1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high-quality educational resources for free. 5 00:00:10,660 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,370 at ocw.mit.edu. 8 00:00:20,815 --> 00:00:23,190 PROFESSOR: So last time we were talking about heavy quark 9 00:00:23,190 --> 00:00:27,450 effective theory, and we phrased several motivations 10 00:00:27,450 --> 00:00:29,700 for talking about heavy quark effective theory related 11 00:00:29,700 --> 00:00:33,180 to things that we hadn't seen in effective theory yet. 12 00:00:33,180 --> 00:00:35,352 We also said that our discussion here 13 00:00:35,352 --> 00:00:37,560 is more general than just talking about heavy quarks. 14 00:00:37,560 --> 00:00:39,435 Heavy quarks is our example, but we're really 15 00:00:39,435 --> 00:00:43,420 thinking about having some very heavy source. 16 00:00:43,420 --> 00:00:45,960 So you can think of it as either a heavy particle 17 00:00:45,960 --> 00:00:48,340 that you want to leave in the theory-- 18 00:00:48,340 --> 00:00:50,150 So let me rewind. 19 00:00:50,150 --> 00:00:52,710 We talked about heavy particles already, and what we did 20 00:00:52,710 --> 00:00:53,850 is we integrated them out. 21 00:00:53,850 --> 00:00:55,220 We got rid of them. 22 00:00:55,220 --> 00:00:56,803 But what if you have a heavy particle, 23 00:00:56,803 --> 00:00:58,410 and you actually want to study it? 24 00:00:58,410 --> 00:01:00,660 That's related to this discussion. 25 00:01:00,660 --> 00:01:02,658 That's one way that this is important. 26 00:01:02,658 --> 00:01:04,200 If you have a heavy particle that you 27 00:01:04,200 --> 00:01:06,030 want to study at low energies. 28 00:01:06,030 --> 00:01:10,920 Or you could just have a source, a static source of some quantum 29 00:01:10,920 --> 00:01:12,720 number, in this case, color. 30 00:01:12,720 --> 00:01:15,105 And this would be the right way of thinking about it. 31 00:01:15,105 --> 00:01:16,980 So those are kind of some of the motivations, 32 00:01:16,980 --> 00:01:19,272 and then there was also some more technical motivations 33 00:01:19,272 --> 00:01:21,610 of things that we wanted to see with this theory. 34 00:01:21,610 --> 00:01:24,000 So we started by saying it's a top-down approach, so we 35 00:01:24,000 --> 00:01:28,480 should be able to just take QCD and take the limit and expand. 36 00:01:28,480 --> 00:01:30,988 But it's a little more tricky than it was so far with, like, 37 00:01:30,988 --> 00:01:33,030 light quark masses for chiral perturbation theory 38 00:01:33,030 --> 00:01:35,070 because the mass is in the numerator, 39 00:01:35,070 --> 00:01:38,370 and we want to take the limit that it goes to infinity, OK? 40 00:01:38,370 --> 00:01:41,430 So we put a question mark beside this equation, 41 00:01:41,430 --> 00:01:45,390 and we started instead thinking about expanding the propagator. 42 00:01:45,390 --> 00:01:48,000 So if we write the propagator out for a heavy quark, 43 00:01:48,000 --> 00:01:50,830 you've got mass terms in the numerator and the denominator. 44 00:01:50,830 --> 00:01:54,420 And when you expand it out, after using 45 00:01:54,420 --> 00:01:58,260 the equation of motion and expanding with 46 00:01:58,260 --> 00:02:04,620 p equals mQ v plus k. 47 00:02:04,620 --> 00:02:09,810 So we introduced a vector, v. v squared was 1. 48 00:02:09,810 --> 00:02:12,270 And then we introduce some fluctuations, k. 49 00:02:16,830 --> 00:02:19,830 So there's m's hiding in this p as well as that p. 50 00:02:19,830 --> 00:02:21,900 The quadratic term in the denominator cancels. 51 00:02:21,900 --> 00:02:23,947 There's a linear term that survives. 52 00:02:23,947 --> 00:02:25,530 There's linear terms in the numerator. 53 00:02:25,530 --> 00:02:28,590 The linear terms, once I just look at what they give, 54 00:02:28,590 --> 00:02:30,492 they give this first contribution. 55 00:02:30,492 --> 00:02:32,200 And then there's some higher-order terms. 56 00:02:32,200 --> 00:02:34,300 Then we drop them. 57 00:02:34,300 --> 00:02:34,800 OK? 58 00:02:34,800 --> 00:02:37,625 So for the propagator we had no problem taking the heavy quark 59 00:02:37,625 --> 00:02:40,590 limit, and we got out this expression. 60 00:02:40,590 --> 00:02:42,930 We also then thought about the vertex. 61 00:02:42,930 --> 00:02:46,980 It's the gluon coupling to a quark, heavy quark. 62 00:02:46,980 --> 00:02:52,050 And if this is the result in QCD, this 1 plus v slash over 2 63 00:02:52,050 --> 00:02:53,670 is a projector. 64 00:02:53,670 --> 00:02:58,610 But that means if you square it, you get back the same thing. 65 00:02:58,610 --> 00:03:01,110 And we said, well, if there's two propagators on either side 66 00:03:01,110 --> 00:03:03,402 of this thing, I can think about it putting a projector 67 00:03:03,402 --> 00:03:05,940 on either side of this thing, and that gives you 68 00:03:05,940 --> 00:03:08,830 this result with just a v mu. 69 00:03:08,830 --> 00:03:09,330 OK? 70 00:03:09,330 --> 00:03:11,190 So the Feynman rule for this guy can 71 00:03:11,190 --> 00:03:13,710 be simplified by using the fact that on either side of it 72 00:03:13,710 --> 00:03:16,170 are sitting these projectors. 73 00:03:16,170 --> 00:03:18,180 And if you take those two facts into account, 74 00:03:18,180 --> 00:03:20,350 we can actually encode them in a Lagrangian. 75 00:03:20,350 --> 00:03:21,890 We could just write it down, OK? 76 00:03:21,890 --> 00:03:23,890 We want a Lagrangian that gives that propagator, 77 00:03:23,890 --> 00:03:27,390 the lowest-order propagator, and that lowest-order vertex. 78 00:03:27,390 --> 00:03:29,770 And that's what this is. 79 00:03:29,770 --> 00:03:38,160 So the v dot D has in it a v dot A. v dot A is giving this guy. 80 00:03:41,499 --> 00:03:44,370 So that's the v dot A inside there. 81 00:03:44,370 --> 00:03:50,060 And then 1 over v dot k, that's the v dot partial, right? 82 00:03:50,060 --> 00:03:53,640 So i v dot D equals i v dot partial. 83 00:03:58,565 --> 00:04:00,190 I don't know which sign I'm using here, 84 00:04:00,190 --> 00:04:01,523 but I think it's the minus sign. 85 00:04:01,523 --> 00:04:03,080 AUDIENCE: I think you're using plus. 86 00:04:03,080 --> 00:04:03,997 PROFESSOR: Using plus? 87 00:04:03,997 --> 00:04:05,372 AUDIENCE: No, you're using minus. 88 00:04:05,372 --> 00:04:06,403 PROFESSOR: Using minus. 89 00:04:06,403 --> 00:04:07,820 I'll switch throughout the course. 90 00:04:07,820 --> 00:04:09,950 In each chapter I'll use a different sign convention 91 00:04:09,950 --> 00:04:10,992 to keep you on your toes. 92 00:04:10,992 --> 00:04:14,740 [CHUCKLES] In this chapter, I will stick with this sign 93 00:04:14,740 --> 00:04:17,329 convention, since the source that we're 94 00:04:17,329 --> 00:04:21,240 using for the reading is using this sign convention. 95 00:04:21,240 --> 00:04:21,740 OK? 96 00:04:21,740 --> 00:04:24,380 So this is the v dot A, and this is the v dot k 97 00:04:24,380 --> 00:04:27,230 that we have over there. 98 00:04:27,230 --> 00:04:29,240 Now, I've encoded-- the way I encoded 99 00:04:29,240 --> 00:04:30,740 the projection relation, as I said, 100 00:04:30,740 --> 00:04:33,510 let's work in terms of a four-component field. 101 00:04:33,510 --> 00:04:35,480 So our field, Qv, still has four components, 102 00:04:35,480 --> 00:04:37,400 but it satisfies this relation. 103 00:04:37,400 --> 00:04:39,860 And that actually projects it down onto two components, 104 00:04:39,860 --> 00:04:41,943 and we'll talk more about what the physics of this 105 00:04:41,943 --> 00:04:42,950 is in a minute. 106 00:04:42,950 --> 00:04:45,740 This is very convenient when talking about heavy quarks 107 00:04:45,740 --> 00:04:48,800 because you often want to couple heavy quarks to light quarks, 108 00:04:48,800 --> 00:04:51,380 and if the light quarks are four-component spinors, 109 00:04:51,380 --> 00:04:53,810 it's easy to couple them to the heavy quarks 110 00:04:53,810 --> 00:04:55,940 if you have heavy quark fields which 111 00:04:55,940 --> 00:04:57,770 are four-component spinors. 112 00:04:57,770 --> 00:05:01,520 So you could work in terms of a two-component Lagrangian that 113 00:05:01,520 --> 00:05:03,950 doesn't have this projection relation, 114 00:05:03,950 --> 00:05:06,170 but for technical reasons it's better 115 00:05:06,170 --> 00:05:08,510 to work in terms of the four-component one. 116 00:05:08,510 --> 00:05:10,640 Makes things simpler. 117 00:05:10,640 --> 00:05:13,850 So that's in some sense an indirect derivation 118 00:05:13,850 --> 00:05:17,390 of the leading-order HQET Lagrangian. 119 00:05:17,390 --> 00:05:19,500 Is there any questions about that? 120 00:05:19,500 --> 00:05:21,458 AUDIENCE: What if you don't have the propagator 121 00:05:21,458 --> 00:05:22,373 on the other side? 122 00:05:22,373 --> 00:05:24,290 PROFESSOR: What if I don't have the propagator 123 00:05:24,290 --> 00:05:25,490 on the other side? 124 00:05:25,490 --> 00:05:28,010 Yeah, so we'll talk about the spinors in a minute, 125 00:05:28,010 --> 00:05:31,155 but it's the same story for the spinors, 126 00:05:31,155 --> 00:05:33,530 that they have a projection in relation, which is exactly 127 00:05:33,530 --> 00:05:35,270 the same projection relation. 128 00:05:35,270 --> 00:05:38,870 So whether it's a line sticking out 129 00:05:38,870 --> 00:05:42,710 or whether it's attached to something else, it's the same. 130 00:05:42,710 --> 00:05:46,475 OK, so let's now go and derive the same thing directly. 131 00:05:52,380 --> 00:05:55,900 And the trick we're going to do is the following. 132 00:05:55,900 --> 00:05:59,370 We need to cancel out this m in the numerator, and the place 133 00:05:59,370 --> 00:06:04,240 that m hides is in the p in this formula here. 134 00:06:04,240 --> 00:06:05,940 So if we think about some field that's 135 00:06:05,940 --> 00:06:12,420 fluctuating in spacetime, we can pull out from that field 136 00:06:12,420 --> 00:06:16,470 the analog of that big piece, p, if we pull out a phase 137 00:06:16,470 --> 00:06:17,790 factor that looks like that. 138 00:06:28,330 --> 00:06:30,090 So let me just write this-- so far 139 00:06:30,090 --> 00:06:34,350 this is just some redefinition of these objects 140 00:06:34,350 --> 00:06:35,770 on the right-hand side. 141 00:06:35,770 --> 00:06:39,570 And I'm setting up two objects here because I want-- 142 00:06:39,570 --> 00:06:46,260 I'm going to break the field into these two pieces that obey 143 00:06:46,260 --> 00:06:47,760 different projection relations. 144 00:06:53,490 --> 00:06:53,990 OK? 145 00:06:53,990 --> 00:06:56,330 So if I sum these two up, the v slashes cancel. 146 00:06:56,330 --> 00:06:59,645 Qv plus-- These are just two components of the-- 147 00:06:59,645 --> 00:07:03,140 I've just split it by two orthogonal projectors, 148 00:07:03,140 --> 00:07:04,250 two pieces. 149 00:07:04,250 --> 00:07:05,960 And I've pulled out this piece here. 150 00:07:05,960 --> 00:07:08,300 And the reason that I pulled out this piece here 151 00:07:08,300 --> 00:07:10,940 is the analog of what I did here with the p, 152 00:07:10,940 --> 00:07:15,105 where I divided it into a big piece and a small piece. 153 00:07:15,105 --> 00:07:17,480 This is going to take care of pulling out that big piece. 154 00:07:21,260 --> 00:07:25,280 OK, so if we want to take the limit here in this equation. 155 00:07:25,280 --> 00:07:27,900 So far we've written out what the Q's will be. 156 00:07:27,900 --> 00:07:31,250 We also have to write out what the D slash is. 157 00:07:31,250 --> 00:07:35,930 So it's convenient to do the following with the D slash. 158 00:07:35,930 --> 00:07:37,055 Break it into two pieces-- 159 00:07:42,430 --> 00:07:46,580 a piece along v and a piece orthogonal to v. 160 00:07:46,580 --> 00:07:50,340 So the definition here of D transverse 161 00:07:50,340 --> 00:07:56,180 is that it's D minus the piece along v, 162 00:07:56,180 --> 00:08:00,570 so if you like that v dot D transverse is 0. 163 00:08:00,570 --> 00:08:02,120 That's what I mean. 164 00:08:07,308 --> 00:08:08,850 With these relations up here, there's 165 00:08:08,850 --> 00:08:11,472 also another way of writing them that I should write down, 166 00:08:11,472 --> 00:08:12,930 as well, which is a little simpler. 167 00:08:12,930 --> 00:08:17,850 If I rearrange these formulas, I can also write as v slash on Qv 168 00:08:17,850 --> 00:08:27,355 is Qv, and v slash on Bv is minus Bv. 169 00:08:27,355 --> 00:08:29,035 So there's a 1 on this side, and if I 170 00:08:29,035 --> 00:08:30,660 come at it with the half and put things 171 00:08:30,660 --> 00:08:31,785 together that's what I get. 172 00:08:34,820 --> 00:08:37,885 OK, so let's put those things together into here, 173 00:08:37,885 --> 00:08:40,010 and then we'll figure out if we can take the limit. 174 00:08:44,520 --> 00:08:47,580 And the crucial thing, really, for being 175 00:08:47,580 --> 00:08:49,740 able to take the limit is to pull out this phase, 176 00:08:49,740 --> 00:08:53,490 and we'll talk about what physically is happening there 177 00:08:53,490 --> 00:08:56,460 in a minute, once we've done the algebra. 178 00:09:18,010 --> 00:09:21,040 OK, so just plugging those pieces together, we have that. 179 00:09:29,650 --> 00:09:30,880 Pull the phase through. 180 00:09:34,630 --> 00:09:38,890 So pull this negative phase through these terms. 181 00:09:38,890 --> 00:09:41,590 With this v dot D, when it hits the v dot x, 182 00:09:41,590 --> 00:09:43,596 we get some extra contribution. 183 00:09:48,840 --> 00:09:51,120 And I can group it together with this mQ term 184 00:09:51,120 --> 00:09:55,200 because what it does is it brings down an mQ. 185 00:09:55,200 --> 00:09:58,558 The D transverse gives no derivative, no contribution, 186 00:09:58,558 --> 00:10:00,600 when it hits that phase because of this relation. 187 00:10:06,720 --> 00:10:09,560 So pulling the phase through, the only term we pick up 188 00:10:09,560 --> 00:10:13,080 is this minus 1 here. 189 00:10:13,080 --> 00:10:16,310 And I just grouped it together-- sorry. 190 00:10:16,310 --> 00:10:19,715 The only term we pick up is this v slash here, 191 00:10:19,715 --> 00:10:22,400 and that comes from this v dot D hitting that term. 192 00:10:28,000 --> 00:10:28,660 All right? 193 00:10:28,660 --> 00:10:34,390 So now we put these pieces together, multiply things out, 194 00:10:34,390 --> 00:10:38,530 and use the fact that I've divided up 195 00:10:38,530 --> 00:10:41,707 the field into these two pieces that have different-- 196 00:10:41,707 --> 00:10:44,290 I know how v slash acts on them, so I can get rid of all the v 197 00:10:44,290 --> 00:10:46,630 slashes by just using the formula. 198 00:10:46,630 --> 00:10:51,340 When v minus 1 acts on Qv, I get 0, OK? 199 00:10:51,340 --> 00:10:54,100 So for that term, for the term that has two Qv's, I just 200 00:10:54,100 --> 00:10:58,060 have the v slash i v dot D. You can also prove that when 201 00:10:58,060 --> 00:11:03,040 you have the Qv and the Qv bar that this term actually is 0, 202 00:11:03,040 --> 00:11:06,040 and to prove that you basically use the projection 203 00:11:06,040 --> 00:11:12,310 relation and the fact that if I have 1 plus v slash over 2 D 204 00:11:12,310 --> 00:11:17,890 transverse, that that can be written as D transverse 205 00:11:17,890 --> 00:11:19,690 1 minus v slash over 2. 206 00:11:22,390 --> 00:11:25,200 Again, using this formula and the fact that gamma matrices 207 00:11:25,200 --> 00:11:27,220 anticommute. 208 00:11:27,220 --> 00:11:29,890 And then 1 minus v slash over 2 kills Qv, 209 00:11:29,890 --> 00:11:34,870 and so a term like this one between these fields is absent, 210 00:11:34,870 --> 00:11:37,480 and the only term with two Qv's is that one. 211 00:11:42,280 --> 00:11:45,430 Likewise, with Bv's it's the same story. 212 00:11:45,430 --> 00:11:48,280 I don't get a D transverse term. 213 00:11:48,280 --> 00:11:53,350 But here, the mQ term does survive, and it doubles, 214 00:11:53,350 --> 00:11:55,190 so the v slash gives another minus 1, 215 00:11:55,190 --> 00:11:57,940 so there's 2mQ sitting there. 216 00:11:57,940 --> 00:11:59,920 And then there's the transverse terms, 217 00:11:59,920 --> 00:12:02,626 and they come in the cross terms. 218 00:12:02,626 --> 00:12:07,060 So Qv Bv. 219 00:12:07,060 --> 00:12:09,190 If I have this, then this projection relation 220 00:12:09,190 --> 00:12:11,690 won't get rid of it because, if I have a 1 plus v slash over 221 00:12:11,690 --> 00:12:13,690 2 on the right, it becomes the correct projector 222 00:12:13,690 --> 00:12:17,380 for the other field on the left, and then vice versa. 223 00:12:26,140 --> 00:12:26,770 OK? 224 00:12:26,770 --> 00:12:33,780 So there's four terms here, and we've simplified things as much 225 00:12:33,780 --> 00:12:36,040 as we can. 226 00:12:36,040 --> 00:12:38,700 So now let's think about what these fields are describing. 227 00:12:45,258 --> 00:12:46,800 In particular, let's think about what 228 00:12:46,800 --> 00:12:49,935 would happen if we only had external Qv fields. 229 00:13:01,230 --> 00:13:13,610 So if we consider only external Qv fields, 230 00:13:13,610 --> 00:13:17,470 then we can think about what happens as mQ goes to infinity. 231 00:13:22,170 --> 00:13:24,630 And if you have a term-- 232 00:13:24,630 --> 00:13:26,983 so as m goes to infinity, nothing happens to this term. 233 00:13:26,983 --> 00:13:28,400 We have to figure out what happens 234 00:13:28,400 --> 00:13:29,930 to those terms over there. 235 00:13:29,930 --> 00:13:31,790 But this guy here, it's clear what happens. 236 00:13:31,790 --> 00:13:36,050 We get a field that has a mass, mQ, and that particles are just 237 00:13:36,050 --> 00:13:38,310 getting heavier and heavier, and they're decoupling. 238 00:13:38,310 --> 00:13:40,685 And when you get particles that get heavy, they decouple. 239 00:13:43,520 --> 00:13:49,410 So Bv decouples in this limit. 240 00:13:49,410 --> 00:13:53,000 So diagrammatically, think about it like this. 241 00:13:56,340 --> 00:13:58,070 You have a Qv. 242 00:13:58,070 --> 00:14:01,640 It could switch into a Bv with one of these D slash vertices, 243 00:14:01,640 --> 00:14:05,000 and then back to a Qv, and this would give some kind of diagram 244 00:14:05,000 --> 00:14:08,720 from this Lagrangian, which has external Qv fields. 245 00:14:08,720 --> 00:14:12,020 But the Bv field has a 1 over m-type propagator, 246 00:14:12,020 --> 00:14:18,072 and so this goes like 1 over m, and hence goes to 0. 247 00:14:18,072 --> 00:14:20,030 And that's one way of talking about decoupling, 248 00:14:20,030 --> 00:14:22,490 that intermediate particles of that type 249 00:14:22,490 --> 00:14:26,550 are giving diagrams that are suppressed by 1 over m. 250 00:14:26,550 --> 00:14:27,980 OK? 251 00:14:27,980 --> 00:14:31,250 And you see here, sitting right here, 252 00:14:31,250 --> 00:14:32,850 you see our Lagrangian that we want, 253 00:14:32,850 --> 00:14:36,045 which is the Qv i v dot D Bv. 254 00:14:36,045 --> 00:14:37,670 So if we just threw away the Bv fields, 255 00:14:37,670 --> 00:14:39,378 we would get the Lagrangian that we want. 256 00:15:01,315 --> 00:15:03,440 So physically what's going on with these two fields 257 00:15:03,440 --> 00:15:07,520 is that Qv corresponds to the particles, OK? 258 00:15:07,520 --> 00:15:08,930 We have a heavy quark. 259 00:15:08,930 --> 00:15:12,170 It's got particles that are of that mass and that flavor that 260 00:15:12,170 --> 00:15:14,630 are heavy, and it has antiparticles, too. 261 00:15:14,630 --> 00:15:17,210 Qv corresponds to the particles, and Bv corresponds 262 00:15:17,210 --> 00:15:18,290 to the antiparticles. 263 00:15:34,160 --> 00:15:37,570 So-- of this flavor. 264 00:15:48,390 --> 00:15:51,090 And by making the phase redefinition that we did, 265 00:15:51,090 --> 00:15:53,970 we chose to focus on on-shell fluctuations that 266 00:15:53,970 --> 00:15:55,500 are close to the particles. 267 00:15:55,500 --> 00:15:58,620 If we'd chosen the opposite phase, plus i, 268 00:15:58,620 --> 00:16:01,367 then we could expand about the antiparticles, 269 00:16:01,367 --> 00:16:03,450 and things would have worked out the opposite way. 270 00:16:08,410 --> 00:16:11,510 So I'll come back to that a little bit more in a second. 271 00:16:11,510 --> 00:16:13,600 So let's have several discussion points here. 272 00:16:18,770 --> 00:16:23,800 So what we did is we made a field redefinition, which 273 00:16:23,800 --> 00:16:26,645 we were always allowed to do. 274 00:16:26,645 --> 00:16:27,520 AUDIENCE: [INAUDIBLE] 275 00:16:27,520 --> 00:16:28,370 PROFESSOR: Yeah? 276 00:16:28,370 --> 00:16:31,360 AUDIENCE: Isn't Qv like a [INAUDIBLE] 277 00:16:31,360 --> 00:16:36,130 eigenstate of positive helicity? 278 00:16:36,130 --> 00:16:37,030 And the other one-- 279 00:16:37,030 --> 00:16:38,530 PROFESSOR: We're talking about heavy particles, 280 00:16:38,530 --> 00:16:40,238 so helicity is not such a good thing to-- 281 00:16:40,238 --> 00:16:44,080 AUDIENCE: [INAUDIBLE] 282 00:16:44,080 --> 00:16:46,200 PROFESSOR: Yeah, I'll talk more about it in a sec. 283 00:16:46,200 --> 00:16:46,700 Yeah. 284 00:16:49,484 --> 00:16:51,480 Yeah, give me a minute, and then if you still 285 00:16:51,480 --> 00:16:53,640 have a question after I talk about it, then ask. 286 00:16:56,430 --> 00:16:59,100 So first of all, what we did here was tree level. 287 00:17:18,450 --> 00:17:18,950 OK? 288 00:17:18,950 --> 00:17:21,650 So when we make this type of redefinition 289 00:17:21,650 --> 00:17:24,319 and we go through this algebra, it's all tree level. 290 00:17:24,319 --> 00:17:26,450 Not thinking about any loops here. 291 00:17:26,450 --> 00:17:28,130 There will be corrections to this story 292 00:17:28,130 --> 00:17:29,630 once we start talking about loops, 293 00:17:29,630 --> 00:17:31,940 and there will be corrections to this story, 294 00:17:31,940 --> 00:17:34,452 namely that we just get the same L 295 00:17:34,452 --> 00:17:38,660 HQET once we want to take into account one of our m 296 00:17:38,660 --> 00:17:42,600 corrections, and will come back to both of those points. 297 00:17:42,600 --> 00:17:45,890 So both of those things to be discussed later. 298 00:17:45,890 --> 00:17:48,380 What we do do with L HQET-- 299 00:17:56,150 --> 00:17:59,750 as we described correctly, the couplings 300 00:17:59,750 --> 00:18:09,530 to gluons at leading order. 301 00:18:14,120 --> 00:18:17,240 And often what people are interested in with k mu here 302 00:18:17,240 --> 00:18:22,190 is k mu of order lambda QCD, separating the scale lambda 303 00:18:22,190 --> 00:18:24,710 QCD that has to do with the confinement into a meson, 304 00:18:24,710 --> 00:18:27,500 like a B meson, from the scale of the heavy quark. 305 00:18:39,340 --> 00:18:43,861 OK, so the antiparticles are being removed, 306 00:18:43,861 --> 00:18:45,070 which is point number 2. 307 00:18:57,370 --> 00:18:59,968 So if you want to ask what is it that we're integrating out 308 00:18:59,968 --> 00:19:02,260 in this theory, we're not integrating out the particle, 309 00:19:02,260 --> 00:19:04,000 because we want to study it. 310 00:19:04,000 --> 00:19:05,960 We're integrating out the antiparticle. 311 00:19:09,180 --> 00:19:11,430 It's easiest to see that if you go to the rest frame. 312 00:19:14,980 --> 00:19:20,520 So taking v rest frame, which is just 1, 0, 0, 0, 313 00:19:20,520 --> 00:19:26,790 just timelike component, then our 1 plus v slash becomes a-- 314 00:19:26,790 --> 00:19:31,170 so if you have 1 plus v slash, then that's just 1 315 00:19:31,170 --> 00:19:33,343 plus gamma 0 over 2. 316 00:19:33,343 --> 00:19:35,760 And if you want to talk about particles and antiparticles, 317 00:19:35,760 --> 00:19:39,360 the best representation to use is the Dirac representation 318 00:19:39,360 --> 00:19:40,710 because then the particles-- 319 00:19:44,900 --> 00:19:48,190 So let's take a spinor in the Dirac representation. 320 00:19:48,190 --> 00:19:50,980 Could be part of this field. 321 00:19:50,980 --> 00:19:53,530 Could just be a spinor for an external state. 322 00:19:53,530 --> 00:20:00,400 In the Dirac representation, what 323 00:20:00,400 --> 00:20:03,280 happens is that you have the guys that 324 00:20:03,280 --> 00:20:06,257 are the particles in the upper two components 325 00:20:06,257 --> 00:20:08,590 and the guys that are the antiparticles in the lower two 326 00:20:08,590 --> 00:20:10,510 components. 327 00:20:10,510 --> 00:20:14,136 So this is particles. 328 00:20:14,136 --> 00:20:15,750 This is antiparticles. 329 00:20:19,250 --> 00:20:21,508 This guy is always projecting in the rest frame 330 00:20:21,508 --> 00:20:23,050 into the particles and antiparticles. 331 00:20:23,050 --> 00:20:25,175 It's just that if you read in other representations 332 00:20:25,175 --> 00:20:26,980 you'd have more trouble seeing that, 333 00:20:26,980 --> 00:20:30,670 but in the Dirac representation we just have a 0, OK? 334 00:20:30,670 --> 00:20:33,690 So this is one way of thinking about what 335 00:20:33,690 --> 00:20:34,690 this projector is doing. 336 00:20:34,690 --> 00:20:36,633 It's getting rid of the antiparticles. 337 00:20:42,310 --> 00:20:44,830 So the way you should think about this physically 338 00:20:44,830 --> 00:20:51,250 is that, if you're studying some process, 339 00:20:51,250 --> 00:20:57,460 you're studying heavy particles close to their mass shell. 340 00:21:09,040 --> 00:21:14,080 What we're doing is we're measuring fluctuations 341 00:21:14,080 --> 00:21:18,150 near the heavy mass, mQ. 342 00:21:18,150 --> 00:21:20,440 We're perturbing about mQ. 343 00:21:20,440 --> 00:21:22,810 If you're perturbing about mQ, the antiparticles 344 00:21:22,810 --> 00:21:26,530 are very far away because the splitting between the particles 345 00:21:26,530 --> 00:21:30,047 and the antiparticles is 2mQ. 346 00:21:30,047 --> 00:21:31,630 So that's why they get integrated out. 347 00:21:37,150 --> 00:21:52,872 So that's why, in this language of being near mQ, 348 00:21:52,872 --> 00:21:54,580 the antiparticles are just very far away, 349 00:21:54,580 --> 00:21:56,950 and we don't want them in our theory. 350 00:21:56,950 --> 00:22:01,180 One way of talking about that is pair creation. 351 00:22:01,180 --> 00:22:03,880 So if we draw some type of Time-Ordered Perturbation 352 00:22:03,880 --> 00:22:08,100 Theory diagram for pair creation-- 353 00:22:08,100 --> 00:22:13,450 so it's not a Feynman diagram, but a TOPT diagram, just so we 354 00:22:13,450 --> 00:22:16,000 can talk about a definite intermediate state 355 00:22:16,000 --> 00:22:19,840 at a definite time, then this intermediate state here 356 00:22:19,840 --> 00:22:21,970 has three heavy particles, and it has-- 357 00:22:21,970 --> 00:22:26,800 If this guy has mQ and this guy has mQ, this guy has 3mQ. 358 00:22:26,800 --> 00:22:34,510 So this intermediate state is off-shell by 2mQ. 359 00:22:44,950 --> 00:22:50,770 Time-ordered perturbation theory diagram. 360 00:22:53,265 --> 00:22:55,390 So I can talk about whether something is definitely 361 00:22:55,390 --> 00:22:58,210 a particle or definitely an antiparticle. 362 00:22:58,210 --> 00:23:00,790 And another way of seeing this 2mQ that I'm talking about 363 00:23:00,790 --> 00:23:04,240 is, like, just to do vacuum polarization, where 364 00:23:04,240 --> 00:23:05,360 you have 2mQ. 365 00:23:05,360 --> 00:23:09,380 You need 2mQ of energy to fluctuate into a heavy quark 366 00:23:09,380 --> 00:23:09,880 pair. 367 00:23:16,040 --> 00:23:19,280 So we have a very light gluon, and it just 368 00:23:19,280 --> 00:23:24,000 doesn't have the energy to do that, 369 00:23:24,000 --> 00:23:30,407 so pair creation is not part of this theory, OK? 370 00:23:30,407 --> 00:23:32,990 So now you can ask your question if you still have a question. 371 00:23:32,990 --> 00:23:35,640 [CHUCKLES] Yeah. 372 00:23:35,640 --> 00:23:36,140 All right? 373 00:23:36,140 --> 00:23:41,232 So the thing we're getting rid of is the antiparticle. 374 00:23:41,232 --> 00:23:43,190 Now, because we're getting really antiparticle, 375 00:23:43,190 --> 00:23:45,985 we no longer have an annihilation process. 376 00:23:45,985 --> 00:23:47,360 So unlike QCD, where you can have 377 00:23:47,360 --> 00:23:50,830 particles and antiparticles annihilate, 378 00:23:50,830 --> 00:23:52,840 here we have actually a conservation rule 379 00:23:52,840 --> 00:23:54,400 for the number of heavy particles. 380 00:24:05,020 --> 00:24:07,740 We're talking here about quarks. 381 00:24:07,740 --> 00:24:11,760 The number of heavy quarks is preserved, 382 00:24:11,760 --> 00:24:13,950 and that's an extra symmetry that this theory 383 00:24:13,950 --> 00:24:29,840 has that QCD didn't have. 384 00:24:35,920 --> 00:24:36,420 OK? 385 00:24:36,420 --> 00:24:42,030 So the conservation law for the number of heavy quarks is u1. 386 00:24:42,030 --> 00:24:45,840 More generally, we can extend this 387 00:24:45,840 --> 00:24:48,338 to something called heavy quark symmetry. 388 00:24:51,760 --> 00:24:54,670 So let's-- since we notice that there's this conservation law, 389 00:24:54,670 --> 00:24:57,850 let's think about what the biggest group we can come up 390 00:24:57,850 --> 00:25:03,160 with is that describes the symmetry of this L HQET. 391 00:25:06,170 --> 00:25:10,420 So this u1 that preserves the heavy quarks 392 00:25:10,420 --> 00:25:11,380 is a flavor symmetry. 393 00:25:14,290 --> 00:25:16,660 And depending on how many heavy particles we have, 394 00:25:16,660 --> 00:25:19,360 it could be bigger than a u1. 395 00:25:19,360 --> 00:25:27,340 So actually, it's a uNh for Nh heavy quarks. 396 00:25:29,990 --> 00:25:32,530 So you could have some heavy sources 397 00:25:32,530 --> 00:25:34,190 that live under some group. 398 00:25:34,190 --> 00:25:36,618 And if you have N types of those sources, 399 00:25:36,618 --> 00:25:37,660 you'd have a uN symmetry. 400 00:25:43,030 --> 00:25:45,505 And the root of this is that we got rid of mQ. 401 00:25:49,810 --> 00:25:53,380 mQ was the thing that would break this symmetry, 402 00:25:53,380 --> 00:25:55,450 but it disappeared. 403 00:25:55,450 --> 00:25:57,570 And once it's gone, we don't care. 404 00:25:57,570 --> 00:26:02,800 We don't know whether the thing was a charm quark or a B 405 00:26:02,800 --> 00:26:05,140 quark or a top quark. 406 00:26:05,140 --> 00:26:08,110 We don't see the flavor of the quark 407 00:26:08,110 --> 00:26:10,210 anymore because the mass term was 408 00:26:10,210 --> 00:26:12,460 what was tracking that flavor, and now it's gone, 409 00:26:12,460 --> 00:26:14,143 and we just end up with this theory that 410 00:26:14,143 --> 00:26:15,560 doesn't remember about the flavor, 411 00:26:15,560 --> 00:26:18,170 so we have a flavor symmetry. 412 00:26:18,170 --> 00:26:18,670 OK? 413 00:26:18,670 --> 00:26:21,340 So that's kind of cool as an example 414 00:26:21,340 --> 00:26:23,320 of having an effective theory where there's 415 00:26:23,320 --> 00:26:25,120 an emergent symmetry, symmetry that 416 00:26:25,120 --> 00:26:29,290 wasn't so apparent in the theory you started with but is 417 00:26:29,290 --> 00:26:31,210 apparent once you get to the effective theory. 418 00:26:39,560 --> 00:26:43,250 The other example is spin symmetry. 419 00:26:43,250 --> 00:26:45,610 There's actually an su2 spin symmetry 420 00:26:45,610 --> 00:26:49,130 because the Lagrangian didn't involve any D slashes anymore. 421 00:26:49,130 --> 00:26:55,840 It just became scalar, the v dot D. So 422 00:26:55,840 --> 00:26:57,290 if we think about our heavy part-- 423 00:26:57,290 --> 00:26:59,080 so we think about a four-component spinor. 424 00:26:59,080 --> 00:27:00,760 Two degrees of freedom have to do with particle 425 00:27:00,760 --> 00:27:01,630 versus antiparticle. 426 00:27:01,630 --> 00:27:04,870 The other two have to do with spin, a half. 427 00:27:04,870 --> 00:27:06,618 And we don't care about the spin, 428 00:27:06,618 --> 00:27:08,785 either, because our Lagrangian is completely scalar. 429 00:27:25,370 --> 00:27:25,870 OK? 430 00:27:25,870 --> 00:27:32,410 We just had Qv i v dot D Qv, and there was no-- 431 00:27:32,410 --> 00:27:35,110 we didn't mention-- the v dot D doesn't have any D slash. 432 00:27:35,110 --> 00:27:38,120 It doesn't have any spin matrices in it, OK? 433 00:27:38,120 --> 00:27:41,210 So that's an su2 symmetry. 434 00:27:41,210 --> 00:27:44,390 So if we want to talk about that again, 435 00:27:44,390 --> 00:27:47,860 it's useful to think about it in the rest frame. 436 00:27:47,860 --> 00:27:53,140 In the rest frame, the heavy quark spin transformation. 437 00:28:01,720 --> 00:28:06,310 Need some Dirac representation, which is gamma 438 00:28:06,310 --> 00:28:07,540 phi, gamma 0, gamma i. 439 00:28:07,540 --> 00:28:10,690 It's just the Pauli matrices, sigma i. 440 00:28:10,690 --> 00:28:15,550 And you could think of making a transformation related 441 00:28:15,550 --> 00:28:16,690 to the spin. 442 00:28:16,690 --> 00:28:18,880 So the infinitesimal version of that transformation 443 00:28:18,880 --> 00:28:19,755 would look like this. 444 00:28:23,050 --> 00:28:24,190 Start with Qv. 445 00:28:24,190 --> 00:28:25,540 Do a rotation. 446 00:28:25,540 --> 00:28:29,860 Ask, does this change the Lagrangian? 447 00:28:29,860 --> 00:28:33,460 And the statement is that it changed 448 00:28:33,460 --> 00:28:35,830 the Lagrangian that you get, which 449 00:28:35,830 --> 00:28:40,750 is i v dot D commutator i epsilon dot sQ. 450 00:28:43,840 --> 00:28:47,410 But i v dot D is just this scalar quantity and just 451 00:28:47,410 --> 00:28:51,555 commutes through this epsilon dot sQ, OK? 452 00:28:51,555 --> 00:28:54,055 So that's the formal statement that there's a spin symmetry. 453 00:29:02,690 --> 00:29:05,780 Since we're using this kind of projected formalism, 454 00:29:05,780 --> 00:29:07,230 you could ask the question, well, 455 00:29:07,230 --> 00:29:11,210 maybe this spin symmetry is mixing up the heavy quark 456 00:29:11,210 --> 00:29:14,960 components with the other components, 457 00:29:14,960 --> 00:29:18,945 these zero components, but you can check that it doesn't. 458 00:29:18,945 --> 00:29:22,910 It's actually acting within the subcategory, which is just 459 00:29:22,910 --> 00:29:25,790 describing the heavy quark. 460 00:29:25,790 --> 00:29:28,400 So you can also check it. 461 00:29:28,400 --> 00:29:29,900 One way of phrasing that is that you 462 00:29:29,900 --> 00:29:32,520 can check that this is true. 463 00:29:32,520 --> 00:29:35,360 So that the rotated field still satisfies the projection 464 00:29:35,360 --> 00:29:35,860 relation. 465 00:29:39,290 --> 00:29:42,320 So really, this is a rotation that's 466 00:29:42,320 --> 00:29:45,870 within the two-component subspace that 467 00:29:45,870 --> 00:29:48,440 are describing the physical degrees of freedom. 468 00:29:52,910 --> 00:29:53,410 OK? 469 00:29:53,410 --> 00:29:56,590 So there's this enhanced symmetry group of this theory. 470 00:30:00,640 --> 00:30:03,340 And we can put it together because nothing stops us 471 00:30:03,340 --> 00:30:06,230 from doing this, or doing this, or doing a bit of both, 472 00:30:06,230 --> 00:30:07,480 so we can build one big group. 473 00:30:15,190 --> 00:30:20,007 So all together, you have a u2Nh, 474 00:30:20,007 --> 00:30:21,340 and that's heavy quark symmetry. 475 00:30:32,600 --> 00:30:36,530 A Qv field is a fundamental in that group. 476 00:30:36,530 --> 00:30:42,860 So [INAUDIBLE] spinors look like this, 477 00:30:42,860 --> 00:30:46,170 with some number of components, and this guy could be-- 478 00:30:46,170 --> 00:30:50,570 for example, maybe this is a B quark with spin up. 479 00:30:55,080 --> 00:30:56,580 And then some other component might 480 00:30:56,580 --> 00:31:01,630 be a charm quark with spin up or a B quark with spin down. 481 00:31:01,630 --> 00:31:04,550 That's what heavy quark symmetry is. 482 00:31:04,550 --> 00:31:05,050 OK? 483 00:31:05,050 --> 00:31:07,610 So any questions about that? 484 00:31:07,610 --> 00:31:11,230 So we'll talk a bit more about how you actually 485 00:31:11,230 --> 00:31:13,570 make predictions related to this heavy quark symmetry. 486 00:31:13,570 --> 00:31:14,987 As you saw, it's kind of emergent, 487 00:31:14,987 --> 00:31:16,450 and so you should build some tools 488 00:31:16,450 --> 00:31:19,780 for seeing what kind of impact that has on observables, 489 00:31:19,780 --> 00:31:22,990 and we'll spend a little bit of time doing that today. 490 00:31:22,990 --> 00:31:25,270 But before we do that, let's continue our list. 491 00:31:28,370 --> 00:31:30,545 So when we made this field redefinition, 492 00:31:30,545 --> 00:31:32,837 we pulled out this phase, and that phase depended on v. 493 00:31:32,837 --> 00:31:38,210 And we also stuck a label v on the field. 494 00:31:38,210 --> 00:31:46,310 So we had this v mu, which appeared on our field, OK? 495 00:31:46,310 --> 00:31:48,873 And that was just a reminder that we pulled out 496 00:31:48,873 --> 00:31:51,040 a phase, and the phase depending on our choice of v, 497 00:31:51,040 --> 00:31:54,970 and we could make actually different choices. 498 00:31:54,970 --> 00:31:57,010 This v mu is actually something quite useful 499 00:31:57,010 --> 00:32:03,040 because it isn't changed when you talk about low-energy QCD 500 00:32:03,040 --> 00:32:05,057 interactions. 501 00:32:05,057 --> 00:32:06,265 So it's a conserved quantity. 502 00:32:20,720 --> 00:32:27,760 So in our notation of dividing up p into mQv plus k, 503 00:32:27,760 --> 00:32:28,390 this changes. 504 00:32:33,050 --> 00:32:36,820 So if you tickle with very soft gluons the heavy quark, 505 00:32:36,820 --> 00:32:39,770 you can't change this term, but you can change that term. 506 00:32:39,770 --> 00:32:42,670 So the v term is not affected by the gluon interactions. 507 00:32:42,670 --> 00:32:45,130 It's a conserved quantity, and hence 508 00:32:45,130 --> 00:32:46,780 having it as a label on our field 509 00:32:46,780 --> 00:32:48,760 tells us what v we're talking about. 510 00:32:48,760 --> 00:32:50,258 Basically, what v signifies is you 511 00:32:50,258 --> 00:32:52,300 could talk about a heavy quark in its rest frame, 512 00:32:52,300 --> 00:32:54,220 and then you could talk about tickling it. 513 00:32:54,220 --> 00:32:56,920 Or it could be moving with some constant velocity, 514 00:32:56,920 --> 00:33:00,250 and you could talk about that moving frame tickling it. 515 00:33:00,250 --> 00:33:02,860 And any such choice is equally valid, 516 00:33:02,860 --> 00:33:04,880 and that's what v is encoding. 517 00:33:04,880 --> 00:33:05,380 OK? 518 00:33:05,380 --> 00:33:09,730 But we can formulate tickling a heavy quark in any given frame 519 00:33:09,730 --> 00:33:12,823 that we'd like, and v is encoding that. 520 00:33:12,823 --> 00:33:14,990 Things are kind of natural to see in the rest frame, 521 00:33:14,990 --> 00:33:16,823 but we don't have to work in the rest frame, 522 00:33:16,823 --> 00:33:19,214 and that's what v is encoding. 523 00:33:19,214 --> 00:33:20,060 OK? 524 00:33:20,060 --> 00:33:22,730 So that's kind of different than we've seen before. 525 00:33:25,670 --> 00:33:28,340 What's the power counting? 526 00:33:28,340 --> 00:33:30,720 Power counting is not so difficult here. 527 00:33:30,720 --> 00:33:32,420 It's just powers of 1 over mQ. 528 00:33:38,050 --> 00:33:40,160 So our leading-order Lagrangian had 529 00:33:40,160 --> 00:33:45,020 no mQ's and then we could formulate corrections 530 00:33:45,020 --> 00:33:47,420 to that leading order, and they would be suppressed 531 00:33:47,420 --> 00:33:48,550 by powers of 1 over mQ. 532 00:33:51,962 --> 00:33:54,045 So let's think about that in a little more detail. 533 00:33:56,980 --> 00:34:01,860 So if you take the field Q of x and you write the mode 534 00:34:01,860 --> 00:34:03,090 expansion for the field-- 535 00:34:18,915 --> 00:34:20,540 but there's this term, and then there's 536 00:34:20,540 --> 00:34:23,840 a term that involves the antiparticles. 537 00:34:28,150 --> 00:34:30,340 What we did with the Q of x field-- 538 00:34:30,340 --> 00:34:32,320 that's the full theory field-- 539 00:34:32,320 --> 00:34:33,865 is we pulled out this phase. 540 00:34:38,420 --> 00:34:40,130 And the phase that we pulled out was just 541 00:34:40,130 --> 00:34:44,114 something that depended on a fixed quantity, m v, 542 00:34:44,114 --> 00:34:45,739 so it would come outside that integral. 543 00:34:52,690 --> 00:34:53,560 So we pulled out-- 544 00:35:00,307 --> 00:35:02,390 so you can think about really literally sticking p 545 00:35:02,390 --> 00:35:05,780 into this formula and then pulling out that phase. 546 00:35:05,780 --> 00:35:15,210 And then that leaves, in Q v, just the e to the minus i k dot 547 00:35:15,210 --> 00:35:15,710 x. 548 00:35:19,020 --> 00:35:21,060 OK? 549 00:35:21,060 --> 00:35:25,620 So if you think about derivatives on Qv of x, 550 00:35:25,620 --> 00:35:27,130 then they are scaling like k. 551 00:35:32,070 --> 00:35:34,860 And this guy here has no mQ's in it. 552 00:35:37,600 --> 00:35:39,728 So there's no mQ's in the derivatives. 553 00:35:43,150 --> 00:35:45,720 So if we build operators out of this Qv field, 554 00:35:45,720 --> 00:35:48,600 if we let derivatives act on that field, 555 00:35:48,600 --> 00:35:51,180 there's no factors of the heavy quark mass hiding anywhere 556 00:35:51,180 --> 00:35:52,710 in those derivatives. 557 00:35:52,710 --> 00:35:56,400 That's the magic of making this kind of phase redefinition. 558 00:35:56,400 --> 00:35:59,610 We've made the mQ's explicit by pulling that out so that we 559 00:35:59,610 --> 00:36:04,170 could get something remaining that doesn't have any mQ's. 560 00:36:04,170 --> 00:36:06,743 Since we want to count mQ's what we want to do 561 00:36:06,743 --> 00:36:08,910 is we want to make all the mQ's very, very explicit. 562 00:36:08,910 --> 00:36:11,220 We don't want any to be hiding anywhere, 563 00:36:11,220 --> 00:36:13,920 and that's what we were doing before with this field 564 00:36:13,920 --> 00:36:18,640 redefinition, was making all the mQ's explicit. 565 00:36:18,640 --> 00:36:21,090 This is one example of why that's important. 566 00:36:21,090 --> 00:36:25,103 AUDIENCE: [INAUDIBLE] 567 00:36:25,103 --> 00:36:25,770 PROFESSOR: Yeah. 568 00:36:31,920 --> 00:36:37,500 So you should-- if you think about the action-- 569 00:36:37,500 --> 00:36:39,203 yeah, if you think about the action 570 00:36:39,203 --> 00:36:41,370 and you thought about making a gauge transformation, 571 00:36:41,370 --> 00:36:43,703 you could worry about making a gauge transformation that 572 00:36:43,703 --> 00:36:46,570 would inject too much momentum into your field 573 00:36:46,570 --> 00:36:49,950 so that it no longer described low-energy degrees of freedom. 574 00:36:49,950 --> 00:36:53,527 But usually people don't talk about that so much in HQET, 575 00:36:53,527 --> 00:36:55,485 so we'll save that discussion for [INAUDIBLE],, 576 00:36:55,485 --> 00:36:58,570 but we will have it. 577 00:36:58,570 --> 00:37:03,300 So the coordinate, x-- one way of thinking about this equation 578 00:37:03,300 --> 00:37:06,090 is that the coordinate x that's in Qv really 579 00:37:06,090 --> 00:37:09,030 corresponds to variations that are the low-energy variations. 580 00:37:16,670 --> 00:37:18,350 So our field is doing what we want. 581 00:37:18,350 --> 00:37:20,420 It's describing low-energy fluctuations. 582 00:37:26,500 --> 00:37:28,750 And that's another way of thinking about the fact 583 00:37:28,750 --> 00:37:32,304 that there's no mQ's in this derivative. 584 00:37:37,250 --> 00:37:46,100 So if we look at subleading operators and-- 585 00:37:46,100 --> 00:37:49,240 if we look at subleading Lagrangians 586 00:37:49,240 --> 00:38:04,910 and external operators, all the powers of m Q 587 00:38:04,910 --> 00:38:07,340 are going to be explicit, and that 588 00:38:07,340 --> 00:38:10,040 makes doing the power counting quite simple because we just 589 00:38:10,040 --> 00:38:11,360 look at them and count. 590 00:38:14,852 --> 00:38:16,935 So as long as we're sure that they're all explicit 591 00:38:16,935 --> 00:38:20,390 and none are hiding anywhere, then it's easy to count. 592 00:38:20,390 --> 00:38:23,210 There will be logs of mQ hiding in Wilson coefficients, 593 00:38:23,210 --> 00:38:24,480 but we're not counting logs. 594 00:38:24,480 --> 00:38:26,480 If we're just counting powers, then what I said 595 00:38:26,480 --> 00:38:29,948 is sufficient, OK? 596 00:38:29,948 --> 00:38:31,490 So is there any questions about that? 597 00:38:35,040 --> 00:38:37,233 So there's actually one mQ that's hiding, 598 00:38:37,233 --> 00:38:40,530 and we should be a little bit careful about it. 599 00:38:40,530 --> 00:38:42,110 So if we talk about states, there's 600 00:38:42,110 --> 00:38:43,430 actually an mQ hiding in there. 601 00:38:50,880 --> 00:38:55,020 So let's think about states of some particle that's 602 00:38:55,020 --> 00:38:56,370 relativistically normalized. 603 00:38:56,370 --> 00:38:58,860 It could be a heavy meson. 604 00:38:58,860 --> 00:39:03,840 Let me just call it H. And the usual normalization 605 00:39:03,840 --> 00:39:08,070 convention for a relativistic particle 606 00:39:08,070 --> 00:39:18,580 is this one for the states. 607 00:39:18,580 --> 00:39:20,330 So this guy, if you look at the dimensions 608 00:39:20,330 --> 00:39:23,030 of the right-hand side here, you've 609 00:39:23,030 --> 00:39:26,000 got minus 3 plus 1 is minus 2, so each of these 610 00:39:26,000 --> 00:39:29,300 is dimension minus 1. 611 00:39:29,300 --> 00:39:31,340 So that's one thing we can [INAUDIBLE].. 612 00:39:31,340 --> 00:39:33,100 But if you think about what this Ep is, 613 00:39:33,100 --> 00:39:37,710 this Ep has the M still hiding in it. 614 00:39:37,710 --> 00:39:41,790 So this Ep is really the physical on-shell energy, 615 00:39:41,790 --> 00:39:43,920 so it's mQ squared. 616 00:39:43,920 --> 00:39:46,340 If this was a heavy hadron, for example, 617 00:39:46,340 --> 00:39:51,290 it'd be mH squared plus the 3-momentum squared. 618 00:39:51,290 --> 00:39:54,650 And mH, for a heavy hadron, has an mQ hiding in it. 619 00:40:30,140 --> 00:40:32,900 OK, so what we have to do if we want to make everything 620 00:40:32,900 --> 00:40:36,380 explicit is use HQET states. 621 00:40:36,380 --> 00:40:39,050 And you can think of HQET states simply as the states 622 00:40:39,050 --> 00:40:42,830 that you get if you define your quantum field 623 00:40:42,830 --> 00:40:45,020 theory using L HQET. 624 00:40:50,520 --> 00:40:52,950 So this is the mQ goes to infinity, 625 00:40:52,950 --> 00:40:54,450 so it's just the leading-order term. 626 00:41:05,130 --> 00:41:08,210 And if you define your states with this Lagrangian, 627 00:41:08,210 --> 00:41:11,540 you can still talk about a bound state like the B meson. 628 00:41:11,540 --> 00:41:13,050 Because the B quark didn't go away. 629 00:41:13,050 --> 00:41:14,690 It just became static. 630 00:41:14,690 --> 00:41:16,970 And all the dynamics of the light degrees of freedom 631 00:41:16,970 --> 00:41:19,100 that are making it into a hadron are still there, 632 00:41:19,100 --> 00:41:21,830 and they're described by a usual QCD Lagrangian. 633 00:41:21,830 --> 00:41:24,380 So you still have hadrons, so you still have a B meson, 634 00:41:24,380 --> 00:41:28,010 but it's not the same because you don't have any mQ's. 635 00:41:28,010 --> 00:41:29,180 They're all gone. 636 00:41:29,180 --> 00:41:33,440 So you can't possibly have this formula for the HQET definition 637 00:41:33,440 --> 00:41:35,622 of the state. 638 00:41:35,622 --> 00:41:37,580 And so the way that you should think about this 639 00:41:37,580 --> 00:41:42,680 is that the relativistic version of the state 640 00:41:42,680 --> 00:41:46,520 is related to the HQET definition 641 00:41:46,520 --> 00:41:47,780 by a formula like this. 642 00:41:55,670 --> 00:41:58,000 So the HQET definition is this Hv, 643 00:41:58,000 --> 00:42:00,580 the relativistic is this Hp. 644 00:42:00,580 --> 00:42:04,310 And there's some normalization factor between them, 645 00:42:04,310 --> 00:42:08,560 and as well as there can be 1 over M type corrections. 646 00:42:08,560 --> 00:42:10,640 Actually, both things are there. 647 00:42:10,640 --> 00:42:18,580 And if we talk about this H v field, 648 00:42:18,580 --> 00:42:24,610 where v is some external label telling us the velocity 649 00:42:24,610 --> 00:42:28,180 and k is describing some residual momentum 650 00:42:28,180 --> 00:42:32,440 of that state, the normalization convention 651 00:42:32,440 --> 00:42:35,980 would be as follows for HQET. 652 00:42:43,910 --> 00:42:45,500 So I've still left it to be 2. 653 00:42:45,500 --> 00:42:49,320 Sometimes people to remove this 2, as well. 654 00:42:49,320 --> 00:42:51,300 So there's several things going on here. 655 00:42:51,300 --> 00:42:53,300 One is that v and v prime-- 656 00:42:53,300 --> 00:42:55,190 in order for these states to have overlap, 657 00:42:55,190 --> 00:42:57,930 we're saying that you should be talking about hadrons that are 658 00:42:57,930 --> 00:42:59,180 moving with the same velocity. 659 00:42:59,180 --> 00:43:01,610 Remember velocity is conserved, so if we 660 00:43:01,610 --> 00:43:03,110 had two things in different sectors, 661 00:43:03,110 --> 00:43:04,490 they just wouldn't overlap. 662 00:43:04,490 --> 00:43:06,350 That's [INAUDIBLE] of delta. 663 00:43:06,350 --> 00:43:09,020 This delta 3 of k minus prime is exactly 664 00:43:09,020 --> 00:43:10,437 the analog of this guy over here, 665 00:43:10,437 --> 00:43:12,395 but now it's with the k, which is the residual. 666 00:43:12,395 --> 00:43:14,330 It has no mQ's. 667 00:43:14,330 --> 00:43:15,890 And I got rid of the mQ-- 668 00:43:15,890 --> 00:43:18,980 I got rid of the thing in the numerator that 669 00:43:18,980 --> 00:43:20,720 was pulling out this root mH. 670 00:43:20,720 --> 00:43:22,428 So there's no mQ's in that formula, 671 00:43:22,428 --> 00:43:23,970 and that's the right way of thinking. 672 00:43:23,970 --> 00:43:26,300 So you have to remember, if you have 673 00:43:26,300 --> 00:43:27,800 some formula for the cross-section 674 00:43:27,800 --> 00:43:29,800 that you derived with relativistic states, which 675 00:43:29,800 --> 00:43:32,382 all of the formulas you're used to are derived that way, 676 00:43:32,382 --> 00:43:34,090 you have to remember about that root of H 677 00:43:34,090 --> 00:43:39,105 when you're going to make predictions with HQET. 678 00:43:39,105 --> 00:43:41,230 And the same thing actually is true of the spinors. 679 00:43:41,230 --> 00:43:44,480 There's a square root of the quark mass hiding in spinors, 680 00:43:44,480 --> 00:43:46,460 and you have to take care of that, as well. 681 00:43:46,460 --> 00:43:48,590 I'll leave that for your reading. 682 00:43:48,590 --> 00:43:49,090 OK? 683 00:43:49,090 --> 00:43:52,807 So up to these little subtleties here, that's 684 00:43:52,807 --> 00:43:54,640 all the mQ's that you have to keep track of, 685 00:43:54,640 --> 00:43:56,682 and once we've kept track of all those mQ's, then 686 00:43:56,682 --> 00:44:00,998 we know how to do the power counting in this theory. 687 00:44:00,998 --> 00:44:02,290 What can we do with the theory? 688 00:44:07,350 --> 00:44:09,100 Well, one thing that we can do immediately 689 00:44:09,100 --> 00:44:10,840 is do some spectroscopy. 690 00:44:13,660 --> 00:44:19,927 So this will actually show us partly what kind of predictions 691 00:44:19,927 --> 00:44:21,010 we can make from symmetry. 692 00:44:24,980 --> 00:44:27,610 So there's light quarks and gluons, 693 00:44:27,610 --> 00:44:30,370 and they are still described, as I said in words 694 00:44:30,370 --> 00:44:35,770 but now I write on the board, by a full L QCD. 695 00:44:35,770 --> 00:44:36,580 Nothing changes. 696 00:44:36,580 --> 00:44:39,520 There was no heavy quark masses in that anyway, 697 00:44:39,520 --> 00:44:43,558 so nothing happens to that, at least for what we're 698 00:44:43,558 --> 00:44:44,350 talking about here. 699 00:44:44,350 --> 00:44:46,142 There could be some loop corrections, but-- 700 00:44:48,600 --> 00:44:50,460 So if we think about mQ going to infinity, 701 00:44:50,460 --> 00:44:52,800 we can ask what happens to these hadrons, which 702 00:44:52,800 --> 00:44:55,740 I've just told you you can still describe by the theory. 703 00:44:58,440 --> 00:45:05,550 So if you think about a Q and a light quark, which is a meson, 704 00:45:05,550 --> 00:45:09,960 it has the quantum numbers of the heavy quark. 705 00:45:09,960 --> 00:45:12,270 And remember that the number of heavy quarks 706 00:45:12,270 --> 00:45:15,552 is conserved in this limit. 707 00:45:15,552 --> 00:45:17,010 And then it has the quantum numbers 708 00:45:17,010 --> 00:45:20,550 of the rest of the stuff, which could be an antiquark. 709 00:45:20,550 --> 00:45:22,890 It could be any number of Q q bar pairs 710 00:45:22,890 --> 00:45:25,080 because we don't know how many are there. 711 00:45:25,080 --> 00:45:27,055 We have nonperturbative interactions. 712 00:45:27,055 --> 00:45:28,680 And it could have any number of gluons. 713 00:45:31,380 --> 00:45:36,090 And generically, we call this stuff light degrees of freedom 714 00:45:36,090 --> 00:45:38,580 because that's what it is. 715 00:45:38,580 --> 00:45:39,340 This is heavy. 716 00:45:39,340 --> 00:45:40,320 This is light. 717 00:45:40,320 --> 00:45:43,780 And it's just a bunch of light stuff. 718 00:45:43,780 --> 00:45:44,280 OK? 719 00:45:44,280 --> 00:45:48,210 So that's kind of how we can think of the state. 720 00:45:48,210 --> 00:45:49,950 It's a bunch of-- it's an infinite number 721 00:45:49,950 --> 00:45:54,510 of single particle states because it's a complicated QCD 722 00:45:54,510 --> 00:45:56,555 bound state. 723 00:45:56,555 --> 00:45:57,930 And what we want to do is we want 724 00:45:57,930 --> 00:46:01,350 to sort of enumerate what symmetry can tell us. 725 00:46:01,350 --> 00:46:04,530 Well, since we have the-- 726 00:46:04,530 --> 00:46:06,360 we can think about total angular momentum, 727 00:46:06,360 --> 00:46:08,530 and that is actually a good quantum number. 728 00:46:08,530 --> 00:46:11,980 It was in QCD, and it remains true in HQET. 729 00:46:11,980 --> 00:46:16,398 The thing that's new is that the heavy quark spin is conserved. 730 00:46:21,880 --> 00:46:24,990 So SQ is conserved. 731 00:46:24,990 --> 00:46:26,610 And that we can use to say something 732 00:46:26,610 --> 00:46:28,260 more about the states. 733 00:46:47,360 --> 00:46:50,120 And usually the way that this is done 734 00:46:50,120 --> 00:46:57,020 is by defining Sl to be J minus SQ. 735 00:46:57,020 --> 00:46:59,250 So if both J and SQ are good things to talk about, 736 00:46:59,250 --> 00:47:02,880 we can also talk about Sl, which is the difference. 737 00:47:02,880 --> 00:47:07,460 And if you want to think of quantum numbers here, 738 00:47:07,460 --> 00:47:10,940 then J squared, right, would be, in terms of quantum numbers, so 739 00:47:10,940 --> 00:47:12,920 J, J plus 1. 740 00:47:12,920 --> 00:47:15,830 And so we can do the same thing with Sl [INAUDIBLE] Sl, 741 00:47:15,830 --> 00:47:18,980 Sl plus 1. 742 00:47:18,980 --> 00:47:22,800 On a state with good quantum numbers of Sl. 743 00:47:22,800 --> 00:47:29,300 So if we organize things in terms of Sl, 744 00:47:29,300 --> 00:47:34,490 then we get symmetry doublets for the mesons. 745 00:47:37,430 --> 00:47:44,330 So let's pick an Sl and some parity, make a little table. 746 00:47:44,330 --> 00:47:50,940 If we take a 1/2 minus, there's two mesons 747 00:47:50,940 --> 00:48:00,470 that fall in that camp, a B and a B star that have little-- 748 00:48:00,470 --> 00:48:04,800 just call it capital J. Let me call it little j. 749 00:48:09,200 --> 00:48:12,080 j is 0 and 1 for these guys. 750 00:48:12,080 --> 00:48:16,470 So this is the scalar guy, and this is a vector guy. 751 00:48:16,470 --> 00:48:19,078 And the parity is negative, so it's a pseudoscalar. 752 00:48:22,670 --> 00:48:24,444 We can keep enumerating. 753 00:48:30,260 --> 00:48:32,270 There's a guy with opposite party. 754 00:48:32,270 --> 00:48:34,940 These guys turned out to be heavier. 755 00:48:34,940 --> 00:48:41,720 This is the lightest ground-state heavy mesons. 756 00:48:41,720 --> 00:48:46,710 3/2 plus called B1 and B2 star. 757 00:48:46,710 --> 00:48:48,210 All these things have been observed. 758 00:48:52,855 --> 00:48:54,480 And you could keep going, and you could 759 00:48:54,480 --> 00:48:57,030 do the same thing for baryons. 760 00:48:57,030 --> 00:48:58,830 So if you do baryons, then the spin 761 00:48:58,830 --> 00:49:00,455 of the light degrees of freedom are not 762 00:49:00,455 --> 00:49:03,150 half-integer, but integer. 763 00:49:03,150 --> 00:49:07,657 0 plus is just the lambda B, and then the only possible j 764 00:49:07,657 --> 00:49:10,410 is to add 1/2 for the SQ. 765 00:49:10,410 --> 00:49:12,360 So what's happening is this is Sl, 766 00:49:12,360 --> 00:49:15,750 and then remember that the Lagrangian and the dynamics 767 00:49:15,750 --> 00:49:17,520 is independent of SQ. 768 00:49:17,520 --> 00:49:21,780 So I can add and subtract the SQ from this Sl, 769 00:49:21,780 --> 00:49:25,950 and I get total j of 0 or 1, but the dynamics of these mesons 770 00:49:25,950 --> 00:49:27,630 doesn't care about the heavy quark spin, 771 00:49:27,630 --> 00:49:30,840 so that's why they come in a symmetry doublet. 772 00:49:30,840 --> 00:49:33,180 So the heavy quark spin symmetry is relating things 773 00:49:33,180 --> 00:49:35,432 in a doublet of given Sl. 774 00:49:35,432 --> 00:49:36,390 That's how you should-- 775 00:49:36,390 --> 00:49:41,550 AUDIENCE: Notion to star just always mean excited [INAUDIBLE] 776 00:49:41,550 --> 00:49:42,050 state? 777 00:49:44,940 --> 00:49:46,560 PROFESSOR: Star is just another-- 778 00:49:46,560 --> 00:49:49,200 star is just something you can put on the B to make it not 779 00:49:49,200 --> 00:49:52,740 look like a B. [CHUCKLE] You could put a twiddle-- 780 00:49:52,740 --> 00:49:56,190 I mean, generically, it started out as star being the vector. 781 00:49:56,190 --> 00:49:58,110 Star is like kind of a notation for vectors. 782 00:49:58,110 --> 00:50:00,600 But then, of course, here B0 star is a scalar, so-- 783 00:50:00,600 --> 00:50:01,100 [CHUCKLES] 784 00:50:01,100 --> 00:50:01,653 AUDIENCE: OK. 785 00:50:01,653 --> 00:50:02,320 PROFESSOR: Yeah. 786 00:50:05,420 --> 00:50:10,460 So we have the lambda B, and then here we have sigmas. 787 00:50:15,980 --> 00:50:18,620 That's telling us that we're already making some predictions 788 00:50:18,620 --> 00:50:21,380 about mesons, and baryons, for that matter, 789 00:50:21,380 --> 00:50:23,520 just from symmetry. 790 00:50:23,520 --> 00:50:26,450 Now, if you want to really make predictions, 791 00:50:26,450 --> 00:50:29,480 you'd like to look at a little more dynamics, 792 00:50:29,480 --> 00:50:31,220 and for that there's something that's 793 00:50:31,220 --> 00:50:35,630 kind of cool called using a covariant representation 794 00:50:35,630 --> 00:50:38,460 of fields. 795 00:50:38,460 --> 00:50:42,110 So I want to show you how that works. 796 00:50:42,110 --> 00:50:44,360 So we're going to see how to make heavy quark symmetry 797 00:50:44,360 --> 00:50:47,270 predictions by building up some object that has 798 00:50:47,270 --> 00:50:49,190 good transformation properties. 799 00:50:49,190 --> 00:50:52,070 And this is in general something that's a more general lesson. 800 00:50:52,070 --> 00:50:53,570 If you have a symmetry, and you want 801 00:50:53,570 --> 00:50:55,208 to make some predictions that are just 802 00:50:55,208 --> 00:50:56,750 based on that symmetry, then you want 803 00:50:56,750 --> 00:50:58,250 to think about how things transform, 804 00:50:58,250 --> 00:51:01,160 and you can do tricks like the ones I'm going to show you. 805 00:51:01,160 --> 00:51:05,300 So we're going to encode heavy quark 806 00:51:05,300 --> 00:51:10,475 symmetry in objects with nice transformation properties. 807 00:51:32,725 --> 00:51:35,350 The moral of this story is that it's much easier to take traces 808 00:51:35,350 --> 00:51:38,647 than to think about Clebsch-Gordan coefficients. 809 00:51:47,096 --> 00:51:48,600 AUDIENCE: [INAUDIBLE] 810 00:51:48,600 --> 00:51:51,000 PROFESSOR: Sure. 811 00:51:51,000 --> 00:51:54,360 AUDIENCE: The way you've done it so far, when 812 00:51:54,360 --> 00:51:56,880 you throw in this v, should I be worried 813 00:51:56,880 --> 00:51:59,300 that you've broken [INAUDIBLE]? 814 00:51:59,300 --> 00:52:00,870 PROFESSOR: Yeah. 815 00:52:00,870 --> 00:52:02,207 Angular momentum is conserved. 816 00:52:02,207 --> 00:52:03,540 It's the boosts that are broken. 817 00:52:03,540 --> 00:52:04,957 But we'll talk about-- we're going 818 00:52:04,957 --> 00:52:06,160 to come to that in a minute. 819 00:52:06,160 --> 00:52:07,470 Yeah, yeah. 820 00:52:07,470 --> 00:52:08,550 But the ang-- 821 00:52:08,550 --> 00:52:11,130 I mean, good question related to this discussion. 822 00:52:11,130 --> 00:52:12,950 The angular momentum is actually conserved. 823 00:52:12,950 --> 00:52:13,450 Yeah. 824 00:52:18,980 --> 00:52:19,480 OK. 825 00:52:23,410 --> 00:52:28,437 But rather than try to make this a kind of derivation, 826 00:52:28,437 --> 00:52:30,020 I'm going to just tell you the answer, 827 00:52:30,020 --> 00:52:32,966 and then we'll talk about its properties. 828 00:52:32,966 --> 00:52:34,910 So I'm going to write down a field that 829 00:52:34,910 --> 00:52:39,995 annihilates the meson doublet for the ground-state mesons. 830 00:52:43,910 --> 00:52:45,320 Q is the heavy quark flavor. 831 00:52:51,200 --> 00:52:57,800 pv mu star Q is something that annihilates 832 00:52:57,800 --> 00:53:00,722 a vector, B star-type meson. 833 00:53:00,722 --> 00:53:05,390 So Q could be charm, as well. 834 00:53:05,390 --> 00:53:10,370 And pv without the mu is annihilating the B meson 835 00:53:10,370 --> 00:53:17,290 if Q was B. This is the flavor, which 836 00:53:17,290 --> 00:53:21,645 you can think of as for this discussion bottom or charm. 837 00:53:24,630 --> 00:53:25,880 And so this guy is a bispinor. 838 00:53:29,340 --> 00:53:33,280 It's got the indices of Q q bar, but it's 839 00:53:33,280 --> 00:53:37,630 turned a vector field and a pseudoscalar field 840 00:53:37,630 --> 00:53:40,240 into a bispinor. 841 00:53:40,240 --> 00:53:42,490 This thing here, this 1 plus v slash over 2, 842 00:53:42,490 --> 00:53:46,440 is projecting out the heavy quark degrees of freedom. 843 00:53:50,250 --> 00:53:52,240 This is a vector. 844 00:53:52,240 --> 00:53:55,110 So this is like something that, if you acted on a state, 845 00:53:55,110 --> 00:53:57,920 would get replaced by the polarization. 846 00:53:57,920 --> 00:54:02,820 Epsilon squared is minus 1, and v dot epsilon is 0, 847 00:54:02,820 --> 00:54:05,115 so that means that v dot this guy is also 0. 848 00:54:08,040 --> 00:54:10,300 And this is pseudoscalar. 849 00:54:10,300 --> 00:54:13,230 So this is the vector meson. 850 00:54:13,230 --> 00:54:14,460 This is the pseudoscalar. 851 00:54:23,810 --> 00:54:26,620 So what are the transformation properties of this? 852 00:54:26,620 --> 00:54:27,760 So it's a bispinor. 853 00:54:27,760 --> 00:54:33,850 If you do a Lorentz, it transforms 854 00:54:33,850 --> 00:54:36,749 like a bispinor should under a Lorentz transformation. 855 00:54:53,715 --> 00:54:55,680 This is a spinor Lorentz transformation, 856 00:54:55,680 --> 00:54:58,440 and v prime is the transformation of v, 857 00:54:58,440 --> 00:55:02,550 and x prime is the transformation of x. 858 00:55:02,550 --> 00:55:04,270 So that's how we want it to transform. 859 00:55:04,270 --> 00:55:05,790 We want something that has-- 860 00:55:05,790 --> 00:55:07,360 it's transforming like this Q q bar. 861 00:55:11,370 --> 00:55:16,950 We've got v slash on Hv equals Hv. 862 00:55:16,950 --> 00:55:19,750 That's one way of writing this 1 plus v slash over 2Hv 863 00:55:19,750 --> 00:55:21,420 is equal to Hv. 864 00:55:21,420 --> 00:55:27,790 And that says that there's no antiquark, so that's good. 865 00:55:27,790 --> 00:55:31,890 We can also, just by studying the properties of this thing, 866 00:55:31,890 --> 00:55:39,757 work out that Hv B slash is actually minus Hv. 867 00:55:39,757 --> 00:55:41,340 That one is not something we built in. 868 00:55:41,340 --> 00:55:42,090 That's an outcome. 869 00:55:45,840 --> 00:55:50,200 So you could work this out, so I won't go through it. 870 00:55:50,200 --> 00:55:53,400 You just push it through the other side, 871 00:55:53,400 --> 00:55:56,260 and you can figure out that that's true. 872 00:55:56,260 --> 00:55:59,190 So we'll use it, but it's not something 873 00:55:59,190 --> 00:56:01,860 that directly-- it's an outcome of putting things together, 874 00:56:01,860 --> 00:56:05,070 not so much something that we built in. 875 00:56:05,070 --> 00:56:12,050 If we talk about heavy quark and light quark symmetries, 876 00:56:12,050 --> 00:56:23,657 so SQ cross Sl, then it's 1/2, 1/2 under that. 877 00:56:23,657 --> 00:56:25,990 So if we work in the rest frame, which is this vr, which 878 00:56:25,990 --> 00:56:32,570 is 1, 0, then you have-- 879 00:56:32,570 --> 00:56:35,750 if you think about doing a symmetry transformation, 880 00:56:35,750 --> 00:56:38,140 you get a commutator, and basically 881 00:56:38,140 --> 00:56:42,340 what happens is that you would get-- 882 00:56:42,340 --> 00:56:45,260 you have two sets of-- you have a bispinor, 883 00:56:45,260 --> 00:56:47,350 so you have two parts of the spinor that 884 00:56:47,350 --> 00:56:49,180 are corresponding to the heavy quark and two parts that 885 00:56:49,180 --> 00:56:50,722 are corresponding to the light quark. 886 00:56:50,722 --> 00:56:56,830 The heavy quark part is on the left, 887 00:56:56,830 --> 00:57:01,120 and then the light quark part is on the right. 888 00:57:13,430 --> 00:57:15,170 That's the spin transformations. 889 00:57:15,170 --> 00:57:17,720 And this is the right spin transformation for something 890 00:57:17,720 --> 00:57:21,605 that's spin 1/2, where the sigma 4 by 4 891 00:57:21,605 --> 00:57:22,985 is the usual spin matrix. 892 00:57:37,790 --> 00:57:40,280 Let me just say a bit more about heavy quark spin 893 00:57:40,280 --> 00:57:48,130 symmetry, which is part of our discussion there. 894 00:57:48,130 --> 00:57:49,870 So Hv is transforming. 895 00:57:49,870 --> 00:57:53,210 If you do a heavy quark spin symmetry transformation, 896 00:57:53,210 --> 00:57:54,460 it's transforming on the left. 897 00:57:54,460 --> 00:57:58,510 That's what I just was discussing. 898 00:57:58,510 --> 00:58:02,110 And if you do some transformation parameterized 899 00:58:02,110 --> 00:58:06,100 by some theta, which I just dot into the S over there, 900 00:58:06,100 --> 00:58:09,340 then this is the way we think about what 901 00:58:09,340 --> 00:58:11,620 the transformation gives. 902 00:58:11,620 --> 00:58:15,730 So you could take this, and you could plug in our Hv field 903 00:58:15,730 --> 00:58:18,610 and then just do the Dirac algebra 904 00:58:18,610 --> 00:58:22,450 with the gamma and the gamma 5 and simplify this thing down 905 00:58:22,450 --> 00:58:31,195 and work out a transformation for delta pv and delta pv star. 906 00:58:34,950 --> 00:58:43,050 And what you'd find is that this guy in the rest frame here-- 907 00:58:43,050 --> 00:58:53,290 everything in the rest frame is that, and this guy, theta cross 908 00:58:53,290 --> 00:59:01,423 pvr star minus 1/2 theta pvr. 909 00:59:01,423 --> 00:59:03,090 So if you talk about the transformations 910 00:59:03,090 --> 00:59:06,330 under heavy quark spin symmetry of the states, 911 00:59:06,330 --> 00:59:09,007 these are actually getting mixed up into each other. 912 00:59:09,007 --> 00:59:10,590 And that's what we already were saying 913 00:59:10,590 --> 00:59:12,548 before when we were talking about spectroscopy, 914 00:59:12,548 --> 00:59:14,850 that spin symmetry was relating the two. 915 00:59:14,850 --> 00:59:17,760 This is us now seeing that very explicitly, 916 00:59:17,760 --> 00:59:20,270 that if you make a heavy quark spin symmetry transformation, 917 00:59:20,270 --> 00:59:22,410 the vector guy-- 918 00:59:22,410 --> 00:59:25,290 so this guy should have a star-- 919 00:59:25,290 --> 00:59:28,740 and the scalar guy get mixed up. 920 00:59:28,740 --> 00:59:30,960 And that's what spin symmetry allows you to do, 921 00:59:30,960 --> 00:59:33,660 is relate any statements about the v 922 00:59:33,660 --> 00:59:35,340 to statements about the v star. 923 00:59:35,340 --> 00:59:37,950 It's one thing it allows you to do. 924 00:59:37,950 --> 00:59:42,180 So the power of this Hvr thing is 925 00:59:42,180 --> 00:59:47,940 that it allows us to make heavy quark spin symmetry 926 00:59:47,940 --> 00:59:53,820 predictions easily because we've encoded 927 00:59:53,820 --> 00:59:55,080 the symmetry in this object. 928 00:59:55,080 --> 00:59:57,420 Kind of like our spurion analysis, 929 00:59:57,420 --> 01:00:01,020 we'll be able to use similar types of ideas 930 01:00:01,020 --> 01:00:04,747 to make spin symmetry predictions easy with this guy. 931 01:00:04,747 --> 01:00:06,330 So let me give you an example of that. 932 01:00:10,900 --> 01:00:16,050 So an example is talking about heavy quark decay constants. 933 01:00:16,050 --> 01:00:18,430 And we can think about B mesons or D mesons. 934 01:00:21,330 --> 01:00:23,220 Correspondingly, B stars and D stars. 935 01:00:30,960 --> 01:00:34,170 So think of the Q field here that I'm writing as a full QCD 936 01:00:34,170 --> 01:00:37,980 field, and the p field here with momentum p, 937 01:00:37,980 --> 01:00:40,350 it's either a B meson or a D meson. 938 01:00:43,020 --> 01:00:46,020 And just by Lorentz symmetry, much like one 939 01:00:46,020 --> 01:00:48,570 works out for the pion, one can write down 940 01:00:48,570 --> 01:00:53,443 a result that just falls from Lorentz invariance and parity 941 01:00:53,443 --> 01:00:55,110 for what the result for this coefficient 942 01:00:55,110 --> 01:00:57,075 is, because you also have to use time reversal. 943 01:01:05,400 --> 01:01:07,410 And you could also write the p as mp times 944 01:01:07,410 --> 01:01:10,500 v if you want to make explicit that there's 945 01:01:10,500 --> 01:01:13,860 some velocity that this thing, that this p squares 946 01:01:13,860 --> 01:01:17,080 to the mass of that state. 947 01:01:17,080 --> 01:01:20,860 So this guy here is some dimension 1 constant. 948 01:01:20,860 --> 01:01:27,750 And if we just, in QCD, do the same thing 949 01:01:27,750 --> 01:01:34,320 for the other guy, which is the vector, 950 01:01:34,320 --> 01:01:36,720 we can do the same game. 951 01:01:36,720 --> 01:01:38,430 Here we have to get a polarization 952 01:01:38,430 --> 01:01:39,818 on the right-hand side. 953 01:01:39,818 --> 01:01:41,610 There has to be linear in the polarization. 954 01:01:41,610 --> 01:01:43,830 That's the statement about a vector. 955 01:01:43,830 --> 01:01:46,980 And because of parity, it's not the gamma mu, gamma phi, 956 01:01:46,980 --> 01:01:49,590 but the gamma mu that comes in because we 957 01:01:49,590 --> 01:01:52,330 have spin 1 instead of spin 0. 958 01:01:52,330 --> 01:01:54,160 So there's an extra minus 1 from that. 959 01:01:54,160 --> 01:01:56,070 So instead of-- so we have a gamma mu here, 960 01:01:56,070 --> 01:01:58,110 and this is how we can parameterize the matrix 961 01:01:58,110 --> 01:01:58,950 element. 962 01:01:58,950 --> 01:02:01,290 And this guy here, if we work out the dimensions, 963 01:02:01,290 --> 01:02:03,090 it would be dimension 2. 964 01:02:03,090 --> 01:02:05,070 So just in QCD, it looks like we have 965 01:02:05,070 --> 01:02:08,400 these two things, fp and fp star, that are different. 966 01:02:08,400 --> 01:02:12,180 But we know heavy quark symmetry relates the p and the p star, 967 01:02:12,180 --> 01:02:14,230 so they actually should be related. 968 01:02:14,230 --> 01:02:16,650 So you could ask the question, how are they related? 969 01:02:16,650 --> 01:02:18,032 Work it out. 970 01:02:18,032 --> 01:02:19,740 And if I asked you that question, the way 971 01:02:19,740 --> 01:02:22,710 to work it out, which I'll show you, is to use this Hv field. 972 01:02:25,890 --> 01:02:28,560 OK, so there's two things we have to work out. 973 01:02:28,560 --> 01:02:31,420 We have to-- we're going to use the Hv field. 974 01:02:31,420 --> 01:02:32,170 I'll show you how. 975 01:02:32,170 --> 01:02:33,930 We also have to think about these currents 976 01:02:33,930 --> 01:02:36,780 that I've talked about here and writing them in terms of the Qv 977 01:02:36,780 --> 01:02:39,090 field, so let's do that first. 978 01:02:51,000 --> 01:02:53,280 So that's pretty easy. 979 01:02:58,650 --> 01:03:01,200 Just replace our field by the Qv field. 980 01:03:22,555 --> 01:03:24,930 Now, if I just want to make a statement about heavy quark 981 01:03:24,930 --> 01:03:27,570 symmetry, then I can look at how this guy transforms 982 01:03:27,570 --> 01:03:29,070 under heavy quark symmetry because I 983 01:03:29,070 --> 01:03:31,350 know how Qv transforms under heavy quark symmetry. 984 01:03:38,190 --> 01:03:46,800 So [INAUDIBLE] D. 985 01:03:46,800 --> 01:03:49,200 And so what I'd like to do is I'd like 986 01:03:49,200 --> 01:03:52,410 to encode in a general object-- 987 01:03:55,925 --> 01:03:57,800 much as we did in chiral perturbation theory, 988 01:03:57,800 --> 01:04:03,860 I'd like to encode in a general object that involves the Hv 989 01:04:03,860 --> 01:04:07,940 field something that has the same transformation 990 01:04:07,940 --> 01:04:10,280 properties as this current. 991 01:04:25,900 --> 01:04:28,540 Much in the same way we could think about putting-- 992 01:04:28,540 --> 01:04:30,760 connecting currents that transformed in a certain way 993 01:04:30,760 --> 01:04:32,260 to a chiral theory, we're doing the same thing 994 01:04:32,260 --> 01:04:33,070 with heavy quarks. 995 01:04:36,570 --> 01:04:40,710 So let's pretend-- in order to do this, 996 01:04:40,710 --> 01:04:43,110 instead of thinking about covariance, 997 01:04:43,110 --> 01:04:44,610 we'd want to think about invariance. 998 01:04:44,610 --> 01:04:49,530 So pretend that the Dirac structure transforms such 999 01:04:49,530 --> 01:04:52,120 that we cancel that D. Then we have an invariant, 1000 01:04:52,120 --> 01:04:55,530 and we can think about building invariance out of the Hv field. 1001 01:04:58,290 --> 01:05:00,900 And it's very simple because if we think about that, 1002 01:05:00,900 --> 01:05:04,060 gamma times Hv is invariant. 1003 01:05:09,450 --> 01:05:12,900 You only have to think about one Hv field 1004 01:05:12,900 --> 01:05:16,059 because the number of heavy quarks is preserved. 1005 01:05:20,450 --> 01:05:20,950 OK? 1006 01:05:20,950 --> 01:05:23,800 So we know there's only one, and then the other thing that 1007 01:05:23,800 --> 01:05:25,308 transforms is to gamma. 1008 01:05:25,308 --> 01:05:27,100 And so that we can form an invariant, which 1009 01:05:27,100 --> 01:05:37,350 is gamma H. So then if you look at this thing-- 1010 01:05:40,810 --> 01:05:44,220 and then the sandwiching with this guy-- 1011 01:05:47,940 --> 01:05:49,770 the indices of-- 1012 01:05:49,770 --> 01:05:51,270 So this guy's got a vector index, 1013 01:05:51,270 --> 01:05:52,687 and this guy's got a scalar index, 1014 01:05:52,687 --> 01:05:56,280 but that's all encoded in our Hv field, right? 1015 01:05:56,280 --> 01:05:58,810 It's got indices for those fields. 1016 01:05:58,810 --> 01:06:00,972 So we're basically constructing things-- 1017 01:06:00,972 --> 01:06:02,430 we're just basically constructing-- 1018 01:06:02,430 --> 01:06:04,748 if we've already accounted for the indices 1019 01:06:04,748 --> 01:06:06,540 here on the left-hand side of the equation, 1020 01:06:06,540 --> 01:06:09,960 then we're just constructing things that are scalars. 1021 01:06:09,960 --> 01:06:18,450 So Lorentz covariance requires us 1022 01:06:18,450 --> 01:06:25,750 to take a trace to get a scalar, and so we basically have trace. 1023 01:06:25,750 --> 01:06:28,190 We can put anything we like here. 1024 01:06:28,190 --> 01:06:33,910 And then we have gamma Hv, which is like the QCD stuff, and then 1025 01:06:33,910 --> 01:06:36,613 the heavy quark part. 1026 01:06:36,613 --> 01:06:38,530 So we just have to write down the most general 1027 01:06:38,530 --> 01:06:39,850 x that we can think of. 1028 01:06:44,380 --> 01:06:47,260 And there's not that much we can do 1029 01:06:47,260 --> 01:06:50,690 because there's not that many vectors available to us. 1030 01:06:50,690 --> 01:06:52,360 So there's two things that we can do. 1031 01:06:52,360 --> 01:06:55,455 We can write down something that just is a diagonal 1032 01:06:55,455 --> 01:06:56,705 or something that's a v slash. 1033 01:06:59,470 --> 01:07:01,300 But then we can simplify this guy 1034 01:07:01,300 --> 01:07:06,020 because Hv v slash is minus Hv. 1035 01:07:06,020 --> 01:07:08,000 See, I can take this, and this is a trace, 1036 01:07:08,000 --> 01:07:09,790 so I can move it over to the other side of the trace, 1037 01:07:09,790 --> 01:07:11,998 and if it's a v slash on the other side of the trace, 1038 01:07:11,998 --> 01:07:13,850 then I can use this formula. 1039 01:07:13,850 --> 01:07:17,380 And so these two guys here actually just become, 1040 01:07:17,380 --> 01:07:21,700 after taking that into account, one thing, which I'll just call 1041 01:07:21,700 --> 01:07:23,390 a. 1042 01:07:23,390 --> 01:07:24,850 So there's basically a unique thing 1043 01:07:24,850 --> 01:07:26,230 that we can write down for the x. 1044 01:07:26,230 --> 01:07:29,680 After we use all the symmetry properties of the Hv, 1045 01:07:29,680 --> 01:07:31,970 we only have one thing. 1046 01:07:31,970 --> 01:07:34,930 And so then you can just work out what that trace is. 1047 01:07:38,830 --> 01:07:40,480 Just take the trace. 1048 01:07:40,480 --> 01:07:46,990 And you get a, and you get minus i v mu pv, 1049 01:07:46,990 --> 01:07:52,930 or you get pv star mu, and this is what you get 1050 01:07:52,930 --> 01:07:55,163 if you have gamma mu, gamma 5. 1051 01:07:55,163 --> 01:07:57,080 And this is what you get if you have gamma mu, 1052 01:07:57,080 --> 01:07:58,960 and if you have the other way-- 1053 01:07:58,960 --> 01:08:01,210 so for gamma mu, you don't get any pseudoscalar, 1054 01:08:01,210 --> 01:08:03,430 and for gamma mu gamma 5, you don't get any vector. 1055 01:08:11,230 --> 01:08:14,190 So the way that you should think about this 1056 01:08:14,190 --> 01:08:22,855 is that in the matrix element and only in the matrix element, 1057 01:08:22,855 --> 01:08:25,140 but between these ground-state mesons, 1058 01:08:25,140 --> 01:08:32,310 I can really think that I've made some connection that 1059 01:08:32,310 --> 01:08:33,540 looks like as follows. 1060 01:08:33,540 --> 01:08:38,279 That Q gamma H v-- 1061 01:08:38,279 --> 01:08:51,830 sorry, Qv is a over 2 times this trace of gamma Hv 1062 01:08:51,830 --> 01:08:54,680 if I'm in the matrix element. 1063 01:08:54,680 --> 01:08:56,840 And then I know how to take matrix elements 1064 01:08:56,840 --> 01:08:58,215 with these fields because they're 1065 01:08:58,215 --> 01:09:00,840 the fields that actually are annihilating those states. 1066 01:09:00,840 --> 01:09:06,890 So then Q bar gamma mu, gamma 5, capital 1067 01:09:06,890 --> 01:09:14,330 Q. Now I'm using HQET states. 1068 01:09:14,330 --> 01:09:17,537 They're actually annihilating HQET states. 1069 01:09:17,537 --> 01:09:19,870 And I just use the formula that I've derived over there. 1070 01:09:34,189 --> 01:09:36,050 And I get these two results. 1071 01:09:36,050 --> 01:09:39,705 So for HQET states, I have only one constant, a. 1072 01:09:39,705 --> 01:09:41,330 And that's because heavy quark symmetry 1073 01:09:41,330 --> 01:09:42,680 related the decay constants. 1074 01:09:46,752 --> 01:09:48,210 And if I look at all the dimensions 1075 01:09:48,210 --> 01:09:51,149 of the things on this side and count them up, 1076 01:09:51,149 --> 01:09:52,607 I find that the dimensions here are 1077 01:09:52,607 --> 01:09:53,982 such that this would be something 1078 01:09:53,982 --> 01:09:55,230 that's lambda QCD to the 3/2. 1079 01:10:02,170 --> 01:10:06,100 So then I can try to go back and connect to my decay constants 1080 01:10:06,100 --> 01:10:08,050 what this a is. 1081 01:10:08,050 --> 01:10:10,120 And we know how to do that because I told you 1082 01:10:10,120 --> 01:10:12,830 how to connect the states, and we 1083 01:10:12,830 --> 01:10:16,060 talked about how to connect the currents, which is very simple. 1084 01:10:16,060 --> 01:10:18,640 These states here are dimension minus 3/2, 1085 01:10:18,640 --> 01:10:20,890 and that's basically what leads to the different power 1086 01:10:20,890 --> 01:10:23,510 on the right-hand side. 1087 01:10:23,510 --> 01:10:25,900 So connect the states. 1088 01:10:25,900 --> 01:10:29,620 That gives some extra factors of meson masses, 1089 01:10:29,620 --> 01:10:33,250 and we find that fp is this universal constant, 1090 01:10:33,250 --> 01:10:40,360 a over square root mp, and fp star is 1091 01:10:40,360 --> 01:10:44,080 this a times the square root of mp star, 1092 01:10:44,080 --> 01:10:46,270 and then we're getting correctly the dimension 2 1093 01:10:46,270 --> 01:10:54,850 or the dimension 1 quantities in terms of one universal a, OK? 1094 01:10:54,850 --> 01:10:56,440 So you can make predictions from this. 1095 01:10:56,440 --> 01:11:00,677 So for example, you could take fB, and you could say, well, 1096 01:11:00,677 --> 01:11:01,640 how big should fB be? 1097 01:11:01,640 --> 01:11:03,580 Well, it should be, by dimensional analysis, 1098 01:11:03,580 --> 01:11:07,960 now that we've figured out how to do dimensional analysis, 1099 01:11:07,960 --> 01:11:09,220 it should be of this size. 1100 01:11:09,220 --> 01:11:13,990 And that gives you something that's not too far off. 1101 01:11:13,990 --> 01:11:19,390 Or you could take fB over fD, and you could predict-- 1102 01:11:19,390 --> 01:11:21,640 because actually we don't care whether it's 1103 01:11:21,640 --> 01:11:24,085 a charm quark or a B quark. 1104 01:11:24,085 --> 01:11:26,710 The a here didn't care about the flavor of the p and the p star 1105 01:11:26,710 --> 01:11:27,080 state. 1106 01:11:27,080 --> 01:11:29,163 So everything I said here could apply equally well 1107 01:11:29,163 --> 01:11:31,340 to B states or D states. 1108 01:11:31,340 --> 01:11:32,140 So if we make-- 1109 01:11:32,140 --> 01:11:33,970 so the only place that it's coming in 1110 01:11:33,970 --> 01:11:37,450 is when we have this explicit square root here. 1111 01:11:37,450 --> 01:11:41,170 And just by power counting, we should say fB over fD 1112 01:11:41,170 --> 01:11:48,120 is like mD over mB, which is about 0.6. 1113 01:11:48,120 --> 01:11:48,620 OK? 1114 01:11:48,620 --> 01:11:51,290 So these are predictions that we can make based on symmetry. 1115 01:11:51,290 --> 01:11:53,655 And then if you have a prediction based on symmetry, 1116 01:11:53,655 --> 01:11:56,030 you could also talk about symmetry-violating corrections, 1117 01:11:56,030 --> 01:11:59,420 and people do that kind of thing to make-- 1118 01:11:59,420 --> 01:12:01,010 a is the lowest prediction, but you 1119 01:12:01,010 --> 01:12:04,070 could talk about violating it with some higher-order a's, and 1120 01:12:04,070 --> 01:12:07,580 those type of things are done. 1121 01:12:07,580 --> 01:12:10,170 The real place that this gives a lot of power, 1122 01:12:10,170 --> 01:12:12,650 I have to mention, although we'll not 1123 01:12:12,650 --> 01:12:14,510 talk about it in detail, is when you 1124 01:12:14,510 --> 01:12:17,373 look at semileptonic decays. 1125 01:12:17,373 --> 01:12:19,040 So if you look at the semileptonic decay 1126 01:12:19,040 --> 01:12:22,160 of a transition of a B meson into a D, 1127 01:12:22,160 --> 01:12:26,240 or a B meson into a D star, then you 1128 01:12:26,240 --> 01:12:28,250 can again use all these symmetry tricks 1129 01:12:28,250 --> 01:12:30,620 and again use this H v field-- 1130 01:12:30,620 --> 01:12:32,990 and some of the reading will talk 1131 01:12:32,990 --> 01:12:35,665 about this, these weak decays. 1132 01:12:35,665 --> 01:12:37,790 If you were to enumerate the number of form factors 1133 01:12:37,790 --> 01:12:40,040 you'd have in QCD, you'd have six form factors. 1134 01:12:45,380 --> 01:12:48,350 And once you go over to the heavy quark limit, 1135 01:12:48,350 --> 01:12:49,070 there's just one. 1136 01:13:00,420 --> 01:13:02,520 And it's actually normalized, too, 1137 01:13:02,520 --> 01:13:05,460 because it's related to a conserved charge. 1138 01:13:05,460 --> 01:13:07,200 [INAUDIBLE] spin symmetry to relate it. 1139 01:13:07,200 --> 01:13:09,033 You could use your heavy quark spin symmetry 1140 01:13:09,033 --> 01:13:10,740 to actually relate it to the-- 1141 01:13:10,740 --> 01:13:13,530 the operator that counts the number of B quarks. 1142 01:13:13,530 --> 01:13:15,600 So these form factors, you don't know anything 1143 01:13:15,600 --> 01:13:17,230 about their normalization. 1144 01:13:17,230 --> 01:13:18,330 There's six of them. 1145 01:13:18,330 --> 01:13:21,220 Take this limit, you get one, and it's normalized. 1146 01:13:21,220 --> 01:13:22,270 It's kind of nice. 1147 01:13:22,270 --> 01:13:23,770 It's called the Isgur-Wise function. 1148 01:13:25,940 --> 01:13:27,430 AUDIENCE: I have a quick question. 1149 01:13:27,430 --> 01:13:28,097 PROFESSOR: Yeah? 1150 01:13:28,097 --> 01:13:31,530 AUDIENCE: [INAUDIBLE] you said that that [INAUDIBLE].. 1151 01:13:31,530 --> 01:13:33,590 PROFESSOR: Yeah. 1152 01:13:33,590 --> 01:13:35,330 AUDIENCE: [INAUDIBLE] 1153 01:13:35,330 --> 01:13:36,580 PROFESSOR: Why did I say that? 1154 01:13:36,580 --> 01:13:39,250 So for example, imagine that I was 1155 01:13:39,250 --> 01:13:42,310 to have done this for not the p and the p star, 1156 01:13:42,310 --> 01:13:45,190 but for these other things that I was calling 1157 01:13:45,190 --> 01:13:49,090 B0 and B1, or B1 and B2. 1158 01:13:49,090 --> 01:13:50,710 Then when I would go through it, I 1159 01:13:50,710 --> 01:13:52,637 would have not used the Hv field. 1160 01:13:52,637 --> 01:13:54,220 I would have had to build a field that 1161 01:13:54,220 --> 01:13:57,580 has those guys in it, which would have spin 1 and spin 2. 1162 01:13:57,580 --> 01:13:59,560 And that would be a different matrix element. 1163 01:13:59,560 --> 01:14:01,910 I can still do the whole same story, 1164 01:14:01,910 --> 01:14:04,540 but really, this formula, this particular version 1165 01:14:04,540 --> 01:14:07,540 of the current, is only true for these states, 1166 01:14:07,540 --> 01:14:10,127 and for other states you'd have to use a different version. 1167 01:14:10,127 --> 01:14:10,960 That's what I meant. 1168 01:14:17,950 --> 01:14:19,407 Any other questions? 1169 01:14:24,550 --> 01:14:29,890 OK, so I want to just, in remaining two minutes, get 1170 01:14:29,890 --> 01:14:33,380 started on something else that I'd like to talk about, 1171 01:14:33,380 --> 01:14:37,780 which is related to the question that we left aside, 1172 01:14:37,780 --> 01:14:40,736 and that is, what about alpha s corrections? 1173 01:14:45,870 --> 01:14:48,510 And there's a couple of things here that are interesting. 1174 01:14:48,510 --> 01:14:51,450 But basically, it boils down to-- 1175 01:14:51,450 --> 01:14:54,720 We've already seen how to do matching and normalization 1176 01:14:54,720 --> 01:15:00,870 group evolution, but now we have these extra labels, 1177 01:15:00,870 --> 01:15:06,450 and so we can ask, what impact does having labels 1178 01:15:06,450 --> 01:15:09,060 on that whole story of matching and normalization group 1179 01:15:09,060 --> 01:15:10,560 evolution, what impact does it have? 1180 01:15:10,560 --> 01:15:12,060 And we'll say that there are examples 1181 01:15:12,060 --> 01:15:13,268 where it does have an impact. 1182 01:15:15,540 --> 01:15:16,040 So-- 1183 01:15:22,680 --> 01:15:24,900 HQET radiative corrections. 1184 01:15:27,570 --> 01:15:29,490 So if you like, we're just thinking 1185 01:15:29,490 --> 01:15:31,710 about the renormalization, as we should 1186 01:15:31,710 --> 01:15:34,290 for any effective theory, think about what's 1187 01:15:34,290 --> 01:15:37,050 the renormalization structure. 1188 01:15:37,050 --> 01:15:40,450 And since it's a top-down, we can also think about matching. 1189 01:15:42,913 --> 01:15:44,580 So we should think about renormalization 1190 01:15:44,580 --> 01:15:46,987 of L. We should think about renormalization 1191 01:15:46,987 --> 01:15:48,570 of external currents, like the ones we 1192 01:15:48,570 --> 01:15:50,800 were talking about over there. 1193 01:15:50,800 --> 01:15:57,110 And if we match a full theory current, 1194 01:15:57,110 --> 01:15:58,860 the way that you should think about what's 1195 01:15:58,860 --> 01:16:00,527 happening from matching is that there'll 1196 01:16:00,527 --> 01:16:05,970 be some Wilson coefficient that's generated 1197 01:16:05,970 --> 01:16:09,600 by radiative corrections. 1198 01:16:09,600 --> 01:16:13,320 So once we think about adding alpha s corrections 1199 01:16:13,320 --> 01:16:16,620 to our story, there could be radiative 1200 01:16:16,620 --> 01:16:19,170 corrections and Lagrangians-- we'll talk about that. 1201 01:16:19,170 --> 01:16:22,890 But there also are nontrivial radiative corrections here 1202 01:16:22,890 --> 01:16:26,310 in the currents, where they get multiplied by something that 1203 01:16:26,310 --> 01:16:29,183 depends on virtual interactions having 1204 01:16:29,183 --> 01:16:31,350 to do with antiparticles that we're integrating out, 1205 01:16:31,350 --> 01:16:33,300 and that just affects these currents, 1206 01:16:33,300 --> 01:16:35,160 the current relations, by multiplying them 1207 01:16:35,160 --> 01:16:36,695 by some coefficients. 1208 01:16:48,720 --> 01:16:49,220 Yeah. 1209 01:16:54,290 --> 01:16:56,150 So the first thing that actually differs, 1210 01:16:56,150 --> 01:16:59,210 and it's because, again, there's no antiparticles-- 1211 01:16:59,210 --> 01:17:01,490 that changes the nature of loop diagrams-- 1212 01:17:01,490 --> 01:17:04,280 is the wavefunction normalization. 1213 01:17:04,280 --> 01:17:05,880 So we have to go through that again. 1214 01:17:10,080 --> 01:17:12,080 So if we do field renormalization, 1215 01:17:12,080 --> 01:17:14,345 it's the same kind of story as it is in QCD. 1216 01:17:14,345 --> 01:17:15,720 We have to consider this diagram, 1217 01:17:15,720 --> 01:17:19,190 but we have to calculate it now in HQET. 1218 01:17:19,190 --> 01:17:21,620 So let there be some external-- 1219 01:17:21,620 --> 01:17:23,150 I'll call it p. 1220 01:17:23,150 --> 01:17:25,455 I should call it k. 1221 01:17:25,455 --> 01:17:26,290 Call it k. 1222 01:17:31,240 --> 01:17:33,640 There'll be some loop momentum, and we have q plus k. 1223 01:17:36,790 --> 01:17:48,120 If we use dimensional regularization, 1224 01:17:48,120 --> 01:17:51,100 then this loop integral looks like this. 1225 01:17:51,100 --> 01:17:55,590 So using [? m ?] [? s ?] bar, there's some usual [? m ?] 1226 01:17:55,590 --> 01:17:57,790 [? s bar-type ?] factors. 1227 01:17:57,790 --> 01:18:00,690 And then there's the relativistic propagator, 1228 01:18:00,690 --> 01:18:03,990 which I'm using Feynman gauge, so it's just q squared, 1229 01:18:03,990 --> 01:18:09,390 and then there's the heavy quark propagator, which 1230 01:18:09,390 --> 01:18:10,718 would look like that. 1231 01:18:10,718 --> 01:18:12,510 So the loop integral that I have to do here 1232 01:18:12,510 --> 01:18:14,700 wouldn't have two relativistic propagators. 1233 01:18:14,700 --> 01:18:16,380 It has one heavy quark propagator 1234 01:18:16,380 --> 01:18:18,570 and one relativistic propagator, and it 1235 01:18:18,570 --> 01:18:20,910 will give something different, and we'll 1236 01:18:20,910 --> 01:18:24,960 talk about it in more detail next time, about evaluating 1237 01:18:24,960 --> 01:18:26,120 this loop integral. 1238 01:18:26,120 --> 01:18:28,283 So I wanted to do one loop integral for you. 1239 01:18:28,283 --> 01:18:30,450 And basically the way I'm going to talk about these, 1240 01:18:30,450 --> 01:18:31,867 the renormalizations and matching, 1241 01:18:31,867 --> 01:18:33,408 is I'll do this loop integral for you 1242 01:18:33,408 --> 01:18:35,910 so you have some idea what would be involved if I was asking 1243 01:18:35,910 --> 01:18:39,165 you to do them yourself, which I don't think I'm going to do. 1244 01:18:39,165 --> 01:18:41,040 And then I'll just be quoting results for you 1245 01:18:41,040 --> 01:18:44,130 and telling you kind of how the structure of the theory looks 1246 01:18:44,130 --> 01:18:47,280 and where the corrections are going and talking 1247 01:18:47,280 --> 01:18:49,980 about the important things that come out of the results, 1248 01:18:49,980 --> 01:18:52,170 rather than dwelling on calculational details. 1249 01:18:52,170 --> 01:18:54,480 But we will talk about one calculation 1250 01:18:54,480 --> 01:18:57,210 because there's one trick that you should know if you ever 1251 01:18:57,210 --> 01:18:59,430 encounter an integral of this type, 1252 01:18:59,430 --> 01:19:02,130 since it's different than just using a Feynman parameter 1253 01:19:02,130 --> 01:19:03,490 trick. 1254 01:19:03,490 --> 01:19:05,608 And that difference comes about because this guy 1255 01:19:05,608 --> 01:19:06,900 is linear in the loop momentum. 1256 01:19:06,900 --> 01:19:08,640 This guy's quadratic. 1257 01:19:08,640 --> 01:19:10,690 You have to know how to deal with that, 1258 01:19:10,690 --> 01:19:13,250 so we'll talk about that next time.