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 --> 00:00:17,520 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,520 --> 00:00:18,370 ocw.mit.edu. 8 00:00:25,742 --> 00:00:27,450 IAIN STEWART: All right, so last time, we 9 00:00:27,450 --> 00:00:30,240 were talking about degrees of freedom in SCET. 10 00:00:30,240 --> 00:00:33,720 And I drew these two pictures and emphasized to you 11 00:00:33,720 --> 00:00:35,670 that the difference between these two pictures 12 00:00:35,670 --> 00:00:38,970 is really where the degrees of freedom live in momentum space. 13 00:00:38,970 --> 00:00:45,040 So this is the P minus P plus plane. 14 00:00:45,040 --> 00:00:47,640 And unlike most effective field theories, 15 00:00:47,640 --> 00:00:49,560 in this effective field theory we 16 00:00:49,560 --> 00:00:51,000 have degrees of freedom that have 17 00:00:51,000 --> 00:00:54,700 to be identified by two variables rather than just one. 18 00:00:54,700 --> 00:00:58,560 So we need two variables to say where collinears are 19 00:00:58,560 --> 00:01:01,560 or where softs are. 20 00:01:01,560 --> 00:01:03,600 And there was generically, I said, 21 00:01:03,600 --> 00:01:05,550 a dependence on the process. 22 00:01:05,550 --> 00:01:09,510 So SCET, or what degrees of freedom you need, 23 00:01:09,510 --> 00:01:12,120 depends what you're studying. 24 00:01:12,120 --> 00:01:14,220 And it's generically not the case 25 00:01:14,220 --> 00:01:15,660 that these two effective theories 26 00:01:15,660 --> 00:01:17,340 that I wrote down for you will describe 27 00:01:17,340 --> 00:01:19,940 every possible process. 28 00:01:19,940 --> 00:01:21,690 In fact, SCET, of course, won't describe 29 00:01:21,690 --> 00:01:22,750 every possible process. 30 00:01:22,750 --> 00:01:24,375 So what effective field theory you need 31 00:01:24,375 --> 00:01:26,310 depends on what you're looking at. 32 00:01:26,310 --> 00:01:30,360 It just turns out that these two sets of degrees of freedom 33 00:01:30,360 --> 00:01:31,860 that I wrote down to you do describe 34 00:01:31,860 --> 00:01:33,120 a lot of different processes. 35 00:01:33,120 --> 00:01:36,720 And that's why the language of SCETI and SCETII is useful, 36 00:01:36,720 --> 00:01:38,040 but there's no guarantees. 37 00:01:38,040 --> 00:01:40,880 You could run into a process that requires 38 00:01:40,880 --> 00:01:41,880 more degrees of freedom. 39 00:01:41,880 --> 00:01:44,160 And there are examples of processes 40 00:01:44,160 --> 00:01:47,070 that require more degrees of freedom than this, 41 00:01:47,070 --> 00:01:48,850 but this is still a useful starting point. 42 00:01:48,850 --> 00:01:51,060 And this is where we'll start. 43 00:01:56,690 --> 00:01:58,850 And then we were talking a little bit about fields 44 00:01:58,850 --> 00:01:59,750 last time. 45 00:01:59,750 --> 00:02:03,980 And one thing I derived for you was that the collinear 46 00:02:03,980 --> 00:02:07,415 field has a power counting that makes it scale like lambda. 47 00:02:11,620 --> 00:02:12,860 This is the collinear quark. 48 00:02:18,780 --> 00:02:21,330 And then at the end of lecture, we also derived a formula 49 00:02:21,330 --> 00:02:23,370 for the collinear gluons. 50 00:02:23,370 --> 00:02:28,000 So if we write down components of the collinear gluon plus, 51 00:02:28,000 --> 00:02:30,720 minus, and perp-- 52 00:02:30,720 --> 00:02:32,820 we remember what these notation means. 53 00:02:32,820 --> 00:02:38,970 This means n dot A. And this means n bar dot A. 54 00:02:38,970 --> 00:02:45,304 And we figured out that the scaling for these guys 55 00:02:45,304 --> 00:02:47,180 is lambda squared 1 lambda. 56 00:02:47,180 --> 00:02:49,550 And we said that that was nice because we could put them 57 00:02:49,550 --> 00:02:57,260 together with a derivative and have a covariant derivative, 58 00:02:57,260 --> 00:02:59,205 which I wrote in terms of momenta just 59 00:02:59,205 --> 00:03:00,080 to sort of emphasize. 60 00:03:07,610 --> 00:03:14,998 So, if you like, this is momentum space, 61 00:03:14,998 --> 00:03:16,790 that these guys had the same power counting 62 00:03:16,790 --> 00:03:19,093 in the different components. 63 00:03:19,093 --> 00:03:21,260 And so when we come and talk about gauge invariance, 64 00:03:21,260 --> 00:03:22,820 this fact that they have the same power counting 65 00:03:22,820 --> 00:03:25,237 in the different components will, of course, be important. 66 00:03:25,237 --> 00:03:29,720 Because what gauge invariance does is ties momenta to gluons. 67 00:03:29,720 --> 00:03:31,970 And if we didn't have the same power counting 68 00:03:31,970 --> 00:03:33,980 in the different components, then we 69 00:03:33,980 --> 00:03:36,860 would not have a gauge field corresponding 70 00:03:36,860 --> 00:03:39,710 to a certain momenta. 71 00:03:39,710 --> 00:03:41,702 OK, so another way of arguing that they should 72 00:03:41,702 --> 00:03:43,910 have the same power counting would have been directly 73 00:03:43,910 --> 00:03:47,430 from gauge invariance. 74 00:03:47,430 --> 00:03:51,430 All right, I'd like to continue there. 75 00:03:51,430 --> 00:03:53,190 The one thing that you'll note, which 76 00:03:53,190 --> 00:03:56,100 is unusual from an effective theory point of view, 77 00:03:56,100 --> 00:04:00,210 is there's a set of fields that have no suppression at all. 78 00:04:00,210 --> 00:04:03,180 These A n minus gluon fields just scale like one. 79 00:04:11,367 --> 00:04:13,200 So when you have dimensional power counting, 80 00:04:13,200 --> 00:04:14,492 all the fields have dimensions. 81 00:04:14,492 --> 00:04:15,900 You add more fields. 82 00:04:15,900 --> 00:04:17,070 You get higher dimensions. 83 00:04:17,070 --> 00:04:18,720 You get suppression. 84 00:04:18,720 --> 00:04:20,760 Here, with this lambda power counting, 85 00:04:20,760 --> 00:04:24,420 there's a component of the gauge field that has no suppression. 86 00:04:24,420 --> 00:04:27,420 We can add as many of them as we want. 87 00:04:27,420 --> 00:04:29,070 And that costs nothing. 88 00:04:42,910 --> 00:04:46,940 So let's do that and see what happens. 89 00:04:58,940 --> 00:05:03,820 So a priori, you might think that this is really bad 90 00:05:03,820 --> 00:05:06,400 because you could have an infinite number 91 00:05:06,400 --> 00:05:08,050 of different operators at lowest order 92 00:05:08,050 --> 00:05:11,770 just with a different number of these A n minus fields. 93 00:05:11,770 --> 00:05:15,895 And we'll see that actually things are not so bad, 94 00:05:15,895 --> 00:05:18,020 but I think it's best to do that by way of example. 95 00:05:22,790 --> 00:05:26,080 So we'll consider an example that just has one collinear 96 00:05:26,080 --> 00:05:31,360 particle in it for simplicity. 97 00:05:31,360 --> 00:05:33,940 And we can do that by having a heavy particle that 98 00:05:33,940 --> 00:05:37,300 carries a lot of energy decay to a light particle, 99 00:05:37,300 --> 00:05:40,360 like a B quark decaying to an up quark. 100 00:05:40,360 --> 00:05:47,650 So in QCD, our current would just be u bar gamma b. 101 00:05:47,650 --> 00:05:51,100 Gamma would be left-handed current. 102 00:05:57,530 --> 00:05:59,855 And we can consider the B quark to be in HQET. 103 00:06:11,060 --> 00:06:16,000 And we have an energetic u quark as what we want to consider. 104 00:06:16,000 --> 00:06:19,275 It's not the only possibility, but we 105 00:06:19,275 --> 00:06:21,400 could pump all the momentum out through the current 106 00:06:21,400 --> 00:06:23,700 into the leptons into the e nu. 107 00:06:23,700 --> 00:06:26,490 But let's pump energy into the up quark, 108 00:06:26,490 --> 00:06:27,510 so that it's energetic. 109 00:06:31,260 --> 00:06:33,234 So here's the current in QCD. 110 00:06:36,100 --> 00:06:38,690 b goes to u. 111 00:06:38,690 --> 00:06:41,540 And the notation that I'll use for this 112 00:06:41,540 --> 00:06:45,110 is a double line for the heavy quark 113 00:06:45,110 --> 00:06:47,270 and then a dashed line for the collinear quark. 114 00:06:50,260 --> 00:06:53,770 So this guy is going to be described by an hv field. 115 00:06:53,770 --> 00:06:56,320 This guy is going to be described by Cn field. 116 00:06:56,320 --> 00:06:58,420 This is production, so this will be a Cn bar. 117 00:07:12,170 --> 00:07:14,680 So I'll reserve this notation of dashed lines 118 00:07:14,680 --> 00:07:17,647 for collinear quark propagators. 119 00:07:20,330 --> 00:07:24,680 OK, so the simple minded thing is just 120 00:07:24,680 --> 00:07:27,047 to write down an effective theory current 121 00:07:27,047 --> 00:07:29,630 where I replace the full theory fields by the effective theory 122 00:07:29,630 --> 00:07:30,130 fields. 123 00:07:30,130 --> 00:07:35,840 I replace the up quark, an up quark that's a Cn collinear 124 00:07:35,840 --> 00:07:38,150 field in some direction. 125 00:07:38,150 --> 00:07:42,230 And I replace the heavy quark by an hv for the b quark. 126 00:07:46,250 --> 00:07:49,130 Then, effectively, that would be the tree level matching 127 00:07:49,130 --> 00:07:51,710 between this diagram and that diagram 128 00:07:51,710 --> 00:07:54,395 or the tree level matching from that particular diagram. 129 00:07:58,310 --> 00:08:01,490 But let's consider adding these A n minus gluons, 130 00:08:01,490 --> 00:08:06,290 since they were things that we could add to the current. 131 00:08:06,290 --> 00:08:08,270 So we can add any number of A n minuses here, 132 00:08:08,270 --> 00:08:10,687 and we'd have something that's the same order in the power 133 00:08:10,687 --> 00:08:12,080 counting. 134 00:08:12,080 --> 00:08:23,420 So let's consider another diagram in QCD, where I just 135 00:08:23,420 --> 00:08:24,867 attach an A n minus gluon. 136 00:08:24,867 --> 00:08:26,450 And I'll attach it to the heavy quark. 137 00:08:31,940 --> 00:08:34,490 So I have a different number of external fields, 138 00:08:34,490 --> 00:08:36,860 but this one costs me nothing when 139 00:08:36,860 --> 00:08:40,159 I go to the effective theory. 140 00:08:40,159 --> 00:08:43,789 Let's call this momentum here k. 141 00:08:43,789 --> 00:08:46,782 We'll call this guy here q. 142 00:08:46,782 --> 00:08:49,980 And this guy here we'll just say is MbV. 143 00:08:54,040 --> 00:08:59,580 So then k is MbV plus q. 144 00:09:05,736 --> 00:09:07,560 And I can write q out in components 145 00:09:07,560 --> 00:09:09,500 since it's a collinear particle. 146 00:09:09,500 --> 00:09:13,500 And if I want to identify the big piece, 147 00:09:13,500 --> 00:09:16,380 then that would be the n bar dot qp's. 148 00:09:16,380 --> 00:09:18,930 So remember last time, from looking at coordinates, 149 00:09:18,930 --> 00:09:21,670 we can write out any vector in terms of the coordinates. 150 00:09:21,670 --> 00:09:23,878 And it's useful to do that since the coordinates have 151 00:09:23,878 --> 00:09:25,845 a different power counter. 152 00:09:25,845 --> 00:09:27,720 Now, what's going to appear in the propagator 153 00:09:27,720 --> 00:09:33,280 here is k squared minus m squared minus Mb squared. 154 00:09:33,280 --> 00:09:36,640 So let's look at k squared. 155 00:09:36,640 --> 00:09:39,960 So if I square this term, I just get Mb squared. 156 00:09:39,960 --> 00:09:42,600 If I square this term, I get 0, but then there's 157 00:09:42,600 --> 00:09:43,700 a cross term as well. 158 00:09:52,095 --> 00:09:53,220 And then there's some dots. 159 00:09:53,220 --> 00:09:55,680 And the dots are indicating terms that are suppressed. 160 00:10:03,490 --> 00:10:09,700 And so if I look at k squared minus M squared, then that's n 161 00:10:09,700 --> 00:10:13,744 dot v Mb n bar dot q. 162 00:10:13,744 --> 00:10:19,125 And this order lambda to the 0 since the momentum n bar 163 00:10:19,125 --> 00:10:22,300 dot q was order lambda to the 0. 164 00:10:22,300 --> 00:10:29,410 So there's no power counting to the propagator. 165 00:10:29,410 --> 00:10:31,525 Another way of saying it is this propagator 166 00:10:31,525 --> 00:10:34,540 is off-shell by a hard amount. 167 00:10:34,540 --> 00:10:35,880 These scales are hard. 168 00:10:35,880 --> 00:10:37,900 And if I go back to my picture, that 169 00:10:37,900 --> 00:10:44,050 means they're up here in this purple region, 170 00:10:44,050 --> 00:10:46,450 purple box around this. 171 00:10:46,450 --> 00:10:50,920 OK, so the degree of freedom, which is this purple propagator 172 00:10:50,920 --> 00:10:55,100 is actually living up in that purple region. 173 00:10:55,100 --> 00:11:02,280 It's something we have to integrate out, OK? 174 00:11:02,280 --> 00:11:03,090 So let's do that. 175 00:11:08,520 --> 00:11:11,262 So integrating it out just means expanding the diagram. 176 00:11:11,262 --> 00:11:12,970 And I've already expanded the propagator, 177 00:11:12,970 --> 00:11:15,760 so there's the denominators. 178 00:11:15,760 --> 00:11:17,957 We just have to expand the numerator. 179 00:11:31,140 --> 00:11:31,640 OK. 180 00:11:34,230 --> 00:11:39,000 Let me write out A n mu, but we replace the spinner 181 00:11:39,000 --> 00:11:39,840 by the field. 182 00:11:42,420 --> 00:11:45,480 And I'll start by just writing in the full propagator, 183 00:11:45,480 --> 00:11:48,000 and then we'll start expanding it. 184 00:11:55,780 --> 00:12:02,170 I'm using a convention here for the gluon interaction in QCD, 185 00:12:02,170 --> 00:12:04,990 which is plus igTA gamma mu. 186 00:12:09,720 --> 00:12:13,927 So there's the full theory. 187 00:12:13,927 --> 00:12:15,510 All I've done is take the full theory. 188 00:12:15,510 --> 00:12:18,472 And rather than write spinners and a polarization vector, 189 00:12:18,472 --> 00:12:19,680 I've just written the fields. 190 00:12:19,680 --> 00:12:21,210 Because I'm, in the end, interested 191 00:12:21,210 --> 00:12:24,300 in looking at what kind of operator I get. 192 00:12:24,300 --> 00:12:26,850 So I'm interested here in taking the piece, which 193 00:12:26,850 --> 00:12:31,140 is not suppressed. 194 00:12:31,140 --> 00:12:34,695 So let's just do this slowly. 195 00:12:38,560 --> 00:12:41,170 So there's an index A on this guy as well. 196 00:12:45,350 --> 00:12:49,040 So let's write out the leading order piece of the numerator. 197 00:12:49,040 --> 00:12:53,170 I've collected some i's, put the M, the g out front. 198 00:13:01,172 --> 00:13:03,380 So if I take all the order 1 pieces of the numerator, 199 00:13:03,380 --> 00:13:04,797 then these are the order 1 pieces. 200 00:13:04,797 --> 00:13:07,280 Mb is order 1. 201 00:13:07,280 --> 00:13:10,931 k I can replace by MbV plus q. 202 00:13:10,931 --> 00:13:14,015 And then there's an MbV slash that's order 1 and an n 203 00:13:14,015 --> 00:13:15,500 slash over 2 n bar dot q. 204 00:13:15,500 --> 00:13:19,100 That's the same decomposition we had over there. 205 00:13:19,100 --> 00:13:24,500 Here, because I'm interested in certain polarization, 206 00:13:24,500 --> 00:13:26,270 let me pull out that polarization. 207 00:13:30,930 --> 00:13:32,960 Then I have my denominator. 208 00:13:32,960 --> 00:13:34,733 And my denominator we already expanded, 209 00:13:34,733 --> 00:13:35,900 so let's just write that in. 210 00:13:43,580 --> 00:13:44,960 OK, so far, so good? 211 00:13:47,570 --> 00:13:51,740 So this piece here is 0 because n squared 212 00:13:51,740 --> 00:13:54,050 is 0. n slash squared is n squared. 213 00:14:04,590 --> 00:14:07,730 So then we have this piece. 214 00:14:07,730 --> 00:14:10,520 For this piece, 1 plus v slash, remember, 215 00:14:10,520 --> 00:14:13,380 is part of a projector that could act on the heavy quark. 216 00:14:13,380 --> 00:14:14,880 But there's this n slash in the way, 217 00:14:14,880 --> 00:14:17,720 so we want to push it through. 218 00:14:17,720 --> 00:14:18,800 So we can do that. 219 00:14:30,832 --> 00:14:33,290 So let me write out the two pieces from pushing it through. 220 00:14:35,960 --> 00:14:40,236 There's an Mb that cancels this Mb once that term's gone. 221 00:14:40,236 --> 00:14:41,450 So let me do that. 222 00:14:44,090 --> 00:14:46,940 When I push it through, I switch to 1 minus v slash. 223 00:14:50,440 --> 00:14:53,740 There's still a 1 over n dot v here. 224 00:14:53,740 --> 00:14:54,730 And then there's an hv. 225 00:14:59,830 --> 00:15:04,450 And 1 minus v slash on the heavy quark field is 0. 226 00:15:04,450 --> 00:15:11,860 And then the n dot v cancels n dot v. 227 00:15:11,860 --> 00:15:14,630 So if I want to write this putting, 228 00:15:14,630 --> 00:15:21,100 now, this A n field inside because the TA is inside, 229 00:15:21,100 --> 00:15:25,900 then I would write it like this. 230 00:15:30,020 --> 00:15:32,800 So the momentum here n bar dot q is order 1. 231 00:15:32,800 --> 00:15:34,670 And this n bar dot A n is order 1. 232 00:15:34,670 --> 00:15:38,680 So this operator is the same size as this operator here. 233 00:15:52,097 --> 00:15:54,430 And that's what I was saying that we should worry about, 234 00:15:54,430 --> 00:15:57,250 the fact that we can add these gluons without any cost. 235 00:15:57,250 --> 00:15:59,123 And we just added one and found out we 236 00:15:59,123 --> 00:16:00,790 have an operative that's the same order. 237 00:16:04,610 --> 00:16:08,230 So the way you might draw this-- 238 00:16:08,230 --> 00:16:10,480 and there's a convention also for collinear gluons 239 00:16:10,480 --> 00:16:12,870 where you put a line through them, 240 00:16:12,870 --> 00:16:14,590 so this is a collinear gluon-- 241 00:16:20,400 --> 00:16:30,177 is that this guy is the same order as this guy, OK? 242 00:16:30,177 --> 00:16:32,760 Well, we'll come back to this, talk more about it in a minute. 243 00:16:35,650 --> 00:16:38,010 Let's consider another diagram. 244 00:16:38,010 --> 00:16:39,780 We attached the gluon on the left. 245 00:16:39,780 --> 00:16:42,180 Let's also attach it on the right, see what happens. 246 00:16:47,620 --> 00:16:54,210 So again-- QCD diagram, doing the same thing, 247 00:16:54,210 --> 00:16:57,330 attaching the gluon here, some q. 248 00:16:57,330 --> 00:16:59,850 This is some momentum p. 249 00:16:59,850 --> 00:17:03,780 And this is p minus q. 250 00:17:03,780 --> 00:17:07,477 This case is different than the previous case. 251 00:17:07,477 --> 00:17:09,060 The reason that this case is different 252 00:17:09,060 --> 00:17:12,119 is that both p and q are collinear. 253 00:17:12,119 --> 00:17:16,369 And so p minus q is collinear, too, same scaling as p and q. 254 00:17:27,187 --> 00:17:29,706 So this propagator is not off-shell. 255 00:17:35,300 --> 00:17:37,940 In our hyperbolas, it would live on the blue hyperbola 256 00:17:37,940 --> 00:17:43,490 that I drew, which is well within the effective theory, 257 00:17:43,490 --> 00:17:45,658 not outside the effective theory. 258 00:17:45,658 --> 00:17:47,950 So we don't want to integrate out this propagator here. 259 00:17:50,950 --> 00:17:52,700 And what that means is that there will be, 260 00:17:52,700 --> 00:17:56,480 in the effective theory itself, an interaction, 261 00:17:56,480 --> 00:17:58,260 a Lagrangian interaction that takes into 262 00:17:58,260 --> 00:17:59,135 account this diagram. 263 00:18:44,670 --> 00:18:47,800 So in the effective theory, if you like, 264 00:18:47,800 --> 00:18:51,640 if I draw an effective theory diagram, 265 00:18:51,640 --> 00:18:54,330 there will be an effective theory diagram 266 00:18:54,330 --> 00:18:56,250 that allows us to attach that gluon 267 00:18:56,250 --> 00:19:00,330 because this propagator here is a propagator 268 00:19:00,330 --> 00:19:01,500 in the effective theory. 269 00:19:06,590 --> 00:19:09,422 Unlike the one that I just erased which was off-shell 270 00:19:09,422 --> 00:19:11,630 and had to be integrated out of the effective theory, 271 00:19:11,630 --> 00:19:17,390 this one is inside the effective theory, OK? 272 00:19:17,390 --> 00:19:18,890 So adding on the left does something 273 00:19:18,890 --> 00:19:20,840 different than adding on the right. 274 00:19:20,840 --> 00:19:23,660 On the left, we've knocked the quark off-shell. 275 00:19:23,660 --> 00:19:27,860 We add on the right, it's close to mass shell. 276 00:19:27,860 --> 00:19:30,710 So we can consider generalizing what we just 277 00:19:30,710 --> 00:19:34,940 did by adding more gluons. 278 00:19:34,940 --> 00:19:41,150 Remember that we don't have any restrictions 279 00:19:41,150 --> 00:19:42,200 on how many we can add. 280 00:19:42,200 --> 00:19:44,030 So let's just consider adding more. 281 00:19:57,630 --> 00:19:58,530 Let's add m of them. 282 00:20:02,769 --> 00:20:10,830 And I'll call the momenta q1, q2, to qm. 283 00:20:10,830 --> 00:20:13,710 And all these propagators here are off-shell. 284 00:20:19,012 --> 00:20:20,390 This is a QCD graph. 285 00:20:23,010 --> 00:20:25,640 When I calculate graphs like this, 286 00:20:25,640 --> 00:20:29,280 if I have these gluons there, I have to also consider 287 00:20:29,280 --> 00:20:30,300 cross-diagrams. 288 00:20:30,300 --> 00:20:35,610 So rather than trying to draw cross-diagrams, 289 00:20:35,610 --> 00:20:38,040 let me just cross-gluon graphs. 290 00:20:45,360 --> 00:20:47,700 And what you would expect from what we've said before 291 00:20:47,700 --> 00:20:49,200 is that in the effective theory this 292 00:20:49,200 --> 00:20:53,490 is going to catch on to some kind of Feynman rule 293 00:20:53,490 --> 00:20:57,360 that looks like this, but then has 294 00:20:57,360 --> 00:21:01,740 a bunch of gluons that can come out of the vertex, 295 00:21:01,740 --> 00:21:03,720 in this case m of them. 296 00:21:06,330 --> 00:21:08,140 That's what we expect. 297 00:21:08,140 --> 00:21:09,690 And so we expect a generalization 298 00:21:09,690 --> 00:21:12,750 of this operator with m fields rather than just 1. 299 00:21:20,020 --> 00:21:23,490 So if we do this calculation, it actually 300 00:21:23,490 --> 00:21:25,200 turns out to not be too complicated. 301 00:21:33,000 --> 00:21:36,780 I can include all cross-diagrams by sum over permutations. 302 00:21:41,050 --> 00:21:47,760 I have one of these n bar dot A gluons for each external gluon 303 00:21:47,760 --> 00:21:51,150 line with a corresponding color factor. 304 00:21:51,150 --> 00:21:55,180 And I keep the largest piece of the denominator. 305 00:21:55,180 --> 00:21:58,230 And I'm always just getting the order 1 momenta. 306 00:22:03,520 --> 00:22:05,220 And there's a slight complication 307 00:22:05,220 --> 00:22:07,540 that, when I have a graph like this, 308 00:22:07,540 --> 00:22:09,450 if I look at the momentum of this, it's q1. 309 00:22:09,450 --> 00:22:11,367 But if I look at the momentum of the next one, 310 00:22:11,367 --> 00:22:13,155 this would be q1 plus q2. 311 00:22:13,155 --> 00:22:16,170 And so the denominators here are not just simply q's, but sums 312 00:22:16,170 --> 00:22:16,740 of q's. 313 00:22:26,120 --> 00:22:26,620 OK. 314 00:22:26,620 --> 00:22:29,380 And these n's would be dotted into fields. 315 00:22:29,380 --> 00:22:30,970 And T's would be dotted into fields. 316 00:22:37,760 --> 00:22:44,110 So if we want to write this as an effective theory operator, 317 00:22:44,110 --> 00:22:46,970 then we should think about what this vertex means 318 00:22:46,970 --> 00:22:48,437 in the effective theory. 319 00:22:48,437 --> 00:22:50,020 And we should be a little bit careful. 320 00:22:50,020 --> 00:22:52,690 Because what this means, if you think about it 321 00:22:52,690 --> 00:22:55,090 in terms of the locality of this diagram, 322 00:22:55,090 --> 00:22:57,910 is that all these gluons are coming out of the same point. 323 00:22:57,910 --> 00:23:01,250 So it's like phi 4 theory where you divide by 4 factorial. 324 00:23:01,250 --> 00:23:02,830 You have to be a little bit careful 325 00:23:02,830 --> 00:23:05,440 because any one of these gluons, from the point of view 326 00:23:05,440 --> 00:23:10,750 of the external states, could be as equally 327 00:23:10,750 --> 00:23:13,190 good as an attachment. 328 00:23:13,190 --> 00:23:16,270 So when we put in the gluon fields, 329 00:23:16,270 --> 00:23:18,453 we ought to be careful about those factors. 330 00:23:18,453 --> 00:23:19,870 So this is the right Feynman rule. 331 00:23:51,800 --> 00:23:56,600 And one way of just thinking about it is we'll 332 00:23:56,600 --> 00:23:59,540 have m of these fields, but then we'll divide by an m factorial. 333 00:24:02,750 --> 00:24:05,970 So we can write down the complete result for the two 334 00:24:05,970 --> 00:24:13,068 level matching as an operator since we just 335 00:24:13,068 --> 00:24:14,235 calculated all the diagrams. 336 00:24:37,770 --> 00:24:39,410 So we started with this current. 337 00:24:39,410 --> 00:24:41,250 We're in QCD. 338 00:24:41,250 --> 00:24:45,330 And we matched it at tree level onto a current 339 00:24:45,330 --> 00:24:48,600 that I can write like this, hiding all the complexity 340 00:24:48,600 --> 00:24:56,930 in this thing I call W. I have to sum 341 00:24:56,930 --> 00:25:01,310 over however many attachments, which I was previously calling 342 00:25:01,310 --> 00:25:02,690 m, but now I'm calling k. 343 00:25:28,700 --> 00:25:34,730 And if you study this guy for a while, actually what is true 344 00:25:34,730 --> 00:25:36,720 is that this guy has a name. 345 00:25:36,720 --> 00:25:38,420 This is a momentum space Wilson line. 346 00:25:52,056 --> 00:25:54,380 So what is a position space Wilson line? 347 00:25:54,380 --> 00:25:56,960 That's where it really looks more like a line. 348 00:26:06,850 --> 00:26:10,810 So a Wilson line is a string of gauge fields 349 00:26:10,810 --> 00:26:12,550 that goes between two points, here 350 00:26:12,550 --> 00:26:16,420 between minus infinity and 0. 351 00:26:16,420 --> 00:26:19,900 There are some ordering to the fields 352 00:26:19,900 --> 00:26:21,160 because they don't commute. 353 00:26:21,160 --> 00:26:22,327 Because they're not abelian. 354 00:26:26,680 --> 00:26:28,720 And the fact that it's a line means 355 00:26:28,720 --> 00:26:32,170 that we go in a straight line between the two points. 356 00:26:35,370 --> 00:26:37,370 So here's a path which is a straight line 357 00:26:37,370 --> 00:26:40,230 s between minus infinity and 0. 358 00:26:40,230 --> 00:26:44,960 And actually, this formula here, the Fourier transform of this, 359 00:26:44,960 --> 00:26:48,010 is giving you the Wilson line that I drew up-- 360 00:26:48,010 --> 00:26:50,625 the Feynman rule that I drew up there or the fields 361 00:26:50,625 --> 00:26:51,500 that I drew up there. 362 00:27:22,702 --> 00:27:24,850 So there's some ordering to the fields. 363 00:27:24,850 --> 00:27:26,860 And what this P does is it orders the fields 364 00:27:26,860 --> 00:27:27,850 in an appropriate way. 365 00:27:27,850 --> 00:27:30,040 Namely, it puts the guys with the larger argument 366 00:27:30,040 --> 00:27:30,640 to the left. 367 00:27:46,510 --> 00:27:48,160 So on your homework, you're going 368 00:27:48,160 --> 00:27:50,260 to do the exercise of thinking about doing 369 00:27:50,260 --> 00:27:55,570 this matching for two gluons and taking the Fourier transform 370 00:27:55,570 --> 00:27:57,670 this and connecting these. 371 00:27:57,670 --> 00:28:00,160 At least I believe that's what I asked. 372 00:28:00,160 --> 00:28:02,950 That's a very good exercise to see everything work out 373 00:28:02,950 --> 00:28:06,830 that I'm just describing to you here in words. 374 00:28:06,830 --> 00:28:09,280 So one way of thinking about what's happened in this 375 00:28:09,280 --> 00:28:11,447 effective theory-- and this actually turns out to be 376 00:28:11,447 --> 00:28:12,490 generically true-- 377 00:28:12,490 --> 00:28:16,540 is that rather than having this embarked on a field, which 378 00:28:16,540 --> 00:28:19,900 was order 1, it actually can be traded for this Wilson line 379 00:28:19,900 --> 00:28:21,305 object. 380 00:28:21,305 --> 00:28:22,930 So I just showed you that that happened 381 00:28:22,930 --> 00:28:25,420 for this particular example, but it turns out actually 382 00:28:25,420 --> 00:28:29,160 to be genetic, that, instead of talking about the n 383 00:28:29,160 --> 00:28:31,990 bar of A field, we can talk about this function 384 00:28:31,990 --> 00:28:33,670 of the field, which is this Wilson line. 385 00:28:44,894 --> 00:28:46,740 Lift that for a sec. 386 00:28:56,660 --> 00:28:58,060 So this is true in this operator. 387 00:28:58,060 --> 00:28:59,620 We just got this Wilson line. 388 00:28:59,620 --> 00:29:02,110 In terms of the Wilson line, things are pretty simple. 389 00:29:04,780 --> 00:29:06,980 And it actually turns out to be generically true, 390 00:29:06,980 --> 00:29:08,170 and we'll see that later on. 391 00:29:12,570 --> 00:29:14,400 And we'll also talk about gauge invariance 392 00:29:14,400 --> 00:29:18,490 later on, which has to do with this story. 393 00:29:18,490 --> 00:29:20,760 It turns out that this Wilson line can also 394 00:29:20,760 --> 00:29:23,160 be understood from the point of view of gauge invariance. 395 00:29:23,160 --> 00:29:24,840 The need for this Wilson line can 396 00:29:24,840 --> 00:29:27,960 be understood from the point of view of gauge invariance. 397 00:29:27,960 --> 00:29:31,120 And I can at least describe that to you in words. 398 00:29:31,120 --> 00:29:35,100 So if you look at this diagram here, 399 00:29:35,100 --> 00:29:38,460 we can't attach collinear gluons to this quark. 400 00:29:38,460 --> 00:29:40,680 But this carries color. 401 00:29:40,680 --> 00:29:43,560 We can attach collinear gluons to this quark, no problem. 402 00:29:43,560 --> 00:29:45,750 It doesn't knock it off-shell or anything. 403 00:29:45,750 --> 00:29:49,620 So we're going to have a problem with gauge symmetry 404 00:29:49,620 --> 00:29:52,080 because we have two things that are colored. 405 00:29:52,080 --> 00:29:54,570 A priori, we're going to have a problem with gauge symmetry 406 00:29:54,570 --> 00:29:56,190 because we have two things of color, one of which 407 00:29:56,190 --> 00:29:58,380 we can't attach effective theory fields to and one 408 00:29:58,380 --> 00:30:00,150 of which we can. 409 00:30:00,150 --> 00:30:03,000 But there's also operators with gluons in them, 410 00:30:03,000 --> 00:30:04,530 which are these ones. 411 00:30:04,530 --> 00:30:09,380 And what effectively happens is that the gauge transformation 412 00:30:09,380 --> 00:30:11,940 that used to be for this heavy quark field, 413 00:30:11,940 --> 00:30:13,650 it gets moved into this Wilson line. 414 00:30:13,650 --> 00:30:16,690 So this Wilson line will transform 415 00:30:16,690 --> 00:30:18,690 in a way that compensates for the transformation 416 00:30:18,690 --> 00:30:21,060 of this collinear quark field. 417 00:30:21,060 --> 00:30:24,480 And that's exactly because the gluons here, which 418 00:30:24,480 --> 00:30:27,660 were kind of the corresponding gluons in QCD for the gauge 419 00:30:27,660 --> 00:30:30,660 transformation, are explicitly integrated out 420 00:30:30,660 --> 00:30:31,860 to give that Wilson line. 421 00:30:31,860 --> 00:30:34,650 So it's all tied together. 422 00:30:34,650 --> 00:30:37,860 We'll come back and talk about that in more detail later. 423 00:30:40,440 --> 00:30:43,590 All right, so let's consider now the effective theory now 424 00:30:43,590 --> 00:30:46,440 that we have some idea of how these calculations are working. 425 00:30:51,340 --> 00:30:54,270 And let's consider this SCETI effective theory. 426 00:30:56,980 --> 00:31:12,210 So we have a collinear mode and an ultra soft mode with momenta 427 00:31:12,210 --> 00:31:14,730 that we're scaling in this way. 428 00:31:14,730 --> 00:31:20,820 And we also know, if we consider the Feynman rules, 429 00:31:20,820 --> 00:31:24,250 we can look at what the propagators would look like. 430 00:31:24,250 --> 00:31:29,040 So let's consider, first of all, having a collinear gluon. 431 00:31:29,040 --> 00:31:33,360 And let's just try to think about what kind of propagators 432 00:31:33,360 --> 00:31:36,180 the effective theory is going to have. 433 00:31:36,180 --> 00:31:39,300 So maybe this comes from a heavy quark. 434 00:31:39,300 --> 00:31:42,990 So we have this possibility of having a collinear gluon, 435 00:31:42,990 --> 00:31:43,890 momentum q. 436 00:31:48,060 --> 00:31:51,130 We expanded the propagator for that last time. 437 00:31:51,130 --> 00:31:53,970 And the important point about this being a collinear gluon 438 00:31:53,970 --> 00:31:55,630 is that p and q are the same size. 439 00:31:55,630 --> 00:31:58,380 So we never drop p relative to q or q relative to p. 440 00:32:05,170 --> 00:32:09,070 So if we write out the components, 441 00:32:09,070 --> 00:32:11,890 we did this last time. 442 00:32:11,890 --> 00:32:13,390 And we found that there was really 443 00:32:13,390 --> 00:32:16,340 no expansion in the denominator. 444 00:32:16,340 --> 00:32:19,330 And in the numerator, we could make an expansion 445 00:32:19,330 --> 00:32:21,940 and just keep the leading order piece. 446 00:32:21,940 --> 00:32:24,935 So the propagator would be proportional to this. 447 00:32:24,935 --> 00:32:27,310 There's some spin structure as well that I'm not writing. 448 00:32:31,180 --> 00:32:35,800 But q of the gluon is of order p of the quark. 449 00:32:35,800 --> 00:32:37,240 So nothing gets dropped. 450 00:32:44,663 --> 00:32:46,330 So this is going to be a propagator that 451 00:32:46,330 --> 00:32:48,870 exists in the effective theory. 452 00:32:48,870 --> 00:32:50,230 That's what I was saying before. 453 00:32:59,545 --> 00:33:01,170 The other thing we could do is we could 454 00:33:01,170 --> 00:33:04,080 attach an ultra soft particle. 455 00:33:10,498 --> 00:33:12,290 We could actually attach it to either side, 456 00:33:12,290 --> 00:33:13,750 but let me focus on the right side. 457 00:33:22,015 --> 00:33:23,890 So there's going to be some ultra soft gluon. 458 00:33:23,890 --> 00:33:26,200 It's not collinear to any direction. 459 00:33:26,200 --> 00:33:31,420 Let's call its momenta k, call this still p. 460 00:33:31,420 --> 00:33:34,340 Now, k is much smaller than p by our power counting. 461 00:33:38,410 --> 00:33:41,970 So the k is of order lambda squared. 462 00:33:41,970 --> 00:33:45,737 So in n bar dot k is much less than n bar dot 463 00:33:45,737 --> 00:33:48,394 p, which is of order lambda 0. 464 00:33:48,394 --> 00:33:53,590 k perp is much less than p perp, which is of order lambda. 465 00:33:53,590 --> 00:33:57,100 And then the n dot k is or order n 466 00:33:57,100 --> 00:34:00,920 dot p because those are both lambda squared. 467 00:34:00,920 --> 00:34:02,990 So here it's not going to be like over there. 468 00:34:02,990 --> 00:34:05,332 There is actually going to be an expansion that 469 00:34:05,332 --> 00:34:07,165 occurs because things are of different size. 470 00:34:11,639 --> 00:34:15,770 So if I only keep the leading order terms and the propagator 471 00:34:15,770 --> 00:34:19,500 here, well, the numerator I just have n bar 472 00:34:19,500 --> 00:34:23,068 dot p because n bar dot k is small. 473 00:34:23,068 --> 00:34:24,860 And the only place that I need to keep both 474 00:34:24,860 --> 00:34:26,350 is in this n dot term. 475 00:34:36,826 --> 00:34:38,409 So there is an expansion that goes on. 476 00:34:47,210 --> 00:34:51,310 And again, if I look at how off-shell this propagator is, 477 00:34:51,310 --> 00:34:54,469 it's perfectly within the effective theory. 478 00:34:54,469 --> 00:35:01,360 So this is a propagator with off-shellness that leaves it 479 00:35:01,360 --> 00:35:02,680 inside the effective theory. 480 00:35:07,160 --> 00:35:07,660 OK. 481 00:35:07,660 --> 00:35:09,730 So it's not far off-shell by any means. 482 00:35:09,730 --> 00:35:13,330 In fact, it's just as off-shell as the propagator over there 483 00:35:13,330 --> 00:35:15,530 for the collinear particles. 484 00:35:15,530 --> 00:35:17,260 So it doesn't really make sense to think 485 00:35:17,260 --> 00:35:19,798 about treating this propagator any different than treating 486 00:35:19,798 --> 00:35:22,090 this propagator from an effective theory point of view. 487 00:35:26,180 --> 00:35:28,390 But you see that the Feynman rule 488 00:35:28,390 --> 00:35:31,780 has to somehow know whether you're in this situation where 489 00:35:31,780 --> 00:35:34,330 you need to keep the momentum of the gluon 490 00:35:34,330 --> 00:35:36,580 or whether you're in this situation where the momentum 491 00:35:36,580 --> 00:35:37,372 should be expanded. 492 00:35:40,760 --> 00:35:44,190 And we'll see how that works. 493 00:35:44,190 --> 00:35:46,820 So this is by way of sort of giving you 494 00:35:46,820 --> 00:35:50,180 a hint as to what's to come and what kind of complications 495 00:35:50,180 --> 00:35:52,010 we'll face in the effective theory 496 00:35:52,010 --> 00:35:54,680 because the effective theory can interact with collinear gluons 497 00:35:54,680 --> 00:35:56,210 or these ultra soft gluons. 498 00:35:56,210 --> 00:35:59,888 And it somehow needs to be only keeping leading order terms. 499 00:35:59,888 --> 00:36:02,180 If it's really the leading order effective theory, then 500 00:36:02,180 --> 00:36:03,240 we should just keep this term. 501 00:36:03,240 --> 00:36:05,210 We shouldn't keep these higher order terms. 502 00:36:05,210 --> 00:36:07,085 But over here, we have to keep all the terms. 503 00:36:11,142 --> 00:36:13,350 So what should the leading order effective theory do? 504 00:36:18,190 --> 00:36:20,410 What should we demand of it? 505 00:36:20,410 --> 00:36:24,330 We've basically, now, set up all the things 506 00:36:24,330 --> 00:36:25,830 that we need to think about in order 507 00:36:25,830 --> 00:36:28,410 to figure out what the leading order effective theory is. 508 00:36:31,600 --> 00:36:32,530 So let's do that. 509 00:36:36,370 --> 00:36:38,770 OK, so what do we demand of this Lagrangian? 510 00:36:41,470 --> 00:36:45,940 It should yield the propagator of course 511 00:36:45,940 --> 00:36:47,560 and, also, interactions. 512 00:36:53,710 --> 00:36:56,050 Within this effective theory, it has interactions 513 00:36:56,050 --> 00:37:02,490 both with collinear gluons in that diagram over there 514 00:37:02,490 --> 00:37:04,350 and with ultra soft gluons. 515 00:37:08,850 --> 00:37:11,130 And you can see even from here why 516 00:37:11,130 --> 00:37:14,460 we need two different fields because there 517 00:37:14,460 --> 00:37:17,430 has to be some way of telling that we 518 00:37:17,430 --> 00:37:20,550 have a different propagator in this case than in that case. 519 00:37:20,550 --> 00:37:23,820 Or put another way, the external field 520 00:37:23,820 --> 00:37:25,290 here has a different power counting 521 00:37:25,290 --> 00:37:26,590 than the external field here. 522 00:37:26,590 --> 00:37:28,740 So already we're kind of seeing that we 523 00:37:28,740 --> 00:37:31,800 need two different gluon fields in this effective theory, 524 00:37:31,800 --> 00:37:34,260 as we said earlier. 525 00:37:34,260 --> 00:37:39,190 So the Lagrangian has to have interactions with both of them. 526 00:37:39,190 --> 00:37:41,790 It has to have both quarks and antiquarks 527 00:37:41,790 --> 00:37:45,090 because remember that collinear quarks and collinear 528 00:37:45,090 --> 00:37:48,870 antiquarks both existed as leading order things. 529 00:37:48,870 --> 00:37:52,470 That's not like HQET where one of them gets integrated out. 530 00:37:52,470 --> 00:37:54,510 And basically, that is just the statement 531 00:37:54,510 --> 00:37:57,310 that, say you have a collinear gluon going along very quickly. 532 00:37:57,310 --> 00:38:01,890 It can para-- create a quark and an antiquark. 533 00:38:01,890 --> 00:38:03,420 And if they're, again, collinear, 534 00:38:03,420 --> 00:38:06,360 that's something that has to be allowed to happen at order 535 00:38:06,360 --> 00:38:08,040 1 in this effective theory. 536 00:38:12,510 --> 00:38:13,010 OK. 537 00:38:13,010 --> 00:38:13,950 So, so far, so good. 538 00:38:13,950 --> 00:38:17,540 This is just saying, what kind of fields 539 00:38:17,540 --> 00:38:19,598 are we going to have in this Lagrangian. 540 00:38:22,340 --> 00:38:26,150 This third bullet is related to what I was just stressing, 541 00:38:26,150 --> 00:38:29,690 that we have to get a leading order propagator 542 00:38:29,690 --> 00:38:32,672 in different situations. 543 00:38:32,672 --> 00:38:34,130 So somehow the effective theory has 544 00:38:34,130 --> 00:38:38,930 to know something about the size of momenta that 545 00:38:38,930 --> 00:38:40,610 are running through it. 546 00:38:45,410 --> 00:38:47,210 And if we want to be strict about defining 547 00:38:47,210 --> 00:38:50,780 the effective theory and just defining the leading order 548 00:38:50,780 --> 00:38:54,120 term, it's not OK that we expand later. 549 00:38:54,120 --> 00:38:57,050 We really have to expand ahead of time. 550 00:39:01,230 --> 00:39:05,120 And so whatever the Lagrangian is, it should just give this. 551 00:39:05,120 --> 00:39:07,100 And it should just give this. 552 00:39:07,100 --> 00:39:10,318 And it should not give these dots, OK? 553 00:39:18,980 --> 00:39:22,610 So that's what I mean by having no additional expansion. 554 00:39:26,740 --> 00:39:29,660 And the final thing is a little more subtle. 555 00:39:29,660 --> 00:39:34,430 And that is that we should think ahead. 556 00:39:34,430 --> 00:39:36,383 So when we design this effective theory, 557 00:39:36,383 --> 00:39:38,800 we're not only going to want to describe the leading order 558 00:39:38,800 --> 00:39:40,330 term, but we're also going to want 559 00:39:40,330 --> 00:39:41,865 to describe power corrections. 560 00:39:41,865 --> 00:39:43,990 Because if we can't describe the power corrections, 561 00:39:43,990 --> 00:39:45,948 we don't really have an effective field theory. 562 00:39:48,450 --> 00:39:50,250 So whatever we do, we should make sure 563 00:39:50,250 --> 00:39:54,210 that we have a notion of what's going on with higher order 564 00:39:54,210 --> 00:39:56,717 terms, that they're well-defined things, that we 565 00:39:56,717 --> 00:39:58,800 can think ahead about what the operators are going 566 00:39:58,800 --> 00:40:00,010 to look like. 567 00:40:00,010 --> 00:40:01,590 And we need that in order to know 568 00:40:01,590 --> 00:40:03,750 that we can formulate those operators and make sure 569 00:40:03,750 --> 00:40:06,810 they're suppressed. 570 00:40:06,810 --> 00:40:14,040 So another way of putting it is that we should set things up, 571 00:40:14,040 --> 00:40:15,780 so we don't have to re-set things up 572 00:40:15,780 --> 00:40:18,248 when we start talking about power corrections. 573 00:40:33,068 --> 00:40:35,610 So we really should think about the effective theory globally 574 00:40:35,610 --> 00:40:38,490 even though we're sort of starting by formulating 575 00:40:38,490 --> 00:40:40,170 a leading order term. 576 00:40:40,170 --> 00:40:42,420 We should think about the power corrections as well 577 00:40:42,420 --> 00:40:45,320 and how those interactions are going to behave. 578 00:40:47,830 --> 00:40:50,152 OK, so that's saying that we want to think about, 579 00:40:50,152 --> 00:40:51,610 at least, what these dots are going 580 00:40:51,610 --> 00:40:53,655 to look like, what this plus dot, 581 00:40:53,655 --> 00:40:56,540 dot, dot is going to look like. 582 00:40:56,540 --> 00:40:57,940 OK, so this is our goal. 583 00:40:57,940 --> 00:41:00,170 And we'll go through it slowly. 584 00:41:06,020 --> 00:41:08,320 So this is a top-down effective theory. 585 00:41:08,320 --> 00:41:10,030 So we can start with QCD, and then we 586 00:41:10,030 --> 00:41:14,420 can integrate out the off-shell degrees of freedom. 587 00:41:14,420 --> 00:41:16,590 We can just formulate from what SCET 588 00:41:16,590 --> 00:41:22,080 is by starting with the QCD Lagrangian 589 00:41:22,080 --> 00:41:26,530 and splitting it into the fields that we want 590 00:41:26,530 --> 00:41:29,660 and manipulating it. 591 00:41:29,660 --> 00:41:34,420 So let's start with psi bar iD slash psi. 592 00:41:34,420 --> 00:41:37,135 And let's write psi in terms of two fields. 593 00:41:47,850 --> 00:41:50,245 And this goes back to our discussion 594 00:41:50,245 --> 00:41:51,745 when we were talking about spinners. 595 00:41:56,350 --> 00:42:01,180 So these two fields have a projector on the full theory 596 00:42:01,180 --> 00:42:10,546 field, and they obey n bar slash, n slash, n slash, 597 00:42:10,546 --> 00:42:11,440 n bar slash. 598 00:42:11,440 --> 00:42:13,266 Cn is Cn. 599 00:42:13,266 --> 00:42:21,430 And the n bar slash n slash by n bar is by n bar. 600 00:42:21,430 --> 00:42:23,800 So I want to write out the QCD Lagrangian in terms 601 00:42:23,800 --> 00:42:26,223 of component because the different derivatives, 602 00:42:26,223 --> 00:42:28,390 the different components of the covariant derivative 603 00:42:28,390 --> 00:42:30,760 and the different regular derivatives 604 00:42:30,760 --> 00:42:35,620 inside that derivative, have different power counting, OK? 605 00:42:35,620 --> 00:42:37,210 So let's do that. 606 00:42:37,210 --> 00:42:39,550 And I'll introduce these two fields in order to do that. 607 00:42:42,280 --> 00:42:47,020 So if you like, I'm just taking QCD and writing it in these 608 00:42:47,020 --> 00:42:49,842 coordinates that we're setting our usual coordinates 609 00:42:49,842 --> 00:42:50,800 for talking about SCET. 610 00:42:56,332 --> 00:43:10,540 So decompose the D slash and just write out 611 00:43:10,540 --> 00:43:12,355 the field as these two pieces. 612 00:43:16,908 --> 00:43:18,325 And then I can multiply these out. 613 00:43:18,325 --> 00:43:19,950 And because of the projection relation, 614 00:43:19,950 --> 00:43:22,060 some of these products will vanish. 615 00:43:22,060 --> 00:43:24,220 So if I only write the pieces that don't vanish, 616 00:43:24,220 --> 00:43:29,140 then I only have four pieces, which are these four. 617 00:43:57,040 --> 00:44:00,790 So all other combinations that I didn't write 618 00:44:00,790 --> 00:44:03,205 vanish, and those four survive. 619 00:44:09,880 --> 00:44:12,880 We'll do one example, so you see what 620 00:44:12,880 --> 00:44:15,880 the strategy is for getting rid of the other terms. 621 00:44:36,342 --> 00:44:38,050 So let's consider one of the other terms. 622 00:44:38,050 --> 00:44:41,440 Let's consider the guy with the D perp slash, 623 00:44:41,440 --> 00:44:43,955 but between two Cn fields. 624 00:44:47,660 --> 00:44:49,250 So I can insert the projector. 625 00:44:58,580 --> 00:45:00,580 And then I can move it through the D perp slash. 626 00:45:06,490 --> 00:45:09,270 I'm moving two things through, and they both 627 00:45:09,270 --> 00:45:11,480 have 0 dot product with D perp. 628 00:45:11,480 --> 00:45:13,620 So there's no additional terms. 629 00:45:13,620 --> 00:45:15,600 I just can push it through. 630 00:45:15,600 --> 00:45:18,600 And n slash on Cn bar is 0. 631 00:45:18,600 --> 00:45:21,090 So n slash on Cn was 0, remember. 632 00:45:21,090 --> 00:45:25,020 And if you take the dagger and make it into a bar relation, 633 00:45:25,020 --> 00:45:27,520 you also have this formula. 634 00:45:27,520 --> 00:45:30,270 So this is 0. 635 00:45:30,270 --> 00:45:31,830 And a similar thing-- 636 00:45:31,830 --> 00:45:33,840 by inserting projectors, we can figure out 637 00:45:33,840 --> 00:45:35,190 which terms are non-0. 638 00:45:35,190 --> 00:45:38,570 And these ones I'm writing are the non-0 ones. 639 00:45:38,570 --> 00:45:39,482 This guy's 0. 640 00:45:39,482 --> 00:45:40,690 That's why I didn't write it. 641 00:45:44,970 --> 00:45:48,240 OK, so any questions so far? 642 00:45:48,240 --> 00:45:49,850 Hot today, drink lots of water. 643 00:45:55,450 --> 00:45:55,960 OK. 644 00:45:55,960 --> 00:45:57,210 This is just QCD. 645 00:45:57,210 --> 00:45:58,210 I haven't done anything. 646 00:46:01,030 --> 00:46:05,034 I just write QCD out and assign some strange coordinates. 647 00:46:29,750 --> 00:46:31,760 So now, I'm going to do one thing, which 648 00:46:31,760 --> 00:46:34,850 is going to end up simplifying our lives a little bit later 649 00:46:34,850 --> 00:46:35,647 on. 650 00:46:35,647 --> 00:46:37,730 And I'm going to use the fact that, when we talked 651 00:46:37,730 --> 00:46:40,550 about production of quarks, we said 652 00:46:40,550 --> 00:46:42,500 we're going to produce this Cn type of quark 653 00:46:42,500 --> 00:46:44,510 and not the phi n bar bar type of quark. 654 00:46:48,370 --> 00:46:54,780 So phi n bar, remember, corresponded 655 00:46:54,780 --> 00:46:56,630 to sub-leading spinner components. 656 00:47:02,170 --> 00:47:04,500 So let's just decree that I don't really 657 00:47:04,500 --> 00:47:06,330 care about this field. 658 00:47:06,330 --> 00:47:09,090 And, therefore, I don't need to have any current in my path 659 00:47:09,090 --> 00:47:10,920 integral that I couple to this guy. 660 00:47:31,990 --> 00:47:33,970 So it's like an auxiliary field if you like. 661 00:47:38,980 --> 00:47:41,530 And because the path integral in this fermionic field 662 00:47:41,530 --> 00:47:44,200 is quadratic, we can just remove it from the path, 663 00:47:44,200 --> 00:47:45,730 just do the path integral over it. 664 00:47:48,770 --> 00:47:50,770 Once we know that there's no source term for it, 665 00:47:50,770 --> 00:47:54,280 we can just do the path integral since all the terms are 666 00:47:54,280 --> 00:47:55,560 explicitly written over there. 667 00:48:04,730 --> 00:48:07,880 So let's see what happens when we do that. 668 00:48:07,880 --> 00:48:10,750 So doing the path integral for this quadratic field just 669 00:48:10,750 --> 00:48:12,640 means solving the equations of motion. 670 00:48:21,200 --> 00:48:24,970 So we take the variational derivative with respect 671 00:48:24,970 --> 00:48:27,717 to phi n bar of the Lagrangian. 672 00:48:27,717 --> 00:48:29,425 And that gives us the following equation. 673 00:48:57,030 --> 00:49:01,740 We could multiply both terms in this equation 674 00:49:01,740 --> 00:49:05,160 by an n bar slash over 2. 675 00:49:05,160 --> 00:49:08,340 And then we can use the projection relation. 676 00:49:08,340 --> 00:49:10,487 Just to simplify the Dirac structure a little, 677 00:49:10,487 --> 00:49:12,570 we can move the Dirac structure from here to here. 678 00:49:34,920 --> 00:49:37,440 And then we could formally solve this equation for the phi n 679 00:49:37,440 --> 00:49:37,940 bar. 680 00:49:42,080 --> 00:49:44,374 It's an inverse covariant derivative. 681 00:49:50,675 --> 00:49:52,947 I can push this through this at the cost of a sign. 682 00:49:52,947 --> 00:49:54,530 So if I move it to the right-hand side 683 00:49:54,530 --> 00:49:58,850 of the equation, that takes care of that sign. 684 00:49:58,850 --> 00:50:00,768 So I did two sign changes. 685 00:50:00,768 --> 00:50:02,060 I moved something to the right. 686 00:50:02,060 --> 00:50:04,790 And then I pushed these through each other 687 00:50:04,790 --> 00:50:07,570 to get that equation. 688 00:50:07,570 --> 00:50:08,070 OK. 689 00:50:08,070 --> 00:50:21,696 So what this is saying, in terms of the original field, 690 00:50:21,696 --> 00:50:23,570 is that I have this equation for psi. 691 00:50:27,450 --> 00:50:30,630 So if we take that result and we plug it 692 00:50:30,630 --> 00:50:43,720 back into our Lagrangian, which I meant to give a star, 693 00:50:43,720 --> 00:50:50,650 then we'll just have an equation in terms of the C. 694 00:50:50,650 --> 00:50:53,080 We already used two terms. 695 00:50:53,080 --> 00:50:55,300 And so those two terms, when you plug it back in, 696 00:50:55,300 --> 00:50:56,140 they just cancel. 697 00:51:02,333 --> 00:51:04,000 So there was two terms in the Lagrangian 698 00:51:04,000 --> 00:51:06,036 that had psi n bar bar. 699 00:51:10,000 --> 00:51:37,930 And the remaining two you have that. 700 00:51:41,620 --> 00:51:45,470 OK, so this is Lagrangian that's just in terms of the Cn field. 701 00:51:45,470 --> 00:51:47,020 We'll call it double star. 702 00:51:51,096 --> 00:51:52,810 Previous guy was supposed to be star. 703 00:51:56,380 --> 00:51:58,480 OK, so several questions arise. 704 00:51:58,480 --> 00:52:00,190 What have we done? 705 00:52:00,190 --> 00:52:02,290 Do we like this inverse covariant derivative? 706 00:52:02,290 --> 00:52:03,400 Are we happy with that? 707 00:52:03,400 --> 00:52:05,480 Are we unhappy with it? 708 00:52:05,480 --> 00:52:08,720 So what does an inverse covariant derivative mean? 709 00:52:08,720 --> 00:52:11,140 Well, actually just what is an inverse derivative mean? 710 00:52:11,140 --> 00:52:12,490 This is an inverse operator. 711 00:52:19,550 --> 00:52:21,030 So what does that mean? 712 00:52:24,040 --> 00:52:25,830 So the way you define an inverse operator 713 00:52:25,830 --> 00:52:27,840 is exactly how you define an inverse operator 714 00:52:27,840 --> 00:52:28,830 in quantum mechanics. 715 00:52:31,670 --> 00:52:34,775 So let me remind you of that. 716 00:52:34,775 --> 00:52:37,150 This is the analog of, in quantum mechanics, having, say, 717 00:52:37,150 --> 00:52:38,950 a 1 over r potential. 718 00:52:38,950 --> 00:52:41,730 But if r is an operator, then you have a 1 over an operator. 719 00:52:46,880 --> 00:52:49,330 And the way that you define that is by the eigenvalues. 720 00:53:07,600 --> 00:53:14,410 So 1 over i in r dot partial of some field 721 00:53:14,410 --> 00:53:19,570 i of x you can write it out in terms of momentum space. 722 00:53:27,620 --> 00:53:30,310 So if we go over to momentum space, that's the eigenbasis. 723 00:53:33,160 --> 00:53:34,570 And then the eigenvalue is just 1 724 00:53:34,570 --> 00:53:38,795 over the momentum, this component of the momentum. 725 00:53:42,840 --> 00:53:46,234 OK, so it's just the Fourier transform of that. 726 00:53:46,234 --> 00:53:47,490 So this is [INAUDIBLE]. 727 00:53:47,490 --> 00:53:47,990 Yup. 728 00:53:54,105 --> 00:53:54,730 AUDIENCE: Yeah. 729 00:53:54,730 --> 00:53:56,710 I have a question about whether or not 730 00:53:56,710 --> 00:53:58,163 you've actually done nothing. 731 00:53:58,163 --> 00:53:59,830 IAIN STEWART: I have done nothing, yeah. 732 00:53:59,830 --> 00:54:01,518 AUDIENCE: Yeah, it seems [INAUDIBLE].. 733 00:54:01,518 --> 00:54:03,310 IAIN STEWART: Yeah, that's my next comment. 734 00:54:03,310 --> 00:54:04,225 AUDIENCE: Right. 735 00:54:04,225 --> 00:54:07,200 And so phi n bar-- 736 00:54:07,200 --> 00:54:08,200 IAIN STEWART: Yeah, so-- 737 00:54:08,200 --> 00:54:10,617 AUDIENCE: Actually corresponding to this other new spinner 738 00:54:10,617 --> 00:54:12,822 would mean that you're only considering interactions 739 00:54:12,822 --> 00:54:13,780 that are all collinear. 740 00:54:13,780 --> 00:54:15,363 In the original piece, the [INAUDIBLE] 741 00:54:15,363 --> 00:54:17,110 does contain that interaction. 742 00:54:17,110 --> 00:54:18,485 IAIN STEWART: The only difference 743 00:54:18,485 --> 00:54:21,640 between what I've done so far in QCD is that, in QCD, 744 00:54:21,640 --> 00:54:23,410 if we were to think about QCD, we'd 745 00:54:23,410 --> 00:54:26,950 couple external currents both to this phi n bar as well 746 00:54:26,950 --> 00:54:29,310 as the Cn. 747 00:54:29,310 --> 00:54:31,580 And that's the only difference. 748 00:54:31,580 --> 00:54:34,452 But there's a small community of people 749 00:54:34,452 --> 00:54:36,160 that actually think about this Lagrangian 750 00:54:36,160 --> 00:54:38,410 as the QCD Lagrangian. 751 00:54:38,410 --> 00:54:41,710 And that's because, if I was willing to a couple 752 00:54:41,710 --> 00:54:44,620 a current to this particular combination of fields, 753 00:54:44,620 --> 00:54:47,810 I could still call it QCD, OK? 754 00:54:47,810 --> 00:54:50,380 So in that sense, if I really demand 755 00:54:50,380 --> 00:54:52,600 that I'm only producing Cn's and I just 756 00:54:52,600 --> 00:54:59,313 have a term that's, like, JCn and I don't have this phi n 757 00:54:59,313 --> 00:55:00,730 bar, then it's something different 758 00:55:00,730 --> 00:55:04,120 because I'm not directly able to produce these guys. 759 00:55:04,120 --> 00:55:08,140 But if I allow myself, also, to couple 760 00:55:08,140 --> 00:55:10,690 to this combination here, then I can do everything 761 00:55:10,690 --> 00:55:11,950 in terms of these Cn fields. 762 00:55:11,950 --> 00:55:13,780 And it's really just QCD. 763 00:55:13,780 --> 00:55:17,150 AUDIENCE: So star star is exactly collinear QCD 764 00:55:17,150 --> 00:55:18,520 if you're only producing in-- 765 00:55:18,520 --> 00:55:18,850 IAIN STEWART: Exactly. 766 00:55:18,850 --> 00:55:20,642 AUDIENCE: --forward direction or something? 767 00:55:20,642 --> 00:55:21,351 [INAUDIBLE] 768 00:55:21,351 --> 00:55:23,530 IAIN STEWART: Just even simpler, it's exactly 769 00:55:23,530 --> 00:55:26,050 QCD if I only produce these Cn. 770 00:55:26,050 --> 00:55:28,450 If I just say I'm only interested 771 00:55:28,450 --> 00:55:31,007 in operators that produce this particular spin component-- 772 00:55:31,007 --> 00:55:32,590 AUDIENCE: But if you try to say that-- 773 00:55:32,590 --> 00:55:33,030 IAIN STEWART: Yeah. 774 00:55:33,030 --> 00:55:33,550 AUDIENCE: [INAUDIBLE] 775 00:55:33,550 --> 00:55:33,970 IAIN STEWART: No, no. 776 00:55:33,970 --> 00:55:35,553 I haven't I haven't made any expansion 777 00:55:35,553 --> 00:55:37,585 to make it collinear yet. 778 00:55:37,585 --> 00:55:41,553 AUDIENCE: But if there was a hard scatter with two totally 779 00:55:41,553 --> 00:55:42,970 different directions [INAUDIBLE],, 780 00:55:42,970 --> 00:55:45,550 you could produce phi n bars, right? 781 00:55:45,550 --> 00:55:48,430 IAIN STEWART: Yeah, but that would be another Lagrangian. 782 00:55:48,430 --> 00:55:52,900 So this Lagrangian, in the end, will only 783 00:55:52,900 --> 00:55:54,850 be useful for producing particles 784 00:55:54,850 --> 00:55:56,620 in the n collinear direction. 785 00:55:56,620 --> 00:55:58,780 That's what I'm going to be after. 786 00:55:58,780 --> 00:56:00,400 And I'd have another copy of this. 787 00:56:00,400 --> 00:56:03,683 For each linear direction, I have one additional copy 788 00:56:03,683 --> 00:56:04,600 of what I'm doing now. 789 00:56:04,600 --> 00:56:06,392 AUDIENCE: Right, so you've lost the ability 790 00:56:06,392 --> 00:56:08,355 to produce hard scattering by doing this. 791 00:56:08,355 --> 00:56:09,730 IAIN STEWART: By just doing this. 792 00:56:09,730 --> 00:56:10,555 AUDIENCE: But as long as you don't talk about 793 00:56:10,555 --> 00:56:10,970 that, it's still [INAUDIBLE]. 794 00:56:10,970 --> 00:56:12,040 IAIN STEWART: Right. 795 00:56:12,040 --> 00:56:14,740 That's right. 796 00:56:14,740 --> 00:56:15,820 OK. 797 00:56:15,820 --> 00:56:18,490 So we're far from done. 798 00:56:18,490 --> 00:56:21,070 And really this is just sort of a warm up. 799 00:56:21,070 --> 00:56:23,890 Really what we've done here is we've just written things out 800 00:56:23,890 --> 00:56:25,152 in terms of components. 801 00:56:25,152 --> 00:56:26,860 And it's true that these derivatives here 802 00:56:26,860 --> 00:56:28,600 have different power counting. 803 00:56:28,600 --> 00:56:30,848 And we're going to make use of that in a second. 804 00:56:30,848 --> 00:56:32,890 And we've also just sort of focused our attention 805 00:56:32,890 --> 00:56:35,620 on this guy that we could produce 806 00:56:35,620 --> 00:56:37,870 at leading order from a hard scattering 807 00:56:37,870 --> 00:56:42,505 in some particular direction, such as in our b to u example. 808 00:56:47,828 --> 00:56:50,120 So if we're interested in just one collinear direction, 809 00:56:50,120 --> 00:56:52,290 then there's two components of the spinner that 810 00:56:52,290 --> 00:56:53,540 are always the big components. 811 00:56:53,540 --> 00:56:56,690 And we focused on a field that gives those components 812 00:56:56,690 --> 00:56:58,190 directly. 813 00:56:58,190 --> 00:56:59,630 So what do we have to do still? 814 00:57:10,083 --> 00:57:11,250 So there's three more steps. 815 00:57:16,210 --> 00:57:18,270 So we have to separate collinear. 816 00:57:18,270 --> 00:57:20,610 So far, we just have one type of covariant derivative. 817 00:57:20,610 --> 00:57:22,950 We have to separate out the fields that are 818 00:57:22,950 --> 00:57:24,700 in that covariant derivative. 819 00:57:24,700 --> 00:57:30,420 In particular, the ultra soft and collinear gauge fields 820 00:57:30,420 --> 00:57:33,455 have to be separated out. 821 00:57:33,455 --> 00:57:34,830 Another thing we have to do is we 822 00:57:34,830 --> 00:57:43,045 have to separate the collinear and ultra soft momenta. 823 00:57:43,045 --> 00:57:44,670 There's two different types of momenta, 824 00:57:44,670 --> 00:57:47,520 as we saw in this simple example of a gluon momenta 825 00:57:47,520 --> 00:57:48,720 flowing into a propagator. 826 00:57:48,720 --> 00:57:52,030 And we actually have to distinguish them. 827 00:57:52,030 --> 00:57:54,870 And then we have to expand. 828 00:57:54,870 --> 00:57:56,940 So until we do these first two steps, 829 00:57:56,940 --> 00:57:59,080 there's not really anything to expand. 830 00:57:59,080 --> 00:58:02,127 And so once we've distinguished those, 831 00:58:02,127 --> 00:58:03,960 we'll be able to say that certain things are 832 00:58:03,960 --> 00:58:04,860 larger than others. 833 00:58:08,580 --> 00:58:10,080 And then we'll be able to figure out 834 00:58:10,080 --> 00:58:11,880 what the leading or Lagrangian actually is. 835 00:58:15,070 --> 00:58:17,610 OK, so let's start with this second step. 836 00:58:22,300 --> 00:58:26,111 So this is step two. 837 00:58:26,111 --> 00:58:28,090 So remember what the power counting was. 838 00:58:33,455 --> 00:58:36,280 The collinear gluons had this scaling. 839 00:58:36,280 --> 00:58:43,798 The ultra soft gluons had this scaling, 840 00:58:43,798 --> 00:58:45,590 which is the same as an ultra soft momenta. 841 00:58:52,540 --> 00:58:54,790 And the right way of doing this kind of thing 842 00:58:54,790 --> 00:58:56,710 would be to take Feynman diagrams 843 00:58:56,710 --> 00:58:59,440 and do some matching calculations, 844 00:58:59,440 --> 00:59:01,270 but sometimes we're able to get away 845 00:59:01,270 --> 00:59:05,170 with just thinking about doing things by writing down 846 00:59:05,170 --> 00:59:07,110 field relations. 847 00:59:07,110 --> 00:59:09,490 And that's almost true here. 848 00:59:09,490 --> 00:59:12,062 When we were drawing our Feynman diagrams before, 849 00:59:12,062 --> 00:59:14,270 if you think about one gluon, this is certainly true. 850 00:59:14,270 --> 00:59:17,380 So we drew Feynman diagrams before. 851 00:59:17,380 --> 00:59:25,960 We had this one, and we had this one, ultra soft collinear. 852 00:59:25,960 --> 00:59:28,760 And we just consider one at a time. 853 00:59:28,760 --> 00:59:31,030 So we could just think of adding them. 854 00:59:31,030 --> 00:59:34,010 And that's kind of almost true. 855 00:59:34,010 --> 00:59:37,060 It turns out that it's true up to some terms 856 00:59:37,060 --> 00:59:41,055 that we don't care about for the moment, OK? 857 00:59:56,480 --> 00:59:58,639 But these terms are power suppressed. 858 01:00:04,870 --> 01:00:07,870 And we'll see where they come from later on. 859 01:00:07,870 --> 01:00:09,370 So the way you can think about this, 860 01:00:09,370 --> 01:00:12,100 if you ignore that complication, is 861 01:00:12,100 --> 01:00:16,060 that you really just divide the gauge field into two pieces. 862 01:00:16,060 --> 01:00:18,370 There's one place that you've perhaps seen that before, 863 01:00:18,370 --> 01:00:20,657 when you talk about a background gauge field, right? 864 01:00:20,657 --> 01:00:22,990 You say you have a quantum field and a background field. 865 01:00:22,990 --> 01:00:25,750 And that's very nice if you go to background field gauge. 866 01:00:25,750 --> 01:00:27,370 And that's kind of physically what's 867 01:00:27,370 --> 01:00:29,800 going on here because we have two different modes that 868 01:00:29,800 --> 01:00:31,660 have different wavelengths. 869 01:00:31,660 --> 01:00:33,250 These guys are really long wavelength. 870 01:00:33,250 --> 01:00:34,570 These guys are shorter wavelength. 871 01:00:34,570 --> 01:00:36,330 Remember that the p squared of these guys 872 01:00:36,330 --> 01:00:39,310 is much larger than the p squared of these guys. 873 01:00:42,550 --> 01:00:45,760 And think about this formula in that way, 874 01:00:45,760 --> 01:00:55,530 that this is like a classical background 875 01:00:55,530 --> 01:01:10,620 field to the Cn and the A n mu since the p ultra soft squared 876 01:01:10,620 --> 01:01:15,810 had a scaling which was lambda to the fourth lambda squared 877 01:01:15,810 --> 01:01:16,920 squared. 878 01:01:16,920 --> 01:01:20,430 And that's much smaller than p collinear squared, which 879 01:01:20,430 --> 01:01:23,920 was q squared lambda squared. 880 01:01:23,920 --> 01:01:27,180 So that's saying that this guy has much larger wavelength, 881 01:01:27,180 --> 01:01:30,800 a long wavelength mode to these short distance modes. 882 01:01:30,800 --> 01:01:33,300 So we can think of this as a slowly varying 883 01:01:33,300 --> 01:01:34,410 classical background. 884 01:01:34,410 --> 01:01:36,493 And that's one way of thinking about this formula. 885 01:01:36,493 --> 01:01:39,120 We'll exploit that a little bit later on as well. 886 01:01:45,270 --> 01:01:47,840 So now that we have the sum of them, 887 01:01:47,840 --> 01:01:50,600 we can just think about how big one is relative to the other. 888 01:01:50,600 --> 01:01:52,220 And actually, the coefficient that's between them 889 01:01:52,220 --> 01:01:53,887 doesn't really matter too much for that. 890 01:02:03,510 --> 01:02:05,870 So let's think about the power counting 891 01:02:05,870 --> 01:02:07,440 I already wrote on the board. 892 01:02:10,380 --> 01:02:11,895 So the power counting is that n bar 893 01:02:11,895 --> 01:02:14,610 dot A n is order lambda to the 0 and so is 894 01:02:14,610 --> 01:02:19,560 much greater than the n bar dot A ultra soft, 895 01:02:19,560 --> 01:02:21,780 which is order lambda squared. 896 01:02:21,780 --> 01:02:25,400 Same for A perp n, that's of order lambda. 897 01:02:25,400 --> 01:02:28,395 That's much greater than A perp ultra soft. 898 01:02:31,770 --> 01:02:34,590 And so those two formulas mean that it would never 899 01:02:34,590 --> 01:02:36,510 have a comparison between this and this. 900 01:02:36,510 --> 01:02:38,070 I can always drop the smaller one. 901 01:02:40,820 --> 01:02:43,640 And just like our momentum, there's 902 01:02:43,640 --> 01:02:45,470 one case where that's not true where 903 01:02:45,470 --> 01:02:47,820 we have to keep both of them. 904 01:02:47,820 --> 01:02:50,499 And that's for the n dot A component. 905 01:02:57,490 --> 01:02:59,170 So here's something we can drop, n bar 906 01:02:59,170 --> 01:03:04,951 dot A ultra soft and A perp ultra soft. 907 01:03:04,951 --> 01:03:10,720 That leading order can be dropped, OK? 908 01:03:10,720 --> 01:03:12,730 So that's what we're going to do and step two. 909 01:03:23,260 --> 01:03:25,360 Step three corresponds to dropping momentum 910 01:03:25,360 --> 01:03:26,830 relative to another momentum. 911 01:03:32,990 --> 01:03:35,150 So there was an expansion of the propagator 912 01:03:35,150 --> 01:03:37,480 in the case where we had an ultra soft and collinear 913 01:03:37,480 --> 01:03:38,290 at the same time. 914 01:03:55,918 --> 01:03:57,460 And let's just write that down again, 915 01:03:57,460 --> 01:03:58,918 but let's keep the next order term. 916 01:04:06,970 --> 01:04:10,390 And I'll ignore the numerator since just looking 917 01:04:10,390 --> 01:04:13,600 at the denominator will give us enough information 918 01:04:13,600 --> 01:04:14,835 for this discussion. 919 01:04:18,720 --> 01:04:20,440 We could do the numerator, too. 920 01:04:20,440 --> 01:04:22,335 It would just lead to more terms. 921 01:04:28,870 --> 01:04:31,410 So there are some terms that I can expand. 922 01:04:31,410 --> 01:04:34,230 And then I get the propagator squared, 923 01:04:34,230 --> 01:04:38,700 or I get a P squared squared, because I expanded something 924 01:04:38,700 --> 01:04:40,827 in the denominator back into the numerator. 925 01:04:44,410 --> 01:04:48,150 So the next order term comes from a dot product 926 01:04:48,150 --> 01:04:52,540 in the perp momenta because the Pn perp is large. 927 01:04:52,540 --> 01:04:56,178 And so we get a term with two k ultra soft dot Pn perp 928 01:04:56,178 --> 01:04:57,720 that corrects the leading order term. 929 01:04:57,720 --> 01:05:02,470 And that's the first sub-leading term. 930 01:05:02,470 --> 01:05:04,680 So if you think about the power counting of these, 931 01:05:04,680 --> 01:05:07,440 this guy is lambda to the minus 2. 932 01:05:07,440 --> 01:05:09,270 And this guy is lambda to the minus 1. 933 01:05:15,770 --> 01:05:18,440 OK, so there has to be something in the effective theory that, 934 01:05:18,440 --> 01:05:20,780 at higher order, will reproduce this second guy. 935 01:05:29,810 --> 01:05:43,370 There will be some power suppressed Feynman rule in SCET 936 01:05:43,370 --> 01:05:45,230 to reproduce this second term. 937 01:05:59,530 --> 01:06:01,740 So what that means is that you can't just 938 01:06:01,740 --> 01:06:06,420 think that you ignore this guy completely, 939 01:06:06,420 --> 01:06:08,850 this k ultra soft perp. 940 01:06:08,850 --> 01:06:12,390 You might just say, well, I just set all the k ultra soft perps 941 01:06:12,390 --> 01:06:16,710 and the k ultra soft minuses to 0. 942 01:06:16,710 --> 01:06:18,210 And that would be something that you 943 01:06:18,210 --> 01:06:22,380 can get away with, if your careful, at lowest order. 944 01:06:22,380 --> 01:06:24,445 But at sub-leading order, you can't. 945 01:06:24,445 --> 01:06:27,540 You need to think that they exist, too. 946 01:06:27,540 --> 01:06:31,900 So whatever formalism we develop should not set them to 0. 947 01:06:31,900 --> 01:06:35,380 And that was one of my bullets that you should 948 01:06:35,380 --> 01:06:37,158 think a little bit ahead. 949 01:06:37,158 --> 01:06:38,950 And this is me thinking ahead, that I'm not 950 01:06:38,950 --> 01:06:41,410 going to be able to just set those momentum exactly to 0. 951 01:06:41,410 --> 01:06:43,202 I'm going to need them when I start talking 952 01:06:43,202 --> 01:06:44,695 about sub-leading terms. 953 01:06:49,890 --> 01:07:00,600 OK, so we expand just like we don't throw away 954 01:07:00,600 --> 01:07:02,850 the sub-leading gauge fields because we need those 955 01:07:02,850 --> 01:07:04,428 at sub-leading order as well. 956 01:07:04,428 --> 01:07:05,970 We neglect them in the leading order, 957 01:07:05,970 --> 01:07:09,182 but we have to sort of think that they exist still. 958 01:07:09,182 --> 01:07:10,890 And with momenta, it's a little different 959 01:07:10,890 --> 01:07:13,230 since fields carry momenta. 960 01:07:13,230 --> 01:07:16,290 And so what you should think is that the fields are allowed 961 01:07:16,290 --> 01:07:18,540 to sort of carry these momenta, but the Feynman rules 962 01:07:18,540 --> 01:07:21,228 of the leading order theory are just not sensitive to them. 963 01:07:21,228 --> 01:07:22,770 And that's the right way of thinking. 964 01:07:58,030 --> 01:08:00,700 OK, so what we need is an expansion 965 01:08:00,700 --> 01:08:02,940 of the theory that will do this. 966 01:08:02,940 --> 01:08:04,690 And that's called the multipole expansion. 967 01:08:26,142 --> 01:08:27,600 So let me remind you of some things 968 01:08:27,600 --> 01:08:30,750 about multipole expansion. 969 01:08:30,750 --> 01:08:33,180 Well, a multipole expansion you see in electromagnetism, 970 01:08:33,180 --> 01:08:34,590 remember? 971 01:08:34,590 --> 01:08:37,410 You look at a charge distribution far away. 972 01:08:37,410 --> 01:08:39,951 You see the total interval over that charge distribution 973 01:08:39,951 --> 01:08:40,784 and then the dipole. 974 01:08:43,817 --> 01:08:45,359 That's the usual thing that you think 975 01:08:45,359 --> 01:08:47,484 of when you think of the words multipole expansion. 976 01:08:47,484 --> 01:08:58,510 You think of E&M. And it's similar here, 977 01:08:58,510 --> 01:09:01,600 except we're doing it for fields. 978 01:09:01,600 --> 01:09:08,010 So in position space, it's similar to this where we expand 979 01:09:08,010 --> 01:09:10,380 and we neglect some coordinates. 980 01:09:10,380 --> 01:09:13,109 So let me show you how a multipole expansion works 981 01:09:13,109 --> 01:09:14,744 in one dimension in position space. 982 01:09:18,920 --> 01:09:20,630 So let's consider the following. 983 01:09:20,630 --> 01:09:26,164 Consider an integral dx psi bar of x A 984 01:09:26,164 --> 01:09:30,890 of 0 psi bar of psi of x. 985 01:09:30,890 --> 01:09:33,500 So one of the fields I stuck at 0 I expanded out 986 01:09:33,500 --> 01:09:37,609 the coordinate around 0, whereas the other ones I kept the x. 987 01:09:37,609 --> 01:09:42,649 Let's see what this does for the momentum space 988 01:09:42,649 --> 01:09:43,910 by Fourier transforming it-- 989 01:09:46,720 --> 01:10:04,187 so some momenta, some phases, and then 990 01:10:04,187 --> 01:10:05,395 some momentum space vehicles. 991 01:10:12,590 --> 01:10:14,360 And I just put 0 because I put one 992 01:10:14,360 --> 01:10:16,730 of the fields in the coordinate space at 0. 993 01:10:20,000 --> 01:10:20,750 Do the x integral. 994 01:10:26,270 --> 01:10:29,810 So I got a delta function that says P1 is equal to P2 995 01:10:29,810 --> 01:10:31,580 because I didn't have anything in the k. 996 01:10:39,860 --> 01:10:42,440 And this is kind of along the lines of what we want. 997 01:10:42,440 --> 01:10:46,490 It says that the momentum P1 is equal to the momentum P2. 998 01:10:46,490 --> 01:10:50,270 And that's like expanding in k. 999 01:10:50,270 --> 01:10:55,020 So if we think about having some Feynman diagram where 1000 01:10:55,020 --> 01:11:04,050 we have P1 and P2 and some k coming in, 1001 01:11:04,050 --> 01:11:07,100 if we want in some component that P1 is equal to P2, 1002 01:11:07,100 --> 01:11:08,300 then this is achieving that. 1003 01:11:19,450 --> 01:11:22,620 OK, so this is the analog of the kind 1004 01:11:22,620 --> 01:11:25,770 of expanding coordinates that you do in E&M. Expanding 1005 01:11:25,770 --> 01:11:28,060 into coordinates of a field is doing the same thing. 1006 01:11:28,060 --> 01:11:30,300 It's giving a momentum conservation 1007 01:11:30,300 --> 01:11:32,370 that ignores the momentum conjugate to that 1008 01:11:32,370 --> 01:11:33,660 coordinate that I expand in. 1009 01:11:42,367 --> 01:11:44,200 So what would the next order term look like? 1010 01:11:59,100 --> 01:12:03,100 So we're still here in one dimension. 1011 01:12:03,100 --> 01:12:04,900 If you think about the expansion, 1012 01:12:04,900 --> 01:12:09,070 the next order term would be a term like this, 1013 01:12:09,070 --> 01:12:12,570 then at x equals 0. 1014 01:12:12,570 --> 01:12:13,070 Sorry. 1015 01:12:15,820 --> 01:12:20,697 I don't really need dot since this is one dimension. 1016 01:12:23,318 --> 01:12:25,360 So that would be the next term in a Taylor series 1017 01:12:25,360 --> 01:12:28,720 of the field about x equals 0. 1018 01:12:28,720 --> 01:12:35,680 And if you do the same thing, Fourier transform this guy, 1019 01:12:35,680 --> 01:12:37,570 you can work out that you get the following. 1020 01:12:53,220 --> 01:12:55,910 OK, so the derivative here on the coordinate, 1021 01:12:55,910 --> 01:12:59,135 that comes with k, momentum. 1022 01:12:59,135 --> 01:13:01,010 Because I should Fourier transform the field, 1023 01:13:01,010 --> 01:13:04,430 then take the derivative, then set the coordinate to 0, 1024 01:13:04,430 --> 01:13:06,890 so I can still get a k. 1025 01:13:06,890 --> 01:13:10,600 And this x actually just ends up leading to a delta prime. 1026 01:13:10,600 --> 01:13:12,080 You can write it as a derivative. 1027 01:13:12,080 --> 01:13:14,850 And x can be written as a derivative in momentum space. 1028 01:13:14,850 --> 01:13:17,630 And then it's a derivative on the delta function. 1029 01:13:17,630 --> 01:13:22,980 OK, so that's what the second order term would look like. 1030 01:13:22,980 --> 01:13:25,400 So if you were to think like this, 1031 01:13:25,400 --> 01:13:27,500 you have some kind of Feynman rule 1032 01:13:27,500 --> 01:13:30,230 that's proportional to a delta prime. 1033 01:13:30,230 --> 01:13:33,710 And then however you deal with a delta prime, 1034 01:13:33,710 --> 01:13:36,197 if you're Feynman rule had a delta prime, what you would do 1035 01:13:36,197 --> 01:13:37,530 is you would integrate by parts. 1036 01:13:37,530 --> 01:13:38,990 And then the delta prime would start taking 1037 01:13:38,990 --> 01:13:40,190 derivatives of other things. 1038 01:13:44,910 --> 01:13:47,098 So that is a possible route, but we're not 1039 01:13:47,098 --> 01:13:48,140 going to take that route. 1040 01:13:48,140 --> 01:13:49,640 We're going to do something simpler. 1041 01:13:53,080 --> 01:13:58,180 So we could formulate the multipole expansion in position 1042 01:13:58,180 --> 01:14:01,180 space like that, but we're actually 1043 01:14:01,180 --> 01:14:03,400 going to formulate it in momentum space. 1044 01:14:20,590 --> 01:14:22,090 So we're going to do the same thing, 1045 01:14:22,090 --> 01:14:23,882 but we're going to do it in momentum space. 1046 01:14:26,310 --> 01:14:28,950 What are some reasons to do that? 1047 01:14:28,950 --> 01:14:30,660 Well, when you do Feynman diagrams, 1048 01:14:30,660 --> 01:14:32,970 you do them in momentum space. 1049 01:14:32,970 --> 01:14:35,650 So if we've already formulated things in momentum space, 1050 01:14:35,650 --> 01:14:37,930 then we'll immediately get the Feynman rules 1051 01:14:37,930 --> 01:14:40,230 we want to do loop integrals and stuff like that. 1052 01:14:44,088 --> 01:14:46,630 So that's an advantage of the momentum space is that you more 1053 01:14:46,630 --> 01:14:48,310 directly get the Feynman rules you want 1054 01:14:48,310 --> 01:14:49,930 and the diagrams in the form you want. 1055 01:14:54,590 --> 01:14:58,340 It simplifies, actually, slightly the formulation 1056 01:14:58,340 --> 01:15:08,000 or somewhat the formulation of gauge transformations, 1057 01:15:08,000 --> 01:15:09,440 which we'll talk about later on. 1058 01:15:12,210 --> 01:15:17,130 And what I think is a nice kind of advantage 1059 01:15:17,130 --> 01:15:21,360 is where the momentum expansion sits is in the propagator, 1060 01:15:21,360 --> 01:15:22,200 not in the vertices. 1061 01:15:30,320 --> 01:15:32,210 We don't get Feynman rules with delta primes, 1062 01:15:32,210 --> 01:15:36,080 but we then have to integrate by parts to understand. 1063 01:15:36,080 --> 01:15:38,330 We just immediately get insertions 1064 01:15:38,330 --> 01:15:40,700 on propagators which would correspond to the derivatives 1065 01:15:40,700 --> 01:15:42,560 that I would take from integrating 1066 01:15:42,560 --> 01:15:45,320 that delta prime by parts. 1067 01:15:51,780 --> 01:15:57,080 So this term that we saw before that looked like a propagator 1068 01:15:57,080 --> 01:15:58,410 squared-- 1069 01:15:58,410 --> 01:16:01,775 which was when we were sitting in momentum space 1070 01:16:01,775 --> 01:16:03,770 and we expanded the propagator to second order. 1071 01:16:07,170 --> 01:16:13,040 So we had this term that was ultra soft perp dot Pn perp. 1072 01:16:13,040 --> 01:16:16,120 And then it had this denominator that was squared. 1073 01:16:16,120 --> 01:16:17,850 If you ask where in the effective theory 1074 01:16:17,850 --> 01:16:21,540 is that, what is going to be the thing that corresponds to that, 1075 01:16:21,540 --> 01:16:27,330 it's going to correspond to some collinear propagator 1076 01:16:27,330 --> 01:16:29,550 with an insertion on it. 1077 01:16:29,550 --> 01:16:31,200 And the insertion gives this numerator. 1078 01:16:31,200 --> 01:16:33,710 And then I have two propagators. 1079 01:16:33,710 --> 01:16:35,220 And that gives the two denominators. 1080 01:16:39,910 --> 01:16:42,690 OK, so that's how that term is going to come about. 1081 01:16:52,497 --> 01:16:54,330 So there will be some sub-leading Lagrangian 1082 01:16:54,330 --> 01:16:57,480 that I could call L1 that I can insert on the propagator. 1083 01:16:57,480 --> 01:17:01,750 And that would give rise to these terms that we were saying 1084 01:17:01,750 --> 01:17:04,240 were higher order. 1085 01:17:04,240 --> 01:17:08,550 OK, so we have to figure out how to do this. 1086 01:17:08,550 --> 01:17:11,050 And I will just give a little bit of a picture for it today, 1087 01:17:11,050 --> 01:17:12,800 and then we'll continue with it next time. 1088 01:17:15,460 --> 01:17:17,905 So I'm going to have to back up a bit. 1089 01:17:17,905 --> 01:17:20,155 So let's back up to the Lagrangian that we had before. 1090 01:17:24,220 --> 01:17:27,520 And I want to just introduce, since we were not 1091 01:17:27,520 --> 01:17:29,890 close to being done-- 1092 01:17:29,890 --> 01:17:33,520 we had sort of started down the path, but we weren't done yet. 1093 01:17:33,520 --> 01:17:37,180 Let me give a slightly different name to this field. 1094 01:17:39,730 --> 01:17:40,810 Let me put a hat on it. 1095 01:17:48,040 --> 01:17:50,710 So I want to do the expansion in momentum space. 1096 01:17:50,710 --> 01:17:53,540 So let me Fourier transform. 1097 01:17:53,540 --> 01:18:02,110 So I'll call psi n twiddle of P, do the full Fourier transform 1098 01:18:02,110 --> 01:18:10,090 of this guy, which was previously just a field x. 1099 01:18:10,090 --> 01:18:11,890 And we'll think about doing expansions 1100 01:18:11,890 --> 01:18:14,350 in this momentum space field. 1101 01:18:14,350 --> 01:18:18,790 And for now, I'm also going to make one other simplification. 1102 01:18:18,790 --> 01:18:21,910 And that is to only consider the quark part. 1103 01:18:21,910 --> 01:18:23,920 And we'll put the antiquarks back in later. 1104 01:18:27,450 --> 01:18:30,550 So if you like, we're considering sort of the a's and 1105 01:18:30,550 --> 01:18:33,550 not the b's in the decomposition of the field. 1106 01:18:36,025 --> 01:18:37,650 OK, so let's think about this and think 1107 01:18:37,650 --> 01:18:39,990 about how we can do an expansion that would correspond 1108 01:18:39,990 --> 01:18:42,810 to this multipole expansion. 1109 01:18:42,810 --> 01:18:46,920 And the analogy that I'm going to exploit here, 1110 01:18:46,920 --> 01:18:50,160 although it's not perfect, is the expansion 1111 01:18:50,160 --> 01:18:52,690 that we did in HQET. 1112 01:18:52,690 --> 01:18:59,678 So in HQET, we said we had a full momentum for the b quark. 1113 01:18:59,678 --> 01:19:02,220 And we split it into a piece, which is kind of the big piece. 1114 01:19:02,220 --> 01:19:05,280 And then we had a piece that was the small piece. 1115 01:19:05,280 --> 01:19:08,400 And I'm going to do something similar in SCET. 1116 01:19:08,400 --> 01:19:13,000 I'm going to say I have a full momentum-- 1117 01:19:13,000 --> 01:19:16,140 I'm going to move the indices up-- 1118 01:19:16,140 --> 01:19:19,560 that I'm going to split into two pieces because of the power 1119 01:19:19,560 --> 01:19:25,820 counting, a big piece and a small piece. 1120 01:19:25,820 --> 01:19:31,900 The big piece here will be the terms which 1121 01:19:31,900 --> 01:19:34,630 were of order 1 or lambda. 1122 01:19:34,630 --> 01:19:36,698 And then the rest here will be something 1123 01:19:36,698 --> 01:19:37,990 that's of order lambda squared. 1124 01:19:45,200 --> 01:19:46,825 And I allow that residual piece not 1125 01:19:46,825 --> 01:19:48,200 just to be in the plus direction, 1126 01:19:48,200 --> 01:19:49,370 but to be in any direction. 1127 01:19:49,370 --> 01:19:51,738 Because remember, at sub-leading orders, 1128 01:19:51,738 --> 01:19:54,280 I'm going to have to care about the other components as well. 1129 01:19:54,280 --> 01:19:57,380 So let me just say that the residual could 1130 01:19:57,380 --> 01:20:00,290 have order lambda squared pieces in any component. 1131 01:20:00,290 --> 01:20:01,790 In the minus and the perp, I'm going 1132 01:20:01,790 --> 01:20:06,440 to expand with these residuals being smaller than these terms 1133 01:20:06,440 --> 01:20:07,790 here. 1134 01:20:07,790 --> 01:20:11,480 And this momentum I'm going to call a label momentum. 1135 01:20:11,480 --> 01:20:14,840 And this momentum I'm going to call residual momentum. 1136 01:20:14,840 --> 01:20:17,540 We'll see why that's a good language next time, 1137 01:20:17,540 --> 01:20:21,140 but it's kind of connected to the idea that, in the HQET, 1138 01:20:21,140 --> 01:20:23,750 we labeled the fields by this V. And we're 1139 01:20:23,750 --> 01:20:26,240 going to see that, for at least some of the interactions, 1140 01:20:26,240 --> 01:20:28,190 it's a reasonable way of thinking. 1141 01:20:28,190 --> 01:20:32,570 These large momentum is just labels on the fields. 1142 01:20:32,570 --> 01:20:35,040 OK, so we'll talk more about that next time. 1143 01:20:35,040 --> 01:20:37,850 I think I don't want to go further with it. 1144 01:20:37,850 --> 01:20:41,180 But basically, we're going to make an expansion in residuals 1145 01:20:41,180 --> 01:20:43,220 being smaller than these guys. 1146 01:20:43,220 --> 01:20:44,720 And in order to make that expansion, 1147 01:20:44,720 --> 01:20:48,290 I have to have some notation for the different pieces. 1148 01:20:48,290 --> 01:20:51,898 And we'll see where that leads us next time. 1149 01:20:51,898 --> 01:20:53,690 It'll effectively lead us to the same thing 1150 01:20:53,690 --> 01:20:56,330 as this multipole expansion in position space, 1151 01:20:56,330 --> 01:20:58,700 but it's a little nicer, I claim. 1152 01:20:58,700 --> 01:20:59,720 I like it a little more. 1153 01:21:03,810 --> 01:21:04,560 Questions? 1154 01:21:09,260 --> 01:21:10,534 Good.