1 00:00:00,000 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,278 at ocw.mit.edu. 8 00:00:21,280 --> 00:00:22,280 IAIN STEWART: All right. 9 00:00:22,280 --> 00:00:23,165 So let's get started. 10 00:00:25,850 --> 00:00:28,900 So last time, we had talked about factorization 11 00:00:28,900 --> 00:00:30,180 in the effective theory. 12 00:00:30,180 --> 00:00:33,110 And there is one type of factorization, 13 00:00:33,110 --> 00:00:36,860 which is this hard-collinear factorization, which 14 00:00:36,860 --> 00:00:39,860 is a factorization between the low energy physics described 15 00:00:39,860 --> 00:00:42,820 by the operator and the Wilson coefficients. 16 00:00:42,820 --> 00:00:44,570 And we decided we wanted to think 17 00:00:44,570 --> 00:00:47,120 about that in the following way, as having convolutions 18 00:00:47,120 --> 00:00:50,300 between variables that appear in the Wilson coefficients 19 00:00:50,300 --> 00:00:52,743 and variables that appear in the operator. 20 00:00:52,743 --> 00:00:54,410 And the way that we can think about that 21 00:00:54,410 --> 00:00:57,050 is just what the most general possible thing is that we can 22 00:00:57,050 --> 00:00:59,660 write down for the Wilson coefficients and of course, 23 00:00:59,660 --> 00:01:02,030 it can depend on large momenta. 24 00:01:02,030 --> 00:01:07,310 And so large momenta here includes the label operator. 25 00:01:07,310 --> 00:01:12,380 So large momenta always show up in Wilson coefficients 26 00:01:12,380 --> 00:01:14,990 and in this case, that includes momenta 27 00:01:14,990 --> 00:01:17,640 that are the large component, the order lambda 0 28 00:01:17,640 --> 00:01:18,890 component of collinear fields. 29 00:01:27,920 --> 00:01:39,680 So this is hard-collinear factorization 30 00:01:39,680 --> 00:01:43,210 and it just comes out in a natural, standard way 31 00:01:43,210 --> 00:01:45,387 in this effective theory because we've 32 00:01:45,387 --> 00:01:47,720 set up the effective theory to have the right low energy 33 00:01:47,720 --> 00:01:49,265 degrees of freedom. 34 00:01:49,265 --> 00:01:51,140 So even though it's a little more complicated 35 00:01:51,140 --> 00:01:53,240 than just a simple product, we see 36 00:01:53,240 --> 00:01:56,750 what the variables are that can connect the two things just 37 00:01:56,750 --> 00:01:58,800 by power counting, essentially. 38 00:01:58,800 --> 00:02:05,600 The large momenta are the ones that are order lambda 0. 39 00:02:05,600 --> 00:02:08,358 We can do this generically if we recall the definition 40 00:02:08,358 --> 00:02:09,150 of our Wilson line. 41 00:02:12,730 --> 00:02:14,690 We can see how this would carry over 42 00:02:14,690 --> 00:02:16,970 to a generic case in the following way. 43 00:02:23,280 --> 00:02:25,760 So that was one way of defining the Wilson line. 44 00:02:30,470 --> 00:02:33,470 And as an operator, we had relations as well. 45 00:02:38,940 --> 00:02:50,910 So we had this relation and in particular, we 46 00:02:50,910 --> 00:02:55,680 could take this to any power and that just takes the p bar 47 00:02:55,680 --> 00:03:00,420 to any power like that. 48 00:03:00,420 --> 00:03:02,100 So you can really think of this as given 49 00:03:02,100 --> 00:03:06,613 any function of this operator i m bar 50 00:03:06,613 --> 00:03:09,950 dot d you can always write that as a Wilson line, 51 00:03:09,950 --> 00:03:12,920 a function of p bar, and a Wilson line. 52 00:03:15,825 --> 00:03:18,200 And what you want to do is you want to stick these Wilson 53 00:03:18,200 --> 00:03:19,940 lines in the operator and you want 54 00:03:19,940 --> 00:03:23,180 to put this function in the Wilson coefficient. 55 00:03:23,180 --> 00:03:26,840 So you could think, lets me start with operator. 56 00:03:26,840 --> 00:03:29,720 I can throw in i m bar dot d's because these guys are 57 00:03:29,720 --> 00:03:33,102 order lambda 0, these are collinear derivatives. 58 00:03:33,102 --> 00:03:35,060 I could just allow it to stick as many of those 59 00:03:35,060 --> 00:03:37,280 as I like in my operator. 60 00:03:37,280 --> 00:03:39,440 But given that I put in any function of them, 61 00:03:39,440 --> 00:03:41,540 I could always do this and the right way 62 00:03:41,540 --> 00:03:46,520 of thinking about it is that this function here 63 00:03:46,520 --> 00:03:49,040 is determined by matching. 64 00:03:49,040 --> 00:03:50,768 It's your Wilson coefficient. 65 00:04:11,210 --> 00:04:14,940 And that's what we were effectively doing up here. 66 00:04:14,940 --> 00:04:17,600 But in some sense of our general discussion 67 00:04:17,600 --> 00:04:20,267 you can see that even if you had multiple places in the operator 68 00:04:20,267 --> 00:04:22,183 where you could insert these derivatives, then 69 00:04:22,183 --> 00:04:23,180 you would just do this. 70 00:04:23,180 --> 00:04:25,138 And what would happen is at the end of the day, 71 00:04:25,138 --> 00:04:28,160 you get a function of all the possible large momenta 72 00:04:28,160 --> 00:04:32,650 that you can form from the operator. 73 00:04:32,650 --> 00:04:36,160 So maybe one more line. 74 00:04:40,670 --> 00:04:43,650 So think of it, you can always separate out. 75 00:04:50,180 --> 00:04:51,940 Make a split like we did over there 76 00:04:51,940 --> 00:04:53,860 with the integral over some variable 77 00:04:53,860 --> 00:04:59,140 and then leave something that you can stick in your operator, 78 00:04:59,140 --> 00:04:59,920 make it like this. 79 00:04:59,920 --> 00:05:04,600 So this could go in the operator and then 80 00:05:04,600 --> 00:05:06,288 that's the coefficient. 81 00:05:09,960 --> 00:05:12,030 AUDIENCE: If I stick this in [INAUDIBLE]?? 82 00:05:14,688 --> 00:05:16,480 IAIN STEWART: Well, here I didn't write it. 83 00:05:16,480 --> 00:05:18,180 So if I wanted to put in i m bar dot 84 00:05:18,180 --> 00:05:20,710 d's, I'd stick them between. 85 00:05:20,710 --> 00:05:21,210 Right? 86 00:05:21,210 --> 00:05:23,270 If I wanted to use the formula. 87 00:05:23,270 --> 00:05:24,870 If I wanted to use this kind of logic 88 00:05:24,870 --> 00:05:27,210 I wouldn't have written this and I 89 00:05:27,210 --> 00:05:29,994 would have said let me stick an arbitrary function of i m bar 90 00:05:29,994 --> 00:05:30,750 dot d in here. 91 00:05:30,750 --> 00:05:32,812 AUDIENCE: But Between before you even write-- 92 00:05:32,812 --> 00:05:34,770 IAIN STEWART: It has to be between them because 93 00:05:34,770 --> 00:05:35,973 of the gauge invariance. 94 00:05:35,973 --> 00:05:36,640 AUDIENCE: Right. 95 00:05:36,640 --> 00:05:38,880 So before you write down the Wilson line. 96 00:05:38,880 --> 00:05:40,320 So say you start and say, OK-- 97 00:05:40,320 --> 00:05:42,070 IAIN STEWART: Oh, yeah, so there's still-- 98 00:05:42,070 --> 00:05:43,177 right. 99 00:05:43,177 --> 00:05:44,010 Right, right, right. 100 00:05:44,010 --> 00:05:46,770 This Wilson line here came from this h. 101 00:05:46,770 --> 00:05:48,500 So that's still going to be true. 102 00:05:48,500 --> 00:05:52,440 And what I'm saying, if you think about this operator, 103 00:05:52,440 --> 00:05:54,642 I could dress it up by putting any function of i 104 00:05:54,642 --> 00:06:00,400 m bar dot d's right in here because that would still be 105 00:06:00,400 --> 00:06:02,945 same order and gauge invariant. 106 00:06:02,945 --> 00:06:04,570 AUDIENCE: OK, but this an alternative-- 107 00:06:04,570 --> 00:06:09,010 IAIN STEWART: But then if I use this formula, 108 00:06:09,010 --> 00:06:13,510 it would sort of push this w. 109 00:06:13,510 --> 00:06:15,910 That would cancel there and then this w comes back 110 00:06:15,910 --> 00:06:16,910 and then you get the if. 111 00:06:21,020 --> 00:06:22,245 OK. 112 00:06:22,245 --> 00:06:24,370 So in general, the right way of thinking about this 113 00:06:24,370 --> 00:06:25,400 is as follows. 114 00:06:25,400 --> 00:06:38,488 We can encode this in some notation 115 00:06:38,488 --> 00:06:42,890 by just setting up a convenient set of building blocks 116 00:06:42,890 --> 00:06:45,310 which are gauge invariant objects 117 00:06:45,310 --> 00:06:47,556 under the collinear gauge transformations. 118 00:07:02,440 --> 00:07:06,160 And so we need to have a fermion field, 119 00:07:06,160 --> 00:07:08,500 but we know that the fermion field, 120 00:07:08,500 --> 00:07:12,370 generically we can make it gauge invariant by multiplying 121 00:07:12,370 --> 00:07:15,770 by Wilson line like that. 122 00:07:15,770 --> 00:07:18,910 And because of this, what we were just discussing here, 123 00:07:18,910 --> 00:07:21,340 generically, we can be in our Wilson coefficient, 124 00:07:21,340 --> 00:07:23,960 sensitive to the momentum of this object. 125 00:07:23,960 --> 00:07:36,020 And so we can denote that by defining the following thing, 126 00:07:36,020 --> 00:07:38,770 which is just a chi field that carries some momentum which 127 00:07:38,770 --> 00:07:41,545 is the overall large momentum of this product of fields. 128 00:07:45,570 --> 00:07:52,910 And sometimes this guy goes by the name of the quark jet field 129 00:07:52,910 --> 00:07:56,790 because if you were to produce a quark in the hard scattering 130 00:07:56,790 --> 00:08:00,240 process, the quark would be represented in your operator 131 00:08:00,240 --> 00:08:01,980 by this w dagger c. 132 00:08:01,980 --> 00:08:04,380 The Wilson coefficients would talk to that quark 133 00:08:04,380 --> 00:08:07,290 through the large momentum and that quark, 134 00:08:07,290 --> 00:08:10,200 in the low energy theory, would evolve into a jet. 135 00:08:10,200 --> 00:08:11,310 So it goes by that name. 136 00:08:11,310 --> 00:08:16,590 You could also think of it as a parton field 137 00:08:16,590 --> 00:08:21,280 because as it turns out and as we'll talk about later on, 138 00:08:21,280 --> 00:08:24,150 if you were to think about the parton distribution function 139 00:08:24,150 --> 00:08:26,460 and what pulls a quark out of the proton, 140 00:08:26,460 --> 00:08:28,750 it's exactly this operator. 141 00:08:28,750 --> 00:08:31,900 And then that quark's momentum, what 142 00:08:31,900 --> 00:08:35,100 momentum that quark carries is exactly picked out 143 00:08:35,100 --> 00:08:36,539 by this delta function as well. 144 00:08:39,140 --> 00:08:41,740 So these are the objects you'll usually 145 00:08:41,740 --> 00:08:44,420 want to work in terms of. 146 00:08:44,420 --> 00:08:54,360 We can do something similar for the gluon, which is a curly b 147 00:08:54,360 --> 00:08:57,520 and I'm going to define it. 148 00:08:57,520 --> 00:09:00,070 If we wanted to get a gluon, the natural thing 149 00:09:00,070 --> 00:09:02,530 is to use a field strength because then you get a gluon 150 00:09:02,530 --> 00:09:04,480 without any derivatives. 151 00:09:04,480 --> 00:09:09,170 But I want my object to be dimension one, 152 00:09:09,170 --> 00:09:11,830 so I'm going to define it in the following way. 153 00:09:17,480 --> 00:09:19,030 So here's a field strength commutator 154 00:09:19,030 --> 00:09:24,220 of covariant derivatives, these are collinear derivatives. 155 00:09:24,220 --> 00:09:26,980 And I throw Wilson lines around it to make it gauge invariant, 156 00:09:26,980 --> 00:09:29,480 and then to make it dimension one I throw in a 1 over p bar. 157 00:09:31,860 --> 00:09:37,690 And if you start expanding this out, 158 00:09:37,690 --> 00:09:40,160 then the first term will be the a n perp gluon 159 00:09:40,160 --> 00:09:43,160 and so you have this b just has a n perp gluon in the same way 160 00:09:43,160 --> 00:09:45,440 that the first kind of order term in this chi 161 00:09:45,440 --> 00:09:49,140 is just to the fermion if I set the Wilson line to 1. 162 00:09:49,140 --> 00:09:49,640 OK? 163 00:09:49,640 --> 00:09:52,743 So it's starting out as a perpendicular gluon 164 00:09:52,743 --> 00:09:55,160 but it's dressed up in a way that makes it gauge invariant 165 00:09:55,160 --> 00:09:55,940 in dimension one. 166 00:10:10,100 --> 00:10:15,410 And then just like we did here, we can put a subscript on it 167 00:10:15,410 --> 00:10:21,370 with a omega to say that we fixed the large momentum of it 168 00:10:21,370 --> 00:10:27,110 and sometimes there's a convention that that's done, 169 00:10:27,110 --> 00:10:29,600 you have to decide if you want the Wilson line-- 170 00:10:29,600 --> 00:10:32,540 I mean, the delta function to be a w minus p bar or w 171 00:10:32,540 --> 00:10:33,340 minus p bar dagger. 172 00:10:36,170 --> 00:10:38,840 Anyway, it doesn't really matter that much because it's just 173 00:10:38,840 --> 00:10:41,580 a sign of what you mean by a w. 174 00:10:41,580 --> 00:10:44,060 If it's an outgoing gluon, this is a more convenient 175 00:10:44,060 --> 00:10:46,760 convention. 176 00:10:46,760 --> 00:10:49,820 OK, and so this delta function here and this p bar 177 00:10:49,820 --> 00:10:51,890 here, when I put these square brackets, what 178 00:10:51,890 --> 00:10:54,590 I mean by that is that they don't know how to act outside. 179 00:10:54,590 --> 00:10:57,740 They just act on the operators inside. 180 00:10:57,740 --> 00:11:05,008 So these are objects that exist by themselves 181 00:11:05,008 --> 00:11:06,800 and they don't care about other things that 182 00:11:06,800 --> 00:11:08,210 are multiplying them later. 183 00:11:17,500 --> 00:11:21,360 So that's the gluon analog of the quark. 184 00:11:21,360 --> 00:11:29,670 So it turns out that we can show the following. 185 00:11:29,670 --> 00:11:31,500 That if you want to build operators 186 00:11:31,500 --> 00:11:33,360 that are subleading order, a complete set 187 00:11:33,360 --> 00:11:35,375 of things to do that is as follows. 188 00:11:35,375 --> 00:11:44,990 It's this chi n, the bn perp, and then you 189 00:11:44,990 --> 00:11:48,930 could also need p perps. 190 00:11:48,930 --> 00:11:53,440 And then you could also have ultrasoft fields, 191 00:11:53,440 --> 00:11:55,150 where it's really kind of similar to how 192 00:11:55,150 --> 00:11:57,317 you're used to building operators in effective field 193 00:11:57,317 --> 00:11:58,460 theories. 194 00:11:58,460 --> 00:12:00,040 But if you just want to talk about the collinear sector, 195 00:12:00,040 --> 00:12:02,290 the only things you need are this chi n this bn field, 196 00:12:02,290 --> 00:12:04,230 and then p perps and no other things. 197 00:12:04,230 --> 00:12:05,815 AUDIENCE: That's to go just one order? 198 00:12:05,815 --> 00:12:07,010 IAIN STEWART: All orders. 199 00:12:07,010 --> 00:12:08,468 AUDIENCE: So why aren't there any-- 200 00:12:08,468 --> 00:12:11,320 IAIN STEWART: I'll talk about it. 201 00:12:11,320 --> 00:12:11,937 Yeah. 202 00:12:11,937 --> 00:12:12,520 I'll show you. 203 00:12:12,520 --> 00:12:13,360 AUDIENCE: OK. 204 00:12:13,360 --> 00:12:14,170 IAIN STEWART: Yeah. 205 00:12:14,170 --> 00:12:16,148 I mean, intuitively you would think 206 00:12:16,148 --> 00:12:18,190 these are the physical degrees of freedom, right? 207 00:12:18,190 --> 00:12:20,147 There's two of them so that's intuitive. 208 00:12:20,147 --> 00:12:22,480 But I'll show you how you get rid of all the other ones. 209 00:12:32,749 --> 00:12:34,517 It kind of matches up with what you want. 210 00:12:34,517 --> 00:12:37,100 You could think that these are the physical gluons that you're 211 00:12:37,100 --> 00:12:42,500 producing and your operators can be set up so that there's not 212 00:12:42,500 --> 00:12:48,210 any spare use components. 213 00:12:48,210 --> 00:12:48,860 OK. 214 00:12:48,860 --> 00:12:53,280 So we'll get there. 215 00:12:53,280 --> 00:12:55,685 So let me introduce a bit of notation. 216 00:12:58,622 --> 00:12:59,600 I have to do. 217 00:13:09,322 --> 00:13:10,530 So let's consider this thing. 218 00:13:10,530 --> 00:13:11,905 A covariant derivative sandwiched 219 00:13:11,905 --> 00:13:15,810 with Wilson lines on either side and this i dn 220 00:13:15,810 --> 00:13:18,360 here is like p bar plus. 221 00:13:25,853 --> 00:13:27,770 So it just involves the collinear gluon field. 222 00:13:31,440 --> 00:13:36,080 So then if I have that operator and I just take n bar i dn, 223 00:13:36,080 --> 00:13:36,830 then that's p bar. 224 00:13:39,470 --> 00:13:46,280 if I take i dn perp mu, then you can 225 00:13:46,280 --> 00:13:54,990 show that this guy is p perp u plus g bn perp mu. 226 00:13:57,740 --> 00:13:59,990 So this one is just a relation we talked about before. 227 00:13:59,990 --> 00:14:03,770 This one is not as obvious, but if you take this guy, 228 00:14:03,770 --> 00:14:08,030 you can let this derivative p perp act on the Wilson line 229 00:14:08,030 --> 00:14:08,990 or act through. 230 00:14:08,990 --> 00:14:11,150 If it acts through, that's this term. 231 00:14:11,150 --> 00:14:14,810 If it acts on, then I can manipulate the operator 232 00:14:14,810 --> 00:14:17,590 so that it is exactly this form. 233 00:14:17,590 --> 00:14:20,420 And part of that comes from the fact that-- 234 00:14:20,420 --> 00:14:21,830 I can explain it to you here. 235 00:14:21,830 --> 00:14:23,955 I have it in my notes and you can look at it later, 236 00:14:23,955 --> 00:14:25,590 but let me just explain it in words. 237 00:14:25,590 --> 00:14:27,423 If you looked at the term in this commutator 238 00:14:27,423 --> 00:14:31,452 that was other order, where the i m bar dot d was sitting here. 239 00:14:31,452 --> 00:14:32,660 It would hit the Wilson line. 240 00:14:32,660 --> 00:14:33,410 You'd get 0. 241 00:14:33,410 --> 00:14:34,910 So that term was just put in to make 242 00:14:34,910 --> 00:14:41,770 it look like a field strength. 243 00:14:41,770 --> 00:14:42,270 OK? 244 00:14:42,270 --> 00:14:46,700 So really you have i m bar dot d dn perp, without the comma, 245 00:14:46,700 --> 00:14:48,020 without the brackets. 246 00:14:48,020 --> 00:14:54,110 But then this combination here, you can use the identity 247 00:14:54,110 --> 00:14:56,653 that we had with it and basically if you 248 00:14:56,653 --> 00:14:58,070 push this guy through here, you're 249 00:14:58,070 --> 00:15:01,640 canceling the p bar so it's just giving you the Wilson line. 250 00:15:01,640 --> 00:15:06,420 So that's basically how you go from here to here. 251 00:15:06,420 --> 00:15:06,920 OK? 252 00:15:06,920 --> 00:15:12,657 So basically what this is saying is 253 00:15:12,657 --> 00:15:14,990 I don't need to consider covariant derivatives because I 254 00:15:14,990 --> 00:15:16,820 can instead consider p perps and b 255 00:15:16,820 --> 00:15:18,840 perps and that's equally good. 256 00:15:23,260 --> 00:15:25,900 OK. 257 00:15:25,900 --> 00:15:34,120 Now if you do a similar type of thing in the n component, 258 00:15:34,120 --> 00:15:36,620 then you can derive a similar type of object. 259 00:15:36,620 --> 00:15:38,200 So this is the same object as we have 260 00:15:38,200 --> 00:15:40,637 over here, but instead of having an i dn perp, 261 00:15:40,637 --> 00:15:41,470 I'd have an n dot d. 262 00:15:50,782 --> 00:15:53,900 So d dot n perp replaced n dot d. 263 00:15:56,482 --> 00:15:58,190 And that looks like it could be something 264 00:15:58,190 --> 00:16:02,450 that you'd build operators out of and and also, furthermore, 265 00:16:02,450 --> 00:16:05,500 why not have operators that depend on this n dot partial? 266 00:16:05,500 --> 00:16:06,000 OK? 267 00:16:09,470 --> 00:16:12,180 So for p bar, we don't have to worry about that. 268 00:16:12,180 --> 00:16:15,797 So OK, I said we'll include this, we'll include that. 269 00:16:15,797 --> 00:16:17,880 I didn't say we have include this, this, and this. 270 00:16:17,880 --> 00:16:19,963 So there's three things I have to argue away here. 271 00:16:22,460 --> 00:16:24,440 The derivatives that are p bars, those just 272 00:16:24,440 --> 00:16:26,250 go into the Wilson coefficient. 273 00:16:26,250 --> 00:16:32,340 So if we had p bar chi n comma omega, for example, 274 00:16:32,340 --> 00:16:35,810 then that's omega chi n comma omega 275 00:16:35,810 --> 00:16:40,940 and this is put into the Wilson coefficient. 276 00:16:40,940 --> 00:16:45,580 So that's why we don't have to worry about having p bars. 277 00:16:45,580 --> 00:16:46,970 In some sense, we do have p bars. 278 00:16:46,970 --> 00:16:49,437 They're all in the Wilson coefficients. 279 00:16:49,437 --> 00:16:51,020 So it's really these other two that we 280 00:16:51,020 --> 00:16:54,270 have to worry more about and those actually 281 00:16:54,270 --> 00:16:56,400 can be simplified using the equation of motion. 282 00:16:56,400 --> 00:17:01,940 So if you have i n dot p partial on chi n, then 283 00:17:01,940 --> 00:17:04,280 equation of motion when you write it out 284 00:17:04,280 --> 00:17:14,110 in terms of these objects, it has some form 285 00:17:14,110 --> 00:17:16,235 and basically, you can just get rid of those terms. 286 00:17:23,650 --> 00:17:27,470 So the equation of motion allows us to get rid of i n dot 287 00:17:27,470 --> 00:17:28,920 partials that are on chi n's. 288 00:17:33,730 --> 00:17:38,440 So that's why we don't have to worry about those. 289 00:17:38,440 --> 00:17:41,110 And this is just like saying that in our leading order 290 00:17:41,110 --> 00:17:45,035 action, there wasn't i n dot partial. 291 00:17:45,035 --> 00:17:46,410 But that's just like saying there 292 00:17:46,410 --> 00:17:48,400 was time derivatives in our leading order 293 00:17:48,400 --> 00:17:51,370 action in some standard effective field theory. 294 00:17:51,370 --> 00:17:53,520 But then in the higher dimension operators, 295 00:17:53,520 --> 00:17:55,270 you can always use the equations of motion 296 00:17:55,270 --> 00:17:56,650 to get rid of those time derivatives. 297 00:17:56,650 --> 00:17:58,120 And here we're using the equation of motion 298 00:17:58,120 --> 00:17:59,560 to get rid of i n dot partial. 299 00:17:59,560 --> 00:18:01,310 So it appears in the leading order action, 300 00:18:01,310 --> 00:18:03,790 but then we don't have to have it in any other subleading 301 00:18:03,790 --> 00:18:08,770 operator and that's why it's not one of the ones that's 302 00:18:08,770 --> 00:18:11,090 included in the list. 303 00:18:11,090 --> 00:18:13,690 And if you had i n dot partial on curly b, 304 00:18:13,690 --> 00:18:15,490 that's also part of the equations 305 00:18:15,490 --> 00:18:16,750 of motion of the gluon field. 306 00:18:20,773 --> 00:18:22,690 Now when you do the gluon equations of motion, 307 00:18:22,690 --> 00:18:25,337 there's components because it's a vector 308 00:18:25,337 --> 00:18:26,920 and one of the other components allows 309 00:18:26,920 --> 00:18:30,370 you to get rid of the n dot b. 310 00:18:30,370 --> 00:18:33,528 So there's another term that you can rearrange, 311 00:18:33,528 --> 00:18:36,070 and I won't write it out because the equation is rather messy 312 00:18:36,070 --> 00:18:42,370 but give you some idea. 313 00:18:45,090 --> 00:18:50,230 There's another component that looks 314 00:18:50,230 --> 00:18:54,078 like this where I can get rid of all the n dot curly b's 315 00:18:54,078 --> 00:18:55,120 using the gluon equation. 316 00:18:55,120 --> 00:18:56,650 So the gluon equations of motions 317 00:18:56,650 --> 00:19:00,480 allows me to get rid of both of these things 318 00:19:00,480 --> 00:19:02,230 and basically after I've done that, I just 319 00:19:02,230 --> 00:19:05,110 have the objects that I've told you we can use. 320 00:19:12,510 --> 00:19:14,150 So after using equations of motion, 321 00:19:14,150 --> 00:19:15,800 we can get down to those objects. 322 00:19:18,850 --> 00:19:22,220 And any other thing that you might dream up 323 00:19:22,220 --> 00:19:26,900 can be reduced to these objects. 324 00:19:26,900 --> 00:19:30,410 So I'm not saying that I went through a complete list here. 325 00:19:30,410 --> 00:19:35,690 For example, what if you had a commutator of 2 d n perps? 326 00:19:35,690 --> 00:19:36,190 Right? 327 00:19:36,190 --> 00:19:39,640 You might say, oh, that's some new thing. 328 00:19:39,640 --> 00:19:40,390 It's one of these. 329 00:19:40,390 --> 00:19:41,598 You can also reduce that too. 330 00:20:03,458 --> 00:20:05,230 So let me list that one just to give you 331 00:20:05,230 --> 00:20:08,650 some idea that there's others you might think of dreaming up. 332 00:20:29,990 --> 00:20:33,310 So you can reduce all of this to be that. 333 00:20:33,310 --> 00:20:35,040 OK, so for the collinear sector this 334 00:20:35,040 --> 00:20:38,010 is enough to build higher dimension operators, just 335 00:20:38,010 --> 00:20:40,350 these three. 336 00:20:40,350 --> 00:20:47,730 And then for the ultrasoft sector, so I guess this is two. 337 00:20:47,730 --> 00:21:06,800 We do need ultrasoft derivatives and ultrasoft field strengths 338 00:21:06,800 --> 00:21:07,745 and ultrasoft quarks. 339 00:21:11,540 --> 00:21:13,700 So this part is really just similar to the story 340 00:21:13,700 --> 00:21:17,240 that you'd have for a standard low energy effective field 341 00:21:17,240 --> 00:21:18,830 theory. 342 00:21:18,830 --> 00:21:20,540 Like integrating out a massive particle, 343 00:21:20,540 --> 00:21:21,998 you can use the equation of motion. 344 00:21:29,160 --> 00:21:32,227 One thing that is worth commenting about 345 00:21:32,227 --> 00:21:33,810 is the connection between one and two. 346 00:21:33,810 --> 00:21:36,180 So one is collinear and two is ultrasoft 347 00:21:36,180 --> 00:21:39,400 and prior you might think, well, they're totally independent. 348 00:21:39,400 --> 00:21:41,730 But we saw last time that reparamaterization invariance 349 00:21:41,730 --> 00:21:43,540 connects them. 350 00:21:43,540 --> 00:21:46,260 So if I've decided that this is the type of basis 351 00:21:46,260 --> 00:21:47,820 I want to use for my operators, then 352 00:21:47,820 --> 00:21:51,360 what is the reparameterization connection? 353 00:21:51,360 --> 00:21:53,970 You can rewrite what we said last time in terms 354 00:21:53,970 --> 00:21:56,850 of these curly d's because it basically just means 355 00:21:56,850 --> 00:22:02,310 moving the Wilson lines that we had in this formula last time. 356 00:22:02,310 --> 00:22:06,558 They were around the ultrasoft operator last time 357 00:22:06,558 --> 00:22:08,850 and if I just move them over to the collinear operator, 358 00:22:08,850 --> 00:22:11,500 then I get this curly d. 359 00:22:11,500 --> 00:22:12,000 Right? 360 00:22:12,000 --> 00:22:17,710 So the RPI connection is connecting 361 00:22:17,710 --> 00:22:21,757 the curly d collinear to the d ultrasoft 362 00:22:21,757 --> 00:22:24,340 and then you can write this guy as the p perp plus the b perp. 363 00:22:26,990 --> 00:22:32,070 And likewise if you do the same for the n bar sector 364 00:22:32,070 --> 00:22:35,580 and you move the Wilson lines from this term to that term, 365 00:22:35,580 --> 00:22:37,730 then it looks like these two combinations. 366 00:22:37,730 --> 00:22:40,230 So that's just rewriting what we had before but now in terms 367 00:22:40,230 --> 00:22:41,915 of this type of notation. 368 00:22:44,210 --> 00:22:44,710 OK? 369 00:22:44,710 --> 00:22:48,250 So that's enough to build operators at higher order 370 00:22:48,250 --> 00:22:51,893 and then we write down Wilson coefficients 371 00:22:51,893 --> 00:22:54,310 for those operators that are functions of a large momentum 372 00:22:54,310 --> 00:22:57,970 and then we start doing physics with them. 373 00:22:57,970 --> 00:22:59,200 So any questions about that? 374 00:23:04,480 --> 00:23:06,160 AUDIENCE: For the equations of motion, 375 00:23:06,160 --> 00:23:09,058 you always just use the leading order equation of motion-- 376 00:23:09,058 --> 00:23:09,850 IAIN STEWART: Yeah. 377 00:23:09,850 --> 00:23:11,350 AUDIENCE: --into a higher order term? 378 00:23:11,350 --> 00:23:13,892 IAIN STEWART: Yeah, these are the L0 equations of the motion. 379 00:23:13,892 --> 00:23:16,070 That's what I'm doing here or writing here. 380 00:23:22,022 --> 00:23:23,630 Yeah, there's one more actually. 381 00:23:23,630 --> 00:23:32,190 There's three but it's only needed at some very high order. 382 00:23:32,190 --> 00:23:32,690 OK. 383 00:23:32,690 --> 00:23:35,780 So the next thing I want to talk about 384 00:23:35,780 --> 00:23:38,300 before we start doing explicit examples 385 00:23:38,300 --> 00:23:40,130 and going through processes is how loops 386 00:23:40,130 --> 00:23:41,690 work in this effective theory. 387 00:23:46,590 --> 00:23:49,360 And we're going to have to come back and talk about our grid 388 00:23:49,360 --> 00:23:51,490 that we have for the split up of momenta and how 389 00:23:51,490 --> 00:23:59,430 should we actually think about it in practice 390 00:23:59,430 --> 00:24:01,990 and then we'll also deal with how matching and running work. 391 00:24:07,180 --> 00:24:09,400 So I'm going to do this in the context of an example 392 00:24:09,400 --> 00:24:11,920 and again, I'm just going to pick the simplest example that 393 00:24:11,920 --> 00:24:16,285 has only one jet just to make our lives a little bit simpler. 394 00:24:19,240 --> 00:24:30,180 So we'll consider our heavy, light current 395 00:24:30,180 --> 00:24:31,750 and I think actually once you see 396 00:24:31,750 --> 00:24:33,970 how it works in this example, you'll 397 00:24:33,970 --> 00:24:38,500 understand what all the general features are of doing loops 398 00:24:38,500 --> 00:24:39,000 in SCET. 399 00:24:47,400 --> 00:24:52,190 So we had operators that we constructed, 400 00:24:52,190 --> 00:24:53,767 lowest order operator. 401 00:24:58,740 --> 00:25:01,520 And let me write it in this way which 402 00:25:01,520 --> 00:25:06,010 was prior to making our field redefinition. 403 00:25:06,010 --> 00:25:09,200 I could just write it this way if I want. 404 00:25:09,200 --> 00:25:17,900 And gamma for beta s gamma is a tensor operator. 405 00:25:17,900 --> 00:25:23,290 And there's a photon field, which is a field strength, 406 00:25:23,290 --> 00:25:24,230 f mu nu. 407 00:25:24,230 --> 00:25:27,708 So let's just think of that as all part of gamma. 408 00:25:27,708 --> 00:25:29,800 OK, and that's also appearing here 409 00:25:29,800 --> 00:25:32,035 and I could use the spin structure properties 410 00:25:32,035 --> 00:25:33,910 of these things to reduce the sigma and mu nu 411 00:25:33,910 --> 00:25:37,220 but that's not really going to be part of our story 412 00:25:37,220 --> 00:25:39,850 so let's not bother with that. 413 00:25:39,850 --> 00:25:40,540 So what I do? 414 00:25:40,540 --> 00:25:43,480 I just compute the QCD loops and the SCET loops 415 00:25:43,480 --> 00:25:45,672 and I compare them. 416 00:25:45,672 --> 00:25:47,380 And if SCET is the right effective theory 417 00:25:47,380 --> 00:25:51,600 for this limit where I have an energetic photon 418 00:25:51,600 --> 00:25:57,310 and am back to back with an energetic strange quark, 419 00:25:57,310 --> 00:25:59,310 then I should match all the infrared divergences 420 00:25:59,310 --> 00:26:01,950 in that QCD computation, I should 421 00:26:01,950 --> 00:26:05,160 be able to extract a matching coefficient, 422 00:26:05,160 --> 00:26:06,720 I should be able to determine what 423 00:26:06,720 --> 00:26:12,123 the C is from that calculation, and I should 424 00:26:12,123 --> 00:26:13,290 be able to run the operator. 425 00:26:13,290 --> 00:26:16,650 I should be able to make this into m s bar, 426 00:26:16,650 --> 00:26:19,690 do some renormalization of this operator, 427 00:26:19,690 --> 00:26:22,530 and then do some renormalization group evolution of that Wilson 428 00:26:22,530 --> 00:26:23,541 coefficient. 429 00:26:25,632 --> 00:26:27,715 Well, let's just think about computing the graphs. 430 00:26:35,920 --> 00:26:38,220 So we have to decide, when we compute the graphs, 431 00:26:38,220 --> 00:26:39,890 how to regulate them? 432 00:26:39,890 --> 00:26:40,500 Right? 433 00:26:40,500 --> 00:26:42,420 And we need to use the same infrared regulator 434 00:26:42,420 --> 00:26:45,720 in the full theory and the effective theory. 435 00:26:45,720 --> 00:26:47,460 So here's how I'm going to regulate them. 436 00:26:47,460 --> 00:26:50,430 We could do this different ways and it's 437 00:26:50,430 --> 00:26:54,120 useful to understand that various answers that we get 438 00:26:54,120 --> 00:26:55,807 are independent of the regulator. 439 00:26:59,710 --> 00:27:02,410 So I'm going to take p squared not equal to 0 440 00:27:02,410 --> 00:27:03,570 for the strange quark. 441 00:27:11,320 --> 00:27:13,330 So there's infrared divergences associated 442 00:27:13,330 --> 00:27:15,300 with the strange quirk and I'm going 443 00:27:15,300 --> 00:27:19,450 to take p squared not equal to 0 for them. 444 00:27:19,450 --> 00:27:22,840 I'm going to use dim reg for the heavy quark. 445 00:27:29,340 --> 00:27:32,010 So I could take the b quark to also be offshell. 446 00:27:32,010 --> 00:27:34,570 That would make the formulas even more complicated. 447 00:27:34,570 --> 00:27:35,987 So instead of doing that, I'm just 448 00:27:35,987 --> 00:27:38,460 going to allow epsilon to regulate the b quark. 449 00:27:38,460 --> 00:27:39,940 These guys are pretty easy to track 450 00:27:39,940 --> 00:27:44,000 so that won't be a problem. 451 00:27:44,000 --> 00:27:45,540 And I'm going to use Feynman gauge. 452 00:27:51,020 --> 00:27:53,830 I don't have to do that, but that's a nice gauge 453 00:27:53,830 --> 00:27:56,480 for doing calculations. 454 00:27:56,480 --> 00:27:59,810 So what are the diagrams where I have my photon 455 00:27:59,810 --> 00:28:02,893 and I have a vertex diagram like this 456 00:28:02,893 --> 00:28:04,810 and then I have wave function renormalization? 457 00:28:04,810 --> 00:28:08,800 And so let me, in the usual kind of cavalier way, 458 00:28:08,800 --> 00:28:11,380 denote the wave function renormalization, which 459 00:28:11,380 --> 00:28:15,220 is just the multiplication by the appropriate z factors. 460 00:28:15,220 --> 00:28:16,350 My diagram's like that. 461 00:28:20,190 --> 00:28:22,910 And this is a standard QCD calculation 462 00:28:22,910 --> 00:28:25,920 and we can carry it out. 463 00:28:29,572 --> 00:28:31,030 And what does the answer look like? 464 00:28:37,960 --> 00:28:51,270 So this diagram here has double logs 465 00:28:51,270 --> 00:28:56,040 and single logs that involve p squared, which 466 00:28:56,040 --> 00:29:02,500 is our IR regulator, and then it has some terms that are finite, 467 00:29:02,500 --> 00:29:04,750 which I'll just note by-- 468 00:29:04,750 --> 00:29:09,390 so p here is the momentum of our strange quark going out 469 00:29:09,390 --> 00:29:15,090 and pb is the momentum of our b quark coming in. 470 00:29:15,090 --> 00:29:19,110 And we're not going to talk much about these finite terms 471 00:29:19,110 --> 00:29:29,970 but what I mean by finite here is 472 00:29:29,970 --> 00:29:32,930 terms with no IR divergences. 473 00:29:32,930 --> 00:29:35,670 The IR divergences are these single logs and double logs 474 00:29:35,670 --> 00:29:41,070 and then there's some remainder that I can write that way. 475 00:29:41,070 --> 00:29:43,060 I'm expanding it for small p squared. 476 00:29:43,060 --> 00:29:45,060 So p squared is not equal to 0 but I 477 00:29:45,060 --> 00:29:46,770 take the limit p squared goes to 0 478 00:29:46,770 --> 00:29:49,230 and then that limit, these are the IR singularities. 479 00:29:49,230 --> 00:29:52,560 And then there's some function of b dot pb, which 480 00:29:52,560 --> 00:29:54,010 is just an order one thing. 481 00:29:54,010 --> 00:29:56,020 And then b squared, of course, is pb squared. 482 00:29:56,020 --> 00:29:58,145 So that's sort of the remaining kinematic variables 483 00:29:58,145 --> 00:30:00,696 this could possibly depend on. 484 00:30:00,696 --> 00:30:03,060 AUDIENCE: [INAUDIBLE]? 485 00:30:03,060 --> 00:30:04,138 IAIN STEWART: No. 486 00:30:04,138 --> 00:30:05,380 AUDIENCE: Is it finite? 487 00:30:05,380 --> 00:30:07,960 IAIN STEWART: It's really finite. 488 00:30:07,960 --> 00:30:11,890 So yeah. 489 00:30:11,890 --> 00:30:15,890 Yeah, so let's see. 490 00:30:15,890 --> 00:30:18,910 Yeah, I guess I carried out the so there 491 00:30:18,910 --> 00:30:26,380 is a z ten for the tensor and when I write this-- 492 00:30:26,380 --> 00:30:29,560 yeah, let's see. 493 00:30:29,560 --> 00:30:30,630 So, OK. 494 00:30:33,490 --> 00:30:34,960 Let me do this. 495 00:30:34,960 --> 00:30:36,130 I think it's better. 496 00:30:36,130 --> 00:30:39,640 So I added up all these graphs and I want to add one more. 497 00:30:42,730 --> 00:30:47,570 So there's a counterterm for the tensor field. 498 00:30:47,570 --> 00:30:49,400 So let me change what I was going 499 00:30:49,400 --> 00:30:50,560 to say and do it this way. 500 00:30:50,560 --> 00:30:53,690 So the sum of these four graphs is this. 501 00:30:53,690 --> 00:30:54,190 OK? 502 00:30:54,190 --> 00:30:56,260 So there's no UV divergences. 503 00:30:56,260 --> 00:30:58,843 And I'll just tell you what the z factors for these three are, 504 00:30:58,843 --> 00:31:00,677 then you could figure out what this graph is 505 00:31:00,677 --> 00:31:01,800 by subtracting the two. 506 00:31:07,140 --> 00:31:15,440 So the tensor current in QCD has a z factor, looks like that. 507 00:31:21,200 --> 00:31:28,070 There is a z for the heavy quark. 508 00:31:28,070 --> 00:31:30,390 And if I include the finite residue as well as 509 00:31:30,390 --> 00:31:37,275 the divergent pieces, this looks like this. 510 00:31:43,498 --> 00:31:45,040 I think I'm going to have to make one 511 00:31:45,040 --> 00:31:46,510 adjustment to my formula here. 512 00:32:11,000 --> 00:32:14,390 Let me just fix something here. 513 00:32:14,390 --> 00:32:15,230 Yeah. 514 00:32:15,230 --> 00:32:21,180 So if I want it to be the way I say, then what do I have to do? 515 00:32:21,180 --> 00:32:28,060 So this guy should be three halves 516 00:32:28,060 --> 00:32:33,000 and there will be one more divergence. 517 00:32:36,904 --> 00:32:39,560 I think that's right. 518 00:32:39,560 --> 00:32:41,990 So this 2 over epsilon ir here is 519 00:32:41,990 --> 00:32:46,160 that 1 over epsilon ir and the UV renormalization 520 00:32:46,160 --> 00:32:49,550 is taken care of once I have the z tensor. 521 00:32:49,550 --> 00:32:51,788 There's divergences in this diagram in these ones. 522 00:32:51,788 --> 00:32:53,330 but there's one left over, and that's 523 00:32:53,330 --> 00:32:54,690 taken care of by z tensor. 524 00:32:54,690 --> 00:33:00,020 So there's no 1 over epsilon UVs after I take care of this guy. 525 00:33:10,170 --> 00:33:15,170 So that's just by the definition of what z tensor should be 526 00:33:15,170 --> 00:33:17,030 and then everything else is as I wrote. 527 00:33:19,680 --> 00:33:21,420 So there's either divergences that 528 00:33:21,420 --> 00:33:24,270 are associated to this strange quirk offshellness, that's 529 00:33:24,270 --> 00:33:24,960 these two terms. 530 00:33:24,960 --> 00:33:27,480 There's an IR divergence associated, a heavy quark going 531 00:33:27,480 --> 00:33:29,698 onshell, that's that term. 532 00:33:29,698 --> 00:33:30,990 And that's the sum of diagrams. 533 00:33:35,810 --> 00:33:36,950 OK, so what about SCET? 534 00:33:42,550 --> 00:33:44,860 So there's going to be, in SCET, collinear diagrams 535 00:33:44,860 --> 00:33:46,510 and ultrasoft diagrams. 536 00:34:09,989 --> 00:34:14,560 So I'm going to use Feynman gauge for everything again. 537 00:34:14,560 --> 00:34:17,020 This is not something I have to do, 538 00:34:17,020 --> 00:34:18,580 but this is what I'm going to do. 539 00:34:21,679 --> 00:34:23,500 So let's start with the ultrasoft loops. 540 00:34:27,989 --> 00:34:29,310 So there's a vertex graph. 541 00:34:31,830 --> 00:34:36,440 So using the notation that we've adopted where the collinear 542 00:34:36,440 --> 00:34:39,940 quarks are dashed and the heavy quarks are double lines, 543 00:34:39,940 --> 00:34:42,790 we have a diagram that looks like that. 544 00:34:42,790 --> 00:34:45,210 There's some free factor. 545 00:34:45,210 --> 00:34:48,940 Let's focus on what the loop looks like. 546 00:34:48,940 --> 00:34:52,590 So this loop here, the k that's going through this loop 547 00:34:52,590 --> 00:34:55,610 is just a residual k. 548 00:34:55,610 --> 00:34:57,690 There's no label k for this loop because it's 549 00:34:57,690 --> 00:34:59,640 an ultrasoft gluon. 550 00:34:59,640 --> 00:35:02,670 So when we write down all the terms in this loop, 551 00:35:02,670 --> 00:35:04,350 it's just standard field theory. 552 00:35:04,350 --> 00:35:06,780 There's nothing special about it. 553 00:35:16,100 --> 00:35:21,890 Since I'm taking the strange quirk offshell, 554 00:35:21,890 --> 00:35:25,340 the propagator that I get is this shifted iconal propagator 555 00:35:25,340 --> 00:35:27,320 where basically the fact that it's offshell 556 00:35:27,320 --> 00:35:29,360 it gives me this extra term there. 557 00:35:29,360 --> 00:35:31,840 That's regulating some minor divergences. 558 00:35:31,840 --> 00:35:34,850 And we want that because we want to regulate the IR divergences 559 00:35:34,850 --> 00:35:39,920 in the same way in the full theory of the effective theory. 560 00:35:39,920 --> 00:35:45,130 This is proportional to that and if I put in all the factors. 561 00:36:16,700 --> 00:36:20,880 So it does have double logs of the p squared, 562 00:36:20,880 --> 00:36:24,128 doesn't have single logs of the p squared, 563 00:36:24,128 --> 00:36:25,920 and actually if we look at the double logs, 564 00:36:25,920 --> 00:36:27,363 the coefficients also don't match 565 00:36:27,363 --> 00:36:28,530 with what we had over there. 566 00:36:28,530 --> 00:36:32,320 And so there's going to be some other diagrams that 567 00:36:32,320 --> 00:36:36,200 are going to involve double logs of p squared as well. 568 00:36:36,200 --> 00:36:37,960 One thing that we can note here is 569 00:36:37,960 --> 00:36:40,578 that if you think about the scales in the problem, 570 00:36:40,578 --> 00:36:42,370 remember that our loop integral was totally 571 00:36:42,370 --> 00:36:43,870 homogeneous in the power counting, 572 00:36:43,870 --> 00:36:46,480 the lambdas were totally homogeneous. 573 00:36:46,480 --> 00:36:49,750 And if you think about p squared scaling 574 00:36:49,750 --> 00:36:52,840 like lambda squared, which is natural size 575 00:36:52,840 --> 00:36:56,740 for an external collinear momentum, all right. 576 00:36:56,740 --> 00:37:00,000 If p squared scales like lambda squared then 577 00:37:00,000 --> 00:37:02,710 so does p squared over m bar dot p. 578 00:37:05,510 --> 00:37:07,270 And this is a dimension one thing 579 00:37:07,270 --> 00:37:10,000 and this is the ultrasoft scale, right? 580 00:37:15,640 --> 00:37:22,445 For some ultra soft momentum and so 581 00:37:22,445 --> 00:37:24,070 if you want to look at these logarithms 582 00:37:24,070 --> 00:37:26,890 and you ask, what scale is this effective field theory 583 00:37:26,890 --> 00:37:28,592 diagram sensitive to? 584 00:37:28,592 --> 00:37:30,175 It's sensitive to the ultrasoft scale. 585 00:37:35,370 --> 00:37:37,710 So the logs are not large logs, they're 586 00:37:37,710 --> 00:37:41,020 order 1 logs as long as mu squared 587 00:37:41,020 --> 00:37:44,010 mu is of order lambda squared, which 588 00:37:44,010 --> 00:37:48,350 is the scale for the ultrasoft momentum. 589 00:37:48,350 --> 00:37:50,653 And that's what we expect from this ultrasoft diagram, 590 00:37:50,653 --> 00:37:53,070 that it's telling us about physics at the ultra soft scale 591 00:37:53,070 --> 00:37:55,980 and that's what we see setting things 592 00:37:55,980 --> 00:37:58,720 up, doing the calculation. 593 00:38:02,502 --> 00:38:04,710 So you could also think about a wave function diagram 594 00:38:04,710 --> 00:38:07,725 with an ultra soft gluon, but this guy 0's 595 00:38:07,725 --> 00:38:14,580 out since in Feynman gauge you just get n mu n mu, which is 0. 596 00:38:14,580 --> 00:38:20,955 So there's no so z for the collinear quark 597 00:38:20,955 --> 00:38:24,540 field from an ultrasoft loop is 0. 598 00:38:24,540 --> 00:38:27,340 That wouldn't necessarily be true in some other gauge, 599 00:38:27,340 --> 00:38:30,090 but in some other gauge this diagram would also change. 600 00:38:30,090 --> 00:38:31,390 All the diagrams would change. 601 00:38:31,390 --> 00:38:35,040 And so if there was a non-zero contribution in this diagram, 602 00:38:35,040 --> 00:38:39,600 it would just be taking care of gauge invariance. 603 00:38:39,600 --> 00:38:43,560 And then finally, there's ultrasoft loop 604 00:38:43,560 --> 00:38:46,560 on the heavy quark. 605 00:38:46,560 --> 00:38:50,490 And that's an HQET diagram, has nothing 606 00:38:50,490 --> 00:38:53,000 to do with the collinear quark. 607 00:38:53,000 --> 00:38:54,450 So from that diagram we would just 608 00:38:54,450 --> 00:39:02,680 get the z factor, which is the appropriate z factor in HQET 609 00:39:02,680 --> 00:39:05,350 with r regulator. 610 00:39:05,350 --> 00:39:08,260 If I specify ultraviolet and infrared divergences, 611 00:39:08,260 --> 00:39:11,020 that's this, and this infrared divergence here 612 00:39:11,020 --> 00:39:15,780 is actually exactly the same as the one that we had over here. 613 00:39:24,240 --> 00:39:26,890 OK, so the ultrasoft sector, there's 614 00:39:26,890 --> 00:39:28,300 nothing really tricky about it. 615 00:39:28,300 --> 00:39:32,423 It's just write down the diagrams, do the loops. 616 00:39:32,423 --> 00:39:33,840 AUDIENCE: So for the [INAUDIBLE],, 617 00:39:33,840 --> 00:39:37,510 isn't that IR divergences of those diagrams 618 00:39:37,510 --> 00:39:40,070 cancel with every automation [INAUDIBLE]?? 619 00:39:43,352 --> 00:39:45,310 IAIN STEWART: So what are we looking at, right? 620 00:39:45,310 --> 00:39:48,560 So it depends on what we're looking at here. 621 00:39:48,560 --> 00:39:52,778 And so, yes, in general that would be true, right? 622 00:39:52,778 --> 00:39:54,820 If you were calculating some cross-section, which 623 00:39:54,820 --> 00:39:57,727 was IR finite cross-section and that, 624 00:39:57,727 --> 00:39:59,560 of course, would depend on defining what you 625 00:39:59,560 --> 00:40:01,580 mean by measuring this quark. 626 00:40:01,580 --> 00:40:02,080 Right? 627 00:40:02,080 --> 00:40:04,360 So the IR divergence would become some physical scale 628 00:40:04,360 --> 00:40:07,330 like the mass of a jet. 629 00:40:10,720 --> 00:40:13,360 The IR divergences would turn into something physical 630 00:40:13,360 --> 00:40:17,420 if you put this into a physical cross-section. 631 00:40:17,420 --> 00:40:21,280 And that's exactly basically what would happen. 632 00:40:21,280 --> 00:40:24,310 These p squareds would become the mx squareds 633 00:40:24,310 --> 00:40:27,665 that we talked about when we talked about beta s gamma. 634 00:40:27,665 --> 00:40:29,290 But here what we're interested in doing 635 00:40:29,290 --> 00:40:30,650 is a matching calculation. 636 00:40:30,650 --> 00:40:32,940 So we fix the external state, still 637 00:40:32,940 --> 00:40:35,097 have a particular number of partons, 638 00:40:35,097 --> 00:40:37,180 and we want to compare the full theory calculation 639 00:40:37,180 --> 00:40:37,930 with the effective theory. 640 00:40:37,930 --> 00:40:39,472 The effective theory should reproduce 641 00:40:39,472 --> 00:40:41,530 the infrared divergences. 642 00:40:41,530 --> 00:40:43,510 So really all we care about is not 643 00:40:43,510 --> 00:40:46,090 that this is infrared finite, but rather 644 00:40:46,090 --> 00:40:52,533 that the effective theory has the same infrared divergences. 645 00:40:52,533 --> 00:40:53,950 And we'll see that in the end when 646 00:40:53,950 --> 00:40:55,908 you can think about is rather than canceling in 647 00:40:55,908 --> 00:40:58,890 for divergences as you're thinking in the full theory, 648 00:40:58,890 --> 00:41:02,080 we've matched the full theory onto the effective theory 649 00:41:02,080 --> 00:41:04,653 and that matching gives us the Wilson coefficient. 650 00:41:04,653 --> 00:41:07,070 And then we take effective theory and all the cancelations 651 00:41:07,070 --> 00:41:09,487 that you're thinking about between real and virtual graphs 652 00:41:09,487 --> 00:41:11,690 will occur in the effective theory too. 653 00:41:11,690 --> 00:41:13,730 So you can just think about the effective theory 654 00:41:13,730 --> 00:41:16,490 virtual-real graphs, then the cancelation will take place 655 00:41:16,490 --> 00:41:17,610 there. 656 00:41:17,610 --> 00:41:19,610 But then you're thinking about that cancelation 657 00:41:19,610 --> 00:41:22,140 later, at a lower scale, which is what you actually 658 00:41:22,140 --> 00:41:22,640 want to do. 659 00:41:22,640 --> 00:41:25,890 Because what I just was telling you about IR 660 00:41:25,890 --> 00:41:29,720 divergences becoming different things in the final state 661 00:41:29,720 --> 00:41:32,840 becomes very trivial once you're in the effective theory 662 00:41:32,840 --> 00:41:38,280 and we'll see sort of exactly how that works later on. 663 00:41:38,280 --> 00:41:42,080 But first we have to talk about linear graphs. 664 00:41:50,350 --> 00:41:52,350 So in the collinear graphs we had 665 00:41:52,350 --> 00:41:54,960 graphs like this one where we can take a gluon out 666 00:41:54,960 --> 00:41:56,560 of the vertex here. 667 00:41:56,560 --> 00:41:59,923 That corresponds to taking it out of the Wilson line. 668 00:41:59,923 --> 00:42:01,840 So let's label the graph in the following way. 669 00:42:01,840 --> 00:42:07,380 This would be k plus p this will be p This will be k. 670 00:42:07,380 --> 00:42:11,880 And if we follow our rules for what this is, 671 00:42:11,880 --> 00:42:20,460 we would write a sum over labels and then an integral 672 00:42:20,460 --> 00:42:23,830 over residuals. 673 00:42:23,830 --> 00:42:25,330 And let me put the residual integral 674 00:42:25,330 --> 00:42:27,550 in dimensional organization. 675 00:42:37,230 --> 00:42:43,800 Let me try to squeeze in everything, 676 00:42:43,800 --> 00:42:44,800 which won't be possible. 677 00:42:47,590 --> 00:42:49,830 So I'm going to write out the components 678 00:42:49,830 --> 00:42:52,450 to make clear whose residual on whose label. 679 00:42:56,930 --> 00:42:58,940 So in the denominator there's one more term. 680 00:43:11,390 --> 00:43:13,220 So the plus guys are always residual, 681 00:43:13,220 --> 00:43:15,980 the perp and the minuses are always labeled. 682 00:43:15,980 --> 00:43:18,940 The short way of saying what I'm trying to squeeze in here. 683 00:43:25,087 --> 00:43:27,420 So the denominator has these three terms, this guy here, 684 00:43:27,420 --> 00:43:28,795 this guy here, and this guy here. 685 00:43:35,660 --> 00:43:37,360 And when I label the diagram like this, 686 00:43:37,360 --> 00:43:42,100 you should think that each of these 687 00:43:42,100 --> 00:43:43,690 has a label and residual part. 688 00:43:48,240 --> 00:43:52,980 So k you can think of as a pair, k label, k residual for now. 689 00:43:55,490 --> 00:43:57,383 And remember the importance of doing this. 690 00:43:57,383 --> 00:43:59,050 The importance of doing this was related 691 00:43:59,050 --> 00:44:01,660 to always being able to identify what the lowest order 692 00:44:01,660 --> 00:44:04,190 term was on the right here. 693 00:44:04,190 --> 00:44:04,690 OK? 694 00:44:04,690 --> 00:44:06,357 And that actually becomes more important 695 00:44:06,357 --> 00:44:08,632 once you start to think about diagrams where 696 00:44:08,632 --> 00:44:10,840 you would add like an extra ultrasoft gluon somewhere 697 00:44:10,840 --> 00:44:12,740 in this picture. 698 00:44:12,740 --> 00:44:13,240 Then, of 699 00:44:13,240 --> 00:44:15,100 Course when that ultrasoft gluon feeds its way 700 00:44:15,100 --> 00:44:16,475 through this loop, you gotta make 701 00:44:16,475 --> 00:44:20,320 sure that it's only the lowest order piece that's showing up. 702 00:44:20,320 --> 00:44:24,370 In this case here, we just have a collinear loop 703 00:44:24,370 --> 00:44:28,960 and we don't have any sort of real ultrasoft 704 00:44:28,960 --> 00:44:34,930 momenta from ultrasoft particles besides the heavy quark, which 705 00:44:34,930 --> 00:44:37,708 is just an external particle to the loop. 706 00:44:37,708 --> 00:44:39,250 So what we want to do with this is we 707 00:44:39,250 --> 00:44:40,940 want to turn it back into an integral. 708 00:45:02,496 --> 00:45:03,880 Just let me say it this way. 709 00:45:07,680 --> 00:45:11,050 And if it wasn't for these restrictions here, 710 00:45:11,050 --> 00:45:15,210 then that's very easy, actually. 711 00:45:15,210 --> 00:45:17,740 and then I'll talk about why it's true. 712 00:45:17,740 --> 00:45:22,270 I claim that if we ignored the restrictions 713 00:45:22,270 --> 00:45:24,610 and those restrictions are ensuring that we don't double 714 00:45:24,610 --> 00:45:28,320 count between our collinear and our ultrasoft degrees 715 00:45:28,320 --> 00:45:28,820 of freedom. 716 00:45:28,820 --> 00:45:32,290 So they are important, but let's ignore them for a minute. 717 00:45:32,290 --> 00:45:44,315 If we do ignore them, we would just get the following. 718 00:45:50,750 --> 00:45:53,060 Where I just basically stop thinking about residuals 719 00:45:53,060 --> 00:45:56,660 and labels, write everything as a full momentum, 720 00:45:56,660 --> 00:45:59,960 and write down exactly the same thing I just wrote. 721 00:45:59,960 --> 00:46:02,450 So this is a full k squared and this 722 00:46:02,450 --> 00:46:04,010 is a full k plus p squared. 723 00:46:07,502 --> 00:46:08,002 OK? 724 00:46:11,000 --> 00:46:14,420 So what I'm going to do is I'm going to first ignore 725 00:46:14,420 --> 00:46:16,300 these restrictions and then I'm going 726 00:46:16,300 --> 00:46:18,710 to tell you how it would work, how this actually 727 00:46:18,710 --> 00:46:20,720 does turn into this. 728 00:46:20,720 --> 00:46:23,212 What are the rules for doing that. 729 00:46:23,212 --> 00:46:24,920 And then I'll come back and I'll tell you 730 00:46:24,920 --> 00:46:27,800 what these extra conditions do. 731 00:46:31,130 --> 00:46:35,270 So really this looks like maybe it's trivial, 732 00:46:35,270 --> 00:46:37,060 but we should think about it. 733 00:46:37,060 --> 00:46:40,410 And it's almost trivial but not quite. 734 00:46:40,410 --> 00:46:42,650 So really what we're doing here is 735 00:46:42,650 --> 00:46:45,620 we're combining back together labels and residual momentum, 736 00:46:45,620 --> 00:46:46,493 right? 737 00:46:46,493 --> 00:46:48,410 And the place that we have to worry about that 738 00:46:48,410 --> 00:46:50,450 is in the perp in the minus space. 739 00:46:50,450 --> 00:46:56,530 And recall we had the grid and the grid 740 00:46:56,530 --> 00:47:00,980 is sort of our way of guiding, our guidance 741 00:47:00,980 --> 00:47:02,820 to see how to put things back together. 742 00:47:09,720 --> 00:47:13,090 So we had vectors that lived in this space 743 00:47:13,090 --> 00:47:14,760 and this is the label and then there's 744 00:47:14,760 --> 00:47:16,500 the residual, if we want to point to some place 745 00:47:16,500 --> 00:47:17,160 in that space. 746 00:47:21,250 --> 00:47:25,710 This picture was like a Wilsonian effective field 747 00:47:25,710 --> 00:47:29,340 theory because the picture makes you think of sharp edges. 748 00:47:36,613 --> 00:47:38,530 But the real effective theory that we're doing 749 00:47:38,530 --> 00:47:42,700 is a continuum one and so you have 750 00:47:42,700 --> 00:47:44,170 to expand your brain a little bit 751 00:47:44,170 --> 00:47:46,170 and think that each of the boxes in this picture 752 00:47:46,170 --> 00:47:49,923 is actually an infinite space as well, 753 00:47:49,923 --> 00:47:52,090 because the residual space doesn't have restrictions 754 00:47:52,090 --> 00:47:54,210 like that that would spoil Lorentz symmetry. 755 00:48:17,540 --> 00:48:27,560 So each grid point really is specifying an infinite space 756 00:48:27,560 --> 00:48:28,600 of residual momenta. 757 00:48:34,780 --> 00:48:40,810 And it's R4 or Minkowski space, so the momenta components 758 00:48:40,810 --> 00:48:46,718 are real numbers and there's some rules. 759 00:48:46,718 --> 00:48:47,635 So what are the rules? 760 00:48:55,890 --> 00:49:03,240 So I'll tell you what the rules are without restrictions 761 00:49:03,240 --> 00:49:06,093 for now and then we'll come back and I'll 762 00:49:06,093 --> 00:49:08,010 tell you what the rules are with restrictions. 763 00:49:15,440 --> 00:49:17,810 So rule number one is the simplest, 764 00:49:17,810 --> 00:49:21,890 and it just says that say I had the following, 765 00:49:21,890 --> 00:49:24,250 I'll use a one dimensional notation. 766 00:49:24,250 --> 00:49:28,580 Say I had a sum over kl's and an integral over all kr's. 767 00:49:28,580 --> 00:49:30,710 Well that's just the same as not having split it up 768 00:49:30,710 --> 00:49:33,110 and doing an integral over everything. 769 00:49:33,110 --> 00:49:36,770 Because we split this thing into boxes and if we really 770 00:49:36,770 --> 00:49:41,238 integrate over all boxes and sum over all labels, 771 00:49:41,238 --> 00:49:43,280 then we should just get back the full integration 772 00:49:43,280 --> 00:49:44,155 over all the momenta. 773 00:49:46,640 --> 00:49:51,185 So that's true for each, for minus and perp momenta. 774 00:49:55,110 --> 00:49:59,010 But really what we have to do is we have to do this 775 00:49:59,010 --> 00:50:00,270 with some integrand, right? 776 00:50:00,270 --> 00:50:05,790 So the type of integrand that we have is following. 777 00:50:05,790 --> 00:50:12,070 If you think about the components 778 00:50:12,070 --> 00:50:15,040 that we're talking about here, which are the minus and perp 779 00:50:15,040 --> 00:50:18,580 ones, then in our integrand up there, 780 00:50:18,580 --> 00:50:23,150 there's no minus or perp residuals. 781 00:50:23,150 --> 00:50:25,555 So it's just a function of the labels. 782 00:50:35,430 --> 00:50:37,080 Right? 783 00:50:37,080 --> 00:50:38,580 And so what that says effectively 784 00:50:38,580 --> 00:50:43,720 is that this is a constant function in this box, 785 00:50:43,720 --> 00:50:44,595 in each of the boxes. 786 00:50:49,675 --> 00:50:52,050 And effectively, the way that you use this formula is you 787 00:50:52,050 --> 00:50:53,830 do the following. 788 00:50:53,830 --> 00:50:56,345 You say, well, if it's a constant function in the box, 789 00:50:56,345 --> 00:50:58,470 I could evaluate it at a different point in the box 790 00:50:58,470 --> 00:51:00,743 and I'd still get the same value. 791 00:51:00,743 --> 00:51:02,160 So in particular, I could evaluate 792 00:51:02,160 --> 00:51:05,340 it k label plus k residual and then I 793 00:51:05,340 --> 00:51:07,560 can use that formula there to say 794 00:51:07,560 --> 00:51:12,810 that this is just an integral with a continuous k 795 00:51:12,810 --> 00:51:14,970 of the function evaluated at that continuous k. 796 00:51:24,500 --> 00:51:28,332 So that's the way that number one works. 797 00:51:28,332 --> 00:51:30,790 We are integrating functions that are constant in the boxes 798 00:51:30,790 --> 00:51:33,130 and so then it's kind of trivial, 799 00:51:33,130 --> 00:51:35,120 how to put the boxes back together. 800 00:51:35,120 --> 00:51:35,620 Yeah? 801 00:51:35,620 --> 00:51:40,600 AUDIENCE: So this d kl division doesn't have kl plus--? 802 00:51:40,600 --> 00:51:42,830 IAIN STEWART: Yeah, so what I mean by it is-- 803 00:51:42,830 --> 00:51:43,330 yeah. 804 00:51:43,330 --> 00:51:48,280 So I'm using here for each minus or perp momenta, 805 00:51:48,280 --> 00:51:50,000 one dimensional notation. 806 00:51:50,000 --> 00:51:51,860 Yeah, and so I have three of these. 807 00:51:51,860 --> 00:51:53,150 I have to do this. 808 00:51:53,150 --> 00:51:56,733 And for each one of them it's true that what I'm saying. 809 00:51:56,733 --> 00:51:58,900 But of course, if I tried to write that on the board 810 00:51:58,900 --> 00:52:01,700 then it would confuse the point, I think, 811 00:52:01,700 --> 00:52:05,070 which is, in some sense, a simple point. 812 00:52:05,070 --> 00:52:05,620 OK? 813 00:52:05,620 --> 00:52:06,910 So this is, in some sense, saying 814 00:52:06,910 --> 00:52:08,680 that this whole split up that we were doing 815 00:52:08,680 --> 00:52:09,597 was not really needed. 816 00:52:09,597 --> 00:52:12,400 We could've just written a continuous integral and that's 817 00:52:12,400 --> 00:52:14,780 kind of what this is saying, right? 818 00:52:14,780 --> 00:52:16,780 We could have just written a continuous interval 819 00:52:16,780 --> 00:52:18,572 and not worried so much about all the split 820 00:52:18,572 --> 00:52:20,080 up that we were doing. 821 00:52:20,080 --> 00:52:22,240 The place that we have to be careful about 822 00:52:22,240 --> 00:52:24,483 is these restrictions, and we'll come back to that. 823 00:52:24,483 --> 00:52:26,650 And the other place that we have to be careful about 824 00:52:26,650 --> 00:52:28,450 is when you have the multiple expansion. 825 00:52:28,450 --> 00:52:32,170 But as long as you take care of that, then basically this 826 00:52:32,170 --> 00:52:36,080 is always happening. 827 00:52:36,080 --> 00:52:41,663 So the reason why this works is the following. 828 00:52:53,020 --> 00:52:57,580 For every label loop momentum that there is in any diagram, 829 00:52:57,580 --> 00:53:09,610 there's always going to be some corresponding residual that's 830 00:53:09,610 --> 00:53:14,950 not specified by delta functions in terms of external momenta. 831 00:53:14,950 --> 00:53:18,850 And effectively, therefore that we can absorb in order 832 00:53:18,850 --> 00:53:19,960 to do what we just said. 833 00:53:25,720 --> 00:53:29,170 So we can always go back from this discrete type notation 834 00:53:29,170 --> 00:53:30,737 back to a continuous notation. 835 00:53:30,737 --> 00:53:32,320 The discrete notation was just helping 836 00:53:32,320 --> 00:53:35,132 us to set up the expansion and be careful about it, 837 00:53:35,132 --> 00:53:37,090 but we can always go back to the continuous one 838 00:53:37,090 --> 00:53:39,805 because there's always a kr that has this property. 839 00:53:45,560 --> 00:53:46,060 OK. 840 00:53:50,280 --> 00:53:55,523 So now in general though, you might have 841 00:53:55,523 --> 00:53:56,690 some more complicated thing. 842 00:53:56,690 --> 00:53:58,315 And if I'm going to give you some rules 843 00:53:58,315 --> 00:54:00,950 I should give you a complete set. 844 00:54:00,950 --> 00:54:06,920 So we have to append our list of rules by the following one. 845 00:54:16,350 --> 00:54:19,110 So if I thought about doing what I just said over here 846 00:54:19,110 --> 00:54:22,060 but I went to some higher order, then what could happen? 847 00:54:22,060 --> 00:54:26,190 Well then these kr minus and kr our perps could show up. 848 00:54:26,190 --> 00:54:28,650 They'd never show up in the denominator of our propagators 849 00:54:28,650 --> 00:54:30,710 if they were just collinear propagators, 850 00:54:30,710 --> 00:54:33,840 but they could show up at some point in the numerator. 851 00:54:33,840 --> 00:54:38,070 And we need actually a rule like this one, which would be clear 852 00:54:38,070 --> 00:54:40,950 if we were regulating d kr in dimensional regularization, 853 00:54:40,950 --> 00:54:43,920 that the power of divergences are getting set to 0. 854 00:54:43,920 --> 00:54:47,190 And that is basically to maintain the run symmetry 855 00:54:47,190 --> 00:54:52,760 in the residual space we need a rule like this for j 856 00:54:52,760 --> 00:54:53,540 greater than 0. 857 00:55:15,380 --> 00:55:18,170 So that doesn't come into our calculation, 858 00:55:18,170 --> 00:55:22,080 but included for completeness. 859 00:55:22,080 --> 00:55:24,860 So when would these integral kr actually do 860 00:55:24,860 --> 00:55:26,300 something non-trivial? 861 00:55:26,300 --> 00:55:27,770 It would do something non-trivial 862 00:55:27,770 --> 00:55:29,960 if we had an ultrasoft loops and collinear 863 00:55:29,960 --> 00:55:31,060 loops at the same time. 864 00:55:44,677 --> 00:55:46,760 So in the case where we just have collinear loops, 865 00:55:46,760 --> 00:55:50,120 it's basically up to this issue about the restrictions 866 00:55:50,120 --> 00:55:51,230 that we'll talk about. 867 00:55:51,230 --> 00:55:52,370 It's basically that we just could 868 00:55:52,370 --> 00:55:53,787 have done everything as continuous 869 00:55:53,787 --> 00:55:58,040 and ignored this split. 870 00:55:58,040 --> 00:56:02,000 But if we have both ultrasoft particles that 871 00:56:02,000 --> 00:56:25,690 are participating through the loops and/or then in general 872 00:56:25,690 --> 00:56:28,930 these will give non-trivial loop momenta 873 00:56:28,930 --> 00:56:32,470 in the residual momenta. 874 00:56:32,470 --> 00:56:35,530 And hence there will be some that we can't just 875 00:56:35,530 --> 00:56:37,210 absorb in the fashion that we just said. 876 00:56:37,210 --> 00:56:41,830 So there will be, in this situation, residual momenta. 877 00:56:41,830 --> 00:56:44,350 Some residual momenta will be absorbed in the same way 878 00:56:44,350 --> 00:56:47,085 to turn the integrals into continuous ones, 879 00:56:47,085 --> 00:56:48,460 but other ones won't be absorbed. 880 00:56:59,078 --> 00:57:01,120 And that's because the ultrasoft propagators rate 881 00:57:01,120 --> 00:57:05,680 would involve the lr plus, the lr minus, and the lr perp 882 00:57:05,680 --> 00:57:09,380 in the denominator, so we don't have this rule to apply. 883 00:57:09,380 --> 00:57:11,590 We can't do what we said here with the constant boxes 884 00:57:11,590 --> 00:57:14,750 because now the functions are depending on that variable. 885 00:57:14,750 --> 00:57:17,600 So we just have to do the integral. 886 00:57:17,600 --> 00:57:23,655 So you could have something that looks like just 887 00:57:23,655 --> 00:57:26,630 in a schematic formula. 888 00:57:26,630 --> 00:57:30,540 Let's have there be 2 kr and an lr. 889 00:57:30,540 --> 00:57:33,950 We're in kind of an obvious notation. 890 00:57:33,950 --> 00:57:36,680 I'm saying that the function could depend on lr residual, 891 00:57:36,680 --> 00:57:38,500 it doesn't depend on kr residual. 892 00:57:38,500 --> 00:57:41,170 So we absorb kr back into the sum to make it continuous 893 00:57:41,170 --> 00:57:42,920 and then this integral we just have to do. 894 00:57:54,076 --> 00:57:58,245 OK, but we're still, in the end, just doing an integral. 895 00:58:05,310 --> 00:58:11,300 So this guy come from ultrasoft propagators, for example. 896 00:58:16,510 --> 00:58:17,330 OK? 897 00:58:17,330 --> 00:58:20,180 So those are the different cases that you can get. 898 00:58:20,180 --> 00:58:23,097 But nevertheless, even if you have ultrasoft particles 899 00:58:23,097 --> 00:58:24,680 and propagators floating around, there 900 00:58:24,680 --> 00:58:27,920 will always be a residual momentum associated to 901 00:58:27,920 --> 00:58:29,990 of what we were doing over here that you 902 00:58:29,990 --> 00:58:35,610 can absorb in the same way I stated, 903 00:58:35,610 --> 00:58:38,262 so let's proceed along these lines and see where it takes us 904 00:58:38,262 --> 00:58:39,720 and then we'll come back and put in 905 00:58:39,720 --> 00:58:42,960 this additional restrictions. 906 00:58:42,960 --> 00:58:45,040 AUDIENCE: Do we ever have to do a discrete sum? 907 00:58:45,040 --> 00:58:48,310 Like is there a sum over ll? 908 00:58:48,310 --> 00:58:51,476 IAIN STEWART: Yeah, so you never have to do a discrete sum. 909 00:58:51,476 --> 00:58:52,393 AUDIENCE: That's good. 910 00:58:52,393 --> 00:58:54,598 IAIN STEWART: Yep. 911 00:58:54,598 --> 00:58:56,640 The discrete sum is really just a way of thinking 912 00:58:56,640 --> 00:58:59,760 and I'll show you in a minute that you even 913 00:58:59,760 --> 00:59:02,620 can avoid thinking about it once you know what to do. 914 00:59:02,620 --> 00:59:04,480 So it's guiding you towards the right answer 915 00:59:04,480 --> 00:59:06,063 but it's not really something that you 916 00:59:06,063 --> 00:59:07,860 have to think about this grid picture. 917 00:59:07,860 --> 00:59:09,490 It's really just if you get confused, 918 00:59:09,490 --> 00:59:11,340 you can always think about it but if you're not confused, 919 00:59:11,340 --> 00:59:12,715 you don't have to think about it. 920 00:59:21,590 --> 00:59:28,300 OK, so this guy is what we said, proportional to what we said. 921 00:59:28,300 --> 00:59:38,030 It's just in this case, the reason why it was so simple 922 00:59:38,030 --> 00:59:42,230 is because k residual is one component of this guy, 923 00:59:42,230 --> 00:59:43,980 and then there's these other three, right? 924 00:59:43,980 --> 00:59:46,677 And the other three, we just absorb 925 00:59:46,677 --> 00:59:48,260 the sum and the integral back together 926 00:59:48,260 --> 00:59:50,510 and there's nothing further to talk about. 927 00:59:50,510 --> 00:59:56,600 So that's why this is, in some sense, very simple. 928 00:59:56,600 --> 01:00:02,480 So we just do that, and what do we get? 929 01:00:06,810 --> 01:00:27,190 Get some 1 over epsilons, we get some logs of p squared, 930 01:00:27,190 --> 01:00:30,678 and I'm putting in even the constant term 931 01:00:30,678 --> 01:00:31,720 since it's pretty simple. 932 01:00:31,720 --> 01:00:34,760 4 minus pi squared over 6. 933 01:00:34,760 --> 01:00:39,160 So if you think about where this came from, 934 01:00:39,160 --> 01:00:41,470 there was a place that this Wilson line 935 01:00:41,470 --> 01:00:43,250 came from was attaching this guy over here 936 01:00:43,250 --> 01:00:47,050 and then integrating out that offshell propagator. 937 01:00:47,050 --> 01:00:48,442 So this is a vertex diagram. 938 01:00:48,442 --> 01:00:50,650 We had a vertex diagram, which is an ultrasoft gluon, 939 01:00:50,650 --> 01:00:53,410 and now we've added an identical type up topology. 940 01:00:53,410 --> 01:00:55,870 Except we drew it different because this line 941 01:00:55,870 --> 01:00:58,515 was offshell so we drew it as a Wilson line 942 01:00:58,515 --> 01:01:00,440 and that's the right way of thinking about it. 943 01:01:00,440 --> 01:01:02,253 But if you think about where it came from, 944 01:01:02,253 --> 01:01:03,670 it was the same topology and we've 945 01:01:03,670 --> 01:01:05,840 added another vertex diagram. 946 01:01:05,840 --> 01:01:07,540 So the effective theory has two type 947 01:01:07,540 --> 01:01:09,142 of vertex diagrams, this collinear 948 01:01:09,142 --> 01:01:10,600 one and the ultrasoft one and we're 949 01:01:10,600 --> 01:01:12,640 going to add them together. 950 01:01:12,640 --> 01:01:15,100 And in this one, you see the logs 951 01:01:15,100 --> 01:01:17,170 are minimized at a different scale, mu squared 952 01:01:17,170 --> 01:01:19,690 of order p squared, which is the right scale for a collinear 953 01:01:19,690 --> 01:01:20,190 loop. 954 01:01:24,032 --> 01:01:25,740 Because remember, the collinear particles 955 01:01:25,740 --> 01:01:32,730 lived at a larger p squared than the ultrasoft ones. 956 01:01:32,730 --> 01:01:37,010 So there's a larger scale that would minimize these loops, 957 01:01:37,010 --> 01:01:38,010 it's a collinear scale. 958 01:01:42,260 --> 01:01:44,530 So the effective theory is capturing the physics 959 01:01:44,530 --> 01:01:45,280 at that scale. 960 01:01:48,577 --> 01:01:50,290 Squeeze in another diagram here. 961 01:01:53,440 --> 01:01:56,590 If we do the collinear wave function renormalization, 962 01:01:56,590 --> 01:01:57,610 then this is non-zero. 963 01:01:57,610 --> 01:02:00,040 This was 0 for the ultrasoft gluon 964 01:02:00,040 --> 01:02:02,890 and but for the collinear gluon, it's non-zero 965 01:02:02,890 --> 01:02:06,190 and it's exactly, actually, the same as the full theory. 966 01:02:10,731 --> 01:02:12,370 It's the same as massless QCD. 967 01:02:17,920 --> 01:02:20,090 So you can do the diagram using the effective theory 968 01:02:20,090 --> 01:02:23,500 Feynman rules, then that's what you find. 969 01:02:23,500 --> 01:02:25,510 But you could also understand why that's true. 970 01:02:28,520 --> 01:02:31,480 And reason why it's true is nothing in this diagram really 971 01:02:31,480 --> 01:02:34,230 specifies a frame. 972 01:02:34,230 --> 01:02:35,880 We called all the particles collinear, 973 01:02:35,880 --> 01:02:38,755 but it's not attached to anything. 974 01:02:38,755 --> 01:02:40,380 So we could just take the whole diagram 975 01:02:40,380 --> 01:02:44,268 and boost back to the frame where everything's kind of soft 976 01:02:44,268 --> 01:02:46,560 and then we would usually be thinking about it in terms 977 01:02:46,560 --> 01:02:47,650 of a full theory field. 978 01:02:47,650 --> 01:02:56,640 So that's effectively why this diagram 979 01:02:56,640 --> 01:03:00,323 like this that doesn't have any reference, unlike this one 980 01:03:00,323 --> 01:03:01,740 which has a reference because it's 981 01:03:01,740 --> 01:03:04,470 attached to the heavy quark. 982 01:03:04,470 --> 01:03:05,850 That's why it's the same as QCD. 983 01:03:12,910 --> 01:03:13,695 Yeah? 984 01:03:13,695 --> 01:03:15,820 AUDIENCE: How do you know what the epsilon position 985 01:03:15,820 --> 01:03:17,460 for the Wilson line is set? 986 01:03:17,460 --> 01:03:18,160 IAIN STEWART: Oh, for this one? 987 01:03:18,160 --> 01:03:19,077 AUDIENCE: m bar dot k. 988 01:03:19,077 --> 01:03:21,270 IAIN STEWART: Yeah, for this one. 989 01:03:21,270 --> 01:03:27,570 Yeah, for this guy it actually doesn't depend on the i epsilon 990 01:03:27,570 --> 01:03:31,160 prescription up to a-- 991 01:03:31,160 --> 01:03:33,247 yeah, if I remember correctly. 992 01:03:37,720 --> 01:03:39,010 Yeah. 993 01:03:39,010 --> 01:03:41,080 That's true, I believe, when I'm being 994 01:03:41,080 --> 01:03:44,710 a little bit cavalier with it but once I put it the zero bin 995 01:03:44,710 --> 01:03:48,922 restrictions, I'm not sure if that's true anymore. 996 01:03:48,922 --> 01:03:50,628 AUDIENCE: OK? 997 01:03:50,628 --> 01:03:51,420 IAIN STEWART: Yeah. 998 01:03:51,420 --> 01:03:53,670 I mean, really what solves that is that in a minute 999 01:03:53,670 --> 01:03:54,870 I'm going to be talking about the fact 1000 01:03:54,870 --> 01:03:56,355 that there is a subtraction term here 1001 01:03:56,355 --> 01:03:58,063 and once you put the subtraction term in, 1002 01:03:58,063 --> 01:03:59,370 that i0 is not relevant. 1003 01:04:02,710 --> 01:04:05,240 But I think even if you do this diagram with arbitrary i0-- 1004 01:04:09,590 --> 01:04:14,308 because what's happening is m bar dot k going to 0 1005 01:04:14,308 --> 01:04:16,100 is related to some of these 1 over epsilons 1006 01:04:16,100 --> 01:04:17,930 and we'll talk about that more in a minute. 1007 01:04:25,710 --> 01:04:27,320 Yeah, I'm not 100% sure. 1008 01:04:27,320 --> 01:04:30,512 It could be that there's some sign that might flip. 1009 01:04:30,512 --> 01:04:32,330 I'm not 100% sure. 1010 01:04:32,330 --> 01:04:34,610 AUDIENCE: Even if they get the wrong side, 1011 01:04:34,610 --> 01:04:36,110 the difference would be subtraction. 1012 01:04:36,110 --> 01:04:39,080 IAIN STEWART: I guess I know that-- 1013 01:04:39,080 --> 01:04:40,100 yeah. 1014 01:04:40,100 --> 01:04:41,750 What I know is that-- 1015 01:04:41,750 --> 01:04:43,520 yeah, let me answer your question later. 1016 01:04:43,520 --> 01:04:44,540 It'll be easier because I'm trying 1017 01:04:44,540 --> 01:04:46,430 to say a bunch of things that depend on something else 1018 01:04:46,430 --> 01:04:47,638 that I haven't explained yet. 1019 01:04:50,580 --> 01:04:54,087 So what are the other possible topologies we could write down? 1020 01:04:54,087 --> 01:04:55,670 So we could write down this one, where 1021 01:04:55,670 --> 01:04:57,500 we take two attachments in the Wilson line 1022 01:04:57,500 --> 01:04:59,870 and just loop them back up but that's 1023 01:04:59,870 --> 01:05:01,620 proportional m bar squared and so that's 1024 01:05:01,620 --> 01:05:02,980 0 in our Feynman gauge. 1025 01:05:05,840 --> 01:05:11,750 And likewise, there's a looping back up in the vertex 1026 01:05:11,750 --> 01:05:13,700 in the wave function renormalization, 1027 01:05:13,700 --> 01:05:21,520 but this guys scale is power law divergent and so we can just 1028 01:05:21,520 --> 01:05:24,735 set it to 0 and dim reg. 1029 01:05:24,735 --> 01:05:26,110 You don't have to worry about it. 1030 01:05:30,400 --> 01:05:32,700 OK, so that's all the diagrams. 1031 01:05:32,700 --> 01:05:35,595 Let's think about doing matching, 1032 01:05:35,595 --> 01:05:39,508 i.e let's think about comparing QCD and SCET by adding up 1033 01:05:39,508 --> 01:05:40,050 the diagrams. 1034 01:05:50,070 --> 01:05:52,050 In QCD we carried out the randomization, 1035 01:05:52,050 --> 01:05:55,670 we added the z for the tensor current. 1036 01:05:55,670 --> 01:05:59,160 Let me just write again the answer looked like. 1037 01:06:19,460 --> 01:06:23,090 In SCET we didn't carry out renormalization yet, 1038 01:06:23,090 --> 01:06:26,450 so let me call this the bare SCET result for now. 1039 01:06:33,860 --> 01:06:36,670 Once we add the ultrasoft and collinear diagrams together, 1040 01:06:36,670 --> 01:06:39,880 the logs of p squared match up exactly with the full theory. 1041 01:06:47,310 --> 01:06:51,243 So this is the first sign really that it makes sense 1042 01:06:51,243 --> 01:06:52,910 to be thinking about adding these loops. 1043 01:06:52,910 --> 01:06:54,830 Even though they were the same topology, 1044 01:06:54,830 --> 01:06:58,235 we are correctly reproducing those logs of p 1045 01:06:58,235 --> 01:06:59,360 squared in the full theory. 1046 01:07:03,677 --> 01:07:05,135 And then there's some other pieces. 1047 01:07:22,222 --> 01:07:23,680 I'll write out all the other pieces 1048 01:07:23,680 --> 01:07:26,262 so you see what they look like. 1049 01:07:26,262 --> 01:07:28,220 Well, maybe I won't I won't write the constant. 1050 01:07:41,390 --> 01:07:43,890 So there's all the effective field theory terms. 1051 01:07:43,890 --> 01:07:49,710 So these terms here we can match up with these terms here. 1052 01:07:53,720 --> 01:07:56,050 So that's good. 1053 01:07:56,050 --> 01:08:00,363 These terms here, which remember in the full theory were finite, 1054 01:08:00,363 --> 01:08:02,530 and these terms here, which in the effective theory, 1055 01:08:02,530 --> 01:08:04,300 are finite, the difference of those 1056 01:08:04,300 --> 01:08:06,207 is going to give the Wilson coefficient. 1057 01:08:23,915 --> 01:08:25,540 Now we said that the Wilson coefficient 1058 01:08:25,540 --> 01:08:29,720 could be a function of p bar, so what's going on with that? 1059 01:08:29,720 --> 01:08:32,109 Well, if you look at momentum conservation 1060 01:08:32,109 --> 01:08:36,024 in this process of beta s gamma-- 1061 01:08:36,024 --> 01:08:39,450 so I probably should've said this earlier. 1062 01:08:39,450 --> 01:08:40,970 So when you look at beta s gamma, 1063 01:08:40,970 --> 01:08:43,399 if you look at momentum conservation then 1064 01:08:43,399 --> 01:08:46,220 the p minus of the strange quark has 1065 01:08:46,220 --> 01:08:48,109 to be equal to the p minus of the b quark, 1066 01:08:48,109 --> 01:08:51,340 but that's just mb. 1067 01:08:51,340 --> 01:08:55,670 OK, so actually p minus is equal to mb by kinematics. 1068 01:09:00,630 --> 01:09:03,080 So mb's in this result here you shouldn't think 1069 01:09:03,080 --> 01:09:07,640 of as p minuses, and that's this p bar that was in our Wilson 1070 01:09:07,640 --> 01:09:09,529 coefficient is just getting set to mb 1071 01:09:09,529 --> 01:09:14,240 because of some delta functions that are specifying kinematics. 1072 01:09:14,240 --> 01:09:17,149 So that leaves the 1 over epsilon terms. 1073 01:09:17,149 --> 01:09:19,370 And so what we'd like is that those terms are 1074 01:09:19,370 --> 01:09:21,412 associated to renormalization. 1075 01:09:27,588 --> 01:09:28,880 Of the effective theory, right? 1076 01:09:28,880 --> 01:09:30,689 I wrote that the effective theory was bare. 1077 01:09:48,310 --> 01:09:49,899 But if I want to do that, then I have 1078 01:09:49,899 --> 01:09:52,540 to ensure that all these epsilons that 1079 01:09:52,540 --> 01:09:56,910 are appearing here are really ultraviolet divergences. 1080 01:09:56,910 --> 01:09:59,610 If they're infrared divergences, then doing that 1081 01:09:59,610 --> 01:10:01,850 doesn't make sense. 1082 01:10:01,850 --> 01:10:04,380 And that's actually the remaining issue 1083 01:10:04,380 --> 01:10:06,080 that we have to deal with. 1084 01:10:06,080 --> 01:10:10,120 I just wrote epsilon, that means I'm ignorant to what they are. 1085 01:10:10,120 --> 01:10:11,692 And if I knew this one was epsilon 1086 01:10:11,692 --> 01:10:13,400 IR because it came from the wave function 1087 01:10:13,400 --> 01:10:14,817 renormalization of the heavy quark 1088 01:10:14,817 --> 01:10:16,830 and that was the same on both sides. 1089 01:10:16,830 --> 01:10:19,200 It was the same diagram, it was the wave function 1090 01:10:19,200 --> 01:10:21,742 renormalization diagram in the full and the effective theory. 1091 01:10:21,742 --> 01:10:23,220 So I could match up that one. 1092 01:10:23,220 --> 01:10:26,130 These ones just came out, but it turns out 1093 01:10:26,130 --> 01:10:28,890 that so far with what we've done, some of these epsilon 1094 01:10:28,890 --> 01:10:31,330 here are IR. 1095 01:10:31,330 --> 01:10:37,912 And so the IR divergences aren't matching up 1096 01:10:37,912 --> 01:10:39,370 and the reason is because we didn't 1097 01:10:39,370 --> 01:10:43,295 put in those restrictions on our sum over labels. 1098 01:11:26,610 --> 01:11:27,120 OK. 1099 01:11:27,120 --> 01:11:32,250 So we have these restrictions, k label not equal to zero 1100 01:11:32,250 --> 01:11:34,080 and k label not equal to minus pl. 1101 01:11:36,780 --> 01:11:39,750 Those are the restrictions that I'm talking about. 1102 01:11:39,750 --> 01:11:41,700 The place that those restrictions came from 1103 01:11:41,700 --> 01:11:45,390 was k was the momentum of the gluon. 1104 01:11:45,390 --> 01:11:49,620 kl not equal to 0 is saying that this is the restriction 1105 01:11:49,620 --> 01:11:51,486 that the gluon is collinear. 1106 01:11:55,140 --> 01:11:59,810 Because kl equals 0 is the ultrasoft gluon 1107 01:11:59,810 --> 01:12:01,900 and this is the restriction that the fermion is 1108 01:12:01,900 --> 01:12:08,240 collinear in the loop and that's why there was two of them. 1109 01:12:13,940 --> 01:12:16,510 So these are called zero bins. 1110 01:12:16,510 --> 01:12:20,710 Zero because it's where the ultrasoft momentum lives 1111 01:12:20,710 --> 01:12:23,590 and from the point of view of collinear, that's zero. 1112 01:12:28,930 --> 01:12:31,840 Imposing these restrictions is removing the zero 1113 01:12:31,840 --> 01:12:34,750 bin, if you like and what these restrictions do 1114 01:12:34,750 --> 01:12:39,220 is they avoid double counting and the way I've 1115 01:12:39,220 --> 01:12:40,860 said it that's, I think, clear. 1116 01:12:44,230 --> 01:12:46,780 So far in our calculation we haven't avoided double counting 1117 01:12:46,780 --> 01:12:47,738 and that's the problem. 1118 01:13:04,240 --> 01:13:04,910 OK. 1119 01:13:04,910 --> 01:13:07,280 So we have to modify our rule or we 1120 01:13:07,280 --> 01:13:10,040 extend our rule to include the case where 1121 01:13:10,040 --> 01:13:11,627 we have these restrictions. 1122 01:13:16,010 --> 01:13:23,243 In an extended version of rule two that house restrictions. 1123 01:13:38,060 --> 01:13:40,616 So really what we want to do is that 1124 01:13:40,616 --> 01:13:44,633 and we want to think about that as an integral. 1125 01:13:44,633 --> 01:13:46,550 So here's how we can manipulate these to think 1126 01:13:46,550 --> 01:13:49,440 about it as an integral. 1127 01:13:49,440 --> 01:13:57,930 That sum over all kl's, but then we'll 1128 01:13:57,930 --> 01:14:12,990 subtract the limit of this f where we take the f 1129 01:14:12,990 --> 01:14:16,140 and we let the kl go to the place. 1130 01:14:16,140 --> 01:14:18,480 So we integrate over everywhere, including the place 1131 01:14:18,480 --> 01:14:22,200 we don't want to go, and then we subtract it back. 1132 01:14:24,870 --> 01:14:26,550 So what is this fl of k? 1133 01:14:34,630 --> 01:14:36,230 This f of kl goes to 0. 1134 01:14:38,750 --> 01:14:43,910 it's defined by taking the scaling limit of the collinear 1135 01:14:43,910 --> 01:14:46,100 momenta towards the ultrasoft. 1136 01:14:53,880 --> 01:14:55,780 So you take your collinear momenta, 1137 01:14:55,780 --> 01:14:58,060 which are the minus in the perp here, 1138 01:14:58,060 --> 01:15:01,720 and you scale them towards an ultrasoft momentum in whatever 1139 01:15:01,720 --> 01:15:06,760 components, i.e you start counting the kn's as order 1140 01:15:06,760 --> 01:15:18,617 lambda squared and you keep the leading order piece 1141 01:15:18,617 --> 01:15:20,950 or you keep the piece that's the same size as this term. 1142 01:15:34,930 --> 01:15:38,530 And then that defines what this f is. 1143 01:15:38,530 --> 01:15:40,510 Once you take that limit and you expand, 1144 01:15:40,510 --> 01:15:41,950 then that's what the f is. 1145 01:15:49,370 --> 01:15:53,120 So what you're doing here by doing this procedure 1146 01:15:53,120 --> 01:15:55,280 is you're basically setting things up 1147 01:15:55,280 --> 01:15:59,540 so that this guy is integrated over in a way 1148 01:15:59,540 --> 01:16:01,460 that we can combine back into an integral. 1149 01:16:01,460 --> 01:16:04,130 But then we have to subtract an overlap 1150 01:16:04,130 --> 01:16:07,645 of when that integral would go into the ultrasoft region. 1151 01:16:07,645 --> 01:16:09,020 But the overlap we're subtracting 1152 01:16:09,020 --> 01:16:20,288 is also an integral, so we have a difference of integrals. 1153 01:16:20,288 --> 01:16:22,080 And you should think of the second integral 1154 01:16:22,080 --> 01:16:27,120 as integrating over the square where the zero bin was. 1155 01:16:27,120 --> 01:16:29,730 So if you think about our picture 1156 01:16:29,730 --> 01:16:33,150 where the collinears were up here, ultrasofts are down here 1157 01:16:33,150 --> 01:16:36,390 and you thought about there being some box, 1158 01:16:36,390 --> 01:16:38,850 you're taking this scaling limit when this guy goes down 1159 01:16:38,850 --> 01:16:41,730 into that box. 1160 01:16:41,730 --> 01:16:44,220 You add up all the boxes, that's this, 1161 01:16:44,220 --> 01:16:46,360 and then you subtract out that box again. 1162 01:16:46,360 --> 01:16:49,350 And that avoids having a double counting in that region. 1163 01:16:54,290 --> 01:16:58,490 So then this guy here, you can do the same kind of trick 1164 01:16:58,490 --> 01:16:59,190 as before. 1165 01:16:59,190 --> 01:17:02,165 So continuing with the equation. 1166 01:17:18,558 --> 01:17:21,100 kl is going to 0, well, we can think of it just as a function 1167 01:17:21,100 --> 01:17:22,867 of kr's. 1168 01:17:22,867 --> 01:17:24,700 And so effectively what we get in the end is 1169 01:17:24,700 --> 01:17:28,600 an integral over all k of f of k, 1170 01:17:28,600 --> 01:17:32,920 the full f, evaluated with a continuous momentum 1171 01:17:32,920 --> 01:17:39,190 minus some f that's expanded and avoids there 1172 01:17:39,190 --> 01:17:41,570 being overlap in this box. 1173 01:17:41,570 --> 01:17:43,600 So rather than having these discrete sum 1174 01:17:43,600 --> 01:17:46,700 with the restriction, we have a difference of integrals 1175 01:17:46,700 --> 01:17:50,260 and this subtraction term avoids the overlap in that region. 1176 01:17:53,380 --> 01:17:55,620 So all the discrete sums are good for is 1177 01:17:55,620 --> 01:17:59,020 a means of figuring out what limits you need to take 1178 01:17:59,020 --> 01:18:00,682 to generate these subtractions. 1179 01:18:00,682 --> 01:18:02,140 Once you've done that, everything's 1180 01:18:02,140 --> 01:18:03,100 a continuous integral. 1181 01:18:22,690 --> 01:18:24,941 And this is called the zero bin subtraction. 1182 01:18:36,010 --> 01:18:39,120 OK, so if you like, one way of phrasing what's going on 1183 01:18:39,120 --> 01:18:41,580 is that the collinear propagators are really 1184 01:18:41,580 --> 01:18:42,750 distributions. 1185 01:18:42,750 --> 01:18:46,560 They're distributions that know that they should 1186 01:18:46,560 --> 01:18:49,290 have a subtraction in order to not overlap the other region 1187 01:18:49,290 --> 01:18:51,423 where we have another degree of freedom. 1188 01:18:51,423 --> 01:18:52,840 So you can think about it that way 1189 01:18:52,840 --> 01:18:57,740 and having this sum it's just a way of encoding that. 1190 01:18:57,740 --> 01:19:00,150 But in the end, it looks like some kind of plus function 1191 01:19:00,150 --> 01:19:02,745 where you have a subtraction. 1192 01:19:02,745 --> 01:19:04,620 AUDIENCE: So it seems like in the second term 1193 01:19:04,620 --> 01:19:08,540 you would have to relearn the expression because initially 1194 01:19:08,540 --> 01:19:12,240 in the [INAUDIBLE] k you ignore kr minus and kr perps. 1195 01:19:12,240 --> 01:19:15,070 IAIN STEWART: I'll show you how it works in an example 1196 01:19:15,070 --> 01:19:15,570 up there. 1197 01:19:18,015 --> 01:19:19,140 So let's go to our example. 1198 01:19:24,740 --> 01:19:27,357 Yeah, so what you just said is not quite the way 1199 01:19:27,357 --> 01:19:28,440 you should think about it. 1200 01:19:28,440 --> 01:19:29,982 You should think about it that you've 1201 01:19:29,982 --> 01:19:31,920 given the effective theory f and now 1202 01:19:31,920 --> 01:19:35,910 I'm saying that that f, as given, with its momentum as 1203 01:19:35,910 --> 01:19:39,510 given still has an overlap with the ultrasoft region 1204 01:19:39,510 --> 01:19:41,380 that I want to subtract. 1205 01:19:41,380 --> 01:19:43,080 So I'm taking a limit of that f, I'm 1206 01:19:43,080 --> 01:19:44,370 not add anything back to it. 1207 01:19:47,530 --> 01:19:50,550 So this guy, integral dk. 1208 01:19:58,952 --> 01:20:00,285 So this is what we wrote before. 1209 01:20:05,290 --> 01:20:09,785 And now if I take the ultrasoft limit of it, of the k, 1210 01:20:09,785 --> 01:20:10,785 then I would write this. 1211 01:20:22,690 --> 01:20:23,370 OK? 1212 01:20:23,370 --> 01:20:26,610 So when I take the scaling limit, 1213 01:20:26,610 --> 01:20:28,560 if you think about k squared, k plus, 1214 01:20:28,560 --> 01:20:31,650 k minus, minus k perp squared, when I take the scaling 1215 01:20:31,650 --> 01:20:33,490 limit of all momentum being ultrasoft, 1216 01:20:33,490 --> 01:20:35,910 the components of the k are still homogeneous. 1217 01:20:35,910 --> 01:20:38,430 So there's nothing there to expand. 1218 01:20:38,430 --> 01:20:41,610 Some expansion happened in this k plus p term. 1219 01:20:41,610 --> 01:20:44,160 One of our n bar dot k changes its power counting, 1220 01:20:44,160 --> 01:20:47,427 but it's still n bar dot k. 1221 01:20:47,427 --> 01:20:48,760 There's nothing to expand there. 1222 01:20:48,760 --> 01:20:50,010 So this is what the subtraction looks 1223 01:20:50,010 --> 01:20:51,576 like and then the numerator, the n bar dot 1224 01:20:51,576 --> 01:20:53,326 k can be dropped relative to the n bar dot 1225 01:20:53,326 --> 01:20:56,695 p, which is large and external. 1226 01:20:56,695 --> 01:20:59,400 OK, so this is taking the ultrasoft limit of this. 1227 01:21:03,176 --> 01:21:04,828 AUDIENCE: Do you have to subtract off 1228 01:21:04,828 --> 01:21:07,385 the the ultrasoft limit of the quark as well? 1229 01:21:07,385 --> 01:21:08,760 IAIN STEWART: Yeah, so in general 1230 01:21:08,760 --> 01:21:11,427 I would have to subtract off the ultrasoft limit of the quark as 1231 01:21:11,427 --> 01:21:12,490 well. 1232 01:21:12,490 --> 01:21:14,220 And when I do that, what I find is 1233 01:21:14,220 --> 01:21:17,940 a term that's power suppressed and so I drop it. 1234 01:21:17,940 --> 01:21:19,940 But in general, I would have to do that as well. 1235 01:21:19,940 --> 01:21:20,482 That's right. 1236 01:21:23,350 --> 01:21:26,450 And so it looks like an ultrasoft diagram 1237 01:21:26,450 --> 01:21:31,100 except it's got this n bar dot k inside of the v dot k 1238 01:21:31,100 --> 01:21:34,082 that we had in the ultrasoft diagram. 1239 01:21:34,082 --> 01:21:36,290 And if you do a power counting with the loop momentum 1240 01:21:36,290 --> 01:21:38,540 scaling as ultrasoft, then you have this piece is of the order 1241 01:21:38,540 --> 01:21:40,123 the same size as this piece and that's 1242 01:21:40,123 --> 01:21:42,287 why you keep only that term. 1243 01:21:42,287 --> 01:21:44,454 AUDIENCE: Do you have to worry about a higher power? 1244 01:21:44,454 --> 01:21:45,290 IAIN STEWART: No. 1245 01:21:45,290 --> 01:21:49,090 So the prescription we have is that we drop the higher powers 1246 01:21:49,090 --> 01:21:50,820 AUDIENCE: But if I were to do say-- 1247 01:21:50,820 --> 01:21:52,778 IAIN STEWART: Oh, if you did the higher power-- 1248 01:21:52,778 --> 01:21:54,890 AUDIENCE: --lower power higher power zero bin? 1249 01:21:54,890 --> 01:21:55,920 IAIN STEWART: No. 1250 01:21:55,920 --> 01:21:57,870 So if we did the higher power, then this 1251 01:21:57,870 --> 01:21:59,707 would start out at higher power. 1252 01:21:59,707 --> 01:22:01,290 And then when we took the limit of it, 1253 01:22:01,290 --> 01:22:04,650 it would end up just starting at that power or higher. 1254 01:22:04,650 --> 01:22:06,210 AUDIENCE: Right, but there was a-- 1255 01:22:06,210 --> 01:22:07,860 IAIN STEWART: Yeah, you don't have to. 1256 01:22:07,860 --> 01:22:08,160 No. 1257 01:22:08,160 --> 01:22:08,702 AUDIENCE: No? 1258 01:22:08,702 --> 01:22:09,720 IAIN STEWART: No. 1259 01:22:09,720 --> 01:22:10,220 Yeah. 1260 01:22:10,220 --> 01:22:11,870 AUDIENCE: Is there a reason? 1261 01:22:11,870 --> 01:22:14,520 IAIN STEWART: Yeah, so really what 1262 01:22:14,520 --> 01:22:18,000 you care about subtracting here are the log divergences 1263 01:22:18,000 --> 01:22:20,640 and that's what this minimal subtraction is doing. 1264 01:22:20,640 --> 01:22:23,910 By keeping the piece that's scaling the same way, 1265 01:22:23,910 --> 01:22:26,490 you're removing the log divergent pieces. 1266 01:22:26,490 --> 01:22:28,260 And it's the log divergent pieces which 1267 01:22:28,260 --> 01:22:31,500 are giving one of our epsilons. 1268 01:22:31,500 --> 01:22:35,050 The pieces that you would get from the higher expansion, 1269 01:22:35,050 --> 01:22:37,467 they would all be kind of like power law divergent terms 1270 01:22:37,467 --> 01:22:39,300 from the point of view of the power counting 1271 01:22:39,300 --> 01:22:42,090 and we just don't have to worry about those. 1272 01:22:42,090 --> 01:22:44,910 And another way of saying it is, it's not 1273 01:22:44,910 --> 01:22:48,330 that I'm removing absolutely this whole integrand 1274 01:22:48,330 --> 01:22:49,620 in that region, right? 1275 01:22:49,620 --> 01:22:52,033 There could still be a constant, for example, 1276 01:22:52,033 --> 01:22:53,200 that comes from that region. 1277 01:22:53,200 --> 01:22:54,750 But if there's a constant that comes from that region, 1278 01:22:54,750 --> 01:22:55,830 I don't care. 1279 01:22:55,830 --> 01:23:00,470 What I care about removing is any spare use IR singularities. 1280 01:23:00,470 --> 01:23:03,120 And for those I can make a minimal subtraction, 1281 01:23:03,120 --> 01:23:06,380 which is just the first term. 1282 01:23:06,380 --> 01:23:09,580 All right, so I want to finish this discussion. 1283 01:23:09,580 --> 01:23:14,100 So if we do this, we get an answer, 1284 01:23:14,100 --> 01:23:21,120 which I will try to write on the board for you. 1285 01:23:24,970 --> 01:23:28,443 So now I'm going to distinguish all the epsilons 1286 01:23:28,443 --> 01:23:30,360 and then we'll see what this subtraction does. 1287 01:23:41,457 --> 01:23:43,790 So if I was careful and I distinguished all the epsilons 1288 01:23:43,790 --> 01:23:49,530 in our original calculation, it'd actually look like this. 1289 01:23:49,530 --> 01:23:56,660 And then the subtraction piece gives an extra contribution 1290 01:23:56,660 --> 01:24:04,270 and it's actually scaleless in the n bar dot k here. 1291 01:24:04,270 --> 01:24:10,190 So there's a scaleless loop in this guy. 1292 01:24:15,025 --> 01:24:20,870 So it actually vanishes if the epsilon IR and the epsilon UV 1293 01:24:20,870 --> 01:24:22,860 are said to be equal. 1294 01:24:22,860 --> 01:24:25,860 But what it does is it converts the epsilon IRs that 1295 01:24:25,860 --> 01:24:28,890 are in the first expression into epsilon UVs, which 1296 01:24:28,890 --> 01:24:30,450 is what we want. 1297 01:24:30,450 --> 01:24:31,950 So once you add up these two things, 1298 01:24:31,950 --> 01:24:34,500 the epsilon IRs are canceling and the epsilon IRs 1299 01:24:34,500 --> 01:24:36,960 that were coming in the original formula, those 1300 01:24:36,960 --> 01:24:41,250 were coming about because of this bad behavior 1301 01:24:41,250 --> 01:24:45,465 as n bar dot k goes into the limit of n bar dot 1302 01:24:45,465 --> 01:24:46,170 k going small. 1303 01:24:46,170 --> 01:24:49,120 You can think about that roughly as where the ultrasoft is. 1304 01:24:49,120 --> 01:24:53,190 This is subtracting off that behavior and the remainder 1305 01:24:53,190 --> 01:24:56,324 then is coming from only having divergences for a n bar 1306 01:24:56,324 --> 01:25:00,480 dot k goes to infinity, which is a proper collinear ultraviolet 1307 01:25:00,480 --> 01:25:03,815 divergence not from n bar dot k going to 0. 1308 01:25:03,815 --> 01:25:05,190 So once you put the two together, 1309 01:25:05,190 --> 01:25:07,680 the epsilon IRs cancel and then we get exactly, 1310 01:25:07,680 --> 01:25:10,470 actually, the same expression we had before but where 1311 01:25:10,470 --> 01:25:14,170 all those 1 over epsilons are 1 over epsilon UVs. 1312 01:25:14,170 --> 01:25:16,480 AUDIENCE: So the epsilon has come from second term? 1313 01:25:16,480 --> 01:25:17,730 IAIN STEWART: From both terms. 1314 01:25:17,730 --> 01:25:21,748 So they both have epsilon IRs but they cancel between them. 1315 01:25:21,748 --> 01:25:22,290 AUDIENCE: OK. 1316 01:25:22,290 --> 01:25:23,160 IAIN STEWART: Yeah. 1317 01:25:23,160 --> 01:25:25,260 And the remainder is just epsilon UVs, so 1318 01:25:25,260 --> 01:25:27,810 all the epsilons that I wrote my earlier formula would 1319 01:25:27,810 --> 01:25:31,530 be now epsilon UVs once I take into account the subtraction. 1320 01:25:31,530 --> 01:25:34,080 So I could have just ignored the subtraction and that's 1321 01:25:34,080 --> 01:25:35,340 often what people do. 1322 01:25:35,340 --> 01:25:37,968 If they know that the zero bins are giving a scaleless 1323 01:25:37,968 --> 01:25:40,260 integral, they say, well, let's ignore the subtraction, 1324 01:25:40,260 --> 01:25:42,932 we'll just say that all the epsilons are UV 1325 01:25:42,932 --> 01:25:45,168 and the zero bin makes them UV. 1326 01:25:45,168 --> 01:25:46,710 But if we really want to look and see 1327 01:25:46,710 --> 01:25:48,300 that things are working properly, 1328 01:25:48,300 --> 01:25:51,690 we should take the subtraction and calculate it and make sure 1329 01:25:51,690 --> 01:25:52,885 that that's true. 1330 01:25:52,885 --> 01:25:54,510 But we could have just taken the answer 1331 01:25:54,510 --> 01:25:57,100 that I wrote down earlier and said those epsilons are 1332 01:25:57,100 --> 01:25:59,700 ultraviolet and let's throw them in a counterterm 1333 01:25:59,700 --> 01:26:03,030 and calculate an anomalous dimension. 1334 01:26:03,030 --> 01:26:04,793 So we'll proceed that way next time, 1335 01:26:04,793 --> 01:26:06,210 but we now know that actually they 1336 01:26:06,210 --> 01:26:10,190 are ultraviolet divergences. 1337 01:26:10,190 --> 01:26:13,610 So next time we'll take the ultraviolet divergences 1338 01:26:13,610 --> 01:26:15,942 and we'll define from them a counterterm 1339 01:26:15,942 --> 01:26:17,900 And we'll see how we get an anomalous dimension 1340 01:26:17,900 --> 01:26:22,272 and what kind of logs we sum by using that anomalous dimension. 1341 01:26:24,990 --> 01:26:28,290 So the zero bin that's scaleless in this particular example 1342 01:26:28,290 --> 01:26:29,400 is not always scaleless. 1343 01:26:29,400 --> 01:26:30,938 So sometimes it could give a nonce. 1344 01:26:30,938 --> 01:26:32,730 Depends on the problem you're dealing with. 1345 01:26:32,730 --> 01:26:37,650 So sometimes you can set things up so that it's scaleless 1346 01:26:37,650 --> 01:26:39,893 and then you just basically can ignore it. 1347 01:26:39,893 --> 01:26:41,310 But that's not always true, so you 1348 01:26:41,310 --> 01:26:43,560 do have to think about whether it's really going to be 1349 01:26:43,560 --> 01:26:45,090 true for what you're doing. 1350 01:26:45,090 --> 01:26:47,250 If it is true, then you can effectively 1351 01:26:47,250 --> 01:26:49,202 ignore it because it's sort of just making 1352 01:26:49,202 --> 01:26:51,660 the physics come out right, making sure there's no overlap. 1353 01:26:51,660 --> 01:26:53,868 But if your regulators set up so that it's scaleless, 1354 01:26:53,868 --> 01:26:56,290 you can just get around it. 1355 01:26:56,290 --> 01:26:58,297 But in general, that might not be true. 1356 01:26:58,297 --> 01:26:59,880 If you had more scales in the problem, 1357 01:26:59,880 --> 01:27:01,422 if you're doing some calculation that 1358 01:27:01,422 --> 01:27:05,160 had some jets of finite size then that 1359 01:27:05,160 --> 01:27:08,690 won't be true typically.