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IAIN STEWART: OK, so
let me remind you where

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00:00:25,880 --> 00:00:28,070
we were before spring break.

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00:00:28,070 --> 00:00:33,650
So first of all, there's a
posting with the project list.

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00:00:33,650 --> 00:00:37,100
Feel free to pick a project
as soon as you're ready.

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00:00:37,100 --> 00:00:38,930
Send me your choice.

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00:00:38,930 --> 00:00:40,550
Problem set number 3 due today.

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00:00:40,550 --> 00:00:43,580
Problem set number 4
is posted this morning.

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00:00:43,580 --> 00:00:46,883
It will be due in two
weeks, plus two days so,

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00:00:46,883 --> 00:00:48,800
two weeks from now there's
actually a holiday,

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00:00:48,800 --> 00:00:52,700
so it will be due
on that Thursday.

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00:00:52,700 --> 00:00:56,000
Last time, we were talking about
two-nucleon effective field

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theory.

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00:00:57,210 --> 00:00:59,720
So this was a
nonrelativistic field theory.

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00:00:59,720 --> 00:01:03,620
Both of these terms here in
the action were the same size.

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00:01:03,620 --> 00:01:04,910
They were the same relevance.

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00:01:07,960 --> 00:01:10,640
And we talked about
that last time.

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00:01:10,640 --> 00:01:13,050
So in HQET, it would
just be the first term,

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00:01:13,050 --> 00:01:15,050
but in this case
of two nucleons,

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00:01:15,050 --> 00:01:16,890
we actually need the
kinetic term, as well,

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00:01:16,890 --> 00:01:18,860
and both of these terms
are equally important.

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00:01:18,860 --> 00:01:21,047
And that basically means
that in this type of system

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00:01:21,047 --> 00:01:22,880
the time derivatives
and spatial derivatives

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00:01:22,880 --> 00:01:24,980
are counted differently.

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00:01:24,980 --> 00:01:26,960
But you can organize
the field theory

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00:01:26,960 --> 00:01:29,240
using equations of motion
so that these are just

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00:01:29,240 --> 00:01:34,220
local operators with
spatial derivatives.

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00:01:34,220 --> 00:01:36,675
So you basically deal with
the time derivatives once

35
00:01:36,675 --> 00:01:39,050
and for all, and you could
always use equations of motion

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00:01:39,050 --> 00:01:40,950
to eliminate them there.

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00:01:40,950 --> 00:01:44,690
And so you just have really
this one time derivative.

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00:01:44,690 --> 00:01:46,680
This theory is
particularly simple

39
00:01:46,680 --> 00:01:49,100
if we neglect the
relativistic corrections,

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00:01:49,100 --> 00:01:51,230
and that's because
it's not-- because it's

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00:01:51,230 --> 00:01:53,330
a nonrelativistic
field theory, we just

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00:01:53,330 --> 00:01:55,238
have these bubble diagrams.

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00:01:55,238 --> 00:01:56,780
And they're just
contact interaction,

44
00:01:56,780 --> 00:01:59,120
so they all decouple from
each other, so at N-loop order

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00:01:59,120 --> 00:02:01,880
we can just write down
what the loops are.

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00:02:01,880 --> 00:02:04,760
We talked about the fact
that, in terms of power

47
00:02:04,760 --> 00:02:07,160
counting things, where it
could be a little subtle,

48
00:02:07,160 --> 00:02:09,632
and that was related also
to a power divergence that

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00:02:09,632 --> 00:02:11,840
occurs in this diagram, and
we could track that power

50
00:02:11,840 --> 00:02:14,450
divergence by using
particular schemes that

51
00:02:14,450 --> 00:02:17,660
gave this extra mu r term, and
I won't repeat all that story.

52
00:02:17,660 --> 00:02:20,660
But that led us to adding
a finite counterterm

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00:02:20,660 --> 00:02:25,040
to this bubble diagram to track
the power law divergent term,

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00:02:25,040 --> 00:02:28,820
and that allowed us to come up
with a power accounting that

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00:02:28,820 --> 00:02:31,580
would take into account systems
where the scattering length is

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00:02:31,580 --> 00:02:34,260
large, or in particular,
the scattering length times

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00:02:34,260 --> 00:02:37,640
momentum is greater than 1, and
in these two-nucleon systems

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00:02:37,640 --> 00:02:40,380
that's exactly the
scenario that we're in.

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00:02:40,380 --> 00:02:42,800
So what we then had
was a power counting

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00:02:42,800 --> 00:02:45,470
that assigned the
very first operator

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00:02:45,470 --> 00:02:48,570
to scale like 1 over
a power of momentum.

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00:02:48,570 --> 00:02:51,440
So not like 1 over the
power of a large scale,

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00:02:51,440 --> 00:02:54,060
but actually 1 over a power of
something small, the momentum

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00:02:54,060 --> 00:02:55,710
but we're interested
in studying.

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00:02:55,710 --> 00:03:02,120
So it's enhanced in
the power counting

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00:03:02,120 --> 00:03:03,230
that we wanted to adopt.

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00:03:03,230 --> 00:03:04,700
And that meant
basically that you

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00:03:04,700 --> 00:03:08,690
had to treat this one coupling,
c 0, nonperturbatively.

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00:03:08,690 --> 00:03:12,290
You summed it up to all
orders, so the term here

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00:03:12,290 --> 00:03:17,090
comes from adding
up all the diagrams

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00:03:17,090 --> 00:03:22,880
with any number of bubbles, and
that all give this first term.

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00:03:22,880 --> 00:03:24,350
And then the rest
of the couplings

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00:03:24,350 --> 00:03:27,480
are treated perturbatively,
the higher derivative term.

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00:03:27,480 --> 00:03:32,060
So a term, c 2, which
comes with grad squared,

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00:03:32,060 --> 00:03:33,590
and that gives this
extra p squared.

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00:03:33,590 --> 00:03:36,242
And the numerator here gets
treated perturbatively,

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00:03:36,242 --> 00:03:38,450
and there's this term, and
then there's higher terms.

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00:03:38,450 --> 00:03:40,790
And that's how we could
organize this theory,

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00:03:40,790 --> 00:03:44,450
and this sums up all
these powers of a times p,

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00:03:44,450 --> 00:03:47,940
these denominators that
I'm showing you there.

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So what I want to do for
the first part of today

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is just really continue with
this effective field theory

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and talk about a few
more of its features.

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First I want to start
with symmetries,

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00:04:00,950 --> 00:04:03,560
and then we'll talk about
something called the deuteron.

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00:04:03,560 --> 00:04:04,306
Yeah?

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00:04:04,306 --> 00:04:07,197
AUDIENCE: I have a question
about the relevance

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00:04:07,197 --> 00:04:08,030
of the kinetic term.

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IAIN STEWART: Yeah.

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00:04:08,822 --> 00:04:11,180
AUDIENCE: So other than
getting pinch singularity

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00:04:11,180 --> 00:04:13,160
in your integrals,
is there, I don't

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know, a more power
counting-friendly kind

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00:04:15,443 --> 00:04:16,610
of argument for [INAUDIBLE]?

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00:04:16,610 --> 00:04:18,589
IAIN STEWART: Yeah, so you
can think about it like--

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00:04:18,589 --> 00:04:20,930
really, what we're doing here
is nonrelativistic quantum

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00:04:20,930 --> 00:04:22,670
mechanics.

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00:04:22,670 --> 00:04:25,352
And if you think about
nonrelativistic quantum

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00:04:25,352 --> 00:04:27,560
mechanics, you think about,
like, the virial theorem.

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00:04:27,560 --> 00:04:29,000
You're used to
thinking about the fact

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00:04:29,000 --> 00:04:30,740
that kinetic energy
and potential energy

101
00:04:30,740 --> 00:04:32,630
are equally relevant, right?

102
00:04:32,630 --> 00:04:34,340
So from a physics
standpoint, if I

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00:04:34,340 --> 00:04:36,530
didn't want to
talk about diagrams

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00:04:36,530 --> 00:04:39,860
or I just wanted to make a
physics argument for why that

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00:04:39,860 --> 00:04:43,005
would be the right thing to
do, I could simply tell you,

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00:04:43,005 --> 00:04:45,380
I want nonrelativistic quantum
mechanics at lowest order.

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00:04:45,380 --> 00:04:47,720
This is a system with
two heavy particles,

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00:04:47,720 --> 00:04:51,230
so that's which the
physics should be.

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00:04:51,230 --> 00:04:52,827
If you start with
the wrong theory,

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00:04:52,827 --> 00:04:54,410
then you run into
problems, and that's

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00:04:54,410 --> 00:04:56,360
kind of what we
saw if we started

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00:04:56,360 --> 00:04:58,640
with an HQET-like reasoning.

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00:04:58,640 --> 00:05:02,330
Then we see this
pinch singularity,

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00:05:02,330 --> 00:05:05,303
which is resolved
by the kinetic term.

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00:05:05,303 --> 00:05:07,720
AUDIENCE: OK, then, why is it
that nonrelativistic quantum

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00:05:07,720 --> 00:05:11,783
mechanics has the form
that it does, then?

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00:05:11,783 --> 00:05:13,700
IAIN STEWART: Why does
nonrelativistic quantum

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00:05:13,700 --> 00:05:15,242
mechanics have the
form that it does?

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00:05:15,242 --> 00:05:15,780
[CHUCKLES]

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00:05:15,780 --> 00:05:16,863
AUDIENCE: I know, I just--

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00:05:16,863 --> 00:05:18,440
IAIN STEWART: Yeah,
so think about it

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00:05:18,440 --> 00:05:19,815
from the following
point of view.

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00:05:19,815 --> 00:05:25,940
So when we just had this term,
what was the particle doing?

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00:05:25,940 --> 00:05:29,030
When we just had the partial
t term, it's like a source.

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00:05:29,030 --> 00:05:31,340
We had a single
source sitting there,

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00:05:31,340 --> 00:05:34,070
and we were interacting with
it with some light stuff.

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00:05:34,070 --> 00:05:35,780
That was HQET.

128
00:05:35,780 --> 00:05:38,240
Here we have two
sources, and they have

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00:05:38,240 --> 00:05:39,500
to interact with each other.

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00:05:39,500 --> 00:05:41,750
And the problem is
that two sources

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00:05:41,750 --> 00:05:43,700
interacting with
each other, really,

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00:05:43,700 --> 00:05:47,360
the kinetic energy
becomes a relevant thing.

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00:05:47,360 --> 00:05:51,950
It's not-- I mean, it's actually
a little more subtle than that,

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00:05:51,950 --> 00:05:54,800
if you really want to go
into it in a gauge theory.

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00:05:54,800 --> 00:05:57,650
There's sort of situations where
these two heavy sources act

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00:05:57,650 --> 00:05:59,480
like two heavy
sources, like HQET,

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00:05:59,480 --> 00:06:01,490
and there's situations
where they wiggle.

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00:06:01,490 --> 00:06:02,990
And the situations
where they wiggle

139
00:06:02,990 --> 00:06:05,157
are exactly the things that
the Schrodinger equation

140
00:06:05,157 --> 00:06:05,870
is giving you.

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00:06:08,450 --> 00:06:12,440
If you like-- yeah, so
to say it precisely,

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00:06:12,440 --> 00:06:15,260
the potential between
the two heavy particles,

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00:06:15,260 --> 00:06:18,020
for the purposes of
figuring out that potential,

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00:06:18,020 --> 00:06:20,570
you can treat them
as static, as fixed.

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00:06:20,570 --> 00:06:23,210
But then, once you
have that potential,

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00:06:23,210 --> 00:06:25,790
these bubble diagrams
are kind of the way

147
00:06:25,790 --> 00:06:29,420
in which that potential
gets iterated into the thing

148
00:06:29,420 --> 00:06:31,950
that you would get from solving
the Schrodinger equation.

149
00:06:31,950 --> 00:06:33,862
So solving the
Schrodinger equation,

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00:06:33,862 --> 00:06:35,570
you're effectively
treating the potential

151
00:06:35,570 --> 00:06:39,410
to all orders in this series,
exactly this bubble series.

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00:06:39,410 --> 00:06:42,320
And when you solve that you
need to include the kinetic,

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00:06:42,320 --> 00:06:44,570
this grad squared over
2m, for the same reason

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00:06:44,570 --> 00:06:49,250
that it's a relevant thing to
include in quantum mechanics.

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00:06:49,250 --> 00:06:52,280
OK, good.

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00:06:52,280 --> 00:06:55,400
Good project choice.

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I urge you-- no.

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00:06:56,545 --> 00:06:58,797
[CHUCKLES]

159
00:06:58,797 --> 00:07:01,380
OK, so let's talk about some of
the symmetries of this theory,

160
00:07:01,380 --> 00:07:02,883
which are kind of interesting.

161
00:07:10,290 --> 00:07:12,380
So the first one is that
this theory actually

162
00:07:12,380 --> 00:07:15,696
has a nonrelativistic
conformal invariance.

163
00:07:31,590 --> 00:07:34,005
So what is a nonrelativistic
conformal invariance?

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00:07:34,005 --> 00:07:39,720
So this is an extension
of the Galilean group,

165
00:07:39,720 --> 00:07:41,160
rather than the Poincaré group.

166
00:07:48,180 --> 00:07:52,140
So we have the
usual translations.

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00:07:52,140 --> 00:07:53,640
We have rotations.

168
00:07:56,460 --> 00:07:57,960
We don't have Lorentz
boosts, but we

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00:07:57,960 --> 00:08:01,740
do Galilean boosts for a
nonrelativistic system,

170
00:08:01,740 --> 00:08:04,180
like this.

171
00:08:04,180 --> 00:08:05,310
So what would those be?

172
00:08:10,200 --> 00:08:12,900
So under a Galilean
boost, time is not changed

173
00:08:12,900 --> 00:08:16,777
and the spatial
coordinates are changed.

174
00:08:16,777 --> 00:08:18,360
And then there's
additional generators

175
00:08:18,360 --> 00:08:22,770
that you have in a
conformal invariance.

176
00:08:22,770 --> 00:08:28,800
One is a scale
transformation, one generator.

177
00:08:28,800 --> 00:08:31,110
And the way that the
scale transformation works

178
00:08:31,110 --> 00:08:32,799
is as follows.

179
00:08:32,799 --> 00:08:35,250
You rescale the coordinates
by some-- let's call

180
00:08:35,250 --> 00:08:37,230
it e to the s.

181
00:08:37,230 --> 00:08:43,945
But you have to rescale time
differently, by e to the 2s.

182
00:08:43,945 --> 00:08:45,570
And the reason that
you have to do that

183
00:08:45,570 --> 00:08:48,000
is exactly because
these two terms

184
00:08:48,000 --> 00:08:50,410
should be treated the same.

185
00:08:50,410 --> 00:08:52,920
So you have to rescale
time twice as much

186
00:08:52,920 --> 00:08:55,890
as you rescale the spatial
derivatives, if you like,

187
00:08:55,890 --> 00:08:59,130
so that's what gives
rise to the 2 there.

188
00:08:59,130 --> 00:09:01,350
And that just goes to the
heart of sort of counting

189
00:09:01,350 --> 00:09:06,340
time and space differently.

190
00:09:06,340 --> 00:09:11,670
And then, finally, there
is a conformal generator.

191
00:09:11,670 --> 00:09:13,560
And there's actually
only one generator

192
00:09:13,560 --> 00:09:15,120
in this nonrelativistic case.

193
00:09:27,520 --> 00:09:29,670
And basically what this
generator corresponds to

194
00:09:29,670 --> 00:09:36,050
is a shift of inverse time and
then a corresponding change

195
00:09:36,050 --> 00:09:38,330
to the coordinates.

196
00:09:38,330 --> 00:09:40,430
And if you take all
these things together,

197
00:09:40,430 --> 00:09:42,638
it's actually something
called the Schrodinger group.

198
00:09:48,375 --> 00:09:50,000
And the theory we've
been talking about

199
00:09:50,000 --> 00:09:52,250
actually has this
nonrelativistic conformal

200
00:09:52,250 --> 00:09:59,100
symmetry, if a goes to infinity.

201
00:09:59,100 --> 00:10:02,180
So I'm not going to spend a
lot of time talking about this,

202
00:10:02,180 --> 00:10:07,770
but just to give you some of
the ideas of what is going on.

203
00:10:07,770 --> 00:10:11,090
If you look back last time at
our solutions for this coupling

204
00:10:11,090 --> 00:10:13,770
constant, and you take
a goes to infinity,

205
00:10:13,770 --> 00:10:15,240
that's a perfectly smooth limit.

206
00:10:15,240 --> 00:10:16,400
You can do that.

207
00:10:16,400 --> 00:10:19,280
And then the coupling, c 0 of
mu, has no free parameters.

208
00:10:19,280 --> 00:10:22,430
It's just given
by 4 pi over m mu.

209
00:10:22,430 --> 00:10:24,440
And that corresponds
to sitting at this--

210
00:10:24,440 --> 00:10:26,300
what I call-- what
we last time saw

211
00:10:26,300 --> 00:10:28,490
was a fixed point of
the beta function.

212
00:10:28,490 --> 00:10:31,040
And at that fixed point,
there's enhanced symmetry, which

213
00:10:31,040 --> 00:10:32,180
is this conformal symmetry.

214
00:10:35,690 --> 00:10:40,940
And basically what you can
show is that, if you scale here

215
00:10:40,940 --> 00:10:46,380
mu just like it was a momentum--

216
00:10:46,380 --> 00:10:49,555
so mu goes to e to the s mu--

217
00:10:49,555 --> 00:10:52,570
then the Lagrangian
is scale-invariant.

218
00:10:52,570 --> 00:10:55,360
My notation up there
would be minus s.

219
00:11:03,660 --> 00:11:04,340
OK?

220
00:11:04,340 --> 00:11:07,610
And that is basically kind of
the same thing as making it

221
00:11:07,610 --> 00:11:09,200
into a relevant coupling.

222
00:11:09,200 --> 00:11:11,630
Once it's got no
free parameters,

223
00:11:11,630 --> 00:11:15,620
then you can sort
of see that, if I

224
00:11:15,620 --> 00:11:19,423
have a certain scaling
that makes the free theory

225
00:11:19,423 --> 00:11:21,590
Schrodinger-invariant, which
is-- so the Schrodinger

226
00:11:21,590 --> 00:11:25,850
group is the symmetry group of
the free Schrodinger equation,

227
00:11:25,850 --> 00:11:27,582
so just this term and this term.

228
00:11:27,582 --> 00:11:29,540
Then the question is,
what interactions can you

229
00:11:29,540 --> 00:11:32,540
add that would
preserve that symmetry?

230
00:11:32,540 --> 00:11:35,360
And I'm just telling you that,
in terms of just the scale

231
00:11:35,360 --> 00:11:38,158
transformation,
you could-- if you

232
00:11:38,158 --> 00:11:39,950
are able to scale the
coupling in some way,

233
00:11:39,950 --> 00:11:43,370
in particular to make it a
relevant interaction, then that

234
00:11:43,370 --> 00:11:45,650
will make it a
scale-invariant theory.

235
00:11:45,650 --> 00:11:47,750
And so the ability to
scale this coupling

236
00:11:47,750 --> 00:11:50,810
because you scale
this mu allows it

237
00:11:50,810 --> 00:11:54,570
to be a scale-invariant theory.

238
00:11:54,570 --> 00:11:56,870
It's a little harder to
see that it's conformal,

239
00:11:56,870 --> 00:11:58,810
but you can work that out, too.

240
00:12:09,170 --> 00:12:11,520
Scale-invariant theories tend
to be conformal-invariant,

241
00:12:11,520 --> 00:12:14,350
and the same is true of this
nonrelativistic conformal

242
00:12:14,350 --> 00:12:15,490
invariance.

243
00:12:21,990 --> 00:12:25,430
So if you sort of
add up the bubbles

244
00:12:25,430 --> 00:12:28,850
that we were talking
about, then you

245
00:12:28,850 --> 00:12:35,330
find in general in amplitude,
which is this thing that

246
00:12:35,330 --> 00:12:38,390
has a square root--

247
00:12:38,390 --> 00:12:40,310
but if I'm not
particular to what

248
00:12:40,310 --> 00:12:43,310
frame I'm in now, if I just
write it for a general frame--

249
00:12:43,310 --> 00:12:45,310
before we write it for
the center of mass frame,

250
00:12:45,310 --> 00:12:47,210
now let me write it
for a general frame.

251
00:12:53,432 --> 00:12:54,640
Then it would look like this.

252
00:13:00,250 --> 00:13:03,790
And that thing
goes like 1 over p,

253
00:13:03,790 --> 00:13:05,980
so it has-- and again,
e's scale differently

254
00:13:05,980 --> 00:13:10,300
than p's, so this thing
has the right scaling

255
00:13:10,300 --> 00:13:11,800
for a scale-invariant amplitude.

256
00:13:17,110 --> 00:13:19,900
And it turns out,
if you do work out

257
00:13:19,900 --> 00:13:23,818
the conformal transformation,
it's also conformal-invariant.

258
00:13:23,818 --> 00:13:25,360
And I'm not going
to go through that,

259
00:13:25,360 --> 00:13:29,080
but I've posted a reference
to show you how it works.

260
00:13:29,080 --> 00:13:33,340
And basically what this leads
to is a cross-section, which

261
00:13:33,340 --> 00:13:36,820
is 4 pi over p squared, which
is the scale-invariant version

262
00:13:36,820 --> 00:13:39,850
of a cross-section, where the
cross-section units are just

263
00:13:39,850 --> 00:13:44,370
set by p, so there's no
other scale in the problem.

264
00:13:44,370 --> 00:13:44,870
OK?

265
00:13:44,870 --> 00:13:46,390
So that's kind of
interesting, and one can make

266
00:13:46,390 --> 00:13:47,515
some predictions with that.

267
00:13:58,945 --> 00:14:00,320
This is actually
something that's

268
00:14:00,320 --> 00:14:05,630
also investigated these days,
both from a string theory

269
00:14:05,630 --> 00:14:07,940
point of view as well as
in cold atoms, so that's

270
00:14:07,940 --> 00:14:12,450
why conformal symmetries
are always fun to play with.

271
00:14:19,412 --> 00:14:21,870
There's also another set of
symmetries that this theory has

272
00:14:21,870 --> 00:14:25,431
in this limit, which
is also interesting.

273
00:14:28,257 --> 00:14:31,655
And that is, like HQET, there's
an enhanced spin symmetry.

274
00:14:36,120 --> 00:14:38,150
This time, the
enhanced spin symmetry

275
00:14:38,150 --> 00:14:40,370
comes about by combining
spin and isospin.

276
00:14:44,220 --> 00:14:49,780
And this is a symmetry that's
known as Wigner's SU(4)

277
00:14:49,780 --> 00:14:50,280
symmetry.

278
00:14:54,560 --> 00:14:58,490
So Wigner-- I don't know
how many years ago--

279
00:14:58,490 --> 00:15:01,040
noticed-- long, long
ago, Wigner noticed

280
00:15:01,040 --> 00:15:04,040
that nuclei tend to
have a symmetry, which

281
00:15:04,040 --> 00:15:08,270
is a combination of spin and
isospin, into a full SU(4).

282
00:15:08,270 --> 00:15:09,980
And we can understand
actually partly

283
00:15:09,980 --> 00:15:13,640
where that comes from this
effective field theory.

284
00:15:13,640 --> 00:15:16,310
So what is this SU(4)?

285
00:15:16,310 --> 00:15:20,660
Think of it as just
transforming the nucleon field

286
00:15:20,660 --> 00:15:23,580
under a way where you could mix
up the spin and the isospin.

287
00:15:23,580 --> 00:15:28,640
So if sigma is for the spin,
tau is for the isospin,

288
00:15:28,640 --> 00:15:30,117
then some combined
transformation

289
00:15:30,117 --> 00:15:31,700
where I can mix up
the mu and nus-- we

290
00:15:31,700 --> 00:15:33,283
just have some set
of generators where

291
00:15:33,283 --> 00:15:36,560
I label them by mu and nu.

292
00:15:36,560 --> 00:15:43,850
Sigma mu here is 1, comma,
sigma vector, tau nu, 1, comma,

293
00:15:43,850 --> 00:15:45,230
tau vector for the isospin.

294
00:15:53,830 --> 00:15:56,590
If you just take the 1 from both
of them, that's baryon number,

295
00:15:56,590 --> 00:16:00,040
and that would turn
the SU(4) into a U(4),

296
00:16:00,040 --> 00:16:04,150
so you can remove that
generator if you like.

297
00:16:04,150 --> 00:16:08,050
So there's a U(1), as well,
but the nontrivial part,

298
00:16:08,050 --> 00:16:11,500
where you take either this
or that or both is an SU(4).

299
00:16:14,590 --> 00:16:17,380
So we can think about this
symmetry with our theory

300
00:16:17,380 --> 00:16:18,920
if we write it as follows.

301
00:16:18,920 --> 00:16:23,380
So take the interaction terms
that we've been talking about

302
00:16:23,380 --> 00:16:25,255
and write them in a
slightly different basis.

303
00:16:27,850 --> 00:16:31,030
Just take these
two-to-two scattering

304
00:16:31,030 --> 00:16:35,290
terms, which had no derivatives.

305
00:16:35,290 --> 00:16:38,290
And I'll write them in a
slightly different basis.

306
00:16:38,290 --> 00:16:40,870
Before, we wrote them in
sort of the physical basis,

307
00:16:40,870 --> 00:16:43,480
where we had 3S1 and 1S0.

308
00:16:43,480 --> 00:16:45,465
That's what we did last time.

309
00:16:45,465 --> 00:16:46,840
But there's other
possible bases,

310
00:16:46,840 --> 00:16:51,850
and I can write them in this
basis, singlet triplet basis.

311
00:16:51,850 --> 00:16:54,130
And the advantage of this
basis is this thing here

312
00:16:54,130 --> 00:16:55,630
is just SU(4) symmetric.

313
00:17:05,970 --> 00:17:09,839
And if you want to write down in
relation to the previous basis,

314
00:17:09,839 --> 00:17:12,650
you can work that out.

315
00:17:12,650 --> 00:17:16,265
I'll just tell you what it
would be by writing down

316
00:17:16,265 --> 00:17:18,390
what the coefficients here
are in terms of the ones

317
00:17:18,390 --> 00:17:18,973
we had before.

318
00:17:44,692 --> 00:17:46,150
So if we go back
to the basis where

319
00:17:46,150 --> 00:17:49,600
we had coefficients for the 3S1
channel and the 1S0 channel,

320
00:17:49,600 --> 00:17:52,330
then this c 0 s becomes
this combination,

321
00:17:52,330 --> 00:17:54,220
and c 0 t is this combination.

322
00:17:56,890 --> 00:18:00,070
Now, in nature-- so one way
you could have SU(4) symmetry

323
00:18:00,070 --> 00:18:02,150
would just be for the
scattering lengths,

324
00:18:02,150 --> 00:18:07,990
the physical parameter, c 0
1S0 and c 0 3S1, to be equal.

325
00:18:07,990 --> 00:18:09,970
But in nature, they're
not equal at all.

326
00:18:09,970 --> 00:18:12,290
In fact, their scattering
lengths have different signs.

327
00:18:12,290 --> 00:18:15,860
There's no sense in
which they're equal.

328
00:18:15,860 --> 00:18:19,043
So what is the sense in
which we have this symmetry?

329
00:18:19,043 --> 00:18:20,710
The sense in which
we have this symmetry

330
00:18:20,710 --> 00:18:23,140
is the fact that
if the scattering

331
00:18:23,140 --> 00:18:28,280
lengths are both infinity, then
both couplings are just this.

332
00:18:28,280 --> 00:18:31,600
So the fact that the scattering
lengths have different signs

333
00:18:31,600 --> 00:18:34,643
but are both large makes them
close to this fixed point,

334
00:18:34,643 --> 00:18:35,935
which then gives this symmetry.

335
00:18:47,920 --> 00:18:56,970
So because of what I just said
for a goes to infinity, c 0 s,

336
00:18:56,970 --> 00:18:58,470
which is sort of equal--

337
00:18:58,470 --> 00:19:02,490
some symmetric combination of
these two that's become equal--

338
00:19:02,490 --> 00:19:09,240
would just become 4 pi over
m mu, and this is for a 1S0

339
00:19:09,240 --> 00:19:13,910
and a 3S1, both
going to infinity.

340
00:19:13,910 --> 00:19:18,120
This goes to minus infinity.

341
00:19:18,120 --> 00:19:19,620
And then the triplet
one, if we just

342
00:19:19,620 --> 00:19:21,330
work out what it is
from our formulas

343
00:19:21,330 --> 00:19:24,857
for those other couplings--

344
00:20:00,150 --> 00:20:02,370
We can just work
out the combination

345
00:20:02,370 --> 00:20:04,770
from formulas we had
last time, and we

346
00:20:04,770 --> 00:20:06,492
see that as the
a's go to infinity,

347
00:20:06,492 --> 00:20:07,450
this coupling vanishes.

348
00:20:16,100 --> 00:20:16,700
OK?

349
00:20:16,700 --> 00:20:19,370
So when we're in the limit
where both of the couplings

350
00:20:19,370 --> 00:20:21,330
become equal, or if we
use this basis, where

351
00:20:21,330 --> 00:20:24,410
this coupling becomes 0,
we just have this N dagger

352
00:20:24,410 --> 00:20:25,600
N squared operator.

353
00:20:25,600 --> 00:20:26,975
Then we have this
SU(4) symmetry.

354
00:20:33,000 --> 00:20:35,370
One of the problems on
this new problem set

355
00:20:35,370 --> 00:20:39,060
is to think about to
what degree you still

356
00:20:39,060 --> 00:20:41,460
have that symmetry when
you think about higher body

357
00:20:41,460 --> 00:20:43,968
operators, three-body
and four-body.

358
00:20:43,968 --> 00:20:46,260
And there's actually some
very nice little group theory

359
00:20:46,260 --> 00:20:49,260
arguments where you can
basically figure out

360
00:20:49,260 --> 00:20:52,350
to what degree you
have that symmetry

361
00:20:52,350 --> 00:20:56,752
in the higher-body
operators, and it turns out

362
00:20:56,752 --> 00:20:57,960
that it's also present there.

363
00:21:05,310 --> 00:21:11,010
And that is one way of thinking
about why Wigner's symmetries

364
00:21:11,010 --> 00:21:11,760
exist in nature.

365
00:21:11,760 --> 00:21:15,990
It's not that we're in any sense
having scattering lengths that

366
00:21:15,990 --> 00:21:17,670
are similar in
these two channels,

367
00:21:17,670 --> 00:21:21,240
but since they're both large
we're close to the same--

368
00:21:21,240 --> 00:21:23,970
we're close to
these fixed points,

369
00:21:23,970 --> 00:21:25,960
which have this symmetry.

370
00:21:25,960 --> 00:21:26,460
OK?

371
00:21:26,460 --> 00:21:28,230
So it's like HQET,
in a sense, where

372
00:21:28,230 --> 00:21:31,080
we didn't have heavy
quark spin symmetry

373
00:21:31,080 --> 00:21:33,735
when the masses are finite,
but if the charm mass and the B

374
00:21:33,735 --> 00:21:35,970
mass are both
infinite, then there's

375
00:21:35,970 --> 00:21:37,772
this new symmetry that pops up.

376
00:21:37,772 --> 00:21:38,730
It's kind of like that.

377
00:22:09,350 --> 00:22:12,170
All right, so rather than
talk about the symmetries

378
00:22:12,170 --> 00:22:14,838
in any more detail than that,
what I wanted to do is actually

379
00:22:14,838 --> 00:22:16,880
do something a little
different with this theory.

380
00:22:16,880 --> 00:22:19,880
So there's lots of things
we could discuss here.

381
00:22:19,880 --> 00:22:22,370
But the thing I want
to discuss is actually

382
00:22:22,370 --> 00:22:26,780
treating a bound state
in a field theory.

383
00:22:26,780 --> 00:22:29,120
This field theory has
a very nice property

384
00:22:29,120 --> 00:22:31,110
that we can
calculate everything.

385
00:22:31,110 --> 00:22:32,630
So we can actually
see how you're

386
00:22:32,630 --> 00:22:36,170
supposed to treat bound states
exactly because we can actually

387
00:22:36,170 --> 00:22:38,270
just look for the pole
that corresponds to that

388
00:22:38,270 --> 00:22:39,660
bound state in this theory.

389
00:22:39,660 --> 00:22:40,910
That's what we're going to do.

390
00:22:43,868 --> 00:22:45,410
So usually if you
have a bound state,

391
00:22:45,410 --> 00:22:49,460
it's either something
that's nonperturbative

392
00:22:49,460 --> 00:22:51,770
that you have trouble
dealing with analytically,

393
00:22:51,770 --> 00:22:55,820
like in QCD, or Coulombic--
even then, the calculations get

394
00:22:55,820 --> 00:22:59,130
kind of hairy, although
you can deal with it there.

395
00:22:59,130 --> 00:23:01,310
But here, the calculations
are very simple,

396
00:23:01,310 --> 00:23:03,980
so we can really follow all
the steps of properly dealing

397
00:23:03,980 --> 00:23:06,853
with a bound state.

398
00:23:06,853 --> 00:23:09,020
And the bound state has
something to do with nature.

399
00:23:09,020 --> 00:23:10,812
It's a state of nature
called the deuteron.

400
00:23:14,362 --> 00:23:16,820
So the deuteron is a bound
state of a neutron and a proton.

401
00:23:22,850 --> 00:23:28,880
It has isospin 0, spin 1,
so it's in this 3S1 state

402
00:23:28,880 --> 00:23:31,810
that we talked about.

403
00:23:31,810 --> 00:23:34,570
And when you want to look for
a bound state in field theory,

404
00:23:34,570 --> 00:23:36,303
what you do--

405
00:23:36,303 --> 00:23:38,720
you may have heard of this
from the lattice point of view.

406
00:23:38,720 --> 00:23:41,210
What you do is you write down
some interpolating field that

407
00:23:41,210 --> 00:23:46,110
has overlap with that state.

408
00:23:46,110 --> 00:23:48,950
So what would that be for us?

409
00:23:48,950 --> 00:23:51,860
That would be this
operator that we

410
00:23:51,860 --> 00:23:56,990
called p i 3S1, which had the
quantum numbers for i equals 0

411
00:23:56,990 --> 00:23:59,490
and s equals 1.

412
00:23:59,490 --> 00:24:01,620
And we can build an
interpolating field out

413
00:24:01,620 --> 00:24:04,230
of our N transpose N.

414
00:24:04,230 --> 00:24:07,140
And actually, any interpolating
field that we write down here

415
00:24:07,140 --> 00:24:08,820
is equally good.

416
00:24:08,820 --> 00:24:10,950
The lattice QCD
people like to tune

417
00:24:10,950 --> 00:24:13,650
what they write down
in order to improve

418
00:24:13,650 --> 00:24:15,460
the overlap with the state.

419
00:24:15,460 --> 00:24:17,520
But for our purposes the
simplest possible thing

420
00:24:17,520 --> 00:24:19,463
will be enough.

421
00:24:19,463 --> 00:24:20,880
And then you can
ask the question,

422
00:24:20,880 --> 00:24:22,380
is the deuteron in the theory?

423
00:24:31,322 --> 00:24:33,030
And to answer that
question, you can just

424
00:24:33,030 --> 00:24:36,270
look for the pole that would
correspond to that state.

425
00:24:36,270 --> 00:24:37,890
And the advantage
of this theory is

426
00:24:37,890 --> 00:24:42,580
that we can just look for
that pole analytically.

427
00:24:42,580 --> 00:24:43,480
So let's do that.

428
00:24:47,080 --> 00:24:50,080
So I'll write down
a Green's function

429
00:24:50,080 --> 00:24:57,730
for the two-particle
state, which

430
00:24:57,730 --> 00:25:00,410
is just the time-order product
of two of these interpolating

431
00:25:00,410 --> 00:25:00,910
fields.

432
00:25:12,000 --> 00:25:14,100
And then ask the
question, if I look

433
00:25:14,100 --> 00:25:15,810
at that time-order
product, is there

434
00:25:15,810 --> 00:25:19,020
a pole that would correspond
to this state, i.e.,

435
00:25:19,020 --> 00:25:25,080
somewhere is there a pole
which would behave as follows?

436
00:25:25,080 --> 00:25:36,150
Some kind of z factor,
and then, in this case,

437
00:25:36,150 --> 00:25:38,280
because of the way
we've scaled energies,

438
00:25:38,280 --> 00:25:40,350
we would just be looking
for a pole that's

439
00:25:40,350 --> 00:25:42,030
when you're in the energy--

440
00:25:42,030 --> 00:25:44,400
at some negative energy
corresponding to the binding

441
00:25:44,400 --> 00:25:46,450
energy of that state.

442
00:25:46,450 --> 00:25:49,110
So remember that we pulled
out the mass of the nucleon.

443
00:25:49,110 --> 00:25:50,640
That wasn't in our Lagrangian.

444
00:25:50,640 --> 00:25:55,470
So really, if you
think about what

445
00:25:55,470 --> 00:25:58,170
the sort of energy of this
state is, it's two nucleons,

446
00:25:58,170 --> 00:26:00,413
so it's got the energy
of those two nucleons,

447
00:26:00,413 --> 00:26:01,830
but then there's
a little bit less

448
00:26:01,830 --> 00:26:03,540
because it's a bound state.

449
00:26:03,540 --> 00:26:07,050
And this is the amount
by how much less.

450
00:26:07,050 --> 00:26:09,870
And so the energy for
the pole is negative.

451
00:26:09,870 --> 00:26:13,425
E bar is less than 0 because
it's much less than the two

452
00:26:13,425 --> 00:26:14,400
nucleon mass.

453
00:26:17,070 --> 00:26:22,150
So this E bar is the
two nuclear center

454
00:26:22,150 --> 00:26:30,090
of mass energy, which
in the conventions

455
00:26:30,090 --> 00:26:31,170
we were using last time--

456
00:26:34,020 --> 00:26:36,990
we had energy E over 2
for each of the nucleons,

457
00:26:36,990 --> 00:26:40,370
so the sum would be E. There's
the kinetic term, as well.

458
00:26:43,380 --> 00:26:47,250
So what are the diagrams that
we would consider to do this?

459
00:26:47,250 --> 00:26:51,020
Well, at lowest order it's just
our friends the bubble sum.

460
00:26:51,020 --> 00:26:53,210
And in fact, when you
go to higher orders,

461
00:26:53,210 --> 00:26:55,190
we also had bubbles there.

462
00:26:55,190 --> 00:26:58,820
So to answer this question,
we insert the field.

463
00:26:58,820 --> 00:27:00,770
We create two nucleons.

464
00:27:00,770 --> 00:27:04,100
And then they can interact
with c 0's, and then we

465
00:27:04,100 --> 00:27:07,890
annihilate them with
our interpolating field.

466
00:27:07,890 --> 00:27:10,010
So these are just
our c 0 bubbles.

467
00:27:15,130 --> 00:27:15,630
OK?

468
00:27:15,630 --> 00:27:17,260
And those are very
easy to compute.

469
00:27:20,655 --> 00:27:24,380
It's just the geometric series
that we already computed.

470
00:27:24,380 --> 00:27:27,800
And instead of having two
nucleons in and two nucleons

471
00:27:27,800 --> 00:27:29,525
out, we just have
this deuteron state in

472
00:27:29,525 --> 00:27:30,590
and deuteron state out.

473
00:27:40,820 --> 00:27:41,320
OK?

474
00:27:41,320 --> 00:27:46,060
So the Green's function
starts with one of these

475
00:27:46,060 --> 00:27:53,470
and then just iterates into
a geometric series, where

476
00:27:53,470 --> 00:27:57,725
sigma I could define as
two-particle irreducible c

477
00:27:57,725 --> 00:28:03,080
0 graphs, so sort of taking
out the c 0 iterations, so

478
00:28:03,080 --> 00:28:06,513
the sigma 1 that I'm
talking about here

479
00:28:06,513 --> 00:28:07,930
is just this bubble
all by itself.

480
00:28:10,498 --> 00:28:12,040
And that's something
that we computed

481
00:28:12,040 --> 00:28:19,040
before and in these
renormalization schemes

482
00:28:19,040 --> 00:28:19,670
that we like.

483
00:28:23,840 --> 00:28:25,040
Just comes out to be this.

484
00:28:27,700 --> 00:28:30,630
So there's an m out
front, m nucleon.

485
00:28:30,630 --> 00:28:33,850
It's not-- that's
an m, not a mu.

486
00:28:33,850 --> 00:28:36,530
And then in terms of this E bar,
it just comes out to be that.

487
00:28:40,670 --> 00:28:42,890
So you could think of
making a positive energy

488
00:28:42,890 --> 00:28:49,190
by taking E bar to minus E
bound, which is less than 0.

489
00:28:49,190 --> 00:28:55,520
And then the square root
would be square root of m E B.

490
00:28:55,520 --> 00:28:58,940
And it's convenient to
define that to be something,

491
00:28:58,940 --> 00:29:02,120
so I'll call it
gamma B. So that's

492
00:29:02,120 --> 00:29:04,460
some quantity greater than 0.

493
00:29:04,460 --> 00:29:07,690
And with this little
notation, this square root

494
00:29:07,690 --> 00:29:17,360
of minus E bar, minus i
0 is equal to minus i p.

495
00:29:17,360 --> 00:29:22,940
And we're looking
for whether that

496
00:29:22,940 --> 00:29:28,820
would be sort of going over
to this gamma B. Sorry,

497
00:29:28,820 --> 00:29:33,480
so this is equal to gamma B.

498
00:29:33,480 --> 00:29:35,400
So gamma B is just
like a momentum

499
00:29:35,400 --> 00:29:39,440
version of the bound state
energy absorbing the extra i.

500
00:29:43,770 --> 00:29:44,270
OK?

501
00:29:44,270 --> 00:29:46,062
This is what it would
have been previously,

502
00:29:46,062 --> 00:29:51,018
but now it's gamma B.
So we know what G was.

503
00:29:51,018 --> 00:29:52,310
We already computed it earlier.

504
00:29:52,310 --> 00:29:59,030
G was just proportional
to 1 over a plus i p.

505
00:29:59,030 --> 00:30:01,550
And now if I just make
that replacement, which

506
00:30:01,550 --> 00:30:04,490
is why I wanted to derive that,
because I wanted to just see

507
00:30:04,490 --> 00:30:08,720
how I would replace the i p, it
becomes 1 over a minus gamma B.

508
00:30:08,720 --> 00:30:10,830
So gamma B is a
kinematic variable.

509
00:30:10,830 --> 00:30:12,488
It's like a momentum.

510
00:30:12,488 --> 00:30:14,030
But it's the right
kinematic variable

511
00:30:14,030 --> 00:30:16,020
to be talking about
for the bound state.

512
00:30:16,020 --> 00:30:17,898
It's a positive thing.

513
00:30:17,898 --> 00:30:19,190
And you can see there's a pole.

514
00:30:32,300 --> 00:30:32,840
OK?

515
00:30:32,840 --> 00:30:33,845
So there's the
pole corresponding

516
00:30:33,845 --> 00:30:35,000
to the deuteron state.

517
00:30:35,000 --> 00:30:40,910
It just corresponds to gamma
equals 1 over a of the 3S1,

518
00:30:40,910 --> 00:30:49,970
which in nature is like 36
m e V is greater than 0.

519
00:30:49,970 --> 00:30:52,520
If you looked at the other
channel, the 1S0 channel,

520
00:30:52,520 --> 00:30:55,110
then the scattering
linked with negative,

521
00:30:55,110 --> 00:30:57,830
so then both terms would be
negative, and there's no pole.

522
00:31:08,130 --> 00:31:08,630
OK?

523
00:31:08,630 --> 00:31:10,280
So for the 3S1 channel,
we have a pole.

524
00:31:10,280 --> 00:31:11,738
That's the physical
deuteron state.

525
00:31:11,738 --> 00:31:15,480
For the other channel,
there is no pole.

526
00:31:15,480 --> 00:31:17,910
And the binding energy that
you would get from this

527
00:31:17,910 --> 00:31:23,520
is gamma B squared over
m, our conventions,

528
00:31:23,520 --> 00:31:28,830
which, if you calculate
it, is 1.4 m e V,

529
00:31:28,830 --> 00:31:30,480
and that's not
ridiculously different

530
00:31:30,480 --> 00:31:34,350
than the real binding energy
of the deuteron, which was 2.2

531
00:31:34,350 --> 00:31:36,600
m e V. OK?

532
00:31:36,600 --> 00:31:41,570
It's not-- it's the
right order of magnitude.

533
00:31:41,570 --> 00:31:43,010
All right, so
that's the deuteron.

534
00:31:43,010 --> 00:31:45,218
It exists in this theory,
and we could perturbatively

535
00:31:45,218 --> 00:31:48,600
correct towards this
correct binding energy.

536
00:31:51,150 --> 00:31:51,650
OK?

537
00:31:51,650 --> 00:31:54,320
So this deuteron really
exists in the field theory,

538
00:31:54,320 --> 00:31:57,260
and that means we can calculate
properties of the deuteron

539
00:31:57,260 --> 00:31:59,630
with this field theory.

540
00:31:59,630 --> 00:32:01,520
We can calculate
scattering processes that

541
00:32:01,520 --> 00:32:03,020
involve this bound
state, and it's

542
00:32:03,020 --> 00:32:05,870
no more difficult than
doing calculations

543
00:32:05,870 --> 00:32:08,087
that we would normally
do with free fields.

544
00:32:11,670 --> 00:32:12,720
And we just have to--

545
00:32:12,720 --> 00:32:14,730
if we want to do something
with the deuteron,

546
00:32:14,730 --> 00:32:16,380
we have to go to the right pole.

547
00:32:16,380 --> 00:32:20,803
That's basically what
it boils down to.

548
00:32:20,803 --> 00:32:22,720
So in order to show you
really how to do that,

549
00:32:22,720 --> 00:32:25,570
I'm going to give one
example of calculating

550
00:32:25,570 --> 00:32:27,550
something nontrivial
that involves a deuteron.

551
00:32:38,450 --> 00:32:41,675
So we'll calculate a form
factor for the deuteron.

552
00:32:46,430 --> 00:32:49,380
So this is a nice
example, if you like,

553
00:32:49,380 --> 00:32:52,910
of really showing you how
to use the LSZ formula,

554
00:32:52,910 --> 00:32:54,650
which you learned
about it in a very sort

555
00:32:54,650 --> 00:33:00,380
of abstract way at some point,
how to use it for bound notes.

556
00:33:00,380 --> 00:33:03,650
Because here we have a bound
state exists in the theory,

557
00:33:03,650 --> 00:33:06,620
and if we want to derive
properties of that bound state,

558
00:33:06,620 --> 00:33:11,338
we'd better properly use the LSZ
formula for that bound state.

559
00:33:11,338 --> 00:33:12,380
Let's see how that works.

560
00:33:15,200 --> 00:33:18,260
OK, so what we'd
like to do, so we'd

561
00:33:18,260 --> 00:33:20,360
like to have-- we have
some deuteron state,

562
00:33:20,360 --> 00:33:22,405
and we'd like to take
the matrix element

563
00:33:22,405 --> 00:33:23,780
with the
electromagnetic current.

564
00:33:30,890 --> 00:33:34,130
And if you work out what
possible form factors you can

565
00:33:34,130 --> 00:33:37,460
have for this matrix element,
there's actually three.

566
00:33:37,460 --> 00:33:39,650
And I'm not going to
spend much time talking

567
00:33:39,650 --> 00:33:43,130
about the sort of
different couplings

568
00:33:43,130 --> 00:33:44,923
with spin and stuff
that correspond

569
00:33:44,923 --> 00:33:46,340
to these different
cases, but I'll

570
00:33:46,340 --> 00:33:49,028
list them for you,
at least, provide

571
00:33:49,028 --> 00:33:50,570
you some additional
references if you

572
00:33:50,570 --> 00:33:53,220
want to dig deeper into this.

573
00:33:53,220 --> 00:33:56,225
So there's a magnetic, electric,
and quadrupole form factor.

574
00:34:05,050 --> 00:34:07,420
The charge of the
deuteron tells us

575
00:34:07,420 --> 00:34:10,449
that this electric
form factor at 2

576
00:34:10,449 --> 00:34:12,699
squared equals 0 should be 1.

577
00:34:12,699 --> 00:34:15,610
The magnetic form factor
is like the magnetic form

578
00:34:15,610 --> 00:34:21,040
factor for an electron in the
sense that, if you go to q

579
00:34:21,040 --> 00:34:23,925
squared equals 0, then that's
giving you the magnetic moment.

580
00:34:23,925 --> 00:34:25,300
But here it's the
magnetic moment

581
00:34:25,300 --> 00:34:27,287
of the deuteron, not
the magnetic moment

582
00:34:27,287 --> 00:34:27,954
of the electron.

583
00:34:31,179 --> 00:34:33,889
That's a nontrivial
number in this case,

584
00:34:33,889 --> 00:34:39,690
whereas here it's just 1
because charge is conserved.

585
00:34:39,690 --> 00:34:43,020
And if we wanted to think
about putting electromagnetism

586
00:34:43,020 --> 00:34:46,222
into the nucleon theory,
which we haven't so far,

587
00:34:46,222 --> 00:34:47,639
then we would turn
our derivatives

588
00:34:47,639 --> 00:34:48,870
into covariant derivatives.

589
00:34:48,870 --> 00:34:51,480
So D mu nucleon--

590
00:34:54,270 --> 00:34:58,680
we have to write down a
charge matrix that couples--

591
00:34:58,680 --> 00:35:00,960
that takes into account
the proton has charge

592
00:35:00,960 --> 00:35:02,070
but the neutron doesn't.

593
00:35:06,660 --> 00:35:12,570
So this is that charge
matrix for electromagnetism.

594
00:35:12,570 --> 00:35:13,110
OK?

595
00:35:13,110 --> 00:35:17,400
So it's not difficult.
So that's easy enough.

596
00:35:17,400 --> 00:35:20,280
Just change your derivatives
into covariant derivatives

597
00:35:20,280 --> 00:35:25,200
for both D 0 and D
partial and D vector.

598
00:35:25,200 --> 00:35:27,150
There's also some
additional operators

599
00:35:27,150 --> 00:35:30,600
that you have to write down.

600
00:35:30,600 --> 00:35:34,990
So there's operators,
for example,

601
00:35:34,990 --> 00:35:37,260
that are just
contact interactions

602
00:35:37,260 --> 00:35:39,270
with the electric
magnetic fields

603
00:35:39,270 --> 00:35:47,880
in principle, and the first such
one that shows up, just by way

604
00:35:47,880 --> 00:35:51,548
of sort of telling you the
range of possibilities,

605
00:35:51,548 --> 00:35:54,090
would be something like this,
where you just have a sigma dot

606
00:35:54,090 --> 00:35:57,090
B magnetic field insertion.

607
00:35:57,090 --> 00:36:03,050
This is the magnetic field
for electromagnetism.

608
00:36:06,900 --> 00:36:09,300
And so if you want to think
about the type of Feynman

609
00:36:09,300 --> 00:36:12,630
diagram you'd draw for this,
you'd have four nucleons,

610
00:36:12,630 --> 00:36:16,760
and then you'd just have a
photon coming out, right?

611
00:36:16,760 --> 00:36:19,260
And if you want to think about
where that could possibly

612
00:36:19,260 --> 00:36:27,510
come from, there is one really
easy thing to think about,

613
00:36:27,510 --> 00:36:30,270
would just be to
have a pion exchange,

614
00:36:30,270 --> 00:36:31,470
and maybe it's pi minus.

615
00:36:34,410 --> 00:36:37,260
And the photon couples
to the pi minus

616
00:36:37,260 --> 00:36:39,630
because it's got
electric charge.

617
00:36:39,630 --> 00:36:42,000
And when I integrate
out the pion,

618
00:36:42,000 --> 00:36:44,160
then I just get something
that looks like this.

619
00:36:44,160 --> 00:36:45,540
It could be any exchange.

620
00:36:45,540 --> 00:36:48,150
It could be a real
minus, or anything

621
00:36:48,150 --> 00:36:50,464
would give an operator
that looks like that.

622
00:36:50,464 --> 00:36:51,240
OK?

623
00:36:51,240 --> 00:36:53,040
So there's additional
contact interactions

624
00:36:53,040 --> 00:36:57,030
involving electromagnetism,
and you gauge the derivatives.

625
00:36:57,030 --> 00:36:59,040
Then you can put
electromagnetism

626
00:36:59,040 --> 00:37:00,847
into the theory.

627
00:37:00,847 --> 00:37:02,430
It gets a little
more nontrivial then.

628
00:37:14,842 --> 00:37:16,300
And that's what we
need to do if we

629
00:37:16,300 --> 00:37:18,010
want to talk about
electromagnetism

630
00:37:18,010 --> 00:37:21,160
coupling for the deuteron.

631
00:37:21,160 --> 00:37:24,740
But I'm mostly interested
here in how the LSZ works.

632
00:37:24,740 --> 00:37:27,400
So what does LSZ say?

633
00:37:27,400 --> 00:37:33,384
So LSZ says that if I have
this state, p prime, J out,

634
00:37:33,384 --> 00:37:39,100
J mu electromagnetism, p i n--

635
00:37:39,100 --> 00:37:44,150
i is like a spin index because
we have a triplet under spin--

636
00:37:44,150 --> 00:37:46,990
then this is how you
should calculate it.

637
00:38:06,150 --> 00:38:09,620
So q is like the--

638
00:38:09,620 --> 00:38:13,280
well, I guess I'm in momentum
space all of a sudden.

639
00:38:13,280 --> 00:38:16,900
Well, no. q is just
p minus p prime.

640
00:38:16,900 --> 00:38:20,790
E and E bar are related to
the p and the p prime also.

641
00:38:20,790 --> 00:38:24,410
And what I do is I have some
z factor out front, which

642
00:38:24,410 --> 00:38:25,910
is the bound state z factor.

643
00:38:30,050 --> 00:38:33,140
LSZ tells me to be careful
about that z factor.

644
00:38:33,140 --> 00:38:35,270
These guys here are just
two-point functions,

645
00:38:35,270 --> 00:38:38,960
and that's because I have
to truncate the lines.

646
00:38:43,310 --> 00:38:47,045
This is truncation by
two-point functions,

647
00:38:47,045 --> 00:38:48,920
but they're two-point
functions for the bound

648
00:38:48,920 --> 00:38:50,628
state, so two lines.

649
00:38:50,628 --> 00:38:52,420
And then this is the
three-point functions.

650
00:38:59,840 --> 00:39:00,530
OK?

651
00:39:00,530 --> 00:39:06,860
So for the
three-point functions,

652
00:39:06,860 --> 00:39:09,020
what would the time-order
product be for that?

653
00:39:15,295 --> 00:39:17,045
It's a time-order
product of three things.

654
00:39:21,875 --> 00:39:23,750
So the two-point function
we already defined.

655
00:39:23,750 --> 00:39:25,167
It was just the
time-order product

656
00:39:25,167 --> 00:39:27,830
of the two deuteron fields.

657
00:39:27,830 --> 00:39:32,060
And the three-point function has
an additional electromagnetic

658
00:39:32,060 --> 00:39:39,472
current, which we would
formulate-- you know,

659
00:39:39,472 --> 00:39:40,930
all the couplings
here are this way

660
00:39:40,930 --> 00:39:44,367
that I was talking
about over there.

661
00:39:44,367 --> 00:39:46,700
So in terms of diagrams, which
are easier to think about

662
00:39:46,700 --> 00:39:50,650
in some ways than
time-order products,

663
00:39:50,650 --> 00:39:53,620
we have the two-point
function, which,

664
00:39:53,620 --> 00:39:58,150
if I just define all possible
two-point functions as G--

665
00:39:58,150 --> 00:40:02,230
and this is the thing that
was our two-particle--

666
00:40:02,230 --> 00:40:04,480
was our irreducible
c 0 contributions,

667
00:40:04,480 --> 00:40:08,287
plus iterations of c0's, right?

668
00:40:08,287 --> 00:40:10,120
Since the c 0 is a
nonperturbative coupling,

669
00:40:10,120 --> 00:40:14,980
we have to iterate, and
this is what we already

670
00:40:14,980 --> 00:40:17,740
said you could
define as the summing

671
00:40:17,740 --> 00:40:20,020
of that geometric series,
and then just define it

672
00:40:20,020 --> 00:40:22,390
as something which is the
full Green's function in terms

673
00:40:22,390 --> 00:40:26,140
of this irreducible piece
in the usual sort of way.

674
00:40:26,140 --> 00:40:29,880
And then, for this
other G, I'll draw

675
00:40:29,880 --> 00:40:33,880
a blob, which I'll also call G,
but with an additional photon

676
00:40:33,880 --> 00:40:35,690
hanging out.

677
00:40:35,690 --> 00:40:38,500
Those are all
three-point functions.

678
00:40:38,500 --> 00:40:41,680
And in a kind of a
similar notation,

679
00:40:41,680 --> 00:40:46,330
I can have kind of
an irreducible piece,

680
00:40:46,330 --> 00:40:50,985
and then I can have
iterations with c 0's.

681
00:41:07,930 --> 00:41:08,590
Et cetera.

682
00:41:13,150 --> 00:41:15,280
And if I sum up those
irreducible diagrams,

683
00:41:15,280 --> 00:41:16,960
I can write the full
Green's function

684
00:41:16,960 --> 00:41:18,460
in terms of the
irreducible pieces,

685
00:41:18,460 --> 00:41:21,520
again, in a similar
sort of notation.

686
00:41:21,520 --> 00:41:25,820
It's just gamma, which
carries the indices,

687
00:41:25,820 --> 00:41:31,990
and then there's just
a couple of-- now

688
00:41:31,990 --> 00:41:34,780
there's geometric series on
both sides, so when I sum it

689
00:41:34,780 --> 00:41:35,470
up I get that.

690
00:41:41,270 --> 00:41:43,668
OK, so this is giving you
the ingredients that we

691
00:41:43,668 --> 00:41:44,960
need to plug into this formula.

692
00:41:44,960 --> 00:41:46,120
We need G inverse.

693
00:41:46,120 --> 00:41:50,600
This is G. The two-point--

694
00:41:50,600 --> 00:41:52,780
we need the G mu
i J, which we can

695
00:41:52,780 --> 00:41:55,170
write in terms of the
irreducible guy, like that.

696
00:41:55,170 --> 00:41:58,227
Then the last thing we need
is this bound state factor.

697
00:41:58,227 --> 00:42:01,030
AUDIENCE: [INAUDIBLE]

698
00:42:01,030 --> 00:42:02,400
IAIN STEWART: Yeah?

699
00:42:02,400 --> 00:42:03,435
AUDIENCE: That cofactor
has a contribution

700
00:42:03,435 --> 00:42:04,380
that you just erased?

701
00:42:06,882 --> 00:42:08,340
IAIN STEWART: Yeah,
what I really--

702
00:42:08,340 --> 00:42:10,030
I'm kind of using
a shorthand here

703
00:42:10,030 --> 00:42:12,240
with this t product
for all the diagrams

704
00:42:12,240 --> 00:42:15,030
that come from what I was
talking about over there.

705
00:42:15,030 --> 00:42:17,513
Yeah, I kind of wrote it as
an electromagnetic current,

706
00:42:17,513 --> 00:42:19,680
but really what I mean by
it is sort of all the ways

707
00:42:19,680 --> 00:42:22,500
that electromagnetism can
appear in the effective theory.

708
00:42:25,052 --> 00:42:26,760
So you can think of
formulating a current

709
00:42:26,760 --> 00:42:31,848
in the effective theory by just
sort of looking for one photon,

710
00:42:31,848 --> 00:42:33,390
lopping it off, and
then writing down

711
00:42:33,390 --> 00:42:35,730
that operator, all the
operators, and that's really

712
00:42:35,730 --> 00:42:37,496
what I mean by this J mu.

713
00:42:43,420 --> 00:42:44,970
Any other questions?

714
00:42:48,730 --> 00:42:50,900
OK, so what's the
final thing we need,

715
00:42:50,900 --> 00:42:54,100
which is the bound
state z factor.

716
00:42:54,100 --> 00:42:59,380
So that's-- let me look back at
this kind of form that we had

717
00:42:59,380 --> 00:43:08,110
for G. We said it's i z, but now
let's denote the fact that z is

718
00:43:08,110 --> 00:43:10,660
really the principal
function of E bar,

719
00:43:10,660 --> 00:43:14,230
and it's really the
residue of the pole.

720
00:43:14,230 --> 00:43:19,850
And what z is, z that appears
in the formula over there,

721
00:43:19,850 --> 00:43:23,380
it's z at the pole,
so that's where

722
00:43:23,380 --> 00:43:29,440
I set E bar equal to minus
B D in the numerator.

723
00:43:29,440 --> 00:43:31,060
So this is the
residue of the pole.

724
00:43:41,620 --> 00:43:44,080
And with a little
bit of manipulation,

725
00:43:44,080 --> 00:43:45,760
we can write that
as the following.

726
00:43:52,578 --> 00:44:04,680
We can relate it to a
derivative at the pole.

727
00:44:04,680 --> 00:44:05,180
OK?

728
00:44:05,180 --> 00:44:08,510
So take the inverse,
take a derivative

729
00:44:08,510 --> 00:44:10,400
to sort of get just this factor.

730
00:44:10,400 --> 00:44:11,630
Take an inverse again.

731
00:44:11,630 --> 00:44:15,470
You can extract off that
residue, killing off

732
00:44:15,470 --> 00:44:18,120
some other pieces by
going to the pole.

733
00:44:18,120 --> 00:44:18,620
OK?

734
00:44:18,620 --> 00:44:20,120
So I won't go through
the derivation

735
00:44:20,120 --> 00:44:24,020
of that little
manipulation, but it's

736
00:44:24,020 --> 00:44:26,780
a straightforward consequence.

737
00:44:26,780 --> 00:44:29,330
And if you put in
kind of the formula

738
00:44:29,330 --> 00:44:32,952
that we had over here for
G, and if you do that,

739
00:44:32,952 --> 00:44:34,910
then you can write a
formula in terms of sigma.

740
00:44:37,730 --> 00:44:40,220
And after that kind of
manipulation, it's--

741
00:44:45,830 --> 00:44:48,660
again, just some algebra
to show that this is true.

742
00:44:51,570 --> 00:44:53,320
So now we can put all
the pieces together.

743
00:44:53,320 --> 00:45:01,201
We have this formula,
we have this formula,

744
00:45:01,201 --> 00:45:02,310
we have this formula.

745
00:45:14,890 --> 00:45:18,200
Put them into the
formula up there.

746
00:45:18,200 --> 00:45:29,260
And what we find is that we can
write the kind of result of it

747
00:45:29,260 --> 00:45:32,560
using LSZ in a kind
of very compact way,

748
00:45:32,560 --> 00:45:36,580
just in terms of irreducible
things, c 0 irreducible things.

749
00:45:41,610 --> 00:45:45,180
And it's not just the
irreducible three-point

750
00:45:45,180 --> 00:45:45,733
functions.

751
00:45:45,733 --> 00:45:48,150
There is a contribution from
the two-point function, which

752
00:45:48,150 --> 00:45:55,230
is in this case D sigma D E bar,
where you take E bar and E bar

753
00:45:55,230 --> 00:45:58,350
prime, incoming and
outgoing energies, to be

754
00:45:58,350 --> 00:46:00,540
the right energies for
the bound states, which

755
00:46:00,540 --> 00:46:03,970
is minus the binding energy.

756
00:46:03,970 --> 00:46:04,470
OK?

757
00:46:04,470 --> 00:46:07,860
So this factor came from here.

758
00:46:07,860 --> 00:46:09,960
There was a sigma
squared in the numerator,

759
00:46:09,960 --> 00:46:12,210
but when I take the inverse
of two of these guys,

760
00:46:12,210 --> 00:46:14,002
there's a sigma squared
in the denominator,

761
00:46:14,002 --> 00:46:15,120
and those canceled.

762
00:46:15,120 --> 00:46:17,220
The inverse of these guys
also had factors of 1

763
00:46:17,220 --> 00:46:19,380
plus i c 0 sigma
in the numerator,

764
00:46:19,380 --> 00:46:21,490
and that canceled
these factors here,

765
00:46:21,490 --> 00:46:27,550
so the only thing left is the
gamma and the D sigma D E bar.

766
00:46:27,550 --> 00:46:28,050
OK?

767
00:46:28,050 --> 00:46:31,270
So that's what LSZ
says in this situation.

768
00:46:36,390 --> 00:46:39,000
And it's actually important to
take this factor into account.

769
00:46:39,000 --> 00:46:40,560
If you didn't take this
factor into account,

770
00:46:40,560 --> 00:46:42,330
you wouldn't even
preserve, for example,

771
00:46:42,330 --> 00:46:45,990
that the charge of
the deuteron is 1.

772
00:46:45,990 --> 00:46:47,735
It's quite important.

773
00:46:51,380 --> 00:46:54,850
If you look at this
at lowest order

774
00:46:54,850 --> 00:46:57,190
and you just look at the
electromagnetic current

775
00:46:57,190 --> 00:47:04,240
for the electric case, and
you can do that with J 0,

776
00:47:04,240 --> 00:47:10,930
looking at the D 0, and we can
use our previous calculations

777
00:47:10,930 --> 00:47:17,260
to figure out what the
two-point function is.

778
00:47:17,260 --> 00:47:28,210
And when I go to
the pole, it's just

779
00:47:28,210 --> 00:47:30,280
this factor, where
gamma B would be

780
00:47:30,280 --> 00:47:35,160
set equal to exactly
the binding energy.

781
00:47:35,160 --> 00:47:35,660
OK?

782
00:47:35,660 --> 00:47:42,520
So that's just from
the trivial diagram.

783
00:47:42,520 --> 00:47:44,200
And then the first
contribution that

784
00:47:44,200 --> 00:47:49,960
comes into the three-point
function can be calculated.

785
00:47:49,960 --> 00:47:51,529
It's pretty straightforward.

786
00:47:55,201 --> 00:47:56,770
It gives a tan
inverse, actually.

787
00:48:04,460 --> 00:48:08,210
And that just
comes from coupling

788
00:48:08,210 --> 00:48:12,110
to the electromagnetic proton
that's inside the deuteron.

789
00:48:12,110 --> 00:48:15,720
So this is the proton, which
has an electromagnetic coupling.

790
00:48:15,720 --> 00:48:17,814
This is the neutron.

791
00:48:17,814 --> 00:48:18,314
OK?

792
00:48:18,314 --> 00:48:20,030
And that gives this formula.

793
00:48:20,030 --> 00:48:20,900
You take the ratio.

794
00:48:20,900 --> 00:48:23,150
You get the lowest-order
form factor.

795
00:48:23,150 --> 00:48:25,940
This is a systematically
provable thing,

796
00:48:25,940 --> 00:48:29,270
so you can do NLO.

797
00:48:29,270 --> 00:48:33,020
You can do NNLO, et cetera, OK?

798
00:48:33,020 --> 00:48:35,848
So we can really calculate
properties of the deuteron

799
00:48:35,848 --> 00:48:37,640
and check whether they
work experimentally,

800
00:48:37,640 --> 00:48:40,332
and this works extremely
well experimentally.

801
00:48:44,110 --> 00:48:46,540
You actually don't
see the violation

802
00:48:46,540 --> 00:48:51,220
of charge conservation in the
form factor until you do NLO.

803
00:48:51,220 --> 00:48:53,830
Then you really see that
you need this guy in order

804
00:48:53,830 --> 00:48:56,412
to make sure that the
charge is conserved.

805
00:48:59,130 --> 00:49:01,240
There would be some
pieces that would show up

806
00:49:01,240 --> 00:49:02,948
that would seem like
they were correcting

807
00:49:02,948 --> 00:49:06,210
the charge, violating
electromagnetism,

808
00:49:06,210 --> 00:49:08,160
magnetic gauge
variance until you--

809
00:49:08,160 --> 00:49:11,975
but then when you
put the z factor in,

810
00:49:11,975 --> 00:49:14,390
the z factor contribution
in, everything is nice.

811
00:49:23,295 --> 00:49:24,670
And this theory
has actually been

812
00:49:24,670 --> 00:49:29,560
used to do phenomenology,
so just to give you

813
00:49:29,560 --> 00:49:30,580
some examples of that.

814
00:49:34,160 --> 00:49:39,340
So you can do a process
like neutron on proton

815
00:49:39,340 --> 00:49:41,360
produces a deuteron
and a photon,

816
00:49:41,360 --> 00:49:44,720
which is something that shows
up in Big Bang nucleosynthesis.

817
00:49:44,720 --> 00:49:48,088
And it's been
calculated to N 4 LO

818
00:49:48,088 --> 00:49:49,380
in this effective field theory.

819
00:49:49,380 --> 00:49:51,490
[CHUCKLES] OK?

820
00:49:51,490 --> 00:49:53,320
And it provides the
most accurate way

821
00:49:53,320 --> 00:49:57,380
of determining the process
for Big Bang nucleosynthesis.

822
00:49:57,380 --> 00:50:01,210
You could reverse it, talk
about deuteron breakup.

823
00:50:01,210 --> 00:50:06,220
You could do processes with
neutrinos on deuterons.

824
00:50:06,220 --> 00:50:08,530
So you could have
neutrino-deuteron scattering

825
00:50:08,530 --> 00:50:11,620
to proton-proton
e minus, which is

826
00:50:11,620 --> 00:50:15,100
the charge current
process at SNO

827
00:50:15,100 --> 00:50:17,620
or the neutral current
process at SNO.

828
00:50:23,997 --> 00:50:26,080
All of these things are
within the realm of things

829
00:50:26,080 --> 00:50:29,800
that you can talk about in this
effective field theory, OK?

830
00:50:29,800 --> 00:50:32,890
So it has some uses.

831
00:50:32,890 --> 00:50:35,065
Let me do one more process.

832
00:50:39,740 --> 00:50:41,260
So you could look at neutron--

833
00:50:41,260 --> 00:50:45,185
nucleon-nucleon to
nucleon-nucleon plus an axion.

834
00:50:49,840 --> 00:50:53,600
And there's sort
of a lesson here.

835
00:50:53,600 --> 00:50:57,820
So if you look at how the
axion couples to the nucleon,

836
00:50:57,820 --> 00:51:01,990
it's derivatively
coupled, so you have

837
00:51:01,990 --> 00:51:03,670
a gradient of the axion field.

838
00:51:06,370 --> 00:51:07,870
And there's two
possible coupling

839
00:51:07,870 --> 00:51:09,730
that you can write
down, just to consider

840
00:51:09,730 --> 00:51:15,100
the sort of lowest dimension
in our power counting.

841
00:51:15,100 --> 00:51:16,570
And this is a
process that actually

842
00:51:16,570 --> 00:51:19,030
is important for
bounding axion physics,

843
00:51:19,030 --> 00:51:26,560
for example, axions in the
sun, to worry about how

844
00:51:26,560 --> 00:51:29,170
they couple to nucleons.

845
00:51:29,170 --> 00:51:31,647
And the thing that's
actually important

846
00:51:31,647 --> 00:51:33,730
is, if you're thinking
about this kind of process,

847
00:51:33,730 --> 00:51:37,300
you have to decide kinematically
what region are you looking at.

848
00:51:37,300 --> 00:51:39,940
If you're interested
in bounds from the sun,

849
00:51:39,940 --> 00:51:42,640
you're interested in a situation
where the energy of the axion

850
00:51:42,640 --> 00:51:45,790
is of order of the
energy of the nucleons,

851
00:51:45,790 --> 00:51:47,920
but the momentum of
the axion is much less,

852
00:51:47,920 --> 00:51:57,520
so I'll call that k axion is
much less than p nucleon, OK?

853
00:51:57,520 --> 00:51:59,150
So from a power
counting perspective,

854
00:51:59,150 --> 00:52:01,733
the energies are the same size,
but the momentum of the axion,

855
00:52:01,733 --> 00:52:03,610
which is a relativistic
particle and is

856
00:52:03,610 --> 00:52:06,240
comparable to the
energy, is much less

857
00:52:06,240 --> 00:52:08,860
since the nucleon momentum is
much bigger than its energy.

858
00:52:08,860 --> 00:52:11,920
Remember that the energy
goes like p squared over m,

859
00:52:11,920 --> 00:52:14,500
so the momentum are bigger.

860
00:52:14,500 --> 00:52:17,410
And if that's the situation
that you're in, then when

861
00:52:17,410 --> 00:52:18,970
you're doing a
Lagrangian like this

862
00:52:18,970 --> 00:52:21,250
you have to be careful
about implementing something

863
00:52:21,250 --> 00:52:22,390
like that.

864
00:52:22,390 --> 00:52:24,370
And the way that
it gets implemented

865
00:52:24,370 --> 00:52:26,740
is by a multipole
expansion, which

866
00:52:26,740 --> 00:52:28,330
we'll have an
opportunity to talk

867
00:52:28,330 --> 00:52:32,170
more about in the near future.

868
00:52:32,170 --> 00:52:33,910
But basically what
that corresponds to

869
00:52:33,910 --> 00:52:38,097
is that you've got to
make sure that you're

870
00:52:38,097 --> 00:52:40,180
sort of exchanging energy
between these operators,

871
00:52:40,180 --> 00:52:41,740
but not momenta.

872
00:52:41,740 --> 00:52:43,370
That's what happens
at lowest order.

873
00:52:43,370 --> 00:52:46,480
So if you make this
expansion in the Lagrangian,

874
00:52:46,480 --> 00:52:49,780
it corresponds to setting the
spatial components of the axion

875
00:52:49,780 --> 00:52:52,360
field to 0, OK?

876
00:52:52,360 --> 00:52:55,210
So this is like
doing an expansion

877
00:52:55,210 --> 00:52:59,095
around x vector equals 0, the
same thing in position space

878
00:52:59,095 --> 00:53:02,570
is this expansion here.

879
00:53:02,570 --> 00:53:13,500
So this is-- implements the
k much less than p expansion.

880
00:53:19,660 --> 00:53:20,160
OK?

881
00:53:20,160 --> 00:53:21,920
So that would actually
be the operator

882
00:53:21,920 --> 00:53:26,060
that you would use to
couple axions to nucleon.

883
00:53:26,060 --> 00:53:28,070
Now, the interesting
thing about this operator,

884
00:53:28,070 --> 00:53:30,740
which ties into what we
started talking about today,

885
00:53:30,740 --> 00:53:33,020
is that once you
take x equals 0,

886
00:53:33,020 --> 00:53:35,150
the things that
you're left with here

887
00:53:35,150 --> 00:53:40,080
are related to conserved
charges at the SU(4) symmetry.

888
00:53:40,080 --> 00:53:49,340
So Q mu nu, which is an integral
over all space of N dagger

889
00:53:49,340 --> 00:53:54,660
sigma mu tau nu N-- that's
the charge of SU(4).

890
00:54:00,000 --> 00:54:02,490
And the reason that
it's related to that

891
00:54:02,490 --> 00:54:05,340
is because, once I
take x equals 0 here,

892
00:54:05,340 --> 00:54:08,670
once I do the integral d 4 x
of the Lagrangian, the time--

893
00:54:08,670 --> 00:54:11,050
this still depends on time
but doesn't depend on space,

894
00:54:11,050 --> 00:54:13,560
so I can move the
d 3 x through, just

895
00:54:13,560 --> 00:54:17,600
let it act on this,
just let it act on that,

896
00:54:17,600 --> 00:54:19,874
and then I get Q mu nu.

897
00:54:19,874 --> 00:54:23,658
Well, Q0j and Q--

898
00:54:23,658 --> 00:54:28,830
or Qj0 and Qj3,
so this guy here.

899
00:54:31,400 --> 00:54:31,900
Color.

900
00:54:37,550 --> 00:54:41,120
Qj3, Qj0.

901
00:54:41,120 --> 00:54:42,840
This guy here.

902
00:54:42,840 --> 00:54:43,340
Qj3.

903
00:54:46,220 --> 00:54:50,540
And that actually has
nontrivial consequences

904
00:54:50,540 --> 00:54:53,570
because charges, if they're
charges of a symmetry,

905
00:54:53,570 --> 00:54:56,300
are time-independent.

906
00:54:56,300 --> 00:55:01,130
So even though in principle
I didn't make any assumptions

907
00:55:01,130 --> 00:55:05,420
about the time dependence
here, and these operators

908
00:55:05,420 --> 00:55:13,310
could exchange energy,
if I know that they

909
00:55:13,310 --> 00:55:16,040
are related to charges of
the effective field theory,

910
00:55:16,040 --> 00:55:17,820
then charges are conserved.

911
00:55:17,820 --> 00:55:19,444
There's no time dependence.

912
00:55:24,550 --> 00:55:31,707
So the charges of the field
theory are time-dependent.

913
00:55:37,790 --> 00:55:41,420
And what that means then is
that this is time-independent,

914
00:55:41,420 --> 00:55:44,690
and there's actually
no energy exchange.

915
00:55:44,690 --> 00:55:47,630
So the axion can only
sort of exchange energy--

916
00:55:47,630 --> 00:55:52,377
the only way the axion can
couple is with zero energy.

917
00:55:52,377 --> 00:55:54,710
And since it's derivatively
coupled, it has zero energy,

918
00:55:54,710 --> 00:55:57,920
it has zero momentum,
and there's no coupling.

919
00:55:57,920 --> 00:56:03,920
So this limit that we talked
about is a symmetry limit.

920
00:56:03,920 --> 00:56:07,002
The axion just doesn't couple.

921
00:56:07,002 --> 00:56:07,502
OK?

922
00:56:07,502 --> 00:56:10,040
So there we're predicting
something falling

923
00:56:10,040 --> 00:56:13,270
through kind of the
consequences of symmetry

924
00:56:13,270 --> 00:56:14,645
on this effective
a field theory.

925
00:56:14,645 --> 00:56:16,145
We're predicting
something about how

926
00:56:16,145 --> 00:56:20,090
axions couple to the nucleons.

927
00:56:20,090 --> 00:56:24,380
So if you look sort of
through the possible processes

928
00:56:24,380 --> 00:56:34,250
that you could have, you could
have two nucleons in a 3S1

929
00:56:34,250 --> 00:56:43,440
state try to produce an axion,
and that actually vanishes

930
00:56:43,440 --> 00:56:46,500
irrespective of the
scattering lengths

931
00:56:46,500 --> 00:56:50,340
because you're
coupling here to spin,

932
00:56:50,340 --> 00:56:54,870
and that is always a good charge
of the nonrelativistic theory.

933
00:56:58,200 --> 00:57:06,300
But there is a way that you
could have the process go,

934
00:57:06,300 --> 00:57:12,630
where you switch from a
1S0 state to a 3S1 state,

935
00:57:12,630 --> 00:57:14,820
and this guy vanishes
if the scattering

936
00:57:14,820 --> 00:57:15,960
lengths go to infinity.

937
00:57:20,470 --> 00:57:23,297
So if you ignored what I just
said and you ignored this here,

938
00:57:23,297 --> 00:57:25,380
and you just went ahead
and calculated the Feynman

939
00:57:25,380 --> 00:57:28,320
diagrams, you would just
find that the amplitude

940
00:57:28,320 --> 00:57:33,408
to this thing is proportional
to this kind of factor

941
00:57:33,408 --> 00:57:34,200
that we saw before.

942
00:57:41,028 --> 00:57:42,820
And since the scattering
lengths are large,

943
00:57:42,820 --> 00:57:47,600
it's vanishing if the scattering
lengths go to infinity.

944
00:57:47,600 --> 00:57:48,530
OK?

945
00:57:48,530 --> 00:57:50,550
So at lowest order,
the effective theory,

946
00:57:50,550 --> 00:57:53,060
it's suppressed, this process,
and that has implications

947
00:57:53,060 --> 00:57:54,828
for of course the phenomenology.

948
00:57:54,828 --> 00:57:56,870
You could also put in
higher-dimension operators,

949
00:57:56,870 --> 00:57:58,617
like the c 2's,
and then of course

950
00:57:58,617 --> 00:58:00,200
you can get a
contribution, but that's

951
00:58:00,200 --> 00:58:01,680
higher order of
the power counting,

952
00:58:01,680 --> 00:58:03,650
and that affects
how the axion could

953
00:58:03,650 --> 00:58:05,910
be constrained in the sun.

954
00:58:05,910 --> 00:58:06,410
OK?

955
00:58:06,410 --> 00:58:08,660
So that gives you a
little bit of a feeling

956
00:58:08,660 --> 00:58:11,090
for what you can do with
this effective field theory

957
00:58:11,090 --> 00:58:13,670
and how you can kind
of exploit symmetry,

958
00:58:13,670 --> 00:58:16,130
as well as the calculability
of the effective theory

959
00:58:16,130 --> 00:58:20,760
to make lots of
interesting predictions.

960
00:58:20,760 --> 00:58:22,020
OK?

961
00:58:22,020 --> 00:58:23,340
So any questions about that?

962
00:58:29,800 --> 00:58:31,240
All right.

963
00:58:31,240 --> 00:58:35,873
So that's all I have to say
about that effective theory.

964
00:58:35,873 --> 00:58:38,290
So we're now going to move on
to our final effective field

965
00:58:38,290 --> 00:58:40,100
theory, [SIGHS] to
soft-collinear effective

966
00:58:40,100 --> 00:58:40,600
theory.

967
00:58:50,750 --> 00:58:53,210
So what I intend to do with
this part of the course, which

968
00:58:53,210 --> 00:58:54,960
will take us through
the rest of the year,

969
00:58:54,960 --> 00:58:58,168
is I'm going to hand out
lecture notes for you to read,

970
00:58:58,168 --> 00:58:59,210
which I'm going to write.

971
00:58:59,210 --> 00:59:00,260
[CHUCKLES]

972
00:59:00,260 --> 00:59:02,940
So I haven't done that yet.

973
00:59:02,940 --> 00:59:04,515
I'm going to continue to post--

974
00:59:04,515 --> 00:59:06,140
I mean, I have some
rough copy of them,

975
00:59:06,140 --> 00:59:08,450
but I'm going to make it
pretty and beautiful for you,

976
00:59:08,450 --> 00:59:11,480
or more beautiful
than it currently is.

977
00:59:11,480 --> 00:59:14,120
I'm also going to continue to
post the lecture notes of what

978
00:59:14,120 --> 00:59:15,620
I present in lecture,
so you'll have

979
00:59:15,620 --> 00:59:17,510
sort of two versions
that you can read,

980
00:59:17,510 --> 00:59:20,330
either the LaTeX version
or the handwritten version,

981
00:59:20,330 --> 00:59:25,230
and depending on your taste you
can look at one or the other.

982
00:59:25,230 --> 00:59:28,652
So what we'll do today is we'll
just sort of briefly introduce

983
00:59:28,652 --> 00:59:30,110
ourselves to this
effective theory.

984
00:59:34,520 --> 00:59:36,890
And by way of introduction
to any effective theory,

985
00:59:36,890 --> 00:59:39,830
the kind of things that we
should talk about or why--

986
00:59:39,830 --> 00:59:41,475
so we'll talk about why.

987
00:59:41,475 --> 00:59:43,850
And then we should talk about
what the degrees of freedom

988
00:59:43,850 --> 00:59:44,350
are.

989
00:59:49,070 --> 00:59:50,720
And in this case,
we'll also talk

990
00:59:50,720 --> 00:59:53,030
about kind of what are
convenient coordinates

991
00:59:53,030 --> 00:59:56,330
to be using for this
effective theory.

992
00:59:56,330 --> 01:00:02,460
So that's what we'll
start covering today.

993
01:00:02,460 --> 01:00:03,255
So what is SCET?

994
01:00:07,670 --> 01:00:09,050
So it's an effective
field theory

995
01:00:09,050 --> 01:00:12,570
that you can use
for multiple things.

996
01:00:12,570 --> 01:00:14,780
One is for energetic hadrons.

997
01:00:22,550 --> 01:00:24,005
So the energy of
the hadron should

998
01:00:24,005 --> 01:00:25,880
be something that's a
large scale, which I'll

999
01:00:25,880 --> 01:00:29,660
call Q, conventional notation.

1000
01:00:29,660 --> 01:00:32,180
And by energetic it
means much more energetic

1001
01:00:32,180 --> 01:00:35,840
than the mass of the hadrons,
which is of order lambda QCD.

1002
01:00:41,720 --> 01:00:44,030
It's also an effective
theory for energetic jets.

1003
01:00:48,230 --> 01:00:52,040
And the situation is in some
ways similar to the hadrons.

1004
01:00:52,040 --> 01:00:56,210
You have a jet energy that's
large, of order some Q,

1005
01:00:56,210 --> 01:00:57,800
and that's much
bigger than the jet

1006
01:00:57,800 --> 01:01:04,970
mass, which is the root squared
of some forward momentum.

1007
01:01:09,240 --> 01:01:11,700
And by the name, you should
also imagine that really

1008
01:01:11,700 --> 01:01:13,635
what it is something
to do with QCD.

1009
01:01:16,285 --> 01:01:23,190
Well, maybe not by the name, but
it is something to do with QCD.

1010
01:01:23,190 --> 01:01:26,280
And it's kind of what you
get if you start with QCD

1011
01:01:26,280 --> 01:01:28,950
and you focus your
attention on a certain set

1012
01:01:28,950 --> 01:01:31,450
of interactions,
which are collinear

1013
01:01:31,450 --> 01:01:33,262
and soft interactions.

1014
01:01:39,163 --> 01:01:41,580
So we've met several different
examples of effective field

1015
01:01:41,580 --> 01:01:42,760
theory so far in the course.

1016
01:01:42,760 --> 01:01:43,927
Some of them were bottom-up.

1017
01:01:43,927 --> 01:01:45,150
Some of them were top-down.

1018
01:01:45,150 --> 01:01:47,220
This is going to be a
top-down effective field

1019
01:01:47,220 --> 01:01:50,100
theory, just like HQET was
a top-down effective field

1020
01:01:50,100 --> 01:01:50,938
theory.

1021
01:01:50,938 --> 01:01:52,980
So we'll be able to start
with the QCD Lagrangian

1022
01:01:52,980 --> 01:01:55,778
and derive this theory
by taking certain limits,

1023
01:01:55,778 --> 01:01:56,820
and that's what we'll do.

1024
01:02:03,680 --> 01:02:06,640
So why study this?

1025
01:02:06,640 --> 01:02:08,120
We'll come to that in a bit.

1026
01:02:08,120 --> 01:02:09,663
But what are the
kind of features

1027
01:02:09,663 --> 01:02:11,830
of this from an effective
field theory point of view

1028
01:02:11,830 --> 01:02:13,349
that make it interesting?

1029
01:02:17,181 --> 01:02:22,640
Well, we'll talk about the
answer to both those questions.

1030
01:02:22,640 --> 01:02:29,160
So first, why study this?

1031
01:02:29,160 --> 01:02:32,600
What's this effective
theory good for?

1032
01:02:32,600 --> 01:02:35,320
So one thing that's good for
is understanding factorization.

1033
01:02:38,680 --> 01:02:40,240
And if you think
about what we're

1034
01:02:40,240 --> 01:02:42,430
doing when we probe for
short-distance physics

1035
01:02:42,430 --> 01:02:46,570
at the LHC, we're probing
for short-distance physics

1036
01:02:46,570 --> 01:02:48,580
by colliding things
at high energy.

1037
01:02:55,390 --> 01:03:00,030
So we're carrying
out hard collisions,

1038
01:03:00,030 --> 01:03:03,987
energetic collisions
of particles.

1039
01:03:03,987 --> 01:03:05,820
So the fact that this
effective field theory

1040
01:03:05,820 --> 01:03:09,360
deals with energetic
particles may ring a bell

1041
01:03:09,360 --> 01:03:11,520
as it being something
useful for understanding

1042
01:03:11,520 --> 01:03:12,510
short-distance physics.

1043
01:03:22,217 --> 01:03:24,300
And so what this effective
theory allows you to do

1044
01:03:24,300 --> 01:03:26,190
is disentangle the
short-distance physics

1045
01:03:26,190 --> 01:03:29,610
that you're interested in from
the longer-distance physics

1046
01:03:29,610 --> 01:03:32,220
that's associated to binding
together into hadrons

1047
01:03:32,220 --> 01:03:34,950
or producing these
energetic jets.

1048
01:03:34,950 --> 01:03:37,350
So it's really disentangling
the long- and short-distance

1049
01:03:37,350 --> 01:03:40,290
physics in the standard
model, or the standard model

1050
01:03:40,290 --> 01:03:42,360
plus new physics.

1051
01:03:42,360 --> 01:03:43,980
And basically what
that amounts to is

1052
01:03:43,980 --> 01:03:46,950
disentangling long- and
short-distance physics in QCD.

1053
01:04:01,390 --> 01:04:03,460
So let me write it this way.

1054
01:04:03,460 --> 01:04:06,010
Disentangling the physics
of QCD with addition

1055
01:04:06,010 --> 01:04:07,675
of electroweak or BSM physics.

1056
01:04:11,498 --> 01:04:13,585
It requires a
separation of scales.

1057
01:04:16,120 --> 01:04:20,340
We have to decide what's short
distance, what's long distance.

1058
01:04:20,340 --> 01:04:22,570
If we're interested in
a process like the LHC,

1059
01:04:22,570 --> 01:04:24,220
where we're colliding
together hadrons,

1060
01:04:24,220 --> 01:04:28,690
then that's of course
absolutely necessary

1061
01:04:28,690 --> 01:04:30,790
because the hadrons are
long-distance things,

1062
01:04:30,790 --> 01:04:33,590
and we're interested in
short-distance physics,

1063
01:04:33,590 --> 01:04:37,030
and we can do that with SCET,
so that's perhaps the biggest

1064
01:04:37,030 --> 01:04:38,100
motivation for us today.

1065
01:04:48,090 --> 01:04:50,150
When we make hard
collisions, these jets

1066
01:04:50,150 --> 01:04:52,790
and energetic hadrons
are very common.

1067
01:04:52,790 --> 01:04:56,690
They're the most common
things that are produced.

1068
01:04:56,690 --> 01:04:58,370
And from a physics
point of view,

1069
01:04:58,370 --> 01:05:00,650
that means that there's many,
many different processes

1070
01:05:00,650 --> 01:05:03,067
that you can actually use this
effective field theory for,

1071
01:05:03,067 --> 01:05:05,750
and that's one of the
reasons it's popular.

1072
01:05:05,750 --> 01:05:08,280
So I want to give
you some examples.

1073
01:05:14,430 --> 01:05:16,140
You could do deep
inelastic scattering.

1074
01:05:16,140 --> 01:05:18,300
In this effective theory,
deep inelastic scattering

1075
01:05:18,300 --> 01:05:22,440
involves a hard collision.

1076
01:05:22,440 --> 01:05:25,710
And even classic textbook
deep inelastic scattering

1077
01:05:25,710 --> 01:05:28,300
is made very much simpler by
this effective field theory,

1078
01:05:28,300 --> 01:05:31,860
and we'll show you that with
one of our examples later on.

1079
01:05:35,970 --> 01:05:40,320
You could do Drell-Yan with
either protons and antiprotons

1080
01:05:40,320 --> 01:05:42,570
or protons and protons.

1081
01:05:42,570 --> 01:05:46,140
You could do Higgs
production, and people

1082
01:05:46,140 --> 01:05:49,080
use this effective field theory
to study Higgs production

1083
01:05:49,080 --> 01:05:51,120
at the LHC.

1084
01:05:51,120 --> 01:05:54,390
You could do, by way of trying
to make the examples more

1085
01:05:54,390 --> 01:05:57,060
varied, e plus e minus to jets.

1086
01:05:57,060 --> 01:06:00,255
This is another thing that the
effective theory is used for.

1087
01:06:00,255 --> 01:06:04,110
You could do quarkonium
physics, e plus or minus,

1088
01:06:04,110 --> 01:06:07,770
producing a J psi
and an X. Here,

1089
01:06:07,770 --> 01:06:09,360
you need a combination
if the J psi

1090
01:06:09,360 --> 01:06:13,110
is produced at large energy of
kind of a nonrelativistic QCD

1091
01:06:13,110 --> 01:06:16,800
for this cc-bar
bound state, as well

1092
01:06:16,800 --> 01:06:21,055
as an SCT for the fact that
it's an energetic state.

1093
01:06:21,055 --> 01:06:22,680
So this would be
actually a combination

1094
01:06:22,680 --> 01:06:25,200
of two different
effective field theories.

1095
01:06:25,200 --> 01:06:30,348
People use it for thinking
about jet substructure.

1096
01:06:30,348 --> 01:06:31,890
So all sorts of
things that you might

1097
01:06:31,890 --> 01:06:33,750
be interested in and
thinking about having

1098
01:06:33,750 --> 01:06:37,385
to do with hard scattering.

1099
01:06:37,385 --> 01:06:38,760
There's also
another way that you

1100
01:06:38,760 --> 01:06:41,250
could get energetic
particles, even

1101
01:06:41,250 --> 01:06:43,075
if you don't force
them to be energetic,

1102
01:06:43,075 --> 01:06:45,325
and that is if you had a
very heavy particle decaying.

1103
01:06:49,390 --> 01:06:53,340
So if you had a B meson decaying
and it decayed to light stuff,

1104
01:06:53,340 --> 01:06:56,710
this would also be the right
effective theory for that.

1105
01:06:56,710 --> 01:06:59,790
So for example, in the
process B to s gamma,

1106
01:06:59,790 --> 01:07:03,060
this effective
theory plays a role,

1107
01:07:03,060 --> 01:07:06,000
in the process B to D pi,
where the pion comes out very

1108
01:07:06,000 --> 01:07:09,600
energetic, or B
to pi pi, which is

1109
01:07:09,600 --> 01:07:12,305
something very interesting
for studying CP violation.

1110
01:07:12,305 --> 01:07:13,680
You can use this
effective theory

1111
01:07:13,680 --> 01:07:16,230
to understand the short-
and long-distance physics

1112
01:07:16,230 --> 01:07:18,670
of those processes, as well.

1113
01:07:18,670 --> 01:07:19,170
OK?

1114
01:07:19,170 --> 01:07:21,840
And this is coming about
because this B meson is heavy.

1115
01:07:24,780 --> 01:07:27,530
So that's the sort
of range of examples,

1116
01:07:27,530 --> 01:07:29,618
and it's not really
complete in any sense.

1117
01:07:29,618 --> 01:07:31,160
There's just lots
of different things

1118
01:07:31,160 --> 01:07:33,465
you can study with this theory.

1119
01:07:59,910 --> 01:08:01,774
So this I already said,
but let me write it.

1120
01:08:06,022 --> 01:08:09,793
I sort of alluded to
separating perturbative physics

1121
01:08:09,793 --> 01:08:15,320
at short distances from
the nonperturbative physics

1122
01:08:15,320 --> 01:08:16,520
at long distances.

1123
01:08:21,937 --> 01:08:23,979
And the nonperturbative
physics, if you're lucky,

1124
01:08:23,979 --> 01:08:26,220
may be only encoded in
parton distribution functions

1125
01:08:26,220 --> 01:08:29,198
or something if you're
doing a p-p collision,

1126
01:08:29,198 --> 01:08:31,740
and you'd like to separate that
from shorter-distance physics

1127
01:08:31,740 --> 01:08:33,069
that you could calculate
perturbatively,

1128
01:08:33,069 --> 01:08:34,402
and this is a way of doing that.

1129
01:08:37,090 --> 01:08:39,990
And then there's a
renormalization group

1130
01:08:39,990 --> 01:08:43,800
in this effective theory, and
that renormalization group

1131
01:08:43,800 --> 01:08:47,069
actually allows you to
do something cool, too,

1132
01:08:47,069 --> 01:08:49,080
and that is that you
can sum up something

1133
01:08:49,080 --> 01:08:54,899
called Sudakov logarithms,
which are algorithms

1134
01:08:54,899 --> 01:08:59,850
that appear in our canonical way
as alpha s times a logarithm.

1135
01:08:59,850 --> 01:09:03,055
But instead of one logarithm,
you actually get two.

1136
01:09:03,055 --> 01:09:05,430
And so the renormalization
group in this effective theory

1137
01:09:05,430 --> 01:09:10,330
will allow you to handle
those Sudakov logarithms.

1138
01:09:10,330 --> 01:09:10,830
OK?

1139
01:09:10,830 --> 01:09:12,540
So that's some set of things.

1140
01:09:16,660 --> 01:09:19,210
What are we going to have to
do that we haven't done before?

1141
01:09:19,210 --> 01:09:21,430
What are the kind
of new ingredients

1142
01:09:21,430 --> 01:09:23,529
that we're going to run into?

1143
01:09:23,529 --> 01:09:30,069
So sort of a prelude
to things to come,

1144
01:09:30,069 --> 01:09:31,649
or another way of
saying what makes

1145
01:09:31,649 --> 01:09:35,710
SCET different from
the effective theories

1146
01:09:35,710 --> 01:09:36,700
we've seen so far.

1147
01:09:48,430 --> 01:09:51,340
Let's have a little
list for that, as well.

1148
01:09:51,340 --> 01:09:54,810
So perhaps the most
interesting one

1149
01:09:54,810 --> 01:09:59,250
is that we are going to
have more than one field

1150
01:09:59,250 --> 01:10:00,970
for each degree of freedom.

1151
01:10:05,200 --> 01:10:07,590
So we're going to have
more than one field--

1152
01:10:07,590 --> 01:10:09,570
that's not quite the
right way of saying it.

1153
01:10:09,570 --> 01:10:11,195
We're going to have
more than one field

1154
01:10:11,195 --> 01:10:12,195
for the same particle.

1155
01:10:15,288 --> 01:10:17,295
Just a slightly
different statement.

1156
01:10:22,630 --> 01:10:28,580
So we're going to have a
field, which we'll talk about,

1157
01:10:28,580 --> 01:10:31,130
that I'll call c sub n,
which is a collinear quark

1158
01:10:31,130 --> 01:10:35,818
field for some quark,
say an up quark.

1159
01:10:35,818 --> 01:10:37,860
And there's also going to
be another field, q sub

1160
01:10:37,860 --> 01:10:41,460
s, which is a soft quark
field for the same up quark.

1161
01:10:45,587 --> 01:10:47,295
So let's say for the
up quark [INAUDIBLE]

1162
01:10:47,295 --> 01:10:48,844
just to make it definite.

1163
01:10:56,958 --> 01:10:59,000
And the reason that we're
going to have these two

1164
01:10:59,000 --> 01:11:00,260
different fields
is because they're

1165
01:11:00,260 --> 01:11:01,718
going to describe
the same particle

1166
01:11:01,718 --> 01:11:03,763
but in different regions
of momentum space.

1167
01:11:03,763 --> 01:11:05,930
So if you like, they're
describing different degrees

1168
01:11:05,930 --> 01:11:07,910
of freedom of the same
particle because they're

1169
01:11:07,910 --> 01:11:09,800
in different regions
of momentum space.

1170
01:11:09,800 --> 01:11:11,250
But it still is
the same particle,

1171
01:11:11,250 --> 01:11:13,760
and that's not something
we've had to deal with in any

1172
01:11:13,760 --> 01:11:14,840
of our examples so far.

1173
01:11:14,840 --> 01:11:16,610
We always had one field
for each particle.

1174
01:11:16,610 --> 01:11:18,920
Now we're going to
have more than one,

1175
01:11:18,920 --> 01:11:20,936
so that's one complication.

1176
01:11:25,040 --> 01:11:29,030
The second one we have
actually encountered before,

1177
01:11:29,030 --> 01:11:32,060
and that is that we're
going to integrate out

1178
01:11:32,060 --> 01:11:45,950
off-shell modes of a field, but
not the entire particle, not

1179
01:11:45,950 --> 01:11:49,400
the entire degree of freedom.

1180
01:11:49,400 --> 01:11:53,140
And we encountered that
when we talked about HQET.

1181
01:11:53,140 --> 01:11:57,260
So the situation here will
be a little more complicated

1182
01:11:57,260 --> 01:12:02,240
than HQET, but this
second point is like HQET,

1183
01:12:02,240 --> 01:12:03,890
and we'll use some
of our experience

1184
01:12:03,890 --> 01:12:05,885
that we gained
from studying HQET.

1185
01:12:15,504 --> 01:12:17,212
AUDIENCE: [INAUDIBLE]
soft and collinear?

1186
01:12:17,212 --> 01:12:18,232
Which one--

1187
01:12:18,232 --> 01:12:19,690
IAIN STEWART: It
never can be both.

1188
01:12:19,690 --> 01:12:20,190
[CHUCKLES]

1189
01:12:20,190 --> 01:12:22,233
AUDIENCE: Why?

1190
01:12:22,233 --> 01:12:23,650
IAIN STEWART: We'll
have to decide

1191
01:12:23,650 --> 01:12:25,640
whether it's one or the other.

1192
01:12:25,640 --> 01:12:27,640
So an important thing--

1193
01:12:27,640 --> 01:12:31,180
exactly this issue of which
it is an important thing

1194
01:12:31,180 --> 01:12:33,130
because you're going
to be describing

1195
01:12:33,130 --> 01:12:35,170
with these fields
certain fluctuations,

1196
01:12:35,170 --> 01:12:36,835
and you don't want overlap.

1197
01:12:36,835 --> 01:12:39,460
So you're going to have to-- if
you point your finger somewhere

1198
01:12:39,460 --> 01:12:41,703
in momentum space,
we'll have to decide

1199
01:12:41,703 --> 01:12:43,370
is it in this category
or this category.

1200
01:12:43,370 --> 01:12:44,770
It can't be in both.

1201
01:12:44,770 --> 01:12:46,510
But that will be
important in a bit.

1202
01:12:46,510 --> 01:12:47,360
That will come back.

1203
01:12:47,360 --> 01:12:48,693
AUDIENCE: So you're partioning--

1204
01:12:48,693 --> 01:12:52,148
IAIN STEWART: Yeah, two
regions, and no overlap.

1205
01:12:52,148 --> 01:12:54,690
But we'll have to work a little
bit to make sure that's true.

1206
01:13:03,270 --> 01:13:04,740
Yeah, we want to
make sure to not

1207
01:13:04,740 --> 01:13:07,320
have double counting in the
effective theory, where you

1208
01:13:07,320 --> 01:13:10,590
could write down something
that looks like it could

1209
01:13:10,590 --> 01:13:11,760
be described by either one.

1210
01:13:11,760 --> 01:13:14,750
And if that was true, then
we would have a problem.

1211
01:13:24,400 --> 01:13:28,860
So another thing that makes
SCET interesting from an EFT

1212
01:13:28,860 --> 01:13:31,320
point of view is that
we have convolutions.

1213
01:13:31,320 --> 01:13:33,900
So usually in an
effective theory

1214
01:13:33,900 --> 01:13:36,300
you think about having
multiple operators that

1215
01:13:36,300 --> 01:13:38,487
have the same quantum numbers.

1216
01:13:38,487 --> 01:13:40,945
We sum over those operators,
say at some order in the power

1217
01:13:40,945 --> 01:13:45,780
counting, sum over i of Wilson
coefficients times operators.

1218
01:13:45,780 --> 01:13:51,206
And this sum becomes a
continuous integral in SCET.

1219
01:13:51,206 --> 01:13:54,370
You'll see how that works.

1220
01:13:54,370 --> 01:13:56,520
And that actually
leads to convolutions.

1221
01:13:56,520 --> 01:13:58,775
In fact, exactly this
little manipulation

1222
01:13:58,775 --> 01:14:00,900
here is what leads to the
convolution of the parton

1223
01:14:00,900 --> 01:14:03,570
distribution function
with hard scattering.

1224
01:14:03,570 --> 01:14:06,090
If you think about this
operator is describing--

1225
01:14:06,090 --> 01:14:07,590
the matrix elements
of this operator

1226
01:14:07,590 --> 01:14:09,990
is describing a parton
distribution function.

1227
01:14:09,990 --> 01:14:12,690
The Wilson's coefficients is
describing the hard scattering.

1228
01:14:12,690 --> 01:14:14,370
Exactly this integral
is what leads

1229
01:14:14,370 --> 01:14:17,730
to the integral the convolutes
together parton distribution

1230
01:14:17,730 --> 01:14:22,760
functions and hard
scattering matrix elements.

1231
01:14:22,760 --> 01:14:23,540
Come back to that.

1232
01:14:34,275 --> 01:14:35,900
So there's going to
be a power counting

1233
01:14:35,900 --> 01:14:36,983
for this effective theory.

1234
01:14:38,923 --> 01:14:41,090
And we're going to call the
power counting parameter

1235
01:14:41,090 --> 01:14:44,490
lambda, and it's not going to
be related to the mass dimension

1236
01:14:44,490 --> 01:14:44,990
of fields.

1237
01:14:56,780 --> 01:14:59,790
So we'll have to figure out
how to do the power counting.

1238
01:15:05,470 --> 01:15:09,460
There's going to be something
called the Wilson line, which

1239
01:15:09,460 --> 01:15:11,980
will show up all over the
place in this effective theory.

1240
01:15:19,420 --> 01:15:22,360
So these are lines in space
that are coupled to the gauge

1241
01:15:22,360 --> 01:15:27,520
field, A. And we'll see
why they're showing up,

1242
01:15:27,520 --> 01:15:29,290
but it's related
actually to the fact

1243
01:15:29,290 --> 01:15:32,920
that this effective theory has
a very interesting and perhaps

1244
01:15:32,920 --> 01:15:36,250
subtle gauge symmetry structure.

1245
01:15:42,940 --> 01:15:45,395
One way of actually already
seeing why that is true

1246
01:15:45,395 --> 01:15:47,020
is because I told
you over here that we

1247
01:15:47,020 --> 01:15:48,620
have two types of quarks.

1248
01:15:48,620 --> 01:15:50,880
Everything I say about quarks
is also true of gluons,

1249
01:15:50,880 --> 01:15:52,672
so there's going to be
two types of gluons,

1250
01:15:52,672 --> 01:15:55,348
an A n gluon and an A s gluon.

1251
01:15:55,348 --> 01:15:56,890
And once there's
two types of gluons,

1252
01:15:56,890 --> 01:15:58,930
you should say, well, what's
the gauge group of this gluon

1253
01:15:58,930 --> 01:15:59,930
and what's the gauge
group for that gluon?

1254
01:15:59,930 --> 01:16:01,597
Well, they're both
the same gauge group.

1255
01:16:01,597 --> 01:16:02,710
It's QCD.

1256
01:16:02,710 --> 01:16:05,485
So we partitioned QCD in
some kind of weird way,

1257
01:16:05,485 --> 01:16:07,360
where we have two fields
for the gauge group,

1258
01:16:07,360 --> 01:16:09,705
rather than just one.

1259
01:16:09,705 --> 01:16:11,080
Part of that story
is going to be

1260
01:16:11,080 --> 01:16:12,330
related to these Wilson lines.

1261
01:16:16,970 --> 01:16:19,430
And finally, one final
kind of interesting thing

1262
01:16:19,430 --> 01:16:21,500
that's going to
happen is we're going

1263
01:16:21,500 --> 01:16:25,670
to see when we do one-loop
calculations in this theory

1264
01:16:25,670 --> 01:16:30,208
that we find 1 over epsilon
squared divergences.

1265
01:16:30,208 --> 01:16:32,250
So usually, when you do
an effective field theory

1266
01:16:32,250 --> 01:16:36,890
calculation, at one loop you
find a 1 over epsilon pole,

1267
01:16:36,890 --> 01:16:39,740
not a 1 over epsilon squared.

1268
01:16:39,740 --> 01:16:44,000
And I denote it by epsilon
u v because both poles here

1269
01:16:44,000 --> 01:16:47,650
require u v renormalization.

1270
01:16:47,650 --> 01:16:51,830
We need to add a counterterm
to cancel this double pole.

1271
01:16:51,830 --> 01:16:53,840
And exactly the presence
of that double pole

1272
01:16:53,840 --> 01:16:58,813
is what leads to these Sudakov
logs that I was mentioning,

1273
01:16:58,813 --> 01:17:00,230
the fact that we
have these double

1274
01:17:00,230 --> 01:17:03,080
logs going along
with each alpha s.

1275
01:17:03,080 --> 01:17:05,895
So we'll see that taking care
of this with renormalization,

1276
01:17:05,895 --> 01:17:08,270
which is little different than
the renormalization you're

1277
01:17:08,270 --> 01:17:13,580
used to, will allow us to sum
up those Sudakov logarithms, OK?

1278
01:17:13,580 --> 01:17:16,190
So that's the kind
of hint of the things

1279
01:17:16,190 --> 01:17:17,450
that we will be studying.

1280
01:17:17,450 --> 01:17:19,325
You're not supposed to
understand any of them

1281
01:17:19,325 --> 01:17:20,930
yet unless you've
study SCET, but we

1282
01:17:20,930 --> 01:17:23,180
will understand all of these
by the end of the course.

1283
01:17:25,930 --> 01:17:27,580
OK?

1284
01:17:27,580 --> 01:17:28,720
Any question?

1285
01:17:28,720 --> 01:17:30,010
Anything I missed on my list?

1286
01:17:30,010 --> 01:17:33,040
[CHUCKLES] All right.

1287
01:17:37,260 --> 01:17:40,080
So let's spend some time talking
about degrees of freedom.

1288
01:17:47,175 --> 01:17:49,387
The way I like to think
about degrees of freedom

1289
01:17:49,387 --> 01:17:50,762
for any effective
field theory is

1290
01:17:50,762 --> 01:17:53,137
this is where you have to
think about the physics of what

1291
01:17:53,137 --> 01:17:55,000
you're doing, OK?

1292
01:17:55,000 --> 01:17:57,190
So let's think about
some physical process.

1293
01:17:57,190 --> 01:18:00,130
I'm going to consider two
different physical processes.

1294
01:18:00,130 --> 01:18:02,590
Actually, today, I'll only
start talking about one of them,

1295
01:18:02,590 --> 01:18:05,320
and then we'll talk about
the other one next time.

1296
01:18:05,320 --> 01:18:06,880
Think about some
physical process

1297
01:18:06,880 --> 01:18:09,310
and try to identify what the
right degrees of freedom are.

1298
01:18:15,860 --> 01:18:18,070
So you could look at
the process of a B meson

1299
01:18:18,070 --> 01:18:20,495
changing into a D
meson and a pion.

1300
01:18:20,495 --> 01:18:22,120
The advantage of
starting with this one

1301
01:18:22,120 --> 01:18:24,500
is that we've already
talked about HQET,

1302
01:18:24,500 --> 01:18:26,620
so we understand how to
do B mesons and D mesons,

1303
01:18:26,620 --> 01:18:28,703
and we just have to
understand how to do the pion.

1304
01:18:34,810 --> 01:18:36,010
So we have a B meson.

1305
01:18:36,010 --> 01:18:38,270
Let's think about it
in the rest frame.

1306
01:18:38,270 --> 01:18:41,260
Decays to a very heavy D
meson and a very light pion.

1307
01:18:44,210 --> 01:18:46,190
So the D meson we
can treat with HQET.

1308
01:18:46,190 --> 01:18:48,680
The B meson we can
treat with HQET.

1309
01:18:48,680 --> 01:18:53,390
The problem is this pion
because it's very energetic.

1310
01:18:53,390 --> 01:18:57,260
If you look at the momentum
of that pion and the way

1311
01:18:57,260 --> 01:19:02,960
I've drawn it, and you
put in some numbers just

1312
01:19:02,960 --> 01:19:13,730
to get a sense of what
we're talking about,

1313
01:19:13,730 --> 01:19:15,163
then it's got an
energy, which is

1314
01:19:15,163 --> 01:19:16,580
slightly bigger
than the momentum,

1315
01:19:16,580 --> 01:19:17,960
and that's because
it has a mass,

1316
01:19:17,960 --> 01:19:19,250
but it's a pretty small mass.

1317
01:19:21,980 --> 01:19:26,585
e squared minus p squared
is 135 m e V squared.

1318
01:19:30,160 --> 01:19:33,570
And so if you look
at that, you see

1319
01:19:33,570 --> 01:19:35,490
that this momentum
of the pion is

1320
01:19:35,490 --> 01:19:36,710
very close to the light cone.

1321
01:19:36,710 --> 01:19:39,100
The light cone would be
if these two were equal.

1322
01:19:39,100 --> 01:19:43,030
That's what a massless
particle would do.

1323
01:19:43,030 --> 01:19:47,140
A pion is pretty light, and
it's moving relativistically,

1324
01:19:47,140 --> 01:19:49,930
and there's no sense in which
we can think about 2.3 G e

1325
01:19:49,930 --> 01:19:51,230
V as a small scale.

1326
01:19:51,230 --> 01:19:54,550
We're not going to expand
in 2.3 G e V over m B

1327
01:19:54,550 --> 01:19:58,610
because that's a
pretty lousy expansion.

1328
01:19:58,610 --> 01:20:02,740
So p pi mu here is
basically some scale

1329
01:20:02,740 --> 01:20:11,260
times a light-like vector,
where n mu is 1, 0, 0, minus 1,

1330
01:20:11,260 --> 01:20:13,960
and that's to a
good approximation.

1331
01:20:13,960 --> 01:20:15,310
We'll see how good over there.

1332
01:20:19,330 --> 01:20:21,850
So we're going to use
light-like vectors to talk

1333
01:20:21,850 --> 01:20:24,550
about particles like a pion.

1334
01:20:24,550 --> 01:20:27,850
So n mu is a light-like vector.

1335
01:20:27,850 --> 01:20:28,845
n squared is 0.

1336
01:20:34,295 --> 01:20:35,920
And what we're going
to be expanding in

1337
01:20:35,920 --> 01:20:39,310
is, instead of expanding
in Q over m B, Q over pion,

1338
01:20:39,310 --> 01:20:44,560
we're going to think about
the dynamics with this scale Q

1339
01:20:44,560 --> 01:20:47,230
scale, 2.3 G e V as being
much bigger than lambda QCD.

1340
01:20:51,090 --> 01:20:52,590
So we're going to
use, for particles

1341
01:20:52,590 --> 01:20:54,215
like this pion that
are very energetic,

1342
01:20:54,215 --> 01:20:55,741
light cone coordinates.

1343
01:21:06,330 --> 01:21:10,320
So when you have a
light cone vector like n

1344
01:21:10,320 --> 01:21:13,470
and you want to decompose
any vector in terms of it,

1345
01:21:13,470 --> 01:21:15,610
you need another
auxiliary vector,

1346
01:21:15,610 --> 01:21:17,840
which I'm going to call n bar.

1347
01:21:17,840 --> 01:21:20,250
So these are two
light-like vectors.

1348
01:21:20,250 --> 01:21:22,410
They both square to 0.

1349
01:21:22,410 --> 01:21:23,790
And the reason I
need two of them

1350
01:21:23,790 --> 01:21:26,190
is because I need some
notion of orthogonality,

1351
01:21:26,190 --> 01:21:30,020
so you should think of n bar as
like the complementary vector

1352
01:21:30,020 --> 01:21:31,830
to n.

1353
01:21:31,830 --> 01:21:35,670
It's the dual vector in
the orthogonal sense.

1354
01:21:38,830 --> 01:21:41,640
So with those two vectors,
you can decompose any vector

1355
01:21:41,640 --> 01:21:42,270
in terms of--

1356
01:21:45,480 --> 01:21:46,470
use them as a basis.

1357
01:21:50,781 --> 01:21:55,820
This is how it works,
my conventions.

1358
01:21:55,820 --> 01:22:00,890
So you have two components that
you can decompose along n and n

1359
01:22:00,890 --> 01:22:05,200
bar, and the value of the
momentum along that direction

1360
01:22:05,200 --> 01:22:06,950
is the complementary
vector dotted into p.

1361
01:22:10,290 --> 01:22:11,510
And so we usually--

1362
01:22:11,510 --> 01:22:16,010
with this notation, we
call n bar dot p p minus.

1363
01:22:16,010 --> 01:22:18,740
So that's just a shorthand.

1364
01:22:18,740 --> 01:22:24,353
And n dot p is p plus, OK?

1365
01:22:24,353 --> 01:22:26,270
So it's going to be
useful to decompose things

1366
01:22:26,270 --> 01:22:28,020
along these directions
for the same reason

1367
01:22:28,020 --> 01:22:30,980
that you kind of saw here,
that the pion was naturally

1368
01:22:30,980 --> 01:22:33,380
decomposing itself along
a particular light-like

1369
01:22:33,380 --> 01:22:33,978
direction.

1370
01:22:33,978 --> 01:22:36,020
So we're going to want to
do that more generally,

1371
01:22:36,020 --> 01:22:37,645
and that's how we'll
do it for momenta.

1372
01:22:40,720 --> 01:22:47,840
If we talk about p squared, and
I just square the components

1373
01:22:47,840 --> 01:22:52,070
here, n dotted into
"perp" is 0, and bar

1374
01:22:52,070 --> 01:22:53,360
dotted into "perp" is 0.

1375
01:22:53,360 --> 01:22:55,450
There's a nontrivial--
that's square root of 0.

1376
01:22:55,450 --> 01:22:56,450
That's square root of 0.

1377
01:22:56,450 --> 01:22:58,533
There's a nontrivial
cross-term between those two,

1378
01:22:58,533 --> 01:22:59,550
which is this.

1379
01:22:59,550 --> 01:23:00,635
This guy can be squared.

1380
01:23:05,610 --> 01:23:09,900
If we use our shorthand,
then it's this.

1381
01:23:09,900 --> 01:23:15,050
And if we want to use Euclidean
notation, then I can--

1382
01:23:15,050 --> 01:23:18,020
I'll switch and I'll say
that p "perp" squared, which

1383
01:23:18,020 --> 01:23:19,640
is a Minkowski
negative quantity,

1384
01:23:19,640 --> 01:23:22,190
is minus p vector
"perp" squared,

1385
01:23:22,190 --> 01:23:24,350
where p vector "perp"
squared is Euclidean.

1386
01:23:30,080 --> 01:23:35,150
So sometimes I'll-- use for
both of those notations.

1387
01:23:35,150 --> 01:23:37,340
All right, so I think we're
out of time for today.

1388
01:23:37,340 --> 01:23:39,620
We'll continue with this
discussion of degrees

1389
01:23:39,620 --> 01:23:43,100
of freedom next time and
talk a little bit more

1390
01:23:43,100 --> 01:23:45,430
about coordinates, as well.