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 --> 00:00:17,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,160 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:22,325 --> 00:00:23,200 PROFESSOR: All right. 9 00:00:23,200 --> 00:00:25,640 So where were we last time? 10 00:00:25,640 --> 00:00:28,480 So we had derived the Fermion Lagrangian 11 00:00:28,480 --> 00:00:30,460 which is at the top of the board, 12 00:00:30,460 --> 00:00:32,320 and really the only difference between this 13 00:00:32,320 --> 00:00:36,280 and what we talked about earlier is that we had to perform 14 00:00:36,280 --> 00:00:37,550 the multipole expansion. 15 00:00:37,550 --> 00:00:41,110 So we distinguish between collinear momenta that 16 00:00:41,110 --> 00:00:43,690 are scaling like a collinear momentum and momentum that 17 00:00:43,690 --> 00:00:47,260 are scaling like an ultrasoft, and only the collinear pieces 18 00:00:47,260 --> 00:00:49,810 were showing up in here, and same with the collinear gauge 19 00:00:49,810 --> 00:00:50,310 field. 20 00:00:50,310 --> 00:00:53,470 It was only showing up in here, whereas both were showing up 21 00:00:53,470 --> 00:00:55,210 in here. 22 00:00:55,210 --> 00:00:57,220 There's only a single type of derivative 23 00:00:57,220 --> 00:00:59,740 here, because ultrasoft and collinear 24 00:00:59,740 --> 00:01:01,900 momenta are the same size in the end component. 25 00:01:01,900 --> 00:01:03,900 But they're different in these other components, 26 00:01:03,900 --> 00:01:06,820 so we had these labels in this label operator 27 00:01:06,820 --> 00:01:08,845 that picked out the collinear momenta. 28 00:01:08,845 --> 00:01:10,970 So we could define collinear covariate derivatives, 29 00:01:10,970 --> 00:01:12,790 if you like, with this subscript n, 30 00:01:12,790 --> 00:01:15,610 whereas this D is like a full derivative that involves 31 00:01:15,610 --> 00:01:18,580 both types of gauge field. 32 00:01:18,580 --> 00:01:20,830 And then at the end of lecture, it was kind of rushed, 33 00:01:20,830 --> 00:01:23,653 but I was talking about the collinear gluon Lagrangian, 34 00:01:23,653 --> 00:01:25,570 and I said there is a set of replacement rules 35 00:01:25,570 --> 00:01:28,690 that we could make to effectively do the same thing 36 00:01:28,690 --> 00:01:29,668 that we did here. 37 00:01:29,668 --> 00:01:31,210 And now I've just written out for you 38 00:01:31,210 --> 00:01:34,690 with that Lagrangian is with those replacement rules. 39 00:01:34,690 --> 00:01:39,940 So curly D is basically just taking all the derivatives 40 00:01:39,940 --> 00:01:41,650 and taking only the leading order pieces. 41 00:01:41,650 --> 00:01:44,210 So we take the collinear pieces in this component 42 00:01:44,210 --> 00:01:45,490 and this component. 43 00:01:45,490 --> 00:01:48,380 We take the ultrasoft pieces in this component. 44 00:01:48,380 --> 00:01:50,650 And then if you wrote the original Lagrangian 45 00:01:50,650 --> 00:01:52,690 as a commutator of two covariant derivatives, 46 00:01:52,690 --> 00:01:54,860 you'd just replace it by the leading order pieces, 47 00:01:54,860 --> 00:01:57,580 and that's what the leading order action will be. 48 00:01:57,580 --> 00:02:00,130 Then, we had to think about gauge fixing. 49 00:02:00,130 --> 00:02:02,830 Since this is the collinear gluon Lagrangian, 50 00:02:02,830 --> 00:02:05,410 we should think about collinear gauge fixing. 51 00:02:05,410 --> 00:02:07,270 This here as a general covariant gauge 52 00:02:07,270 --> 00:02:09,610 fixing with parameter tau, and then there's 53 00:02:09,610 --> 00:02:11,470 a corresponding ghost term. 54 00:02:11,470 --> 00:02:14,800 And in this Lagrangian, the usual way it would look, 55 00:02:14,800 --> 00:02:20,740 this would be i partial, and then 56 00:02:20,740 --> 00:02:24,850 I said that, because we don't want the collinear gauge fixing 57 00:02:24,850 --> 00:02:27,910 term to break ultrasoft gauge invariance, 58 00:02:27,910 --> 00:02:29,710 we're going to turn that i partial into 59 00:02:29,710 --> 00:02:31,321 a covariant derivative under-- 60 00:02:34,217 --> 00:02:35,800 we're going to include this piece here 61 00:02:35,800 --> 00:02:40,150 at the n dot a ultrasoft to make it ultrasoft gauge 62 00:02:40,150 --> 00:02:41,550 invariant at lowest order. 63 00:02:41,550 --> 00:02:42,050 OK? 64 00:02:42,050 --> 00:02:44,200 So that's really the only-- 65 00:02:44,200 --> 00:02:47,800 other than just doing this most naive thing by just replacing 66 00:02:47,800 --> 00:02:49,450 derivatives by covariant derivatives, 67 00:02:49,450 --> 00:02:51,250 and you might think, well, I'll just keep 68 00:02:51,250 --> 00:02:52,458 this as a partial derivative. 69 00:02:52,458 --> 00:02:54,190 Since it's to doing gauge fixing, 70 00:02:54,190 --> 00:02:56,140 I don't need to make it covariant, 71 00:02:56,140 --> 00:02:59,110 but we do want to make it covariant under the ultrasofts, 72 00:02:59,110 --> 00:03:04,690 and that's why I write this curly D ultrasoft. 73 00:03:04,690 --> 00:03:05,190 OK. 74 00:03:05,190 --> 00:03:08,580 So that gives you the leading order Lagrangian. 75 00:03:08,580 --> 00:03:12,180 Once you put together this with what 76 00:03:12,180 --> 00:03:13,680 you want for the ultrasoft, then you 77 00:03:13,680 --> 00:03:16,650 would have the full leading order Lagrangian, 78 00:03:16,650 --> 00:03:18,780 and the ultrasoft part is actually very simple. 79 00:03:23,040 --> 00:03:26,430 So for the ultrasoft part, we just 80 00:03:26,430 --> 00:03:29,830 take a full QCD action for the ultrasoft field. 81 00:03:29,830 --> 00:03:35,085 So this is just q ultrasoft bar i 82 00:03:35,085 --> 00:03:39,000 D slash ultrasoft q ultrasoft. 83 00:03:39,000 --> 00:03:46,350 And likewise for the gluon piece, it's just QCD, 84 00:03:46,350 --> 00:03:50,970 and we would do the gauge fixing without thinking 85 00:03:50,970 --> 00:03:55,440 about any complications, like these ones, 86 00:03:55,440 --> 00:03:59,620 and these are just involving ultrasoft fields, so just 87 00:03:59,620 --> 00:04:01,320 QCD, just ultrasoft fields. 88 00:04:07,270 --> 00:04:07,770 OK? 89 00:04:07,770 --> 00:04:08,800 So the only place-- 90 00:04:08,800 --> 00:04:10,930 and you take everything here together-- 91 00:04:10,930 --> 00:04:13,330 the only place that the ultrasoft and collinear fields 92 00:04:13,330 --> 00:04:15,370 talk to each other in the Lagrangian 93 00:04:15,370 --> 00:04:19,490 is in this single component n dot partial, n dot a ultrasoft. 94 00:04:19,490 --> 00:04:24,180 And that comes about basically because of the power counting, 95 00:04:24,180 --> 00:04:26,680 that this is really the only place that these two things can 96 00:04:26,680 --> 00:04:27,220 interact. 97 00:04:27,220 --> 00:04:31,623 And we'll talk about the implications of that later on, 98 00:04:31,623 --> 00:04:33,290 but that's the leading order Lagrangian. 99 00:04:33,290 --> 00:04:36,970 So once you have this piece, and this piece is pretty straight 100 00:04:36,970 --> 00:04:38,907 forward with this additional complication 101 00:04:38,907 --> 00:04:40,990 of worrying about what gauge symmetry means, which 102 00:04:40,990 --> 00:04:43,392 we'll talk more about later. 103 00:04:43,392 --> 00:04:45,100 But we had to be careful not to break it, 104 00:04:45,100 --> 00:04:47,410 when we introduced this term, because we wanted 105 00:04:47,410 --> 00:04:49,930 to in some sense have a gauge fixing both for the collinear 106 00:04:49,930 --> 00:04:52,900 gluon and a separate gauge fixing for the ultrasoft gluon, 107 00:04:52,900 --> 00:04:56,380 and that's why we wanted to do this. 108 00:04:56,380 --> 00:04:58,130 And then this piece here is simple. 109 00:04:58,130 --> 00:05:01,000 It's just QCD, because it's, if you'd like, 110 00:05:01,000 --> 00:05:02,710 it's the lowest energy mode, and it 111 00:05:02,710 --> 00:05:04,690 doesn't know about any of the complications 112 00:05:04,690 --> 00:05:07,780 that we had for the collinear modes. 113 00:05:07,780 --> 00:05:09,460 OK? 114 00:05:09,460 --> 00:05:11,680 So any questions about this so far? 115 00:05:14,120 --> 00:05:14,620 OK. 116 00:05:14,620 --> 00:05:17,440 So everything that we did in deriving these actions 117 00:05:17,440 --> 00:05:19,630 is tree level. 118 00:05:19,630 --> 00:05:22,120 All the steps that we did were tree level. 119 00:05:22,120 --> 00:05:24,478 So you can ask, if I start to do loops, 120 00:05:24,478 --> 00:05:26,020 will there be some Wilson coefficient 121 00:05:26,020 --> 00:05:27,580 that shows up somewhere here? 122 00:05:27,580 --> 00:05:29,380 Will I generate some new operators 123 00:05:29,380 --> 00:05:31,360 that I don't see here? 124 00:05:31,360 --> 00:05:33,135 Those are reasonable questions, and that's 125 00:05:33,135 --> 00:05:34,510 what we're going to address next. 126 00:05:46,020 --> 00:05:57,870 So to go further, we'll use symmetries, 127 00:05:57,870 --> 00:06:00,240 and we're actually going to consider 128 00:06:00,240 --> 00:06:02,550 three different symmetries, gauge symmetry, which I've 129 00:06:02,550 --> 00:06:21,850 been promising you for a while, reparameterization invariance, 130 00:06:21,850 --> 00:06:23,560 and spin symmetry. 131 00:06:30,400 --> 00:06:32,855 Where I'll put a question mark by this last one, 132 00:06:32,855 --> 00:06:35,230 because we need to answer the question whether there even 133 00:06:35,230 --> 00:06:37,892 is a spin symmetry. 134 00:06:37,892 --> 00:06:39,850 These two here, this number one and number two, 135 00:06:39,850 --> 00:06:42,970 will turn out to be quite important. 136 00:06:42,970 --> 00:06:44,490 Number three is not so important. 137 00:06:50,500 --> 00:06:52,110 So reparameterization invariance here 138 00:06:52,110 --> 00:06:54,510 will be like reparameterization invariance in HQET, 139 00:06:54,510 --> 00:06:56,190 except now it's different. 140 00:06:56,190 --> 00:06:58,530 We've introduced the parameters and then n bar, 141 00:06:58,530 --> 00:07:00,660 and we'll have to see what kind of symmetries 142 00:07:00,660 --> 00:07:05,610 we have with respect to that choice of basis factors 143 00:07:05,610 --> 00:07:06,893 that we made. 144 00:07:06,893 --> 00:07:09,060 But it otherwise will be analogous to our discussion 145 00:07:09,060 --> 00:07:09,585 of HQET. 146 00:07:12,970 --> 00:07:15,730 So let's actually first dispense with the one that in some sense 147 00:07:15,730 --> 00:07:18,430 is the least important, this number three. 148 00:07:26,460 --> 00:07:31,780 So first, let's revisit our spinners a little bit. 149 00:07:31,780 --> 00:07:34,440 So if I put together the information 150 00:07:34,440 --> 00:07:36,840 that we talked about when we derived 151 00:07:36,840 --> 00:07:41,340 the equation at the top of the board there for the Lagrangian, 152 00:07:41,340 --> 00:07:48,360 we worked out at a tree level we have this formula that 153 00:07:48,360 --> 00:07:49,170 relates the fields. 154 00:07:57,700 --> 00:08:00,630 So from that formula, if we just project onto the spinner 155 00:08:00,630 --> 00:08:02,580 pieces, we can write down a formula 156 00:08:02,580 --> 00:08:04,680 that relates the spinners. 157 00:08:04,680 --> 00:08:07,710 So throwing away the gauge fields, 158 00:08:07,710 --> 00:08:18,490 we have in momentum space the u of p, the spinner, 159 00:08:18,490 --> 00:08:20,050 would be related to whatever spinner 160 00:08:20,050 --> 00:08:24,640 we have for this cn field which I call Un by that formula. 161 00:08:24,640 --> 00:08:26,680 And then if you take this formula, 162 00:08:26,680 --> 00:08:30,250 you also see that if I hit it with a projector, 163 00:08:30,250 --> 00:08:36,460 n slash n bar slash over 4, if I hit the u with a projector, 164 00:08:36,460 --> 00:08:37,840 it's going to kill this piece. 165 00:08:37,840 --> 00:08:40,749 Because the n slash n bar slash can be pushed through the p 166 00:08:40,749 --> 00:08:45,280 perp slash, then n bar squared is 0, 167 00:08:45,280 --> 00:08:47,210 so that kills that second piece. 168 00:08:47,210 --> 00:08:49,780 So we also have this formula. 169 00:08:49,780 --> 00:08:51,700 And then once you have this formula, 170 00:08:51,700 --> 00:08:57,230 you have the formulas that we wanted for that. 171 00:09:03,940 --> 00:09:04,600 OK? 172 00:09:04,600 --> 00:09:07,180 But this actually, this spinner here, 173 00:09:07,180 --> 00:09:09,640 this Un is not exactly the same as the spinner 174 00:09:09,640 --> 00:09:12,527 that we talked about earlier. 175 00:09:12,527 --> 00:09:13,360 So let me come back. 176 00:09:13,360 --> 00:09:16,425 Let me come to that in a minute. 177 00:09:16,425 --> 00:09:17,800 So first thing you might consider 178 00:09:17,800 --> 00:09:21,640 is whether, when I take this cn field, 179 00:09:21,640 --> 00:09:24,400 and I take the Lagrangian up at the top of the board, 180 00:09:24,400 --> 00:09:27,370 do I just get the collinear propagator that we 181 00:09:27,370 --> 00:09:28,270 talked about? 182 00:09:28,270 --> 00:09:29,350 And indeed, you do. 183 00:09:35,380 --> 00:09:38,470 It's kind of obvious for the momentum-dependent parts, 184 00:09:38,470 --> 00:09:42,470 and really you might only worry about the spin. 185 00:09:42,470 --> 00:09:45,670 And so if you consider this formula, 186 00:09:45,670 --> 00:09:49,480 and you consider the sum over spins of u u bar which 187 00:09:49,480 --> 00:09:52,000 is what's going to appear in the numerator of the propagator 188 00:09:52,000 --> 00:09:54,458 for the Fermion when you're driving the Fermion propagator, 189 00:09:54,458 --> 00:09:57,400 you get a sum over the physical spins. 190 00:09:57,400 --> 00:10:01,630 Then, from this formula, this is a projector on something 191 00:10:01,630 --> 00:10:10,345 you know how to do the spin sum for which is the full theory 192 00:10:10,345 --> 00:10:10,845 spinner. 193 00:10:15,190 --> 00:10:19,120 And that spin sum is p slash, so this 194 00:10:19,120 --> 00:10:21,550 is like p slash sandwiched between projectors. 195 00:10:28,110 --> 00:10:32,610 And you could work out that that's exactly the numerator 196 00:10:32,610 --> 00:10:38,745 that we had before after a little bit of Dirac algebra. 197 00:10:44,150 --> 00:10:46,040 So that part works as expected. 198 00:10:46,040 --> 00:10:48,610 If we take this Lagrangian, and we work out 199 00:10:48,610 --> 00:10:51,610 what the propagator is, we get exactly the propagator 200 00:10:51,610 --> 00:10:52,690 we got from expansion. 201 00:11:13,108 --> 00:11:17,390 So quantizing lc0 gives us that propagator. 202 00:11:26,570 --> 00:11:28,970 But the situation is not quite the same for the spinner. 203 00:11:39,590 --> 00:11:42,440 And in some sense, this is not-- 204 00:11:42,440 --> 00:11:45,110 this point is not absolutely crucial, 205 00:11:45,110 --> 00:11:48,830 but actually, there's a little simplification 206 00:11:48,830 --> 00:11:51,590 I want to do in order to discuss the spin symmetry, 207 00:11:51,590 --> 00:11:54,170 and in order to do that at this point, 208 00:11:54,170 --> 00:11:56,585 it's important to understand this point. 209 00:11:56,585 --> 00:11:58,460 So that I'm going through this in some sense, 210 00:11:58,460 --> 00:12:01,593 because then it'll be very easy to discuss what spin symmetry 211 00:12:01,593 --> 00:12:02,510 we have in the theory. 212 00:12:06,240 --> 00:12:10,890 So this guy is actually not equal to our expanded spinner 213 00:12:10,890 --> 00:12:15,990 which putting in some normalization I 214 00:12:15,990 --> 00:12:21,390 could write like this. 215 00:12:21,390 --> 00:12:22,920 So this is what we got by expanding, 216 00:12:22,920 --> 00:12:25,860 something like this which is very simple where this u is 217 00:12:25,860 --> 00:12:30,606 equal to 1, 0 or 0, 1. 218 00:12:30,606 --> 00:12:33,492 in the Dirac representation. 219 00:12:33,492 --> 00:12:34,950 But that's actually not what we get 220 00:12:34,950 --> 00:12:38,970 if we just take the formula up here and use this. 221 00:12:43,573 --> 00:12:44,740 So let's see what we do get. 222 00:13:02,940 --> 00:13:06,830 So if you use the formula up there, 223 00:13:06,830 --> 00:13:08,447 then you have the following. 224 00:13:14,660 --> 00:13:17,020 So here's the full theory spinner 225 00:13:17,020 --> 00:13:19,391 with a conventional normalization. 226 00:13:22,550 --> 00:13:27,790 So this thing and this thing here is the projector. 227 00:13:27,790 --> 00:13:30,040 OK? 228 00:13:30,040 --> 00:13:33,970 So you could work out what that product is, and it turns out 229 00:13:33,970 --> 00:13:36,040 that you can write it in the following way which 230 00:13:36,040 --> 00:13:42,700 is very closely related but not precisely the same as what 231 00:13:42,700 --> 00:13:44,930 we had before. 232 00:13:44,930 --> 00:13:49,720 So you can write it in what looks like the same form 233 00:13:49,720 --> 00:13:56,200 as we have over here, except this curly U, curly capital U, 234 00:13:56,200 --> 00:13:57,505 is a more complicated object. 235 00:14:10,580 --> 00:14:13,070 And it's just whatever I get by multiplying these two 236 00:14:13,070 --> 00:14:15,620 things out which turns out to be something 237 00:14:15,620 --> 00:14:16,790 I can write in that form. 238 00:14:28,220 --> 00:14:32,260 So it's some two-component spinner, 239 00:14:32,260 --> 00:14:34,720 but it's got momentum dependence, unlike our simple 240 00:14:34,720 --> 00:14:41,690 1, 0 and 0,1. 241 00:14:41,690 --> 00:14:44,133 But everything we said about Un really depended only 242 00:14:44,133 --> 00:14:46,550 on the fact that it could be written in this form in terms 243 00:14:46,550 --> 00:14:48,530 of some two-component spinner-- 244 00:14:48,530 --> 00:14:50,540 the fact that n slash on it was 0, 245 00:14:50,540 --> 00:14:53,990 the fact that it had projection relation. 246 00:14:53,990 --> 00:14:59,600 So these formulas here, if you have a formula like these, 247 00:14:59,600 --> 00:15:02,240 these formulas here are true. 248 00:15:02,240 --> 00:15:04,100 OK? 249 00:15:04,100 --> 00:15:07,230 So actually, it would be true whatever spinner we have there. 250 00:15:07,230 --> 00:15:10,640 So why should I want this U twiddle spinner 251 00:15:10,640 --> 00:15:13,175 and rather than the U spinner? 252 00:15:17,393 --> 00:15:19,560 That actually has to do with this reparameterization 253 00:15:19,560 --> 00:15:26,410 invariant, so it'll become clear when we talk about that. 254 00:15:26,410 --> 00:15:30,000 But these extra terms in U relative to those 255 00:15:30,000 --> 00:15:40,120 for the other guy, the simple U, actually 256 00:15:40,120 --> 00:15:45,162 ensure the proper structure under reparameterizations. 257 00:15:50,470 --> 00:15:55,317 And basically, it'll become clear in a moment, 258 00:15:55,317 --> 00:15:57,400 but basically, if we wanted to get this U spinner, 259 00:15:57,400 --> 00:16:11,910 we should have a slightly different projector, 260 00:16:11,910 --> 00:16:13,610 which I'll call Pn prime. 261 00:16:20,960 --> 00:16:23,390 So we could have used a different projector which 262 00:16:23,390 --> 00:16:28,820 is this one, and then we could have come up 263 00:16:28,820 --> 00:16:31,880 with another projector which was the Pn bar projector which 264 00:16:31,880 --> 00:16:45,230 would satisfy that the sum is 1, if we wanted. 265 00:16:45,230 --> 00:16:49,165 And this projector here, when acting on the full theory 266 00:16:49,165 --> 00:16:50,540 field, would have given something 267 00:16:50,540 --> 00:16:52,640 that would have been proportional 268 00:16:52,640 --> 00:16:54,460 to this combination over here. 269 00:16:54,460 --> 00:16:54,960 OK? 270 00:16:54,960 --> 00:16:55,880 So you just have to believe me. 271 00:16:55,880 --> 00:16:57,530 I don't want to go through the algebra, 272 00:16:57,530 --> 00:16:59,420 or you can check it yourself. 273 00:16:59,420 --> 00:17:02,810 But this projector here is not invariant under that symmetry 274 00:17:02,810 --> 00:17:04,165 of reparameterization variance. 275 00:17:09,310 --> 00:17:11,950 When we talk about RPI, it'll be clear 276 00:17:11,950 --> 00:17:17,619 why we want a projector which is this projector and not 277 00:17:17,619 --> 00:17:20,140 the slightly different projector which 278 00:17:20,140 --> 00:17:21,790 has this extra n slash over 2. 279 00:17:25,530 --> 00:17:26,130 OK. 280 00:17:26,130 --> 00:17:28,050 But nevertheless, the important point 281 00:17:28,050 --> 00:17:30,360 that I wanted to emphasize is really 282 00:17:30,360 --> 00:17:33,150 that we have this way of decomposing 283 00:17:33,150 --> 00:17:37,080 the spinner, the true spinner, for our field cn 284 00:17:37,080 --> 00:17:40,770 in terms of a two-component object, U twiddle. 285 00:17:40,770 --> 00:17:41,400 OK? 286 00:17:41,400 --> 00:17:45,083 So we are able to do that. 287 00:17:45,083 --> 00:17:47,500 And if you want to talk about spin symmetry of the theory, 288 00:17:47,500 --> 00:17:50,223 it's easier if you use a two-component notation. 289 00:17:50,223 --> 00:17:52,140 So the reason that I wanted to go through this 290 00:17:52,140 --> 00:17:55,030 is really to have on the board this equation, 291 00:17:55,030 --> 00:17:57,240 which I can then use to motivate writing down 292 00:17:57,240 --> 00:17:59,385 to two-component version of cn. 293 00:17:59,385 --> 00:18:01,962 And once I have a two-component version of SCET, 294 00:18:01,962 --> 00:18:04,170 then it's very easy to see what the spin symmetry is. 295 00:18:09,330 --> 00:18:11,220 So let's write down a two-component version 296 00:18:11,220 --> 00:18:12,960 of our collinear quark Lagrangian. 297 00:18:20,835 --> 00:18:22,960 You can do the same thing, of course, in HQET right 298 00:18:22,960 --> 00:18:26,320 down the two-component version rather than 299 00:18:26,320 --> 00:18:27,580 a four-component version. 300 00:18:27,580 --> 00:18:28,810 If you have a four-component version, 301 00:18:28,810 --> 00:18:30,160 it has this projection relation. 302 00:18:30,160 --> 00:18:31,380 If you have the two-component version, 303 00:18:31,380 --> 00:18:32,880 the projection relation is built in. 304 00:18:37,028 --> 00:18:39,070 The reason to consider the four-component version 305 00:18:39,070 --> 00:18:41,715 is if you want a couple this object 306 00:18:41,715 --> 00:18:43,840 to four-component fields, like the ultrasoft field, 307 00:18:43,840 --> 00:18:45,257 then it's, of course, a nice thing 308 00:18:45,257 --> 00:18:47,920 to have a four-component version, 309 00:18:47,920 --> 00:18:50,389 but some things are easier in two components. 310 00:19:01,880 --> 00:19:12,730 So take cn, and write it as follows, where 311 00:19:12,730 --> 00:19:14,850 this phi n has two components. 312 00:19:14,850 --> 00:19:18,780 And I've set things up, so the dimensions of cn 313 00:19:18,780 --> 00:19:20,390 are equal to the dimensions of phi n. 314 00:19:24,390 --> 00:19:24,890 OK? 315 00:19:24,890 --> 00:19:33,800 So I can take that formula, plug it into our SCET Lagrangian, 316 00:19:33,800 --> 00:19:37,235 and then I can, using the Dirac representation for the gamma 317 00:19:37,235 --> 00:19:44,090 matrices, write out a Lagrangian for this phi n. 318 00:19:49,830 --> 00:19:52,120 So that requires doing a bit of algebra, 319 00:19:52,120 --> 00:19:55,680 which I will take you through. 320 00:20:00,730 --> 00:20:03,620 And then we get an equivalent Lagrangian 321 00:20:03,620 --> 00:20:06,210 but in terms of this field, and it looks as follows. 322 00:20:51,200 --> 00:20:51,900 OK. 323 00:20:51,900 --> 00:20:55,530 So it's almost independent of spin but not quite. 324 00:20:55,530 --> 00:20:57,750 There's this sigma 3 sitting there. 325 00:20:57,750 --> 00:20:59,640 If sigma 3 wasn't there, it'd be like HQET, 326 00:20:59,640 --> 00:21:01,770 where you have an SU2 symmetry. 327 00:21:01,770 --> 00:21:03,300 The fact that sigma 3 is there means 328 00:21:03,300 --> 00:21:05,745 you don't have an SU2 symmetry. 329 00:21:20,370 --> 00:21:27,650 And really, all you have is a U1 symmetry, 330 00:21:27,650 --> 00:21:30,220 and that U1 really corresponds just to helicity. 331 00:21:33,110 --> 00:21:39,770 So in four-component notation, that U1 would be the following. 332 00:21:43,706 --> 00:21:47,110 It's projecting a spin operator onto perpendicular indices 333 00:21:47,110 --> 00:21:51,490 anti-symmetric in both of them, and in the two components, 334 00:21:51,490 --> 00:21:53,950 that just becomes a sigma 3. 335 00:21:53,950 --> 00:21:56,860 So obviously, if we do an exponential rotation 336 00:21:56,860 --> 00:21:59,780 with respect to sigma 3, sigma 3 commutes with sigma 3, 337 00:21:59,780 --> 00:22:01,160 so does the identity. 338 00:22:01,160 --> 00:22:06,550 And so we could have a rotation of this guy by that, 339 00:22:06,550 --> 00:22:09,340 and that's the only spin symmetry 340 00:22:09,340 --> 00:22:10,780 that you have is this helicity. 341 00:22:28,263 --> 00:22:30,180 And so because of the coordinates we're using, 342 00:22:30,180 --> 00:22:32,980 this corresponds to the spin along the direction of motion 343 00:22:32,980 --> 00:22:37,604 which is the three direction, if you like, 344 00:22:37,604 --> 00:22:39,340 which sometimes we denote by just saying 345 00:22:39,340 --> 00:22:42,790 it's along the direction n which is then more independent of how 346 00:22:42,790 --> 00:22:45,310 we pick our axes. 347 00:22:45,310 --> 00:22:46,990 So this symmetry here is actually 348 00:22:46,990 --> 00:22:50,140 related to what you would call the chirality in QCD. 349 00:22:50,140 --> 00:22:54,205 So it's not really a new symmetry. 350 00:23:00,207 --> 00:23:02,165 So this is just related to the chiral rotation. 351 00:23:06,980 --> 00:23:12,180 So if you look at gamma 5 times c and gamma 5 352 00:23:12,180 --> 00:23:16,950 in our representation would be 0, 1, 1 0. 353 00:23:16,950 --> 00:23:19,710 And then if I write it out in terms 354 00:23:19,710 --> 00:23:24,960 of this two-component thing, then that 355 00:23:24,960 --> 00:23:33,200 is just giving me 1 over root 2, and it's swapping up and down. 356 00:23:33,200 --> 00:23:38,750 And so that just means that phi n has gone to sigma 3 phi n. 357 00:23:38,750 --> 00:23:41,060 So multiplying by gamma 5 is actually the same 358 00:23:41,060 --> 00:23:45,840 as multiplying by sigma 3 in the two-component notation. 359 00:23:45,840 --> 00:23:46,340 OK? 360 00:23:46,340 --> 00:23:50,060 So this is not really new symmetry, and actually 361 00:23:50,060 --> 00:23:51,770 all the usual things that you would 362 00:23:51,770 --> 00:23:57,380 say about chiral rotations in QCD would apply here too. 363 00:23:57,380 --> 00:23:59,630 So chiral rotations, of course, are not 364 00:23:59,630 --> 00:24:02,270 exact chiral rotations are broken by masses. 365 00:24:02,270 --> 00:24:06,130 Chiral rotations are broken by non-perturbative effects. 366 00:24:06,130 --> 00:24:09,830 You can worry about anomalies. 367 00:24:09,830 --> 00:24:12,980 They are useful in perturbation theory for quantifying 368 00:24:12,980 --> 00:24:16,290 operators, and that remains true here, 369 00:24:16,290 --> 00:24:20,248 but if the collinear fields were non-perturbative, 370 00:24:20,248 --> 00:24:22,540 then you should worry about those other things as well. 371 00:24:25,700 --> 00:24:26,200 OK. 372 00:24:26,200 --> 00:24:30,250 So spin symmetry, there's not really anything new 373 00:24:30,250 --> 00:24:33,560 there to talk about, but along the way, 374 00:24:33,560 --> 00:24:35,560 we saw we could write SCET in two-component form 375 00:24:35,560 --> 00:24:36,477 which is kind of nice. 376 00:24:38,790 --> 00:24:41,615 So let's talk about something that's more important which 377 00:24:41,615 --> 00:24:42,365 is gauge symmetry. 378 00:24:44,930 --> 00:24:47,670 Is there any questions about the spin before we talk about 379 00:24:47,670 --> 00:24:48,295 gauge symmetry? 380 00:24:51,780 --> 00:24:56,945 AUDIENCE: So the extra term you got in determining the theta n 381 00:24:56,945 --> 00:25:01,460 bar, that is why you had the minus n slash [INAUDIBLE] 382 00:25:01,460 --> 00:25:02,960 and why you don't have the full SU2? 383 00:25:12,338 --> 00:25:13,005 PROFESSOR: Yeah. 384 00:25:18,040 --> 00:25:19,930 I looked at it once, but it's hard for me 385 00:25:19,930 --> 00:25:20,888 to remember the answer. 386 00:25:20,888 --> 00:25:23,868 I think not, but I think if you do the other version, 387 00:25:23,868 --> 00:25:25,910 then you just get something more complicated here 388 00:25:25,910 --> 00:25:28,720 but still breaks SU2. 389 00:25:28,720 --> 00:25:30,010 Yeah. 390 00:25:30,010 --> 00:25:30,910 AUDIENCE: Is it-- 391 00:25:30,910 --> 00:25:32,900 I'm just looking for [INAUDIBLE].. 392 00:25:34,250 --> 00:25:37,220 PROFESSOR: So they're two disconnected fact. 393 00:25:37,220 --> 00:25:40,008 The thing that I wanted to motivate 394 00:25:40,008 --> 00:25:41,800 that I could use this formula, and I wanted 395 00:25:41,800 --> 00:25:44,600 to be honest with you about where that came from, 396 00:25:44,600 --> 00:25:47,253 and that's why I told you this other fact. 397 00:25:47,253 --> 00:25:48,670 But I could have jumped right here 398 00:25:48,670 --> 00:25:51,147 and said, remember, and glossed over it, 399 00:25:51,147 --> 00:25:53,230 and that would have been fine for this discussion. 400 00:25:53,230 --> 00:25:55,180 So they're in some sense two disconnected things, at least 401 00:25:55,180 --> 00:25:55,690 in my mind. 402 00:25:57,910 --> 00:25:59,410 AUDIENCE: Do you have some intuition 403 00:25:59,410 --> 00:26:03,190 for what theta n bar is? 404 00:26:03,190 --> 00:26:05,040 PROFESSOR: For theta n bar? 405 00:26:05,040 --> 00:26:07,120 AUDIENCE: [INAUDIBLE] 406 00:26:07,120 --> 00:26:08,380 PROFESSOR: Oh. 407 00:26:08,380 --> 00:26:10,150 Yeah. 408 00:26:10,150 --> 00:26:12,700 It's really just saying-- 409 00:26:12,700 --> 00:26:14,320 my intuition for it is really just 410 00:26:14,320 --> 00:26:17,950 that, when you're producing energetic particles, 411 00:26:17,950 --> 00:26:19,510 it's what we did at the beginning. 412 00:26:19,510 --> 00:26:22,930 When you're producing energetic particles, 413 00:26:22,930 --> 00:26:25,990 these are the spin components that you have at lowest order, 414 00:26:25,990 --> 00:26:28,700 and that leads to, if you like-- 415 00:26:28,700 --> 00:26:30,560 the fact that you have this form, 416 00:26:30,560 --> 00:26:32,350 and you have this projection relation 417 00:26:32,350 --> 00:26:34,630 is kind of non-trivial, even though you 418 00:26:34,630 --> 00:26:37,030 don't have a spin symmetry. 419 00:26:37,030 --> 00:26:38,110 There is something to it. 420 00:26:38,110 --> 00:26:42,730 It's not like a SU2 symmetry of the spin, 421 00:26:42,730 --> 00:26:46,420 but it does, for example, when you go and look 422 00:26:46,420 --> 00:26:48,820 at form factors, it does lead to non-trivial relations 423 00:26:48,820 --> 00:26:50,390 for those form factors. 424 00:26:50,390 --> 00:26:51,970 So I don't call it a symmetry. 425 00:26:51,970 --> 00:26:55,590 I don't think of it as a symmetry, 426 00:26:55,590 --> 00:26:59,220 because it's not a formal group theory statement of Lagrangian, 427 00:26:59,220 --> 00:27:02,070 but I think it's basically-- 428 00:27:02,070 --> 00:27:03,150 well, OK. 429 00:27:03,150 --> 00:27:05,490 Let me not speculate any further. 430 00:27:10,110 --> 00:27:14,780 It caused some confusion in the literature actually as well. 431 00:27:14,780 --> 00:27:17,120 All right. 432 00:27:17,120 --> 00:27:20,780 So what about gauge symmetry? 433 00:27:20,780 --> 00:27:23,940 So we're doing a non-abelian gauge theory. 434 00:27:23,940 --> 00:27:30,070 A gauge transformation is something like this. 435 00:27:30,070 --> 00:27:32,273 And in order to just talk about gauge symmetry, what 436 00:27:32,273 --> 00:27:34,940 we need to do is we need to talk about the gauge symmetries that 437 00:27:34,940 --> 00:27:37,087 are related to the two type of gluons that we have. 438 00:27:37,087 --> 00:27:38,170 We have a collinear gluon. 439 00:27:38,170 --> 00:27:39,680 We have an ultrasoft gluon. 440 00:27:39,680 --> 00:27:43,790 You know that the field acting like the connection 441 00:27:43,790 --> 00:27:44,870 for that gauge symmetry. 442 00:27:44,870 --> 00:27:46,400 So there should be, in some sense, 443 00:27:46,400 --> 00:27:48,710 a gauge symmetry associated to the ultrasoft gluon 444 00:27:48,710 --> 00:27:50,870 and a separate one associated with collinear gluon, 445 00:27:50,870 --> 00:27:53,060 if the meaning of these things as gauge fields 446 00:27:53,060 --> 00:27:55,060 is going to have any sense to it. 447 00:27:55,060 --> 00:27:58,423 So rather than have just this set of gauge transformations 448 00:27:58,423 --> 00:27:59,840 that we can make in QCD, we should 449 00:27:59,840 --> 00:28:02,762 have some more complicated structure of gauge symmetry 450 00:28:02,762 --> 00:28:04,220 in the effective theory, because we 451 00:28:04,220 --> 00:28:06,303 have a more complicated structure of gauge fields. 452 00:28:12,610 --> 00:28:15,073 So in order to talk about U's, we're 453 00:28:15,073 --> 00:28:16,740 going to want to divide them into camps. 454 00:28:16,740 --> 00:28:19,530 And the useful thing to think about 455 00:28:19,530 --> 00:28:23,250 is that you want to have U's that, first of all, 456 00:28:23,250 --> 00:28:25,170 leave you inside the effective theory. 457 00:28:28,290 --> 00:28:30,180 And really what this means is that you 458 00:28:30,180 --> 00:28:32,400 want talking about gauge symmetry 459 00:28:32,400 --> 00:28:34,230 to commute with your discussion of talking 460 00:28:34,230 --> 00:28:37,270 about power counting. 461 00:28:37,270 --> 00:28:41,040 So by way of example of this, say 462 00:28:41,040 --> 00:28:48,350 you had a gauge transformation that was large, 463 00:28:48,350 --> 00:28:51,590 that made a formula like that one, that i partial on it 464 00:28:51,590 --> 00:28:53,194 was scaling like q. 465 00:28:53,194 --> 00:28:54,020 OK? 466 00:28:54,020 --> 00:28:57,270 So this would be a very large momentum associated 467 00:28:57,270 --> 00:29:00,560 to that spacetime dependence. 468 00:29:00,560 --> 00:29:03,110 Then, if you talked about the gauge transformation which 469 00:29:03,110 --> 00:29:09,530 would be U times c, even if this guy has collinear momentum 470 00:29:09,530 --> 00:29:13,460 scaling, the new guy would not because of this fact. 471 00:29:33,530 --> 00:29:34,130 OK? 472 00:29:34,130 --> 00:29:36,140 So if we want to talk about gauge symmetries, 473 00:29:36,140 --> 00:29:38,390 we want to talk about symmetries that take us to the fields 474 00:29:38,390 --> 00:29:40,580 that we've already decided to have a certain power counting. 475 00:29:40,580 --> 00:29:42,413 And we don't want the gauge symmetry to mess 476 00:29:42,413 --> 00:29:44,930 that up, so we're going to demand that that's true. 477 00:29:48,030 --> 00:29:50,570 So given that, we can think about dividing up the gauge 478 00:29:50,570 --> 00:29:52,550 symmetry into pieces, into pieces 479 00:29:52,550 --> 00:29:54,500 that have different scaling in terms 480 00:29:54,500 --> 00:29:56,750 of how big the transformation are, 481 00:29:56,750 --> 00:29:59,180 how the coordinates are behaving. 482 00:29:59,180 --> 00:30:02,510 And it's convenient for this discussion 483 00:30:02,510 --> 00:30:05,150 to divide them into three parts. 484 00:30:05,150 --> 00:30:09,090 A global transformation, where this guy is not 485 00:30:09,090 --> 00:30:14,930 spacetime dependent, a collinear transformation, 486 00:30:14,930 --> 00:30:22,430 which I'll call Uc, where i partial on Uc 487 00:30:22,430 --> 00:30:29,240 scales like a collinear momentum, 488 00:30:29,240 --> 00:30:41,688 and then an ultrasoft U Us, where the i partial scales like 489 00:30:41,688 --> 00:30:42,605 an ultrasoft momentum. 490 00:30:50,540 --> 00:30:51,920 And basically, what we want to do 491 00:30:51,920 --> 00:30:56,900 is we want to connect this gauge transformation to the gauge 492 00:30:56,900 --> 00:30:59,950 field for the collinears and this transformation 493 00:30:59,950 --> 00:31:01,325 to the gauge field for the softs. 494 00:31:08,310 --> 00:31:10,710 OK? 495 00:31:10,710 --> 00:31:13,500 So one complication that we have to deal with 496 00:31:13,500 --> 00:31:16,842 is this momentum labels that we were talking about. 497 00:31:16,842 --> 00:31:18,300 Because the way that we distinguish 498 00:31:18,300 --> 00:31:20,490 between these types of momenta and these types of momenta 499 00:31:20,490 --> 00:31:22,770 was dividing things into these labels and residuals. 500 00:31:26,540 --> 00:31:28,100 Gauge transformations are something 501 00:31:28,100 --> 00:31:30,200 that's very nice in position space but not so nice 502 00:31:30,200 --> 00:31:40,040 in momentum space, and that's because gauge transformations 503 00:31:40,040 --> 00:31:41,870 are local transformations. 504 00:31:41,870 --> 00:31:46,160 Local means local in x not in P. So 505 00:31:46,160 --> 00:31:48,890 what is simple multiplication in position space 506 00:31:48,890 --> 00:31:51,740 becomes convolutions in momentum space. 507 00:31:56,610 --> 00:32:06,800 So if you go over to this hybrid notation that we developed, 508 00:32:06,800 --> 00:32:08,556 then you would have the following. 509 00:32:11,820 --> 00:32:28,970 At this field, if I do a transformation, 510 00:32:28,970 --> 00:32:31,650 it goes into something like that. 511 00:32:31,650 --> 00:32:35,940 Where the way that you should think of what this sum over q 512 00:32:35,940 --> 00:32:37,890 is in the following way. 513 00:32:42,627 --> 00:32:44,710 But if you have a transformation in position space 514 00:32:44,710 --> 00:32:49,780 that looks like that, then in a momentum space, 515 00:32:49,780 --> 00:32:51,820 that simple product is convolution. 516 00:33:00,850 --> 00:33:01,570 OK? 517 00:33:01,570 --> 00:33:09,280 So this is one dimension, and you'd have that correspondence. 518 00:33:09,280 --> 00:33:12,668 And so this complication that we have here just correspond 519 00:33:12,668 --> 00:33:14,710 to the complication that you'd have for any gauge 520 00:33:14,710 --> 00:33:16,793 symmetry that you want to write in momentum space. 521 00:33:19,770 --> 00:33:23,130 But we can be slick about this and introduce a notation 522 00:33:23,130 --> 00:33:25,860 and then dispense with it, because it's really 523 00:33:25,860 --> 00:33:27,520 not a technical complication. 524 00:33:27,520 --> 00:33:30,300 It's just a notational complication. 525 00:33:30,300 --> 00:33:33,750 So let's do that in the following way. 526 00:33:33,750 --> 00:33:43,410 Let's let Uc which is a matrix with two indices 527 00:33:43,410 --> 00:33:49,030 be defined by this Uc of bl minus ql. 528 00:33:54,580 --> 00:33:58,280 So I'll say define a matrix. 529 00:33:58,280 --> 00:34:01,890 Then we can use a matrix notation for the convolutions, 530 00:34:01,890 --> 00:34:03,570 where we sum over repeated indices, 531 00:34:03,570 --> 00:34:08,256 and that takes care of this notational complication. 532 00:34:12,639 --> 00:34:19,900 So what is meant by this is that the Pl, ql entry of this matrix 533 00:34:19,900 --> 00:34:24,190 is a number, and that number is given by this thing 534 00:34:24,190 --> 00:34:27,790 that we call Uc Pl minus ql. 535 00:34:33,560 --> 00:34:34,060 OK. 536 00:34:34,060 --> 00:34:35,770 So then the formula up here could just 537 00:34:35,770 --> 00:34:37,449 be written in terms of that matrix 538 00:34:37,449 --> 00:34:39,512 as a kind of matrix multiplication. 539 00:34:44,060 --> 00:34:44,560 OK. 540 00:34:44,560 --> 00:34:47,320 So what are the transformations once we adopt that? 541 00:34:54,880 --> 00:34:57,053 Let me write them down and then talk about them. 542 00:35:01,000 --> 00:35:10,090 So cn of x goes to Uc of x cn of x, where you should understand 543 00:35:10,090 --> 00:35:15,290 this thing is a matrix that has two indices, like this, 544 00:35:15,290 --> 00:35:19,030 also dependent on spacetime. 545 00:35:19,030 --> 00:35:22,570 And I just dot over the repeated indices in this thing which 546 00:35:22,570 --> 00:35:24,072 is like a vector. 547 00:35:24,072 --> 00:35:28,420 So this is like a matrix, and this like a vector. 548 00:35:35,350 --> 00:35:39,310 That makes it look like it's the usual notation which 549 00:35:39,310 --> 00:35:42,430 is the point of introducing this notation. 550 00:35:55,800 --> 00:35:58,490 So for the collinear gauge field, 551 00:35:58,490 --> 00:36:00,530 it's going to be basically that it, 552 00:36:00,530 --> 00:36:03,020 as we said over here, that it becomes the gauge 553 00:36:03,020 --> 00:36:06,560 field of that transformation, and so that's 554 00:36:06,560 --> 00:36:07,370 what this would be. 555 00:36:07,370 --> 00:36:09,375 That would be the standard gauge transformation, 556 00:36:09,375 --> 00:36:11,750 and then there's one little complication here, which is I 557 00:36:11,750 --> 00:36:15,050 wrote curly D ultrasoft, the ultrasoft derivative, 558 00:36:15,050 --> 00:36:17,790 the thing that has the n dot a ultrasoft field in it. 559 00:36:23,277 --> 00:36:24,860 So the reason that I wanted to do that 560 00:36:24,860 --> 00:36:27,112 is because I'm thinking here of the ultrasoft field. 561 00:36:27,112 --> 00:36:28,820 When I'm thinking about collinear fields, 562 00:36:28,820 --> 00:36:32,190 I'm thinking of the ultrasoft field as a background field. 563 00:36:32,190 --> 00:36:35,420 And if I was to do a background field gauge transformation, 564 00:36:35,420 --> 00:36:37,100 then I would have a covariant derivative 565 00:36:37,100 --> 00:36:38,870 here under the background. 566 00:36:55,010 --> 00:36:58,390 So this n dot a ultrasoft is like a background field 567 00:36:58,390 --> 00:36:59,950 to the collinear gauge field. 568 00:37:04,390 --> 00:37:20,980 So this, if you want to make an analogy of what this is like, 569 00:37:20,980 --> 00:37:25,600 it's like the quantum gauge transformation of a field that 570 00:37:25,600 --> 00:37:26,995 has a background. 571 00:37:38,400 --> 00:37:40,290 If you look up how background field gauge 572 00:37:40,290 --> 00:37:44,007 works and background field transformations, 573 00:37:44,007 --> 00:37:45,090 you'd have the same thing. 574 00:37:45,090 --> 00:37:47,342 You'd have something covariant under the background, 575 00:37:47,342 --> 00:37:49,300 and that's what we're going to do here as well. 576 00:37:49,300 --> 00:37:50,400 Yep? 577 00:37:50,400 --> 00:37:52,030 AUDIENCE: Kind of a silly question. 578 00:37:52,030 --> 00:37:54,600 So UC of x is a matrix. 579 00:37:54,600 --> 00:37:57,220 So what exactly UC of p [INAUDIBLE]?? 580 00:38:00,217 --> 00:38:01,050 PROFESSOR: Well, no. 581 00:38:01,050 --> 00:38:02,830 This is just a number. 582 00:38:02,830 --> 00:38:06,580 So just think of this as like this is some vector, 583 00:38:06,580 --> 00:38:07,830 and this is some other vector. 584 00:38:07,830 --> 00:38:11,310 The difference of these are some label on this thing. 585 00:38:11,310 --> 00:38:11,920 Right? 586 00:38:11,920 --> 00:38:14,610 And for each value of this, I get a different number, 587 00:38:14,610 --> 00:38:16,450 and then for each value it's based on, 588 00:38:16,450 --> 00:38:18,283 I get a the different number, but this thing 589 00:38:18,283 --> 00:38:19,330 is just a number. 590 00:38:19,330 --> 00:38:22,470 And I'm saying think of it like a matrix in the sense of, 591 00:38:22,470 --> 00:38:25,320 because we are thinking of these as discrete-- 592 00:38:25,320 --> 00:38:27,907 AUDIENCE: But isn't UC [INAUDIBLE]?? 593 00:38:27,907 --> 00:38:28,740 PROFESSOR: Oh, yeah. 594 00:38:28,740 --> 00:38:29,240 Yeah. 595 00:38:29,240 --> 00:38:30,360 That's another matrix. 596 00:38:30,360 --> 00:38:31,690 It's an orthogonal matrix. 597 00:38:31,690 --> 00:38:33,120 It's also a matrix in that space. 598 00:38:33,120 --> 00:38:33,420 Yeah. 599 00:38:33,420 --> 00:38:33,620 Yeah. 600 00:38:33,620 --> 00:38:34,120 Yeah. 601 00:38:34,120 --> 00:38:37,860 So here, I'm defining a matrix in a new space which 602 00:38:37,860 --> 00:38:40,440 is this momentum space, and that's just really 603 00:38:40,440 --> 00:38:43,560 my way of making transformations in momentum space look 604 00:38:43,560 --> 00:38:47,340 like as simple as they do in position space 605 00:38:47,340 --> 00:38:51,148 by just having a matrix notation for the convolutions. 606 00:38:51,148 --> 00:38:53,190 AUDIENCE: So that number is essentially a matrix. 607 00:38:53,190 --> 00:38:56,280 PROFESSOR: It's a matrix, yeah, in that space. 608 00:38:56,280 --> 00:38:56,790 Good point. 609 00:39:01,780 --> 00:39:05,553 What about the ultrasoft fields? 610 00:39:05,553 --> 00:39:08,220 Well, ultrasoft fields shouldn't transform under collinear gauge 611 00:39:08,220 --> 00:39:13,120 transformations because of exactly the same logic 612 00:39:13,120 --> 00:39:15,280 that I gave you a minute ago about why I shouldn't 613 00:39:15,280 --> 00:39:17,710 be able to inject a hard momentum like this one 614 00:39:17,710 --> 00:39:19,840 into the effective theory. 615 00:39:19,840 --> 00:39:23,710 If I want my ultrasoft fields to remain ultrasoft, 616 00:39:23,710 --> 00:39:26,440 then I can't let them transform, because that would 617 00:39:26,440 --> 00:39:29,780 spoil their power counting. 618 00:39:29,780 --> 00:39:34,550 So q ultrasoft under this Uc transformation 619 00:39:34,550 --> 00:39:40,430 just goes to q ultrasoft, and A ultrasoft 620 00:39:40,430 --> 00:39:42,928 has to go to A ultrasoft. 621 00:39:42,928 --> 00:39:45,470 So these things should not be touched by that transformation. 622 00:39:45,470 --> 00:39:50,090 AUDIENCE: So why cannot make the ultrasoft and the collinear 623 00:39:50,090 --> 00:39:50,840 talk to the-- 624 00:39:50,840 --> 00:39:52,340 PROFESSOR: Well, we don't want that. 625 00:39:52,340 --> 00:39:55,130 You could think of trying to develop effective theory, where 626 00:39:55,130 --> 00:39:56,460 they would mix. 627 00:39:56,460 --> 00:39:56,960 Right? 628 00:39:56,960 --> 00:39:58,640 But whenever you have a Lagrangian that 629 00:39:58,640 --> 00:40:00,598 mixes two fields, the first thing you try to do 630 00:40:00,598 --> 00:40:02,900 is to force them not to mix. 631 00:40:02,900 --> 00:40:06,793 So it's better, say I set the theory up so that they mixed. 632 00:40:06,793 --> 00:40:08,960 The first thing I would try to do to that Lagrangian 633 00:40:08,960 --> 00:40:10,520 is make field redefinitions to go 634 00:40:10,520 --> 00:40:13,910 to some form of the Lagrangian that wouldn't mix. 635 00:40:13,910 --> 00:40:14,930 OK? 636 00:40:14,930 --> 00:40:19,675 So I prefer, if you like, to develop the one that 637 00:40:19,675 --> 00:40:21,050 doesn't mix right from the start, 638 00:40:21,050 --> 00:40:25,040 and you could view it that way. 639 00:40:25,040 --> 00:40:26,990 So you could always think of I'm going 640 00:40:26,990 --> 00:40:28,940 to tell you like two sets of transformations. 641 00:40:28,940 --> 00:40:31,378 You could always think of formulating linear combinations 642 00:40:31,378 --> 00:40:32,420 of those transformations. 643 00:40:32,420 --> 00:40:34,520 Say these are the fundamental ones, not the ones 644 00:40:34,520 --> 00:40:36,680 you wrote down but these other ones. 645 00:40:36,680 --> 00:40:37,580 Right? 646 00:40:37,580 --> 00:40:40,245 And then by some manipulations, one 647 00:40:40,245 --> 00:40:41,870 could go to the ones I'm talking about. 648 00:41:12,170 --> 00:41:12,670 All right. 649 00:41:16,540 --> 00:41:20,425 So now, the other set of transformations that we want 650 00:41:20,425 --> 00:41:24,640 are these ultrasoft ones, and for that type 651 00:41:24,640 --> 00:41:28,498 of transformation, we don't have the same type of problem 652 00:41:28,498 --> 00:41:29,290 with the collinear. 653 00:41:29,290 --> 00:41:32,380 We can't say the collinear fields don't transform, 654 00:41:32,380 --> 00:41:35,847 because that type of transformation 655 00:41:35,847 --> 00:41:37,930 doesn't spoil the collinear field being collinear. 656 00:41:37,930 --> 00:41:39,750 We can always add ultrasoft momentum to it. 657 00:41:43,410 --> 00:41:47,900 So these fields cn and An will transform 658 00:41:47,900 --> 00:41:51,410 like background fields-- 659 00:41:51,410 --> 00:41:54,336 sorry, like quantum fields in a background. 660 00:42:04,770 --> 00:42:06,980 So from the point of view of the cn fields 661 00:42:06,980 --> 00:42:08,870 which are the shorter distance fields, 662 00:42:08,870 --> 00:42:10,298 you can really think of like what 663 00:42:10,298 --> 00:42:11,840 we're doing here is exactly like what 664 00:42:11,840 --> 00:42:14,510 you would do if you had quantum and background gauge 665 00:42:14,510 --> 00:42:15,833 transformations. 666 00:42:19,220 --> 00:42:21,800 And the physics of thinking of the background as a longer 667 00:42:21,800 --> 00:42:24,380 wavelength mode is exactly the right way 668 00:42:24,380 --> 00:42:26,720 of thinking about how these fields which 669 00:42:26,720 --> 00:42:29,450 are shorter wavelength think about the ultrasoft fields. 670 00:42:37,382 --> 00:42:39,090 So what that means is that they transform 671 00:42:39,090 --> 00:42:41,460 like matter fields in the appropriate representation. 672 00:42:55,000 --> 00:42:55,500 OK? 673 00:42:55,500 --> 00:42:57,680 So with that in hand, we can write down 674 00:42:57,680 --> 00:43:00,680 how things transform under the ultrasofts. 675 00:43:08,060 --> 00:43:11,780 A Fermion just transforms like that on the left, 676 00:43:11,780 --> 00:43:14,270 and this, unlike our notation over there, 677 00:43:14,270 --> 00:43:18,440 this is one number for all entries in the vector. 678 00:43:21,093 --> 00:43:22,510 This guy doesn't have any indices. 679 00:43:30,572 --> 00:43:31,550 OK? 680 00:43:31,550 --> 00:43:34,480 So this is like transforming all the terms in the vector 681 00:43:34,480 --> 00:43:38,830 by the same overall color matrix, overall number 682 00:43:38,830 --> 00:43:41,140 in the momentum matrix base. 683 00:43:50,890 --> 00:43:54,310 So this guy should transform like an adjoint, 684 00:43:54,310 --> 00:43:58,960 and you can read the adjoint as a U on each side. 685 00:43:58,960 --> 00:44:01,660 So there's two ways of writing the transformation here. 686 00:44:01,660 --> 00:44:07,350 You could write it as like this. 687 00:44:07,350 --> 00:44:09,580 If you wrote down an adjoint transformation, 688 00:44:09,580 --> 00:44:16,490 then this guy would have two indices A and B, 689 00:44:16,490 --> 00:44:18,240 and you'd have a transformation like that, 690 00:44:18,240 --> 00:44:20,340 where this is in the adjoint. 691 00:44:20,340 --> 00:44:23,310 But if you want to write it in terms of the U ultrasoft 692 00:44:23,310 --> 00:44:25,920 that's e to the ita, which is in the fundamental, 693 00:44:25,920 --> 00:44:30,338 then you write it like this, and those are equivalent. 694 00:44:38,640 --> 00:44:40,980 So why are they equivalent? 695 00:44:40,980 --> 00:44:42,710 So they're equivalent because-- 696 00:44:42,710 --> 00:44:43,700 well, OK. 697 00:44:43,700 --> 00:44:46,090 Maybe I'll tell you later. 698 00:44:46,090 --> 00:44:46,590 Yeah? 699 00:44:49,980 --> 00:44:53,890 AUDIENCE: I'm always confused about why n dot is a background 700 00:44:53,890 --> 00:44:58,200 field to A ultrasoft, because it actually 701 00:44:58,200 --> 00:45:00,579 has the same size of momentum, so its wavelength 702 00:45:00,579 --> 00:45:01,940 is the same size. 703 00:45:01,940 --> 00:45:06,090 PROFESSOR: The wavelength is not the same size. 704 00:45:06,090 --> 00:45:08,430 The component's power counting is the same size, 705 00:45:08,430 --> 00:45:12,060 but the n dot An field carries momentum which have 706 00:45:12,060 --> 00:45:13,473 P squared which is much larger. 707 00:45:13,473 --> 00:45:14,140 AUDIENCE: Right. 708 00:45:14,140 --> 00:45:16,013 P squared [INAUDIBLE] wavelength, right? 709 00:45:16,013 --> 00:45:18,180 PROFESSOR: So I'm talking about the wavelength which 710 00:45:18,180 --> 00:45:19,660 is related to that P squared. 711 00:45:19,660 --> 00:45:22,920 So you think about the n dot An field carries a large P minus 712 00:45:22,920 --> 00:45:23,700 momentum still. 713 00:45:23,700 --> 00:45:24,200 Right? 714 00:45:24,200 --> 00:45:26,400 Even though it has a small-- 715 00:45:26,400 --> 00:45:30,220 it carries in order a 1 P minus momentum. 716 00:45:30,220 --> 00:45:32,280 So if it's confusing, think about just the P 717 00:45:32,280 --> 00:45:33,780 minus momentum. 718 00:45:33,780 --> 00:45:39,360 The n dot An field, right, has P minus, which is 719 00:45:39,360 --> 00:45:42,840 the what are lambda to the 0. 720 00:45:42,840 --> 00:45:45,030 So just let's think about wavelength with respect 721 00:45:45,030 --> 00:45:46,030 to P minus. 722 00:45:46,030 --> 00:45:46,980 AUDIENCE: That's definitely true, yeah. 723 00:45:46,980 --> 00:45:47,688 PROFESSOR: Right. 724 00:45:47,688 --> 00:45:52,770 Whereas the n dot A ultrasoft field carries P minus, which 725 00:45:52,770 --> 00:45:56,370 is what are lambda squared, so just thinking about those two. 726 00:45:56,370 --> 00:45:59,863 AUDIENCE: You were saying conflating [INAUDIBLE] 727 00:45:59,863 --> 00:46:00,530 PROFESSOR: Yeah. 728 00:46:00,530 --> 00:46:01,030 Yeah. 729 00:46:04,230 --> 00:46:05,670 All right. 730 00:46:05,670 --> 00:46:09,700 We have room here to squeeze the final transformation. 731 00:46:09,700 --> 00:46:12,810 So the final transformations are just what you'd expect, 732 00:46:12,810 --> 00:46:16,090 that the ultrasoft fields transform as regular gauge 733 00:46:16,090 --> 00:46:16,590 fields. 734 00:46:16,590 --> 00:46:19,220 So let me just squeeze that in. 735 00:46:39,590 --> 00:46:40,090 All right. 736 00:46:40,090 --> 00:46:41,740 So there's the regular transformations 737 00:46:41,740 --> 00:46:44,990 that you'd have for a gauge field. 738 00:46:44,990 --> 00:46:46,310 Yeah? 739 00:46:46,310 --> 00:46:49,790 AUDIENCE: So I just want to clarify. 740 00:46:49,790 --> 00:46:53,200 These two analogies for background field 741 00:46:53,200 --> 00:46:54,950 transformations, they're not disconnected. 742 00:46:54,950 --> 00:46:55,450 Right? 743 00:46:55,450 --> 00:46:57,020 PROFESSOR: They're not disconnected 744 00:46:57,020 --> 00:46:58,800 in the sense-- that's right. 745 00:46:58,800 --> 00:46:59,300 Yeah. 746 00:46:59,300 --> 00:47:02,168 So if you take the background field notation, 747 00:47:02,168 --> 00:47:03,710 and you just really think about that, 748 00:47:03,710 --> 00:47:05,960 then this is how you would transform the quantum 749 00:47:05,960 --> 00:47:08,630 field into the background gauge transformation. 750 00:47:08,630 --> 00:47:12,650 And this is how you would transform the quantum field 751 00:47:12,650 --> 00:47:15,840 under the quantum gauge transformation. 752 00:47:15,840 --> 00:47:16,340 Yeah. 753 00:47:16,340 --> 00:47:17,990 That's right. 754 00:47:17,990 --> 00:47:18,540 Yeah. 755 00:47:18,540 --> 00:47:22,550 AUDIENCE: So even if you made these to separate the gauge 756 00:47:22,550 --> 00:47:24,260 transformation for just [INAUDIBLE] 757 00:47:24,260 --> 00:47:26,443 they still have the same quantum numbers? 758 00:47:26,443 --> 00:47:27,110 PROFESSOR: Yeah. 759 00:47:27,110 --> 00:47:27,652 That's right. 760 00:47:27,652 --> 00:47:30,462 AUDIENCE: Are they going to meet at some level? 761 00:47:30,462 --> 00:47:32,420 PROFESSOR: We're going to make sure they don't. 762 00:47:32,420 --> 00:47:34,010 AUDIENCE: OK. 763 00:47:34,010 --> 00:47:34,770 PROFESSOR: Yeah. 764 00:47:34,770 --> 00:47:37,490 In some sense, we really want-- the picture is this one 765 00:47:37,490 --> 00:47:40,250 that we had earlier, where we really 766 00:47:40,250 --> 00:47:43,730 want the collinear fields to live up here, 767 00:47:43,730 --> 00:47:44,930 so we had these hyperbolas. 768 00:47:44,930 --> 00:47:45,240 Right? 769 00:47:45,240 --> 00:47:47,115 We want this guy to live up here and this guy 770 00:47:47,115 --> 00:47:51,380 to live down there in momentum space. 771 00:47:51,380 --> 00:47:53,660 So this was P minus. 772 00:47:53,660 --> 00:47:54,407 This is P plus. 773 00:47:54,407 --> 00:47:56,240 So we want them to live in different places. 774 00:47:56,240 --> 00:47:58,010 They have the same quantum numbers, 775 00:47:58,010 --> 00:48:00,770 except we want them in momentum space 776 00:48:00,770 --> 00:48:02,600 to live in different places and to describe 777 00:48:02,600 --> 00:48:04,550 different degrees of freedom. 778 00:48:04,550 --> 00:48:05,330 Right? 779 00:48:05,330 --> 00:48:07,760 So everything that we did here is exactly related 780 00:48:07,760 --> 00:48:10,520 to this picture, that we want to talk about this guy 781 00:48:10,520 --> 00:48:13,190 as having larger momentum, talk about gauge transformations 782 00:48:13,190 --> 00:48:15,600 that shuffle things around with components 783 00:48:15,600 --> 00:48:17,705 that have large momentum versus small momentum. 784 00:48:29,280 --> 00:48:31,260 All right. 785 00:48:31,260 --> 00:48:34,260 So we're going to hypothesize that these gauge 786 00:48:34,260 --> 00:48:36,470 transformations are the fundamental symmetry 787 00:48:36,470 --> 00:48:37,470 of the effective theory. 788 00:48:41,880 --> 00:48:43,950 And I tried to motivate them physically 789 00:48:43,950 --> 00:48:46,770 by this analogy with background field gauge 790 00:48:46,770 --> 00:48:49,410 and thinking about a longer wavelength mode with respect 791 00:48:49,410 --> 00:48:52,560 to a shorter wavelength mode which is in this picture 792 00:48:52,560 --> 00:48:54,390 exactly the fact that this hyperbola is 793 00:48:54,390 --> 00:48:56,820 higher than this one. 794 00:48:56,820 --> 00:49:00,350 Maybe the hyperbola should stay on the board. 795 00:49:00,350 --> 00:49:00,850 OK? 796 00:49:00,850 --> 00:49:04,680 So they multipole, where the multipole expansion 797 00:49:04,680 --> 00:49:07,770 that we did between these fields and the statement 798 00:49:07,770 --> 00:49:11,610 that this is a longer wavelength mode 799 00:49:11,610 --> 00:49:13,560 is exactly the fact that there's a separation 800 00:49:13,560 --> 00:49:14,430 between these guys. 801 00:49:32,730 --> 00:49:40,050 So we'll take these to be the fundamental transformations 802 00:49:40,050 --> 00:49:42,715 that do not get corrected by power corrections. 803 00:49:51,750 --> 00:49:53,910 And I'm not claiming that that's necessarily 804 00:49:53,910 --> 00:49:56,770 a unique way of getting to the effective theory. 805 00:49:56,770 --> 00:50:00,403 You could think of there might be 806 00:50:00,403 --> 00:50:02,820 an example in the literature of where people were thinking 807 00:50:02,820 --> 00:50:04,893 of other gauge transformations. 808 00:50:04,893 --> 00:50:06,810 These are the ones that people talk about now, 809 00:50:06,810 --> 00:50:08,352 but in the early days, I think, there 810 00:50:08,352 --> 00:50:09,935 is an example in the literature, where 811 00:50:09,935 --> 00:50:11,820 people are thinking of other transformations, 812 00:50:11,820 --> 00:50:13,278 and you could prove that they could 813 00:50:13,278 --> 00:50:16,060 be connected to these ones. 814 00:50:16,060 --> 00:50:19,343 But we'll just take the ones that 815 00:50:19,343 --> 00:50:20,760 are the most useful from the start 816 00:50:20,760 --> 00:50:22,968 and not worry too much about what we could have done. 817 00:50:27,450 --> 00:50:27,990 All right. 818 00:50:27,990 --> 00:50:31,200 So there's no mixing in the power counting, 819 00:50:31,200 --> 00:50:34,560 and that's something that we like. 820 00:50:34,560 --> 00:50:35,480 Set up that way. 821 00:50:47,990 --> 00:50:50,720 If there was a mixing, if we connected things 822 00:50:50,720 --> 00:50:53,210 that were different sizes in lambda, 823 00:50:53,210 --> 00:50:56,420 then you could say, well, that the power counting, 824 00:50:56,420 --> 00:50:59,150 that the gauge transformation induces power suppressed terms. 825 00:50:59,150 --> 00:51:00,817 And you could end up with some situation 826 00:51:00,817 --> 00:51:03,350 where you're trying to connect things at different orders, 827 00:51:03,350 --> 00:51:06,680 and we don't have that here. 828 00:51:11,005 --> 00:51:12,630 And that's because, if we want to think 829 00:51:12,630 --> 00:51:14,670 about this thing as a symmetry, it should really 830 00:51:14,670 --> 00:51:16,378 be a symmetry of the leading order action 831 00:51:16,378 --> 00:51:19,680 that shouldn't require some other sub-leading terms 832 00:51:19,680 --> 00:51:22,110 necessarily to compensate for it, at least 833 00:51:22,110 --> 00:51:24,360 for a gauge symmetry like this. 834 00:51:24,360 --> 00:51:27,060 We'd like to think that what the meaning of gauge symmetry is 835 00:51:27,060 --> 00:51:28,980 and the redundancy encoded in gauge symmetry 836 00:51:28,980 --> 00:51:30,180 is all something that you're talking 837 00:51:30,180 --> 00:51:31,710 about at the order of the leading order action. 838 00:51:31,710 --> 00:51:33,750 And you're not talking about redundancies 839 00:51:33,750 --> 00:51:35,900 in fields in sub-leading order, et cetera. 840 00:51:40,600 --> 00:51:41,100 OK. 841 00:51:41,100 --> 00:51:42,990 So let's do an example, now that we 842 00:51:42,990 --> 00:51:47,580 have this, of how it comes in. 843 00:51:47,580 --> 00:51:49,170 And I want to come back to our example 844 00:51:49,170 --> 00:51:54,170 of our heavy-to-light current and talk about gauge symmetry 845 00:51:54,170 --> 00:51:54,670 there. 846 00:51:59,290 --> 00:52:00,940 If you just took the tree level result 847 00:52:00,940 --> 00:52:05,750 that we had in that case, then we 848 00:52:05,750 --> 00:52:09,440 had something that just involved these two guys, 849 00:52:09,440 --> 00:52:12,800 and this hv field was ultrasoft. 850 00:52:12,800 --> 00:52:15,170 This guy was collinear. 851 00:52:15,170 --> 00:52:17,010 So if you did a Uc transformation, 852 00:52:17,010 --> 00:52:22,700 then under our rules, what you'd find is that the cn transforms, 853 00:52:22,700 --> 00:52:25,670 and this guy doesn't, the h doesn't. 854 00:52:25,670 --> 00:52:27,750 So that would not be gauge invariant. 855 00:52:39,020 --> 00:52:42,150 But actually, we saw that this also wasn't the lowest order 856 00:52:42,150 --> 00:52:42,650 current. 857 00:52:42,650 --> 00:52:45,350 There was this additional thing, the Wilson line showed up, 858 00:52:45,350 --> 00:52:47,660 and that Wilson line is going to fix this issue. 859 00:52:47,660 --> 00:52:49,367 It's going to make it gauge invariant. 860 00:52:54,240 --> 00:52:56,260 So we have to ask, how do Wilson lines transform 861 00:52:56,260 --> 00:52:57,695 under gauge symmetries? 862 00:53:05,940 --> 00:53:10,390 And that Wilson line in position space, 863 00:53:10,390 --> 00:53:13,620 I called it something like this. 864 00:53:13,620 --> 00:53:15,700 I can't remember if I used a twiddle or a bar. 865 00:53:15,700 --> 00:53:18,430 We'll use a bar here for the position space Wilson line. 866 00:53:21,070 --> 00:53:24,880 So in general, in just a regular gauge theory, 867 00:53:24,880 --> 00:53:28,360 the Wilson line between two points 868 00:53:28,360 --> 00:53:31,060 transforms at the endpoints. 869 00:53:31,060 --> 00:53:32,380 So you get U of x. 870 00:53:35,710 --> 00:53:37,015 Then, you get a U dagger of y. 871 00:53:41,340 --> 00:53:43,560 So if you like, what we would say 872 00:53:43,560 --> 00:53:46,350 for this Wilson line is that we have a U of x on one side 873 00:53:46,350 --> 00:53:48,590 and a U dagger of minus infinity on the other side. 874 00:53:57,120 --> 00:54:01,650 Minus infinity is really long wavelength physics, 875 00:54:01,650 --> 00:54:03,420 and so we have to worry a little bit 876 00:54:03,420 --> 00:54:06,600 about the overlap between our different types of gauge 877 00:54:06,600 --> 00:54:09,860 transformations. 878 00:54:09,860 --> 00:54:13,340 I divided them into three camps-- collinear, ultrasoft, 879 00:54:13,340 --> 00:54:18,006 and the global, so let me do the following. 880 00:54:29,920 --> 00:54:33,760 To avoid some double counting with what I called U global, 881 00:54:33,760 --> 00:54:38,290 let's just simply take, since U global acts the same 882 00:54:38,290 --> 00:54:41,950 everywhere, let's just simply take Uc dagger at minus 883 00:54:41,950 --> 00:54:44,251 infinity to be 1. 884 00:54:44,251 --> 00:54:46,810 So this is at some particular spacetime point. 885 00:54:46,810 --> 00:54:50,140 We take it to be 1, and this U global 886 00:54:50,140 --> 00:54:51,530 will transform at that point. 887 00:54:51,530 --> 00:54:56,400 We then know there's no overlap between Uc and U global. 888 00:54:56,400 --> 00:54:57,930 OK? 889 00:54:57,930 --> 00:55:01,890 Then, that's enough to ensure that. 890 00:55:01,890 --> 00:55:07,290 So with that, our guy under a collinear gauge transformation 891 00:55:07,290 --> 00:55:08,700 just transforms on one side. 892 00:55:12,085 --> 00:55:14,460 If you like, what's happening is that the coordinate here 893 00:55:14,460 --> 00:55:17,580 corresponds to this very large momentum, 894 00:55:17,580 --> 00:55:22,200 and we're stretching ourselves out to long distance 895 00:55:22,200 --> 00:55:25,450 physics which is corresponding to the smaller momentum. 896 00:55:25,450 --> 00:55:27,660 So when I think about transformations 897 00:55:27,660 --> 00:55:31,170 that should be collinear, I want them to be associated to the x. 898 00:55:31,170 --> 00:55:33,420 And I want to associate other types of transformations 899 00:55:33,420 --> 00:55:35,190 to what's going on at infinity which is really 900 00:55:35,190 --> 00:55:37,857 a spacetime of its own, which is this ultrasoft spacetime that's 901 00:55:37,857 --> 00:55:41,680 sitting at the collinear infinity. 902 00:55:41,680 --> 00:55:44,318 So that's why I want, actually, it to look like this. 903 00:55:44,318 --> 00:55:48,878 AUDIENCE: [INAUDIBLE] just with the collinear fields? 904 00:55:48,878 --> 00:55:49,545 PROFESSOR: Yeah. 905 00:55:49,545 --> 00:55:50,087 AUDIENCE: OK. 906 00:55:50,087 --> 00:55:50,840 PROFESSOR: Yeah. 907 00:55:50,840 --> 00:55:55,084 So this guy here is a path-ordered exponential, 908 00:55:55,084 --> 00:56:01,640 and it just involved the n bar guy. 909 00:56:01,640 --> 00:56:02,140 Yeah. 910 00:56:05,270 --> 00:56:07,640 So that's what it was in position space, 911 00:56:07,640 --> 00:56:13,250 and in momentum space, they called it W, 912 00:56:13,250 --> 00:56:16,770 and it was a sum over a bunch of things. 913 00:56:21,940 --> 00:56:22,690 I'm not going to-- 914 00:56:27,540 --> 00:56:30,450 we had this formula. 915 00:56:30,450 --> 00:56:31,940 Let me not write it all out again. 916 00:56:38,210 --> 00:56:41,520 As a short form for that formula, 917 00:56:41,520 --> 00:56:46,060 we can write the following formula 918 00:56:46,060 --> 00:56:47,850 which is given our notation that we've 919 00:56:47,850 --> 00:56:52,540 developed a useful way of thinking about it. 920 00:56:52,540 --> 00:56:55,680 So if we formally expand out this exponential, 921 00:56:55,680 --> 00:56:58,350 and we just keep the order of that expansion, where the P 922 00:56:58,350 --> 00:57:01,290 bar acts on all the fields to the right, that will correctly 923 00:57:01,290 --> 00:57:04,770 reproduce this denominator, the sum over permutations 924 00:57:04,770 --> 00:57:06,990 is something that we still have to do. 925 00:57:06,990 --> 00:57:12,280 It was 1 over n factorial, k factorial in here. 926 00:57:12,280 --> 00:57:13,735 M factorial with this notation. 927 00:57:16,130 --> 00:57:16,630 OK? 928 00:57:16,630 --> 00:57:18,160 But this is like a little shorthand 929 00:57:18,160 --> 00:57:20,326 for that messy formula. 930 00:57:20,326 --> 00:57:21,730 AUDIENCE: [INAUDIBLE] 931 00:57:21,730 --> 00:57:22,550 PROFESSOR: Yeah? 932 00:57:22,550 --> 00:57:25,310 AUDIENCE: I think it's related to the gauge transformation 933 00:57:25,310 --> 00:57:26,230 equation before. 934 00:57:26,230 --> 00:57:27,520 PROFESSOR: Yeah. 935 00:57:27,520 --> 00:57:30,958 AUDIENCE: How do you get the i epsilon prescription? 936 00:57:30,958 --> 00:57:32,837 [INAUDIBLE] 937 00:57:32,837 --> 00:57:35,170 PROFESSOR: So that actually is related to the fact that, 938 00:57:35,170 --> 00:57:36,940 when you look at the-- 939 00:57:36,940 --> 00:57:39,460 for this Wilson line with the i epsilon, 940 00:57:39,460 --> 00:57:43,310 it's not related to the gauge transformation at infinity, 941 00:57:43,310 --> 00:57:46,040 but it's related to the fact that the path goes that way. 942 00:57:46,040 --> 00:57:49,060 So that when you do the Fourier integral, 943 00:57:49,060 --> 00:57:52,750 the fact that it goes from minus infinity to x will give the i0. 944 00:57:52,750 --> 00:57:53,710 OK? 945 00:57:53,710 --> 00:57:56,952 But for these Wilson lines here, if you 946 00:57:56,952 --> 00:57:58,660 think about what's happening at the place 947 00:57:58,660 --> 00:58:00,970 where the i0 would actually matter, 948 00:58:00,970 --> 00:58:03,820 that's exactly the 0 bin. 949 00:58:03,820 --> 00:58:05,290 So it's going to be a little more-- 950 00:58:05,290 --> 00:58:05,790 Yeah. 951 00:58:05,790 --> 00:58:08,308 So you don't really care. 952 00:58:08,308 --> 00:58:09,260 AUDIENCE: OK. 953 00:58:09,260 --> 00:58:12,120 PROFESSOR: Yeah. 954 00:58:12,120 --> 00:58:13,320 I can only say that-- 955 00:58:13,320 --> 00:58:15,960 we'll explain what I just said to Ilia more 956 00:58:15,960 --> 00:58:17,650 when we come to it later on. 957 00:58:21,160 --> 00:58:21,660 OK. 958 00:58:21,660 --> 00:58:24,690 So this field here has-- 959 00:58:24,690 --> 00:58:30,510 I said, it's a momentum space, but then I wrote x. 960 00:58:30,510 --> 00:58:32,310 So really what I mean by momentum space 961 00:58:32,310 --> 00:58:36,310 is momentum is in the label momentum space. 962 00:58:36,310 --> 00:58:39,290 There's some guy, and the labels are these labels on the fields 963 00:58:39,290 --> 00:58:39,790 here. 964 00:58:39,790 --> 00:58:40,590 Right? 965 00:58:40,590 --> 00:58:42,975 So there's still an x, and that x corresponds 966 00:58:42,975 --> 00:58:44,850 to the residual coordinate that we were using 967 00:58:44,850 --> 00:58:47,110 to do the multipole expansion. 968 00:58:47,110 --> 00:58:49,980 So in some sense, this isn't simply the Fourier transform 969 00:58:49,980 --> 00:58:52,170 of that, because of that. 970 00:58:52,170 --> 00:58:54,300 It's still knows about the separation 971 00:58:54,300 --> 00:58:56,550 between the large momentum and the soft momentum 972 00:58:56,550 --> 00:58:59,680 from that multipole expansion. 973 00:58:59,680 --> 00:59:01,350 So this dependence on x allows us 974 00:59:01,350 --> 00:59:04,830 to think about the multipole expansion on these Wilson 975 00:59:04,830 --> 00:59:06,990 lines. 976 00:59:06,990 --> 00:59:10,230 It encodes all the residual momenta, 977 00:59:10,230 --> 00:59:12,930 and the fields in the Wilson line carry a residual momenta. 978 00:59:16,020 --> 00:59:18,720 So we're just putting our notation to use here. 979 00:59:25,523 --> 00:59:27,940 And if you really strictly want to connect to that formula 980 00:59:27,940 --> 00:59:33,630 up there, then you would take the residual momenta to be 0. 981 00:59:33,630 --> 00:59:40,350 And then the Fourier transform with respect 982 00:59:40,350 --> 00:59:46,560 to just the large momentum Pl minus 983 00:59:46,560 --> 00:59:51,820 gives a line which is just depending 984 00:59:51,820 --> 00:59:53,629 on that one single coordinate. 985 01:00:02,610 --> 01:00:04,110 So if we just had a coordinate which 986 01:00:04,110 --> 01:00:05,910 is conjugate to the large momenta, which 987 01:00:05,910 --> 01:00:07,920 is the thing that's showing up here 988 01:00:07,920 --> 01:00:11,990 are the large momenta, then that would be the analog of what 989 01:00:11,990 --> 01:00:13,240 we were talking about up here. 990 01:00:13,240 --> 01:00:15,090 Where in general, this thing also 991 01:00:15,090 --> 01:00:17,660 has a dependence on the residual x, which 992 01:00:17,660 --> 01:00:18,947 we'll make use of in a minute. 993 01:00:24,420 --> 01:00:24,920 OK. 994 01:00:24,920 --> 01:00:32,350 So let's think about, so I told you 995 01:00:32,350 --> 01:00:35,200 how this Wilson line transforms under Uc see. 996 01:00:35,200 --> 01:00:40,120 How does it transform under U ultrasoft? 997 01:00:40,120 --> 01:00:49,410 So under Uc, again using this matrix-type notation, 998 01:00:49,410 --> 01:00:50,635 it's just Uc. 999 01:00:57,110 --> 01:00:58,610 So this is a matrix multiplication 1000 01:00:58,610 --> 01:00:59,600 in momentum space. 1001 01:01:07,850 --> 01:01:12,350 What about under an ultrasoft transformation? 1002 01:01:12,350 --> 01:01:14,390 Well, under an ultrasoft transformation, 1003 01:01:14,390 --> 01:01:15,850 this should be a derived-- 1004 01:01:15,850 --> 01:01:17,850 in some sense, under a collinear transformation, 1005 01:01:17,850 --> 01:01:19,600 it was a derived quantity, because we knew 1006 01:01:19,600 --> 01:01:20,948 of the gauge field transform. 1007 01:01:20,948 --> 01:01:23,240 The thing that gave us the transformation of the Wilson 1008 01:01:23,240 --> 01:01:25,680 line is just the transformation of the gauge field. 1009 01:01:25,680 --> 01:01:26,180 OK? 1010 01:01:26,180 --> 01:01:29,030 So that's what led to, actually, this form, once we 1011 01:01:29,030 --> 01:01:30,240 put in this fact. 1012 01:01:30,240 --> 01:01:31,970 So under the ultrasoft transformation, 1013 01:01:31,970 --> 01:01:34,553 we just take the transformation that we have for the collinear 1014 01:01:34,553 --> 01:01:36,530 gauge field, and then we derive how 1015 01:01:36,530 --> 01:01:41,360 the Wilson line should transform under the ultrasoft gauge 1016 01:01:41,360 --> 01:01:42,510 symmetry. 1017 01:01:42,510 --> 01:01:43,010 OK? 1018 01:01:43,010 --> 01:01:45,800 So how does this guy transform? 1019 01:01:45,800 --> 01:01:49,160 This guy transforms by putting U's on the left 1020 01:01:49,160 --> 01:01:57,190 and the right that are at x. 1021 01:01:57,190 --> 01:01:58,840 But every field in this exponential 1022 01:01:58,840 --> 01:02:02,260 has the same, U U dagger, multiply it, 1023 01:02:02,260 --> 01:02:04,120 think about expanding that out. 1024 01:02:04,120 --> 01:02:06,470 All the U U daggers that are next to each other cancel. 1025 01:02:06,470 --> 01:02:08,595 You just get an overall U U dagger that comes right 1026 01:02:08,595 --> 01:02:10,670 outside the exponential. 1027 01:02:10,670 --> 01:02:16,960 So this guy is just transforming with U ultrasoft of x. 1028 01:02:16,960 --> 01:02:23,440 Wilson line back again, U ultrasoft dagger of x, 1029 01:02:23,440 --> 01:02:26,200 and that's because this Wilson line with our notation 1030 01:02:26,200 --> 01:02:29,470 is really a local thing with respect to the coordinate x. 1031 01:02:29,470 --> 01:02:31,640 All the fields sit at x. 1032 01:02:31,640 --> 01:02:32,140 OK? 1033 01:02:32,140 --> 01:02:35,500 So it just transforms like that, because that's 1034 01:02:35,500 --> 01:02:39,359 how the gauge field inside it transforms. 1035 01:02:48,840 --> 01:02:50,310 So in some sense, both of these are 1036 01:02:50,310 --> 01:02:52,030 derived from the transformation of An, 1037 01:02:52,030 --> 01:02:56,277 but it's easier to think about how a Wilson line transforms 1038 01:02:56,277 --> 01:02:58,110 and then think about how it should transform 1039 01:02:58,110 --> 01:03:00,410 under collinear than just go through the An route 1040 01:03:00,410 --> 01:03:03,070 and impose this condition. 1041 01:03:03,070 --> 01:03:03,570 OK. 1042 01:03:03,570 --> 01:03:10,020 So let's come back now to our full current which was this, 1043 01:03:10,020 --> 01:03:13,590 that had the Wilson line in it. 1044 01:03:13,590 --> 01:03:16,410 And now if you ask how it transforms under Uc, 1045 01:03:16,410 --> 01:03:19,560 you see what's going to happen. 1046 01:03:19,560 --> 01:03:24,720 You get Uc dagger, Uc, and the ultrasoft guy 1047 01:03:24,720 --> 01:03:27,720 doesn't transform, and then these cancel. 1048 01:03:27,720 --> 01:03:31,050 So it's invariant under the transformations 1049 01:03:31,050 --> 01:03:33,380 that have momenta of order of the ultrasoft. 1050 01:03:33,380 --> 01:03:35,610 It'll have a momentum of what are the linear scale. 1051 01:03:35,610 --> 01:03:38,070 There's two objects in this thing that have momenta 1052 01:03:38,070 --> 01:03:39,210 that can be of that size. 1053 01:03:39,210 --> 01:03:42,180 The gluons, they're in W in this collinear quark field. 1054 01:03:42,180 --> 01:03:46,950 Both of those things transform, but that transformation 1055 01:03:46,950 --> 01:03:48,826 cancels between the two things. 1056 01:03:48,826 --> 01:03:51,310 AUDIENCE: [INAUDIBLE] 1057 01:03:51,310 --> 01:03:54,040 PROFESSOR: Whoops, that's important. 1058 01:03:54,040 --> 01:03:57,040 Thank you. 1059 01:03:57,040 --> 01:04:01,330 And then under an ultrasoft transformation, 1060 01:04:01,330 --> 01:04:03,610 we can also transform this guy, and then everybody's 1061 01:04:03,610 --> 01:04:04,853 transforming. 1062 01:04:12,250 --> 01:04:19,330 Gamma, of course, has not got color indices, 1063 01:04:19,330 --> 01:04:21,740 and again, it's gauge invariant. 1064 01:04:21,740 --> 01:04:22,240 OK? 1065 01:04:26,668 --> 01:04:27,780 Missed a gamma. 1066 01:04:30,910 --> 01:04:34,970 Gamma's not so important. 1067 01:04:34,970 --> 01:04:35,470 All right? 1068 01:04:35,470 --> 01:04:37,990 So this current that we derived earlier 1069 01:04:37,990 --> 01:04:40,690 is actually invariant under two different types of symmetries 1070 01:04:40,690 --> 01:04:44,230 which are really momentum space splitting of a standard gauge 1071 01:04:44,230 --> 01:04:47,380 symmetry into pieces where we have a large momenta and pieces 1072 01:04:47,380 --> 01:04:49,390 where you have these larger momenta, collinear 1073 01:04:49,390 --> 01:04:51,370 momenta and pieces where we have a smaller momenta, 1074 01:04:51,370 --> 01:04:52,360 this ultrasoft momenta. 1075 01:04:52,360 --> 01:04:53,777 But both of these types of momenta 1076 01:04:53,777 --> 01:04:57,790 are momenta that we're allowing inside the effective theory. 1077 01:04:57,790 --> 01:04:58,440 Yeah? 1078 01:04:58,440 --> 01:04:59,815 AUDIENCE: Just trying to remember 1079 01:04:59,815 --> 01:05:03,080 that the ultrasoft scale was the same for the [INAUDIBLE].. 1080 01:05:03,080 --> 01:05:04,180 PROFESSOR: Yeah. 1081 01:05:04,180 --> 01:05:07,770 The momentum of the ultrasoft-- 1082 01:05:07,770 --> 01:05:10,610 the ultrasoft momentum of the collinear 1083 01:05:10,610 --> 01:05:12,700 quark, you pulled out mv, and then 1084 01:05:12,700 --> 01:05:15,580 the remaining part was ultrasoft in the discussion 1085 01:05:15,580 --> 01:05:16,780 for this particular thing. 1086 01:05:20,150 --> 01:05:20,710 All right. 1087 01:05:24,650 --> 01:05:26,720 So it's the Wilson line, the moral of this 1088 01:05:26,720 --> 01:05:29,180 is that what we were doing before, where we're integrating 1089 01:05:29,180 --> 01:05:31,460 out those collinear gluons, is actually 1090 01:05:31,460 --> 01:05:32,660 connected to gauge symmetry. 1091 01:05:32,660 --> 01:05:35,180 Because without that Wilson line, 1092 01:05:35,180 --> 01:05:38,518 we wouldn't have a result that's invariant under the type 1093 01:05:38,518 --> 01:05:40,310 of gauge symmetry we've been talking about. 1094 01:05:45,540 --> 01:05:50,120 So this Wilson line carries these n collinear gluons, 1095 01:05:50,120 --> 01:05:52,910 and those gluons would be the gluons that in the full theory 1096 01:05:52,910 --> 01:06:01,910 would give you a gauge symmetric result. They would combine. 1097 01:06:01,910 --> 01:06:04,550 They would be the attachments to the heavy quark that 1098 01:06:04,550 --> 01:06:06,560 combined with attachments to the light quark 1099 01:06:06,560 --> 01:06:09,878 to give you a result that's invariant, 1100 01:06:09,878 --> 01:06:11,795 if you looked at word identities, for example. 1101 01:06:18,100 --> 01:06:20,320 But in the effective theory, these are just 1102 01:06:20,320 --> 01:06:23,650 put into a Wilson line, because that's the leading order 1103 01:06:23,650 --> 01:06:26,080 way that they can show up, but still we 1104 01:06:26,080 --> 01:06:29,680 have a gauge invariant answer. 1105 01:06:29,680 --> 01:06:34,150 So the Wilson line is really needed for the gauge symmetry. 1106 01:06:34,150 --> 01:06:39,490 And finally, the ultrasofts, as can be clear from the notation, 1107 01:06:39,490 --> 01:06:41,290 the ultrasofts can just as well include 1108 01:06:41,290 --> 01:06:46,680 the global transformation, if you like. 1109 01:06:46,680 --> 01:06:47,930 So we don't really need three. 1110 01:06:47,930 --> 01:06:50,140 We could put the global back with the ultrasofts. 1111 01:06:57,450 --> 01:06:59,970 Then, we just have two. 1112 01:06:59,970 --> 01:07:01,590 All right. 1113 01:07:01,590 --> 01:07:03,030 So that's our symmetry. 1114 01:07:03,030 --> 01:07:06,000 That's an example of how it works. 1115 01:07:06,000 --> 01:07:08,415 Now, let's talk about covariant derivatives. 1116 01:07:29,710 --> 01:07:32,410 So gauge symmetry ties together regular derivatives 1117 01:07:32,410 --> 01:07:36,090 with the gauge field, and here, what it ties together 1118 01:07:36,090 --> 01:07:37,090 at the following things. 1119 01:07:44,630 --> 01:07:46,980 So because of the treatment of the A ultrasoft 1120 01:07:46,980 --> 01:07:48,980 field as a background, it actually ties together 1121 01:07:48,980 --> 01:07:49,910 both of these things. 1122 01:08:03,910 --> 01:08:07,750 And then it does what you expect in a collinear sector. 1123 01:08:07,750 --> 01:08:11,620 It ties together the derivative with the gauge field. 1124 01:08:15,660 --> 01:08:17,819 And if you're acting on ultrasoft fields, 1125 01:08:17,819 --> 01:08:20,620 then there's an ultrasoft covariant derivative 1126 01:08:20,620 --> 01:08:32,510 which is just like that. 1127 01:08:35,920 --> 01:08:37,979 So now, you can ask the question if I just 1128 01:08:37,979 --> 01:08:39,660 had this gauge symmetry, which is going 1129 01:08:39,660 --> 01:08:42,038 to force me to use covariant derivatives-- which I was 1130 01:08:42,038 --> 01:08:43,830 already thinking ahead to when I wrote down 1131 01:08:43,830 --> 01:08:45,439 the Lagrangian earlier. 1132 01:08:45,439 --> 01:08:45,939 Right? 1133 01:08:45,939 --> 01:08:48,420 We already wrote it in terms of these kind of objects, 1134 01:08:48,420 --> 01:08:49,878 so it seems like we're good to go. 1135 01:08:49,878 --> 01:08:51,420 So you can ask the question if I just 1136 01:08:51,420 --> 01:08:55,080 have power counting, the cn that we 1137 01:08:55,080 --> 01:08:58,590 decided we're going to use, and gauge symmetry, is that enough? 1138 01:08:58,590 --> 01:08:59,250 Am I done? 1139 01:08:59,250 --> 01:09:02,053 Do I have everything I need for the leading order Lagrangian? 1140 01:09:24,760 --> 01:09:29,200 So power counting told us that we had this type of derivative, 1141 01:09:29,200 --> 01:09:35,810 and that we could have two of these perp derivatives. 1142 01:09:35,810 --> 01:09:36,310 OK? 1143 01:09:36,310 --> 01:09:38,435 So those are the two type of terms we were getting. 1144 01:09:44,590 --> 01:09:51,399 There's no other order lambda squared field structures that 1145 01:09:51,399 --> 01:09:58,100 would have the right dimensions and that could do the job. 1146 01:09:58,100 --> 01:10:01,580 So these are really the only two that we have to think about. 1147 01:10:01,580 --> 01:10:03,130 So you could try to think about maybe 1148 01:10:03,130 --> 01:10:06,880 I can have something with some additional n bars floating 1149 01:10:06,880 --> 01:10:10,450 around, but really, it boils down-- 1150 01:10:10,450 --> 01:10:12,460 since you need to get a lambda squared, 1151 01:10:12,460 --> 01:10:15,940 you need either two perp derivatives. 1152 01:10:15,940 --> 01:10:18,310 You need two of them, because you still have a rotation 1153 01:10:18,310 --> 01:10:22,690 symmetry around the direction of motion, 1154 01:10:22,690 --> 01:10:24,970 or you could have an n dot D. 1155 01:10:24,970 --> 01:10:35,110 But there is one operator that is not rolled out, 1156 01:10:35,110 --> 01:10:40,860 by gauge symmetry alone, and that is this operator. 1157 01:10:46,470 --> 01:10:53,325 When we wrote the operator, we had D perp slash D perp slash. 1158 01:10:53,325 --> 01:10:56,370 But nothing stops me from doing something similar in some way 1159 01:10:56,370 --> 01:11:02,590 by just taking the D perps and contracting them, like that. 1160 01:11:02,590 --> 01:11:05,290 That's a different operator than the D perp slash D perp 1161 01:11:05,290 --> 01:11:08,380 operator that we got before. 1162 01:11:08,380 --> 01:11:12,140 Oops, only one of those. 1163 01:11:12,140 --> 01:11:12,640 OK? 1164 01:11:12,640 --> 01:11:14,230 So this operator is a different operator. 1165 01:11:14,230 --> 01:11:15,640 In a priori, you could say, well, 1166 01:11:15,640 --> 01:11:17,800 do loop connections generate this operator which 1167 01:11:17,800 --> 01:11:22,420 is different than the D perp slash D perp slash operator. 1168 01:11:22,420 --> 01:11:26,743 The answer is no, actually, but so far what we've done, 1169 01:11:26,743 --> 01:11:27,910 it's not enough to see that. 1170 01:11:38,293 --> 01:11:40,710 So we need actually this other symmetry that I alluded to, 1171 01:11:40,710 --> 01:11:43,230 the reparameterization invariance, 1172 01:11:43,230 --> 01:11:45,000 and that will actually rule out this guy. 1173 01:11:49,370 --> 01:11:51,578 So what's RPI? 1174 01:11:51,578 --> 01:11:53,120 Well, when we formulated this theory, 1175 01:11:53,120 --> 01:11:55,280 we needed a direction for the collinear line 1176 01:11:55,280 --> 01:11:57,380 or for the collinear particles. 1177 01:11:57,380 --> 01:11:59,990 And then we needed an auxiliary vector n bar, 1178 01:11:59,990 --> 01:12:01,550 and we formulated the whole set up 1179 01:12:01,550 --> 01:12:03,710 in terms of this n and n bar. 1180 01:12:03,710 --> 01:12:06,320 But having vectors like that that we write down, 1181 01:12:06,320 --> 01:12:09,140 that breaks Lorentz invariance in the same way that 1182 01:12:09,140 --> 01:12:16,820 specifying v mu in HQET breaks Lorentz invariance. 1183 01:12:16,820 --> 01:12:19,240 So if you say that m mu nu is the set of Lorentz 1184 01:12:19,240 --> 01:12:21,760 transformations, the usual six generators that 1185 01:12:21,760 --> 01:12:26,800 are anti-symmetric, then the ones that are broken 1186 01:12:26,800 --> 01:12:27,730 are these ones. 1187 01:12:35,550 --> 01:12:44,160 And there's five of them, and the one that's not broken 1188 01:12:44,160 --> 01:12:49,260 is this one which corresponds to rotations about the three, 1189 01:12:49,260 --> 01:12:52,110 axis with the axis specified by n. 1190 01:13:07,210 --> 01:13:07,710 OK? 1191 01:13:07,710 --> 01:13:11,190 So those rotations would act in the components of these guys 1192 01:13:11,190 --> 01:13:12,840 that are 0, if you like. 1193 01:13:12,840 --> 01:13:14,850 So there's no issue there, and these guys 1194 01:13:14,850 --> 01:13:17,940 are the guys that are connected to the pieces that 1195 01:13:17,940 --> 01:13:21,160 were non-0 in general. 1196 01:13:21,160 --> 01:13:23,550 So there's going to be a larger, because there's 1197 01:13:23,550 --> 01:13:26,580 five things here and because there's two vectors, 1198 01:13:26,580 --> 01:13:29,250 it's going to be a larger set of reparameterization symmetries 1199 01:13:29,250 --> 01:13:29,792 than in HQET. 1200 01:13:36,360 --> 01:13:40,120 First, talk about just the n and n bar themselves. 1201 01:13:40,120 --> 01:13:42,570 There's three types of reparameterization invariance 1202 01:13:42,570 --> 01:13:48,900 that would leave n squared equals 0 1203 01:13:48,900 --> 01:13:53,450 and bar squared equals 0 and n dot n bar equals 2. 1204 01:13:53,450 --> 01:13:55,950 And those were the formulas that we were using over and over 1205 01:13:55,950 --> 01:13:57,210 again really. 1206 01:13:57,210 --> 01:13:59,610 Everything else was just convention. 1207 01:13:59,610 --> 01:14:01,590 OK? 1208 01:14:01,590 --> 01:14:04,710 So what are those three types? 1209 01:14:04,710 --> 01:14:06,750 Well, we could take n, and we could 1210 01:14:06,750 --> 01:14:10,290 change it by some amount in the perp direction 1211 01:14:10,290 --> 01:14:12,690 and leave n bar unchanged. 1212 01:14:12,690 --> 01:14:15,300 And since n bar dot something perp is 0, 1213 01:14:15,300 --> 01:14:21,900 that would satisfy this, or we could do the opposite. 1214 01:14:21,900 --> 01:14:23,955 That would be type two. 1215 01:14:23,955 --> 01:14:27,600 So don't change n, and change n bar. 1216 01:14:30,300 --> 01:14:31,920 And so this is two transformations 1217 01:14:31,920 --> 01:14:37,020 there's two things specified by this perp guy and two here. 1218 01:14:37,020 --> 01:14:41,790 And then there's one more which is called three, 1219 01:14:41,790 --> 01:14:46,450 and this one let me write it like this. 1220 01:14:46,450 --> 01:14:48,210 It's a simultaneous transformation 1221 01:14:48,210 --> 01:14:54,960 of these two guys, where I just do a multiplicative factor. 1222 01:14:54,960 --> 01:14:56,460 Well, a multiplicative factor is not 1223 01:14:56,460 --> 01:14:59,550 going to change the fact that's the square of something 0. 1224 01:14:59,550 --> 01:15:02,250 The place where the normalization comes in 1225 01:15:02,250 --> 01:15:04,470 is this n dot n bar, and if I just rescale them 1226 01:15:04,470 --> 01:15:09,550 both by an opposite amount, then that remains satisfied as well. 1227 01:15:09,550 --> 01:15:11,550 So I formulated this in terms of an infinite-- 1228 01:15:11,550 --> 01:15:15,120 I'm going to think about this is an infinitesimal transformation 1229 01:15:15,120 --> 01:15:18,510 for delta perp, infinitesimal for epsilon perp. 1230 01:15:18,510 --> 01:15:20,160 I could also expand this exponential 1231 01:15:20,160 --> 01:15:21,535 and think of it as infinitesimal, 1232 01:15:21,535 --> 01:15:24,010 but the finite one's easier. 1233 01:15:24,010 --> 01:15:26,850 So let's just think of the finite transformation 1234 01:15:26,850 --> 01:15:27,540 in the last one. 1235 01:15:30,810 --> 01:15:33,220 Now, this is an effective theory. 1236 01:15:33,220 --> 01:15:35,220 So whenever we think about transformations, 1237 01:15:35,220 --> 01:15:36,480 we should think about power counting. 1238 01:15:36,480 --> 01:15:38,040 We just spent a lot of time thinking 1239 01:15:38,040 --> 01:15:39,690 about that for gauge symmetry, and we should think 1240 01:15:39,690 --> 01:15:41,275 about power counting here too. 1241 01:15:41,275 --> 01:15:43,500 Did you have a question? 1242 01:15:43,500 --> 01:15:45,530 AUDIENCE: [INAUDIBLE] 1243 01:15:46,853 --> 01:15:47,520 PROFESSOR: Yeah. 1244 01:15:47,520 --> 01:15:53,790 So what I was just about to talk about, how big things can be. 1245 01:15:53,790 --> 01:15:58,980 So the power counting that's the right power counting 1246 01:15:58,980 --> 01:16:01,860 for these guys, so these things are infinitesimal or finite, 1247 01:16:01,860 --> 01:16:04,980 but they also have some counting in a different space which 1248 01:16:04,980 --> 01:16:08,640 is the power counting space. 1249 01:16:08,640 --> 01:16:13,330 And the right power counting for these guys is as follows. 1250 01:16:13,330 --> 01:16:16,200 So this means I can make arbitrarily 1251 01:16:16,200 --> 01:16:19,410 large transformations in a power counting sense of type 1252 01:16:19,410 --> 01:16:21,000 two and type three. 1253 01:16:21,000 --> 01:16:21,990 There's no constraint. 1254 01:16:21,990 --> 01:16:25,650 That's what this means, so that maybe that's easier to swallow. 1255 01:16:25,650 --> 01:16:28,290 This one, there is a constraint, and the reason 1256 01:16:28,290 --> 01:16:31,116 that you can think of there being a constraint, 1257 01:16:31,116 --> 01:16:35,010 I can give you a simple example, and then you'll 1258 01:16:35,010 --> 01:16:37,580 see how you would derive the other things as well. 1259 01:16:37,580 --> 01:16:39,735 So think of n dot P. If you transform 1260 01:16:39,735 --> 01:16:42,180 n dot P, that becomes n dot-- 1261 01:16:42,180 --> 01:16:45,120 think of this is n mu, P mu. 1262 01:16:45,120 --> 01:16:49,770 That becomes n mu, P mu plus delta perp dot P perp. 1263 01:16:52,432 --> 01:16:54,390 So this is order lambda squared, so this better 1264 01:16:54,390 --> 01:16:55,390 be order lambda squared. 1265 01:16:55,390 --> 01:16:57,310 But this momentum here is order lambda, 1266 01:16:57,310 --> 01:16:59,602 so you need to have a power counting for the delta perp 1267 01:16:59,602 --> 01:17:01,890 that makes it of order lambda. 1268 01:17:01,890 --> 01:17:04,625 In order to keep the transform guy the same size as the guy 1269 01:17:04,625 --> 01:17:06,000 you started with, because I don't 1270 01:17:06,000 --> 01:17:08,333 want to consider transformations that would take me away 1271 01:17:08,333 --> 01:17:10,720 from my power counting, that would violate it. 1272 01:17:10,720 --> 01:17:14,490 So imposing that this thing is of order lambda squared 1273 01:17:14,490 --> 01:17:15,780 says that this is true. 1274 01:17:15,780 --> 01:17:19,530 If you go through the same logic for the other guys, 1275 01:17:19,530 --> 01:17:21,750 the n bar component was order lambda 0, 1276 01:17:21,750 --> 01:17:24,430 so you can make the epsilon perp of order lambda 0. 1277 01:17:24,430 --> 01:17:28,071 It doesn't cost you anything, because there's no constraint. 1278 01:17:34,670 --> 01:17:36,590 You're not making something that would mess up 1279 01:17:36,590 --> 01:17:37,596 the power counting. 1280 01:17:44,550 --> 01:17:47,340 All right. 1281 01:17:47,340 --> 01:17:49,637 So what does this correspond to physically? 1282 01:17:53,380 --> 01:17:55,660 So type three is actually pretty simple. 1283 01:17:55,660 --> 01:17:57,460 Type three is like a boost along-- if you 1284 01:17:57,460 --> 01:17:59,230 think about our back-to-back vectors, 1285 01:17:59,230 --> 01:18:01,450 type three is like the analog of a boost, 1286 01:18:01,450 --> 01:18:04,030 but you're transforming it in the passive sense 1287 01:18:04,030 --> 01:18:05,095 of transforming the axis. 1288 01:18:10,230 --> 01:18:13,380 And what the outcome is is very simple. 1289 01:18:13,380 --> 01:18:20,130 It just implies that any operator that has an n 1290 01:18:20,130 --> 01:18:22,360 must have a corresponding n bar sitting next 1291 01:18:22,360 --> 01:18:27,278 to it, effectively. 1292 01:18:27,278 --> 01:18:28,570 So I'll give you some examples. 1293 01:18:32,070 --> 01:18:33,570 So really, if you have-- 1294 01:18:33,570 --> 01:18:36,780 let's just say it this way-- if you have an n bar mu 1295 01:18:36,780 --> 01:18:44,610 in the numerator, then you either have a corresponding n 1296 01:18:44,610 --> 01:18:51,070 in the numerator, or-- 1297 01:18:57,860 --> 01:19:03,390 well, you have an n bar in the denominator. 1298 01:19:03,390 --> 01:19:06,260 Since we could have n bars in the denominator and that 1299 01:19:06,260 --> 01:19:07,730 was showing up in some places. 1300 01:19:07,730 --> 01:19:10,040 Those are the two possible ways of compensating 1301 01:19:10,040 --> 01:19:11,610 for the transformation. 1302 01:19:11,610 --> 01:19:14,610 So it's just like a simple counting that you have. 1303 01:19:14,610 --> 01:19:24,370 So if you look at our lc 0, we had various terms. 1304 01:19:24,370 --> 01:19:29,462 One of them was n bar slash 1 over i n bar over D. 1305 01:19:29,462 --> 01:19:31,420 So here, we have an n bar in both the numerator 1306 01:19:31,420 --> 01:19:33,370 and the denominator, and that compensates 1307 01:19:33,370 --> 01:19:34,840 for the transformation. 1308 01:19:34,840 --> 01:19:40,030 And then in another term, we had an n bar slash n dot D, 1309 01:19:40,030 --> 01:19:43,425 and then that again is invariant in this type of transformation. 1310 01:19:50,220 --> 01:19:53,550 What about type one and two? 1311 01:19:53,550 --> 01:19:57,460 So type one is the following. 1312 01:19:57,460 --> 01:20:01,140 So think about these guys as in this kind of language 1313 01:20:01,140 --> 01:20:03,720 they would be back to back. 1314 01:20:03,720 --> 01:20:05,220 We're trying to describe the physics 1315 01:20:05,220 --> 01:20:08,310 in this cone for the collinear particles, 1316 01:20:08,310 --> 01:20:12,360 and n was the vector that was pointing inside that cone. 1317 01:20:12,360 --> 01:20:15,750 What this is saying, type one, is that I can rotate that guy, 1318 01:20:15,750 --> 01:20:18,895 and as long as I'm not rotating it by too much-- 1319 01:20:18,895 --> 01:20:19,395 i.e. 1320 01:20:19,395 --> 01:20:21,000 I'm rotating it by a small amount 1321 01:20:21,000 --> 01:20:23,670 of order lambda, which you can think 1322 01:20:23,670 --> 01:20:25,980 of as staying inside the cone-- 1323 01:20:25,980 --> 01:20:27,795 then I can describe everything equally well 1324 01:20:27,795 --> 01:20:29,670 by some other vector that lives in that cone, 1325 01:20:29,670 --> 01:20:32,760 and I can decompose the momenta and the modes 1326 01:20:32,760 --> 01:20:36,820 in terms of that vector, and it will work equally well. 1327 01:20:36,820 --> 01:20:37,320 OK. 1328 01:20:37,320 --> 01:20:41,050 So that's what type one is physically doing, 1329 01:20:41,050 --> 01:20:43,260 and so you're not making a large transformation here, 1330 01:20:43,260 --> 01:20:44,820 because you want to still be inside the cone. 1331 01:20:44,820 --> 01:20:47,195 You still want something that's pointing in the collinear 1332 01:20:47,195 --> 01:20:48,030 direction. 1333 01:20:48,030 --> 01:20:51,930 Type two is related to the fact that this n bar 1334 01:20:51,930 --> 01:20:53,790 vector was just an auxiliary vector that we 1335 01:20:53,790 --> 01:20:54,838 used to decompose things. 1336 01:20:54,838 --> 01:20:56,130 We didn't really care about it. 1337 01:20:56,130 --> 01:21:03,810 It didn't have a strong physical motivation. 1338 01:21:03,810 --> 01:21:06,460 It was just needed because we're using light cone coordinates. 1339 01:21:06,460 --> 01:21:10,485 So I can make a very large transformation of that guy. 1340 01:21:10,485 --> 01:21:13,942 So here's a large transformation, 1341 01:21:13,942 --> 01:21:15,150 and I can use some other guy. 1342 01:21:15,150 --> 01:21:17,150 And we gave you an example earlier, which was 3, 1343 01:21:17,150 --> 01:21:20,670 2, 2, 1 as a possible value for the n bar, 1344 01:21:20,670 --> 01:21:22,710 and that's something that you could think 1345 01:21:22,710 --> 01:21:25,620 of getting to by a finite type two reparameterization 1346 01:21:25,620 --> 01:21:27,520 transformation. 1347 01:21:27,520 --> 01:21:28,020 OK? 1348 01:21:28,020 --> 01:21:32,070 So that's the picture for what these transformations are, 1349 01:21:32,070 --> 01:21:34,830 and we will finish up talking about them next time 1350 01:21:34,830 --> 01:21:38,400 and will show that actually this additional term that we could 1351 01:21:38,400 --> 01:21:40,830 write down in the Lagrangian is actually 1352 01:21:40,830 --> 01:21:44,660 ruled out reparameterization symmetries.