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 00:00:24,850 --> 00:00:28,220 IAIN STEWART: All right, so this is where we got to last time. 9 00:00:28,220 --> 00:00:30,970 So we were on the road to deriving the leading order SCET 10 00:00:30,970 --> 00:00:32,140 Lagrangian. 11 00:00:32,140 --> 00:00:34,000 We said that there were some things 12 00:00:34,000 --> 00:00:38,080 that we had to deal with, in particular expanding it. 13 00:00:38,080 --> 00:00:41,140 And we started out, as a first step, 14 00:00:41,140 --> 00:00:44,200 getting rid of two of the components of the field 15 00:00:44,200 --> 00:00:47,200 that we said are not going to be ones that we project 16 00:00:47,200 --> 00:00:49,240 onto in the high energy limit. 17 00:00:49,240 --> 00:00:53,100 And that led us to this Lagrangian right here, 18 00:00:53,100 --> 00:00:54,920 but we haven't yet expanded it. 19 00:00:54,920 --> 00:00:58,070 We haven't yet separated the ultra soft and collinear 20 00:00:58,070 --> 00:01:02,050 momenta or the ultra soft and collinear gauge fields. 21 00:01:02,050 --> 00:01:04,569 And so in order to do that, I argued 22 00:01:04,569 --> 00:01:06,700 that you have to make multipole expansion. 23 00:01:06,700 --> 00:01:08,200 And I talked a little bit about what 24 00:01:08,200 --> 00:01:10,915 the multipole expansion would look like in position space. 25 00:01:10,915 --> 00:01:13,540 And then I said, it's better for us to do it in momentum space. 26 00:01:13,540 --> 00:01:14,623 And we started to do that. 27 00:01:14,623 --> 00:01:16,833 And we're going to continue to do that today. 28 00:01:16,833 --> 00:01:18,875 So first of all, let's just go to momentum space. 29 00:01:18,875 --> 00:01:21,640 So I've called the field here-- since it's not the final field 30 00:01:21,640 --> 00:01:25,600 that we want, I introduced a crummy notation for it, Cn hat. 31 00:01:25,600 --> 00:01:28,870 And once we get to the final field, it won't have a hat. 32 00:01:28,870 --> 00:01:32,600 And that'll be the one that we keep using from then on. 33 00:01:32,600 --> 00:01:35,380 So let's take this hatted guy, which is not the final guy, 34 00:01:35,380 --> 00:01:38,560 Fourier transform him and talk about doing things 35 00:01:38,560 --> 00:01:40,120 in momentum space. 36 00:01:40,120 --> 00:01:42,123 In momentum space, I'm going to make an analogy 37 00:01:42,123 --> 00:01:43,165 with what we did in HQET. 38 00:01:43,165 --> 00:01:46,535 In HQET, we split the momentum into two pieces. 39 00:01:46,535 --> 00:01:48,910 We had a piece that was large and a piece that was small. 40 00:01:48,910 --> 00:01:50,140 And then we could expand. 41 00:01:50,140 --> 00:01:52,060 I'm going to do the same thing here, 42 00:01:52,060 --> 00:01:54,190 split it into a large piece and a small piece 43 00:01:54,190 --> 00:01:56,110 where I'll call the large pieces the one 44 00:01:56,110 --> 00:01:57,933 in the lambda components. 45 00:01:57,933 --> 00:01:59,350 And then the small piece will just 46 00:01:59,350 --> 00:02:02,990 be all the pieces that are of order lambda squared. 47 00:02:02,990 --> 00:02:06,850 So if you draw a picture for this, it looks as follows. 48 00:02:09,729 --> 00:02:12,250 How should I think about what I'm doing here? 49 00:02:12,250 --> 00:02:14,900 Here's how I want you to think about it. 50 00:02:14,900 --> 00:02:19,030 So let's draw a two-dimensional picture in minus perp space. 51 00:02:19,030 --> 00:02:23,510 In plus space, there's no split because there's no plus here. 52 00:02:23,510 --> 00:02:25,652 So it's really in the minus and the perp space 53 00:02:25,652 --> 00:02:26,860 that we have to make a split. 54 00:02:31,330 --> 00:02:33,880 Yeah, let's see. 55 00:02:33,880 --> 00:02:39,385 All right, so I'm going to draw a grid. 56 00:02:52,010 --> 00:02:54,290 And the way I want to make this split precise 57 00:02:54,290 --> 00:02:55,970 is I'm going to think-- 58 00:02:55,970 --> 00:02:58,910 OK, so I should have maybe made my axes in a different color 59 00:02:58,910 --> 00:03:01,392 to distinguish them from the grid. 60 00:03:01,392 --> 00:03:02,435 Let's make them orange. 61 00:03:09,120 --> 00:03:12,090 So the way I want to think about these two types of momenta 62 00:03:12,090 --> 00:03:14,520 is I'm going to think about these Pl's as discrete 63 00:03:14,520 --> 00:03:16,870 and the Pr's as continuous. 64 00:03:16,870 --> 00:03:18,930 So let's think about the Pl's as labeling 65 00:03:18,930 --> 00:03:21,960 these points at the center of each 66 00:03:21,960 --> 00:03:24,810 of these boxes in the grid. 67 00:03:32,192 --> 00:03:33,400 This one was supposed to be-- 68 00:03:40,264 --> 00:03:42,110 that's fine. 69 00:03:42,110 --> 00:03:43,730 My squares are not the same size, 70 00:03:43,730 --> 00:03:45,680 but they roughly should be. 71 00:03:45,680 --> 00:03:48,140 And so then we can think about, if we specify 72 00:03:48,140 --> 00:03:49,650 some momentum in the space-- 73 00:03:49,650 --> 00:03:52,550 let's say we specify momentum that points to this point right 74 00:03:52,550 --> 00:03:53,307 here. 75 00:03:53,307 --> 00:03:54,890 The way that I can point to that point 76 00:03:54,890 --> 00:03:59,810 is I take one vector that points to the center. 77 00:03:59,810 --> 00:04:02,150 And then I take another small vector 78 00:04:02,150 --> 00:04:04,400 that points to that point. 79 00:04:04,400 --> 00:04:05,990 The large vector is the Pl. 80 00:04:05,990 --> 00:04:08,990 The small vector is the Pr. 81 00:04:08,990 --> 00:04:10,970 OK, so this guy here is Pl. 82 00:04:14,060 --> 00:04:22,730 Maybe I should make the other one a different color, Pr. 83 00:04:26,450 --> 00:04:28,700 And so the way that we should think about this 84 00:04:28,700 --> 00:04:29,902 is in the following sense. 85 00:04:29,902 --> 00:04:31,610 We should think about the average spacing 86 00:04:31,610 --> 00:04:34,697 in between these grid points here 87 00:04:34,697 --> 00:04:36,530 is something that gets a power counting that 88 00:04:36,530 --> 00:04:39,170 makes it of order Q lambda. 89 00:04:39,170 --> 00:04:43,460 And the average spacing between grid points in this direction 90 00:04:43,460 --> 00:04:47,360 is something that makes it order 1. 91 00:04:47,360 --> 00:04:49,880 But if I ask about distances inside a box, 92 00:04:49,880 --> 00:04:52,153 if I ask about a distance like this one, 93 00:04:52,153 --> 00:04:53,445 that's of order lambda squared. 94 00:05:00,780 --> 00:05:02,750 So to get between boxes, you need 95 00:05:02,750 --> 00:05:05,480 to make a big jump to the larger momentum. 96 00:05:05,480 --> 00:05:08,510 But to move around in a box, that's only small momenta. 97 00:05:08,510 --> 00:05:11,360 AUDIENCE: So why can't you have order lambda 98 00:05:11,360 --> 00:05:15,668 basically inside the box in the P minus dimension? 99 00:05:15,668 --> 00:05:16,460 IAIN STEWART: Yeah. 100 00:05:16,460 --> 00:05:19,280 So you could. 101 00:05:19,280 --> 00:05:22,785 So you can ask, where is in order lambda momentum in the P 102 00:05:22,785 --> 00:05:23,285 minus? 103 00:05:23,285 --> 00:05:24,650 AUDIENCE: Yeah. 104 00:05:24,650 --> 00:05:26,420 IAIN STEWART: And it's in a nether land. 105 00:05:31,513 --> 00:05:32,930 Either way, you can think about it 106 00:05:32,930 --> 00:05:34,550 as being part of this one or that one. 107 00:05:34,550 --> 00:05:35,300 It doesn't matter. 108 00:05:35,300 --> 00:05:40,730 Because technically, if I just talked about minus, 109 00:05:40,730 --> 00:05:42,740 I'd only be interested in separating a 1 110 00:05:42,740 --> 00:05:44,240 and a lambda squared. 111 00:05:44,240 --> 00:05:46,760 And whether I call this lambda squared or I call it 112 00:05:46,760 --> 00:05:48,630 lambda or I call it eta-- 113 00:05:48,630 --> 00:05:49,880 let eta equal lambda squared-- 114 00:05:49,880 --> 00:05:52,950 I'm only trying to separate two things. 115 00:05:52,950 --> 00:05:56,082 So if you talk about some momentum that's in between, 116 00:05:56,082 --> 00:05:58,040 you could ask the same question actually about, 117 00:05:58,040 --> 00:06:02,572 what about a lambda to the 3/2 momentum in the perp space? 118 00:06:02,572 --> 00:06:05,030 And that's the same question you just asked about the minus 119 00:06:05,030 --> 00:06:06,862 space, right? 120 00:06:06,862 --> 00:06:08,570 And that's something that we don't really 121 00:06:08,570 --> 00:06:10,850 care about because there's not going to be momenta 122 00:06:10,850 --> 00:06:12,797 that are scaling like that. 123 00:06:12,797 --> 00:06:14,630 So we don't really care about separating it. 124 00:06:17,760 --> 00:06:25,275 OK, so Pl are discrete grid points. 125 00:06:28,230 --> 00:06:29,760 And Pr-- continuous. 126 00:06:37,720 --> 00:06:41,500 And this picture also makes clear that, if you take any P, 127 00:06:41,500 --> 00:06:45,750 it uniquely is given by some Pl and some Pr. 128 00:06:45,750 --> 00:06:48,300 If I point at any place in this grid, 129 00:06:48,300 --> 00:06:50,760 you can tell me exactly which Pl I should pick 130 00:06:50,760 --> 00:06:52,190 and then what Pr I should pick. 131 00:06:57,820 --> 00:07:00,252 This is, in some ways, just going to be a tool. 132 00:07:00,252 --> 00:07:01,960 And in the end of the day, we can kind of 133 00:07:01,960 --> 00:07:03,085 dispense with this picture. 134 00:07:03,085 --> 00:07:07,030 But it'll be a useful way of thinking about what's going on 135 00:07:07,030 --> 00:07:09,680 and getting to sort of the final results that allow us 136 00:07:09,680 --> 00:07:10,930 to dispense with this picture. 137 00:07:13,580 --> 00:07:16,060 So what about integration? 138 00:07:16,060 --> 00:07:19,750 So if I integrate over all P and P is collinear, 139 00:07:19,750 --> 00:07:21,890 what does that mean? 140 00:07:21,890 --> 00:07:25,210 So it corresponds to summing over all the grid points 141 00:07:25,210 --> 00:07:33,280 and then integrating over all the residuals 142 00:07:33,280 --> 00:07:37,720 except for one location. 143 00:07:37,720 --> 00:07:43,210 And that's the special box here that has label equal 0 144 00:07:43,210 --> 00:07:45,580 because we know that the power counting of the collinear 145 00:07:45,580 --> 00:07:48,940 momentum is such that it always has to have a non-0 Pl. 146 00:07:48,940 --> 00:07:55,000 And so the one box where Pl is equal to 0, 0, 147 00:07:55,000 --> 00:07:58,225 that particular box doesn't have a collinear momentum. 148 00:08:09,820 --> 00:08:11,710 And actually, that box is exactly what 149 00:08:11,710 --> 00:08:15,590 you think of as the ultra soft momentum. 150 00:08:15,590 --> 00:08:21,010 So if you have an integral d4P and P is ultra soft, 151 00:08:21,010 --> 00:08:22,870 there's no sum. 152 00:08:22,870 --> 00:08:25,240 And you just have interval d4Pr. 153 00:08:25,240 --> 00:08:29,590 And that's like integrating over the 0, 0 location in the grid. 154 00:08:42,169 --> 00:08:43,960 OK, so you can see that together we 155 00:08:43,960 --> 00:08:47,620 cover all possible places for an integration. 156 00:08:47,620 --> 00:08:51,280 And we actually don't do any double counting 157 00:08:51,280 --> 00:08:51,970 with this setup. 158 00:08:54,890 --> 00:08:56,600 What am I going to do with my fields? 159 00:08:56,600 --> 00:08:59,360 Well, because of this split into two different pieces, 160 00:08:59,360 --> 00:09:03,020 I'm going to say that my fields actually can be labeled by Pl 161 00:09:03,020 --> 00:09:12,320 and carry Pr also as an index or as a function of Pr. 162 00:09:12,320 --> 00:09:14,300 So in order to make this explicit, 163 00:09:14,300 --> 00:09:18,260 we're going to write our Fourier transform 164 00:09:18,260 --> 00:09:28,340 guy with a label Pl and an argument Pr like that. 165 00:09:28,340 --> 00:09:33,630 So the name label comes from the fact that we do this. 166 00:09:33,630 --> 00:09:36,660 And there will be some analogies with why 167 00:09:36,660 --> 00:09:38,380 we called this a label in HQET. 168 00:09:38,380 --> 00:09:41,220 We'll see to what extent that analogy carries through 169 00:09:41,220 --> 00:09:43,360 in a minute. 170 00:09:43,360 --> 00:09:46,260 Another thing you should note from this grid picture 171 00:09:46,260 --> 00:09:48,300 is that you have separate momentum conservation 172 00:09:48,300 --> 00:09:49,550 in the label and the residual. 173 00:10:10,500 --> 00:10:17,150 So if you think about some integral d4x of something 174 00:10:17,150 --> 00:10:27,202 that you split between label pieces and residual pieces, 175 00:10:27,202 --> 00:10:28,910 then the right way of thinking about this 176 00:10:28,910 --> 00:10:35,780 is that you have a discrete [INAUDIBLE] delta for the label 177 00:10:35,780 --> 00:10:39,650 and then a continuous Dirac delta 178 00:10:39,650 --> 00:10:42,584 for the continuous variables. 179 00:10:42,584 --> 00:10:44,060 And there's some 2 pis. 180 00:10:50,740 --> 00:10:52,870 And so this is what would be meant by-- 181 00:10:52,870 --> 00:10:54,760 if you have two momenta that are equal, 182 00:10:54,760 --> 00:10:57,460 you know that they're in the same grid square. 183 00:10:57,460 --> 00:11:00,790 And then they have the same little green arrow as well. 184 00:11:00,790 --> 00:11:03,700 That's what that's saying. 185 00:11:03,700 --> 00:11:07,240 So when we talked about what happens 186 00:11:07,240 --> 00:11:09,820 with the multipole expansion in position space, 187 00:11:09,820 --> 00:11:11,200 we said that effectively you have 188 00:11:11,200 --> 00:11:13,730 a non-conservation of momenta. 189 00:11:13,730 --> 00:11:14,230 OK. 190 00:11:14,230 --> 00:11:15,470 We had a field at 0. 191 00:11:15,470 --> 00:11:16,992 We worked through what that meant. 192 00:11:16,992 --> 00:11:18,700 And it meant, basically, that the momenta 193 00:11:18,700 --> 00:11:20,825 was conserved on the other line, but not on the one 194 00:11:20,825 --> 00:11:22,880 that you were doing the expansion on. 195 00:11:22,880 --> 00:11:24,970 The nice thing about this type of setup 196 00:11:24,970 --> 00:11:27,190 is that we have conservation of momenta, 197 00:11:27,190 --> 00:11:29,900 but we actually have two conservations of momenta. 198 00:11:29,900 --> 00:11:31,687 So we don't give up momentum conservation. 199 00:11:31,687 --> 00:11:33,520 It's just that we go over to a picture where 200 00:11:33,520 --> 00:11:35,240 we have two of them. 201 00:11:35,240 --> 00:11:38,840 So let me show you how that works. 202 00:11:38,840 --> 00:11:41,110 Let's say we have an ultra soft particle, 203 00:11:41,110 --> 00:11:43,900 and it's hitting a collinear quark. 204 00:11:43,900 --> 00:11:45,880 The collinear quark, because it's collinear, 205 00:11:45,880 --> 00:11:47,770 carries two types of momentum. 206 00:11:47,770 --> 00:11:51,670 It carries a label P and a residual. 207 00:11:51,670 --> 00:11:55,235 But the ultra soft guy, it's only residual. 208 00:11:57,817 --> 00:11:59,275 So let's just call that k residual. 209 00:12:02,700 --> 00:12:05,350 There's no label to that ultra soft guy. 210 00:12:05,350 --> 00:12:08,490 So if I just look then what this guy's momentum is, 211 00:12:08,490 --> 00:12:13,958 it's P label for the large piece because that is not 212 00:12:13,958 --> 00:12:15,250 affected by the ultra soft guy. 213 00:12:15,250 --> 00:12:18,025 And then it's P residual plus k residual. 214 00:12:20,940 --> 00:12:25,740 OK, so we have a conservation in the residual space, 215 00:12:25,740 --> 00:12:27,750 which is the standard one. 216 00:12:27,750 --> 00:12:29,760 And we have a conservation in the label space, 217 00:12:29,760 --> 00:12:33,580 but the ultra soft particles just don't contribute to that. 218 00:12:33,580 --> 00:12:38,400 So we have two conservation laws. 219 00:13:14,150 --> 00:13:18,867 And we just know that certain fields can carry these label 220 00:13:18,867 --> 00:13:20,200 momentum and other fields can't. 221 00:13:24,950 --> 00:13:32,390 All right, well, since Pr here and kr, 222 00:13:32,390 --> 00:13:35,990 the residual components, is just standard quantum field theory-- 223 00:13:35,990 --> 00:13:38,930 the only thing that's special is the Pl. 224 00:13:38,930 --> 00:13:42,530 For the Pr, what we're going to do for the residual momenta is 225 00:13:42,530 --> 00:13:44,660 we're going to transform back to position space. 226 00:13:49,010 --> 00:13:51,730 And that's kind of what we do in HQET as well, right? 227 00:13:51,730 --> 00:13:55,040 We treat this guy in momentum space as a label and this guy 228 00:13:55,040 --> 00:13:58,170 as a position space for the residual. 229 00:14:00,740 --> 00:14:03,590 And the reason we really want to do that is related to locality. 230 00:14:03,590 --> 00:14:08,545 And I'll say some more words about that. 231 00:14:08,545 --> 00:14:09,420 AUDIENCE: [INAUDIBLE] 232 00:14:09,420 --> 00:14:09,750 IAIN STEWART: Yeah. 233 00:14:09,750 --> 00:14:10,917 AUDIENCE: I have a question. 234 00:14:10,917 --> 00:14:13,800 IAIN STEWART: Sure. 235 00:14:13,800 --> 00:14:15,498 AUDIENCE: So this top equation-- 236 00:14:15,498 --> 00:14:16,290 IAIN STEWART: Yeah. 237 00:14:16,290 --> 00:14:20,400 AUDIENCE: So are you bounding the d4Pr by lambda squared, 238 00:14:20,400 --> 00:14:23,760 or is there some reason why the integral should fall off 239 00:14:23,760 --> 00:14:25,330 before you get there or something? 240 00:14:25,330 --> 00:14:27,455 IAIN STEWART: Yeah, we're going to talk about that. 241 00:14:27,455 --> 00:14:30,925 So, so far, this is just notation. 242 00:14:30,925 --> 00:14:32,800 If you think about it from this grid picture, 243 00:14:32,800 --> 00:14:35,483 you'd think some hard walls put in. 244 00:14:35,483 --> 00:14:37,650 That's not quite the right way of thinking about it. 245 00:14:37,650 --> 00:14:40,920 It's actually not right for several reasons. 246 00:14:40,920 --> 00:14:42,870 And we will come back and talk much more 247 00:14:42,870 --> 00:14:45,390 about what the right way of thinking about it is. 248 00:14:45,390 --> 00:14:47,447 But the physical picture that you're in this box 249 00:14:47,447 --> 00:14:49,030 is the right way of thinking about it. 250 00:14:49,030 --> 00:14:51,600 The tricky thing is that these boxes are infinite. 251 00:14:51,600 --> 00:14:54,060 Each of these boxes is infinite. 252 00:14:54,060 --> 00:14:55,900 And we'll have to see why that is. 253 00:14:55,900 --> 00:14:57,450 In what I've told you so far, you'd 254 00:14:57,450 --> 00:15:01,890 think that the d4Pr would really be a box with some fixed edges. 255 00:15:01,890 --> 00:15:03,640 And it's going to turn out that the d4Pr-- 256 00:15:03,640 --> 00:15:05,807 we're going to use dimensional regularization-- it's 257 00:15:05,807 --> 00:15:07,147 going to be an infinite box. 258 00:15:07,147 --> 00:15:08,730 But we're still going to be able to do 259 00:15:08,730 --> 00:15:11,280 exactly what I'm telling you. 260 00:15:11,280 --> 00:15:12,893 And it's going to make sense. 261 00:15:12,893 --> 00:15:14,310 We're going to want infinite boxes 262 00:15:14,310 --> 00:15:17,640 to avoid breaking symmetries, like gauge symmetry and Lorentz 263 00:15:17,640 --> 00:15:19,420 symmetry. 264 00:15:19,420 --> 00:15:22,150 And we'll come back and see how that works. 265 00:15:22,150 --> 00:15:24,603 So in some sense, what I've told you is a Wilsonian story. 266 00:15:24,603 --> 00:15:26,520 And we're going to have to make it into a more 267 00:15:26,520 --> 00:15:28,890 continuum friendly story. 268 00:15:28,890 --> 00:15:31,440 But the moral of the story, the way 269 00:15:31,440 --> 00:15:33,630 you think about where these momentum modes live, 270 00:15:33,630 --> 00:15:37,410 what kind of momentum they carry, that is just true. 271 00:15:37,410 --> 00:15:38,910 So that part of the story, which you 272 00:15:38,910 --> 00:15:41,490 can think of so far as Wilsonian, is true. 273 00:15:41,490 --> 00:15:44,190 And it's OK to think like that. 274 00:15:48,430 --> 00:15:52,680 So we'll see actually what we do in practice a little later on. 275 00:16:01,560 --> 00:16:02,948 Where this will really come in is 276 00:16:02,948 --> 00:16:04,990 when we start wanting to think about doing loops, 277 00:16:04,990 --> 00:16:10,330 but we won't get there for a little while yet. 278 00:16:16,740 --> 00:16:20,100 So the conservation law in the residual 279 00:16:20,100 --> 00:16:21,600 just corresponds to locality. 280 00:16:24,210 --> 00:16:25,970 We learned all about locality yesterday 281 00:16:25,970 --> 00:16:27,180 in [INAUDIBLE] seminar. 282 00:16:35,930 --> 00:16:41,680 And so we can make things local by just going back 283 00:16:41,680 --> 00:16:53,140 to the x base for the residual momenta. 284 00:16:53,140 --> 00:16:55,900 OK, so we go back to the x base, which 285 00:16:55,900 --> 00:16:57,820 is the Fourier transform conjugate variable 286 00:16:57,820 --> 00:16:59,170 to the residual momenta. 287 00:17:02,720 --> 00:17:06,298 So the fields we're actually going to use 288 00:17:06,298 --> 00:17:14,050 are these ones, where we Fourier transform back 289 00:17:14,050 --> 00:17:23,050 in the residual, these guys here. 290 00:17:31,820 --> 00:17:33,550 So it's kind of like HQET, where you 291 00:17:33,550 --> 00:17:35,230 think about this part as momentum space 292 00:17:35,230 --> 00:17:37,960 and that part in position space. 293 00:17:37,960 --> 00:17:43,200 OK, so if you talk about ultra soft interactions-- 294 00:17:43,200 --> 00:17:45,010 you can see it from over there-- 295 00:17:45,010 --> 00:17:46,060 they conserve the labels. 296 00:17:48,590 --> 00:17:56,710 So there's label conservation of a field 297 00:17:56,710 --> 00:17:58,510 for ultra soft interactions. 298 00:17:58,510 --> 00:18:00,430 The label of the field here is the same 299 00:18:00,430 --> 00:18:03,672 as the label of the field there, so that ultra soft guys do not 300 00:18:03,672 --> 00:18:04,255 change labels. 301 00:18:08,992 --> 00:18:10,920 So this is Pl. 302 00:18:10,920 --> 00:18:12,770 This is Pl. 303 00:18:12,770 --> 00:18:16,090 And if this is ultra soft, it's still labeled by Pl. 304 00:18:16,090 --> 00:18:19,110 And the thing that can change is the residual momenta. 305 00:18:19,110 --> 00:18:21,030 And that's now encoded in locality 306 00:18:21,030 --> 00:18:24,600 in saying these fields all sit at the same point x. 307 00:18:31,950 --> 00:18:34,320 On the other hand, the collinear gluons cause us 308 00:18:34,320 --> 00:18:36,790 to hop between boxes. 309 00:18:36,790 --> 00:18:40,710 So the collinear gluons are label changing. 310 00:18:40,710 --> 00:18:43,830 And that's what's different than HQET. 311 00:18:43,830 --> 00:18:46,110 In HQET, we don't really have interactions 312 00:18:46,110 --> 00:18:48,870 in the theory in the Lagrangian that do that, but here we do. 313 00:18:53,150 --> 00:18:55,070 So if we have a collinear gluon come in, 314 00:18:55,070 --> 00:18:58,010 the collinear gluon, again, carries both labels 315 00:18:58,010 --> 00:19:00,980 and residuals if I just draw the labels. 316 00:19:00,980 --> 00:19:04,190 And this is some Ql coming in. 317 00:19:04,190 --> 00:19:06,470 And this will be Pl plus Ql. 318 00:19:06,470 --> 00:19:08,960 And so the label of the outgoing field has changed. 319 00:19:12,380 --> 00:19:14,390 There's one more label. 320 00:19:14,390 --> 00:19:16,200 And that's this n, right? 321 00:19:16,200 --> 00:19:21,440 We also label these guys by a direction. 322 00:19:21,440 --> 00:19:24,937 And that is unchanged also by the collinear interactions. 323 00:19:42,830 --> 00:19:46,380 So it's preserved by both ultra soft and collinear 324 00:19:46,380 --> 00:19:47,510 interactions. 325 00:19:56,630 --> 00:19:58,613 It can only be changed by hard interactions 326 00:19:58,613 --> 00:20:00,530 where you would have some hard production that 327 00:20:00,530 --> 00:20:03,420 could produce, say, two guys in two different directions. 328 00:20:03,420 --> 00:20:05,480 And those type of interactions are typically 329 00:20:05,480 --> 00:20:08,090 only going to occur once in your quantum field theory, 330 00:20:08,090 --> 00:20:10,880 kind of like a decay, a weak decay where it occurs once. 331 00:20:10,880 --> 00:20:13,010 And then you do perturbation theory around that. 332 00:20:13,010 --> 00:20:14,690 Hard interactions are that way, too. 333 00:20:14,690 --> 00:20:16,460 So as far as dynamics is concerned, 334 00:20:16,460 --> 00:20:19,400 this label n is something that you 335 00:20:19,400 --> 00:20:21,590 can think of as being conserved for the field. 336 00:20:21,590 --> 00:20:23,548 It's not going to be changed by the Lagrangian, 337 00:20:23,548 --> 00:20:26,210 for example, all right? 338 00:20:28,740 --> 00:20:30,672 OK, so, now, we have kind of what we want. 339 00:20:30,672 --> 00:20:32,130 And now, I want to write the action 340 00:20:32,130 --> 00:20:33,795 that we had before in terms of this guy 341 00:20:33,795 --> 00:20:34,920 and see what it looks like. 342 00:20:43,630 --> 00:20:48,340 This split of these guys into a Pl and Pr or Pl and x 343 00:20:48,340 --> 00:20:53,560 allows us to expand very easily in momenta. 344 00:20:53,560 --> 00:20:58,240 It's allowing us to do the multipole expansion because we 345 00:20:58,240 --> 00:20:59,490 have two things to talk about. 346 00:20:59,490 --> 00:21:00,700 And we just can compare them. 347 00:21:05,440 --> 00:21:08,170 So let's put the pieces together. 348 00:21:08,170 --> 00:21:17,820 So Cn hat of x is the full Fourier transform 349 00:21:17,820 --> 00:21:21,120 of this Cn twiddle. 350 00:21:21,120 --> 00:21:26,520 And that can be split for a collinear guy. 351 00:21:26,520 --> 00:21:31,025 We can split the measure, and we can split the phase. 352 00:21:35,370 --> 00:21:43,320 And we can write the field in this way. 353 00:21:43,320 --> 00:21:45,720 And then we decided that we wanted to use this field. 354 00:21:52,095 --> 00:21:53,850 Let me see, [INAUDIBLE] two pis here. 355 00:22:00,550 --> 00:22:04,170 So that means we take this measure, we take this phase, 356 00:22:04,170 --> 00:22:08,030 we take this guy, and we write it back in terms of that field. 357 00:22:08,030 --> 00:22:12,270 So that leaves, if we want to relate the field 358 00:22:12,270 --> 00:22:18,450 that their Lagrangian was in terms of in terms of this field 359 00:22:18,450 --> 00:22:27,930 that I told you we're going to use, it's this equation, OK? 360 00:22:27,930 --> 00:22:32,822 All right-- a little bit more formalism. 361 00:22:32,822 --> 00:22:35,280 So we're going to actually want to have two different types 362 00:22:35,280 --> 00:22:37,740 of derivatives, derivatives that can talk to the momentum 363 00:22:37,740 --> 00:22:39,825 conjugate to x and derivatives that give this Pl. 364 00:22:45,940 --> 00:22:48,460 So we define two types of derivative operators. 365 00:22:57,090 --> 00:22:59,910 One is just, if you like, a standard derivative 366 00:22:59,910 --> 00:23:05,970 in the x base of this guy, acts only on that coordinate. 367 00:23:05,970 --> 00:23:08,550 And that derivative scales like lambda squared. 368 00:23:11,840 --> 00:23:16,000 So that's residual, picking out talking to residual momenta. 369 00:23:22,690 --> 00:23:25,450 Then we need something that can give us the labels. 370 00:23:25,450 --> 00:23:26,920 So we define an operator that does 371 00:23:26,920 --> 00:23:29,380 that, which we'll call curly p. 372 00:23:33,100 --> 00:23:36,010 It doesn't care about x. 373 00:23:36,010 --> 00:23:37,820 And it's just defined to give the Pl 374 00:23:37,820 --> 00:23:40,810 mu of the field it acts on. 375 00:23:45,460 --> 00:23:48,670 So there's no plus component to this. 376 00:23:48,670 --> 00:23:51,395 There's an order 1 minus component. 377 00:23:51,395 --> 00:23:52,645 And then perp is order lambda. 378 00:23:57,190 --> 00:24:00,010 I want to say how big that guy is. 379 00:24:00,010 --> 00:24:03,065 So then we have a power counting that tells us how big the p is 380 00:24:03,065 --> 00:24:04,190 and how big the partial is. 381 00:24:04,190 --> 00:24:08,540 And we can compare them, and we can expand in them, OK? 382 00:24:08,540 --> 00:24:12,910 So the bottom line is that we now 383 00:24:12,910 --> 00:24:24,070 can just say that, like we had a comparison for the gauge field, 384 00:24:24,070 --> 00:24:26,411 we have a comparison for derivatives. 385 00:24:35,900 --> 00:24:38,420 And so this is exactly analogous to what 386 00:24:38,420 --> 00:24:41,600 we said before where we had M bar dot A ultra soft 387 00:24:41,600 --> 00:24:48,320 is much less than N bar dot A collinear and A perp ultra 388 00:24:48,320 --> 00:24:52,820 soft being much less than A perp collinear. 389 00:24:52,820 --> 00:24:55,340 And in fact, the way we're setting things up 390 00:24:55,340 --> 00:24:57,890 is going to make the gauge symmetry also easy. 391 00:24:57,890 --> 00:25:01,548 Because by gauge symmetry, these derivatives and these fields 392 00:25:01,548 --> 00:25:02,340 should go together. 393 00:25:02,340 --> 00:25:04,580 They are the same size in the power counting. 394 00:25:04,580 --> 00:25:08,610 And these guys should go with those guys. 395 00:25:08,610 --> 00:25:12,732 So we're kind of doing this same thing we thought of for fields 396 00:25:12,732 --> 00:25:13,940 we're doing for momenta here. 397 00:25:18,650 --> 00:25:20,710 So when we talk about gauge symmetry, 398 00:25:20,710 --> 00:25:32,360 this will make gauge symmetry easier, 399 00:25:32,360 --> 00:25:36,237 this homogeneity between the derivatives and the fields. 400 00:25:36,237 --> 00:25:38,320 So that's one of the powers of thinking like this. 401 00:25:49,890 --> 00:25:52,032 OK, so let's see that this notation is actually 402 00:25:52,032 --> 00:25:52,740 kind of friendly. 403 00:25:57,500 --> 00:26:02,010 So let me try to convince you of that. 404 00:26:02,010 --> 00:26:05,070 It looks kind of daunting, but maybe it's not so daunting. 405 00:26:09,830 --> 00:26:18,980 So let's look at this guy, which was equal to this. 406 00:26:23,910 --> 00:26:27,860 So what I can do is I can take this and replace it 407 00:26:27,860 --> 00:26:34,500 by e to the minus i label operator dot x, right? 408 00:26:34,500 --> 00:26:37,140 But the label operator then doesn't care about the sum. 409 00:26:37,140 --> 00:26:38,270 So I can push it through. 410 00:26:41,895 --> 00:26:45,290 So as an operator, it goes through the sum. 411 00:26:53,765 --> 00:26:54,760 I have that. 412 00:26:58,220 --> 00:27:06,650 And for a lot of purposes, in a sense of suppressing indices, 413 00:27:06,650 --> 00:27:08,420 we can just use a simplified notation 414 00:27:08,420 --> 00:27:10,100 for this guy where we suppress the sum. 415 00:27:13,640 --> 00:27:18,500 So we were thinking about this as a discrete sum. 416 00:27:18,500 --> 00:27:21,600 And we can always write out that sum if we get confused. 417 00:27:21,600 --> 00:27:23,120 But we could also adopt a notation 418 00:27:23,120 --> 00:27:24,470 where we suppress the sum. 419 00:27:24,470 --> 00:27:26,467 And for many manipulations, just like 420 00:27:26,467 --> 00:27:28,550 you might suppress some indices in a quantum field 421 00:27:28,550 --> 00:27:30,530 theory, that's perfectly fine. 422 00:27:48,450 --> 00:27:51,360 And having this label operator makes it easy to do that. 423 00:28:01,320 --> 00:28:03,890 So if you have, for example, field products, 424 00:28:03,890 --> 00:28:10,640 say I had two fermions, we try to make the notation work 425 00:28:10,640 --> 00:28:12,870 for us as nice as possible. 426 00:28:12,870 --> 00:28:15,980 Well, if you have a sum of momenta 427 00:28:15,980 --> 00:28:19,370 and then it's exponentiated, if you have a product 428 00:28:19,370 --> 00:28:20,830 and it's a product of exponentials, 429 00:28:20,830 --> 00:28:25,080 then it just becomes the sum in the exponential. 430 00:28:25,080 --> 00:28:30,543 So this is true if we do it for two fields, right? 431 00:28:30,543 --> 00:28:31,960 I'd have the label operator acting 432 00:28:31,960 --> 00:28:34,230 on this guy and that guy. 433 00:28:34,230 --> 00:28:37,760 And if I were to write out this twice with two sums, 434 00:28:37,760 --> 00:28:39,913 then because the product of those two exponentials 435 00:28:39,913 --> 00:28:42,330 can be just written as a single exponential, this is true. 436 00:28:45,270 --> 00:28:46,700 So this guy acts on both fields. 437 00:28:52,920 --> 00:28:55,850 So in this sense, the notation is friendly. 438 00:28:55,850 --> 00:28:58,297 And what this phase factor here is really doing for you, 439 00:28:58,297 --> 00:29:00,130 if you think about it, if you think about it 440 00:29:00,130 --> 00:29:02,660 in a whole product of fields, it's 441 00:29:02,660 --> 00:29:06,540 just when you Fourier transform giving the label conservation, 442 00:29:06,540 --> 00:29:07,040 OK? 443 00:29:07,040 --> 00:29:09,740 So there's going to be this kind of overall e to the minus 444 00:29:09,740 --> 00:29:11,000 i p dot x. 445 00:29:11,000 --> 00:29:13,170 And it's just sitting there to conserve momentum 446 00:29:13,170 --> 00:29:14,090 in the label space. 447 00:29:31,110 --> 00:29:42,980 So that's its purpose. 448 00:29:42,980 --> 00:29:45,260 OK, so there's one more step that we 449 00:29:45,260 --> 00:29:48,110 have to do before we start talking about the Lagrangian. 450 00:29:48,110 --> 00:29:49,610 And that's this little caveat that I 451 00:29:49,610 --> 00:29:51,277 had right at the beginning where I said, 452 00:29:51,277 --> 00:29:53,330 let's consider only quarks. 453 00:29:53,330 --> 00:29:55,490 So far, we've only considered the quark part, not 454 00:29:55,490 --> 00:29:56,420 the antiquark part. 455 00:29:59,898 --> 00:30:01,940 So let's see what we want to do about antiquarks. 456 00:30:05,250 --> 00:30:08,520 Technically, the way that the field theory wants to work 457 00:30:08,520 --> 00:30:10,110 is that the quarks and the antiquarks 458 00:30:10,110 --> 00:30:12,420 kind of want to be different fields, 459 00:30:12,420 --> 00:30:14,880 but that's very cumbersome. 460 00:30:14,880 --> 00:30:16,500 And so we're not going to do that. 461 00:30:16,500 --> 00:30:19,785 And we're going to find, again, sneaky notation to treat them 462 00:30:19,785 --> 00:30:21,660 as one field even though they want to be two. 463 00:30:25,852 --> 00:30:27,310 The reason that they want to be two 464 00:30:27,310 --> 00:30:29,868 is because, if you look at the phase factor for the quarks 465 00:30:29,868 --> 00:30:32,410 in the antiquarks in the kind of standard mode decomposition, 466 00:30:32,410 --> 00:30:35,680 it has the opposite sign. 467 00:30:35,680 --> 00:30:41,080 So look at the mode expansion in QCD. 468 00:30:45,100 --> 00:30:52,651 I can write it in a covariant type notation or with a d4p p 469 00:30:52,651 --> 00:30:55,990 if I put in the proper projection. 470 00:30:55,990 --> 00:31:01,555 And then we have the quark part with the e to the minus ip 471 00:31:01,555 --> 00:31:02,620 dot x. 472 00:31:02,620 --> 00:31:08,560 And then we have antiquark part, our b daggers, 473 00:31:08,560 --> 00:31:18,370 and the e to the plus ip dot x, OK? 474 00:31:18,370 --> 00:31:22,660 So let me think of that as a psi plus plus a psi minus. 475 00:31:22,660 --> 00:31:24,670 And we've so far only talked about psi plus. 476 00:31:29,190 --> 00:31:36,960 So for psi plus of x, we have a sum over Pl 477 00:31:36,960 --> 00:31:41,200 not equal to 0 e to the minus i Pl dot 478 00:31:41,200 --> 00:31:47,745 x of sort of a Cn Pl plus of x. 479 00:31:50,310 --> 00:31:51,480 So this is the case we did. 480 00:31:51,480 --> 00:31:54,810 If we were to repeat everything we did for this case, then what 481 00:31:54,810 --> 00:31:56,970 we would get for psi minus-- 482 00:31:56,970 --> 00:32:01,740 and this is why I say that they want to kind of look 483 00:32:01,740 --> 00:32:03,780 like separate fields-- 484 00:32:03,780 --> 00:32:06,600 is we'd get a plus i Pl dot x. 485 00:32:16,540 --> 00:32:20,770 And in this notation, if you think about the P0, 486 00:32:20,770 --> 00:32:23,890 the P0 is the sum of the P minus and the P plus. 487 00:32:23,890 --> 00:32:26,260 So saying that, in this mode decomposition, 488 00:32:26,260 --> 00:32:31,920 P0 is greater than 0 for these two equations 489 00:32:31,920 --> 00:32:34,735 and when we go over to sort of a collinear scaling, 490 00:32:34,735 --> 00:32:37,090 it becomes that Pl minus is greater than 0 491 00:32:37,090 --> 00:32:37,960 in these equations. 492 00:32:40,740 --> 00:32:43,320 OK, so that's the sense in which these are different. 493 00:32:43,320 --> 00:32:46,260 With a positive Pl minus, we have an opposite phase factor 494 00:32:46,260 --> 00:32:47,100 for these two guys. 495 00:32:49,840 --> 00:32:51,970 And in both of these cases, it's true 496 00:32:51,970 --> 00:32:57,250 that just as kind of a way of remark, we still have this. 497 00:32:57,250 --> 00:33:02,620 That didn't change, remember. 498 00:33:02,620 --> 00:33:05,260 OK, but talking about minus and plus fields would just 499 00:33:05,260 --> 00:33:07,360 make our life really difficult. Actually, 500 00:33:07,360 --> 00:33:11,160 when we first derived SCET, sometimes you're bone-headed. 501 00:33:11,160 --> 00:33:13,750 And we went through everything in a much more detailed way 502 00:33:13,750 --> 00:33:15,417 that I'm going to present it to you now, 503 00:33:15,417 --> 00:33:17,680 wrote out everything in terms of plus fields and minus 504 00:33:17,680 --> 00:33:19,480 fields because of this little issue. 505 00:33:19,480 --> 00:33:21,010 Then we noticed the following thing 506 00:33:21,010 --> 00:33:23,650 that I'm going to tell you, that we could just put everything 507 00:33:23,650 --> 00:33:26,560 back together with a simple definition. 508 00:33:26,560 --> 00:33:30,050 And that's because really the only difference is this sign. 509 00:33:30,050 --> 00:33:33,460 So if we want a kind of convenient notation, 510 00:33:33,460 --> 00:33:37,160 we can do the following. 511 00:33:37,160 --> 00:33:40,480 So far, if you like, we have this guy with Pl's 512 00:33:40,480 --> 00:33:43,270 not equal to 0, but the Pl minus should be greater than 0. 513 00:33:47,800 --> 00:33:50,590 So we can make use of kind of negative labels 514 00:33:50,590 --> 00:33:53,950 by the following trick. 515 00:33:53,950 --> 00:34:00,070 Just define a composite field that 516 00:34:00,070 --> 00:34:02,830 looks like this where I put the minuses and the pluses 517 00:34:02,830 --> 00:34:04,720 back together. 518 00:34:04,720 --> 00:34:06,740 If I have a positive Pl minus here, 519 00:34:06,740 --> 00:34:07,990 then that's giving the pluses. 520 00:34:07,990 --> 00:34:09,550 If I have a negative Pl minus here, 521 00:34:09,550 --> 00:34:10,989 then that's giving the minuses because 522 00:34:10,989 --> 00:34:11,989 of what I just told you. 523 00:34:20,909 --> 00:34:24,389 We don't just think about this guy as having plus Pl minus. 524 00:34:24,389 --> 00:34:25,949 We allow it to have negative. 525 00:34:25,949 --> 00:34:27,780 And we take the negative Pl minuses 526 00:34:27,780 --> 00:34:30,659 as a way of encoding what's going on with this guy. 527 00:34:36,360 --> 00:34:41,830 So we have different possibilities for M bar dot Pl. 528 00:34:41,830 --> 00:34:45,590 And if you like, what the plus guy is doing 529 00:34:45,590 --> 00:34:51,489 is destroying particles. 530 00:34:51,489 --> 00:34:59,610 And so let's say this is for the-- 531 00:34:59,610 --> 00:35:01,260 if I think about Cn of-- 532 00:35:01,260 --> 00:35:02,790 let me make my chart like this. 533 00:35:02,790 --> 00:35:05,940 If I think about what Cn of Pl is doing, 534 00:35:05,940 --> 00:35:10,560 it's destroying particles and the antiparticles are created. 535 00:35:17,490 --> 00:35:19,950 But the antiparticles are going to be 536 00:35:19,950 --> 00:35:23,742 created for the opposite sign of the Pl. 537 00:35:23,742 --> 00:35:25,950 And then if I think about what the barred field would 538 00:35:25,950 --> 00:35:29,720 be doing with this notation, if the M 539 00:35:29,720 --> 00:35:33,960 bar dot Pl is greater than 0, then the particles are created. 540 00:35:33,960 --> 00:35:35,460 So this guy is really just the field 541 00:35:35,460 --> 00:35:37,870 we were talking about before if Pl is greater than 0. 542 00:35:37,870 --> 00:35:39,630 And if Pl is less than 0, it just 543 00:35:39,630 --> 00:35:41,760 repeats the whole story for the antiparticles. 544 00:35:51,700 --> 00:35:54,197 OK, so I could write everything out 545 00:35:54,197 --> 00:35:55,780 in terms of this plus and minus field. 546 00:35:55,780 --> 00:35:58,705 But if I just make this simple definition, 547 00:35:58,705 --> 00:36:00,448 I could put everything back together 548 00:36:00,448 --> 00:36:02,740 and just work in terms of one field, which kind of does 549 00:36:02,740 --> 00:36:03,970 what we're used to. 550 00:36:03,970 --> 00:36:07,240 And it's just a sign convention to remember for this Pl. 551 00:36:07,240 --> 00:36:09,610 And that sign convention is also easy to remember 552 00:36:09,610 --> 00:36:13,060 because the Pl is really following the fermion flow. 553 00:36:13,060 --> 00:36:19,630 If you think about the fermion, then 554 00:36:19,630 --> 00:36:24,220 if you think about the physical momentum of that fermion, 555 00:36:24,220 --> 00:36:26,380 then the physical momentum follows the fermion 556 00:36:26,380 --> 00:36:27,730 in the case of the fermion. 557 00:36:27,730 --> 00:36:32,650 And it's opposite to the fermion in the case of the antifermion. 558 00:36:32,650 --> 00:36:35,155 So here's a fermion coming into some vertex. 559 00:36:37,740 --> 00:36:41,850 There's a antiparticle coming into the vertex. 560 00:36:41,850 --> 00:36:44,580 And the Pl goes this way. 561 00:36:44,580 --> 00:36:49,500 It follows the fermion flow, whereas the physical momentum 562 00:36:49,500 --> 00:36:53,310 is going this way. 563 00:36:53,310 --> 00:36:56,200 And that's the sign we're just keeping track of. 564 00:36:56,200 --> 00:36:58,590 And so the P labels are always following the arrows 565 00:36:58,590 --> 00:37:00,150 of the fermion lines. 566 00:37:00,150 --> 00:37:03,180 They're always along the fermion number with the conventions 567 00:37:03,180 --> 00:37:05,100 we've set up. 568 00:37:05,100 --> 00:37:10,920 OK, so with that little piece of notation, 569 00:37:10,920 --> 00:37:13,130 we take into account the antiparticles as well. 570 00:37:16,940 --> 00:37:20,960 There's no real additional complications 571 00:37:20,960 --> 00:37:25,005 since any way you draw a picture it's 572 00:37:25,005 --> 00:37:26,630 completely clear what's an antiparticle 573 00:37:26,630 --> 00:37:27,765 and what's a particle. 574 00:37:27,765 --> 00:37:29,709 And it just follows your intuition. 575 00:37:35,810 --> 00:37:42,470 And furthermore, because the only difference in the mode 576 00:37:42,470 --> 00:37:45,980 decomposition was that sign and we've now 577 00:37:45,980 --> 00:37:49,100 accounted for it by just making the labels negative, 578 00:37:49,100 --> 00:37:53,240 we have this equation, too, for both particles 579 00:37:53,240 --> 00:37:54,685 and antiparticles. 580 00:37:57,415 --> 00:37:58,790 So we get the opposite phase here 581 00:37:58,790 --> 00:38:00,550 because this guy has the opposite sign. 582 00:38:06,080 --> 00:38:08,137 So there's really no additional complications 583 00:38:08,137 --> 00:38:10,595 from adding the antiparticles once we set up this notation. 584 00:38:20,930 --> 00:38:22,570 OK, any questions so far? 585 00:38:27,018 --> 00:38:28,185 What about collinear gluons? 586 00:38:32,910 --> 00:38:37,610 So the gluon field is Hermitian, the original one. 587 00:38:37,610 --> 00:38:40,710 And what does that translate into? 588 00:38:40,710 --> 00:38:43,050 Well, it translates into the B's and the A's, right? 589 00:38:43,050 --> 00:38:45,660 They're both appearing in the decomposition of the field. 590 00:38:45,660 --> 00:38:46,380 There's no B's. 591 00:38:46,380 --> 00:38:47,190 There are just A's. 592 00:38:49,800 --> 00:38:52,942 And if you work out, using the type of body decomposition 593 00:38:52,942 --> 00:38:54,900 that we're doing and using the type of notation 594 00:38:54,900 --> 00:38:57,000 that we're doing, what that means, 595 00:38:57,000 --> 00:39:01,500 it means that the star of this guy just flips the sign. 596 00:39:06,440 --> 00:39:09,300 So that's what Hermiticity of the original field 597 00:39:09,300 --> 00:39:12,490 becomes for this notation. 598 00:39:12,490 --> 00:39:17,340 And again, because of the theta functions 599 00:39:17,340 --> 00:39:19,350 that show up in the mode decomposition, 600 00:39:19,350 --> 00:39:23,775 you can think of associating ql minus greater than 0 601 00:39:23,775 --> 00:39:29,190 as sort of destruction and then ql minus less than 0 602 00:39:29,190 --> 00:39:31,020 as creation. 603 00:39:31,020 --> 00:39:34,750 If you want, although it's not absolutely necessary, 604 00:39:34,750 --> 00:39:36,150 you could adopt this convention. 605 00:39:36,150 --> 00:39:39,440 And then that would just be putting a particular direction 606 00:39:39,440 --> 00:39:40,065 to your gluons. 607 00:39:43,740 --> 00:39:48,010 And everything that we said before goes through. 608 00:39:48,010 --> 00:39:54,270 And in particular, if we think of decomposing 609 00:39:54,270 --> 00:40:01,500 sort of the hatted version of the fields, then this-- 610 00:40:01,500 --> 00:40:02,265 just like before. 611 00:40:09,240 --> 00:40:12,690 And we can just also, again as before, 612 00:40:12,690 --> 00:40:16,140 if we like, adopt the notation where we suppress the labels 613 00:40:16,140 --> 00:40:18,400 if we don't need them. 614 00:40:18,400 --> 00:40:21,113 So the gluons really, in some sense, 615 00:40:21,113 --> 00:40:22,530 it's just simpler because we don't 616 00:40:22,530 --> 00:40:24,480 have to worry about the particles and the antiparticles 617 00:40:24,480 --> 00:40:25,688 because of their Hermiticity. 618 00:40:25,688 --> 00:40:29,560 It's sort of they're all the same thing. 619 00:40:29,560 --> 00:40:32,318 And for most purposes, there's no sense in even keeping 620 00:40:32,318 --> 00:40:33,360 track of this convention. 621 00:40:33,360 --> 00:40:40,508 So that's why I went through it quickly 622 00:40:40,508 --> 00:40:42,050 because the convention is always kind 623 00:40:42,050 --> 00:40:44,690 of obvious from the context. 624 00:40:44,690 --> 00:40:47,390 If you're really talking about production of a real gluon 625 00:40:47,390 --> 00:40:49,868 or annihilation of a real gluon, then you 626 00:40:49,868 --> 00:40:51,410 can keep track of the signs that way. 627 00:40:54,110 --> 00:40:55,809 The fermions are a little different. 628 00:41:01,080 --> 00:41:04,347 So because any starred field is just the negative of this guy, 629 00:41:04,347 --> 00:41:06,680 we could always write everything in terms of fields that 630 00:41:06,680 --> 00:41:10,730 have no daggers for the gluons. 631 00:41:10,730 --> 00:41:12,470 But for the fermions, some of the fields 632 00:41:12,470 --> 00:41:13,610 are barred or daggered. 633 00:41:18,350 --> 00:41:23,540 OK, so what would be the form of having a general label 634 00:41:23,540 --> 00:41:26,452 operator act on some combination of daggered fields 635 00:41:26,452 --> 00:41:28,160 and some combination of undaggered fields 636 00:41:28,160 --> 00:41:31,445 where the gluons are always undaggered? 637 00:41:31,445 --> 00:41:33,320 Well, it just gives the sum of these momenta. 638 00:41:36,440 --> 00:41:39,530 And the way it does it-- so sum of the label of momenta. 639 00:41:43,370 --> 00:41:46,986 So I'm thinking of all these denominators as label momenta. 640 00:41:46,986 --> 00:41:50,780 I'll maybe not put the l. 641 00:41:50,780 --> 00:41:55,880 And the way that the convention works 642 00:41:55,880 --> 00:41:58,055 is that we get a minus sign for the daggered guys. 643 00:42:07,280 --> 00:42:10,839 That's our sign convention. 644 00:42:18,260 --> 00:42:20,760 And the reason for that sign convention is effectively that, 645 00:42:20,760 --> 00:42:24,330 if you think about e to the minus i 646 00:42:24,330 --> 00:42:28,230 curly p dot x on sum 5 p1 and then you 647 00:42:28,230 --> 00:42:35,265 take the dagger of that, then it becomes 5 p1 dagger of e 648 00:42:35,265 --> 00:42:41,730 top the plus i kind of p dagger dot x. 649 00:42:41,730 --> 00:42:44,130 And basically, I'm saying that I want to be 650 00:42:44,130 --> 00:42:47,340 able to write that like this. 651 00:42:53,190 --> 00:42:56,020 And that convention is useful. 652 00:42:56,020 --> 00:42:58,170 So if I want to act forward on this field, which 653 00:42:58,170 --> 00:42:59,628 is a daggered field, then I'm going 654 00:42:59,628 --> 00:43:01,210 to take the opposite side. 655 00:43:01,210 --> 00:43:06,390 And if I have a daggered p, then it would take the plus sign. 656 00:43:06,390 --> 00:43:08,100 And that convention is the useful one 657 00:43:08,100 --> 00:43:14,580 for keeping track of momentum conservation 658 00:43:14,580 --> 00:43:17,040 because then every field has the same convention. 659 00:43:17,040 --> 00:43:20,040 We just have e to the minus i p dot x for every single field. 660 00:43:31,620 --> 00:43:39,230 All right, so if we started with some partial derivative-- 661 00:43:39,230 --> 00:43:41,390 I mean some regular derivative. 662 00:43:41,390 --> 00:43:44,420 And it was acting on one of our hatted fields. 663 00:43:49,480 --> 00:43:52,585 So if I write it out in terms of the labels-- 664 00:43:55,710 --> 00:43:57,790 put a Pl on it if you like-- 665 00:44:01,820 --> 00:44:02,570 we'd have that. 666 00:44:02,570 --> 00:44:04,820 And so there's two contributions from this derivative. 667 00:44:04,820 --> 00:44:07,610 It either hits the phase or it hits the field. 668 00:44:07,610 --> 00:44:10,220 And so we can write it like-- 669 00:44:16,996 --> 00:44:20,630 if we go over to the label operator, 670 00:44:20,630 --> 00:44:22,790 we can write it that this partial derivative 671 00:44:22,790 --> 00:44:31,860 is the sum of two derivatives, which then we 672 00:44:31,860 --> 00:44:34,350 can take the sum over Pl inside and write it as-- 673 00:44:37,630 --> 00:44:41,940 so every derivative for the purposes of this exercise 674 00:44:41,940 --> 00:44:46,140 is to convince you of the sanity of the notation. 675 00:44:46,140 --> 00:44:48,780 Every regular derivative acting on one of these hatted fields 676 00:44:48,780 --> 00:44:50,760 where we hadn't yet decomposed momenta 677 00:44:50,760 --> 00:44:53,340 just becomes a sum of these two derivatives. 678 00:44:53,340 --> 00:44:56,190 And that's just like the fact that we split the full momentum 679 00:44:56,190 --> 00:44:59,880 into the sum of the label in the residual piece. 680 00:44:59,880 --> 00:45:12,240 OK, this is the field version of P mu equals Pl mu plus P 681 00:45:12,240 --> 00:45:13,010 residual mu. 682 00:45:17,490 --> 00:45:19,230 OK, but the advantage of this notation 683 00:45:19,230 --> 00:45:23,130 is that we can then drop the partials if they're smaller. 684 00:45:23,130 --> 00:45:27,810 That's the whole reason for this setup. 685 00:45:27,810 --> 00:45:31,080 So now, we can expand because we've 686 00:45:31,080 --> 00:45:36,280 split up both the fields and the momentum. 687 00:45:36,280 --> 00:45:38,567 So we're able to expand. 688 00:45:38,567 --> 00:45:40,650 So let's go back to the thing we wanted to expand. 689 00:45:43,170 --> 00:45:47,130 So we had this guy. 690 00:45:55,660 --> 00:45:56,870 And we wanted to expand it. 691 00:45:56,870 --> 00:46:00,550 And now, we know how to expand it. 692 00:46:00,550 --> 00:46:03,370 We've set up a nice way of doing the multipole expansion 693 00:46:03,370 --> 00:46:07,940 by introducing some additional notation. 694 00:46:07,940 --> 00:46:21,340 So because of what I just said and because of what I 695 00:46:21,340 --> 00:46:36,020 said last night for the gauge field, 696 00:46:36,020 --> 00:46:38,030 we have a very simple decomposition 697 00:46:38,030 --> 00:46:39,470 for the covariant derivative. 698 00:46:39,470 --> 00:46:41,240 It's the sum of a collinear kind of piece 699 00:46:41,240 --> 00:46:43,010 and an ultra soft kind of piece. 700 00:46:43,010 --> 00:46:45,860 So this P mu plus i partial mu is just this. 701 00:46:45,860 --> 00:46:48,380 And the split of the gauge field into these two pieces 702 00:46:48,380 --> 00:46:50,120 is what we talked about last time. 703 00:46:50,120 --> 00:46:51,662 And that split of the gauge field 704 00:46:51,662 --> 00:46:53,870 actually led to some things that we didn't talk about 705 00:46:53,870 --> 00:46:55,953 that we'll talk about later. 706 00:46:55,953 --> 00:46:57,620 So if we now look at the components that 707 00:46:57,620 --> 00:47:01,400 are appearing in here, we can figure out 708 00:47:01,400 --> 00:47:02,602 what to keep in each case. 709 00:47:02,602 --> 00:47:03,560 And it's pretty simple. 710 00:47:11,890 --> 00:47:14,980 Well, for the plus, there's nothing to drop. 711 00:47:14,980 --> 00:47:16,535 We just keep everything. 712 00:47:16,535 --> 00:47:18,410 And there's also only one type of derivative, 713 00:47:18,410 --> 00:47:20,365 so we just keep this. 714 00:47:20,365 --> 00:47:21,400 This is exact. 715 00:47:25,840 --> 00:47:27,790 All terms here are order lambda squared. 716 00:47:31,420 --> 00:47:36,170 For the perp, we just keep the collinear parts. 717 00:47:36,170 --> 00:47:44,560 So if I write out the other part, 718 00:47:44,560 --> 00:47:48,420 just for completeness we can drop those terms 719 00:47:48,420 --> 00:47:49,420 relative to these terms. 720 00:47:49,420 --> 00:47:52,870 Because these terms here are lambda, 721 00:47:52,870 --> 00:47:57,020 and these terms here are order lambda squared. 722 00:47:57,020 --> 00:48:03,070 So we drop these guys from the leading order and then, 723 00:48:03,070 --> 00:48:04,300 similarly, for the minus. 724 00:48:35,930 --> 00:48:38,720 So here, this guy is order lambda squared. 725 00:48:38,720 --> 00:48:41,480 And this guy is order lambda 0. 726 00:48:41,480 --> 00:48:44,400 So we drop this guy. 727 00:48:44,400 --> 00:48:44,900 OK. 728 00:48:44,900 --> 00:48:47,420 So now, just plug those things into our Lagrangian. 729 00:48:54,640 --> 00:48:56,849 And then we get the leading order Lagrangian. 730 00:49:07,840 --> 00:49:11,795 All of these phase factors can be encoded in an overall phase 731 00:49:11,795 --> 00:49:12,295 factor. 732 00:49:17,860 --> 00:49:22,010 We get the full n dot D derivative. 733 00:49:22,010 --> 00:49:23,620 But for the collinear parts, we only 734 00:49:23,620 --> 00:49:27,350 get these piece, this piece, and the piece over there. 735 00:49:27,350 --> 00:49:30,100 And so you can think of that as a collinear covariant 736 00:49:30,100 --> 00:49:31,096 derivative. 737 00:49:34,690 --> 00:49:35,190 No. 738 00:49:45,810 --> 00:49:48,930 So sort of in the obvious notation, 739 00:49:48,930 --> 00:49:54,480 we take the two pieces that are collinear and put them 740 00:49:54,480 --> 00:49:59,010 into a collinear D and likewise in the n bar direction. 741 00:50:06,220 --> 00:50:10,370 OK, so this Lagrangian here is the leading order Lagrangian 742 00:50:10,370 --> 00:50:11,930 with this additional notation. 743 00:50:11,930 --> 00:50:14,130 And if you like, we have this kind of D 744 00:50:14,130 --> 00:50:17,850 that has both pieces in it for that n dot direction. 745 00:50:17,850 --> 00:50:20,660 And then for the other two, we just have the collinear pieces. 746 00:50:20,660 --> 00:50:23,010 And that's the leading order action. 747 00:50:23,010 --> 00:50:26,630 Where we've carried out the multipole expansion, 748 00:50:26,630 --> 00:50:29,840 after we introduced notation, it became something trivial. 749 00:50:29,840 --> 00:50:33,120 So the power of the notation is that, once you've swallowed it, 750 00:50:33,120 --> 00:50:34,695 everything else is simple. 751 00:50:34,695 --> 00:50:36,320 If I want to go to higher order, I just 752 00:50:36,320 --> 00:50:38,518 work out what these terms are. 753 00:50:38,518 --> 00:50:40,310 Then I should also worry about these terms. 754 00:50:40,310 --> 00:50:41,435 And we'll do that later on. 755 00:50:48,950 --> 00:50:51,720 OK, so what can we note about this? 756 00:50:51,720 --> 00:50:54,260 Well, one thing that we worked out earlier 757 00:50:54,260 --> 00:50:58,070 was that this should be of order lambda. 758 00:50:58,070 --> 00:51:00,485 And that actually implies that, our new field, 759 00:51:00,485 --> 00:51:02,000 we were just pulling out phases. 760 00:51:02,000 --> 00:51:04,125 And that doesn't change the overall power counting. 761 00:51:04,125 --> 00:51:06,770 So our new field is of order lambda as well. 762 00:51:06,770 --> 00:51:09,060 And likewise, before we sort of had a power 763 00:51:09,060 --> 00:51:11,480 counting for the measure, which you 764 00:51:11,480 --> 00:51:13,300 can think of in our new notation as a power 765 00:51:13,300 --> 00:51:16,890 counting for this thing, and that just 766 00:51:16,890 --> 00:51:18,835 scales like lambda to the minus 4 as before. 767 00:51:18,835 --> 00:51:20,210 So all the power counting that we 768 00:51:20,210 --> 00:51:23,390 were doing even before we sort of figured out 769 00:51:23,390 --> 00:51:25,640 what the right field to use, that all carries through. 770 00:51:25,640 --> 00:51:30,140 And that's why I did it earlier. 771 00:51:30,140 --> 00:51:33,950 OK, so what's going on here, which I didn't really 772 00:51:33,950 --> 00:51:36,620 spell out, but I can spell out now, 773 00:51:36,620 --> 00:51:39,590 is since this phase is order 1, the power counting 774 00:51:39,590 --> 00:51:41,930 from momenta induces a power counting on the x. 775 00:51:51,150 --> 00:51:54,450 And it's kind of the larger momentum that dominates. 776 00:51:54,450 --> 00:52:01,530 So x minus scales like 1 over p plus x plus-- 777 00:52:01,530 --> 00:52:03,300 remember in the dot product, it's 778 00:52:03,300 --> 00:52:06,660 always plus times minus, a confusing thing 779 00:52:06,660 --> 00:52:08,690 about light cone coordinates. 780 00:52:12,537 --> 00:52:14,370 And so if you count up the powers of lambda, 781 00:52:14,370 --> 00:52:16,200 you get minus 4 of them because you 782 00:52:16,200 --> 00:52:21,750 get 2 here, none here, and then 2 there. 783 00:52:21,750 --> 00:52:24,250 So that's all going through as before. 784 00:52:24,250 --> 00:52:27,750 And then if you ask, what's the order of the Lagrangian, 785 00:52:27,750 --> 00:52:30,570 the Lagrangian density here is order lambda to the 4. 786 00:52:45,900 --> 00:52:48,040 Because we have two fields, and then 787 00:52:48,040 --> 00:52:52,730 we have a lambda squared here and then a lambda here, 788 00:52:52,730 --> 00:52:58,890 a lambda 0 here, and a lambda there. 789 00:52:58,890 --> 00:53:00,640 And our power counting made that explicit. 790 00:53:00,640 --> 00:53:02,830 So the whole thing is lambda to the 4. 791 00:53:02,830 --> 00:53:04,720 This lambda to the 4 of the Lagrange density 792 00:53:04,720 --> 00:53:07,630 cancels the lambda to the minus 4 of the measure. 793 00:53:07,630 --> 00:53:16,680 And you get that, when you integrate, 794 00:53:16,680 --> 00:53:17,760 this is lambda to the 0. 795 00:53:17,760 --> 00:53:19,740 And that's what we wanted. 796 00:53:19,740 --> 00:53:23,880 That was our convention for the leading order term, OK? 797 00:53:23,880 --> 00:53:25,980 So the power counting, nothing really changes. 798 00:53:25,980 --> 00:53:27,850 We had a power counting for momenta. 799 00:53:27,850 --> 00:53:29,730 We just now have a power counting for all these objects 800 00:53:29,730 --> 00:53:30,772 that we're talking about. 801 00:53:33,500 --> 00:53:34,670 What about locality? 802 00:53:37,940 --> 00:53:40,160 Well, all the fields in this Lagrangian are at x. 803 00:53:45,400 --> 00:53:47,590 So it is local in residual momenta. 804 00:53:47,590 --> 00:53:51,362 We said that already when we were drawing Feynman diagrams. 805 00:53:51,362 --> 00:53:52,820 And that's just true of the action. 806 00:54:16,870 --> 00:54:18,370 And so what that means is the action 807 00:54:18,370 --> 00:54:23,810 is local at the smallest momentum scale in our problem. 808 00:54:23,810 --> 00:54:26,020 And we would have been surprised if we'd encountered 809 00:54:26,020 --> 00:54:27,228 anything different than that. 810 00:54:33,930 --> 00:54:36,225 The action is actually also local at the scale lambda. 811 00:54:45,020 --> 00:54:48,230 Even though we adopted this label convention just 812 00:54:48,230 --> 00:54:52,400 to separate the momenta, there's no inverse perp derivatives 813 00:54:52,400 --> 00:54:54,383 in this action. 814 00:54:54,383 --> 00:54:56,550 And so the perp derivatives appear in the numerator. 815 00:54:56,550 --> 00:54:58,790 And so it's actually local at that scale, too. 816 00:55:05,120 --> 00:55:06,620 And so if you like this, we've just 817 00:55:06,620 --> 00:55:08,552 written something that's in momentum space. 818 00:55:08,552 --> 00:55:10,010 And it's the momentum space version 819 00:55:10,010 --> 00:55:11,585 of locality at the scale lambda. 820 00:55:14,450 --> 00:55:16,700 And we've hidden all the momentum space 821 00:55:16,700 --> 00:55:19,640 so it even looks local. 822 00:55:19,640 --> 00:55:24,562 But if we were to write out the indices again, 823 00:55:24,562 --> 00:55:26,270 we would see that it's the momentum space 824 00:55:26,270 --> 00:55:28,620 version of locality. 825 00:55:28,620 --> 00:55:30,540 And so the only scale that it's non-local at 826 00:55:30,540 --> 00:55:32,540 is actually scale that we want to integrate out, 827 00:55:32,540 --> 00:55:33,290 the hard scale. 828 00:55:44,037 --> 00:55:46,370 And that's due to sort of the presence of this kind of 1 829 00:55:46,370 --> 00:55:48,950 over M bar dot Pl type factor. 830 00:55:52,900 --> 00:55:55,450 So that we already sort of saw earlier 831 00:55:55,450 --> 00:55:57,820 when we were talking about building up the Wilson line, 832 00:55:57,820 --> 00:55:58,120 right? 833 00:55:58,120 --> 00:55:59,650 We said, well, there's some off-shell fields. 834 00:55:59,650 --> 00:56:01,030 Let's start integrating them out. 835 00:56:01,030 --> 00:56:03,518 And we got inverse powers of this M bar dot Pl, 836 00:56:03,518 --> 00:56:04,810 but that was at the hard scale. 837 00:56:04,810 --> 00:56:07,660 So we had no choice but to integrate them out. 838 00:56:07,660 --> 00:56:10,775 And you can think about this factor here. 839 00:56:10,775 --> 00:56:13,150 If it bothers you, you can think about it in the same way 840 00:56:13,150 --> 00:56:15,870 that it's not any worse than what we already did. 841 00:56:19,830 --> 00:56:24,760 OK, let's see if this Lagrangian does what we want. 842 00:56:24,760 --> 00:56:25,980 Let's look at propagators. 843 00:56:30,750 --> 00:56:37,200 If we have a collinear propagator with a collinear 844 00:56:37,200 --> 00:56:42,060 gluon, say, coming in, let's look at this diagram. 845 00:56:42,060 --> 00:56:44,420 These are fields in the n direction. 846 00:56:44,420 --> 00:56:47,800 This is some guy that's Pl Pr. 847 00:56:47,800 --> 00:56:52,626 And let me just take it to be Pr plus for convenience. 848 00:56:56,520 --> 00:56:59,100 Then if you look at this guy and you 849 00:56:59,100 --> 00:57:09,212 ask what that propagator is, it comes out exactly how 850 00:57:09,212 --> 00:57:10,170 we want it to come out. 851 00:57:17,330 --> 00:57:19,120 So there's nothing that gets dropped 852 00:57:19,120 --> 00:57:27,430 in the denominator in particular in the sense 853 00:57:27,430 --> 00:57:31,928 that all components of the momentum are showing up. 854 00:57:31,928 --> 00:57:33,220 We're still dropping residuals. 855 00:57:33,220 --> 00:57:35,952 But if we're just talking about collinear particles, 856 00:57:35,952 --> 00:57:37,660 then the residuals are kind of redundant. 857 00:57:37,660 --> 00:57:39,460 The purpose of the residual's momenta 858 00:57:39,460 --> 00:57:41,530 is when we're talking about both collinear 859 00:57:41,530 --> 00:57:44,930 and ultra soft particles. 860 00:57:44,930 --> 00:57:47,520 So the other case that we need to treat 861 00:57:47,520 --> 00:57:49,817 is when we have an ultra soft particle. 862 00:57:49,817 --> 00:57:52,150 We could, of course, have both of them at the same time, 863 00:57:52,150 --> 00:57:54,166 but let's treat this case. 864 00:57:54,166 --> 00:58:02,240 So we have Pl, Pl plus some k, which is purely residual. 865 00:58:02,240 --> 00:58:04,780 And now, this guy, because of the derivatives, 866 00:58:04,780 --> 00:58:08,140 these derivatives don't see the k. 867 00:58:08,140 --> 00:58:10,052 None of these derivatives see the k. 868 00:58:16,250 --> 00:58:19,500 So in the n bar in the perp, we don't see the k. 869 00:58:28,340 --> 00:58:29,952 It would look like that. 870 00:58:29,952 --> 00:58:32,160 If you like, if you want to make it kind of look more 871 00:58:32,160 --> 00:58:33,743 like the Lagrangian, convince yourself 872 00:58:33,743 --> 00:58:37,170 that this is the inverse, just divide by this factor. 873 00:58:37,170 --> 00:58:40,057 Divide by this factor, then it puts it under here. 874 00:58:40,057 --> 00:58:41,890 And that's exactly what this is doing if you 875 00:58:41,890 --> 00:58:44,830 ignore the Dirac structure. 876 00:58:44,830 --> 00:58:50,400 It's not playing a role really for the propagator. 877 00:58:50,400 --> 00:58:52,650 And same here, if you take this factor, divide it out, 878 00:58:52,650 --> 00:58:56,310 then you got the plus piece which is this derivative. 879 00:58:56,310 --> 00:58:59,440 And then you get perp squared over minus. 880 00:58:59,440 --> 00:59:03,390 And that's what happens if you put this down stairs, OK? 881 00:59:03,390 --> 00:59:05,670 For linear, all the components show up 882 00:59:05,670 --> 00:59:07,530 because we didn't drop anything. 883 00:59:07,530 --> 00:59:11,210 For ultra soft, only the plus shows up 884 00:59:11,210 --> 00:59:13,720 because it only changes the plus momentum of the collinear 885 00:59:13,720 --> 00:59:16,200 field. 886 00:59:16,200 --> 00:59:18,820 So the Lagrangian is smart. 887 00:59:18,820 --> 00:59:20,237 It knows how to deal with the fact 888 00:59:20,237 --> 00:59:22,445 that the propagator should be different for these two 889 00:59:22,445 --> 00:59:23,010 situations. 890 00:59:27,743 --> 00:59:29,660 And that was one of the things that we wanted, 891 00:59:29,660 --> 00:59:30,790 and we achieved it. 892 00:59:33,248 --> 00:59:34,790 So even though the Lagrangian doesn't 893 00:59:34,790 --> 00:59:36,190 know what's going to happen, it's 894 00:59:36,190 --> 00:59:38,420 smart enough to deal with any possible thing that 895 00:59:38,420 --> 00:59:41,485 could happen and give the right propagator. 896 00:59:52,380 --> 00:59:55,370 So we don't have to expand any further. 897 00:59:55,370 --> 00:59:57,940 The Lagrangian knows how to give the leading order propagator 898 00:59:57,940 --> 00:59:59,890 for any situation that we might be interested 899 00:59:59,890 --> 01:00:04,390 in at leading order, which are these two situations. 900 01:00:09,610 --> 01:00:13,150 What about Feynman rules for interactions? 901 01:00:22,380 --> 01:00:25,330 So that's also interesting. 902 01:00:25,330 --> 01:00:28,800 So if we have an ultra soft gluon, 903 01:00:28,800 --> 01:00:31,290 and let's say it has an index mu, 904 01:00:31,290 --> 01:00:38,730 there's only one term that comes in 905 01:00:38,730 --> 01:00:42,240 because it's only the n dot D that had an ultra soft field. 906 01:00:42,240 --> 01:00:45,030 All the other ultra soft fields got dropped. 907 01:00:47,700 --> 01:00:51,420 So there's only n dot A ultra soft in the Lcc0. 908 01:00:54,300 --> 01:00:55,530 And so there's only one term. 909 01:00:59,188 --> 01:01:01,480 Things are a little more complicated for the collinear. 910 01:01:01,480 --> 01:01:03,022 And this is kind of the price that we 911 01:01:03,022 --> 01:01:08,590 pay for setting up the notation that we did the way we did. 912 01:01:08,590 --> 01:01:12,070 So this guy here has all terms. 913 01:01:20,530 --> 01:01:27,625 Just by pulling a gluon field out of any one of the terms 914 01:01:27,625 --> 01:01:30,970 it can couple to any component of the gluon field 915 01:01:30,970 --> 01:01:32,972 just like in QCD, but our Feynman rule 916 01:01:32,972 --> 01:01:34,930 is a little more complicated because of the way 917 01:01:34,930 --> 01:01:36,250 we set things up. 918 01:01:36,250 --> 01:01:38,410 And in some ways, what's happened 919 01:01:38,410 --> 01:01:41,680 is that we've made the propagator simpler 920 01:01:41,680 --> 01:01:44,050 for the collinear fields at the expense of complicating 921 01:01:44,050 --> 01:01:45,226 the interaction. 922 01:01:55,440 --> 01:01:57,570 OK, so we couple to all components-- 923 01:01:57,570 --> 01:01:59,280 the perp component, the n component, 924 01:01:59,280 --> 01:02:04,228 and the n bar component if this is a collinear field. 925 01:02:04,228 --> 01:02:05,645 So these are all collinear fields. 926 01:02:09,480 --> 01:02:14,250 So one way of thinking about why that kind of had to happen 927 01:02:14,250 --> 01:02:15,660 is the following. 928 01:02:15,660 --> 01:02:17,320 What if we didn't have ultra softs? 929 01:02:17,320 --> 01:02:19,487 Imagine that we were going through this whole story, 930 01:02:19,487 --> 01:02:21,150 but we just had collinears. 931 01:02:21,150 --> 01:02:23,040 Then you wouldn't really need to go 932 01:02:23,040 --> 01:02:25,582 through this multipole expansion because there would never be 933 01:02:25,582 --> 01:02:26,880 two momentum of different size. 934 01:02:26,880 --> 01:02:29,580 And you could actually, just for the collinears, use QCD. 935 01:02:29,580 --> 01:02:31,740 The purpose of having this multipole expansion 936 01:02:31,740 --> 01:02:34,240 is really because we want to talk about both of these things 937 01:02:34,240 --> 01:02:34,948 at the same time. 938 01:02:34,948 --> 01:02:36,448 In fact, if you only had collinears, 939 01:02:36,448 --> 01:02:38,460 you just boost to the frame where they're not 940 01:02:38,460 --> 01:02:40,620 having this boosted scaling. 941 01:02:40,620 --> 01:02:42,930 It's really relative scalings that the effective theory 942 01:02:42,930 --> 01:02:45,660 is trying to encode, relative scaling between collinear 943 01:02:45,660 --> 01:02:49,770 and ultra soft or between, say, two collinears going off 944 01:02:49,770 --> 01:02:50,727 back to back. 945 01:02:50,727 --> 01:02:52,810 That's where the power of the effective theory is, 946 01:02:52,810 --> 01:02:53,730 in relative thing. 947 01:02:53,730 --> 01:02:55,980 So it's only when you have two things at the same time 948 01:02:55,980 --> 01:02:57,668 and you want to describe both of them 949 01:02:57,668 --> 01:02:59,460 that you get into the kind of complications 950 01:02:59,460 --> 01:03:00,293 we're talking about. 951 01:03:00,293 --> 01:03:03,810 If you just have one thing, then things are simple. 952 01:03:03,810 --> 01:03:05,850 You could just use QCD. 953 01:03:05,850 --> 01:03:08,220 But if you just had QCD for the collinear sector, 954 01:03:08,220 --> 01:03:12,260 you'd know that you have a very simple Feynman rule, igTA gamma 955 01:03:12,260 --> 01:03:15,090 mu, and a more complicated propagator 956 01:03:15,090 --> 01:03:17,588 because you'd have a P slash. 957 01:03:17,588 --> 01:03:19,130 Well, here we don't have any P slash. 958 01:03:19,130 --> 01:03:23,220 The propagator, if you write out the spin, 959 01:03:23,220 --> 01:03:25,710 is simply n slash over 2. 960 01:03:25,710 --> 01:03:27,570 But the P slash has to come back somewhere. 961 01:03:27,570 --> 01:03:29,250 And it's coming back because it gets 962 01:03:29,250 --> 01:03:32,140 encoded in the way we set things up in this vertex. 963 01:03:32,140 --> 01:03:35,710 So these factors of P slash here are just reproducing P slashes. 964 01:03:35,710 --> 01:03:39,930 Then in the full QCD context, it would be in the propagator. 965 01:03:39,930 --> 01:03:41,640 You can actually work out, if you 966 01:03:41,640 --> 01:03:43,640 start writing down Feynman diagrams, that that's 967 01:03:43,640 --> 01:03:46,032 exactly what's going on, OK? 968 01:03:46,032 --> 01:03:47,490 So that's kind of the price that we 969 01:03:47,490 --> 01:03:50,335 paid for sort of making a Lagrangian that 970 01:03:50,335 --> 01:03:51,960 dealt with both things at the same time 971 01:03:51,960 --> 01:03:53,700 and made this part really simple. 972 01:03:53,700 --> 01:03:55,747 That part became slightly more complicated. 973 01:03:55,747 --> 01:03:57,330 But once we know what I just told you, 974 01:03:57,330 --> 01:04:00,330 we can oftentimes, you know, carry out calculations here 975 01:04:00,330 --> 01:04:04,260 by making correspondences with the full QCD Lagrangian. 976 01:04:04,260 --> 01:04:06,675 Sometimes that's done anyway. 977 01:04:12,600 --> 01:04:15,900 They couple to all four components of the field. 978 01:04:15,900 --> 01:04:17,640 Actually, once you start learning 979 01:04:17,640 --> 01:04:21,583 how to deal with numerators like this, 980 01:04:21,583 --> 01:04:23,250 it's different than what you're used to, 981 01:04:23,250 --> 01:04:25,740 but it's just, in some sense, as complicated 982 01:04:25,740 --> 01:04:26,833 using this Lagrangian. 983 01:04:26,833 --> 01:04:28,500 It's just that you're writing things out 984 01:04:28,500 --> 01:04:31,740 in components, which you might not want to do, right? 985 01:04:31,740 --> 01:04:34,110 Why decompose things into components, 986 01:04:34,110 --> 01:04:35,220 which we're doing here? 987 01:04:35,220 --> 01:04:38,130 Why do that if you don't have sort of something that's 988 01:04:38,130 --> 01:04:40,110 projecting onto a component? 989 01:04:40,110 --> 01:04:43,440 That's, in some sense, the only complication. 990 01:04:43,440 --> 01:04:44,010 OK. 991 01:04:44,010 --> 01:04:47,170 And there's one other thing, of course, 992 01:04:47,170 --> 01:04:51,340 which is that the Lagrangian also had terms like this. 993 01:04:51,340 --> 01:04:54,510 So it's not simply just one gluon interaction. 994 01:04:54,510 --> 01:04:57,642 We could have, for example, two covariant derivatives, 995 01:04:57,642 --> 01:04:59,850 two of these perp covariant derivatives, for example. 996 01:04:59,850 --> 01:05:05,610 And there's a Feynman rule for that, too, 997 01:05:05,610 --> 01:05:09,370 which comes out of the action that we wrote down. 998 01:05:09,370 --> 01:05:12,240 OK, questions? 999 01:05:16,160 --> 01:05:18,248 AUDIENCE: So I guess this, I mean, 1000 01:05:18,248 --> 01:05:20,788 the effective theory is reproducing the collinear 1001 01:05:20,788 --> 01:05:22,220 [INAUDIBLE] divergences of QCD? 1002 01:05:22,220 --> 01:05:23,700 IAIN STEWART: It is, yeah. 1003 01:05:23,700 --> 01:05:27,570 AUDIENCE: So in [INAUDIBLE] QCD, the higher divergence 1004 01:05:27,570 --> 01:05:30,540 cancel with interference [INAUDIBLE]?? 1005 01:05:30,540 --> 01:05:32,460 IAIN STEWART: And that'll happen here, too. 1006 01:05:32,460 --> 01:05:33,950 AUDIENCE: And you have to loops? 1007 01:05:33,950 --> 01:05:36,117 Or it's happening like a leading order [INAUDIBLE]?? 1008 01:05:36,117 --> 01:05:38,158 IAIN STEWART: No, you have to look at loops, too. 1009 01:05:38,158 --> 01:05:39,870 So, yeah, there can be cancellations 1010 01:05:39,870 --> 01:05:41,820 between real and virtual graphs. 1011 01:05:41,820 --> 01:05:44,070 And there actually can be such cancellations 1012 01:05:44,070 --> 01:05:46,680 separately for ultra soft gluons and for collinear gluons. 1013 01:05:46,680 --> 01:05:48,930 So we'll talk about what you're talking about a little 1014 01:05:48,930 --> 01:05:50,440 later on. 1015 01:05:50,440 --> 01:05:52,770 But there is an important point that I'll emphasize 1016 01:05:52,770 --> 01:05:54,180 since you ask the question. 1017 01:05:54,180 --> 01:05:56,310 If you ask about the divergences, 1018 01:05:56,310 --> 01:06:00,240 the infrared divergences of an ultra soft gluon, so 1019 01:06:00,240 --> 01:06:02,790 the ultra soft gluon, if we go back to sort of where it was 1020 01:06:02,790 --> 01:06:05,770 living, it was littered here. 1021 01:06:05,770 --> 01:06:08,440 Collinear gluon was living here. 1022 01:06:08,440 --> 01:06:10,410 And if you ask about divergences, 1023 01:06:10,410 --> 01:06:11,910 what the divergences are coming from 1024 01:06:11,910 --> 01:06:15,390 are momenta going to 0 or momentum going collinear. 1025 01:06:15,390 --> 01:06:18,240 And this guy here that lives kind of here 1026 01:06:18,240 --> 01:06:22,980 can go both collinear, and it can go soft. 1027 01:06:22,980 --> 01:06:25,320 So the names collinear and soft, which 1028 01:06:25,320 --> 01:06:28,200 correspond to the names collinear and soft divergences, 1029 01:06:28,200 --> 01:06:30,610 are not a one-to-one correspondence. 1030 01:06:30,610 --> 01:06:34,800 An ultra soft particle can actually 1031 01:06:34,800 --> 01:06:38,940 have both collinear and soft divergences. 1032 01:06:38,940 --> 01:06:40,770 The collinear particle can also actually 1033 01:06:40,770 --> 01:06:43,500 have both types of divergences. 1034 01:06:43,500 --> 01:06:45,450 And so the names are associated to where 1035 01:06:45,450 --> 01:06:48,030 they live in momentum space, not the type of divergences 1036 01:06:48,030 --> 01:06:49,140 that they're encoding. 1037 01:06:49,140 --> 01:06:51,630 If I wanted to do something along the lines of setting up 1038 01:06:51,630 --> 01:06:55,685 the field theory to just exactly correspond to the divergences, 1039 01:06:55,685 --> 01:06:57,060 well, first of all, no one's ever 1040 01:06:57,060 --> 01:06:58,750 figured out how to do that. 1041 01:06:58,750 --> 01:07:00,292 But you could imagine how it would be 1042 01:07:00,292 --> 01:07:01,542 different than what I'm doing. 1043 01:07:01,542 --> 01:07:03,420 Because if I wanted to do that, then somehow 1044 01:07:03,420 --> 01:07:06,570 my collinear field would sort of be sort of sensitive to angles. 1045 01:07:06,570 --> 01:07:08,580 It should be like an angular variable 1046 01:07:08,580 --> 01:07:11,940 because the divergences are coming from angles going to 0. 1047 01:07:11,940 --> 01:07:13,630 And it's not an angular variable. 1048 01:07:13,630 --> 01:07:15,402 It's just a regular field. 1049 01:07:15,402 --> 01:07:16,860 So setting things up in this way is 1050 01:07:16,860 --> 01:07:19,590 kind of how we usually think of an effective field theory. 1051 01:07:19,590 --> 01:07:22,140 We're just talking about momenta small or big. 1052 01:07:22,140 --> 01:07:24,870 And that's what we're doing, but it comes in the sense 1053 01:07:24,870 --> 01:07:27,720 that you don't have a direct correspondence 1054 01:07:27,720 --> 01:07:31,520 with the divergences. 1055 01:07:31,520 --> 01:07:35,213 AUDIENCE: So you have more divergences [INAUDIBLE]?? 1056 01:07:35,213 --> 01:07:36,630 IAIN STEWART: The way of saying it 1057 01:07:36,630 --> 01:07:41,010 is that you have more divergences in a way, 1058 01:07:41,010 --> 01:07:43,460 but they're the same divergences-- 1059 01:07:43,460 --> 01:07:44,760 yeah. 1060 01:07:44,760 --> 01:07:45,960 Yeah. 1061 01:07:45,960 --> 01:07:48,480 So you don't have more divergences, 1062 01:07:48,480 --> 01:07:52,020 but really what's happening is-- 1063 01:07:52,020 --> 01:07:56,220 so if you look at this kind of picture, 1064 01:07:56,220 --> 01:07:58,200 if you look at the kind of infrared divergences 1065 01:07:58,200 --> 01:08:01,740 that you could have, so let's regulate them 1066 01:08:01,740 --> 01:08:03,660 with an off-shellness. 1067 01:08:03,660 --> 01:08:05,160 You could have infrared divergences 1068 01:08:05,160 --> 01:08:07,530 that would be like that. 1069 01:08:07,530 --> 01:08:10,830 But when you do some diagrams, there's 1070 01:08:10,830 --> 01:08:16,620 kind of infrared divergences that could look like this, OK? 1071 01:08:16,620 --> 01:08:19,350 I'm just telling you something that you have to trust me. 1072 01:08:19,350 --> 01:08:22,140 And this is coming from an ultra soft scale. 1073 01:08:22,140 --> 01:08:25,380 P squared over P minus, that's this hyperbola. 1074 01:08:25,380 --> 01:08:27,720 That's the scale of this hyperbola. 1075 01:08:27,720 --> 01:08:30,630 And the P squared here is the scale of this hyperbola. 1076 01:08:30,630 --> 01:08:33,540 What the effective theory does is, any ultra soft diagram, 1077 01:08:33,540 --> 01:08:35,819 that'll give all these logs. 1078 01:08:35,819 --> 01:08:40,560 Any collinear diagram will give all these logs. 1079 01:08:40,560 --> 01:08:42,930 You could say, well, P minus is a hard scale. 1080 01:08:42,930 --> 01:08:46,696 So really the only the IR divergences are just P squared, 1081 01:08:46,696 --> 01:08:48,779 but the effective theory is kind of dividing it up 1082 01:08:48,779 --> 01:08:50,069 into these two types of logs. 1083 01:08:50,069 --> 01:08:51,986 And that's actually going to be useful for us. 1084 01:08:51,986 --> 01:08:53,700 We'll see that that allows us to do 1085 01:08:53,700 --> 01:08:55,325 some things that are not so obvious how 1086 01:08:55,325 --> 01:08:56,399 to do in the full theory. 1087 01:09:00,370 --> 01:09:04,154 So, yes, it is kind of splitting and IR divergence in these two 1088 01:09:04,154 --> 01:09:06,279 different types of terms, but that's actually going 1089 01:09:06,279 --> 01:09:08,080 to be something that's useful. 1090 01:09:08,080 --> 01:09:10,390 Because in some sense, there's two physical scales 1091 01:09:10,390 --> 01:09:11,109 in the infrared. 1092 01:09:11,109 --> 01:09:12,981 There's a sort of collinear hyperbola 1093 01:09:12,981 --> 01:09:14,189 and the ultra soft hyperbola. 1094 01:09:14,189 --> 01:09:16,231 And the effective theory is making that explicit. 1095 01:09:19,479 --> 01:09:21,130 Other questions? 1096 01:09:21,130 --> 01:09:23,840 All questions have been good so far. 1097 01:09:23,840 --> 01:09:26,302 Keep asking them. 1098 01:09:26,302 --> 01:09:30,100 AUDIENCE: Could you try using two different lambdas 1099 01:09:30,100 --> 01:09:32,998 for the ultra soft and the collinear? 1100 01:09:32,998 --> 01:09:33,790 IAIN STEWART: Yeah. 1101 01:09:36,790 --> 01:09:40,210 So you could if there was a physical reason why 1102 01:09:40,210 --> 01:09:43,000 you wanted to. 1103 01:09:43,000 --> 01:09:47,439 And the thing is that, if the lambdas order each other, 1104 01:09:47,439 --> 01:09:49,569 then there's no point. 1105 01:09:49,569 --> 01:09:51,910 If there's a hierarchy, then what that would change 1106 01:09:51,910 --> 01:09:56,530 would be that the n dot D should get expanded, too. 1107 01:09:56,530 --> 01:09:58,412 But if the n dot D gets expanded, 1108 01:09:58,412 --> 01:10:00,370 then kind of what is happening is you're really 1109 01:10:00,370 --> 01:10:02,264 decoupling everything. 1110 01:10:02,264 --> 01:10:03,230 AUDIENCE: [INAUDIBLE] 1111 01:10:03,230 --> 01:10:04,022 IAIN STEWART: Yeah. 1112 01:10:04,022 --> 01:10:06,970 And so you're kind of losing the dynamical connection that we 1113 01:10:06,970 --> 01:10:08,170 have so far in this theory. 1114 01:10:08,170 --> 01:10:10,210 So I think it's not so interesting, 1115 01:10:10,210 --> 01:10:12,600 but it's worth thinking about. 1116 01:10:12,600 --> 01:10:13,870 OK. 1117 01:10:13,870 --> 01:10:15,130 We're going to erase this. 1118 01:10:24,710 --> 01:10:25,210 OK. 1119 01:10:25,210 --> 01:10:27,400 So now, that we have a leading order Lagrangian, 1120 01:10:27,400 --> 01:10:29,730 it does everything that we want it to do. 1121 01:10:29,730 --> 01:10:31,720 We can use it to calculate Feynman diagrams. 1122 01:10:31,720 --> 01:10:34,360 It defines the leading order collinear 1123 01:10:34,360 --> 01:10:36,460 sector in the effective theory. 1124 01:10:36,460 --> 01:10:38,380 There's one more thing I want to do. 1125 01:10:38,380 --> 01:10:42,490 And that's come back to the fact that I told you before, 1126 01:10:42,490 --> 01:10:47,860 but now that we can actually make more explicit. 1127 01:10:47,860 --> 01:10:51,312 And that fact was, when we were talking about currents 1128 01:10:51,312 --> 01:10:53,020 and integrating that off-shell particles, 1129 01:10:53,020 --> 01:10:54,768 we saw that you start with these guys. 1130 01:10:54,768 --> 01:10:55,810 You start attaching them. 1131 01:10:55,810 --> 01:10:57,430 And you end up with a Wilson line. 1132 01:10:57,430 --> 01:11:01,870 I'd now like to make it obvious that that kind of manipulation 1133 01:11:01,870 --> 01:11:04,630 always is true, that we can always get rid of this field 1134 01:11:04,630 --> 01:11:07,750 and put it in terms of a Wilson line. 1135 01:11:07,750 --> 01:11:10,990 And the notations that we've introduced 1136 01:11:10,990 --> 01:11:14,680 are going to allow us to do that. 1137 01:11:14,680 --> 01:11:20,320 So let's have a little aside on that fact. 1138 01:11:20,320 --> 01:11:26,410 So if you are in momentum space, the Wilson line 1139 01:11:26,410 --> 01:11:32,281 has an equation of motion, or a defining equation if you like, 1140 01:11:32,281 --> 01:11:33,660 that's this. 1141 01:11:33,660 --> 01:11:36,470 This defines the direction, that the Wilson line is 1142 01:11:36,470 --> 01:11:40,120 long n bar dot n bar direction. 1143 01:11:40,120 --> 01:11:42,760 And the x and the minus infinity are telling you 1144 01:11:42,760 --> 01:11:44,830 what the end points are. 1145 01:11:44,830 --> 01:11:47,680 So this is the defining equation. 1146 01:11:47,680 --> 01:11:49,330 You can look back at Peskin if you 1147 01:11:49,330 --> 01:11:53,430 want to see how that all works. 1148 01:11:53,430 --> 01:11:55,555 And that's the defining equation in position space. 1149 01:11:59,740 --> 01:12:02,620 But we've just advocated for a kind of momentum space 1150 01:12:02,620 --> 01:12:03,920 notation. 1151 01:12:03,920 --> 01:12:08,140 So what's the corresponding equation to momentum space? 1152 01:12:08,140 --> 01:12:09,740 It's pretty simple. 1153 01:12:09,740 --> 01:12:12,700 If we go over to our momentum space Wilson line, 1154 01:12:12,700 --> 01:12:15,220 we just go over to our momentum space covariant derivative. 1155 01:12:15,220 --> 01:12:16,780 And the defining equation is this. 1156 01:12:25,120 --> 01:12:27,170 So the notation, in some sense, is pretty simple. 1157 01:12:27,170 --> 01:12:29,620 We just have to switch objects. 1158 01:12:29,620 --> 01:12:37,500 If we write out what that derivative is, 1159 01:12:37,500 --> 01:12:44,430 then it's that, OK? 1160 01:12:44,430 --> 01:12:46,920 So now, consider the following thing. 1161 01:12:46,920 --> 01:12:50,770 Consider this combination, but let 1162 01:12:50,770 --> 01:12:53,170 me think about it as an operator equation. 1163 01:12:53,170 --> 01:12:55,680 So let me think about it as if there could 1164 01:12:55,680 --> 01:12:58,168 be something to the right. 1165 01:12:58,168 --> 01:13:00,210 If there's something to the right, than the P bar 1166 01:13:00,210 --> 01:13:02,910 can act on that thing, too. 1167 01:13:02,910 --> 01:13:05,830 So I get 0 if the P bar acts on this thing. 1168 01:13:05,830 --> 01:13:08,100 But if the P bar acts through, then I 1169 01:13:08,100 --> 01:13:10,050 get just P bar on the operator. 1170 01:13:14,470 --> 01:13:16,090 So as an operator equation, I can 1171 01:13:16,090 --> 01:13:28,860 trace the O. This is true for any O. 1172 01:13:28,860 --> 01:13:30,630 And we have this operator equation. 1173 01:13:40,600 --> 01:13:45,370 But I know something else I know that the Wilson line is 1174 01:13:45,370 --> 01:13:47,440 at e to the i something. 1175 01:13:47,440 --> 01:13:49,390 And actually, it's unitary. 1176 01:13:49,390 --> 01:13:54,580 So W dagger W is 1. 1177 01:13:59,340 --> 01:14:03,450 And again, in momentum space notation, 1178 01:14:03,450 --> 01:14:05,440 that's kind of the obvious thing. 1179 01:14:05,440 --> 01:14:07,860 W dagger W is 1. 1180 01:14:07,860 --> 01:14:10,020 And that means I can multiply this equation here 1181 01:14:10,020 --> 01:14:13,830 on the left or the right by W daggers. 1182 01:14:13,830 --> 01:14:23,310 And so if I do that, let me multiply it by W dagger. 1183 01:14:23,310 --> 01:14:33,000 And I can write i n bar dot Dn is Wn P bar Wn dagger. 1184 01:14:38,910 --> 01:14:44,340 So the gauge field, g n bar dot An, because of gauge symmetry, 1185 01:14:44,340 --> 01:14:47,400 it's always going to show up in the covariant derivative. 1186 01:14:47,400 --> 01:14:50,010 So it's always going to show up in this combination. 1187 01:14:50,010 --> 01:14:51,390 And this equation is telling me I 1188 01:14:51,390 --> 01:14:56,850 can switch that for Wilson lines in this P bar if I want to. 1189 01:14:56,850 --> 01:14:59,910 I could have also multiplied things the other way around 1190 01:14:59,910 --> 01:15:02,640 and written it P bar. 1191 01:15:02,640 --> 01:15:06,263 So sort of by way of completeness, 1192 01:15:06,263 --> 01:15:07,680 I could have written that as well. 1193 01:15:27,260 --> 01:15:30,650 It's also easy to convince yourself of the following. 1194 01:15:30,650 --> 01:15:41,930 1 over P bar is Wn dagger 1 over i n bar dot D W. 1195 01:15:41,930 --> 01:15:44,540 And 1 over i n bar dot D-- 1196 01:15:47,650 --> 01:15:50,830 I'll check that I got the daggers in the right place. 1197 01:15:53,450 --> 01:15:55,360 But these two equations are true, kind 1198 01:15:55,360 --> 01:15:57,290 of the inverse of those equations. 1199 01:15:57,290 --> 01:15:58,340 So how do we check that? 1200 01:15:58,340 --> 01:15:59,840 Well, just multiply this times this. 1201 01:15:59,840 --> 01:16:02,230 You should get 1, right? 1202 01:16:02,230 --> 01:16:04,880 And if I multiply, say, on the right here, 1203 01:16:04,880 --> 01:16:07,450 then the W dagger here kills the W dagger. 1204 01:16:07,450 --> 01:16:08,800 Then the P bar kills the P bar. 1205 01:16:08,800 --> 01:16:11,270 And then the W kills the W dagger. 1206 01:16:14,320 --> 01:16:15,190 So those are true. 1207 01:16:27,700 --> 01:16:30,860 And so we can put this into the Lagrangian if we want. 1208 01:16:30,860 --> 01:16:33,790 And we can do what I said, that we can actually, 1209 01:16:33,790 --> 01:16:36,740 if we want to, get rid of the n bar 1210 01:16:36,740 --> 01:16:45,970 dot A field for the collinears in terms of Wilson lines. 1211 01:16:45,970 --> 01:16:52,690 And we'll use this from time to time when it's convenient. 1212 01:16:52,690 --> 01:16:57,410 If we use the D Lagrangian, the Lagrangian becomes this thing. 1213 01:17:03,910 --> 01:17:07,190 So that 1 over i n bar dot D becomes this thing. 1214 01:17:07,190 --> 01:17:08,680 And in some sense, you might think 1215 01:17:08,680 --> 01:17:10,450 that this is easier to expand if you 1216 01:17:10,450 --> 01:17:12,170 know the expansion of the Wilson line. 1217 01:17:12,170 --> 01:17:16,480 And we just have to let a 1 over P bar act 1218 01:17:16,480 --> 01:17:19,180 on this full combination of fields. 1219 01:17:19,180 --> 01:17:21,460 And it just gives the total n bar 1220 01:17:21,460 --> 01:17:24,270 dot P momentum of that full combination of fields. 1221 01:17:27,070 --> 01:17:28,610 I claimed this was true. 1222 01:17:28,610 --> 01:17:30,730 And now, I show you in a kind of non-trivial case 1223 01:17:30,730 --> 01:17:32,980 how, with these operator manipulations, 1224 01:17:32,980 --> 01:17:35,290 we could just quickly go back and forth. 1225 01:17:35,290 --> 01:17:36,790 And it's related to the fact that it 1226 01:17:36,790 --> 01:17:40,000 appears in this combination. 1227 01:17:40,000 --> 01:17:42,010 We'll talk more about gauge symmetry later on. 1228 01:17:42,010 --> 01:17:43,552 I'm alluding to it from time to time, 1229 01:17:43,552 --> 01:17:46,180 but we'll talk about it more precisely later on. 1230 01:17:52,260 --> 01:17:57,390 We have to do this story for the collinear gluon Lagrangian. 1231 01:17:57,390 --> 01:18:01,980 And much of this is just repeating 1232 01:18:01,980 --> 01:18:05,190 what we've said for the quarks for the gluons. 1233 01:18:05,190 --> 01:18:08,640 And once we know how to write down the covariant derivatives, 1234 01:18:08,640 --> 01:18:12,540 then it's actually not so complicated. 1235 01:18:12,540 --> 01:18:14,220 And there's just one little complication 1236 01:18:14,220 --> 01:18:18,510 that we should be careful about, which 1237 01:18:18,510 --> 01:18:19,960 I want to emphasize to you. 1238 01:18:19,960 --> 01:18:22,320 And so we'll go through it. 1239 01:18:25,200 --> 01:18:29,025 Let me write the Lagrangian in QCD in terms of traces. 1240 01:18:32,730 --> 01:18:34,740 And let me use a general covariant gauge. 1241 01:18:38,440 --> 01:18:40,750 So this is a general covariant gauge fixing term. 1242 01:18:44,152 --> 01:18:46,620 That should be called the gauge parameter tau. 1243 01:18:46,620 --> 01:18:50,460 Then we have some ghost fields as well. 1244 01:18:50,460 --> 01:18:57,180 And of course, g mu nu is the usual g mu nu A TA. 1245 01:18:57,180 --> 01:19:02,160 And that's my sign convention commutator of two derivatives 1246 01:19:02,160 --> 01:19:06,760 with the pi over g. 1247 01:19:06,760 --> 01:19:08,310 So we do the same steps to get SCET. 1248 01:19:11,830 --> 01:19:14,340 And so really, if you think about writing 1249 01:19:14,340 --> 01:19:16,133 all the g's in terms of D's, we already 1250 01:19:16,133 --> 01:19:17,550 know how to split the D's in terms 1251 01:19:17,550 --> 01:19:19,680 of ultra soft and collinear parts. 1252 01:19:19,680 --> 01:19:20,820 And so that's what we do. 1253 01:19:28,740 --> 01:19:32,880 Let me write out a D, which is kind of a leading order D. 1254 01:19:32,880 --> 01:19:35,633 And we'll just set up some notation. 1255 01:19:35,633 --> 01:19:37,800 And then I can just basically write down the answer. 1256 01:19:44,770 --> 01:19:47,670 So let's set up a curly D that has all the sort of pieces that 1257 01:19:47,670 --> 01:19:49,110 will survive at leading order. 1258 01:19:59,405 --> 01:20:01,780 So all the pieces that survived were the collinear pieces 1259 01:20:01,780 --> 01:20:04,705 and perp in n bar and then, in the n direction, both pieces. 1260 01:20:07,270 --> 01:20:10,030 And effectively, I'm just changing all the regular Roman 1261 01:20:10,030 --> 01:20:13,420 D's in my gluon action into these curly D's. 1262 01:20:13,420 --> 01:20:16,240 And that's almost the entire story. 1263 01:20:35,110 --> 01:20:38,410 So from a kind of power counting perspective, once 1264 01:20:38,410 --> 01:20:41,020 you take into account that, when you dot things, 1265 01:20:41,020 --> 01:20:42,800 n's are always dotting with the n bars, 1266 01:20:42,800 --> 01:20:43,930 so it's this times this. 1267 01:20:43,930 --> 01:20:45,160 So this is lambda squared. 1268 01:20:45,160 --> 01:20:46,310 This is order lambda 0. 1269 01:20:46,310 --> 01:20:49,143 This guy dots into himself. 1270 01:20:49,143 --> 01:20:50,810 So you get lambda squared there as well. 1271 01:20:50,810 --> 01:20:54,130 So for power counting purposes, when you start squaring things, 1272 01:20:54,130 --> 01:20:56,630 they're all going to be homogeneous with this setup. 1273 01:20:59,140 --> 01:21:01,470 So basically, that's what I'm saying here. 1274 01:21:05,350 --> 01:21:06,790 And then there's one complication, 1275 01:21:06,790 --> 01:21:08,597 and that's this partial derivative. 1276 01:21:08,597 --> 01:21:10,180 Well, there's two partial derivatives. 1277 01:21:10,180 --> 01:21:13,690 There's one there, and there's one there. 1278 01:21:13,690 --> 01:21:15,190 So this is in the gauge fixing term. 1279 01:21:15,190 --> 01:21:18,790 These terms are both related to the gauge fixing. 1280 01:21:18,790 --> 01:21:28,780 And then those terms, actually, I want to make them covariant. 1281 01:21:28,780 --> 01:21:31,270 So let me define one more thing. 1282 01:21:31,270 --> 01:21:32,770 I want to make them covariant, but I 1283 01:21:32,770 --> 01:21:36,597 want to make them only covariant under the ultra soft part 1284 01:21:36,597 --> 01:21:37,180 of the action. 1285 01:21:44,590 --> 01:21:46,518 So here is just pulling out the pieces 1286 01:21:46,518 --> 01:21:48,310 that would have the ultra soft field making 1287 01:21:48,310 --> 01:21:52,930 a derivative that I'll call curly D ultra soft. 1288 01:21:52,930 --> 01:21:55,300 So one way of thinking about that 1289 01:21:55,300 --> 01:21:57,640 is that it would be like background field gauge. 1290 01:21:57,640 --> 01:22:00,610 In background field gauge, you'd make this partial view 1291 01:22:00,610 --> 01:22:03,910 into the D mu of the background involving the background field. 1292 01:22:03,910 --> 01:22:05,860 And that would be a background field gauge. 1293 01:22:05,860 --> 01:22:10,720 So from the point of view of the ultra soft field 1294 01:22:10,720 --> 01:22:12,880 being a background field, it's very natural 1295 01:22:12,880 --> 01:22:16,870 to have a replacement that looks like this. 1296 01:22:16,870 --> 01:22:19,720 And that's one way of arguing. 1297 01:22:19,720 --> 01:22:21,910 Really what's going on in the effective theory 1298 01:22:21,910 --> 01:22:26,020 is that this is general covariant gauge for A n. 1299 01:22:26,020 --> 01:22:29,620 That's what we want it to be for A n mu. 1300 01:22:29,620 --> 01:22:33,257 This should be the gauge fixing term for the collinear gluons. 1301 01:22:33,257 --> 01:22:34,840 And it shouldn't be breaking any gauge 1302 01:22:34,840 --> 01:22:36,700 symmetry of ultra soft gluons. 1303 01:22:36,700 --> 01:22:38,590 So we'd actually like the Lagrangian 1304 01:22:38,590 --> 01:22:41,110 to both have gauge symmetry for collinear gluons 1305 01:22:41,110 --> 01:22:43,705 and a separate gauge symmetry for the ultra soft gluons. 1306 01:22:43,705 --> 01:22:46,330 Since this guy is supposed to be gauge fixing for the collinear 1307 01:22:46,330 --> 01:22:48,205 gluons, we don't want it to be doing anything 1308 01:22:48,205 --> 01:22:49,810 to the ultra soft gluons. 1309 01:22:49,810 --> 01:22:52,720 And that's accomplished if we make this replacement 1310 01:22:52,720 --> 01:22:57,250 that i partial goes to i curly D, all right? 1311 01:22:57,250 --> 01:23:01,540 So we could write out what Lcg0 is, 1312 01:23:01,540 --> 01:23:05,460 but it's just these two replacements. 1313 01:23:09,220 --> 01:23:15,880 Since we're out of time, I'll just say that 1314 01:23:15,880 --> 01:23:18,820 and, you know, sort of write out the fields and the notation 1315 01:23:18,820 --> 01:23:22,450 that we developed, OK? 1316 01:23:22,450 --> 01:23:26,412 So we'll talk more next time. 1317 01:23:26,412 --> 01:23:28,950 What we'll do next time is we'll talk about symmetries. 1318 01:23:28,950 --> 01:23:30,790 We'll finally talk about gauge symmetry. 1319 01:23:30,790 --> 01:23:32,430 And we'll talk about other symmetries. 1320 01:23:32,430 --> 01:23:33,805 And in particular, symmetries are 1321 01:23:33,805 --> 01:23:36,120 going to be important for understanding 1322 01:23:36,120 --> 01:23:39,270 how much of this story that I've told you really carries through 1323 01:23:39,270 --> 01:23:40,830 once you start doing loops. 1324 01:23:40,830 --> 01:23:44,190 Symmetries protect you from certain loop corrections. 1325 01:23:44,190 --> 01:23:46,050 Gauge symmetry, for example, connects terms 1326 01:23:46,050 --> 01:23:47,592 to all orders in perturbation theory. 1327 01:23:47,592 --> 01:23:51,750 And we'll see how those things kind of work next time. 1328 01:23:51,750 --> 01:23:55,200 Really, the story I've told you so far is all tree level. 1329 01:23:55,200 --> 01:23:57,090 We haven't done any loops. 1330 01:23:57,090 --> 01:23:59,190 One way of handling what loops can do 1331 01:23:59,190 --> 01:24:01,140 is by looking at symmetries and saying, 1332 01:24:01,140 --> 01:24:02,772 once we impose those symmetries, what's 1333 01:24:02,772 --> 01:24:04,980 the most general possible things that loops could do. 1334 01:24:04,980 --> 01:24:07,250 And we'll do that next time.