1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high-quality educational resources for free. 5 00:00:10,660 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,322 at ocw.mit.edu. 8 00:00:23,600 --> 00:00:26,390 PROFESSOR: So here's where we were. 9 00:00:26,390 --> 00:00:30,110 So we were talking about this example of b to s gamma 10 00:00:30,110 --> 00:00:31,370 and a heavy to light current. 11 00:00:35,640 --> 00:00:37,830 In terms of this chi field, our current 12 00:00:37,830 --> 00:00:43,470 was chi, which is going to have some label, 13 00:00:43,470 --> 00:00:45,630 in a heavy quark field. 14 00:00:45,630 --> 00:00:48,060 And we did the one with diagrams in SET. 15 00:00:48,060 --> 00:00:52,020 We saw that after we take into account also the 0 bin 16 00:00:52,020 --> 00:00:53,950 contribution that it breaks up as follows. 17 00:00:53,950 --> 00:00:55,450 There's a piece that I've underlined 18 00:00:55,450 --> 00:00:58,395 in orange that matches exactly with the IR of QCD. 19 00:01:04,285 --> 00:01:05,660 There's a piece that I underlined 20 00:01:05,660 --> 00:01:08,390 in blue which is the ultraviolet divergences 21 00:01:08,390 --> 00:01:09,740 in the effective theory. 22 00:01:09,740 --> 00:01:11,150 And so what we do with that is we 23 00:01:11,150 --> 00:01:13,290 add a counter-term for the operators 24 00:01:13,290 --> 00:01:16,257 and the effective theory to cancel these divergences. 25 00:01:16,257 --> 00:01:17,840 And then there's whatever is left over 26 00:01:17,840 --> 00:01:19,940 after doing that, which is this pink. 27 00:01:19,940 --> 00:01:22,520 And the difference of the analog pink 28 00:01:22,520 --> 00:01:25,610 in the full theory and this pink will give you the matching. 29 00:01:25,610 --> 00:01:26,990 So when I say "matching," this is 30 00:01:26,990 --> 00:01:29,500 part of what goes into that matching. 31 00:01:29,500 --> 00:01:31,250 So what I want to spend today's lecture on 32 00:01:31,250 --> 00:01:33,560 is really discussing renormalization 33 00:01:33,560 --> 00:01:35,840 in the effective theory and summing logarithms 34 00:01:35,840 --> 00:01:37,300 in the effective theory. 35 00:01:37,300 --> 00:01:40,670 It's kind of interesting and different than things 36 00:01:40,670 --> 00:01:43,280 we've encountered before, because we see even 37 00:01:43,280 --> 00:01:45,290 at one loop order, we have 1 over 38 00:01:45,290 --> 00:01:46,880 epsilon squared divergence. 39 00:01:46,880 --> 00:01:49,250 And that wasn't something that we encountered before. 40 00:01:49,250 --> 00:01:51,710 And so we have to see how that plays out. 41 00:02:06,050 --> 00:02:08,660 But just at the simplistic level, 42 00:02:08,660 --> 00:02:14,930 let's just say or record for the record 43 00:02:14,930 --> 00:02:17,390 that we could write the following. 44 00:02:17,390 --> 00:02:20,730 There was a Wilson coefficient of this operator. 45 00:02:20,730 --> 00:02:23,690 And we can divide it into renormalize Wilson coefficient 46 00:02:23,690 --> 00:02:25,292 in a counterterm. 47 00:02:25,292 --> 00:02:27,125 And then we just think that this counterterm 48 00:02:27,125 --> 00:02:29,084 is canceling that divergence. 49 00:02:32,880 --> 00:02:39,327 And so the counterterm we need is the following. 50 00:02:39,327 --> 00:02:41,410 So with this minus sign canceling that minus sign, 51 00:02:41,410 --> 00:02:42,640 you get plus. 52 00:02:42,640 --> 00:02:46,510 And so the counterterm for the operator therefore 53 00:02:46,510 --> 00:02:51,280 should have a minus and just all-- 54 00:02:51,280 --> 00:02:52,330 exactly these terms. 55 00:03:03,910 --> 00:03:06,970 Because of kinematics in this problem, 56 00:03:06,970 --> 00:03:10,570 this omega, which is a delta function in this operator, 57 00:03:10,570 --> 00:03:12,910 is equal to actually mb. 58 00:03:12,910 --> 00:03:17,480 And I'll go through why that is a little later today. 59 00:03:17,480 --> 00:03:20,290 So take it for granted for now. 60 00:03:20,290 --> 00:03:23,180 It comes about just simply from kinematics. 61 00:03:23,180 --> 00:03:25,840 But I'm going to sometimes write the mb as omega 62 00:03:25,840 --> 00:03:29,160 because that reminds you that it was this label in the colinear 63 00:03:29,160 --> 00:03:31,368 field. 64 00:03:31,368 --> 00:03:33,160 So this is the type of counterterm we have. 65 00:03:33,160 --> 00:03:34,702 It's got this 1 over epsilon squared. 66 00:03:34,702 --> 00:03:39,790 It's got this log, so it looks kind of different. 67 00:03:39,790 --> 00:03:52,240 And the notation here is that we have 68 00:03:52,240 --> 00:04:02,080 a coefficient that's inside an integral, where 69 00:04:02,080 --> 00:04:10,160 this thing meant that. 70 00:04:16,750 --> 00:04:20,290 So basically what I'm saying when I say omega is equal to mb 71 00:04:20,290 --> 00:04:26,250 is that this delta function becomes omega minus mb. 72 00:04:26,250 --> 00:04:29,220 We'll come back to that. 73 00:04:29,220 --> 00:04:32,970 So if we have a counterterm like that, then we can do running. 74 00:04:38,190 --> 00:04:40,320 So we'll come back to this example. 75 00:04:40,320 --> 00:04:42,450 Let me first make a few general comments, 76 00:04:42,450 --> 00:04:45,510 and then we'll come back and see where this z leads us. 77 00:04:52,840 --> 00:04:56,500 So in general, you're going to have this structure-- 78 00:04:56,500 --> 00:04:58,670 some integral, some Wilson coefficients that depend 79 00:04:58,670 --> 00:05:00,388 on some omegas-- 80 00:05:00,388 --> 00:05:02,680 could be more than one of them-- and then some operator 81 00:05:02,680 --> 00:05:05,305 that has some delta functions in it that pick out those omegas. 82 00:05:07,850 --> 00:05:11,690 And because of that integral, in general, 83 00:05:11,690 --> 00:05:21,250 you have to be careful, because that integral could also play 84 00:05:21,250 --> 00:05:23,560 a role in the renormalization. 85 00:05:39,280 --> 00:05:43,030 So that delta function is picking out 86 00:05:43,030 --> 00:05:45,180 the total momentum of the product of the quark 87 00:05:45,180 --> 00:05:47,860 field times the Wilson line. 88 00:05:47,860 --> 00:05:51,520 And in our example, it's fixed by external kinematics. 89 00:05:51,520 --> 00:05:53,590 So I'll come back to this and show you 90 00:05:53,590 --> 00:05:57,340 that not in every example is it fixed by external kinematics 91 00:05:57,340 --> 00:05:58,480 in a minute. 92 00:05:58,480 --> 00:06:00,850 But our example is special because the integral actually 93 00:06:00,850 --> 00:06:02,590 doesn't play an important role. 94 00:06:32,870 --> 00:06:36,850 So there's in some sense two parts of this statement. 95 00:06:36,850 --> 00:06:38,500 There is the fact that it's external, 96 00:06:38,500 --> 00:06:40,930 and then there's the fact that it's fixed to something. 97 00:06:40,930 --> 00:06:45,520 So let me first deal with the issue of it being external. 98 00:06:45,520 --> 00:06:51,100 And what that means is it does not involve loop momenta. 99 00:06:51,100 --> 00:06:58,120 You see, when we did the calculation, 100 00:06:58,120 --> 00:07:00,958 we were basically doing a calculation, 101 00:07:00,958 --> 00:07:03,250 not worrying so much about this delta function that was 102 00:07:03,250 --> 00:07:06,250 sitting inside our operator. 103 00:07:06,250 --> 00:07:07,180 We ignored it. 104 00:07:07,180 --> 00:07:09,232 We didn't write it down in our diagrams. 105 00:07:09,232 --> 00:07:10,690 Technically, we should have written 106 00:07:10,690 --> 00:07:13,150 that delta function, this delta function, down, too, 107 00:07:13,150 --> 00:07:15,400 and included it in our calculation. 108 00:07:15,400 --> 00:07:18,040 But actually that's OK because that delta function just comes 109 00:07:18,040 --> 00:07:19,630 outside the loop integrals. 110 00:07:19,630 --> 00:07:21,840 That's what I want to show you. 111 00:07:26,250 --> 00:07:30,930 So it does not involve loop momenta. 112 00:07:30,930 --> 00:07:35,370 So in all the diagrams we consider, 113 00:07:35,370 --> 00:07:38,220 that's trivial, except for one non-trivial example. 114 00:07:40,890 --> 00:07:42,990 And that non-trivial example is the example 115 00:07:42,990 --> 00:07:48,000 where we had a colinear vertex-type diagram like this. 116 00:07:51,760 --> 00:07:55,080 So we called the external momentum here p. 117 00:07:55,080 --> 00:07:56,360 This was p plus k. 118 00:07:56,360 --> 00:08:00,270 And if I let this guy go this way, we can call it minus k. 119 00:08:00,270 --> 00:08:04,710 You see, the Wilson coefficient sits here at the vertex. 120 00:08:04,710 --> 00:08:06,420 And it acts on the colinear fields 121 00:08:06,420 --> 00:08:08,730 that come out of that vertex. 122 00:08:08,730 --> 00:08:11,340 But it gives some of the momentum of those colinear 123 00:08:11,340 --> 00:08:11,860 fields. 124 00:08:11,860 --> 00:08:22,850 So we get this guy, and then we get this guy. 125 00:08:22,850 --> 00:08:24,700 And so we just kept the external guy. 126 00:08:30,200 --> 00:08:33,620 So from this notation, the same thing as saying this delta 127 00:08:33,620 --> 00:08:35,700 function. 128 00:08:35,700 --> 00:08:39,919 We can see that it only depends on the external p, 129 00:08:39,919 --> 00:08:41,240 on the n bar dot k. 130 00:08:49,840 --> 00:08:53,590 And that's because of the structure of this operator. 131 00:08:58,290 --> 00:08:59,500 So that won't always be true. 132 00:08:59,500 --> 00:09:02,010 It could be the case that the Wilson-- in a more complicated 133 00:09:02,010 --> 00:09:03,990 example, that the Wilson coefficient 134 00:09:03,990 --> 00:09:05,000 could depend on the k. 135 00:09:05,000 --> 00:09:06,750 And then that would have some implications 136 00:09:06,750 --> 00:09:08,160 for renormalization. 137 00:09:08,160 --> 00:09:11,640 And we will treat an example of that sort in the near future. 138 00:09:15,780 --> 00:09:17,870 The second thing to ask is, what does it fix to? 139 00:09:20,660 --> 00:09:23,030 So if we put a little more into our diagram, 140 00:09:23,030 --> 00:09:24,980 let's put the photon in. 141 00:09:24,980 --> 00:09:26,390 This is a b quark. 142 00:09:26,390 --> 00:09:31,140 This is a strange quark b to s gamma, for example. 143 00:09:31,140 --> 00:09:35,120 Then the kinematics of the external lines there is we 144 00:09:35,120 --> 00:09:38,840 have mbv coming in for the b quark, p of the photon, 145 00:09:38,840 --> 00:09:42,800 and p of the external strange quark. 146 00:09:42,800 --> 00:09:46,190 And the photon is light-like. p gamma squared is 0. 147 00:09:46,190 --> 00:09:49,850 And in the setup here, we can just 148 00:09:49,850 --> 00:09:53,630 think of it as e gamma times n bar, 149 00:09:53,630 --> 00:09:57,230 because p is colinear to the n direction, and it's going-- 150 00:09:57,230 --> 00:10:00,200 we have the photon going this way. 151 00:10:00,200 --> 00:10:04,630 And it's back to back with a jet that's 152 00:10:04,630 --> 00:10:10,580 described by the colinear n direction, 153 00:10:10,580 --> 00:10:12,070 just as the beam goes on decaying 154 00:10:12,070 --> 00:10:14,710 to a photon in a back-to-back jet. 155 00:10:14,710 --> 00:10:20,281 And so if you dot into this equation n bar, 156 00:10:20,281 --> 00:10:24,430 since n bar dot v is 1 in conventional coordinates, 157 00:10:24,430 --> 00:10:30,940 we just get that mb is 0 plus n bar dot p. 158 00:10:30,940 --> 00:10:34,570 And that's the statement that omega was just mb. 159 00:10:34,570 --> 00:10:36,850 So when we were doing our loop calculations last time, 160 00:10:36,850 --> 00:10:38,290 we kept writing mb. 161 00:10:38,290 --> 00:10:40,750 We could have been more careful and written-- 162 00:10:40,750 --> 00:10:44,140 well, we couldn't have been more careful, because kinematics was 163 00:10:44,140 --> 00:10:46,000 demanding that this was true. 164 00:10:46,000 --> 00:10:48,940 But just in order to track what the Wilson coefficients can 165 00:10:48,940 --> 00:10:51,040 depend on, really what they can depend on 166 00:10:51,040 --> 00:10:53,680 is these labels of these colinear fields. 167 00:10:53,680 --> 00:10:57,250 And so I'm going to start writing omega for that reason 168 00:10:57,250 --> 00:10:59,837 because that's the more generic thing that happens. 169 00:10:59,837 --> 00:11:01,420 Even if we're not in a situation where 170 00:11:01,420 --> 00:11:03,660 it's fixed to be some mass, we'll 171 00:11:03,660 --> 00:11:05,500 be able to get dependence on the values 172 00:11:05,500 --> 00:11:07,300 of these parameters of omega. 173 00:11:12,900 --> 00:11:14,330 So any questions about that? 174 00:11:18,532 --> 00:11:20,740 All right, so this Wilson coefficient just comes out, 175 00:11:20,740 --> 00:11:23,980 and we didn't have to worry about this delta function. 176 00:11:23,980 --> 00:11:28,110 So how do we calculate the anomalous dimension? 177 00:11:28,110 --> 00:11:30,300 It's the usual story. 178 00:11:30,300 --> 00:11:34,110 So mu d by dbu of the bare coefficient is 0. 179 00:11:34,110 --> 00:11:36,450 That implies a renormalization group equation 180 00:11:36,450 --> 00:11:40,326 for the ms bar coefficient. 181 00:11:52,490 --> 00:11:57,590 And it's got this familiar form, where the gamma is products 182 00:11:57,590 --> 00:11:59,486 of z's and z inverses-- 183 00:12:06,300 --> 00:12:07,230 just this. 184 00:12:11,660 --> 00:12:13,550 So this is an all-orders formula. 185 00:12:13,550 --> 00:12:15,230 If we plug in our one loop result 186 00:12:15,230 --> 00:12:20,653 here, then on the left, if we worked order by order in alpha, 187 00:12:20,653 --> 00:12:22,070 on the left-hand side, we can just 188 00:12:22,070 --> 00:12:25,580 take the 1 in the zc universe here. 189 00:12:25,580 --> 00:12:28,940 And the zc, when we take b by d by d mu, 190 00:12:28,940 --> 00:12:32,010 there's mu in the alpha s, and there's mu in this log. 191 00:12:32,010 --> 00:12:35,850 And we're going to have to differentiate both of those. 192 00:12:35,850 --> 00:12:39,500 So let me write it out. 193 00:12:59,260 --> 00:13:01,090 So the minus sign canceled the minus sign. 194 00:13:01,090 --> 00:13:02,772 The zc inverse is just 1. 195 00:13:02,772 --> 00:13:05,230 And then I have b db dbu over the rest of this stuff, which 196 00:13:05,230 --> 00:13:10,210 includes both the alpha and the log mu over w. 197 00:13:39,135 --> 00:13:54,510 These are the type of contributions we get, 198 00:13:54,510 --> 00:13:58,860 where these guys here are coming from using the fact that 199 00:13:58,860 --> 00:14:04,650 mu d by d mu of alpha when you're in d dimensions is minus 200 00:14:04,650 --> 00:14:07,290 2 epsilon alpha plus-- 201 00:14:07,290 --> 00:14:09,660 order alpha squared term's related to the beta function, 202 00:14:09,660 --> 00:14:11,580 which we can drop here. 203 00:14:11,580 --> 00:14:16,770 And then this guy comes from the explicit-- 204 00:14:16,770 --> 00:14:22,110 differentiating the explicit log mu over w. 205 00:14:22,110 --> 00:14:23,910 The divergent terms cancel, and we're 206 00:14:23,910 --> 00:14:26,170 left with a finite anomalous dimension. 207 00:14:36,670 --> 00:14:38,380 And it's got two types of terms in it. 208 00:14:42,770 --> 00:14:44,320 So the type of anomalous dimensions 209 00:14:44,320 --> 00:14:48,760 that we're used to are like this guy, here-- this 5 times cf. 210 00:14:48,760 --> 00:14:50,920 That would be like a standard anomalous dimension 211 00:14:50,920 --> 00:14:52,157 for an operator. 212 00:14:52,157 --> 00:14:53,740 In this case, we have this extra piece 213 00:14:53,740 --> 00:14:56,500 that has this log of mu over w. 214 00:14:56,500 --> 00:15:03,370 And this guy is what's called cusp anomalous dimension, 215 00:15:03,370 --> 00:15:05,140 the guy that multiplies the log. 216 00:15:09,530 --> 00:15:11,480 So with that language, you could call this guy 217 00:15:11,480 --> 00:15:14,690 the regular anomalous dimension. 218 00:15:22,370 --> 00:15:24,440 Now, if you're going to do renormalization group 219 00:15:24,440 --> 00:15:26,648 evolution, you're going to sum logs. 220 00:15:26,648 --> 00:15:28,940 You could think that you're starting at some hard scale 221 00:15:28,940 --> 00:15:30,470 like w. w is mb. 222 00:15:30,470 --> 00:15:32,030 That's the hard scale. 223 00:15:32,030 --> 00:15:35,750 And then you want to run down to some small-scale mu. 224 00:15:35,750 --> 00:15:38,040 When you run down, if you take mu small, 225 00:15:38,040 --> 00:15:41,060 this term is larger than this term. 226 00:15:41,060 --> 00:15:44,390 So actually, this term here gives you 227 00:15:44,390 --> 00:15:47,300 the leading-- what are called the leading logs in this case. 228 00:15:47,300 --> 00:15:52,670 And this term here is actually only needed 229 00:15:52,670 --> 00:15:55,250 for next leading log. 230 00:15:55,250 --> 00:15:57,320 Because this term here grows when you go down, 231 00:15:57,320 --> 00:16:00,770 and that term doesn't, this term here 232 00:16:00,770 --> 00:16:03,467 is actually giving you the leading logs. 233 00:16:03,467 --> 00:16:05,300 So I'll come back and talk a little bit more 234 00:16:05,300 --> 00:16:07,220 about leading log versus next-to-leading log. 235 00:16:07,220 --> 00:16:11,410 And also, I'll explain to you what this word "cusp" in this-- 236 00:16:11,410 --> 00:16:14,450 where's the cusp, since we haven't seen it yet? 237 00:16:14,450 --> 00:16:16,670 I'll come back and explain where that is in a minute. 238 00:16:22,180 --> 00:16:24,680 For now, let's just take the leading log anomalous dimension 239 00:16:24,680 --> 00:16:29,673 and see just to solve the anomalous dimension equation. 240 00:16:29,673 --> 00:16:31,340 So we'll just take the leading long term 241 00:16:31,340 --> 00:16:42,140 and not worry about the other term, 242 00:16:42,140 --> 00:16:45,610 so just taking the cusp term. 243 00:16:45,610 --> 00:16:48,520 So plugging that back into our equation for c, 244 00:16:48,520 --> 00:16:52,240 we have this differential equation, which we can also 245 00:16:52,240 --> 00:16:58,010 write as d log c by d log mu. 246 00:17:11,319 --> 00:17:13,210 And that's an equation that we can basically 247 00:17:13,210 --> 00:17:19,450 just integrate since there's no c on the right-hand side. 248 00:17:19,450 --> 00:17:20,990 It's a homogeneous equation. 249 00:17:20,990 --> 00:17:23,235 We can integrate it with some boundary condition. 250 00:17:27,990 --> 00:17:30,230 So we could specify the boundary condition 251 00:17:30,230 --> 00:17:32,570 at an arbitrary scale. 252 00:17:32,570 --> 00:17:36,530 Let me, for simplicity, use the following tree-level boundary 253 00:17:36,530 --> 00:17:37,561 condition. 254 00:17:41,490 --> 00:17:44,300 So at tree level, the Wilson coefficient was 1. 255 00:17:44,300 --> 00:17:48,110 That we have to pick some scale where 256 00:17:48,110 --> 00:17:51,770 we make that statement true. 257 00:17:51,770 --> 00:17:54,710 And for convenience, I'm just going to take that scale to bw. 258 00:17:54,710 --> 00:17:57,260 That makes things a little simpler. 259 00:17:57,260 --> 00:18:01,760 In general, you could make this any hard scale, ie, 260 00:18:01,760 --> 00:18:04,430 any scale mu of order w. 261 00:18:04,430 --> 00:18:07,610 So you could call it something mu 0 262 00:18:07,610 --> 00:18:11,750 and consider the renormalization group from that scale. 263 00:18:11,750 --> 00:18:15,230 But for convenience, I'm just taking that mu 0 to be w here. 264 00:18:29,740 --> 00:18:32,098 So we take that down [INAUDIBLE] It's 265 00:18:32,098 --> 00:18:33,640 actually useful to look at what would 266 00:18:33,640 --> 00:18:35,057 happen if we didn't have a running 267 00:18:35,057 --> 00:18:37,250 coupling on the right-hand side. 268 00:18:37,250 --> 00:18:40,570 So say we were doing QED, and we didn't have any massless light 269 00:18:40,570 --> 00:18:43,240 fermions, so there was no running. 270 00:18:43,240 --> 00:18:46,325 Then this coupling would just be alpha, and it wouldn't run. 271 00:18:46,325 --> 00:18:47,950 And then you'd just integrate this log, 272 00:18:47,950 --> 00:18:49,240 and you'd get a double log. 273 00:18:49,240 --> 00:18:53,830 You integrate one log by d log, you get log squared. 274 00:18:53,830 --> 00:18:55,930 For each coupling, you get log squared. 275 00:18:55,930 --> 00:18:58,480 This is the double log series that I promised you-- 276 00:18:58,480 --> 00:19:00,340 the Sudakov double logarithms that 277 00:19:00,340 --> 00:19:05,440 were going to come out of the renormalization group of SET. 278 00:19:05,440 --> 00:19:13,307 So QED with alpha fixed-- 279 00:19:13,307 --> 00:19:15,790 I'm going to take cf to be 1. 280 00:19:18,820 --> 00:19:31,190 When you take the exponential of what I just said, 281 00:19:31,190 --> 00:19:34,935 then this is your result for the Wilson coefficient. 282 00:19:40,990 --> 00:19:43,140 And this is what's known as a Sudakov exponential. 283 00:19:43,140 --> 00:19:49,530 The leading log Wilson coefficient 284 00:19:49,530 --> 00:19:51,648 is the sum of these alpha times double log. 285 00:19:51,648 --> 00:19:53,940 So if I expanded this out, I'd have an infinite series, 286 00:19:53,940 --> 00:19:57,450 where for each pair of logarithms, I get one coupling. 287 00:20:11,060 --> 00:20:15,170 And that's the Sudakov double logarithms. 288 00:20:15,170 --> 00:20:16,670 So if we expand out the exponential, 289 00:20:16,670 --> 00:20:18,840 that's what we would get. 290 00:20:18,840 --> 00:20:23,360 So really, I should make my line go under the whole thing. 291 00:20:27,410 --> 00:20:30,270 So why did we get something like this? 292 00:20:30,270 --> 00:20:34,760 So physically the reason that we got this Sudakov exponential 293 00:20:34,760 --> 00:20:37,790 or Sudakov form factor is related to the fact 294 00:20:37,790 --> 00:20:40,370 that the whole theory that we've developed 295 00:20:40,370 --> 00:20:42,020 put restrictions on radiation. 296 00:20:58,614 --> 00:21:00,540 So there was kinematic restrictions 297 00:21:00,540 --> 00:21:02,875 put on the radiation by the whole setup that we had. 298 00:21:02,875 --> 00:21:04,500 We had this jet, and it was colonnaded. 299 00:21:04,500 --> 00:21:07,860 And the radiation was forced to go inside that jet. 300 00:21:21,080 --> 00:21:23,840 In the effective theory, we're seeing the Sudakov exponential 301 00:21:23,840 --> 00:21:25,770 come out of UV renormalization. 302 00:21:25,770 --> 00:21:27,440 The usual picture for it is that you're 303 00:21:27,440 --> 00:21:29,388 thinking about IR divergences. 304 00:21:29,388 --> 00:21:31,430 And you're thinking about the real radiation, not 305 00:21:31,430 --> 00:21:33,320 the virtual radiation. 306 00:21:33,320 --> 00:21:36,710 But as you may be familiar with what the effective theory does, 307 00:21:36,710 --> 00:21:40,790 is if you have an IR divergence in full QCD or full QED, 308 00:21:40,790 --> 00:21:42,440 then what the effective theory does 309 00:21:42,440 --> 00:21:45,110 is it takes the scale corresponding to-- 310 00:21:45,110 --> 00:21:48,050 so the IR divergences are logarithms. 311 00:21:48,050 --> 00:21:50,240 And they're logarithms of something over something. 312 00:21:50,240 --> 00:21:52,160 And we've done in the effective theory 313 00:21:52,160 --> 00:21:54,620 is we've introduced a scale mu. 314 00:21:54,620 --> 00:21:56,360 We've introduced-- effectively, we've 315 00:21:56,360 --> 00:21:59,120 taken the part of that logarithm that was UV 316 00:21:59,120 --> 00:22:00,860 and taken it to infinity. 317 00:22:00,860 --> 00:22:02,810 And in this standard way, we've introduced 318 00:22:02,810 --> 00:22:07,640 a scale mu that splits things between Wilson coefficients 319 00:22:07,640 --> 00:22:09,260 and operators. 320 00:22:09,260 --> 00:22:11,580 So there's this factorization scale 321 00:22:11,580 --> 00:22:13,220 that's splitting the hard physics 322 00:22:13,220 --> 00:22:16,880 from-- which is part of the logarithm that we're capturing. 323 00:22:16,880 --> 00:22:19,998 It's separating the hard physics from the infrared physics. 324 00:22:19,998 --> 00:22:22,040 And here, we're just looking at the hard physics. 325 00:22:22,040 --> 00:22:23,180 And we can see the Sudakov. 326 00:22:23,180 --> 00:22:24,847 We could also look at the real radiation 327 00:22:24,847 --> 00:22:28,100 and see the Sudakov by studying real radiation diagrams. 328 00:22:28,100 --> 00:22:29,973 But in some sense, this is easier, 329 00:22:29,973 --> 00:22:32,390 because this is just coming out of a renormalization group 330 00:22:32,390 --> 00:22:34,453 equation. 331 00:22:34,453 --> 00:22:35,870 So the usual way of doing it would 332 00:22:35,870 --> 00:22:37,610 be to just look at in the full theory, 333 00:22:37,610 --> 00:22:41,060 and you'd see Sudakov logarithms from ratios of two things 334 00:22:41,060 --> 00:22:42,680 that you were thinking of as IR. 335 00:22:42,680 --> 00:22:45,890 Because the hard scale, which is mbw here, 336 00:22:45,890 --> 00:22:49,280 would also be an IR scale from the point of view of QCD. 337 00:22:49,280 --> 00:22:52,100 But here, it's become part of the Wilson coefficient. 338 00:22:52,100 --> 00:22:57,290 And now it can be treated by the renormalization group. 339 00:22:57,290 --> 00:23:01,082 So QCD is not really any more complicated. 340 00:23:01,082 --> 00:23:03,040 We just have to deal with the running coupling. 341 00:23:07,190 --> 00:23:09,900 We've done that before. 342 00:23:09,900 --> 00:23:15,310 Let me remind you how it works because it's slightly 343 00:23:15,310 --> 00:23:17,450 more involved in this case. 344 00:23:17,450 --> 00:23:21,220 So for the leading logs, we just need the beta 0 term. 345 00:23:21,220 --> 00:23:23,380 And we can switch variables from d log to d 346 00:23:23,380 --> 00:23:25,787 alpha in this equation, here. 347 00:23:25,787 --> 00:23:28,120 And so then we have to integrate this thing with respect 348 00:23:28,120 --> 00:23:31,120 to d alpha. 349 00:23:31,120 --> 00:23:34,630 And so what we can do is we can take this log, 350 00:23:34,630 --> 00:23:37,930 and we can also write the log as an integral over alpha. 351 00:23:49,340 --> 00:23:52,790 Alpha in this equation is some w variable. 352 00:23:52,790 --> 00:23:58,760 And so then just integrating both sides of this equation 353 00:23:58,760 --> 00:24:30,770 here, I can write it out like that. 354 00:24:30,770 --> 00:24:34,330 So the log became this thing. 355 00:24:34,330 --> 00:24:36,030 And then one of these factors-- 356 00:24:36,030 --> 00:24:37,390 there was a minus sign. 357 00:24:37,390 --> 00:24:39,900 But then when I use this formula twice, 358 00:24:39,900 --> 00:24:42,850 the minus sign goes away, and I get two of those factors. 359 00:24:42,850 --> 00:24:46,302 This is the other measure here from the explicit d log mu. 360 00:24:46,302 --> 00:24:47,910 So we could underline things-- 361 00:24:52,200 --> 00:24:59,565 half of this guy, all of that guy, d 362 00:24:59,565 --> 00:25:04,640 log mu, other half of this guy and that guy. 363 00:25:04,640 --> 00:25:07,270 And then there's this explicit alpha cf 364 00:25:07,270 --> 00:25:10,150 over pi, which I've written as cf over pi out front, 365 00:25:10,150 --> 00:25:11,410 and this alpha here. 366 00:25:23,150 --> 00:25:25,020 So we do that integral. 367 00:25:25,020 --> 00:25:25,910 We do this integral. 368 00:25:25,910 --> 00:25:27,410 Well, all the intervals are trivial. 369 00:25:27,410 --> 00:25:30,290 We do this integral, and then we get integration variable 370 00:25:30,290 --> 00:25:31,080 at fixed quantity. 371 00:25:31,080 --> 00:25:33,305 And then we can do this other integral. 372 00:25:33,305 --> 00:25:34,055 We get a solution. 373 00:25:57,600 --> 00:25:59,510 So I'll show you what the answer looks like. 374 00:25:59,510 --> 00:26:02,060 It's, again, an exponential, because we're solving for a log 375 00:26:02,060 --> 00:26:02,560 c. 376 00:26:13,090 --> 00:26:14,330 I'll write it like this. 377 00:26:28,150 --> 00:26:30,070 So we're familiar that renormalization group 378 00:26:30,070 --> 00:26:31,903 equations, when you have a running coupling, 379 00:26:31,903 --> 00:26:33,980 can give you ratios of alphas. 380 00:26:33,980 --> 00:26:38,077 We saw that when we were running for fermion operators. 381 00:26:38,077 --> 00:26:39,910 Here, it's just a more complicated function, 382 00:26:39,910 --> 00:26:42,500 but it's again a function of that ratio. 383 00:26:42,500 --> 00:26:44,842 And this extra structure is coming about just 384 00:26:44,842 --> 00:26:45,925 from the running coupling. 385 00:26:51,690 --> 00:26:55,480 So the only difference between this and QED Sudakov 386 00:26:55,480 --> 00:26:56,480 is the running coupling. 387 00:27:07,500 --> 00:27:14,370 So if we were to take this now, and we were to expand z, 388 00:27:14,370 --> 00:27:17,100 we were, say, to expand about alpha of w, 389 00:27:17,100 --> 00:27:20,430 then we would get logs of mu over w, right? 390 00:27:20,430 --> 00:27:22,920 And so what you should think that this z dependence 391 00:27:22,920 --> 00:27:23,610 encodes-- 392 00:27:23,610 --> 00:27:29,490 is an infinite series in this exponential. 393 00:27:29,490 --> 00:27:31,720 And so the structure of that infinite series, 394 00:27:31,720 --> 00:27:38,490 if I was to expand, would be alpha squared u over w alpha 395 00:27:38,490 --> 00:27:42,960 squared log cubed u over w, et cetera. 396 00:27:46,850 --> 00:27:49,850 So in the exponent, a running of the coupling 397 00:27:49,850 --> 00:27:52,620 is only giving you one log for each alpha. 398 00:27:52,620 --> 00:27:55,280 So the series that I'm solving in the exponent 399 00:27:55,280 --> 00:27:57,270 by taking into account the running coupling 400 00:27:57,270 --> 00:27:59,720 is a series of this form, where I go down by one alpha 401 00:27:59,720 --> 00:28:00,950 and down by one log. 402 00:28:04,614 --> 00:28:09,533 So from the structure of what we've been talking about here, 403 00:28:09,533 --> 00:28:11,950 it should become clear that the right thing to talk about, 404 00:28:11,950 --> 00:28:15,400 if you want to do counting, is log c, 405 00:28:15,400 --> 00:28:18,250 because the anomalous dimension equation was simple for log c. 406 00:28:18,250 --> 00:28:22,180 It was mu d by mu or d by d log mu of log c 407 00:28:22,180 --> 00:28:23,620 is equal to something. 408 00:28:23,620 --> 00:28:26,945 That something on the right-hand side is an expansion in alpha. 409 00:28:26,945 --> 00:28:28,570 And we could include higher-order terms 410 00:28:28,570 --> 00:28:31,090 in that expansion, but it would always be of that structure. 411 00:28:43,220 --> 00:28:47,440 So to talk about what terms we're summing, 412 00:28:47,440 --> 00:28:50,830 we can write the following schematic equation 413 00:28:50,830 --> 00:28:54,080 for different orders. 414 00:28:54,080 --> 00:28:57,040 So if we have alpha s to the k log 415 00:28:57,040 --> 00:29:00,430 to the k plus 1, which is this series I've just denoted, 416 00:29:00,430 --> 00:29:01,810 we'll call that leading log. 417 00:29:04,660 --> 00:29:07,360 And if we add some higher-order term, 418 00:29:07,360 --> 00:29:10,295 like alpha s to the k log to the k, 419 00:29:10,295 --> 00:29:11,920 then that would be next-to-leading log. 420 00:29:19,910 --> 00:29:22,800 And if we had more alphas than logs, 421 00:29:22,800 --> 00:29:24,740 then we call that next-to-next-to leading log. 422 00:29:28,072 --> 00:29:30,280 And the structure of the anomalous dimension equation 423 00:29:30,280 --> 00:29:34,180 always guarantees that if you solve it for this log, 424 00:29:34,180 --> 00:29:36,580 that this is what the higher-order terms would 425 00:29:36,580 --> 00:29:38,810 give you. 426 00:29:38,810 --> 00:29:42,010 So you should think of it that really, log counting 427 00:29:42,010 --> 00:29:44,140 is kind of like normal log counting, except for two 428 00:29:44,140 --> 00:29:46,720 caveats in this theory. 429 00:29:46,720 --> 00:29:49,900 You're doing it in the exponential. 430 00:29:49,900 --> 00:29:51,790 That's what taking the log did. 431 00:29:51,790 --> 00:29:55,270 And you start out with one extra log, one more log 432 00:29:55,270 --> 00:29:57,200 than you're used to. 433 00:29:57,200 --> 00:29:59,440 But after that, it's kind of normal, 434 00:29:59,440 --> 00:30:02,560 because I'm summing here alpha times log, 435 00:30:02,560 --> 00:30:05,100 and I just increment for each alpha an extra log. 436 00:30:05,100 --> 00:30:06,940 And that's all the running coupling effects. 437 00:30:06,940 --> 00:30:08,865 When I go to next-to-leading log, of course, 438 00:30:08,865 --> 00:30:10,990 we write in coupling effects, too, and other things 439 00:30:10,990 --> 00:30:13,390 that are causing the series. 440 00:30:13,390 --> 00:30:15,100 But it's down from this one because it 441 00:30:15,100 --> 00:30:17,050 doesn't have that enhanced log. 442 00:30:17,050 --> 00:30:20,530 And then this one is down again because it's, again, 443 00:30:20,530 --> 00:30:22,828 one less log. 444 00:30:22,828 --> 00:30:24,370 So this is the kind of terms that you 445 00:30:24,370 --> 00:30:26,890 could sum by having higher-order corrections 446 00:30:26,890 --> 00:30:29,080 in the anomalous dimensions. 447 00:30:29,080 --> 00:30:31,010 And for example, this cusp anomalous dimension 448 00:30:31,010 --> 00:30:32,260 is known at three loop orders. 449 00:30:32,260 --> 00:30:37,880 So certainly, next-to-next-to leading log is well within 450 00:30:37,880 --> 00:30:40,760 the realm of things that people talk about. 451 00:30:40,760 --> 00:30:42,110 So let's ask that question. 452 00:30:42,110 --> 00:30:45,220 What coefficients, if we were to carry out this one 453 00:30:45,220 --> 00:30:48,423 new calculation, and we were to do higher loops, 454 00:30:48,423 --> 00:30:50,090 what coefficients do we need to compute? 455 00:30:56,050 --> 00:30:58,860 So remember the story when we were summing single log series 456 00:30:58,860 --> 00:31:00,480 was we would do tree-level matching 457 00:31:00,480 --> 00:31:02,018 one loop anomalous dimension. 458 00:31:02,018 --> 00:31:04,560 Then we'd do one loop matching, two loop anomalous dimension. 459 00:31:04,560 --> 00:31:06,330 What's the analog of that story here? 460 00:31:12,857 --> 00:31:13,940 Let's make a little table. 461 00:31:18,120 --> 00:31:22,220 So what information do we need at tree level? 462 00:31:22,220 --> 00:31:25,580 What information do we need at one loop, 463 00:31:25,580 --> 00:31:29,000 two loops, three loops? 464 00:31:34,860 --> 00:31:36,390 So for a leading log, what we did is 465 00:31:36,390 --> 00:31:38,340 we put in tree-level matching. 466 00:31:38,340 --> 00:31:40,308 And at one loop, we only actually took 467 00:31:40,308 --> 00:31:42,600 the information that came from the coefficient of the 1 468 00:31:42,600 --> 00:31:47,082 over epsilon squared, which is the cusp anomalous dimension. 469 00:31:47,082 --> 00:31:49,350 [INAUDIBLE] here. 470 00:31:49,350 --> 00:31:51,270 If you go to one higher order, it 471 00:31:51,270 --> 00:31:56,580 turns out that this is the story to get this series. 472 00:32:01,970 --> 00:32:05,140 We would need the two-loop cusp anomalous dimension, the one 473 00:32:05,140 --> 00:32:08,320 loop non-cusp, and still the tree-level matching. 474 00:32:08,320 --> 00:32:11,500 So this next-to-leading log coefficient, this 5cf, 475 00:32:11,500 --> 00:32:14,320 that's this 1 over epsilon. 476 00:32:14,320 --> 00:32:17,800 In our example, this was the 5cf. 477 00:32:17,800 --> 00:32:21,100 And only when you get to next-to-next-to-leading log, 478 00:32:21,100 --> 00:32:22,540 then the matching is one loop. 479 00:32:26,508 --> 00:32:30,880 We have the two-loop regular anomalous dimension, 480 00:32:30,880 --> 00:32:32,380 and then the 1 over epsilon squared 481 00:32:32,380 --> 00:32:34,780 cusp anomalous dimension from three loop. 482 00:32:34,780 --> 00:32:37,330 So it's kind of like it's the usual story, 483 00:32:37,330 --> 00:32:39,132 but we have an enhanced-- because 484 00:32:39,132 --> 00:32:41,590 of the double logarithms, we have an enhanced thing that we 485 00:32:41,590 --> 00:32:43,603 can talk about here, where we're not 486 00:32:43,603 --> 00:32:45,520 talking about the regular anomalous dimension, 487 00:32:45,520 --> 00:32:48,372 but this cusp anomalous dimension. 488 00:32:48,372 --> 00:32:50,080 So this is the information that you would 489 00:32:50,080 --> 00:32:52,030 need to go to higher orders. 490 00:32:52,030 --> 00:32:53,500 And this cusp anomalous dimension 491 00:32:53,500 --> 00:32:54,865 is actually a universal thing. 492 00:32:54,865 --> 00:33:01,680 So given that it's been calculated, you can use it. 493 00:33:01,680 --> 00:33:03,933 And you don't have to recalculate it every time. 494 00:33:03,933 --> 00:33:05,850 These pieces you have to recalculate-- these 1 495 00:33:05,850 --> 00:33:11,541 over epsilons, in general. 496 00:33:11,541 --> 00:33:12,041 Yeah? 497 00:33:12,041 --> 00:33:16,190 STUDENT: Can you comment on the seemingly 498 00:33:16,190 --> 00:33:20,690 arbitrariness of the boundary condition you chose? 499 00:33:20,690 --> 00:33:24,040 PROFESSOR: Yeah, so if I hadn't chose that boundary condition-- 500 00:33:24,040 --> 00:33:24,910 sure. 501 00:33:24,910 --> 00:33:26,325 Let me write something. 502 00:33:41,802 --> 00:33:43,760 So the way that you could think about the rg is 503 00:33:43,760 --> 00:33:46,520 as follows-- that you have c. 504 00:33:46,520 --> 00:33:49,610 And really what the rg is doing is determining some u. 505 00:33:54,120 --> 00:33:59,130 It allows you to run c from some point to some other point, 506 00:33:59,130 --> 00:34:00,610 like that. 507 00:34:00,610 --> 00:34:04,620 So in this formula, mu 0 in some scale that's 508 00:34:04,620 --> 00:34:08,530 of order omega or w-- it doesn't have to be equal to w. 509 00:34:08,530 --> 00:34:10,433 It could be 2w, 1/2w. 510 00:34:10,433 --> 00:34:12,600 And the way that you're thinking about this equation 511 00:34:12,600 --> 00:34:13,590 is you're going to do perturbation 512 00:34:13,590 --> 00:34:14,507 theory for this thing. 513 00:34:19,940 --> 00:34:22,330 And then in that perturbation theory, you'll have alpha, 514 00:34:22,330 --> 00:34:26,480 and you'll have log squared of mu 0 over w. 515 00:34:26,480 --> 00:34:28,659 But as long as you're saying mu 0's of order w, 516 00:34:28,659 --> 00:34:30,610 those aren't large logs. 517 00:34:30,610 --> 00:34:33,280 And this thing here in this formulation 518 00:34:33,280 --> 00:34:36,880 would have the double log series of alpha log squared 519 00:34:36,880 --> 00:34:39,370 of mu over mu0. 520 00:34:39,370 --> 00:34:40,864 But then mu0's of order w. 521 00:34:40,864 --> 00:34:42,239 And mu you can take much smaller, 522 00:34:42,239 --> 00:34:43,464 and that's the large logs. 523 00:34:47,510 --> 00:34:51,659 So for simplicity, I just took mu0 equal to w. 524 00:34:51,659 --> 00:34:53,560 Everything I've said, I could repeat, 525 00:34:53,560 --> 00:34:55,893 and this story would really be the same if I just 526 00:34:55,893 --> 00:34:56,810 used an arbitrary mu0. 527 00:35:00,950 --> 00:35:03,140 And the reason you actually want to use an arbitrary 528 00:35:03,140 --> 00:35:06,560 mu0 in general is that you want to do scale variation 529 00:35:06,560 --> 00:35:08,430 to think about uncertainties. 530 00:35:08,430 --> 00:35:11,060 And so the mu0 dependence, just like in our-- 531 00:35:11,060 --> 00:35:13,370 when we were talking about electroweak operators 532 00:35:13,370 --> 00:35:16,190 and electric Hamiltonian, the mu0 dependence 533 00:35:16,190 --> 00:35:18,200 cancels between these things here. 534 00:35:18,200 --> 00:35:21,770 But that calculation is in order by order an alpha cancellation. 535 00:35:21,770 --> 00:35:25,410 There's no large logs associated with that cancellation. 536 00:35:25,410 --> 00:35:26,915 And so what it allows you to do is 537 00:35:26,915 --> 00:35:28,790 if you've truncated this to some fixed order, 538 00:35:28,790 --> 00:35:31,070 and you've worked to some fixed order over there, 539 00:35:31,070 --> 00:35:33,590 then you can probe how much uncertainty you 540 00:35:33,590 --> 00:35:36,230 have by varying the mu0. 541 00:35:36,230 --> 00:35:40,880 This mu here will get tied up with the mu in the operator. 542 00:35:40,880 --> 00:35:44,270 So we can talk more about where that mu goes later on. 543 00:35:44,270 --> 00:35:47,660 But at the level of this formula, it is-- in general, 544 00:35:47,660 --> 00:35:50,450 it's the case that you would do it with an arbitrary mu0, 545 00:35:50,450 --> 00:35:52,670 and then only fix mu0 at the end. 546 00:35:52,670 --> 00:35:55,935 And you'd even vary mu0 to get an idea of how much uncertainty 547 00:35:55,935 --> 00:35:58,310 you have in your leading log calculation, because varying 548 00:35:58,310 --> 00:36:02,640 mu0 would probe the next term in the series, here. 549 00:36:02,640 --> 00:36:06,620 STUDENT: So when you say tree-level matching, you mean-- 550 00:36:06,620 --> 00:36:07,880 PROFESSOR: I mean this c. 551 00:36:07,880 --> 00:36:08,797 STUDENT: --equal to 1. 552 00:36:08,797 --> 00:36:10,220 PROFESSOR: That's right. 553 00:36:10,220 --> 00:36:11,520 I mean this guy, here. 554 00:36:11,520 --> 00:36:12,020 Right. 555 00:36:12,020 --> 00:36:12,645 STUDENT: Right. 556 00:36:12,645 --> 00:36:15,007 And if you want to go to nnll, you'd take 1-- 557 00:36:15,007 --> 00:36:16,340 PROFESSOR: Then you'd have this. 558 00:36:16,340 --> 00:36:17,650 That wouldn't have this term. 559 00:36:17,650 --> 00:36:18,150 Yeah. 560 00:36:22,140 --> 00:36:26,020 All right, any other questions? 561 00:36:26,020 --> 00:36:28,020 OK, so what is this cusp that we've 562 00:36:28,020 --> 00:36:29,640 been saying the words for? 563 00:36:36,290 --> 00:36:39,870 I'll answer a couple of questions that have come up. 564 00:36:39,870 --> 00:36:40,760 Where's the cusp? 565 00:36:44,970 --> 00:36:57,540 So if we look back at our operator, we had this operator. 566 00:36:57,540 --> 00:37:03,100 And we could make a field redefinition, if you remember, 567 00:37:03,100 --> 00:37:06,310 and put the ultrasoft effects into a Wilson line. 568 00:37:10,800 --> 00:37:14,250 Now for this heavy quark here, remember what that-- 569 00:37:14,250 --> 00:37:17,070 the theory for that was with hqet. 570 00:37:17,070 --> 00:37:23,560 So the field theory for this heavy quark was an iv dot dhv. 571 00:37:23,560 --> 00:37:26,940 And if we actually make a similar field redefinition 572 00:37:26,940 --> 00:37:29,430 to the one we talked about to get this y, 573 00:37:29,430 --> 00:37:32,520 but on the heavy quark field, then we 574 00:37:32,520 --> 00:37:37,050 can actually, in terms of this guy 575 00:37:37,050 --> 00:37:40,540 here, get a free Lagrangian as well. 576 00:37:40,540 --> 00:37:43,770 So actually, all the effects from ultra soft gluons 577 00:37:43,770 --> 00:37:47,390 in this operator can be encoded in Wilson lines. 578 00:37:51,872 --> 00:37:53,130 So let me do that. 579 00:37:57,560 --> 00:37:59,990 So that just says that actually, when 580 00:37:59,990 --> 00:38:02,330 we talked about static sources, static sources 581 00:38:02,330 --> 00:38:05,870 could also be encoded as Wilson lines 582 00:38:05,870 --> 00:38:07,310 by a kind of similar manipulation 583 00:38:07,310 --> 00:38:11,210 to what we did when we talked about SET. 584 00:38:11,210 --> 00:38:13,010 So if we look at the ultrasoft sector here, 585 00:38:13,010 --> 00:38:15,230 we have a Wilson line that actually has a path. 586 00:38:28,380 --> 00:38:30,020 So there's a Wilson line that comes 587 00:38:30,020 --> 00:38:33,080 from minus infinity along v to whatever position 588 00:38:33,080 --> 00:38:34,070 we put our operator at. 589 00:38:34,070 --> 00:38:35,810 Let's take it to be 0. 590 00:38:35,810 --> 00:38:37,490 And then we have a yn dagger which 591 00:38:37,490 --> 00:38:43,340 is extending out to plus infinity from 0, like that. 592 00:38:43,340 --> 00:38:45,345 And the cusp is this fact that there's 593 00:38:45,345 --> 00:38:46,470 a kink in this Wilson line. 594 00:38:46,470 --> 00:38:47,660 It's not a smooth path. 595 00:38:47,660 --> 00:38:49,640 It actually has a sharp angle. 596 00:38:49,640 --> 00:38:50,660 And that's the cusp. 597 00:39:05,370 --> 00:39:07,580 So there's actually a general renormalization theory 598 00:39:07,580 --> 00:39:10,670 for renormalizing Wilson lines with cusps. 599 00:39:10,670 --> 00:39:12,890 So this is something that at some point 600 00:39:12,890 --> 00:39:15,770 in the history of QCD, people tried to reformulate QCD 601 00:39:15,770 --> 00:39:17,750 in terms of Wilson lines, entirely 602 00:39:17,750 --> 00:39:19,280 in terms of Wilson lines. 603 00:39:19,280 --> 00:39:21,890 Then they realized that they could have these cusps 604 00:39:21,890 --> 00:39:22,760 in the Wilson lines. 605 00:39:22,760 --> 00:39:24,380 And the renormalization of that theory 606 00:39:24,380 --> 00:39:26,588 became much more complicated than the renormalization 607 00:39:26,588 --> 00:39:28,460 of the QCD action because they couldn't 608 00:39:28,460 --> 00:39:30,627 prove that in general, these cusps wouldn't come out 609 00:39:30,627 --> 00:39:31,880 of the dynamics. 610 00:39:31,880 --> 00:39:33,260 And then people dropped it. 611 00:39:33,260 --> 00:39:36,560 But along the way, anyway, they formulated 612 00:39:36,560 --> 00:39:41,085 a general renormalization theory of cusps in Wilson lines. 613 00:39:41,085 --> 00:39:42,710 And whenever you have a cusp like this, 614 00:39:42,710 --> 00:39:45,450 you end up having, in our language, this log, 615 00:39:45,450 --> 00:39:48,350 single log, in the anomalous dimension. 616 00:39:48,350 --> 00:39:59,880 So if one of the lines here is light-like, and one of them 617 00:39:59,880 --> 00:40:06,390 is, because n squared was 0, then the anomalous dimension 618 00:40:06,390 --> 00:40:14,520 for this cusp will have a single log like our log of mu over w. 619 00:40:14,520 --> 00:40:16,770 So that actually holds not only to one loop, 620 00:40:16,770 --> 00:40:19,650 as we talked about, but actually to all orders 621 00:40:19,650 --> 00:40:20,730 in perturbation theory. 622 00:40:20,730 --> 00:40:24,210 If we'd gone to higher orders in perturbation theory, 623 00:40:24,210 --> 00:40:29,167 then what would happen is that the coefficient of that log 624 00:40:29,167 --> 00:40:30,000 would get corrected. 625 00:40:30,000 --> 00:40:33,750 But there'd still be only one log in our anomalous dimension. 626 00:40:33,750 --> 00:40:39,900 That's also something that you can argue from SET directly. 627 00:40:39,900 --> 00:40:43,920 But originally, it falls from this general theory 628 00:40:43,920 --> 00:40:46,500 of the renormalization of cusps of Wilson lines. 629 00:40:52,550 --> 00:40:55,160 So if you were thinking about our calculation, 630 00:40:55,160 --> 00:40:57,830 you may remember actually that we got 1 over epsilon squared. 631 00:40:57,830 --> 00:40:59,450 It's from the ultrasoft diagrams. 632 00:40:59,450 --> 00:41:02,630 And that's in some sense what we've just talked about, 633 00:41:02,630 --> 00:41:05,090 because these are the ultrasoft diagrams, 634 00:41:05,090 --> 00:41:07,197 and they have 1 over epsilon squareds. 635 00:41:07,197 --> 00:41:09,530 There was also 1 over epsilon squareds from the colinear 636 00:41:09,530 --> 00:41:12,350 diagrams, right? 637 00:41:12,350 --> 00:41:17,750 So if you like, this piece here also in some sense 638 00:41:17,750 --> 00:41:19,910 has kind of a cusp. 639 00:41:19,910 --> 00:41:21,625 And in this case, it's not-- 640 00:41:21,625 --> 00:41:23,000 it's actually what we're doing is 641 00:41:23,000 --> 00:41:26,510 we're taking a Wilson line along the n bar direction, which 642 00:41:26,510 --> 00:41:27,085 is our wn. 643 00:41:29,870 --> 00:41:34,610 And then we're attaching it to a quark field. 644 00:41:34,610 --> 00:41:37,760 So this is a full cn quark field. 645 00:41:37,760 --> 00:41:38,820 It's not a Wilson line. 646 00:41:38,820 --> 00:41:41,300 So you can think of it as just ending on a quark field. 647 00:41:41,300 --> 00:41:43,160 But even just ending on a quark field, 648 00:41:43,160 --> 00:41:44,780 the quark field also has dynamics. 649 00:41:44,780 --> 00:41:46,640 It's not as simple as Wilson line dynamics, 650 00:41:46,640 --> 00:41:48,350 but it has dynamics. 651 00:41:48,350 --> 00:41:51,080 And we have interactions between these. 652 00:41:51,080 --> 00:41:53,040 And this is also kind of a cusp. 653 00:41:53,040 --> 00:41:55,013 Although it's not a simple Wilson line cusp, 654 00:41:55,013 --> 00:41:56,055 it's also kind of a cusp. 655 00:42:07,120 --> 00:42:09,250 So even the colinear graphs with the Wilson lines 656 00:42:09,250 --> 00:42:12,940 ended on quarks also produced these q over epsilon squareds 657 00:42:12,940 --> 00:42:15,010 and have this kind of structure. 658 00:42:15,010 --> 00:42:17,870 And they actually have similar-- 659 00:42:17,870 --> 00:42:20,660 very similar relations to the cusps in regular Wilson lines. 660 00:42:23,390 --> 00:42:26,540 So that was one question-- what was this cusp? 661 00:42:26,540 --> 00:42:29,420 Another question that came up was this fact 662 00:42:29,420 --> 00:42:32,390 that the w got fixed. 663 00:42:32,390 --> 00:42:34,010 And so we could ask the question, 664 00:42:34,010 --> 00:42:35,600 in general, when will that happen? 665 00:42:46,630 --> 00:42:48,390 So when will the w's that are showing up 666 00:42:48,390 --> 00:42:53,790 in our Wilson coefficients be fixed by external kinematics? 667 00:43:04,660 --> 00:43:08,660 And actually, that's going to happen in the following case. 668 00:43:08,660 --> 00:43:12,975 We can actually state when it will happen quite generally. 669 00:43:18,310 --> 00:43:21,250 So imagine that we had an effective theory SET that had 670 00:43:21,250 --> 00:43:23,800 multiple colinear directions. 671 00:43:23,800 --> 00:43:25,300 So then we could build operators out 672 00:43:25,300 --> 00:43:28,540 of all those different colinear fields. 673 00:43:28,540 --> 00:43:33,980 And we would do it with our building blocks, which are 674 00:43:33,980 --> 00:43:41,120 chi n or curley bn for each n. 675 00:43:41,120 --> 00:43:45,800 And if our operator only involves one building block 676 00:43:45,800 --> 00:43:51,170 for each direction, then it's going 677 00:43:51,170 --> 00:43:56,630 to be the case that the value of those labels or those omegas 678 00:43:56,630 --> 00:43:58,850 are always going to be fixed by external kinematics. 679 00:44:05,100 --> 00:44:09,520 So let me just write down another example. 680 00:44:09,520 --> 00:44:11,880 Imagine we had something like an LHC process, 681 00:44:11,880 --> 00:44:16,800 where we had two gluons coming in and two quarks going out. 682 00:44:16,800 --> 00:44:20,863 But we'll think about this as protons colliding, 683 00:44:20,863 --> 00:44:22,530 coming in from two different directions. 684 00:44:22,530 --> 00:44:25,140 Let's call them n2 and n3-- 685 00:44:25,140 --> 00:44:28,980 and then going to two jets transverse to the axis 686 00:44:28,980 --> 00:44:29,800 by a large amount. 687 00:44:29,800 --> 00:44:32,560 So there are also two different directions, 688 00:44:32,560 --> 00:44:34,680 which I can call n1 and n4. 689 00:44:34,680 --> 00:44:40,430 So this process of glu-glu goes to qq bar, which 690 00:44:40,430 --> 00:44:45,110 is really pp goes to dijets. 691 00:44:45,110 --> 00:44:48,110 As long as those dijets are well-separated from the b 692 00:44:48,110 --> 00:44:50,750 maxes, which is the case we're interested in, then we have 693 00:44:50,750 --> 00:44:54,080 this situation, where we have four different n's. 694 00:44:54,080 --> 00:44:56,750 And so what kind of operator would you write down for that? 695 00:44:56,750 --> 00:45:04,550 Well, you have dw1, dw2, dw3, dw4. 696 00:45:04,550 --> 00:45:08,270 You'd have a Wilson coefficient that could depend on four 697 00:45:08,270 --> 00:45:09,350 different w's. 698 00:45:09,350 --> 00:45:12,140 And you'd then write down some operator 699 00:45:12,140 --> 00:45:13,730 out of building blocks, n1w1. 700 00:45:17,570 --> 00:45:20,583 I'm not going to worry about all the indices, 701 00:45:20,583 --> 00:45:22,000 but I'll worry about some of them. 702 00:45:32,177 --> 00:45:33,260 That would look like that. 703 00:45:33,260 --> 00:45:37,050 So I just put down some operator that's got the right structure. 704 00:45:37,050 --> 00:45:38,450 It's got two quarks, two gluons. 705 00:45:38,450 --> 00:45:41,330 I make the gluons these b [INAUDIBLE].. 706 00:45:41,330 --> 00:45:43,555 I've chosen to contract them. 707 00:45:43,555 --> 00:45:45,180 Each of them has a different direction. 708 00:45:45,180 --> 00:45:47,630 This is in n1. 709 00:45:47,630 --> 00:45:51,260 And each of them also gets a corresponding large momentum. 710 00:45:51,260 --> 00:45:55,183 And the Wilson coefficient can depend on those large momenta. 711 00:45:55,183 --> 00:45:56,600 So this would be the operator that 712 00:45:56,600 --> 00:45:59,260 would describe that process. 713 00:45:59,260 --> 00:46:02,930 But again, if you think about the renormalization, 714 00:46:02,930 --> 00:46:06,290 then if you think about the renormalization for a minute, 715 00:46:06,290 --> 00:46:07,983 the colinear diagrams aren't going 716 00:46:07,983 --> 00:46:09,650 to involve contractions between this guy 717 00:46:09,650 --> 00:46:11,150 and any of these other guys because these guys are 718 00:46:11,150 --> 00:46:11,930 totally independent. 719 00:46:11,930 --> 00:46:13,010 They're a different Lagrangian. 720 00:46:13,010 --> 00:46:15,093 So the contractions are, again, just like the kind 721 00:46:15,093 --> 00:46:16,550 of calculation that we did. 722 00:46:16,550 --> 00:46:20,400 The colinear diagrams are just coming from this thing alone, 723 00:46:20,400 --> 00:46:22,268 and they don't care about that stuff. 724 00:46:22,268 --> 00:46:23,810 So it's actually the same calculation 725 00:46:23,810 --> 00:46:26,120 that we already did for b to s gamma 726 00:46:26,120 --> 00:46:28,110 if I wanted to do the colinear diagrams here. 727 00:46:28,110 --> 00:46:29,777 And then I would have to do the colinear 728 00:46:29,777 --> 00:46:32,600 diagrams for each of these guys, but it's kind of independent. 729 00:46:32,600 --> 00:46:35,970 And again, the w1 is external for that calculation. 730 00:46:35,970 --> 00:46:38,330 So the anomalous dimension would just-- 731 00:46:38,330 --> 00:46:39,800 it would be outside it. 732 00:46:39,800 --> 00:46:41,480 It wouldn't involve-- it wouldn't 733 00:46:41,480 --> 00:46:47,030 appear and get thrown into some loop integral momentum. 734 00:46:47,030 --> 00:46:49,730 And so if you think about what can fix these w's, it's 735 00:46:49,730 --> 00:46:51,920 really only external information. 736 00:46:51,920 --> 00:46:55,580 So what are the momentum fractions 737 00:46:55,580 --> 00:46:58,460 of the incoming quarks from the PDFs? 738 00:46:58,460 --> 00:47:00,800 What are the energies of the outgoing jets? 739 00:47:00,800 --> 00:47:03,620 That's the type of information that, even in this case, 740 00:47:03,620 --> 00:47:04,520 would fix the w's. 741 00:47:20,208 --> 00:47:22,250 So if we were to calculate the Wilson coefficient 742 00:47:22,250 --> 00:47:24,710 for this operator and do its renormalization group, 743 00:47:24,710 --> 00:47:25,580 there would again-- 744 00:47:25,580 --> 00:47:27,260 it would again be of a product form. 745 00:47:38,100 --> 00:47:40,340 There's no convolution, just a product 746 00:47:40,340 --> 00:47:43,900 for this Wilson coefficient. 747 00:47:43,900 --> 00:47:47,080 So when could that not be true? 748 00:47:47,080 --> 00:47:49,900 It could not be true if, for example, we 749 00:47:49,900 --> 00:47:52,862 had a chi bar and a chi that were in the same direction. 750 00:47:57,590 --> 00:48:02,320 So if we're in a situation where all the objects are 751 00:48:02,320 --> 00:48:07,660 different directions, that's what happens. 752 00:48:07,660 --> 00:48:08,710 And it's pretty simple. 753 00:48:13,250 --> 00:48:20,320 But let's also do an example when is it not true. 754 00:48:20,320 --> 00:48:22,670 It's always good to have a counterexample. 755 00:48:22,670 --> 00:48:25,045 And it would not be true if we had an operator like this. 756 00:48:38,710 --> 00:48:41,070 So here's an operator with just two quark 757 00:48:41,070 --> 00:48:42,950 building-block fields. 758 00:48:42,950 --> 00:48:45,450 I gave them different labels, because in general, the Wilson 759 00:48:45,450 --> 00:48:47,890 coefficient could depend on their large momenta. 760 00:48:47,890 --> 00:48:54,662 But now, they're in the same direction n, 761 00:48:54,662 --> 00:48:58,410 so they belong to the same colinear sector. 762 00:48:58,410 --> 00:49:01,380 And in this operator here, we would actually no longer 763 00:49:01,380 --> 00:49:05,100 be in the case where this thing would totally decouple 764 00:49:05,100 --> 00:49:06,650 from the loop integrals. 765 00:49:20,595 --> 00:49:22,470 So if we think about inserting this operator, 766 00:49:22,470 --> 00:49:25,100 then some of these guys here will involve loop momenta. 767 00:49:30,300 --> 00:49:33,960 Actually, it turns out that one combination of the w's is still 768 00:49:33,960 --> 00:49:35,970 fixed by external kinematics. 769 00:49:35,970 --> 00:49:38,640 And so there's an overall-- 770 00:49:38,640 --> 00:49:41,550 think about the overall delta function on this product. 771 00:49:41,550 --> 00:49:42,730 That guy would be fixed. 772 00:49:42,730 --> 00:49:47,130 And then some difference of these w's would not be fixed. 773 00:49:47,130 --> 00:49:51,970 That's the true story about at least one combination 774 00:49:51,970 --> 00:49:53,220 that involve the loop momenta. 775 00:50:13,790 --> 00:50:15,050 And what does that lead to? 776 00:50:15,050 --> 00:50:17,600 How does it complicate what we've already said? 777 00:50:17,600 --> 00:50:23,980 What that leads to is that when we formulate 778 00:50:23,980 --> 00:50:27,870 the anomalous dimension equations, 779 00:50:27,870 --> 00:50:29,802 they also involve integrals. 780 00:50:48,520 --> 00:50:53,205 So for this guy here, for the one momentum that's not fixed-- 781 00:50:53,205 --> 00:50:54,330 so in general, there's two. 782 00:50:54,330 --> 00:50:56,940 And this is the one that's not fixed. 783 00:50:56,940 --> 00:50:58,890 We would get an equation that looks like this. 784 00:51:06,420 --> 00:51:08,310 Rather than the simple product form, 785 00:51:08,310 --> 00:51:11,615 there'd be an integral on the right-hand side. 786 00:51:11,615 --> 00:51:13,740 And indeed, actually, if you do the renormalization 787 00:51:13,740 --> 00:51:17,280 of this operator that I wrote up there, just that simple quark 788 00:51:17,280 --> 00:51:23,370 operator, you actually reproduce a bunch 789 00:51:23,370 --> 00:51:31,140 of classical evolution equations that are of this form. 790 00:51:39,660 --> 00:51:41,730 So even just the simplest possible case 791 00:51:41,730 --> 00:51:44,780 we can think of where things are not fixed 792 00:51:44,780 --> 00:51:49,010 gives us a bunch of interesting results. 793 00:51:49,010 --> 00:51:52,330 So for deep inelastic scattering, 794 00:51:52,330 --> 00:51:56,530 then what you're getting is the Altarelli-Parisi equation 795 00:51:56,530 --> 00:51:57,640 or DGLAP equation. 796 00:52:07,700 --> 00:52:09,140 And the evolution of the operator 797 00:52:09,140 --> 00:52:11,960 would be the evolution of the part-time distribution 798 00:52:11,960 --> 00:52:13,528 function. 799 00:52:13,528 --> 00:52:16,070 That's one thing that would be encoded in the renormalization 800 00:52:16,070 --> 00:52:18,030 of that equation. 801 00:52:18,030 --> 00:52:21,410 There's other processes that are actually-- 802 00:52:21,410 --> 00:52:23,780 some of which we may talk about later on-- 803 00:52:23,780 --> 00:52:26,107 that are also encoded in the same type 804 00:52:26,107 --> 00:52:28,190 of operator, kind of a different projection of it. 805 00:52:31,142 --> 00:52:32,850 And one of them leads to something called 806 00:52:32,850 --> 00:52:34,020 the Brodsky-Lepage equation. 807 00:52:40,800 --> 00:52:43,890 And then there's another one that's showing up 808 00:52:43,890 --> 00:52:45,270 in something called-- 809 00:52:45,270 --> 00:52:48,407 a process called deeply virtual Compton scattering. 810 00:52:55,260 --> 00:52:57,540 So the one that's most familiar is the renormalization 811 00:52:57,540 --> 00:52:58,147 of the PDF. 812 00:52:58,147 --> 00:53:00,480 And you may remember that the renormalization of the PDF 813 00:53:00,480 --> 00:53:03,990 has this form of having an integral. 814 00:53:03,990 --> 00:53:07,750 That'll come out of renormalization this operator. 815 00:53:07,750 --> 00:53:08,790 And we'll do that case. 816 00:53:20,500 --> 00:53:25,870 So in general, the structure of the effective theory 817 00:53:25,870 --> 00:53:29,890 is leading us to find out what the structure 818 00:53:29,890 --> 00:53:32,620 of the renormalization is. 819 00:53:32,620 --> 00:53:36,010 And it can reproduce some well-known, classic things, 820 00:53:36,010 --> 00:53:37,920 but it comes out actually pretty easy. 821 00:53:44,050 --> 00:53:47,430 So I'm going to do this case, but I'm 822 00:53:47,430 --> 00:53:49,710 going to do this case in complete-- with all 823 00:53:49,710 --> 00:53:51,450 of the details filled in. 824 00:53:51,450 --> 00:53:53,550 So first, I have to convince you that 825 00:53:53,550 --> 00:53:57,010 in deep inelastic scattering, you actually get that operator. 826 00:53:57,010 --> 00:53:59,130 And we'll see precisely what kind 827 00:53:59,130 --> 00:54:00,600 of matrix elements of that operator 828 00:54:00,600 --> 00:54:02,308 show up in deep inelastic scattering, why 829 00:54:02,308 --> 00:54:03,455 we get that operator. 830 00:54:03,455 --> 00:54:05,830 And then once we're convinced that we have that operator, 831 00:54:05,830 --> 00:54:07,830 and we'll actually drive a factorization theorem 832 00:54:07,830 --> 00:54:09,420 that involves that operator. 833 00:54:09,420 --> 00:54:11,730 Then we'll talk about its renormalization group 834 00:54:11,730 --> 00:54:12,330 evolution. 835 00:54:12,330 --> 00:54:14,520 And I'll show you that it has this form that I'm 836 00:54:14,520 --> 00:54:16,320 writing here. 837 00:54:16,320 --> 00:54:19,650 So probably for the rest of today's lecture, certainly 838 00:54:19,650 --> 00:54:22,350 for the rest of today's lecture, we won't get to the running. 839 00:54:22,350 --> 00:54:23,350 We'll do that next time. 840 00:54:23,350 --> 00:54:27,540 But we'll at least get to the point, I think, 841 00:54:27,540 --> 00:54:30,315 where you'll see why that operator is showing up in DIS. 842 00:54:34,540 --> 00:54:35,480 So let's do DIS. 843 00:54:40,090 --> 00:54:52,900 So I'm only going to talk in DIS about factorization, 844 00:54:52,900 --> 00:54:58,060 and then the renormalization group evolution. 845 00:54:58,060 --> 00:55:00,280 So no phenomenology, nothing like that-- we'll just 846 00:55:00,280 --> 00:55:03,770 talk about these two concepts. 847 00:55:03,770 --> 00:55:09,760 So DIS is electron-proton to electron anything, so just 848 00:55:09,760 --> 00:55:10,600 kinematics. 849 00:55:13,832 --> 00:55:15,415 Think about a virtual photon exchange. 850 00:55:19,270 --> 00:55:21,640 Proton comes in. 851 00:55:21,640 --> 00:55:23,290 I'll say that the proton's momentum is 852 00:55:23,290 --> 00:55:29,720 a capital P. Gets blown apart. 853 00:55:29,720 --> 00:55:31,765 Call the stuff that's blown apart Px. 854 00:55:34,300 --> 00:55:40,420 And so Px is the sum of all the particles, 855 00:55:40,420 --> 00:55:41,635 all the final-state hadrons. 856 00:55:48,060 --> 00:55:49,950 Q squared of the virtual photon-- 857 00:55:49,950 --> 00:55:50,880 so this is q. 858 00:55:53,700 --> 00:55:56,400 Little q squared is minus capital Q squared. 859 00:55:56,400 --> 00:55:58,260 And this thing is much bigger than on QCD. 860 00:56:02,530 --> 00:56:08,140 mu arc in x is capital Q squared divided by this dot product. 861 00:56:12,820 --> 00:56:17,680 And you can talk about Px, which is the Px mu. 862 00:56:17,680 --> 00:56:19,450 And by momentum conservation, that's 863 00:56:19,450 --> 00:56:21,910 the proton momentum, plus whatever momentum came in 864 00:56:21,910 --> 00:56:23,080 from the leptons q. 865 00:56:26,990 --> 00:56:30,650 So Px squared-- if you square it, 866 00:56:30,650 --> 00:56:33,750 well, this guy is giving you mass of the protons squared. 867 00:56:33,750 --> 00:56:36,570 And then there's a dot-- a cross term, and this guy squared. 868 00:56:36,570 --> 00:56:44,319 And if you put those things together like this, 869 00:56:44,319 --> 00:56:47,912 so this is an exact equation. 870 00:56:47,912 --> 00:56:50,180 Px squared is that. 871 00:56:50,180 --> 00:56:52,430 And so there's actually different regions of DIS, 872 00:56:52,430 --> 00:56:54,388 and we're only going to talk about one of them. 873 00:56:56,870 --> 00:57:00,230 I have to enumerate what I'm talking about carefully. 874 00:57:00,230 --> 00:57:06,990 And I can do that by looking at Px squared or this factor 875 00:57:06,990 --> 00:57:12,140 1 minus x over x, which I'll call 1 over x minus 1. 876 00:57:12,140 --> 00:57:14,652 If this thing is of order q squared, 877 00:57:14,652 --> 00:57:16,610 then that means that this thing you're counting 878 00:57:16,610 --> 00:57:18,320 is of quarter 1. 879 00:57:18,320 --> 00:57:21,590 And in that case, it's what's called the inclusive operator 880 00:57:21,590 --> 00:57:23,790 product expansion. 881 00:57:23,790 --> 00:57:28,010 So this is the case that most books would deal with. 882 00:57:28,010 --> 00:57:30,920 And that's the one we'll deal with, actually, 883 00:57:30,920 --> 00:57:33,320 where effectively, we're not putting restrictions on x. 884 00:57:33,320 --> 00:57:35,570 We're just saying it's generic, and it's not 885 00:57:35,570 --> 00:57:37,430 approaching any endpoints. 886 00:57:37,430 --> 00:57:42,740 There's also a situation where the Px gets smaller. 887 00:57:42,740 --> 00:57:47,307 And then this thing is close to an endpoint, lambda QCD over q. 888 00:57:47,307 --> 00:57:48,890 And that's called the endpoint region. 889 00:57:52,070 --> 00:57:54,530 And people talk about that. 890 00:57:54,530 --> 00:57:56,740 Usually when people say x goes to 1, 891 00:57:56,740 --> 00:57:59,110 this is what they mean-- that the 1 minus x minus 1 892 00:57:59,110 --> 00:58:03,730 is of that size on the QCD over q. 893 00:58:03,730 --> 00:58:07,000 There's even a third region, where the Px squared becomes 894 00:58:07,000 --> 00:58:12,410 hadronic lambda QCD squared. 895 00:58:12,410 --> 00:58:16,690 And that's like here, two powers. 896 00:58:16,690 --> 00:58:18,670 So taking that factor to be really small. 897 00:58:18,670 --> 00:58:21,980 And that's the resonance region. 898 00:58:21,980 --> 00:58:25,300 And that's the case where the final state x is just 899 00:58:25,300 --> 00:58:27,445 another proton or an excited state of a proton. 900 00:58:31,977 --> 00:58:33,810 So that's where elastic scattering would be. 901 00:58:33,810 --> 00:58:39,750 And that's an exclusive process. 902 00:58:39,750 --> 00:58:41,393 It's not inclusive anymore. 903 00:58:47,070 --> 00:58:50,660 So actually, all three of these cases can be done with SET. 904 00:58:50,660 --> 00:58:52,963 And the way that it works is different in each case. 905 00:58:52,963 --> 00:58:54,380 And the case that we'll do is just 906 00:58:54,380 --> 00:58:57,950 the first one, which is kind of a classic one, and also 907 00:58:57,950 --> 00:58:58,730 the simplest one. 908 00:59:04,618 --> 00:59:06,410 So our Px squared is going to be of ordered 909 00:59:06,410 --> 00:59:12,970 q squared for our analysis. 910 00:59:25,600 --> 00:59:27,520 We also need some partonic variables. 911 00:59:40,830 --> 00:59:44,848 So the struck quark carries some momentum fraction 912 00:59:44,848 --> 00:59:45,515 from the proton. 913 00:59:49,240 --> 00:59:50,620 This is the familiar language. 914 00:59:53,260 --> 00:59:57,010 And for our analysis, what we're going to do is just 915 00:59:57,010 --> 01:00:01,180 take m bar dot little p of the quark 916 01:00:01,180 --> 01:00:04,945 to be something times n bar dot big P of the proton. 917 01:00:07,480 --> 01:00:10,600 So the fraction is this c variable, 918 01:00:10,600 --> 01:00:18,080 and it's the ratio of the quark momentum to the proton. 919 01:00:18,080 --> 01:00:20,570 But we'll do it in a very particular component, 920 01:00:20,570 --> 01:00:25,930 and we'll see why that's the right thing to do. 921 01:00:25,930 --> 01:00:28,200 And if we have our picture here of the quark 922 01:00:28,200 --> 01:00:30,960 kinematics, then this P is the incoming P. 923 01:00:30,960 --> 01:00:34,950 And the outgoing P would be P prime, let's say. 924 01:00:34,950 --> 01:00:37,320 So I could think about an analog of this equation, where 925 01:00:37,320 --> 01:00:39,750 I square P prime. 926 01:00:39,750 --> 01:00:43,360 And if you do that, it's kind of similar to the hadronic case. 927 01:00:43,360 --> 01:00:45,170 The only difference is that the c shows up. 928 01:00:47,988 --> 01:00:49,905 And, of course, there's no mass of the proton. 929 01:00:52,650 --> 01:00:56,100 So P prime squared would be that. 930 01:01:03,950 --> 01:01:08,390 So we'll see how this variable c shows up. 931 01:01:08,390 --> 01:01:10,310 Now, one thing that we have to decide about 932 01:01:10,310 --> 01:01:12,110 is frames of reference, because remember, 933 01:01:12,110 --> 01:01:15,080 when we were talking about degrees of freedom in SET, 934 01:01:15,080 --> 01:01:17,480 we had picked a frame to do that discussion. 935 01:01:17,480 --> 01:01:21,050 And it was almost always a center of mass frame so far 936 01:01:21,050 --> 01:01:26,960 in our discussions, or the rest frame of the initial state. 937 01:01:26,960 --> 01:01:30,900 And here, we're going to use a slightly different frame, which 938 01:01:30,900 --> 01:01:34,067 is the most convenient frame for deep elastic scattering. 939 01:01:34,067 --> 01:01:35,150 It's called a Breit frame. 940 01:01:40,950 --> 01:01:44,540 So we're going to do our analysis in this frame. 941 01:01:44,540 --> 01:01:46,070 So what defines this frame? 942 01:01:50,210 --> 01:01:53,380 This frame is defined by taking q mu 943 01:01:53,380 --> 01:01:57,040 to just have a z component. 944 01:01:57,040 --> 01:01:59,800 Remember, it's space-like, so it has to be somewhere 945 01:01:59,800 --> 01:02:01,180 in the space-like column. 946 01:02:01,180 --> 01:02:02,680 And we can choose it such that it's 947 01:02:02,680 --> 01:02:04,870 entirely in the z component, and nowhere else. 948 01:02:04,870 --> 01:02:06,525 And that's the Breit frame. 949 01:02:06,525 --> 01:02:07,900 If we want to write that in terms 950 01:02:07,900 --> 01:02:10,600 of our classic decomposition of nnn bar, is we 951 01:02:10,600 --> 01:02:13,450 can write it as a difference divided by 2. 952 01:02:13,450 --> 01:02:16,620 And that's giving the z component. 953 01:02:19,620 --> 01:02:22,190 So in this frame what's happening 954 01:02:22,190 --> 01:02:24,710 is that your initial state proton is coming 955 01:02:24,710 --> 01:02:26,750 in with a very large momentum. 956 01:02:26,750 --> 01:02:29,606 And then it's being-- 957 01:02:29,606 --> 01:02:32,270 you're killing that momentum and then spitting it back out 958 01:02:32,270 --> 01:02:34,380 in a different direction. 959 01:02:34,380 --> 01:02:40,470 So you're spitting back out stuff in a different direction. 960 01:02:40,470 --> 01:02:44,480 So the initial state proton is coming in with a large momentum 961 01:02:44,480 --> 01:02:47,940 in some direction. 962 01:02:47,940 --> 01:02:50,990 So if you work out the kinematics given that for what 963 01:02:50,990 --> 01:02:56,080 the proton would be, the proton's momentum 964 01:02:56,080 --> 01:02:58,960 would be in the following form. 965 01:02:58,960 --> 01:03:00,080 So it's got a large-- 966 01:03:00,080 --> 01:03:02,800 this is large, and this is small. 967 01:03:05,750 --> 01:03:07,150 And if you have a large component 968 01:03:07,150 --> 01:03:09,442 in some light-like direction, that means it's colinear. 969 01:03:11,560 --> 01:03:16,450 So you could actually write it as-- 970 01:03:16,450 --> 01:03:20,300 using momentum conservation, you could write it like this. 971 01:03:20,300 --> 01:03:24,190 And it has a colinear scaling. 972 01:03:24,190 --> 01:03:26,135 So what we have in the deep inelastic 973 01:03:26,135 --> 01:03:28,510 scattering in the Breit frame is that the incoming proton 974 01:03:28,510 --> 01:03:29,345 is a linear proton. 975 01:03:32,200 --> 01:03:36,310 And if we look at Px, and we just-- 976 01:03:36,310 --> 01:03:38,980 again, I'm not going through the details of this. 977 01:03:38,980 --> 01:03:46,910 But if we just decompose Px in terms of these coordinates, 978 01:03:46,910 --> 01:03:48,590 then we get this. 979 01:03:48,590 --> 01:03:52,400 And so as long as this factor 1 minus x over x is of order 1, 980 01:03:52,400 --> 01:03:54,340 you see that there's a large component in n 981 01:03:54,340 --> 01:03:56,630 and a large component in n bar. 982 01:03:56,630 --> 01:03:57,770 And that means it's hard. 983 01:04:03,550 --> 01:04:06,270 So what would happen in these other cases here 984 01:04:06,270 --> 01:04:07,930 is that you would change that, right? 985 01:04:07,930 --> 01:04:09,100 It would no longer be hard. 986 01:04:09,100 --> 01:04:12,300 And that's why these cases here are different. 987 01:04:12,300 --> 01:04:14,550 But as long as we're in this first case, this is hard. 988 01:04:14,550 --> 01:04:16,890 And we can say that we have colinear modes and hard modes. 989 01:04:16,890 --> 01:04:18,720 And then we just want to write down an effective theory 990 01:04:18,720 --> 01:04:19,595 for those two things. 991 01:04:22,130 --> 01:04:23,810 You could also do an analysis of DIS 992 01:04:23,810 --> 01:04:26,660 in the rest frame of the proton. 993 01:04:26,660 --> 01:04:29,360 That's another case. 994 01:04:29,360 --> 01:04:32,150 And actually, the final result that you would get 995 01:04:32,150 --> 01:04:33,067 would be the same. 996 01:04:33,067 --> 01:04:34,650 That's what we'll get from this frame. 997 01:04:34,650 --> 01:04:36,442 But this frame is actually a little easier. 998 01:04:47,280 --> 01:04:49,980 So we're really talking about hard colinear factorization 999 01:04:49,980 --> 01:04:50,670 in some sense. 1000 01:04:54,920 --> 01:04:57,160 Colinear describes the low-energy degree of freedom, 1001 01:04:57,160 --> 01:04:58,800 which is the proton. 1002 01:04:58,800 --> 01:05:03,450 And hard describes the off-shell final state 1003 01:05:03,450 --> 01:05:05,550 and the hard fluctuations. 1004 01:05:05,550 --> 01:05:09,210 And really, what we want to do in DIS in this classic case is 1005 01:05:09,210 --> 01:05:11,372 just separate hard and colinear fluctuations, 1006 01:05:11,372 --> 01:05:12,330 at least in this frame. 1007 01:05:17,660 --> 01:05:19,875 So we can do that for the cross-section. 1008 01:05:23,970 --> 01:05:26,800 So let me remind you something about the cross-section in DIS. 1009 01:05:29,630 --> 01:05:34,190 So just using nothing more than the fact 1010 01:05:34,190 --> 01:05:38,910 that we're treating the leptons order by order in the photon, 1011 01:05:38,910 --> 01:05:41,600 we can work the first-order in the electromagnetic coupling. 1012 01:05:41,600 --> 01:05:43,100 And then we can write down a formula 1013 01:05:43,100 --> 01:05:47,345 like this, where we split it into electronic and hadronic 1014 01:05:47,345 --> 01:05:47,845 tensor. 1015 01:05:53,280 --> 01:05:55,050 And the hadronic tensor can be written 1016 01:05:55,050 --> 01:06:06,480 as the imaginary part of some T, where T is the following thing. 1017 01:06:06,480 --> 01:06:09,180 So this doesn't use anything about-- 1018 01:06:09,180 --> 01:06:11,700 we haven't used any sort of perturbation theory or anything 1019 01:06:11,700 --> 01:06:13,580 to write this-- what I'm telling you-- down. 1020 01:06:19,846 --> 01:06:22,370 We just worked all orders, and basically used 1021 01:06:22,370 --> 01:06:23,210 the optical theorem. 1022 01:06:28,238 --> 01:06:30,280 So T is the time order product of the two curves. 1023 01:06:36,248 --> 01:06:37,040 Let me call this z. 1024 01:06:40,893 --> 01:06:42,310 Actually, I don't know if that's-- 1025 01:06:46,980 --> 01:06:49,110 before you start doing anything in QCD, 1026 01:06:49,110 --> 01:06:50,940 that's how you could write this. 1027 01:06:50,940 --> 01:06:54,620 And these are the electromagnetic currents 1028 01:06:54,620 --> 01:06:56,082 for a quark. 1029 01:06:56,082 --> 01:06:58,290 STUDENT: But wouldn't you write it over that, though? 1030 01:06:58,290 --> 01:06:59,250 [INAUDIBLE] 1031 01:06:59,250 --> 01:07:07,676 PROFESSOR: Thanks, yeah, because x was something else. 1032 01:07:07,676 --> 01:07:08,710 Don't want to use x. 1033 01:07:12,040 --> 01:07:15,790 All right, so for this T mu nu, we 1034 01:07:15,790 --> 01:07:18,190 can also use current conservation 1035 01:07:18,190 --> 01:07:20,990 and decompose it into two pieces. 1036 01:07:20,990 --> 01:07:25,120 So this is a classic thing that we do in DIS. 1037 01:07:25,120 --> 01:07:26,740 And nothing about the fact that we're 1038 01:07:26,740 --> 01:07:31,073 using the effective theory really changes during this. 1039 01:07:31,073 --> 01:07:32,740 What you're after-- the effective theory 1040 01:07:32,740 --> 01:07:39,100 is calculating these T's, which would be coefficients. 1041 01:07:39,100 --> 01:07:44,455 And this part is all standard stuff for any analysis of DIS. 1042 01:07:48,610 --> 01:07:51,280 So if you're not familiar with it, or you don't remember it, 1043 01:07:51,280 --> 01:07:53,640 it's actually not that important. 1044 01:07:53,640 --> 01:07:56,170 But using the current conservation of T, 1045 01:07:56,170 --> 01:07:58,060 the fact that when I dot a q into the T, 1046 01:07:58,060 --> 01:08:02,230 you should get 0, because you're dotting a q into the current. 1047 01:08:02,230 --> 01:08:03,512 The current is conserved. 1048 01:08:03,512 --> 01:08:04,720 It's electromagnetic current. 1049 01:08:04,720 --> 01:08:07,002 That tells you the possible structure of this thing 1050 01:08:07,002 --> 01:08:08,710 and that there's two general terms if I'm 1051 01:08:08,710 --> 01:08:10,060 doing a spin sum, which I am. 1052 01:08:18,151 --> 01:08:19,609 There's the sum over spin up there. 1053 01:08:22,680 --> 01:08:25,080 If we weren't summing over spin, if we were picking out 1054 01:08:25,080 --> 01:08:27,270 particular spins of the proton, then the formula 1055 01:08:27,270 --> 01:08:31,439 here could be a little more complicated. 1056 01:08:31,439 --> 01:08:34,398 So this satisfies all the symmetries. 1057 01:08:39,271 --> 01:08:40,979 And in general, what you know is that you 1058 01:08:40,979 --> 01:08:43,529 want the imaginary part of forward scattering graphs. 1059 01:08:50,210 --> 01:08:53,168 So pictorially, it's sometimes useful to draw something. 1060 01:08:56,450 --> 01:08:59,040 So here's a forward scattering graph. 1061 01:08:59,040 --> 01:09:01,170 And the thing that's different that we have-- 1062 01:09:01,170 --> 01:09:05,700 know now-- and the information that we're going to use 1063 01:09:05,700 --> 01:09:09,990 is that we're assigning the external guys 1064 01:09:09,990 --> 01:09:12,910 to be colinear and the intermediate guy to be hard. 1065 01:09:12,910 --> 01:09:15,570 So we want to integrate out the pink guy, as usual. 1066 01:09:15,570 --> 01:09:18,000 The guy we want to integrate out is always pink. 1067 01:09:18,000 --> 01:09:21,120 And we want to keep these colinear guys. 1068 01:09:21,120 --> 01:09:25,170 And so the operators in the effective theory-- 1069 01:09:25,170 --> 01:09:27,420 we can already intuit what they should look like. 1070 01:09:32,858 --> 01:09:34,150 They're just going to involve-- 1071 01:09:37,024 --> 01:09:38,560 I have a current with two photons 1072 01:09:38,560 --> 01:09:42,490 hanging out of it, and then colinear quarks. 1073 01:09:42,490 --> 01:09:45,790 Those guys-- those are the external lines there. 1074 01:09:45,790 --> 01:09:49,029 And there's actually also an analogous thing 1075 01:09:49,029 --> 01:09:52,990 with colinear gluons. 1076 01:09:52,990 --> 01:09:54,550 And that's what the type of-- so I 1077 01:09:54,550 --> 01:09:56,620 could have an operator at higher orders, where 1078 01:09:56,620 --> 01:09:58,232 the external states here were gluons. 1079 01:09:58,232 --> 01:10:00,440 And that's what the operators of the effective theory 1080 01:10:00,440 --> 01:10:02,823 are going to look like. 1081 01:10:02,823 --> 01:10:04,490 Just contract that pink line to a point, 1082 01:10:04,490 --> 01:10:05,782 and that's what they look like. 1083 01:10:20,550 --> 01:10:22,300 So given that we know what they look like, 1084 01:10:22,300 --> 01:10:24,258 we just have to write down the lowest dimension 1085 01:10:24,258 --> 01:10:25,337 operators of that form. 1086 01:10:25,337 --> 01:10:27,295 And the lowest dimension operators of that form 1087 01:10:27,295 --> 01:10:28,840 are exactly the operator that I told 1088 01:10:28,840 --> 01:10:31,030 you was going to be the one that comes in. 1089 01:10:36,780 --> 01:10:38,570 So we can enumerate the lowest dimension, 1090 01:10:38,570 --> 01:10:41,150 which where dimension here is counted as lambdas, right? 1091 01:10:41,150 --> 01:10:44,720 So it's the lowest order in the power counting operators, so 1092 01:10:44,720 --> 01:10:46,855 not the lowest order in mass dimension. 1093 01:10:51,410 --> 01:10:57,320 Let me write out a few things. 1094 01:11:05,340 --> 01:11:07,375 Yeah, I already switched to this notation. 1095 01:11:11,840 --> 01:11:12,340 Yeah. 1096 01:11:32,393 --> 01:11:34,060 So the type of operator we'd have that's 1097 01:11:34,060 --> 01:11:35,520 order lambda squared-- 1098 01:11:35,520 --> 01:11:36,520 this guy's order lambda. 1099 01:11:36,520 --> 01:11:37,520 This guy's order lambda. 1100 01:11:37,520 --> 01:11:38,770 That's lambda squared. 1101 01:11:38,770 --> 01:11:42,610 And the most general thing we can think of is this. 1102 01:11:42,610 --> 01:11:44,860 The reason why I wrote this out rather than writing it 1103 01:11:44,860 --> 01:11:46,540 as a chi field is I wanted to emphasize 1104 01:11:46,540 --> 01:11:47,707 that there's a flavor index. 1105 01:11:51,202 --> 01:11:57,030 So i is up quarks, down quarks, strange quirks, et cetera. 1106 01:11:57,030 --> 01:12:06,660 And then we could also have gluons, 1107 01:12:06,660 --> 01:12:19,485 and gluons are similar except in terms of our curly v field. 1108 01:12:26,010 --> 01:12:35,460 And it is actually just a contraction of two curly v's, 1109 01:12:35,460 --> 01:12:38,555 and then traced over. 1110 01:12:38,555 --> 01:12:40,430 And then there's some Wilson coefficient here 1111 01:12:40,430 --> 01:12:47,750 which is also just the Wilson coefficient for the gluon. 1112 01:12:47,750 --> 01:12:50,000 So this is, again, order lambda squared. 1113 01:12:50,000 --> 01:12:51,810 The b perp field is order lambda squared. 1114 01:12:51,810 --> 01:12:53,990 So we just write down the lowest dimension operators 1115 01:12:53,990 --> 01:12:55,352 that have this form. 1116 01:12:55,352 --> 01:12:57,060 And that's going to be the right answer-- 1117 01:12:57,060 --> 01:12:59,240 the lowest order in lambda. 1118 01:13:15,660 --> 01:13:18,000 So it turns out actually that in this case, 1119 01:13:18,000 --> 01:13:21,380 our lambda counting is exactly the corresponding twist 1120 01:13:21,380 --> 01:13:22,350 expansion. 1121 01:13:26,490 --> 01:13:27,410 I'm not going to-- 1122 01:13:27,410 --> 01:13:29,830 STUDENT: All j's correspond to the operators that go 1123 01:13:29,830 --> 01:13:31,442 into the T1's and T2's? 1124 01:13:31,442 --> 01:13:32,400 PROFESSOR: Yeah, right. 1125 01:13:32,400 --> 01:13:35,630 So now what's the index j exactly? 1126 01:13:35,630 --> 01:13:37,860 There's a T1 and a T2. 1127 01:13:37,860 --> 01:13:39,480 And the T1 and the T2, although they 1128 01:13:39,480 --> 01:13:41,940 have the same kind of quark and gluon field structure, 1129 01:13:41,940 --> 01:13:45,480 they get different coefficients, and that's what the j is. 1130 01:13:45,480 --> 01:13:53,270 So when we think about doing a similar kind of decomposition, 1131 01:13:53,270 --> 01:13:56,210 the effective theory, we can write it as follows. 1132 01:14:01,080 --> 01:14:05,030 So as an 01 and an 02. 1133 01:14:05,030 --> 01:14:10,010 And I think if I'm getting my mass detentions right, 1134 01:14:10,010 --> 01:14:10,880 it looks like this. 1135 01:14:28,080 --> 01:14:31,070 So there's a piece that multiplies a spin structure 1136 01:14:31,070 --> 01:14:32,630 that's this g perp mu nu. 1137 01:14:32,630 --> 01:14:34,052 That's transverse to Q. 1138 01:14:34,052 --> 01:14:35,510 And there's a piece that multiplies 1139 01:14:35,510 --> 01:14:37,790 this guy, which is also transverse to Q in the Breit 1140 01:14:37,790 --> 01:14:39,920 frame. 1141 01:14:39,920 --> 01:14:41,410 So that's the kind of decomposition 1142 01:14:41,410 --> 01:14:42,610 we would do in the effective theory. 1143 01:14:42,610 --> 01:14:44,170 There'd be two different types of operators 1144 01:14:44,170 --> 01:14:45,215 we could think about. 1145 01:14:45,215 --> 01:14:46,840 But the only thing that actually tracks 1146 01:14:46,840 --> 01:14:48,850 that-- because of the spin relations, 1147 01:14:48,850 --> 01:14:51,667 the only thing that tracks that is the Wilson coefficient. 1148 01:14:51,667 --> 01:14:53,875 And so that will differ in these two different cases. 1149 01:14:57,370 --> 01:15:01,550 So this guy here is going to be-- 1150 01:15:01,550 --> 01:15:04,260 this will give the quark PDFs. 1151 01:15:10,380 --> 01:15:11,970 And this Wilson coefficient will then 1152 01:15:11,970 --> 01:15:14,852 be the thing that you convolute with the quark PDF in DIS. 1153 01:15:14,852 --> 01:15:17,310 And then likewise, this guy is going to give the gluon PDF. 1154 01:15:22,870 --> 01:15:24,280 So let's do the quark-- 1155 01:15:24,280 --> 01:15:28,060 we'll do the quark contribution in detail. 1156 01:15:28,060 --> 01:15:30,310 The gluon contribution is not really any harder. 1157 01:15:34,983 --> 01:15:36,400 So what I'm going to do is I'm not 1158 01:15:36,400 --> 01:15:38,680 going to think about it in perturbation theory. 1159 01:15:38,680 --> 01:15:40,570 I'm just going to think about it to all orders in perturbation 1160 01:15:40,570 --> 01:15:41,308 theory. 1161 01:15:41,308 --> 01:15:42,850 And really what that means is I'm not 1162 01:15:42,850 --> 01:15:46,000 going to think about this c as expanded in perturbation theory 1163 01:15:46,000 --> 01:15:48,565 or think about any of the diagrams here as expanded 1164 01:15:48,565 --> 01:15:50,260 in perturbation theory. 1165 01:15:50,260 --> 01:15:52,690 I'm just going to see, if I manipulate things, 1166 01:15:52,690 --> 01:15:55,150 what does it lead to, using things 1167 01:15:55,150 --> 01:15:58,610 like momentum conservation and stuff like that? 1168 01:15:58,610 --> 01:16:00,130 So let me write-- 1169 01:16:00,130 --> 01:16:02,980 in order to do that, let me write it 1170 01:16:02,980 --> 01:16:10,420 slightly differently than I just did, which I apologize for. 1171 01:16:10,420 --> 01:16:11,170 But I'm going to-- 1172 01:16:11,170 --> 01:16:17,080 I want the arguments of the c, just for a later convenience, 1173 01:16:17,080 --> 01:16:19,450 instead of being w1 and w1 to be w plus 1174 01:16:19,450 --> 01:16:22,570 and w-- minus the sum of the w's and the difference 1175 01:16:22,570 --> 01:16:28,420 of the w's, but everything else the same. 1176 01:16:32,470 --> 01:16:34,390 It's convenient to just talk about 1177 01:16:34,390 --> 01:16:37,900 the sum and the difference rather than the individuals. 1178 01:16:48,818 --> 01:16:50,610 So what's the parton distribution function? 1179 01:16:54,737 --> 01:16:56,320 Let me convince you that it's actually 1180 01:16:56,320 --> 01:16:59,800 related to this operator. 1181 01:16:59,800 --> 01:17:09,700 If you were in coordinate space, the coordinate space for us 1182 01:17:09,700 --> 01:17:11,950 was z. 1183 01:17:11,950 --> 01:17:15,110 We're just leaving x for something else. 1184 01:17:15,110 --> 01:17:22,110 Then you could define the parton distribution function 1185 01:17:22,110 --> 01:17:26,857 as a proton matrix element of quark fields 1186 01:17:26,857 --> 01:17:28,190 with a Wilson line between them. 1187 01:17:33,890 --> 01:17:39,720 So they're on the light cone, and we have a formula 1188 01:17:39,720 --> 01:17:41,790 like that. 1189 01:17:41,790 --> 01:17:45,180 And also, one can convince oneself 1190 01:17:45,180 --> 01:17:49,470 that if you want to write down for the antiquarks what 1191 01:17:49,470 --> 01:17:51,690 the part-time distribution function is, 1192 01:17:51,690 --> 01:17:55,710 it's the same formula as the quark formula, 1193 01:17:55,710 --> 01:17:57,520 just with an overall minus sign. 1194 01:17:57,520 --> 01:17:59,529 And z goes to minus z. 1195 01:18:09,510 --> 01:18:12,540 So that's an operator definition of the part-time distribution 1196 01:18:12,540 --> 01:18:14,020 function. 1197 01:18:14,020 --> 01:18:17,010 You can also define it by moments of the operator. 1198 01:18:17,010 --> 01:18:20,370 That's the way that, for example, Peskin does it. 1199 01:18:20,370 --> 01:18:22,620 But if you put all the information in those moments 1200 01:18:22,620 --> 01:18:25,140 back into a single operator, then it becomes this thing. 1201 01:18:30,420 --> 01:18:32,705 And if we Fourier transform this, 1202 01:18:32,705 --> 01:18:34,830 then it becomes the operator that we have up there. 1203 01:18:48,802 --> 01:18:51,010 So there's something special about the matrix element 1204 01:18:51,010 --> 01:18:52,870 we're taking here. 1205 01:18:52,870 --> 01:18:55,900 And that is that this matrix element is forward. 1206 01:18:55,900 --> 01:18:58,360 It has the same kinematics for the-- 1207 01:18:58,360 --> 01:19:01,330 so this is a colinear proton, but we have the same momentum 1208 01:19:01,330 --> 01:19:03,820 here and here. 1209 01:19:03,820 --> 01:19:06,290 In the in state and the out state, 1210 01:19:06,290 --> 01:19:08,830 you have the same momentum. 1211 01:19:08,830 --> 01:19:11,445 And that leads to one kinematic restriction 1212 01:19:11,445 --> 01:19:12,570 on what I'm about to write. 1213 01:19:15,500 --> 01:19:21,650 So what that imposes is that w1 should be equal to w2. 1214 01:19:21,650 --> 01:19:23,630 So there's a delta function of w minus 1215 01:19:23,630 --> 01:19:27,110 that comes from the momentum conservation or the restriction 1216 01:19:27,110 --> 01:19:30,110 that it's a forward matrix element. 1217 01:19:30,110 --> 01:19:33,170 But the sum of the two-- 1218 01:19:33,170 --> 01:19:35,780 with my sign conventions, it's w minus. 1219 01:19:35,780 --> 01:19:39,960 The sum of the two is unconstrained. 1220 01:19:39,960 --> 01:19:42,645 Well, it's bounded, but unconstrained. 1221 01:20:02,480 --> 01:20:04,100 So we can write the unconstrained part 1222 01:20:04,100 --> 01:20:06,680 as this interval of c parameter. 1223 01:20:06,680 --> 01:20:08,390 The bounding just comes from the fact 1224 01:20:08,390 --> 01:20:10,130 that the quark can't carry more momentum 1225 01:20:10,130 --> 01:20:13,055 than the overall momentum of the proton. 1226 01:20:13,055 --> 01:20:14,180 So that's why it's bounded. 1227 01:20:14,180 --> 01:20:16,610 It can't carry negative momentum, and it can't carry-- 1228 01:20:19,078 --> 01:20:20,870 negative physical momentum, it can't carry, 1229 01:20:20,870 --> 01:20:23,600 and it can't carry less momentum in the proton. 1230 01:20:23,600 --> 01:20:25,350 That's where the limits come from. 1231 01:20:25,350 --> 01:20:26,630 And then there's two pieces. 1232 01:20:26,630 --> 01:20:28,550 The w plus could either be positive, 1233 01:20:28,550 --> 01:20:30,240 or it could be negative. 1234 01:20:30,240 --> 01:20:32,120 If it's positive, remember, positive 1235 01:20:32,120 --> 01:20:34,340 labels-- that was our quarks. 1236 01:20:34,340 --> 01:20:35,690 So this is the quark piece. 1237 01:20:38,770 --> 01:20:40,020 And these are the antiquarks. 1238 01:20:40,020 --> 01:20:43,140 They come with negative labels. 1239 01:20:43,140 --> 01:20:47,470 Remember that we talked about that earlier. 1240 01:20:47,470 --> 01:20:55,292 And so both the f and the f bar are hiding inside this formula. 1241 01:20:55,292 --> 01:20:57,000 So there is a simplification that there's 1242 01:20:57,000 --> 01:20:58,230 this delta function here. 1243 01:21:01,590 --> 01:21:04,510 And so basically what we're talking about then-- 1244 01:21:04,510 --> 01:21:06,600 you can think of just doing the integral 1245 01:21:06,600 --> 01:21:08,440 over that delta function. 1246 01:21:08,440 --> 01:21:13,447 So then you're talking about effectively an operator 1247 01:21:13,447 --> 01:21:15,780 where you don't worry about putting a label on this one. 1248 01:21:15,780 --> 01:21:17,410 You have only one delta function left, 1249 01:21:17,410 --> 01:21:19,640 which I can denote by putting a label on this one. 1250 01:21:19,640 --> 01:21:23,340 And this operator is actually exactly the operator 1251 01:21:23,340 --> 01:21:26,760 that gives you the PDF. 1252 01:21:26,760 --> 01:21:30,510 This operator is like a number operator 1253 01:21:30,510 --> 01:21:43,900 for quarks, where you're thinking 1254 01:21:43,900 --> 01:21:52,550 about momentum, a number operator with momentum omega. 1255 01:21:52,550 --> 01:21:54,300 And if you want to think about there being 1256 01:21:54,300 --> 01:21:56,760 some kind of field for a parton, this is about as close 1257 01:21:56,760 --> 01:21:57,450 as you can get. 1258 01:22:01,990 --> 01:22:03,930 So this quark field dressed by Wilson line 1259 01:22:03,930 --> 01:22:07,820 is kind of like a parton in the parton model. 1260 01:22:07,820 --> 01:22:11,630 All right, so next time, we'll take this, put it together 1261 01:22:11,630 --> 01:22:13,880 with this formula here, with the c, 1262 01:22:13,880 --> 01:22:15,230 and just see where it leads us. 1263 01:22:15,230 --> 01:22:17,300 And it will lead us directly to a factorization 1264 01:22:17,300 --> 01:22:19,820 theorem for deep inelastic scattering 1265 01:22:19,820 --> 01:22:22,280 involving parton distribution functions 1266 01:22:22,280 --> 01:22:24,800 and some hard perturbatively calculable thing, which 1267 01:22:24,800 --> 01:22:28,940 is actually a cross-section in the parton model. 1268 01:22:28,940 --> 01:22:32,343 And then we'll talk about renormalization [INAUDIBLE].. 1269 01:22:32,343 --> 01:22:34,760 You can already see that we're getting the operator that I 1270 01:22:34,760 --> 01:22:37,160 promised you-- that DIS is described 1271 01:22:37,160 --> 01:22:40,010 by this, by a linear operator with two different quark 1272 01:22:40,010 --> 01:22:42,620 fields, and particular labeling of the momenta. 1273 01:22:47,980 --> 01:22:50,130 Any questions?