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 00:00:22,050 --> 00:00:24,360 IAIN STEWART: All right, well, it's time to start. 9 00:00:27,617 --> 00:00:29,325 So last time we were talking about e plus 10 00:00:29,325 --> 00:00:31,350 e minus to jets, and-- 11 00:00:34,353 --> 00:00:35,520 I should have written that-- 12 00:00:38,070 --> 00:00:40,920 in particular, e plus e minus to dijets. 13 00:00:40,920 --> 00:00:43,740 And we were talking about-- we talked about factorization 14 00:00:43,740 --> 00:00:45,352 and how to derive it. 15 00:00:45,352 --> 00:00:47,310 And I've schematically written the result here. 16 00:00:47,310 --> 00:00:49,320 Last time we had a more definite formula 17 00:00:49,320 --> 00:00:51,600 with all the arguments made explicit. 18 00:00:51,600 --> 00:00:54,630 So there were hard functions, jet functions, 19 00:00:54,630 --> 00:00:58,110 and the soft function, and we could either think about 20 00:00:58,110 --> 00:01:00,450 measuring invariant masses in two hemispheres 21 00:01:00,450 --> 00:01:03,450 and constraining them to be small to know we have dijets, 22 00:01:03,450 --> 00:01:06,000 or measuring kind of the sum of the two, which we called-- 23 00:01:06,000 --> 00:01:09,750 which is effectively the thrust, and then we get a slightly 24 00:01:09,750 --> 00:01:13,200 simpler formula because we can project these guys down onto 25 00:01:13,200 --> 00:01:14,850 guys that'd only have-- 26 00:01:14,850 --> 00:01:17,610 or functions of one variable. 27 00:01:17,610 --> 00:01:20,255 But either way, in either case, we 28 00:01:20,255 --> 00:01:21,630 have a factorization theorem that 29 00:01:21,630 --> 00:01:25,170 separates the degrees of freedom and kind of the picture 30 00:01:25,170 --> 00:01:33,010 is that we have these kind of scales in the problem. 31 00:01:33,010 --> 00:01:39,900 So this is new jet, new hard, and new soft, 32 00:01:39,900 --> 00:01:41,760 and we can use this factorization theorem 33 00:01:41,760 --> 00:01:43,975 to separate those scales into these functions. 34 00:01:43,975 --> 00:01:44,850 And then we can use-- 35 00:01:44,850 --> 00:01:46,975 I started talking about using renormalization group 36 00:01:46,975 --> 00:01:49,845 equations in order to sum the logs between these scales. 37 00:01:54,850 --> 00:01:57,800 And I wanted to finish up that discussion. 38 00:01:57,800 --> 00:02:00,850 So for the Wilson coefficient of the operator 39 00:02:00,850 --> 00:02:02,803 or, for that matter, for the hard function, 40 00:02:02,803 --> 00:02:04,720 which is the square of the Wilson coefficient, 41 00:02:04,720 --> 00:02:07,510 you have a very simple renormalization group equation. 42 00:02:07,510 --> 00:02:09,280 And that's because momentum conservation 43 00:02:09,280 --> 00:02:13,960 is enough to fix all the variables that 44 00:02:13,960 --> 00:02:15,250 label collinear fields. 45 00:02:15,250 --> 00:02:17,620 So you really just have this overall Q, 46 00:02:17,620 --> 00:02:19,150 which is the center of mass energy, 47 00:02:19,150 --> 00:02:22,300 and that's the only variable that it's fixed by kinematics. 48 00:02:22,300 --> 00:02:24,178 There's no convolutions. 49 00:02:24,178 --> 00:02:26,470 In the case of these things called jet functions, which 50 00:02:26,470 --> 00:02:28,420 we talked a little bit about last time, 51 00:02:28,420 --> 00:02:31,300 you end up with something like the Altarelli-Parisi 52 00:02:31,300 --> 00:02:32,903 where there's an integral. 53 00:02:32,903 --> 00:02:34,570 It's a little bit different in the sense 54 00:02:34,570 --> 00:02:37,030 that we actually know its structure-- tolerance 55 00:02:37,030 --> 00:02:39,200 and perturbation theory-- it has this structure. 56 00:02:39,200 --> 00:02:42,070 So it only has a very particular dependence on S, 57 00:02:42,070 --> 00:02:44,080 has to scale like 1 over S, so one thing 58 00:02:44,080 --> 00:02:48,010 that scales like 1 over S is a delta function 59 00:02:48,010 --> 00:02:50,740 or a plus function, and basically, there's 60 00:02:50,740 --> 00:02:54,190 only this single type of plus function that can show up, 61 00:02:54,190 --> 00:02:56,343 and that's the analog actually of something that 62 00:02:56,343 --> 00:02:57,760 happened to this guy where we said 63 00:02:57,760 --> 00:03:01,870 there was only a possibility of a single logarithm showing up. 64 00:03:01,870 --> 00:03:04,240 OK. 65 00:03:04,240 --> 00:03:07,240 So this is how far we got. 66 00:03:07,240 --> 00:03:10,715 Whenever you have something-- whenever you have equations 67 00:03:10,715 --> 00:03:13,090 where you have convolutions like this, what you should do 68 00:03:13,090 --> 00:03:16,152 is Fourier transform because if you Fourier 69 00:03:16,152 --> 00:03:17,860 transform something that's a convolution, 70 00:03:17,860 --> 00:03:18,735 it becomes a product. 71 00:03:27,730 --> 00:03:29,440 So we'll Fourier transform, and in order 72 00:03:29,440 --> 00:03:34,450 to be careful about convergence of our Fourier transform, 73 00:03:34,450 --> 00:03:37,600 we give it a small imaginary part. 74 00:03:37,600 --> 00:03:46,470 And we define a position space anomalous dimension, 75 00:03:46,470 --> 00:03:48,890 as the anomalous dimension up there that depend on S, 76 00:03:48,890 --> 00:03:51,900 but now Fourier transformed and likewise, we 77 00:03:51,900 --> 00:04:02,120 define a position space jet function the same way, 78 00:04:02,120 --> 00:04:03,920 and then this formula here, which 79 00:04:03,920 --> 00:04:07,400 is a convolution, if you use these in the inverse 80 00:04:07,400 --> 00:04:12,000 transforms, you arrive at a very simple formula for the position 81 00:04:12,000 --> 00:04:12,500 space. 82 00:04:19,392 --> 00:04:30,640 Gamma J. 83 00:04:30,640 --> 00:04:32,530 So the Fourier transform of a delta function 84 00:04:32,530 --> 00:04:35,910 is just the identity so there's this term here, 85 00:04:35,910 --> 00:04:37,525 becomes that term there. 86 00:04:37,525 --> 00:04:39,790 The Fourier transform of a plus function 87 00:04:39,790 --> 00:04:43,750 is actually a logarithm, and in order to get that result, 88 00:04:43,750 --> 00:04:47,410 you do need to have this convergence. 89 00:04:47,410 --> 00:04:50,180 This little i0 in there. 90 00:04:50,180 --> 00:04:52,810 So in general, if you have there's 91 00:04:52,810 --> 00:05:02,950 a general kind of relation between logs and plus 92 00:05:02,950 --> 00:05:07,540 functions like that, and in Fourier space, 93 00:05:07,540 --> 00:05:11,170 the kind of highest logarithm you get is L to the K plus 1. 94 00:05:17,350 --> 00:05:21,705 OK, so these logs here, you're basically-- 95 00:05:21,705 --> 00:05:23,080 if you think about counting them, 96 00:05:23,080 --> 00:05:25,540 you should count this one over S as an extra log, 97 00:05:25,540 --> 00:05:27,460 and in Fourier space, that becomes explicit. 98 00:05:27,460 --> 00:05:29,440 You really just get a log to the K plus 1. 99 00:05:33,160 --> 00:05:35,770 All right, but this formula here, 100 00:05:35,770 --> 00:05:38,320 is something that, again, is of this kind 101 00:05:38,320 --> 00:05:39,630 of multiplicative form. 102 00:05:39,630 --> 00:05:43,510 So what happens to the anomalous dimension in the Fourier space 103 00:05:43,510 --> 00:05:47,050 is that it has a multiplicative form. 104 00:05:47,050 --> 00:05:54,630 All right so we have mu debugging mu of J Y 105 00:05:54,630 --> 00:06:03,790 comma mu is simply gamma J Y comma mu J Y comma mu. 106 00:06:03,790 --> 00:06:07,790 So it's no more difficult than this formula up here. 107 00:06:07,790 --> 00:06:12,160 Just have to find the right space, which is Fourier space. 108 00:06:12,160 --> 00:06:15,970 So then we can make an all order solution of that formula. 109 00:06:15,970 --> 00:06:18,970 Given that we know this, we can just 110 00:06:18,970 --> 00:06:22,420 plug-in this and integrate, and we've 111 00:06:22,420 --> 00:06:23,710 done this a few times now. 112 00:06:29,250 --> 00:06:31,597 Let me just do it formally by not doing the integrals, 113 00:06:31,597 --> 00:06:33,180 but writing them out in a way that you 114 00:06:33,180 --> 00:06:37,300 could do them to show you what the all order solution would 115 00:06:37,300 --> 00:06:37,800 look like. 116 00:06:44,020 --> 00:06:47,580 So you can write the solution using this fact 117 00:06:47,580 --> 00:06:50,970 and using this fact. 118 00:07:03,090 --> 00:07:05,670 So we have to use that fact to convert this logarithm here 119 00:07:05,670 --> 00:07:11,580 into something that is just in terms of alphas. 120 00:07:11,580 --> 00:07:14,640 But using those two facts, we can do the same kind of things 121 00:07:14,640 --> 00:07:17,760 that we've done before and write the all order solution 122 00:07:17,760 --> 00:07:18,690 in the following way. 123 00:08:02,820 --> 00:08:22,440 The integrals are contained in this W and this K. 124 00:08:22,440 --> 00:08:25,160 Here's the noncusp piece, and then 125 00:08:25,160 --> 00:08:26,840 the cusp piece comes with two integrals 126 00:08:26,840 --> 00:08:30,320 because it had a logarithm, and we turned one of the logarithms 127 00:08:30,320 --> 00:08:31,400 into an integral as well. 128 00:08:41,068 --> 00:08:42,860 So hopefully I've gotten all my twos right. 129 00:08:53,322 --> 00:08:54,780 So we saw something similar when we 130 00:08:54,780 --> 00:08:57,325 were talking about the running of the Wilson 131 00:08:57,325 --> 00:09:00,820 coefficient of the C, because it also had this kind of form 132 00:09:00,820 --> 00:09:03,850 with just a logarithm and a constant term. 133 00:09:03,850 --> 00:09:06,072 And it's a similar thing here. 134 00:09:06,072 --> 00:09:08,530 So this is an all order solution for the running of the jet 135 00:09:08,530 --> 00:09:09,370 function. 136 00:09:09,370 --> 00:09:12,190 Of course, we don't know these functions to all orders. 137 00:09:12,190 --> 00:09:15,760 We only know them up to some given order, 138 00:09:15,760 --> 00:09:17,695 and so you plug those-- whatever order you 139 00:09:17,695 --> 00:09:19,570 want to work-- you plug it into this formula, 140 00:09:19,570 --> 00:09:20,987 and then you can do this integral, 141 00:09:20,987 --> 00:09:22,930 and then you figure out these factors here. 142 00:09:22,930 --> 00:09:25,420 And those factors are summing the logs. 143 00:09:25,420 --> 00:09:29,665 In the case of the jet function, what you would-- 144 00:09:29,665 --> 00:09:31,540 the way that you can think about this picture 145 00:09:31,540 --> 00:09:32,500 is that you want to-- 146 00:09:32,500 --> 00:09:35,800 you have to put the guys in the factorization theorem 147 00:09:35,800 --> 00:09:40,780 at a common scale, and so one way of thinking about it 148 00:09:40,780 --> 00:09:42,760 is as follows. 149 00:09:42,760 --> 00:09:46,105 Let's take our hard function and do perturbation theory at mu 150 00:09:46,105 --> 00:09:49,540 equals mu age, and then we'll sum logs down 151 00:09:49,540 --> 00:09:50,890 to say the soft scale. 152 00:09:50,890 --> 00:09:53,680 That's one way of doing it, and we'll do perturbation theory 153 00:09:53,680 --> 00:09:57,130 for the jet function, at a scale mu equals mu J, 154 00:09:57,130 --> 00:10:01,850 and then we'll use this renormalization group here, 155 00:10:01,850 --> 00:10:09,170 which will just say gamma J to run this guy down 156 00:10:09,170 --> 00:10:10,370 to the soft scale. 157 00:10:10,370 --> 00:10:12,260 In that kind of scenario, you wouldn't 158 00:10:12,260 --> 00:10:15,380 have to run the soft function. 159 00:10:15,380 --> 00:10:17,210 This is like a top down kind of picture 160 00:10:17,210 --> 00:10:19,880 where you're just running the objects at the higher scales 161 00:10:19,880 --> 00:10:21,200 down to the lowest scale. 162 00:10:21,200 --> 00:10:22,860 That's one way of doing it. 163 00:10:22,860 --> 00:10:24,770 You could also do it in an equivalent way 164 00:10:24,770 --> 00:10:26,990 where you run the soft function, for example, 165 00:10:26,990 --> 00:10:29,660 up to the jet scale, and you don't run the jet function. 166 00:10:29,660 --> 00:10:35,280 And that'll give you actually the same result. 167 00:10:35,280 --> 00:10:36,410 But in the way-- 168 00:10:36,410 --> 00:10:38,120 with the information I've presented you, 169 00:10:38,120 --> 00:10:39,470 this is the way that you would think 170 00:10:39,470 --> 00:10:41,120 about doing the renormalization group, 171 00:10:41,120 --> 00:10:45,913 and that would sum up logarithms in the cross section. 172 00:10:45,913 --> 00:10:48,080 And it's very easy if you write it in position space 173 00:10:48,080 --> 00:10:50,163 to see what type of logarithms you're summing out. 174 00:10:53,280 --> 00:10:55,140 So in position space, if we go back 175 00:10:55,140 --> 00:10:58,320 to a formula like the one at the top, 176 00:10:58,320 --> 00:11:00,930 basically, what happens is that in position space, 177 00:11:00,930 --> 00:11:02,640 you, again, have a product. 178 00:11:02,640 --> 00:11:08,610 So these convolutions here, which were also this form, 179 00:11:08,610 --> 00:11:11,400 if you Fourier transform, then you end up with a product. 180 00:11:11,400 --> 00:11:16,050 D sigma D, so schematically, D sigma 181 00:11:16,050 --> 00:11:22,230 Dy for, say, where y is the Fourier transform for thrust, 182 00:11:22,230 --> 00:11:27,540 would just be h times j in position space 183 00:11:27,540 --> 00:11:30,960 times the soft function in position space as well. 184 00:11:30,960 --> 00:11:38,370 And there's Q somewhere, but it would be a very simple formula 185 00:11:38,370 --> 00:11:40,210 which just has a product. 186 00:11:40,210 --> 00:11:43,680 And so you can figure out what logs you're summing. 187 00:11:43,680 --> 00:11:45,582 The logs that you're summing, again, 188 00:11:45,582 --> 00:11:47,040 just as we talked about before, are 189 00:11:47,040 --> 00:11:51,570 simplest to describe in when you take the log because you're 190 00:11:51,570 --> 00:11:54,480 really summing logs in an exponent. 191 00:11:54,480 --> 00:12:00,000 So if we talk about the log of D sigma Dy, 192 00:12:00,000 --> 00:12:03,900 then we can enumerate the types of logs 193 00:12:03,900 --> 00:12:05,145 that one sums as follows. 194 00:12:17,940 --> 00:12:19,550 So these are the leading logs. 195 00:12:19,550 --> 00:12:22,206 This is next leading log, and this 196 00:12:22,206 --> 00:12:26,270 is next to next leading log, just like before. 197 00:12:26,270 --> 00:12:28,400 When we were talking about enumerating the logs, 198 00:12:28,400 --> 00:12:29,910 we talked about before an example, 199 00:12:29,910 --> 00:12:32,780 we were enumerating logs in the Wilson coefficient C. 200 00:12:32,780 --> 00:12:35,090 Now we're doing it for a full cross section, 201 00:12:35,090 --> 00:12:39,040 but because in position space, the cross section is simply 202 00:12:39,040 --> 00:12:43,873 a product of objects, you can think about just-- 203 00:12:43,873 --> 00:12:46,040 if you take the logarithm, they kind of split apart. 204 00:12:46,040 --> 00:12:48,680 And it's very simple to enumerate 205 00:12:48,680 --> 00:12:50,600 what corresponds to the things that you 206 00:12:50,600 --> 00:12:52,760 would get by putting in anomalous dimensions 207 00:12:52,760 --> 00:12:54,180 at a given order. 208 00:12:54,180 --> 00:12:56,960 So if you use the leading log anomalous dimension, just 209 00:12:56,960 --> 00:13:00,270 the one loop cusp, then you get these terms. 210 00:13:00,270 --> 00:13:03,050 Get the higher terms from the running of the coupling. 211 00:13:03,050 --> 00:13:05,090 If you put in the next leading log terms, then 212 00:13:05,090 --> 00:13:07,720 you get those terms. 213 00:13:07,720 --> 00:13:09,080 OK. 214 00:13:09,080 --> 00:13:13,940 And so you supplement that resummation which 215 00:13:13,940 --> 00:13:16,640 comes from this K and this omega, 216 00:13:16,640 --> 00:13:18,680 in general, you supplement that with sort 217 00:13:18,680 --> 00:13:22,010 of fixed order calculations of the H and the J and the S. 218 00:13:22,010 --> 00:13:24,910 And that gives you a complete cross-section at some order 219 00:13:24,910 --> 00:13:26,990 and resum perturbation theory. 220 00:13:30,190 --> 00:13:32,708 So you could do that also in momentum space. 221 00:13:32,708 --> 00:13:34,750 You could write out the formula in momentum space 222 00:13:34,750 --> 00:13:40,090 just because after all that's what you want in the end. 223 00:13:40,090 --> 00:13:42,970 Position space is just really a nice way 224 00:13:42,970 --> 00:13:47,220 of deriving the results that then you eventually put 225 00:13:47,220 --> 00:13:55,290 back into momentum space in some way or another 226 00:13:55,290 --> 00:13:56,970 because that's where the data is. 227 00:14:06,750 --> 00:14:10,520 And let me just show you what the formula would look like, 228 00:14:10,520 --> 00:14:17,170 just so you get a kind of picture with all the arguments. 229 00:14:17,170 --> 00:14:20,590 So if I were to Fourier transform the resum formula, 230 00:14:20,590 --> 00:14:23,430 over here I didn't write the resummation in. 231 00:14:23,430 --> 00:14:26,340 I didn't write in the evolution factors 232 00:14:26,340 --> 00:14:28,340 that correspond to this. 233 00:14:28,340 --> 00:14:31,470 So let me do that, but let me do it in momentum space. 234 00:14:31,470 --> 00:14:39,630 So I guess I made a slightly different choice here. 235 00:14:50,370 --> 00:14:51,890 So I'm doing it for the thrust case. 236 00:15:01,190 --> 00:15:03,770 These convolutions just being integrals over the variables 237 00:15:03,770 --> 00:15:09,000 that are like this S prime, for example. 238 00:15:09,000 --> 00:15:10,330 So this is S prime integral. 239 00:15:20,120 --> 00:15:22,330 And this is an L prime integral. 240 00:15:37,440 --> 00:15:40,650 OK so there's three integrals in this formula. 241 00:15:40,650 --> 00:15:42,730 I guess this is-- it's kind of a funny notation, 242 00:15:42,730 --> 00:15:45,690 but I could have written integral instead 243 00:15:45,690 --> 00:15:48,152 of writing these tensor signs. 244 00:15:48,152 --> 00:15:49,860 That would have been probably more clear. 245 00:15:58,260 --> 00:15:58,770 Do that. 246 00:16:01,410 --> 00:16:05,730 OK so there's the formula in momentum space. 247 00:16:05,730 --> 00:16:09,060 And it's doing what I said. 248 00:16:09,060 --> 00:16:14,310 This factor here is running the hard function 249 00:16:14,310 --> 00:16:21,840 down to the soft scale, and this other factor here 250 00:16:21,840 --> 00:16:25,710 is running from the jet scale to the soft scale, 251 00:16:25,710 --> 00:16:28,830 and I didn't have to run the soft function. 252 00:16:28,830 --> 00:16:31,590 I should have written in here that I evaluate 253 00:16:31,590 --> 00:16:34,620 this guy at the soft scale. 254 00:16:34,620 --> 00:16:38,610 And this guy here remember is the non-perturbative guy which 255 00:16:38,610 --> 00:16:41,410 we also talked about a little last time. 256 00:16:41,410 --> 00:16:45,373 OK so that's kind of the basic structure these use as usual 257 00:16:45,373 --> 00:16:47,040 are summing the logarithms, and then you 258 00:16:47,040 --> 00:16:55,110 put fixed order results in for this guy, for that guy, 259 00:16:55,110 --> 00:16:59,090 and for this guy here. 260 00:16:59,090 --> 00:17:07,579 And in doing so you get the [INAUDIBLE] result. 261 00:17:07,579 --> 00:17:10,843 All right, so this fact that I could 262 00:17:10,843 --> 00:17:12,260 have done the running differently, 263 00:17:12,260 --> 00:17:14,359 that I could have run the soft function 264 00:17:14,359 --> 00:17:16,640 is a kind of consistency. 265 00:17:16,640 --> 00:17:23,470 So there are other ways of doing the r g, 266 00:17:23,470 --> 00:17:26,753 and they all lead to the same answer. 267 00:17:26,753 --> 00:17:28,420 And when I say the same answer, I really 268 00:17:28,420 --> 00:17:33,150 mean the same precisely exactly the same number. 269 00:17:33,150 --> 00:17:36,280 So I really mean the same, not just 270 00:17:36,280 --> 00:17:40,330 sort of approximately the same but exactly the same. 271 00:17:40,330 --> 00:17:42,250 And the other ways of doing the RGE 272 00:17:42,250 --> 00:17:46,570 are the basic idea of why there are more than one way is again 273 00:17:46,570 --> 00:17:51,100 this fact that if I do coefficient renormalization, 274 00:17:51,100 --> 00:17:54,790 that that's the inverse of doing operator renormalization. 275 00:17:54,790 --> 00:17:57,520 So there's two ways of thinking about doing the run. 276 00:17:57,520 --> 00:18:00,209 You run the operators, you run the coefficients. 277 00:18:05,200 --> 00:18:07,480 OK in this picture it's a little bit more complicated 278 00:18:07,480 --> 00:18:09,820 because we have three scales, but when 279 00:18:09,820 --> 00:18:11,770 I'm running the hard function, I quivalently 280 00:18:11,770 --> 00:18:13,990 could have run both the jet and the soft function 281 00:18:13,990 --> 00:18:16,550 and not run the hard function. 282 00:18:16,550 --> 00:18:18,370 So let me maybe give one other way 283 00:18:18,370 --> 00:18:21,170 of doing this just to show you. 284 00:18:21,170 --> 00:18:24,730 So I could have run the hard function say, just 285 00:18:24,730 --> 00:18:26,605 down to the jet scale. 286 00:18:26,605 --> 00:18:28,480 And then instead of running the jet function, 287 00:18:28,480 --> 00:18:31,690 I could have run the soft function up to the jet scale. 288 00:18:31,690 --> 00:18:34,330 That would be another way of doing the RGE, 289 00:18:34,330 --> 00:18:37,010 and that would lead to an equivalent picture. 290 00:18:37,010 --> 00:18:40,600 And there's more than two, because there's 291 00:18:40,600 --> 00:18:41,708 three scales in this. 292 00:18:41,708 --> 00:18:43,250 There's more than two natural things. 293 00:18:43,250 --> 00:18:45,172 There's three possible natural things, 294 00:18:45,172 --> 00:18:47,380 where you would run to either this scale, this scale, 295 00:18:47,380 --> 00:18:50,410 or this scale, but at the base of it, 296 00:18:50,410 --> 00:18:52,480 it all has to do with this equivalence 297 00:18:52,480 --> 00:18:57,340 that we talked about in simpler scenarios earlier. 298 00:18:57,340 --> 00:19:01,412 And you can write this also as a relation 299 00:19:01,412 --> 00:19:02,620 between anomalous dimensions. 300 00:19:02,620 --> 00:19:05,022 So it implies a non-trivial things 301 00:19:05,022 --> 00:19:07,480 about the formula that have to be trivial in order for this 302 00:19:07,480 --> 00:19:09,740 to work. 303 00:19:09,740 --> 00:19:16,120 So for example, it implies that the coefficients 304 00:19:16,120 --> 00:19:19,300 of the cusp anomalous dimension in the various renormalization 305 00:19:19,300 --> 00:19:22,660 groups would be in a certain way that they would be related, 306 00:19:22,660 --> 00:19:25,090 and even in the non-cusp anomalous dimensions, which 307 00:19:25,090 --> 00:19:27,460 is this formula, there's a relation 308 00:19:27,460 --> 00:19:31,810 between the jet, hard, and soft anomalous dimensions. 309 00:19:31,810 --> 00:19:32,310 OK? 310 00:19:32,310 --> 00:19:36,162 And that's needed-- this is an expression of this fact 311 00:19:36,162 --> 00:19:37,870 that you can do the renormalization group 312 00:19:37,870 --> 00:19:40,690 by running different objects. 313 00:19:40,690 --> 00:19:42,610 So you can derive this formula by just 314 00:19:42,610 --> 00:19:45,670 saying mu D divided by the cross section is 0. 315 00:19:45,670 --> 00:19:49,060 And going through it, if you were to allow, 316 00:19:49,060 --> 00:19:52,180 then you would find formulas like this one. 317 00:19:52,180 --> 00:19:54,470 OK? 318 00:19:54,470 --> 00:19:55,430 So questions? 319 00:19:59,100 --> 00:20:01,975 I went quick, because most of the concepts 320 00:20:01,975 --> 00:20:04,350 here are things that we've seen before in simpler guises. 321 00:20:08,347 --> 00:20:09,930 The new complication is really that we 322 00:20:09,930 --> 00:20:13,080 have this dependence on a variable which ended up 323 00:20:13,080 --> 00:20:15,750 being y, but by going to Fourier space, 324 00:20:15,750 --> 00:20:17,550 it looked just like a simple product 325 00:20:17,550 --> 00:20:20,980 again, and y behaved like a simple kinematic variable. 326 00:20:20,980 --> 00:20:23,230 The end of the day, we want to Fourier transform back, 327 00:20:23,230 --> 00:20:26,025 but we have a space where things look familiar. 328 00:20:30,540 --> 00:20:33,470 OK. 329 00:20:33,470 --> 00:20:36,170 So that's e plus minus the dijets, 330 00:20:36,170 --> 00:20:38,510 and that's the last SCETI example 331 00:20:38,510 --> 00:20:39,920 I'm going to do for a while. 332 00:20:39,920 --> 00:20:42,140 I'm now going to turn to doing SCETII, 333 00:20:42,140 --> 00:20:44,120 and we'll spend some time talking about SCETII. 334 00:20:55,720 --> 00:20:58,750 So in SCETII, instead of having ultrasoft interactions 335 00:20:58,750 --> 00:21:01,390 with collinear particles, we have soft interactions 336 00:21:01,390 --> 00:21:05,340 with collinear particles. 337 00:21:05,340 --> 00:21:09,380 So we have to call how that's going to work. 338 00:21:09,380 --> 00:21:16,680 So if we take a soft particle, consider a soft particle 339 00:21:16,680 --> 00:21:20,740 interacting with some collinear particle, and ask what 340 00:21:20,740 --> 00:21:21,420 do we get out? 341 00:21:25,933 --> 00:21:27,850 So if we just ask about momentum conservation, 342 00:21:27,850 --> 00:21:31,450 and we call this q, then q is equal to q 343 00:21:31,450 --> 00:21:35,118 soft plus q collinear. 344 00:21:35,118 --> 00:21:36,910 And since we know the scaling of these two, 345 00:21:36,910 --> 00:21:39,250 we know the scaling of q. 346 00:21:39,250 --> 00:21:42,790 It's just given by whatever the larger of the scalings is. 347 00:21:47,440 --> 00:21:50,760 So soft and collinear both have the same-- so 348 00:21:50,760 --> 00:21:52,135 let me remind you of the scaling. 349 00:21:52,135 --> 00:21:57,340 So this guy was q, lambda, lambda, lambda, and this guy 350 00:21:57,340 --> 00:22:01,550 was q lambda squared 1 lambda. 351 00:22:01,550 --> 00:22:03,610 So the difference between ultrasoft and collinear 352 00:22:03,610 --> 00:22:05,860 and soft and collinear is that soft and collinear have 353 00:22:05,860 --> 00:22:07,990 the same size of perp momenta. 354 00:22:07,990 --> 00:22:11,170 So the perp momenta are just of order lambda. 355 00:22:11,170 --> 00:22:13,480 Obviously, in the minus momenta, the 1 356 00:22:13,480 --> 00:22:16,210 wins, so the minus momentum of order 1. 357 00:22:16,210 --> 00:22:18,547 And then the plus momentum, it's the soft 358 00:22:18,547 --> 00:22:20,630 that wins, because it's bigger than the collinear. 359 00:22:20,630 --> 00:22:23,470 So this is order lambda. 360 00:22:23,470 --> 00:22:25,450 OK. 361 00:22:25,450 --> 00:22:28,360 And that's actually much different than what 362 00:22:28,360 --> 00:22:31,525 we found when we added ultrasoft and collinear. 363 00:22:31,525 --> 00:22:33,400 Because when we added ultrasoft and collinear 364 00:22:33,400 --> 00:22:35,380 we got back collinear. 365 00:22:35,380 --> 00:22:37,320 Here, we're not getting back collinear. 366 00:22:37,320 --> 00:22:40,420 This is actually off-shell, from the perspective 367 00:22:40,420 --> 00:22:41,530 of our low-energy modes. 368 00:22:44,950 --> 00:22:47,320 Because q squared, the biggest part of q squared 369 00:22:47,320 --> 00:22:48,820 would be 1 times lambda. 370 00:22:48,820 --> 00:22:52,030 q squared's of order lambda which is much bigger 371 00:22:52,030 --> 00:22:52,960 than lambda squared. 372 00:22:58,590 --> 00:23:02,180 So adding a soft and a collinear particle in SCETII immediately 373 00:23:02,180 --> 00:23:03,850 gives you something off-shell. 374 00:23:03,850 --> 00:23:06,350 That's going to make some things easier and some things more 375 00:23:06,350 --> 00:23:08,695 complicated. 376 00:23:08,695 --> 00:23:10,070 Mostly, it's going to make things 377 00:23:10,070 --> 00:23:11,153 a little more complicated. 378 00:23:21,497 --> 00:23:23,330 At least at the start, it'll looking easier. 379 00:23:32,970 --> 00:23:40,020 So you could think about this from our mode picture, 380 00:23:40,020 --> 00:23:42,870 where we had softs that live here 381 00:23:42,870 --> 00:23:45,120 and collinears that live there. 382 00:23:45,120 --> 00:23:48,982 In order for them to interact, they have to go up to a place 383 00:23:48,982 --> 00:23:50,940 where we can have a common momentum, and that's 384 00:23:50,940 --> 00:23:54,210 a higher hyperbola, like this. 385 00:23:54,210 --> 00:23:58,390 It's not all the way up to the hard scale which is up here. 386 00:23:58,390 --> 00:24:02,070 That was hard, but there's an intermediate scale 387 00:24:02,070 --> 00:24:07,050 that comes in which is the scale of this q squared. 388 00:24:07,050 --> 00:24:09,882 So this is q squared. 389 00:24:09,882 --> 00:24:12,300 This is the order of the q squared there, 390 00:24:12,300 --> 00:24:16,410 and sometimes this is called a hard collinear scale. 391 00:24:16,410 --> 00:24:18,930 This would be called a hard collinear mode, hc. 392 00:24:30,398 --> 00:24:31,940 And you could even write down, if you 393 00:24:31,940 --> 00:24:34,482 wanted, an on-shell degree of freedom for this hard collinear 394 00:24:34,482 --> 00:24:37,280 mode. 395 00:24:37,280 --> 00:24:38,870 So an on-shell version of it, you 396 00:24:38,870 --> 00:24:42,590 could think of it as a mode, in theory. 397 00:24:42,590 --> 00:24:44,570 An on-shell version of it would have a scaling 398 00:24:44,570 --> 00:24:47,135 that's q 1 lambda root lambda. 399 00:24:52,540 --> 00:24:55,620 So you can think of one way of approaching it which we'll 400 00:24:55,620 --> 00:24:57,042 take this attitude in a minute. 401 00:24:57,042 --> 00:24:58,500 One way of approaching this problem 402 00:24:58,500 --> 00:25:00,780 would be to first think about doing 403 00:25:00,780 --> 00:25:03,240 some matching onto a theory with this hard collinear mode 404 00:25:03,240 --> 00:25:06,180 and then trying to match down onto a theory that just has 405 00:25:06,180 --> 00:25:08,400 the collinear and the soft mode, and we'll 406 00:25:08,400 --> 00:25:10,760 exploit that a little later. 407 00:25:10,760 --> 00:25:13,200 But first, let me do a different thing 408 00:25:13,200 --> 00:25:16,050 and just ignore this dashed line and just think 409 00:25:16,050 --> 00:25:20,130 about matching directly from the QCD 410 00:25:20,130 --> 00:25:24,060 onto the SCETII which just has these two degrees of freedom. 411 00:25:47,090 --> 00:25:48,860 So what's going to happen is that we're 412 00:25:48,860 --> 00:25:51,440 going to have to integrate out more things, 413 00:25:51,440 --> 00:25:54,407 because any type of interactions that we write down 414 00:25:54,407 --> 00:25:56,240 are basically giving us something off-shell. 415 00:25:56,240 --> 00:25:59,310 So let's do an example of this. 416 00:25:59,310 --> 00:26:04,880 So we'll do an example of a heavy-to-light current, 417 00:26:04,880 --> 00:26:10,620 but now, it won't be an SCETII current but rather an SCETI. 418 00:26:17,810 --> 00:26:20,570 So I just want to have one soft particle and one collinear 419 00:26:20,570 --> 00:26:22,130 particle. 420 00:26:22,130 --> 00:26:24,730 Simplest possible example, but now this is soft, 421 00:26:24,730 --> 00:26:28,081 and this is collinear, not ultrasoft and collinear. 422 00:26:32,950 --> 00:26:33,450 All right. 423 00:26:33,450 --> 00:26:35,520 So we have some current. 424 00:26:35,520 --> 00:26:39,630 I'm just going to label the lines with c's and s's. 425 00:26:39,630 --> 00:26:45,180 So imagine that you attached a soft gluon here. 426 00:26:45,180 --> 00:26:47,880 That would give you something that's off-shelf. 427 00:26:47,880 --> 00:26:51,420 So this line here is off-shelf, and likewise, 428 00:26:51,420 --> 00:26:52,950 on this side, if this is soft coming 429 00:26:52,950 --> 00:26:57,150 in but you attach something collinear here, 430 00:26:57,150 --> 00:27:00,030 then this is off-shelf. 431 00:27:00,030 --> 00:27:01,890 Before, when we had this picture, 432 00:27:01,890 --> 00:27:04,440 and we were doing it for SCETI, one 433 00:27:04,440 --> 00:27:06,510 set of attachments which still need to on-shell, 434 00:27:06,510 --> 00:27:08,535 in particular on this side. 435 00:27:08,535 --> 00:27:10,410 And on this side, we got something off-shell. 436 00:27:10,410 --> 00:27:13,170 We integrated it out, and we got a Wilson line. 437 00:27:13,170 --> 00:27:14,190 Right? 438 00:27:14,190 --> 00:27:15,750 Here, it's a little more complicated, 439 00:27:15,750 --> 00:27:18,645 because you touch alternate modes on either side. 440 00:27:18,645 --> 00:27:20,520 And you get something off-shell, and you just 441 00:27:20,520 --> 00:27:25,135 have to put out this pink line all at once, 442 00:27:25,135 --> 00:27:27,510 and that's going to give you another type of Wilson line. 443 00:27:31,410 --> 00:27:33,360 In this calculation, we're going to get 444 00:27:33,360 --> 00:27:38,430 W. It's both of the collinear modes, 445 00:27:38,430 --> 00:27:43,650 and another type of Wilson line from the soft modes. 446 00:27:54,880 --> 00:27:57,640 Let's call it Sn. 447 00:27:57,640 --> 00:27:59,860 So how do you build up a Wilson line? 448 00:27:59,860 --> 00:28:02,890 Well, first of all, think about attaching more gluons 449 00:28:02,890 --> 00:28:04,910 onto this line. 450 00:28:04,910 --> 00:28:05,560 Right? 451 00:28:05,560 --> 00:28:09,050 That's kind of the analog of what we did before. 452 00:28:09,050 --> 00:28:14,980 So just attach more soft gluons here, 453 00:28:14,980 --> 00:28:18,137 more collinear gluons here. 454 00:28:18,137 --> 00:28:19,970 This might be the first thing you would try, 455 00:28:19,970 --> 00:28:22,270 just adding those guys up, and that 456 00:28:22,270 --> 00:28:25,217 means you have a whole line of things that is off-shell, 457 00:28:25,217 --> 00:28:26,800 and you just calculate these diagrams. 458 00:28:29,560 --> 00:28:35,518 And if you do that calculation, it will give you Wilson, lines. 459 00:28:35,518 --> 00:28:37,060 And what it'll give is something that 460 00:28:37,060 --> 00:28:51,050 looks like this, where Sn dagger is a function of the n dot 461 00:28:51,050 --> 00:28:55,150 As component, and W as a function of 462 00:28:55,150 --> 00:28:58,920 said n bar dot Ac component. 463 00:28:58,920 --> 00:29:03,800 So there are Wilson lines along some direction, 464 00:29:03,800 --> 00:29:07,320 and it looks like this. 465 00:29:07,320 --> 00:29:08,240 This is c. 466 00:29:08,240 --> 00:29:11,180 This is S. 467 00:29:11,180 --> 00:29:14,510 Now, there's something wrong with this. 468 00:29:14,510 --> 00:29:16,532 It's not quite the right answer. 469 00:29:16,532 --> 00:29:18,740 The reason that the Wilson lines are the way they are 470 00:29:18,740 --> 00:29:22,250 is because I got the collinear ones from integrating them out 471 00:29:22,250 --> 00:29:25,100 from the next to the heavy quark which is the soft particle. 472 00:29:25,100 --> 00:29:27,560 So that's why W sits next to h. 473 00:29:27,560 --> 00:29:30,050 S dagger sits next to c because of the same thing 474 00:29:30,050 --> 00:29:31,820 on the collinear side. 475 00:29:31,820 --> 00:29:34,460 But if I wanted to make them into a gauge invariant thing, 476 00:29:34,460 --> 00:29:37,280 I want the W to sit next to the c and the S dagger 477 00:29:37,280 --> 00:29:39,320 to sit next to the h. 478 00:29:39,320 --> 00:29:42,260 That would be the analog of what we had in SCETI, 479 00:29:42,260 --> 00:29:43,940 and that's not what we got. 480 00:29:43,940 --> 00:29:46,910 If this was QED, then that would be fine, because this and this, 481 00:29:46,910 --> 00:29:49,385 I could just commute them. 482 00:29:49,385 --> 00:29:50,510 Well, I can always do that. 483 00:29:50,510 --> 00:29:52,260 I can just push this guy through that guy. 484 00:29:52,260 --> 00:29:55,130 But in QCD I can't, because these guys have color matrices, 485 00:29:55,130 --> 00:29:57,340 and they don't commute with each other. 486 00:29:57,340 --> 00:30:00,920 So that means that this isn't quite the whole story here. 487 00:30:00,920 --> 00:30:02,480 There's some diagrams that we missed. 488 00:30:07,612 --> 00:30:09,070 So there must be some diagrams that 489 00:30:09,070 --> 00:30:24,185 are non-abelian that we missed, and what are those diagrams? 490 00:30:28,160 --> 00:30:31,095 These are diagrams that involve triple gluon and four gluon 491 00:30:31,095 --> 00:30:31,595 vertices. 492 00:30:43,790 --> 00:30:45,530 And what these guys do, it turns out, 493 00:30:45,530 --> 00:30:47,730 is they do one thing in this calculation. 494 00:30:47,730 --> 00:30:58,380 They really just flip the order of the W and the S. 495 00:30:58,380 --> 00:31:02,220 So why do we have to consider those diagrams? 496 00:31:05,900 --> 00:31:11,270 So think about, instead of attaching gluons to quarks, 497 00:31:11,270 --> 00:31:13,190 attach gluons to gluons. 498 00:31:13,190 --> 00:31:22,090 So say we have soft collinear, and we attach them 499 00:31:22,090 --> 00:31:24,410 to each other but through a three-gluon vortex. 500 00:31:24,410 --> 00:31:25,848 Then, this guy's off-shell. 501 00:31:25,848 --> 00:31:27,640 If that off-shell guy attaches to this guy, 502 00:31:27,640 --> 00:31:29,230 then it's off-shell. 503 00:31:29,230 --> 00:31:31,210 And if you integrate out diagrams like this, 504 00:31:31,210 --> 00:31:36,010 that's going to change what our result over there look like. 505 00:31:36,010 --> 00:31:39,190 If you go to the same order on the other side, 506 00:31:39,190 --> 00:31:40,450 exactly analogous thing. 507 00:31:46,810 --> 00:31:47,310 OK. 508 00:31:47,310 --> 00:31:49,643 So these are also things that you have to integrate out. 509 00:31:49,643 --> 00:31:53,020 You have to integrate all these pink things out. 510 00:31:53,020 --> 00:31:57,645 And if you do that, then it does what I said. 511 00:31:57,645 --> 00:32:01,650 If you do that to all orders, what 512 00:32:01,650 --> 00:32:09,360 you can do with a axillary Lagrangian-type approach, 513 00:32:09,360 --> 00:32:11,735 obviously, you're not going to start calculating diagrams 514 00:32:11,735 --> 00:32:15,050 to all this, but there are some tricks to doing it. 515 00:32:15,050 --> 00:32:17,560 Then, you get them in the right order. 516 00:32:22,090 --> 00:32:24,190 And it's easy to check, for example, 517 00:32:24,190 --> 00:32:26,920 that at the order of two gluons that this guy does 518 00:32:26,920 --> 00:32:29,630 exactly what you want. 519 00:32:29,630 --> 00:32:35,530 So this is then a collinear gauge invariant object, 520 00:32:35,530 --> 00:32:40,000 and this is then something that's soft gauge invariant, 521 00:32:40,000 --> 00:32:41,620 and that's nice. 522 00:32:41,620 --> 00:32:42,790 That's what you're hoping. 523 00:32:42,790 --> 00:32:46,360 That's what you're expecting. 524 00:32:46,360 --> 00:32:49,255 AUDIENCE: What about diagrams like collinear soft 525 00:32:49,255 --> 00:32:52,600 all from the collinear line or something? 526 00:32:52,600 --> 00:32:54,930 IAIN STEWART: You want to add one more gluon where? 527 00:32:54,930 --> 00:32:56,680 AUDIENCE: I want to stick like a collinear 528 00:32:56,680 --> 00:32:58,588 right in between some softs. 529 00:32:58,588 --> 00:32:59,380 IAIN STEWART: Yeah. 530 00:32:59,380 --> 00:33:04,865 You mean like on here, like that? 531 00:33:04,865 --> 00:33:05,740 AUDIENCE: Well, yeah. 532 00:33:05,740 --> 00:33:07,950 Essentially, it'd be over on that diagram over there, 533 00:33:07,950 --> 00:33:10,618 like not even [INAUDIBLE]. 534 00:33:10,618 --> 00:33:11,410 IAIN STEWART: Yeah. 535 00:33:11,410 --> 00:33:12,618 It's the same thing, I think. 536 00:33:12,618 --> 00:33:15,700 So if you add this guy here, then I 537 00:33:15,700 --> 00:33:17,740 think it's power suppressed. 538 00:33:17,740 --> 00:33:22,330 Because it ends up being a power suppressed term 539 00:33:22,330 --> 00:33:25,220 that you don't need, if I remember correctly. 540 00:33:25,220 --> 00:33:25,720 Yeah. 541 00:33:25,720 --> 00:33:26,860 This guy's power suppressed. 542 00:33:26,860 --> 00:33:28,410 AUDIENCE: You have to define like a different lambda 543 00:33:28,410 --> 00:33:29,350 for each one? 544 00:33:29,350 --> 00:33:30,522 It seems like it would be-- 545 00:33:30,522 --> 00:33:31,230 IAIN STEWART: No. 546 00:33:33,880 --> 00:33:35,840 You can think of this as a full theory diagram. 547 00:33:35,840 --> 00:33:36,340 Right? 548 00:33:36,340 --> 00:33:39,400 Where this guy is off-shell, and you haven't 549 00:33:39,400 --> 00:33:40,800 changed how he's off-shell. 550 00:33:40,800 --> 00:33:42,550 You just change the value of the momentum, 551 00:33:42,550 --> 00:33:45,400 but one thing you've done is you've doubled 552 00:33:45,400 --> 00:33:46,870 the number of hard propagators. 553 00:33:46,870 --> 00:33:49,580 So if I remember correctly, this guy just 554 00:33:49,580 --> 00:33:50,830 gives you something off-shell. 555 00:33:50,830 --> 00:33:53,650 I'm pretty confident, something that's power 556 00:33:53,650 --> 00:33:54,700 suppressed in lambda. 557 00:33:58,680 --> 00:34:00,900 All right. 558 00:34:00,900 --> 00:34:05,160 So this is how soft collinear factorization works. 559 00:34:05,160 --> 00:34:07,410 So soft collinear factorization, if you think about it 560 00:34:07,410 --> 00:34:11,130 from the point of view of going from QCD to SCETII, 561 00:34:11,130 --> 00:34:14,159 it's just integrate out these pink lines, as usual, 562 00:34:14,159 --> 00:34:16,020 and you end up with something that's where 563 00:34:16,020 --> 00:34:17,393 things are splitting apart. 564 00:34:17,393 --> 00:34:18,810 And the reason that it's happening 565 00:34:18,810 --> 00:34:21,239 is because these lines are off-shell 566 00:34:21,239 --> 00:34:23,800 that are where you would try to interact, have interactions, 567 00:34:23,800 --> 00:34:27,420 and so your theory is forced apart by the fact 568 00:34:27,420 --> 00:34:30,179 that these things can't interact in an on-shell way. 569 00:34:30,179 --> 00:34:35,350 So it's different than SCETI that in that sense. 570 00:34:35,350 --> 00:34:37,500 So this is kind of cumbersome, as you can see, 571 00:34:37,500 --> 00:34:40,590 if you wanted to think about doing it arbitrarily. 572 00:34:40,590 --> 00:34:43,050 Because it seems like you just have to calculate diagrams, 573 00:34:43,050 --> 00:34:46,139 and who wants to calculate infinite classes of diagrams 574 00:34:46,139 --> 00:34:47,969 for arbitrarily complicated scenarios? 575 00:34:47,969 --> 00:34:49,199 Right? 576 00:34:49,199 --> 00:34:50,670 Maybe in this case, we can do it, 577 00:34:50,670 --> 00:34:52,440 but actually even in this case, we 578 00:34:52,440 --> 00:34:55,270 have to resort to some tricks to do it to all orders. 579 00:34:55,270 --> 00:34:57,090 And in more complicated scenarios, 580 00:34:57,090 --> 00:34:59,290 it would just get even more and more cumbersome. 581 00:34:59,290 --> 00:35:02,670 So we'd like to have a trick that's generic, 582 00:35:02,670 --> 00:35:04,630 where we could get the same answer. 583 00:35:04,630 --> 00:35:08,760 And the way that we can do that is by using this dashed line up 584 00:35:08,760 --> 00:35:10,390 in the picture. 585 00:35:10,390 --> 00:35:13,920 Think about formulating rather than directly this SCETII, 586 00:35:13,920 --> 00:35:16,500 formulate first an SCETI, and then we'll 587 00:35:16,500 --> 00:35:18,885 match that SCETI onto QCD. 588 00:35:21,850 --> 00:35:36,120 So another, in some ways, better method is to do QCD to SCETI 589 00:35:36,120 --> 00:35:45,380 and then SCETI to SCETII, so three steps. 590 00:35:50,010 --> 00:35:54,110 So we're going to first use a SCETI that 591 00:35:54,110 --> 00:35:55,790 doesn't have the collinear, just has 592 00:35:55,790 --> 00:35:58,280 the soft mode and the hard collinear mode. 593 00:35:58,280 --> 00:36:03,510 So it has what I call the hc in the picture and s. 594 00:36:03,510 --> 00:36:04,010 OK? 595 00:36:04,010 --> 00:36:06,140 So that would be a mode that has P 596 00:36:06,140 --> 00:36:11,400 squared in this picture of order, say let's call it lambda 597 00:36:11,400 --> 00:36:11,900 squared. 598 00:36:14,850 --> 00:36:20,653 So if I say that this is for some lambda squared, 599 00:36:20,653 --> 00:36:22,070 then you'd have the soft mode that 600 00:36:22,070 --> 00:36:23,360 has P squared of order lambda squared, 601 00:36:23,360 --> 00:36:24,985 and this hard collinear mode that has P 602 00:36:24,985 --> 00:36:26,150 squared of order q lambda. 603 00:36:26,150 --> 00:36:30,530 And that's exactly an SCETI-type situation, where 604 00:36:30,530 --> 00:36:35,040 we were calling these c and us. 605 00:36:35,040 --> 00:36:38,220 So that's step one. 606 00:36:38,220 --> 00:36:40,310 Step two is to factorize that theory, which 607 00:36:40,310 --> 00:36:43,495 we know how to do. 608 00:36:43,495 --> 00:36:48,830 In particular, the ultrasoft which is soft 609 00:36:48,830 --> 00:36:51,597 can be factorized with a field redefinition. 610 00:36:55,010 --> 00:36:57,770 So this has the advantage that at this first stage 611 00:36:57,770 --> 00:37:00,620 we still have a locality that protects us 612 00:37:00,620 --> 00:37:03,140 and helps us to understand the theory. 613 00:37:03,140 --> 00:37:07,650 We then factorize, and then in the third step, 614 00:37:07,650 --> 00:37:09,490 we match SCETI onto SCETII. 615 00:37:20,830 --> 00:37:26,560 And then, we're getting rid of these hard collinear modes, 616 00:37:26,560 --> 00:37:31,650 matching them onto some collinear modes, 617 00:37:31,650 --> 00:37:35,340 and then we still have our soft modes. 618 00:37:39,880 --> 00:37:42,870 So we think that the hard collinear 619 00:37:42,870 --> 00:37:45,610 modes contain the collinear modes in the first stage. 620 00:37:45,610 --> 00:37:47,270 But they also contain some other stuff 621 00:37:47,270 --> 00:37:50,540 which is unwanted baggage that we have to get rid of, 622 00:37:50,540 --> 00:37:54,835 and that's why we have this second stage of matching. 623 00:37:54,835 --> 00:37:56,710 But you can really think of it in the picture 624 00:37:56,710 --> 00:37:59,260 as doing first the matching, where you integrate out 625 00:37:59,260 --> 00:38:02,610 the hard scale, but you contain in your effective theory 626 00:38:02,610 --> 00:38:06,340 this scale associated to the dashed line. 627 00:38:06,340 --> 00:38:10,462 And then in a second stage, you integrate out the dashed line, 628 00:38:10,462 --> 00:38:13,180 and that gets you to the final thing 629 00:38:13,180 --> 00:38:15,670 you want which is just low energy modes on this line here. 630 00:38:18,970 --> 00:38:20,890 OK. 631 00:38:20,890 --> 00:38:33,030 So one thing this does is give a simple procedure 632 00:38:33,030 --> 00:38:36,420 of constructing SCETII operators, 633 00:38:36,420 --> 00:38:39,630 even though there's more non-locality in SCETII 634 00:38:39,630 --> 00:38:40,710 than there was in SCETI. 635 00:38:48,260 --> 00:38:50,870 Because there's non-locality, because you don't just 636 00:38:50,870 --> 00:38:53,060 have this one scale that could cause you, 637 00:38:53,060 --> 00:38:56,150 even have this smaller scale that's causing non-locality. 638 00:38:56,150 --> 00:38:59,360 And that's explicit in the soft Wilson line. 639 00:38:59,360 --> 00:39:02,630 The thing that's giving the soft Wilson line-- 640 00:39:02,630 --> 00:39:04,940 well, the things that are producing the Wilson lines 641 00:39:04,940 --> 00:39:06,370 are really modes of this scale. 642 00:39:09,670 --> 00:39:12,188 OK so it's more non-local. 643 00:39:12,188 --> 00:39:13,730 One way of saying it's more non-local 644 00:39:13,730 --> 00:39:15,880 is simply that there'll be 1 over l pluses, 645 00:39:15,880 --> 00:39:17,360 as well as 1 over l minuses. 646 00:39:17,360 --> 00:39:22,200 That's another way of saying why it's more non-local. 647 00:39:22,200 --> 00:39:26,580 OK so that's just an example of something 648 00:39:26,580 --> 00:39:28,710 that makes it look trivial. 649 00:39:28,710 --> 00:39:32,970 So let's say we wanted to do this calculation, this way. 650 00:39:32,970 --> 00:39:36,450 So then in step one, we would simply write down 651 00:39:36,450 --> 00:39:37,860 the current in SCETI. 652 00:39:41,850 --> 00:39:44,070 We would integrate out the off-shell pink lines, 653 00:39:44,070 --> 00:39:45,750 but there'd only be lines on one side, 654 00:39:45,750 --> 00:39:50,460 and we get this which is our SCETI result 655 00:39:50,460 --> 00:39:52,200 for heavy-to-light current. 656 00:39:52,200 --> 00:39:58,920 Then, in step two, we would do the field redefinition in order 657 00:39:58,920 --> 00:40:07,890 to factorize this theory, and we get that result. 658 00:40:07,890 --> 00:40:11,540 And in step three, in order to match this result 659 00:40:11,540 --> 00:40:17,570 onto a current in SCETII, it's really simply a renaming here. 660 00:40:30,050 --> 00:40:32,876 So why is this step so easy? 661 00:40:32,876 --> 00:40:35,700 It looks completely trivial. 662 00:40:35,700 --> 00:40:37,370 The reason that this step is so easy 663 00:40:37,370 --> 00:40:40,198 is that any T-product, any time-order product that you 664 00:40:40,198 --> 00:40:41,990 write down here, will have a correspondence 665 00:40:41,990 --> 00:40:42,860 in this other case. 666 00:40:53,400 --> 00:40:53,900 OK? 667 00:40:53,900 --> 00:40:54,692 So you can really-- 668 00:41:02,330 --> 00:41:03,470 it really is that simple. 669 00:41:06,630 --> 00:41:09,080 So when so if it's really that simple, then 670 00:41:09,080 --> 00:41:12,260 you can see the advantages of this procedure. 671 00:41:12,260 --> 00:41:16,460 It's not always that simple, and let 672 00:41:16,460 --> 00:41:20,900 me present as a kind of theorem or just a bit 673 00:41:20,900 --> 00:41:24,290 of a statement when it will be this simple 674 00:41:24,290 --> 00:41:27,812 and when it will not be this simple. 675 00:41:27,812 --> 00:41:29,270 So basically, you have to know when 676 00:41:29,270 --> 00:41:32,930 will the T-products between these two theories match up? 677 00:41:32,930 --> 00:41:35,930 And the T-products will match up under the following 678 00:41:35,930 --> 00:41:42,650 circumstances, or they won't match up under the following 679 00:41:42,650 --> 00:41:43,953 circumstances. 680 00:41:49,880 --> 00:41:54,040 So if in the SCETI one theory you 681 00:41:54,040 --> 00:41:57,580 had time-order products with greater than or equal to two 682 00:41:57,580 --> 00:42:17,180 operators that had soft and collinear fields, 683 00:42:17,180 --> 00:42:19,910 then you can generate some non-trivial matching. 684 00:42:51,373 --> 00:42:52,790 So I think that's best illustrated 685 00:42:52,790 --> 00:42:55,140 by an example again. 686 00:42:55,140 --> 00:42:57,260 So in this particular example, up here, we only 687 00:42:57,260 --> 00:42:59,720 had one operator that had both soft and collinear fields, 688 00:42:59,720 --> 00:43:01,500 this external current. 689 00:43:01,500 --> 00:43:04,400 And then we had Lagrangians, but they were totally decoupled 690 00:43:04,400 --> 00:43:06,590 after we made this field redefinition, the L0 691 00:43:06,590 --> 00:43:07,160 Lagrangian. 692 00:43:07,160 --> 00:43:09,467 So they don't count as something that's mixing 693 00:43:09,467 --> 00:43:10,550 soft and collinear fields. 694 00:43:10,550 --> 00:43:12,350 We just had this operator. 695 00:43:12,350 --> 00:43:13,940 But if you had two of these operators, 696 00:43:13,940 --> 00:43:17,210 and you wanted to go through the same procedure, 697 00:43:17,210 --> 00:43:19,010 then you could get something non-trivial. 698 00:43:19,010 --> 00:43:20,302 So let's imagine that scenario. 699 00:43:24,356 --> 00:43:30,950 We have two operators, so let's think 700 00:43:30,950 --> 00:43:36,890 of having in the SCETI, two interactions 701 00:43:36,890 --> 00:43:41,785 that are like this, then we could string them together 702 00:43:41,785 --> 00:43:42,285 as follows. 703 00:43:56,760 --> 00:43:59,690 So this is a T-product between two different interactions that 704 00:43:59,690 --> 00:44:00,950 both had soft and collinear. 705 00:44:00,950 --> 00:44:03,960 I'm just taking two of these guys and putting them together. 706 00:44:03,960 --> 00:44:06,680 And if you look at the off-shell-ness of this line 707 00:44:06,680 --> 00:44:10,880 here, then P squared is of order, in our counting, 708 00:44:10,880 --> 00:44:17,830 it's like a k plus times a q minus which is of q times 709 00:44:17,830 --> 00:44:19,916 a lambda times a q. 710 00:44:19,916 --> 00:44:22,430 So it's really something that lives at that hard collinear 711 00:44:22,430 --> 00:44:24,230 scale. 712 00:44:24,230 --> 00:44:26,770 OK? 713 00:44:26,770 --> 00:44:30,298 But you need two terms in the time-order product that, 714 00:44:30,298 --> 00:44:32,590 in order for there really to be a propagator like this, 715 00:44:32,590 --> 00:44:34,280 that you're integrating out. 716 00:44:34,280 --> 00:44:37,390 So this guy here will match onto something 717 00:44:37,390 --> 00:44:40,960 when you do this matching, where you 718 00:44:40,960 --> 00:44:46,390 have two soft fields and two collinear fields, like that. 719 00:44:46,390 --> 00:44:48,220 Because this guy here really is something 720 00:44:48,220 --> 00:44:53,560 that you would want to integrate out at that dashed line. 721 00:44:53,560 --> 00:44:56,230 Really, it's an off-shell, hard, collinear mode. 722 00:44:59,120 --> 00:45:00,850 But if you didn't have two T-products, 723 00:45:00,850 --> 00:45:03,170 then effectively what happens is, 724 00:45:03,170 --> 00:45:05,170 when you change the external kinematics in order 725 00:45:05,170 --> 00:45:08,770 to do the matching, all what were called hard collinear 726 00:45:08,770 --> 00:45:11,200 lines become collinear lines, and then 727 00:45:11,200 --> 00:45:13,755 the matching is trivial, as it was up here. 728 00:45:13,755 --> 00:45:15,380 But if you're in a situation like this, 729 00:45:15,380 --> 00:45:17,440 then there is a line that, by changing 730 00:45:17,440 --> 00:45:19,390 the scaling of these external guys, 731 00:45:19,390 --> 00:45:22,390 it doesn't change the scaling of that internal line, 732 00:45:22,390 --> 00:45:25,244 and then you get some non-trivial Wilson coefficient. 733 00:45:29,116 --> 00:45:32,504 AUDIENCE: [INAUDIBLE] 734 00:45:36,730 --> 00:45:39,810 IAIN STEWART: So let's call these hc. 735 00:45:39,810 --> 00:45:47,010 Yeah, and so I start by calling them hc. 736 00:45:47,010 --> 00:45:48,300 Right? 737 00:45:48,300 --> 00:45:52,440 And then, what I want to do, when I calculate this diagram 738 00:45:52,440 --> 00:45:56,520 to do the matching, is I want to assign them a different scale. 739 00:45:56,520 --> 00:45:59,650 I want to define them to be c instead of hc. 740 00:45:59,650 --> 00:46:01,830 So I start out thinking of them as hc. 741 00:46:01,830 --> 00:46:05,070 That's what I do in step one. 742 00:46:05,070 --> 00:46:06,390 I can do a field redefinition. 743 00:46:06,390 --> 00:46:08,110 It doesn't matter. 744 00:46:08,110 --> 00:46:10,890 But now, I want to match from SCETI onto II. 745 00:46:10,890 --> 00:46:14,070 So therefore, what I do is I take my full theory, which 746 00:46:14,070 --> 00:46:17,430 is SCETI, and I evaluate it with fields 747 00:46:17,430 --> 00:46:20,010 that don't have the right power counting for that theory. 748 00:46:20,010 --> 00:46:23,460 And then I do a Taylor-series expansion of all the diagrams, 749 00:46:23,460 --> 00:46:24,960 so that's this. 750 00:46:24,960 --> 00:46:27,720 I change my external fields, and I call them c instead of hc. 751 00:46:27,720 --> 00:46:29,762 AUDIENCE: Oh, because you're doing the same thing 752 00:46:29,762 --> 00:46:30,445 with ultrasoft. 753 00:46:30,445 --> 00:46:30,900 So ultrasoft, it really means-- 754 00:46:30,900 --> 00:46:32,567 IAIN STEWART: And now I make these soft, 755 00:46:32,567 --> 00:46:36,540 but soft and ultrasoft, that's just really a renaming. 756 00:46:36,540 --> 00:46:39,480 AUDIENCE: Ultrasoft in the SCETI [INAUDIBLE].. 757 00:46:39,480 --> 00:46:41,730 IAIN STEWART: Ultrasoft is equal soft. 758 00:46:41,730 --> 00:46:42,810 Sorry. 759 00:46:42,810 --> 00:46:44,340 Maybe this makes it clearer. 760 00:46:44,340 --> 00:46:46,650 Ultrasoft and soft just two different names 761 00:46:46,650 --> 00:46:48,150 for the same thing. 762 00:46:48,150 --> 00:46:51,210 But hc and c aren't, because hc really 763 00:46:51,210 --> 00:46:53,820 lived on the upper hyperbola. 764 00:46:53,820 --> 00:46:56,940 I put the external particles onto the lower hyperbola, 765 00:46:56,940 --> 00:46:59,210 but I'm frozen in here with one particle that 766 00:46:59,210 --> 00:47:00,960 stays in the upper hyperbola and therefore 767 00:47:00,960 --> 00:47:02,002 has to be integrated out. 768 00:47:07,700 --> 00:47:09,950 OK. 769 00:47:09,950 --> 00:47:21,150 So the kind of thing that you would get by doing that 770 00:47:21,150 --> 00:47:27,330 is that you would get convolutions 771 00:47:27,330 --> 00:47:29,342 with some Wilson coefficient. 772 00:47:36,100 --> 00:47:39,050 In this case, they were-- 773 00:47:39,050 --> 00:47:43,770 well, let me just say, it depends on some P minuses. 774 00:47:47,860 --> 00:47:52,760 This is P1, this is P2, and then in my example over there, 775 00:47:52,760 --> 00:47:56,180 I guess I had some heavy quarks. 776 00:48:05,320 --> 00:48:07,301 OK, and there's some Dirac structures. 777 00:48:09,782 --> 00:48:11,240 So it would be something like this, 778 00:48:11,240 --> 00:48:14,410 where you'd get some function, whatever it is, 779 00:48:14,410 --> 00:48:16,120 that's coming exactly from integrating 780 00:48:16,120 --> 00:48:17,090 all the purple stuff. 781 00:48:17,090 --> 00:48:19,630 So this is the Wilson coefficient, 782 00:48:19,630 --> 00:48:21,640 and usually, you would call these things 783 00:48:21,640 --> 00:48:26,380 jet functions, because they look like jet functions. 784 00:48:29,740 --> 00:48:48,980 So this comes from the SCETI modes that got integrated out. 785 00:48:48,980 --> 00:48:53,168 So the difference between this and the SCETI matching, where 786 00:48:53,168 --> 00:48:54,710 you're integrating out the hard scale 787 00:48:54,710 --> 00:48:56,460 is you're already getting sensitivity here 788 00:48:56,460 --> 00:48:58,700 to the plus momentum of the soft. 789 00:48:58,700 --> 00:49:00,261 That's one difference. 790 00:49:04,190 --> 00:49:05,088 OK. 791 00:49:05,088 --> 00:49:07,380 So this is a general procedure for constructing SCETII. 792 00:49:12,830 --> 00:49:14,450 Let's see how much I want to say. 793 00:49:14,450 --> 00:49:18,260 So let's say a few words about power counting. 794 00:49:18,260 --> 00:49:23,690 So if you take some T-product of terms in SCETI, 795 00:49:23,690 --> 00:49:28,160 that will scale like some power of lambda to the 2K OK? 796 00:49:28,160 --> 00:49:32,690 So in SCETI, you have to be a little bit careful about power 797 00:49:32,690 --> 00:49:34,620 counting when you do this procedure. 798 00:49:34,620 --> 00:49:37,640 So in SCETI, you could just assign some power. 799 00:49:37,640 --> 00:49:40,340 Let's call it lambda to the 2K. 800 00:49:40,340 --> 00:49:45,200 And when you do a matching onto an operator in SCETII, what 801 00:49:45,200 --> 00:49:46,940 you're generically going to find is 802 00:49:46,940 --> 00:49:51,920 that there'll be a relation between the power counting here 803 00:49:51,920 --> 00:49:54,830 and the power counting there which is good, 804 00:49:54,830 --> 00:49:58,790 but there's also one thing to be aware of. 805 00:50:01,860 --> 00:50:04,340 So if I define eta to be the power counting 806 00:50:04,340 --> 00:50:05,570 parameter for SCETII-- 807 00:50:05,570 --> 00:50:06,980 just to give it a different name, 808 00:50:06,980 --> 00:50:09,170 so I can talk about a correspondence-- 809 00:50:09,170 --> 00:50:12,350 then basically, eta is like lambda squared. 810 00:50:12,350 --> 00:50:20,350 And that, well, that just follows from this formula, 811 00:50:20,350 --> 00:50:23,170 that if you want to identify lambda squared 812 00:50:23,170 --> 00:50:25,360 is lambda over q for the collinear modes, 813 00:50:25,360 --> 00:50:28,330 then this would be the right way of doing it. 814 00:50:28,330 --> 00:50:30,880 Little lambda would be square root of lambda over q. 815 00:50:30,880 --> 00:50:32,490 But in SCETI for the collinear modes, 816 00:50:32,490 --> 00:50:34,240 you'd say there's some parameter eta which 817 00:50:34,240 --> 00:50:36,590 is just lambda over q. 818 00:50:36,590 --> 00:50:37,090 OK? 819 00:50:37,090 --> 00:50:39,600 So basically, it looks like you just take 2K and go over 820 00:50:39,600 --> 00:50:42,100 to k, because you just changed the definition of what you're 821 00:50:42,100 --> 00:50:45,940 calling the parameter, but there's also this plus E. 822 00:50:45,940 --> 00:50:49,000 That means that you can get additional suppression, 823 00:50:49,000 --> 00:50:51,065 and that plus E comes about from the fact 824 00:50:51,065 --> 00:50:52,690 that you also change the power counting 825 00:50:52,690 --> 00:50:56,302 for your external fields, when you do this procedure. 826 00:50:56,302 --> 00:50:58,510 So something that was scaling like lambda in the hard 827 00:50:58,510 --> 00:51:02,560 collinear theory might become lambda squared-- i.e. eta-- 828 00:51:02,560 --> 00:51:04,045 in the SCETII theory. 829 00:51:08,656 --> 00:51:12,560 So this E which is greater than 0 830 00:51:12,560 --> 00:51:22,510 comes from changing the power counting of external fields 831 00:51:22,510 --> 00:51:23,290 in the matching. 832 00:51:40,010 --> 00:51:42,050 One example is just having an external collinear 833 00:51:42,050 --> 00:51:44,570 quark or external collinear perp gluon, 834 00:51:44,570 --> 00:51:47,720 for example, where you would have c of order lambda 835 00:51:47,720 --> 00:51:50,150 in SCETII, and it becomes of order eta in SCETII 836 00:51:50,150 --> 00:51:53,120 which is lambda squared. 837 00:51:53,120 --> 00:51:57,350 Those are extra factors of the power counting parameter. 838 00:51:57,350 --> 00:51:58,070 OK. 839 00:51:58,070 --> 00:52:03,210 So what do we learn, if we put all these things together? 840 00:52:03,210 --> 00:52:07,850 Well, one thing since soft and the collinear fields 841 00:52:07,850 --> 00:52:10,422 don't talk to each other, when you write down 842 00:52:10,422 --> 00:52:12,380 Lagrangians for them, they're totally decoupled 843 00:52:12,380 --> 00:52:14,130 right from the start. 844 00:52:14,130 --> 00:52:20,630 And so there's actually no mixed soft collinear Lagrangian 845 00:52:20,630 --> 00:52:21,380 at leading order. 846 00:52:31,020 --> 00:52:33,150 And so all the meat of this theory 847 00:52:33,150 --> 00:52:34,800 is coming about when you integrate out 848 00:52:34,800 --> 00:52:36,467 these off-shell modes, and you construct 849 00:52:36,467 --> 00:52:39,250 the external operators. 850 00:52:39,250 --> 00:52:39,750 OK. 851 00:52:44,570 --> 00:52:46,760 I'll say little bit more about power counting. 852 00:52:46,760 --> 00:52:48,635 So there's no mixed soft collinear Lagrangian 853 00:52:48,635 --> 00:52:53,115 at leading order, and you can get mixed things 854 00:52:53,115 --> 00:52:53,990 at sub-leading order. 855 00:53:01,180 --> 00:53:04,827 So in that way, it's like SCETI after the field redefinition. 856 00:53:10,342 --> 00:53:11,550 But here, there is no analog. 857 00:53:11,550 --> 00:53:13,740 If you just want to go directly to SCETII, 858 00:53:13,740 --> 00:53:16,217 there wouldn't be an analog of the field redefinition. 859 00:53:20,200 --> 00:53:22,360 All the Wilson lines are coming from integrating 860 00:53:22,360 --> 00:53:23,778 off-shell particles. 861 00:53:33,050 --> 00:53:33,680 OK. 862 00:53:33,680 --> 00:53:37,860 So we're going to speed up a little. 863 00:53:37,860 --> 00:53:40,310 I'm going to skip over one thing, which 864 00:53:40,310 --> 00:53:41,490 I don't think I need. 865 00:53:41,490 --> 00:53:43,970 So you can write down power counting formulas, 866 00:53:43,970 --> 00:53:45,720 like we did in chiral perturbation theory. 867 00:53:45,720 --> 00:53:48,020 Maybe I'll just write those formulas down. 868 00:53:48,020 --> 00:53:54,020 So in general, in chiral perturbation theory, 869 00:53:54,020 --> 00:53:55,395 we had a formula that said, if we 870 00:53:55,395 --> 00:53:57,853 want to figure out the power counting of any given diagram, 871 00:53:57,853 --> 00:53:59,540 we can have a counting for that diagram, 872 00:53:59,540 --> 00:54:01,830 and we know what size it is. 873 00:54:01,830 --> 00:54:04,460 And there's analogous formulas like that 874 00:54:04,460 --> 00:54:07,730 in both SCETI and SCETII. 875 00:54:07,730 --> 00:54:12,350 So in SCETI it's pretty simple. 876 00:54:12,350 --> 00:54:13,850 You say you have something that's 877 00:54:13,850 --> 00:54:16,010 order lambda to the delta. 878 00:54:16,010 --> 00:54:19,115 Then delta, the formula for it follows. 879 00:54:23,183 --> 00:54:25,100 This is very analogous to what we talked about 880 00:54:25,100 --> 00:54:26,392 for chiral perturbation theory. 881 00:54:32,930 --> 00:54:40,470 So these guys here are vertices that are purely ultrasoft, 882 00:54:40,470 --> 00:54:42,350 and if you have purely ultrasoft vertices, 883 00:54:42,350 --> 00:54:44,360 you're subtracting 8 because the measure 884 00:54:44,360 --> 00:54:47,100 of the ultrasoft particles is 8. 885 00:54:47,100 --> 00:54:48,350 I'm not deriving this for you. 886 00:54:48,350 --> 00:54:52,050 I could, but I think it's fairly intuitive. 887 00:54:52,050 --> 00:54:56,180 So if you think about the lowest order Lagrangian, 888 00:54:56,180 --> 00:55:00,320 say psi bar ID slash psi for ultrasofts. 889 00:55:00,320 --> 00:55:03,900 This would already be something that's lambda to the 8, 890 00:55:03,900 --> 00:55:08,270 and this is then leading order, so you subtract 8. 891 00:55:08,270 --> 00:55:12,410 And this guy here is the rest, anything mixed 892 00:55:12,410 --> 00:55:15,740 or purely collinear. 893 00:55:15,740 --> 00:55:17,540 And then for that guy, you subtract 4, 894 00:55:17,540 --> 00:55:19,623 because as soon as you have at least one collinear 895 00:55:19,623 --> 00:55:22,430 particle, then you have to use the collinear measure. 896 00:55:22,430 --> 00:55:26,240 And so here, some operator like c bar, 897 00:55:26,240 --> 00:55:28,970 and dot dc was order lambda to the 4. 898 00:55:28,970 --> 00:55:31,672 So you subtract 4 and that's leading order. 899 00:55:31,672 --> 00:55:33,380 And so what this formula allows you to do 900 00:55:33,380 --> 00:55:36,260 is really just you do power counting entirely 901 00:55:36,260 --> 00:55:37,380 in terms of vertices. 902 00:55:37,380 --> 00:55:40,400 You never have to power count measures for loops or power 903 00:55:40,400 --> 00:55:41,870 count propagators. 904 00:55:41,870 --> 00:55:43,920 You just count vertices. 905 00:55:43,920 --> 00:55:46,790 So if you want to know how big a time-order product is, and it's 906 00:55:46,790 --> 00:55:54,830 a time-order product of an L3 with an L2 with this kind 907 00:55:54,830 --> 00:55:59,180 of setup, this will be basically you count this as lambda phi. 908 00:56:04,580 --> 00:56:07,910 And you could even get the absolute size right, 909 00:56:07,910 --> 00:56:10,990 taking into account the scaling of external particles, 910 00:56:10,990 --> 00:56:13,040 and that's what this constant 3 factor does. 911 00:56:16,204 --> 00:56:21,350 And U is equal to 1 only for pure ultrasoft. 912 00:56:26,650 --> 00:56:30,095 Otherwise, it's 0. 913 00:56:30,095 --> 00:56:31,070 OK? 914 00:56:31,070 --> 00:56:33,230 So there's formulas like this, and there's 915 00:56:33,230 --> 00:56:36,840 an analogous one in SCETII which is a little more complicated. 916 00:56:36,840 --> 00:56:39,440 But it allows you just to have a power counting where you can 917 00:56:39,440 --> 00:56:46,490 just power count Lagrangians or operators 01 with L1, 918 00:56:46,490 --> 00:56:50,420 and that's lambda squared without having to worry about 919 00:56:50,420 --> 00:56:53,270 what propagators do I have in this time-order product? 920 00:56:53,270 --> 00:56:54,807 What loops do I have? 921 00:56:54,807 --> 00:56:56,390 You never have to ask those questions. 922 00:56:56,390 --> 00:56:58,098 You just have power counting at vertices, 923 00:56:58,098 --> 00:57:00,508 and that makes things quite easy. 924 00:57:00,508 --> 00:57:02,300 AUDIENCE: When you write these expressions, 925 00:57:02,300 --> 00:57:05,363 one insertion of Lagrangian corresponds to a single vertex. 926 00:57:05,363 --> 00:57:06,530 [INAUDIBLE] that Lagrangian. 927 00:57:06,530 --> 00:57:07,728 IAIN STEWART: Yeah. 928 00:57:07,728 --> 00:57:08,270 That's right. 929 00:57:10,575 --> 00:57:11,075 OK. 930 00:57:17,780 --> 00:57:18,290 All right. 931 00:57:22,722 --> 00:57:24,680 So I want to do a couple of examples in SCETII. 932 00:57:33,950 --> 00:57:36,900 So let's do an example. 933 00:57:36,900 --> 00:57:40,760 Let's do an example which is in some ways simple, 934 00:57:40,760 --> 00:57:48,710 and it's an exclusive analog of our DIS example, 935 00:57:48,710 --> 00:57:51,980 and it's called the photon pion form factor. 936 00:57:51,980 --> 00:57:54,290 So it's exclusive. 937 00:57:54,290 --> 00:57:55,500 It's clearly exclusive. 938 00:57:55,500 --> 00:57:58,970 There's a pion, and really, all it's going to come in here 939 00:57:58,970 --> 00:58:01,730 is hard collinear factorization. 940 00:58:01,730 --> 00:58:05,360 So it's simple in the way that it actually is not really 941 00:58:05,360 --> 00:58:08,720 exploiting the full complications of SCETII. 942 00:58:08,720 --> 00:58:10,220 It's really just another example, 943 00:58:10,220 --> 00:58:12,470 where we have hard modes factoring from cleaner modes. 944 00:58:12,470 --> 00:58:14,060 But they're clearly SCETII modes, 945 00:58:14,060 --> 00:58:17,840 because they're going to be modes for this pion. 946 00:58:17,840 --> 00:58:20,360 So we're going to use again for this calculation 947 00:58:20,360 --> 00:58:24,560 a bright frame, which you'll see, I guess, 948 00:58:24,560 --> 00:58:26,040 when I write down some momenta. 949 00:58:26,040 --> 00:58:30,980 So what would we write down for this process in QCD, first 950 00:58:30,980 --> 00:58:32,130 of all? 951 00:58:32,130 --> 00:58:33,890 So you'd say, I have a pi 0. 952 00:58:33,890 --> 00:58:37,850 It has some momentum P pi. 953 00:58:37,850 --> 00:58:40,460 I have some current which I can take at 0, 954 00:58:40,460 --> 00:58:42,533 and I want to make a transition from the pi 0 955 00:58:42,533 --> 00:58:43,325 to a single photon. 956 00:59:01,662 --> 00:59:03,120 So you could write that as follows. 957 00:59:08,360 --> 00:59:26,510 I can always replace the photon by current, 958 00:59:26,510 --> 00:59:28,508 and then I can parameterized that matrix on it 959 00:59:28,508 --> 00:59:29,300 like a form factor. 960 00:59:41,610 --> 00:59:46,190 And if we go through things like parity, charge conjugation, 961 00:59:46,190 --> 00:59:50,060 stuff like that, time reversal, we 962 00:59:50,060 --> 00:59:52,420 find out that there's one real form factor, 963 00:59:52,420 --> 00:59:55,490 and there's an epsilon symbol in this guy. 964 00:59:55,490 --> 00:59:57,650 It has to be linear in the polarization vector. 965 00:59:57,650 --> 01:00:00,260 That was already true here, and there's 966 01:00:00,260 --> 01:00:02,090 one way of getting the indices to work out 967 01:00:02,090 --> 01:00:03,950 which is with this epsilon. 968 01:00:03,950 --> 01:00:06,410 So what you're really after in this process 969 01:00:06,410 --> 01:00:09,370 would be some understanding of this form factor. 970 01:00:09,370 --> 01:00:12,002 That's the one piece of non-perturbative information. 971 01:00:12,002 --> 01:00:13,460 Everything else, Lorentz invariance 972 01:00:13,460 --> 01:00:15,713 is enough to tell you about. 973 01:00:15,713 --> 01:00:17,630 So we'd like to see if there's a factorization 974 01:00:17,630 --> 01:00:23,630 theorem for that form factor, if we take the limit where 975 01:00:23,630 --> 01:00:24,470 q squared is large. 976 01:00:35,450 --> 01:00:37,100 So we have an electromagnetic current. 977 01:00:43,457 --> 01:00:47,535 You can think about it is up and down quarks for the pion. 978 01:00:53,070 --> 01:00:56,280 So there's some matrix in the 2 by 2 space 979 01:00:56,280 --> 01:01:02,940 which is the charge, which you can write as 2/3 minus 1/3, 980 01:01:02,940 --> 01:01:03,840 like that. 981 01:01:03,840 --> 01:01:06,330 Or if you wanted to write it in terms of polymatrices, 982 01:01:06,330 --> 01:01:12,660 you could write it as identity over 6 plus a tau 3 983 01:01:12,660 --> 01:01:17,208 for isospin polymatrix over 2. 984 01:01:17,208 --> 01:01:21,600 So there's some charge matrix that's going to show up. 985 01:01:21,600 --> 01:01:24,660 What happens if we expand q squared much greater 986 01:01:24,660 --> 01:01:27,060 than lambda QCD squared? 987 01:01:27,060 --> 01:01:29,712 In this formula over here, this form factor 988 01:01:29,712 --> 01:01:31,920 knows about lambda QCD, and it knows about q squared, 989 01:01:31,920 --> 01:01:33,600 and we haven't expanded. 990 01:01:33,600 --> 01:01:36,420 So what happens if we do expand? 991 01:01:36,420 --> 01:01:39,490 Can we simplify that form factor? 992 01:01:39,490 --> 01:01:41,690 And it does. 993 01:01:41,690 --> 01:01:45,500 So I'm going to do this in the bright frame again, 994 01:01:45,500 --> 01:01:50,020 and that effectively means I'm taking the momentum that 995 01:01:50,020 --> 01:01:55,990 corresponds to the off-shell photon as the same form 996 01:01:55,990 --> 01:01:58,010 as we did before for our calculation 997 01:01:58,010 --> 01:02:05,662 in DIS, purely in the z coordinate. 998 01:02:08,620 --> 01:02:11,470 The momentum of the photon, it's an on-shell thing, 999 01:02:11,470 --> 01:02:14,102 so I can just write it as E the photons say 1000 01:02:14,102 --> 01:02:15,185 times a light-like vector. 1001 01:02:15,185 --> 01:02:19,370 Then, P gamma squared will be 0, as we want it to be. 1002 01:02:19,370 --> 01:02:24,220 And if I pick this kinematics, then P pi is just P plus P 1003 01:02:24,220 --> 01:02:37,960 gamma, so that would be a P pi. 1004 01:02:44,650 --> 01:02:45,150 OK? 1005 01:02:45,150 --> 01:02:49,750 So in Feynman diagrams, what am I talking about? 1006 01:02:49,750 --> 01:02:53,850 I'm talking about making a transition 1007 01:02:53,850 --> 01:02:56,970 with an off-shell photon, through a diagram 1008 01:02:56,970 --> 01:03:01,470 like this one, to a pi 0 or plus the cross graph. 1009 01:03:06,230 --> 01:03:09,980 And with this setup with this kinematics, 1010 01:03:09,980 --> 01:03:13,160 this intermediate line here, if we go through the scaling, 1011 01:03:13,160 --> 01:03:14,730 it's going to be hard. 1012 01:03:14,730 --> 01:03:18,560 So these guys with this kinematics, 1013 01:03:18,560 --> 01:03:21,890 we make the pion collinear. 1014 01:03:21,890 --> 01:03:27,170 In order to see that, you can impose P pi 1015 01:03:27,170 --> 01:03:29,780 squared equals M pi squared. 1016 01:03:29,780 --> 01:03:30,950 Right? 1017 01:03:30,950 --> 01:03:33,530 This is never going to be made small, 1018 01:03:33,530 --> 01:03:37,460 but E is going to have to be tuned to be basically q over 2. 1019 01:03:37,460 --> 01:03:39,350 So this thing is going to become, 1020 01:03:39,350 --> 01:03:42,050 if you impose this condition, something 1021 01:03:42,050 --> 01:03:47,990 like M pi squared over 2 q. 1022 01:03:47,990 --> 01:03:52,130 Then, you find out that the pion is collinear. 1023 01:03:52,130 --> 01:03:54,290 So the pion is collinear. 1024 01:03:54,290 --> 01:03:55,850 The photon is a photon. 1025 01:03:55,850 --> 01:03:59,432 This line here is hard, and so we 1026 01:03:59,432 --> 01:04:00,890 want to integrate out the hard line 1027 01:04:00,890 --> 01:04:05,540 and match this guy in the effective theory 1028 01:04:05,540 --> 01:04:09,405 onto some effective theory operators that 1029 01:04:09,405 --> 01:04:10,280 would look like this. 1030 01:04:21,016 --> 01:04:22,170 This guy is pink. 1031 01:04:27,790 --> 01:04:28,290 OK. 1032 01:04:28,290 --> 01:04:31,230 So we just have to write down what type of effective theory 1033 01:04:31,230 --> 01:04:34,410 operators that could be, and again, it's an effective theory 1034 01:04:34,410 --> 01:04:36,230 operator of two quark fields. 1035 01:04:36,230 --> 01:04:37,980 So it's very much like what we did before. 1036 01:04:47,320 --> 01:04:49,720 I'm not going to be going through all the indices 1037 01:04:49,720 --> 01:04:51,220 that I have to go through in order 1038 01:04:51,220 --> 01:04:54,188 to keep track of charge conjugation and all 1039 01:04:54,188 --> 01:04:54,730 those things. 1040 01:04:54,730 --> 01:04:56,950 I'll just write down the right answers for that part. 1041 01:05:15,680 --> 01:05:19,090 So I could write like this. 1042 01:05:19,090 --> 01:05:21,780 So after doing hard collinear factorization, 1043 01:05:21,780 --> 01:05:24,960 there's again operators which is two quark's. 1044 01:05:24,960 --> 01:05:27,180 Some hard Wilson coefficient which 1045 01:05:27,180 --> 01:05:31,200 is this pink thing, integrating out the pink line that 1046 01:05:31,200 --> 01:05:33,330 sits in the middle. 1047 01:05:33,330 --> 01:05:37,110 This obey current conservation. 1048 01:05:37,110 --> 01:05:40,440 Dimensional analysis fixes this 1 over q. 1049 01:05:40,440 --> 01:05:44,250 Charge conjugation, actually, also provides 1050 01:05:44,250 --> 01:05:47,070 constraints on the Wilson coefficients, 1051 01:05:47,070 --> 01:05:49,770 just like it did in DIS. 1052 01:05:49,770 --> 01:06:05,060 So charge conjugation tells us that there's 1053 01:06:05,060 --> 01:06:06,605 a relation between flipping signs. 1054 01:06:10,100 --> 01:06:12,230 But we just impose on this operator 1055 01:06:12,230 --> 01:06:14,480 all the symmetries and things that we can think of 1056 01:06:14,480 --> 01:06:16,160 and see what it says about the operator. 1057 01:06:16,160 --> 01:06:18,243 You can think about just writing the operator down 1058 01:06:18,243 --> 01:06:21,320 based on this picture without ever calculating any diagrams, 1059 01:06:21,320 --> 01:06:23,790 just knowing what you're after. 1060 01:06:23,790 --> 01:06:25,640 And then imposing on it all the symmetries 1061 01:06:25,640 --> 01:06:27,330 that should be conserved. 1062 01:06:29,910 --> 01:06:31,393 So one is dimensional analysis. 1063 01:06:31,393 --> 01:06:32,060 That gave the q. 1064 01:06:32,060 --> 01:06:34,970 Current conservation, that's partly responsible 1065 01:06:34,970 --> 01:06:37,805 for the epsilon which is also like a parody. 1066 01:06:44,240 --> 01:06:50,340 Go through flavor and spin, and you 1067 01:06:50,340 --> 01:06:55,260 find that this has this structure, 1068 01:06:55,260 --> 01:06:57,360 and there's some constant that I threw in here. 1069 01:07:04,660 --> 01:07:06,310 And you actually know from the flavor 1070 01:07:06,310 --> 01:07:08,320 that it's got two photons. 1071 01:07:08,320 --> 01:07:09,670 Right? 1072 01:07:09,670 --> 01:07:11,770 And I'm not ever adding any more photons, 1073 01:07:11,770 --> 01:07:15,160 so that means there's two factors of q hat. 1074 01:07:15,160 --> 01:07:17,950 And so there's a q hat squared, which 1075 01:07:17,950 --> 01:07:20,440 I can stick inside this gamma. 1076 01:07:20,440 --> 01:07:25,738 And it has to be a color singlet, 1077 01:07:25,738 --> 01:07:28,030 because it's going to have to have a non-trivial matrix 1078 01:07:28,030 --> 01:07:30,400 element with pi 0 which is a color singlet. 1079 01:07:34,540 --> 01:07:35,890 And since it has to be-- 1080 01:07:35,890 --> 01:07:41,650 so there's no TA inside the gamma that's what I mean. 1081 01:07:41,650 --> 01:07:44,860 There'd be nothing for the index A to contract with. 1082 01:07:44,860 --> 01:07:46,390 And since it's a color singlet, that 1083 01:07:46,390 --> 01:07:49,180 means if you were to think about soft interactions here, 1084 01:07:49,180 --> 01:07:50,560 they would just cancel. 1085 01:07:50,560 --> 01:07:51,590 Right? 1086 01:07:51,590 --> 01:07:54,085 If I were to think about putting in soft interactions 1087 01:07:54,085 --> 01:07:55,960 and integrating them out, I'd get an S dagger 1088 01:07:55,960 --> 01:07:58,510 S that would cancel out. 1089 01:07:58,510 --> 01:08:00,760 So that's the sense in which this is a simple example. 1090 01:08:15,910 --> 01:08:16,569 OK. 1091 01:08:16,569 --> 01:08:18,760 So then, we would have a formula where 1092 01:08:18,760 --> 01:08:20,859 we could equate the effective theory 1093 01:08:20,859 --> 01:08:24,810 result with the full theory result. 1094 01:08:24,810 --> 01:08:30,790 So we can equate matrix elements, 1095 01:08:30,790 --> 01:08:33,370 and this is one way in which this example is 1096 01:08:33,370 --> 01:08:35,569 different than dependent elastic scattering. 1097 01:08:35,569 --> 01:08:40,600 We're really doing a matching at the amplitude level. 1098 01:08:40,600 --> 01:08:42,790 Remember, the form factor was a parameterization 1099 01:08:42,790 --> 01:08:48,189 of the amplitude, and if we do that-- 1100 01:08:58,810 --> 01:09:02,609 If you like, if you think about the two currents that we had, 1101 01:09:02,609 --> 01:09:04,960 and we've integrated them out. 1102 01:09:04,960 --> 01:09:07,210 Now, think about just this operator here. 1103 01:09:07,210 --> 01:09:09,160 The matrix element pi 0 to vacuum 1104 01:09:09,160 --> 01:09:12,819 that we had to finding this thing which had two currents 1105 01:09:12,819 --> 01:09:16,158 just becomes a matrix element of that operator. 1106 01:09:20,550 --> 01:09:24,240 Again, like our DIS example, we form the sum and difference 1107 01:09:24,240 --> 01:09:25,410 at the momenta. 1108 01:09:25,410 --> 01:09:28,170 We can think of forming P bar plus which 1109 01:09:28,170 --> 01:09:35,640 is P dagger plus or minus P. And one of these guys just 1110 01:09:35,640 --> 01:09:39,944 gives the total momentum, just the minus 1111 01:09:39,944 --> 01:09:43,319 with our sign conventions. 1112 01:09:43,319 --> 01:09:49,270 So that gets fixed which, in this case, 1113 01:09:49,270 --> 01:09:58,160 it just gets fixed to the pion momentum which is q. 1114 01:10:03,490 --> 01:10:08,350 So if you like, you've inserted the operator here. 1115 01:10:08,350 --> 01:10:11,230 You have some collinear lines, and then you 1116 01:10:11,230 --> 01:10:14,860 have a matrix element of the pi 0, and then you have gluons. 1117 01:10:20,670 --> 01:10:22,592 But you know that whatever momentum comes in 1118 01:10:22,592 --> 01:10:24,050 has to be the momentum of the pion, 1119 01:10:24,050 --> 01:10:28,180 and that's the one momentum constraint. 1120 01:10:28,180 --> 01:10:30,390 So that means that the answer just 1121 01:10:30,390 --> 01:10:36,030 involves one convolution again which is 1122 01:10:36,030 --> 01:10:37,730 the one that's unconstrained. 1123 01:10:41,670 --> 01:10:43,170 So there's a Wilson coefficient that 1124 01:10:43,170 --> 01:10:45,660 depends on the other unconstrained guy which 1125 01:10:45,660 --> 01:10:51,480 was P plus, and then there's a matrix element, 1126 01:10:51,480 --> 01:10:54,870 where I can write in a delta function with that P 1127 01:10:54,870 --> 01:10:56,970 plus, the usual kind of way that we've done. 1128 01:11:01,290 --> 01:11:03,480 But here, it's a little different matrix element 1129 01:11:03,480 --> 01:11:05,820 than the one we saw on DIS, because it's not forward. 1130 01:11:05,820 --> 01:11:07,745 It's vacuum to pi 0. 1131 01:11:19,722 --> 01:11:21,180 But it's actually the same operator 1132 01:11:21,180 --> 01:11:24,960 that we were talking about in DIS. 1133 01:11:24,960 --> 01:11:28,470 It's just a bilinear operator with two collinear quark 1134 01:11:28,470 --> 01:11:32,070 fields, two collinear quarks in dress with Wilson lines. 1135 01:11:38,270 --> 01:11:41,270 So one can go through a similar type of matrix element 1136 01:11:41,270 --> 01:11:43,400 analysis for this operator, and it 1137 01:11:43,400 --> 01:11:46,610 gives something that's called the light cone function. 1138 01:11:46,610 --> 01:11:48,250 So let me define that for you. 1139 01:12:04,690 --> 01:12:10,720 This matrix element can be written 1140 01:12:10,720 --> 01:12:14,560 in terms of an object that has a dimensionless variable. 1141 01:12:25,090 --> 01:12:28,030 It's an analog of the part-time distribution function 1142 01:12:28,030 --> 01:12:29,890 but for this matrix element that we're 1143 01:12:29,890 --> 01:12:32,316 dealing with here which is vacuum to pion. 1144 01:12:35,172 --> 01:12:38,078 So this has some similarity to the formula we had in DIS. 1145 01:12:38,078 --> 01:12:39,370 There's a delta function there. 1146 01:12:39,370 --> 01:12:41,360 There's a delta function here. 1147 01:12:41,360 --> 01:12:43,810 There's a dimensionless variable z, dimensionless 1148 01:12:43,810 --> 01:12:47,110 variable z there, and this is a non-perturbative function. 1149 01:12:57,160 --> 01:12:59,510 So this is what's called a light cone distribution 1150 01:12:59,510 --> 01:13:00,640 function for the pion. 1151 01:13:05,870 --> 01:13:08,540 So generically, when you have an exclusive process, 1152 01:13:08,540 --> 01:13:12,350 and you're producing some hadron that's very energetic, 1153 01:13:12,350 --> 01:13:15,330 like a pion, this is the type of thing that's going to show up, 1154 01:13:15,330 --> 01:13:17,481 one of these light cone distribution functions. 1155 01:13:21,540 --> 01:13:23,980 This example of photon to pion is in some sense 1156 01:13:23,980 --> 01:13:28,600 a very, very simple, exclusive process, 1157 01:13:28,600 --> 01:13:31,640 the simplest one in some sense. 1158 01:13:31,640 --> 01:13:32,140 OK. 1159 01:13:32,140 --> 01:13:34,430 So we could take this formula, plug it back in there, 1160 01:13:34,430 --> 01:13:36,520 and then we'd have a factorization theorem. 1161 01:13:36,520 --> 01:13:39,290 And I think that you can imagine what that would look like. 1162 01:13:39,290 --> 01:13:41,290 I could write it, instead of an integral over W, 1163 01:13:41,290 --> 01:13:45,250 as an integral over z, and that would be the factorization 1164 01:13:45,250 --> 01:13:49,090 theorem involving this 5 pi. 1165 01:13:49,090 --> 01:13:52,282 AUDIENCE: [INAUDIBLE] 1166 01:13:53,278 --> 01:13:54,070 IAIN STEWART: Yeah. 1167 01:13:54,070 --> 01:13:54,640 AUDIENCE: [INAUDIBLE] 1168 01:13:54,640 --> 01:13:55,765 IAIN STEWART: That's right. 1169 01:13:55,765 --> 01:13:58,960 So you should think of the z as like-- 1170 01:13:58,960 --> 01:14:01,430 so the way to think about the z is as follows. 1171 01:14:01,430 --> 01:14:03,430 So think about like when you initially 1172 01:14:03,430 --> 01:14:07,780 produced these guys here, think about all the momentum 1173 01:14:07,780 --> 01:14:09,970 going this way. 1174 01:14:09,970 --> 01:14:13,120 When you initially produced them, 1175 01:14:13,120 --> 01:14:16,510 after you integrated out the hard interactions, you had z 1176 01:14:16,510 --> 01:14:19,540 and 1 minus z is the possible split of the quarks fields 1177 01:14:19,540 --> 01:14:21,070 in the operator. 1178 01:14:21,070 --> 01:14:23,110 So one of these guys carries z. 1179 01:14:23,110 --> 01:14:25,330 Effectively, what this 2z minus 1 is doing 1180 01:14:25,330 --> 01:14:27,455 is one of these guys is carrying z, and one of them 1181 01:14:27,455 --> 01:14:28,930 is carrying 1 minus z. 1182 01:14:28,930 --> 01:14:30,220 OK? 1183 01:14:30,220 --> 01:14:31,750 The sum of these is 1, and that's 1184 01:14:31,750 --> 01:14:34,420 the analog of the statement that the whole total momentum should 1185 01:14:34,420 --> 01:14:36,175 be the pi 0 momentum. 1186 01:14:36,175 --> 01:14:38,050 But you don't know how to split how much goes 1187 01:14:38,050 --> 01:14:40,240 into each one of those. 1188 01:14:40,240 --> 01:14:42,190 And what the wave function is, it's 1189 01:14:42,190 --> 01:14:45,460 all the linear interactions that subsequently 1190 01:14:45,460 --> 01:14:49,070 rearrange this thing before you annihilate it with the state. 1191 01:14:49,070 --> 01:14:52,627 So all these things are dressing up the pion state. 1192 01:14:52,627 --> 01:14:53,960 They're producing the pion pole. 1193 01:14:53,960 --> 01:14:56,950 So this is like 5 pi. 1194 01:14:56,950 --> 01:15:00,910 And so what you have is an operator that 1195 01:15:00,910 --> 01:15:03,495 depends on z, a wave function that depends on z. 1196 01:15:03,495 --> 01:15:06,100 So this is a Wilson coefficient that depends on z, 1197 01:15:06,100 --> 01:15:08,020 and your final factorization theorem 1198 01:15:08,020 --> 01:15:13,570 is exactly of that type, that you sort of-- 1199 01:15:17,410 --> 01:15:21,910 of an integral of z of c of z Which also 1200 01:15:21,910 --> 01:15:31,285 can depend on q and mu and then 5 pi of z and mu. 1201 01:15:31,285 --> 01:15:33,760 AUDIENCE: Is there a sense of the 5 pi z as universal? 1202 01:15:33,760 --> 01:15:33,940 IAIN STEWART: Yeah. 1203 01:15:33,940 --> 01:15:34,565 It's universal. 1204 01:15:34,565 --> 01:15:36,640 AUDIENCE: You can use it for other [INAUDIBLE]?? 1205 01:15:36,640 --> 01:15:37,750 IAIN STEWART: Absolutely. 1206 01:15:37,750 --> 01:15:40,400 Yeah. 1207 01:15:40,400 --> 01:15:42,590 So as long as you can factor it so 1208 01:15:42,590 --> 01:15:45,770 that it's these fields ans that pion, 1209 01:15:45,770 --> 01:15:48,200 then you have this matrix element, you get this guy. 1210 01:15:51,588 --> 01:15:53,630 Some people try to measure things about this guy, 1211 01:15:53,630 --> 01:15:56,630 it's moments and stuff. 1212 01:15:56,630 --> 01:15:57,130 All right. 1213 01:15:59,700 --> 01:16:03,180 One thing that happens here has to do with this integral over z 1214 01:16:03,180 --> 01:16:10,290 which is still in some sense an unsolved problem in SCET. 1215 01:16:10,290 --> 01:16:12,300 So I have to mention it. 1216 01:16:12,300 --> 01:16:14,820 So when you do this integral, you 1217 01:16:14,820 --> 01:16:17,380 could ask, what does the c of z look like? 1218 01:16:17,380 --> 01:16:21,780 And it turns out that c of z will have in it terms that 1219 01:16:21,780 --> 01:16:24,803 go like 1 over z. 1220 01:16:24,803 --> 01:16:26,220 And so you get integrals that look 1221 01:16:26,220 --> 01:16:30,960 like dz over z of 5 pi of z. 1222 01:16:30,960 --> 01:16:34,500 So at lowest order, this integral would show up. 1223 01:16:34,500 --> 01:16:36,595 And it turns out that, for this matrix element 1224 01:16:36,595 --> 01:16:38,470 here, you don't get anything worse than that. 1225 01:16:38,470 --> 01:16:40,470 You never get 1 over z squared. 1226 01:16:40,470 --> 01:16:44,760 And this integral here, because of properties of this 5 pi, 1227 01:16:44,760 --> 01:16:45,840 is finite. 1228 01:16:45,840 --> 01:16:47,970 There's no problems. 1229 01:16:47,970 --> 01:16:51,390 But there are examples known in the literature, where that's 1230 01:16:51,390 --> 01:16:54,743 not the case, where you actually get 1 over z squared, 1231 01:16:54,743 --> 01:16:56,160 and people have some understanding 1232 01:16:56,160 --> 01:16:59,640 of the physics that's happening in those cases. 1233 01:16:59,640 --> 01:17:01,380 But there's not a complete understanding 1234 01:17:01,380 --> 01:17:04,560 of how factorization works in those cases. 1235 01:17:04,560 --> 01:17:05,130 OK? 1236 01:17:05,130 --> 01:17:07,270 So that doesn't happen in this example, 1237 01:17:07,270 --> 01:17:10,210 but there are other examples of exclusive processes 1238 01:17:10,210 --> 01:17:11,710 that would lead to 1 over z squares, 1239 01:17:11,710 --> 01:17:13,730 and then this integral is not well-defined. 1240 01:17:21,730 --> 01:17:24,100 And people understand that there's 1241 01:17:24,100 --> 01:17:25,918 a cut-off that's coming in, and they 1242 01:17:25,918 --> 01:17:27,460 understand that that cut-off actually 1243 01:17:27,460 --> 01:17:29,080 has to do with some rapidity. 1244 01:17:29,080 --> 01:17:33,100 But how to explicitly write down an analog factorization 1245 01:17:33,100 --> 01:17:35,320 theorem that involves those cut-offs 1246 01:17:35,320 --> 01:17:38,950 and has renormalization group is an unsolved problem, 1247 01:17:38,950 --> 01:17:42,110 unsolved SCETII problem. 1248 01:17:42,110 --> 01:17:42,610 OK? 1249 01:17:42,610 --> 01:17:44,440 But for the example we did, everything's 1250 01:17:44,440 --> 01:17:48,140 kosher and beautiful. 1251 01:17:48,140 --> 01:17:49,130 All right. 1252 01:17:49,130 --> 01:17:52,550 So I think what I'll do next time, 1253 01:17:52,550 --> 01:17:54,290 since we're out of time I'm not going 1254 01:17:54,290 --> 01:17:56,570 to start my second example now. 1255 01:17:56,570 --> 01:17:58,010 So next time, we'll do an example 1256 01:17:58,010 --> 01:18:01,020 that does involve soft fields, both soft and collinear fields 1257 01:18:01,020 --> 01:18:01,520 in SCETII. 1258 01:18:04,600 --> 01:18:09,400 That's where we're going, but we'll leave that to next time.