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PROFESSOR: All right,
so last time we

00:00:28.940 --> 00:00:30.830
were talking about
symmetries of SCET.

00:00:30.830 --> 00:00:33.800
We talked in detail
about gauge symmetry,

00:00:33.800 --> 00:00:36.350
and we started to talk about
reparameterization invariance,

00:00:36.350 --> 00:00:38.225
and we are going to
continue with that today.

00:00:40.620 --> 00:00:42.960
So reparameterization
invariance in SCET

00:00:42.960 --> 00:00:45.150
is kind of rich,
because we have lots

00:00:45.150 --> 00:00:47.610
of things that are messing
up Lorentz invariance.

00:00:47.610 --> 00:00:50.970
But we started out talking about
three different types here.

00:00:50.970 --> 00:00:53.843
One where we rotate the n,
the physical n vector that

00:00:53.843 --> 00:00:55.260
was the direction
of our collinear

00:00:55.260 --> 00:00:58.230
particles to some other vector
that's equally good, which

00:00:58.230 --> 00:01:00.140
I call n prime in the figure.

00:01:00.140 --> 00:01:02.460
So a type one, transformation.

00:01:02.460 --> 00:01:04.650
Or we could change this
auxiliary vector, n bar,

00:01:04.650 --> 00:01:06.240
by a large amount.

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That was type two.

00:01:07.470 --> 00:01:10.560
Or we could change both
at n and n bar in a way

00:01:10.560 --> 00:01:13.050
that dot products
remain invariant.

00:01:13.050 --> 00:01:14.430
That's type three.

00:01:14.430 --> 00:01:16.980
So all three of these
preserved these relations,

00:01:16.980 --> 00:01:19.500
n squared equals 0, n
bar squared equals 0.

00:01:19.500 --> 00:01:21.030
n dot n bar equals 2.

00:01:21.030 --> 00:01:22.530
And really, these
transformations

00:01:22.530 --> 00:01:25.900
are just giving us a different
basis for our decomposition,

00:01:25.900 --> 00:01:28.500
which satisfies all the things
that we wanted that basis

00:01:28.500 --> 00:01:29.670
to satisfy.

00:01:29.670 --> 00:01:32.500
Doesn't change any of
our power counting,

00:01:32.500 --> 00:01:34.860
and so is an equally
good description

00:01:34.860 --> 00:01:36.940
for the effective theory.

00:01:36.940 --> 00:01:39.270
And therefore, we want
to have invariance

00:01:39.270 --> 00:01:43.450
under these transformations.

00:01:43.450 --> 00:01:46.480
So we're restoring Lorentz
symmetry in this way.

00:01:46.480 --> 00:01:49.260
We break it by
introducing these vectors,

00:01:49.260 --> 00:01:52.830
but we're restoring it in a
way by having transformations

00:01:52.830 --> 00:01:54.960
on these factors.

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But it's not, you're
not restoring,

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if you like, it's still
different than Lorentz

00:01:59.970 --> 00:02:01.800
symmetry, for example,
because you're not

00:02:01.800 --> 00:02:03.750
making huge transformations
of the vector n

00:02:03.750 --> 00:02:07.860
here for any arbitrary
transformation.

00:02:07.860 --> 00:02:09.620
And it is a
reparameterization not

00:02:09.620 --> 00:02:14.230
a Lorentz symmetry in general.

00:02:14.230 --> 00:02:17.970
So we'll talk more
about how this connects

00:02:17.970 --> 00:02:19.390
to the HQET one in a minute.

00:02:19.390 --> 00:02:24.522
But if we have a vector, then
obviously the vector, p mu--

00:02:24.522 --> 00:02:25.980
I don't write the
right hand side--

00:02:25.980 --> 00:02:30.120
is invariant to what
choice of basis I use.

00:02:30.120 --> 00:02:33.010
So that means that this
p mu should not change.

00:02:33.010 --> 00:02:36.150
And that means if
I change n here,

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let's say I do a type one,
then I'll change n here

00:02:38.430 --> 00:02:40.500
and I'll change n here.

00:02:40.500 --> 00:02:43.350
But I must compensate those
changes by changing what--

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perp also depended
on the meaning of n,

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so there will be a compensating
change for the perp

00:02:48.390 --> 00:02:51.930
So we can figure out from
statements like this what

00:02:51.930 --> 00:02:54.400
the transformation laws are.

00:02:54.400 --> 00:02:56.670
So this guy is invariant
to the decomposition.

00:03:05.310 --> 00:03:13.370
So the transformation
of p perp mu

00:03:13.370 --> 00:03:24.777
compensates for n in type
one, or n bar in type two.

00:03:24.777 --> 00:03:26.360
Type 3 is already
invariant because we

00:03:26.360 --> 00:03:29.930
have an n and an n bar here,
an n and an n bar here.

00:03:29.930 --> 00:03:33.570
So that's type three
invariant already.

00:03:33.570 --> 00:03:35.630
So if we go through
that, we can make

00:03:35.630 --> 00:03:39.560
a table of all the
different transformations,

00:03:39.560 --> 00:03:41.930
and we can derive
the transformations.

00:03:41.930 --> 00:03:46.040
We also have to do the same
thing that we did in HQET,

00:03:46.040 --> 00:03:48.080
because we also-- the
other important fact

00:03:48.080 --> 00:03:52.430
is that we have a projection
relation out here.

00:03:58.510 --> 00:04:01.308
So we also have this,
and you'll remember

00:04:01.308 --> 00:04:03.600
that when we talked about
reparameterization invariance

00:04:03.600 --> 00:04:05.880
and HQET, the
projection relation was

00:04:05.880 --> 00:04:09.962
part of the discussion,
and so that's true too.

00:04:09.962 --> 00:04:11.670
But let me just quote
to you the results,

00:04:11.670 --> 00:04:13.860
that you could kind of get
an impression for where

00:04:13.860 --> 00:04:17.980
these various terms
are coming from.

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So let's summarize,
first, type one.

00:04:20.880 --> 00:04:24.900
So type one was n goes
to n plus delta perp.

00:04:24.900 --> 00:04:26.670
So if you have
something like n dot D,

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That goes to n dot D
plus delta perp dot D,

00:04:32.400 --> 00:04:35.160
there's a transformation
of p perp or D perp.

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Let me write it as D perp.

00:04:37.700 --> 00:04:44.270
This is really for
any vector which

00:04:44.270 --> 00:04:48.350
has to compensate for the
transformation of the n

00:04:48.350 --> 00:04:49.370
in the way I described.

00:04:55.890 --> 00:04:59.770
There's a transformation
like that.

00:04:59.770 --> 00:05:04.610
The n bar component under
type one does not transform.

00:05:04.610 --> 00:05:08.120
And if you go
through the spinner,

00:05:08.120 --> 00:05:13.100
or you go through the field,
the fermion field, it does get--

00:05:13.100 --> 00:05:16.050
because the projection
relation has a transformation,

00:05:16.050 --> 00:05:21.770
there's a transformation
of this guy.

00:05:21.770 --> 00:05:24.230
The Wilson line,
which we talked about,

00:05:24.230 --> 00:05:26.930
was only a function
of M bar dot A,

00:05:26.930 --> 00:05:28.550
and so it doesn't
get transformed.

00:05:31.070 --> 00:05:32.720
So that's what type
one transformations

00:05:32.720 --> 00:05:33.387
would look like.

00:05:37.390 --> 00:05:38.280
And type two--

00:05:51.540 --> 00:05:54.238
Summarize.

00:05:54.238 --> 00:05:56.280
They start out looking
the same, and then there's

00:05:56.280 --> 00:05:57.180
some differences.

00:06:17.450 --> 00:06:21.100
So everything so far looks
exactly the same as type one,

00:06:21.100 --> 00:06:23.890
just with a sort of
suitable replacement.

00:06:23.890 --> 00:06:30.245
But the fermion ends up looking
a little more complicated here.

00:06:44.750 --> 00:06:47.258
And the Wilson line
also transforms

00:06:47.258 --> 00:06:48.800
because the Wilson
line was built out

00:06:48.800 --> 00:06:50.660
of n bar dot D field, so.

00:06:55.760 --> 00:06:58.580
Working the first order
in that transformation,

00:06:58.580 --> 00:07:00.200
there's an additional term.

00:07:00.200 --> 00:07:02.074
Epsilon perp dot D perp.

00:07:08.370 --> 00:07:11.790
OK, so I didn't want to take you
through the derivation of all

00:07:11.790 --> 00:07:12.480
these equations.

00:07:12.480 --> 00:07:16.260
I have provided references.

00:07:16.260 --> 00:07:18.570
There's a reference list at
the beginning of the notes,

00:07:18.570 --> 00:07:21.270
and there's a reference with
a paper that talks about how

00:07:21.270 --> 00:07:24.060
to derive these equations.

00:07:24.060 --> 00:07:27.220
We won't be spending too
much time talking about it.

00:07:27.220 --> 00:07:29.280
So I won't go through
the derivation,

00:07:29.280 --> 00:07:33.520
but it's very analogous to
what you would do in HQET.

00:07:33.520 --> 00:07:35.020
So how do we use these results?

00:07:35.020 --> 00:07:37.350
Let's take them as given.

00:07:37.350 --> 00:07:39.450
We can use these
results to close out

00:07:39.450 --> 00:07:41.910
our discussion of what we
were talking about last time,

00:07:41.910 --> 00:07:46.290
and show what the leading
order Lagrangian is,

00:07:46.290 --> 00:07:49.080
imposing all the symmetries.

00:07:49.080 --> 00:07:50.910
And so the first
thing that we can note

00:07:50.910 --> 00:08:06.460
is that if we do a type one
transformation of this guy,

00:08:06.460 --> 00:08:10.280
and you simplify
the result, then

00:08:10.280 --> 00:08:13.610
at the end of the day that boils
down to just the following.

00:08:23.210 --> 00:08:26.280
Well, it can be simplified
to a single term.

00:08:26.280 --> 00:08:39.440
And if one does the
same thing on this guy,

00:08:39.440 --> 00:08:43.169
then that gives us the same
turn, with the opposite sign.

00:08:43.169 --> 00:08:45.870
So these two operators are
connected by reparameterization

00:08:45.870 --> 00:08:46.856
invariance.

00:08:50.258 --> 00:08:53.990
The sum of them is
reparameterization invariant,

00:08:53.990 --> 00:08:54.940
and that looks good.

00:08:59.430 --> 00:09:03.573
So the sum is 0,
connected by RPI.

00:09:03.573 --> 00:09:05.490
And that means, for
example, that you couldn't

00:09:05.490 --> 00:09:09.120
have any non-trivial Wilson
coefficient between them

00:09:09.120 --> 00:09:11.580
or anything like that.

00:09:11.580 --> 00:09:14.108
Now, if you go through
the same argument

00:09:14.108 --> 00:09:16.650
with the other operator that we
talked about last time, where

00:09:16.650 --> 00:09:20.670
it was D perp mu, D perp mu.

00:09:20.670 --> 00:09:22.755
So delta 1.

00:09:36.760 --> 00:09:40.040
So same thing as above, but just
having D perp mu, D perp mu.

00:09:40.040 --> 00:09:41.740
D n perp mu.

00:09:41.740 --> 00:09:44.790
You actually find that it
gives the same result as this.

00:09:52.220 --> 00:09:55.940
OK, so if you just have
reparameterization of type one,

00:09:55.940 --> 00:09:57.320
then you can't really--

00:09:57.320 --> 00:09:59.422
the sum of this and
this would be invariant.

00:09:59.422 --> 00:10:01.130
Or the sum of this
and this is invariant,

00:10:01.130 --> 00:10:03.900
or the sum of any combination
of 1 minus a times that,

00:10:03.900 --> 00:10:06.170
and a times this
would be invariant.

00:10:06.170 --> 00:10:08.940
But type two does distinguish
between this operator

00:10:08.940 --> 00:10:10.197
and this operator.

00:10:27.750 --> 00:10:29.530
If you do a type
two transformation,

00:10:29.530 --> 00:10:31.680
you get terms from this
that you can't possibly

00:10:31.680 --> 00:10:34.080
cancel by any other term
that you could write down.

00:10:34.080 --> 00:10:37.927
And those terms are such that
the power counting, there's

00:10:37.927 --> 00:10:40.260
not a subleading operative
that could compensate either.

00:10:46.280 --> 00:10:49.790
OK, so going through all
the type two transformations

00:10:49.790 --> 00:10:50.640
is kind of messy.

00:10:50.640 --> 00:10:52.830
So again, I don't
want to do that.

00:10:52.830 --> 00:10:55.010
But I would refer you to
some reading for that.

00:10:58.320 --> 00:11:02.380
OK, but if you take these
two things together,

00:11:02.380 --> 00:11:06.370
then you do rule out
that additional operator.

00:11:06.370 --> 00:11:08.820
And so then, taking
everything together,

00:11:08.820 --> 00:11:11.534
we have a unique leading
order Lagrangian.

00:11:17.470 --> 00:11:19.780
It's not so easy, actually.

00:11:23.930 --> 00:11:27.210
It requires, it requires
looking at the transformations

00:11:27.210 --> 00:11:30.560
carefully.

00:11:30.560 --> 00:11:31.280
Yeah.

00:11:31.280 --> 00:11:34.260
So it is-- so even though I
made it sound like it's simple,

00:11:34.260 --> 00:11:35.300
but it's not.

00:11:35.300 --> 00:11:37.163
There's a reason why
I don't want to spend

00:11:37.163 --> 00:11:38.330
the rest of the class on it.

00:11:41.280 --> 00:11:41.780
OK.

00:11:49.940 --> 00:11:54.290
So if we put all the
things that we've

00:11:54.290 --> 00:12:02.120
talked about last time and this
time together, what we find

00:12:02.120 --> 00:12:10.130
is that the Lagrangian
that we've been discussing

00:12:10.130 --> 00:12:15.020
is unique for the CN field.

00:12:15.020 --> 00:12:21.230
So it's the unique leading order
Lagrangian for the CN field.

00:12:21.230 --> 00:12:23.875
And the things that we've used
are power counting in order

00:12:23.875 --> 00:12:25.250
to say that the
terms here should

00:12:25.250 --> 00:12:29.750
be lambda to the fourth, that's
what leading order meant.

00:12:29.750 --> 00:12:33.230
Gauge invariance, make
these covariant derivatives

00:12:33.230 --> 00:12:35.220
of appropriate types.

00:12:35.220 --> 00:12:41.340
And now also reparameterization
of symmetries.

00:12:41.340 --> 00:12:45.720
So as we kind of
discussed last time,

00:12:45.720 --> 00:12:47.220
the reparameterization
of symmetries

00:12:47.220 --> 00:12:49.710
here can do different things.

00:12:49.710 --> 00:12:52.560
They can connect leading order
operators to leading order

00:12:52.560 --> 00:12:54.960
operators and hence, constrain
the form of leading order

00:12:54.960 --> 00:12:57.690
operators that you write down,
and that's what we just did.

00:12:57.690 --> 00:12:59.940
And in particular,
the type two, remember

00:12:59.940 --> 00:13:01.980
the epsilon perp was
order of lambda to the 0,

00:13:01.980 --> 00:13:05.850
so maybe it's not surprising
that that's what it's doing.

00:13:05.850 --> 00:13:08.880
The type one transformation,
the delta perp

00:13:08.880 --> 00:13:11.550
took you down a power in lambda,
but if you had some subleading

00:13:11.550 --> 00:13:13.140
operator, then it
would also take you

00:13:13.140 --> 00:13:16.120
down a power in lambda, so
that's one of the things

00:13:16.120 --> 00:13:18.203
that you have to be careful
about when you're sort

00:13:18.203 --> 00:13:20.100
of counting powers of lambda.

00:13:20.100 --> 00:13:21.922
But there is a
sense, also, in which

00:13:21.922 --> 00:13:23.880
reparameterization symmetry
does the same thing

00:13:23.880 --> 00:13:26.100
as HQET, which is it connects.

00:13:26.100 --> 00:13:27.552
So it can do two
different things.

00:13:27.552 --> 00:13:29.760
It can constrain the form
of operators you write down

00:13:29.760 --> 00:13:33.270
at lowest order, or it
could connect the Wilson

00:13:33.270 --> 00:13:36.330
coefficients, for example,
of leading order operators

00:13:36.330 --> 00:13:38.280
to subleading operators.

00:13:38.280 --> 00:13:41.970
And that's what we saw on the
problem set in the HQET example

00:13:41.970 --> 00:13:44.400
it can do that here too.

00:13:44.400 --> 00:13:47.760
And so if we want to
think about that freedom,

00:13:47.760 --> 00:13:49.920
then we should note
that there's actually

00:13:49.920 --> 00:13:54.720
more kind of reparameterization
than we've talked

00:13:54.720 --> 00:13:56.080
about just with this example.

00:13:56.080 --> 00:13:58.680
So far, what we've talked
about is changing the basis.

00:13:58.680 --> 00:14:01.770
But we could also change
the amount of the momentum

00:14:01.770 --> 00:14:04.260
that we stick in the
label versus the residual.

00:14:04.260 --> 00:14:06.960
That was also a choice that
we made in doing things.

00:14:12.630 --> 00:14:17.700
And we should be careful
to think about that freedom

00:14:17.700 --> 00:14:19.330
as well.

00:14:19.330 --> 00:14:21.240
And that's also a
reparameterization freedom.

00:14:26.663 --> 00:14:28.080
And this one looks,
in some sense,

00:14:28.080 --> 00:14:29.770
more like what we
talked about in HQET.

00:14:29.770 --> 00:14:31.500
In some sense, what
HQET it is doing

00:14:31.500 --> 00:14:34.560
is a combination of
these two things.

00:14:34.560 --> 00:14:36.640
Because of the
decomposition of momentum

00:14:36.640 --> 00:14:38.415
there is a bit simpler.

00:14:41.900 --> 00:14:49.205
So we had this split of
label and residual momenta,

00:14:49.205 --> 00:14:51.592
and we thought about
the labels as discrete.

00:14:51.592 --> 00:14:53.050
But still, even if
they're discrete

00:14:53.050 --> 00:14:55.420
there's a freedom in
how we make this split.

00:14:55.420 --> 00:14:59.740
And so we could, if you
like, take our label momentum

00:14:59.740 --> 00:15:04.900
or our label momentum
operator, and transform it

00:15:04.900 --> 00:15:09.130
by adding some beta, which
has a power counting of order

00:15:09.130 --> 00:15:11.270
lambda squared.

00:15:11.270 --> 00:15:15.190
And if we make a
compensating transformation

00:15:15.190 --> 00:15:18.730
in the residual momentum,
or the residual derivative,

00:15:18.730 --> 00:15:24.100
then that would be a symmetry
of the decomposition.

00:15:24.100 --> 00:15:28.240
Wouldn't change anything,
end up beta being 0.

00:15:28.240 --> 00:15:29.575
Projects onto these two cases.

00:15:33.530 --> 00:15:36.970
So if you wrote that as
a change to the field,

00:15:36.970 --> 00:15:43.120
then you'd be saying that
the field picks up a phase.

00:15:43.120 --> 00:15:46.180
That's the transformation
for the derivatives.

00:15:46.180 --> 00:15:49.030
And then, you could
shift the label

00:15:49.030 --> 00:15:51.430
by beta, that's the
transformation for the beta,

00:15:51.430 --> 00:15:54.072
for the label.

00:15:54.072 --> 00:15:55.780
OK, so this is a
different transformation

00:15:55.780 --> 00:15:57.738
than the ones we were
talking about over there,

00:15:57.738 --> 00:16:01.000
and it's a transformation that
has to do with this freedom.

00:16:01.000 --> 00:16:03.730
But basically, what
this freedom does

00:16:03.730 --> 00:16:07.490
is it connects the
derivatives to each other.

00:16:07.490 --> 00:16:10.580
So it connects this combination,
it does something very simple,

00:16:10.580 --> 00:16:13.120
so after you know what
it does it's simpler

00:16:13.120 --> 00:16:15.782
just to say what it does
and forget about thinking

00:16:15.782 --> 00:16:16.990
about it as a transformation.

00:16:16.990 --> 00:16:19.210
It simply connects
the label operator

00:16:19.210 --> 00:16:22.930
to be label operator plus
the derivative operator.

00:16:22.930 --> 00:16:23.470
OK.

00:16:23.470 --> 00:16:26.500
But these two have different
orders in the power counting.

00:16:26.500 --> 00:16:28.558
And so what this
will do, is it'll

00:16:28.558 --> 00:16:30.100
connect things that
are leading order

00:16:30.100 --> 00:16:33.115
to things that are subleading
order, for example.

00:16:39.270 --> 00:16:53.350
So one way of saying that
is it that it connects

00:16:53.350 --> 00:16:55.780
leaving and subleading
Wilson coefficients,

00:16:55.780 --> 00:16:57.970
because you can always
think of the coefficient

00:16:57.970 --> 00:17:00.400
as the thing being fixed.

00:17:00.400 --> 00:17:07.089
And then it would do it in both
something like a leading order

00:17:07.089 --> 00:17:12.310
to subleading order
Lagrangian, or if you also

00:17:12.310 --> 00:17:14.887
wrote down operators.

00:17:14.887 --> 00:17:17.619
If you wrote down a series
of some external operator,

00:17:17.619 --> 00:17:20.920
maybe it's a weak
interaction, then

00:17:20.920 --> 00:17:23.290
it could connect as we
saw on the problems set.

00:17:23.290 --> 00:17:28.060
It could connect C1 to
C0 like it did in HQET.

00:17:28.060 --> 00:17:31.480
OK, so this one
clearly does that.

00:17:40.600 --> 00:17:43.977
So after you see that,
then you should think

00:17:43.977 --> 00:17:45.060
about the following thing.

00:17:45.060 --> 00:17:46.810
Well, this is
reparameterization symmetry.

00:17:46.810 --> 00:17:49.180
What did reparameterization
symmetry do here?

00:17:49.180 --> 00:17:51.633
It connected label and
residuals to each other.

00:17:51.633 --> 00:17:53.800
It said, well, they were
both part of the same thing

00:17:53.800 --> 00:17:56.800
in the beginning,
so there should

00:17:56.800 --> 00:18:00.430
be some connection later
on in the effective theory,

00:18:00.430 --> 00:18:03.780
and that remains true.

00:18:03.780 --> 00:18:08.320
It's encoded in this
reparameterization symmetry.

00:18:08.320 --> 00:18:10.620
But you can also now
try to think about,

00:18:10.620 --> 00:18:12.370
given that we have
these derivatives, what

00:18:12.370 --> 00:18:13.495
about gauging that formula?

00:18:16.310 --> 00:18:18.230
Gauge symmetry
basically was telling us

00:18:18.230 --> 00:18:20.360
how to turn p mu into
a covariant derivative

00:18:20.360 --> 00:18:22.880
and partial mu into a
convenient derivative.

00:18:22.880 --> 00:18:25.310
So can we think
about combining what

00:18:25.310 --> 00:18:26.870
we know from gauge
symmetry together

00:18:26.870 --> 00:18:28.760
with reparameterization?

00:18:28.760 --> 00:18:31.910
And we actually can,
if we're careful.

00:18:35.660 --> 00:18:37.697
And where this is
leading is, we'll

00:18:37.697 --> 00:18:39.780
find that there's basically
in the end of the day,

00:18:39.780 --> 00:18:41.000
there's very simple--

00:18:41.000 --> 00:18:43.400
I don't know if we'll get
there today, but we'll try.

00:18:43.400 --> 00:18:44.990
There's very simple
building blocks

00:18:44.990 --> 00:18:46.657
that in the end of
the day you can build

00:18:46.657 --> 00:18:48.420
all the SCET operators out of.

00:18:48.420 --> 00:18:52.940
And we're kind of moving
our way in that direction.

00:18:52.940 --> 00:18:54.620
OK, so let's try to gauge this.

00:18:57.470 --> 00:19:00.450
n dot label operator is 0.

00:19:00.450 --> 00:19:02.840
So for that guy, you just
have i n dot partial,

00:19:02.840 --> 00:19:05.000
there's no label operator piece.

00:19:05.000 --> 00:19:06.950
And gauging it
just takes you to i

00:19:06.950 --> 00:19:09.440
n dot D, which is
the full D that

00:19:09.440 --> 00:19:12.690
had both an ultra soft
and collinear piece to it.

00:19:12.690 --> 00:19:16.700
And if you look back at how our
transformations were defined

00:19:16.700 --> 00:19:27.580
for U c and U ultra
soft, then this guy

00:19:27.580 --> 00:19:29.770
is basically just
transforming in the way

00:19:29.770 --> 00:19:32.110
that you would want a covariant
derivative to transform,

00:19:32.110 --> 00:19:34.810
and it does that under both
types of symmetries, because

00:19:34.810 --> 00:19:37.756
of the way we set it up like
a background field for the n

00:19:37.756 --> 00:19:40.660
dot a ultra soft.

00:19:40.660 --> 00:19:43.790
OK, but that's actually not
really related to this story,

00:19:43.790 --> 00:19:47.840
because there is no split
in the n dot p component.

00:19:47.840 --> 00:19:51.388
So it's really the other
components, the and the n bar,

00:19:51.388 --> 00:19:53.680
where we have to pay more
attention to what's going on.

00:19:56.300 --> 00:20:01.180
So if we look at what the
gauge transformations mean

00:20:01.180 --> 00:20:03.828
for those components,
it's the same story

00:20:03.828 --> 00:20:05.370
but I'm going to
write it out anyway.

00:20:30.530 --> 00:20:32.600
OK, so this isn't quite
how we wrote it before.

00:20:32.600 --> 00:20:35.810
We wrote it before as a
transformation on the A,

00:20:35.810 --> 00:20:37.670
but if you put the
transformation on the A

00:20:37.670 --> 00:20:39.212
together with the
derivative then you

00:20:39.212 --> 00:20:41.605
can write it like this.

00:20:41.605 --> 00:20:44.210
For this piece
here, for example,

00:20:44.210 --> 00:20:45.500
remember what this thing is.

00:20:45.500 --> 00:20:50.510
This thing here is a label
operator p bar plus g n bar

00:20:50.510 --> 00:20:51.860
dot A n.

00:20:51.860 --> 00:20:53.790
Label operator
doesn't hit this guy,

00:20:53.790 --> 00:20:57.830
there's no label momentum
in the ultra soft U,

00:20:57.830 --> 00:21:02.570
so it just goes through that guy
or gives 0, and UU dagger is 1.

00:21:02.570 --> 00:21:05.840
So this piece doesn't
contribute there,

00:21:05.840 --> 00:21:08.600
and this piece transformed
in this way already.

00:21:08.600 --> 00:21:13.130
These ones are slightly more
involved, but what I'm saying

00:21:13.130 --> 00:21:15.890
is just a summary of what
we already learned earlier

00:21:15.890 --> 00:21:17.720
or talked about earlier.

00:21:17.720 --> 00:21:21.830
And then finally, there's also
the ultra soft transformation.

00:21:21.830 --> 00:21:28.250
And the ultra soft didn't
transform under the collinear.

00:21:28.250 --> 00:21:32.310
And of course it did transform
under the ultra soft.

00:21:32.310 --> 00:21:36.783
So this is a summary of
what we said earlier.

00:21:36.783 --> 00:21:38.450
So the simplest idea
that you would say,

00:21:38.450 --> 00:21:40.735
well, is just like,
take these guys

00:21:40.735 --> 00:21:42.860
and replace this by a
covariant derivative and that

00:21:42.860 --> 00:21:45.380
by a covariant
derivative, right?

00:21:57.400 --> 00:22:01.250
Always useful to think about
the simplest thing first.

00:22:01.250 --> 00:22:11.625
So this would be the simplest
thing, and that doesn't work.

00:22:11.625 --> 00:22:13.000
The reason that
that doesn't work

00:22:13.000 --> 00:22:15.850
is that if you look
at these two pieces,

00:22:15.850 --> 00:22:17.422
they don't transform
in the same way.

00:22:17.422 --> 00:22:18.880
So if I were to
build operators out

00:22:18.880 --> 00:22:21.520
of this guy that are
gauge invariant, then when

00:22:21.520 --> 00:22:23.680
I looked at the
collinear transformation,

00:22:23.680 --> 00:22:26.260
because this guy doesn't
transform and this guy does,

00:22:26.260 --> 00:22:28.030
this whole thing is
not transforming.

00:22:28.030 --> 00:22:31.420
So if I just stuck this guy
in, if this guy was invariant,

00:22:31.420 --> 00:22:34.750
this guy wouldn't
be, and vise versa.

00:22:34.750 --> 00:22:36.070
OK, so this doesn't work.

00:22:59.990 --> 00:23:03.005
AUDIENCE: Can you remind me how
the ultra soft field transforms

00:23:03.005 --> 00:23:09.321
under collinear with its
label summing conventions?

00:23:09.321 --> 00:23:11.220
PROFESSOR: Ultra
soft under collinear?

00:23:11.220 --> 00:23:15.035
So this guy didn't transform.

00:23:15.035 --> 00:23:18.510
AUDIENCE: I'm thinking of the
top equation, in the middle.

00:23:18.510 --> 00:23:19.397
i n D goes under--

00:23:19.397 --> 00:23:20.230
PROFESSOR: Oh, yeah.

00:23:20.230 --> 00:23:27.000
So that one's a
little, you know, yeah.

00:23:27.000 --> 00:23:30.630
So, you want to know which one?

00:23:30.630 --> 00:23:32.730
I can look it up for you.

00:23:32.730 --> 00:23:34.750
AUDIENCE: Well the one
you were pointing at,

00:23:34.750 --> 00:23:39.660
so that's how the D and the
A n part that transforms,

00:23:39.660 --> 00:23:41.232
but the A U soft--

00:23:41.232 --> 00:23:41.940
PROFESSOR: Right.

00:23:41.940 --> 00:23:44.040
AUDIENCE: Is there
something, like,

00:23:44.040 --> 00:23:46.388
PROFESSOR: Me remind myself,
then I'll remind you.

00:23:59.880 --> 00:24:01.580
So the A ultra slot
never transformed

00:24:01.580 --> 00:24:03.405
under the collinear, right?

00:24:03.405 --> 00:24:04.030
AUDIENCE: Yeah.

00:24:04.030 --> 00:24:05.210
I guess I don't know how--

00:24:05.210 --> 00:24:06.890
PROFESSOR: And
then, so then, yeah.

00:24:06.890 --> 00:24:07.900
So right.

00:24:07.900 --> 00:24:08.990
So how does this work out?

00:24:08.990 --> 00:24:10.790
The reason that
this works out is

00:24:10.790 --> 00:24:15.860
because there was an n dot A
ultra soft field in the gauge

00:24:15.860 --> 00:24:20.300
transformation of the
calendar n dot a field.

00:24:20.300 --> 00:24:21.590
That's why it works out.

00:24:24.120 --> 00:24:27.830
Good, because it would
take me forever to find it.

00:24:27.830 --> 00:24:30.780
All right, so what
are we going to do?

00:24:30.780 --> 00:24:32.870
So we need some
kind of compensating

00:24:32.870 --> 00:24:35.450
object to make these things
transform in the same way.

00:24:35.450 --> 00:24:38.240
And we have such a compensating
object, that's our Wilson line.

00:24:54.420 --> 00:24:58.750
So the Wilson line
transformed on one side.

00:24:58.750 --> 00:25:02.760
And so we can use it to
modify those formulas

00:25:02.760 --> 00:25:05.700
and get a result that,
where both terms transform

00:25:05.700 --> 00:25:06.410
in the same way.

00:25:45.450 --> 00:25:57.990
I just stick to the W in where
I need it in these two terms.

00:25:57.990 --> 00:26:03.360
And then I have set up things
so that the transformation

00:26:03.360 --> 00:26:06.727
of the two terms are the same.

00:26:06.727 --> 00:26:08.310
And you could call
this, if you wanted

00:26:08.310 --> 00:26:11.550
to define some kind
of full D, then this

00:26:11.550 --> 00:26:13.710
is the kind of
closest that you can

00:26:13.710 --> 00:26:24.740
get to defining a
sort of full D. OK,

00:26:24.740 --> 00:26:30.470
so these W's ensure both terms--

00:26:40.813 --> 00:26:42.980
it would ensure that under
collinear transformation,

00:26:42.980 --> 00:26:48.110
both terms transform at
the U c n, on the outside.

00:26:48.110 --> 00:26:50.060
OK, so we could
just stick anywhere

00:26:50.060 --> 00:26:53.120
that we have i n
bar dot D perp, we

00:26:53.120 --> 00:26:55.790
could just replace it by
this and then expand out.

00:26:55.790 --> 00:26:56.500
I.e.

00:26:56.500 --> 00:26:58.580
we could get a
subleading term by just

00:26:58.580 --> 00:27:00.560
using this operator here.

00:27:00.560 --> 00:27:03.140
And we know that that
is going to be there

00:27:03.140 --> 00:27:06.380
because of reparameterization
symmetry, which

00:27:06.380 --> 00:27:07.880
was connecting the derivatives.

00:27:07.880 --> 00:27:09.320
And therefore
connects these two.

00:27:09.320 --> 00:27:10.737
And once we put
in gauge cemetery,

00:27:10.737 --> 00:27:12.270
then it's-- this is the formula.

00:27:15.950 --> 00:27:18.740
So if you go back to what
we talked about earlier,

00:27:18.740 --> 00:27:25.075
we said earlier, I said A
mu, for the full theory,

00:27:25.075 --> 00:27:26.450
that you could
just sort of think

00:27:26.450 --> 00:27:29.180
of it as splitting
between a collinear

00:27:29.180 --> 00:27:30.650
and an ultra soft field.

00:27:30.650 --> 00:27:34.390
And then I said plus
dot dot dot, OK?

00:27:34.390 --> 00:27:37.560
And I said, we'll talk about
what plus dot dot dot is later.

00:27:37.560 --> 00:27:39.470
So now is later.

00:27:39.470 --> 00:27:41.480
What plus dot dot
dot is, is terms

00:27:41.480 --> 00:27:43.260
that you would generate
from, for example,

00:27:43.260 --> 00:27:45.135
the derivatives here
hitting the Wilson line,

00:27:45.135 --> 00:27:47.630
or expanding out these Wilson
lines, or the gauge field.

00:27:47.630 --> 00:27:50.450
And so those are
making it have terms

00:27:50.450 --> 00:27:53.040
with more fields
than just one field.

00:27:53.040 --> 00:27:55.100
And so, if you thought
about the A in here

00:27:55.100 --> 00:27:57.380
as sort of a full theory
A, and the A's in here

00:27:57.380 --> 00:27:59.900
as the effective theory,
you could drive a formula

00:27:59.900 --> 00:28:03.330
like this one but it would
involve more complicated stuff.

00:28:03.330 --> 00:28:05.457
It's not good to think
about it this way,

00:28:05.457 --> 00:28:07.040
it's better to think
about it in terms

00:28:07.040 --> 00:28:09.415
of the covariant derivatives,
because then you're already

00:28:09.415 --> 00:28:12.323
thinking about gauge invariance,
so this formula was useful

00:28:12.323 --> 00:28:14.240
earlier, but it's actually
these formulas here

00:28:14.240 --> 00:28:16.430
that are the more
useful way of thinking

00:28:16.430 --> 00:28:18.500
about taking full
theory derivatives

00:28:18.500 --> 00:28:20.510
and turning them into
effective theory results.

00:28:37.440 --> 00:28:39.923
All right, so we can
dispense with thinking

00:28:39.923 --> 00:28:42.090
about the plus dot dot dot
there because it's better

00:28:42.090 --> 00:28:43.590
just to think about
it in this form,

00:28:43.590 --> 00:28:47.190
where you can write a
nice closed expression.

00:28:47.190 --> 00:28:52.590
OK, so let's do one example
of a subleading term.

00:28:57.250 --> 00:29:00.150
So we had an operator that
involved covariate derivatives

00:29:00.150 --> 00:29:01.244
that were collinear.

00:29:04.210 --> 00:29:04.750
This one.

00:29:07.960 --> 00:29:11.710
And if we just use the top
formula there, then we know--

00:29:11.710 --> 00:29:14.920
so this is a term
that was in L0.

00:29:17.710 --> 00:29:24.515
Then we know that there's a term
in L1 that looks as follows.

00:29:29.645 --> 00:29:31.520
And I want to write it
out so you sort of see

00:29:31.520 --> 00:29:34.365
how also the gauge
symmetry works out nicely.

00:29:57.310 --> 00:30:07.690
OK, so using the fact that I
can write this guy as a Wilson

00:30:07.690 --> 00:30:09.210
line.

00:30:09.210 --> 00:30:12.600
1 over p bar times
the Wilson line.

00:30:12.600 --> 00:30:15.000
And the fact that I get
some Wilson lines when

00:30:15.000 --> 00:30:16.900
I look at the subleading term.

00:30:16.900 --> 00:30:20.040
So some of the Wilson lines,
I get W-W dagger is 1.

00:30:20.040 --> 00:30:21.917
And then the
remaining Wilson lines

00:30:21.917 --> 00:30:24.000
are sitting next to collinear
fields in such a way

00:30:24.000 --> 00:30:26.430
that this guy here is
collinear gauge invariant,

00:30:26.430 --> 00:30:28.530
this guy here is
collinear gauge invariant,

00:30:28.530 --> 00:30:31.113
and then there's an ultra soft
transformation that connects up

00:30:31.113 --> 00:30:32.820
all the pieces, OK?

00:30:32.820 --> 00:30:34.500
If we didn't have
these W's here,

00:30:34.500 --> 00:30:37.350
we wouldn't have got those
W's in the right spots there,

00:30:37.350 --> 00:30:39.030
and this operator
that I wrote down here

00:30:39.030 --> 00:30:41.880
would not be collinear
gauge invariant.

00:30:41.880 --> 00:30:44.280
So this operator is both
collinear and ultra soft gauge

00:30:44.280 --> 00:30:45.218
invariant.

00:30:49.800 --> 00:30:53.340
But most importantly,
it's collinear gauge

00:30:53.340 --> 00:30:59.250
invariant because of
what we did there.

00:30:59.250 --> 00:31:01.377
And there's no Wilson
coefficient to this operator.

00:31:01.377 --> 00:31:03.710
It's got the same coefficient
as the leading order term,

00:31:03.710 --> 00:31:05.010
so there's nothing non-trivial.

00:31:05.010 --> 00:31:08.630
We could just use it to
get subleading stuff.

00:31:08.630 --> 00:31:10.640
OK.

00:31:10.640 --> 00:31:14.150
And you could also do
similar things for currents.

00:31:14.150 --> 00:31:15.990
I'm not going to give
examples of that,

00:31:15.990 --> 00:31:19.490
but it's also a powerful
tool for connecting

00:31:19.490 --> 00:31:21.080
coefficients of
leading and subleading

00:31:21.080 --> 00:31:23.990
operators and currents.

00:31:23.990 --> 00:31:26.270
Not every operator in
the subleading Lagrangian

00:31:26.270 --> 00:31:27.450
is connected.

00:31:27.450 --> 00:31:34.220
So there are some that could
not be connected in some order,

00:31:34.220 --> 00:31:44.790
but there's actually
more connections

00:31:44.790 --> 00:31:49.670
than there are in HQET because
we have more symmetries.

00:31:49.670 --> 00:31:51.740
OK, so any questions about that?

00:31:57.040 --> 00:31:58.630
OK.

00:31:58.630 --> 00:32:01.090
So so far, all of
our discussions

00:32:01.090 --> 00:32:02.980
have been about one
collinear field.

00:32:02.980 --> 00:32:05.118
When we SCET, we
actually in general

00:32:05.118 --> 00:32:07.160
want to talk about more
than one collinear field.

00:32:07.160 --> 00:32:08.952
So how would we generalize
everything we've

00:32:08.952 --> 00:32:11.140
discussed to more than one?

00:32:25.690 --> 00:32:29.200
So we would have, in this case,
more than one energetic hadron,

00:32:29.200 --> 00:32:30.470
more than one energetic jet.

00:32:30.470 --> 00:32:33.220
So far we've been talking about
one energetic hadron, or one

00:32:33.220 --> 00:32:34.368
energetic jet.

00:32:34.368 --> 00:32:35.410
What if we have two jets?

00:32:35.410 --> 00:32:37.577
Then we would need two types
of collinear field, one

00:32:37.577 --> 00:32:39.280
for each of those jets.

00:32:39.280 --> 00:32:41.380
And basically,
what we have to do

00:32:41.380 --> 00:32:44.870
is take our collinear
Lagrangian--

00:32:44.870 --> 00:32:51.550
well let me call it L n
0, which is the fermion

00:32:51.550 --> 00:32:55.090
piece plus the gluon piece.

00:32:58.850 --> 00:33:00.080
And we sum over n.

00:33:00.080 --> 00:33:02.570
We have to sum
over all n's which

00:33:02.570 --> 00:33:05.510
are corresponding to individual
distinguishable collinear

00:33:05.510 --> 00:33:06.950
fields.

00:33:06.950 --> 00:33:09.320
So the question is, what
does it mean, sum over n?

00:33:09.320 --> 00:33:12.625
What is the distinction
between collinear fields?

00:33:15.710 --> 00:33:17.690
And so here's the
words, and then we'll

00:33:17.690 --> 00:33:20.780
explain what they mean.

00:33:20.780 --> 00:33:24.705
This sum is over
inequivalent n's,

00:33:24.705 --> 00:33:26.330
though that should
be obvious that they

00:33:26.330 --> 00:33:28.030
have to be equivalent.

00:33:28.030 --> 00:33:31.460
But what makes them
inequivalent is the fact

00:33:31.460 --> 00:33:37.920
that they are RPI
equivalence classes.

00:33:37.920 --> 00:33:39.740
So that's a funny sentence.

00:33:39.740 --> 00:33:43.470
Inequivalent RPI
equivalence classes.

00:33:43.470 --> 00:33:45.830
So, two ends are
the same if they

00:33:45.830 --> 00:33:48.700
belong in an equivalence class
that could be connected to--

00:33:48.700 --> 00:33:51.705
where they could be connected by
reparameterization invariance.

00:33:51.705 --> 00:33:54.080
So you should think of the
n's that I'm summing over here

00:33:54.080 --> 00:33:56.470
as just members,
one of each class.

00:33:56.470 --> 00:34:01.790
They're kind of just picking
out what that class is, just

00:34:01.790 --> 00:34:04.070
labeling it by one member.

00:34:04.070 --> 00:34:06.440
And then I sum over
an inequivalent set.

00:34:09.540 --> 00:34:12.840
So let's just imagine
that we have some n's.

00:34:12.840 --> 00:34:16.739
n1, n2, n3.

00:34:16.739 --> 00:34:18.300
And we can ask
the question, what

00:34:18.300 --> 00:34:20.703
makes them equivalent
or inequivalent?

00:34:34.270 --> 00:34:42.580
So let me call them, that the
n i collinear modes for any i

00:34:42.580 --> 00:34:43.150
are distinct.

00:34:48.420 --> 00:34:52.110
And there's a condition that
if I dot two of these ends

00:34:52.110 --> 00:34:55.020
together, they should
not be close together.

00:34:55.020 --> 00:34:57.030
And in fact that they
should be some value

00:34:57.030 --> 00:35:00.750
that's much bigger
than lambda squared.

00:35:00.750 --> 00:35:03.910
Obviously, if i was equal
to j, we would get 0.

00:35:03.910 --> 00:35:06.510
But for any i not
equal to j, we will

00:35:06.510 --> 00:35:09.420
say the n's are equivalent
if the dot product is much

00:35:09.420 --> 00:35:12.810
bigger than lambda squared.

00:35:12.810 --> 00:35:16.812
So let's see why it's lambda
squared by doing an example.

00:35:16.812 --> 00:35:18.270
So that's, if you
like, how you can

00:35:18.270 --> 00:35:19.562
define the equivalence classes.

00:35:25.390 --> 00:35:28.480
Let's imagine you have
some momentum, p2, which

00:35:28.480 --> 00:35:31.270
is a large piece times n2.

00:35:31.270 --> 00:35:34.610
And then you dot n1 into it.

00:35:34.610 --> 00:35:39.400
So n1 dot p2 is Q n1 dot n2.

00:35:39.400 --> 00:35:41.860
And that would be
of order lambda

00:35:41.860 --> 00:35:48.650
squared if n1 dot n2 were
of order lambda squared.

00:35:48.650 --> 00:35:49.600
Right?

00:35:49.600 --> 00:35:51.790
But if n1 dot n2 are
order lambda squared,

00:35:51.790 --> 00:35:54.160
and therefore n1 dot p2
is order lambda squared,

00:35:54.160 --> 00:35:56.980
you would say p2 is an
n1 collinear particle,

00:35:56.980 --> 00:35:59.510
because this is the right power
counting for an n1 collinear

00:35:59.510 --> 00:36:00.010
particle.

00:36:07.860 --> 00:36:10.520
So it's both n1 collinear
and n2 collinear,

00:36:10.520 --> 00:36:12.040
and that just means
n1 n2 are just

00:36:12.040 --> 00:36:13.790
two members of the
same equivalence class.

00:36:22.980 --> 00:36:26.150
So if this is true, that the
dot products of order lambda

00:36:26.150 --> 00:36:27.770
squared, then n1
and n2 are within

00:36:27.770 --> 00:36:29.030
the same equivalence class.

00:36:36.710 --> 00:36:39.760
Which if you wanted some
notation, you could say n2

00:36:39.760 --> 00:36:42.293
is in the class defined by n1.

00:36:42.293 --> 00:36:43.960
So you could really
think of this as sum

00:36:43.960 --> 00:36:45.340
over classes but its--

00:36:45.340 --> 00:36:48.370
usually people just
write sum over n.

00:36:48.370 --> 00:36:50.993
OK, so that is in
some sense clear,

00:36:50.993 --> 00:36:53.410
that you want to be summing
over things that are distinct.

00:36:53.410 --> 00:36:55.660
In this case of back to back
jets that we talked about,

00:36:55.660 --> 00:36:56.660
they're pretty distinct.

00:36:56.660 --> 00:36:59.290
One's going this way,
one's going that way.

00:36:59.290 --> 00:37:01.120
The n's dotted into
each other are 2,

00:37:01.120 --> 00:37:05.960
so that's certainly much
greater the lambda squared.

00:37:05.960 --> 00:37:09.230
But in general, this is what
you have to have in order

00:37:09.230 --> 00:37:12.210
to make them distinct fields.

00:37:12.210 --> 00:37:15.500
So then, everything basically
that we've talked about kind of

00:37:15.500 --> 00:37:19.310
goes through again, and I'm
not going to dwell on it,

00:37:19.310 --> 00:37:21.410
but I will just kind
of repeat some things.

00:37:59.160 --> 00:38:02.350
So collinear gauge
transformations, for example.

00:38:02.350 --> 00:38:04.530
You would have now a
new type of scaling

00:38:04.530 --> 00:38:06.505
that you can have
for different fields,

00:38:06.505 --> 00:38:08.880
and you could have two different
types of collinear gauge

00:38:08.880 --> 00:38:09.570
transformations.

00:38:09.570 --> 00:38:13.140
One for your n1 collinear
field, one for your n2.

00:38:13.140 --> 00:38:15.360
If n1 and n2 are
distinct, then those

00:38:15.360 --> 00:38:19.090
will have distinct scalings
for the corresponding momenta,

00:38:19.090 --> 00:38:22.470
so they'll be distinct
transformations.

00:38:22.470 --> 00:38:25.620
And fields won't transform
under the other guys'--

00:38:25.620 --> 00:38:28.512
n1 collinear fields won't
transform under the n2 gauge

00:38:28.512 --> 00:38:29.970
transformation
because, again, that

00:38:29.970 --> 00:38:32.760
would spoil the power
counting for the momenta.

00:38:35.290 --> 00:38:36.930
So at some level
it's very intuitive

00:38:36.930 --> 00:38:42.170
to figure out how
the results are.

00:38:42.170 --> 00:38:44.082
I'm not going to go through it.

00:38:44.082 --> 00:38:45.540
But suffice it to
say that we would

00:38:45.540 --> 00:38:48.720
have collinear gauge
transformations

00:38:48.720 --> 00:38:52.800
for each collinear guy.

00:38:52.800 --> 00:38:56.610
Reparameterization, same story.

00:38:56.610 --> 00:39:06.396
We have separate invariances
for each pair of n's.

00:39:06.396 --> 00:39:10.090
So n1 and n1 bar,
for the n1 sector.

00:39:10.090 --> 00:39:12.520
We have a
reparameterization symmetry.

00:39:12.520 --> 00:39:15.250
n2 and n2 bar for the n2 sector.

00:39:15.250 --> 00:39:19.803
Reparameterization
symmetry, et cetera.

00:39:19.803 --> 00:39:21.220
And here is actually
where you see

00:39:21.220 --> 00:39:23.553
that there's something that
looks different than Lorentz

00:39:23.553 --> 00:39:24.910
invariance that's going on.

00:39:24.910 --> 00:39:26.680
Because the
reparameterization are only

00:39:26.680 --> 00:39:28.960
acting within a sector.

00:39:28.960 --> 00:39:31.960
So if you do an n1 type
reparameterization,

00:39:31.960 --> 00:39:35.590
there's no transformation
of an n2 type Wilson line,

00:39:35.590 --> 00:39:37.740
or an n2 type gauge field.

00:39:37.740 --> 00:39:40.480
So an n1 transformation
affects the n1 collinear fields

00:39:40.480 --> 00:39:44.410
and objects, n2 type,
which could affect that.

00:39:44.410 --> 00:39:47.020
And it's more like a
Lorentz transformation

00:39:47.020 --> 00:39:49.495
that sort of acts within
the sector all by itself.

00:39:49.495 --> 00:39:51.370
But it's not really a
Lorentz transformation,

00:39:51.370 --> 00:39:53.440
it's just
reparameterization symmetry.

00:39:56.470 --> 00:39:58.300
OK?

00:39:58.300 --> 00:40:01.360
But it's exactly the
transformations we wrote down,

00:40:01.360 --> 00:40:05.410
just you don't transform n2
type fields when you do an n1.

00:40:09.570 --> 00:40:15.090
And, just like we had before,
if you do matching calculations

00:40:15.090 --> 00:40:16.770
you get Wilson
lines, but now there

00:40:16.770 --> 00:40:18.000
can be more than one type.

00:40:23.280 --> 00:40:26.130
So we had this W Wilson
line that showed up,

00:40:26.130 --> 00:40:27.780
and we did matching
calculations.

00:40:27.780 --> 00:40:30.276
And I want to give
you here one example

00:40:30.276 --> 00:40:35.730
which we'll come back and
talk about more later on,

00:40:35.730 --> 00:40:37.230
and which we've
already mentioned.

00:40:37.230 --> 00:40:39.720
So consider our
example of e plus e

00:40:39.720 --> 00:40:41.430
minus producing two jets.

00:40:46.870 --> 00:40:49.360
So in the full
theory, you would just

00:40:49.360 --> 00:40:51.910
have a vector current
from the photon.

00:40:51.910 --> 00:40:55.992
And if you want to match that
onto the two jet operator,

00:40:55.992 --> 00:40:57.700
you can go through
the same type of thing

00:40:57.700 --> 00:41:03.685
that we did when we were doing
the B to S gamma example.

00:41:12.220 --> 00:41:15.010
And the difference is here
that we get two different types

00:41:15.010 --> 00:41:16.060
of Wilson lines.

00:41:16.060 --> 00:41:21.550
So this n1 Wilson line, W n1.

00:41:21.550 --> 00:41:23.543
My notation here is
that the subscript

00:41:23.543 --> 00:41:25.210
is supposed to indicate
to you that it's

00:41:25.210 --> 00:41:30.250
n1 bar of A n1 that shows up.

00:41:30.250 --> 00:41:37.480
And then likewise, W n2 is a
function of n2 bar dot A n2.

00:41:37.480 --> 00:41:38.980
So you have to
decide whether you're

00:41:38.980 --> 00:41:42.250
going to call it W
sub n2 bar or n2,

00:41:42.250 --> 00:41:45.200
but anyway this is my notation.

00:41:45.200 --> 00:41:48.490
So you get Wilson lines that
are built out of the n1 bar

00:41:48.490 --> 00:41:51.720
dot A n1 field, which are order
of lambda 0, or the n2 bar

00:41:51.720 --> 00:41:54.760
dot A n2 fields, which
were order lambda 0.

00:41:54.760 --> 00:42:01.503
So this is lambda 0,
and this is lambda 0.

00:42:01.503 --> 00:42:02.920
By power counting
we can certainly

00:42:02.920 --> 00:42:04.257
get objects like that.

00:42:04.257 --> 00:42:06.340
And when you go through
the process of integrating

00:42:06.340 --> 00:42:09.880
our off shell particles, just
like we did for B to S gamma

00:42:09.880 --> 00:42:11.823
where we attach
gluons and we found

00:42:11.823 --> 00:42:13.240
that some lines
were off shell, so

00:42:13.240 --> 00:42:14.620
we had to integrate them out.

00:42:14.620 --> 00:42:17.423
If you do that for this
process, you get this operator.

00:42:40.450 --> 00:42:42.200
So when we construct
the effective theory,

00:42:42.200 --> 00:42:46.520
we have to integrate out off
shell particles and doing so

00:42:46.520 --> 00:42:48.418
generates this
Wilson mine operator.

00:42:48.418 --> 00:42:50.210
It's a little more
complicated in this case

00:42:50.210 --> 00:42:52.460
because we get these
two Wilson lines.

00:42:52.460 --> 00:42:55.640
And I'll talk a little bit more
later on in a different context

00:42:55.640 --> 00:42:58.340
about what type of diagrams are
involved in getting these two

00:42:58.340 --> 00:42:59.970
different Wilson lines.

00:42:59.970 --> 00:43:02.870
But the result
is, in some sense,

00:43:02.870 --> 00:43:05.510
more intuitive than a
way of getting there.

00:43:05.510 --> 00:43:08.840
What's happening is, you're
getting this W n1 Wilson line

00:43:08.840 --> 00:43:11.540
next to the C n1 field, and
then this form of combination

00:43:11.540 --> 00:43:14.660
here is gauge invariant
under the n1 collinear gauge

00:43:14.660 --> 00:43:16.460
transformations.

00:43:16.460 --> 00:43:17.930
And the same thing here.

00:43:17.930 --> 00:43:20.280
This guy doesn't transform
under the n1 collinear gauge

00:43:20.280 --> 00:43:21.290
transformations.

00:43:21.290 --> 00:43:24.305
This guy does, this
guy's invariant.

00:43:24.305 --> 00:43:26.180
This guy's invariant
under n2 collinear gauge

00:43:26.180 --> 00:43:27.710
transformations.

00:43:27.710 --> 00:43:30.360
They both transform under ultra
soft gauge transformation,

00:43:30.360 --> 00:43:32.280
so they get connected
in that way.

00:43:32.280 --> 00:43:33.897
And again, you have--

00:43:33.897 --> 00:43:35.480
if you just think
about gauge symmetry

00:43:35.480 --> 00:43:37.438
and how it should come
out, then you would have

00:43:37.438 --> 00:43:39.410
guessed that it should be this.

00:43:39.410 --> 00:43:41.600
But you can also
derive it this way.

00:43:44.640 --> 00:43:45.435
Yeah.

00:43:45.435 --> 00:43:48.280
AUDIENCE: So for the
one collinear receptor,

00:43:48.280 --> 00:43:51.630
it's been very top-down.

00:43:51.630 --> 00:43:57.242
Is this, when you start to
throw in [INAUDIBLE] would you--

00:43:57.242 --> 00:43:59.970
PROFESSOR: So this is the
top-down way of thinking,

00:43:59.970 --> 00:44:02.250
that you just generate
it by integrating out.

00:44:02.250 --> 00:44:03.590
And you can do that.

00:44:03.590 --> 00:44:05.190
Do It to all orders
of the tree level

00:44:05.190 --> 00:44:07.710
diagrams, that's possible.

00:44:07.710 --> 00:44:11.010
Or you can-- but we're starting
to see a picture emerge

00:44:11.010 --> 00:44:12.278
from the bottom up, right?

00:44:12.278 --> 00:44:12.945
AUDIENCE: Right.

00:44:12.945 --> 00:44:14.610
I'm talking about the top order.

00:44:14.610 --> 00:44:19.320
So that's pretty bottom-up
only, is there a way of,

00:44:19.320 --> 00:44:23.480
because you're saying
let's now state

00:44:23.480 --> 00:44:27.790
that the effective theory has
many copies of the collinear

00:44:27.790 --> 00:44:28.290
gauge.

00:44:28.290 --> 00:44:29.970
PROFESSOR: So.

00:44:29.970 --> 00:44:30.570
Right.

00:44:30.570 --> 00:44:33.780
So I mean, you could try to
think of writing a formula,

00:44:33.780 --> 00:44:34.380
right?

00:44:34.380 --> 00:44:37.080
You could start, try to think
of it like, let's take A,

00:44:37.080 --> 00:44:40.910
and let's write A.
Let's just do two.

00:44:40.910 --> 00:44:41.410
Right.

00:44:46.350 --> 00:44:47.800
You could start
trying to do that.

00:44:47.800 --> 00:44:51.180
But at some point, it just
loses its friendliness.

00:44:51.180 --> 00:44:52.890
It's not really
buying you anything.

00:44:52.890 --> 00:44:54.570
So, starting to think
from the bottom up

00:44:54.570 --> 00:44:58.820
is actually a good way
of going at this point.

00:44:58.820 --> 00:45:00.600
You could still do
it from the top down,

00:45:00.600 --> 00:45:02.350
but it just gets more
and more cumbersome.

00:45:02.350 --> 00:45:02.892
AUDIENCE: OK.

00:45:05.480 --> 00:45:08.130
PROFESSOR: Any other questions?

00:45:08.130 --> 00:45:09.990
All right.

00:45:09.990 --> 00:45:14.955
So let's come back and study are
our leading order Lagrangian.

00:45:14.955 --> 00:45:16.830
And actually, we've
already learned something

00:45:16.830 --> 00:45:20.950
about factorization although
we don't know it yet.

00:45:20.950 --> 00:45:26.370
So what is this
word, factorization?

00:45:26.370 --> 00:45:28.860
One way of thinking about
what factorization is, is it's

00:45:28.860 --> 00:45:31.800
how different degrees of
freedom talk to each other.

00:45:31.800 --> 00:45:33.720
And given that we
have a leading order

00:45:33.720 --> 00:45:36.030
Lagrangian for the collinear
and ultra soft fields,

00:45:36.030 --> 00:45:38.280
we should know something
about how collinear and ultra

00:45:38.280 --> 00:45:41.080
soft fields talk to each other.

00:45:51.600 --> 00:45:56.810
So let's come back
and study Lcc 0.

00:46:05.370 --> 00:46:15.630
So the propagator, if we read
out what the propagator is,

00:46:15.630 --> 00:46:19.500
if we're careful about
signs of i epsilons--

00:46:29.520 --> 00:46:31.560
We had both particles
and antiparticles.

00:46:38.230 --> 00:46:39.820
You can think of
the antiparticles

00:46:39.820 --> 00:46:42.760
as the guys that
have the minus i0.

00:46:42.760 --> 00:46:46.580
And if you combine these
two things together,

00:46:46.580 --> 00:46:49.240
then that's just giving
you the thing that we

00:46:49.240 --> 00:46:54.970
got when we expanded QCD.

00:46:54.970 --> 00:46:56.440
So when you think
about deriving it

00:46:56.440 --> 00:46:58.090
from the [? SCET ?]
Lagrangian, you

00:46:58.090 --> 00:46:59.463
think about getting this piece.

00:46:59.463 --> 00:47:00.880
But then you have
to also sum over

00:47:00.880 --> 00:47:03.370
the other pieces with the
other side of the n bar dot p,

00:47:03.370 --> 00:47:07.976
and you get this piece and
they come back to that thing.

00:47:07.976 --> 00:47:10.615
So this is the particles.

00:47:10.615 --> 00:47:13.150
If I want to split out the
particles and antiparticles

00:47:13.150 --> 00:47:15.324
I can do this.

00:47:15.324 --> 00:47:18.190
This is the particles that have
n bar dot p greater than 0,

00:47:18.190 --> 00:47:19.480
and this is the antiparticles.

00:47:23.620 --> 00:47:28.700
In this notation where n bar dot
p falls, the fermion line flow,

00:47:28.700 --> 00:47:32.240
we have n bar dot p less than 0.

00:47:32.240 --> 00:47:34.030
OK, so that, in some
sense, we already

00:47:34.030 --> 00:47:37.160
alluded to, that the
propagator works out correctly.

00:47:37.160 --> 00:47:39.820
This is showing it explicitly.

00:47:39.820 --> 00:47:43.150
What about interactions?

00:47:43.150 --> 00:47:45.460
Well, I want to be
interested in a minute

00:47:45.460 --> 00:47:49.420
about ultra soft
interactions because they're

00:47:49.420 --> 00:47:51.850
kind of special.

00:47:51.850 --> 00:47:55.595
For the ultra soft
gluons, only n

00:47:55.595 --> 00:47:58.510
dot A ultra soft showed
up at leading order.

00:48:01.580 --> 00:48:05.080
In the LCC 0.

00:48:05.080 --> 00:48:09.110
So if we look at the Feynman
rule there, we have this--

00:48:09.110 --> 00:48:11.290
let's give this guy an index mu.

00:48:11.290 --> 00:48:12.840
And this guy is ultra soft.

00:48:12.840 --> 00:48:14.836
This guy is collinear.

00:48:21.650 --> 00:48:25.190
Then the Feynman rule just
has n mu in it, not gamma mu.

00:48:25.190 --> 00:48:27.360
OK?

00:48:27.360 --> 00:48:31.320
So that's an observation
that we know from our LCC 0

00:48:31.320 --> 00:48:34.390
and that just comes from--

00:48:34.390 --> 00:48:36.897
remember this just
comes from the i n

00:48:36.897 --> 00:48:38.730
dot D term because
that's the only term that

00:48:38.730 --> 00:48:41.930
had the ultra soft
gauge field in it.

00:48:41.930 --> 00:48:45.080
But there's another fact
that our Lagrangian told us.

00:48:45.080 --> 00:48:48.560
And that is that only the n
dot k ultra soft momentum,

00:48:48.560 --> 00:48:52.622
so if I call this momentum
k, only the component n

00:48:52.622 --> 00:48:55.090
dot k was also showing
up in the propagators.

00:49:05.650 --> 00:49:11.310
So there's a gauge field
statement as well as

00:49:11.310 --> 00:49:14.067
a statement about the
ultra soft momenta.

00:49:14.067 --> 00:49:15.900
And that was due to the
multi-pole expansion

00:49:15.900 --> 00:49:16.440
that we did.

00:49:23.960 --> 00:49:27.010
So if we do that at
the level of thinking

00:49:27.010 --> 00:49:29.695
about some kind of diagram--

00:49:29.695 --> 00:49:33.670
so let's think about some
type of diagram like this,

00:49:33.670 --> 00:49:36.820
and let's look at
this propagator here.

00:49:36.820 --> 00:49:46.000
That propagator-- we
brought this up before

00:49:46.000 --> 00:49:47.250
but let me write it out again.

00:49:53.790 --> 00:49:58.980
So if I say p is this
guy, and k is this guy,

00:49:58.980 --> 00:50:02.280
then this guy is p plus
k, but k being ultra soft,

00:50:02.280 --> 00:50:05.790
these guys are supposed to
be ultra soft, let's say.

00:50:05.790 --> 00:50:08.340
k being ultra soft, only
the n dot k shows up

00:50:08.340 --> 00:50:10.350
in that propagator.

00:50:10.350 --> 00:50:16.170
And if we work on shell,
for p squared of 0,

00:50:16.170 --> 00:50:21.280
so let me first rewrite this
so you can see where I'm going.

00:50:21.280 --> 00:50:25.140
I can form a full p
squared in the denominator

00:50:25.140 --> 00:50:26.970
from the terms that depend on p.

00:50:26.970 --> 00:50:30.000
That's just the full p
squared of the particle.

00:50:30.000 --> 00:50:35.040
And if this guy is an external
particle as it is in my figure,

00:50:35.040 --> 00:50:37.980
then p squared would be 0.

00:50:37.980 --> 00:50:44.250
OK, so if I go to the on shell
case, p squared equals zero,

00:50:44.250 --> 00:50:48.990
then this just becomes n
bar dot p, n bar dot p,

00:50:48.990 --> 00:50:52.995
n dot k, plus i0, which is
looking like just 1 over n

00:50:52.995 --> 00:50:55.150
dot k.

00:50:55.150 --> 00:50:56.490
So it's becoming very simple.

00:51:11.315 --> 00:51:12.940
For cleaner gluons
it would, of course,

00:51:12.940 --> 00:51:14.760
not be-- it would
be no simplification

00:51:14.760 --> 00:51:20.700
like that possible, but for the
ultra soft particles there is.

00:51:20.700 --> 00:51:29.429
And so what has just happened
is that the propagator

00:51:29.429 --> 00:51:30.262
is becoming eikonal.

00:51:36.550 --> 00:51:38.410
So our collinear
propagator reduces

00:51:38.410 --> 00:51:41.950
to the eikonal approximation,
which is just 1

00:51:41.950 --> 00:51:47.290
over n dot k, when appropriate.

00:51:47.290 --> 00:51:51.160
And when appropriate means
when it's interacting

00:51:51.160 --> 00:51:53.990
with ultra soft fields.

00:51:53.990 --> 00:51:56.260
So we can kind of
summarize that for all

00:51:56.260 --> 00:52:00.410
the different possible
cases in the following way.

00:52:00.410 --> 00:52:09.040
So you could have, if you want
to be careful about signs,

00:52:09.040 --> 00:52:13.480
you can think about
attaching ultra soft gluons

00:52:13.480 --> 00:52:17.110
to a fermion coming in
or a fermion going out.

00:52:17.110 --> 00:52:24.910
Or likewise, to a antiparticle
coming in or going out.

00:52:29.480 --> 00:52:31.800
We always take k going up.

00:52:35.310 --> 00:52:39.690
And if I work with the
external particle on shell,

00:52:39.690 --> 00:52:43.500
then combining together the
Feynman rule for the vertex

00:52:43.500 --> 00:52:46.050
and the rule for
the propagator, I'm

00:52:46.050 --> 00:52:51.060
getting these what are called
sometimes eikonal vertices,

00:52:51.060 --> 00:52:52.470
or propagator vertices.

00:53:14.410 --> 00:53:15.940
So these are eikonal.

00:53:15.940 --> 00:53:19.780
Eikonal in both the
interactions and in the vertex.

00:53:19.780 --> 00:53:23.088
And that's what should happen
for having a very soft particle

00:53:23.088 --> 00:53:24.880
talking to something
very energetic, that's

00:53:24.880 --> 00:53:27.650
called the eikonal approximation
and it just falls out

00:53:27.650 --> 00:53:28.900
of our effective field theory.

00:53:32.190 --> 00:53:34.040
So this can actually
lead us to something

00:53:34.040 --> 00:53:38.270
deeper, which is called ultra
soft collinear factorization.

00:53:51.660 --> 00:53:53.400
So let's consider
more than one gluon.

00:54:00.020 --> 00:54:02.220
He'll have one collinear
fermion and we'll just

00:54:02.220 --> 00:54:04.910
touch a bunch of ultra
soft gluons to it.

00:54:19.420 --> 00:54:22.530
Kind of obvious notation.

00:54:22.530 --> 00:54:25.053
Let's call it m.

00:54:28.760 --> 00:54:31.250
So we could sum
up those diagrams,

00:54:31.250 --> 00:54:32.930
and if we sum up
those diagrams we

00:54:32.930 --> 00:54:37.010
get something that
looks familiar.

00:54:37.010 --> 00:54:44.450
So sum over m, sum
over permutation.

00:54:44.450 --> 00:54:47.630
Factors of g.

00:54:47.630 --> 00:54:48.295
Gauge fields.

00:54:54.810 --> 00:54:56.472
OK.

00:54:56.472 --> 00:54:57.555
Be careful about ordering.

00:55:10.310 --> 00:55:12.020
And I'm working here on shell.

00:55:16.800 --> 00:55:18.660
So that external
collinear particle

00:55:18.660 --> 00:55:19.860
has p squared equals 0.

00:55:23.497 --> 00:55:26.080
So if you think about what this
is, it's just our Wilson line.

00:55:26.080 --> 00:55:28.413
It's not the same Wilson line
that we were talking about

00:55:28.413 --> 00:55:29.440
before.

00:55:29.440 --> 00:55:32.480
This Wilson line is built
from ultra soft fields, not

00:55:32.480 --> 00:55:34.570
collinear fields.

00:55:34.570 --> 00:55:36.580
And it doesn't even point
in the same direction,

00:55:36.580 --> 00:55:40.230
it points in the n direction,
not the n bar direction.

00:55:40.230 --> 00:55:41.230
But it is a Wilson line.

00:55:44.138 --> 00:55:45.346
AUDIENCE: Are these incoming?

00:55:49.390 --> 00:55:50.835
PROFESSOR: Yeah, I think so.

00:55:50.835 --> 00:55:54.420
AUDIENCE: I think you put
a minus sign in the top.

00:55:58.270 --> 00:56:02.770
PROFESSOR: So for this guy,
there's no minus sign there

00:56:02.770 --> 00:56:07.210
because, well, I didn't
tell you what the--

00:56:07.210 --> 00:56:10.780
there could be a minus g
or a plus g here, right?

00:56:10.780 --> 00:56:11.650
Yeah.

00:56:11.650 --> 00:56:12.160
Yeah.

00:56:12.160 --> 00:56:14.560
That's minus g, probably.

00:56:14.560 --> 00:56:15.060
Yeah.

00:56:28.610 --> 00:56:30.092
OK.

00:56:30.092 --> 00:56:31.550
So actually, what
this motivates us

00:56:31.550 --> 00:56:33.425
to do from the effective
theory point of view

00:56:33.425 --> 00:56:36.120
is to consider the following.

00:56:36.120 --> 00:56:51.440
It motivates us to think about
making a field redefinition

00:56:51.440 --> 00:56:54.680
because what we just did is we
iterated the leading order L 0

00:56:54.680 --> 00:56:58.520
Lagrangian over and over again
to get these vertices as well

00:56:58.520 --> 00:56:59.900
as the propagators.

00:56:59.900 --> 00:57:02.990
And we ended up with something
that was just a Wilson line.

00:57:02.990 --> 00:57:06.780
Could we capture that
somehow, in a simpler way?

00:57:06.780 --> 00:57:09.410
And the answer is yes, if
we make the following field

00:57:09.410 --> 00:57:10.625
redefinition.

00:57:21.530 --> 00:57:24.320
We take our original
CNP field and we

00:57:24.320 --> 00:57:27.020
pull out a Wilson
line, Y. And we

00:57:27.020 --> 00:57:29.230
can do a similar thing
for the gauge field.

00:57:44.220 --> 00:57:49.920
This is just the adjoint version
of the same field redefinition,

00:57:49.920 --> 00:57:54.180
where Y is a Wilson line, which
corresponds to that in momentum

00:57:54.180 --> 00:57:57.330
space and in position space.

00:57:57.330 --> 00:57:58.560
Pathway to exponential.

00:58:13.180 --> 00:58:14.760
So when I say that
the Wilson line's

00:58:14.760 --> 00:58:18.930
at x, that means I've
shifted the whole line by x.

00:58:18.930 --> 00:58:23.290
And I could just denote
that by putting an x here.

00:58:23.290 --> 00:58:26.040
And then from that
point in space time,

00:58:26.040 --> 00:58:29.450
there's a line going
out in the n direction.

00:58:29.450 --> 00:58:31.900
That's what this
formula is saying.

00:58:31.900 --> 00:58:34.180
So at S equals zero,
you just sit at x.

00:58:39.020 --> 00:58:41.520
This should be a picture.

00:58:41.520 --> 00:58:44.040
Oh, this is minus infinity so
it's not a very good picture.

00:58:50.560 --> 00:58:54.670
And so this Wilson line,
like any Wilson line,

00:58:54.670 --> 00:58:57.730
has an equation of
motion, which is this.

00:58:57.730 --> 00:59:02.210
It satisfies Y dagger Y is one.

00:59:02.210 --> 00:59:04.510
And if you go through
with our transformation

00:59:04.510 --> 00:59:07.540
for the collinear gluon, which
I could have also motivated

00:59:07.540 --> 00:59:08.470
in this way.

00:59:08.470 --> 00:59:10.180
If I'd gone through
the calculation where

00:59:10.180 --> 00:59:11.847
instead of having a
collinear quark here

00:59:11.847 --> 00:59:13.780
I had a collinear
gluon, I would have

00:59:13.780 --> 00:59:16.270
found exactly the same thing,
except all these A's would be

00:59:16.270 --> 00:59:18.131
in the adjoint representation.

00:59:24.850 --> 00:59:27.650
I didn't write the
t's anyway, so.

00:59:27.650 --> 00:59:28.150
All right.

00:59:28.150 --> 00:59:29.920
So either the fundamental
representation

00:59:29.920 --> 00:59:32.917
or the adjoint
representation, and that

00:59:32.917 --> 00:59:35.500
would motivate you to make the
same type of field redefinition

00:59:35.500 --> 00:59:38.090
to try to capture
what's going on there.

00:59:38.090 --> 00:59:40.220
OK.

00:59:40.220 --> 00:59:42.170
So let's see what that
does to our Lagrangian.

00:59:50.100 --> 00:59:51.994
So our original Lagrangian--

00:59:55.870 --> 00:59:59.080
I'm not going to write
out all the terms.

00:59:59.080 --> 01:00:01.390
Just enough terms
to get you the idea.

01:00:01.390 --> 01:00:05.110
If I make this
field redefinition,

01:00:05.110 --> 01:00:07.920
it goes to this guy with a
superscript 0 that I'm writing.

01:00:10.480 --> 01:00:23.630
And then-- let me
write out here.

01:00:23.630 --> 01:00:25.700
Let me split this
derivative into two pieces.

01:00:35.300 --> 01:00:38.350
So I'm still not writing in
the terms there in the dot.

01:00:41.690 --> 01:00:44.090
So what I did is I took the
Y from this guy and the Y

01:00:44.090 --> 01:00:46.730
from that guy and I put them
inside the square bracket.

01:00:46.730 --> 01:00:48.950
This n dot D, remember,
has two pieces.

01:00:48.950 --> 01:00:54.900
It's i n dot D is sort of an i
n dot D ultra soft, which just

01:00:54.900 --> 01:00:56.900
has the ultra soft gauge
field, but then there's

01:00:56.900 --> 01:00:59.413
a piece that involves the
collinear gauge field.

01:00:59.413 --> 01:01:00.830
But then the
collinear gauge field

01:01:00.830 --> 01:01:03.170
also transforms, as
I wrote up there.

01:01:03.170 --> 01:01:05.960
So I get this Y, Y
dagger for that guy.

01:01:05.960 --> 01:01:07.670
So for the collinear
gauge field,

01:01:07.670 --> 01:01:09.810
these Y's are cancelling.

01:01:09.810 --> 01:01:12.270
And this guy here, if
you push the derivative

01:01:12.270 --> 01:01:15.710
through the Y using this
formula, then the Y's are also

01:01:15.710 --> 01:01:19.310
canceling and this is just
becoming i n dot partial.

01:01:22.150 --> 01:01:23.920
So that's becoming
i n dot partial,

01:01:23.920 --> 01:01:27.385
and this is becoming
n dot a n of 0.

01:01:30.010 --> 01:01:32.230
So what's happening is
that the ultra soft gauge

01:01:32.230 --> 01:01:35.140
field is dropping out.

01:01:35.140 --> 01:01:37.300
Now there's terms in
the plus dot dot dot,

01:01:37.300 --> 01:01:39.790
those terms involve the D perp
slash and the 1 over n bar

01:01:39.790 --> 01:01:42.100
dot D. And if I had
written out those terms,

01:01:42.100 --> 01:01:44.283
too, then the same thing
would happen, actually.

01:01:44.283 --> 01:01:46.450
All the Y's would cancel
out in those terms as well.

01:01:57.620 --> 01:02:06.010
So what I'm getting
here is something

01:02:06.010 --> 01:02:09.640
that I might call an n dot Dn.

01:02:09.640 --> 01:02:13.810
It's a covariant derivative that
only involves the A n0 field.

01:02:16.910 --> 01:02:18.580
And there's actually
no soft fields

01:02:18.580 --> 01:02:20.870
at all left in the Lagrangian
after I make this field

01:02:20.870 --> 01:02:22.044
redefinition.

01:02:33.410 --> 01:02:38.330
So that's obviously giving us
a very much simpler Lagrangian.

01:02:38.330 --> 01:02:42.068
OK, so it looks like
a good thing to do.

01:02:42.068 --> 01:02:43.860
And that, if you go
through it, is actually

01:02:43.860 --> 01:02:47.430
true also for the gluon action.

01:02:47.430 --> 01:02:51.732
Gluon action was also built
of n dot D type fields, type

01:02:51.732 --> 01:02:53.190
covariant derivatives,
that's where

01:02:53.190 --> 01:02:58.200
the ultra soft gluon showed up,
and basically the same thing

01:02:58.200 --> 01:02:59.880
happens.

01:02:59.880 --> 01:03:05.378
This same relation
right there means

01:03:05.378 --> 01:03:07.920
that the ultra soft gluon would
decouple from that Lagrangian

01:03:07.920 --> 01:03:09.640
as well.

01:03:09.640 --> 01:03:11.100
So making this
field redefinition,

01:03:11.100 --> 01:03:14.790
we actually decoupled in the
Lagrangians the ultra soft

01:03:14.790 --> 01:03:16.545
and the collinear fields.

01:03:22.900 --> 01:03:25.740
So even though the effective
theory allowed these modes

01:03:25.740 --> 01:03:29.380
to couple to each other, they
coupled in a very simple way.

01:03:29.380 --> 01:03:32.520
And so what we're saying is that
it's convenient because they

01:03:32.520 --> 01:03:33.775
couple in such a simple way.

01:03:33.775 --> 01:03:36.150
If they didn't couple in such
a simple way, we would stop

01:03:36.150 --> 01:03:38.700
and we'd just use that
Lagrangian, we'd do physics.

01:03:38.700 --> 01:03:40.830
Since they do a couple
in such a simple way,

01:03:40.830 --> 01:03:44.610
we can do physics in terms of
some reparameterized variables,

01:03:44.610 --> 01:03:48.390
or redefined variables, which
are these new variables.

01:03:48.390 --> 01:03:52.080
And that's convenient
because now,

01:03:52.080 --> 01:03:55.890
if I think about our
original Lagrangian,

01:03:55.890 --> 01:03:58.740
it goes over to
something that looks

01:03:58.740 --> 01:04:07.150
simpler because it's going
over to something that has no--

01:04:07.150 --> 01:04:10.110
the L0 has no ultra soft fields.

01:04:10.110 --> 01:04:13.500
This wouldn't be, of course,
true for the L1 or higher order

01:04:13.500 --> 01:04:15.600
Lagrangians, it
actually would still

01:04:15.600 --> 01:04:18.065
simplify all those
Lagrangians a certain way.

01:04:18.065 --> 01:04:19.440
But for the leading
order one, it

01:04:19.440 --> 01:04:21.622
seems to make a dramatic
simplification because we

01:04:21.622 --> 01:04:22.830
no longer have this coupling.

01:04:26.075 --> 01:04:28.200
So you might think, well,
the interactions have all

01:04:28.200 --> 01:04:31.290
disappeared.

01:04:31.290 --> 01:04:34.440
But they haven't disappeared,
because when you make the field

01:04:34.440 --> 01:04:37.470
redefinition, you can't just
make it to the Lagrangian.

01:04:37.470 --> 01:04:40.410
You also have to make it on
operators in their theory.

01:04:44.910 --> 01:04:49.200
And what this does is it
moves some interactions out

01:04:49.200 --> 01:04:50.745
of the Lagrangian
and into currents.

01:05:10.290 --> 01:05:11.820
So let's do some
examples about it.

01:05:13.993 --> 01:05:16.160
Since it's for the Lagrangian,
it's always the same.

01:05:16.160 --> 01:05:17.720
That's one of the reasons
why this is powerful,

01:05:17.720 --> 01:05:19.400
because if we do it
once and for all,

01:05:19.400 --> 01:05:22.220
we just did it for
the Lagrangian.

01:05:22.220 --> 01:05:24.470
And then we can see
what happens for a bunch

01:05:24.470 --> 01:05:27.690
of different currents, so
I'll do three examples.

01:05:27.690 --> 01:05:31.670
So one example that we had
was this current for B to S

01:05:31.670 --> 01:05:35.030
gamma, which had a heavy quark
field and a one light collinear

01:05:35.030 --> 01:05:36.590
up quark, and a Wilson line.

01:05:40.640 --> 01:05:42.830
I should have said
another thing that I'm

01:05:42.830 --> 01:05:45.230
using, which is important.

01:05:45.230 --> 01:05:51.265
It's that if you
go through what--

01:05:51.265 --> 01:05:53.780
let me make sure I got
my Y, or my Y dagger's

01:05:53.780 --> 01:05:56.540
on the right side.

01:05:56.540 --> 01:06:00.740
If you go through what happens
for the collinear Wilson line

01:06:00.740 --> 01:06:02.795
after the field redefinition,
this guy remember,

01:06:02.795 --> 01:06:05.810
is written in terms
of these fields.

01:06:05.810 --> 01:06:08.420
And this guy here is written
in terms of those fields.

01:06:08.420 --> 01:06:11.397
And it also gets
Y's on the outside.

01:06:11.397 --> 01:06:13.730
That's important when you
start considering, for example

01:06:13.730 --> 01:06:16.880
these dots or--

01:06:16.880 --> 01:06:17.570
OK.

01:06:17.570 --> 01:06:18.690
But I need that here.

01:06:18.690 --> 01:06:23.360
So given that that's also true,
if I want to look at this guy,

01:06:23.360 --> 01:06:25.338
I just make the
field redefinition.

01:06:28.206 --> 01:06:30.570
So I got this.

01:06:30.570 --> 01:06:35.370
I got a Y, Y dagger on both
sides of the Wilson line.

01:06:42.100 --> 01:06:45.130
These guys cancel each other.

01:06:45.130 --> 01:06:46.560
This is a Wilson line 0.

01:06:50.408 --> 01:06:54.334
I have C bar 0,
Wilson line zero.

01:06:54.334 --> 01:06:56.460
And I have gamma.

01:06:56.460 --> 01:06:59.680
And I have Y dagger, hv.

01:06:59.680 --> 01:07:01.198
So now I have a
Wilson line sitting

01:07:01.198 --> 01:07:03.490
next to the heavy quark and
another Wilson line sitting

01:07:03.490 --> 01:07:04.657
next to the collinear quark.

01:07:08.360 --> 01:07:10.730
So all the, if you
think in the diagram

01:07:10.730 --> 01:07:12.830
that we drew, the
interactions that

01:07:12.830 --> 01:07:15.470
were those gluons attaching
to the collinear quark,

01:07:15.470 --> 01:07:18.950
they're now just all represented
by this Wilson line here.

01:07:18.950 --> 01:07:21.560
But it's even more than
that, because we also

01:07:21.560 --> 01:07:23.820
transformed the Wilson line.

01:07:23.820 --> 01:07:27.320
So even if we had attached ultra
gluons to that Wilson line,

01:07:27.320 --> 01:07:29.840
they all magically
simplify into a simple,

01:07:29.840 --> 01:07:32.450
one simple Wilson line Y dagger.

01:07:32.450 --> 01:07:34.010
So from a diagrammatic
point of view,

01:07:34.010 --> 01:07:36.427
if we had actually tried to
carry out the calculation that

01:07:36.427 --> 01:07:38.900
would give this formula, it
would be kind of horrendous

01:07:38.900 --> 01:07:44.180
because there's an enormous
amount of calculations going on

01:07:44.180 --> 01:07:45.680
to give this.

01:07:45.680 --> 01:07:47.450
It just looks complicated
in the diagrams,

01:07:47.450 --> 01:07:49.790
but here it looks very simple.

01:07:49.790 --> 01:07:52.610
And the fact that the leading
order Lagrangian gave us 0

01:07:52.610 --> 01:07:56.615
there tells us that it really
is the sum of the diagrams.

01:07:59.323 --> 01:08:00.740
Let's have a couple
more examples.

01:08:03.960 --> 01:08:07.710
So you could take our
example for two jets,

01:08:07.710 --> 01:08:11.430
and you could ask what
would happen in that case.

01:08:11.430 --> 01:08:14.000
What's the generalization if
I have more than one collinear

01:08:14.000 --> 01:08:16.010
direction?

01:08:16.010 --> 01:08:18.830
And the generalization is that
you just make the same field

01:08:18.830 --> 01:08:21.290
redefinition, but it's
a different component

01:08:21.290 --> 01:08:24.859
of the ultra soft field that
couples to different collinear

01:08:24.859 --> 01:08:25.470
fields.

01:08:25.470 --> 01:08:27.402
So you're making a different--

01:08:27.402 --> 01:08:28.819
you're making a
field redefinition

01:08:28.819 --> 01:08:32.025
that's appropriate to each
of the different collinear

01:08:32.025 --> 01:08:32.525
sectors.

01:08:41.250 --> 01:08:44.830
But other than that it goes
through in the same way.

01:08:44.830 --> 01:08:48.729
And so here, this Y
n1 dagger involves

01:08:48.729 --> 01:08:51.720
n1 dot A ultra soft
fields, and this guy here

01:08:51.720 --> 01:08:55.109
involves n2 dot A
ultra soft fields.

01:08:55.109 --> 01:08:56.910
And they don't
cancel, but they do

01:08:56.910 --> 01:08:59.580
simplify all the interactions,
in this case simplify

01:08:59.580 --> 01:09:04.200
to the simple Y
dagger Y combination.

01:09:04.200 --> 01:09:08.939
OK, so having more than one
clear direction is not--

01:09:08.939 --> 01:09:12.630
I mean, it means that you
have more than one type

01:09:12.630 --> 01:09:13.647
of Y showing up.

01:09:13.647 --> 01:09:15.689
But again, it's just the
leading order Lagrangian

01:09:15.689 --> 01:09:17.147
for each of those
sectors that tell

01:09:17.147 --> 01:09:18.810
you what's going to show up.

01:09:27.260 --> 01:09:30.679
Type of Y. Let's do
one more example.

01:09:33.279 --> 01:09:39.979
Let's do an example where we
have collinear fields that

01:09:39.979 --> 01:09:42.770
are in the following form.

01:09:42.770 --> 01:09:46.729
Where both directions are n.

01:09:46.729 --> 01:09:48.347
And we have an
operator like that.

01:09:48.347 --> 01:09:50.430
So it's the same operator
I was writing down here,

01:09:50.430 --> 01:09:53.300
but in this case, I
had n1 n2 for two jets.

01:09:56.210 --> 01:09:58.190
I haven't really told
you about an example that

01:09:58.190 --> 01:09:59.972
would involve this
operator here,

01:09:59.972 --> 01:10:01.430
but it turns out
this operator here

01:10:01.430 --> 01:10:03.097
is something that
shows up, for example,

01:10:03.097 --> 01:10:06.660
in a part time distribution
function and other places.

01:10:06.660 --> 01:10:10.250
So this operator also does
make an appearance in physics.

01:10:10.250 --> 01:10:12.380
And if you look at what
happens for this operator,

01:10:12.380 --> 01:10:13.325
all the Y's cancel.

01:10:18.600 --> 01:10:21.240
So when you go through
the transformation,

01:10:21.240 --> 01:10:24.560
you have Y, Y dagger, but
then the Y's exactly cancel.

01:10:29.850 --> 01:10:31.560
So in that case,
it's really true

01:10:31.560 --> 01:10:34.998
that the ultra soft gluons
are dropping out effectively

01:10:34.998 --> 01:10:36.540
when you add up
diagrams that there's

01:10:36.540 --> 01:10:38.520
no ultra soft gluons left.

01:10:38.520 --> 01:10:42.688
In these cases, the leftover
is these Wilson lines that

01:10:42.688 --> 01:10:43.980
are showing up in the operator.

01:10:43.980 --> 01:10:47.686
In this case,
there's no left over.

01:10:47.686 --> 01:10:50.070
AUDIENCE: Are those both
values in the same position?

01:10:50.070 --> 01:10:50.862
PROFESSOR: Yeah.

01:10:50.862 --> 01:10:53.640
AUDIENCE: Y to the w?

01:10:53.640 --> 01:10:55.590
PROFESSOR: Because, yeah.

01:10:55.590 --> 01:10:57.720
So they would if
I just wrote this.

01:10:57.720 --> 01:11:01.050
In general, what you could have
inserted in here is some kind

01:11:01.050 --> 01:11:04.560
of something that
picks out, like,

01:11:04.560 --> 01:11:06.760
the momentum of
one of those w's.

01:11:06.760 --> 01:11:09.070
So let me just throw that
delta function in there

01:11:09.070 --> 01:11:10.630
so they don't cancel.

01:11:10.630 --> 01:11:13.440
We'll talk about that
in a minute or two.

01:11:13.440 --> 01:11:13.950
Yeah.

01:11:13.950 --> 01:11:15.930
But in general, I
could cancel them

01:11:15.930 --> 01:11:17.970
in this particular formula.

01:11:17.970 --> 01:11:20.418
But there's reasons
why, actually, we

01:11:20.418 --> 01:11:21.960
won't want to cancel
them, because we

01:11:21.960 --> 01:11:24.000
will be putting other
things in between that

01:11:24.000 --> 01:11:26.105
won't change what I just said.

01:11:26.105 --> 01:11:27.480
So imagine that
you had something

01:11:27.480 --> 01:11:29.520
that measured the
momentum of one

01:11:29.520 --> 01:11:30.960
of these products of fields.

01:11:30.960 --> 01:11:33.450
It would still be true that
these Y's, which are not

01:11:33.450 --> 01:11:37.320
having labels, would cancel
in the way that we said,

01:11:37.320 --> 01:11:40.710
but the w's wouldn't cancel.

01:11:40.710 --> 01:11:43.560
OK, so this is called the
BPS field redefinition,

01:11:43.560 --> 01:11:47.640
and S is me.

01:11:47.640 --> 01:11:53.625
And this thing sums up an
infinite class of diagrams.

01:12:02.510 --> 01:12:05.720
What it does in
example one, if you

01:12:05.720 --> 01:12:08.840
want to think about what the
diagrams would look like,

01:12:08.840 --> 01:12:12.120
let me draw kind of
an example for you.

01:12:12.120 --> 01:12:14.180
So let's have some
collinear particles.

01:12:18.130 --> 01:12:19.810
And just to make
it look nontrivial,

01:12:19.810 --> 01:12:23.750
let me draw something it
seems kind of nontrivial.

01:12:23.750 --> 01:12:25.840
So there's a bunch of
collinear particles,

01:12:25.840 --> 01:12:28.360
and then we could add
ultra soft gluons to them.

01:12:28.360 --> 01:12:29.860
And the ultra soft
gluons can couple

01:12:29.860 --> 01:12:31.735
to all those collinear
particles, the gluons,

01:12:31.735 --> 01:12:33.287
the quarks, everybody.

01:12:36.030 --> 01:12:37.920
If we consider all
the ultra soft gluons

01:12:37.920 --> 01:12:44.040
coupling to the collinear
particles everywhere,

01:12:44.040 --> 01:12:46.560
and we add up all
those attachments,

01:12:46.560 --> 01:12:49.530
then it just becomes
a single Wilson line.

01:12:49.530 --> 01:12:55.950
So this thing becomes
a single Wilson line.

01:12:55.950 --> 01:13:00.060
And then times exactly the
same collinear structure.

01:13:03.780 --> 01:13:06.450
So what I'm saying doesn't
rely, since we did it

01:13:06.450 --> 01:13:09.233
at the level of Lagrangian, it
doesn't rely on whether or not

01:13:09.233 --> 01:13:11.400
they're loops, or tree
level, or anything like that.

01:13:14.690 --> 01:13:17.650
So this diagram there, if we
add up all possible attachments,

01:13:17.650 --> 01:13:18.970
will be equal to that one.

01:13:18.970 --> 01:13:23.880
Where this is a Y. In this
case, it's a Y dagger.

01:13:27.880 --> 01:13:30.910
OK, so that's the simplicity
encoded in this formula right

01:13:30.910 --> 01:13:32.620
here.

01:13:32.620 --> 01:13:33.775
This is an example of J1.

01:13:38.450 --> 01:13:46.100
In example three, then,
it's a little simpler, even.

01:13:49.940 --> 01:14:00.420
Because the ultra soft gluons
are decoupling at lowest order

01:14:00.420 --> 01:14:02.080
from any graph that
you might consider.

01:14:02.080 --> 01:14:03.580
So you go through
the same exercise,

01:14:03.580 --> 01:14:08.130
but now that the Y's
are all canceling out.

01:14:08.130 --> 01:14:12.450
And this has a name that
sometimes people use,

01:14:12.450 --> 01:14:14.040
called color transparency.

01:14:19.920 --> 01:14:22.970
So one place that it
shows up is the following.

01:14:22.970 --> 01:14:25.760
Let's take our
current J3 and imagine

01:14:25.760 --> 01:14:28.490
that we produce from that
current an energetic pion.

01:14:28.490 --> 01:14:30.560
That was one of the examples
we talked about when

01:14:30.560 --> 01:14:33.170
we were talking about BDD pi.

01:14:33.170 --> 01:14:37.460
So there's a collinear pion,
and these are collinear quarks.

01:14:37.460 --> 01:14:39.840
collinear quark and anti quark.

01:14:39.840 --> 01:14:44.120
Supposed to be inside the pion,
and there's collinear gluons.

01:14:44.120 --> 01:14:46.820
If we attach all the ultra
soft gluons to this object,

01:14:46.820 --> 01:14:48.485
then the Wilson lines cancel.

01:14:55.810 --> 01:15:04.340
The ultra soft
gluons are decoupling

01:15:04.340 --> 01:15:05.750
from energetic particles.

01:15:05.750 --> 01:15:08.630
And the reason that
they're decoupling

01:15:08.630 --> 01:15:11.420
is because the partons
here are in a color singlet

01:15:11.420 --> 01:15:14.360
state, which is this pion.

01:15:24.220 --> 01:15:26.620
OK, so here we're producing
a color neutral pion

01:15:26.620 --> 01:15:29.920
out of color contracted
collinear quark fields,

01:15:29.920 --> 01:15:31.780
and there's these
Wilson lines for reasons

01:15:31.780 --> 01:15:33.610
we'll discuss in a minute.

01:15:33.610 --> 01:15:36.620
How they kind of play
a role in physics here.

01:15:36.620 --> 01:15:38.920
But the reason that
Wilson lines are canceling

01:15:38.920 --> 01:15:41.590
is because the collinear
things were already contracted.

01:15:41.590 --> 01:15:43.390
All the collinear
things in n direction

01:15:43.390 --> 01:15:49.300
were contracted in a color
singlet, global color singlet.

01:15:49.300 --> 01:15:52.840
So the words that go along with
this phrase color transparency

01:15:52.840 --> 01:15:55.420
is that you have these
very soft gluons,

01:15:55.420 --> 01:15:58.252
and they're trying to come
in and see this thing.

01:15:58.252 --> 01:15:59.710
But they can't
really, all they can

01:15:59.710 --> 01:16:03.340
see is sort of the overall
color charge of the whole thing.

01:16:03.340 --> 01:16:06.667
They couple, of the
multi-pole expansion,

01:16:06.667 --> 01:16:08.500
they're only coupling
to a single component,

01:16:08.500 --> 01:16:10.420
and you can think of
that as if they're only

01:16:10.420 --> 01:16:12.700
seeing an overall color charge.

01:16:12.700 --> 01:16:14.620
And therefore, since
it's overall color

01:16:14.620 --> 01:16:17.530
charge is neutral, you don't
see, they just cancel out.

01:16:21.380 --> 01:16:22.520
OK.

01:16:22.520 --> 01:16:24.440
In my notes, I also
have a page talking

01:16:24.440 --> 01:16:27.590
about how you could think
of gauge transformations

01:16:27.590 --> 01:16:29.740
after making this
field redefinition,

01:16:29.740 --> 01:16:31.490
but it's kind of an
aside so I'm not going

01:16:31.490 --> 01:16:33.881
to talk about it in lecture.

01:16:33.881 --> 01:16:35.050
But I will post it.

01:16:38.500 --> 01:16:40.090
OK, so what about these?

01:16:40.090 --> 01:16:42.700
What about this
additional kind of thing

01:16:42.700 --> 01:16:45.300
that I was alluding to here?

01:16:45.300 --> 01:16:48.995
How does that come
in to our story?

01:16:52.430 --> 01:16:54.410
So, so far in our
story, we haven't really

01:16:54.410 --> 01:16:56.480
talked about Wilson
coefficients except

01:16:56.480 --> 01:16:58.283
to say that they
could be constrained

01:16:58.283 --> 01:17:00.200
by reparameterization
invariance to be absent.

01:17:07.070 --> 01:17:09.020
So let's think about
Wilson coefficients now.

01:17:27.290 --> 01:17:30.030
Obviously that's
something important.

01:17:30.030 --> 01:17:32.600
And the way that Wilson
coefficients can come in

01:17:32.600 --> 01:17:35.060
is the following way.

01:17:35.060 --> 01:17:39.845
They can depend on the large
momenta that we're at order 1.

01:17:39.845 --> 01:17:41.970
And one way of denoting
that is by saying that they

01:17:41.970 --> 01:17:43.910
depend on label operators.

01:17:52.670 --> 01:17:55.500
OK, so nothing stops that.

01:17:55.500 --> 01:17:58.335
But if we want the momentum
that's picked up by this label

01:17:58.335 --> 01:17:59.960
operator to be gauge
invariant, then we

01:17:59.960 --> 01:18:01.970
should act on products
of fields that

01:18:01.970 --> 01:18:03.710
are collinear gauge invariant.

01:18:03.710 --> 01:18:07.370
So the way that we should set
it up is to have an operator.

01:18:11.000 --> 01:18:14.090
Here's a kind of notation
that's sometimes used.

01:18:14.090 --> 01:18:25.070
Where the operator acts on both
fields, the C bar and the W.

01:18:25.070 --> 01:18:27.620
Because of our formulas
for the label operator,

01:18:27.620 --> 01:18:31.070
we could also write this
as a label operator acting

01:18:31.070 --> 01:18:34.640
to the left if we wanted.

01:18:34.640 --> 01:18:37.550
So, what this Wilson
coefficient is the function.

01:18:37.550 --> 01:18:39.672
It's not just simply a number.

01:18:39.672 --> 01:18:42.005
And it picks out the momentum
of this product of fields.

01:18:54.490 --> 01:18:59.698
And that's because that product
is collinear gauge invariant.

01:18:59.698 --> 01:19:01.490
So it's a well-defined
thing to talk about.

01:19:06.280 --> 01:19:08.720
OK, so that's actually
the general structure,

01:19:08.720 --> 01:19:11.090
that whenever we have these
products of fields that

01:19:11.090 --> 01:19:14.060
are collinear gauge invariant,
if we ask what the Wilson

01:19:14.060 --> 01:19:16.100
coefficient could
be a function of,

01:19:16.100 --> 01:19:18.665
it can be a function of the
momentum of those products.

01:19:22.260 --> 01:19:29.565
So one way of writing this in a
sort of more elegant fashion as

01:19:29.565 --> 01:19:30.065
follows.

01:19:34.440 --> 01:19:40.670
So take this guy, and write
it as a delta function

01:19:40.670 --> 01:19:41.740
in the following way.

01:19:53.280 --> 01:19:55.440
So if I do the integral
over this omega,

01:19:55.440 --> 01:19:57.930
then I would just get back
that I stick the p bar

01:19:57.930 --> 01:19:59.760
dagger inside the
Wilson coefficient,

01:19:59.760 --> 01:20:02.310
and that it acts on
this product of fields.

01:20:02.310 --> 01:20:05.430
But if I write it
this way, what I get

01:20:05.430 --> 01:20:07.950
is that my Wilson
coefficients are just

01:20:07.950 --> 01:20:11.380
functions of a number, not
functions of an operator.

01:20:11.380 --> 01:20:14.430
And my operators have these
delta functions in them.

01:20:14.430 --> 01:20:16.530
But then could depend
on some variables

01:20:16.530 --> 01:20:19.600
that are distinguishing
those delta functions.

01:20:19.600 --> 01:20:21.630
OK.

01:20:21.630 --> 01:20:25.230
So in general, products
of fields like this,

01:20:25.230 --> 01:20:28.710
we have to think about their
momenta as being something

01:20:28.710 --> 01:20:29.915
that we could label.

01:20:29.915 --> 01:20:32.460
If you like, we could
label it by omega.

01:20:32.460 --> 01:20:34.710
Because it's a linear
gauge invariant concept,

01:20:34.710 --> 01:20:38.160
the Wilson coefficients can
depend on those momenta,

01:20:38.160 --> 01:20:40.320
and then we have
Wilson coefficients

01:20:40.320 --> 01:20:43.110
that depend on those momenta
and operators that are just

01:20:43.110 --> 01:20:45.810
labeled by those momenta.

01:20:45.810 --> 01:20:47.310
And this is the
convolution formula

01:20:47.310 --> 01:20:50.130
that I sort of promised you at
the beginning of the discussion

01:20:50.130 --> 01:20:54.480
of SCET that was going to show
up, and now it's shown up.

01:20:54.480 --> 01:20:57.780
OK, so Wilson coefficients can
depend on those large order 1

01:20:57.780 --> 01:20:59.400
momenta.

01:20:59.400 --> 01:21:01.525
And traditionally they're
written as integrals,

01:21:01.525 --> 01:21:03.900
even though you could think
of them as sums at this point

01:21:03.900 --> 01:21:06.730
and it wouldn't make
much difference.

01:21:06.730 --> 01:21:09.360
So this here is what's called
hard collinear factorization.

01:21:12.290 --> 01:21:16.460
In the traditional
QCD literature.

01:21:16.460 --> 01:21:18.980
Because it's telling you how
hard degrees of freedom, which

01:21:18.980 --> 01:21:21.380
are encoded in our
Wilson coefficients,

01:21:21.380 --> 01:21:23.130
can talk to collinear
degrees of freedom,

01:21:23.130 --> 01:21:25.220
which are encoded
in our operators.

01:21:25.220 --> 01:21:27.770
In SCET, that's just
come out of the formalism

01:21:27.770 --> 01:21:29.520
in a very sort of simple way.

01:21:29.520 --> 01:21:31.670
In QCD, you'd have to
use word identities

01:21:31.670 --> 01:21:34.130
and work hard to get
what I just derived

01:21:34.130 --> 01:21:36.750
for you in a couple of lines.

01:21:36.750 --> 01:21:40.460
So we're kind of done
for today, and we've

01:21:40.460 --> 01:21:43.580
seen kind of two examples
of mode factorization

01:21:43.580 --> 01:21:44.960
in the effective theory.

01:21:44.960 --> 01:21:48.080
collinear and ultra soft fields
and collinear and hard degrees

01:21:48.080 --> 01:21:50.090
of freedom, and
how it simplifies

01:21:50.090 --> 01:21:55.243
the discussion of factorization
which is kind of traditional,

01:21:55.243 --> 01:21:57.660
in a more traditional language
which I haven't taught you,

01:21:57.660 --> 01:21:59.520
but you can believe
me is more complicated

01:21:59.520 --> 01:22:01.230
than what we've discussed.

01:22:01.230 --> 01:22:03.640
So we'll talk a little bit
more about this next time.

01:22:03.640 --> 01:22:06.780
And then we'll talk about how
the ideas that we have here

01:22:06.780 --> 01:22:08.610
lead us just to define
a set of objects

01:22:08.610 --> 01:22:10.183
that we build operators from.

01:22:10.183 --> 01:22:11.850
And once we know what
those objects are,

01:22:11.850 --> 01:22:14.017
then we can kind of dispense
with a lot of the steps

01:22:14.017 --> 01:22:17.910
that we've done and just jump
right to building operators out

01:22:17.910 --> 01:22:19.620
of those objects.

01:22:19.620 --> 01:22:21.540
But the steps are
necessary to understand

01:22:21.540 --> 01:22:26.040
why it's those objects that we
want to build operators from.

01:22:26.040 --> 01:22:26.850
OK.

01:22:26.850 --> 01:22:27.900
So any questions?

01:22:31.912 --> 01:22:34.120
So we'll talk a little bit
about how this generalizes

01:22:34.120 --> 01:22:36.280
to other operators next time.

01:22:36.280 --> 01:22:39.870
So that's just the
one time I did here.