110
to emphasize by saying
that you could derive it
01:06:37.110 --> 01:06:38.295
by considering axioms.
01:06:46.740 --> 01:06:49.800
So d was equal to
4 minus 2 epsilon.
01:06:49.800 --> 01:06:53.070
Epsilon greater
than 0 is what you
01:06:53.070 --> 01:06:57.554
need to lower the powers of
p, and therefore tame the uv.
01:06:57.554 --> 01:07:03.630
Epsilon less than 0
can be used to regulate
01:07:03.630 --> 01:07:05.307
infrared singularities.
01:07:10.542 --> 01:07:12.000
There's some
counterintuitive facts
01:07:12.000 --> 01:07:13.710
about dimensional
regularization,
01:07:13.710 --> 01:07:15.730
and I want to mention a
couple of them to you.
01:07:19.160 --> 01:07:23.280
One of them is that, if I
have p to an arbitrary power--
01:07:23.280 --> 01:07:24.650
think of it as Euclidean--
01:07:27.270 --> 01:07:27.870
that's 0.
01:07:38.970 --> 01:07:43.950
So Collins constructs
a proof of this
01:07:43.950 --> 01:07:47.970
on page 71, which is actually
a little more involved,
01:07:47.970 --> 01:07:50.280
in general.
01:07:50.280 --> 01:07:54.270
I'll just give you an idea
of how you can see that,
01:07:54.270 --> 01:07:56.850
from using our axioms,
that something like this
01:07:56.850 --> 01:07:58.810
better be true.
01:07:58.810 --> 01:08:01.980
So let's consider
a special example
01:08:01.980 --> 01:08:04.530
that won't be enough to
prove it for arbitrary alpha.
01:08:04.530 --> 01:08:07.800
This is any alpha.
01:08:07.800 --> 01:08:09.450
Let's consider a
special example that
01:08:09.450 --> 01:08:13.140
at least will be enough
to prove it for integers.
01:08:16.359 --> 01:08:19.729
So we'll consider k's
that are integers.
01:08:19.729 --> 01:08:23.550
And we'll think of k's
that are greater than 0.
01:08:23.550 --> 01:08:32.240
So if I just expand out
this p plus q squared,
01:08:32.240 --> 01:08:35.640
then the first term
is p to the 2k.
01:08:35.640 --> 01:08:39.900
Then I get some coefficient,
p to the 2k minus 2,
01:08:39.900 --> 01:08:46.500
q squared, some coefficient,
p to the 2k minus 4,
01:08:46.500 --> 01:08:50.550
q to the fourth, et cetera.
01:08:50.550 --> 01:08:52.600
In general, there's p
dot q terms as well.
01:08:52.600 --> 01:08:54.830
But then I could do integral--
01:08:54.830 --> 01:08:56.880
angular average and
combine those together
01:08:56.880 --> 01:08:58.840
with these terms.
01:08:58.840 --> 01:09:01.140
And that's why I'm not being
very explicit about what
01:09:01.140 --> 01:09:03.069
the coefficients are.
01:09:03.069 --> 01:09:06.367
But they're some
positive numbers.
01:09:06.367 --> 01:09:08.700
Now, I could also take this
integral, and I could shift.
01:09:08.700 --> 01:09:09.825
That was one of our axioms.
01:09:18.192 --> 01:09:21.359
And that is p to the 2k.
01:09:21.359 --> 01:09:25.200
So that means that all these
terms here better be 0.
01:09:25.200 --> 01:09:41.279
And they have to be 0 for
arbitrary q and arbitrary k,
01:09:41.279 --> 01:09:43.478
or any k under the
assumptions that we used,
01:09:43.478 --> 01:09:45.520
which are that it's an
integer and it's positive.
01:09:45.520 --> 01:09:47.979
So I could expand
it in this way.
01:09:47.979 --> 01:09:50.790
And so therefore, we have
all these integrals over p
01:09:50.790 --> 01:09:51.450
to the powers.
01:09:51.450 --> 01:09:52.920
And they better be 0.
01:09:52.920 --> 01:09:56.610
And that's enough to prove
this for integer alphas--
01:09:56.610 --> 01:09:57.780
positive integer alphas.
01:10:08.660 --> 01:10:10.280
OK.
01:10:10.280 --> 01:10:12.380
And so if you want to fill
in between the integers
01:10:12.380 --> 01:10:14.030
and you want to do
the negative cases,
01:10:14.030 --> 01:10:15.405
then you have to
look at Collins.
01:10:15.405 --> 01:10:18.575
But you could do that, too.
01:10:18.575 --> 01:10:20.450
Then it requires a little
more heavy lifting.
01:10:28.000 --> 01:10:31.000
There's one fact about this--
01:10:31.000 --> 01:10:33.490
fact number one--
which is a little bit
01:10:33.490 --> 01:10:36.580
subtle, and you
have to be careful.
01:10:36.580 --> 01:10:39.500
And it's worth noting.
01:10:39.500 --> 01:10:42.910
So let me do another
example, which
01:10:42.910 --> 01:10:46.120
is by way of warning you
that this can sometimes
01:10:46.120 --> 01:10:48.080
be dangerous.
01:10:48.080 --> 01:10:50.740
So let's think of a
scalar-field theory
01:10:50.740 --> 01:10:52.810
and a simple loop
diagram like this.
01:10:52.810 --> 01:10:57.055
But let's take 0
momentum and 0 mass.
01:11:00.032 --> 01:11:01.990
So if you do that, you'll
encounter an integral
01:11:01.990 --> 01:11:03.620
that looks like this.
01:11:03.620 --> 01:11:05.290
There's two propagators,
so I get a p
01:11:05.290 --> 01:11:10.240
to the fourth downstairs,
and I get d d p.
01:11:10.240 --> 01:11:15.820
And that integral is 0, but
it's 0 in a special way.
01:11:15.820 --> 01:11:18.820
It's 0 due to a cancellation
between ultraviolet and
01:11:18.820 --> 01:11:21.850
infrared physics.
01:11:21.850 --> 01:11:23.830
I said that epsilon
could be regulating
01:11:23.830 --> 01:11:25.450
both infrared
divergences as well
01:11:25.450 --> 01:11:27.790
as ultraviolet divergences.
01:11:27.790 --> 01:11:29.620
If I only used
epsilon to regulate
01:11:29.620 --> 01:11:33.190
ultraviolet divergences,
I'd get a 1-over-epsilon uv.
01:11:33.190 --> 01:11:36.152
But in this integral, I'm
actually using it to do both.
01:11:36.152 --> 01:11:38.110
It's regulating an infrared
divergence as well.
01:11:38.110 --> 01:11:39.770
And it just comes in
with the opposite sign.
01:11:39.770 --> 01:11:41.645
And since epsilon uv is
equal to epsilon IR--
01:11:44.950 --> 01:11:50.080
they're just notation to
signify what region of physics
01:11:50.080 --> 01:11:51.730
is giving the divergence--
01:11:51.730 --> 01:11:54.405
you get 0.
01:11:54.405 --> 01:11:55.780
But even though
that's true, that
01:11:55.780 --> 01:11:59.140
doesn't mean you don't
have to add a counterterm
01:11:59.140 --> 01:12:01.690
for this diagram, because
counterterms are supposed
01:12:01.690 --> 01:12:06.550
to cancel ultraviolet
divergences, not infrared ones,
01:12:06.550 --> 01:12:08.800
OK?
01:12:08.800 --> 01:12:16.870
So even though it's 0, you
still need to add a counterterm,
01:12:16.870 --> 01:12:18.910
because the 0 is
actually a cancellation
01:12:18.910 --> 01:12:22.760
between ultraviolet
and infrared physics.
01:12:22.760 --> 01:12:25.435
So there's some counterterm.
01:12:28.230 --> 01:12:31.620
And it would be
exactly of this sort,
01:12:31.620 --> 01:12:34.670
because this is the epsilon uv.
01:12:38.220 --> 01:12:44.160
And then if you add the bare
diagram plus the counterterm,
01:12:44.160 --> 01:12:47.460
the answer is non-zero.
01:12:47.460 --> 01:12:50.040
You've canceled, if
you like, the uv pole,
01:12:50.040 --> 01:12:53.130
and you've left
over the IR pole.
01:12:53.130 --> 01:12:53.630
OK.
01:12:53.630 --> 01:12:55.490
So you have to be a
bit careful about using
01:12:55.490 --> 01:12:57.140
dimensional regularization,
because if you encounter
01:12:57.140 --> 01:12:59.265
scaleless integrals, it
could be that they actually
01:12:59.265 --> 01:13:01.410
are affecting counterterm.
01:13:01.410 --> 01:13:03.740
And if you want to do
some renormalization-group
01:13:03.740 --> 01:13:06.020
improvement of the theory
or something like that,
01:13:06.020 --> 01:13:08.970
you have to be aware of this.
01:13:08.970 --> 01:13:11.420
If you know that all the
infrared poles are going
01:13:11.420 --> 01:13:14.555
to cancel because you're looking
at some infrared safe quantity,
01:13:14.555 --> 01:13:16.430
then you can be a little
bit glib about this,
01:13:16.430 --> 01:13:18.347
because if they're going
to cancel in the end,
01:13:18.347 --> 01:13:21.560
that means that any of
these corresponding uv poles
01:13:21.560 --> 01:13:23.570
will also cancel.
01:13:23.570 --> 01:13:26.180
But it's not always the case
that you're renormalizing
01:13:26.180 --> 01:13:31.010
operators that have no
1-over-epsilon IR's [INAUDIBLE]
01:13:31.010 --> 01:13:33.300
have to be careful about this.
01:13:33.300 --> 01:13:35.990
So this is a subtlety that
sometimes people get wrong
01:13:35.990 --> 01:13:38.960
when they write papers.
01:13:38.960 --> 01:13:41.450
So dim reg is beautiful, but
there are some things about it
01:13:41.450 --> 01:13:42.658
that are a little bit tricky.
01:13:46.600 --> 01:13:51.070
Another thing that can be
confusing about dim reg
01:13:51.070 --> 01:13:58.053
is that it does this, that it
regulates both uv and IR poles.
01:13:58.053 --> 01:14:00.220
And even though it's doing
that, and even though you
01:14:00.220 --> 01:14:04.090
need different
values of epsilon,
01:14:04.090 --> 01:14:08.500
if you want to do
that, it's actually
01:14:08.500 --> 01:14:10.450
still a well-defined
procedure, even
01:14:10.450 --> 01:14:13.260
in the presence of
uv and IR poles,
01:14:13.260 --> 01:14:15.010
even if they're both
in the same integral.
01:14:17.680 --> 01:14:24.190
And basically, you're using
analytic continuation here.
01:14:28.930 --> 01:14:31.590
So let me give you a
little example which
01:14:31.590 --> 01:14:37.570
is not exactly related to this
but will allow me to show you
01:14:37.570 --> 01:14:39.670
both how you use
analytic continuation
01:14:39.670 --> 01:14:43.630
and how you could think about
separating uv and IR poles.
01:14:43.630 --> 01:14:45.520
So we'll start out
just by thinking
01:14:45.520 --> 01:14:47.710
about analytic continuation.
01:14:47.710 --> 01:14:49.240
So suppose I had some integral.
01:14:57.320 --> 01:14:59.620
So what does analytic
continuation mean?
01:14:59.620 --> 01:15:02.340
Or, how should I construct it?
01:15:02.340 --> 01:15:04.850
So let me, again, write it
in a way where I've separated
01:15:04.850 --> 01:15:05.870
out the radial integral.
01:15:10.250 --> 01:15:12.290
And let's suppose that
this integral here
01:15:12.290 --> 01:15:17.130
is perfectly well-defined
for d in some range.
01:15:17.130 --> 01:15:29.360
So this is well-defined for
some range of positive d.
01:15:29.360 --> 01:15:32.390
And then let's say that
we wanted to continue
01:15:32.390 --> 01:15:33.680
that integral to negative d.
01:15:42.705 --> 01:15:44.450
Now, the problem
with negative d may
01:15:44.450 --> 01:15:46.160
be that, when you
get to negative d,
01:15:46.160 --> 01:15:48.438
you're getting some
infrared divergences,
01:15:48.438 --> 01:15:50.480
and you have to figure
out how to deal with them.
01:15:58.340 --> 01:16:01.600
And you can do this, if
you'd like, step by step.
01:16:04.340 --> 01:16:06.820
So if we wanted to extend the
lower limit down to minus 2
01:16:06.820 --> 01:16:09.959
from 0, then we would
do the following.
01:16:14.650 --> 01:16:17.740
We take our integral,
write it out
01:16:17.740 --> 01:16:27.730
in this angular/radial
separation,
01:16:27.730 --> 01:16:33.700
split up, in the radial
variable, the piece that's
01:16:33.700 --> 01:16:36.623
ultraviolet, which is
the high-momentum piece,
01:16:36.623 --> 01:16:37.915
from the piece that's infrared.
01:16:50.975 --> 01:16:52.350
And in the piece
that's infrared,
01:16:52.350 --> 01:16:55.290
we could also just do some
addition and some subtraction
01:16:55.290 --> 01:16:57.840
to make it more
convergent as p goes to 0.
01:16:57.840 --> 01:17:00.600
So for example, we
could subtract f at 0.
01:17:00.600 --> 01:17:03.810
This thing would fall off
faster and hence, give
01:17:03.810 --> 01:17:05.110
more powers of p.
01:17:05.110 --> 01:17:09.300
And we could make it
more convergent at 0.
01:17:09.300 --> 01:17:15.300
And then we just integrate the
subtraction up to the cutoff c.
01:17:17.950 --> 01:17:20.950
And the idea of
introducing this cutoff c
01:17:20.950 --> 01:17:23.680
is that we split the uv piece
and the ultraviolet piece.
01:17:27.470 --> 01:17:29.650
And so we can do one
kind of continuation
01:17:29.650 --> 01:17:32.860
for d up here, making
epsilon positive,
01:17:32.860 --> 01:17:34.700
one kind of
continuation down here.
01:17:34.700 --> 01:17:36.700
And the result, when we
put these back together,
01:17:36.700 --> 01:17:39.370
is independent of c, OK?
01:17:39.370 --> 01:17:43.210
So that's the sense, actually,
in which what I said up here
01:17:43.210 --> 01:17:43.750
is--
01:17:43.750 --> 01:17:46.090
that you can use it for
both uv and IR divergences,
01:17:46.090 --> 01:17:48.132
because you could always
introduce some parameter
01:17:48.132 --> 01:17:48.932
c to split--
01:17:48.932 --> 01:17:51.140
they're occurring in different
regions of space base,
01:17:51.140 --> 01:17:56.740
so you could always split
them up, regulate each one
01:17:56.740 --> 01:18:00.680
with different values of d, and
then put them back together.
01:18:00.680 --> 01:18:03.370
And the answer, when you
put them back together,
01:18:03.370 --> 01:18:04.360
is independent of c.
01:18:08.040 --> 01:18:08.970
OK.
01:18:08.970 --> 01:18:11.100
Now, if you wanted to define--
01:18:11.100 --> 01:18:12.953
that was one of our goals.
01:18:12.953 --> 01:18:14.370
The other one was
just to show you
01:18:14.370 --> 01:18:16.980
what we mean by
analytic continuation.
01:18:16.980 --> 01:18:19.290
So since it's independent
of c, then you
01:18:19.290 --> 01:18:21.970
could do the following.
01:18:21.970 --> 01:18:27.390
So for minus 2 less
than d less than 0,
01:18:27.390 --> 01:18:30.410
let's take c goes to infinity.
01:18:33.490 --> 01:18:35.640
And so for that
particular range, what you
01:18:35.640 --> 01:18:38.340
find, then, if you
take that limit--
01:18:55.540 --> 01:18:57.430
Well, because of the
c to the minus d,
01:18:57.430 --> 01:18:59.320
the c is downstairs
if d is negative.
01:18:59.320 --> 01:19:03.550
So that term goes
away, and you're just
01:19:03.550 --> 01:19:06.860
left with this term.
01:19:06.860 --> 01:19:08.870
This term here goes
away, too, because c
01:19:08.870 --> 01:19:12.080
approaches the upper limit,
and everything is regulated.
01:19:12.080 --> 01:19:15.750
And so I just would
be left with that.
01:19:15.750 --> 01:19:16.250
OK.
01:19:16.250 --> 01:19:17.480
Making d negative is--
01:19:20.220 --> 01:19:20.720
OK.
01:19:20.720 --> 01:19:22.480
So that would be the definition.
01:19:22.480 --> 01:19:24.188
So you can see some
of the kind of tricks
01:19:24.188 --> 01:19:26.272
that you could use with
dimensional regularization
01:19:26.272 --> 01:19:27.680
or adding and subtracting terms.
01:19:30.780 --> 01:19:33.050
And these are the things
that are valid things to do.
01:19:39.580 --> 01:19:40.800
Any questions about that?
01:19:43.970 --> 01:19:45.410
OK.
01:19:45.410 --> 01:19:48.740
So when we do dimensional
regularization in MS bar,
01:19:48.740 --> 01:19:50.720
you're used to doing
that for a gauge theory.
01:19:50.720 --> 01:19:53.420
That's what you've
learned about.
01:19:53.420 --> 01:19:55.670
But you can also do it for
any effective field theory.
01:19:55.670 --> 01:19:57.870
And the logic is the same.
01:19:57.870 --> 01:20:00.770
So let me remind you of
the gauge-theory logic
01:20:00.770 --> 01:20:04.490
and then just tell you how you
would define MS bar precisely
01:20:04.490 --> 01:20:08.690
for the fermionic effective
theory with the "psi bar psi
01:20:08.690 --> 01:20:12.380
squared" operator that we had.
01:20:12.380 --> 01:20:16.560
So we talked about
dimensional regularization.
01:20:16.560 --> 01:20:18.190
Let's talk about the MS scheme.
01:20:30.530 --> 01:20:34.550
So if we've set up our
effective theory in a way
01:20:34.550 --> 01:20:39.510
where we've made the
mass scale explicit,
01:20:39.510 --> 01:20:41.135
which is often a
nice thing to do--
01:20:41.135 --> 01:20:43.260
and we did that when we
set up the effective theory
01:20:43.260 --> 01:20:46.303
where you have the capital
M showing up explicitly.
01:20:46.303 --> 01:20:47.720
If you do that,
then the couplings
01:20:47.720 --> 01:20:49.370
start out dimensionless.
01:20:49.370 --> 01:20:52.640
And that's a nice thing, to
have dimensionless coupling.
01:20:52.640 --> 01:20:54.290
And the MS scheme
is simply the scheme
01:20:54.290 --> 01:20:56.150
where you want to
introduce a scale
01:20:56.150 --> 01:20:59.520
to keep the renormalized
couplings dimensionless.
01:20:59.520 --> 01:21:04.490
So the example
you're familiar with
01:21:04.490 --> 01:21:06.370
is just having a gauge coupling.
01:21:09.817 --> 01:21:11.900
And if you go through the
dimensions of the fields
01:21:11.900 --> 01:21:14.450
here, which I do in
my notes, but I'm
01:21:14.450 --> 01:21:17.900
going to assume that you've
got some familiarity with this,
01:21:17.900 --> 01:21:22.520
you find that the bare
coupling has dimension epsilon.
01:21:22.520 --> 01:21:25.618
And so you define a renormalized
coupling as dimensionless.
01:21:29.675 --> 01:21:31.675
And you introduce a factor
of mu to the epsilon.
01:21:35.367 --> 01:21:39.790
So you say g bare,
which has dimensions,
01:21:39.790 --> 01:21:43.540
some dimensionless z factor,
some mu to the epsilon
01:21:43.540 --> 01:21:44.750
to make up those dimensions.
01:21:44.750 --> 01:21:47.710
And then left over is the
renormalized coupling.
01:21:54.210 --> 01:21:56.520
And the idea and the strategy
for any other coupling
01:21:56.520 --> 01:21:58.020
in the effective
theory is the same,
01:21:58.020 --> 01:22:00.400
so let's do one other example.
01:22:00.400 --> 01:22:07.698
So take our [? dimension-six ?]
a bare over capital M squared,
01:22:07.698 --> 01:22:10.980
psi bar psi squared.
01:22:10.980 --> 01:22:17.817
Do dimension counting
on this guy, which
01:22:17.817 --> 01:22:18.900
I do, again, in the notes.
01:22:18.900 --> 01:22:21.600
But if you go through
that dimension
01:22:21.600 --> 01:22:24.270
counting, and you remember that
the dimensions for the fermions
01:22:24.270 --> 01:22:26.498
are assigned by
the kinetic term--
01:22:26.498 --> 01:22:28.540
so we have a dimension
counting for the fermions.
01:22:28.540 --> 01:22:31.050
We know that this is
dimension minus 2.
01:22:31.050 --> 01:22:32.670
The whole thing
has to add up to d.
01:22:32.670 --> 01:22:34.950
So that tells us
what the a bare is.
01:22:34.950 --> 01:22:38.390
And we get 4 minus
d, in this case.
01:22:38.390 --> 01:22:45.000
And so then we can write down
a formula analogous to that one
01:22:45.000 --> 01:22:47.490
but for the a coefficient.
01:22:47.490 --> 01:22:48.705
And it's mu to the 2 epsilon.
01:22:53.540 --> 01:22:54.050
OK.
01:22:54.050 --> 01:22:56.060
So it's as simple as that.
01:23:14.010 --> 01:23:15.560
So in looking at
the action where
01:23:15.560 --> 01:23:17.180
this is a term in
the Lagrangian,
01:23:17.180 --> 01:23:19.520
we want to ensure that,
when we go to the--
01:23:19.520 --> 01:23:21.440
that we figure out the
dimensions of this guy
01:23:21.440 --> 01:23:23.120
in dimensional regularization.
01:23:23.120 --> 01:23:26.233
That's 4 minus d, which
we determine from knowing
01:23:26.233 --> 01:23:27.150
the other pieces here.
01:23:27.150 --> 01:23:30.840
And then we just
make a redefinition
01:23:30.840 --> 01:23:33.810
to give a
dimensionless coupling.
01:23:33.810 --> 01:23:36.450
So that's how MS bar will work.
01:23:36.450 --> 01:23:40.920
Well, this is MS, but this
is how minimal subtraction
01:23:40.920 --> 01:23:44.100
works for defining all the
operators that you may have.
01:23:49.990 --> 01:23:55.510
And then minimal subtraction
is simply a rescaling.
01:23:55.510 --> 01:24:00.070
And that's the same as it is in
gauge theory, where we get rid
01:24:00.070 --> 01:24:03.690
of some annoying factors,
and [? g is ?] a slightly
01:24:03.690 --> 01:24:05.930
different definition.
01:24:05.930 --> 01:24:09.670
So this was MS,
and this is MS bar.
01:24:09.670 --> 01:24:10.290
OK.
01:24:10.290 --> 01:24:13.410
So it really works in a very
similar way to gauge theory.
01:24:13.410 --> 01:24:15.450
And you figure out
the factors of mu
01:24:15.450 --> 01:24:18.270
to the epsilon to include in
your calculation this way.
01:24:18.270 --> 01:24:20.520
Sometimes you see
books do it by saying
01:24:20.520 --> 01:24:24.060
the loop measure is continued
within mu to the epsilon.
01:24:24.060 --> 01:24:25.650
That's not right.
01:24:25.650 --> 01:24:27.810
This is right.
01:24:27.810 --> 01:24:31.200
If you do that, you'll
get into trouble--
01:24:31.200 --> 01:24:32.040
not always.
01:24:32.040 --> 01:24:33.790
That's why the books
can get away with it.
01:24:33.790 --> 01:24:37.020
But in general, you'll
get into trouble.
01:24:37.020 --> 01:24:37.590
All right.
01:24:37.590 --> 01:24:39.607
So we should stop there.