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IAIN STEWART: So
last time, we were

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00:00:24,310 --> 00:00:27,790
talking about the standard model
as an effective field theory.

10
00:00:27,790 --> 00:00:29,410
And we decided that
the power counting

11
00:00:29,410 --> 00:00:32,908
would be this, epsilon,
the masses of the particles

12
00:00:32,908 --> 00:00:35,200
in the standard model, the
scales in the standard model

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00:00:35,200 --> 00:00:37,750
divided by some new
physics scale, scale

14
00:00:37,750 --> 00:00:39,843
outside the standard model.

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00:00:39,843 --> 00:00:41,260
And I made the
statement that this

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00:00:41,260 --> 00:00:43,480
was connected to
operator dimension,

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00:00:43,480 --> 00:00:45,520
but I didn't make that precise.

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00:00:45,520 --> 00:00:49,120
And I want to do that now as
the first thing we do today.

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00:01:11,300 --> 00:01:12,900
So let's spend a
few minutes and talk

20
00:01:12,900 --> 00:01:16,865
about marginal, irrelevant,
and relevant operators

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00:01:16,865 --> 00:01:18,490
and their connection
to power counting.

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00:01:43,650 --> 00:01:45,900
I'm going to write power
counting over and over again.

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00:01:45,900 --> 00:01:48,880
And I'm going to
abbreviate it p.c.

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00:01:48,880 --> 00:01:49,710
from now on.

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00:01:54,377 --> 00:01:56,210
So let's consider an
effective field theory.

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00:01:56,210 --> 00:02:07,030
It'll be a scalar effective
field theory in d dimensions,

27
00:02:07,030 --> 00:02:19,240
standard kinetic term,
mass term, phi 4 term.

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00:02:19,240 --> 00:02:22,530
So it'll be a phi 4
scalar field theory.

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00:02:22,530 --> 00:02:25,230
It will be an effective
theory, so we won't stop there.

30
00:02:28,480 --> 00:02:30,240
And I'll just write
down up to phi 6.

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00:02:30,240 --> 00:02:32,032
And then, in principle,
I could keep going.

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00:02:36,110 --> 00:02:39,030
So we can look at the dimensions
of the various objects here.

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00:02:39,030 --> 00:02:42,580
The action with our
units is dimensionless.

34
00:02:42,580 --> 00:02:46,230
h bar and c are 1.

35
00:02:46,230 --> 00:02:49,320
So the mass dimensions of
the field in d dimensions

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00:02:49,320 --> 00:02:55,506
are d minus 2 over 2 since the
dimensions of ddx or minus d.

37
00:02:55,506 --> 00:02:57,077
We have to compensate for that.

38
00:02:57,077 --> 00:02:59,160
And we have to compensate
for the two derivatives.

39
00:02:59,160 --> 00:03:00,990
The canonically
normalized kinetic term

40
00:03:00,990 --> 00:03:03,902
tells us what the
dimensions of phi are.

41
00:03:03,902 --> 00:03:06,360
And then we can work out the
dimensions of everything else,

42
00:03:06,360 --> 00:03:09,540
so mass squared dimension 2.

43
00:03:23,040 --> 00:03:25,130
Tau dimension-- 6 minus 2d.

44
00:03:25,130 --> 00:03:28,860
Lambda could be
dimension 0 if d is 4.

45
00:03:28,860 --> 00:03:29,360
OK.

46
00:03:29,360 --> 00:03:33,000
So hopefully-- somewhat
familiar stuff.

47
00:03:33,000 --> 00:03:35,720
So let's say we want
to study a correlation

48
00:03:35,720 --> 00:03:48,075
function of a bunch of phis at
different space time points.

49
00:03:55,380 --> 00:03:58,380
And we want to look at
it at long distance.

50
00:03:58,380 --> 00:03:59,735
Long distance is small momenta.

51
00:04:04,470 --> 00:04:07,765
So the way I'm going to
make the distance long

52
00:04:07,765 --> 00:04:10,140
is I'm going to say that all
these x's that are appearing

53
00:04:10,140 --> 00:04:13,050
in my phis, x1
through xn, I'm going

54
00:04:13,050 --> 00:04:17,640
to redefine them as some s
common parameter times x prime.

55
00:04:17,640 --> 00:04:22,380
And then I'm just going to take
s goes to infinity with the x

56
00:04:22,380 --> 00:04:23,010
prime fixed.

57
00:04:26,310 --> 00:04:27,765
So that makes all the x's large.

58
00:04:35,940 --> 00:04:40,830
So when I do that, if I make
a redefinition like that,

59
00:04:40,830 --> 00:04:43,050
I can mess up the normalization
of my kinetic term.

60
00:04:43,050 --> 00:04:47,580
It'll no longer be
canonical, but I can fix that

61
00:04:47,580 --> 00:04:50,770
by just redefining my field.

62
00:04:50,770 --> 00:04:53,140
And the way to do that
is to do the following.

63
00:04:58,410 --> 00:05:01,580
To find a new, phi prime,
it's equal to the old field,

64
00:05:01,580 --> 00:05:02,690
but rescaled by an s.

65
00:05:13,860 --> 00:05:17,370
And the outcome
of that is that we

66
00:05:17,370 --> 00:05:22,860
get an action for the
phi prime field written

67
00:05:22,860 --> 00:05:28,650
in terms of prime coordinates,
which has a kinetic term that's

68
00:05:28,650 --> 00:05:31,217
the same form.

69
00:05:31,217 --> 00:05:33,300
But then s's start showing
up in the other places.

70
00:06:00,670 --> 00:06:03,530
And if you look at the powers
of the s's that are showing up,

71
00:06:03,530 --> 00:06:07,600
it's related also to the
powers of these parameters.

72
00:06:07,600 --> 00:06:08,860
Yeah.

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00:06:08,860 --> 00:06:11,750
AUDIENCE: Are you sure about
the powers of lambda tau,

74
00:06:11,750 --> 00:06:13,678
for the dimension lambda tau?

75
00:06:13,678 --> 00:06:15,220
IAIN STEWART: Did
I get it backwards?

76
00:06:15,220 --> 00:06:16,542
Should it be d minus 4?

77
00:06:16,542 --> 00:06:18,250
AUDIENCE: I think it
should be d minus 4.

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00:06:18,250 --> 00:06:20,583
And then the other one should
be d minus 6 [INAUDIBLE]..

79
00:06:23,350 --> 00:06:24,550
IAIN STEWART: Yeah.

80
00:06:24,550 --> 00:06:27,680
That looks right.

81
00:06:27,680 --> 00:06:28,630
Oh, you have to be--

82
00:06:28,630 --> 00:06:29,800
so let's see.

83
00:06:29,800 --> 00:06:32,200
There's d's here, right?

84
00:06:32,200 --> 00:06:33,910
So it's not d.

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00:06:33,910 --> 00:06:34,530
AUDIENCE: Oh.

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00:06:34,530 --> 00:06:34,940
IAIN STEWART:
You've got to keep--

87
00:06:34,940 --> 00:06:36,110
AUDIENCE: [INAUDIBLE]

88
00:06:36,110 --> 00:06:37,390
IAIN STEWART: Yeah.

89
00:06:37,390 --> 00:06:38,420
I stick by what I wrote.

90
00:06:38,420 --> 00:06:38,920
Check it.

91
00:06:42,928 --> 00:06:45,470
All right, so let's look at the
correlation function in terms

92
00:06:45,470 --> 00:06:48,290
of the phi prime because
the phi prime is just

93
00:06:48,290 --> 00:06:49,400
a function of x primes.

94
00:06:49,400 --> 00:06:51,540
And the x primes
are holding fixed.

95
00:06:51,540 --> 00:06:54,440
So if we rescale everything
in terms of the x primes,

96
00:06:54,440 --> 00:06:58,460
then we have some matrix element
that's not growing with s.

97
00:06:58,460 --> 00:07:00,020
We can make all
the s's explicit.

98
00:07:05,840 --> 00:07:14,580
So we take our
original guy, which

99
00:07:14,580 --> 00:07:17,300
in terms of our new
variables looks like that.

100
00:07:17,300 --> 00:07:19,900
We make this redefinition.

101
00:07:19,900 --> 00:07:22,230
We get some powers
of s out front.

102
00:07:25,032 --> 00:07:38,490
And then we get something which
is just in terms of the x prime

103
00:07:38,490 --> 00:07:39,510
and won't grow with s.

104
00:07:45,380 --> 00:07:49,180
So we can study this in various
dimensions if we wanted to.

105
00:07:49,180 --> 00:07:53,550
Let's, for simplicity and also
since it's the most common case

106
00:07:53,550 --> 00:07:57,080
we're interested
in, take d equals 4.

107
00:07:57,080 --> 00:08:03,510
I still may write d's, but let's
from here on take d equals 4

108
00:08:03,510 --> 00:08:07,840
and ask the question, what
happens is s gets large?

109
00:08:07,840 --> 00:08:09,550
So now, we've made
all the s's explicit.

110
00:08:09,550 --> 00:08:11,040
This is something that
you often do when you're

111
00:08:11,040 --> 00:08:12,248
doing effective field theory.

112
00:08:12,248 --> 00:08:15,930
You can figure out how
you're going to study

113
00:08:15,930 --> 00:08:18,570
the large distance behavior.

114
00:08:18,570 --> 00:08:20,610
You want to make
the parameter that's

115
00:08:20,610 --> 00:08:22,860
controlling that limit
explicit, so you can see it.

116
00:08:22,860 --> 00:08:24,510
So it's not hiding anywhere.

117
00:08:24,510 --> 00:08:26,385
And that's what we've
done with this algebra.

118
00:08:31,360 --> 00:08:34,392
So as s goes to
infinity, because we

119
00:08:34,392 --> 00:08:35,850
have this explicit
s squared there,

120
00:08:35,850 --> 00:08:39,085
the m squared term is becoming
more and more important.

121
00:08:52,435 --> 00:08:53,310
It's called relevant.

122
00:08:57,110 --> 00:08:59,495
Child term is becoming
less important.

123
00:09:06,780 --> 00:09:09,540
Because if I put in d equals 4,
then this is s to the minus 2.

124
00:09:09,540 --> 00:09:15,690
It's tracking the s's, making
it less important as s grows.

125
00:09:15,690 --> 00:09:19,710
And the lambda term is equally
as important as it was before.

126
00:09:28,650 --> 00:09:31,040
And the terminology that
goes along with this

127
00:09:31,040 --> 00:09:32,660
is an association.

128
00:09:32,660 --> 00:09:34,663
So that was a statement
about parameters.

129
00:09:34,663 --> 00:09:36,580
We could also make a
statement about operators

130
00:09:36,580 --> 00:09:38,840
since obviously they were
part of the story here

131
00:09:38,840 --> 00:09:40,625
that gave the s factors.

132
00:09:43,440 --> 00:09:46,295
So we would say that phi
squared is a relevant operator.

133
00:09:49,410 --> 00:09:50,815
The phi 4 is marginal.

134
00:09:55,240 --> 00:09:57,297
And phi 6 is irrelevant.

135
00:10:06,610 --> 00:10:08,860
And you can see, because of
the argument that we made,

136
00:10:08,860 --> 00:10:32,638
that this was just directly
connected to dimension,

137
00:10:32,638 --> 00:10:34,430
so either to the
dimension of the operators

138
00:10:34,430 --> 00:10:36,013
or to the dimension
of the parameters.

139
00:10:38,417 --> 00:10:40,500
OK, so we're connecting
something that we can say,

140
00:10:40,500 --> 00:10:41,490
which is the power counting.

141
00:10:41,490 --> 00:10:42,990
In this case, we're
controlling that

142
00:10:42,990 --> 00:10:45,560
with s using s as our
control parameters

143
00:10:45,560 --> 00:10:47,152
to look at long distances.

144
00:10:47,152 --> 00:10:48,860
And we're seeing that
that gets connected

145
00:10:48,860 --> 00:10:50,330
to dimensions of operators.

146
00:11:18,972 --> 00:11:19,930
Is there any questions?

147
00:11:24,530 --> 00:11:27,880
So let's take s
finite, but large.

148
00:11:35,525 --> 00:11:37,150
Usually, we're not
interested in taking

149
00:11:37,150 --> 00:11:39,107
it all the way to infinity.

150
00:11:39,107 --> 00:11:40,690
Although we may make
it as large as we

151
00:11:40,690 --> 00:11:48,710
want to study some
long distance behavior.

152
00:11:48,710 --> 00:11:54,550
And so what I just said
is that we can see,

153
00:11:54,550 --> 00:11:56,620
from the powers of
s, the importance

154
00:11:56,620 --> 00:11:57,550
of the various terms.

155
00:12:04,150 --> 00:12:06,490
Relevant terms are more
important than marginal terms.

156
00:12:06,490 --> 00:12:08,698
And marginal terms are more
important than irrelevant

157
00:12:08,698 --> 00:12:09,815
terms.

158
00:12:09,815 --> 00:12:10,690
The words say it all.

159
00:12:19,220 --> 00:12:23,270
So that means that, if you want
to think of how to do the power

160
00:12:23,270 --> 00:12:25,790
counting and you don't want to
think of introducing this s,

161
00:12:25,790 --> 00:12:28,100
since that was kind
of just our choice--

162
00:12:28,100 --> 00:12:31,078
we introduced it as a way of
thinking about this question.

163
00:12:31,078 --> 00:12:32,870
But if we went back to
the original action,

164
00:12:32,870 --> 00:12:34,870
we should have a way of
doing the power counting

165
00:12:34,870 --> 00:12:37,368
from that without having
to do this rescaling.

166
00:12:37,368 --> 00:12:38,660
And we know how to do that now.

167
00:12:38,660 --> 00:12:41,240
This exercise teaches
us that we can just

168
00:12:41,240 --> 00:12:43,768
look at mass dimensions
of the parameters

169
00:12:43,768 --> 00:12:44,810
to do the power counting.

170
00:12:49,790 --> 00:13:03,340
So if we just associate a
power to the parameters,

171
00:13:03,340 --> 00:13:06,470
we're still in d equals 4.

172
00:13:06,470 --> 00:13:08,690
Then we would get
this association,

173
00:13:08,690 --> 00:13:10,940
this being the statement
that it's relevant, marginal,

174
00:13:10,940 --> 00:13:14,228
and irrelevant.

175
00:13:14,228 --> 00:13:16,270
And we can do a power
counting in this lambda nu.

176
00:13:40,420 --> 00:13:42,965
And we can then say,
in a language which

177
00:13:42,965 --> 00:13:44,590
would be familiar
from Feynman diagrams

178
00:13:44,590 --> 00:13:46,612
where we do everything
in momentum space,

179
00:13:46,612 --> 00:13:48,070
that the momentum
we want to study,

180
00:13:48,070 --> 00:13:51,640
p, has to be much less
than this lambda nu.

181
00:13:51,640 --> 00:13:53,820
And we'll do the power
counting to lambda nu.

182
00:13:53,820 --> 00:13:57,060
And that will make the,
for example, tau term

183
00:13:57,060 --> 00:14:00,863
an irrelevant, less
important, operator.

184
00:14:06,420 --> 00:14:08,270
So there's one comment here.

185
00:14:08,270 --> 00:14:12,080
We did the scalar field theory
just because it's simplest.

186
00:14:12,080 --> 00:14:15,530
It also has a relevant operator.

187
00:14:15,530 --> 00:14:18,350
And we see that relevant
operators actually

188
00:14:18,350 --> 00:14:26,750
can be dangerous
because we'd like

189
00:14:26,750 --> 00:14:29,060
to set the power counting
for the whole problem

190
00:14:29,060 --> 00:14:29,900
by the kinetic term.

191
00:14:29,900 --> 00:14:32,180
We'd like to say that
the kinetic term, which

192
00:14:32,180 --> 00:14:35,450
was canonically normalized
and had no s's in it,

193
00:14:35,450 --> 00:14:37,130
we'd like to say that
that was relevant,

194
00:14:37,130 --> 00:14:39,613
that that's part of the
leading order Lagrangian.

195
00:14:39,613 --> 00:14:41,780
But when we went through
it, we found something that

196
00:14:41,780 --> 00:14:43,270
was more relevant, the master.

197
00:14:43,270 --> 00:14:47,385
Phi squared could become even
larger than the kinetic term.

198
00:14:52,340 --> 00:14:55,771
So we have to be careful
about relevant operators.

199
00:14:55,771 --> 00:14:58,820
And then this is, of course,
related to the Higgs fine

200
00:14:58,820 --> 00:14:59,320
tuning.

201
00:15:05,530 --> 00:15:08,230
So even though I'm using
a scalar field theory,

202
00:15:08,230 --> 00:15:10,880
I'm, for the most part, going
to just fine tune and ignore

203
00:15:10,880 --> 00:15:13,910
this problem, since if I was
using something fermionic field

204
00:15:13,910 --> 00:15:15,710
theory, I could
set things up so I

205
00:15:15,710 --> 00:15:17,030
could ignore it from the start.

206
00:15:17,030 --> 00:15:21,500
But still using a scalar
field theory is convenient.

207
00:15:21,500 --> 00:15:24,500
So I want to also come
back to something else

208
00:15:24,500 --> 00:15:27,340
that we mentioned last time and
go into a little more detail.

209
00:15:27,340 --> 00:15:29,090
And that is the
discussion of divergences.

210
00:15:29,090 --> 00:15:31,700
So last time, we
said that there was

211
00:15:31,700 --> 00:15:34,390
two different ways of thinking
about renormalizability,

212
00:15:34,390 --> 00:15:36,330
a traditional sense
of renormalizability,

213
00:15:36,330 --> 00:15:39,170
renormalizability of
the standard model,

214
00:15:39,170 --> 00:15:41,450
or an effective field
theory way of thinking

215
00:15:41,450 --> 00:15:43,580
about renormalizability.

216
00:15:43,580 --> 00:15:46,430
So I want to come back to that
with our example of this scalar

217
00:15:46,430 --> 00:15:49,160
field theory.

218
00:15:49,160 --> 00:15:54,260
So let's get rid of this
issue of having something that

219
00:15:54,260 --> 00:15:56,690
can upset the power
counting either

220
00:15:56,690 --> 00:16:01,460
by taking m to be 0 or just
fine tuning it to be small.

221
00:16:01,460 --> 00:16:06,380
And what that means is I
just demand that, as s grows,

222
00:16:06,380 --> 00:16:07,760
I shrink m.

223
00:16:07,760 --> 00:16:13,880
And if I do that, then I can,
by hand, tune term and this term

224
00:16:13,880 --> 00:16:14,925
to be the same size.

225
00:16:14,925 --> 00:16:16,550
So if you like, I'm
assigning a scaling

226
00:16:16,550 --> 00:16:19,760
to m in order to make the
mass term be always as

227
00:16:19,760 --> 00:16:21,080
important as the kinetic term.

228
00:16:25,882 --> 00:16:27,590
So with that little
proviso, we can start

229
00:16:27,590 --> 00:16:29,120
thinking about divergences.

230
00:16:29,120 --> 00:16:31,910
And when we start
drawing Feynman diagrams,

231
00:16:31,910 --> 00:16:34,460
they will generically
have divergences.

232
00:16:34,460 --> 00:16:36,870
So we could have two
four-point interactions,

233
00:16:36,870 --> 00:16:39,590
which I label by lambda, because
that's the parameter that

234
00:16:39,590 --> 00:16:42,240
shows up in the Feynman rule.

235
00:16:42,240 --> 00:16:46,285
If this is k and this
is some k plus p,

236
00:16:46,285 --> 00:16:48,410
then this guy is going to
have two pairs of lambda.

237
00:16:51,090 --> 00:17:05,775
And it's going to be
some integral like that.

238
00:17:05,775 --> 00:17:09,740
We won't worry about
overall factors here.

239
00:17:09,740 --> 00:17:12,259
I'm regulating with
dimensional regularization.

240
00:17:17,150 --> 00:17:22,220
I'll often do that when
it's convenient for us.

241
00:17:22,220 --> 00:17:24,510
If you ask how this
integral diverges,

242
00:17:24,510 --> 00:17:25,970
you could ask how
it diverges just

243
00:17:25,970 --> 00:17:28,910
in terms of thinking about it in
terms of some parameter that's

244
00:17:28,910 --> 00:17:32,030
controlling the
ultraviolet, like a cutoff.

245
00:17:32,030 --> 00:17:35,000
So even if I am using dim
reg, I could ask, what's

246
00:17:35,000 --> 00:17:36,950
the power of the divergence?

247
00:17:36,950 --> 00:17:40,420
And it diverges as, in
d dimensions, lambda

248
00:17:40,420 --> 00:17:42,512
to d minus 4.

249
00:17:42,512 --> 00:17:49,510
You say d minus 4 is the
degree of the divergence.

250
00:17:49,510 --> 00:17:53,450
And that's because you have d
powers of k from the measure

251
00:17:53,450 --> 00:17:55,620
and minus 4 from
the propagators.

252
00:17:58,700 --> 00:18:03,290
So if d is equal to 4, you
say degree of divergence is 0,

253
00:18:03,290 --> 00:18:05,112
but that means log diversion.

254
00:18:20,370 --> 00:18:26,630
So if you take d equals 4 in a
UV, 4 powers of k downstairs,

255
00:18:26,630 --> 00:18:29,330
d upstairs-- but
if d is 4, that's

256
00:18:29,330 --> 00:18:30,800
4 upstairs and 4 downstairs.

257
00:18:30,800 --> 00:18:33,110
So it's scaling
length dk over k.

258
00:18:33,110 --> 00:18:34,500
So I made it Euclidean.

259
00:18:34,500 --> 00:18:36,990
That's exactly what
it would become.

260
00:18:36,990 --> 00:18:37,910
And that's like a log.

261
00:18:37,910 --> 00:18:40,940
So it's a log of the cut off.

262
00:18:40,940 --> 00:18:48,476
So it's a 1 over
epsilon in dim reg

263
00:18:48,476 --> 00:18:50,460
where d is 4 minus 2 epsilon.

264
00:18:53,760 --> 00:18:56,380
And if you just want to
think about what this does,

265
00:18:56,380 --> 00:18:59,750
well, it's something that
renormalizes the lambda phi 4

266
00:18:59,750 --> 00:19:00,360
operator.

267
00:19:00,360 --> 00:19:04,540
So you need a counter
term for the lambda phi 4.

268
00:19:04,540 --> 00:19:07,700
So you add to your
theory that counter term.

269
00:19:07,700 --> 00:19:11,670
And you could get rid
of this divergence.

270
00:19:11,670 --> 00:19:12,170
OK.

271
00:19:12,170 --> 00:19:17,910
So, so far-- hopefully
standard stuff.

272
00:19:17,910 --> 00:19:22,050
Let's keep going, think
about other diagrams.

273
00:19:22,050 --> 00:19:27,822
So what if I put in a tau
term and a lambda term?

274
00:19:27,822 --> 00:19:29,280
This integral is
the same integral.

275
00:19:29,280 --> 00:19:32,550
I just have different
fields on the outside.

276
00:19:32,550 --> 00:19:41,850
So it's got the same
divergence, but now

277
00:19:41,850 --> 00:19:44,032
the operator it's
renormalizing is an operator

278
00:19:44,032 --> 00:19:45,240
with 6 points on the outside.

279
00:19:55,810 --> 00:19:58,660
So it's renormalizing
the tau phi 6 term.

280
00:19:58,660 --> 00:20:01,060
So I insert one
tau and one lambda,

281
00:20:01,060 --> 00:20:04,470
and I have to get back the
renormalization of tau.

282
00:20:04,470 --> 00:20:05,470
Well, that's not so bad.

283
00:20:05,470 --> 00:20:10,120
We had tau from the start, if
we include the tau term, so not

284
00:20:10,120 --> 00:20:12,270
really a problem from
the point of view

285
00:20:12,270 --> 00:20:14,790
of a standard
renormalization program.

286
00:20:14,790 --> 00:20:23,980
But we could also include
two taus, like this, again,

287
00:20:23,980 --> 00:20:29,950
same integral, so
same divergence.

288
00:20:29,950 --> 00:20:31,780
And now, this
renormalizes something

289
00:20:31,780 --> 00:20:38,770
that we haven't included yet,
something with 8 points a phi 8

290
00:20:38,770 --> 00:20:39,370
operator.

291
00:20:42,890 --> 00:20:44,530
So in order to
renormalize that diagram

292
00:20:44,530 --> 00:20:46,180
and make the theory
renormalizable

293
00:20:46,180 --> 00:20:47,680
and an effective
field theory sense,

294
00:20:47,680 --> 00:20:50,230
we need to include
the phi 8 operator.

295
00:20:55,290 --> 00:20:58,160
So if I'd ignored the
dots that I wrote down--

296
00:21:04,300 --> 00:21:05,675
so let me say,
without the dots--

297
00:21:09,260 --> 00:21:12,132
then phi 8 wasn't there.

298
00:21:12,132 --> 00:21:14,340
And so then, therefore, I
would say the theory is not

299
00:21:14,340 --> 00:21:16,490
renormalizable.

300
00:21:16,490 --> 00:21:19,640
That's what makes the
tau operator the phi 6

301
00:21:19,640 --> 00:21:24,530
operator via non-renormalizable
theory in the traditional sense

302
00:21:24,530 --> 00:21:26,147
if we include that operator.

303
00:21:34,270 --> 00:21:36,550
That's the classic
way of thinking.

304
00:21:40,193 --> 00:21:41,860
And the effective
theory way of thinking

305
00:21:41,860 --> 00:21:43,480
is just that we have
to add that operator

306
00:21:43,480 --> 00:21:45,400
as soon as this diagram
would become relevant.

307
00:21:50,810 --> 00:21:53,240
So we just determined
a minute ago

308
00:21:53,240 --> 00:21:55,440
that tau goes like
lambda nu to the minus 2.

309
00:22:00,040 --> 00:22:02,900
So tau goes like 1
over lambda nu squared.

310
00:22:02,900 --> 00:22:05,590
And in this diagram, we
have two powers of tau.

311
00:22:05,590 --> 00:22:07,110
So it's even less important.

312
00:22:07,110 --> 00:22:08,830
So lambda nu to the fourth--

313
00:22:08,830 --> 00:22:09,550
downstairs.

314
00:22:19,660 --> 00:22:22,408
And so when it becomes
relevant to us,

315
00:22:22,408 --> 00:22:23,950
we want that kind
of accuracy that we

316
00:22:23,950 --> 00:22:26,290
want to include things that
go like 1 over lambda nu

317
00:22:26,290 --> 00:22:27,310
to the fourth.

318
00:22:27,310 --> 00:22:28,690
We have to consider
this diagram.

319
00:22:28,690 --> 00:22:30,690
And we have to consider
adding that operator

320
00:22:30,690 --> 00:22:31,690
to the effective theory.

321
00:22:44,827 --> 00:22:47,160
And that's the sense in which
we say that the theory can

322
00:22:47,160 --> 00:22:57,560
be renormalized order by order
in its power kinetic parameter,

323
00:22:57,560 --> 00:23:00,430
which is 1 over lambda nu.

324
00:23:06,322 --> 00:23:14,299
So I could order this order, but
we have to add this operator.

325
00:23:32,762 --> 00:23:33,760
OK?

326
00:23:33,760 --> 00:23:34,760
Make sense?

327
00:23:39,390 --> 00:23:41,970
Silence means that
it makes sense.

328
00:23:41,970 --> 00:23:44,010
Jumping up and down saying
it doesn't make sense

329
00:23:44,010 --> 00:23:47,280
means that it doesn't make
sense or puzzled looks

330
00:23:47,280 --> 00:23:49,305
from everybody, but
that's harder to discern.

331
00:24:08,390 --> 00:24:14,463
So we can summarize this way of
thinking in the following way.

332
00:24:14,463 --> 00:24:15,880
Remember with the
effective theory

333
00:24:15,880 --> 00:24:17,505
that we're only
interested in computing

334
00:24:17,505 --> 00:24:19,510
things to some accuracy.

335
00:24:19,510 --> 00:24:23,300
And the accuracy controls
where we stop in the series.

336
00:24:23,300 --> 00:24:28,450
So if we're interested in
stopping at lambda nu to the r

337
00:24:28,450 --> 00:24:31,300
or 1 over s to
the r-- s was big.

338
00:24:31,300 --> 00:24:33,510
And lambda nu is also
much bigger than the p.

339
00:24:37,290 --> 00:24:40,110
But let's stick to
talking about lambda nu.

340
00:24:44,800 --> 00:24:50,230
And we include all operators
that have dimensions up

341
00:24:50,230 --> 00:24:52,950
to a certain level.

342
00:24:52,950 --> 00:24:58,407
And since the power counting
is connected to dimensions,

343
00:24:58,407 --> 00:25:00,990
we're kind of guaranteed that
we will have everything we need.

344
00:25:09,900 --> 00:25:16,220
So as I promised, what this
little argument or discussion

345
00:25:16,220 --> 00:25:21,618
tells us, is how power counting
is connected to dimensions.

346
00:25:21,618 --> 00:25:24,160
And this is the classic way of
thinking about effective field

347
00:25:24,160 --> 00:25:25,660
theory is that the
power counting is

348
00:25:25,660 --> 00:25:26,864
connected to dimensions.

349
00:25:30,660 --> 00:25:33,600
So this seems pretty
generic actually.

350
00:25:33,600 --> 00:25:35,790
You could imagine that,
if I did scalars, I mean,

351
00:25:35,790 --> 00:25:38,640
fermions and scalars
of gauge theory,

352
00:25:38,640 --> 00:25:41,250
then I could still go through
the same type of arguments,

353
00:25:41,250 --> 00:25:43,410
write down higher
dimensional operators,

354
00:25:43,410 --> 00:25:45,680
go through all these arguments.

355
00:25:45,680 --> 00:25:47,400
And so it seems
that I'd actually

356
00:25:47,400 --> 00:25:51,090
shown you something more
powerful than what I claimed.

357
00:25:51,090 --> 00:25:54,360
Because I said, here, it seems
like it's almost this, right,

358
00:25:54,360 --> 00:25:57,510
that power counting is always
connected to dimensions.

359
00:25:57,510 --> 00:26:01,050
So can anyone spot where there
was an assumption in what we

360
00:26:01,050 --> 00:26:07,140
did that where in some case the
power counting might not have

361
00:26:07,140 --> 00:26:08,999
been related to dimensions?

362
00:26:13,310 --> 00:26:14,747
It's a tough question.

363
00:26:14,747 --> 00:26:19,488
AUDIENCE: If the power
counting isn't a ratio scale?

364
00:26:19,488 --> 00:26:20,280
IAIN STEWART: Yeah.

365
00:26:20,280 --> 00:26:25,810
But I want a little bit
more than that, right track.

366
00:26:25,810 --> 00:26:27,690
So going back to this
example that we did,

367
00:26:27,690 --> 00:26:30,300
what did we assume at the
beginning that led us here?

368
00:26:34,150 --> 00:26:35,910
AUDIENCE: That we
changed the mass point?

369
00:26:35,910 --> 00:26:38,780
IAIN STEWART: No, the mass
wasn't so much the issue.

370
00:26:41,520 --> 00:26:43,710
So what it was is that we
scaled all the coordinates

371
00:26:43,710 --> 00:26:44,910
by the same amount.

372
00:26:44,910 --> 00:26:47,550
We said all the coordinates are
getting large in the same way.

373
00:26:47,550 --> 00:26:50,050
And we could have done something
more complicated than that.

374
00:26:50,050 --> 00:26:52,500
We could have said some
components of this coordinate

375
00:26:52,500 --> 00:26:54,600
are getting larger, faster
than other components.

376
00:26:54,600 --> 00:26:56,693
That's what you do in a
non-relativistic theory

377
00:26:56,693 --> 00:26:58,860
where the time component
and the spatial coordinates

378
00:26:58,860 --> 00:27:00,915
would scale in different ways.

379
00:27:00,915 --> 00:27:02,040
So that was the assumption.

380
00:27:02,040 --> 00:27:04,770
We assumed, basically, that
everything was getting large.

381
00:27:04,770 --> 00:27:08,490
All the coordinates were
getting large uniformly with s.

382
00:27:08,490 --> 00:27:12,360
And we said x mu i was
equal to s x prime mu i.

383
00:27:23,460 --> 00:27:26,130
With a universal s for
all the components,

384
00:27:26,130 --> 00:27:28,210
that was an assumption
that led us here.

385
00:27:28,210 --> 00:27:29,885
And we may not always do that.

386
00:27:29,885 --> 00:27:32,010
In fact, in some of our
examples, we won't do that.

387
00:27:34,925 --> 00:27:36,300
But here, for the
standard model,

388
00:27:36,300 --> 00:27:37,425
that's what you want to do.

389
00:27:52,690 --> 00:27:56,230
So if we have the standard
model, which is just L0,

390
00:27:56,230 --> 00:28:00,797
the usual standard
model, then we

391
00:28:00,797 --> 00:28:02,630
know, as part of the
way of constructing it,

392
00:28:02,630 --> 00:28:08,300
that we wrote down all the
operators with dimension

393
00:28:08,300 --> 00:28:09,500
less than or equal to 4.

394
00:28:16,780 --> 00:28:21,730
And we also know that
it was renormalizable

395
00:28:21,730 --> 00:28:22,810
in a traditional sense.

396
00:28:35,188 --> 00:28:36,980
So now, let's talk
about the standard model

397
00:28:36,980 --> 00:28:40,367
corrections, i.e. terms
in the standard model

398
00:28:40,367 --> 00:28:42,200
from an effective field
theory point of view

399
00:28:42,200 --> 00:28:45,335
that are operators we
can write down, like L1.

400
00:28:50,936 --> 00:28:56,910
So L1, I can write it in the
following way, which is kind

401
00:28:56,910 --> 00:29:00,300
of a convenient thing to do.

402
00:29:00,300 --> 00:29:03,750
Pull out the scale lambda nu.

403
00:29:03,750 --> 00:29:07,090
Leave over some
dimensionless constant.

404
00:29:07,090 --> 00:29:10,103
So I'll just use some scale
lambda nu for all the operators

405
00:29:10,103 --> 00:29:10,770
that I consider.

406
00:29:10,770 --> 00:29:12,330
And I'll just allow
for differences

407
00:29:12,330 --> 00:29:14,205
between the various
scales that the operators

408
00:29:14,205 --> 00:29:18,600
could have to be taken up
by dimensionless constants.

409
00:29:18,600 --> 00:29:22,006
And O5 here is a
dimension 5 operator.

410
00:29:32,430 --> 00:29:33,760
The dimension of c is 0.

411
00:29:36,860 --> 00:29:40,290
In a power counting notation,
we say c is of order 1.

412
00:29:40,290 --> 00:29:42,580
That means that we
don't count any powers

413
00:29:42,580 --> 00:29:44,280
of lambda nu associated to c.

414
00:29:44,280 --> 00:29:47,460
What we've done here is we've
made the power explicit by just

415
00:29:47,460 --> 00:29:48,050
writing it in.

416
00:29:55,892 --> 00:29:57,850
So that's often convenient
just in the same way

417
00:29:57,850 --> 00:30:00,520
it was convenient to make the
s's explicit in that argument.

418
00:30:00,520 --> 00:30:03,670
Now, we're just building up the
theory, writing down operators,

419
00:30:03,670 --> 00:30:06,010
making the lambda nu's,
which are our power counting

420
00:30:06,010 --> 00:30:07,210
parameter, very explicit.

421
00:30:11,820 --> 00:30:14,580
Now, the statement that,
in the standard model,

422
00:30:14,580 --> 00:30:17,130
it was renormalizable in
the traditional sense,

423
00:30:17,130 --> 00:30:22,740
I told you that what that meant
is that nothing in lambda 0,

424
00:30:22,740 --> 00:30:26,520
nothing in Lagrangian 0, really
tells us about lambda nu.

425
00:30:35,828 --> 00:30:37,620
So we're free to take
it as big as we want.

426
00:30:47,520 --> 00:30:52,862
There's no constraint on it from
our leading order Lagrangian.

427
00:30:52,862 --> 00:30:54,570
In particular, we can
take it much bigger

428
00:30:54,570 --> 00:30:56,610
than things like the
taught mass or the w mass.

429
00:31:01,640 --> 00:31:03,390
And we can make these
corrections as small

430
00:31:03,390 --> 00:31:03,900
as we want.

431
00:31:08,050 --> 00:31:11,560
So L1 is, therefore, really
we can think of it as really

432
00:31:11,560 --> 00:31:12,810
giving some small corrections.

433
00:31:12,810 --> 00:31:17,330
And we can adjust how small
they are by just dialing up

434
00:31:17,330 --> 00:31:18,390
the scale of lambda nu.

435
00:31:40,000 --> 00:31:44,040
All right, so let's
get down to business

436
00:31:44,040 --> 00:31:47,460
and actually talk about
what this Lagrangian is.

437
00:31:55,654 --> 00:32:03,370
Our notation that we index the
Lagrangian in a series, this

438
00:32:03,370 --> 00:32:05,977
is our sum over n that we
talked about last time.

439
00:32:05,977 --> 00:32:07,810
And our power counting
is that we associated

440
00:32:07,810 --> 00:32:12,640
this guy here with no
powers of lambda nu,

441
00:32:12,640 --> 00:32:15,480
this guy here with
one inverse power,

442
00:32:15,480 --> 00:32:20,290
this guy here with two
inverse power, et cetera.

443
00:32:20,290 --> 00:32:24,460
And we want to think about
using this for some p, which you

444
00:32:24,460 --> 00:32:27,340
could say is of
order m top squared,

445
00:32:27,340 --> 00:32:33,490
some scale that's much
less than the lambda nu.

446
00:32:33,490 --> 00:32:35,380
It could be larger than on top.

447
00:32:35,380 --> 00:32:38,830
It could be 10 TeV,
whatever we decide.

448
00:32:38,830 --> 00:32:41,290
But it's just, in order
for me to write something

449
00:32:41,290 --> 00:32:44,480
on the board, let
me write m top.

450
00:32:44,480 --> 00:32:44,980
OK.

451
00:32:44,980 --> 00:32:47,350
So how do we
construct L1 and L2?

452
00:32:47,350 --> 00:32:49,630
What do we assume?

453
00:32:49,630 --> 00:32:52,540
Well, one thing we assume,
or we are free to assume

454
00:32:52,540 --> 00:32:55,840
and is a reasonable
assumption, is

455
00:32:55,840 --> 00:32:59,980
that there are not going to be
any Lorentz invariant or gauge

456
00:32:59,980 --> 00:33:01,145
invariant violating terms.

457
00:33:07,020 --> 00:33:07,520
OK.

458
00:33:07,520 --> 00:33:09,860
So we can maintain these as
symmetries of our theory.

459
00:33:17,400 --> 00:33:20,290
So we assume that
they're unbroken.

460
00:33:20,290 --> 00:33:21,910
So that means,
when we write down

461
00:33:21,910 --> 00:33:27,480
these L1 and L2, or
generically each Li,

462
00:33:27,480 --> 00:33:30,528
that we're going to have to do
it by writing down operators

463
00:33:30,528 --> 00:33:31,570
that are gauge invariant.

464
00:33:31,570 --> 00:33:33,112
Even though they're
higher dimension,

465
00:33:33,112 --> 00:33:38,130
we still have to satisfy gauge
variance of the standard model

466
00:33:38,130 --> 00:33:39,677
and Lorentz invariance.

467
00:33:45,530 --> 00:33:49,120
So that's going to
restrict what we can do.

468
00:33:56,610 --> 00:34:00,918
We also construct Li from
the same degrees of freedom

469
00:34:00,918 --> 00:34:01,710
that we have in L0.

470
00:34:10,370 --> 00:34:11,710
So we know what fields to use.

471
00:34:15,170 --> 00:34:16,639
So that's important.

472
00:34:16,639 --> 00:34:18,639
It means that, once you've
got the leading order

473
00:34:18,639 --> 00:34:20,469
effective field
theory, you know where

474
00:34:20,469 --> 00:34:23,409
to go for the higher order
terms because you're just

475
00:34:23,409 --> 00:34:24,429
using the same fields.

476
00:34:33,834 --> 00:34:36,730
I'll also make the assumption
that the Higgs vacuum

477
00:34:36,730 --> 00:34:49,650
expectation value is going to
stay to be the value in L0.

478
00:34:54,230 --> 00:34:59,200
So the way that the gauge group
is broken by the Higgs vacuum

479
00:34:59,200 --> 00:35:01,833
expectation value is
spontaneously broken.

480
00:35:01,833 --> 00:35:03,250
We're not going
to mess with that.

481
00:35:15,950 --> 00:35:19,550
And we built into this
idea that we do this

482
00:35:19,550 --> 00:35:23,800
because there's no new particles
that are produced at p.

483
00:35:23,800 --> 00:35:25,810
If there was a new
particle produced at p,

484
00:35:25,810 --> 00:35:28,210
then they would have
to have a mass that

485
00:35:28,210 --> 00:35:30,220
would allow us to produce it.

486
00:35:30,220 --> 00:35:33,970
And that would mean that
it doesn't have a mass up

487
00:35:33,970 --> 00:35:35,020
at this lambda nu scale.

488
00:35:35,020 --> 00:35:38,320
And we'd have to include it
in our effective Lagrangian.

489
00:35:38,320 --> 00:35:40,068
So by taking this
point of view, we're

490
00:35:40,068 --> 00:35:41,860
assuming that there's
no new particles that

491
00:35:41,860 --> 00:35:45,610
are produced at the scale
p, only at lambda nu.

492
00:35:49,450 --> 00:35:51,158
And effectively,
we've integrated out,

493
00:35:51,158 --> 00:35:53,200
if you want to use that
language-- although we're

494
00:35:53,200 --> 00:35:55,680
doing this from the bottom up.

495
00:35:55,680 --> 00:35:57,430
If you wanted to have
a top-down language,

496
00:35:57,430 --> 00:35:59,690
you'd say we integrated
out the particles

497
00:35:59,690 --> 00:36:02,270
at the scale lambda nu.

498
00:36:02,270 --> 00:36:03,640
OK, so that's our logic.

499
00:36:07,780 --> 00:36:12,430
So let me just start seeing
what we can write down.

500
00:36:12,430 --> 00:36:16,065
And for the dimension 5, it's
actually very restrictive.

501
00:36:20,250 --> 00:36:24,342
Gauge symmetry is very
restrictive for dimension 5.

502
00:36:24,342 --> 00:36:26,666
And there's basically
only one operator.

503
00:36:35,098 --> 00:36:37,000
So we won't stop at dimension 5.

504
00:36:37,000 --> 00:36:38,340
We'll go up to the dimension 6.

505
00:36:42,316 --> 00:36:48,970
At dimension 5, it turns
out that, once you satisfy

506
00:36:48,970 --> 00:36:52,090
the gauge symmetry, the unique
term that you can write down

507
00:36:52,090 --> 00:36:52,840
looks like this.

508
00:36:58,200 --> 00:37:05,110
And my notation is
that this guy is

509
00:37:05,110 --> 00:37:11,933
our left-handed lepton with a
charge conjugation operator.

510
00:37:11,933 --> 00:37:13,100
And these guys are doublets.

511
00:37:13,100 --> 00:37:17,210
So the Higgs doublet
is a doublet like that.

512
00:37:17,210 --> 00:37:22,730
And the left-handed leptons,
neutrino, an electron

513
00:37:22,730 --> 00:37:25,270
is a doublet like that.

514
00:37:25,270 --> 00:37:27,810
So these are doublets.

515
00:37:27,810 --> 00:37:31,640
And I'm figuring out how to
contract with doublet indices,

516
00:37:31,640 --> 00:37:35,960
but I have to satisfy the U1
hyper charged gauge invariance.

517
00:37:35,960 --> 00:37:38,640
This has no color.

518
00:37:38,640 --> 00:37:40,820
So that's automatically
satisfied.

519
00:37:40,820 --> 00:37:42,653
And then I have to
satisfy the SU2.

520
00:37:42,653 --> 00:37:44,570
And I've done that by
the way that the indices

521
00:37:44,570 --> 00:37:45,665
are contracted.

522
00:37:50,780 --> 00:37:53,275
One thing I didn't write
down in this operator

523
00:37:53,275 --> 00:37:54,200
is flavor indices.

524
00:37:57,098 --> 00:37:58,640
So if you were to
add flavor indices,

525
00:37:58,640 --> 00:37:59,810
you could do a bit more.

526
00:37:59,810 --> 00:38:06,690
But in some sense, that's a
pretty simple generalization.

527
00:38:06,690 --> 00:38:10,160
So we'll still count it as
one even though we could think

528
00:38:10,160 --> 00:38:11,320
about having more flavors.

529
00:38:11,320 --> 00:38:13,790
We're contracting things
with more flavors.

530
00:38:13,790 --> 00:38:16,280
And that would, of
course, affect arguments

531
00:38:16,280 --> 00:38:17,600
made on gauge symmetry alone.

532
00:38:20,670 --> 00:38:23,420
So in all my
counting today, I'll

533
00:38:23,420 --> 00:38:25,775
be agnostic about
flavor matrices.

534
00:38:28,750 --> 00:38:33,870
So when I say only, that
proviso is hidden there, OK?

535
00:38:33,870 --> 00:38:35,430
So this guy is
kind of interesting

536
00:38:35,430 --> 00:38:38,500
from an phenomenological
point of view.

537
00:38:38,500 --> 00:38:42,660
Because if you replace the Higgs
field by its vacuum expectation

538
00:38:42,660 --> 00:38:45,300
value, which means
getting rid of H+ ?

539
00:38:45,300 --> 00:38:50,160
And taking h0 to be a
constant, which is v,

540
00:38:50,160 --> 00:38:51,860
then that gives a
Majorana mass term.

541
00:39:06,370 --> 00:39:10,590
So the observed
left-handed neutrino

542
00:39:10,590 --> 00:39:13,850
would get a term
in its Lagrangian

543
00:39:13,850 --> 00:39:20,850
that, after we do that,
it looks like that.

544
00:39:20,850 --> 00:39:34,780
And that's a Majorana
mass term, where

545
00:39:34,780 --> 00:39:36,630
this m nu is a
parameter that shows up,

546
00:39:36,630 --> 00:39:38,338
but it's built out of
the parameters that

547
00:39:38,338 --> 00:39:39,230
were in this thing.

548
00:39:39,230 --> 00:39:41,972
So once I replace
these by [INAUDIBLE]

549
00:39:41,972 --> 00:39:43,430
and go through the
various factors,

550
00:39:43,430 --> 00:39:44,830
we get something like that.

551
00:39:48,577 --> 00:39:50,910
So just the fact that we know
that the observed neutrino

552
00:39:50,910 --> 00:39:57,640
masses are less
than, say, 3.5 eV

553
00:39:57,640 --> 00:39:59,220
tells us something
about the scale.

554
00:39:59,220 --> 00:40:00,640
So this is small.

555
00:40:00,640 --> 00:40:04,310
If c5 is of order 1, I
told you we know what v is.

556
00:40:04,310 --> 00:40:05,980
It's 246 GeV.

557
00:40:05,980 --> 00:40:13,380
That tells us something about
lambda nu that it's big.

558
00:40:20,186 --> 00:40:22,800
But we could try to
think up some reasons

559
00:40:22,800 --> 00:40:24,700
why the c5 maybe have
some suppression in it

560
00:40:24,700 --> 00:40:26,200
and make the c a
little bit smaller.

561
00:40:26,200 --> 00:40:29,650
But if C5 is order 1, then
we get a very large lambda nu

562
00:40:29,650 --> 00:40:30,150
scale.

563
00:40:35,090 --> 00:40:37,340
I'll give you a problem on
your problem set to explore

564
00:40:37,340 --> 00:40:38,548
this in a little more detail.

565
00:40:40,860 --> 00:40:43,730
I should also note that the
Majorana mass term, having

566
00:40:43,730 --> 00:40:47,200
two neutrinos like this,
violates lepton number, which

567
00:40:47,200 --> 00:40:55,390
is a global symmetry of the
standard model, at least

568
00:40:55,390 --> 00:40:56,464
classically.

569
00:41:02,880 --> 00:41:04,840
So this guy violates
lepton numbers.

570
00:41:04,840 --> 00:41:07,540
You can also write down
dimension 6 operators

571
00:41:07,540 --> 00:41:08,830
that violate baryon number.

572
00:41:14,110 --> 00:41:17,723
And I'm going to leave that,
also, as a problem set problem.

573
00:41:17,723 --> 00:41:19,640
So you'll figure out
what those operators are.

574
00:41:28,220 --> 00:41:35,670
So example 3, if I conserve
lepton number and baryon

575
00:41:35,670 --> 00:41:36,170
number--

576
00:41:39,050 --> 00:41:41,120
which are things that,
if they're broken,

577
00:41:41,120 --> 00:41:44,058
that's obviously
having a big impact.

578
00:41:44,058 --> 00:41:46,100
And there's obviously
strong constraints on that.

579
00:41:49,580 --> 00:41:53,230
So you can ask then, if
you go to dimension 6

580
00:41:53,230 --> 00:41:55,370
and we conserve those
things because they're

581
00:41:55,370 --> 00:41:58,070
highly constrained, how many
operators are there left over?

582
00:41:58,070 --> 00:41:58,855
And there's 80.

583
00:42:08,300 --> 00:42:14,840
So L2, we can account exactly
how many operators there are.

584
00:42:14,840 --> 00:42:22,190
And there's i from 1 to 80, some
coefficients and some operators

585
00:42:22,190 --> 00:42:23,090
that are dimension 6.

586
00:42:23,090 --> 00:42:26,612
And if I'm going to make
a lambda nu explicit,

587
00:42:26,612 --> 00:42:28,070
I put a lambda nu
squared in there.

588
00:42:31,280 --> 00:42:36,900
So 80 sounds like a big number,
but big is always relative.

589
00:42:36,900 --> 00:42:38,580
So if you think
about 80 relative to,

590
00:42:38,580 --> 00:42:41,480
for example, how many soft
SUSY breaking parameters

591
00:42:41,480 --> 00:42:43,630
you have in the MSSM,
is greater than 100,

592
00:42:43,630 --> 00:42:47,255
then 80 doesn't sound so bad.

593
00:42:47,255 --> 00:42:49,400
Also, you should
remember that, if you're

594
00:42:49,400 --> 00:42:52,280
going to do some
phenomenology with this L2,

595
00:42:52,280 --> 00:42:54,320
that many of the 80 are
not going to contribute.

596
00:42:54,320 --> 00:42:55,820
If you look at a
particular process,

597
00:42:55,820 --> 00:42:58,760
only some small subset
of them will contribute.

598
00:42:58,760 --> 00:43:01,220
And there's many,
many, many observables

599
00:43:01,220 --> 00:43:04,020
in the standard model with
all the different particles.

600
00:43:04,020 --> 00:43:06,620
So 80, once you
start dividing it up

601
00:43:06,620 --> 00:43:09,350
into camps that contribute
to different observables,

602
00:43:09,350 --> 00:43:11,940
is not such a large number.

603
00:43:11,940 --> 00:43:13,990
Or at least it's not
an unmanageable number,

604
00:43:13,990 --> 00:43:15,820
and people do
phenomenology with this.

605
00:43:32,200 --> 00:43:36,892
So for any observable, only a
manageable number contribute.

606
00:43:39,490 --> 00:43:44,980
And I should also say that, if
you have a top-down perspective

607
00:43:44,980 --> 00:43:48,460
where you have a new physics
theory that you've constructed

608
00:43:48,460 --> 00:43:51,070
that has the scale
lambda nu in it,

609
00:43:51,070 --> 00:43:53,470
then from this point of view
what that theory predicts is

610
00:43:53,470 --> 00:43:55,385
a particular
pattern for the c's.

611
00:44:04,830 --> 00:44:07,600
Hopefully, if it has
less parameters than 80,

612
00:44:07,600 --> 00:44:10,990
then you get some patterns of
connections between the c's.

613
00:44:10,990 --> 00:44:14,140
So if you have some new physics
model that has the number

614
00:44:14,140 --> 00:44:15,918
of parameters that
are less than 80,

615
00:44:15,918 --> 00:44:17,710
then you get connections
between these c's.

616
00:44:17,710 --> 00:44:20,050
And you could think that,
if you use this Lagrangian,

617
00:44:20,050 --> 00:44:23,980
constrain the c's, that
you could test generically

618
00:44:23,980 --> 00:44:26,860
for classes of new
physics theories

619
00:44:26,860 --> 00:44:28,870
that are ruled in or ruled out.

620
00:44:28,870 --> 00:44:31,030
Because they have to, if
you match on to these c's

621
00:44:31,030 --> 00:44:34,300
from those theories,
obey whatever constraints

622
00:44:34,300 --> 00:44:35,980
you would derive
from this logic.

623
00:44:35,980 --> 00:44:39,670
Of course, that assumes
that the new physics

624
00:44:39,670 --> 00:44:41,290
particles are at a high scale.

625
00:44:41,290 --> 00:44:46,450
So we can make
this expansion, OK?

626
00:44:46,450 --> 00:44:48,610
Questions about that?

627
00:44:48,610 --> 00:44:49,532
Yeah.

628
00:44:49,532 --> 00:44:51,612
AUDIENCE: [INAUDIBLE]
assumption that you

629
00:44:51,612 --> 00:44:56,408
don't have [INAUDIBLE] freedom
is returning to the high order.

630
00:44:56,408 --> 00:44:57,200
IAIN STEWART: Yeah.

631
00:44:57,200 --> 00:44:59,117
AUDIENCE: If you want
them to do a [INAUDIBLE]

632
00:44:59,117 --> 00:45:04,060
or [INAUDIBLE] for high
order, would [INAUDIBLE] they

633
00:45:04,060 --> 00:45:05,890
have to be at
higher [INAUDIBLE]..

634
00:45:05,890 --> 00:45:10,347
And why is it at a higher
energy than [INAUDIBLE]??

635
00:45:10,347 --> 00:45:11,930
IAIN STEWART: So
it's not to say that,

636
00:45:11,930 --> 00:45:14,840
at higher energy, that
no new degrees of freedom

637
00:45:14,840 --> 00:45:18,240
would show up if I really
probe those energies directly.

638
00:45:18,240 --> 00:45:20,750
But what I'm doing is I'm
saying that I'm probing

639
00:45:20,750 --> 00:45:23,352
the physics at small energy.

640
00:45:23,352 --> 00:45:25,310
And the way that high
energy degrees of freedom

641
00:45:25,310 --> 00:45:28,940
would show up is by
a contribution to one

642
00:45:28,940 --> 00:45:29,780
of these operators.

643
00:45:29,780 --> 00:45:32,150
So say I have added in
some new particle at 10

644
00:45:32,150 --> 00:45:35,300
TeV, mass of 10 TeV.

645
00:45:35,300 --> 00:45:39,320
If I expand in momentum
over that mass,

646
00:45:39,320 --> 00:45:41,450
then what will happen is
you'll get an operator

647
00:45:41,450 --> 00:45:43,790
like this O6, where
that particle is

648
00:45:43,790 --> 00:45:45,680
removed because I expanded it.

649
00:45:45,680 --> 00:45:47,180
It got removed.

650
00:45:47,180 --> 00:45:50,120
And its mass will be
exactly in this denominator.

651
00:45:50,120 --> 00:45:51,470
It'll show up here as lambda nu.

652
00:45:51,470 --> 00:45:53,930
So think about it as lambda
nu squared could just

653
00:45:53,930 --> 00:45:55,520
be the propagator.

654
00:45:55,520 --> 00:46:01,516
If I had 1 over p squared
minus some massive--

655
00:46:01,516 --> 00:46:04,550
I don't know, some
gluino, right?

656
00:46:04,550 --> 00:46:07,220
And then I start expanding this.

657
00:46:07,220 --> 00:46:11,210
The first term where I drop
the momentum is just that.

658
00:46:11,210 --> 00:46:14,700
And that could be exactly
this lambda nu squared.

659
00:46:14,700 --> 00:46:19,310
So the new physics particles
don't show up in Oi6.

660
00:46:19,310 --> 00:46:20,870
What they affect
is the pre-factor.

661
00:46:27,770 --> 00:46:30,350
So you don't need to add new
physics particles in order

662
00:46:30,350 --> 00:46:31,800
could construct the operators.

663
00:46:31,800 --> 00:46:33,467
You're just building
those operators out

664
00:46:33,467 --> 00:46:35,390
of the standard model
degrees of freedom.

665
00:46:35,390 --> 00:46:36,878
AUDIENCE: So you're
still working

666
00:46:36,878 --> 00:46:38,180
in p but just [INAUDIBLE].

667
00:46:38,180 --> 00:46:40,650
IAIN STEWART: Yes, yes,
then the lambda nu.

668
00:46:40,650 --> 00:46:43,630
Is there another question?

669
00:46:43,630 --> 00:46:44,130
OK.

670
00:46:44,130 --> 00:46:46,730
So I've been assuming
some familiarity

671
00:46:46,730 --> 00:46:48,620
with the standard model here.

672
00:46:48,620 --> 00:46:50,948
And I've posted also,
as I said last time,

673
00:46:50,948 --> 00:46:52,740
my lecture notes on
quantum field theory 3.

674
00:46:52,740 --> 00:46:56,660
And there's some review
reading there if some of this

675
00:46:56,660 --> 00:46:59,750
is unfamiliar to you, discussing
[INAUDIBLE] and things

676
00:46:59,750 --> 00:47:02,480
like that.

677
00:47:02,480 --> 00:47:02,980
OK.

678
00:47:02,980 --> 00:47:07,305
So what kind of operators
can we have at dimension 6?

679
00:47:10,420 --> 00:47:13,840
I'm not going to list
all 80, obviously,

680
00:47:13,840 --> 00:47:16,370
but I'll list a few of them.

681
00:47:16,370 --> 00:47:21,430
So we could take an operator
that's the following, built out

682
00:47:21,430 --> 00:47:23,390
of gluon field strengths.

683
00:47:32,250 --> 00:47:35,170
So making it Lorentz invariant
and contracting up the indices

684
00:47:35,170 --> 00:47:43,630
and contracting up color
indices with an FABC, that's

685
00:47:43,630 --> 00:47:47,900
an operator that's dimension 6
because the Gs are dimension 2.

686
00:47:47,900 --> 00:47:49,000
That's one of the 80.

687
00:47:58,870 --> 00:48:01,150
You could also do
something with fermions.

688
00:48:01,150 --> 00:48:03,210
AUDIENCE: Sorry, what
are the [INAUDIBLE]??

689
00:48:03,210 --> 00:48:04,460
IAIN STEWART: Here?

690
00:48:04,460 --> 00:48:04,960
Whoops.

691
00:48:04,960 --> 00:48:06,880
Yeah, what happened?

692
00:48:06,880 --> 00:48:08,780
All right, there we go.

693
00:48:08,780 --> 00:48:09,280
Thanks.

694
00:48:20,600 --> 00:48:22,920
So here's something with a
lepton doublet and a quark

695
00:48:22,920 --> 00:48:23,420
doublet.

696
00:48:23,420 --> 00:48:26,420
We already introduced
the notation for LL.

697
00:48:26,420 --> 00:48:29,020
And QL is similar,
but just up and down.

698
00:48:32,160 --> 00:48:34,390
So that's each
fermion dimension 3/2.

699
00:48:34,390 --> 00:48:35,920
So four of them
is dimension six.

700
00:48:39,990 --> 00:48:42,290
There's something called
magnetic operators.

701
00:48:45,090 --> 00:48:46,550
So there's lots of
different things

702
00:48:46,550 --> 00:48:48,290
that you can do
with four fermions.

703
00:48:48,290 --> 00:48:50,980
I've only given you one example.

704
00:48:50,980 --> 00:48:54,350
And I've posted the
reference for the paper

705
00:48:54,350 --> 00:48:57,175
that lists all 80.

706
00:48:57,175 --> 00:48:58,550
So you can look
at it if you want

707
00:48:58,550 --> 00:48:59,758
to look at the complete list.

708
00:49:05,090 --> 00:49:08,320
So we can do something where
we have leptons, a Higgs field,

709
00:49:08,320 --> 00:49:10,570
as well as a field
strength for the SU2.

710
00:49:19,653 --> 00:49:20,570
So this is in the SU2.

711
00:49:24,023 --> 00:49:25,440
And these are all
gauge invariant,

712
00:49:25,440 --> 00:49:27,982
as you can convince yourself by
looking at the standard model

713
00:49:27,982 --> 00:49:30,110
gauge transformations
of these operators.

714
00:49:30,110 --> 00:49:32,160
And so if I write something like
this, where this is a doublet

715
00:49:32,160 --> 00:49:33,535
and this is a
doublet and I don't

716
00:49:33,535 --> 00:49:35,803
write the doublet
contraction, then I'm

717
00:49:35,803 --> 00:49:37,095
just contracting those indices.

718
00:49:40,120 --> 00:49:43,460
These two here contribute to
the mu on the magnetic moment,

719
00:49:43,460 --> 00:49:44,280
anomalous moment.

720
00:49:47,992 --> 00:49:49,920
So they contribute to g minus 2.

721
00:50:02,867 --> 00:50:04,700
So g minus 2 at the
muon, which is something

722
00:50:04,700 --> 00:50:09,850
that we've measured
to very high precision

723
00:50:09,850 --> 00:50:13,820
has what sometimes you
would call standard model

724
00:50:13,820 --> 00:50:16,940
contributions,
which people usually

725
00:50:16,940 --> 00:50:18,830
mean as the contributions
in our notation

726
00:50:18,830 --> 00:50:21,680
from L0 and then plus
some contributions

727
00:50:21,680 --> 00:50:24,470
from these higher dimension
operators, whatever

728
00:50:24,470 --> 00:50:28,092
coefficients these
operators have.

729
00:50:28,092 --> 00:50:30,050
And again, I would replace
the Higgs field here

730
00:50:30,050 --> 00:50:31,168
by a [INAUDIBLE].

731
00:50:34,100 --> 00:50:35,870
I know they're
dimension 6, so there's

732
00:50:35,870 --> 00:50:37,430
lambda nu squared downstairs.

733
00:50:37,430 --> 00:50:40,100
One factor of dimension is
made up by the [INAUDIBLE]..

734
00:50:40,100 --> 00:50:42,830
And then the next
scale that comes in

735
00:50:42,830 --> 00:50:43,880
is the mass of the muon.

736
00:50:43,880 --> 00:50:47,270
So this is the generic size
of those contributions.

737
00:50:47,270 --> 00:50:51,590
And again, if you take into
account experimentally how well

738
00:50:51,590 --> 00:50:54,010
we've measured this, it
puts a pretty strong,

739
00:50:54,010 --> 00:50:58,040
or at least it puts a
constraint, on lambda nu.

740
00:50:58,040 --> 00:51:02,750
Actually, it's probably stronger
than this number, some number

741
00:51:02,750 --> 00:51:03,800
greater than 100 TeV.

742
00:51:03,800 --> 00:51:06,142
Maybe it's even 1,000 TeV.

743
00:51:06,142 --> 00:51:07,600
AUDIENCE: [INAUDIBLE]
contribution,

744
00:51:07,600 --> 00:51:09,010
it was [INAUDIBLE]?

745
00:51:09,010 --> 00:51:11,310
IAIN STEWART: Yeah.

746
00:51:11,310 --> 00:51:12,810
So the standard
model contribution

747
00:51:12,810 --> 00:51:15,250
makes up all the
digits we've measured.

748
00:51:15,250 --> 00:51:18,210
And then there's some digits
where there's some uncertainty.

749
00:51:18,210 --> 00:51:19,240
We haven't [INAUDIBLE].

750
00:51:19,240 --> 00:51:21,870
And you can constrain, based
on experimental uncertainty,

751
00:51:21,870 --> 00:51:25,470
how big this possible
contribution could be.

752
00:51:25,470 --> 00:51:30,728
And there's 2 and 1/2 deviations
from the standard model

753
00:51:30,728 --> 00:51:31,520
in this observable.

754
00:51:31,520 --> 00:51:35,990
So you can make up for them
with an operator like that.

755
00:51:35,990 --> 00:51:39,230
But I never really pay
attention to things

756
00:51:39,230 --> 00:51:43,820
that are less than
4 sigma personally.

757
00:51:43,820 --> 00:51:46,160
Though sometimes it's
interesting to get

758
00:51:46,160 --> 00:51:47,450
excited about 3 sigma.

759
00:51:50,859 --> 00:51:53,867
AUDIENCE: [INAUDIBLE]

760
00:51:53,867 --> 00:51:54,950
IAIN STEWART: No, I'm not.

761
00:51:54,950 --> 00:51:55,750
AUDIENCE: Oh.

762
00:51:55,750 --> 00:51:56,860
IAIN STEWART: Yeah.

763
00:51:56,860 --> 00:51:59,100
So there's also
flavor on top of that.

764
00:51:59,100 --> 00:52:00,592
That's right.

765
00:52:00,592 --> 00:52:02,300
So flavor is actually
highly constrained.

766
00:52:02,300 --> 00:52:05,720
And you could put in some
assumptions about flavor

767
00:52:05,720 --> 00:52:09,210
and then you get
to the [INAUDIBLE]..

768
00:52:09,210 --> 00:52:09,720
OK.

769
00:52:09,720 --> 00:52:15,320
So for the remaining 76, see
the reference I've posted.

770
00:52:31,480 --> 00:52:33,662
So this paper actually
wasn't the first to try

771
00:52:33,662 --> 00:52:34,870
to enumerate these operators.

772
00:52:34,870 --> 00:52:37,780
As you can imagine, this would
be a pretty standard thing

773
00:52:37,780 --> 00:52:38,830
to do.

774
00:52:38,830 --> 00:52:40,780
But it was the first to get 80.

775
00:52:40,780 --> 00:52:44,140
And the reason that they got
80 where other people got more

776
00:52:44,140 --> 00:52:47,680
is because they used
the equations of motion

777
00:52:47,680 --> 00:52:49,000
to simplify the operators.

778
00:52:52,410 --> 00:52:56,530
So let me phrase that
as, is there a caveat?

779
00:53:02,870 --> 00:53:05,980
So when they did their
counting of the operators,

780
00:53:05,980 --> 00:53:18,750
they took the equations
of motion from L0,

781
00:53:18,750 --> 00:53:20,490
and they used that
to simplify L1.

782
00:53:25,800 --> 00:53:29,340
So they worked out the standard
model equations of motion

783
00:53:29,340 --> 00:53:31,552
that tree level.

784
00:53:31,552 --> 00:53:33,510
And then they applied
those equations of motion

785
00:53:33,510 --> 00:53:41,965
to reduce the form of
the operators down to 80.

786
00:53:41,965 --> 00:53:44,530
So that got rid of a lot
of operators for them

787
00:53:44,530 --> 00:53:46,210
that other people had
considered as part

788
00:53:46,210 --> 00:53:47,377
of the counting in the past.

789
00:53:50,230 --> 00:53:54,430
So for example-- so
you get some idea,

790
00:53:54,430 --> 00:53:56,220
if I have a covariant
derivative acting

791
00:53:56,220 --> 00:54:00,240
on a right-handed electron.

792
00:54:00,240 --> 00:54:02,340
And the equation of motion
in the standard model

793
00:54:02,340 --> 00:54:05,040
relates that to
[INAUDIBLE] couplings.

794
00:54:05,040 --> 00:54:09,590
Here, I'm writing
the flavor indices

795
00:54:09,590 --> 00:54:11,490
in the left-handed
doublet Higgs field

796
00:54:11,490 --> 00:54:14,010
and the left-handed doublet.

797
00:54:14,010 --> 00:54:17,726
This would be like a mass term
if I put in the [INAUDIBLE]..

798
00:54:21,460 --> 00:54:25,920
So that's like the analog, if I
write it in terms of spinners,

799
00:54:25,920 --> 00:54:31,455
of p slash u for the
right-hand electron is m u,

800
00:54:31,455 --> 00:54:33,330
but just written as
an equation of motion.

801
00:54:37,180 --> 00:54:38,730
So they're using
things like this

802
00:54:38,730 --> 00:54:40,530
to get rid of covariant
derivatives acting

803
00:54:40,530 --> 00:54:43,580
on the right-handed electron and
just replace it by an operator

804
00:54:43,580 --> 00:54:47,100
that a priori doesn't look
like it's equivalent, which

805
00:54:47,100 --> 00:54:50,280
is H dagger L.

806
00:54:50,280 --> 00:54:55,650
Now, if you are stuck at tree
level-- which is actually

807
00:54:55,650 --> 00:54:57,780
the language in which
they constructed

808
00:54:57,780 --> 00:55:02,190
their paper is just
to think that you

809
00:55:02,190 --> 00:55:04,890
apply this in a
way that is valid

810
00:55:04,890 --> 00:55:10,870
when you look at lowest order,
the operators at lowest order.

811
00:55:10,870 --> 00:55:13,110
Then it's pretty obvious,
actually that this is OK.

812
00:55:16,798 --> 00:55:18,840
If the lines that are
coming out of your operator

813
00:55:18,840 --> 00:55:21,600
are always external
lines, then when

814
00:55:21,600 --> 00:55:25,032
you look at the final
rule for those lines,

815
00:55:25,032 --> 00:55:26,240
you're putting them on shell.

816
00:55:33,960 --> 00:55:38,310
So for example, let's think
that we had an operator that

817
00:55:38,310 --> 00:55:39,910
was not part of their list.

818
00:55:39,910 --> 00:55:45,570
So H dagger H e right
id slash e right,

819
00:55:45,570 --> 00:55:48,300
you won't find that as
one of the ones that's

820
00:55:48,300 --> 00:55:49,170
listed in the 80.

821
00:55:54,760 --> 00:55:56,760
You could think about the
Feynman rule for that.

822
00:55:56,760 --> 00:55:58,260
So there's two Higgs
particles, say,

823
00:55:58,260 --> 00:56:02,900
and two right-handed electrons.

824
00:56:02,900 --> 00:56:07,130
And if I just take
the derivative out

825
00:56:07,130 --> 00:56:13,790
of here and then I get a p slash
and if I use p slash as m u,

826
00:56:13,790 --> 00:56:17,000
then I get just u bar u.

827
00:56:17,000 --> 00:56:20,660
And so this is, of
course, connecting us

828
00:56:20,660 --> 00:56:24,110
to the left-handed doublet,
the left-handed guy.

829
00:56:24,110 --> 00:56:28,377
So this operator, in
that sense of thinking

830
00:56:28,377 --> 00:56:30,710
about using the equation of
motion, which are connecting

831
00:56:30,710 --> 00:56:33,350
left and right fields,
putting them together

832
00:56:33,350 --> 00:56:36,170
in the standard model, is
connecting this operator

833
00:56:36,170 --> 00:56:38,930
to the one that we
just have H digger L.

834
00:56:38,930 --> 00:56:42,328
And you could just immediately
get the right result just

835
00:56:42,328 --> 00:56:43,620
by starting with that operator.

836
00:56:43,620 --> 00:56:45,440
And that's the logic
that they used.

837
00:56:45,440 --> 00:56:48,180
So you don't have to write this
down because it's redundant.

838
00:56:48,180 --> 00:56:52,730
You get the same result
from writing this down.

839
00:56:52,730 --> 00:56:56,180
So that seems fairly
straightforward at tree level

840
00:56:56,180 --> 00:57:00,080
and at lowest order where
all the fields are external.

841
00:57:00,080 --> 00:57:05,280
But I actually claim that it's
true regardless of what I do.

842
00:57:05,280 --> 00:57:08,810
Whether I have loops, whether
I have propagators, it's OK.

843
00:57:08,810 --> 00:57:11,250
We can do this.

844
00:57:11,250 --> 00:57:13,330
And that seems a
lot less trivial.

845
00:57:19,604 --> 00:57:22,320
So they didn't know that
when they wrote their paper.

846
00:57:22,320 --> 00:57:26,190
At least I don't think
they did, but it is true.

847
00:57:33,927 --> 00:57:35,760
So that's what I actually
want to talk about

848
00:57:35,760 --> 00:57:39,450
for the rest of today's
lecture because it's

849
00:57:39,450 --> 00:57:41,050
a pretty powerful thing to do.

850
00:57:41,050 --> 00:57:43,290
If we are allowed to use
these equations of motion

851
00:57:43,290 --> 00:57:46,350
to simplify the form of the
higher dimensional operators

852
00:57:46,350 --> 00:57:48,930
in our effective field
theory, it certainly

853
00:57:48,930 --> 00:57:52,290
helps us to reduce the
number of operators.

854
00:58:08,410 --> 00:58:11,560
So one way of phrasing
this is what's

855
00:58:11,560 --> 00:58:15,730
called the representation
independence theorem, sometimes

856
00:58:15,730 --> 00:58:16,720
called that.

857
00:58:16,720 --> 00:58:18,220
I'll phrase it a
few different ways,

858
00:58:18,220 --> 00:58:19,428
but I'll start with this way.

859
00:58:24,350 --> 00:58:27,400
So if we have some
field, phi, we

860
00:58:27,400 --> 00:58:31,610
can set it equal to some
combination of other fields.

861
00:58:31,610 --> 00:58:35,270
So phi and chi can be
scalars, let's say.

862
00:58:35,270 --> 00:58:38,680
And this function, f, which
could be fairly complicated,

863
00:58:38,680 --> 00:58:41,380
has to have at least one
property that, when chi is 0,

864
00:58:41,380 --> 00:58:42,250
it's 1.

865
00:58:42,250 --> 00:58:46,630
So phi and chi both
show up linearly

866
00:58:46,630 --> 00:58:50,283
in some term in the
function if you'd

867
00:58:50,283 --> 00:58:52,450
like, if you want to think
of it as a Taylor series,

868
00:58:52,450 --> 00:58:52,950
for example.

869
00:58:55,560 --> 00:59:01,150
So if that's true, the
statement of this theorem

870
00:59:01,150 --> 00:59:07,960
is that calculations
of observable

871
00:59:07,960 --> 00:59:14,210
that are done with phi or the
Lagrangian that's made of phi--

872
00:59:14,210 --> 00:59:16,780
and what that really means
is that I've quantized phi--

873
00:59:20,030 --> 00:59:32,140
will give the same results
as those with a Lagrangian

874
00:59:32,140 --> 00:59:37,620
where I quantize chi
where I construct

875
00:59:37,620 --> 00:59:40,692
that Lagrangian by just making
this change of variables.

876
00:59:49,550 --> 00:59:52,200
And we're going to
exploit that fact in order

877
00:59:52,200 --> 00:59:54,990
to argue that we're allowed
to do what I just said.

878
00:59:57,758 --> 00:59:59,550
So we'll start slowly,
then we'll build up.

879
00:59:59,550 --> 01:00:02,390
Then we'll state a more
general theorem than this one.

880
01:00:02,390 --> 01:00:03,890
Then we'll show you
how to prove it.

881
01:00:10,540 --> 01:00:14,100
So let's start by
thinking of an example.

882
01:00:14,100 --> 01:00:16,315
Examples are always good.

883
01:00:16,315 --> 01:00:18,190
And we'll stick with
our scalar field theory.

884
01:00:22,285 --> 01:00:23,910
I'm getting tired of
writing factorial,

885
01:00:23,910 --> 01:00:27,150
so I'm choosing a little bit of
different normalization here.

886
01:00:37,500 --> 01:00:39,660
So I'll consider
an effective theory

887
01:00:39,660 --> 01:00:42,300
that has these terms
at least to start.

888
01:00:42,300 --> 01:00:44,310
And eta has dimensions.

889
01:00:44,310 --> 01:00:45,280
And it's a small thing.

890
01:00:45,280 --> 01:00:47,760
It's like 1 over lambda nu.

891
01:00:54,730 --> 01:00:56,560
And the statement that
we want to explore

892
01:00:56,560 --> 01:01:00,250
is the fact that we can
use the equation of motion

893
01:01:00,250 --> 01:01:18,730
to effectively drop
the last term by using

894
01:01:18,730 --> 01:01:21,200
the equation of motion.

895
01:01:21,200 --> 01:01:25,880
So how do we make use
of this statement?

896
01:01:25,880 --> 01:01:29,620
Well, this tells us that we
can make changes to variable

897
01:01:29,620 --> 01:01:31,360
and that we won't
change anything.

898
01:01:31,360 --> 01:01:33,860
So we try to make a change of
variable to make this go away.

899
01:01:36,268 --> 01:01:37,557
AUDIENCE: Wait, sorry.

900
01:01:37,557 --> 01:01:39,890
But that's not even the tree
level operation [INAUDIBLE]

901
01:01:39,890 --> 01:01:40,395
right?

902
01:01:40,395 --> 01:01:42,920
You still have the lambda term.

903
01:01:42,920 --> 01:01:44,640
IAIN STEWART: Yeah, that's true.

904
01:01:44,640 --> 01:01:47,090
Yeah.

905
01:01:47,090 --> 01:01:47,610
Yeah.

906
01:01:47,610 --> 01:01:52,600
I should write the
lambda term in there,

907
01:01:52,600 --> 01:01:54,220
plus, minus, whatever it is.

908
01:01:56,600 --> 01:01:57,100
Thanks.

909
01:02:03,880 --> 01:02:04,530
There's a 2.

910
01:02:10,650 --> 01:02:11,932
Thanks.

911
01:02:11,932 --> 01:02:14,140
All right, so how do we to
get rid of this last term?

912
01:02:21,410 --> 01:02:25,402
Well, let's just use our theorem
by making a field redefinition.

913
01:02:35,250 --> 01:02:38,910
So I claim, after making
this field redefinition here,

914
01:02:38,910 --> 01:02:41,327
that something
magical will happen,

915
01:02:41,327 --> 01:02:42,910
or something that
we want will happen.

916
01:02:51,874 --> 01:02:55,598
So we're going to integrate by
parts at will, which is often

917
01:02:55,598 --> 01:02:57,890
a convenient thing to do when
you're making these field

918
01:02:57,890 --> 01:02:58,800
redefinitions.

919
01:03:01,340 --> 01:03:09,150
So the term that's a 1/2 del mu
phi squared goes to /2 del mu

920
01:03:09,150 --> 01:03:12,070
phi squared from the first term.

921
01:03:12,070 --> 01:03:17,620
And then we pick
up a term that's

922
01:03:17,620 --> 01:03:21,760
exactly in the form we want
to kill this extra term here

923
01:03:21,760 --> 01:03:24,360
that was proportional
to del squared.

924
01:03:24,360 --> 01:03:27,590
And then there would be
order eta squared term.

925
01:03:31,113 --> 01:03:33,280
We have to consistently
make that field redefinition

926
01:03:33,280 --> 01:03:36,490
everywhere, so do it at
the mass term as well.

927
01:03:46,540 --> 01:03:50,230
Well, let me write
out the eta term here,

928
01:03:50,230 --> 01:03:56,005
just a eta squared term just
so you have all the terms.

929
01:03:56,005 --> 01:03:57,320
And we keep doing that.

930
01:03:57,320 --> 01:03:59,690
So we would do it also
for the lambda phi 4 term.

931
01:03:59,690 --> 01:04:03,670
And we do it for
this phi 6 term.

932
01:04:03,670 --> 01:04:07,920
And what you get out when we do
that is that you can write down

933
01:04:07,920 --> 01:04:09,037
the Lagrangian.

934
01:04:12,240 --> 01:04:14,290
And you can group
together the terms.

935
01:04:18,790 --> 01:04:20,620
And that operator is gone.

936
01:04:28,810 --> 01:04:31,120
So it got cancelled by this guy.

937
01:04:31,120 --> 01:04:32,620
There's also these
other things that

938
01:04:32,620 --> 01:04:34,870
are induced, but
really what that

939
01:04:34,870 --> 01:04:38,740
does is it gives you
more terms that are phi 4

940
01:04:38,740 --> 01:04:40,230
and more terms that are phi 6.

941
01:04:40,230 --> 01:04:41,980
So you can really think
of the other terms

942
01:04:41,980 --> 01:04:43,780
as just this guy here
is just adjusting

943
01:04:43,780 --> 01:04:45,790
the constant of the phi 4.

944
01:04:45,790 --> 01:04:49,630
So if it's no longer lambda,
it becomes a term that

945
01:04:49,630 --> 01:04:52,600
has this extra piece to it.

946
01:04:52,600 --> 01:04:55,413
So I'm going to give you more
on this on the problem set.

947
01:04:55,413 --> 01:04:57,330
On the problem set,
you'll work out explicitly

948
01:04:57,330 --> 01:04:58,773
with these relations are.

949
01:04:58,773 --> 01:05:00,190
And I'll also, on
the problem set,

950
01:05:00,190 --> 01:05:01,660
let you think about
what's going on here when

951
01:05:01,660 --> 01:05:02,827
you start considering loops.

952
01:05:06,950 --> 01:05:13,160
In this particular example,
I'll ask you what's going on.

953
01:05:13,160 --> 01:05:15,160
I'll ask you to
show what's going on

954
01:05:15,160 --> 01:05:16,910
and that it's still
OK when we have loops.

955
01:05:21,120 --> 01:05:23,910
So rather than explore this
example in more detail, which

956
01:05:23,910 --> 01:05:26,460
I'm asking you to do
on your problem set,

957
01:05:26,460 --> 01:05:28,950
let's state a more
precise definition

958
01:05:28,950 --> 01:05:31,445
of what we're doing here.

959
01:05:31,445 --> 01:05:33,820
So that's the idea, that I
can make a field redefinition.

960
01:05:33,820 --> 01:05:35,195
When I make a
field redefinition,

961
01:05:35,195 --> 01:05:37,740
it's always going to do
something from the kinetic term

962
01:05:37,740 --> 01:05:38,850
because the kinetic
term is there.

963
01:05:38,850 --> 01:05:40,410
If there's a higher
order term that's

964
01:05:40,410 --> 01:05:42,760
proportional to
that kinetic piece,

965
01:05:42,760 --> 01:05:44,850
then I can set up my
field redefinition

966
01:05:44,850 --> 01:05:45,990
in order to cancel it off.

967
01:05:51,590 --> 01:05:55,002
So there's some things
that are important here.

968
01:05:55,002 --> 01:05:56,710
We have to make a
field redefinition that

969
01:05:56,710 --> 01:05:57,502
preserves symmetry.

970
01:05:57,502 --> 01:05:59,290
So if we make a field
redefinition that

971
01:05:59,290 --> 01:06:01,900
breaks Lorentz
invariance, you can't

972
01:06:01,900 --> 01:06:05,050
expect that you're going
to have a Lorentz invariant

973
01:06:05,050 --> 01:06:07,390
description after that.

974
01:06:07,390 --> 01:06:11,730
And this statement
of f of 0 being 1

975
01:06:11,730 --> 01:06:13,480
is the statement that
you have to preserve

976
01:06:13,480 --> 01:06:17,740
the same one-particle states
before and after the field

977
01:06:17,740 --> 01:06:19,628
redefinition.

978
01:06:19,628 --> 01:06:21,670
And such field redefinition
definitions basically

979
01:06:21,670 --> 01:06:33,860
allow the classical
equations of motion

980
01:06:33,860 --> 01:06:36,400
to be used to
simplify the theory.

981
01:06:44,763 --> 01:06:46,680
I'll also put the proviso
in that it should be

982
01:06:46,680 --> 01:06:47,970
a local quantum field theory.

983
01:07:02,670 --> 01:07:04,663
And there is no
statement in this theorem

984
01:07:04,663 --> 01:07:06,580
that we have to stop
without considering loops

985
01:07:06,580 --> 01:07:08,663
or without considering
propagators that can really

986
01:07:08,663 --> 01:07:13,600
make this argument hold even
beyond the level that we've

987
01:07:13,600 --> 01:07:21,800
showed it, but with loops and
with propagators in as well.

988
01:07:21,800 --> 01:07:23,140
So there's some references.

989
01:07:23,140 --> 01:07:26,040
Again, I've posted
the one that is

990
01:07:26,040 --> 01:07:29,940
closest to our discussion,
which is this paper by Arzt.

991
01:07:37,850 --> 01:07:39,950
There's also a classic
paper by Howard Georgi.

992
01:07:45,665 --> 01:07:47,790
The title of the paper is
"On-Shell Effective Field

993
01:07:47,790 --> 01:07:48,290
Theory."

994
01:07:56,220 --> 01:07:58,720
But I'll mostly follow the
notations in this paper

995
01:07:58,720 --> 01:08:00,737
by Arzt in our discussion here.

996
01:08:00,737 --> 01:08:02,320
We'll go a little
further than he does

997
01:08:02,320 --> 01:08:04,780
and elaborate a little
more, but it's basically

998
01:08:04,780 --> 01:08:05,830
the same notation.

999
01:08:09,500 --> 01:08:10,000
OK.

1000
01:08:10,000 --> 01:08:12,130
So how do we prove
something like that?

1001
01:08:16,979 --> 01:08:19,319
Well, there are some
lessons in this proof.

1002
01:08:19,319 --> 01:08:22,260
So it is something that's
worth going through.

1003
01:08:25,660 --> 01:08:27,930
So the way that I set
up my example up there,

1004
01:08:27,930 --> 01:08:31,060
I was power counting in eta.

1005
01:08:31,060 --> 01:08:34,899
Eta was 1 over lambda nu.

1006
01:08:34,899 --> 01:08:40,189
So let me just write our
effective theory organized

1007
01:08:40,189 --> 01:08:41,189
as in a series in eta.

1008
01:08:45,660 --> 01:08:49,950
And let's consider
removing some operator that

1009
01:08:49,950 --> 01:08:53,010
looks like we should
be able to remove it

1010
01:08:53,010 --> 01:08:56,776
by making a field redefinition.

1011
01:08:56,776 --> 01:08:58,859
And I'll try to be a little
bit generic about what

1012
01:08:58,859 --> 01:09:00,067
the form of that operator is.

1013
01:09:04,740 --> 01:09:07,020
So I'll say it's covariant
derivative squared

1014
01:09:07,020 --> 01:09:11,700
acting on a scalar field just,
again, to make things simpler.

1015
01:09:11,700 --> 01:09:13,350
But then multiply
by any function

1016
01:09:13,350 --> 01:09:16,850
of all the other fields in the
problem, and that's what t is.

1017
01:09:16,850 --> 01:09:18,510
So phi is a complex scalar.

1018
01:09:22,720 --> 01:09:27,229
And t is any
function that sort of

1019
01:09:27,229 --> 01:09:29,779
meets the needs of our
symmetries of the problem.

1020
01:09:34,279 --> 01:09:40,010
And I'm just using
this other phi

1021
01:09:40,010 --> 01:09:43,020
as a shorthand for all the other
fields that we might consider.

1022
01:09:43,020 --> 01:09:52,364
So that could be fermions,
other scalars, gauge fields.

1023
01:09:55,952 --> 01:10:00,830
So let's say we want to get
rid of an operator like that.

1024
01:10:00,830 --> 01:10:02,600
Oh, I should also
say that it's local.

1025
01:10:08,710 --> 01:10:11,990
All right, well,
let's write down

1026
01:10:11,990 --> 01:10:20,690
the generating function for
this theory, path integral

1027
01:10:20,690 --> 01:10:25,090
over the fields exponential.

1028
01:10:25,090 --> 01:10:28,071
I'll regulate the problem with
dimensional regularization.

1029
01:10:31,228 --> 01:10:31,980
It's convenient.

1030
01:10:31,980 --> 01:10:33,360
It preserves the cemeteries.

1031
01:10:33,360 --> 01:10:36,890
That's something we want to do.

1032
01:10:36,890 --> 01:10:39,000
I write out some terms
in the Lagrangian.

1033
01:10:39,000 --> 01:10:39,870
We won't need L2.

1034
01:10:39,870 --> 01:10:41,840
We'll stop at L1.

1035
01:10:41,840 --> 01:10:43,880
But the idea, if we
were to think about L2,

1036
01:10:43,880 --> 01:10:45,680
would be similar.

1037
01:10:45,680 --> 01:10:48,961
And let me write it as adding
and subtracting something.

1038
01:10:52,570 --> 01:10:58,460
So I'll subtract td squared
phi and then add it back.

1039
01:10:58,460 --> 01:11:01,390
So this is, if you
like, you can think

1040
01:11:01,390 --> 01:11:03,175
that L1 had a td squared phi.

1041
01:11:03,175 --> 01:11:04,800
And what I'm doing
here is removing it.

1042
01:11:07,820 --> 01:11:11,130
So this is what we want.

1043
01:11:11,130 --> 01:11:11,860
OK.

1044
01:11:11,860 --> 01:11:14,860
So I'm just making it explicit,
but still writing things

1045
01:11:14,860 --> 01:11:16,750
in terms of L1.

1046
01:11:16,750 --> 01:11:17,970
So there's that.

1047
01:11:17,970 --> 01:11:21,430
And then there's the
coupling to the source.

1048
01:11:21,430 --> 01:11:26,860
And kind of in a generic
notation for each field phi k,

1049
01:11:26,860 --> 01:11:29,142
I have a source jk.

1050
01:11:29,142 --> 01:11:31,910
And I truncate
everything at order eta.

1051
01:11:35,450 --> 01:11:39,102
OK, so that's my starting point.

1052
01:11:39,102 --> 01:11:41,310
And then Green's functions
are obtained by functional

1053
01:11:41,310 --> 01:11:42,840
derivatives with
respect to the j's.

1054
01:12:23,560 --> 01:12:25,370
All right, so what
we're going to do,

1055
01:12:25,370 --> 01:12:28,040
we're going make a change of
variable in the path integral.

1056
01:12:32,803 --> 01:12:34,720
So of course, there's
additional complications

1057
01:12:34,720 --> 01:12:37,307
beyond what we were doing when
we were just thinking about it

1058
01:12:37,307 --> 01:12:38,140
at Lagrangian level.

1059
01:12:42,790 --> 01:12:45,910
There's basically two
additional complications.

1060
01:12:45,910 --> 01:12:49,090
When we make a change a
variable in the path integral,

1061
01:12:49,090 --> 01:12:51,670
we're going to change
the Lagrangian.

1062
01:12:51,670 --> 01:12:54,080
We're also going to
change the measure.

1063
01:12:54,080 --> 01:12:55,808
So there could be a Jacobian.

1064
01:12:55,808 --> 01:12:58,350
And we have to worry about what
happens with the source term.

1065
01:13:05,160 --> 01:13:09,120
So let me just think of it as
a change of variable on the phi

1066
01:13:09,120 --> 01:13:09,620
dagger.

1067
01:13:15,940 --> 01:13:18,512
That's why I wanted to think
about a complex scalar,

1068
01:13:18,512 --> 01:13:20,470
so I could think about
it as phi and phi dagger

1069
01:13:20,470 --> 01:13:23,790
and just make the change of
variable in the phi dagger.

1070
01:13:23,790 --> 01:13:27,760
But doing a real
scalar is just--

1071
01:13:27,760 --> 01:13:32,030
everything would go
through just as well.

1072
01:13:32,030 --> 01:13:37,030
So there's one term, which
is the phi dagger term where

1073
01:13:37,030 --> 01:13:38,530
there's a Jacobian factor.

1074
01:13:38,530 --> 01:13:42,180
So here's that promised
Jacobian for making

1075
01:13:42,180 --> 01:13:43,180
that change of variable.

1076
01:13:50,770 --> 01:13:53,560
Since the change of
variable is order eta,

1077
01:13:53,560 --> 01:13:54,540
it's not affecting L0.

1078
01:13:59,750 --> 01:14:04,220
And I can write, in a
kind of nice notation,

1079
01:14:04,220 --> 01:14:07,300
the way that these eta
t terms from this change

1080
01:14:07,300 --> 01:14:08,410
of variable show up.

1081
01:14:08,410 --> 01:14:13,780
I can write them as taking
all the terms in L0,

1082
01:14:13,780 --> 01:14:16,120
looking for a phi dagger.

1083
01:14:16,120 --> 01:14:17,320
That's the derivative.

1084
01:14:17,320 --> 01:14:19,110
And then I replace
it by an eta t.

1085
01:14:21,850 --> 01:14:24,590
Now, some places in L0, there
won't be just a straight phi

1086
01:14:24,590 --> 01:14:25,090
dagger.

1087
01:14:25,090 --> 01:14:29,210
There might be a del
mu phi dagger, right?

1088
01:14:29,210 --> 01:14:35,090
So I can take that
into account as well

1089
01:14:35,090 --> 01:14:37,090
by taking the functional
derivative with respect

1090
01:14:37,090 --> 01:14:38,790
to del mu phi dagger.

1091
01:14:38,790 --> 01:14:42,550
So any del mu phi dagger
term, if I integrate by parts,

1092
01:14:42,550 --> 01:14:44,560
and then take the
derivative, you

1093
01:14:44,560 --> 01:14:47,500
can see why this
is the right form.

1094
01:14:47,500 --> 01:14:50,200
Integration by parts is
what gives the minus sign.

1095
01:14:50,200 --> 01:14:53,920
OK, so this finds all
the phi daggers in my L0

1096
01:14:53,920 --> 01:14:58,825
and sticks in an eta
t to order eta, which

1097
01:14:58,825 --> 01:14:59,950
is the order we're working.

1098
01:15:08,370 --> 01:15:12,180
So anytime we had in eta,
which is these terms here,

1099
01:15:12,180 --> 01:15:15,288
then we would induce
something ordering eta square.

1100
01:15:15,288 --> 01:15:17,580
And we're dropping those
terms, so that's not something

1101
01:15:17,580 --> 01:15:18,538
we have to worry about.

1102
01:15:22,083 --> 01:15:23,250
And then there's the source.

1103
01:15:23,250 --> 01:15:26,070
And one of the sources
is 4 phi dagger.

1104
01:15:26,070 --> 01:15:30,260
So for that particular
source, j phi dagger,

1105
01:15:30,260 --> 01:15:34,635
we induce a term
that's j phi eta t

1106
01:15:34,635 --> 01:15:36,720
and then plus order eta squared.

1107
01:15:49,750 --> 01:15:52,500
So as I said, there's
three types of changes.

1108
01:15:52,500 --> 01:15:55,170
There's a change to the
Lagrangian, like in the example

1109
01:15:55,170 --> 01:15:57,360
we did before.

1110
01:15:57,360 --> 01:16:00,906
There's a change to the
Jacobian through the Jacobian.

1111
01:16:00,906 --> 01:16:03,285
And then there's a change
through the source.

1112
01:16:06,377 --> 01:16:08,460
We have to worry about
whether any of these things

1113
01:16:08,460 --> 01:16:08,980
will matter.

1114
01:16:12,840 --> 01:16:16,810
And basically, what the claim
is and where we're going

1115
01:16:16,810 --> 01:16:22,560
is considering the Lagrangian
is actually enough,

1116
01:16:22,560 --> 01:16:25,110
so that 2 and 3 are something
that we can kind of deal

1117
01:16:25,110 --> 01:16:26,180
with generically.

1118
01:16:26,180 --> 01:16:28,680
And then we only have to check
that changing Lagrangian does

1119
01:16:28,680 --> 01:16:31,330
what we want.

1120
01:16:31,330 --> 01:16:34,420
So the claim is and
what we'll show,

1121
01:16:34,420 --> 01:16:38,100
without changing
the s matrix, we

1122
01:16:38,100 --> 01:16:46,858
can remove considering 2
and 3 rather generically.

1123
01:16:52,128 --> 01:16:53,670
So we don't have to
worry about them.

1124
01:17:10,980 --> 01:17:20,010
So some of this we'll
do next time, but let's

1125
01:17:20,010 --> 01:17:20,860
just start today.

1126
01:17:20,860 --> 01:17:24,990
So let's first
look at del L, kind

1127
01:17:24,990 --> 01:17:29,610
of analogous to what we were
doing in our example before.

1128
01:17:33,070 --> 01:17:41,860
So what we need is a
change of variable,

1129
01:17:41,860 --> 01:17:44,430
which is this one up here.

1130
01:17:44,430 --> 01:17:47,310
And one restriction
on del L, which

1131
01:17:47,310 --> 01:17:48,960
I mentioned as part
of the assumption,

1132
01:17:48,960 --> 01:17:53,910
was that this change of
variable should transform

1133
01:17:53,910 --> 01:17:58,830
in the same way as the original
phi dagger, so same Lorentz

1134
01:17:58,830 --> 01:18:03,930
index structure, same gauge
index structure, et cetera.

1135
01:18:03,930 --> 01:18:09,090
So that's kind of an
assumption that we said.

1136
01:18:09,090 --> 01:18:10,680
In order to respect
the symmetries

1137
01:18:10,680 --> 01:18:14,010
and not mess them up with
our field redefinition,

1138
01:18:14,010 --> 01:18:14,640
we assume that.

1139
01:18:18,967 --> 01:18:20,300
So let's see what happens to L0.

1140
01:18:32,565 --> 01:18:33,190
So there is L0.

1141
01:18:36,597 --> 01:18:38,180
Maybe it has some
other terms as well,

1142
01:18:38,180 --> 01:18:41,797
but let's just
consider these pieces

1143
01:18:41,797 --> 01:18:43,630
and ask what happens
with them since they're

1144
01:18:43,630 --> 01:18:44,440
the relevant ones.

1145
01:18:50,400 --> 01:18:54,080
So when I switch to prime
fields, I get that back again.

1146
01:18:54,080 --> 01:18:58,380
I get m squared phi
dagger prime phi prime.

1147
01:18:58,380 --> 01:19:03,165
And then I get some
new terms, which

1148
01:19:03,165 --> 01:19:06,736
I integrate the covariant
derivative by parts,

1149
01:19:06,736 --> 01:19:09,270
which I'm always able to do.

1150
01:19:09,270 --> 01:19:12,070
It's nice what gauge
symmetry allows you to do.

1151
01:19:18,860 --> 01:19:21,090
Then I get a term
like that, and I've

1152
01:19:21,090 --> 01:19:28,790
set things up so that this term
here does exactly what we need.

1153
01:19:28,790 --> 01:19:31,490
I go back over here.

1154
01:19:31,490 --> 01:19:33,128
Those terms are cancelling.

1155
01:19:49,560 --> 01:19:51,150
OK, so that's good.

1156
01:19:59,120 --> 01:20:01,460
Now, you may be worried about
all the other terms that

1157
01:20:01,460 --> 01:20:02,060
get induced.

1158
01:20:02,060 --> 01:20:04,898
You've removed something, but
you've induced a lot of stuff.

1159
01:20:04,898 --> 01:20:06,440
But the point of
the effective theory

1160
01:20:06,440 --> 01:20:08,990
is that you already wrote down
every possible operator that

1161
01:20:08,990 --> 01:20:11,150
was consistent with
the symmetries.

1162
01:20:11,150 --> 01:20:13,600
So even if you induce
a bunch of other terms,

1163
01:20:13,600 --> 01:20:15,650
they should be terms
you already have.

1164
01:20:15,650 --> 01:20:17,120
So all you're
doing with inducing

1165
01:20:17,120 --> 01:20:19,490
those other terms is shifting
the coefficients of the theory

1166
01:20:19,490 --> 01:20:19,990
around.

1167
01:20:23,450 --> 01:20:29,270
So L1 was already
complete in the sense

1168
01:20:29,270 --> 01:20:33,020
of having all the terms
allowed by the symmetry.

1169
01:20:33,020 --> 01:20:36,730
And we've respected the
symmetry in the way we've

1170
01:20:36,730 --> 01:20:38,371
made this field redefinition.

1171
01:20:52,140 --> 01:20:53,623
So any terms that
I didn't write,

1172
01:20:53,623 --> 01:20:55,290
which are all these
terms in the da, da,

1173
01:20:55,290 --> 01:21:03,660
da prime are already
operators that

1174
01:21:03,660 --> 01:21:06,510
are present in the other
dots that I didn't write.

1175
01:21:09,070 --> 01:21:10,520
So there were some dots here.

1176
01:21:10,520 --> 01:21:11,640
There was some dots there.

1177
01:21:11,640 --> 01:21:14,107
Even if I include
operators in those dots,

1178
01:21:14,107 --> 01:21:16,690
the operators in these dots are
already present in those dots.

1179
01:21:19,900 --> 01:21:23,940
And so all I'm doing really
is shifting couplings.

1180
01:21:30,060 --> 01:21:32,150
OK.

1181
01:21:32,150 --> 01:21:35,410
And that's same effective
theory just with a new name

1182
01:21:35,410 --> 01:21:37,160
for the coefficients,
but we haven't fixed

1183
01:21:37,160 --> 01:21:38,330
the coefficients yet anyway.

1184
01:21:38,330 --> 01:21:40,287
We're setting up our
effect effective theory.

1185
01:21:40,287 --> 01:21:41,870
So whether we fix
the new coefficients

1186
01:21:41,870 --> 01:21:45,190
or the old coefficients,
it's perfectly fine.

1187
01:21:56,830 --> 01:21:59,550
So number two is the Jacobian.

1188
01:21:59,550 --> 01:22:01,690
And I won't go all
the way through this,

1189
01:22:01,690 --> 01:22:05,042
but what are we
going to do there?

1190
01:22:05,042 --> 01:22:06,750
We're going to use
the same kind of trick

1191
01:22:06,750 --> 01:22:09,270
that we did in gauge theory.

1192
01:22:09,270 --> 01:22:12,000
We're going to write the
Jacobian as a Lagrangian

1193
01:22:12,000 --> 01:22:15,000
involving ghosts.

1194
01:22:15,000 --> 01:22:17,610
So remember, when you
talked about Faddeev-Popov

1195
01:22:17,610 --> 01:22:20,610
in some field theory
course prior to this one,

1196
01:22:20,610 --> 01:22:25,590
you saw that you could write a
determinant as an exponential

1197
01:22:25,590 --> 01:22:27,105
involving some ghost fields.

1198
01:22:30,150 --> 01:22:37,072
And the way that it worked
is that, up to a sign,

1199
01:22:37,072 --> 01:22:39,030
whatever was sitting here
ended up just sitting

1200
01:22:39,030 --> 01:22:40,080
between the ghost fields.

1201
01:22:55,140 --> 01:23:01,931
So this is the ghost of
Faddeev-Popov procedure.

1202
01:23:04,438 --> 01:23:06,230
And we're going to just
use the same thing,

1203
01:23:06,230 --> 01:23:09,900
although [INAUDIBLE] we're
talking about scalar field

1204
01:23:09,900 --> 01:23:12,270
theory.

1205
01:23:12,270 --> 01:23:23,700
So for us, we take this
guy, which is the Jacobian,

1206
01:23:23,700 --> 01:23:25,945
I'll take that derivative.

1207
01:23:33,560 --> 01:23:35,540
And if you take
this thing, which

1208
01:23:35,540 --> 01:23:38,120
is sitting in
functional determinant,

1209
01:23:38,120 --> 01:23:41,240
and you turned it into
ghosts, then what you get

1210
01:23:41,240 --> 01:23:51,680
is a terminal Lagrangian
that looks like that.

1211
01:24:00,860 --> 01:24:04,850
So that looks like it
could do something.

1212
01:24:04,850 --> 01:24:06,750
And I think I'll take
this up next time.

1213
01:24:06,750 --> 01:24:09,350
It turns out that
this term is actually

1214
01:24:09,350 --> 01:24:12,890
a ghost that has mass that's
of order the high scale, lambda

1215
01:24:12,890 --> 01:24:14,830
nu.

1216
01:24:14,830 --> 01:24:17,300
And so, again, because of our
logic of the effective theory

1217
01:24:17,300 --> 01:24:20,058
only being valid
below that high scale,

1218
01:24:20,058 --> 01:24:22,100
we can just effectively
integrate out this ghost.

1219
01:24:22,100 --> 01:24:25,190
And then it shifts
coefficients again,

1220
01:24:25,190 --> 01:24:27,620
but I'll continue
with that next time.

1221
01:24:27,620 --> 01:24:31,760
So we have to figure out what to
do with this ghost Lagrangian.

1222
01:24:31,760 --> 01:24:34,590
And we'll deal with
that next time.

1223
01:24:34,590 --> 01:24:37,520
So any questions about
our halfway done proof?

1224
01:24:42,833 --> 01:24:44,748
No?

1225
01:24:44,748 --> 01:24:45,248
OK.

1226
01:24:48,630 --> 01:24:50,090
So this is a pretty
powerful thing

1227
01:24:50,090 --> 01:24:52,070
that you can use in
effective field theory.

1228
01:24:52,070 --> 01:24:54,300
And it also keeps
your eye on the ball

1229
01:24:54,300 --> 01:24:57,247
because it tells you
to think about physics.

1230
01:24:57,247 --> 01:24:58,830
Because this leaves
physics invariant.

1231
01:24:58,830 --> 01:25:00,800
It doesn't leave
something like if you--

1232
01:25:00,800 --> 01:25:02,858
say you had a theory that
had some long distance

1233
01:25:02,858 --> 01:25:03,650
degrees of freedom.

1234
01:25:03,650 --> 01:25:05,630
And it had some short
distance potential.

1235
01:25:05,630 --> 01:25:08,060
The short distance
potential could be changed

1236
01:25:08,060 --> 01:25:09,140
by a field redefinition.

1237
01:25:09,140 --> 01:25:11,660
It might not be
a physical thing.

1238
01:25:11,660 --> 01:25:14,900
It keeps your eye on the
ball as to what is physical

1239
01:25:14,900 --> 01:25:17,470
and how to talk about things.