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IAIN STEWART: So
last time, we were

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00:00:24,140 --> 00:00:27,530
talking about the chiral
Lagrangian as another example

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00:00:27,530 --> 00:00:29,368
of an effective field
theory to illustrate

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00:00:29,368 --> 00:00:31,160
some of the tools of
effective field theory

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00:00:31,160 --> 00:00:32,660
that we didn't meet
when integrating

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00:00:32,660 --> 00:00:33,955
our heavy particles.

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It was an example of a bottom-up
effective field theory.

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00:00:36,820 --> 00:00:38,570
So we were constructing
it from the bottom

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00:00:38,570 --> 00:00:42,320
up for the most part.

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00:00:42,320 --> 00:00:44,810
That's how one should
think about it.

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00:00:44,810 --> 00:00:46,563
There was several
things in our program

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00:00:46,563 --> 00:00:48,480
to understand from chiral
perturbation theory.

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00:00:48,480 --> 00:00:50,240
And we kind of got
halfway through.

21
00:00:50,240 --> 00:00:52,160
So we want to understand
nonlinear symmetry

22
00:00:52,160 --> 00:00:53,065
representations.

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00:00:53,065 --> 00:00:54,440
And we went through
how you could

24
00:00:54,440 --> 00:00:56,630
think about linear
representations

25
00:00:56,630 --> 00:00:59,780
and making changes of
variable and putting

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00:00:59,780 --> 00:01:02,000
the Lagrangian in a form
where you could see where

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00:01:02,000 --> 00:01:05,910
the nonlinear realization was
coming from very explicitly,

28
00:01:05,910 --> 00:01:08,540
as well as another way
where we constructed it

29
00:01:08,540 --> 00:01:12,800
from the bottom up thinking
about parameterizing

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00:01:12,800 --> 00:01:14,720
the co-set.

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00:01:14,720 --> 00:01:16,650
And then we started
talking about loops

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00:01:16,650 --> 00:01:17,900
in chiral perturbation theory.

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00:01:17,900 --> 00:01:20,883
And I gave you this
example of this loop here.

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00:01:20,883 --> 00:01:22,550
So we constructed the
chiral Lagrangian.

35
00:01:22,550 --> 00:01:24,260
That's what I've written here.

36
00:01:24,260 --> 00:01:25,640
In terms of the
sigma field, this

37
00:01:25,640 --> 00:01:30,110
is the nonlinear version, sigma
field, exponential of M field.

38
00:01:30,110 --> 00:01:32,690
M field has the
pi n fields in it.

39
00:01:32,690 --> 00:01:35,360
I rewrote it in a little bit
of a different notation, which

40
00:01:35,360 --> 00:01:37,550
will be useful today,
where I introduced

41
00:01:37,550 --> 00:01:39,200
this thing called chi.

42
00:01:39,200 --> 00:01:41,840
But f squared over
8 chi is just Mq.

43
00:01:41,840 --> 00:01:44,042
So it's just a rescaling.

44
00:01:44,042 --> 00:01:46,250
And I just wanted to have
the same kind of pre-factor

45
00:01:46,250 --> 00:01:47,750
here and here.

46
00:01:47,750 --> 00:01:49,430
That's why I did that.

47
00:01:49,430 --> 00:01:54,260
The power counting is in p
squared derivatives or M pi

48
00:01:54,260 --> 00:01:56,480
squared, which is
the same as Mq.

49
00:01:56,480 --> 00:01:59,690
And the reason that M pi
squared and Mq count the same

50
00:01:59,690 --> 00:02:03,260
is there's this relation
between pi squared and the quark

51
00:02:03,260 --> 00:02:04,910
masses.

52
00:02:04,910 --> 00:02:06,890
And then we started
talking about loops.

53
00:02:06,890 --> 00:02:09,710
And we said that, if
we looked at this loop,

54
00:02:09,710 --> 00:02:12,350
the result for this loop diagram
would be a factor of p squared

55
00:02:12,350 --> 00:02:14,142
or M pi squared times
a factor of p squared

56
00:02:14,142 --> 00:02:17,000
or M pi squared
divided by 4 pi f.

57
00:02:24,140 --> 00:02:26,450
This is the same size
as the lowest order,

58
00:02:26,450 --> 00:02:28,190
so the suppression
factor is this.

59
00:02:45,755 --> 00:02:47,380
So we can say that
loops are suppressed

60
00:02:47,380 --> 00:02:49,600
by p squared over some
scale lambda chi squared.

61
00:02:49,600 --> 00:02:51,760
So this is scale
of our expansion.

62
00:02:51,760 --> 00:02:54,460
And we can associate that
scale to 4 pi times f.

63
00:02:57,590 --> 00:02:59,870
OK, so we're going to
continue with that story

64
00:02:59,870 --> 00:03:03,970
unless there's any
questions from last time.

65
00:03:03,970 --> 00:03:08,920
OK, so we want to use
dimensional regularization.

66
00:03:08,920 --> 00:03:12,670
And if we use dimensional
regularization,

67
00:03:12,670 --> 00:03:14,710
we can think about
having an MS bar scheme.

68
00:03:17,620 --> 00:03:19,890
So how do we do that?

69
00:03:19,890 --> 00:03:21,640
It's a little different
than gauge theory,

70
00:03:21,640 --> 00:03:23,810
but the idea is
exactly the same.

71
00:03:23,810 --> 00:03:25,540
So we look at the
dimensions of objects.

72
00:03:25,540 --> 00:03:30,910
This capital M-- supposed
to be a capital M--

73
00:03:30,910 --> 00:03:36,310
is like a scalar field, so it
has dimension 1 minus epsilon.

74
00:03:36,310 --> 00:03:39,520
The decay constant
then, if you look

75
00:03:39,520 --> 00:03:42,760
at it, which is the coupling,
has to have dimension

76
00:03:42,760 --> 00:03:43,953
1 minus epsilon 2.

77
00:03:43,953 --> 00:03:46,120
And you can see that because
this exponential better

78
00:03:46,120 --> 00:03:47,900
be the exponential of
something dimensionless

79
00:03:47,900 --> 00:03:49,442
so that, whatever
the dimension of M,

80
00:03:49,442 --> 00:03:54,220
should cancel the
dimension of f.

81
00:03:54,220 --> 00:03:56,740
And then that works out with
the look with the measure

82
00:03:56,740 --> 00:03:57,820
as well, which has a ddx.

83
00:04:01,180 --> 00:04:07,880
So therefore, just like you
did for a gauge coupling,

84
00:04:07,880 --> 00:04:11,590
you can say f bare in some mu
to the minus epsilon times f.

85
00:04:24,770 --> 00:04:27,830
And if you want to do the same
thing for the second term,

86
00:04:27,830 --> 00:04:30,500
though I've put
the V0 inside the--

87
00:04:30,500 --> 00:04:33,200
sorry, there's something
missing in my equation.

88
00:04:33,200 --> 00:04:36,300
There was a V0 here.

89
00:04:36,300 --> 00:04:38,300
So before last time, we've
written this equation

90
00:04:38,300 --> 00:04:40,790
as V0 out front
and then Mq inside.

91
00:04:40,790 --> 00:04:43,940
And I just rescaled
it to this chi.

92
00:04:43,940 --> 00:04:47,030
OK, so you can think
either in terms of chi.

93
00:04:47,030 --> 00:04:53,960
Chi is kind of absorbing the
parameter along with the Mq,

94
00:04:53,960 --> 00:04:55,400
but they're equivalent things.

95
00:04:55,400 --> 00:04:58,340
So you can think of also
doing a rescaling that

96
00:04:58,340 --> 00:05:00,410
gets the dimension
of this term right.

97
00:05:00,410 --> 00:05:03,980
And the thing that
you want is this.

98
00:05:03,980 --> 00:05:06,290
The P0 would then be mu
to the minus 2 epsilon

99
00:05:06,290 --> 00:05:08,960
to compensate for the fact that
the first term had an f squared

100
00:05:08,960 --> 00:05:10,402
and the second
term just had a V0.

101
00:05:13,097 --> 00:05:14,680
But that's just to
dimension counting.

102
00:05:32,820 --> 00:05:35,820
There's also no mus in physical
quantities like the pi n

103
00:05:35,820 --> 00:05:38,850
mass, which is an
observable, which

104
00:05:38,850 --> 00:05:41,520
has just a definite value.

105
00:05:41,520 --> 00:05:45,680
And there's also, in
chiral perturbation theory,

106
00:05:45,680 --> 00:05:48,360
no mus in the quark mass.

107
00:05:48,360 --> 00:05:51,780
So when you look at
V0 over f squared,

108
00:05:51,780 --> 00:05:58,491
if you look at it bare, that's
the same as renormalized.

109
00:06:02,340 --> 00:06:05,340
The other thing that's different
about this than in a gauge

110
00:06:05,340 --> 00:06:07,470
theory is, in the gauge
theory, you'd have a z.

111
00:06:07,470 --> 00:06:11,250
You'd have g bare is equal
to e to the epsilon times g,

112
00:06:11,250 --> 00:06:13,343
but then you'd have a zg.

113
00:06:13,343 --> 00:06:14,760
But in chiral
perturbation theory,

114
00:06:14,760 --> 00:06:16,800
the way that the
loops work, the loops

115
00:06:16,800 --> 00:06:20,940
aren't renormalizing the
leading order Lagrangian.

116
00:06:20,940 --> 00:06:23,070
The loops are renormalizing
something else

117
00:06:23,070 --> 00:06:25,560
because they ended
up being suppressed.

118
00:06:25,560 --> 00:06:29,265
So you don't need counter-terms
here for the leading order

119
00:06:29,265 --> 00:06:30,840
Lagrangian.

120
00:06:30,840 --> 00:06:33,510
And that's why I didn't
put a zf or a zV0 here.

121
00:06:37,680 --> 00:06:39,210
OK, but when you
do this loop, you

122
00:06:39,210 --> 00:06:41,377
do get ultraviolet divergences.

123
00:06:53,070 --> 00:06:54,758
So you get 1 over
epsilon divergence

124
00:06:54,758 --> 00:06:56,550
if you're using
dimensional regularization.

125
00:06:56,550 --> 00:06:59,460
And it comes along with
the log of mu squared.

126
00:06:59,460 --> 00:07:01,710
And then it'll be
divided by some scale

127
00:07:01,710 --> 00:07:05,400
that you have in your loop, like
an external momentum p squared,

128
00:07:05,400 --> 00:07:11,490
or it could be the pi n mass.

129
00:07:11,490 --> 00:07:13,862
Both of those will
show up generically.

130
00:07:24,960 --> 00:07:35,592
And in the way we're doing
counting, the way the loops are

131
00:07:35,592 --> 00:07:37,800
showing up, if I can bind
together the factors that I

132
00:07:37,800 --> 00:07:42,480
told you about here, it's either
p to the fourth, 4 powers of p,

133
00:07:42,480 --> 00:07:46,920
or 2 of p and 2 of M
pi or 4 powers of M pi.

134
00:07:46,920 --> 00:07:49,020
So to cancel that
1 over epsilon,

135
00:07:49,020 --> 00:07:50,542
we need a counter-term.

136
00:07:50,542 --> 00:07:52,500
But it's not a counter-term
in that Lagrangian.

137
00:07:52,500 --> 00:07:54,843
It's a counter term in the
higher order Lagrangian.

138
00:08:00,640 --> 00:08:03,010
So we're not done by just
calculating the loop.

139
00:08:03,010 --> 00:08:05,527
We have to actually include
some higher dimension

140
00:08:05,527 --> 00:08:07,860
operators that are the same
order in our power counting.

141
00:08:19,240 --> 00:08:21,130
So we'll talk about
SU(3) a little later.

142
00:08:21,130 --> 00:08:24,010
I'll start out by
talking about SU(2).

143
00:08:24,010 --> 00:08:28,240
In SU(2), the form of that
Lagrangian with higher order

144
00:08:28,240 --> 00:08:32,140
terms is just taking what
we had before and just

145
00:08:32,140 --> 00:08:33,991
going to a higher dimension.

146
00:08:37,090 --> 00:08:40,330
So we could have two
more derivatives.

147
00:08:40,330 --> 00:08:43,809
One way of doing that is taking
what we had at lowest order

148
00:08:43,809 --> 00:08:44,680
and squaring it.

149
00:08:51,880 --> 00:08:54,820
Another way of doing it
would be to take and contract

150
00:08:54,820 --> 00:08:56,420
the indices a little
bit differently.

151
00:08:56,420 --> 00:09:00,490
So I could do it like this.

152
00:09:03,520 --> 00:09:07,360
The trace is cyclic, so I can
move things around from back

153
00:09:07,360 --> 00:09:08,530
to front.

154
00:09:08,530 --> 00:09:10,330
But I could have
mu nu in one trace

155
00:09:10,330 --> 00:09:12,902
and mu nu in another trace
rather than just contracting

156
00:09:12,902 --> 00:09:13,610
within the trace.

157
00:09:13,610 --> 00:09:17,050
That's another possible term.

158
00:09:17,050 --> 00:09:18,550
And there's a bunch
more terms which

159
00:09:18,550 --> 00:09:22,570
I decided to wait and enumerate
when we do SU(3) to give you

160
00:09:22,570 --> 00:09:24,550
a full enumeration.

161
00:09:24,550 --> 00:09:32,800
These additional
terms you can build

162
00:09:32,800 --> 00:09:39,250
from one Mq, which means one
of these chi fields or chi

163
00:09:39,250 --> 00:09:45,490
objects and two partials.

164
00:09:45,490 --> 00:09:54,340
That's one possibility or
two Mq's because each Mq

165
00:09:54,340 --> 00:09:56,222
counts like two derivatives.

166
00:09:56,222 --> 00:09:58,180
So there's some other
terms that we could build

167
00:09:58,180 --> 00:09:59,873
that are analogs of this term.

168
00:09:59,873 --> 00:10:01,540
You could square that
term, for example.

169
00:10:01,540 --> 00:10:04,420
That would be a possible
higher order term.

170
00:10:04,420 --> 00:10:07,360
And so these coefficients
of this Lagrangian

171
00:10:07,360 --> 00:10:10,300
are what we need to cancel
off the divergences.

172
00:10:10,300 --> 00:10:12,490
So the counter-terms
that we have

173
00:10:12,490 --> 00:10:14,200
that renormalize
these loop graphs

174
00:10:14,200 --> 00:10:15,862
come from this Lagrangian.

175
00:10:38,020 --> 00:10:42,220
So the theory is renormalizable
in a EFT sense, order

176
00:10:42,220 --> 00:10:44,800
by order in its power
counting expansion.

177
00:10:44,800 --> 00:10:46,740
When you go to order
p to the fourth,

178
00:10:46,740 --> 00:10:48,490
you have to consistently
put in everything

179
00:10:48,490 --> 00:10:50,110
that's order p to the fourth.

180
00:10:50,110 --> 00:10:53,110
That includes the loops,
which are p to the fourth,

181
00:10:53,110 --> 00:10:59,260
as well as new local
Lagrangian interactions.

182
00:10:59,260 --> 00:11:01,580
AUDIENCE: Is that a
choice to not renormalize

183
00:11:01,580 --> 00:11:03,370
the f bare and [? d ?] bare?

184
00:11:03,370 --> 00:11:05,620
IAIN STEWART: They just
don't get renormalized.

185
00:11:05,620 --> 00:11:07,870
AUDIENCE: But why couldn't
I just multiply by 1 plus--

186
00:11:07,870 --> 00:11:08,620
IAIN STEWART: Oh--

187
00:11:08,620 --> 00:11:12,072
AUDIENCE: --order p squared
by times delta [INAUDIBLE]..

188
00:11:12,072 --> 00:11:13,780
IAIN STEWART: Oh, you
want to try to make

189
00:11:13,780 --> 00:11:15,010
a different scheme for them?

190
00:11:15,010 --> 00:11:17,302
AUDIENCE: I'd screw up the
power counting [INAUDIBLE]..

191
00:11:19,930 --> 00:11:25,890
IAIN STEWART: So
effectively, you

192
00:11:25,890 --> 00:11:27,890
could screw up the power
counting by doing that.

193
00:11:27,890 --> 00:11:31,690
So let me tell you
what you can do.

194
00:11:31,690 --> 00:11:33,400
You could put a 2 here, right?

195
00:11:36,670 --> 00:11:38,530
But you don't really
have much more freedom

196
00:11:38,530 --> 00:11:40,270
than just multiplying
by a number.

197
00:11:40,270 --> 00:11:41,895
AUDIENCE: So I can't
multiply by 1 plus

198
00:11:41,895 --> 00:11:43,798
p squared over s squared.

199
00:11:43,798 --> 00:11:44,590
IAIN STEWART: Yeah.

200
00:11:44,590 --> 00:11:46,548
That would be a screwing
up the power counting.

201
00:11:46,548 --> 00:11:50,945
And effectively, you wouldn't
be multiplying by p squared.

202
00:11:50,945 --> 00:11:52,570
You'd be putting
derivatives in, right?

203
00:11:52,570 --> 00:11:53,860
Because there's no p.

204
00:11:53,860 --> 00:11:58,893
So p is not something that
you're allowed to multiply by.

205
00:11:58,893 --> 00:12:00,310
AUDIENCE: So that's
the only thing

206
00:12:00,310 --> 00:12:02,680
that's stopping me-- is what
is derivatives [INAUDIBLE]??

207
00:12:02,680 --> 00:12:03,850
IAIN STEWART: You could
put derivatives in,

208
00:12:03,850 --> 00:12:05,440
but then that's equivalent
to something here.

209
00:12:05,440 --> 00:12:08,160
So you have to ask what you're
doing because you're mixing up

210
00:12:08,160 --> 00:12:08,660
things.

211
00:12:08,660 --> 00:12:09,310
AUDIENCE: OK.

212
00:12:09,310 --> 00:12:09,720
IAIN STEWART: Yeah.

213
00:12:09,720 --> 00:12:11,030
AUDIENCE: [INAUDIBLE]
definitely a better way to do.

214
00:12:11,030 --> 00:12:13,180
I was just wondering if
there was sort of freedom.

215
00:12:13,180 --> 00:12:15,097
IAIN STEWART: Yeah,
there's no freedom really.

216
00:12:15,097 --> 00:12:15,890
AUDIENCE: Good.

217
00:12:15,890 --> 00:12:18,432
AUDIENCE: Why you don't get the
renormalization [INAUDIBLE],,

218
00:12:18,432 --> 00:12:21,312
like from [INAUDIBLE]
[? loop? ?]

219
00:12:21,312 --> 00:12:23,770
IAIN STEWART: Because all the
loops end up being suppressed

220
00:12:23,770 --> 00:12:27,980
by p squared or M pi squared.

221
00:12:27,980 --> 00:12:33,220
So if you write down any diagram
and you think it might give

222
00:12:33,220 --> 00:12:38,210
something here, even if
it's a two-point function,

223
00:12:38,210 --> 00:12:40,825
it just doesn't.

224
00:12:40,825 --> 00:12:43,870
AUDIENCE: [INAUDIBLE]
to 4 [INAUDIBLE]..

225
00:12:43,870 --> 00:12:45,560
IAIN STEWART: That's right.

226
00:12:45,560 --> 00:12:48,940
Oh, so if you want, you could
think that some terms here

227
00:12:48,940 --> 00:12:51,670
are kind of corrections
to the kinetic term

228
00:12:51,670 --> 00:12:54,200
if you derive the
equation of motion

229
00:12:54,200 --> 00:13:03,730
and you include L and that L
and this L. But this Lagrangian,

230
00:13:03,730 --> 00:13:06,130
there's no loop corrections
to this Lagrangian.

231
00:13:06,130 --> 00:13:09,820
There's no corrections to
this lowest order Lagrangian.

232
00:13:09,820 --> 00:13:12,460
And that's because the
loops are all suppressed.

233
00:13:12,460 --> 00:13:13,960
Totally different
then gauge theory,

234
00:13:13,960 --> 00:13:16,310
loops are suppressed by p
squared or M pi squared.

235
00:13:16,310 --> 00:13:18,460
So they only renormalize
some high order things.

236
00:13:18,460 --> 00:13:24,670
And we don't have to worry
about really thinking about f.

237
00:13:24,670 --> 00:13:27,850
I mean, the way I've
talked about f here

238
00:13:27,850 --> 00:13:29,650
and the factors of
mu is just in order

239
00:13:29,650 --> 00:13:32,530
to see where these factors
and mu come out here.

240
00:13:32,530 --> 00:13:38,702
But what actually happens is
that you get 1 over epsilons

241
00:13:38,702 --> 00:13:40,660
that are cancelled like
the counter-terms here.

242
00:13:40,660 --> 00:13:45,700
And there's no need from a
renormalization perspective

243
00:13:45,700 --> 00:13:47,655
to sort of change
your definitions here.

244
00:13:47,655 --> 00:13:49,280
People play with
different definitions,

245
00:13:49,280 --> 00:13:50,905
but it's like they
use f squared over 4

246
00:13:50,905 --> 00:13:52,420
instead of f squared over 8.

247
00:13:52,420 --> 00:13:54,700
That's the extent
of what people do

248
00:13:54,700 --> 00:13:56,965
with playing with the
leading order Lagrangian.

249
00:13:56,965 --> 00:13:58,420
AUDIENCE: So you have
[INAUDIBLE] the same thing with

250
00:13:58,420 --> 00:13:59,830
the [INAUDIBLE] that
[INAUDIBLE] [? is not ?]

251
00:13:59,830 --> 00:14:01,212
[? renormalizing ?] [INAUDIBLE].

252
00:14:01,212 --> 00:14:02,920
IAIN STEWART: Both of
them are not, yeah.

253
00:14:02,920 --> 00:14:03,970
Yeah.

254
00:14:03,970 --> 00:14:07,270
So loops that you build
out of these interactions

255
00:14:07,270 --> 00:14:16,418
end up renormalizing these
terms, which is kind of neat.

256
00:14:16,418 --> 00:14:19,830
AUDIENCE: That's just
because [INAUDIBLE]..

257
00:14:19,830 --> 00:14:22,746
That's just because there's
[INAUDIBLE] [? sort of ?]

258
00:14:22,746 --> 00:14:24,210
[INAUDIBLE].

259
00:14:24,210 --> 00:14:25,840
IAIN STEWART: Yeah.

260
00:14:25,840 --> 00:14:29,160
Well, so we'll prove in a
minute a general power counting

261
00:14:29,160 --> 00:14:32,570
formula that tells us
how to organize all this.

262
00:14:32,570 --> 00:14:34,920
So for now, you can think
of it as an observation,

263
00:14:34,920 --> 00:14:38,970
but we'll build it into a
general formula that tells you

264
00:14:38,970 --> 00:14:41,573
sort of how you would
organize the power counting

265
00:14:41,573 --> 00:14:42,990
and renormalization
of this theory

266
00:14:42,990 --> 00:14:46,810
to all orders in its
expansion in a minute.

267
00:14:46,810 --> 00:14:49,060
AUDIENCE: So when you write
down all these [INAUDIBLE]

268
00:14:49,060 --> 00:14:51,680
at [? power ?] [? p4, ?]
does that include all

269
00:14:51,680 --> 00:14:52,800
the [INAUDIBLE] you need?

270
00:14:52,800 --> 00:14:53,610
IAIN STEWART: Yeah.

271
00:14:53,610 --> 00:14:55,530
That's right.

272
00:14:55,530 --> 00:14:57,270
Once I've enumerated
all the terms,

273
00:14:57,270 --> 00:14:59,310
which I will do for
you with SU(3)--

274
00:14:59,310 --> 00:15:02,580
so it includes things like this
and things like this as well--

275
00:15:02,580 --> 00:15:04,710
then those are all the
possible counter-terms

276
00:15:04,710 --> 00:15:07,278
that you could actually
need to renormalize

277
00:15:07,278 --> 00:15:08,820
any of the one-loop
diagrams that you

278
00:15:08,820 --> 00:15:12,420
could generate with the
leading order Lagrangian.

279
00:15:12,420 --> 00:15:15,067
And you know that
by power counting.

280
00:15:15,067 --> 00:15:16,650
Once you know the
loops are that order

281
00:15:16,650 --> 00:15:19,830
and you've included all local
interactions, then you're done.

282
00:15:23,160 --> 00:15:28,860
So there's things I didn't
write down there, so just

283
00:15:28,860 --> 00:15:30,930
comment about that.

284
00:15:30,930 --> 00:15:34,747
So what is the equation
of motion here?

285
00:15:34,747 --> 00:15:37,080
It's a little more complicated
because we have the sigma

286
00:15:37,080 --> 00:15:40,680
field, but you can work out
that the equation of motion

287
00:15:40,680 --> 00:15:41,460
is the following.

288
00:15:57,580 --> 00:16:01,440
And I've given you
some reading, so you

289
00:16:01,440 --> 00:16:06,000
can read about a derivation
of this equation which

290
00:16:06,000 --> 00:16:07,590
takes a little effort.

291
00:16:07,590 --> 00:16:09,330
So your leading order
equation of motion

292
00:16:09,330 --> 00:16:12,120
basically allows you to
get rid of partial squareds

293
00:16:12,120 --> 00:16:13,590
on the sigma.

294
00:16:13,590 --> 00:16:16,350
And that's why
I've always written

295
00:16:16,350 --> 00:16:18,887
one partial on each sigma.

296
00:16:18,887 --> 00:16:20,970
So that's what you should
think of this guy doing.

297
00:16:29,860 --> 00:16:31,500
There's also some
SU(2) identities

298
00:16:31,500 --> 00:16:37,380
that have been used, even
in what I've written,

299
00:16:37,380 --> 00:16:40,230
because you could think of
having other types of traces

300
00:16:40,230 --> 00:16:42,300
like this one, for example.

301
00:16:42,300 --> 00:16:46,925
Instead of having trace squared,
I could have the following.

302
00:16:46,925 --> 00:16:48,300
Just put everything
in one trace.

303
00:16:55,110 --> 00:16:57,840
I always have to alternate
sigma sigma dagger,

304
00:16:57,840 --> 00:17:04,170
sigma sigma dagger because del
mu sigma sigma dagger is 0.

305
00:17:04,170 --> 00:17:06,990
Because sigma sigma dagger is 1.

306
00:17:06,990 --> 00:17:11,160
And that provides an identity
that I'm also able to use.

307
00:17:11,160 --> 00:17:13,035
And so I want to
have sigmas and sigma

308
00:17:13,035 --> 00:17:14,160
daggers next to each other.

309
00:17:18,733 --> 00:17:21,150
Sorry, chiral symmetry means
that sigmas and sigma daggers

310
00:17:21,150 --> 00:17:23,290
should be next to each other.

311
00:17:23,290 --> 00:17:26,290
This is an identity that I can
also use to simplify things.

312
00:17:26,290 --> 00:17:28,380
So that allows me to
move things around.

313
00:17:28,380 --> 00:17:31,230
And this operator,
though, in SU(2)

314
00:17:31,230 --> 00:17:33,330
is actually related
to these two.

315
00:17:33,330 --> 00:17:34,740
It's not unique.

316
00:17:34,740 --> 00:17:37,500
Because of the structure
of Pauli matrices,

317
00:17:37,500 --> 00:17:40,500
you have a formula
which you can work out,

318
00:17:40,500 --> 00:17:43,350
which says that
this guy is actually

319
00:17:43,350 --> 00:17:44,865
half of the trace squared.

320
00:17:51,240 --> 00:17:54,240
And there's another
one if you did.

321
00:18:03,550 --> 00:18:09,615
This guy, he also can be related
to those two guys over there.

322
00:18:09,615 --> 00:18:10,990
So there's a bunch
of things that

323
00:18:10,990 --> 00:18:13,780
went into even writing down
the level of information

324
00:18:13,780 --> 00:18:14,860
that I told you.

325
00:18:14,860 --> 00:18:16,900
You could think
there's more things,

326
00:18:16,900 --> 00:18:19,130
but then, when you enumerate
all the possible things

327
00:18:19,130 --> 00:18:21,130
that you can do, you find
that those more things

328
00:18:21,130 --> 00:18:23,030
are related to these things.

329
00:18:23,030 --> 00:18:25,030
So you really want to
construct a minimal basis.

330
00:18:25,030 --> 00:18:27,030
And there's some things
that go into doing that.

331
00:18:35,280 --> 00:18:41,060
So at order p to the fourth, as
I said, we include both loops.

332
00:18:41,060 --> 00:18:45,170
And those loops will have
terms like p to the fourth log

333
00:18:45,170 --> 00:18:46,370
mu squared over p squared.

334
00:18:52,370 --> 00:18:55,970
And we include terms
like the L1 and L2,

335
00:18:55,970 --> 00:19:02,150
which are p of the
fourth type interactions.

336
00:19:02,150 --> 00:19:04,010
Once I take out
the counter-term,

337
00:19:04,010 --> 00:19:06,520
the renormalized coupling
is mu dependent just

338
00:19:06,520 --> 00:19:07,520
like the gauge coupling.

339
00:19:14,075 --> 00:19:15,950
This just comes from a
four-point interaction

340
00:19:15,950 --> 00:19:17,780
with the Li.

341
00:19:17,780 --> 00:19:20,960
And the mu dependence,
by construction,

342
00:19:20,960 --> 00:19:22,640
was canceled between
these two things.

343
00:19:25,310 --> 00:19:28,400
The divergence from the loop is
cancelled by the counter-term.

344
00:19:28,400 --> 00:19:30,500
And correspondingly, the
corresponding statement,

345
00:19:30,500 --> 00:19:32,083
if you want to make
it in terms of mu,

346
00:19:32,083 --> 00:19:35,600
is that the mu dependence that
is tightly tied to that 1 over

347
00:19:35,600 --> 00:19:39,065
epsilon is cancelled in the
renormalized quantities.

348
00:19:50,172 --> 00:19:51,630
So there's these
two contributions.

349
00:19:51,630 --> 00:19:53,070
They're both mu dependent.

350
00:19:53,070 --> 00:19:55,853
And that mu dependence cancels.

351
00:19:55,853 --> 00:19:57,520
So the way that you
should think of this

352
00:19:57,520 --> 00:20:01,190
is you should think
mu is a cut off.

353
00:20:01,190 --> 00:20:03,550
It's not a hard cut off,
but it is a cut off.

354
00:20:08,350 --> 00:20:10,455
And what the cut off
is doing is dividing up

355
00:20:10,455 --> 00:20:11,830
infrared and
ultraviolet physics.

356
00:20:16,210 --> 00:20:18,610
In this case, the
low energy physics

357
00:20:18,610 --> 00:20:26,050
is in the matrix
elements in the loops

358
00:20:26,050 --> 00:20:28,730
where we have our propagating
low energy degrees of freedom,

359
00:20:28,730 --> 00:20:31,060
which are the pions.

360
00:20:31,060 --> 00:20:34,645
And the high energy physics is
in the coefficients, as usual.

361
00:20:43,730 --> 00:20:44,980
And they're both mu dependent.

362
00:20:44,980 --> 00:20:47,270
And you can think of that
mu dependent as a cut off

363
00:20:47,270 --> 00:20:49,310
that divides up how
much of the physics

364
00:20:49,310 --> 00:20:52,010
goes into those low energy
loops, how much of the physics

365
00:20:52,010 --> 00:20:55,630
goes into the couplings.

366
00:20:55,630 --> 00:20:59,050
So the difference between
this and integrating

367
00:20:59,050 --> 00:21:01,030
on a massive particle
is not the physics

368
00:21:01,030 --> 00:21:03,550
of where things go because
the low energy physics always

369
00:21:03,550 --> 00:21:04,717
goes in the matrix elements.

370
00:21:04,717 --> 00:21:06,790
The high energy physics
goes in the couplings.

371
00:21:06,790 --> 00:21:10,638
The difference is that, here,
the way that you should think

372
00:21:10,638 --> 00:21:13,180
of it is that we can calculate
the matrix elements explicitly

373
00:21:13,180 --> 00:21:16,510
because our theory is in terms
of the right degrees of freedom

374
00:21:16,510 --> 00:21:21,910
to describe long distance
physics, which are the pions.

375
00:21:21,910 --> 00:21:24,010
And then the coefficients
are the unknowns.

376
00:21:24,010 --> 00:21:28,093
That's the way in
which it's different.

377
00:21:28,093 --> 00:21:29,760
And that's really
kind of the difference

378
00:21:29,760 --> 00:21:31,500
between bottom-up and top-down.

379
00:21:42,740 --> 00:21:47,500
So with that in
mind, if you want

380
00:21:47,500 --> 00:21:51,670
to think of what these
couplings are, if we just

381
00:21:51,670 --> 00:21:59,470
divide by f squared, then you
can think about dimensionally

382
00:21:59,470 --> 00:22:01,030
what they are.

383
00:22:01,030 --> 00:22:02,860
In order for them
to sort of match up

384
00:22:02,860 --> 00:22:05,260
with what we got
from the loops is

385
00:22:05,260 --> 00:22:10,800
that they have some dependence
that looks like this.

386
00:22:10,800 --> 00:22:12,400
You can think of
it as parameterized

387
00:22:12,400 --> 00:22:14,590
by some coefficients a and b.

388
00:22:14,590 --> 00:22:17,200
If I want to match up with the
4 pi f squared that comes from

389
00:22:17,200 --> 00:22:19,750
the loops, then I have to sort
of say that there's a 4 pi

390
00:22:19,750 --> 00:22:22,150
hiding inside the Li's.

391
00:22:22,150 --> 00:22:24,200
So I can do that.

392
00:22:24,200 --> 00:22:26,300
And then there's some
coefficients here.

393
00:22:26,300 --> 00:22:27,730
There's some mu dependence.

394
00:22:27,730 --> 00:22:29,770
And the scales that
are in the coefficients

395
00:22:29,770 --> 00:22:32,350
are the high scales, like
lambda chi and higher

396
00:22:32,350 --> 00:22:34,970
and rho, those types of things.

397
00:22:34,970 --> 00:22:36,940
So let me just
call it lambda chi.

398
00:22:36,940 --> 00:22:38,920
And then there's
some numbers here a

399
00:22:38,920 --> 00:22:44,650
and b, which encode
the high mass physics.

400
00:22:44,650 --> 00:22:47,560
Now, the point of thinking
about power counting

401
00:22:47,560 --> 00:22:49,570
is that thinking about
the fact that the loops

402
00:22:49,570 --> 00:22:51,610
and these coefficients
are the same size

403
00:22:51,610 --> 00:22:54,130
actually tells you that
these ai's and bi's should

404
00:22:54,130 --> 00:22:54,820
be order 1.

405
00:23:04,760 --> 00:23:07,310
And this actually goes under
the rubric of something called

406
00:23:07,310 --> 00:23:08,780
naive dimensional analysis.

407
00:23:21,900 --> 00:23:23,613
So what naive
dimensional analysis says

408
00:23:23,613 --> 00:23:26,030
is that this cut off that we've
put between the low energy

409
00:23:26,030 --> 00:23:28,310
physics and the high energy
physics is arbitrary.

410
00:23:28,310 --> 00:23:29,300
And we could change it.

411
00:23:29,300 --> 00:23:31,250
We could change it
by a factor of 2.

412
00:23:31,250 --> 00:23:35,070
And what changing it does is
it moves pieces back and forth.

413
00:23:35,070 --> 00:23:37,100
But if we could move
pieces back and forth,

414
00:23:37,100 --> 00:23:39,800
then you wouldn't expect that
the two things that you're

415
00:23:39,800 --> 00:23:43,130
talking about would be
different in magnitude

416
00:23:43,130 --> 00:23:45,760
because we're allowed to
move pieces back and forth.

417
00:23:45,760 --> 00:23:47,840
So you expect that the
size of contributions

418
00:23:47,840 --> 00:23:51,680
from the coefficients are about
the same size as the loops.

419
00:23:51,680 --> 00:23:58,320
And that's this naive
dimensional analysis.

420
00:23:58,320 --> 00:24:00,410
So changing mu moves
pieces back and forth.

421
00:24:05,450 --> 00:24:14,610
The sum is mu independent, but
each individual thing is not.

422
00:24:21,330 --> 00:24:26,130
And because we're able to
move things back and forth,

423
00:24:26,130 --> 00:24:28,860
we expect them to be the
same order of magnitude.

424
00:24:44,300 --> 00:24:50,030
And that is this statement
here, that the ai's and bi's,

425
00:24:50,030 --> 00:24:53,300
once I account for the 4 pis--

426
00:24:53,300 --> 00:24:56,202
so with this argument,
I can figure out

427
00:24:56,202 --> 00:24:58,160
where there's 4 pis hiding
in the coefficients.

428
00:24:58,160 --> 00:25:00,108
Because I can identify
4 pis in loops.

429
00:25:00,108 --> 00:25:02,150
And then if they're supposed
to be the same size,

430
00:25:02,150 --> 00:25:06,460
I can also identify 4
pis in coefficients.

431
00:25:06,460 --> 00:25:10,150
So if you had figured this out,
I don't know, 20 years ago,

432
00:25:10,150 --> 00:25:13,430
then you could have got a PhD
thesis like Aneesh Manohar did,

433
00:25:13,430 --> 00:25:14,710
which was 25 pages long.

434
00:25:20,540 --> 00:25:22,400
Short PhD theses do exist.

435
00:25:27,605 --> 00:25:28,105
OK.

436
00:25:37,160 --> 00:25:39,110
So what do we do in practice?

437
00:25:39,110 --> 00:25:41,210
In practice, we have
to pick a value of mu.

438
00:25:41,210 --> 00:25:43,680
Just like when we were
talking about gauge theory,

439
00:25:43,680 --> 00:25:44,930
we need to pick a value of mu.

440
00:25:44,930 --> 00:25:47,780
There we were picking
things like mu equals Mb.

441
00:25:47,780 --> 00:25:50,630
Here, what people do
is they typically pick

442
00:25:50,630 --> 00:25:52,700
mus that are high.

443
00:25:52,700 --> 00:25:54,710
So maybe they would
pick the rho mass.

444
00:25:54,710 --> 00:25:58,640
Maybe they would
pick lambda chi.

445
00:25:58,640 --> 00:26:02,390
Or they pick just something
in between, like a GeV.

446
00:26:02,390 --> 00:26:04,190
These are typical
values that people use.

447
00:26:06,710 --> 00:26:12,820
So what that means is that
you've removed all large logs

448
00:26:12,820 --> 00:26:13,930
from your coefficients.

449
00:26:13,930 --> 00:26:15,930
And that's because you
want dimensional analysis

450
00:26:15,930 --> 00:26:18,830
to hold for your coefficients
because you don't know them.

451
00:26:18,830 --> 00:26:20,740
So you would like to
have some power counting

452
00:26:20,740 --> 00:26:21,430
estimate for them.

453
00:26:21,430 --> 00:26:22,990
Then you'd like to go
out and fit them to data.

454
00:26:22,990 --> 00:26:25,115
And you'd like the result
that you get from fitting

455
00:26:25,115 --> 00:26:27,880
to the data to agree with
your power counting estimate,

456
00:26:27,880 --> 00:26:29,290
so that you're happy.

457
00:26:29,290 --> 00:26:33,460
It turns out that it
does, so you are happy.

458
00:26:33,460 --> 00:26:37,300
And so to avoid large
logs in the story,

459
00:26:37,300 --> 00:26:39,310
you put the large logs
into the matrix elements.

460
00:26:54,655 --> 00:26:57,030
So that's different than our
story in gauge theory, where

461
00:26:57,030 --> 00:26:59,613
we were thinking that we would
re-sum all the large logarithms

462
00:26:59,613 --> 00:27:00,780
in the coefficients.

463
00:27:00,780 --> 00:27:02,490
And then our matrix
elements would also

464
00:27:02,490 --> 00:27:03,750
have no large logarithms.

465
00:27:03,750 --> 00:27:05,250
Here, we're just
saying, well, let's

466
00:27:05,250 --> 00:27:07,800
allow for large logs
in the matrix element.

467
00:27:07,800 --> 00:27:09,992
But the story is also
different from gauge theory

468
00:27:09,992 --> 00:27:11,700
in another way, which
is that there's not

469
00:27:11,700 --> 00:27:14,810
an infinite series here
of large logarithms

470
00:27:14,810 --> 00:27:15,810
that you need to re-sum.

471
00:27:31,190 --> 00:27:33,710
And that's related to the fact
that the kinetic term didn't

472
00:27:33,710 --> 00:27:35,237
get renormalized.

473
00:27:35,237 --> 00:27:37,070
There's no mu dependence
in the coefficients

474
00:27:37,070 --> 00:27:37,960
of the kinetic term.

475
00:27:37,960 --> 00:27:41,030
So when the loop
graphs aren't having

476
00:27:41,030 --> 00:27:44,450
coefficient squared depending
on mu, that doesn't happen.

477
00:27:44,450 --> 00:27:47,300
We simply have one log
in our loop graphs,

478
00:27:47,300 --> 00:27:50,060
one log from our
counter-term diagrams.

479
00:27:50,060 --> 00:27:51,840
They explicitly cancel.

480
00:27:51,840 --> 00:27:53,060
There's no high order terms.

481
00:27:53,060 --> 00:27:55,268
The renormalization group
here is completely trivial.

482
00:27:55,268 --> 00:27:57,900
If you integrate,
you just get one log.

483
00:27:57,900 --> 00:28:01,900
So there's not really even
a reason to talk about it.

484
00:28:01,900 --> 00:28:04,790
OK, so there's certainly some
differences in this theory

485
00:28:04,790 --> 00:28:06,200
than there are in gauge theory.

486
00:28:29,370 --> 00:28:31,650
And so typically, the
paradigm that you have is you

487
00:28:31,650 --> 00:28:33,030
can calculate matrix elements.

488
00:28:33,030 --> 00:28:34,830
You're going to fit
the coefficients.

489
00:28:34,830 --> 00:28:36,468
You think of enough
experiments such

490
00:28:36,468 --> 00:28:38,010
that you can get
all that information

491
00:28:38,010 --> 00:28:39,052
about those coefficients.

492
00:28:39,052 --> 00:28:40,980
And then you can think
of other experiments

493
00:28:40,980 --> 00:28:43,833
and make predictions.

494
00:28:43,833 --> 00:28:45,750
Now, as you go to higher
orders in the theory,

495
00:28:45,750 --> 00:28:47,292
you get more
coefficients because you

496
00:28:47,292 --> 00:28:48,780
keep having to go
to higher orders

497
00:28:48,780 --> 00:28:51,630
and construct higher
dimension operators.

498
00:28:51,630 --> 00:28:54,480
And so the paradigm of figuring
out how to fit the coefficients

499
00:28:54,480 --> 00:28:55,800
gets harder and harder.

500
00:28:55,800 --> 00:28:57,240
And at some point,
you effectively

501
00:28:57,240 --> 00:28:58,410
lose predictive
power because you

502
00:28:58,410 --> 00:28:59,993
can't think of enough
observables that

503
00:28:59,993 --> 00:29:04,110
can be measured in order to
fit all your coefficients.

504
00:29:04,110 --> 00:29:07,410
But certainly, at
order p to the fourth,

505
00:29:07,410 --> 00:29:09,502
people can think of
many more observables.

506
00:29:09,502 --> 00:29:11,085
And actually, people
have worked out p

507
00:29:11,085 --> 00:29:13,470
to the sixth as well here.

508
00:29:19,217 --> 00:29:20,800
So two-loop chiral
perturbation theory

509
00:29:20,800 --> 00:29:22,133
is kind of the state of the art.

510
00:29:26,933 --> 00:29:28,100
AUDIENCE: I have a question.

511
00:29:28,100 --> 00:29:30,500
IAIN STEWART: Yeah.

512
00:29:30,500 --> 00:29:33,464
AUDIENCE: So why allow log in
this case [? is not ?] screwing

513
00:29:33,464 --> 00:29:35,630
up the [INAUDIBLE]?

514
00:29:35,630 --> 00:29:38,123
IAIN STEWART: Yeah.

515
00:29:38,123 --> 00:29:40,290
It doesn't screw it up
because it only happens once.

516
00:29:44,390 --> 00:29:46,565
So basically, what you
could say is you can say,

517
00:29:46,565 --> 00:29:47,690
I have this matrix element.

518
00:29:47,690 --> 00:29:49,880
It's got a large log
and a coefficient.

519
00:29:49,880 --> 00:29:52,580
The coupling, it's not
like large log times

520
00:29:52,580 --> 00:29:55,190
the coupling is something that
you could count as order 1

521
00:29:55,190 --> 00:29:57,260
because that's not
going to reappear

522
00:29:57,260 --> 00:30:00,140
in the higher order when
you go to the higher order.

523
00:30:00,140 --> 00:30:02,030
So you still have a large log.

524
00:30:02,030 --> 00:30:04,460
And you are allowed to
say that large log is

525
00:30:04,460 --> 00:30:07,070
the most important piece
of my matrix element,

526
00:30:07,070 --> 00:30:10,130
but it doesn't repeat
itself in the higher orders.

527
00:30:10,130 --> 00:30:11,900
Because once you
go to higher loops,

528
00:30:11,900 --> 00:30:13,275
you're actually
getting something

529
00:30:13,275 --> 00:30:16,960
that's power suppressed,
not loop suppressed.

530
00:30:16,960 --> 00:30:18,710
So the loops are giving
power suppression.

531
00:30:18,710 --> 00:30:20,900
And that changes
a lot of things.

532
00:30:28,575 --> 00:30:29,075
OK.

533
00:30:32,920 --> 00:30:35,050
What would happen if
we used a hard cut off?

534
00:30:39,880 --> 00:30:43,900
Maybe some of this story would
be a bit more transparent

535
00:30:43,900 --> 00:30:46,600
if we'd done that because we'd
see that we were explicitly

536
00:30:46,600 --> 00:30:49,450
dividing up low energy
and high energy physics.

537
00:30:49,450 --> 00:30:52,540
But there's a price to
pay for that transparency.

538
00:30:52,540 --> 00:30:55,450
And here, effectively
everybody decides

539
00:30:55,450 --> 00:30:56,720
it's too high a price to pay.

540
00:30:56,720 --> 00:30:59,680
And so they use dim reg.

541
00:30:59,680 --> 00:31:02,170
So what type of
prices would you pay?

542
00:31:02,170 --> 00:31:03,610
So if you did that,
this loop here

543
00:31:03,610 --> 00:31:05,027
would actually
have terms that are

544
00:31:05,027 --> 00:31:07,675
cut-off to the fourth over
lambda chi to the fourth.

545
00:31:07,675 --> 00:31:09,175
And that guy breaks
chiral symmetry.

546
00:31:12,190 --> 00:31:14,710
I mean, it's giving
a constant term.

547
00:31:14,710 --> 00:31:17,710
And you're not seeing
the derivative coupling.

548
00:31:17,710 --> 00:31:21,010
And that means, effectively,
that there's no counter-term

549
00:31:21,010 --> 00:31:23,260
to absorb this
dependence in our L chi

550
00:31:23,260 --> 00:31:26,690
because our L chi
respected chiral symmetry.

551
00:31:26,690 --> 00:31:30,190
So that's kind of bad.

552
00:31:30,190 --> 00:31:31,690
If you had a cut-off,
you'd also get

553
00:31:31,690 --> 00:31:34,750
terms that went cut-off
squared times momentum squared

554
00:31:34,750 --> 00:31:37,720
where two powers of the p
are replaced by cut-off.

555
00:31:37,720 --> 00:31:41,710
And that, in this language
that we used earlier,

556
00:31:41,710 --> 00:31:43,720
would break the power
counting in the sense

557
00:31:43,720 --> 00:31:46,060
that we have a
power counting that

558
00:31:46,060 --> 00:31:48,880
says that the loops should be
suppressed by d to the fourth.

559
00:31:48,880 --> 00:31:51,370
And it's broken by this.

560
00:31:51,370 --> 00:31:55,870
And then you would need to
absorb that and renormalize

561
00:31:55,870 --> 00:31:58,090
you're leading order coupling.

562
00:31:58,090 --> 00:31:59,810
But effectively, all
the renormalization

563
00:31:59,810 --> 00:32:01,810
would be doing is restoring
your power counting.

564
00:32:01,810 --> 00:32:03,940
It's not doing something
that you should interpret

565
00:32:03,940 --> 00:32:05,380
as really a physical thing.

566
00:32:08,760 --> 00:32:11,500
So it's just something
to be avoided.

567
00:32:11,500 --> 00:32:16,860
And then finally, the physics
that you actually want,

568
00:32:16,860 --> 00:32:23,830
which is the logs of the
cut-off, would also be there.

569
00:32:23,830 --> 00:32:33,520
And you would do the same
thing as you do in dim reg.

570
00:32:33,520 --> 00:32:38,530
You would absorb that in
higher dimension operators, OK?

571
00:32:38,530 --> 00:32:40,990
So we won't use a
cut-off because we

572
00:32:40,990 --> 00:32:42,640
don't want to think
about things that

573
00:32:42,640 --> 00:32:45,057
are breaking power counting
or messing up chiral symmetry.

574
00:32:52,510 --> 00:32:55,000
Another thing that's different
about chiral theories

575
00:32:55,000 --> 00:32:56,770
versus gauge theories
is the structure

576
00:32:56,770 --> 00:32:58,390
of infrared divergences.

577
00:32:58,390 --> 00:33:00,460
And that's because you
have derivative couplings.

578
00:33:08,480 --> 00:33:11,650
So you have many fewer
infrared singularities

579
00:33:11,650 --> 00:33:13,160
than you have in gauge theory.

580
00:33:13,160 --> 00:33:15,700
And in fact, you
usually don't have any.

581
00:33:19,360 --> 00:33:21,580
And one way of describing
that is that you usually

582
00:33:21,580 --> 00:33:24,940
have a good M pi
squared goes to 0

583
00:33:24,940 --> 00:33:28,570
or p squared goes to 0
limit of your results.

584
00:33:33,230 --> 00:33:36,760
So you can just explicitly
take these limits

585
00:33:36,760 --> 00:33:39,680
and talk about the
results in those limits.

586
00:33:39,680 --> 00:33:42,430
And if we look back at
what we were talking about,

587
00:33:42,430 --> 00:33:44,310
you get p to the
fourth log p squared.

588
00:33:44,310 --> 00:33:46,770
So there's a p to the fourth
multiplying the log p squared.

589
00:33:46,770 --> 00:33:48,520
So you're not seeing
log p squared blow up

590
00:33:48,520 --> 00:33:51,280
because it's got so many
powers of p multiplying it.

591
00:33:57,100 --> 00:34:00,040
Although our focus
is on formalism,

592
00:34:00,040 --> 00:34:03,040
I have to at least give you one
example of something predictive

593
00:34:03,040 --> 00:34:04,780
and phenomenological.

594
00:34:04,780 --> 00:34:12,219
So pi pi scattering is a nice
example of phenomenology.

595
00:34:12,219 --> 00:34:14,800
And it's particularly
nice if you look at it

596
00:34:14,800 --> 00:34:17,770
below an elastic threshold.

597
00:34:17,770 --> 00:34:19,769
So you just have
elastic scattering.

598
00:34:25,870 --> 00:34:28,795
So you could look at
pi pi goes to 4 pi.

599
00:34:28,795 --> 00:34:30,420
And that's something
you could actually

600
00:34:30,420 --> 00:34:32,760
look at in chiral
perturbation theory.

601
00:34:32,760 --> 00:34:37,440
But if we look at just
pi pi goes to pi pi

602
00:34:37,440 --> 00:34:40,739
where we have not enough
energy to produce 4 pions,

603
00:34:40,739 --> 00:34:44,500
then the scattering is
particularly simple.

604
00:34:44,500 --> 00:34:48,449
It's just described by
an S matrix, which we can

605
00:34:48,449 --> 00:34:50,969
enumerate channel by channel.

606
00:34:50,969 --> 00:35:04,680
And it's just a phase where
this L is a partial wave phase.

607
00:35:04,680 --> 00:35:05,700
And the i is an isospin.

608
00:35:08,820 --> 00:35:12,800
So for each isospin and for
each angular momentum state,

609
00:35:12,800 --> 00:35:13,800
we get different phases.

610
00:35:13,800 --> 00:35:17,190
But that's encoding
all the scattering.

611
00:35:17,190 --> 00:35:19,490
That's like doing
non-relativistic scattering

612
00:35:19,490 --> 00:35:24,230
or very simple quantum
mechanical scattering.

613
00:35:35,657 --> 00:35:37,740
And when you have an elastic
scattering like that,

614
00:35:37,740 --> 00:35:44,880
there's something called the
effective range expansion,

615
00:35:44,880 --> 00:35:47,875
which is a derivative expansion
for the phase shift delta.

616
00:35:50,540 --> 00:35:52,500
And if you do it for an
arbitrary partial wave,

617
00:35:52,500 --> 00:35:54,350
this is how it looks.

618
00:35:54,350 --> 00:35:56,060
So this is, again,
something that you

619
00:35:56,060 --> 00:36:00,410
would find in the discussion
of scattering theory in quantum

620
00:36:00,410 --> 00:36:01,418
mechanics.

621
00:36:04,700 --> 00:36:06,500
And you have a
derivative expansion

622
00:36:06,500 --> 00:36:09,860
of this quantity p to
the 2L plus 1 cotan

623
00:36:09,860 --> 00:36:13,070
of that phase shift.

624
00:36:13,070 --> 00:36:19,790
And the a's here and the
r0 depend on what channel

625
00:36:19,790 --> 00:36:22,880
you're talking about.

626
00:36:22,880 --> 00:36:26,150
And if we actually just
take the fact that we can--

627
00:36:26,150 --> 00:36:28,220
so this is a description
from quantum mechanics.

628
00:36:28,220 --> 00:36:29,990
This is true irrespective
of what theory

629
00:36:29,990 --> 00:36:32,750
you're talking about as
long as you don't have

630
00:36:32,750 --> 00:36:34,802
other channels you can produce.

631
00:36:34,802 --> 00:36:37,010
If you talk about it from
chiral perturbation theory,

632
00:36:37,010 --> 00:36:40,790
though, we can just calculate
pi pi goes to pi pi.

633
00:36:40,790 --> 00:36:46,520
And if you do that,
which [? Weinberg ?] did,

634
00:36:46,520 --> 00:36:49,175
then you just get results
for these coefficients.

635
00:36:53,708 --> 00:36:55,500
So I'll just quote a
couple of them to you.

636
00:37:12,200 --> 00:37:14,470
So you just get
parameter-free results,

637
00:37:14,470 --> 00:37:16,220
parameter-free in the
sense that they just

638
00:37:16,220 --> 00:37:18,560
involve M pi and this thing f.

639
00:37:18,560 --> 00:37:21,490
This thing f is actually
measured by pi on decay,

640
00:37:21,490 --> 00:37:23,390
so it's not an unknown.

641
00:37:23,390 --> 00:37:25,490
And then from that,
at lowest order

642
00:37:25,490 --> 00:37:27,260
in chiral perturbation
theory, you

643
00:37:27,260 --> 00:37:30,800
have no parameters once
you fixed M pi and f pi.

644
00:37:30,800 --> 00:37:32,450
And you can just
make predictions

645
00:37:32,450 --> 00:37:35,690
for these scattering lengths,
and those are the predictions.

646
00:37:35,690 --> 00:37:44,570
And they're parameter-free in
the sense that I just said.

647
00:37:47,520 --> 00:37:49,340
OK, so that gives
you some of the idea

648
00:37:49,340 --> 00:37:51,460
of what chiral perturbation
theory can do for you.

649
00:37:58,330 --> 00:38:01,968
OK, so let's talk about
back to formalism.

650
00:38:01,968 --> 00:38:03,760
There's many more
phenomenological examples

651
00:38:03,760 --> 00:38:06,880
you could do, but
that's not our focus.

652
00:38:06,880 --> 00:38:12,820
Back to formalism and
back to this general power

653
00:38:12,820 --> 00:38:25,020
counting discussion, so let's
consider an arbitrary diagram

654
00:38:25,020 --> 00:38:27,300
in this theory, this
chiral perturbation theory.

655
00:38:32,340 --> 00:38:34,470
And let's enumerate some
pieces of that diagram.

656
00:38:37,600 --> 00:38:40,550
So we'll say that it has
some number of vertices.

657
00:38:40,550 --> 00:38:42,360
We just count how
many times I've

658
00:38:42,360 --> 00:38:45,660
inserted vertices
from the Lagrangians,

659
00:38:45,660 --> 00:38:48,360
and I'll call that NV.

660
00:38:48,360 --> 00:38:49,755
Some number of internal lines--

661
00:38:55,410 --> 00:38:56,420
call NI.

662
00:38:56,420 --> 00:38:58,005
Some number of external lines--

663
00:39:02,730 --> 00:39:04,350
the external lines
are all pions.

664
00:39:08,700 --> 00:39:10,695
That's what our
theory is describing.

665
00:39:10,695 --> 00:39:14,920
It could be kaons and etas
if we're doing SU(3)--

666
00:39:14,920 --> 00:39:16,290
and. then some number of loops.

667
00:39:18,890 --> 00:39:22,510
So we have an integer associated
with each of these things.

668
00:39:22,510 --> 00:39:24,000
And when I talk
about vertices, I

669
00:39:24,000 --> 00:39:25,920
don't want to restrict myself
just to the leading order

670
00:39:25,920 --> 00:39:26,420
vertices.

671
00:39:26,420 --> 00:39:29,310
I want to talk about also
these Li's and higher order

672
00:39:29,310 --> 00:39:31,030
vertices as well.

673
00:39:31,030 --> 00:39:35,190
So let me have a notation for
that, where I take this integer

674
00:39:35,190 --> 00:39:37,560
NV, which counts all
vertices, and split it

675
00:39:37,560 --> 00:39:39,480
into pieces that count
the vertices at each

676
00:39:39,480 --> 00:39:42,600
of those different orders
in the expansion in p.

677
00:39:51,280 --> 00:39:54,100
So Nn is the number
of vertices that

678
00:39:54,100 --> 00:40:01,195
are order p to the n or M pi
to the n or combinations of p

679
00:40:01,195 --> 00:40:07,230
and M pi to the n, but
n of them, all right?

680
00:40:07,230 --> 00:40:08,790
So hopefully, that's clear.

681
00:40:08,790 --> 00:40:11,030
So if I have two insertions
of the leading order

682
00:40:11,030 --> 00:40:13,580
Lagrangian and one insertion
of the sub-leading,

683
00:40:13,580 --> 00:40:16,910
then n sub 0 would be 2.

684
00:40:16,910 --> 00:40:18,710
And N sub 1 would be 1.

685
00:40:18,710 --> 00:40:20,677
And the total would be 3.

686
00:40:20,677 --> 00:40:22,010
That's what this notation means.

687
00:40:40,890 --> 00:40:42,540
So we'll assume we're
using a regulator

688
00:40:42,540 --> 00:40:43,830
like dimensional regularization.

689
00:40:43,830 --> 00:40:45,900
So we don't have to worry
about the regulator messing up

690
00:40:45,900 --> 00:40:46,525
power counting.

691
00:40:46,525 --> 00:40:48,210
And basically,
that means we could

692
00:40:48,210 --> 00:40:52,000
ignore the regulator as far as
this discussion is concerned.

693
00:40:52,000 --> 00:40:52,860
And we just count.

694
00:40:57,570 --> 00:41:00,462
And effectively, we can
just count mass dimension.

695
00:41:05,190 --> 00:41:06,755
So we'll count
lambda chi factors.

696
00:41:09,700 --> 00:41:10,950
Let's think about it that way.

697
00:41:13,920 --> 00:41:21,450
And we'll think about counting
lambda chi factors for a matrix

698
00:41:21,450 --> 00:41:27,255
element that I'll call curly
M, which has NE external pions.

699
00:41:30,080 --> 00:41:32,550
So NE lines are
poking out of it.

700
00:41:38,740 --> 00:41:42,180
So then if we look
at the vertices,

701
00:41:42,180 --> 00:41:45,450
we can count how many factors
of lambda chi there is.

702
00:41:45,450 --> 00:41:47,010
And I'm counting
all dimensionful

703
00:41:47,010 --> 00:41:49,680
and turning all dimensionful
things into lambda chi.

704
00:41:55,320 --> 00:42:02,820
And so each different order in
N gets a number of lambda chis

705
00:42:02,820 --> 00:42:05,910
because we even saw that
already in the examples

706
00:42:05,910 --> 00:42:08,730
we treated where we got f
squared from the leading order.

707
00:42:08,730 --> 00:42:11,250
But from the
sub-leading, we got Li.

708
00:42:11,250 --> 00:42:14,110
And that was dimensionless.

709
00:42:14,110 --> 00:42:18,540
So for n equals 2, which
is the leading order here,

710
00:42:18,540 --> 00:42:20,200
leading order was p squared.

711
00:42:20,200 --> 00:42:23,840
I said that not quite
right a minute ago.

712
00:42:23,840 --> 00:42:27,100
So lowest order is p
squared n equals 2.

713
00:42:27,100 --> 00:42:28,020
That's L0.

714
00:42:30,650 --> 00:42:31,880
That had an f squared.

715
00:42:31,880 --> 00:42:35,960
And that comes out, if I just
have 4 minus 2, that's 2.

716
00:42:35,960 --> 00:42:42,250
n equals 4, that was giving our
Li's, which were dimensionless,

717
00:42:42,250 --> 00:42:42,750
OK?

718
00:42:42,750 --> 00:42:44,208
So you can see the
formula working.

719
00:42:44,208 --> 00:42:47,440
And if we went to higher orders
in the derivative and chiral

720
00:42:47,440 --> 00:42:49,690
expansion, then we'd
start getting lambda chis

721
00:42:49,690 --> 00:42:50,975
in the denominator.

722
00:42:53,650 --> 00:42:56,160
So this is just counting
from the vertices, which

723
00:42:56,160 --> 00:42:57,900
you should think
of as counting just

724
00:42:57,900 --> 00:43:00,660
from the pre-factors
in the Lagrangian.

725
00:43:00,660 --> 00:43:13,290
There's also f's that
come with the pions

726
00:43:13,290 --> 00:43:18,340
because every factor of the pion
field comes with a factor of f.

727
00:43:18,340 --> 00:43:19,980
You always have pi over f.

728
00:43:19,980 --> 00:43:25,200
And I'm turning f into 4
pi f for this discussion.

729
00:43:25,200 --> 00:43:26,855
I'm not worrying
about the 4 pis.

730
00:43:26,855 --> 00:43:28,230
You could do a
more fancy version

731
00:43:28,230 --> 00:43:29,897
of this where you
worry about the 4 pis,

732
00:43:29,897 --> 00:43:33,700
but let's just focus
on the dimensions.

733
00:43:33,700 --> 00:43:35,890
So if you have an
internal line, that's

734
00:43:35,890 --> 00:43:37,660
a contraction of 2 pion fields.

735
00:43:37,660 --> 00:43:38,640
So that gets 2f's.

736
00:43:38,640 --> 00:43:40,687
And the external line
is just 1 pion field,

737
00:43:40,687 --> 00:43:41,520
so that just gets 1.

738
00:43:49,000 --> 00:43:51,910
Topologically, these different
things that we enumerated

739
00:43:51,910 --> 00:43:55,186
are not unrelated.

740
00:43:55,186 --> 00:43:57,760
So the order identity
tells us that the number

741
00:43:57,760 --> 00:44:01,450
of internal lines is
the number of loops

742
00:44:01,450 --> 00:44:04,330
plus the number of
vertices minus 1.

743
00:44:06,880 --> 00:44:08,710
And so we use that
to get rid of NI.

744
00:44:17,190 --> 00:44:19,230
So then we can just put
these things together.

745
00:44:34,395 --> 00:44:38,520
So if I get rid of NI here
and I replace it by NL--

746
00:44:38,520 --> 00:44:40,050
so that's that term--

747
00:44:40,050 --> 00:44:44,940
I replace it by the sum over
the Nn's which are the vertices.

748
00:44:44,940 --> 00:44:47,340
And then this gives me a plus 2.

749
00:44:51,600 --> 00:44:52,920
OK.

750
00:44:52,920 --> 00:44:56,295
Now, that's not the dimension
of the left-hand side.

751
00:44:56,295 --> 00:44:58,170
That's just the dimension
of the ingredients.

752
00:44:58,170 --> 00:45:00,555
There's also things that are
coming from factors of M pi

753
00:45:00,555 --> 00:45:02,940
or factors of p.

754
00:45:02,940 --> 00:45:07,170
And let me just call that
something, E to the D

755
00:45:07,170 --> 00:45:10,290
where D is just some
number, integer,

756
00:45:10,290 --> 00:45:17,420
and then some function of
logarithms of p over mu

757
00:45:17,420 --> 00:45:18,480
or M pi over mu.

758
00:45:18,480 --> 00:45:24,030
So E could be M pi or p.

759
00:45:24,030 --> 00:45:25,772
And for the purpose
of power counting,

760
00:45:25,772 --> 00:45:26,730
I'm not distinguishing.

761
00:45:26,730 --> 00:45:30,840
So let's just call it E
just to have a notation that

762
00:45:30,840 --> 00:45:34,290
could be either M pi or p.

763
00:45:34,290 --> 00:45:36,040
And then there's one
more thing we can do,

764
00:45:36,040 --> 00:45:38,010
which is we can look
at the left-hand side.

765
00:45:38,010 --> 00:45:40,740
And we can say, just by
dimensional analysis,

766
00:45:40,740 --> 00:45:43,920
the matrix element, what
should be its dimension?

767
00:45:43,920 --> 00:45:46,320
And depending on how many
bosons you have sticking out,

768
00:45:46,320 --> 00:45:48,487
your matrix element should
have a certain dimension,

769
00:45:48,487 --> 00:45:50,290
which you could figure out.

770
00:45:50,290 --> 00:45:56,190
And that dimension
is just 4 minus NE.

771
00:45:59,130 --> 00:46:02,820
Two-point function would scale
like p squared, et cetera.

772
00:46:02,820 --> 00:46:06,060
So now, I have different
things that are giving mass

773
00:46:06,060 --> 00:46:08,445
dimensions, the lambda
chis and the E's.

774
00:46:08,445 --> 00:46:11,430
But because I know what the
answer has to be, 4 minus NE,

775
00:46:11,430 --> 00:46:27,280
I can solve for D.

776
00:46:27,280 --> 00:46:31,110
And that's what I got.

777
00:46:31,110 --> 00:46:31,800
OK.

778
00:46:31,800 --> 00:46:34,770
So the answer for D, in order
to get the dimensions right,

779
00:46:34,770 --> 00:46:36,930
we have to compensate for
the number of lambda chis

780
00:46:36,930 --> 00:46:39,690
by factors of M pi and p.

781
00:46:39,690 --> 00:46:42,600
And when we do that, then
we need this many of them

782
00:46:42,600 --> 00:46:45,247
in order to get the
dimensions right.

783
00:46:45,247 --> 00:46:46,830
So one thing you see
from this formula

784
00:46:46,830 --> 00:46:48,782
is that D is greater
than or equal to 2.

785
00:46:48,782 --> 00:46:50,490
That's because these
things are positive.

786
00:46:50,490 --> 00:46:53,370
The Lagrangian starts with n
equals 2 and then goes higher.

787
00:47:00,520 --> 00:47:05,740
And when I add either of these
terms, I cause suppression,

788
00:47:05,740 --> 00:47:06,790
or I stay the same.

789
00:47:16,820 --> 00:47:17,770
So this could be 0.

790
00:47:17,770 --> 00:47:19,850
This could be 0, or
it could be bigger.

791
00:47:19,850 --> 00:47:20,845
But it can be smaller.

792
00:47:27,900 --> 00:47:35,355
So you always get
more E's, which

793
00:47:35,355 --> 00:47:39,005
are M pis or p's, by adding
vertices or adding loops.

794
00:47:39,005 --> 00:47:41,380
So when you looked at the loop
graphs that were built out

795
00:47:41,380 --> 00:47:43,530
of the leading order
Lagrangian, those terms

796
00:47:43,530 --> 00:47:45,780
had this be 0 because n was 2.

797
00:47:45,780 --> 00:47:48,780
But then you got suppression
because we built loop graphs.

798
00:47:48,780 --> 00:47:50,718
So you got suppression
from this term.

799
00:47:50,718 --> 00:47:53,010
And when we looked at the
higher dimensional operators,

800
00:47:53,010 --> 00:47:53,820
there was no loop.

801
00:47:53,820 --> 00:47:55,487
So we didn't have
this term, but then we

802
00:47:55,487 --> 00:47:57,800
had a contribution
from this guy.

803
00:47:57,800 --> 00:47:59,250
But those were trading off.

804
00:48:04,700 --> 00:48:07,283
So having this is,
effectively, part

805
00:48:07,283 --> 00:48:09,950
of what we need in order to make
sure the theory is well-defined

806
00:48:09,950 --> 00:48:12,350
because it tells us how
to organize the theory

807
00:48:12,350 --> 00:48:15,440
and what parts of
the theory we need

808
00:48:15,440 --> 00:48:18,560
to worry about if we
want a certain accuracy

809
00:48:18,560 --> 00:48:21,628
and what parts we can ignore.

810
00:48:21,628 --> 00:48:23,420
We need to know that
we don't need to think

811
00:48:23,420 --> 00:48:24,740
about two-loop diagrams.

812
00:48:24,740 --> 00:48:28,490
And this tells us that we don't.

813
00:48:28,490 --> 00:48:29,422
Yeah.

814
00:48:29,422 --> 00:48:31,880
AUDIENCE: I didn't understand
[? where you ?] [INAUDIBLE]..

815
00:48:31,880 --> 00:48:33,547
IAIN STEWART: Yeah,
let me say it again.

816
00:48:33,547 --> 00:48:38,150
So we figured out the lambda
chis by this stuff up here.

817
00:48:38,150 --> 00:48:39,950
Then I said, let
there be an arbitrary

818
00:48:39,950 --> 00:48:44,300
E to the D, some
parameter which we haven't

819
00:48:44,300 --> 00:48:45,860
figured out anything yet.

820
00:48:45,860 --> 00:48:49,010
But I know by just
what possibly this

821
00:48:49,010 --> 00:48:51,840
could depend on that it could
also depend on an M pis or pis.

822
00:48:51,840 --> 00:48:55,160
So let me put some polynomial
power in and then some function

823
00:48:55,160 --> 00:48:56,770
that could be non-polynomial.

824
00:48:56,770 --> 00:48:58,437
AUDIENCE: But where
would that-- can you

825
00:48:58,437 --> 00:49:00,738
give an example of a
calculation where I

826
00:49:00,738 --> 00:49:02,030
would see exactly what that is?

827
00:49:02,030 --> 00:49:02,900
IAIN STEWART:
Yeah, so if you did

828
00:49:02,900 --> 00:49:04,820
this loop calculation
we did a minute ago,

829
00:49:04,820 --> 00:49:06,530
we got a p to the fourth.

830
00:49:06,530 --> 00:49:07,820
And then D would be 4.

831
00:49:07,820 --> 00:49:10,040
AUDIENCE: And those
p's came from where?

832
00:49:10,040 --> 00:49:13,770
IAIN STEWART: Oh, they came
from the momentum going--

833
00:49:13,770 --> 00:49:16,580
so if you looked
at this diagram,

834
00:49:16,580 --> 00:49:18,060
there's p coming in here.

835
00:49:18,060 --> 00:49:21,020
And then it goes
into the loop, right?

836
00:49:21,020 --> 00:49:22,730
And this is
derivatively coupled,

837
00:49:22,730 --> 00:49:24,545
so you get p's in the numerator.

838
00:49:24,545 --> 00:49:27,597
AUDIENCE: But that wasn't
taken care of with [INAUDIBLE]..

839
00:49:27,597 --> 00:49:28,430
IAIN STEWART: Right.

840
00:49:28,430 --> 00:49:30,380
Because NV is just
counting lambda chis,

841
00:49:30,380 --> 00:49:33,335
which are constants,
not the p's.

842
00:49:33,335 --> 00:49:37,088
It's just counting the
constants, the f's.

843
00:49:37,088 --> 00:49:37,630
AUDIENCE: OK.

844
00:49:37,630 --> 00:49:38,422
IAIN STEWART: Yeah.

845
00:49:38,422 --> 00:49:39,008
AUDIENCE: OK.

846
00:49:39,008 --> 00:49:39,800
IAIN STEWART: Yeah.

847
00:49:39,800 --> 00:49:44,213
And then you equate it to this,
and then you get the D. Yeah.

848
00:49:44,213 --> 00:49:45,880
So you could have
tried to set things up

849
00:49:45,880 --> 00:49:48,730
by thinking about counting p's
instead of counting lambdas.

850
00:49:48,730 --> 00:49:50,470
But it's-- yeah, anyway.

851
00:49:54,950 --> 00:49:58,190
All right, so what people
often refer to this

852
00:49:58,190 --> 00:50:16,450
is they say it's p
counting because you're

853
00:50:16,450 --> 00:50:18,400
counting momenta.

854
00:50:18,400 --> 00:50:21,820
And that includes p or M
pi, but sometimes people

855
00:50:21,820 --> 00:50:24,160
call it p counting.

856
00:50:24,160 --> 00:50:29,080
And just to do some examples,
when we had the lowest order

857
00:50:29,080 --> 00:50:33,760
Lagrangian, this guy
comes out as 2 powers of p

858
00:50:33,760 --> 00:50:35,421
because there's two derivatives.

859
00:50:38,650 --> 00:50:42,940
And that, in our
formula, is just the fact

860
00:50:42,940 --> 00:50:46,150
that D is 2 for that.

861
00:50:46,150 --> 00:50:51,520
When we thought about this loop
with leading order Lagrangians,

862
00:50:51,520 --> 00:50:56,260
which are scaling
like p squared,

863
00:50:56,260 --> 00:50:58,183
we ended up having D equals 4.

864
00:50:58,183 --> 00:51:00,100
And the way that that
comes out of the formula

865
00:51:00,100 --> 00:51:03,160
is because of this
loop term, which is 1.

866
00:51:03,160 --> 00:51:06,070
And then there's 2 plus 2 is 4.

867
00:51:10,270 --> 00:51:15,478
And this guy is 4 because
of the explicit suppression.

868
00:51:21,330 --> 00:51:22,220
OK.

869
00:51:22,220 --> 00:51:25,340
So the theory is organized
as an expansion of this p.

870
00:51:30,520 --> 00:51:32,470
All right, so I
want to come back

871
00:51:32,470 --> 00:51:35,080
to SU(3) partly
because the problem

872
00:51:35,080 --> 00:51:37,570
that I gave you is in SU(3).

873
00:51:37,570 --> 00:51:41,230
So we'll do some discussion
of the SU(3) case.

874
00:51:53,400 --> 00:51:55,510
I'll go into a few things
in a little more detail

875
00:51:55,510 --> 00:51:58,570
than we did for
SU(2), where we were

876
00:51:58,570 --> 00:52:00,070
focusing on more formal things.

877
00:52:02,670 --> 00:52:06,280
So SU(3) would have gamma
matrices instead of the Pauli

878
00:52:06,280 --> 00:52:08,470
matrices.

879
00:52:08,470 --> 00:52:10,270
There's two bases
that you can use.

880
00:52:10,270 --> 00:52:12,100
You can either use
this basis, which

881
00:52:12,100 --> 00:52:15,040
is like enumerated
1 to 8, or you

882
00:52:15,040 --> 00:52:16,820
could use the charged basis.

883
00:52:16,820 --> 00:52:19,630
And if you use
the charge basis--

884
00:52:19,630 --> 00:52:22,030
and often you write it
out as a matrix like this.

885
00:52:28,790 --> 00:52:31,100
On your problem set, you're
free to pick which basis

886
00:52:31,100 --> 00:52:31,880
you want to use.

887
00:52:37,220 --> 00:52:39,950
It may be that one or
the other is easier,

888
00:52:39,950 --> 00:52:41,962
but I can't even tell
you which one is easier

889
00:52:41,962 --> 00:52:42,920
since I don't remember.

890
00:52:46,105 --> 00:52:48,185
So sometimes one or the
other is easier to use.

891
00:52:48,185 --> 00:52:50,060
And you have a freedom
of what basis to pick.

892
00:52:54,520 --> 00:53:01,960
If you expand, in this case,
the trace of sigma Mq dagger

893
00:53:01,960 --> 00:53:08,890
plus Mq sigma dagger, which
is that term that had a V0,

894
00:53:08,890 --> 00:53:10,870
then that gives
mass to the mesons,

895
00:53:10,870 --> 00:53:11,935
as it did for the pions.

896
00:53:18,308 --> 00:53:19,850
Because the symmetry
group is bigger,

897
00:53:19,850 --> 00:53:21,700
you get more predictions.

898
00:53:21,700 --> 00:53:25,060
Here, you get predictions
for the kaon masses.

899
00:53:25,060 --> 00:53:35,120
And you get things like the
fact that the neutral kaons have

900
00:53:35,120 --> 00:53:41,180
massless order Md plus
Ms. And you get things

901
00:53:41,180 --> 00:53:46,950
like eta pi 0 mixing, where
there's a mixing matrix.

902
00:53:46,950 --> 00:53:49,270
So if you look at the
masses of eta and pi 0,

903
00:53:49,270 --> 00:53:51,050
they're actually non-diagonal.

904
00:53:51,050 --> 00:53:52,890
So for the eta in
the pi 0 system,

905
00:53:52,890 --> 00:53:54,980
you actually get a matrix.

906
00:53:54,980 --> 00:53:56,315
So M squared is a matrix.

907
00:53:59,430 --> 00:54:03,800
So for example, for the pi
0, it was just M up plus Md.

908
00:54:06,980 --> 00:54:11,060
But then once you're in SU(3),
there's actually a mixing term.

909
00:54:11,060 --> 00:54:13,895
And there's like an M up
minus M down term here

910
00:54:13,895 --> 00:54:20,220
that is mixing between, and
then same thing over here.

911
00:54:20,220 --> 00:54:22,190
So the etas and the pi
0s actually are mixing.

912
00:54:22,190 --> 00:54:25,310
And there's something
in this n tree.

913
00:54:25,310 --> 00:54:27,560
And the mixing is isospin
violating in the sense

914
00:54:27,560 --> 00:54:29,050
that it's M up minus M down.

915
00:54:29,050 --> 00:54:33,230
So it's a small effect,
but this is something

916
00:54:33,230 --> 00:54:36,440
that you can predict from
chiral Lagrangian, something

917
00:54:36,440 --> 00:54:39,560
about because is describes
isospin violating effects

918
00:54:39,560 --> 00:54:40,550
from the quark masses.

919
00:54:46,100 --> 00:54:48,320
So the reason that
I mention that is

920
00:54:48,320 --> 00:54:50,810
because, often when you
do calculations, keeping

921
00:54:50,810 --> 00:54:53,420
track of Mu's,
Md's, and Ms's all

922
00:54:53,420 --> 00:54:56,820
as separate independent
parameters is a little much.

923
00:54:56,820 --> 00:54:58,620
And so you want to
make an approximation.

924
00:54:58,620 --> 00:55:01,430
And so if you make
an approximation that

925
00:55:01,430 --> 00:55:03,110
ignores isospin violation--

926
00:55:09,090 --> 00:55:15,400
so we often ignore
isospin violation.

927
00:55:15,400 --> 00:55:17,680
Isospin violation is very small.

928
00:55:17,680 --> 00:55:19,720
And if you remember,
for the SU(3) case,

929
00:55:19,720 --> 00:55:23,460
you're expanding in Ms
over lambda QCD or Mk

930
00:55:23,460 --> 00:55:24,180
over lambda chi.

931
00:55:24,180 --> 00:55:25,440
So it's not a great expansion.

932
00:55:25,440 --> 00:55:27,840
You have something like a third.

933
00:55:27,840 --> 00:55:30,180
So ignoring isospin
is perfectly valid

934
00:55:30,180 --> 00:55:31,820
if you're expanding in a third.

935
00:55:31,820 --> 00:55:33,570
So basically, because
there is a hierarchy

936
00:55:33,570 --> 00:55:36,792
between the strange quark
mass and the down and the up,

937
00:55:36,792 --> 00:55:39,000
you want to focus on places
where you get the largest

938
00:55:39,000 --> 00:55:39,570
corrections.

939
00:55:39,570 --> 00:55:41,975
And one way of making
an approximation that

940
00:55:41,975 --> 00:55:44,100
allows you to do that is
to ignore isospin and take

941
00:55:44,100 --> 00:55:47,400
Mu equal to Md.

942
00:55:47,400 --> 00:55:52,200
So if we take Mu at
Md to be some M hat--

943
00:55:52,200 --> 00:55:55,800
which if you want an exact
definition, you could say

944
00:55:55,800 --> 00:55:58,185
it's the average.

945
00:55:58,185 --> 00:55:59,868
And you'd drop the difference.

946
00:56:04,350 --> 00:56:06,750
And then you can think
of the strange quark mass

947
00:56:06,750 --> 00:56:08,280
as being somewhat
bigger than M hat.

948
00:56:14,450 --> 00:56:17,175
That's an approximation that
you can use on your problem set,

949
00:56:17,175 --> 00:56:17,675
for example.

950
00:56:23,760 --> 00:56:26,420
So let me write out
here with all the terms

951
00:56:26,420 --> 00:56:28,130
in the chiral Lagrangian is.

952
00:56:28,130 --> 00:56:32,120
And I'll do it for
a case where we

953
00:56:32,120 --> 00:56:37,582
include in our chiral Lagrangian
one other type of coupling,

954
00:56:37,582 --> 00:56:39,290
which is this left-handed
current that we

955
00:56:39,290 --> 00:56:42,000
talked about last time.

956
00:56:42,000 --> 00:56:44,665
So we talked about a spurion
analysis for the chi term.

957
00:56:44,665 --> 00:56:46,040
And I said you
could do something

958
00:56:46,040 --> 00:56:49,160
similar to a couple in
a left-handed current.

959
00:56:49,160 --> 00:56:53,120
And we had this,
where D mu sigma

960
00:56:53,120 --> 00:56:57,680
was partial mu sigma times
the left-handed current sigma.

961
00:56:57,680 --> 00:56:59,180
So we thought of
putting it together

962
00:56:59,180 --> 00:57:00,950
into a coherent derivative.

963
00:57:00,950 --> 00:57:03,650
And if I do that, it modifies
the leading order Lagrangian

964
00:57:03,650 --> 00:57:06,465
and just makes these partials
into covariant derivatives.

965
00:57:15,250 --> 00:57:19,760
Power counting, for
counting purposes,

966
00:57:19,760 --> 00:57:22,250
you count sigmas of order 1.

967
00:57:22,250 --> 00:57:27,320
You count D mu sigma as order
p, which means you count

968
00:57:27,320 --> 00:57:29,700
L mu, the source, as order p.

969
00:57:29,700 --> 00:57:31,070
This is a left-handed source.

970
00:57:33,620 --> 00:57:37,513
And you count chis and
Mq's as of order p squared.

971
00:57:37,513 --> 00:57:39,430
That's just repeating
what we've already said.

972
00:57:44,640 --> 00:57:46,950
And these higher order
terms, which where L's, we

973
00:57:46,950 --> 00:57:48,250
can then enumerate.

974
00:57:52,850 --> 00:57:57,650
And I now just use
D's, covariant D's

975
00:57:57,650 --> 00:58:00,590
when I write them down.

976
00:58:00,590 --> 00:58:03,200
I'm now in SU(3).

977
00:58:03,200 --> 00:58:06,860
And it turns out that one of the
relations that we used in SU(2)

978
00:58:06,860 --> 00:58:08,960
doesn't carry over to SU(3).

979
00:58:08,960 --> 00:58:14,930
So if I just think of guys with
four covariant derivatives,

980
00:58:14,930 --> 00:58:18,820
it turns out that there's
one more operator there.

981
00:58:22,400 --> 00:58:27,350
So there's those two, which
we talked about, L1 and L2.

982
00:58:27,350 --> 00:58:30,620
And then there's also L3.

983
00:58:30,620 --> 00:58:32,180
So we can't get rid of this guy.

984
00:58:35,540 --> 00:58:37,910
I'm always writing sigmas
next to sigma daggers because

985
00:58:37,910 --> 00:58:41,240
of the chiral transformation.

986
00:58:41,240 --> 00:58:43,967
I'm also imposing
parity, though I'm not

987
00:58:43,967 --> 00:58:45,800
going to spend much
time talking about that.

988
00:58:49,780 --> 00:58:52,000
And really I want to
enumerate, also, for you

989
00:58:52,000 --> 00:58:59,090
some of the terms that involve
the quark mass, the chi guy.

990
00:58:59,090 --> 00:59:01,860
So you could have a guy
that's a cross-term.

991
00:59:01,860 --> 00:59:06,630
This would take care of the
renormalization of things

992
00:59:06,630 --> 00:59:09,320
like p squared and pi
squared times 1 of epsilon.

993
00:59:13,560 --> 00:59:16,290
That's L4, L5.

994
00:59:41,610 --> 00:59:44,115
We could also have the quark
mass type term just squared.

995
00:59:48,657 --> 00:59:49,990
So take the trace and square it.

996
00:59:49,990 --> 00:59:52,560
That's L6.

997
00:59:52,560 --> 00:59:54,570
It turns out that we
could also build something

998
00:59:54,570 --> 00:59:56,880
with the right parity by just
having the difference instead

999
00:59:56,880 --> 00:59:57,380
of the sum.

1000
00:59:57,380 --> 00:59:58,918
And that's a different operator.

1001
01:00:05,610 --> 01:00:06,585
So that's L7.

1002
01:00:10,080 --> 01:00:11,190
We're going to go up to 9.

1003
01:00:16,200 --> 01:00:18,398
Don't be afraid.

1004
01:00:18,398 --> 01:00:20,190
You can have something
with 2 sigma daggers

1005
01:00:20,190 --> 01:00:25,440
and 2 chis, which is another way
of building a chiral invariant.

1006
01:00:25,440 --> 01:00:30,190
And then for parity,
you need the other way.

1007
01:00:30,190 --> 01:00:33,510
And then finally--
something called

1008
01:00:33,510 --> 01:00:39,030
L9, which involves a
trace that involves

1009
01:00:39,030 --> 01:00:43,042
L mu nu, where L mu nu is built
out of this external current.

1010
01:00:43,042 --> 01:00:45,000
So we can take two of
our covariant derivatives

1011
01:00:45,000 --> 01:00:46,477
and take a commutator.

1012
01:00:51,840 --> 01:00:53,826
That's giving the
final operator.

1013
01:01:10,850 --> 01:01:16,190
So there's a complete basis for
SU(3) for a left-handed current

1014
01:01:16,190 --> 01:01:17,428
in the chi.

1015
01:01:17,428 --> 01:01:19,220
AUDIENCE: [? So does ?]
[? L mu ?] [? nu ?]

1016
01:01:19,220 --> 01:01:20,018
[? have a field ?] [INAUDIBLE]
[? associated with ?]

1017
01:01:20,018 --> 01:01:20,180
[INAUDIBLE]?

1018
01:01:20,180 --> 01:01:21,200
IAIN STEWART: Yeah,
I'll write it down.

1019
01:01:21,200 --> 01:01:21,742
AUDIENCE: Oh.

1020
01:01:45,698 --> 01:01:47,240
IAIN STEWART: So if
L mu is something

1021
01:01:47,240 --> 01:01:50,690
that has SU(3) indices, has an
SU(3) matrix hiding inside it,

1022
01:01:50,690 --> 01:01:53,660
then there's a
commutator term as well.

1023
01:01:53,660 --> 01:01:57,590
And then it's just
this combination.

1024
01:01:57,590 --> 01:02:00,998
I also use the
equation of motion.

1025
01:02:00,998 --> 01:02:03,540
And I didn't talk about it, but
I use the equation of motion.

1026
01:02:03,540 --> 01:02:05,370
And I used SU(3) relations.

1027
01:02:05,370 --> 01:02:06,950
Much as I talked
about for SU(2),

1028
01:02:06,950 --> 01:02:09,448
there's some SU(3)
relations that survive.

1029
01:02:09,448 --> 01:02:11,240
And I got rid of some
operators doing that.

1030
01:02:16,200 --> 01:02:19,620
Now, you could say, well, I
have this SU(3) and SU(2),

1031
01:02:19,620 --> 01:02:21,990
so why don't I try
to relate them?

1032
01:02:21,990 --> 01:02:23,340
One of them has a kaon.

1033
01:02:23,340 --> 01:02:24,420
The other one doesn't.

1034
01:02:24,420 --> 01:02:27,000
The one without the kaon
thinks about the kaon

1035
01:02:27,000 --> 01:02:28,298
as a heavy particle.

1036
01:02:28,298 --> 01:02:30,090
The one with the kaon
thinks about the kaon

1037
01:02:30,090 --> 01:02:31,530
as a light particle.

1038
01:02:31,530 --> 01:02:32,520
Those are two theories.

1039
01:02:32,520 --> 01:02:35,820
I could try to make them
match up with each other.

1040
01:02:35,820 --> 01:02:37,890
And that's something that
you can do, actually.

1041
01:02:41,860 --> 01:02:48,300
So there's a little
bit of a lesson there.

1042
01:02:48,300 --> 01:02:50,220
So that was why I
want to mention it.

1043
01:02:56,790 --> 01:02:58,540
So SU(2) and SU(3)
seem like they're

1044
01:02:58,540 --> 01:02:59,690
describing similar physics.

1045
01:02:59,690 --> 01:03:01,560
They both could describe pions.

1046
01:03:01,560 --> 01:03:03,130
They both have pions in them.

1047
01:03:03,130 --> 01:03:04,090
But the SU(3) has more.

1048
01:03:04,090 --> 01:03:05,770
It's got the kaon.

1049
01:03:05,770 --> 01:03:08,500
That means that, in
the SU(2) theory,

1050
01:03:08,500 --> 01:03:13,540
the kaon is in the
coefficients, OK?

1051
01:03:13,540 --> 01:03:16,090
So if you do a
correspondence, you

1052
01:03:16,090 --> 01:03:18,650
get relations that
are like this.

1053
01:03:18,650 --> 01:03:20,260
So the 2 here means SU(2).

1054
01:03:25,380 --> 01:03:28,180
And where I don't put any
subscript, it's SU(3).

1055
01:03:32,080 --> 01:03:35,410
And something
that-- this is a 96.

1056
01:03:44,140 --> 01:03:48,010
And if you really do that, what
I said, compare observables,

1057
01:03:48,010 --> 01:03:50,710
you get relations like
this one where you actually

1058
01:03:50,710 --> 01:03:54,190
see the kaon is
showing up on this side

1059
01:03:54,190 --> 01:03:56,470
and is being encoded in
coefficients in the SU(2)

1060
01:03:56,470 --> 01:03:58,190
theory.

1061
01:03:58,190 --> 01:03:59,868
So what you think of
as the coefficients

1062
01:03:59,868 --> 01:04:01,660
in your chiral theory
depends on the matter

1063
01:04:01,660 --> 01:04:03,783
that you've put in,
including things

1064
01:04:03,783 --> 01:04:05,950
like what particles like
the kaon, what group you're

1065
01:04:05,950 --> 01:04:06,640
talking about.

1066
01:04:09,500 --> 01:04:18,680
This is an explicit example of
the kaon being the coefficients

1067
01:04:18,680 --> 01:04:21,890
if we use SU(2).

1068
01:04:25,690 --> 01:04:31,510
OK, so just like in SU(2),
we have to go through

1069
01:04:31,510 --> 01:04:34,420
a renormalization of the Li's.

1070
01:04:39,280 --> 01:04:41,890
And you can think of
that by writing a formula

1071
01:04:41,890 --> 01:04:42,520
like this one.

1072
01:04:45,410 --> 01:04:47,690
Bare Li is equal to
some renormalized one,

1073
01:04:47,690 --> 01:04:51,800
which I put a bar on top
of and then a counter-term.

1074
01:04:51,800 --> 01:04:55,850
And the counter-term
will be some coefficient,

1075
01:04:55,850 --> 01:05:01,760
some number of 4 pis,
and then some epsilons.

1076
01:05:01,760 --> 01:05:05,570
And just like in
gauge theory, we

1077
01:05:05,570 --> 01:05:09,070
get rid of the Euler
gamma and the log 4 pi.

1078
01:05:09,070 --> 01:05:11,330
We have an Ms bar
type definition.

1079
01:05:11,330 --> 01:05:13,790
And in chiral
perturbation theory,

1080
01:05:13,790 --> 01:05:19,010
people often take an extra
1 along with the ride

1081
01:05:19,010 --> 01:05:21,510
just because they're allowed to.

1082
01:05:21,510 --> 01:05:23,510
Because it tends to show
up in the kind of loops

1083
01:05:23,510 --> 01:05:24,530
that you encounter.

1084
01:05:24,530 --> 01:05:28,940
So sometimes that's
included, sometimes it's not.

1085
01:05:28,940 --> 01:05:35,780
So for example, as an
example of a diagram,

1086
01:05:35,780 --> 01:05:38,060
you could think about this one.

1087
01:05:38,060 --> 01:05:42,022
And that does cause
mass renormalization

1088
01:05:42,022 --> 01:05:42,980
for the physical boson.

1089
01:05:48,840 --> 01:05:51,120
So if you think about
our lowest order relation

1090
01:05:51,120 --> 01:05:57,570
as where we started, let me
write that as M0 squared is

1091
01:05:57,570 --> 01:06:04,320
4V0 over f squared Mu plus Md.

1092
01:06:07,950 --> 01:06:10,060
Sometimes people call this--

1093
01:06:10,060 --> 01:06:18,330
well, making up
some more notation,

1094
01:06:18,330 --> 01:06:21,758
yeah, which I'll
use in a minute.

1095
01:06:21,758 --> 01:06:23,550
And so if you actually
calculate this loop,

1096
01:06:23,550 --> 01:06:25,300
then you get a correction
to that formula.

1097
01:06:25,300 --> 01:06:29,820
This would have be M pi squared,
but here we get a correction.

1098
01:06:29,820 --> 01:06:34,500
If you did it in
SU(2), just to keep

1099
01:06:34,500 --> 01:06:38,760
the formula a little simpler,
then it would look like this.

1100
01:06:43,270 --> 01:06:46,710
So M0 is like the pi n mass, but
the pi n mass at lowest order

1101
01:06:46,710 --> 01:06:47,952
in the chiral expansion.

1102
01:06:47,952 --> 01:06:49,410
So it's not the
physical pi n mass.

1103
01:06:52,320 --> 01:07:01,110
So 2L 4 bar in theory two with
two flavors with two SU(2) plus

1104
01:07:01,110 --> 01:07:04,770
an L5 bar minus 4 n L6 bar--

1105
01:07:04,770 --> 01:07:06,570
so these various
coefficients are coming in

1106
01:07:06,570 --> 01:07:10,883
with some numbers in front,
some combination of them.

1107
01:07:10,883 --> 01:07:12,300
And then there's
some contribution

1108
01:07:12,300 --> 01:07:13,870
from a chiral loop.

1109
01:07:13,870 --> 01:07:19,600
And if I take away that 1,
it's just a chiral logarithm.

1110
01:07:19,600 --> 01:07:28,470
So this is 4 pi f squared log
M0 squared over mu squared.

1111
01:07:31,890 --> 01:07:35,190
There would be an extra term
of just plus 1 times this

1112
01:07:35,190 --> 01:07:38,580
if I hadn't gotten rid of that,
the minus 1 times that, OK?

1113
01:07:38,580 --> 01:07:39,720
So this is from the loop.

1114
01:07:39,720 --> 01:07:43,500
This is from the explicit
dimension operators

1115
01:07:43,500 --> 01:07:46,200
that were of the higher
dimension, some combination

1116
01:07:46,200 --> 01:07:47,700
of them.

1117
01:07:47,700 --> 01:07:51,230
The UV divergence is absorbed
in some combination of them.

1118
01:07:51,230 --> 01:07:53,750
And if you want to figure out
exactly how UV divergences go

1119
01:07:53,750 --> 01:07:56,430
in between this, then you've
got to think of renormalizing

1120
01:07:56,430 --> 01:07:59,470
more than just this diagram.

1121
01:07:59,470 --> 01:08:02,310
And you can generically
think of observables

1122
01:08:02,310 --> 01:08:07,090
as having this kind of
expansion in Mu and Md.

1123
01:08:07,090 --> 01:08:10,440
So if you like, you should think
of M0 here as really just an up

1124
01:08:10,440 --> 01:08:11,462
plus M down.

1125
01:08:11,462 --> 01:08:13,920
This is saying, at lowest order
chiral perturbation theory,

1126
01:08:13,920 --> 01:08:15,720
there's a linear term.

1127
01:08:15,720 --> 01:08:19,439
But then here there's an M0
to the fourth term, which is

1128
01:08:19,439 --> 01:08:21,750
quadratic in the quark masses.

1129
01:08:21,750 --> 01:08:24,779
And there's also a quadratic
term from the loops.

1130
01:08:24,779 --> 01:08:27,660
So you have an expansion
in the quark masses.

1131
01:08:27,660 --> 01:08:32,880
Think of M0 as the quark masses
and M pi as the meson mass.

1132
01:08:32,880 --> 01:08:34,859
I just gave you the
example of the pion mass,

1133
01:08:34,859 --> 01:08:38,910
but this is generically
true for observables

1134
01:08:38,910 --> 01:08:42,210
that you might calculate that
they have this type of result

1135
01:08:42,210 --> 01:08:44,218
where you have an expansion.

1136
01:08:44,218 --> 01:08:47,880
AUDIENCE: Is there any physical
reason for why this mass square

1137
01:08:47,880 --> 01:08:52,390
of the meson space [? would ?]
[? be ?] [INAUDIBLE]??

1138
01:08:52,390 --> 01:08:54,859
IAIN STEWART: So one
way of saying it is--

1139
01:08:54,859 --> 01:08:55,359
yeah.

1140
01:08:55,359 --> 01:08:59,189
So you might think, well, why
is it not M pi equals Mu and Md?

1141
01:08:59,189 --> 01:09:00,910
And the glib way of
saying it is, well,

1142
01:09:00,910 --> 01:09:02,535
if you think about
M pi squared and you

1143
01:09:02,535 --> 01:09:04,450
think about it
having an expansion,

1144
01:09:04,450 --> 01:09:06,990
then it could have a
constant plus linear term

1145
01:09:06,990 --> 01:09:08,117
plus quadratic term.

1146
01:09:08,117 --> 01:09:10,575
You don't have the constant
because of the chiral symmetry.

1147
01:09:10,575 --> 01:09:14,069
So the linear term is the
first thing that's allowed.

1148
01:09:14,069 --> 01:09:16,939
That's one way of saying it.

1149
01:09:16,939 --> 01:09:19,380
I mean, if you think about
it in the bosonic theory,

1150
01:09:19,380 --> 01:09:23,100
we have the Lagrangian
is quadratic in masses,

1151
01:09:23,100 --> 01:09:25,047
right, whereas in
the fermionic theory,

1152
01:09:25,047 --> 01:09:26,130
you have linear in masses.

1153
01:09:26,130 --> 01:09:29,670
And that's also part of
what it had to do with,

1154
01:09:29,670 --> 01:09:33,330
but it's linear combination
of symmetry breaking and that.

1155
01:09:37,779 --> 01:09:41,680
It's allowed, so it happens
is kind of the bottom line.

1156
01:09:41,680 --> 01:09:43,628
But it's allowed
by the symmetries.

1157
01:09:47,319 --> 01:09:47,819
OK.

1158
01:09:47,819 --> 01:09:50,611
And so finally, this is
the example I'm actually

1159
01:09:50,611 --> 01:09:51,569
going to get you to do.

1160
01:09:54,960 --> 01:09:58,410
So I'll have you look on the
problem set at the k constants.

1161
01:09:58,410 --> 01:10:01,270
And they have an analogous
result to that one.

1162
01:10:01,270 --> 01:10:04,470
You'll do the
calculation in SU(3).

1163
01:10:04,470 --> 01:10:07,860
And the kind of result that
you should expect to get

1164
01:10:07,860 --> 01:10:09,450
is something that
looks like this.

1165
01:10:22,340 --> 01:10:24,250
So I'm using this
B0 notation that I

1166
01:10:24,250 --> 01:10:28,900
introduced over there just so I
could write it all on one line.

1167
01:10:33,170 --> 01:10:36,410
And these mus with
the subscript i

1168
01:10:36,410 --> 01:10:39,110
are just some shorthand
for some contributions

1169
01:10:39,110 --> 01:10:39,920
coming from loops.

1170
01:10:51,320 --> 01:10:57,800
And f is the result in L0.

1171
01:11:00,920 --> 01:11:06,940
So f is the parameter
in L0, whereas f pi

1172
01:11:06,940 --> 01:11:10,360
is the physical decay
constant of the pion.

1173
01:11:16,830 --> 01:11:21,240
So one result which is
encoded in this formula

1174
01:11:21,240 --> 01:11:23,160
is that the Lagrangian
parameter is actually

1175
01:11:23,160 --> 01:11:25,200
equal to the decay
constant at lowest order

1176
01:11:25,200 --> 01:11:25,950
in chiral perturbation theory.

1177
01:11:25,950 --> 01:11:27,460
That's something I didn't cover.

1178
01:11:27,460 --> 01:11:29,810
I cover it when I teach
quantum field theory three.

1179
01:11:29,810 --> 01:11:32,880
You can look at my notes
to see that derivation.

1180
01:11:32,880 --> 01:11:36,640
Those of you that have
taken QFT3 from me,

1181
01:11:36,640 --> 01:11:37,902
you've already seen that.

1182
01:11:37,902 --> 01:11:40,110
And basically, what I'm
asking for in the problem set

1183
01:11:40,110 --> 01:11:41,820
is I guide you
with several parts,

1184
01:11:41,820 --> 01:11:45,067
but how to think about
these higher order terms.

1185
01:11:45,067 --> 01:11:46,650
What are the loop
graph contributions?

1186
01:11:46,650 --> 01:11:49,890
How do you get these terms from
the higher order Lagrangians?

1187
01:11:49,890 --> 01:11:51,870
How you put it all together?

1188
01:11:51,870 --> 01:11:55,950
It's a nice example of this use
of chiral perturbation theory.

1189
01:11:55,950 --> 01:11:57,840
And again, we see that
a physical observable,

1190
01:11:57,840 --> 01:12:00,660
which is this decay constant,
has a chiral expansion.

1191
01:12:00,660 --> 01:12:03,330
And the thing that chiral
perturbation theory is actually

1192
01:12:03,330 --> 01:12:06,720
doing is allowing you
to predict both the form

1193
01:12:06,720 --> 01:12:09,390
of that expansion, as well
as these things here, which

1194
01:12:09,390 --> 01:12:12,450
are chiral logarithms.

1195
01:12:12,450 --> 01:12:16,270
So if you ask about predictive
power, the polynomial terms

1196
01:12:16,270 --> 01:12:18,450
in terms of higher order
in the quark masses

1197
01:12:18,450 --> 01:12:21,630
that are polynomial, you end
up having unknown coefficients.

1198
01:12:21,630 --> 01:12:23,550
But the terms with
logarithms have

1199
01:12:23,550 --> 01:12:26,670
coefficients which are fixed
by your lower order Lagrangian.

1200
01:12:26,670 --> 01:12:28,170
So those are things
that you predict

1201
01:12:28,170 --> 01:12:31,800
with the chiral
perturbation theory.

1202
01:12:31,800 --> 01:12:33,360
When people do
lattice calculations,

1203
01:12:33,360 --> 01:12:34,950
they need to do
chiral extrapolations.

1204
01:12:34,950 --> 01:12:36,867
And then they're using
formulas like this one.

1205
01:12:40,140 --> 01:12:43,384
OK, so any questions about that?

1206
01:12:43,384 --> 01:12:45,283
AUDIENCE: Did the
M pi [INAUDIBLE]??

1207
01:12:45,283 --> 01:12:46,700
IAIN STEWART: Yeah,
physical mass.

1208
01:12:46,700 --> 01:12:47,870
This is the physical mass.

1209
01:12:47,870 --> 01:12:49,670
This is the quark masses.

1210
01:12:49,670 --> 01:12:54,610
M0 is just this combination
with M up and M down.

1211
01:12:54,610 --> 01:12:56,690
AUDIENCE: So the mu
dependence cancel between--

1212
01:12:56,690 --> 01:12:58,148
IAIN STEWART: And
the mu dependence

1213
01:12:58,148 --> 01:13:01,040
cancels because these
guys all depend on mu.

1214
01:13:01,040 --> 01:13:03,230
Yeah.

1215
01:13:03,230 --> 01:13:06,600
And same thing here,
these guys depend on mu.

1216
01:13:09,668 --> 01:13:12,190
Here, I have just enough
room to make that explicit.

1217
01:13:15,440 --> 01:13:17,828
OK, so that actually
covers all the goals

1218
01:13:17,828 --> 01:13:19,620
that we had for chiral
perturbation theory,

1219
01:13:19,620 --> 01:13:21,410
though it's a fun topic.

1220
01:13:21,410 --> 01:13:23,870
And there's many more
things we could discuss,

1221
01:13:23,870 --> 01:13:27,420
but that's what we
needed to discuss.

1222
01:13:27,420 --> 01:13:30,000
And so we're going to move on.

1223
01:13:30,000 --> 01:13:31,920
So the next thing I
want to talk about

1224
01:13:31,920 --> 01:13:33,920
as an example of effective
field theory is heavy

1225
01:13:33,920 --> 01:13:34,970
quark effective theory.

1226
01:13:48,160 --> 01:13:50,580
So again, what are our goals
with this effective theory?

1227
01:13:58,730 --> 01:14:00,500
We will see some new
features showing up

1228
01:14:00,500 --> 01:14:01,667
that we haven't seen before.

1229
01:14:06,887 --> 01:14:09,220
We will find out what it means
to take a Lagrangian that

1230
01:14:09,220 --> 01:14:10,280
has labeled fields.

1231
01:14:16,315 --> 01:14:18,190
We'll spend a little
bit of time on symmetry.

1232
01:14:18,190 --> 01:14:21,465
Because in this heavy
quark effective theory,

1233
01:14:21,465 --> 01:14:23,840
there's actually something
called heavy quark's symmetry,

1234
01:14:23,840 --> 01:14:26,140
which is not apparent in
QCD, but becomes apparent

1235
01:14:26,140 --> 01:14:29,110
in this theory.

1236
01:14:29,110 --> 01:14:35,590
And there's a trick known as
using covariate representations

1237
01:14:35,590 --> 01:14:37,900
in order to encode
symmetry predictions.

1238
01:14:37,900 --> 01:14:40,000
And it's a very
powerful thing that's

1239
01:14:40,000 --> 01:14:42,750
not special to this theory, but
you could use it in general.

1240
01:14:42,750 --> 01:14:44,000
And I want to teach it to you.

1241
01:14:44,000 --> 01:14:45,208
So that's one thing we'll do.

1242
01:14:52,558 --> 01:14:54,100
It's kind of like
a spurion analysis,

1243
01:14:54,100 --> 01:14:59,050
but a little bit more powerful.

1244
01:14:59,050 --> 01:15:03,190
And finally, anomalous
dimensions that are functions

1245
01:15:03,190 --> 01:15:09,790
is something that we'll show
up here, not just numbers.

1246
01:15:09,790 --> 01:15:20,180
There's something called
reparameterization invariance,

1247
01:15:20,180 --> 01:15:25,490
which you can think of a kind of
symmetry that we'll talk about.

1248
01:15:25,490 --> 01:15:31,180
And finally, if that
list is not long enough,

1249
01:15:31,180 --> 01:15:36,820
we'll add one more
thing which I hinted at.

1250
01:15:36,820 --> 01:15:40,090
So I said, when we
talked about Ms bar,

1251
01:15:40,090 --> 01:15:42,387
that it wasn't a perfect
scheme for doing things,

1252
01:15:42,387 --> 01:15:43,720
but there were some limitations.

1253
01:15:43,720 --> 01:15:45,512
And we'll come, at the
end of this chapter,

1254
01:15:45,512 --> 01:15:47,080
to what those limitations are.

1255
01:15:54,140 --> 01:15:58,310
The limitations come from
power-like scale separation.

1256
01:15:58,310 --> 01:16:02,210
And that's related to
something called renormalons.

1257
01:16:02,210 --> 01:16:05,030
So we'll learn what
a renormalon is

1258
01:16:05,030 --> 01:16:07,640
and why it has something to
do with the failure of Ms bar

1259
01:16:07,640 --> 01:16:10,893
and how one can get
around that failure

1260
01:16:10,893 --> 01:16:12,560
and how one actually
needs to get around

1261
01:16:12,560 --> 01:16:15,000
that failure in some cases.

1262
01:16:15,000 --> 01:16:17,840
OK, so that's the list of goals.

1263
01:16:17,840 --> 01:16:22,220
It's kind of, in some ways, in
order of importance, actually.

1264
01:16:24,750 --> 01:16:27,000
So when we think about heavy
quark effective theory,

1265
01:16:27,000 --> 01:16:29,137
you shouldn't think about
it as just something

1266
01:16:29,137 --> 01:16:31,470
that you would need to do if
you wanted to do heavy work

1267
01:16:31,470 --> 01:16:34,200
physics because the idea
of what we're doing here

1268
01:16:34,200 --> 01:16:35,915
is much more general than that.

1269
01:16:35,915 --> 01:16:37,290
The idea of what
we're doing here

1270
01:16:37,290 --> 01:16:39,990
is we're saying,
take a heavy particle

1271
01:16:39,990 --> 01:16:43,770
and think about what
happens if I tickle it.

1272
01:16:43,770 --> 01:16:49,120
And that heavy particle could be
a heavy source to some theory.

1273
01:16:49,120 --> 01:16:51,120
We'll talk about it in
the context of them being

1274
01:16:51,120 --> 01:16:52,578
heavy quarks, but
you should really

1275
01:16:52,578 --> 01:16:56,100
think of it as any
heavy particle tickled

1276
01:16:56,100 --> 01:16:57,060
by light particles.

1277
01:17:15,010 --> 01:17:18,370
And what we really
mean is that we want

1278
01:17:18,370 --> 01:17:21,037
to study the heavy particle.

1279
01:17:21,037 --> 01:17:22,870
And since we want to
study it, we better not

1280
01:17:22,870 --> 01:17:23,980
remove it from the theory.

1281
01:17:27,532 --> 01:17:29,740
So we want to tickle it with
light degrees of freedom

1282
01:17:29,740 --> 01:17:32,830
with small momentum
transfer, but we don't want

1283
01:17:32,830 --> 01:17:35,578
to integrate that particle out.

1284
01:17:35,578 --> 01:17:37,120
So another way of
saying this, if you

1285
01:17:37,120 --> 01:17:39,037
don't want to think of
it as a heavy particle,

1286
01:17:39,037 --> 01:17:40,810
is that you have some source.

1287
01:17:40,810 --> 01:17:45,550
And that source can be
tickled and could wiggle,

1288
01:17:45,550 --> 01:17:47,718
but it's sort of mostly
just a static source.

1289
01:17:47,718 --> 01:17:48,760
And then it could wiggle.

1290
01:17:48,760 --> 01:17:51,663
And that's what this is an
effective field theory for.

1291
01:17:51,663 --> 01:17:53,830
So we'll talk about it in
the context of heavy quark

1292
01:17:53,830 --> 01:17:54,460
effective theory.

1293
01:17:54,460 --> 01:17:56,543
And some of the things,
like heavy quark symmetry,

1294
01:17:56,543 --> 01:17:57,892
will be special to that theory.

1295
01:17:57,892 --> 01:18:00,100
But more generally, the kind
of approach we're taking

1296
01:18:00,100 --> 01:18:01,100
is a more general thing.

1297
01:18:04,330 --> 01:18:07,270
So what's heavy quark
effective theory?

1298
01:18:07,270 --> 01:18:08,837
So you have a heavy quark.

1299
01:18:08,837 --> 01:18:11,295
And you have it sitting in a
bound state, which is a meson.

1300
01:18:14,770 --> 01:18:18,070
So that means it's surrounded
by light degrees of freedom,

1301
01:18:18,070 --> 01:18:21,520
one heavy quark,
lots of light junk.

1302
01:18:21,520 --> 01:18:24,430
In the quark model, you'd say
it's a Q bar q, heavy cork

1303
01:18:24,430 --> 01:18:31,270
capital Q and a light cork
little q, like b bar d,

1304
01:18:31,270 --> 01:18:33,490
which is the B0.

1305
01:18:33,490 --> 01:18:36,850
But that's just a quark
model approximation

1306
01:18:36,850 --> 01:18:38,470
for what the degrees
of freedom are.

1307
01:18:38,470 --> 01:18:41,137
And there's many more
pairs of Qq bars and gluons

1308
01:18:41,137 --> 01:18:42,970
that are really forming
this hydronic state.

1309
01:18:45,547 --> 01:18:48,130
So you'd like to be able to make
model independent predictions

1310
01:18:48,130 --> 01:18:51,370
for that without having
to worry about the fact

1311
01:18:51,370 --> 01:18:54,790
that you're not parameterizing
properly that stuff

1312
01:18:54,790 --> 01:18:55,990
that you can't calculate.

1313
01:18:55,990 --> 01:19:00,140
And you can do that using
effective field theory.

1314
01:19:00,140 --> 01:19:01,820
There's two scales
in the problem.

1315
01:19:01,820 --> 01:19:04,540
There's the size of this
thing, the inverse size

1316
01:19:04,540 --> 01:19:05,650
of order lambda QCD.

1317
01:19:05,650 --> 01:19:09,317
That's the hadronization
that scale.

1318
01:19:09,317 --> 01:19:11,650
And that's a scale that's
much less than the quark mass.

1319
01:19:11,650 --> 01:19:14,190
So you're really just expanding
in lambda QCD divided by Mq.

1320
01:19:14,190 --> 01:19:15,940
That's what this
effective theory will be.

1321
01:19:21,610 --> 01:19:27,250
And what you want
to describe are

1322
01:19:27,250 --> 01:19:31,882
fluctuations of the heavy
quark due to the light quarks.

1323
01:19:31,882 --> 01:19:33,340
So you could think
that this guy is

1324
01:19:33,340 --> 01:19:36,550
so heavy he just sits in
the middle of the state,

1325
01:19:36,550 --> 01:19:39,087
and he's static.

1326
01:19:39,087 --> 01:19:41,170
And then he gets tickled
by the light stuff that's

1327
01:19:41,170 --> 01:19:42,295
flying and whizzing around.

1328
01:19:42,295 --> 01:19:43,660
And that's a reasonable picture.

1329
01:19:47,200 --> 01:19:48,760
So what we really
want to do here

1330
01:19:48,760 --> 01:19:52,090
is an example of top-down
effective field theory again.

1331
01:19:52,090 --> 01:19:58,210
So going back to our
like integrating out

1332
01:19:58,210 --> 01:20:02,860
heavy particles, but now we
want to keep the heavy particles

1333
01:20:02,860 --> 01:20:04,540
in the theory, not remove them.

1334
01:20:04,540 --> 01:20:06,790
But still, we want to take
a low energy limit of them.

1335
01:20:09,710 --> 01:20:14,050
So that means that we have to
take a low energy limit of QCD,

1336
01:20:14,050 --> 01:20:19,438
which is id slash
minus Mq Q. And you

1337
01:20:19,438 --> 01:20:20,980
can see that part
of the problem here

1338
01:20:20,980 --> 01:20:23,560
is that the Mq is
upstairs, which

1339
01:20:23,560 --> 01:20:26,530
makes taking the limit
not completely obvious.

1340
01:20:26,530 --> 01:20:28,810
In the case where we were
doing the heavy bosons,

1341
01:20:28,810 --> 01:20:30,892
we sort of saw that it
was always in propagators.

1342
01:20:30,892 --> 01:20:33,100
And we could just think of,
since it's always they're

1343
01:20:33,100 --> 01:20:35,500
internal, we just expand.

1344
01:20:35,500 --> 01:20:37,640
Here, we want to keep
the Q in the theory.

1345
01:20:37,640 --> 01:20:41,930
So we have to really think
about how we take this limit.

1346
01:20:41,930 --> 01:20:48,490
So let's start slowly and
consider the propagator

1347
01:20:48,490 --> 01:20:50,987
for a heavy quark.

1348
01:20:50,987 --> 01:20:52,570
And we'll come back
to the Lagrangian.

1349
01:20:56,890 --> 01:21:06,990
And we'll consider it with some
on-shell momentum, which we'll

1350
01:21:06,990 --> 01:21:09,570
consider an on-shell
momentum to be parameterized

1351
01:21:09,570 --> 01:21:11,560
in the following way.

1352
01:21:11,560 --> 01:21:13,200
So if I have an
on-shell momentum,

1353
01:21:13,200 --> 01:21:17,640
I'll say that p is equal to
MqV, so that p squared is

1354
01:21:17,640 --> 01:21:23,310
equal to Mq squared
and this V is 1.

1355
01:21:23,310 --> 01:21:25,590
So once I pull out
the mass dimension Mq,

1356
01:21:25,590 --> 01:21:28,440
the remaining thing I call V.
And that's just some parameter

1357
01:21:28,440 --> 01:21:32,850
which squares to 1 on-shell.

1358
01:21:32,850 --> 01:21:36,990
Now, if I have kicks, which are
from light degrees of freedom,

1359
01:21:36,990 --> 01:21:39,780
then I don't have
exactly on-shell.

1360
01:21:39,780 --> 01:21:45,240
So P mu is MqV, which is like
on-shell piece plus something

1361
01:21:45,240 --> 01:21:48,480
small, k mu.

1362
01:21:48,480 --> 01:21:52,620
And k mu is of order lambda QCD.

1363
01:21:52,620 --> 01:21:56,280
So this is like saying the
on-shell piece plus the tickle.

1364
01:22:01,380 --> 01:22:03,240
And if I want to
construct the propagator,

1365
01:22:03,240 --> 01:22:09,120
then the propagator is encoding
the optional off-shellness

1366
01:22:09,120 --> 01:22:10,740
of the degree of freedom.

1367
01:22:10,740 --> 01:22:13,510
That's 1 over the off-shellness.

1368
01:22:13,510 --> 01:22:14,760
So it'll depend on the tickle.

1369
01:22:24,710 --> 01:22:33,200
OK, so we could just take QCD
propagator, which is this,

1370
01:22:33,200 --> 01:22:34,530
and just plug in that formula.

1371
01:22:50,100 --> 01:22:52,920
There's Mq squared, which
cancels this Mq squared.

1372
01:22:56,900 --> 01:23:00,650
Then there's some cross-terms
that don't get cancelled.

1373
01:23:00,650 --> 01:23:04,760
So there's one that comes from
the dot product of the V dot M

1374
01:23:04,760 --> 01:23:05,540
with the k term.

1375
01:23:05,540 --> 01:23:06,050
That's this.

1376
01:23:06,050 --> 01:23:07,550
And then there's
the k squared term.

1377
01:23:07,550 --> 01:23:10,130
So I have these.

1378
01:23:10,130 --> 01:23:11,420
And then I can expand that.

1379
01:23:19,700 --> 01:23:22,310
M is a positive
quantity, so it doesn't

1380
01:23:22,310 --> 01:23:25,070
change the sign of the i0.

1381
01:23:25,070 --> 01:23:27,650
And if I expand in M, then the
leading term looks like that.

1382
01:23:27,650 --> 01:23:30,350
It's M independent.

1383
01:23:30,350 --> 01:23:32,750
And then there's some order
1 over M terms, which I could

1384
01:23:32,750 --> 01:23:33,917
also work out what they are.

1385
01:23:38,150 --> 01:23:40,610
OK, so that we could expand
even though we don't, a priori,

1386
01:23:40,610 --> 01:23:42,426
know how to expand
the Lagrangian.

1387
01:23:46,400 --> 01:23:48,562
We could also think about
vertices in this theory.

1388
01:23:48,562 --> 01:23:49,520
This is the propagator.

1389
01:23:49,520 --> 01:23:51,380
What about vertices?

1390
01:23:51,380 --> 01:23:53,090
And again, this is a top-down.

1391
01:23:53,090 --> 01:23:55,820
So we can think about
vertices in QCD.

1392
01:23:55,820 --> 01:23:58,340
So if this is a
heavy particle here,

1393
01:23:58,340 --> 01:24:00,890
we could think about
what would happen.

1394
01:24:00,890 --> 01:24:02,000
How do we expand those?

1395
01:24:06,460 --> 01:24:08,210
And there doesn't look
like there's really

1396
01:24:08,210 --> 01:24:11,390
anything going on here because
there's nothing to expand.

1397
01:24:11,390 --> 01:24:14,540
But because these things
are heavy particles here,

1398
01:24:14,540 --> 01:24:17,870
you realize from the
propagator formula

1399
01:24:17,870 --> 01:24:20,090
here that you're going
to have 1 plus V slashes

1400
01:24:20,090 --> 01:24:23,670
on each side of this guy.

1401
01:24:23,670 --> 01:24:26,880
So for a propagator
on each side,

1402
01:24:26,880 --> 01:24:28,807
you can make a simplification.

1403
01:24:35,350 --> 01:24:38,740
And that's because 1
plus V slash over 2 gamma

1404
01:24:38,740 --> 01:24:44,110
mu 1 plus V slash over 2,
after some Dirac algebra,

1405
01:24:44,110 --> 01:24:48,850
is just V mu times 1
plus V slash over 2.

1406
01:24:51,610 --> 01:24:54,460
So the gamma mu becomes a V mu.

1407
01:24:54,460 --> 01:24:58,520
So the vertice, once you
take that into account,

1408
01:24:58,520 --> 01:25:01,840
is just minus igTA V mu.

1409
01:25:06,700 --> 01:25:08,980
Even if we don't think
about starting with the QCD

1410
01:25:08,980 --> 01:25:10,690
Lagrangian, if this
is our Feynman rule

1411
01:25:10,690 --> 01:25:12,850
and that's our propagator,
we can write down

1412
01:25:12,850 --> 01:25:18,820
an effective theory for that
that gives those Feynman rules.

1413
01:25:18,820 --> 01:25:21,377
And that actually is
the HQET Lagrangian.

1414
01:25:30,640 --> 01:25:37,360
So sorry for going
a little bit over,

1415
01:25:37,360 --> 01:25:40,930
but I wanted to at least
get a little bit into this.

1416
01:25:40,930 --> 01:25:43,990
OK, so by construction
this way, we

1417
01:25:43,990 --> 01:25:45,947
can arrive at this Lagrangian.

1418
01:25:45,947 --> 01:25:47,530
And next time, I'll
come back and I'll

1419
01:25:47,530 --> 01:25:49,660
show you how you
can really properly

1420
01:25:49,660 --> 01:25:52,300
take this limit of
the QCD Lagrangian

1421
01:25:52,300 --> 01:25:55,010
to get the same thing.

1422
01:25:55,010 --> 01:25:56,710
OK, but that is the HQET.

1423
01:25:56,710 --> 01:25:59,170
The lowest order of the
HQET Lagrangian is this.

1424
01:26:02,810 --> 01:26:05,120
So it's a linear V
dot D decoupling.

1425
01:26:05,120 --> 01:26:07,403
V is a parameter,
and so I've put it

1426
01:26:07,403 --> 01:26:08,570
as a parameter on the field.

1427
01:26:08,570 --> 01:26:11,780
We'll see more why that
was done next time.

1428
01:26:11,780 --> 01:26:14,000
And there's a projection
relation on the field.

1429
01:26:14,000 --> 01:26:15,650
If I'm using a
four-component field,

1430
01:26:15,650 --> 01:26:17,192
then I have this
projection relation.

1431
01:26:17,192 --> 01:26:19,590
I could also read it in
a two-component notation,

1432
01:26:19,590 --> 01:26:21,920
and then I wouldn't need that.

1433
01:26:21,920 --> 01:26:24,695
But for four-component
notation, you do need that.

1434
01:26:24,695 --> 01:26:27,610
We'll talk about
that next time, too.