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PROFESSOR: So here's
where we were.

9
00:00:26,390 --> 00:00:30,110
So we were talking about
this example of b to s gamma

10
00:00:30,110 --> 00:00:31,370
and a heavy to light current.

11
00:00:35,640 --> 00:00:37,830
In terms of this chi
field, our current

12
00:00:37,830 --> 00:00:43,470
was chi, which is going
to have some label,

13
00:00:43,470 --> 00:00:45,630
in a heavy quark field.

14
00:00:45,630 --> 00:00:48,060
And we did the one
with diagrams in SET.

15
00:00:48,060 --> 00:00:52,020
We saw that after we take
into account also the 0 bin

16
00:00:52,020 --> 00:00:53,950
contribution that it
breaks up as follows.

17
00:00:53,950 --> 00:00:55,450
There's a piece
that I've underlined

18
00:00:55,450 --> 00:00:58,395
in orange that matches
exactly with the IR of QCD.

19
00:01:04,285 --> 00:01:05,660
There's a piece
that I underlined

20
00:01:05,660 --> 00:01:08,390
in blue which is the
ultraviolet divergences

21
00:01:08,390 --> 00:01:09,740
in the effective theory.

22
00:01:09,740 --> 00:01:11,150
And so what we do
with that is we

23
00:01:11,150 --> 00:01:13,290
add a counter-term
for the operators

24
00:01:13,290 --> 00:01:16,257
and the effective theory to
cancel these divergences.

25
00:01:16,257 --> 00:01:17,840
And then there's
whatever is left over

26
00:01:17,840 --> 00:01:19,940
after doing that,
which is this pink.

27
00:01:19,940 --> 00:01:22,520
And the difference
of the analog pink

28
00:01:22,520 --> 00:01:25,610
in the full theory and this
pink will give you the matching.

29
00:01:25,610 --> 00:01:26,990
So when I say
"matching," this is

30
00:01:26,990 --> 00:01:29,500
part of what goes
into that matching.

31
00:01:29,500 --> 00:01:31,250
So what I want to spend
today's lecture on

32
00:01:31,250 --> 00:01:33,560
is really discussing
renormalization

33
00:01:33,560 --> 00:01:35,840
in the effective theory
and summing logarithms

34
00:01:35,840 --> 00:01:37,300
in the effective theory.

35
00:01:37,300 --> 00:01:40,670
It's kind of interesting
and different than things

36
00:01:40,670 --> 00:01:43,280
we've encountered before,
because we see even

37
00:01:43,280 --> 00:01:45,290
at one loop order,
we have 1 over

38
00:01:45,290 --> 00:01:46,880
epsilon squared divergence.

39
00:01:46,880 --> 00:01:49,250
And that wasn't something
that we encountered before.

40
00:01:49,250 --> 00:01:51,710
And so we have to see
how that plays out.

41
00:02:06,050 --> 00:02:08,660
But just at the
simplistic level,

42
00:02:08,660 --> 00:02:14,930
let's just say or
record for the record

43
00:02:14,930 --> 00:02:17,390
that we could write
the following.

44
00:02:17,390 --> 00:02:20,730
There was a Wilson
coefficient of this operator.

45
00:02:20,730 --> 00:02:23,690
And we can divide it into
renormalize Wilson coefficient

46
00:02:23,690 --> 00:02:25,292
in a counterterm.

47
00:02:25,292 --> 00:02:27,125
And then we just think
that this counterterm

48
00:02:27,125 --> 00:02:29,084
is canceling that divergence.

49
00:02:32,880 --> 00:02:39,327
And so the counterterm
we need is the following.

50
00:02:39,327 --> 00:02:41,410
So with this minus sign
canceling that minus sign,

51
00:02:41,410 --> 00:02:42,640
you get plus.

52
00:02:42,640 --> 00:02:46,510
And so the counterterm
for the operator therefore

53
00:02:46,510 --> 00:02:51,280
should have a minus
and just all--

54
00:02:51,280 --> 00:02:52,330
exactly these terms.

55
00:03:03,910 --> 00:03:06,970
Because of kinematics
in this problem,

56
00:03:06,970 --> 00:03:10,570
this omega, which is a delta
function in this operator,

57
00:03:10,570 --> 00:03:12,910
is equal to actually mb.

58
00:03:12,910 --> 00:03:17,480
And I'll go through why that
is a little later today.

59
00:03:17,480 --> 00:03:20,290
So take it for granted for now.

60
00:03:20,290 --> 00:03:23,180
It comes about just
simply from kinematics.

61
00:03:23,180 --> 00:03:25,840
But I'm going to sometimes
write the mb as omega

62
00:03:25,840 --> 00:03:29,160
because that reminds you that it
was this label in the colinear

63
00:03:29,160 --> 00:03:31,368
field.

64
00:03:31,368 --> 00:03:33,160
So this is the type of
counterterm we have.

65
00:03:33,160 --> 00:03:34,702
It's got this 1 over
epsilon squared.

66
00:03:34,702 --> 00:03:39,790
It's got this log, so it
looks kind of different.

67
00:03:39,790 --> 00:03:52,240
And the notation
here is that we have

68
00:03:52,240 --> 00:04:02,080
a coefficient that's
inside an integral, where

69
00:04:02,080 --> 00:04:10,160
this thing meant that.

70
00:04:16,750 --> 00:04:20,290
So basically what I'm saying
when I say omega is equal to mb

71
00:04:20,290 --> 00:04:26,250
is that this delta function
becomes omega minus mb.

72
00:04:26,250 --> 00:04:29,220
We'll come back to that.

73
00:04:29,220 --> 00:04:32,970
So if we have a counterterm like
that, then we can do running.

74
00:04:38,190 --> 00:04:40,320
So we'll come back
to this example.

75
00:04:40,320 --> 00:04:42,450
Let me first make a
few general comments,

76
00:04:42,450 --> 00:04:45,510
and then we'll come back and
see where this z leads us.

77
00:04:52,840 --> 00:04:56,500
So in general, you're going
to have this structure--

78
00:04:56,500 --> 00:04:58,670
some integral, some Wilson
coefficients that depend

79
00:04:58,670 --> 00:05:00,388
on some omegas--

80
00:05:00,388 --> 00:05:02,680
could be more than one of
them-- and then some operator

81
00:05:02,680 --> 00:05:05,305
that has some delta functions in
it that pick out those omegas.

82
00:05:07,850 --> 00:05:11,690
And because of that
integral, in general,

83
00:05:11,690 --> 00:05:21,250
you have to be careful, because
that integral could also play

84
00:05:21,250 --> 00:05:23,560
a role in the renormalization.

85
00:05:39,280 --> 00:05:43,030
So that delta function
is picking out

86
00:05:43,030 --> 00:05:45,180
the total momentum of
the product of the quark

87
00:05:45,180 --> 00:05:47,860
field times the Wilson line.

88
00:05:47,860 --> 00:05:51,520
And in our example, it's
fixed by external kinematics.

89
00:05:51,520 --> 00:05:53,590
So I'll come back
to this and show you

90
00:05:53,590 --> 00:05:57,340
that not in every example is
it fixed by external kinematics

91
00:05:57,340 --> 00:05:58,480
in a minute.

92
00:05:58,480 --> 00:06:00,850
But our example is special
because the integral actually

93
00:06:00,850 --> 00:06:02,590
doesn't play an important role.

94
00:06:32,870 --> 00:06:36,850
So there's in some sense
two parts of this statement.

95
00:06:36,850 --> 00:06:38,500
There is the fact
that it's external,

96
00:06:38,500 --> 00:06:40,930
and then there's the fact
that it's fixed to something.

97
00:06:40,930 --> 00:06:45,520
So let me first deal with the
issue of it being external.

98
00:06:45,520 --> 00:06:51,100
And what that means is it
does not involve loop momenta.

99
00:06:51,100 --> 00:06:58,120
You see, when we
did the calculation,

100
00:06:58,120 --> 00:07:00,958
we were basically
doing a calculation,

101
00:07:00,958 --> 00:07:03,250
not worrying so much about
this delta function that was

102
00:07:03,250 --> 00:07:06,250
sitting inside our operator.

103
00:07:06,250 --> 00:07:07,180
We ignored it.

104
00:07:07,180 --> 00:07:09,232
We didn't write it
down in our diagrams.

105
00:07:09,232 --> 00:07:10,690
Technically, we
should have written

106
00:07:10,690 --> 00:07:13,150
that delta function, this
delta function, down, too,

107
00:07:13,150 --> 00:07:15,400
and included it in
our calculation.

108
00:07:15,400 --> 00:07:18,040
But actually that's OK because
that delta function just comes

109
00:07:18,040 --> 00:07:19,630
outside the loop integrals.

110
00:07:19,630 --> 00:07:21,840
That's what I want to show you.

111
00:07:26,250 --> 00:07:30,930
So it does not
involve loop momenta.

112
00:07:30,930 --> 00:07:35,370
So in all the
diagrams we consider,

113
00:07:35,370 --> 00:07:38,220
that's trivial, except for
one non-trivial example.

114
00:07:40,890 --> 00:07:42,990
And that non-trivial
example is the example

115
00:07:42,990 --> 00:07:48,000
where we had a colinear
vertex-type diagram like this.

116
00:07:51,760 --> 00:07:55,080
So we called the
external momentum here p.

117
00:07:55,080 --> 00:07:56,360
This was p plus k.

118
00:07:56,360 --> 00:08:00,270
And if I let this guy go this
way, we can call it minus k.

119
00:08:00,270 --> 00:08:04,710
You see, the Wilson coefficient
sits here at the vertex.

120
00:08:04,710 --> 00:08:06,420
And it acts on the
colinear fields

121
00:08:06,420 --> 00:08:08,730
that come out of that vertex.

122
00:08:08,730 --> 00:08:11,340
But it gives some of the
momentum of those colinear

123
00:08:11,340 --> 00:08:11,860
fields.

124
00:08:11,860 --> 00:08:22,850
So we get this guy, and
then we get this guy.

125
00:08:22,850 --> 00:08:24,700
And so we just kept
the external guy.

126
00:08:30,200 --> 00:08:33,620
So from this notation, the
same thing as saying this delta

127
00:08:33,620 --> 00:08:35,700
function.

128
00:08:35,700 --> 00:08:39,919
We can see that it only
depends on the external p,

129
00:08:39,919 --> 00:08:41,240
on the n bar dot k.

130
00:08:49,840 --> 00:08:53,590
And that's because of the
structure of this operator.

131
00:08:58,290 --> 00:08:59,500
So that won't always be true.

132
00:08:59,500 --> 00:09:02,010
It could be the case that the
Wilson-- in a more complicated

133
00:09:02,010 --> 00:09:03,990
example, that the
Wilson coefficient

134
00:09:03,990 --> 00:09:05,000
could depend on the k.

135
00:09:05,000 --> 00:09:06,750
And then that would
have some implications

136
00:09:06,750 --> 00:09:08,160
for renormalization.

137
00:09:08,160 --> 00:09:11,640
And we will treat an example of
that sort in the near future.

138
00:09:15,780 --> 00:09:17,870
The second thing to ask
is, what does it fix to?

139
00:09:20,660 --> 00:09:23,030
So if we put a little
more into our diagram,

140
00:09:23,030 --> 00:09:24,980
let's put the photon in.

141
00:09:24,980 --> 00:09:26,390
This is a b quark.

142
00:09:26,390 --> 00:09:31,140
This is a strange quark b
to s gamma, for example.

143
00:09:31,140 --> 00:09:35,120
Then the kinematics of the
external lines there is we

144
00:09:35,120 --> 00:09:38,840
have mbv coming in for the
b quark, p of the photon,

145
00:09:38,840 --> 00:09:42,800
and p of the external
strange quark.

146
00:09:42,800 --> 00:09:46,190
And the photon is light-like.
p gamma squared is 0.

147
00:09:46,190 --> 00:09:49,850
And in the setup
here, we can just

148
00:09:49,850 --> 00:09:53,630
think of it as e
gamma times n bar,

149
00:09:53,630 --> 00:09:57,230
because p is colinear to the
n direction, and it's going--

150
00:09:57,230 --> 00:10:00,200
we have the photon
going this way.

151
00:10:00,200 --> 00:10:04,630
And it's back to back
with a jet that's

152
00:10:04,630 --> 00:10:10,580
described by the
colinear n direction,

153
00:10:10,580 --> 00:10:12,070
just as the beam
goes on decaying

154
00:10:12,070 --> 00:10:14,710
to a photon in a
back-to-back jet.

155
00:10:14,710 --> 00:10:20,281
And so if you dot into
this equation n bar,

156
00:10:20,281 --> 00:10:24,430
since n bar dot v is 1 in
conventional coordinates,

157
00:10:24,430 --> 00:10:30,940
we just get that mb
is 0 plus n bar dot p.

158
00:10:30,940 --> 00:10:34,570
And that's the statement
that omega was just mb.

159
00:10:34,570 --> 00:10:36,850
So when we were doing our
loop calculations last time,

160
00:10:36,850 --> 00:10:38,290
we kept writing mb.

161
00:10:38,290 --> 00:10:40,750
We could have been more
careful and written--

162
00:10:40,750 --> 00:10:44,140
well, we couldn't have been more
careful, because kinematics was

163
00:10:44,140 --> 00:10:46,000
demanding that this was true.

164
00:10:46,000 --> 00:10:48,940
But just in order to track what
the Wilson coefficients can

165
00:10:48,940 --> 00:10:51,040
depend on, really what
they can depend on

166
00:10:51,040 --> 00:10:53,680
is these labels of
these colinear fields.

167
00:10:53,680 --> 00:10:57,250
And so I'm going to start
writing omega for that reason

168
00:10:57,250 --> 00:10:59,837
because that's the more
generic thing that happens.

169
00:10:59,837 --> 00:11:01,420
Even if we're not
in a situation where

170
00:11:01,420 --> 00:11:03,660
it's fixed to be
some mass, we'll

171
00:11:03,660 --> 00:11:05,500
be able to get
dependence on the values

172
00:11:05,500 --> 00:11:07,300
of these parameters of omega.

173
00:11:12,900 --> 00:11:14,330
So any questions about that?

174
00:11:18,532 --> 00:11:20,740
All right, so this Wilson
coefficient just comes out,

175
00:11:20,740 --> 00:11:23,980
and we didn't have to worry
about this delta function.

176
00:11:23,980 --> 00:11:28,110
So how do we calculate
the anomalous dimension?

177
00:11:28,110 --> 00:11:30,300
It's the usual story.

178
00:11:30,300 --> 00:11:34,110
So mu d by dbu of the
bare coefficient is 0.

179
00:11:34,110 --> 00:11:36,450
That implies a
renormalization group equation

180
00:11:36,450 --> 00:11:40,326
for the ms bar coefficient.

181
00:11:52,490 --> 00:11:57,590
And it's got this familiar form,
where the gamma is products

182
00:11:57,590 --> 00:11:59,486
of z's and z inverses--

183
00:12:06,300 --> 00:12:07,230
just this.

184
00:12:11,660 --> 00:12:13,550
So this is an
all-orders formula.

185
00:12:13,550 --> 00:12:15,230
If we plug in our
one loop result

186
00:12:15,230 --> 00:12:20,653
here, then on the left, if we
worked order by order in alpha,

187
00:12:20,653 --> 00:12:22,070
on the left-hand
side, we can just

188
00:12:22,070 --> 00:12:25,580
take the 1 in the
zc universe here.

189
00:12:25,580 --> 00:12:28,940
And the zc, when we
take b by d by d mu,

190
00:12:28,940 --> 00:12:32,010
there's mu in the alpha s,
and there's mu in this log.

191
00:12:32,010 --> 00:12:35,850
And we're going to have to
differentiate both of those.

192
00:12:35,850 --> 00:12:39,500
So let me write it out.

193
00:12:59,260 --> 00:13:01,090
So the minus sign
canceled the minus sign.

194
00:13:01,090 --> 00:13:02,772
The zc inverse is just 1.

195
00:13:02,772 --> 00:13:05,230
And then I have b db dbu over
the rest of this stuff, which

196
00:13:05,230 --> 00:13:10,210
includes both the alpha
and the log mu over w.

197
00:13:39,135 --> 00:13:54,510
These are the type of
contributions we get,

198
00:13:54,510 --> 00:13:58,860
where these guys here are
coming from using the fact that

199
00:13:58,860 --> 00:14:04,650
mu d by d mu of alpha when
you're in d dimensions is minus

200
00:14:04,650 --> 00:14:07,290
2 epsilon alpha plus--

201
00:14:07,290 --> 00:14:09,660
order alpha squared term's
related to the beta function,

202
00:14:09,660 --> 00:14:11,580
which we can drop here.

203
00:14:11,580 --> 00:14:16,770
And then this guy comes
from the explicit--

204
00:14:16,770 --> 00:14:22,110
differentiating the
explicit log mu over w.

205
00:14:22,110 --> 00:14:23,910
The divergent terms
cancel, and we're

206
00:14:23,910 --> 00:14:26,170
left with a finite
anomalous dimension.

207
00:14:36,670 --> 00:14:38,380
And it's got two
types of terms in it.

208
00:14:42,770 --> 00:14:44,320
So the type of
anomalous dimensions

209
00:14:44,320 --> 00:14:48,760
that we're used to are like this
guy, here-- this 5 times cf.

210
00:14:48,760 --> 00:14:50,920
That would be like a
standard anomalous dimension

211
00:14:50,920 --> 00:14:52,157
for an operator.

212
00:14:52,157 --> 00:14:53,740
In this case, we
have this extra piece

213
00:14:53,740 --> 00:14:56,500
that has this log of mu over w.

214
00:14:56,500 --> 00:15:03,370
And this guy is what's called
cusp anomalous dimension,

215
00:15:03,370 --> 00:15:05,140
the guy that multiplies the log.

216
00:15:09,530 --> 00:15:11,480
So with that language,
you could call this guy

217
00:15:11,480 --> 00:15:14,690
the regular anomalous dimension.

218
00:15:22,370 --> 00:15:24,440
Now, if you're going to
do renormalization group

219
00:15:24,440 --> 00:15:26,648
evolution, you're
going to sum logs.

220
00:15:26,648 --> 00:15:28,940
You could think that you're
starting at some hard scale

221
00:15:28,940 --> 00:15:30,470
like w. w is mb.

222
00:15:30,470 --> 00:15:32,030
That's the hard scale.

223
00:15:32,030 --> 00:15:35,750
And then you want to run
down to some small-scale mu.

224
00:15:35,750 --> 00:15:38,040
When you run down,
if you take mu small,

225
00:15:38,040 --> 00:15:41,060
this term is larger
than this term.

226
00:15:41,060 --> 00:15:44,390
So actually, this
term here gives you

227
00:15:44,390 --> 00:15:47,300
the leading-- what are called
the leading logs in this case.

228
00:15:47,300 --> 00:15:52,670
And this term here is
actually only needed

229
00:15:52,670 --> 00:15:55,250
for next leading log.

230
00:15:55,250 --> 00:15:57,320
Because this term here
grows when you go down,

231
00:15:57,320 --> 00:16:00,770
and that term doesn't,
this term here

232
00:16:00,770 --> 00:16:03,467
is actually giving
you the leading logs.

233
00:16:03,467 --> 00:16:05,300
So I'll come back and
talk a little bit more

234
00:16:05,300 --> 00:16:07,220
about leading log versus
next-to-leading log.

235
00:16:07,220 --> 00:16:11,410
And also, I'll explain to you
what this word "cusp" in this--

236
00:16:11,410 --> 00:16:14,450
where's the cusp, since
we haven't seen it yet?

237
00:16:14,450 --> 00:16:16,670
I'll come back and explain
where that is in a minute.

238
00:16:22,180 --> 00:16:24,680
For now, let's just take the
leading log anomalous dimension

239
00:16:24,680 --> 00:16:29,673
and see just to solve the
anomalous dimension equation.

240
00:16:29,673 --> 00:16:31,340
So we'll just take
the leading long term

241
00:16:31,340 --> 00:16:42,140
and not worry about
the other term,

242
00:16:42,140 --> 00:16:45,610
so just taking the cusp term.

243
00:16:45,610 --> 00:16:48,520
So plugging that back
into our equation for c,

244
00:16:48,520 --> 00:16:52,240
we have this differential
equation, which we can also

245
00:16:52,240 --> 00:16:58,010
write as d log c by d log mu.

246
00:17:11,319 --> 00:17:13,210
And that's an equation
that we can basically

247
00:17:13,210 --> 00:17:19,450
just integrate since there's
no c on the right-hand side.

248
00:17:19,450 --> 00:17:20,990
It's a homogeneous equation.

249
00:17:20,990 --> 00:17:23,235
We can integrate it with
some boundary condition.

250
00:17:27,990 --> 00:17:30,230
So we could specify
the boundary condition

251
00:17:30,230 --> 00:17:32,570
at an arbitrary scale.

252
00:17:32,570 --> 00:17:36,530
Let me, for simplicity, use the
following tree-level boundary

253
00:17:36,530 --> 00:17:37,561
condition.

254
00:17:41,490 --> 00:17:44,300
So at tree level, the
Wilson coefficient was 1.

255
00:17:44,300 --> 00:17:48,110
That we have to pick
some scale where

256
00:17:48,110 --> 00:17:51,770
we make that statement true.

257
00:17:51,770 --> 00:17:54,710
And for convenience, I'm just
going to take that scale to bw.

258
00:17:54,710 --> 00:17:57,260
That makes things
a little simpler.

259
00:17:57,260 --> 00:18:01,760
In general, you could make
this any hard scale, ie,

260
00:18:01,760 --> 00:18:04,430
any scale mu of order w.

261
00:18:04,430 --> 00:18:07,610
So you could call
it something mu 0

262
00:18:07,610 --> 00:18:11,750
and consider the renormalization
group from that scale.

263
00:18:11,750 --> 00:18:15,230
But for convenience, I'm just
taking that mu 0 to be w here.

264
00:18:29,740 --> 00:18:32,098
So we take that down
[INAUDIBLE] It's

265
00:18:32,098 --> 00:18:33,640
actually useful to
look at what would

266
00:18:33,640 --> 00:18:35,057
happen if we didn't
have a running

267
00:18:35,057 --> 00:18:37,250
coupling on the right-hand side.

268
00:18:37,250 --> 00:18:40,570
So say we were doing QED, and we
didn't have any massless light

269
00:18:40,570 --> 00:18:43,240
fermions, so there
was no running.

270
00:18:43,240 --> 00:18:46,325
Then this coupling would just
be alpha, and it wouldn't run.

271
00:18:46,325 --> 00:18:47,950
And then you'd just
integrate this log,

272
00:18:47,950 --> 00:18:49,240
and you'd get a double log.

273
00:18:49,240 --> 00:18:53,830
You integrate one log by d
log, you get log squared.

274
00:18:53,830 --> 00:18:55,930
For each coupling,
you get log squared.

275
00:18:55,930 --> 00:18:58,480
This is the double log
series that I promised you--

276
00:18:58,480 --> 00:19:00,340
the Sudakov double
logarithms that

277
00:19:00,340 --> 00:19:05,440
were going to come out of the
renormalization group of SET.

278
00:19:05,440 --> 00:19:13,307
So QED with alpha fixed--

279
00:19:13,307 --> 00:19:15,790
I'm going to take cf to be 1.

280
00:19:18,820 --> 00:19:31,190
When you take the exponential
of what I just said,

281
00:19:31,190 --> 00:19:34,935
then this is your result
for the Wilson coefficient.

282
00:19:40,990 --> 00:19:43,140
And this is what's known
as a Sudakov exponential.

283
00:19:43,140 --> 00:19:49,530
The leading log
Wilson coefficient

284
00:19:49,530 --> 00:19:51,648
is the sum of these
alpha times double log.

285
00:19:51,648 --> 00:19:53,940
So if I expanded this out,
I'd have an infinite series,

286
00:19:53,940 --> 00:19:57,450
where for each pair of
logarithms, I get one coupling.

287
00:20:11,060 --> 00:20:15,170
And that's the Sudakov
double logarithms.

288
00:20:15,170 --> 00:20:16,670
So if we expand out
the exponential,

289
00:20:16,670 --> 00:20:18,840
that's what we would get.

290
00:20:18,840 --> 00:20:23,360
So really, I should make my
line go under the whole thing.

291
00:20:27,410 --> 00:20:30,270
So why did we get
something like this?

292
00:20:30,270 --> 00:20:34,760
So physically the reason that
we got this Sudakov exponential

293
00:20:34,760 --> 00:20:37,790
or Sudakov form factor
is related to the fact

294
00:20:37,790 --> 00:20:40,370
that the whole theory
that we've developed

295
00:20:40,370 --> 00:20:42,020
put restrictions on radiation.

296
00:20:58,614 --> 00:21:00,540
So there was
kinematic restrictions

297
00:21:00,540 --> 00:21:02,875
put on the radiation by the
whole setup that we had.

298
00:21:02,875 --> 00:21:04,500
We had this jet, and
it was colonnaded.

299
00:21:04,500 --> 00:21:07,860
And the radiation was forced
to go inside that jet.

300
00:21:21,080 --> 00:21:23,840
In the effective theory, we're
seeing the Sudakov exponential

301
00:21:23,840 --> 00:21:25,770
come out of UV renormalization.

302
00:21:25,770 --> 00:21:27,440
The usual picture
for it is that you're

303
00:21:27,440 --> 00:21:29,388
thinking about IR divergences.

304
00:21:29,388 --> 00:21:31,430
And you're thinking about
the real radiation, not

305
00:21:31,430 --> 00:21:33,320
the virtual radiation.

306
00:21:33,320 --> 00:21:36,710
But as you may be familiar with
what the effective theory does,

307
00:21:36,710 --> 00:21:40,790
is if you have an IR divergence
in full QCD or full QED,

308
00:21:40,790 --> 00:21:42,440
then what the
effective theory does

309
00:21:42,440 --> 00:21:45,110
is it takes the scale
corresponding to--

310
00:21:45,110 --> 00:21:48,050
so the IR divergences
are logarithms.

311
00:21:48,050 --> 00:21:50,240
And they're logarithms of
something over something.

312
00:21:50,240 --> 00:21:52,160
And we've done in
the effective theory

313
00:21:52,160 --> 00:21:54,620
is we've introduced a scale mu.

314
00:21:54,620 --> 00:21:56,360
We've introduced--
effectively, we've

315
00:21:56,360 --> 00:21:59,120
taken the part of that
logarithm that was UV

316
00:21:59,120 --> 00:22:00,860
and taken it to infinity.

317
00:22:00,860 --> 00:22:02,810
And in this standard
way, we've introduced

318
00:22:02,810 --> 00:22:07,640
a scale mu that splits things
between Wilson coefficients

319
00:22:07,640 --> 00:22:09,260
and operators.

320
00:22:09,260 --> 00:22:11,580
So there's this
factorization scale

321
00:22:11,580 --> 00:22:13,220
that's splitting
the hard physics

322
00:22:13,220 --> 00:22:16,880
from-- which is part of the
logarithm that we're capturing.

323
00:22:16,880 --> 00:22:19,998
It's separating the hard physics
from the infrared physics.

324
00:22:19,998 --> 00:22:22,040
And here, we're just
looking at the hard physics.

325
00:22:22,040 --> 00:22:23,180
And we can see the Sudakov.

326
00:22:23,180 --> 00:22:24,847
We could also look
at the real radiation

327
00:22:24,847 --> 00:22:28,100
and see the Sudakov by studying
real radiation diagrams.

328
00:22:28,100 --> 00:22:29,973
But in some sense,
this is easier,

329
00:22:29,973 --> 00:22:32,390
because this is just coming
out of a renormalization group

330
00:22:32,390 --> 00:22:34,453
equation.

331
00:22:34,453 --> 00:22:35,870
So the usual way
of doing it would

332
00:22:35,870 --> 00:22:37,610
be to just look at
in the full theory,

333
00:22:37,610 --> 00:22:41,060
and you'd see Sudakov logarithms
from ratios of two things

334
00:22:41,060 --> 00:22:42,680
that you were thinking of as IR.

335
00:22:42,680 --> 00:22:45,890
Because the hard scale,
which is mbw here,

336
00:22:45,890 --> 00:22:49,280
would also be an IR scale
from the point of view of QCD.

337
00:22:49,280 --> 00:22:52,100
But here, it's become part
of the Wilson coefficient.

338
00:22:52,100 --> 00:22:57,290
And now it can be treated by
the renormalization group.

339
00:22:57,290 --> 00:23:01,082
So QCD is not really
any more complicated.

340
00:23:01,082 --> 00:23:03,040
We just have to deal with
the running coupling.

341
00:23:07,190 --> 00:23:09,900
We've done that before.

342
00:23:09,900 --> 00:23:15,310
Let me remind you how it
works because it's slightly

343
00:23:15,310 --> 00:23:17,450
more involved in this case.

344
00:23:17,450 --> 00:23:21,220
So for the leading logs, we
just need the beta 0 term.

345
00:23:21,220 --> 00:23:23,380
And we can switch
variables from d log to d

346
00:23:23,380 --> 00:23:25,787
alpha in this equation, here.

347
00:23:25,787 --> 00:23:28,120
And so then we have to integrate
this thing with respect

348
00:23:28,120 --> 00:23:31,120
to d alpha.

349
00:23:31,120 --> 00:23:34,630
And so what we can do
is we can take this log,

350
00:23:34,630 --> 00:23:37,930
and we can also write the log
as an integral over alpha.

351
00:23:49,340 --> 00:23:52,790
Alpha in this equation
is some w variable.

352
00:23:52,790 --> 00:23:58,760
And so then just integrating
both sides of this equation

353
00:23:58,760 --> 00:24:30,770
here, I can write
it out like that.

354
00:24:30,770 --> 00:24:34,330
So the log became this thing.

355
00:24:34,330 --> 00:24:36,030
And then one of these factors--

356
00:24:36,030 --> 00:24:37,390
there was a minus sign.

357
00:24:37,390 --> 00:24:39,900
But then when I use
this formula twice,

358
00:24:39,900 --> 00:24:42,850
the minus sign goes away, and
I get two of those factors.

359
00:24:42,850 --> 00:24:46,302
This is the other measure here
from the explicit d log mu.

360
00:24:46,302 --> 00:24:47,910
So we could underline things--

361
00:24:52,200 --> 00:24:59,565
half of this guy,
all of that guy, d

362
00:24:59,565 --> 00:25:04,640
log mu, other half of
this guy and that guy.

363
00:25:04,640 --> 00:25:07,270
And then there's this
explicit alpha cf

364
00:25:07,270 --> 00:25:10,150
over pi, which I've written
as cf over pi out front,

365
00:25:10,150 --> 00:25:11,410
and this alpha here.

366
00:25:23,150 --> 00:25:25,020
So we do that integral.

367
00:25:25,020 --> 00:25:25,910
We do this integral.

368
00:25:25,910 --> 00:25:27,410
Well, all the
intervals are trivial.

369
00:25:27,410 --> 00:25:30,290
We do this integral, and then
we get integration variable

370
00:25:30,290 --> 00:25:31,080
at fixed quantity.

371
00:25:31,080 --> 00:25:33,305
And then we can do
this other integral.

372
00:25:33,305 --> 00:25:34,055
We get a solution.

373
00:25:57,600 --> 00:25:59,510
So I'll show you what
the answer looks like.

374
00:25:59,510 --> 00:26:02,060
It's, again, an exponential,
because we're solving for a log

375
00:26:02,060 --> 00:26:02,560
c.

376
00:26:13,090 --> 00:26:14,330
I'll write it like this.

377
00:26:28,150 --> 00:26:30,070
So we're familiar that
renormalization group

378
00:26:30,070 --> 00:26:31,903
equations, when you
have a running coupling,

379
00:26:31,903 --> 00:26:33,980
can give you ratios of alphas.

380
00:26:33,980 --> 00:26:38,077
We saw that when we were
running for fermion operators.

381
00:26:38,077 --> 00:26:39,910
Here, it's just a more
complicated function,

382
00:26:39,910 --> 00:26:42,500
but it's again a
function of that ratio.

383
00:26:42,500 --> 00:26:44,842
And this extra structure
is coming about just

384
00:26:44,842 --> 00:26:45,925
from the running coupling.

385
00:26:51,690 --> 00:26:55,480
So the only difference
between this and QED Sudakov

386
00:26:55,480 --> 00:26:56,480
is the running coupling.

387
00:27:07,500 --> 00:27:14,370
So if we were to take this
now, and we were to expand z,

388
00:27:14,370 --> 00:27:17,100
we were, say, to expand
about alpha of w,

389
00:27:17,100 --> 00:27:20,430
then we would get logs
of mu over w, right?

390
00:27:20,430 --> 00:27:22,920
And so what you should
think that this z dependence

391
00:27:22,920 --> 00:27:23,610
encodes--

392
00:27:23,610 --> 00:27:29,490
is an infinite series
in this exponential.

393
00:27:29,490 --> 00:27:31,720
And so the structure of
that infinite series,

394
00:27:31,720 --> 00:27:38,490
if I was to expand, would be
alpha squared u over w alpha

395
00:27:38,490 --> 00:27:42,960
squared log cubed u
over w, et cetera.

396
00:27:46,850 --> 00:27:49,850
So in the exponent, a
running of the coupling

397
00:27:49,850 --> 00:27:52,620
is only giving you one
log for each alpha.

398
00:27:52,620 --> 00:27:55,280
So the series that I'm
solving in the exponent

399
00:27:55,280 --> 00:27:57,270
by taking into account
the running coupling

400
00:27:57,270 --> 00:27:59,720
is a series of this form,
where I go down by one alpha

401
00:27:59,720 --> 00:28:00,950
and down by one log.

402
00:28:04,614 --> 00:28:09,533
So from the structure of what
we've been talking about here,

403
00:28:09,533 --> 00:28:11,950
it should become clear that
the right thing to talk about,

404
00:28:11,950 --> 00:28:15,400
if you want to do
counting, is log c,

405
00:28:15,400 --> 00:28:18,250
because the anomalous dimension
equation was simple for log c.

406
00:28:18,250 --> 00:28:22,180
It was mu d by mu or
d by d log mu of log c

407
00:28:22,180 --> 00:28:23,620
is equal to something.

408
00:28:23,620 --> 00:28:26,945
That something on the right-hand
side is an expansion in alpha.

409
00:28:26,945 --> 00:28:28,570
And we could include
higher-order terms

410
00:28:28,570 --> 00:28:31,090
in that expansion, but it would
always be of that structure.

411
00:28:43,220 --> 00:28:47,440
So to talk about what
terms we're summing,

412
00:28:47,440 --> 00:28:50,830
we can write the following
schematic equation

413
00:28:50,830 --> 00:28:54,080
for different orders.

414
00:28:54,080 --> 00:28:57,040
So if we have alpha
s to the k log

415
00:28:57,040 --> 00:29:00,430
to the k plus 1, which is
this series I've just denoted,

416
00:29:00,430 --> 00:29:01,810
we'll call that leading log.

417
00:29:04,660 --> 00:29:07,360
And if we add some
higher-order term,

418
00:29:07,360 --> 00:29:10,295
like alpha s to
the k log to the k,

419
00:29:10,295 --> 00:29:11,920
then that would be
next-to-leading log.

420
00:29:19,910 --> 00:29:22,800
And if we had more
alphas than logs,

421
00:29:22,800 --> 00:29:24,740
then we call that
next-to-next-to leading log.

422
00:29:28,072 --> 00:29:30,280
And the structure of the
anomalous dimension equation

423
00:29:30,280 --> 00:29:34,180
always guarantees that if
you solve it for this log,

424
00:29:34,180 --> 00:29:36,580
that this is what the
higher-order terms would

425
00:29:36,580 --> 00:29:38,810
give you.

426
00:29:38,810 --> 00:29:42,010
So you should think of it
that really, log counting

427
00:29:42,010 --> 00:29:44,140
is kind of like normal log
counting, except for two

428
00:29:44,140 --> 00:29:46,720
caveats in this theory.

429
00:29:46,720 --> 00:29:49,900
You're doing it in
the exponential.

430
00:29:49,900 --> 00:29:51,790
That's what taking the log did.

431
00:29:51,790 --> 00:29:55,270
And you start out with one
extra log, one more log

432
00:29:55,270 --> 00:29:57,200
than you're used to.

433
00:29:57,200 --> 00:29:59,440
But after that,
it's kind of normal,

434
00:29:59,440 --> 00:30:02,560
because I'm summing
here alpha times log,

435
00:30:02,560 --> 00:30:05,100
and I just increment for
each alpha an extra log.

436
00:30:05,100 --> 00:30:06,940
And that's all the
running coupling effects.

437
00:30:06,940 --> 00:30:08,865
When I go to next-to-leading
log, of course,

438
00:30:08,865 --> 00:30:10,990
we write in coupling effects,
too, and other things

439
00:30:10,990 --> 00:30:13,390
that are causing the series.

440
00:30:13,390 --> 00:30:15,100
But it's down from
this one because it

441
00:30:15,100 --> 00:30:17,050
doesn't have that enhanced log.

442
00:30:17,050 --> 00:30:20,530
And then this one is down
again because it's, again,

443
00:30:20,530 --> 00:30:22,828
one less log.

444
00:30:22,828 --> 00:30:24,370
So this is the kind
of terms that you

445
00:30:24,370 --> 00:30:26,890
could sum by having
higher-order corrections

446
00:30:26,890 --> 00:30:29,080
in the anomalous dimensions.

447
00:30:29,080 --> 00:30:31,010
And for example, this
cusp anomalous dimension

448
00:30:31,010 --> 00:30:32,260
is known at three loop orders.

449
00:30:32,260 --> 00:30:37,880
So certainly, next-to-next-to
leading log is well within

450
00:30:37,880 --> 00:30:40,760
the realm of things
that people talk about.

451
00:30:40,760 --> 00:30:42,110
So let's ask that question.

452
00:30:42,110 --> 00:30:45,220
What coefficients, if we
were to carry out this one

453
00:30:45,220 --> 00:30:48,423
new calculation, and we
were to do higher loops,

454
00:30:48,423 --> 00:30:50,090
what coefficients do
we need to compute?

455
00:30:56,050 --> 00:30:58,860
So remember the story when we
were summing single log series

456
00:30:58,860 --> 00:31:00,480
was we would do
tree-level matching

457
00:31:00,480 --> 00:31:02,018
one loop anomalous dimension.

458
00:31:02,018 --> 00:31:04,560
Then we'd do one loop matching,
two loop anomalous dimension.

459
00:31:04,560 --> 00:31:06,330
What's the analog
of that story here?

460
00:31:12,857 --> 00:31:13,940
Let's make a little table.

461
00:31:18,120 --> 00:31:22,220
So what information do
we need at tree level?

462
00:31:22,220 --> 00:31:25,580
What information do
we need at one loop,

463
00:31:25,580 --> 00:31:29,000
two loops, three loops?

464
00:31:34,860 --> 00:31:36,390
So for a leading
log, what we did is

465
00:31:36,390 --> 00:31:38,340
we put in tree-level matching.

466
00:31:38,340 --> 00:31:40,308
And at one loop, we
only actually took

467
00:31:40,308 --> 00:31:42,600
the information that came
from the coefficient of the 1

468
00:31:42,600 --> 00:31:47,082
over epsilon squared, which is
the cusp anomalous dimension.

469
00:31:47,082 --> 00:31:49,350
[INAUDIBLE] here.

470
00:31:49,350 --> 00:31:51,270
If you go to one
higher order, it

471
00:31:51,270 --> 00:31:56,580
turns out that this is the
story to get this series.

472
00:32:01,970 --> 00:32:05,140
We would need the two-loop cusp
anomalous dimension, the one

473
00:32:05,140 --> 00:32:08,320
loop non-cusp, and still
the tree-level matching.

474
00:32:08,320 --> 00:32:11,500
So this next-to-leading
log coefficient, this 5cf,

475
00:32:11,500 --> 00:32:14,320
that's this 1 over epsilon.

476
00:32:14,320 --> 00:32:17,800
In our example,
this was the 5cf.

477
00:32:17,800 --> 00:32:21,100
And only when you get to
next-to-next-to-leading log,

478
00:32:21,100 --> 00:32:22,540
then the matching is one loop.

479
00:32:26,508 --> 00:32:30,880
We have the two-loop
regular anomalous dimension,

480
00:32:30,880 --> 00:32:32,380
and then the 1 over
epsilon squared

481
00:32:32,380 --> 00:32:34,780
cusp anomalous dimension
from three loop.

482
00:32:34,780 --> 00:32:37,330
So it's kind of like
it's the usual story,

483
00:32:37,330 --> 00:32:39,132
but we have an
enhanced-- because

484
00:32:39,132 --> 00:32:41,590
of the double logarithms, we
have an enhanced thing that we

485
00:32:41,590 --> 00:32:43,603
can talk about here,
where we're not

486
00:32:43,603 --> 00:32:45,520
talking about the regular
anomalous dimension,

487
00:32:45,520 --> 00:32:48,372
but this cusp
anomalous dimension.

488
00:32:48,372 --> 00:32:50,080
So this is the
information that you would

489
00:32:50,080 --> 00:32:52,030
need to go to higher orders.

490
00:32:52,030 --> 00:32:53,500
And this cusp
anomalous dimension

491
00:32:53,500 --> 00:32:54,865
is actually a universal thing.

492
00:32:54,865 --> 00:33:01,680
So given that it's been
calculated, you can use it.

493
00:33:01,680 --> 00:33:03,933
And you don't have to
recalculate it every time.

494
00:33:03,933 --> 00:33:05,850
These pieces you have
to recalculate-- these 1

495
00:33:05,850 --> 00:33:11,541
over epsilons, in general.

496
00:33:11,541 --> 00:33:12,041
Yeah?

497
00:33:12,041 --> 00:33:16,190
STUDENT: Can you
comment on the seemingly

498
00:33:16,190 --> 00:33:20,690
arbitrariness of the
boundary condition you chose?

499
00:33:20,690 --> 00:33:24,040
PROFESSOR: Yeah, so if I hadn't
chose that boundary condition--

500
00:33:24,040 --> 00:33:24,910
sure.

501
00:33:24,910 --> 00:33:26,325
Let me write something.

502
00:33:41,802 --> 00:33:43,760
So the way that you could
think about the rg is

503
00:33:43,760 --> 00:33:46,520
as follows-- that you have c.

504
00:33:46,520 --> 00:33:49,610
And really what the rg is
doing is determining some u.

505
00:33:54,120 --> 00:33:59,130
It allows you to run c from
some point to some other point,

506
00:33:59,130 --> 00:34:00,610
like that.

507
00:34:00,610 --> 00:34:04,620
So in this formula, mu
0 in some scale that's

508
00:34:04,620 --> 00:34:08,530
of order omega or w-- it
doesn't have to be equal to w.

509
00:34:08,530 --> 00:34:10,433
It could be 2w, 1/2w.

510
00:34:10,433 --> 00:34:12,600
And the way that you're
thinking about this equation

511
00:34:12,600 --> 00:34:13,590
is you're going
to do perturbation

512
00:34:13,590 --> 00:34:14,507
theory for this thing.

513
00:34:19,940 --> 00:34:22,330
And then in that perturbation
theory, you'll have alpha,

514
00:34:22,330 --> 00:34:26,480
and you'll have log
squared of mu 0 over w.

515
00:34:26,480 --> 00:34:28,659
But as long as you're
saying mu 0's of order w,

516
00:34:28,659 --> 00:34:30,610
those aren't large logs.

517
00:34:30,610 --> 00:34:33,280
And this thing here
in this formulation

518
00:34:33,280 --> 00:34:36,880
would have the double log
series of alpha log squared

519
00:34:36,880 --> 00:34:39,370
of mu over mu0.

520
00:34:39,370 --> 00:34:40,864
But then mu0's of order w.

521
00:34:40,864 --> 00:34:42,239
And mu you can
take much smaller,

522
00:34:42,239 --> 00:34:43,464
and that's the large logs.

523
00:34:47,510 --> 00:34:51,659
So for simplicity, I
just took mu0 equal to w.

524
00:34:51,659 --> 00:34:53,560
Everything I've
said, I could repeat,

525
00:34:53,560 --> 00:34:55,893
and this story would really
be the same if I just

526
00:34:55,893 --> 00:34:56,810
used an arbitrary mu0.

527
00:35:00,950 --> 00:35:03,140
And the reason you actually
want to use an arbitrary

528
00:35:03,140 --> 00:35:06,560
mu0 in general is that you
want to do scale variation

529
00:35:06,560 --> 00:35:08,430
to think about uncertainties.

530
00:35:08,430 --> 00:35:11,060
And so the mu0 dependence,
just like in our--

531
00:35:11,060 --> 00:35:13,370
when we were talking about
electroweak operators

532
00:35:13,370 --> 00:35:16,190
and electric Hamiltonian,
the mu0 dependence

533
00:35:16,190 --> 00:35:18,200
cancels between
these things here.

534
00:35:18,200 --> 00:35:21,770
But that calculation is in order
by order an alpha cancellation.

535
00:35:21,770 --> 00:35:25,410
There's no large logs associated
with that cancellation.

536
00:35:25,410 --> 00:35:26,915
And so what it
allows you to do is

537
00:35:26,915 --> 00:35:28,790
if you've truncated this
to some fixed order,

538
00:35:28,790 --> 00:35:31,070
and you've worked to some
fixed order over there,

539
00:35:31,070 --> 00:35:33,590
then you can probe how
much uncertainty you

540
00:35:33,590 --> 00:35:36,230
have by varying the mu0.

541
00:35:36,230 --> 00:35:40,880
This mu here will get tied up
with the mu in the operator.

542
00:35:40,880 --> 00:35:44,270
So we can talk more about
where that mu goes later on.

543
00:35:44,270 --> 00:35:47,660
But at the level of this
formula, it is-- in general,

544
00:35:47,660 --> 00:35:50,450
it's the case that you would
do it with an arbitrary mu0,

545
00:35:50,450 --> 00:35:52,670
and then only fix
mu0 at the end.

546
00:35:52,670 --> 00:35:55,935
And you'd even vary mu0 to get
an idea of how much uncertainty

547
00:35:55,935 --> 00:35:58,310
you have in your leading log
calculation, because varying

548
00:35:58,310 --> 00:36:02,640
mu0 would probe the next
term in the series, here.

549
00:36:02,640 --> 00:36:06,620
STUDENT: So when you say
tree-level matching, you mean--

550
00:36:06,620 --> 00:36:07,880
PROFESSOR: I mean this c.

551
00:36:07,880 --> 00:36:08,797
STUDENT: --equal to 1.

552
00:36:08,797 --> 00:36:10,220
PROFESSOR: That's right.

553
00:36:10,220 --> 00:36:11,520
I mean this guy, here.

554
00:36:11,520 --> 00:36:12,020
Right.

555
00:36:12,020 --> 00:36:12,645
STUDENT: Right.

556
00:36:12,645 --> 00:36:15,007
And if you want to go
to nnll, you'd take 1--

557
00:36:15,007 --> 00:36:16,340
PROFESSOR: Then you'd have this.

558
00:36:16,340 --> 00:36:17,650
That wouldn't have this term.

559
00:36:17,650 --> 00:36:18,150
Yeah.

560
00:36:22,140 --> 00:36:26,020
All right, any other questions?

561
00:36:26,020 --> 00:36:28,020
OK, so what is this
cusp that we've

562
00:36:28,020 --> 00:36:29,640
been saying the words for?

563
00:36:36,290 --> 00:36:39,870
I'll answer a couple of
questions that have come up.

564
00:36:39,870 --> 00:36:40,760
Where's the cusp?

565
00:36:44,970 --> 00:36:57,540
So if we look back at our
operator, we had this operator.

566
00:36:57,540 --> 00:37:03,100
And we could make a field
redefinition, if you remember,

567
00:37:03,100 --> 00:37:06,310
and put the ultrasoft
effects into a Wilson line.

568
00:37:10,800 --> 00:37:14,250
Now for this heavy quark
here, remember what that--

569
00:37:14,250 --> 00:37:17,070
the theory for
that was with hqet.

570
00:37:17,070 --> 00:37:23,560
So the field theory for this
heavy quark was an iv dot dhv.

571
00:37:23,560 --> 00:37:26,940
And if we actually make a
similar field redefinition

572
00:37:26,940 --> 00:37:29,430
to the one we talked
about to get this y,

573
00:37:29,430 --> 00:37:32,520
but on the heavy
quark field, then we

574
00:37:32,520 --> 00:37:37,050
can actually, in
terms of this guy

575
00:37:37,050 --> 00:37:40,540
here, get a free
Lagrangian as well.

576
00:37:40,540 --> 00:37:43,770
So actually, all the effects
from ultra soft gluons

577
00:37:43,770 --> 00:37:47,390
in this operator can be
encoded in Wilson lines.

578
00:37:51,872 --> 00:37:53,130
So let me do that.

579
00:37:57,560 --> 00:37:59,990
So that just says
that actually, when

580
00:37:59,990 --> 00:38:02,330
we talked about static
sources, static sources

581
00:38:02,330 --> 00:38:05,870
could also be encoded
as Wilson lines

582
00:38:05,870 --> 00:38:07,310
by a kind of
similar manipulation

583
00:38:07,310 --> 00:38:11,210
to what we did when
we talked about SET.

584
00:38:11,210 --> 00:38:13,010
So if we look at the
ultrasoft sector here,

585
00:38:13,010 --> 00:38:15,230
we have a Wilson line
that actually has a path.

586
00:38:28,380 --> 00:38:30,020
So there's a Wilson
line that comes

587
00:38:30,020 --> 00:38:33,080
from minus infinity along
v to whatever position

588
00:38:33,080 --> 00:38:34,070
we put our operator at.

589
00:38:34,070 --> 00:38:35,810
Let's take it to be 0.

590
00:38:35,810 --> 00:38:37,490
And then we have
a yn dagger which

591
00:38:37,490 --> 00:38:43,340
is extending out to plus
infinity from 0, like that.

592
00:38:43,340 --> 00:38:45,345
And the cusp is this
fact that there's

593
00:38:45,345 --> 00:38:46,470
a kink in this Wilson line.

594
00:38:46,470 --> 00:38:47,660
It's not a smooth path.

595
00:38:47,660 --> 00:38:49,640
It actually has a sharp angle.

596
00:38:49,640 --> 00:38:50,660
And that's the cusp.

597
00:39:05,370 --> 00:39:07,580
So there's actually a general
renormalization theory

598
00:39:07,580 --> 00:39:10,670
for renormalizing
Wilson lines with cusps.

599
00:39:10,670 --> 00:39:12,890
So this is something
that at some point

600
00:39:12,890 --> 00:39:15,770
in the history of QCD, people
tried to reformulate QCD

601
00:39:15,770 --> 00:39:17,750
in terms of Wilson
lines, entirely

602
00:39:17,750 --> 00:39:19,280
in terms of Wilson lines.

603
00:39:19,280 --> 00:39:21,890
Then they realized that
they could have these cusps

604
00:39:21,890 --> 00:39:22,760
in the Wilson lines.

605
00:39:22,760 --> 00:39:24,380
And the renormalization
of that theory

606
00:39:24,380 --> 00:39:26,588
became much more complicated
than the renormalization

607
00:39:26,588 --> 00:39:28,460
of the QCD action
because they couldn't

608
00:39:28,460 --> 00:39:30,627
prove that in general, these
cusps wouldn't come out

609
00:39:30,627 --> 00:39:31,880
of the dynamics.

610
00:39:31,880 --> 00:39:33,260
And then people dropped it.

611
00:39:33,260 --> 00:39:36,560
But along the way,
anyway, they formulated

612
00:39:36,560 --> 00:39:41,085
a general renormalization
theory of cusps in Wilson lines.

613
00:39:41,085 --> 00:39:42,710
And whenever you have
a cusp like this,

614
00:39:42,710 --> 00:39:45,450
you end up having, in
our language, this log,

615
00:39:45,450 --> 00:39:48,350
single log, in the
anomalous dimension.

616
00:39:48,350 --> 00:39:59,880
So if one of the lines here
is light-like, and one of them

617
00:39:59,880 --> 00:40:06,390
is, because n squared was 0,
then the anomalous dimension

618
00:40:06,390 --> 00:40:14,520
for this cusp will have a single
log like our log of mu over w.

619
00:40:14,520 --> 00:40:16,770
So that actually holds
not only to one loop,

620
00:40:16,770 --> 00:40:19,650
as we talked about, but
actually to all orders

621
00:40:19,650 --> 00:40:20,730
in perturbation theory.

622
00:40:20,730 --> 00:40:24,210
If we'd gone to higher orders
in perturbation theory,

623
00:40:24,210 --> 00:40:29,167
then what would happen is that
the coefficient of that log

624
00:40:29,167 --> 00:40:30,000
would get corrected.

625
00:40:30,000 --> 00:40:33,750
But there'd still be only one
log in our anomalous dimension.

626
00:40:33,750 --> 00:40:39,900
That's also something that you
can argue from SET directly.

627
00:40:39,900 --> 00:40:43,920
But originally, it falls
from this general theory

628
00:40:43,920 --> 00:40:46,500
of the renormalization
of cusps of Wilson lines.

629
00:40:52,550 --> 00:40:55,160
So if you were thinking
about our calculation,

630
00:40:55,160 --> 00:40:57,830
you may remember actually that
we got 1 over epsilon squared.

631
00:40:57,830 --> 00:40:59,450
It's from the
ultrasoft diagrams.

632
00:40:59,450 --> 00:41:02,630
And that's in some sense
what we've just talked about,

633
00:41:02,630 --> 00:41:05,090
because these are the
ultrasoft diagrams,

634
00:41:05,090 --> 00:41:07,197
and they have 1 over
epsilon squareds.

635
00:41:07,197 --> 00:41:09,530
There was also 1 over epsilon
squareds from the colinear

636
00:41:09,530 --> 00:41:12,350
diagrams, right?

637
00:41:12,350 --> 00:41:17,750
So if you like, this piece
here also in some sense

638
00:41:17,750 --> 00:41:19,910
has kind of a cusp.

639
00:41:19,910 --> 00:41:21,625
And in this case, it's not--

640
00:41:21,625 --> 00:41:23,000
it's actually what
we're doing is

641
00:41:23,000 --> 00:41:26,510
we're taking a Wilson line
along the n bar direction, which

642
00:41:26,510 --> 00:41:27,085
is our wn.

643
00:41:29,870 --> 00:41:34,610
And then we're attaching
it to a quark field.

644
00:41:34,610 --> 00:41:37,760
So this is a full
cn quark field.

645
00:41:37,760 --> 00:41:38,820
It's not a Wilson line.

646
00:41:38,820 --> 00:41:41,300
So you can think of it as
just ending on a quark field.

647
00:41:41,300 --> 00:41:43,160
But even just ending
on a quark field,

648
00:41:43,160 --> 00:41:44,780
the quark field
also has dynamics.

649
00:41:44,780 --> 00:41:46,640
It's not as simple as
Wilson line dynamics,

650
00:41:46,640 --> 00:41:48,350
but it has dynamics.

651
00:41:48,350 --> 00:41:51,080
And we have interactions
between these.

652
00:41:51,080 --> 00:41:53,040
And this is also kind of a cusp.

653
00:41:53,040 --> 00:41:55,013
Although it's not a
simple Wilson line cusp,

654
00:41:55,013 --> 00:41:56,055
it's also kind of a cusp.

655
00:42:07,120 --> 00:42:09,250
So even the colinear graphs
with the Wilson lines

656
00:42:09,250 --> 00:42:12,940
ended on quarks also produced
these q over epsilon squareds

657
00:42:12,940 --> 00:42:15,010
and have this kind of structure.

658
00:42:15,010 --> 00:42:17,870
And they actually have similar--

659
00:42:17,870 --> 00:42:20,660
very similar relations to the
cusps in regular Wilson lines.

660
00:42:23,390 --> 00:42:26,540
So that was one question--
what was this cusp?

661
00:42:26,540 --> 00:42:29,420
Another question that
came up was this fact

662
00:42:29,420 --> 00:42:32,390
that the w got fixed.

663
00:42:32,390 --> 00:42:34,010
And so we could
ask the question,

664
00:42:34,010 --> 00:42:35,600
in general, when
will that happen?

665
00:42:46,630 --> 00:42:48,390
So when will the w's
that are showing up

666
00:42:48,390 --> 00:42:53,790
in our Wilson coefficients be
fixed by external kinematics?

667
00:43:04,660 --> 00:43:08,660
And actually, that's going to
happen in the following case.

668
00:43:08,660 --> 00:43:12,975
We can actually state when it
will happen quite generally.

669
00:43:18,310 --> 00:43:21,250
So imagine that we had an
effective theory SET that had

670
00:43:21,250 --> 00:43:23,800
multiple colinear directions.

671
00:43:23,800 --> 00:43:25,300
So then we could
build operators out

672
00:43:25,300 --> 00:43:28,540
of all those different
colinear fields.

673
00:43:28,540 --> 00:43:33,980
And we would do it with our
building blocks, which are

674
00:43:33,980 --> 00:43:41,120
chi n or curley bn for each n.

675
00:43:41,120 --> 00:43:45,800
And if our operator only
involves one building block

676
00:43:45,800 --> 00:43:51,170
for each direction,
then it's going

677
00:43:51,170 --> 00:43:56,630
to be the case that the value
of those labels or those omegas

678
00:43:56,630 --> 00:43:58,850
are always going to be fixed
by external kinematics.

679
00:44:05,100 --> 00:44:09,520
So let me just write
down another example.

680
00:44:09,520 --> 00:44:11,880
Imagine we had something
like an LHC process,

681
00:44:11,880 --> 00:44:16,800
where we had two gluons coming
in and two quarks going out.

682
00:44:16,800 --> 00:44:20,863
But we'll think about
this as protons colliding,

683
00:44:20,863 --> 00:44:22,530
coming in from two
different directions.

684
00:44:22,530 --> 00:44:25,140
Let's call them n2 and n3--

685
00:44:25,140 --> 00:44:28,980
and then going to two jets
transverse to the axis

686
00:44:28,980 --> 00:44:29,800
by a large amount.

687
00:44:29,800 --> 00:44:32,560
So there are also two
different directions,

688
00:44:32,560 --> 00:44:34,680
which I can call n1 and n4.

689
00:44:34,680 --> 00:44:40,430
So this process of glu-glu
goes to qq bar, which

690
00:44:40,430 --> 00:44:45,110
is really pp goes to dijets.

691
00:44:45,110 --> 00:44:48,110
As long as those dijets are
well-separated from the b

692
00:44:48,110 --> 00:44:50,750
maxes, which is the case we're
interested in, then we have

693
00:44:50,750 --> 00:44:54,080
this situation, where we
have four different n's.

694
00:44:54,080 --> 00:44:56,750
And so what kind of operator
would you write down for that?

695
00:44:56,750 --> 00:45:04,550
Well, you have
dw1, dw2, dw3, dw4.

696
00:45:04,550 --> 00:45:08,270
You'd have a Wilson coefficient
that could depend on four

697
00:45:08,270 --> 00:45:09,350
different w's.

698
00:45:09,350 --> 00:45:12,140
And you'd then write
down some operator

699
00:45:12,140 --> 00:45:13,730
out of building blocks, n1w1.

700
00:45:17,570 --> 00:45:20,583
I'm not going to worry
about all the indices,

701
00:45:20,583 --> 00:45:22,000
but I'll worry
about some of them.

702
00:45:32,177 --> 00:45:33,260
That would look like that.

703
00:45:33,260 --> 00:45:37,050
So I just put down some operator
that's got the right structure.

704
00:45:37,050 --> 00:45:38,450
It's got two quarks, two gluons.

705
00:45:38,450 --> 00:45:41,330
I make the gluons
these b [INAUDIBLE]..

706
00:45:41,330 --> 00:45:43,555
I've chosen to contract them.

707
00:45:43,555 --> 00:45:45,180
Each of them has a
different direction.

708
00:45:45,180 --> 00:45:47,630
This is in n1.

709
00:45:47,630 --> 00:45:51,260
And each of them also gets a
corresponding large momentum.

710
00:45:51,260 --> 00:45:55,183
And the Wilson coefficient can
depend on those large momenta.

711
00:45:55,183 --> 00:45:56,600
So this would be
the operator that

712
00:45:56,600 --> 00:45:59,260
would describe that process.

713
00:45:59,260 --> 00:46:02,930
But again, if you think
about the renormalization,

714
00:46:02,930 --> 00:46:06,290
then if you think about the
renormalization for a minute,

715
00:46:06,290 --> 00:46:07,983
the colinear
diagrams aren't going

716
00:46:07,983 --> 00:46:09,650
to involve contractions
between this guy

717
00:46:09,650 --> 00:46:11,150
and any of these other
guys because these guys are

718
00:46:11,150 --> 00:46:11,930
totally independent.

719
00:46:11,930 --> 00:46:13,010
They're a different Lagrangian.

720
00:46:13,010 --> 00:46:15,093
So the contractions are,
again, just like the kind

721
00:46:15,093 --> 00:46:16,550
of calculation that we did.

722
00:46:16,550 --> 00:46:20,400
The colinear diagrams are just
coming from this thing alone,

723
00:46:20,400 --> 00:46:22,268
and they don't care
about that stuff.

724
00:46:22,268 --> 00:46:23,810
So it's actually
the same calculation

725
00:46:23,810 --> 00:46:26,120
that we already did
for b to s gamma

726
00:46:26,120 --> 00:46:28,110
if I wanted to do the
colinear diagrams here.

727
00:46:28,110 --> 00:46:29,777
And then I would have
to do the colinear

728
00:46:29,777 --> 00:46:32,600
diagrams for each of these guys,
but it's kind of independent.

729
00:46:32,600 --> 00:46:35,970
And again, the w1 is external
for that calculation.

730
00:46:35,970 --> 00:46:38,330
So the anomalous
dimension would just--

731
00:46:38,330 --> 00:46:39,800
it would be outside it.

732
00:46:39,800 --> 00:46:41,480
It wouldn't
involve-- it wouldn't

733
00:46:41,480 --> 00:46:47,030
appear and get thrown into
some loop integral momentum.

734
00:46:47,030 --> 00:46:49,730
And so if you think about
what can fix these w's, it's

735
00:46:49,730 --> 00:46:51,920
really only external
information.

736
00:46:51,920 --> 00:46:55,580
So what are the
momentum fractions

737
00:46:55,580 --> 00:46:58,460
of the incoming
quarks from the PDFs?

738
00:46:58,460 --> 00:47:00,800
What are the energies
of the outgoing jets?

739
00:47:00,800 --> 00:47:03,620
That's the type of information
that, even in this case,

740
00:47:03,620 --> 00:47:04,520
would fix the w's.

741
00:47:20,208 --> 00:47:22,250
So if we were to calculate
the Wilson coefficient

742
00:47:22,250 --> 00:47:24,710
for this operator and do
its renormalization group,

743
00:47:24,710 --> 00:47:25,580
there would again--

744
00:47:25,580 --> 00:47:27,260
it would again be
of a product form.

745
00:47:38,100 --> 00:47:40,340
There's no convolution,
just a product

746
00:47:40,340 --> 00:47:43,900
for this Wilson coefficient.

747
00:47:43,900 --> 00:47:47,080
So when could that not be true?

748
00:47:47,080 --> 00:47:49,900
It could not be true
if, for example, we

749
00:47:49,900 --> 00:47:52,862
had a chi bar and a chi that
were in the same direction.

750
00:47:57,590 --> 00:48:02,320
So if we're in a situation
where all the objects are

751
00:48:02,320 --> 00:48:07,660
different directions,
that's what happens.

752
00:48:07,660 --> 00:48:08,710
And it's pretty simple.

753
00:48:13,250 --> 00:48:20,320
But let's also do an
example when is it not true.

754
00:48:20,320 --> 00:48:22,670
It's always good to
have a counterexample.

755
00:48:22,670 --> 00:48:25,045
And it would not be true if
we had an operator like this.

756
00:48:38,710 --> 00:48:41,070
So here's an operator
with just two quark

757
00:48:41,070 --> 00:48:42,950
building-block fields.

758
00:48:42,950 --> 00:48:45,450
I gave them different labels,
because in general, the Wilson

759
00:48:45,450 --> 00:48:47,890
coefficient could depend
on their large momenta.

760
00:48:47,890 --> 00:48:54,662
But now, they're in
the same direction n,

761
00:48:54,662 --> 00:48:58,410
so they belong to the
same colinear sector.

762
00:48:58,410 --> 00:49:01,380
And in this operator here,
we would actually no longer

763
00:49:01,380 --> 00:49:05,100
be in the case where this
thing would totally decouple

764
00:49:05,100 --> 00:49:06,650
from the loop integrals.

765
00:49:20,595 --> 00:49:22,470
So if we think about
inserting this operator,

766
00:49:22,470 --> 00:49:25,100
then some of these guys here
will involve loop momenta.

767
00:49:30,300 --> 00:49:33,960
Actually, it turns out that one
combination of the w's is still

768
00:49:33,960 --> 00:49:35,970
fixed by external kinematics.

769
00:49:35,970 --> 00:49:38,640
And so there's an overall--

770
00:49:38,640 --> 00:49:41,550
think about the overall delta
function on this product.

771
00:49:41,550 --> 00:49:42,730
That guy would be fixed.

772
00:49:42,730 --> 00:49:47,130
And then some difference of
these w's would not be fixed.

773
00:49:47,130 --> 00:49:51,970
That's the true story about
at least one combination

774
00:49:51,970 --> 00:49:53,220
that involve the loop momenta.

775
00:50:13,790 --> 00:50:15,050
And what does that lead to?

776
00:50:15,050 --> 00:50:17,600
How does it complicate
what we've already said?

777
00:50:17,600 --> 00:50:23,980
What that leads to is
that when we formulate

778
00:50:23,980 --> 00:50:27,870
the anomalous
dimension equations,

779
00:50:27,870 --> 00:50:29,802
they also involve integrals.

780
00:50:48,520 --> 00:50:53,205
So for this guy here, for the
one momentum that's not fixed--

781
00:50:53,205 --> 00:50:54,330
so in general, there's two.

782
00:50:54,330 --> 00:50:56,940
And this is the one
that's not fixed.

783
00:50:56,940 --> 00:50:58,890
We would get an equation
that looks like this.

784
00:51:06,420 --> 00:51:08,310
Rather than the
simple product form,

785
00:51:08,310 --> 00:51:11,615
there'd be an integral
on the right-hand side.

786
00:51:11,615 --> 00:51:13,740
And indeed, actually, if
you do the renormalization

787
00:51:13,740 --> 00:51:17,280
of this operator that I wrote
up there, just that simple quark

788
00:51:17,280 --> 00:51:23,370
operator, you actually
reproduce a bunch

789
00:51:23,370 --> 00:51:31,140
of classical evolution
equations that are of this form.

790
00:51:39,660 --> 00:51:41,730
So even just the
simplest possible case

791
00:51:41,730 --> 00:51:44,780
we can think of where
things are not fixed

792
00:51:44,780 --> 00:51:49,010
gives us a bunch of
interesting results.

793
00:51:49,010 --> 00:51:52,330
So for deep
inelastic scattering,

794
00:51:52,330 --> 00:51:56,530
then what you're getting is
the Altarelli-Parisi equation

795
00:51:56,530 --> 00:51:57,640
or DGLAP equation.

796
00:52:07,700 --> 00:52:09,140
And the evolution
of the operator

797
00:52:09,140 --> 00:52:11,960
would be the evolution of
the part-time distribution

798
00:52:11,960 --> 00:52:13,528
function.

799
00:52:13,528 --> 00:52:16,070
That's one thing that would be
encoded in the renormalization

800
00:52:16,070 --> 00:52:18,030
of that equation.

801
00:52:18,030 --> 00:52:21,410
There's other processes
that are actually--

802
00:52:21,410 --> 00:52:23,780
some of which we may
talk about later on--

803
00:52:23,780 --> 00:52:26,107
that are also encoded
in the same type

804
00:52:26,107 --> 00:52:28,190
of operator, kind of a
different projection of it.

805
00:52:31,142 --> 00:52:32,850
And one of them leads
to something called

806
00:52:32,850 --> 00:52:34,020
the Brodsky-Lepage equation.

807
00:52:40,800 --> 00:52:43,890
And then there's another
one that's showing up

808
00:52:43,890 --> 00:52:45,270
in something called--

809
00:52:45,270 --> 00:52:48,407
a process called deeply
virtual Compton scattering.

810
00:52:55,260 --> 00:52:57,540
So the one that's most
familiar is the renormalization

811
00:52:57,540 --> 00:52:58,147
of the PDF.

812
00:52:58,147 --> 00:53:00,480
And you may remember that the
renormalization of the PDF

813
00:53:00,480 --> 00:53:03,990
has this form of
having an integral.

814
00:53:03,990 --> 00:53:07,750
That'll come out of
renormalization this operator.

815
00:53:07,750 --> 00:53:08,790
And we'll do that case.

816
00:53:20,500 --> 00:53:25,870
So in general, the structure
of the effective theory

817
00:53:25,870 --> 00:53:29,890
is leading us to find
out what the structure

818
00:53:29,890 --> 00:53:32,620
of the renormalization is.

819
00:53:32,620 --> 00:53:36,010
And it can reproduce some
well-known, classic things,

820
00:53:36,010 --> 00:53:37,920
but it comes out
actually pretty easy.

821
00:53:44,050 --> 00:53:47,430
So I'm going to do
this case, but I'm

822
00:53:47,430 --> 00:53:49,710
going to do this case
in complete-- with all

823
00:53:49,710 --> 00:53:51,450
of the details filled in.

824
00:53:51,450 --> 00:53:53,550
So first, I have to
convince you that

825
00:53:53,550 --> 00:53:57,010
in deep inelastic scattering,
you actually get that operator.

826
00:53:57,010 --> 00:53:59,130
And we'll see
precisely what kind

827
00:53:59,130 --> 00:54:00,600
of matrix elements
of that operator

828
00:54:00,600 --> 00:54:02,308
show up in deep
inelastic scattering, why

829
00:54:02,308 --> 00:54:03,455
we get that operator.

830
00:54:03,455 --> 00:54:05,830
And then once we're convinced
that we have that operator,

831
00:54:05,830 --> 00:54:07,830
and we'll actually drive
a factorization theorem

832
00:54:07,830 --> 00:54:09,420
that involves that operator.

833
00:54:09,420 --> 00:54:11,730
Then we'll talk about
its renormalization group

834
00:54:11,730 --> 00:54:12,330
evolution.

835
00:54:12,330 --> 00:54:14,520
And I'll show you that
it has this form that I'm

836
00:54:14,520 --> 00:54:16,320
writing here.

837
00:54:16,320 --> 00:54:19,650
So probably for the rest of
today's lecture, certainly

838
00:54:19,650 --> 00:54:22,350
for the rest of today's lecture,
we won't get to the running.

839
00:54:22,350 --> 00:54:23,350
We'll do that next time.

840
00:54:23,350 --> 00:54:27,540
But we'll at least get
to the point, I think,

841
00:54:27,540 --> 00:54:30,315
where you'll see why that
operator is showing up in DIS.

842
00:54:34,540 --> 00:54:35,480
So let's do DIS.

843
00:54:40,090 --> 00:54:52,900
So I'm only going to talk
in DIS about factorization,

844
00:54:52,900 --> 00:54:58,060
and then the renormalization
group evolution.

845
00:54:58,060 --> 00:55:00,280
So no phenomenology, nothing
like that-- we'll just

846
00:55:00,280 --> 00:55:03,770
talk about these two concepts.

847
00:55:03,770 --> 00:55:09,760
So DIS is electron-proton to
electron anything, so just

848
00:55:09,760 --> 00:55:10,600
kinematics.

849
00:55:13,832 --> 00:55:15,415
Think about a virtual
photon exchange.

850
00:55:19,270 --> 00:55:21,640
Proton comes in.

851
00:55:21,640 --> 00:55:23,290
I'll say that the
proton's momentum is

852
00:55:23,290 --> 00:55:29,720
a capital P. Gets blown apart.

853
00:55:29,720 --> 00:55:31,765
Call the stuff that's
blown apart Px.

854
00:55:34,300 --> 00:55:40,420
And so Px is the sum
of all the particles,

855
00:55:40,420 --> 00:55:41,635
all the final-state hadrons.

856
00:55:48,060 --> 00:55:49,950
Q squared of the
virtual photon--

857
00:55:49,950 --> 00:55:50,880
so this is q.

858
00:55:53,700 --> 00:55:56,400
Little q squared is
minus capital Q squared.

859
00:55:56,400 --> 00:55:58,260
And this thing is much
bigger than on QCD.

860
00:56:02,530 --> 00:56:08,140
mu arc in x is capital Q squared
divided by this dot product.

861
00:56:12,820 --> 00:56:17,680
And you can talk about
Px, which is the Px mu.

862
00:56:17,680 --> 00:56:19,450
And by momentum
conservation, that's

863
00:56:19,450 --> 00:56:21,910
the proton momentum, plus
whatever momentum came in

864
00:56:21,910 --> 00:56:23,080
from the leptons q.

865
00:56:26,990 --> 00:56:30,650
So Px squared--
if you square it,

866
00:56:30,650 --> 00:56:33,750
well, this guy is giving you
mass of the protons squared.

867
00:56:33,750 --> 00:56:36,570
And then there's a dot-- a cross
term, and this guy squared.

868
00:56:36,570 --> 00:56:44,319
And if you put those
things together like this,

869
00:56:44,319 --> 00:56:47,912
so this is an exact equation.

870
00:56:47,912 --> 00:56:50,180
Px squared is that.

871
00:56:50,180 --> 00:56:52,430
And so there's actually
different regions of DIS,

872
00:56:52,430 --> 00:56:54,388
and we're only going to
talk about one of them.

873
00:56:56,870 --> 00:57:00,230
I have to enumerate what
I'm talking about carefully.

874
00:57:00,230 --> 00:57:06,990
And I can do that by looking
at Px squared or this factor

875
00:57:06,990 --> 00:57:12,140
1 minus x over x, which
I'll call 1 over x minus 1.

876
00:57:12,140 --> 00:57:14,652
If this thing is
of order q squared,

877
00:57:14,652 --> 00:57:16,610
then that means that this
thing you're counting

878
00:57:16,610 --> 00:57:18,320
is of quarter 1.

879
00:57:18,320 --> 00:57:21,590
And in that case, it's what's
called the inclusive operator

880
00:57:21,590 --> 00:57:23,790
product expansion.

881
00:57:23,790 --> 00:57:28,010
So this is the case that
most books would deal with.

882
00:57:28,010 --> 00:57:30,920
And that's the one we'll
deal with, actually,

883
00:57:30,920 --> 00:57:33,320
where effectively, we're not
putting restrictions on x.

884
00:57:33,320 --> 00:57:35,570
We're just saying it's
generic, and it's not

885
00:57:35,570 --> 00:57:37,430
approaching any endpoints.

886
00:57:37,430 --> 00:57:42,740
There's also a situation
where the Px gets smaller.

887
00:57:42,740 --> 00:57:47,307
And then this thing is close to
an endpoint, lambda QCD over q.

888
00:57:47,307 --> 00:57:48,890
And that's called
the endpoint region.

889
00:57:52,070 --> 00:57:54,530
And people talk about that.

890
00:57:54,530 --> 00:57:56,740
Usually when people
say x goes to 1,

891
00:57:56,740 --> 00:57:59,110
this is what they mean--
that the 1 minus x minus 1

892
00:57:59,110 --> 00:58:03,730
is of that size
on the QCD over q.

893
00:58:03,730 --> 00:58:07,000
There's even a third region,
where the Px squared becomes

894
00:58:07,000 --> 00:58:12,410
hadronic lambda QCD squared.

895
00:58:12,410 --> 00:58:16,690
And that's like
here, two powers.

896
00:58:16,690 --> 00:58:18,670
So taking that factor
to be really small.

897
00:58:18,670 --> 00:58:21,980
And that's the resonance region.

898
00:58:21,980 --> 00:58:25,300
And that's the case where
the final state x is just

899
00:58:25,300 --> 00:58:27,445
another proton or an
excited state of a proton.

900
00:58:31,977 --> 00:58:33,810
So that's where elastic
scattering would be.

901
00:58:33,810 --> 00:58:39,750
And that's an exclusive process.

902
00:58:39,750 --> 00:58:41,393
It's not inclusive anymore.

903
00:58:47,070 --> 00:58:50,660
So actually, all three of these
cases can be done with SET.

904
00:58:50,660 --> 00:58:52,963
And the way that it works
is different in each case.

905
00:58:52,963 --> 00:58:54,380
And the case that
we'll do is just

906
00:58:54,380 --> 00:58:57,950
the first one, which is kind
of a classic one, and also

907
00:58:57,950 --> 00:58:58,730
the simplest one.

908
00:59:04,618 --> 00:59:06,410
So our Px squared is
going to be of ordered

909
00:59:06,410 --> 00:59:12,970
q squared for our analysis.

910
00:59:25,600 --> 00:59:27,520
We also need some
partonic variables.

911
00:59:40,830 --> 00:59:44,848
So the struck quark carries
some momentum fraction

912
00:59:44,848 --> 00:59:45,515
from the proton.

913
00:59:49,240 --> 00:59:50,620
This is the familiar language.

914
00:59:53,260 --> 00:59:57,010
And for our analysis, what
we're going to do is just

915
00:59:57,010 --> 01:00:01,180
take m bar dot
little p of the quark

916
01:00:01,180 --> 01:00:04,945
to be something times n bar
dot big P of the proton.

917
01:00:07,480 --> 01:00:10,600
So the fraction is
this c variable,

918
01:00:10,600 --> 01:00:18,080
and it's the ratio of the
quark momentum to the proton.

919
01:00:18,080 --> 01:00:20,570
But we'll do it in a very
particular component,

920
01:00:20,570 --> 01:00:25,930
and we'll see why that's
the right thing to do.

921
01:00:25,930 --> 01:00:28,200
And if we have our
picture here of the quark

922
01:00:28,200 --> 01:00:30,960
kinematics, then this
P is the incoming P.

923
01:00:30,960 --> 01:00:34,950
And the outgoing P would
be P prime, let's say.

924
01:00:34,950 --> 01:00:37,320
So I could think about an
analog of this equation, where

925
01:00:37,320 --> 01:00:39,750
I square P prime.

926
01:00:39,750 --> 01:00:43,360
And if you do that, it's kind
of similar to the hadronic case.

927
01:00:43,360 --> 01:00:45,170
The only difference is
that the c shows up.

928
01:00:47,988 --> 01:00:49,905
And, of course, there's
no mass of the proton.

929
01:00:52,650 --> 01:00:56,100
So P prime squared
would be that.

930
01:01:03,950 --> 01:01:08,390
So we'll see how this
variable c shows up.

931
01:01:08,390 --> 01:01:10,310
Now, one thing that we
have to decide about

932
01:01:10,310 --> 01:01:12,110
is frames of reference,
because remember,

933
01:01:12,110 --> 01:01:15,080
when we were talking about
degrees of freedom in SET,

934
01:01:15,080 --> 01:01:17,480
we had picked a frame
to do that discussion.

935
01:01:17,480 --> 01:01:21,050
And it was almost always a
center of mass frame so far

936
01:01:21,050 --> 01:01:26,960
in our discussions, or the rest
frame of the initial state.

937
01:01:26,960 --> 01:01:30,900
And here, we're going to use a
slightly different frame, which

938
01:01:30,900 --> 01:01:34,067
is the most convenient frame
for deep elastic scattering.

939
01:01:34,067 --> 01:01:35,150
It's called a Breit frame.

940
01:01:40,950 --> 01:01:44,540
So we're going to do our
analysis in this frame.

941
01:01:44,540 --> 01:01:46,070
So what defines this frame?

942
01:01:50,210 --> 01:01:53,380
This frame is defined
by taking q mu

943
01:01:53,380 --> 01:01:57,040
to just have a z component.

944
01:01:57,040 --> 01:01:59,800
Remember, it's space-like,
so it has to be somewhere

945
01:01:59,800 --> 01:02:01,180
in the space-like column.

946
01:02:01,180 --> 01:02:02,680
And we can choose
it such that it's

947
01:02:02,680 --> 01:02:04,870
entirely in the z
component, and nowhere else.

948
01:02:04,870 --> 01:02:06,525
And that's the Breit frame.

949
01:02:06,525 --> 01:02:07,900
If we want to
write that in terms

950
01:02:07,900 --> 01:02:10,600
of our classic decomposition
of nnn bar, is we

951
01:02:10,600 --> 01:02:13,450
can write it as a
difference divided by 2.

952
01:02:13,450 --> 01:02:16,620
And that's giving
the z component.

953
01:02:19,620 --> 01:02:22,190
So in this frame
what's happening

954
01:02:22,190 --> 01:02:24,710
is that your initial
state proton is coming

955
01:02:24,710 --> 01:02:26,750
in with a very large momentum.

956
01:02:26,750 --> 01:02:29,606
And then it's being--

957
01:02:29,606 --> 01:02:32,270
you're killing that momentum
and then spitting it back out

958
01:02:32,270 --> 01:02:34,380
in a different direction.

959
01:02:34,380 --> 01:02:40,470
So you're spitting back out
stuff in a different direction.

960
01:02:40,470 --> 01:02:44,480
So the initial state proton is
coming in with a large momentum

961
01:02:44,480 --> 01:02:47,940
in some direction.

962
01:02:47,940 --> 01:02:50,990
So if you work out the
kinematics given that for what

963
01:02:50,990 --> 01:02:56,080
the proton would be,
the proton's momentum

964
01:02:56,080 --> 01:02:58,960
would be in the following form.

965
01:02:58,960 --> 01:03:00,080
So it's got a large--

966
01:03:00,080 --> 01:03:02,800
this is large,
and this is small.

967
01:03:05,750 --> 01:03:07,150
And if you have
a large component

968
01:03:07,150 --> 01:03:09,442
in some light-like direction,
that means it's colinear.

969
01:03:11,560 --> 01:03:16,450
So you could actually
write it as--

970
01:03:16,450 --> 01:03:20,300
using momentum conservation,
you could write it like this.

971
01:03:20,300 --> 01:03:24,190
And it has a colinear scaling.

972
01:03:24,190 --> 01:03:26,135
So what we have in
the deep inelastic

973
01:03:26,135 --> 01:03:28,510
scattering in the Breit frame
is that the incoming proton

974
01:03:28,510 --> 01:03:29,345
is a linear proton.

975
01:03:32,200 --> 01:03:36,310
And if we look at
Px, and we just--

976
01:03:36,310 --> 01:03:38,980
again, I'm not going
through the details of this.

977
01:03:38,980 --> 01:03:46,910
But if we just decompose Px
in terms of these coordinates,

978
01:03:46,910 --> 01:03:48,590
then we get this.

979
01:03:48,590 --> 01:03:52,400
And so as long as this factor
1 minus x over x is of order 1,

980
01:03:52,400 --> 01:03:54,340
you see that there's
a large component in n

981
01:03:54,340 --> 01:03:56,630
and a large component in n bar.

982
01:03:56,630 --> 01:03:57,770
And that means it's hard.

983
01:04:03,550 --> 01:04:06,270
So what would happen in
these other cases here

984
01:04:06,270 --> 01:04:07,930
is that you would
change that, right?

985
01:04:07,930 --> 01:04:09,100
It would no longer be hard.

986
01:04:09,100 --> 01:04:12,300
And that's why these
cases here are different.

987
01:04:12,300 --> 01:04:14,550
But as long as we're in this
first case, this is hard.

988
01:04:14,550 --> 01:04:16,890
And we can say that we have
colinear modes and hard modes.

989
01:04:16,890 --> 01:04:18,720
And then we just want to
write down an effective theory

990
01:04:18,720 --> 01:04:19,595
for those two things.

991
01:04:22,130 --> 01:04:23,810
You could also do
an analysis of DIS

992
01:04:23,810 --> 01:04:26,660
in the rest frame of the proton.

993
01:04:26,660 --> 01:04:29,360
That's another case.

994
01:04:29,360 --> 01:04:32,150
And actually, the final
result that you would get

995
01:04:32,150 --> 01:04:33,067
would be the same.

996
01:04:33,067 --> 01:04:34,650
That's what we'll
get from this frame.

997
01:04:34,650 --> 01:04:36,442
But this frame is
actually a little easier.

998
01:04:47,280 --> 01:04:49,980
So we're really talking about
hard colinear factorization

999
01:04:49,980 --> 01:04:50,670
in some sense.

1000
01:04:54,920 --> 01:04:57,160
Colinear describes the
low-energy degree of freedom,

1001
01:04:57,160 --> 01:04:58,800
which is the proton.

1002
01:04:58,800 --> 01:05:03,450
And hard describes the
off-shell final state

1003
01:05:03,450 --> 01:05:05,550
and the hard fluctuations.

1004
01:05:05,550 --> 01:05:09,210
And really, what we want to do
in DIS in this classic case is

1005
01:05:09,210 --> 01:05:11,372
just separate hard and
colinear fluctuations,

1006
01:05:11,372 --> 01:05:12,330
at least in this frame.

1007
01:05:17,660 --> 01:05:19,875
So we can do that for
the cross-section.

1008
01:05:23,970 --> 01:05:26,800
So let me remind you something
about the cross-section in DIS.

1009
01:05:29,630 --> 01:05:34,190
So just using nothing
more than the fact

1010
01:05:34,190 --> 01:05:38,910
that we're treating the leptons
order by order in the photon,

1011
01:05:38,910 --> 01:05:41,600
we can work the first-order in
the electromagnetic coupling.

1012
01:05:41,600 --> 01:05:43,100
And then we can
write down a formula

1013
01:05:43,100 --> 01:05:47,345
like this, where we split it
into electronic and hadronic

1014
01:05:47,345 --> 01:05:47,845
tensor.

1015
01:05:53,280 --> 01:05:55,050
And the hadronic
tensor can be written

1016
01:05:55,050 --> 01:06:06,480
as the imaginary part of some T,
where T is the following thing.

1017
01:06:06,480 --> 01:06:09,180
So this doesn't use
anything about--

1018
01:06:09,180 --> 01:06:11,700
we haven't used any sort of
perturbation theory or anything

1019
01:06:11,700 --> 01:06:13,580
to write this-- what
I'm telling you-- down.

1020
01:06:19,846 --> 01:06:22,370
We just worked all
orders, and basically used

1021
01:06:22,370 --> 01:06:23,210
the optical theorem.

1022
01:06:28,238 --> 01:06:30,280
So T is the time order
product of the two curves.

1023
01:06:36,248 --> 01:06:37,040
Let me call this z.

1024
01:06:40,893 --> 01:06:42,310
Actually, I don't
know if that's--

1025
01:06:46,980 --> 01:06:49,110
before you start
doing anything in QCD,

1026
01:06:49,110 --> 01:06:50,940
that's how you could write this.

1027
01:06:50,940 --> 01:06:54,620
And these are the
electromagnetic currents

1028
01:06:54,620 --> 01:06:56,082
for a quark.

1029
01:06:56,082 --> 01:06:58,290
STUDENT: But wouldn't you
write it over that, though?

1030
01:06:58,290 --> 01:06:59,250
[INAUDIBLE]

1031
01:06:59,250 --> 01:07:07,676
PROFESSOR: Thanks, yeah,
because x was something else.

1032
01:07:07,676 --> 01:07:08,710
Don't want to use x.

1033
01:07:12,040 --> 01:07:15,790
All right, so for
this T mu nu, we

1034
01:07:15,790 --> 01:07:18,190
can also use
current conservation

1035
01:07:18,190 --> 01:07:20,990
and decompose it
into two pieces.

1036
01:07:20,990 --> 01:07:25,120
So this is a classic
thing that we do in DIS.

1037
01:07:25,120 --> 01:07:26,740
And nothing about
the fact that we're

1038
01:07:26,740 --> 01:07:31,073
using the effective theory
really changes during this.

1039
01:07:31,073 --> 01:07:32,740
What you're after--
the effective theory

1040
01:07:32,740 --> 01:07:39,100
is calculating these T's,
which would be coefficients.

1041
01:07:39,100 --> 01:07:44,455
And this part is all standard
stuff for any analysis of DIS.

1042
01:07:48,610 --> 01:07:51,280
So if you're not familiar with
it, or you don't remember it,

1043
01:07:51,280 --> 01:07:53,640
it's actually not
that important.

1044
01:07:53,640 --> 01:07:56,170
But using the current
conservation of T,

1045
01:07:56,170 --> 01:07:58,060
the fact that when I
dot a q into the T,

1046
01:07:58,060 --> 01:08:02,230
you should get 0, because you're
dotting a q into the current.

1047
01:08:02,230 --> 01:08:03,512
The current is conserved.

1048
01:08:03,512 --> 01:08:04,720
It's electromagnetic current.

1049
01:08:04,720 --> 01:08:07,002
That tells you the possible
structure of this thing

1050
01:08:07,002 --> 01:08:08,710
and that there's two
general terms if I'm

1051
01:08:08,710 --> 01:08:10,060
doing a spin sum, which I am.

1052
01:08:18,151 --> 01:08:19,609
There's the sum
over spin up there.

1053
01:08:22,680 --> 01:08:25,080
If we weren't summing over
spin, if we were picking out

1054
01:08:25,080 --> 01:08:27,270
particular spins of the
proton, then the formula

1055
01:08:27,270 --> 01:08:31,439
here could be a little
more complicated.

1056
01:08:31,439 --> 01:08:34,398
So this satisfies
all the symmetries.

1057
01:08:39,271 --> 01:08:40,979
And in general, what
you know is that you

1058
01:08:40,979 --> 01:08:43,529
want the imaginary part of
forward scattering graphs.

1059
01:08:50,210 --> 01:08:53,168
So pictorially, it's sometimes
useful to draw something.

1060
01:08:56,450 --> 01:08:59,040
So here's a forward
scattering graph.

1061
01:08:59,040 --> 01:09:01,170
And the thing that's
different that we have--

1062
01:09:01,170 --> 01:09:05,700
know now-- and the information
that we're going to use

1063
01:09:05,700 --> 01:09:09,990
is that we're assigning
the external guys

1064
01:09:09,990 --> 01:09:12,910
to be colinear and the
intermediate guy to be hard.

1065
01:09:12,910 --> 01:09:15,570
So we want to integrate
out the pink guy, as usual.

1066
01:09:15,570 --> 01:09:18,000
The guy we want to integrate
out is always pink.

1067
01:09:18,000 --> 01:09:21,120
And we want to keep
these colinear guys.

1068
01:09:21,120 --> 01:09:25,170
And so the operators in
the effective theory--

1069
01:09:25,170 --> 01:09:27,420
we can already intuit what
they should look like.

1070
01:09:32,858 --> 01:09:34,150
They're just going to involve--

1071
01:09:37,024 --> 01:09:38,560
I have a current
with two photons

1072
01:09:38,560 --> 01:09:42,490
hanging out of it, and
then colinear quarks.

1073
01:09:42,490 --> 01:09:45,790
Those guys-- those are
the external lines there.

1074
01:09:45,790 --> 01:09:49,029
And there's actually
also an analogous thing

1075
01:09:49,029 --> 01:09:52,990
with colinear gluons.

1076
01:09:52,990 --> 01:09:54,550
And that's what
the type of-- so I

1077
01:09:54,550 --> 01:09:56,620
could have an operator
at higher orders, where

1078
01:09:56,620 --> 01:09:58,232
the external states
here were gluons.

1079
01:09:58,232 --> 01:10:00,440
And that's what the operators
of the effective theory

1080
01:10:00,440 --> 01:10:02,823
are going to look like.

1081
01:10:02,823 --> 01:10:04,490
Just contract that
pink line to a point,

1082
01:10:04,490 --> 01:10:05,782
and that's what they look like.

1083
01:10:20,550 --> 01:10:22,300
So given that we know
what they look like,

1084
01:10:22,300 --> 01:10:24,258
we just have to write
down the lowest dimension

1085
01:10:24,258 --> 01:10:25,337
operators of that form.

1086
01:10:25,337 --> 01:10:27,295
And the lowest dimension
operators of that form

1087
01:10:27,295 --> 01:10:28,840
are exactly the
operator that I told

1088
01:10:28,840 --> 01:10:31,030
you was going to be
the one that comes in.

1089
01:10:36,780 --> 01:10:38,570
So we can enumerate
the lowest dimension,

1090
01:10:38,570 --> 01:10:41,150
which where dimension here
is counted as lambdas, right?

1091
01:10:41,150 --> 01:10:44,720
So it's the lowest order in the
power counting operators, so

1092
01:10:44,720 --> 01:10:46,855
not the lowest order
in mass dimension.

1093
01:10:51,410 --> 01:10:57,320
Let me write out a few things.

1094
01:11:05,340 --> 01:11:07,375
Yeah, I already switched
to this notation.

1095
01:11:11,840 --> 01:11:12,340
Yeah.

1096
01:11:32,393 --> 01:11:34,060
So the type of operator
we'd have that's

1097
01:11:34,060 --> 01:11:35,520
order lambda squared--

1098
01:11:35,520 --> 01:11:36,520
this guy's order lambda.

1099
01:11:36,520 --> 01:11:37,520
This guy's order lambda.

1100
01:11:37,520 --> 01:11:38,770
That's lambda squared.

1101
01:11:38,770 --> 01:11:42,610
And the most general thing
we can think of is this.

1102
01:11:42,610 --> 01:11:44,860
The reason why I wrote this
out rather than writing it

1103
01:11:44,860 --> 01:11:46,540
as a chi field is I
wanted to emphasize

1104
01:11:46,540 --> 01:11:47,707
that there's a flavor index.

1105
01:11:51,202 --> 01:11:57,030
So i is up quarks, down quarks,
strange quirks, et cetera.

1106
01:11:57,030 --> 01:12:06,660
And then we could
also have gluons,

1107
01:12:06,660 --> 01:12:19,485
and gluons are similar except
in terms of our curly v field.

1108
01:12:26,010 --> 01:12:35,460
And it is actually just a
contraction of two curly v's,

1109
01:12:35,460 --> 01:12:38,555
and then traced over.

1110
01:12:38,555 --> 01:12:40,430
And then there's some
Wilson coefficient here

1111
01:12:40,430 --> 01:12:47,750
which is also just the Wilson
coefficient for the gluon.

1112
01:12:47,750 --> 01:12:50,000
So this is, again,
order lambda squared.

1113
01:12:50,000 --> 01:12:51,810
The b perp field is
order lambda squared.

1114
01:12:51,810 --> 01:12:53,990
So we just write down the
lowest dimension operators

1115
01:12:53,990 --> 01:12:55,352
that have this form.

1116
01:12:55,352 --> 01:12:57,060
And that's going to
be the right answer--

1117
01:12:57,060 --> 01:12:59,240
the lowest order in lambda.

1118
01:13:15,660 --> 01:13:18,000
So it turns out actually
that in this case,

1119
01:13:18,000 --> 01:13:21,380
our lambda counting is exactly
the corresponding twist

1120
01:13:21,380 --> 01:13:22,350
expansion.

1121
01:13:26,490 --> 01:13:27,410
I'm not going to--

1122
01:13:27,410 --> 01:13:29,830
STUDENT: All j's correspond
to the operators that go

1123
01:13:29,830 --> 01:13:31,442
into the T1's and T2's?

1124
01:13:31,442 --> 01:13:32,400
PROFESSOR: Yeah, right.

1125
01:13:32,400 --> 01:13:35,630
So now what's the
index j exactly?

1126
01:13:35,630 --> 01:13:37,860
There's a T1 and a T2.

1127
01:13:37,860 --> 01:13:39,480
And the T1 and the
T2, although they

1128
01:13:39,480 --> 01:13:41,940
have the same kind of quark
and gluon field structure,

1129
01:13:41,940 --> 01:13:45,480
they get different coefficients,
and that's what the j is.

1130
01:13:45,480 --> 01:13:53,270
So when we think about doing a
similar kind of decomposition,

1131
01:13:53,270 --> 01:13:56,210
the effective theory, we
can write it as follows.

1132
01:14:01,080 --> 01:14:05,030
So as an 01 and an 02.

1133
01:14:05,030 --> 01:14:10,010
And I think if I'm getting
my mass detentions right,

1134
01:14:10,010 --> 01:14:10,880
it looks like this.

1135
01:14:28,080 --> 01:14:31,070
So there's a piece that
multiplies a spin structure

1136
01:14:31,070 --> 01:14:32,630
that's this g perp mu nu.

1137
01:14:32,630 --> 01:14:34,052
That's transverse to Q.

1138
01:14:34,052 --> 01:14:35,510
And there's a piece
that multiplies

1139
01:14:35,510 --> 01:14:37,790
this guy, which is also
transverse to Q in the Breit

1140
01:14:37,790 --> 01:14:39,920
frame.

1141
01:14:39,920 --> 01:14:41,410
So that's the kind
of decomposition

1142
01:14:41,410 --> 01:14:42,610
we would do in the
effective theory.

1143
01:14:42,610 --> 01:14:44,170
There'd be two different
types of operators

1144
01:14:44,170 --> 01:14:45,215
we could think about.

1145
01:14:45,215 --> 01:14:46,840
But the only thing
that actually tracks

1146
01:14:46,840 --> 01:14:48,850
that-- because of
the spin relations,

1147
01:14:48,850 --> 01:14:51,667
the only thing that tracks
that is the Wilson coefficient.

1148
01:14:51,667 --> 01:14:53,875
And so that will differ in
these two different cases.

1149
01:14:57,370 --> 01:15:01,550
So this guy here
is going to be--

1150
01:15:01,550 --> 01:15:04,260
this will give the quark PDFs.

1151
01:15:10,380 --> 01:15:11,970
And this Wilson
coefficient will then

1152
01:15:11,970 --> 01:15:14,852
be the thing that you convolute
with the quark PDF in DIS.

1153
01:15:14,852 --> 01:15:17,310
And then likewise, this guy is
going to give the gluon PDF.

1154
01:15:22,870 --> 01:15:24,280
So let's do the quark--

1155
01:15:24,280 --> 01:15:28,060
we'll do the quark
contribution in detail.

1156
01:15:28,060 --> 01:15:30,310
The gluon contribution
is not really any harder.

1157
01:15:34,983 --> 01:15:36,400
So what I'm going
to do is I'm not

1158
01:15:36,400 --> 01:15:38,680
going to think about it
in perturbation theory.

1159
01:15:38,680 --> 01:15:40,570
I'm just going to think about
it to all orders in perturbation

1160
01:15:40,570 --> 01:15:41,308
theory.

1161
01:15:41,308 --> 01:15:42,850
And really what that
means is I'm not

1162
01:15:42,850 --> 01:15:46,000
going to think about this c as
expanded in perturbation theory

1163
01:15:46,000 --> 01:15:48,565
or think about any of the
diagrams here as expanded

1164
01:15:48,565 --> 01:15:50,260
in perturbation theory.

1165
01:15:50,260 --> 01:15:52,690
I'm just going to see,
if I manipulate things,

1166
01:15:52,690 --> 01:15:55,150
what does it lead
to, using things

1167
01:15:55,150 --> 01:15:58,610
like momentum conservation
and stuff like that?

1168
01:15:58,610 --> 01:16:00,130
So let me write--

1169
01:16:00,130 --> 01:16:02,980
in order to do that,
let me write it

1170
01:16:02,980 --> 01:16:10,420
slightly differently than I
just did, which I apologize for.

1171
01:16:10,420 --> 01:16:11,170
But I'm going to--

1172
01:16:11,170 --> 01:16:17,080
I want the arguments of the c,
just for a later convenience,

1173
01:16:17,080 --> 01:16:19,450
instead of being w1
and w1 to be w plus

1174
01:16:19,450 --> 01:16:22,570
and w-- minus the sum of
the w's and the difference

1175
01:16:22,570 --> 01:16:28,420
of the w's, but
everything else the same.

1176
01:16:32,470 --> 01:16:34,390
It's convenient
to just talk about

1177
01:16:34,390 --> 01:16:37,900
the sum and the difference
rather than the individuals.

1178
01:16:48,818 --> 01:16:50,610
So what's the parton
distribution function?

1179
01:16:54,737 --> 01:16:56,320
Let me convince you
that it's actually

1180
01:16:56,320 --> 01:16:59,800
related to this operator.

1181
01:16:59,800 --> 01:17:09,700
If you were in coordinate space,
the coordinate space for us

1182
01:17:09,700 --> 01:17:11,950
was z.

1183
01:17:11,950 --> 01:17:15,110
We're just leaving x
for something else.

1184
01:17:15,110 --> 01:17:22,110
Then you could define the
parton distribution function

1185
01:17:22,110 --> 01:17:26,857
as a proton matrix
element of quark fields

1186
01:17:26,857 --> 01:17:28,190
with a Wilson line between them.

1187
01:17:33,890 --> 01:17:39,720
So they're on the light
cone, and we have a formula

1188
01:17:39,720 --> 01:17:41,790
like that.

1189
01:17:41,790 --> 01:17:45,180
And also, one can
convince oneself

1190
01:17:45,180 --> 01:17:49,470
that if you want to write
down for the antiquarks what

1191
01:17:49,470 --> 01:17:51,690
the part-time
distribution function is,

1192
01:17:51,690 --> 01:17:55,710
it's the same formula
as the quark formula,

1193
01:17:55,710 --> 01:17:57,520
just with an overall minus sign.

1194
01:17:57,520 --> 01:17:59,529
And z goes to minus z.

1195
01:18:09,510 --> 01:18:12,540
So that's an operator definition
of the part-time distribution

1196
01:18:12,540 --> 01:18:14,020
function.

1197
01:18:14,020 --> 01:18:17,010
You can also define it by
moments of the operator.

1198
01:18:17,010 --> 01:18:20,370
That's the way that, for
example, Peskin does it.

1199
01:18:20,370 --> 01:18:22,620
But if you put all the
information in those moments

1200
01:18:22,620 --> 01:18:25,140
back into a single operator,
then it becomes this thing.

1201
01:18:30,420 --> 01:18:32,705
And if we Fourier
transform this,

1202
01:18:32,705 --> 01:18:34,830
then it becomes the operator
that we have up there.

1203
01:18:48,802 --> 01:18:51,010
So there's something special
about the matrix element

1204
01:18:51,010 --> 01:18:52,870
we're taking here.

1205
01:18:52,870 --> 01:18:55,900
And that is that this
matrix element is forward.

1206
01:18:55,900 --> 01:18:58,360
It has the same
kinematics for the--

1207
01:18:58,360 --> 01:19:01,330
so this is a colinear proton,
but we have the same momentum

1208
01:19:01,330 --> 01:19:03,820
here and here.

1209
01:19:03,820 --> 01:19:06,290
In the in state
and the out state,

1210
01:19:06,290 --> 01:19:08,830
you have the same momentum.

1211
01:19:08,830 --> 01:19:11,445
And that leads to one
kinematic restriction

1212
01:19:11,445 --> 01:19:12,570
on what I'm about to write.

1213
01:19:15,500 --> 01:19:21,650
So what that imposes is that
w1 should be equal to w2.

1214
01:19:21,650 --> 01:19:23,630
So there's a delta
function of w minus

1215
01:19:23,630 --> 01:19:27,110
that comes from the momentum
conservation or the restriction

1216
01:19:27,110 --> 01:19:30,110
that it's a forward
matrix element.

1217
01:19:30,110 --> 01:19:33,170
But the sum of the two--

1218
01:19:33,170 --> 01:19:35,780
with my sign conventions,
it's w minus.

1219
01:19:35,780 --> 01:19:39,960
The sum of the two
is unconstrained.

1220
01:19:39,960 --> 01:19:42,645
Well, it's bounded,
but unconstrained.

1221
01:20:02,480 --> 01:20:04,100
So we can write the
unconstrained part

1222
01:20:04,100 --> 01:20:06,680
as this interval of c parameter.

1223
01:20:06,680 --> 01:20:08,390
The bounding just
comes from the fact

1224
01:20:08,390 --> 01:20:10,130
that the quark can't
carry more momentum

1225
01:20:10,130 --> 01:20:13,055
than the overall
momentum of the proton.

1226
01:20:13,055 --> 01:20:14,180
So that's why it's bounded.

1227
01:20:14,180 --> 01:20:16,610
It can't carry negative
momentum, and it can't carry--

1228
01:20:19,078 --> 01:20:20,870
negative physical
momentum, it can't carry,

1229
01:20:20,870 --> 01:20:23,600
and it can't carry less
momentum in the proton.

1230
01:20:23,600 --> 01:20:25,350
That's where the
limits come from.

1231
01:20:25,350 --> 01:20:26,630
And then there's two pieces.

1232
01:20:26,630 --> 01:20:28,550
The w plus could
either be positive,

1233
01:20:28,550 --> 01:20:30,240
or it could be negative.

1234
01:20:30,240 --> 01:20:32,120
If it's positive,
remember, positive

1235
01:20:32,120 --> 01:20:34,340
labels-- that was our quarks.

1236
01:20:34,340 --> 01:20:35,690
So this is the quark piece.

1237
01:20:38,770 --> 01:20:40,020
And these are the antiquarks.

1238
01:20:40,020 --> 01:20:43,140
They come with negative labels.

1239
01:20:43,140 --> 01:20:47,470
Remember that we talked
about that earlier.

1240
01:20:47,470 --> 01:20:55,292
And so both the f and the f bar
are hiding inside this formula.

1241
01:20:55,292 --> 01:20:57,000
So there is a
simplification that there's

1242
01:20:57,000 --> 01:20:58,230
this delta function here.

1243
01:21:01,590 --> 01:21:04,510
And so basically what
we're talking about then--

1244
01:21:04,510 --> 01:21:06,600
you can think of just
doing the integral

1245
01:21:06,600 --> 01:21:08,440
over that delta function.

1246
01:21:08,440 --> 01:21:13,447
So then you're talking about
effectively an operator

1247
01:21:13,447 --> 01:21:15,780
where you don't worry about
putting a label on this one.

1248
01:21:15,780 --> 01:21:17,410
You have only one
delta function left,

1249
01:21:17,410 --> 01:21:19,640
which I can denote by
putting a label on this one.

1250
01:21:19,640 --> 01:21:23,340
And this operator is
actually exactly the operator

1251
01:21:23,340 --> 01:21:26,760
that gives you the PDF.

1252
01:21:26,760 --> 01:21:30,510
This operator is like
a number operator

1253
01:21:30,510 --> 01:21:43,900
for quarks, where
you're thinking

1254
01:21:43,900 --> 01:21:52,550
about momentum, a number
operator with momentum omega.

1255
01:21:52,550 --> 01:21:54,300
And if you want to
think about there being

1256
01:21:54,300 --> 01:21:56,760
some kind of field for a
parton, this is about as close

1257
01:21:56,760 --> 01:21:57,450
as you can get.

1258
01:22:01,990 --> 01:22:03,930
So this quark field
dressed by Wilson line

1259
01:22:03,930 --> 01:22:07,820
is kind of like a parton
in the parton model.

1260
01:22:07,820 --> 01:22:11,630
All right, so next time, we'll
take this, put it together

1261
01:22:11,630 --> 01:22:13,880
with this formula
here, with the c,

1262
01:22:13,880 --> 01:22:15,230
and just see where it leads us.

1263
01:22:15,230 --> 01:22:17,300
And it will lead us
directly to a factorization

1264
01:22:17,300 --> 01:22:19,820
theorem for deep
inelastic scattering

1265
01:22:19,820 --> 01:22:22,280
involving parton
distribution functions

1266
01:22:22,280 --> 01:22:24,800
and some hard perturbatively
calculable thing, which

1267
01:22:24,800 --> 01:22:28,940
is actually a cross-section
in the parton model.

1268
01:22:28,940 --> 01:22:32,343
And then we'll talk about
renormalization [INAUDIBLE]..

1269
01:22:32,343 --> 01:22:34,760
You can already see that we're
getting the operator that I

1270
01:22:34,760 --> 01:22:37,160
promised you-- that
DIS is described

1271
01:22:37,160 --> 01:22:40,010
by this, by a linear operator
with two different quark

1272
01:22:40,010 --> 01:22:42,620
fields, and particular
labeling of the momenta.

1273
01:22:47,980 --> 01:22:50,130
Any questions?