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IAIN STEWART: OK,
so let me remind
00:00:27.430 --> 00:00:30.350
you of what we were
talking about last time.
00:00:30.350 --> 00:00:32.878
So we were discussing
the example of DIS
00:00:32.878 --> 00:00:33.670
in the Breit frame.
00:00:33.670 --> 00:00:36.370
And the way we led
into this example
00:00:36.370 --> 00:00:39.160
is we talked about
renormalization group evolution
00:00:39.160 --> 00:00:40.285
with a heavy light current.
00:00:40.285 --> 00:00:43.570
And we saw that it had
this [INAUDIBLE] dimension.
00:00:43.570 --> 00:00:45.760
But it was a multiplicative
renormalization group
00:00:45.760 --> 00:00:48.970
evolution, and I said that
that happened because we only
00:00:48.970 --> 00:00:52.900
had one collinear
gauge-invariant
00:00:52.900 --> 00:00:54.710
object in our operator.
00:00:54.710 --> 00:00:57.280
And then I just wrote down an
operator that looked like this,
00:00:57.280 --> 00:00:59.470
and I said, there's
one that has two.
00:00:59.470 --> 00:01:03.520
If we run that object, we will
get a renormalization group
00:01:03.520 --> 00:01:05.813
equation that
involves convolutions.
00:01:05.813 --> 00:01:07.480
And I said that that's
going to give you
00:01:07.480 --> 00:01:09.438
the renormalization group
evolution of a parton
00:01:09.438 --> 00:01:11.050
distribution function.
00:01:11.050 --> 00:01:12.580
And we wanted to explore that.
00:01:12.580 --> 00:01:14.175
And so in order to
explore that, we
00:01:14.175 --> 00:01:15.550
should think of
some process that
00:01:15.550 --> 00:01:17.620
has the parton
distribution function in it
00:01:17.620 --> 00:01:19.630
so we can really
make sure we know
00:01:19.630 --> 00:01:21.790
precisely what the operator is.
00:01:21.790 --> 00:01:23.155
And that process is DIS.
00:01:23.155 --> 00:01:25.270
That's the simplest process.
00:01:25.270 --> 00:01:27.520
So we started thinking about
deep inelastic scattering
00:01:27.520 --> 00:01:29.950
in the Breit frame, which
is this framework, Q,
00:01:29.950 --> 00:01:30.550
of the photon.
00:01:30.550 --> 00:01:34.810
It has that form, just a
component in the z-direction.
00:01:34.810 --> 00:01:37.480
And in that frame,
the incoming quarks
00:01:37.480 --> 00:01:41.025
in the proton, quarks and
gluons, are collinear.
00:01:41.025 --> 00:01:42.550
Intermediate state,
the outstate,
00:01:42.550 --> 00:01:45.700
the x-state that's
going out, is hard.
00:01:45.700 --> 00:01:47.230
So you can think
of a-- if you were
00:01:47.230 --> 00:01:49.923
to draw perturbative diagrams,
you'd draw them like this.
00:01:49.923 --> 00:01:51.340
And this propagator
would be hard.
00:01:51.340 --> 00:01:52.780
It would have a hard momentum.
00:01:52.780 --> 00:01:54.700
And then you would
have loop corrections
00:01:54.700 --> 00:01:56.082
that could also be hard.
00:01:56.082 --> 00:01:57.790
In the effective
theory, you don't really
00:01:57.790 --> 00:01:59.710
have to think about
what diagrams.
00:01:59.710 --> 00:02:03.290
You just write down the lowest
possible dimension operator,
00:02:03.290 --> 00:02:06.040
and everything that's
a loop that's hard
00:02:06.040 --> 00:02:07.870
goes to the x,
which is the Wilson
00:02:07.870 --> 00:02:09.789
coefficient, if you like.
00:02:09.789 --> 00:02:13.420
And likewise, we also get
not just external quarks,
00:02:13.420 --> 00:02:15.640
but external gluons
from diagrams.
00:02:15.640 --> 00:02:19.180
In the full theory, it would
involve a quark loop like that.
00:02:19.180 --> 00:02:22.130
OK, so this is going to
lead to the quark PDF,
00:02:22.130 --> 00:02:24.700
and this is going to
lead to the gluon PDF.
00:02:24.700 --> 00:02:27.310
And we decided we would do
the quark one in some detail.
00:02:27.310 --> 00:02:29.860
So this is kind of writing
out now the operator
00:02:29.860 --> 00:02:32.920
and the Wilson coefficient in
kind of a combined notation
00:02:32.920 --> 00:02:43.380
where this w plus and minus
are w1 plus and minus w2.
00:02:43.380 --> 00:02:46.470
And then we had
one more formula,
00:02:46.470 --> 00:02:49.540
which is where we ended.
00:02:49.540 --> 00:02:52.500
So we have a collinear proton,
and then we have this operator.
00:03:06.310 --> 00:03:09.840
And then we have the
collinear proton again.
00:03:09.840 --> 00:03:16.440
And this matrix element
can be written as follows.
00:03:24.180 --> 00:03:26.740
So this is the last
formula we had last time.
00:03:29.990 --> 00:03:33.280
So some things here
are just conventions,
00:03:33.280 --> 00:03:37.165
but other things are important.
00:03:37.165 --> 00:03:39.040
Well, everything's
important, but some things
00:03:39.040 --> 00:03:40.582
are more important
than other things.
00:03:56.150 --> 00:03:58.770
So this quark here has a flavor.
00:03:58.770 --> 00:03:59.770
It could be an up quark.
00:03:59.770 --> 00:04:01.990
It could be down quark.
00:04:01.990 --> 00:04:03.910
Let me denote that
by an index i.
00:04:06.760 --> 00:04:08.590
This proton here is collinear.
00:04:08.590 --> 00:04:11.860
And really, all that
matters for this example
00:04:11.860 --> 00:04:12.970
is that we have some--
00:04:12.970 --> 00:04:16.390
we can think of it as a
massless proton, even.
00:04:16.390 --> 00:04:19.008
And as far as its
momentum is concerned,
00:04:19.008 --> 00:04:20.300
we can think of it as massless.
00:04:20.300 --> 00:04:22.630
So really, the only momentum
that matters in here
00:04:22.630 --> 00:04:24.950
is the minus momentum--
00:04:24.950 --> 00:04:33.306
so minus, which is n bar dot
P, and that's what this is--
00:04:33.306 --> 00:04:38.980
n bar dot P, n bar
dot P. It's capital P.
00:04:38.980 --> 00:04:40.730
So capital P was
the proton momentum.
00:04:40.730 --> 00:04:43.390
And we can think of this state
as just carrying large P minus.
00:04:43.390 --> 00:04:46.710
All the other components don't
matter for this matrix element.
00:04:46.710 --> 00:04:48.430
But it's a forward
matrix element,
00:04:48.430 --> 00:04:51.850
so both states carry
the same large momentum.
00:04:51.850 --> 00:04:53.710
And that's what led
to this delta function
00:04:53.710 --> 00:04:57.490
here that says that w1 and
w2, with the sign conventions
00:04:57.490 --> 00:04:59.920
we have, this guy has the
opposite sign convention.
00:04:59.920 --> 00:05:02.850
And so if it's
forward, these two guys
00:05:02.850 --> 00:05:07.660
have to have equal momentum
so that the sum is 0.
00:05:07.660 --> 00:05:09.640
And if you take into
account the sign,
00:05:09.640 --> 00:05:12.220
then that means w minus is 0.
00:05:12.220 --> 00:05:14.600
So that's what that
delta function is doing.
00:05:14.600 --> 00:05:17.650
And then the sum is something
that's not constrained
00:05:17.650 --> 00:05:19.550
by the matrix element.
00:05:19.550 --> 00:05:22.760
And so the sum could either
be positive or negative.
00:05:22.760 --> 00:05:25.390
If it's positive,
we can say that it's
00:05:25.390 --> 00:05:27.340
some fraction of
the proton momentum
00:05:27.340 --> 00:05:29.770
because this is a
quark inside the proton
00:05:29.770 --> 00:05:31.550
and it carries some
momentum, but it
00:05:31.550 --> 00:05:32.800
can't be more than the proton.
00:05:32.800 --> 00:05:35.560
Otherwise, we would get 0.
00:05:35.560 --> 00:05:37.360
So it's some fraction,
and that fraction
00:05:37.360 --> 00:05:40.422
is defined to be
xi in this formula.
00:05:40.422 --> 00:05:42.130
And the reason there's
a 2 is because I'm
00:05:42.130 --> 00:05:46.990
adding the w1 and the
w2, which are equal.
00:05:46.990 --> 00:05:49.260
So this is the
momentum fraction.
00:05:49.260 --> 00:05:51.010
And then we can have
an arbitrary function
00:05:51.010 --> 00:05:52.330
of that momentum fraction.
00:05:52.330 --> 00:05:54.160
Nothing stops us from
writing that down.
00:05:57.310 --> 00:05:59.170
And that's kind of
where we got to.
00:06:01.870 --> 00:06:07.553
Now, so on general
grounds, you can
00:06:07.553 --> 00:06:09.220
argue that that's the
most general thing
00:06:09.220 --> 00:06:11.725
that you can write down
for this matrix element.
00:06:11.725 --> 00:06:17.200
And I tried to argue
about why that's true.
00:06:17.200 --> 00:06:19.900
From charge conjugation, you
can actually do something more.
00:06:19.900 --> 00:06:21.610
So you can let charge
conjugation act
00:06:21.610 --> 00:06:23.260
on these operators.
00:06:23.260 --> 00:06:28.870
And since charge conjugation's
a good symmetry of QCD,
00:06:28.870 --> 00:06:31.240
you can prove that
that relates, actually,
00:06:31.240 --> 00:06:37.480
the quark and antiquark
operators in the following way.
00:06:42.410 --> 00:06:46.620
So quark and antiquark operators
are switching signs of w plus--
00:06:46.620 --> 00:06:48.190
so if I switch
the sign of w plus
00:06:48.190 --> 00:06:50.470
that's going from
quark to antiquark.
00:06:50.470 --> 00:06:54.580
And basically what happens
in the operator when
00:06:54.580 --> 00:06:56.080
you do charge
conjugation, remember
00:06:56.080 --> 00:06:59.890
that [? chi ?] goes over
here to [? chi ?] transpose
00:06:59.890 --> 00:07:02.110
and the w switches sign.
00:07:02.110 --> 00:07:05.320
So basically, what charge
conjugation is doing
00:07:05.320 --> 00:07:11.650
is taking w1 to minus
w2 and w2 to minus w1.
00:07:11.650 --> 00:07:14.380
And that's why the w plus,
which is signed by the w minus,
00:07:14.380 --> 00:07:15.220
doesn't.
00:07:15.220 --> 00:07:18.445
And then there's an overall
sign just from the fields--
00:07:21.502 --> 00:07:23.560
so from the usual
charge conjugation
00:07:23.560 --> 00:07:27.100
transformation of the
fields for a vector current.
00:07:27.100 --> 00:07:30.790
OK, so these are all orders
of relation between the Wilson
00:07:30.790 --> 00:07:32.080
coefficients.
00:07:32.080 --> 00:07:33.580
So really, when you
do the matching,
00:07:33.580 --> 00:07:35.872
you really only need to do
the matching for the quarks.
00:07:44.293 --> 00:07:46.210
So if you want to do the
matching calculation,
00:07:46.210 --> 00:07:48.627
you'd do a matching calculation
for the Wilson coefficient
00:07:48.627 --> 00:07:49.780
with positive w plus.
00:07:55.600 --> 00:08:01.370
And you could do it for
the antiquarks, as well,
00:08:01.370 --> 00:08:05.020
but you would just be
basically wasting time.
00:08:10.690 --> 00:08:13.440
Now, last time, we went through
the kinematics of the Breit
00:08:13.440 --> 00:08:14.580
frame a little bit.
00:08:14.580 --> 00:08:18.090
And this n bar dot proton
momentum is actually Q over x.
00:08:18.090 --> 00:08:20.920
So we could also write
this formula like that.
00:08:20.920 --> 00:08:22.740
And so you see that
w plus is actually
00:08:22.740 --> 00:08:27.000
something that's [? xi ?]
over x, Bjorken x, which is
00:08:27.000 --> 00:08:28.680
an external leptonic variable.
00:08:31.410 --> 00:08:34.950
Now, there was another
index over here, j, which
00:08:34.950 --> 00:08:36.059
we talked about last time.
00:08:36.059 --> 00:08:37.851
And that had to do with
the fact that we're
00:08:37.851 --> 00:08:42.270
taking the forward scattering
graphs with a tensor
00:08:42.270 --> 00:08:47.550
and we decompose that
into two scalars,
00:08:47.550 --> 00:08:50.015
multiplying things
that had indices.
00:08:50.015 --> 00:08:51.390
So there were two
possible things
00:08:51.390 --> 00:08:53.760
that we could write down.
00:08:53.760 --> 00:09:00.270
And the index j is
just this 1 or 2.
00:09:00.270 --> 00:09:02.100
And there was a
similar decomposition.
00:09:02.100 --> 00:09:05.310
In the effective theory, we
could think of decomposing--
00:09:05.310 --> 00:09:08.130
the effective theory
is a scalar in terms
00:09:08.130 --> 00:09:09.630
of the scalar
operators, which are
00:09:09.630 --> 00:09:12.210
these guys, with some
coefficients that
00:09:12.210 --> 00:09:15.420
have some indices, then
multiplied by some tensor.
00:09:15.420 --> 00:09:16.830
So these guys
don't have indices,
00:09:16.830 --> 00:09:19.770
but I could just multiply
them by effectively
00:09:19.770 --> 00:09:22.680
the effective theory
versions of these tensors.
00:09:22.680 --> 00:09:26.130
And so that that's
why there's a j here.
00:09:26.130 --> 00:09:29.630
Is that clear to everybody?
00:09:29.630 --> 00:09:32.270
OK, so there's various
indices-- i flavor,
00:09:32.270 --> 00:09:35.540
j for tensor decomposition,
and then a bunch
00:09:35.540 --> 00:09:40.300
of momentum indices.
00:09:40.300 --> 00:09:43.480
So when you go
through the analysis
00:09:43.480 --> 00:09:47.890
of trying to find a
formula for, say, T1,
00:09:47.890 --> 00:09:49.840
it's going to be related to C1.
00:09:49.840 --> 00:09:52.768
And T2 will be
related to C2, OK?
00:09:52.768 --> 00:09:54.310
Because this guy's
a scalar operator.
00:09:54.310 --> 00:09:57.310
It doesn't have any indices.
00:09:57.310 --> 00:10:05.140
So the way that that works, if
you just look at the two bases
00:10:05.140 --> 00:10:14.800
and write down the formula,
you'd have an integral over
00:10:14.800 --> 00:10:15.310
these w's.
00:10:18.050 --> 00:10:20.830
There are some prefactors
which just come about
00:10:20.830 --> 00:10:24.820
from being careful,
and then the thing that
00:10:24.820 --> 00:10:33.105
has an imaginary part of
these Wilson coefficients,
00:10:33.105 --> 00:10:34.480
and then you have
matrix elements
00:10:34.480 --> 00:10:43.120
of operators, which
have a flavor index
00:10:43.120 --> 00:10:46.600
but don't have a subscript j.
00:10:46.600 --> 00:10:48.340
So in general, this
has a flavor index.
00:10:52.750 --> 00:10:53.650
Keep track of things.
00:10:56.570 --> 00:10:59.210
And then there's
another one for T2--
00:11:08.667 --> 00:11:13.678
so kinematic prefactors that
are easy to work out that
00:11:13.678 --> 00:11:15.220
just come about from
the fact that we
00:11:15.220 --> 00:11:17.050
wrote the tensors and
the effective theory
00:11:17.050 --> 00:11:19.067
and the full theory
slightly differently.
00:11:32.630 --> 00:11:37.640
But these two guys have
the same matrix elements
00:11:37.640 --> 00:11:38.790
of the same operator.
00:11:38.790 --> 00:11:41.402
And all the sort of
tensor stuff is just
00:11:41.402 --> 00:11:43.610
saying that there's two
different Wilson coefficients
00:11:43.610 --> 00:11:46.572
that you have to
compute, and that's
00:11:46.572 --> 00:11:48.905
because you have these vector
currents from the photons.
00:11:52.750 --> 00:11:54.125
OK, so this is what we're after.
00:11:58.740 --> 00:12:03.000
These show up in
the cross-section.
00:12:03.000 --> 00:12:06.192
And what we're doing is we're
writing, at lowest order,
00:12:06.192 --> 00:12:08.400
the things that show up in
the cross-section in terms
00:12:08.400 --> 00:12:11.055
of effective theory objects,
the Wilson coefficients and then
00:12:11.055 --> 00:12:13.530
the matrix elements
of our operators
00:12:13.530 --> 00:12:17.250
here, which is this
thing in square brackets.
00:12:17.250 --> 00:12:20.490
And we're almost at what you
would call a factorization
00:12:20.490 --> 00:12:22.080
theorem.
00:12:22.080 --> 00:12:26.730
Factorization theorem is a
result for the cross-section,
00:12:26.730 --> 00:12:29.500
in our language, in terms of
effective theory quantities,
00:12:29.500 --> 00:12:31.950
and that's going to factor
the hard stuff, which
00:12:31.950 --> 00:12:38.130
is the pink stuff, which is
in these Wilson coefficients,
00:12:38.130 --> 00:12:41.372
from the low-energy stuff,
which is in these operators.
00:12:41.372 --> 00:12:44.640
AUDIENCE: So those pink elements
are [INAUDIBLE] equation.
00:12:44.640 --> 00:12:45.960
IAIN STEWART: Yeah.
00:12:45.960 --> 00:12:49.820
AUDIENCE: And what is
square bracket [INAUDIBLE]??
00:12:49.820 --> 00:12:50.820
IAIN STEWART: It's both.
00:12:50.820 --> 00:12:51.820
AUDIENCE: It's both, OK.
00:12:51.820 --> 00:12:53.500
IAIN STEWART: Yeah.
00:12:53.500 --> 00:12:55.290
I can write it like
this, if you like.
00:12:58.290 --> 00:13:01.050
Any other questions?
00:13:01.050 --> 00:13:06.030
OK, so literally what I
do is I take this formula
00:13:06.030 --> 00:13:08.402
and I plug it into that formula.
00:13:08.402 --> 00:13:10.860
And when I do that, I can do
one of the integrals trivially
00:13:10.860 --> 00:13:13.740
because it's a delta function.
00:13:13.740 --> 00:13:15.030
This one's just trivial.
00:13:15.030 --> 00:13:18.220
And then I do this one with
the other delta function.
00:13:18.220 --> 00:13:20.880
So both integrals
are actually trivial.
00:13:20.880 --> 00:13:25.080
And I can write the result
in terms of something
00:13:25.080 --> 00:13:33.300
that I'll call the
hard function, which
00:13:33.300 --> 00:13:35.881
is just the imaginary part
of the Wilson coefficient.
00:13:50.200 --> 00:13:53.120
And I'm going to denote
it in the following way.
00:13:53.120 --> 00:13:54.145
So this is w--
00:13:54.145 --> 00:13:55.733
the Wilson cost
efficient can depend
00:13:55.733 --> 00:13:56.900
on various different things.
00:13:56.900 --> 00:13:58.720
It can depend on w plus.
00:13:58.720 --> 00:14:00.490
It can depend on w minus.
00:14:00.490 --> 00:14:03.010
It can depend on the hard
scale, which is q squared.
00:14:03.010 --> 00:14:05.570
Or it could depend
on mu squared.
00:14:05.570 --> 00:14:08.740
So w minus, when you do the
delta function, gets set to 0.
00:14:08.740 --> 00:14:10.720
w plus gets set to something.
00:14:10.720 --> 00:14:13.480
And it's convenient
because of the way
00:14:13.480 --> 00:14:16.660
this delta function is with
the-- it kind of has a ratio.
00:14:16.660 --> 00:14:18.550
Because this function
is a function of xi
00:14:18.550 --> 00:14:20.810
which is the ratio
of two things,
00:14:20.810 --> 00:14:24.580
it's convenient to
define a dimensionless z
00:14:24.580 --> 00:14:28.570
and only talk about a function
of that dimensionless thing.
00:14:28.570 --> 00:14:32.560
And if you do that, then the
final result for these kind
00:14:32.560 --> 00:14:35.680
of T's, which are
imaginary parts of T's--
00:14:46.820 --> 00:14:49.720
you can just put the
formula together.
00:14:49.720 --> 00:14:50.720
I'll write one of them--
00:15:03.750 --> 00:15:04.340
is that.
00:15:08.200 --> 00:15:15.310
And then there's a
similar formula for in T2
00:15:15.310 --> 00:15:19.090
that involves H2 and H1.
00:15:19.090 --> 00:15:21.520
OK, so this is the
factorization theorem.
00:15:21.520 --> 00:15:24.280
And it came about, in some
sense, just trivially.
00:15:24.280 --> 00:15:26.350
Once we knew how to
write down the operators
00:15:26.350 --> 00:15:28.780
in the effective theory,
we were basically done,
00:15:28.780 --> 00:15:32.020
and then the rest was just sort
of algebraic manipulations,
00:15:32.020 --> 00:15:35.050
being careful about
what momenta go where,
00:15:35.050 --> 00:15:40.060
and knowing what the sign of
this formula for the matrix
00:15:40.060 --> 00:15:41.650
element is.
00:15:41.650 --> 00:15:44.470
This is a kind of
important point.
00:15:44.470 --> 00:15:47.140
But in some sense, the effective
theory, from the get-go,
00:15:47.140 --> 00:15:48.940
was already designed
to factorize
00:15:48.940 --> 00:15:51.398
because we were integrating
out the hard degrees of freedom
00:15:51.398 --> 00:15:52.370
right at the start.
00:15:52.370 --> 00:15:56.850
And so knowing what operators
and knowing their matrix
00:15:56.850 --> 00:16:00.520
element is really all we
needed to do to get to the DIS
00:16:00.520 --> 00:16:01.510
factorization theorem.
00:16:08.680 --> 00:16:12.110
So if you ever look
up the original way
00:16:12.110 --> 00:16:15.500
that this was derived,
it was not that easy.
00:16:15.500 --> 00:16:17.420
This is actually
something that's
00:16:17.420 --> 00:16:21.770
very complicated in a sort
of traditional approach.
00:16:21.770 --> 00:16:23.880
But in the effective
theory approach,
00:16:23.880 --> 00:16:27.680
it becomes almost trivial.
00:16:27.680 --> 00:16:29.810
And this is an all-orders
result because we never
00:16:29.810 --> 00:16:31.520
expanded in alpha s.
00:16:31.520 --> 00:16:34.120
We just used symmetries,
and we used the fact
00:16:34.120 --> 00:16:35.870
that we knew what form
the operators would
00:16:35.870 --> 00:16:38.203
take when we integrated out
the hard degrees of freedom.
00:16:45.000 --> 00:16:48.410
So any alpha s corrections
that one might want to add
00:16:48.410 --> 00:16:49.535
will fit into this formula.
00:16:52.220 --> 00:16:55.700
And this gives a perturbative
result for this H,
00:16:55.700 --> 00:16:59.060
which you would compute in
perturbation theory, which
00:16:59.060 --> 00:17:01.730
people do [INAUDIBLE]
these days.
00:17:07.050 --> 00:17:08.579
Now, if you ask
about things like--
00:17:08.579 --> 00:17:11.010
I didn't write
all the possible--
00:17:11.010 --> 00:17:14.380
I suppressed some things, right,
like Q squared and mu squared.
00:17:14.380 --> 00:17:16.500
If you ask about the Q
squared and the mu squared,
00:17:16.500 --> 00:17:19.020
then your Wilson coefficients
do depend on Q squared
00:17:19.020 --> 00:17:20.430
and mu squared.
00:17:20.430 --> 00:17:22.050
And the Wilson
coefficients, H here,
00:17:22.050 --> 00:17:24.092
are actually dimension--
the Wilson coefficients,
00:17:24.092 --> 00:17:26.135
the original one, [? xi, ?]
were dimensionless.
00:17:30.020 --> 00:17:31.737
So the H is dimensionless.
00:17:35.940 --> 00:17:37.995
I just pulled out the
dimensionable factor
00:17:37.995 --> 00:17:40.270
so that that would be true.
00:17:40.270 --> 00:17:43.320
And so this guy can depend
on Q squared over mu squared.
00:17:43.320 --> 00:17:45.240
The fact that Q squared
only shows up there,
00:17:45.240 --> 00:17:47.700
that's Bjorken scaling.
00:17:47.700 --> 00:17:51.630
And if you look at the
perturbative result for T2,
00:17:51.630 --> 00:17:53.950
then it vanishes
at lowest order.
00:17:53.950 --> 00:17:58.950
And so that's the
Callan-Gross relation.
00:17:58.950 --> 00:18:01.260
So there's various things
that are sort of encoded
00:18:01.260 --> 00:18:06.780
in this that come out, from the
effective theory point of view,
00:18:06.780 --> 00:18:10.380
in a very simple way.
00:18:10.380 --> 00:18:11.330
OK, so let me write.
00:18:32.380 --> 00:18:33.910
So there's logarithmic
corrections
00:18:33.910 --> 00:18:39.940
that involve Q in the
Wilson coefficients
00:18:39.940 --> 00:18:41.500
that will show up like that.
00:18:46.240 --> 00:18:50.360
So there's a mu also that you
could add to this formula.
00:18:50.360 --> 00:18:52.810
So the way that I
described it, we
00:18:52.810 --> 00:18:58.210
didn't think too hard about
bare versus normalized, right?
00:18:58.210 --> 00:18:59.770
We just take these operators.
00:18:59.770 --> 00:19:02.110
So far, they could
have been bare.
00:19:02.110 --> 00:19:09.100
But remember that when you have
C bare, O bare in [INAUDIBLE]
00:19:09.100 --> 00:19:12.090
Hamiltonian, for example,
that's C mu, O mu.
00:19:15.690 --> 00:19:17.635
So switching from bare
and renormalized--
00:19:17.635 --> 00:19:19.840
I mean bare operators
and coefficients
00:19:19.840 --> 00:19:21.880
to renormalized operators
and coefficients
00:19:21.880 --> 00:19:24.145
is simply a matter of
sticking in a mu here,
00:19:24.145 --> 00:19:26.020
and then you imagine
that the renormalization
00:19:26.020 --> 00:19:28.640
has taking place.
00:19:28.640 --> 00:19:32.680
So we could equally well
insert in these formulas
00:19:32.680 --> 00:19:35.290
a mu for that.
00:19:38.810 --> 00:19:41.725
And then what I'm saying is that
there being logs of mu over Q
00:19:41.725 --> 00:19:44.720
will make a little more sense.
00:19:44.720 --> 00:19:53.540
So there's also a Q. Squeeze
everything in here [INAUDIBLE]..
00:19:53.540 --> 00:19:56.840
OK, now we're being
completely honest
00:19:56.840 --> 00:19:59.880
about what it depends on.
00:19:59.880 --> 00:20:01.320
All right.
00:20:01.320 --> 00:20:08.640
So traditionally what happens
in the traditional literature,
00:20:08.640 --> 00:20:11.190
people talk about factorization
scales and renormalization
00:20:11.190 --> 00:20:12.720
group scales.
00:20:12.720 --> 00:20:15.240
So factorization
scales is the fact
00:20:15.240 --> 00:20:18.307
that this parton distribution
function is mu-dependent-- so
00:20:18.307 --> 00:20:19.890
operator that you
have to renormalize,
00:20:19.890 --> 00:20:22.270
and we're going to
do that in a minute.
00:20:22.270 --> 00:20:24.210
And so there has to
be a cancellation.
00:20:24.210 --> 00:20:26.370
Since this thing here
is a physical observable
00:20:26.370 --> 00:20:28.230
and is independent
of mu, there has
00:20:28.230 --> 00:20:30.180
to be a cancellation of
the mu-dependence here
00:20:30.180 --> 00:20:33.570
and the mu-dependence
here, all right?
00:20:33.570 --> 00:20:35.680
And that's this mu.
00:20:35.680 --> 00:20:39.330
So the thing that's
multiplying this result here
00:20:39.330 --> 00:20:41.310
would involve a cancellation
of mu-dependence
00:20:41.310 --> 00:20:42.477
here and mu-dependence here.
00:20:42.477 --> 00:20:44.490
So the same anomalous
dimension would show up
00:20:44.490 --> 00:20:46.470
in both the H and the f.
00:20:46.470 --> 00:20:50.520
And then sometimes people
also talk about mu-dependence
00:20:50.520 --> 00:20:53.640
that's just cancelling
within H itself.
00:20:53.640 --> 00:20:56.460
And they call that
renormalization group
00:20:56.460 --> 00:20:58.740
renormalization group mu.
00:20:58.740 --> 00:21:00.720
Sometimes people vary
these independently.
00:21:00.720 --> 00:21:03.160
In the effective theory,
it's really simple.
00:21:03.160 --> 00:21:05.340
You really just have
the classic setup of you
00:21:05.340 --> 00:21:08.310
have some hard
degrees of freedom.
00:21:08.310 --> 00:21:11.808
In this case, you can even
think of it one-dimensional.
00:21:11.808 --> 00:21:13.350
You have some hard
degrees of freedom
00:21:13.350 --> 00:21:14.642
that you want to integrate out.
00:21:14.642 --> 00:21:16.830
You have some scale
which we could
00:21:16.830 --> 00:21:18.750
call mu 0 that's
of order 2 where
00:21:18.750 --> 00:21:20.340
we do that integrating out.
00:21:20.340 --> 00:21:26.640
And then you can run
down or you could
00:21:26.640 --> 00:21:28.680
run in a more complicated way.
00:21:28.680 --> 00:21:32.070
So you could run the PDFs
which are sitting here
00:21:32.070 --> 00:21:38.398
at the collinear scale,
which is lambda QCD.
00:21:38.398 --> 00:21:40.440
You could think of evolving
them up to some scale
00:21:40.440 --> 00:21:42.990
and evolving the Wilson
coefficients down and meeting
00:21:42.990 --> 00:21:46.110
somewhere, OK?
00:21:46.110 --> 00:21:49.890
And so, yeah, it's just really
a sort of classic running
00:21:49.890 --> 00:21:51.720
and matching picture.
00:21:51.720 --> 00:21:53.790
Here, I've just
used the fact that I
00:21:53.790 --> 00:21:56.130
could run either one of them
or I could run both of them
00:21:56.130 --> 00:21:57.960
to a common scale.
00:21:57.960 --> 00:22:00.600
So I usually would pick mu
to be either something small
00:22:00.600 --> 00:22:04.803
or something large rather
than running both things.
00:22:04.803 --> 00:22:07.220
But in general, you could think
about running both things.
00:22:07.220 --> 00:22:09.480
And we've talked about
having anomalous dimensions
00:22:09.480 --> 00:22:12.180
for either one of these.
00:22:12.180 --> 00:22:15.180
And usually, we just
run one of them, OK?
00:22:15.180 --> 00:22:17.430
But it's no more complicated
than the standard picture
00:22:17.430 --> 00:22:19.770
of integrating out modes and
doing renormalization group
00:22:19.770 --> 00:22:20.816
evolution.
00:22:24.150 --> 00:22:27.870
So if we want to do tree-level
matching or one-loop matching
00:22:27.870 --> 00:22:31.112
or any kind of matching--
00:22:31.112 --> 00:22:32.820
let me just show you
tree-level matching.
00:22:39.785 --> 00:22:41.160
So tree-level
matching, you would
00:22:41.160 --> 00:22:47.120
compute this forward
scattering graph,
00:22:47.120 --> 00:22:51.770
and that will give you the
other diagram that we drew.
00:22:51.770 --> 00:22:55.280
And so you'd want to match
this guy onto that guy.
00:22:58.060 --> 00:23:06.988
And what you find is you
find one tensor structure
00:23:06.988 --> 00:23:08.900
at lowest order.
00:23:08.900 --> 00:23:11.610
So C1 is not equal to 0.
00:23:11.610 --> 00:23:13.010
C2 is equal to 0.
00:23:13.010 --> 00:23:16.730
And that's the
Callan-Gross relation
00:23:16.730 --> 00:23:19.340
which tells you about
the spin of the object
00:23:19.340 --> 00:23:20.840
that you're scattering
off, and this
00:23:20.840 --> 00:23:27.380
is how we know that quarks are
spin 1/2, or one way we know.
00:23:27.380 --> 00:23:33.620
And then you can calculate
C. And so that way
00:23:33.620 --> 00:23:35.660
that I set things
up, C was complex,
00:23:35.660 --> 00:23:37.740
and then I had to take
the imaginary part.
00:23:37.740 --> 00:23:41.240
So C is just this
propagator, basically.
00:23:41.240 --> 00:23:44.550
And it's only a nontrivial
function of w plus.
00:23:44.550 --> 00:23:49.100
There are some charges
that sit out front.
00:23:49.100 --> 00:23:51.363
And so the only way
that this guy depends on
00:23:51.363 --> 00:23:53.030
whether it's an up
quark or a down quark
00:23:53.030 --> 00:23:55.100
is you have 2/3
squared or 1/3 squared.
00:24:03.260 --> 00:24:07.280
And then there's something that
comes about from the propagator
00:24:07.280 --> 00:24:10.210
that looks like this.
00:24:10.210 --> 00:24:14.520
And then I take
the imaginary part
00:24:14.520 --> 00:24:21.650
and then I get H1, which
is a function of z, which
00:24:21.650 --> 00:24:23.600
is the xi over x.
00:24:23.600 --> 00:24:25.910
So if I write it as xi
over x like it shows up
00:24:25.910 --> 00:24:28.370
in the factorization
theorem, then I'm
00:24:28.370 --> 00:24:31.278
getting a delta
function of xi over x,
00:24:31.278 --> 00:24:32.570
which is this coming from this.
00:24:38.230 --> 00:24:41.900
So the lowest-order H1
is just a delta function.
00:24:41.900 --> 00:24:44.230
And that's where the
parton model picture
00:24:44.230 --> 00:24:46.660
comes from because
the parton model
00:24:46.660 --> 00:24:53.780
picture is that you think of xi
and x as being the same thing.
00:24:53.780 --> 00:24:56.653
And that's the tree-level
way of thinking,
00:24:56.653 --> 00:24:58.570
and that's just satisfying
this delta function
00:24:58.570 --> 00:25:00.640
and the hard function.
00:25:00.640 --> 00:25:02.860
And then you would
get that the T is just
00:25:02.860 --> 00:25:05.200
given by the parton
distribution at x, which is
00:25:05.200 --> 00:25:09.570
the external measurable thing.
00:25:09.570 --> 00:25:10.070
OK?
00:25:10.070 --> 00:25:12.480
So this is how all these
classic things come about
00:25:12.480 --> 00:25:16.000
in the effective
theory language.
00:25:16.000 --> 00:25:17.170
Any questions about that?
00:25:24.600 --> 00:25:28.840
All right, so let's
renormalize this operator
00:25:28.840 --> 00:25:32.250
and see how the classic result
for the renormalization group
00:25:32.250 --> 00:25:35.940
evolution of a PDF comes about.
00:25:35.940 --> 00:25:38.730
And again, the way that
you should think about this
00:25:38.730 --> 00:25:44.240
is you have an operator, and
you should just renormalize it.
00:25:44.240 --> 00:25:47.480
And once you've got
the effective theory,
00:25:47.480 --> 00:25:50.188
you shouldn't have to think too
deeply about what you're doing.
00:25:50.188 --> 00:25:51.980
You should just be able
to follow your nose
00:25:51.980 --> 00:25:53.360
and do the renormalization.
00:25:53.360 --> 00:25:55.880
You may have to be careful
because these operators are
00:25:55.880 --> 00:25:57.750
kind of complicated.
00:25:57.750 --> 00:25:59.743
They have this
dependence on these w's
00:25:59.743 --> 00:26:01.160
that you have to
be careful about.
00:26:01.160 --> 00:26:03.950
But really, it's just
follow your nose,
00:26:03.950 --> 00:26:05.510
compute the one-loop graphs.
00:26:20.035 --> 00:26:24.210
If you look up how Peskin would
do one-loop renormalization,
00:26:24.210 --> 00:26:25.960
there'd be an infinite
number of operators
00:26:25.960 --> 00:26:29.080
you'd have to derive in a
renormalization group, result
00:26:29.080 --> 00:26:29.920
for all of them.
00:26:29.920 --> 00:26:31.930
Here, we only have one
operator and we're just
00:26:31.930 --> 00:26:34.890
going to renormalize it.
00:26:34.890 --> 00:26:36.850
Our operator is
nonlocal in the sense
00:26:36.850 --> 00:26:38.770
that it depends on
these omegas, and that's
00:26:38.770 --> 00:26:42.190
what's encoding this
infinite number of operators
00:26:42.190 --> 00:26:43.150
that Peskin has.
00:27:03.620 --> 00:27:09.860
OK, so solving, if you
like, for f from the formula
00:27:09.860 --> 00:27:11.990
that we had before,
I can do that
00:27:11.990 --> 00:27:16.550
by integrating over the w minus.
00:27:16.550 --> 00:27:18.980
That sets these
guys to be equal.
00:27:18.980 --> 00:27:20.330
And then if I--
00:27:20.330 --> 00:27:24.920
so I can think of it as that
there's one free momentum, xi.
00:27:24.920 --> 00:27:27.410
And that free momentum xi
is one of these labels,
00:27:27.410 --> 00:27:29.220
which is this guy here.
00:27:29.220 --> 00:27:34.310
So xi is w over
Pn minus, and this
00:27:34.310 --> 00:27:38.690
is the proton which is carrying
some momentum Pn minus.
00:27:48.770 --> 00:27:52.670
This is the proton state, which
carries momentum Pn minus.
00:27:52.670 --> 00:27:54.080
And there's one delta function.
00:27:56.900 --> 00:27:58.550
I could put it either
place, but I only
00:27:58.550 --> 00:28:01.518
need one because the other
one's kind of trivial.
00:28:01.518 --> 00:28:03.560
So the first thing you
can think about doing here
00:28:03.560 --> 00:28:05.120
is looking at mass dimensions.
00:28:05.120 --> 00:28:07.010
And I already told you that
this guy was dimensionless,
00:28:07.010 --> 00:28:08.427
but let's check
that that's true--
00:28:10.830 --> 00:28:14.790
so a mass dimension.
00:28:14.790 --> 00:28:17.540
So relativistically
normalized states
00:28:17.540 --> 00:28:20.370
have mass dimension minus 1.
00:28:20.370 --> 00:28:22.890
Quark fields that don't
have a delta function
00:28:22.890 --> 00:28:26.310
have mass dimension 3/2.
00:28:26.310 --> 00:28:31.150
The delta function gives a
minus 1, and then a minus 1,
00:28:31.150 --> 00:28:33.660
so you get 0.
00:28:33.660 --> 00:28:37.560
3/2 plus 3/2 minus 3/1 is 0.
00:28:37.560 --> 00:28:40.033
So that means this f is a
really dimensionless function,
00:28:40.033 --> 00:28:42.450
and that's why it makes sense
that we defined it to depend
00:28:42.450 --> 00:28:45.030
on this dimensionless ratio.
00:28:45.030 --> 00:28:50.880
You can also look at
the lambda dimension,
00:28:50.880 --> 00:28:51.990
and here's how that works.
00:28:59.730 --> 00:29:00.810
That's also 0.
00:29:04.240 --> 00:29:07.540
So the only thing that's-- so
we already had power counting
00:29:07.540 --> 00:29:08.540
for our kai fields.
00:29:08.540 --> 00:29:10.630
Remember, the c field
inside the chi field scale
00:29:10.630 --> 00:29:15.070
like [? 1. ?] So this is just
coming about because this guy's
00:29:15.070 --> 00:29:17.260
order lambda.
00:29:17.260 --> 00:29:19.878
The delta function just
involves large momentum,
00:29:19.878 --> 00:29:21.045
so it has no power counting.
00:29:27.940 --> 00:29:29.440
And the only thing
that's nontrivial
00:29:29.440 --> 00:29:33.010
is that the states have
power counting minus 1.
00:29:33.010 --> 00:29:36.200
So here's how we
can derive that.
00:29:36.200 --> 00:29:38.410
So if you think about
relativistically normalized
00:29:38.410 --> 00:29:41.050
states, what you're
doing is you're
00:29:41.050 --> 00:29:44.050
defining sort of the
inverse of this d3 p
00:29:44.050 --> 00:29:47.920
over e, which you
can write actually,
00:29:47.920 --> 00:29:50.680
which is more convenient
for power counting,
00:29:50.680 --> 00:29:58.270
in terms of things that we
can power count more simply.
00:29:58.270 --> 00:30:02.320
So this is an exact relation
for a nontrivial particle
00:30:02.320 --> 00:30:06.760
between p minuses and pz.
00:30:06.760 --> 00:30:10.090
So then I can write,
because of that,
00:30:10.090 --> 00:30:12.190
the standard relativistic
normalization
00:30:12.190 --> 00:30:17.204
formula for a state with
two different momenta
00:30:17.204 --> 00:30:27.970
as kind of the inverse,
which would be this.
00:30:27.970 --> 00:30:33.645
So the usual formula would have
2e and then delta 3, right?
00:30:33.645 --> 00:30:35.020
Because it's the
inverse of this.
00:30:35.020 --> 00:30:38.230
But I can write
it also this way.
00:30:38.230 --> 00:30:40.720
This guy is lambda 0.
00:30:40.720 --> 00:30:42.730
This guy is lambda minus 2.
00:30:42.730 --> 00:30:47.292
Therefore, each of these
guys must be lambda minus 1.
00:30:47.292 --> 00:30:48.750
That's where the
minus 1 came from.
00:30:52.710 --> 00:30:56.070
All right, so we want to
renormalize that thing,
00:30:56.070 --> 00:30:58.140
that matrix element.
00:30:58.140 --> 00:31:05.850
And what loops can do is
that they can change omega.
00:31:05.850 --> 00:31:11.457
So you might-- or xi,
which are equivalent.
00:31:11.457 --> 00:31:13.290
And so the way that you
should think of that
00:31:13.290 --> 00:31:15.330
is in the following
sense, and it's actually
00:31:15.330 --> 00:31:18.270
something you're familiar
with, although you're
00:31:18.270 --> 00:31:20.200
familiar with it for
discrete quantum numbers.
00:31:20.200 --> 00:31:23.670
And here, in some sense,
we have a continuous one.
00:31:23.670 --> 00:31:26.040
So you have some
function, fq, that
00:31:26.040 --> 00:31:27.960
depends on some variable xi.
00:31:27.960 --> 00:31:30.810
And it can mix, under the
renormalization group,
00:31:30.810 --> 00:31:35.712
with an operator at a
different value of xi.
00:31:35.712 --> 00:31:37.170
So you can really
think of the fact
00:31:37.170 --> 00:31:40.785
that loops can change this
omega as just a mixing.
00:31:40.785 --> 00:31:42.910
You're used to mixing for
discrete quantum numbers.
00:31:42.910 --> 00:31:45.510
You write down all the operators
that have the same quantum
00:31:45.510 --> 00:31:47.850
numbers, and they can mix
under renormalization.
00:31:47.850 --> 00:31:50.010
Here, there's kind of
an additional thing
00:31:50.010 --> 00:31:53.490
that the object can depend
on, which is the xi parameter.
00:31:53.490 --> 00:31:55.560
And in general, when you
do the renormalization,
00:31:55.560 --> 00:31:57.660
that can change too.
00:31:57.660 --> 00:32:00.150
Because the operators
or matrix elements here
00:32:00.150 --> 00:32:02.370
are labeled by this
xi, in general, there's
00:32:02.370 --> 00:32:04.740
no reason that it should
stay the same under the loop
00:32:04.740 --> 00:32:05.340
corrections.
00:32:05.340 --> 00:32:09.060
And it was really a special case
that we dealt with last time
00:32:09.060 --> 00:32:10.080
where that did happen.
00:32:10.080 --> 00:32:11.205
But in general, it doesn't.
00:32:15.270 --> 00:32:21.270
OK, so this is
actually what we expect
00:32:21.270 --> 00:32:24.450
to happen in general unless
we can argue that it doesn't
00:32:24.450 --> 00:32:32.770
happen because you should
think of each value of xi
00:32:32.770 --> 00:32:37.360
as giving a different operator
or different matrix element.
00:32:40.570 --> 00:32:42.900
So I could write formulas
here just for the operator.
00:32:42.900 --> 00:32:44.983
It's actually the operator
that gets renormalized,
00:32:44.983 --> 00:32:46.650
not the matrix element.
00:32:46.650 --> 00:32:49.500
So I'm going to keep
writing f's just
00:32:49.500 --> 00:32:52.290
to avoid too much notation,
but we could always actually
00:32:52.290 --> 00:32:55.762
replace the f's by
just the operator.
00:32:55.762 --> 00:32:57.720
And we could do everything
in terms of actually
00:32:57.720 --> 00:33:01.510
just the w instead
of the xi variable.
00:33:01.510 --> 00:33:04.740
But I'll just keep using f.
00:33:04.740 --> 00:33:08.520
So what does this mean
in terms of the operator
00:33:08.520 --> 00:33:09.250
is the following.
00:33:09.250 --> 00:33:16.480
We can think of, if we
have some bare operator
00:33:16.480 --> 00:33:21.000
and we want to split that into
a piece that has divergences
00:33:21.000 --> 00:33:27.900
and a piece that is
just the finite pieces,
00:33:27.900 --> 00:33:32.740
the general formula for doing
that involves an integral.
00:33:32.740 --> 00:33:34.450
So this guy here--
00:33:34.450 --> 00:33:36.630
so there's also these
indices, i and j,
00:33:36.630 --> 00:33:40.890
and that's the flavor, if you
like, or quarks and gluons.
00:33:40.890 --> 00:33:45.270
So i is quark or gluon.
00:33:45.270 --> 00:33:48.570
And in general, you can also
have a mixing in the quark
00:33:48.570 --> 00:33:49.440
and gluon operators.
00:33:49.440 --> 00:33:51.120
We started with these
two different operators,
00:33:51.120 --> 00:33:53.078
and they can mix under
renormalization as well.
00:34:02.640 --> 00:34:05.540
So there's two operators in the
effective theory, same order
00:34:05.540 --> 00:34:08.830
in lambda, and they can mix
when you do the renormalization.
00:34:08.830 --> 00:34:12.139
And I'll draw a
diagram in a minute.
00:34:12.139 --> 00:34:16.540
So this thing here
is mu-independent.
00:34:16.540 --> 00:34:21.100
This thing here in MS bar has
all the 1 over epsilon UV's.
00:34:21.100 --> 00:34:23.679
And it also depends
on alpha of mu,
00:34:23.679 --> 00:34:26.920
and it depends on
these xi and xi prime.
00:34:26.920 --> 00:34:28.840
And this guy here is UV-finite.
00:34:34.238 --> 00:34:36.030
So this guy here is
really the thing that's
00:34:36.030 --> 00:34:37.350
the low-energy matrix element.
00:34:37.350 --> 00:34:39.510
But remember what low
energy meant here.
00:34:39.510 --> 00:34:42.300
Low energy was physics
at lambda QCD, physics
00:34:42.300 --> 00:34:44.880
of the initial-state proton.
00:34:44.880 --> 00:34:48.715
So actually, in this guy,
there are IR divergences.
00:34:48.715 --> 00:34:51.090
This is just some matrix
element in the effective theory,
00:34:51.090 --> 00:34:52.757
and in general, it
could be IR-divergent
00:34:52.757 --> 00:34:53.810
if you calculate it.
00:34:53.810 --> 00:34:59.010
And this guy actually is.
00:34:59.010 --> 00:35:01.922
And it really encodes--
00:35:01.922 --> 00:35:04.380
that's not going to bother us
at all because this is really
00:35:04.380 --> 00:35:08.370
some universal thing that
encodes lambda QCD effects,
00:35:08.370 --> 00:35:11.320
and that's what parton
distribution functions are.
00:35:11.320 --> 00:35:13.320
Then from the point of
view of what we're doing,
00:35:13.320 --> 00:35:16.630
it doesn't really matter that
it has this extra IR divergence
00:35:16.630 --> 00:35:19.080
so that we will have
to regulate diagrams
00:35:19.080 --> 00:35:22.633
in order to separate UV and IR
divergences because of that.
00:35:22.633 --> 00:35:24.300
Really, in terms of
the renormalization,
00:35:24.300 --> 00:35:26.791
what we're after is
getting the UV divergences.
00:35:31.610 --> 00:35:33.510
OK, so the usual kind
of formula that you'd
00:35:33.510 --> 00:35:36.010
have where you just
write o is z times o
00:35:36.010 --> 00:35:37.983
is slightly more
complicated here.
00:35:37.983 --> 00:35:39.150
There's this extra integral.
00:35:52.590 --> 00:35:55.980
And now, remember how you
derive a renormalization group
00:35:55.980 --> 00:35:57.450
equation.
00:35:57.450 --> 00:35:59.955
What you do is you say mu
d by u mu is this guy is 0.
00:36:03.490 --> 00:36:07.430
And so if I take mu d by d
mu, on the right-hand side,
00:36:07.430 --> 00:36:10.690
I get mu d by d mu of z
and mu d by d mu of f.
00:36:10.690 --> 00:36:14.980
And I can rearrange
that in the usual way,
00:36:14.980 --> 00:36:19.640
except for keeping track of
these integrals, as follows.
00:36:19.640 --> 00:36:21.880
So I imagine that there's
a z and a z inverse.
00:36:21.880 --> 00:36:27.440
And the relation between z
and z inverse is as follows.
00:36:27.440 --> 00:36:31.540
Let's just call
this double prime.
00:36:31.540 --> 00:36:34.010
It's matrix multiplication
except in the function space,
00:36:34.010 --> 00:36:34.510
right?
00:36:34.510 --> 00:36:37.885
So this is like
a delta function.
00:36:41.570 --> 00:36:43.810
So if you like,
you can just think
00:36:43.810 --> 00:36:46.060
that there's more indices.
00:36:46.060 --> 00:36:48.730
In some sense, what we have in
terms of the quark and gluon
00:36:48.730 --> 00:36:50.860
operators mixing is a
matrix equation, right?
00:36:50.860 --> 00:36:51.830
This is a vector.
00:36:51.830 --> 00:36:52.660
This is a matrix.
00:36:52.660 --> 00:36:55.330
This is a vector for
the indices i and j.
00:36:55.330 --> 00:36:58.780
And you can think of this
integral here as just another--
00:36:58.780 --> 00:37:00.825
it really looks, the
way I've drawn it,
00:37:00.825 --> 00:37:02.200
like this is
contracted with that
00:37:02.200 --> 00:37:04.300
and this is summing
over the indices.
00:37:04.300 --> 00:37:06.320
And really, that's what it is.
00:37:06.320 --> 00:37:09.490
So really, this idea that it's
just mixing of quantum numbers
00:37:09.490 --> 00:37:11.478
is kind of a good way of
thinking about things.
00:37:11.478 --> 00:37:12.895
And when you think
about formulas,
00:37:12.895 --> 00:37:16.030
you know you're just
summing over these indices,
00:37:16.030 --> 00:37:20.230
and the Kronecker delta
becomes a regular delta.
00:37:20.230 --> 00:37:26.720
So in that sense, it's
not that hard to do this.
00:37:26.720 --> 00:37:36.920
And so we get an anomalous
dimension equation
00:37:36.920 --> 00:37:38.800
which, again, has
that kind of form
00:37:38.800 --> 00:37:44.620
of just an integral for
the renormalized guy,
00:37:44.620 --> 00:37:46.390
and it has mixing.
00:37:46.390 --> 00:37:53.650
And this gamma ij, if
we go through the steps
00:37:53.650 --> 00:37:58.510
and use this formula,
looks like this.
00:38:04.750 --> 00:38:06.250
So I'm kind of
skipping steps, but I
00:38:06.250 --> 00:38:10.215
hope that you can
kind of picture
00:38:10.215 --> 00:38:11.590
where this result
will come from.
00:38:11.590 --> 00:38:13.240
And it's actually not--
00:38:13.240 --> 00:38:15.790
it's pretty easy to go from
that line with this formula
00:38:15.790 --> 00:38:16.840
to this line.
00:38:16.840 --> 00:38:19.230
This is one line, but I just
split it into two things
00:38:19.230 --> 00:38:20.980
and defined this
quantity, gamma ij, which
00:38:20.980 --> 00:38:23.170
is the anomalous dimension.
00:38:23.170 --> 00:38:25.850
AUDIENCE: So this mu in QCD
[INAUDIBLE] factorization
00:38:25.850 --> 00:38:26.570
scale?
00:38:26.570 --> 00:38:28.890
IAIN STEWART:
Yeah, that's right.
00:38:32.150 --> 00:38:36.020
OK, so at one loop,
things are simpler.
00:38:36.020 --> 00:38:37.490
Because at one
loop, this thing, we
00:38:37.490 --> 00:38:44.750
can just replace it by delta
ii prime, Kronecker delta
00:38:44.750 --> 00:38:47.210
at one loop.
00:38:47.210 --> 00:38:54.880
Because at one loop, we just
need the order alpha piece
00:38:54.880 --> 00:38:57.670
from this guy, and then
we can set the tree level
00:38:57.670 --> 00:38:59.020
for that guy.
00:38:59.020 --> 00:39:04.765
So at one loop, which is
all we're going to do,
00:39:04.765 --> 00:39:05.890
we get the simpler formula.
00:39:22.940 --> 00:39:25.850
OK, so that's our
setup, and now we
00:39:25.850 --> 00:39:28.850
want to calculate this
one-loop anomalous dimension
00:39:28.850 --> 00:39:31.940
by calculating the 1
over epsilon alpha s term
00:39:31.940 --> 00:39:32.490
and the zij.
00:39:39.830 --> 00:39:44.130
Before I do that, is
there any questions?
00:39:44.130 --> 00:39:45.410
All right, so tree level--
00:39:52.910 --> 00:39:56.020
so think about there being an
external p for whatever state
00:39:56.020 --> 00:40:00.580
I'm considering, and then
the operator is labeled by w.
00:40:00.580 --> 00:40:06.250
And so we're summing over spin.
00:40:06.250 --> 00:40:07.780
I've kind of somehow--
00:40:07.780 --> 00:40:09.640
sometimes I've dropped that.
00:40:09.640 --> 00:40:12.140
I said it last time.
00:40:12.140 --> 00:40:17.330
And so we get some spinners,
and we got a delta function.
00:40:17.330 --> 00:40:19.450
So what the delta
function in the operator
00:40:19.450 --> 00:40:22.870
is it's delta function
of w minus this label,
00:40:22.870 --> 00:40:23.587
momentum p bar.
00:40:23.587 --> 00:40:25.920
And in something like this
where it's completely trivial
00:40:25.920 --> 00:40:28.087
and there's just one state,
we just get the momentum
00:40:28.087 --> 00:40:29.890
of that state, which is p.
00:40:29.890 --> 00:40:33.730
This sum over spin
here is a p minus.
00:40:33.730 --> 00:40:38.170
And so the result is
a delta function of 1
00:40:38.170 --> 00:40:43.960
minus omega over p minus for
this tree-level matrix element.
00:40:51.060 --> 00:40:53.760
One loop-- now we have
to think about how
00:40:53.760 --> 00:40:55.980
we're going to regulate the IR.
00:40:55.980 --> 00:40:58.030
And I'll do it with
an off-shellness.
00:41:00.970 --> 00:41:04.270
So I'll introduce
a nonzero p class,
00:41:04.270 --> 00:41:08.050
and that will be enough to
regulate IR divergences.
00:41:12.400 --> 00:41:14.010
And we're really
after the UV one,
00:41:14.010 --> 00:41:16.480
so we just want to
separate these guys out.
00:41:19.587 --> 00:41:21.045
So there's some
different diagrams.
00:41:24.617 --> 00:41:27.105
We insert our operator,
and we just attach gluons.
00:41:27.105 --> 00:41:28.980
So one thing we can do
is just string a gluon
00:41:28.980 --> 00:41:35.145
across kind of like a standard
vertex renormalization diagram.
00:41:37.952 --> 00:41:39.160
So there's some loop momenta.
00:41:39.160 --> 00:41:40.890
Let me label it
on the quark line.
00:41:40.890 --> 00:41:44.390
And then the gluon here,
which is a collinear gluon,
00:41:44.390 --> 00:41:46.500
has momentum p minus l.
00:41:46.500 --> 00:41:50.640
And it's forward so
it's kind of set up.
00:41:53.340 --> 00:41:55.338
There are some
numerator to deal with.
00:41:55.338 --> 00:41:56.880
And I'm not going
to go through that,
00:41:56.880 --> 00:42:00.960
but it simplifies to
something kind of simple.
00:42:00.960 --> 00:42:03.660
After some [INAUDIBLE] algebra,
it simplifies down just
00:42:03.660 --> 00:42:05.410
to an l perp squared.
00:42:05.410 --> 00:42:11.730
For this diagram, there's
two l squared propagators,
00:42:11.730 --> 00:42:14.675
and there's one l minus
p squared propagator.
00:42:19.530 --> 00:42:21.330
And then there's
a delta function
00:42:21.330 --> 00:42:23.503
from the insertion
of the operator,
00:42:23.503 --> 00:42:25.920
but now the delta function
doesn't involve the [INAUDIBLE]
00:42:25.920 --> 00:42:27.000
momentum as it did there.
00:42:27.000 --> 00:42:29.640
It involves the loop
momentum, and that
00:42:29.640 --> 00:42:32.740
was kind of the whole
point of this example.
00:42:32.740 --> 00:42:39.210
So we have a delta function
of l minus minus w.
00:42:39.210 --> 00:42:40.832
And then there's
some dimreg factors,
00:42:40.832 --> 00:42:42.540
which we can be careful
about if we want.
00:42:48.030 --> 00:42:50.850
So in MS bar, we'd have
some factor like that.
00:42:50.850 --> 00:42:53.100
So this is some loop integral
that we just have to do,
00:42:53.100 --> 00:42:56.284
and we can do it with kind
of standard techniques.
00:43:10.820 --> 00:43:15.290
So in my, notes I wrote it
as a function of epsilon,
00:43:15.290 --> 00:43:17.755
and then epsilon is just
regulating the ultraviolet,
00:43:17.755 --> 00:43:19.190
and we expand in epsilon.
00:43:19.190 --> 00:43:22.238
So let me just write down
the result after expanding.
00:43:48.134 --> 00:43:50.260
So this is an
ultraviolet divergence,
00:43:50.260 --> 00:43:54.430
and A here has the
infrared regulator--
00:43:54.430 --> 00:43:55.870
p plus, p minus.
00:43:55.870 --> 00:43:57.320
And it also has
a z and a 1 minus
00:43:57.320 --> 00:43:59.890
z, which you can
group all together.
00:43:59.890 --> 00:44:02.620
And z is just this ratio.
00:44:02.620 --> 00:44:06.448
That thing is dependent on a
tree-level omega over p minus.
00:44:06.448 --> 00:44:07.990
Now, when I'm doing
this calculation,
00:44:07.990 --> 00:44:10.240
this is a small p, not
a big P, because I'm
00:44:10.240 --> 00:44:15.830
using quark states,
not a proton state.
00:44:15.830 --> 00:44:18.648
So really, if I wanted to
think about this as an f,
00:44:18.648 --> 00:44:20.440
I should say it's an
f for the quark state.
00:44:20.440 --> 00:44:24.370
But I think that you
can remember that.
00:44:24.370 --> 00:44:26.207
But the renormalization
of the operator
00:44:26.207 --> 00:44:27.790
doesn't depend on
the state, remember.
00:44:27.790 --> 00:44:29.780
We always take the
simplest states possible
00:44:29.780 --> 00:44:31.720
when we're doing
the renormalization
00:44:31.720 --> 00:44:32.620
or doing matching.
00:44:32.620 --> 00:44:34.550
And so we're free
to use quark states,
00:44:34.550 --> 00:44:35.675
so that's what we're doing.
00:44:38.320 --> 00:44:43.430
OK, that's one diagram.
00:44:43.430 --> 00:44:45.165
Now there's another diagram.
00:44:45.165 --> 00:44:48.980
I think that should be
B. Sometimes in my notes,
00:44:48.980 --> 00:44:52.345
I'll call it 1, which
doesn't make any sense.
00:44:52.345 --> 00:44:54.470
And we can contract the
gluon with the Wilson line.
00:44:54.470 --> 00:44:58.580
So there's that graph, and
there's a symmetric friend.
00:45:10.600 --> 00:45:12.880
And each of these actually
has two contractions
00:45:12.880 --> 00:45:16.930
because there was two
Wilson lines in the way we
00:45:16.930 --> 00:45:17.800
wrote our operators.
00:45:17.800 --> 00:45:21.670
So our operator, as we
wrote, is like this.
00:45:27.625 --> 00:45:29.000
And you can think--
so let's just
00:45:29.000 --> 00:45:30.583
think of a contraction
with the quark.
00:45:30.583 --> 00:45:32.780
You can think that there's
a contraction like that
00:45:32.780 --> 00:45:35.960
and there's a contraction
like that of a gluon--
00:45:35.960 --> 00:45:38.102
OK, I'm contracting
gluons with quarks.
00:45:38.102 --> 00:45:39.560
But really, what
I mean is that I'm
00:45:39.560 --> 00:45:42.810
contracting to the
Lagrangian, right,
00:45:42.810 --> 00:45:44.270
that this quark
is evolving under.
00:45:44.270 --> 00:45:47.510
So hopefully that's clear.
00:45:47.510 --> 00:45:50.270
All right, so there's two
different ways in which--
00:45:50.270 --> 00:45:52.790
when I work out the Feynman
rule for this thing where
00:45:52.790 --> 00:45:55.680
I attach the gluon,
you can either
00:45:55.680 --> 00:45:57.680
get the gluon from here
or the gluon from there.
00:45:57.680 --> 00:46:00.820
That's all I'm saying.
00:46:00.820 --> 00:46:02.570
But these actually
have different physical
00:46:02.570 --> 00:46:06.020
interpretations because
this delta function
00:46:06.020 --> 00:46:08.360
here, if you think
about what it's doing,
00:46:08.360 --> 00:46:12.410
it's really-- in the original
diagram, it's like the cut.
00:46:12.410 --> 00:46:15.800
So in the original diagrams
that we were drawing,
00:46:15.800 --> 00:46:19.040
we would cut them because
we'd take the imaginary part.
00:46:19.040 --> 00:46:21.030
And this delta function
is in the middle.
00:46:21.030 --> 00:46:22.730
We have kind of a
parton on this side
00:46:22.730 --> 00:46:25.160
and a squared
parton on that side.
00:46:25.160 --> 00:46:27.150
This delta function is the cut.
00:46:27.150 --> 00:46:28.790
So this contraction
here actually
00:46:28.790 --> 00:46:34.580
corresponds to a virtual
graph, and this guy here
00:46:34.580 --> 00:46:37.670
corresponds to real
emission because you're
00:46:37.670 --> 00:46:39.920
doing a contraction
across the cut, right?
00:46:39.920 --> 00:46:41.990
So one of these guys would
be a graph like this,
00:46:41.990 --> 00:46:43.865
and the other one would
be a graph like that.
00:46:46.940 --> 00:46:50.636
I can label them 1 and 2--
00:46:50.636 --> 00:46:52.010
1, 2.
00:46:55.706 --> 00:46:59.440
But we'll just keep them
and treat them all together.
00:46:59.440 --> 00:47:03.124
These two graphs give
an overall factor of 2.
00:47:03.124 --> 00:47:04.600
So that's simple.
00:47:08.854 --> 00:47:16.295
There's some
spinner stuff, which
00:47:16.295 --> 00:47:18.640
is even simpler in this
case, so I write it out.
00:47:23.510 --> 00:47:28.680
There are some stuff
from the Wilson line.
00:47:28.680 --> 00:47:31.470
And then there's
two propagators.
00:47:31.470 --> 00:47:34.720
Let me not write all the i0's.
00:47:34.720 --> 00:47:37.890
And then there's two
different delta functions.
00:47:37.890 --> 00:47:43.120
So either we have the real
graph where the w is inside,
00:47:43.120 --> 00:47:49.090
or we have the virtual
graph where the w--
00:47:49.090 --> 00:47:50.350
sorry.
00:47:50.350 --> 00:47:52.270
Either we have the
real graph where
00:47:52.270 --> 00:47:55.270
the loop goes around
the delta function,
00:47:55.270 --> 00:47:57.580
or we have the virtual graph
where this guy is overall
00:47:57.580 --> 00:47:58.540
on that thing.
00:47:58.540 --> 00:48:01.870
So in the overall one,
it's just a p minus minus w
00:48:01.870 --> 00:48:03.640
like it was at tree level.
00:48:03.640 --> 00:48:08.180
And in the real emission,
it's an l minus minus w.
00:48:08.180 --> 00:48:09.380
And one's a w.
00:48:09.380 --> 00:48:10.250
One's a w dagger.
00:48:10.250 --> 00:48:11.480
So there's a relative sign.
00:48:14.850 --> 00:48:17.245
So the sign is just
easier to understand
00:48:17.245 --> 00:48:22.120
as w versus w dagger,
which has a relative sign.
00:48:22.120 --> 00:48:23.800
OK, so if we just
followed our nose
00:48:23.800 --> 00:48:26.358
with what the Feynman
rule for this thing is,
00:48:26.358 --> 00:48:27.400
that's what we would get.
00:48:34.310 --> 00:48:36.560
And this is, again, some
loop integral that we can do.
00:48:50.700 --> 00:48:54.410
One way of writing the
result is as follows.
00:48:54.410 --> 00:48:59.482
And there's one thing we have to
be careful about here which is
00:48:59.482 --> 00:49:00.690
why I'm writing this all out.
00:49:11.680 --> 00:49:13.120
So there's actually
a cancellation
00:49:13.120 --> 00:49:15.400
between the virtual
and the real diagrams
00:49:15.400 --> 00:49:19.470
of an infrared divergence, so I
want to be careful about that.
00:49:30.982 --> 00:49:36.260
So that's why I'm writing this
guy out in epsilon dimensions
00:49:36.260 --> 00:49:38.660
fully without expanding first.
00:49:38.660 --> 00:49:41.720
OK, so this is the
real contribution,
00:49:41.720 --> 00:49:44.120
and this is the virtual.
00:49:44.120 --> 00:49:47.465
So in order to sort
of deal with this,
00:49:47.465 --> 00:49:50.090
we have to make use of something
that's called the distribution
00:49:50.090 --> 00:49:51.040
identity.
00:49:57.630 --> 00:50:00.177
If you know what the result is
for the anomalous dimension,
00:50:00.177 --> 00:50:02.010
you'll be aware of the
fact that it involves
00:50:02.010 --> 00:50:04.977
something called a plus
function because splitting
00:50:04.977 --> 00:50:06.435
functions for a
parton distribution
00:50:06.435 --> 00:50:08.185
involves something
called a plus function.
00:50:30.960 --> 00:50:34.265
So the way that we can deal
with that is as follows.
00:50:38.405 --> 00:50:39.780
The way we can
deal with the fact
00:50:39.780 --> 00:50:42.030
that actually the result's
going to be a distribution,
00:50:42.030 --> 00:50:46.830
we have to be careful
because you see, z goes to 1
00:50:46.830 --> 00:50:48.360
is being regulated by epsilon.
00:50:51.030 --> 00:50:53.490
And so if we integrate
over z, for example,
00:50:53.490 --> 00:50:55.230
it's epsilon that's
going to allow us
00:50:55.230 --> 00:50:58.620
to integrate all the way to 1.
00:50:58.620 --> 00:51:02.115
And we'd like to
encode that in some way
00:51:02.115 --> 00:51:03.990
where we can expand in
epsilon because that's
00:51:03.990 --> 00:51:05.840
what we need to do
in order to extract
00:51:05.840 --> 00:51:06.840
the anomalous dimension.
00:51:06.840 --> 00:51:08.757
And this formula is what
allows us to do that.
00:51:12.640 --> 00:51:15.070
So I'll tell you
how to derive it
00:51:15.070 --> 00:51:18.010
after I tell you what the l is.
00:51:18.010 --> 00:51:22.780
So ln of anything is defined
to be a plus function
00:51:22.780 --> 00:51:25.180
with a log to that power.
00:51:31.660 --> 00:51:33.400
And the plus function
is defined so
00:51:33.400 --> 00:51:38.830
that if you integrate
from 0 to 1, you get 0.
00:51:38.830 --> 00:51:44.557
And if you integrate
with a test function,
00:51:44.557 --> 00:51:48.700
which is the more general result
that you need to define it--
00:51:48.700 --> 00:51:52.750
so you can define it by this
result with a test function.
00:51:52.750 --> 00:51:55.850
And it just gives you
the normal function,
00:51:55.850 --> 00:51:58.510
but the test function
with a subtraction that
00:51:58.510 --> 00:52:00.190
makes the test function
more convergent
00:52:00.190 --> 00:52:03.700
so that you can
integrate through 0.
00:52:03.700 --> 00:52:05.920
OK, so that's the definition
of a plus function.
00:52:05.920 --> 00:52:08.860
You could also define
it with a limit.
00:52:08.860 --> 00:52:12.610
This will be sufficient.
00:52:12.610 --> 00:52:15.970
OK, so these things are
like delta functions.
00:52:15.970 --> 00:52:17.950
The way that you would
derive this formula
00:52:17.950 --> 00:52:21.220
is you would say, well,
if z is away from 1,
00:52:21.220 --> 00:52:24.400
then I can expand because
then there's no problem.
00:52:24.400 --> 00:52:26.470
And if z is away
from 1, it turns out
00:52:26.470 --> 00:52:28.930
that this plus function is
just the regular function.
00:52:28.930 --> 00:52:32.210
It's only at 1 that something
special is happening.
00:52:32.210 --> 00:52:35.390
And so the standard
expansion is what you'd
00:52:35.390 --> 00:52:37.930
get if you took z away from 1.
00:52:37.930 --> 00:52:39.850
And to see what's
happening at z equals 1,
00:52:39.850 --> 00:52:41.785
you'd just integrate
both sides from 0 to 1,
00:52:41.785 --> 00:52:43.660
and that's how you can
derive the coefficient
00:52:43.660 --> 00:52:44.577
of the delta function.
00:52:49.320 --> 00:52:54.630
All right, so if I plug this
formula in here for this thing,
00:52:54.630 --> 00:52:57.632
then I actually get another
1 over epsilon in this guy.
00:52:57.632 --> 00:53:00.090
There's a gamma of epsilon out
front, and that guy is good.
00:53:00.090 --> 00:53:01.350
This is our UV divergence.
00:53:01.350 --> 00:53:04.650
This is our 1 over epsilon UV.
00:53:04.650 --> 00:53:06.570
But there's also a
gamma of minus epsilon
00:53:06.570 --> 00:53:09.160
here, which is an IR divergence.
00:53:09.160 --> 00:53:11.310
So even though I tried
to regulate all the IR
00:53:11.310 --> 00:53:13.810
by off-shellness,
it didn't quite work
00:53:13.810 --> 00:53:16.380
and there was one that
was regulated by dimreg.
00:53:16.380 --> 00:53:19.890
And that one actually cancels
between these two pieces
00:53:19.890 --> 00:53:25.620
once I use this identity
and take into account
00:53:25.620 --> 00:53:29.070
that that's an IR divergence.
00:53:29.070 --> 00:53:36.360
So there's a 1 over epsilon
IR times 1 over epsilon UV.
00:53:36.360 --> 00:53:40.260
And that cancels between
the real and virtual graphs.
00:53:45.710 --> 00:53:48.450
So this is like a
standard 1 over epsilon IR
00:53:48.450 --> 00:53:50.200
canceling between real
and virtual graphs.
00:53:50.200 --> 00:53:51.790
And since it's only
the 1 over epsilon UV
00:53:51.790 --> 00:53:53.920
that we're interested in,
we're really only worried
00:53:53.920 --> 00:53:56.080
about that part of it canceling.
00:53:56.080 --> 00:53:58.415
There's a piece actually that--
00:53:58.415 --> 00:53:58.915
anyway.
00:54:06.640 --> 00:54:10.930
And then the 1 over
epsilon that's left
00:54:10.930 --> 00:54:17.290
is the guy that
we're after in order
00:54:17.290 --> 00:54:18.932
to get the anomalous dimension.
00:54:24.600 --> 00:54:28.950
All right, so let me not--
00:54:28.950 --> 00:54:30.600
so in my notes, I
write one more line
00:54:30.600 --> 00:54:32.310
where I expand this guy out.
00:54:32.310 --> 00:54:36.630
And I think just because of the
time, I'm going to skip that.
00:54:36.630 --> 00:54:38.670
And I'll just write
the final result.
00:54:38.670 --> 00:54:40.710
When we do the final
result, we also
00:54:40.710 --> 00:54:43.770
have to include wave
function renormalization.
00:54:43.770 --> 00:54:49.380
So you can think of this
graph as a wave function
00:54:49.380 --> 00:54:51.990
renormalization term.
00:54:51.990 --> 00:54:53.937
And it just involves
the delta function again
00:54:53.937 --> 00:54:55.020
like the tree-level graph.
00:55:16.533 --> 00:55:18.080
[INAUDIBLE] like that.
00:55:18.080 --> 00:55:21.830
So in general, if I wanted to
do this calculation at one loop,
00:55:21.830 --> 00:55:25.820
there's one more type of
diagram I should consider, OK?
00:55:25.820 --> 00:55:28.840
And that's a graph where
I could have mixing.
00:55:28.840 --> 00:55:33.890
This guy should be dashed since
we're in the effective theory.
00:55:33.890 --> 00:55:36.350
So how does the
mixing graph work?
00:55:36.350 --> 00:55:38.720
Well, there's a graph where
I have external gluons,
00:55:38.720 --> 00:55:42.236
but I still am renormalizing
the same operator.
00:55:42.236 --> 00:55:45.800
I've still inserted the
quark operator here,
00:55:45.800 --> 00:55:49.130
but now we have
antiquarks in this theory.
00:55:49.130 --> 00:55:50.900
We can draw a
triangle like that.
00:55:50.900 --> 00:55:54.537
And this graph here would
give a mixing that involves--
00:55:54.537 --> 00:55:56.870
that would give a mixing term
in the anomalous dimension
00:55:56.870 --> 00:56:00.250
where you're mixing
gluons and quarks.
00:56:00.250 --> 00:56:07.550
So this mix is what we
sort of called O glue.
00:56:07.550 --> 00:56:11.830
Let me just say this, that
it mixes O glue with O quark.
00:56:11.830 --> 00:56:13.760
And we could compute
this graph too,
00:56:13.760 --> 00:56:17.680
but I'm going to neglect
it just for simplicity.
00:56:17.680 --> 00:56:20.320
I just won't write it down.
00:56:20.320 --> 00:56:23.650
One way of doing
that rigorously would
00:56:23.650 --> 00:56:26.770
be to consider operators where
the flavors of these guys
00:56:26.770 --> 00:56:28.660
are different, OK?
00:56:28.660 --> 00:56:31.630
That's what would happen, for
example, if you were having
00:56:31.630 --> 00:56:32.830
a w exchange or something.
00:56:32.830 --> 00:56:36.220
So we could look at
nonflavored diagonal operators
00:56:36.220 --> 00:56:39.043
with, like, a u
quark and a d quark.
00:56:39.043 --> 00:56:41.210
And then you would not have
this mixing with O glue.
00:56:41.210 --> 00:56:42.793
It's only if the
flavors of the quarks
00:56:42.793 --> 00:56:45.800
are the same that you can
write down this diagram.
00:56:45.800 --> 00:56:50.718
But just think about it as I'm
focusing on the quark piece,
00:56:50.718 --> 00:56:52.510
and in general, there's
also a gluon piece.
00:57:04.360 --> 00:57:05.830
So we have all our
one-loop graphs.
00:57:05.830 --> 00:57:07.480
We know how to expand
them in epsilon.
00:57:07.480 --> 00:57:09.953
And so we just proceed,
expand them in epsilon,
00:57:09.953 --> 00:57:10.620
and add them up.
00:57:28.340 --> 00:57:30.400
So you could think
that what we derive
00:57:30.400 --> 00:57:35.680
by doing that is a distribution
for a quark inside a quark.
00:57:35.680 --> 00:57:39.010
So here, I'm being--
this is the state
00:57:39.010 --> 00:57:41.250
and this is what
type of operator.
00:57:47.360 --> 00:57:51.880
And it's a function of some z.
00:57:51.880 --> 00:57:57.690
And if I go up to one
loop, then the tree level
00:57:57.690 --> 00:58:02.920
was just a delta function
of that fraction z.
00:58:02.920 --> 00:58:05.290
And then at one loop, we
had all these other terms.
00:58:12.880 --> 00:58:19.810
So if I collect all the pieces,
I had some delta functions.
00:58:19.810 --> 00:58:24.070
The graph with the Wilson
lines actually gives me
00:58:24.070 --> 00:58:26.810
one of these L0 functions.
00:58:26.810 --> 00:58:27.950
And then the graph--
00:58:27.950 --> 00:58:29.860
so there's wave
function normalization
00:58:29.860 --> 00:58:33.727
plus some other terms that
involve delta function.
00:58:33.727 --> 00:58:35.185
And then there's
some other pieces.
00:58:39.820 --> 00:58:42.880
And then this is all
times 1 over epsilon.
00:58:42.880 --> 00:58:44.668
And then there's other pieces.
00:58:44.668 --> 00:58:46.960
But if we're interested in
ultraviolet renormalization,
00:58:46.960 --> 00:58:50.050
we only care about
the 1 over epsilon.
00:58:50.050 --> 00:58:55.390
And all those terms
can be written
00:58:55.390 --> 00:58:57.070
in a kind of more
compact form, which
00:58:57.070 --> 00:59:09.130
is the more standard form
for the anomalous dimension.
00:59:09.130 --> 00:59:12.940
You can actually group them
all together into a single plus
00:59:12.940 --> 00:59:17.030
function like this.
00:59:17.030 --> 00:59:19.900
So just in terms
of distributions,
00:59:19.900 --> 00:59:22.840
this distribution is equal
to the sum of these pieces.
00:59:35.190 --> 00:59:37.380
You can see, as z goes
to 1, that there'd be a 2
00:59:37.380 --> 00:59:38.340
here and a 2 here.
00:59:38.340 --> 00:59:40.110
And this would be
1 over 1 minus z.
00:59:40.110 --> 00:59:42.433
And as z goes to 1
here, that would be 1
00:59:42.433 --> 00:59:43.975
and this would be
a 1 over 1 minus z.
00:59:43.975 --> 00:59:48.110
So you see some pieces
of it matching up.
00:59:50.890 --> 00:59:53.710
Basically, the way that you
would derive this is you'd
00:59:53.710 --> 01:00:00.550
write 1 plus z squared
is a plus b, 1 minus z
01:00:00.550 --> 01:00:02.800
plus c, 1 minus z squared.
01:00:02.800 --> 01:00:04.760
You'd work out what
a, b, and c are,
01:00:04.760 --> 01:00:07.070
just relating two polynomials.
01:00:07.070 --> 01:00:11.482
And then this guy here, the
1 minus z in the numerator
01:00:11.482 --> 01:00:12.940
cancels the one in
the denominator,
01:00:12.940 --> 01:00:14.482
and it's not a plus
function anymore.
01:00:14.482 --> 01:00:15.810
It's just a number.
01:00:15.810 --> 01:00:19.790
And that's how you would
connect the two formulas.
01:00:19.790 --> 01:00:23.110
All right, so we were
after determining the z.
01:00:23.110 --> 01:00:27.170
The z has to cancel
this 1 over epsilon.
01:00:27.170 --> 01:00:30.550
So let's go back to our
formula which connected
01:00:30.550 --> 01:00:34.960
those, which was this.
01:00:34.960 --> 01:00:39.370
Our general formula was
that the bare guy could
01:00:39.370 --> 01:00:48.370
be written in terms of
split into UV pieces
01:00:48.370 --> 01:00:50.320
and finite pieces in
the following ways,
01:00:50.320 --> 01:00:53.600
is with this integral.
01:00:53.600 --> 01:00:56.800
Now, this looks like it could
be an arbitrary function of xi
01:00:56.800 --> 01:00:59.860
and xi prime, but our result
here was only a function of z,
01:00:59.860 --> 01:01:02.000
which is actually a ratio.
01:01:02.000 --> 01:01:05.350
And that's actually something
that we can argue in general,
01:01:05.350 --> 01:01:08.710
that this thing here is
actually only a function
01:01:08.710 --> 01:01:10.750
of one variable, not two.
01:01:23.330 --> 01:01:25.190
So that follows from
two different things.
01:01:25.190 --> 01:01:27.440
It follows from
RPI III invariance.
01:01:30.170 --> 01:01:32.060
So remember that
RPI III invariance
01:01:32.060 --> 01:01:35.300
said that you should have the
same number of n's and n bars.
01:01:35.300 --> 01:01:40.220
And remember-- OK, so that's
one thing that you have to use.
01:01:40.220 --> 01:01:42.080
That tells you that
you need to get ratios.
01:01:42.080 --> 01:01:43.690
Well, the z's are
already ratios.
01:01:43.690 --> 01:01:48.290
So you might say, well,
that should be fine.
01:01:48.290 --> 01:01:52.525
The z's are already ratios
between the momentum
01:01:52.525 --> 01:01:53.900
and the operator
and the momentum
01:01:53.900 --> 01:01:55.942
and the state, the minus
momentum of the operator
01:01:55.942 --> 01:02:01.970
over the minus momentum
of the state, right?
01:02:01.970 --> 01:02:03.230
And this is a minus momentum.
01:02:03.230 --> 01:02:04.320
That's a minus momentum.
01:02:04.320 --> 01:02:06.425
So the z's are
RPI III invariant.
01:02:06.425 --> 01:02:08.948
So that doesn't seem
like it would imply this.
01:02:08.948 --> 01:02:10.490
But there's one
other thing you know,
01:02:10.490 --> 01:02:13.790
and that is that it can't
depend on the state momentum.
01:02:13.790 --> 01:02:15.500
I could have taken a proton.
01:02:15.500 --> 01:02:17.180
I could have taken a quark.
01:02:17.180 --> 01:02:18.800
And the result for
the renormalization
01:02:18.800 --> 01:02:22.580
shouldn't depend on
what state I'm taking.
01:02:22.580 --> 01:02:24.110
And this combination
where I have
01:02:24.110 --> 01:02:28.220
d xi prime xi prime with a xi
over xi prime, the p minuses
01:02:28.220 --> 01:02:28.970
cancel out.
01:02:48.410 --> 01:02:50.960
So if I were to do the
whole thing with a proton
01:02:50.960 --> 01:02:53.393
state rather than a
quark state, then I
01:02:53.393 --> 01:02:55.310
should still get the
same anomalous dimension.
01:02:55.310 --> 01:02:58.100
And in order for
that to be true,
01:02:58.100 --> 01:03:00.600
it has to depend on the ratio.
01:03:00.600 --> 01:03:07.050
And that ratio is then just
a ratio of the bare operator
01:03:07.050 --> 01:03:09.340
and the renormalized operator.
01:03:09.340 --> 01:03:12.960
It's like saying, if
you had O of omega,
01:03:12.960 --> 01:03:17.610
there is a convolution
of z with an omega
01:03:17.610 --> 01:03:24.530
over omega prime or
something, with O omega
01:03:24.530 --> 01:03:27.140
prime renormalized.
01:03:27.140 --> 01:03:29.540
And if I had done it in
an operator level and not
01:03:29.540 --> 01:03:32.840
even written states,
then it would really just
01:03:32.840 --> 01:03:35.487
be RPI III invariance, OK?
01:03:35.487 --> 01:03:37.070
Because I wrote it
in terms of states,
01:03:37.070 --> 01:03:38.737
there was this other
momentum available,
01:03:38.737 --> 01:03:42.890
but I'm not allowed
to have that really
01:03:42.890 --> 01:03:45.400
be playing a part
of the discussion.
01:03:48.880 --> 01:03:52.550
So given that formula, then
I can expand to one loop.
01:03:52.550 --> 01:03:55.450
So this guy I think of as
having a tree-level result.
01:03:55.450 --> 01:03:57.220
This guy is a
matrix element that
01:03:57.220 --> 01:04:00.500
has a tree-level and
one-loop result as well.
01:04:00.500 --> 01:04:05.380
So if they're both tree
level, I get delta, 1 minus z.
01:04:10.730 --> 01:04:38.620
And in some kind of obvious
notation, up to one-loop order,
01:04:38.620 --> 01:04:41.558
I can write it out
formally like that.
01:04:41.558 --> 01:04:43.600
And then I know what these
tree-level things are.
01:04:43.600 --> 01:04:45.392
This guy's a delta
function, and this guy's
01:04:45.392 --> 01:04:46.670
also a delta function.
01:04:46.670 --> 01:04:47.920
So I can just do the integral.
01:05:00.570 --> 01:05:03.180
And it really is pretty simple.
01:05:03.180 --> 01:05:04.950
All the 1 over epsilon
terms are just z.
01:05:08.090 --> 01:05:13.710
And what's left would be
associated to this guy
01:05:13.710 --> 01:05:15.392
in perturbation theory.
01:05:15.392 --> 01:05:17.100
But if we want to do
the renormalization,
01:05:17.100 --> 01:05:21.600
we just need the z and
not worry about that.
01:05:21.600 --> 01:05:23.580
So we read off
from over here what
01:05:23.580 --> 01:05:28.120
z is because z is
just this right there.
01:05:36.130 --> 01:05:42.050
So z-- OK, z is just this thing.
01:05:47.878 --> 01:05:49.670
So when I put the
tree-level piece together
01:05:49.670 --> 01:05:51.795
with the one-loop piece,
then this thing is just z.
01:05:54.290 --> 01:05:56.240
And then I compute the
anomalous dimension
01:05:56.240 --> 01:05:57.890
by taking mu d by d mu of it--
01:06:00.960 --> 01:06:09.050
and that hits the alpha,
so that kills the epsilon
01:06:09.050 --> 01:06:12.220
and gives me a factor
of 2 and a minus sign.
01:06:12.220 --> 01:06:15.665
But the anomalous
dimension, gamma qq--
01:06:20.030 --> 01:06:23.800
so there was a 1 over xi prime.
01:06:23.800 --> 01:06:25.925
And then it was
minus mu d by d mu.
01:06:31.060 --> 01:06:33.550
And if I plug it in the
formula that we have,
01:06:33.550 --> 01:06:38.120
zqq of [? xi over ?]
[? xi ?] prime.
01:06:38.120 --> 01:06:39.460
So there's an a minus here.
01:06:39.460 --> 01:06:41.650
There's a minus there.
01:06:41.650 --> 01:06:44.290
And the 2 epsilon cancels
this 2 and that epsilon.
01:06:46.778 --> 01:06:48.320
And this 1 over xi
prime is the thing
01:06:48.320 --> 01:06:50.605
that we needed to make
the measure RPI invariant.
01:06:53.220 --> 01:07:00.150
So in this notation,
our original notation,
01:07:00.150 --> 01:07:03.510
putting all the pieces
together and being careful
01:07:03.510 --> 01:07:17.260
about beta functions, which I
was mostly suppressing, that's
01:07:17.260 --> 01:07:21.010
the result. OK, so this
is the function of xi
01:07:21.010 --> 01:07:23.440
over xi prime, which
I've just written as z.
01:07:23.440 --> 01:07:26.950
And then there's
some beta functions
01:07:26.950 --> 01:07:29.020
that are setting the
boundaries for the integral.
01:07:29.020 --> 01:07:31.400
And that comes also
out of the calculation.
01:07:31.400 --> 01:07:33.490
And that's the quark
one-loop splitting function.
01:07:36.680 --> 01:07:39.070
So if we've done the gluon
from that other diagram,
01:07:39.070 --> 01:07:40.510
that we've got the mixing term.
01:07:48.642 --> 01:07:50.600
OK, so this is the one-loop
anomalous dimension
01:07:50.600 --> 01:07:51.975
for the PDF, and
it's really just
01:07:51.975 --> 01:07:56.010
doing operator renormalization,
calculating one-loop diagrams
01:07:56.010 --> 01:07:57.010
in the effective theory.
01:08:00.180 --> 01:08:00.810
Questions?
01:08:09.035 --> 01:08:09.535
OK.
01:08:12.090 --> 01:08:15.660
So one question
that you can ask,
01:08:15.660 --> 01:08:17.609
which is an
interesting question,
01:08:17.609 --> 01:08:22.649
is when we did this
result for the DIS,
01:08:22.649 --> 01:08:25.050
we got this convolution
between the hard function
01:08:25.050 --> 01:08:27.359
and the Parton
distribution function.
01:08:27.359 --> 01:08:29.250
And you can ask,
why did that happen
01:08:29.250 --> 01:08:31.470
and, in general, is there
a way of characterizing
01:08:31.470 --> 01:08:32.760
when it could possibly happen?
01:08:36.547 --> 01:08:38.630
Because if you think about
the answer that we got,
01:08:38.630 --> 01:08:41.250
it was just Wilson
coefficient times operator.
01:08:41.250 --> 01:08:43.191
And the really only
nontrivial thing about it
01:08:43.191 --> 01:08:44.899
was that there was
this one momentum that
01:08:44.899 --> 01:08:46.899
could kind of trade back
and forth between them.
01:08:46.899 --> 01:08:50.450
There was an integral
in the answer.
01:08:50.450 --> 01:08:52.069
And actually,
power counting even
01:08:52.069 --> 01:08:57.270
constrains how those integrals
can, in general, show up.
01:08:57.270 --> 01:09:02.040
So if you ask most generally
what could possibly happen--
01:09:02.040 --> 01:09:04.160
and just thinking about
the power counting
01:09:04.160 --> 01:09:06.020
for the degrees of
freedom actually
01:09:06.020 --> 01:09:08.180
tells us what type of
integrals can show up
01:09:08.180 --> 01:09:09.470
in factorization theorems.
01:09:33.396 --> 01:09:34.979
This is constrained
by power counting.
01:09:42.920 --> 01:09:45.170
I keep forgetting to say
that there's a makeup lecture
01:09:45.170 --> 01:09:45.560
tomorrow.
01:09:45.560 --> 01:09:47.359
I sent around an email,
but I should have--
01:09:52.569 --> 01:09:54.279
tomorrow, this room at 10:00 AM.
01:10:00.580 --> 01:10:02.890
And the lecture next week
is canceled on Tuesday.
01:10:02.890 --> 01:10:05.740
That's why we have a
makeup lecture tomorrow.
01:10:05.740 --> 01:10:11.510
OK, so in what way is it
constrained by power counting?
01:10:11.510 --> 01:10:14.420
So if you think about the
degrees of freedom that we had,
01:10:14.420 --> 01:10:20.942
say, for SCET I, then we had
hard, collinear, and soft--
01:10:20.942 --> 01:10:23.080
so let's just take a
simple case with only one
01:10:23.080 --> 01:10:25.180
type of collinear--
01:10:25.180 --> 01:10:27.970
hard, collinear, and ultrasoft.
01:10:27.970 --> 01:10:32.260
And the p mu of these guys in
terms of plus, minus, and perp
01:10:32.260 --> 01:10:33.422
components--
01:10:36.254 --> 01:10:38.552
I should be more
fancy about this.
01:10:42.172 --> 01:10:43.160
[INAUDIBLE]
01:10:46.130 --> 01:10:48.620
So if you think about just
power counting for the momentum,
01:10:48.620 --> 01:10:49.385
it was as follows.
01:10:52.850 --> 01:10:56.810
Factorization was separating
these different things
01:10:56.810 --> 01:10:58.100
into different objects.
01:10:58.100 --> 01:11:00.050
We had a Wilson
coefficient for the hard.
01:11:00.050 --> 01:11:02.600
In the case we just did, we only
had a proton matrix element.
01:11:02.600 --> 01:11:04.642
For the collinear, we
didn't have any ultrasofts.
01:11:04.642 --> 01:11:07.100
If we put the ultrasofts in,
they would have all cancelled
01:11:07.100 --> 01:11:07.730
away.
01:11:07.730 --> 01:11:10.250
We wouldn't have seen any
ultrasofts showing up.
01:11:10.250 --> 01:11:12.620
And that's because the
operator we were dealing with,
01:11:12.620 --> 01:11:15.530
the Wilson lines would just have
cancelled completely out of it.
01:11:15.530 --> 01:11:17.570
But actually, it turns out,
for deep inelastic scattering,
01:11:17.570 --> 01:11:19.362
that you shouldn't even
include ultrasofts.
01:11:19.362 --> 01:11:22.140
They're not a good degree
of freedom to include there.
01:11:22.140 --> 01:11:24.410
So really, for the process
that we were talking about,
01:11:24.410 --> 01:11:27.830
you really should
only take those two.
01:11:27.830 --> 01:11:31.430
But anyway, more generally
in some other process,
01:11:31.430 --> 01:11:33.590
you would have these
three different things.
01:11:33.590 --> 01:11:35.360
And the way that
convolutions can show up
01:11:35.360 --> 01:11:39.080
is simply who can trade
momentum with who.
01:11:39.080 --> 01:11:43.940
So this is plus momenta, minus
momenta, and perp momentum.
01:11:43.940 --> 01:11:46.230
And in order for
momentum to be exchanged,
01:11:46.230 --> 01:11:48.160
they have to be
of the same size.
01:11:48.160 --> 01:11:54.020
So these guys here are the same
size and they can be exchanged,
01:11:54.020 --> 01:11:56.910
and that's exactly what showed
up in our DIS factorization
01:11:56.910 --> 01:11:57.410
theorem.
01:11:57.410 --> 01:12:00.617
The hard Wilson coefficients
exchanged minus momentum
01:12:00.617 --> 01:12:03.200
with the collinear [INAUDIBLE]
because they are the same order
01:12:03.200 --> 01:12:04.940
in the power counting.
01:12:04.940 --> 01:12:07.213
In another case,
in a more general
01:12:07.213 --> 01:12:08.630
or in some other
example, we might
01:12:08.630 --> 01:12:13.222
find that there was
nontrivial ultrasoft stuff,
01:12:13.222 --> 01:12:14.680
and then we could
get a convolution
01:12:14.680 --> 01:12:17.100
in the plus momentum and
collinear and ultrasoft
01:12:17.100 --> 01:12:18.350
because they're the same size.
01:12:22.240 --> 01:12:24.910
And so that's a
pretty simple way
01:12:24.910 --> 01:12:27.400
of thinking about why those
integrals can possibly show up.
01:12:27.400 --> 01:12:29.680
It's just because the two
sectors can talk to each other
01:12:29.680 --> 01:12:31.722
because they have momenta
that are the same size.
01:12:31.722 --> 01:12:34.300
And then the rest is about
momentum conservation
01:12:34.300 --> 01:12:37.660
because momentum conservation
places nontrivial constraints.
01:12:37.660 --> 01:12:39.340
And we saw in the DIS
example that there
01:12:39.340 --> 01:12:41.080
were two omegas to
start, but one of them
01:12:41.080 --> 01:12:43.497
was projected to 0 because it
was a forward matrix element
01:12:43.497 --> 01:12:47.990
and we only had one integral.
01:12:47.990 --> 01:12:51.710
And that also has
analogs elsewhere.
01:12:51.710 --> 01:12:56.240
If we do SCET II, which we
haven't talked about yet--
01:12:56.240 --> 01:12:59.330
we did talk about what the
degrees of freedom were.
01:12:59.330 --> 01:13:05.210
And again, if I try
to make it completely
01:13:05.210 --> 01:13:08.487
generic for some examples
that we'll treat later,
01:13:08.487 --> 01:13:10.070
then I can write
down something that's
01:13:10.070 --> 01:13:15.600
a slightly extended version of
what we talked about so far.
01:13:15.600 --> 01:13:19.850
So I can have Q, Q,
Q again for the hard.
01:13:19.850 --> 01:13:22.050
And then I can
have my collinear,
01:13:22.050 --> 01:13:28.580
which is Q lambda squared, Q, Q
lambda, and then soft, which is
01:13:28.580 --> 01:13:32.943
Q lambda, Q lambda, Q lambda.
01:13:32.943 --> 01:13:34.610
And it turns out that
sometimes, there's
01:13:34.610 --> 01:13:38.600
also another mode which
we haven't talked about,
01:13:38.600 --> 01:13:46.395
but I'll include it
for completeness, which
01:13:46.395 --> 01:13:48.430
is kind of a
collinear mode that's
01:13:48.430 --> 01:13:50.440
in between the
low-energy collinear
01:13:50.440 --> 01:13:52.896
mode and the high-energy
collinear mode.
01:13:52.896 --> 01:13:55.030
AUDIENCE: Do you mean the
square root of lambda?
01:13:55.030 --> 01:13:58.570
IAIN STEWART: Yeah, square
root of lambda, sorry.
01:13:58.570 --> 01:14:01.375
Yeah, otherwise my
dimensions are wrong.
01:14:05.830 --> 01:14:07.030
Yeah.
01:14:07.030 --> 01:14:09.160
So again here, you can just--
01:14:09.160 --> 01:14:11.998
I mean, the reason I was
extending this is just to,
01:14:11.998 --> 01:14:14.540
again, argue that it's kind of
simple to see what can happen.
01:14:14.540 --> 01:14:17.593
So in general, when you
think about convolutions
01:14:17.593 --> 01:14:19.510
in the hard momentum,
it could be in this case
01:14:19.510 --> 01:14:20.950
between these three modes.
01:14:20.950 --> 01:14:22.600
There could be some integrals.
01:14:22.600 --> 01:14:25.810
And then look where else
there can be something.
01:14:25.810 --> 01:14:29.035
So this and this
are the same size.
01:14:31.610 --> 01:14:35.090
And this and this
are the same size.
01:14:35.090 --> 01:14:41.740
So in general, we
can have convolutions
01:14:41.740 --> 01:14:44.830
in general between
all these things,
01:14:44.830 --> 01:14:47.170
between the guys that
are the same size.
01:14:47.170 --> 01:14:49.420
But that's the most complicated
thing that can happen.
01:14:49.420 --> 01:14:51.760
It can't be more
complicated than that.
01:14:51.760 --> 01:14:54.080
And you see that
in some examples,
01:14:54.080 --> 01:14:57.628
you either have the purple
or the orange, but not both.
01:14:57.628 --> 01:14:59.170
That's kind of
typical that you don't
01:14:59.170 --> 01:15:02.920
get the most complicated thing.
01:15:02.920 --> 01:15:05.955
So when you have results from
observables that tell you
01:15:05.955 --> 01:15:07.330
how these things
couple together,
01:15:07.330 --> 01:15:09.050
those are called
factorization theorems.
01:15:09.050 --> 01:15:11.050
And in the effective
theory, because you sort of
01:15:11.050 --> 01:15:13.973
define the modes, separated
them at the start,
01:15:13.973 --> 01:15:16.390
you're kind of very quickly
getting to these factorization
01:15:16.390 --> 01:15:16.890
theorems.
01:15:27.090 --> 01:15:29.870
Let's see.
01:15:29.870 --> 01:15:33.560
So we're going to deal with a
bunch of different examples.
01:15:33.560 --> 01:15:37.380
And I decided that I'm going to
do it in the following order.
01:15:37.380 --> 01:15:39.870
So we're going to
do the next exam--
01:15:39.870 --> 01:15:42.560
so we're going to do
a bunch of examples
01:15:42.560 --> 01:15:50.030
in order to see the range of
possibilities that can happen.
01:15:53.710 --> 01:15:58.540
And so I'm going to stick
with SCET I for next lecture.
01:15:58.540 --> 01:16:02.670
So the next example we'll do,
which I'll call example one,
01:16:02.670 --> 01:16:06.650
is we'll do our
[? dijet ?] production.
01:16:16.650 --> 01:16:20.540
So this is a SCET I situation.
01:16:20.540 --> 01:16:22.410
And the difference
between the example--
01:16:22.410 --> 01:16:24.670
so so far, what we
did is we did DIS.
01:16:24.670 --> 01:16:26.990
DIS actually it was so
simple because it only
01:16:26.990 --> 01:16:29.420
had two degrees of freedom
that you could kind of either
01:16:29.420 --> 01:16:32.000
think of it as
SCET I or SCET II.
01:16:32.000 --> 01:16:34.010
I mean, technically,
it's more like SCET II,
01:16:34.010 --> 01:16:35.450
but it behaves like SCET I.
01:16:35.450 --> 01:16:36.735
But there's no ultrasofts.
01:16:36.735 --> 01:16:38.360
And remember, it's
ultrasofts and softs
01:16:38.360 --> 01:16:41.120
that are making the
distinction between the two.
01:16:41.120 --> 01:16:43.578
So if you just have
this mode and this mode,
01:16:43.578 --> 01:16:45.620
it's not really any
difference between calling it
01:16:45.620 --> 01:16:46.790
SCET I or SCET II.
01:16:46.790 --> 01:16:48.650
So it's just SCET.
01:16:48.650 --> 01:16:51.110
So e plus, e minus
to [? dijets ?]
01:16:51.110 --> 01:16:54.020
will be an SCET I example
which has ultrasofts.
01:16:58.330 --> 01:17:00.340
And actually, what
we'll find in this case
01:17:00.340 --> 01:17:02.695
is that it will be the purple.
01:17:05.220 --> 01:17:09.040
We'll get a purple convolution.
01:17:09.040 --> 01:17:10.420
We'll see how that happens.
01:17:10.420 --> 01:17:12.370
And we won't actually--
momentum conservation
01:17:12.370 --> 01:17:14.370
will rule out the possibility
of the orange one.
01:17:17.360 --> 01:17:19.370
So we'll see the
opposite situation
01:17:19.370 --> 01:17:24.470
where it could be that we
have ultrasoft modes as well,
01:17:24.470 --> 01:17:26.600
but then we only
get a convolution
01:17:26.600 --> 01:17:29.060
with those ultrasoft in
the factorization theorem
01:17:29.060 --> 01:17:30.830
and not with the [INAUDIBLE].
01:17:36.760 --> 01:17:40.540
And then we'll turn to SCET II.
01:17:40.540 --> 01:17:46.320
And I haven't totally decided
what processes I'll do,
01:17:46.320 --> 01:17:47.990
but I think I'll do
the following ones.
01:17:52.700 --> 01:17:55.000
So one thing that you can
do, which is pretty simple,
01:17:55.000 --> 01:17:59.110
is to look at something called
the photon-pion form factor.
01:17:59.110 --> 01:18:01.960
So real photon-to-pion
transition
01:18:01.960 --> 01:18:04.810
through another
virtual photon, but you
01:18:04.810 --> 01:18:08.320
can think of this happening
through a diagram like this
01:18:08.320 --> 01:18:11.990
with two quarks,
one of them that's
01:18:11.990 --> 01:18:14.510
off-shell and one of
them that's on-shell.
01:18:14.510 --> 01:18:17.110
This is pi 0.
01:18:17.110 --> 01:18:18.772
So this is a SCET II example.
01:18:18.772 --> 01:18:20.980
But again, it's pretty simple
because it's just going
01:18:20.980 --> 01:18:25.180
to involve one hadronic object.
01:18:25.180 --> 01:18:27.580
And actually, it will
just, in this case,
01:18:27.580 --> 01:18:28.945
have collinear modes.
01:18:32.740 --> 01:18:36.820
We can set things up so
the pion is n-collinear,
01:18:36.820 --> 01:18:41.170
and then we have hard modes.
01:18:47.710 --> 01:18:50.250
So that's one example we'll do.
01:18:50.250 --> 01:18:55.240
Another example we'll
do is B to D pi.
01:18:55.240 --> 01:18:57.190
And here, the B
and the D are soft.
01:18:57.190 --> 01:19:02.980
Remember, we talked about this
one, and the pion is collinear.
01:19:02.980 --> 01:19:05.950
And then we have hard modes.
01:19:05.950 --> 01:19:09.280
So this is kind of like,
in some sense, a DIS,
01:19:09.280 --> 01:19:12.490
but it's an exclusive
process, not an inclusive one.
01:19:12.490 --> 01:19:14.740
And so actually, all
the tools that we use,
01:19:14.740 --> 01:19:15.820
which were kind of--
01:19:15.820 --> 01:19:18.700
in DIS, it's the most inclusive
process you can think of.
01:19:18.700 --> 01:19:20.920
It's deep inelastic scattering.
01:19:20.920 --> 01:19:22.750
The I is for "inclusive."
01:19:22.750 --> 01:19:24.983
Here, we're doing something
completely exclusive,
01:19:24.983 --> 01:19:27.400
but we'll see that all the
things that we've been thinking
01:19:27.400 --> 01:19:29.233
about, which is just
separation of collinear
01:19:29.233 --> 01:19:31.750
modes and hard modes, will just
go through for that process
01:19:31.750 --> 01:19:33.220
too.
01:19:33.220 --> 01:19:35.470
So the effective
theory, the difference
01:19:35.470 --> 01:19:40.630
is that in this case, you will
not be taking the amplitude
01:19:40.630 --> 01:19:41.130
squared.
01:19:41.130 --> 01:19:42.922
You won't be looking
at forward scattering.
01:19:42.922 --> 01:19:45.310
The forward scattering was
what was making it inclusive.
01:19:45.310 --> 01:19:47.440
We were summing over
all the final states.
01:19:47.440 --> 01:19:49.687
Here, there's only
one final state.
01:19:49.687 --> 01:19:52.270
So the difference between this
example and the one we just did
01:19:52.270 --> 01:20:04.210
is that in this case, we'll
factor the amplitude, not
01:20:04.210 --> 01:20:05.260
the squared amplitude.
01:20:08.442 --> 01:20:10.900
But other than that, it'll look
very similar to the example
01:20:10.900 --> 01:20:12.880
that we did for DIS.
01:20:12.880 --> 01:20:14.538
B to D pi, then, is
an SCET II example
01:20:14.538 --> 01:20:17.080
where we make things a little
more complicated because now we
01:20:17.080 --> 01:20:19.720
have soft, collinear,
and hard modes.
01:20:19.720 --> 01:20:23.150
And we'll see what
happens there.
01:20:23.150 --> 01:20:25.120
And then I'll do some
more examples after that.
01:20:25.120 --> 01:20:30.055
But let me say we'll
do some LHC examples.
01:20:32.860 --> 01:20:44.210
And I think we'll
also do broadening,
01:20:44.210 --> 01:20:47.240
which is another e plus,
e minus observable.
01:20:47.240 --> 01:20:49.400
All right, so that's
where we're going,
01:20:49.400 --> 01:20:53.280
and we'll start
going there next time
01:20:53.280 --> 01:20:57.355
by talking about
[? dijets ?] and SCET I.