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 00:00:20,960 --> 00:00:22,900 PROFESSOR: --the back of the room. 9 00:00:22,900 --> 00:00:24,910 This course is being taped. 10 00:00:24,910 --> 00:00:28,110 It may appear in the future on EdX or MITx. 11 00:00:28,110 --> 00:00:30,272 It will appear on OpenCourseWare. 12 00:00:30,272 --> 00:00:32,439 So that means that by just sitting in the classroom, 13 00:00:32,439 --> 00:00:35,590 you're choosing to participate in this adventure of having 14 00:00:35,590 --> 00:00:36,940 the course videotaped. 15 00:00:36,940 --> 00:00:39,340 If you want to hide, you can sit in the back row. 16 00:00:39,340 --> 00:00:42,892 If you want to be part of the-- 17 00:00:42,892 --> 00:00:44,850 if you want to be famous, sit in the front row. 18 00:00:52,330 --> 00:00:53,910 So I handed out the syllabus. 19 00:00:53,910 --> 00:00:57,100 And I'll just go through some of the things on here. 20 00:00:57,100 --> 00:00:58,800 So as a prerequisite-- 21 00:00:58,800 --> 00:01:02,070 so at MIT, we have three quantum field theory courses, 1, 2, 3. 22 00:01:02,070 --> 00:01:05,099 As a prerequisite, I've listed quantum field theory two. 23 00:01:05,099 --> 00:01:07,770 I don't want to stop people that are taking quantum field 24 00:01:07,770 --> 00:01:10,920 theory 3 this term from also registering for this course. 25 00:01:10,920 --> 00:01:13,020 But some of the things that you'd usually 26 00:01:13,020 --> 00:01:15,480 see in quantum field theory 3 are 27 00:01:15,480 --> 00:01:18,340 kind of important background things for this course. 28 00:01:18,340 --> 00:01:22,078 So what I've done is I've posted on the website for the course 29 00:01:22,078 --> 00:01:24,120 my lecture notes from when I taught quantum field 30 00:01:24,120 --> 00:01:25,730 theory 3 last year. 31 00:01:25,730 --> 00:01:26,730 So you can look at that. 32 00:01:26,730 --> 00:01:29,340 And I'll assign some reading for that as background material 33 00:01:29,340 --> 00:01:31,410 for this. 34 00:01:31,410 --> 00:01:34,440 Some things actually were taught in quantum field theory 2 35 00:01:34,440 --> 00:01:36,600 last term, some things like, for example, 36 00:01:36,600 --> 00:01:39,828 the beta function of QCD for normalization group. 37 00:01:39,828 --> 00:01:42,120 So those things are going to be important prerequisites 38 00:01:42,120 --> 00:01:43,770 for this course. 39 00:01:43,770 --> 00:01:46,170 And I'll assign some background reading 40 00:01:46,170 --> 00:01:49,900 so that you can remind yourself about those things. 41 00:01:49,900 --> 00:01:51,900 If you have any concerns about your preparation, 42 00:01:51,900 --> 00:01:54,390 please feel free to talk to me. 43 00:01:54,390 --> 00:01:57,040 In terms of grading, this is pretty low key. 44 00:01:57,040 --> 00:02:00,300 We have five problem sets that will 45 00:02:00,300 --> 00:02:01,920 be roughly every two weeks. 46 00:02:01,920 --> 00:02:06,120 We won't start the first problem set until next week. 47 00:02:06,120 --> 00:02:07,979 And then there'll be a presentation 48 00:02:07,979 --> 00:02:08,979 at the end of the class. 49 00:02:08,979 --> 00:02:11,520 There's no exams, no tests. 50 00:02:11,520 --> 00:02:14,370 The purpose of the presentation is basically, 51 00:02:14,370 --> 00:02:16,527 effective field theory is a very broad topic 52 00:02:16,527 --> 00:02:18,360 and there's no chance we'll be able to cover 53 00:02:18,360 --> 00:02:20,110 every possible thing about effective field 54 00:02:20,110 --> 00:02:21,300 theory in this course. 55 00:02:21,300 --> 00:02:23,010 In particular, a lot of applications 56 00:02:23,010 --> 00:02:27,610 we'll have to be brief on or we won't touch on at all. 57 00:02:27,610 --> 00:02:31,080 So if you look at the syllabus on page three, 58 00:02:31,080 --> 00:02:33,810 there's a list of possible presentation topics. 59 00:02:33,810 --> 00:02:35,010 Each one of you will pick-- 60 00:02:35,010 --> 00:02:36,900 each one of you that registers for the course 61 00:02:36,900 --> 00:02:40,380 and wants to give a presentation will give a presentation 62 00:02:40,380 --> 00:02:41,820 at the end on one of these topics, 63 00:02:41,820 --> 00:02:45,323 prepare it, and stand up and lecture to the class. 64 00:02:45,323 --> 00:02:46,740 And the idea is that basically you 65 00:02:46,740 --> 00:02:49,428 will teach the class for half an hour or so about one 66 00:02:49,428 --> 00:02:50,970 of these topics, and then that way we 67 00:02:50,970 --> 00:02:55,500 will broaden the scope of things that we'll be able to cover. 68 00:02:55,500 --> 00:02:59,430 Even this long list is not complete. 69 00:02:59,430 --> 00:03:01,740 And you can also pick a topic that's not on this list 70 00:03:01,740 --> 00:03:03,790 if you talk to me about it. 71 00:03:03,790 --> 00:03:07,622 So a little later I'll also give you some suggested references 72 00:03:07,622 --> 00:03:08,580 for reading about this. 73 00:03:08,580 --> 00:03:10,410 I just, at this point, give you the list. 74 00:03:10,410 --> 00:03:13,090 And we'll talk about where you might read about these things 75 00:03:13,090 --> 00:03:13,590 later on. 76 00:03:13,590 --> 00:03:17,340 I'll give you a more detailed handout. 77 00:03:17,340 --> 00:03:18,970 So even if you just look at this list, 78 00:03:18,970 --> 00:03:21,480 you kind of see the breadth of-- the idea of effective field 79 00:03:21,480 --> 00:03:27,240 theory, how broadly it impacts physics, and particle physics, 80 00:03:27,240 --> 00:03:28,230 in particular. 81 00:03:28,230 --> 00:03:31,200 So you know, everything from finite temperature QCD, 82 00:03:31,200 --> 00:03:35,280 finite density, effective field theories in inflation, 83 00:03:35,280 --> 00:03:36,215 popular these days. 84 00:03:36,215 --> 00:03:37,840 Effective field theories of cold atoms, 85 00:03:37,840 --> 00:03:39,810 are also very popular these days. 86 00:03:39,810 --> 00:03:43,470 Non-relativistic QCD production of quarkonia, both in medium, 87 00:03:43,470 --> 00:03:44,820 out of medium. 88 00:03:44,820 --> 00:03:46,860 Relativistic super fluids. 89 00:03:46,860 --> 00:03:48,320 Conformal effective field theories. 90 00:03:48,320 --> 00:03:49,320 The list goes on and on. 91 00:03:49,320 --> 00:03:51,120 Lattice effective field theories. 92 00:03:51,120 --> 00:03:52,650 So it's a very broad topic. 93 00:03:52,650 --> 00:03:54,540 And what I'm going to do in this course 94 00:03:54,540 --> 00:03:57,180 is I'm going to start out with the first half of the course 95 00:03:57,180 --> 00:03:59,693 teaching you about the ideas of effective field theory, 96 00:03:59,693 --> 00:04:01,110 some of the basic ingredients that 97 00:04:01,110 --> 00:04:03,060 go into all effective field theories 98 00:04:03,060 --> 00:04:05,645 and some more technical things that only show up 99 00:04:05,645 --> 00:04:07,020 in some effective field theories, 100 00:04:07,020 --> 00:04:08,702 but that are kind of providing you 101 00:04:08,702 --> 00:04:10,410 with lessons if you had to ever construct 102 00:04:10,410 --> 00:04:11,880 your own effective field theory. 103 00:04:11,880 --> 00:04:14,153 That's the idea of the first half of the course. 104 00:04:14,153 --> 00:04:15,570 The second half of the course, I'm 105 00:04:15,570 --> 00:04:17,862 going to focus on a particular effective field theory-- 106 00:04:17,862 --> 00:04:19,860 soft-colinear effective field theory, which 107 00:04:19,860 --> 00:04:25,860 is an effective field theory that's close to my heart that's 108 00:04:25,860 --> 00:04:28,350 exciting these days because of its applications to the jet 109 00:04:28,350 --> 00:04:30,690 physics in the LHC. 110 00:04:30,690 --> 00:04:33,270 So we'll talk about that as the second half of the course. 111 00:04:38,150 --> 00:04:41,060 So as I say on the outline, there's not really 112 00:04:41,060 --> 00:04:44,330 a textbook for this course. 113 00:04:44,330 --> 00:04:46,830 I suggest some text that may be useful, 114 00:04:46,830 --> 00:04:48,710 but they're not any one of them will 115 00:04:48,710 --> 00:04:51,740 be useful for more than 20% of the material, many of them 116 00:04:51,740 --> 00:04:53,640 for less than 10% of the material. 117 00:04:53,640 --> 00:04:55,640 So I don't necessarily recommend that you go out 118 00:04:55,640 --> 00:04:57,890 and buy all of them-- or buy any of them. 119 00:04:57,890 --> 00:05:01,040 I will try to post chapters of things 120 00:05:01,040 --> 00:05:03,020 on the website when it's required reading. 121 00:05:03,020 --> 00:05:05,270 And these books will be available in the reading room, 122 00:05:05,270 --> 00:05:07,580 so you can do it that way, if you like. 123 00:05:10,767 --> 00:05:13,100 If you're really serious about phenomenology, of course, 124 00:05:13,100 --> 00:05:14,392 you should own all these books. 125 00:05:20,630 --> 00:05:24,300 So on the course outline, I also listed office hours. 126 00:05:24,300 --> 00:05:29,130 And I will adjust those when I find out all your schedules. 127 00:05:29,130 --> 00:05:30,710 And I also listed that [INAUDIBLE] 128 00:05:30,710 --> 00:05:34,760 who's sitting in the back, is a 10% TA for the course. 129 00:05:34,760 --> 00:05:36,590 He'll be doing some grading for us. 130 00:05:36,590 --> 00:05:37,715 He won't have office hours. 131 00:05:42,604 --> 00:05:43,437 AUDIENCE: Thank you. 132 00:05:46,898 --> 00:05:48,440 PROFESSOR: So I know that some of you 133 00:05:48,440 --> 00:05:49,732 have registered for the course. 134 00:05:49,732 --> 00:05:51,320 And some of you are here as listeners. 135 00:05:51,320 --> 00:05:54,410 If you are here is a listener, I have an add form 136 00:05:54,410 --> 00:05:56,450 and I want you to add it as a listener, 137 00:05:56,450 --> 00:05:58,640 just so that we have a record of you, 138 00:05:58,640 --> 00:06:00,770 partly because that affects the ability 139 00:06:00,770 --> 00:06:02,600 to offer this course in the future, 140 00:06:02,600 --> 00:06:05,090 also because we're going to use online tools 141 00:06:05,090 --> 00:06:07,790 and we may even use some online MITx 142 00:06:07,790 --> 00:06:11,010 tools if I find out that they're very useful for the course. 143 00:06:11,010 --> 00:06:12,600 So I want you to be in the system. 144 00:06:12,600 --> 00:06:16,070 And I want you to be able to access that information if we 145 00:06:16,070 --> 00:06:17,000 end up going that way. 146 00:06:21,790 --> 00:06:24,110 So I think that's it. 147 00:06:24,110 --> 00:06:27,520 So as far as preamble, does anybody have any questions? 148 00:06:32,110 --> 00:06:32,610 OK. 149 00:06:48,110 --> 00:06:48,610 All right. 150 00:06:48,610 --> 00:06:50,965 So let's start. 151 00:06:50,965 --> 00:06:52,465 So let's start with the big picture. 152 00:06:59,820 --> 00:07:01,370 I like to say that the big picture is 153 00:07:01,370 --> 00:07:03,328 that there's interesting physics at all scales. 154 00:07:16,010 --> 00:07:17,690 What effective field theory lets you do 155 00:07:17,690 --> 00:07:19,760 is it lets you tease out this interesting physics 156 00:07:19,760 --> 00:07:21,080 at all scales. 157 00:07:21,080 --> 00:07:25,190 So in particular, you can focus on a particular scale 158 00:07:25,190 --> 00:07:27,470 and find the interesting physics there using 159 00:07:27,470 --> 00:07:30,620 the tools of effective field theory. 160 00:07:30,620 --> 00:07:33,200 Now that's a little different than how we teach physics. 161 00:07:33,200 --> 00:07:36,860 If you think about drawing a diagram from your freshman year 162 00:07:36,860 --> 00:07:40,490 to now many of you in graduate studies about how you learned 163 00:07:40,490 --> 00:07:43,400 physics, it was not by kind of focusing in 164 00:07:43,400 --> 00:07:45,980 on individual particular scales, but more 165 00:07:45,980 --> 00:07:49,230 from a kind of bottom-up point of view. 166 00:07:49,230 --> 00:07:52,980 So I would draw a picture of how we teach 167 00:07:52,980 --> 00:07:55,870 physics in the following way. 168 00:07:55,870 --> 00:07:57,540 So you start out as a freshman. 169 00:07:57,540 --> 00:07:58,980 You learn classical mechanics. 170 00:08:06,480 --> 00:08:17,640 You learn E&M, classical E&M. You learn Newtonian gravity. 171 00:08:22,932 --> 00:08:24,390 And then you build on these things. 172 00:08:24,390 --> 00:08:25,530 These things are your starting point 173 00:08:25,530 --> 00:08:26,655 and then you build on them. 174 00:08:26,655 --> 00:08:35,429 So you learn quantum mechanics and that 175 00:08:35,429 --> 00:08:38,440 builds on classical mechanics. 176 00:08:38,440 --> 00:08:46,590 You learn special relativity, which 177 00:08:46,590 --> 00:08:53,220 is building on both classical mechanics and electromagnetism. 178 00:08:53,220 --> 00:09:09,900 At some point, you learn general relativity and quantum field 179 00:09:09,900 --> 00:09:10,400 theory. 180 00:09:17,510 --> 00:09:19,250 And quantum field theory is synthesizing 181 00:09:19,250 --> 00:09:23,880 quantum mechanics and special relativity, and then these two 182 00:09:23,880 --> 00:09:24,380 here. 183 00:09:35,065 --> 00:09:37,190 So you think about learning physics from the bottom 184 00:09:37,190 --> 00:09:38,562 up in this picture. 185 00:09:38,562 --> 00:09:39,770 You learn these things first. 186 00:09:39,770 --> 00:09:40,700 And then you learn these things. 187 00:09:40,700 --> 00:09:42,140 And you sort of keep synthesizing, 188 00:09:42,140 --> 00:09:44,257 keep putting things together. 189 00:09:44,257 --> 00:09:45,840 We're going to be doing in this course 190 00:09:45,840 --> 00:09:47,930 the exact opposite of that. 191 00:09:47,930 --> 00:09:49,910 We're going to be taking one of these blobs-- 192 00:09:49,910 --> 00:09:51,523 in particular, this one-- 193 00:09:51,523 --> 00:09:52,940 and we're going to be looking deep 194 00:09:52,940 --> 00:09:55,665 inside it trying to make more and more specific field 195 00:09:55,665 --> 00:09:56,165 theories. 196 00:10:06,700 --> 00:10:09,870 So if you like, if you want to draw it as a blob, 197 00:10:09,870 --> 00:10:13,500 we'll be taking quantum field theory and we'll be 198 00:10:13,500 --> 00:10:16,410 looking for derivatives of it-- 199 00:10:16,410 --> 00:10:19,425 or perhaps derivatives of derivatives of it, 200 00:10:19,425 --> 00:10:22,050 like this-- and figuring out how to take a very general theory, 201 00:10:22,050 --> 00:10:24,630 like quantum field theory for the standard model, 202 00:10:24,630 --> 00:10:27,960 and finding things that are more specific and in some ways more 203 00:10:27,960 --> 00:10:31,040 powerful than just having the original theory 204 00:10:31,040 --> 00:10:33,390 we started with-- more powerful in the sense of being 205 00:10:33,390 --> 00:10:34,620 able to do calculations. 206 00:11:05,300 --> 00:11:06,520 So why do we want to do that? 207 00:11:11,470 --> 00:11:13,820 There's a couple of different reasons-- 208 00:11:13,820 --> 00:11:16,390 or why do we do that, since, as I've 209 00:11:16,390 --> 00:11:18,610 tried to convince you with the outline of the course 210 00:11:18,610 --> 00:11:20,230 and some of the presentation topics, 211 00:11:20,230 --> 00:11:23,170 that this is something that happens all over the place. 212 00:11:27,820 --> 00:11:31,260 So as you go up in this chart, it actually becomes-- 213 00:11:31,260 --> 00:11:33,010 even though you have a more general theory 214 00:11:33,010 --> 00:11:35,177 and it becomes more beautiful and you can write down 215 00:11:35,177 --> 00:11:37,710 a synthesis of physics in fewer lines, 216 00:11:37,710 --> 00:11:39,910 it also becomes harder to compute things. 217 00:11:47,260 --> 00:11:50,080 So just as an example, if you just 218 00:11:50,080 --> 00:11:54,820 wanted to compute the energy spectrum of hydrogen, 219 00:11:54,820 --> 00:11:56,740 and you know very well that you can 220 00:11:56,740 --> 00:11:58,060 do that in quantum mechanics. 221 00:11:58,060 --> 00:12:02,443 And particularly it's a classic example, and fairly easy. 222 00:12:02,443 --> 00:12:04,360 If you try to do that in quantum field theory, 223 00:12:04,360 --> 00:12:06,130 it's much harder, because quantum field 224 00:12:06,130 --> 00:12:09,143 theory, in some sense, has too much for that problem. 225 00:12:09,143 --> 00:12:10,060 So that's one example. 226 00:12:13,760 --> 00:12:15,760 Another is the elliptical orbits of the planets, 227 00:12:15,760 --> 00:12:18,490 which are easier in Newtonian gravity 228 00:12:18,490 --> 00:12:22,360 than in general relativity. 229 00:12:22,360 --> 00:12:23,790 And these are just two examples. 230 00:12:23,790 --> 00:12:26,800 There's many more where the field-- 231 00:12:26,800 --> 00:12:29,440 one of these blobs may be too general for actually 232 00:12:29,440 --> 00:12:34,040 tackling the problem that you want to deal with. 233 00:12:34,040 --> 00:12:37,060 And by focusing in, which is what effective field 234 00:12:37,060 --> 00:12:41,710 theory allows you to do, you can get more ability 235 00:12:41,710 --> 00:12:45,730 to compute more accurately and in a simpler fashion. 236 00:12:54,220 --> 00:13:00,030 So what we want when we think about effective field theory 237 00:13:00,030 --> 00:13:02,280 is we want the simplest framework that 238 00:13:02,280 --> 00:13:03,690 captures the essential physics. 239 00:13:07,920 --> 00:13:09,720 We don't want to carry along for the ride 240 00:13:09,720 --> 00:13:11,220 a whole bunch of superfluous things 241 00:13:11,220 --> 00:13:13,053 that are not important for the problem we're 242 00:13:13,053 --> 00:13:15,730 trying to deal with. 243 00:13:15,730 --> 00:13:17,970 But we also don't want to give up anything. 244 00:13:17,970 --> 00:13:22,170 So we're very demanding. 245 00:13:22,170 --> 00:13:26,180 So even if we're giving up something in our leading order 246 00:13:26,180 --> 00:13:28,400 description, we want to retain the ability 247 00:13:28,400 --> 00:13:30,770 to correct that leading order description-- 248 00:13:30,770 --> 00:13:34,070 order by order in some expansions-- 249 00:13:34,070 --> 00:13:36,375 so that we can make it as precise as we desire. 250 00:13:42,260 --> 00:13:48,800 So I'll say that we can correct it, in principle, to arbitrary 251 00:13:48,800 --> 00:13:49,430 precision. 252 00:13:52,322 --> 00:13:53,780 So if you like, what we're doing is 253 00:13:53,780 --> 00:13:56,690 we're taking quantum field theory and we're expanding it. 254 00:13:56,690 --> 00:13:58,940 And the lowest order determine that description 255 00:13:58,940 --> 00:14:00,800 is an effective field theory. 256 00:14:00,800 --> 00:14:04,400 And that effective field theory may have different fields. 257 00:14:04,400 --> 00:14:07,670 It may have different symmetries. 258 00:14:07,670 --> 00:14:10,112 And it will certainly have ability 259 00:14:10,112 --> 00:14:11,570 to calculate in a different fashion 260 00:14:11,570 --> 00:14:13,100 than the original theory. 261 00:14:13,100 --> 00:14:15,615 But we will keep higher order terms in that expansion. 262 00:14:15,615 --> 00:14:17,240 And therefore, we'll be able to correct 263 00:14:17,240 --> 00:14:20,750 it to arbitrary precision just by expanding 264 00:14:20,750 --> 00:14:23,180 to higher and higher order. 265 00:14:23,180 --> 00:14:26,540 So examples of this that even are familiar here-- 266 00:14:26,540 --> 00:14:28,520 non relativistic expansion, getting back 267 00:14:28,520 --> 00:14:30,860 to non relativistic quantum mechanics, 268 00:14:30,860 --> 00:14:34,400 or doing a post-Newtonian expansion in general relativity 269 00:14:34,400 --> 00:14:36,352 to go back towards Newtonian gravity. 270 00:14:36,352 --> 00:14:38,060 You don't have to stop at the first term. 271 00:14:38,060 --> 00:14:39,740 You can keep higher-order terms. 272 00:14:39,740 --> 00:14:41,310 And in that way, you could calculate, 273 00:14:41,310 --> 00:14:43,010 for example, energy levels in hydrogen 274 00:14:43,010 --> 00:14:45,110 using a non-relativistic framework that 275 00:14:45,110 --> 00:14:48,260 encodes all the ingredients of quantum field theory. 276 00:14:48,260 --> 00:14:49,730 Or you could use-- 277 00:14:49,730 --> 00:14:52,142 you could look at the orbit of planets 278 00:14:52,142 --> 00:14:54,350 and relativistic corrections and general relativistic 279 00:14:54,350 --> 00:14:57,557 corrections by expanding this theory. 280 00:14:57,557 --> 00:14:58,640 So those are two examples. 281 00:15:11,980 --> 00:15:14,670 So most of the examples that I've listed on the project list 282 00:15:14,670 --> 00:15:17,010 are the type of taking quantum field 283 00:15:17,010 --> 00:15:21,040 theory for the standard model. 284 00:15:21,040 --> 00:15:23,110 Some of them involve gravity, but most of them 285 00:15:23,110 --> 00:15:28,960 involve the standard model and expanding it 286 00:15:28,960 --> 00:15:33,270 and focusing in on particular degrees of freedom. 287 00:15:33,270 --> 00:15:35,270 So let's say that we've picked a physical system 288 00:15:35,270 --> 00:15:36,920 and we want to describe it. 289 00:15:36,920 --> 00:15:38,390 What are the things that we should 290 00:15:38,390 --> 00:15:41,000 do in order to develop an effective field theory? 291 00:15:51,828 --> 00:15:52,620 What are the steps? 292 00:15:56,050 --> 00:15:58,330 I should say that I'm going to post my lecture notes. 293 00:15:58,330 --> 00:16:00,580 So if you don't want to take notes, you don't have to. 294 00:16:00,580 --> 00:16:04,470 If you'd like to take notes, feel free. 295 00:16:04,470 --> 00:16:06,310 But I will scan and post my notes. 296 00:16:22,305 --> 00:16:23,680 So the first thing you need to do 297 00:16:23,680 --> 00:16:25,972 is figure out what the relevant degrees of freedom are. 298 00:16:29,865 --> 00:16:31,240 What are the things that actually 299 00:16:31,240 --> 00:16:33,850 matter for the problem you want to study? 300 00:16:33,850 --> 00:16:36,500 Sometimes that is easier than other times. 301 00:16:36,500 --> 00:16:39,070 Sometimes it's completely obvious. 302 00:16:39,070 --> 00:16:41,063 You want to study some low energy properties 303 00:16:41,063 --> 00:16:41,980 of the standard model. 304 00:16:41,980 --> 00:16:43,438 You get rid of the heavy particles. 305 00:16:43,438 --> 00:16:44,740 You keep the light ones. 306 00:16:44,740 --> 00:16:46,270 Fairly straightforward. 307 00:16:46,270 --> 00:16:48,370 Other times it may be tricky to actually determine 308 00:16:48,370 --> 00:16:50,230 what the relevant degrees of freedom are. 309 00:16:50,230 --> 00:16:53,330 And people in the field may even argue about what they are. 310 00:16:53,330 --> 00:16:55,090 So we'll talk about examples of both types 311 00:16:55,090 --> 00:16:59,420 here throughout the course. 312 00:16:59,420 --> 00:17:02,296 So it sounds trivial, but it may not be. 313 00:17:02,296 --> 00:17:04,588 You also want to think about the symmetries. 314 00:17:04,588 --> 00:17:06,130 Sometimes that guides you in thinking 315 00:17:06,130 --> 00:17:07,713 about the relevant degrees of freedom. 316 00:17:07,713 --> 00:17:10,240 Sometimes these things go hand in hand. 317 00:17:10,240 --> 00:17:12,190 But that's certainly an important ingredient 318 00:17:12,190 --> 00:17:16,119 in developing the effective field theory. 319 00:17:16,119 --> 00:17:18,670 And you also have to be careful here, because sometimes you 320 00:17:18,670 --> 00:17:22,030 might have a theory that has no symmetry, 321 00:17:22,030 --> 00:17:24,280 or doesn't have an apparent symmetry. 322 00:17:24,280 --> 00:17:26,829 But when you start expanding-- which is what I've argued 323 00:17:26,829 --> 00:17:28,000 you're going to be doing-- 324 00:17:28,000 --> 00:17:30,910 when you start expanding, you may have a symmetry suddenly 325 00:17:30,910 --> 00:17:31,870 appear. 326 00:17:31,870 --> 00:17:34,270 And we'll actually talk about several examples 327 00:17:34,270 --> 00:17:38,690 of that happening throughout the course, as well. 328 00:17:38,690 --> 00:17:40,990 So your effective field theory may have more symmetry 329 00:17:40,990 --> 00:17:43,402 than the theory you started with. 330 00:17:43,402 --> 00:17:44,860 Because you've neglected something, 331 00:17:44,860 --> 00:17:46,068 you could have more symmetry. 332 00:17:48,763 --> 00:17:50,680 And the other important thing is to figure out 333 00:17:50,680 --> 00:17:51,722 what you're expanding in. 334 00:17:58,267 --> 00:18:00,100 And at the same time, what the leading order 335 00:18:00,100 --> 00:18:01,690 description of the theory is. 336 00:18:01,690 --> 00:18:04,622 What is the lowest order Lagrangian? 337 00:18:04,622 --> 00:18:06,080 And basically, these are the things 338 00:18:06,080 --> 00:18:08,560 you have to do to get started. 339 00:18:08,560 --> 00:18:11,620 If this is true, actually, independent of kind 340 00:18:11,620 --> 00:18:13,180 of what theory you're talking about, 341 00:18:13,180 --> 00:18:16,240 if you're doing quantum field theory, what this first one 342 00:18:16,240 --> 00:18:17,620 means is figuring out what fields 343 00:18:17,620 --> 00:18:18,662 you're going to be using. 344 00:18:25,000 --> 00:18:27,040 The symmetry is basically guiding you 345 00:18:27,040 --> 00:18:28,750 about the interactions. 346 00:18:32,320 --> 00:18:34,210 If you have gauge symmetry, then of course 347 00:18:34,210 --> 00:18:36,070 you're going to write down something 348 00:18:36,070 --> 00:18:37,277 that respects that symmetry. 349 00:18:37,277 --> 00:18:39,610 That's going to tell you something about the interaction 350 00:18:39,610 --> 00:18:41,050 terms. 351 00:18:41,050 --> 00:18:44,380 And then finally, this expansion parameters 352 00:18:44,380 --> 00:18:47,050 goes under the rubric of what is called power counting. 353 00:18:59,940 --> 00:19:01,920 And if you take those three things together 354 00:19:01,920 --> 00:19:03,962 and you figure out the leading order description, 355 00:19:03,962 --> 00:19:05,640 then you have an effective field theory. 356 00:19:05,640 --> 00:19:07,140 You write down he Lagrangian for it. 357 00:19:17,736 --> 00:19:19,370 You should try to use some color. 358 00:19:29,950 --> 00:19:32,340 So if you thought about regular quantum field theory-- 359 00:19:32,340 --> 00:19:34,098 for example, for the standard model-- 360 00:19:34,098 --> 00:19:35,640 you'd also do these first two things. 361 00:19:35,640 --> 00:19:36,840 You figure out what the fields are. 362 00:19:36,840 --> 00:19:38,040 You figure out what the interactions are. 363 00:19:38,040 --> 00:19:40,332 But you don't think about too much about this question, 364 00:19:40,332 --> 00:19:42,060 about the power counting. 365 00:19:42,060 --> 00:19:45,420 And actually, more broadly, in the field of effective field 366 00:19:45,420 --> 00:19:49,690 theory, this idea here of power counting is very important. 367 00:19:49,690 --> 00:19:52,650 It's as important as something like gauge symmetry. 368 00:19:52,650 --> 00:19:55,530 It's really a fundamental thing about the whole framework 369 00:19:55,530 --> 00:19:56,280 that you're doing. 370 00:20:03,230 --> 00:20:17,720 So in an effective field theory, power counting 371 00:20:17,720 --> 00:20:23,090 is just as important as figuring out things like symmetries. 372 00:20:23,090 --> 00:20:24,590 And in particular, just to make it 373 00:20:24,590 --> 00:20:27,283 sort of clear that it's important, 374 00:20:27,283 --> 00:20:28,700 I'll compare it to gauge symmetry. 375 00:20:31,400 --> 00:20:35,030 It's really a fundamental ingredient in what you're doing 376 00:20:35,030 --> 00:20:38,343 and in the whole theory, because the power counting 377 00:20:38,343 --> 00:20:40,760 being consistent is actually necessary for the whole field 378 00:20:40,760 --> 00:20:41,270 theory-- 379 00:20:41,270 --> 00:20:42,895 effective field theory-- to make sense. 380 00:20:46,200 --> 00:20:47,910 OK, so what's the key principle? 381 00:20:50,640 --> 00:20:53,640 Well, there's a key principle of quantum field theory 382 00:20:53,640 --> 00:20:58,500 that we're using when we design effective field theories. 383 00:20:58,500 --> 00:21:01,385 And that is that we're insensitive to physics 384 00:21:01,385 --> 00:21:02,385 at higher energy scales. 385 00:21:10,654 --> 00:21:15,630 So going back to Wilson, if we're 386 00:21:15,630 --> 00:21:18,210 interested in describing the physics at some scale 387 00:21:18,210 --> 00:21:24,660 m squared, we don't need to know the details of physics 388 00:21:24,660 --> 00:21:25,410 at higher energy. 389 00:21:48,978 --> 00:21:53,028 So at scales lambda squared that are much bigger than m squared, 390 00:21:53,028 --> 00:21:55,570 we don't really need to know the details of the dynamics that 391 00:21:55,570 --> 00:21:56,962 are going on there. 392 00:21:56,962 --> 00:21:59,170 We don't need to know the field content, necessarily, 393 00:21:59,170 --> 00:22:03,505 at those scales, or anything else about the dynamics. 394 00:22:08,110 --> 00:22:10,575 And that is, in some sense, a key idea 395 00:22:10,575 --> 00:22:12,700 that makes the whole idea of effective field theory 396 00:22:12,700 --> 00:22:13,200 possible. 397 00:22:18,730 --> 00:22:20,470 I want you to make sure that you don't 398 00:22:20,470 --> 00:22:22,020 think of this too narrowly. 399 00:22:22,020 --> 00:22:24,520 In the way that I've written, it's actually a little bit too 400 00:22:24,520 --> 00:22:27,100 narrow, because I said that-- 401 00:22:27,100 --> 00:22:28,940 I've described it in terms of mass scale. 402 00:22:28,940 --> 00:22:31,120 So I've kind of intuited in your mind something 403 00:22:31,120 --> 00:22:33,940 like a Z' boson or a W' boson, which 404 00:22:33,940 --> 00:22:36,850 is a heavy particle from some perspective. 405 00:22:36,850 --> 00:22:39,490 And maybe they treat that as a heavy particle and I'm 406 00:22:39,490 --> 00:22:41,760 interested-- 407 00:22:41,760 --> 00:22:43,925 and let me say it differently. 408 00:22:43,925 --> 00:22:45,550 Say I'm interested in a light particle, 409 00:22:45,550 --> 00:22:47,860 like the bottom quark. 410 00:22:47,860 --> 00:22:51,460 Then I can get rid of physics at a heavy scale, like the w mass. 411 00:22:51,460 --> 00:22:55,630 Or if I think about how new physics impacts precision 412 00:22:55,630 --> 00:22:58,390 electroweak data and that new physics is heavy, 413 00:22:58,390 --> 00:23:00,580 then I can think about effective operators. 414 00:23:00,580 --> 00:23:01,955 That's kind of the intuition I've 415 00:23:01,955 --> 00:23:04,080 given you with the sentence that I've written here. 416 00:23:04,080 --> 00:23:05,200 But it's actually not-- 417 00:23:05,200 --> 00:23:07,670 that's not quite general enough. 418 00:23:07,670 --> 00:23:09,610 And the reason it's not quite general enough 419 00:23:09,610 --> 00:23:13,780 is that this mentality doesn't always apply just strictly 420 00:23:13,780 --> 00:23:15,820 to math scales. 421 00:23:15,820 --> 00:23:18,555 If I say this-- m squared much less than lambda squared-- then 422 00:23:18,555 --> 00:23:19,930 you immediately think that you're 423 00:23:19,930 --> 00:23:22,480 expanding an m squared over lambda squared. 424 00:23:22,480 --> 00:23:24,470 And that, of classic effective field theory, 425 00:23:24,470 --> 00:23:26,150 that's exactly what you do. 426 00:23:26,150 --> 00:23:28,990 You expand in light scales divided by heavy scales. 427 00:23:28,990 --> 00:23:29,920 They're mass scales. 428 00:23:29,920 --> 00:23:31,030 They're invariant masses. 429 00:23:31,030 --> 00:23:33,100 They're Lorentz invariant quantities. 430 00:23:33,100 --> 00:23:35,092 That's the most common thing that you do. 431 00:23:35,092 --> 00:23:37,550 We're going to be doing much more than that in this course. 432 00:23:37,550 --> 00:23:39,675 We'll be doing-- we'll be talking about cases where 433 00:23:39,675 --> 00:23:43,150 the power counting is in situations that are not simply 434 00:23:43,150 --> 00:23:46,300 m squared much less than lambda squared, but other things-- 435 00:23:46,300 --> 00:23:48,010 dimensionless parameters. 436 00:23:48,010 --> 00:23:50,025 So sometimes it will become more complicated. 437 00:23:50,025 --> 00:23:51,400 But still, this guiding principle 438 00:23:51,400 --> 00:23:55,450 that there's physics that's far away, in some sense, 439 00:23:55,450 --> 00:23:57,820 from the physics that you're interested in, that it can 440 00:23:57,820 --> 00:24:01,900 be removed from the theory when you're just talking about it, 441 00:24:01,900 --> 00:24:03,640 in that sense, this is more general. 442 00:24:03,640 --> 00:24:06,370 It's really not just a strictly one-dimensional thing, 443 00:24:06,370 --> 00:24:09,310 as it would be if I describe it in terms of mass scales, 444 00:24:09,310 --> 00:24:11,890 but a more general principle. 445 00:24:11,890 --> 00:24:14,260 And I think that will become clearer 446 00:24:14,260 --> 00:24:16,720 when we actually do examples where it's not simply 447 00:24:16,720 --> 00:24:17,980 mass scales. 448 00:24:17,980 --> 00:24:21,040 We will, of course, start out by talking about mass scale, 449 00:24:21,040 --> 00:24:23,835 since that's the classic thing that most people think 450 00:24:23,835 --> 00:24:25,710 of when they think of effective field theory. 451 00:24:28,420 --> 00:24:29,950 So can anyone give me an example-- 452 00:24:29,950 --> 00:24:32,180 just to try to get the class engaged here-- 453 00:24:32,180 --> 00:24:35,170 can anyone give me an example of an effective field theory that 454 00:24:35,170 --> 00:24:38,815 doesn't involve expanding strictly in mass scales, 455 00:24:38,815 --> 00:24:41,477 involves some other type of expansion? 456 00:24:41,477 --> 00:24:43,690 AUDIENCE: [INAUDIBLE] 457 00:24:43,690 --> 00:24:46,270 PROFESSOR: Well, OK. 458 00:24:46,270 --> 00:24:49,150 HQT you can say is just an expansion 459 00:24:49,150 --> 00:24:55,564 in lambda QCD over MB or MC, so not quite. 460 00:24:55,564 --> 00:24:56,440 AUDIENCE: [INAUDIBLE] 461 00:24:56,440 --> 00:25:01,480 PROFESSOR: [INAUDIBLE] Well, you've studied it. 462 00:25:01,480 --> 00:25:04,630 But correct answer. 463 00:25:04,630 --> 00:25:08,088 Any other thoughts? 464 00:25:08,088 --> 00:25:09,630 So another example would be something 465 00:25:09,630 --> 00:25:12,000 like non-relativistic QED. 466 00:25:12,000 --> 00:25:13,710 If you do non-relativistic QED, you're 467 00:25:13,710 --> 00:25:15,480 actually expanding in the velocity, 468 00:25:15,480 --> 00:25:17,760 not strictly in the ratio of mass scales, 469 00:25:17,760 --> 00:25:20,400 although you would think, for example, non-relativistic 470 00:25:20,400 --> 00:25:23,230 QED would be for a heavy electron, which is true-- 471 00:25:23,230 --> 00:25:25,230 a heavy, massive particle. 472 00:25:25,230 --> 00:25:28,410 That's not what the power counting expansion is in. 473 00:25:28,410 --> 00:25:30,180 It'll actually be in the velocity 474 00:25:30,180 --> 00:25:31,930 being much less than the speed of light. 475 00:25:31,930 --> 00:25:33,638 And that actually plays an important role 476 00:25:33,638 --> 00:25:35,400 in designing the effective field theory, 477 00:25:35,400 --> 00:25:38,710 determining what the leading operators are. 478 00:25:38,710 --> 00:25:41,790 So even something as simple as going 479 00:25:41,790 --> 00:25:44,010 to higher order corrections in hydrogen 480 00:25:44,010 --> 00:25:46,018 involves thinking about-- 481 00:25:46,018 --> 00:25:47,810 thinking beyond this simple statement here. 482 00:25:51,684 --> 00:25:55,360 Well, since we're on the topic of hydrogen, 483 00:25:55,360 --> 00:25:57,990 let me go into a little more detail there. 484 00:26:04,060 --> 00:26:11,070 So just to flesh this out a bit and to talk about some 485 00:26:11,070 --> 00:26:15,990 of the things that you have to be careful about, 486 00:26:15,990 --> 00:26:18,840 I'll phrase an example of this statement as the fact 487 00:26:18,840 --> 00:26:21,420 that we don't need to know about bottom quarks 488 00:26:21,420 --> 00:26:28,550 to describe hydrogen. 489 00:26:28,550 --> 00:26:29,300 Well, that's good. 490 00:26:29,300 --> 00:26:31,230 When you took quantum mechanics as an undergraduate, 491 00:26:31,230 --> 00:26:33,540 you didn't have bottom quarks in your description. 492 00:26:33,540 --> 00:26:36,779 So if you did need them, you would have missed something. 493 00:26:41,180 --> 00:26:43,192 What did you have in your description? 494 00:26:43,192 --> 00:26:44,900 Well, in a quantum field theory language, 495 00:26:44,900 --> 00:26:45,910 you have this diagram. 496 00:26:45,910 --> 00:26:49,760 You had an electron and a proton with photon exchange. 497 00:26:53,810 --> 00:26:56,300 And you also, when you thought about the binding energy, 498 00:26:56,300 --> 00:26:59,300 if we work in units where h bar and c is 1, which I will always 499 00:26:59,300 --> 00:27:04,480 do, then the binding energy is 1/2 ME alpha squared. 500 00:27:04,480 --> 00:27:06,470 And if you ask about the bottom quarks, 501 00:27:06,470 --> 00:27:08,600 the reason that you didn't need them 502 00:27:08,600 --> 00:27:11,900 is because they were suppressed. 503 00:27:11,900 --> 00:27:14,870 They weren't negligible-- completely negligible-- 504 00:27:14,870 --> 00:27:17,450 at least, at the level of how accurately we 505 00:27:17,450 --> 00:27:19,070 can measure this thing. 506 00:27:19,070 --> 00:27:20,295 Well, they're pretty small. 507 00:27:20,295 --> 00:27:21,920 They're at the 10 to the minus 8 level. 508 00:27:25,340 --> 00:27:27,110 And that's because they come in suppressed 509 00:27:27,110 --> 00:27:29,030 by the mass of the electron squared 510 00:27:29,030 --> 00:27:31,070 over the mass of the bottom quarks squared. 511 00:27:34,200 --> 00:27:36,000 So how would you think of them coming in? 512 00:27:36,000 --> 00:27:38,660 Well you'd think of them coming in through some diagram, 513 00:27:38,660 --> 00:27:45,350 for example, where the bottom quark couples to the photon 514 00:27:45,350 --> 00:27:48,860 through a vacuum polarization like that. 515 00:27:48,860 --> 00:27:50,840 And this diagram, indeed, will give you 516 00:27:50,840 --> 00:27:53,390 corrections of this type. 517 00:27:53,390 --> 00:27:55,400 Now, it's a bit more subtle than that. 518 00:27:55,400 --> 00:27:57,980 And that's because a diagram like this 519 00:27:57,980 --> 00:28:00,860 also has other contributions besides just these ones that I 520 00:28:00,860 --> 00:28:01,550 mentioned here. 521 00:28:08,490 --> 00:28:10,158 So the basic picture is, indeed, correct 522 00:28:10,158 --> 00:28:12,200 that we can neglect the bottom quark because it's 523 00:28:12,200 --> 00:28:13,580 giving small corrections. 524 00:28:13,580 --> 00:28:15,110 But there is one subtlety. 525 00:28:15,110 --> 00:28:17,700 And that has to do with the fact that we have to decide 526 00:28:17,700 --> 00:28:18,950 what we mean by this coupling. 527 00:28:21,530 --> 00:28:22,370 OK. 528 00:28:22,370 --> 00:28:29,510 So from your previous courses in quantum field theory, 529 00:28:29,510 --> 00:28:32,148 when you learned about running couplings, 530 00:28:32,148 --> 00:28:34,190 you learned the diagrams like that one contribute 531 00:28:34,190 --> 00:28:37,610 to running couplings. 532 00:28:37,610 --> 00:28:42,200 And so the B quark, therefore, can affect the coupling 533 00:28:42,200 --> 00:28:45,873 if you worked in, for example, the MS bar scheme, 534 00:28:45,873 --> 00:28:48,040 since it contributes to the running of the coupling. 535 00:28:51,080 --> 00:28:55,910 And in particular, you know for the electromagnetic coupling, 536 00:28:55,910 --> 00:28:58,250 if you ask about what that coupling is, 537 00:28:58,250 --> 00:29:00,420 it has a different value because it runs-- 538 00:29:00,420 --> 00:29:04,100 if you evaluate it at a scale like the W mass, 539 00:29:04,100 --> 00:29:09,710 then it's like 1/128, versus if you evaluate it at a very low 540 00:29:09,710 --> 00:29:13,170 energy, the electron mass or below, 541 00:29:13,170 --> 00:29:18,270 then it's the classic 1/137.036. 542 00:29:18,270 --> 00:29:18,770 OK. 543 00:29:18,770 --> 00:29:20,540 So there's some change. 544 00:29:20,540 --> 00:29:22,295 And the bottom quark is part of what 545 00:29:22,295 --> 00:29:23,420 contributes to that change. 546 00:29:23,420 --> 00:29:25,462 Of course, other particles are contributing, too. 547 00:29:31,940 --> 00:29:36,440 So if we want to say this statement about bottom quarks 548 00:29:36,440 --> 00:29:40,130 and we want to state the conclusions more precisely, 549 00:29:40,130 --> 00:29:41,450 then we would do it this way. 550 00:29:41,450 --> 00:29:58,970 We would say if alpha is a parameter of the standard model 551 00:29:58,970 --> 00:30:03,320 and we imagine that we fix it at a high energy-- 552 00:30:03,320 --> 00:30:04,820 so we could imagine that we fixed it 553 00:30:04,820 --> 00:30:08,960 by doing [INAUDIBLE] on physics in an E plus E minus collider. 554 00:30:08,960 --> 00:30:12,290 But some process that's a high energy process, 555 00:30:12,290 --> 00:30:14,180 we determine, say, for example, this value. 556 00:30:19,760 --> 00:30:23,430 If we take that attitude as how we define the parameter, 557 00:30:23,430 --> 00:30:25,250 then the parameter that actually matters 558 00:30:25,250 --> 00:30:27,950 for hydrogen, which is this parameter at the low scale, 559 00:30:27,950 --> 00:30:36,718 does depend on the bottom quark, because how 560 00:30:36,718 --> 00:30:38,510 we get from the high scale to the low scale 561 00:30:38,510 --> 00:30:41,600 depends on the fact that the bottom quark exists. 562 00:30:49,790 --> 00:30:53,180 But we could also take a different attitude. 563 00:30:53,180 --> 00:30:55,250 And that is we could take a low energy attitude. 564 00:31:06,860 --> 00:31:09,830 So we could say, let's forget about doing high energy 565 00:31:09,830 --> 00:31:10,800 physics. 566 00:31:10,800 --> 00:31:13,370 Let's just do low energy physics and extract 567 00:31:13,370 --> 00:31:18,839 alpha of 0 from some low energy atomic experiments. 568 00:31:29,600 --> 00:31:34,700 And if that's the way that we define the parameter, 569 00:31:34,700 --> 00:31:36,700 then the value can be used in other experiments. 570 00:31:36,700 --> 00:31:38,533 And we never have to know anything about MB. 571 00:31:54,810 --> 00:31:58,500 So we didn't really have to know about the high energy-- 572 00:31:58,500 --> 00:32:00,558 about the higher energy theory unless we actually 573 00:32:00,558 --> 00:32:02,100 were doing some experiments up there. 574 00:32:09,080 --> 00:32:11,660 Is there any questions about that? 575 00:32:11,660 --> 00:32:12,160 Good. 576 00:32:17,040 --> 00:32:20,847 So if we want to write an equation for that, 577 00:32:20,847 --> 00:32:22,680 what it means is that when you integrate out 578 00:32:22,680 --> 00:32:25,830 particles like the B quark, remove them from your theory, 579 00:32:25,830 --> 00:32:31,470 stop considering them, that it's not simply the case 580 00:32:31,470 --> 00:32:34,830 that you generate higher order terms in the series. 581 00:32:34,830 --> 00:32:37,080 You can also affect what you mean by the leading order 582 00:32:37,080 --> 00:32:39,750 term in the sense of changing what you mean by the coupling. 583 00:32:48,450 --> 00:32:51,300 So if I write it in terms of Lagrangian, 584 00:32:51,300 --> 00:32:54,690 I would say that the Lagrangian for hydrogen, 585 00:32:54,690 --> 00:32:57,090 if we include the B quark, well, it's 586 00:32:57,090 --> 00:32:59,820 got our proton, electron, and photon. 587 00:32:59,820 --> 00:33:02,280 And let's keep the B quark. 588 00:33:02,280 --> 00:33:06,490 Alpha and MB are parameters. 589 00:33:06,490 --> 00:33:10,770 If we drop the B quark because it's giving small effects, 590 00:33:10,770 --> 00:33:15,450 we just have these fields-- proton, electron, and photon. 591 00:33:15,450 --> 00:33:24,240 We get a different coupling in practice-- in principle. 592 00:33:24,240 --> 00:33:26,730 So you think of this as being a higher energy coupling 593 00:33:26,730 --> 00:33:30,223 and over there alpha prime being the low energy coupling. 594 00:33:30,223 --> 00:33:31,890 There's a lot of other things, actually, 595 00:33:31,890 --> 00:33:34,360 that if you think about hydrogen for a minute, 596 00:33:34,360 --> 00:33:36,930 there's a lot of other expansions that you've done. 597 00:33:36,930 --> 00:33:39,360 Hydrogen is a very fertile ground 598 00:33:39,360 --> 00:33:41,480 for effective field theory. 599 00:33:41,480 --> 00:33:43,142 So let's do that. 600 00:33:43,142 --> 00:33:45,600 Let's make a little list of what we dropped when we thought 601 00:33:45,600 --> 00:33:48,960 about hydrogen. How much did we lie to you when we first 602 00:33:48,960 --> 00:33:54,295 taught you the hydrogen atom in a quantum mechanics course? 603 00:33:54,295 --> 00:33:55,920 Well, we didn't teach you about quarks. 604 00:33:58,770 --> 00:34:02,480 Why didn't we teach you about quarks? 605 00:34:02,480 --> 00:34:04,907 And the reason we didn't teach you about quarks 606 00:34:04,907 --> 00:34:06,990 is because if you think about the typical momentum 607 00:34:06,990 --> 00:34:12,302 transfer in hydrogen, three momentum transfer, 608 00:34:12,302 --> 00:34:14,010 it's [INAUDIBLE] the mass of the electron 609 00:34:14,010 --> 00:34:16,679 times the fine structure constant. 610 00:34:16,679 --> 00:34:20,190 And that's much less than the proton size. 611 00:34:20,190 --> 00:34:23,250 So the typical photons that are involved in binding together 612 00:34:23,250 --> 00:34:28,920 the hydrogen atom just have much lower energy and they can't see 613 00:34:28,920 --> 00:34:29,760 inside the proton. 614 00:34:29,760 --> 00:34:31,425 They just see it as one overall object. 615 00:34:31,425 --> 00:34:33,300 And so we don't need to know about the quarks 616 00:34:33,300 --> 00:34:36,730 inside the proton. 617 00:34:36,730 --> 00:34:37,830 So that was an expansion. 618 00:34:43,889 --> 00:34:50,769 It's also an insensitive to the proton mass itself. 619 00:34:50,769 --> 00:34:57,840 So the proton we keep as an object, but the mass of the-- 620 00:34:57,840 --> 00:35:01,170 again, the momentum transfer, ME alpha, 621 00:35:01,170 --> 00:35:03,330 is much less than the mass of the proton, which 622 00:35:03,330 --> 00:35:06,930 is of order of GeV. 623 00:35:06,930 --> 00:35:10,230 And so we expand in our treatment of the proton, 624 00:35:10,230 --> 00:35:11,830 as well. 625 00:35:11,830 --> 00:35:17,960 And basically what this means is that the proton 626 00:35:17,960 --> 00:35:20,975 acts like a static charge. 627 00:35:28,700 --> 00:35:32,150 But proton mass wasn't showing up in our lowest order 628 00:35:32,150 --> 00:35:33,950 description of the energy here. 629 00:35:33,950 --> 00:35:37,233 It would show up in higher order corrections that we neglect. 630 00:35:40,480 --> 00:35:42,610 And again, it's because we're expanding. 631 00:35:42,610 --> 00:35:45,400 And that actually affects, when we design an effective field 632 00:35:45,400 --> 00:35:48,040 theory for this situation, how we would treat 633 00:35:48,040 --> 00:35:49,570 the proton, what type of Lagrangian 634 00:35:49,570 --> 00:35:52,210 we would write down for it. 635 00:35:52,210 --> 00:35:54,010 And that will be one of our topics 636 00:35:54,010 --> 00:35:57,040 is to figure out how we treat heavy particles, like a proton, 637 00:35:57,040 --> 00:35:57,670 in this case. 638 00:36:02,200 --> 00:36:04,265 Another expansion that we did is we used the fact 639 00:36:04,265 --> 00:36:06,640 that the momentum transfer is much less than the electron 640 00:36:06,640 --> 00:36:08,980 mass, not just the proton mass. 641 00:36:08,980 --> 00:36:12,943 And that meant that the theory is non-relativistic 642 00:36:12,943 --> 00:36:14,860 and that's why we did non-relativistic quantum 643 00:36:14,860 --> 00:36:16,520 mechanics. 644 00:36:16,520 --> 00:36:18,520 If we wanted to do it as a quantum field theory, 645 00:36:18,520 --> 00:36:20,710 we would do a non-relativistic quantum field theory. 646 00:36:25,930 --> 00:36:28,510 So already in something as simple as hydrogen, 647 00:36:28,510 --> 00:36:31,720 we have here three expansions plus many more, 648 00:36:31,720 --> 00:36:33,220 thinking about the particles that we 649 00:36:33,220 --> 00:36:37,112 neglected in the description. 650 00:36:37,112 --> 00:36:38,820 AUDIENCE: So the second point [INAUDIBLE] 651 00:36:38,820 --> 00:36:42,176 and the alpha, how do you know that's not just the ratio of ME 652 00:36:42,176 --> 00:36:45,630 by MP rather than ME alpha? 653 00:36:45,630 --> 00:36:46,320 PROFESSOR: So-- 654 00:36:46,320 --> 00:36:47,153 [INTERPOSING VOICES] 655 00:36:47,153 --> 00:36:49,140 AUDIENCE: --electron was 1 [? GeV. ?] 656 00:36:49,140 --> 00:36:49,690 PROFESSOR: That's right. 657 00:36:49,690 --> 00:36:50,050 So-- 658 00:36:50,050 --> 00:36:50,850 AUDIENCE: We would care about it. 659 00:36:50,850 --> 00:36:51,767 PROFESSOR: Absolutely. 660 00:36:51,767 --> 00:36:54,780 So in some sense, I could have written ME here, 661 00:36:54,780 --> 00:36:56,373 and that would have been fine. 662 00:36:56,373 --> 00:36:58,290 If you take these together, then of course you 663 00:36:58,290 --> 00:37:02,310 could take a ratio and you'd get ME over M proton. 664 00:37:02,310 --> 00:37:04,260 The reason I wrote ME alpha is I was really 665 00:37:04,260 --> 00:37:07,560 thinking about the momentum, sort 666 00:37:07,560 --> 00:37:10,200 of the non-static properties, the dynamics. 667 00:37:10,200 --> 00:37:14,220 And the momentum of the photons, the largest energy is this. 668 00:37:14,220 --> 00:37:15,720 That's why I was thinking about it. 669 00:37:15,720 --> 00:37:19,210 But you're right that I should write ME over M proton, 670 00:37:19,210 --> 00:37:19,710 as well. 671 00:37:22,410 --> 00:37:25,660 As any other comments, questions? 672 00:37:39,530 --> 00:37:43,280 So another point that I just want to briefly comment, 673 00:37:43,280 --> 00:37:46,850 which has to be true if everything I'm telling you 674 00:37:46,850 --> 00:37:49,460 is right, which has to be true because it's 675 00:37:49,460 --> 00:37:55,910 what we taught you, is that this whole description is true 676 00:37:55,910 --> 00:37:58,660 even though there's ultraviolet divergences. 677 00:37:58,660 --> 00:38:00,410 When you start doing quantum field theory, 678 00:38:00,410 --> 00:38:02,493 even if you do quantum field theory, in this case, 679 00:38:02,493 --> 00:38:05,990 for the hydrogen atom, you run into ultraviolet divergences 680 00:38:05,990 --> 00:38:08,160 where things are blowing up. 681 00:38:08,160 --> 00:38:11,066 Actually, even this diagram has ultraviolet divergences. 682 00:38:14,960 --> 00:38:17,900 So before you regulate the theory, 683 00:38:17,900 --> 00:38:20,970 the bottom quark group is infinity. 684 00:38:20,970 --> 00:38:22,970 Once you regulate the theory, it's well-defined. 685 00:38:22,970 --> 00:38:26,300 And you can make everything well-defined. 686 00:38:26,300 --> 00:38:28,187 But you may worry that this diagram seems 687 00:38:28,187 --> 00:38:30,020 to be contributing an infinite amount rather 688 00:38:30,020 --> 00:38:31,310 than a finite amount. 689 00:38:31,310 --> 00:38:33,230 And the whole story goes through even 690 00:38:33,230 --> 00:38:36,080 in the context of having ultraviolet divergences. 691 00:38:39,230 --> 00:38:41,543 That better be true, because, for example, 692 00:38:41,543 --> 00:38:43,460 if we had graviton loops, they would also lead 693 00:38:43,460 --> 00:38:45,500 to ultraviolet divergences. 694 00:38:45,500 --> 00:38:51,650 And so we're neglecting gravity. 695 00:38:51,650 --> 00:38:54,320 So it better that these ideas of effective field theory 696 00:38:54,320 --> 00:38:57,980 are not changed by having ultraviolet divergences. 697 00:38:57,980 --> 00:39:00,620 And we'll encounter that in our discussion later on. 698 00:39:04,950 --> 00:39:05,450 OK. 699 00:39:05,450 --> 00:39:07,880 So that gives you a bit of a sense 700 00:39:07,880 --> 00:39:10,370 for how these ideas of effective field theory, you've 701 00:39:10,370 --> 00:39:13,160 been using them all along, whether or not you knew it. 702 00:39:18,050 --> 00:39:20,990 And we will, in this course, flesh out 703 00:39:20,990 --> 00:39:23,030 how we figure out some of these corrections, 704 00:39:23,030 --> 00:39:25,280 how we would actually compute them, 705 00:39:25,280 --> 00:39:28,070 how we would actually figure out how to even the leading order-- 706 00:39:28,070 --> 00:39:29,487 what the leading order description 707 00:39:29,487 --> 00:39:32,630 of a theory in cases where we may not know it or someone else 708 00:39:32,630 --> 00:39:33,890 hasn't figured it out yet. 709 00:39:33,890 --> 00:39:35,598 Those are the type of things we're after. 710 00:39:37,887 --> 00:39:40,220 Now, if you talk about the categories of effective field 711 00:39:40,220 --> 00:39:41,595 theories, if you look at the list 712 00:39:41,595 --> 00:39:44,030 that I handed out to you or the list of things 713 00:39:44,030 --> 00:39:46,010 we're going to do in the course, then 714 00:39:46,010 --> 00:39:48,440 there's really, in general, two ways 715 00:39:48,440 --> 00:39:50,090 that effective field theories are used. 716 00:40:09,810 --> 00:40:14,335 So the two ways are from the top down or from the bottom up. 717 00:40:14,335 --> 00:40:15,710 So we'll start with the top down. 718 00:40:18,590 --> 00:40:21,920 So in the top down situation, you 719 00:40:21,920 --> 00:40:24,020 know what the high energy theory is. 720 00:40:24,020 --> 00:40:26,270 I'm going to keep using this language of masses, 721 00:40:26,270 --> 00:40:28,665 where I have a high energy and low energy theory. 722 00:40:28,665 --> 00:40:30,290 And we'll think about it more generally 723 00:40:30,290 --> 00:40:32,248 when we come to examples where we need to think 724 00:40:32,248 --> 00:40:34,020 about it more generally. 725 00:40:34,020 --> 00:40:37,790 So in this top-down case, we have a high energy theory-- 726 00:40:37,790 --> 00:40:39,440 say, the standard model-- 727 00:40:39,440 --> 00:40:41,330 and that theory is understood in the sense 728 00:40:41,330 --> 00:40:45,560 that we can write down the Lagrangian for it. 729 00:40:45,560 --> 00:40:49,550 But we're not satisfied with that. 730 00:40:49,550 --> 00:41:01,190 We find it useful to have a simpler theory 731 00:41:01,190 --> 00:41:09,013 to do some low energy physics, or even to do some high energy 732 00:41:09,013 --> 00:41:10,430 physics, where not all the degrees 733 00:41:10,430 --> 00:41:13,220 of freedom of this high energy theory are relevant. 734 00:41:17,640 --> 00:41:20,360 So we're in a situation where we have some theory, which we'll 735 00:41:20,360 --> 00:41:23,030 call theory one, which is this high energy theory 736 00:41:23,030 --> 00:41:25,340 that we understand it that we can think 737 00:41:25,340 --> 00:41:27,037 about doing calculations in it. 738 00:41:27,037 --> 00:41:29,120 But we want to go over to some other theory, which 739 00:41:29,120 --> 00:41:33,230 I'll call theory two, which has less degrees of freedom. 740 00:41:33,230 --> 00:41:35,300 And we're making expansions. 741 00:41:35,300 --> 00:41:37,040 And that's a low energy theory. 742 00:41:37,040 --> 00:41:39,581 This is the high theory. 743 00:41:39,581 --> 00:41:42,450 This is the low theory. 744 00:41:42,450 --> 00:41:45,080 So that's what we are in this situation of what 745 00:41:45,080 --> 00:41:48,320 we call top down, coming from the top, from high energy, 746 00:41:48,320 --> 00:41:50,220 down. 747 00:41:50,220 --> 00:41:51,387 So what do we do? 748 00:41:51,387 --> 00:41:52,970 Well, in this case, it's kind of nice, 749 00:41:52,970 --> 00:41:55,640 because we can actually use the fact that we know this theory 750 00:41:55,640 --> 00:41:58,820 one and can do calculations in that theory one 751 00:41:58,820 --> 00:42:00,830 to even think about constructing theory two. 752 00:42:09,900 --> 00:42:13,080 So what we can do is we could just start calculating things 753 00:42:13,080 --> 00:42:21,450 in theory one and integrate out-- i.e. remove-- 754 00:42:21,450 --> 00:42:22,590 the heavier particles. 755 00:42:30,550 --> 00:42:33,700 And in doing that, we can do what's 756 00:42:33,700 --> 00:42:38,680 called matching onto the low energy theory. 757 00:42:48,140 --> 00:42:49,700 That means we can use this ability 758 00:42:49,700 --> 00:42:53,060 to do calculations in the high energy theory 759 00:42:53,060 --> 00:42:58,320 to find what the operators are of the low energy theory, 760 00:42:58,320 --> 00:43:01,740 just by direct calculation. 761 00:43:01,740 --> 00:43:06,170 And also if there's new low energy constants that show up, 762 00:43:06,170 --> 00:43:08,810 we can calculate the values of those constants 763 00:43:08,810 --> 00:43:12,590 by using information and connecting them 764 00:43:12,590 --> 00:43:15,300 to the high energy theory. 765 00:43:15,300 --> 00:43:18,740 So in this case, we're able to use calculations really 766 00:43:18,740 --> 00:43:20,330 to construct the low energy theory. 767 00:43:26,130 --> 00:43:28,500 So just schematically, I start with some high energy 768 00:43:28,500 --> 00:43:36,930 Lagrangian and I go over to some low energy Lagrangians where 769 00:43:36,930 --> 00:43:39,360 there's an infinite series, which I've indexed by n. 770 00:43:39,360 --> 00:43:41,970 And that index is to denote higher order 771 00:43:41,970 --> 00:43:44,608 terms that are less relevant in whatever expansion 772 00:43:44,608 --> 00:43:45,150 you're doing. 773 00:43:54,930 --> 00:43:56,630 So just to be general, I'll say it's 774 00:43:56,630 --> 00:44:00,245 an expansion in decreasing relevance of the terms. 775 00:44:05,760 --> 00:44:08,360 So if you're in this situation, then these two theories 776 00:44:08,360 --> 00:44:11,090 are, in some sense, describing common things. 777 00:44:11,090 --> 00:44:13,580 The high energy theory describes more than the low energy 778 00:44:13,580 --> 00:44:15,247 theory, because you've removed something 779 00:44:15,247 --> 00:44:17,270 in constructing a low energy theory. 780 00:44:17,270 --> 00:44:22,430 But the two theories have to at least agree where they overlap. 781 00:44:22,430 --> 00:44:47,050 And so they have to agree on certain infrared observables, 782 00:44:47,050 --> 00:44:52,050 which I will also often denote as IR for infrared. 783 00:44:52,050 --> 00:44:55,680 The place where they differ is in the ultraviolet. 784 00:44:55,680 --> 00:44:58,440 So they might have different ultraviolet divergences. 785 00:44:58,440 --> 00:45:01,410 Most often, they will have different ultraviolet 786 00:45:01,410 --> 00:45:02,910 divergences. 787 00:45:02,910 --> 00:45:05,340 They don't have to agree in the ultraviolet. 788 00:45:05,340 --> 00:45:07,290 And actually, you exploit that to do things 789 00:45:07,290 --> 00:45:09,510 with the effective theory that would be hard to do 790 00:45:09,510 --> 00:45:11,770 with the full theory. 791 00:45:11,770 --> 00:45:14,467 We'll talk about how we do that later on. 792 00:45:14,467 --> 00:45:16,800 So the fact that they differ is actually not necessarily 793 00:45:16,800 --> 00:45:17,590 a negative thing. 794 00:45:17,590 --> 00:45:18,390 It can be a bonus. 795 00:45:30,680 --> 00:45:34,040 And finally, you have to ask about this sum over n. 796 00:45:34,040 --> 00:45:35,870 Well, this sum over n is infinite. 797 00:45:35,870 --> 00:45:37,390 Goes on forever. 798 00:45:37,390 --> 00:45:41,135 And so you have to ask the question, when should I stop? 799 00:45:41,135 --> 00:45:43,010 And therefore you have to look to experiments 800 00:45:43,010 --> 00:45:49,940 and see how precise they are, or just to your own perseverance 801 00:45:49,940 --> 00:45:52,160 and figure out what level do you want, 802 00:45:52,160 --> 00:45:56,790 what precision do you want in your description? 803 00:45:56,790 --> 00:46:00,520 So what n do you want to stop at? 804 00:46:00,520 --> 00:46:03,020 Sometimes experiment tells you to only do the first two n's. 805 00:46:03,020 --> 00:46:04,478 Sometimes you have to decide, maybe 806 00:46:04,478 --> 00:46:05,970 I only want the first one. 807 00:46:05,970 --> 00:46:07,553 If you're doing it for the first time, 808 00:46:07,553 --> 00:46:09,090 I suggest you stop at the beginning 809 00:46:09,090 --> 00:46:13,380 and let someone else to do the corrections. 810 00:46:13,380 --> 00:46:18,113 This idea of doing this can be important for separating 811 00:46:18,113 --> 00:46:19,530 physics of the high energy theory. 812 00:46:19,530 --> 00:46:22,120 One example of this is in QCD. 813 00:46:22,120 --> 00:46:25,290 If you take QCD for just about any process, 814 00:46:25,290 --> 00:46:27,600 there will be some parts of it which were perturbative 815 00:46:27,600 --> 00:46:30,150 and some parts of it that were non-perturbative. 816 00:46:30,150 --> 00:46:33,150 And by doing this kind of thing where you expand, 817 00:46:33,150 --> 00:46:34,950 you could construct a low energy theory 818 00:46:34,950 --> 00:46:37,860 that only has the non-perturbative scale in it 819 00:46:37,860 --> 00:46:39,870 and removes all the perturbative scales. 820 00:46:39,870 --> 00:46:42,540 If you did that, then just doing this procedure would allow you 821 00:46:42,540 --> 00:46:44,970 to figure out, what is the non-perturbative physics 822 00:46:44,970 --> 00:46:46,810 and what is the perturbative physics? 823 00:46:46,810 --> 00:46:49,860 You'd separate out, in that case, into operators. 824 00:46:49,860 --> 00:46:51,930 You'd separate out the infrared physics. 825 00:46:51,930 --> 00:46:54,540 You'd have operators built out of the infrared fields. 826 00:46:54,540 --> 00:46:56,010 And those operators would describe 827 00:46:56,010 --> 00:46:57,533 the non-perturbative physics. 828 00:46:57,533 --> 00:46:58,950 And you'd have some new low energy 829 00:46:58,950 --> 00:47:01,980 constants, which would describe the perturbative physics. 830 00:47:01,980 --> 00:47:07,915 And some of the examples that we'll do will make use of that 831 00:47:07,915 --> 00:47:08,415 probably. 832 00:47:12,020 --> 00:47:14,600 So that's kind of a motivation, actually, 833 00:47:14,600 --> 00:47:21,492 for some of the examples in the standard model. 834 00:47:21,492 --> 00:47:22,950 So if you think about, for example, 835 00:47:22,950 --> 00:47:32,220 integrating out heavy particles like the w 836 00:47:32,220 --> 00:47:37,233 or the z or the top quark, one of the motivations 837 00:47:37,233 --> 00:47:39,150 is sometimes what I just said, to separate out 838 00:47:39,150 --> 00:47:41,770 perturbative and non-perturbative physics. 839 00:47:41,770 --> 00:47:44,190 So that's one example. 840 00:47:44,190 --> 00:47:45,810 Heavy quark effective theory is also 841 00:47:45,810 --> 00:47:51,397 an example of this top-down effective field theory. 842 00:47:51,397 --> 00:47:52,980 In heavy quark effective field theory, 843 00:47:52,980 --> 00:47:57,210 you have a field theory for the B quark or for the charm quark 844 00:47:57,210 --> 00:48:02,800 but you want to describe things like the B meson or the charm 845 00:48:02,800 --> 00:48:03,300 mesons. 846 00:48:07,245 --> 00:48:09,120 Objects have non-perturbative physics as well 847 00:48:09,120 --> 00:48:11,942 as perturbative-- have mostly non-perturbative physics. 848 00:48:11,942 --> 00:48:14,400 And in order to do that, you want to actually integrate out 849 00:48:14,400 --> 00:48:17,490 the mass scale of the charm quark or the bottom quark. 850 00:48:17,490 --> 00:48:19,380 If you integrate out the mass of the charm 851 00:48:19,380 --> 00:48:20,640 quark and the bottom quark, you go over 852 00:48:20,640 --> 00:48:22,998 to something called heavy quark effective field theory. 853 00:48:22,998 --> 00:48:25,290 But it can be done in exactly this way that I described 854 00:48:25,290 --> 00:48:28,402 to you, where do you start with the theory with the full B 855 00:48:28,402 --> 00:48:30,360 quark relativistic description and you actually 856 00:48:30,360 --> 00:48:32,318 just expand and figure out what the heavy quark 857 00:48:32,318 --> 00:48:33,675 effective theory is. 858 00:48:36,930 --> 00:48:46,740 So non relativistic QED and non-relativistic QCD 859 00:48:46,740 --> 00:48:48,240 are also examples here. 860 00:48:48,240 --> 00:48:50,820 And soft-collinear effective theory, 861 00:48:50,820 --> 00:48:54,812 one of our main subjects, is also an example of this type 862 00:48:54,812 --> 00:48:57,972 where we can just start in QCD, do an expansion, 863 00:48:57,972 --> 00:48:59,430 and get the effective field theory. 864 00:49:07,350 --> 00:49:09,160 So what's the other category? 865 00:49:09,160 --> 00:49:12,990 So the other category is from the bottom up. 866 00:49:12,990 --> 00:49:15,300 And typically in this case, you're 867 00:49:15,300 --> 00:49:18,120 interested in using effective theory logic. 868 00:49:18,120 --> 00:49:20,923 But maybe you don't know the high energy theory. 869 00:49:20,923 --> 00:49:22,590 You don't really know anything about it. 870 00:49:22,590 --> 00:49:24,810 Maybe we've never probed it. 871 00:49:24,810 --> 00:49:27,330 That's one way in which bottom-up effective field 872 00:49:27,330 --> 00:49:30,270 theory shows up. 873 00:49:30,270 --> 00:49:34,920 Or it could be that the high energy theory is known 874 00:49:34,920 --> 00:49:39,510 but actually doing the matching calculations to integrate out 875 00:49:39,510 --> 00:49:42,330 the degrees of freedom to do those calculations explicitly 876 00:49:42,330 --> 00:49:48,060 could be just very, very difficult. 877 00:49:48,060 --> 00:49:51,480 Maybe it would be non-perturbative, for example. 878 00:49:51,480 --> 00:49:52,980 So if the matching is too difficult, 879 00:49:52,980 --> 00:49:54,750 then you may also want to be thinking 880 00:49:54,750 --> 00:49:59,550 in this bottom-up framework where you really just start 881 00:49:59,550 --> 00:50:01,860 by thinking about the low energy theory 882 00:50:01,860 --> 00:50:06,210 without worrying about what the high energy theory was, 883 00:50:06,210 --> 00:50:09,363 or without thinking too hard about the high energy theory, 884 00:50:09,363 --> 00:50:12,030 and in particular, without doing calculations in the high energy 885 00:50:12,030 --> 00:50:17,860 theory in order to motivate the low energy theory. 886 00:50:17,860 --> 00:50:21,538 You need to know some things about the high energy theory, 887 00:50:21,538 --> 00:50:23,580 like you may need to know that it's [? llorens ?] 888 00:50:23,580 --> 00:50:27,378 invariant, that it has certain gauge symmetry, 889 00:50:27,378 --> 00:50:28,545 that it's not totally crazy. 890 00:50:33,040 --> 00:50:34,790 But you don't need to know it at the level 891 00:50:34,790 --> 00:50:36,980 where you would actually carry out calculations with it 892 00:50:36,980 --> 00:50:38,890 in order to construct the low energy theory. 893 00:50:38,890 --> 00:50:40,890 Instead, you think about the lower energy theory 894 00:50:40,890 --> 00:50:45,170 from the bottom up, where you can just devise it 895 00:50:45,170 --> 00:50:48,080 based on the symmetries, based on your power counting, 896 00:50:48,080 --> 00:50:50,780 and based on identifying the degrees of freedom. 897 00:50:54,500 --> 00:51:10,950 So construct the series simply by writing down the most 898 00:51:10,950 --> 00:51:12,750 general operators that we can think 899 00:51:12,750 --> 00:51:32,870 of consistent with whatever degrees of freedom we have, 900 00:51:32,870 --> 00:51:35,142 and of course, consistent with the symmetries 901 00:51:35,142 --> 00:51:35,975 that we're imposing. 902 00:51:41,120 --> 00:51:41,620 OK. 903 00:51:44,750 --> 00:51:47,450 So the picture is that you don't know 904 00:51:47,450 --> 00:51:50,930 or you want to remain agnostic about theory one, 905 00:51:50,930 --> 00:51:54,320 but you still are interested in constructing theory two. 906 00:51:57,230 --> 00:51:59,180 If you do this, then unlike the other case, 907 00:51:59,180 --> 00:52:01,520 the couplings that you have when you write down these operators, 908 00:52:01,520 --> 00:52:03,228 they all are multiplied by some couplings 909 00:52:03,228 --> 00:52:06,330 if they're higher dimensional operators 910 00:52:06,330 --> 00:52:09,500 and they're not constrained by gauge symmetry. 911 00:52:09,500 --> 00:52:12,440 Then all of those couplings are unknown. 912 00:52:12,440 --> 00:52:15,165 But you can fit them to experiment. 913 00:52:18,970 --> 00:52:22,210 So the effective theory may still be powerful 914 00:52:22,210 --> 00:52:25,810 because you can make more predictions than the number 915 00:52:25,810 --> 00:52:31,351 of parameters that you have, like for hydrogen, 916 00:52:31,351 --> 00:52:32,920 where we have very few parameters 917 00:52:32,920 --> 00:52:35,462 and the effective theory, but we can make lots of predictions 918 00:52:35,462 --> 00:52:38,020 from non-relativistic quantum mechanics. 919 00:52:38,020 --> 00:52:40,145 Or it could be in the case where you 920 00:52:40,145 --> 00:52:41,770 imagine is too difficult that maybe you 921 00:52:41,770 --> 00:52:43,562 have to carry out the matching numerically, 922 00:52:43,562 --> 00:52:46,090 like with a lot of QCD. 923 00:52:46,090 --> 00:52:49,630 And so that would be another possible way 924 00:52:49,630 --> 00:52:52,400 of determining couplings. 925 00:52:52,400 --> 00:52:57,150 And again, the desired precision tells us when to stop. 926 00:52:57,150 --> 00:53:00,500 So it's important that we have a power counting for this theory. 927 00:53:00,500 --> 00:53:02,840 But that power counting is in some sense defined 928 00:53:02,840 --> 00:53:06,800 irrespective of what the full theory was so that we can stop, 929 00:53:06,800 --> 00:53:09,620 even in the bottom-up case. 930 00:53:18,380 --> 00:53:20,510 So what are examples here? 931 00:53:20,510 --> 00:53:22,100 Well, the classic example of this type 932 00:53:22,100 --> 00:53:24,890 is chiral perturbation theory, when 933 00:53:24,890 --> 00:53:28,325 you're thinking about a field theory for kaons and pions. 934 00:53:31,730 --> 00:53:36,050 And doing the matching onto kaons and pions from QCD 935 00:53:36,050 --> 00:53:38,257 is a non-perturbative process, so you 936 00:53:38,257 --> 00:53:40,340 think about constructing the effective theory just 937 00:53:40,340 --> 00:53:42,350 from low energy and from symmetries, 938 00:53:42,350 --> 00:53:45,510 knowing the symmetry breaking pattern, in particular. 939 00:53:45,510 --> 00:53:47,510 And you construct the chiral perturbation theory 940 00:53:47,510 --> 00:53:51,530 without thinking about doing the matching explicitly. 941 00:53:51,530 --> 00:53:53,640 So that's one example. 942 00:53:53,640 --> 00:53:55,442 Another example of this type is actually 943 00:53:55,442 --> 00:53:56,525 the standard model itself. 944 00:54:00,980 --> 00:54:03,320 If you think about the logic that we 945 00:54:03,320 --> 00:54:05,330 used when we construct the standard model, 946 00:54:05,330 --> 00:54:07,870 it was exactly this effective field theory logic. 947 00:54:07,870 --> 00:54:10,790 We said, what are the relevant degrees of freedom? 948 00:54:10,790 --> 00:54:11,600 Electron. 949 00:54:11,600 --> 00:54:12,560 Quarks. 950 00:54:12,560 --> 00:54:14,150 W bosons. 951 00:54:14,150 --> 00:54:15,260 Listed them. 952 00:54:15,260 --> 00:54:16,105 We wrote down. 953 00:54:16,105 --> 00:54:18,230 We said, what are the important guiding principles? 954 00:54:18,230 --> 00:54:20,107 Gauge symmetry. 955 00:54:20,107 --> 00:54:22,190 And then we wrote down the most general Lagrangian 956 00:54:22,190 --> 00:54:23,180 that we could think of. 957 00:54:23,180 --> 00:54:25,640 That was the standard model. 958 00:54:25,640 --> 00:54:29,090 OK, so it's an example of a bottom-up effective field 959 00:54:29,090 --> 00:54:29,670 theory. 960 00:54:29,670 --> 00:54:34,200 We don't ask questions about what's higher up. 961 00:54:34,200 --> 00:54:36,960 Let me write down the leading order Lagrangian. 962 00:54:36,960 --> 00:54:39,090 And we can actually construct higher order terms 963 00:54:39,090 --> 00:54:41,430 in the standard model expanding in the idea 964 00:54:41,430 --> 00:54:44,820 that there's physics above the scales of the standard model 965 00:54:44,820 --> 00:54:46,560 and write down higher dimension operators 966 00:54:46,560 --> 00:54:50,867 and have a real standard model that has an infinite series. 967 00:54:50,867 --> 00:54:52,950 So the standard model is an effective field theory 968 00:54:52,950 --> 00:54:55,200 that has an infinite number of operators. 969 00:54:55,200 --> 00:55:00,257 And we'll talk about that momentarily as well next time. 970 00:55:00,257 --> 00:55:02,340 So that'll be the first example we actually treat, 971 00:55:02,340 --> 00:55:05,297 is the standard model as an effective field theory. 972 00:55:09,280 --> 00:55:11,050 Another example is quantum gravity. 973 00:55:18,250 --> 00:55:20,050 If you take Einstein gravity and you 974 00:55:20,050 --> 00:55:23,198 make it quantum and you allow yourself to expand-- 975 00:55:23,198 --> 00:55:25,240 i.e. you say you're only interested in low energy 976 00:55:25,240 --> 00:55:25,870 physics. 977 00:55:25,870 --> 00:55:28,120 So you allow yourself to write down an infinite number 978 00:55:28,120 --> 00:55:29,710 of operators. 979 00:55:29,710 --> 00:55:32,770 Then you can also renormalize that theory order 980 00:55:32,770 --> 00:55:35,740 by ordering those infinite number of operators. 981 00:55:35,740 --> 00:55:37,600 And so it's also an example of something 982 00:55:37,600 --> 00:55:40,040 that you can treat from this effective field theory 983 00:55:40,040 --> 00:55:40,540 paradigm. 984 00:55:43,365 --> 00:55:45,240 That was the last topic that I would actually 985 00:55:45,240 --> 00:55:48,117 be cover in this-- if we had a couple extra weeks of lectures, 986 00:55:48,117 --> 00:55:48,950 I would get to this. 987 00:55:48,950 --> 00:55:50,590 But probably we won't. 988 00:55:50,590 --> 00:55:53,870 So that's a topic that somebody might pick for their project 989 00:55:53,870 --> 00:55:54,890 at the end. 990 00:55:54,890 --> 00:55:57,920 That's a good presentation. 991 00:55:57,920 --> 00:55:59,440 OK, so important stuff. 992 00:56:04,705 --> 00:56:06,080 Is there any question about that? 993 00:56:09,600 --> 00:56:13,360 Sits well with everybody, makes them feel good inside? 994 00:56:13,360 --> 00:56:13,860 OK. 995 00:56:24,530 --> 00:56:28,430 So far when we've been talking about this sum over n, 996 00:56:28,430 --> 00:56:32,300 we've been really thinking about expansions in powers. 997 00:56:32,300 --> 00:56:34,580 Some mass scale divided by some other scale 998 00:56:34,580 --> 00:56:36,730 being much less than 1. 999 00:56:36,730 --> 00:56:38,710 OK, that's what we've meant by it. 1000 00:56:43,560 --> 00:56:46,550 But when you have two scales like this, m and lambda, 1001 00:56:46,550 --> 00:56:47,930 you also get logarithms. 1002 00:56:54,500 --> 00:56:55,618 So it's not always powers. 1003 00:56:55,618 --> 00:56:56,910 There's also logs that show up. 1004 00:57:00,510 --> 00:57:02,010 And this comment I meant about-- 1005 00:57:02,010 --> 00:57:05,550 that I made about ultraviolet divergences 1006 00:57:05,550 --> 00:57:08,640 in the low energy theory, it can actually 1007 00:57:08,640 --> 00:57:10,470 help you to understand those logarithms. 1008 00:57:13,886 --> 00:57:17,150 Let's get you over here. 1009 00:57:17,150 --> 00:57:18,760 So when you treat the renormalizaiton 1010 00:57:18,760 --> 00:57:27,810 of the low energy theory, as you know from quantum filtering, 1011 00:57:27,810 --> 00:57:30,840 you have different types of divergences, power divergences. 1012 00:57:30,840 --> 00:57:33,105 But the logarithmic divergences, in particular, 1013 00:57:33,105 --> 00:57:36,390 are things that are playing an important role, often, 1014 00:57:36,390 --> 00:57:37,800 in quantum field theory. 1015 00:57:37,800 --> 00:57:40,080 And in effective field theory, it's the same thing. 1016 00:57:40,080 --> 00:57:42,870 Logarithms can be tied to the re-normalization 1017 00:57:42,870 --> 00:57:48,100 of the low energy effective theory 1018 00:57:48,100 --> 00:57:54,460 and allow us to sum infinite series of those logarithms. 1019 00:58:01,660 --> 00:58:06,280 So often just the power counting and the re-normalization 1020 00:58:06,280 --> 00:58:08,830 of the low energy theory will actually 1021 00:58:08,830 --> 00:58:10,840 allow us not only to calculate the logarithms, 1022 00:58:10,840 --> 00:58:13,150 but to think about summing up infinite series 1023 00:58:13,150 --> 00:58:17,050 of those logarithms. 1024 00:58:17,050 --> 00:58:17,550 OK. 1025 00:58:17,550 --> 00:58:19,080 So that was kind of just elaborating 1026 00:58:19,080 --> 00:58:20,910 on a point I made earlier. 1027 00:58:20,910 --> 00:58:23,070 And again, I should say here that I've said this 1028 00:58:23,070 --> 00:58:26,430 in the language of there being two masses, m1 and m2 and w 1029 00:58:26,430 --> 00:58:27,720 over MB. 1030 00:58:27,720 --> 00:58:31,080 But this is actually true more generally again. 1031 00:58:31,080 --> 00:58:35,160 So I'd actually make a claim that there's not any log 1032 00:58:35,160 --> 00:58:37,830 that you've seen in quantum field theory that shouldn't 1033 00:58:37,830 --> 00:58:40,650 be possible to figure out an effective field theory that 1034 00:58:40,650 --> 00:58:44,280 allows you to understand those logs and predict logarithms 1035 00:58:44,280 --> 00:58:46,680 at higher orders in perturbation theory. 1036 00:58:46,680 --> 00:58:48,840 There's not an example in quantum field theory 1037 00:58:48,840 --> 00:58:51,347 that I've met that hasn't fallen into that rubric where 1038 00:58:51,347 --> 00:58:53,430 some effective field theory description allows you 1039 00:58:53,430 --> 00:58:56,170 to understand the logarithms. 1040 00:58:56,170 --> 00:58:56,670 OK. 1041 00:58:59,200 --> 00:59:00,728 So that's kind of a bonus. 1042 00:59:00,728 --> 00:59:02,020 It's not the guiding principle. 1043 00:59:02,020 --> 00:59:05,020 It's not what we're doing when we're expanding in powers. 1044 00:59:05,020 --> 00:59:09,010 But it's something that we get along for the ride. 1045 00:59:09,010 --> 00:59:10,510 And maybe it would be the motivation 1046 00:59:10,510 --> 00:59:12,790 if you see some logarithms and some process 1047 00:59:12,790 --> 00:59:15,150 and you want to understand them. 1048 00:59:15,150 --> 00:59:16,625 Maybe we would say, well, I'd like 1049 00:59:16,625 --> 00:59:19,000 to understand what effective field theory would give rise 1050 00:59:19,000 --> 00:59:20,708 to a description where I could understand 1051 00:59:20,708 --> 00:59:23,745 those logarithms from a re-normalization perspective. 1052 00:59:23,745 --> 00:59:25,120 And sometimes that's very useful, 1053 00:59:25,120 --> 00:59:27,910 because maybe those logarithms are phenomenologically 1054 00:59:27,910 --> 00:59:29,680 important and you want to make predictions 1055 00:59:29,680 --> 00:59:34,870 about higher order logarithms, or maybe there's controversy. 1056 00:59:34,870 --> 00:59:37,090 When I was a postdoc, there was some controversy 1057 00:59:37,090 --> 00:59:43,120 about a term that was alpha to the 8 log cubed alpha 1058 00:59:43,120 --> 00:59:46,990 in hydrogen energy levels. 1059 00:59:46,990 --> 00:59:49,720 8 powers of alpha, 3 powers of logs. 1060 00:59:49,720 --> 00:59:50,918 There was four groups. 1061 00:59:50,918 --> 00:59:52,210 Two of them had got one answer. 1062 00:59:52,210 --> 00:59:54,310 Two of them had got another answer. 1063 00:59:54,310 --> 00:59:56,227 And using the ideas of effective field theory, 1064 00:59:56,227 --> 00:59:58,768 we were able to figure out that one of those groups was right 1065 00:59:58,768 --> 01:00:00,550 and the other was wrong very clearly, 1066 01:00:00,550 --> 01:00:04,322 because you could connect these logarithms 1067 01:00:04,322 --> 01:00:05,530 to an effective field theory. 1068 01:00:05,530 --> 01:00:08,072 And then the whole consistency of that effective field theory 1069 01:00:08,072 --> 01:00:10,510 really allows you to connect this logarithm 1070 01:00:10,510 --> 01:00:14,260 to other logarithms and really to build a picture for what's 1071 01:00:14,260 --> 01:00:17,440 going on with the physics that makes it totally clear what 1072 01:00:17,440 --> 01:00:19,470 the answer must be. 1073 01:00:19,470 --> 01:00:19,970 OK. 1074 01:00:19,970 --> 01:00:22,360 So just to give you an example from my own past. 1075 01:00:39,760 --> 01:00:43,030 So let's now turn to this question of the standard model 1076 01:00:43,030 --> 01:00:44,667 as an effective field theory. 1077 01:00:48,650 --> 01:00:51,520 So we have sum over n and we're treating this 1078 01:00:51,520 --> 01:00:52,490 from the bottom up. 1079 01:00:52,490 --> 01:00:54,700 So we're going to just talk about what 1080 01:00:54,700 --> 01:00:57,580 the degrees of freedom are and then think 1081 01:00:57,580 --> 01:00:58,870 about constructing operators. 1082 01:01:02,648 --> 01:01:04,690 And part of the job has already been done for us, 1083 01:01:04,690 --> 01:01:06,988 because I'm assuming you have a background 1084 01:01:06,988 --> 01:01:09,280 in the standard model, at least at the level of knowing 1085 01:01:09,280 --> 01:01:10,732 what the Lagrangian is. 1086 01:01:10,732 --> 01:01:12,190 And if you haven't, then you should 1087 01:01:12,190 --> 01:01:16,210 look at the quantum field theory three lecture notes. 1088 01:01:16,210 --> 01:01:19,540 So the L0 here is the standard model, 1089 01:01:19,540 --> 01:01:21,370 as taught in quantum field theory three. 1090 01:01:25,900 --> 01:01:28,600 So [INAUDIBLE] be interested in as the higher order terms. 1091 01:01:34,290 --> 01:01:35,953 But let me nevertheless remind you 1092 01:01:35,953 --> 01:01:38,370 of what the degrees of freedom were in the standard model. 1093 01:01:45,370 --> 01:01:47,310 So you at least know what the players are 1094 01:01:47,310 --> 01:01:48,780 when we go to talk about L1. 1095 01:01:53,252 --> 01:01:54,210 So it's a gauge theory. 1096 01:01:57,609 --> 01:02:05,200 So we have color across Su 2 weak across the un 1097 01:02:05,200 --> 01:02:07,726 of hypercharge. 1098 01:02:07,726 --> 01:02:20,650 And so we have eight gluons here, three week bosons here, 1099 01:02:20,650 --> 01:02:23,710 and one guy here. 1100 01:02:28,380 --> 01:02:31,290 So just to introduce some notation for fields, 1101 01:02:31,290 --> 01:02:34,200 I'll call these guys with an index capital A running 1102 01:02:34,200 --> 01:02:36,390 from 1 to 8, these guys with an index 1103 01:02:36,390 --> 01:02:38,850 lower a running from 1 to 3. 1104 01:02:38,850 --> 01:02:42,780 And B here would be the analog of a photon 1105 01:02:42,780 --> 01:02:44,770 field for U1 electromagnetism. 1106 01:02:44,770 --> 01:02:49,300 But this is the U1 of hypercharge, so it's B mu. 1107 01:02:49,300 --> 01:02:52,110 So we have gauge bosons. 1108 01:02:52,110 --> 01:02:55,210 We have fermions. 1109 01:02:55,210 --> 01:02:56,610 Let me do the fermions over here. 1110 01:03:04,120 --> 01:03:05,620 So an important thing and thinking 1111 01:03:05,620 --> 01:03:07,330 about this as an effective field theory 1112 01:03:07,330 --> 01:03:08,920 is to note what the mass scales are. 1113 01:03:13,600 --> 01:03:16,060 So maybe I should do that already here. 1114 01:03:16,060 --> 01:03:17,530 Protons are massless. 1115 01:03:17,530 --> 01:03:21,820 That's one combination of the weak and U1 boson. 1116 01:03:21,820 --> 01:03:24,160 Gluons are massless. 1117 01:03:24,160 --> 01:03:26,440 That's these guys. 1118 01:03:26,440 --> 01:03:35,188 And then there's the mass of the W. 80.42 GB the mass 1119 01:03:35,188 --> 01:03:40,390 of the Z, 91.19. 1120 01:03:40,390 --> 01:03:44,348 And for the first time in me teaching this course, 1121 01:03:44,348 --> 01:03:46,140 we also know what the mass of the Higgs is. 1122 01:03:46,140 --> 01:03:47,170 So let me just-- 1123 01:03:47,170 --> 01:03:48,688 that's not part of the gauge theory, 1124 01:03:48,688 --> 01:03:50,230 but I'll just list it there, as well, 1125 01:03:50,230 --> 01:03:51,980 since it doesn't fit in with the fermions. 1126 01:03:56,660 --> 01:04:04,660 So fermions-- so you can see that these scales here 1127 01:04:04,660 --> 01:04:06,110 are kind of similar. 1128 01:04:06,110 --> 01:04:08,360 For the fermions, there's a broad spectrum of scales. 1129 01:04:08,360 --> 01:04:10,568 And that's why I wanted to put them all on one board. 1130 01:04:16,720 --> 01:04:24,280 So quarks-- up quarks, down quarks, strange quarks-- 1131 01:04:24,280 --> 01:04:27,400 they all come in left and right handed [INAUDIBLE] 1132 01:04:27,400 --> 01:04:29,350 and the gauge couplings are different for left 1133 01:04:29,350 --> 01:04:33,088 and right handed, for the electroweak and U1 parts 1134 01:04:33,088 --> 01:04:33,880 of the gauge group. 1135 01:04:36,620 --> 01:04:41,830 So there are six different flavors and both right 1136 01:04:41,830 --> 01:04:44,050 and left handed. 1137 01:04:44,050 --> 01:04:46,180 What masses do we have? 1138 01:04:46,180 --> 01:04:51,070 Up quarks and down quarks are rather light, 1139 01:04:51,070 --> 01:04:52,525 about a couple of MeV. 1140 01:04:56,320 --> 01:04:57,820 It's hard to measure the light ones. 1141 01:05:00,720 --> 01:05:02,827 It's a little easier to measure. 1142 01:05:02,827 --> 01:05:04,160 Everything's going to be in MeV. 1143 01:05:04,160 --> 01:05:06,370 I'm going to stop writing MeV. 1144 01:05:06,370 --> 01:05:08,050 Oh, that's not true. 1145 01:05:08,050 --> 01:05:10,240 I switched to GeV. 1146 01:05:10,240 --> 01:05:13,000 Sorry. 1147 01:05:13,000 --> 01:05:15,250 Everything is going to be in GeV. 1148 01:05:24,850 --> 01:05:25,505 Oh, sorry. 1149 01:05:25,505 --> 01:05:26,380 That's the top quark. 1150 01:05:40,340 --> 01:05:41,510 OK. 1151 01:05:41,510 --> 01:05:43,220 So there's a pretty wide range of scales 1152 01:05:43,220 --> 01:05:47,150 here from an MeV to 100 GeV. 1153 01:05:47,150 --> 01:05:49,022 That's just the quarks. 1154 01:05:49,022 --> 01:05:50,480 And then we also have the leptons-- 1155 01:05:59,220 --> 01:06:03,150 three types of charge leptons, again, 1156 01:06:03,150 --> 01:06:04,815 with a fairly wide range of scales. 1157 01:06:10,300 --> 01:06:12,790 So now I'm switching back to MeV, 1158 01:06:12,790 --> 01:06:14,040 just to keep you on your toes. 1159 01:06:18,270 --> 01:06:18,770 Whoops. 1160 01:06:25,300 --> 01:06:27,490 Then we have neutrinos. 1161 01:06:27,490 --> 01:06:29,830 The left-handed ones we've studied much more 1162 01:06:29,830 --> 01:06:33,600 than anything else. 1163 01:06:33,600 --> 01:06:35,350 And in particular, what we know most about 1164 01:06:35,350 --> 01:06:39,280 the left-handed guys is mass splittings from neutrino 1165 01:06:39,280 --> 01:06:41,245 oscillations. 1166 01:06:41,245 --> 01:06:42,790 And these are pretty small numbers. 1167 01:06:46,425 --> 01:06:48,050 We also know that overall, these things 1168 01:06:48,050 --> 01:06:51,530 are quite light, from cosmological constraints 1169 01:06:51,530 --> 01:06:52,659 and otherwise. 1170 01:06:57,820 --> 01:07:01,150 And we don't really know about anything 1171 01:07:01,150 --> 01:07:07,570 like a sterile neutrino but that we can put bounds on its mass. 1172 01:07:07,570 --> 01:07:09,110 So even within the standard model, 1173 01:07:09,110 --> 01:07:10,527 there's a lot of different scales. 1174 01:07:12,297 --> 01:07:14,630 And if you think about it from an effective field theory 1175 01:07:14,630 --> 01:07:17,210 point of view and you think about it from the top down, 1176 01:07:17,210 --> 01:07:19,850 the first thing you'd get rid of would be the top quark. 1177 01:07:19,850 --> 01:07:23,660 And then you'd get rid of the W and the Z and the Higgs. 1178 01:07:23,660 --> 01:07:25,380 And then you would proceed down. 1179 01:07:25,380 --> 01:07:29,595 The next thing to go would be the bottom quark, et cetera. 1180 01:07:29,595 --> 01:07:31,970 And you could think about constructing an effective field 1181 01:07:31,970 --> 01:07:33,865 theory by integrating out one at a time, 1182 01:07:33,865 --> 01:07:35,990 getting a new effective field theory every time you 1183 01:07:35,990 --> 01:07:38,216 remove a degree of freedom. 1184 01:07:38,216 --> 01:07:39,680 You could take the standard model 1185 01:07:39,680 --> 01:07:41,858 and expand in that fashion. 1186 01:07:41,858 --> 01:07:44,150 That's not the sense in which we are thinking about it. 1187 01:07:44,150 --> 01:07:46,692 That would be the top-down sense of taking the standard model 1188 01:07:46,692 --> 01:07:48,200 and deriving something else. 1189 01:07:48,200 --> 01:07:50,780 We're thinking of it here in a different context 1190 01:07:50,780 --> 01:07:52,162 where we have all this stuff. 1191 01:07:52,162 --> 01:07:53,870 And we're actually interested in thinking 1192 01:07:53,870 --> 01:07:55,670 about physics at higher energy scales, 1193 01:07:55,670 --> 01:07:57,930 beyond the scale of the weak bosons, 1194 01:07:57,930 --> 01:08:00,140 beyond the scale of the top quark, the things 1195 01:08:00,140 --> 01:08:04,143 we're trying to figure out at the LHC, scales 1196 01:08:04,143 --> 01:08:05,060 we're trying to probe. 1197 01:08:07,950 --> 01:08:09,910 That's the attitude in this bottom-up approach. 1198 01:08:21,800 --> 01:08:22,300 OK. 1199 01:08:22,300 --> 01:08:23,800 So the lowest order Lagrangian would 1200 01:08:23,800 --> 01:08:26,170 be the gauge sector of the thermionic Lagrangian, 1201 01:08:26,170 --> 01:08:27,189 the Higgs Lagrangian. 1202 01:08:27,189 --> 01:08:29,098 And if we have right-handed neutrinos, 1203 01:08:29,098 --> 01:08:30,640 we'd need a Lagrangian for them, too. 1204 01:08:36,479 --> 01:08:42,790 So these are topics that come up in [INAUDIBLE].. 1205 01:08:42,790 --> 01:08:49,229 I'm not even going to touch them at the moment. 1206 01:08:49,229 --> 01:08:57,540 I can't give you a complete review, but just a taste, 1207 01:08:57,540 --> 01:09:00,725 emphasizing things that are important. 1208 01:09:00,725 --> 01:09:07,838 So to give you a taste, I just write the other two down, 1209 01:09:07,838 --> 01:09:09,380 which are the prettier parts, anyway. 1210 01:09:15,979 --> 01:09:18,620 So we have field strengths for the kinetic terms 1211 01:09:18,620 --> 01:09:19,495 for our gauge bosons. 1212 01:09:27,040 --> 01:09:31,540 And the thermionic Lagrangian, I can write it 1213 01:09:31,540 --> 01:09:34,609 as a sum over the left handed fields-- 1214 01:09:34,609 --> 01:09:38,060 fermion, covariant derivative fermion. 1215 01:09:38,060 --> 01:09:41,440 Add a sum over right handed fields. 1216 01:09:41,440 --> 01:09:47,620 Fermion covariant derivative fermion 1217 01:09:47,620 --> 01:09:51,220 where this covariant derivative is a covariant derivative 1218 01:09:51,220 --> 01:09:53,750 with these gauge fields. 1219 01:09:53,750 --> 01:10:00,960 So there's some gauge coupling, G1, for hypercharge, 1220 01:10:00,960 --> 01:10:06,460 some gauge coupling, G2, for [INAUDIBLE] weak, 1221 01:10:06,460 --> 01:10:09,490 and some gauge coupling, G, for QCD. 1222 01:10:17,110 --> 01:10:20,064 So what is the power counting? 1223 01:10:20,064 --> 01:10:24,940 So we've just said what the degrees of freedom are 1224 01:10:24,940 --> 01:10:28,060 and what kind of some of the guiding principles 1225 01:10:28,060 --> 01:10:31,282 are-- the symmetries, the gauge symmetry. 1226 01:10:31,282 --> 01:10:33,490 You learn much more about symmetries in quantum field 1227 01:10:33,490 --> 01:10:35,780 theory three, so I'm not going to go into that. 1228 01:10:35,780 --> 01:10:39,250 But those are basically the guiding principles 1229 01:10:39,250 --> 01:10:41,470 in figuring out this L0. 1230 01:10:41,470 --> 01:10:44,980 What is it that we would do a power counting in here? 1231 01:10:44,980 --> 01:10:47,110 So the power counting in this bottom up approach 1232 01:10:47,110 --> 01:10:48,730 is related to what we left out. 1233 01:10:59,890 --> 01:11:02,650 So we're expanding an epsilon here, 1234 01:11:02,650 --> 01:11:06,580 where epsilon is mass scales in this standard model divided 1235 01:11:06,580 --> 01:11:09,800 by things that we've left out of our description. 1236 01:11:09,800 --> 01:11:12,220 So in the numerator would be things like the top quark 1237 01:11:12,220 --> 01:11:17,140 mass, the W mass, Z mass, Higgs mass, all the mass scales 1238 01:11:17,140 --> 01:11:18,640 of the standard model. 1239 01:11:18,640 --> 01:11:21,040 In the denominator, well, certainly something 1240 01:11:21,040 --> 01:11:25,750 like M plank is left out of our description here. 1241 01:11:25,750 --> 01:11:28,120 If we had some grand unified theory, 1242 01:11:28,120 --> 01:11:30,010 that goes in the denominator. 1243 01:11:30,010 --> 01:11:32,920 If we have supersymmetry and we broke it, 1244 01:11:32,920 --> 01:11:34,310 that would go in the denominator. 1245 01:11:34,310 --> 01:11:36,010 So from this effective field theory point of view, 1246 01:11:36,010 --> 01:11:38,260 any physics that we've left out of the standard model 1247 01:11:38,260 --> 01:11:42,050 description is anything that generates a higher energy 1248 01:11:42,050 --> 01:11:42,550 scale. 1249 01:11:42,550 --> 01:11:44,246 That goes in the denominator. 1250 01:11:53,680 --> 01:11:56,515 And this is what we expanded. 1251 01:11:56,515 --> 01:11:58,890 So even not knowing something about what this physics is, 1252 01:11:58,890 --> 01:12:02,430 we can come up with a universal description-- a universal L1-- 1253 01:12:02,430 --> 01:12:04,650 that describes corrections beyond the standard model. 1254 01:12:09,060 --> 01:12:11,370 And what will be describing that physics is higher 1255 01:12:11,370 --> 01:12:14,340 dimension operators-- operators beyond dimension four. 1256 01:12:23,570 --> 01:12:25,670 But they'll be built out of standard model fields. 1257 01:12:37,260 --> 01:12:42,060 So kind of from your teaching of quantum field theory, maybe 1258 01:12:42,060 --> 01:12:45,150 perhaps the idea that these two things are connected 1259 01:12:45,150 --> 01:12:47,580 may be clear to you, but it's something 1260 01:12:47,580 --> 01:12:55,530 that we will actually cover mostly next class, actually. 1261 01:12:55,530 --> 01:12:57,670 Some part of the beginning of next class, 1262 01:12:57,670 --> 01:13:00,250 we'll make this connection between the fact 1263 01:13:00,250 --> 01:13:03,790 that we want to expand in that epsilon and the fact 1264 01:13:03,790 --> 01:13:05,522 that we can do-- in doing so, we get 1265 01:13:05,522 --> 01:13:06,730 higher dimensional operators. 1266 01:13:06,730 --> 01:13:08,682 We'll make that absolutely clear. 1267 01:13:08,682 --> 01:13:09,640 That'll come next time. 1268 01:13:14,375 --> 01:13:15,750 So in the remainder of today, let 1269 01:13:15,750 --> 01:13:22,620 me just address one final point, and that 1270 01:13:22,620 --> 01:13:25,590 is the idea of what it means to have a renormalizable field 1271 01:13:25,590 --> 01:13:27,210 theory. 1272 01:13:27,210 --> 01:13:29,820 So in our description of the standard model that I gave 1273 01:13:29,820 --> 01:13:31,708 here, I mentioned symmetries. 1274 01:13:31,708 --> 01:13:33,000 I mentioned degrees of freedom. 1275 01:13:33,000 --> 01:13:35,436 I didn't mention renormalizability. 1276 01:13:43,212 --> 01:13:46,280 So what does re-normalizable mean? 1277 01:13:46,280 --> 01:13:49,070 So the traditional definition of what renormalizable will mean 1278 01:13:49,070 --> 01:13:51,540 would be the following. 1279 01:13:51,540 --> 01:13:59,130 You would say a theory is renormalizable 1280 01:13:59,130 --> 01:14:06,620 if at any order in perturbation theory in this quantum field 1281 01:14:06,620 --> 01:14:20,956 theory the UV divergences can be absorbed-- 1282 01:14:20,956 --> 01:14:24,215 so there's UV divergences from loop integrals. 1283 01:14:24,215 --> 01:14:26,340 If they can always be absorbed into a finite number 1284 01:14:26,340 --> 01:14:30,146 of parameters, then you'd say the theory is renormalizable. 1285 01:14:33,930 --> 01:14:35,430 But that's a traditional definition. 1286 01:14:35,430 --> 01:14:37,388 And we will use a more general definition here. 1287 01:14:46,177 --> 01:14:47,760 Certainly this was a guiding principle 1288 01:14:47,760 --> 01:14:49,552 when people constructed the standard model. 1289 01:14:52,500 --> 01:14:55,390 What is the effective field theory definition of this? 1290 01:14:55,390 --> 01:15:00,360 It's a little more general because it 1291 01:15:00,360 --> 01:15:04,950 brings in the idea of doing power counting. 1292 01:15:04,950 --> 01:15:08,160 So the effective field theory's definition 1293 01:15:08,160 --> 01:15:10,620 allows for the possibility of having an infinite number 1294 01:15:10,620 --> 01:15:12,210 of parameters. 1295 01:15:12,210 --> 01:15:14,388 But at any order that you truncate the theory, 1296 01:15:14,388 --> 01:15:15,930 there should only be a finite number. 1297 01:15:20,120 --> 01:15:26,770 So it says that a theory must be renormalizable order by order 1298 01:15:26,770 --> 01:15:28,654 in its expansion parameter. 1299 01:15:32,450 --> 01:15:35,588 Well, if there's more than one, it's expansion parameters. 1300 01:15:40,960 --> 01:15:42,820 So even just this sentence alone tells you 1301 01:15:42,820 --> 01:15:46,000 why power counting is such an important part 1302 01:15:46,000 --> 01:15:48,970 of the effective theory, because the effective theory 1303 01:15:48,970 --> 01:15:51,010 to make sense as a renormalizable quantum field 1304 01:15:51,010 --> 01:15:53,560 theory needs to know about its expansion parameter. 1305 01:15:53,560 --> 01:15:55,498 We're saying that it's renormalizable, 1306 01:15:55,498 --> 01:15:57,040 that we can make sense of the theory, 1307 01:15:57,040 --> 01:16:00,370 absorb all the infinities, only order by order in expansion 1308 01:16:00,370 --> 01:16:02,920 parameters in general. 1309 01:16:10,190 --> 01:16:12,185 So it could be that you do some calculation, 1310 01:16:12,185 --> 01:16:13,715 you counter some divergences. 1311 01:16:13,715 --> 01:16:15,590 But if they're higher order in the expansion, 1312 01:16:15,590 --> 01:16:18,590 then what you need-- and you have a power counting that 1313 01:16:18,590 --> 01:16:19,760 tells you that-- 1314 01:16:19,760 --> 01:16:21,892 they would be absorbable into some operators 1315 01:16:21,892 --> 01:16:24,350 that you haven't even written down, you can just drop them. 1316 01:16:30,580 --> 01:16:41,140 So this definition allows for an infinite number 1317 01:16:41,140 --> 01:16:45,430 of parameters that are needed, for example, 1318 01:16:45,430 --> 01:16:54,170 for normalizability, but only a finite number at some fixed 1319 01:16:54,170 --> 01:16:54,670 order. 1320 01:17:05,500 --> 01:17:07,785 Now, if you take this logic that I just 1321 01:17:07,785 --> 01:17:09,910 said to you, that you could think about things more 1322 01:17:09,910 --> 01:17:11,530 generally, then you can ask, well, 1323 01:17:11,530 --> 01:17:14,250 what was the point of thinking about the standard model, where 1324 01:17:14,250 --> 01:17:16,000 the theory turned out to be renormalizable 1325 01:17:16,000 --> 01:17:17,900 in the traditional sense? 1326 01:17:17,900 --> 01:17:21,100 How does this fact, which is just a subset of this case, 1327 01:17:21,100 --> 01:17:23,500 but an important one, how does it 1328 01:17:23,500 --> 01:17:26,170 fit into this rubric from an effective field 1329 01:17:26,170 --> 01:17:28,640 theory point of view? 1330 01:17:28,640 --> 01:17:31,660 And so the way that that fits in is as follows. 1331 01:17:31,660 --> 01:17:36,310 It could turn out that your L0 in your expansion 1332 01:17:36,310 --> 01:17:39,040 is renormalizable in the traditional sense rather 1333 01:17:39,040 --> 01:17:41,140 than this more general sense. 1334 01:17:43,930 --> 01:17:46,150 And if that's true, what it means 1335 01:17:46,150 --> 01:17:52,930 is that you don't see the higher energy scales from your lowest 1336 01:17:52,930 --> 01:17:53,740 order Lagrangian. 1337 01:17:59,260 --> 01:18:11,920 So we do not know directly about lambda 1338 01:18:11,920 --> 01:18:14,590 new from just looking at L0. 1339 01:18:17,233 --> 01:18:19,150 And that's what happens in the standard model. 1340 01:18:19,150 --> 01:18:22,593 We don't really know precisely what the high energy 1341 01:18:22,593 --> 01:18:24,010 scale should be just from studying 1342 01:18:24,010 --> 01:18:26,650 the effect-- from studying the leading order Lagrangian. 1343 01:18:26,650 --> 01:18:28,070 This will not always be the case. 1344 01:18:28,070 --> 01:18:29,470 Sometimes we'll be in a situation 1345 01:18:29,470 --> 01:18:32,613 where when we study the effective theory at lowest 1346 01:18:32,613 --> 01:18:34,030 order in the Lagrangian, we really 1347 01:18:34,030 --> 01:18:36,910 find that even in order to make sense of that as a quantum 1348 01:18:36,910 --> 01:18:39,550 field theory that we need to-- 1349 01:18:39,550 --> 01:18:41,530 that there's a scale that gets generated 1350 01:18:41,530 --> 01:18:43,310 and it's part of our expansion. 1351 01:18:43,310 --> 01:18:45,430 And there's some terms that we calculate 1352 01:18:45,430 --> 01:18:48,400 with L0 that end up being higher order in our expansion. 1353 01:18:48,400 --> 01:18:51,430 And we can't renormalize the theory 1354 01:18:51,430 --> 01:18:55,570 unless we actually include higher dimension operators. 1355 01:18:55,570 --> 01:18:58,090 Chiral perturbation theory is an example of that type. 1356 01:18:58,090 --> 01:19:00,560 And that's an example that we'll treat. 1357 01:19:00,560 --> 01:19:03,560 So it's not always the case in the standard model, 1358 01:19:03,560 --> 01:19:05,830 whether it's renormalizable in a traditional sense. 1359 01:19:05,830 --> 01:19:11,030 That's a special case, though it's an important one. 1360 01:19:11,030 --> 01:19:11,860 So questions? 1361 01:19:17,710 --> 01:19:21,575 OK, so hopefully this has been partly a review of some things 1362 01:19:21,575 --> 01:19:23,950 that you've thought of before, but putting them together, 1363 01:19:23,950 --> 01:19:25,690 perhaps in a nicer package. 1364 01:19:25,690 --> 01:19:28,450 And we'll continue next time talking 1365 01:19:28,450 --> 01:19:30,702 about the standard model as an effective field theory. 1366 01:19:30,702 --> 01:19:31,910 What can be gained from that? 1367 01:19:31,910 --> 01:19:34,750 How do we construct the operators? 1368 01:19:34,750 --> 01:19:37,260 And we'll keep going from there.