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PROFESSOR: Apparently.
9
00:00:28,190 --> 00:00:29,210
We'll start slow.
10
00:00:29,210 --> 00:00:31,088
So last time we were
talking, we just
11
00:00:31,088 --> 00:00:32,630
started talking
about effective field
12
00:00:32,630 --> 00:00:34,310
theory with a fine tuning.
13
00:00:34,310 --> 00:00:36,620
And what actually
that means takes
14
00:00:36,620 --> 00:00:38,060
a little bit of discussion.
15
00:00:41,840 --> 00:00:44,380
So what you could
mean by a fine tuning
16
00:00:44,380 --> 00:00:47,350
is that you have something
that's irrelevant.
17
00:00:47,350 --> 00:00:49,610
You look at the operator,
you think it's irrelevant,
18
00:00:49,610 --> 00:00:50,530
but it's not.
19
00:00:50,530 --> 00:00:51,340
It's relevant.
20
00:00:51,340 --> 00:00:53,800
Something that you should
include even at lowest order
21
00:00:53,800 --> 00:00:55,300
in your power accounting.
22
00:00:55,300 --> 00:00:57,815
But saying something
is irrelevant
23
00:00:57,815 --> 00:01:00,190
means that you have a power
counting, that you understand
24
00:01:00,190 --> 00:01:02,650
the power counting for the
theory, what the correct power
25
00:01:02,650 --> 00:01:04,010
counting is.
26
00:01:04,010 --> 00:01:05,710
So in this example
that I'll give,
27
00:01:05,710 --> 00:01:07,960
what "irrelevant" will
mean is that you basically
28
00:01:07,960 --> 00:01:10,480
do a dimensional
power counting, which
29
00:01:10,480 --> 00:01:11,980
is how we usually
think of defining
30
00:01:11,980 --> 00:01:13,450
irrelevant and relevant.
31
00:01:13,450 --> 00:01:15,520
Do a dimensional power
counting and you end up
32
00:01:15,520 --> 00:01:17,047
finding an operator that--
33
00:01:17,047 --> 00:01:18,880
you find that the
operator looks irrelevant,
34
00:01:18,880 --> 00:01:20,920
but when you do
calculations you can
35
00:01:20,920 --> 00:01:22,618
see that should be relevant.
36
00:01:22,618 --> 00:01:24,160
And that means,
really, what it means
37
00:01:24,160 --> 00:01:26,433
is that this natural power
counting of dimensions
38
00:01:26,433 --> 00:01:27,850
is not the right
one, and you have
39
00:01:27,850 --> 00:01:29,692
to do something
more complicated.
40
00:01:29,692 --> 00:01:31,150
But it is still is
a sense in which
41
00:01:31,150 --> 00:01:37,110
it can be thought of as a fine
tuning, because as you'll see,
42
00:01:37,110 --> 00:01:39,330
the changing of the power
counting from the naive one
43
00:01:39,330 --> 00:01:41,700
to the more complicated
power counting
44
00:01:41,700 --> 00:01:46,330
involves some kind of
tuning, if you like.
45
00:01:46,330 --> 00:01:47,850
And it really, in
this case, we'll
46
00:01:47,850 --> 00:01:49,730
see actually that
it corresponds not
47
00:01:49,730 --> 00:01:52,230
to expanding around the trivial
fixed point, where you would
48
00:01:52,230 --> 00:01:54,060
have a free theory,
but expanding
49
00:01:54,060 --> 00:01:55,560
around an interactive
fixed point.
50
00:01:58,455 --> 00:01:59,830
So it'll be a
little non-trivial,
51
00:01:59,830 --> 00:02:02,080
but we'll be doing this
in the context of two
52
00:02:02,080 --> 00:02:03,490
nucleon effective field theory.
53
00:02:03,490 --> 00:02:05,710
And the advantage of this
is that the nucleons are
54
00:02:05,710 --> 00:02:07,390
going to be non-relativistic.
55
00:02:07,390 --> 00:02:09,857
So P is going to be
much less than M pi.
56
00:02:09,857 --> 00:02:11,440
It's going to be a
very simple theory.
57
00:02:11,440 --> 00:02:14,620
Everything that's an exchange
particle gets integrated out.
58
00:02:14,620 --> 00:02:16,690
It's just a theory of
contact interactions,
59
00:02:16,690 --> 00:02:19,210
and derivatives of
contact interactions.
60
00:02:19,210 --> 00:02:20,800
And because it's
non-relativistic,
61
00:02:20,800 --> 00:02:22,510
we can actually
calculate all the loops
62
00:02:22,510 --> 00:02:24,093
to all orders and
perturbation theory,
63
00:02:24,093 --> 00:02:25,490
and we'll do that in a minute.
64
00:02:25,490 --> 00:02:28,820
So this theory, we can
calculate a lot of stuff.
65
00:02:28,820 --> 00:02:32,980
And so we'll actually be able
to see how this non-trivial fine
66
00:02:32,980 --> 00:02:35,713
tuning works, and explore
it from multiple directions,
67
00:02:35,713 --> 00:02:37,380
and we'll be sure
that what we're saying
68
00:02:37,380 --> 00:02:38,500
is actually correct.
69
00:02:41,340 --> 00:02:43,720
So it can be a lesson
for understanding
70
00:02:43,720 --> 00:02:46,330
some of the concepts and
other effective field theories
71
00:02:46,330 --> 00:02:49,360
with the fine tuning, which
you might want to design,
72
00:02:49,360 --> 00:02:53,407
where you don't have as
much ability to calculate.
73
00:02:53,407 --> 00:02:55,240
All right, so let's
start off with something
74
00:02:55,240 --> 00:02:59,650
simple, which is
elastic scattering.
75
00:02:59,650 --> 00:03:01,930
And that's mostly what
we're going to talk about.
76
00:03:07,350 --> 00:03:12,300
Two particles in,
two particles out.
77
00:03:12,300 --> 00:03:15,430
Center of mass frame.
78
00:03:15,430 --> 00:03:21,183
They scatter to some P's
coming in, P primes going out.
79
00:03:21,183 --> 00:03:23,100
And this is basically a
problem that you could
80
00:03:23,100 --> 00:03:24,390
treat with quantum mechanics.
81
00:03:27,342 --> 00:03:31,810
It's like non-relativistic
scattering.
82
00:03:31,810 --> 00:03:41,650
So if you have a
single partial wave,
83
00:03:41,650 --> 00:03:43,290
then this scattering
is described
84
00:03:43,290 --> 00:03:45,930
by a phase shift, delta.
85
00:03:45,930 --> 00:03:51,870
And the relation of the
phase shift to the amplitude,
86
00:03:51,870 --> 00:03:57,600
with our normalization for
the amplitude, is this.
87
00:03:57,600 --> 00:03:59,700
So this is the S-matrix,
it's just a phase,
88
00:03:59,700 --> 00:04:02,670
and that's the relation of
the S-matrix to the amplitude.
89
00:04:02,670 --> 00:04:05,922
And this thing is the amplitude.
90
00:04:05,922 --> 00:04:07,380
And I guess the
other thing we know
91
00:04:07,380 --> 00:04:12,960
is that, by energy
conservation, the magnitude of P
92
00:04:12,960 --> 00:04:16,529
is equal to the
magnitude of P prime.
93
00:04:16,529 --> 00:04:19,860
All right, so if we
rearrange this equation,
94
00:04:19,860 --> 00:04:23,760
and we write it as A and
put the phase, solve for a.
95
00:04:29,736 --> 00:04:30,810
Could do that.
96
00:04:34,550 --> 00:04:37,363
So that gives that
equation, which we
97
00:04:37,363 --> 00:04:38,655
can rearrange a little further.
98
00:04:47,954 --> 00:05:00,690
Now, there's kind of a
conventional way of writing it,
99
00:05:00,690 --> 00:05:02,420
which is that the
amplitude should
100
00:05:02,420 --> 00:05:06,440
be given by 1 over something
that's P cotangent delta.
101
00:05:06,440 --> 00:05:10,370
Scattering angle, or
that S-matrix angle,
102
00:05:10,370 --> 00:05:13,440
and then minus this IP.
103
00:05:13,440 --> 00:05:14,000
OK?
104
00:05:14,000 --> 00:05:15,860
So the I of--
105
00:05:15,860 --> 00:05:19,250
the I's just-- shows
up here in this part,
106
00:05:19,250 --> 00:05:21,660
and that's the complex
part of the amplitude.
107
00:05:21,660 --> 00:05:23,810
That's basically going to
be related to unitarity,
108
00:05:23,810 --> 00:05:26,960
that that IP is there.
109
00:05:26,960 --> 00:05:30,326
This S-matrix is
obviously unitary.
110
00:05:30,326 --> 00:05:33,830
This digress is 1.
111
00:05:33,830 --> 00:05:37,760
All right, so let me
tell you something
112
00:05:37,760 --> 00:05:44,000
about non-relativistic
scattering,
113
00:05:44,000 --> 00:05:46,700
which on the face of it,
looks kind of non-trivial.
114
00:06:10,990 --> 00:06:13,690
So this thing, P
cotangent delta.
115
00:06:13,690 --> 00:06:16,450
So here, I was doing
a single partial wave.
116
00:06:16,450 --> 00:06:18,460
Yeah, OK, no, it's fine.
117
00:06:18,460 --> 00:06:20,590
So what is this L?
118
00:06:20,590 --> 00:06:22,360
This L is the partial
I am considering.
119
00:06:22,360 --> 00:06:23,690
S wave, P wave.
120
00:06:23,690 --> 00:06:27,310
So L is 0 for the S wave,
L is 1 for the P wave.
121
00:06:27,310 --> 00:06:30,100
And the statement is that if you
have a short range potential,
122
00:06:30,100 --> 00:06:35,860
and you pick a wave, then
this P cotangent delta
123
00:06:35,860 --> 00:06:38,590
with the appropriate power
of P can be a Taylor series
124
00:06:38,590 --> 00:06:41,398
expansion of P. OK?
125
00:06:41,398 --> 00:06:42,940
And this is actually
something that's
126
00:06:42,940 --> 00:06:46,000
quite difficult to prove
in quantum mechanics,
127
00:06:46,000 --> 00:06:49,043
this particular fact.
128
00:06:49,043 --> 00:06:50,960
And it's called the
effective range expansion.
129
00:07:00,807 --> 00:07:03,390
It's difficult to prove because
when you do quantum mechanics,
130
00:07:03,390 --> 00:07:05,250
you pick a potential.
131
00:07:05,250 --> 00:07:08,140
And I'm saying that this
is true for any potential.
132
00:07:08,140 --> 00:07:11,140
So if you start doing quantum
mechanics with some potential,
133
00:07:11,140 --> 00:07:13,380
you've got to prove that
you can put it in this form
134
00:07:13,380 --> 00:07:16,510
irrespective of what the choice
of that potential would be.
135
00:07:16,510 --> 00:07:18,360
And that makes it
a little bit tricky
136
00:07:18,360 --> 00:07:22,002
to do in a quantum
mechanical setup.
137
00:07:22,002 --> 00:07:23,460
But we'll see
actually that this is
138
00:07:23,460 --> 00:07:25,950
very easy to prove from an
effective field theory setup.
139
00:07:34,104 --> 00:07:36,280
So, as a way of getting
into this effective field
140
00:07:36,280 --> 00:07:40,450
theory of two nucleons,
let's prove this fact.
141
00:07:40,450 --> 00:07:42,320
So what is the Lagrangian
for this theory?
142
00:07:46,930 --> 00:07:49,980
There's no-- it's
not a gauge theory,
143
00:07:49,980 --> 00:07:53,920
so we just have
ordinary derivatives.
144
00:07:53,920 --> 00:07:56,170
So if you like, you can think
of what I'm writing here
145
00:07:56,170 --> 00:07:58,780
as kind of like our IV.D
term, except I think
146
00:07:58,780 --> 00:08:00,790
the center of mass
frame, and this
147
00:08:00,790 --> 00:08:02,560
would be like the
kinetic energy term,
148
00:08:02,560 --> 00:08:06,040
but now it's just partial
squared with no D, et cetera.
149
00:08:20,370 --> 00:08:23,028
And then there's a bunch
of contact interactions.
150
00:08:23,028 --> 00:08:24,570
So there's a whole
bunch of operators
151
00:08:24,570 --> 00:08:27,540
that are involved in
nucleon and fields
152
00:08:27,540 --> 00:08:30,450
with some Wilson coefficients.
153
00:08:30,450 --> 00:08:34,350
The notation here
is this S is kind
154
00:08:34,350 --> 00:08:36,850
of a pseudonym for the channel.
155
00:08:36,850 --> 00:08:38,909
So this S here--
156
00:08:38,909 --> 00:08:41,970
maybe it should be a
script S or something,
157
00:08:41,970 --> 00:08:43,679
is telling me what
channel I'm in,
158
00:08:43,679 --> 00:08:46,980
and if in a
spectroscopic notation,
159
00:08:46,980 --> 00:08:52,717
you'd say you're in the
2S plus 1 LJ channel.
160
00:08:52,717 --> 00:08:54,300
So this would be the
angular momentum,
161
00:08:54,300 --> 00:08:55,470
total momentum in the spin.
162
00:08:58,970 --> 00:09:04,010
And these operators
here, for our purposes,
163
00:09:04,010 --> 00:09:07,892
are four nuclear fields
with 2M derivatives.
164
00:09:14,300 --> 00:09:17,650
Now, this is not really
the complete theory
165
00:09:17,650 --> 00:09:18,650
for a couple of reasons.
166
00:09:18,650 --> 00:09:20,210
Well, there's higher order
relativistic corrections
167
00:09:20,210 --> 00:09:21,560
indicated by the dots.
168
00:09:21,560 --> 00:09:24,230
There would also
be dots over here
169
00:09:24,230 --> 00:09:26,660
That could have to do with
having more nucleon fields.
170
00:09:26,660 --> 00:09:28,340
For example, I could have--
171
00:09:28,340 --> 00:09:30,612
instead of just four,
I could have six.
172
00:09:30,612 --> 00:09:32,570
But I don't need to worry
about those operators
173
00:09:32,570 --> 00:09:37,113
for two-to-two scattering,
so I'm leaving them out.
174
00:09:37,113 --> 00:09:38,780
So this is actually
the complete theory,
175
00:09:38,780 --> 00:09:45,060
if we include these dots here,
for two-to-two scattering.
176
00:09:45,060 --> 00:09:55,100
And this nucleon field,
[? its ?] spin [? at ?] half,
177
00:09:55,100 --> 00:09:56,630
and it's isospin
been at half too,
178
00:09:56,630 --> 00:09:58,610
so it includes both the
proton and the neutron.
179
00:10:04,680 --> 00:10:13,260
Nucleons are fermions, and that
implies, actually, a relation,
180
00:10:13,260 --> 00:10:15,850
because the wave function
has to be anti-symmetric.
181
00:10:15,850 --> 00:10:23,040
And so actually, you know
that you can associate isospin
182
00:10:23,040 --> 00:10:25,410
and the angular momentum
in the following
183
00:10:25,410 --> 00:10:27,370
way because of this fact.
184
00:10:27,370 --> 00:10:31,770
So all the isotriplets have
-1 to the S plus L even,
185
00:10:31,770 --> 00:10:37,770
and the isosinglets have
-1 to the S plus L odd.
186
00:10:37,770 --> 00:10:40,230
So that cuts down
by a factor of two
187
00:10:40,230 --> 00:10:42,990
the number of combinations
you have to consider.
188
00:10:42,990 --> 00:10:45,000
And basically, what
this theory has
189
00:10:45,000 --> 00:10:49,320
is for some given channel
in and some given channel
190
00:10:49,320 --> 00:11:01,700
out, which I could denote
in general different,
191
00:11:01,700 --> 00:11:04,220
we get operators that
just will have some power
192
00:11:04,220 --> 00:11:05,810
of the center of mass momentum.
193
00:11:05,810 --> 00:11:06,750
P to the 2M.
194
00:11:09,500 --> 00:11:13,670
And actually, just by angular
momentum conservation,
195
00:11:13,670 --> 00:11:16,460
that has to be the same as that.
196
00:11:16,460 --> 00:11:19,010
And so all that can
change is the L's.
197
00:11:19,010 --> 00:11:24,680
And so if S is 0,
S is either 0 or 1,
198
00:11:24,680 --> 00:11:27,500
because we have 2
spin half particles.
199
00:11:27,500 --> 00:11:31,010
If S is zero, L will
be equal to L prime
200
00:11:31,010 --> 00:11:34,130
because J is equal to
J prime, and there's
201
00:11:34,130 --> 00:11:36,110
no shift of the spin.
202
00:11:36,110 --> 00:11:39,920
So that's one possibility,
and if S is 1, then--
203
00:11:42,630 --> 00:11:45,450
so S here is the
same, if S is 1,
204
00:11:45,450 --> 00:11:55,480
then you can have L minus L
prime which is 2, so, or 0.
205
00:11:55,480 --> 00:11:56,220
OK?
206
00:11:56,220 --> 00:11:57,720
You're shifting by
one unit, and you
207
00:11:57,720 --> 00:12:03,390
can compensate either by
having minus L prime be 0 or 2.
208
00:12:08,880 --> 00:12:14,930
OK, so we conserve J.
So we can enumerate
209
00:12:14,930 --> 00:12:16,340
all the possible partial waves.
210
00:12:16,340 --> 00:12:19,530
We'll mostly focus
on the S wave.
211
00:12:19,530 --> 00:12:26,360
So, let me write out some
of these operators for you.
212
00:12:33,540 --> 00:12:36,290
So the first operator
has no derivatives,
213
00:12:36,290 --> 00:12:42,220
and I can write it in a way that
makes the partial wave explicit
214
00:12:42,220 --> 00:12:43,786
if I do the following.
215
00:12:50,940 --> 00:12:54,135
And then there would be
some derivative operator.
216
00:12:54,135 --> 00:12:55,760
And I'm going to pick
the normalization
217
00:12:55,760 --> 00:12:59,930
to make our lives
as easy as possible,
218
00:12:59,930 --> 00:13:02,440
as we usually do when we're
setting up the operator basis.
219
00:13:22,660 --> 00:13:29,242
There's the first two guys
where this derivative operator
220
00:13:29,242 --> 00:13:33,736
is like, [? grad squared ?]
to the left.
221
00:13:33,736 --> 00:13:36,857
[? Grad ?] left dot
[? grad ?] right.
222
00:13:36,857 --> 00:13:38,190
That [? grad ?] go to the right.
223
00:13:41,980 --> 00:13:44,720
And these P's, if
you look at them,
224
00:13:44,720 --> 00:13:48,080
they're just matrices in
the spin and isospin space.
225
00:13:48,080 --> 00:13:53,260
So the two we're focusing
on are the S waves.
226
00:13:53,260 --> 00:13:55,570
And in the S wave you
either have 1S0 or 3S1.
227
00:14:16,310 --> 00:14:18,350
And so we've encoded
all the, sort of,
228
00:14:18,350 --> 00:14:21,230
complexity in just these
matrices, which kind of just
229
00:14:21,230 --> 00:14:26,460
go along for the ride, and
I'll tell you what they are.
230
00:14:26,460 --> 00:14:33,980
So I sigma 2 projects you onto
a spin singlet, and I tau 2 tau
231
00:14:33,980 --> 00:14:37,290
I projects you
onto an isotriplet.
232
00:14:37,290 --> 00:14:43,700
And then likewise, 3S1,
which is a spin triplet,
233
00:14:43,700 --> 00:14:50,540
you put I sigma 2
sigma I. I tau 2.
234
00:14:50,540 --> 00:14:52,815
I tau 2 and I sigma 2 are
just because of the way
235
00:14:52,815 --> 00:14:53,690
I wrote the operator.
236
00:14:53,690 --> 00:14:56,060
I wrote it, instead
of writing N dagger N,
237
00:14:56,060 --> 00:14:58,958
I wrote N transpose
N, all dagger.
238
00:14:58,958 --> 00:15:00,500
And that means,
basically, you should
239
00:15:00,500 --> 00:15:02,167
think about the way
this operator works,
240
00:15:02,167 --> 00:15:05,960
is it annihilates two nucleons
in a particular spin wave,
241
00:15:05,960 --> 00:15:09,680
or a particular spin,
isospin channel,
242
00:15:09,680 --> 00:15:13,180
and then creates them again.
243
00:15:13,180 --> 00:15:18,340
So annihilate, create.
244
00:15:18,340 --> 00:15:20,080
So I just put the
two fields that
245
00:15:20,080 --> 00:15:21,587
are doing the
annihilating together,
246
00:15:21,587 --> 00:15:23,920
and the two fields that are
doing the creating together.
247
00:15:23,920 --> 00:15:26,560
And that's nice because
you're annihilating them
248
00:15:26,560 --> 00:15:27,570
in a particular channel.
249
00:15:32,110 --> 00:15:36,190
So with those conventions,
our Feynman Rules
250
00:15:36,190 --> 00:15:39,500
are particularly simple.
251
00:15:39,500 --> 00:15:43,600
If we just have a
C0 in some channel,
252
00:15:43,600 --> 00:15:47,020
then the Feynman Rule
is just minus IC0,
253
00:15:47,020 --> 00:15:52,870
and if we have one of
these higher C2 operators
254
00:15:52,870 --> 00:15:56,320
in the center of mass frame,
it's just minus IC2P squared.
255
00:16:02,172 --> 00:16:03,630
In the center of
mass frame, that's
256
00:16:03,630 --> 00:16:04,620
the center of mass momentum.
257
00:16:04,620 --> 00:16:06,370
And remember, in the
center of mass frame,
258
00:16:06,370 --> 00:16:08,670
P squared was equal
to P prime squared.
259
00:16:08,670 --> 00:16:12,120
And so we can actually just
write down the Feynman Rule
260
00:16:12,120 --> 00:16:13,890
for the complete
set of operators
261
00:16:13,890 --> 00:16:16,129
there if we adopt
this convention.
262
00:16:20,350 --> 00:16:22,693
So if you insert a guy
with 2M derivatives--
263
00:16:22,693 --> 00:16:24,360
derivatives always
have to come in pairs
264
00:16:24,360 --> 00:16:27,140
because of angular momentum.
265
00:16:35,077 --> 00:16:36,660
You just have that
Feynman Rule, sum's
266
00:16:36,660 --> 00:16:43,300
over the number of derivatives.
267
00:16:43,300 --> 00:16:45,080
So this is the complete,
in this theory,
268
00:16:45,080 --> 00:16:53,682
this is the complete
tree-level amplitude
269
00:16:53,682 --> 00:16:55,140
from those interactions
over there.
270
00:16:57,970 --> 00:17:01,285
It's very nice theory.
271
00:17:01,285 --> 00:17:01,785
Simple.
272
00:17:04,660 --> 00:17:05,847
What about loops?
273
00:17:05,847 --> 00:17:07,180
We are going to need some loops.
274
00:17:10,950 --> 00:17:14,609
So let's look at
the following loop,
275
00:17:14,609 --> 00:17:17,251
and I'll start by looking
at just E equals 0.
276
00:17:22,069 --> 00:17:24,050
Let's just take it,
take the nuclear arms
277
00:17:24,050 --> 00:17:26,599
and then scatter them again.
278
00:17:26,599 --> 00:17:28,460
So, in terms of
the momentum flow
279
00:17:28,460 --> 00:17:31,070
I have some Q going
this way, and then I
280
00:17:31,070 --> 00:17:36,800
have minus Q going that
way, that's my loop momenta.
281
00:17:36,800 --> 00:17:39,453
So I get 2 coupling C0.
282
00:17:39,453 --> 00:17:40,800
Want these to be 0's.
283
00:17:43,680 --> 00:17:44,180
Ergo, DDQ.
284
00:17:48,061 --> 00:17:56,580
And if I just kept the HQET
type term in my kinetic term,
285
00:17:56,580 --> 00:17:59,220
then it would look
like that, OK?
286
00:17:59,220 --> 00:18:00,060
So this is just--
287
00:18:03,090 --> 00:18:06,840
for the moment, if we
just keep partial DT
288
00:18:06,840 --> 00:18:11,970
in the kinetic term, which is
what we were doing in HQET,
289
00:18:11,970 --> 00:18:14,200
then we would get that.
290
00:18:14,200 --> 00:18:15,750
And that's a bad integral.
291
00:18:15,750 --> 00:18:17,912
It's got a pinch singularity.
292
00:18:17,912 --> 00:18:19,120
It's an ill-defined interval.
293
00:18:26,400 --> 00:18:28,620
So just from that
little algebra,
294
00:18:28,620 --> 00:18:30,855
we see that actually
keeping just the partial D
295
00:18:30,855 --> 00:18:32,940
by DT in the kinetic term
is not the right thing
296
00:18:32,940 --> 00:18:34,920
to do for this theory.
297
00:18:34,920 --> 00:18:39,330
And that's because the kinetic
energy is a relevant operator
298
00:18:39,330 --> 00:18:40,743
in quantum mechanics.
299
00:18:45,090 --> 00:18:47,160
Whenever you write down
the Schrodinger equation,
300
00:18:47,160 --> 00:18:47,700
you kept it.
301
00:18:54,417 --> 00:18:56,000
So the right power
counting and should
302
00:18:56,000 --> 00:19:00,100
have E, which is of
order P squared over 2M.
303
00:19:00,100 --> 00:19:02,675
So the partial T term and the
[? grad ?] scored over M term
304
00:19:02,675 --> 00:19:03,675
should be the same size.
305
00:19:13,580 --> 00:19:16,720
So this is generically true
of two heavy particles.
306
00:19:25,030 --> 00:19:32,350
They have a different power
counting for the kinetic term,
307
00:19:32,350 --> 00:19:33,120
than HQET.
308
00:19:39,452 --> 00:19:41,410
Any two heavy particles,
whether it's too heavy
309
00:19:41,410 --> 00:19:44,350
quarks, two heavy nucleons,
two heavy anything.
310
00:19:44,350 --> 00:19:46,580
All right, P should be
[? order ?] P squared over 2M.
311
00:19:46,580 --> 00:19:49,157
AUDIENCE: E being
the kinetic energy?
312
00:19:49,157 --> 00:19:50,740
PROFESSOR: So really,
what I mean here
313
00:19:50,740 --> 00:19:54,400
is just the partial
T term in the action
314
00:19:54,400 --> 00:19:56,990
over there should be the same
size as the [? grad ?] squared
315
00:19:56,990 --> 00:19:57,490
over 2M.
316
00:20:04,140 --> 00:20:05,620
Yes, I have pulled out the mass.
317
00:20:05,620 --> 00:20:06,870
Yeah.
318
00:20:06,870 --> 00:20:10,260
So just like in HQET,
to get this partial T
319
00:20:10,260 --> 00:20:12,750
we pulled out the
mass, and we have
320
00:20:12,750 --> 00:20:15,192
a kind of non-relativistic
type expansion.
321
00:20:15,192 --> 00:20:17,400
And the difference here is
that we need the partial T
322
00:20:17,400 --> 00:20:18,775
to be of
[? order grad squared ?]
323
00:20:18,775 --> 00:20:22,410
over 2M, and that leads to
effectively counting velocities
324
00:20:22,410 --> 00:20:25,260
rather than--
325
00:20:25,260 --> 00:20:29,190
because you have to count
energies different than P's.
326
00:20:29,190 --> 00:20:31,410
We won't spend too
long talking about
327
00:20:31,410 --> 00:20:34,080
that because we have other
things to discuss here,
328
00:20:34,080 --> 00:20:36,870
but this is a whole
interesting subject in itself.
329
00:20:36,870 --> 00:20:39,172
The power counting
and what it means.
330
00:20:39,172 --> 00:20:41,130
One thing that's kind of
interesting here which
331
00:20:41,130 --> 00:20:44,370
we won't cover, which I
can't help but mention,
332
00:20:44,370 --> 00:20:47,050
is that say you did quarks,
which was a gauge theory.
333
00:20:47,050 --> 00:20:48,720
This is not a gauge theory
that we're talking about,
334
00:20:48,720 --> 00:20:51,303
but let's-- but you could do a
gauge theory that has this type
335
00:20:51,303 --> 00:20:54,090
of power counting and it has
exactly the same problem.
336
00:20:54,090 --> 00:20:58,140
Just replace these heavy
nucleons by quarks,
337
00:20:58,140 --> 00:21:01,410
and replace this dot here
by cooling potential.
338
00:21:01,410 --> 00:21:05,130
Exactly the same problem
if you try to use HQET.
339
00:21:05,130 --> 00:21:07,170
And in that theory,
too, you need
340
00:21:07,170 --> 00:21:08,670
E to be of order
P squared over 2,
341
00:21:08,670 --> 00:21:11,120
and that's called
non-relativistic QCD.
342
00:21:11,120 --> 00:21:12,870
Or you could do heavy
electrons, where you
343
00:21:12,870 --> 00:21:15,990
have QED as the gauge theory.
344
00:21:15,990 --> 00:21:19,200
Non-relativistic
QED, same issue.
345
00:21:19,200 --> 00:21:22,650
E has to be of order P squared
over M. And in gauge theory,
346
00:21:22,650 --> 00:21:24,780
there's even a further
complication, which
347
00:21:24,780 --> 00:21:27,330
is basically that there's
gauge particles that
348
00:21:27,330 --> 00:21:29,940
want to talk to E, and
there's gauge particles that
349
00:21:29,940 --> 00:21:33,010
want to talk to P, and
those are different sizes.
350
00:21:33,010 --> 00:21:35,820
So you have something
called ultra-soft photons
351
00:21:35,820 --> 00:21:38,700
and soft photons that are
the gauge particles for E,
352
00:21:38,700 --> 00:21:42,660
and the gauge particles for P.
Kind of an interesting theory.
353
00:21:42,660 --> 00:21:44,320
We want to have time
to talk about it.
354
00:21:44,320 --> 00:21:47,240
Somebody wants to talk
about it for their project,
355
00:21:47,240 --> 00:21:48,090
it's kind of fun.
356
00:21:52,980 --> 00:21:56,790
OK, so we have to keep
P squared over 2M.
357
00:21:56,790 --> 00:21:59,610
At least knowing a
little quantum mechanics,
358
00:21:59,610 --> 00:22:01,650
we know that that's true.
359
00:22:01,650 --> 00:22:03,450
Or running into this
pinch singularity,
360
00:22:03,450 --> 00:22:08,810
we see that trying to do
something different than that
361
00:22:08,810 --> 00:22:09,560
leads to problems.
362
00:22:21,080 --> 00:22:23,380
So let's redo our calculation
here, but now keeping
363
00:22:23,380 --> 00:22:27,380
that term, and see what we get.
364
00:22:27,380 --> 00:22:29,740
Same bubble diagram.
365
00:22:29,740 --> 00:22:34,060
Let me send in on each
of these legs E over 2,
366
00:22:34,060 --> 00:22:36,070
so E is the total energy
that I'm sending in.
367
00:22:40,528 --> 00:22:42,228
Convenient normalization.
368
00:22:47,710 --> 00:22:51,400
Let's see how having the kinetic
energy fixes this pinch pole.
369
00:23:16,390 --> 00:23:18,610
OK, so if you look at the
poles in the complex plane
370
00:23:18,610 --> 00:23:22,010
here, what's happened is
you've split them like this.
371
00:23:22,010 --> 00:23:25,160
So that you've moved
them along the real axis.
372
00:23:25,160 --> 00:23:27,280
And so now you can just
think of a contour,
373
00:23:27,280 --> 00:23:32,860
for example, if you want to
think in the complex Q0 plane,
374
00:23:32,860 --> 00:23:37,200
you can think of doing a
contour integral like that.
375
00:23:37,200 --> 00:23:40,640
Everything is well defined,
convergence at infinity,
376
00:23:40,640 --> 00:23:42,370
everybody's happy.
377
00:23:42,370 --> 00:23:46,200
So we can close
above, pick the polar.
378
00:23:46,200 --> 00:23:55,227
Q0 is E over 2, minus Q
squared over 2M, plus I0.
379
00:23:55,227 --> 00:23:56,560
Plug it back into the other one.
380
00:24:03,760 --> 00:24:08,110
My notation is that N is
going to be D minus 1.
381
00:24:08,110 --> 00:24:10,150
So when I do one of the
integrals by contour,
382
00:24:10,150 --> 00:24:11,140
I go down a dimension.
383
00:24:11,140 --> 00:24:13,480
I'll call that N.
So it's N here.
384
00:24:24,630 --> 00:24:25,740
This integral we can do.
385
00:24:33,905 --> 00:24:35,655
You have to be careful
about the epsilons.
386
00:24:45,930 --> 00:24:48,240
Because they tell us tell
us whether it's minus IP
387
00:24:48,240 --> 00:24:50,805
or plus IP.
388
00:24:50,805 --> 00:24:53,870
ME is set up in my convention.
389
00:24:53,870 --> 00:24:56,450
ME is P squared.
390
00:24:56,450 --> 00:24:59,948
So this is giving a P, but
it's giving either a minus
391
00:24:59,948 --> 00:25:01,990
IP or a plus IP, depending
on the sign of the I0,
392
00:25:01,990 --> 00:25:05,800
but I know it's a minus IP.
393
00:25:05,800 --> 00:25:07,240
So I used dimreg here.
394
00:25:11,878 --> 00:25:13,420
Because if you look
at this integral,
395
00:25:13,420 --> 00:25:16,240
there's three pairs of Q
upstairs, two downstairs.
396
00:25:16,240 --> 00:25:18,458
So it's power law divergent.
397
00:25:18,458 --> 00:25:20,500
But we don't see power
law divergences in dimreg,
398
00:25:20,500 --> 00:25:23,155
we just get this finite answer.
399
00:25:29,120 --> 00:25:32,030
And actually, that finite answer
is exactly the imaginary part
400
00:25:32,030 --> 00:25:34,670
that you need if you
want to cut to graph,
401
00:25:34,670 --> 00:25:37,970
and say that the cut of
the forward scattering
402
00:25:37,970 --> 00:25:42,360
is the same as this
amplitude squared.
403
00:25:42,360 --> 00:25:44,390
So it's exactly,
in essence, this
404
00:25:44,390 --> 00:25:47,090
is the piece that you need
to be there by unitary.
405
00:25:50,210 --> 00:25:51,867
If you were trying
to keep the E,
406
00:25:51,867 --> 00:25:53,450
you could think,
well, maybe if I just
407
00:25:53,450 --> 00:25:54,950
kept the E in this
calculation, it
408
00:25:54,950 --> 00:25:56,610
would solve this pinch problem.
409
00:25:56,610 --> 00:26:00,050
But it doesn't really do
it because if you really
410
00:26:00,050 --> 00:26:03,290
stick with the partial D by DT
as your leading order action,
411
00:26:03,290 --> 00:26:06,830
then the equations of
motion are equal 0, so.
412
00:26:06,830 --> 00:26:10,145
You have to take equal
0 to go on [INAUDIBLE]..
413
00:26:10,145 --> 00:26:12,020
So you can't really
avoid the pinch that way,
414
00:26:12,020 --> 00:26:15,980
you really have to
take this kinetic term.
415
00:26:15,980 --> 00:26:18,470
you have to take the kinetic
term to have both the partial
416
00:26:18,470 --> 00:26:21,440
DT and the [? grad ?]
[? squared ?] over 2M.
417
00:26:21,440 --> 00:26:23,510
OK, any questions so far?
418
00:26:28,417 --> 00:26:30,000
All right, well
there's something here
419
00:26:30,000 --> 00:26:31,680
that might bother you.
420
00:26:31,680 --> 00:26:34,350
We've got an M upstairs.
421
00:26:34,350 --> 00:26:35,730
M is big.
422
00:26:35,730 --> 00:26:39,970
M is the mass the nucleon,
and it's appearing upstairs.
423
00:26:39,970 --> 00:26:41,250
That's always a bad sign.
424
00:26:43,920 --> 00:26:45,450
Usually a bad sign.
425
00:26:45,450 --> 00:26:48,720
Well, at least that's something
we should worry about.
426
00:26:48,720 --> 00:26:50,490
So let's figure out
where all the M's are.
427
00:26:53,910 --> 00:26:57,540
Let's count all the
M's in the theory,
428
00:26:57,540 --> 00:26:59,025
holding the momentum fixed.
429
00:27:04,960 --> 00:27:07,410
So if we hold the
momentum fixed,
430
00:27:07,410 --> 00:27:11,520
then [? grad ?] has no M's.
431
00:27:11,520 --> 00:27:17,340
But partial T scales
like 1 over M.
432
00:27:17,340 --> 00:27:25,360
And correspondingly,
T scales like M. OK.
433
00:27:25,360 --> 00:27:30,050
X and [? grad ?] don't scale
but partial T and T do.
434
00:27:30,050 --> 00:27:32,950
And that's exactly what makes
these two terms the same size.
435
00:27:38,300 --> 00:27:42,640
So we can ask, what is the
scaling of our nucleon field?
436
00:27:42,640 --> 00:27:47,860
And if we want to do that,
we go back to our action
437
00:27:47,860 --> 00:27:52,410
and we just say the action
shouldn't have any scaling.
438
00:27:52,410 --> 00:27:55,900
T has a scaling, so
there's an M in here.
439
00:27:55,900 --> 00:27:58,690
This guy here, the whole
thing scales nicely, 1 over M.
440
00:27:58,690 --> 00:28:00,770
From what we just
said over there.
441
00:28:00,770 --> 00:28:04,630
This M cancels that
1 over M, so this guy
442
00:28:04,630 --> 00:28:08,390
is therefore M to the 0.
443
00:28:08,390 --> 00:28:13,000
The nucleon field has
no scaling with them.
444
00:28:13,000 --> 00:28:15,370
So then, once you've done the
kinetic term to figure out
445
00:28:15,370 --> 00:28:17,260
the scaling of the
nucleon field, you can do,
446
00:28:17,260 --> 00:28:20,875
then, any other interaction.
447
00:28:20,875 --> 00:28:22,750
So that the power
counting, the kinetic term,
448
00:28:22,750 --> 00:28:26,080
is always which determines the
power counting of the field.
449
00:28:26,080 --> 00:28:28,443
That's true in any theory,
any effective theory,
450
00:28:28,443 --> 00:28:29,110
because you're--
451
00:28:32,080 --> 00:28:34,180
that's the basis of
the fluctuations you're
452
00:28:34,180 --> 00:28:37,578
describing by that field.
453
00:28:37,578 --> 00:28:39,370
And once you've fixed
that counting, you've
454
00:28:39,370 --> 00:28:41,120
got all the counting
you need, and now you
455
00:28:41,120 --> 00:28:44,590
can go count other operators
like the one with 2M
456
00:28:44,590 --> 00:28:48,100
derivatives.
457
00:28:48,100 --> 00:28:50,290
And this guy here just has--
458
00:28:50,290 --> 00:28:53,410
you can have just vector
derivatives, no time
459
00:28:53,410 --> 00:28:55,840
derivatives, just vector
derivatives, 2M of them,
460
00:28:55,840 --> 00:28:56,830
and nucleon fields.
461
00:28:56,830 --> 00:29:00,640
So there's no M's in
there, that's M to the 0.
462
00:29:00,640 --> 00:29:05,260
And this guy here is
an M. So this guy here
463
00:29:05,260 --> 00:29:17,380
is C2M, must be a 1 over M. So
therefore, any C2M scales like
464
00:29:17,380 --> 00:29:24,490
1 over M. And now
you see it's not
465
00:29:24,490 --> 00:29:26,680
such a problem, because
here I have 2 C0's,
466
00:29:26,680 --> 00:29:28,270
each one of them
scales like 1 over M,
467
00:29:28,270 --> 00:29:30,730
I pick up one more
M in the numerator,
468
00:29:30,730 --> 00:29:33,790
and that just makes the whole
thing feel like one over M.
469
00:29:33,790 --> 00:29:36,220
Which is the same order that
the tree level was scaling,
470
00:29:36,220 --> 00:29:39,550
the tree level with
scaling like 1 over M.
471
00:29:39,550 --> 00:29:44,230
So loop diagrams and the tree
level both scale like 1 over M,
472
00:29:44,230 --> 00:29:46,030
and that's actually
generically true
473
00:29:46,030 --> 00:29:48,560
because every time I add
a loop I add a coupling,
474
00:29:48,560 --> 00:29:51,350
and so those two M's can
compensate each other.
475
00:29:51,350 --> 00:29:52,690
So there's no issue with M's.
476
00:30:05,090 --> 00:30:10,700
So the one that graph is
the same size as tree level,
477
00:30:10,700 --> 00:30:16,190
and they both go like
1 over M. If you now
478
00:30:16,190 --> 00:30:22,730
do dimension counting, you can
say given that we've identified
479
00:30:22,730 --> 00:30:26,990
that the C has an
M in it, if you
480
00:30:26,990 --> 00:30:31,880
ask about the dimensions
of C, -2 minus 2M.
481
00:30:31,880 --> 00:30:36,390
Because the dimensions of the
nucleon field are 3 halves.
482
00:30:36,390 --> 00:30:39,080
And we're doing some expansion
in P much less than lambda,
483
00:30:39,080 --> 00:30:40,850
so just by dimensional
power counting
484
00:30:40,850 --> 00:30:47,710
we expect that the coefficients
would be of the following size.
485
00:30:47,710 --> 00:30:51,200
We know that there's an M
and that it's the only M,
486
00:30:51,200 --> 00:30:52,910
and all the rest
of the dimensions
487
00:30:52,910 --> 00:30:54,980
you should think
of as being made up
488
00:30:54,980 --> 00:30:56,910
by the stuff you
integrated it out.
489
00:30:56,910 --> 00:30:58,700
So it could be the
pion, for example,
490
00:30:58,700 --> 00:31:00,540
setting the scale lambda.
491
00:31:00,540 --> 00:31:02,300
So what you'd expect
for the C2M's is
492
00:31:02,300 --> 00:31:05,060
that this is how big they are,
and that the derivatives are
493
00:31:05,060 --> 00:31:07,998
then being suppressed
because of these lambdas
494
00:31:07,998 --> 00:31:10,040
and you're expanding for
P much less than lambda.
495
00:31:13,540 --> 00:31:14,590
OK.
496
00:31:14,590 --> 00:31:15,690
Are we happy so far?
497
00:31:20,470 --> 00:31:22,120
All right, and
we'll see when you
498
00:31:22,120 --> 00:31:24,740
do matching calculations, this
M [? law's ?] always there.
499
00:31:24,740 --> 00:31:29,770
And this is just a nice, elegant
way of figuring that out.
500
00:31:29,770 --> 00:31:34,445
Because this is a
very simple argument.
501
00:31:34,445 --> 00:31:34,945
All right.
502
00:31:34,945 --> 00:31:36,903
AUDIENCE: I don't know
if I totally understand.
503
00:31:36,903 --> 00:31:39,820
How do you know that C2M--
504
00:31:39,820 --> 00:31:41,380
PROFESSOR: Goes like 1 over M?
505
00:31:41,380 --> 00:31:42,160
AUDIENCE: Yeah--
506
00:31:42,160 --> 00:31:43,860
PROFESSOR: So the kinetic term--
507
00:31:43,860 --> 00:31:47,350
so at first you
know this, right?
508
00:31:47,350 --> 00:31:48,550
AUDIENCE: Yeah.
509
00:31:48,550 --> 00:31:49,690
PROFESSOR: Yeah, so I see.
510
00:31:49,690 --> 00:31:50,860
You're worried about
whether there could be
511
00:31:50,860 --> 00:31:52,080
1 over M squared corrections?
512
00:31:52,080 --> 00:31:52,705
AUDIENCE: Yeah.
513
00:31:52,705 --> 00:31:53,020
PROFESSOR: Yeah.
514
00:31:53,020 --> 00:31:55,240
In principle, there could be
1 over M squared corrections
515
00:31:55,240 --> 00:31:57,282
from relativistic corrections,
so that this would
516
00:31:57,282 --> 00:31:58,810
be the leading order term.
517
00:31:58,810 --> 00:31:59,500
Yeah.
518
00:31:59,500 --> 00:32:01,750
Not worse than that.
519
00:32:01,750 --> 00:32:05,560
If you looked at the P cubed,
P to the fourth over 8M
520
00:32:05,560 --> 00:32:08,133
cubed term, that term would
have a higher power of M,
521
00:32:08,133 --> 00:32:09,550
and you could
imagine that there's
522
00:32:09,550 --> 00:32:12,840
some relativistic corrections
in the four nucleons as well.
523
00:32:12,840 --> 00:32:14,590
Yeah.
524
00:32:14,590 --> 00:32:16,803
So we'll work basically
here, at lowest order
525
00:32:16,803 --> 00:32:18,220
in the relativistic
corrections so
526
00:32:18,220 --> 00:32:20,470
that you could put in the
relativistic corrections,
527
00:32:20,470 --> 00:32:20,970
as well.
528
00:32:24,060 --> 00:32:25,800
All right.
529
00:32:25,800 --> 00:32:27,990
So we're basically stopping
at that order, which
530
00:32:27,990 --> 00:32:29,448
is equivalent to
quantum mechanics,
531
00:32:29,448 --> 00:32:31,320
but we could go
further if we wanted to
532
00:32:31,320 --> 00:32:33,330
in this effective theory.
533
00:32:33,330 --> 00:32:37,980
So now, let's think about other
loop diagrams in the theory,
534
00:32:37,980 --> 00:32:39,480
without relativistic
corrections,
535
00:32:39,480 --> 00:32:40,800
just with these interactions.
536
00:32:40,800 --> 00:32:43,890
And let's insert the 2N and
the 2M derivative operator
537
00:32:43,890 --> 00:32:45,510
on these vertices.
538
00:32:45,510 --> 00:32:48,720
You see all the loops look
like this, they're all bubbles.
539
00:32:48,720 --> 00:32:50,820
And that's the
same reason in HQET
540
00:32:50,820 --> 00:32:53,070
that you don't have any
diagrams that kind of-- you
541
00:32:53,070 --> 00:32:55,770
don't have diagrams that look
like this, because these types
542
00:32:55,770 --> 00:32:58,365
of diagrams involve
antiparticles
543
00:32:58,365 --> 00:32:59,490
and we only have particles.
544
00:33:03,717 --> 00:33:05,800
So that's the beauty of a
non-relativistic theory,
545
00:33:05,800 --> 00:33:07,650
you don't have any
diagrams like that.
546
00:33:07,650 --> 00:33:09,830
You just have
diagrams like this.
547
00:33:09,830 --> 00:33:12,610
And so basically, the
whole theory is bubbles.
548
00:33:12,610 --> 00:33:13,590
The theory of bubbles.
549
00:33:16,290 --> 00:33:18,180
If you go through
this loop diagram,
550
00:33:18,180 --> 00:33:20,190
you do the pole the same way.
551
00:33:20,190 --> 00:33:24,570
It's exactly the
same two propagators.
552
00:33:24,570 --> 00:33:33,443
Then you get the same Q
squared minus ME, minus I0.
553
00:33:33,443 --> 00:33:34,860
And the only
difference is you get
554
00:33:34,860 --> 00:33:39,435
powers of Q in the numerator.
555
00:33:39,435 --> 00:33:41,310
And this, so this integral
is one of the ones
556
00:33:41,310 --> 00:33:42,840
that shows up in here.
557
00:33:42,840 --> 00:33:46,230
And this integral, you can
do the same kind of trick
558
00:33:46,230 --> 00:33:49,290
that you used when we were
discussing field definitions,
559
00:33:49,290 --> 00:33:52,020
where you basically take
the top and organize it
560
00:33:52,020 --> 00:33:55,150
by adding and subtracting.
561
00:33:55,150 --> 00:34:09,530
So add and subtract
any like that.
562
00:34:09,530 --> 00:34:11,600
And now, you can think,
N and M are integers,
563
00:34:11,600 --> 00:34:13,969
just expand this thing out.
564
00:34:13,969 --> 00:34:16,940
Some number of these factors,
some number of these factors.
565
00:34:16,940 --> 00:34:19,310
But any time you get one
or more of these factors,
566
00:34:19,310 --> 00:34:20,780
it cancels the
denominator and then
567
00:34:20,780 --> 00:34:23,058
you end up with
scaleless integrals.
568
00:34:23,058 --> 00:34:25,350
So basically, that means you
can throw this piece away.
569
00:34:41,150 --> 00:34:43,580
So higher order
derivatives actually just
570
00:34:43,580 --> 00:34:47,330
lead to ME's, whether
they act inside or outside
571
00:34:47,330 --> 00:34:49,850
on the nucleon fields,
they lead to P squared.
572
00:34:53,449 --> 00:34:54,415
Yeah, sorry.
573
00:34:54,415 --> 00:34:55,790
This M is a little
M, and this is
574
00:34:55,790 --> 00:35:00,040
supposed to be a big
M. Oh, N. Oh yeah,
575
00:35:00,040 --> 00:35:02,750
I am also using that too.
576
00:35:02,750 --> 00:35:03,800
Sorry.
577
00:35:03,800 --> 00:35:04,770
Yes.
578
00:35:04,770 --> 00:35:05,810
This is an integer.
579
00:35:05,810 --> 00:35:08,450
I should call it J or something.
580
00:35:08,450 --> 00:35:13,295
And the other M is
3 minus 2 epsilon.
581
00:35:28,100 --> 00:35:30,570
Yeah, it's dangerous.
582
00:35:30,570 --> 00:35:31,610
All right.
583
00:35:31,610 --> 00:35:34,220
But this means that basically,
the graphs in this theory
584
00:35:34,220 --> 00:35:35,010
are very simple.
585
00:35:35,010 --> 00:35:38,390
So if we actually now consider
the complete set of diagrams,
586
00:35:38,390 --> 00:35:39,365
or we impose--
587
00:35:42,530 --> 00:35:46,640
we put the full
amplitude in there
588
00:35:46,640 --> 00:35:47,990
and we consider these bubbles.
589
00:35:53,060 --> 00:35:54,620
Because of this fact
that I just told
590
00:35:54,620 --> 00:35:57,770
you, which doesn't change
when you insert more bubbles,
591
00:35:57,770 --> 00:36:01,640
the bubbles all decouple
from each other.
592
00:36:01,640 --> 00:36:03,470
The contact interaction
is decoupling them
593
00:36:03,470 --> 00:36:05,730
from each other.
594
00:36:05,730 --> 00:36:14,180
So here's a complete
amplitude with K insertions
595
00:36:14,180 --> 00:36:16,430
of the coupling,
and K minus 1 loops.
596
00:36:24,990 --> 00:36:28,245
OK, so we just completed all
the loop diagrams the theory,
597
00:36:28,245 --> 00:36:30,120
and it's giving us
something that we can sum,
598
00:36:30,120 --> 00:36:31,320
its a geometric series.
599
00:36:35,740 --> 00:36:40,540
So this is for K minus 1
loops, we can sum them up.
600
00:36:40,540 --> 00:36:46,150
So if you like I should have
called this the K-th term.
601
00:36:46,150 --> 00:37:07,830
And if I sum up, cancel the
I on each side, you get that.
602
00:37:16,270 --> 00:37:17,410
Let me round it again.
603
00:37:24,890 --> 00:37:25,610
Like this.
604
00:37:30,235 --> 00:37:34,390
So dividing out the
numerator, rearranging
605
00:37:34,390 --> 00:37:38,150
the four pi over M's little
bit, I can write it like that.
606
00:37:38,150 --> 00:37:40,030
And then we can
identify this thing here
607
00:37:40,030 --> 00:37:41,320
as P cotangent delta.
608
00:37:45,950 --> 00:37:51,020
OK, so P cotangent delta,
from our effective theory,
609
00:37:51,020 --> 00:37:55,925
we calculate it to
be the sum over this,
610
00:37:55,925 --> 00:37:59,870
P2M, where now I've
conveniently defined it,
611
00:37:59,870 --> 00:38:04,160
hatted coefficients, which
are the following things.
612
00:38:07,540 --> 00:38:09,040
And that just
cancels the M that's
613
00:38:09,040 --> 00:38:13,990
in the C2M's, so C2 hat
scales like M to the 0.
614
00:38:16,960 --> 00:38:18,040
So that's just a--
615
00:38:18,040 --> 00:38:22,060
is an obvious way of
reorganizing things given
616
00:38:22,060 --> 00:38:24,610
the factors of 4 pi
over M that we knew
617
00:38:24,610 --> 00:38:27,310
had to sit out front
of this S matrix
618
00:38:27,310 --> 00:38:30,050
calculation of the amplitude.
619
00:38:30,050 --> 00:38:33,190
So we can identify P
cotangent delta as something
620
00:38:33,190 --> 00:38:35,500
that doesn't involve any M's,
and that actually is also
621
00:38:35,500 --> 00:38:39,440
what we would expect for
non-relativistic scattering.
622
00:38:39,440 --> 00:38:41,200
So then you can look
at different waves,
623
00:38:41,200 --> 00:38:45,725
and you can look at this
formula, and you can--
624
00:38:45,725 --> 00:38:47,600
doing two of them will
show you how it works.
625
00:38:47,600 --> 00:38:50,260
So S wave is L equals 0.
626
00:39:14,030 --> 00:39:18,140
So that's something we can do
a Taylor expansion in P in.
627
00:39:18,140 --> 00:39:20,460
And this is what the
Taylor series looks like.
628
00:39:20,460 --> 00:39:22,530
And this has exactly
the form that we wanted.
629
00:39:22,530 --> 00:39:24,800
This is 1-- some
constant 1 over A,
630
00:39:24,800 --> 00:39:27,290
which is the scattering length.
631
00:39:27,290 --> 00:39:29,090
Some constant times
P squared, which
632
00:39:29,090 --> 00:39:31,580
is called the effective range.
633
00:39:31,580 --> 00:39:33,535
And we see that
expansion coming out,
634
00:39:33,535 --> 00:39:35,660
and we never had to specify
what the potential was,
635
00:39:35,660 --> 00:39:38,270
because the effective
theory was agnostic to what
636
00:39:38,270 --> 00:39:39,050
the potential was.
637
00:39:39,050 --> 00:39:40,490
That's the whole power
of effective theory,
638
00:39:40,490 --> 00:39:42,890
that you don't need to know
what the particles were that you
639
00:39:42,890 --> 00:39:45,057
were integrating out, they
just give you some values
640
00:39:45,057 --> 00:39:46,490
for these coefficients.
641
00:39:46,490 --> 00:39:49,598
And those values exactly
become the effective range
642
00:39:49,598 --> 00:39:51,140
and scattering length
in this theory.
643
00:39:53,930 --> 00:39:56,620
AUDIENCE: So this seems
very [INAUDIBLE] dependent.
644
00:39:56,620 --> 00:40:00,130
PROFESSOR: Yeah, we're
going to talk about that.
645
00:40:00,130 --> 00:40:02,110
Yes.
646
00:40:02,110 --> 00:40:06,220
Yeah, that will encompass
the second half of lecture.
647
00:40:18,230 --> 00:40:20,185
So we'll come to
that momentarily.
648
00:40:23,510 --> 00:40:27,065
Let me just do one more
way, just so you see.
649
00:40:27,065 --> 00:40:29,220
L equals 1.
650
00:40:29,220 --> 00:40:31,820
in the L equals 1 case,
there's no C0 hat,
651
00:40:31,820 --> 00:40:35,970
you need the derivatives
to correspond to the wave.
652
00:40:35,970 --> 00:40:43,220
And so in this case, you
look at PQ cotangent delta,
653
00:40:43,220 --> 00:40:44,750
the denominator starts at C2.
654
00:40:47,390 --> 00:40:49,760
And the reason why there
is this P to the 2L plus 1
655
00:40:49,760 --> 00:40:52,380
is just so that you
get the extra P here,
656
00:40:52,380 --> 00:40:54,217
which compensates
the P's downstairs,
657
00:40:54,217 --> 00:40:55,925
and again this thing
has a Taylor series.
658
00:41:04,900 --> 00:41:07,990
Same story as over there, we
can identify the scattering link
659
00:41:07,990 --> 00:41:10,750
for the P wave as
the 1 over C2 hat.
660
00:41:13,977 --> 00:41:17,680
So all the higher partial
ways work the same way.
661
00:41:17,680 --> 00:41:20,050
And you just have P to
the 2 L plus 1 in front
662
00:41:20,050 --> 00:41:22,450
of your cotangent delta.
663
00:41:22,450 --> 00:41:22,950
OK.
664
00:41:27,080 --> 00:41:30,332
Here you can see how this
generalizes [INAUDIBLE]..
665
00:41:35,160 --> 00:41:37,020
So that proves this
non-relativistic quantum
666
00:41:37,020 --> 00:41:38,990
mechanics theorem.
667
00:41:52,160 --> 00:41:55,010
And we did it without having to
specify what the potential was,
668
00:41:55,010 --> 00:41:58,010
because the potential is
encoded in BC's and effectively
669
00:41:58,010 --> 00:42:00,647
that's like a basis
expansion of the potential.
670
00:42:00,647 --> 00:42:02,730
But it's a fun one because
it's in delta functions
671
00:42:02,730 --> 00:42:04,740
and derivatives of
delta functions.
672
00:42:04,740 --> 00:42:07,010
So we're doing a
local effective field
673
00:42:07,010 --> 00:42:09,200
theory, which is not
something you'd think
674
00:42:09,200 --> 00:42:11,230
of ever using for a basis--
675
00:42:11,230 --> 00:42:13,310
well, maybe you
would, but most people
676
00:42:13,310 --> 00:42:16,190
wouldn't think of using basis of
derivatives of delta functions
677
00:42:16,190 --> 00:42:18,120
for quantum mechanics.
678
00:42:18,120 --> 00:42:18,620
OK.
679
00:42:21,662 --> 00:42:23,870
Well, you'd hope we can do
a little more than quantum
680
00:42:23,870 --> 00:42:24,578
mechanics, right?
681
00:42:29,000 --> 00:42:31,607
So you can think, if
you like, in terms
682
00:42:31,607 --> 00:42:33,690
of determining the values
of the C's, you can say,
683
00:42:33,690 --> 00:42:36,232
well, experiment tells me the
values of the scattering length
684
00:42:36,232 --> 00:42:38,510
are 0, and that's indeed true.
685
00:42:42,300 --> 00:42:45,268
So this equation that
C0 hat is equal to A,
686
00:42:45,268 --> 00:42:47,060
you could view this as
a matching equation.
687
00:42:49,980 --> 00:42:54,770
Putting back the
M, you have that.
688
00:42:54,770 --> 00:42:59,030
And then for C2 hat, we
have r0 over 2, a squared.
689
00:43:07,980 --> 00:43:10,370
So pretend that
experiment gives r0.
690
00:43:15,660 --> 00:43:17,913
And a, which they
do measure, and then
691
00:43:17,913 --> 00:43:19,830
you know the value of
your Wilson coefficients
692
00:43:19,830 --> 00:43:22,230
and you can start
using this theory.
693
00:43:22,230 --> 00:43:27,750
Now, if you think about
the power counting, if a
694
00:43:27,750 --> 00:43:30,570
and r0 are order 1
over lambda, that's
695
00:43:30,570 --> 00:43:32,408
the natural thing you'd expect.
696
00:43:32,408 --> 00:43:33,825
Then you reproduce
what we expect.
697
00:43:43,230 --> 00:43:45,270
Whatever it was a minute ago.
698
00:43:50,340 --> 00:43:51,440
2M plus one.
699
00:43:56,810 --> 00:44:00,410
So if all the constants
are scaling like whatever
700
00:44:00,410 --> 00:44:01,980
their dimensions
are, in this case
701
00:44:01,980 --> 00:44:05,235
they're both dimension
minus 1 in momentum units,
702
00:44:05,235 --> 00:44:07,610
then you would reproduce
exactly with this power counting
703
00:44:07,610 --> 00:44:09,340
that we said.
704
00:44:09,340 --> 00:44:11,330
OK, so everything would be nice.
705
00:44:11,330 --> 00:44:13,050
But nature is not so nice.
706
00:44:13,050 --> 00:44:16,460
Or nature threw us in
a different direction.
707
00:44:16,460 --> 00:44:19,293
When we actually look at the
value of these constants,
708
00:44:19,293 --> 00:44:20,585
the scattering length is large.
709
00:44:27,970 --> 00:44:34,100
So C0 is large, from
this point of view.
710
00:44:37,303 --> 00:44:38,720
Let me quote some
numbers for you.
711
00:44:58,820 --> 00:45:03,320
So in the 1S0 channel,
which is the larger one,
712
00:45:03,320 --> 00:45:04,700
this guy is 23 fermis.
713
00:45:11,990 --> 00:45:17,288
And in the 3S1 channel, it's
5 fermis, and both of these
714
00:45:17,288 --> 00:45:18,080
are actually large.
715
00:45:23,260 --> 00:45:25,010
We know they're large,
look at the errors.
716
00:45:27,650 --> 00:45:29,570
If you want to think
about momentum units,
717
00:45:29,570 --> 00:45:32,480
you could take 1 over a.
718
00:45:32,480 --> 00:45:38,940
And for this guy here, taking
1 over a is giving him, like,
719
00:45:38,940 --> 00:45:43,910
if I did the calculation
right, minus 8.30 MeV.
720
00:45:43,910 --> 00:45:45,170
Oh, sorry, no.
721
00:45:45,170 --> 00:45:45,920
That's this guy.
722
00:45:48,750 --> 00:45:55,740
And this guy here is
giving 1 over a, is 36 MeV.
723
00:45:55,740 --> 00:45:57,950
So if you thought the
natural size was the pion,
724
00:45:57,950 --> 00:46:00,860
then you'd say, well, these
constants should be 1 over pi--
725
00:46:00,860 --> 00:46:04,130
that these numbers that
are in MeV should be in pi.
726
00:46:04,130 --> 00:46:07,970
And 8 MeV is a much smaller
number than 138 MeV.
727
00:46:07,970 --> 00:46:10,830
36 is also a smaller
number than 130 MeV.
728
00:46:10,830 --> 00:46:17,600
So both of these
guys are not natural.
729
00:46:17,600 --> 00:46:20,935
In particular, this one you
see it's very not natural.
730
00:46:24,080 --> 00:46:26,620
So there's a fine
tuning going on.
731
00:46:26,620 --> 00:46:29,082
Some kind of fine tuning
from the perspective--
732
00:46:37,460 --> 00:46:40,080
from our dimensional
counting EFT point of view.
733
00:46:45,120 --> 00:46:49,870
There's a fine tuning
that's making the a big.
734
00:46:49,870 --> 00:46:53,460
And if you look at
the other guys, the r0
735
00:46:53,460 --> 00:46:55,770
and the other guys,
they are exactly
736
00:46:55,770 --> 00:46:56,890
of the size you'd expect.
737
00:46:56,890 --> 00:46:58,223
So there's no fine tuning there.
738
00:47:01,440 --> 00:47:04,560
The only fine tuning is in a.
739
00:47:04,560 --> 00:47:06,660
And not the other ones.
740
00:47:06,660 --> 00:47:08,880
Just by comparing two data.
741
00:47:08,880 --> 00:47:10,300
OK, so how do we deal with that?
742
00:47:10,300 --> 00:47:12,092
It looks like we set
up a defective theory.
743
00:47:12,092 --> 00:47:13,660
It has seems like
a beautiful theory,
744
00:47:13,660 --> 00:47:15,910
it could describe some things
about quantum mechanics,
745
00:47:15,910 --> 00:47:20,050
but then we learned that
our power counting sucks.
746
00:47:20,050 --> 00:47:22,790
So what we want is
actually a power counting
747
00:47:22,790 --> 00:47:24,188
it's a little different.
748
00:47:32,010 --> 00:47:38,190
Where AP is of order 1, or
even AP much greater than 1,
749
00:47:38,190 --> 00:47:40,200
we'd like that to be allowed.
750
00:47:40,200 --> 00:47:44,520
Where our 0P is
much less than 1.
751
00:47:44,520 --> 00:47:47,160
We'd like to be able to take a
scattering length effectively
752
00:47:47,160 --> 00:47:50,310
into account to all orders,
and that means basically
753
00:47:50,310 --> 00:47:55,170
that we want to
treat C0 as relevant.
754
00:47:55,170 --> 00:47:58,537
We don't want 8MeV to be
the limit of the lowered--
755
00:47:58,537 --> 00:48:00,870
the thing that goes downstairs
when we're making a power
756
00:48:00,870 --> 00:48:02,340
expansion, right?
757
00:48:02,340 --> 00:48:05,610
We'd like something like
M pi to be downstairs.
758
00:48:05,610 --> 00:48:06,990
If we want M pi
to be downstairs,
759
00:48:06,990 --> 00:48:09,180
you got to treat
AP to all orders.
760
00:48:09,180 --> 00:48:12,570
And then you're just limited
by M pi which is the r0 term.
761
00:48:12,570 --> 00:48:14,340
And that means you've
got to promote C0
762
00:48:14,340 --> 00:48:18,060
from being irrelevant,
scaling like 1 over M lambda,
763
00:48:18,060 --> 00:48:21,580
to something that's relevant.
764
00:48:21,580 --> 00:48:28,080
So this is actually
a problem that
765
00:48:28,080 --> 00:48:30,630
occurred because we just
proceeded and started
766
00:48:30,630 --> 00:48:34,820
calculating, and we used
effectively the MS bar scheme
767
00:48:34,820 --> 00:48:36,780
in dimensional regularization.
768
00:48:36,780 --> 00:48:39,720
We saw powers and
fractions, we did nothing.
769
00:48:39,720 --> 00:48:41,970
So let's try another scheme.
770
00:48:48,440 --> 00:48:51,260
It's a little more physical.
771
00:48:51,260 --> 00:48:54,800
Called offshell
momentum subtraction.
772
00:48:54,800 --> 00:48:57,530
So what is offshell
momentum subtraction?
773
00:48:57,530 --> 00:49:01,220
It says take the amplitude,
and in this non-relativistic
774
00:49:01,220 --> 00:49:05,990
theory, you take P to be
at some imaginary point
775
00:49:05,990 --> 00:49:10,302
so that you avoid any cuts.
776
00:49:10,302 --> 00:49:11,135
And you can define--
777
00:49:15,160 --> 00:49:21,520
and whatever channel we're
in, in this new r scheme,
778
00:49:21,520 --> 00:49:24,190
you can define the amplitude
of that particular point
779
00:49:24,190 --> 00:49:26,730
to be the tree level result.
780
00:49:26,730 --> 00:49:29,540
OK, so any loops, if you
take them at this point,
781
00:49:29,540 --> 00:49:35,830
you should get back
that result. So this
782
00:49:35,830 --> 00:49:38,080
is the analog of what you
do in a relativistic theory
783
00:49:38,080 --> 00:49:41,050
where you would take P squared
to be minus mu squared.
784
00:49:41,050 --> 00:49:42,550
In the non-relativistic
theory, it's
785
00:49:42,550 --> 00:49:46,000
always P that's showing up,
which is the three vector P,
786
00:49:46,000 --> 00:49:49,200
magnitude of the three vector.
787
00:49:49,200 --> 00:49:50,910
So we should assign
some rule for that.
788
00:49:50,910 --> 00:49:53,990
And it's just, we take
it to be I times mu.
789
00:50:00,620 --> 00:50:03,450
So how does this
change our calculation?
790
00:50:03,450 --> 00:50:06,750
So let's go back to
this one calculation.
791
00:50:06,750 --> 00:50:11,210
So now there's going to be
some counterterm needed.
792
00:50:11,210 --> 00:50:14,630
It's going to be a
finite counterterm.
793
00:50:14,630 --> 00:50:15,620
Let's see what it does.
794
00:50:21,990 --> 00:50:27,830
So the loop graph gave this IP,
that's what this graph gave.
795
00:50:27,830 --> 00:50:32,000
And this has to be just such
that if I set P equal to I mu,
796
00:50:32,000 --> 00:50:32,960
that I get 0.
797
00:50:32,960 --> 00:50:35,370
So this has two plus mu.
798
00:50:35,370 --> 00:50:39,050
So the counterterm is
giving mu r, C0 squared.
799
00:50:39,050 --> 00:50:42,455
And that exactly makes this
amplitude vanish at that point,
800
00:50:42,455 --> 00:50:45,080
and that's what you want because
the tree level graph of the C0
801
00:50:45,080 --> 00:50:47,690
is already giving
the right condition.
802
00:50:47,690 --> 00:50:48,470
OK?
803
00:50:48,470 --> 00:50:50,150
Is that clear to everybody?
804
00:50:50,150 --> 00:50:52,880
This is the correction
to that, to this.
805
00:50:52,880 --> 00:50:56,210
The tree level graph in the
amplitude here gave minus IC0,
806
00:50:56,210 --> 00:50:57,710
so we already got what we want.
807
00:50:57,710 --> 00:50:59,210
When we go to our
loop, we just want
808
00:50:59,210 --> 00:51:01,190
to make sure it
doesn't contribute,
809
00:51:01,190 --> 00:51:05,190
and that forces us to
put the plus mu there.
810
00:51:05,190 --> 00:51:10,940
And so what this mu is doing is
tracking the power divergence
811
00:51:10,940 --> 00:51:14,630
that dimreg in MS
bar did not see.
812
00:51:17,810 --> 00:51:20,450
There was an integral is power
law divergence and our cut
813
00:51:20,450 --> 00:51:22,910
off, mu r, has appeared
in the numerator
814
00:51:22,910 --> 00:51:24,981
and has tracked that divergence.
815
00:51:28,140 --> 00:51:29,970
So we're doing a
standard thing here where
816
00:51:29,970 --> 00:51:36,240
we split the coefficient
into the bare coefficient
817
00:51:36,240 --> 00:51:38,910
and to our more renormalized
and countertrend piece.
818
00:51:38,910 --> 00:51:41,310
And unlike MS bar in this
particular normalization
819
00:51:41,310 --> 00:51:44,085
scheme, there's a
finite correction there.
820
00:51:44,085 --> 00:51:45,960
And we have a renormalization
group equation.
821
00:52:00,540 --> 00:52:08,290
And if you work out what
that is from the counterterm,
822
00:52:08,290 --> 00:52:12,480
it turns out that only this
one loop, it's one loop exact.
823
00:52:12,480 --> 00:52:15,150
Showing that is a little
more work than I've done,
824
00:52:15,150 --> 00:52:18,690
but it turns out that the beta
function is one loop exact,
825
00:52:18,690 --> 00:52:20,520
higher bubbles don't
give any contribution
826
00:52:20,520 --> 00:52:24,540
to this beta function, and this
is the anomalous dimension.
827
00:52:24,540 --> 00:52:26,820
So we could just calculate
all those bubbles,
828
00:52:26,820 --> 00:52:28,840
and pose the same type of thing.
829
00:52:28,840 --> 00:52:31,898
And I've given you the
reference that does that.
830
00:52:31,898 --> 00:52:33,690
So there's a renormalization
group equation
831
00:52:33,690 --> 00:52:36,060
in this scheme which we
didn't have in MS bar.
832
00:52:36,060 --> 00:52:38,820
In MS bar it was
scale independent.
833
00:52:38,820 --> 00:52:41,250
We also have a connection
to the MS bar scheme
834
00:52:41,250 --> 00:52:44,970
because if mu r was 0,
that corresponds to what
835
00:52:44,970 --> 00:52:48,090
the MS bar result, right?
836
00:52:48,090 --> 00:52:53,010
So C0 of 0 is where you can
think of putting your boundary
837
00:52:53,010 --> 00:52:57,900
condition, which is matching to
the experiment, which is the a.
838
00:52:57,900 --> 00:53:00,210
And the advantage of
this offshell subtraction
839
00:53:00,210 --> 00:53:03,190
is that we have a mu, and
we can go somewhere else.
840
00:53:03,190 --> 00:53:07,740
And if you look at the
solution, of the RG
841
00:53:07,740 --> 00:53:11,557
with that boundary condition,
you see something interesting.
842
00:53:20,330 --> 00:53:22,840
So this is the result
for the coefficient.
843
00:53:22,840 --> 00:53:24,570
One over mu r, minus 1 over a.
844
00:53:28,240 --> 00:53:31,760
So if mu is of order P,
which is much greater than 1
845
00:53:31,760 --> 00:53:36,460
over a, like we want, then
the right counting for the C,
846
00:53:36,460 --> 00:53:40,210
which is a function
of mu, is 1 over M mu.
847
00:53:43,700 --> 00:53:45,950
OK?
848
00:53:45,950 --> 00:53:48,140
And so what we've
done here is we've
849
00:53:48,140 --> 00:53:54,770
swapped 1 over the physics
scale that we're integrating out
850
00:53:54,770 --> 00:53:56,960
for 1 over the scale
mu, which we're
851
00:53:56,960 --> 00:54:00,100
taking to be of order the
physics we're keeping.
852
00:54:00,100 --> 00:54:02,120
OK?
853
00:54:02,120 --> 00:54:03,440
And this is relevant now.
854
00:54:03,440 --> 00:54:05,680
This is a relevant
coupling with that change.
855
00:54:09,730 --> 00:54:11,620
We've made it much
bigger by just
856
00:54:11,620 --> 00:54:13,970
switching to the physical
renormalization scheme.
857
00:54:13,970 --> 00:54:16,210
And if we take mu of
order p in that scheme,
858
00:54:16,210 --> 00:54:19,030
this all of a sudden
becomes an order 1 effect
859
00:54:19,030 --> 00:54:20,740
with leading order
in the Lagrangian,
860
00:54:20,740 --> 00:54:25,710
because we have effectively
P's downstairs in the coupling.
861
00:54:25,710 --> 00:54:26,570
OK?
862
00:54:26,570 --> 00:54:30,100
So this scheme allows
us a way of thinking
863
00:54:30,100 --> 00:54:31,890
in the effective
theory way of thinking
864
00:54:31,890 --> 00:54:33,640
of having a power
counting where we'd have
865
00:54:33,640 --> 00:54:35,843
to keep the C0 to all orders.
866
00:54:35,843 --> 00:54:37,260
AUDIENCE: I thought
it was order--
867
00:54:37,260 --> 00:54:39,350
AP was order 1.
868
00:54:39,350 --> 00:54:40,820
PROFESSOR: Yeah, AP is order 1.
869
00:54:44,990 --> 00:54:46,760
So if you look at
the Lagrangian,
870
00:54:46,760 --> 00:54:53,180
then you have to go through
the counting of how many powers
871
00:54:53,180 --> 00:54:55,100
of P the nucleon field has.
872
00:54:55,100 --> 00:54:59,210
Which you could do in the same
way that we did with the mass.
873
00:54:59,210 --> 00:55:01,340
And if you do that with
the four-nucleon operator
874
00:55:01,340 --> 00:55:03,530
with an extra power
of P downstairs,
875
00:55:03,530 --> 00:55:08,600
you will find it has the same
P scaling as the kinetic term.
876
00:55:08,600 --> 00:55:09,230
OK?
877
00:55:09,230 --> 00:55:10,933
I didn't go through that, but--
878
00:55:10,933 --> 00:55:13,100
AUDIENCE: So that justifies
this, even when P is not
879
00:55:13,100 --> 00:55:14,555
much bigger than 1 over a?
880
00:55:14,555 --> 00:55:15,680
Is that what you're saying?
881
00:55:15,680 --> 00:55:18,350
PROFESSOR: Yeah, so if P is
much bigger than 1 over a, or P
882
00:55:18,350 --> 00:55:24,980
is even order or 1 over a,
which is a sort of also--
883
00:55:24,980 --> 00:55:27,560
it's also fine
with this counting.
884
00:55:27,560 --> 00:55:29,570
If P is of order 1
over a, these two terms
885
00:55:29,570 --> 00:55:32,750
are sort of comparably
big, but you could also
886
00:55:32,750 --> 00:55:34,940
use this approach for
that case, as long
887
00:55:34,940 --> 00:55:40,700
as you're not in the case where
P is much less than 1 over a.
888
00:55:40,700 --> 00:55:42,890
So if you're in
that case, then you
889
00:55:42,890 --> 00:55:45,420
should really think about
expanding this out some sense.
890
00:55:45,420 --> 00:55:47,670
And then you're getting back
to the end MS bar result,
891
00:55:47,670 --> 00:55:50,640
the MS bar result
would have been fine.
892
00:55:50,640 --> 00:55:52,650
But to get away from
the MS bar result,
893
00:55:52,650 --> 00:55:56,068
and think about
physics, mu of order P,
894
00:55:56,068 --> 00:55:57,860
we could use this scheme
instead of MS bar,
895
00:55:57,860 --> 00:56:00,650
and then we actually see that
we get a reasonable power
896
00:56:00,650 --> 00:56:01,450
counting.
897
00:56:01,450 --> 00:56:04,800
AUDIENCE: OK, Because my concern
is, what about when mu is like,
898
00:56:04,800 --> 00:56:07,092
what about this pole
that you're going to--
899
00:56:07,092 --> 00:56:10,181
PROFESSOR: Yeah, we'll
talk about the pole, yeah.
900
00:56:10,181 --> 00:56:12,580
It's coming up.
901
00:56:12,580 --> 00:56:14,800
All right.
902
00:56:14,800 --> 00:56:16,780
So it's interesting
to think about this
903
00:56:16,780 --> 00:56:19,075
from our renormalization
group point of view, which
904
00:56:19,075 --> 00:56:20,950
is kind of what I was
doing when I wrote down
905
00:56:20,950 --> 00:56:22,630
a beta function that I was--
906
00:56:22,630 --> 00:56:25,870
here I was just
after this solution,
907
00:56:25,870 --> 00:56:30,740
because from that solution I got
the power counting that I want.
908
00:56:30,740 --> 00:56:33,413
Which is that the C0 term is the
same size as the kinetic term,
909
00:56:33,413 --> 00:56:34,330
and both are relevant.
910
00:56:43,980 --> 00:56:46,910
So we can do that just like
we counted P's for [INAUDIBLE]
911
00:56:46,910 --> 00:56:48,500
theory, or just
like we counted--
912
00:56:51,430 --> 00:57:08,950
[? just comment. ?] These
guys are all relevant
913
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.
914
00:57:13,780 --> 00:57:15,700
So there's another way
of thinking about this,
915
00:57:15,700 --> 00:57:17,408
and thinking about
these different cases,
916
00:57:17,408 --> 00:57:20,690
and that's just to think about
the beta function itself.
917
00:57:20,690 --> 00:57:24,730
So if you look at the beta
function for the C0 coupling,
918
00:57:24,730 --> 00:57:27,430
and we just put in the
solution, plug this back
919
00:57:27,430 --> 00:57:37,243
into that equation here,
with some constant out front,
920
00:57:37,243 --> 00:57:38,410
and then it looks like this.
921
00:57:38,410 --> 00:57:43,430
8 times mu, 1 minus
a mu, squared.
922
00:57:43,430 --> 00:57:45,880
If we want to talk about
all possible values of a,
923
00:57:45,880 --> 00:57:49,580
well a can go from minus
infinity to plus infinity.
924
00:57:49,580 --> 00:57:52,030
So it's useful if we want
to draw this to map it
925
00:57:52,030 --> 00:57:53,110
to a compact interval.
926
00:57:55,760 --> 00:57:56,560
So let's do that.
927
00:58:04,680 --> 00:58:06,950
Move the tangent.
928
00:58:06,950 --> 00:58:09,800
And let's just plot
beta as a function of x.
929
00:58:21,555 --> 00:58:23,805
So there's three values,
actually where beta vanishes.
930
00:58:27,150 --> 00:58:28,650
There's one value
where it blows up.
931
00:58:32,330 --> 00:58:38,150
So this is-- here
is x equals minus 1.
932
00:58:38,150 --> 00:58:43,170
This is x equals 0,
this is x equals plus 1.
933
00:58:45,840 --> 00:58:53,640
And what it looks like dips
down here, goes there, blows up,
934
00:58:53,640 --> 00:58:57,930
then it comes back
down, it goes like that.
935
00:58:57,930 --> 00:59:00,930
So that's what the beta
function looks like.
936
00:59:00,930 --> 00:59:03,290
So this point here
corresponds if you
937
00:59:03,290 --> 00:59:05,630
think of being at fixed mu.
938
00:59:05,630 --> 00:59:08,240
Say you're studying the
physics at fixed mu.
939
00:59:08,240 --> 00:59:11,720
This point corresponds to
a equals minus infinity,
940
00:59:11,720 --> 00:59:14,870
this point corresponds
to a equals zero,
941
00:59:14,870 --> 00:59:18,002
and this point corresponds
to a equals plus infinity.
942
00:59:24,400 --> 00:59:26,980
So there's three points where
the beta function vanishes,
943
00:59:26,980 --> 00:59:29,168
if you think about a
space for fixed mu.
944
00:59:41,590 --> 00:59:42,370
Use some color.
945
00:59:53,250 --> 00:59:55,450
So you can ask about nature.
946
00:59:55,450 --> 00:59:57,450
So nature told us the
value of a, so what
947
00:59:57,450 --> 00:59:59,200
does that correspond to?
948
00:59:59,200 --> 01:00:01,830
So taking some value
of P and mapping it
949
01:00:01,830 --> 01:00:04,823
to some value of x, kind of
generically the kind of point
950
01:00:04,823 --> 01:00:06,240
that you're
interested in is here.
951
01:00:06,240 --> 01:00:09,120
This is a1S0, sits there.
952
01:00:09,120 --> 01:00:16,420
And then a for the 3S1
makes it kind of here.
953
01:00:16,420 --> 01:00:19,350
So what's going on in this
case is that you're not
954
01:00:19,350 --> 01:00:22,800
near this fixed point, you're
actually close to this one.
955
01:00:22,800 --> 01:00:24,630
And you're not near
this fixed point,
956
01:00:24,630 --> 01:00:26,505
well actually there's
an infinity in between,
957
01:00:26,505 --> 01:00:28,510
you're closer to this one.
958
01:00:28,510 --> 01:00:32,940
This size you should think
of as 8 MeV, generically,
959
01:00:32,940 --> 01:00:35,580
and then this is like the 8 but
there's also a pole in between.
960
01:00:38,440 --> 01:00:39,670
So three fixed points.
961
01:00:42,778 --> 01:00:44,320
When we do perturbation
theory and we
962
01:00:44,320 --> 01:00:46,445
expand about fixed
points, and one way
963
01:00:46,445 --> 01:00:48,820
of saying what was wrong with
dimensional analysis was it
964
01:00:48,820 --> 01:00:51,340
was just expanding about
the wrong fixed point.
965
01:00:51,340 --> 01:00:51,910
The pink one.
966
01:00:55,300 --> 01:01:02,730
So a equals zero, was
the non-interactive one
967
01:01:02,730 --> 01:01:04,480
where we just had the
relevant interaction
968
01:01:04,480 --> 01:01:08,080
being the kinetic term, but
none of the interaction terms
969
01:01:08,080 --> 01:01:09,580
are relevant.
970
01:01:09,580 --> 01:01:13,060
And a equals plus
or minus infinity
971
01:01:13,060 --> 01:01:15,040
are interacting
fixed points, where
972
01:01:15,040 --> 01:01:17,170
you have an interaction
that is relevant.
973
01:01:22,370 --> 01:01:24,550
So you can think about that
in the following sense.
974
01:01:24,550 --> 01:01:27,070
Classically, what a is measuring
is kind of the interaction
975
01:01:27,070 --> 01:01:30,250
size, if you have classical
scattering across sections
976
01:01:30,250 --> 01:01:31,700
4 pi a squared.
977
01:01:31,700 --> 01:01:34,858
And if a is small--
978
01:01:34,858 --> 01:01:38,420
if a is either very
small or very big,
979
01:01:38,420 --> 01:01:41,290
then basically it's
the same on all scales
980
01:01:41,290 --> 01:01:44,290
because it's either
infinity or 0
981
01:01:44,290 --> 01:01:46,510
and it looks the
same to particles
982
01:01:46,510 --> 01:01:48,190
of all different momenta.
983
01:01:48,190 --> 01:01:49,150
OK?
984
01:01:49,150 --> 01:01:51,940
And that's one way of thinking
about these fixed points.
985
01:01:51,940 --> 01:01:53,680
That the physics can't--
986
01:01:53,680 --> 01:01:57,220
the physics-- yeah.
987
01:01:57,220 --> 01:01:58,720
Just what I said.
988
01:01:58,720 --> 01:02:00,880
So what about this infinity?
989
01:02:00,880 --> 01:02:02,518
That's also interesting.
990
01:02:10,486 --> 01:02:16,370
So when a is one over mu,
beta goes to infinity.
991
01:02:16,370 --> 01:02:19,910
And this actually--
there is a reflection
992
01:02:19,910 --> 01:02:23,480
of this in the theory, it
corresponds to a bound state.
993
01:02:28,580 --> 01:02:30,920
Which I think we'll talk
about next time, but.
994
01:02:33,440 --> 01:02:35,090
So there's a bound
state in the theory,
995
01:02:35,090 --> 01:02:37,310
and actually if you
start from this side,
996
01:02:37,310 --> 01:02:39,110
you can never see that
bound because you're
997
01:02:39,110 --> 01:02:39,980
doing perturbation theory.
998
01:02:39,980 --> 01:02:42,200
You don't see perturbation
theory and bound states.
999
01:02:42,200 --> 01:02:44,210
If you start from this
side, the bound state
1000
01:02:44,210 --> 01:02:46,160
is actually just in the theory.
1001
01:02:46,160 --> 01:02:48,605
We'll talk about that next time.
1002
01:02:48,605 --> 01:02:50,480
And so it's a state in
the theory, you can go
1003
01:02:50,480 --> 01:02:52,400
and you could find the
pole in your amplitude,
1004
01:02:52,400 --> 01:02:53,900
it's just there.
1005
01:02:53,900 --> 01:02:56,180
Corresponds to a physical
state of the spectrum,
1006
01:02:56,180 --> 01:02:57,388
and it's called the deuteron.
1007
01:03:14,620 --> 01:03:16,120
So we have a
non-interactive theory,
1008
01:03:16,120 --> 01:03:19,990
we have some
non-trivial amplitude,
1009
01:03:19,990 --> 01:03:22,540
and this deuteron is a
pole in that amplitude.
1010
01:04:09,310 --> 01:04:13,890
And you never see a pole if you
use the perturbation theory.
1011
01:04:13,890 --> 01:04:16,350
And if you actually look at
the energy, the binding energy
1012
01:04:16,350 --> 01:04:18,773
of this state, it's also small.
1013
01:04:18,773 --> 01:04:20,190
Characteristically
small, and it's
1014
01:04:20,190 --> 01:04:24,030
actually related to the fact
that the binding energy is
1015
01:04:24,030 --> 01:04:26,820
small just related to sort of
the natural size of this a.
1016
01:04:26,820 --> 01:04:28,260
We'll talk about that next time.
1017
01:04:34,850 --> 01:04:38,290
So what I was saying before
about the theory at all scales
1018
01:04:38,290 --> 01:04:41,170
looking the same, when you
have a equals 0 of course
1019
01:04:41,170 --> 01:04:42,550
it's not interacting.
1020
01:04:42,550 --> 01:04:44,770
Or when a is equal to
plus minus infinity,
1021
01:04:44,770 --> 01:04:47,770
looks the same at all scales.
1022
01:04:47,770 --> 01:04:50,998
That means that scale invariant.
1023
01:04:50,998 --> 01:04:52,540
So it's a scale
invariant theory when
1024
01:04:52,540 --> 01:04:55,540
a is at these fixed points.
1025
01:04:55,540 --> 01:04:58,600
And something I
worked on was the fact
1026
01:04:58,600 --> 01:05:02,650
that these points are actually
conformal fixed points.
1027
01:05:12,570 --> 01:05:16,600
So that-- there's a conformal
symmetry of the theory that
1028
01:05:16,600 --> 01:05:20,090
exists at those points, we'll
talk about that a little bit.
1029
01:05:20,090 --> 01:05:22,750
There's another symmetry,
too, which I have to mention.
1030
01:05:25,480 --> 01:05:29,140
And not only because I
also worked on this one,
1031
01:05:29,140 --> 01:05:32,070
because it's good to
enumerate all the symmetries.
1032
01:05:35,360 --> 01:05:37,880
There's actually a combined
spin, isospin symmetry,
1033
01:05:37,880 --> 01:05:41,027
that turns into an
Su4 in this limit.
1034
01:05:41,027 --> 01:05:42,860
So much like in heavy
quark effective theory
1035
01:05:42,860 --> 01:05:45,818
where the mass was big,
new symmetries popped up.
1036
01:05:45,818 --> 01:05:48,110
Same thing happens here, when
the scattering lengths go
1037
01:05:48,110 --> 01:05:50,552
to infinity, and you go
over to those fixed points,
1038
01:05:50,552 --> 01:05:52,010
there's new symmetries
that pop up.
1039
01:05:52,010 --> 01:05:53,927
One's a conformal symmetry
and the other one's
1040
01:05:53,927 --> 01:05:55,190
a spin, isospin symmetry.
1041
01:06:09,810 --> 01:06:12,060
All right.
1042
01:06:12,060 --> 01:06:16,200
So this looks interesting.
1043
01:06:16,200 --> 01:06:18,398
You could ask the
question, did this pick--
1044
01:06:18,398 --> 01:06:20,190
did this physical
picture that we developed
1045
01:06:20,190 --> 01:06:22,050
depend on picking
this renormalization
1046
01:06:22,050 --> 01:06:24,240
scheme that I told you about?
1047
01:06:24,240 --> 01:06:26,670
We kind of gave up on dimreg,
we went over to-- well,
1048
01:06:26,670 --> 01:06:28,110
we gave up on MS
bar, we went over
1049
01:06:28,110 --> 01:06:31,502
to this scheme which was
offshell momentum subtraction.
1050
01:06:31,502 --> 01:06:33,960
In general, people don't like
offshell momentum subtraction
1051
01:06:33,960 --> 01:06:35,793
because it makes
calculations more difficult
1052
01:06:35,793 --> 01:06:37,001
once you go to higher orders.
1053
01:06:37,001 --> 01:06:39,210
Well here, are the calculations
are not so difficult,
1054
01:06:39,210 --> 01:06:40,320
so we could do them, but--
1055
01:06:40,320 --> 01:06:42,750
you might be interested in
adding pions to this theory,
1056
01:06:42,750 --> 01:06:45,420
or coupling external
currents like photons,
1057
01:06:45,420 --> 01:06:47,910
and then the calculations
would get more difficult.
1058
01:06:47,910 --> 01:06:51,180
And you'd like, for example,
to have a dimreg MS bar type
1059
01:06:51,180 --> 01:06:53,190
description of
this power counting
1060
01:06:53,190 --> 01:06:55,620
rather than a kind of
minimal subtraction.
1061
01:06:55,620 --> 01:06:57,210
Could I get, could
I kind of dress up
1062
01:06:57,210 --> 01:07:00,300
minimal subtraction to get
the same physical picture?
1063
01:07:00,300 --> 01:07:03,210
That's a reasonable
question to ask.
1064
01:07:03,210 --> 01:07:04,290
The answer is you can.
1065
01:07:10,737 --> 01:07:13,070
So there's something called
power divergence subtraction
1066
01:07:13,070 --> 01:07:15,650
scheme, different
scheme than MS bar.
1067
01:07:22,244 --> 01:07:25,780
So the PDS scheme.
1068
01:07:25,780 --> 01:07:31,060
And what it says is, don't just
subtract poles at D equals 4,
1069
01:07:31,060 --> 01:07:37,510
like you do in MS bar, which
are corresponding to logs
1070
01:07:37,510 --> 01:07:42,580
at the cut off, but also
subtract poles and D equals 3.
1071
01:07:42,580 --> 01:07:45,090
And dimreg knows about
power law divergences
1072
01:07:45,090 --> 01:07:46,840
and they're just poles
at different places
1073
01:07:46,840 --> 01:07:48,530
in the dimensions.
1074
01:07:48,530 --> 01:07:50,560
And so if we subtract
poles at D equals three,
1075
01:07:50,560 --> 01:07:52,700
we can track the power law
divergences in that way.
1076
01:07:52,700 --> 01:07:54,700
And it's the power of
divergence that's actually
1077
01:07:54,700 --> 01:07:56,410
causing, if you
want to think of it
1078
01:07:56,410 --> 01:07:58,495
as a change to the
anomalous dimension, where
1079
01:07:58,495 --> 01:08:00,370
the anomalous dimension
was saying this thing
1080
01:08:00,370 --> 01:08:03,250
was irrelevant, to changing
it to something relevant,
1081
01:08:03,250 --> 01:08:05,000
you need a big change
for that to happen.
1082
01:08:05,000 --> 01:08:06,730
And the big change
that's occurring
1083
01:08:06,730 --> 01:08:09,100
is coming from a power
law divergence here.
1084
01:08:09,100 --> 01:08:11,650
That's what sort of allowed
you to jump, if you like,
1085
01:08:11,650 --> 01:08:15,730
from this fixed
point to this one.
1086
01:08:15,730 --> 01:08:17,950
The renormalization group,
including the power law
1087
01:08:17,950 --> 01:08:20,937
divergence, allows you to even
flow between those points.
1088
01:08:20,937 --> 01:08:23,020
Usually we think that power
law divergences aren't
1089
01:08:23,020 --> 01:08:25,562
doing anything, here's an
example where they are.
1090
01:08:25,562 --> 01:08:26,979
They're not doing
anything as long
1091
01:08:26,979 --> 01:08:28,812
as you know you're at
the right fixed point.
1092
01:08:28,812 --> 01:08:31,078
If you're describing the
right physics around one
1093
01:08:31,078 --> 01:08:33,370
of these fixed points, you
can concentrate on the logs,
1094
01:08:33,370 --> 01:08:35,715
but if you don't know where
you are then the power
1095
01:08:35,715 --> 01:08:37,090
law divergences
could be crucial.
1096
01:08:39,609 --> 01:08:41,490
All right, how does
this scheme work?
1097
01:08:45,250 --> 01:08:49,240
So this is a dimreg-type scheme.
1098
01:08:49,240 --> 01:08:52,930
So we're going to
get the power of mu
1099
01:08:52,930 --> 01:08:55,510
from the mu to the two epsilon
that we have out front.
1100
01:09:07,520 --> 01:09:11,587
So if I just write this
guy down in D dimensions,
1101
01:09:11,587 --> 01:09:12,670
here's what it looks like.
1102
01:09:31,140 --> 01:09:33,770
And I've normalized mu slightly
differently than we usually
1103
01:09:33,770 --> 01:09:37,680
do just because it's convenient
for this scheme to do that.
1104
01:09:37,680 --> 01:09:40,160
So it's not exactly the same
as MS bar, it's mu over 2
1105
01:09:40,160 --> 01:09:41,270
that I'm putting in.
1106
01:09:41,270 --> 01:09:43,640
Other than that, it's the
same kind of set up as MS bar.
1107
01:09:51,584 --> 01:09:54,269
You'll see why I want to put
that 2 there in a minute.
1108
01:09:59,030 --> 01:10:01,608
OK, so this is just the result
that we would write down
1109
01:10:01,608 --> 01:10:02,900
for dimensional regularization.
1110
01:10:02,900 --> 01:10:04,865
Dimensional regularization
is not a scheme.
1111
01:10:04,865 --> 01:10:08,930
A scheme has to do
with what we subtract.
1112
01:10:08,930 --> 01:10:10,613
Dimreg is just how we regulate.
1113
01:10:21,740 --> 01:10:25,160
So now, look at D equals 4.
1114
01:10:25,160 --> 01:10:31,370
So in D equals 4, OK.
1115
01:10:31,370 --> 01:10:32,495
There's a bunch of factors.
1116
01:10:38,315 --> 01:10:40,940
This is just giving, this is the
answer I quoted to you before.
1117
01:10:44,480 --> 01:10:46,700
Something finite.
1118
01:10:46,700 --> 01:10:54,160
And if we look at
D equals three,
1119
01:10:54,160 --> 01:10:56,285
then we have a pole because
of that gamma function.
1120
01:11:02,030 --> 01:11:04,370
And I've put the 2 in here
just to cancel that 2.
1121
01:11:07,520 --> 01:11:11,990
And so, what this scheme says
is to add a part subtraction
1122
01:11:11,990 --> 01:11:14,090
for this guy.
1123
01:11:14,090 --> 01:11:16,950
So what we do is, we
add a counter term.
1124
01:11:16,950 --> 01:11:25,640
It looks like minus IM over 4
pi, mu, got one power of mu.
1125
01:11:25,640 --> 01:11:29,480
Over 3 minus D. C0 squared.
1126
01:11:32,430 --> 01:11:36,710
And then if we take the graph,
plus the counterterm and we
1127
01:11:36,710 --> 01:11:38,840
set D equal four, which
is where we actually want
1128
01:11:38,840 --> 01:11:45,320
to do physics, lo and behold.
1129
01:11:45,320 --> 01:11:51,500
In this approach, you get
actually the same answer
1130
01:11:51,500 --> 01:11:53,510
as in our offshell momentum
subtraction scheme.
1131
01:11:53,510 --> 01:11:55,610
And that's just really because
this scheme tracks the power
1132
01:11:55,610 --> 01:11:57,170
correction, the
power divergence,
1133
01:11:57,170 --> 01:12:00,250
just like the offshell
momentum subtraction did.
1134
01:12:00,250 --> 01:12:03,500
So we just invented
a dimreg style
1135
01:12:03,500 --> 01:12:05,965
of looking for poles that
can track the same physics,
1136
01:12:05,965 --> 01:12:07,340
and we just have
to look at poles
1137
01:12:07,340 --> 01:12:11,440
in D equals 3 rather than
poles in D equals 4, OK?
1138
01:12:13,887 --> 01:12:16,220
And this is easier in general
than the offshell momentum
1139
01:12:16,220 --> 01:12:19,880
subtraction scheme, although
for basically everything
1140
01:12:19,880 --> 01:12:22,650
we're talking about today
you could do either one.
1141
01:12:22,650 --> 01:12:25,848
Now, where is the predictive
power of this effective theory?
1142
01:12:25,848 --> 01:12:27,890
So far, we've just kind
of cooked things together
1143
01:12:27,890 --> 01:12:30,350
to make the C0 do what we want.
1144
01:12:30,350 --> 01:12:32,420
Well, we didn't completely
cook things together.
1145
01:12:32,420 --> 01:12:34,370
We switched to another scheme
and it kind of popped out
1146
01:12:34,370 --> 01:12:35,745
naturally, but
you can say, well,
1147
01:12:35,745 --> 01:12:39,178
why not explore three other
schemes and see if they work?
1148
01:12:39,178 --> 01:12:41,720
But let's just imagine that we
got things to work, as we just
1149
01:12:41,720 --> 01:12:44,210
did in two different ways
by tracking the power law
1150
01:12:44,210 --> 01:12:45,320
divergence.
1151
01:12:45,320 --> 01:12:47,750
The predictive power
becomes from now the fact
1152
01:12:47,750 --> 01:12:51,710
that if I say that's my fine
tuning, that C0 got enhanced,
1153
01:12:51,710 --> 01:12:54,230
I can now predict the size
of all other operators
1154
01:12:54,230 --> 01:12:55,310
in the theory.
1155
01:12:55,310 --> 01:12:57,887
And other operators
like C2 and C4,
1156
01:12:57,887 --> 01:13:00,470
the power counting we assigned
to them previously is not true.
1157
01:13:00,470 --> 01:13:01,980
We have to figure it out.
1158
01:13:01,980 --> 01:13:04,610
But we can figure
that out once we know
1159
01:13:04,610 --> 01:13:09,160
what approach we should use.
1160
01:13:09,160 --> 01:13:15,667
OK, so this is same as above.
1161
01:13:15,667 --> 01:13:18,250
I won't go through it, but you
know, same anomalous dimension,
1162
01:13:18,250 --> 01:13:19,420
et cetera.
1163
01:13:19,420 --> 01:13:22,240
And it's easier in general.
1164
01:13:22,240 --> 01:13:25,990
Let's think about C2 mu.
1165
01:13:25,990 --> 01:13:29,170
So if you look at
C2, the first kind
1166
01:13:29,170 --> 01:13:33,160
of diagram you could think about
would be a guy with one C2,
1167
01:13:33,160 --> 01:13:36,070
and then this is the
first type of loop diagram
1168
01:13:36,070 --> 01:13:38,240
you might think about.
1169
01:13:38,240 --> 01:13:41,960
And these guys have a P squared
because C2 gave a P squared.
1170
01:13:41,960 --> 01:13:43,840
So they have an extra P squared.
1171
01:13:43,840 --> 01:13:45,010
And they diverge.
1172
01:13:45,010 --> 01:13:46,690
They also have a
power divergence.
1173
01:13:46,690 --> 01:13:48,280
So if you calculate
in either one
1174
01:13:48,280 --> 01:13:50,320
of these schemes,
offshell momentum
1175
01:13:50,320 --> 01:13:56,927
subtraction, or this PDS scheme,
these guys get a divergence.
1176
01:13:56,927 --> 01:13:58,510
And again, it's a
power law divergence
1177
01:13:58,510 --> 01:14:01,190
so there's a mu here.
1178
01:14:01,190 --> 01:14:06,190
And you get a beta
function, that's that.
1179
01:14:06,190 --> 01:14:08,800
2C0 C2.
1180
01:14:08,800 --> 01:14:11,040
So if you take the
boundary condition
1181
01:14:11,040 --> 01:14:15,670
C2 of 0 which is our MS
bar result. 4 pi over M. A
1182
01:14:15,670 --> 01:14:17,320
squared, r0.
1183
01:14:17,320 --> 01:14:30,010
And you solve this, you find
C2 of mu is 4 pi over M. 1
1184
01:14:30,010 --> 01:14:34,870
over mu minus 1 over a, squared.
1185
01:14:34,870 --> 01:14:36,010
r0 over 2.
1186
01:14:41,700 --> 01:14:44,165
So there's two-- we
previously, with our counting,
1187
01:14:44,165 --> 01:14:45,540
when we were
counting dimensions,
1188
01:14:45,540 --> 01:14:49,080
we would have said C0
goes like 1 over lambda,
1189
01:14:49,080 --> 01:14:51,900
C2 goes like one
over lambda cubed.
1190
01:14:51,900 --> 01:14:54,480
We had 2M plus 1 lambdas.
1191
01:14:54,480 --> 01:14:57,300
What we've just
discovered is that yes,
1192
01:14:57,300 --> 01:15:00,700
there's a lambda from this
r0, that's a 1 over lambda,
1193
01:15:00,700 --> 01:15:03,570
but the other two
lambdas are really mu's.
1194
01:15:03,570 --> 01:15:06,900
So this operator
is also enhanced,
1195
01:15:06,900 --> 01:15:09,797
and it's enhanced by two powers.
1196
01:15:09,797 --> 01:15:11,380
So once you know
leading order theory,
1197
01:15:11,380 --> 01:15:13,770
you should be able to determine
the power counting of all
1198
01:15:13,770 --> 01:15:15,352
of the other
operators, especially
1199
01:15:15,352 --> 01:15:16,560
if they're not relevant ones.
1200
01:15:16,560 --> 01:15:18,450
You have to get the
relevant part right,
1201
01:15:18,450 --> 01:15:20,040
and then you can
use that Lagrangian
1202
01:15:20,040 --> 01:15:22,697
to predict all the scaling
for everything else.
1203
01:15:22,697 --> 01:15:24,030
And that's what we've just done.
1204
01:15:38,970 --> 01:15:41,850
Gone to 1 over mu
squared lambda.
1205
01:15:41,850 --> 01:15:42,360
OK?
1206
01:15:42,360 --> 01:15:45,223
And those powers of mu
we see in the scheme,
1207
01:15:45,223 --> 01:15:46,890
and the 1 over lambda
comes from the r0.
1208
01:15:54,000 --> 01:15:54,500
OK.
1209
01:15:54,500 --> 01:15:59,780
So what the RGE actually
does is it tells us--
1210
01:15:59,780 --> 01:16:02,990
one way of thinking about
it is that it tells us
1211
01:16:02,990 --> 01:16:04,640
the enhancement,
due to fine tuning,
1212
01:16:04,640 --> 01:16:05,990
of all operators in the theory.
1213
01:16:09,607 --> 01:16:11,690
And that's really because
the fine tuning was just
1214
01:16:11,690 --> 01:16:14,180
a change of our power
counting, and we
1215
01:16:14,180 --> 01:16:16,930
have to propagate that
change everywhere.
1216
01:16:22,930 --> 01:16:25,360
And we can do that, and
it's the beta functions
1217
01:16:25,360 --> 01:16:28,870
that tell us how to
propagate the fine tuning.
1218
01:16:28,870 --> 01:16:34,246
So if you keep going,
you can do C2K of mu.
1219
01:16:34,246 --> 01:16:36,070
You find an anomalous dimension.
1220
01:16:39,680 --> 01:16:43,060
This theory is kind of
nice, you can basically
1221
01:16:43,060 --> 01:16:44,490
do all the calculations, so.
1222
01:16:48,076 --> 01:16:50,500
When you go to CK,
you get contributions
1223
01:16:50,500 --> 01:16:54,250
from various lower
order coefficients,
1224
01:16:54,250 --> 01:16:56,365
and it's one loop exact
so you only have pairs.
1225
01:16:59,365 --> 01:17:00,740
And then you can
kind of contrast
1226
01:17:00,740 --> 01:17:04,880
what's going on in a
naive power counting where
1227
01:17:04,880 --> 01:17:06,230
P is much less than 1 over a.
1228
01:17:14,720 --> 01:17:17,750
Let's just go up to C4.
1229
01:17:17,750 --> 01:17:21,200
Versus this kind
of improved power
1230
01:17:21,200 --> 01:17:26,540
counting, which is valid when
PA is greater than our order 1.
1231
01:17:30,995 --> 01:17:34,820
So C0 hat it went like 1 over
mu, no suppression there.
1232
01:17:34,820 --> 01:17:37,370
C2 hat goes like one
over mu squared lambda,
1233
01:17:37,370 --> 01:17:39,840
and that actually is irrelevant.
1234
01:17:39,840 --> 01:17:41,480
But it's just
irrelevant by one power.
1235
01:17:46,300 --> 01:17:49,950
So relative to this guy,
it's down by a P over lambda.
1236
01:17:49,950 --> 01:17:52,020
And then interesting
things start
1237
01:17:52,020 --> 01:17:54,660
to happen with the
higher ones, at least
1238
01:17:54,660 --> 01:17:56,386
from an RGE perspective.
1239
01:17:59,493 --> 01:18:01,410
So these guys start to
get more than one term.
1240
01:18:05,358 --> 01:18:07,650
But this guy actually doesn't
introduce a new constant.
1241
01:18:13,458 --> 01:18:16,000
There's a piece of the anomalous
dimension of this guy that's
1242
01:18:16,000 --> 01:18:17,458
actually just fixed
by the constant
1243
01:18:17,458 --> 01:18:20,170
that you already had here,
and then there's a new piece.
1244
01:18:20,170 --> 01:18:22,350
So the new piece is down
by two powers of lambda.
1245
01:18:26,545 --> 01:18:29,170
And that's encoding things about
the amplitude, actually, but--
1246
01:18:32,240 --> 01:18:33,800
OK, so that's just
a little table
1247
01:18:33,800 --> 01:18:35,342
to kind of convince
you that once you
1248
01:18:35,342 --> 01:18:37,370
have a beta function
that you can compute
1249
01:18:37,370 --> 01:18:39,260
for the coefficients,
you can quickly
1250
01:18:39,260 --> 01:18:41,570
propagate this enhancement
from the fine tuning
1251
01:18:41,570 --> 01:18:42,860
to the rest of your theory.
1252
01:18:42,860 --> 01:18:45,320
I.e., figure out what
the power counting
1253
01:18:45,320 --> 01:18:47,020
is for all the operators.
1254
01:18:47,020 --> 01:18:49,640
AUDIENCE: So every time there
is a power law divergence,
1255
01:18:49,640 --> 01:18:51,482
should I be worried
if I'm using MS bar,
1256
01:18:51,482 --> 01:18:53,940
should I be worried that the
power counting could be wrong?
1257
01:18:53,940 --> 01:18:56,160
PROFESSOR: Every time--
1258
01:18:56,160 --> 01:18:58,160
Yeah, every time there's
a power law divergence,
1259
01:18:58,160 --> 01:18:59,660
it's worth thinking
about whether it
1260
01:18:59,660 --> 01:19:06,220
had some physical impact on
what you're doing, I think.
1261
01:19:11,042 --> 01:19:13,500
If you know you're expanding--
if you can convince yourself
1262
01:19:13,500 --> 01:19:15,930
that you're expanding around
the right fixed point then
1263
01:19:15,930 --> 01:19:16,530
you're OK.
1264
01:19:16,530 --> 01:19:20,520
That's my equivalence claim.
1265
01:19:20,520 --> 01:19:22,020
But you don't
necessarily know that.
1266
01:19:22,020 --> 01:19:24,110
So let's go back
to our amplitude
1267
01:19:24,110 --> 01:19:26,930
and see what's going
on here, and see
1268
01:19:26,930 --> 01:19:29,890
what it looks like with
this power counting.
1269
01:19:29,890 --> 01:19:33,290
And so it's really just
a different expansion
1270
01:19:33,290 --> 01:19:34,760
of that amplitude that we had.
1271
01:19:41,420 --> 01:19:48,380
And in either the PDS scheme or
the power diversion subtraction
1272
01:19:48,380 --> 01:19:54,620
scheme, we end up with this
amplitude in the case where
1273
01:19:54,620 --> 01:19:56,670
we would use that scheme.
1274
01:19:56,670 --> 01:19:59,948
So you can see in PDS that if
I set this coefficient to 0,
1275
01:19:59,948 --> 01:20:01,490
the denominator
becomes 1, and then I
1276
01:20:01,490 --> 01:20:04,580
get with the offshell
momentum subtraction scheme.
1277
01:20:04,580 --> 01:20:07,640
In PDS is a little harder
see that it gives just that,
1278
01:20:07,640 --> 01:20:10,080
but it actually gives the
same thing in either scheme.
1279
01:20:10,080 --> 01:20:14,270
And if you think about what type
of expansion you're doing here,
1280
01:20:14,270 --> 01:20:15,930
you're keeping C0 to all orders.
1281
01:20:15,930 --> 01:20:23,720
So your amplitude at
lowest order is just this,
1282
01:20:23,720 --> 01:20:38,510
and then the C2 term
looks like that.
1283
01:20:38,510 --> 01:20:40,040
And then there's
some higher terms
1284
01:20:40,040 --> 01:20:43,770
which I wrote in my notes,
but I want right here.
1285
01:20:43,770 --> 01:20:47,270
And what this is, this here
is some kind of interaction.
1286
01:20:47,270 --> 01:20:50,030
I'll make it a
bigger circle, which
1287
01:20:50,030 --> 01:20:53,900
sums up all the bubbles
with C0's in them.
1288
01:20:53,900 --> 01:20:56,780
That's what's happened here.
1289
01:20:56,780 --> 01:21:01,610
And this here, if you like,
is like taking C2 and then
1290
01:21:01,610 --> 01:21:04,535
dressing it with
bubbles on either side.
1291
01:21:04,535 --> 01:21:07,700
So there's two, there's bubbles.
1292
01:21:07,700 --> 01:21:11,523
The bubbles on the other side.
1293
01:21:11,523 --> 01:21:12,815
And then bubbles on both sides.
1294
01:21:16,500 --> 01:21:18,480
So we calculate
these loop graphs
1295
01:21:18,480 --> 01:21:19,980
and these are the
amplitudes we get,
1296
01:21:19,980 --> 01:21:21,355
and that's because
we're treating
1297
01:21:21,355 --> 01:21:23,645
the C0 coupling to all
orders, we're summing it up.
1298
01:21:23,645 --> 01:21:25,770
And actually, each of these
amplitudes, if you look
1299
01:21:25,770 --> 01:21:28,590
at the RGE, is mu independent.
1300
01:21:28,590 --> 01:21:31,115
Explicitly mu independent.
1301
01:21:31,115 --> 01:21:32,490
So it's like
perturbation theory,
1302
01:21:32,490 --> 01:21:33,960
where we're doing a
momentum expansion,
1303
01:21:33,960 --> 01:21:35,877
and order by order in
that momentum expansion,
1304
01:21:35,877 --> 01:21:39,300
the amplitude is
independent of the scale mu.
1305
01:21:39,300 --> 01:21:41,160
The only purpose
of the scale mu is
1306
01:21:41,160 --> 01:21:45,155
to help us think about power
counting of these operators.
1307
01:21:45,155 --> 01:21:47,530
In the end of the day, when
we make physical predictions,
1308
01:21:47,530 --> 01:21:50,790
then getting mu
independent answers.
1309
01:21:50,790 --> 01:21:51,840
OK.
1310
01:21:51,840 --> 01:21:55,813
And this is like
organizing, if you like.
1311
01:21:55,813 --> 01:21:56,980
And you put it in terms of--
1312
01:22:01,380 --> 01:22:07,255
put it back in
terms of a, this is
1313
01:22:07,255 --> 01:22:08,880
like organizing the
theory in this way,
1314
01:22:08,880 --> 01:22:12,900
where you keep all powers of
AP, and that makes it very clear
1315
01:22:12,900 --> 01:22:14,910
that it's mu independent.
1316
01:22:14,910 --> 01:22:17,100
Now, this part of
the theory is so
1317
01:22:17,100 --> 01:22:19,380
simple you could have
figured that out just
1318
01:22:19,380 --> 01:22:22,380
by writing the top line down in
terms of a's and P's and just
1319
01:22:22,380 --> 01:22:24,180
writing this line down.
1320
01:22:24,180 --> 01:22:26,730
But you could also use what
I've been talking about
1321
01:22:26,730 --> 01:22:31,260
to figure out, for example, say
I coupled an external photon
1322
01:22:31,260 --> 01:22:33,150
to my four nucleon operators.
1323
01:22:33,150 --> 01:22:34,830
How big is this?
1324
01:22:34,830 --> 01:22:35,640
OK.
1325
01:22:35,640 --> 01:22:38,370
Well, it actually gets enhanced
from the fact the scattering
1326
01:22:38,370 --> 01:22:39,995
length is large, and
you can figure out
1327
01:22:39,995 --> 01:22:43,650
how important this effect
is, and when people do things
1328
01:22:43,650 --> 01:22:48,030
like deuteron formation in
the sun and stuff like that,
1329
01:22:48,030 --> 01:22:51,330
they use this effective theory
to do higher order calculations
1330
01:22:51,330 --> 01:22:55,040
and make precision predictions
for deuteron physics.
1331
01:22:55,040 --> 01:22:56,910
So it's not just a toy model.
1332
01:22:56,910 --> 01:23:00,750
It's actually something that has
a real impact on some physics.
1333
01:23:00,750 --> 01:23:02,813
We'll talk a little bit
more about it next time.
1334
01:23:02,813 --> 01:23:04,980
We'll talk a little bit
about the conformal symmetry
1335
01:23:04,980 --> 01:23:07,105
and I'll talk a little bit
more about the deuterons
1336
01:23:07,105 --> 01:23:09,210
since that's something
interesting in this theory,
1337
01:23:09,210 --> 01:23:13,330
and then we'll go on from there.
1338
01:23:13,330 --> 01:23:14,760
So, any questions?
1339
01:23:17,388 --> 01:23:18,880
It's cool stuff.
1340
01:23:18,880 --> 01:23:21,310
Simple to do calculations.
1341
01:23:21,310 --> 01:23:23,850
It's kind of interesting
to think about.