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PROFESSOR: Apparently.
00:00:28.190 --> 00:00:29.210
We'll start slow.
00:00:29.210 --> 00:00:31.088
So last time we were
talking, we just
00:00:31.088 --> 00:00:32.630
started talking
about effective field
00:00:32.630 --> 00:00:34.310
theory with a fine tuning.
00:00:34.310 --> 00:00:36.620
And what actually
that means takes
00:00:36.620 --> 00:00:38.060
a little bit of discussion.
00:00:41.840 --> 00:00:44.380
So what you could
mean by a fine tuning
00:00:44.380 --> 00:00:47.350
is that you have something
that's irrelevant.
00:00:47.350 --> 00:00:49.610
You look at the operator,
you think it's irrelevant,
00:00:49.610 --> 00:00:50.530
but it's not.
00:00:50.530 --> 00:00:51.340
It's relevant.
00:00:51.340 --> 00:00:53.800
Something that you should
include even at lowest order
00:00:53.800 --> 00:00:55.300
in your power accounting.
00:00:55.300 --> 00:00:57.815
But saying something
is irrelevant
00:00:57.815 --> 00:01:00.190
means that you have a power
counting, that you understand
00:01:00.190 --> 00:01:02.650
the power counting for the
theory, what the correct power
00:01:02.650 --> 00:01:04.010
counting is.
00:01:04.010 --> 00:01:05.710
So in this example
that I'll give,
00:01:05.710 --> 00:01:07.960
what "irrelevant" will
mean is that you basically
00:01:07.960 --> 00:01:10.480
do a dimensional
power counting, which
00:01:10.480 --> 00:01:11.980
is how we usually
think of defining
00:01:11.980 --> 00:01:13.450
irrelevant and relevant.
00:01:13.450 --> 00:01:15.520
Do a dimensional power
counting and you end up
00:01:15.520 --> 00:01:17.047
finding an operator that--
00:01:17.047 --> 00:01:18.880
you find that the
operator looks irrelevant,
00:01:18.880 --> 00:01:20.920
but when you do
calculations you can
00:01:20.920 --> 00:01:22.618
see that should be relevant.
00:01:22.618 --> 00:01:24.160
And that means,
really, what it means
00:01:24.160 --> 00:01:26.433
is that this natural power
counting of dimensions
00:01:26.433 --> 00:01:27.850
is not the right
one, and you have
00:01:27.850 --> 00:01:29.692
to do something
more complicated.
00:01:29.692 --> 00:01:31.150
But it is still is
a sense in which
00:01:31.150 --> 00:01:37.110
it can be thought of as a fine
tuning, because as you'll see,
00:01:37.110 --> 00:01:39.330
the changing of the power
counting from the naive one
00:01:39.330 --> 00:01:41.700
to the more complicated
power counting
00:01:41.700 --> 00:01:46.330
involves some kind of
tuning, if you like.
00:01:46.330 --> 00:01:47.850
And it really, in
this case, we'll
00:01:47.850 --> 00:01:49.730
see actually that
it corresponds not
00:01:49.730 --> 00:01:52.230
to expanding around the trivial
fixed point, where you would
00:01:52.230 --> 00:01:54.060
have a free theory,
but expanding
00:01:54.060 --> 00:01:55.560
around an interactive
fixed point.
00:01:58.455 --> 00:01:59.830
So it'll be a
little non-trivial,
00:01:59.830 --> 00:02:02.080
but we'll be doing this
in the context of two
00:02:02.080 --> 00:02:03.490
nucleon effective field theory.
00:02:03.490 --> 00:02:05.710
And the advantage of this
is that the nucleons are
00:02:05.710 --> 00:02:07.390
going to be non-relativistic.
00:02:07.390 --> 00:02:09.857
So P is going to be
much less than M pi.
00:02:09.857 --> 00:02:11.440
It's going to be a
very simple theory.
00:02:11.440 --> 00:02:14.620
Everything that's an exchange
particle gets integrated out.
00:02:14.620 --> 00:02:16.690
It's just a theory of
contact interactions,
00:02:16.690 --> 00:02:19.210
and derivatives of
contact interactions.
00:02:19.210 --> 00:02:20.800
And because it's
non-relativistic,
00:02:20.800 --> 00:02:22.510
we can actually
calculate all the loops
00:02:22.510 --> 00:02:24.093
to all orders and
perturbation theory,
00:02:24.093 --> 00:02:25.490
and we'll do that in a minute.
00:02:25.490 --> 00:02:28.820
So this theory, we can
calculate a lot of stuff.
00:02:28.820 --> 00:02:32.980
And so we'll actually be able
to see how this non-trivial fine
00:02:32.980 --> 00:02:35.713
tuning works, and explore
it from multiple directions,
00:02:35.713 --> 00:02:37.380
and we'll be sure
that what we're saying
00:02:37.380 --> 00:02:38.500
is actually correct.
00:02:41.340 --> 00:02:43.720
So it can be a lesson
for understanding
00:02:43.720 --> 00:02:46.330
some of the concepts and
other effective field theories
00:02:46.330 --> 00:02:49.360
with the fine tuning, which
you might want to design,
00:02:49.360 --> 00:02:53.407
where you don't have as
much ability to calculate.
00:02:53.407 --> 00:02:55.240
All right, so let's
start off with something
00:02:55.240 --> 00:02:59.650
simple, which is
elastic scattering.
00:02:59.650 --> 00:03:01.930
And that's mostly what
we're going to talk about.
00:03:07.350 --> 00:03:12.300
Two particles in,
two particles out.
00:03:12.300 --> 00:03:15.430
Center of mass frame.
00:03:15.430 --> 00:03:21.183
They scatter to some P's
coming in, P primes going out.
00:03:21.183 --> 00:03:23.100
And this is basically a
problem that you could
00:03:23.100 --> 00:03:24.390
treat with quantum mechanics.
00:03:27.342 --> 00:03:31.810
It's like non-relativistic
scattering.
00:03:31.810 --> 00:03:41.650
So if you have a
single partial wave,
00:03:41.650 --> 00:03:43.290
then this scattering
is described
00:03:43.290 --> 00:03:45.930
by a phase shift, delta.
00:03:45.930 --> 00:03:51.870
And the relation of the
phase shift to the amplitude,
00:03:51.870 --> 00:03:57.600
with our normalization for
the amplitude, is this.
00:03:57.600 --> 00:03:59.700
So this is the S-matrix,
it's just a phase,
00:03:59.700 --> 00:04:02.670
and that's the relation of
the S-matrix to the amplitude.
00:04:02.670 --> 00:04:05.922
And this thing is the amplitude.
00:04:05.922 --> 00:04:07.380
And I guess the
other thing we know
00:04:07.380 --> 00:04:12.960
is that, by energy
conservation, the magnitude of P
00:04:12.960 --> 00:04:16.529
is equal to the
magnitude of P prime.
00:04:16.529 --> 00:04:19.860
All right, so if we
rearrange this equation,
00:04:19.860 --> 00:04:23.760
and we write it as A and
put the phase, solve for a.
00:04:29.736 --> 00:04:30.810
Could do that.
00:04:34.550 --> 00:04:37.363
So that gives that
equation, which we
00:04:37.363 --> 00:04:38.655
can rearrange a little further.
00:04:47.954 --> 00:05:00.690
Now, there's kind of a
conventional way of writing it,
00:05:00.690 --> 00:05:02.420
which is that the
amplitude should
00:05:02.420 --> 00:05:06.440
be given by 1 over something
that's P cotangent delta.
00:05:06.440 --> 00:05:10.370
Scattering angle, or
that S-matrix angle,
00:05:10.370 --> 00:05:13.440
and then minus this IP.
00:05:13.440 --> 00:05:14.000
OK?
00:05:14.000 --> 00:05:15.860
So the I of--
00:05:15.860 --> 00:05:19.250
the I's just-- shows
up here in this part,
00:05:19.250 --> 00:05:21.660
and that's the complex
part of the amplitude.
00:05:21.660 --> 00:05:23.810
That's basically going to
be related to unitarity,
00:05:23.810 --> 00:05:26.960
that that IP is there.
00:05:26.960 --> 00:05:30.326
This S-matrix is
obviously unitary.
00:05:30.326 --> 00:05:33.830
This digress is 1.
00:05:33.830 --> 00:05:37.760
All right, so let me
tell you something
00:05:37.760 --> 00:05:44.000
about non-relativistic
scattering,
00:05:44.000 --> 00:05:46.700
which on the face of it,
looks kind of non-trivial.
00:06:10.990 --> 00:06:13.690
So this thing, P
cotangent delta.
00:06:13.690 --> 00:06:16.450
So here, I was doing
a single partial wave.
00:06:16.450 --> 00:06:18.460
Yeah, OK, no, it's fine.
00:06:18.460 --> 00:06:20.590
So what is this L?
00:06:20.590 --> 00:06:22.360
This L is the partial
I am considering.
00:06:22.360 --> 00:06:23.690
S wave, P wave.
00:06:23.690 --> 00:06:27.310
So L is 0 for the S wave,
L is 1 for the P wave.
00:06:27.310 --> 00:06:30.100
And the statement is that if you
have a short range potential,
00:06:30.100 --> 00:06:35.860
and you pick a wave, then
this P cotangent delta
00:06:35.860 --> 00:06:38.590
with the appropriate power
of P can be a Taylor series
00:06:38.590 --> 00:06:41.398
expansion of P. OK?
00:06:41.398 --> 00:06:42.940
And this is actually
something that's
00:06:42.940 --> 00:06:46.000
quite difficult to prove
in quantum mechanics,
00:06:46.000 --> 00:06:49.043
this particular fact.
00:06:49.043 --> 00:06:50.960
And it's called the
effective range expansion.
00:07:00.807 --> 00:07:03.390
It's difficult to prove because
when you do quantum mechanics,
00:07:03.390 --> 00:07:05.250
you pick a potential.
00:07:05.250 --> 00:07:08.140
And I'm saying that this
is true for any potential.
00:07:08.140 --> 00:07:11.140
So if you start doing quantum
mechanics with some potential,
00:07:11.140 --> 00:07:13.380
you've got to prove that
you can put it in this form
00:07:13.380 --> 00:07:16.510
irrespective of what the choice
of that potential would be.
00:07:16.510 --> 00:07:18.360
And that makes it
a little bit tricky
00:07:18.360 --> 00:07:22.002
to do in a quantum
mechanical setup.
00:07:22.002 --> 00:07:23.460
But we'll see
actually that this is
00:07:23.460 --> 00:07:25.950
very easy to prove from an
effective field theory setup.
00:07:34.104 --> 00:07:36.280
So, as a way of getting
into this effective field
00:07:36.280 --> 00:07:40.450
theory of two nucleons,
let's prove this fact.
00:07:40.450 --> 00:07:42.320
So what is the Lagrangian
for this theory?
00:07:46.930 --> 00:07:49.980
There's no-- it's
not a gauge theory,
00:07:49.980 --> 00:07:53.920
so we just have
ordinary derivatives.
00:07:53.920 --> 00:07:56.170
So if you like, you can think
of what I'm writing here
00:07:56.170 --> 00:07:58.780
as kind of like our IV.D
term, except I think
00:07:58.780 --> 00:08:00.790
the center of mass
frame, and this
00:08:00.790 --> 00:08:02.560
would be like the
kinetic energy term,
00:08:02.560 --> 00:08:06.040
but now it's just partial
squared with no D, et cetera.
00:08:20.370 --> 00:08:23.028
And then there's a bunch
of contact interactions.
00:08:23.028 --> 00:08:24.570
So there's a whole
bunch of operators
00:08:24.570 --> 00:08:27.540
that are involved in
nucleon and fields
00:08:27.540 --> 00:08:30.450
with some Wilson coefficients.
00:08:30.450 --> 00:08:34.350
The notation here
is this S is kind
00:08:34.350 --> 00:08:36.850
of a pseudonym for the channel.
00:08:36.850 --> 00:08:38.909
So this S here--
00:08:38.909 --> 00:08:41.970
maybe it should be a
script S or something,
00:08:41.970 --> 00:08:43.679
is telling me what
channel I'm in,
00:08:43.679 --> 00:08:46.980
and if in a
spectroscopic notation,
00:08:46.980 --> 00:08:52.717
you'd say you're in the
2S plus 1 LJ channel.
00:08:52.717 --> 00:08:54.300
So this would be the
angular momentum,
00:08:54.300 --> 00:08:55.470
total momentum in the spin.
00:08:58.970 --> 00:09:04.010
And these operators
here, for our purposes,
00:09:04.010 --> 00:09:07.892
are four nuclear fields
with 2M derivatives.
00:09:14.300 --> 00:09:17.650
Now, this is not really
the complete theory
00:09:17.650 --> 00:09:18.650
for a couple of reasons.
00:09:18.650 --> 00:09:20.210
Well, there's higher order
relativistic corrections
00:09:20.210 --> 00:09:21.560
indicated by the dots.
00:09:21.560 --> 00:09:24.230
There would also
be dots over here
00:09:24.230 --> 00:09:26.660
That could have to do with
having more nucleon fields.
00:09:26.660 --> 00:09:28.340
For example, I could have--
00:09:28.340 --> 00:09:30.612
instead of just four,
I could have six.
00:09:30.612 --> 00:09:32.570
But I don't need to worry
about those operators
00:09:32.570 --> 00:09:37.113
for two-to-two scattering,
so I'm leaving them out.
00:09:37.113 --> 00:09:38.780
So this is actually
the complete theory,
00:09:38.780 --> 00:09:45.060
if we include these dots here,
for two-to-two scattering.
00:09:45.060 --> 00:09:55.100
And this nucleon field,
[? its ?] spin [? at ?] half,
00:09:55.100 --> 00:09:56.630
and it's isospin
been at half too,
00:09:56.630 --> 00:09:58.610
so it includes both the
proton and the neutron.
00:10:04.680 --> 00:10:13.260
Nucleons are fermions, and that
implies, actually, a relation,
00:10:13.260 --> 00:10:15.850
because the wave function
has to be anti-symmetric.
00:10:15.850 --> 00:10:23.040
And so actually, you know
that you can associate isospin
00:10:23.040 --> 00:10:25.410
and the angular momentum
in the following
00:10:25.410 --> 00:10:27.370
way because of this fact.
00:10:27.370 --> 00:10:31.770
So all the isotriplets have
-1 to the S plus L even,
00:10:31.770 --> 00:10:37.770
and the isosinglets have
-1 to the S plus L odd.
00:10:37.770 --> 00:10:40.230
So that cuts down
by a factor of two
00:10:40.230 --> 00:10:42.990
the number of combinations
you have to consider.
00:10:42.990 --> 00:10:45.000
And basically, what
this theory has
00:10:45.000 --> 00:10:49.320
is for some given channel
in and some given channel
00:10:49.320 --> 00:11:01.700
out, which I could denote
in general different,
00:11:01.700 --> 00:11:04.220
we get operators that
just will have some power
00:11:04.220 --> 00:11:05.810
of the center of mass momentum.
00:11:05.810 --> 00:11:06.750
P to the 2M.
00:11:09.500 --> 00:11:13.670
And actually, just by angular
momentum conservation,
00:11:13.670 --> 00:11:16.460
that has to be the same as that.
00:11:16.460 --> 00:11:19.010
And so all that can
change is the L's.
00:11:19.010 --> 00:11:24.680
And so if S is 0,
S is either 0 or 1,
00:11:24.680 --> 00:11:27.500
because we have 2
spin half particles.
00:11:27.500 --> 00:11:31.010
If S is zero, L will
be equal to L prime
00:11:31.010 --> 00:11:34.130
because J is equal to
J prime, and there's
00:11:34.130 --> 00:11:36.110
no shift of the spin.
00:11:36.110 --> 00:11:39.920
So that's one possibility,
and if S is 1, then--
00:11:42.630 --> 00:11:45.450
so S here is the
same, if S is 1,
00:11:45.450 --> 00:11:55.480
then you can have L minus L
prime which is 2, so, or 0.
00:11:55.480 --> 00:11:56.220
OK?
00:11:56.220 --> 00:11:57.720
You're shifting by
one unit, and you
00:11:57.720 --> 00:12:03.390
can compensate either by
having minus L prime be 0 or 2.
00:12:08.880 --> 00:12:14.930
OK, so we conserve J.
So we can enumerate
00:12:14.930 --> 00:12:16.340
all the possible partial waves.
00:12:16.340 --> 00:12:19.530
We'll mostly focus
on the S wave.
00:12:19.530 --> 00:12:26.360
So, let me write out some
of these operators for you.
00:12:33.540 --> 00:12:36.290
So the first operator
has no derivatives,
00:12:36.290 --> 00:12:42.220
and I can write it in a way that
makes the partial wave explicit
00:12:42.220 --> 00:12:43.786
if I do the following.
00:12:50.940 --> 00:12:54.135
And then there would be
some derivative operator.
00:12:54.135 --> 00:12:55.760
And I'm going to pick
the normalization
00:12:55.760 --> 00:12:59.930
to make our lives
as easy as possible,
00:12:59.930 --> 00:13:02.440
as we usually do when we're
setting up the operator basis.
00:13:22.660 --> 00:13:29.242
There's the first two guys
where this derivative operator
00:13:29.242 --> 00:13:33.736
is like, [? grad squared ?]
to the left.
00:13:33.736 --> 00:13:36.857
[? Grad ?] left dot
[? grad ?] right.
00:13:36.857 --> 00:13:38.190
That [? grad ?] go to the right.
00:13:41.980 --> 00:13:44.720
And these P's, if
you look at them,
00:13:44.720 --> 00:13:48.080
they're just matrices in
the spin and isospin space.
00:13:48.080 --> 00:13:53.260
So the two we're focusing
on are the S waves.
00:13:53.260 --> 00:13:55.570
And in the S wave you
either have 1S0 or 3S1.
00:14:16.310 --> 00:14:18.350
And so we've encoded
all the, sort of,
00:14:18.350 --> 00:14:21.230
complexity in just these
matrices, which kind of just
00:14:21.230 --> 00:14:26.460
go along for the ride, and
I'll tell you what they are.
00:14:26.460 --> 00:14:33.980
So I sigma 2 projects you onto
a spin singlet, and I tau 2 tau
00:14:33.980 --> 00:14:37.290
I projects you
onto an isotriplet.
00:14:37.290 --> 00:14:43.700
And then likewise, 3S1,
which is a spin triplet,
00:14:43.700 --> 00:14:50.540
you put I sigma 2
sigma I. I tau 2.
00:14:50.540 --> 00:14:52.815
I tau 2 and I sigma 2 are
just because of the way
00:14:52.815 --> 00:14:53.690
I wrote the operator.
00:14:53.690 --> 00:14:56.060
I wrote it, instead
of writing N dagger N,
00:14:56.060 --> 00:14:58.958
I wrote N transpose
N, all dagger.
00:14:58.958 --> 00:15:00.500
And that means,
basically, you should
00:15:00.500 --> 00:15:02.167
think about the way
this operator works,
00:15:02.167 --> 00:15:05.960
is it annihilates two nucleons
in a particular spin wave,
00:15:05.960 --> 00:15:09.680
or a particular spin,
isospin channel,
00:15:09.680 --> 00:15:13.180
and then creates them again.
00:15:13.180 --> 00:15:18.340
So annihilate, create.
00:15:18.340 --> 00:15:20.080
So I just put the
two fields that
00:15:20.080 --> 00:15:21.587
are doing the
annihilating together,
00:15:21.587 --> 00:15:23.920
and the two fields that are
doing the creating together.
00:15:23.920 --> 00:15:26.560
And that's nice because
you're annihilating them
00:15:26.560 --> 00:15:27.570
in a particular channel.
00:15:32.110 --> 00:15:36.190
So with those conventions,
our Feynman Rules
00:15:36.190 --> 00:15:39.500
are particularly simple.
00:15:39.500 --> 00:15:43.600
If we just have a
C0 in some channel,
00:15:43.600 --> 00:15:47.020
then the Feynman Rule
is just minus IC0,
00:15:47.020 --> 00:15:52.870
and if we have one of
these higher C2 operators
00:15:52.870 --> 00:15:56.320
in the center of mass frame,
it's just minus IC2P squared.
00:16:02.172 --> 00:16:03.630
In the center of
mass frame, that's
00:16:03.630 --> 00:16:04.620
the center of mass momentum.
00:16:04.620 --> 00:16:06.370
And remember, in the
center of mass frame,
00:16:06.370 --> 00:16:08.670
P squared was equal
to P prime squared.
00:16:08.670 --> 00:16:12.120
And so we can actually just
write down the Feynman Rule
00:16:12.120 --> 00:16:13.890
for the complete
set of operators
00:16:13.890 --> 00:16:16.129
there if we adopt
this convention.
00:16:20.350 --> 00:16:22.693
So if you insert a guy
with 2M derivatives--
00:16:22.693 --> 00:16:24.360
derivatives always
have to come in pairs
00:16:24.360 --> 00:16:27.140
because of angular momentum.
00:16:35.077 --> 00:16:36.660
You just have that
Feynman Rule, sum's
00:16:36.660 --> 00:16:43.300
over the number of derivatives.
00:16:43.300 --> 00:16:45.080
So this is the complete,
in this theory,
00:16:45.080 --> 00:16:53.682
this is the complete
tree-level amplitude
00:16:53.682 --> 00:16:55.140
from those interactions
over there.
00:16:57.970 --> 00:17:01.285
It's very nice theory.
00:17:01.285 --> 00:17:01.785
Simple.
00:17:04.660 --> 00:17:05.847
What about loops?
00:17:05.847 --> 00:17:07.180
We are going to need some loops.
00:17:10.950 --> 00:17:14.609
So let's look at
the following loop,
00:17:14.609 --> 00:17:17.251
and I'll start by looking
at just E equals 0.
00:17:22.069 --> 00:17:24.050
Let's just take it,
take the nuclear arms
00:17:24.050 --> 00:17:26.599
and then scatter them again.
00:17:26.599 --> 00:17:28.460
So, in terms of
the momentum flow
00:17:28.460 --> 00:17:31.070
I have some Q going
this way, and then I
00:17:31.070 --> 00:17:36.800
have minus Q going that
way, that's my loop momenta.
00:17:36.800 --> 00:17:39.453
So I get 2 coupling C0.
00:17:39.453 --> 00:17:40.800
Want these to be 0's.
00:17:43.680 --> 00:17:44.180
Ergo, DDQ.
00:17:48.061 --> 00:17:56.580
And if I just kept the HQET
type term in my kinetic term,
00:17:56.580 --> 00:17:59.220
then it would look
like that, OK?
00:17:59.220 --> 00:18:00.060
So this is just--
00:18:03.090 --> 00:18:06.840
for the moment, if we
just keep partial DT
00:18:06.840 --> 00:18:11.970
in the kinetic term, which is
what we were doing in HQET,
00:18:11.970 --> 00:18:14.200
then we would get that.
00:18:14.200 --> 00:18:15.750
And that's a bad integral.
00:18:15.750 --> 00:18:17.912
It's got a pinch singularity.
00:18:17.912 --> 00:18:19.120
It's an ill-defined interval.
00:18:26.400 --> 00:18:28.620
So just from that
little algebra,
00:18:28.620 --> 00:18:30.855
we see that actually
keeping just the partial D
00:18:30.855 --> 00:18:32.940
by DT in the kinetic term
is not the right thing
00:18:32.940 --> 00:18:34.920
to do for this theory.
00:18:34.920 --> 00:18:39.330
And that's because the kinetic
energy is a relevant operator
00:18:39.330 --> 00:18:40.743
in quantum mechanics.
00:18:45.090 --> 00:18:47.160
Whenever you write down
the Schrodinger equation,
00:18:47.160 --> 00:18:47.700
you kept it.
00:18:54.417 --> 00:18:56.000
So the right power
counting and should
00:18:56.000 --> 00:19:00.100
have E, which is of
order P squared over 2M.
00:19:00.100 --> 00:19:02.675
So the partial T term and the
[? grad ?] scored over M term
00:19:02.675 --> 00:19:03.675
should be the same size.
00:19:13.580 --> 00:19:16.720
So this is generically true
of two heavy particles.
00:19:25.030 --> 00:19:32.350
They have a different power
counting for the kinetic term,
00:19:32.350 --> 00:19:33.120
than HQET.
00:19:39.452 --> 00:19:41.410
Any two heavy particles,
whether it's too heavy
00:19:41.410 --> 00:19:44.350
quarks, two heavy nucleons,
two heavy anything.
00:19:44.350 --> 00:19:46.580
All right, P should be
[? order ?] P squared over 2M.
00:19:46.580 --> 00:19:49.157
AUDIENCE: E being
the kinetic energy?
00:19:49.157 --> 00:19:50.740
PROFESSOR: So really,
what I mean here
00:19:50.740 --> 00:19:54.400
is just the partial
T term in the action
00:19:54.400 --> 00:19:56.990
over there should be the same
size as the [? grad ?] squared
00:19:56.990 --> 00:19:57.490
over 2M.
00:20:04.140 --> 00:20:05.620
Yes, I have pulled out the mass.
00:20:05.620 --> 00:20:06.870
Yeah.
00:20:06.870 --> 00:20:10.260
So just like in HQET,
to get this partial T
00:20:10.260 --> 00:20:12.750
we pulled out the
mass, and we have
00:20:12.750 --> 00:20:15.192
a kind of non-relativistic
type expansion.
00:20:15.192 --> 00:20:17.400
And the difference here is
that we need the partial T
00:20:17.400 --> 00:20:18.775
to be of
[? order grad squared ?]
00:20:18.775 --> 00:20:22.410
over 2M, and that leads to
effectively counting velocities
00:20:22.410 --> 00:20:25.260
rather than--
00:20:25.260 --> 00:20:29.190
because you have to count
energies different than P's.
00:20:29.190 --> 00:20:31.410
We won't spend too
long talking about
00:20:31.410 --> 00:20:34.080
that because we have other
things to discuss here,
00:20:34.080 --> 00:20:36.870
but this is a whole
interesting subject in itself.
00:20:36.870 --> 00:20:39.172
The power counting
and what it means.
00:20:39.172 --> 00:20:41.130
One thing that's kind of
interesting here which
00:20:41.130 --> 00:20:44.370
we won't cover, which I
can't help but mention,
00:20:44.370 --> 00:20:47.050
is that say you did quarks,
which was a gauge theory.
00:20:47.050 --> 00:20:48.720
This is not a gauge theory
that we're talking about,
00:20:48.720 --> 00:20:51.303
but let's-- but you could do a
gauge theory that has this type
00:20:51.303 --> 00:20:54.090
of power counting and it has
exactly the same problem.
00:20:54.090 --> 00:20:58.140
Just replace these heavy
nucleons by quarks,
00:20:58.140 --> 00:21:01.410
and replace this dot here
by cooling potential.
00:21:01.410 --> 00:21:05.130
Exactly the same problem
if you try to use HQET.
00:21:05.130 --> 00:21:07.170
And in that theory,
too, you need
00:21:07.170 --> 00:21:08.670
E to be of order
P squared over 2,
00:21:08.670 --> 00:21:11.120
and that's called
non-relativistic QCD.
00:21:11.120 --> 00:21:12.870
Or you could do heavy
electrons, where you
00:21:12.870 --> 00:21:15.990
have QED as the gauge theory.
00:21:15.990 --> 00:21:19.200
Non-relativistic
QED, same issue.
00:21:19.200 --> 00:21:22.650
E has to be of order P squared
over M. And in gauge theory,
00:21:22.650 --> 00:21:24.780
there's even a further
complication, which
00:21:24.780 --> 00:21:27.330
is basically that there's
gauge particles that
00:21:27.330 --> 00:21:29.940
want to talk to E, and
there's gauge particles that
00:21:29.940 --> 00:21:33.010
want to talk to P, and
those are different sizes.
00:21:33.010 --> 00:21:35.820
So you have something
called ultra-soft photons
00:21:35.820 --> 00:21:38.700
and soft photons that are
the gauge particles for E,
00:21:38.700 --> 00:21:42.660
and the gauge particles for P.
Kind of an interesting theory.
00:21:42.660 --> 00:21:44.320
We want to have time
to talk about it.
00:21:44.320 --> 00:21:47.240
Somebody wants to talk
about it for their project,
00:21:47.240 --> 00:21:48.090
it's kind of fun.
00:21:52.980 --> 00:21:56.790
OK, so we have to keep
P squared over 2M.
00:21:56.790 --> 00:21:59.610
At least knowing a
little quantum mechanics,
00:21:59.610 --> 00:22:01.650
we know that that's true.
00:22:01.650 --> 00:22:03.450
Or running into this
pinch singularity,
00:22:03.450 --> 00:22:08.810
we see that trying to do
something different than that
00:22:08.810 --> 00:22:09.560
leads to problems.
00:22:21.080 --> 00:22:23.380
So let's redo our calculation
here, but now keeping
00:22:23.380 --> 00:22:27.380
that term, and see what we get.
00:22:27.380 --> 00:22:29.740
Same bubble diagram.
00:22:29.740 --> 00:22:34.060
Let me send in on each
of these legs E over 2,
00:22:34.060 --> 00:22:36.070
so E is the total energy
that I'm sending in.
00:22:40.528 --> 00:22:42.228
Convenient normalization.
00:22:47.710 --> 00:22:51.400
Let's see how having the kinetic
energy fixes this pinch pole.
00:23:16.390 --> 00:23:18.610
OK, so if you look at the
poles in the complex plane
00:23:18.610 --> 00:23:22.010
here, what's happened is
you've split them like this.
00:23:22.010 --> 00:23:25.160
So that you've moved
them along the real axis.
00:23:25.160 --> 00:23:27.280
And so now you can just
think of a contour,
00:23:27.280 --> 00:23:32.860
for example, if you want to
think in the complex Q0 plane,
00:23:32.860 --> 00:23:37.200
you can think of doing a
contour integral like that.
00:23:37.200 --> 00:23:40.640
Everything is well defined,
convergence at infinity,
00:23:40.640 --> 00:23:42.370
everybody's happy.
00:23:42.370 --> 00:23:46.200
So we can close
above, pick the polar.
00:23:46.200 --> 00:23:55.227
Q0 is E over 2, minus Q
squared over 2M, plus I0.
00:23:55.227 --> 00:23:56.560
Plug it back into the other one.
00:24:03.760 --> 00:24:08.110
My notation is that N is
going to be D minus 1.
00:24:08.110 --> 00:24:10.150
So when I do one of the
integrals by contour,
00:24:10.150 --> 00:24:11.140
I go down a dimension.
00:24:11.140 --> 00:24:13.480
I'll call that N.
So it's N here.
00:24:24.630 --> 00:24:25.740
This integral we can do.
00:24:33.905 --> 00:24:35.655
You have to be careful
about the epsilons.
00:24:45.930 --> 00:24:48.240
Because they tell us tell
us whether it's minus IP
00:24:48.240 --> 00:24:50.805
or plus IP.
00:24:50.805 --> 00:24:53.870
ME is set up in my convention.
00:24:53.870 --> 00:24:56.450
ME is P squared.
00:24:56.450 --> 00:24:59.948
So this is giving a P, but
it's giving either a minus
00:24:59.948 --> 00:25:01.990
IP or a plus IP, depending
on the sign of the I0,
00:25:01.990 --> 00:25:05.800
but I know it's a minus IP.
00:25:05.800 --> 00:25:07.240
So I used dimreg here.
00:25:11.878 --> 00:25:13.420
Because if you look
at this integral,
00:25:13.420 --> 00:25:16.240
there's three pairs of Q
upstairs, two downstairs.
00:25:16.240 --> 00:25:18.458
So it's power law divergent.
00:25:18.458 --> 00:25:20.500
But we don't see power
law divergences in dimreg,
00:25:20.500 --> 00:25:23.155
we just get this finite answer.
00:25:29.120 --> 00:25:32.030
And actually, that finite answer
is exactly the imaginary part
00:25:32.030 --> 00:25:34.670
that you need if you
want to cut to graph,
00:25:34.670 --> 00:25:37.970
and say that the cut of
the forward scattering
00:25:37.970 --> 00:25:42.360
is the same as this
amplitude squared.
00:25:42.360 --> 00:25:44.390
So it's exactly,
in essence, this
00:25:44.390 --> 00:25:47.090
is the piece that you need
to be there by unitary.
00:25:50.210 --> 00:25:51.867
If you were trying
to keep the E,
00:25:51.867 --> 00:25:53.450
you could think,
well, maybe if I just
00:25:53.450 --> 00:25:54.950
kept the E in this
calculation, it
00:25:54.950 --> 00:25:56.610
would solve this pinch problem.
00:25:56.610 --> 00:26:00.050
But it doesn't really do
it because if you really
00:26:00.050 --> 00:26:03.290
stick with the partial D by DT
as your leading order action,
00:26:03.290 --> 00:26:06.830
then the equations of
motion are equal 0, so.
00:26:06.830 --> 00:26:10.145
You have to take equal
0 to go on [INAUDIBLE]..
00:26:10.145 --> 00:26:12.020
So you can't really
avoid the pinch that way,
00:26:12.020 --> 00:26:15.980
you really have to
take this kinetic term.
00:26:15.980 --> 00:26:18.470
you have to take the kinetic
term to have both the partial
00:26:18.470 --> 00:26:21.440
DT and the [? grad ?]
[? squared ?] over 2M.
00:26:21.440 --> 00:26:23.510
OK, any questions so far?
00:26:28.417 --> 00:26:30.000
All right, well
there's something here
00:26:30.000 --> 00:26:31.680
that might bother you.
00:26:31.680 --> 00:26:34.350
We've got an M upstairs.
00:26:34.350 --> 00:26:35.730
M is big.
00:26:35.730 --> 00:26:39.970
M is the mass the nucleon,
and it's appearing upstairs.
00:26:39.970 --> 00:26:41.250
That's always a bad sign.
00:26:43.920 --> 00:26:45.450
Usually a bad sign.
00:26:45.450 --> 00:26:48.720
Well, at least that's something
we should worry about.
00:26:48.720 --> 00:26:50.490
So let's figure out
where all the M's are.
00:26:53.910 --> 00:26:57.540
Let's count all the
M's in the theory,
00:26:57.540 --> 00:26:59.025
holding the momentum fixed.
00:27:04.960 --> 00:27:07.410
So if we hold the
momentum fixed,
00:27:07.410 --> 00:27:11.520
then [? grad ?] has no M's.
00:27:11.520 --> 00:27:17.340
But partial T scales
like 1 over M.
00:27:17.340 --> 00:27:25.360
And correspondingly,
T scales like M. OK.
00:27:25.360 --> 00:27:30.050
X and [? grad ?] don't scale
but partial T and T do.
00:27:30.050 --> 00:27:32.950
And that's exactly what makes
these two terms the same size.
00:27:38.300 --> 00:27:42.640
So we can ask, what is the
scaling of our nucleon field?
00:27:42.640 --> 00:27:47.860
And if we want to do that,
we go back to our action
00:27:47.860 --> 00:27:52.410
and we just say the action
shouldn't have any scaling.
00:27:52.410 --> 00:27:55.900
T has a scaling, so
there's an M in here.
00:27:55.900 --> 00:27:58.690
This guy here, the whole
thing scales nicely, 1 over M.
00:27:58.690 --> 00:28:00.770
From what we just
said over there.
00:28:00.770 --> 00:28:04.630
This M cancels that
1 over M, so this guy
00:28:04.630 --> 00:28:08.390
is therefore M to the 0.
00:28:08.390 --> 00:28:13.000
The nucleon field has
no scaling with them.
00:28:13.000 --> 00:28:15.370
So then, once you've done the
kinetic term to figure out
00:28:15.370 --> 00:28:17.260
the scaling of the
nucleon field, you can do,
00:28:17.260 --> 00:28:20.875
then, any other interaction.
00:28:20.875 --> 00:28:22.750
So that the power
counting, the kinetic term,
00:28:22.750 --> 00:28:26.080
is always which determines the
power counting of the field.
00:28:26.080 --> 00:28:28.443
That's true in any theory,
any effective theory,
00:28:28.443 --> 00:28:29.110
because you're--
00:28:32.080 --> 00:28:34.180
that's the basis of
the fluctuations you're
00:28:34.180 --> 00:28:37.578
describing by that field.
00:28:37.578 --> 00:28:39.370
And once you've fixed
that counting, you've
00:28:39.370 --> 00:28:41.120
got all the counting
you need, and now you
00:28:41.120 --> 00:28:44.590
can go count other operators
like the one with 2M
00:28:44.590 --> 00:28:48.100
derivatives.
00:28:48.100 --> 00:28:50.290
And this guy here just has--
00:28:50.290 --> 00:28:53.410
you can have just vector
derivatives, no time
00:28:53.410 --> 00:28:55.840
derivatives, just vector
derivatives, 2M of them,
00:28:55.840 --> 00:28:56.830
and nucleon fields.
00:28:56.830 --> 00:29:00.640
So there's no M's in
there, that's M to the 0.
00:29:00.640 --> 00:29:05.260
And this guy here is
an M. So this guy here
00:29:05.260 --> 00:29:17.380
is C2M, must be a 1 over M. So
therefore, any C2M scales like
00:29:17.380 --> 00:29:24.490
1 over M. And now
you see it's not
00:29:24.490 --> 00:29:26.680
such a problem, because
here I have 2 C0's,
00:29:26.680 --> 00:29:28.270
each one of them
scales like 1 over M,
00:29:28.270 --> 00:29:30.730
I pick up one more
M in the numerator,
00:29:30.730 --> 00:29:33.790
and that just makes the whole
thing feel like one over M.
00:29:33.790 --> 00:29:36.220
Which is the same order that
the tree level was scaling,
00:29:36.220 --> 00:29:39.550
the tree level with
scaling like 1 over M.
00:29:39.550 --> 00:29:44.230
So loop diagrams and the tree
level both scale like 1 over M,
00:29:44.230 --> 00:29:46.030
and that's actually
generically true
00:29:46.030 --> 00:29:48.560
because every time I add
a loop I add a coupling,
00:29:48.560 --> 00:29:51.350
and so those two M's can
compensate each other.
00:29:51.350 --> 00:29:52.690
So there's no issue with M's.
00:30:05.090 --> 00:30:10.700
So the one that graph is
the same size as tree level,
00:30:10.700 --> 00:30:16.190
and they both go like
1 over M. If you now
00:30:16.190 --> 00:30:22.730
do dimension counting, you can
say given that we've identified
00:30:22.730 --> 00:30:26.990
that the C has an
M in it, if you
00:30:26.990 --> 00:30:31.880
ask about the dimensions
of C, -2 minus 2M.
00:30:31.880 --> 00:30:36.390
Because the dimensions of the
nucleon field are 3 halves.
00:30:36.390 --> 00:30:39.080
And we're doing some expansion
in P much less than lambda,
00:30:39.080 --> 00:30:40.850
so just by dimensional
power counting
00:30:40.850 --> 00:30:47.710
we expect that the coefficients
would be of the following size.
00:30:47.710 --> 00:30:51.200
We know that there's an M
and that it's the only M,
00:30:51.200 --> 00:30:52.910
and all the rest
of the dimensions
00:30:52.910 --> 00:30:54.980
you should think
of as being made up
00:30:54.980 --> 00:30:56.910
by the stuff you
integrated it out.
00:30:56.910 --> 00:30:58.700
So it could be the
pion, for example,
00:30:58.700 --> 00:31:00.540
setting the scale lambda.
00:31:00.540 --> 00:31:02.300
So what you'd expect
for the C2M's is
00:31:02.300 --> 00:31:05.060
that this is how big they are,
and that the derivatives are
00:31:05.060 --> 00:31:07.998
then being suppressed
because of these lambdas
00:31:07.998 --> 00:31:10.040
and you're expanding for
P much less than lambda.
00:31:13.540 --> 00:31:14.590
OK.
00:31:14.590 --> 00:31:15.690
Are we happy so far?
00:31:20.470 --> 00:31:22.120
All right, and
we'll see when you
00:31:22.120 --> 00:31:24.740
do matching calculations, this
M [? law's ?] always there.
00:31:24.740 --> 00:31:29.770
And this is just a nice, elegant
way of figuring that out.
00:31:29.770 --> 00:31:34.445
Because this is a
very simple argument.
00:31:34.445 --> 00:31:34.945
All right.
00:31:34.945 --> 00:31:36.903
AUDIENCE: I don't know
if I totally understand.
00:31:36.903 --> 00:31:39.820
How do you know that C2M--
00:31:39.820 --> 00:31:41.380
PROFESSOR: Goes like 1 over M?
00:31:41.380 --> 00:31:42.160
AUDIENCE: Yeah--
00:31:42.160 --> 00:31:43.860
PROFESSOR: So the kinetic term--
00:31:43.860 --> 00:31:47.350
so at first you
know this, right?
00:31:47.350 --> 00:31:48.550
AUDIENCE: Yeah.
00:31:48.550 --> 00:31:49.690
PROFESSOR: Yeah, so I see.
00:31:49.690 --> 00:31:50.860
You're worried about
whether there could be
00:31:50.860 --> 00:31:52.080
1 over M squared corrections?
00:31:52.080 --> 00:31:52.705
AUDIENCE: Yeah.
00:31:52.705 --> 00:31:53.020
PROFESSOR: Yeah.
00:31:53.020 --> 00:31:55.240
In principle, there could be
1 over M squared corrections
00:31:55.240 --> 00:31:57.282
from relativistic corrections,
so that this would
00:31:57.282 --> 00:31:58.810
be the leading order term.
00:31:58.810 --> 00:31:59.500
Yeah.
00:31:59.500 --> 00:32:01.750
Not worse than that.
00:32:01.750 --> 00:32:05.560
If you looked at the P cubed,
P to the fourth over 8M
00:32:05.560 --> 00:32:08.133
cubed term, that term would
have a higher power of M,
00:32:08.133 --> 00:32:09.550
and you could
imagine that there's
00:32:09.550 --> 00:32:12.840
some relativistic corrections
in the four nucleons as well.
00:32:12.840 --> 00:32:14.590
Yeah.
00:32:14.590 --> 00:32:16.803
So we'll work basically
here, at lowest order
00:32:16.803 --> 00:32:18.220
in the relativistic
corrections so
00:32:18.220 --> 00:32:20.470
that you could put in the
relativistic corrections,
00:32:20.470 --> 00:32:20.970
as well.
00:32:24.060 --> 00:32:25.800
All right.
00:32:25.800 --> 00:32:27.990
So we're basically stopping
at that order, which
00:32:27.990 --> 00:32:29.448
is equivalent to
quantum mechanics,
00:32:29.448 --> 00:32:31.320
but we could go
further if we wanted to
00:32:31.320 --> 00:32:33.330
in this effective theory.
00:32:33.330 --> 00:32:37.980
So now, let's think about other
loop diagrams in the theory,
00:32:37.980 --> 00:32:39.480
without relativistic
corrections,
00:32:39.480 --> 00:32:40.800
just with these interactions.
00:32:40.800 --> 00:32:43.890
And let's insert the 2N and
the 2M derivative operator
00:32:43.890 --> 00:32:45.510
on these vertices.
00:32:45.510 --> 00:32:48.720
You see all the loops look
like this, they're all bubbles.
00:32:48.720 --> 00:32:50.820
And that's the
same reason in HQET
00:32:50.820 --> 00:32:53.070
that you don't have any
diagrams that kind of-- you
00:32:53.070 --> 00:32:55.770
don't have diagrams that look
like this, because these types
00:32:55.770 --> 00:32:58.365
of diagrams involve
antiparticles
00:32:58.365 --> 00:32:59.490
and we only have particles.
00:33:03.717 --> 00:33:05.800
So that's the beauty of a
non-relativistic theory,
00:33:05.800 --> 00:33:07.650
you don't have any
diagrams like that.
00:33:07.650 --> 00:33:09.830
You just have
diagrams like this.
00:33:09.830 --> 00:33:12.610
And so basically, the
whole theory is bubbles.
00:33:12.610 --> 00:33:13.590
The theory of bubbles.
00:33:16.290 --> 00:33:18.180
If you go through
this loop diagram,
00:33:18.180 --> 00:33:20.190
you do the pole the same way.
00:33:20.190 --> 00:33:24.570
It's exactly the
same two propagators.
00:33:24.570 --> 00:33:33.443
Then you get the same Q
squared minus ME, minus I0.
00:33:33.443 --> 00:33:34.860
And the only
difference is you get
00:33:34.860 --> 00:33:39.435
powers of Q in the numerator.
00:33:39.435 --> 00:33:41.310
And this, so this integral
is one of the ones
00:33:41.310 --> 00:33:42.840
that shows up in here.
00:33:42.840 --> 00:33:46.230
And this integral, you can
do the same kind of trick
00:33:46.230 --> 00:33:49.290
that you used when we were
discussing field definitions,
00:33:49.290 --> 00:33:52.020
where you basically take
the top and organize it
00:33:52.020 --> 00:33:55.150
by adding and subtracting.
00:33:55.150 --> 00:34:09.530
So add and subtract
any like that.
00:34:09.530 --> 00:34:11.600
And now, you can think,
N and M are integers,
00:34:11.600 --> 00:34:13.969
just expand this thing out.
00:34:13.969 --> 00:34:16.940
Some number of these factors,
some number of these factors.
00:34:16.940 --> 00:34:19.310
But any time you get one
or more of these factors,
00:34:19.310 --> 00:34:20.780
it cancels the
denominator and then
00:34:20.780 --> 00:34:23.058
you end up with
scaleless integrals.
00:34:23.058 --> 00:34:25.350
So basically, that means you
can throw this piece away.
00:34:41.150 --> 00:34:43.580
So higher order
derivatives actually just
00:34:43.580 --> 00:34:47.330
lead to ME's, whether
they act inside or outside
00:34:47.330 --> 00:34:49.850
on the nucleon fields,
they lead to P squared.
00:34:53.449 --> 00:34:54.415
Yeah, sorry.
00:34:54.415 --> 00:34:55.790
This M is a little
M, and this is
00:34:55.790 --> 00:35:00.040
supposed to be a big
M. Oh, N. Oh yeah,
00:35:00.040 --> 00:35:02.750
I am also using that too.
00:35:02.750 --> 00:35:03.800
Sorry.
00:35:03.800 --> 00:35:04.770
Yes.
00:35:04.770 --> 00:35:05.810
This is an integer.
00:35:05.810 --> 00:35:08.450
I should call it J or something.
00:35:08.450 --> 00:35:13.295
And the other M is
3 minus 2 epsilon.
00:35:28.100 --> 00:35:30.570
Yeah, it's dangerous.
00:35:30.570 --> 00:35:31.610
All right.
00:35:31.610 --> 00:35:34.220
But this means that basically,
the graphs in this theory
00:35:34.220 --> 00:35:35.010
are very simple.
00:35:35.010 --> 00:35:38.390
So if we actually now consider
the complete set of diagrams,
00:35:38.390 --> 00:35:39.365
or we impose--
00:35:42.530 --> 00:35:46.640
we put the full
amplitude in there
00:35:46.640 --> 00:35:47.990
and we consider these bubbles.
00:35:53.060 --> 00:35:54.620
Because of this fact
that I just told
00:35:54.620 --> 00:35:57.770
you, which doesn't change
when you insert more bubbles,
00:35:57.770 --> 00:36:01.640
the bubbles all decouple
from each other.
00:36:01.640 --> 00:36:03.470
The contact interaction
is decoupling them
00:36:03.470 --> 00:36:05.730
from each other.
00:36:05.730 --> 00:36:14.180
So here's a complete
amplitude with K insertions
00:36:14.180 --> 00:36:16.430
of the coupling,
and K minus 1 loops.
00:36:24.990 --> 00:36:28.245
OK, so we just completed all
the loop diagrams the theory,
00:36:28.245 --> 00:36:30.120
and it's giving us
something that we can sum,
00:36:30.120 --> 00:36:31.320
its a geometric series.
00:36:35.740 --> 00:36:40.540
So this is for K minus 1
loops, we can sum them up.
00:36:40.540 --> 00:36:46.150
So if you like I should have
called this the K-th term.
00:36:46.150 --> 00:37:07.830
And if I sum up, cancel the
I on each side, you get that.
00:37:16.270 --> 00:37:17.410
Let me round it again.
00:37:24.890 --> 00:37:25.610
Like this.
00:37:30.235 --> 00:37:34.390
So dividing out the
numerator, rearranging
00:37:34.390 --> 00:37:38.150
the four pi over M's little
bit, I can write it like that.
00:37:38.150 --> 00:37:40.030
And then we can
identify this thing here
00:37:40.030 --> 00:37:41.320
as P cotangent delta.
00:37:45.950 --> 00:37:51.020
OK, so P cotangent delta,
from our effective theory,
00:37:51.020 --> 00:37:55.925
we calculate it to
be the sum over this,
00:37:55.925 --> 00:37:59.870
P2M, where now I've
conveniently defined it,
00:37:59.870 --> 00:38:04.160
hatted coefficients, which
are the following things.
00:38:07.540 --> 00:38:09.040
And that just
cancels the M that's
00:38:09.040 --> 00:38:13.990
in the C2M's, so C2 hat
scales like M to the 0.
00:38:16.960 --> 00:38:18.040
So that's just a--
00:38:18.040 --> 00:38:22.060
is an obvious way of
reorganizing things given
00:38:22.060 --> 00:38:24.610
the factors of 4 pi
over M that we knew
00:38:24.610 --> 00:38:27.310
had to sit out front
of this S matrix
00:38:27.310 --> 00:38:30.050
calculation of the amplitude.
00:38:30.050 --> 00:38:33.190
So we can identify P
cotangent delta as something
00:38:33.190 --> 00:38:35.500
that doesn't involve any M's,
and that actually is also
00:38:35.500 --> 00:38:39.440
what we would expect for
non-relativistic scattering.
00:38:39.440 --> 00:38:41.200
So then you can look
at different waves,
00:38:41.200 --> 00:38:45.725
and you can look at this
formula, and you can--
00:38:45.725 --> 00:38:47.600
doing two of them will
show you how it works.
00:38:47.600 --> 00:38:50.260
So S wave is L equals 0.
00:39:14.030 --> 00:39:18.140
So that's something we can do
a Taylor expansion in P in.
00:39:18.140 --> 00:39:20.460
And this is what the
Taylor series looks like.
00:39:20.460 --> 00:39:22.530
And this has exactly
the form that we wanted.
00:39:22.530 --> 00:39:24.800
This is 1-- some
constant 1 over A,
00:39:24.800 --> 00:39:27.290
which is the scattering length.
00:39:27.290 --> 00:39:29.090
Some constant times
P squared, which
00:39:29.090 --> 00:39:31.580
is called the effective range.
00:39:31.580 --> 00:39:33.535
And we see that
expansion coming out,
00:39:33.535 --> 00:39:35.660
and we never had to specify
what the potential was,
00:39:35.660 --> 00:39:38.270
because the effective
theory was agnostic to what
00:39:38.270 --> 00:39:39.050
the potential was.
00:39:39.050 --> 00:39:40.490
That's the whole power
of effective theory,
00:39:40.490 --> 00:39:42.890
that you don't need to know
what the particles were that you
00:39:42.890 --> 00:39:45.057
were integrating out, they
just give you some values
00:39:45.057 --> 00:39:46.490
for these coefficients.
00:39:46.490 --> 00:39:49.598
And those values exactly
become the effective range
00:39:49.598 --> 00:39:51.140
and scattering length
in this theory.
00:39:53.930 --> 00:39:56.620
AUDIENCE: So this seems
very [INAUDIBLE] dependent.
00:39:56.620 --> 00:40:00.130
PROFESSOR: Yeah, we're
going to talk about that.
00:40:00.130 --> 00:40:02.110
Yes.
00:40:02.110 --> 00:40:06.220
Yeah, that will encompass
the second half of lecture.
00:40:18.230 --> 00:40:20.185
So we'll come to
that momentarily.
00:40:23.510 --> 00:40:27.065
Let me just do one more
way, just so you see.
00:40:27.065 --> 00:40:29.220
L equals 1.
00:40:29.220 --> 00:40:31.820
in the L equals 1 case,
there's no C0 hat,
00:40:31.820 --> 00:40:35.970
you need the derivatives
to correspond to the wave.
00:40:35.970 --> 00:40:43.220
And so in this case, you
look at PQ cotangent delta,
00:40:43.220 --> 00:40:44.750
the denominator starts at C2.
00:40:47.390 --> 00:40:49.760
And the reason why there
is this P to the 2L plus 1
00:40:49.760 --> 00:40:52.380
is just so that you
get the extra P here,
00:40:52.380 --> 00:40:54.217
which compensates
the P's downstairs,
00:40:54.217 --> 00:40:55.925
and again this thing
has a Taylor series.
00:41:04.900 --> 00:41:07.990
Same story as over there, we
can identify the scattering link
00:41:07.990 --> 00:41:10.750
for the P wave as
the 1 over C2 hat.
00:41:13.977 --> 00:41:17.680
So all the higher partial
ways work the same way.
00:41:17.680 --> 00:41:20.050
And you just have P to
the 2 L plus 1 in front
00:41:20.050 --> 00:41:22.450
of your cotangent delta.
00:41:22.450 --> 00:41:22.950
OK.
00:41:27.080 --> 00:41:30.332
Here you can see how this
generalizes [INAUDIBLE]..
00:41:35.160 --> 00:41:37.020
So that proves this
non-relativistic quantum
00:41:37.020 --> 00:41:38.990
mechanics theorem.
00:41:52.160 --> 00:41:55.010
And we did it without having to
specify what the potential was,
00:41:55.010 --> 00:41:58.010
because the potential is
encoded in BC's and effectively
00:41:58.010 --> 00:42:00.647
that's like a basis
expansion of the potential.
00:42:00.647 --> 00:42:02.730
But it's a fun one because
it's in delta functions
00:42:02.730 --> 00:42:04.740
and derivatives of
delta functions.
00:42:04.740 --> 00:42:07.010
So we're doing a
local effective field
00:42:07.010 --> 00:42:09.200
theory, which is not
something you'd think
00:42:09.200 --> 00:42:11.230
of ever using for a basis--
00:42:11.230 --> 00:42:13.310
well, maybe you
would, but most people
00:42:13.310 --> 00:42:16.190
wouldn't think of using basis of
derivatives of delta functions
00:42:16.190 --> 00:42:18.120
for quantum mechanics.
00:42:18.120 --> 00:42:18.620
OK.
00:42:21.662 --> 00:42:23.870
Well, you'd hope we can do
a little more than quantum
00:42:23.870 --> 00:42:24.578
mechanics, right?
00:42:29.000 --> 00:42:31.607
So you can think, if
you like, in terms
00:42:31.607 --> 00:42:33.690
of determining the values
of the C's, you can say,
00:42:33.690 --> 00:42:36.232
well, experiment tells me the
values of the scattering length
00:42:36.232 --> 00:42:38.510
are 0, and that's indeed true.
00:42:42.300 --> 00:42:45.268
So this equation that
C0 hat is equal to A,
00:42:45.268 --> 00:42:47.060
you could view this as
a matching equation.
00:42:49.980 --> 00:42:54.770
Putting back the
M, you have that.
00:42:54.770 --> 00:42:59.030
And then for C2 hat, we
have r0 over 2, a squared.
00:43:07.980 --> 00:43:10.370
So pretend that
experiment gives r0.
00:43:15.660 --> 00:43:17.913
And a, which they
do measure, and then
00:43:17.913 --> 00:43:19.830
you know the value of
your Wilson coefficients
00:43:19.830 --> 00:43:22.230
and you can start
using this theory.
00:43:22.230 --> 00:43:27.750
Now, if you think about
the power counting, if a
00:43:27.750 --> 00:43:30.570
and r0 are order 1
over lambda, that's
00:43:30.570 --> 00:43:32.408
the natural thing you'd expect.
00:43:32.408 --> 00:43:33.825
Then you reproduce
what we expect.
00:43:43.230 --> 00:43:45.270
Whatever it was a minute ago.
00:43:50.340 --> 00:43:51.440
2M plus one.
00:43:56.810 --> 00:44:00.410
So if all the constants
are scaling like whatever
00:44:00.410 --> 00:44:01.980
their dimensions
are, in this case
00:44:01.980 --> 00:44:05.235
they're both dimension
minus 1 in momentum units,
00:44:05.235 --> 00:44:07.610
then you would reproduce
exactly with this power counting
00:44:07.610 --> 00:44:09.340
that we said.
00:44:09.340 --> 00:44:11.330
OK, so everything would be nice.
00:44:11.330 --> 00:44:13.050
But nature is not so nice.
00:44:13.050 --> 00:44:16.460
Or nature threw us in
a different direction.
00:44:16.460 --> 00:44:19.293
When we actually look at the
value of these constants,
00:44:19.293 --> 00:44:20.585
the scattering length is large.
00:44:27.970 --> 00:44:34.100
So C0 is large, from
this point of view.
00:44:37.303 --> 00:44:38.720
Let me quote some
numbers for you.
00:44:58.820 --> 00:45:03.320
So in the 1S0 channel,
which is the larger one,
00:45:03.320 --> 00:45:04.700
this guy is 23 fermis.
00:45:11.990 --> 00:45:17.288
And in the 3S1 channel, it's
5 fermis, and both of these
00:45:17.288 --> 00:45:18.080
are actually large.
00:45:23.260 --> 00:45:25.010
We know they're large,
look at the errors.
00:45:27.650 --> 00:45:29.570
If you want to think
about momentum units,
00:45:29.570 --> 00:45:32.480
you could take 1 over a.
00:45:32.480 --> 00:45:38.940
And for this guy here, taking
1 over a is giving him, like,
00:45:38.940 --> 00:45:43.910
if I did the calculation
right, minus 8.30 MeV.
00:45:43.910 --> 00:45:45.170
Oh, sorry, no.
00:45:45.170 --> 00:45:45.920
That's this guy.
00:45:48.750 --> 00:45:55.740
And this guy here is
giving 1 over a, is 36 MeV.
00:45:55.740 --> 00:45:57.950
So if you thought the
natural size was the pion,
00:45:57.950 --> 00:46:00.860
then you'd say, well, these
constants should be 1 over pi--
00:46:00.860 --> 00:46:04.130
that these numbers that
are in MeV should be in pi.
00:46:04.130 --> 00:46:07.970
And 8 MeV is a much smaller
number than 138 MeV.
00:46:07.970 --> 00:46:10.830
36 is also a smaller
number than 130 MeV.
00:46:10.830 --> 00:46:17.600
So both of these
guys are not natural.
00:46:17.600 --> 00:46:20.935
In particular, this one you
see it's very not natural.
00:46:24.080 --> 00:46:26.620
So there's a fine
tuning going on.
00:46:26.620 --> 00:46:29.082
Some kind of fine tuning
from the perspective--
00:46:37.460 --> 00:46:40.080
from our dimensional
counting EFT point of view.
00:46:45.120 --> 00:46:49.870
There's a fine tuning
that's making the a big.
00:46:49.870 --> 00:46:53.460
And if you look at
the other guys, the r0
00:46:53.460 --> 00:46:55.770
and the other guys,
they are exactly
00:46:55.770 --> 00:46:56.890
of the size you'd expect.
00:46:56.890 --> 00:46:58.223
So there's no fine tuning there.
00:47:01.440 --> 00:47:04.560
The only fine tuning is in a.
00:47:04.560 --> 00:47:06.660
And not the other ones.
00:47:06.660 --> 00:47:08.880
Just by comparing two data.
00:47:08.880 --> 00:47:10.300
OK, so how do we deal with that?
00:47:10.300 --> 00:47:12.092
It looks like we set
up a defective theory.
00:47:12.092 --> 00:47:13.660
It has seems like
a beautiful theory,
00:47:13.660 --> 00:47:15.910
it could describe some things
about quantum mechanics,
00:47:15.910 --> 00:47:20.050
but then we learned that
our power counting sucks.
00:47:20.050 --> 00:47:22.790
So what we want is
actually a power counting
00:47:22.790 --> 00:47:24.188
it's a little different.
00:47:32.010 --> 00:47:38.190
Where AP is of order 1, or
even AP much greater than 1,
00:47:38.190 --> 00:47:40.200
we'd like that to be allowed.
00:47:40.200 --> 00:47:44.520
Where our 0P is
much less than 1.
00:47:44.520 --> 00:47:47.160
We'd like to be able to take a
scattering length effectively
00:47:47.160 --> 00:47:50.310
into account to all orders,
and that means basically
00:47:50.310 --> 00:47:55.170
that we want to
treat C0 as relevant.
00:47:55.170 --> 00:47:58.537
We don't want 8MeV to be
the limit of the lowered--
00:47:58.537 --> 00:48:00.870
the thing that goes downstairs
when we're making a power
00:48:00.870 --> 00:48:02.340
expansion, right?
00:48:02.340 --> 00:48:05.610
We'd like something like
M pi to be downstairs.
00:48:05.610 --> 00:48:06.990
If we want M pi
to be downstairs,
00:48:06.990 --> 00:48:09.180
you got to treat
AP to all orders.
00:48:09.180 --> 00:48:12.570
And then you're just limited
by M pi which is the r0 term.
00:48:12.570 --> 00:48:14.340
And that means you've
got to promote C0
00:48:14.340 --> 00:48:18.060
from being irrelevant,
scaling like 1 over M lambda,
00:48:18.060 --> 00:48:21.580
to something that's relevant.
00:48:21.580 --> 00:48:28.080
So this is actually
a problem that
00:48:28.080 --> 00:48:30.630
occurred because we just
proceeded and started
00:48:30.630 --> 00:48:34.820
calculating, and we used
effectively the MS bar scheme
00:48:34.820 --> 00:48:36.780
in dimensional regularization.
00:48:36.780 --> 00:48:39.720
We saw powers and
fractions, we did nothing.
00:48:39.720 --> 00:48:41.970
So let's try another scheme.
00:48:48.440 --> 00:48:51.260
It's a little more physical.
00:48:51.260 --> 00:48:54.800
Called offshell
momentum subtraction.
00:48:54.800 --> 00:48:57.530
So what is offshell
momentum subtraction?
00:48:57.530 --> 00:49:01.220
It says take the amplitude,
and in this non-relativistic
00:49:01.220 --> 00:49:05.990
theory, you take P to be
at some imaginary point
00:49:05.990 --> 00:49:10.302
so that you avoid any cuts.
00:49:10.302 --> 00:49:11.135
And you can define--
00:49:15.160 --> 00:49:21.520
and whatever channel we're
in, in this new r scheme,
00:49:21.520 --> 00:49:24.190
you can define the amplitude
of that particular point
00:49:24.190 --> 00:49:26.730
to be the tree level result.
00:49:26.730 --> 00:49:29.540
OK, so any loops, if you
take them at this point,
00:49:29.540 --> 00:49:35.830
you should get back
that result. So this
00:49:35.830 --> 00:49:38.080
is the analog of what you
do in a relativistic theory
00:49:38.080 --> 00:49:41.050
where you would take P squared
to be minus mu squared.
00:49:41.050 --> 00:49:42.550
In the non-relativistic
theory, it's
00:49:42.550 --> 00:49:46.000
always P that's showing up,
which is the three vector P,
00:49:46.000 --> 00:49:49.200
magnitude of the three vector.
00:49:49.200 --> 00:49:50.910
So we should assign
some rule for that.
00:49:50.910 --> 00:49:53.990
And it's just, we take
it to be I times mu.
00:50:00.620 --> 00:50:03.450
So how does this
change our calculation?
00:50:03.450 --> 00:50:06.750
So let's go back to
this one calculation.
00:50:06.750 --> 00:50:11.210
So now there's going to be
some counterterm needed.
00:50:11.210 --> 00:50:14.630
It's going to be a
finite counterterm.
00:50:14.630 --> 00:50:15.620
Let's see what it does.
00:50:21.990 --> 00:50:27.830
So the loop graph gave this IP,
that's what this graph gave.
00:50:27.830 --> 00:50:32.000
And this has to be just such
that if I set P equal to I mu,
00:50:32.000 --> 00:50:32.960
that I get 0.
00:50:32.960 --> 00:50:35.370
So this has two plus mu.
00:50:35.370 --> 00:50:39.050
So the counterterm is
giving mu r, C0 squared.
00:50:39.050 --> 00:50:42.455
And that exactly makes this
amplitude vanish at that point,
00:50:42.455 --> 00:50:45.080
and that's what you want because
the tree level graph of the C0
00:50:45.080 --> 00:50:47.690
is already giving
the right condition.
00:50:47.690 --> 00:50:48.470
OK?
00:50:48.470 --> 00:50:50.150
Is that clear to everybody?
00:50:50.150 --> 00:50:52.880
This is the correction
to that, to this.
00:50:52.880 --> 00:50:56.210
The tree level graph in the
amplitude here gave minus IC0,
00:50:56.210 --> 00:50:57.710
so we already got what we want.
00:50:57.710 --> 00:50:59.210
When we go to our
loop, we just want
00:50:59.210 --> 00:51:01.190
to make sure it
doesn't contribute,
00:51:01.190 --> 00:51:05.190
and that forces us to
put the plus mu there.
00:51:05.190 --> 00:51:10.940
And so what this mu is doing is
tracking the power divergence
00:51:10.940 --> 00:51:14.630
that dimreg in MS
bar did not see.
00:51:17.810 --> 00:51:20.450
There was an integral is power
law divergence and our cut
00:51:20.450 --> 00:51:22.910
off, mu r, has appeared
in the numerator
00:51:22.910 --> 00:51:24.981
and has tracked that divergence.
00:51:28.140 --> 00:51:29.970
So we're doing a
standard thing here where
00:51:29.970 --> 00:51:36.240
we split the coefficient
into the bare coefficient
00:51:36.240 --> 00:51:38.910
and to our more renormalized
and countertrend piece.
00:51:38.910 --> 00:51:41.310
And unlike MS bar in this
particular normalization
00:51:41.310 --> 00:51:44.085
scheme, there's a
finite correction there.
00:51:44.085 --> 00:51:45.960
And we have a renormalization
group equation.
00:52:00.540 --> 00:52:08.290
And if you work out what
that is from the counterterm,
00:52:08.290 --> 00:52:12.480
it turns out that only this
one loop, it's one loop exact.
00:52:12.480 --> 00:52:15.150
Showing that is a little
more work than I've done,
00:52:15.150 --> 00:52:18.690
but it turns out that the beta
function is one loop exact,
00:52:18.690 --> 00:52:20.520
higher bubbles don't
give any contribution
00:52:20.520 --> 00:52:24.540
to this beta function, and this
is the anomalous dimension.
00:52:24.540 --> 00:52:26.820
So we could just calculate
all those bubbles,
00:52:26.820 --> 00:52:28.840
and pose the same type of thing.
00:52:28.840 --> 00:52:31.898
And I've given you the
reference that does that.
00:52:31.898 --> 00:52:33.690
So there's a renormalization
group equation
00:52:33.690 --> 00:52:36.060
in this scheme which we
didn't have in MS bar.
00:52:36.060 --> 00:52:38.820
In MS bar it was
scale independent.
00:52:38.820 --> 00:52:41.250
We also have a connection
to the MS bar scheme
00:52:41.250 --> 00:52:44.970
because if mu r was 0,
that corresponds to what
00:52:44.970 --> 00:52:48.090
the MS bar result, right?
00:52:48.090 --> 00:52:53.010
So C0 of 0 is where you can
think of putting your boundary
00:52:53.010 --> 00:52:57.900
condition, which is matching to
the experiment, which is the a.
00:52:57.900 --> 00:53:00.210
And the advantage of
this offshell subtraction
00:53:00.210 --> 00:53:03.190
is that we have a mu, and
we can go somewhere else.
00:53:03.190 --> 00:53:07.740
And if you look at the
solution, of the RG
00:53:07.740 --> 00:53:11.557
with that boundary condition,
you see something interesting.
00:53:20.330 --> 00:53:22.840
So this is the result
for the coefficient.
00:53:22.840 --> 00:53:24.570
One over mu r, minus 1 over a.
00:53:28.240 --> 00:53:31.760
So if mu is of order P,
which is much greater than 1
00:53:31.760 --> 00:53:36.460
over a, like we want, then
the right counting for the C,
00:53:36.460 --> 00:53:40.210
which is a function
of mu, is 1 over M mu.
00:53:43.700 --> 00:53:45.950
OK?
00:53:45.950 --> 00:53:48.140
And so what we've
done here is we've
00:53:48.140 --> 00:53:54.770
swapped 1 over the physics
scale that we're integrating out
00:53:54.770 --> 00:53:56.960
for 1 over the scale
mu, which we're
00:53:56.960 --> 00:54:00.100
taking to be of order the
physics we're keeping.
00:54:00.100 --> 00:54:02.120
OK?
00:54:02.120 --> 00:54:03.440
And this is relevant now.
00:54:03.440 --> 00:54:05.680
This is a relevant
coupling with that change.
00:54:09.730 --> 00:54:11.620
We've made it much
bigger by just
00:54:11.620 --> 00:54:13.970
switching to the physical
renormalization scheme.
00:54:13.970 --> 00:54:16.210
And if we take mu of
order p in that scheme,
00:54:16.210 --> 00:54:19.030
this all of a sudden
becomes an order 1 effect
00:54:19.030 --> 00:54:20.740
with leading order
in the Lagrangian,
00:54:20.740 --> 00:54:25.710
because we have effectively
P's downstairs in the coupling.
00:54:25.710 --> 00:54:26.570
OK?
00:54:26.570 --> 00:54:30.100
So this scheme allows
us a way of thinking
00:54:30.100 --> 00:54:31.890
in the effective
theory way of thinking
00:54:31.890 --> 00:54:33.640
of having a power
counting where we'd have
00:54:33.640 --> 00:54:35.843
to keep the C0 to all orders.
00:54:35.843 --> 00:54:37.260
AUDIENCE: I thought
it was order--
00:54:37.260 --> 00:54:39.350
AP was order 1.
00:54:39.350 --> 00:54:40.820
PROFESSOR: Yeah, AP is order 1.
00:54:44.990 --> 00:54:46.760
So if you look at
the Lagrangian,
00:54:46.760 --> 00:54:53.180
then you have to go through
the counting of how many powers
00:54:53.180 --> 00:54:55.100
of P the nucleon field has.
00:54:55.100 --> 00:54:59.210
Which you could do in the same
way that we did with the mass.
00:54:59.210 --> 00:55:01.340
And if you do that with
the four-nucleon operator
00:55:01.340 --> 00:55:03.530
with an extra power
of P downstairs,
00:55:03.530 --> 00:55:08.600
you will find it has the same
P scaling as the kinetic term.
00:55:08.600 --> 00:55:09.230
OK?
00:55:09.230 --> 00:55:10.933
I didn't go through that, but--
00:55:10.933 --> 00:55:13.100
AUDIENCE: So that justifies
this, even when P is not
00:55:13.100 --> 00:55:14.555
much bigger than 1 over a?
00:55:14.555 --> 00:55:15.680
Is that what you're saying?
00:55:15.680 --> 00:55:18.350
PROFESSOR: Yeah, so if P is
much bigger than 1 over a, or P
00:55:18.350 --> 00:55:24.980
is even order or 1 over a,
which is a sort of also--
00:55:24.980 --> 00:55:27.560
it's also fine
with this counting.
00:55:27.560 --> 00:55:29.570
If P is of order 1
over a, these two terms
00:55:29.570 --> 00:55:32.750
are sort of comparably
big, but you could also
00:55:32.750 --> 00:55:34.940
use this approach for
that case, as long
00:55:34.940 --> 00:55:40.700
as you're not in the case where
P is much less than 1 over a.
00:55:40.700 --> 00:55:42.890
So if you're in
that case, then you
00:55:42.890 --> 00:55:45.420
should really think about
expanding this out some sense.
00:55:45.420 --> 00:55:47.670
And then you're getting back
to the end MS bar result,
00:55:47.670 --> 00:55:50.640
the MS bar result
would have been fine.
00:55:50.640 --> 00:55:52.650
But to get away from
the MS bar result,
00:55:52.650 --> 00:55:56.068
and think about
physics, mu of order P,
00:55:56.068 --> 00:55:57.860
we could use this scheme
instead of MS bar,
00:55:57.860 --> 00:56:00.650
and then we actually see that
we get a reasonable power
00:56:00.650 --> 00:56:01.450
counting.
00:56:01.450 --> 00:56:04.800
AUDIENCE: OK, Because my concern
is, what about when mu is like,
00:56:04.800 --> 00:56:07.092
what about this pole
that you're going to--
00:56:07.092 --> 00:56:10.181
PROFESSOR: Yeah, we'll
talk about the pole, yeah.
00:56:10.181 --> 00:56:12.580
It's coming up.
00:56:12.580 --> 00:56:14.800
All right.
00:56:14.800 --> 00:56:16.780
So it's interesting
to think about this
00:56:16.780 --> 00:56:19.075
from our renormalization
group point of view, which
00:56:19.075 --> 00:56:20.950
is kind of what I was
doing when I wrote down
00:56:20.950 --> 00:56:22.630
a beta function that I was--
00:56:22.630 --> 00:56:25.870
here I was just
after this solution,
00:56:25.870 --> 00:56:30.740
because from that solution I got
the power counting that I want.
00:56:30.740 --> 00:56:33.413
Which is that the C0 term is the
same size as the kinetic term,
00:56:33.413 --> 00:56:34.330
and both are relevant.
00:56:43.980 --> 00:56:46.910
So we can do that just like
we counted P's for [INAUDIBLE]
00:56:46.910 --> 00:56:48.500
theory, or just
like we counted--
00:56:51.430 --> 00:57:08.950
[? just comment. ?] These
guys are all relevant
00:57:08.950 --> 00:57:13.780
as long as we count this mu as
1 over P. 1 over mu is 1 over P.
00:57:13.780 --> 00:57:15.700
So there's another way
of thinking about this,
00:57:15.700 --> 00:57:17.408
and thinking about
these different cases,
00:57:17.408 --> 00:57:20.690
and that's just to think about
the beta function itself.
00:57:20.690 --> 00:57:24.730
So if you look at the beta
function for the C0 coupling,
00:57:24.730 --> 00:57:27.430
and we just put in the
solution, plug this back
00:57:27.430 --> 00:57:37.243
into that equation here,
with some constant out front,
00:57:37.243 --> 00:57:38.410
and then it looks like this.
00:57:38.410 --> 00:57:43.430
8 times mu, 1 minus
a mu, squared.
00:57:43.430 --> 00:57:45.880
If we want to talk about
all possible values of a,
00:57:45.880 --> 00:57:49.580
well a can go from minus
infinity to plus infinity.
00:57:49.580 --> 00:57:52.030
So it's useful if we want
to draw this to map it
00:57:52.030 --> 00:57:53.110
to a compact interval.
00:57:55.760 --> 00:57:56.560
So let's do that.
00:58:04.680 --> 00:58:06.950
Move the tangent.
00:58:06.950 --> 00:58:09.800
And let's just plot
beta as a function of x.
00:58:21.555 --> 00:58:23.805
So there's three values,
actually where beta vanishes.
00:58:27.150 --> 00:58:28.650
There's one value
where it blows up.
00:58:32.330 --> 00:58:38.150
So this is-- here
is x equals minus 1.
00:58:38.150 --> 00:58:43.170
This is x equals 0,
this is x equals plus 1.
00:58:45.840 --> 00:58:53.640
And what it looks like dips
down here, goes there, blows up,
00:58:53.640 --> 00:58:57.930
then it comes back
down, it goes like that.
00:58:57.930 --> 00:59:00.930
So that's what the beta
function looks like.
00:59:00.930 --> 00:59:03.290
So this point here
corresponds if you
00:59:03.290 --> 00:59:05.630
think of being at fixed mu.
00:59:05.630 --> 00:59:08.240
Say you're studying the
physics at fixed mu.
00:59:08.240 --> 00:59:11.720
This point corresponds to
a equals minus infinity,
00:59:11.720 --> 00:59:14.870
this point corresponds
to a equals zero,
00:59:14.870 --> 00:59:18.002
and this point corresponds
to a equals plus infinity.
00:59:24.400 --> 00:59:26.980
So there's three points where
the beta function vanishes,
00:59:26.980 --> 00:59:29.168
if you think about a
space for fixed mu.
00:59:41.590 --> 00:59:42.370
Use some color.
00:59:53.250 --> 00:59:55.450
So you can ask about nature.
00:59:55.450 --> 00:59:57.450
So nature told us the
value of a, so what
00:59:57.450 --> 00:59:59.200
does that correspond to?
00:59:59.200 --> 01:00:01.830
So taking some value
of P and mapping it
01:00:01.830 --> 01:00:04.823
to some value of x, kind of
generically the kind of point
01:00:04.823 --> 01:00:06.240
that you're
interested in is here.
01:00:06.240 --> 01:00:09.120
This is a1S0, sits there.
01:00:09.120 --> 01:00:16.420
And then a for the 3S1
makes it kind of here.
01:00:16.420 --> 01:00:19.350
So what's going on in this
case is that you're not
01:00:19.350 --> 01:00:22.800
near this fixed point, you're
actually close to this one.
01:00:22.800 --> 01:00:24.630
And you're not near
this fixed point,
01:00:24.630 --> 01:00:26.505
well actually there's
an infinity in between,
01:00:26.505 --> 01:00:28.510
you're closer to this one.
01:00:28.510 --> 01:00:32.940
This size you should think
of as 8 MeV, generically,
01:00:32.940 --> 01:00:35.580
and then this is like the 8 but
there's also a pole in between.
01:00:38.440 --> 01:00:39.670
So three fixed points.
01:00:42.778 --> 01:00:44.320
When we do perturbation
theory and we
01:00:44.320 --> 01:00:46.445
expand about fixed
points, and one way
01:00:46.445 --> 01:00:48.820
of saying what was wrong with
dimensional analysis was it
01:00:48.820 --> 01:00:51.340
was just expanding about
the wrong fixed point.
01:00:51.340 --> 01:00:51.910
The pink one.
01:00:55.300 --> 01:01:02.730
So a equals zero, was
the non-interactive one
01:01:02.730 --> 01:01:04.480
where we just had the
relevant interaction
01:01:04.480 --> 01:01:08.080
being the kinetic term, but
none of the interaction terms
01:01:08.080 --> 01:01:09.580
are relevant.
01:01:09.580 --> 01:01:13.060
And a equals plus
or minus infinity
01:01:13.060 --> 01:01:15.040
are interacting
fixed points, where
01:01:15.040 --> 01:01:17.170
you have an interaction
that is relevant.
01:01:22.370 --> 01:01:24.550
So you can think about that
in the following sense.
01:01:24.550 --> 01:01:27.070
Classically, what a is measuring
is kind of the interaction
01:01:27.070 --> 01:01:30.250
size, if you have classical
scattering across sections
01:01:30.250 --> 01:01:31.700
4 pi a squared.
01:01:31.700 --> 01:01:34.858
And if a is small--
01:01:34.858 --> 01:01:38.420
if a is either very
small or very big,
01:01:38.420 --> 01:01:41.290
then basically it's
the same on all scales
01:01:41.290 --> 01:01:44.290
because it's either
infinity or 0
01:01:44.290 --> 01:01:46.510
and it looks the
same to particles
01:01:46.510 --> 01:01:48.190
of all different momenta.
01:01:48.190 --> 01:01:49.150
OK?
01:01:49.150 --> 01:01:51.940
And that's one way of thinking
about these fixed points.
01:01:51.940 --> 01:01:53.680
That the physics can't--
01:01:53.680 --> 01:01:57.220
the physics-- yeah.
01:01:57.220 --> 01:01:58.720
Just what I said.
01:01:58.720 --> 01:02:00.880
So what about this infinity?
01:02:00.880 --> 01:02:02.518
That's also interesting.
01:02:10.486 --> 01:02:16.370
So when a is one over mu,
beta goes to infinity.
01:02:16.370 --> 01:02:19.910
And this actually--
there is a reflection
01:02:19.910 --> 01:02:23.480
of this in the theory, it
corresponds to a bound state.
01:02:28.580 --> 01:02:30.920
Which I think we'll talk
about next time, but.
01:02:33.440 --> 01:02:35.090
So there's a bound
state in the theory,
01:02:35.090 --> 01:02:37.310
and actually if you
start from this side,
01:02:37.310 --> 01:02:39.110
you can never see that
bound because you're
01:02:39.110 --> 01:02:39.980
doing perturbation theory.
01:02:39.980 --> 01:02:42.200
You don't see perturbation
theory and bound states.
01:02:42.200 --> 01:02:44.210
If you start from this
side, the bound state
01:02:44.210 --> 01:02:46.160
is actually just in the theory.
01:02:46.160 --> 01:02:48.605
We'll talk about that next time.
01:02:48.605 --> 01:02:50.480
And so it's a state in
the theory, you can go
01:02:50.480 --> 01:02:52.400
and you could find the
pole in your amplitude,
01:02:52.400 --> 01:02:53.900
it's just there.
01:02:53.900 --> 01:02:56.180
Corresponds to a physical
state of the spectrum,
01:02:56.180 --> 01:02:57.388
and it's called the deuteron.
01:03:14.620 --> 01:03:16.120
So we have a
non-interactive theory,
01:03:16.120 --> 01:03:19.990
we have some
non-trivial amplitude,
01:03:19.990 --> 01:03:22.540
and this deuteron is a
pole in that amplitude.
01:04:09.310 --> 01:04:13.890
And you never see a pole if you
use the perturbation theory.
01:04:13.890 --> 01:04:16.350
And if you actually look at
the energy, the binding energy
01:04:16.350 --> 01:04:18.773
of this state, it's also small.
01:04:18.773 --> 01:04:20.190
Characteristically
small, and it's
01:04:20.190 --> 01:04:24.030
actually related to the fact
that the binding energy is
01:04:24.030 --> 01:04:26.820
small just related to sort of
the natural size of this a.
01:04:26.820 --> 01:04:28.260
We'll talk about that next time.
01:04:34.850 --> 01:04:38.290
So what I was saying before
about the theory at all scales
01:04:38.290 --> 01:04:41.170
looking the same, when you
have a equals 0 of course
01:04:41.170 --> 01:04:42.550
it's not interacting.
01:04:42.550 --> 01:04:44.770
Or when a is equal to
plus minus infinity,
01:04:44.770 --> 01:04:47.770
looks the same at all scales.
01:04:47.770 --> 01:04:50.998
That means that scale invariant.
01:04:50.998 --> 01:04:52.540
So it's a scale
invariant theory when
01:04:52.540 --> 01:04:55.540
a is at these fixed points.
01:04:55.540 --> 01:04:58.600
And something I
worked on was the fact
01:04:58.600 --> 01:05:02.650
that these points are actually
conformal fixed points.
01:05:12.570 --> 01:05:16.600
So that-- there's a conformal
symmetry of the theory that
01:05:16.600 --> 01:05:20.090
exists at those points, we'll
talk about that a little bit.
01:05:20.090 --> 01:05:22.750
There's another symmetry,
too, which I have to mention.
01:05:25.480 --> 01:05:29.140
And not only because I
also worked on this one,
01:05:29.140 --> 01:05:32.070
because it's good to
enumerate all the symmetries.
01:05:35.360 --> 01:05:37.880
There's actually a combined
spin, isospin symmetry,
01:05:37.880 --> 01:05:41.027
that turns into an
Su4 in this limit.
01:05:41.027 --> 01:05:42.860
So much like in heavy
quark effective theory
01:05:42.860 --> 01:05:45.818
where the mass was big,
new symmetries popped up.
01:05:45.818 --> 01:05:48.110
Same thing happens here, when
the scattering lengths go
01:05:48.110 --> 01:05:50.552
to infinity, and you go
over to those fixed points,
01:05:50.552 --> 01:05:52.010
there's new symmetries
that pop up.
01:05:52.010 --> 01:05:53.927
One's a conformal symmetry
and the other one's
01:05:53.927 --> 01:05:55.190
a spin, isospin symmetry.
01:06:09.810 --> 01:06:12.060
All right.
01:06:12.060 --> 01:06:16.200
So this looks interesting.
01:06:16.200 --> 01:06:18.398
You could ask the
question, did this pick--
01:06:18.398 --> 01:06:20.190
did this physical
picture that we developed
01:06:20.190 --> 01:06:22.050
depend on picking
this renormalization
01:06:22.050 --> 01:06:24.240
scheme that I told you about?
01:06:24.240 --> 01:06:26.670
We kind of gave up on dimreg,
we went over to-- well,
01:06:26.670 --> 01:06:28.110
we gave up on MS
bar, we went over
01:06:28.110 --> 01:06:31.502
to this scheme which was
offshell momentum subtraction.
01:06:31.502 --> 01:06:33.960
In general, people don't like
offshell momentum subtraction
01:06:33.960 --> 01:06:35.793
because it makes
calculations more difficult
01:06:35.793 --> 01:06:37.001
once you go to higher orders.
01:06:37.001 --> 01:06:39.210
Well here, are the calculations
are not so difficult,
01:06:39.210 --> 01:06:40.320
so we could do them, but--
01:06:40.320 --> 01:06:42.750
you might be interested in
adding pions to this theory,
01:06:42.750 --> 01:06:45.420
or coupling external
currents like photons,
01:06:45.420 --> 01:06:47.910
and then the calculations
would get more difficult.
01:06:47.910 --> 01:06:51.180
And you'd like, for example,
to have a dimreg MS bar type
01:06:51.180 --> 01:06:53.190
description of
this power counting
01:06:53.190 --> 01:06:55.620
rather than a kind of
minimal subtraction.
01:06:55.620 --> 01:06:57.210
Could I get, could
I kind of dress up
01:06:57.210 --> 01:07:00.300
minimal subtraction to get
the same physical picture?
01:07:00.300 --> 01:07:03.210
That's a reasonable
question to ask.
01:07:03.210 --> 01:07:04.290
The answer is you can.
01:07:10.737 --> 01:07:13.070
So there's something called
power divergence subtraction
01:07:13.070 --> 01:07:15.650
scheme, different
scheme than MS bar.
01:07:22.244 --> 01:07:25.780
So the PDS scheme.
01:07:25.780 --> 01:07:31.060
And what it says is, don't just
subtract poles at D equals 4,
01:07:31.060 --> 01:07:37.510
like you do in MS bar, which
are corresponding to logs
01:07:37.510 --> 01:07:42.580
at the cut off, but also
subtract poles and D equals 3.
01:07:42.580 --> 01:07:45.090
And dimreg knows about
power law divergences
01:07:45.090 --> 01:07:46.840
and they're just poles
at different places
01:07:46.840 --> 01:07:48.530
in the dimensions.
01:07:48.530 --> 01:07:50.560
And so if we subtract
poles at D equals three,
01:07:50.560 --> 01:07:52.700
we can track the power law
divergences in that way.
01:07:52.700 --> 01:07:54.700
And it's the power of
divergence that's actually
01:07:54.700 --> 01:07:56.410
causing, if you
want to think of it
01:07:56.410 --> 01:07:58.495
as a change to the
anomalous dimension, where
01:07:58.495 --> 01:08:00.370
the anomalous dimension
was saying this thing
01:08:00.370 --> 01:08:03.250
was irrelevant, to changing
it to something relevant,
01:08:03.250 --> 01:08:05.000
you need a big change
for that to happen.
01:08:05.000 --> 01:08:06.730
And the big change
that's occurring
01:08:06.730 --> 01:08:09.100
is coming from a power
law divergence here.
01:08:09.100 --> 01:08:11.650
That's what sort of allowed
you to jump, if you like,
01:08:11.650 --> 01:08:15.730
from this fixed
point to this one.
01:08:15.730 --> 01:08:17.950
The renormalization group,
including the power law
01:08:17.950 --> 01:08:20.937
divergence, allows you to even
flow between those points.
01:08:20.937 --> 01:08:23.020
Usually we think that power
law divergences aren't
01:08:23.020 --> 01:08:25.562
doing anything, here's an
example where they are.
01:08:25.562 --> 01:08:26.979
They're not doing
anything as long
01:08:26.979 --> 01:08:28.812
as you know you're at
the right fixed point.
01:08:28.812 --> 01:08:31.078
If you're describing the
right physics around one
01:08:31.078 --> 01:08:33.370
of these fixed points, you
can concentrate on the logs,
01:08:33.370 --> 01:08:35.715
but if you don't know where
you are then the power
01:08:35.715 --> 01:08:37.090
law divergences
could be crucial.
01:08:39.609 --> 01:08:41.490
All right, how does
this scheme work?
01:08:45.250 --> 01:08:49.240
So this is a dimreg-type scheme.
01:08:49.240 --> 01:08:52.930
So we're going to
get the power of mu
01:08:52.930 --> 01:08:55.510
from the mu to the two epsilon
that we have out front.
01:09:07.520 --> 01:09:11.587
So if I just write this
guy down in D dimensions,
01:09:11.587 --> 01:09:12.670
here's what it looks like.
01:09:31.140 --> 01:09:33.770
And I've normalized mu slightly
differently than we usually
01:09:33.770 --> 01:09:37.680
do just because it's convenient
for this scheme to do that.
01:09:37.680 --> 01:09:40.160
So it's not exactly the same
as MS bar, it's mu over 2
01:09:40.160 --> 01:09:41.270
that I'm putting in.
01:09:41.270 --> 01:09:43.640
Other than that, it's the
same kind of set up as MS bar.
01:09:51.584 --> 01:09:54.269
You'll see why I want to put
that 2 there in a minute.
01:09:59.030 --> 01:10:01.608
OK, so this is just the result
that we would write down
01:10:01.608 --> 01:10:02.900
for dimensional regularization.
01:10:02.900 --> 01:10:04.865
Dimensional regularization
is not a scheme.
01:10:04.865 --> 01:10:08.930
A scheme has to do
with what we subtract.
01:10:08.930 --> 01:10:10.613
Dimreg is just how we regulate.
01:10:21.740 --> 01:10:25.160
So now, look at D equals 4.
01:10:25.160 --> 01:10:31.370
So in D equals 4, OK.
01:10:31.370 --> 01:10:32.495
There's a bunch of factors.
01:10:38.315 --> 01:10:40.940
This is just giving, this is the
answer I quoted to you before.
01:10:44.480 --> 01:10:46.700
Something finite.
01:10:46.700 --> 01:10:54.160
And if we look at
D equals three,
01:10:54.160 --> 01:10:56.285
then we have a pole because
of that gamma function.
01:11:02.030 --> 01:11:04.370
And I've put the 2 in here
just to cancel that 2.
01:11:07.520 --> 01:11:11.990
And so, what this scheme says
is to add a part subtraction
01:11:11.990 --> 01:11:14.090
for this guy.
01:11:14.090 --> 01:11:16.950
So what we do is, we
add a counter term.
01:11:16.950 --> 01:11:25.640
It looks like minus IM over 4
pi, mu, got one power of mu.
01:11:25.640 --> 01:11:29.480
Over 3 minus D. C0 squared.
01:11:32.430 --> 01:11:36.710
And then if we take the graph,
plus the counterterm and we
01:11:36.710 --> 01:11:38.840
set D equal four, which
is where we actually want
01:11:38.840 --> 01:11:45.320
to do physics, lo and behold.
01:11:45.320 --> 01:11:51.500
In this approach, you get
actually the same answer
01:11:51.500 --> 01:11:53.510
as in our offshell momentum
subtraction scheme.
01:11:53.510 --> 01:11:55.610
And that's just really because
this scheme tracks the power
01:11:55.610 --> 01:11:57.170
correction, the
power divergence,
01:11:57.170 --> 01:12:00.250
just like the offshell
momentum subtraction did.
01:12:00.250 --> 01:12:03.500
So we just invented
a dimreg style
01:12:03.500 --> 01:12:05.965
of looking for poles that
can track the same physics,
01:12:05.965 --> 01:12:07.340
and we just have
to look at poles
01:12:07.340 --> 01:12:11.440
in D equals 3 rather than
poles in D equals 4, OK?
01:12:13.887 --> 01:12:16.220
And this is easier in general
than the offshell momentum
01:12:16.220 --> 01:12:19.880
subtraction scheme, although
for basically everything
01:12:19.880 --> 01:12:22.650
we're talking about today
you could do either one.
01:12:22.650 --> 01:12:25.848
Now, where is the predictive
power of this effective theory?
01:12:25.848 --> 01:12:27.890
So far, we've just kind
of cooked things together
01:12:27.890 --> 01:12:30.350
to make the C0 do what we want.
01:12:30.350 --> 01:12:32.420
Well, we didn't completely
cook things together.
01:12:32.420 --> 01:12:34.370
We switched to another scheme
and it kind of popped out
01:12:34.370 --> 01:12:35.745
naturally, but
you can say, well,
01:12:35.745 --> 01:12:39.178
why not explore three other
schemes and see if they work?
01:12:39.178 --> 01:12:41.720
But let's just imagine that we
got things to work, as we just
01:12:41.720 --> 01:12:44.210
did in two different ways
by tracking the power law
01:12:44.210 --> 01:12:45.320
divergence.
01:12:45.320 --> 01:12:47.750
The predictive power
becomes from now the fact
01:12:47.750 --> 01:12:51.710
that if I say that's my fine
tuning, that C0 got enhanced,
01:12:51.710 --> 01:12:54.230
I can now predict the size
of all other operators
01:12:54.230 --> 01:12:55.310
in the theory.
01:12:55.310 --> 01:12:57.887
And other operators
like C2 and C4,
01:12:57.887 --> 01:13:00.470
the power counting we assigned
to them previously is not true.
01:13:00.470 --> 01:13:01.980
We have to figure it out.
01:13:01.980 --> 01:13:04.610
But we can figure
that out once we know
01:13:04.610 --> 01:13:09.160
what approach we should use.
01:13:09.160 --> 01:13:15.667
OK, so this is same as above.
01:13:15.667 --> 01:13:18.250
I won't go through it, but you
know, same anomalous dimension,
01:13:18.250 --> 01:13:19.420
et cetera.
01:13:19.420 --> 01:13:22.240
And it's easier in general.
01:13:22.240 --> 01:13:25.990
Let's think about C2 mu.
01:13:25.990 --> 01:13:29.170
So if you look at
C2, the first kind
01:13:29.170 --> 01:13:33.160
of diagram you could think about
would be a guy with one C2,
01:13:33.160 --> 01:13:36.070
and then this is the
first type of loop diagram
01:13:36.070 --> 01:13:38.240
you might think about.
01:13:38.240 --> 01:13:41.960
And these guys have a P squared
because C2 gave a P squared.
01:13:41.960 --> 01:13:43.840
So they have an extra P squared.
01:13:43.840 --> 01:13:45.010
And they diverge.
01:13:45.010 --> 01:13:46.690
They also have a
power divergence.
01:13:46.690 --> 01:13:48.280
So if you calculate
in either one
01:13:48.280 --> 01:13:50.320
of these schemes,
offshell momentum
01:13:50.320 --> 01:13:56.927
subtraction, or this PDS scheme,
these guys get a divergence.
01:13:56.927 --> 01:13:58.510
And again, it's a
power law divergence
01:13:58.510 --> 01:14:01.190
so there's a mu here.
01:14:01.190 --> 01:14:06.190
And you get a beta
function, that's that.
01:14:06.190 --> 01:14:08.800
2C0 C2.
01:14:08.800 --> 01:14:11.040
So if you take the
boundary condition
01:14:11.040 --> 01:14:15.670
C2 of 0 which is our MS
bar result. 4 pi over M. A
01:14:15.670 --> 01:14:17.320
squared, r0.
01:14:17.320 --> 01:14:30.010
And you solve this, you find
C2 of mu is 4 pi over M. 1
01:14:30.010 --> 01:14:34.870
over mu minus 1 over a, squared.
01:14:34.870 --> 01:14:36.010
r0 over 2.
01:14:41.700 --> 01:14:44.165
So there's two-- we
previously, with our counting,
01:14:44.165 --> 01:14:45.540
when we were
counting dimensions,
01:14:45.540 --> 01:14:49.080
we would have said C0
goes like 1 over lambda,
01:14:49.080 --> 01:14:51.900
C2 goes like one
over lambda cubed.
01:14:51.900 --> 01:14:54.480
We had 2M plus 1 lambdas.
01:14:54.480 --> 01:14:57.300
What we've just
discovered is that yes,
01:14:57.300 --> 01:15:00.700
there's a lambda from this
r0, that's a 1 over lambda,
01:15:00.700 --> 01:15:03.570
but the other two
lambdas are really mu's.
01:15:03.570 --> 01:15:06.900
So this operator
is also enhanced,
01:15:06.900 --> 01:15:09.797
and it's enhanced by two powers.
01:15:09.797 --> 01:15:11.380
So once you know
leading order theory,
01:15:11.380 --> 01:15:13.770
you should be able to determine
the power counting of all
01:15:13.770 --> 01:15:15.352
of the other
operators, especially
01:15:15.352 --> 01:15:16.560
if they're not relevant ones.
01:15:16.560 --> 01:15:18.450
You have to get the
relevant part right,
01:15:18.450 --> 01:15:20.040
and then you can
use that Lagrangian
01:15:20.040 --> 01:15:22.697
to predict all the scaling
for everything else.
01:15:22.697 --> 01:15:24.030
And that's what we've just done.
01:15:38.970 --> 01:15:41.850
Gone to 1 over mu
squared lambda.
01:15:41.850 --> 01:15:42.360
OK?
01:15:42.360 --> 01:15:45.223
And those powers of mu
we see in the scheme,
01:15:45.223 --> 01:15:46.890
and the 1 over lambda
comes from the r0.
01:15:54.000 --> 01:15:54.500
OK.
01:15:54.500 --> 01:15:59.780
So what the RGE actually
does is it tells us--
01:15:59.780 --> 01:16:02.990
one way of thinking about
it is that it tells us
01:16:02.990 --> 01:16:04.640
the enhancement,
due to fine tuning,
01:16:04.640 --> 01:16:05.990
of all operators in the theory.
01:16:09.607 --> 01:16:11.690
And that's really because
the fine tuning was just
01:16:11.690 --> 01:16:14.180
a change of our power
counting, and we
01:16:14.180 --> 01:16:16.930
have to propagate that
change everywhere.
01:16:22.930 --> 01:16:25.360
And we can do that, and
it's the beta functions
01:16:25.360 --> 01:16:28.870
that tell us how to
propagate the fine tuning.
01:16:28.870 --> 01:16:34.246
So if you keep going,
you can do C2K of mu.
01:16:34.246 --> 01:16:36.070
You find an anomalous dimension.
01:16:39.680 --> 01:16:43.060
This theory is kind of
nice, you can basically
01:16:43.060 --> 01:16:44.490
do all the calculations, so.
01:16:48.076 --> 01:16:50.500
When you go to CK,
you get contributions
01:16:50.500 --> 01:16:54.250
from various lower
order coefficients,
01:16:54.250 --> 01:16:56.365
and it's one loop exact
so you only have pairs.
01:16:59.365 --> 01:17:00.740
And then you can
kind of contrast
01:17:00.740 --> 01:17:04.880
what's going on in a
naive power counting where
01:17:04.880 --> 01:17:06.230
P is much less than 1 over a.
01:17:14.720 --> 01:17:17.750
Let's just go up to C4.
01:17:17.750 --> 01:17:21.200
Versus this kind
of improved power
01:17:21.200 --> 01:17:26.540
counting, which is valid when
PA is greater than our order 1.
01:17:30.995 --> 01:17:34.820
So C0 hat it went like 1 over
mu, no suppression there.
01:17:34.820 --> 01:17:37.370
C2 hat goes like one
over mu squared lambda,
01:17:37.370 --> 01:17:39.840
and that actually is irrelevant.
01:17:39.840 --> 01:17:41.480
But it's just
irrelevant by one power.
01:17:46.300 --> 01:17:49.950
So relative to this guy,
it's down by a P over lambda.
01:17:49.950 --> 01:17:52.020
And then interesting
things start
01:17:52.020 --> 01:17:54.660
to happen with the
higher ones, at least
01:17:54.660 --> 01:17:56.386
from an RGE perspective.
01:17:59.493 --> 01:18:01.410
So these guys start to
get more than one term.
01:18:05.358 --> 01:18:07.650
But this guy actually doesn't
introduce a new constant.
01:18:13.458 --> 01:18:16.000
There's a piece of the anomalous
dimension of this guy that's
01:18:16.000 --> 01:18:17.458
actually just fixed
by the constant
01:18:17.458 --> 01:18:20.170
that you already had here,
and then there's a new piece.
01:18:20.170 --> 01:18:22.350
So the new piece is down
by two powers of lambda.
01:18:26.545 --> 01:18:29.170
And that's encoding things about
the amplitude, actually, but--
01:18:32.240 --> 01:18:33.800
OK, so that's just
a little table
01:18:33.800 --> 01:18:35.342
to kind of convince
you that once you
01:18:35.342 --> 01:18:37.370
have a beta function
that you can compute
01:18:37.370 --> 01:18:39.260
for the coefficients,
you can quickly
01:18:39.260 --> 01:18:41.570
propagate this enhancement
from the fine tuning
01:18:41.570 --> 01:18:42.860
to the rest of your theory.
01:18:42.860 --> 01:18:45.320
I.e., figure out what
the power counting
01:18:45.320 --> 01:18:47.020
is for all the operators.
01:18:47.020 --> 01:18:49.640
AUDIENCE: So every time there
is a power law divergence,
01:18:49.640 --> 01:18:51.482
should I be worried
if I'm using MS bar,
01:18:51.482 --> 01:18:53.940
should I be worried that the
power counting could be wrong?
01:18:53.940 --> 01:18:56.160
PROFESSOR: Every time--
01:18:56.160 --> 01:18:58.160
Yeah, every time there's
a power law divergence,
01:18:58.160 --> 01:18:59.660
it's worth thinking
about whether it
01:18:59.660 --> 01:19:06.220
had some physical impact on
what you're doing, I think.
01:19:11.042 --> 01:19:13.500
If you know you're expanding--
if you can convince yourself
01:19:13.500 --> 01:19:15.930
that you're expanding around
the right fixed point then
01:19:15.930 --> 01:19:16.530
you're OK.
01:19:16.530 --> 01:19:20.520
That's my equivalence claim.
01:19:20.520 --> 01:19:22.020
But you don't
necessarily know that.
01:19:22.020 --> 01:19:24.110
So let's go back
to our amplitude
01:19:24.110 --> 01:19:26.930
and see what's going
on here, and see
01:19:26.930 --> 01:19:29.890
what it looks like with
this power counting.
01:19:29.890 --> 01:19:33.290
And so it's really just
a different expansion
01:19:33.290 --> 01:19:34.760
of that amplitude that we had.
01:19:41.420 --> 01:19:48.380
And in either the PDS scheme or
the power diversion subtraction
01:19:48.380 --> 01:19:54.620
scheme, we end up with this
amplitude in the case where
01:19:54.620 --> 01:19:56.670
we would use that scheme.
01:19:56.670 --> 01:19:59.948
So you can see in PDS that if
I set this coefficient to 0,
01:19:59.948 --> 01:20:01.490
the denominator
becomes 1, and then I
01:20:01.490 --> 01:20:04.580
get with the offshell
momentum subtraction scheme.
01:20:04.580 --> 01:20:07.640
In PDS is a little harder
see that it gives just that,
01:20:07.640 --> 01:20:10.080
but it actually gives the
same thing in either scheme.
01:20:10.080 --> 01:20:14.270
And if you think about what type
of expansion you're doing here,
01:20:14.270 --> 01:20:15.930
you're keeping C0 to all orders.
01:20:15.930 --> 01:20:23.720
So your amplitude at
lowest order is just this,
01:20:23.720 --> 01:20:38.510
and then the C2 term
looks like that.
01:20:38.510 --> 01:20:40.040
And then there's
some higher terms
01:20:40.040 --> 01:20:43.770
which I wrote in my notes,
but I want right here.
01:20:43.770 --> 01:20:47.270
And what this is, this here
is some kind of interaction.
01:20:47.270 --> 01:20:50.030
I'll make it a
bigger circle, which
01:20:50.030 --> 01:20:53.900
sums up all the bubbles
with C0's in them.
01:20:53.900 --> 01:20:56.780
That's what's happened here.
01:20:56.780 --> 01:21:01.610
And this here, if you like,
is like taking C2 and then
01:21:01.610 --> 01:21:04.535
dressing it with
bubbles on either side.
01:21:04.535 --> 01:21:07.700
So there's two, there's bubbles.
01:21:07.700 --> 01:21:11.523
The bubbles on the other side.
01:21:11.523 --> 01:21:12.815
And then bubbles on both sides.
01:21:16.500 --> 01:21:18.480
So we calculate
these loop graphs
01:21:18.480 --> 01:21:19.980
and these are the
amplitudes we get,
01:21:19.980 --> 01:21:21.355
and that's because
we're treating
01:21:21.355 --> 01:21:23.645
the C0 coupling to all
orders, we're summing it up.
01:21:23.645 --> 01:21:25.770
And actually, each of these
amplitudes, if you look
01:21:25.770 --> 01:21:28.590
at the RGE, is mu independent.
01:21:28.590 --> 01:21:31.115
Explicitly mu independent.
01:21:31.115 --> 01:21:32.490
So it's like
perturbation theory,
01:21:32.490 --> 01:21:33.960
where we're doing a
momentum expansion,
01:21:33.960 --> 01:21:35.877
and order by order in
that momentum expansion,
01:21:35.877 --> 01:21:39.300
the amplitude is
independent of the scale mu.
01:21:39.300 --> 01:21:41.160
The only purpose
of the scale mu is
01:21:41.160 --> 01:21:45.155
to help us think about power
counting of these operators.
01:21:45.155 --> 01:21:47.530
In the end of the day, when
we make physical predictions,
01:21:47.530 --> 01:21:50.790
then getting mu
independent answers.
01:21:50.790 --> 01:21:51.840
OK.
01:21:51.840 --> 01:21:55.813
And this is like
organizing, if you like.
01:21:55.813 --> 01:21:56.980
And you put it in terms of--
01:22:01.380 --> 01:22:07.255
put it back in
terms of a, this is
01:22:07.255 --> 01:22:08.880
like organizing the
theory in this way,
01:22:08.880 --> 01:22:12.900
where you keep all powers of
AP, and that makes it very clear
01:22:12.900 --> 01:22:14.910
that it's mu independent.
01:22:14.910 --> 01:22:17.100
Now, this part of
the theory is so
01:22:17.100 --> 01:22:19.380
simple you could have
figured that out just
01:22:19.380 --> 01:22:22.380
by writing the top line down in
terms of a's and P's and just
01:22:22.380 --> 01:22:24.180
writing this line down.
01:22:24.180 --> 01:22:26.730
But you could also use what
I've been talking about
01:22:26.730 --> 01:22:31.260
to figure out, for example, say
I coupled an external photon
01:22:31.260 --> 01:22:33.150
to my four nucleon operators.
01:22:33.150 --> 01:22:34.830
How big is this?
01:22:34.830 --> 01:22:35.640
OK.
01:22:35.640 --> 01:22:38.370
Well, it actually gets enhanced
from the fact the scattering
01:22:38.370 --> 01:22:39.995
length is large, and
you can figure out
01:22:39.995 --> 01:22:43.650
how important this effect
is, and when people do things
01:22:43.650 --> 01:22:48.030
like deuteron formation in
the sun and stuff like that,
01:22:48.030 --> 01:22:51.330
they use this effective theory
to do higher order calculations
01:22:51.330 --> 01:22:55.040
and make precision predictions
for deuteron physics.
01:22:55.040 --> 01:22:56.910
So it's not just a toy model.
01:22:56.910 --> 01:23:00.750
It's actually something that has
a real impact on some physics.
01:23:00.750 --> 01:23:02.813
We'll talk a little bit
more about it next time.
01:23:02.813 --> 01:23:04.980
We'll talk a little bit
about the conformal symmetry
01:23:04.980 --> 01:23:07.105
and I'll talk a little bit
more about the deuterons
01:23:07.105 --> 01:23:09.210
since that's something
interesting in this theory,
01:23:09.210 --> 01:23:13.330
and then we'll go on from there.
01:23:13.330 --> 01:23:14.760
So, any questions?
01:23:17.388 --> 01:23:18.880
It's cool stuff.
01:23:18.880 --> 01:23:21.310
Simple to do calculations.
01:23:21.310 --> 01:23:23.850
It's kind of interesting
to think about.