1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:22,250 --> 00:00:27,340 PROFESSOR: So this is an example in SCET one 9 00:00:27,340 --> 00:00:33,715 where the degrees of freedom in the p plus p minus plane 10 00:00:33,715 --> 00:00:34,465 looked as follows. 11 00:00:45,260 --> 00:00:48,470 So we had hard modes that lived out here at p 12 00:00:48,470 --> 00:00:52,880 squared in order some large scale q squared, 13 00:00:52,880 --> 00:00:56,590 then we had collinear modes for the two jets 14 00:00:56,590 --> 00:00:58,660 that are going to be going back to back, 15 00:00:58,660 --> 00:01:00,580 and I'll draw the picture in a second. 16 00:01:00,580 --> 00:01:02,530 Then there were some ultra soft modes 17 00:01:02,530 --> 00:01:05,500 as well that describe radiation between the jets. 18 00:01:05,500 --> 00:01:09,100 And then there can be also some lambda QCD modes, 19 00:01:09,100 --> 00:01:11,545 and we'll talk about them a bit today too. 20 00:01:11,545 --> 00:01:13,670 So there's a lot of different things going on here, 21 00:01:13,670 --> 00:01:16,460 but we'll see that actually we can understand everything 22 00:01:16,460 --> 00:01:19,210 and organize things using SCET. 23 00:01:19,210 --> 00:01:22,720 So what's the picture for these modes? 24 00:01:22,720 --> 00:01:26,350 We have back-to-back jets. 25 00:01:26,350 --> 00:01:30,655 Let me draw one jet this way, one jet this way. 26 00:01:30,655 --> 00:01:31,750 I should use some color. 27 00:01:39,730 --> 00:01:40,990 So there's one of our jets. 28 00:01:48,690 --> 00:01:52,800 There's another, and then there can be soft radiation. 29 00:01:58,260 --> 00:02:00,180 And between these jets, there can also 30 00:02:00,180 --> 00:02:03,498 be soft radiation physically within the jets, 31 00:02:03,498 --> 00:02:05,040 but there's no directional dependence 32 00:02:05,040 --> 00:02:06,150 to the soft radiation. 33 00:02:14,390 --> 00:02:18,660 And then set that this is blue. 34 00:02:21,460 --> 00:02:25,384 This and this so far are orange in my picture. 35 00:02:25,384 --> 00:02:31,200 This is green, and then there's some hard interactions, 36 00:02:31,200 --> 00:02:35,590 which we've already sort of localized 37 00:02:35,590 --> 00:02:37,480 into a dot in this picture. 38 00:02:42,860 --> 00:02:44,500 So what we're doing is we're colliding 39 00:02:44,500 --> 00:02:48,130 e plus e minus producing a virtual photon or z 40 00:02:48,130 --> 00:02:55,490 and then from that producing, if you like, 41 00:02:55,490 --> 00:02:58,100 a state which is full of collinear modes, state which 42 00:02:58,100 --> 00:03:00,770 is full of the other type of collinear 43 00:03:00,770 --> 00:03:04,130 modes and ultra soft particles. 44 00:03:04,130 --> 00:03:09,020 And this is in the center of mass frame, 45 00:03:09,020 --> 00:03:12,512 or that's kind of how you could think about it in SCET, 46 00:03:12,512 --> 00:03:13,970 or physically what you're producing 47 00:03:13,970 --> 00:03:24,200 is you're producing jet 1 plus jet 2 plus soft radiation 48 00:03:24,200 --> 00:03:29,165 or ultra soft hadrons, soft hadrons. 49 00:03:39,525 --> 00:03:41,650 OK, so the first scale that you want to think about 50 00:03:41,650 --> 00:03:46,270 is the hard scale, and so that is just 51 00:03:46,270 --> 00:03:48,820 the q squared to the virtual photon or z. 52 00:03:48,820 --> 00:03:51,640 So if you think about the Feynman diagram, 53 00:03:51,640 --> 00:03:53,430 there's e plus e minus. 54 00:03:53,430 --> 00:03:55,660 Here's q mu. 55 00:03:55,660 --> 00:03:57,310 You're going to have a pair of quarks 56 00:03:57,310 --> 00:04:00,358 that that virtual particle can couple to, 57 00:04:00,358 --> 00:04:01,900 and there could be additional gluons. 58 00:04:05,300 --> 00:04:08,320 But the hard scale is set by the virtual photon or z. 59 00:04:13,050 --> 00:04:15,750 So there's going to be some scale associated 60 00:04:15,750 --> 00:04:18,209 to that hard scale q. 61 00:04:18,209 --> 00:04:20,529 If you want to think about it as a normalization scale, 62 00:04:20,529 --> 00:04:22,029 there'll be some scale where we want 63 00:04:22,029 --> 00:04:24,105 to match from QCD onto SCET, and that's 64 00:04:24,105 --> 00:04:25,230 going to be the hard scale. 65 00:04:35,126 --> 00:04:36,940 So I'll call that scale mu h. 66 00:04:41,150 --> 00:04:43,820 So then something physical has to set these other scales 67 00:04:43,820 --> 00:04:46,790 in the picture, and the next thing 68 00:04:46,790 --> 00:04:51,050 that we need to talk about is what we're going to measure. 69 00:04:51,050 --> 00:04:53,570 In principle, if you collide e plus e minus 70 00:04:53,570 --> 00:04:56,150 and you produce hadrons, you could have lots 71 00:04:56,150 --> 00:04:57,650 of different possibilities. 72 00:04:57,650 --> 00:05:00,380 In particular, you could have not dijets but trijets, 73 00:05:00,380 --> 00:05:03,650 so you could have a third jet and made it from a gluon. 74 00:05:03,650 --> 00:05:06,560 And some kind of measurement that we do on the final state 75 00:05:06,560 --> 00:05:09,590 is going to restrict us to this configuration with just two 76 00:05:09,590 --> 00:05:10,790 jets. 77 00:05:10,790 --> 00:05:12,830 And the one we'll talk about to start 78 00:05:12,830 --> 00:05:16,460 is what's called hemisphere invariant masses, which 79 00:05:16,460 --> 00:05:17,670 we already mentioned earlier. 80 00:05:21,598 --> 00:05:23,390 So the way that you should think about this 81 00:05:23,390 --> 00:05:27,680 is you take all the final state particles, 82 00:05:27,680 --> 00:05:30,320 and you could just take the sum of all their momenta 83 00:05:30,320 --> 00:05:36,970 and divide it into two parts, those that are in hemisphere A 84 00:05:36,970 --> 00:05:40,410 and those that are in hemisphere B. 85 00:05:40,410 --> 00:05:47,960 So let me augment my figure here by saying 86 00:05:47,960 --> 00:05:51,718 that there's two hemispheres. 87 00:05:51,718 --> 00:05:53,760 And then what these masses are, which I'll call m 88 00:05:53,760 --> 00:05:57,350 squared and then bar squared are just 89 00:05:57,350 --> 00:06:02,580 the four vectors squared for p, for the a, and the b guy. 90 00:06:02,580 --> 00:06:06,660 So written in terms of the individual particles 91 00:06:06,660 --> 00:06:10,200 in that hemisphere, you sum up all their four 92 00:06:10,200 --> 00:06:13,920 vectors for the particles in that hemisphere and square it, 93 00:06:13,920 --> 00:06:30,050 and then likewise for B. 94 00:06:30,050 --> 00:06:33,170 And if you're measuring these two things, 95 00:06:33,170 --> 00:06:35,060 we talked about earlier the fact that you 96 00:06:35,060 --> 00:06:37,250 could specify that it was a jet by demanding 97 00:06:37,250 --> 00:06:38,730 that they're small. 98 00:06:38,730 --> 00:06:45,990 So the dijets is going to be m squared 99 00:06:45,990 --> 00:06:49,908 and then bar squared much less than q squared. 100 00:06:49,908 --> 00:06:52,200 And if you had trijets, you wouldn't-- that wouldn't be 101 00:06:52,200 --> 00:06:54,030 the case because if you had trijets, 102 00:06:54,030 --> 00:06:58,760 you'd have two particle-- two jets in this hemisphere at wide 103 00:06:58,760 --> 00:07:01,520 angles, and those-- that wide angle would produce a large q 104 00:07:01,520 --> 00:07:05,350 squared, a large and variant mass of m squared. 105 00:07:05,350 --> 00:07:06,830 That's of order q squared. 106 00:07:06,830 --> 00:07:08,770 But if you just have a single jet, 107 00:07:08,770 --> 00:07:10,970 then you can have a small m squared, 108 00:07:10,970 --> 00:07:13,735 so the small m squared limit is what forces you to have dijets. 109 00:07:16,800 --> 00:07:20,160 And these guys here, it also puts us 110 00:07:20,160 --> 00:07:23,850 in the kinematic situation where we have our n collinear 111 00:07:23,850 --> 00:07:27,330 modes for one jet and n bar collinear modes 112 00:07:27,330 --> 00:07:29,953 for the other jet. 113 00:07:29,953 --> 00:07:32,370 So those are two of the degrees of freedom in our picture. 114 00:07:37,650 --> 00:07:40,660 So I'm reminding you of some things we talked about before, 115 00:07:40,660 --> 00:07:44,280 but it was a while ago so-- 116 00:07:44,280 --> 00:07:47,857 and the power counting parameter here is m over q. 117 00:07:47,857 --> 00:07:49,440 And so if you want to say that there's 118 00:07:49,440 --> 00:07:52,470 a scale in this picture associated to the collinear 119 00:07:52,470 --> 00:07:57,000 modes, then you would say it's on a kind of jet scale, 120 00:07:57,000 --> 00:07:58,394 and that's of order m. 121 00:08:02,840 --> 00:08:08,860 OK, any questions so far? 122 00:08:08,860 --> 00:08:17,360 So then there's the ultra soft radiation, 123 00:08:17,360 --> 00:08:19,900 which is being ultra soft, it's uniform. 124 00:08:19,900 --> 00:08:22,221 It's not collimated in any direction. 125 00:08:25,562 --> 00:08:27,520 It's the thing that gives you the communication 126 00:08:27,520 --> 00:08:28,360 between the jets. 127 00:08:32,919 --> 00:08:35,710 The jets are going to decouple from each other in the sense 128 00:08:35,710 --> 00:08:38,539 that interactions directly between the n collinear and n 129 00:08:38,539 --> 00:08:42,270 bar collinear modes is all going into the pink dot, 130 00:08:42,270 --> 00:08:45,280 but there can be long distance communication between the jets 131 00:08:45,280 --> 00:08:47,170 caused by the ultra soft modes. 132 00:08:47,170 --> 00:08:53,410 So, really, what I mean here is long distance communication. 133 00:08:53,410 --> 00:08:55,810 Short distance communication can happen too. 134 00:08:55,810 --> 00:08:59,350 That's what the pink dot is, but that's, in some sense, simpler. 135 00:08:59,350 --> 00:09:00,940 Long distance communication can only 136 00:09:00,940 --> 00:09:04,630 happen by this ultra soft radiation. 137 00:09:04,630 --> 00:09:08,500 And we know about how ultra soft radiation 138 00:09:08,500 --> 00:09:10,180 interacts with energetic particles, 139 00:09:10,180 --> 00:09:13,060 and in particular, what we know is that it's eikonal. 140 00:09:13,060 --> 00:09:16,000 That came out of our Lagrangian description 141 00:09:16,000 --> 00:09:18,520 of ultra soft radiation. 142 00:09:18,520 --> 00:09:22,340 And that will have implications here. 143 00:09:22,340 --> 00:09:23,950 So what is this ultra soft radiation? 144 00:09:23,950 --> 00:09:26,182 It's radiation that has energy that's 145 00:09:26,182 --> 00:09:27,265 of order q lambda squared. 146 00:09:30,110 --> 00:09:36,010 And so that's m squared over q energy radiation, 147 00:09:36,010 --> 00:09:39,370 and so the soft scale, which is the next scale down 148 00:09:39,370 --> 00:09:44,890 in the picture, ultra soft scale, is m squared over q. 149 00:09:44,890 --> 00:09:47,590 Often, people will-- because there's 150 00:09:47,590 --> 00:09:51,760 no soft radiation in this setup, sometimes people 151 00:09:51,760 --> 00:09:54,250 will just drop the ultra and just call it soft. 152 00:09:54,250 --> 00:09:56,760 That's very common. 153 00:09:56,760 --> 00:09:58,810 I'll try to always call it ultra soft here 154 00:09:58,810 --> 00:10:01,750 because we'll be drawing some distinctions with SCET 2 155 00:10:01,750 --> 00:10:04,810 examples, and in those examples, we have soft and collinear. 156 00:10:04,810 --> 00:10:07,120 In this example, we have ultra soft and collinear, 157 00:10:07,120 --> 00:10:09,520 so I'll keep trying to call it ultra soft. 158 00:10:09,520 --> 00:10:11,050 But you can see that I've already 159 00:10:11,050 --> 00:10:12,910 abbreviated it to u soft to make it 160 00:10:12,910 --> 00:10:15,280 look more and more like soft. 161 00:10:15,280 --> 00:10:17,200 Anyway, but in the literature, people 162 00:10:17,200 --> 00:10:20,470 often just drop the u completely and call it soft. 163 00:10:20,470 --> 00:10:23,520 Something to be aware of. 164 00:10:23,520 --> 00:10:27,550 OK, so this actually-- this radiation here, 165 00:10:27,550 --> 00:10:30,040 this is kind of the largest scale that 166 00:10:30,040 --> 00:10:32,930 can show up in the soft function, 167 00:10:32,930 --> 00:10:34,450 and there's also lambda QCD. 168 00:10:39,310 --> 00:10:41,470 And that's a scale that, at least to start, 169 00:10:41,470 --> 00:10:43,220 we're going to leave in the soft function. 170 00:10:51,148 --> 00:10:53,190 So if we go back to our picture over here, that's 171 00:10:53,190 --> 00:10:57,870 why I put both the ultra soft mode side up on this hyperbola 172 00:10:57,870 --> 00:11:03,210 there, which is this m squared over q line, mu over m 173 00:11:03,210 --> 00:11:04,800 squared over q line. 174 00:11:04,800 --> 00:11:06,810 That's why I made it orange, and I also 175 00:11:06,810 --> 00:11:10,320 made orange this lambda QCD because they're together. 176 00:11:10,320 --> 00:11:12,480 And you can think of it as just that there's 177 00:11:12,480 --> 00:11:16,320 some modes that are capturing this entire region, which 178 00:11:16,320 --> 00:11:18,930 are the ultra soft modes. 179 00:11:18,930 --> 00:11:20,430 Now there are two-- 180 00:11:20,430 --> 00:11:25,500 if m squared over q was of order lambda QCD, then 181 00:11:25,500 --> 00:11:26,570 those are the same thing. 182 00:11:26,570 --> 00:11:29,190 So there's kind of two possibilities here. 183 00:11:29,190 --> 00:11:35,250 You could have m squared over q of order lambda QCD, 184 00:11:35,250 --> 00:11:39,466 and that means that this scale is not non-perturbative. 185 00:11:43,200 --> 00:11:45,180 And this is what's called, for reasons 186 00:11:45,180 --> 00:11:50,137 that will become apparent by the end of lecture, the peak region 187 00:11:50,137 --> 00:11:52,470 because this is actually the region of the cross section 188 00:11:52,470 --> 00:11:53,952 where there's a peak. 189 00:11:53,952 --> 00:11:56,890 OK, that's why it's called the peak region. 190 00:11:56,890 --> 00:12:01,290 And in this region, you have the following hierarchy. 191 00:12:01,290 --> 00:12:05,110 You have mu h is much greater than mu j. 192 00:12:05,110 --> 00:12:07,920 Mu j is much greater than mu s, but mu 193 00:12:07,920 --> 00:12:10,350 s is of order lambda QCD. 194 00:12:10,350 --> 00:12:15,590 OK, so if we take this to always be true, 195 00:12:15,590 --> 00:12:18,020 then we're in this situation. 196 00:12:18,020 --> 00:12:19,860 And there is another possibility, 197 00:12:19,860 --> 00:12:23,510 and that is that there's really a separation 198 00:12:23,510 --> 00:12:27,530 between these hyperbolas that's just as large or hierarchical 199 00:12:27,530 --> 00:12:30,030 as the previous separations. 200 00:12:30,030 --> 00:12:34,130 So if this thing is much bigger than lambda QCD, 201 00:12:34,130 --> 00:12:37,370 that's another possibility, and there'll 202 00:12:37,370 --> 00:12:39,260 be a region of phase based where this is true 203 00:12:39,260 --> 00:12:42,090 and a region of phase based where this is true. 204 00:12:42,090 --> 00:12:43,340 The second one is-- 205 00:12:43,340 --> 00:12:46,430 here, the soft, there's-- 206 00:12:49,990 --> 00:12:59,500 here there is perturbative ultra soft radiation, 207 00:12:59,500 --> 00:13:01,960 so you can just calculate it order by order 208 00:13:01,960 --> 00:13:05,860 in the ultra soft scale, which is this scale, which 209 00:13:05,860 --> 00:13:09,910 is perturbative because it's much bigger than lambda QCD. 210 00:13:09,910 --> 00:13:11,740 And this is what's called the tail region. 211 00:13:16,880 --> 00:13:19,030 And the analog of this statement here 212 00:13:19,030 --> 00:13:21,835 is that we have a hierarchy between everybody. 213 00:13:34,230 --> 00:13:36,840 So when we do this analysis, what we're going to do 214 00:13:36,840 --> 00:13:39,190 is we're going to do a power expansion. 215 00:13:39,190 --> 00:13:41,580 And you can say, well, the power expansion 216 00:13:41,580 --> 00:13:43,600 is just what you told me, m squared over-- 217 00:13:43,600 --> 00:13:45,060 m squared much less than q squared, 218 00:13:45,060 --> 00:13:47,700 but you also have to worry about this when you're 219 00:13:47,700 --> 00:13:50,320 doing the power expansion. 220 00:13:50,320 --> 00:13:52,710 So one way of thinking about it is just exactly 221 00:13:52,710 --> 00:13:56,490 as I'm writing here, and that's how I want to advocate. 222 00:13:56,490 --> 00:13:59,683 The things you're expanding in are these assumptions. 223 00:13:59,683 --> 00:14:01,350 And you can see that in these two cases, 224 00:14:01,350 --> 00:14:03,000 there's a slightly different set up. 225 00:14:17,920 --> 00:14:20,110 So usually, when you have an effective theory, 226 00:14:20,110 --> 00:14:23,860 you have to define this from the beginning because the way 227 00:14:23,860 --> 00:14:27,340 you proceed and how you set up your theory is going 228 00:14:27,340 --> 00:14:30,010 to depend on whether you're in this situation 229 00:14:30,010 --> 00:14:32,260 or this situation. 230 00:14:32,260 --> 00:14:35,560 But part of this story is the same in these two situations, 231 00:14:35,560 --> 00:14:38,000 namely the first two greater thans, much greater thans 232 00:14:38,000 --> 00:14:39,470 are the same. 233 00:14:39,470 --> 00:14:41,830 And so that, we can proceed without worrying about this. 234 00:14:45,372 --> 00:14:47,330 And then we can make this distinction later on, 235 00:14:47,330 --> 00:14:49,038 and that's what my picture was advocating 236 00:14:49,038 --> 00:14:56,167 for by having these guys in the same category. 237 00:14:56,167 --> 00:14:57,250 So hopefully that's clear. 238 00:15:05,140 --> 00:15:17,590 OK, so if you're in the tail region, 239 00:15:17,590 --> 00:15:24,310 there's going to be power corrections, 240 00:15:24,310 --> 00:15:28,120 and these are actually the most important power corrections 241 00:15:28,120 --> 00:15:35,350 that come as powers of lambda QCD over the soft scale 242 00:15:35,350 --> 00:15:36,040 to some power. 243 00:15:36,040 --> 00:15:42,220 But you're going to have a power expansion in that. 244 00:15:42,220 --> 00:15:46,000 So that's the-- if you like in this situation 245 00:15:46,000 --> 00:15:47,500 when you're in the tail region, you 246 00:15:47,500 --> 00:15:50,710 can have power corrections that are non-perturbative by having 247 00:15:50,710 --> 00:15:52,330 ratios of these things, lambda QCD 248 00:15:52,330 --> 00:15:55,340 over this over that over this. 249 00:15:55,340 --> 00:15:58,966 But the biggest one are these guys over the soft scale, 250 00:15:58,966 --> 00:16:00,700 so that's the next smallest scale. 251 00:16:05,618 --> 00:16:07,410 But if you neglect these, the leading order 252 00:16:07,410 --> 00:16:09,750 cross section in this region is perturbative. 253 00:16:19,580 --> 00:16:25,460 Now you know you can have any power here, and those exist. 254 00:16:25,460 --> 00:16:28,100 What happens in the peak region is 255 00:16:28,100 --> 00:16:31,490 that thinking about those as an expansion is no longer good. 256 00:16:38,120 --> 00:16:41,960 And you have to take all of them and treat them 257 00:16:41,960 --> 00:16:47,930 as if they're order one and for any k, 258 00:16:47,930 --> 00:16:50,368 and that ends up meaning that there's 259 00:16:50,368 --> 00:16:52,160 going to be some non-perturbative function, 260 00:16:52,160 --> 00:16:54,160 like a Parton distribution function in the sense 261 00:16:54,160 --> 00:16:58,490 that it's non-perturbative, that describes part of what's 262 00:16:58,490 --> 00:17:04,800 going on here in this region. 263 00:17:04,800 --> 00:17:06,740 And that, we just see from our power counting, 264 00:17:06,740 --> 00:17:08,365 and our power counting already tells us 265 00:17:08,365 --> 00:17:11,677 that that's what's going to happen. 266 00:17:11,677 --> 00:17:13,260 So we can learn a lot just by thinking 267 00:17:13,260 --> 00:17:16,030 about the scales and the problem and thinking about the power 268 00:17:16,030 --> 00:17:16,530 counting. 269 00:17:25,970 --> 00:17:32,463 So you could ask about other power corrections. 270 00:17:36,170 --> 00:17:37,730 So there's a set of power connections 271 00:17:37,730 --> 00:17:41,660 where you would expand and say mu s over mu j. 272 00:17:41,660 --> 00:17:50,350 And let me label these as kinematic because these-- 273 00:17:50,350 --> 00:17:53,120 the things that setting the scale for mu s and mu 274 00:17:53,120 --> 00:17:56,530 j, say in this picture where mu s is this 275 00:17:56,530 --> 00:18:00,280 and I always distinguished lambda QCD as a separate thing, 276 00:18:00,280 --> 00:18:02,140 these scales are just perturbative scales, 277 00:18:02,140 --> 00:18:03,670 and they just correspond to if you 278 00:18:03,670 --> 00:18:05,503 think about having some function that you're 279 00:18:05,503 --> 00:18:08,150 making an expansion of it. 280 00:18:08,150 --> 00:18:09,670 So they come about from expansion 281 00:18:09,670 --> 00:18:11,380 of kinematic variables basically. 282 00:18:19,846 --> 00:18:23,086 So it's the m squared much less than q squared. 283 00:18:23,086 --> 00:18:27,320 So I mean, there's nothing non-perturbative about that. 284 00:18:30,040 --> 00:18:31,910 And then you could have power corrections 285 00:18:31,910 --> 00:18:36,620 that are lambda QCD over mu, which, if you like, 286 00:18:36,620 --> 00:18:38,720 are really hard power corrections. 287 00:18:38,720 --> 00:18:42,500 They're kind of a traditional power corrections, which are 288 00:18:42,500 --> 00:18:45,590 lambda QCD over the hard scale. 289 00:18:45,590 --> 00:18:49,910 And the exercise that I gave in the homework, in some sense, 290 00:18:49,910 --> 00:18:54,950 is going towards figuring out what the sort of soft function 291 00:18:54,950 --> 00:18:56,630 matrix elements would be. 292 00:18:56,630 --> 00:18:58,800 Part of the problem-- there's other parts to it-- 293 00:18:58,800 --> 00:19:00,782 but would be going towards figuring out 294 00:19:00,782 --> 00:19:02,990 what these guys actually are and defining them, which 295 00:19:02,990 --> 00:19:06,770 is not known in the literature. 296 00:19:06,770 --> 00:19:12,680 And then finally, you could have lambda QCD over mu j, 297 00:19:12,680 --> 00:19:14,150 which, of course, you could write. 298 00:19:14,150 --> 00:19:15,080 You have to be a little bit careful 299 00:19:15,080 --> 00:19:17,120 because one way that this could happen 300 00:19:17,120 --> 00:19:22,250 is that you just have one of these guys times one 301 00:19:22,250 --> 00:19:24,230 of those guys. 302 00:19:24,230 --> 00:19:25,520 OK, so they're not-- 303 00:19:25,520 --> 00:19:27,523 it's not necessarily an independent category, 304 00:19:27,523 --> 00:19:29,690 but you could still think about these types of power 305 00:19:29,690 --> 00:19:33,170 corrections as kind of something that might show up, 306 00:19:33,170 --> 00:19:36,240 either this way or maybe somewhat maybe directly. 307 00:19:36,240 --> 00:19:40,640 And anyway, that's another category. 308 00:19:40,640 --> 00:19:43,400 It turns out that kind of-- 309 00:19:43,400 --> 00:19:46,100 you can get it this way, which is something 310 00:19:46,100 --> 00:19:47,930 that one can deal with. 311 00:19:47,930 --> 00:19:51,380 And if it happens directly, then it actually 312 00:19:51,380 --> 00:19:56,630 is down by two powers, so these guys 313 00:19:56,630 --> 00:19:58,159 are kind of less important. 314 00:20:05,830 --> 00:20:08,710 OK, so people deal with these guys in the literature. 315 00:20:08,710 --> 00:20:11,950 These guys haven't really been fully treated yet. 316 00:20:11,950 --> 00:20:14,000 Not very much is known about them, 317 00:20:14,000 --> 00:20:16,780 and this is fully treated in the literature. 318 00:20:16,780 --> 00:20:22,230 These ones that I said are the most important ones. 319 00:20:22,230 --> 00:20:24,690 OK, so even-- we can already even outline 320 00:20:24,690 --> 00:20:28,377 kind of what people are doing because we just 321 00:20:28,377 --> 00:20:30,210 think about higher order terms of expansion, 322 00:20:30,210 --> 00:20:33,420 and this is an example, unlike B to C decays, 323 00:20:33,420 --> 00:20:35,310 where we have people going to four orders 324 00:20:35,310 --> 00:20:36,840 down in the expansion. 325 00:20:36,840 --> 00:20:40,230 Here even the first non-trivial order down hasn't really been 326 00:20:40,230 --> 00:20:41,590 done-- 327 00:20:41,590 --> 00:20:42,590 hasn't been done at all. 328 00:20:45,210 --> 00:20:47,318 In the second, place various expansions, 329 00:20:47,318 --> 00:20:48,360 so it's more complicated. 330 00:20:48,360 --> 00:20:51,030 And some of them have been treated, but this one hasn't. 331 00:21:02,296 --> 00:21:05,280 And I should say that the way that this one is treated 332 00:21:05,280 --> 00:21:08,487 is really just perturbatively so it's not-- 333 00:21:08,487 --> 00:21:10,320 this one, in some sense, is also not treated 334 00:21:10,320 --> 00:21:13,470 fully in the language of the effective theory. 335 00:21:13,470 --> 00:21:15,330 It's just put in by hand. 336 00:21:19,130 --> 00:21:21,670 So in some sense, the only one that's treated properly 337 00:21:21,670 --> 00:21:22,450 is the top one. 338 00:21:29,670 --> 00:21:32,610 OK, well, we're going to work mostly to leading order, 339 00:21:32,610 --> 00:21:36,330 and we're going to call these guys power corrections 340 00:21:36,330 --> 00:21:38,760 and work with this stuff. 341 00:21:46,100 --> 00:21:50,730 OK, so start with the full theory current. 342 00:21:50,730 --> 00:21:52,310 I'm not going to distinguish too much 343 00:21:52,310 --> 00:21:55,460 between pieces that come from charges, like the electrically 344 00:21:55,460 --> 00:21:56,720 charged and stuff like that. 345 00:21:56,720 --> 00:21:59,460 That's not going to be our main focus here. 346 00:21:59,460 --> 00:22:02,360 We can keep track of all that. 347 00:22:02,360 --> 00:22:04,520 Our main focus is going to be understanding the QCD 348 00:22:04,520 --> 00:22:06,560 effects for the jets. 349 00:22:06,560 --> 00:22:08,120 So we've already done this. 350 00:22:08,120 --> 00:22:10,520 The current that has a psi bar psi 351 00:22:10,520 --> 00:22:14,570 gets matched in SCET onto something 352 00:22:14,570 --> 00:22:16,325 with two Wilson lines. 353 00:22:16,325 --> 00:22:17,075 There's the label. 354 00:22:23,990 --> 00:22:26,870 There's another label, and we can also 355 00:22:26,870 --> 00:22:28,568 make a field redefinition. 356 00:22:41,030 --> 00:22:43,210 And let me not put zeros on the field 357 00:22:43,210 --> 00:22:46,300 after the field redefinition because that makes 358 00:22:46,300 --> 00:22:47,630 it notationally cumbersome. 359 00:22:47,630 --> 00:22:54,640 So this goes to this after making that field redefinition. 360 00:22:54,640 --> 00:22:58,560 OK, so this is something we talked about earlier, 361 00:22:58,560 --> 00:23:05,370 and the Wilson lines here are capturing diagrams, 362 00:23:05,370 --> 00:23:08,040 as I said earlier, where you could have, 363 00:23:08,040 --> 00:23:10,110 say, two n collinear is one n bar, 364 00:23:10,110 --> 00:23:12,290 and then this guy is off shell. 365 00:23:12,290 --> 00:23:21,140 So that goes into a Wilson line once you match it on to the-- 366 00:23:21,140 --> 00:23:22,685 so this is the [? fall ?] theory. 367 00:23:22,685 --> 00:23:24,560 Once you match it on to the effective theory, 368 00:23:24,560 --> 00:23:28,053 you get various extra gluons coming out of your operator, 369 00:23:28,053 --> 00:23:29,720 which is coming from these Wilson lines, 370 00:23:29,720 --> 00:23:31,580 and it's because you're integrating off 371 00:23:31,580 --> 00:23:32,660 shell particles. 372 00:23:37,230 --> 00:23:39,290 So if you just look at the kinematics, 373 00:23:39,290 --> 00:23:43,520 then there's already actually some fairly powerful 374 00:23:43,520 --> 00:23:45,150 restrictions. 375 00:23:45,150 --> 00:23:47,390 We're in the center of mass frame. 376 00:23:47,390 --> 00:24:00,763 Momentum conservation says that if we 377 00:24:00,763 --> 00:24:03,180 think about all particles in the final stages either being 378 00:24:03,180 --> 00:24:11,520 n collinear or n bar collinear for ultra soft, 379 00:24:11,520 --> 00:24:16,080 then the initial momentum of the virtual photon or virtual C 380 00:24:16,080 --> 00:24:18,970 has to add up to all the final stage momenta. 381 00:24:18,970 --> 00:24:25,110 And if we just look at the large part of that, 382 00:24:25,110 --> 00:24:30,750 in the center of mass frame, n bar dot q is just capital Q, 383 00:24:30,750 --> 00:24:34,560 and that's n bar dot pxn plus things that are small. 384 00:24:37,270 --> 00:24:41,790 And likewise, n dot q equals capital q equals 385 00:24:41,790 --> 00:24:46,540 n dot pxn bar, which is large order one plus small. 386 00:24:46,540 --> 00:24:49,290 So these guys here are lowest-- 387 00:24:49,290 --> 00:24:50,790 leading order in the power counting, 388 00:24:50,790 --> 00:24:52,290 and these guys are suppressed. 389 00:24:56,380 --> 00:24:59,530 And what happens is like in the example 390 00:24:59,530 --> 00:25:02,470 we talked about when we were doing [? bs ?] gamma. 391 00:25:02,470 --> 00:25:04,060 We have these labels, omega and omega 392 00:25:04,060 --> 00:25:06,790 bar on our collinear fields up there. 393 00:25:06,790 --> 00:25:10,060 They just get fixed to the qs by momentum conservation. 394 00:25:23,980 --> 00:25:25,620 So I said this was generically going 395 00:25:25,620 --> 00:25:28,800 to happen whenever we had operators that had only one 396 00:25:28,800 --> 00:25:32,070 type of building block in each collinear sector, 397 00:25:32,070 --> 00:25:34,800 and that's exactly the situation we have. 398 00:25:34,800 --> 00:25:36,930 Some momentum conservation is strong enough 399 00:25:36,930 --> 00:25:44,040 to fix that omega is equal to q and so is omega bar. 400 00:25:56,370 --> 00:25:58,940 So the first thing I want to do is show you 401 00:25:58,940 --> 00:26:02,130 how to factorize to the cross section in this case. 402 00:26:02,130 --> 00:26:05,960 So what would you do in QCD for this cross section? 403 00:26:05,960 --> 00:26:08,690 Well, you'd say the cross section 404 00:26:08,690 --> 00:26:12,020 is a sum over some restricted set of final states 405 00:26:12,020 --> 00:26:14,660 that satisfy the kinematic criteria that I'm 406 00:26:14,660 --> 00:26:15,350 interested in. 407 00:26:19,390 --> 00:26:22,480 You'd have a momentum conserving delta function, 408 00:26:22,480 --> 00:26:26,350 and you could write it as sort of leptonic or-- 409 00:26:26,350 --> 00:26:31,270 yeah, leptonic tensor, and then kind of a hadronic tensor, 410 00:26:31,270 --> 00:26:40,380 the same way we do for [? DIS. ?] 411 00:26:40,380 --> 00:26:42,810 So you would start-- could start with that formula, which 412 00:26:42,810 --> 00:26:46,080 is really true for anything that you might want to measure, 413 00:26:46,080 --> 00:26:49,470 which is just imposed by kind of what states you include here. 414 00:26:54,900 --> 00:26:58,560 And I haven't yet specified exactly what 415 00:26:58,560 --> 00:26:59,730 I'm going to measure. 416 00:26:59,730 --> 00:27:02,850 We'll still have to put that in. 417 00:27:02,850 --> 00:27:06,270 So part of what SCET does is make these restrictions 418 00:27:06,270 --> 00:27:08,463 into something that shows up in the operators 419 00:27:08,463 --> 00:27:10,005 rather than showing up in the states. 420 00:27:22,678 --> 00:27:24,970 And ideally, we'd like to sort of get rid of the states 421 00:27:24,970 --> 00:27:28,270 eventually and be able to calculate-- 422 00:27:28,270 --> 00:27:29,780 see how to calculate things. 423 00:27:29,780 --> 00:27:35,620 So in SCET, once we know that we're in a dijet configuration, 424 00:27:35,620 --> 00:27:37,960 we can think of the state as having 425 00:27:37,960 --> 00:27:43,000 been composed of sort of pieces for the different sectors. 426 00:27:43,000 --> 00:27:45,580 And we know that this is an OK picture because our Lagrangian 427 00:27:45,580 --> 00:27:47,548 for these guys factored. 428 00:27:47,548 --> 00:27:49,090 And when the Lagrangian factors, that 429 00:27:49,090 --> 00:28:01,040 means the Hamiltonian factors and the Hilbert space factors 430 00:28:01,040 --> 00:28:02,936 after the field redefinition. 431 00:28:12,180 --> 00:28:16,092 OK, so we can put in our expansion of the current 432 00:28:16,092 --> 00:28:17,550 at the top of the board over there, 433 00:28:17,550 --> 00:28:21,640 and then put it into this formula, and see what we get. 434 00:28:21,640 --> 00:28:23,010 And there's some pre-factors. 435 00:28:27,210 --> 00:28:31,028 I'm now summing over just these states. 436 00:28:31,028 --> 00:28:32,820 There are still some restrictions depending 437 00:28:32,820 --> 00:28:35,160 on what I'm measuring, but they are different restrictions, 438 00:28:35,160 --> 00:28:35,993 and they're simpler. 439 00:28:39,088 --> 00:28:41,130 There's still momentum conserving delta function, 440 00:28:41,130 --> 00:28:47,160 but I can write it out as pxn pxn bar px ultra soft. 441 00:28:50,680 --> 00:28:52,090 And because of the states factor, 442 00:28:52,090 --> 00:28:56,870 I can factor also with a little bit of work the operators. 443 00:28:56,870 --> 00:28:59,350 So I can write the operators. 444 00:28:59,350 --> 00:29:00,370 I'll just write it down. 445 00:29:00,370 --> 00:29:02,470 I think it's clear. 446 00:29:05,770 --> 00:29:07,430 And I'll tell you some of the things 447 00:29:07,430 --> 00:29:08,597 you have to do to get there. 448 00:29:31,730 --> 00:29:35,740 So I am skipping some steps here because if we do it-- 449 00:29:35,740 --> 00:29:38,710 we can do it very carefully, and we come write out each step 450 00:29:38,710 --> 00:29:40,183 one at a time. 451 00:29:40,183 --> 00:29:42,100 But there's a lot of writing if one does that, 452 00:29:42,100 --> 00:29:44,920 so I'm kind of trying to write steps 453 00:29:44,920 --> 00:29:48,940 where I think it's intuitive what the results are 454 00:29:48,940 --> 00:29:52,990 and skip steps that some level can 455 00:29:52,990 --> 00:29:58,210 be filled in either by looking at the literature 456 00:29:58,210 --> 00:30:00,820 or you just believe me. 457 00:30:00,820 --> 00:30:04,840 So we have these operators that were still tied together 458 00:30:04,840 --> 00:30:07,150 in some sense because there is a color contraction 459 00:30:07,150 --> 00:30:09,310 of an index between them. 460 00:30:09,310 --> 00:30:12,460 But we can get around that basically using some-- 461 00:30:12,460 --> 00:30:15,670 once we realized that the matrix elements factor like this, 462 00:30:15,670 --> 00:30:18,230 we can use some color identities and put it in this form. 463 00:30:18,230 --> 00:30:21,250 So there's an overall pre-factor 1 464 00:30:21,250 --> 00:30:24,090 over nc, which I didn't write. 465 00:30:24,090 --> 00:30:25,690 There's some overall pre-factors, 466 00:30:25,690 --> 00:30:27,410 but the color is separate. 467 00:30:27,410 --> 00:30:30,460 There's a trace, if you like, over color here, 468 00:30:30,460 --> 00:30:34,460 and these guys here are traced into color singlets as well. 469 00:30:34,460 --> 00:30:37,340 Ray trace in each case. 470 00:30:37,340 --> 00:30:41,340 So just using some color identities, we can do that. 471 00:30:41,340 --> 00:30:44,028 So once-- remember the fact that the states factor 472 00:30:44,028 --> 00:30:46,070 in that the operators can be written in a product 473 00:30:46,070 --> 00:30:48,650 means that all contractions are happening between these guys 474 00:30:48,650 --> 00:30:49,275 and these guys. 475 00:30:49,275 --> 00:30:51,908 That's why I can separate the matrix elements the way I have. 476 00:30:51,908 --> 00:30:53,450 And the only thing is then that there 477 00:30:53,450 --> 00:30:55,492 could be color indices tying these guys together, 478 00:30:55,492 --> 00:30:58,757 but it turns out you could deal with that as well. 479 00:30:58,757 --> 00:31:00,340 When you deal with the color theories, 480 00:31:00,340 --> 00:31:03,860 you could get things like-- you could get a ta in here or-- 481 00:31:03,860 --> 00:31:07,260 well, you could get, for example, ta ta or something, 482 00:31:07,260 --> 00:31:09,440 but then there's some ways of getting 483 00:31:09,440 --> 00:31:14,072 rid of those terms related to the fact that this has to be-- 484 00:31:14,072 --> 00:31:16,820 you have to argue that certain things are color singlets. 485 00:31:16,820 --> 00:31:18,930 I'm not going to go through that. 486 00:31:18,930 --> 00:31:20,690 So there's terms that we're dropping here, 487 00:31:20,690 --> 00:31:24,290 and those are the other high order-- 488 00:31:24,290 --> 00:31:28,140 those are the other power corrections. 489 00:31:28,140 --> 00:31:31,440 So this keeps all the power corrections of the first type, 490 00:31:31,440 --> 00:31:34,830 mu s, lambda QCD over mu s because those can still 491 00:31:34,830 --> 00:31:37,230 be encoded in our soft function, which 492 00:31:37,230 --> 00:31:41,820 is going to be this thing involving the soft states. 493 00:31:41,820 --> 00:31:44,310 But it drops the other power corrections. 494 00:31:44,310 --> 00:31:46,083 AUDIENCE: How do you know that since you 495 00:31:46,083 --> 00:31:48,120 derived the [INAUDIBLE] by looking 496 00:31:48,120 --> 00:31:51,500 at the perturbative diagrams? 497 00:31:51,500 --> 00:31:53,945 PROFESSOR: No, I didn't. 498 00:31:53,945 --> 00:31:55,235 I can derive-- 499 00:31:55,235 --> 00:31:57,110 I don't need to look at perturbative diagrams 500 00:31:57,110 --> 00:31:58,540 to draw the Wilson lines. 501 00:31:58,540 --> 00:32:01,560 I just derived them from the SCET Lagrangian 502 00:32:01,560 --> 00:32:04,371 by making the field redefinition. 503 00:32:04,371 --> 00:32:07,260 AUDIENCE: So you're saying that because the Langrangian 504 00:32:07,260 --> 00:32:11,250 decoupled, that's good for all-- 505 00:32:11,250 --> 00:32:12,915 PROFESSOR: All soft particles. 506 00:32:19,230 --> 00:32:21,300 So far, we actually haven't needed 507 00:32:21,300 --> 00:32:23,430 to use perturbation theory, and the idea here 508 00:32:23,430 --> 00:32:25,680 is that we're not going to use perturbation theory. 509 00:32:25,680 --> 00:32:27,240 We're going to write down some formula that's 510 00:32:27,240 --> 00:32:28,990 true to all orders in perturbation theory, 511 00:32:28,990 --> 00:32:32,550 and we're really only trading the power expansion. 512 00:32:32,550 --> 00:32:34,860 That's what we're focusing on here 513 00:32:34,860 --> 00:32:37,377 without thinking about things as perturbative things. 514 00:32:37,377 --> 00:32:39,210 Although, obviously, some of these functions 515 00:32:39,210 --> 00:32:40,620 will be perturbative functions that we 516 00:32:40,620 --> 00:32:41,912 will end up wanting to compute. 517 00:32:45,480 --> 00:32:49,810 OK, so let's think about what res prime means. 518 00:32:49,810 --> 00:32:53,490 And it's really a reminder that this formula is only 519 00:32:53,490 --> 00:32:59,700 really valid if we're making a measurement 520 00:32:59,700 --> 00:33:03,285 on the final state that really puts us in this configuration. 521 00:33:08,740 --> 00:33:11,477 So it's not enough to say we are in this situation. 522 00:33:11,477 --> 00:33:13,810 You're going to have to measure something that makes you 523 00:33:13,810 --> 00:33:19,160 in that dijet configuration, and that's important to remember. 524 00:33:23,090 --> 00:33:25,630 And so we're going to measure the hemisphere invariant 525 00:33:25,630 --> 00:33:26,510 masses. 526 00:33:26,510 --> 00:33:28,780 So we need to cook that into our formula, 527 00:33:28,780 --> 00:33:31,540 and there's a very easy way of doing that. 528 00:33:31,540 --> 00:33:35,570 We say that the following is true. 529 00:33:35,570 --> 00:33:39,070 I can write one, which is equal to an integral of two 530 00:33:39,070 --> 00:33:39,970 delta functions. 531 00:33:49,590 --> 00:33:51,210 And this formula is obviously true. 532 00:34:00,990 --> 00:34:02,670 And then what I'm going to do is I'm 533 00:34:02,670 --> 00:34:04,530 going to instead of integrating this, 534 00:34:04,530 --> 00:34:07,162 I'm going to move it as a d sigma d these things, 535 00:34:07,162 --> 00:34:09,120 and that's going to leave these delta functions 536 00:34:09,120 --> 00:34:10,889 inside the formula here. 537 00:34:10,889 --> 00:34:13,800 But since they depend on collinear and soft momenta, 538 00:34:13,800 --> 00:34:16,830 which are the momenta that are in these states, 539 00:34:16,830 --> 00:34:18,840 they're going to be sort of tightly connected 540 00:34:18,840 --> 00:34:22,290 to the whole thing, whatever is going on in this formula. 541 00:34:22,290 --> 00:34:24,742 It's not like they commute with this xn. 542 00:34:24,742 --> 00:34:26,159 They only commute with the xn if I 543 00:34:26,159 --> 00:34:28,534 can make them into the identity by integrating over this. 544 00:34:28,534 --> 00:34:31,500 But once I pull this through and put it over 545 00:34:31,500 --> 00:34:32,530 on the right hand side-- 546 00:34:32,530 --> 00:34:35,174 left hand side, then you have to leave these delta functions 547 00:34:35,174 --> 00:34:35,730 there. 548 00:34:35,730 --> 00:34:37,439 They're specifying the measurement. 549 00:34:40,980 --> 00:34:47,699 So this is a total n collinear momentum, 550 00:34:47,699 --> 00:34:54,819 and this is the total ultra soft in hemisphere A. 551 00:34:54,819 --> 00:34:57,480 That's what the A means. 552 00:34:57,480 --> 00:35:00,840 So I divided everything into hemisphere A and B. Obviously, 553 00:35:00,840 --> 00:35:02,340 all the n collinear particles, which 554 00:35:02,340 --> 00:35:04,860 are going in that direction, are in that hemisphere, 555 00:35:04,860 --> 00:35:06,940 but the soft ones could go in either case. 556 00:35:06,940 --> 00:35:09,690 And so then there's a soft momentum in hemisphere B 557 00:35:09,690 --> 00:35:15,240 and soft momentum in hemisphere A. Now this is unexpanded. 558 00:35:15,240 --> 00:35:20,720 I can also expand these delta functions, 559 00:35:20,720 --> 00:35:23,418 and I only need to keep the leading order piece if we're 560 00:35:23,418 --> 00:35:24,460 working at leading order. 561 00:35:38,100 --> 00:35:40,570 And so the leading order pieces are as follows. 562 00:35:40,570 --> 00:35:43,410 There's an order of lambda squared piece. 563 00:35:43,410 --> 00:35:44,383 This is lambda squared. 564 00:35:44,383 --> 00:35:46,050 There's an order of lambda squared piece 565 00:35:46,050 --> 00:35:47,950 from the collinear momentum squared, 566 00:35:47,950 --> 00:35:50,220 and since this is a total collinear momentum, 567 00:35:50,220 --> 00:35:52,050 it can involve multiple massless particles. 568 00:35:52,050 --> 00:35:53,610 It doesn't have to be zero. 569 00:35:53,610 --> 00:35:56,040 And then there's a cross term between the softs 570 00:35:56,040 --> 00:35:56,790 and the collinear. 571 00:35:56,790 --> 00:35:58,082 It's also order lambda squared. 572 00:35:58,082 --> 00:36:00,440 This order one. 573 00:36:00,440 --> 00:36:02,850 And this is order lambda squared, 574 00:36:02,850 --> 00:36:04,293 and that term's the same size. 575 00:36:04,293 --> 00:36:06,210 And then all the other terms are higher order. 576 00:36:08,770 --> 00:36:11,560 These terms are suppressed, so this is really 577 00:36:11,560 --> 00:36:12,970 all I need to keep. 578 00:36:12,970 --> 00:36:17,620 There's also something else that effectively we've 579 00:36:17,620 --> 00:36:20,448 done by setting things up, and that is, in general, you 580 00:36:20,448 --> 00:36:22,990 could think that the collinear particles would have some perp 581 00:36:22,990 --> 00:36:24,070 momentum. 582 00:36:24,070 --> 00:36:27,650 But we aligned our axes with the jet axis, 583 00:36:27,650 --> 00:36:30,190 so there's actually no perp momentum there either. 584 00:36:30,190 --> 00:36:33,550 So that guy is really just-- 585 00:36:33,550 --> 00:36:45,620 we can pull out a pn minus and then we have pn plus k sa plus. 586 00:36:45,620 --> 00:36:49,970 And we also know that pn minus is q. 587 00:36:49,970 --> 00:36:52,100 That was fixed by kinematics. 588 00:36:52,100 --> 00:36:54,640 OK, so this simplifies quite a bit. 589 00:36:57,340 --> 00:36:59,840 And we're going to drop all the powers of [INAUDIBLE] terms. 590 00:37:17,620 --> 00:37:21,770 All right, so that is one thing. 591 00:37:21,770 --> 00:37:28,660 So then I can do what I said in words, write this guy, 592 00:37:28,660 --> 00:37:34,930 and it will have these two deltas that 593 00:37:34,930 --> 00:37:37,850 are under similar acts. 594 00:37:37,850 --> 00:37:40,210 They don't commute through with it-- without it. 595 00:37:40,210 --> 00:37:43,010 They depend on x. 596 00:37:43,010 --> 00:37:46,210 Now another thing we would like to do 597 00:37:46,210 --> 00:37:50,560 is the factorize the measurement. 598 00:37:50,560 --> 00:37:52,090 So this measurement here involves 599 00:37:52,090 --> 00:37:55,630 the sum of a plus momentum collinear and a plus momentum 600 00:37:55,630 --> 00:37:57,340 in soft. 601 00:37:57,340 --> 00:37:58,810 And it's all in one delta function. 602 00:37:58,810 --> 00:38:00,852 We'd like to write it as separate delta functions 603 00:38:00,852 --> 00:38:03,760 that we can associate with those different parts here, 604 00:38:03,760 --> 00:38:07,150 one delta-- we'd like to have a separate thing here 605 00:38:07,150 --> 00:38:08,560 from these guys. 606 00:38:08,560 --> 00:38:10,880 And that's actually very easy to do. 607 00:38:10,880 --> 00:38:16,596 We simply write the following. 608 00:38:27,530 --> 00:38:29,632 Just introduce some more delta functions. 609 00:38:38,800 --> 00:38:42,270 So there's this tying together delta function. 610 00:38:42,270 --> 00:38:45,960 K plus is just some dummy variable. 611 00:38:45,960 --> 00:38:47,580 It's not the momentum of any state. 612 00:38:47,580 --> 00:38:54,340 It's just a dummy variable, whereas pn plus and ksa plus 613 00:38:54,340 --> 00:38:57,560 were momenta of particles in the state. 614 00:38:57,560 --> 00:39:01,340 So this guy can then associate with the-- 615 00:39:01,340 --> 00:39:11,100 put them together with the n collinear matrix element, 616 00:39:11,100 --> 00:39:12,150 which is this guy here. 617 00:39:15,454 --> 00:39:17,040 I forget what color he was. 618 00:39:21,990 --> 00:39:27,360 And then for the ultra soft, we can 619 00:39:27,360 --> 00:39:29,730 associated this delta function. 620 00:39:32,123 --> 00:39:33,540 So that's what I mean by factoring 621 00:39:33,540 --> 00:39:35,332 the measurement, that we can put the piece, 622 00:39:35,332 --> 00:39:36,990 depended on the state, together. 623 00:39:36,990 --> 00:39:39,810 And then you can see that we can move our-- 624 00:39:39,810 --> 00:39:42,660 well, OK, we have to also factor this delta function. 625 00:39:42,660 --> 00:39:45,360 I should say that too. 626 00:39:45,360 --> 00:39:48,760 That's the part that I'm not going to go through, 627 00:39:48,760 --> 00:39:53,100 but we can play similar games expanding and factoring 628 00:39:53,100 --> 00:39:56,250 this delta four as well. 629 00:39:56,250 --> 00:39:59,730 And that's besides just being a little bit tedious, 630 00:39:59,730 --> 00:40:04,970 it's not really any more difficult. Maybe 631 00:40:04,970 --> 00:40:07,440 a little more difficult. 632 00:40:07,440 --> 00:40:08,660 So we factor that guy too. 633 00:40:14,920 --> 00:40:17,900 And then there's one other trick that we want to do, 634 00:40:17,900 --> 00:40:23,320 which is useful, and that is that we can write 635 00:40:23,320 --> 00:40:24,910 some deltas in Fourier's space. 636 00:40:37,135 --> 00:40:38,760 So we can always write a delta function 637 00:40:38,760 --> 00:40:42,750 as an integral over a phase, and that's convenient 638 00:40:42,750 --> 00:40:44,677 because in the phase, the momenta that 639 00:40:44,677 --> 00:40:46,260 are in the delta function also factor. 640 00:40:49,930 --> 00:40:52,870 So I can write it as a product of two separate factors 641 00:40:52,870 --> 00:40:53,410 like this. 642 00:40:56,002 --> 00:40:58,150 There's some halves. 643 00:40:58,150 --> 00:41:01,060 It's conventional. 644 00:41:01,060 --> 00:41:05,230 OK, so this guy here, which is the guy that 645 00:41:05,230 --> 00:41:07,150 is associated to the state, it looks 646 00:41:07,150 --> 00:41:09,250 like a translation operator. 647 00:41:09,250 --> 00:41:13,960 if you have an e to the i x dot p, that you can use. 648 00:41:13,960 --> 00:41:16,030 You can put into your matrix element, 649 00:41:16,030 --> 00:41:19,000 and you can translate the fields, which 650 00:41:19,000 --> 00:41:21,802 are in this formula all at 0. 651 00:41:21,802 --> 00:41:25,000 They're all at space time 0. 652 00:41:25,000 --> 00:41:27,370 What you can do with this guy here 653 00:41:27,370 --> 00:41:31,300 is you can put it into the matrix element 654 00:41:31,300 --> 00:41:33,790 and translate the fields to point x. 655 00:41:45,037 --> 00:41:49,240 And then you'd have a chi and x minus. 656 00:41:49,240 --> 00:41:51,910 So that's how we can deal with that guy. 657 00:41:51,910 --> 00:41:55,570 And so if we do a bunch of stuff like that I'm not 658 00:41:55,570 --> 00:41:58,870 going to go through on the board for you, after some work, 659 00:41:58,870 --> 00:42:02,200 we get something just kind of an intermediate step 660 00:42:02,200 --> 00:42:10,260 of our factorization, which I'll write out 661 00:42:10,260 --> 00:42:13,320 so you can get some idea of-- 662 00:42:13,320 --> 00:42:16,140 if I jump to the final answer, it's kind of too simple 663 00:42:16,140 --> 00:42:18,850 to see what's going on. 664 00:42:18,850 --> 00:42:20,460 So let's write this one out. 665 00:42:32,360 --> 00:42:36,423 So I was focusing on the n collinear matrix element, 666 00:42:36,423 --> 00:42:38,840 and everything I was writing, I wrote this delta function. 667 00:42:38,840 --> 00:42:41,540 But there's, of course, another one for the delta n bar, 668 00:42:41,540 --> 00:42:43,850 and I do the same thing for him. 669 00:42:43,850 --> 00:42:46,740 So we got k plus and l plus by factoring the measurement 670 00:42:46,740 --> 00:42:47,240 there. 671 00:42:47,240 --> 00:42:49,310 You get a k minus and an l minus by factoring 672 00:42:49,310 --> 00:42:50,495 the other measurement. 673 00:43:02,880 --> 00:43:05,880 One thing that comes into-- 674 00:43:05,880 --> 00:43:08,060 that makes this guy here a little more complicated 675 00:43:08,060 --> 00:43:10,477 is you have to think a little bit about residual and label 676 00:43:10,477 --> 00:43:11,420 momentum. 677 00:43:11,420 --> 00:43:16,640 But at the end of the day, the result is what I'm writing. 678 00:43:36,230 --> 00:43:43,640 So this is going to be completely factorizing to three 679 00:43:43,640 --> 00:43:44,495 independent things. 680 00:44:02,570 --> 00:44:05,900 There's one set of collinear fields for the quark, 681 00:44:05,900 --> 00:44:09,830 and there's another for the antiquark if you like. 682 00:44:09,830 --> 00:44:11,480 You're specifying whether it's a quark 683 00:44:11,480 --> 00:44:13,290 or an antiquark with a label. 684 00:44:13,290 --> 00:44:16,350 So the fact that this label here is positive 685 00:44:16,350 --> 00:44:20,298 means that this is a quark, and maybe this should be a minus q. 686 00:44:26,860 --> 00:44:31,538 And then this guy is specifying the antiquark, 687 00:44:31,538 --> 00:44:32,830 and then there's the soft part. 688 00:45:04,660 --> 00:45:10,760 OK, so this is a fairly messy one board expression, 689 00:45:10,760 --> 00:45:14,140 but this guy here is factored. 690 00:45:14,140 --> 00:45:16,570 We have the hard modes here, collinear modes, 691 00:45:16,570 --> 00:45:19,450 n collinear here, and bar collinear there, 692 00:45:19,450 --> 00:45:26,920 and the ultra soft goes here, which I've already 693 00:45:26,920 --> 00:45:30,100 dropped in my ultra soft. 694 00:45:30,100 --> 00:45:35,020 So this last guy if you think about what this matrix element 695 00:45:35,020 --> 00:45:39,090 is it's just some function. 696 00:45:39,090 --> 00:45:40,268 We sum over all x of this. 697 00:45:40,268 --> 00:45:42,060 So we sum over all the intermediate states. 698 00:45:42,060 --> 00:45:44,185 The only thing we're fixing are l plus and l minus. 699 00:45:48,555 --> 00:45:50,180 So when we sum over states, we actually 700 00:45:50,180 --> 00:45:52,990 integrate the momentum of particles in the state 701 00:45:52,990 --> 00:45:54,090 if you can think of that. 702 00:45:54,090 --> 00:45:55,010 Those are different states. 703 00:45:55,010 --> 00:45:57,427 You have to sum over them too, but it's a continuous label 704 00:45:57,427 --> 00:46:01,467 and therefore you integrate over the phase base. 705 00:46:01,467 --> 00:46:03,800 And that's what we mean when we write the sum over axis. 706 00:46:03,800 --> 00:46:06,025 That includes phase based intervals. 707 00:46:08,940 --> 00:46:10,890 And so the only-- so this momentum 708 00:46:10,890 --> 00:46:12,990 gets integrated over if you like, 709 00:46:12,990 --> 00:46:15,570 but then it gets-- there's a component that 710 00:46:15,570 --> 00:46:18,960 gets fixed, which is the total momentum of plus or minus 711 00:46:18,960 --> 00:46:21,055 momentum in each hemisphere. 712 00:46:21,055 --> 00:46:22,305 So this is just some function. 713 00:46:29,900 --> 00:46:34,880 So that we color the same. 714 00:46:34,880 --> 00:46:41,367 It's just some function of l plus or l minus, 715 00:46:41,367 --> 00:46:42,700 and it's called a soft function. 716 00:46:49,210 --> 00:46:51,220 And at this stage of the game, it's 717 00:46:51,220 --> 00:46:53,905 actually encoding two different momentum scales. 718 00:46:57,560 --> 00:47:01,600 The l plus and minus as well as sort 719 00:47:01,600 --> 00:47:03,790 of lambda QCD, which I didn't write explicitly 720 00:47:03,790 --> 00:47:04,420 as an argument. 721 00:47:07,060 --> 00:47:11,230 And l plus and minus, well, we'll 722 00:47:11,230 --> 00:47:14,260 see later, but you can think of l plus and minus 723 00:47:14,260 --> 00:47:19,210 roughly as encoding the scale that's m squared over q. 724 00:47:19,210 --> 00:47:22,180 This will become more clear in a minute. 725 00:47:26,510 --> 00:47:30,181 So again, we need to figure out what these guys here are. 726 00:47:30,181 --> 00:47:32,830 That's a little bit more work but not too much. 727 00:47:37,550 --> 00:47:41,270 And they're both, of course, mirrors of each other. 728 00:47:41,270 --> 00:47:44,560 So really, we just have to be one of them, and if you like, 729 00:47:44,560 --> 00:47:47,238 the other is just kind of charge congregation. 730 00:47:56,510 --> 00:47:59,170 So this guy here, we're going to write in Fourier's space. 731 00:48:01,870 --> 00:48:09,530 That's a convenient thing to do because what 732 00:48:09,530 --> 00:48:11,870 we know about the Feynman rules in Fourier's space 733 00:48:11,870 --> 00:48:13,580 when we thought about the Feynman rules 734 00:48:13,580 --> 00:48:17,690 in Fourier's space, we knew that the collinear propagators would 735 00:48:17,690 --> 00:48:20,520 only depend on the small plus momentum. 736 00:48:20,520 --> 00:48:23,270 So the x-coordinates here, the x-coordinates 737 00:48:23,270 --> 00:48:25,840 of our fields, those corresponded 738 00:48:25,840 --> 00:48:30,020 to residual momenta, but when we do Feynman diagrams, 739 00:48:30,020 --> 00:48:32,720 there's only the plus momentum showing up. 740 00:48:32,720 --> 00:48:34,250 That's the multipole expansion. 741 00:48:39,248 --> 00:48:41,540 So you could think about it from the [? phonographs, ?] 742 00:48:41,540 --> 00:48:44,060 but it's really a general property 743 00:48:44,060 --> 00:48:45,490 from the multipole expansion. 744 00:48:50,550 --> 00:48:54,260 So that means that some of these integrals here are just trivial 745 00:48:54,260 --> 00:48:55,550 and give me a delta function. 746 00:48:59,805 --> 00:49:02,180 So I'm getting delta functions in some directions because 747 00:49:02,180 --> 00:49:05,015 of the multipole expansion in this formula. 748 00:49:12,320 --> 00:49:14,460 Because I aligned the axis, there 749 00:49:14,460 --> 00:49:17,210 was no perpendicular momentum, so it's really just this r plus 750 00:49:17,210 --> 00:49:19,710 that we get, and this thing here is called the jet function. 751 00:49:24,602 --> 00:49:26,810 So this is the non-trivial function that can show up. 752 00:49:31,190 --> 00:49:32,300 We do the same thing. 753 00:49:35,190 --> 00:49:38,480 We do the same thing for the case with the sum over xn bar, 754 00:49:38,480 --> 00:49:42,530 and it gives us another jet function. 755 00:49:42,530 --> 00:49:47,390 And by charge conjugation, it's the same jet function. 756 00:49:47,390 --> 00:49:53,200 But in this case, it would be q of sum r prime minus 757 00:49:53,200 --> 00:49:55,880 and some other momentum. 758 00:49:55,880 --> 00:49:58,760 And so what we then do is we take this formula, 759 00:49:58,760 --> 00:50:03,560 plug it back into here, do all the integrals, and at the end 760 00:50:03,560 --> 00:50:05,330 of the day, we can actually do-- 761 00:50:05,330 --> 00:50:08,000 we have lots of delta functions and lots of integrals. 762 00:50:08,000 --> 00:50:10,770 And we can boil everything down to just two integrals left 763 00:50:10,770 --> 00:50:11,270 over. 764 00:50:16,230 --> 00:50:21,810 And that gives us our final result. So I'll write this way. 765 00:50:39,710 --> 00:50:42,800 So it boils down to just involving 766 00:50:42,800 --> 00:50:44,845 these jet functions and the soft function, 767 00:50:44,845 --> 00:50:46,220 and then there's something that's 768 00:50:46,220 --> 00:50:50,330 called the hard function, which is just our Wilson coefficient 769 00:50:50,330 --> 00:50:50,985 squared. 770 00:50:50,985 --> 00:50:52,610 So rather than write Wilson coefficient 771 00:50:52,610 --> 00:50:56,300 squared all the time, we just call it another function h. 772 00:50:56,300 --> 00:50:59,430 So this is the factorization theorem. 773 00:50:59,430 --> 00:51:01,790 So you see in this case, the hard function was just 774 00:51:01,790 --> 00:51:04,520 a multiplicative factor unlike dis, 775 00:51:04,520 --> 00:51:07,653 and what was-- where things are talking to each other. 776 00:51:07,653 --> 00:51:09,320 Again, things are talking to each other, 777 00:51:09,320 --> 00:51:10,760 but it's the soft modes-- 778 00:51:10,760 --> 00:51:13,370 ultra soft modes that are talking to the jets. 779 00:51:13,370 --> 00:51:16,250 And ultra soft momenta can change the mass, 780 00:51:16,250 --> 00:51:19,460 and that's what kind of the correct mass-- 781 00:51:19,460 --> 00:51:20,900 this is the total mass. 782 00:51:20,900 --> 00:51:23,840 The correct mass for the jet function is the collinear mass, 783 00:51:23,840 --> 00:51:25,637 and that's this thing minus this thing. 784 00:51:25,637 --> 00:51:28,220 So if you wanted to guess this formula, this is how you do it. 785 00:51:28,220 --> 00:51:30,800 You'd say m squared, just from the kinematic relation, 786 00:51:30,800 --> 00:51:34,212 m squared had a collinear piece and a soft piece. 787 00:51:34,212 --> 00:51:36,170 The right thing to evaluate the jet function at 788 00:51:36,170 --> 00:51:40,050 would be just the collinear, so that's the difference. 789 00:51:40,050 --> 00:51:43,340 And then the soft function could depend on these momentum, 790 00:51:43,340 --> 00:51:46,580 and you're basically led to a formula like this one. 791 00:51:46,580 --> 00:51:48,560 But we can also just, with the field theory, 792 00:51:48,560 --> 00:51:50,060 go through and derive it, and that's 793 00:51:50,060 --> 00:51:53,600 what I'm convincing you of even if I was skipping some steps. 794 00:51:56,860 --> 00:51:59,200 All right, so this is the dijet factorization theorem 795 00:51:59,200 --> 00:52:01,855 for hemisphere invariant masses. 796 00:52:06,880 --> 00:52:08,920 And a lot of event shapes kind of 797 00:52:08,920 --> 00:52:11,555 go along the pattern that we've done here. 798 00:52:11,555 --> 00:52:13,930 If you think about what would be the difference if I pick 799 00:52:13,930 --> 00:52:16,870 some other observable both sides of these masses, 800 00:52:16,870 --> 00:52:19,930 pick some other thing that you could measure that would tell 801 00:52:19,930 --> 00:52:23,350 you that there's dijets, we just swap out the measurement part, 802 00:52:23,350 --> 00:52:26,470 and that would lead to-- and just see where it leads us. 803 00:52:26,470 --> 00:52:30,068 A lot of the steps would be exactly the same steps. 804 00:52:30,068 --> 00:52:31,860 So we did it for one particular observable, 805 00:52:31,860 --> 00:52:35,260 but steps are the same if you do others. 806 00:52:39,820 --> 00:52:41,980 If you're in this situation where you have dijets, 807 00:52:41,980 --> 00:52:44,740 there's more than one way of measuring an event 808 00:52:44,740 --> 00:52:46,300 to ensure you have dijets. 809 00:52:46,300 --> 00:52:52,780 There's something called dijet event shapes. 810 00:52:52,780 --> 00:52:54,740 They go under the names of things like thrust, 811 00:52:54,740 --> 00:52:57,460 C parameter, heavy jet mass. 812 00:53:00,580 --> 00:53:03,423 All these things actually have factorization theorems 813 00:53:03,423 --> 00:53:05,590 that you could derive in a similar way to what we're 814 00:53:05,590 --> 00:53:06,100 doing here. 815 00:53:09,510 --> 00:53:12,860 All right, now this is kind of like-- 816 00:53:12,860 --> 00:53:14,540 I didn't put any mus in, but just 817 00:53:14,540 --> 00:53:17,850 like we did before, we can always put the mus in. 818 00:53:17,850 --> 00:53:21,050 So if we put the mus in, everybody gets a mu. 819 00:53:21,050 --> 00:53:24,020 The h is like the mu of the Wilson coefficient, 820 00:53:24,020 --> 00:53:25,460 and then the Js are like-- 821 00:53:25,460 --> 00:53:27,645 and Ss are like mus of-- 822 00:53:27,645 --> 00:53:29,270 from the point of view of SCET, they're 823 00:53:29,270 --> 00:53:30,593 like mus of the operator. 824 00:53:30,593 --> 00:53:32,510 We're just switching to renormalized operators 825 00:53:32,510 --> 00:53:34,494 and renormalized Wilson coefficients. 826 00:53:38,130 --> 00:53:42,740 Now if we draw our scale diagram again 827 00:53:42,740 --> 00:53:47,300 but draw it a little bigger this time 828 00:53:47,300 --> 00:53:49,820 and put in where these various functions are-- 829 00:54:00,970 --> 00:54:02,956 so somebody was blue. 830 00:54:11,530 --> 00:54:13,050 Not going to label everybody again. 831 00:54:15,710 --> 00:54:17,290 So this is the hard function. 832 00:54:17,290 --> 00:54:20,120 Our jet functions are kind of sitting at this scale. 833 00:54:20,120 --> 00:54:22,360 They had to do with the collinear modes, 834 00:54:22,360 --> 00:54:25,870 and then our soft function is sitting at this scale or this 835 00:54:25,870 --> 00:54:27,512 and this scale. 836 00:54:27,512 --> 00:54:29,845 So it's clear that these things sit at different scales. 837 00:54:36,935 --> 00:54:39,310 And what that means from the point of view of the formula 838 00:54:39,310 --> 00:54:44,670 is that if you associate what scale these things want 839 00:54:44,670 --> 00:54:47,790 to live at, which from the point of view of perturbation theory 840 00:54:47,790 --> 00:54:51,030 means at what scale should I expect-- would I 841 00:54:51,030 --> 00:54:52,680 be able to calculate these things given 842 00:54:52,680 --> 00:54:53,470 that they're perturbative? 843 00:54:53,470 --> 00:54:55,178 Would I be able to calculate these things 844 00:54:55,178 --> 00:54:57,150 without encountering large logarithms? 845 00:54:57,150 --> 00:55:00,580 And that would be these scales that I'm writing here. 846 00:55:00,580 --> 00:55:04,560 So these are the scales where we could 847 00:55:04,560 --> 00:55:07,980 do perturbative theory for the functions 848 00:55:07,980 --> 00:55:10,950 without encountering large logarithms. 849 00:55:10,950 --> 00:55:13,380 So we can figure out what these skills 850 00:55:13,380 --> 00:55:15,330 are by just looking at the perturbative theory 851 00:55:15,330 --> 00:55:16,580 and looking at the logarithms. 852 00:55:26,340 --> 00:55:28,340 But you see that it's a different scale for each 853 00:55:28,340 --> 00:55:29,480 of the different functions. 854 00:55:33,720 --> 00:55:36,590 So in this formula, there's a common mu, which 855 00:55:36,590 --> 00:55:38,570 is like a factorization scale. 856 00:55:38,570 --> 00:55:41,060 But each of the functions wants to live at a different mu, 857 00:55:41,060 --> 00:55:43,393 and we're going to have to do some renormalization group 858 00:55:43,393 --> 00:55:46,520 evolution to put things at kind of at the scales they 859 00:55:46,520 --> 00:55:47,430 want to be at. 860 00:55:47,430 --> 00:55:49,820 And then there'll be some resummation going on, 861 00:55:49,820 --> 00:55:53,610 and that's going to sum logs of m squared over q squared. 862 00:55:53,610 --> 00:56:05,910 So we need normalization group, and that's 863 00:56:05,910 --> 00:56:15,870 trying to sum up logarithms, which 864 00:56:15,870 --> 00:56:17,570 are ratios of these scales, which 865 00:56:17,570 --> 00:56:19,940 expressed in terms of some physical thing 866 00:56:19,940 --> 00:56:23,340 is, in this case, logs of m squared over q squared. 867 00:56:28,380 --> 00:56:35,767 All right, what about this situation here? 868 00:56:35,767 --> 00:56:36,600 Well, it turns out-- 869 00:56:40,510 --> 00:56:42,580 and I won't go into it in too much detail, 870 00:56:42,580 --> 00:56:45,130 but we can factor that guy. 871 00:56:45,130 --> 00:56:46,660 If these two are hierarchical, we 872 00:56:46,660 --> 00:56:51,940 can actually factor this in something called 873 00:56:51,940 --> 00:56:56,930 the soft function OPE, and even if they're not hierarchical, 874 00:56:56,930 --> 00:56:59,762 we can actually write it in the following way. 875 00:56:59,762 --> 00:57:01,720 So we can always, in some sense, whether or not 876 00:57:01,720 --> 00:57:03,820 we're in a situation where these are comparable 877 00:57:03,820 --> 00:57:05,920 or whether they're hierarchical, we can always 878 00:57:05,920 --> 00:57:07,900 use the following formula. 879 00:57:07,900 --> 00:57:13,490 We're not guaranteed that that is possible, 880 00:57:13,490 --> 00:57:17,050 but in this situation, it is possible to sort of make 881 00:57:17,050 --> 00:57:19,430 those two situations compatible with each other 882 00:57:19,430 --> 00:57:21,850 with a single formula. 883 00:57:21,850 --> 00:57:23,740 Usually, when you design effective theories, 884 00:57:23,740 --> 00:57:26,470 you kind of for each different expansion, 885 00:57:26,470 --> 00:57:29,170 you have a different result. And making them-- 886 00:57:29,170 --> 00:57:32,618 whether you can put them together into a single result 887 00:57:32,618 --> 00:57:33,660 is not always guaranteed. 888 00:57:33,660 --> 00:57:35,993 But this is a situation where we can put things together 889 00:57:35,993 --> 00:57:38,170 in a single result where these are 890 00:57:38,170 --> 00:57:45,740 the perturbative part of the soft function, 891 00:57:45,740 --> 00:57:49,625 and this is the non-perturbative lambda QCD effects. 892 00:57:54,110 --> 00:57:56,450 If you imagine that the function-- if you imagine 893 00:57:56,450 --> 00:57:59,300 that these two axes are hierarchical, 894 00:57:59,300 --> 00:58:04,130 then this perturbative guy here gives you some ls corrections, 895 00:58:04,130 --> 00:58:06,890 and it basically, in terms of the l plus, 896 00:58:06,890 --> 00:58:09,110 in terms of the momentum variable in this formula, 897 00:58:09,110 --> 00:58:12,240 it's giving something that's, like, a power-- what's called 898 00:58:12,240 --> 00:58:14,270 the power law tail. 899 00:58:14,270 --> 00:58:17,120 So the kind of dependence that you're getting in 900 00:58:17,120 --> 00:58:26,420 is some logs over an l plus, and that's 901 00:58:26,420 --> 00:58:31,730 called a power law since it goes like 1 over l plus, 902 00:58:31,730 --> 00:58:34,550 whereas these effects here that are non-perturbative 903 00:58:34,550 --> 00:58:37,310 live down at non-perturbative momenta, 904 00:58:37,310 --> 00:58:42,590 and you can actually prove that they have an exponential tail. 905 00:58:47,110 --> 00:58:51,430 So this is some plot of f in a one dimensional projection, 906 00:58:51,430 --> 00:58:53,980 so let's just think of it as one. 907 00:58:53,980 --> 00:58:56,800 And this is some scale of order lambda QCD. 908 00:58:56,800 --> 00:58:59,950 So at some blob down lambda QCD, this 909 00:58:59,950 --> 00:59:03,820 describing the distribution of soft hadrons, 910 00:59:03,820 --> 00:59:05,410 and it's non-perturbative. 911 00:59:05,410 --> 00:59:08,450 It's not like I'm drawing it because I know it. 912 00:59:08,450 --> 00:59:11,680 So it looks something like this, whereas this part, we 913 00:59:11,680 --> 00:59:13,977 can calculate and it has a different dependence. 914 00:59:13,977 --> 00:59:14,810 This is a power law. 915 00:59:14,810 --> 00:59:16,384 This is an exponential. 916 00:59:20,650 --> 00:59:23,290 OK, so that's kind of further factorization 917 00:59:23,290 --> 00:59:27,535 of the s and to get rid to make those two axis separated. 918 00:59:27,535 --> 00:59:29,080 Can separate the perturbative and 919 00:59:29,080 --> 00:59:33,190 non-perturbative corrections. 920 00:59:33,190 --> 00:59:34,870 In some sense, if you're in a situation 921 00:59:34,870 --> 00:59:39,010 where they're comparable, then whether you 922 00:59:39,010 --> 00:59:42,065 take a non-perturbative function and if l plus is small 923 00:59:42,065 --> 00:59:43,690 and you integrated against this, you're 924 00:59:43,690 --> 00:59:46,370 just getting back some other non-perturbative function. 925 00:59:46,370 --> 00:59:48,055 So this formula is not really-- 926 00:59:48,055 --> 00:59:50,680 there are some things that this formula is actually doing, even 927 00:59:50,680 --> 00:59:53,200 in that situation, is ensuring that you're in the ms bar 928 00:59:53,200 --> 00:59:53,992 scheme for example. 929 00:59:57,130 --> 01:00:02,230 But the sort of more important in some ways 930 01:00:02,230 --> 01:00:04,360 is when they're hierarchical, and then you 931 01:00:04,360 --> 01:00:06,850 can start to expand this thing and-- 932 01:00:06,850 --> 01:00:08,620 because in that case, this l plus prime 933 01:00:08,620 --> 01:00:11,680 would be localized down here, and the l plus-- if the l plus 934 01:00:11,680 --> 01:00:14,140 got big, then you can start doing a Taylor series, 935 01:00:14,140 --> 01:00:16,720 and this formula would tell you what the-- how 936 01:00:16,720 --> 01:00:18,280 to put the corrections together. 937 01:00:24,160 --> 01:00:26,700 So there's kind of a lot of physics in that formula 938 01:00:26,700 --> 01:00:28,760 that I'm not going into in detail. 939 01:00:32,010 --> 01:00:32,715 Yeah. 940 01:00:32,715 --> 01:00:37,440 AUDIENCE: So I don't really totally understand 941 01:00:37,440 --> 01:00:39,960 why this is valid. 942 01:00:39,960 --> 01:00:42,960 Are you allowed to-- does the soft function code all those 943 01:00:42,960 --> 01:00:44,750 below that hyperbola? 944 01:00:44,750 --> 01:00:45,510 PROFESSOR: Right. 945 01:00:45,510 --> 01:00:46,302 AUDIENCE: Is that-- 946 01:00:46,302 --> 01:00:47,660 PROFESSOR: That's right. 947 01:00:47,660 --> 01:00:49,330 Below the above one and-- 948 01:00:49,330 --> 01:00:50,600 AUDIENCE: Yeah, below the s. 949 01:00:50,600 --> 01:00:51,950 PROFESSOR: Yeah, that's right. 950 01:00:51,950 --> 01:00:52,492 That's right. 951 01:00:52,492 --> 01:00:53,360 The soft function. 952 01:00:53,360 --> 01:00:55,310 You should always think of these things 953 01:00:55,310 --> 01:01:01,680 as kind of extending down to the axes in some kind of fashion. 954 01:01:01,680 --> 01:01:02,900 So let's drop like this. 955 01:01:05,810 --> 01:01:08,568 Since they're infrared modes and the infrared is sort of 956 01:01:08,568 --> 01:01:10,610 down at the axis, you should always think of them 957 01:01:10,610 --> 01:01:11,990 as extending down to the axis. 958 01:01:11,990 --> 01:01:18,500 Now exactly how you want to think about how the soft modes 959 01:01:18,500 --> 01:01:22,470 and whether the soft modes are capturing the entire axis 960 01:01:22,470 --> 01:01:24,030 or the collinear-- how the collinear 961 01:01:24,030 --> 01:01:27,630 modes and the soft mouths kind of what the edges are 962 01:01:27,630 --> 01:01:29,070 is related to the power counting. 963 01:01:29,070 --> 01:01:31,080 And you can think of the collinear modes 964 01:01:31,080 --> 01:01:33,510 as kind of being held away from the axis 965 01:01:33,510 --> 01:01:36,270 because the soft modes are really deeper in the infrared. 966 01:01:36,270 --> 01:01:40,050 AUDIENCE: But at higher power, are you going to screw these up 967 01:01:40,050 --> 01:01:44,550 if you say-- like suppose the soft modes have an invariant 968 01:01:44,550 --> 01:01:48,217 mass of lambda to the fifth, right, or they're-- 969 01:01:48,217 --> 01:01:49,050 PROFESSOR: Oh, yeah. 970 01:01:49,050 --> 01:01:53,400 No, that's-- you never have to worry about-- 971 01:01:53,400 --> 01:01:56,500 you could worry about putting in more hyperbolas, right, 972 01:01:56,500 --> 01:01:58,535 with lambda QCD over q or something. 973 01:01:58,535 --> 01:01:59,160 AUDIENCE: Yeah. 974 01:01:59,160 --> 01:02:00,210 PROFESSOR: And you don't have to do that. 975 01:02:00,210 --> 01:02:00,930 AUDIENCE: OK, so-- 976 01:02:00,930 --> 01:02:01,980 PROFESSOR: You never have to do that. 977 01:02:01,980 --> 01:02:02,940 AUDIENCE: [INAUDIBLE]. 978 01:02:02,940 --> 01:02:03,840 Why is that? 979 01:02:03,840 --> 01:02:06,750 PROFESSOR: I mean, essentially, that's because those-- 980 01:02:06,750 --> 01:02:09,540 any lambda QCDs, what this x means 981 01:02:09,540 --> 01:02:13,140 is that lambda QCD is encoded in the soft-- in this kind 982 01:02:13,140 --> 01:02:14,910 of non-perturbative soft mode. 983 01:02:14,910 --> 01:02:19,830 And if you had something that at higher powers like lambda QCD, 984 01:02:19,830 --> 01:02:22,560 to take your example, to the sixth, 985 01:02:22,560 --> 01:02:26,010 the one over q to the sixth is going to come out 986 01:02:26,010 --> 01:02:27,540 in some coefficient. 987 01:02:27,540 --> 01:02:29,488 And the lambda QCD to the sixth is just 988 01:02:29,488 --> 01:02:31,530 going to be some matrix element of this mode that 989 01:02:31,530 --> 01:02:33,810 has that dimension. 990 01:02:33,810 --> 01:02:36,980 That's how it's going to work. 991 01:02:36,980 --> 01:02:39,057 So in some sense, you could say, well, 992 01:02:39,057 --> 01:02:40,390 what if I put a mode down there? 993 01:02:40,390 --> 01:02:42,340 And I would just say, well, that mode's inside this mode. 994 01:02:42,340 --> 01:02:44,757 You don't have to write down something different for that. 995 01:02:44,757 --> 01:02:47,590 It's all encoded in the Lagrangian for that mode, 996 01:02:47,590 --> 01:02:49,735 and that mode has all the non-perturbative physics. 997 01:02:57,313 --> 01:02:58,730 AUDIENCE: Yeah, at a higher power, 998 01:02:58,730 --> 01:03:00,365 I would think you're comparing-- 999 01:03:00,365 --> 01:03:01,470 PROFESSOR: So this is-- 1000 01:03:01,470 --> 01:03:01,980 AUDIENCE: --two powers by-- 1001 01:03:01,980 --> 01:03:03,205 PROFESSOR: Yeah, this is a good question. 1002 01:03:03,205 --> 01:03:03,660 AUDIENCE: [INAUDIBLE] 1003 01:03:03,660 --> 01:03:04,460 PROFESSOR: Right, so-- 1004 01:03:04,460 --> 01:03:05,555 AUDIENCE: [INAUDIBLE] actually significantly much smaller. 1005 01:03:05,555 --> 01:03:09,170 PROFESSOR: Yeah, so one way of thinking about it is like this. 1006 01:03:09,170 --> 01:03:10,880 When you design the effective theory, 1007 01:03:10,880 --> 01:03:13,338 you actually have to figure out what the degrees of freedom 1008 01:03:13,338 --> 01:03:14,790 are at lowest order. 1009 01:03:14,790 --> 01:03:16,490 And when you go to higher order, you 1010 01:03:16,490 --> 01:03:19,590 shouldn't be having new degrees of freedom popping up at you. 1011 01:03:19,590 --> 01:03:23,900 Yeah, and so that's kind of at the heart of what 1012 01:03:23,900 --> 01:03:24,890 I'm saying here. 1013 01:03:24,890 --> 01:03:29,927 That's a kind of an even deeper way of saying it 1014 01:03:29,927 --> 01:03:32,510 because what you're doing with the power connections is you're 1015 01:03:32,510 --> 01:03:34,882 kind of figuring out how to put factors in the numerator 1016 01:03:34,882 --> 01:03:36,590 if you want to think in Feynman diagrams. 1017 01:03:36,590 --> 01:03:37,880 You're not changing the propagators. 1018 01:03:37,880 --> 01:03:38,920 You've already figured out the propagators 1019 01:03:38,920 --> 01:03:40,860 with your leading order theory. 1020 01:03:40,860 --> 01:03:43,610 So you have sort of where all the poles can occur, 1021 01:03:43,610 --> 01:03:45,890 and now you're just sort of doing perturbation theory. 1022 01:03:45,890 --> 01:03:47,515 Perturbation theory shouldn't introduce 1023 01:03:47,515 --> 01:03:50,030 new degrees of freedom in the-- 1024 01:03:50,030 --> 01:03:54,510 when you're expanding in the power expansion. 1025 01:03:54,510 --> 01:03:58,778 All right, so to draw a picture, I 1026 01:03:58,778 --> 01:04:01,320 want to tell you about one other observable, which is thrust. 1027 01:04:12,480 --> 01:04:16,080 So thrust can be defined by this formula 1028 01:04:16,080 --> 01:04:19,530 here in general, which is that you 1029 01:04:19,530 --> 01:04:21,330 find some axis, which is actually 1030 01:04:21,330 --> 01:04:22,440 an axis that's going to-- 1031 01:04:22,440 --> 01:04:26,595 in our dijet configuration in the jet axis. 1032 01:04:29,580 --> 01:04:32,470 You find the axis that maximizes this thing, 1033 01:04:32,470 --> 01:04:34,710 and in the case of a dijet configuration, 1034 01:04:34,710 --> 01:04:36,990 that's going to align with the jet axis. 1035 01:04:36,990 --> 01:04:40,050 And then you do this sum, and that calculate some variable t. 1036 01:04:43,770 --> 01:04:47,400 T turns out to have kinematic limits between a half and one, 1037 01:04:47,400 --> 01:04:50,463 and if you want to talk about something that's 1038 01:04:50,463 --> 01:04:52,380 a little more convenient, you define something 1039 01:04:52,380 --> 01:04:55,140 called tau, which is 1 minus t. 1040 01:04:55,140 --> 01:04:59,850 And the dijet limit of tau is zero. 1041 01:04:59,850 --> 01:05:01,120 That's why this is more. 1042 01:05:01,120 --> 01:05:02,940 So the dijet limit of t is the limit 1043 01:05:02,940 --> 01:05:06,250 where you just have kind of two pencil-like back-to-back 1044 01:05:06,250 --> 01:05:06,750 things. 1045 01:05:06,750 --> 01:05:08,610 In that case, this goes to one. 1046 01:05:08,610 --> 01:05:10,860 And if you wanted to find something that goes to zero, 1047 01:05:10,860 --> 01:05:12,870 then you define it as tau. 1048 01:05:12,870 --> 01:05:16,240 Just a little more convenient. 1049 01:05:16,240 --> 01:05:21,040 And so it's possible to work out that when you're 1050 01:05:21,040 --> 01:05:26,320 in a dijet configuration that tau with a bit of work 1051 01:05:26,320 --> 01:05:30,010 you can show it's actually simply the sum of our two 1052 01:05:30,010 --> 01:05:32,300 hemisphere invariant masses. 1053 01:05:32,300 --> 01:05:34,270 So obviously, in a dijet configuration, 1054 01:05:34,270 --> 01:05:36,880 there's lots of simplifications to the kinematics. 1055 01:05:36,880 --> 01:05:40,700 And with some work, one can show that that's true. 1056 01:05:40,700 --> 01:05:42,850 So what this is an observable, which is simply 1057 01:05:42,850 --> 01:05:46,300 a symmetric projection of our previous two variable 1058 01:05:46,300 --> 01:05:48,730 observable onto one variable observable. 1059 01:05:48,730 --> 01:05:51,070 But if we demand that, for example, tau is small, 1060 01:05:51,070 --> 01:05:53,563 that's demanding both m squared and n bar squared 1061 01:05:53,563 --> 01:05:55,480 are small because they're positive quantities, 1062 01:05:55,480 --> 01:05:56,170 and we're just summing. 1063 01:05:56,170 --> 01:05:58,212 So demanding that this is small is demanding both 1064 01:05:58,212 --> 01:06:00,230 of those variables is small. 1065 01:06:00,230 --> 01:06:06,670 So tau much less than one also is a way of getting dijets, 1066 01:06:06,670 --> 01:06:08,743 and that's a way of getting dijets just 1067 01:06:08,743 --> 01:06:10,160 with one variable rather than two. 1068 01:06:10,160 --> 01:06:12,100 So that's, of course, easier to draw a picture 1069 01:06:12,100 --> 01:06:14,890 with one thing rather than two. 1070 01:06:14,890 --> 01:06:17,080 So if we take our formula for the two variables 1071 01:06:17,080 --> 01:06:26,176 and we project onto one, then we can do that. 1072 01:06:26,176 --> 01:06:27,520 It looks like this. 1073 01:06:41,060 --> 01:06:43,600 So what happens to the jet-- the two jet functions 1074 01:06:43,600 --> 01:06:45,490 and the soft function that had two variables 1075 01:06:45,490 --> 01:06:47,290 is that they get projected kind of 1076 01:06:47,290 --> 01:06:50,780 on to their symmetric projection of the two variables. 1077 01:06:50,780 --> 01:06:52,810 So this is like-- 1078 01:06:52,810 --> 01:06:55,120 these two are symmetric projections. 1079 01:07:05,510 --> 01:07:09,870 So we had two variables before, and now we have one variable. 1080 01:07:09,870 --> 01:07:11,940 And it's a symmetric projection, and that just 1081 01:07:11,940 --> 01:07:14,015 follows from this formula. 1082 01:07:14,015 --> 01:07:15,390 So literally, to get this result, 1083 01:07:15,390 --> 01:07:17,790 we would just do the same trick we did before. 1084 01:07:25,060 --> 01:07:26,570 We'd say that this is true. 1085 01:07:26,570 --> 01:07:29,080 Now let's move the integral d m squared 1086 01:07:29,080 --> 01:07:31,607 and d n bar squared onto the right hand side 1087 01:07:31,607 --> 01:07:33,190 and think about doing those integrals. 1088 01:07:33,190 --> 01:07:35,040 And we follow through, and it would lead 1089 01:07:35,040 --> 01:07:37,004 to a formula like this one. 1090 01:07:37,004 --> 01:07:38,870 AUDIENCE: J tau is j squared? 1091 01:07:38,870 --> 01:07:41,800 PROFESSOR: J tau is j squared. 1092 01:07:41,800 --> 01:07:47,270 J tau is literally j j integrated 1093 01:07:47,270 --> 01:07:50,390 over so fixing kind some of the momentum 1094 01:07:50,390 --> 01:07:53,111 and integrating over the difference. 1095 01:07:53,111 --> 01:07:55,220 So this guy depends on one momentum, 1096 01:07:55,220 --> 01:07:59,690 and this guy-- these guys depend on 2, and there's one integral. 1097 01:07:59,690 --> 01:08:03,560 And then s tau is like one integral 1098 01:08:03,560 --> 01:08:08,240 of kind of something like this. 1099 01:08:08,240 --> 01:08:09,940 So let's call the integral l prime. 1100 01:08:09,940 --> 01:08:15,420 It's kind of like that. 1101 01:08:19,270 --> 01:08:21,271 So that's what I mean by asymmetric projection. 1102 01:08:21,271 --> 01:08:22,979 AUDIENCE: And so how do you know that tau 1103 01:08:22,979 --> 01:08:26,410 can be written as a function n bar, 1104 01:08:26,410 --> 01:08:30,220 or could another kinematic variable [INAUDIBLE]?? 1105 01:08:30,220 --> 01:08:32,800 PROFESSOR: No, it can be written as a function of n bar. 1106 01:08:32,800 --> 01:08:35,859 Yeah, because you're still-- if you're in subleading power, 1107 01:08:35,859 --> 01:08:37,865 as long as you're in the dijet's configuration, 1108 01:08:37,865 --> 01:08:39,490 we've got the right degrees of freedom. 1109 01:08:39,490 --> 01:08:41,670 We don't need another jet degree of freedom. 1110 01:08:44,300 --> 01:08:46,090 So if you want to describe-- 1111 01:08:46,090 --> 01:08:48,970 you could think about the thrust distribution. 1112 01:08:48,970 --> 01:08:51,490 Well, let me answer your question in more detail. 1113 01:08:51,490 --> 01:08:52,990 Let me tell you where you would have 1114 01:08:52,990 --> 01:08:55,870 to worry about more degrees of freedom, 1115 01:08:55,870 --> 01:08:59,000 but let me do it in a second. 1116 01:08:59,000 --> 01:09:16,660 I'll give you-- all right, so in the thrust case 1117 01:09:16,660 --> 01:09:18,760 and what we said about masses carries over 1118 01:09:18,760 --> 01:09:24,830 to a formula like this, where this mu h squared 1119 01:09:24,830 --> 01:09:30,580 being much bigger than mu j squared, much bigger 1120 01:09:30,580 --> 01:09:31,899 than mu s squared. 1121 01:09:31,899 --> 01:09:39,790 And then that could be much greater than or of order lambda 1122 01:09:39,790 --> 01:09:41,050 QCD squared. 1123 01:09:41,050 --> 01:09:45,100 And the type of terms that our factorization theorem would 1124 01:09:45,100 --> 01:09:49,300 be able to resum if we did the renormalization group evolution 1125 01:09:49,300 --> 01:09:54,340 would be sums of powers of alpha and logs of tau. 1126 01:09:54,340 --> 01:09:57,490 Tau is dimensionless variable, and so the kind 1127 01:09:57,490 --> 01:10:00,370 of terms that would show up in the cross section 1128 01:10:00,370 --> 01:10:05,245 are terms like that plus non-perturbative effects. 1129 01:10:08,980 --> 01:10:11,700 So if you want to think the base level, what 1130 01:10:11,700 --> 01:10:15,300 is inside our formula, it's all these terms. 1131 01:10:15,300 --> 01:10:18,000 We're able to determine them from our factorization theorem, 1132 01:10:18,000 --> 01:10:20,334 and then there's also some non-perturbative, 1133 01:10:20,334 --> 01:10:23,730 which would be kind of-- and non-perturbative effects that 1134 01:10:23,730 --> 01:10:26,388 in this f function. 1135 01:10:26,388 --> 01:10:28,380 And the factorization theorem is telling us 1136 01:10:28,380 --> 01:10:32,350 how to compute these things and include them. 1137 01:10:32,350 --> 01:10:35,367 So what does it look like as a picture? 1138 01:10:35,367 --> 01:10:36,950 What does the cross-section look like? 1139 01:10:39,630 --> 01:10:48,600 So there's a peak as I promised, and there's kind of-- 1140 01:10:48,600 --> 01:10:49,650 and then there's a tail. 1141 01:10:54,030 --> 01:10:59,670 And so this is tau. 1142 01:10:59,670 --> 01:11:02,670 This is d sigma d tau. 1143 01:11:02,670 --> 01:11:06,083 You can always normalize. 1144 01:11:06,083 --> 01:11:07,500 This peak is happening in the kind 1145 01:11:07,500 --> 01:11:08,670 of non-perturbative region. 1146 01:11:08,670 --> 01:11:14,225 So because it's a symmetric projection, 1147 01:11:14,225 --> 01:11:15,600 there's a factor of two, but it's 1148 01:11:15,600 --> 01:11:18,510 kind of basically happening when taus of order lambda 1149 01:11:18,510 --> 01:11:19,440 QCD over q. 1150 01:11:19,440 --> 01:11:22,630 And we could figure out that just by power counting. 1151 01:11:22,630 --> 01:11:26,520 And then depending on the value of cue, 1152 01:11:26,520 --> 01:11:31,050 that sort of for a typical configuration, tau 1153 01:11:31,050 --> 01:11:32,520 equals 0.1 would be over somewhere 1154 01:11:32,520 --> 01:11:36,210 over there, et cetera. 1155 01:11:36,210 --> 01:11:39,650 So this is the peak, aptly named because it looks like a peak. 1156 01:11:39,650 --> 01:11:44,120 And in this peak, there's non-perturbative effects 1157 01:11:44,120 --> 01:11:46,820 that come from the f. 1158 01:11:46,820 --> 01:11:49,270 This is the tail. 1159 01:11:49,270 --> 01:11:53,270 It could be called the shoulder, but it's 1160 01:11:53,270 --> 01:11:56,180 kind of once you're out here and down here, 1161 01:11:56,180 --> 01:11:58,280 this is what you call the tail where 1162 01:11:58,280 --> 01:12:01,670 you have a perturbative s tau. 1163 01:12:01,670 --> 01:12:07,880 And then you would include power corrections by expanding. 1164 01:12:07,880 --> 01:12:11,280 And the expansion would give you lambda QCD over the soft scale, 1165 01:12:11,280 --> 01:12:13,385 which is q tau. 1166 01:12:13,385 --> 01:12:19,430 So this is-- that's that expansion that we talked about, 1167 01:12:19,430 --> 01:12:22,864 and that comes from expanding it, 1168 01:12:22,864 --> 01:12:28,862 f, or an expansion that involves f. 1169 01:12:28,862 --> 01:12:30,530 It's not expanding f. 1170 01:12:34,460 --> 01:12:35,960 Basically, what these numerators are 1171 01:12:35,960 --> 01:12:39,830 are moments of f once you do the expansion. 1172 01:12:39,830 --> 01:12:41,930 And then there's a region out here, 1173 01:12:41,930 --> 01:12:45,110 and in both of these regions, you have dijets. 1174 01:12:45,110 --> 01:12:47,750 But in this region out here, you don't have dijets anymore. 1175 01:12:47,750 --> 01:12:50,235 That's where tau is starting to get large 1176 01:12:50,235 --> 01:12:51,485 and you no longer have dijets. 1177 01:12:57,320 --> 01:12:59,690 So if you wanted to think about sort of higher order 1178 01:12:59,690 --> 01:13:02,480 corrections in the distribution out here, 1179 01:13:02,480 --> 01:13:05,030 then you would need to perhaps think about three jets 1180 01:13:05,030 --> 01:13:06,230 and other things. 1181 01:13:06,230 --> 01:13:08,480 In particular, there's also a kind of shoulder 1182 01:13:08,480 --> 01:13:10,260 here that if you really want to think 1183 01:13:10,260 --> 01:13:12,260 about how power corrections are kind of properly 1184 01:13:12,260 --> 01:13:17,930 dealing with that region, you think about more than two jets. 1185 01:13:17,930 --> 01:13:21,390 But if you just kind of restrict yourself to here and here, 1186 01:13:21,390 --> 01:13:25,222 which is all we're going to do, then you don't-- then every 1187 01:13:25,222 --> 01:13:26,930 mode, even if you go to subleading order, 1188 01:13:26,930 --> 01:13:28,460 is exactly the ones that we just said. 1189 01:13:28,460 --> 01:13:31,002 And you just have to deal with them by constructing operators 1190 01:13:31,002 --> 01:13:33,320 in subleading order. 1191 01:13:33,320 --> 01:13:38,430 So that's the more detailed answer to your question. 1192 01:13:38,430 --> 01:13:39,852 Any other questions? 1193 01:13:43,710 --> 01:13:50,490 OK, so I'll say a few things about the perturbation theory. 1194 01:13:50,490 --> 01:13:53,400 So so far we've kind of argued that we 1195 01:13:53,400 --> 01:13:56,610 could get pretty far without doing perturbation theory. 1196 01:13:56,610 --> 01:13:59,640 So we got here without doing perturbation theory. 1197 01:13:59,640 --> 01:14:02,670 What if we actually now want to calculate these series, 1198 01:14:02,670 --> 01:14:05,258 do the normalization group evolution? 1199 01:14:05,258 --> 01:14:06,300 How's that going to work? 1200 01:14:10,970 --> 01:14:13,360 And it's actually-- in some ways, 1201 01:14:13,360 --> 01:14:17,350 it's a combination of the two examples that we treated. 1202 01:14:17,350 --> 01:14:20,080 We did an example where we had a Wilson coefficient that 1203 01:14:20,080 --> 01:14:21,520 was just running multiplicatively, 1204 01:14:21,520 --> 01:14:22,990 and we did another example where we 1205 01:14:22,990 --> 01:14:25,642 had a function for deep and elastic scattering. 1206 01:14:25,642 --> 01:14:27,100 It had an integral, and we're going 1207 01:14:27,100 --> 01:14:29,620 to have both of those situations in this case. 1208 01:14:34,370 --> 01:14:37,030 So let me tell you how it works. 1209 01:14:37,030 --> 01:14:41,500 So to get the hard function, you would do some matching, 1210 01:14:41,500 --> 01:14:44,470 and the matching is actually for what people call the quark form 1211 01:14:44,470 --> 01:14:46,578 factor. 1212 01:14:46,578 --> 01:14:49,120 So if you wonder why people care about the quark form factor, 1213 01:14:49,120 --> 01:14:50,830 the quark form factor is basically 1214 01:14:50,830 --> 01:14:54,040 the h in our formula, not exactly, 1215 01:14:54,040 --> 01:14:56,430 but closely related to the h in our formula. 1216 01:15:00,470 --> 01:15:06,640 So you would do some one loop matching by taking QCD graphs 1217 01:15:06,640 --> 01:15:10,030 and subtracting the SCET graphs in exactly the kind of way 1218 01:15:10,030 --> 01:15:13,482 we were already doing for the examples 1219 01:15:13,482 --> 01:15:14,440 that we treated before. 1220 01:15:24,030 --> 01:15:25,430 So those are collinear gluons. 1221 01:15:25,430 --> 01:15:26,690 This is an ultra soft gluon. 1222 01:15:29,650 --> 01:15:33,290 And there's some wave function graphs that you can consider. 1223 01:15:33,290 --> 01:15:35,180 We form the difference. 1224 01:15:35,180 --> 01:15:38,390 And at one loop, we check that all the IR divergences 1225 01:15:38,390 --> 01:15:41,330 between this and this cancel, and then we 1226 01:15:41,330 --> 01:15:42,925 get the Wilson coefficient. 1227 01:15:48,270 --> 01:15:51,230 So this is very, very analogous to the example 1228 01:15:51,230 --> 01:15:53,810 that we did when we were doing a heavy light current. 1229 01:16:12,180 --> 01:16:15,870 And you see here that this thing, which depends on mu, 1230 01:16:15,870 --> 01:16:18,060 it's mu over q. 1231 01:16:18,060 --> 01:16:20,100 And that's what I was saying that the Wilson 1232 01:16:20,100 --> 01:16:22,050 coefficient or the hard function should be-- 1233 01:16:22,050 --> 01:16:25,996 the log should be minimized from mu of order q. 1234 01:16:25,996 --> 01:16:27,870 And we see that by doing the calculation. 1235 01:16:32,910 --> 01:16:36,600 H is the square of this thing. 1236 01:16:36,600 --> 01:16:39,390 So there's imaginary parts on these negative logarithms. 1237 01:16:39,390 --> 01:16:41,640 These are minus q squared minus i 0. 1238 01:16:41,640 --> 01:16:43,860 But when I take the mod square, this thing is real. 1239 01:16:50,850 --> 01:16:53,550 What about the jet function? 1240 01:16:53,550 --> 01:16:56,280 Well, if you go back to what the formula for the jet function 1241 01:16:56,280 --> 01:17:00,360 was, the jet function is basically 1242 01:17:00,360 --> 01:17:06,090 two fields like spit out a quark and absorb a quark. 1243 01:17:06,090 --> 01:17:07,760 But it's a vacuum matrix element, 1244 01:17:07,760 --> 01:17:10,207 so the quarks are just contracted. 1245 01:17:10,207 --> 01:17:11,790 So you're basically calculating graphs 1246 01:17:11,790 --> 01:17:14,290 like this or, really, actually the imaginary parts of graphs 1247 01:17:14,290 --> 01:17:16,092 like this because the way I drew it, 1248 01:17:16,092 --> 01:17:17,550 you were summing over final states, 1249 01:17:17,550 --> 01:17:20,770 so it would be the imaginary part. 1250 01:17:20,770 --> 01:17:24,210 So we could just calculate the graphs without putting a cut in 1251 01:17:24,210 --> 01:17:26,430 and take the imaginary part, and that would actually 1252 01:17:26,430 --> 01:17:28,700 be giving us the jet function. 1253 01:17:28,700 --> 01:17:32,760 So the jet function comes from some Feynman diagrams 1254 01:17:32,760 --> 01:17:33,870 that look like this. 1255 01:17:37,980 --> 01:17:44,480 And then there's one more looks like that. 1256 01:17:44,480 --> 01:17:46,850 So at one move, we have those three Feynman diagrams. 1257 01:17:46,850 --> 01:17:48,110 And then we have the [INAUDIBLE] little guy. 1258 01:17:48,110 --> 01:17:49,100 We take the imaginary part. 1259 01:17:49,100 --> 01:17:50,100 We get the jet function. 1260 01:17:56,810 --> 01:17:58,850 At lowest order, it gives a delta function 1261 01:17:58,850 --> 01:18:00,020 just to cut propagator. 1262 01:18:07,020 --> 01:18:09,390 And then at one [INAUDIBLE],, you get a delta function, 1263 01:18:09,390 --> 01:18:10,860 and you also get plus functions. 1264 01:18:17,420 --> 01:18:24,640 That may not worry so much about what the numbers are. 1265 01:18:24,640 --> 01:18:28,015 I can just tell you what the result looks like. 1266 01:18:28,015 --> 01:18:29,170 It looks like that. 1267 01:18:29,170 --> 01:18:32,028 There's three different types of terms that we could get. 1268 01:18:32,028 --> 01:18:33,820 They all kind of have a power counting that 1269 01:18:33,820 --> 01:18:35,517 makes them go like one over s. 1270 01:18:35,517 --> 01:18:36,850 And that you know ahead of time. 1271 01:18:36,850 --> 01:18:39,760 If you power count the operator here, 1272 01:18:39,760 --> 01:18:40,990 it should scale one over s. 1273 01:18:40,990 --> 01:18:42,740 These are the different kind of structures 1274 01:18:42,740 --> 01:18:45,317 that you can get at one loop that scale one over s. 1275 01:18:45,317 --> 01:18:46,900 And this is kind of a symptom of there 1276 01:18:46,900 --> 01:18:49,275 being 1 over epsilon squared divergences, 1277 01:18:49,275 --> 01:18:50,650 and this guy here are [INAUDIBLE] 1278 01:18:50,650 --> 01:18:51,692 the renormalized results. 1279 01:18:51,692 --> 01:18:55,490 So we're taking care of the renormalization. 1280 01:18:55,490 --> 01:18:57,380 So just like in our example-- 1281 01:18:57,380 --> 01:18:59,450 I mean, it's the same diagram really. 1282 01:18:59,450 --> 01:19:02,420 Just this diagram was familiar because this diagram 1283 01:19:02,420 --> 01:19:04,130 was showing up in beta s gamma, right? 1284 01:19:04,130 --> 01:19:06,800 So we saw it had 1 over epsilon squared poles. 1285 01:19:06,800 --> 01:19:10,862 Here, the 1 over epsilon squared poles lead to this. 1286 01:19:10,862 --> 01:19:12,320 And, really, you can actually think 1287 01:19:12,320 --> 01:19:13,820 of that very closely related to what 1288 01:19:13,820 --> 01:19:16,970 we did because we were finding logs of mu squared over p 1289 01:19:16,970 --> 01:19:20,040 squared, but here, p squared is a physical thing. 1290 01:19:20,040 --> 01:19:23,780 It's the s, the invariant mass that we pump into the operator. 1291 01:19:23,780 --> 01:19:28,810 S is kind of what we put in through the-- 1292 01:19:28,810 --> 01:19:33,830 we put in a momentum, if you like, q, where-- 1293 01:19:33,830 --> 01:19:34,820 I shouldn't call it q-- 1294 01:19:37,622 --> 01:19:41,990 t mu where s is t squared, right? 1295 01:19:41,990 --> 01:19:45,890 And so before we were having logs of p squared, which 1296 01:19:45,890 --> 01:19:48,740 were an IR regulator, but in this calculation for this jet 1297 01:19:48,740 --> 01:19:50,990 function, it's actually a physical thing. 1298 01:19:50,990 --> 01:19:54,840 And it's giving the momentum dependents of the jet function, 1299 01:19:54,840 --> 01:19:58,400 but it's the right thing to stick into the factorization 1300 01:19:58,400 --> 01:20:00,200 theorem. 1301 01:20:00,200 --> 01:20:02,990 And then there's the soft function 1302 01:20:02,990 --> 01:20:08,180 where if it's perturbative, you can calculate it. 1303 01:20:08,180 --> 01:20:12,320 And you can draw these kind of in some notation for the Wilson 1304 01:20:12,320 --> 01:20:13,800 lines. 1305 01:20:13,800 --> 01:20:21,680 So here's our Wilson lines in different directions. 1306 01:20:21,680 --> 01:20:23,970 And then want this matrix element squared, 1307 01:20:23,970 --> 01:20:26,060 and, again, you can sort of think 1308 01:20:26,060 --> 01:20:30,590 of as kind of cut graphs like that if you like. 1309 01:20:30,590 --> 01:20:35,540 And if we look at the soft function, 1310 01:20:35,540 --> 01:20:39,720 it kind of has a similar structure to the jet function 1311 01:20:39,720 --> 01:20:43,200 but now with the-- 1312 01:20:43,200 --> 01:20:46,730 so, again, it's got a delta function and then 1313 01:20:46,730 --> 01:20:47,660 plus functions. 1314 01:20:55,541 --> 01:20:56,041 Whoops. 1315 01:21:00,500 --> 01:21:03,830 And it turns out there's no single plus function, 1316 01:21:03,830 --> 01:21:09,470 but there is a plus function with a logarithm in this case. 1317 01:21:09,470 --> 01:21:17,920 And then same for l minus. 1318 01:21:17,920 --> 01:21:19,480 Same structure. 1319 01:21:19,480 --> 01:21:21,560 It's just a product if you like. 1320 01:21:21,560 --> 01:21:24,020 And the reason that happens is if you only have one gluon, 1321 01:21:24,020 --> 01:21:26,030 it's either in hemisphere A or hemisphere 1322 01:21:26,030 --> 01:21:27,900 B. It can't be in both. 1323 01:21:27,900 --> 01:21:30,500 So the alpha s corrections are either a function of l plus 1324 01:21:30,500 --> 01:21:32,540 or a function of l minus, and that's 1325 01:21:32,540 --> 01:21:34,860 why it has a kind of very simple structure at one loop. 1326 01:21:52,235 --> 01:21:53,610 So that gives you an idea of what 1327 01:21:53,610 --> 01:21:55,980 these perturbative functions look like. 1328 01:21:55,980 --> 01:21:58,380 C, if you talk about renormalization, 1329 01:21:58,380 --> 01:22:01,067 C renormalizes multiplicatively. 1330 01:22:05,050 --> 01:22:08,320 And so the renormalization group equation for C 1331 01:22:08,320 --> 01:22:13,320 is just like the one we had before for beta s gamma. 1332 01:22:17,600 --> 01:22:19,100 There's no integrals. 1333 01:22:19,100 --> 01:22:20,900 That, again, came about from the kinematics 1334 01:22:20,900 --> 01:22:22,970 fixing the variables. 1335 01:22:22,970 --> 01:22:30,950 But the jet function and the soft function 1336 01:22:30,950 --> 01:22:34,310 have convolutions in this case. 1337 01:22:41,232 --> 01:22:43,190 Well, they depend on this non-trivial momentum, 1338 01:22:43,190 --> 01:22:45,170 and it's-- 1339 01:22:45,170 --> 01:22:48,620 you can see in the factorization theorem, that is convoluting 1340 01:22:48,620 --> 01:22:49,880 between two different sectors. 1341 01:22:49,880 --> 01:22:52,310 And it kind of generically a hint 1342 01:22:52,310 --> 01:22:57,332 that you're going to get out a formula like this one, which 1343 01:22:57,332 --> 01:23:01,140 is like the PDF, but now it's a different formula. 1344 01:23:01,140 --> 01:23:05,040 It was kind of an almost dimension for the jet function. 1345 01:23:09,350 --> 01:23:11,620 So we could go through that, but I was not 1346 01:23:11,620 --> 01:23:14,792 writing down for you what the 1 over epsilons look like. 1347 01:23:14,792 --> 01:23:18,940 But we could go through the renormalization 1348 01:23:18,940 --> 01:23:20,310 and find these results. 1349 01:23:31,520 --> 01:23:33,470 And actually, in this jet function case, 1350 01:23:33,470 --> 01:23:36,140 we even know more. 1351 01:23:36,140 --> 01:23:39,680 The general structure of this anomalous dimension 1352 01:23:39,680 --> 01:23:42,350 is actually simpler than the Parton distribution case. 1353 01:23:46,370 --> 01:23:49,795 And it's the following. 1354 01:23:56,590 --> 01:24:00,155 There's two types of terms that can show up. 1355 01:24:00,155 --> 01:24:02,280 So the general structure of the anomalous dimension 1356 01:24:02,280 --> 01:24:05,700 is that there's a single plus function 1357 01:24:05,700 --> 01:24:07,710 in it or a delta function. 1358 01:24:07,710 --> 01:24:10,620 And this single plus function is the analog 1359 01:24:10,620 --> 01:24:13,050 in the jet function of the single logarithm that 1360 01:24:13,050 --> 01:24:14,290 was showing up. 1361 01:24:14,290 --> 01:24:17,370 Remember that when we decompose this guy, 1362 01:24:17,370 --> 01:24:20,370 there could be a log mu over q term 1363 01:24:20,370 --> 01:24:24,690 or a one term with no log of mu over q. 1364 01:24:24,690 --> 01:24:28,080 This is like an analog of a log, this plus function. 1365 01:24:28,080 --> 01:24:30,210 If you integrate over s, then it's 1366 01:24:30,210 --> 01:24:31,960 like ds over s, which is like a log. 1367 01:24:31,960 --> 01:24:35,220 So this is a log, and integrating delta of s 1368 01:24:35,220 --> 01:24:37,370 is like one. 1369 01:24:37,370 --> 01:24:38,980 So the analog statement that there 1370 01:24:38,980 --> 01:24:42,090 was two possibilities there. 1371 01:24:42,090 --> 01:24:43,930 In this case, there's an analog of that, 1372 01:24:43,930 --> 01:24:45,670 and there's two possibilities here. 1373 01:24:45,670 --> 01:24:47,250 And what perturbation theory is doing 1374 01:24:47,250 --> 01:24:50,100 is actually just computing the coefficients of these two 1375 01:24:50,100 --> 01:24:52,430 different structures. 1376 01:24:52,430 --> 01:24:57,110 OK, and we're out of time, so I'll say a few more words 1377 01:24:57,110 --> 01:25:00,050 about how you would solve, for example, an equation 1378 01:25:00,050 --> 01:25:01,730 like this one. 1379 01:25:01,730 --> 01:25:03,650 Next time I'll tell you how to solve it, 1380 01:25:03,650 --> 01:25:06,928 and then we'll basically be done with our example. 1381 01:25:06,928 --> 01:25:09,470 We'll put things back together and write down a factorization 1382 01:25:09,470 --> 01:25:13,953 theorem that includes the resummation 1383 01:25:13,953 --> 01:25:15,620 and then we'll go on to another example. 1384 01:25:18,750 --> 01:25:21,000 Moving on, the next example we'll treat after this one 1385 01:25:21,000 --> 01:25:25,650 is SCET two where we'll be dealing 1386 01:25:25,650 --> 01:25:28,830 with energetic hadrons, some other types of examples 1387 01:25:28,830 --> 01:25:30,770 besides jets.