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IAIN STEWART: OK,
so last time, we

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00:00:24,100 --> 00:00:26,920
were talking about the
[? massive ?] Sudakov form

10
00:00:26,920 --> 00:00:28,090
factor.

11
00:00:28,090 --> 00:00:30,490
We're going to continue
that discussion today.

12
00:00:30,490 --> 00:00:33,467
And we saw that in order
to do this calculation,

13
00:00:33,467 --> 00:00:35,050
we needed to have
additional regulator

14
00:00:35,050 --> 00:00:37,900
besides dimensional
regularization.

15
00:00:37,900 --> 00:00:40,930
This was an example in SCT2
where we had these rapidity

16
00:00:40,930 --> 00:00:42,310
divergences.

17
00:00:42,310 --> 00:00:44,140
And so there is this
additional regulator

18
00:00:44,140 --> 00:00:47,260
that led to these 1 over
8 eta poles in our answer

19
00:00:47,260 --> 00:00:50,320
and these logs of nu as well.

20
00:00:53,803 --> 00:00:55,720
And we said that if you
stare at these results

21
00:00:55,720 --> 00:00:59,230
that you could see where you
would need to take those scale

22
00:00:59,230 --> 00:01:02,230
parameters in order to
minimize these logarithms.

23
00:01:02,230 --> 00:01:05,290
And it is such that in
the collinear diagrams,

24
00:01:05,290 --> 00:01:06,925
you need to take nu
of order p minus.

25
00:01:11,640 --> 00:01:13,860
And p minus is the hard scale.

26
00:01:13,860 --> 00:01:16,710
And you want you mu
of order m, which

27
00:01:16,710 --> 00:01:19,350
is this scale of the hyperbola.

28
00:01:19,350 --> 00:01:22,890
And in the soft,
you want nu and mu

29
00:01:22,890 --> 00:01:25,530
to be the same size and
then both of order m.

30
00:01:30,330 --> 00:01:32,310
So what we'll do in
a minute is we'll--

31
00:01:32,310 --> 00:01:33,900
so you can imagine
what happens next.

32
00:01:33,900 --> 00:01:37,140
You add some counter terms,
remove these 1 over epsilon

33
00:01:37,140 --> 00:01:38,880
poles and 1 over eta poles.

34
00:01:38,880 --> 00:01:41,730
You're left with something
that's a finite result that's

35
00:01:41,730 --> 00:01:43,530
a function of the cutoffs.

36
00:01:43,530 --> 00:01:45,362
And you put things together.

37
00:01:45,362 --> 00:01:47,070
When you put things
together, the cutoffs

38
00:01:47,070 --> 00:01:49,543
should cancel out of
the physical observable,

39
00:01:49,543 --> 00:01:50,835
which here was the form factor.

40
00:01:53,910 --> 00:01:58,970
So the form of the
factorization theorem for this--

41
00:01:58,970 --> 00:02:01,150
so you have some hard function.

42
00:02:01,150 --> 00:02:06,870
It's just the
Wilson coefficient,

43
00:02:06,870 --> 00:02:13,180
some collinear function
which I'll can cn,

44
00:02:13,180 --> 00:02:20,055
cn bar, and then
some soft function.

45
00:02:25,110 --> 00:02:27,120
And so it's the usual
story in some sense,

46
00:02:27,120 --> 00:02:30,330
but we have this additional
dependence on a parameter nu,

47
00:02:30,330 --> 00:02:32,690
not just mu.

48
00:02:32,690 --> 00:02:35,613
OK, so this same statements
I was making up there kind

49
00:02:35,613 --> 00:02:36,780
of encoded in the arguments.

50
00:02:36,780 --> 00:02:37,988
You want mu to be of order m.

51
00:02:37,988 --> 00:02:40,800
You want nu to be of order
q, mu to be of order m nu

52
00:02:40,800 --> 00:02:43,380
to be of order q, mu to
be of order m, nu to be of

53
00:02:43,380 --> 00:02:45,400
[? order mu, ?]
which is order m.

54
00:02:45,400 --> 00:02:49,860
So I could have written
nu over m here as well.

55
00:02:49,860 --> 00:02:52,080
OK, and then there's also
some hard factor that

56
00:02:52,080 --> 00:02:55,440
really looks at the hard scale.

57
00:02:55,440 --> 00:03:08,310
So renormalization, and they
have two cut off parameters.

58
00:03:15,760 --> 00:03:17,423
So what we'll
have, therefore, is

59
00:03:17,423 --> 00:03:18,840
we'll have a
renormalization group

60
00:03:18,840 --> 00:03:21,895
for these objects in a two
dimensional space-- mu and nu

61
00:03:21,895 --> 00:03:22,395
space.

62
00:03:42,510 --> 00:03:47,940
And we'll need both of those
to sum the large logarithms.

63
00:03:50,610 --> 00:03:52,738
So just having
renormalization group in mu

64
00:03:52,738 --> 00:03:54,780
would not be enough to
solve the large logarithms

65
00:03:54,780 --> 00:03:55,490
in this problem.

66
00:04:02,800 --> 00:04:04,925
So let's-- even before we--
so we'll go through it,

67
00:04:04,925 --> 00:04:07,175
and we'll write down anomalous
dimensions in a minute.

68
00:04:07,175 --> 00:04:08,550
But even before
we do that, let's

69
00:04:08,550 --> 00:04:12,630
just picture what we
want from the running.

70
00:04:12,630 --> 00:04:17,635
So here's a two
dimensional space where

71
00:04:17,635 --> 00:04:18,760
we want to do that running.

72
00:04:18,760 --> 00:04:23,287
So let's put mu on this
axis and nu on this axis.

73
00:04:23,287 --> 00:04:24,120
Break out the color.

74
00:04:28,450 --> 00:04:30,383
So we have some hard
degrees of freedom

75
00:04:30,383 --> 00:04:31,800
that we're integrating
out, right?

76
00:04:31,800 --> 00:04:33,970
But they live somewhere
in this picture.

77
00:04:33,970 --> 00:04:37,320
So let's say here, we
have mu of order 2.

78
00:04:37,320 --> 00:04:40,260
The hard degrees of freedom,
as I wrote over here,

79
00:04:40,260 --> 00:04:42,240
they don't depend on nu.

80
00:04:42,240 --> 00:04:45,510
We talked about that last time,
and one way of seeing it is nu

81
00:04:45,510 --> 00:04:47,760
is just a parameter
that distinguishes modes

82
00:04:47,760 --> 00:04:48,930
within the effective theory.

83
00:04:48,930 --> 00:04:51,420
It's a regulator that was
needed to distinguish soft

84
00:04:51,420 --> 00:04:54,450
from collinear, but if you
add it up these contributions,

85
00:04:54,450 --> 00:04:58,530
the nu dependence and the
1 over etas all cancel.

86
00:04:58,530 --> 00:05:02,640
So the hard degrees of freedom
only depend on mu and q.

87
00:05:02,640 --> 00:05:04,330
And so they don't care about nu.

88
00:05:04,330 --> 00:05:07,950
So we can draw them as a line.

89
00:05:07,950 --> 00:05:08,480
It's hard.

90
00:05:11,070 --> 00:05:13,600
They don't live
localized in that space.

91
00:05:13,600 --> 00:05:16,260
But the other modes, we
could put a dot associated

92
00:05:16,260 --> 00:05:18,970
with each one of them.

93
00:05:18,970 --> 00:05:24,797
So you have nu of order m here.

94
00:05:24,797 --> 00:05:25,630
[INAUDIBLE] equal m.

95
00:05:33,680 --> 00:05:35,460
Mu equals q.

96
00:05:35,460 --> 00:05:40,790
So then we need to have a low
energy mu scale, mu equals m.

97
00:05:40,790 --> 00:05:42,740
So the soft modes
are going to live

98
00:05:42,740 --> 00:05:45,110
at mu equals m, mu equals m.

99
00:05:48,090 --> 00:05:52,620
And then the collinear mode's
going to live over here.

100
00:05:52,620 --> 00:05:56,190
And the way we've set up
our calculation, we didn't--

101
00:05:56,190 --> 00:06:00,690
we sort of had one cut off
for both the cn and cn bar.

102
00:06:00,690 --> 00:06:03,210
So they both live
at the same point.

103
00:06:03,210 --> 00:06:06,810
It could have been
more fancy, but that

104
00:06:06,810 --> 00:06:10,570
suffices for doing
the calculations here.

105
00:06:10,570 --> 00:06:12,240
So they both live
at nu equals q.

106
00:06:12,240 --> 00:06:13,950
We basically had a--

107
00:06:13,950 --> 00:06:16,620
we didn't care about whether
it was p minus or p plus.

108
00:06:16,620 --> 00:06:19,830
We regulated them both with nu.

109
00:06:19,830 --> 00:06:21,960
OK, so to do the
renormalization group,

110
00:06:21,960 --> 00:06:24,150
this is where the
modes want to live.

111
00:06:24,150 --> 00:06:27,570
But we need to connect them
because in this formula,

112
00:06:27,570 --> 00:06:29,970
they all have the
same mu and nu.

113
00:06:29,970 --> 00:06:31,860
So the general thing
that you could imagine

114
00:06:31,860 --> 00:06:37,440
is that there's some point mu
and nu in this space, right,

115
00:06:37,440 --> 00:06:39,833
and that you write down
this formula there.

116
00:06:39,833 --> 00:06:42,000
But then everybody has large
logs because you're not

117
00:06:42,000 --> 00:06:43,488
at the right point.

118
00:06:43,488 --> 00:06:45,780
And so you have to do a
renormalization group evolution

119
00:06:45,780 --> 00:06:47,368
to that point.

120
00:06:47,368 --> 00:06:49,410
So the most general thing
that you could think of

121
00:06:49,410 --> 00:06:54,810
would be that there would
be a renormalization

122
00:06:54,810 --> 00:06:57,620
group of the hard function
down to that point.

123
00:06:57,620 --> 00:07:00,437
And that's just the
whole line moves down.

124
00:07:00,437 --> 00:07:02,020
So this would be
some evolution kernel

125
00:07:02,020 --> 00:07:05,850
uh that would run from some
initial condition, which

126
00:07:05,850 --> 00:07:07,730
I'll call uh.

127
00:07:07,730 --> 00:07:11,160
I'll call this mu
equals mu h, and we'll

128
00:07:11,160 --> 00:07:12,660
call this guy mu light.

129
00:07:15,510 --> 00:07:17,650
Call this guy nu light.

130
00:07:17,650 --> 00:07:20,220
Call this guy nu heavy.

131
00:07:20,220 --> 00:07:22,240
Just have some names for them.

132
00:07:22,240 --> 00:07:24,840
And then these guys here would
have running in two dimensions,

133
00:07:24,840 --> 00:07:28,190
and you have to take a path
from this point to that point.

134
00:07:28,190 --> 00:07:30,630
And so here's a simple path.

135
00:07:30,630 --> 00:07:34,560
Just go over and then up.

136
00:07:34,560 --> 00:07:39,330
Just go over, up.

137
00:07:39,330 --> 00:07:41,250
And there'd be an
evolution kernel associated

138
00:07:41,250 --> 00:07:42,640
with each one of these things.

139
00:07:42,640 --> 00:07:46,005
So this mu 1 would
be some kernel mu

140
00:07:46,005 --> 00:07:51,690
n that would run for
mu light to some mu.

141
00:07:51,690 --> 00:07:53,305
And then it's at fixed new.

142
00:07:56,910 --> 00:07:59,670
And this guy here will
be some other evolution

143
00:07:59,670 --> 00:08:03,300
kernel which I'll call v because
it's running in nu space--

144
00:08:03,300 --> 00:08:05,850
so some linear evolution
kernel that's running in

145
00:08:05,850 --> 00:08:09,630
nu from some nu
heavy down to nu.

146
00:08:09,630 --> 00:08:11,730
And it's at fixed nu light.

147
00:08:15,380 --> 00:08:17,630
And then there would likewise
be kernels on this side.

148
00:08:17,630 --> 00:08:25,330
There would be a
mu s and a vs. OK,

149
00:08:25,330 --> 00:08:28,060
so that would be the
sort of general thing

150
00:08:28,060 --> 00:08:30,040
that you could possibly imagine.

151
00:08:30,040 --> 00:08:33,270
Now the choice at this
point is arbitrary,

152
00:08:33,270 --> 00:08:35,020
and you know that
effectively that there's

153
00:08:35,020 --> 00:08:37,419
consistency in the sense
that you can move that point

154
00:08:37,419 --> 00:08:39,492
around and nothing changes.

155
00:08:39,492 --> 00:08:41,409
So you can use your
freedom to pick this point

156
00:08:41,409 --> 00:08:43,720
to make your life as
simple as possible.

157
00:08:43,720 --> 00:08:45,590
If you pick it up here
at the hard scale,

158
00:08:45,590 --> 00:08:47,190
you don't have to
do any uh running.

159
00:08:47,190 --> 00:08:49,190
If you pick it over here
at the collinear scale,

160
00:08:49,190 --> 00:08:51,400
then you don't have to
do the collinear running.

161
00:08:51,400 --> 00:08:56,095
So we can do that
to simplify things.

162
00:09:04,170 --> 00:09:05,780
And that's again
just the freedom--

163
00:09:05,780 --> 00:09:09,590
the usual, simple freedom-- of
either running the coefficients

164
00:09:09,590 --> 00:09:11,120
or running the operators.

165
00:09:11,120 --> 00:09:12,800
And it's just a little
more complicated.

166
00:09:12,800 --> 00:09:15,050
That's what this freedom to
move this point around is.

167
00:09:20,490 --> 00:09:22,910
So this is like running
coefficients versus operators.

168
00:09:25,382 --> 00:09:26,840
It's just a little
more complicated

169
00:09:26,840 --> 00:09:28,257
because we have
these two cutoffs.

170
00:09:32,050 --> 00:09:39,090
So let's just pick,
for example, mu and nu

171
00:09:39,090 --> 00:09:44,140
to be mu light and nu heavy.

172
00:09:44,140 --> 00:09:47,850
So that's putting it over
here at the orange point.

173
00:09:50,400 --> 00:09:53,077
Then you would just
have evolution kernels.

174
00:09:53,077 --> 00:09:55,410
The softs would just have to
run all the way over there.

175
00:09:55,410 --> 00:09:57,250
So you wouldn't
need mu us either.

176
00:09:57,250 --> 00:10:04,390
You would just need nu s, vs.

177
00:10:04,390 --> 00:10:12,550
So you have vs, and vs would
run from starting at nu light

178
00:10:12,550 --> 00:10:13,810
over to nu heavy.

179
00:10:13,810 --> 00:10:25,420
So this is the
initial condition,

180
00:10:25,420 --> 00:10:26,980
and this is the final.

181
00:10:26,980 --> 00:10:30,770
And it's at fixed mu s.

182
00:10:30,770 --> 00:10:35,020
And then you'd have
a running from uh

183
00:10:35,020 --> 00:10:42,170
down to mu light from mu heavy.

184
00:10:42,170 --> 00:10:44,420
OK, so then you just have
these two evolution kernels.

185
00:10:44,420 --> 00:10:47,960
So it would be pretty simple.

186
00:10:47,960 --> 00:10:52,148
So there's a consistency of both
moving around in the nu space.

187
00:10:52,148 --> 00:10:54,690
When I move this point around,
there's a cancellation between

188
00:10:54,690 --> 00:10:57,970
vn and vs, and there's also a
cancellation between the three

189
00:10:57,970 --> 00:11:05,820
u's.

190
00:11:05,820 --> 00:11:09,970
So there's also in this thing,
there's a path independence.

191
00:11:09,970 --> 00:11:13,410
So I drew a particular path
where I went like that.

192
00:11:13,410 --> 00:11:14,820
I could have drawn another path.

193
00:11:14,820 --> 00:11:17,670
I could have gone up
first and then over.

194
00:11:17,670 --> 00:11:19,530
And because the
parameters nu and mu

195
00:11:19,530 --> 00:11:24,240
are independent parameters,
the choice of the path

196
00:11:24,240 --> 00:11:24,960
doesn't matter.

197
00:11:29,400 --> 00:11:31,140
We'll see that from
our calculations

198
00:11:31,140 --> 00:11:37,634
as well in a minute when I write
down the anomalous dimensions.

199
00:11:41,590 --> 00:11:43,330
But this is just
following from the fact

200
00:11:43,330 --> 00:11:46,060
that they're
independent parameters.

201
00:11:46,060 --> 00:11:47,860
And effectively,
what that means is

202
00:11:47,860 --> 00:11:50,650
if you think about
taking derivatives,

203
00:11:50,650 --> 00:11:54,340
you can switch the order of the
integration of the derivatives.

204
00:11:54,340 --> 00:12:00,700
So mu d by d mu and nu d by d
nu is the same as nu d by d nu,

205
00:12:00,700 --> 00:12:02,960
mu d by d mu.

206
00:12:02,960 --> 00:12:03,460
OK.

207
00:12:06,173 --> 00:12:08,090
There are examples of
effective field theories

208
00:12:08,090 --> 00:12:10,100
where you would set
up cutoff parameters,

209
00:12:10,100 --> 00:12:12,033
and this wouldn't be true.

210
00:12:12,033 --> 00:12:13,700
You'd think maybe you
have two cut offs,

211
00:12:13,700 --> 00:12:15,033
but actually, you only have one.

212
00:12:15,033 --> 00:12:18,310
There's an example of
this in our QCD, where

213
00:12:18,310 --> 00:12:20,270
if you try to draw
things in two dimensions,

214
00:12:20,270 --> 00:12:21,978
you find out that
there's really only one

215
00:12:21,978 --> 00:12:24,240
path that the effective
theory picks out.

216
00:12:24,240 --> 00:12:26,900
But in this case, there really
are two cutoff parameters.

217
00:12:29,510 --> 00:12:33,050
OK, so let's do that in
detail by going back over here

218
00:12:33,050 --> 00:12:35,822
to these formulas, writing
down the counter terms,

219
00:12:35,822 --> 00:12:38,280
deriving from those counter
terms the anomalous dimensions,

220
00:12:38,280 --> 00:12:40,820
and then we can see how you
would derive these evolution

221
00:12:40,820 --> 00:12:42,740
kernels in this picture.

222
00:12:42,740 --> 00:12:43,385
Yeah.

223
00:12:43,385 --> 00:12:49,340
AUDIENCE: Can you comment
on why sort of this

224
00:12:49,340 --> 00:12:52,356
works even though the mu and
the nu and the collinear cycle

225
00:12:52,356 --> 00:12:55,032
aren't, like, factorizing?

226
00:12:55,032 --> 00:12:57,490
Like, for example, why couldn't
I, just right off the bat--

227
00:12:57,490 --> 00:12:57,810
IAIN STEWART: Yeah.

228
00:12:57,810 --> 00:13:00,520
AUDIENCE: --regulators and never
have a factorization theorem

229
00:13:00,520 --> 00:13:02,030
but run in different scales?

230
00:13:02,030 --> 00:13:06,057
Like, this seems like too easy.

231
00:13:06,057 --> 00:13:07,640
IAIN STEWART: Yeah,
it seems too easy.

232
00:13:11,000 --> 00:13:13,790
It's really because
if you like the--

233
00:13:13,790 --> 00:13:17,540
there's a certain-- if you think
about renormalization theorems,

234
00:13:17,540 --> 00:13:19,610
which we know much
better from mu than--

235
00:13:19,610 --> 00:13:21,680
there's not really
that much known for nu.

236
00:13:21,680 --> 00:13:23,930
There is kind of a
renormalization theorem here

237
00:13:23,930 --> 00:13:27,230
that is related to the fact
that this divergence is

238
00:13:27,230 --> 00:13:30,860
really related to that iconal
propagator that we had.

239
00:13:30,860 --> 00:13:33,200
And even when you go
to higher loop orders,

240
00:13:33,200 --> 00:13:35,720
there's a certain universality
to that propagator that's

241
00:13:35,720 --> 00:13:37,620
making this whole thing work.

242
00:13:37,620 --> 00:13:40,820
So yeah.

243
00:13:40,820 --> 00:13:43,520
I think this is a
relatively new thing,

244
00:13:43,520 --> 00:13:47,290
and I think it hasn't
really fully been explored.

245
00:13:47,290 --> 00:13:49,540
So there's probably some
interesting work to do there.

246
00:13:54,160 --> 00:13:55,540
All right, so let's do it.

247
00:13:55,540 --> 00:13:58,420
Let's see what counter terms
we get and how things work out.

248
00:14:13,870 --> 00:14:15,610
So we're going to do
the standard thing.

249
00:14:15,610 --> 00:14:17,703
Think about renormalization
the objects.

250
00:14:17,703 --> 00:14:19,870
So we'll have bare objects
and renormalized objects.

251
00:14:25,750 --> 00:14:27,580
Each one of these
guys corresponded

252
00:14:27,580 --> 00:14:29,210
to a single collinear field.

253
00:14:29,210 --> 00:14:32,170
So there's some wave function
renormalization factor,

254
00:14:32,170 --> 00:14:36,790
and then there's
some renormalization

255
00:14:36,790 --> 00:14:39,660
for the operator.

256
00:14:39,660 --> 00:14:41,620
And together, those
two things will take us

257
00:14:41,620 --> 00:14:43,750
from the bare to
the renormalization

258
00:14:43,750 --> 00:14:46,300
and likewise for
[INAUDIBLE] soft guy.

259
00:14:52,510 --> 00:14:54,160
In this case, there's no--

260
00:14:54,160 --> 00:14:55,300
we just have Wilson lines.

261
00:14:55,300 --> 00:15:03,320
So there's just a
renormalization of the object.

262
00:15:03,320 --> 00:15:07,460
And this guy here, it's
the usual wave function

263
00:15:07,460 --> 00:15:08,690
renormalization.

264
00:15:08,690 --> 00:15:10,880
It's just the same in SET
for a collinear particle

265
00:15:10,880 --> 00:15:12,856
as it is in QCD.

266
00:15:12,856 --> 00:15:18,770
So just for the
record, there's a 1

267
00:15:18,770 --> 00:15:21,950
over epsilon coming from
that that will modify our--

268
00:15:21,950 --> 00:15:23,630
this 1 over 2 epsilon here.

269
00:15:23,630 --> 00:15:26,015
Let me put it together
for the z factors.

270
00:15:29,270 --> 00:15:31,940
[? So what ?] [? of ?]
[? the z's? ?] Taking those

271
00:15:31,940 --> 00:15:36,505
graphs and these formulas,
find the following.

272
00:15:42,820 --> 00:15:44,686
Think I dropped something.

273
00:15:44,686 --> 00:15:45,670
[INAUDIBLE]

274
00:15:49,610 --> 00:15:51,840
There was this pesky
factor of w squared

275
00:15:51,840 --> 00:15:53,480
that we talked about last time.

276
00:15:58,010 --> 00:15:59,660
But we'll need it
for this calculation.

277
00:16:11,772 --> 00:16:13,480
So we simply just
subtract all the poles.

278
00:16:21,410 --> 00:16:22,956
Minimal subtraction.

279
00:16:27,150 --> 00:16:28,910
So I'm always going
to only just write zn

280
00:16:28,910 --> 00:16:30,740
because zn bar is
really the same.

281
00:16:36,380 --> 00:16:38,120
We set up our kinematics
symmetrically.

282
00:16:38,120 --> 00:16:45,210
So it's really no difference
between zn and zn bar.

283
00:16:50,030 --> 00:16:51,980
So there's a minus
sign plus sign there.

284
00:16:55,040 --> 00:16:56,330
Comes to 3/8 epsilon.

285
00:17:05,270 --> 00:17:08,134
So those are the counter
terms that we get.

286
00:17:08,134 --> 00:17:09,384
AUDIENCE: Did you [INAUDIBLE]?

287
00:17:12,208 --> 00:17:13,250
IAIN STEWART: Yes, I did.

288
00:17:13,250 --> 00:17:13,849
Thank you.

289
00:17:19,680 --> 00:17:21,960
And we get the anomalous
dimensions again

290
00:17:21,960 --> 00:17:26,099
just by demanding the usual
thing-- that the bare objects

291
00:17:26,099 --> 00:17:27,329
don't depend on mu and nu.

292
00:17:31,010 --> 00:17:32,990
And we can do that
separately in the mu and nu.

293
00:17:40,700 --> 00:17:44,540
So if you like, actually,
kind of along the lines

294
00:17:44,540 --> 00:17:48,230
of your question,
[INAUDIBLE],, there's

295
00:17:48,230 --> 00:17:51,020
a assumption built into this,
the way we wrote this, right?

296
00:17:51,020 --> 00:17:53,510
I've kept the full
epsilon dependence

297
00:17:53,510 --> 00:17:55,320
in my 1 over eta pole.

298
00:17:55,320 --> 00:17:57,320
And that's related to
saying that I could really

299
00:17:57,320 --> 00:18:01,370
think of doing all the
eta renormalization first

300
00:18:01,370 --> 00:18:05,100
and then think about
doing epsilon second.

301
00:18:05,100 --> 00:18:06,182
So the kind of--

302
00:18:06,182 --> 00:18:08,390
we have built something into
this that's non-trivial.

303
00:18:11,140 --> 00:18:15,150
AUDIENCE: [INAUDIBLE] the
order is the same [INAUDIBLE]..

304
00:18:15,150 --> 00:18:17,853
IAIN STEWART: You mean
that the order being--

305
00:18:17,853 --> 00:18:19,770
AUDIENCE: Like that you
can do one [INAUDIBLE]

306
00:18:19,770 --> 00:18:20,630
before the other.

307
00:18:20,630 --> 00:18:22,520
IAIN STEWART: Yeah, oh, that
the order doesn't matter.

308
00:18:22,520 --> 00:18:22,830
Yeah.

309
00:18:22,830 --> 00:18:23,660
AUDIENCE: That seems like--

310
00:18:23,660 --> 00:18:24,040
IAIN STEWART: Yeah.

311
00:18:24,040 --> 00:18:25,880
AUDIENCE: --something that you
would get from a factorized

312
00:18:25,880 --> 00:18:26,630
expression--

313
00:18:26,630 --> 00:18:27,463
IAIN STEWART: Right.

314
00:18:27,463 --> 00:18:28,625
AUDIENCE: [INAUDIBLE]

315
00:18:28,625 --> 00:18:30,500
IAIN STEWART: Yeah, it
kind of is factorized.

316
00:18:30,500 --> 00:18:33,140
But it's hidden,
right, because we

317
00:18:33,140 --> 00:18:35,420
are saying that this is kind
of a multiplicative thing

318
00:18:35,420 --> 00:18:38,570
on that whole thing.

319
00:18:38,570 --> 00:18:40,190
And that's the kind
of factorization,

320
00:18:40,190 --> 00:18:44,190
although it looks kind of
like a non-factorized thing.

321
00:18:44,190 --> 00:18:47,900
So let's talk about mu
anomalous dimensions.

322
00:18:47,900 --> 00:18:50,630
And you'll see that in
order for this to be true,

323
00:18:50,630 --> 00:18:53,300
you also don't-- it's not
the case that mu anomalous

324
00:18:53,300 --> 00:18:57,630
dimension needs to be
independent of nu, for example.

325
00:18:57,630 --> 00:18:59,730
So this kind of looks
like it's not factoring,

326
00:18:59,730 --> 00:19:03,440
but magically, it is in
the sense of this path

327
00:19:03,440 --> 00:19:04,760
independence.

328
00:19:04,760 --> 00:19:09,130
So what is gamma mu
for the soft function?

329
00:19:09,130 --> 00:19:17,000
So we just say that mu d by
d mu on bare things is 0 is--

330
00:19:17,000 --> 00:19:18,705
so is nu d by d nu.

331
00:19:18,705 --> 00:19:22,010
So that's the way
that this works.

332
00:19:22,010 --> 00:19:24,830
And that gives us this
kind of standard formulas

333
00:19:24,830 --> 00:19:31,110
that we're used to for
the anomalous dimensions.

334
00:19:31,110 --> 00:19:34,490
So zs inverse mu d by d mu
with the appropriate sign

335
00:19:34,490 --> 00:19:37,100
is the anomalous
dimension in mu.

336
00:19:37,100 --> 00:19:42,510
That comes from this
equation for soft function.

337
00:19:42,510 --> 00:19:47,160
So if we look at that
and we work it out,

338
00:19:47,160 --> 00:19:51,920
so if you look at this term,
it actually doesn't contribute,

339
00:19:51,920 --> 00:19:57,080
because in this term, alpha s
of mu times mu to the 2 epsilon

340
00:19:57,080 --> 00:19:58,610
is mu independent.

341
00:19:58,610 --> 00:20:02,600
So that term has no contribution
to the new anomalous dimension.

342
00:20:02,600 --> 00:20:05,060
And then this term
does contribute,

343
00:20:05,060 --> 00:20:07,610
and there's a contribution
from this term.

344
00:20:07,610 --> 00:20:10,250
Where you differentiate
the alpha, you get a 1

345
00:20:10,250 --> 00:20:11,710
over 2 epsilon.

346
00:20:11,710 --> 00:20:18,771
You differentiate this
guy, and there's a--

347
00:20:18,771 --> 00:20:20,820
if you differentiate
the explicit log mu,

348
00:20:20,820 --> 00:20:22,260
there's also a 1 over epsilon.

349
00:20:22,260 --> 00:20:25,440
Those 1 over
epsilons cancel, OK?

350
00:20:25,440 --> 00:20:27,330
So there's no contribution
from this guy.

351
00:20:27,330 --> 00:20:29,970
There's no contribution
from differentiating

352
00:20:29,970 --> 00:20:33,940
the alpha in this guy or the
explicit log in this guy.

353
00:20:33,940 --> 00:20:37,050
So the only contribution
is taking the alpha,

354
00:20:37,050 --> 00:20:39,000
differentiating it,
getting a 2 epsilon,

355
00:20:39,000 --> 00:20:41,950
and multiplying
that 1 over epsilon.

356
00:20:41,950 --> 00:20:52,100
So this line switches,
and there's a 2.

357
00:20:56,830 --> 00:20:58,240
AUDIENCE: Set w to 1?

358
00:20:58,240 --> 00:20:59,782
IAIN STEWART: And
I set w to 1, yeah.

359
00:21:02,620 --> 00:21:05,445
w-- renormalize w.

360
00:21:08,175 --> 00:21:08,980
[INAUDIBLE] 1.

361
00:21:22,710 --> 00:21:26,590
So likewise, for the other
case, for the collinear--

362
00:21:34,440 --> 00:21:39,300
and this is actually the same
as [INAUDIBLE] mu and bar.

363
00:21:42,000 --> 00:21:46,200
And what this gives if you
solve the anomalous dimension

364
00:21:46,200 --> 00:21:48,990
equation--

365
00:21:48,990 --> 00:21:50,490
so the anomalous
dimension equations

366
00:21:50,490 --> 00:21:54,060
are like [INAUDIBLE]
sign convention

367
00:21:54,060 --> 00:22:02,750
like this or et cetera.

368
00:22:02,750 --> 00:22:05,120
So you would solve
those equations.

369
00:22:05,120 --> 00:22:09,396
And those equations would give
you the kernels us and un.

370
00:22:13,590 --> 00:22:15,290
And it's a simple
multiplicative rge.

371
00:22:15,290 --> 00:22:17,292
So if you just-- this
is mu d by dmu of log

372
00:22:17,292 --> 00:22:23,350
s, and you just integrate
the way we've done before.

373
00:22:23,350 --> 00:22:28,630
And consistency
says that you could

374
00:22:28,630 --> 00:22:30,680
run in this picture either way.

375
00:22:30,680 --> 00:22:35,602
And that is a relation between
the anomalous dimensions.

376
00:22:35,602 --> 00:22:37,060
There's also an
anomalous dimension

377
00:22:37,060 --> 00:22:39,320
for the hard function.

378
00:22:39,320 --> 00:22:42,384
And if we add them
all up, we get zero.

379
00:22:42,384 --> 00:22:45,800
We pick the sign
conventions accordingly.

380
00:22:45,800 --> 00:22:47,860
So you could write
the relation that way.

381
00:22:50,980 --> 00:22:52,480
And so you can
calculate the gamma h

382
00:22:52,480 --> 00:22:55,000
by knowing these three--

383
00:22:55,000 --> 00:22:58,330
gamma for the hard function.

384
00:22:58,330 --> 00:23:01,150
Or you could calculate
it by the counter term

385
00:23:01,150 --> 00:23:02,605
for the hard function.

386
00:23:05,460 --> 00:23:09,120
And as expected, it only
depends on a log of mu over q.

387
00:23:09,120 --> 00:23:10,230
There's no nus in that.

388
00:23:13,128 --> 00:23:14,670
So that's the mu
anomalous dimension.

389
00:23:14,670 --> 00:23:16,440
It works as usual.

390
00:23:16,440 --> 00:23:21,170
And the interesting thing
is that there's also

391
00:23:21,170 --> 00:23:24,230
a new anomalous dimension
because the bear functions are

392
00:23:24,230 --> 00:23:25,190
independent of nu.

393
00:23:53,800 --> 00:23:55,050
It's the same type of formula.

394
00:23:55,050 --> 00:23:58,680
We just have nu d by d nu
instead of mu d by d mu.

395
00:23:58,680 --> 00:24:01,830
There's no nu dependence
in the coupling alpha,

396
00:24:01,830 --> 00:24:04,740
but there is a nu
dependence in this w.

397
00:24:04,740 --> 00:24:07,620
When you differentiate
the w, you get an eta

398
00:24:07,620 --> 00:24:14,860
And that's where basically the
contribution is coming from.

399
00:24:14,860 --> 00:24:18,690
So what happens is you
differentiate the w.

400
00:24:18,690 --> 00:24:19,558
That kills this eta.

401
00:24:19,558 --> 00:24:21,600
But then it looks like
your result would have a 1

402
00:24:21,600 --> 00:24:23,595
over epsilon pole.

403
00:24:23,595 --> 00:24:25,470
But you also have to
differentiate explicitly

404
00:24:25,470 --> 00:24:27,510
this nu here.

405
00:24:27,510 --> 00:24:29,340
Again, that has a 1
over epsilon pole.

406
00:24:29,340 --> 00:24:31,250
So those two cancel.

407
00:24:31,250 --> 00:24:33,000
Once you take the
epsilon goes to 0 limit,

408
00:24:33,000 --> 00:24:34,375
they cancel, and
you're just left

409
00:24:34,375 --> 00:24:36,300
with a finite
anomalous dimension.

410
00:24:36,300 --> 00:24:40,710
So here, I'm always
sending epsilon to 0,

411
00:24:40,710 --> 00:24:42,600
and eta goes to 0.

412
00:24:42,600 --> 00:24:48,330
And that's what sets w
to 1 and stuff like that.

413
00:24:48,330 --> 00:24:51,820
OK, so this guy--

414
00:24:51,820 --> 00:24:53,650
so [INAUDIBLE]
cancellations [INAUDIBLE]

415
00:24:53,650 --> 00:24:55,470
1 over epsilon poles
to get those results,

416
00:24:55,470 --> 00:24:57,780
and there's also cancellations
of 1 over epsilon poles

417
00:24:57,780 --> 00:24:59,320
to get these results.

418
00:25:04,790 --> 00:25:13,070
And equations like nu d by d
nu of s [INAUDIBLE] gamma nu ss

419
00:25:13,070 --> 00:25:18,760
would give the
kernel vs, et cetera.

420
00:25:23,870 --> 00:25:26,963
OK, so there's the explicit
anomalous dimensions.

421
00:25:26,963 --> 00:25:28,130
Let me keep that picture up.

422
00:25:40,650 --> 00:25:44,270
So if you ask what path
independence means,

423
00:25:44,270 --> 00:25:48,530
you could say path independence
could be phrased by the fact

424
00:25:48,530 --> 00:25:52,370
that I could take
z inverse, and I

425
00:25:52,370 --> 00:25:54,320
can take a commutator
of derivatives,

426
00:25:54,320 --> 00:25:56,450
right, because I'm saying
that either order should

427
00:25:56,450 --> 00:26:01,580
get the same result. And
if either order gives

428
00:26:01,580 --> 00:26:04,250
the same result,
that must be zero.

429
00:26:04,250 --> 00:26:07,940
And what this formula says that
the order that we're working

430
00:26:07,940 --> 00:26:15,770
is that you can take
mu d by d mu of some,

431
00:26:15,770 --> 00:26:17,930
say, nu anomalous
dimension, and that

432
00:26:17,930 --> 00:26:23,870
should be equal to nu d by d nu
of the mu anomalous dimension.

433
00:26:23,870 --> 00:26:26,857
That there's a connection
between the two things.

434
00:26:26,857 --> 00:26:28,190
And if you look at the factors--

435
00:26:28,190 --> 00:26:29,930
and if I wrote everything
down correctly,

436
00:26:29,930 --> 00:26:39,900
then that's true and
likewise for the collinear.

437
00:26:39,900 --> 00:26:42,660
So this is the
statement of there

438
00:26:42,660 --> 00:26:46,530
not being a dependence
on the path.

439
00:26:46,530 --> 00:26:48,330
These are formulas
that have to be true

440
00:26:48,330 --> 00:26:49,800
if that's going to be true.

441
00:26:56,965 --> 00:26:59,340
So I'm not going to write down
all these kernels for you.

442
00:26:59,340 --> 00:27:00,500
There's a lot of them.

443
00:27:00,500 --> 00:27:02,120
But just to give you
a flavor for what

444
00:27:02,120 --> 00:27:04,790
the solutions look like,
they kind of look familiar.

445
00:27:04,790 --> 00:27:07,320
Let me write down
a couple of them.

446
00:27:07,320 --> 00:27:08,515
So I'll write down one.

447
00:27:08,515 --> 00:27:09,890
I'll write the
ones for the soft.

448
00:27:17,120 --> 00:27:19,430
So at leading log order,
what would they look like?

449
00:27:21,950 --> 00:27:27,500
They're exponentials, Sudakov
logarithm type formulas.

450
00:27:30,650 --> 00:27:37,605
They're not the same precise
formulas that we had earlier,

451
00:27:37,605 --> 00:27:38,855
but they look kind of similar.

452
00:28:00,057 --> 00:28:02,140
Let me write them down,
then I'll talk about them.

453
00:28:10,860 --> 00:28:12,610
So these are running
along straight paths.

454
00:28:19,500 --> 00:28:22,150
And they involve ratios
of alpha, basically.

455
00:28:22,150 --> 00:28:24,720
[INAUDIBLE] that's what the
running coupling is doing.

456
00:28:39,950 --> 00:28:43,580
So you could always write
things in such a way

457
00:28:43,580 --> 00:28:47,000
that you don't have alphas
that involve nu, right,

458
00:28:47,000 --> 00:28:50,325
because the rga just
has alphas of mu.

459
00:28:50,325 --> 00:28:51,950
And so if you like,
when you're solving

460
00:28:51,950 --> 00:28:53,722
the nu anomalous
dimension, this equation

461
00:28:53,722 --> 00:28:55,430
is very easy to
integrate because there's

462
00:28:55,430 --> 00:28:56,720
no nus on the right hand side.

463
00:28:56,720 --> 00:28:58,640
You just get a log of nu.

464
00:28:58,640 --> 00:29:01,190
And that's why this is--

465
00:29:01,190 --> 00:29:03,590
that's why we have this
kind of simple log nu here.

466
00:29:06,770 --> 00:29:09,350
Here I wrote alpha,
1 over alpha of nu s.

467
00:29:09,350 --> 00:29:13,430
But I could have write that
back in terms of a log.

468
00:29:13,430 --> 00:29:15,020
So it really isn't.

469
00:29:20,510 --> 00:29:24,002
That was just convenient to
make it a simpler formula.

470
00:29:24,002 --> 00:29:26,210
So if you write down the
evolution kernels like that,

471
00:29:26,210 --> 00:29:29,120
they'll satisfy these
multiplication properties

472
00:29:29,120 --> 00:29:31,460
that this figure implies.

473
00:29:31,460 --> 00:29:34,700
And I won't go through
that, but it's kind of

474
00:29:34,700 --> 00:29:36,020
neat to see how it works out.

475
00:29:39,610 --> 00:29:42,610
All right, so that gives you
an idea of how we would sum

476
00:29:42,610 --> 00:29:45,520
logs for this Sudakov form
factor with these two cut offs.

477
00:29:45,520 --> 00:29:49,512
We just have the evolution
kernels and use them as usual.

478
00:29:49,512 --> 00:29:51,220
But we have this more
complicated picture

479
00:29:51,220 --> 00:29:53,350
of what type of
renormalization we have to do.

480
00:29:59,720 --> 00:30:01,950
And you can think about
also, if you wanted to--

481
00:30:01,950 --> 00:30:06,360
for example, say you wanted to
do this in some calculation,

482
00:30:06,360 --> 00:30:08,522
and then you calculated
up to some order

483
00:30:08,522 --> 00:30:09,980
solving these
anomalous dimensions.

484
00:30:09,980 --> 00:30:12,200
And then you wanted
to vary scales.

485
00:30:12,200 --> 00:30:14,180
Well, now you have a
two dimensional plane

486
00:30:14,180 --> 00:30:15,410
to vary scales in, right?

487
00:30:15,410 --> 00:30:17,630
So if you're varying
the soft scales,

488
00:30:17,630 --> 00:30:21,680
you can kind of move around
in a box around this guy

489
00:30:21,680 --> 00:30:24,620
where you're varying
both nu l and mu l

490
00:30:24,620 --> 00:30:27,080
by kind of factors of 2.

491
00:30:27,080 --> 00:30:29,840
And so they're
doing uncertainties

492
00:30:29,840 --> 00:30:31,744
in this kind of
setup would have--

493
00:30:31,744 --> 00:30:34,730
you'd have more parameters to
vary than you would usually

494
00:30:34,730 --> 00:30:36,800
have just with mus.

495
00:30:36,800 --> 00:30:40,280
But really, it's just a
straightforward generalization

496
00:30:40,280 --> 00:30:43,220
of the one dimensional
picture of mu evolution

497
00:30:43,220 --> 00:30:45,637
to a two dimensional
picture of nu evolution.

498
00:30:48,980 --> 00:30:51,380
And the interesting
part is the kind

499
00:30:51,380 --> 00:30:53,611
of overlaps between
these parameters.

500
00:31:00,000 --> 00:31:03,890
So let me give you one other
physics example of where

501
00:31:03,890 --> 00:31:05,600
this comes in just
so that you see

502
00:31:05,600 --> 00:31:08,270
it's not just this one example.

503
00:31:08,270 --> 00:31:18,850
So I'll just go through
one of them in detail,

504
00:31:18,850 --> 00:31:20,350
and I'll just mention
one other one.

505
00:31:23,380 --> 00:31:27,220
So it'll be two examples, but
we'll only really cover one.

506
00:31:27,220 --> 00:31:31,390
So one thing you can
do is do gg to Higgs,

507
00:31:31,390 --> 00:31:33,970
so Higgs production,
where I measure

508
00:31:33,970 --> 00:31:37,075
a particular distribution,
which is the pt distribution,

509
00:31:37,075 --> 00:31:37,960
pt of the Higgs.

510
00:31:43,210 --> 00:31:45,490
And it turns out
that this process

511
00:31:45,490 --> 00:31:48,850
involves rapidity divergences.

512
00:31:48,850 --> 00:31:51,130
So let me try to
draw one picture that

513
00:31:51,130 --> 00:31:54,710
allows me to capture all the
different degrees of freedom.

514
00:31:54,710 --> 00:31:58,810
So here's-- you could imagine
that this is your top loop,

515
00:31:58,810 --> 00:32:01,450
but it's some short
distance thing.

516
00:32:01,450 --> 00:32:03,437
And you can even
integrate out the top.

517
00:32:03,437 --> 00:32:05,770
Think of it as an effective
operator coupling two gluons

518
00:32:05,770 --> 00:32:07,570
to the Higgs.

519
00:32:07,570 --> 00:32:11,730
And so I'm in the center of
mass frame of the collision.

520
00:32:11,730 --> 00:32:13,090
So these guys are back to back.

521
00:32:13,090 --> 00:32:15,673
So that means that one of them
is n collinear, and one of them

522
00:32:15,673 --> 00:32:17,170
is n bar collinear.

523
00:32:17,170 --> 00:32:20,950
So if n and n bar collinear
is coming in, annihilating

524
00:32:20,950 --> 00:32:22,870
and producing a Higgs--

525
00:32:22,870 --> 00:32:25,750
and because of what we're
measuring about the Higgs,

526
00:32:25,750 --> 00:32:27,820
we're only measuring the pt.

527
00:32:27,820 --> 00:32:29,195
If you think about
what radiation

528
00:32:29,195 --> 00:32:30,945
you can have in the
final state, well, you

529
00:32:30,945 --> 00:32:32,600
could have collinear radiation.

530
00:32:32,600 --> 00:32:35,230
So here's some
collinear radiation.

531
00:32:35,230 --> 00:32:38,128
And that radiation
has a small pt.

532
00:32:38,128 --> 00:32:39,670
So that's allowed
in the final state.

533
00:32:39,670 --> 00:32:44,210
If we have a pth distribution
and we think about the limit

534
00:32:44,210 --> 00:32:48,370
pth much less than the mass
of the Higgs, so there's some

535
00:32:48,370 --> 00:32:51,795
logs that you would want
to sum, for example.

536
00:32:51,795 --> 00:32:53,170
So you could have
collinear modes

537
00:32:53,170 --> 00:32:55,480
in the final state
that would fit

538
00:32:55,480 --> 00:32:58,120
within this kind
of kinematic setup.

539
00:32:58,120 --> 00:33:00,970
But you could also
have soft modes.

540
00:33:00,970 --> 00:33:06,270
So soft modes have the same size
of pt as the collinear modes.

541
00:33:06,270 --> 00:33:09,340
So they would be allowed,
and this propagator here

542
00:33:09,340 --> 00:33:11,060
would be off shell.

543
00:33:11,060 --> 00:33:12,580
So it's an [? set ?] 2 problem.

544
00:33:12,580 --> 00:33:15,250
Because you're only
constraining a pt,

545
00:33:15,250 --> 00:33:19,150
then it's [? set ?] 2
problem with n bar, n, and s.

546
00:33:22,012 --> 00:33:27,230
So pt Higgs [INAUDIBLE]
order mh lambda,

547
00:33:27,230 --> 00:33:31,040
and modes are these ones.

548
00:33:31,040 --> 00:33:34,550
So we'll SCET two.

549
00:33:34,550 --> 00:33:36,500
And you'd want to sum--

550
00:33:36,500 --> 00:33:42,300
in this case, you'd want to sum
up double logs of pth over mh.

551
00:33:42,300 --> 00:33:44,765
That's what you might
be interested in using

552
00:33:44,765 --> 00:33:45,890
the effective theory to do.

553
00:33:49,280 --> 00:33:52,220
OK, so you would go through
the procedure of factorizing

554
00:33:52,220 --> 00:33:53,060
the cross section.

555
00:33:57,020 --> 00:34:00,770
It's an inclusive calculation
in the sense that basically,

556
00:34:00,770 --> 00:34:02,750
in the final state,
it's Higgs plus x,

557
00:34:02,750 --> 00:34:06,350
right, where x is collinear
radiation or soft radiation.

558
00:34:06,350 --> 00:34:11,330
And really, the process
is proton proton,

559
00:34:11,330 --> 00:34:12,830
for example, the Higgs plus x.

560
00:34:18,880 --> 00:34:21,370
So we would want to factorize
the cross section, amplitude

561
00:34:21,370 --> 00:34:23,257
squared.

562
00:34:23,257 --> 00:34:25,340
And here is kind of a
sketch of how that would go.

563
00:34:25,340 --> 00:34:27,020
I won't go through the details.

564
00:34:27,020 --> 00:34:29,728
So you could think of starting
with some operator where you've

565
00:34:29,728 --> 00:34:31,020
already integrated out the top.

566
00:34:31,020 --> 00:34:33,550
So you have a coupling
of a Higgs to two gluons,

567
00:34:33,550 --> 00:34:35,770
and that happens, and this
gives you variant operator

568
00:34:35,770 --> 00:34:37,156
with the field strengths.

569
00:34:40,820 --> 00:34:43,383
And then you would do the
factorization procedure.

570
00:34:47,179 --> 00:34:51,620
After you shift momenta
to some other states,

571
00:34:51,620 --> 00:34:54,139
you can basically right
that is the factorization

572
00:34:54,139 --> 00:34:56,750
for a matrix element
of two currents.

573
00:34:56,750 --> 00:35:00,620
And you get some
hard function, which

574
00:35:00,620 --> 00:35:03,870
is the Wilson coefficient
squared of this operator.

575
00:35:03,870 --> 00:35:09,160
And then you get
some operators that

576
00:35:09,160 --> 00:35:14,350
look like this, matrix
elements that look like this.

577
00:35:19,977 --> 00:35:21,560
So there's some
proton matrix elements

578
00:35:21,560 --> 00:35:23,685
where one of the proteins
is collinear, one of them

579
00:35:23,685 --> 00:35:25,640
is n bar collinear.

580
00:35:25,640 --> 00:35:28,310
And then there's a
soft matrix element

581
00:35:28,310 --> 00:35:29,750
of some soft Wilson lines.

582
00:35:33,870 --> 00:35:37,870
They're actually in the
adjoint representation.

583
00:35:37,870 --> 00:35:41,351
So I wrote transpose
rather than dagger.

584
00:35:41,351 --> 00:35:44,430
So these are adjoint
representation.

585
00:35:44,430 --> 00:35:47,455
They should be in the same
representation as the collinear

586
00:35:47,455 --> 00:35:49,830
fields, and the collinear
fields here were gluons, right?

587
00:35:49,830 --> 00:35:52,620
So when you go through
the field redefinition,

588
00:35:52,620 --> 00:35:55,500
you would get adjoint
Wilson lines for the soft.

589
00:35:55,500 --> 00:35:58,950
If you went through an [? set ?]
1, for example, picture,

590
00:35:58,950 --> 00:36:03,870
and then the g [? mu nu is ?]
here would become

591
00:36:03,870 --> 00:36:08,477
the perpendicular
polarization of the gluons.

592
00:36:08,477 --> 00:36:10,560
So these things here are
going to give gluon PDFs.

593
00:36:13,113 --> 00:36:15,030
So basically, you have
a factorization theorem

594
00:36:15,030 --> 00:36:22,340
that involves gluon
PDFs and a soft function

595
00:36:22,340 --> 00:36:25,460
and a hard function.

596
00:36:25,460 --> 00:36:27,170
Now if you look at
this calculation,

597
00:36:27,170 --> 00:36:28,190
you have three modes.

598
00:36:28,190 --> 00:36:29,180
There's an [? set ?] 2.

599
00:36:29,180 --> 00:36:30,590
They live on the same hyperbola.

600
00:36:30,590 --> 00:36:32,480
And you do have
rapidity divergences

601
00:36:32,480 --> 00:36:34,350
in this calculation.

602
00:36:34,350 --> 00:36:35,950
So it's a story like this one.

603
00:36:38,710 --> 00:36:41,650
And I'm not going to
go in too much detail,

604
00:36:41,650 --> 00:36:44,298
but just let me write down the
factorization theorem for you

605
00:36:44,298 --> 00:36:45,715
with all the mus
and nus explicit.

606
00:36:45,715 --> 00:36:50,880
And it's really the
same picture where

607
00:36:50,880 --> 00:36:52,455
the soft modes and
collinear modes

608
00:36:52,455 --> 00:36:55,110
need to live at different nus.

609
00:36:55,110 --> 00:36:57,900
Same mu, and then
there's the hard function

610
00:36:57,900 --> 00:37:01,557
where you have to run
from one side to the next.

611
00:37:01,557 --> 00:37:03,890
So it's really just-- you
keep the picture on the board.

612
00:37:03,890 --> 00:37:05,390
It's applicable to
this example too.

613
00:37:08,850 --> 00:37:12,270
So if you calculate d
sigma d, if you measure

614
00:37:12,270 --> 00:37:14,955
rapidity of the Higgs boson--

615
00:37:14,955 --> 00:37:17,430
so you can call this yh.

616
00:37:17,430 --> 00:37:21,330
And you measure the pth
squared the magnitude

617
00:37:21,330 --> 00:37:23,670
of the transverse momentum.

618
00:37:23,670 --> 00:37:28,022
There's some normalization
factor, just kinematics.

619
00:37:28,022 --> 00:37:29,730
There's a hard function
that only depends

620
00:37:29,730 --> 00:37:32,100
on the Higgs mass and mu.

621
00:37:32,100 --> 00:37:36,515
And then your other functions
can exchange [? perp ?] momenta

622
00:37:36,515 --> 00:37:38,640
because the [? perp ?]
momenta of the soft function

623
00:37:38,640 --> 00:37:41,098
and the [? perp momenta ?] of
these collinear functions are

624
00:37:41,098 --> 00:37:42,300
the same size.

625
00:37:42,300 --> 00:37:44,460
And what you're constraining
is just the total.

626
00:37:44,460 --> 00:37:48,840
So the pt of the Higgs, if
the initial guys coming in

627
00:37:48,840 --> 00:37:50,550
have zero [? perp ?]
momenta, there's

628
00:37:50,550 --> 00:37:53,820
zero [? perp ?] momenta
for the protons by design.

629
00:37:53,820 --> 00:37:58,260
So really, there's a balancing
between the final state

630
00:37:58,260 --> 00:38:02,580
radiation and the Higgs,
which I can write like this.

631
00:38:09,877 --> 00:38:11,460
OK, there's a delta
function, and then

632
00:38:11,460 --> 00:38:18,000
you just have objects
for the different guys.

633
00:38:18,000 --> 00:38:26,950
So do this in the proton
center of mass frame.

634
00:38:41,830 --> 00:38:44,380
Because of the fact
that we're measuring

635
00:38:44,380 --> 00:38:48,500
perpendicular momentum, it
allows the PDF to be a tensor.

636
00:38:48,500 --> 00:38:54,550
But that kind of means it
has two scalar PDFs in it.

637
00:39:07,500 --> 00:39:08,406
[INAUDIBLE] h.

638
00:39:11,980 --> 00:39:13,597
And then there's
a soft function.

639
00:39:20,760 --> 00:39:23,900
OK, so it's kind of got
the same type of structure

640
00:39:23,900 --> 00:39:27,180
that we were seeing in
our previous example.

641
00:39:27,180 --> 00:39:32,010
The external kinematics actually
fixes these variables here,

642
00:39:32,010 --> 00:39:33,048
which are like the x.

643
00:39:33,048 --> 00:39:34,340
This is like the [INAUDIBLE] x.

644
00:39:41,510 --> 00:39:44,180
And the kinematics
of the process

645
00:39:44,180 --> 00:39:45,680
actually end up
fixing that when you

646
00:39:45,680 --> 00:39:46,888
go through the factorization.

647
00:39:46,888 --> 00:39:50,820
This should be a plus sign here.

648
00:39:50,820 --> 00:39:53,000
So the only thing that can
change from the dynamics

649
00:39:53,000 --> 00:39:57,410
is the [? perp ?] momentum.

650
00:39:57,410 --> 00:39:59,530
That's what gets exchanged
between these guys.

651
00:39:59,530 --> 00:40:02,870
So you would sum logs by having
normalization group equations

652
00:40:02,870 --> 00:40:04,400
for these objects.

653
00:40:04,400 --> 00:40:06,680
And you need them in mu and nu.

654
00:40:06,680 --> 00:40:09,080
The nu one is always kind of
simple because of the alpha

655
00:40:09,080 --> 00:40:10,020
doesn't depend on nu.

656
00:40:10,020 --> 00:40:11,395
So it's very simple
to integrate.

657
00:40:11,395 --> 00:40:12,685
You just get one log.

658
00:40:12,685 --> 00:40:14,060
But it's still an
important thing

659
00:40:14,060 --> 00:40:15,500
because that one
log is something

660
00:40:15,500 --> 00:40:19,970
that you can't get
from the mu evolution.

661
00:40:19,970 --> 00:40:23,670
OK, so these are called
transverse momentum dependent

662
00:40:23,670 --> 00:40:24,170
PDFs.

663
00:40:37,910 --> 00:40:40,012
So they're not the
standard PDFs that you're

664
00:40:40,012 --> 00:40:40,970
used to thinking about.

665
00:40:40,970 --> 00:40:43,310
They have this dependence
on transverse momentum.

666
00:40:43,310 --> 00:40:45,860
And because of that dependence
on transverse momentum,

667
00:40:45,860 --> 00:40:48,470
they also have these
rapidity divergences.

668
00:40:48,470 --> 00:40:51,450
And these are the renormalized
ones at this nu scale.

669
00:40:51,450 --> 00:40:53,630
So there's a long, sordid
history of these guys.

670
00:40:53,630 --> 00:40:56,360
They were introduced in
the early days of QCD,

671
00:40:56,360 --> 00:40:59,090
and people have really only
sort it out very recently,

672
00:40:59,090 --> 00:41:01,730
both in the QCD literature
and the SET literature

673
00:41:01,730 --> 00:41:05,090
kind of simultaneously
with different regulators,

674
00:41:05,090 --> 00:41:06,830
exactly how to make
sense of these guys

675
00:41:06,830 --> 00:41:09,390
and define them properly and
renormalize them properly.

676
00:41:09,390 --> 00:41:12,350
So this is kind of the last
couple of years type stuff.

677
00:41:16,250 --> 00:41:18,190
So another example
that we could do which

678
00:41:18,190 --> 00:41:21,250
I won't go through in detail--

679
00:41:21,250 --> 00:41:23,800
so that was example one.

680
00:41:23,800 --> 00:41:28,360
We could do an example that
is not involving protons

681
00:41:28,360 --> 00:41:32,330
but involving jets
in e plus e minus.

682
00:41:32,330 --> 00:41:35,060
So we could do e plus e
minus to [? dijets. ?]

683
00:41:35,060 --> 00:41:37,240
And if we did something
similar to what we did here

684
00:41:37,240 --> 00:41:40,000
where we only
measured a pt, then we

685
00:41:40,000 --> 00:41:43,220
would also be in this
[? set ?] 2 situation.

686
00:41:43,220 --> 00:41:48,010
So jet broadening
is a variable b

687
00:41:48,010 --> 00:41:52,280
for broadening where it's like
an event shaped like thrust.

688
00:41:52,280 --> 00:41:55,300
But you only measure
perpendicular momentum.

689
00:41:55,300 --> 00:41:58,750
And what you
measure are actually

690
00:41:58,750 --> 00:42:02,830
perpendicular to
the thrust axis.

691
00:42:02,830 --> 00:42:06,500
So you still use thrust to
get the axis for the jet.

692
00:42:06,500 --> 00:42:08,560
But then you don't
measure anything like

693
00:42:08,560 --> 00:42:11,200
the minus or plus momentum.

694
00:42:11,200 --> 00:42:13,060
You just measure pts.

695
00:42:13,060 --> 00:42:15,760
This would be another
example, which is SCET 2,

696
00:42:15,760 --> 00:42:20,080
and it actually would have,
again, cn, cn bar type modes.

697
00:42:20,080 --> 00:42:21,850
And it-- you'd
[? read ?] a formula

698
00:42:21,850 --> 00:42:24,340
not exactly the same as
this one but again involving

699
00:42:24,340 --> 00:42:27,580
similar types of things--
collinear jet functions that

700
00:42:27,580 --> 00:42:34,382
depend on pt, soft function,
some constraint between them.

701
00:42:34,382 --> 00:42:36,090
It's a little more
complicated, actually,

702
00:42:36,090 --> 00:42:38,790
than this pt Higgs one,
which is why I chose

703
00:42:38,790 --> 00:42:40,500
to write down the pt Higgs one.

704
00:42:40,500 --> 00:42:43,450
So I write that in my notes,
but just because of time,

705
00:42:43,450 --> 00:42:47,260
I'm going to skip it
for our discussion.

706
00:42:47,260 --> 00:42:50,980
So questions so far.

707
00:42:50,980 --> 00:42:54,040
So really, it's turn the
crank once you believe some

708
00:42:54,040 --> 00:42:55,480
of the things I've told you.

709
00:42:55,480 --> 00:42:57,100
So something that
might be interesting

710
00:42:57,100 --> 00:42:59,320
would be to work out
with mu evolution,

711
00:42:59,320 --> 00:43:01,630
we have constraints on the
counter terms at all higher

712
00:43:01,630 --> 00:43:05,450
orders that you can write
down by consistency,

713
00:43:05,450 --> 00:43:06,700
anomalous dimension equations.

714
00:43:06,700 --> 00:43:11,380
And I don't think there's to
my knowledge in the literature

715
00:43:11,380 --> 00:43:14,140
an expression like that
for two loops, three loops,

716
00:43:14,140 --> 00:43:17,343
what would the various
1 over etas and 1

717
00:43:17,343 --> 00:43:18,760
over epsilons, how
would they have

718
00:43:18,760 --> 00:43:21,650
to work out based on the
consistency of the picture I've

719
00:43:21,650 --> 00:43:22,150
told you.

720
00:43:22,150 --> 00:43:24,430
I don't think
anyone's done that.

721
00:43:24,430 --> 00:43:27,070
That would be kind
of interesting.

722
00:43:27,070 --> 00:43:31,630
All right, so questions-- none.

723
00:43:31,630 --> 00:43:32,500
Good.

724
00:43:32,500 --> 00:43:34,010
All right, so one final example.

725
00:43:43,710 --> 00:43:46,370
So presentations Monday--
they're not in this room.

726
00:43:46,370 --> 00:43:50,880
They're in the seminar room,
the large CPT seminar room.

727
00:43:50,880 --> 00:43:52,220
It's on the fourth floor.

728
00:44:00,440 --> 00:44:02,060
And I have also
made a note that you

729
00:44:02,060 --> 00:44:08,100
should use the blackboard, which
will stop you from preparing

730
00:44:08,100 --> 00:44:11,940
too much information.

731
00:44:11,940 --> 00:44:13,830
OK, so let's do
one final example.

732
00:44:20,620 --> 00:44:22,240
The final example I
want to talk about

733
00:44:22,240 --> 00:44:27,670
is Drell-Yan, which we almost
kind of talked about already.

734
00:44:27,670 --> 00:44:30,700
But I'd like to talk about it
in a little different context.

735
00:44:30,700 --> 00:44:34,210
So we talked about pp
to xh, Higgs boson.

736
00:44:34,210 --> 00:44:36,850
Classic Drell-Yan is
pp to xl plus l minus.

737
00:44:36,850 --> 00:44:40,120
But kinematically, that doesn't
really make too much difference

738
00:44:40,120 --> 00:44:43,328
from having a Higgs boson here.

739
00:44:43,328 --> 00:44:45,370
So I will talk about this
in a little more detail

740
00:44:45,370 --> 00:44:46,835
than I did the Higgs example.

741
00:44:46,835 --> 00:44:48,460
We'll go through some
of the kinematics

742
00:44:48,460 --> 00:44:50,283
in a little more detail.

743
00:44:50,283 --> 00:44:52,450
So the reason that I want
to talk about this process

744
00:44:52,450 --> 00:44:54,910
is there's kind of one
function that we haven't yet

745
00:44:54,910 --> 00:45:00,460
seen that will show up
in our discussion here,

746
00:45:00,460 --> 00:45:03,040
one kind spread function
that's ubiquitous

747
00:45:03,040 --> 00:45:05,980
and shows up in all sorts
of factorization theorems.

748
00:45:05,980 --> 00:45:08,530
And we haven't seen it yet.

749
00:45:08,530 --> 00:45:10,720
So what are the kinematics?

750
00:45:10,720 --> 00:45:13,640
We have some momentum.

751
00:45:13,640 --> 00:45:14,830
Let me do it this way.

752
00:45:14,830 --> 00:45:17,290
Let me write it below this guy.

753
00:45:17,290 --> 00:45:21,280
So pa for the proton,
pb for the other proton,

754
00:45:21,280 --> 00:45:29,440
goes equals p for the x plus
q for the l plus l minus pair.

755
00:45:29,440 --> 00:45:31,660
So what are the
important variables?

756
00:45:31,660 --> 00:45:34,150
Well, there's the center of
mass, energy of the collision.

757
00:45:34,150 --> 00:45:38,410
That's just s in
Mandelstam variables,

758
00:45:38,410 --> 00:45:43,450
but I'll call it ECM so
you remember what it is.

759
00:45:43,450 --> 00:45:45,760
pa plus pb all squared--
that's the collision energy.

760
00:45:45,760 --> 00:45:51,730
That's ATV [INAUDIBLE]
[? LHC, ?] soon to be higher.

761
00:45:51,730 --> 00:45:53,920
There's q squared,
and that's the scale

762
00:45:53,920 --> 00:45:55,348
of the hard collision.

763
00:45:58,898 --> 00:46:01,440
So the hard collision scale is
not the center of mass energy.

764
00:46:01,440 --> 00:46:04,150
You take a parton out of each
proton, and those collide.

765
00:46:04,150 --> 00:46:08,090
And they carry a
fraction of that energy.

766
00:46:08,090 --> 00:46:09,640
You can see how
much you have been

767
00:46:09,640 --> 00:46:14,290
looking at how hard the leptons
are, and that's q squared.

768
00:46:14,290 --> 00:46:16,780
It's useful to talk about the
dimensionless variable, which

769
00:46:16,780 --> 00:46:18,340
is [INAUDIBLE].

770
00:46:18,340 --> 00:46:22,060
And that's the ratio of these
two variables, which is always

771
00:46:22,060 --> 00:46:25,180
less than or equal to 1.

772
00:46:25,180 --> 00:46:29,320
And then you can talk about
the rapidity of the leptons.

773
00:46:29,320 --> 00:46:34,870
And you can write that in a kind
of Lorentz invariant looking

774
00:46:34,870 --> 00:46:41,332
way by having a pb
dot q and a pa dot q.

775
00:46:41,332 --> 00:46:42,460
Start using capitals.

776
00:46:47,070 --> 00:46:53,332
So this variable is like
the theta for the leptons.

777
00:46:53,332 --> 00:46:54,480
So theta q.

778
00:46:59,140 --> 00:47:02,270
So again, it's just
like our Higgs example,

779
00:47:02,270 --> 00:47:04,300
but let me draw it in
a little different way.

780
00:47:04,300 --> 00:47:07,810
You can think about doing
everything in a cm frame.

781
00:47:07,810 --> 00:47:10,210
Protons are coming
in back to back.

782
00:47:10,210 --> 00:47:12,130
So then you have
collinear particles

783
00:47:12,130 --> 00:47:15,400
because your protons
are very energetic.

784
00:47:15,400 --> 00:47:19,480
So if this is a quark and
this is an anti-quark,

785
00:47:19,480 --> 00:47:21,250
you produce a virtual photon.

786
00:47:21,250 --> 00:47:25,210
And the virtual photon
produces a lepton pair.

787
00:47:25,210 --> 00:47:27,740
So here's an n collinear
quark and an n bar collinear

788
00:47:27,740 --> 00:47:28,240
anti-quark.

789
00:47:28,240 --> 00:47:31,480
So the anti-quark is
coming in that way,

790
00:47:31,480 --> 00:47:32,680
quark is coming in that way.

791
00:47:32,680 --> 00:47:35,530
They annihilate, produce
a virtual photon.

792
00:47:35,530 --> 00:47:38,350
That produces a lepton pair.

793
00:47:38,350 --> 00:47:40,570
That's Drell-Yan.

794
00:47:40,570 --> 00:47:42,280
OK, so in the cn
frame, we're going

795
00:47:42,280 --> 00:47:50,930
to have cn modes, cn bar
modes, for the protons

796
00:47:50,930 --> 00:47:54,700
and the anti-proton.

797
00:47:54,700 --> 00:47:58,000
Some other variables that
are useful to talk about

798
00:47:58,000 --> 00:48:00,430
are something that people
call xa and xb, which

799
00:48:00,430 --> 00:48:02,200
are the Bjorken variables.

800
00:48:02,200 --> 00:48:05,320
And they already showed up
in our previous example,

801
00:48:05,320 --> 00:48:07,990
but let me just define them.

802
00:48:10,730 --> 00:48:16,930
So these are taking tau and
taking the square root of it

803
00:48:16,930 --> 00:48:19,810
and then splitting up
the rapidity e to the y

804
00:48:19,810 --> 00:48:20,740
and e to the minus y.

805
00:48:20,740 --> 00:48:22,930
And these are like analogs
of the Bjorken variable

806
00:48:22,930 --> 00:48:23,638
for this problem.

807
00:48:34,650 --> 00:48:36,810
And it's actually
these combinations

808
00:48:36,810 --> 00:48:39,730
that are showing up here.

809
00:48:39,730 --> 00:48:41,910
So if you put in
what tau is, that's

810
00:48:41,910 --> 00:48:43,470
explaining what
these arguments were.

811
00:48:43,470 --> 00:48:47,070
They're just the xa and xb.

812
00:48:47,070 --> 00:48:51,090
And kinematics, you
can work out that tau

813
00:48:51,090 --> 00:48:54,630
is less than or equal to xa or
b and that they're less than

814
00:48:54,630 --> 00:48:55,260
or equal to 1.

815
00:48:55,260 --> 00:48:58,710
So there's some
simple bounds on them.

816
00:48:58,710 --> 00:49:00,240
And then finally,
you can work out

817
00:49:00,240 --> 00:49:03,150
that x squared,
if you square it,

818
00:49:03,150 --> 00:49:07,860
is less than or equal to ecm
squared 1 minus square root

819
00:49:07,860 --> 00:49:11,400
of tau all squared.

820
00:49:11,400 --> 00:49:13,770
And one more thing
that you have are

821
00:49:13,770 --> 00:49:18,210
sort of the parton
distribution fractions.

822
00:49:21,730 --> 00:49:23,350
So parton time
distribution variables,

823
00:49:23,350 --> 00:49:25,170
and they're bounded
just like dis.

824
00:49:25,170 --> 00:49:32,448
So xa and xb are things that
are, if you like leptonic,

825
00:49:32,448 --> 00:49:33,490
you might call them that.

826
00:49:33,490 --> 00:49:36,465
But they're things that are
external to the QCD, right?

827
00:49:36,465 --> 00:49:37,840
They're just
measuring properties

828
00:49:37,840 --> 00:49:44,530
of the leptons, q squared, the
center of mass of the equation,

829
00:49:44,530 --> 00:49:46,840
and y, which is a y
of the leptons too.

830
00:49:46,840 --> 00:49:50,590
So everything here is
not a QCD variable.

831
00:49:50,590 --> 00:49:52,330
And then there's
these QCD variables ca

832
00:49:52,330 --> 00:49:54,670
and cb, which are inside
the parton distributions.

833
00:49:54,670 --> 00:49:55,720
And they're bounded.

834
00:49:55,720 --> 00:49:57,575
Just like we had a bound
in dis, x less than

835
00:49:57,575 --> 00:49:59,200
or equal to c, less
than or equal to 1,

836
00:49:59,200 --> 00:50:01,810
here, we have two
analogs of that formula.

837
00:50:01,810 --> 00:50:03,044
AUDIENCE: So [INAUDIBLE].

838
00:50:06,310 --> 00:50:07,690
IAIN STEWART:
Yeah, that's right.

839
00:50:07,690 --> 00:50:10,000
The rapidity of the q--

840
00:50:10,000 --> 00:50:12,040
so really, this is like--

841
00:50:12,040 --> 00:50:15,760
I mean, this is my kind of--

842
00:50:15,760 --> 00:50:18,303
this is like log of--

843
00:50:18,303 --> 00:50:19,720
the thing that's
important here is

844
00:50:19,720 --> 00:50:21,880
the log of q plus or q minus.

845
00:50:21,880 --> 00:50:23,220
It's the rapidity of the q.

846
00:50:42,690 --> 00:50:45,430
Yeah, you could talk about
measuring individual things

847
00:50:45,430 --> 00:50:47,800
about the leptons, but
then you would just

848
00:50:47,800 --> 00:50:50,110
be tacking something
onto the q and taking

849
00:50:50,110 --> 00:50:52,300
something else out of it.

850
00:50:52,300 --> 00:50:54,040
All right, so what kind of--

851
00:50:54,040 --> 00:50:55,543
so this is just some kinematics.

852
00:50:55,543 --> 00:50:57,460
What kind of limits do
we want to think about?

853
00:51:01,900 --> 00:51:03,610
So we already talked
about one example

854
00:51:03,610 --> 00:51:05,980
where we would measure q perp.

855
00:51:05,980 --> 00:51:07,900
That's what we were
doing in the Higgs case.

856
00:51:07,900 --> 00:51:09,650
And I'm not going to
talk about that case.

857
00:51:09,650 --> 00:51:11,067
I'll talk about
three other cases.

858
00:51:15,140 --> 00:51:18,010
So if we didn't measure
q perp, then there's

859
00:51:18,010 --> 00:51:19,960
kind of three
different things we

860
00:51:19,960 --> 00:51:22,510
could do that I'll talk about.

861
00:51:27,922 --> 00:51:29,900
Let's see.

862
00:51:29,900 --> 00:51:31,325
Organize my board better.

863
00:51:35,780 --> 00:51:37,460
I'll tell you about
the kinematics

864
00:51:37,460 --> 00:51:38,540
of each one of
these, and then I'll

865
00:51:38,540 --> 00:51:39,998
draw a little
picture for each one.

866
00:51:50,390 --> 00:51:52,390
So in the inclusive
process, this

867
00:51:52,390 --> 00:51:56,090
is the analog of what we did
for deep elastic scattering.

868
00:51:56,090 --> 00:51:58,630
So deep elastic scattering,
we set the Bjorken x variables

869
00:51:58,630 --> 00:52:00,100
of order 1.

870
00:52:00,100 --> 00:52:02,005
Here we could say
that some sense, tau

871
00:52:02,005 --> 00:52:09,010
is of order 1 as well as
xa and xb are of order 1.

872
00:52:09,010 --> 00:52:12,720
And in this case, what
you're saying about px

873
00:52:12,720 --> 00:52:14,690
squared is that it's hard.

874
00:52:14,690 --> 00:52:17,200
It's of order 2 squared,
and if tau is of order 1,

875
00:52:17,200 --> 00:52:18,725
than that's of
order ecm squared.

876
00:52:18,725 --> 00:52:20,350
So these things are
all hard variables.

877
00:52:26,140 --> 00:52:27,790
And in this case,
the way you should

878
00:52:27,790 --> 00:52:29,582
think about what's
happening in the process

879
00:52:29,582 --> 00:52:32,240
is the following picture.

880
00:52:32,240 --> 00:52:38,620
So you have your partons coming
in or your protons coming in.

881
00:52:38,620 --> 00:52:42,010
And basically, you're allowing
radiation that's hard anywhere.

882
00:52:42,010 --> 00:52:44,687
So hard is supposed to be pink.

883
00:52:44,687 --> 00:52:46,270
You're not constraining
the radiation.

884
00:52:46,270 --> 00:52:52,570
You're really allowing hard
radiation [INAUDIBLE] hard.

885
00:52:52,570 --> 00:52:55,030
Your x is hard and you
allow jets in any direction,

886
00:52:55,030 --> 00:52:56,170
for example.

887
00:52:56,170 --> 00:52:58,794
And then somewhere,
there's a lepton pair.

888
00:52:58,794 --> 00:53:02,170
[INAUDIBLE] purple.

889
00:53:02,170 --> 00:53:04,370
But it's not constrained either.

890
00:53:04,370 --> 00:53:06,130
It's really fairly general.

891
00:53:12,960 --> 00:53:13,960
My picture is too big.

892
00:53:13,960 --> 00:53:15,293
I'm going to run it [INAUDIBLE].

893
00:53:15,293 --> 00:53:16,137
Oh well.

894
00:53:16,137 --> 00:53:16,720
We'll do this.

895
00:53:20,660 --> 00:53:23,690
So I want to draw analogous
pictures for the other cases.

896
00:53:23,690 --> 00:53:26,290
So in the end point, you're
taking a different limit.

897
00:53:26,290 --> 00:53:30,190
What you're doing is you're
taking this tau goes to 1.

898
00:53:30,190 --> 00:53:34,450
And you can see from over
here that if tau goes to 1,

899
00:53:34,450 --> 00:53:36,700
that forces xa
and xb to go to 1.

900
00:53:41,700 --> 00:53:46,260
And if xa and xb go to 1, that
forces ca and cb to go to 1.

901
00:53:49,867 --> 00:53:51,450
So you're really
talking about probing

902
00:53:51,450 --> 00:53:54,118
the proton in a very
special kinematics

903
00:53:54,118 --> 00:53:55,410
where everything is going to 1.

904
00:53:55,410 --> 00:53:57,750
And you're basically
in the hard collision,

905
00:53:57,750 --> 00:54:00,630
you're forcing all the energy
to go into the parton that's

906
00:54:00,630 --> 00:54:02,410
colliding.

907
00:54:02,410 --> 00:54:08,790
So the proton, the
full proton momentum,

908
00:54:08,790 --> 00:54:12,537
goes into the active parton.

909
00:54:12,537 --> 00:54:14,370
Let me just say it that
way-- active parton.

910
00:54:17,950 --> 00:54:22,210
So that changes the
picture because it also--

911
00:54:22,210 --> 00:54:25,450
if you look at it,
when tau goes to 1,

912
00:54:25,450 --> 00:54:28,990
it says the outgoing energy,
the full ecm squared,

913
00:54:28,990 --> 00:54:32,650
is going into q squared, which
is the leptonic variable.

914
00:54:32,650 --> 00:54:34,840
So all the energy is
coming in on the partons

915
00:54:34,840 --> 00:54:36,790
and going out on the leptons.

916
00:54:36,790 --> 00:54:39,318
So you don't have hard
radiation like this anymore.

917
00:54:39,318 --> 00:54:41,110
The only thing that
you could possibly have

918
00:54:41,110 --> 00:54:43,550
is soft radiation.

919
00:54:43,550 --> 00:54:47,560
So in this case, what
happens is this picture--

920
00:54:47,560 --> 00:54:50,210
make my lepton a
little shorter--

921
00:54:50,210 --> 00:54:50,710
changes.

922
00:54:54,330 --> 00:54:59,340
You still, of course, have these
incoming guys, but what happens

923
00:54:59,340 --> 00:55:01,650
is that everything
outgoing is soft.

924
00:55:01,650 --> 00:55:03,660
So make it green.

925
00:55:03,660 --> 00:55:11,190
You have soft
radiation like that.

926
00:55:11,190 --> 00:55:13,750
And then it turns out
also that in this case,

927
00:55:13,750 --> 00:55:20,250
the leptons end up having to be
back to back because you can't

928
00:55:20,250 --> 00:55:21,632
have any transverse momentum.

929
00:55:21,632 --> 00:55:24,090
That would be-- there's nothing
for the transverse momentum

930
00:55:24,090 --> 00:55:25,080
to recoil against.

931
00:55:30,130 --> 00:55:31,420
So that's a possibility.

932
00:55:35,630 --> 00:55:38,390
And then there's a third
thing that we'll talk about,

933
00:55:38,390 --> 00:55:41,980
which is what I call isolated.

934
00:55:41,980 --> 00:55:45,310
And it in some sense is trying
to combine these two pictures

935
00:55:45,310 --> 00:55:48,670
here without taking
a limit on tau.

936
00:55:48,670 --> 00:55:51,970
So you might say, well,
what's the most typical event

937
00:55:51,970 --> 00:55:54,790
at the LHC?

938
00:55:54,790 --> 00:55:57,550
Where is most of the
cross-section for this process?

939
00:55:57,550 --> 00:55:59,170
And that would be
in a situation where

940
00:55:59,170 --> 00:56:01,160
tau is not in the endpoint.

941
00:56:01,160 --> 00:56:03,280
It doesn't go to 1.

942
00:56:03,280 --> 00:56:05,140
It's kind of an order
1 quantity, actually.

943
00:56:05,140 --> 00:56:09,460
You typically get xa's and
xb's that are like 0.1 or 0.01,

944
00:56:09,460 --> 00:56:12,980
small x's for--

945
00:56:12,980 --> 00:56:14,660
depends on what q
squared you look at.

946
00:56:14,660 --> 00:56:18,140
But the typical ones you're
interested in are small x's.

947
00:56:18,140 --> 00:56:19,850
So you don't want x to go to 1.

948
00:56:19,850 --> 00:56:23,210
So tau can be of order 1.

949
00:56:23,210 --> 00:56:25,970
But if you ask what the most
probable thing for the px

950
00:56:25,970 --> 00:56:28,610
to do, if you have these
collinear particles coming in

951
00:56:28,610 --> 00:56:30,110
and they start
radiating, well, they

952
00:56:30,110 --> 00:56:32,362
like to radiate
collinear particles.

953
00:56:32,362 --> 00:56:34,070
So the most likely
thing that will happen

954
00:56:34,070 --> 00:56:35,690
is that you'll get
collinear radiation

955
00:56:35,690 --> 00:56:37,580
from the incoming particles.

956
00:56:37,580 --> 00:56:40,700
And you can look
at that by studying

957
00:56:40,700 --> 00:56:42,700
the following situation
where you constrain

958
00:56:42,700 --> 00:56:47,510
px squared to be 2 ISR jets.

959
00:56:47,510 --> 00:56:50,450
So the picture--
we'll talk about how

960
00:56:50,450 --> 00:56:51,840
we do that in a minute.

961
00:56:51,840 --> 00:56:55,440
But-- or one way of doing it.

962
00:56:55,440 --> 00:56:57,580
So the picture would
then be as follows.

963
00:56:57,580 --> 00:56:59,330
We have these incoming
guys, which now I'm

964
00:56:59,330 --> 00:57:01,400
going to try to draw
some radiation for.

965
00:57:01,400 --> 00:57:05,060
So here are some colors.

966
00:57:05,060 --> 00:57:08,900
So here's a collinear
guy in one direction.

967
00:57:08,900 --> 00:57:11,660
Here's a collinear guy
in the other direction.

968
00:57:11,660 --> 00:57:13,370
And they can radiate.

969
00:57:13,370 --> 00:57:15,950
They radiate prior to
the hard collision here.

970
00:57:15,950 --> 00:57:23,190
So you're getting some jets from
these guys that look like this

971
00:57:23,190 --> 00:57:30,540
and then from this guy a
symmetric thing like that.

972
00:57:30,540 --> 00:57:33,390
And then in the central
part of the collision,

973
00:57:33,390 --> 00:57:34,940
you just allow soft radiation.

974
00:57:40,020 --> 00:57:43,760
So this is the
isolated scenario.

975
00:57:43,760 --> 00:57:48,200
And if I draw the
leptons, it turns out

976
00:57:48,200 --> 00:57:49,760
they don't have to
be back to back,

977
00:57:49,760 --> 00:57:52,790
but there is some constraint
on their kinematics.

978
00:57:52,790 --> 00:57:55,520
Basically, they're back to back
in the transverse plane but not

979
00:57:55,520 --> 00:57:58,200
longitudinally.

980
00:57:58,200 --> 00:58:00,770
But we won't dwell on that.

981
00:58:00,770 --> 00:58:03,420
OK, so this is the sort of
third kinematic configuration.

982
00:58:03,420 --> 00:58:05,747
So even though we're
interested in one process,

983
00:58:05,747 --> 00:58:07,205
we've already just
described to you

984
00:58:07,205 --> 00:58:08,930
four different ways
that you could think

985
00:58:08,930 --> 00:58:10,273
about looking at it.

986
00:58:10,273 --> 00:58:11,690
And those four
different ways will

987
00:58:11,690 --> 00:58:13,565
lead to four different
factorization theorems

988
00:58:13,565 --> 00:58:15,890
because it's a different
kinematic setup.

989
00:58:15,890 --> 00:58:17,900
The first one was
pt of the Higgs.

990
00:58:17,900 --> 00:58:20,390
That led to this rapidly
divergent type factorization

991
00:58:20,390 --> 00:58:21,290
theorem.

992
00:58:21,290 --> 00:58:23,970
Inclusive we'll talk about in
a minute, what it looks like.

993
00:58:23,970 --> 00:58:27,502
Endpoint and isolated, they all
will have different formulas

994
00:58:27,502 --> 00:58:29,210
for the factorization
theorem, and that's

995
00:58:29,210 --> 00:58:32,690
because they look different.

996
00:58:32,690 --> 00:58:36,110
I really should say that
this is ultra soft, not soft.

997
00:58:39,470 --> 00:58:42,326
And so this is
ultra soft too here.

998
00:58:42,326 --> 00:58:45,350
This is ultra soft.

999
00:58:45,350 --> 00:58:51,200
This is cn radiation,
and this is cn bar.

1000
00:58:55,720 --> 00:59:00,610
All right, so let's
see how far we get.

1001
00:59:00,610 --> 00:59:02,470
OK, so let's start
with inclusive.

1002
00:59:15,470 --> 00:59:17,008
So the x is hard.

1003
00:59:17,008 --> 00:59:18,550
And so the way you
can think about it

1004
00:59:18,550 --> 00:59:22,630
is that what you're
doing is that you have--

1005
00:59:22,630 --> 00:59:25,840
you can use kind of an
optical theorem type picture

1006
00:59:25,840 --> 00:59:28,570
where you're cutting
these forward graphs.

1007
00:59:28,570 --> 00:59:32,650
These are the leptons
here, which are

1008
00:59:32,650 --> 00:59:35,140
the things in the final state--

1009
00:59:35,140 --> 00:59:36,640
so vertical photon.

1010
00:59:36,640 --> 00:59:37,930
And I'm squaring it.

1011
00:59:37,930 --> 00:59:42,470
And in comes qq bar
and then squaring it,

1012
00:59:42,470 --> 00:59:46,100
so qq bar on that side as well.

1013
00:59:46,100 --> 00:59:49,480
And then if you think about it,
every kind of radiation gluon

1014
00:59:49,480 --> 00:59:52,700
that I would put across this cut
I can think of as kind of hard.

1015
00:59:52,700 --> 00:59:54,350
And so I can integrate it out.

1016
00:59:54,350 --> 00:59:58,030
And so what you're going to get
is if you just think about this

1017
00:59:58,030 --> 01:00:00,040
process, when you match
the cross-section,

1018
01:00:00,040 --> 01:00:05,290
you're going to get some four
quark operator in [? SCT. ?] So

1019
01:00:05,290 --> 01:00:06,710
this will match
on to an operator,

1020
01:00:06,710 --> 01:00:10,540
which is a four quark operator
where two of the quarks are n

1021
01:00:10,540 --> 01:00:13,090
collinear--

1022
01:00:13,090 --> 01:00:14,440
I'll draw it like this--

1023
01:00:14,440 --> 01:00:18,550
and two of them are
n bar because that's

1024
01:00:18,550 --> 01:00:19,870
the external particles.

1025
01:00:19,870 --> 01:00:24,280
We have qn, q bar, n bar.

1026
01:00:24,280 --> 01:00:25,570
So we're going to get that.

1027
01:00:25,570 --> 01:00:29,800
And that we know how to
write down the lowest order

1028
01:00:29,800 --> 01:00:30,710
operator for that.

1029
01:00:30,710 --> 01:00:34,390
That's just going to be chi bar
chi, chi bar chi, chi bar chi.

1030
01:00:48,560 --> 01:00:59,497
So we have a four
quark operator in SCET,

1031
01:00:59,497 --> 01:01:01,330
which you can ride after
doing a [INAUDIBLE]

1032
01:01:01,330 --> 01:01:06,640
in the spin with all the n
collinear fields together.

1033
01:01:06,640 --> 01:01:10,240
You can work out constraints
on the [? rack ?] structure.

1034
01:01:10,240 --> 01:01:11,980
There's only one
operator, basically.

1035
01:01:24,720 --> 01:01:25,630
It's not quite true.

1036
01:01:25,630 --> 01:01:28,290
So you could have
a gluon operator,

1037
01:01:28,290 --> 01:01:30,120
an operator with
four gluon fields,

1038
01:01:30,120 --> 01:01:32,530
that's the analog of this one.

1039
01:01:32,530 --> 01:01:35,790
Then you would have to have some
higher order diagram in order

1040
01:01:35,790 --> 01:01:37,590
to take those gluons
and attach a photon.

1041
01:01:37,590 --> 01:01:41,920
It would have to be a quark
loop inside the hard function.

1042
01:01:41,920 --> 01:01:45,940
But you could also have a
gluon operator with four b's.

1043
01:01:48,800 --> 01:01:58,108
So let me just say there's
also bn, bn, bn bar, bn bar.

1044
01:01:58,108 --> 01:02:02,430
So you don't have a color
octet structure-- no ta, ta.

1045
01:02:02,430 --> 01:02:04,260
And again, that's like
our bdd pi example

1046
01:02:04,260 --> 01:02:05,718
where if you had
that structure, it

1047
01:02:05,718 --> 01:02:08,580
would vanish when you
take matrix elements.

1048
01:02:08,580 --> 01:02:12,300
So there's no ta,
ta, just constraints

1049
01:02:12,300 --> 01:02:13,800
on the color contractions.

1050
01:02:13,800 --> 01:02:16,740
Here, these guys would
have to be contracted.

1051
01:02:16,740 --> 01:02:19,000
Those guys would have
to be contracted.

1052
01:02:19,000 --> 01:02:20,850
You can make the
field redefinition.

1053
01:02:20,850 --> 01:02:23,200
The field redefinition
the y's would cancel out.

1054
01:02:23,200 --> 01:02:26,040
So there's no ultrasofts here.

1055
01:02:29,190 --> 01:02:31,710
[INAUDIBLE] leading order.

1056
01:02:31,710 --> 01:02:35,130
And if you take these matrix
elements of these objects,

1057
01:02:35,130 --> 01:02:37,770
you can kind of guess what
they're going to give.

1058
01:02:37,770 --> 01:02:39,780
This is just giving a PDF.

1059
01:02:39,780 --> 01:02:41,400
Each of these are giving a PDF.

1060
01:02:41,400 --> 01:02:46,170
And they're the regular
PDFs, the standard ones.

1061
01:02:46,170 --> 01:02:49,590
Because we didn't measure
a [? perp ?] momentum,

1062
01:02:49,590 --> 01:02:51,000
we just get standard PDFs.

1063
01:02:54,090 --> 01:02:56,255
So some parton
inside the proton,

1064
01:02:56,255 --> 01:02:58,380
which would be a gluon from
these guys-- gluon PDFs

1065
01:02:58,380 --> 01:03:00,330
are quarks from these guys.

1066
01:03:00,330 --> 01:03:04,800
And it depends on
some x [INAUDIBLE] mu.

1067
01:03:04,800 --> 01:03:06,510
x could be xa or x--

1068
01:03:06,510 --> 01:03:08,040
x could be ca or cb.

1069
01:03:10,820 --> 01:03:11,925
So let me just write ca.

1070
01:03:16,610 --> 01:03:17,860
And that's really all you get.

1071
01:03:17,860 --> 01:03:20,790
So you have a hard function
and then two collinear parton

1072
01:03:20,790 --> 01:03:22,530
distribution functions.

1073
01:03:22,530 --> 01:03:23,550
So the cross section--

1074
01:03:32,095 --> 01:03:37,410
[INAUDIBLE] these limits
that we talked about

1075
01:03:37,410 --> 01:03:40,530
come from the kinematics.

1076
01:03:40,530 --> 01:03:42,900
[INAUDIBLE] some hard
function, which is an inclusive

1077
01:03:42,900 --> 01:03:44,550
hard function.

1078
01:03:44,550 --> 01:03:47,410
Like in DIS, it depends on
the ratio of these variables.

1079
01:03:47,410 --> 01:03:49,620
But now there's two of them.

1080
01:03:49,620 --> 01:03:53,520
just depends on q
squared, depends on mu.

1081
01:03:53,520 --> 01:03:55,830
And then there's times PDFs.

1082
01:04:08,490 --> 01:04:11,260
There's an important
caveat here,

1083
01:04:11,260 --> 01:04:13,648
which is if you want
to derive this result,

1084
01:04:13,648 --> 01:04:15,690
you have to make sure that
the degrees of freedom

1085
01:04:15,690 --> 01:04:18,452
I've told you with the degrees
of freedom are the right ones.

1086
01:04:18,452 --> 01:04:19,410
So what did I tell you?

1087
01:04:19,410 --> 01:04:24,480
I told you we have cn, cn
bar, and ultra soft, right?

1088
01:04:24,480 --> 01:04:29,412
And that from the
kinematics, it's a SCET one.

1089
01:04:29,412 --> 01:04:31,620
Turns out there's one other
type of degree of freedom

1090
01:04:31,620 --> 01:04:33,818
that you could worry
about, and that's

1091
01:04:33,818 --> 01:04:34,860
called the Glauber gluon?

1092
01:04:37,440 --> 01:04:39,990
And to derive this, we must
know that that doesn't matter.

1093
01:04:49,140 --> 01:04:51,930
What is a Glauber gluon?

1094
01:04:51,930 --> 01:05:00,360
It's a Coulombic gluon
between n and m bar particles.

1095
01:05:00,360 --> 01:05:06,970
So it's a Coulombic
type potential

1096
01:05:06,970 --> 01:05:11,800
that goes like 1 over
pt vector squared.

1097
01:05:11,800 --> 01:05:12,700
And that's a Glauber.

1098
01:05:15,638 --> 01:05:16,930
It's called a Glauber exchange.

1099
01:05:16,930 --> 01:05:18,340
So it's not an
on-shell particle.

1100
01:05:18,340 --> 01:05:21,220
It's like a potential.

1101
01:05:21,220 --> 01:05:24,010
But you have to know that those
actually are relevant in order

1102
01:05:24,010 --> 01:05:25,080
to get to this formula.

1103
01:05:25,080 --> 01:05:28,453
So there's a little
more work involved which

1104
01:05:28,453 --> 01:05:29,620
I'm not going to talk about.

1105
01:05:33,910 --> 01:05:36,580
OK, but-- and that's going
to be true, actually,

1106
01:05:36,580 --> 01:05:40,490
of all the other cases
that we do as well.

1107
01:05:40,490 --> 01:05:42,070
So what about this
threshold limit?

1108
01:05:42,070 --> 01:05:44,050
How does this formula change?

1109
01:05:44,050 --> 01:05:46,960
Well, you could think about
how the formula changes.

1110
01:05:46,960 --> 01:05:51,100
It's really a change of this
h in the threshold limit

1111
01:05:51,100 --> 01:05:52,750
because what's
happening is that you're

1112
01:05:52,750 --> 01:05:55,090
constraining the pink
radiation and making it green.

1113
01:05:59,540 --> 01:06:00,790
[INAUDIBLE] keep my pictures.

1114
01:06:09,820 --> 01:06:20,810
So if we want to go from here
to here for the threshold limit,

1115
01:06:20,810 --> 01:06:24,605
then only certain terms, if you
like, in the HIJ are included.

1116
01:06:28,350 --> 01:06:31,010
So if you wrote out that
function, really what

1117
01:06:31,010 --> 01:06:33,140
it corresponds to is that
only the most singular

1118
01:06:33,140 --> 01:06:39,200
terms in the variable 1 minus
tau in these 1 minus type

1119
01:06:39,200 --> 01:06:42,380
variables are included.

1120
01:06:42,380 --> 01:06:44,540
So delta functions
of 1 minus tau,

1121
01:06:44,540 --> 01:06:47,090
plus functions of
1 minus tau, those

1122
01:06:47,090 --> 01:06:51,740
are the terms that you
would include from the h

1123
01:06:51,740 --> 01:06:56,930
whereas the inclusive
one has much more in it.

1124
01:06:56,930 --> 01:07:00,540
This keeps only
particular terms.

1125
01:07:00,540 --> 01:07:02,540
But there's also different
hierarchies of scales

1126
01:07:02,540 --> 01:07:05,600
because now 1 minus tau is
small, and you want to--

1127
01:07:05,600 --> 01:07:11,210
for example, some logs of 1
minus tau in this situation.

1128
01:07:16,427 --> 01:07:18,135
And that's what the
factorization theorem

1129
01:07:18,135 --> 01:07:20,940
in this threshold
region does for you.

1130
01:07:20,940 --> 01:07:31,620
So the HIJ inclusive gets
turned into the following.

1131
01:07:31,620 --> 01:07:35,720
This is the modification of
the factorization theorem.

1132
01:07:35,720 --> 01:07:38,870
So there's a soft function
for that soft radiation.

1133
01:07:43,460 --> 01:07:52,280
And then there's a different
heart function, which is only

1134
01:07:52,280 --> 01:07:53,480
a function of q squared.

1135
01:07:53,480 --> 01:07:54,920
And it's like the square--

1136
01:07:54,920 --> 01:07:58,160
this guy here is the square
of a Wilson coefficient again,

1137
01:07:58,160 --> 01:08:01,985
and i and j are just pairs
of quarks-- uu bar, dd bar.

1138
01:08:01,985 --> 01:08:07,090
There's actually no
gluons in this case.

1139
01:08:07,090 --> 01:08:10,100
So the terms of the gluon
PDFs actually get suppressed.

1140
01:08:10,100 --> 01:08:12,350
They're not singular
in 1 minus tau,

1141
01:08:12,350 --> 01:08:14,550
and we get this formula here.

1142
01:08:14,550 --> 01:08:16,220
And now there's going
to be some running

1143
01:08:16,220 --> 01:08:18,803
that you have to do between the
hard scale and the soft scale,

1144
01:08:18,803 --> 01:08:21,620
and that running is going to
sum up these logarithms of 1

1145
01:08:21,620 --> 01:08:23,990
minus tau.

1146
01:08:23,990 --> 01:08:32,109
OK, so I'm not going to go
through that in any more

1147
01:08:32,109 --> 01:08:32,620
detail.

1148
01:08:32,620 --> 01:08:34,370
I'll spend a little
more time on this one.

1149
01:08:38,920 --> 01:08:41,770
So what about
isolated Drell-Yan?

1150
01:08:41,770 --> 01:08:44,020
This one is a little bit
different than the other ones

1151
01:08:44,020 --> 01:08:46,689
because we have to measure
something about the process

1152
01:08:46,689 --> 01:08:49,300
to know that it looks like
this if we really want to drive

1153
01:08:49,300 --> 01:08:50,859
a factorization theorem.

1154
01:08:50,859 --> 01:08:52,990
Here, we were measuring tau.

1155
01:08:52,990 --> 01:08:55,479
But now I'm telling you I
don't want to measure tau.

1156
01:08:55,479 --> 01:08:56,918
Yet I still want
to distinguish--

1157
01:08:56,918 --> 01:08:58,210
so I don't want to measure tau.

1158
01:08:58,210 --> 01:09:00,700
I don't want to constrain
it to be close to 1.

1159
01:09:00,700 --> 01:09:03,460
But I still want to distinguish
between these two, right?

1160
01:09:03,460 --> 01:09:06,069
And if I'm going to distinguish
between those two situations,

1161
01:09:06,069 --> 01:09:08,479
I need to measure
something else.

1162
01:09:08,479 --> 01:09:13,090
And we can measure something
that's exactly like what we did

1163
01:09:13,090 --> 01:09:17,080
when we did e plus e minus to
[? dijets ?] because this is

1164
01:09:17,080 --> 01:09:19,819
really like [? dijets. ?] But
they're just initial stage jets

1165
01:09:19,819 --> 01:09:21,189
rather than final stage jets.

1166
01:09:25,170 --> 01:09:28,649
So that's what
we're going to do.

1167
01:09:28,649 --> 01:09:32,130
We only have four jets.

1168
01:09:32,130 --> 01:09:35,790
We have tau that's generic.

1169
01:09:35,790 --> 01:09:38,609
So the c's are not
being forced to go to 1.

1170
01:09:38,609 --> 01:09:42,750
And so we need something that
we can observe to do that.

1171
01:09:42,750 --> 01:09:46,365
And here's what we can do.

1172
01:09:46,365 --> 01:09:48,660
Let's do something
that's the analog of e

1173
01:09:48,660 --> 01:09:54,900
plus e minus to [? dijets ?] but
for these initial stage jets.

1174
01:09:54,900 --> 01:10:00,150
So we say that px is a sum of
momentum in two hemispheres.

1175
01:10:00,150 --> 01:10:06,750
The hemispheres a and b
are just the dividing line

1176
01:10:06,750 --> 01:10:10,260
of the center of mass of the
collision-- so very simple,

1177
01:10:10,260 --> 01:10:12,480
perpendicular to the beam axis.

1178
01:10:12,480 --> 01:10:15,630
And then we just
say that ba plus--

1179
01:10:15,630 --> 01:10:19,320
so this is just a [? four ?]
momentum decomposition.

1180
01:10:19,320 --> 01:10:22,680
But we can look
at ba plus, which

1181
01:10:22,680 --> 01:10:27,880
is like dotting the n vector
for one axis into ba--

1182
01:10:27,880 --> 01:10:34,600
and that's a sum over particles
in one of the hemispheres

1183
01:10:34,600 --> 01:10:35,850
and then likewise for b.

1184
01:10:40,390 --> 01:10:42,630
So this is if I
write it in terms

1185
01:10:42,630 --> 01:10:45,933
of energies and rapidities.

1186
01:10:45,933 --> 01:10:47,240
It'll look like this.

1187
01:10:51,670 --> 01:10:53,490
So this is some
observable ba plus.

1188
01:10:53,490 --> 01:10:56,730
And then I can constrain that
ba plus, which is exactly

1189
01:10:56,730 --> 01:10:59,518
like constraining the
plus momentum in one

1190
01:10:59,518 --> 01:11:00,810
of the hemispheres to be small.

1191
01:11:00,810 --> 01:11:03,233
And what that does, if you
constrain the plus momentum

1192
01:11:03,233 --> 01:11:04,650
in this hemisphere
to be small, it

1193
01:11:04,650 --> 01:11:07,200
will allow ultra soft radiation
because they have small plus

1194
01:11:07,200 --> 01:11:07,950
momentum.

1195
01:11:07,950 --> 01:11:10,080
And it'll allow
collinear radiation

1196
01:11:10,080 --> 01:11:12,160
because they have
small plus momentum.

1197
01:11:12,160 --> 01:11:14,040
So constraining
ba plus constrains

1198
01:11:14,040 --> 01:11:16,740
all the ba plus momentum
in that hemisphere.

1199
01:11:16,740 --> 01:11:20,718
And that puts you in
an SCET one situation

1200
01:11:20,718 --> 01:11:22,260
where we have a
figure like that one.

1201
01:11:27,800 --> 01:11:30,603
So rather than constrain
the [? perp ?] momentum,

1202
01:11:30,603 --> 01:11:31,520
we constrain the plus.

1203
01:11:31,520 --> 01:11:32,570
And then we get SCET one.

1204
01:11:38,460 --> 01:11:47,280
So take ba plus to be
less than or something,

1205
01:11:47,280 --> 01:11:51,820
which you could say, well,
less than or equal to some cut.

1206
01:11:51,820 --> 01:11:54,560
Let's just call it b cut.

1207
01:11:54,560 --> 01:11:57,410
And that is much less than q.

1208
01:11:57,410 --> 01:12:01,040
And then we do the
same thing for b plus,

1209
01:12:01,040 --> 01:12:05,910
which we define as
nb, which is na bar.

1210
01:12:05,910 --> 01:12:09,080
So the notation here
is a little awkward,

1211
01:12:09,080 --> 01:12:10,680
but we do the same thing here.

1212
01:12:10,680 --> 01:12:13,760
So this is less than or
equal to some cut and that

1213
01:12:13,760 --> 01:12:15,950
that's much less than q.

1214
01:12:15,950 --> 01:12:17,450
And by demanding
small plus momentum

1215
01:12:17,450 --> 01:12:19,285
in both hemispheres,
appropriate momentum

1216
01:12:19,285 --> 01:12:21,470
with the small components--

1217
01:12:21,470 --> 01:12:25,670
these are the small components--

1218
01:12:25,670 --> 01:12:31,230
that puts us in a
SCET one situation.

1219
01:12:31,230 --> 01:12:34,220
So we have cn, cn
bar, and ultra soft.

1220
01:12:34,220 --> 01:12:37,200
And those are the allowed
types of radiation,

1221
01:12:37,200 --> 01:12:38,840
and that's what I
drew in my figure.

1222
01:12:38,840 --> 01:12:40,070
So hopefully that's clear.

1223
01:12:45,090 --> 01:12:49,660
All right, so what does
the factorization look

1224
01:12:49,660 --> 01:12:52,060
like in this case?

1225
01:12:52,060 --> 01:12:53,560
Well, you have two
types of things

1226
01:12:53,560 --> 01:12:56,140
that are collinear if
you look at the figure.

1227
01:12:56,140 --> 01:12:58,480
There's the initial
state proton, right,

1228
01:12:58,480 --> 01:13:01,810
which came in here, and
it was collinear here.

1229
01:13:01,810 --> 01:13:03,700
So here's a proton.

1230
01:13:03,700 --> 01:13:07,450
And then it's cruising
along, and at some point,

1231
01:13:07,450 --> 01:13:11,170
it widens out like
this and becomes a jet.

1232
01:13:11,170 --> 01:13:14,800
So this here is collinear at a
smaller scale than this here.

1233
01:13:14,800 --> 01:13:16,720
This is a jet, right?

1234
01:13:16,720 --> 01:13:21,667
It's got large invariant mass
relative to the proton mass.

1235
01:13:21,667 --> 01:13:23,500
So there's actually two
types of collinears,

1236
01:13:23,500 --> 01:13:27,068
and if you really want to draw
the mode picture for this,

1237
01:13:27,068 --> 01:13:28,360
here's what it would look like.

1238
01:13:33,620 --> 01:13:38,170
So here's our collinear
hyperbola cn cn bar.

1239
01:13:38,170 --> 01:13:42,100
We have an ultra soft hyperbola.

1240
01:13:42,100 --> 01:13:45,250
And then somewhere, we have
a lambda qcd hyperbola.

1241
01:13:45,250 --> 01:13:47,590
And there's protons that
sit on the lambda qcd.

1242
01:13:47,590 --> 01:13:50,436
So this is like pn bar,
if you like, or let

1243
01:13:50,436 --> 01:13:52,310
me give it a different name--

1244
01:13:52,310 --> 01:13:53,530
pn.

1245
01:13:53,530 --> 01:13:54,530
Then there's hard modes.

1246
01:13:58,570 --> 01:14:00,465
So what we'll do
is we'll not really

1247
01:14:00,465 --> 01:14:01,840
worry about
distinguishing these.

1248
01:14:01,840 --> 01:14:04,860
We'll think of
them all together.

1249
01:14:04,860 --> 01:14:06,280
So we'll just have
[? sct ?] one,

1250
01:14:06,280 --> 01:14:10,000
where we have ultra softs
here, a cn, and a cn bar.

1251
01:14:10,000 --> 01:14:12,310
But then we'll later have
to worry about factoring.

1252
01:14:12,310 --> 01:14:14,330
This includes the jet.

1253
01:14:14,330 --> 01:14:16,270
This is the collinear jet.

1254
01:14:16,270 --> 01:14:19,360
And this is the proton, right?

1255
01:14:19,360 --> 01:14:24,173
So both objects will be in the
same function at the beginning,

1256
01:14:24,173 --> 01:14:25,840
and then we'll have
to factor them later

1257
01:14:25,840 --> 01:14:28,450
if we want to
separate those scales.

1258
01:14:28,450 --> 01:14:31,630
It's a similar trick to what we
used before in another example.

1259
01:14:31,630 --> 01:14:34,250
It was for soft in that case.

1260
01:14:34,250 --> 01:14:36,620
So if we just keep
these guys together,

1261
01:14:36,620 --> 01:14:37,960
then it's just a SCET one.

1262
01:14:37,960 --> 01:14:41,050
Just have these modes go
through the factorization.

1263
01:14:41,050 --> 01:14:42,210
And this is what we get.

1264
01:14:46,300 --> 01:14:49,690
So we can measure q
squared, rapidity,

1265
01:14:49,690 --> 01:14:51,880
and we can measure these
two hemisphere momenta.

1266
01:14:57,190 --> 01:15:05,470
The hard function is again the
square of a Wilson coefficient

1267
01:15:05,470 --> 01:15:07,210
just as it was in
the threshold case.

1268
01:15:13,970 --> 01:15:19,920
We get some collinear functions
that are called beam functions,

1269
01:15:19,920 --> 01:15:23,160
and they're like, they
have an argument that's

1270
01:15:23,160 --> 01:15:26,200
a plus times a minus momentum.

1271
01:15:26,200 --> 01:15:30,290
So this wa is like
a minus momentum.

1272
01:15:30,290 --> 01:15:32,460
We have a Bjorken x variable.

1273
01:15:32,460 --> 01:15:34,710
And we have mu.

1274
01:15:34,710 --> 01:15:37,920
There's one of these
for each direction.

1275
01:15:37,920 --> 01:15:39,840
So there's one for
the b direction too.

1276
01:15:47,810 --> 01:15:50,390
And then there's
a soft function.

1277
01:15:50,390 --> 01:15:52,380
And it's a hemisphere
soft function.

1278
01:15:52,380 --> 01:15:56,465
It's really exactly analogous
to the soft function that we had

1279
01:15:56,465 --> 01:15:58,250
in e plus e minus,
the [? dijets, ?]

1280
01:15:58,250 --> 01:16:02,264
except that the Wilson lines are
incoming instead of outgoing.

1281
01:16:02,264 --> 01:16:07,010
So it depends on two
plus momenta in mu.

1282
01:16:07,010 --> 01:16:09,500
OK, so the factorization theorem
involves these b's, which

1283
01:16:09,500 --> 01:16:10,640
are called beam functions.

1284
01:16:14,250 --> 01:16:18,230
So this is a beam
function, four parton i.

1285
01:16:18,230 --> 01:16:22,940
And parton i could be a quark
or a gluon in this case.

1286
01:16:22,940 --> 01:16:29,750
Well, sorry, parton i here
will be a quark, actually,

1287
01:16:29,750 --> 01:16:30,750
in this case--

1288
01:16:30,750 --> 01:16:32,360
quark or a q bar.

1289
01:16:34,960 --> 01:16:37,625
You have analogous
functions for gluons as well

1290
01:16:37,625 --> 01:16:39,250
that would come in,
for example, if you

1291
01:16:39,250 --> 01:16:42,110
were doing Higgs production.

1292
01:16:42,110 --> 01:16:43,690
So what is this beam function?

1293
01:16:43,690 --> 01:16:45,700
It's the one object
that we haven't

1294
01:16:45,700 --> 01:16:48,595
seen an example of before.

1295
01:16:52,120 --> 01:16:54,760
And it's really a matrix
element of an operator

1296
01:16:54,760 --> 01:16:58,450
that we've studied in
many different situations.

1297
01:16:58,450 --> 01:17:00,460
It's just the chi
bar chi operator

1298
01:17:00,460 --> 01:17:04,730
with different arguments than
we've studied previously.

1299
01:17:04,730 --> 01:17:07,330
So it's a slightly different
matrix element of the chi bar

1300
01:17:07,330 --> 01:17:11,920
chi operator because
what we're measuring

1301
01:17:11,920 --> 01:17:15,850
is both the large momentum
and the small momentum

1302
01:17:15,850 --> 01:17:16,840
of that operator.

1303
01:17:16,840 --> 01:17:21,190
When we measured just the large
momentum, that gave us the PDF.

1304
01:17:21,190 --> 01:17:25,950
But now we're measuring
also the small momentum,

1305
01:17:25,950 --> 01:17:27,610
which I can write like this.

1306
01:17:39,020 --> 01:17:43,060
So it's a protein matrix element
of a chi bar chi operator

1307
01:17:43,060 --> 01:17:45,880
where if it was a PDF,
this would be 0 and 0,

1308
01:17:45,880 --> 01:17:47,710
and we just have
the minus momentum.

1309
01:17:47,710 --> 01:17:50,050
But now we're
measuring, if you like,

1310
01:17:50,050 --> 01:17:53,290
the other momentum, which
is like a plus momentum.

1311
01:17:53,290 --> 01:17:54,940
And I wrote it in Fourier space.

1312
01:17:54,940 --> 01:17:58,630
So the plus momentum that
you're measuring is here.

1313
01:17:58,630 --> 01:18:01,450
Those are the two variables
that are showing up

1314
01:18:01,450 --> 01:18:06,280
on the right hand side as
well as the minus momentum

1315
01:18:06,280 --> 01:18:08,470
of the proton.

1316
01:18:08,470 --> 01:18:11,890
So it's just a different
matrix element than one

1317
01:18:11,890 --> 01:18:13,060
we've had before.

1318
01:18:13,060 --> 01:18:18,070
The jet function, remember,
would be vacuum matrix element

1319
01:18:18,070 --> 01:18:23,830
of chi bar chi, and
the PDF, standard PDF,

1320
01:18:23,830 --> 01:18:29,200
would be proton matrix
element of chi bar delta chi.

1321
01:18:29,200 --> 01:18:31,090
And it's just really--

1322
01:18:31,090 --> 01:18:32,830
the arguments are
slightly different,

1323
01:18:32,830 --> 01:18:35,290
and that allows this
thing to contain both kind

1324
01:18:35,290 --> 01:18:37,960
of a jet, initial state jet--

1325
01:18:37,960 --> 01:18:39,790
in this case here,
the jet function,

1326
01:18:39,790 --> 01:18:42,987
you'd have some variable zero.

1327
01:18:42,987 --> 01:18:44,570
In this case here,
you'd have 0 and 0.

1328
01:18:44,570 --> 01:18:45,945
And so if you
like, what this guy

1329
01:18:45,945 --> 01:18:48,490
is is just a combination of the
two things we studied before.

1330
01:18:48,490 --> 01:18:49,657
We studied the jet function.

1331
01:18:49,657 --> 01:18:50,470
We studied the PDF.

1332
01:18:50,470 --> 01:18:54,460
Now this has both inside it.

1333
01:18:54,460 --> 01:18:57,940
And if we go through this final
factorization between there

1334
01:18:57,940 --> 01:19:01,735
and there, then that's like a
SCET one to SCET two matching.

1335
01:19:07,440 --> 01:19:11,460
And so we can integrate out the
perturbative radiation, which

1336
01:19:11,460 --> 01:19:12,840
is the jet function, right?

1337
01:19:12,840 --> 01:19:14,190
That's perturbative.

1338
01:19:17,070 --> 01:19:18,630
And this is non-perturbative.

1339
01:19:18,630 --> 01:19:22,620
We can separate those by
doing another matching,

1340
01:19:22,620 --> 01:19:27,450
and that gives a factorization
theorem for the b alone

1341
01:19:27,450 --> 01:19:28,470
that looks like this.

1342
01:19:40,130 --> 01:19:42,700
So there's some perturbative
matching coefficients

1343
01:19:42,700 --> 01:19:43,660
which are called i.

1344
01:19:47,230 --> 01:19:48,715
And then you get a standard PDF.

1345
01:19:53,170 --> 01:19:57,980
So this i is the thing that
has in it the jet radiation.

1346
01:19:57,980 --> 01:20:00,490
This is sort of
an i for the jet.

1347
01:20:00,490 --> 01:20:01,570
And then this is the PDF.

1348
01:20:04,090 --> 01:20:06,022
OK, and the T scale
here is large.

1349
01:20:06,022 --> 01:20:07,480
And what you're
basically expanding

1350
01:20:07,480 --> 01:20:12,550
in is you're expanding
in lambda QCD over t.

1351
01:20:12,550 --> 01:20:15,730
So if you ask what am I doing
to make that separation,

1352
01:20:15,730 --> 01:20:17,980
I'm expanding in
lambda QCD over t.

1353
01:20:17,980 --> 01:20:19,120
There's no t here.

1354
01:20:19,120 --> 01:20:20,480
The t is perturbative.

1355
01:20:20,480 --> 01:20:22,240
That's the scale for the jet.

1356
01:20:22,240 --> 01:20:26,050
And this f is a non-perturbative
distribution function.

1357
01:20:26,050 --> 01:20:28,060
And that's then the full
factorization theorem

1358
01:20:28,060 --> 01:20:30,050
putting those things together.

1359
01:20:30,050 --> 01:20:32,668
So these beam
functions are kind of--

1360
01:20:32,668 --> 01:20:34,210
they are things that
show up whenever

1361
01:20:34,210 --> 01:20:37,330
you have a process where you
ask for a particular number

1362
01:20:37,330 --> 01:20:38,380
of jets.

1363
01:20:38,380 --> 01:20:42,290
And if you asked, for
example, for a particular--

1364
01:20:42,290 --> 01:20:44,890
if you ask for two--

1365
01:20:44,890 --> 01:20:47,340
if you ask for this
setup plus one more jet,

1366
01:20:47,340 --> 01:20:48,340
how would things change?

1367
01:20:48,340 --> 01:20:50,290
Well, you'd have the
same beam radiation,

1368
01:20:50,290 --> 01:20:52,210
the same initial
state radiation.

1369
01:20:52,210 --> 01:20:55,040
And you'd add one more jet
function to this formula.

1370
01:20:55,040 --> 01:20:57,925
So that would be exclusive
one jet production.

1371
01:20:57,925 --> 01:20:59,050
Or you could have two jets.

1372
01:20:59,050 --> 01:21:00,430
Then you'd have
two jet functions.

1373
01:21:00,430 --> 01:21:02,230
In each case, you have a
different soft function.

1374
01:21:02,230 --> 01:21:03,310
But these beam
functions are going

1375
01:21:03,310 --> 01:21:05,350
to be always showing up
because they're describing

1376
01:21:05,350 --> 01:21:06,558
this initial state radiation.

1377
01:21:06,558 --> 01:21:10,780
So any time that's an allowed
thing, then it'll show up.

1378
01:21:10,780 --> 01:21:12,925
If you start doing
[? perp ?] measurements,

1379
01:21:12,925 --> 01:21:14,800
then you might have
[? perp-dependent ?] beam

1380
01:21:14,800 --> 01:21:17,260
functions in the same
way that we could get

1381
01:21:17,260 --> 01:21:19,330
[? perp-dependent ?] PDF.

1382
01:21:19,330 --> 01:21:21,280
In some sense, a
[? perp-dependent ?] PDF is

1383
01:21:21,280 --> 01:21:23,300
like a beam function.

1384
01:21:23,300 --> 01:21:25,753
So that's kind of
the last function

1385
01:21:25,753 --> 01:21:28,420
that you need to know about that
is kind of a generic thing that

1386
01:21:28,420 --> 01:21:30,670
shows up with these
factorizations theorems.

1387
01:21:30,670 --> 01:21:32,390
And we're out of time.

1388
01:21:32,390 --> 01:21:35,050
So let's stop there.

1389
01:21:35,050 --> 01:21:36,790
And that's it.

1390
01:21:36,790 --> 01:21:38,820
See you on Monday.