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PROFESSOR: All right.

00:00:26.100 --> 00:00:31.110
So, so far we've recently been
talking about examples in SET2,

00:00:31.110 --> 00:00:34.030
and we're going to
continue to do so today.

00:00:34.030 --> 00:00:36.090
So the example that
we did last time

00:00:36.090 --> 00:00:38.070
was the plan photon form factor.

00:00:38.070 --> 00:00:40.140
That did not have any
soft degrees of freedom.

00:00:40.140 --> 00:00:43.060
It just had colinear and
higher degrees of freedom.

00:00:43.060 --> 00:00:45.570
So it was a particularly
simple example of something

00:00:45.570 --> 00:00:47.790
we could think of in SET2.

00:00:47.790 --> 00:00:50.370
We'll start with a slightly
more complicated example,

00:00:50.370 --> 00:00:54.480
this decay, B to D pi,
where we have both colinear

00:00:54.480 --> 00:00:55.860
and soft degrees of freedom.

00:00:55.860 --> 00:00:57.690
This was an example
that we mentioned

00:00:57.690 --> 00:00:59.940
at the very beginning of
our discussion of SET,

00:00:59.940 --> 00:01:03.462
and now we're going to see how
factorization looks for it.

00:01:03.462 --> 00:01:04.920
And then we'll talk
about something

00:01:04.920 --> 00:01:07.920
called the rapidity
renormalization group,

00:01:07.920 --> 00:01:10.860
which has to do with
situations in SET2

00:01:10.860 --> 00:01:13.020
where the separation
of degrees of freedom

00:01:13.020 --> 00:01:15.570
is a little more complicated
than in the previous examples.

00:01:15.570 --> 00:01:18.660
And we'll see that there can be
a new type of divergence that

00:01:18.660 --> 00:01:19.650
shows up.

00:01:19.650 --> 00:01:22.050
And that new type of
divergence leads to a new type

00:01:22.050 --> 00:01:25.800
of renormalization group.

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So B to D pie.

00:01:32.340 --> 00:01:39.270
So there's going to be, in
some sense, three hard scales

00:01:39.270 --> 00:01:41.790
of this problem.

00:01:41.790 --> 00:01:44.275
The mass of the B quark and
the mass of the charm quark

00:01:44.275 --> 00:01:46.650
are going to be taken to be
much greater than lambda QCD,

00:01:46.650 --> 00:01:48.960
so we'll have an HQET
type description of the B

00:01:48.960 --> 00:01:50.670
quark and the charm quark.

00:01:50.670 --> 00:01:53.423
And also the energy
of the pion, which

00:01:53.423 --> 00:01:55.590
is in some sense, related
to the difference of the B

00:01:55.590 --> 00:01:58.080
quark and the charm
quark mass, will also

00:01:58.080 --> 00:02:02.050
take that to be much
greater than the lambda QCD.

00:02:02.050 --> 00:02:11.190
So just by kinematics this thing
is proportional to MB minus MC

00:02:11.190 --> 00:02:12.353
roughly.

00:02:12.353 --> 00:02:13.770
You could say,
it's the difference

00:02:13.770 --> 00:02:17.500
of the squares of
the hadron masses.

00:02:17.500 --> 00:02:18.000
OK.

00:02:18.000 --> 00:02:21.480
So let's first-- we know how
to treat this decay if we were

00:02:21.480 --> 00:02:23.490
integrating out the W.
This is a weak decay,

00:02:23.490 --> 00:02:25.540
so B is changing to charm.

00:02:25.540 --> 00:02:28.650
Integrate of the W boson,
run down to the scale MB,

00:02:28.650 --> 00:02:30.720
which is the larger
scale here, that's

00:02:30.720 --> 00:02:32.040
the electroweak Hamiltonian.

00:02:34.590 --> 00:02:39.690
So that's what we'll call
the QCD operators, which

00:02:39.690 --> 00:02:42.920
are the relevant description
at the scale of order MB.

00:02:54.030 --> 00:02:54.990
Some pre-factor.

00:03:02.430 --> 00:03:05.610
And I'll write the operators
in the following way--

00:03:05.610 --> 00:03:07.440
a slightly different
basis than we

00:03:07.440 --> 00:03:13.480
used previously,
or a long time ago

00:03:13.480 --> 00:03:19.910
when we were talking about
this particular case.

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So just a different color
basis, singlet and octet.

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OK, so that's our
description, where

00:03:41.780 --> 00:03:48.340
p left is projecting us onto,
the left-handed components.

00:03:48.340 --> 00:03:51.160
So what we want to do is we
want to factorize the amplitude.

00:03:51.160 --> 00:03:52.840
This is an exclusive
process where

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we make a transition
between specific states.

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So we'd like to separate
scales in the D pi,

00:04:06.580 --> 00:04:12.380
and then we have O0 or O8
in this matrix element.

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So we have two matrix elements,
one with O0 and one with O8.

00:04:16.779 --> 00:04:19.370
And so, what could it
possibly look like?

00:04:19.370 --> 00:04:21.790
Well, we already talked about
the degrees of freedom here.

00:04:21.790 --> 00:04:25.330
The D is going to be soft and
the B is going to be soft.

00:04:25.330 --> 00:04:30.350
So this is soft, this is soft.

00:04:30.350 --> 00:04:31.570
This is going to be colinear.

00:04:31.570 --> 00:04:33.280
And so, if it's
going to factorize,

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and the soft degrees
of freedom are not

00:04:34.905 --> 00:04:36.550
going to talk to the
colinear, the kind

00:04:36.550 --> 00:04:39.890
of thing that you
would expect to show

00:04:39.890 --> 00:04:44.740
at leading order in
the lambda expansion

00:04:44.740 --> 00:04:48.100
is that you have the
following kind of process.

00:04:55.240 --> 00:04:56.430
Let me reclaim this space.

00:05:11.550 --> 00:05:13.280
So here's a heavy quark.

00:05:13.280 --> 00:05:16.880
Here's one of these operators.

00:05:16.880 --> 00:05:19.760
Here's the valence
quarks in the pion.

00:05:19.760 --> 00:05:36.930
Another heavy quark--
this was B charm, U and D.

00:05:36.930 --> 00:05:38.610
And there's an
anti-quark, and we

00:05:38.610 --> 00:05:40.620
have to address this by gluons.

00:05:40.620 --> 00:05:43.410
And if it's going to
factorize, then the way

00:05:43.410 --> 00:05:46.990
that we should dress it
by gluons is as follows.

00:05:46.990 --> 00:05:50.850
We would have soft
gluons here, and they

00:05:50.850 --> 00:05:53.730
could interact, if you
like, between things

00:05:53.730 --> 00:05:57.990
in the B and the D, because
the B and the D are both soft.

00:05:57.990 --> 00:06:02.370
We can also have back and
polarization diagrams.

00:06:02.370 --> 00:06:09.180
And that is going to factorize
from things in the pion which

00:06:09.180 --> 00:06:10.240
are going to be colinear.

00:06:10.240 --> 00:06:14.790
So we have our colinear
gluons and colinear quarks

00:06:14.790 --> 00:06:19.007
inside here, and maybe
there's some Wilson lines too.

00:06:19.007 --> 00:06:21.090
So we would expect some
kind of picture like that.

00:06:21.090 --> 00:06:24.570
And that's actually going
to be what we do find.

00:06:24.570 --> 00:06:27.300
But exactly what happens
at this vertex, what kind

00:06:27.300 --> 00:06:31.530
of convolutions there are,
that we have to work out.

00:06:31.530 --> 00:06:32.250
All right.

00:06:32.250 --> 00:06:35.700
So what factorization
in this context means

00:06:35.700 --> 00:06:40.650
is that there's no
gluons that are directly

00:06:40.650 --> 00:06:43.500
exchanged between the B to
D part and the pion part.

00:06:52.100 --> 00:06:55.490
So that it effectively
factorizes into a matrix

00:06:55.490 --> 00:06:57.260
element that's like
a B to D transition

00:06:57.260 --> 00:07:00.770
and a vacuum to pion transition.

00:07:00.770 --> 00:07:03.500
So you can even guess
what kind of objects

00:07:03.500 --> 00:07:05.180
this would depend on.

00:07:05.180 --> 00:07:07.550
If you have something
like this green thing,

00:07:07.550 --> 00:07:10.670
that's a B to D form factor.

00:07:10.670 --> 00:07:17.780
So we expect a B
to D form factor.

00:07:22.520 --> 00:07:24.770
And for the pion, if you
have something like this,

00:07:24.770 --> 00:07:26.812
well, we already talked
about something like this

00:07:26.812 --> 00:07:29.900
when we were talking about
gamma star, gamma to pi zero.

00:07:29.900 --> 00:07:33.500
So for the pion, we
expect the pi zero--

00:07:33.500 --> 00:07:42.800
the light cone
distribution, which

00:07:42.800 --> 00:07:47.210
is the sort of leading
order operator for the pion.

00:07:47.210 --> 00:07:50.330
So we'd expect a 5pi of x.

00:07:50.330 --> 00:07:53.210
And we'll see that we
do indeed find that.

00:08:03.160 --> 00:08:07.010
So the B and D have P
squared of order lambda QCD

00:08:07.010 --> 00:08:08.600
squared for their constituents.

00:08:08.600 --> 00:08:13.790
The pion is colinear,
and its constituents

00:08:13.790 --> 00:08:16.670
have P squared of order lambda
QCD2, but they're boosted.

00:08:19.330 --> 00:08:20.950
We can again use SET--

00:08:20.950 --> 00:08:25.650
this is SET2, but we can use
SET1 as an intermediate step,

00:08:25.650 --> 00:08:27.190
just like we did--

00:08:27.190 --> 00:08:28.450
we talked about last time.

00:08:42.770 --> 00:08:45.130
So let's do that again.

00:08:45.130 --> 00:08:49.390
So match-- step one was to
match QCD onto the SCET1.

00:08:52.670 --> 00:08:53.990
So there was some hard scale.

00:08:53.990 --> 00:08:55.365
And the harder
scale in this case

00:08:55.365 --> 00:08:57.990
could be any one of these three.

00:08:57.990 --> 00:09:05.933
So collectively I
just denote them by Q.

00:09:05.933 --> 00:09:07.850
And so what's going to
happen in that matching

00:09:07.850 --> 00:09:10.040
is we take these
operators O0 and O8,

00:09:10.040 --> 00:09:12.200
and we have to match
onto SET operators.

00:09:20.720 --> 00:09:22.490
So let me call the
SET operators Q0.

00:09:26.510 --> 00:09:35.950
I just there's-- because of the
fact that there's a heavy quark

00:09:35.950 --> 00:09:40.960
in the way that that works,
there's two possible spin

00:09:40.960 --> 00:09:43.730
structures here.

00:09:43.730 --> 00:09:46.630
But we'll see that
actually only one of them

00:09:46.630 --> 00:09:49.900
has the quantum
numbers in the end.

00:09:49.900 --> 00:09:56.650
So we have heavy quark
fields, charm, and bottom,

00:09:56.650 --> 00:09:59.120
that we can imagine
that type of operator.

00:09:59.120 --> 00:10:02.380
And then the colinear part, we
can dress it with the Wilson

00:10:02.380 --> 00:10:04.240
line, as always.

00:10:04.240 --> 00:10:05.800
And let me put the
flavor upstairs.

00:10:15.282 --> 00:10:17.490
And let me put in the most
general Wilson coefficient

00:10:17.490 --> 00:10:20.980
that I can think of
for this process.

00:10:20.980 --> 00:10:24.900
This could also depend
on V dot V prime.

00:10:24.900 --> 00:10:26.530
I didn't denote that.

00:10:26.530 --> 00:10:27.540
But in general, it will.

00:10:27.540 --> 00:10:31.530
And it could depend on
any of the scales Q.

00:10:31.530 --> 00:10:35.400
So it can depend on the large
momentum of the colinear

00:10:35.400 --> 00:10:36.478
fields, as always.

00:10:36.478 --> 00:10:38.520
And there's one combination
that's not restricted

00:10:38.520 --> 00:10:41.075
by momentum conservation.

00:10:41.075 --> 00:10:42.450
So there's one
combination that's

00:10:42.450 --> 00:10:44.880
not related to these
scales, and that's

00:10:44.880 --> 00:10:49.673
the, in my notation,
the P bar plus operator.

00:10:49.673 --> 00:10:51.090
And then there's
likewise, there's

00:10:51.090 --> 00:10:52.350
another thing with the TA.

00:10:52.350 --> 00:11:00.780
So same thing, TA,
everything the same, TA.

00:11:00.780 --> 00:11:06.420
And then this has got
a different coefficient

00:11:06.420 --> 00:11:09.400
like that.

00:11:09.400 --> 00:11:11.895
And so, what are the Dirac
structures just to be explicit?

00:11:14.640 --> 00:11:22.620
The heavy one-- so the light
one is going to be M bar

00:11:22.620 --> 00:11:28.680
slash over 4 in my notation,
the 1 minus gamma 5.

00:11:28.680 --> 00:11:32.440
And for the heavy one, you
could have either 1 or gamma 5.

00:11:32.440 --> 00:11:37.020
And that's because you
originally have a left handed--

00:11:37.020 --> 00:11:39.420
originally here you
have a left handed guy

00:11:39.420 --> 00:11:42.930
between the charm and the B.
But remember that a mass term

00:11:42.930 --> 00:11:47.070
connects left and right.

00:11:47.070 --> 00:11:49.800
So after you integrate out
the mass of these quarks,

00:11:49.800 --> 00:11:51.330
you don't know whether--

00:11:51.330 --> 00:11:54.278
you don't know the
chirality anymore in this.

00:11:54.278 --> 00:11:56.820
So that's why you can have both
the possibilities 1 and gamma

00:11:56.820 --> 00:11:57.320
5.

00:12:01.330 --> 00:12:03.820
So if you put any other
Dirac structure-- so here

00:12:03.820 --> 00:12:04.900
you could use chirality.

00:12:04.900 --> 00:12:07.767
And so, you know that these guys
should be left handed still,

00:12:07.767 --> 00:12:09.850
and that's why that this
should be this structure.

00:12:09.850 --> 00:12:11.320
You know it should
be an M bar slash,

00:12:11.320 --> 00:12:13.320
because any other structure
that you would stick

00:12:13.320 --> 00:12:16.510
in here between them
would give you something

00:12:16.510 --> 00:12:18.640
that's power suppressed.

00:12:18.640 --> 00:12:25.160
Because you know that N
slash, when CN 0, and also CN

00:12:25.160 --> 00:12:30.744
bar gamma [? per mu ?] with
any kind of P left or whatever,

00:12:30.744 --> 00:12:32.530
is also 0.

00:12:32.530 --> 00:12:38.450
So you'd have to have
something more complicated.

00:12:38.450 --> 00:12:40.310
All right, so there's
those operators.

00:12:40.310 --> 00:12:43.280
And when you do this matching,
it is non-trivial in the sense

00:12:43.280 --> 00:12:45.590
that these two operators--
it's not diagonal.

00:12:45.590 --> 00:12:48.560
It's not like O0 goes to
Q0, and O8 goes to Q8.

00:12:48.560 --> 00:12:51.830
As soon as you start adding
loop corrections these two mix,

00:12:51.830 --> 00:12:54.320
and then they give
you some contributions

00:12:54.320 --> 00:12:57.390
to these coefficients C0 and C8.

00:12:57.390 --> 00:12:57.950
OK?

00:12:57.950 --> 00:13:01.423
So what you mean by octet
operator in the electroweak

00:13:01.423 --> 00:13:02.840
Hamiltonian is
different than what

00:13:02.840 --> 00:13:10.360
you mean by octet operator in
the SET1 factorized result.

00:13:10.360 --> 00:13:14.320
But that's just a complication
that you deal with when

00:13:14.320 --> 00:13:15.700
you are doing the matching.

00:13:15.700 --> 00:13:18.100
This guy can be proportional
to the Wilson coefficient

00:13:18.100 --> 00:13:25.420
C0F and C08, and that's
not really a big deal.

00:13:25.420 --> 00:13:28.440
Any questions so far?

00:13:28.440 --> 00:13:29.790
STUDENT: [INAUDIBLE]

00:13:29.790 --> 00:13:31.000
PROFESSOR: Yeah

00:13:31.000 --> 00:13:33.920
STUDENT: So is the point of
matching to get one of them

00:13:33.920 --> 00:13:35.930
as being [INAUDIBLE]
to get the softs?

00:13:35.930 --> 00:13:36.650
PROFESSOR: Yeah.

00:13:36.650 --> 00:13:37.400
STUDENT: So you're going
to distribute them--

00:13:37.400 --> 00:13:37.710
PROFESSOR: That's right.

00:13:37.710 --> 00:13:38.690
I'm going to do that right now.

00:13:38.690 --> 00:13:39.330
STUDENT: OK, so that's--

00:13:39.330 --> 00:13:39.997
PROFESSOR: Yeah.

00:13:39.997 --> 00:13:49.490
So then step two, field
redefinition in the SCET1.

00:13:55.560 --> 00:13:59.660
And let me not put
superscript zeros,

00:13:59.660 --> 00:14:02.450
let me just make it
as a replacement.

00:14:02.450 --> 00:14:08.600
So then we get RQ0 again.

00:14:08.600 --> 00:14:11.960
And OK.

00:14:11.960 --> 00:14:15.470
For this guy, it's
exactly the same,

00:14:15.470 --> 00:14:21.200
because in the Q01 comma
5, all the Y's cancel.

00:14:25.630 --> 00:14:29.260
In the octet guy, it's not quite
that way, because there is--

00:14:29.260 --> 00:14:32.680
because we do have--

00:14:32.680 --> 00:14:35.530
in that case, we do have
the Wilson and lines

00:14:35.530 --> 00:14:37.030
getting trapped by the TA.

00:14:37.030 --> 00:14:38.470
So let me write out that case.

00:14:42.691 --> 00:15:21.210
So we have-- so that's what
we would get after the field

00:15:21.210 --> 00:15:24.780
redefinition for two operators.

00:15:24.780 --> 00:15:25.290
OK.

00:15:25.290 --> 00:15:29.640
So the next thing to do would
be to instead of calling them

00:15:29.640 --> 00:15:31.140
Y's call them S's.

00:15:31.140 --> 00:15:34.740
So, but there's one
more thing I can do too.

00:15:34.740 --> 00:15:37.430
So these are, remember,
these are soft fields

00:15:37.430 --> 00:15:38.680
and these are colinear fields.

00:15:38.680 --> 00:15:40.470
So this isn't
factorized, because we

00:15:40.470 --> 00:15:44.430
have contractions between these
Y's in the fields over here.

00:15:44.430 --> 00:15:47.010
Gluons can attach
to heavy quarks.

00:15:47.010 --> 00:15:50.400
So in order to factorize we want
to move those Y's from there

00:15:50.400 --> 00:15:52.450
over to here.

00:15:52.450 --> 00:15:53.690
And we can do that.

00:15:53.690 --> 00:15:55.440
So here's how that works.

00:15:55.440 --> 00:15:59.530
This is a formula that I
could have told you earlier.

00:15:59.530 --> 00:16:02.820
So if you have a Y
that lies around a TA,

00:16:02.820 --> 00:16:06.030
that's just actually
the adjoint Wilson line.

00:16:14.445 --> 00:16:19.450
So this is a formula that
relates fundamental Wilson

00:16:19.450 --> 00:16:21.210
lines and an
adjoint Wilson line.

00:16:26.820 --> 00:16:31.200
So in the adjoint Wilson line,
you'd build it out of matrices.

00:16:31.200 --> 00:16:32.670
If you like, they're like this.

00:16:39.500 --> 00:16:42.616
So the matrix indices
would be B and C,

00:16:42.616 --> 00:16:46.220
and instead of having
fundamental indices for the TA

00:16:46.220 --> 00:16:51.053
alpha beta, you have an FABC,
and this is the kind of thing

00:16:51.053 --> 00:16:52.220
that you would exponentiate.

00:16:55.970 --> 00:16:57.810
But other than that
it's the same thing.

00:16:57.810 --> 00:17:01.372
And there's just a color
identity relating them.

00:17:01.372 --> 00:17:03.080
So because of this
identity, you can also

00:17:03.080 --> 00:17:10.220
write down another identity,
which is Y dagger TAY.

00:17:10.220 --> 00:17:18.475
This guy is-- so Y dagger
TAY is just the other YAB--

00:17:18.475 --> 00:17:23.790
this guy's an
orthogonal matrix, TB.

00:17:23.790 --> 00:17:29.700
So if you reverse the indices
then you get the opposite way.

00:17:29.700 --> 00:17:32.700
And also, this guy is--

00:17:32.700 --> 00:17:35.860
remember this is a matrix
just in the AB space.

00:17:35.860 --> 00:17:38.150
So if I use this
formula in here,

00:17:38.150 --> 00:17:40.650
that allows me to take these
Wilson lines here and move them

00:17:40.650 --> 00:17:42.310
over here.

00:17:42.310 --> 00:17:42.810
Right?

00:17:42.810 --> 00:17:45.870
Because I can take
them, write them as a Y,

00:17:45.870 --> 00:17:47.483
and then the Y is
just something that

00:17:47.483 --> 00:17:49.650
doesn't care about-- it
just moves over because it's

00:17:49.650 --> 00:17:52.000
contracted with that index A.

00:17:52.000 --> 00:17:54.570
I guess I've got
some problems with

00:17:54.570 --> 00:17:57.303
capitals and small letters.

00:17:57.303 --> 00:17:58.470
Let's make them all capital.

00:18:03.955 --> 00:18:05.580
So then not going to
move it over here.

00:18:05.580 --> 00:18:08.850
And then I can convert it back
to a Y dagger Y if I want to.

00:18:08.850 --> 00:18:09.350
OK.

00:18:09.350 --> 00:18:14.200
So we can take this
guy and this thing

00:18:14.200 --> 00:18:26.310
and write it over there as
HV Y TAY dagger HV prime.

00:18:29.130 --> 00:18:31.050
I was careful about that.

00:18:31.050 --> 00:18:32.220
This was the prime.

00:18:36.790 --> 00:18:39.370
And then all the soft gluons--

00:18:39.370 --> 00:18:43.150
all the ultra soft ones are
over here in this matrix element

00:18:43.150 --> 00:18:46.150
and all the colinear
fields are over there.

00:18:46.150 --> 00:18:49.730
So if you like, you
could say, we get this,

00:18:49.730 --> 00:18:52.300
and then we get our colinear
matrix element that has TA,

00:18:52.300 --> 00:19:00.700
but it has now no ultra softs.

00:19:00.700 --> 00:19:02.770
And so we have a product
of colinear and ultra

00:19:02.770 --> 00:19:07.610
soft things tied together
by one index A. OK,

00:19:07.610 --> 00:19:12.330
so that's just a little
color rearrangement.

00:19:12.330 --> 00:19:14.880
It's useful, because now
they're really factorized.

00:19:14.880 --> 00:19:16.560
And now when you
take matrix elements,

00:19:16.560 --> 00:19:18.090
the matrix elements
will factorize.

00:19:41.330 --> 00:19:44.460
Oh, sorry, before we
take matrix elements,

00:19:44.460 --> 00:19:46.090
let's switch to SCET 2.

00:19:54.080 --> 00:19:56.560
So this is, again, an
example where it's trivial.

00:19:56.560 --> 00:19:58.582
Because what we have
is, we have one type

00:19:58.582 --> 00:20:00.040
of operator that
we're considering,

00:20:00.040 --> 00:20:02.140
this weak transition,
and we don't

00:20:02.140 --> 00:20:04.240
have a time limited
product of any type

00:20:04.240 --> 00:20:06.490
of two operators in SET2.

00:20:06.490 --> 00:20:09.290
We just have a single operator
that has both types of fields,

00:20:09.290 --> 00:20:13.105
and then we have the Lagrangians
So this is, again, simple.

00:20:18.640 --> 00:20:28.210
So we have one mixed operator
plus L0 colinear and L0 soft.

00:20:28.210 --> 00:20:29.920
And those things are
already decoupled,

00:20:29.920 --> 00:20:30.970
and so this is simple.

00:20:34.560 --> 00:20:39.320
And so, we simply replace
Y's by S, renaming

00:20:39.320 --> 00:20:40.940
it soft instead of ultra soft.

00:20:40.940 --> 00:20:42.470
Really nothing is changing.

00:20:42.470 --> 00:20:44.990
And these colinears,
we just put them down

00:20:44.990 --> 00:20:49.430
onto SET2 colinears
from SET1 colinears.

00:20:49.430 --> 00:21:00.610
OK, so to make it look
like I've done something,

00:21:00.610 --> 00:21:01.830
I'll write it out again.

00:21:01.830 --> 00:21:06.950
But there's really
nothing happening

00:21:06.950 --> 00:21:10.440
except that now the fields are
in SET2, with the correct SET2

00:21:10.440 --> 00:21:10.940
scaling.

00:21:17.940 --> 00:21:19.820
So there's no Wilson
coefficient that's

00:21:19.820 --> 00:21:21.230
generated by this stuff--

00:21:21.230 --> 00:21:23.540
there's no additional
Wilson coefficient

00:21:23.540 --> 00:21:25.790
because of that fact.

00:21:25.790 --> 00:21:29.010
So these are SCET2 now.

00:21:29.010 --> 00:21:30.500
And similarly for the octet.

00:21:34.690 --> 00:21:36.340
So here we would
really have the--

00:21:50.680 --> 00:21:51.180
OK.

00:21:53.930 --> 00:21:55.827
So now we can take
matrix elements.

00:21:55.827 --> 00:21:56.660
STUDENT: [INAUDIBLE]

00:21:56.660 --> 00:21:56.870
PROFESSOR: Yeah?

00:21:56.870 --> 00:21:58.580
STUDENT: What do
the coefficients

00:21:58.580 --> 00:22:01.530
look like when
they're not just 1?

00:22:01.530 --> 00:22:04.160
PROFESSOR: So they would
be functions of say,

00:22:04.160 --> 00:22:06.530
plus times minus momenta.

00:22:06.530 --> 00:22:08.460
So we could have--
if it wasn't 1,

00:22:08.460 --> 00:22:11.417
what would happen is
effectively-- so yeah,

00:22:11.417 --> 00:22:13.250
we talked a little bit
about this last time,

00:22:13.250 --> 00:22:14.960
but let me remind you.

00:22:14.960 --> 00:22:17.510
If it wasn't 1, that would
happen in a situation

00:22:17.510 --> 00:22:20.660
where you had
something like this.

00:22:20.660 --> 00:22:27.820
Some off shell field,
O, say like this.

00:22:27.820 --> 00:22:30.890
So this is a T
product of two things

00:22:30.890 --> 00:22:34.220
rather than just one thing
that mix off the colinear.

00:22:34.220 --> 00:22:37.910
And this field here
was off shell in a way

00:22:37.910 --> 00:22:42.560
that basically, this field
here is an off shell field that

00:22:42.560 --> 00:22:45.170
would be a product of the
plus and minus momentum

00:22:45.170 --> 00:22:45.817
of these guys.

00:22:45.817 --> 00:22:47.900
And so, you could get
something that's effectively

00:22:47.900 --> 00:22:50.600
living at this
hard colinear scale

00:22:50.600 --> 00:22:53.270
below the hard scale
in the problem,

00:22:53.270 --> 00:22:57.200
from sort of T products of
soft and colinear operators.

00:22:57.200 --> 00:23:00.680
This is getting a
little sketchy, but--

00:23:00.680 --> 00:23:03.470
since here we only have one
operator, that couldn't happen,

00:23:03.470 --> 00:23:05.950
because you just
start attaching softs.

00:23:05.950 --> 00:23:08.163
And colinear is been--
it's already factored.

00:23:08.163 --> 00:23:09.830
So there's no way
that you could sort of

00:23:09.830 --> 00:23:12.620
get this intermediate
off shell guy.

00:23:12.620 --> 00:23:13.460
STUDENT: [INAUDIBLE]

00:23:13.460 --> 00:23:15.460
PROFESSOR: So this guy
would be colinear, right?

00:23:15.460 --> 00:23:18.920
And the way to think
about this is like this.

00:23:18.920 --> 00:23:21.270
This is just a
diagram that exists.

00:23:21.270 --> 00:23:22.940
But then you go
over to the SET2,

00:23:22.940 --> 00:23:26.750
and then you have your change of
colinear to the right colinear.

00:23:26.750 --> 00:23:28.260
But this guy doesn't change.

00:23:28.260 --> 00:23:31.252
He's still hard colinear.

00:23:31.252 --> 00:23:33.920
STUDENT: OK, so it's a matching
when the material actually

00:23:33.920 --> 00:23:35.563
has a different
scale than matching--

00:23:35.563 --> 00:23:36.230
PROFESSOR: Yeah.

00:23:36.230 --> 00:23:38.180
This scale would
show actually up.

00:23:38.180 --> 00:23:40.830
And so if it doesn't show up--

00:23:40.830 --> 00:23:42.395
so this would be
kind of a situation,

00:23:42.395 --> 00:23:48.210
an going from step one.

00:23:48.210 --> 00:23:50.358
And actually, we could
have done some examples

00:23:50.358 --> 00:23:51.150
where that happens.

00:23:53.760 --> 00:23:57.120
But I'm choosing to do this
rapidity renormalization

00:23:57.120 --> 00:23:58.590
group instead.

00:23:58.590 --> 00:24:00.888
Basically this happens if
you look at matrix elements,

00:24:00.888 --> 00:24:02.430
and you could look
at matrix elements

00:24:02.430 --> 00:24:04.890
where you have some
subleading interactions.

00:24:04.890 --> 00:24:07.470
And there are examples
in exclusive decays

00:24:07.470 --> 00:24:09.480
where you could
have this happen.

00:24:09.480 --> 00:24:14.100
One example is if you
look at B0 to D0, pi 0,

00:24:14.100 --> 00:24:18.203
just having all neutral charges,
then actually this will happen.

00:24:18.203 --> 00:24:19.870
It'll be-- and it'll
be more complicated

00:24:19.870 --> 00:24:20.610
than what I'm telling you.

00:24:20.610 --> 00:24:23.085
But you can derive a
factorization theorem for this.

00:24:23.085 --> 00:24:26.100
It's power suppressed relative
to the one we're talking about,

00:24:26.100 --> 00:24:29.610
because the one we're talking
about always has a charge pion,

00:24:29.610 --> 00:24:31.410
and it turns out
that that happens

00:24:31.410 --> 00:24:33.690
at leading order,
whereas the neutral pion

00:24:33.690 --> 00:24:36.630
process with the
neutral B and neutral D

00:24:36.630 --> 00:24:39.362
is something that's
power suppressed.

00:24:49.484 --> 00:24:52.020
Hope I'm remembering that right.

00:24:52.020 --> 00:24:53.500
Yeah.

00:24:53.500 --> 00:24:55.450
I am.

00:24:55.450 --> 00:24:55.950
All right.

00:24:55.950 --> 00:25:00.990
So number four,
this is an aside.

00:25:00.990 --> 00:25:09.025
Number four, take
matrix elements,

00:25:09.025 --> 00:25:11.400
and here we find actually that
one of the matrix elements

00:25:11.400 --> 00:25:13.560
is just 0, the one
with the octet.

00:25:16.200 --> 00:25:18.090
So let me write the
nonzero ones first.

00:25:42.860 --> 00:25:45.256
I wasn't too careful
about the two different--

00:25:53.830 --> 00:25:55.530
about this.

00:25:55.530 --> 00:25:57.170
So there's some guy
that's just giving

00:25:57.170 --> 00:26:00.350
a convolution between the
Wilson coefficient and 5 pi.

00:26:00.350 --> 00:26:12.620
And then this guy, which
is some normalization

00:26:12.620 --> 00:26:14.840
factor times a form factor.

00:26:18.800 --> 00:26:22.520
These things are all mu
dependent in general.

00:26:22.520 --> 00:26:26.010
And this thing here is
the Isgur-Wise function,

00:26:26.010 --> 00:26:27.440
which is the HQT form factor.

00:26:32.320 --> 00:26:37.210
And W0 is kind of the
kinematic variable

00:26:37.210 --> 00:26:38.720
that that form
factor can depend on,

00:26:38.720 --> 00:26:41.020
which is V dot V prime,
the labels on the fields,

00:26:41.020 --> 00:26:43.570
and that encodes the
momentum transfer.

00:26:43.570 --> 00:26:45.520
Which here is just
related to the kinematics.

00:26:45.520 --> 00:26:52.878
So W0 is some
function of MB and MC,

00:26:52.878 --> 00:26:54.670
which I'm not going to
bother writing down.

00:27:03.454 --> 00:27:06.640
STUDENT: [INAUDIBLE]

00:27:06.640 --> 00:27:07.890
PROFESSOR: Just some constant.

00:27:07.890 --> 00:27:11.460
I mean, it could have some
kinematic factors that makes up

00:27:11.460 --> 00:27:13.590
the dimension, like some--

00:27:13.590 --> 00:27:15.075
depends on my gamma.

00:27:15.075 --> 00:27:16.320
Yeah, it's just some number.

00:27:16.320 --> 00:27:19.230
STUDENT: [INAUDIBLE]

00:27:31.870 --> 00:27:34.540
PROFESSOR: OK so these
are the singlet operators.

00:27:38.020 --> 00:27:41.940
So in the singlet case, we have
initial state and final state.

00:27:41.940 --> 00:27:44.130
In all cases we have initial
state and final states,

00:27:44.130 --> 00:27:46.050
which are color singlets.

00:27:46.050 --> 00:27:48.210
And these operators
are color singlets.

00:27:48.210 --> 00:27:51.780
In the case of the octet, we
also have color singlet states.

00:27:51.780 --> 00:27:54.720
And we factorize
such that we do have

00:27:54.720 --> 00:27:58.090
a matrix element for example.

00:27:58.090 --> 00:28:01.920
So let me just write
one of them, and no.

00:28:10.700 --> 00:28:13.790
And this is 0, because
there's nothing

00:28:13.790 --> 00:28:17.000
that could carry the index
A in this matrix element.

00:28:20.750 --> 00:28:22.120
So the octet matrix element's 0.

00:28:22.120 --> 00:28:23.620
It's important that
we factorized it

00:28:23.620 --> 00:28:24.745
for that to be true, right?

00:28:24.745 --> 00:28:29.320
If we had D pi, then we'd have
a color singlet operator here,

00:28:29.320 --> 00:28:31.750
B. So we wouldn't have been
able to make this statement

00:28:31.750 --> 00:28:34.273
in the original operator in
the electroweak Hamiltonian.

00:28:34.273 --> 00:28:35.440
This would just not be true.

00:28:35.440 --> 00:28:39.220
But once we factored it and put
all the ultra soft fields here,

00:28:39.220 --> 00:28:41.540
that everything that's going
to be contracted together,

00:28:41.540 --> 00:28:42.820
then we can make this statement.

00:28:42.820 --> 00:28:44.612
We couldn't even really
make this statement

00:28:44.612 --> 00:28:46.280
when the Y's were
on the other side.

00:28:46.280 --> 00:28:50.200
We had to move them over here
to ensure that this statement is

00:28:50.200 --> 00:28:50.980
completely true.

00:28:53.650 --> 00:28:54.150
OK.

00:28:54.150 --> 00:29:03.365
So color octet operator
is color singlet states.

00:29:11.800 --> 00:29:13.870
OK, so then you just
put things together

00:29:13.870 --> 00:29:22.640
and we can multiply these two
things to get the final result.

00:29:22.640 --> 00:29:27.040
So if we write it as a
matching from the electroweak

00:29:27.040 --> 00:29:30.680
Hamiltonian, there are
some normalization factors.

00:29:30.680 --> 00:29:33.640
So grouping together
these factors of F pi,

00:29:33.640 --> 00:29:38.750
E pi, and M prime, which I'm
not worrying so much about,

00:29:38.750 --> 00:29:42.550
there's a Isgur-Wise
function, and then there's

00:29:42.550 --> 00:29:47.390
a single convolutional between
the hard coefficient, which

00:29:47.390 --> 00:29:51.020
is kind of like our example of
the photon pi on form factor.

00:29:53.895 --> 00:29:55.770
And then the slight
[INAUDIBLE] distribution.

00:29:55.770 --> 00:29:57.537
So we see an example
where it showed up

00:29:57.537 --> 00:29:59.120
in a totally different
type of process

00:29:59.120 --> 00:30:03.500
from the one we were considering
previously, thereby showing us

00:30:03.500 --> 00:30:05.873
kind of the universality
of that function.

00:30:05.873 --> 00:30:07.790
And then there would be
some power corrections

00:30:07.790 --> 00:30:11.517
to this whole thing
that we're neglecting,

00:30:11.517 --> 00:30:13.475
that go like lambda QCD
over those hard scales.

00:30:18.080 --> 00:30:20.870
OK, so this is the
Isgur-Wise function.

00:30:20.870 --> 00:30:28.400
And actually I did write
down what the W would be.

00:30:28.400 --> 00:30:35.520
So this W would be that, some--

00:30:35.520 --> 00:30:38.060
you can write it in terms of
the meson masses like that,

00:30:38.060 --> 00:30:40.100
so it's the Isgur-Wise
function at max recoil.

00:30:43.960 --> 00:30:47.860
And this function is
measured, for example,

00:30:47.860 --> 00:30:49.600
in a semi-leptonic transition.

00:30:52.330 --> 00:30:54.400
So you can imagine that
the pion distribution

00:30:54.400 --> 00:30:56.800
function, or properties
of it, were measured

00:30:56.800 --> 00:30:59.618
in the photon pion transition.

00:30:59.618 --> 00:31:01.410
Is Isgur-Wise function
is measured in the B

00:31:01.410 --> 00:31:03.145
to DL new transition.

00:31:03.145 --> 00:31:05.620
And then you can make
predictions for this B to D pi.

00:31:09.180 --> 00:31:11.910
OK, so that gives you an
example of how you would use

00:31:11.910 --> 00:31:14.820
these factorization theorems.

00:31:14.820 --> 00:31:16.650
So this applies to basically--

00:31:16.650 --> 00:31:20.160
this type of
factorization that we just

00:31:20.160 --> 00:31:24.150
talked about, it applies to a
lot of different things with

00:31:24.150 --> 00:31:28.950
charged pi minuses.

00:31:28.950 --> 00:31:31.020
Or you can make the
pi minus a rho minus,

00:31:31.020 --> 00:31:34.530
and that wouldn't
really change anything.

00:31:34.530 --> 00:31:37.040
So you could have B0.

00:31:37.040 --> 00:31:38.415
If you wanted to
look at charges,

00:31:38.415 --> 00:31:41.400
you could have B0
to D plus pi minus.

00:31:41.400 --> 00:31:46.475
Or you could have B
minus to D0 pi minus.

00:31:46.475 --> 00:31:47.850
And there's a
third one, which is

00:31:47.850 --> 00:31:51.060
the one we were talking about
over here, B0 to D0 pi 0.

00:31:51.060 --> 00:31:52.800
So there's three
different ways--

00:31:52.800 --> 00:31:55.080
three different B
to D pi transitions

00:31:55.080 --> 00:31:56.650
depending on the charges.

00:31:56.650 --> 00:32:02.130
And what we've derived applies
to charge pions or charge rhos.

00:32:02.130 --> 00:32:05.070
The neutral ones end up
being power suppressed.

00:32:05.070 --> 00:32:08.640
And you can see that
kind from our discussion

00:32:08.640 --> 00:32:10.110
there was kind of never a--

00:32:10.110 --> 00:32:14.130
there was-- it just writing down
the leading order operators,

00:32:14.130 --> 00:32:16.620
well, maybe you have to
work a little harder.

00:32:16.620 --> 00:32:19.290
But effectively, the
leading order operators

00:32:19.290 --> 00:32:22.183
don't make this transition
with the two charges being

00:32:22.183 --> 00:32:23.850
the same such that
you could get a pi 0,

00:32:23.850 --> 00:32:28.980
so you have to do something
more in order to get that case.

00:32:28.980 --> 00:32:30.420
All right.

00:32:30.420 --> 00:32:33.406
So questions?

00:32:33.406 --> 00:32:35.830
STUDENT: Can you [INAUDIBLE]?

00:32:35.830 --> 00:32:37.960
PROFESSOR: You can if you want.

00:32:37.960 --> 00:32:39.580
Oh, MB over MC.

00:32:39.580 --> 00:32:41.590
So MB and MC, we're
treating both of them

00:32:41.590 --> 00:32:44.560
as avoiding the same
in what we've done.

00:32:44.560 --> 00:32:46.660
So if we wanted to some
logs of MB over MC,

00:32:46.660 --> 00:32:48.910
we'd have to do something a
little different than what

00:32:48.910 --> 00:32:49.750
we did.

00:32:49.750 --> 00:32:52.690
You'd have to first integrate
[? O to MV, ?] treat the charm

00:32:52.690 --> 00:32:53.830
quark as a light quark.

00:32:53.830 --> 00:32:55.150
You could do that.

00:32:55.150 --> 00:32:58.010
STUDENT: Would it make
the SCET [INAUDIBLE]----

00:32:58.010 --> 00:32:59.260
PROFESSOR: It turns out that--

00:32:59.260 --> 00:32:59.750
STUDENT: --analysis different?

00:32:59.750 --> 00:33:00.500
PROFESSOR: --yeah.

00:33:00.500 --> 00:33:02.620
So those are single
logs, actually, they're

00:33:02.620 --> 00:33:04.018
not double logs.

00:33:04.018 --> 00:33:05.560
And it's related to
the fact that you

00:33:05.560 --> 00:33:06.560
have massive particles.

00:33:06.560 --> 00:33:08.018
When you have
massive particles you

00:33:08.018 --> 00:33:09.850
don't get the extra singularity.

00:33:09.850 --> 00:33:12.610
And people in HQT
worried about something--

00:33:12.610 --> 00:33:14.950
logs of MB over MC for a while--

00:33:14.950 --> 00:33:17.140
and then after a flight of
doing enough calculations

00:33:17.140 --> 00:33:18.878
they realized it was
totally irrelevant,

00:33:18.878 --> 00:33:20.170
and you should just not bother.

00:33:20.170 --> 00:33:22.360
You should just calculate
the alpha S corrections,

00:33:22.360 --> 00:33:24.648
treating MB and
MC as comparable,

00:33:24.648 --> 00:33:26.440
and summing the logs--
if you sort of think

00:33:26.440 --> 00:33:28.910
of leading log as being more
important than the order FS

00:33:28.910 --> 00:33:30.880
calculation, that misleads you.

00:33:30.880 --> 00:33:32.390
Sometimes the sign
is even wrong.

00:33:32.390 --> 00:33:34.900
And so there's sort of
a general experience

00:33:34.900 --> 00:33:39.440
that something logs of MB
over MC and HQT is not even--

00:33:39.440 --> 00:33:41.190
STUDENT: Just because
they're single logs?

00:33:41.190 --> 00:33:42.730
PROFESSOR: Just because
they're single logs.

00:33:42.730 --> 00:33:44.800
I mean, that's one thing
that makes it different

00:33:44.800 --> 00:33:47.230
than, say, the
double logs that you

00:33:47.230 --> 00:33:50.320
would resum in this process.

00:33:50.320 --> 00:33:53.560
Actually, these double
logs are also single logs.

00:33:53.560 --> 00:33:56.140
So you could decide whether
or not to resum of them.

00:33:56.140 --> 00:34:00.285
But either-- whether or
not you do resummation,

00:34:00.285 --> 00:34:01.660
this is still
useful, because you

00:34:01.660 --> 00:34:04.178
could make a prediction
for this decay rate,

00:34:04.178 --> 00:34:05.220
and it works really well.

00:34:08.909 --> 00:34:11.409
All right.

00:34:11.409 --> 00:34:13.679
You can actually
also make predictions

00:34:13.679 --> 00:34:16.590
for these decay rates.

00:34:16.590 --> 00:34:19.590
You can predict actually the
relations between the D0 and D

00:34:19.590 --> 00:34:22.440
star 0 using the factorization--
subleading factorization

00:34:22.440 --> 00:34:24.750
theorem.

00:34:24.750 --> 00:34:29.899
OK, so let's move on to
our second topic, which

00:34:29.899 --> 00:34:32.980
will take the rest of today.

00:34:32.980 --> 00:34:35.190
And that's rapidity divergences.

00:34:46.480 --> 00:34:46.980
OK.

00:34:46.980 --> 00:34:48.989
So when we were
talking about SET1,

00:34:48.989 --> 00:34:51.179
and we were talking
about loop calculations,

00:34:51.179 --> 00:34:53.400
we saw that there
was a subtlety where

00:34:53.400 --> 00:34:55.050
when we were doing
our colinear loops,

00:34:55.050 --> 00:34:56.889
that could double count
the ultra soft loop,

00:34:56.889 --> 00:34:57.570
if you remember.

00:35:05.640 --> 00:35:07.230
So kind of
schematically, I could

00:35:07.230 --> 00:35:16.630
say, that this true CN was sort
of a CN naive minus a CN 0 bin.

00:35:16.630 --> 00:35:18.540
So you could do a
calculation ignoring

00:35:18.540 --> 00:35:20.130
that, but then you
have to be careful,

00:35:20.130 --> 00:35:21.240
and there's a subtraction.

00:35:21.240 --> 00:35:24.930
And that subtraction avoids the
double counting with the ultra

00:35:24.930 --> 00:35:25.788
soft.

00:35:25.788 --> 00:35:28.080
So if you think about there
being ultra soft amplitudes

00:35:28.080 --> 00:35:34.785
and colinear amplitudes, this
avoids a double counting.

00:35:39.590 --> 00:35:41.340
Now, we never talked
about whether there's

00:35:41.340 --> 00:35:43.200
something analogous
to that in SET2,

00:35:43.200 --> 00:35:45.330
and we just did a
lot of SET2 examples

00:35:45.330 --> 00:35:47.970
without ever even
saying those words.

00:35:47.970 --> 00:35:51.810
So why it's actually-- for what
we've talked about so far--

00:35:51.810 --> 00:35:53.210
OK to ignore this issue.

00:35:59.530 --> 00:36:01.260
But in general, it's not OK.

00:36:12.110 --> 00:36:17.300
So if we go back to our picture
of the degrees of freedom

00:36:17.300 --> 00:36:24.200
in SET2, have this hyperbola,
and you could have softs,

00:36:24.200 --> 00:36:25.640
you could have some colinears.

00:36:25.640 --> 00:36:27.380
And then the example
that we just did,

00:36:27.380 --> 00:36:30.440
it's like these were kind
of the relevant modes.

00:36:30.440 --> 00:36:34.740
And in general, you might have
some guy down here as well.

00:36:34.740 --> 00:36:37.670
So these are the degrees
of freedom in the SET.

00:36:37.670 --> 00:36:40.250
And effectively
what's happened is,

00:36:40.250 --> 00:36:42.350
if you want to think
about double counting,

00:36:42.350 --> 00:36:44.300
you're sliding
down this hyperbola

00:36:44.300 --> 00:36:48.740
so this hyperbola is kind of at
a constant invariant mass, say,

00:36:48.740 --> 00:36:52.880
lambda QCD squared, or
it could be lambda QCD.

00:36:55.610 --> 00:36:58.485
And unlike the
case in SET1, where

00:36:58.485 --> 00:37:00.110
this guy lived in a
different hyperbola

00:37:00.110 --> 00:37:01.700
here, to get between
them you would

00:37:01.700 --> 00:37:06.770
be sliding down the hyperbola
at fixed invariant mass.

00:37:06.770 --> 00:37:08.248
So that's a little different.

00:37:12.640 --> 00:37:16.320
But in general in SCET2,
there are also 0 bins.

00:37:21.880 --> 00:37:24.730
So in general, you would
have something like,

00:37:24.730 --> 00:37:29.890
met me denote it this
way, CN minus CN soft.

00:37:29.890 --> 00:37:33.310
And what I mean by this, is this
is my original amplitude, where

00:37:33.310 --> 00:37:37.595
P mu was scaling like Q
lambda squared 1 lambda.

00:37:37.595 --> 00:37:39.370
This was the original.

00:37:39.370 --> 00:37:42.290
And this would be a subtraction
where you take that amplitude,

00:37:42.290 --> 00:37:45.220
and you'd make it scale
like in the soft regime.

00:37:45.220 --> 00:37:51.550
So it would be P mu,
lambda, lambda, lambda.

00:37:51.550 --> 00:37:57.320
So that's different than
the example of SET1.

00:37:57.320 --> 00:37:57.820
In

00:37:57.820 --> 00:38:00.220
The SET1 case, we
would really just

00:38:00.220 --> 00:38:02.205
be scaling down the
1 in the lambda,

00:38:02.205 --> 00:38:04.330
so that they would be both
of order lambda squared.

00:38:04.330 --> 00:38:05.920
Here we're actually
scaling down the 1

00:38:05.920 --> 00:38:09.040
and scaling up the lambda
squared to a lambda.

00:38:09.040 --> 00:38:12.307
And that's the right thing to
do to go from here to here.

00:38:12.307 --> 00:38:14.390
So we're just taking the
amplitudes in this region

00:38:14.390 --> 00:38:16.550
and subtracting
them in this region.

00:38:16.550 --> 00:38:19.610
And in general, we do have that.

00:38:19.610 --> 00:38:22.030
But actually that's not
the real complication

00:38:22.030 --> 00:38:25.720
that shows up in the SET2.

00:38:25.720 --> 00:38:28.000
The real complication has
to do with whether there's

00:38:28.000 --> 00:38:30.970
any divergences
associated to that.

00:38:30.970 --> 00:38:33.460
If this amplitude here
didn't have any divergences,

00:38:33.460 --> 00:38:36.208
it wasn't kind of
log singular, then

00:38:36.208 --> 00:38:38.500
you wouldn't really care
about doing this subtractions,

00:38:38.500 --> 00:38:40.810
because then there would be
no infrared singularities

00:38:40.810 --> 00:38:43.780
that you're double counting and
it would just be effectively

00:38:43.780 --> 00:38:44.660
a constant.

00:38:44.660 --> 00:38:46.360
And the constants
are always ambiguous.

00:38:46.360 --> 00:38:49.000
So whatever mistake you
make in constants here

00:38:49.000 --> 00:38:51.680
you just make up by
changing your hard matching.

00:38:51.680 --> 00:38:54.370
So you don't have to worry if
when the colinear goes down

00:38:54.370 --> 00:38:57.340
into the soft region
there's no divergences.

00:38:57.340 --> 00:38:59.680
And that is actually what's
happened in all the examples

00:38:59.680 --> 00:39:01.150
we've treated so far.

00:39:01.150 --> 00:39:04.180
That when the colinear goes
into some region where it's not

00:39:04.180 --> 00:39:06.760
supposed to have
singularities, that you just

00:39:06.760 --> 00:39:08.710
end up with no singularities.

00:39:08.710 --> 00:39:13.040
There's no log singularities.

00:39:13.040 --> 00:39:24.038
OK, so, so far there's
no log singularities

00:39:24.038 --> 00:39:25.205
from the overlapped regions.

00:39:29.570 --> 00:39:31.520
But that's not in
general the situation.

00:39:31.520 --> 00:39:32.978
And we'll do an
example in a minute

00:39:32.978 --> 00:39:36.760
where there are
singularities that overlap.

00:39:36.760 --> 00:39:40.130
And the true difficulty
here is the following.

00:39:40.130 --> 00:39:43.120
If you think about what's
separating these modes,

00:39:43.120 --> 00:39:46.240
you might draw lines like this.

00:39:46.240 --> 00:39:49.060
Just to draw some straight
lines separating the modes.

00:39:49.060 --> 00:39:50.710
And remember that
we're plotting here

00:39:50.710 --> 00:39:53.650
in the P minus, P plus plain.

00:39:53.650 --> 00:39:57.670
And that fixed P squared is
like fixed product of P minus P

00:39:57.670 --> 00:39:59.090
plus.

00:39:59.090 --> 00:40:02.980
So P squared is P plus P
minus, up to the P perp

00:40:02.980 --> 00:40:04.880
squared piece, which
we're ignoring.

00:40:04.880 --> 00:40:07.930
So you can think of these
lines as lines of constant P

00:40:07.930 --> 00:40:08.935
plus over P minus.

00:40:12.940 --> 00:40:15.570
And if I-- this is the--

00:40:19.840 --> 00:40:24.920
so something orthogonal
to P squared.

00:40:24.920 --> 00:40:25.810
All right.

00:40:25.810 --> 00:40:28.357
So that would be one
way of thinking about--

00:40:28.357 --> 00:40:30.190
so you need something
that's orthogonal to P

00:40:30.190 --> 00:40:32.350
squared in order to
distinguish these modes.

00:40:32.350 --> 00:40:36.010
And the real issue with that
is related to the regulators.

00:40:36.010 --> 00:40:38.230
When you use dimensional
regularization

00:40:38.230 --> 00:40:40.480
it turns out the dimensional
regularization is not

00:40:40.480 --> 00:40:43.480
sufficient to regulate a
divergence that would happen

00:40:43.480 --> 00:40:46.680
when the CN comes
down on top of the S.

00:40:46.680 --> 00:40:49.000
And the reason is, because
dimensional regularization

00:40:49.000 --> 00:40:50.830
regulates P squared.

00:40:50.830 --> 00:40:55.540
It regulates-- remember, it's
a Lorentz invariant regulator.

00:40:55.540 --> 00:40:58.150
So it's regulating
Lorentz invariant things

00:40:58.150 --> 00:41:01.330
like P squared, not something
like the rapidity, which

00:41:01.330 --> 00:41:04.000
is this P plus over P
minus that you would need

00:41:04.000 --> 00:41:06.560
to distinguish these modes.

00:41:06.560 --> 00:41:25.190
So invariant mass does not
distinguish the low energy

00:41:25.190 --> 00:41:25.690
modes.

00:41:46.570 --> 00:41:51.570
So rapidity, you could define--
is usually defined this way.

00:41:54.820 --> 00:42:01.150
So exponent of 2Y,
where Y is the rapidity,

00:42:01.150 --> 00:42:03.220
is P minus over P plus.

00:42:03.220 --> 00:42:06.100
And if you look at
the scaling of that,

00:42:06.100 --> 00:42:10.780
that scaling either lambda
minus 2, lambda 0, or lambda

00:42:10.780 --> 00:42:16.570
squared for the different
cases for CN, S, and CN bar.

00:42:16.570 --> 00:42:19.510
So it's this variable
that's really distinguishing

00:42:19.510 --> 00:42:21.200
the different modes.

00:42:21.200 --> 00:42:24.670
OK, and that's these
lines-- these orange lines

00:42:24.670 --> 00:42:29.551
are just putting dividing lines
between these in rapidity.

00:42:29.551 --> 00:42:30.340
All right.

00:42:36.200 --> 00:42:36.700
OK.

00:42:49.770 --> 00:42:52.403
So there's a complication that
dimensional regularization

00:42:52.403 --> 00:42:53.070
doesn't suffice.

00:43:05.460 --> 00:43:06.960
So you can think
of it as regulating

00:43:06.960 --> 00:43:11.300
P Euclidean squared once you do
the Wick rotation, for example.

00:43:11.300 --> 00:43:13.450
So it regulates-- it
separates hyperbolas,

00:43:13.450 --> 00:43:17.230
but it does not separate
modes along a hyperbola.

00:43:17.230 --> 00:43:18.850
It's a way of
regulating singularities

00:43:18.850 --> 00:43:21.310
between hyperbolas, but
not along hyperbola.

00:43:24.070 --> 00:43:25.390
So that's one complication.

00:43:25.390 --> 00:43:28.480
We'll need an
additional regulator.

00:43:28.480 --> 00:43:31.840
And we'll see that that
regulator will eventually

00:43:31.840 --> 00:43:36.370
lead to a new type of a
normalization group flow, which

00:43:36.370 --> 00:43:38.470
is flow along a hyperbola.

00:43:38.470 --> 00:43:41.500
It's not a flow in invariant
mass, but a flow in rapidity.

00:43:51.390 --> 00:43:51.890
OK.

00:43:51.890 --> 00:43:55.430
So let's explore what
can happen in an example

00:43:55.430 --> 00:44:00.170
where there are these
divergences in sort

00:44:00.170 --> 00:44:01.625
of the simplest
possible example.

00:44:10.604 --> 00:44:12.740
So there's enough
going on that we

00:44:12.740 --> 00:44:14.980
want to make our lives
as simple as possible.

00:44:14.980 --> 00:44:20.540
So what I'll talk
about is something

00:44:20.540 --> 00:44:22.385
called the massive
Sudakov form factor.

00:44:32.670 --> 00:44:37.630
So you should think of
it set up as follows.

00:44:37.630 --> 00:44:39.810
We're going to
consider a form factor,

00:44:39.810 --> 00:44:42.890
and it's going to be a
space-like form factor.

00:44:42.890 --> 00:44:45.210
So it's a space-like
quark-quark form factor.

00:44:49.020 --> 00:44:53.970
Q of the photon
here is space-like.

00:44:53.970 --> 00:44:57.690
And we're going to think about,
rather than having photons

00:44:57.690 --> 00:45:01.560
or gluons, we're going to think
about massive gauge bosons.

00:45:01.560 --> 00:45:03.715
So this is going to be some
kind of Z, if you like.

00:45:03.715 --> 00:45:05.010
It could be a Z boson.

00:45:09.810 --> 00:45:15.780
And I'll just call
the mass M. OK.

00:45:15.780 --> 00:45:19.350
So the thing that I'm going to
want to iterate is the mass--

00:45:19.350 --> 00:45:22.590
rather than doing QCD, I'm
doing electroweak corrections--

00:45:22.590 --> 00:45:24.840
electroweak corrections
from a massive gauge boson.

00:45:31.180 --> 00:45:35.080
So this is relevant if we
have electroweak corrections

00:45:35.080 --> 00:45:36.730
in a situation
where Q squared is

00:45:36.730 --> 00:45:39.713
much greater than at M squared.

00:45:39.713 --> 00:45:41.380
And we're not going
to be having gluons.

00:45:41.380 --> 00:45:44.530
Instead we'll be talking about
these massive gauge bosons--

00:45:44.530 --> 00:45:47.410
multiple Z bosons, if you like.

00:45:47.410 --> 00:45:51.130
We could also put the W's
in, but let's make it simple

00:45:51.130 --> 00:45:53.680
and just talk about Z's.

00:45:53.680 --> 00:45:55.570
OK, so let's do this example.

00:45:58.310 --> 00:46:01.380
So in the full theory, you would
start with a vector current,

00:46:01.380 --> 00:46:05.950
say, and you'd want to
match that onto SET.

00:46:05.950 --> 00:46:08.557
Before we do that, let's just--

00:46:08.557 --> 00:46:10.390
let me just write down
a kind of full theory

00:46:10.390 --> 00:46:13.180
object using Lorentz invariance.

00:46:13.180 --> 00:46:19.720
So you could think about
the quark form factor.

00:46:19.720 --> 00:46:24.250
And four massive gauge bosons,
this is just some form factor

00:46:24.250 --> 00:46:28.480
that you can calculate that's
a function of Q squared and M

00:46:28.480 --> 00:46:32.350
squared, and then
there's some spinners.

00:46:32.350 --> 00:46:34.840
And so, really kind
of the dependence

00:46:34.840 --> 00:46:37.690
is encoded in this F, which is
a function of Q squared and M

00:46:37.690 --> 00:46:38.950
squared.

00:46:38.950 --> 00:46:41.650
M squared is acting kind of
like an infrared regulator.

00:46:41.650 --> 00:46:44.170
So this is Z boson, there's
not a soft singularity

00:46:44.170 --> 00:46:45.030
associated to it.

00:46:48.650 --> 00:46:51.340
And so, what you'd like to
do-- and in this process--

00:46:51.340 --> 00:46:53.890
is factorize Q squared
and M squared, i.e.

00:46:53.890 --> 00:46:55.910
expand this thing
in Q squared and M

00:46:55.910 --> 00:46:59.590
squared, and maybe some logs
of Q squared and M squared.

00:46:59.590 --> 00:47:13.480
So we want to factorize
some logs, et cetera.

00:47:13.480 --> 00:47:15.580
OK, so what are the type
of degrees of freedom

00:47:15.580 --> 00:47:17.440
that we could have here?

00:47:17.440 --> 00:47:21.050
So lambda is going to
be M over Q, massive Z

00:47:21.050 --> 00:47:24.220
boson over the energy
scale of the collision,

00:47:24.220 --> 00:47:27.280
of the gamma star.

00:47:27.280 --> 00:47:31.000
And if you just
look at the Z boson,

00:47:31.000 --> 00:47:34.045
then it could be colinear
or it could be soft.

00:47:36.770 --> 00:47:44.113
So it could be actually three
different possibilities.

00:47:44.113 --> 00:47:46.030
And I could think about
doing this effectively

00:47:46.030 --> 00:47:46.960
in a bright frame.

00:47:46.960 --> 00:47:48.650
Just like we did earlier.

00:47:48.650 --> 00:47:52.060
And then what you would have
is that the colinear guy

00:47:52.060 --> 00:47:56.860
is like the quark, and
then the anti-colinear guys

00:47:56.860 --> 00:47:59.700
is like the outgoing quark.

00:47:59.700 --> 00:48:00.700
[INAUDIBLE]

00:48:00.700 --> 00:48:04.440
So you'd be making a transition
in this diagram from N colinear

00:48:04.440 --> 00:48:06.580
of objects to N bar
colinear of objects.

00:48:06.580 --> 00:48:09.400
And you could likewise have a Z
boson which could be colinear.

00:48:18.850 --> 00:48:22.060
Or you could have a
Z boson that's soft.

00:48:22.060 --> 00:48:26.020
And you have a soft rather than
ultra soft because of the mass.

00:48:26.020 --> 00:48:30.730
If Q times lambda is M, and so
if we want a propagator that's

00:48:30.730 --> 00:48:34.370
like P squared minus M squared,
P squared better be of order

00:48:34.370 --> 00:48:34.870
M squared.

00:48:34.870 --> 00:48:37.750
That happens for softs,
not for ultra softs.

00:48:37.750 --> 00:48:38.980
So that's why we have softs.

00:48:38.980 --> 00:48:41.140
And the same thing for
colinears, P squared of order M

00:48:41.140 --> 00:48:42.265
squared for these colinear.

00:48:42.265 --> 00:48:45.520
So it's really an
SET2 type situation,

00:48:45.520 --> 00:48:49.030
where the hyperbola is just
set by P squared of order M

00:48:49.030 --> 00:48:50.280
squared.

00:48:50.280 --> 00:48:54.630
We have our mode sitting
on that hyperbola.

00:48:54.630 --> 00:48:57.040
OK, so it's exactly of
this type over there.

00:48:57.040 --> 00:48:59.080
And the thing that's
new in this example

00:48:59.080 --> 00:49:02.536
is that we're going to encounter
these rapidity divergences.

00:49:02.536 --> 00:49:06.310
STUDENT: You mentioned there's
the only form for vertical

00:49:06.310 --> 00:49:07.443
in your theories, is that?

00:49:07.443 --> 00:49:08.110
PROFESSOR: Yeah.

00:49:13.460 --> 00:49:16.520
Make things as
simple as possible.

00:49:16.520 --> 00:49:20.120
So you could talk about mixed
sort of electroweak in QCD,

00:49:20.120 --> 00:49:22.160
but yeah, we don't.

00:49:22.160 --> 00:49:24.037
Let's make it-- in
some sense that's

00:49:24.037 --> 00:49:26.120
kind of just like mixing
a problem that we already

00:49:26.120 --> 00:49:28.080
would know how to deal
with, with this one.

00:49:28.080 --> 00:49:32.340
So let's just deal
with this one.

00:49:32.340 --> 00:49:32.840
All right.

00:49:32.840 --> 00:49:36.920
So in terms of the
external court momenta,

00:49:36.920 --> 00:49:40.760
we can therefore kind of
treat them as follows.

00:49:40.760 --> 00:49:45.020
Let's just let them be
of a large component.

00:49:45.020 --> 00:49:46.370
They're massless particles.

00:49:52.510 --> 00:49:56.600
So this is P, and this is PR.

00:49:56.600 --> 00:49:58.603
And these guys are massless--

00:49:58.603 --> 00:49:59.520
should have said that.

00:50:04.670 --> 00:50:06.220
And if you go through
the kinematics,

00:50:06.220 --> 00:50:09.970
Q squared, which is minus
P, minus P prime squared.

00:50:09.970 --> 00:50:13.150
If you square that, you just
find it to P minus P plus.

00:50:13.150 --> 00:50:15.670
And we're just effectively,
if we pick the bright frame--

00:50:19.420 --> 00:50:21.520
which is what
we're going to do--

00:50:21.520 --> 00:50:25.930
each one of those is
separately keep going at prime.

00:50:25.930 --> 00:50:27.470
Call it bar.

00:50:27.470 --> 00:50:31.270
Each one of those is
separately Q. The large--

00:50:31.270 --> 00:50:36.310
these guys are just fixed--
both would be Q. All right.

00:50:36.310 --> 00:50:39.220
So we could factor
is this current

00:50:39.220 --> 00:50:40.900
with these degrees of freedom.

00:50:40.900 --> 00:50:42.310
The quarks are colinear.

00:50:42.310 --> 00:50:46.180
So at lowest order they've
just become a CN and a CN bar.

00:50:46.180 --> 00:50:50.090
And then we have to address
that with Wilson lines.

00:50:50.090 --> 00:50:51.270
And we know how to do that.

00:50:51.270 --> 00:50:52.812
So let me just break
down the answer.

00:50:58.080 --> 00:50:59.940
We could again
follow our procedure

00:50:59.940 --> 00:51:01.500
of going through
SCET1, but it's now

00:51:01.500 --> 00:51:05.610
so familiar we just
know what to write down.

00:51:05.610 --> 00:51:06.110
OK.

00:51:06.110 --> 00:51:07.630
So the current would
look like that.

00:51:07.630 --> 00:51:10.222
And I'm not to worry too much
about the Dirac structure.

00:51:13.390 --> 00:51:16.030
So I won't worry, for
example, about gamma fives.

00:51:16.030 --> 00:51:23.794
And we could put
that in, it's easy.

00:51:27.970 --> 00:51:29.470
So that's the leading
order current,

00:51:29.470 --> 00:51:31.480
and then we'd have
leading order Lagrangians,

00:51:31.480 --> 00:51:33.515
and we need to
start calculating.

00:51:36.630 --> 00:51:39.300
And if we calculate what you
would expect from a major

00:51:39.300 --> 00:51:43.860
from that is, you'd expect that
F of Q squared and M squared

00:51:43.860 --> 00:51:45.570
is going to split up--

00:51:45.570 --> 00:51:47.220
given the degrees
of freedom we have

00:51:47.220 --> 00:51:51.090
into some kind of
hard function--

00:51:51.090 --> 00:51:54.570
and then some kind of amplitude
for the colinear parts,

00:51:54.570 --> 00:51:58.080
and then some kind of
amplitude for the soft part.

00:51:58.080 --> 00:52:02.130
OK, so you'd expect some hard
times colinear factorization

00:52:02.130 --> 00:52:03.000
of the form factor.

00:52:03.000 --> 00:52:05.541
And this is what we'll be after.

00:52:05.541 --> 00:52:07.740
So it's always good to
sort of have an idea

00:52:07.740 --> 00:52:08.850
where you're going.

00:52:08.850 --> 00:52:11.080
And that's where we're going.

00:52:11.080 --> 00:52:14.310
So let's consider just
one loop diagrams.

00:52:14.310 --> 00:52:17.190
And it suffices, in
order to make the point,

00:52:17.190 --> 00:52:19.680
just to consider the
most singular one.

00:52:19.680 --> 00:52:21.254
So I'm going to consider--

00:52:38.775 --> 00:52:40.150
so there's various
loop intervals

00:52:40.150 --> 00:52:42.483
that you could have to do
when you're doing the diagrams

00:52:42.483 --> 00:52:44.080
if they're fermions.

00:52:44.080 --> 00:52:45.670
Let's just take
the simplest, which

00:52:45.670 --> 00:52:48.370
is a scalar loop interval.

00:52:48.370 --> 00:52:51.010
And I'm going to contrast how
that scalar loop interval would

00:52:51.010 --> 00:52:52.593
look if you were
doing the full theory

00:52:52.593 --> 00:52:55.450
calculation with how it would
look in the effective theory.

00:52:55.450 --> 00:52:59.140
And then we'll see where
the divergences come from.

00:53:02.700 --> 00:53:04.820
So you can think about
this as kind of--

00:53:04.820 --> 00:53:08.280
the piece where the numerator
is independent of the loop

00:53:08.280 --> 00:53:10.860
momenta, so the numerator
just factors out.

00:53:21.000 --> 00:53:25.110
OK, so if I took our vertex
triangle diagram over there,

00:53:25.110 --> 00:53:27.345
then a piece of it-- where
the numerator is trivial

00:53:27.345 --> 00:53:28.950
and factors out--

00:53:28.950 --> 00:53:30.310
would look like this interval.

00:53:30.310 --> 00:53:32.220
So let's just study this guy.

00:53:32.220 --> 00:53:34.380
If we did this interval
in the full theory,

00:53:34.380 --> 00:53:37.700
this would be both
UV and IR finite.

00:53:37.700 --> 00:53:40.840
So this is just
giving us some result

00:53:40.840 --> 00:53:44.310
that involves logs of Q
squared over M squared.

00:53:48.840 --> 00:53:51.830
So it does have double logs
of Q squared over M squared,

00:53:51.830 --> 00:53:55.440
and single logs of Q
squared over M squared.

00:53:55.440 --> 00:53:59.150
But it's perfectly--
there's no 1 over epsilons.

00:54:03.280 --> 00:54:04.360
So now let's see what--

00:54:04.360 --> 00:54:08.840
let's think about what would
happen in the effective theory.

00:54:08.840 --> 00:54:12.910
So we have a kind of analogous
loop interval for colinear,

00:54:12.910 --> 00:54:14.680
where the gauge
boson is colinear.

00:54:18.340 --> 00:54:20.860
There's some numerator
that again I'm

00:54:20.860 --> 00:54:22.920
not going to worry about.

00:54:22.920 --> 00:54:25.780
If this numerator is constant,
it doesn't-- it's effectively

00:54:25.780 --> 00:54:27.867
the same constant.

00:54:27.867 --> 00:54:29.950
In the case of the-- if
you take the leading order

00:54:29.950 --> 00:54:31.408
numerator in the
full theory, it'll

00:54:31.408 --> 00:54:33.910
be the leading order numerator
the effective theory as well.

00:54:33.910 --> 00:54:37.030
But the denominators do change.

00:54:37.030 --> 00:54:39.340
And so, if we took
the N colinear,

00:54:39.340 --> 00:54:46.230
then yeah, so this
guy doesn't change.

00:54:46.230 --> 00:54:48.990
Because that's just like
saying P minus and K are--

00:54:48.990 --> 00:54:51.960
P minus and K minus
are the same size.

00:54:51.960 --> 00:54:54.630
But this guy does change.

00:54:54.630 --> 00:54:59.820
OK, so this guy would be K minus
P bar plus, because K minus is

00:54:59.820 --> 00:55:02.260
big and B plus is big.

00:55:02.260 --> 00:55:04.860
Both of these are big.

00:55:04.860 --> 00:55:10.690
So both of those are
big in this diagram.

00:55:10.690 --> 00:55:17.040
And so that's effectively
the Wilson line diagram.

00:55:17.040 --> 00:55:19.253
OK, where the propagator
here was off shell,

00:55:19.253 --> 00:55:21.045
got integrated out,
and just became iconal.

00:55:24.040 --> 00:55:26.560
And the K squared is
smaller, so we don't keep it

00:55:26.560 --> 00:55:28.801
in a leading order term.

00:55:28.801 --> 00:55:33.940
And then analogously, for
IN bar, it's the other way.

00:55:41.230 --> 00:55:49.300
So both of these are big,
and this one remains.

00:55:49.300 --> 00:55:52.690
And then they're soft.

00:55:57.860 --> 00:56:00.470
And in the soft
case, what happens

00:56:00.470 --> 00:56:03.470
is that both of the propagators
end up being iconal.

00:56:12.180 --> 00:56:17.280
And in our SET operator,
that's a diagram where

00:56:17.280 --> 00:56:20.910
we have our colinear lines,
and then we have kind

00:56:20.910 --> 00:56:24.600
of a self contraction of the S.

00:56:24.600 --> 00:56:26.815
But we're taking
an SN with an SN--

00:56:26.815 --> 00:56:28.440
we have a contraction
that's like this.

00:56:30.965 --> 00:56:32.340
We have two of
the Wilson lines--

00:56:32.340 --> 00:56:34.507
soft Wilson lines that are
sitting in that operator.

00:56:34.507 --> 00:56:36.883
That's a non-zero contraction.

00:56:36.883 --> 00:56:38.550
That would lead to a
diagram like-- that

00:56:38.550 --> 00:56:41.580
would lead to this amplitude.

00:56:41.580 --> 00:56:42.175
All right.

00:56:42.175 --> 00:56:43.800
I'm going to leave
a little space here,

00:56:43.800 --> 00:56:45.758
because I'm going to add
something in a minute.

00:56:48.900 --> 00:56:49.680
All right.

00:56:49.680 --> 00:56:53.220
So how do we see that there's
a problem with these intervals

00:56:53.220 --> 00:56:55.352
that they're not
regulated by dim reg?

00:56:55.352 --> 00:56:57.060
Well, you could look
at the soft integral

00:56:57.060 --> 00:56:58.770
and you could just do the perp.

00:56:58.770 --> 00:57:01.742
The perp is only showing up in
this K squared minus M squared.

00:57:01.742 --> 00:57:03.450
So you would get
something by doing that.

00:57:08.030 --> 00:57:13.000
And so, if we do the
perp with dim reg and IS,

00:57:13.000 --> 00:57:15.010
we would end up
proportional to something

00:57:15.010 --> 00:57:23.260
that's DK plus DK minus K
plus K minus, minus M squared

00:57:23.260 --> 00:57:26.860
to the some power of
epsilon, divided still

00:57:26.860 --> 00:57:30.310
by the factors of
K plus and K minus.

00:57:30.310 --> 00:57:34.540
So you see that the invariant
mass is being regulated.

00:57:34.540 --> 00:57:35.680
We just did the perp.

00:57:35.680 --> 00:57:37.090
Perp is gone.

00:57:37.090 --> 00:57:39.610
Plus times minus
is being regulated,

00:57:39.610 --> 00:57:42.460
because plus times minus-- if
plus times minus grows larger,

00:57:42.460 --> 00:57:45.910
or goes small, and
regulated by this epsilon--

00:57:45.910 --> 00:57:50.230
but either one, plus going large
or minus going large, or plus

00:57:50.230 --> 00:57:54.790
going small minus going small,
with plus and times minus fixed

00:57:54.790 --> 00:57:56.410
is not regulated.

00:57:56.410 --> 00:57:58.120
And that's the
rapidity divergence.

00:57:58.120 --> 00:58:02.330
If the invariant mass is
fixed and K plus over K

00:58:02.330 --> 00:58:03.610
minus goes large or small.

00:58:11.610 --> 00:58:15.320
So let me write it as K
minus over K plus going to 0,

00:58:15.320 --> 00:58:16.580
or going to infinity.

00:58:20.750 --> 00:58:26.236
Let me say, with K plus
times K minus fixed, then

00:58:26.236 --> 00:58:28.925
it diverges as
these things happen.

00:58:28.925 --> 00:58:30.550
And if you think
about what's happening

00:58:30.550 --> 00:58:34.870
in our picture over there
within these limits,

00:58:34.870 --> 00:58:37.600
it's exactly a situation
where this X here would

00:58:37.600 --> 00:58:41.650
be sliding up or sliding down.

00:58:41.650 --> 00:58:49.230
So in one of these limits, this
one's going towards the CN bar,

00:58:49.230 --> 00:58:52.690
and this one would
be going towards CN.

00:58:52.690 --> 00:58:55.220
So it's exactly a region where
you would be overlapping--

00:58:55.220 --> 00:58:56.720
sliding down the hyperbola.

00:58:56.720 --> 00:59:00.740
And the interval has
log singularities.

00:59:00.740 --> 00:59:02.510
So this is exactly
a situation where

00:59:02.510 --> 00:59:06.680
we can't ignore the overlaps
and we have to worry about them.

00:59:27.390 --> 00:59:30.060
OK so we need another regulator.

00:59:30.060 --> 00:59:31.200
Dim reg is not enough.

00:59:31.200 --> 00:59:33.210
We have to do something else.

00:59:33.210 --> 00:59:34.130
So what could we do?

00:59:34.130 --> 00:59:36.380
So there's lots of different
things that you could do.

00:59:36.380 --> 00:59:38.360
One thing is, you could
just sort of put it

00:59:38.360 --> 00:59:41.300
in some plus something in
these denominators-- that's

00:59:41.300 --> 00:59:43.190
called the delta
regulated, K plus,

00:59:43.190 --> 00:59:46.490
plus delta-- that's one choice.

00:59:46.490 --> 00:59:50.243
We'll do something a little
bit more dim reg-like.

00:59:50.243 --> 00:59:52.910
Which makes it sort of easier to
think about the renormalization

00:59:52.910 --> 00:59:54.470
group.

00:59:54.470 --> 01:00:05.520
So one choice for an additional
regulator is the following.

01:00:05.520 --> 01:00:08.417
So if you think about where
these divergences came from,

01:00:08.417 --> 01:00:09.750
they came from the Wilson lines.

01:00:09.750 --> 01:00:12.380
So what you really need to do
is regulate the Wilson lines.

01:00:15.110 --> 01:00:18.150
And you can do that as follows.

01:00:18.150 --> 01:00:19.730
Let me write out
the Wilson lines

01:00:19.730 --> 01:00:21.830
in our kind of momentum
space notation.

01:00:24.840 --> 01:00:30.980
So we have some N dot P type
momentum for the soft Wilson

01:00:30.980 --> 01:00:35.690
line, and an N dot AS field.

01:00:35.690 --> 01:00:38.570
And really, it's this one over
this iconal denominator that's

01:00:38.570 --> 01:00:40.880
giving rise to these
denominators here

01:00:40.880 --> 01:00:43.070
that are giving rise
to the singularity.

01:00:43.070 --> 01:00:44.870
So if we want to
regulate that singularity

01:00:44.870 --> 01:00:48.900
we need to add something, and
we could do that as follows.

01:00:48.900 --> 01:00:53.040
So this is the
regulator we'll pick.

01:00:56.232 --> 01:01:02.940
And I'll just write everything
as kind of a momentum operator.

01:01:06.620 --> 01:01:08.570
So I've just tucked
the Z momentum in

01:01:08.570 --> 01:01:09.920
and raised it to some power.

01:01:12.560 --> 01:01:18.040
So PZ is the difference
between P minus and P plus.

01:01:18.040 --> 01:01:19.450
And that seems
kind of arbitrary,

01:01:19.450 --> 01:01:21.880
but that'll do the job for us.

01:01:21.880 --> 01:01:24.070
You can motivate why
you want to do PZ--

01:01:24.070 --> 01:01:26.590
so this is 2PZ actually.

01:01:26.590 --> 01:01:29.110
You can motivate why you want
to do PZ rather than something

01:01:29.110 --> 01:01:31.190
else in the following way.

01:01:31.190 --> 01:01:34.480
And it's a true fact, that
once you have enough experience

01:01:34.480 --> 01:01:39.965
you realize it's good to use PZ,
because PZ doesn't involve P0.

01:01:39.965 --> 01:01:42.340
And the softs don't really
make a distinction between any

01:01:42.340 --> 01:01:43.970
of the different components.

01:01:43.970 --> 01:01:47.770
And if you put in P0,
something that involved P0,

01:01:47.770 --> 01:01:49.180
that would be dangerous.

01:01:53.052 --> 01:02:00.880
So this is nice,
because there is no P0.

01:02:00.880 --> 01:02:04.090
So it's the combination
of P plus and P

01:02:04.090 --> 01:02:05.590
minus that you can
form that doesn't

01:02:05.590 --> 01:02:07.790
have the P0, which is energy.

01:02:07.790 --> 01:02:09.610
And remember that
the polls in P0

01:02:09.610 --> 01:02:12.520
are related to things like
quarks and anti-quarks.

01:02:12.520 --> 01:02:14.350
They're related to unitarity.

01:02:14.350 --> 01:02:17.620
So not messing up
the structure in P0

01:02:17.620 --> 01:02:21.610
means that you'll be fine with
unitarity, fine with causality,

01:02:21.610 --> 01:02:26.280
you're not messing up a lot of
nice things about the theory.

01:02:26.280 --> 01:02:32.935
So if you do put P0
in, then you have

01:02:32.935 --> 01:02:34.310
to be careful
about those things.

01:02:42.340 --> 01:02:45.090
So if you just arbitrarily
put in some power of P0,

01:02:45.090 --> 01:02:47.005
then you'd have more trouble.

01:02:47.005 --> 01:02:49.380
And so that's kind of why
we're avoiding and just putting

01:02:49.380 --> 01:02:49.992
in PZ.

01:02:52.950 --> 01:02:54.960
For the colinears, we
can do something similar.

01:02:54.960 --> 01:02:57.420
But for the colinears we
also have a power counting

01:02:57.420 --> 01:02:59.530
between the minus and the plus.

01:02:59.530 --> 01:03:05.460
So for the colinear,
we can still

01:03:05.460 --> 01:03:10.050
make the power counting OK by
thinking about putting in PZ,

01:03:10.050 --> 01:03:16.000
but then just expanding
it to be a P minus.

01:03:16.000 --> 01:03:18.210
And that's true up
to power corrections.

01:03:18.210 --> 01:03:20.790
And we don't really need to
worry about power corrections

01:03:20.790 --> 01:03:22.740
when we're regulating
these divergences.

01:03:26.630 --> 01:03:28.700
So it's just putting
in the large momentum.

01:03:28.700 --> 01:03:33.035
And so W written in
a similar notation.

01:03:43.432 --> 01:03:45.640
So I'll explain what the
other things in this formula

01:03:45.640 --> 01:03:48.180
are in a minute.

01:03:48.180 --> 01:03:50.890
But the important
thing for regulating

01:03:50.890 --> 01:03:52.240
is that we have some factor.

01:03:52.240 --> 01:03:55.756
In this case, it would be
a factor of N bar dot P.

01:03:55.756 --> 01:03:58.372
STUDENT: Dot [INAUDIBLE]?

01:03:58.372 --> 01:03:59.080
PROFESSOR: Sorry?

01:03:59.080 --> 01:04:00.807
STUDENT: Dot eta to bar dot P?

01:04:00.807 --> 01:04:01.390
PROFESSOR: No.

01:04:01.390 --> 01:04:05.710
It's supposed to be
an N. Looks like eta.

01:04:05.710 --> 01:04:07.000
Too many variables.

01:04:07.000 --> 01:04:08.196
There's the eta.

01:04:14.755 --> 01:04:16.630
So there's some factor
raising of the-- again

01:04:16.630 --> 01:04:19.105
the iconal propagator
sort of mixing up

01:04:19.105 --> 01:04:20.230
with the iconal propagator.

01:04:20.230 --> 01:04:21.730
In this case, it's
even more obvious

01:04:21.730 --> 01:04:25.130
that it's just regulating
that iconal propagator.

01:04:25.130 --> 01:04:25.630
OK.

01:04:25.630 --> 01:04:28.390
So if we were to do that,
and go back over here,

01:04:28.390 --> 01:04:30.790
and put the regulators
into these integrals,

01:04:30.790 --> 01:04:32.430
what would happen?

01:04:32.430 --> 01:04:38.990
So here, we'd get
an extra factor--

01:04:38.990 --> 01:04:40.255
K minus to the eta.

01:04:40.255 --> 01:04:42.340
And so, that would
regulate this K minus.

01:04:42.340 --> 01:04:45.310
And these integrals will
also have polls from the 1

01:04:45.310 --> 01:04:46.570
over K minus.

01:04:46.570 --> 01:04:47.650
You could think about--

01:04:47.650 --> 01:04:48.203
well, OK.

01:04:48.203 --> 01:04:49.870
If we did those
integrals, we would also

01:04:49.870 --> 01:04:54.073
have rapidity divergences that
are kind of the analog ones,

01:04:54.073 --> 01:04:55.990
and the colinear SECT
are still the soft ones.

01:04:59.020 --> 01:05:00.700
Here we have two propagators.

01:05:00.700 --> 01:05:04.360
And so if I have two
soft Wilson lines, and so

01:05:04.360 --> 01:05:06.010
I get two factors.

01:05:06.010 --> 01:05:09.470
But I've conveniently chose
it to be the square root.

01:05:09.470 --> 01:05:11.590
So it comes out kind of
looking the same here.

01:05:16.970 --> 01:05:19.998
So one thing that is just a
part of this regulator-- which

01:05:19.998 --> 01:05:21.790
actually I don't know
a good argument for--

01:05:21.790 --> 01:05:24.670
is kind of a priority from
the symmetries of the theory,

01:05:24.670 --> 01:05:30.190
you might like to argue that
that should be eta over 2,

01:05:30.190 --> 01:05:33.430
and this should be eta.

01:05:33.430 --> 01:05:34.960
But it's really
just part of-- it's

01:05:34.960 --> 01:05:39.668
just a choice, a convention that
we've made, as far as I know.

01:05:39.668 --> 01:05:41.710
There probably is some
nice deep argument for it,

01:05:41.710 --> 01:05:44.420
but I don't know it.

01:05:44.420 --> 01:05:45.920
So what are these other factors?

01:05:45.920 --> 01:05:49.810
So nu is going to
play the role of mu.

01:05:49.810 --> 01:05:52.210
We've changed the
dimension of the operator.

01:05:52.210 --> 01:05:54.730
We've compensated
it back with nu,

01:05:54.730 --> 01:05:58.190
just like we were doing with mu.

01:05:58.190 --> 01:06:01.450
We're going to get 1 over
eta divergences, which

01:06:01.450 --> 01:06:04.120
are like our 1 over
epsilon divergences.

01:06:04.120 --> 01:06:06.850
And we're going to
get logs of nu, which

01:06:06.850 --> 01:06:09.568
are the analogs of logs of mu.

01:06:09.568 --> 01:06:11.110
And that's the sense
in which there's

01:06:11.110 --> 01:06:16.000
kind of an analog of this M up
with our usual dim reg setup.

01:06:16.000 --> 01:06:18.250
In order to have
a full analog, we

01:06:18.250 --> 01:06:21.710
should think about
having a coupling.

01:06:21.710 --> 01:06:23.830
And so, that's what
this W factor is.

01:06:26.265 --> 01:06:27.640
You can think
about it like there

01:06:27.640 --> 01:06:30.680
was some bare pseudo coupling,
which is really just 1.

01:06:30.680 --> 01:06:32.680
But just imagine that
you're switching from bare

01:06:32.680 --> 01:06:35.800
to renormalized, in order to
set up a renormalization group

01:06:35.800 --> 01:06:36.880
equation.

01:06:36.880 --> 01:06:42.460
And then this guy here, which is
in eta dimensions, if you like,

01:06:42.460 --> 01:06:44.260
would have a
renormalization group which

01:06:44.260 --> 01:06:51.910
would say nu D by D nu
of this W of eta and nu

01:06:51.910 --> 01:06:55.030
is minus eta over 2.

01:06:55.030 --> 01:06:58.300
Sorry, this is eta over 2.

01:07:04.070 --> 01:07:07.790
So that's the analog of
saying mu by D mu of alpha

01:07:07.790 --> 01:07:10.585
is minus 2 epsilon alpha.

01:07:10.585 --> 01:07:12.905
So an analog statement.

01:07:15.695 --> 01:07:17.750
I think this is OK.

01:07:17.750 --> 01:07:20.030
So this guy here is
like a dummy coupling.

01:07:20.030 --> 01:07:22.130
And the boundary
condition for it

01:07:22.130 --> 01:07:24.000
after you've carried out these--

01:07:24.000 --> 01:07:26.780
this is just to set
it to back to 1.

01:07:26.780 --> 01:07:30.210
So it's identically 1, it's
really just a bookkeeping

01:07:30.210 --> 01:07:30.710
device.

01:07:33.300 --> 01:07:42.930
It's just--e it's a dummy
coupling once you go to the eta

01:07:42.930 --> 01:07:43.460
dimensions.

01:07:43.460 --> 01:07:45.627
But you just set it always
the renormalized coupling

01:07:45.627 --> 01:07:47.840
is just identically set to 1.

01:07:47.840 --> 01:07:49.520
And identically
setting it to 1 is

01:07:49.520 --> 01:07:53.210
what you need to keep gauge
invariance in these Wilson

01:07:53.210 --> 01:07:53.840
lines.

01:07:53.840 --> 01:07:55.820
It turns out actually
that this regulator here

01:07:55.820 --> 01:07:58.752
is gauge invariant, though
it doesn't look like it.

01:07:58.752 --> 01:08:00.710
We've modified the
structure of the Wilson line

01:08:00.710 --> 01:08:06.230
in some kind of way that looks
like it might be drastic.

01:08:06.230 --> 01:08:08.480
But actually these
factors here are gauge--

01:08:08.480 --> 01:08:10.410
still leave a gauge
invariant object.

01:08:10.410 --> 01:08:12.970
So--

01:08:12.970 --> 01:08:15.560
STUDENT: Can you write
[INAUDIBLE] space, I assume?

01:08:15.560 --> 01:08:18.180
PROFESSOR: Not that I know of.

01:08:18.180 --> 01:08:18.908
Yeah.

01:08:18.908 --> 01:08:20.116
STUDENT: I think [INAUDIBLE].

01:08:20.116 --> 01:08:21.680
PROFESSOR: Maybe you can.

01:08:21.680 --> 01:08:22.819
Yeah.

01:08:22.819 --> 01:08:25.515
But it's not-- since it's not--

01:08:25.515 --> 01:08:26.640
yeah, I don't know how to--

01:08:26.640 --> 01:08:28.182
I don't know what
it would look like.

01:08:28.182 --> 01:08:32.744
You could probably transform
that power, and it--

01:08:32.744 --> 01:08:35.753
STUDENT: [INAUDIBLE].

01:08:35.753 --> 01:08:36.420
PROFESSOR: Yeah.

01:08:42.187 --> 01:08:43.770
I'm sure you can
probably just try out

01:08:43.770 --> 01:08:45.145
before you
[? transplant ?] that.

01:08:45.145 --> 01:08:47.090
I'm just not sure if
it would look nice.

01:08:47.090 --> 01:08:47.754
Yeah.

01:08:47.754 --> 01:08:49.029
It might not look too bad.

01:08:51.502 --> 01:08:53.210
Yeah, and it might
actually be a nice way

01:08:53.210 --> 01:08:57.380
of saying what I'm about to
say in a less nice way, which

01:08:57.380 --> 01:08:59.569
is, if you look at
the gauge symmetry,

01:08:59.569 --> 01:09:02.035
why is this not messing it up?

01:09:02.035 --> 01:09:03.410
So one way of
thinking about that

01:09:03.410 --> 01:09:06.410
is just to look at
general covariant gauge.

01:09:06.410 --> 01:09:19.470
So note, the 1 over eta and eta
0 terms are gauge invariant.

01:09:19.470 --> 01:09:21.750
And you can think about
that by just going

01:09:21.750 --> 01:09:24.420
to a general covariant gauge and
seeing the parameter dependence

01:09:24.420 --> 01:09:26.100
drop out.

01:09:26.100 --> 01:09:31.020
So for example, at
1 loop, you would

01:09:31.020 --> 01:09:33.640
take [? g mu ?] nu
in the contractions

01:09:33.640 --> 01:09:35.520
and replace it in
general covariant gauge

01:09:35.520 --> 01:09:39.195
by some gauge parameter--

01:09:39.195 --> 01:09:44.760
of general covariant gauge,
K mu, K nu over K squared.

01:09:44.760 --> 01:09:47.460
And you'd like to see
it independent of this.

01:09:47.460 --> 01:09:50.700
But this eta to the 0 piece
is kind of independent

01:09:50.700 --> 01:09:52.590
of that for the usual reasons.

01:09:52.590 --> 01:09:55.470
And the 1 over eta term
is independent of that,

01:09:55.470 --> 01:09:58.500
because this guy actually
doesn't deuce any rapidity

01:09:58.500 --> 01:09:59.130
singularities.

01:09:59.130 --> 01:10:03.240
What happens is that if
you have an N dot K, then

01:10:03.240 --> 01:10:05.620
you have a corresponding
N mu in the numerator.

01:10:05.620 --> 01:10:08.745
And so, basically what happens
is, you get an extra N dot

01:10:08.745 --> 01:10:10.840
K in the numerator.

01:10:10.840 --> 01:10:14.070
So any time you
have 1 over N dot K,

01:10:14.070 --> 01:10:18.690
you would get for this piece
multiplied by an N dot K

01:10:18.690 --> 01:10:21.480
upstairs.

01:10:21.480 --> 01:10:24.360
And so this is cancelling,
you don't have a rapidity

01:10:24.360 --> 01:10:27.730
divergence in the
C-dependent part.

01:10:27.730 --> 01:10:31.980
So that's why this is invariant
under the gauge symmetry.

01:10:31.980 --> 01:10:36.990
And then, because of
this boundary condition,

01:10:36.990 --> 01:10:38.850
the kind of cancellation
of the C-dependence

01:10:38.850 --> 01:10:41.142
in the order [? A ?] to the
0 piece, this kind of works

01:10:41.142 --> 01:10:45.262
out in the standard way.

01:10:45.262 --> 01:10:46.720
So it gives you an
idea of why it's

01:10:46.720 --> 01:10:49.680
gauge invariant without giving
you a kind of full proof

01:10:49.680 --> 01:10:50.240
or anything.

01:10:53.210 --> 01:10:55.720
So we have both 1
over epsilon polls

01:10:55.720 --> 01:10:57.550
and 1 over eta polls
in general, and we

01:10:57.550 --> 01:10:59.835
have to understand
what to do with them.

01:10:59.835 --> 01:11:01.210
So here's what
we're going to do.

01:11:03.880 --> 01:11:07.815
For any fixed invariant
mass, it turns out

01:11:07.815 --> 01:11:09.565
that we can have these
one over eta polls.

01:11:18.650 --> 01:11:20.690
And the right procedure
for dealing with them

01:11:20.690 --> 01:11:22.140
is as follows.

01:11:22.140 --> 01:11:25.160
First you take eta goes to 0
and deal with these new polls

01:11:25.160 --> 01:11:28.730
that you have introduced
in your amplitude.

01:11:28.730 --> 01:11:31.010
In order to deal with them,
because you can have them

01:11:31.010 --> 01:11:33.613
for any invariant
mass, you actually

01:11:33.613 --> 01:11:35.030
have to add counter
terms that can

01:11:35.030 --> 01:11:36.920
be a whole function
of epsilon, where

01:11:36.920 --> 01:11:39.800
you have an expanded in epsilon
and then divide it by eta.

01:11:43.027 --> 01:11:47.900
So let me abbreviate
counterterm as CT dot.

01:11:47.900 --> 01:11:52.040
Then, after you've done that,
you take epsilon goes to 0,

01:11:52.040 --> 01:11:54.834
and you find your 1 over
epsilon counterterms.

01:12:00.650 --> 01:12:04.050
And this is the correct
way of doing it.

01:12:04.050 --> 01:12:07.780
And we'll see how that works
in practice in a minute.

01:12:07.780 --> 01:12:10.690
So let's go back to our
integrals that I've now erased

01:12:10.690 --> 01:12:12.750
and just write out the answers.

01:12:12.750 --> 01:12:15.480
We're doing those integrals
with this regulator.

01:12:15.480 --> 01:12:17.100
And I'll also make
them fermions,

01:12:17.100 --> 01:12:19.610
so I'm putting in
the numerators.

01:12:19.610 --> 01:12:21.540
We wrote them down for scalars.

01:12:21.540 --> 01:12:24.748
The scalars where the most
divergent integrals actually.

01:12:24.748 --> 01:12:26.790
I can include the numerators,
that doesn't really

01:12:26.790 --> 01:12:28.950
change the story.

01:12:28.950 --> 01:12:31.050
And I can include the
pre-factors as well.

01:12:41.520 --> 01:12:44.340
And I'll kind of write things
in a QCD type notation,

01:12:44.340 --> 01:12:48.210
even we can imagine that it's
a non-abelian group, just so CF

01:12:48.210 --> 01:12:50.040
is the whatever
group it is, it's

01:12:50.040 --> 01:12:52.575
the Casimir of the fundamental.

01:12:58.030 --> 01:13:01.450
Whatever group our
gauge boson's in.

01:13:01.450 --> 01:13:04.360
So here's the eta poll.

01:13:04.360 --> 01:13:06.980
It has a whole function of
epsilon in the numerator,

01:13:06.980 --> 01:13:09.070
and it's even divergent.

01:13:09.070 --> 01:13:11.413
So this is 2 eta.

01:13:11.413 --> 01:13:12.955
And then the rest
of it I can expand.

01:13:23.960 --> 01:13:27.830
So there's going to be 1
over epsilon times the log.

01:13:27.830 --> 01:13:29.930
When the log replaces
that 1 over epsilon

01:13:29.930 --> 01:13:33.080
then I can start to expand,
and I get another 1 over--

01:13:33.080 --> 01:13:35.930
when the log nu replaces
the one over eta,

01:13:35.930 --> 01:13:41.360
I can expand this gamma, and
it gives me 1 over epsilon.

01:13:41.360 --> 01:13:44.681
And there's also
some other pieces.

01:13:44.681 --> 01:13:49.496
So over 2 epsilon
there's a log mu over M.

01:13:49.496 --> 01:13:50.548
And there's a constant.

01:13:50.548 --> 01:13:52.340
And I'm never going to
write the constants.

01:13:55.437 --> 01:13:58.020
So let me read all the results
and then we'll talk about them.

01:13:58.020 --> 01:14:01.300
So ICN bar is the same.

01:14:01.300 --> 01:14:04.080
The only difference between
this is that P minus

01:14:04.080 --> 01:14:05.790
close to P bar plus.

01:14:05.790 --> 01:14:08.340
It was really symmetric.

01:14:08.340 --> 01:14:11.742
And then IS is different.

01:14:36.360 --> 01:14:37.615
So the 1 over epsilon--

01:14:37.615 --> 01:14:39.490
1 over eta poll comes
with the opposite sign,

01:14:39.490 --> 01:14:41.795
and it also comes to a
factor of 2 different.

01:14:49.543 --> 01:14:50.960
And in this case,
there's actually

01:14:50.960 --> 01:14:53.450
1 over 2 epsilons squared term.

01:14:56.520 --> 01:15:02.568
So there's also a
double log of mu or M.

01:15:02.568 --> 01:15:03.860
And then there's plus constant.

01:15:07.415 --> 01:15:07.915
OK?

01:15:10.470 --> 01:15:12.940
And so, you could think
about adding them up.

01:15:12.940 --> 01:15:15.300
And what happens when
you add them up is,

01:15:15.300 --> 01:15:18.150
you have 1 over 2 eta, 1
over 2 eta, minus 1 over eta,

01:15:18.150 --> 01:15:21.040
and so that 1 over
eta polls cancel.

01:15:21.040 --> 01:15:22.620
And that's exactly
what you'd expect,

01:15:22.620 --> 01:15:26.315
because in the
full theory the eta

01:15:26.315 --> 01:15:28.440
was something we introduced
in order to distinguish

01:15:28.440 --> 01:15:29.670
these effective theory modes.

01:15:29.670 --> 01:15:32.040
It wasn't something
that was there needed

01:15:32.040 --> 01:15:34.020
for the full theory integral.

01:15:34.020 --> 01:15:35.830
And so you don't really--

01:15:35.830 --> 01:15:38.560
you'd expect that it's sort of--

01:15:38.560 --> 01:15:41.550
that there's a
corresponding regulator

01:15:41.550 --> 01:15:42.550
between the two sectors.

01:15:42.550 --> 01:15:44.008
So that when you
add them together,

01:15:44.008 --> 01:15:46.810
that the dependence on that
parameter is canceling away,

01:15:46.810 --> 01:15:49.630
because it was just an
artificial separation,

01:15:49.630 --> 01:15:52.970
if you like, or separation
that we're doing.

01:15:52.970 --> 01:15:58.885
So if I add them up, 1
over eta is cancelled,

01:15:58.885 --> 01:16:02.386
and so do all the logs of nu.

01:16:02.386 --> 01:16:03.195
We have alpha--

01:16:12.160 --> 01:16:17.515
I'm left with a log of mu
over Q, 1 over epsilon poll,

01:16:17.515 --> 01:16:24.410
double log, some
types of single logs,

01:16:24.410 --> 01:16:26.567
and some other
type of double log.

01:16:38.260 --> 01:16:39.400
So it would look like that.

01:16:39.400 --> 01:16:42.700
All the nu dependence
is canceling away.

01:16:42.700 --> 01:16:53.020
So sort of various things which
we'll start talking about now,

01:16:53.020 --> 01:16:56.000
and we'll continue
talking about next time.

01:16:56.000 --> 01:17:00.380
So the rapidity 1
over eta divergence,

01:17:00.380 --> 01:17:07.510
which we can call a rapidity
divergence, cancels in sum.

01:17:07.510 --> 01:17:09.340
And of course, so
does the log nu's.

01:17:14.070 --> 01:17:15.045
And that's as expected.

01:17:22.067 --> 01:17:23.650
And if you add an
overall counterterm,

01:17:23.650 --> 01:17:27.190
for the entire thing it just
involves the hard scale log

01:17:27.190 --> 01:17:34.670
mu over Q. So if you were
to think about there being

01:17:34.670 --> 01:17:37.800
some Wilson coefficient,
which is sort of C

01:17:37.800 --> 01:17:47.770
bare is ZC minus 1, Z bare
is ZC, C renormalized,

01:17:47.770 --> 01:17:55.640
then ZC and the C renormalized
only involve logs of mu

01:17:55.640 --> 01:17:58.510
over Q, which is the hard scale.

01:17:58.510 --> 01:17:59.870
OK?

01:17:59.870 --> 01:18:03.110
And that means that our
hard function, which

01:18:03.110 --> 01:18:05.870
is the Wilson coefficient
squared, or just the Wilson

01:18:05.870 --> 01:18:10.970
coefficient in this case, is
only a function of Q and mu.

01:18:10.970 --> 01:18:13.010
OK, so integrating out
the hard scale physics

01:18:13.010 --> 01:18:14.522
didn't know about
the separation.

01:18:14.522 --> 01:18:15.980
The separation was
really something

01:18:15.980 --> 01:18:18.800
that we needed to do
in the effective theory

01:18:18.800 --> 01:18:22.640
to distinguish the
CN and S modes.

01:18:22.640 --> 01:18:24.420
And you can see why
we needed to do it

01:18:24.420 --> 01:18:29.570
if you look at these
answers, because if you look

01:18:29.570 --> 01:18:34.080
at the types of logs that are
showing up here, in this case,

01:18:34.080 --> 01:18:35.930
we have a nu over P minus.

01:18:35.930 --> 01:18:37.460
And in this case, we have a--

01:18:37.460 --> 01:18:38.740
is it nu over mu?

01:18:42.013 --> 01:18:44.710
Just make sure I got that right.

01:18:44.710 --> 01:18:46.130
I guess it is.

01:18:46.130 --> 01:18:50.415
In this case, we
have a nu over M.

01:18:50.415 --> 01:18:53.620
And we also have a mu over nu.

01:18:53.620 --> 01:18:56.668
And so, the sort of
right scale to-- in order

01:18:56.668 --> 01:18:58.210
to minimize the
logarithms here we're

01:18:58.210 --> 01:18:59.793
going to have to,
again, as usual take

01:18:59.793 --> 01:19:02.500
different values of mu and nu.

01:19:02.500 --> 01:19:04.180
Well, it's the same value of mu.

01:19:04.180 --> 01:19:07.720
All of them are M. But it's
a different value of nu,

01:19:07.720 --> 01:19:11.380
because it's the nu that would
need to be of order Q here.

01:19:11.380 --> 01:19:15.940
P minus is Q. And the nu would
need to be of order M here.

01:19:15.940 --> 01:19:19.280
So it's the nu that
distinguishes the modes.

01:19:19.280 --> 01:19:25.600
So the logs NCN are minimized.

01:19:29.870 --> 01:19:33.200
Or mu of order M, which
says being on the hyperbola,

01:19:33.200 --> 01:19:37.730
but the nu should be of
order P minus, which is Q.

01:19:37.730 --> 01:19:39.500
And that's precisely
actually where

01:19:39.500 --> 01:19:48.890
we put the X in our picture,
if you think about it.

01:19:48.890 --> 01:19:54.840
OK, so that's saying that you
have a large P minus momentum,

01:19:54.840 --> 01:19:57.623
and we have-- and we're on this
hyperbola where P squared is

01:19:57.623 --> 01:19:59.928
an order M squared.

01:19:59.928 --> 01:20:06.750
So this is-- and it's
likewise for the other pieces.

01:20:06.750 --> 01:20:08.160
So for the soft piece--

01:20:12.520 --> 01:20:20.030
so for the-- say for the anti-
for the other colinear piece,

01:20:20.030 --> 01:20:21.170
we need the same thing.

01:20:25.840 --> 01:20:35.200
And then for the soft we
need a different value

01:20:35.200 --> 01:20:37.140
for this new parameter.

01:20:37.140 --> 01:20:38.837
So having this
regulator is behaving

01:20:38.837 --> 01:20:40.920
like dim reg, where we
needed different mu's, when

01:20:40.920 --> 01:20:42.087
we had different hyperbolas.

01:20:42.087 --> 01:20:44.122
Now we have different
places on the hyperbola,

01:20:44.122 --> 01:20:46.080
and we're tracking that
with the new parameter,

01:20:46.080 --> 01:20:47.922
and that's showing
up in the logarithms.

01:20:47.922 --> 01:20:50.130
And if you think about what
the logarithms are doing,

01:20:50.130 --> 01:20:57.420
you can see that when you
combine terms, let's see--

01:20:57.420 --> 01:20:59.077
if you look at the
1 over epsilon,

01:20:59.077 --> 01:21:00.660
and the mu's are
canceling out, you're

01:21:00.660 --> 01:21:04.530
getting a mu over
Q. Yeah, that's

01:21:04.530 --> 01:21:06.660
maybe not the best example.

01:21:06.660 --> 01:21:08.070
Look over here.

01:21:08.070 --> 01:21:12.180
You have this log of
M over Q In this kind

01:21:12.180 --> 01:21:13.860
of complete decomposition.

01:21:13.860 --> 01:21:16.950
The way that that
logarithm here gets made up

01:21:16.950 --> 01:21:21.960
is by having m over nu
and nu over Q. All right?

01:21:21.960 --> 01:21:24.650
So in order to get this log that
doesn't have any mu's in it,

01:21:24.650 --> 01:21:26.950
mu is not telling you that
there's that large log.

01:21:26.950 --> 01:21:28.630
But there is a large log.

01:21:28.630 --> 01:21:32.460
So there's large logs associated
to these rapidity divergences.

01:21:32.460 --> 01:21:35.760
And what we'll talk
about next time

01:21:35.760 --> 01:21:38.820
is how you do the
renormalization group

01:21:38.820 --> 01:21:40.500
with the diagrams like this.

01:21:40.500 --> 01:21:42.570
How you write down in
almost dimension equations.

01:21:42.570 --> 01:21:45.180
There'll be an almost dimension
equations in both mu and nu

01:21:45.180 --> 01:21:46.190
space.

01:21:46.190 --> 01:21:47.940
So we'll have to move
around in that space

01:21:47.940 --> 01:21:50.982
and see how it works to sum
of the large logarithms.

01:21:50.982 --> 01:21:54.300
But I'll postpone
that to next time.

01:21:57.770 --> 01:21:58.790
So any questions?

01:22:06.430 --> 01:22:08.440
So the general idea
is really, as usual,

01:22:08.440 --> 01:22:10.630
it's just that now we're
dealing with a situation

01:22:10.630 --> 01:22:12.490
where there's two regulators.

01:22:12.490 --> 01:22:15.310
And they're actually
independent regulators.

01:22:15.310 --> 01:22:18.250
One's, if you like, is
regulating invariant mass,

01:22:18.250 --> 01:22:21.310
and the other is regulating
these extra divergences.

01:22:21.310 --> 01:22:24.580
And so, we'll be able to
move around in the space

01:22:24.580 --> 01:22:28.920
without any worrying about
path dependence, for example.

01:22:28.920 --> 01:22:31.690
We'll talk a little bit
about that next time--

01:22:31.690 --> 01:22:34.600
in this two dimensional
space of mu and nu.

01:22:34.600 --> 01:22:37.370
But you can see just from
looking at the logs, as usual,

01:22:37.370 --> 01:22:38.872
you can see where
you need to be.

01:22:38.872 --> 01:22:40.330
And you can see
that you need to be

01:22:40.330 --> 01:22:42.122
in different places
for the different modes

01:22:42.122 --> 01:22:44.578
in order to minimize the
logs of these amplitudes.

01:22:44.578 --> 01:22:46.120
And if you do that,
then there should

01:22:46.120 --> 01:22:52.460
be some renormalization group
that would connect these guys.

01:22:52.460 --> 01:22:58.390
So there should do something RGE
that goes between these guys,

01:22:58.390 --> 01:23:01.410
and it'll be an RGE
in this new parameter.