1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,520 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,520 --> 00:00:18,370 ocw.mit.edu. 8 00:00:23,100 --> 00:00:25,350 IAIN STEWART: All right, that's roughly where we were. 9 00:00:25,350 --> 00:00:28,530 So last time, we started talking about SCET. 10 00:00:28,530 --> 00:00:30,480 We said it's going to be a theory that 11 00:00:30,480 --> 00:00:34,100 can describe energetic hadrons and energetic jets. 12 00:00:34,100 --> 00:00:37,080 Our first example was discussing about a process 13 00:00:37,080 --> 00:00:40,560 with an energetic hadron, which is this pink pion. 14 00:00:40,560 --> 00:00:44,540 So the pion has a large momentum, large energy. 15 00:00:44,540 --> 00:00:45,990 It's much bigger than lambda QCD. 16 00:00:45,990 --> 00:00:47,400 It's much bigger than in m pi. 17 00:00:47,400 --> 00:00:50,470 And it moves basically along a light cone direction. 18 00:00:50,470 --> 00:00:53,560 So that was a motivation for us to use light cone coordinates. 19 00:00:53,560 --> 00:00:56,010 So we introduce an n and an n bar. 20 00:00:56,010 --> 00:01:02,550 And with that n and n bar, which satisfy n squared 0, 21 00:01:02,550 --> 00:01:08,520 n bar squared 0, and n dot n bar equals 2 as a normalization 22 00:01:08,520 --> 00:01:11,752 convention, we can decompose any momentum P 23 00:01:11,752 --> 00:01:16,380 mu in terms of components along n, components along n bar, 24 00:01:16,380 --> 00:01:18,630 and then the remaining two components which we call 25 00:01:18,630 --> 00:01:22,450 the perpendicular components. 26 00:01:22,450 --> 00:01:27,224 So we can also write the metric out in these coordinates. 27 00:01:32,164 --> 00:01:34,500 And this kind of makes explicit that you 28 00:01:34,500 --> 00:01:36,570 have this off-diagonal nature to the basis, 29 00:01:36,570 --> 00:01:38,830 that you have n mu with n bar mu. 30 00:01:46,500 --> 00:01:49,020 So unlike Cartesian coordinates, where 31 00:01:49,020 --> 00:01:50,880 the component along a direction is just 32 00:01:50,880 --> 00:01:53,850 given by dotting that vector into the vector 33 00:01:53,850 --> 00:01:56,640 you start with, here the component along n 34 00:01:56,640 --> 00:01:59,603 is given by dotting n bar into the vector. 35 00:01:59,603 --> 00:02:01,770 And that's reflected in the metric here in the sense 36 00:02:01,770 --> 00:02:04,320 that you have these terms n within n bar. 37 00:02:08,160 --> 00:02:12,060 So you can do this with any tensor if you have an epsilon. 38 00:02:12,060 --> 00:02:16,290 You can find an epsilon perp tensor, for example, 39 00:02:16,290 --> 00:02:24,180 by taking epsilon and putting in an n bar and an n. 40 00:02:24,180 --> 00:02:26,070 And then this would be a two-component tensor 41 00:02:26,070 --> 00:02:27,960 that behaves in the perpendicular direction 42 00:02:27,960 --> 00:02:30,060 as an antisymmetric tensor. 43 00:02:30,060 --> 00:02:33,570 And this g perp mu nu would be effectively living 44 00:02:33,570 --> 00:02:37,470 in the little subspace of the perp coordinates. 45 00:02:37,470 --> 00:02:39,490 And again, it's a metric tensor there. 46 00:02:39,490 --> 00:02:43,440 And this would be the antisymmetric tensor there. 47 00:02:43,440 --> 00:02:46,710 OK, so these are the coordinates we're going to use. 48 00:02:53,610 --> 00:02:56,990 So n here had a physical motivation, 49 00:02:56,990 --> 00:02:59,150 as you saw from my picture. 50 00:02:59,150 --> 00:03:01,280 The pion was moving in the n direction. 51 00:03:01,280 --> 00:03:03,350 And bar was just a vector that we 52 00:03:03,350 --> 00:03:05,600 decided that we needed in order to define things. 53 00:03:08,760 --> 00:03:11,540 So if you have some vector where n squared is 0 54 00:03:11,540 --> 00:03:14,360 and you want to make a decomposition of coordinates, 55 00:03:14,360 --> 00:03:15,770 then you're required to introduce 56 00:03:15,770 --> 00:03:26,110 a complimentary vector, which is this n bar 57 00:03:26,110 --> 00:03:31,570 to make the decomposition for the reasons I said. 58 00:03:38,123 --> 00:03:39,790 The simplest choice that you could make, 59 00:03:39,790 --> 00:03:46,540 if you made this choice for n, so if we choose n 60 00:03:46,540 --> 00:03:58,990 to be 1, 0, 0, minus 1, as I did, 61 00:03:58,990 --> 00:04:03,260 then the simplest thing you could do for n bar is to pick n 62 00:04:03,260 --> 00:04:07,616 bar to be 1, 0, 0, plus 1. 63 00:04:07,616 --> 00:04:10,600 So then that would be a lightlike vector. 64 00:04:10,600 --> 00:04:13,660 When you dot it into this vector, you get 2. 65 00:04:13,660 --> 00:04:19,459 And it satisfies all the criteria that we would need. 66 00:04:19,459 --> 00:04:21,709 If I choose these two vectors, then I 67 00:04:21,709 --> 00:04:23,420 have to find what perp is because perp 68 00:04:23,420 --> 00:04:26,030 is the space that's orthogonal to these vectors. 69 00:04:26,030 --> 00:04:31,490 So that's these two coordinates, OK? 70 00:04:31,490 --> 00:04:36,440 So perp in general is defined such that n dot P perp is 0 71 00:04:36,440 --> 00:04:40,040 and n bar dot P perp is 0. 72 00:04:40,040 --> 00:04:41,540 It's the orthogonal two directions 73 00:04:41,540 --> 00:04:44,642 to the ones that are picked out by an n bar. 74 00:04:44,642 --> 00:04:46,850 And so you need to know what n and n bar are in order 75 00:04:46,850 --> 00:04:47,808 to define what perp is. 76 00:04:50,550 --> 00:04:51,620 So this is one choice. 77 00:04:51,620 --> 00:04:54,140 You could make other choices. 78 00:04:54,140 --> 00:04:57,330 And we'll come back to this later on. 79 00:04:57,330 --> 00:05:04,910 So just by way of example, if I have the same choice for n, 80 00:05:04,910 --> 00:05:11,210 but I choose n bar to be 3, 2, 2, 1, 81 00:05:11,210 --> 00:05:13,790 that would also be a choice that's equally good. 82 00:05:18,130 --> 00:05:19,360 I need this to be minus 1. 83 00:05:26,944 --> 00:05:31,420 I'll do this, make my choice work. 84 00:05:31,420 --> 00:05:34,180 I guess, well, OK, if we want the same sign, 85 00:05:34,180 --> 00:05:35,200 then I have to do this. 86 00:05:39,910 --> 00:05:42,820 OK, so 9 minus 8, this thing still squares to 0. 87 00:05:42,820 --> 00:05:47,740 You dot it, you get 3 minus 1 is 2, OK? 88 00:05:47,740 --> 00:05:50,320 So it still satisfies the criteria that we had here. 89 00:05:53,440 --> 00:05:56,360 And it points in some other weird direction. 90 00:05:56,360 --> 00:05:58,480 So the point is that this is an auxiliary vector, 91 00:05:58,480 --> 00:06:03,585 and there's some freedom in what you pick for it. 92 00:06:03,585 --> 00:06:05,710 And once you've picked this, if you pick these two, 93 00:06:05,710 --> 00:06:08,230 you would have a different definition of perp. 94 00:06:08,230 --> 00:06:08,740 OK. 95 00:06:08,740 --> 00:06:12,250 But it's an equally valid possible choice. 96 00:06:12,250 --> 00:06:14,840 And we'll actually exploit this freedom later on. 97 00:06:14,840 --> 00:06:18,430 But for now, we'll mostly focus on picking the simplest choice. 98 00:06:21,150 --> 00:06:23,610 OK, what we're actually interested in describing 99 00:06:23,610 --> 00:06:26,310 in these processes is not just the plan, 100 00:06:26,310 --> 00:06:27,870 but what goes on inside the plan. 101 00:06:27,870 --> 00:06:30,730 What is the quark level process? 102 00:06:30,730 --> 00:06:34,050 So we're interested in the constituents. 103 00:06:34,050 --> 00:06:35,430 That's where the dynamics are. 104 00:06:43,522 --> 00:06:46,548 Is there any questions before I keep going? 105 00:06:46,548 --> 00:06:48,840 No. 106 00:06:48,840 --> 00:06:52,710 So in this process, B to D pi, if you think about it 107 00:06:52,710 --> 00:06:54,100 in the rest frame of the B meson, 108 00:06:54,100 --> 00:06:56,168 which is the most natural frame, then the B meson 109 00:06:56,168 --> 00:06:57,960 we've already learned how to describe that. 110 00:06:57,960 --> 00:07:03,900 We can describe that with HQET, same with the D meson. 111 00:07:03,900 --> 00:07:06,390 And we know that the things that are inside the B and the D 112 00:07:06,390 --> 00:07:10,930 meson are one heavy quark and then a bunch of soft stuff. 113 00:07:10,930 --> 00:07:13,945 So I'll call these guys soft because the dynamical part is 114 00:07:13,945 --> 00:07:14,445 soft. 115 00:07:23,600 --> 00:07:33,720 And so we can use HQET for them as we did before. 116 00:07:43,930 --> 00:07:47,590 And that means we're describing gluons and quarks that 117 00:07:47,590 --> 00:07:54,205 are inside these hadrons where the forward momentum are 118 00:07:54,205 --> 00:07:57,370 of order lambda QCD. 119 00:07:57,370 --> 00:08:00,280 The pion, on the other hand, is what we would collinear. 120 00:08:16,620 --> 00:08:18,590 So as I already described, the pion's energy 121 00:08:18,590 --> 00:08:21,060 is much greater than its mass. 122 00:08:21,060 --> 00:08:21,905 Its highly boosted. 123 00:08:29,720 --> 00:08:32,600 If you were to talk about it in the rest frame, 124 00:08:32,600 --> 00:08:34,880 then like the B and the D meson, then the constituents 125 00:08:34,880 --> 00:08:40,215 of the pion would have momentum of order lambda QCD. 126 00:08:40,215 --> 00:08:42,590 But if you were to talk about the pion in the rest frame, 127 00:08:42,590 --> 00:08:47,050 you'd have to talk about the B and the D in the boosted frame. 128 00:08:47,050 --> 00:08:49,150 So let's stick with describing the B and the D 129 00:08:49,150 --> 00:08:52,790 in the rest frame or close to the rest frame. 130 00:08:52,790 --> 00:08:55,150 The B meson is in its rest frame. 131 00:08:55,150 --> 00:08:57,610 In the D meson is slow. 132 00:08:57,610 --> 00:09:00,700 And in that case, we're stuck with the pion being energetic. 133 00:09:03,820 --> 00:09:12,640 So in rest frame, our pion would be-- 134 00:09:17,890 --> 00:09:25,698 it would also have quarks and gluons 135 00:09:25,698 --> 00:09:27,560 our P mu is order lambda QCD. 136 00:09:32,780 --> 00:09:35,360 And we can actually just take that result, once we know that, 137 00:09:35,360 --> 00:09:38,720 and boost it to another frame. 138 00:09:38,720 --> 00:09:43,520 So let's just boost along z hat by some kappa that's 139 00:09:43,520 --> 00:09:46,695 much greater than 1 as the boost. 140 00:09:46,695 --> 00:09:48,570 And the way that light cone coordinates boost 141 00:09:48,570 --> 00:09:49,290 is very simple. 142 00:09:49,290 --> 00:09:52,040 If you're along the axis of the light cone coordinates, 143 00:09:52,040 --> 00:09:55,410 it's multiplicative. 144 00:09:55,410 --> 00:09:57,500 So P minus gets enhanced by some amount. 145 00:09:57,500 --> 00:09:59,390 P plus gets suppressed by the same amount. 146 00:10:02,100 --> 00:10:05,268 That's one nice thing about these coordinates. 147 00:10:05,268 --> 00:10:07,310 And of course, P perp doesn't change because it's 148 00:10:07,310 --> 00:10:08,510 perpendicular to the boost. 149 00:10:11,570 --> 00:10:14,925 So now, we can get our pion, which is moving, 150 00:10:14,925 --> 00:10:15,800 which should be pink. 151 00:10:25,340 --> 00:10:27,020 And if we ask about its constituents, 152 00:10:27,020 --> 00:10:29,060 we just boost the components of this for vector. 153 00:10:38,430 --> 00:10:41,330 So we ask about how they scale. 154 00:10:41,330 --> 00:10:44,810 And we look at the different components. 155 00:10:44,810 --> 00:10:47,870 The plus, minus, and perp scale differently now, 156 00:10:47,870 --> 00:10:50,320 so we have to break it up by that. 157 00:10:50,320 --> 00:10:53,240 And if we boost it by this amount Q or lambda 158 00:10:53,240 --> 00:11:02,970 over Q, Q over lambda, then that's 159 00:11:02,970 --> 00:11:06,440 the scaling for this boosted pion. 160 00:11:06,440 --> 00:11:06,940 OK. 161 00:11:06,940 --> 00:11:10,110 So now, it's got a component in the minus direction. 162 00:11:10,110 --> 00:11:13,650 n bar dot P is order Q. That's what 163 00:11:13,650 --> 00:11:15,360 we saw before when we decomposed the P, 164 00:11:15,360 --> 00:11:18,570 that it was basically Q times a lightlike vector. 165 00:11:18,570 --> 00:11:20,430 But that was the pion. 166 00:11:20,430 --> 00:11:23,130 Now, we're talking about the constituents inside the pion. 167 00:11:23,130 --> 00:11:25,638 Constituents inside the pion fill it out. 168 00:11:25,638 --> 00:11:27,180 They fill it out in the perpendicular 169 00:11:27,180 --> 00:11:30,210 by an amount lambda QCD that's perpendicular to the direction 170 00:11:30,210 --> 00:11:31,350 of its motion. 171 00:11:31,350 --> 00:11:34,440 And then the plus momentum got correspondingly smaller 172 00:11:34,440 --> 00:11:37,310 as the minus momentum have got bigger, 173 00:11:37,310 --> 00:11:38,310 so we have that scaling. 174 00:11:40,830 --> 00:11:42,510 And so the relative scaling here is 175 00:11:42,510 --> 00:11:45,110 what actually defines something being collinear. 176 00:11:51,000 --> 00:11:53,610 So the relative scaling of this vector 177 00:11:53,610 --> 00:11:58,230 here is that the P minus is much bigger than the P perp is 178 00:11:58,230 --> 00:12:00,480 much bigger than the P plus. 179 00:12:00,480 --> 00:12:02,472 And that's what we mean by collinear. 180 00:12:06,900 --> 00:12:08,490 It's collimated in some direction, 181 00:12:08,490 --> 00:12:09,990 and that direction is the direction 182 00:12:09,990 --> 00:12:10,907 of the large momentum. 183 00:12:13,353 --> 00:12:14,770 You always have to be careful when 184 00:12:14,770 --> 00:12:16,810 you say things like that because the component 185 00:12:16,810 --> 00:12:19,610 along the direction is the opposite lightlike vector, 186 00:12:19,610 --> 00:12:22,600 but I think you'll always know what I mean. 187 00:12:22,600 --> 00:12:27,730 OK, so in the n mu direction, we have a large component P minus. 188 00:12:27,730 --> 00:12:29,837 And that defines this thing is collimated 189 00:12:29,837 --> 00:12:30,920 in a particular direction. 190 00:12:30,920 --> 00:12:34,065 It's perpendicular fluctuations to that direction are small. 191 00:12:34,065 --> 00:12:35,440 And so all the degrees of freedom 192 00:12:35,440 --> 00:12:40,330 that are in this boosted pion have that type of scaling. 193 00:12:40,330 --> 00:12:42,070 So what we're describing, or what 194 00:12:42,070 --> 00:12:45,370 we want to describe if we have a field theory for this, 195 00:12:45,370 --> 00:12:47,110 is we want to describe, if you like, 196 00:12:47,110 --> 00:12:57,340 fluctuations about the pion momentum, which, 197 00:12:57,340 --> 00:13:06,580 ignoring the pion mass, we could just take it to be like this. 198 00:13:06,580 --> 00:13:09,700 And the size of the fluctuations we need to treat 199 00:13:09,700 --> 00:13:12,250 are things that can fluctuate by amounts of this size. 200 00:13:16,110 --> 00:13:17,610 So the field theory is going to have 201 00:13:17,610 --> 00:13:23,153 to describe fluctuations about some kind of canonical scaling. 202 00:13:23,153 --> 00:13:24,570 And the field theory for this pion 203 00:13:24,570 --> 00:13:26,520 is going to have to be describing 204 00:13:26,520 --> 00:13:29,100 collinear fluctuations that are of this type. 205 00:13:29,100 --> 00:13:33,560 Just like the HQET had to describe soft fluctuations, 206 00:13:33,560 --> 00:13:36,690 P mu's of order lambda QCD ignorant of the heavy quark 207 00:13:36,690 --> 00:13:40,730 mass, here it's a little bit more complicated. 208 00:13:40,730 --> 00:13:43,230 But that's the kind of thing we want the field theory to do. 209 00:13:47,300 --> 00:13:48,520 Any questions about that? 210 00:13:57,220 --> 00:14:02,340 So the way that we write this is we say that P plus, P minus, 211 00:14:02,340 --> 00:14:08,460 P perp has a particular scaling that we 212 00:14:08,460 --> 00:14:13,200 call lambda squared 1 lambda where 213 00:14:13,200 --> 00:14:14,640 lambda is some small parameter. 214 00:14:20,510 --> 00:14:22,750 And if we have a momentum that scales that way, 215 00:14:22,750 --> 00:14:24,066 we call it collinear. 216 00:14:35,320 --> 00:14:39,630 So this is generic any case with any lambda. 217 00:14:39,630 --> 00:14:43,116 And our allowed here was just lambda QCD over Q. 218 00:14:43,116 --> 00:14:46,680 But if we encounter another physical problem 219 00:14:46,680 --> 00:14:49,620 where the lambda was different, we 220 00:14:49,620 --> 00:14:51,224 would also call that collinear. 221 00:14:55,870 --> 00:15:00,970 All right, so what's a nice way of picturing this, 222 00:15:00,970 --> 00:15:02,980 what we're doing here? 223 00:15:02,980 --> 00:15:04,990 Because it's a little bit different than you're 224 00:15:04,990 --> 00:15:07,223 used to with an effective field theory. 225 00:15:07,223 --> 00:15:09,640 Usually, with an effective field theory, what you're doing 226 00:15:09,640 --> 00:15:12,593 is you're separating modes by their invariant mass. 227 00:15:12,593 --> 00:15:15,010 You have things with large invariant mass, small invariant 228 00:15:15,010 --> 00:15:16,618 mass. 229 00:15:16,618 --> 00:15:18,160 If you think about massive particles, 230 00:15:18,160 --> 00:15:20,950 well, that's just the invariant mass squared. 231 00:15:20,950 --> 00:15:22,900 So if you're separating massive particles 232 00:15:22,900 --> 00:15:25,780 from massless particles or less massive particles, 233 00:15:25,780 --> 00:15:30,370 you're really separating things along an invariant mass curve. 234 00:15:30,370 --> 00:15:33,940 Just an invariant mass variable is used for the separation. 235 00:15:33,940 --> 00:15:35,740 And that doesn't quite suffice here. 236 00:15:35,740 --> 00:15:38,830 Because as you saw, the pion in the B and the D meson, 237 00:15:38,830 --> 00:15:41,620 they both had P squared of order lambda QCD. 238 00:15:41,620 --> 00:15:44,710 What separates the pion from the B or the D meson 239 00:15:44,710 --> 00:15:47,350 is this morphyne structure. 240 00:16:05,290 --> 00:16:07,720 So SCET is actually an example of an effective field 241 00:16:07,720 --> 00:16:12,340 theory that requires at least more than one variable 242 00:16:12,340 --> 00:16:14,830 to describe where the degrees of freedom live. 243 00:16:19,413 --> 00:16:21,580 So we can draw a picture for what we've been talking 244 00:16:21,580 --> 00:16:24,880 about here in two variables. 245 00:16:24,880 --> 00:16:26,470 Let's just pick P minus and P plus. 246 00:16:31,600 --> 00:16:35,980 And essentially, what's going on in this space is 247 00:16:35,980 --> 00:16:38,260 you can think that there's degrees of freedom that 248 00:16:38,260 --> 00:16:40,840 live in this space at different locations. 249 00:16:47,820 --> 00:16:51,570 So out here, if I draw a hyperbola like this, 250 00:16:51,570 --> 00:16:58,938 then remember that P squared was P plus times P minus minus P 251 00:16:58,938 --> 00:16:59,480 perp squared. 252 00:16:59,480 --> 00:17:02,460 But let's ignore P perp squared for this picture. 253 00:17:02,460 --> 00:17:05,839 So if I draw a curve of constant P squared in this plane, then 254 00:17:05,839 --> 00:17:07,010 it's I hyperbola. 255 00:17:07,010 --> 00:17:08,930 So these are curves of constant P squared. 256 00:17:16,540 --> 00:17:18,670 And this one here has P squared of order Q 257 00:17:18,670 --> 00:17:21,800 squared, which might be Mb squared or some hard scale. 258 00:17:21,800 --> 00:17:25,420 So this any degrees of freedom that live on this curve, 259 00:17:25,420 --> 00:17:27,640 or in particularly these ones, would 260 00:17:27,640 --> 00:17:29,752 be what we would call hard degrees of freedom. 261 00:17:29,752 --> 00:17:31,960 And those are something that we want to integrate out 262 00:17:31,960 --> 00:17:34,210 of the effective theory. 263 00:17:34,210 --> 00:17:37,180 And the other degrees of freedom that we've been talking about 264 00:17:37,180 --> 00:17:38,680 have smaller invariant mass. 265 00:17:38,680 --> 00:17:42,460 So this hyperbola down here has P squared of order lambda 266 00:17:42,460 --> 00:17:44,410 QCD squared. 267 00:17:44,410 --> 00:17:46,570 But there's two different degrees of freedom 268 00:17:46,570 --> 00:17:48,340 that live on this curve. 269 00:17:48,340 --> 00:17:50,020 One of them has a large P minus. 270 00:17:50,020 --> 00:17:53,110 That's the collinear one, so it should be pink. 271 00:18:02,030 --> 00:18:08,090 And then the soft one lives down there. 272 00:18:08,090 --> 00:18:12,680 So P minus here is scaling, if you like, lambda to the 0. 273 00:18:12,680 --> 00:18:17,135 And here, for this soft mode, which also exists in this case, 274 00:18:17,135 --> 00:18:18,635 this is actually going to be lambda. 275 00:18:24,060 --> 00:18:27,210 So you can contrast that type of picture 276 00:18:27,210 --> 00:18:31,260 with a more usual picture where you would just have one line. 277 00:18:31,260 --> 00:18:32,940 And you'd say there's some modes up here 278 00:18:32,940 --> 00:18:33,900 and some modes down there. 279 00:18:33,900 --> 00:18:35,400 And you'd integrate out these modes. 280 00:18:35,400 --> 00:18:36,790 And you keep those modes. 281 00:18:36,790 --> 00:18:38,582 This is a little different because you want 282 00:18:38,582 --> 00:18:39,798 to integrate out these modes. 283 00:18:39,798 --> 00:18:41,340 You want to keep both of those modes, 284 00:18:41,340 --> 00:18:43,465 but they live in a little bit of a different place. 285 00:18:43,465 --> 00:18:45,960 And that's actually going to be important to formulating 286 00:18:45,960 --> 00:18:47,760 the effective theory. 287 00:18:47,760 --> 00:18:50,910 So the way that you should think about this, physically the way 288 00:18:50,910 --> 00:18:53,490 you should think about it, is that these modes are kind of 289 00:18:53,490 --> 00:18:54,840 localized in that region. 290 00:18:57,360 --> 00:19:05,320 This is the right physical picture, 291 00:19:05,320 --> 00:19:08,440 which requires another variable besides just invariant mass 292 00:19:08,440 --> 00:19:15,940 in order to specify that, right? 293 00:19:15,940 --> 00:19:18,790 The reason we don't have to draw a third direction for P perp 294 00:19:18,790 --> 00:19:25,410 is because it was just redundant information. 295 00:19:25,410 --> 00:19:28,455 P perp squared is always of P plus P 296 00:19:28,455 --> 00:19:30,330 minus if you're talking about fluctuations 297 00:19:30,330 --> 00:19:31,538 that are near the mass shell. 298 00:19:40,490 --> 00:19:43,400 For a massless mode, that mass shell is P squared equals 0. 299 00:19:47,030 --> 00:19:49,730 And so P perp would just be providing redundant information 300 00:19:49,730 --> 00:19:51,605 to our picture, and we just can leave it out. 301 00:19:54,870 --> 00:19:57,080 Now, the boundaries of the regions 302 00:19:57,080 --> 00:20:01,700 between soft and collinear here seems like an interesting thing 303 00:20:01,700 --> 00:20:03,550 to worry about. 304 00:20:03,550 --> 00:20:07,400 And that is, indeed, true. 305 00:20:07,400 --> 00:20:09,470 You have to think about how you want to set up 306 00:20:09,470 --> 00:20:11,060 this effective theory. 307 00:20:11,060 --> 00:20:13,425 And of course, as I have been emphasizing earlier 308 00:20:13,425 --> 00:20:15,050 in the course, the easiest way to think 309 00:20:15,050 --> 00:20:17,630 about momentum degrees of freedom 310 00:20:17,630 --> 00:20:19,430 is with a Wilsonian picture. 311 00:20:19,430 --> 00:20:22,348 That makes physically what's going on very clear. 312 00:20:22,348 --> 00:20:24,140 So the simplest thing would be to introduce 313 00:20:24,140 --> 00:20:35,270 a Wilsonian cut-off and set this up 314 00:20:35,270 --> 00:20:38,750 as a Wilsonian effective field theory. 315 00:20:38,750 --> 00:20:40,910 And then we would just take these regions. 316 00:20:40,910 --> 00:20:44,180 And I would literally carve them out in the way that I drew. 317 00:20:44,180 --> 00:20:46,160 I would carve out some cut-off between them. 318 00:20:46,160 --> 00:20:48,890 And I would decide who's in the soft region, who's 319 00:20:48,890 --> 00:20:53,720 in the collinear region based on those hard cut-offs. 320 00:20:53,720 --> 00:20:55,700 But we don't want to do that, actually, 321 00:20:55,700 --> 00:20:59,240 because it would mess up all sorts of symmetries. 322 00:20:59,240 --> 00:21:01,515 In particular, it would mess up gauge cemetery 323 00:21:01,515 --> 00:21:03,140 which is an important thing when you're 324 00:21:03,140 --> 00:21:05,400 talking about gauge theory. 325 00:21:05,400 --> 00:21:08,000 So we're going to use dimensional regularization, 326 00:21:08,000 --> 00:21:09,515 as we have for other problems. 327 00:21:29,360 --> 00:21:31,280 And that actually will still leave us 328 00:21:31,280 --> 00:21:33,860 with this picture, which I drew as a cartoon. 329 00:21:33,860 --> 00:21:35,000 You'll still be correct. 330 00:21:35,000 --> 00:21:37,400 Think about the modes live in those places 331 00:21:37,400 --> 00:21:39,200 in dimensional regularization. 332 00:21:39,200 --> 00:21:42,020 What's a little bit harder is how to think about the cut-off. 333 00:21:42,020 --> 00:21:46,121 And we'll treat that in some detail later on. 334 00:21:46,121 --> 00:21:59,260 So it's still the correct picture, 335 00:21:59,260 --> 00:22:04,740 but treating the region overlaps with dimensional regularization 336 00:22:04,740 --> 00:22:05,740 is a little more tricky. 337 00:22:08,285 --> 00:22:09,910 But at least we can do it in a way that 338 00:22:09,910 --> 00:22:11,692 preserves the gauge invariance. 339 00:22:17,050 --> 00:22:22,529 So that's going to be our mode of operation. 340 00:22:43,500 --> 00:22:44,790 This theory has a name. 341 00:22:44,790 --> 00:22:48,020 It goes by the name SCET2. 342 00:22:48,020 --> 00:22:49,260 That's why I called it SCET2. 343 00:22:54,510 --> 00:22:59,780 We'll come back in a moment to what SCET1 is. 344 00:22:59,780 --> 00:23:02,120 So I can say that the degrees of freedom 345 00:23:02,120 --> 00:23:05,758 in this theory, the one that we've been talking about, 346 00:23:05,758 --> 00:23:07,550 are some collinear degree of freedom that's 347 00:23:07,550 --> 00:23:09,920 associated to some direction, some collinear 348 00:23:09,920 --> 00:23:14,450 degrees of freedom as well as some soft degrees of freedom. 349 00:23:14,450 --> 00:23:16,568 And when you have effective theories that 350 00:23:16,568 --> 00:23:18,860 are like this one, where the soft and collinear degrees 351 00:23:18,860 --> 00:23:21,230 of freedom live on the same mass hyperbola, 352 00:23:21,230 --> 00:23:22,670 they're called SCET2 theories. 353 00:23:50,720 --> 00:23:53,090 And these are really the kind of theories 354 00:23:53,090 --> 00:23:56,360 that you get when you're talking about energetic hadron 355 00:23:56,360 --> 00:23:57,425 production. 356 00:24:09,820 --> 00:24:11,730 So any questions so far? 357 00:24:17,730 --> 00:24:21,420 OK, so if that's energetic hadrons, 358 00:24:21,420 --> 00:24:23,520 then SCET1 will be energetic jets. 359 00:24:23,520 --> 00:24:25,740 And that's what we'll talk about next. 360 00:24:28,270 --> 00:24:31,950 So let's do another example which has jets in it. 361 00:24:31,950 --> 00:24:34,530 And we'll see what the similarities are 362 00:24:34,530 --> 00:24:39,690 to this SCET2 set up in terms of just identifying still what 363 00:24:39,690 --> 00:24:43,630 the right degrees of freedom are. 364 00:24:43,630 --> 00:24:45,810 So let's look at e plus e minus to two jets. 365 00:24:53,730 --> 00:24:57,750 So e plus e minus collide. 366 00:24:57,750 --> 00:25:01,170 They produce a virtual photon, say. 367 00:25:01,170 --> 00:25:05,250 The virtual photon produces a quark-antiquark pair. 368 00:25:05,250 --> 00:25:08,430 The quark-antiquark pair starts to radiate. 369 00:25:08,430 --> 00:25:12,690 And we get jets, two of them. 370 00:25:21,360 --> 00:25:23,243 So again, there's a kind of natural frame 371 00:25:23,243 --> 00:25:24,410 to describe this scattering. 372 00:25:24,410 --> 00:25:28,460 And that's the center of mass frame of the e plus e minus. 373 00:25:28,460 --> 00:25:30,860 Most e plus e minus colliders are built in that frame. 374 00:25:37,120 --> 00:25:40,270 And if you're in the center of mass frame 375 00:25:40,270 --> 00:25:44,530 and you call the for momentum of this photon Q, 376 00:25:44,530 --> 00:25:47,650 then it just has an energy component. 377 00:25:47,650 --> 00:25:50,320 So Q here is not the same as the Q in our previous example, 378 00:25:50,320 --> 00:25:52,690 but we're always going to identify the hard scale as Q. 379 00:25:52,690 --> 00:25:54,550 So here, the hard scale is the scale 380 00:25:54,550 --> 00:25:58,823 of the energy of the collision in the center of mass frame. 381 00:25:58,823 --> 00:26:00,490 And if you ask, what does the event look 382 00:26:00,490 --> 00:26:02,080 like in the center of mass frame, 383 00:26:02,080 --> 00:26:04,000 then these two jets which are going out 384 00:26:04,000 --> 00:26:05,480 have to balance each other. 385 00:26:05,480 --> 00:26:07,270 And so you have two back to back jets. 386 00:26:10,300 --> 00:26:19,160 So we draw it like this, one jet going this way, one jet going 387 00:26:19,160 --> 00:26:20,010 that way. 388 00:26:20,010 --> 00:26:21,890 Our original e plus e minus might have come 389 00:26:21,890 --> 00:26:23,100 in from some other direction. 390 00:26:23,100 --> 00:26:29,030 So maybe e plus e minus were coming in from here and here. 391 00:26:32,550 --> 00:26:36,440 And then we have these two jets going out this way. 392 00:26:36,440 --> 00:26:38,690 Since they're jets, that means they're 393 00:26:38,690 --> 00:26:40,460 collimated sprays of radiation. 394 00:26:46,745 --> 00:26:47,870 So they're not featureless. 395 00:26:47,870 --> 00:26:53,550 They have some size to them like this. 396 00:26:53,550 --> 00:26:57,920 And in this process, we can quickly 397 00:26:57,920 --> 00:27:00,560 identify that there's two relevant directions. 398 00:27:00,560 --> 00:27:03,950 It's back to back, but let's define 399 00:27:03,950 --> 00:27:07,460 this direction of this jet to be n1, 400 00:27:07,460 --> 00:27:09,960 some lightlike vector that points in that direction, 401 00:27:09,960 --> 00:27:11,870 and this one to be n2. 402 00:27:11,870 --> 00:27:16,940 So generically, we could say you n is some 1, n hat. 403 00:27:16,940 --> 00:27:18,900 And if we choose this to be the z-axis, 404 00:27:18,900 --> 00:27:24,120 then this must be 1, 0, 0 1, just like we had before. 405 00:27:24,120 --> 00:27:26,780 But we can always pick a lightlike vector that points 406 00:27:26,780 --> 00:27:28,082 along some direction n hat. 407 00:27:28,082 --> 00:27:29,540 So when I draw a picture like this, 408 00:27:29,540 --> 00:27:32,480 really what I mean when I say n1 points in this direction 409 00:27:32,480 --> 00:27:35,360 is that the hat part of it points in that direction 410 00:27:35,360 --> 00:27:38,520 in through space. 411 00:27:38,520 --> 00:27:41,160 OK, so just like before, what happens 412 00:27:41,160 --> 00:27:44,040 with the jet is, as I said last time actually, 413 00:27:44,040 --> 00:27:47,310 we have a large energy flow in this direction and a smaller 414 00:27:47,310 --> 00:27:49,540 perpendicular flow. 415 00:27:49,540 --> 00:27:52,380 So if we measure perpendicular momentum, which 416 00:27:52,380 --> 00:27:57,040 we can think of as perpendicular to that axis, free axis, 417 00:27:57,040 --> 00:28:01,110 then that's the perpendicular flow inside the jet. 418 00:28:01,110 --> 00:28:06,090 And so if we want to talk about constituents inside the jet, 419 00:28:06,090 --> 00:28:08,670 then we'll be interested in smaller perpendicular 420 00:28:08,670 --> 00:28:10,650 flow than flow in the forward direction. 421 00:28:10,650 --> 00:28:13,960 And that's what makes it into a collimated jet. 422 00:28:13,960 --> 00:28:14,710 You have question? 423 00:28:14,710 --> 00:28:15,390 AUDIENCE: [INAUDIBLE] 424 00:28:15,390 --> 00:28:15,840 IAIN STEWART: Sure. 425 00:28:15,840 --> 00:28:17,423 AUDIENCE: So you're not using anything 426 00:28:17,423 --> 00:28:19,440 about the QCD [INAUDIBLE] to tell you 427 00:28:19,440 --> 00:28:20,828 the things is [INAUDIBLE]. 428 00:28:20,828 --> 00:28:23,370 You're just saying we know that we have to [INAUDIBLE] topic. 429 00:28:23,370 --> 00:28:25,740 IAIN STEWART: Yeah, that's the attitude, right. 430 00:28:25,740 --> 00:28:27,022 So I mean, we'll see why. 431 00:28:27,022 --> 00:28:28,480 You know, you can ask the question, 432 00:28:28,480 --> 00:28:30,063 why do we get jets in the first place? 433 00:28:30,063 --> 00:28:31,470 We haven't talked about that. 434 00:28:31,470 --> 00:28:33,130 And we could talk about that. 435 00:28:33,130 --> 00:28:35,910 I'm taking the attitude here, this is what we observe. 436 00:28:35,910 --> 00:28:38,220 How do we design an effective field theory for it? 437 00:28:38,220 --> 00:28:40,320 And we'll see then, from that effective theory, 438 00:28:40,320 --> 00:28:41,850 we can go back and understand why 439 00:28:41,850 --> 00:28:43,475 it is that we get these objects and why 440 00:28:43,475 --> 00:28:45,225 actually, when you look at cross-sections, 441 00:28:45,225 --> 00:28:47,835 that this is the leading order description of what happens. 442 00:28:47,835 --> 00:28:50,790 We'll come to that later. 443 00:28:50,790 --> 00:28:52,710 Good question. 444 00:28:52,710 --> 00:28:54,107 Any other questions? 445 00:28:56,830 --> 00:28:57,330 All right. 446 00:29:02,310 --> 00:29:06,480 Yeah, in the hadron case, you just say this process exists. 447 00:29:06,480 --> 00:29:09,530 And I didn't explain to you why the process could exist. 448 00:29:09,530 --> 00:29:11,280 But in the jet case, there's more dynamics 449 00:29:11,280 --> 00:29:13,800 going into the fact that the process exists, 450 00:29:13,800 --> 00:29:17,850 the fact that QCD likes to radiate collinear 451 00:29:17,850 --> 00:29:20,400 to a direction, which has to do with the infrared structure 452 00:29:20,400 --> 00:29:22,200 of the theory. 453 00:29:22,200 --> 00:29:23,640 And we'll come back to that. 454 00:29:28,730 --> 00:29:30,690 I mean, the short answer, of course, 455 00:29:30,690 --> 00:29:33,200 is that things like to radiate in that direction 456 00:29:33,200 --> 00:29:37,160 because there's large logarithms that 457 00:29:37,160 --> 00:29:41,670 enhance splittings that are collinear to the direction 458 00:29:41,670 --> 00:29:42,210 of motion. 459 00:29:45,030 --> 00:29:46,880 And you also get smaller coupling consonants 460 00:29:46,880 --> 00:29:51,963 when you do that rather than a wide angle emission. 461 00:29:51,963 --> 00:29:53,630 So we're saying here, what we're saying, 462 00:29:53,630 --> 00:29:56,600 is we measure e plus e minus to two jets. 463 00:29:56,600 --> 00:29:58,758 If I had an extra wide angle emission, 464 00:29:58,758 --> 00:30:00,550 that would be e plus e minus to three jets. 465 00:30:00,550 --> 00:30:03,041 So I ruled that out just right at the beginning. 466 00:30:05,690 --> 00:30:07,790 OK, so what you can do with this picture 467 00:30:07,790 --> 00:30:11,400 in order to define what's going on is you can say, 468 00:30:11,400 --> 00:30:13,490 well they're to back-to-back jets. 469 00:30:13,490 --> 00:30:14,750 Let's draw a hemisphere. 470 00:30:17,978 --> 00:30:19,520 So this is supposed to be kind of out 471 00:30:19,520 --> 00:30:23,380 of the board point at you. 472 00:30:23,380 --> 00:30:25,130 Let's draw a hemisphere between these two. 473 00:30:25,130 --> 00:30:28,370 And then we can call one side a and the other side b. 474 00:30:28,370 --> 00:30:29,840 And we can talk about momenta that 475 00:30:29,840 --> 00:30:32,048 are flowing in the a hemisphere and the b hemisphere. 476 00:30:32,048 --> 00:30:34,318 And you see that we have one jet in each hemisphere. 477 00:30:43,600 --> 00:30:46,590 So we have a jet of hadrons in hemisphere a. 478 00:30:50,910 --> 00:30:56,540 And we have another one in hemisphere b. 479 00:31:01,110 --> 00:31:04,431 So in some ways, we can talk about a and b independently. 480 00:31:10,560 --> 00:31:12,270 So let's start off by talking about a, 481 00:31:12,270 --> 00:31:15,350 which is what I called the n1 collinear jet. 482 00:31:17,930 --> 00:31:26,550 And if we ask about constituents in the jet, 483 00:31:26,550 --> 00:31:28,110 then the perpendicular momentum will 484 00:31:28,110 --> 00:31:31,980 be of some size, which let me just call it delta. 485 00:31:31,980 --> 00:31:37,290 And that'll be much smaller than P minus, which is of size Q. 486 00:31:37,290 --> 00:31:40,500 So the energy that we pump in through the photon 487 00:31:40,500 --> 00:31:41,310 has to leave. 488 00:31:41,310 --> 00:31:44,760 And the only place that it can leave is that half of it 489 00:31:44,760 --> 00:31:46,260 has to kind of leave this direction. 490 00:31:46,260 --> 00:31:48,570 Half of it has to leave that direction 491 00:31:48,570 --> 00:31:50,850 by energy conservation. 492 00:31:50,850 --> 00:31:55,090 And so we have a large energy flow in the P minus component. 493 00:31:55,090 --> 00:31:57,510 And then we have a much smaller amount in the P perp. 494 00:31:57,510 --> 00:31:59,820 And that's what I already said. 495 00:31:59,820 --> 00:32:01,740 And given those two facts, you can 496 00:32:01,740 --> 00:32:04,260 ask, what about the constituents of the jet? 497 00:32:04,260 --> 00:32:07,575 And again, they're collinear because you have a hierarchy. 498 00:32:16,485 --> 00:32:19,050 So the plus minus and perpendicular momentum 499 00:32:19,050 --> 00:32:21,540 of constituents would scale in that way. 500 00:32:21,540 --> 00:32:26,668 And that is Q lambda squared 1 lambda, just as 501 00:32:26,668 --> 00:32:27,960 before with a different lambda. 502 00:32:34,670 --> 00:32:37,650 So here, lambda is how much spread and perp 503 00:32:37,650 --> 00:32:43,230 do we have over Q. This delta doesn't have to be lambda QCD. 504 00:32:43,230 --> 00:32:46,350 It could be something much bigger. 505 00:32:46,350 --> 00:32:48,855 Another way of thinking about physically what this delta is 506 00:32:48,855 --> 00:32:52,570 is to calculate something called the jet mass. 507 00:32:52,570 --> 00:33:01,260 So you could define the mass of the jet as the sum of the four 508 00:33:01,260 --> 00:33:10,510 vectors in hemisphere a of all the particles squared. 509 00:33:10,510 --> 00:33:14,010 And if you ask about how big that is, in these coordinates 510 00:33:14,010 --> 00:33:16,650 that we're using, it's P plus times P minus minus P 511 00:33:16,650 --> 00:33:18,460 perp squared. 512 00:33:18,460 --> 00:33:22,980 If we align things so that this thing is really 513 00:33:22,980 --> 00:33:26,260 aligned with the jet, there won't be any P perp squared. 514 00:33:26,260 --> 00:33:29,910 But that wouldn't change us scaling our given anyway. 515 00:33:29,910 --> 00:33:32,820 So basically, if you ask about what this jet mass is, 516 00:33:32,820 --> 00:33:35,970 it's scaling like a P plus times a P minus. 517 00:33:35,970 --> 00:33:38,630 And it's scaling like delta squared. 518 00:33:38,630 --> 00:33:43,230 So the jet mass is something of order delta squared. 519 00:33:43,230 --> 00:33:45,490 And that's much less than Q squared. 520 00:33:45,490 --> 00:33:48,030 So another way of characterizing that you have a jet 521 00:33:48,030 --> 00:33:50,670 is to measure the invariant mass of all the particles 522 00:33:50,670 --> 00:33:52,020 in this hemisphere. 523 00:33:52,020 --> 00:33:54,390 And if that invariant mass is small relative 524 00:33:54,390 --> 00:33:58,320 to the hard scale, then it's collimated, OK? 525 00:33:58,320 --> 00:34:05,040 So MJ squared Q squared much less than 1 also 526 00:34:05,040 --> 00:34:06,060 means collimated. 527 00:34:11,342 --> 00:34:13,050 So we could talk about it either in terms 528 00:34:13,050 --> 00:34:15,060 of perpendicular spread, or we could talk about it 529 00:34:15,060 --> 00:34:15,960 as an invariant mass. 530 00:34:19,830 --> 00:34:22,183 OK. 531 00:34:22,183 --> 00:34:25,620 So this is much less than 1. 532 00:34:25,620 --> 00:34:29,639 Now, with this delta, we didn't specify what it is. 533 00:34:29,639 --> 00:34:31,980 So what are possible values of delta? 534 00:34:31,980 --> 00:34:35,400 Well, let's first talk about what it's not. 535 00:34:35,400 --> 00:34:37,739 If delta was of order Q, then obviously 536 00:34:37,739 --> 00:34:40,530 we'd break the kind of scaling that we have here that this 537 00:34:40,530 --> 00:34:42,525 should be much less than 1. 538 00:34:42,525 --> 00:34:44,775 And what happens in that case is we don't have dijets. 539 00:34:48,989 --> 00:34:52,600 So if either the mass MJ squared of the particles 540 00:34:52,600 --> 00:34:56,130 becomes of order Q squared or the perpendicular spread 541 00:34:56,130 --> 00:34:59,751 becomes of order Q, then we don't have dijets anymore. 542 00:35:02,460 --> 00:35:04,230 And basically, in this case, you would 543 00:35:04,230 --> 00:35:14,490 be talking about inclusive sum over our jets 544 00:35:14,490 --> 00:35:16,740 And that would be something that you would actually 545 00:35:16,740 --> 00:35:19,115 describe in a different way in the effective field theory 546 00:35:19,115 --> 00:35:22,500 because you wouldn't then have collinear degrees of freedom 547 00:35:22,500 --> 00:35:23,550 for this jet. 548 00:35:23,550 --> 00:35:25,570 It's really picking out the dijet process 549 00:35:25,570 --> 00:35:27,653 that means that these collinear degrees of freedom 550 00:35:27,653 --> 00:35:28,500 are relevant. 551 00:35:28,500 --> 00:35:31,410 If you did just e plus e minus to hadrons, that's something 552 00:35:31,410 --> 00:35:33,720 you could do with an operator product expansion 553 00:35:33,720 --> 00:35:35,940 without ever talking about SCET. 554 00:35:35,940 --> 00:35:38,520 And that would be valid if you were really 555 00:35:38,520 --> 00:35:40,860 doing an inclusive sum over jets in all directions 556 00:35:40,860 --> 00:35:43,950 without any restrictions that tell you it's a dijet. 557 00:35:43,950 --> 00:35:46,560 And then the effect of power counting in your OPE 558 00:35:46,560 --> 00:35:49,290 would be such that delta is of order Q. 559 00:35:49,290 --> 00:35:50,970 So this is actually the OPE region. 560 00:35:54,040 --> 00:36:01,170 Let me call it the OPE region of Peskin or any other field 561 00:36:01,170 --> 00:36:01,680 theory book. 562 00:36:06,650 --> 00:36:08,710 Another thing we could do is we could take delta 563 00:36:08,710 --> 00:36:09,460 to be very small. 564 00:36:12,550 --> 00:36:14,830 We could take delta all the way down to lambda QCD. 565 00:36:21,550 --> 00:36:22,870 And that's also not a jet. 566 00:36:29,810 --> 00:36:31,610 If delta is of order lambda QCD, what 567 00:36:31,610 --> 00:36:33,770 happens with the spray of radiation 568 00:36:33,770 --> 00:36:35,960 is that it gets bound into a hadron. 569 00:36:35,960 --> 00:36:37,130 It just can't separate. 570 00:36:37,130 --> 00:36:44,490 Confinement grabs it, and you get an energetic hadron, not 571 00:36:44,490 --> 00:36:44,990 a jet. 572 00:36:55,860 --> 00:37:04,498 So if the jets get too narrow, in particular this narrow, 573 00:37:04,498 --> 00:37:06,415 then the constituents are bound into a hadron. 574 00:37:11,440 --> 00:37:13,920 And that might be something you want to talk about, 575 00:37:13,920 --> 00:37:16,420 but it wouldn't be talking about e plus e minus, the dijets. 576 00:37:16,420 --> 00:37:18,545 You'd be talking about e plus e minus to pi plus pi 577 00:37:18,545 --> 00:37:19,789 minus or something. 578 00:37:33,858 --> 00:37:35,650 And then you'd actually use this other SCET 579 00:37:35,650 --> 00:37:37,450 that we were talking about a moment ago, 580 00:37:37,450 --> 00:37:39,790 not the one for jets. 581 00:37:39,790 --> 00:37:42,130 OK, so anything kind of in between these two regions 582 00:37:42,130 --> 00:37:45,010 much greater than lambda QCD, much less than Q, 583 00:37:45,010 --> 00:37:47,050 then we can talk about it as being jets. 584 00:38:02,520 --> 00:38:03,020 OK. 585 00:38:06,223 --> 00:38:07,640 So we figure out what region we're 586 00:38:07,640 --> 00:38:09,500 interested in by figuring out what region we're not 587 00:38:09,500 --> 00:38:10,083 interested in. 588 00:38:14,530 --> 00:38:16,770 So that was one jet. 589 00:38:16,770 --> 00:38:20,680 And kind of by symmetry we can talk about the other one. 590 00:38:20,680 --> 00:38:23,700 So there's this one that I called n2. 591 00:38:23,700 --> 00:38:26,850 The simplest way of invoking symmetry 592 00:38:26,850 --> 00:38:30,300 is to take n1 to point along n, which is our, 593 00:38:30,300 --> 00:38:32,790 say 0, 0, minus 1. 594 00:38:32,790 --> 00:38:38,820 And then just take n2 to be n bar, which is 1, 0, 0, plus 1. 595 00:38:38,820 --> 00:38:40,920 And then the description of the n2 jet 596 00:38:40,920 --> 00:38:42,810 is the same as the description of the n1 jet. 597 00:38:42,810 --> 00:38:44,477 It's just you switch pluses and minuses. 598 00:38:54,790 --> 00:38:57,140 Remember that what defines plus and minus 599 00:38:57,140 --> 00:38:59,440 depends on the choice of this n bar. 600 00:38:59,440 --> 00:39:02,120 A priori, that choice of n bar has nothing to do with n2. 601 00:39:02,120 --> 00:39:04,160 And this is just a different physical vector. 602 00:39:04,160 --> 00:39:06,770 Both n1 and n2 are physical. 603 00:39:06,770 --> 00:39:08,210 And bar is an auxiliary vector. 604 00:39:08,210 --> 00:39:10,010 But if I just happened to choose that n2 605 00:39:10,010 --> 00:39:12,050 is equal to that auxiliary vector, 606 00:39:12,050 --> 00:39:15,480 then it makes things simple. 607 00:39:15,480 --> 00:39:17,660 Because we just have a relation between the two 608 00:39:17,660 --> 00:39:20,495 sets of degrees of freedom, this swapping pluses and minuses. 609 00:39:30,555 --> 00:39:31,055 OK. 610 00:39:35,850 --> 00:39:37,890 So that's actually not the end of the story 611 00:39:37,890 --> 00:39:41,400 of the degrees of freedom here. 612 00:39:41,400 --> 00:39:44,100 And that is because, even if we make restrictions 613 00:39:44,100 --> 00:39:47,040 to getting these dijets, we can still have soft radiation that 614 00:39:47,040 --> 00:39:49,427 is between the jets. 615 00:39:49,427 --> 00:39:51,510 And so this one's a little less intuitive perhaps. 616 00:40:02,730 --> 00:40:06,780 So I'll call these guys ultra soft modes. 617 00:40:06,780 --> 00:40:09,330 And I'll label them by US for Ultra Soft. 618 00:40:13,412 --> 00:40:15,870 And one way of thinking about physically what they're doing 619 00:40:15,870 --> 00:40:18,412 is that they're allowing you to communicate between the jets. 620 00:40:23,920 --> 00:40:26,770 There can be radiation that's radiated from one jet that 621 00:40:26,770 --> 00:40:28,450 interferes with the other jet. 622 00:40:40,410 --> 00:40:42,427 This is a homogeneous type of radiation, 623 00:40:42,427 --> 00:40:44,760 so it has the same scaling in the plus, minus, and perp. 624 00:40:50,250 --> 00:40:52,860 So this is for these ultra soft modes. 625 00:40:52,860 --> 00:40:56,970 If I compare this to the scaling that we have for, say, 626 00:40:56,970 --> 00:40:58,430 the n collinear modes-- 627 00:40:58,430 --> 00:41:00,450 so let me call that Pn-- 628 00:41:00,450 --> 00:41:05,310 that was delta squared over Q Q delta. 629 00:41:05,310 --> 00:41:09,930 And if I make the mirror for the other jet, 630 00:41:09,930 --> 00:41:15,910 then I would switch these two delta squared over Q delta. 631 00:41:15,910 --> 00:41:17,790 So when I say communicate, what I mean 632 00:41:17,790 --> 00:41:20,760 is that these guys can talk to both of these guys 633 00:41:20,760 --> 00:41:24,503 without interfering with their scaling. 634 00:41:24,503 --> 00:41:26,670 So if they're not going to interfere with this guy's 635 00:41:26,670 --> 00:41:29,523 scaling, they better have plus momentum that's the same size. 636 00:41:29,523 --> 00:41:31,440 If it was any bigger, then we'd have a problem 637 00:41:31,440 --> 00:41:33,190 because they'd interfere with this scaling 638 00:41:33,190 --> 00:41:34,800 when they tried to communicate. 639 00:41:34,800 --> 00:41:37,500 And then likewise for the minus, they better 640 00:41:37,500 --> 00:41:38,972 have scaling of the same size. 641 00:41:38,972 --> 00:41:39,930 And then perp is fixed. 642 00:41:44,770 --> 00:41:53,930 So the word communicate here in the way I'm defining it 643 00:41:53,930 --> 00:42:01,740 means sharing momenta of a common size-- 644 00:42:04,820 --> 00:42:20,760 well, means sharing momenta and not taking the other particle 645 00:42:20,760 --> 00:42:21,782 off-shell. 646 00:42:30,750 --> 00:42:31,250 OK. 647 00:42:31,250 --> 00:42:33,290 So we can draw a picture for this one, too. 648 00:42:33,290 --> 00:42:36,890 Like we have our SCET2 picture, we could draw an SCET1 picture. 649 00:42:36,890 --> 00:42:38,600 And that helps. 650 00:42:38,600 --> 00:42:45,320 Pictures always help to make words more palatable, more 651 00:42:45,320 --> 00:42:46,446 absorbable. 652 00:42:53,900 --> 00:43:04,220 So same type of picture where you have P minus and P plus, 653 00:43:04,220 --> 00:43:06,540 we also can think about things in terms of hyperbola 654 00:43:06,540 --> 00:43:09,235 of constant invariant mass. 655 00:43:09,235 --> 00:43:15,530 We now have some collinear modes for the n1 direction. 656 00:43:15,530 --> 00:43:22,990 There's going to be some purple collinear 657 00:43:22,990 --> 00:43:25,618 modes for the n2 directions. 658 00:43:25,618 --> 00:43:27,160 There's going to be these soft modes. 659 00:43:30,070 --> 00:43:32,380 And then we could also have some hard modes 660 00:43:32,380 --> 00:43:35,290 that we want to integrate out. 661 00:43:40,220 --> 00:43:42,700 So these are the ultra soft modes. 662 00:43:42,700 --> 00:43:45,700 So the reason that I call them ultra soft rather than soft 663 00:43:45,700 --> 00:43:48,100 is because, by soft, we meant something 664 00:43:48,100 --> 00:43:50,530 that sat on the same hyperbola. 665 00:43:50,530 --> 00:43:52,330 And here it sits on a lower hyperbola, 666 00:43:52,330 --> 00:43:53,320 so it should be softer. 667 00:43:53,320 --> 00:43:54,550 So we call it ultra soft. 668 00:43:59,288 --> 00:44:00,830 This is the hyperbola where P squared 669 00:44:00,830 --> 00:44:01,913 is of order delta squared. 670 00:44:01,913 --> 00:44:05,060 This is the hyperbola where P squared is of order delta 671 00:44:05,060 --> 00:44:06,620 4 over Q squared. 672 00:44:06,620 --> 00:44:09,230 If you square any one of these guys, 673 00:44:09,230 --> 00:44:11,780 you get delta 4 over Q squared. 674 00:44:11,780 --> 00:44:16,550 And up here, is P squared over Q squared. 675 00:44:16,550 --> 00:44:18,710 And this kind of thing is called SCET1. 676 00:44:23,200 --> 00:44:24,850 So what does it mean to communicate? 677 00:44:24,850 --> 00:44:27,680 Well, it means that there's two momenta that are the same size. 678 00:44:27,680 --> 00:44:35,440 So the fact that I tried to line this up as best I could, 679 00:44:35,440 --> 00:44:39,880 partially succeeding, is what I mean by communicate. 680 00:44:39,880 --> 00:44:41,737 These things are the same size in the plus. 681 00:44:41,737 --> 00:44:43,570 These things are the same size in the minus. 682 00:44:47,130 --> 00:44:59,062 And if we want to put in kind of how big these things are, 683 00:44:59,062 --> 00:45:00,000 we would say that. 684 00:45:00,000 --> 00:45:02,070 So the scaling of the ultra soft, 685 00:45:02,070 --> 00:45:03,750 unlike the scaling the soft, the ultra 686 00:45:03,750 --> 00:45:06,120 soft in all component scales like lambda squared. 687 00:45:24,900 --> 00:45:28,290 So in terms of scaling parameters, 688 00:45:28,290 --> 00:45:29,850 we would say that it goes like this. 689 00:45:33,450 --> 00:45:35,670 And here, we're not talking about soft. 690 00:45:45,600 --> 00:45:47,808 It would have lambda in all components. 691 00:45:51,193 --> 00:45:53,110 So this is the effective theory that turns out 692 00:45:53,110 --> 00:45:54,318 to be the right one for jets. 693 00:46:00,090 --> 00:46:02,070 And whenever you have this kind of situation 694 00:46:02,070 --> 00:46:04,680 where you've got collinear modes that are living on a higher 695 00:46:04,680 --> 00:46:07,230 hyperbola than the soft modes, which 696 00:46:07,230 --> 00:46:10,920 you call ultra soft modes, that's an SCET1 type theory. 697 00:46:16,540 --> 00:46:18,480 And these two actually cover a wide range 698 00:46:18,480 --> 00:46:21,330 of phenomenology, these two particular cases, SCET1 699 00:46:21,330 --> 00:46:21,830 and SCET2. 700 00:46:26,510 --> 00:46:30,070 OK, questions? 701 00:46:30,070 --> 00:46:32,843 We'll talk a little bit more about this picture. 702 00:46:32,843 --> 00:46:34,510 AUDIENCE: Do you think of the soft modes 703 00:46:34,510 --> 00:46:40,288 as being smaller than or equal to lambda squared? 704 00:46:40,288 --> 00:46:41,080 IAIN STEWART: Yeah. 705 00:46:45,220 --> 00:46:47,200 So you can ask, now, how should I 706 00:46:47,200 --> 00:46:50,920 think about drawing the blobs around these things, right? 707 00:46:50,920 --> 00:46:54,400 And you still should think about these things 708 00:46:54,400 --> 00:46:56,750 with blobs like that. 709 00:46:56,750 --> 00:47:02,420 But then it becomes a question which we didn't have, 710 00:47:02,420 --> 00:47:05,900 which was kind of more obvious actually in the SCET1 case. 711 00:47:05,900 --> 00:47:09,570 And that is kind of how these things overlap with the axes. 712 00:47:09,570 --> 00:47:11,730 And we'll come back and talk about that later. 713 00:47:11,730 --> 00:47:13,730 But I think, for now, just think of them 714 00:47:13,730 --> 00:47:17,270 as being localized in that way. 715 00:47:17,270 --> 00:47:20,390 In the case of the softs in the collinear we had before, 716 00:47:20,390 --> 00:47:22,640 you can think of them as uniformly kind of coming down 717 00:47:22,640 --> 00:47:23,970 into the infrared. 718 00:47:23,970 --> 00:47:25,820 Here, it's not like that because this guy 719 00:47:25,820 --> 00:47:29,380 is more infrared to begin with. 720 00:47:29,380 --> 00:47:33,520 And that will have some impact later on. 721 00:47:33,520 --> 00:47:35,510 OK. 722 00:47:35,510 --> 00:47:37,820 So what are the important features here? 723 00:47:43,230 --> 00:47:45,515 Well, we see this idea that I mentioned 724 00:47:45,515 --> 00:47:49,310 would occur that we have multiple modes 725 00:47:49,310 --> 00:47:52,100 for the infrared. 726 00:47:52,100 --> 00:47:54,440 And you can try to get away with not doing 727 00:47:54,440 --> 00:47:56,540 that, but then you would have a lot of trouble 728 00:47:56,540 --> 00:47:58,400 with your power counting. 729 00:47:58,400 --> 00:48:01,310 And since power counting, as I've convinced you hopefully 730 00:48:01,310 --> 00:48:03,890 by now, is just as important as other things 731 00:48:03,890 --> 00:48:06,260 when you're designing an effective theory, 732 00:48:06,260 --> 00:48:10,470 you really don't want to mess with that. 733 00:48:10,470 --> 00:48:12,290 And so you're forced into a situation where 734 00:48:12,290 --> 00:48:14,240 you start talking about having multiple fields 735 00:48:14,240 --> 00:48:16,850 for the same degrees of freedom because the scaling 736 00:48:16,850 --> 00:48:19,160 of the momentum in different regions of the space 737 00:48:19,160 --> 00:48:21,333 you're interested in is just different. 738 00:48:21,333 --> 00:48:23,750 So the derivatives that are corresponding to those momenta 739 00:48:23,750 --> 00:48:25,340 are going to scale differently. 740 00:48:25,340 --> 00:48:27,320 And if that's the case, you're going 741 00:48:27,320 --> 00:48:29,360 to need to have multiple fields to describe 742 00:48:29,360 --> 00:48:30,360 those different regions. 743 00:48:33,060 --> 00:48:35,250 We'll also see the power counting is different. 744 00:48:38,610 --> 00:48:41,530 So you can ask, what are you integrating out? 745 00:48:41,530 --> 00:48:44,430 And we're still taking the attitude 746 00:48:44,430 --> 00:48:54,300 that we integrate out modes that are off-shell and so 747 00:48:54,300 --> 00:48:55,830 above a given hyperbola. 748 00:48:58,920 --> 00:49:01,320 So that part of our story of effective field theory 749 00:49:01,320 --> 00:49:02,230 is really the same. 750 00:49:02,230 --> 00:49:04,720 It's just that, when we describe the low energy modes, 751 00:49:04,720 --> 00:49:05,970 we need more than one of them. 752 00:49:14,863 --> 00:49:16,530 So what we mean by off-shell is the same 753 00:49:16,530 --> 00:49:17,910 as it always has meant. 754 00:49:22,590 --> 00:49:24,420 Off-shell modes get integrated out. 755 00:49:24,420 --> 00:49:29,654 Those are modes with large P squared, like the hard modes. 756 00:49:58,330 --> 00:50:00,037 So one natural question when you're 757 00:50:00,037 --> 00:50:01,870 talking about these effective field theories 758 00:50:01,870 --> 00:50:05,440 is, how do I know I have all the modes? 759 00:50:05,440 --> 00:50:09,320 And that's a good question. 760 00:50:09,320 --> 00:50:11,317 And there's been examples in the literature 761 00:50:11,317 --> 00:50:13,900 where people wrote papers where they didn't know all the modes 762 00:50:13,900 --> 00:50:16,480 and modes were missing. 763 00:50:16,480 --> 00:50:18,130 So it's not even an academic question. 764 00:50:18,130 --> 00:50:21,710 It's really a pitfall in some sense that you can fall into. 765 00:50:21,710 --> 00:50:24,790 So I take the following attitude towards that question. 766 00:50:24,790 --> 00:50:27,110 You should attack it from all sides. 767 00:50:27,110 --> 00:50:29,260 So one is that you should really physically 768 00:50:29,260 --> 00:50:30,760 think about what the modes are doing 769 00:50:30,760 --> 00:50:32,650 and have a physical description of why 770 00:50:32,650 --> 00:50:34,900 those modes are something relevant for your effective 771 00:50:34,900 --> 00:50:35,300 field theory. 772 00:50:35,300 --> 00:50:36,883 After all, your effective field theory 773 00:50:36,883 --> 00:50:40,160 is supposed to be a description of nature and infrared physics. 774 00:50:40,160 --> 00:50:42,160 So there should be something physical associated 775 00:50:42,160 --> 00:50:44,020 to what those models are doing. 776 00:50:44,020 --> 00:50:45,730 That's the physics side. 777 00:50:45,730 --> 00:50:47,530 Calculationally, there's various ways 778 00:50:47,530 --> 00:50:48,975 that you could approach this. 779 00:50:48,975 --> 00:50:50,350 So one thing you can do, which is 780 00:50:50,350 --> 00:50:52,420 sort of an order by order thing, is 781 00:50:52,420 --> 00:50:55,450 you can just calculate results at one loop 782 00:50:55,450 --> 00:50:58,300 and make sure you match up infrared divergences. 783 00:50:58,300 --> 00:51:00,910 Because if you don't match up the infrared divergences, 784 00:51:00,910 --> 00:51:03,670 then you're missing some infrared degree of freedom 785 00:51:03,670 --> 00:51:05,810 in the effective field theory. 786 00:51:05,810 --> 00:51:06,310 OK. 787 00:51:06,310 --> 00:51:18,800 So that's one way you can check for the modes. 788 00:51:18,800 --> 00:51:22,790 So if you can either do physics, or you can calculate. 789 00:51:25,490 --> 00:51:26,990 And when you calculate, there's also 790 00:51:26,990 --> 00:51:28,710 something called the method of regions, 791 00:51:28,710 --> 00:51:32,720 which is a nice way of thinking about trying to discover modes, 792 00:51:32,720 --> 00:51:35,240 which basically says that, any full theory calculation I can 793 00:51:35,240 --> 00:51:38,540 do, I can do that calculation by dividing up 794 00:51:38,540 --> 00:51:40,520 the integrand into regions. 795 00:51:40,520 --> 00:51:43,610 And if I'm using dimensional regularization, 796 00:51:43,610 --> 00:51:48,390 then the full theory answer is just the sum over regions. 797 00:51:48,390 --> 00:51:53,030 So you can just calculate using some EFT, 798 00:51:53,030 --> 00:51:56,000 as I described to you, and check it. 799 00:51:58,790 --> 00:52:00,620 Or could do a different way. 800 00:52:00,620 --> 00:52:07,940 You could say, calculate the full theory result 801 00:52:07,940 --> 00:52:11,090 with something called the method of regions. 802 00:52:11,090 --> 00:52:12,590 And that's just a way of calculating 803 00:52:12,590 --> 00:52:15,110 the full theory of result, but it's 804 00:52:15,110 --> 00:52:21,080 a way of telling you what regions are important. 805 00:52:21,080 --> 00:52:23,030 There's all-other theorems in QCD 806 00:52:23,030 --> 00:52:26,843 about what regions can give infrared divergences. 807 00:52:26,843 --> 00:52:28,385 And that's another thing you can use. 808 00:52:40,180 --> 00:52:44,650 And I won't talk about this last one or actually about this one. 809 00:52:48,940 --> 00:52:50,440 So these are different ways that you 810 00:52:50,440 --> 00:52:53,890 could look for what are the relevant degrees of freedom. 811 00:52:53,890 --> 00:52:56,200 And actually, when we started SCET, 812 00:52:56,200 --> 00:52:59,140 we didn't separate between ultra soft and soft. 813 00:52:59,140 --> 00:53:02,070 We had both of them at the same time. 814 00:53:02,070 --> 00:53:04,070 And the reason was we knew that in some examples 815 00:53:04,070 --> 00:53:06,670 we needed soft and some examples we needed ultra soft. 816 00:53:06,670 --> 00:53:09,130 And we were thinking of it as one theory, not SCET1 817 00:53:09,130 --> 00:53:10,415 and SCET2. 818 00:53:10,415 --> 00:53:12,790 And then at some point, we realized that all the examples 819 00:53:12,790 --> 00:53:14,860 we were doing, as we did more and more examples, 820 00:53:14,860 --> 00:53:16,630 they broke into these two categories, one 821 00:53:16,630 --> 00:53:18,760 for jets, one for hadrons. 822 00:53:18,760 --> 00:53:21,380 And so we were finding that, for example, 823 00:53:21,380 --> 00:53:23,232 if you add a soft mode to this picture, 824 00:53:23,232 --> 00:53:24,940 it just ends up being totally irrelevant. 825 00:53:24,940 --> 00:53:26,607 And you can just absorb it or remove it. 826 00:53:26,607 --> 00:53:28,210 You don't need it. 827 00:53:28,210 --> 00:53:30,640 And likewise, if you tried to put an ultra soft mode 828 00:53:30,640 --> 00:53:33,430 into the example of energetic hadrons, 829 00:53:33,430 --> 00:53:35,110 you'd find you don't need it. 830 00:53:35,110 --> 00:53:38,210 So you could put too many degrees of freedom in as well. 831 00:53:38,210 --> 00:53:40,995 It's not just that you could have too few. 832 00:53:40,995 --> 00:53:42,370 You could put more than you need, 833 00:53:42,370 --> 00:53:44,412 and then you would see from your calculations 834 00:53:44,412 --> 00:53:46,120 that you actually had more than you need. 835 00:53:49,660 --> 00:53:51,160 The reason I'm spending time on this 836 00:53:51,160 --> 00:53:53,390 is because this is, in some sense, the tricky part. 837 00:53:53,390 --> 00:53:54,973 Once we know what the modes are, we'll 838 00:53:54,973 --> 00:53:57,190 just get the effective field theory and go. 839 00:54:00,010 --> 00:54:02,415 AUDIENCE: Is your second point here the method 840 00:54:02,415 --> 00:54:05,400 or regions [INAUDIBLE],, is that any different than the check it 841 00:54:05,400 --> 00:54:06,828 of the first point? 842 00:54:06,828 --> 00:54:07,620 IAIN STEWART: Yeah. 843 00:54:07,620 --> 00:54:09,750 Because, here, what I'm saying is hypothesize 844 00:54:09,750 --> 00:54:11,160 some effective field theory. 845 00:54:11,160 --> 00:54:12,510 AUDIENCE: But check it means full theory. 846 00:54:12,510 --> 00:54:14,300 IAIN STEWART: And then check it against the full theory. 847 00:54:14,300 --> 00:54:14,590 AUDIENCE: [INAUDIBLE] 848 00:54:14,590 --> 00:54:16,320 IAIN STEWART: And so the direction here 849 00:54:16,320 --> 00:54:18,480 is that you write down something, 850 00:54:18,480 --> 00:54:21,630 and you do calculations in two theories. 851 00:54:21,630 --> 00:54:24,240 And you make sure they agree. 852 00:54:24,240 --> 00:54:25,950 Here, it's a little bit different 853 00:54:25,950 --> 00:54:29,310 because there may not be a one-to-one correspondence 854 00:54:29,310 --> 00:54:31,140 of the modes of the method of regions. 855 00:54:31,140 --> 00:54:32,730 Method of regions is not technically 856 00:54:32,730 --> 00:54:35,100 exactly what the effective theory is. 857 00:54:35,100 --> 00:54:38,005 So this can give you hints about what the effective theory is. 858 00:54:38,005 --> 00:54:39,630 Then you basically go back and do this, 859 00:54:39,630 --> 00:54:42,360 but it sort of helps you because you have most 860 00:54:42,360 --> 00:54:44,470 of the calculations to do this. 861 00:54:44,470 --> 00:54:44,970 Yeah. 862 00:54:44,970 --> 00:54:46,380 So technically, they're a little bit different, 863 00:54:46,380 --> 00:54:47,505 but they're pretty related. 864 00:54:56,440 --> 00:54:58,330 OK. 865 00:54:58,330 --> 00:55:01,060 So in the example we had here, it 866 00:55:01,060 --> 00:55:03,400 was a little more complicated than our B to D pi example 867 00:55:03,400 --> 00:55:06,550 because we had three things in the infrared. 868 00:55:06,550 --> 00:55:08,710 And you could ask the question, is there something 869 00:55:08,710 --> 00:55:09,670 with just one jet? 870 00:55:09,670 --> 00:55:12,550 The reason we had three things is because we had two jets. 871 00:55:12,550 --> 00:55:15,760 We had the purple jet and the pink jet. 872 00:55:15,760 --> 00:55:19,450 And we can come up with a process with only one jet. 873 00:55:19,450 --> 00:55:20,980 So let me give you a third example. 874 00:55:27,980 --> 00:55:31,150 If we have something recoiling against the jet that's 875 00:55:31,150 --> 00:55:32,980 electromagnetic, like a photon, then we 876 00:55:32,980 --> 00:55:34,600 could just have one jet. 877 00:55:34,600 --> 00:55:36,670 So one way of doing that is to look at a process 878 00:55:36,670 --> 00:55:41,080 like b to s gamma, where you say that really what you want 879 00:55:41,080 --> 00:55:47,920 is B to one jet plus a photon. 880 00:55:47,920 --> 00:55:49,600 And that's a region of phase space 881 00:55:49,600 --> 00:55:54,460 in b to s gamma, where the picture over there 882 00:55:54,460 --> 00:55:57,190 just has one jet. 883 00:55:57,190 --> 00:56:01,360 And so it's the same picture, but now we just 884 00:56:01,360 --> 00:56:02,140 have our pink jet. 885 00:56:13,110 --> 00:56:15,410 So the right modes for this picture 886 00:56:15,410 --> 00:56:19,175 for this process in the region where you just have one jet 887 00:56:19,175 --> 00:56:21,920 would be like this where this is hard and you integrate it out. 888 00:56:21,920 --> 00:56:25,590 And you just have two things in the infrared, a Cn and an ultra 889 00:56:25,590 --> 00:56:26,090 soft. 890 00:56:29,450 --> 00:56:33,140 In this case, you also have to ask the question, what 891 00:56:33,140 --> 00:56:34,160 is the hydronic process? 892 00:56:34,160 --> 00:56:35,870 Well, it has to B meson. 893 00:56:35,870 --> 00:56:37,910 What are the things inside the B meson 894 00:56:37,910 --> 00:56:39,380 that are binding it together? 895 00:56:39,380 --> 00:56:42,320 And you would make those the ultra soft modes. 896 00:56:42,320 --> 00:56:44,630 So in this process here, it's very natural 897 00:56:44,630 --> 00:56:50,960 to take P squared of this lowest line to be lambda QCD squared. 898 00:56:50,960 --> 00:56:54,440 And then these guys are binding the B meson there, 899 00:56:54,440 --> 00:56:55,695 the soft modes of the B meson. 900 00:57:09,980 --> 00:57:12,620 This collinear hyperbola for the jet lives somewhere between. 901 00:57:12,620 --> 00:57:16,790 And this here, in this case, would be something of order 902 00:57:16,790 --> 00:57:19,512 the B quark mass squared, OK? 903 00:57:19,512 --> 00:57:21,470 So that's a little bit different than over here 904 00:57:21,470 --> 00:57:24,470 where there was kind of a no natural-- 905 00:57:24,470 --> 00:57:26,300 this thing was set by kind of what 906 00:57:26,300 --> 00:57:29,210 we chose to do with the jets was setting this. 907 00:57:29,210 --> 00:57:31,195 Here, there's kind of a natural scale 908 00:57:31,195 --> 00:57:32,570 where you know that there's going 909 00:57:32,570 --> 00:57:34,095 to be some degrees of freedom. 910 00:57:34,095 --> 00:57:35,720 And so it's very natural, in this case, 911 00:57:35,720 --> 00:57:38,060 to take that as an input. 912 00:57:38,060 --> 00:57:40,370 And then you could actually figure out, 913 00:57:40,370 --> 00:57:45,890 given that what the jets should be and the jets would have, 914 00:57:45,890 --> 00:57:49,640 lambda QCD times Mb, which is in the middle. 915 00:57:54,190 --> 00:57:56,880 So that's kind of a natural scaling for the b to s 916 00:57:56,880 --> 00:58:00,000 gamma process, OK? 917 00:58:00,000 --> 00:58:03,840 So just to give you an idea how, if I have a jet, 918 00:58:03,840 --> 00:58:05,480 it's going to look something like this, 919 00:58:05,480 --> 00:58:06,980 it might not look exactly like that. 920 00:58:06,980 --> 00:58:10,117 It depends on how many jets you have. 921 00:58:10,117 --> 00:58:11,700 And of course, these pictures actually 922 00:58:11,700 --> 00:58:13,908 get more complicated if you try to start drawing them 923 00:58:13,908 --> 00:58:16,500 when you have three jets because then the plane 924 00:58:16,500 --> 00:58:19,590 is no longer enough. 925 00:58:19,590 --> 00:58:24,490 All right, so any more questions? 926 00:58:24,490 --> 00:58:25,150 OK. 927 00:58:25,150 --> 00:58:29,020 So when we did HQET, the first thing that we did 928 00:58:29,020 --> 00:58:30,940 is we started to expand. 929 00:58:30,940 --> 00:58:33,220 Before we designed the effective Lagrangian, 930 00:58:33,220 --> 00:58:36,580 we just said, well, what happens if I expand the full theory? 931 00:58:36,580 --> 00:58:39,493 And I'm going to take the same attitude here. 932 00:58:39,493 --> 00:58:41,410 Let's just write down some full theory objects 933 00:58:41,410 --> 00:58:47,920 and expand them in the limits that we've been talking about. 934 00:58:47,920 --> 00:58:49,920 And then we'll see what kind of effective theory 935 00:58:49,920 --> 00:58:52,950 we want based on the results from those expansions. 936 00:58:58,890 --> 00:59:02,840 So let's start with spinners in a collinear limit. 937 00:59:14,680 --> 00:59:16,800 So let me start with some massless QCD spinners 938 00:59:16,800 --> 00:59:18,270 in the Dirac representation. 939 00:59:21,685 --> 00:59:23,310 We could use some other representation, 940 00:59:23,310 --> 00:59:25,170 but let's just use Dirac. 941 00:59:43,630 --> 00:59:45,430 So we have spinners for the quarks. 942 00:59:45,430 --> 00:59:56,890 We have spinners for the antiquarks, V, 943 00:59:56,890 --> 00:59:58,750 where this curly V and this curly U 944 00:59:58,750 --> 00:59:59,935 are two-component objects. 945 01:00:11,700 --> 01:00:13,370 I'll make sure it looks curly enough. 946 01:00:17,500 --> 01:00:19,540 So what we can do here is we can expand. 947 01:00:19,540 --> 01:00:21,970 And you see that what happens when you expand 948 01:00:21,970 --> 01:00:24,880 is that you can think about the P3 vector being larger 949 01:00:24,880 --> 01:00:26,740 than P1 and P2. 950 01:00:26,740 --> 01:00:33,430 So let's just let our n be 1, 0, 0, 1 and our n bar 951 01:00:33,430 --> 01:00:35,693 be 1, 0, 0, minus 1. 952 01:00:35,693 --> 01:00:37,360 So they're back to back with each other. 953 01:00:41,230 --> 01:00:43,790 Whether I put the plus or minus 1 there or there 954 01:00:43,790 --> 01:00:46,070 doesn't really matter. 955 01:00:46,070 --> 01:00:52,660 Let's expand in n bar dot P, which in this case 956 01:00:52,660 --> 01:00:58,240 is P0 plus P3 being much greater than P perp, just P1 and P2. 957 01:00:58,240 --> 01:01:00,760 And then that's much greater than n 958 01:01:00,760 --> 01:01:04,220 dot P, which is P0 minus P3. 959 01:01:07,900 --> 01:01:11,930 And what that means is that you can approximate sigma dot P 960 01:01:11,930 --> 01:01:14,870 over P0 from these massless particles. 961 01:01:14,870 --> 01:01:18,498 It's just sigma 3 because you pick out the P3. 962 01:01:18,498 --> 01:01:19,540 That's the big component. 963 01:01:19,540 --> 01:01:20,920 The P1 and P2, you can drop. 964 01:01:20,920 --> 01:01:22,330 Then it kicks out the sigma 3. 965 01:01:22,330 --> 01:01:24,548 And then P3 is also the same size as P0, 966 01:01:24,548 --> 01:01:25,840 so you're just getting sigma 3. 967 01:01:30,700 --> 01:01:36,530 So what you get from this then would 968 01:01:36,530 --> 01:01:39,830 be guys that look like this. 969 01:01:43,340 --> 01:01:45,350 Just so, I put it in the two possibilities 970 01:01:45,350 --> 01:01:49,670 for the curly U are four-component spinners that 971 01:01:49,670 --> 01:02:22,830 look like this for U and then likewise for V of P. 972 01:02:22,830 --> 01:02:24,550 We can work out what we get. 973 01:02:29,800 --> 01:02:31,878 We get that. 974 01:02:31,878 --> 01:02:33,920 So this is actually a little different than HQET. 975 01:02:33,920 --> 01:02:35,450 In HQET, what you would have found 976 01:02:35,450 --> 01:02:39,330 is that the antiquarks would have been just left out. 977 01:02:39,330 --> 01:02:42,110 And the quarks would've been there in the theory. 978 01:02:42,110 --> 01:02:43,892 Here, both of them survive. 979 01:02:43,892 --> 01:02:45,350 And actually two degrees of freedom 980 01:02:45,350 --> 01:02:49,190 survive for both the particles and the antiparticles. 981 01:02:49,190 --> 01:02:50,810 So it's not like we're integrating out 982 01:02:50,810 --> 01:02:57,260 of something like the antiparticle, like in HQET. 983 01:02:57,260 --> 01:02:59,090 By this expansion, we still have all four 984 01:02:59,090 --> 01:03:02,330 of these degrees of freedom if you count degrees of freedom 985 01:03:02,330 --> 01:03:05,420 by whether you have particles, antiparticles, and spin states. 986 01:03:09,000 --> 01:03:12,870 Nevertheless, there is a simplification that occurs. 987 01:03:12,870 --> 01:03:18,870 And that is the fact that the spinners that you have 988 01:03:18,870 --> 01:03:19,995 have a projection relation. 989 01:03:26,390 --> 01:03:29,510 So if you look in this basis that we're 990 01:03:29,510 --> 01:03:32,810 talking about here, what n slash is if you write down 991 01:03:32,810 --> 01:03:38,048 what the gamma matrices are in the Dirac representation, 992 01:03:38,048 --> 01:03:40,760 then n slash is this. 993 01:03:40,760 --> 01:03:43,250 And another useful thing is n slash n bar 994 01:03:43,250 --> 01:03:50,530 slash over 4, which you can work out as just this. 995 01:03:54,780 --> 01:04:00,420 And these spinners here, which I need a name for-- 996 01:04:00,420 --> 01:04:03,920 so let's call this Un and call this Vn. 997 01:04:08,220 --> 01:04:15,540 They satisfy n slash Un is n slash Vn is 0. 998 01:04:15,540 --> 01:04:21,252 And they also satisfy n slash n bar slash 999 01:04:21,252 --> 01:04:25,460 Un is Un for both of them. 1000 01:04:42,015 --> 01:04:43,890 So what we can do with that is the following. 1001 01:04:46,780 --> 01:04:51,000 We can take the identity in this 4 by 4 space. 1002 01:04:51,000 --> 01:04:53,550 And we can actually write it as n 1003 01:04:53,550 --> 01:04:59,190 slash n bar slash over 4 plus n bar slash n slash over 4. 1004 01:04:59,190 --> 01:05:02,770 And that's because, remember, that gamma mu 1005 01:05:02,770 --> 01:05:09,300 gamma nu plus 2 g mu nu and n dot n bar is 2. 1006 01:05:09,300 --> 01:05:12,510 So this is just one way of using those. 1007 01:05:12,510 --> 01:05:15,480 This is the anticommutator dotted into n and n bar. 1008 01:05:15,480 --> 01:05:17,370 So I can write it out that way. 1009 01:05:17,370 --> 01:05:20,340 And this kind of formula here is the formula 1010 01:05:20,340 --> 01:05:24,090 that is for projection operators, right? 1011 01:05:24,090 --> 01:05:27,640 So as you act with the operator, you get that guy back again. 1012 01:05:27,640 --> 01:05:31,540 So you could act twice with that operator. 1013 01:05:31,540 --> 01:05:34,080 And so what you can do with this one 1014 01:05:34,080 --> 01:05:36,255 is you could let one act on psi of QCD. 1015 01:05:39,210 --> 01:05:44,720 And if you did that, you'd get n slash n bar slash over 4 psi 1016 01:05:44,720 --> 01:05:48,780 plus n bar slash n slash over 4 psi. 1017 01:05:48,780 --> 01:05:51,390 And you could define these two pieces 1018 01:05:51,390 --> 01:05:53,670 as being two different components of the full theory 1019 01:05:53,670 --> 01:05:59,230 field that I'll call Cn and psi n. 1020 01:05:59,230 --> 01:06:00,760 And what happens at high energies, 1021 01:06:00,760 --> 01:06:03,070 because of the type of thing we were doing over there, 1022 01:06:03,070 --> 01:06:04,960 is that we only like to produce Cn's. 1023 01:06:04,960 --> 01:06:06,730 We don't like to produce psi n bars. 1024 01:06:13,620 --> 01:06:18,530 So if you look at some high energy process, 1025 01:06:18,530 --> 01:06:20,540 this sort of thing I was doing at the spinners 1026 01:06:20,540 --> 01:06:24,320 basically boils down to one sentence. 1027 01:06:24,320 --> 01:06:27,770 And that is that we produce or annihilate 1028 01:06:27,770 --> 01:06:29,690 the components, the guys that live 1029 01:06:29,690 --> 01:06:44,890 in this Cn, not the so-called small components, which 1030 01:06:44,890 --> 01:06:47,380 live in this other guy. 1031 01:06:47,380 --> 01:06:49,532 This language of calling them the small components 1032 01:06:49,532 --> 01:06:52,115 is something that goes back to the early days of QCD actually. 1033 01:06:56,290 --> 01:06:58,450 We may say that word a few times, 1034 01:06:58,450 --> 01:07:03,630 but the history won't be so important to us. 1035 01:07:03,630 --> 01:07:05,457 OK, so, so much for the spinners. 1036 01:07:05,457 --> 01:07:07,290 There is some simplification in the spinners 1037 01:07:07,290 --> 01:07:11,340 because we do like to produce certain combinations. 1038 01:07:11,340 --> 01:07:18,360 But it didn't really teach us too much beyond that. 1039 01:07:24,610 --> 01:07:29,140 And we didn't see that we lost a degree of freedom 1040 01:07:29,140 --> 01:07:30,020 like we did in HQET. 1041 01:07:35,510 --> 01:07:38,900 But nevertheless, there was some simplification. 1042 01:07:38,900 --> 01:07:43,715 Let's do the same thing for the propagator of the quarks. 1043 01:07:50,080 --> 01:07:52,640 Take the propagator of the full theory, expand in this limit. 1044 01:07:57,170 --> 01:07:59,200 So first of all, propagators always 1045 01:07:59,200 --> 01:08:01,690 involve P squared plus i0. 1046 01:08:01,690 --> 01:08:04,120 In our decomposition, that's in n bar 1047 01:08:04,120 --> 01:08:11,197 dot P n dot P plus Minkowski P perp squared plus i0. 1048 01:08:11,197 --> 01:08:13,030 And if you look at the size of these things, 1049 01:08:13,030 --> 01:08:14,450 they're all the same size. 1050 01:08:14,450 --> 01:08:18,939 This guy is lambda 0, and this guy is lambda squared. 1051 01:08:18,939 --> 01:08:21,260 And this guy is just lambda squared by itself. 1052 01:08:21,260 --> 01:08:22,689 So the two guys are the same size, 1053 01:08:22,689 --> 01:08:25,858 although they become the same size for different regions. 1054 01:08:25,858 --> 01:08:28,779 This is lambda 1 squared if you like. 1055 01:08:28,779 --> 01:08:32,050 So I don't drop anything in that propagator. 1056 01:08:36,770 --> 01:08:40,040 And you see that actually that's not entirely true. 1057 01:08:40,040 --> 01:08:42,430 And it does depend on what type of things 1058 01:08:42,430 --> 01:08:43,430 you're interacting with. 1059 01:08:43,430 --> 01:08:48,319 But if I just have P's that are collinear, as I've drawn here, 1060 01:08:48,319 --> 01:08:49,729 then there's nothing to drop. 1061 01:08:59,450 --> 01:09:04,525 So if you look at fermions that are collinear 1062 01:09:04,525 --> 01:09:10,540 and you look at i P slash over P squared plus i0, 1063 01:09:10,540 --> 01:09:13,547 you can decompose P slash out in terms of n slash 1064 01:09:13,547 --> 01:09:15,130 and n bar slash, write it out in terms 1065 01:09:15,130 --> 01:09:17,649 of the coordinates we're using. 1066 01:09:17,649 --> 01:09:23,722 And then in the numerator, you keep the full denominator. 1067 01:09:23,722 --> 01:09:25,930 But in the numerator, there is one momentum component 1068 01:09:25,930 --> 01:09:27,939 that's larger than the others. 1069 01:09:27,939 --> 01:09:31,090 And that is the n bar dot P piece, which is order one. 1070 01:09:41,026 --> 01:09:42,859 So there is a simplification of the P slash. 1071 01:09:42,859 --> 01:09:45,073 And it's related in some ways to the simplification 1072 01:09:45,073 --> 01:09:45,740 of the spinners. 1073 01:09:45,740 --> 01:09:50,149 If you take two of these guys and you do the sum over spin, 1074 01:09:50,149 --> 01:09:57,202 you'll actually get n slash over 2 n bar dot P. 1075 01:09:57,202 --> 01:10:00,660 AUDIENCE: Is it n slash on the side of the 0? 1076 01:10:00,660 --> 01:10:01,470 IAIN STEWART: Yeah. 1077 01:10:01,470 --> 01:10:02,345 AUDIENCE: [INAUDIBLE] 1078 01:10:02,345 --> 01:10:04,928 IAIN STEWART: So the right way of thinking about the numerator 1079 01:10:04,928 --> 01:10:07,470 here is that, if you do the sum over spins of sort of two 1080 01:10:07,470 --> 01:10:12,210 of these spinners, like this, that's 1081 01:10:12,210 --> 01:10:14,200 giving you the numerator. 1082 01:10:14,200 --> 01:10:18,330 And if you look at sort of the overlap of this, 1083 01:10:18,330 --> 01:10:19,980 if you look at any amplitude, you also 1084 01:10:19,980 --> 01:10:22,770 have to take into account, but I haven't written yet-- 1085 01:10:22,770 --> 01:10:25,250 which is this. 1086 01:10:25,250 --> 01:10:28,910 And that's going to go like n bar slash. 1087 01:10:28,910 --> 01:10:31,960 So what happens is you have n slash, n bar slash. 1088 01:10:31,960 --> 01:10:33,920 And that gives you the projector, 1089 01:10:33,920 --> 01:10:37,480 which then overlaps order 1 with the spinner. 1090 01:10:37,480 --> 01:10:38,145 Yeah. 1091 01:10:38,145 --> 01:10:39,270 But that's a good question. 1092 01:10:44,970 --> 01:10:48,880 Later-- but that's a good point. 1093 01:11:13,240 --> 01:11:13,840 OK. 1094 01:11:13,840 --> 01:11:19,540 So we can ask about, if we have some propagator for a fermion, 1095 01:11:19,540 --> 01:11:21,770 then what does it look like? 1096 01:11:21,770 --> 01:11:25,240 And you can take that first order term, 1097 01:11:25,240 --> 01:11:29,440 and we can write it in a way that kind of is reminiscent 1098 01:11:29,440 --> 01:11:33,280 of something more like NRQCD or this nucleon theory 1099 01:11:33,280 --> 01:11:34,515 or even HQET. 1100 01:11:37,650 --> 01:11:40,870 It's just another way of writing the same formula that 1101 01:11:40,870 --> 01:11:43,810 is sometimes useful. 1102 01:11:43,810 --> 01:11:46,300 We just divide through by the n bar dot P. 1103 01:11:46,300 --> 01:11:49,585 And then it's n dot P plus P perp squared over n bar 1104 01:11:49,585 --> 01:11:53,050 dot P. And then there's the i0, but the sign of the i0 1105 01:11:53,050 --> 01:11:56,740 will depend on the sign of the n bar dot P. 1106 01:11:56,740 --> 01:11:59,890 The fact that this could be both plus i0 or minus i0 1107 01:11:59,890 --> 01:12:02,740 is the same thing as saying that there's antiparticles 1108 01:12:02,740 --> 01:12:04,540 and particles in the theory. 1109 01:12:04,540 --> 01:12:06,880 If it was just plus i0, as it was in HQET, 1110 01:12:06,880 --> 01:12:08,930 we only had the particles. 1111 01:12:08,930 --> 01:12:10,180 Here, it could be either sign. 1112 01:12:10,180 --> 01:12:11,420 We have both. 1113 01:12:11,420 --> 01:12:13,510 And so this thing, again, has both particles 1114 01:12:13,510 --> 01:12:17,650 and antiparticles, which we saw when we were 1115 01:12:17,650 --> 01:12:18,820 doing the spinners as well. 1116 01:12:31,965 --> 01:12:34,340 And if we want to think about particles and antiparticles 1117 01:12:34,340 --> 01:12:36,840 separately, then we could have a definite sign for the i0's. 1118 01:12:36,840 --> 01:12:39,340 But if we want to think about them in a combined propagator, 1119 01:12:39,340 --> 01:12:41,690 then we have to write it this way. 1120 01:12:41,690 --> 01:12:46,430 All right, so once we know what the propagator is, 1121 01:12:46,430 --> 01:12:50,480 then we can also figure out what the power counting 1122 01:12:50,480 --> 01:12:53,390 of the fields are because the propagator tells us 1123 01:12:53,390 --> 01:12:57,890 what the kinetic term should look like of the Lagrangian. 1124 01:13:02,730 --> 01:13:05,280 So let me show you how that works. 1125 01:13:12,330 --> 01:13:14,190 So where does the propagator come from? 1126 01:13:14,190 --> 01:13:17,220 The propagator comes from the time order 1127 01:13:17,220 --> 01:13:19,560 product of two fields. 1128 01:13:19,560 --> 01:13:23,160 And in this case, it comes from the time order product 1129 01:13:23,160 --> 01:13:27,990 of a field for this Cn component that we were talking about. 1130 01:13:34,010 --> 01:13:37,960 And if we just take the free kinetic term, which 1131 01:13:37,960 --> 01:13:41,200 is the Lagrangian that we'd give that propagator, 1132 01:13:41,200 --> 01:13:48,130 that's enough to determine the power counting for fields, 1133 01:13:48,130 --> 01:13:50,390 as is always the case in any effective field theory. 1134 01:13:55,060 --> 01:13:58,600 So we haven't figured out what the Lagrangian is, 1135 01:13:58,600 --> 01:14:02,870 but we know something about what it's going to look like. 1136 01:14:02,870 --> 01:14:05,200 So let me write down enough of that 1137 01:14:05,200 --> 01:14:07,840 to determine for you what the power counting would look like. 1138 01:14:12,440 --> 01:14:14,270 So when I read it this way, as I wrote it, 1139 01:14:14,270 --> 01:14:17,150 where it's linear in this n dot P derivative, 1140 01:14:17,150 --> 01:14:19,150 you know that what that's going to correspond to 1141 01:14:19,150 --> 01:14:22,780 in the Lagrangian is some n dot partial. 1142 01:14:22,780 --> 01:14:25,960 And I have to have an n bar slash here 1143 01:14:25,960 --> 01:14:29,110 because that always comes along with an n decomposing 1144 01:14:29,110 --> 01:14:30,610 the metric. 1145 01:14:30,610 --> 01:14:32,680 If you ask about the power counting here, 1146 01:14:32,680 --> 01:14:36,730 well, d4x has all 4 components of k. 1147 01:14:36,730 --> 01:14:39,040 And x is the inverse of k. 1148 01:14:39,040 --> 01:14:41,410 So the way that you assign a power counting 1149 01:14:41,410 --> 01:14:45,370 for x is that you say the phase should be of order 1. 1150 01:14:45,370 --> 01:14:49,210 So the scaling for x is the opposite of k. 1151 01:14:49,210 --> 01:14:54,550 So that fixes that d4x should be lambda to the minus 4. 1152 01:14:54,550 --> 01:15:02,590 So x plus times k minus is of order 1, et cetera. 1153 01:15:02,590 --> 01:15:05,260 And that tells you how many powers 1154 01:15:05,260 --> 01:15:08,110 to associate with the d4x. 1155 01:15:08,110 --> 01:15:10,360 We know how many powers to associate with this partial 1156 01:15:10,360 --> 01:15:13,480 because that was our momentum. 1157 01:15:13,480 --> 01:15:15,363 That's lambda squared. 1158 01:15:15,363 --> 01:15:16,780 And actually, the other terms here 1159 01:15:16,780 --> 01:15:19,367 will also be lambda squared. 1160 01:15:19,367 --> 01:15:20,950 And then we just say, well, that's let 1161 01:15:20,950 --> 01:15:25,240 the power counting in this field be arbitrary lambda to the a. 1162 01:15:25,240 --> 01:15:28,150 And so if we do that, then we get an overall scaling 1163 01:15:28,150 --> 01:15:33,850 for this Lagrangian that's lambda to the 2a minus 2. 1164 01:15:37,000 --> 01:15:39,940 2 to the power of the lambda minus 4 cancel by the partial n 1165 01:15:39,940 --> 01:15:42,640 dot partial and we get that. 1166 01:15:42,640 --> 01:15:45,070 Now, the way that we do effective field theory is we 1167 01:15:45,070 --> 01:15:47,620 look at the lowest order term, and we count everything 1168 01:15:47,620 --> 01:15:49,460 relative to that. 1169 01:15:49,460 --> 01:15:55,600 So what we do is we say, we want the lowest order Lagrangian 1170 01:15:55,600 --> 01:15:58,330 to scale like lambda to the 0. 1171 01:15:58,330 --> 01:16:00,960 And you can think of that roughly in a power 1172 01:16:00,960 --> 01:16:04,240 counting sense as normalizing the free kinetic term 1173 01:16:04,240 --> 01:16:06,190 or normalizing the kinetic term in general. 1174 01:16:15,080 --> 01:16:17,660 And then, once you do that, then you 1175 01:16:17,660 --> 01:16:19,430 fix what the scaling of the field 1176 01:16:19,430 --> 01:16:22,750 is, Cn to the order lambda. 1177 01:16:27,300 --> 01:16:31,020 And that's different than the mass dimension. 1178 01:16:31,020 --> 01:16:32,550 So I said that we were going to be 1179 01:16:32,550 --> 01:16:34,175 doing a power counting that's different 1180 01:16:34,175 --> 01:16:36,240 than the mass dimension. 1181 01:16:36,240 --> 01:16:39,900 If we looked at the mass dimension of this field, 1182 01:16:39,900 --> 01:16:46,380 Cn would have mass dimension that's 3/2, whereas it has, 1183 01:16:46,380 --> 01:16:49,740 if you like, a lambda power counting dimension which is 1. 1184 01:16:53,980 --> 01:16:57,048 So if I look for the powers of lambda, that's 1. 1185 01:16:57,048 --> 01:16:59,590 And that was one of the things I told you was going to happen 1186 01:16:59,590 --> 01:17:01,087 is that, in this effective theory, 1187 01:17:01,087 --> 01:17:02,920 we wouldn't just be counting mass dimension. 1188 01:17:02,920 --> 01:17:04,300 We'd be counting something else. 1189 01:17:14,040 --> 01:17:14,540 OK. 1190 01:17:14,540 --> 01:17:16,550 So that's collinear quarks. 1191 01:17:16,550 --> 01:17:20,738 We can do a similar thing for collinear gluons. 1192 01:17:20,738 --> 01:17:23,030 And we may not get to the end of that discussion today, 1193 01:17:23,030 --> 01:17:23,822 but let's start it. 1194 01:17:32,660 --> 01:17:34,040 The momenta for a collinear gluon 1195 01:17:34,040 --> 01:17:37,023 scales the same as the momenta for a collinear quark. 1196 01:17:37,023 --> 01:17:38,690 That means collinear doesn't distinguish 1197 01:17:38,690 --> 01:17:40,970 between quarks and gluons. 1198 01:17:40,970 --> 01:17:42,590 So P squared, the full P squared, 1199 01:17:42,590 --> 01:17:47,510 is still something that we're going to leave together. 1200 01:17:47,510 --> 01:17:51,740 And let's just consider looking at the propagator 1201 01:17:51,740 --> 01:18:00,320 in a general covariant gauge and asking 1202 01:18:00,320 --> 01:18:03,860 kind of the same type of thing that we did over here 1203 01:18:03,860 --> 01:18:05,330 about the scaling of the field. 1204 01:18:11,350 --> 01:18:17,760 So the propagator for two collinear gluons 1205 01:18:17,760 --> 01:18:19,950 and the general covariant gauge-- 1206 01:18:19,950 --> 01:18:23,160 time order product vacuum matrix element of two fields. 1207 01:18:26,010 --> 01:18:28,440 Ignore the subscript ends right now. 1208 01:18:28,440 --> 01:18:35,983 I'm just writing down a full theory result. 1209 01:18:35,983 --> 01:18:37,650 We called C something else, so we better 1210 01:18:37,650 --> 01:18:39,060 not make that gauge parameter. 1211 01:18:39,060 --> 01:18:40,740 So let me call the gauge parameter tau. 1212 01:18:53,723 --> 01:18:56,140 That's the gauge parameter of the general covariant gauge. 1213 01:18:56,140 --> 01:18:58,790 And this is a full three result. 1214 01:18:58,790 --> 01:19:00,880 The thing that makes it collinear 1215 01:19:00,880 --> 01:19:02,980 is if we say that k has a collinear scaling. 1216 01:19:07,400 --> 01:19:14,340 So as above, k squared is k plus k minus plus k perp squared. 1217 01:19:14,340 --> 01:19:16,970 That's of order lambda squared, and there's 1218 01:19:16,970 --> 01:19:20,347 no expansion in there. 1219 01:19:20,347 --> 01:19:21,680 And there's two k squareds here. 1220 01:19:21,680 --> 01:19:25,230 There's one there, and there's one there. 1221 01:19:25,230 --> 01:19:28,220 And if you start looking at g mu nu minus k mu k nu 1222 01:19:28,220 --> 01:19:31,550 over k squared, you also find that the terms there 1223 01:19:31,550 --> 01:19:34,670 are the same size. 1224 01:19:34,670 --> 01:19:38,500 So there's actually-- let's see how that works. 1225 01:19:52,550 --> 01:19:54,830 So let's do an example of that. 1226 01:19:54,830 --> 01:19:58,210 So g perp you knew was just 1. 1227 01:19:58,210 --> 01:20:01,310 So that's obviously something that has no scaling. 1228 01:20:01,310 --> 01:20:05,400 And if you compare that to k perp mu k perp nu over k 1229 01:20:05,400 --> 01:20:08,620 squared, k perp scale like lambda. k 1230 01:20:08,620 --> 01:20:10,200 squared scales like lambda squared. 1231 01:20:10,200 --> 01:20:12,117 So this is lambda squared over lambda squared. 1232 01:20:12,117 --> 01:20:14,140 So that's also lambda to the 0. 1233 01:20:14,140 --> 01:20:17,610 So both the g perp mu nu and the k perp nu term 1234 01:20:17,610 --> 01:20:18,600 are the same size. 1235 01:20:18,600 --> 01:20:19,100 Yeah. 1236 01:20:19,100 --> 01:20:20,950 AUDIENCE: If it happened that it didn't work out, 1237 01:20:20,950 --> 01:20:23,375 could you choose the gauge parameter have a [INAUDIBLE] 1238 01:20:23,375 --> 01:20:23,875 with lambda? 1239 01:20:23,875 --> 01:20:25,500 IAIN STEWART: You could, but then you'd 1240 01:20:25,500 --> 01:20:27,040 be restricted to the classic gauges 1241 01:20:27,040 --> 01:20:29,290 that you would be able to use in the effective theory. 1242 01:20:29,290 --> 01:20:31,707 And then you have to ask what gauge invariance would mean. 1243 01:20:34,750 --> 01:20:38,350 If I don't make any restrictions on tau, 1244 01:20:38,350 --> 01:20:40,010 I don't assign a power counting to it, 1245 01:20:40,010 --> 01:20:43,960 that means my effective theory should allow all these gauges. 1246 01:20:43,960 --> 01:20:46,652 And I actually want that. 1247 01:20:46,652 --> 01:20:47,610 That's a good question. 1248 01:20:51,580 --> 01:20:53,920 You can do the same thing looking at, 1249 01:20:53,920 --> 01:20:56,280 for example, g plus g minus. 1250 01:20:56,280 --> 01:21:00,720 That's also 1, so lambda 0. 1251 01:21:00,720 --> 01:21:03,802 Then you get k plus k minus over k squared. 1252 01:21:03,802 --> 01:21:05,760 That's also lambda squared over lambda squared. 1253 01:21:05,760 --> 01:21:07,510 So this is what I was saying, that the two 1254 01:21:07,510 --> 01:21:09,330 terms are the same size. 1255 01:21:09,330 --> 01:21:14,730 If you dot in n mu n nu, that kills the g mu nu. 1256 01:21:14,730 --> 01:21:18,090 g plus plus is 0. 1257 01:21:18,090 --> 01:21:21,180 And then you just get k plus squared 1258 01:21:21,180 --> 01:21:26,670 over k squared, which is lambda to the 1/4 1259 01:21:26,670 --> 01:21:30,010 over lambda squared, which is lambda squared. 1260 01:21:30,010 --> 01:21:33,450 So when the g mu nu is not 1, then the k mu k nu term 1261 01:21:33,450 --> 01:21:35,220 can still determine how big something is. 1262 01:21:35,220 --> 01:21:39,735 And that's what would happen for these off-diagonal terms. 1263 01:21:39,735 --> 01:21:41,610 So if we go through the same type of exercise 1264 01:21:41,610 --> 01:21:45,947 that we did for the fermion, d4x scales like 1 over k 1265 01:21:45,947 --> 01:21:46,530 to the fourth. 1266 01:21:51,200 --> 01:21:54,950 So that's lambda to the minus 4, just as before. 1267 01:21:54,950 --> 01:21:58,170 And actually, if we were to do the following, 1268 01:21:58,170 --> 01:22:01,220 if we were to write this as-- 1269 01:22:01,220 --> 01:22:02,810 I should have done that up there. 1270 01:22:02,810 --> 01:22:04,910 If we were to write it as minus i over k 1271 01:22:04,910 --> 01:22:12,410 to the fourth k squared g mu nu minus tau k mu k nu, 1272 01:22:12,410 --> 01:22:15,290 the k to the fourth just matches up with the d4x. 1273 01:22:15,290 --> 01:22:17,180 So those take care of each other. 1274 01:22:17,180 --> 01:22:20,930 And then the fields here have to match up with the rest. 1275 01:22:20,930 --> 01:22:26,420 So the scaling of this should be the scaling of the A mu A nu. 1276 01:22:26,420 --> 01:22:29,335 And that basically means that A mu in A nu scale 1277 01:22:29,335 --> 01:22:29,960 like a momenta. 1278 01:22:38,860 --> 01:22:43,300 So A mu n for this collinear gluon scales like k mu, scales 1279 01:22:43,300 --> 01:22:44,605 like lambda squared 1 lambda. 1280 01:22:54,907 --> 01:22:56,240 Let me write that one more time. 1281 01:23:09,160 --> 01:23:12,220 And that's also a nice thing because it also 1282 01:23:12,220 --> 01:23:14,770 means you could form a covariate derivative that's 1283 01:23:14,770 --> 01:23:19,610 homogeneous by combining together, 1284 01:23:19,610 --> 01:23:23,410 if I write it this way, k mu and g A mu. 1285 01:23:23,410 --> 01:23:27,640 I can get a covariant derivative where, for each component of k, 1286 01:23:27,640 --> 01:23:29,390 I also have a component of the gauge field 1287 01:23:29,390 --> 01:23:30,440 that's the same size. 1288 01:23:30,440 --> 01:23:32,530 So you could have argued it, originally 1289 01:23:32,530 --> 01:23:36,010 just from gauge invariance, that you want to sort of have fields 1290 01:23:36,010 --> 01:23:38,450 that are of the same size as your momenta, 1291 01:23:38,450 --> 01:23:40,360 but we did it a little bit differently here. 1292 01:23:55,150 --> 01:23:58,090 So it's nice that that comes out. 1293 01:23:58,090 --> 01:23:59,945 This all hangs together. 1294 01:23:59,945 --> 01:24:01,570 So next time, we'll talk about the fact 1295 01:24:01,570 --> 01:24:03,730 that what does it mean that the gauge field has components that 1296 01:24:03,730 --> 01:24:04,940 are scaling in a different way. 1297 01:24:04,940 --> 01:24:07,273 And particularly, there's a component of the gauge field 1298 01:24:07,273 --> 01:24:08,200 that's order 1. 1299 01:24:08,200 --> 01:24:10,510 There's no power suppression for that component. 1300 01:24:10,510 --> 01:24:14,126 And we'll talk about what implications that has. 1301 01:24:14,126 --> 01:24:16,210 Yeah. 1302 01:24:16,210 --> 01:24:20,194 AUDIENCE: So what about in the non-variant case, like n bar 1303 01:24:20,194 --> 01:24:21,190 of A equals 0? 1304 01:24:21,190 --> 01:24:21,982 IAIN STEWART: Yeah. 1305 01:24:21,982 --> 01:24:25,270 So you could go through this argument in that gauge as well, 1306 01:24:25,270 --> 01:24:27,210 and it will work. 1307 01:24:27,210 --> 01:24:29,677 AUDIENCE: But you have n bar to the A equals 0, so-- 1308 01:24:29,677 --> 01:24:31,010 IAIN STEWART: Yeah, that's fine. 1309 01:24:31,010 --> 01:24:32,620 That's just a restriction on the-- you 1310 01:24:32,620 --> 01:24:34,150 can still assign a scaling to it. 1311 01:24:34,150 --> 01:24:36,550 And a special choice if it is 0. 1312 01:24:36,550 --> 01:24:40,600 Scaling and values of things are not the same thing, right? 1313 01:24:40,600 --> 01:24:43,900 Like, you could have a field that scales like 1 in the power 1314 01:24:43,900 --> 01:24:46,610 company, but happens to be 0. 1315 01:24:46,610 --> 01:24:47,480 And that's OK. 1316 01:24:50,870 --> 01:24:52,096 Yeah. 1317 01:24:52,096 --> 01:24:54,198 AUDIENCE: I had a question about SCET1 and SCET2. 1318 01:24:54,198 --> 01:24:54,990 IAIN STEWART: Sure. 1319 01:24:54,990 --> 01:24:59,874 AUDIENCE: So in the example adding one jab, and one hadron, 1320 01:24:59,874 --> 01:25:01,827 like the [INAUDIBLE],, why do I need 1321 01:25:01,827 --> 01:25:03,982 to have different degrees of freedom to describe 1322 01:25:03,982 --> 01:25:04,798 [INAUDIBLE]? 1323 01:25:04,798 --> 01:25:05,590 IAIN STEWART: Yeah. 1324 01:25:05,590 --> 01:25:07,690 It's really because of the way the modes 1325 01:25:07,690 --> 01:25:11,690 sat on those hyperbolas that, in the case of the hadrons, 1326 01:25:11,690 --> 01:25:13,190 you had the soft modes and collinear 1327 01:25:13,190 --> 01:25:14,320 modes on the same hyperbola. 1328 01:25:14,320 --> 01:25:16,510 And that's going to change how the effective theory looks. 1329 01:25:16,510 --> 01:25:17,135 AUDIENCE: Yeah. 1330 01:25:17,135 --> 01:25:19,430 But from a physically point of view, why the hadrons-- 1331 01:25:19,430 --> 01:25:21,469 [INAUDIBLE] the hadrons behaving in a different way 1332 01:25:21,469 --> 01:25:22,180 than [INAUDIBLE]? 1333 01:25:22,180 --> 01:25:22,972 IAIN STEWART: Yeah. 1334 01:25:22,972 --> 01:25:24,942 So one way of thinking about it is, 1335 01:25:24,942 --> 01:25:26,650 if you just had the collinear modes alone 1336 01:25:26,650 --> 01:25:28,040 and you didn't have the soft modes, 1337 01:25:28,040 --> 01:25:29,290 then it would be very similar. 1338 01:25:29,290 --> 01:25:32,020 It would just be that the hyperbola moved down. 1339 01:25:32,020 --> 01:25:34,720 And you think maybe those two theories are the same. 1340 01:25:34,720 --> 01:25:36,280 There's examples of that actually. 1341 01:25:36,280 --> 01:25:39,220 There's one example that we'll cover where 1342 01:25:39,220 --> 01:25:40,630 you don't need soft modes. 1343 01:25:40,630 --> 01:25:42,505 And then you can't really tell whether you're 1344 01:25:42,505 --> 01:25:44,410 using SCET1 or SCET2. 1345 01:25:44,410 --> 01:25:47,350 But if you have a process that has both soft and collinear 1346 01:25:47,350 --> 01:25:50,410 modes, then it's the kind of way that those modes 1347 01:25:50,410 --> 01:25:52,180 talk to each other that distinguishes 1348 01:25:52,180 --> 01:25:54,310 the cases of hadrons and jets in the cases-- 1349 01:25:54,310 --> 01:25:56,270 AUDIENCE: For example, for the case of the jet, 1350 01:25:56,270 --> 01:25:58,742 [INAUDIBLE] soft modes like as a need for the-- 1351 01:25:58,742 --> 01:25:59,575 IAIN STEWART: Right. 1352 01:25:59,575 --> 01:26:01,960 You could try, yes. 1353 01:26:01,960 --> 01:26:04,870 And it turns out that those modes aren't relevant. 1354 01:26:04,870 --> 01:26:06,580 So you have to be a little bit faithful 1355 01:26:06,580 --> 01:26:08,107 from what I've told you so far. 1356 01:26:08,107 --> 01:26:10,190 AUDIENCE: But there is no physical picture for why 1357 01:26:10,190 --> 01:26:11,520 it can understand [INAUDIBLE]. 1358 01:26:11,520 --> 01:26:12,430 IAIN STEWART: You can understand it 1359 01:26:12,430 --> 01:26:14,513 because what would happen with those soft modes is 1360 01:26:14,513 --> 01:26:18,400 that they would take the collinear modes far off-shell. 1361 01:26:18,400 --> 01:26:20,695 If you just had one of those soft modes interacting 1362 01:26:20,695 --> 01:26:23,320 with the collinear mode, you end up not with the collinear mode 1363 01:26:23,320 --> 01:26:25,930 back again, but something that's further off-shell that you 1364 01:26:25,930 --> 01:26:27,430 actually want to integrate out. 1365 01:26:27,430 --> 01:26:31,330 So that thing that has the scaling that allows you to just 1366 01:26:31,330 --> 01:26:34,960 have collinear mode in something which we call ultra soft in 1367 01:26:34,960 --> 01:26:38,370 and still have collinear, that's the ultra soft mode. 1368 01:26:38,370 --> 01:26:40,120 So if you want to communicate with the jet 1369 01:26:40,120 --> 01:26:42,760 without disturbing it and blowing it apart, 1370 01:26:42,760 --> 01:26:45,430 then you really need the ultra soft mode. 1371 01:26:45,430 --> 01:26:46,000 Yeah. 1372 01:26:46,000 --> 01:26:48,100 But these are all good questions, 1373 01:26:48,100 --> 01:26:50,700 and we'll talk much more about it.