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IAIN STEWART: --play
with each other.
9
00:00:22,540 --> 00:00:24,350
We did the standard model
as an effective field
10
00:00:24,350 --> 00:00:25,780
theory, higher
dimension operators
11
00:00:25,780 --> 00:00:27,220
in the standard model.
12
00:00:27,220 --> 00:00:29,950
And then we started talking
about taking the standard model
13
00:00:29,950 --> 00:00:33,340
as a theory one and
removing things from it,
14
00:00:33,340 --> 00:00:35,380
in particular
constructing what's
15
00:00:35,380 --> 00:00:38,590
called the weak Hamiltonian
by removing the top W, Z,
16
00:00:38,590 --> 00:00:41,260
and Higgs from the
standard model.
17
00:00:41,260 --> 00:00:45,670
And last time, we were focusing
on the anomalous dimensions
18
00:00:45,670 --> 00:00:48,640
and things about
renormalization.
19
00:00:48,640 --> 00:00:51,790
So we had this equation
for the weak Hamiltonian
20
00:00:51,790 --> 00:00:57,220
for a particular case
of b goes to c u bar d.
21
00:00:57,220 --> 00:00:58,900
That was the case
that we decided
22
00:00:58,900 --> 00:01:01,660
to study rather than
the full Hamiltonian.
23
00:01:01,660 --> 00:01:03,130
So there was some pre-factor.
24
00:01:03,130 --> 00:01:06,680
We had two operators with Wilson
coefficients and operators.
25
00:01:06,680 --> 00:01:10,790
These are four-fermion
operators.
26
00:01:10,790 --> 00:01:12,040
And there was different bases.
27
00:01:12,040 --> 00:01:14,577
We could write them
in the bare form,
28
00:01:14,577 --> 00:01:16,660
or we could write them in
renormalized coefficient
29
00:01:16,660 --> 00:01:18,650
and renormalized operator.
30
00:01:18,650 --> 00:01:20,800
And then we could do it
either in the 1, 2 basis
31
00:01:20,800 --> 00:01:22,970
or the plus minus basis.
32
00:01:22,970 --> 00:01:25,330
So the plus minus basis is
just linear combinations
33
00:01:25,330 --> 00:01:26,050
of the 1, 2.
34
00:01:29,410 --> 00:01:31,000
If you're in the 1,
2 basis, then you
35
00:01:31,000 --> 00:01:32,300
have this mixing matrix.
36
00:01:32,300 --> 00:01:33,325
So it's a 2 by 2 matrix.
37
00:01:33,325 --> 00:01:36,760
So if you're in the
plus minus basis,
38
00:01:36,760 --> 00:01:42,100
then at least, at
the lowest order,
39
00:01:42,100 --> 00:01:44,540
it's a simple product equation.
40
00:01:44,540 --> 00:01:46,690
So plus doesn't
interfere with minus.
41
00:01:50,610 --> 00:01:51,110
OK.
42
00:01:51,110 --> 00:01:56,730
So that's where we got to,
and we'll just continue today.
43
00:01:56,730 --> 00:01:59,390
So we have these anomalous
dimension equations
44
00:01:59,390 --> 00:02:00,493
for the operators.
45
00:02:00,493 --> 00:02:02,660
We can also write down
anomalous dimension equations
46
00:02:02,660 --> 00:02:04,770
for the Wilson coefficients.
47
00:02:04,770 --> 00:02:06,396
How do we do that?
48
00:02:06,396 --> 00:02:10,542
Well, the way that we do that
is we make use of the fact
49
00:02:10,542 --> 00:02:14,060
that, if we look at the first
line here, there's no mu.
50
00:02:14,060 --> 00:02:17,173
So the Hamiltonian
is mu independent.
51
00:02:17,173 --> 00:02:19,340
That means that the mu
dependence of the coefficient
52
00:02:19,340 --> 00:02:23,420
cancels the mu dependence
of the operator.
53
00:02:23,420 --> 00:02:26,458
And you can use that to
take an anomalous dimension
54
00:02:26,458 --> 00:02:28,000
equation for the
operator and turn it
55
00:02:28,000 --> 00:02:29,729
into one for the coefficient.
56
00:02:40,210 --> 00:02:42,365
So last time, we
talked about the fact
57
00:02:42,365 --> 00:02:43,990
that the normalization
of the operators
58
00:02:43,990 --> 00:02:45,310
was equivalent
to-- you can think
59
00:02:45,310 --> 00:02:46,330
about it two different ways.
60
00:02:46,330 --> 00:02:48,538
There's renormalization of
operators, renormalization
61
00:02:48,538 --> 00:02:49,240
of coefficients.
62
00:02:49,240 --> 00:02:51,633
Likewise, you can think
of anomalous dimensions
63
00:02:51,633 --> 00:02:53,050
is either running
the coefficients
64
00:02:53,050 --> 00:02:54,160
or running the operators.
65
00:02:54,160 --> 00:02:56,320
So those are equivalent things.
66
00:02:56,320 --> 00:02:59,260
And the thing that
makes them equivalent
67
00:02:59,260 --> 00:03:04,720
is just imposing that the
derivative with respect
68
00:03:04,720 --> 00:03:06,310
to mu of the Hamiltonian is 0.
69
00:03:15,860 --> 00:03:18,890
So if I use my equation up
here for anomalous dimension
70
00:03:18,890 --> 00:03:27,080
of the operator, I get, with my
sign convention, a minus sign
71
00:03:27,080 --> 00:03:28,715
here.
72
00:03:28,715 --> 00:03:30,560
Or I had a minus sign here.
73
00:03:30,560 --> 00:03:31,640
I had no minus sign here.
74
00:03:31,640 --> 00:03:34,010
These things are conventions.
75
00:03:34,010 --> 00:03:37,650
I picked some convention,
and I'll stick with it.
76
00:03:37,650 --> 00:03:41,235
So from this
equation, we basically
77
00:03:41,235 --> 00:03:43,610
have an equation that has to
be true for the coefficients
78
00:03:43,610 --> 00:03:45,830
if we think of just
stripping off the operator.
79
00:04:05,540 --> 00:04:08,737
So we could write it this
way, just reading it off right
80
00:04:08,737 --> 00:04:10,820
from here, just putting
this guy on the other side
81
00:04:10,820 --> 00:04:13,640
and then reading it off,
dropping the operator.
82
00:04:13,640 --> 00:04:15,800
Or we could write
it this way if we
83
00:04:15,800 --> 00:04:17,630
wanted to write
it in a way that's
84
00:04:17,630 --> 00:04:19,610
more similar to this
equation up here, where
85
00:04:19,610 --> 00:04:22,780
you have the anomalous dimension
matrix times the coefficient.
86
00:04:22,780 --> 00:04:24,620
It's either from the
right or from the left,
87
00:04:24,620 --> 00:04:27,593
and it's just a
matter of transposing.
88
00:04:27,593 --> 00:04:29,510
So the anomalous dimension
for the coefficient
89
00:04:29,510 --> 00:04:32,660
is determined from the
one, this one here.
90
00:04:32,660 --> 00:04:34,160
If you know it,
then you immediately
91
00:04:34,160 --> 00:04:37,490
know the one for the
coefficient, which
92
00:04:37,490 --> 00:04:40,040
shouldn't be surprising given
that we could think of the Z
93
00:04:40,040 --> 00:04:43,820
factors as being related
to renormalization factors.
94
00:04:43,820 --> 00:04:47,970
Anomalous dimensions, therefore,
should also be related.
95
00:04:47,970 --> 00:04:51,030
OK, so we'll solve
this equation.
96
00:04:51,030 --> 00:04:53,080
It's a little bit
simpler to think about.
97
00:04:53,080 --> 00:04:54,538
Although we could
have equivalently
98
00:04:54,538 --> 00:04:56,687
solve the operator equation.
99
00:04:56,687 --> 00:04:58,020
So how do we solve the equation?
100
00:05:03,490 --> 00:05:07,300
Well, we go over to
our plus minus basis.
101
00:05:07,300 --> 00:05:11,850
So take the coefficient
of either C+ or C-.
102
00:05:11,850 --> 00:05:13,740
And the equation
that we need to solve
103
00:05:13,740 --> 00:05:17,115
can be written as follows.
104
00:05:30,510 --> 00:05:31,750
It's a simple equation.
105
00:05:31,750 --> 00:05:34,790
And I can even take the C,
which is on the right-hand side,
106
00:05:34,790 --> 00:05:36,950
move it over to the left,
if I write log C here.
107
00:05:36,950 --> 00:05:39,560
Because the derivative of the
log is giving me a 1 over C.
108
00:05:39,560 --> 00:05:42,560
If I put it back over here,
it would just multiply.
109
00:05:42,560 --> 00:05:44,330
OK, so that's the analog.
110
00:05:44,330 --> 00:05:46,550
This equation here is the
analog of this equation
111
00:05:46,550 --> 00:05:50,260
here for the operators, but
now for the coefficients.
112
00:05:50,260 --> 00:05:52,760
Obviously, if these are numbers,
there's no transpose to do.
113
00:05:56,050 --> 00:05:58,810
So you have to solve this
equation simultaneously
114
00:05:58,810 --> 00:06:00,350
with another
differential equation.
115
00:06:00,350 --> 00:06:02,698
This is a coupled equation
because alpha also
116
00:06:02,698 --> 00:06:03,865
has a differential equation.
117
00:06:20,150 --> 00:06:23,350
So at lowest order, this is
the beta function equation.
118
00:06:23,350 --> 00:06:27,340
And so if we want to solve,
take into account this equation
119
00:06:27,340 --> 00:06:30,610
and integrate this equation,
the simple trick for doing that
120
00:06:30,610 --> 00:06:34,250
is to make a change of variable.
121
00:06:34,250 --> 00:06:39,040
So we use this equation here
to make a change of variable.
122
00:06:39,040 --> 00:06:41,350
This is a very useful
trick because it actually
123
00:06:41,350 --> 00:06:43,360
works to whatever
order in the expansion
124
00:06:43,360 --> 00:06:46,896
that you might want to work.
125
00:06:46,896 --> 00:06:49,780
So let me explain what the
trick is, and then I'll
126
00:06:49,780 --> 00:06:51,480
explain why that is.
127
00:06:51,480 --> 00:06:53,890
So we're going to change
variables in this equation
128
00:06:53,890 --> 00:06:54,880
from mu to alpha.
129
00:07:01,658 --> 00:07:03,950
You see we could think about
solving this equation here
130
00:07:03,950 --> 00:07:07,820
by integrating, just move
this thing to the other side
131
00:07:07,820 --> 00:07:09,230
and integrate.
132
00:07:09,230 --> 00:07:12,080
But we'd be integrating,
in mu, a function that's
133
00:07:12,080 --> 00:07:14,910
a function of alpha of mu.
134
00:07:14,910 --> 00:07:16,985
So if we can switch
from mu to alpha,
135
00:07:16,985 --> 00:07:19,110
then we'll be just integrating
a function of alpha.
136
00:07:19,110 --> 00:07:21,770
And that's what we're going
to do using this equation.
137
00:07:38,240 --> 00:07:41,050
So if I write it in general,
that's this equality here.
138
00:07:41,050 --> 00:07:45,610
For any function of alpha,
I say that d mu over mu--
139
00:07:45,610 --> 00:07:49,730
just rearranging this equation--
is d alpha over beta of alpha.
140
00:07:49,730 --> 00:07:51,940
So I can switch the
integration d mu over mu
141
00:07:51,940 --> 00:07:55,040
to d alpha over beta of alpha.
142
00:07:55,040 --> 00:07:56,800
And that's exactly
what I want to do
143
00:07:56,800 --> 00:07:59,650
if I want to move this
operator to the other side,
144
00:07:59,650 --> 00:08:01,082
this operation.
145
00:08:01,082 --> 00:08:03,040
If I work at lowest order
in the beta function,
146
00:08:03,040 --> 00:08:06,710
then that's just I just
plug in this result.
147
00:08:06,710 --> 00:08:10,180
So we can switch variables
from mu to alpha.
148
00:08:10,180 --> 00:08:13,360
And if I had high order terms
in this equation and high order
149
00:08:13,360 --> 00:08:16,810
terms in this equation, I
could use the same trick.
150
00:08:16,810 --> 00:08:18,575
I just have other
integrals to do.
151
00:08:18,575 --> 00:08:20,450
And the integrals are
pretty straightforward,
152
00:08:20,450 --> 00:08:24,260
so this is a useful
way of proceeding.
153
00:08:24,260 --> 00:08:24,760
OK.
154
00:08:24,760 --> 00:08:25,750
So what does that do?
155
00:08:25,750 --> 00:08:26,770
So now, let's do that.
156
00:08:26,770 --> 00:08:28,764
Let's move this
over and integrate.
157
00:08:36,669 --> 00:08:41,400
Well, we'll do a definite
integral from mu w up to mu.
158
00:08:46,846 --> 00:08:49,300
So here, we just
have d log this.
159
00:08:49,300 --> 00:08:54,610
Integrate that-- just gives
log between the limits.
160
00:09:06,200 --> 00:09:09,010
And if I didn't change
variable, I would have that.
161
00:09:09,010 --> 00:09:11,950
But if I make the
change variable,
162
00:09:11,950 --> 00:09:13,600
then it becomes a
very simple integral.
163
00:09:36,400 --> 00:09:38,890
And remember that
this guy here is also
164
00:09:38,890 --> 00:09:46,150
just a number times alpha
that we worked out last time.
165
00:09:48,830 --> 00:09:51,590
So this is just d
alpha over alpha.
166
00:09:51,590 --> 00:09:53,410
And that's a simple
logarithmic integral.
167
00:09:53,410 --> 00:09:55,030
Yeah.
168
00:09:55,030 --> 00:09:56,530
AUDIENCE: I don't
know if it matter,
169
00:09:56,530 --> 00:10:00,340
but mu w should be
greater than mu, right?
170
00:10:00,340 --> 00:10:02,380
IAIN STEWART: Mu w should
be greater than mu.
171
00:10:02,380 --> 00:10:03,040
That's right.
172
00:10:03,040 --> 00:10:04,957
AUDIENCE: OK, so you're
just writing integrals
173
00:10:04,957 --> 00:10:06,324
like that to avoid signs?
174
00:10:06,324 --> 00:10:07,596
OK.
175
00:10:07,596 --> 00:10:08,708
I have another question.
176
00:10:08,708 --> 00:10:09,500
IAIN STEWART: Yeah.
177
00:10:09,500 --> 00:10:12,830
AUDIENCE: How do you know
that the anomalous dimensions,
178
00:10:12,830 --> 00:10:15,474
including the beta function,
are only functions of alpha S
179
00:10:15,474 --> 00:10:16,630
rather than [INAUDIBLE].
180
00:10:16,630 --> 00:10:18,530
IAIN STEWART: Ah, yeah.
181
00:10:18,530 --> 00:10:21,750
So I'm sneaking that in here.
182
00:10:21,750 --> 00:10:25,580
So it follows from the
renormalization structure
183
00:10:25,580 --> 00:10:28,070
of this effective field
theory that there's only
184
00:10:28,070 --> 00:10:31,250
single logarithmic divergences.
185
00:10:31,250 --> 00:10:33,890
So in the standard model,
if you're at one loop,
186
00:10:33,890 --> 00:10:36,500
you only have 1 over epsilon
poles for the renormalization.
187
00:10:36,500 --> 00:10:38,470
And you're renormalizing
the coupling.
188
00:10:38,470 --> 00:10:40,220
The same is true of
this effective theory.
189
00:10:40,220 --> 00:10:43,640
At one loop, you only have
1 over epsilon divergences.
190
00:10:43,640 --> 00:10:47,450
And that implies that
your anomalous dimensions
191
00:10:47,450 --> 00:10:50,340
won't depend on anything
more complicated.
192
00:10:50,340 --> 00:10:53,180
We will discuss more
complicated cases in the future,
193
00:10:53,180 --> 00:10:54,740
as you know.
194
00:10:54,740 --> 00:11:00,080
But the structure of this
effective theory and its UV
195
00:11:00,080 --> 00:11:04,460
structure, which I didn't go
into on a lot of detail about,
196
00:11:04,460 --> 00:11:06,510
implies that fact.
197
00:11:06,510 --> 00:11:07,010
Yeah.
198
00:11:07,010 --> 00:11:08,300
AUDIENCE: [INAUDIBLE]
199
00:11:08,300 --> 00:11:09,092
IAIN STEWART: Yeah.
200
00:11:11,540 --> 00:11:12,040
OK.
201
00:11:12,040 --> 00:11:13,860
So do the integral.
202
00:11:13,860 --> 00:11:19,120
There's some pre-factor,
which I'll call a+-.
203
00:11:19,120 --> 00:11:22,100
And then I get a
log, as I mentioned.
204
00:11:27,160 --> 00:11:29,620
And just for the
record, this a+-,
205
00:11:29,620 --> 00:11:32,020
if I put all the factors
together and put in what this
206
00:11:32,020 --> 00:11:43,990
number is, these would be
some factors like this.
207
00:11:43,990 --> 00:11:47,350
And I've put in that Nc is 3.
208
00:11:51,540 --> 00:11:55,340
So this alpha at mu w
and this C+ of mu w,
209
00:11:55,340 --> 00:11:57,960
you should think of mu w as
the boundary condition scale.
210
00:12:02,202 --> 00:12:03,660
So this is a
differential equation.
211
00:12:03,660 --> 00:12:06,330
We needed a boundary
condition to solve it.
212
00:12:06,330 --> 00:12:08,220
And the boundary
condition is the value
213
00:12:08,220 --> 00:12:13,195
of the coefficients
at the scale uw, which
214
00:12:13,195 --> 00:12:14,445
is supposed to be of order Mw.
215
00:12:23,760 --> 00:12:26,100
Typically, what
that means is you
216
00:12:26,100 --> 00:12:28,110
could take a common
choice, which
217
00:12:28,110 --> 00:12:31,020
would be just to take it equal.
218
00:12:31,020 --> 00:12:34,740
Or you could pick twice or half.
219
00:12:34,740 --> 00:12:38,220
And these are the most common
choices that people pick.
220
00:12:41,400 --> 00:12:44,490
So the way that you
should think of that,
221
00:12:44,490 --> 00:12:48,060
this guy in the
denominator, is you
222
00:12:48,060 --> 00:12:52,860
should think that he's really
a fixed order series in alpha
223
00:12:52,860 --> 00:13:02,970
of mu w, something that you
would calculate order by order
224
00:13:02,970 --> 00:13:06,180
and perturbation theory.
225
00:13:06,180 --> 00:13:09,060
And you'd be determining
the boundary condition.
226
00:13:09,060 --> 00:13:13,420
We'll talk about how you would
do that a little later today.
227
00:13:13,420 --> 00:13:16,770
But for now, just think of it
as a series in alpha of mu w.
228
00:13:16,770 --> 00:13:19,500
And it doesn't have
any large logarithms.
229
00:13:19,500 --> 00:13:21,480
And as Elia said,
you want to think
230
00:13:21,480 --> 00:13:26,040
of this mu as some small scale,
some scale that's less than uw
231
00:13:26,040 --> 00:13:27,690
because you're
thinking of evolving
232
00:13:27,690 --> 00:13:30,145
the operators to a scale
less than the scale
233
00:13:30,145 --> 00:13:31,770
where you integrated
out the particles.
234
00:13:34,322 --> 00:13:35,780
I'll draw that
picture in a second.
235
00:13:53,030 --> 00:13:56,470
So we can take the
exponential of that equation,
236
00:13:56,470 --> 00:14:00,220
and then we can write C of
mu is equal to something
237
00:14:00,220 --> 00:14:01,638
that we've determined.
238
00:14:10,610 --> 00:14:12,090
Take the exponential.
239
00:14:12,090 --> 00:14:13,520
Move this guy to the other side.
240
00:14:41,280 --> 00:14:44,910
Remember the a+-
are just numbers.
241
00:14:44,910 --> 00:14:47,730
We can write the solution in
this way, where we determine
242
00:14:47,730 --> 00:14:50,250
this guy by thinking about
doing a matching calculation
243
00:14:50,250 --> 00:14:51,090
at the high scale.
244
00:14:51,090 --> 00:14:54,810
We determine it to be 1/2
already at the lowest order.
245
00:14:54,810 --> 00:14:57,100
So think about
sticking in 1/2 here.
246
00:14:57,100 --> 00:15:00,030
And then this factor
here is what you get
247
00:15:00,030 --> 00:15:02,192
from the renormalization group.
248
00:15:02,192 --> 00:15:03,900
And you can see from
this form right here
249
00:15:03,900 --> 00:15:06,192
that you've summed up an
infinite number of logarithms.
250
00:15:06,192 --> 00:15:07,860
It's exponential of
a number times log.
251
00:15:07,860 --> 00:15:14,280
And if I were to expand that out
in alpha of some fixed scale,
252
00:15:14,280 --> 00:15:16,280
there would be an infinite
series in logarithms.
253
00:15:20,310 --> 00:15:20,810
OK.
254
00:15:20,810 --> 00:15:22,610
So what do we want to
pick this mu to be?
255
00:15:22,610 --> 00:15:29,180
Well, we're thinking about the
process mu goes to c u bar d.
256
00:15:29,180 --> 00:15:31,700
If you think about
this process in nature,
257
00:15:31,700 --> 00:15:36,350
the scale in the initial state
here is the b has a mass.
258
00:15:36,350 --> 00:15:39,668
So you'd like to take this
scale here not at Mw, but down
259
00:15:39,668 --> 00:15:40,460
at the b core mass.
260
00:15:48,290 --> 00:15:50,350
So we want mu to be of order Mb.
261
00:15:55,776 --> 00:15:58,430
And then you have a large
hierarchy because this
262
00:15:58,430 --> 00:16:01,280
is much less than Mw--
263
00:16:01,280 --> 00:16:04,160
5 g of e-ish, 80 g of e.
264
00:16:12,150 --> 00:16:23,620
So our result here sums what are
called the leading logarithms,
265
00:16:23,620 --> 00:16:27,360
which is denoted by LL.
266
00:16:27,360 --> 00:16:30,960
And schematically, the
lowest order term was 1/2.
267
00:16:30,960 --> 00:16:37,130
And if we were to expand
higher order terms
268
00:16:37,130 --> 00:16:39,380
and think about what
logarithms we're talking about,
269
00:16:39,380 --> 00:16:43,130
we're talking about
logarithms of Mw over Mb.
270
00:16:45,930 --> 00:16:47,810
And the series, if
we were to expand it,
271
00:16:47,810 --> 00:16:53,195
would look like
this schematically
272
00:16:53,195 --> 00:16:58,100
without worrying about
the coefficients,
273
00:16:58,100 --> 00:17:02,330
an infinite series where each
term has one alpha and one log.
274
00:17:05,900 --> 00:17:07,480
So the counting
that you're doing
275
00:17:07,480 --> 00:17:11,500
in this type of setup,
where we would think
276
00:17:11,500 --> 00:17:25,020
of using this equation to
go down to the scale Mb,
277
00:17:25,020 --> 00:17:30,845
is that you're counting
this parameter as order 1.
278
00:17:30,845 --> 00:17:32,220
And you're saying,
any time I see
279
00:17:32,220 --> 00:17:34,388
a log of Mw over
Mb times an alpha,
280
00:17:34,388 --> 00:17:36,180
I'm not going to count
that as order alpha.
281
00:17:36,180 --> 00:17:38,010
I'm going to count
that as order 1.
282
00:17:38,010 --> 00:17:40,170
That's why I have to sum
up this infinite series.
283
00:17:46,820 --> 00:17:49,390
So the physical picture
of what we've said here
284
00:17:49,390 --> 00:17:50,617
is the following.
285
00:17:54,395 --> 00:17:55,853
So the basic physical
picture would
286
00:17:55,853 --> 00:17:58,990
be that there's two
scales, Mw an Mb, which
287
00:17:58,990 --> 00:18:00,220
are physical scales.
288
00:18:00,220 --> 00:18:02,590
You want to get rid
of the scale Mw.
289
00:18:02,590 --> 00:18:05,290
You do that by going over to
this electroweak Hamiltonian.
290
00:18:05,290 --> 00:18:06,850
But then you have
to renormalization
291
00:18:06,850 --> 00:18:10,300
group evolve Hamiltonian down to
the scale where you want to do
292
00:18:10,300 --> 00:18:13,300
physics, which is the scale Mb.
293
00:18:13,300 --> 00:18:15,890
And when you do that, there's
some choice in the matter.
294
00:18:15,890 --> 00:18:19,400
And we've been careful to
parameterize that choice.
295
00:18:19,400 --> 00:18:22,420
We said that you
pick a scale that's
296
00:18:22,420 --> 00:18:25,943
of order Mw, which
we called mu w.
297
00:18:25,943 --> 00:18:27,610
And then we said, you
pick another scale
298
00:18:27,610 --> 00:18:30,355
that's of order Mb,
which we called mu.
299
00:18:30,355 --> 00:18:32,230
And you actually do the
renormalization group
300
00:18:32,230 --> 00:18:34,570
between these two.
301
00:18:34,570 --> 00:18:39,400
You could pick Mw and
mu equal exactly Mb.
302
00:18:39,400 --> 00:18:42,580
That would be another
simpler story.
303
00:18:42,580 --> 00:18:45,100
But it is actually
important that,
304
00:18:45,100 --> 00:18:46,960
once you go beyond
the lowest order,
305
00:18:46,960 --> 00:18:49,900
to keep track of the fact
that you have this freedom.
306
00:18:49,900 --> 00:18:53,240
And that's why I've
kept track over here.
307
00:18:53,240 --> 00:18:55,660
So what that means is
that, in terms of counting,
308
00:18:55,660 --> 00:19:05,280
you've counted this,
but logs of mu or Mw
309
00:19:05,280 --> 00:19:07,380
were counted as order 1.
310
00:19:07,380 --> 00:19:11,760
And then down here,
logs of mu over B
311
00:19:11,760 --> 00:19:15,900
are counted as order 1 numbers.
312
00:19:15,900 --> 00:19:21,810
It could be 0, but 0 is
order 1, not enhanced
313
00:19:21,810 --> 00:19:25,230
such that they would compensate
for a factor of alpha.
314
00:19:25,230 --> 00:19:30,810
And this is the renormalization
group evolution or the running
315
00:19:30,810 --> 00:19:34,560
that sums up these logarithms
here, which are the large logs.
316
00:19:38,760 --> 00:19:39,940
And that's pretty simple.
317
00:19:39,940 --> 00:19:42,580
It just gave this factor.
318
00:19:42,580 --> 00:19:45,253
And that's pretty common
in QCD to get factors
319
00:19:45,253 --> 00:19:46,920
like that, alpha at
one scale over alpha
320
00:19:46,920 --> 00:19:49,260
at another scale
raised to a power.
321
00:19:49,260 --> 00:19:51,780
That's a very common thing to
get from renormalization group
322
00:19:51,780 --> 00:19:52,817
evolution.
323
00:19:55,800 --> 00:19:57,860
OK, any questions so far?
324
00:20:14,600 --> 00:20:17,720
How many people have
done the calculation
325
00:20:17,720 --> 00:20:20,480
of the anomalous dimension
for four-fermion operators
326
00:20:20,480 --> 00:20:21,680
in some other course?
327
00:20:21,680 --> 00:20:23,870
It's a common problem.
328
00:20:23,870 --> 00:20:24,560
Nobody?
329
00:20:24,560 --> 00:20:25,905
All right.
330
00:20:25,905 --> 00:20:27,530
That means you'll
see it on a homework.
331
00:20:41,210 --> 00:20:46,510
So let's come back
here and think
332
00:20:46,510 --> 00:20:49,960
about what the general
structure of what we've done is.
333
00:20:55,937 --> 00:20:57,020
And I'll put back indices.
334
00:21:09,738 --> 00:21:12,030
So you can think about taking
the solution that we have
335
00:21:12,030 --> 00:21:16,320
at the top of the board
here and generalizing it
336
00:21:16,320 --> 00:21:18,937
to a form that would be
valid at higher orders.
337
00:21:18,937 --> 00:21:21,270
And basically, it says that
the coefficient at one scale
338
00:21:21,270 --> 00:21:23,310
is connected to the
coefficient at another scale
339
00:21:23,310 --> 00:21:25,350
times some evolution factor.
340
00:21:25,350 --> 00:21:26,850
In this case, the
evolution factor
341
00:21:26,850 --> 00:21:29,010
is just the ratio of
these alphas to a power.
342
00:21:29,010 --> 00:21:31,140
It could be some more
complicated function
343
00:21:31,140 --> 00:21:32,100
at higher orders.
344
00:21:32,100 --> 00:21:33,520
And it could even be a matrix.
345
00:21:33,520 --> 00:21:34,895
That's why I've
given it indices.
346
00:21:57,950 --> 00:22:01,420
So we can put our
results back together
347
00:22:01,420 --> 00:22:03,520
using this higher order
form, so that they're
348
00:22:03,520 --> 00:22:06,430
generally true,
into our Hamiltonian
349
00:22:06,430 --> 00:22:08,035
and see what we've achieved.
350
00:22:21,740 --> 00:22:24,840
And let me call this scale
that I was calling mu a minute
351
00:22:24,840 --> 00:22:28,440
ago mu b just to remind you
that it's a scale of order Mb.
352
00:22:34,140 --> 00:22:36,390
So previously, we had the
coefficient and the operator
353
00:22:36,390 --> 00:22:38,247
at the same scale.
354
00:22:38,247 --> 00:22:40,580
But now, using this equation,
I can move the coefficient
355
00:22:40,580 --> 00:22:42,510
to a different scale.
356
00:22:42,510 --> 00:22:45,140
And so let me think of
sticking this equation in.
357
00:22:45,140 --> 00:22:46,700
And then I have mu w.
358
00:22:46,700 --> 00:22:48,810
I've called mu equals mu b.
359
00:22:48,810 --> 00:22:50,860
So now, this is mu w mu b.
360
00:22:50,860 --> 00:22:57,890
So this here is the
coefficient Ci at mu b,
361
00:22:57,890 --> 00:22:59,540
but I find it useful
to write it out.
362
00:23:05,968 --> 00:23:07,760
So the thing in square
brackets is Ci mu b,
363
00:23:07,760 --> 00:23:11,120
but I write it out using
the renormalization group
364
00:23:11,120 --> 00:23:12,800
equation that way.
365
00:23:12,800 --> 00:23:15,560
And this tells you how
you're doing the calculation.
366
00:23:15,560 --> 00:23:17,250
This is a fixed
order calculation.
367
00:23:22,130 --> 00:23:26,330
This comes from
anomalous dimensions
368
00:23:26,330 --> 00:23:28,940
and gives you the evolution.
369
00:23:28,940 --> 00:23:31,730
And then you have operators
involving the B quark
370
00:23:31,730 --> 00:23:38,960
that you would calculate
matrix elements of at mu
371
00:23:38,960 --> 00:23:41,330
b of order Mb.
372
00:23:41,330 --> 00:23:45,500
And there's no dependence at all
in those operators on the scale
373
00:23:45,500 --> 00:23:47,270
Mw.
374
00:23:47,270 --> 00:23:49,338
All the Mw's are in
the pre-factors here.
375
00:23:49,338 --> 00:23:50,630
You've taken that into account.
376
00:23:50,630 --> 00:23:53,720
You've calculated it.
377
00:23:53,720 --> 00:23:58,180
OK, so that's how this
organizes the physics
378
00:23:58,180 --> 00:24:01,030
of the different scales.
379
00:24:01,030 --> 00:24:03,117
So you could ask,
if I had this story,
380
00:24:03,117 --> 00:24:04,450
how would I go to higher orders?
381
00:24:04,450 --> 00:24:06,730
And we will have some
discussion of what
382
00:24:06,730 --> 00:24:09,400
goes on at higher orders because
there are some things that
383
00:24:09,400 --> 00:24:11,912
happen at higher orders that
you don't see at leading order.
384
00:24:11,912 --> 00:24:13,870
And they're actually
important physical things,
385
00:24:13,870 --> 00:24:16,147
so important things
to know about and keep
386
00:24:16,147 --> 00:24:18,230
track of if you ever want
to use things like this.
387
00:24:18,230 --> 00:24:21,040
So let's talk a little
bit about what it
388
00:24:21,040 --> 00:24:22,270
takes to go to higher orders.
389
00:24:33,970 --> 00:24:35,550
So let's just first
think about what
390
00:24:35,550 --> 00:24:39,990
it would look like if we
went to higher orders.
391
00:24:39,990 --> 00:24:43,680
Well, leading order was a series
that I could schematically say
392
00:24:43,680 --> 00:24:46,500
is alpha times large logs.
393
00:24:46,500 --> 00:24:49,920
And I summed them all up, and
I called that leading log.
394
00:24:49,920 --> 00:24:51,750
When I go to higher
orders, I am going
395
00:24:51,750 --> 00:24:57,315
to continue to get series, but
I got extra factors of alpha.
396
00:25:05,880 --> 00:25:10,490
So something that you call
Next to Leading Log, or NLL,
397
00:25:10,490 --> 00:25:13,400
is the same type of thing, a
different series than that 1
398
00:25:13,400 --> 00:25:14,970
times an extra factor of alpha.
399
00:25:14,970 --> 00:25:18,110
So it's down compared
to this by alpha S.
400
00:25:18,110 --> 00:25:19,160
And then you keep going.
401
00:25:23,680 --> 00:25:28,370
This is the general structure
of the renormalization group
402
00:25:28,370 --> 00:25:29,690
improved perturbation theory.
403
00:25:32,690 --> 00:25:36,380
Just keep adding Ns and
keep adding alphas, always
404
00:25:36,380 --> 00:25:40,250
summing up some series which
changes from order to order.
405
00:25:40,250 --> 00:25:41,960
And that summation
of that series
406
00:25:41,960 --> 00:25:43,700
is determined by
determining higher order
407
00:25:43,700 --> 00:25:45,710
anomalous dimensions.
408
00:25:45,710 --> 00:25:56,690
So this kind of thing is called
renormalization group improved
409
00:25:56,690 --> 00:25:57,620
perturbation theory.
410
00:26:00,680 --> 00:26:04,490
Every time you take alpha S
and you take it at some scale,
411
00:26:04,490 --> 00:26:06,680
you're already doing
renormalization group
412
00:26:06,680 --> 00:26:08,120
improved perturbation theory.
413
00:26:08,120 --> 00:26:10,247
It's just that, once
you have theories
414
00:26:10,247 --> 00:26:12,830
that have other things that run
and have anomalous dimensions,
415
00:26:12,830 --> 00:26:16,340
then it can be more complicated
than just simply picking alpha
416
00:26:16,340 --> 00:26:18,380
S at the appropriate scale.
417
00:26:18,380 --> 00:26:20,870
Here, in this theory, we
have these coefficients.
418
00:26:20,870 --> 00:26:21,835
We have to run them.
419
00:26:21,835 --> 00:26:23,960
Then we have to pick them
at the appropriate scale.
420
00:26:23,960 --> 00:26:25,670
And that's what we're
doing by solving
421
00:26:25,670 --> 00:26:29,750
these renormalization
group equations.
422
00:26:29,750 --> 00:26:30,250
OK.
423
00:26:30,250 --> 00:26:31,520
So what do we need to do?
424
00:26:31,520 --> 00:26:32,410
We determined this.
425
00:26:32,410 --> 00:26:34,660
I showed you what you
needed to do to get that.
426
00:26:34,660 --> 00:26:37,210
What would we need to do to get
the next term in the series?
427
00:26:37,210 --> 00:26:39,340
How much would we
have to compute?
428
00:26:43,980 --> 00:26:45,930
Well, we just have to
go to one higher order
429
00:26:45,930 --> 00:26:48,340
in the perturbation theory.
430
00:26:48,340 --> 00:27:00,290
So let's make a
little table of what
431
00:27:00,290 --> 00:27:02,900
it takes to get leading
log, next leading log.
432
00:27:05,470 --> 00:27:07,350
Maybe we'll even
add one more term.
433
00:27:12,813 --> 00:27:14,480
So there's two parts
to the calculation.
434
00:27:14,480 --> 00:27:16,910
There's the boundary
condition, and then there's
435
00:27:16,910 --> 00:27:23,150
the differential equation, which
is the anomalous dimension.
436
00:27:23,150 --> 00:27:26,480
At leading log, we had
tree level matching.
437
00:27:26,480 --> 00:27:29,480
We determined the C plus
and minus where 1/2.
438
00:27:29,480 --> 00:27:32,990
C1 and C2 were 1 and 0.
439
00:27:32,990 --> 00:27:36,980
And we just needed the one
loop anomalous dimension.
440
00:27:36,980 --> 00:27:39,090
And then we just keep
going in this pattern.
441
00:27:39,090 --> 00:27:44,330
So next leading log, we
need to match it one loop.
442
00:27:44,330 --> 00:27:49,220
And we would need the
two-loop anomalous dimension
443
00:27:49,220 --> 00:27:50,050
and et cetera.
444
00:27:53,790 --> 00:27:55,647
So the order in which
you need the running
445
00:27:55,647 --> 00:27:57,980
is one higher order than what
you need for the matching.
446
00:28:01,760 --> 00:28:02,820
That's the rule.
447
00:28:02,820 --> 00:28:04,400
And given those
ingredients, we would
448
00:28:04,400 --> 00:28:09,860
be able to determine exactly
these series here, OK?
449
00:28:15,080 --> 00:28:18,012
So there's some things
that happen at this order
450
00:28:18,012 --> 00:28:19,970
that aren't really apparent
yet at leading log,
451
00:28:19,970 --> 00:28:21,980
and so I want to talk a
little bit about that.
452
00:28:34,980 --> 00:28:37,590
Before we get there, let me
add one other little note.
453
00:28:41,080 --> 00:28:44,290
This operator O2, we didn't see
it when we thought originally
454
00:28:44,290 --> 00:28:44,980
about matching.
455
00:28:44,980 --> 00:28:48,100
It had Wilson coefficient
that was 0 at tree level.
456
00:28:53,040 --> 00:28:55,410
So at leading order, you
could say that this Wilson
457
00:28:55,410 --> 00:28:58,020
coefficient is 0.
458
00:28:58,020 --> 00:28:59,760
But at leading log, it's not 0.
459
00:29:05,160 --> 00:29:08,070
So I have these two different
types of perturbation theory.
460
00:29:08,070 --> 00:29:10,200
Just order by order
and alpha or doing
461
00:29:10,200 --> 00:29:11,847
renormalization
group improvement,
462
00:29:11,847 --> 00:29:12,930
you get different results.
463
00:29:17,970 --> 00:29:21,120
And that's because you've
included some higher order
464
00:29:21,120 --> 00:29:25,110
terms by using the
renormalization group improved
465
00:29:25,110 --> 00:29:27,720
version.
466
00:29:27,720 --> 00:29:30,960
But you can argue, if alpha
times the large log is order 1,
467
00:29:30,960 --> 00:29:35,410
then this is the right type
of perturbation theory to do.
468
00:29:35,410 --> 00:29:45,030
So if you think about it as
a picture where this is mu,
469
00:29:45,030 --> 00:29:51,720
this is Mw, this is Mb, then
you have two coefficients, C1
470
00:29:51,720 --> 00:29:52,470
and C2.
471
00:29:52,470 --> 00:29:55,890
We call them C+ and C-,
but they're just related.
472
00:29:55,890 --> 00:29:58,200
And the results that we
derived at leading order
473
00:29:58,200 --> 00:30:04,590
were that, for C1, it started
at 1 at the high scale,
474
00:30:04,590 --> 00:30:07,740
basically, if we
mu w equal to Mw.
475
00:30:07,740 --> 00:30:09,930
And it would evolve,
actually, this direction
476
00:30:09,930 --> 00:30:14,820
if we put in all the signs that
came out of our calculations.
477
00:30:14,820 --> 00:30:18,360
And for C2, it starts at 0 here,
and then it evolves this way
478
00:30:18,360 --> 00:30:21,248
to a negative value.
479
00:30:21,248 --> 00:30:22,212
[INAUDIBLE]
480
00:30:32,340 --> 00:30:42,330
So roughly putting
in some numbers,
481
00:30:42,330 --> 00:30:46,470
the kind of thing that
we would get is this.
482
00:30:46,470 --> 00:30:49,860
So a coefficient which was 0 all
of a sudden becomes minus 0.3
483
00:30:49,860 --> 00:30:52,320
and becomes something that
you have to keep track of.
484
00:30:52,320 --> 00:30:53,280
That's at leading log.
485
00:30:56,793 --> 00:30:58,460
Obviously, when you
go to higher orders,
486
00:30:58,460 --> 00:31:00,755
those numbers will be
perturbatively improved.
487
00:31:04,370 --> 00:31:05,600
OK.
488
00:31:05,600 --> 00:31:08,510
So is the physical picture
here clear of what's happening
489
00:31:08,510 --> 00:31:12,180
with these operators?
490
00:31:12,180 --> 00:31:14,855
So what is the application?
491
00:31:14,855 --> 00:31:18,292
Since we spent all this
time deriving these results,
492
00:31:18,292 --> 00:31:20,000
we should have some
applications in mind.
493
00:31:42,630 --> 00:31:47,570
So for b to c u bar d, if you
ask about what process that
494
00:31:47,570 --> 00:31:52,550
gives, well, one
process that it gives
495
00:31:52,550 --> 00:31:56,285
is just a B to D pi transition.
496
00:31:56,285 --> 00:32:00,230
B bar is built over u bar b.
497
00:32:00,230 --> 00:32:04,220
And this guy is u bar c.
498
00:32:04,220 --> 00:32:06,050
And the pi is the u bar d.
499
00:32:08,783 --> 00:32:11,200
So we can think of the reason
we're studying this is maybe
500
00:32:11,200 --> 00:32:13,880
we want to calculate B of D pi.
501
00:32:13,880 --> 00:32:15,850
So if we wanted to
calculate B of D pi,
502
00:32:15,850 --> 00:32:20,650
we take matrix elements
involving our Hamiltonian
503
00:32:20,650 --> 00:32:23,270
with a B [INAUDIBLE] in the
in state and a D pi in the out
504
00:32:23,270 --> 00:32:23,770
state.
505
00:32:31,400 --> 00:32:33,670
And if we just use the
original Hamiltonian
506
00:32:33,670 --> 00:32:35,740
that we wrote down with
the renormalization group
507
00:32:35,740 --> 00:32:40,810
improvement, then
we would have that.
508
00:32:40,810 --> 00:32:43,760
That's with that renormalization
group improvement.
509
00:32:43,760 --> 00:32:46,450
So this is at mu equals Mw.
510
00:32:46,450 --> 00:32:47,890
And the problem
with this formula
511
00:32:47,890 --> 00:32:49,990
is that this matrix
element has large logs.
512
00:32:52,900 --> 00:32:54,190
It depends on Mw.
513
00:32:54,190 --> 00:32:55,450
It also depends on Mb.
514
00:33:00,342 --> 00:33:02,050
And if it's something
we can't calculate,
515
00:33:02,050 --> 00:33:04,420
then that's kind of bad news.
516
00:33:04,420 --> 00:33:06,170
In particular, having
large logs like that
517
00:33:06,170 --> 00:33:08,087
would also make it hard
to calculate something
518
00:33:08,087 --> 00:33:09,140
like this on the lattice.
519
00:33:09,140 --> 00:33:12,030
So it's not just a--
520
00:33:12,030 --> 00:33:13,290
It's really a problem.
521
00:33:13,290 --> 00:33:15,267
If you have multiple
scales tied together,
522
00:33:15,267 --> 00:33:16,850
it just makes the
calculations harder.
523
00:33:20,445 --> 00:33:22,320
It's also a problem for
dimensional analysis.
524
00:33:22,320 --> 00:33:23,700
Because if you have
large logs, that
525
00:33:23,700 --> 00:33:24,900
means you've got large numbers.
526
00:33:24,900 --> 00:33:27,090
And something that you
thought was of a certain size
527
00:33:27,090 --> 00:33:29,280
might be bigger or smaller.
528
00:33:29,280 --> 00:33:31,830
So what we do is, instead, we
work in the renormalization
529
00:33:31,830 --> 00:33:39,630
group improved version where we
take this down at the scale Mb.
530
00:33:39,630 --> 00:33:41,580
So this guy includes the
renormalization group
531
00:33:41,580 --> 00:33:44,160
evolution.
532
00:33:44,160 --> 00:33:45,690
We use our results over there.
533
00:33:49,500 --> 00:33:51,870
And then we've got the
operators at the scale Mb,
534
00:33:51,870 --> 00:33:53,070
and there's no large logs.
535
00:33:58,820 --> 00:34:00,600
OK, so you'd want to
calculate something
536
00:34:00,600 --> 00:34:03,090
like this on the lattice
or some other way.
537
00:34:03,090 --> 00:34:05,250
There's other ways of doing it.
538
00:34:05,250 --> 00:34:07,930
And we'll talk about other
ways of doing it later on.
539
00:34:07,930 --> 00:34:10,630
But so far, we've
separated out the scale Mw
540
00:34:10,630 --> 00:34:13,659
into this coefficient
that's evaluated
541
00:34:13,659 --> 00:34:15,070
at the scale of mu equals Mb.
542
00:34:28,730 --> 00:34:29,230
OK.
543
00:34:29,230 --> 00:34:31,000
So the one way of
thinking about this
544
00:34:31,000 --> 00:34:33,670
is, if you want to do
physics at the scale Mb,
545
00:34:33,670 --> 00:34:37,989
the right couplings to use in
your theory are these ones.
546
00:34:37,989 --> 00:34:40,030
Forget about what's going
on at the high scale.
547
00:34:40,030 --> 00:34:41,565
You have to determine
the low energy
548
00:34:41,565 --> 00:34:43,690
couplings that are appropriate
to the theory you're
549
00:34:43,690 --> 00:34:44,739
dealing with.
550
00:34:44,739 --> 00:34:47,800
And those are the C's at Mb, OK?
551
00:35:11,040 --> 00:35:13,880
All right.
552
00:35:13,880 --> 00:35:14,380
OK.
553
00:35:14,380 --> 00:35:16,990
So now, I want to come back
to this question of thinking
554
00:35:16,990 --> 00:35:19,390
about the next leading log.
555
00:35:19,390 --> 00:35:21,850
And I'm going to do
that by going back
556
00:35:21,850 --> 00:35:26,230
to our comparison of full
theory to effective theory.
557
00:35:26,230 --> 00:35:29,620
We'll do a comparison of
results in the full theory
558
00:35:29,620 --> 00:35:31,120
and results in the
effective theory.
559
00:35:31,120 --> 00:35:35,230
And I'll show you how, by
making that comparison,
560
00:35:35,230 --> 00:35:40,090
we can determine the ingredients
that we need for this one loop
561
00:35:40,090 --> 00:35:41,652
matching here.
562
00:35:41,652 --> 00:35:43,210
So we'll focus on this.
563
00:35:47,990 --> 00:35:50,080
So we already renormalized
the effective theory.
564
00:35:50,080 --> 00:35:52,450
So we can compare the
renormalized effective theory
565
00:35:52,450 --> 00:35:53,170
and full theory.
566
00:36:00,580 --> 00:36:02,080
And that's the right
way to proceed.
567
00:36:06,200 --> 00:36:09,070
So in our parlance of
theory one and theory two,
568
00:36:09,070 --> 00:36:12,490
the effective theory
would be theory two.
569
00:36:12,490 --> 00:36:15,310
We have to think about the full
theory, which in our parlance
570
00:36:15,310 --> 00:36:19,800
would be theory one,
the full theory being
571
00:36:19,800 --> 00:36:20,800
the standard model here.
572
00:36:24,370 --> 00:36:28,390
We have to think about
renormalizing that theory.
573
00:36:28,390 --> 00:36:30,580
But in the standard
model, our calculation
574
00:36:30,580 --> 00:36:31,795
involves conserved currents.
575
00:36:36,070 --> 00:36:38,350
These are just a weak currents.
576
00:36:38,350 --> 00:36:41,522
And so there's actually
no extra UV divergences
577
00:36:41,522 --> 00:36:42,730
associated to those currents.
578
00:36:42,730 --> 00:36:44,819
We just have coupling
renormalization.
579
00:36:50,920 --> 00:36:53,860
And one way of saying
this is that what happens
580
00:36:53,860 --> 00:36:55,600
is that the vertex
in the wave function
581
00:36:55,600 --> 00:37:03,728
graphs, the UV
divergences cancel
582
00:37:03,728 --> 00:37:04,770
or the conserved current.
583
00:37:08,880 --> 00:37:13,170
So the result for
the full theory
584
00:37:13,170 --> 00:37:18,170
will be some result that
is independent of having--
585
00:37:18,170 --> 00:37:20,402
it doesn't have any
ultraviolet divergences.
586
00:37:25,190 --> 00:37:26,690
And like the effective
theory, where
587
00:37:26,690 --> 00:37:28,815
you had to carry out a
renormalization of operators
588
00:37:28,815 --> 00:37:30,342
in that theory, for
the full theory,
589
00:37:30,342 --> 00:37:32,050
coupling renormalization
is all there is.
590
00:37:39,040 --> 00:37:40,750
So let's draw the
full theory graphs.
591
00:37:47,610 --> 00:37:51,470
Gluons should be green.
592
00:37:51,470 --> 00:37:52,880
Maybe my w should be pink.
593
00:38:29,010 --> 00:38:36,030
Six permutations--
and then there's also
594
00:38:36,030 --> 00:38:37,682
wave function normalization.
595
00:38:40,930 --> 00:38:41,430
OK.
596
00:38:41,430 --> 00:38:43,472
So if you want to do the
full theory calculation,
597
00:38:43,472 --> 00:38:45,000
these are the
graphs you compute.
598
00:38:45,000 --> 00:38:47,070
It's triangles as
well as box integrals.
599
00:38:47,070 --> 00:38:50,074
It's actually a much
harder calculation
600
00:38:50,074 --> 00:38:51,580
than in the effective theory.
601
00:38:54,750 --> 00:38:56,910
And I'm not going to
do the calculation,
602
00:38:56,910 --> 00:38:58,785
but I'll tell you what
the results look like.
603
00:39:06,760 --> 00:39:09,090
So let's start by
thinking about the logs.
604
00:39:13,017 --> 00:39:14,850
And then we'll talk
about the constants that
605
00:39:14,850 --> 00:39:15,975
are under the logs as well.
606
00:39:21,610 --> 00:39:26,100
So if we look at
this calculation,
607
00:39:26,100 --> 00:39:27,390
it has the following form.
608
00:39:52,730 --> 00:39:56,810
So there's S1, which
was some spinners.
609
00:39:56,810 --> 00:39:59,443
We defined it in
an earlier lecture.
610
00:39:59,443 --> 00:40:00,860
There's something
involving a log,
611
00:40:00,860 --> 00:40:03,260
and it has a p squared. p
squared was the off-shellness
612
00:40:03,260 --> 00:40:04,790
associated to these guys.
613
00:40:04,790 --> 00:40:09,000
And we regulated the infrared
divergences with p squared.
614
00:40:09,000 --> 00:40:15,710
So p squared not equal to
0 regulates IR divergences.
615
00:40:15,710 --> 00:40:19,160
And there are IR divergences
in these diagrams.
616
00:40:19,160 --> 00:40:20,750
Even though I said
they are UV finite,
617
00:40:20,750 --> 00:40:23,000
they're not finite
in the infrared.
618
00:40:23,000 --> 00:40:25,282
And that's what leads to
these logs of p squared.
619
00:40:32,370 --> 00:40:33,750
OK.
620
00:40:33,750 --> 00:40:35,250
Now, I didn't write everything.
621
00:40:35,250 --> 00:40:38,250
I only wrote the pieces
proportional to S1.
622
00:40:38,250 --> 00:40:42,390
There's pieces proportional
to the other spinner, S2.
623
00:40:42,390 --> 00:40:44,802
And there's mod log terms.
624
00:40:44,802 --> 00:40:46,260
And they're all
hiding in the dots.
625
00:40:50,430 --> 00:40:53,070
So let's compare this result
to a similar expression
626
00:40:53,070 --> 00:40:58,020
in the effective
theory that we just
627
00:40:58,020 --> 00:41:00,690
set the coefficients to the
values at the high scale.
628
00:41:05,530 --> 00:41:08,520
So then we have 1
for the coefficient
629
00:41:08,520 --> 00:41:12,150
times the one-loop
matrix element of O1,
630
00:41:12,150 --> 00:41:14,730
which we wrote down earlier.
631
00:41:14,730 --> 00:41:18,030
And it looks kind of similar to
this, but not exactly the same.
632
00:41:25,700 --> 00:41:28,400
It's very similar, but
not precisely the same.
633
00:41:38,476 --> 00:41:44,480
A similar statement applies
to these guys over here, OK?
634
00:41:44,480 --> 00:41:46,940
So the difference is
really that, instead
635
00:41:46,940 --> 00:41:49,027
of Mw squared in this
log, we have a mu squared.
636
00:41:49,027 --> 00:41:51,110
That's really the only
difference for these terms.
637
00:41:56,840 --> 00:41:59,000
With constant terms, the
non-logarithmic terms
638
00:41:59,000 --> 00:42:00,470
here and here won't agree.
639
00:42:00,470 --> 00:42:03,360
And we'll talk about
those things in a minute.
640
00:42:03,360 --> 00:42:06,200
So what do we learn by thinking
about the physics of these two
641
00:42:06,200 --> 00:42:08,130
equations?
642
00:42:08,130 --> 00:42:09,980
Well, one comment is
the comment I already
643
00:42:09,980 --> 00:42:14,240
said, that the effective theory
computation for this line
644
00:42:14,240 --> 00:42:17,730
here is much, much
easier than this one.
645
00:42:17,730 --> 00:42:19,740
So one reason to use
effective field theory
646
00:42:19,740 --> 00:42:21,590
is just that it makes
computations easier.
647
00:42:25,550 --> 00:42:27,422
And the reason it makes
computations easier
648
00:42:27,422 --> 00:42:28,880
is because you're
basically dealing
649
00:42:28,880 --> 00:42:30,750
with one scale at a time.
650
00:42:30,750 --> 00:42:33,530
And whenever you have integrals
involving only one scale,
651
00:42:33,530 --> 00:42:36,740
that's always much easier
than having multiple scales.
652
00:42:41,908 --> 00:42:44,450
But if you want to encode all
the physics of the full theory,
653
00:42:44,450 --> 00:42:46,700
you'll still have to do that
calculation at some point
654
00:42:46,700 --> 00:42:47,480
as well.
655
00:42:47,480 --> 00:42:50,030
Although you may be able to do
it in a simpler configuration
656
00:42:50,030 --> 00:42:51,738
to get off the
information that you need.
657
00:42:56,750 --> 00:42:59,083
Furthermore, if you really
only cared about the logs,
658
00:42:59,083 --> 00:43:01,250
then all you really need
is the 1 over epsilon term.
659
00:43:01,250 --> 00:43:06,320
And that's even easier than
the full triangle diagrams.
660
00:43:06,320 --> 00:43:10,250
So to compute the
anomalous dimensions,
661
00:43:10,250 --> 00:43:13,200
you just have to
keep the divergences.
662
00:43:13,200 --> 00:43:17,210
And you can throw away
all the finite pieces.
663
00:43:17,210 --> 00:43:18,568
And that's even easier.
664
00:43:18,568 --> 00:43:21,110
So you can organize things by
thinking about doing the easier
665
00:43:21,110 --> 00:43:22,443
calculations first.
666
00:43:22,443 --> 00:43:23,360
That's what people do.
667
00:43:23,360 --> 00:43:24,818
They calculate
anomalous dimensions
668
00:43:24,818 --> 00:43:29,110
before they calculate matching
because it's just easier.
669
00:43:29,110 --> 00:43:31,610
And it's also the thing you
need for the leading log result.
670
00:43:31,610 --> 00:43:34,068
You don't need the matching at
one loop for the leading log
671
00:43:34,068 --> 00:43:36,440
result. So there's a
conservation of ease
672
00:43:36,440 --> 00:43:37,370
and what you need.
673
00:43:41,795 --> 00:43:43,670
These two things play
nicely with each other.
674
00:43:50,270 --> 00:43:53,860
Second point-- in
the effective theory,
675
00:43:53,860 --> 00:43:57,200
you're supposed to think
that Mw goes to infinity.
676
00:43:57,200 --> 00:43:59,540
And that's why this thing
doesn't know about Mw.
677
00:43:59,540 --> 00:44:01,810
So how could it
possibly get an Mw?
678
00:44:01,810 --> 00:44:04,630
And so what happens is that
Mw gets replaced by the cut
679
00:44:04,630 --> 00:44:05,680
off, which is mu here.
680
00:44:26,840 --> 00:44:29,080
Another point that we can
make about this-- if you
681
00:44:29,080 --> 00:44:31,090
look at the logs
of minus p squared,
682
00:44:31,090 --> 00:44:33,550
they're all the same
between the two equations.
683
00:44:37,870 --> 00:44:41,118
That's actually
important because what
684
00:44:41,118 --> 00:44:43,660
that means is that the infrared
structure of the two theories
685
00:44:43,660 --> 00:44:44,160
agree.
686
00:44:46,990 --> 00:44:49,120
The logs of p squared are
the infrared divergences.
687
00:44:49,120 --> 00:44:51,642
They agree between the higher
theory and the lower theory.
688
00:44:51,642 --> 00:44:52,600
And they have to agree.
689
00:44:56,503 --> 00:44:58,420
What this tells you about
the effective theory
690
00:44:58,420 --> 00:45:01,523
is that you're doing
something right.
691
00:45:01,523 --> 00:45:03,940
In particular, it tells you
that your effective theory has
692
00:45:03,940 --> 00:45:05,148
the right degrees of freedom.
693
00:45:08,050 --> 00:45:10,642
That's almost trivial
in this example.
694
00:45:10,642 --> 00:45:12,850
What other degrees of freedom
could we possibly think
695
00:45:12,850 --> 00:45:13,990
that would be missing?
696
00:45:18,460 --> 00:45:20,660
But in more
complicated examples--
697
00:45:20,660 --> 00:45:24,400
and we will deal with at
least one such example later
698
00:45:24,400 --> 00:45:25,820
in the course--
699
00:45:25,820 --> 00:45:28,510
it's not so trivial to see
that these things match up.
700
00:45:28,510 --> 00:45:30,880
And people have discovered
new degrees of freedom
701
00:45:30,880 --> 00:45:32,860
by doing matching
computations like this.
702
00:45:32,860 --> 00:45:35,027
They said, oh, this is a
relevant degree of freedom.
703
00:45:35,027 --> 00:45:35,980
And it's needed.
704
00:45:35,980 --> 00:45:39,370
Because if I do a
one-loop calculation,
705
00:45:39,370 --> 00:45:41,997
I need it to get the
infrared divergences right.
706
00:45:41,997 --> 00:45:44,330
So this can really teach you
about the effective theory,
707
00:45:44,330 --> 00:45:46,780
doing a matching
computation, teach you
708
00:45:46,780 --> 00:45:50,550
about the physics of
the effective theory.
709
00:45:50,550 --> 00:45:53,455
So if you made a mistake,
this would be a place
710
00:45:53,455 --> 00:45:54,580
where you'd catch yourself.
711
00:46:06,820 --> 00:46:09,670
Now, that you've analyzed
fully what the differences are,
712
00:46:09,670 --> 00:46:10,870
you can subtract them.
713
00:46:15,270 --> 00:46:19,290
You take the difference of
the renormalized calculations.
714
00:46:19,290 --> 00:46:23,850
And that is what gives
you one-loop matching.
715
00:46:23,850 --> 00:46:25,770
Just like we compared
tree level calculations
716
00:46:25,770 --> 00:46:28,500
to get tree level matching, we
compare one-loop calculations
717
00:46:28,500 --> 00:46:31,770
to get one-loop matching.
718
00:46:37,620 --> 00:46:48,630
So [INAUDIBLE] we do that.
719
00:46:48,630 --> 00:46:53,580
At tree level, if we're just
looking at the S1 pieces,
720
00:46:53,580 --> 00:46:56,760
we have these two terms.
721
00:46:56,760 --> 00:46:58,250
And then at one-loop--
722
00:47:00,920 --> 00:47:02,120
let me use this notation.
723
00:47:02,120 --> 00:47:03,980
This is the full A.
So it has a tree level
724
00:47:03,980 --> 00:47:05,150
piece and one-loop piece.
725
00:47:08,210 --> 00:47:10,400
Then we take the piece of
the C1 that's at one loop.
726
00:47:14,000 --> 00:47:19,520
And we take the matrix
element of the operator,
727
00:47:19,520 --> 00:47:21,490
evaluate it order alpha.
728
00:47:28,430 --> 00:47:29,660
And then there's C2 terms.
729
00:47:29,660 --> 00:47:30,590
Let me just put dots.
730
00:47:37,440 --> 00:47:40,692
OK, so there's two ways
that I can get an alpha.
731
00:47:40,692 --> 00:47:42,150
There's an order
alpha coefficient.
732
00:47:42,150 --> 00:47:45,210
That's what I want to know,
what you want to determine.
733
00:47:45,210 --> 00:47:48,120
And then there's an order
alpha matrix element.
734
00:47:48,120 --> 00:47:52,110
So by subtracting, putting
this guy on the left-hand side,
735
00:47:52,110 --> 00:47:55,800
I get the value of C1 of 1.
736
00:47:55,800 --> 00:47:59,550
So we use this equation
to determine this.
737
00:47:59,550 --> 00:48:02,250
And the story would be similar
if I kept all the C2 terms.
738
00:48:05,070 --> 00:48:07,260
So the matching,
i.e. the difference
739
00:48:07,260 --> 00:48:10,620
of the full and effective
theory and calculations,
740
00:48:10,620 --> 00:48:12,120
determines that
coefficient for you.
741
00:48:19,930 --> 00:48:21,460
So the notation
here is that we'd
742
00:48:21,460 --> 00:48:27,530
write the full coefficient as 1
plus C1 plus higher order terms
743
00:48:27,530 --> 00:48:30,560
where this is order alpha.
744
00:48:30,560 --> 00:48:31,840
That's the notation I'm using.
745
00:48:36,130 --> 00:48:37,470
So let's do that for the logs.
746
00:48:37,470 --> 00:48:38,460
It's pretty simple.
747
00:48:38,460 --> 00:48:40,030
These things here
are just cancel.
748
00:48:40,030 --> 00:48:43,200
And this one we can
just subtract them.
749
00:48:43,200 --> 00:48:45,040
The p squareds will cancel.
750
00:48:45,040 --> 00:48:47,220
And we'll get a log of my
squared over Mw squared.
751
00:49:22,180 --> 00:49:24,910
So just focusing on those
terms that we have in S1--
752
00:49:27,595 --> 00:49:29,470
and rearranging the
equation in the way
753
00:49:29,470 --> 00:49:33,730
I said and then plugging in
the values for the things that
754
00:49:33,730 --> 00:49:35,630
don't cancel.
755
00:49:50,140 --> 00:49:54,460
So the terms that were
slightly different--
756
00:49:54,460 --> 00:49:55,690
and dropping the S1.
757
00:50:09,090 --> 00:50:11,316
CF is 4/3.
758
00:50:11,316 --> 00:50:13,489
It's Casimir of the fundamental.
759
00:50:18,540 --> 00:50:21,420
We see that, since we've
only kept the log terms, what
760
00:50:21,420 --> 00:50:24,420
we find is the one-loop
correction in this guy that's
761
00:50:24,420 --> 00:50:26,130
got a logarithm.
762
00:50:26,130 --> 00:50:30,510
but it'd also be a term here
that's a number times alpha.
763
00:50:33,240 --> 00:50:38,670
But we haven't kept
those terms in what
764
00:50:38,670 --> 00:50:40,478
I've written on the board.
765
00:50:40,478 --> 00:50:42,270
So the way that you
should think about this
766
00:50:42,270 --> 00:50:44,270
is that we've got these
Wilson coefficients that
767
00:50:44,270 --> 00:50:46,560
depend on the scale Mw.
768
00:50:46,560 --> 00:50:51,240
And what the matching is doing,
it's taking the full theory.
769
00:50:51,240 --> 00:50:54,813
And it's dividing it into
large momentum pieces
770
00:50:54,813 --> 00:50:55,980
times small momentum pieces.
771
00:51:04,160 --> 00:51:07,660
So the large momentum
pieces are in C. Small
772
00:51:07,660 --> 00:51:12,440
momentum pieces are in the
matrix element of the operator.
773
00:51:12,440 --> 00:51:16,330
And this statement, we see an
explicit realization of it.
774
00:51:16,330 --> 00:51:21,830
The full theory knows about high
scale Mw's and the p squared.
775
00:51:21,830 --> 00:51:32,687
And we can write this
as a split like this.
776
00:51:32,687 --> 00:51:35,270
The effective theory knows about
p squared, doesn't know about
777
00:51:35,270 --> 00:51:35,870
Mw squared.
778
00:51:35,870 --> 00:51:37,820
The Wilson coefficient
knows about Mw squared,
779
00:51:37,820 --> 00:51:39,350
doesn't know about p squared.
780
00:51:39,350 --> 00:51:41,267
The additional thing
that they both know about
781
00:51:41,267 --> 00:51:42,230
is the scale mu.
782
00:51:42,230 --> 00:51:44,000
And that's providing
a cutoff for where
783
00:51:44,000 --> 00:52:04,800
you split between large
momentum and small momentum
784
00:52:04,800 --> 00:52:07,980
p squared was the small scale.
785
00:52:07,980 --> 00:52:10,353
Now, if you look
at this equation,
786
00:52:10,353 --> 00:52:11,520
you may wonder for a minute.
787
00:52:11,520 --> 00:52:13,380
Why is it additive?
788
00:52:13,380 --> 00:52:16,170
Up here, I just said times.
789
00:52:16,170 --> 00:52:17,670
And then immediately
below times,
790
00:52:17,670 --> 00:52:22,410
I wrote something that was a
sum, seemed a little weird.
791
00:52:22,410 --> 00:52:26,190
That's just because, if you take
something that includes the 1,
792
00:52:26,190 --> 00:52:31,890
then the product becomes a sum.
793
00:52:31,890 --> 00:52:34,890
So if I write it this way, I
can write it in product form
794
00:52:34,890 --> 00:52:37,790
if I include the one tree level.
795
00:52:41,880 --> 00:52:44,360
So it really is a product.
796
00:52:44,360 --> 00:52:46,880
It's just that, if you look
at the order alpha pieces,
797
00:52:46,880 --> 00:52:49,920
it breaks into the sum
where we can nicely see how
798
00:52:49,920 --> 00:52:51,750
things are combining together.
799
00:52:51,750 --> 00:52:53,795
But really it has this
product structure,
800
00:52:53,795 --> 00:52:55,920
and there's non-trivial
relations between these two
801
00:52:55,920 --> 00:52:58,440
series that make it
all work out even when
802
00:52:58,440 --> 00:53:00,520
you go to higher orders.
803
00:53:00,520 --> 00:53:02,250
So if I expand that
to order alpha,
804
00:53:02,250 --> 00:53:03,750
look at the order
alpha coefficient.
805
00:53:03,750 --> 00:53:05,635
I get this equation back again.
806
00:53:05,635 --> 00:53:08,010
And this is how you would
think about it in product form.
807
00:53:11,070 --> 00:53:12,360
OK.
808
00:53:12,360 --> 00:53:16,560
So the other thing you see
here is that order by order
809
00:53:16,560 --> 00:53:20,640
in our expansion, as we
kind of already stated,
810
00:53:20,640 --> 00:53:23,070
the mu dependence between
these coefficients and these
811
00:53:23,070 --> 00:53:26,010
operators is exactly cancelling
because the full theory here
812
00:53:26,010 --> 00:53:27,810
didn't involve that mu.
813
00:53:31,760 --> 00:53:35,390
That's another little piece
of information that we get
814
00:53:35,390 --> 00:53:37,110
or that we knew, but
we see explicitly
815
00:53:37,110 --> 00:53:38,450
from looking at this.
816
00:53:48,650 --> 00:53:51,320
So I think, if I'm
counting right,
817
00:53:51,320 --> 00:53:53,000
this is comment number five.
818
00:53:57,335 --> 00:53:58,210
Was there a question?
819
00:54:04,550 --> 00:54:06,940
So not surprisingly,
the cut off dependence
820
00:54:06,940 --> 00:54:10,960
cancels in the product
of C of mu O of mu
821
00:54:10,960 --> 00:54:13,627
because the cut off
is what we introduced
822
00:54:13,627 --> 00:54:15,460
to split up the physics
in these two things.
823
00:54:18,990 --> 00:54:23,810
Now, if you look at that
in a little more detail,
824
00:54:23,810 --> 00:54:26,390
it's only mu independent of the
order in perturbation theory
825
00:54:26,390 --> 00:54:27,223
that you're working.
826
00:54:31,343 --> 00:54:34,690
If you've worked at a fixed
order in some expansion,
827
00:54:34,690 --> 00:54:38,290
then you shouldn't be surprised
that everything you've derived
828
00:54:38,290 --> 00:54:39,580
is only true at that order.
829
00:54:43,820 --> 00:54:45,550
So if you stopped
at one loop, then
830
00:54:45,550 --> 00:54:50,440
it's mu independent
at order alpha S.
831
00:54:50,440 --> 00:54:52,390
What that technically
means is that terms that
832
00:54:52,390 --> 00:54:55,480
are alpha S mu log mu cancel.
833
00:54:58,060 --> 00:55:00,610
The log mu here cancels, but
there's mu dependence also
834
00:55:00,610 --> 00:55:01,540
here.
835
00:55:01,540 --> 00:55:03,760
And that mu dependence
in the alpha
836
00:55:03,760 --> 00:55:06,100
is something that would
be related to terms
837
00:55:06,100 --> 00:55:10,970
that are alpha squared log mu.
838
00:55:10,970 --> 00:55:12,470
And that cancels
at higher order.
839
00:55:16,910 --> 00:55:18,410
So some of the mu
dependents cancel.
840
00:55:18,410 --> 00:55:20,300
Some of the mu dependents
doesn't cancel.
841
00:55:20,300 --> 00:55:22,640
And people actually use the fact
that some of the mu dependence
842
00:55:22,640 --> 00:55:24,680
doesn't cancel as getting a
handle on the higher order
843
00:55:24,680 --> 00:55:25,180
terms.
844
00:55:25,180 --> 00:55:27,770
It's doing a kind of
theory uncertainty.
845
00:55:52,800 --> 00:55:54,810
If we just think
about the logarithms,
846
00:55:54,810 --> 00:56:00,090
then actually the one-loop
results in the full theory
847
00:56:00,090 --> 00:56:02,299
has actually less information.
848
00:56:09,235 --> 00:56:11,610
And the reason is that, if
you wanted to get higher order
849
00:56:11,610 --> 00:56:14,027
terms in this leading log
series that we talked about,
850
00:56:14,027 --> 00:56:16,110
if you wanted to derive
those from the full theory
851
00:56:16,110 --> 00:56:19,410
point of view, you'd have to
do a two-loop computation.
852
00:56:24,220 --> 00:56:29,590
So if you wanted to get alpha
squared log squared of Mw
853
00:56:29,590 --> 00:56:34,060
squared over minus p
squared, then you'd
854
00:56:34,060 --> 00:56:41,125
have to look at
diagrams, two gluons.
855
00:56:43,995 --> 00:56:45,370
On the full theory
point of view,
856
00:56:45,370 --> 00:56:50,513
that's what you'd have to
do to find those terms.
857
00:56:50,513 --> 00:56:52,180
From the effective
theory point of view,
858
00:56:52,180 --> 00:56:53,763
all you have to do
to find those terms
859
00:56:53,763 --> 00:56:56,380
is renormalize the
effective theory properly.
860
00:56:56,380 --> 00:56:58,420
And then you get those terms.
861
00:57:09,630 --> 00:57:12,750
So we just needed the
one-loop anomalous dimension.
862
00:57:12,750 --> 00:57:14,660
So in that sense,
the effective theory,
863
00:57:14,660 --> 00:57:16,640
because of the
renormalization properties
864
00:57:16,640 --> 00:57:19,280
of the effective
theory, know something
865
00:57:19,280 --> 00:57:22,550
that the full theory
doesn't know so easily.
866
00:57:22,550 --> 00:57:24,290
And that kind of shows
you the advantage
867
00:57:24,290 --> 00:57:27,710
of taking something that's
a constant, Mw squared,
868
00:57:27,710 --> 00:57:29,132
and turning it into a scale.
869
00:57:29,132 --> 00:57:30,590
Because by turning
it into a scale,
870
00:57:30,590 --> 00:57:32,798
you have the whole power of
the renormalization group
871
00:57:32,798 --> 00:57:34,820
at your disposal to
predict higher order
872
00:57:34,820 --> 00:57:38,030
things, like the higher
order coefficients.
873
00:57:38,030 --> 00:57:39,800
And that's one way
of phrasing what
874
00:57:39,800 --> 00:57:42,163
the example is of
splitting scales
875
00:57:42,163 --> 00:57:43,580
and going to the
effective theory.
876
00:57:51,290 --> 00:57:53,750
So the final thing that
I want to talk about here
877
00:57:53,750 --> 00:57:56,360
has to do with the fact that--
878
00:57:56,360 --> 00:57:59,600
well, actually, there's two more
things I want to talk about,
879
00:57:59,600 --> 00:58:01,340
but let me make the
final comment here.
880
00:58:04,020 --> 00:58:06,920
So the final comment I
want to make in my list,
881
00:58:06,920 --> 00:58:10,745
which is number seven, has
to do with scheme dependence.
882
00:58:14,000 --> 00:58:15,530
So scheme dependence
means that we
883
00:58:15,530 --> 00:58:18,537
pick the renormalization
scheme MS bar.
884
00:58:18,537 --> 00:58:20,120
And we could have
done the calculation
885
00:58:20,120 --> 00:58:23,438
in a different
renormalization scheme.
886
00:58:23,438 --> 00:58:25,355
And we should ask what
depends on that choice.
887
00:58:28,850 --> 00:58:31,490
You may know, if you've taken
a course on the beta function
888
00:58:31,490 --> 00:58:34,660
or if you've taken QFD3,
that the beta function of QCD
889
00:58:34,660 --> 00:58:36,800
is scheme independent
for the first two orders.
890
00:58:40,580 --> 00:58:42,680
The analog of that
statement here
891
00:58:42,680 --> 00:58:45,920
is that the one-loop anomalous
dimension for our operators
892
00:58:45,920 --> 00:58:47,328
is scheme independent.
893
00:58:50,674 --> 00:58:55,360
It doesn't depend on which mass
independent scheme you pick.
894
00:58:55,360 --> 00:58:58,270
So in the class of mass
independence schemes,
895
00:58:58,270 --> 00:59:00,535
the result is what we derived.
896
00:59:06,000 --> 00:59:08,600
We'll come back and study
that in a little more detail.
897
00:59:11,430 --> 00:59:14,958
OK, so let's go back now
and establish some notation
898
00:59:14,958 --> 00:59:17,000
where we actually just
put the constants back in.
899
00:59:20,260 --> 00:59:22,010
And again, I'm not
going to write numbers.
900
00:59:22,010 --> 00:59:24,408
I'll just give them names.
901
00:59:24,408 --> 00:59:25,950
And we'll track what
happens to them.
902
00:59:28,530 --> 00:59:35,180
So let's think about the
full one-loop matching
903
00:59:35,180 --> 00:59:39,780
and how we get the next
leading log result.
904
00:59:39,780 --> 00:59:42,200
And really what I want to
focus on, or at least one thing
905
00:59:42,200 --> 00:59:46,580
I want to focus on, is
the scheme dependence.
906
00:59:46,580 --> 00:59:50,630
Because the coefficients, once
you get to next leading log,
907
00:59:50,630 --> 00:59:52,940
are totally scheme dependent.
908
00:59:52,940 --> 00:59:54,920
So you can ask,
what physical sense
909
00:59:54,920 --> 00:59:57,943
do they make if they're
scheme dependent?
910
00:59:57,943 --> 01:00:00,110
Well, it turns out that the
matrix elements are also
911
01:00:00,110 --> 01:00:02,000
scheme dependent.
912
01:00:02,000 --> 01:00:04,760
And the anomalous dimensions
are scheme dependent.
913
01:00:04,760 --> 01:00:08,180
So basically, everything
is scheme dependent.
914
01:00:08,180 --> 01:00:10,940
And when we put it all
together, we get a scheme
915
01:00:10,940 --> 01:00:25,965
independent result. So
you might think, well,
916
01:00:25,965 --> 01:00:27,590
if we can get scheme
dependent results,
917
01:00:27,590 --> 01:00:30,050
we should just stop
because maybe we can't
918
01:00:30,050 --> 01:00:32,030
understand what's going on.
919
01:00:32,030 --> 01:00:41,360
But C of mu times O of mu is
independent of the scheme.
920
01:00:44,900 --> 01:00:46,070
It's a physical observable.
921
01:00:46,070 --> 01:00:47,840
And physical
observables don't depend
922
01:00:47,840 --> 01:00:49,940
on our definitions of things.
923
01:00:54,620 --> 01:00:56,090
Nature gets to decide, not us.
924
01:01:01,020 --> 01:01:04,400
So one way of
thinking about this
925
01:01:04,400 --> 01:01:07,910
is that we already saw some
kind of scheme independence
926
01:01:07,910 --> 01:01:11,060
in a statement that
C of mu times O of mu
927
01:01:11,060 --> 01:01:12,260
is independent of mu.
928
01:01:12,260 --> 01:01:15,080
But there's even a deeper
scheme independence to it
929
01:01:15,080 --> 01:01:18,890
that it's independent of whether
we chose MS bar or some others
930
01:01:18,890 --> 01:01:21,440
scheme.
931
01:01:21,440 --> 01:01:23,910
So for the context
of this discussion,
932
01:01:23,910 --> 01:01:27,560
I'm going to start dropping
all the matrix indices.
933
01:01:27,560 --> 01:01:29,540
And we're not going
to write i and j
934
01:01:29,540 --> 01:01:33,470
just because I want to keep
things a little bit simple.
935
01:01:33,470 --> 01:01:37,580
So we'll write that the
effective theory is simply
936
01:01:37,580 --> 01:01:41,510
one coefficient times the
matrix of one operator.
937
01:01:47,940 --> 01:01:54,050
So let's think about,
in that context, trying
938
01:01:54,050 --> 01:01:56,570
to understand where all this
scheme dependence is floating
939
01:01:56,570 --> 01:02:05,500
around and how the
matching works.
940
01:02:10,360 --> 01:02:13,430
So we just do the same
thing we did before.
941
01:02:13,430 --> 01:02:16,970
I'm leaving off
some pre-factors,
942
01:02:16,970 --> 01:02:18,310
leaving off the pre-factors.
943
01:02:18,310 --> 01:02:20,102
I don't have the write
the spinners anymore
944
01:02:20,102 --> 01:02:22,540
since there's only
one structure.
945
01:02:22,540 --> 01:02:28,700
And let me introduce some
notation for the results.
946
01:02:28,700 --> 01:02:31,090
So we had this Mw squared
over p squared type term.
947
01:02:33,620 --> 01:02:36,140
And let me just focus on these
terms and not the terms that
948
01:02:36,140 --> 01:02:39,290
just cancelled away.
949
01:02:39,290 --> 01:02:41,900
So let me focus on the
terms that are different.
950
01:02:41,900 --> 01:02:44,000
But now, I'm also going
to include the constants.
951
01:03:03,860 --> 01:03:05,920
So the constants that we
get in the full theory
952
01:03:05,920 --> 01:03:07,545
and the effective
theory are different.
953
01:03:07,545 --> 01:03:09,850
So I'll call one of them
A and the other one B.
954
01:03:09,850 --> 01:03:12,310
So I call the A the
full theory result
955
01:03:12,310 --> 01:03:14,723
and the B the effective
theory result.
956
01:03:14,723 --> 01:03:16,390
So you should think
of this as a number,
957
01:03:16,390 --> 01:03:21,280
like 3, just some
number, same thing here.
958
01:03:21,280 --> 01:03:23,830
But just to avoid
talking about numbers
959
01:03:23,830 --> 01:03:26,500
and to track also where
the scheme dependence is--
960
01:03:26,500 --> 01:03:28,240
like this 3 could
be 2 in one scheme
961
01:03:28,240 --> 01:03:33,413
and this 4 could be 2 in one
scheme and 5 in another scheme.
962
01:03:33,413 --> 01:03:35,830
In order to keep track of that,
let me call it a variable.
963
01:03:35,830 --> 01:03:39,520
Let me call it B.
964
01:03:39,520 --> 01:03:41,310
So then the Wilson
coefficient is just
965
01:03:41,310 --> 01:03:42,635
we construct the difference.
966
01:03:42,635 --> 01:03:44,260
And then we'll have
an A minus B in it.
967
01:04:07,133 --> 01:04:09,300
So if you like what the
Wilson coefficient is doing,
968
01:04:09,300 --> 01:04:11,400
it's compensating for the
fact that the effective theory
969
01:04:11,400 --> 01:04:12,930
has the wrong value
for this constant.
970
01:04:12,930 --> 01:04:13,680
It should be this.
971
01:04:13,680 --> 01:04:15,960
That's what the full
theory told you it was.
972
01:04:15,960 --> 01:04:20,730
So the effective theory
Wilson coefficient
973
01:04:20,730 --> 01:04:22,740
has minus the effective
theory matrix element
974
01:04:22,740 --> 01:04:25,620
result plus the
correct result. So this
975
01:04:25,620 --> 01:04:29,842
is the thing that's correcting
the effective theory.
976
01:04:29,842 --> 01:04:31,050
So it has the right constant.
977
01:04:43,510 --> 01:04:47,280
And if we just take
C at Mw, then it
978
01:04:47,280 --> 01:04:51,107
would simply be equal to that.
979
01:04:51,107 --> 01:04:52,190
And the log would go away.
980
01:05:08,130 --> 01:05:11,180
So in order to do the
renormalization group
981
01:05:11,180 --> 01:05:13,520
improved perturbation
theory at next leading log,
982
01:05:13,520 --> 01:05:15,290
we also need to do a
two-loop computation.
983
01:05:18,530 --> 01:05:22,235
We're not going to do the
two-loop consultation,
984
01:05:22,235 --> 01:05:24,110
but I'll tell you the
structure of the series
985
01:05:24,110 --> 01:05:26,240
that you get if you
did that computation.
986
01:05:31,780 --> 01:05:33,940
So this equation is true.
987
01:05:33,940 --> 01:05:36,420
Therefore, we can write the
anomalous dimension equation
988
01:05:36,420 --> 01:05:37,922
again as log C.
989
01:05:37,922 --> 01:05:39,630
And the right-hand
side will be a series.
990
01:05:43,113 --> 01:05:45,030
And the structure of the
series that was there
991
01:05:45,030 --> 01:05:48,690
is the 0-th order term.
992
01:05:48,690 --> 01:05:50,640
And then there's some
higher order terms.
993
01:05:55,500 --> 01:06:00,770
And we need this guy,
the two-loop coefficient,
994
01:06:00,770 --> 01:06:01,610
[INAUDIBLE] gamma 1.
995
01:06:04,580 --> 01:06:06,870
Again, this is a coupled
differential equation.
996
01:06:06,870 --> 01:06:09,260
And we would solve it by
using the kind of thing
997
01:06:09,260 --> 01:06:10,080
that we did before.
998
01:06:10,080 --> 01:06:17,540
So d mu over mu is d
alpha over beta of alpha.
999
01:06:17,540 --> 01:06:20,030
And we would write
down beta to one higher
1000
01:06:20,030 --> 01:06:24,410
order, which I do in my notes.
1001
01:06:24,410 --> 01:06:29,090
But it's the same idea is
I just expand it in alpha.
1002
01:06:29,090 --> 01:06:31,990
And I keep not just
the coefficient beta 0,
1003
01:06:31,990 --> 01:06:33,615
but I also keep the
coefficient beta 1.
1004
01:06:47,910 --> 01:06:50,370
I want to kind of not focus
so much on the calculations,
1005
01:06:50,370 --> 01:06:52,710
but more the results
and the implications
1006
01:06:52,710 --> 01:06:56,610
of the calculations.
1007
01:06:56,610 --> 01:07:01,850
So do some renormalization
group evolution.
1008
01:07:01,850 --> 01:07:07,658
You can write the all-order
solution as an integral,
1009
01:07:07,658 --> 01:07:08,450
like we did before.
1010
01:07:15,460 --> 01:07:19,410
And if I just keep it in terms
of these all-order objects,
1011
01:07:19,410 --> 01:07:23,760
then it's just the ratio, which
I expand that ratio in alpha.
1012
01:07:23,760 --> 01:07:26,225
And if I want to do
it to second order,
1013
01:07:26,225 --> 01:07:27,600
I don't just keep
the first time.
1014
01:07:27,600 --> 01:07:28,558
I keep the second term.
1015
01:07:38,220 --> 01:07:41,170
So the first term
was a 1 over alpha.
1016
01:07:41,170 --> 01:07:45,930
So we're going to keep the
order alpha to the 0 term.
1017
01:07:51,140 --> 01:07:53,840
And if we use our notation
that we established before,
1018
01:07:53,840 --> 01:07:57,050
where we call this guy here,
we call the exponential
1019
01:07:57,050 --> 01:08:09,020
of this guy u, so C of u C of
mu 0, of mu w mu of mu w mu.
1020
01:08:18,140 --> 01:08:20,644
Then we can write the
solution of that guy
1021
01:08:20,644 --> 01:08:24,830
as an exponential of an integral
of d alpha gamma over beta.
1022
01:08:28,904 --> 01:08:31,279
So some of the steps that we
were doing at one-loop, just
1023
01:08:31,279 --> 01:08:33,319
like the exponentiation,
the separation,
1024
01:08:33,319 --> 01:08:34,802
they just all go through.
1025
01:08:34,802 --> 01:08:36,260
And the only thing
we do have to do
1026
01:08:36,260 --> 01:08:38,177
is evaluate this integral
at one higher order.
1027
01:08:43,290 --> 01:08:50,270
Let me take mu w equal to
Mw and then do the integral.
1028
01:08:50,270 --> 01:08:53,420
And what you get is
a result that we can
1029
01:08:53,420 --> 01:08:54,850
organize in the following way.
1030
01:09:02,248 --> 01:09:04,040
Try to get my arguments
in the right order.
1031
01:09:14,640 --> 01:09:18,060
In this particular case, the
next leading log solution
1032
01:09:18,060 --> 01:09:19,770
looks as follows.
1033
01:09:19,770 --> 01:09:23,460
Our leading log solution is
obviously buried inside it.
1034
01:09:23,460 --> 01:09:35,660
So we have this ratio of alphas,
something which is a number.
1035
01:09:35,660 --> 01:09:39,700
And then there's
these extra factors
1036
01:09:39,700 --> 01:09:41,020
that depend on this then j.
1037
01:09:44,620 --> 01:09:48,189
And I can write
the result this way
1038
01:09:48,189 --> 01:09:51,310
where j involves all the things
that are the higher order
1039
01:09:51,310 --> 01:09:52,170
ingredients.
1040
01:09:52,170 --> 01:09:55,480
So it involves the lowest order
anomalous dimension, but now
1041
01:09:55,480 --> 01:09:57,245
times beta 1.
1042
01:09:57,245 --> 01:09:59,620
That's like taking the leading
order anomalous dimension,
1043
01:09:59,620 --> 01:10:02,500
but now running the coupling
with the second order term
1044
01:10:02,500 --> 01:10:03,980
as well.
1045
01:10:03,980 --> 01:10:07,480
And then there's a term that
involves the second order
1046
01:10:07,480 --> 01:10:11,170
anomalous dimension.
1047
01:10:11,170 --> 01:10:14,930
So it encodes that information.
1048
01:10:14,930 --> 01:10:17,380
So this is the U. We can
combine that together
1049
01:10:17,380 --> 01:10:22,810
with our equation for the
C over here, or this one.
1050
01:10:22,810 --> 01:10:24,300
So let me keep that one.
1051
01:10:44,610 --> 01:10:47,910
So I take this equation,
multiply by that equation.
1052
01:10:47,910 --> 01:10:49,476
That gives us C of mu.
1053
01:10:52,194 --> 01:10:53,910
So I have to write
this lone more time.
1054
01:11:09,830 --> 01:11:12,500
And basically, I can group that
together with these other terms
1055
01:11:12,500 --> 01:11:13,870
that depend on an alpha of Mw.
1056
01:11:27,250 --> 01:11:28,240
OK.
1057
01:11:28,240 --> 01:11:31,330
So j is the anomalous
dimension piece.
1058
01:11:31,330 --> 01:11:33,010
A and B are the matching.
1059
01:11:33,010 --> 01:11:35,680
A minus B is the matching piece.
1060
01:11:35,680 --> 01:11:38,710
And I can write the
result this way.
1061
01:11:38,710 --> 01:11:43,590
So this is next
leading order matching,
1062
01:11:43,590 --> 01:11:48,470
which is A minus
B and next leading
1063
01:11:48,470 --> 01:12:00,340
log running to get the full
next leading log result.
1064
01:12:00,340 --> 01:12:02,530
So this is the kind of
structure that you could get.
1065
01:12:02,530 --> 01:12:03,970
That's what
renormalization group
1066
01:12:03,970 --> 01:12:05,803
improved perturbation
theory looks like when
1067
01:12:05,803 --> 01:12:07,180
you go to higher orders.
1068
01:12:07,180 --> 01:12:08,758
You basically have logs.
1069
01:12:08,758 --> 01:12:10,300
But then the higher
order terms, when
1070
01:12:10,300 --> 01:12:12,940
you expand out this
integral, are just giving you
1071
01:12:12,940 --> 01:12:14,853
polynomials in alpha.
1072
01:12:14,853 --> 01:12:16,270
So when you integrate
polynomials,
1073
01:12:16,270 --> 01:12:17,390
you get back polynomials.
1074
01:12:17,390 --> 01:12:19,070
So if you integrate
1, you get alpha.
1075
01:12:19,070 --> 01:12:23,665
If you integrate alpha squared,
you get alpha cubed, et cetera.
1076
01:12:23,665 --> 01:12:25,040
So you just get
back polynomials.
1077
01:12:25,040 --> 01:12:28,420
And that's why you
can write it this way.
1078
01:12:28,420 --> 01:12:30,520
What are the terms
in this result
1079
01:12:30,520 --> 01:12:31,966
that are scheme dependent?
1080
01:12:36,430 --> 01:12:40,630
I claim that beta 1 gamma 1--
1081
01:12:44,454 --> 01:12:48,000
oh, why did I-- not beta 1, B1.
1082
01:12:52,440 --> 01:12:59,850
B1 gamma 1 J, C, O, these
are all scheme dependent.
1083
01:12:59,850 --> 01:13:01,920
They depend on what
renormalization scheme
1084
01:13:01,920 --> 01:13:03,690
I use to define my
effective theory.
1085
01:13:09,545 --> 01:13:10,920
And then there's
a list of things
1086
01:13:10,920 --> 01:13:13,230
that are scheme independent.
1087
01:13:13,230 --> 01:13:16,890
So beta 0 and beta 1
are scheme independent.
1088
01:13:16,890 --> 01:13:20,153
I told you that gamma 0
is scheme independent.
1089
01:13:20,153 --> 01:13:21,570
You could think
of that like, when
1090
01:13:21,570 --> 01:13:24,237
you do the one-loop calculation,
the ultraviolet divergences are
1091
01:13:24,237 --> 01:13:25,403
always going to be the same.
1092
01:13:25,403 --> 01:13:26,370
You get 1 over epsilon.
1093
01:13:26,370 --> 01:13:28,662
And it's only the constant
that depends on your scheme.
1094
01:13:31,980 --> 01:13:34,200
A1 is scheme independent.
1095
01:13:34,200 --> 01:13:36,425
That's because A1 was the
full theory calculation.
1096
01:13:36,425 --> 01:13:37,800
So how could it
possibly know how
1097
01:13:37,800 --> 01:13:40,930
we define the effective theory?
1098
01:13:40,930 --> 01:13:43,800
So that's scheme independent.
1099
01:13:43,800 --> 01:13:51,417
And a non-trivial one is that
B1 plus J is scheme independent.
1100
01:13:51,417 --> 01:13:54,000
So there's scheme dependence in
B1 and scheme dependence in J,
1101
01:13:54,000 --> 01:13:56,220
but it cancels in exactly
the combination that's
1102
01:13:56,220 --> 01:14:00,220
showing up in this result.
1103
01:14:00,220 --> 01:14:07,300
And as I mentioned, C times
O is scheme independent
1104
01:14:07,300 --> 01:14:09,798
because that's related
to observables.
1105
01:14:16,150 --> 01:14:18,340
So I have a little proof
of that in my notes,
1106
01:14:18,340 --> 01:14:21,490
which, because of time,
I'm going to skip.
1107
01:14:21,490 --> 01:14:24,493
But I encourage you, when I post
my notes through the website,
1108
01:14:24,493 --> 01:14:26,410
that you take a look at
where that comes from.
1109
01:14:30,123 --> 01:14:31,540
So the only
non-trivial one really
1110
01:14:31,540 --> 01:14:38,012
is this B1 plus J being
scheme independent, OK?
1111
01:14:38,012 --> 01:14:39,970
I have a little proof of
that in my notes here.
1112
01:14:44,490 --> 01:14:47,190
OK, so let's go
back to the equation
1113
01:14:47,190 --> 01:14:50,520
at the top in the
middle there and see
1114
01:14:50,520 --> 01:14:55,810
what conclusions we can
draw once we believe this.
1115
01:14:55,810 --> 01:14:58,650
So if B1 plus J is
scheme independent, then
1116
01:14:58,650 --> 01:15:03,180
this thing that's showing up in
that term, A1 minus B minus J,
1117
01:15:03,180 --> 01:15:07,237
is scheme independent, as A was.
1118
01:15:07,237 --> 01:15:08,570
It was just a full theory thing.
1119
01:15:12,982 --> 01:15:14,940
And there's a cancellation
of scheme dependence
1120
01:15:14,940 --> 01:15:19,170
between the one-loop
anomalous dimension.
1121
01:15:19,170 --> 01:15:25,135
There's a cancellation
here between
1122
01:15:25,135 --> 01:15:26,760
the two-loop anomalous
dimension, which
1123
01:15:26,760 --> 01:15:30,570
we called gamma 1, and the B1.
1124
01:15:30,570 --> 01:15:32,610
That's where the scheme
dependence cancels.
1125
01:15:32,610 --> 01:15:34,797
So the scheme you pick,
you have to be consistent.
1126
01:15:34,797 --> 01:15:35,880
You have to keep using it.
1127
01:15:35,880 --> 01:15:38,005
If you do a matching
calculation or if someone else
1128
01:15:38,005 --> 01:15:40,002
did a matching calculation,
you want to use it.
1129
01:15:40,002 --> 01:15:42,210
You better figure out what
scheme they're working in.
1130
01:15:42,210 --> 01:15:43,800
Because if you start working
in a different scheme,
1131
01:15:43,800 --> 01:15:44,967
you're just making mistakes.
1132
01:15:48,220 --> 01:15:50,700
So this is the statement
that the matching is scheme
1133
01:15:50,700 --> 01:15:54,580
dependent, the anomalous
dimension scheme dependent,
1134
01:15:54,580 --> 01:15:56,490
but there's a cancellation
between those two.
1135
01:16:07,540 --> 01:16:09,787
If we look at the gamma
0 over beta 0 term,
1136
01:16:09,787 --> 01:16:10,870
that's scheme independent.
1137
01:16:10,870 --> 01:16:11,590
So that's good.
1138
01:16:11,590 --> 01:16:16,215
If we look over here, J
was not scheme dependent.
1139
01:16:16,215 --> 01:16:19,168
J is scheme dependent.
1140
01:16:19,168 --> 01:16:20,710
So we still have to
worry about that.
1141
01:16:35,181 --> 01:16:39,170
So leading log result
was scheme independent,
1142
01:16:39,170 --> 01:16:46,080
but we still have
scheme dependence
1143
01:16:46,080 --> 01:16:57,310
of this factor 1 plus alpha of
mu J over 4 pi in our C of mu.
1144
01:16:57,310 --> 01:17:01,920
And the thing that cancels
that scheme dependence
1145
01:17:01,920 --> 01:17:04,150
is the fact that the
Wilson coefficient alone
1146
01:17:04,150 --> 01:17:05,890
is not a physical observable.
1147
01:17:05,890 --> 01:17:08,985
It's really the Wilson
coefficient times the operator.
1148
01:17:08,985 --> 01:17:10,360
And so there is
scheme dependence
1149
01:17:10,360 --> 01:17:12,834
in the matrix element
of the operator.
1150
01:17:28,650 --> 01:17:31,430
So a matrix element of the
operator at the scale mu
1151
01:17:31,430 --> 01:17:34,610
is scheme dependent.
1152
01:17:34,610 --> 01:17:37,720
And this is at the lower
end of our integration.
1153
01:17:42,220 --> 01:17:43,775
So this is the final
matrix element,
1154
01:17:43,775 --> 01:17:45,400
like the matrix
element at the B scale.
1155
01:17:45,400 --> 01:17:46,730
That's a scheme dependent thing.
1156
01:17:46,730 --> 01:17:48,480
So if you think of
these things as numbers
1157
01:17:48,480 --> 01:17:51,040
that you want to determine
from data, one way of thinking
1158
01:17:51,040 --> 01:17:52,630
about it, those
numbers are going
1159
01:17:52,630 --> 01:17:54,490
to depend on what
scheme you're using.
1160
01:17:54,490 --> 01:17:56,650
If you extract some
numbers in one scheme
1161
01:17:56,650 --> 01:17:58,400
and your friend does
it in another scheme,
1162
01:17:58,400 --> 01:18:00,560
you could get totally
different numbers.
1163
01:18:00,560 --> 01:18:02,080
So you have to know what
scheme you're working in.
1164
01:18:02,080 --> 01:18:03,538
And you have to
combine it together
1165
01:18:03,538 --> 01:18:06,247
with the Wilson coefficient
in the same scheme.
1166
01:18:06,247 --> 01:18:08,080
If you take some numbers
from the literature
1167
01:18:08,080 --> 01:18:09,788
and you don't know
what scheme they're in
1168
01:18:09,788 --> 01:18:11,450
and you're working
at next leading log,
1169
01:18:11,450 --> 01:18:13,280
you have a problem.
1170
01:18:13,280 --> 01:18:14,950
You got to know
what the scheme is
1171
01:18:14,950 --> 01:18:17,995
because you have to work in
the same scheme consistently.
1172
01:18:17,995 --> 01:18:19,120
And that's the lesson here.
1173
01:18:25,960 --> 01:18:28,270
If you really want to do
this whole program that I've
1174
01:18:28,270 --> 01:18:30,970
talked about, which is done in
this 250 page review article--
1175
01:18:30,970 --> 01:18:33,790
and I'm not asking
you to read that.
1176
01:18:33,790 --> 01:18:36,160
If you really want to
do this whole program,
1177
01:18:36,160 --> 01:18:37,480
there are some subtleties.
1178
01:18:37,480 --> 01:18:43,270
And I should at least mention
them to you since maybe you'll
1179
01:18:43,270 --> 01:18:44,395
encounter the word someday.
1180
01:18:57,270 --> 01:19:01,050
So we've sketched the physics
and the basic stuff that
1181
01:19:01,050 --> 01:19:02,590
would be involved
in the analysis,
1182
01:19:02,590 --> 01:19:04,715
but we haven't written down
the full operator basis
1183
01:19:04,715 --> 01:19:06,750
with the full set
of mixing and dozens
1184
01:19:06,750 --> 01:19:08,820
and dozens of diagrams,
which people have done.
1185
01:19:12,450 --> 01:19:15,730
Mostly what you should be
thinking of this is as a user.
1186
01:19:15,730 --> 01:19:17,520
So I'm teaching you
the things that you
1187
01:19:17,520 --> 01:19:19,505
need to be able to
use results like that.
1188
01:19:19,505 --> 01:19:20,880
In an effective
theory, if you're
1189
01:19:20,880 --> 01:19:22,422
using a higher order
result, you have
1190
01:19:22,422 --> 01:19:25,060
to worry about scheme dependent.
1191
01:19:25,060 --> 01:19:26,370
So what are the subtleties?
1192
01:19:26,370 --> 01:19:28,530
Well, one of them is
that there's gamma 5s.
1193
01:19:31,265 --> 01:19:34,200
This theory is chiral.
1194
01:19:34,200 --> 01:19:37,076
And gamma 5 is inherently
four-dimensional.
1195
01:19:44,700 --> 01:19:46,680
And you have to
worry about that.
1196
01:19:46,680 --> 01:19:48,930
And you have to create
that carefully in dim reg.
1197
01:19:52,120 --> 01:19:54,550
And when people originally
did these calculations,
1198
01:19:54,550 --> 01:19:57,880
that caused some confusion.
1199
01:19:57,880 --> 01:20:00,820
Be careful enough.
1200
01:20:00,820 --> 01:20:03,850
Obviously, dim reg is a powerful
way of doing the calculation,
1201
01:20:03,850 --> 01:20:06,513
but you do have to be
careful about gamma 5.
1202
01:20:06,513 --> 01:20:07,930
And there's another
thing you have
1203
01:20:07,930 --> 01:20:10,553
to be careful about in dim reg.
1204
01:20:10,553 --> 01:20:12,970
And that's something that are
called evanescent operators.
1205
01:20:18,513 --> 01:20:20,430
You see, part of our
arguments, and originally
1206
01:20:20,430 --> 01:20:22,430
when we were writing down
the basis of operators
1207
01:20:22,430 --> 01:20:24,210
for our calculation,
were actually
1208
01:20:24,210 --> 01:20:25,881
inherently four-dimensional.
1209
01:20:29,460 --> 01:20:32,280
When we wrote down
the operators,
1210
01:20:32,280 --> 01:20:40,920
we said we effectively used that
these Dirac structures, which
1211
01:20:40,920 --> 01:20:47,430
are 16 of them, we used
completeness over those 16.
1212
01:20:47,430 --> 01:20:51,090
And the problem is
that, in d dimensions,
1213
01:20:51,090 --> 01:20:53,220
that's not a complete set.
1214
01:21:05,860 --> 01:21:09,130
And any opinions
that are outside that
1215
01:21:09,130 --> 01:21:10,900
set that are additional
operators that you
1216
01:21:10,900 --> 01:21:16,434
need in d dimensions are
called evanescent operators.
1217
01:21:21,080 --> 01:21:23,120
So they involve
Dirac structures that
1218
01:21:23,120 --> 01:21:33,330
vanish as epsilon goes to 0,
but are technically needed
1219
01:21:33,330 --> 01:21:36,150
to get some calculations right.
1220
01:21:36,150 --> 01:21:37,680
OK, so those are
two subtle things
1221
01:21:37,680 --> 01:21:41,610
to be aware of in
the full calculation.
1222
01:21:41,610 --> 01:21:43,448
And I think we'll
stop there for today.
1223
01:21:43,448 --> 01:21:45,240
And we'll do something
different next time.
1224
01:21:47,860 --> 01:21:51,460
So homework is due next Tuesday.
1225
01:21:51,460 --> 01:21:54,160
And as I said in my
original handout,
1226
01:21:54,160 --> 01:21:56,230
you should talk to each
other about the homework.
1227
01:21:56,230 --> 01:21:58,680
That's how you learn.