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IAIN STEWART: OK,
so last time, we

00:00:24.100 --> 00:00:26.920
were talking about the
[? massive ?] Sudakov form

00:00:26.920 --> 00:00:28.090
factor.

00:00:28.090 --> 00:00:30.490
We're going to continue
that discussion today.

00:00:30.490 --> 00:00:33.467
And we saw that in order
to do this calculation,

00:00:33.467 --> 00:00:35.050
we needed to have
additional regulator

00:00:35.050 --> 00:00:37.900
besides dimensional
regularization.

00:00:37.900 --> 00:00:40.930
This was an example in SCT2
where we had these rapidity

00:00:40.930 --> 00:00:42.310
divergences.

00:00:42.310 --> 00:00:44.140
And so there is this
additional regulator

00:00:44.140 --> 00:00:47.260
that led to these 1 over
8 eta poles in our answer

00:00:47.260 --> 00:00:50.320
and these logs of nu as well.

00:00:53.803 --> 00:00:55.720
And we said that if you
stare at these results

00:00:55.720 --> 00:00:59.230
that you could see where you
would need to take those scale

00:00:59.230 --> 00:01:02.230
parameters in order to
minimize these logarithms.

00:01:02.230 --> 00:01:05.290
And it is such that in
the collinear diagrams,

00:01:05.290 --> 00:01:06.925
you need to take nu
of order p minus.

00:01:11.640 --> 00:01:13.860
And p minus is the hard scale.

00:01:13.860 --> 00:01:16.710
And you want you mu
of order m, which

00:01:16.710 --> 00:01:19.350
is this scale of the hyperbola.

00:01:19.350 --> 00:01:22.890
And in the soft,
you want nu and mu

00:01:22.890 --> 00:01:25.530
to be the same size and
then both of order m.

00:01:30.330 --> 00:01:32.310
So what we'll do in
a minute is we'll--

00:01:32.310 --> 00:01:33.900
so you can imagine
what happens next.

00:01:33.900 --> 00:01:37.140
You add some counter terms,
remove these 1 over epsilon

00:01:37.140 --> 00:01:38.880
poles and 1 over eta poles.

00:01:38.880 --> 00:01:41.730
You're left with something
that's a finite result that's

00:01:41.730 --> 00:01:43.530
a function of the cutoffs.

00:01:43.530 --> 00:01:45.362
And you put things together.

00:01:45.362 --> 00:01:47.070
When you put things
together, the cutoffs

00:01:47.070 --> 00:01:49.543
should cancel out of
the physical observable,

00:01:49.543 --> 00:01:50.835
which here was the form factor.

00:01:53.910 --> 00:01:58.970
So the form of the
factorization theorem for this--

00:01:58.970 --> 00:02:01.150
so you have some hard function.

00:02:01.150 --> 00:02:06.870
It's just the
Wilson coefficient,

00:02:06.870 --> 00:02:13.180
some collinear function
which I'll can cn,

00:02:13.180 --> 00:02:20.055
cn bar, and then
some soft function.

00:02:25.110 --> 00:02:27.120
And so it's the usual
story in some sense,

00:02:27.120 --> 00:02:30.330
but we have this additional
dependence on a parameter nu,

00:02:30.330 --> 00:02:32.690
not just mu.

00:02:32.690 --> 00:02:35.613
OK, so this same statements
I was making up there kind

00:02:35.613 --> 00:02:36.780
of encoded in the arguments.

00:02:36.780 --> 00:02:37.988
You want mu to be of order m.

00:02:37.988 --> 00:02:40.800
You want nu to be of order
q, mu to be of order m nu

00:02:40.800 --> 00:02:43.380
to be of order q, mu to
be of order m, nu to be of

00:02:43.380 --> 00:02:45.400
[? order mu, ?]
which is order m.

00:02:45.400 --> 00:02:49.860
So I could have written
nu over m here as well.

00:02:49.860 --> 00:02:52.080
OK, and then there's also
some hard factor that

00:02:52.080 --> 00:02:55.440
really looks at the hard scale.

00:02:55.440 --> 00:03:08.310
So renormalization, and they
have two cut off parameters.

00:03:15.760 --> 00:03:17.423
So what we'll
have, therefore, is

00:03:17.423 --> 00:03:18.840
we'll have a
renormalization group

00:03:18.840 --> 00:03:21.895
for these objects in a two
dimensional space-- mu and nu

00:03:21.895 --> 00:03:22.395
space.

00:03:42.510 --> 00:03:47.940
And we'll need both of those
to sum the large logarithms.

00:03:50.610 --> 00:03:52.738
So just having
renormalization group in mu

00:03:52.738 --> 00:03:54.780
would not be enough to
solve the large logarithms

00:03:54.780 --> 00:03:55.490
in this problem.

00:04:02.800 --> 00:04:04.925
So let's-- even before we--
so we'll go through it,

00:04:04.925 --> 00:04:07.175
and we'll write down anomalous
dimensions in a minute.

00:04:07.175 --> 00:04:08.550
But even before
we do that, let's

00:04:08.550 --> 00:04:12.630
just picture what we
want from the running.

00:04:12.630 --> 00:04:17.635
So here's a two
dimensional space where

00:04:17.635 --> 00:04:18.760
we want to do that running.

00:04:18.760 --> 00:04:23.287
So let's put mu on this
axis and nu on this axis.

00:04:23.287 --> 00:04:24.120
Break out the color.

00:04:28.450 --> 00:04:30.383
So we have some hard
degrees of freedom

00:04:30.383 --> 00:04:31.800
that we're integrating
out, right?

00:04:31.800 --> 00:04:33.970
But they live somewhere
in this picture.

00:04:33.970 --> 00:04:37.320
So let's say here, we
have mu of order 2.

00:04:37.320 --> 00:04:40.260
The hard degrees of freedom,
as I wrote over here,

00:04:40.260 --> 00:04:42.240
they don't depend on nu.

00:04:42.240 --> 00:04:45.510
We talked about that last time,
and one way of seeing it is nu

00:04:45.510 --> 00:04:47.760
is just a parameter
that distinguishes modes

00:04:47.760 --> 00:04:48.930
within the effective theory.

00:04:48.930 --> 00:04:51.420
It's a regulator that was
needed to distinguish soft

00:04:51.420 --> 00:04:54.450
from collinear, but if you
add it up these contributions,

00:04:54.450 --> 00:04:58.530
the nu dependence and the
1 over etas all cancel.

00:04:58.530 --> 00:05:02.640
So the hard degrees of freedom
only depend on mu and q.

00:05:02.640 --> 00:05:04.330
And so they don't care about nu.

00:05:04.330 --> 00:05:07.950
So we can draw them as a line.

00:05:07.950 --> 00:05:08.480
It's hard.

00:05:11.070 --> 00:05:13.600
They don't live
localized in that space.

00:05:13.600 --> 00:05:16.260
But the other modes, we
could put a dot associated

00:05:16.260 --> 00:05:18.970
with each one of them.

00:05:18.970 --> 00:05:24.797
So you have nu of order m here.

00:05:24.797 --> 00:05:25.630
[INAUDIBLE] equal m.

00:05:33.680 --> 00:05:35.460
Mu equals q.

00:05:35.460 --> 00:05:40.790
So then we need to have a low
energy mu scale, mu equals m.

00:05:40.790 --> 00:05:42.740
So the soft modes
are going to live

00:05:42.740 --> 00:05:45.110
at mu equals m, mu equals m.

00:05:48.090 --> 00:05:52.620
And then the collinear mode's
going to live over here.

00:05:52.620 --> 00:05:56.190
And the way we've set up
our calculation, we didn't--

00:05:56.190 --> 00:06:00.690
we sort of had one cut off
for both the cn and cn bar.

00:06:00.690 --> 00:06:03.210
So they both live
at the same point.

00:06:03.210 --> 00:06:06.810
It could have been
more fancy, but that

00:06:06.810 --> 00:06:10.570
suffices for doing
the calculations here.

00:06:10.570 --> 00:06:12.240
So they both live
at nu equals q.

00:06:12.240 --> 00:06:13.950
We basically had a--

00:06:13.950 --> 00:06:16.620
we didn't care about whether
it was p minus or p plus.

00:06:16.620 --> 00:06:19.830
We regulated them both with nu.

00:06:19.830 --> 00:06:21.960
OK, so to do the
renormalization group,

00:06:21.960 --> 00:06:24.150
this is where the
modes want to live.

00:06:24.150 --> 00:06:27.570
But we need to connect them
because in this formula,

00:06:27.570 --> 00:06:29.970
they all have the
same mu and nu.

00:06:29.970 --> 00:06:31.860
So the general thing
that you could imagine

00:06:31.860 --> 00:06:37.440
is that there's some point mu
and nu in this space, right,

00:06:37.440 --> 00:06:39.833
and that you write down
this formula there.

00:06:39.833 --> 00:06:42.000
But then everybody has large
logs because you're not

00:06:42.000 --> 00:06:43.488
at the right point.

00:06:43.488 --> 00:06:45.780
And so you have to do a
renormalization group evolution

00:06:45.780 --> 00:06:47.368
to that point.

00:06:47.368 --> 00:06:49.410
So the most general thing
that you could think of

00:06:49.410 --> 00:06:54.810
would be that there would
be a renormalization

00:06:54.810 --> 00:06:57.620
group of the hard function
down to that point.

00:06:57.620 --> 00:07:00.437
And that's just the
whole line moves down.

00:07:00.437 --> 00:07:02.020
So this would be
some evolution kernel

00:07:02.020 --> 00:07:05.850
uh that would run from some
initial condition, which

00:07:05.850 --> 00:07:07.730
I'll call uh.

00:07:07.730 --> 00:07:11.160
I'll call this mu
equals mu h, and we'll

00:07:11.160 --> 00:07:12.660
call this guy mu light.

00:07:15.510 --> 00:07:17.650
Call this guy nu light.

00:07:17.650 --> 00:07:20.220
Call this guy nu heavy.

00:07:20.220 --> 00:07:22.240
Just have some names for them.

00:07:22.240 --> 00:07:24.840
And then these guys here would
have running in two dimensions,

00:07:24.840 --> 00:07:28.190
and you have to take a path
from this point to that point.

00:07:28.190 --> 00:07:30.630
And so here's a simple path.

00:07:30.630 --> 00:07:34.560
Just go over and then up.

00:07:34.560 --> 00:07:39.330
Just go over, up.

00:07:39.330 --> 00:07:41.250
And there'd be an
evolution kernel associated

00:07:41.250 --> 00:07:42.640
with each one of these things.

00:07:42.640 --> 00:07:46.005
So this mu 1 would
be some kernel mu

00:07:46.005 --> 00:07:51.690
n that would run for
mu light to some mu.

00:07:51.690 --> 00:07:53.305
And then it's at fixed new.

00:07:56.910 --> 00:07:59.670
And this guy here will
be some other evolution

00:07:59.670 --> 00:08:03.300
kernel which I'll call v because
it's running in nu space--

00:08:03.300 --> 00:08:05.850
so some linear evolution
kernel that's running in

00:08:05.850 --> 00:08:09.630
nu from some nu
heavy down to nu.

00:08:09.630 --> 00:08:11.730
And it's at fixed nu light.

00:08:15.380 --> 00:08:17.630
And then there would likewise
be kernels on this side.

00:08:17.630 --> 00:08:25.330
There would be a
mu s and a vs. OK,

00:08:25.330 --> 00:08:28.060
so that would be the
sort of general thing

00:08:28.060 --> 00:08:30.040
that you could possibly imagine.

00:08:30.040 --> 00:08:33.270
Now the choice at this
point is arbitrary,

00:08:33.270 --> 00:08:35.020
and you know that
effectively that there's

00:08:35.020 --> 00:08:37.419
consistency in the sense
that you can move that point

00:08:37.419 --> 00:08:39.492
around and nothing changes.

00:08:39.492 --> 00:08:41.409
So you can use your
freedom to pick this point

00:08:41.409 --> 00:08:43.720
to make your life as
simple as possible.

00:08:43.720 --> 00:08:45.590
If you pick it up here
at the hard scale,

00:08:45.590 --> 00:08:47.190
you don't have to
do any uh running.

00:08:47.190 --> 00:08:49.190
If you pick it over here
at the collinear scale,

00:08:49.190 --> 00:08:51.400
then you don't have to
do the collinear running.

00:08:51.400 --> 00:08:56.095
So we can do that
to simplify things.

00:09:04.170 --> 00:09:05.780
And that's again
just the freedom--

00:09:05.780 --> 00:09:09.590
the usual, simple freedom-- of
either running the coefficients

00:09:09.590 --> 00:09:11.120
or running the operators.

00:09:11.120 --> 00:09:12.800
And it's just a little
more complicated.

00:09:12.800 --> 00:09:15.050
That's what this freedom to
move this point around is.

00:09:20.490 --> 00:09:22.910
So this is like running
coefficients versus operators.

00:09:25.382 --> 00:09:26.840
It's just a little
more complicated

00:09:26.840 --> 00:09:28.257
because we have
these two cutoffs.

00:09:32.050 --> 00:09:39.090
So let's just pick,
for example, mu and nu

00:09:39.090 --> 00:09:44.140
to be mu light and nu heavy.

00:09:44.140 --> 00:09:47.850
So that's putting it over
here at the orange point.

00:09:50.400 --> 00:09:53.077
Then you would just
have evolution kernels.

00:09:53.077 --> 00:09:55.410
The softs would just have to
run all the way over there.

00:09:55.410 --> 00:09:57.250
So you wouldn't
need mu us either.

00:09:57.250 --> 00:10:04.390
You would just need nu s, vs.

00:10:04.390 --> 00:10:12.550
So you have vs, and vs would
run from starting at nu light

00:10:12.550 --> 00:10:13.810
over to nu heavy.

00:10:13.810 --> 00:10:25.420
So this is the
initial condition,

00:10:25.420 --> 00:10:26.980
and this is the final.

00:10:26.980 --> 00:10:30.770
And it's at fixed mu s.

00:10:30.770 --> 00:10:35.020
And then you'd have
a running from uh

00:10:35.020 --> 00:10:42.170
down to mu light from mu heavy.

00:10:42.170 --> 00:10:44.420
OK, so then you just have
these two evolution kernels.

00:10:44.420 --> 00:10:47.960
So it would be pretty simple.

00:10:47.960 --> 00:10:52.148
So there's a consistency of both
moving around in the nu space.

00:10:52.148 --> 00:10:54.690
When I move this point around,
there's a cancellation between

00:10:54.690 --> 00:10:57.970
vn and vs, and there's also a
cancellation between the three

00:10:57.970 --> 00:11:05.820
u's.

00:11:05.820 --> 00:11:09.970
So there's also in this thing,
there's a path independence.

00:11:09.970 --> 00:11:13.410
So I drew a particular path
where I went like that.

00:11:13.410 --> 00:11:14.820
I could have drawn another path.

00:11:14.820 --> 00:11:17.670
I could have gone up
first and then over.

00:11:17.670 --> 00:11:19.530
And because the
parameters nu and mu

00:11:19.530 --> 00:11:24.240
are independent parameters,
the choice of the path

00:11:24.240 --> 00:11:24.960
doesn't matter.

00:11:29.400 --> 00:11:31.140
We'll see that from
our calculations

00:11:31.140 --> 00:11:37.634
as well in a minute when I write
down the anomalous dimensions.

00:11:41.590 --> 00:11:43.330
But this is just
following from the fact

00:11:43.330 --> 00:11:46.060
that they're
independent parameters.

00:11:46.060 --> 00:11:47.860
And effectively,
what that means is

00:11:47.860 --> 00:11:50.650
if you think about
taking derivatives,

00:11:50.650 --> 00:11:54.340
you can switch the order of the
integration of the derivatives.

00:11:54.340 --> 00:12:00.700
So mu d by d mu and nu d by d
nu is the same as nu d by d nu,

00:12:00.700 --> 00:12:02.960
mu d by d mu.

00:12:02.960 --> 00:12:03.460
OK.

00:12:06.173 --> 00:12:08.090
There are examples of
effective field theories

00:12:08.090 --> 00:12:10.100
where you would set
up cutoff parameters,

00:12:10.100 --> 00:12:12.033
and this wouldn't be true.

00:12:12.033 --> 00:12:13.700
You'd think maybe you
have two cut offs,

00:12:13.700 --> 00:12:15.033
but actually, you only have one.

00:12:15.033 --> 00:12:18.310
There's an example of
this in our QCD, where

00:12:18.310 --> 00:12:20.270
if you try to draw
things in two dimensions,

00:12:20.270 --> 00:12:21.978
you find out that
there's really only one

00:12:21.978 --> 00:12:24.240
path that the effective
theory picks out.

00:12:24.240 --> 00:12:26.900
But in this case, there really
are two cutoff parameters.

00:12:29.510 --> 00:12:33.050
OK, so let's do that in
detail by going back over here

00:12:33.050 --> 00:12:35.822
to these formulas, writing
down the counter terms,

00:12:35.822 --> 00:12:38.280
deriving from those counter
terms the anomalous dimensions,

00:12:38.280 --> 00:12:40.820
and then we can see how you
would derive these evolution

00:12:40.820 --> 00:12:42.740
kernels in this picture.

00:12:42.740 --> 00:12:43.385
Yeah.

00:12:43.385 --> 00:12:49.340
AUDIENCE: Can you comment
on why sort of this

00:12:49.340 --> 00:12:52.356
works even though the mu and
the nu and the collinear cycle

00:12:52.356 --> 00:12:55.032
aren't, like, factorizing?

00:12:55.032 --> 00:12:57.490
Like, for example, why couldn't
I, just right off the bat--

00:12:57.490 --> 00:12:57.810
IAIN STEWART: Yeah.

00:12:57.810 --> 00:13:00.520
AUDIENCE: --regulators and never
have a factorization theorem

00:13:00.520 --> 00:13:02.030
but run in different scales?

00:13:02.030 --> 00:13:06.057
Like, this seems like too easy.

00:13:06.057 --> 00:13:07.640
IAIN STEWART: Yeah,
it seems too easy.

00:13:11.000 --> 00:13:13.790
It's really because
if you like the--

00:13:13.790 --> 00:13:17.540
there's a certain-- if you think
about renormalization theorems,

00:13:17.540 --> 00:13:19.610
which we know much
better from mu than--

00:13:19.610 --> 00:13:21.680
there's not really
that much known for nu.

00:13:21.680 --> 00:13:23.930
There is kind of a
renormalization theorem here

00:13:23.930 --> 00:13:27.230
that is related to the fact
that this divergence is

00:13:27.230 --> 00:13:30.860
really related to that iconal
propagator that we had.

00:13:30.860 --> 00:13:33.200
And even when you go
to higher loop orders,

00:13:33.200 --> 00:13:35.720
there's a certain universality
to that propagator that's

00:13:35.720 --> 00:13:37.620
making this whole thing work.

00:13:37.620 --> 00:13:40.820
So yeah.

00:13:40.820 --> 00:13:43.520
I think this is a
relatively new thing,

00:13:43.520 --> 00:13:47.290
and I think it hasn't
really fully been explored.

00:13:47.290 --> 00:13:49.540
So there's probably some
interesting work to do there.

00:13:54.160 --> 00:13:55.540
All right, so let's do it.

00:13:55.540 --> 00:13:58.420
Let's see what counter terms
we get and how things work out.

00:14:13.870 --> 00:14:15.610
So we're going to do
the standard thing.

00:14:15.610 --> 00:14:17.703
Think about renormalization
the objects.

00:14:17.703 --> 00:14:19.870
So we'll have bare objects
and renormalized objects.

00:14:25.750 --> 00:14:27.580
Each one of these
guys corresponded

00:14:27.580 --> 00:14:29.210
to a single collinear field.

00:14:29.210 --> 00:14:32.170
So there's some wave function
renormalization factor,

00:14:32.170 --> 00:14:36.790
and then there's
some renormalization

00:14:36.790 --> 00:14:39.660
for the operator.

00:14:39.660 --> 00:14:41.620
And together, those
two things will take us

00:14:41.620 --> 00:14:43.750
from the bare to
the renormalization

00:14:43.750 --> 00:14:46.300
and likewise for
[INAUDIBLE] soft guy.

00:14:52.510 --> 00:14:54.160
In this case, there's no--

00:14:54.160 --> 00:14:55.300
we just have Wilson lines.

00:14:55.300 --> 00:15:03.320
So there's just a
renormalization of the object.

00:15:03.320 --> 00:15:07.460
And this guy here, it's
the usual wave function

00:15:07.460 --> 00:15:08.690
renormalization.

00:15:08.690 --> 00:15:10.880
It's just the same in SET
for a collinear particle

00:15:10.880 --> 00:15:12.856
as it is in QCD.

00:15:12.856 --> 00:15:18.770
So just for the
record, there's a 1

00:15:18.770 --> 00:15:21.950
over epsilon coming from
that that will modify our--

00:15:21.950 --> 00:15:23.630
this 1 over 2 epsilon here.

00:15:23.630 --> 00:15:26.015
Let me put it together
for the z factors.

00:15:29.270 --> 00:15:31.940
[? So what ?] [? of ?]
[? the z's? ?] Taking those

00:15:31.940 --> 00:15:36.505
graphs and these formulas,
find the following.

00:15:42.820 --> 00:15:44.686
Think I dropped something.

00:15:44.686 --> 00:15:45.670
[INAUDIBLE]

00:15:49.610 --> 00:15:51.840
There was this pesky
factor of w squared

00:15:51.840 --> 00:15:53.480
that we talked about last time.

00:15:58.010 --> 00:15:59.660
But we'll need it
for this calculation.

00:16:11.772 --> 00:16:13.480
So we simply just
subtract all the poles.

00:16:21.410 --> 00:16:22.956
Minimal subtraction.

00:16:27.150 --> 00:16:28.910
So I'm always going
to only just write zn

00:16:28.910 --> 00:16:30.740
because zn bar is
really the same.

00:16:36.380 --> 00:16:38.120
We set up our kinematics
symmetrically.

00:16:38.120 --> 00:16:45.210
So it's really no difference
between zn and zn bar.

00:16:50.030 --> 00:16:51.980
So there's a minus
sign plus sign there.

00:16:55.040 --> 00:16:56.330
Comes to 3/8 epsilon.

00:17:05.270 --> 00:17:08.134
So those are the counter
terms that we get.

00:17:08.134 --> 00:17:09.384
AUDIENCE: Did you [INAUDIBLE]?

00:17:12.208 --> 00:17:13.250
IAIN STEWART: Yes, I did.

00:17:13.250 --> 00:17:13.849
Thank you.

00:17:19.680 --> 00:17:21.960
And we get the anomalous
dimensions again

00:17:21.960 --> 00:17:26.099
just by demanding the usual
thing-- that the bare objects

00:17:26.099 --> 00:17:27.329
don't depend on mu and nu.

00:17:31.010 --> 00:17:32.990
And we can do that
separately in the mu and nu.

00:17:40.700 --> 00:17:44.540
So if you like, actually,
kind of along the lines

00:17:44.540 --> 00:17:48.230
of your question,
[INAUDIBLE],, there's

00:17:48.230 --> 00:17:51.020
a assumption built into this,
the way we wrote this, right?

00:17:51.020 --> 00:17:53.510
I've kept the full
epsilon dependence

00:17:53.510 --> 00:17:55.320
in my 1 over eta pole.

00:17:55.320 --> 00:17:57.320
And that's related to
saying that I could really

00:17:57.320 --> 00:18:01.370
think of doing all the
eta renormalization first

00:18:01.370 --> 00:18:05.100
and then think about
doing epsilon second.

00:18:05.100 --> 00:18:06.182
So the kind of--

00:18:06.182 --> 00:18:08.390
we have built something into
this that's non-trivial.

00:18:11.140 --> 00:18:15.150
AUDIENCE: [INAUDIBLE] the
order is the same [INAUDIBLE]..

00:18:15.150 --> 00:18:17.853
IAIN STEWART: You mean
that the order being--

00:18:17.853 --> 00:18:19.770
AUDIENCE: Like that you
can do one [INAUDIBLE]

00:18:19.770 --> 00:18:20.630
before the other.

00:18:20.630 --> 00:18:22.520
IAIN STEWART: Yeah, oh, that
the order doesn't matter.

00:18:22.520 --> 00:18:22.830
Yeah.

00:18:22.830 --> 00:18:23.660
AUDIENCE: That seems like--

00:18:23.660 --> 00:18:24.040
IAIN STEWART: Yeah.

00:18:24.040 --> 00:18:25.880
AUDIENCE: --something that you
would get from a factorized

00:18:25.880 --> 00:18:26.630
expression--

00:18:26.630 --> 00:18:27.463
IAIN STEWART: Right.

00:18:27.463 --> 00:18:28.625
AUDIENCE: [INAUDIBLE]

00:18:28.625 --> 00:18:30.500
IAIN STEWART: Yeah, it
kind of is factorized.

00:18:30.500 --> 00:18:33.140
But it's hidden,
right, because we

00:18:33.140 --> 00:18:35.420
are saying that this is kind
of a multiplicative thing

00:18:35.420 --> 00:18:38.570
on that whole thing.

00:18:38.570 --> 00:18:40.190
And that's the kind
of factorization,

00:18:40.190 --> 00:18:44.190
although it looks kind of
like a non-factorized thing.

00:18:44.190 --> 00:18:47.900
So let's talk about mu
anomalous dimensions.

00:18:47.900 --> 00:18:50.630
And you'll see that in
order for this to be true,

00:18:50.630 --> 00:18:53.300
you also don't-- it's not
the case that mu anomalous

00:18:53.300 --> 00:18:57.630
dimension needs to be
independent of nu, for example.

00:18:57.630 --> 00:18:59.730
So this kind of looks
like it's not factoring,

00:18:59.730 --> 00:19:03.440
but magically, it is in
the sense of this path

00:19:03.440 --> 00:19:04.760
independence.

00:19:04.760 --> 00:19:09.130
So what is gamma mu
for the soft function?

00:19:09.130 --> 00:19:17.000
So we just say that mu d by
d mu on bare things is 0 is--

00:19:17.000 --> 00:19:18.705
so is nu d by d nu.

00:19:18.705 --> 00:19:22.010
So that's the way
that this works.

00:19:22.010 --> 00:19:24.830
And that gives us this
kind of standard formulas

00:19:24.830 --> 00:19:31.110
that we're used to for
the anomalous dimensions.

00:19:31.110 --> 00:19:34.490
So zs inverse mu d by d mu
with the appropriate sign

00:19:34.490 --> 00:19:37.100
is the anomalous
dimension in mu.

00:19:37.100 --> 00:19:42.510
That comes from this
equation for soft function.

00:19:42.510 --> 00:19:47.160
So if we look at that
and we work it out,

00:19:47.160 --> 00:19:51.920
so if you look at this term,
it actually doesn't contribute,

00:19:51.920 --> 00:19:57.080
because in this term, alpha s
of mu times mu to the 2 epsilon

00:19:57.080 --> 00:19:58.610
is mu independent.

00:19:58.610 --> 00:20:02.600
So that term has no contribution
to the new anomalous dimension.

00:20:02.600 --> 00:20:05.060
And then this term
does contribute,

00:20:05.060 --> 00:20:07.610
and there's a contribution
from this term.

00:20:07.610 --> 00:20:10.250
Where you differentiate
the alpha, you get a 1

00:20:10.250 --> 00:20:11.710
over 2 epsilon.

00:20:11.710 --> 00:20:18.771
You differentiate this
guy, and there's a--

00:20:18.771 --> 00:20:20.820
if you differentiate
the explicit log mu,

00:20:20.820 --> 00:20:22.260
there's also a 1 over epsilon.

00:20:22.260 --> 00:20:25.440
Those 1 over
epsilons cancel, OK?

00:20:25.440 --> 00:20:27.330
So there's no contribution
from this guy.

00:20:27.330 --> 00:20:29.970
There's no contribution
from differentiating

00:20:29.970 --> 00:20:33.940
the alpha in this guy or the
explicit log in this guy.

00:20:33.940 --> 00:20:37.050
So the only contribution
is taking the alpha,

00:20:37.050 --> 00:20:39.000
differentiating it,
getting a 2 epsilon,

00:20:39.000 --> 00:20:41.950
and multiplying
that 1 over epsilon.

00:20:41.950 --> 00:20:52.100
So this line switches,
and there's a 2.

00:20:56.830 --> 00:20:58.240
AUDIENCE: Set w to 1?

00:20:58.240 --> 00:20:59.782
IAIN STEWART: And
I set w to 1, yeah.

00:21:02.620 --> 00:21:05.445
w-- renormalize w.

00:21:08.175 --> 00:21:08.980
[INAUDIBLE] 1.

00:21:22.710 --> 00:21:26.590
So likewise, for the other
case, for the collinear--

00:21:34.440 --> 00:21:39.300
and this is actually the same
as [INAUDIBLE] mu and bar.

00:21:42.000 --> 00:21:46.200
And what this gives if you
solve the anomalous dimension

00:21:46.200 --> 00:21:48.990
equation--

00:21:48.990 --> 00:21:50.490
so the anomalous
dimension equations

00:21:50.490 --> 00:21:54.060
are like [INAUDIBLE]
sign convention

00:21:54.060 --> 00:22:02.750
like this or et cetera.

00:22:02.750 --> 00:22:05.120
So you would solve
those equations.

00:22:05.120 --> 00:22:09.396
And those equations would give
you the kernels us and un.

00:22:13.590 --> 00:22:15.290
And it's a simple
multiplicative rge.

00:22:15.290 --> 00:22:17.292
So if you just-- this
is mu d by dmu of log

00:22:17.292 --> 00:22:23.350
s, and you just integrate
the way we've done before.

00:22:23.350 --> 00:22:28.630
And consistency
says that you could

00:22:28.630 --> 00:22:30.680
run in this picture either way.

00:22:30.680 --> 00:22:35.602
And that is a relation between
the anomalous dimensions.

00:22:35.602 --> 00:22:37.060
There's also an
anomalous dimension

00:22:37.060 --> 00:22:39.320
for the hard function.

00:22:39.320 --> 00:22:42.384
And if we add them
all up, we get zero.

00:22:42.384 --> 00:22:45.800
We pick the sign
conventions accordingly.

00:22:45.800 --> 00:22:47.860
So you could write
the relation that way.

00:22:50.980 --> 00:22:52.480
And so you can
calculate the gamma h

00:22:52.480 --> 00:22:55.000
by knowing these three--

00:22:55.000 --> 00:22:58.330
gamma for the hard function.

00:22:58.330 --> 00:23:01.150
Or you could calculate
it by the counter term

00:23:01.150 --> 00:23:02.605
for the hard function.

00:23:05.460 --> 00:23:09.120
And as expected, it only
depends on a log of mu over q.

00:23:09.120 --> 00:23:10.230
There's no nus in that.

00:23:13.128 --> 00:23:14.670
So that's the mu
anomalous dimension.

00:23:14.670 --> 00:23:16.440
It works as usual.

00:23:16.440 --> 00:23:21.170
And the interesting thing
is that there's also

00:23:21.170 --> 00:23:24.230
a new anomalous dimension
because the bear functions are

00:23:24.230 --> 00:23:25.190
independent of nu.

00:23:53.800 --> 00:23:55.050
It's the same type of formula.

00:23:55.050 --> 00:23:58.680
We just have nu d by d nu
instead of mu d by d mu.

00:23:58.680 --> 00:24:01.830
There's no nu dependence
in the coupling alpha,

00:24:01.830 --> 00:24:04.740
but there is a nu
dependence in this w.

00:24:04.740 --> 00:24:07.620
When you differentiate
the w, you get an eta

00:24:07.620 --> 00:24:14.860
And that's where basically the
contribution is coming from.

00:24:14.860 --> 00:24:18.690
So what happens is you
differentiate the w.

00:24:18.690 --> 00:24:19.558
That kills this eta.

00:24:19.558 --> 00:24:21.600
But then it looks like
your result would have a 1

00:24:21.600 --> 00:24:23.595
over epsilon pole.

00:24:23.595 --> 00:24:25.470
But you also have to
differentiate explicitly

00:24:25.470 --> 00:24:27.510
this nu here.

00:24:27.510 --> 00:24:29.340
Again, that has a 1
over epsilon pole.

00:24:29.340 --> 00:24:31.250
So those two cancel.

00:24:31.250 --> 00:24:33.000
Once you take the
epsilon goes to 0 limit,

00:24:33.000 --> 00:24:34.375
they cancel, and
you're just left

00:24:34.375 --> 00:24:36.300
with a finite
anomalous dimension.

00:24:36.300 --> 00:24:40.710
So here, I'm always
sending epsilon to 0,

00:24:40.710 --> 00:24:42.600
and eta goes to 0.

00:24:42.600 --> 00:24:48.330
And that's what sets w
to 1 and stuff like that.

00:24:48.330 --> 00:24:51.820
OK, so this guy--

00:24:51.820 --> 00:24:53.650
so [INAUDIBLE]
cancellations [INAUDIBLE]

00:24:53.650 --> 00:24:55.470
1 over epsilon poles
to get those results,

00:24:55.470 --> 00:24:57.780
and there's also cancellations
of 1 over epsilon poles

00:24:57.780 --> 00:24:59.320
to get these results.

00:25:04.790 --> 00:25:13.070
And equations like nu d by d
nu of s [INAUDIBLE] gamma nu ss

00:25:13.070 --> 00:25:18.760
would give the
kernel vs, et cetera.

00:25:23.870 --> 00:25:26.963
OK, so there's the explicit
anomalous dimensions.

00:25:26.963 --> 00:25:28.130
Let me keep that picture up.

00:25:40.650 --> 00:25:44.270
So if you ask what path
independence means,

00:25:44.270 --> 00:25:48.530
you could say path independence
could be phrased by the fact

00:25:48.530 --> 00:25:52.370
that I could take
z inverse, and I

00:25:52.370 --> 00:25:54.320
can take a commutator
of derivatives,

00:25:54.320 --> 00:25:56.450
right, because I'm saying
that either order should

00:25:56.450 --> 00:26:01.580
get the same result. And
if either order gives

00:26:01.580 --> 00:26:04.250
the same result,
that must be zero.

00:26:04.250 --> 00:26:07.940
And what this formula says that
the order that we're working

00:26:07.940 --> 00:26:15.770
is that you can take
mu d by d mu of some,

00:26:15.770 --> 00:26:17.930
say, nu anomalous
dimension, and that

00:26:17.930 --> 00:26:23.870
should be equal to nu d by d nu
of the mu anomalous dimension.

00:26:23.870 --> 00:26:26.857
That there's a connection
between the two things.

00:26:26.857 --> 00:26:28.190
And if you look at the factors--

00:26:28.190 --> 00:26:29.930
and if I wrote everything
down correctly,

00:26:29.930 --> 00:26:39.900
then that's true and
likewise for the collinear.

00:26:39.900 --> 00:26:42.660
So this is the
statement of there

00:26:42.660 --> 00:26:46.530
not being a dependence
on the path.

00:26:46.530 --> 00:26:48.330
These are formulas
that have to be true

00:26:48.330 --> 00:26:49.800
if that's going to be true.

00:26:56.965 --> 00:26:59.340
So I'm not going to write down
all these kernels for you.

00:26:59.340 --> 00:27:00.500
There's a lot of them.

00:27:00.500 --> 00:27:02.120
But just to give you
a flavor for what

00:27:02.120 --> 00:27:04.790
the solutions look like,
they kind of look familiar.

00:27:04.790 --> 00:27:07.320
Let me write down
a couple of them.

00:27:07.320 --> 00:27:08.515
So I'll write down one.

00:27:08.515 --> 00:27:09.890
I'll write the
ones for the soft.

00:27:17.120 --> 00:27:19.430
So at leading log order,
what would they look like?

00:27:21.950 --> 00:27:27.500
They're exponentials, Sudakov
logarithm type formulas.

00:27:30.650 --> 00:27:37.605
They're not the same precise
formulas that we had earlier,

00:27:37.605 --> 00:27:38.855
but they look kind of similar.

00:28:00.057 --> 00:28:02.140
Let me write them down,
then I'll talk about them.

00:28:10.860 --> 00:28:12.610
So these are running
along straight paths.

00:28:19.500 --> 00:28:22.150
And they involve ratios
of alpha, basically.

00:28:22.150 --> 00:28:24.720
[INAUDIBLE] that's what the
running coupling is doing.

00:28:39.950 --> 00:28:43.580
So you could always write
things in such a way

00:28:43.580 --> 00:28:47.000
that you don't have alphas
that involve nu, right,

00:28:47.000 --> 00:28:50.325
because the rga just
has alphas of mu.

00:28:50.325 --> 00:28:51.950
And so if you like,
when you're solving

00:28:51.950 --> 00:28:53.722
the nu anomalous
dimension, this equation

00:28:53.722 --> 00:28:55.430
is very easy to
integrate because there's

00:28:55.430 --> 00:28:56.720
no nus on the right hand side.

00:28:56.720 --> 00:28:58.640
You just get a log of nu.

00:28:58.640 --> 00:29:01.190
And that's why this is--

00:29:01.190 --> 00:29:03.590
that's why we have this
kind of simple log nu here.

00:29:06.770 --> 00:29:09.350
Here I wrote alpha,
1 over alpha of nu s.

00:29:09.350 --> 00:29:13.430
But I could have write that
back in terms of a log.

00:29:13.430 --> 00:29:15.020
So it really isn't.

00:29:20.510 --> 00:29:24.002
That was just convenient to
make it a simpler formula.

00:29:24.002 --> 00:29:26.210
So if you write down the
evolution kernels like that,

00:29:26.210 --> 00:29:29.120
they'll satisfy these
multiplication properties

00:29:29.120 --> 00:29:31.460
that this figure implies.

00:29:31.460 --> 00:29:34.700
And I won't go through
that, but it's kind of

00:29:34.700 --> 00:29:36.020
neat to see how it works out.

00:29:39.610 --> 00:29:42.610
All right, so that gives you
an idea of how we would sum

00:29:42.610 --> 00:29:45.520
logs for this Sudakov form
factor with these two cut offs.

00:29:45.520 --> 00:29:49.512
We just have the evolution
kernels and use them as usual.

00:29:49.512 --> 00:29:51.220
But we have this more
complicated picture

00:29:51.220 --> 00:29:53.350
of what type of
renormalization we have to do.

00:29:59.720 --> 00:30:01.950
And you can think about
also, if you wanted to--

00:30:01.950 --> 00:30:06.360
for example, say you wanted to
do this in some calculation,

00:30:06.360 --> 00:30:08.522
and then you calculated
up to some order

00:30:08.522 --> 00:30:09.980
solving these
anomalous dimensions.

00:30:09.980 --> 00:30:12.200
And then you wanted
to vary scales.

00:30:12.200 --> 00:30:14.180
Well, now you have a
two dimensional plane

00:30:14.180 --> 00:30:15.410
to vary scales in, right?

00:30:15.410 --> 00:30:17.630
So if you're varying
the soft scales,

00:30:17.630 --> 00:30:21.680
you can kind of move around
in a box around this guy

00:30:21.680 --> 00:30:24.620
where you're varying
both nu l and mu l

00:30:24.620 --> 00:30:27.080
by kind of factors of 2.

00:30:27.080 --> 00:30:29.840
And so they're
doing uncertainties

00:30:29.840 --> 00:30:31.744
in this kind of
setup would have--

00:30:31.744 --> 00:30:34.730
you'd have more parameters to
vary than you would usually

00:30:34.730 --> 00:30:36.800
have just with mus.

00:30:36.800 --> 00:30:40.280
But really, it's just a
straightforward generalization

00:30:40.280 --> 00:30:43.220
of the one dimensional
picture of mu evolution

00:30:43.220 --> 00:30:45.637
to a two dimensional
picture of nu evolution.

00:30:48.980 --> 00:30:51.380
And the interesting
part is the kind

00:30:51.380 --> 00:30:53.611
of overlaps between
these parameters.

00:31:00.000 --> 00:31:03.890
So let me give you one other
physics example of where

00:31:03.890 --> 00:31:05.600
this comes in just
so that you see

00:31:05.600 --> 00:31:08.270
it's not just this one example.

00:31:08.270 --> 00:31:18.850
So I'll just go through
one of them in detail,

00:31:18.850 --> 00:31:20.350
and I'll just mention
one other one.

00:31:23.380 --> 00:31:27.220
So it'll be two examples, but
we'll only really cover one.

00:31:27.220 --> 00:31:31.390
So one thing you can
do is do gg to Higgs,

00:31:31.390 --> 00:31:33.970
so Higgs production,
where I measure

00:31:33.970 --> 00:31:37.075
a particular distribution,
which is the pt distribution,

00:31:37.075 --> 00:31:37.960
pt of the Higgs.

00:31:43.210 --> 00:31:45.490
And it turns out
that this process

00:31:45.490 --> 00:31:48.850
involves rapidity divergences.

00:31:48.850 --> 00:31:51.130
So let me try to
draw one picture that

00:31:51.130 --> 00:31:54.710
allows me to capture all the
different degrees of freedom.

00:31:54.710 --> 00:31:58.810
So here's-- you could imagine
that this is your top loop,

00:31:58.810 --> 00:32:01.450
but it's some short
distance thing.

00:32:01.450 --> 00:32:03.437
And you can even
integrate out the top.

00:32:03.437 --> 00:32:05.770
Think of it as an effective
operator coupling two gluons

00:32:05.770 --> 00:32:07.570
to the Higgs.

00:32:07.570 --> 00:32:11.730
And so I'm in the center of
mass frame of the collision.

00:32:11.730 --> 00:32:13.090
So these guys are back to back.

00:32:13.090 --> 00:32:15.673
So that means that one of them
is n collinear, and one of them

00:32:15.673 --> 00:32:17.170
is n bar collinear.

00:32:17.170 --> 00:32:20.950
So if n and n bar collinear
is coming in, annihilating

00:32:20.950 --> 00:32:22.870
and producing a Higgs--

00:32:22.870 --> 00:32:25.750
and because of what we're
measuring about the Higgs,

00:32:25.750 --> 00:32:27.820
we're only measuring the pt.

00:32:27.820 --> 00:32:29.195
If you think about
what radiation

00:32:29.195 --> 00:32:30.945
you can have in the
final state, well, you

00:32:30.945 --> 00:32:32.600
could have collinear radiation.

00:32:32.600 --> 00:32:35.230
So here's some
collinear radiation.

00:32:35.230 --> 00:32:38.128
And that radiation
has a small pt.

00:32:38.128 --> 00:32:39.670
So that's allowed
in the final state.

00:32:39.670 --> 00:32:44.210
If we have a pth distribution
and we think about the limit

00:32:44.210 --> 00:32:48.370
pth much less than the mass
of the Higgs, so there's some

00:32:48.370 --> 00:32:51.795
logs that you would want
to sum, for example.

00:32:51.795 --> 00:32:53.170
So you could have
collinear modes

00:32:53.170 --> 00:32:55.480
in the final state
that would fit

00:32:55.480 --> 00:32:58.120
within this kind
of kinematic setup.

00:32:58.120 --> 00:33:00.970
But you could also
have soft modes.

00:33:00.970 --> 00:33:06.270
So soft modes have the same size
of pt as the collinear modes.

00:33:06.270 --> 00:33:09.340
So they would be allowed,
and this propagator here

00:33:09.340 --> 00:33:11.060
would be off shell.

00:33:11.060 --> 00:33:12.580
So it's an [? set ?] 2 problem.

00:33:12.580 --> 00:33:15.250
Because you're only
constraining a pt,

00:33:15.250 --> 00:33:19.150
then it's [? set ?] 2
problem with n bar, n, and s.

00:33:22.012 --> 00:33:27.230
So pt Higgs [INAUDIBLE]
order mh lambda,

00:33:27.230 --> 00:33:31.040
and modes are these ones.

00:33:31.040 --> 00:33:34.550
So we'll SCET two.

00:33:34.550 --> 00:33:36.500
And you'd want to sum--

00:33:36.500 --> 00:33:42.300
in this case, you'd want to sum
up double logs of pth over mh.

00:33:42.300 --> 00:33:44.765
That's what you might
be interested in using

00:33:44.765 --> 00:33:45.890
the effective theory to do.

00:33:49.280 --> 00:33:52.220
OK, so you would go through
the procedure of factorizing

00:33:52.220 --> 00:33:53.060
the cross section.

00:33:57.020 --> 00:34:00.770
It's an inclusive calculation
in the sense that basically,

00:34:00.770 --> 00:34:02.750
in the final state,
it's Higgs plus x,

00:34:02.750 --> 00:34:06.350
right, where x is collinear
radiation or soft radiation.

00:34:06.350 --> 00:34:11.330
And really, the process
is proton proton,

00:34:11.330 --> 00:34:12.830
for example, the Higgs plus x.

00:34:18.880 --> 00:34:21.370
So we would want to factorize
the cross section, amplitude

00:34:21.370 --> 00:34:23.257
squared.

00:34:23.257 --> 00:34:25.340
And here is kind of a
sketch of how that would go.

00:34:25.340 --> 00:34:27.020
I won't go through the details.

00:34:27.020 --> 00:34:29.728
So you could think of starting
with some operator where you've

00:34:29.728 --> 00:34:31.020
already integrated out the top.

00:34:31.020 --> 00:34:33.550
So you have a coupling
of a Higgs to two gluons,

00:34:33.550 --> 00:34:35.770
and that happens, and this
gives you variant operator

00:34:35.770 --> 00:34:37.156
with the field strengths.

00:34:40.820 --> 00:34:43.383
And then you would do the
factorization procedure.

00:34:47.179 --> 00:34:51.620
After you shift momenta
to some other states,

00:34:51.620 --> 00:34:54.139
you can basically right
that is the factorization

00:34:54.139 --> 00:34:56.750
for a matrix element
of two currents.

00:34:56.750 --> 00:35:00.620
And you get some
hard function, which

00:35:00.620 --> 00:35:03.870
is the Wilson coefficient
squared of this operator.

00:35:03.870 --> 00:35:09.160
And then you get
some operators that

00:35:09.160 --> 00:35:14.350
look like this, matrix
elements that look like this.

00:35:19.977 --> 00:35:21.560
So there's some
proton matrix elements

00:35:21.560 --> 00:35:23.685
where one of the proteins
is collinear, one of them

00:35:23.685 --> 00:35:25.640
is n bar collinear.

00:35:25.640 --> 00:35:28.310
And then there's a
soft matrix element

00:35:28.310 --> 00:35:29.750
of some soft Wilson lines.

00:35:33.870 --> 00:35:37.870
They're actually in the
adjoint representation.

00:35:37.870 --> 00:35:41.351
So I wrote transpose
rather than dagger.

00:35:41.351 --> 00:35:44.430
So these are adjoint
representation.

00:35:44.430 --> 00:35:47.455
They should be in the same
representation as the collinear

00:35:47.455 --> 00:35:49.830
fields, and the collinear
fields here were gluons, right?

00:35:49.830 --> 00:35:52.620
So when you go through
the field redefinition,

00:35:52.620 --> 00:35:55.500
you would get adjoint
Wilson lines for the soft.

00:35:55.500 --> 00:35:58.950
If you went through an [? set ?]
1, for example, picture,

00:35:58.950 --> 00:36:03.870
and then the g [? mu nu is ?]
here would become

00:36:03.870 --> 00:36:08.477
the perpendicular
polarization of the gluons.

00:36:08.477 --> 00:36:10.560
So these things here are
going to give gluon PDFs.

00:36:13.113 --> 00:36:15.030
So basically, you have
a factorization theorem

00:36:15.030 --> 00:36:22.340
that involves gluon
PDFs and a soft function

00:36:22.340 --> 00:36:25.460
and a hard function.

00:36:25.460 --> 00:36:27.170
Now if you look at
this calculation,

00:36:27.170 --> 00:36:28.190
you have three modes.

00:36:28.190 --> 00:36:29.180
There's an [? set ?] 2.

00:36:29.180 --> 00:36:30.590
They live on the same hyperbola.

00:36:30.590 --> 00:36:32.480
And you do have
rapidity divergences

00:36:32.480 --> 00:36:34.350
in this calculation.

00:36:34.350 --> 00:36:35.950
So it's a story like this one.

00:36:38.710 --> 00:36:41.650
And I'm not going to
go in too much detail,

00:36:41.650 --> 00:36:44.298
but just let me write down the
factorization theorem for you

00:36:44.298 --> 00:36:45.715
with all the mus
and nus explicit.

00:36:45.715 --> 00:36:50.880
And it's really the
same picture where

00:36:50.880 --> 00:36:52.455
the soft modes and
collinear modes

00:36:52.455 --> 00:36:55.110
need to live at different nus.

00:36:55.110 --> 00:36:57.900
Same mu, and then
there's the hard function

00:36:57.900 --> 00:37:01.557
where you have to run
from one side to the next.

00:37:01.557 --> 00:37:03.890
So it's really just-- you
keep the picture on the board.

00:37:03.890 --> 00:37:05.390
It's applicable to
this example too.

00:37:08.850 --> 00:37:12.270
So if you calculate d
sigma d, if you measure

00:37:12.270 --> 00:37:14.955
rapidity of the Higgs boson--

00:37:14.955 --> 00:37:17.430
so you can call this yh.

00:37:17.430 --> 00:37:21.330
And you measure the pth
squared the magnitude

00:37:21.330 --> 00:37:23.670
of the transverse momentum.

00:37:23.670 --> 00:37:28.022
There's some normalization
factor, just kinematics.

00:37:28.022 --> 00:37:29.730
There's a hard function
that only depends

00:37:29.730 --> 00:37:32.100
on the Higgs mass and mu.

00:37:32.100 --> 00:37:36.515
And then your other functions
can exchange [? perp ?] momenta

00:37:36.515 --> 00:37:38.640
because the [? perp ?]
momenta of the soft function

00:37:38.640 --> 00:37:41.098
and the [? perp momenta ?] of
these collinear functions are

00:37:41.098 --> 00:37:42.300
the same size.

00:37:42.300 --> 00:37:44.460
And what you're constraining
is just the total.

00:37:44.460 --> 00:37:48.840
So the pt of the Higgs, if
the initial guys coming in

00:37:48.840 --> 00:37:50.550
have zero [? perp ?]
momenta, there's

00:37:50.550 --> 00:37:53.820
zero [? perp ?] momenta
for the protons by design.

00:37:53.820 --> 00:37:58.260
So really, there's a balancing
between the final state

00:37:58.260 --> 00:38:02.580
radiation and the Higgs,
which I can write like this.

00:38:09.877 --> 00:38:11.460
OK, there's a delta
function, and then

00:38:11.460 --> 00:38:18.000
you just have objects
for the different guys.

00:38:18.000 --> 00:38:26.950
So do this in the proton
center of mass frame.

00:38:41.830 --> 00:38:44.380
Because of the fact
that we're measuring

00:38:44.380 --> 00:38:48.500
perpendicular momentum, it
allows the PDF to be a tensor.

00:38:48.500 --> 00:38:54.550
But that kind of means it
has two scalar PDFs in it.

00:39:07.500 --> 00:39:08.406
[INAUDIBLE] h.

00:39:11.980 --> 00:39:13.597
And then there's
a soft function.

00:39:20.760 --> 00:39:23.900
OK, so it's kind of got
the same type of structure

00:39:23.900 --> 00:39:27.180
that we were seeing in
our previous example.

00:39:27.180 --> 00:39:32.010
The external kinematics actually
fixes these variables here,

00:39:32.010 --> 00:39:33.048
which are like the x.

00:39:33.048 --> 00:39:34.340
This is like the [INAUDIBLE] x.

00:39:41.510 --> 00:39:44.180
And the kinematics
of the process

00:39:44.180 --> 00:39:45.680
actually end up
fixing that when you

00:39:45.680 --> 00:39:46.888
go through the factorization.

00:39:46.888 --> 00:39:50.820
This should be a plus sign here.

00:39:50.820 --> 00:39:53.000
So the only thing that can
change from the dynamics

00:39:53.000 --> 00:39:57.410
is the [? perp ?] momentum.

00:39:57.410 --> 00:39:59.530
That's what gets exchanged
between these guys.

00:39:59.530 --> 00:40:02.870
So you would sum logs by having
normalization group equations

00:40:02.870 --> 00:40:04.400
for these objects.

00:40:04.400 --> 00:40:06.680
And you need them in mu and nu.

00:40:06.680 --> 00:40:09.080
The nu one is always kind of
simple because of the alpha

00:40:09.080 --> 00:40:10.020
doesn't depend on nu.

00:40:10.020 --> 00:40:11.395
So it's very simple
to integrate.

00:40:11.395 --> 00:40:12.685
You just get one log.

00:40:12.685 --> 00:40:14.060
But it's still an
important thing

00:40:14.060 --> 00:40:15.500
because that one
log is something

00:40:15.500 --> 00:40:19.970
that you can't get
from the mu evolution.

00:40:19.970 --> 00:40:23.670
OK, so these are called
transverse momentum dependent

00:40:23.670 --> 00:40:24.170
PDFs.

00:40:37.910 --> 00:40:40.012
So they're not the
standard PDFs that you're

00:40:40.012 --> 00:40:40.970
used to thinking about.

00:40:40.970 --> 00:40:43.310
They have this dependence
on transverse momentum.

00:40:43.310 --> 00:40:45.860
And because of that dependence
on transverse momentum,

00:40:45.860 --> 00:40:48.470
they also have these
rapidity divergences.

00:40:48.470 --> 00:40:51.450
And these are the renormalized
ones at this nu scale.

00:40:51.450 --> 00:40:53.630
So there's a long, sordid
history of these guys.

00:40:53.630 --> 00:40:56.360
They were introduced in
the early days of QCD,

00:40:56.360 --> 00:40:59.090
and people have really only
sort it out very recently,

00:40:59.090 --> 00:41:01.730
both in the QCD literature
and the SET literature

00:41:01.730 --> 00:41:05.090
kind of simultaneously
with different regulators,

00:41:05.090 --> 00:41:06.830
exactly how to make
sense of these guys

00:41:06.830 --> 00:41:09.390
and define them properly and
renormalize them properly.

00:41:09.390 --> 00:41:12.350
So this is kind of the last
couple of years type stuff.

00:41:16.250 --> 00:41:18.190
So another example
that we could do which

00:41:18.190 --> 00:41:21.250
I won't go through in detail--

00:41:21.250 --> 00:41:23.800
so that was example one.

00:41:23.800 --> 00:41:28.360
We could do an example that
is not involving protons

00:41:28.360 --> 00:41:32.330
but involving jets
in e plus e minus.

00:41:32.330 --> 00:41:35.060
So we could do e plus e
minus to [? dijets. ?]

00:41:35.060 --> 00:41:37.240
And if we did something
similar to what we did here

00:41:37.240 --> 00:41:40.000
where we only
measured a pt, then we

00:41:40.000 --> 00:41:43.220
would also be in this
[? set ?] 2 situation.

00:41:43.220 --> 00:41:48.010
So jet broadening
is a variable b

00:41:48.010 --> 00:41:52.280
for broadening where it's like
an event shaped like thrust.

00:41:52.280 --> 00:41:55.300
But you only measure
perpendicular momentum.

00:41:55.300 --> 00:41:58.750
And what you
measure are actually

00:41:58.750 --> 00:42:02.830
perpendicular to
the thrust axis.

00:42:02.830 --> 00:42:06.500
So you still use thrust to
get the axis for the jet.

00:42:06.500 --> 00:42:08.560
But then you don't
measure anything like

00:42:08.560 --> 00:42:11.200
the minus or plus momentum.

00:42:11.200 --> 00:42:13.060
You just measure pts.

00:42:13.060 --> 00:42:15.760
This would be another
example, which is SCET 2,

00:42:15.760 --> 00:42:20.080
and it actually would have,
again, cn, cn bar type modes.

00:42:20.080 --> 00:42:21.850
And it-- you'd
[? read ?] a formula

00:42:21.850 --> 00:42:24.340
not exactly the same as
this one but again involving

00:42:24.340 --> 00:42:27.580
similar types of things--
collinear jet functions that

00:42:27.580 --> 00:42:34.382
depend on pt, soft function,
some constraint between them.

00:42:34.382 --> 00:42:36.090
It's a little more
complicated, actually,

00:42:36.090 --> 00:42:38.790
than this pt Higgs one,
which is why I chose

00:42:38.790 --> 00:42:40.500
to write down the pt Higgs one.

00:42:40.500 --> 00:42:43.450
So I write that in my notes,
but just because of time,

00:42:43.450 --> 00:42:47.260
I'm going to skip it
for our discussion.

00:42:47.260 --> 00:42:50.980
So questions so far.

00:42:50.980 --> 00:42:54.040
So really, it's turn the
crank once you believe some

00:42:54.040 --> 00:42:55.480
of the things I've told you.

00:42:55.480 --> 00:42:57.100
So something that
might be interesting

00:42:57.100 --> 00:42:59.320
would be to work out
with mu evolution,

00:42:59.320 --> 00:43:01.630
we have constraints on the
counter terms at all higher

00:43:01.630 --> 00:43:05.450
orders that you can write
down by consistency,

00:43:05.450 --> 00:43:06.700
anomalous dimension equations.

00:43:06.700 --> 00:43:11.380
And I don't think there's to
my knowledge in the literature

00:43:11.380 --> 00:43:14.140
an expression like that
for two loops, three loops,

00:43:14.140 --> 00:43:17.343
what would the various
1 over etas and 1

00:43:17.343 --> 00:43:18.760
over epsilons, how
would they have

00:43:18.760 --> 00:43:21.650
to work out based on the
consistency of the picture I've

00:43:21.650 --> 00:43:22.150
told you.

00:43:22.150 --> 00:43:24.430
I don't think
anyone's done that.

00:43:24.430 --> 00:43:27.070
That would be kind
of interesting.

00:43:27.070 --> 00:43:31.630
All right, so questions-- none.

00:43:31.630 --> 00:43:32.500
Good.

00:43:32.500 --> 00:43:34.010
All right, so one final example.

00:43:43.710 --> 00:43:46.370
So presentations Monday--
they're not in this room.

00:43:46.370 --> 00:43:50.880
They're in the seminar room,
the large CPT seminar room.

00:43:50.880 --> 00:43:52.220
It's on the fourth floor.

00:44:00.440 --> 00:44:02.060
And I have also
made a note that you

00:44:02.060 --> 00:44:08.100
should use the blackboard, which
will stop you from preparing

00:44:08.100 --> 00:44:11.940
too much information.

00:44:11.940 --> 00:44:13.830
OK, so let's do
one final example.

00:44:20.620 --> 00:44:22.240
The final example I
want to talk about

00:44:22.240 --> 00:44:27.670
is Drell-Yan, which we almost
kind of talked about already.

00:44:27.670 --> 00:44:30.700
But I'd like to talk about it
in a little different context.

00:44:30.700 --> 00:44:34.210
So we talked about pp
to xh, Higgs boson.

00:44:34.210 --> 00:44:36.850
Classic Drell-Yan is
pp to xl plus l minus.

00:44:36.850 --> 00:44:40.120
But kinematically, that doesn't
really make too much difference

00:44:40.120 --> 00:44:43.328
from having a Higgs boson here.

00:44:43.328 --> 00:44:45.370
So I will talk about this
in a little more detail

00:44:45.370 --> 00:44:46.835
than I did the Higgs example.

00:44:46.835 --> 00:44:48.460
We'll go through some
of the kinematics

00:44:48.460 --> 00:44:50.283
in a little more detail.

00:44:50.283 --> 00:44:52.450
So the reason that I want
to talk about this process

00:44:52.450 --> 00:44:54.910
is there's kind of one
function that we haven't yet

00:44:54.910 --> 00:45:00.460
seen that will show up
in our discussion here,

00:45:00.460 --> 00:45:03.040
one kind spread function
that's ubiquitous

00:45:03.040 --> 00:45:05.980
and shows up in all sorts
of factorization theorems.

00:45:05.980 --> 00:45:08.530
And we haven't seen it yet.

00:45:08.530 --> 00:45:10.720
So what are the kinematics?

00:45:10.720 --> 00:45:13.640
We have some momentum.

00:45:13.640 --> 00:45:14.830
Let me do it this way.

00:45:14.830 --> 00:45:17.290
Let me write it below this guy.

00:45:17.290 --> 00:45:21.280
So pa for the proton,
pb for the other proton,

00:45:21.280 --> 00:45:29.440
goes equals p for the x plus
q for the l plus l minus pair.

00:45:29.440 --> 00:45:31.660
So what are the
important variables?

00:45:31.660 --> 00:45:34.150
Well, there's the center of
mass, energy of the collision.

00:45:34.150 --> 00:45:38.410
That's just s in
Mandelstam variables,

00:45:38.410 --> 00:45:43.450
but I'll call it ECM so
you remember what it is.

00:45:43.450 --> 00:45:45.760
pa plus pb all squared--
that's the collision energy.

00:45:45.760 --> 00:45:51.730
That's ATV [INAUDIBLE]
[? LHC, ?] soon to be higher.

00:45:51.730 --> 00:45:53.920
There's q squared,
and that's the scale

00:45:53.920 --> 00:45:55.348
of the hard collision.

00:45:58.898 --> 00:46:01.440
So the hard collision scale is
not the center of mass energy.

00:46:01.440 --> 00:46:04.150
You take a parton out of each
proton, and those collide.

00:46:04.150 --> 00:46:08.090
And they carry a
fraction of that energy.

00:46:08.090 --> 00:46:09.640
You can see how
much you have been

00:46:09.640 --> 00:46:14.290
looking at how hard the leptons
are, and that's q squared.

00:46:14.290 --> 00:46:16.780
It's useful to talk about the
dimensionless variable, which

00:46:16.780 --> 00:46:18.340
is [INAUDIBLE].

00:46:18.340 --> 00:46:22.060
And that's the ratio of these
two variables, which is always

00:46:22.060 --> 00:46:25.180
less than or equal to 1.

00:46:25.180 --> 00:46:29.320
And then you can talk about
the rapidity of the leptons.

00:46:29.320 --> 00:46:34.870
And you can write that in a kind
of Lorentz invariant looking

00:46:34.870 --> 00:46:41.332
way by having a pb
dot q and a pa dot q.

00:46:41.332 --> 00:46:42.460
Start using capitals.

00:46:47.070 --> 00:46:53.332
So this variable is like
the theta for the leptons.

00:46:53.332 --> 00:46:54.480
So theta q.

00:46:59.140 --> 00:47:02.270
So again, it's just
like our Higgs example,

00:47:02.270 --> 00:47:04.300
but let me draw it in
a little different way.

00:47:04.300 --> 00:47:07.810
You can think about doing
everything in a cm frame.

00:47:07.810 --> 00:47:10.210
Protons are coming
in back to back.

00:47:10.210 --> 00:47:12.130
So then you have
collinear particles

00:47:12.130 --> 00:47:15.400
because your protons
are very energetic.

00:47:15.400 --> 00:47:19.480
So if this is a quark and
this is an anti-quark,

00:47:19.480 --> 00:47:21.250
you produce a virtual photon.

00:47:21.250 --> 00:47:25.210
And the virtual photon
produces a lepton pair.

00:47:25.210 --> 00:47:27.740
So here's an n collinear
quark and an n bar collinear

00:47:27.740 --> 00:47:28.240
anti-quark.

00:47:28.240 --> 00:47:31.480
So the anti-quark is
coming in that way,

00:47:31.480 --> 00:47:32.680
quark is coming in that way.

00:47:32.680 --> 00:47:35.530
They annihilate, produce
a virtual photon.

00:47:35.530 --> 00:47:38.350
That produces a lepton pair.

00:47:38.350 --> 00:47:40.570
That's Drell-Yan.

00:47:40.570 --> 00:47:42.280
OK, so in the cn
frame, we're going

00:47:42.280 --> 00:47:50.930
to have cn modes, cn bar
modes, for the protons

00:47:50.930 --> 00:47:54.700
and the anti-proton.

00:47:54.700 --> 00:47:58.000
Some other variables that
are useful to talk about

00:47:58.000 --> 00:48:00.430
are something that people
call xa and xb, which

00:48:00.430 --> 00:48:02.200
are the Bjorken variables.

00:48:02.200 --> 00:48:05.320
And they already showed up
in our previous example,

00:48:05.320 --> 00:48:07.990
but let me just define them.

00:48:10.730 --> 00:48:16.930
So these are taking tau and
taking the square root of it

00:48:16.930 --> 00:48:19.810
and then splitting up
the rapidity e to the y

00:48:19.810 --> 00:48:20.740
and e to the minus y.

00:48:20.740 --> 00:48:22.930
And these are like analogs
of the Bjorken variable

00:48:22.930 --> 00:48:23.638
for this problem.

00:48:34.650 --> 00:48:36.810
And it's actually
these combinations

00:48:36.810 --> 00:48:39.730
that are showing up here.

00:48:39.730 --> 00:48:41.910
So if you put in
what tau is, that's

00:48:41.910 --> 00:48:43.470
explaining what
these arguments were.

00:48:43.470 --> 00:48:47.070
They're just the xa and xb.

00:48:47.070 --> 00:48:51.090
And kinematics, you
can work out that tau

00:48:51.090 --> 00:48:54.630
is less than or equal to xa or
b and that they're less than

00:48:54.630 --> 00:48:55.260
or equal to 1.

00:48:55.260 --> 00:48:58.710
So there's some
simple bounds on them.

00:48:58.710 --> 00:49:00.240
And then finally,
you can work out

00:49:00.240 --> 00:49:03.150
that x squared,
if you square it,

00:49:03.150 --> 00:49:07.860
is less than or equal to ecm
squared 1 minus square root

00:49:07.860 --> 00:49:11.400
of tau all squared.

00:49:11.400 --> 00:49:13.770
And one more thing
that you have are

00:49:13.770 --> 00:49:18.210
sort of the parton
distribution fractions.

00:49:21.730 --> 00:49:23.350
So parton time
distribution variables,

00:49:23.350 --> 00:49:25.170
and they're bounded
just like dis.

00:49:25.170 --> 00:49:32.448
So xa and xb are things that
are, if you like leptonic,

00:49:32.448 --> 00:49:33.490
you might call them that.

00:49:33.490 --> 00:49:36.465
But they're things that are
external to the QCD, right?

00:49:36.465 --> 00:49:37.840
They're just
measuring properties

00:49:37.840 --> 00:49:44.530
of the leptons, q squared, the
center of mass of the equation,

00:49:44.530 --> 00:49:46.840
and y, which is a y
of the leptons too.

00:49:46.840 --> 00:49:50.590
So everything here is
not a QCD variable.

00:49:50.590 --> 00:49:52.330
And then there's
these QCD variables ca

00:49:52.330 --> 00:49:54.670
and cb, which are inside
the parton distributions.

00:49:54.670 --> 00:49:55.720
And they're bounded.

00:49:55.720 --> 00:49:57.575
Just like we had a bound
in dis, x less than

00:49:57.575 --> 00:49:59.200
or equal to c, less
than or equal to 1,

00:49:59.200 --> 00:50:01.810
here, we have two
analogs of that formula.

00:50:01.810 --> 00:50:03.044
AUDIENCE: So [INAUDIBLE].

00:50:06.310 --> 00:50:07.690
IAIN STEWART:
Yeah, that's right.

00:50:07.690 --> 00:50:10.000
The rapidity of the q--

00:50:10.000 --> 00:50:12.040
so really, this is like--

00:50:12.040 --> 00:50:15.760
I mean, this is my kind of--

00:50:15.760 --> 00:50:18.303
this is like log of--

00:50:18.303 --> 00:50:19.720
the thing that's
important here is

00:50:19.720 --> 00:50:21.880
the log of q plus or q minus.

00:50:21.880 --> 00:50:23.220
It's the rapidity of the q.

00:50:42.690 --> 00:50:45.430
Yeah, you could talk about
measuring individual things

00:50:45.430 --> 00:50:47.800
about the leptons, but
then you would just

00:50:47.800 --> 00:50:50.110
be tacking something
onto the q and taking

00:50:50.110 --> 00:50:52.300
something else out of it.

00:50:52.300 --> 00:50:54.040
All right, so what kind of--

00:50:54.040 --> 00:50:55.543
so this is just some kinematics.

00:50:55.543 --> 00:50:57.460
What kind of limits do
we want to think about?

00:51:01.900 --> 00:51:03.610
So we already talked
about one example

00:51:03.610 --> 00:51:05.980
where we would measure q perp.

00:51:05.980 --> 00:51:07.900
That's what we were
doing in the Higgs case.

00:51:07.900 --> 00:51:09.650
And I'm not going to
talk about that case.

00:51:09.650 --> 00:51:11.067
I'll talk about
three other cases.

00:51:15.140 --> 00:51:18.010
So if we didn't measure
q perp, then there's

00:51:18.010 --> 00:51:19.960
kind of three
different things we

00:51:19.960 --> 00:51:22.510
could do that I'll talk about.

00:51:27.922 --> 00:51:29.900
Let's see.

00:51:29.900 --> 00:51:31.325
Organize my board better.

00:51:35.780 --> 00:51:37.460
I'll tell you about
the kinematics

00:51:37.460 --> 00:51:38.540
of each one of
these, and then I'll

00:51:38.540 --> 00:51:39.998
draw a little
picture for each one.

00:51:50.390 --> 00:51:52.390
So in the inclusive
process, this

00:51:52.390 --> 00:51:56.090
is the analog of what we did
for deep elastic scattering.

00:51:56.090 --> 00:51:58.630
So deep elastic scattering,
we set the Bjorken x variables

00:51:58.630 --> 00:52:00.100
of order 1.

00:52:00.100 --> 00:52:02.005
Here we could say
that some sense, tau

00:52:02.005 --> 00:52:09.010
is of order 1 as well as
xa and xb are of order 1.

00:52:09.010 --> 00:52:12.720
And in this case, what
you're saying about px

00:52:12.720 --> 00:52:14.690
squared is that it's hard.

00:52:14.690 --> 00:52:17.200
It's of order 2 squared,
and if tau is of order 1,

00:52:17.200 --> 00:52:18.725
than that's of
order ecm squared.

00:52:18.725 --> 00:52:20.350
So these things are
all hard variables.

00:52:26.140 --> 00:52:27.790
And in this case,
the way you should

00:52:27.790 --> 00:52:29.582
think about what's
happening in the process

00:52:29.582 --> 00:52:32.240
is the following picture.

00:52:32.240 --> 00:52:38.620
So you have your partons coming
in or your protons coming in.

00:52:38.620 --> 00:52:42.010
And basically, you're allowing
radiation that's hard anywhere.

00:52:42.010 --> 00:52:44.687
So hard is supposed to be pink.

00:52:44.687 --> 00:52:46.270
You're not constraining
the radiation.

00:52:46.270 --> 00:52:52.570
You're really allowing hard
radiation [INAUDIBLE] hard.

00:52:52.570 --> 00:52:55.030
Your x is hard and you
allow jets in any direction,

00:52:55.030 --> 00:52:56.170
for example.

00:52:56.170 --> 00:52:58.794
And then somewhere,
there's a lepton pair.

00:52:58.794 --> 00:53:02.170
[INAUDIBLE] purple.

00:53:02.170 --> 00:53:04.370
But it's not constrained either.

00:53:04.370 --> 00:53:06.130
It's really fairly general.

00:53:12.960 --> 00:53:13.960
My picture is too big.

00:53:13.960 --> 00:53:15.293
I'm going to run it [INAUDIBLE].

00:53:15.293 --> 00:53:16.137
Oh well.

00:53:16.137 --> 00:53:16.720
We'll do this.

00:53:20.660 --> 00:53:23.690
So I want to draw analogous
pictures for the other cases.

00:53:23.690 --> 00:53:26.290
So in the end point, you're
taking a different limit.

00:53:26.290 --> 00:53:30.190
What you're doing is you're
taking this tau goes to 1.

00:53:30.190 --> 00:53:34.450
And you can see from over
here that if tau goes to 1,

00:53:34.450 --> 00:53:36.700
that forces xa
and xb to go to 1.

00:53:41.700 --> 00:53:46.260
And if xa and xb go to 1, that
forces ca and cb to go to 1.

00:53:49.867 --> 00:53:51.450
So you're really
talking about probing

00:53:51.450 --> 00:53:54.118
the proton in a very
special kinematics

00:53:54.118 --> 00:53:55.410
where everything is going to 1.

00:53:55.410 --> 00:53:57.750
And you're basically
in the hard collision,

00:53:57.750 --> 00:54:00.630
you're forcing all the energy
to go into the parton that's

00:54:00.630 --> 00:54:02.410
colliding.

00:54:02.410 --> 00:54:08.790
So the proton, the
full proton momentum,

00:54:08.790 --> 00:54:12.537
goes into the active parton.

00:54:12.537 --> 00:54:14.370
Let me just say it that
way-- active parton.

00:54:17.950 --> 00:54:22.210
So that changes the
picture because it also--

00:54:22.210 --> 00:54:25.450
if you look at it,
when tau goes to 1,

00:54:25.450 --> 00:54:28.990
it says the outgoing energy,
the full ecm squared,

00:54:28.990 --> 00:54:32.650
is going into q squared, which
is the leptonic variable.

00:54:32.650 --> 00:54:34.840
So all the energy is
coming in on the partons

00:54:34.840 --> 00:54:36.790
and going out on the leptons.

00:54:36.790 --> 00:54:39.318
So you don't have hard
radiation like this anymore.

00:54:39.318 --> 00:54:41.110
The only thing that
you could possibly have

00:54:41.110 --> 00:54:43.550
is soft radiation.

00:54:43.550 --> 00:54:47.560
So in this case, what
happens is this picture--

00:54:47.560 --> 00:54:50.210
make my lepton a
little shorter--

00:54:50.210 --> 00:54:50.710
changes.

00:54:54.330 --> 00:54:59.340
You still, of course, have these
incoming guys, but what happens

00:54:59.340 --> 00:55:01.650
is that everything
outgoing is soft.

00:55:01.650 --> 00:55:03.660
So make it green.

00:55:03.660 --> 00:55:11.190
You have soft
radiation like that.

00:55:11.190 --> 00:55:13.750
And then it turns out
also that in this case,

00:55:13.750 --> 00:55:20.250
the leptons end up having to be
back to back because you can't

00:55:20.250 --> 00:55:21.632
have any transverse momentum.

00:55:21.632 --> 00:55:24.090
That would be-- there's nothing
for the transverse momentum

00:55:24.090 --> 00:55:25.080
to recoil against.

00:55:30.130 --> 00:55:31.420
So that's a possibility.

00:55:35.630 --> 00:55:38.390
And then there's a third
thing that we'll talk about,

00:55:38.390 --> 00:55:41.980
which is what I call isolated.

00:55:41.980 --> 00:55:45.310
And it in some sense is trying
to combine these two pictures

00:55:45.310 --> 00:55:48.670
here without taking
a limit on tau.

00:55:48.670 --> 00:55:51.970
So you might say, well,
what's the most typical event

00:55:51.970 --> 00:55:54.790
at the LHC?

00:55:54.790 --> 00:55:57.550
Where is most of the
cross-section for this process?

00:55:57.550 --> 00:55:59.170
And that would be
in a situation where

00:55:59.170 --> 00:56:01.160
tau is not in the endpoint.

00:56:01.160 --> 00:56:03.280
It doesn't go to 1.

00:56:03.280 --> 00:56:05.140
It's kind of an order
1 quantity, actually.

00:56:05.140 --> 00:56:09.460
You typically get xa's and
xb's that are like 0.1 or 0.01,

00:56:09.460 --> 00:56:12.980
small x's for--

00:56:12.980 --> 00:56:14.660
depends on what q
squared you look at.

00:56:14.660 --> 00:56:18.140
But the typical ones you're
interested in are small x's.

00:56:18.140 --> 00:56:19.850
So you don't want x to go to 1.

00:56:19.850 --> 00:56:23.210
So tau can be of order 1.

00:56:23.210 --> 00:56:25.970
But if you ask what the most
probable thing for the px

00:56:25.970 --> 00:56:28.610
to do, if you have these
collinear particles coming in

00:56:28.610 --> 00:56:30.110
and they start
radiating, well, they

00:56:30.110 --> 00:56:32.362
like to radiate
collinear particles.

00:56:32.362 --> 00:56:34.070
So the most likely
thing that will happen

00:56:34.070 --> 00:56:35.690
is that you'll get
collinear radiation

00:56:35.690 --> 00:56:37.580
from the incoming particles.

00:56:37.580 --> 00:56:40.700
And you can look
at that by studying

00:56:40.700 --> 00:56:42.700
the following situation
where you constrain

00:56:42.700 --> 00:56:47.510
px squared to be 2 ISR jets.

00:56:47.510 --> 00:56:50.450
So the picture--
we'll talk about how

00:56:50.450 --> 00:56:51.840
we do that in a minute.

00:56:51.840 --> 00:56:55.440
But-- or one way of doing it.

00:56:55.440 --> 00:56:57.580
So the picture would
then be as follows.

00:56:57.580 --> 00:56:59.330
We have these incoming
guys, which now I'm

00:56:59.330 --> 00:57:01.400
going to try to draw
some radiation for.

00:57:01.400 --> 00:57:05.060
So here are some colors.

00:57:05.060 --> 00:57:08.900
So here's a collinear
guy in one direction.

00:57:08.900 --> 00:57:11.660
Here's a collinear guy
in the other direction.

00:57:11.660 --> 00:57:13.370
And they can radiate.

00:57:13.370 --> 00:57:15.950
They radiate prior to
the hard collision here.

00:57:15.950 --> 00:57:23.190
So you're getting some jets from
these guys that look like this

00:57:23.190 --> 00:57:30.540
and then from this guy a
symmetric thing like that.

00:57:30.540 --> 00:57:33.390
And then in the central
part of the collision,

00:57:33.390 --> 00:57:34.940
you just allow soft radiation.

00:57:40.020 --> 00:57:43.760
So this is the
isolated scenario.

00:57:43.760 --> 00:57:48.200
And if I draw the
leptons, it turns out

00:57:48.200 --> 00:57:49.760
they don't have to
be back to back,

00:57:49.760 --> 00:57:52.790
but there is some constraint
on their kinematics.

00:57:52.790 --> 00:57:55.520
Basically, they're back to back
in the transverse plane but not

00:57:55.520 --> 00:57:58.200
longitudinally.

00:57:58.200 --> 00:58:00.770
But we won't dwell on that.

00:58:00.770 --> 00:58:03.420
OK, so this is the sort of
third kinematic configuration.

00:58:03.420 --> 00:58:05.747
So even though we're
interested in one process,

00:58:05.747 --> 00:58:07.205
we've already just
described to you

00:58:07.205 --> 00:58:08.930
four different ways
that you could think

00:58:08.930 --> 00:58:10.273
about looking at it.

00:58:10.273 --> 00:58:11.690
And those four
different ways will

00:58:11.690 --> 00:58:13.565
lead to four different
factorization theorems

00:58:13.565 --> 00:58:15.890
because it's a different
kinematic setup.

00:58:15.890 --> 00:58:17.900
The first one was
pt of the Higgs.

00:58:17.900 --> 00:58:20.390
That led to this rapidly
divergent type factorization

00:58:20.390 --> 00:58:21.290
theorem.

00:58:21.290 --> 00:58:23.970
Inclusive we'll talk about in
a minute, what it looks like.

00:58:23.970 --> 00:58:27.502
Endpoint and isolated, they all
will have different formulas

00:58:27.502 --> 00:58:29.210
for the factorization
theorem, and that's

00:58:29.210 --> 00:58:32.690
because they look different.

00:58:32.690 --> 00:58:36.110
I really should say that
this is ultra soft, not soft.

00:58:39.470 --> 00:58:42.326
And so this is
ultra soft too here.

00:58:42.326 --> 00:58:45.350
This is ultra soft.

00:58:45.350 --> 00:58:51.200
This is cn radiation,
and this is cn bar.

00:58:55.720 --> 00:59:00.610
All right, so let's
see how far we get.

00:59:00.610 --> 00:59:02.470
OK, so let's start
with inclusive.

00:59:15.470 --> 00:59:17.008
So the x is hard.

00:59:17.008 --> 00:59:18.550
And so the way you
can think about it

00:59:18.550 --> 00:59:22.630
is that what you're
doing is that you have--

00:59:22.630 --> 00:59:25.840
you can use kind of an
optical theorem type picture

00:59:25.840 --> 00:59:28.570
where you're cutting
these forward graphs.

00:59:28.570 --> 00:59:32.650
These are the leptons
here, which are

00:59:32.650 --> 00:59:35.140
the things in the final state--

00:59:35.140 --> 00:59:36.640
so vertical photon.

00:59:36.640 --> 00:59:37.930
And I'm squaring it.

00:59:37.930 --> 00:59:42.470
And in comes qq bar
and then squaring it,

00:59:42.470 --> 00:59:46.100
so qq bar on that side as well.

00:59:46.100 --> 00:59:49.480
And then if you think about it,
every kind of radiation gluon

00:59:49.480 --> 00:59:52.700
that I would put across this cut
I can think of as kind of hard.

00:59:52.700 --> 00:59:54.350
And so I can integrate it out.

00:59:54.350 --> 00:59:58.030
And so what you're going to get
is if you just think about this

00:59:58.030 --> 01:00:00.040
process, when you match
the cross-section,

01:00:00.040 --> 01:00:05.290
you're going to get some four
quark operator in [? SCT. ?] So

01:00:05.290 --> 01:00:06.710
this will match
on to an operator,

01:00:06.710 --> 01:00:10.540
which is a four quark operator
where two of the quarks are n

01:00:10.540 --> 01:00:13.090
collinear--

01:00:13.090 --> 01:00:14.440
I'll draw it like this--

01:00:14.440 --> 01:00:18.550
and two of them are
n bar because that's

01:00:18.550 --> 01:00:19.870
the external particles.

01:00:19.870 --> 01:00:24.280
We have qn, q bar, n bar.

01:00:24.280 --> 01:00:25.570
So we're going to get that.

01:00:25.570 --> 01:00:29.800
And that we know how to
write down the lowest order

01:00:29.800 --> 01:00:30.710
operator for that.

01:00:30.710 --> 01:00:34.390
That's just going to be chi bar
chi, chi bar chi, chi bar chi.

01:00:48.560 --> 01:00:59.497
So we have a four
quark operator in SCET,

01:00:59.497 --> 01:01:01.330
which you can ride after
doing a [INAUDIBLE]

01:01:01.330 --> 01:01:06.640
in the spin with all the n
collinear fields together.

01:01:06.640 --> 01:01:10.240
You can work out constraints
on the [? rack ?] structure.

01:01:10.240 --> 01:01:11.980
There's only one
operator, basically.

01:01:24.720 --> 01:01:25.630
It's not quite true.

01:01:25.630 --> 01:01:28.290
So you could have
a gluon operator,

01:01:28.290 --> 01:01:30.120
an operator with
four gluon fields,

01:01:30.120 --> 01:01:32.530
that's the analog of this one.

01:01:32.530 --> 01:01:35.790
Then you would have to have some
higher order diagram in order

01:01:35.790 --> 01:01:37.590
to take those gluons
and attach a photon.

01:01:37.590 --> 01:01:41.920
It would have to be a quark
loop inside the hard function.

01:01:41.920 --> 01:01:45.940
But you could also have a
gluon operator with four b's.

01:01:48.800 --> 01:01:58.108
So let me just say there's
also bn, bn, bn bar, bn bar.

01:01:58.108 --> 01:02:02.430
So you don't have a color
octet structure-- no ta, ta.

01:02:02.430 --> 01:02:04.260
And again, that's like
our bdd pi example

01:02:04.260 --> 01:02:05.718
where if you had
that structure, it

01:02:05.718 --> 01:02:08.580
would vanish when you
take matrix elements.

01:02:08.580 --> 01:02:12.300
So there's no ta,
ta, just constraints

01:02:12.300 --> 01:02:13.800
on the color contractions.

01:02:13.800 --> 01:02:16.740
Here, these guys would
have to be contracted.

01:02:16.740 --> 01:02:19.000
Those guys would have
to be contracted.

01:02:19.000 --> 01:02:20.850
You can make the
field redefinition.

01:02:20.850 --> 01:02:23.200
The field redefinition
the y's would cancel out.

01:02:23.200 --> 01:02:26.040
So there's no ultrasofts here.

01:02:29.190 --> 01:02:31.710
[INAUDIBLE] leading order.

01:02:31.710 --> 01:02:35.130
And if you take these matrix
elements of these objects,

01:02:35.130 --> 01:02:37.770
you can kind of guess what
they're going to give.

01:02:37.770 --> 01:02:39.780
This is just giving a PDF.

01:02:39.780 --> 01:02:41.400
Each of these are giving a PDF.

01:02:41.400 --> 01:02:46.170
And they're the regular
PDFs, the standard ones.

01:02:46.170 --> 01:02:49.590
Because we didn't measure
a [? perp ?] momentum,

01:02:49.590 --> 01:02:51.000
we just get standard PDFs.

01:02:54.090 --> 01:02:56.255
So some parton
inside the proton,

01:02:56.255 --> 01:02:58.380
which would be a gluon from
these guys-- gluon PDFs

01:02:58.380 --> 01:03:00.330
are quarks from these guys.

01:03:00.330 --> 01:03:04.800
And it depends on
some x [INAUDIBLE] mu.

01:03:04.800 --> 01:03:06.510
x could be xa or x--

01:03:06.510 --> 01:03:08.040
x could be ca or cb.

01:03:10.820 --> 01:03:11.925
So let me just write ca.

01:03:16.610 --> 01:03:17.860
And that's really all you get.

01:03:17.860 --> 01:03:20.790
So you have a hard function
and then two collinear parton

01:03:20.790 --> 01:03:22.530
distribution functions.

01:03:22.530 --> 01:03:23.550
So the cross section--

01:03:32.095 --> 01:03:37.410
[INAUDIBLE] these limits
that we talked about

01:03:37.410 --> 01:03:40.530
come from the kinematics.

01:03:40.530 --> 01:03:42.900
[INAUDIBLE] some hard
function, which is an inclusive

01:03:42.900 --> 01:03:44.550
hard function.

01:03:44.550 --> 01:03:47.410
Like in DIS, it depends on
the ratio of these variables.

01:03:47.410 --> 01:03:49.620
But now there's two of them.

01:03:49.620 --> 01:03:53.520
just depends on q
squared, depends on mu.

01:03:53.520 --> 01:03:55.830
And then there's times PDFs.

01:04:08.490 --> 01:04:11.260
There's an important
caveat here,

01:04:11.260 --> 01:04:13.648
which is if you want
to derive this result,

01:04:13.648 --> 01:04:15.690
you have to make sure that
the degrees of freedom

01:04:15.690 --> 01:04:18.452
I've told you with the degrees
of freedom are the right ones.

01:04:18.452 --> 01:04:19.410
So what did I tell you?

01:04:19.410 --> 01:04:24.480
I told you we have cn, cn
bar, and ultra soft, right?

01:04:24.480 --> 01:04:29.412
And that from the
kinematics, it's a SCET one.

01:04:29.412 --> 01:04:31.620
Turns out there's one other
type of degree of freedom

01:04:31.620 --> 01:04:33.818
that you could worry
about, and that's

01:04:33.818 --> 01:04:34.860
called the Glauber gluon?

01:04:37.440 --> 01:04:39.990
And to derive this, we must
know that that doesn't matter.

01:04:49.140 --> 01:04:51.930
What is a Glauber gluon?

01:04:51.930 --> 01:05:00.360
It's a Coulombic gluon
between n and m bar particles.

01:05:00.360 --> 01:05:06.970
So it's a Coulombic
type potential

01:05:06.970 --> 01:05:11.800
that goes like 1 over
pt vector squared.

01:05:11.800 --> 01:05:12.700
And that's a Glauber.

01:05:15.638 --> 01:05:16.930
It's called a Glauber exchange.

01:05:16.930 --> 01:05:18.340
So it's not an
on-shell particle.

01:05:18.340 --> 01:05:21.220
It's like a potential.

01:05:21.220 --> 01:05:24.010
But you have to know that those
actually are relevant in order

01:05:24.010 --> 01:05:25.080
to get to this formula.

01:05:25.080 --> 01:05:28.453
So there's a little
more work involved which

01:05:28.453 --> 01:05:29.620
I'm not going to talk about.

01:05:33.910 --> 01:05:36.580
OK, but-- and that's going
to be true, actually,

01:05:36.580 --> 01:05:40.490
of all the other cases
that we do as well.

01:05:40.490 --> 01:05:42.070
So what about this
threshold limit?

01:05:42.070 --> 01:05:44.050
How does this formula change?

01:05:44.050 --> 01:05:46.960
Well, you could think about
how the formula changes.

01:05:46.960 --> 01:05:51.100
It's really a change of this
h in the threshold limit

01:05:51.100 --> 01:05:52.750
because what's
happening is that you're

01:05:52.750 --> 01:05:55.090
constraining the pink
radiation and making it green.

01:05:59.540 --> 01:06:00.790
[INAUDIBLE] keep my pictures.

01:06:09.820 --> 01:06:20.810
So if we want to go from here
to here for the threshold limit,

01:06:20.810 --> 01:06:24.605
then only certain terms, if you
like, in the HIJ are included.

01:06:28.350 --> 01:06:31.010
So if you wrote out that
function, really what

01:06:31.010 --> 01:06:33.140
it corresponds to is that
only the most singular

01:06:33.140 --> 01:06:39.200
terms in the variable 1 minus
tau in these 1 minus type

01:06:39.200 --> 01:06:42.380
variables are included.

01:06:42.380 --> 01:06:44.540
So delta functions
of 1 minus tau,

01:06:44.540 --> 01:06:47.090
plus functions of
1 minus tau, those

01:06:47.090 --> 01:06:51.740
are the terms that you
would include from the h

01:06:51.740 --> 01:06:56.930
whereas the inclusive
one has much more in it.

01:06:56.930 --> 01:07:00.540
This keeps only
particular terms.

01:07:00.540 --> 01:07:02.540
But there's also different
hierarchies of scales

01:07:02.540 --> 01:07:05.600
because now 1 minus tau is
small, and you want to--

01:07:05.600 --> 01:07:11.210
for example, some logs of 1
minus tau in this situation.

01:07:16.427 --> 01:07:18.135
And that's what the
factorization theorem

01:07:18.135 --> 01:07:20.940
in this threshold
region does for you.

01:07:20.940 --> 01:07:31.620
So the HIJ inclusive gets
turned into the following.

01:07:31.620 --> 01:07:35.720
This is the modification of
the factorization theorem.

01:07:35.720 --> 01:07:38.870
So there's a soft function
for that soft radiation.

01:07:43.460 --> 01:07:52.280
And then there's a different
heart function, which is only

01:07:52.280 --> 01:07:53.480
a function of q squared.

01:07:53.480 --> 01:07:54.920
And it's like the square--

01:07:54.920 --> 01:07:58.160
this guy here is the square
of a Wilson coefficient again,

01:07:58.160 --> 01:08:01.985
and i and j are just pairs
of quarks-- uu bar, dd bar.

01:08:01.985 --> 01:08:07.090
There's actually no
gluons in this case.

01:08:07.090 --> 01:08:10.100
So the terms of the gluon
PDFs actually get suppressed.

01:08:10.100 --> 01:08:12.350
They're not singular
in 1 minus tau,

01:08:12.350 --> 01:08:14.550
and we get this formula here.

01:08:14.550 --> 01:08:16.220
And now there's going
to be some running

01:08:16.220 --> 01:08:18.803
that you have to do between the
hard scale and the soft scale,

01:08:18.803 --> 01:08:21.620
and that running is going to
sum up these logarithms of 1

01:08:21.620 --> 01:08:23.990
minus tau.

01:08:23.990 --> 01:08:32.109
OK, so I'm not going to go
through that in any more

01:08:32.109 --> 01:08:32.620
detail.

01:08:32.620 --> 01:08:34.370
I'll spend a little
more time on this one.

01:08:38.920 --> 01:08:41.770
So what about
isolated Drell-Yan?

01:08:41.770 --> 01:08:44.020
This one is a little bit
different than the other ones

01:08:44.020 --> 01:08:46.689
because we have to measure
something about the process

01:08:46.689 --> 01:08:49.300
to know that it looks like
this if we really want to drive

01:08:49.300 --> 01:08:50.859
a factorization theorem.

01:08:50.859 --> 01:08:52.990
Here, we were measuring tau.

01:08:52.990 --> 01:08:55.479
But now I'm telling you I
don't want to measure tau.

01:08:55.479 --> 01:08:56.918
Yet I still want
to distinguish--

01:08:56.918 --> 01:08:58.210
so I don't want to measure tau.

01:08:58.210 --> 01:09:00.700
I don't want to constrain
it to be close to 1.

01:09:00.700 --> 01:09:03.460
But I still want to distinguish
between these two, right?

01:09:03.460 --> 01:09:06.069
And if I'm going to distinguish
between those two situations,

01:09:06.069 --> 01:09:08.479
I need to measure
something else.

01:09:08.479 --> 01:09:13.090
And we can measure something
that's exactly like what we did

01:09:13.090 --> 01:09:17.080
when we did e plus e minus to
[? dijets ?] because this is

01:09:17.080 --> 01:09:19.819
really like [? dijets. ?] But
they're just initial stage jets

01:09:19.819 --> 01:09:21.189
rather than final stage jets.

01:09:25.170 --> 01:09:28.649
So that's what
we're going to do.

01:09:28.649 --> 01:09:32.130
We only have four jets.

01:09:32.130 --> 01:09:35.790
We have tau that's generic.

01:09:35.790 --> 01:09:38.609
So the c's are not
being forced to go to 1.

01:09:38.609 --> 01:09:42.750
And so we need something that
we can observe to do that.

01:09:42.750 --> 01:09:46.365
And here's what we can do.

01:09:46.365 --> 01:09:48.660
Let's do something
that's the analog of e

01:09:48.660 --> 01:09:54.900
plus e minus to [? dijets ?] but
for these initial stage jets.

01:09:54.900 --> 01:10:00.150
So we say that px is a sum of
momentum in two hemispheres.

01:10:00.150 --> 01:10:06.750
The hemispheres a and b
are just the dividing line

01:10:06.750 --> 01:10:10.260
of the center of mass of the
collision-- so very simple,

01:10:10.260 --> 01:10:12.480
perpendicular to the beam axis.

01:10:12.480 --> 01:10:15.630
And then we just
say that ba plus--

01:10:15.630 --> 01:10:19.320
so this is just a [? four ?]
momentum decomposition.

01:10:19.320 --> 01:10:22.680
But we can look
at ba plus, which

01:10:22.680 --> 01:10:27.880
is like dotting the n vector
for one axis into ba--

01:10:27.880 --> 01:10:34.600
and that's a sum over particles
in one of the hemispheres

01:10:34.600 --> 01:10:35.850
and then likewise for b.

01:10:40.390 --> 01:10:42.630
So this is if I
write it in terms

01:10:42.630 --> 01:10:45.933
of energies and rapidities.

01:10:45.933 --> 01:10:47.240
It'll look like this.

01:10:51.670 --> 01:10:53.490
So this is some
observable ba plus.

01:10:53.490 --> 01:10:56.730
And then I can constrain that
ba plus, which is exactly

01:10:56.730 --> 01:10:59.518
like constraining the
plus momentum in one

01:10:59.518 --> 01:11:00.810
of the hemispheres to be small.

01:11:00.810 --> 01:11:03.233
And what that does, if you
constrain the plus momentum

01:11:03.233 --> 01:11:04.650
in this hemisphere
to be small, it

01:11:04.650 --> 01:11:07.200
will allow ultra soft radiation
because they have small plus

01:11:07.200 --> 01:11:07.950
momentum.

01:11:07.950 --> 01:11:10.080
And it'll allow
collinear radiation

01:11:10.080 --> 01:11:12.160
because they have
small plus momentum.

01:11:12.160 --> 01:11:14.040
So constraining
ba plus constrains

01:11:14.040 --> 01:11:16.740
all the ba plus momentum
in that hemisphere.

01:11:16.740 --> 01:11:20.718
And that puts you in
an SCET one situation

01:11:20.718 --> 01:11:22.260
where we have a
figure like that one.

01:11:27.800 --> 01:11:30.603
So rather than constrain
the [? perp ?] momentum,

01:11:30.603 --> 01:11:31.520
we constrain the plus.

01:11:31.520 --> 01:11:32.570
And then we get SCET one.

01:11:38.460 --> 01:11:47.280
So take ba plus to be
less than or something,

01:11:47.280 --> 01:11:51.820
which you could say, well,
less than or equal to some cut.

01:11:51.820 --> 01:11:54.560
Let's just call it b cut.

01:11:54.560 --> 01:11:57.410
And that is much less than q.

01:11:57.410 --> 01:12:01.040
And then we do the
same thing for b plus,

01:12:01.040 --> 01:12:05.910
which we define as
nb, which is na bar.

01:12:05.910 --> 01:12:09.080
So the notation here
is a little awkward,

01:12:09.080 --> 01:12:10.680
but we do the same thing here.

01:12:10.680 --> 01:12:13.760
So this is less than or
equal to some cut and that

01:12:13.760 --> 01:12:15.950
that's much less than q.

01:12:15.950 --> 01:12:17.450
And by demanding
small plus momentum

01:12:17.450 --> 01:12:19.285
in both hemispheres,
appropriate momentum

01:12:19.285 --> 01:12:21.470
with the small components--

01:12:21.470 --> 01:12:25.670
these are the small components--

01:12:25.670 --> 01:12:31.230
that puts us in a
SCET one situation.

01:12:31.230 --> 01:12:34.220
So we have cn, cn
bar, and ultra soft.

01:12:34.220 --> 01:12:37.200
And those are the allowed
types of radiation,

01:12:37.200 --> 01:12:38.840
and that's what I
drew in my figure.

01:12:38.840 --> 01:12:40.070
So hopefully that's clear.

01:12:45.090 --> 01:12:49.660
All right, so what does
the factorization look

01:12:49.660 --> 01:12:52.060
like in this case?

01:12:52.060 --> 01:12:53.560
Well, you have two
types of things

01:12:53.560 --> 01:12:56.140
that are collinear if
you look at the figure.

01:12:56.140 --> 01:12:58.480
There's the initial
state proton, right,

01:12:58.480 --> 01:13:01.810
which came in here, and
it was collinear here.

01:13:01.810 --> 01:13:03.700
So here's a proton.

01:13:03.700 --> 01:13:07.450
And then it's cruising
along, and at some point,

01:13:07.450 --> 01:13:11.170
it widens out like
this and becomes a jet.

01:13:11.170 --> 01:13:14.800
So this here is collinear at a
smaller scale than this here.

01:13:14.800 --> 01:13:16.720
This is a jet, right?

01:13:16.720 --> 01:13:21.667
It's got large invariant mass
relative to the proton mass.

01:13:21.667 --> 01:13:23.500
So there's actually two
types of collinears,

01:13:23.500 --> 01:13:27.068
and if you really want to draw
the mode picture for this,

01:13:27.068 --> 01:13:28.360
here's what it would look like.

01:13:33.620 --> 01:13:38.170
So here's our collinear
hyperbola cn cn bar.

01:13:38.170 --> 01:13:42.100
We have an ultra soft hyperbola.

01:13:42.100 --> 01:13:45.250
And then somewhere, we have
a lambda qcd hyperbola.

01:13:45.250 --> 01:13:47.590
And there's protons that
sit on the lambda qcd.

01:13:47.590 --> 01:13:50.436
So this is like pn bar,
if you like, or let

01:13:50.436 --> 01:13:52.310
me give it a different name--

01:13:52.310 --> 01:13:53.530
pn.

01:13:53.530 --> 01:13:54.530
Then there's hard modes.

01:13:58.570 --> 01:14:00.465
So what we'll do
is we'll not really

01:14:00.465 --> 01:14:01.840
worry about
distinguishing these.

01:14:01.840 --> 01:14:04.860
We'll think of
them all together.

01:14:04.860 --> 01:14:06.280
So we'll just have
[? sct ?] one,

01:14:06.280 --> 01:14:10.000
where we have ultra softs
here, a cn, and a cn bar.

01:14:10.000 --> 01:14:12.310
But then we'll later have
to worry about factoring.

01:14:12.310 --> 01:14:14.330
This includes the jet.

01:14:14.330 --> 01:14:16.270
This is the collinear jet.

01:14:16.270 --> 01:14:19.360
And this is the proton, right?

01:14:19.360 --> 01:14:24.173
So both objects will be in the
same function at the beginning,

01:14:24.173 --> 01:14:25.840
and then we'll have
to factor them later

01:14:25.840 --> 01:14:28.450
if we want to
separate those scales.

01:14:28.450 --> 01:14:31.630
It's a similar trick to what we
used before in another example.

01:14:31.630 --> 01:14:34.250
It was for soft in that case.

01:14:34.250 --> 01:14:36.620
So if we just keep
these guys together,

01:14:36.620 --> 01:14:37.960
then it's just a SCET one.

01:14:37.960 --> 01:14:41.050
Just have these modes go
through the factorization.

01:14:41.050 --> 01:14:42.210
And this is what we get.

01:14:46.300 --> 01:14:49.690
So we can measure q
squared, rapidity,

01:14:49.690 --> 01:14:51.880
and we can measure these
two hemisphere momenta.

01:14:57.190 --> 01:15:05.470
The hard function is again the
square of a Wilson coefficient

01:15:05.470 --> 01:15:07.210
just as it was in
the threshold case.

01:15:13.970 --> 01:15:19.920
We get some collinear functions
that are called beam functions,

01:15:19.920 --> 01:15:23.160
and they're like, they
have an argument that's

01:15:23.160 --> 01:15:26.200
a plus times a minus momentum.

01:15:26.200 --> 01:15:30.290
So this wa is like
a minus momentum.

01:15:30.290 --> 01:15:32.460
We have a Bjorken x variable.

01:15:32.460 --> 01:15:34.710
And we have mu.

01:15:34.710 --> 01:15:37.920
There's one of these
for each direction.

01:15:37.920 --> 01:15:39.840
So there's one for
the b direction too.

01:15:47.810 --> 01:15:50.390
And then there's
a soft function.

01:15:50.390 --> 01:15:52.380
And it's a hemisphere
soft function.

01:15:52.380 --> 01:15:56.465
It's really exactly analogous
to the soft function that we had

01:15:56.465 --> 01:15:58.250
in e plus e minus,
the [? dijets, ?]

01:15:58.250 --> 01:16:02.264
except that the Wilson lines are
incoming instead of outgoing.

01:16:02.264 --> 01:16:07.010
So it depends on two
plus momenta in mu.

01:16:07.010 --> 01:16:09.500
OK, so the factorization theorem
involves these b's, which

01:16:09.500 --> 01:16:10.640
are called beam functions.

01:16:14.250 --> 01:16:18.230
So this is a beam
function, four parton i.

01:16:18.230 --> 01:16:22.940
And parton i could be a quark
or a gluon in this case.

01:16:22.940 --> 01:16:29.750
Well, sorry, parton i here
will be a quark, actually,

01:16:29.750 --> 01:16:30.750
in this case--

01:16:30.750 --> 01:16:32.360
quark or a q bar.

01:16:34.960 --> 01:16:37.625
You have analogous
functions for gluons as well

01:16:37.625 --> 01:16:39.250
that would come in,
for example, if you

01:16:39.250 --> 01:16:42.110
were doing Higgs production.

01:16:42.110 --> 01:16:43.690
So what is this beam function?

01:16:43.690 --> 01:16:45.700
It's the one object
that we haven't

01:16:45.700 --> 01:16:48.595
seen an example of before.

01:16:52.120 --> 01:16:54.760
And it's really a matrix
element of an operator

01:16:54.760 --> 01:16:58.450
that we've studied in
many different situations.

01:16:58.450 --> 01:17:00.460
It's just the chi
bar chi operator

01:17:00.460 --> 01:17:04.730
with different arguments than
we've studied previously.

01:17:04.730 --> 01:17:07.330
So it's a slightly different
matrix element of the chi bar

01:17:07.330 --> 01:17:11.920
chi operator because
what we're measuring

01:17:11.920 --> 01:17:15.850
is both the large momentum
and the small momentum

01:17:15.850 --> 01:17:16.840
of that operator.

01:17:16.840 --> 01:17:21.190
When we measured just the large
momentum, that gave us the PDF.

01:17:21.190 --> 01:17:25.950
But now we're measuring
also the small momentum,

01:17:25.950 --> 01:17:27.610
which I can write like this.

01:17:39.020 --> 01:17:43.060
So it's a protein matrix element
of a chi bar chi operator

01:17:43.060 --> 01:17:45.880
where if it was a PDF,
this would be 0 and 0,

01:17:45.880 --> 01:17:47.710
and we just have
the minus momentum.

01:17:47.710 --> 01:17:50.050
But now we're
measuring, if you like,

01:17:50.050 --> 01:17:53.290
the other momentum, which
is like a plus momentum.

01:17:53.290 --> 01:17:54.940
And I wrote it in Fourier space.

01:17:54.940 --> 01:17:58.630
So the plus momentum that
you're measuring is here.

01:17:58.630 --> 01:18:01.450
Those are the two variables
that are showing up

01:18:01.450 --> 01:18:06.280
on the right hand side as
well as the minus momentum

01:18:06.280 --> 01:18:08.470
of the proton.

01:18:08.470 --> 01:18:11.890
So it's just a different
matrix element than one

01:18:11.890 --> 01:18:13.060
we've had before.

01:18:13.060 --> 01:18:18.070
The jet function, remember,
would be vacuum matrix element

01:18:18.070 --> 01:18:23.830
of chi bar chi, and
the PDF, standard PDF,

01:18:23.830 --> 01:18:29.200
would be proton matrix
element of chi bar delta chi.

01:18:29.200 --> 01:18:31.090
And it's just really--

01:18:31.090 --> 01:18:32.830
the arguments are
slightly different,

01:18:32.830 --> 01:18:35.290
and that allows this
thing to contain both kind

01:18:35.290 --> 01:18:37.960
of a jet, initial state jet--

01:18:37.960 --> 01:18:39.790
in this case here,
the jet function,

01:18:39.790 --> 01:18:42.987
you'd have some variable zero.

01:18:42.987 --> 01:18:44.570
In this case here,
you'd have 0 and 0.

01:18:44.570 --> 01:18:45.945
And so if you
like, what this guy

01:18:45.945 --> 01:18:48.490
is is just a combination of the
two things we studied before.

01:18:48.490 --> 01:18:49.657
We studied the jet function.

01:18:49.657 --> 01:18:50.470
We studied the PDF.

01:18:50.470 --> 01:18:54.460
Now this has both inside it.

01:18:54.460 --> 01:18:57.940
And if we go through this final
factorization between there

01:18:57.940 --> 01:19:01.735
and there, then that's like a
SCET one to SCET two matching.

01:19:07.440 --> 01:19:11.460
And so we can integrate out the
perturbative radiation, which

01:19:11.460 --> 01:19:12.840
is the jet function, right?

01:19:12.840 --> 01:19:14.190
That's perturbative.

01:19:17.070 --> 01:19:18.630
And this is non-perturbative.

01:19:18.630 --> 01:19:22.620
We can separate those by
doing another matching,

01:19:22.620 --> 01:19:27.450
and that gives a factorization
theorem for the b alone

01:19:27.450 --> 01:19:28.470
that looks like this.

01:19:40.130 --> 01:19:42.700
So there's some perturbative
matching coefficients

01:19:42.700 --> 01:19:43.660
which are called i.

01:19:47.230 --> 01:19:48.715
And then you get a standard PDF.

01:19:53.170 --> 01:19:57.980
So this i is the thing that
has in it the jet radiation.

01:19:57.980 --> 01:20:00.490
This is sort of
an i for the jet.

01:20:00.490 --> 01:20:01.570
And then this is the PDF.

01:20:04.090 --> 01:20:06.022
OK, and the T scale
here is large.

01:20:06.022 --> 01:20:07.480
And what you're
basically expanding

01:20:07.480 --> 01:20:12.550
in is you're expanding
in lambda QCD over t.

01:20:12.550 --> 01:20:15.730
So if you ask what am I doing
to make that separation,

01:20:15.730 --> 01:20:17.980
I'm expanding in
lambda QCD over t.

01:20:17.980 --> 01:20:19.120
There's no t here.

01:20:19.120 --> 01:20:20.480
The t is perturbative.

01:20:20.480 --> 01:20:22.240
That's the scale for the jet.

01:20:22.240 --> 01:20:26.050
And this f is a non-perturbative
distribution function.

01:20:26.050 --> 01:20:28.060
And that's then the full
factorization theorem

01:20:28.060 --> 01:20:30.050
putting those things together.

01:20:30.050 --> 01:20:32.668
So these beam
functions are kind of--

01:20:32.668 --> 01:20:34.210
they are things that
show up whenever

01:20:34.210 --> 01:20:37.330
you have a process where you
ask for a particular number

01:20:37.330 --> 01:20:38.380
of jets.

01:20:38.380 --> 01:20:42.290
And if you asked, for
example, for a particular--

01:20:42.290 --> 01:20:44.890
if you ask for two--

01:20:44.890 --> 01:20:47.340
if you ask for this
setup plus one more jet,

01:20:47.340 --> 01:20:48.340
how would things change?

01:20:48.340 --> 01:20:50.290
Well, you'd have the
same beam radiation,

01:20:50.290 --> 01:20:52.210
the same initial
state radiation.

01:20:52.210 --> 01:20:55.040
And you'd add one more jet
function to this formula.

01:20:55.040 --> 01:20:57.925
So that would be exclusive
one jet production.

01:20:57.925 --> 01:20:59.050
Or you could have two jets.

01:20:59.050 --> 01:21:00.430
Then you'd have
two jet functions.

01:21:00.430 --> 01:21:02.230
In each case, you have a
different soft function.

01:21:02.230 --> 01:21:03.310
But these beam
functions are going

01:21:03.310 --> 01:21:05.350
to be always showing up
because they're describing

01:21:05.350 --> 01:21:06.558
this initial state radiation.

01:21:06.558 --> 01:21:10.780
So any time that's an allowed
thing, then it'll show up.

01:21:10.780 --> 01:21:12.925
If you start doing
[? perp ?] measurements,

01:21:12.925 --> 01:21:14.800
then you might have
[? perp-dependent ?] beam

01:21:14.800 --> 01:21:17.260
functions in the same
way that we could get

01:21:17.260 --> 01:21:19.330
[? perp-dependent ?] PDF.

01:21:19.330 --> 01:21:21.280
In some sense, a
[? perp-dependent ?] PDF is

01:21:21.280 --> 01:21:23.300
like a beam function.

01:21:23.300 --> 01:21:25.753
So that's kind of
the last function

01:21:25.753 --> 01:21:28.420
that you need to know about that
is kind of a generic thing that

01:21:28.420 --> 01:21:30.670
shows up with these
factorizations theorems.

01:21:30.670 --> 01:21:32.390
And we're out of time.

01:21:32.390 --> 01:21:35.050
So let's stop there.

01:21:35.050 --> 01:21:36.790
And that's it.

01:21:36.790 --> 01:21:38.820
See you on Monday.