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PROFESSOR: All right,
so last time we

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00:00:28,940 --> 00:00:30,830
were talking about
symmetries of SCET.

10
00:00:30,830 --> 00:00:33,800
We talked in detail
about gauge symmetry,

11
00:00:33,800 --> 00:00:36,350
and we started to talk about
reparameterization invariance,

12
00:00:36,350 --> 00:00:38,225
and we are going to
continue with that today.

13
00:00:40,620 --> 00:00:42,960
So reparameterization
invariance in SCET

14
00:00:42,960 --> 00:00:45,150
is kind of rich,
because we have lots

15
00:00:45,150 --> 00:00:47,610
of things that are messing
up Lorentz invariance.

16
00:00:47,610 --> 00:00:50,970
But we started out talking about
three different types here.

17
00:00:50,970 --> 00:00:53,843
One where we rotate the n,
the physical n vector that

18
00:00:53,843 --> 00:00:55,260
was the direction
of our collinear

19
00:00:55,260 --> 00:00:58,230
particles to some other vector
that's equally good, which

20
00:00:58,230 --> 00:01:00,140
I call n prime in the figure.

21
00:01:00,140 --> 00:01:02,460
So a type one, transformation.

22
00:01:02,460 --> 00:01:04,650
Or we could change this
auxiliary vector, n bar,

23
00:01:04,650 --> 00:01:06,240
by a large amount.

24
00:01:06,240 --> 00:01:07,470
That was type two.

25
00:01:07,470 --> 00:01:10,560
Or we could change both
at n and n bar in a way

26
00:01:10,560 --> 00:01:13,050
that dot products
remain invariant.

27
00:01:13,050 --> 00:01:14,430
That's type three.

28
00:01:14,430 --> 00:01:16,980
So all three of these
preserved these relations,

29
00:01:16,980 --> 00:01:19,500
n squared equals 0, n
bar squared equals 0.

30
00:01:19,500 --> 00:01:21,030
n dot n bar equals 2.

31
00:01:21,030 --> 00:01:22,530
And really, these
transformations

32
00:01:22,530 --> 00:01:25,900
are just giving us a different
basis for our decomposition,

33
00:01:25,900 --> 00:01:28,500
which satisfies all the things
that we wanted that basis

34
00:01:28,500 --> 00:01:29,670
to satisfy.

35
00:01:29,670 --> 00:01:32,500
Doesn't change any of
our power counting,

36
00:01:32,500 --> 00:01:34,860
and so is an equally
good description

37
00:01:34,860 --> 00:01:36,940
for the effective theory.

38
00:01:36,940 --> 00:01:39,270
And therefore, we want
to have invariance

39
00:01:39,270 --> 00:01:43,450
under these transformations.

40
00:01:43,450 --> 00:01:46,480
So we're restoring Lorentz
symmetry in this way.

41
00:01:46,480 --> 00:01:49,260
We break it by
introducing these vectors,

42
00:01:49,260 --> 00:01:52,830
but we're restoring it in a
way by having transformations

43
00:01:52,830 --> 00:01:54,960
on these factors.

44
00:01:54,960 --> 00:01:57,690
But it's not, you're
not restoring,

45
00:01:57,690 --> 00:01:59,970
if you like, it's still
different than Lorentz

46
00:01:59,970 --> 00:02:01,800
symmetry, for example,
because you're not

47
00:02:01,800 --> 00:02:03,750
making huge transformations
of the vector n

48
00:02:03,750 --> 00:02:07,860
here for any arbitrary
transformation.

49
00:02:07,860 --> 00:02:09,620
And it is a
reparameterization not

50
00:02:09,620 --> 00:02:14,230
a Lorentz symmetry in general.

51
00:02:14,230 --> 00:02:17,970
So we'll talk more
about how this connects

52
00:02:17,970 --> 00:02:19,390
to the HQET one in a minute.

53
00:02:19,390 --> 00:02:24,522
But if we have a vector, then
obviously the vector, p mu--

54
00:02:24,522 --> 00:02:25,980
I don't write the
right hand side--

55
00:02:25,980 --> 00:02:30,120
is invariant to what
choice of basis I use.

56
00:02:30,120 --> 00:02:33,010
So that means that this
p mu should not change.

57
00:02:33,010 --> 00:02:36,150
And that means if
I change n here,

58
00:02:36,150 --> 00:02:38,430
let's say I do a type one,
then I'll change n here

59
00:02:38,430 --> 00:02:40,500
and I'll change n here.

60
00:02:40,500 --> 00:02:43,350
But I must compensate those
changes by changing what--

61
00:02:43,350 --> 00:02:45,510
perp also depended
on the meaning of n,

62
00:02:45,510 --> 00:02:48,390
so there will be a compensating
change for the perp

63
00:02:48,390 --> 00:02:51,930
So we can figure out from
statements like this what

64
00:02:51,930 --> 00:02:54,400
the transformation laws are.

65
00:02:54,400 --> 00:02:56,670
So this guy is invariant
to the decomposition.

66
00:03:05,310 --> 00:03:13,370
So the transformation
of p perp mu

67
00:03:13,370 --> 00:03:24,777
compensates for n in type
one, or n bar in type two.

68
00:03:24,777 --> 00:03:26,360
Type 3 is already
invariant because we

69
00:03:26,360 --> 00:03:29,930
have an n and an n bar here,
an n and an n bar here.

70
00:03:29,930 --> 00:03:33,570
So that's type three
invariant already.

71
00:03:33,570 --> 00:03:35,630
So if we go through
that, we can make

72
00:03:35,630 --> 00:03:39,560
a table of all the
different transformations,

73
00:03:39,560 --> 00:03:41,930
and we can derive
the transformations.

74
00:03:41,930 --> 00:03:46,040
We also have to do the same
thing that we did in HQET,

75
00:03:46,040 --> 00:03:48,080
because we also-- the
other important fact

76
00:03:48,080 --> 00:03:52,430
is that we have a projection
relation out here.

77
00:03:58,510 --> 00:04:01,308
So we also have this,
and you'll remember

78
00:04:01,308 --> 00:04:03,600
that when we talked about
reparameterization invariance

79
00:04:03,600 --> 00:04:05,880
and HQET, the
projection relation was

80
00:04:05,880 --> 00:04:09,962
part of the discussion,
and so that's true too.

81
00:04:09,962 --> 00:04:11,670
But let me just quote
to you the results,

82
00:04:11,670 --> 00:04:13,860
that you could kind of get
an impression for where

83
00:04:13,860 --> 00:04:17,980
these various terms
are coming from.

84
00:04:17,980 --> 00:04:20,880
So let's summarize,
first, type one.

85
00:04:20,880 --> 00:04:24,900
So type one was n goes
to n plus delta perp.

86
00:04:24,900 --> 00:04:26,670
So if you have
something like n dot D,

87
00:04:26,670 --> 00:04:32,400
That goes to n dot D
plus delta perp dot D,

88
00:04:32,400 --> 00:04:35,160
there's a transformation
of p perp or D perp.

89
00:04:35,160 --> 00:04:37,700
Let me write it as D perp.

90
00:04:37,700 --> 00:04:44,270
This is really for
any vector which

91
00:04:44,270 --> 00:04:48,350
has to compensate for the
transformation of the n

92
00:04:48,350 --> 00:04:49,370
in the way I described.

93
00:04:55,890 --> 00:04:59,770
There's a transformation
like that.

94
00:04:59,770 --> 00:05:04,610
The n bar component under
type one does not transform.

95
00:05:04,610 --> 00:05:08,120
And if you go
through the spinner,

96
00:05:08,120 --> 00:05:13,100
or you go through the field,
the fermion field, it does get--

97
00:05:13,100 --> 00:05:16,050
because the projection
relation has a transformation,

98
00:05:16,050 --> 00:05:21,770
there's a transformation
of this guy.

99
00:05:21,770 --> 00:05:24,230
The Wilson line,
which we talked about,

100
00:05:24,230 --> 00:05:26,930
was only a function
of M bar dot A,

101
00:05:26,930 --> 00:05:28,550
and so it doesn't
get transformed.

102
00:05:31,070 --> 00:05:32,720
So that's what type
one transformations

103
00:05:32,720 --> 00:05:33,387
would look like.

104
00:05:37,390 --> 00:05:38,280
And type two--

105
00:05:51,540 --> 00:05:54,238
Summarize.

106
00:05:54,238 --> 00:05:56,280
They start out looking
the same, and then there's

107
00:05:56,280 --> 00:05:57,180
some differences.

108
00:06:17,450 --> 00:06:21,100
So everything so far looks
exactly the same as type one,

109
00:06:21,100 --> 00:06:23,890
just with a sort of
suitable replacement.

110
00:06:23,890 --> 00:06:30,245
But the fermion ends up looking
a little more complicated here.

111
00:06:44,750 --> 00:06:47,258
And the Wilson line
also transforms

112
00:06:47,258 --> 00:06:48,800
because the Wilson
line was built out

113
00:06:48,800 --> 00:06:50,660
of n bar dot D field, so.

114
00:06:55,760 --> 00:06:58,580
Working the first order
in that transformation,

115
00:06:58,580 --> 00:07:00,200
there's an additional term.

116
00:07:00,200 --> 00:07:02,074
Epsilon perp dot D perp.

117
00:07:08,370 --> 00:07:11,790
OK, so I didn't want to take you
through the derivation of all

118
00:07:11,790 --> 00:07:12,480
these equations.

119
00:07:12,480 --> 00:07:16,260
I have provided references.

120
00:07:16,260 --> 00:07:18,570
There's a reference list at
the beginning of the notes,

121
00:07:18,570 --> 00:07:21,270
and there's a reference with
a paper that talks about how

122
00:07:21,270 --> 00:07:24,060
to derive these equations.

123
00:07:24,060 --> 00:07:27,220
We won't be spending too
much time talking about it.

124
00:07:27,220 --> 00:07:29,280
So I won't go through
the derivation,

125
00:07:29,280 --> 00:07:33,520
but it's very analogous to
what you would do in HQET.

126
00:07:33,520 --> 00:07:35,020
So how do we use these results?

127
00:07:35,020 --> 00:07:37,350
Let's take them as given.

128
00:07:37,350 --> 00:07:39,450
We can use these
results to close out

129
00:07:39,450 --> 00:07:41,910
our discussion of what we
were talking about last time,

130
00:07:41,910 --> 00:07:46,290
and show what the leading
order Lagrangian is,

131
00:07:46,290 --> 00:07:49,080
imposing all the symmetries.

132
00:07:49,080 --> 00:07:50,910
And so the first
thing that we can note

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00:07:50,910 --> 00:08:06,460
is that if we do a type one
transformation of this guy,

134
00:08:06,460 --> 00:08:10,280
and you simplify
the result, then

135
00:08:10,280 --> 00:08:13,610
at the end of the day that boils
down to just the following.

136
00:08:23,210 --> 00:08:26,280
Well, it can be simplified
to a single term.

137
00:08:26,280 --> 00:08:39,440
And if one does the
same thing on this guy,

138
00:08:39,440 --> 00:08:43,169
then that gives us the same
turn, with the opposite sign.

139
00:08:43,169 --> 00:08:45,870
So these two operators are
connected by reparameterization

140
00:08:45,870 --> 00:08:46,856
invariance.

141
00:08:50,258 --> 00:08:53,990
The sum of them is
reparameterization invariant,

142
00:08:53,990 --> 00:08:54,940
and that looks good.

143
00:08:59,430 --> 00:09:03,573
So the sum is 0,
connected by RPI.

144
00:09:03,573 --> 00:09:05,490
And that means, for
example, that you couldn't

145
00:09:05,490 --> 00:09:09,120
have any non-trivial Wilson
coefficient between them

146
00:09:09,120 --> 00:09:11,580
or anything like that.

147
00:09:11,580 --> 00:09:14,108
Now, if you go through
the same argument

148
00:09:14,108 --> 00:09:16,650
with the other operator that we
talked about last time, where

149
00:09:16,650 --> 00:09:20,670
it was D perp mu, D perp mu.

150
00:09:20,670 --> 00:09:22,755
So delta 1.

151
00:09:36,760 --> 00:09:40,040
So same thing as above, but just
having D perp mu, D perp mu.

152
00:09:40,040 --> 00:09:41,740
D n perp mu.

153
00:09:41,740 --> 00:09:44,790
You actually find that it
gives the same result as this.

154
00:09:52,220 --> 00:09:55,940
OK, so if you just have
reparameterization of type one,

155
00:09:55,940 --> 00:09:57,320
then you can't really--

156
00:09:57,320 --> 00:09:59,422
the sum of this and
this would be invariant.

157
00:09:59,422 --> 00:10:01,130
Or the sum of this
and this is invariant,

158
00:10:01,130 --> 00:10:03,900
or the sum of any combination
of 1 minus a times that,

159
00:10:03,900 --> 00:10:06,170
and a times this
would be invariant.

160
00:10:06,170 --> 00:10:08,940
But type two does distinguish
between this operator

161
00:10:08,940 --> 00:10:10,197
and this operator.

162
00:10:27,750 --> 00:10:29,530
If you do a type
two transformation,

163
00:10:29,530 --> 00:10:31,680
you get terms from this
that you can't possibly

164
00:10:31,680 --> 00:10:34,080
cancel by any other term
that you could write down.

165
00:10:34,080 --> 00:10:37,927
And those terms are such that
the power counting, there's

166
00:10:37,927 --> 00:10:40,260
not a subleading operative
that could compensate either.

167
00:10:46,280 --> 00:10:49,790
OK, so going through all
the type two transformations

168
00:10:49,790 --> 00:10:50,640
is kind of messy.

169
00:10:50,640 --> 00:10:52,830
So again, I don't
want to do that.

170
00:10:52,830 --> 00:10:55,010
But I would refer you to
some reading for that.

171
00:10:58,320 --> 00:11:02,380
OK, but if you take these
two things together,

172
00:11:02,380 --> 00:11:06,370
then you do rule out
that additional operator.

173
00:11:06,370 --> 00:11:08,820
And so then, taking
everything together,

174
00:11:08,820 --> 00:11:11,534
we have a unique leading
order Lagrangian.

175
00:11:17,470 --> 00:11:19,780
It's not so easy, actually.

176
00:11:23,930 --> 00:11:27,210
It requires, it requires
looking at the transformations

177
00:11:27,210 --> 00:11:30,560
carefully.

178
00:11:30,560 --> 00:11:31,280
Yeah.

179
00:11:31,280 --> 00:11:34,260
So it is-- so even though I
made it sound like it's simple,

180
00:11:34,260 --> 00:11:35,300
but it's not.

181
00:11:35,300 --> 00:11:37,163
There's a reason why
I don't want to spend

182
00:11:37,163 --> 00:11:38,330
the rest of the class on it.

183
00:11:41,280 --> 00:11:41,780
OK.

184
00:11:49,940 --> 00:11:54,290
So if we put all the
things that we've

185
00:11:54,290 --> 00:12:02,120
talked about last time and this
time together, what we find

186
00:12:02,120 --> 00:12:10,130
is that the Lagrangian
that we've been discussing

187
00:12:10,130 --> 00:12:15,020
is unique for the CN field.

188
00:12:15,020 --> 00:12:21,230
So it's the unique leading order
Lagrangian for the CN field.

189
00:12:21,230 --> 00:12:23,875
And the things that we've used
are power counting in order

190
00:12:23,875 --> 00:12:25,250
to say that the
terms here should

191
00:12:25,250 --> 00:12:29,750
be lambda to the fourth, that's
what leading order meant.

192
00:12:29,750 --> 00:12:33,230
Gauge invariance, make
these covariant derivatives

193
00:12:33,230 --> 00:12:35,220
of appropriate types.

194
00:12:35,220 --> 00:12:41,340
And now also reparameterization
of symmetries.

195
00:12:41,340 --> 00:12:45,720
So as we kind of
discussed last time,

196
00:12:45,720 --> 00:12:47,220
the reparameterization
of symmetries

197
00:12:47,220 --> 00:12:49,710
here can do different things.

198
00:12:49,710 --> 00:12:52,560
They can connect leading order
operators to leading order

199
00:12:52,560 --> 00:12:54,960
operators and hence, constrain
the form of leading order

200
00:12:54,960 --> 00:12:57,690
operators that you write down,
and that's what we just did.

201
00:12:57,690 --> 00:12:59,940
And in particular,
the type two, remember

202
00:12:59,940 --> 00:13:01,980
the epsilon perp was
order of lambda to the 0,

203
00:13:01,980 --> 00:13:05,850
so maybe it's not surprising
that that's what it's doing.

204
00:13:05,850 --> 00:13:08,880
The type one transformation,
the delta perp

205
00:13:08,880 --> 00:13:11,550
took you down a power in lambda,
but if you had some subleading

206
00:13:11,550 --> 00:13:13,140
operator, then it
would also take you

207
00:13:13,140 --> 00:13:16,120
down a power in lambda, so
that's one of the things

208
00:13:16,120 --> 00:13:18,203
that you have to be careful
about when you're sort

209
00:13:18,203 --> 00:13:20,100
of counting powers of lambda.

210
00:13:20,100 --> 00:13:21,922
But there is a
sense, also, in which

211
00:13:21,922 --> 00:13:23,880
reparameterization symmetry
does the same thing

212
00:13:23,880 --> 00:13:26,100
as HQET, which is it connects.

213
00:13:26,100 --> 00:13:27,552
So it can do two
different things.

214
00:13:27,552 --> 00:13:29,760
It can constrain the form
of operators you write down

215
00:13:29,760 --> 00:13:33,270
at lowest order, or it
could connect the Wilson

216
00:13:33,270 --> 00:13:36,330
coefficients, for example,
of leading order operators

217
00:13:36,330 --> 00:13:38,280
to subleading operators.

218
00:13:38,280 --> 00:13:41,970
And that's what we saw on the
problem set in the HQET example

219
00:13:41,970 --> 00:13:44,400
it can do that here too.

220
00:13:44,400 --> 00:13:47,760
And so if we want to
think about that freedom,

221
00:13:47,760 --> 00:13:49,920
then we should note
that there's actually

222
00:13:49,920 --> 00:13:54,720
more kind of reparameterization
than we've talked

223
00:13:54,720 --> 00:13:56,080
about just with this example.

224
00:13:56,080 --> 00:13:58,680
So far, what we've talked
about is changing the basis.

225
00:13:58,680 --> 00:14:01,770
But we could also change
the amount of the momentum

226
00:14:01,770 --> 00:14:04,260
that we stick in the
label versus the residual.

227
00:14:04,260 --> 00:14:06,960
That was also a choice that
we made in doing things.

228
00:14:12,630 --> 00:14:17,700
And we should be careful
to think about that freedom

229
00:14:17,700 --> 00:14:19,330
as well.

230
00:14:19,330 --> 00:14:21,240
And that's also a
reparameterization freedom.

231
00:14:26,663 --> 00:14:28,080
And this one looks,
in some sense,

232
00:14:28,080 --> 00:14:29,770
more like what we
talked about in HQET.

233
00:14:29,770 --> 00:14:31,500
In some sense, what
HQET it is doing

234
00:14:31,500 --> 00:14:34,560
is a combination of
these two things.

235
00:14:34,560 --> 00:14:36,640
Because of the
decomposition of momentum

236
00:14:36,640 --> 00:14:38,415
there is a bit simpler.

237
00:14:41,900 --> 00:14:49,205
So we had this split of
label and residual momenta,

238
00:14:49,205 --> 00:14:51,592
and we thought about
the labels as discrete.

239
00:14:51,592 --> 00:14:53,050
But still, even if
they're discrete

240
00:14:53,050 --> 00:14:55,420
there's a freedom in
how we make this split.

241
00:14:55,420 --> 00:14:59,740
And so we could, if you
like, take our label momentum

242
00:14:59,740 --> 00:15:04,900
or our label momentum
operator, and transform it

243
00:15:04,900 --> 00:15:09,130
by adding some beta, which
has a power counting of order

244
00:15:09,130 --> 00:15:11,270
lambda squared.

245
00:15:11,270 --> 00:15:15,190
And if we make a
compensating transformation

246
00:15:15,190 --> 00:15:18,730
in the residual momentum,
or the residual derivative,

247
00:15:18,730 --> 00:15:24,100
then that would be a symmetry
of the decomposition.

248
00:15:24,100 --> 00:15:28,240
Wouldn't change anything,
end up beta being 0.

249
00:15:28,240 --> 00:15:29,575
Projects onto these two cases.

250
00:15:33,530 --> 00:15:36,970
So if you wrote that as
a change to the field,

251
00:15:36,970 --> 00:15:43,120
then you'd be saying that
the field picks up a phase.

252
00:15:43,120 --> 00:15:46,180
That's the transformation
for the derivatives.

253
00:15:46,180 --> 00:15:49,030
And then, you could
shift the label

254
00:15:49,030 --> 00:15:51,430
by beta, that's the
transformation for the beta,

255
00:15:51,430 --> 00:15:54,072
for the label.

256
00:15:54,072 --> 00:15:55,780
OK, so this is a
different transformation

257
00:15:55,780 --> 00:15:57,738
than the ones we were
talking about over there,

258
00:15:57,738 --> 00:16:01,000
and it's a transformation that
has to do with this freedom.

259
00:16:01,000 --> 00:16:03,730
But basically, what
this freedom does

260
00:16:03,730 --> 00:16:07,490
is it connects the
derivatives to each other.

261
00:16:07,490 --> 00:16:10,580
So it connects this combination,
it does something very simple,

262
00:16:10,580 --> 00:16:13,120
so after you know what
it does it's simpler

263
00:16:13,120 --> 00:16:15,782
just to say what it does
and forget about thinking

264
00:16:15,782 --> 00:16:16,990
about it as a transformation.

265
00:16:16,990 --> 00:16:19,210
It simply connects
the label operator

266
00:16:19,210 --> 00:16:22,930
to be label operator plus
the derivative operator.

267
00:16:22,930 --> 00:16:23,470
OK.

268
00:16:23,470 --> 00:16:26,500
But these two have different
orders in the power counting.

269
00:16:26,500 --> 00:16:28,558
And so what this
will do, is it'll

270
00:16:28,558 --> 00:16:30,100
connect things that
are leading order

271
00:16:30,100 --> 00:16:33,115
to things that are subleading
order, for example.

272
00:16:39,270 --> 00:16:53,350
So one way of saying that
is it that it connects

273
00:16:53,350 --> 00:16:55,780
leaving and subleading
Wilson coefficients,

274
00:16:55,780 --> 00:16:57,970
because you can always
think of the coefficient

275
00:16:57,970 --> 00:17:00,400
as the thing being fixed.

276
00:17:00,400 --> 00:17:07,089
And then it would do it in both
something like a leading order

277
00:17:07,089 --> 00:17:12,310
to subleading order
Lagrangian, or if you also

278
00:17:12,310 --> 00:17:14,887
wrote down operators.

279
00:17:14,887 --> 00:17:17,619
If you wrote down a series
of some external operator,

280
00:17:17,619 --> 00:17:20,920
maybe it's a weak
interaction, then

281
00:17:20,920 --> 00:17:23,290
it could connect as we
saw on the problems set.

282
00:17:23,290 --> 00:17:28,060
It could connect C1 to
C0 like it did in HQET.

283
00:17:28,060 --> 00:17:31,480
OK, so this one
clearly does that.

284
00:17:40,600 --> 00:17:43,977
So after you see that,
then you should think

285
00:17:43,977 --> 00:17:45,060
about the following thing.

286
00:17:45,060 --> 00:17:46,810
Well, this is
reparameterization symmetry.

287
00:17:46,810 --> 00:17:49,180
What did reparameterization
symmetry do here?

288
00:17:49,180 --> 00:17:51,633
It connected label and
residuals to each other.

289
00:17:51,633 --> 00:17:53,800
It said, well, they were
both part of the same thing

290
00:17:53,800 --> 00:17:56,800
in the beginning,
so there should

291
00:17:56,800 --> 00:18:00,430
be some connection later
on in the effective theory,

292
00:18:00,430 --> 00:18:03,780
and that remains true.

293
00:18:03,780 --> 00:18:08,320
It's encoded in this
reparameterization symmetry.

294
00:18:08,320 --> 00:18:10,620
But you can also now
try to think about,

295
00:18:10,620 --> 00:18:12,370
given that we have
these derivatives, what

296
00:18:12,370 --> 00:18:13,495
about gauging that formula?

297
00:18:16,310 --> 00:18:18,230
Gauge symmetry
basically was telling us

298
00:18:18,230 --> 00:18:20,360
how to turn p mu into
a covariant derivative

299
00:18:20,360 --> 00:18:22,880
and partial mu into a
convenient derivative.

300
00:18:22,880 --> 00:18:25,310
So can we think
about combining what

301
00:18:25,310 --> 00:18:26,870
we know from gauge
symmetry together

302
00:18:26,870 --> 00:18:28,760
with reparameterization?

303
00:18:28,760 --> 00:18:31,910
And we actually can,
if we're careful.

304
00:18:35,660 --> 00:18:37,697
And where this is
leading is, we'll

305
00:18:37,697 --> 00:18:39,780
find that there's basically
in the end of the day,

306
00:18:39,780 --> 00:18:41,000
there's very simple--

307
00:18:41,000 --> 00:18:43,400
I don't know if we'll get
there today, but we'll try.

308
00:18:43,400 --> 00:18:44,990
There's very simple
building blocks

309
00:18:44,990 --> 00:18:46,657
that in the end of
the day you can build

310
00:18:46,657 --> 00:18:48,420
all the SCET operators out of.

311
00:18:48,420 --> 00:18:52,940
And we're kind of moving
our way in that direction.

312
00:18:52,940 --> 00:18:54,620
OK, so let's try to gauge this.

313
00:18:57,470 --> 00:19:00,450
n dot label operator is 0.

314
00:19:00,450 --> 00:19:02,840
So for that guy, you just
have i n dot partial,

315
00:19:02,840 --> 00:19:05,000
there's no label operator piece.

316
00:19:05,000 --> 00:19:06,950
And gauging it
just takes you to i

317
00:19:06,950 --> 00:19:09,440
n dot D, which is
the full D that

318
00:19:09,440 --> 00:19:12,690
had both an ultra soft
and collinear piece to it.

319
00:19:12,690 --> 00:19:16,700
And if you look back at how our
transformations were defined

320
00:19:16,700 --> 00:19:27,580
for U c and U ultra
soft, then this guy

321
00:19:27,580 --> 00:19:29,770
is basically just
transforming in the way

322
00:19:29,770 --> 00:19:32,110
that you would want a covariant
derivative to transform,

323
00:19:32,110 --> 00:19:34,810
and it does that under both
types of symmetries, because

324
00:19:34,810 --> 00:19:37,756
of the way we set it up like
a background field for the n

325
00:19:37,756 --> 00:19:40,660
dot a ultra soft.

326
00:19:40,660 --> 00:19:43,790
OK, but that's actually not
really related to this story,

327
00:19:43,790 --> 00:19:47,840
because there is no split
in the n dot p component.

328
00:19:47,840 --> 00:19:51,388
So it's really the other
components, the and the n bar,

329
00:19:51,388 --> 00:19:53,680
where we have to pay more
attention to what's going on.

330
00:19:56,300 --> 00:20:01,180
So if we look at what the
gauge transformations mean

331
00:20:01,180 --> 00:20:03,828
for those components,
it's the same story

332
00:20:03,828 --> 00:20:05,370
but I'm going to
write it out anyway.

333
00:20:30,530 --> 00:20:32,600
OK, so this isn't quite
how we wrote it before.

334
00:20:32,600 --> 00:20:35,810
We wrote it before as a
transformation on the A,

335
00:20:35,810 --> 00:20:37,670
but if you put the
transformation on the A

336
00:20:37,670 --> 00:20:39,212
together with the
derivative then you

337
00:20:39,212 --> 00:20:41,605
can write it like this.

338
00:20:41,605 --> 00:20:44,210
For this piece
here, for example,

339
00:20:44,210 --> 00:20:45,500
remember what this thing is.

340
00:20:45,500 --> 00:20:50,510
This thing here is a label
operator p bar plus g n bar

341
00:20:50,510 --> 00:20:51,860
dot A n.

342
00:20:51,860 --> 00:20:53,790
Label operator
doesn't hit this guy,

343
00:20:53,790 --> 00:20:57,830
there's no label momentum
in the ultra soft U,

344
00:20:57,830 --> 00:21:02,570
so it just goes through that guy
or gives 0, and UU dagger is 1.

345
00:21:02,570 --> 00:21:05,840
So this piece doesn't
contribute there,

346
00:21:05,840 --> 00:21:08,600
and this piece transformed
in this way already.

347
00:21:08,600 --> 00:21:13,130
These ones are slightly more
involved, but what I'm saying

348
00:21:13,130 --> 00:21:15,890
is just a summary of what
we already learned earlier

349
00:21:15,890 --> 00:21:17,720
or talked about earlier.

350
00:21:17,720 --> 00:21:21,830
And then finally, there's also
the ultra soft transformation.

351
00:21:21,830 --> 00:21:28,250
And the ultra soft didn't
transform under the collinear.

352
00:21:28,250 --> 00:21:32,310
And of course it did transform
under the ultra soft.

353
00:21:32,310 --> 00:21:36,783
So this is a summary of
what we said earlier.

354
00:21:36,783 --> 00:21:38,450
So the simplest idea
that you would say,

355
00:21:38,450 --> 00:21:40,735
well, is just like,
take these guys

356
00:21:40,735 --> 00:21:42,860
and replace this by a
covariant derivative and that

357
00:21:42,860 --> 00:21:45,380
by a covariant
derivative, right?

358
00:21:57,400 --> 00:22:01,250
Always useful to think about
the simplest thing first.

359
00:22:01,250 --> 00:22:11,625
So this would be the simplest
thing, and that doesn't work.

360
00:22:11,625 --> 00:22:13,000
The reason that
that doesn't work

361
00:22:13,000 --> 00:22:15,850
is that if you look
at these two pieces,

362
00:22:15,850 --> 00:22:17,422
they don't transform
in the same way.

363
00:22:17,422 --> 00:22:18,880
So if I were to
build operators out

364
00:22:18,880 --> 00:22:21,520
of this guy that are
gauge invariant, then when

365
00:22:21,520 --> 00:22:23,680
I looked at the
collinear transformation,

366
00:22:23,680 --> 00:22:26,260
because this guy doesn't
transform and this guy does,

367
00:22:26,260 --> 00:22:28,030
this whole thing is
not transforming.

368
00:22:28,030 --> 00:22:31,420
So if I just stuck this guy
in, if this guy was invariant,

369
00:22:31,420 --> 00:22:34,750
this guy wouldn't
be, and vise versa.

370
00:22:34,750 --> 00:22:36,070
OK, so this doesn't work.

371
00:22:59,990 --> 00:23:03,005
AUDIENCE: Can you remind me how
the ultra soft field transforms

372
00:23:03,005 --> 00:23:09,321
under collinear with its
label summing conventions?

373
00:23:09,321 --> 00:23:11,220
PROFESSOR: Ultra
soft under collinear?

374
00:23:11,220 --> 00:23:15,035
So this guy didn't transform.

375
00:23:15,035 --> 00:23:18,510
AUDIENCE: I'm thinking of the
top equation, in the middle.

376
00:23:18,510 --> 00:23:19,397
i n D goes under--

377
00:23:19,397 --> 00:23:20,230
PROFESSOR: Oh, yeah.

378
00:23:20,230 --> 00:23:27,000
So that one's a
little, you know, yeah.

379
00:23:27,000 --> 00:23:30,630
So, you want to know which one?

380
00:23:30,630 --> 00:23:32,730
I can look it up for you.

381
00:23:32,730 --> 00:23:34,750
AUDIENCE: Well the one
you were pointing at,

382
00:23:34,750 --> 00:23:39,660
so that's how the D and the
A n part that transforms,

383
00:23:39,660 --> 00:23:41,232
but the A U soft--

384
00:23:41,232 --> 00:23:41,940
PROFESSOR: Right.

385
00:23:41,940 --> 00:23:44,040
AUDIENCE: Is there
something, like,

386
00:23:44,040 --> 00:23:46,388
PROFESSOR: Me remind myself,
then I'll remind you.

387
00:23:59,880 --> 00:24:01,580
So the A ultra slot
never transformed

388
00:24:01,580 --> 00:24:03,405
under the collinear, right?

389
00:24:03,405 --> 00:24:04,030
AUDIENCE: Yeah.

390
00:24:04,030 --> 00:24:05,210
I guess I don't know how--

391
00:24:05,210 --> 00:24:06,890
PROFESSOR: And
then, so then, yeah.

392
00:24:06,890 --> 00:24:07,900
So right.

393
00:24:07,900 --> 00:24:08,990
So how does this work out?

394
00:24:08,990 --> 00:24:10,790
The reason that
this works out is

395
00:24:10,790 --> 00:24:15,860
because there was an n dot A
ultra soft field in the gauge

396
00:24:15,860 --> 00:24:20,300
transformation of the
calendar n dot a field.

397
00:24:20,300 --> 00:24:21,590
That's why it works out.

398
00:24:24,120 --> 00:24:27,830
Good, because it would
take me forever to find it.

399
00:24:27,830 --> 00:24:30,780
All right, so what
are we going to do?

400
00:24:30,780 --> 00:24:32,870
So we need some
kind of compensating

401
00:24:32,870 --> 00:24:35,450
object to make these things
transform in the same way.

402
00:24:35,450 --> 00:24:38,240
And we have such a compensating
object, that's our Wilson line.

403
00:24:54,420 --> 00:24:58,750
So the Wilson line
transformed on one side.

404
00:24:58,750 --> 00:25:02,760
And so we can use it to
modify those formulas

405
00:25:02,760 --> 00:25:05,700
and get a result that,
where both terms transform

406
00:25:05,700 --> 00:25:06,410
in the same way.

407
00:25:45,450 --> 00:25:57,990
I just stick to the W in where
I need it in these two terms.

408
00:25:57,990 --> 00:26:03,360
And then I have set up things
so that the transformation

409
00:26:03,360 --> 00:26:06,727
of the two terms are the same.

410
00:26:06,727 --> 00:26:08,310
And you could call
this, if you wanted

411
00:26:08,310 --> 00:26:11,550
to define some kind
of full D, then this

412
00:26:11,550 --> 00:26:13,710
is the kind of
closest that you can

413
00:26:13,710 --> 00:26:24,740
get to defining a
sort of full D. OK,

414
00:26:24,740 --> 00:26:30,470
so these W's ensure both terms--

415
00:26:40,813 --> 00:26:42,980
it would ensure that under
collinear transformation,

416
00:26:42,980 --> 00:26:48,110
both terms transform at
the U c n, on the outside.

417
00:26:48,110 --> 00:26:50,060
OK, so we could
just stick anywhere

418
00:26:50,060 --> 00:26:53,120
that we have i n
bar dot D perp, we

419
00:26:53,120 --> 00:26:55,790
could just replace it by
this and then expand out.

420
00:26:55,790 --> 00:26:56,500
I.e.

421
00:26:56,500 --> 00:26:58,580
we could get a
subleading term by just

422
00:26:58,580 --> 00:27:00,560
using this operator here.

423
00:27:00,560 --> 00:27:03,140
And we know that that
is going to be there

424
00:27:03,140 --> 00:27:06,380
because of reparameterization
symmetry, which

425
00:27:06,380 --> 00:27:07,880
was connecting the derivatives.

426
00:27:07,880 --> 00:27:09,320
And therefore
connects these two.

427
00:27:09,320 --> 00:27:10,737
And once we put
in gauge cemetery,

428
00:27:10,737 --> 00:27:12,270
then it's-- this is the formula.

429
00:27:15,950 --> 00:27:18,740
So if you go back to what
we talked about earlier,

430
00:27:18,740 --> 00:27:25,075
we said earlier, I said A
mu, for the full theory,

431
00:27:25,075 --> 00:27:26,450
that you could
just sort of think

432
00:27:26,450 --> 00:27:29,180
of it as splitting
between a collinear

433
00:27:29,180 --> 00:27:30,650
and an ultra soft field.

434
00:27:30,650 --> 00:27:34,390
And then I said plus
dot dot dot, OK?

435
00:27:34,390 --> 00:27:37,560
And I said, we'll talk about
what plus dot dot dot is later.

436
00:27:37,560 --> 00:27:39,470
So now is later.

437
00:27:39,470 --> 00:27:41,480
What plus dot dot
dot is, is terms

438
00:27:41,480 --> 00:27:43,260
that you would generate
from, for example,

439
00:27:43,260 --> 00:27:45,135
the derivatives here
hitting the Wilson line,

440
00:27:45,135 --> 00:27:47,630
or expanding out these Wilson
lines, or the gauge field.

441
00:27:47,630 --> 00:27:50,450
And so those are
making it have terms

442
00:27:50,450 --> 00:27:53,040
with more fields
than just one field.

443
00:27:53,040 --> 00:27:55,100
And so, if you thought
about the A in here

444
00:27:55,100 --> 00:27:57,380
as sort of a full theory
A, and the A's in here

445
00:27:57,380 --> 00:27:59,900
as the effective theory,
you could drive a formula

446
00:27:59,900 --> 00:28:03,330
like this one but it would
involve more complicated stuff.

447
00:28:03,330 --> 00:28:05,457
It's not good to think
about it this way,

448
00:28:05,457 --> 00:28:07,040
it's better to think
about it in terms

449
00:28:07,040 --> 00:28:09,415
of the covariant derivatives,
because then you're already

450
00:28:09,415 --> 00:28:12,323
thinking about gauge invariance,
so this formula was useful

451
00:28:12,323 --> 00:28:14,240
earlier, but it's actually
these formulas here

452
00:28:14,240 --> 00:28:16,430
that are the more
useful way of thinking

453
00:28:16,430 --> 00:28:18,500
about taking full
theory derivatives

454
00:28:18,500 --> 00:28:20,510
and turning them into
effective theory results.

455
00:28:37,440 --> 00:28:39,923
All right, so we can
dispense with thinking

456
00:28:39,923 --> 00:28:42,090
about the plus dot dot dot
there because it's better

457
00:28:42,090 --> 00:28:43,590
just to think about
it in this form,

458
00:28:43,590 --> 00:28:47,190
where you can write a
nice closed expression.

459
00:28:47,190 --> 00:28:52,590
OK, so let's do one example
of a subleading term.

460
00:28:57,250 --> 00:29:00,150
So we had an operator that
involved covariate derivatives

461
00:29:00,150 --> 00:29:01,244
that were collinear.

462
00:29:04,210 --> 00:29:04,750
This one.

463
00:29:07,960 --> 00:29:11,710
And if we just use the top
formula there, then we know--

464
00:29:11,710 --> 00:29:14,920
so this is a term
that was in L0.

465
00:29:17,710 --> 00:29:24,515
Then we know that there's a term
in L1 that looks as follows.

466
00:29:29,645 --> 00:29:31,520
And I want to write it
out so you sort of see

467
00:29:31,520 --> 00:29:34,365
how also the gauge
symmetry works out nicely.

468
00:29:57,310 --> 00:30:07,690
OK, so using the fact that I
can write this guy as a Wilson

469
00:30:07,690 --> 00:30:09,210
line.

470
00:30:09,210 --> 00:30:12,600
1 over p bar times
the Wilson line.

471
00:30:12,600 --> 00:30:15,000
And the fact that I get
some Wilson lines when

472
00:30:15,000 --> 00:30:16,900
I look at the subleading term.

473
00:30:16,900 --> 00:30:20,040
So some of the Wilson lines,
I get W-W dagger is 1.

474
00:30:20,040 --> 00:30:21,917
And then the
remaining Wilson lines

475
00:30:21,917 --> 00:30:24,000
are sitting next to collinear
fields in such a way

476
00:30:24,000 --> 00:30:26,430
that this guy here is
collinear gauge invariant,

477
00:30:26,430 --> 00:30:28,530
this guy here is
collinear gauge invariant,

478
00:30:28,530 --> 00:30:31,113
and then there's an ultra soft
transformation that connects up

479
00:30:31,113 --> 00:30:32,820
all the pieces, OK?

480
00:30:32,820 --> 00:30:34,500
If we didn't have
these W's here,

481
00:30:34,500 --> 00:30:37,350
we wouldn't have got those
W's in the right spots there,

482
00:30:37,350 --> 00:30:39,030
and this operator
that I wrote down here

483
00:30:39,030 --> 00:30:41,880
would not be collinear
gauge invariant.

484
00:30:41,880 --> 00:30:44,280
So this operator is both
collinear and ultra soft gauge

485
00:30:44,280 --> 00:30:45,218
invariant.

486
00:30:49,800 --> 00:30:53,340
But most importantly,
it's collinear gauge

487
00:30:53,340 --> 00:30:59,250
invariant because of
what we did there.

488
00:30:59,250 --> 00:31:01,377
And there's no Wilson
coefficient to this operator.

489
00:31:01,377 --> 00:31:03,710
It's got the same coefficient
as the leading order term,

490
00:31:03,710 --> 00:31:05,010
so there's nothing non-trivial.

491
00:31:05,010 --> 00:31:08,630
We could just use it to
get subleading stuff.

492
00:31:08,630 --> 00:31:10,640
OK.

493
00:31:10,640 --> 00:31:14,150
And you could also do
similar things for currents.

494
00:31:14,150 --> 00:31:15,990
I'm not going to give
examples of that,

495
00:31:15,990 --> 00:31:19,490
but it's also a powerful
tool for connecting

496
00:31:19,490 --> 00:31:21,080
coefficients of
leading and subleading

497
00:31:21,080 --> 00:31:23,990
operators and currents.

498
00:31:23,990 --> 00:31:26,270
Not every operator in
the subleading Lagrangian

499
00:31:26,270 --> 00:31:27,450
is connected.

500
00:31:27,450 --> 00:31:34,220
So there are some that could
not be connected in some order,

501
00:31:34,220 --> 00:31:44,790
but there's actually
more connections

502
00:31:44,790 --> 00:31:49,670
than there are in HQET because
we have more symmetries.

503
00:31:49,670 --> 00:31:51,740
OK, so any questions about that?

504
00:31:57,040 --> 00:31:58,630
OK.

505
00:31:58,630 --> 00:32:01,090
So so far, all of
our discussions

506
00:32:01,090 --> 00:32:02,980
have been about one
collinear field.

507
00:32:02,980 --> 00:32:05,118
When we SCET, we
actually in general

508
00:32:05,118 --> 00:32:07,160
want to talk about more
than one collinear field.

509
00:32:07,160 --> 00:32:08,952
So how would we generalize
everything we've

510
00:32:08,952 --> 00:32:11,140
discussed to more than one?

511
00:32:25,690 --> 00:32:29,200
So we would have, in this case,
more than one energetic hadron,

512
00:32:29,200 --> 00:32:30,470
more than one energetic jet.

513
00:32:30,470 --> 00:32:33,220
So far we've been talking about
one energetic hadron, or one

514
00:32:33,220 --> 00:32:34,368
energetic jet.

515
00:32:34,368 --> 00:32:35,410
What if we have two jets?

516
00:32:35,410 --> 00:32:37,577
Then we would need two types
of collinear field, one

517
00:32:37,577 --> 00:32:39,280
for each of those jets.

518
00:32:39,280 --> 00:32:41,380
And basically,
what we have to do

519
00:32:41,380 --> 00:32:44,870
is take our collinear
Lagrangian--

520
00:32:44,870 --> 00:32:51,550
well let me call it L n
0, which is the fermion

521
00:32:51,550 --> 00:32:55,090
piece plus the gluon piece.

522
00:32:58,850 --> 00:33:00,080
And we sum over n.

523
00:33:00,080 --> 00:33:02,570
We have to sum
over all n's which

524
00:33:02,570 --> 00:33:05,510
are corresponding to individual
distinguishable collinear

525
00:33:05,510 --> 00:33:06,950
fields.

526
00:33:06,950 --> 00:33:09,320
So the question is, what
does it mean, sum over n?

527
00:33:09,320 --> 00:33:12,625
What is the distinction
between collinear fields?

528
00:33:15,710 --> 00:33:17,690
And so here's the
words, and then we'll

529
00:33:17,690 --> 00:33:20,780
explain what they mean.

530
00:33:20,780 --> 00:33:24,705
This sum is over
inequivalent n's,

531
00:33:24,705 --> 00:33:26,330
though that should
be obvious that they

532
00:33:26,330 --> 00:33:28,030
have to be equivalent.

533
00:33:28,030 --> 00:33:31,460
But what makes them
inequivalent is the fact

534
00:33:31,460 --> 00:33:37,920
that they are RPI
equivalence classes.

535
00:33:37,920 --> 00:33:39,740
So that's a funny sentence.

536
00:33:39,740 --> 00:33:43,470
Inequivalent RPI
equivalence classes.

537
00:33:43,470 --> 00:33:45,830
So, two ends are
the same if they

538
00:33:45,830 --> 00:33:48,700
belong in an equivalence class
that could be connected to--

539
00:33:48,700 --> 00:33:51,705
where they could be connected by
reparameterization invariance.

540
00:33:51,705 --> 00:33:54,080
So you should think of the
n's that I'm summing over here

541
00:33:54,080 --> 00:33:56,470
as just members,
one of each class.

542
00:33:56,470 --> 00:34:01,790
They're kind of just picking
out what that class is, just

543
00:34:01,790 --> 00:34:04,070
labeling it by one member.

544
00:34:04,070 --> 00:34:06,440
And then I sum over
an inequivalent set.

545
00:34:09,540 --> 00:34:12,840
So let's just imagine
that we have some n's.

546
00:34:12,840 --> 00:34:16,739
n1, n2, n3.

547
00:34:16,739 --> 00:34:18,300
And we can ask
the question, what

548
00:34:18,300 --> 00:34:20,703
makes them equivalent
or inequivalent?

549
00:34:34,270 --> 00:34:42,580
So let me call them, that the
n i collinear modes for any i

550
00:34:42,580 --> 00:34:43,150
are distinct.

551
00:34:48,420 --> 00:34:52,110
And there's a condition that
if I dot two of these ends

552
00:34:52,110 --> 00:34:55,020
together, they should
not be close together.

553
00:34:55,020 --> 00:34:57,030
And in fact that they
should be some value

554
00:34:57,030 --> 00:35:00,750
that's much bigger
than lambda squared.

555
00:35:00,750 --> 00:35:03,910
Obviously, if i was equal
to j, we would get 0.

556
00:35:03,910 --> 00:35:06,510
But for any i not
equal to j, we will

557
00:35:06,510 --> 00:35:09,420
say the n's are equivalent
if the dot product is much

558
00:35:09,420 --> 00:35:12,810
bigger than lambda squared.

559
00:35:12,810 --> 00:35:16,812
So let's see why it's lambda
squared by doing an example.

560
00:35:16,812 --> 00:35:18,270
So that's, if you
like, how you can

561
00:35:18,270 --> 00:35:19,562
define the equivalence classes.

562
00:35:25,390 --> 00:35:28,480
Let's imagine you have
some momentum, p2, which

563
00:35:28,480 --> 00:35:31,270
is a large piece times n2.

564
00:35:31,270 --> 00:35:34,610
And then you dot n1 into it.

565
00:35:34,610 --> 00:35:39,400
So n1 dot p2 is Q n1 dot n2.

566
00:35:39,400 --> 00:35:41,860
And that would be
of order lambda

567
00:35:41,860 --> 00:35:48,650
squared if n1 dot n2 were
of order lambda squared.

568
00:35:48,650 --> 00:35:49,600
Right?

569
00:35:49,600 --> 00:35:51,790
But if n1 dot n2 are
order lambda squared,

570
00:35:51,790 --> 00:35:54,160
and therefore n1 dot p2
is order lambda squared,

571
00:35:54,160 --> 00:35:56,980
you would say p2 is an
n1 collinear particle,

572
00:35:56,980 --> 00:35:59,510
because this is the right power
counting for an n1 collinear

573
00:35:59,510 --> 00:36:00,010
particle.

574
00:36:07,860 --> 00:36:10,520
So it's both n1 collinear
and n2 collinear,

575
00:36:10,520 --> 00:36:12,040
and that just means
n1 n2 are just

576
00:36:12,040 --> 00:36:13,790
two members of the
same equivalence class.

577
00:36:22,980 --> 00:36:26,150
So if this is true, that the
dot products of order lambda

578
00:36:26,150 --> 00:36:27,770
squared, then n1
and n2 are within

579
00:36:27,770 --> 00:36:29,030
the same equivalence class.

580
00:36:36,710 --> 00:36:39,760
Which if you wanted some
notation, you could say n2

581
00:36:39,760 --> 00:36:42,293
is in the class defined by n1.

582
00:36:42,293 --> 00:36:43,960
So you could really
think of this as sum

583
00:36:43,960 --> 00:36:45,340
over classes but its--

584
00:36:45,340 --> 00:36:48,370
usually people just
write sum over n.

585
00:36:48,370 --> 00:36:50,993
OK, so that is in
some sense clear,

586
00:36:50,993 --> 00:36:53,410
that you want to be summing
over things that are distinct.

587
00:36:53,410 --> 00:36:55,660
In this case of back to back
jets that we talked about,

588
00:36:55,660 --> 00:36:56,660
they're pretty distinct.

589
00:36:56,660 --> 00:36:59,290
One's going this way,
one's going that way.

590
00:36:59,290 --> 00:37:01,120
The n's dotted into
each other are 2,

591
00:37:01,120 --> 00:37:05,960
so that's certainly much
greater the lambda squared.

592
00:37:05,960 --> 00:37:09,230
But in general, this is what
you have to have in order

593
00:37:09,230 --> 00:37:12,210
to make them distinct fields.

594
00:37:12,210 --> 00:37:15,500
So then, everything basically
that we've talked about kind of

595
00:37:15,500 --> 00:37:19,310
goes through again, and I'm
not going to dwell on it,

596
00:37:19,310 --> 00:37:21,410
but I will just kind
of repeat some things.

597
00:37:59,160 --> 00:38:02,350
So collinear gauge
transformations, for example.

598
00:38:02,350 --> 00:38:04,530
You would have now a
new type of scaling

599
00:38:04,530 --> 00:38:06,505
that you can have
for different fields,

600
00:38:06,505 --> 00:38:08,880
and you could have two different
types of collinear gauge

601
00:38:08,880 --> 00:38:09,570
transformations.

602
00:38:09,570 --> 00:38:13,140
One for your n1 collinear
field, one for your n2.

603
00:38:13,140 --> 00:38:15,360
If n1 and n2 are
distinct, then those

604
00:38:15,360 --> 00:38:19,090
will have distinct scalings
for the corresponding momenta,

605
00:38:19,090 --> 00:38:22,470
so they'll be distinct
transformations.

606
00:38:22,470 --> 00:38:25,620
And fields won't transform
under the other guys'--

607
00:38:25,620 --> 00:38:28,512
n1 collinear fields won't
transform under the n2 gauge

608
00:38:28,512 --> 00:38:29,970
transformation
because, again, that

609
00:38:29,970 --> 00:38:32,760
would spoil the power
counting for the momenta.

610
00:38:35,290 --> 00:38:36,930
So at some level
it's very intuitive

611
00:38:36,930 --> 00:38:42,170
to figure out how
the results are.

612
00:38:42,170 --> 00:38:44,082
I'm not going to go through it.

613
00:38:44,082 --> 00:38:45,540
But suffice it to
say that we would

614
00:38:45,540 --> 00:38:48,720
have collinear gauge
transformations

615
00:38:48,720 --> 00:38:52,800
for each collinear guy.

616
00:38:52,800 --> 00:38:56,610
Reparameterization, same story.

617
00:38:56,610 --> 00:39:06,396
We have separate invariances
for each pair of n's.

618
00:39:06,396 --> 00:39:10,090
So n1 and n1 bar,
for the n1 sector.

619
00:39:10,090 --> 00:39:12,520
We have a
reparameterization symmetry.

620
00:39:12,520 --> 00:39:15,250
n2 and n2 bar for the n2 sector.

621
00:39:15,250 --> 00:39:19,803
Reparameterization
symmetry, et cetera.

622
00:39:19,803 --> 00:39:21,220
And here is actually
where you see

623
00:39:21,220 --> 00:39:23,553
that there's something that
looks different than Lorentz

624
00:39:23,553 --> 00:39:24,910
invariance that's going on.

625
00:39:24,910 --> 00:39:26,680
Because the
reparameterization are only

626
00:39:26,680 --> 00:39:28,960
acting within a sector.

627
00:39:28,960 --> 00:39:31,960
So if you do an n1 type
reparameterization,

628
00:39:31,960 --> 00:39:35,590
there's no transformation
of an n2 type Wilson line,

629
00:39:35,590 --> 00:39:37,740
or an n2 type gauge field.

630
00:39:37,740 --> 00:39:40,480
So an n1 transformation
affects the n1 collinear fields

631
00:39:40,480 --> 00:39:44,410
and objects, n2 type,
which could affect that.

632
00:39:44,410 --> 00:39:47,020
And it's more like a
Lorentz transformation

633
00:39:47,020 --> 00:39:49,495
that sort of acts within
the sector all by itself.

634
00:39:49,495 --> 00:39:51,370
But it's not really a
Lorentz transformation,

635
00:39:51,370 --> 00:39:53,440
it's just
reparameterization symmetry.

636
00:39:56,470 --> 00:39:58,300
OK?

637
00:39:58,300 --> 00:40:01,360
But it's exactly the
transformations we wrote down,

638
00:40:01,360 --> 00:40:05,410
just you don't transform n2
type fields when you do an n1.

639
00:40:09,570 --> 00:40:15,090
And, just like we had before,
if you do matching calculations

640
00:40:15,090 --> 00:40:16,770
you get Wilson
lines, but now there

641
00:40:16,770 --> 00:40:18,000
can be more than one type.

642
00:40:23,280 --> 00:40:26,130
So we had this W Wilson
line that showed up,

643
00:40:26,130 --> 00:40:27,780
and we did matching
calculations.

644
00:40:27,780 --> 00:40:30,276
And I want to give
you here one example

645
00:40:30,276 --> 00:40:35,730
which we'll come back and
talk about more later on,

646
00:40:35,730 --> 00:40:37,230
and which we've
already mentioned.

647
00:40:37,230 --> 00:40:39,720
So consider our
example of e plus e

648
00:40:39,720 --> 00:40:41,430
minus producing two jets.

649
00:40:46,870 --> 00:40:49,360
So in the full
theory, you would just

650
00:40:49,360 --> 00:40:51,910
have a vector current
from the photon.

651
00:40:51,910 --> 00:40:55,992
And if you want to match that
onto the two jet operator,

652
00:40:55,992 --> 00:40:57,700
you can go through
the same type of thing

653
00:40:57,700 --> 00:41:03,685
that we did when we were doing
the B to S gamma example.

654
00:41:12,220 --> 00:41:15,010
And the difference is here
that we get two different types

655
00:41:15,010 --> 00:41:16,060
of Wilson lines.

656
00:41:16,060 --> 00:41:21,550
So this n1 Wilson line, W n1.

657
00:41:21,550 --> 00:41:23,543
My notation here is
that the subscript

658
00:41:23,543 --> 00:41:25,210
is supposed to indicate
to you that it's

659
00:41:25,210 --> 00:41:30,250
n1 bar of A n1 that shows up.

660
00:41:30,250 --> 00:41:37,480
And then likewise, W n2 is a
function of n2 bar dot A n2.

661
00:41:37,480 --> 00:41:38,980
So you have to
decide whether you're

662
00:41:38,980 --> 00:41:42,250
going to call it W
sub n2 bar or n2,

663
00:41:42,250 --> 00:41:45,200
but anyway this is my notation.

664
00:41:45,200 --> 00:41:48,490
So you get Wilson lines that
are built out of the n1 bar

665
00:41:48,490 --> 00:41:51,720
dot A n1 field, which are order
of lambda 0, or the n2 bar

666
00:41:51,720 --> 00:41:54,760
dot A n2 fields, which
were order lambda 0.

667
00:41:54,760 --> 00:42:01,503
So this is lambda 0,
and this is lambda 0.

668
00:42:01,503 --> 00:42:02,920
By power counting
we can certainly

669
00:42:02,920 --> 00:42:04,257
get objects like that.

670
00:42:04,257 --> 00:42:06,340
And when you go through
the process of integrating

671
00:42:06,340 --> 00:42:09,880
our off shell particles, just
like we did for B to S gamma

672
00:42:09,880 --> 00:42:11,823
where we attach
gluons and we found

673
00:42:11,823 --> 00:42:13,240
that some lines
were off shell, so

674
00:42:13,240 --> 00:42:14,620
we had to integrate them out.

675
00:42:14,620 --> 00:42:17,423
If you do that for this
process, you get this operator.

676
00:42:40,450 --> 00:42:42,200
So when we construct
the effective theory,

677
00:42:42,200 --> 00:42:46,520
we have to integrate out off
shell particles and doing so

678
00:42:46,520 --> 00:42:48,418
generates this
Wilson mine operator.

679
00:42:48,418 --> 00:42:50,210
It's a little more
complicated in this case

680
00:42:50,210 --> 00:42:52,460
because we get these
two Wilson lines.

681
00:42:52,460 --> 00:42:55,640
And I'll talk a little bit more
later on in a different context

682
00:42:55,640 --> 00:42:58,340
about what type of diagrams are
involved in getting these two

683
00:42:58,340 --> 00:42:59,970
different Wilson lines.

684
00:42:59,970 --> 00:43:02,870
But the result
is, in some sense,

685
00:43:02,870 --> 00:43:05,510
more intuitive than a
way of getting there.

686
00:43:05,510 --> 00:43:08,840
What's happening is, you're
getting this W n1 Wilson line

687
00:43:08,840 --> 00:43:11,540
next to the C n1 field, and
then this form of combination

688
00:43:11,540 --> 00:43:14,660
here is gauge invariant
under the n1 collinear gauge

689
00:43:14,660 --> 00:43:16,460
transformations.

690
00:43:16,460 --> 00:43:17,930
And the same thing here.

691
00:43:17,930 --> 00:43:20,280
This guy doesn't transform
under the n1 collinear gauge

692
00:43:20,280 --> 00:43:21,290
transformations.

693
00:43:21,290 --> 00:43:24,305
This guy does, this
guy's invariant.

694
00:43:24,305 --> 00:43:26,180
This guy's invariant
under n2 collinear gauge

695
00:43:26,180 --> 00:43:27,710
transformations.

696
00:43:27,710 --> 00:43:30,360
They both transform under ultra
soft gauge transformation,

697
00:43:30,360 --> 00:43:32,280
so they get connected
in that way.

698
00:43:32,280 --> 00:43:33,897
And again, you have--

699
00:43:33,897 --> 00:43:35,480
if you just think
about gauge symmetry

700
00:43:35,480 --> 00:43:37,438
and how it should come
out, then you would have

701
00:43:37,438 --> 00:43:39,410
guessed that it should be this.

702
00:43:39,410 --> 00:43:41,600
But you can also
derive it this way.

703
00:43:44,640 --> 00:43:45,435
Yeah.

704
00:43:45,435 --> 00:43:48,280
AUDIENCE: So for the
one collinear receptor,

705
00:43:48,280 --> 00:43:51,630
it's been very top-down.

706
00:43:51,630 --> 00:43:57,242
Is this, when you start to
throw in [INAUDIBLE] would you--

707
00:43:57,242 --> 00:43:59,970
PROFESSOR: So this is the
top-down way of thinking,

708
00:43:59,970 --> 00:44:02,250
that you just generate
it by integrating out.

709
00:44:02,250 --> 00:44:03,590
And you can do that.

710
00:44:03,590 --> 00:44:05,190
Do It to all orders
of the tree level

711
00:44:05,190 --> 00:44:07,710
diagrams, that's possible.

712
00:44:07,710 --> 00:44:11,010
Or you can-- but we're starting
to see a picture emerge

713
00:44:11,010 --> 00:44:12,278
from the bottom up, right?

714
00:44:12,278 --> 00:44:12,945
AUDIENCE: Right.

715
00:44:12,945 --> 00:44:14,610
I'm talking about the top order.

716
00:44:14,610 --> 00:44:19,320
So that's pretty bottom-up
only, is there a way of,

717
00:44:19,320 --> 00:44:23,480
because you're saying
let's now state

718
00:44:23,480 --> 00:44:27,790
that the effective theory has
many copies of the collinear

719
00:44:27,790 --> 00:44:28,290
gauge.

720
00:44:28,290 --> 00:44:29,970
PROFESSOR: So.

721
00:44:29,970 --> 00:44:30,570
Right.

722
00:44:30,570 --> 00:44:33,780
So I mean, you could try to
think of writing a formula,

723
00:44:33,780 --> 00:44:34,380
right?

724
00:44:34,380 --> 00:44:37,080
You could start, try to think
of it like, let's take A,

725
00:44:37,080 --> 00:44:40,910
and let's write A.
Let's just do two.

726
00:44:40,910 --> 00:44:41,410
Right.

727
00:44:46,350 --> 00:44:47,800
You could start
trying to do that.

728
00:44:47,800 --> 00:44:51,180
But at some point, it just
loses its friendliness.

729
00:44:51,180 --> 00:44:52,890
It's not really
buying you anything.

730
00:44:52,890 --> 00:44:54,570
So, starting to think
from the bottom up

731
00:44:54,570 --> 00:44:58,820
is actually a good way
of going at this point.

732
00:44:58,820 --> 00:45:00,600
You could still do
it from the top down,

733
00:45:00,600 --> 00:45:02,350
but it just gets more
and more cumbersome.

734
00:45:02,350 --> 00:45:02,892
AUDIENCE: OK.

735
00:45:05,480 --> 00:45:08,130
PROFESSOR: Any other questions?

736
00:45:08,130 --> 00:45:09,990
All right.

737
00:45:09,990 --> 00:45:14,955
So let's come back and study are
our leading order Lagrangian.

738
00:45:14,955 --> 00:45:16,830
And actually, we've
already learned something

739
00:45:16,830 --> 00:45:20,950
about factorization although
we don't know it yet.

740
00:45:20,950 --> 00:45:26,370
So what is this
word, factorization?

741
00:45:26,370 --> 00:45:28,860
One way of thinking about
what factorization is, is it's

742
00:45:28,860 --> 00:45:31,800
how different degrees of
freedom talk to each other.

743
00:45:31,800 --> 00:45:33,720
And given that we
have a leading order

744
00:45:33,720 --> 00:45:36,030
Lagrangian for the collinear
and ultra soft fields,

745
00:45:36,030 --> 00:45:38,280
we should know something
about how collinear and ultra

746
00:45:38,280 --> 00:45:41,080
soft fields talk to each other.

747
00:45:51,600 --> 00:45:56,810
So let's come back
and study Lcc 0.

748
00:46:05,370 --> 00:46:15,630
So the propagator, if we read
out what the propagator is,

749
00:46:15,630 --> 00:46:19,500
if we're careful about
signs of i epsilons--

750
00:46:29,520 --> 00:46:31,560
We had both particles
and antiparticles.

751
00:46:38,230 --> 00:46:39,820
You can think of
the antiparticles

752
00:46:39,820 --> 00:46:42,760
as the guys that
have the minus i0.

753
00:46:42,760 --> 00:46:46,580
And if you combine these
two things together,

754
00:46:46,580 --> 00:46:49,240
then that's just giving
you the thing that we

755
00:46:49,240 --> 00:46:54,970
got when we expanded QCD.

756
00:46:54,970 --> 00:46:56,440
So when you think
about deriving it

757
00:46:56,440 --> 00:46:58,090
from the [? SCET ?]
Lagrangian, you

758
00:46:58,090 --> 00:46:59,463
think about getting this piece.

759
00:46:59,463 --> 00:47:00,880
But then you have
to also sum over

760
00:47:00,880 --> 00:47:03,370
the other pieces with the
other side of the n bar dot p,

761
00:47:03,370 --> 00:47:07,976
and you get this piece and
they come back to that thing.

762
00:47:07,976 --> 00:47:10,615
So this is the particles.

763
00:47:10,615 --> 00:47:13,150
If I want to split out the
particles and antiparticles

764
00:47:13,150 --> 00:47:15,324
I can do this.

765
00:47:15,324 --> 00:47:18,190
This is the particles that have
n bar dot p greater than 0,

766
00:47:18,190 --> 00:47:19,480
and this is the antiparticles.

767
00:47:23,620 --> 00:47:28,700
In this notation where n bar dot
p falls, the fermion line flow,

768
00:47:28,700 --> 00:47:32,240
we have n bar dot p less than 0.

769
00:47:32,240 --> 00:47:34,030
OK, so that, in some
sense, we already

770
00:47:34,030 --> 00:47:37,160
alluded to, that the
propagator works out correctly.

771
00:47:37,160 --> 00:47:39,820
This is showing it explicitly.

772
00:47:39,820 --> 00:47:43,150
What about interactions?

773
00:47:43,150 --> 00:47:45,460
Well, I want to be
interested in a minute

774
00:47:45,460 --> 00:47:49,420
about ultra soft
interactions because they're

775
00:47:49,420 --> 00:47:51,850
kind of special.

776
00:47:51,850 --> 00:47:55,595
For the ultra soft
gluons, only n

777
00:47:55,595 --> 00:47:58,510
dot A ultra soft showed
up at leading order.

778
00:48:01,580 --> 00:48:05,080
In the LCC 0.

779
00:48:05,080 --> 00:48:09,110
So if we look at the Feynman
rule there, we have this--

780
00:48:09,110 --> 00:48:11,290
let's give this guy an index mu.

781
00:48:11,290 --> 00:48:12,840
And this guy is ultra soft.

782
00:48:12,840 --> 00:48:14,836
This guy is collinear.

783
00:48:21,650 --> 00:48:25,190
Then the Feynman rule just
has n mu in it, not gamma mu.

784
00:48:25,190 --> 00:48:27,360
OK?

785
00:48:27,360 --> 00:48:31,320
So that's an observation
that we know from our LCC 0

786
00:48:31,320 --> 00:48:34,390
and that just comes from--

787
00:48:34,390 --> 00:48:36,897
remember this just
comes from the i n

788
00:48:36,897 --> 00:48:38,730
dot D term because
that's the only term that

789
00:48:38,730 --> 00:48:41,930
had the ultra soft
gauge field in it.

790
00:48:41,930 --> 00:48:45,080
But there's another fact
that our Lagrangian told us.

791
00:48:45,080 --> 00:48:48,560
And that is that only the n
dot k ultra soft momentum,

792
00:48:48,560 --> 00:48:52,622
so if I call this momentum
k, only the component n

793
00:48:52,622 --> 00:48:55,090
dot k was also showing
up in the propagators.

794
00:49:05,650 --> 00:49:11,310
So there's a gauge field
statement as well as

795
00:49:11,310 --> 00:49:14,067
a statement about the
ultra soft momenta.

796
00:49:14,067 --> 00:49:15,900
And that was due to the
multi-pole expansion

797
00:49:15,900 --> 00:49:16,440
that we did.

798
00:49:23,960 --> 00:49:27,010
So if we do that at
the level of thinking

799
00:49:27,010 --> 00:49:29,695
about some kind of diagram--

800
00:49:29,695 --> 00:49:33,670
so let's think about some
type of diagram like this,

801
00:49:33,670 --> 00:49:36,820
and let's look at
this propagator here.

802
00:49:36,820 --> 00:49:46,000
That propagator-- we
brought this up before

803
00:49:46,000 --> 00:49:47,250
but let me write it out again.

804
00:49:53,790 --> 00:49:58,980
So if I say p is this
guy, and k is this guy,

805
00:49:58,980 --> 00:50:02,280
then this guy is p plus
k, but k being ultra soft,

806
00:50:02,280 --> 00:50:05,790
these guys are supposed to
be ultra soft, let's say.

807
00:50:05,790 --> 00:50:08,340
k being ultra soft, only
the n dot k shows up

808
00:50:08,340 --> 00:50:10,350
in that propagator.

809
00:50:10,350 --> 00:50:16,170
And if we work on shell,
for p squared of 0,

810
00:50:16,170 --> 00:50:21,280
so let me first rewrite this
so you can see where I'm going.

811
00:50:21,280 --> 00:50:25,140
I can form a full p
squared in the denominator

812
00:50:25,140 --> 00:50:26,970
from the terms that depend on p.

813
00:50:26,970 --> 00:50:30,000
That's just the full p
squared of the particle.

814
00:50:30,000 --> 00:50:35,040
And if this guy is an external
particle as it is in my figure,

815
00:50:35,040 --> 00:50:37,980
then p squared would be 0.

816
00:50:37,980 --> 00:50:44,250
OK, so if I go to the on shell
case, p squared equals zero,

817
00:50:44,250 --> 00:50:48,990
then this just becomes n
bar dot p, n bar dot p,

818
00:50:48,990 --> 00:50:52,995
n dot k, plus i0, which is
looking like just 1 over n

819
00:50:52,995 --> 00:50:55,150
dot k.

820
00:50:55,150 --> 00:50:56,490
So it's becoming very simple.

821
00:51:11,315 --> 00:51:12,940
For cleaner gluons
it would, of course,

822
00:51:12,940 --> 00:51:14,760
not be-- it would
be no simplification

823
00:51:14,760 --> 00:51:20,700
like that possible, but for the
ultra soft particles there is.

824
00:51:20,700 --> 00:51:29,429
And so what has just happened
is that the propagator

825
00:51:29,429 --> 00:51:30,262
is becoming eikonal.

826
00:51:36,550 --> 00:51:38,410
So our collinear
propagator reduces

827
00:51:38,410 --> 00:51:41,950
to the eikonal approximation,
which is just 1

828
00:51:41,950 --> 00:51:47,290
over n dot k, when appropriate.

829
00:51:47,290 --> 00:51:51,160
And when appropriate means
when it's interacting

830
00:51:51,160 --> 00:51:53,990
with ultra soft fields.

831
00:51:53,990 --> 00:51:56,260
So we can kind of
summarize that for all

832
00:51:56,260 --> 00:52:00,410
the different possible
cases in the following way.

833
00:52:00,410 --> 00:52:09,040
So you could have, if you want
to be careful about signs,

834
00:52:09,040 --> 00:52:13,480
you can think about
attaching ultra soft gluons

835
00:52:13,480 --> 00:52:17,110
to a fermion coming in
or a fermion going out.

836
00:52:17,110 --> 00:52:24,910
Or likewise, to a antiparticle
coming in or going out.

837
00:52:29,480 --> 00:52:31,800
We always take k going up.

838
00:52:35,310 --> 00:52:39,690
And if I work with the
external particle on shell,

839
00:52:39,690 --> 00:52:43,500
then combining together the
Feynman rule for the vertex

840
00:52:43,500 --> 00:52:46,050
and the rule for
the propagator, I'm

841
00:52:46,050 --> 00:52:51,060
getting these what are called
sometimes eikonal vertices,

842
00:52:51,060 --> 00:52:52,470
or propagator vertices.

843
00:53:14,410 --> 00:53:15,940
So these are eikonal.

844
00:53:15,940 --> 00:53:19,780
Eikonal in both the
interactions and in the vertex.

845
00:53:19,780 --> 00:53:23,088
And that's what should happen
for having a very soft particle

846
00:53:23,088 --> 00:53:24,880
talking to something
very energetic, that's

847
00:53:24,880 --> 00:53:27,650
called the eikonal approximation
and it just falls out

848
00:53:27,650 --> 00:53:28,900
of our effective field theory.

849
00:53:32,190 --> 00:53:34,040
So this can actually
lead us to something

850
00:53:34,040 --> 00:53:38,270
deeper, which is called ultra
soft collinear factorization.

851
00:53:51,660 --> 00:53:53,400
So let's consider
more than one gluon.

852
00:54:00,020 --> 00:54:02,220
He'll have one collinear
fermion and we'll just

853
00:54:02,220 --> 00:54:04,910
touch a bunch of ultra
soft gluons to it.

854
00:54:19,420 --> 00:54:22,530
Kind of obvious notation.

855
00:54:22,530 --> 00:54:25,053
Let's call it m.

856
00:54:28,760 --> 00:54:31,250
So we could sum
up those diagrams,

857
00:54:31,250 --> 00:54:32,930
and if we sum up
those diagrams we

858
00:54:32,930 --> 00:54:37,010
get something that
looks familiar.

859
00:54:37,010 --> 00:54:44,450
So sum over m, sum
over permutation.

860
00:54:44,450 --> 00:54:47,630
Factors of g.

861
00:54:47,630 --> 00:54:48,295
Gauge fields.

862
00:54:54,810 --> 00:54:56,472
OK.

863
00:54:56,472 --> 00:54:57,555
Be careful about ordering.

864
00:55:10,310 --> 00:55:12,020
And I'm working here on shell.

865
00:55:16,800 --> 00:55:18,660
So that external
collinear particle

866
00:55:18,660 --> 00:55:19,860
has p squared equals 0.

867
00:55:23,497 --> 00:55:26,080
So if you think about what this
is, it's just our Wilson line.

868
00:55:26,080 --> 00:55:28,413
It's not the same Wilson line
that we were talking about

869
00:55:28,413 --> 00:55:29,440
before.

870
00:55:29,440 --> 00:55:32,480
This Wilson line is built
from ultra soft fields, not

871
00:55:32,480 --> 00:55:34,570
collinear fields.

872
00:55:34,570 --> 00:55:36,580
And it doesn't even point
in the same direction,

873
00:55:36,580 --> 00:55:40,230
it points in the n direction,
not the n bar direction.

874
00:55:40,230 --> 00:55:41,230
But it is a Wilson line.

875
00:55:44,138 --> 00:55:45,346
AUDIENCE: Are these incoming?

876
00:55:49,390 --> 00:55:50,835
PROFESSOR: Yeah, I think so.

877
00:55:50,835 --> 00:55:54,420
AUDIENCE: I think you put
a minus sign in the top.

878
00:55:58,270 --> 00:56:02,770
PROFESSOR: So for this guy,
there's no minus sign there

879
00:56:02,770 --> 00:56:07,210
because, well, I didn't
tell you what the--

880
00:56:07,210 --> 00:56:10,780
there could be a minus g
or a plus g here, right?

881
00:56:10,780 --> 00:56:11,650
Yeah.

882
00:56:11,650 --> 00:56:12,160
Yeah.

883
00:56:12,160 --> 00:56:14,560
That's minus g, probably.

884
00:56:14,560 --> 00:56:15,060
Yeah.

885
00:56:28,610 --> 00:56:30,092
OK.

886
00:56:30,092 --> 00:56:31,550
So actually, what
this motivates us

887
00:56:31,550 --> 00:56:33,425
to do from the effective
theory point of view

888
00:56:33,425 --> 00:56:36,120
is to consider the following.

889
00:56:36,120 --> 00:56:51,440
It motivates us to think about
making a field redefinition

890
00:56:51,440 --> 00:56:54,680
because what we just did is we
iterated the leading order L 0

891
00:56:54,680 --> 00:56:58,520
Lagrangian over and over again
to get these vertices as well

892
00:56:58,520 --> 00:56:59,900
as the propagators.

893
00:56:59,900 --> 00:57:02,990
And we ended up with something
that was just a Wilson line.

894
00:57:02,990 --> 00:57:06,780
Could we capture that
somehow, in a simpler way?

895
00:57:06,780 --> 00:57:09,410
And the answer is yes, if
we make the following field

896
00:57:09,410 --> 00:57:10,625
redefinition.

897
00:57:21,530 --> 00:57:24,320
We take our original
CNP field and we

898
00:57:24,320 --> 00:57:27,020
pull out a Wilson
line, Y. And we

899
00:57:27,020 --> 00:57:29,230
can do a similar thing
for the gauge field.

900
00:57:44,220 --> 00:57:49,920
This is just the adjoint version
of the same field redefinition,

901
00:57:49,920 --> 00:57:54,180
where Y is a Wilson line, which
corresponds to that in momentum

902
00:57:54,180 --> 00:57:57,330
space and in position space.

903
00:57:57,330 --> 00:57:58,560
Pathway to exponential.

904
00:58:13,180 --> 00:58:14,760
So when I say that
the Wilson line's

905
00:58:14,760 --> 00:58:18,930
at x, that means I've
shifted the whole line by x.

906
00:58:18,930 --> 00:58:23,290
And I could just denote
that by putting an x here.

907
00:58:23,290 --> 00:58:26,040
And then from that
point in space time,

908
00:58:26,040 --> 00:58:29,450
there's a line going
out in the n direction.

909
00:58:29,450 --> 00:58:31,900
That's what this
formula is saying.

910
00:58:31,900 --> 00:58:34,180
So at S equals zero,
you just sit at x.

911
00:58:39,020 --> 00:58:41,520
This should be a picture.

912
00:58:41,520 --> 00:58:44,040
Oh, this is minus infinity so
it's not a very good picture.

913
00:58:50,560 --> 00:58:54,670
And so this Wilson line,
like any Wilson line,

914
00:58:54,670 --> 00:58:57,730
has an equation of
motion, which is this.

915
00:58:57,730 --> 00:59:02,210
It satisfies Y dagger Y is one.

916
00:59:02,210 --> 00:59:04,510
And if you go through
with our transformation

917
00:59:04,510 --> 00:59:07,540
for the collinear gluon, which
I could have also motivated

918
00:59:07,540 --> 00:59:08,470
in this way.

919
00:59:08,470 --> 00:59:10,180
If I'd gone through
the calculation where

920
00:59:10,180 --> 00:59:11,847
instead of having a
collinear quark here

921
00:59:11,847 --> 00:59:13,780
I had a collinear
gluon, I would have

922
00:59:13,780 --> 00:59:16,270
found exactly the same thing,
except all these A's would be

923
00:59:16,270 --> 00:59:18,131
in the adjoint representation.

924
00:59:24,850 --> 00:59:27,650
I didn't write the
t's anyway, so.

925
00:59:27,650 --> 00:59:28,150
All right.

926
00:59:28,150 --> 00:59:29,920
So either the fundamental
representation

927
00:59:29,920 --> 00:59:32,917
or the adjoint
representation, and that

928
00:59:32,917 --> 00:59:35,500
would motivate you to make the
same type of field redefinition

929
00:59:35,500 --> 00:59:38,090
to try to capture
what's going on there.

930
00:59:38,090 --> 00:59:40,220
OK.

931
00:59:40,220 --> 00:59:42,170
So let's see what that
does to our Lagrangian.

932
00:59:50,100 --> 00:59:51,994
So our original Lagrangian--

933
00:59:55,870 --> 00:59:59,080
I'm not going to write
out all the terms.

934
00:59:59,080 --> 01:00:01,390
Just enough terms
to get you the idea.

935
01:00:01,390 --> 01:00:05,110
If I make this
field redefinition,

936
01:00:05,110 --> 01:00:07,920
it goes to this guy with a
superscript 0 that I'm writing.

937
01:00:10,480 --> 01:00:23,630
And then-- let me
write out here.

938
01:00:23,630 --> 01:00:25,700
Let me split this
derivative into two pieces.

939
01:00:35,300 --> 01:00:38,350
So I'm still not writing in
the terms there in the dot.

940
01:00:41,690 --> 01:00:44,090
So what I did is I took the
Y from this guy and the Y

941
01:00:44,090 --> 01:00:46,730
from that guy and I put them
inside the square bracket.

942
01:00:46,730 --> 01:00:48,950
This n dot D, remember,
has two pieces.

943
01:00:48,950 --> 01:00:54,900
It's i n dot D is sort of an i
n dot D ultra soft, which just

944
01:00:54,900 --> 01:00:56,900
has the ultra soft gauge
field, but then there's

945
01:00:56,900 --> 01:00:59,413
a piece that involves the
collinear gauge field.

946
01:00:59,413 --> 01:01:00,830
But then the
collinear gauge field

947
01:01:00,830 --> 01:01:03,170
also transforms, as
I wrote up there.

948
01:01:03,170 --> 01:01:05,960
So I get this Y, Y
dagger for that guy.

949
01:01:05,960 --> 01:01:07,670
So for the collinear
gauge field,

950
01:01:07,670 --> 01:01:09,810
these Y's are cancelling.

951
01:01:09,810 --> 01:01:12,270
And this guy here, if
you push the derivative

952
01:01:12,270 --> 01:01:15,710
through the Y using this
formula, then the Y's are also

953
01:01:15,710 --> 01:01:19,310
canceling and this is just
becoming i n dot partial.

954
01:01:22,150 --> 01:01:23,920
So that's becoming
i n dot partial,

955
01:01:23,920 --> 01:01:27,385
and this is becoming
n dot a n of 0.

956
01:01:30,010 --> 01:01:32,230
So what's happening is
that the ultra soft gauge

957
01:01:32,230 --> 01:01:35,140
field is dropping out.

958
01:01:35,140 --> 01:01:37,300
Now there's terms in
the plus dot dot dot,

959
01:01:37,300 --> 01:01:39,790
those terms involve the D perp
slash and the 1 over n bar

960
01:01:39,790 --> 01:01:42,100
dot D. And if I had
written out those terms,

961
01:01:42,100 --> 01:01:44,283
too, then the same thing
would happen, actually.

962
01:01:44,283 --> 01:01:46,450
All the Y's would cancel
out in those terms as well.

963
01:01:57,620 --> 01:02:06,010
So what I'm getting
here is something

964
01:02:06,010 --> 01:02:09,640
that I might call an n dot Dn.

965
01:02:09,640 --> 01:02:13,810
It's a covariant derivative that
only involves the A n0 field.

966
01:02:16,910 --> 01:02:18,580
And there's actually
no soft fields

967
01:02:18,580 --> 01:02:20,870
at all left in the Lagrangian
after I make this field

968
01:02:20,870 --> 01:02:22,044
redefinition.

969
01:02:33,410 --> 01:02:38,330
So that's obviously giving us
a very much simpler Lagrangian.

970
01:02:38,330 --> 01:02:42,068
OK, so it looks like
a good thing to do.

971
01:02:42,068 --> 01:02:43,860
And that, if you go
through it, is actually

972
01:02:43,860 --> 01:02:47,430
true also for the gluon action.

973
01:02:47,430 --> 01:02:51,732
Gluon action was also built
of n dot D type fields, type

974
01:02:51,732 --> 01:02:53,190
covariant derivatives,
that's where

975
01:02:53,190 --> 01:02:58,200
the ultra soft gluon showed up,
and basically the same thing

976
01:02:58,200 --> 01:02:59,880
happens.

977
01:02:59,880 --> 01:03:05,378
This same relation
right there means

978
01:03:05,378 --> 01:03:07,920
that the ultra soft gluon would
decouple from that Lagrangian

979
01:03:07,920 --> 01:03:09,640
as well.

980
01:03:09,640 --> 01:03:11,100
So making this
field redefinition,

981
01:03:11,100 --> 01:03:14,790
we actually decoupled in the
Lagrangians the ultra soft

982
01:03:14,790 --> 01:03:16,545
and the collinear fields.

983
01:03:22,900 --> 01:03:25,740
So even though the effective
theory allowed these modes

984
01:03:25,740 --> 01:03:29,380
to couple to each other, they
coupled in a very simple way.

985
01:03:29,380 --> 01:03:32,520
And so what we're saying is that
it's convenient because they

986
01:03:32,520 --> 01:03:33,775
couple in such a simple way.

987
01:03:33,775 --> 01:03:36,150
If they didn't couple in such
a simple way, we would stop

988
01:03:36,150 --> 01:03:38,700
and we'd just use that
Lagrangian, we'd do physics.

989
01:03:38,700 --> 01:03:40,830
Since they do a couple
in such a simple way,

990
01:03:40,830 --> 01:03:44,610
we can do physics in terms of
some reparameterized variables,

991
01:03:44,610 --> 01:03:48,390
or redefined variables, which
are these new variables.

992
01:03:48,390 --> 01:03:52,080
And that's convenient
because now,

993
01:03:52,080 --> 01:03:55,890
if I think about our
original Lagrangian,

994
01:03:55,890 --> 01:03:58,740
it goes over to
something that looks

995
01:03:58,740 --> 01:04:07,150
simpler because it's going
over to something that has no--

996
01:04:07,150 --> 01:04:10,110
the L0 has no ultra soft fields.

997
01:04:10,110 --> 01:04:13,500
This wouldn't be, of course,
true for the L1 or higher order

998
01:04:13,500 --> 01:04:15,600
Lagrangians, it
actually would still

999
01:04:15,600 --> 01:04:18,065
simplify all those
Lagrangians a certain way.

1000
01:04:18,065 --> 01:04:19,440
But for the leading
order one, it

1001
01:04:19,440 --> 01:04:21,622
seems to make a dramatic
simplification because we

1002
01:04:21,622 --> 01:04:22,830
no longer have this coupling.

1003
01:04:26,075 --> 01:04:28,200
So you might think, well,
the interactions have all

1004
01:04:28,200 --> 01:04:31,290
disappeared.

1005
01:04:31,290 --> 01:04:34,440
But they haven't disappeared,
because when you make the field

1006
01:04:34,440 --> 01:04:37,470
redefinition, you can't just
make it to the Lagrangian.

1007
01:04:37,470 --> 01:04:40,410
You also have to make it on
operators in their theory.

1008
01:04:44,910 --> 01:04:49,200
And what this does is it
moves some interactions out

1009
01:04:49,200 --> 01:04:50,745
of the Lagrangian
and into currents.

1010
01:05:10,290 --> 01:05:11,820
So let's do some
examples about it.

1011
01:05:13,993 --> 01:05:16,160
Since it's for the Lagrangian,
it's always the same.

1012
01:05:16,160 --> 01:05:17,720
That's one of the reasons
why this is powerful,

1013
01:05:17,720 --> 01:05:19,400
because if we do it
once and for all,

1014
01:05:19,400 --> 01:05:22,220
we just did it for
the Lagrangian.

1015
01:05:22,220 --> 01:05:24,470
And then we can see
what happens for a bunch

1016
01:05:24,470 --> 01:05:27,690
of different currents, so
I'll do three examples.

1017
01:05:27,690 --> 01:05:31,670
So one example that we had
was this current for B to S

1018
01:05:31,670 --> 01:05:35,030
gamma, which had a heavy quark
field and a one light collinear

1019
01:05:35,030 --> 01:05:36,590
up quark, and a Wilson line.

1020
01:05:40,640 --> 01:05:42,830
I should have said
another thing that I'm

1021
01:05:42,830 --> 01:05:45,230
using, which is important.

1022
01:05:45,230 --> 01:05:51,265
It's that if you
go through what--

1023
01:05:51,265 --> 01:05:53,780
let me make sure I got
my Y, or my Y dagger's

1024
01:05:53,780 --> 01:05:56,540
on the right side.

1025
01:05:56,540 --> 01:06:00,740
If you go through what happens
for the collinear Wilson line

1026
01:06:00,740 --> 01:06:02,795
after the field redefinition,
this guy remember,

1027
01:06:02,795 --> 01:06:05,810
is written in terms
of these fields.

1028
01:06:05,810 --> 01:06:08,420
And this guy here is written
in terms of those fields.

1029
01:06:08,420 --> 01:06:11,397
And it also gets
Y's on the outside.

1030
01:06:11,397 --> 01:06:13,730
That's important when you
start considering, for example

1031
01:06:13,730 --> 01:06:16,880
these dots or--

1032
01:06:16,880 --> 01:06:17,570
OK.

1033
01:06:17,570 --> 01:06:18,690
But I need that here.

1034
01:06:18,690 --> 01:06:23,360
So given that that's also true,
if I want to look at this guy,

1035
01:06:23,360 --> 01:06:25,338
I just make the
field redefinition.

1036
01:06:28,206 --> 01:06:30,570
So I got this.

1037
01:06:30,570 --> 01:06:35,370
I got a Y, Y dagger on both
sides of the Wilson line.

1038
01:06:42,100 --> 01:06:45,130
These guys cancel each other.

1039
01:06:45,130 --> 01:06:46,560
This is a Wilson line 0.

1040
01:06:50,408 --> 01:06:54,334
I have C bar 0,
Wilson line zero.

1041
01:06:54,334 --> 01:06:56,460
And I have gamma.

1042
01:06:56,460 --> 01:06:59,680
And I have Y dagger, hv.

1043
01:06:59,680 --> 01:07:01,198
So now I have a
Wilson line sitting

1044
01:07:01,198 --> 01:07:03,490
next to the heavy quark and
another Wilson line sitting

1045
01:07:03,490 --> 01:07:04,657
next to the collinear quark.

1046
01:07:08,360 --> 01:07:10,730
So all the, if you
think in the diagram

1047
01:07:10,730 --> 01:07:12,830
that we drew, the
interactions that

1048
01:07:12,830 --> 01:07:15,470
were those gluons attaching
to the collinear quark,

1049
01:07:15,470 --> 01:07:18,950
they're now just all represented
by this Wilson line here.

1050
01:07:18,950 --> 01:07:21,560
But it's even more than
that, because we also

1051
01:07:21,560 --> 01:07:23,820
transformed the Wilson line.

1052
01:07:23,820 --> 01:07:27,320
So even if we had attached ultra
gluons to that Wilson line,

1053
01:07:27,320 --> 01:07:29,840
they all magically
simplify into a simple,

1054
01:07:29,840 --> 01:07:32,450
one simple Wilson line Y dagger.

1055
01:07:32,450 --> 01:07:34,010
So from a diagrammatic
point of view,

1056
01:07:34,010 --> 01:07:36,427
if we had actually tried to
carry out the calculation that

1057
01:07:36,427 --> 01:07:38,900
would give this formula, it
would be kind of horrendous

1058
01:07:38,900 --> 01:07:44,180
because there's an enormous
amount of calculations going on

1059
01:07:44,180 --> 01:07:45,680
to give this.

1060
01:07:45,680 --> 01:07:47,450
It just looks complicated
in the diagrams,

1061
01:07:47,450 --> 01:07:49,790
but here it looks very simple.

1062
01:07:49,790 --> 01:07:52,610
And the fact that the leading
order Lagrangian gave us 0

1063
01:07:52,610 --> 01:07:56,615
there tells us that it really
is the sum of the diagrams.

1064
01:07:59,323 --> 01:08:00,740
Let's have a couple
more examples.

1065
01:08:03,960 --> 01:08:07,710
So you could take our
example for two jets,

1066
01:08:07,710 --> 01:08:11,430
and you could ask what
would happen in that case.

1067
01:08:11,430 --> 01:08:14,000
What's the generalization if
I have more than one collinear

1068
01:08:14,000 --> 01:08:16,010
direction?

1069
01:08:16,010 --> 01:08:18,830
And the generalization is that
you just make the same field

1070
01:08:18,830 --> 01:08:21,290
redefinition, but it's
a different component

1071
01:08:21,290 --> 01:08:24,859
of the ultra soft field that
couples to different collinear

1072
01:08:24,859 --> 01:08:25,470
fields.

1073
01:08:25,470 --> 01:08:27,402
So you're making a different--

1074
01:08:27,402 --> 01:08:28,819
you're making a
field redefinition

1075
01:08:28,819 --> 01:08:32,025
that's appropriate to each
of the different collinear

1076
01:08:32,025 --> 01:08:32,525
sectors.

1077
01:08:41,250 --> 01:08:44,830
But other than that it goes
through in the same way.

1078
01:08:44,830 --> 01:08:48,729
And so here, this Y
n1 dagger involves

1079
01:08:48,729 --> 01:08:51,720
n1 dot A ultra soft
fields, and this guy here

1080
01:08:51,720 --> 01:08:55,109
involves n2 dot A
ultra soft fields.

1081
01:08:55,109 --> 01:08:56,910
And they don't
cancel, but they do

1082
01:08:56,910 --> 01:08:59,580
simplify all the interactions,
in this case simplify

1083
01:08:59,580 --> 01:09:04,200
to the simple Y
dagger Y combination.

1084
01:09:04,200 --> 01:09:08,939
OK, so having more than one
clear direction is not--

1085
01:09:08,939 --> 01:09:12,630
I mean, it means that you
have more than one type

1086
01:09:12,630 --> 01:09:13,647
of Y showing up.

1087
01:09:13,647 --> 01:09:15,689
But again, it's just the
leading order Lagrangian

1088
01:09:15,689 --> 01:09:17,147
for each of those
sectors that tell

1089
01:09:17,147 --> 01:09:18,810
you what's going to show up.

1090
01:09:27,260 --> 01:09:30,679
Type of Y. Let's do
one more example.

1091
01:09:33,279 --> 01:09:39,979
Let's do an example where we
have collinear fields that

1092
01:09:39,979 --> 01:09:42,770
are in the following form.

1093
01:09:42,770 --> 01:09:46,729
Where both directions are n.

1094
01:09:46,729 --> 01:09:48,347
And we have an
operator like that.

1095
01:09:48,347 --> 01:09:50,430
So it's the same operator
I was writing down here,

1096
01:09:50,430 --> 01:09:53,300
but in this case, I
had n1 n2 for two jets.

1097
01:09:56,210 --> 01:09:58,190
I haven't really told
you about an example that

1098
01:09:58,190 --> 01:09:59,972
would involve this
operator here,

1099
01:09:59,972 --> 01:10:01,430
but it turns out
this operator here

1100
01:10:01,430 --> 01:10:03,097
is something that
shows up, for example,

1101
01:10:03,097 --> 01:10:06,660
in a part time distribution
function and other places.

1102
01:10:06,660 --> 01:10:10,250
So this operator also does
make an appearance in physics.

1103
01:10:10,250 --> 01:10:12,380
And if you look at what
happens for this operator,

1104
01:10:12,380 --> 01:10:13,325
all the Y's cancel.

1105
01:10:18,600 --> 01:10:21,240
So when you go through
the transformation,

1106
01:10:21,240 --> 01:10:24,560
you have Y, Y dagger, but
then the Y's exactly cancel.

1107
01:10:29,850 --> 01:10:31,560
So in that case,
it's really true

1108
01:10:31,560 --> 01:10:34,998
that the ultra soft gluons
are dropping out effectively

1109
01:10:34,998 --> 01:10:36,540
when you add up
diagrams that there's

1110
01:10:36,540 --> 01:10:38,520
no ultra soft gluons left.

1111
01:10:38,520 --> 01:10:42,688
In these cases, the leftover
is these Wilson lines that

1112
01:10:42,688 --> 01:10:43,980
are showing up in the operator.

1113
01:10:43,980 --> 01:10:47,686
In this case,
there's no left over.

1114
01:10:47,686 --> 01:10:50,070
AUDIENCE: Are those both
values in the same position?

1115
01:10:50,070 --> 01:10:50,862
PROFESSOR: Yeah.

1116
01:10:50,862 --> 01:10:53,640
AUDIENCE: Y to the w?

1117
01:10:53,640 --> 01:10:55,590
PROFESSOR: Because, yeah.

1118
01:10:55,590 --> 01:10:57,720
So they would if
I just wrote this.

1119
01:10:57,720 --> 01:11:01,050
In general, what you could have
inserted in here is some kind

1120
01:11:01,050 --> 01:11:04,560
of something that
picks out, like,

1121
01:11:04,560 --> 01:11:06,760
the momentum of
one of those w's.

1122
01:11:06,760 --> 01:11:09,070
So let me just throw that
delta function in there

1123
01:11:09,070 --> 01:11:10,630
so they don't cancel.

1124
01:11:10,630 --> 01:11:13,440
We'll talk about that
in a minute or two.

1125
01:11:13,440 --> 01:11:13,950
Yeah.

1126
01:11:13,950 --> 01:11:15,930
But in general, I
could cancel them

1127
01:11:15,930 --> 01:11:17,970
in this particular formula.

1128
01:11:17,970 --> 01:11:20,418
But there's reasons
why, actually, we

1129
01:11:20,418 --> 01:11:21,960
won't want to cancel
them, because we

1130
01:11:21,960 --> 01:11:24,000
will be putting other
things in between that

1131
01:11:24,000 --> 01:11:26,105
won't change what I just said.

1132
01:11:26,105 --> 01:11:27,480
So imagine that
you had something

1133
01:11:27,480 --> 01:11:29,520
that measured the
momentum of one

1134
01:11:29,520 --> 01:11:30,960
of these products of fields.

1135
01:11:30,960 --> 01:11:33,450
It would still be true that
these Y's, which are not

1136
01:11:33,450 --> 01:11:37,320
having labels, would cancel
in the way that we said,

1137
01:11:37,320 --> 01:11:40,710
but the w's wouldn't cancel.

1138
01:11:40,710 --> 01:11:43,560
OK, so this is called the
BPS field redefinition,

1139
01:11:43,560 --> 01:11:47,640
and S is me.

1140
01:11:47,640 --> 01:11:53,625
And this thing sums up an
infinite class of diagrams.

1141
01:12:02,510 --> 01:12:05,720
What it does in
example one, if you

1142
01:12:05,720 --> 01:12:08,840
want to think about what the
diagrams would look like,

1143
01:12:08,840 --> 01:12:12,120
let me draw kind of
an example for you.

1144
01:12:12,120 --> 01:12:14,180
So let's have some
collinear particles.

1145
01:12:18,130 --> 01:12:19,810
And just to make
it look nontrivial,

1146
01:12:19,810 --> 01:12:23,750
let me draw something it
seems kind of nontrivial.

1147
01:12:23,750 --> 01:12:25,840
So there's a bunch of
collinear particles,

1148
01:12:25,840 --> 01:12:28,360
and then we could add
ultra soft gluons to them.

1149
01:12:28,360 --> 01:12:29,860
And the ultra soft
gluons can couple

1150
01:12:29,860 --> 01:12:31,735
to all those collinear
particles, the gluons,

1151
01:12:31,735 --> 01:12:33,287
the quarks, everybody.

1152
01:12:36,030 --> 01:12:37,920
If we consider all
the ultra soft gluons

1153
01:12:37,920 --> 01:12:44,040
coupling to the collinear
particles everywhere,

1154
01:12:44,040 --> 01:12:46,560
and we add up all
those attachments,

1155
01:12:46,560 --> 01:12:49,530
then it just becomes
a single Wilson line.

1156
01:12:49,530 --> 01:12:55,950
So this thing becomes
a single Wilson line.

1157
01:12:55,950 --> 01:13:00,060
And then times exactly the
same collinear structure.

1158
01:13:03,780 --> 01:13:06,450
So what I'm saying doesn't
rely, since we did it

1159
01:13:06,450 --> 01:13:09,233
at the level of Lagrangian, it
doesn't rely on whether or not

1160
01:13:09,233 --> 01:13:11,400
they're loops, or tree
level, or anything like that.

1161
01:13:14,690 --> 01:13:17,650
So this diagram there, if we
add up all possible attachments,

1162
01:13:17,650 --> 01:13:18,970
will be equal to that one.

1163
01:13:18,970 --> 01:13:23,880
Where this is a Y. In this
case, it's a Y dagger.

1164
01:13:27,880 --> 01:13:30,910
OK, so that's the simplicity
encoded in this formula right

1165
01:13:30,910 --> 01:13:32,620
here.

1166
01:13:32,620 --> 01:13:33,775
This is an example of J1.

1167
01:13:38,450 --> 01:13:46,100
In example three, then,
it's a little simpler, even.

1168
01:13:49,940 --> 01:14:00,420
Because the ultra soft gluons
are decoupling at lowest order

1169
01:14:00,420 --> 01:14:02,080
from any graph that
you might consider.

1170
01:14:02,080 --> 01:14:03,580
So you go through
the same exercise,

1171
01:14:03,580 --> 01:14:08,130
but now that the Y's
are all canceling out.

1172
01:14:08,130 --> 01:14:12,450
And this has a name that
sometimes people use,

1173
01:14:12,450 --> 01:14:14,040
called color transparency.

1174
01:14:19,920 --> 01:14:22,970
So one place that it
shows up is the following.

1175
01:14:22,970 --> 01:14:25,760
Let's take our
current J3 and imagine

1176
01:14:25,760 --> 01:14:28,490
that we produce from that
current an energetic pion.

1177
01:14:28,490 --> 01:14:30,560
That was one of the examples
we talked about when

1178
01:14:30,560 --> 01:14:33,170
we were talking about BDD pi.

1179
01:14:33,170 --> 01:14:37,460
So there's a collinear pion,
and these are collinear quarks.

1180
01:14:37,460 --> 01:14:39,840
collinear quark and anti quark.

1181
01:14:39,840 --> 01:14:44,120
Supposed to be inside the pion,
and there's collinear gluons.

1182
01:14:44,120 --> 01:14:46,820
If we attach all the ultra
soft gluons to this object,

1183
01:14:46,820 --> 01:14:48,485
then the Wilson lines cancel.

1184
01:14:55,810 --> 01:15:04,340
The ultra soft
gluons are decoupling

1185
01:15:04,340 --> 01:15:05,750
from energetic particles.

1186
01:15:05,750 --> 01:15:08,630
And the reason that
they're decoupling

1187
01:15:08,630 --> 01:15:11,420
is because the partons
here are in a color singlet

1188
01:15:11,420 --> 01:15:14,360
state, which is this pion.

1189
01:15:24,220 --> 01:15:26,620
OK, so here we're producing
a color neutral pion

1190
01:15:26,620 --> 01:15:29,920
out of color contracted
collinear quark fields,

1191
01:15:29,920 --> 01:15:31,780
and there's these
Wilson lines for reasons

1192
01:15:31,780 --> 01:15:33,610
we'll discuss in a minute.

1193
01:15:33,610 --> 01:15:36,620
How they kind of play
a role in physics here.

1194
01:15:36,620 --> 01:15:38,920
But the reason that
Wilson lines are canceling

1195
01:15:38,920 --> 01:15:41,590
is because the collinear
things were already contracted.

1196
01:15:41,590 --> 01:15:43,390
All the collinear
things in n direction

1197
01:15:43,390 --> 01:15:49,300
were contracted in a color
singlet, global color singlet.

1198
01:15:49,300 --> 01:15:52,840
So the words that go along with
this phrase color transparency

1199
01:15:52,840 --> 01:15:55,420
is that you have these
very soft gluons,

1200
01:15:55,420 --> 01:15:58,252
and they're trying to come
in and see this thing.

1201
01:15:58,252 --> 01:15:59,710
But they can't
really, all they can

1202
01:15:59,710 --> 01:16:03,340
see is sort of the overall
color charge of the whole thing.

1203
01:16:03,340 --> 01:16:06,667
They couple, of the
multi-pole expansion,

1204
01:16:06,667 --> 01:16:08,500
they're only coupling
to a single component,

1205
01:16:08,500 --> 01:16:10,420
and you can think of
that as if they're only

1206
01:16:10,420 --> 01:16:12,700
seeing an overall color charge.

1207
01:16:12,700 --> 01:16:14,620
And therefore, since
it's overall color

1208
01:16:14,620 --> 01:16:17,530
charge is neutral, you don't
see, they just cancel out.

1209
01:16:21,380 --> 01:16:22,520
OK.

1210
01:16:22,520 --> 01:16:24,440
In my notes, I also
have a page talking

1211
01:16:24,440 --> 01:16:27,590
about how you could think
of gauge transformations

1212
01:16:27,590 --> 01:16:29,740
after making this
field redefinition,

1213
01:16:29,740 --> 01:16:31,490
but it's kind of an
aside so I'm not going

1214
01:16:31,490 --> 01:16:33,881
to talk about it in lecture.

1215
01:16:33,881 --> 01:16:35,050
But I will post it.

1216
01:16:38,500 --> 01:16:40,090
OK, so what about these?

1217
01:16:40,090 --> 01:16:42,700
What about this
additional kind of thing

1218
01:16:42,700 --> 01:16:45,300
that I was alluding to here?

1219
01:16:45,300 --> 01:16:48,995
How does that come
in to our story?

1220
01:16:52,430 --> 01:16:54,410
So, so far in our
story, we haven't really

1221
01:16:54,410 --> 01:16:56,480
talked about Wilson
coefficients except

1222
01:16:56,480 --> 01:16:58,283
to say that they
could be constrained

1223
01:16:58,283 --> 01:17:00,200
by reparameterization
invariance to be absent.

1224
01:17:07,070 --> 01:17:09,020
So let's think about
Wilson coefficients now.

1225
01:17:27,290 --> 01:17:30,030
Obviously that's
something important.

1226
01:17:30,030 --> 01:17:32,600
And the way that Wilson
coefficients can come in

1227
01:17:32,600 --> 01:17:35,060
is the following way.

1228
01:17:35,060 --> 01:17:39,845
They can depend on the large
momenta that we're at order 1.

1229
01:17:39,845 --> 01:17:41,970
And one way of denoting
that is by saying that they

1230
01:17:41,970 --> 01:17:43,910
depend on label operators.

1231
01:17:52,670 --> 01:17:55,500
OK, so nothing stops that.

1232
01:17:55,500 --> 01:17:58,335
But if we want the momentum
that's picked up by this label

1233
01:17:58,335 --> 01:17:59,960
operator to be gauge
invariant, then we

1234
01:17:59,960 --> 01:18:01,970
should act on products
of fields that

1235
01:18:01,970 --> 01:18:03,710
are collinear gauge invariant.

1236
01:18:03,710 --> 01:18:07,370
So the way that we should set
it up is to have an operator.

1237
01:18:11,000 --> 01:18:14,090
Here's a kind of notation
that's sometimes used.

1238
01:18:14,090 --> 01:18:25,070
Where the operator acts on both
fields, the C bar and the W.

1239
01:18:25,070 --> 01:18:27,620
Because of our formulas
for the label operator,

1240
01:18:27,620 --> 01:18:31,070
we could also write this
as a label operator acting

1241
01:18:31,070 --> 01:18:34,640
to the left if we wanted.

1242
01:18:34,640 --> 01:18:37,550
So, what this Wilson
coefficient is the function.

1243
01:18:37,550 --> 01:18:39,672
It's not just simply a number.

1244
01:18:39,672 --> 01:18:42,005
And it picks out the momentum
of this product of fields.

1245
01:18:54,490 --> 01:18:59,698
And that's because that product
is collinear gauge invariant.

1246
01:18:59,698 --> 01:19:01,490
So it's a well-defined
thing to talk about.

1247
01:19:06,280 --> 01:19:08,720
OK, so that's actually
the general structure,

1248
01:19:08,720 --> 01:19:11,090
that whenever we have these
products of fields that

1249
01:19:11,090 --> 01:19:14,060
are collinear gauge invariant,
if we ask what the Wilson

1250
01:19:14,060 --> 01:19:16,100
coefficient could
be a function of,

1251
01:19:16,100 --> 01:19:18,665
it can be a function of the
momentum of those products.

1252
01:19:22,260 --> 01:19:29,565
So one way of writing this in a
sort of more elegant fashion as

1253
01:19:29,565 --> 01:19:30,065
follows.

1254
01:19:34,440 --> 01:19:40,670
So take this guy, and write
it as a delta function

1255
01:19:40,670 --> 01:19:41,740
in the following way.

1256
01:19:53,280 --> 01:19:55,440
So if I do the integral
over this omega,

1257
01:19:55,440 --> 01:19:57,930
then I would just get back
that I stick the p bar

1258
01:19:57,930 --> 01:19:59,760
dagger inside the
Wilson coefficient,

1259
01:19:59,760 --> 01:20:02,310
and that it acts on
this product of fields.

1260
01:20:02,310 --> 01:20:05,430
But if I write it
this way, what I get

1261
01:20:05,430 --> 01:20:07,950
is that my Wilson
coefficients are just

1262
01:20:07,950 --> 01:20:11,380
functions of a number, not
functions of an operator.

1263
01:20:11,380 --> 01:20:14,430
And my operators have these
delta functions in them.

1264
01:20:14,430 --> 01:20:16,530
But then could depend
on some variables

1265
01:20:16,530 --> 01:20:19,600
that are distinguishing
those delta functions.

1266
01:20:19,600 --> 01:20:21,630
OK.

1267
01:20:21,630 --> 01:20:25,230
So in general, products
of fields like this,

1268
01:20:25,230 --> 01:20:28,710
we have to think about their
momenta as being something

1269
01:20:28,710 --> 01:20:29,915
that we could label.

1270
01:20:29,915 --> 01:20:32,460
If you like, we could
label it by omega.

1271
01:20:32,460 --> 01:20:34,710
Because it's a linear
gauge invariant concept,

1272
01:20:34,710 --> 01:20:38,160
the Wilson coefficients can
depend on those momenta,

1273
01:20:38,160 --> 01:20:40,320
and then we have
Wilson coefficients

1274
01:20:40,320 --> 01:20:43,110
that depend on those momenta
and operators that are just

1275
01:20:43,110 --> 01:20:45,810
labeled by those momenta.

1276
01:20:45,810 --> 01:20:47,310
And this is the
convolution formula

1277
01:20:47,310 --> 01:20:50,130
that I sort of promised you at
the beginning of the discussion

1278
01:20:50,130 --> 01:20:54,480
of SCET that was going to show
up, and now it's shown up.

1279
01:20:54,480 --> 01:20:57,780
OK, so Wilson coefficients can
depend on those large order 1

1280
01:20:57,780 --> 01:20:59,400
momenta.

1281
01:20:59,400 --> 01:21:01,525
And traditionally they're
written as integrals,

1282
01:21:01,525 --> 01:21:03,900
even though you could think
of them as sums at this point

1283
01:21:03,900 --> 01:21:06,730
and it wouldn't make
much difference.

1284
01:21:06,730 --> 01:21:09,360
So this here is what's called
hard collinear factorization.

1285
01:21:12,290 --> 01:21:16,460
In the traditional
QCD literature.

1286
01:21:16,460 --> 01:21:18,980
Because it's telling you how
hard degrees of freedom, which

1287
01:21:18,980 --> 01:21:21,380
are encoded in our
Wilson coefficients,

1288
01:21:21,380 --> 01:21:23,130
can talk to collinear
degrees of freedom,

1289
01:21:23,130 --> 01:21:25,220
which are encoded
in our operators.

1290
01:21:25,220 --> 01:21:27,770
In SCET, that's just
come out of the formalism

1291
01:21:27,770 --> 01:21:29,520
in a very sort of simple way.

1292
01:21:29,520 --> 01:21:31,670
In QCD, you'd have to
use word identities

1293
01:21:31,670 --> 01:21:34,130
and work hard to get
what I just derived

1294
01:21:34,130 --> 01:21:36,750
for you in a couple of lines.

1295
01:21:36,750 --> 01:21:40,460
So we're kind of done
for today, and we've

1296
01:21:40,460 --> 01:21:43,580
seen kind of two examples
of mode factorization

1297
01:21:43,580 --> 01:21:44,960
in the effective theory.

1298
01:21:44,960 --> 01:21:48,080
collinear and ultra soft fields
and collinear and hard degrees

1299
01:21:48,080 --> 01:21:50,090
of freedom, and
how it simplifies

1300
01:21:50,090 --> 01:21:55,243
the discussion of factorization
which is kind of traditional,

1301
01:21:55,243 --> 01:21:57,660
in a more traditional language
which I haven't taught you,

1302
01:21:57,660 --> 01:21:59,520
but you can believe
me is more complicated

1303
01:21:59,520 --> 01:22:01,230
than what we've discussed.

1304
01:22:01,230 --> 01:22:03,640
So we'll talk a little bit
more about this next time.

1305
01:22:03,640 --> 01:22:06,780
And then we'll talk about how
the ideas that we have here

1306
01:22:06,780 --> 01:22:08,610
lead us just to define
a set of objects

1307
01:22:08,610 --> 01:22:10,183
that we build operators from.

1308
01:22:10,183 --> 01:22:11,850
And once we know what
those objects are,

1309
01:22:11,850 --> 01:22:14,017
then we can kind of dispense
with a lot of the steps

1310
01:22:14,017 --> 01:22:17,910
that we've done and just jump
right to building operators out

1311
01:22:17,910 --> 01:22:19,620
of those objects.

1312
01:22:19,620 --> 01:22:21,540
But the steps are
necessary to understand

1313
01:22:21,540 --> 01:22:26,040
why it's those objects that we
want to build operators from.

1314
01:22:26,040 --> 01:22:26,850
OK.

1315
01:22:26,850 --> 01:22:27,900
So any questions?

1316
01:22:31,912 --> 01:22:34,120
So we'll talk a little bit
about how this generalizes

1317
01:22:34,120 --> 01:22:36,280
to other operators next time.

1318
01:22:36,280 --> 01:22:39,870
So that's just the
one time I did here.