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IAIN STEWART: OK,
so let me remind
9
00:00:27,430 --> 00:00:30,350
you of what we were
talking about last time.
10
00:00:30,350 --> 00:00:32,878
So we were discussing
the example of DIS
11
00:00:32,878 --> 00:00:33,670
in the Breit frame.
12
00:00:33,670 --> 00:00:36,370
And the way we led
into this example
13
00:00:36,370 --> 00:00:39,160
is we talked about
renormalization group evolution
14
00:00:39,160 --> 00:00:40,285
with a heavy light current.
15
00:00:40,285 --> 00:00:43,570
And we saw that it had
this [INAUDIBLE] dimension.
16
00:00:43,570 --> 00:00:45,760
But it was a multiplicative
renormalization group
17
00:00:45,760 --> 00:00:48,970
evolution, and I said that
that happened because we only
18
00:00:48,970 --> 00:00:52,900
had one collinear
gauge-invariant
19
00:00:52,900 --> 00:00:54,710
object in our operator.
20
00:00:54,710 --> 00:00:57,280
And then I just wrote down an
operator that looked like this,
21
00:00:57,280 --> 00:00:59,470
and I said, there's
one that has two.
22
00:00:59,470 --> 00:01:03,520
If we run that object, we will
get a renormalization group
23
00:01:03,520 --> 00:01:05,813
equation that
involves convolutions.
24
00:01:05,813 --> 00:01:07,480
And I said that that's
going to give you
25
00:01:07,480 --> 00:01:09,438
the renormalization group
evolution of a parton
26
00:01:09,438 --> 00:01:11,050
distribution function.
27
00:01:11,050 --> 00:01:12,580
And we wanted to explore that.
28
00:01:12,580 --> 00:01:14,175
And so in order to
explore that, we
29
00:01:14,175 --> 00:01:15,550
should think of
some process that
30
00:01:15,550 --> 00:01:17,620
has the parton
distribution function in it
31
00:01:17,620 --> 00:01:19,630
so we can really
make sure we know
32
00:01:19,630 --> 00:01:21,790
precisely what the operator is.
33
00:01:21,790 --> 00:01:23,155
And that process is DIS.
34
00:01:23,155 --> 00:01:25,270
That's the simplest process.
35
00:01:25,270 --> 00:01:27,520
So we started thinking about
deep inelastic scattering
36
00:01:27,520 --> 00:01:29,950
in the Breit frame, which
is this framework, Q,
37
00:01:29,950 --> 00:01:30,550
of the photon.
38
00:01:30,550 --> 00:01:34,810
It has that form, just a
component in the z-direction.
39
00:01:34,810 --> 00:01:37,480
And in that frame,
the incoming quarks
40
00:01:37,480 --> 00:01:41,025
in the proton, quarks and
gluons, are collinear.
41
00:01:41,025 --> 00:01:42,550
Intermediate state,
the outstate,
42
00:01:42,550 --> 00:01:45,700
the x-state that's
going out, is hard.
43
00:01:45,700 --> 00:01:47,230
So you can think
of a-- if you were
44
00:01:47,230 --> 00:01:49,923
to draw perturbative diagrams,
you'd draw them like this.
45
00:01:49,923 --> 00:01:51,340
And this propagator
would be hard.
46
00:01:51,340 --> 00:01:52,780
It would have a hard momentum.
47
00:01:52,780 --> 00:01:54,700
And then you would
have loop corrections
48
00:01:54,700 --> 00:01:56,082
that could also be hard.
49
00:01:56,082 --> 00:01:57,790
In the effective
theory, you don't really
50
00:01:57,790 --> 00:01:59,710
have to think about
what diagrams.
51
00:01:59,710 --> 00:02:03,290
You just write down the lowest
possible dimension operator,
52
00:02:03,290 --> 00:02:06,040
and everything that's
a loop that's hard
53
00:02:06,040 --> 00:02:07,870
goes to the x,
which is the Wilson
54
00:02:07,870 --> 00:02:09,789
coefficient, if you like.
55
00:02:09,789 --> 00:02:13,420
And likewise, we also get
not just external quarks,
56
00:02:13,420 --> 00:02:15,640
but external gluons
from diagrams.
57
00:02:15,640 --> 00:02:19,180
In the full theory, it would
involve a quark loop like that.
58
00:02:19,180 --> 00:02:22,130
OK, so this is going to
lead to the quark PDF,
59
00:02:22,130 --> 00:02:24,700
and this is going to
lead to the gluon PDF.
60
00:02:24,700 --> 00:02:27,310
And we decided we would do
the quark one in some detail.
61
00:02:27,310 --> 00:02:29,860
So this is kind of writing
out now the operator
62
00:02:29,860 --> 00:02:32,920
and the Wilson coefficient in
kind of a combined notation
63
00:02:32,920 --> 00:02:43,380
where this w plus and minus
are w1 plus and minus w2.
64
00:02:43,380 --> 00:02:46,470
And then we had
one more formula,
65
00:02:46,470 --> 00:02:49,540
which is where we ended.
66
00:02:49,540 --> 00:02:52,500
So we have a collinear proton,
and then we have this operator.
67
00:03:06,310 --> 00:03:09,840
And then we have the
collinear proton again.
68
00:03:09,840 --> 00:03:16,440
And this matrix element
can be written as follows.
69
00:03:24,180 --> 00:03:26,740
So this is the last
formula we had last time.
70
00:03:29,990 --> 00:03:33,280
So some things here
are just conventions,
71
00:03:33,280 --> 00:03:37,165
but other things are important.
72
00:03:37,165 --> 00:03:39,040
Well, everything's
important, but some things
73
00:03:39,040 --> 00:03:40,582
are more important
than other things.
74
00:03:56,150 --> 00:03:58,770
So this quark here has a flavor.
75
00:03:58,770 --> 00:03:59,770
It could be an up quark.
76
00:03:59,770 --> 00:04:01,990
It could be down quark.
77
00:04:01,990 --> 00:04:03,910
Let me denote that
by an index i.
78
00:04:06,760 --> 00:04:08,590
This proton here is collinear.
79
00:04:08,590 --> 00:04:11,860
And really, all that
matters for this example
80
00:04:11,860 --> 00:04:12,970
is that we have some--
81
00:04:12,970 --> 00:04:16,390
we can think of it as a
massless proton, even.
82
00:04:16,390 --> 00:04:19,008
And as far as its
momentum is concerned,
83
00:04:19,008 --> 00:04:20,300
we can think of it as massless.
84
00:04:20,300 --> 00:04:22,630
So really, the only momentum
that matters in here
85
00:04:22,630 --> 00:04:24,950
is the minus momentum--
86
00:04:24,950 --> 00:04:33,306
so minus, which is n bar dot
P, and that's what this is--
87
00:04:33,306 --> 00:04:38,980
n bar dot P, n bar
dot P. It's capital P.
88
00:04:38,980 --> 00:04:40,730
So capital P was
the proton momentum.
89
00:04:40,730 --> 00:04:43,390
And we can think of this state
as just carrying large P minus.
90
00:04:43,390 --> 00:04:46,710
All the other components don't
matter for this matrix element.
91
00:04:46,710 --> 00:04:48,430
But it's a forward
matrix element,
92
00:04:48,430 --> 00:04:51,850
so both states carry
the same large momentum.
93
00:04:51,850 --> 00:04:53,710
And that's what led
to this delta function
94
00:04:53,710 --> 00:04:57,490
here that says that w1 and
w2, with the sign conventions
95
00:04:57,490 --> 00:04:59,920
we have, this guy has the
opposite sign convention.
96
00:04:59,920 --> 00:05:02,850
And so if it's
forward, these two guys
97
00:05:02,850 --> 00:05:07,660
have to have equal momentum
so that the sum is 0.
98
00:05:07,660 --> 00:05:09,640
And if you take into
account the sign,
99
00:05:09,640 --> 00:05:12,220
then that means w minus is 0.
100
00:05:12,220 --> 00:05:14,600
So that's what that
delta function is doing.
101
00:05:14,600 --> 00:05:17,650
And then the sum is something
that's not constrained
102
00:05:17,650 --> 00:05:19,550
by the matrix element.
103
00:05:19,550 --> 00:05:22,760
And so the sum could either
be positive or negative.
104
00:05:22,760 --> 00:05:25,390
If it's positive,
we can say that it's
105
00:05:25,390 --> 00:05:27,340
some fraction of
the proton momentum
106
00:05:27,340 --> 00:05:29,770
because this is a
quark inside the proton
107
00:05:29,770 --> 00:05:31,550
and it carries some
momentum, but it
108
00:05:31,550 --> 00:05:32,800
can't be more than the proton.
109
00:05:32,800 --> 00:05:35,560
Otherwise, we would get 0.
110
00:05:35,560 --> 00:05:37,360
So it's some fraction,
and that fraction
111
00:05:37,360 --> 00:05:40,422
is defined to be
xi in this formula.
112
00:05:40,422 --> 00:05:42,130
And the reason there's
a 2 is because I'm
113
00:05:42,130 --> 00:05:46,990
adding the w1 and the
w2, which are equal.
114
00:05:46,990 --> 00:05:49,260
So this is the
momentum fraction.
115
00:05:49,260 --> 00:05:51,010
And then we can have
an arbitrary function
116
00:05:51,010 --> 00:05:52,330
of that momentum fraction.
117
00:05:52,330 --> 00:05:54,160
Nothing stops us from
writing that down.
118
00:05:57,310 --> 00:05:59,170
And that's kind of
where we got to.
119
00:06:01,870 --> 00:06:07,553
Now, so on general
grounds, you can
120
00:06:07,553 --> 00:06:09,220
argue that that's the
most general thing
121
00:06:09,220 --> 00:06:11,725
that you can write down
for this matrix element.
122
00:06:11,725 --> 00:06:17,200
And I tried to argue
about why that's true.
123
00:06:17,200 --> 00:06:19,900
From charge conjugation, you
can actually do something more.
124
00:06:19,900 --> 00:06:21,610
So you can let charge
conjugation act
125
00:06:21,610 --> 00:06:23,260
on these operators.
126
00:06:23,260 --> 00:06:28,870
And since charge conjugation's
a good symmetry of QCD,
127
00:06:28,870 --> 00:06:31,240
you can prove that
that relates, actually,
128
00:06:31,240 --> 00:06:37,480
the quark and antiquark
operators in the following way.
129
00:06:42,410 --> 00:06:46,620
So quark and antiquark operators
are switching signs of w plus--
130
00:06:46,620 --> 00:06:48,190
so if I switch
the sign of w plus
131
00:06:48,190 --> 00:06:50,470
that's going from
quark to antiquark.
132
00:06:50,470 --> 00:06:54,580
And basically what happens
in the operator when
133
00:06:54,580 --> 00:06:56,080
you do charge
conjugation, remember
134
00:06:56,080 --> 00:06:59,890
that [? chi ?] goes over
here to [? chi ?] transpose
135
00:06:59,890 --> 00:07:02,110
and the w switches sign.
136
00:07:02,110 --> 00:07:05,320
So basically, what charge
conjugation is doing
137
00:07:05,320 --> 00:07:11,650
is taking w1 to minus
w2 and w2 to minus w1.
138
00:07:11,650 --> 00:07:14,380
And that's why the w plus,
which is signed by the w minus,
139
00:07:14,380 --> 00:07:15,220
doesn't.
140
00:07:15,220 --> 00:07:18,445
And then there's an overall
sign just from the fields--
141
00:07:21,502 --> 00:07:23,560
so from the usual
charge conjugation
142
00:07:23,560 --> 00:07:27,100
transformation of the
fields for a vector current.
143
00:07:27,100 --> 00:07:30,790
OK, so these are all orders
of relation between the Wilson
144
00:07:30,790 --> 00:07:32,080
coefficients.
145
00:07:32,080 --> 00:07:33,580
So really, when you
do the matching,
146
00:07:33,580 --> 00:07:35,872
you really only need to do
the matching for the quarks.
147
00:07:44,293 --> 00:07:46,210
So if you want to do the
matching calculation,
148
00:07:46,210 --> 00:07:48,627
you'd do a matching calculation
for the Wilson coefficient
149
00:07:48,627 --> 00:07:49,780
with positive w plus.
150
00:07:55,600 --> 00:08:01,370
And you could do it for
the antiquarks, as well,
151
00:08:01,370 --> 00:08:05,020
but you would just be
basically wasting time.
152
00:08:10,690 --> 00:08:13,440
Now, last time, we went through
the kinematics of the Breit
153
00:08:13,440 --> 00:08:14,580
frame a little bit.
154
00:08:14,580 --> 00:08:18,090
And this n bar dot proton
momentum is actually Q over x.
155
00:08:18,090 --> 00:08:20,920
So we could also write
this formula like that.
156
00:08:20,920 --> 00:08:22,740
And so you see that
w plus is actually
157
00:08:22,740 --> 00:08:27,000
something that's [? xi ?]
over x, Bjorken x, which is
158
00:08:27,000 --> 00:08:28,680
an external leptonic variable.
159
00:08:31,410 --> 00:08:34,950
Now, there was another
index over here, j, which
160
00:08:34,950 --> 00:08:36,059
we talked about last time.
161
00:08:36,059 --> 00:08:37,851
And that had to do with
the fact that we're
162
00:08:37,851 --> 00:08:42,270
taking the forward scattering
graphs with a tensor
163
00:08:42,270 --> 00:08:47,550
and we decompose that
into two scalars,
164
00:08:47,550 --> 00:08:50,015
multiplying things
that had indices.
165
00:08:50,015 --> 00:08:51,390
So there were two
possible things
166
00:08:51,390 --> 00:08:53,760
that we could write down.
167
00:08:53,760 --> 00:09:00,270
And the index j is
just this 1 or 2.
168
00:09:00,270 --> 00:09:02,100
And there was a
similar decomposition.
169
00:09:02,100 --> 00:09:05,310
In the effective theory, we
could think of decomposing--
170
00:09:05,310 --> 00:09:08,130
the effective theory
is a scalar in terms
171
00:09:08,130 --> 00:09:09,630
of the scalar
operators, which are
172
00:09:09,630 --> 00:09:12,210
these guys, with some
coefficients that
173
00:09:12,210 --> 00:09:15,420
have some indices, then
multiplied by some tensor.
174
00:09:15,420 --> 00:09:16,830
So these guys
don't have indices,
175
00:09:16,830 --> 00:09:19,770
but I could just multiply
them by effectively
176
00:09:19,770 --> 00:09:22,680
the effective theory
versions of these tensors.
177
00:09:22,680 --> 00:09:26,130
And so that that's
why there's a j here.
178
00:09:26,130 --> 00:09:29,630
Is that clear to everybody?
179
00:09:29,630 --> 00:09:32,270
OK, so there's various
indices-- i flavor,
180
00:09:32,270 --> 00:09:35,540
j for tensor decomposition,
and then a bunch
181
00:09:35,540 --> 00:09:40,300
of momentum indices.
182
00:09:40,300 --> 00:09:43,480
So when you go
through the analysis
183
00:09:43,480 --> 00:09:47,890
of trying to find a
formula for, say, T1,
184
00:09:47,890 --> 00:09:49,840
it's going to be related to C1.
185
00:09:49,840 --> 00:09:52,768
And T2 will be
related to C2, OK?
186
00:09:52,768 --> 00:09:54,310
Because this guy's
a scalar operator.
187
00:09:54,310 --> 00:09:57,310
It doesn't have any indices.
188
00:09:57,310 --> 00:10:05,140
So the way that that works, if
you just look at the two bases
189
00:10:05,140 --> 00:10:14,800
and write down the formula,
you'd have an integral over
190
00:10:14,800 --> 00:10:15,310
these w's.
191
00:10:18,050 --> 00:10:20,830
There are some prefactors
which just come about
192
00:10:20,830 --> 00:10:24,820
from being careful,
and then the thing that
193
00:10:24,820 --> 00:10:33,105
has an imaginary part of
these Wilson coefficients,
194
00:10:33,105 --> 00:10:34,480
and then you have
matrix elements
195
00:10:34,480 --> 00:10:43,120
of operators, which
have a flavor index
196
00:10:43,120 --> 00:10:46,600
but don't have a subscript j.
197
00:10:46,600 --> 00:10:48,340
So in general, this
has a flavor index.
198
00:10:52,750 --> 00:10:53,650
Keep track of things.
199
00:10:56,570 --> 00:10:59,210
And then there's
another one for T2--
200
00:11:08,667 --> 00:11:13,678
so kinematic prefactors that
are easy to work out that
201
00:11:13,678 --> 00:11:15,220
just come about from
the fact that we
202
00:11:15,220 --> 00:11:17,050
wrote the tensors and
the effective theory
203
00:11:17,050 --> 00:11:19,067
and the full theory
slightly differently.
204
00:11:32,630 --> 00:11:37,640
But these two guys have
the same matrix elements
205
00:11:37,640 --> 00:11:38,790
of the same operator.
206
00:11:38,790 --> 00:11:41,402
And all the sort of
tensor stuff is just
207
00:11:41,402 --> 00:11:43,610
saying that there's two
different Wilson coefficients
208
00:11:43,610 --> 00:11:46,572
that you have to
compute, and that's
209
00:11:46,572 --> 00:11:48,905
because you have these vector
currents from the photons.
210
00:11:52,750 --> 00:11:54,125
OK, so this is what we're after.
211
00:11:58,740 --> 00:12:03,000
These show up in
the cross-section.
212
00:12:03,000 --> 00:12:06,192
And what we're doing is we're
writing, at lowest order,
213
00:12:06,192 --> 00:12:08,400
the things that show up in
the cross-section in terms
214
00:12:08,400 --> 00:12:11,055
of effective theory objects,
the Wilson coefficients and then
215
00:12:11,055 --> 00:12:13,530
the matrix elements
of our operators
216
00:12:13,530 --> 00:12:17,250
here, which is this
thing in square brackets.
217
00:12:17,250 --> 00:12:20,490
And we're almost at what you
would call a factorization
218
00:12:20,490 --> 00:12:22,080
theorem.
219
00:12:22,080 --> 00:12:26,730
Factorization theorem is a
result for the cross-section,
220
00:12:26,730 --> 00:12:29,500
in our language, in terms of
effective theory quantities,
221
00:12:29,500 --> 00:12:31,950
and that's going to factor
the hard stuff, which
222
00:12:31,950 --> 00:12:38,130
is the pink stuff, which is
in these Wilson coefficients,
223
00:12:38,130 --> 00:12:41,372
from the low-energy stuff,
which is in these operators.
224
00:12:41,372 --> 00:12:44,640
AUDIENCE: So those pink elements
are [INAUDIBLE] equation.
225
00:12:44,640 --> 00:12:45,960
IAIN STEWART: Yeah.
226
00:12:45,960 --> 00:12:49,820
AUDIENCE: And what is
square bracket [INAUDIBLE]??
227
00:12:49,820 --> 00:12:50,820
IAIN STEWART: It's both.
228
00:12:50,820 --> 00:12:51,820
AUDIENCE: It's both, OK.
229
00:12:51,820 --> 00:12:53,500
IAIN STEWART: Yeah.
230
00:12:53,500 --> 00:12:55,290
I can write it like
this, if you like.
231
00:12:58,290 --> 00:13:01,050
Any other questions?
232
00:13:01,050 --> 00:13:06,030
OK, so literally what I
do is I take this formula
233
00:13:06,030 --> 00:13:08,402
and I plug it into that formula.
234
00:13:08,402 --> 00:13:10,860
And when I do that, I can do
one of the integrals trivially
235
00:13:10,860 --> 00:13:13,740
because it's a delta function.
236
00:13:13,740 --> 00:13:15,030
This one's just trivial.
237
00:13:15,030 --> 00:13:18,220
And then I do this one with
the other delta function.
238
00:13:18,220 --> 00:13:20,880
So both integrals
are actually trivial.
239
00:13:20,880 --> 00:13:25,080
And I can write the result
in terms of something
240
00:13:25,080 --> 00:13:33,300
that I'll call the
hard function, which
241
00:13:33,300 --> 00:13:35,881
is just the imaginary part
of the Wilson coefficient.
242
00:13:50,200 --> 00:13:53,120
And I'm going to denote
it in the following way.
243
00:13:53,120 --> 00:13:54,145
So this is w--
244
00:13:54,145 --> 00:13:55,733
the Wilson cost
efficient can depend
245
00:13:55,733 --> 00:13:56,900
on various different things.
246
00:13:56,900 --> 00:13:58,720
It can depend on w plus.
247
00:13:58,720 --> 00:14:00,490
It can depend on w minus.
248
00:14:00,490 --> 00:14:03,010
It can depend on the hard
scale, which is q squared.
249
00:14:03,010 --> 00:14:05,570
Or it could depend
on mu squared.
250
00:14:05,570 --> 00:14:08,740
So w minus, when you do the
delta function, gets set to 0.
251
00:14:08,740 --> 00:14:10,720
w plus gets set to something.
252
00:14:10,720 --> 00:14:13,480
And it's convenient
because of the way
253
00:14:13,480 --> 00:14:16,660
this delta function is with
the-- it kind of has a ratio.
254
00:14:16,660 --> 00:14:18,550
Because this function
is a function of xi
255
00:14:18,550 --> 00:14:20,810
which is the ratio
of two things,
256
00:14:20,810 --> 00:14:24,580
it's convenient to
define a dimensionless z
257
00:14:24,580 --> 00:14:28,570
and only talk about a function
of that dimensionless thing.
258
00:14:28,570 --> 00:14:32,560
And if you do that, then the
final result for these kind
259
00:14:32,560 --> 00:14:35,680
of T's, which are
imaginary parts of T's--
260
00:14:46,820 --> 00:14:49,720
you can just put the
formula together.
261
00:14:49,720 --> 00:14:50,720
I'll write one of them--
262
00:15:03,750 --> 00:15:04,340
is that.
263
00:15:08,200 --> 00:15:15,310
And then there's a
similar formula for in T2
264
00:15:15,310 --> 00:15:19,090
that involves H2 and H1.
265
00:15:19,090 --> 00:15:21,520
OK, so this is the
factorization theorem.
266
00:15:21,520 --> 00:15:24,280
And it came about, in some
sense, just trivially.
267
00:15:24,280 --> 00:15:26,350
Once we knew how to
write down the operators
268
00:15:26,350 --> 00:15:28,780
in the effective theory,
we were basically done,
269
00:15:28,780 --> 00:15:32,020
and then the rest was just sort
of algebraic manipulations,
270
00:15:32,020 --> 00:15:35,050
being careful about
what momenta go where,
271
00:15:35,050 --> 00:15:40,060
and knowing what the sign of
this formula for the matrix
272
00:15:40,060 --> 00:15:41,650
element is.
273
00:15:41,650 --> 00:15:44,470
This is a kind of
important point.
274
00:15:44,470 --> 00:15:47,140
But in some sense, the effective
theory, from the get-go,
275
00:15:47,140 --> 00:15:48,940
was already designed
to factorize
276
00:15:48,940 --> 00:15:51,398
because we were integrating
out the hard degrees of freedom
277
00:15:51,398 --> 00:15:52,370
right at the start.
278
00:15:52,370 --> 00:15:56,850
And so knowing what operators
and knowing their matrix
279
00:15:56,850 --> 00:16:00,520
element is really all we
needed to do to get to the DIS
280
00:16:00,520 --> 00:16:01,510
factorization theorem.
281
00:16:08,680 --> 00:16:12,110
So if you ever look
up the original way
282
00:16:12,110 --> 00:16:15,500
that this was derived,
it was not that easy.
283
00:16:15,500 --> 00:16:17,420
This is actually
something that's
284
00:16:17,420 --> 00:16:21,770
very complicated in a sort
of traditional approach.
285
00:16:21,770 --> 00:16:23,880
But in the effective
theory approach,
286
00:16:23,880 --> 00:16:27,680
it becomes almost trivial.
287
00:16:27,680 --> 00:16:29,810
And this is an all-orders
result because we never
288
00:16:29,810 --> 00:16:31,520
expanded in alpha s.
289
00:16:31,520 --> 00:16:34,120
We just used symmetries,
and we used the fact
290
00:16:34,120 --> 00:16:35,870
that we knew what form
the operators would
291
00:16:35,870 --> 00:16:38,203
take when we integrated out
the hard degrees of freedom.
292
00:16:45,000 --> 00:16:48,410
So any alpha s corrections
that one might want to add
293
00:16:48,410 --> 00:16:49,535
will fit into this formula.
294
00:16:52,220 --> 00:16:55,700
And this gives a perturbative
result for this H,
295
00:16:55,700 --> 00:16:59,060
which you would compute in
perturbation theory, which
296
00:16:59,060 --> 00:17:01,730
people do [INAUDIBLE]
these days.
297
00:17:07,050 --> 00:17:08,579
Now, if you ask
about things like--
298
00:17:08,579 --> 00:17:11,010
I didn't write
all the possible--
299
00:17:11,010 --> 00:17:14,380
I suppressed some things, right,
like Q squared and mu squared.
300
00:17:14,380 --> 00:17:16,500
If you ask about the Q
squared and the mu squared,
301
00:17:16,500 --> 00:17:19,020
then your Wilson coefficients
do depend on Q squared
302
00:17:19,020 --> 00:17:20,430
and mu squared.
303
00:17:20,430 --> 00:17:22,050
And the Wilson
coefficients, H here,
304
00:17:22,050 --> 00:17:24,092
are actually dimension--
the Wilson coefficients,
305
00:17:24,092 --> 00:17:26,135
the original one, [? xi, ?]
were dimensionless.
306
00:17:30,020 --> 00:17:31,737
So the H is dimensionless.
307
00:17:35,940 --> 00:17:37,995
I just pulled out the
dimensionable factor
308
00:17:37,995 --> 00:17:40,270
so that that would be true.
309
00:17:40,270 --> 00:17:43,320
And so this guy can depend
on Q squared over mu squared.
310
00:17:43,320 --> 00:17:45,240
The fact that Q squared
only shows up there,
311
00:17:45,240 --> 00:17:47,700
that's Bjorken scaling.
312
00:17:47,700 --> 00:17:51,630
And if you look at the
perturbative result for T2,
313
00:17:51,630 --> 00:17:53,950
then it vanishes
at lowest order.
314
00:17:53,950 --> 00:17:58,950
And so that's the
Callan-Gross relation.
315
00:17:58,950 --> 00:18:01,260
So there's various things
that are sort of encoded
316
00:18:01,260 --> 00:18:06,780
in this that come out, from the
effective theory point of view,
317
00:18:06,780 --> 00:18:10,380
in a very simple way.
318
00:18:10,380 --> 00:18:11,330
OK, so let me write.
319
00:18:32,380 --> 00:18:33,910
So there's logarithmic
corrections
320
00:18:33,910 --> 00:18:39,940
that involve Q in the
Wilson coefficients
321
00:18:39,940 --> 00:18:41,500
that will show up like that.
322
00:18:46,240 --> 00:18:50,360
So there's a mu also that you
could add to this formula.
323
00:18:50,360 --> 00:18:52,810
So the way that I
described it, we
324
00:18:52,810 --> 00:18:58,210
didn't think too hard about
bare versus normalized, right?
325
00:18:58,210 --> 00:18:59,770
We just take these operators.
326
00:18:59,770 --> 00:19:02,110
So far, they could
have been bare.
327
00:19:02,110 --> 00:19:09,100
But remember that when you have
C bare, O bare in [INAUDIBLE]
328
00:19:09,100 --> 00:19:12,090
Hamiltonian, for example,
that's C mu, O mu.
329
00:19:15,690 --> 00:19:17,635
So switching from bare
and renormalized--
330
00:19:17,635 --> 00:19:19,840
I mean bare operators
and coefficients
331
00:19:19,840 --> 00:19:21,880
to renormalized operators
and coefficients
332
00:19:21,880 --> 00:19:24,145
is simply a matter of
sticking in a mu here,
333
00:19:24,145 --> 00:19:26,020
and then you imagine
that the renormalization
334
00:19:26,020 --> 00:19:28,640
has taking place.
335
00:19:28,640 --> 00:19:32,680
So we could equally well
insert in these formulas
336
00:19:32,680 --> 00:19:35,290
a mu for that.
337
00:19:38,810 --> 00:19:41,725
And then what I'm saying is that
there being logs of mu over Q
338
00:19:41,725 --> 00:19:44,720
will make a little more sense.
339
00:19:44,720 --> 00:19:53,540
So there's also a Q. Squeeze
everything in here [INAUDIBLE]..
340
00:19:53,540 --> 00:19:56,840
OK, now we're being
completely honest
341
00:19:56,840 --> 00:19:59,880
about what it depends on.
342
00:19:59,880 --> 00:20:01,320
All right.
343
00:20:01,320 --> 00:20:08,640
So traditionally what happens
in the traditional literature,
344
00:20:08,640 --> 00:20:11,190
people talk about factorization
scales and renormalization
345
00:20:11,190 --> 00:20:12,720
group scales.
346
00:20:12,720 --> 00:20:15,240
So factorization
scales is the fact
347
00:20:15,240 --> 00:20:18,307
that this parton distribution
function is mu-dependent-- so
348
00:20:18,307 --> 00:20:19,890
operator that you
have to renormalize,
349
00:20:19,890 --> 00:20:22,270
and we're going to
do that in a minute.
350
00:20:22,270 --> 00:20:24,210
And so there has to
be a cancellation.
351
00:20:24,210 --> 00:20:26,370
Since this thing here
is a physical observable
352
00:20:26,370 --> 00:20:28,230
and is independent
of mu, there has
353
00:20:28,230 --> 00:20:30,180
to be a cancellation of
the mu-dependence here
354
00:20:30,180 --> 00:20:33,570
and the mu-dependence
here, all right?
355
00:20:33,570 --> 00:20:35,680
And that's this mu.
356
00:20:35,680 --> 00:20:39,330
So the thing that's
multiplying this result here
357
00:20:39,330 --> 00:20:41,310
would involve a cancellation
of mu-dependence
358
00:20:41,310 --> 00:20:42,477
here and mu-dependence here.
359
00:20:42,477 --> 00:20:44,490
So the same anomalous
dimension would show up
360
00:20:44,490 --> 00:20:46,470
in both the H and the f.
361
00:20:46,470 --> 00:20:50,520
And then sometimes people
also talk about mu-dependence
362
00:20:50,520 --> 00:20:53,640
that's just cancelling
within H itself.
363
00:20:53,640 --> 00:20:56,460
And they call that
renormalization group
364
00:20:56,460 --> 00:20:58,740
renormalization group mu.
365
00:20:58,740 --> 00:21:00,720
Sometimes people vary
these independently.
366
00:21:00,720 --> 00:21:03,160
In the effective theory,
it's really simple.
367
00:21:03,160 --> 00:21:05,340
You really just have
the classic setup of you
368
00:21:05,340 --> 00:21:08,310
have some hard
degrees of freedom.
369
00:21:08,310 --> 00:21:11,808
In this case, you can even
think of it one-dimensional.
370
00:21:11,808 --> 00:21:13,350
You have some hard
degrees of freedom
371
00:21:13,350 --> 00:21:14,642
that you want to integrate out.
372
00:21:14,642 --> 00:21:16,830
You have some scale
which we could
373
00:21:16,830 --> 00:21:18,750
call mu 0 that's
of order 2 where
374
00:21:18,750 --> 00:21:20,340
we do that integrating out.
375
00:21:20,340 --> 00:21:26,640
And then you can run
down or you could
376
00:21:26,640 --> 00:21:28,680
run in a more complicated way.
377
00:21:28,680 --> 00:21:32,070
So you could run the PDFs
which are sitting here
378
00:21:32,070 --> 00:21:38,398
at the collinear scale,
which is lambda QCD.
379
00:21:38,398 --> 00:21:40,440
You could think of evolving
them up to some scale
380
00:21:40,440 --> 00:21:42,990
and evolving the Wilson
coefficients down and meeting
381
00:21:42,990 --> 00:21:46,110
somewhere, OK?
382
00:21:46,110 --> 00:21:49,890
And so, yeah, it's just really
a sort of classic running
383
00:21:49,890 --> 00:21:51,720
and matching picture.
384
00:21:51,720 --> 00:21:53,790
Here, I've just
used the fact that I
385
00:21:53,790 --> 00:21:56,130
could run either one of them
or I could run both of them
386
00:21:56,130 --> 00:21:57,960
to a common scale.
387
00:21:57,960 --> 00:22:00,600
So I usually would pick mu
to be either something small
388
00:22:00,600 --> 00:22:04,803
or something large rather
than running both things.
389
00:22:04,803 --> 00:22:07,220
But in general, you could think
about running both things.
390
00:22:07,220 --> 00:22:09,480
And we've talked about
having anomalous dimensions
391
00:22:09,480 --> 00:22:12,180
for either one of these.
392
00:22:12,180 --> 00:22:15,180
And usually, we just
run one of them, OK?
393
00:22:15,180 --> 00:22:17,430
But it's no more complicated
than the standard picture
394
00:22:17,430 --> 00:22:19,770
of integrating out modes and
doing renormalization group
395
00:22:19,770 --> 00:22:20,816
evolution.
396
00:22:24,150 --> 00:22:27,870
So if we want to do tree-level
matching or one-loop matching
397
00:22:27,870 --> 00:22:31,112
or any kind of matching--
398
00:22:31,112 --> 00:22:32,820
let me just show you
tree-level matching.
399
00:22:39,785 --> 00:22:41,160
So tree-level
matching, you would
400
00:22:41,160 --> 00:22:47,120
compute this forward
scattering graph,
401
00:22:47,120 --> 00:22:51,770
and that will give you the
other diagram that we drew.
402
00:22:51,770 --> 00:22:55,280
And so you'd want to match
this guy onto that guy.
403
00:22:58,060 --> 00:23:06,988
And what you find is you
find one tensor structure
404
00:23:06,988 --> 00:23:08,900
at lowest order.
405
00:23:08,900 --> 00:23:11,610
So C1 is not equal to 0.
406
00:23:11,610 --> 00:23:13,010
C2 is equal to 0.
407
00:23:13,010 --> 00:23:16,730
And that's the
Callan-Gross relation
408
00:23:16,730 --> 00:23:19,340
which tells you about
the spin of the object
409
00:23:19,340 --> 00:23:20,840
that you're scattering
off, and this
410
00:23:20,840 --> 00:23:27,380
is how we know that quarks are
spin 1/2, or one way we know.
411
00:23:27,380 --> 00:23:33,620
And then you can calculate
C. And so that way
412
00:23:33,620 --> 00:23:35,660
that I set things
up, C was complex,
413
00:23:35,660 --> 00:23:37,740
and then I had to take
the imaginary part.
414
00:23:37,740 --> 00:23:41,240
So C is just this
propagator, basically.
415
00:23:41,240 --> 00:23:44,550
And it's only a nontrivial
function of w plus.
416
00:23:44,550 --> 00:23:49,100
There are some charges
that sit out front.
417
00:23:49,100 --> 00:23:51,363
And so the only way
that this guy depends on
418
00:23:51,363 --> 00:23:53,030
whether it's an up
quark or a down quark
419
00:23:53,030 --> 00:23:55,100
is you have 2/3
squared or 1/3 squared.
420
00:24:03,260 --> 00:24:07,280
And then there's something that
comes about from the propagator
421
00:24:07,280 --> 00:24:10,210
that looks like this.
422
00:24:10,210 --> 00:24:14,520
And then I take
the imaginary part
423
00:24:14,520 --> 00:24:21,650
and then I get H1, which
is a function of z, which
424
00:24:21,650 --> 00:24:23,600
is the xi over x.
425
00:24:23,600 --> 00:24:25,910
So if I write it as xi
over x like it shows up
426
00:24:25,910 --> 00:24:28,370
in the factorization
theorem, then I'm
427
00:24:28,370 --> 00:24:31,278
getting a delta
function of xi over x,
428
00:24:31,278 --> 00:24:32,570
which is this coming from this.
429
00:24:38,230 --> 00:24:41,900
So the lowest-order H1
is just a delta function.
430
00:24:41,900 --> 00:24:44,230
And that's where the
parton model picture
431
00:24:44,230 --> 00:24:46,660
comes from because
the parton model
432
00:24:46,660 --> 00:24:53,780
picture is that you think of xi
and x as being the same thing.
433
00:24:53,780 --> 00:24:56,653
And that's the tree-level
way of thinking,
434
00:24:56,653 --> 00:24:58,570
and that's just satisfying
this delta function
435
00:24:58,570 --> 00:25:00,640
and the hard function.
436
00:25:00,640 --> 00:25:02,860
And then you would
get that the T is just
437
00:25:02,860 --> 00:25:05,200
given by the parton
distribution at x, which is
438
00:25:05,200 --> 00:25:09,570
the external measurable thing.
439
00:25:09,570 --> 00:25:10,070
OK?
440
00:25:10,070 --> 00:25:12,480
So this is how all these
classic things come about
441
00:25:12,480 --> 00:25:16,000
in the effective
theory language.
442
00:25:16,000 --> 00:25:17,170
Any questions about that?
443
00:25:24,600 --> 00:25:28,840
All right, so let's
renormalize this operator
444
00:25:28,840 --> 00:25:32,250
and see how the classic result
for the renormalization group
445
00:25:32,250 --> 00:25:35,940
evolution of a PDF comes about.
446
00:25:35,940 --> 00:25:38,730
And again, the way that
you should think about this
447
00:25:38,730 --> 00:25:44,240
is you have an operator, and
you should just renormalize it.
448
00:25:44,240 --> 00:25:47,480
And once you've got
the effective theory,
449
00:25:47,480 --> 00:25:50,188
you shouldn't have to think too
deeply about what you're doing.
450
00:25:50,188 --> 00:25:51,980
You should just be able
to follow your nose
451
00:25:51,980 --> 00:25:53,360
and do the renormalization.
452
00:25:53,360 --> 00:25:55,880
You may have to be careful
because these operators are
453
00:25:55,880 --> 00:25:57,750
kind of complicated.
454
00:25:57,750 --> 00:25:59,743
They have this
dependence on these w's
455
00:25:59,743 --> 00:26:01,160
that you have to
be careful about.
456
00:26:01,160 --> 00:26:03,950
But really, it's just
follow your nose,
457
00:26:03,950 --> 00:26:05,510
compute the one-loop graphs.
458
00:26:20,035 --> 00:26:24,210
If you look up how Peskin would
do one-loop renormalization,
459
00:26:24,210 --> 00:26:25,960
there'd be an infinite
number of operators
460
00:26:25,960 --> 00:26:29,080
you'd have to derive in a
renormalization group, result
461
00:26:29,080 --> 00:26:29,920
for all of them.
462
00:26:29,920 --> 00:26:31,930
Here, we only have one
operator and we're just
463
00:26:31,930 --> 00:26:34,890
going to renormalize it.
464
00:26:34,890 --> 00:26:36,850
Our operator is
nonlocal in the sense
465
00:26:36,850 --> 00:26:38,770
that it depends on
these omegas, and that's
466
00:26:38,770 --> 00:26:42,190
what's encoding this
infinite number of operators
467
00:26:42,190 --> 00:26:43,150
that Peskin has.
468
00:27:03,620 --> 00:27:09,860
OK, so solving, if you
like, for f from the formula
469
00:27:09,860 --> 00:27:11,990
that we had before,
I can do that
470
00:27:11,990 --> 00:27:16,550
by integrating over the w minus.
471
00:27:16,550 --> 00:27:18,980
That sets these
guys to be equal.
472
00:27:18,980 --> 00:27:20,330
And then if I--
473
00:27:20,330 --> 00:27:24,920
so I can think of it as that
there's one free momentum, xi.
474
00:27:24,920 --> 00:27:27,410
And that free momentum xi
is one of these labels,
475
00:27:27,410 --> 00:27:29,220
which is this guy here.
476
00:27:29,220 --> 00:27:34,310
So xi is w over
Pn minus, and this
477
00:27:34,310 --> 00:27:38,690
is the proton which is carrying
some momentum Pn minus.
478
00:27:48,770 --> 00:27:52,670
This is the proton state, which
carries momentum Pn minus.
479
00:27:52,670 --> 00:27:54,080
And there's one delta function.
480
00:27:56,900 --> 00:27:58,550
I could put it either
place, but I only
481
00:27:58,550 --> 00:28:01,518
need one because the other
one's kind of trivial.
482
00:28:01,518 --> 00:28:03,560
So the first thing you
can think about doing here
483
00:28:03,560 --> 00:28:05,120
is looking at mass dimensions.
484
00:28:05,120 --> 00:28:07,010
And I already told you that
this guy was dimensionless,
485
00:28:07,010 --> 00:28:08,427
but let's check
that that's true--
486
00:28:10,830 --> 00:28:14,790
so a mass dimension.
487
00:28:14,790 --> 00:28:17,540
So relativistically
normalized states
488
00:28:17,540 --> 00:28:20,370
have mass dimension minus 1.
489
00:28:20,370 --> 00:28:22,890
Quark fields that don't
have a delta function
490
00:28:22,890 --> 00:28:26,310
have mass dimension 3/2.
491
00:28:26,310 --> 00:28:31,150
The delta function gives a
minus 1, and then a minus 1,
492
00:28:31,150 --> 00:28:33,660
so you get 0.
493
00:28:33,660 --> 00:28:37,560
3/2 plus 3/2 minus 3/1 is 0.
494
00:28:37,560 --> 00:28:40,033
So that means this f is a
really dimensionless function,
495
00:28:40,033 --> 00:28:42,450
and that's why it makes sense
that we defined it to depend
496
00:28:42,450 --> 00:28:45,030
on this dimensionless ratio.
497
00:28:45,030 --> 00:28:50,880
You can also look at
the lambda dimension,
498
00:28:50,880 --> 00:28:51,990
and here's how that works.
499
00:28:59,730 --> 00:29:00,810
That's also 0.
500
00:29:04,240 --> 00:29:07,540
So the only thing that's-- so
we already had power counting
501
00:29:07,540 --> 00:29:08,540
for our kai fields.
502
00:29:08,540 --> 00:29:10,630
Remember, the c field
inside the chi field scale
503
00:29:10,630 --> 00:29:15,070
like [? 1. ?] So this is just
coming about because this guy's
504
00:29:15,070 --> 00:29:17,260
order lambda.
505
00:29:17,260 --> 00:29:19,878
The delta function just
involves large momentum,
506
00:29:19,878 --> 00:29:21,045
so it has no power counting.
507
00:29:27,940 --> 00:29:29,440
And the only thing
that's nontrivial
508
00:29:29,440 --> 00:29:33,010
is that the states have
power counting minus 1.
509
00:29:33,010 --> 00:29:36,200
So here's how we
can derive that.
510
00:29:36,200 --> 00:29:38,410
So if you think about
relativistically normalized
511
00:29:38,410 --> 00:29:41,050
states, what you're
doing is you're
512
00:29:41,050 --> 00:29:44,050
defining sort of the
inverse of this d3 p
513
00:29:44,050 --> 00:29:47,920
over e, which you
can write actually,
514
00:29:47,920 --> 00:29:50,680
which is more convenient
for power counting,
515
00:29:50,680 --> 00:29:58,270
in terms of things that we
can power count more simply.
516
00:29:58,270 --> 00:30:02,320
So this is an exact relation
for a nontrivial particle
517
00:30:02,320 --> 00:30:06,760
between p minuses and pz.
518
00:30:06,760 --> 00:30:10,090
So then I can write,
because of that,
519
00:30:10,090 --> 00:30:12,190
the standard relativistic
normalization
520
00:30:12,190 --> 00:30:17,204
formula for a state with
two different momenta
521
00:30:17,204 --> 00:30:27,970
as kind of the inverse,
which would be this.
522
00:30:27,970 --> 00:30:33,645
So the usual formula would have
2e and then delta 3, right?
523
00:30:33,645 --> 00:30:35,020
Because it's the
inverse of this.
524
00:30:35,020 --> 00:30:38,230
But I can write
it also this way.
525
00:30:38,230 --> 00:30:40,720
This guy is lambda 0.
526
00:30:40,720 --> 00:30:42,730
This guy is lambda minus 2.
527
00:30:42,730 --> 00:30:47,292
Therefore, each of these
guys must be lambda minus 1.
528
00:30:47,292 --> 00:30:48,750
That's where the
minus 1 came from.
529
00:30:52,710 --> 00:30:56,070
All right, so we want to
renormalize that thing,
530
00:30:56,070 --> 00:30:58,140
that matrix element.
531
00:30:58,140 --> 00:31:05,850
And what loops can do is
that they can change omega.
532
00:31:05,850 --> 00:31:11,457
So you might-- or xi,
which are equivalent.
533
00:31:11,457 --> 00:31:13,290
And so the way that you
should think of that
534
00:31:13,290 --> 00:31:15,330
is in the following
sense, and it's actually
535
00:31:15,330 --> 00:31:18,270
something you're familiar
with, although you're
536
00:31:18,270 --> 00:31:20,200
familiar with it for
discrete quantum numbers.
537
00:31:20,200 --> 00:31:23,670
And here, in some sense,
we have a continuous one.
538
00:31:23,670 --> 00:31:26,040
So you have some
function, fq, that
539
00:31:26,040 --> 00:31:27,960
depends on some variable xi.
540
00:31:27,960 --> 00:31:30,810
And it can mix, under the
renormalization group,
541
00:31:30,810 --> 00:31:35,712
with an operator at a
different value of xi.
542
00:31:35,712 --> 00:31:37,170
So you can really
think of the fact
543
00:31:37,170 --> 00:31:40,785
that loops can change this
omega as just a mixing.
544
00:31:40,785 --> 00:31:42,910
You're used to mixing for
discrete quantum numbers.
545
00:31:42,910 --> 00:31:45,510
You write down all the operators
that have the same quantum
546
00:31:45,510 --> 00:31:47,850
numbers, and they can mix
under renormalization.
547
00:31:47,850 --> 00:31:50,010
Here, there's kind of
an additional thing
548
00:31:50,010 --> 00:31:53,490
that the object can depend
on, which is the xi parameter.
549
00:31:53,490 --> 00:31:55,560
And in general, when you
do the renormalization,
550
00:31:55,560 --> 00:31:57,660
that can change too.
551
00:31:57,660 --> 00:32:00,150
Because the operators
or matrix elements here
552
00:32:00,150 --> 00:32:02,370
are labeled by this
xi, in general, there's
553
00:32:02,370 --> 00:32:04,740
no reason that it should
stay the same under the loop
554
00:32:04,740 --> 00:32:05,340
corrections.
555
00:32:05,340 --> 00:32:09,060
And it was really a special case
that we dealt with last time
556
00:32:09,060 --> 00:32:10,080
where that did happen.
557
00:32:10,080 --> 00:32:11,205
But in general, it doesn't.
558
00:32:15,270 --> 00:32:21,270
OK, so this is
actually what we expect
559
00:32:21,270 --> 00:32:24,450
to happen in general unless
we can argue that it doesn't
560
00:32:24,450 --> 00:32:32,770
happen because you should
think of each value of xi
561
00:32:32,770 --> 00:32:37,360
as giving a different operator
or different matrix element.
562
00:32:40,570 --> 00:32:42,900
So I could write formulas
here just for the operator.
563
00:32:42,900 --> 00:32:44,983
It's actually the operator
that gets renormalized,
564
00:32:44,983 --> 00:32:46,650
not the matrix element.
565
00:32:46,650 --> 00:32:49,500
So I'm going to keep
writing f's just
566
00:32:49,500 --> 00:32:52,290
to avoid too much notation,
but we could always actually
567
00:32:52,290 --> 00:32:55,762
replace the f's by
just the operator.
568
00:32:55,762 --> 00:32:57,720
And we could do everything
in terms of actually
569
00:32:57,720 --> 00:33:01,510
just the w instead
of the xi variable.
570
00:33:01,510 --> 00:33:04,740
But I'll just keep using f.
571
00:33:04,740 --> 00:33:08,520
So what does this mean
in terms of the operator
572
00:33:08,520 --> 00:33:09,250
is the following.
573
00:33:09,250 --> 00:33:16,480
We can think of, if we
have some bare operator
574
00:33:16,480 --> 00:33:21,000
and we want to split that into
a piece that has divergences
575
00:33:21,000 --> 00:33:27,900
and a piece that is
just the finite pieces,
576
00:33:27,900 --> 00:33:32,740
the general formula for doing
that involves an integral.
577
00:33:32,740 --> 00:33:34,450
So this guy here--
578
00:33:34,450 --> 00:33:36,630
so there's also these
indices, i and j,
579
00:33:36,630 --> 00:33:40,890
and that's the flavor, if you
like, or quarks and gluons.
580
00:33:40,890 --> 00:33:45,270
So i is quark or gluon.
581
00:33:45,270 --> 00:33:48,570
And in general, you can also
have a mixing in the quark
582
00:33:48,570 --> 00:33:49,440
and gluon operators.
583
00:33:49,440 --> 00:33:51,120
We started with these
two different operators,
584
00:33:51,120 --> 00:33:53,078
and they can mix under
renormalization as well.
585
00:34:02,640 --> 00:34:05,540
So there's two operators in the
effective theory, same order
586
00:34:05,540 --> 00:34:08,830
in lambda, and they can mix
when you do the renormalization.
587
00:34:08,830 --> 00:34:12,139
And I'll draw a
diagram in a minute.
588
00:34:12,139 --> 00:34:16,540
So this thing here
is mu-independent.
589
00:34:16,540 --> 00:34:21,100
This thing here in MS bar has
all the 1 over epsilon UV's.
590
00:34:21,100 --> 00:34:23,679
And it also depends
on alpha of mu,
591
00:34:23,679 --> 00:34:26,920
and it depends on
these xi and xi prime.
592
00:34:26,920 --> 00:34:28,840
And this guy here is UV-finite.
593
00:34:34,238 --> 00:34:36,030
So this guy here is
really the thing that's
594
00:34:36,030 --> 00:34:37,350
the low-energy matrix element.
595
00:34:37,350 --> 00:34:39,510
But remember what low
energy meant here.
596
00:34:39,510 --> 00:34:42,300
Low energy was physics
at lambda QCD, physics
597
00:34:42,300 --> 00:34:44,880
of the initial-state proton.
598
00:34:44,880 --> 00:34:48,715
So actually, in this guy,
there are IR divergences.
599
00:34:48,715 --> 00:34:51,090
This is just some matrix
element in the effective theory,
600
00:34:51,090 --> 00:34:52,757
and in general, it
could be IR-divergent
601
00:34:52,757 --> 00:34:53,810
if you calculate it.
602
00:34:53,810 --> 00:34:59,010
And this guy actually is.
603
00:34:59,010 --> 00:35:01,922
And it really encodes--
604
00:35:01,922 --> 00:35:04,380
that's not going to bother us
at all because this is really
605
00:35:04,380 --> 00:35:08,370
some universal thing that
encodes lambda QCD effects,
606
00:35:08,370 --> 00:35:11,320
and that's what parton
distribution functions are.
607
00:35:11,320 --> 00:35:13,320
Then from the point of
view of what we're doing,
608
00:35:13,320 --> 00:35:16,630
it doesn't really matter that
it has this extra IR divergence
609
00:35:16,630 --> 00:35:19,080
so that we will have
to regulate diagrams
610
00:35:19,080 --> 00:35:22,633
in order to separate UV and IR
divergences because of that.
611
00:35:22,633 --> 00:35:24,300
Really, in terms of
the renormalization,
612
00:35:24,300 --> 00:35:26,791
what we're after is
getting the UV divergences.
613
00:35:31,610 --> 00:35:33,510
OK, so the usual kind
of formula that you'd
614
00:35:33,510 --> 00:35:36,010
have where you just
write o is z times o
615
00:35:36,010 --> 00:35:37,983
is slightly more
complicated here.
616
00:35:37,983 --> 00:35:39,150
There's this extra integral.
617
00:35:52,590 --> 00:35:55,980
And now, remember how you
derive a renormalization group
618
00:35:55,980 --> 00:35:57,450
equation.
619
00:35:57,450 --> 00:35:59,955
What you do is you say mu
d by u mu is this guy is 0.
620
00:36:03,490 --> 00:36:07,430
And so if I take mu d by d
mu, on the right-hand side,
621
00:36:07,430 --> 00:36:10,690
I get mu d by d mu of z
and mu d by d mu of f.
622
00:36:10,690 --> 00:36:14,980
And I can rearrange
that in the usual way,
623
00:36:14,980 --> 00:36:19,640
except for keeping track of
these integrals, as follows.
624
00:36:19,640 --> 00:36:21,880
So I imagine that there's
a z and a z inverse.
625
00:36:21,880 --> 00:36:27,440
And the relation between z
and z inverse is as follows.
626
00:36:27,440 --> 00:36:31,540
Let's just call
this double prime.
627
00:36:31,540 --> 00:36:34,010
It's matrix multiplication
except in the function space,
628
00:36:34,010 --> 00:36:34,510
right?
629
00:36:34,510 --> 00:36:37,885
So this is like
a delta function.
630
00:36:41,570 --> 00:36:43,810
So if you like,
you can just think
631
00:36:43,810 --> 00:36:46,060
that there's more indices.
632
00:36:46,060 --> 00:36:48,730
In some sense, what we have in
terms of the quark and gluon
633
00:36:48,730 --> 00:36:50,860
operators mixing is a
matrix equation, right?
634
00:36:50,860 --> 00:36:51,830
This is a vector.
635
00:36:51,830 --> 00:36:52,660
This is a matrix.
636
00:36:52,660 --> 00:36:55,330
This is a vector for
the indices i and j.
637
00:36:55,330 --> 00:36:58,780
And you can think of this
integral here as just another--
638
00:36:58,780 --> 00:37:00,825
it really looks, the
way I've drawn it,
639
00:37:00,825 --> 00:37:02,200
like this is
contracted with that
640
00:37:02,200 --> 00:37:04,300
and this is summing
over the indices.
641
00:37:04,300 --> 00:37:06,320
And really, that's what it is.
642
00:37:06,320 --> 00:37:09,490
So really, this idea that it's
just mixing of quantum numbers
643
00:37:09,490 --> 00:37:11,478
is kind of a good way of
thinking about things.
644
00:37:11,478 --> 00:37:12,895
And when you think
about formulas,
645
00:37:12,895 --> 00:37:16,030
you know you're just
summing over these indices,
646
00:37:16,030 --> 00:37:20,230
and the Kronecker delta
becomes a regular delta.
647
00:37:20,230 --> 00:37:26,720
So in that sense, it's
not that hard to do this.
648
00:37:26,720 --> 00:37:36,920
And so we get an anomalous
dimension equation
649
00:37:36,920 --> 00:37:38,800
which, again, has
that kind of form
650
00:37:38,800 --> 00:37:44,620
of just an integral for
the renormalized guy,
651
00:37:44,620 --> 00:37:46,390
and it has mixing.
652
00:37:46,390 --> 00:37:53,650
And this gamma ij, if
we go through the steps
653
00:37:53,650 --> 00:37:58,510
and use this formula,
looks like this.
654
00:38:04,750 --> 00:38:06,250
So I'm kind of
skipping steps, but I
655
00:38:06,250 --> 00:38:10,215
hope that you can
kind of picture
656
00:38:10,215 --> 00:38:11,590
where this result
will come from.
657
00:38:11,590 --> 00:38:13,240
And it's actually not--
658
00:38:13,240 --> 00:38:15,790
it's pretty easy to go from
that line with this formula
659
00:38:15,790 --> 00:38:16,840
to this line.
660
00:38:16,840 --> 00:38:19,230
This is one line, but I just
split it into two things
661
00:38:19,230 --> 00:38:20,980
and defined this
quantity, gamma ij, which
662
00:38:20,980 --> 00:38:23,170
is the anomalous dimension.
663
00:38:23,170 --> 00:38:25,850
AUDIENCE: So this mu in QCD
[INAUDIBLE] factorization
664
00:38:25,850 --> 00:38:26,570
scale?
665
00:38:26,570 --> 00:38:28,890
IAIN STEWART:
Yeah, that's right.
666
00:38:32,150 --> 00:38:36,020
OK, so at one loop,
things are simpler.
667
00:38:36,020 --> 00:38:37,490
Because at one
loop, this thing, we
668
00:38:37,490 --> 00:38:44,750
can just replace it by delta
ii prime, Kronecker delta
669
00:38:44,750 --> 00:38:47,210
at one loop.
670
00:38:47,210 --> 00:38:54,880
Because at one loop, we just
need the order alpha piece
671
00:38:54,880 --> 00:38:57,670
from this guy, and then
we can set the tree level
672
00:38:57,670 --> 00:38:59,020
for that guy.
673
00:38:59,020 --> 00:39:04,765
So at one loop, which is
all we're going to do,
674
00:39:04,765 --> 00:39:05,890
we get the simpler formula.
675
00:39:22,940 --> 00:39:25,850
OK, so that's our
setup, and now we
676
00:39:25,850 --> 00:39:28,850
want to calculate this
one-loop anomalous dimension
677
00:39:28,850 --> 00:39:31,940
by calculating the 1
over epsilon alpha s term
678
00:39:31,940 --> 00:39:32,490
and the zij.
679
00:39:39,830 --> 00:39:44,130
Before I do that, is
there any questions?
680
00:39:44,130 --> 00:39:45,410
All right, so tree level--
681
00:39:52,910 --> 00:39:56,020
so think about there being an
external p for whatever state
682
00:39:56,020 --> 00:40:00,580
I'm considering, and then
the operator is labeled by w.
683
00:40:00,580 --> 00:40:06,250
And so we're summing over spin.
684
00:40:06,250 --> 00:40:07,780
I've kind of somehow--
685
00:40:07,780 --> 00:40:09,640
sometimes I've dropped that.
686
00:40:09,640 --> 00:40:12,140
I said it last time.
687
00:40:12,140 --> 00:40:17,330
And so we get some spinners,
and we got a delta function.
688
00:40:17,330 --> 00:40:19,450
So what the delta
function in the operator
689
00:40:19,450 --> 00:40:22,870
is it's delta function
of w minus this label,
690
00:40:22,870 --> 00:40:23,587
momentum p bar.
691
00:40:23,587 --> 00:40:25,920
And in something like this
where it's completely trivial
692
00:40:25,920 --> 00:40:28,087
and there's just one state,
we just get the momentum
693
00:40:28,087 --> 00:40:29,890
of that state, which is p.
694
00:40:29,890 --> 00:40:33,730
This sum over spin
here is a p minus.
695
00:40:33,730 --> 00:40:38,170
And so the result is
a delta function of 1
696
00:40:38,170 --> 00:40:43,960
minus omega over p minus for
this tree-level matrix element.
697
00:40:51,060 --> 00:40:53,760
One loop-- now we have
to think about how
698
00:40:53,760 --> 00:40:55,980
we're going to regulate the IR.
699
00:40:55,980 --> 00:40:58,030
And I'll do it with
an off-shellness.
700
00:41:00,970 --> 00:41:04,270
So I'll introduce
a nonzero p class,
701
00:41:04,270 --> 00:41:08,050
and that will be enough to
regulate IR divergences.
702
00:41:12,400 --> 00:41:14,010
And we're really
after the UV one,
703
00:41:14,010 --> 00:41:16,480
so we just want to
separate these guys out.
704
00:41:19,587 --> 00:41:21,045
So there's some
different diagrams.
705
00:41:24,617 --> 00:41:27,105
We insert our operator,
and we just attach gluons.
706
00:41:27,105 --> 00:41:28,980
So one thing we can do
is just string a gluon
707
00:41:28,980 --> 00:41:35,145
across kind of like a standard
vertex renormalization diagram.
708
00:41:37,952 --> 00:41:39,160
So there's some loop momenta.
709
00:41:39,160 --> 00:41:40,890
Let me label it
on the quark line.
710
00:41:40,890 --> 00:41:44,390
And then the gluon here,
which is a collinear gluon,
711
00:41:44,390 --> 00:41:46,500
has momentum p minus l.
712
00:41:46,500 --> 00:41:50,640
And it's forward so
it's kind of set up.
713
00:41:53,340 --> 00:41:55,338
There are some
numerator to deal with.
714
00:41:55,338 --> 00:41:56,880
And I'm not going
to go through that,
715
00:41:56,880 --> 00:42:00,960
but it simplifies to
something kind of simple.
716
00:42:00,960 --> 00:42:03,660
After some [INAUDIBLE] algebra,
it simplifies down just
717
00:42:03,660 --> 00:42:05,410
to an l perp squared.
718
00:42:05,410 --> 00:42:11,730
For this diagram, there's
two l squared propagators,
719
00:42:11,730 --> 00:42:14,675
and there's one l minus
p squared propagator.
720
00:42:19,530 --> 00:42:21,330
And then there's
a delta function
721
00:42:21,330 --> 00:42:23,503
from the insertion
of the operator,
722
00:42:23,503 --> 00:42:25,920
but now the delta function
doesn't involve the [INAUDIBLE]
723
00:42:25,920 --> 00:42:27,000
momentum as it did there.
724
00:42:27,000 --> 00:42:29,640
It involves the loop
momentum, and that
725
00:42:29,640 --> 00:42:32,740
was kind of the whole
point of this example.
726
00:42:32,740 --> 00:42:39,210
So we have a delta function
of l minus minus w.
727
00:42:39,210 --> 00:42:40,832
And then there's
some dimreg factors,
728
00:42:40,832 --> 00:42:42,540
which we can be careful
about if we want.
729
00:42:48,030 --> 00:42:50,850
So in MS bar, we'd have
some factor like that.
730
00:42:50,850 --> 00:42:53,100
So this is some loop integral
that we just have to do,
731
00:42:53,100 --> 00:42:56,284
and we can do it with kind
of standard techniques.
732
00:43:10,820 --> 00:43:15,290
So in my, notes I wrote it
as a function of epsilon,
733
00:43:15,290 --> 00:43:17,755
and then epsilon is just
regulating the ultraviolet,
734
00:43:17,755 --> 00:43:19,190
and we expand in epsilon.
735
00:43:19,190 --> 00:43:22,238
So let me just write down
the result after expanding.
736
00:43:48,134 --> 00:43:50,260
So this is an
ultraviolet divergence,
737
00:43:50,260 --> 00:43:54,430
and A here has the
infrared regulator--
738
00:43:54,430 --> 00:43:55,870
p plus, p minus.
739
00:43:55,870 --> 00:43:57,320
And it also has
a z and a 1 minus
740
00:43:57,320 --> 00:43:59,890
z, which you can
group all together.
741
00:43:59,890 --> 00:44:02,620
And z is just this ratio.
742
00:44:02,620 --> 00:44:06,448
That thing is dependent on a
tree-level omega over p minus.
743
00:44:06,448 --> 00:44:07,990
Now, when I'm doing
this calculation,
744
00:44:07,990 --> 00:44:10,240
this is a small p, not
a big P, because I'm
745
00:44:10,240 --> 00:44:15,830
using quark states,
not a proton state.
746
00:44:15,830 --> 00:44:18,648
So really, if I wanted to
think about this as an f,
747
00:44:18,648 --> 00:44:20,440
I should say it's an
f for the quark state.
748
00:44:20,440 --> 00:44:24,370
But I think that you
can remember that.
749
00:44:24,370 --> 00:44:26,207
But the renormalization
of the operator
750
00:44:26,207 --> 00:44:27,790
doesn't depend on
the state, remember.
751
00:44:27,790 --> 00:44:29,780
We always take the
simplest states possible
752
00:44:29,780 --> 00:44:31,720
when we're doing
the renormalization
753
00:44:31,720 --> 00:44:32,620
or doing matching.
754
00:44:32,620 --> 00:44:34,550
And so we're free
to use quark states,
755
00:44:34,550 --> 00:44:35,675
so that's what we're doing.
756
00:44:38,320 --> 00:44:43,430
OK, that's one diagram.
757
00:44:43,430 --> 00:44:45,165
Now there's another diagram.
758
00:44:45,165 --> 00:44:48,980
I think that should be
B. Sometimes in my notes,
759
00:44:48,980 --> 00:44:52,345
I'll call it 1, which
doesn't make any sense.
760
00:44:52,345 --> 00:44:54,470
And we can contract the
gluon with the Wilson line.
761
00:44:54,470 --> 00:44:58,580
So there's that graph, and
there's a symmetric friend.
762
00:45:10,600 --> 00:45:12,880
And each of these actually
has two contractions
763
00:45:12,880 --> 00:45:16,930
because there was two
Wilson lines in the way we
764
00:45:16,930 --> 00:45:17,800
wrote our operators.
765
00:45:17,800 --> 00:45:21,670
So our operator, as we
wrote, is like this.
766
00:45:27,625 --> 00:45:29,000
And you can think--
so let's just
767
00:45:29,000 --> 00:45:30,583
think of a contraction
with the quark.
768
00:45:30,583 --> 00:45:32,780
You can think that there's
a contraction like that
769
00:45:32,780 --> 00:45:35,960
and there's a contraction
like that of a gluon--
770
00:45:35,960 --> 00:45:38,102
OK, I'm contracting
gluons with quarks.
771
00:45:38,102 --> 00:45:39,560
But really, what
I mean is that I'm
772
00:45:39,560 --> 00:45:42,810
contracting to the
Lagrangian, right,
773
00:45:42,810 --> 00:45:44,270
that this quark
is evolving under.
774
00:45:44,270 --> 00:45:47,510
So hopefully that's clear.
775
00:45:47,510 --> 00:45:50,270
All right, so there's two
different ways in which--
776
00:45:50,270 --> 00:45:52,790
when I work out the Feynman
rule for this thing where
777
00:45:52,790 --> 00:45:55,680
I attach the gluon,
you can either
778
00:45:55,680 --> 00:45:57,680
get the gluon from here
or the gluon from there.
779
00:45:57,680 --> 00:46:00,820
That's all I'm saying.
780
00:46:00,820 --> 00:46:02,570
But these actually
have different physical
781
00:46:02,570 --> 00:46:06,020
interpretations because
this delta function
782
00:46:06,020 --> 00:46:08,360
here, if you think
about what it's doing,
783
00:46:08,360 --> 00:46:12,410
it's really-- in the original
diagram, it's like the cut.
784
00:46:12,410 --> 00:46:15,800
So in the original diagrams
that we were drawing,
785
00:46:15,800 --> 00:46:19,040
we would cut them because
we'd take the imaginary part.
786
00:46:19,040 --> 00:46:21,030
And this delta function
is in the middle.
787
00:46:21,030 --> 00:46:22,730
We have kind of a
parton on this side
788
00:46:22,730 --> 00:46:25,160
and a squared
parton on that side.
789
00:46:25,160 --> 00:46:27,150
This delta function is the cut.
790
00:46:27,150 --> 00:46:28,790
So this contraction
here actually
791
00:46:28,790 --> 00:46:34,580
corresponds to a virtual
graph, and this guy here
792
00:46:34,580 --> 00:46:37,670
corresponds to real
emission because you're
793
00:46:37,670 --> 00:46:39,920
doing a contraction
across the cut, right?
794
00:46:39,920 --> 00:46:41,990
So one of these guys would
be a graph like this,
795
00:46:41,990 --> 00:46:43,865
and the other one would
be a graph like that.
796
00:46:46,940 --> 00:46:50,636
I can label them 1 and 2--
797
00:46:50,636 --> 00:46:52,010
1, 2.
798
00:46:55,706 --> 00:46:59,440
But we'll just keep them
and treat them all together.
799
00:46:59,440 --> 00:47:03,124
These two graphs give
an overall factor of 2.
800
00:47:03,124 --> 00:47:04,600
So that's simple.
801
00:47:08,854 --> 00:47:16,295
There's some
spinner stuff, which
802
00:47:16,295 --> 00:47:18,640
is even simpler in this
case, so I write it out.
803
00:47:23,510 --> 00:47:28,680
There are some stuff
from the Wilson line.
804
00:47:28,680 --> 00:47:31,470
And then there's
two propagators.
805
00:47:31,470 --> 00:47:34,720
Let me not write all the i0's.
806
00:47:34,720 --> 00:47:37,890
And then there's two
different delta functions.
807
00:47:37,890 --> 00:47:43,120
So either we have the real
graph where the w is inside,
808
00:47:43,120 --> 00:47:49,090
or we have the virtual
graph where the w--
809
00:47:49,090 --> 00:47:50,350
sorry.
810
00:47:50,350 --> 00:47:52,270
Either we have the
real graph where
811
00:47:52,270 --> 00:47:55,270
the loop goes around
the delta function,
812
00:47:55,270 --> 00:47:57,580
or we have the virtual graph
where this guy is overall
813
00:47:57,580 --> 00:47:58,540
on that thing.
814
00:47:58,540 --> 00:48:01,870
So in the overall one,
it's just a p minus minus w
815
00:48:01,870 --> 00:48:03,640
like it was at tree level.
816
00:48:03,640 --> 00:48:08,180
And in the real emission,
it's an l minus minus w.
817
00:48:08,180 --> 00:48:09,380
And one's a w.
818
00:48:09,380 --> 00:48:10,250
One's a w dagger.
819
00:48:10,250 --> 00:48:11,480
So there's a relative sign.
820
00:48:14,850 --> 00:48:17,245
So the sign is just
easier to understand
821
00:48:17,245 --> 00:48:22,120
as w versus w dagger,
which has a relative sign.
822
00:48:22,120 --> 00:48:23,800
OK, so if we just
followed our nose
823
00:48:23,800 --> 00:48:26,358
with what the Feynman
rule for this thing is,
824
00:48:26,358 --> 00:48:27,400
that's what we would get.
825
00:48:34,310 --> 00:48:36,560
And this is, again, some
loop integral that we can do.
826
00:48:50,700 --> 00:48:54,410
One way of writing the
result is as follows.
827
00:48:54,410 --> 00:48:59,482
And there's one thing we have to
be careful about here which is
828
00:48:59,482 --> 00:49:00,690
why I'm writing this all out.
829
00:49:11,680 --> 00:49:13,120
So there's actually
a cancellation
830
00:49:13,120 --> 00:49:15,400
between the virtual
and the real diagrams
831
00:49:15,400 --> 00:49:19,470
of an infrared divergence, so I
want to be careful about that.
832
00:49:30,982 --> 00:49:36,260
So that's why I'm writing this
guy out in epsilon dimensions
833
00:49:36,260 --> 00:49:38,660
fully without expanding first.
834
00:49:38,660 --> 00:49:41,720
OK, so this is the
real contribution,
835
00:49:41,720 --> 00:49:44,120
and this is the virtual.
836
00:49:44,120 --> 00:49:47,465
So in order to sort
of deal with this,
837
00:49:47,465 --> 00:49:50,090
we have to make use of something
that's called the distribution
838
00:49:50,090 --> 00:49:51,040
identity.
839
00:49:57,630 --> 00:50:00,177
If you know what the result is
for the anomalous dimension,
840
00:50:00,177 --> 00:50:02,010
you'll be aware of the
fact that it involves
841
00:50:02,010 --> 00:50:04,977
something called a plus
function because splitting
842
00:50:04,977 --> 00:50:06,435
functions for a
parton distribution
843
00:50:06,435 --> 00:50:08,185
involves something
called a plus function.
844
00:50:30,960 --> 00:50:34,265
So the way that we can deal
with that is as follows.
845
00:50:38,405 --> 00:50:39,780
The way we can
deal with the fact
846
00:50:39,780 --> 00:50:42,030
that actually the result's
going to be a distribution,
847
00:50:42,030 --> 00:50:46,830
we have to be careful
because you see, z goes to 1
848
00:50:46,830 --> 00:50:48,360
is being regulated by epsilon.
849
00:50:51,030 --> 00:50:53,490
And so if we integrate
over z, for example,
850
00:50:53,490 --> 00:50:55,230
it's epsilon that's
going to allow us
851
00:50:55,230 --> 00:50:58,620
to integrate all the way to 1.
852
00:50:58,620 --> 00:51:02,115
And we'd like to
encode that in some way
853
00:51:02,115 --> 00:51:03,990
where we can expand in
epsilon because that's
854
00:51:03,990 --> 00:51:05,840
what we need to do
in order to extract
855
00:51:05,840 --> 00:51:06,840
the anomalous dimension.
856
00:51:06,840 --> 00:51:08,757
And this formula is what
allows us to do that.
857
00:51:12,640 --> 00:51:15,070
So I'll tell you
how to derive it
858
00:51:15,070 --> 00:51:18,010
after I tell you what the l is.
859
00:51:18,010 --> 00:51:22,780
So ln of anything is defined
to be a plus function
860
00:51:22,780 --> 00:51:25,180
with a log to that power.
861
00:51:31,660 --> 00:51:33,400
And the plus function
is defined so
862
00:51:33,400 --> 00:51:38,830
that if you integrate
from 0 to 1, you get 0.
863
00:51:38,830 --> 00:51:44,557
And if you integrate
with a test function,
864
00:51:44,557 --> 00:51:48,700
which is the more general result
that you need to define it--
865
00:51:48,700 --> 00:51:52,750
so you can define it by this
result with a test function.
866
00:51:52,750 --> 00:51:55,850
And it just gives you
the normal function,
867
00:51:55,850 --> 00:51:58,510
but the test function
with a subtraction that
868
00:51:58,510 --> 00:52:00,190
makes the test function
more convergent
869
00:52:00,190 --> 00:52:03,700
so that you can
integrate through 0.
870
00:52:03,700 --> 00:52:05,920
OK, so that's the definition
of a plus function.
871
00:52:05,920 --> 00:52:08,860
You could also define
it with a limit.
872
00:52:08,860 --> 00:52:12,610
This will be sufficient.
873
00:52:12,610 --> 00:52:15,970
OK, so these things are
like delta functions.
874
00:52:15,970 --> 00:52:17,950
The way that you would
derive this formula
875
00:52:17,950 --> 00:52:21,220
is you would say, well,
if z is away from 1,
876
00:52:21,220 --> 00:52:24,400
then I can expand because
then there's no problem.
877
00:52:24,400 --> 00:52:26,470
And if z is away
from 1, it turns out
878
00:52:26,470 --> 00:52:28,930
that this plus function is
just the regular function.
879
00:52:28,930 --> 00:52:32,210
It's only at 1 that something
special is happening.
880
00:52:32,210 --> 00:52:35,390
And so the standard
expansion is what you'd
881
00:52:35,390 --> 00:52:37,930
get if you took z away from 1.
882
00:52:37,930 --> 00:52:39,850
And to see what's
happening at z equals 1,
883
00:52:39,850 --> 00:52:41,785
you'd just integrate
both sides from 0 to 1,
884
00:52:41,785 --> 00:52:43,660
and that's how you can
derive the coefficient
885
00:52:43,660 --> 00:52:44,577
of the delta function.
886
00:52:49,320 --> 00:52:54,630
All right, so if I plug this
formula in here for this thing,
887
00:52:54,630 --> 00:52:57,632
then I actually get another
1 over epsilon in this guy.
888
00:52:57,632 --> 00:53:00,090
There's a gamma of epsilon out
front, and that guy is good.
889
00:53:00,090 --> 00:53:01,350
This is our UV divergence.
890
00:53:01,350 --> 00:53:04,650
This is our 1 over epsilon UV.
891
00:53:04,650 --> 00:53:06,570
But there's also a
gamma of minus epsilon
892
00:53:06,570 --> 00:53:09,160
here, which is an IR divergence.
893
00:53:09,160 --> 00:53:11,310
So even though I tried
to regulate all the IR
894
00:53:11,310 --> 00:53:13,810
by off-shellness,
it didn't quite work
895
00:53:13,810 --> 00:53:16,380
and there was one that
was regulated by dimreg.
896
00:53:16,380 --> 00:53:19,890
And that one actually cancels
between these two pieces
897
00:53:19,890 --> 00:53:25,620
once I use this identity
and take into account
898
00:53:25,620 --> 00:53:29,070
that that's an IR divergence.
899
00:53:29,070 --> 00:53:36,360
So there's a 1 over epsilon
IR times 1 over epsilon UV.
900
00:53:36,360 --> 00:53:40,260
And that cancels between
the real and virtual graphs.
901
00:53:45,710 --> 00:53:48,450
So this is like a
standard 1 over epsilon IR
902
00:53:48,450 --> 00:53:50,200
canceling between real
and virtual graphs.
903
00:53:50,200 --> 00:53:51,790
And since it's only
the 1 over epsilon UV
904
00:53:51,790 --> 00:53:53,920
that we're interested in,
we're really only worried
905
00:53:53,920 --> 00:53:56,080
about that part of it canceling.
906
00:53:56,080 --> 00:53:58,415
There's a piece actually that--
907
00:53:58,415 --> 00:53:58,915
anyway.
908
00:54:06,640 --> 00:54:10,930
And then the 1 over
epsilon that's left
909
00:54:10,930 --> 00:54:17,290
is the guy that
we're after in order
910
00:54:17,290 --> 00:54:18,932
to get the anomalous dimension.
911
00:54:24,600 --> 00:54:28,950
All right, so let me not--
912
00:54:28,950 --> 00:54:30,600
so in my notes, I
write one more line
913
00:54:30,600 --> 00:54:32,310
where I expand this guy out.
914
00:54:32,310 --> 00:54:36,630
And I think just because of the
time, I'm going to skip that.
915
00:54:36,630 --> 00:54:38,670
And I'll just write
the final result.
916
00:54:38,670 --> 00:54:40,710
When we do the final
result, we also
917
00:54:40,710 --> 00:54:43,770
have to include wave
function renormalization.
918
00:54:43,770 --> 00:54:49,380
So you can think of this
graph as a wave function
919
00:54:49,380 --> 00:54:51,990
renormalization term.
920
00:54:51,990 --> 00:54:53,937
And it just involves
the delta function again
921
00:54:53,937 --> 00:54:55,020
like the tree-level graph.
922
00:55:16,533 --> 00:55:18,080
[INAUDIBLE] like that.
923
00:55:18,080 --> 00:55:21,830
So in general, if I wanted to
do this calculation at one loop,
924
00:55:21,830 --> 00:55:25,820
there's one more type of
diagram I should consider, OK?
925
00:55:25,820 --> 00:55:28,840
And that's a graph where
I could have mixing.
926
00:55:28,840 --> 00:55:33,890
This guy should be dashed since
we're in the effective theory.
927
00:55:33,890 --> 00:55:36,350
So how does the
mixing graph work?
928
00:55:36,350 --> 00:55:38,720
Well, there's a graph where
I have external gluons,
929
00:55:38,720 --> 00:55:42,236
but I still am renormalizing
the same operator.
930
00:55:42,236 --> 00:55:45,800
I've still inserted the
quark operator here,
931
00:55:45,800 --> 00:55:49,130
but now we have
antiquarks in this theory.
932
00:55:49,130 --> 00:55:50,900
We can draw a
triangle like that.
933
00:55:50,900 --> 00:55:54,537
And this graph here would
give a mixing that involves--
934
00:55:54,537 --> 00:55:56,870
that would give a mixing term
in the anomalous dimension
935
00:55:56,870 --> 00:56:00,250
where you're mixing
gluons and quarks.
936
00:56:00,250 --> 00:56:07,550
So this mix is what we
sort of called O glue.
937
00:56:07,550 --> 00:56:11,830
Let me just say this, that
it mixes O glue with O quark.
938
00:56:11,830 --> 00:56:13,760
And we could compute
this graph too,
939
00:56:13,760 --> 00:56:17,680
but I'm going to neglect
it just for simplicity.
940
00:56:17,680 --> 00:56:20,320
I just won't write it down.
941
00:56:20,320 --> 00:56:23,650
One way of doing
that rigorously would
942
00:56:23,650 --> 00:56:26,770
be to consider operators where
the flavors of these guys
943
00:56:26,770 --> 00:56:28,660
are different, OK?
944
00:56:28,660 --> 00:56:31,630
That's what would happen, for
example, if you were having
945
00:56:31,630 --> 00:56:32,830
a w exchange or something.
946
00:56:32,830 --> 00:56:36,220
So we could look at
nonflavored diagonal operators
947
00:56:36,220 --> 00:56:39,043
with, like, a u
quark and a d quark.
948
00:56:39,043 --> 00:56:41,210
And then you would not have
this mixing with O glue.
949
00:56:41,210 --> 00:56:42,793
It's only if the
flavors of the quarks
950
00:56:42,793 --> 00:56:45,800
are the same that you can
write down this diagram.
951
00:56:45,800 --> 00:56:50,718
But just think about it as I'm
focusing on the quark piece,
952
00:56:50,718 --> 00:56:52,510
and in general, there's
also a gluon piece.
953
00:57:04,360 --> 00:57:05,830
So we have all our
one-loop graphs.
954
00:57:05,830 --> 00:57:07,480
We know how to expand
them in epsilon.
955
00:57:07,480 --> 00:57:09,953
And so we just proceed,
expand them in epsilon,
956
00:57:09,953 --> 00:57:10,620
and add them up.
957
00:57:28,340 --> 00:57:30,400
So you could think
that what we derive
958
00:57:30,400 --> 00:57:35,680
by doing that is a distribution
for a quark inside a quark.
959
00:57:35,680 --> 00:57:39,010
So here, I'm being--
this is the state
960
00:57:39,010 --> 00:57:41,250
and this is what
type of operator.
961
00:57:47,360 --> 00:57:51,880
And it's a function of some z.
962
00:57:51,880 --> 00:57:57,690
And if I go up to one
loop, then the tree level
963
00:57:57,690 --> 00:58:02,920
was just a delta function
of that fraction z.
964
00:58:02,920 --> 00:58:05,290
And then at one loop, we
had all these other terms.
965
00:58:12,880 --> 00:58:19,810
So if I collect all the pieces,
I had some delta functions.
966
00:58:19,810 --> 00:58:24,070
The graph with the Wilson
lines actually gives me
967
00:58:24,070 --> 00:58:26,810
one of these L0 functions.
968
00:58:26,810 --> 00:58:27,950
And then the graph--
969
00:58:27,950 --> 00:58:29,860
so there's wave
function normalization
970
00:58:29,860 --> 00:58:33,727
plus some other terms that
involve delta function.
971
00:58:33,727 --> 00:58:35,185
And then there's
some other pieces.
972
00:58:39,820 --> 00:58:42,880
And then this is all
times 1 over epsilon.
973
00:58:42,880 --> 00:58:44,668
And then there's other pieces.
974
00:58:44,668 --> 00:58:46,960
But if we're interested in
ultraviolet renormalization,
975
00:58:46,960 --> 00:58:50,050
we only care about
the 1 over epsilon.
976
00:58:50,050 --> 00:58:55,390
And all those terms
can be written
977
00:58:55,390 --> 00:58:57,070
in a kind of more
compact form, which
978
00:58:57,070 --> 00:59:09,130
is the more standard form
for the anomalous dimension.
979
00:59:09,130 --> 00:59:12,940
You can actually group them
all together into a single plus
980
00:59:12,940 --> 00:59:17,030
function like this.
981
00:59:17,030 --> 00:59:19,900
So just in terms
of distributions,
982
00:59:19,900 --> 00:59:22,840
this distribution is equal
to the sum of these pieces.
983
00:59:35,190 --> 00:59:37,380
You can see, as z goes
to 1, that there'd be a 2
984
00:59:37,380 --> 00:59:38,340
here and a 2 here.
985
00:59:38,340 --> 00:59:40,110
And this would be
1 over 1 minus z.
986
00:59:40,110 --> 00:59:42,433
And as z goes to 1
here, that would be 1
987
00:59:42,433 --> 00:59:43,975
and this would be
a 1 over 1 minus z.
988
00:59:43,975 --> 00:59:48,110
So you see some pieces
of it matching up.
989
00:59:50,890 --> 00:59:53,710
Basically, the way that you
would derive this is you'd
990
00:59:53,710 --> 01:00:00,550
write 1 plus z squared
is a plus b, 1 minus z
991
01:00:00,550 --> 01:00:02,800
plus c, 1 minus z squared.
992
01:00:02,800 --> 01:00:04,760
You'd work out what
a, b, and c are,
993
01:00:04,760 --> 01:00:07,070
just relating two polynomials.
994
01:00:07,070 --> 01:00:11,482
And then this guy here, the
1 minus z in the numerator
995
01:00:11,482 --> 01:00:12,940
cancels the one in
the denominator,
996
01:00:12,940 --> 01:00:14,482
and it's not a plus
function anymore.
997
01:00:14,482 --> 01:00:15,810
It's just a number.
998
01:00:15,810 --> 01:00:19,790
And that's how you would
connect the two formulas.
999
01:00:19,790 --> 01:00:23,110
All right, so we were
after determining the z.
1000
01:00:23,110 --> 01:00:27,170
The z has to cancel
this 1 over epsilon.
1001
01:00:27,170 --> 01:00:30,550
So let's go back to our
formula which connected
1002
01:00:30,550 --> 01:00:34,960
those, which was this.
1003
01:00:34,960 --> 01:00:39,370
Our general formula was
that the bare guy could
1004
01:00:39,370 --> 01:00:48,370
be written in terms of
split into UV pieces
1005
01:00:48,370 --> 01:00:50,320
and finite pieces in
the following ways,
1006
01:00:50,320 --> 01:00:53,600
is with this integral.
1007
01:00:53,600 --> 01:00:56,800
Now, this looks like it could
be an arbitrary function of xi
1008
01:00:56,800 --> 01:00:59,860
and xi prime, but our result
here was only a function of z,
1009
01:00:59,860 --> 01:01:02,000
which is actually a ratio.
1010
01:01:02,000 --> 01:01:05,350
And that's actually something
that we can argue in general,
1011
01:01:05,350 --> 01:01:08,710
that this thing here is
actually only a function
1012
01:01:08,710 --> 01:01:10,750
of one variable, not two.
1013
01:01:23,330 --> 01:01:25,190
So that follows from
two different things.
1014
01:01:25,190 --> 01:01:27,440
It follows from
RPI III invariance.
1015
01:01:30,170 --> 01:01:32,060
So remember that
RPI III invariance
1016
01:01:32,060 --> 01:01:35,300
said that you should have the
same number of n's and n bars.
1017
01:01:35,300 --> 01:01:40,220
And remember-- OK, so that's
one thing that you have to use.
1018
01:01:40,220 --> 01:01:42,080
That tells you that
you need to get ratios.
1019
01:01:42,080 --> 01:01:43,690
Well, the z's are
already ratios.
1020
01:01:43,690 --> 01:01:48,290
So you might say, well,
that should be fine.
1021
01:01:48,290 --> 01:01:52,525
The z's are already ratios
between the momentum
1022
01:01:52,525 --> 01:01:53,900
and the operator
and the momentum
1023
01:01:53,900 --> 01:01:55,942
and the state, the minus
momentum of the operator
1024
01:01:55,942 --> 01:02:01,970
over the minus momentum
of the state, right?
1025
01:02:01,970 --> 01:02:03,230
And this is a minus momentum.
1026
01:02:03,230 --> 01:02:04,320
That's a minus momentum.
1027
01:02:04,320 --> 01:02:06,425
So the z's are
RPI III invariant.
1028
01:02:06,425 --> 01:02:08,948
So that doesn't seem
like it would imply this.
1029
01:02:08,948 --> 01:02:10,490
But there's one
other thing you know,
1030
01:02:10,490 --> 01:02:13,790
and that is that it can't
depend on the state momentum.
1031
01:02:13,790 --> 01:02:15,500
I could have taken a proton.
1032
01:02:15,500 --> 01:02:17,180
I could have taken a quark.
1033
01:02:17,180 --> 01:02:18,800
And the result for
the renormalization
1034
01:02:18,800 --> 01:02:22,580
shouldn't depend on
what state I'm taking.
1035
01:02:22,580 --> 01:02:24,110
And this combination
where I have
1036
01:02:24,110 --> 01:02:28,220
d xi prime xi prime with a xi
over xi prime, the p minuses
1037
01:02:28,220 --> 01:02:28,970
cancel out.
1038
01:02:48,410 --> 01:02:50,960
So if I were to do the
whole thing with a proton
1039
01:02:50,960 --> 01:02:53,393
state rather than a
quark state, then I
1040
01:02:53,393 --> 01:02:55,310
should still get the
same anomalous dimension.
1041
01:02:55,310 --> 01:02:58,100
And in order for
that to be true,
1042
01:02:58,100 --> 01:03:00,600
it has to depend on the ratio.
1043
01:03:00,600 --> 01:03:07,050
And that ratio is then just
a ratio of the bare operator
1044
01:03:07,050 --> 01:03:09,340
and the renormalized operator.
1045
01:03:09,340 --> 01:03:12,960
It's like saying, if
you had O of omega,
1046
01:03:12,960 --> 01:03:17,610
there is a convolution
of z with an omega
1047
01:03:17,610 --> 01:03:24,530
over omega prime or
something, with O omega
1048
01:03:24,530 --> 01:03:27,140
prime renormalized.
1049
01:03:27,140 --> 01:03:29,540
And if I had done it in
an operator level and not
1050
01:03:29,540 --> 01:03:32,840
even written states,
then it would really just
1051
01:03:32,840 --> 01:03:35,487
be RPI III invariance, OK?
1052
01:03:35,487 --> 01:03:37,070
Because I wrote it
in terms of states,
1053
01:03:37,070 --> 01:03:38,737
there was this other
momentum available,
1054
01:03:38,737 --> 01:03:42,890
but I'm not allowed
to have that really
1055
01:03:42,890 --> 01:03:45,400
be playing a part
of the discussion.
1056
01:03:48,880 --> 01:03:52,550
So given that formula, then
I can expand to one loop.
1057
01:03:52,550 --> 01:03:55,450
So this guy I think of as
having a tree-level result.
1058
01:03:55,450 --> 01:03:57,220
This guy is a
matrix element that
1059
01:03:57,220 --> 01:04:00,500
has a tree-level and
one-loop result as well.
1060
01:04:00,500 --> 01:04:05,380
So if they're both tree
level, I get delta, 1 minus z.
1061
01:04:10,730 --> 01:04:38,620
And in some kind of obvious
notation, up to one-loop order,
1062
01:04:38,620 --> 01:04:41,558
I can write it out
formally like that.
1063
01:04:41,558 --> 01:04:43,600
And then I know what these
tree-level things are.
1064
01:04:43,600 --> 01:04:45,392
This guy's a delta
function, and this guy's
1065
01:04:45,392 --> 01:04:46,670
also a delta function.
1066
01:04:46,670 --> 01:04:47,920
So I can just do the integral.
1067
01:05:00,570 --> 01:05:03,180
And it really is pretty simple.
1068
01:05:03,180 --> 01:05:04,950
All the 1 over epsilon
terms are just z.
1069
01:05:08,090 --> 01:05:13,710
And what's left would be
associated to this guy
1070
01:05:13,710 --> 01:05:15,392
in perturbation theory.
1071
01:05:15,392 --> 01:05:17,100
But if we want to do
the renormalization,
1072
01:05:17,100 --> 01:05:21,600
we just need the z and
not worry about that.
1073
01:05:21,600 --> 01:05:23,580
So we read off
from over here what
1074
01:05:23,580 --> 01:05:28,120
z is because z is
just this right there.
1075
01:05:36,130 --> 01:05:42,050
So z-- OK, z is just this thing.
1076
01:05:47,878 --> 01:05:49,670
So when I put the
tree-level piece together
1077
01:05:49,670 --> 01:05:51,795
with the one-loop piece,
then this thing is just z.
1078
01:05:54,290 --> 01:05:56,240
And then I compute the
anomalous dimension
1079
01:05:56,240 --> 01:05:57,890
by taking mu d by d mu of it--
1080
01:06:00,960 --> 01:06:09,050
and that hits the alpha,
so that kills the epsilon
1081
01:06:09,050 --> 01:06:12,220
and gives me a factor
of 2 and a minus sign.
1082
01:06:12,220 --> 01:06:15,665
But the anomalous
dimension, gamma qq--
1083
01:06:20,030 --> 01:06:23,800
so there was a 1 over xi prime.
1084
01:06:23,800 --> 01:06:25,925
And then it was
minus mu d by d mu.
1085
01:06:31,060 --> 01:06:33,550
And if I plug it in the
formula that we have,
1086
01:06:33,550 --> 01:06:38,120
zqq of [? xi over ?]
[? xi ?] prime.
1087
01:06:38,120 --> 01:06:39,460
So there's an a minus here.
1088
01:06:39,460 --> 01:06:41,650
There's a minus there.
1089
01:06:41,650 --> 01:06:44,290
And the 2 epsilon cancels
this 2 and that epsilon.
1090
01:06:46,778 --> 01:06:48,320
And this 1 over xi
prime is the thing
1091
01:06:48,320 --> 01:06:50,605
that we needed to make
the measure RPI invariant.
1092
01:06:53,220 --> 01:07:00,150
So in this notation,
our original notation,
1093
01:07:00,150 --> 01:07:03,510
putting all the pieces
together and being careful
1094
01:07:03,510 --> 01:07:17,260
about beta functions, which I
was mostly suppressing, that's
1095
01:07:17,260 --> 01:07:21,010
the result. OK, so this
is the function of xi
1096
01:07:21,010 --> 01:07:23,440
over xi prime, which
I've just written as z.
1097
01:07:23,440 --> 01:07:26,950
And then there's
some beta functions
1098
01:07:26,950 --> 01:07:29,020
that are setting the
boundaries for the integral.
1099
01:07:29,020 --> 01:07:31,400
And that comes also
out of the calculation.
1100
01:07:31,400 --> 01:07:33,490
And that's the quark
one-loop splitting function.
1101
01:07:36,680 --> 01:07:39,070
So if we've done the gluon
from that other diagram,
1102
01:07:39,070 --> 01:07:40,510
that we've got the mixing term.
1103
01:07:48,642 --> 01:07:50,600
OK, so this is the one-loop
anomalous dimension
1104
01:07:50,600 --> 01:07:51,975
for the PDF, and
it's really just
1105
01:07:51,975 --> 01:07:56,010
doing operator renormalization,
calculating one-loop diagrams
1106
01:07:56,010 --> 01:07:57,010
in the effective theory.
1107
01:08:00,180 --> 01:08:00,810
Questions?
1108
01:08:09,035 --> 01:08:09,535
OK.
1109
01:08:12,090 --> 01:08:15,660
So one question
that you can ask,
1110
01:08:15,660 --> 01:08:17,609
which is an
interesting question,
1111
01:08:17,609 --> 01:08:22,649
is when we did this
result for the DIS,
1112
01:08:22,649 --> 01:08:25,050
we got this convolution
between the hard function
1113
01:08:25,050 --> 01:08:27,359
and the Parton
distribution function.
1114
01:08:27,359 --> 01:08:29,250
And you can ask,
why did that happen
1115
01:08:29,250 --> 01:08:31,470
and, in general, is there
a way of characterizing
1116
01:08:31,470 --> 01:08:32,760
when it could possibly happen?
1117
01:08:36,547 --> 01:08:38,630
Because if you think about
the answer that we got,
1118
01:08:38,630 --> 01:08:41,250
it was just Wilson
coefficient times operator.
1119
01:08:41,250 --> 01:08:43,191
And the really only
nontrivial thing about it
1120
01:08:43,191 --> 01:08:44,899
was that there was
this one momentum that
1121
01:08:44,899 --> 01:08:46,899
could kind of trade back
and forth between them.
1122
01:08:46,899 --> 01:08:50,450
There was an integral
in the answer.
1123
01:08:50,450 --> 01:08:52,069
And actually,
power counting even
1124
01:08:52,069 --> 01:08:57,270
constrains how those integrals
can, in general, show up.
1125
01:08:57,270 --> 01:09:02,040
So if you ask most generally
what could possibly happen--
1126
01:09:02,040 --> 01:09:04,160
and just thinking about
the power counting
1127
01:09:04,160 --> 01:09:06,020
for the degrees of
freedom actually
1128
01:09:06,020 --> 01:09:08,180
tells us what type of
integrals can show up
1129
01:09:08,180 --> 01:09:09,470
in factorization theorems.
1130
01:09:33,396 --> 01:09:34,979
This is constrained
by power counting.
1131
01:09:42,920 --> 01:09:45,170
I keep forgetting to say
that there's a makeup lecture
1132
01:09:45,170 --> 01:09:45,560
tomorrow.
1133
01:09:45,560 --> 01:09:47,359
I sent around an email,
but I should have--
1134
01:09:52,569 --> 01:09:54,279
tomorrow, this room at 10:00 AM.
1135
01:10:00,580 --> 01:10:02,890
And the lecture next week
is canceled on Tuesday.
1136
01:10:02,890 --> 01:10:05,740
That's why we have a
makeup lecture tomorrow.
1137
01:10:05,740 --> 01:10:11,510
OK, so in what way is it
constrained by power counting?
1138
01:10:11,510 --> 01:10:14,420
So if you think about the
degrees of freedom that we had,
1139
01:10:14,420 --> 01:10:20,942
say, for SCET I, then we had
hard, collinear, and soft--
1140
01:10:20,942 --> 01:10:23,080
so let's just take a
simple case with only one
1141
01:10:23,080 --> 01:10:25,180
type of collinear--
1142
01:10:25,180 --> 01:10:27,970
hard, collinear, and ultrasoft.
1143
01:10:27,970 --> 01:10:32,260
And the p mu of these guys in
terms of plus, minus, and perp
1144
01:10:32,260 --> 01:10:33,422
components--
1145
01:10:36,254 --> 01:10:38,552
I should be more
fancy about this.
1146
01:10:42,172 --> 01:10:43,160
[INAUDIBLE]
1147
01:10:46,130 --> 01:10:48,620
So if you think about just
power counting for the momentum,
1148
01:10:48,620 --> 01:10:49,385
it was as follows.
1149
01:10:52,850 --> 01:10:56,810
Factorization was separating
these different things
1150
01:10:56,810 --> 01:10:58,100
into different objects.
1151
01:10:58,100 --> 01:11:00,050
We had a Wilson
coefficient for the hard.
1152
01:11:00,050 --> 01:11:02,600
In the case we just did, we only
had a proton matrix element.
1153
01:11:02,600 --> 01:11:04,642
For the collinear, we
didn't have any ultrasofts.
1154
01:11:04,642 --> 01:11:07,100
If we put the ultrasofts in,
they would have all cancelled
1155
01:11:07,100 --> 01:11:07,730
away.
1156
01:11:07,730 --> 01:11:10,250
We wouldn't have seen any
ultrasofts showing up.
1157
01:11:10,250 --> 01:11:12,620
And that's because the
operator we were dealing with,
1158
01:11:12,620 --> 01:11:15,530
the Wilson lines would just have
cancelled completely out of it.
1159
01:11:15,530 --> 01:11:17,570
But actually, it turns out,
for deep inelastic scattering,
1160
01:11:17,570 --> 01:11:19,362
that you shouldn't even
include ultrasofts.
1161
01:11:19,362 --> 01:11:22,140
They're not a good degree
of freedom to include there.
1162
01:11:22,140 --> 01:11:24,410
So really, for the process
that we were talking about,
1163
01:11:24,410 --> 01:11:27,830
you really should
only take those two.
1164
01:11:27,830 --> 01:11:31,430
But anyway, more generally
in some other process,
1165
01:11:31,430 --> 01:11:33,590
you would have these
three different things.
1166
01:11:33,590 --> 01:11:35,360
And the way that
convolutions can show up
1167
01:11:35,360 --> 01:11:39,080
is simply who can trade
momentum with who.
1168
01:11:39,080 --> 01:11:43,940
So this is plus momenta, minus
momenta, and perp momentum.
1169
01:11:43,940 --> 01:11:46,230
And in order for
momentum to be exchanged,
1170
01:11:46,230 --> 01:11:48,160
they have to be
of the same size.
1171
01:11:48,160 --> 01:11:54,020
So these guys here are the same
size and they can be exchanged,
1172
01:11:54,020 --> 01:11:56,910
and that's exactly what showed
up in our DIS factorization
1173
01:11:56,910 --> 01:11:57,410
theorem.
1174
01:11:57,410 --> 01:12:00,617
The hard Wilson coefficients
exchanged minus momentum
1175
01:12:00,617 --> 01:12:03,200
with the collinear [INAUDIBLE]
because they are the same order
1176
01:12:03,200 --> 01:12:04,940
in the power counting.
1177
01:12:04,940 --> 01:12:07,213
In another case,
in a more general
1178
01:12:07,213 --> 01:12:08,630
or in some other
example, we might
1179
01:12:08,630 --> 01:12:13,222
find that there was
nontrivial ultrasoft stuff,
1180
01:12:13,222 --> 01:12:14,680
and then we could
get a convolution
1181
01:12:14,680 --> 01:12:17,100
in the plus momentum and
collinear and ultrasoft
1182
01:12:17,100 --> 01:12:18,350
because they're the same size.
1183
01:12:22,240 --> 01:12:24,910
And so that's a
pretty simple way
1184
01:12:24,910 --> 01:12:27,400
of thinking about why those
integrals can possibly show up.
1185
01:12:27,400 --> 01:12:29,680
It's just because the two
sectors can talk to each other
1186
01:12:29,680 --> 01:12:31,722
because they have momenta
that are the same size.
1187
01:12:31,722 --> 01:12:34,300
And then the rest is about
momentum conservation
1188
01:12:34,300 --> 01:12:37,660
because momentum conservation
places nontrivial constraints.
1189
01:12:37,660 --> 01:12:39,340
And we saw in the DIS
example that there
1190
01:12:39,340 --> 01:12:41,080
were two omegas to
start, but one of them
1191
01:12:41,080 --> 01:12:43,497
was projected to 0 because it
was a forward matrix element
1192
01:12:43,497 --> 01:12:47,990
and we only had one integral.
1193
01:12:47,990 --> 01:12:51,710
And that also has
analogs elsewhere.
1194
01:12:51,710 --> 01:12:56,240
If we do SCET II, which we
haven't talked about yet--
1195
01:12:56,240 --> 01:12:59,330
we did talk about what the
degrees of freedom were.
1196
01:12:59,330 --> 01:13:05,210
And again, if I try
to make it completely
1197
01:13:05,210 --> 01:13:08,487
generic for some examples
that we'll treat later,
1198
01:13:08,487 --> 01:13:10,070
then I can write
down something that's
1199
01:13:10,070 --> 01:13:15,600
a slightly extended version of
what we talked about so far.
1200
01:13:15,600 --> 01:13:19,850
So I can have Q, Q,
Q again for the hard.
1201
01:13:19,850 --> 01:13:22,050
And then I can
have my collinear,
1202
01:13:22,050 --> 01:13:28,580
which is Q lambda squared, Q, Q
lambda, and then soft, which is
1203
01:13:28,580 --> 01:13:32,943
Q lambda, Q lambda, Q lambda.
1204
01:13:32,943 --> 01:13:34,610
And it turns out that
sometimes, there's
1205
01:13:34,610 --> 01:13:38,600
also another mode which
we haven't talked about,
1206
01:13:38,600 --> 01:13:46,395
but I'll include it
for completeness, which
1207
01:13:46,395 --> 01:13:48,430
is kind of a
collinear mode that's
1208
01:13:48,430 --> 01:13:50,440
in between the
low-energy collinear
1209
01:13:50,440 --> 01:13:52,896
mode and the high-energy
collinear mode.
1210
01:13:52,896 --> 01:13:55,030
AUDIENCE: Do you mean the
square root of lambda?
1211
01:13:55,030 --> 01:13:58,570
IAIN STEWART: Yeah, square
root of lambda, sorry.
1212
01:13:58,570 --> 01:14:01,375
Yeah, otherwise my
dimensions are wrong.
1213
01:14:05,830 --> 01:14:07,030
Yeah.
1214
01:14:07,030 --> 01:14:09,160
So again here, you can just--
1215
01:14:09,160 --> 01:14:11,998
I mean, the reason I was
extending this is just to,
1216
01:14:11,998 --> 01:14:14,540
again, argue that it's kind of
simple to see what can happen.
1217
01:14:14,540 --> 01:14:17,593
So in general, when you
think about convolutions
1218
01:14:17,593 --> 01:14:19,510
in the hard momentum,
it could be in this case
1219
01:14:19,510 --> 01:14:20,950
between these three modes.
1220
01:14:20,950 --> 01:14:22,600
There could be some integrals.
1221
01:14:22,600 --> 01:14:25,810
And then look where else
there can be something.
1222
01:14:25,810 --> 01:14:29,035
So this and this
are the same size.
1223
01:14:31,610 --> 01:14:35,090
And this and this
are the same size.
1224
01:14:35,090 --> 01:14:41,740
So in general, we
can have convolutions
1225
01:14:41,740 --> 01:14:44,830
in general between
all these things,
1226
01:14:44,830 --> 01:14:47,170
between the guys that
are the same size.
1227
01:14:47,170 --> 01:14:49,420
But that's the most complicated
thing that can happen.
1228
01:14:49,420 --> 01:14:51,760
It can't be more
complicated than that.
1229
01:14:51,760 --> 01:14:54,080
And you see that
in some examples,
1230
01:14:54,080 --> 01:14:57,628
you either have the purple
or the orange, but not both.
1231
01:14:57,628 --> 01:14:59,170
That's kind of
typical that you don't
1232
01:14:59,170 --> 01:15:02,920
get the most complicated thing.
1233
01:15:02,920 --> 01:15:05,955
So when you have results from
observables that tell you
1234
01:15:05,955 --> 01:15:07,330
how these things
couple together,
1235
01:15:07,330 --> 01:15:09,050
those are called
factorization theorems.
1236
01:15:09,050 --> 01:15:11,050
And in the effective
theory, because you sort of
1237
01:15:11,050 --> 01:15:13,973
define the modes, separated
them at the start,
1238
01:15:13,973 --> 01:15:16,390
you're kind of very quickly
getting to these factorization
1239
01:15:16,390 --> 01:15:16,890
theorems.
1240
01:15:27,090 --> 01:15:29,870
Let's see.
1241
01:15:29,870 --> 01:15:33,560
So we're going to deal with a
bunch of different examples.
1242
01:15:33,560 --> 01:15:37,380
And I decided that I'm going to
do it in the following order.
1243
01:15:37,380 --> 01:15:39,870
So we're going to
do the next exam--
1244
01:15:39,870 --> 01:15:42,560
so we're going to do
a bunch of examples
1245
01:15:42,560 --> 01:15:50,030
in order to see the range of
possibilities that can happen.
1246
01:15:53,710 --> 01:15:58,540
And so I'm going to stick
with SCET I for next lecture.
1247
01:15:58,540 --> 01:16:02,670
So the next example we'll do,
which I'll call example one,
1248
01:16:02,670 --> 01:16:06,650
is we'll do our
[? dijet ?] production.
1249
01:16:16,650 --> 01:16:20,540
So this is a SCET I situation.
1250
01:16:20,540 --> 01:16:22,410
And the difference
between the example--
1251
01:16:22,410 --> 01:16:24,670
so so far, what we
did is we did DIS.
1252
01:16:24,670 --> 01:16:26,990
DIS actually it was so
simple because it only
1253
01:16:26,990 --> 01:16:29,420
had two degrees of freedom
that you could kind of either
1254
01:16:29,420 --> 01:16:32,000
think of it as
SCET I or SCET II.
1255
01:16:32,000 --> 01:16:34,010
I mean, technically,
it's more like SCET II,
1256
01:16:34,010 --> 01:16:35,450
but it behaves like SCET I.
1257
01:16:35,450 --> 01:16:36,735
But there's no ultrasofts.
1258
01:16:36,735 --> 01:16:38,360
And remember, it's
ultrasofts and softs
1259
01:16:38,360 --> 01:16:41,120
that are making the
distinction between the two.
1260
01:16:41,120 --> 01:16:43,578
So if you just have
this mode and this mode,
1261
01:16:43,578 --> 01:16:45,620
it's not really any
difference between calling it
1262
01:16:45,620 --> 01:16:46,790
SCET I or SCET II.
1263
01:16:46,790 --> 01:16:48,650
So it's just SCET.
1264
01:16:48,650 --> 01:16:51,110
So e plus, e minus
to [? dijets ?]
1265
01:16:51,110 --> 01:16:54,020
will be an SCET I example
which has ultrasofts.
1266
01:16:58,330 --> 01:17:00,340
And actually, what
we'll find in this case
1267
01:17:00,340 --> 01:17:02,695
is that it will be the purple.
1268
01:17:05,220 --> 01:17:09,040
We'll get a purple convolution.
1269
01:17:09,040 --> 01:17:10,420
We'll see how that happens.
1270
01:17:10,420 --> 01:17:12,370
And we won't actually--
momentum conservation
1271
01:17:12,370 --> 01:17:14,370
will rule out the possibility
of the orange one.
1272
01:17:17,360 --> 01:17:19,370
So we'll see the
opposite situation
1273
01:17:19,370 --> 01:17:24,470
where it could be that we
have ultrasoft modes as well,
1274
01:17:24,470 --> 01:17:26,600
but then we only
get a convolution
1275
01:17:26,600 --> 01:17:29,060
with those ultrasoft in
the factorization theorem
1276
01:17:29,060 --> 01:17:30,830
and not with the [INAUDIBLE].
1277
01:17:36,760 --> 01:17:40,540
And then we'll turn to SCET II.
1278
01:17:40,540 --> 01:17:46,320
And I haven't totally decided
what processes I'll do,
1279
01:17:46,320 --> 01:17:47,990
but I think I'll do
the following ones.
1280
01:17:52,700 --> 01:17:55,000
So one thing that you can
do, which is pretty simple,
1281
01:17:55,000 --> 01:17:59,110
is to look at something called
the photon-pion form factor.
1282
01:17:59,110 --> 01:18:01,960
So real photon-to-pion
transition
1283
01:18:01,960 --> 01:18:04,810
through another
virtual photon, but you
1284
01:18:04,810 --> 01:18:08,320
can think of this happening
through a diagram like this
1285
01:18:08,320 --> 01:18:11,990
with two quarks,
one of them that's
1286
01:18:11,990 --> 01:18:14,510
off-shell and one of
them that's on-shell.
1287
01:18:14,510 --> 01:18:17,110
This is pi 0.
1288
01:18:17,110 --> 01:18:18,772
So this is a SCET II example.
1289
01:18:18,772 --> 01:18:20,980
But again, it's pretty simple
because it's just going
1290
01:18:20,980 --> 01:18:25,180
to involve one hadronic object.
1291
01:18:25,180 --> 01:18:27,580
And actually, it will
just, in this case,
1292
01:18:27,580 --> 01:18:28,945
have collinear modes.
1293
01:18:32,740 --> 01:18:36,820
We can set things up so
the pion is n-collinear,
1294
01:18:36,820 --> 01:18:41,170
and then we have hard modes.
1295
01:18:47,710 --> 01:18:50,250
So that's one example we'll do.
1296
01:18:50,250 --> 01:18:55,240
Another example we'll
do is B to D pi.
1297
01:18:55,240 --> 01:18:57,190
And here, the B
and the D are soft.
1298
01:18:57,190 --> 01:19:02,980
Remember, we talked about this
one, and the pion is collinear.
1299
01:19:02,980 --> 01:19:05,950
And then we have hard modes.
1300
01:19:05,950 --> 01:19:09,280
So this is kind of like,
in some sense, a DIS,
1301
01:19:09,280 --> 01:19:12,490
but it's an exclusive
process, not an inclusive one.
1302
01:19:12,490 --> 01:19:14,740
And so actually, all
the tools that we use,
1303
01:19:14,740 --> 01:19:15,820
which were kind of--
1304
01:19:15,820 --> 01:19:18,700
in DIS, it's the most inclusive
process you can think of.
1305
01:19:18,700 --> 01:19:20,920
It's deep inelastic scattering.
1306
01:19:20,920 --> 01:19:22,750
The I is for "inclusive."
1307
01:19:22,750 --> 01:19:24,983
Here, we're doing something
completely exclusive,
1308
01:19:24,983 --> 01:19:27,400
but we'll see that all the
things that we've been thinking
1309
01:19:27,400 --> 01:19:29,233
about, which is just
separation of collinear
1310
01:19:29,233 --> 01:19:31,750
modes and hard modes, will just
go through for that process
1311
01:19:31,750 --> 01:19:33,220
too.
1312
01:19:33,220 --> 01:19:35,470
So the effective
theory, the difference
1313
01:19:35,470 --> 01:19:40,630
is that in this case, you will
not be taking the amplitude
1314
01:19:40,630 --> 01:19:41,130
squared.
1315
01:19:41,130 --> 01:19:42,922
You won't be looking
at forward scattering.
1316
01:19:42,922 --> 01:19:45,310
The forward scattering was
what was making it inclusive.
1317
01:19:45,310 --> 01:19:47,440
We were summing over
all the final states.
1318
01:19:47,440 --> 01:19:49,687
Here, there's only
one final state.
1319
01:19:49,687 --> 01:19:52,270
So the difference between this
example and the one we just did
1320
01:19:52,270 --> 01:20:04,210
is that in this case, we'll
factor the amplitude, not
1321
01:20:04,210 --> 01:20:05,260
the squared amplitude.
1322
01:20:08,442 --> 01:20:10,900
But other than that, it'll look
very similar to the example
1323
01:20:10,900 --> 01:20:12,880
that we did for DIS.
1324
01:20:12,880 --> 01:20:14,538
B to D pi, then, is
an SCET II example
1325
01:20:14,538 --> 01:20:17,080
where we make things a little
more complicated because now we
1326
01:20:17,080 --> 01:20:19,720
have soft, collinear,
and hard modes.
1327
01:20:19,720 --> 01:20:23,150
And we'll see what
happens there.
1328
01:20:23,150 --> 01:20:25,120
And then I'll do some
more examples after that.
1329
01:20:25,120 --> 01:20:30,055
But let me say we'll
do some LHC examples.
1330
01:20:32,860 --> 01:20:44,210
And I think we'll
also do broadening,
1331
01:20:44,210 --> 01:20:47,240
which is another e plus,
e minus observable.
1332
01:20:47,240 --> 01:20:49,400
All right, so that's
where we're going,
1333
01:20:49,400 --> 01:20:53,280
and we'll start
going there next time
1334
01:20:53,280 --> 01:20:57,355
by talking about
[? dijets ?] and SCET I.