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 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:26,013 --> 00:00:27,430 IAIN STEWART: OK, so let me remind 9 00:00:27,430 --> 00:00:30,350 you of what we were talking about last time. 10 00:00:30,350 --> 00:00:32,878 So we were discussing the example of DIS 11 00:00:32,878 --> 00:00:33,670 in the Breit frame. 12 00:00:33,670 --> 00:00:36,370 And the way we led into this example 13 00:00:36,370 --> 00:00:39,160 is we talked about renormalization group evolution 14 00:00:39,160 --> 00:00:40,285 with a heavy light current. 15 00:00:40,285 --> 00:00:43,570 And we saw that it had this [INAUDIBLE] dimension. 16 00:00:43,570 --> 00:00:45,760 But it was a multiplicative renormalization group 17 00:00:45,760 --> 00:00:48,970 evolution, and I said that that happened because we only 18 00:00:48,970 --> 00:00:52,900 had one collinear gauge-invariant 19 00:00:52,900 --> 00:00:54,710 object in our operator. 20 00:00:54,710 --> 00:00:57,280 And then I just wrote down an operator that looked like this, 21 00:00:57,280 --> 00:00:59,470 and I said, there's one that has two. 22 00:00:59,470 --> 00:01:03,520 If we run that object, we will get a renormalization group 23 00:01:03,520 --> 00:01:05,813 equation that involves convolutions. 24 00:01:05,813 --> 00:01:07,480 And I said that that's going to give you 25 00:01:07,480 --> 00:01:09,438 the renormalization group evolution of a parton 26 00:01:09,438 --> 00:01:11,050 distribution function. 27 00:01:11,050 --> 00:01:12,580 And we wanted to explore that. 28 00:01:12,580 --> 00:01:14,175 And so in order to explore that, we 29 00:01:14,175 --> 00:01:15,550 should think of some process that 30 00:01:15,550 --> 00:01:17,620 has the parton distribution function in it 31 00:01:17,620 --> 00:01:19,630 so we can really make sure we know 32 00:01:19,630 --> 00:01:21,790 precisely what the operator is. 33 00:01:21,790 --> 00:01:23,155 And that process is DIS. 34 00:01:23,155 --> 00:01:25,270 That's the simplest process. 35 00:01:25,270 --> 00:01:27,520 So we started thinking about deep inelastic scattering 36 00:01:27,520 --> 00:01:29,950 in the Breit frame, which is this framework, Q, 37 00:01:29,950 --> 00:01:30,550 of the photon. 38 00:01:30,550 --> 00:01:34,810 It has that form, just a component in the z-direction. 39 00:01:34,810 --> 00:01:37,480 And in that frame, the incoming quarks 40 00:01:37,480 --> 00:01:41,025 in the proton, quarks and gluons, are collinear. 41 00:01:41,025 --> 00:01:42,550 Intermediate state, the outstate, 42 00:01:42,550 --> 00:01:45,700 the x-state that's going out, is hard. 43 00:01:45,700 --> 00:01:47,230 So you can think of a-- if you were 44 00:01:47,230 --> 00:01:49,923 to draw perturbative diagrams, you'd draw them like this. 45 00:01:49,923 --> 00:01:51,340 And this propagator would be hard. 46 00:01:51,340 --> 00:01:52,780 It would have a hard momentum. 47 00:01:52,780 --> 00:01:54,700 And then you would have loop corrections 48 00:01:54,700 --> 00:01:56,082 that could also be hard. 49 00:01:56,082 --> 00:01:57,790 In the effective theory, you don't really 50 00:01:57,790 --> 00:01:59,710 have to think about what diagrams. 51 00:01:59,710 --> 00:02:03,290 You just write down the lowest possible dimension operator, 52 00:02:03,290 --> 00:02:06,040 and everything that's a loop that's hard 53 00:02:06,040 --> 00:02:07,870 goes to the x, which is the Wilson 54 00:02:07,870 --> 00:02:09,789 coefficient, if you like. 55 00:02:09,789 --> 00:02:13,420 And likewise, we also get not just external quarks, 56 00:02:13,420 --> 00:02:15,640 but external gluons from diagrams. 57 00:02:15,640 --> 00:02:19,180 In the full theory, it would involve a quark loop like that. 58 00:02:19,180 --> 00:02:22,130 OK, so this is going to lead to the quark PDF, 59 00:02:22,130 --> 00:02:24,700 and this is going to lead to the gluon PDF. 60 00:02:24,700 --> 00:02:27,310 And we decided we would do the quark one in some detail. 61 00:02:27,310 --> 00:02:29,860 So this is kind of writing out now the operator 62 00:02:29,860 --> 00:02:32,920 and the Wilson coefficient in kind of a combined notation 63 00:02:32,920 --> 00:02:43,380 where this w plus and minus are w1 plus and minus w2. 64 00:02:43,380 --> 00:02:46,470 And then we had one more formula, 65 00:02:46,470 --> 00:02:49,540 which is where we ended. 66 00:02:49,540 --> 00:02:52,500 So we have a collinear proton, and then we have this operator. 67 00:03:06,310 --> 00:03:09,840 And then we have the collinear proton again. 68 00:03:09,840 --> 00:03:16,440 And this matrix element can be written as follows. 69 00:03:24,180 --> 00:03:26,740 So this is the last formula we had last time. 70 00:03:29,990 --> 00:03:33,280 So some things here are just conventions, 71 00:03:33,280 --> 00:03:37,165 but other things are important. 72 00:03:37,165 --> 00:03:39,040 Well, everything's important, but some things 73 00:03:39,040 --> 00:03:40,582 are more important than other things. 74 00:03:56,150 --> 00:03:58,770 So this quark here has a flavor. 75 00:03:58,770 --> 00:03:59,770 It could be an up quark. 76 00:03:59,770 --> 00:04:01,990 It could be down quark. 77 00:04:01,990 --> 00:04:03,910 Let me denote that by an index i. 78 00:04:06,760 --> 00:04:08,590 This proton here is collinear. 79 00:04:08,590 --> 00:04:11,860 And really, all that matters for this example 80 00:04:11,860 --> 00:04:12,970 is that we have some-- 81 00:04:12,970 --> 00:04:16,390 we can think of it as a massless proton, even. 82 00:04:16,390 --> 00:04:19,008 And as far as its momentum is concerned, 83 00:04:19,008 --> 00:04:20,300 we can think of it as massless. 84 00:04:20,300 --> 00:04:22,630 So really, the only momentum that matters in here 85 00:04:22,630 --> 00:04:24,950 is the minus momentum-- 86 00:04:24,950 --> 00:04:33,306 so minus, which is n bar dot P, and that's what this is-- 87 00:04:33,306 --> 00:04:38,980 n bar dot P, n bar dot P. It's capital P. 88 00:04:38,980 --> 00:04:40,730 So capital P was the proton momentum. 89 00:04:40,730 --> 00:04:43,390 And we can think of this state as just carrying large P minus. 90 00:04:43,390 --> 00:04:46,710 All the other components don't matter for this matrix element. 91 00:04:46,710 --> 00:04:48,430 But it's a forward matrix element, 92 00:04:48,430 --> 00:04:51,850 so both states carry the same large momentum. 93 00:04:51,850 --> 00:04:53,710 And that's what led to this delta function 94 00:04:53,710 --> 00:04:57,490 here that says that w1 and w2, with the sign conventions 95 00:04:57,490 --> 00:04:59,920 we have, this guy has the opposite sign convention. 96 00:04:59,920 --> 00:05:02,850 And so if it's forward, these two guys 97 00:05:02,850 --> 00:05:07,660 have to have equal momentum so that the sum is 0. 98 00:05:07,660 --> 00:05:09,640 And if you take into account the sign, 99 00:05:09,640 --> 00:05:12,220 then that means w minus is 0. 100 00:05:12,220 --> 00:05:14,600 So that's what that delta function is doing. 101 00:05:14,600 --> 00:05:17,650 And then the sum is something that's not constrained 102 00:05:17,650 --> 00:05:19,550 by the matrix element. 103 00:05:19,550 --> 00:05:22,760 And so the sum could either be positive or negative. 104 00:05:22,760 --> 00:05:25,390 If it's positive, we can say that it's 105 00:05:25,390 --> 00:05:27,340 some fraction of the proton momentum 106 00:05:27,340 --> 00:05:29,770 because this is a quark inside the proton 107 00:05:29,770 --> 00:05:31,550 and it carries some momentum, but it 108 00:05:31,550 --> 00:05:32,800 can't be more than the proton. 109 00:05:32,800 --> 00:05:35,560 Otherwise, we would get 0. 110 00:05:35,560 --> 00:05:37,360 So it's some fraction, and that fraction 111 00:05:37,360 --> 00:05:40,422 is defined to be xi in this formula. 112 00:05:40,422 --> 00:05:42,130 And the reason there's a 2 is because I'm 113 00:05:42,130 --> 00:05:46,990 adding the w1 and the w2, which are equal. 114 00:05:46,990 --> 00:05:49,260 So this is the momentum fraction. 115 00:05:49,260 --> 00:05:51,010 And then we can have an arbitrary function 116 00:05:51,010 --> 00:05:52,330 of that momentum fraction. 117 00:05:52,330 --> 00:05:54,160 Nothing stops us from writing that down. 118 00:05:57,310 --> 00:05:59,170 And that's kind of where we got to. 119 00:06:01,870 --> 00:06:07,553 Now, so on general grounds, you can 120 00:06:07,553 --> 00:06:09,220 argue that that's the most general thing 121 00:06:09,220 --> 00:06:11,725 that you can write down for this matrix element. 122 00:06:11,725 --> 00:06:17,200 And I tried to argue about why that's true. 123 00:06:17,200 --> 00:06:19,900 From charge conjugation, you can actually do something more. 124 00:06:19,900 --> 00:06:21,610 So you can let charge conjugation act 125 00:06:21,610 --> 00:06:23,260 on these operators. 126 00:06:23,260 --> 00:06:28,870 And since charge conjugation's a good symmetry of QCD, 127 00:06:28,870 --> 00:06:31,240 you can prove that that relates, actually, 128 00:06:31,240 --> 00:06:37,480 the quark and antiquark operators in the following way. 129 00:06:42,410 --> 00:06:46,620 So quark and antiquark operators are switching signs of w plus-- 130 00:06:46,620 --> 00:06:48,190 so if I switch the sign of w plus 131 00:06:48,190 --> 00:06:50,470 that's going from quark to antiquark. 132 00:06:50,470 --> 00:06:54,580 And basically what happens in the operator when 133 00:06:54,580 --> 00:06:56,080 you do charge conjugation, remember 134 00:06:56,080 --> 00:06:59,890 that [? chi ?] goes over here to [? chi ?] transpose 135 00:06:59,890 --> 00:07:02,110 and the w switches sign. 136 00:07:02,110 --> 00:07:05,320 So basically, what charge conjugation is doing 137 00:07:05,320 --> 00:07:11,650 is taking w1 to minus w2 and w2 to minus w1. 138 00:07:11,650 --> 00:07:14,380 And that's why the w plus, which is signed by the w minus, 139 00:07:14,380 --> 00:07:15,220 doesn't. 140 00:07:15,220 --> 00:07:18,445 And then there's an overall sign just from the fields-- 141 00:07:21,502 --> 00:07:23,560 so from the usual charge conjugation 142 00:07:23,560 --> 00:07:27,100 transformation of the fields for a vector current. 143 00:07:27,100 --> 00:07:30,790 OK, so these are all orders of relation between the Wilson 144 00:07:30,790 --> 00:07:32,080 coefficients. 145 00:07:32,080 --> 00:07:33,580 So really, when you do the matching, 146 00:07:33,580 --> 00:07:35,872 you really only need to do the matching for the quarks. 147 00:07:44,293 --> 00:07:46,210 So if you want to do the matching calculation, 148 00:07:46,210 --> 00:07:48,627 you'd do a matching calculation for the Wilson coefficient 149 00:07:48,627 --> 00:07:49,780 with positive w plus. 150 00:07:55,600 --> 00:08:01,370 And you could do it for the antiquarks, as well, 151 00:08:01,370 --> 00:08:05,020 but you would just be basically wasting time. 152 00:08:10,690 --> 00:08:13,440 Now, last time, we went through the kinematics of the Breit 153 00:08:13,440 --> 00:08:14,580 frame a little bit. 154 00:08:14,580 --> 00:08:18,090 And this n bar dot proton momentum is actually Q over x. 155 00:08:18,090 --> 00:08:20,920 So we could also write this formula like that. 156 00:08:20,920 --> 00:08:22,740 And so you see that w plus is actually 157 00:08:22,740 --> 00:08:27,000 something that's [? xi ?] over x, Bjorken x, which is 158 00:08:27,000 --> 00:08:28,680 an external leptonic variable. 159 00:08:31,410 --> 00:08:34,950 Now, there was another index over here, j, which 160 00:08:34,950 --> 00:08:36,059 we talked about last time. 161 00:08:36,059 --> 00:08:37,851 And that had to do with the fact that we're 162 00:08:37,851 --> 00:08:42,270 taking the forward scattering graphs with a tensor 163 00:08:42,270 --> 00:08:47,550 and we decompose that into two scalars, 164 00:08:47,550 --> 00:08:50,015 multiplying things that had indices. 165 00:08:50,015 --> 00:08:51,390 So there were two possible things 166 00:08:51,390 --> 00:08:53,760 that we could write down. 167 00:08:53,760 --> 00:09:00,270 And the index j is just this 1 or 2. 168 00:09:00,270 --> 00:09:02,100 And there was a similar decomposition. 169 00:09:02,100 --> 00:09:05,310 In the effective theory, we could think of decomposing-- 170 00:09:05,310 --> 00:09:08,130 the effective theory is a scalar in terms 171 00:09:08,130 --> 00:09:09,630 of the scalar operators, which are 172 00:09:09,630 --> 00:09:12,210 these guys, with some coefficients that 173 00:09:12,210 --> 00:09:15,420 have some indices, then multiplied by some tensor. 174 00:09:15,420 --> 00:09:16,830 So these guys don't have indices, 175 00:09:16,830 --> 00:09:19,770 but I could just multiply them by effectively 176 00:09:19,770 --> 00:09:22,680 the effective theory versions of these tensors. 177 00:09:22,680 --> 00:09:26,130 And so that that's why there's a j here. 178 00:09:26,130 --> 00:09:29,630 Is that clear to everybody? 179 00:09:29,630 --> 00:09:32,270 OK, so there's various indices-- i flavor, 180 00:09:32,270 --> 00:09:35,540 j for tensor decomposition, and then a bunch 181 00:09:35,540 --> 00:09:40,300 of momentum indices. 182 00:09:40,300 --> 00:09:43,480 So when you go through the analysis 183 00:09:43,480 --> 00:09:47,890 of trying to find a formula for, say, T1, 184 00:09:47,890 --> 00:09:49,840 it's going to be related to C1. 185 00:09:49,840 --> 00:09:52,768 And T2 will be related to C2, OK? 186 00:09:52,768 --> 00:09:54,310 Because this guy's a scalar operator. 187 00:09:54,310 --> 00:09:57,310 It doesn't have any indices. 188 00:09:57,310 --> 00:10:05,140 So the way that that works, if you just look at the two bases 189 00:10:05,140 --> 00:10:14,800 and write down the formula, you'd have an integral over 190 00:10:14,800 --> 00:10:15,310 these w's. 191 00:10:18,050 --> 00:10:20,830 There are some prefactors which just come about 192 00:10:20,830 --> 00:10:24,820 from being careful, and then the thing that 193 00:10:24,820 --> 00:10:33,105 has an imaginary part of these Wilson coefficients, 194 00:10:33,105 --> 00:10:34,480 and then you have matrix elements 195 00:10:34,480 --> 00:10:43,120 of operators, which have a flavor index 196 00:10:43,120 --> 00:10:46,600 but don't have a subscript j. 197 00:10:46,600 --> 00:10:48,340 So in general, this has a flavor index. 198 00:10:52,750 --> 00:10:53,650 Keep track of things. 199 00:10:56,570 --> 00:10:59,210 And then there's another one for T2-- 200 00:11:08,667 --> 00:11:13,678 so kinematic prefactors that are easy to work out that 201 00:11:13,678 --> 00:11:15,220 just come about from the fact that we 202 00:11:15,220 --> 00:11:17,050 wrote the tensors and the effective theory 203 00:11:17,050 --> 00:11:19,067 and the full theory slightly differently. 204 00:11:32,630 --> 00:11:37,640 But these two guys have the same matrix elements 205 00:11:37,640 --> 00:11:38,790 of the same operator. 206 00:11:38,790 --> 00:11:41,402 And all the sort of tensor stuff is just 207 00:11:41,402 --> 00:11:43,610 saying that there's two different Wilson coefficients 208 00:11:43,610 --> 00:11:46,572 that you have to compute, and that's 209 00:11:46,572 --> 00:11:48,905 because you have these vector currents from the photons. 210 00:11:52,750 --> 00:11:54,125 OK, so this is what we're after. 211 00:11:58,740 --> 00:12:03,000 These show up in the cross-section. 212 00:12:03,000 --> 00:12:06,192 And what we're doing is we're writing, at lowest order, 213 00:12:06,192 --> 00:12:08,400 the things that show up in the cross-section in terms 214 00:12:08,400 --> 00:12:11,055 of effective theory objects, the Wilson coefficients and then 215 00:12:11,055 --> 00:12:13,530 the matrix elements of our operators 216 00:12:13,530 --> 00:12:17,250 here, which is this thing in square brackets. 217 00:12:17,250 --> 00:12:20,490 And we're almost at what you would call a factorization 218 00:12:20,490 --> 00:12:22,080 theorem. 219 00:12:22,080 --> 00:12:26,730 Factorization theorem is a result for the cross-section, 220 00:12:26,730 --> 00:12:29,500 in our language, in terms of effective theory quantities, 221 00:12:29,500 --> 00:12:31,950 and that's going to factor the hard stuff, which 222 00:12:31,950 --> 00:12:38,130 is the pink stuff, which is in these Wilson coefficients, 223 00:12:38,130 --> 00:12:41,372 from the low-energy stuff, which is in these operators. 224 00:12:41,372 --> 00:12:44,640 AUDIENCE: So those pink elements are [INAUDIBLE] equation. 225 00:12:44,640 --> 00:12:45,960 IAIN STEWART: Yeah. 226 00:12:45,960 --> 00:12:49,820 AUDIENCE: And what is square bracket [INAUDIBLE]?? 227 00:12:49,820 --> 00:12:50,820 IAIN STEWART: It's both. 228 00:12:50,820 --> 00:12:51,820 AUDIENCE: It's both, OK. 229 00:12:51,820 --> 00:12:53,500 IAIN STEWART: Yeah. 230 00:12:53,500 --> 00:12:55,290 I can write it like this, if you like. 231 00:12:58,290 --> 00:13:01,050 Any other questions? 232 00:13:01,050 --> 00:13:06,030 OK, so literally what I do is I take this formula 233 00:13:06,030 --> 00:13:08,402 and I plug it into that formula. 234 00:13:08,402 --> 00:13:10,860 And when I do that, I can do one of the integrals trivially 235 00:13:10,860 --> 00:13:13,740 because it's a delta function. 236 00:13:13,740 --> 00:13:15,030 This one's just trivial. 237 00:13:15,030 --> 00:13:18,220 And then I do this one with the other delta function. 238 00:13:18,220 --> 00:13:20,880 So both integrals are actually trivial. 239 00:13:20,880 --> 00:13:25,080 And I can write the result in terms of something 240 00:13:25,080 --> 00:13:33,300 that I'll call the hard function, which 241 00:13:33,300 --> 00:13:35,881 is just the imaginary part of the Wilson coefficient. 242 00:13:50,200 --> 00:13:53,120 And I'm going to denote it in the following way. 243 00:13:53,120 --> 00:13:54,145 So this is w-- 244 00:13:54,145 --> 00:13:55,733 the Wilson cost efficient can depend 245 00:13:55,733 --> 00:13:56,900 on various different things. 246 00:13:56,900 --> 00:13:58,720 It can depend on w plus. 247 00:13:58,720 --> 00:14:00,490 It can depend on w minus. 248 00:14:00,490 --> 00:14:03,010 It can depend on the hard scale, which is q squared. 249 00:14:03,010 --> 00:14:05,570 Or it could depend on mu squared. 250 00:14:05,570 --> 00:14:08,740 So w minus, when you do the delta function, gets set to 0. 251 00:14:08,740 --> 00:14:10,720 w plus gets set to something. 252 00:14:10,720 --> 00:14:13,480 And it's convenient because of the way 253 00:14:13,480 --> 00:14:16,660 this delta function is with the-- it kind of has a ratio. 254 00:14:16,660 --> 00:14:18,550 Because this function is a function of xi 255 00:14:18,550 --> 00:14:20,810 which is the ratio of two things, 256 00:14:20,810 --> 00:14:24,580 it's convenient to define a dimensionless z 257 00:14:24,580 --> 00:14:28,570 and only talk about a function of that dimensionless thing. 258 00:14:28,570 --> 00:14:32,560 And if you do that, then the final result for these kind 259 00:14:32,560 --> 00:14:35,680 of T's, which are imaginary parts of T's-- 260 00:14:46,820 --> 00:14:49,720 you can just put the formula together. 261 00:14:49,720 --> 00:14:50,720 I'll write one of them-- 262 00:15:03,750 --> 00:15:04,340 is that. 263 00:15:08,200 --> 00:15:15,310 And then there's a similar formula for in T2 264 00:15:15,310 --> 00:15:19,090 that involves H2 and H1. 265 00:15:19,090 --> 00:15:21,520 OK, so this is the factorization theorem. 266 00:15:21,520 --> 00:15:24,280 And it came about, in some sense, just trivially. 267 00:15:24,280 --> 00:15:26,350 Once we knew how to write down the operators 268 00:15:26,350 --> 00:15:28,780 in the effective theory, we were basically done, 269 00:15:28,780 --> 00:15:32,020 and then the rest was just sort of algebraic manipulations, 270 00:15:32,020 --> 00:15:35,050 being careful about what momenta go where, 271 00:15:35,050 --> 00:15:40,060 and knowing what the sign of this formula for the matrix 272 00:15:40,060 --> 00:15:41,650 element is. 273 00:15:41,650 --> 00:15:44,470 This is a kind of important point. 274 00:15:44,470 --> 00:15:47,140 But in some sense, the effective theory, from the get-go, 275 00:15:47,140 --> 00:15:48,940 was already designed to factorize 276 00:15:48,940 --> 00:15:51,398 because we were integrating out the hard degrees of freedom 277 00:15:51,398 --> 00:15:52,370 right at the start. 278 00:15:52,370 --> 00:15:56,850 And so knowing what operators and knowing their matrix 279 00:15:56,850 --> 00:16:00,520 element is really all we needed to do to get to the DIS 280 00:16:00,520 --> 00:16:01,510 factorization theorem. 281 00:16:08,680 --> 00:16:12,110 So if you ever look up the original way 282 00:16:12,110 --> 00:16:15,500 that this was derived, it was not that easy. 283 00:16:15,500 --> 00:16:17,420 This is actually something that's 284 00:16:17,420 --> 00:16:21,770 very complicated in a sort of traditional approach. 285 00:16:21,770 --> 00:16:23,880 But in the effective theory approach, 286 00:16:23,880 --> 00:16:27,680 it becomes almost trivial. 287 00:16:27,680 --> 00:16:29,810 And this is an all-orders result because we never 288 00:16:29,810 --> 00:16:31,520 expanded in alpha s. 289 00:16:31,520 --> 00:16:34,120 We just used symmetries, and we used the fact 290 00:16:34,120 --> 00:16:35,870 that we knew what form the operators would 291 00:16:35,870 --> 00:16:38,203 take when we integrated out the hard degrees of freedom. 292 00:16:45,000 --> 00:16:48,410 So any alpha s corrections that one might want to add 293 00:16:48,410 --> 00:16:49,535 will fit into this formula. 294 00:16:52,220 --> 00:16:55,700 And this gives a perturbative result for this H, 295 00:16:55,700 --> 00:16:59,060 which you would compute in perturbation theory, which 296 00:16:59,060 --> 00:17:01,730 people do [INAUDIBLE] these days. 297 00:17:07,050 --> 00:17:08,579 Now, if you ask about things like-- 298 00:17:08,579 --> 00:17:11,010 I didn't write all the possible-- 299 00:17:11,010 --> 00:17:14,380 I suppressed some things, right, like Q squared and mu squared. 300 00:17:14,380 --> 00:17:16,500 If you ask about the Q squared and the mu squared, 301 00:17:16,500 --> 00:17:19,020 then your Wilson coefficients do depend on Q squared 302 00:17:19,020 --> 00:17:20,430 and mu squared. 303 00:17:20,430 --> 00:17:22,050 And the Wilson coefficients, H here, 304 00:17:22,050 --> 00:17:24,092 are actually dimension-- the Wilson coefficients, 305 00:17:24,092 --> 00:17:26,135 the original one, [? xi, ?] were dimensionless. 306 00:17:30,020 --> 00:17:31,737 So the H is dimensionless. 307 00:17:35,940 --> 00:17:37,995 I just pulled out the dimensionable factor 308 00:17:37,995 --> 00:17:40,270 so that that would be true. 309 00:17:40,270 --> 00:17:43,320 And so this guy can depend on Q squared over mu squared. 310 00:17:43,320 --> 00:17:45,240 The fact that Q squared only shows up there, 311 00:17:45,240 --> 00:17:47,700 that's Bjorken scaling. 312 00:17:47,700 --> 00:17:51,630 And if you look at the perturbative result for T2, 313 00:17:51,630 --> 00:17:53,950 then it vanishes at lowest order. 314 00:17:53,950 --> 00:17:58,950 And so that's the Callan-Gross relation. 315 00:17:58,950 --> 00:18:01,260 So there's various things that are sort of encoded 316 00:18:01,260 --> 00:18:06,780 in this that come out, from the effective theory point of view, 317 00:18:06,780 --> 00:18:10,380 in a very simple way. 318 00:18:10,380 --> 00:18:11,330 OK, so let me write. 319 00:18:32,380 --> 00:18:33,910 So there's logarithmic corrections 320 00:18:33,910 --> 00:18:39,940 that involve Q in the Wilson coefficients 321 00:18:39,940 --> 00:18:41,500 that will show up like that. 322 00:18:46,240 --> 00:18:50,360 So there's a mu also that you could add to this formula. 323 00:18:50,360 --> 00:18:52,810 So the way that I described it, we 324 00:18:52,810 --> 00:18:58,210 didn't think too hard about bare versus normalized, right? 325 00:18:58,210 --> 00:18:59,770 We just take these operators. 326 00:18:59,770 --> 00:19:02,110 So far, they could have been bare. 327 00:19:02,110 --> 00:19:09,100 But remember that when you have C bare, O bare in [INAUDIBLE] 328 00:19:09,100 --> 00:19:12,090 Hamiltonian, for example, that's C mu, O mu. 329 00:19:15,690 --> 00:19:17,635 So switching from bare and renormalized-- 330 00:19:17,635 --> 00:19:19,840 I mean bare operators and coefficients 331 00:19:19,840 --> 00:19:21,880 to renormalized operators and coefficients 332 00:19:21,880 --> 00:19:24,145 is simply a matter of sticking in a mu here, 333 00:19:24,145 --> 00:19:26,020 and then you imagine that the renormalization 334 00:19:26,020 --> 00:19:28,640 has taking place. 335 00:19:28,640 --> 00:19:32,680 So we could equally well insert in these formulas 336 00:19:32,680 --> 00:19:35,290 a mu for that. 337 00:19:38,810 --> 00:19:41,725 And then what I'm saying is that there being logs of mu over Q 338 00:19:41,725 --> 00:19:44,720 will make a little more sense. 339 00:19:44,720 --> 00:19:53,540 So there's also a Q. Squeeze everything in here [INAUDIBLE].. 340 00:19:53,540 --> 00:19:56,840 OK, now we're being completely honest 341 00:19:56,840 --> 00:19:59,880 about what it depends on. 342 00:19:59,880 --> 00:20:01,320 All right. 343 00:20:01,320 --> 00:20:08,640 So traditionally what happens in the traditional literature, 344 00:20:08,640 --> 00:20:11,190 people talk about factorization scales and renormalization 345 00:20:11,190 --> 00:20:12,720 group scales. 346 00:20:12,720 --> 00:20:15,240 So factorization scales is the fact 347 00:20:15,240 --> 00:20:18,307 that this parton distribution function is mu-dependent-- so 348 00:20:18,307 --> 00:20:19,890 operator that you have to renormalize, 349 00:20:19,890 --> 00:20:22,270 and we're going to do that in a minute. 350 00:20:22,270 --> 00:20:24,210 And so there has to be a cancellation. 351 00:20:24,210 --> 00:20:26,370 Since this thing here is a physical observable 352 00:20:26,370 --> 00:20:28,230 and is independent of mu, there has 353 00:20:28,230 --> 00:20:30,180 to be a cancellation of the mu-dependence here 354 00:20:30,180 --> 00:20:33,570 and the mu-dependence here, all right? 355 00:20:33,570 --> 00:20:35,680 And that's this mu. 356 00:20:35,680 --> 00:20:39,330 So the thing that's multiplying this result here 357 00:20:39,330 --> 00:20:41,310 would involve a cancellation of mu-dependence 358 00:20:41,310 --> 00:20:42,477 here and mu-dependence here. 359 00:20:42,477 --> 00:20:44,490 So the same anomalous dimension would show up 360 00:20:44,490 --> 00:20:46,470 in both the H and the f. 361 00:20:46,470 --> 00:20:50,520 And then sometimes people also talk about mu-dependence 362 00:20:50,520 --> 00:20:53,640 that's just cancelling within H itself. 363 00:20:53,640 --> 00:20:56,460 And they call that renormalization group 364 00:20:56,460 --> 00:20:58,740 renormalization group mu. 365 00:20:58,740 --> 00:21:00,720 Sometimes people vary these independently. 366 00:21:00,720 --> 00:21:03,160 In the effective theory, it's really simple. 367 00:21:03,160 --> 00:21:05,340 You really just have the classic setup of you 368 00:21:05,340 --> 00:21:08,310 have some hard degrees of freedom. 369 00:21:08,310 --> 00:21:11,808 In this case, you can even think of it one-dimensional. 370 00:21:11,808 --> 00:21:13,350 You have some hard degrees of freedom 371 00:21:13,350 --> 00:21:14,642 that you want to integrate out. 372 00:21:14,642 --> 00:21:16,830 You have some scale which we could 373 00:21:16,830 --> 00:21:18,750 call mu 0 that's of order 2 where 374 00:21:18,750 --> 00:21:20,340 we do that integrating out. 375 00:21:20,340 --> 00:21:26,640 And then you can run down or you could 376 00:21:26,640 --> 00:21:28,680 run in a more complicated way. 377 00:21:28,680 --> 00:21:32,070 So you could run the PDFs which are sitting here 378 00:21:32,070 --> 00:21:38,398 at the collinear scale, which is lambda QCD. 379 00:21:38,398 --> 00:21:40,440 You could think of evolving them up to some scale 380 00:21:40,440 --> 00:21:42,990 and evolving the Wilson coefficients down and meeting 381 00:21:42,990 --> 00:21:46,110 somewhere, OK? 382 00:21:46,110 --> 00:21:49,890 And so, yeah, it's just really a sort of classic running 383 00:21:49,890 --> 00:21:51,720 and matching picture. 384 00:21:51,720 --> 00:21:53,790 Here, I've just used the fact that I 385 00:21:53,790 --> 00:21:56,130 could run either one of them or I could run both of them 386 00:21:56,130 --> 00:21:57,960 to a common scale. 387 00:21:57,960 --> 00:22:00,600 So I usually would pick mu to be either something small 388 00:22:00,600 --> 00:22:04,803 or something large rather than running both things. 389 00:22:04,803 --> 00:22:07,220 But in general, you could think about running both things. 390 00:22:07,220 --> 00:22:09,480 And we've talked about having anomalous dimensions 391 00:22:09,480 --> 00:22:12,180 for either one of these. 392 00:22:12,180 --> 00:22:15,180 And usually, we just run one of them, OK? 393 00:22:15,180 --> 00:22:17,430 But it's no more complicated than the standard picture 394 00:22:17,430 --> 00:22:19,770 of integrating out modes and doing renormalization group 395 00:22:19,770 --> 00:22:20,816 evolution. 396 00:22:24,150 --> 00:22:27,870 So if we want to do tree-level matching or one-loop matching 397 00:22:27,870 --> 00:22:31,112 or any kind of matching-- 398 00:22:31,112 --> 00:22:32,820 let me just show you tree-level matching. 399 00:22:39,785 --> 00:22:41,160 So tree-level matching, you would 400 00:22:41,160 --> 00:22:47,120 compute this forward scattering graph, 401 00:22:47,120 --> 00:22:51,770 and that will give you the other diagram that we drew. 402 00:22:51,770 --> 00:22:55,280 And so you'd want to match this guy onto that guy. 403 00:22:58,060 --> 00:23:06,988 And what you find is you find one tensor structure 404 00:23:06,988 --> 00:23:08,900 at lowest order. 405 00:23:08,900 --> 00:23:11,610 So C1 is not equal to 0. 406 00:23:11,610 --> 00:23:13,010 C2 is equal to 0. 407 00:23:13,010 --> 00:23:16,730 And that's the Callan-Gross relation 408 00:23:16,730 --> 00:23:19,340 which tells you about the spin of the object 409 00:23:19,340 --> 00:23:20,840 that you're scattering off, and this 410 00:23:20,840 --> 00:23:27,380 is how we know that quarks are spin 1/2, or one way we know. 411 00:23:27,380 --> 00:23:33,620 And then you can calculate C. And so that way 412 00:23:33,620 --> 00:23:35,660 that I set things up, C was complex, 413 00:23:35,660 --> 00:23:37,740 and then I had to take the imaginary part. 414 00:23:37,740 --> 00:23:41,240 So C is just this propagator, basically. 415 00:23:41,240 --> 00:23:44,550 And it's only a nontrivial function of w plus. 416 00:23:44,550 --> 00:23:49,100 There are some charges that sit out front. 417 00:23:49,100 --> 00:23:51,363 And so the only way that this guy depends on 418 00:23:51,363 --> 00:23:53,030 whether it's an up quark or a down quark 419 00:23:53,030 --> 00:23:55,100 is you have 2/3 squared or 1/3 squared. 420 00:24:03,260 --> 00:24:07,280 And then there's something that comes about from the propagator 421 00:24:07,280 --> 00:24:10,210 that looks like this. 422 00:24:10,210 --> 00:24:14,520 And then I take the imaginary part 423 00:24:14,520 --> 00:24:21,650 and then I get H1, which is a function of z, which 424 00:24:21,650 --> 00:24:23,600 is the xi over x. 425 00:24:23,600 --> 00:24:25,910 So if I write it as xi over x like it shows up 426 00:24:25,910 --> 00:24:28,370 in the factorization theorem, then I'm 427 00:24:28,370 --> 00:24:31,278 getting a delta function of xi over x, 428 00:24:31,278 --> 00:24:32,570 which is this coming from this. 429 00:24:38,230 --> 00:24:41,900 So the lowest-order H1 is just a delta function. 430 00:24:41,900 --> 00:24:44,230 And that's where the parton model picture 431 00:24:44,230 --> 00:24:46,660 comes from because the parton model 432 00:24:46,660 --> 00:24:53,780 picture is that you think of xi and x as being the same thing. 433 00:24:53,780 --> 00:24:56,653 And that's the tree-level way of thinking, 434 00:24:56,653 --> 00:24:58,570 and that's just satisfying this delta function 435 00:24:58,570 --> 00:25:00,640 and the hard function. 436 00:25:00,640 --> 00:25:02,860 And then you would get that the T is just 437 00:25:02,860 --> 00:25:05,200 given by the parton distribution at x, which is 438 00:25:05,200 --> 00:25:09,570 the external measurable thing. 439 00:25:09,570 --> 00:25:10,070 OK? 440 00:25:10,070 --> 00:25:12,480 So this is how all these classic things come about 441 00:25:12,480 --> 00:25:16,000 in the effective theory language. 442 00:25:16,000 --> 00:25:17,170 Any questions about that? 443 00:25:24,600 --> 00:25:28,840 All right, so let's renormalize this operator 444 00:25:28,840 --> 00:25:32,250 and see how the classic result for the renormalization group 445 00:25:32,250 --> 00:25:35,940 evolution of a PDF comes about. 446 00:25:35,940 --> 00:25:38,730 And again, the way that you should think about this 447 00:25:38,730 --> 00:25:44,240 is you have an operator, and you should just renormalize it. 448 00:25:44,240 --> 00:25:47,480 And once you've got the effective theory, 449 00:25:47,480 --> 00:25:50,188 you shouldn't have to think too deeply about what you're doing. 450 00:25:50,188 --> 00:25:51,980 You should just be able to follow your nose 451 00:25:51,980 --> 00:25:53,360 and do the renormalization. 452 00:25:53,360 --> 00:25:55,880 You may have to be careful because these operators are 453 00:25:55,880 --> 00:25:57,750 kind of complicated. 454 00:25:57,750 --> 00:25:59,743 They have this dependence on these w's 455 00:25:59,743 --> 00:26:01,160 that you have to be careful about. 456 00:26:01,160 --> 00:26:03,950 But really, it's just follow your nose, 457 00:26:03,950 --> 00:26:05,510 compute the one-loop graphs. 458 00:26:20,035 --> 00:26:24,210 If you look up how Peskin would do one-loop renormalization, 459 00:26:24,210 --> 00:26:25,960 there'd be an infinite number of operators 460 00:26:25,960 --> 00:26:29,080 you'd have to derive in a renormalization group, result 461 00:26:29,080 --> 00:26:29,920 for all of them. 462 00:26:29,920 --> 00:26:31,930 Here, we only have one operator and we're just 463 00:26:31,930 --> 00:26:34,890 going to renormalize it. 464 00:26:34,890 --> 00:26:36,850 Our operator is nonlocal in the sense 465 00:26:36,850 --> 00:26:38,770 that it depends on these omegas, and that's 466 00:26:38,770 --> 00:26:42,190 what's encoding this infinite number of operators 467 00:26:42,190 --> 00:26:43,150 that Peskin has. 468 00:27:03,620 --> 00:27:09,860 OK, so solving, if you like, for f from the formula 469 00:27:09,860 --> 00:27:11,990 that we had before, I can do that 470 00:27:11,990 --> 00:27:16,550 by integrating over the w minus. 471 00:27:16,550 --> 00:27:18,980 That sets these guys to be equal. 472 00:27:18,980 --> 00:27:20,330 And then if I-- 473 00:27:20,330 --> 00:27:24,920 so I can think of it as that there's one free momentum, xi. 474 00:27:24,920 --> 00:27:27,410 And that free momentum xi is one of these labels, 475 00:27:27,410 --> 00:27:29,220 which is this guy here. 476 00:27:29,220 --> 00:27:34,310 So xi is w over Pn minus, and this 477 00:27:34,310 --> 00:27:38,690 is the proton which is carrying some momentum Pn minus. 478 00:27:48,770 --> 00:27:52,670 This is the proton state, which carries momentum Pn minus. 479 00:27:52,670 --> 00:27:54,080 And there's one delta function. 480 00:27:56,900 --> 00:27:58,550 I could put it either place, but I only 481 00:27:58,550 --> 00:28:01,518 need one because the other one's kind of trivial. 482 00:28:01,518 --> 00:28:03,560 So the first thing you can think about doing here 483 00:28:03,560 --> 00:28:05,120 is looking at mass dimensions. 484 00:28:05,120 --> 00:28:07,010 And I already told you that this guy was dimensionless, 485 00:28:07,010 --> 00:28:08,427 but let's check that that's true-- 486 00:28:10,830 --> 00:28:14,790 so a mass dimension. 487 00:28:14,790 --> 00:28:17,540 So relativistically normalized states 488 00:28:17,540 --> 00:28:20,370 have mass dimension minus 1. 489 00:28:20,370 --> 00:28:22,890 Quark fields that don't have a delta function 490 00:28:22,890 --> 00:28:26,310 have mass dimension 3/2. 491 00:28:26,310 --> 00:28:31,150 The delta function gives a minus 1, and then a minus 1, 492 00:28:31,150 --> 00:28:33,660 so you get 0. 493 00:28:33,660 --> 00:28:37,560 3/2 plus 3/2 minus 3/1 is 0. 494 00:28:37,560 --> 00:28:40,033 So that means this f is a really dimensionless function, 495 00:28:40,033 --> 00:28:42,450 and that's why it makes sense that we defined it to depend 496 00:28:42,450 --> 00:28:45,030 on this dimensionless ratio. 497 00:28:45,030 --> 00:28:50,880 You can also look at the lambda dimension, 498 00:28:50,880 --> 00:28:51,990 and here's how that works. 499 00:28:59,730 --> 00:29:00,810 That's also 0. 500 00:29:04,240 --> 00:29:07,540 So the only thing that's-- so we already had power counting 501 00:29:07,540 --> 00:29:08,540 for our kai fields. 502 00:29:08,540 --> 00:29:10,630 Remember, the c field inside the chi field scale 503 00:29:10,630 --> 00:29:15,070 like [? 1. ?] So this is just coming about because this guy's 504 00:29:15,070 --> 00:29:17,260 order lambda. 505 00:29:17,260 --> 00:29:19,878 The delta function just involves large momentum, 506 00:29:19,878 --> 00:29:21,045 so it has no power counting. 507 00:29:27,940 --> 00:29:29,440 And the only thing that's nontrivial 508 00:29:29,440 --> 00:29:33,010 is that the states have power counting minus 1. 509 00:29:33,010 --> 00:29:36,200 So here's how we can derive that. 510 00:29:36,200 --> 00:29:38,410 So if you think about relativistically normalized 511 00:29:38,410 --> 00:29:41,050 states, what you're doing is you're 512 00:29:41,050 --> 00:29:44,050 defining sort of the inverse of this d3 p 513 00:29:44,050 --> 00:29:47,920 over e, which you can write actually, 514 00:29:47,920 --> 00:29:50,680 which is more convenient for power counting, 515 00:29:50,680 --> 00:29:58,270 in terms of things that we can power count more simply. 516 00:29:58,270 --> 00:30:02,320 So this is an exact relation for a nontrivial particle 517 00:30:02,320 --> 00:30:06,760 between p minuses and pz. 518 00:30:06,760 --> 00:30:10,090 So then I can write, because of that, 519 00:30:10,090 --> 00:30:12,190 the standard relativistic normalization 520 00:30:12,190 --> 00:30:17,204 formula for a state with two different momenta 521 00:30:17,204 --> 00:30:27,970 as kind of the inverse, which would be this. 522 00:30:27,970 --> 00:30:33,645 So the usual formula would have 2e and then delta 3, right? 523 00:30:33,645 --> 00:30:35,020 Because it's the inverse of this. 524 00:30:35,020 --> 00:30:38,230 But I can write it also this way. 525 00:30:38,230 --> 00:30:40,720 This guy is lambda 0. 526 00:30:40,720 --> 00:30:42,730 This guy is lambda minus 2. 527 00:30:42,730 --> 00:30:47,292 Therefore, each of these guys must be lambda minus 1. 528 00:30:47,292 --> 00:30:48,750 That's where the minus 1 came from. 529 00:30:52,710 --> 00:30:56,070 All right, so we want to renormalize that thing, 530 00:30:56,070 --> 00:30:58,140 that matrix element. 531 00:30:58,140 --> 00:31:05,850 And what loops can do is that they can change omega. 532 00:31:05,850 --> 00:31:11,457 So you might-- or xi, which are equivalent. 533 00:31:11,457 --> 00:31:13,290 And so the way that you should think of that 534 00:31:13,290 --> 00:31:15,330 is in the following sense, and it's actually 535 00:31:15,330 --> 00:31:18,270 something you're familiar with, although you're 536 00:31:18,270 --> 00:31:20,200 familiar with it for discrete quantum numbers. 537 00:31:20,200 --> 00:31:23,670 And here, in some sense, we have a continuous one. 538 00:31:23,670 --> 00:31:26,040 So you have some function, fq, that 539 00:31:26,040 --> 00:31:27,960 depends on some variable xi. 540 00:31:27,960 --> 00:31:30,810 And it can mix, under the renormalization group, 541 00:31:30,810 --> 00:31:35,712 with an operator at a different value of xi. 542 00:31:35,712 --> 00:31:37,170 So you can really think of the fact 543 00:31:37,170 --> 00:31:40,785 that loops can change this omega as just a mixing. 544 00:31:40,785 --> 00:31:42,910 You're used to mixing for discrete quantum numbers. 545 00:31:42,910 --> 00:31:45,510 You write down all the operators that have the same quantum 546 00:31:45,510 --> 00:31:47,850 numbers, and they can mix under renormalization. 547 00:31:47,850 --> 00:31:50,010 Here, there's kind of an additional thing 548 00:31:50,010 --> 00:31:53,490 that the object can depend on, which is the xi parameter. 549 00:31:53,490 --> 00:31:55,560 And in general, when you do the renormalization, 550 00:31:55,560 --> 00:31:57,660 that can change too. 551 00:31:57,660 --> 00:32:00,150 Because the operators or matrix elements here 552 00:32:00,150 --> 00:32:02,370 are labeled by this xi, in general, there's 553 00:32:02,370 --> 00:32:04,740 no reason that it should stay the same under the loop 554 00:32:04,740 --> 00:32:05,340 corrections. 555 00:32:05,340 --> 00:32:09,060 And it was really a special case that we dealt with last time 556 00:32:09,060 --> 00:32:10,080 where that did happen. 557 00:32:10,080 --> 00:32:11,205 But in general, it doesn't. 558 00:32:15,270 --> 00:32:21,270 OK, so this is actually what we expect 559 00:32:21,270 --> 00:32:24,450 to happen in general unless we can argue that it doesn't 560 00:32:24,450 --> 00:32:32,770 happen because you should think of each value of xi 561 00:32:32,770 --> 00:32:37,360 as giving a different operator or different matrix element. 562 00:32:40,570 --> 00:32:42,900 So I could write formulas here just for the operator. 563 00:32:42,900 --> 00:32:44,983 It's actually the operator that gets renormalized, 564 00:32:44,983 --> 00:32:46,650 not the matrix element. 565 00:32:46,650 --> 00:32:49,500 So I'm going to keep writing f's just 566 00:32:49,500 --> 00:32:52,290 to avoid too much notation, but we could always actually 567 00:32:52,290 --> 00:32:55,762 replace the f's by just the operator. 568 00:32:55,762 --> 00:32:57,720 And we could do everything in terms of actually 569 00:32:57,720 --> 00:33:01,510 just the w instead of the xi variable. 570 00:33:01,510 --> 00:33:04,740 But I'll just keep using f. 571 00:33:04,740 --> 00:33:08,520 So what does this mean in terms of the operator 572 00:33:08,520 --> 00:33:09,250 is the following. 573 00:33:09,250 --> 00:33:16,480 We can think of, if we have some bare operator 574 00:33:16,480 --> 00:33:21,000 and we want to split that into a piece that has divergences 575 00:33:21,000 --> 00:33:27,900 and a piece that is just the finite pieces, 576 00:33:27,900 --> 00:33:32,740 the general formula for doing that involves an integral. 577 00:33:32,740 --> 00:33:34,450 So this guy here-- 578 00:33:34,450 --> 00:33:36,630 so there's also these indices, i and j, 579 00:33:36,630 --> 00:33:40,890 and that's the flavor, if you like, or quarks and gluons. 580 00:33:40,890 --> 00:33:45,270 So i is quark or gluon. 581 00:33:45,270 --> 00:33:48,570 And in general, you can also have a mixing in the quark 582 00:33:48,570 --> 00:33:49,440 and gluon operators. 583 00:33:49,440 --> 00:33:51,120 We started with these two different operators, 584 00:33:51,120 --> 00:33:53,078 and they can mix under renormalization as well. 585 00:34:02,640 --> 00:34:05,540 So there's two operators in the effective theory, same order 586 00:34:05,540 --> 00:34:08,830 in lambda, and they can mix when you do the renormalization. 587 00:34:08,830 --> 00:34:12,139 And I'll draw a diagram in a minute. 588 00:34:12,139 --> 00:34:16,540 So this thing here is mu-independent. 589 00:34:16,540 --> 00:34:21,100 This thing here in MS bar has all the 1 over epsilon UV's. 590 00:34:21,100 --> 00:34:23,679 And it also depends on alpha of mu, 591 00:34:23,679 --> 00:34:26,920 and it depends on these xi and xi prime. 592 00:34:26,920 --> 00:34:28,840 And this guy here is UV-finite. 593 00:34:34,238 --> 00:34:36,030 So this guy here is really the thing that's 594 00:34:36,030 --> 00:34:37,350 the low-energy matrix element. 595 00:34:37,350 --> 00:34:39,510 But remember what low energy meant here. 596 00:34:39,510 --> 00:34:42,300 Low energy was physics at lambda QCD, physics 597 00:34:42,300 --> 00:34:44,880 of the initial-state proton. 598 00:34:44,880 --> 00:34:48,715 So actually, in this guy, there are IR divergences. 599 00:34:48,715 --> 00:34:51,090 This is just some matrix element in the effective theory, 600 00:34:51,090 --> 00:34:52,757 and in general, it could be IR-divergent 601 00:34:52,757 --> 00:34:53,810 if you calculate it. 602 00:34:53,810 --> 00:34:59,010 And this guy actually is. 603 00:34:59,010 --> 00:35:01,922 And it really encodes-- 604 00:35:01,922 --> 00:35:04,380 that's not going to bother us at all because this is really 605 00:35:04,380 --> 00:35:08,370 some universal thing that encodes lambda QCD effects, 606 00:35:08,370 --> 00:35:11,320 and that's what parton distribution functions are. 607 00:35:11,320 --> 00:35:13,320 Then from the point of view of what we're doing, 608 00:35:13,320 --> 00:35:16,630 it doesn't really matter that it has this extra IR divergence 609 00:35:16,630 --> 00:35:19,080 so that we will have to regulate diagrams 610 00:35:19,080 --> 00:35:22,633 in order to separate UV and IR divergences because of that. 611 00:35:22,633 --> 00:35:24,300 Really, in terms of the renormalization, 612 00:35:24,300 --> 00:35:26,791 what we're after is getting the UV divergences. 613 00:35:31,610 --> 00:35:33,510 OK, so the usual kind of formula that you'd 614 00:35:33,510 --> 00:35:36,010 have where you just write o is z times o 615 00:35:36,010 --> 00:35:37,983 is slightly more complicated here. 616 00:35:37,983 --> 00:35:39,150 There's this extra integral. 617 00:35:52,590 --> 00:35:55,980 And now, remember how you derive a renormalization group 618 00:35:55,980 --> 00:35:57,450 equation. 619 00:35:57,450 --> 00:35:59,955 What you do is you say mu d by u mu is this guy is 0. 620 00:36:03,490 --> 00:36:07,430 And so if I take mu d by d mu, on the right-hand side, 621 00:36:07,430 --> 00:36:10,690 I get mu d by d mu of z and mu d by d mu of f. 622 00:36:10,690 --> 00:36:14,980 And I can rearrange that in the usual way, 623 00:36:14,980 --> 00:36:19,640 except for keeping track of these integrals, as follows. 624 00:36:19,640 --> 00:36:21,880 So I imagine that there's a z and a z inverse. 625 00:36:21,880 --> 00:36:27,440 And the relation between z and z inverse is as follows. 626 00:36:27,440 --> 00:36:31,540 Let's just call this double prime. 627 00:36:31,540 --> 00:36:34,010 It's matrix multiplication except in the function space, 628 00:36:34,010 --> 00:36:34,510 right? 629 00:36:34,510 --> 00:36:37,885 So this is like a delta function. 630 00:36:41,570 --> 00:36:43,810 So if you like, you can just think 631 00:36:43,810 --> 00:36:46,060 that there's more indices. 632 00:36:46,060 --> 00:36:48,730 In some sense, what we have in terms of the quark and gluon 633 00:36:48,730 --> 00:36:50,860 operators mixing is a matrix equation, right? 634 00:36:50,860 --> 00:36:51,830 This is a vector. 635 00:36:51,830 --> 00:36:52,660 This is a matrix. 636 00:36:52,660 --> 00:36:55,330 This is a vector for the indices i and j. 637 00:36:55,330 --> 00:36:58,780 And you can think of this integral here as just another-- 638 00:36:58,780 --> 00:37:00,825 it really looks, the way I've drawn it, 639 00:37:00,825 --> 00:37:02,200 like this is contracted with that 640 00:37:02,200 --> 00:37:04,300 and this is summing over the indices. 641 00:37:04,300 --> 00:37:06,320 And really, that's what it is. 642 00:37:06,320 --> 00:37:09,490 So really, this idea that it's just mixing of quantum numbers 643 00:37:09,490 --> 00:37:11,478 is kind of a good way of thinking about things. 644 00:37:11,478 --> 00:37:12,895 And when you think about formulas, 645 00:37:12,895 --> 00:37:16,030 you know you're just summing over these indices, 646 00:37:16,030 --> 00:37:20,230 and the Kronecker delta becomes a regular delta. 647 00:37:20,230 --> 00:37:26,720 So in that sense, it's not that hard to do this. 648 00:37:26,720 --> 00:37:36,920 And so we get an anomalous dimension equation 649 00:37:36,920 --> 00:37:38,800 which, again, has that kind of form 650 00:37:38,800 --> 00:37:44,620 of just an integral for the renormalized guy, 651 00:37:44,620 --> 00:37:46,390 and it has mixing. 652 00:37:46,390 --> 00:37:53,650 And this gamma ij, if we go through the steps 653 00:37:53,650 --> 00:37:58,510 and use this formula, looks like this. 654 00:38:04,750 --> 00:38:06,250 So I'm kind of skipping steps, but I 655 00:38:06,250 --> 00:38:10,215 hope that you can kind of picture 656 00:38:10,215 --> 00:38:11,590 where this result will come from. 657 00:38:11,590 --> 00:38:13,240 And it's actually not-- 658 00:38:13,240 --> 00:38:15,790 it's pretty easy to go from that line with this formula 659 00:38:15,790 --> 00:38:16,840 to this line. 660 00:38:16,840 --> 00:38:19,230 This is one line, but I just split it into two things 661 00:38:19,230 --> 00:38:20,980 and defined this quantity, gamma ij, which 662 00:38:20,980 --> 00:38:23,170 is the anomalous dimension. 663 00:38:23,170 --> 00:38:25,850 AUDIENCE: So this mu in QCD [INAUDIBLE] factorization 664 00:38:25,850 --> 00:38:26,570 scale? 665 00:38:26,570 --> 00:38:28,890 IAIN STEWART: Yeah, that's right. 666 00:38:32,150 --> 00:38:36,020 OK, so at one loop, things are simpler. 667 00:38:36,020 --> 00:38:37,490 Because at one loop, this thing, we 668 00:38:37,490 --> 00:38:44,750 can just replace it by delta ii prime, Kronecker delta 669 00:38:44,750 --> 00:38:47,210 at one loop. 670 00:38:47,210 --> 00:38:54,880 Because at one loop, we just need the order alpha piece 671 00:38:54,880 --> 00:38:57,670 from this guy, and then we can set the tree level 672 00:38:57,670 --> 00:38:59,020 for that guy. 673 00:38:59,020 --> 00:39:04,765 So at one loop, which is all we're going to do, 674 00:39:04,765 --> 00:39:05,890 we get the simpler formula. 675 00:39:22,940 --> 00:39:25,850 OK, so that's our setup, and now we 676 00:39:25,850 --> 00:39:28,850 want to calculate this one-loop anomalous dimension 677 00:39:28,850 --> 00:39:31,940 by calculating the 1 over epsilon alpha s term 678 00:39:31,940 --> 00:39:32,490 and the zij. 679 00:39:39,830 --> 00:39:44,130 Before I do that, is there any questions? 680 00:39:44,130 --> 00:39:45,410 All right, so tree level-- 681 00:39:52,910 --> 00:39:56,020 so think about there being an external p for whatever state 682 00:39:56,020 --> 00:40:00,580 I'm considering, and then the operator is labeled by w. 683 00:40:00,580 --> 00:40:06,250 And so we're summing over spin. 684 00:40:06,250 --> 00:40:07,780 I've kind of somehow-- 685 00:40:07,780 --> 00:40:09,640 sometimes I've dropped that. 686 00:40:09,640 --> 00:40:12,140 I said it last time. 687 00:40:12,140 --> 00:40:17,330 And so we get some spinners, and we got a delta function. 688 00:40:17,330 --> 00:40:19,450 So what the delta function in the operator 689 00:40:19,450 --> 00:40:22,870 is it's delta function of w minus this label, 690 00:40:22,870 --> 00:40:23,587 momentum p bar. 691 00:40:23,587 --> 00:40:25,920 And in something like this where it's completely trivial 692 00:40:25,920 --> 00:40:28,087 and there's just one state, we just get the momentum 693 00:40:28,087 --> 00:40:29,890 of that state, which is p. 694 00:40:29,890 --> 00:40:33,730 This sum over spin here is a p minus. 695 00:40:33,730 --> 00:40:38,170 And so the result is a delta function of 1 696 00:40:38,170 --> 00:40:43,960 minus omega over p minus for this tree-level matrix element. 697 00:40:51,060 --> 00:40:53,760 One loop-- now we have to think about how 698 00:40:53,760 --> 00:40:55,980 we're going to regulate the IR. 699 00:40:55,980 --> 00:40:58,030 And I'll do it with an off-shellness. 700 00:41:00,970 --> 00:41:04,270 So I'll introduce a nonzero p class, 701 00:41:04,270 --> 00:41:08,050 and that will be enough to regulate IR divergences. 702 00:41:12,400 --> 00:41:14,010 And we're really after the UV one, 703 00:41:14,010 --> 00:41:16,480 so we just want to separate these guys out. 704 00:41:19,587 --> 00:41:21,045 So there's some different diagrams. 705 00:41:24,617 --> 00:41:27,105 We insert our operator, and we just attach gluons. 706 00:41:27,105 --> 00:41:28,980 So one thing we can do is just string a gluon 707 00:41:28,980 --> 00:41:35,145 across kind of like a standard vertex renormalization diagram. 708 00:41:37,952 --> 00:41:39,160 So there's some loop momenta. 709 00:41:39,160 --> 00:41:40,890 Let me label it on the quark line. 710 00:41:40,890 --> 00:41:44,390 And then the gluon here, which is a collinear gluon, 711 00:41:44,390 --> 00:41:46,500 has momentum p minus l. 712 00:41:46,500 --> 00:41:50,640 And it's forward so it's kind of set up. 713 00:41:53,340 --> 00:41:55,338 There are some numerator to deal with. 714 00:41:55,338 --> 00:41:56,880 And I'm not going to go through that, 715 00:41:56,880 --> 00:42:00,960 but it simplifies to something kind of simple. 716 00:42:00,960 --> 00:42:03,660 After some [INAUDIBLE] algebra, it simplifies down just 717 00:42:03,660 --> 00:42:05,410 to an l perp squared. 718 00:42:05,410 --> 00:42:11,730 For this diagram, there's two l squared propagators, 719 00:42:11,730 --> 00:42:14,675 and there's one l minus p squared propagator. 720 00:42:19,530 --> 00:42:21,330 And then there's a delta function 721 00:42:21,330 --> 00:42:23,503 from the insertion of the operator, 722 00:42:23,503 --> 00:42:25,920 but now the delta function doesn't involve the [INAUDIBLE] 723 00:42:25,920 --> 00:42:27,000 momentum as it did there. 724 00:42:27,000 --> 00:42:29,640 It involves the loop momentum, and that 725 00:42:29,640 --> 00:42:32,740 was kind of the whole point of this example. 726 00:42:32,740 --> 00:42:39,210 So we have a delta function of l minus minus w. 727 00:42:39,210 --> 00:42:40,832 And then there's some dimreg factors, 728 00:42:40,832 --> 00:42:42,540 which we can be careful about if we want. 729 00:42:48,030 --> 00:42:50,850 So in MS bar, we'd have some factor like that. 730 00:42:50,850 --> 00:42:53,100 So this is some loop integral that we just have to do, 731 00:42:53,100 --> 00:42:56,284 and we can do it with kind of standard techniques. 732 00:43:10,820 --> 00:43:15,290 So in my, notes I wrote it as a function of epsilon, 733 00:43:15,290 --> 00:43:17,755 and then epsilon is just regulating the ultraviolet, 734 00:43:17,755 --> 00:43:19,190 and we expand in epsilon. 735 00:43:19,190 --> 00:43:22,238 So let me just write down the result after expanding. 736 00:43:48,134 --> 00:43:50,260 So this is an ultraviolet divergence, 737 00:43:50,260 --> 00:43:54,430 and A here has the infrared regulator-- 738 00:43:54,430 --> 00:43:55,870 p plus, p minus. 739 00:43:55,870 --> 00:43:57,320 And it also has a z and a 1 minus 740 00:43:57,320 --> 00:43:59,890 z, which you can group all together. 741 00:43:59,890 --> 00:44:02,620 And z is just this ratio. 742 00:44:02,620 --> 00:44:06,448 That thing is dependent on a tree-level omega over p minus. 743 00:44:06,448 --> 00:44:07,990 Now, when I'm doing this calculation, 744 00:44:07,990 --> 00:44:10,240 this is a small p, not a big P, because I'm 745 00:44:10,240 --> 00:44:15,830 using quark states, not a proton state. 746 00:44:15,830 --> 00:44:18,648 So really, if I wanted to think about this as an f, 747 00:44:18,648 --> 00:44:20,440 I should say it's an f for the quark state. 748 00:44:20,440 --> 00:44:24,370 But I think that you can remember that. 749 00:44:24,370 --> 00:44:26,207 But the renormalization of the operator 750 00:44:26,207 --> 00:44:27,790 doesn't depend on the state, remember. 751 00:44:27,790 --> 00:44:29,780 We always take the simplest states possible 752 00:44:29,780 --> 00:44:31,720 when we're doing the renormalization 753 00:44:31,720 --> 00:44:32,620 or doing matching. 754 00:44:32,620 --> 00:44:34,550 And so we're free to use quark states, 755 00:44:34,550 --> 00:44:35,675 so that's what we're doing. 756 00:44:38,320 --> 00:44:43,430 OK, that's one diagram. 757 00:44:43,430 --> 00:44:45,165 Now there's another diagram. 758 00:44:45,165 --> 00:44:48,980 I think that should be B. Sometimes in my notes, 759 00:44:48,980 --> 00:44:52,345 I'll call it 1, which doesn't make any sense. 760 00:44:52,345 --> 00:44:54,470 And we can contract the gluon with the Wilson line. 761 00:44:54,470 --> 00:44:58,580 So there's that graph, and there's a symmetric friend. 762 00:45:10,600 --> 00:45:12,880 And each of these actually has two contractions 763 00:45:12,880 --> 00:45:16,930 because there was two Wilson lines in the way we 764 00:45:16,930 --> 00:45:17,800 wrote our operators. 765 00:45:17,800 --> 00:45:21,670 So our operator, as we wrote, is like this. 766 00:45:27,625 --> 00:45:29,000 And you can think-- so let's just 767 00:45:29,000 --> 00:45:30,583 think of a contraction with the quark. 768 00:45:30,583 --> 00:45:32,780 You can think that there's a contraction like that 769 00:45:32,780 --> 00:45:35,960 and there's a contraction like that of a gluon-- 770 00:45:35,960 --> 00:45:38,102 OK, I'm contracting gluons with quarks. 771 00:45:38,102 --> 00:45:39,560 But really, what I mean is that I'm 772 00:45:39,560 --> 00:45:42,810 contracting to the Lagrangian, right, 773 00:45:42,810 --> 00:45:44,270 that this quark is evolving under. 774 00:45:44,270 --> 00:45:47,510 So hopefully that's clear. 775 00:45:47,510 --> 00:45:50,270 All right, so there's two different ways in which-- 776 00:45:50,270 --> 00:45:52,790 when I work out the Feynman rule for this thing where 777 00:45:52,790 --> 00:45:55,680 I attach the gluon, you can either 778 00:45:55,680 --> 00:45:57,680 get the gluon from here or the gluon from there. 779 00:45:57,680 --> 00:46:00,820 That's all I'm saying. 780 00:46:00,820 --> 00:46:02,570 But these actually have different physical 781 00:46:02,570 --> 00:46:06,020 interpretations because this delta function 782 00:46:06,020 --> 00:46:08,360 here, if you think about what it's doing, 783 00:46:08,360 --> 00:46:12,410 it's really-- in the original diagram, it's like the cut. 784 00:46:12,410 --> 00:46:15,800 So in the original diagrams that we were drawing, 785 00:46:15,800 --> 00:46:19,040 we would cut them because we'd take the imaginary part. 786 00:46:19,040 --> 00:46:21,030 And this delta function is in the middle. 787 00:46:21,030 --> 00:46:22,730 We have kind of a parton on this side 788 00:46:22,730 --> 00:46:25,160 and a squared parton on that side. 789 00:46:25,160 --> 00:46:27,150 This delta function is the cut. 790 00:46:27,150 --> 00:46:28,790 So this contraction here actually 791 00:46:28,790 --> 00:46:34,580 corresponds to a virtual graph, and this guy here 792 00:46:34,580 --> 00:46:37,670 corresponds to real emission because you're 793 00:46:37,670 --> 00:46:39,920 doing a contraction across the cut, right? 794 00:46:39,920 --> 00:46:41,990 So one of these guys would be a graph like this, 795 00:46:41,990 --> 00:46:43,865 and the other one would be a graph like that. 796 00:46:46,940 --> 00:46:50,636 I can label them 1 and 2-- 797 00:46:50,636 --> 00:46:52,010 1, 2. 798 00:46:55,706 --> 00:46:59,440 But we'll just keep them and treat them all together. 799 00:46:59,440 --> 00:47:03,124 These two graphs give an overall factor of 2. 800 00:47:03,124 --> 00:47:04,600 So that's simple. 801 00:47:08,854 --> 00:47:16,295 There's some spinner stuff, which 802 00:47:16,295 --> 00:47:18,640 is even simpler in this case, so I write it out. 803 00:47:23,510 --> 00:47:28,680 There are some stuff from the Wilson line. 804 00:47:28,680 --> 00:47:31,470 And then there's two propagators. 805 00:47:31,470 --> 00:47:34,720 Let me not write all the i0's. 806 00:47:34,720 --> 00:47:37,890 And then there's two different delta functions. 807 00:47:37,890 --> 00:47:43,120 So either we have the real graph where the w is inside, 808 00:47:43,120 --> 00:47:49,090 or we have the virtual graph where the w-- 809 00:47:49,090 --> 00:47:50,350 sorry. 810 00:47:50,350 --> 00:47:52,270 Either we have the real graph where 811 00:47:52,270 --> 00:47:55,270 the loop goes around the delta function, 812 00:47:55,270 --> 00:47:57,580 or we have the virtual graph where this guy is overall 813 00:47:57,580 --> 00:47:58,540 on that thing. 814 00:47:58,540 --> 00:48:01,870 So in the overall one, it's just a p minus minus w 815 00:48:01,870 --> 00:48:03,640 like it was at tree level. 816 00:48:03,640 --> 00:48:08,180 And in the real emission, it's an l minus minus w. 817 00:48:08,180 --> 00:48:09,380 And one's a w. 818 00:48:09,380 --> 00:48:10,250 One's a w dagger. 819 00:48:10,250 --> 00:48:11,480 So there's a relative sign. 820 00:48:14,850 --> 00:48:17,245 So the sign is just easier to understand 821 00:48:17,245 --> 00:48:22,120 as w versus w dagger, which has a relative sign. 822 00:48:22,120 --> 00:48:23,800 OK, so if we just followed our nose 823 00:48:23,800 --> 00:48:26,358 with what the Feynman rule for this thing is, 824 00:48:26,358 --> 00:48:27,400 that's what we would get. 825 00:48:34,310 --> 00:48:36,560 And this is, again, some loop integral that we can do. 826 00:48:50,700 --> 00:48:54,410 One way of writing the result is as follows. 827 00:48:54,410 --> 00:48:59,482 And there's one thing we have to be careful about here which is 828 00:48:59,482 --> 00:49:00,690 why I'm writing this all out. 829 00:49:11,680 --> 00:49:13,120 So there's actually a cancellation 830 00:49:13,120 --> 00:49:15,400 between the virtual and the real diagrams 831 00:49:15,400 --> 00:49:19,470 of an infrared divergence, so I want to be careful about that. 832 00:49:30,982 --> 00:49:36,260 So that's why I'm writing this guy out in epsilon dimensions 833 00:49:36,260 --> 00:49:38,660 fully without expanding first. 834 00:49:38,660 --> 00:49:41,720 OK, so this is the real contribution, 835 00:49:41,720 --> 00:49:44,120 and this is the virtual. 836 00:49:44,120 --> 00:49:47,465 So in order to sort of deal with this, 837 00:49:47,465 --> 00:49:50,090 we have to make use of something that's called the distribution 838 00:49:50,090 --> 00:49:51,040 identity. 839 00:49:57,630 --> 00:50:00,177 If you know what the result is for the anomalous dimension, 840 00:50:00,177 --> 00:50:02,010 you'll be aware of the fact that it involves 841 00:50:02,010 --> 00:50:04,977 something called a plus function because splitting 842 00:50:04,977 --> 00:50:06,435 functions for a parton distribution 843 00:50:06,435 --> 00:50:08,185 involves something called a plus function. 844 00:50:30,960 --> 00:50:34,265 So the way that we can deal with that is as follows. 845 00:50:38,405 --> 00:50:39,780 The way we can deal with the fact 846 00:50:39,780 --> 00:50:42,030 that actually the result's going to be a distribution, 847 00:50:42,030 --> 00:50:46,830 we have to be careful because you see, z goes to 1 848 00:50:46,830 --> 00:50:48,360 is being regulated by epsilon. 849 00:50:51,030 --> 00:50:53,490 And so if we integrate over z, for example, 850 00:50:53,490 --> 00:50:55,230 it's epsilon that's going to allow us 851 00:50:55,230 --> 00:50:58,620 to integrate all the way to 1. 852 00:50:58,620 --> 00:51:02,115 And we'd like to encode that in some way 853 00:51:02,115 --> 00:51:03,990 where we can expand in epsilon because that's 854 00:51:03,990 --> 00:51:05,840 what we need to do in order to extract 855 00:51:05,840 --> 00:51:06,840 the anomalous dimension. 856 00:51:06,840 --> 00:51:08,757 And this formula is what allows us to do that. 857 00:51:12,640 --> 00:51:15,070 So I'll tell you how to derive it 858 00:51:15,070 --> 00:51:18,010 after I tell you what the l is. 859 00:51:18,010 --> 00:51:22,780 So ln of anything is defined to be a plus function 860 00:51:22,780 --> 00:51:25,180 with a log to that power. 861 00:51:31,660 --> 00:51:33,400 And the plus function is defined so 862 00:51:33,400 --> 00:51:38,830 that if you integrate from 0 to 1, you get 0. 863 00:51:38,830 --> 00:51:44,557 And if you integrate with a test function, 864 00:51:44,557 --> 00:51:48,700 which is the more general result that you need to define it-- 865 00:51:48,700 --> 00:51:52,750 so you can define it by this result with a test function. 866 00:51:52,750 --> 00:51:55,850 And it just gives you the normal function, 867 00:51:55,850 --> 00:51:58,510 but the test function with a subtraction that 868 00:51:58,510 --> 00:52:00,190 makes the test function more convergent 869 00:52:00,190 --> 00:52:03,700 so that you can integrate through 0. 870 00:52:03,700 --> 00:52:05,920 OK, so that's the definition of a plus function. 871 00:52:05,920 --> 00:52:08,860 You could also define it with a limit. 872 00:52:08,860 --> 00:52:12,610 This will be sufficient. 873 00:52:12,610 --> 00:52:15,970 OK, so these things are like delta functions. 874 00:52:15,970 --> 00:52:17,950 The way that you would derive this formula 875 00:52:17,950 --> 00:52:21,220 is you would say, well, if z is away from 1, 876 00:52:21,220 --> 00:52:24,400 then I can expand because then there's no problem. 877 00:52:24,400 --> 00:52:26,470 And if z is away from 1, it turns out 878 00:52:26,470 --> 00:52:28,930 that this plus function is just the regular function. 879 00:52:28,930 --> 00:52:32,210 It's only at 1 that something special is happening. 880 00:52:32,210 --> 00:52:35,390 And so the standard expansion is what you'd 881 00:52:35,390 --> 00:52:37,930 get if you took z away from 1. 882 00:52:37,930 --> 00:52:39,850 And to see what's happening at z equals 1, 883 00:52:39,850 --> 00:52:41,785 you'd just integrate both sides from 0 to 1, 884 00:52:41,785 --> 00:52:43,660 and that's how you can derive the coefficient 885 00:52:43,660 --> 00:52:44,577 of the delta function. 886 00:52:49,320 --> 00:52:54,630 All right, so if I plug this formula in here for this thing, 887 00:52:54,630 --> 00:52:57,632 then I actually get another 1 over epsilon in this guy. 888 00:52:57,632 --> 00:53:00,090 There's a gamma of epsilon out front, and that guy is good. 889 00:53:00,090 --> 00:53:01,350 This is our UV divergence. 890 00:53:01,350 --> 00:53:04,650 This is our 1 over epsilon UV. 891 00:53:04,650 --> 00:53:06,570 But there's also a gamma of minus epsilon 892 00:53:06,570 --> 00:53:09,160 here, which is an IR divergence. 893 00:53:09,160 --> 00:53:11,310 So even though I tried to regulate all the IR 894 00:53:11,310 --> 00:53:13,810 by off-shellness, it didn't quite work 895 00:53:13,810 --> 00:53:16,380 and there was one that was regulated by dimreg. 896 00:53:16,380 --> 00:53:19,890 And that one actually cancels between these two pieces 897 00:53:19,890 --> 00:53:25,620 once I use this identity and take into account 898 00:53:25,620 --> 00:53:29,070 that that's an IR divergence. 899 00:53:29,070 --> 00:53:36,360 So there's a 1 over epsilon IR times 1 over epsilon UV. 900 00:53:36,360 --> 00:53:40,260 And that cancels between the real and virtual graphs. 901 00:53:45,710 --> 00:53:48,450 So this is like a standard 1 over epsilon IR 902 00:53:48,450 --> 00:53:50,200 canceling between real and virtual graphs. 903 00:53:50,200 --> 00:53:51,790 And since it's only the 1 over epsilon UV 904 00:53:51,790 --> 00:53:53,920 that we're interested in, we're really only worried 905 00:53:53,920 --> 00:53:56,080 about that part of it canceling. 906 00:53:56,080 --> 00:53:58,415 There's a piece actually that-- 907 00:53:58,415 --> 00:53:58,915 anyway. 908 00:54:06,640 --> 00:54:10,930 And then the 1 over epsilon that's left 909 00:54:10,930 --> 00:54:17,290 is the guy that we're after in order 910 00:54:17,290 --> 00:54:18,932 to get the anomalous dimension. 911 00:54:24,600 --> 00:54:28,950 All right, so let me not-- 912 00:54:28,950 --> 00:54:30,600 so in my notes, I write one more line 913 00:54:30,600 --> 00:54:32,310 where I expand this guy out. 914 00:54:32,310 --> 00:54:36,630 And I think just because of the time, I'm going to skip that. 915 00:54:36,630 --> 00:54:38,670 And I'll just write the final result. 916 00:54:38,670 --> 00:54:40,710 When we do the final result, we also 917 00:54:40,710 --> 00:54:43,770 have to include wave function renormalization. 918 00:54:43,770 --> 00:54:49,380 So you can think of this graph as a wave function 919 00:54:49,380 --> 00:54:51,990 renormalization term. 920 00:54:51,990 --> 00:54:53,937 And it just involves the delta function again 921 00:54:53,937 --> 00:54:55,020 like the tree-level graph. 922 00:55:16,533 --> 00:55:18,080 [INAUDIBLE] like that. 923 00:55:18,080 --> 00:55:21,830 So in general, if I wanted to do this calculation at one loop, 924 00:55:21,830 --> 00:55:25,820 there's one more type of diagram I should consider, OK? 925 00:55:25,820 --> 00:55:28,840 And that's a graph where I could have mixing. 926 00:55:28,840 --> 00:55:33,890 This guy should be dashed since we're in the effective theory. 927 00:55:33,890 --> 00:55:36,350 So how does the mixing graph work? 928 00:55:36,350 --> 00:55:38,720 Well, there's a graph where I have external gluons, 929 00:55:38,720 --> 00:55:42,236 but I still am renormalizing the same operator. 930 00:55:42,236 --> 00:55:45,800 I've still inserted the quark operator here, 931 00:55:45,800 --> 00:55:49,130 but now we have antiquarks in this theory. 932 00:55:49,130 --> 00:55:50,900 We can draw a triangle like that. 933 00:55:50,900 --> 00:55:54,537 And this graph here would give a mixing that involves-- 934 00:55:54,537 --> 00:55:56,870 that would give a mixing term in the anomalous dimension 935 00:55:56,870 --> 00:56:00,250 where you're mixing gluons and quarks. 936 00:56:00,250 --> 00:56:07,550 So this mix is what we sort of called O glue. 937 00:56:07,550 --> 00:56:11,830 Let me just say this, that it mixes O glue with O quark. 938 00:56:11,830 --> 00:56:13,760 And we could compute this graph too, 939 00:56:13,760 --> 00:56:17,680 but I'm going to neglect it just for simplicity. 940 00:56:17,680 --> 00:56:20,320 I just won't write it down. 941 00:56:20,320 --> 00:56:23,650 One way of doing that rigorously would 942 00:56:23,650 --> 00:56:26,770 be to consider operators where the flavors of these guys 943 00:56:26,770 --> 00:56:28,660 are different, OK? 944 00:56:28,660 --> 00:56:31,630 That's what would happen, for example, if you were having 945 00:56:31,630 --> 00:56:32,830 a w exchange or something. 946 00:56:32,830 --> 00:56:36,220 So we could look at nonflavored diagonal operators 947 00:56:36,220 --> 00:56:39,043 with, like, a u quark and a d quark. 948 00:56:39,043 --> 00:56:41,210 And then you would not have this mixing with O glue. 949 00:56:41,210 --> 00:56:42,793 It's only if the flavors of the quarks 950 00:56:42,793 --> 00:56:45,800 are the same that you can write down this diagram. 951 00:56:45,800 --> 00:56:50,718 But just think about it as I'm focusing on the quark piece, 952 00:56:50,718 --> 00:56:52,510 and in general, there's also a gluon piece. 953 00:57:04,360 --> 00:57:05,830 So we have all our one-loop graphs. 954 00:57:05,830 --> 00:57:07,480 We know how to expand them in epsilon. 955 00:57:07,480 --> 00:57:09,953 And so we just proceed, expand them in epsilon, 956 00:57:09,953 --> 00:57:10,620 and add them up. 957 00:57:28,340 --> 00:57:30,400 So you could think that what we derive 958 00:57:30,400 --> 00:57:35,680 by doing that is a distribution for a quark inside a quark. 959 00:57:35,680 --> 00:57:39,010 So here, I'm being-- this is the state 960 00:57:39,010 --> 00:57:41,250 and this is what type of operator. 961 00:57:47,360 --> 00:57:51,880 And it's a function of some z. 962 00:57:51,880 --> 00:57:57,690 And if I go up to one loop, then the tree level 963 00:57:57,690 --> 00:58:02,920 was just a delta function of that fraction z. 964 00:58:02,920 --> 00:58:05,290 And then at one loop, we had all these other terms. 965 00:58:12,880 --> 00:58:19,810 So if I collect all the pieces, I had some delta functions. 966 00:58:19,810 --> 00:58:24,070 The graph with the Wilson lines actually gives me 967 00:58:24,070 --> 00:58:26,810 one of these L0 functions. 968 00:58:26,810 --> 00:58:27,950 And then the graph-- 969 00:58:27,950 --> 00:58:29,860 so there's wave function normalization 970 00:58:29,860 --> 00:58:33,727 plus some other terms that involve delta function. 971 00:58:33,727 --> 00:58:35,185 And then there's some other pieces. 972 00:58:39,820 --> 00:58:42,880 And then this is all times 1 over epsilon. 973 00:58:42,880 --> 00:58:44,668 And then there's other pieces. 974 00:58:44,668 --> 00:58:46,960 But if we're interested in ultraviolet renormalization, 975 00:58:46,960 --> 00:58:50,050 we only care about the 1 over epsilon. 976 00:58:50,050 --> 00:58:55,390 And all those terms can be written 977 00:58:55,390 --> 00:58:57,070 in a kind of more compact form, which 978 00:58:57,070 --> 00:59:09,130 is the more standard form for the anomalous dimension. 979 00:59:09,130 --> 00:59:12,940 You can actually group them all together into a single plus 980 00:59:12,940 --> 00:59:17,030 function like this. 981 00:59:17,030 --> 00:59:19,900 So just in terms of distributions, 982 00:59:19,900 --> 00:59:22,840 this distribution is equal to the sum of these pieces. 983 00:59:35,190 --> 00:59:37,380 You can see, as z goes to 1, that there'd be a 2 984 00:59:37,380 --> 00:59:38,340 here and a 2 here. 985 00:59:38,340 --> 00:59:40,110 And this would be 1 over 1 minus z. 986 00:59:40,110 --> 00:59:42,433 And as z goes to 1 here, that would be 1 987 00:59:42,433 --> 00:59:43,975 and this would be a 1 over 1 minus z. 988 00:59:43,975 --> 00:59:48,110 So you see some pieces of it matching up. 989 00:59:50,890 --> 00:59:53,710 Basically, the way that you would derive this is you'd 990 00:59:53,710 --> 01:00:00,550 write 1 plus z squared is a plus b, 1 minus z 991 01:00:00,550 --> 01:00:02,800 plus c, 1 minus z squared. 992 01:00:02,800 --> 01:00:04,760 You'd work out what a, b, and c are, 993 01:00:04,760 --> 01:00:07,070 just relating two polynomials. 994 01:00:07,070 --> 01:00:11,482 And then this guy here, the 1 minus z in the numerator 995 01:00:11,482 --> 01:00:12,940 cancels the one in the denominator, 996 01:00:12,940 --> 01:00:14,482 and it's not a plus function anymore. 997 01:00:14,482 --> 01:00:15,810 It's just a number. 998 01:00:15,810 --> 01:00:19,790 And that's how you would connect the two formulas. 999 01:00:19,790 --> 01:00:23,110 All right, so we were after determining the z. 1000 01:00:23,110 --> 01:00:27,170 The z has to cancel this 1 over epsilon. 1001 01:00:27,170 --> 01:00:30,550 So let's go back to our formula which connected 1002 01:00:30,550 --> 01:00:34,960 those, which was this. 1003 01:00:34,960 --> 01:00:39,370 Our general formula was that the bare guy could 1004 01:00:39,370 --> 01:00:48,370 be written in terms of split into UV pieces 1005 01:00:48,370 --> 01:00:50,320 and finite pieces in the following ways, 1006 01:00:50,320 --> 01:00:53,600 is with this integral. 1007 01:00:53,600 --> 01:00:56,800 Now, this looks like it could be an arbitrary function of xi 1008 01:00:56,800 --> 01:00:59,860 and xi prime, but our result here was only a function of z, 1009 01:00:59,860 --> 01:01:02,000 which is actually a ratio. 1010 01:01:02,000 --> 01:01:05,350 And that's actually something that we can argue in general, 1011 01:01:05,350 --> 01:01:08,710 that this thing here is actually only a function 1012 01:01:08,710 --> 01:01:10,750 of one variable, not two. 1013 01:01:23,330 --> 01:01:25,190 So that follows from two different things. 1014 01:01:25,190 --> 01:01:27,440 It follows from RPI III invariance. 1015 01:01:30,170 --> 01:01:32,060 So remember that RPI III invariance 1016 01:01:32,060 --> 01:01:35,300 said that you should have the same number of n's and n bars. 1017 01:01:35,300 --> 01:01:40,220 And remember-- OK, so that's one thing that you have to use. 1018 01:01:40,220 --> 01:01:42,080 That tells you that you need to get ratios. 1019 01:01:42,080 --> 01:01:43,690 Well, the z's are already ratios. 1020 01:01:43,690 --> 01:01:48,290 So you might say, well, that should be fine. 1021 01:01:48,290 --> 01:01:52,525 The z's are already ratios between the momentum 1022 01:01:52,525 --> 01:01:53,900 and the operator and the momentum 1023 01:01:53,900 --> 01:01:55,942 and the state, the minus momentum of the operator 1024 01:01:55,942 --> 01:02:01,970 over the minus momentum of the state, right? 1025 01:02:01,970 --> 01:02:03,230 And this is a minus momentum. 1026 01:02:03,230 --> 01:02:04,320 That's a minus momentum. 1027 01:02:04,320 --> 01:02:06,425 So the z's are RPI III invariant. 1028 01:02:06,425 --> 01:02:08,948 So that doesn't seem like it would imply this. 1029 01:02:08,948 --> 01:02:10,490 But there's one other thing you know, 1030 01:02:10,490 --> 01:02:13,790 and that is that it can't depend on the state momentum. 1031 01:02:13,790 --> 01:02:15,500 I could have taken a proton. 1032 01:02:15,500 --> 01:02:17,180 I could have taken a quark. 1033 01:02:17,180 --> 01:02:18,800 And the result for the renormalization 1034 01:02:18,800 --> 01:02:22,580 shouldn't depend on what state I'm taking. 1035 01:02:22,580 --> 01:02:24,110 And this combination where I have 1036 01:02:24,110 --> 01:02:28,220 d xi prime xi prime with a xi over xi prime, the p minuses 1037 01:02:28,220 --> 01:02:28,970 cancel out. 1038 01:02:48,410 --> 01:02:50,960 So if I were to do the whole thing with a proton 1039 01:02:50,960 --> 01:02:53,393 state rather than a quark state, then I 1040 01:02:53,393 --> 01:02:55,310 should still get the same anomalous dimension. 1041 01:02:55,310 --> 01:02:58,100 And in order for that to be true, 1042 01:02:58,100 --> 01:03:00,600 it has to depend on the ratio. 1043 01:03:00,600 --> 01:03:07,050 And that ratio is then just a ratio of the bare operator 1044 01:03:07,050 --> 01:03:09,340 and the renormalized operator. 1045 01:03:09,340 --> 01:03:12,960 It's like saying, if you had O of omega, 1046 01:03:12,960 --> 01:03:17,610 there is a convolution of z with an omega 1047 01:03:17,610 --> 01:03:24,530 over omega prime or something, with O omega 1048 01:03:24,530 --> 01:03:27,140 prime renormalized. 1049 01:03:27,140 --> 01:03:29,540 And if I had done it in an operator level and not 1050 01:03:29,540 --> 01:03:32,840 even written states, then it would really just 1051 01:03:32,840 --> 01:03:35,487 be RPI III invariance, OK? 1052 01:03:35,487 --> 01:03:37,070 Because I wrote it in terms of states, 1053 01:03:37,070 --> 01:03:38,737 there was this other momentum available, 1054 01:03:38,737 --> 01:03:42,890 but I'm not allowed to have that really 1055 01:03:42,890 --> 01:03:45,400 be playing a part of the discussion. 1056 01:03:48,880 --> 01:03:52,550 So given that formula, then I can expand to one loop. 1057 01:03:52,550 --> 01:03:55,450 So this guy I think of as having a tree-level result. 1058 01:03:55,450 --> 01:03:57,220 This guy is a matrix element that 1059 01:03:57,220 --> 01:04:00,500 has a tree-level and one-loop result as well. 1060 01:04:00,500 --> 01:04:05,380 So if they're both tree level, I get delta, 1 minus z. 1061 01:04:10,730 --> 01:04:38,620 And in some kind of obvious notation, up to one-loop order, 1062 01:04:38,620 --> 01:04:41,558 I can write it out formally like that. 1063 01:04:41,558 --> 01:04:43,600 And then I know what these tree-level things are. 1064 01:04:43,600 --> 01:04:45,392 This guy's a delta function, and this guy's 1065 01:04:45,392 --> 01:04:46,670 also a delta function. 1066 01:04:46,670 --> 01:04:47,920 So I can just do the integral. 1067 01:05:00,570 --> 01:05:03,180 And it really is pretty simple. 1068 01:05:03,180 --> 01:05:04,950 All the 1 over epsilon terms are just z. 1069 01:05:08,090 --> 01:05:13,710 And what's left would be associated to this guy 1070 01:05:13,710 --> 01:05:15,392 in perturbation theory. 1071 01:05:15,392 --> 01:05:17,100 But if we want to do the renormalization, 1072 01:05:17,100 --> 01:05:21,600 we just need the z and not worry about that. 1073 01:05:21,600 --> 01:05:23,580 So we read off from over here what 1074 01:05:23,580 --> 01:05:28,120 z is because z is just this right there. 1075 01:05:36,130 --> 01:05:42,050 So z-- OK, z is just this thing. 1076 01:05:47,878 --> 01:05:49,670 So when I put the tree-level piece together 1077 01:05:49,670 --> 01:05:51,795 with the one-loop piece, then this thing is just z. 1078 01:05:54,290 --> 01:05:56,240 And then I compute the anomalous dimension 1079 01:05:56,240 --> 01:05:57,890 by taking mu d by d mu of it-- 1080 01:06:00,960 --> 01:06:09,050 and that hits the alpha, so that kills the epsilon 1081 01:06:09,050 --> 01:06:12,220 and gives me a factor of 2 and a minus sign. 1082 01:06:12,220 --> 01:06:15,665 But the anomalous dimension, gamma qq-- 1083 01:06:20,030 --> 01:06:23,800 so there was a 1 over xi prime. 1084 01:06:23,800 --> 01:06:25,925 And then it was minus mu d by d mu. 1085 01:06:31,060 --> 01:06:33,550 And if I plug it in the formula that we have, 1086 01:06:33,550 --> 01:06:38,120 zqq of [? xi over ?] [? xi ?] prime. 1087 01:06:38,120 --> 01:06:39,460 So there's an a minus here. 1088 01:06:39,460 --> 01:06:41,650 There's a minus there. 1089 01:06:41,650 --> 01:06:44,290 And the 2 epsilon cancels this 2 and that epsilon. 1090 01:06:46,778 --> 01:06:48,320 And this 1 over xi prime is the thing 1091 01:06:48,320 --> 01:06:50,605 that we needed to make the measure RPI invariant. 1092 01:06:53,220 --> 01:07:00,150 So in this notation, our original notation, 1093 01:07:00,150 --> 01:07:03,510 putting all the pieces together and being careful 1094 01:07:03,510 --> 01:07:17,260 about beta functions, which I was mostly suppressing, that's 1095 01:07:17,260 --> 01:07:21,010 the result. OK, so this is the function of xi 1096 01:07:21,010 --> 01:07:23,440 over xi prime, which I've just written as z. 1097 01:07:23,440 --> 01:07:26,950 And then there's some beta functions 1098 01:07:26,950 --> 01:07:29,020 that are setting the boundaries for the integral. 1099 01:07:29,020 --> 01:07:31,400 And that comes also out of the calculation. 1100 01:07:31,400 --> 01:07:33,490 And that's the quark one-loop splitting function. 1101 01:07:36,680 --> 01:07:39,070 So if we've done the gluon from that other diagram, 1102 01:07:39,070 --> 01:07:40,510 that we've got the mixing term. 1103 01:07:48,642 --> 01:07:50,600 OK, so this is the one-loop anomalous dimension 1104 01:07:50,600 --> 01:07:51,975 for the PDF, and it's really just 1105 01:07:51,975 --> 01:07:56,010 doing operator renormalization, calculating one-loop diagrams 1106 01:07:56,010 --> 01:07:57,010 in the effective theory. 1107 01:08:00,180 --> 01:08:00,810 Questions? 1108 01:08:09,035 --> 01:08:09,535 OK. 1109 01:08:12,090 --> 01:08:15,660 So one question that you can ask, 1110 01:08:15,660 --> 01:08:17,609 which is an interesting question, 1111 01:08:17,609 --> 01:08:22,649 is when we did this result for the DIS, 1112 01:08:22,649 --> 01:08:25,050 we got this convolution between the hard function 1113 01:08:25,050 --> 01:08:27,359 and the Parton distribution function. 1114 01:08:27,359 --> 01:08:29,250 And you can ask, why did that happen 1115 01:08:29,250 --> 01:08:31,470 and, in general, is there a way of characterizing 1116 01:08:31,470 --> 01:08:32,760 when it could possibly happen? 1117 01:08:36,547 --> 01:08:38,630 Because if you think about the answer that we got, 1118 01:08:38,630 --> 01:08:41,250 it was just Wilson coefficient times operator. 1119 01:08:41,250 --> 01:08:43,191 And the really only nontrivial thing about it 1120 01:08:43,191 --> 01:08:44,899 was that there was this one momentum that 1121 01:08:44,899 --> 01:08:46,899 could kind of trade back and forth between them. 1122 01:08:46,899 --> 01:08:50,450 There was an integral in the answer. 1123 01:08:50,450 --> 01:08:52,069 And actually, power counting even 1124 01:08:52,069 --> 01:08:57,270 constrains how those integrals can, in general, show up. 1125 01:08:57,270 --> 01:09:02,040 So if you ask most generally what could possibly happen-- 1126 01:09:02,040 --> 01:09:04,160 and just thinking about the power counting 1127 01:09:04,160 --> 01:09:06,020 for the degrees of freedom actually 1128 01:09:06,020 --> 01:09:08,180 tells us what type of integrals can show up 1129 01:09:08,180 --> 01:09:09,470 in factorization theorems. 1130 01:09:33,396 --> 01:09:34,979 This is constrained by power counting. 1131 01:09:42,920 --> 01:09:45,170 I keep forgetting to say that there's a makeup lecture 1132 01:09:45,170 --> 01:09:45,560 tomorrow. 1133 01:09:45,560 --> 01:09:47,359 I sent around an email, but I should have-- 1134 01:09:52,569 --> 01:09:54,279 tomorrow, this room at 10:00 AM. 1135 01:10:00,580 --> 01:10:02,890 And the lecture next week is canceled on Tuesday. 1136 01:10:02,890 --> 01:10:05,740 That's why we have a makeup lecture tomorrow. 1137 01:10:05,740 --> 01:10:11,510 OK, so in what way is it constrained by power counting? 1138 01:10:11,510 --> 01:10:14,420 So if you think about the degrees of freedom that we had, 1139 01:10:14,420 --> 01:10:20,942 say, for SCET I, then we had hard, collinear, and soft-- 1140 01:10:20,942 --> 01:10:23,080 so let's just take a simple case with only one 1141 01:10:23,080 --> 01:10:25,180 type of collinear-- 1142 01:10:25,180 --> 01:10:27,970 hard, collinear, and ultrasoft. 1143 01:10:27,970 --> 01:10:32,260 And the p mu of these guys in terms of plus, minus, and perp 1144 01:10:32,260 --> 01:10:33,422 components-- 1145 01:10:36,254 --> 01:10:38,552 I should be more fancy about this. 1146 01:10:42,172 --> 01:10:43,160 [INAUDIBLE] 1147 01:10:46,130 --> 01:10:48,620 So if you think about just power counting for the momentum, 1148 01:10:48,620 --> 01:10:49,385 it was as follows. 1149 01:10:52,850 --> 01:10:56,810 Factorization was separating these different things 1150 01:10:56,810 --> 01:10:58,100 into different objects. 1151 01:10:58,100 --> 01:11:00,050 We had a Wilson coefficient for the hard. 1152 01:11:00,050 --> 01:11:02,600 In the case we just did, we only had a proton matrix element. 1153 01:11:02,600 --> 01:11:04,642 For the collinear, we didn't have any ultrasofts. 1154 01:11:04,642 --> 01:11:07,100 If we put the ultrasofts in, they would have all cancelled 1155 01:11:07,100 --> 01:11:07,730 away. 1156 01:11:07,730 --> 01:11:10,250 We wouldn't have seen any ultrasofts showing up. 1157 01:11:10,250 --> 01:11:12,620 And that's because the operator we were dealing with, 1158 01:11:12,620 --> 01:11:15,530 the Wilson lines would just have cancelled completely out of it. 1159 01:11:15,530 --> 01:11:17,570 But actually, it turns out, for deep inelastic scattering, 1160 01:11:17,570 --> 01:11:19,362 that you shouldn't even include ultrasofts. 1161 01:11:19,362 --> 01:11:22,140 They're not a good degree of freedom to include there. 1162 01:11:22,140 --> 01:11:24,410 So really, for the process that we were talking about, 1163 01:11:24,410 --> 01:11:27,830 you really should only take those two. 1164 01:11:27,830 --> 01:11:31,430 But anyway, more generally in some other process, 1165 01:11:31,430 --> 01:11:33,590 you would have these three different things. 1166 01:11:33,590 --> 01:11:35,360 And the way that convolutions can show up 1167 01:11:35,360 --> 01:11:39,080 is simply who can trade momentum with who. 1168 01:11:39,080 --> 01:11:43,940 So this is plus momenta, minus momenta, and perp momentum. 1169 01:11:43,940 --> 01:11:46,230 And in order for momentum to be exchanged, 1170 01:11:46,230 --> 01:11:48,160 they have to be of the same size. 1171 01:11:48,160 --> 01:11:54,020 So these guys here are the same size and they can be exchanged, 1172 01:11:54,020 --> 01:11:56,910 and that's exactly what showed up in our DIS factorization 1173 01:11:56,910 --> 01:11:57,410 theorem. 1174 01:11:57,410 --> 01:12:00,617 The hard Wilson coefficients exchanged minus momentum 1175 01:12:00,617 --> 01:12:03,200 with the collinear [INAUDIBLE] because they are the same order 1176 01:12:03,200 --> 01:12:04,940 in the power counting. 1177 01:12:04,940 --> 01:12:07,213 In another case, in a more general 1178 01:12:07,213 --> 01:12:08,630 or in some other example, we might 1179 01:12:08,630 --> 01:12:13,222 find that there was nontrivial ultrasoft stuff, 1180 01:12:13,222 --> 01:12:14,680 and then we could get a convolution 1181 01:12:14,680 --> 01:12:17,100 in the plus momentum and collinear and ultrasoft 1182 01:12:17,100 --> 01:12:18,350 because they're the same size. 1183 01:12:22,240 --> 01:12:24,910 And so that's a pretty simple way 1184 01:12:24,910 --> 01:12:27,400 of thinking about why those integrals can possibly show up. 1185 01:12:27,400 --> 01:12:29,680 It's just because the two sectors can talk to each other 1186 01:12:29,680 --> 01:12:31,722 because they have momenta that are the same size. 1187 01:12:31,722 --> 01:12:34,300 And then the rest is about momentum conservation 1188 01:12:34,300 --> 01:12:37,660 because momentum conservation places nontrivial constraints. 1189 01:12:37,660 --> 01:12:39,340 And we saw in the DIS example that there 1190 01:12:39,340 --> 01:12:41,080 were two omegas to start, but one of them 1191 01:12:41,080 --> 01:12:43,497 was projected to 0 because it was a forward matrix element 1192 01:12:43,497 --> 01:12:47,990 and we only had one integral. 1193 01:12:47,990 --> 01:12:51,710 And that also has analogs elsewhere. 1194 01:12:51,710 --> 01:12:56,240 If we do SCET II, which we haven't talked about yet-- 1195 01:12:56,240 --> 01:12:59,330 we did talk about what the degrees of freedom were. 1196 01:12:59,330 --> 01:13:05,210 And again, if I try to make it completely 1197 01:13:05,210 --> 01:13:08,487 generic for some examples that we'll treat later, 1198 01:13:08,487 --> 01:13:10,070 then I can write down something that's 1199 01:13:10,070 --> 01:13:15,600 a slightly extended version of what we talked about so far. 1200 01:13:15,600 --> 01:13:19,850 So I can have Q, Q, Q again for the hard. 1201 01:13:19,850 --> 01:13:22,050 And then I can have my collinear, 1202 01:13:22,050 --> 01:13:28,580 which is Q lambda squared, Q, Q lambda, and then soft, which is 1203 01:13:28,580 --> 01:13:32,943 Q lambda, Q lambda, Q lambda. 1204 01:13:32,943 --> 01:13:34,610 And it turns out that sometimes, there's 1205 01:13:34,610 --> 01:13:38,600 also another mode which we haven't talked about, 1206 01:13:38,600 --> 01:13:46,395 but I'll include it for completeness, which 1207 01:13:46,395 --> 01:13:48,430 is kind of a collinear mode that's 1208 01:13:48,430 --> 01:13:50,440 in between the low-energy collinear 1209 01:13:50,440 --> 01:13:52,896 mode and the high-energy collinear mode. 1210 01:13:52,896 --> 01:13:55,030 AUDIENCE: Do you mean the square root of lambda? 1211 01:13:55,030 --> 01:13:58,570 IAIN STEWART: Yeah, square root of lambda, sorry. 1212 01:13:58,570 --> 01:14:01,375 Yeah, otherwise my dimensions are wrong. 1213 01:14:05,830 --> 01:14:07,030 Yeah. 1214 01:14:07,030 --> 01:14:09,160 So again here, you can just-- 1215 01:14:09,160 --> 01:14:11,998 I mean, the reason I was extending this is just to, 1216 01:14:11,998 --> 01:14:14,540 again, argue that it's kind of simple to see what can happen. 1217 01:14:14,540 --> 01:14:17,593 So in general, when you think about convolutions 1218 01:14:17,593 --> 01:14:19,510 in the hard momentum, it could be in this case 1219 01:14:19,510 --> 01:14:20,950 between these three modes. 1220 01:14:20,950 --> 01:14:22,600 There could be some integrals. 1221 01:14:22,600 --> 01:14:25,810 And then look where else there can be something. 1222 01:14:25,810 --> 01:14:29,035 So this and this are the same size. 1223 01:14:31,610 --> 01:14:35,090 And this and this are the same size. 1224 01:14:35,090 --> 01:14:41,740 So in general, we can have convolutions 1225 01:14:41,740 --> 01:14:44,830 in general between all these things, 1226 01:14:44,830 --> 01:14:47,170 between the guys that are the same size. 1227 01:14:47,170 --> 01:14:49,420 But that's the most complicated thing that can happen. 1228 01:14:49,420 --> 01:14:51,760 It can't be more complicated than that. 1229 01:14:51,760 --> 01:14:54,080 And you see that in some examples, 1230 01:14:54,080 --> 01:14:57,628 you either have the purple or the orange, but not both. 1231 01:14:57,628 --> 01:14:59,170 That's kind of typical that you don't 1232 01:14:59,170 --> 01:15:02,920 get the most complicated thing. 1233 01:15:02,920 --> 01:15:05,955 So when you have results from observables that tell you 1234 01:15:05,955 --> 01:15:07,330 how these things couple together, 1235 01:15:07,330 --> 01:15:09,050 those are called factorization theorems. 1236 01:15:09,050 --> 01:15:11,050 And in the effective theory, because you sort of 1237 01:15:11,050 --> 01:15:13,973 define the modes, separated them at the start, 1238 01:15:13,973 --> 01:15:16,390 you're kind of very quickly getting to these factorization 1239 01:15:16,390 --> 01:15:16,890 theorems. 1240 01:15:27,090 --> 01:15:29,870 Let's see. 1241 01:15:29,870 --> 01:15:33,560 So we're going to deal with a bunch of different examples. 1242 01:15:33,560 --> 01:15:37,380 And I decided that I'm going to do it in the following order. 1243 01:15:37,380 --> 01:15:39,870 So we're going to do the next exam-- 1244 01:15:39,870 --> 01:15:42,560 so we're going to do a bunch of examples 1245 01:15:42,560 --> 01:15:50,030 in order to see the range of possibilities that can happen. 1246 01:15:53,710 --> 01:15:58,540 And so I'm going to stick with SCET I for next lecture. 1247 01:15:58,540 --> 01:16:02,670 So the next example we'll do, which I'll call example one, 1248 01:16:02,670 --> 01:16:06,650 is we'll do our [? dijet ?] production. 1249 01:16:16,650 --> 01:16:20,540 So this is a SCET I situation. 1250 01:16:20,540 --> 01:16:22,410 And the difference between the example-- 1251 01:16:22,410 --> 01:16:24,670 so so far, what we did is we did DIS. 1252 01:16:24,670 --> 01:16:26,990 DIS actually it was so simple because it only 1253 01:16:26,990 --> 01:16:29,420 had two degrees of freedom that you could kind of either 1254 01:16:29,420 --> 01:16:32,000 think of it as SCET I or SCET II. 1255 01:16:32,000 --> 01:16:34,010 I mean, technically, it's more like SCET II, 1256 01:16:34,010 --> 01:16:35,450 but it behaves like SCET I. 1257 01:16:35,450 --> 01:16:36,735 But there's no ultrasofts. 1258 01:16:36,735 --> 01:16:38,360 And remember, it's ultrasofts and softs 1259 01:16:38,360 --> 01:16:41,120 that are making the distinction between the two. 1260 01:16:41,120 --> 01:16:43,578 So if you just have this mode and this mode, 1261 01:16:43,578 --> 01:16:45,620 it's not really any difference between calling it 1262 01:16:45,620 --> 01:16:46,790 SCET I or SCET II. 1263 01:16:46,790 --> 01:16:48,650 So it's just SCET. 1264 01:16:48,650 --> 01:16:51,110 So e plus, e minus to [? dijets ?] 1265 01:16:51,110 --> 01:16:54,020 will be an SCET I example which has ultrasofts. 1266 01:16:58,330 --> 01:17:00,340 And actually, what we'll find in this case 1267 01:17:00,340 --> 01:17:02,695 is that it will be the purple. 1268 01:17:05,220 --> 01:17:09,040 We'll get a purple convolution. 1269 01:17:09,040 --> 01:17:10,420 We'll see how that happens. 1270 01:17:10,420 --> 01:17:12,370 And we won't actually-- momentum conservation 1271 01:17:12,370 --> 01:17:14,370 will rule out the possibility of the orange one. 1272 01:17:17,360 --> 01:17:19,370 So we'll see the opposite situation 1273 01:17:19,370 --> 01:17:24,470 where it could be that we have ultrasoft modes as well, 1274 01:17:24,470 --> 01:17:26,600 but then we only get a convolution 1275 01:17:26,600 --> 01:17:29,060 with those ultrasoft in the factorization theorem 1276 01:17:29,060 --> 01:17:30,830 and not with the [INAUDIBLE]. 1277 01:17:36,760 --> 01:17:40,540 And then we'll turn to SCET II. 1278 01:17:40,540 --> 01:17:46,320 And I haven't totally decided what processes I'll do, 1279 01:17:46,320 --> 01:17:47,990 but I think I'll do the following ones. 1280 01:17:52,700 --> 01:17:55,000 So one thing that you can do, which is pretty simple, 1281 01:17:55,000 --> 01:17:59,110 is to look at something called the photon-pion form factor. 1282 01:17:59,110 --> 01:18:01,960 So real photon-to-pion transition 1283 01:18:01,960 --> 01:18:04,810 through another virtual photon, but you 1284 01:18:04,810 --> 01:18:08,320 can think of this happening through a diagram like this 1285 01:18:08,320 --> 01:18:11,990 with two quarks, one of them that's 1286 01:18:11,990 --> 01:18:14,510 off-shell and one of them that's on-shell. 1287 01:18:14,510 --> 01:18:17,110 This is pi 0. 1288 01:18:17,110 --> 01:18:18,772 So this is a SCET II example. 1289 01:18:18,772 --> 01:18:20,980 But again, it's pretty simple because it's just going 1290 01:18:20,980 --> 01:18:25,180 to involve one hadronic object. 1291 01:18:25,180 --> 01:18:27,580 And actually, it will just, in this case, 1292 01:18:27,580 --> 01:18:28,945 have collinear modes. 1293 01:18:32,740 --> 01:18:36,820 We can set things up so the pion is n-collinear, 1294 01:18:36,820 --> 01:18:41,170 and then we have hard modes. 1295 01:18:47,710 --> 01:18:50,250 So that's one example we'll do. 1296 01:18:50,250 --> 01:18:55,240 Another example we'll do is B to D pi. 1297 01:18:55,240 --> 01:18:57,190 And here, the B and the D are soft. 1298 01:18:57,190 --> 01:19:02,980 Remember, we talked about this one, and the pion is collinear. 1299 01:19:02,980 --> 01:19:05,950 And then we have hard modes. 1300 01:19:05,950 --> 01:19:09,280 So this is kind of like, in some sense, a DIS, 1301 01:19:09,280 --> 01:19:12,490 but it's an exclusive process, not an inclusive one. 1302 01:19:12,490 --> 01:19:14,740 And so actually, all the tools that we use, 1303 01:19:14,740 --> 01:19:15,820 which were kind of-- 1304 01:19:15,820 --> 01:19:18,700 in DIS, it's the most inclusive process you can think of. 1305 01:19:18,700 --> 01:19:20,920 It's deep inelastic scattering. 1306 01:19:20,920 --> 01:19:22,750 The I is for "inclusive." 1307 01:19:22,750 --> 01:19:24,983 Here, we're doing something completely exclusive, 1308 01:19:24,983 --> 01:19:27,400 but we'll see that all the things that we've been thinking 1309 01:19:27,400 --> 01:19:29,233 about, which is just separation of collinear 1310 01:19:29,233 --> 01:19:31,750 modes and hard modes, will just go through for that process 1311 01:19:31,750 --> 01:19:33,220 too. 1312 01:19:33,220 --> 01:19:35,470 So the effective theory, the difference 1313 01:19:35,470 --> 01:19:40,630 is that in this case, you will not be taking the amplitude 1314 01:19:40,630 --> 01:19:41,130 squared. 1315 01:19:41,130 --> 01:19:42,922 You won't be looking at forward scattering. 1316 01:19:42,922 --> 01:19:45,310 The forward scattering was what was making it inclusive. 1317 01:19:45,310 --> 01:19:47,440 We were summing over all the final states. 1318 01:19:47,440 --> 01:19:49,687 Here, there's only one final state. 1319 01:19:49,687 --> 01:19:52,270 So the difference between this example and the one we just did 1320 01:19:52,270 --> 01:20:04,210 is that in this case, we'll factor the amplitude, not 1321 01:20:04,210 --> 01:20:05,260 the squared amplitude. 1322 01:20:08,442 --> 01:20:10,900 But other than that, it'll look very similar to the example 1323 01:20:10,900 --> 01:20:12,880 that we did for DIS. 1324 01:20:12,880 --> 01:20:14,538 B to D pi, then, is an SCET II example 1325 01:20:14,538 --> 01:20:17,080 where we make things a little more complicated because now we 1326 01:20:17,080 --> 01:20:19,720 have soft, collinear, and hard modes. 1327 01:20:19,720 --> 01:20:23,150 And we'll see what happens there. 1328 01:20:23,150 --> 01:20:25,120 And then I'll do some more examples after that. 1329 01:20:25,120 --> 01:20:30,055 But let me say we'll do some LHC examples. 1330 01:20:32,860 --> 01:20:44,210 And I think we'll also do broadening, 1331 01:20:44,210 --> 01:20:47,240 which is another e plus, e minus observable. 1332 01:20:47,240 --> 01:20:49,400 All right, so that's where we're going, 1333 01:20:49,400 --> 01:20:53,280 and we'll start going there next time 1334 01:20:53,280 --> 01:20:57,355 by talking about [? dijets ?] and SCET I.