WEBVTT 00:00:00.000 --> 00:00:02.520 The following content is provided under a Creative 00:00:02.520 --> 00:00:03.970 Commons license. 00:00:03.970 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.660 continue to offer high quality educational resources for free. 00:00:10.660 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.190 --> 00:00:18.370 at ocw.mit.edu. 00:00:20.960 --> 00:00:22.900 PROFESSOR: --the back of the room. 00:00:22.900 --> 00:00:24.910 This course is being taped. 00:00:24.910 --> 00:00:28.110 It may appear in the future on EdX or MITx. 00:00:28.110 --> 00:00:30.272 It will appear on OpenCourseWare. 00:00:30.272 --> 00:00:32.439 So that means that by just sitting in the classroom, 00:00:32.439 --> 00:00:35.590 you're choosing to participate in this adventure of having 00:00:35.590 --> 00:00:36.940 the course videotaped. 00:00:36.940 --> 00:00:39.340 If you want to hide, you can sit in the back row. 00:00:39.340 --> 00:00:42.892 If you want to be part of the-- 00:00:42.892 --> 00:00:44.850 if you want to be famous, sit in the front row. 00:00:52.330 --> 00:00:53.910 So I handed out the syllabus. 00:00:53.910 --> 00:00:57.100 And I'll just go through some of the things on here. 00:00:57.100 --> 00:00:58.800 So as a prerequisite-- 00:00:58.800 --> 00:01:02.070 so at MIT, we have three quantum field theory courses, 1, 2, 3. 00:01:02.070 --> 00:01:05.099 As a prerequisite, I've listed quantum field theory two. 00:01:05.099 --> 00:01:07.770 I don't want to stop people that are taking quantum field 00:01:07.770 --> 00:01:10.920 theory 3 this term from also registering for this course. 00:01:10.920 --> 00:01:13.020 But some of the things that you'd usually 00:01:13.020 --> 00:01:15.480 see in quantum field theory 3 are 00:01:15.480 --> 00:01:18.340 kind of important background things for this course. 00:01:18.340 --> 00:01:22.078 So what I've done is I've posted on the website for the course 00:01:22.078 --> 00:01:24.120 my lecture notes from when I taught quantum field 00:01:24.120 --> 00:01:25.730 theory 3 last year. 00:01:25.730 --> 00:01:26.730 So you can look at that. 00:01:26.730 --> 00:01:29.340 And I'll assign some reading for that as background material 00:01:29.340 --> 00:01:31.410 for this. 00:01:31.410 --> 00:01:34.440 Some things actually were taught in quantum field theory 2 00:01:34.440 --> 00:01:36.600 last term, some things like, for example, 00:01:36.600 --> 00:01:39.828 the beta function of QCD for normalization group. 00:01:39.828 --> 00:01:42.120 So those things are going to be important prerequisites 00:01:42.120 --> 00:01:43.770 for this course. 00:01:43.770 --> 00:01:46.170 And I'll assign some background reading 00:01:46.170 --> 00:01:49.900 so that you can remind yourself about those things. 00:01:49.900 --> 00:01:51.900 If you have any concerns about your preparation, 00:01:51.900 --> 00:01:54.390 please feel free to talk to me. 00:01:54.390 --> 00:01:57.040 In terms of grading, this is pretty low key. 00:01:57.040 --> 00:02:00.300 We have five problem sets that will 00:02:00.300 --> 00:02:01.920 be roughly every two weeks. 00:02:01.920 --> 00:02:06.120 We won't start the first problem set until next week. 00:02:06.120 --> 00:02:07.979 And then there'll be a presentation 00:02:07.979 --> 00:02:08.979 at the end of the class. 00:02:08.979 --> 00:02:11.520 There's no exams, no tests. 00:02:11.520 --> 00:02:14.370 The purpose of the presentation is basically, 00:02:14.370 --> 00:02:16.527 effective field theory is a very broad topic 00:02:16.527 --> 00:02:18.360 and there's no chance we'll be able to cover 00:02:18.360 --> 00:02:20.110 every possible thing about effective field 00:02:20.110 --> 00:02:21.300 theory in this course. 00:02:21.300 --> 00:02:23.010 In particular, a lot of applications 00:02:23.010 --> 00:02:27.610 we'll have to be brief on or we won't touch on at all. 00:02:27.610 --> 00:02:31.080 So if you look at the syllabus on page three, 00:02:31.080 --> 00:02:33.810 there's a list of possible presentation topics. 00:02:33.810 --> 00:02:35.010 Each one of you will pick-- 00:02:35.010 --> 00:02:36.900 each one of you that registers for the course 00:02:36.900 --> 00:02:40.380 and wants to give a presentation will give a presentation 00:02:40.380 --> 00:02:41.820 at the end on one of these topics, 00:02:41.820 --> 00:02:45.323 prepare it, and stand up and lecture to the class. 00:02:45.323 --> 00:02:46.740 And the idea is that basically you 00:02:46.740 --> 00:02:49.428 will teach the class for half an hour or so about one 00:02:49.428 --> 00:02:50.970 of these topics, and then that way we 00:02:50.970 --> 00:02:55.500 will broaden the scope of things that we'll be able to cover. 00:02:55.500 --> 00:02:59.430 Even this long list is not complete. 00:02:59.430 --> 00:03:01.740 And you can also pick a topic that's not on this list 00:03:01.740 --> 00:03:03.790 if you talk to me about it. 00:03:03.790 --> 00:03:07.622 So a little later I'll also give you some suggested references 00:03:07.622 --> 00:03:08.580 for reading about this. 00:03:08.580 --> 00:03:10.410 I just, at this point, give you the list. 00:03:10.410 --> 00:03:13.090 And we'll talk about where you might read about these things 00:03:13.090 --> 00:03:13.590 later on. 00:03:13.590 --> 00:03:17.340 I'll give you a more detailed handout. 00:03:17.340 --> 00:03:18.970 So even if you just look at this list, 00:03:18.970 --> 00:03:21.480 you kind of see the breadth of-- the idea of effective field 00:03:21.480 --> 00:03:27.240 theory, how broadly it impacts physics, and particle physics, 00:03:27.240 --> 00:03:28.230 in particular. 00:03:28.230 --> 00:03:31.200 So you know, everything from finite temperature QCD, 00:03:31.200 --> 00:03:35.280 finite density, effective field theories in inflation, 00:03:35.280 --> 00:03:36.215 popular these days. 00:03:36.215 --> 00:03:37.840 Effective field theories of cold atoms, 00:03:37.840 --> 00:03:39.810 are also very popular these days. 00:03:39.810 --> 00:03:43.470 Non-relativistic QCD production of quarkonia, both in medium, 00:03:43.470 --> 00:03:44.820 out of medium. 00:03:44.820 --> 00:03:46.860 Relativistic super fluids. 00:03:46.860 --> 00:03:48.320 Conformal effective field theories. 00:03:48.320 --> 00:03:49.320 The list goes on and on. 00:03:49.320 --> 00:03:51.120 Lattice effective field theories. 00:03:51.120 --> 00:03:52.650 So it's a very broad topic. 00:03:52.650 --> 00:03:54.540 And what I'm going to do in this course 00:03:54.540 --> 00:03:57.180 is I'm going to start out with the first half of the course 00:03:57.180 --> 00:03:59.693 teaching you about the ideas of effective field theory, 00:03:59.693 --> 00:04:01.110 some of the basic ingredients that 00:04:01.110 --> 00:04:03.060 go into all effective field theories 00:04:03.060 --> 00:04:05.645 and some more technical things that only show up 00:04:05.645 --> 00:04:07.020 in some effective field theories, 00:04:07.020 --> 00:04:08.702 but that are kind of providing you 00:04:08.702 --> 00:04:10.410 with lessons if you had to ever construct 00:04:10.410 --> 00:04:11.880 your own effective field theory. 00:04:11.880 --> 00:04:14.153 That's the idea of the first half of the course. 00:04:14.153 --> 00:04:15.570 The second half of the course, I'm 00:04:15.570 --> 00:04:17.862 going to focus on a particular effective field theory-- 00:04:17.862 --> 00:04:19.860 soft-colinear effective field theory, which 00:04:19.860 --> 00:04:25.860 is an effective field theory that's close to my heart that's 00:04:25.860 --> 00:04:28.350 exciting these days because of its applications to the jet 00:04:28.350 --> 00:04:30.690 physics in the LHC. 00:04:30.690 --> 00:04:33.270 So we'll talk about that as the second half of the course. 00:04:38.150 --> 00:04:41.060 So as I say on the outline, there's not really 00:04:41.060 --> 00:04:44.330 a textbook for this course. 00:04:44.330 --> 00:04:46.830 I suggest some text that may be useful, 00:04:46.830 --> 00:04:48.710 but they're not any one of them will 00:04:48.710 --> 00:04:51.740 be useful for more than 20% of the material, many of them 00:04:51.740 --> 00:04:53.640 for less than 10% of the material. 00:04:53.640 --> 00:04:55.640 So I don't necessarily recommend that you go out 00:04:55.640 --> 00:04:57.890 and buy all of them-- or buy any of them. 00:04:57.890 --> 00:05:01.040 I will try to post chapters of things 00:05:01.040 --> 00:05:03.020 on the website when it's required reading. 00:05:03.020 --> 00:05:05.270 And these books will be available in the reading room, 00:05:05.270 --> 00:05:07.580 so you can do it that way, if you like. 00:05:10.767 --> 00:05:13.100 If you're really serious about phenomenology, of course, 00:05:13.100 --> 00:05:14.392 you should own all these books. 00:05:20.630 --> 00:05:24.300 So on the course outline, I also listed office hours. 00:05:24.300 --> 00:05:29.130 And I will adjust those when I find out all your schedules. 00:05:29.130 --> 00:05:30.710 And I also listed that [INAUDIBLE] 00:05:30.710 --> 00:05:34.760 who's sitting in the back, is a 10% TA for the course. 00:05:34.760 --> 00:05:36.590 He'll be doing some grading for us. 00:05:36.590 --> 00:05:37.715 He won't have office hours. 00:05:42.604 --> 00:05:43.437 AUDIENCE: Thank you. 00:05:46.898 --> 00:05:48.440 PROFESSOR: So I know that some of you 00:05:48.440 --> 00:05:49.732 have registered for the course. 00:05:49.732 --> 00:05:51.320 And some of you are here as listeners. 00:05:51.320 --> 00:05:54.410 If you are here is a listener, I have an add form 00:05:54.410 --> 00:05:56.450 and I want you to add it as a listener, 00:05:56.450 --> 00:05:58.640 just so that we have a record of you, 00:05:58.640 --> 00:06:00.770 partly because that affects the ability 00:06:00.770 --> 00:06:02.600 to offer this course in the future, 00:06:02.600 --> 00:06:05.090 also because we're going to use online tools 00:06:05.090 --> 00:06:07.790 and we may even use some online MITx 00:06:07.790 --> 00:06:11.010 tools if I find out that they're very useful for the course. 00:06:11.010 --> 00:06:12.600 So I want you to be in the system. 00:06:12.600 --> 00:06:16.070 And I want you to be able to access that information if we 00:06:16.070 --> 00:06:17.000 end up going that way. 00:06:21.790 --> 00:06:24.110 So I think that's it. 00:06:24.110 --> 00:06:27.520 So as far as preamble, does anybody have any questions? 00:06:32.110 --> 00:06:32.610 OK. 00:06:48.110 --> 00:06:48.610 All right. 00:06:48.610 --> 00:06:50.965 So let's start. 00:06:50.965 --> 00:06:52.465 So let's start with the big picture. 00:06:59.820 --> 00:07:01.370 I like to say that the big picture is 00:07:01.370 --> 00:07:03.328 that there's interesting physics at all scales. 00:07:16.010 --> 00:07:17.690 What effective field theory lets you do 00:07:17.690 --> 00:07:19.760 is it lets you tease out this interesting physics 00:07:19.760 --> 00:07:21.080 at all scales. 00:07:21.080 --> 00:07:25.190 So in particular, you can focus on a particular scale 00:07:25.190 --> 00:07:27.470 and find the interesting physics there using 00:07:27.470 --> 00:07:30.620 the tools of effective field theory. 00:07:30.620 --> 00:07:33.200 Now that's a little different than how we teach physics. 00:07:33.200 --> 00:07:36.860 If you think about drawing a diagram from your freshman year 00:07:36.860 --> 00:07:40.490 to now many of you in graduate studies about how you learned 00:07:40.490 --> 00:07:43.400 physics, it was not by kind of focusing in 00:07:43.400 --> 00:07:45.980 on individual particular scales, but more 00:07:45.980 --> 00:07:49.230 from a kind of bottom-up point of view. 00:07:49.230 --> 00:07:52.980 So I would draw a picture of how we teach 00:07:52.980 --> 00:07:55.870 physics in the following way. 00:07:55.870 --> 00:07:57.540 So you start out as a freshman. 00:07:57.540 --> 00:07:58.980 You learn classical mechanics. 00:08:06.480 --> 00:08:17.640 You learn E&M, classical E&M. You learn Newtonian gravity. 00:08:22.932 --> 00:08:24.390 And then you build on these things. 00:08:24.390 --> 00:08:25.530 These things are your starting point 00:08:25.530 --> 00:08:26.655 and then you build on them. 00:08:26.655 --> 00:08:35.429 So you learn quantum mechanics and that 00:08:35.429 --> 00:08:38.440 builds on classical mechanics. 00:08:38.440 --> 00:08:46.590 You learn special relativity, which 00:08:46.590 --> 00:08:53.220 is building on both classical mechanics and electromagnetism. 00:08:53.220 --> 00:09:09.900 At some point, you learn general relativity and quantum field 00:09:09.900 --> 00:09:10.400 theory. 00:09:17.510 --> 00:09:19.250 And quantum field theory is synthesizing 00:09:19.250 --> 00:09:23.880 quantum mechanics and special relativity, and then these two 00:09:23.880 --> 00:09:24.380 here. 00:09:35.065 --> 00:09:37.190 So you think about learning physics from the bottom 00:09:37.190 --> 00:09:38.562 up in this picture. 00:09:38.562 --> 00:09:39.770 You learn these things first. 00:09:39.770 --> 00:09:40.700 And then you learn these things. 00:09:40.700 --> 00:09:42.140 And you sort of keep synthesizing, 00:09:42.140 --> 00:09:44.257 keep putting things together. 00:09:44.257 --> 00:09:45.840 We're going to be doing in this course 00:09:45.840 --> 00:09:47.930 the exact opposite of that. 00:09:47.930 --> 00:09:49.910 We're going to be taking one of these blobs-- 00:09:49.910 --> 00:09:51.523 in particular, this one-- 00:09:51.523 --> 00:09:52.940 and we're going to be looking deep 00:09:52.940 --> 00:09:55.665 inside it trying to make more and more specific field 00:09:55.665 --> 00:09:56.165 theories. 00:10:06.700 --> 00:10:09.870 So if you like, if you want to draw it as a blob, 00:10:09.870 --> 00:10:13.500 we'll be taking quantum field theory and we'll be 00:10:13.500 --> 00:10:16.410 looking for derivatives of it-- 00:10:16.410 --> 00:10:19.425 or perhaps derivatives of derivatives of it, 00:10:19.425 --> 00:10:22.050 like this-- and figuring out how to take a very general theory, 00:10:22.050 --> 00:10:24.630 like quantum field theory for the standard model, 00:10:24.630 --> 00:10:27.960 and finding things that are more specific and in some ways more 00:10:27.960 --> 00:10:31.040 powerful than just having the original theory 00:10:31.040 --> 00:10:33.390 we started with-- more powerful in the sense of being 00:10:33.390 --> 00:10:34.620 able to do calculations. 00:11:05.300 --> 00:11:06.520 So why do we want to do that? 00:11:11.470 --> 00:11:13.820 There's a couple of different reasons-- 00:11:13.820 --> 00:11:16.390 or why do we do that, since, as I've 00:11:16.390 --> 00:11:18.610 tried to convince you with the outline of the course 00:11:18.610 --> 00:11:20.230 and some of the presentation topics, 00:11:20.230 --> 00:11:23.170 that this is something that happens all over the place. 00:11:27.820 --> 00:11:31.260 So as you go up in this chart, it actually becomes-- 00:11:31.260 --> 00:11:33.010 even though you have a more general theory 00:11:33.010 --> 00:11:35.177 and it becomes more beautiful and you can write down 00:11:35.177 --> 00:11:37.710 a synthesis of physics in fewer lines, 00:11:37.710 --> 00:11:39.910 it also becomes harder to compute things. 00:11:47.260 --> 00:11:50.080 So just as an example, if you just 00:11:50.080 --> 00:11:54.820 wanted to compute the energy spectrum of hydrogen, 00:11:54.820 --> 00:11:56.740 and you know very well that you can 00:11:56.740 --> 00:11:58.060 do that in quantum mechanics. 00:11:58.060 --> 00:12:02.443 And particularly it's a classic example, and fairly easy. 00:12:02.443 --> 00:12:04.360 If you try to do that in quantum field theory, 00:12:04.360 --> 00:12:06.130 it's much harder, because quantum field 00:12:06.130 --> 00:12:09.143 theory, in some sense, has too much for that problem. 00:12:09.143 --> 00:12:10.060 So that's one example. 00:12:13.760 --> 00:12:15.760 Another is the elliptical orbits of the planets, 00:12:15.760 --> 00:12:18.490 which are easier in Newtonian gravity 00:12:18.490 --> 00:12:22.360 than in general relativity. 00:12:22.360 --> 00:12:23.790 And these are just two examples. 00:12:23.790 --> 00:12:26.800 There's many more where the field-- 00:12:26.800 --> 00:12:29.440 one of these blobs may be too general for actually 00:12:29.440 --> 00:12:34.040 tackling the problem that you want to deal with. 00:12:34.040 --> 00:12:37.060 And by focusing in, which is what effective field 00:12:37.060 --> 00:12:41.710 theory allows you to do, you can get more ability 00:12:41.710 --> 00:12:45.730 to compute more accurately and in a simpler fashion. 00:12:54.220 --> 00:13:00.030 So what we want when we think about effective field theory 00:13:00.030 --> 00:13:02.280 is we want the simplest framework that 00:13:02.280 --> 00:13:03.690 captures the essential physics. 00:13:07.920 --> 00:13:09.720 We don't want to carry along for the ride 00:13:09.720 --> 00:13:11.220 a whole bunch of superfluous things 00:13:11.220 --> 00:13:13.053 that are not important for the problem we're 00:13:13.053 --> 00:13:15.730 trying to deal with. 00:13:15.730 --> 00:13:17.970 But we also don't want to give up anything. 00:13:17.970 --> 00:13:22.170 So we're very demanding. 00:13:22.170 --> 00:13:26.180 So even if we're giving up something in our leading order 00:13:26.180 --> 00:13:28.400 description, we want to retain the ability 00:13:28.400 --> 00:13:30.770 to correct that leading order description-- 00:13:30.770 --> 00:13:34.070 order by order in some expansions-- 00:13:34.070 --> 00:13:36.375 so that we can make it as precise as we desire. 00:13:42.260 --> 00:13:48.800 So I'll say that we can correct it, in principle, to arbitrary 00:13:48.800 --> 00:13:49.430 precision. 00:13:52.322 --> 00:13:53.780 So if you like, what we're doing is 00:13:53.780 --> 00:13:56.690 we're taking quantum field theory and we're expanding it. 00:13:56.690 --> 00:13:58.940 And the lowest order determine that description 00:13:58.940 --> 00:14:00.800 is an effective field theory. 00:14:00.800 --> 00:14:04.400 And that effective field theory may have different fields. 00:14:04.400 --> 00:14:07.670 It may have different symmetries. 00:14:07.670 --> 00:14:10.112 And it will certainly have ability 00:14:10.112 --> 00:14:11.570 to calculate in a different fashion 00:14:11.570 --> 00:14:13.100 than the original theory. 00:14:13.100 --> 00:14:15.615 But we will keep higher order terms in that expansion. 00:14:15.615 --> 00:14:17.240 And therefore, we'll be able to correct 00:14:17.240 --> 00:14:20.750 it to arbitrary precision just by expanding 00:14:20.750 --> 00:14:23.180 to higher and higher order. 00:14:23.180 --> 00:14:26.540 So examples of this that even are familiar here-- 00:14:26.540 --> 00:14:28.520 non relativistic expansion, getting back 00:14:28.520 --> 00:14:30.860 to non relativistic quantum mechanics, 00:14:30.860 --> 00:14:34.400 or doing a post-Newtonian expansion in general relativity 00:14:34.400 --> 00:14:36.352 to go back towards Newtonian gravity. 00:14:36.352 --> 00:14:38.060 You don't have to stop at the first term. 00:14:38.060 --> 00:14:39.740 You can keep higher-order terms. 00:14:39.740 --> 00:14:41.310 And in that way, you could calculate, 00:14:41.310 --> 00:14:43.010 for example, energy levels in hydrogen 00:14:43.010 --> 00:14:45.110 using a non-relativistic framework that 00:14:45.110 --> 00:14:48.260 encodes all the ingredients of quantum field theory. 00:14:48.260 --> 00:14:49.730 Or you could use-- 00:14:49.730 --> 00:14:52.142 you could look at the orbit of planets 00:14:52.142 --> 00:14:54.350 and relativistic corrections and general relativistic 00:14:54.350 --> 00:14:57.557 corrections by expanding this theory. 00:14:57.557 --> 00:14:58.640 So those are two examples. 00:15:11.980 --> 00:15:14.670 So most of the examples that I've listed on the project list 00:15:14.670 --> 00:15:17.010 are the type of taking quantum field 00:15:17.010 --> 00:15:21.040 theory for the standard model. 00:15:21.040 --> 00:15:23.110 Some of them involve gravity, but most of them 00:15:23.110 --> 00:15:28.960 involve the standard model and expanding it 00:15:28.960 --> 00:15:33.270 and focusing in on particular degrees of freedom. 00:15:33.270 --> 00:15:35.270 So let's say that we've picked a physical system 00:15:35.270 --> 00:15:36.920 and we want to describe it. 00:15:36.920 --> 00:15:38.390 What are the things that we should 00:15:38.390 --> 00:15:41.000 do in order to develop an effective field theory? 00:15:51.828 --> 00:15:52.620 What are the steps? 00:15:56.050 --> 00:15:58.330 I should say that I'm going to post my lecture notes. 00:15:58.330 --> 00:16:00.580 So if you don't want to take notes, you don't have to. 00:16:00.580 --> 00:16:04.470 If you'd like to take notes, feel free. 00:16:04.470 --> 00:16:06.310 But I will scan and post my notes. 00:16:22.305 --> 00:16:23.680 So the first thing you need to do 00:16:23.680 --> 00:16:25.972 is figure out what the relevant degrees of freedom are. 00:16:29.865 --> 00:16:31.240 What are the things that actually 00:16:31.240 --> 00:16:33.850 matter for the problem you want to study? 00:16:33.850 --> 00:16:36.500 Sometimes that is easier than other times. 00:16:36.500 --> 00:16:39.070 Sometimes it's completely obvious. 00:16:39.070 --> 00:16:41.063 You want to study some low energy properties 00:16:41.063 --> 00:16:41.980 of the standard model. 00:16:41.980 --> 00:16:43.438 You get rid of the heavy particles. 00:16:43.438 --> 00:16:44.740 You keep the light ones. 00:16:44.740 --> 00:16:46.270 Fairly straightforward. 00:16:46.270 --> 00:16:48.370 Other times it may be tricky to actually determine 00:16:48.370 --> 00:16:50.230 what the relevant degrees of freedom are. 00:16:50.230 --> 00:16:53.330 And people in the field may even argue about what they are. 00:16:53.330 --> 00:16:55.090 So we'll talk about examples of both types 00:16:55.090 --> 00:16:59.420 here throughout the course. 00:16:59.420 --> 00:17:02.296 So it sounds trivial, but it may not be. 00:17:02.296 --> 00:17:04.588 You also want to think about the symmetries. 00:17:04.588 --> 00:17:06.130 Sometimes that guides you in thinking 00:17:06.130 --> 00:17:07.713 about the relevant degrees of freedom. 00:17:07.713 --> 00:17:10.240 Sometimes these things go hand in hand. 00:17:10.240 --> 00:17:12.190 But that's certainly an important ingredient 00:17:12.190 --> 00:17:16.119 in developing the effective field theory. 00:17:16.119 --> 00:17:18.670 And you also have to be careful here, because sometimes you 00:17:18.670 --> 00:17:22.030 might have a theory that has no symmetry, 00:17:22.030 --> 00:17:24.280 or doesn't have an apparent symmetry. 00:17:24.280 --> 00:17:26.829 But when you start expanding-- which is what I've argued 00:17:26.829 --> 00:17:28.000 you're going to be doing-- 00:17:28.000 --> 00:17:30.910 when you start expanding, you may have a symmetry suddenly 00:17:30.910 --> 00:17:31.870 appear. 00:17:31.870 --> 00:17:34.270 And we'll actually talk about several examples 00:17:34.270 --> 00:17:38.690 of that happening throughout the course, as well. 00:17:38.690 --> 00:17:40.990 So your effective field theory may have more symmetry 00:17:40.990 --> 00:17:43.402 than the theory you started with. 00:17:43.402 --> 00:17:44.860 Because you've neglected something, 00:17:44.860 --> 00:17:46.068 you could have more symmetry. 00:17:48.763 --> 00:17:50.680 And the other important thing is to figure out 00:17:50.680 --> 00:17:51.722 what you're expanding in. 00:17:58.267 --> 00:18:00.100 And at the same time, what the leading order 00:18:00.100 --> 00:18:01.690 description of the theory is. 00:18:01.690 --> 00:18:04.622 What is the lowest order Lagrangian? 00:18:04.622 --> 00:18:06.080 And basically, these are the things 00:18:06.080 --> 00:18:08.560 you have to do to get started. 00:18:08.560 --> 00:18:11.620 If this is true, actually, independent of kind 00:18:11.620 --> 00:18:13.180 of what theory you're talking about, 00:18:13.180 --> 00:18:16.240 if you're doing quantum field theory, what this first one 00:18:16.240 --> 00:18:17.620 means is figuring out what fields 00:18:17.620 --> 00:18:18.662 you're going to be using. 00:18:25.000 --> 00:18:27.040 The symmetry is basically guiding you 00:18:27.040 --> 00:18:28.750 about the interactions. 00:18:32.320 --> 00:18:34.210 If you have gauge symmetry, then of course 00:18:34.210 --> 00:18:36.070 you're going to write down something 00:18:36.070 --> 00:18:37.277 that respects that symmetry. 00:18:37.277 --> 00:18:39.610 That's going to tell you something about the interaction 00:18:39.610 --> 00:18:41.050 terms. 00:18:41.050 --> 00:18:44.380 And then finally, this expansion parameters 00:18:44.380 --> 00:18:47.050 goes under the rubric of what is called power counting. 00:18:59.940 --> 00:19:01.920 And if you take those three things together 00:19:01.920 --> 00:19:03.962 and you figure out the leading order description, 00:19:03.962 --> 00:19:05.640 then you have an effective field theory. 00:19:05.640 --> 00:19:07.140 You write down he Lagrangian for it. 00:19:17.736 --> 00:19:19.370 You should try to use some color. 00:19:29.950 --> 00:19:32.340 So if you thought about regular quantum field theory-- 00:19:32.340 --> 00:19:34.098 for example, for the standard model-- 00:19:34.098 --> 00:19:35.640 you'd also do these first two things. 00:19:35.640 --> 00:19:36.840 You figure out what the fields are. 00:19:36.840 --> 00:19:38.040 You figure out what the interactions are. 00:19:38.040 --> 00:19:40.332 But you don't think about too much about this question, 00:19:40.332 --> 00:19:42.060 about the power counting. 00:19:42.060 --> 00:19:45.420 And actually, more broadly, in the field of effective field 00:19:45.420 --> 00:19:49.690 theory, this idea here of power counting is very important. 00:19:49.690 --> 00:19:52.650 It's as important as something like gauge symmetry. 00:19:52.650 --> 00:19:55.530 It's really a fundamental thing about the whole framework 00:19:55.530 --> 00:19:56.280 that you're doing. 00:20:03.230 --> 00:20:17.720 So in an effective field theory, power counting 00:20:17.720 --> 00:20:23.090 is just as important as figuring out things like symmetries. 00:20:23.090 --> 00:20:24.590 And in particular, just to make it 00:20:24.590 --> 00:20:27.283 sort of clear that it's important, 00:20:27.283 --> 00:20:28.700 I'll compare it to gauge symmetry. 00:20:31.400 --> 00:20:35.030 It's really a fundamental ingredient in what you're doing 00:20:35.030 --> 00:20:38.343 and in the whole theory, because the power counting 00:20:38.343 --> 00:20:40.760 being consistent is actually necessary for the whole field 00:20:40.760 --> 00:20:41.270 theory-- 00:20:41.270 --> 00:20:42.895 effective field theory-- to make sense. 00:20:46.200 --> 00:20:47.910 OK, so what's the key principle? 00:20:50.640 --> 00:20:53.640 Well, there's a key principle of quantum field theory 00:20:53.640 --> 00:20:58.500 that we're using when we design effective field theories. 00:20:58.500 --> 00:21:01.385 And that is that we're insensitive to physics 00:21:01.385 --> 00:21:02.385 at higher energy scales. 00:21:10.654 --> 00:21:15.630 So going back to Wilson, if we're 00:21:15.630 --> 00:21:18.210 interested in describing the physics at some scale 00:21:18.210 --> 00:21:24.660 m squared, we don't need to know the details of physics 00:21:24.660 --> 00:21:25.410 at higher energy. 00:21:48.978 --> 00:21:53.028 So at scales lambda squared that are much bigger than m squared, 00:21:53.028 --> 00:21:55.570 we don't really need to know the details of the dynamics that 00:21:55.570 --> 00:21:56.962 are going on there. 00:21:56.962 --> 00:21:59.170 We don't need to know the field content, necessarily, 00:21:59.170 --> 00:22:03.505 at those scales, or anything else about the dynamics. 00:22:08.110 --> 00:22:10.575 And that is, in some sense, a key idea 00:22:10.575 --> 00:22:12.700 that makes the whole idea of effective field theory 00:22:12.700 --> 00:22:13.200 possible. 00:22:18.730 --> 00:22:20.470 I want you to make sure that you don't 00:22:20.470 --> 00:22:22.020 think of this too narrowly. 00:22:22.020 --> 00:22:24.520 In the way that I've written, it's actually a little bit too 00:22:24.520 --> 00:22:27.100 narrow, because I said that-- 00:22:27.100 --> 00:22:28.940 I've described it in terms of mass scale. 00:22:28.940 --> 00:22:31.120 So I've kind of intuited in your mind something 00:22:31.120 --> 00:22:33.940 like a Z' boson or a W' boson, which 00:22:33.940 --> 00:22:36.850 is a heavy particle from some perspective. 00:22:36.850 --> 00:22:39.490 And maybe they treat that as a heavy particle and I'm 00:22:39.490 --> 00:22:41.760 interested-- 00:22:41.760 --> 00:22:43.925 and let me say it differently. 00:22:43.925 --> 00:22:45.550 Say I'm interested in a light particle, 00:22:45.550 --> 00:22:47.860 like the bottom quark. 00:22:47.860 --> 00:22:51.460 Then I can get rid of physics at a heavy scale, like the w mass. 00:22:51.460 --> 00:22:55.630 Or if I think about how new physics impacts precision 00:22:55.630 --> 00:22:58.390 electroweak data and that new physics is heavy, 00:22:58.390 --> 00:23:00.580 then I can think about effective operators. 00:23:00.580 --> 00:23:01.955 That's kind of the intuition I've 00:23:01.955 --> 00:23:04.080 given you with the sentence that I've written here. 00:23:04.080 --> 00:23:05.200 But it's actually not-- 00:23:05.200 --> 00:23:07.670 that's not quite general enough. 00:23:07.670 --> 00:23:09.610 And the reason it's not quite general enough 00:23:09.610 --> 00:23:13.780 is that this mentality doesn't always apply just strictly 00:23:13.780 --> 00:23:15.820 to math scales. 00:23:15.820 --> 00:23:18.555 If I say this-- m squared much less than lambda squared-- then 00:23:18.555 --> 00:23:19.930 you immediately think that you're 00:23:19.930 --> 00:23:22.480 expanding an m squared over lambda squared. 00:23:22.480 --> 00:23:24.470 And that, of classic effective field theory, 00:23:24.470 --> 00:23:26.150 that's exactly what you do. 00:23:26.150 --> 00:23:28.990 You expand in light scales divided by heavy scales. 00:23:28.990 --> 00:23:29.920 They're mass scales. 00:23:29.920 --> 00:23:31.030 They're invariant masses. 00:23:31.030 --> 00:23:33.100 They're Lorentz invariant quantities. 00:23:33.100 --> 00:23:35.092 That's the most common thing that you do. 00:23:35.092 --> 00:23:37.550 We're going to be doing much more than that in this course. 00:23:37.550 --> 00:23:39.675 We'll be doing-- we'll be talking about cases where 00:23:39.675 --> 00:23:43.150 the power counting is in situations that are not simply 00:23:43.150 --> 00:23:46.300 m squared much less than lambda squared, but other things-- 00:23:46.300 --> 00:23:48.010 dimensionless parameters. 00:23:48.010 --> 00:23:50.025 So sometimes it will become more complicated. 00:23:50.025 --> 00:23:51.400 But still, this guiding principle 00:23:51.400 --> 00:23:55.450 that there's physics that's far away, in some sense, 00:23:55.450 --> 00:23:57.820 from the physics that you're interested in, that it can 00:23:57.820 --> 00:24:01.900 be removed from the theory when you're just talking about it, 00:24:01.900 --> 00:24:03.640 in that sense, this is more general. 00:24:03.640 --> 00:24:06.370 It's really not just a strictly one-dimensional thing, 00:24:06.370 --> 00:24:09.310 as it would be if I describe it in terms of mass scales, 00:24:09.310 --> 00:24:11.890 but a more general principle. 00:24:11.890 --> 00:24:14.260 And I think that will become clearer 00:24:14.260 --> 00:24:16.720 when we actually do examples where it's not simply 00:24:16.720 --> 00:24:17.980 mass scales. 00:24:17.980 --> 00:24:21.040 We will, of course, start out by talking about mass scale, 00:24:21.040 --> 00:24:23.835 since that's the classic thing that most people think 00:24:23.835 --> 00:24:25.710 of when they think of effective field theory. 00:24:28.420 --> 00:24:29.950 So can anyone give me an example-- 00:24:29.950 --> 00:24:32.180 just to try to get the class engaged here-- 00:24:32.180 --> 00:24:35.170 can anyone give me an example of an effective field theory that 00:24:35.170 --> 00:24:38.815 doesn't involve expanding strictly in mass scales, 00:24:38.815 --> 00:24:41.477 involves some other type of expansion? 00:24:41.477 --> 00:24:43.690 AUDIENCE: [INAUDIBLE] 00:24:43.690 --> 00:24:46.270 PROFESSOR: Well, OK. 00:24:46.270 --> 00:24:49.150 HQT you can say is just an expansion 00:24:49.150 --> 00:24:55.564 in lambda QCD over MB or MC, so not quite. 00:24:55.564 --> 00:24:56.440 AUDIENCE: [INAUDIBLE] 00:24:56.440 --> 00:25:01.480 PROFESSOR: [INAUDIBLE] Well, you've studied it. 00:25:01.480 --> 00:25:04.630 But correct answer. 00:25:04.630 --> 00:25:08.088 Any other thoughts? 00:25:08.088 --> 00:25:09.630 So another example would be something 00:25:09.630 --> 00:25:12.000 like non-relativistic QED. 00:25:12.000 --> 00:25:13.710 If you do non-relativistic QED, you're 00:25:13.710 --> 00:25:15.480 actually expanding in the velocity, 00:25:15.480 --> 00:25:17.760 not strictly in the ratio of mass scales, 00:25:17.760 --> 00:25:20.400 although you would think, for example, non-relativistic 00:25:20.400 --> 00:25:23.230 QED would be for a heavy electron, which is true-- 00:25:23.230 --> 00:25:25.230 a heavy, massive particle. 00:25:25.230 --> 00:25:28.410 That's not what the power counting expansion is in. 00:25:28.410 --> 00:25:30.180 It'll actually be in the velocity 00:25:30.180 --> 00:25:31.930 being much less than the speed of light. 00:25:31.930 --> 00:25:33.638 And that actually plays an important role 00:25:33.638 --> 00:25:35.400 in designing the effective field theory, 00:25:35.400 --> 00:25:38.710 determining what the leading operators are. 00:25:38.710 --> 00:25:41.790 So even something as simple as going 00:25:41.790 --> 00:25:44.010 to higher order corrections in hydrogen 00:25:44.010 --> 00:25:46.018 involves thinking about-- 00:25:46.018 --> 00:25:47.810 thinking beyond this simple statement here. 00:25:51.684 --> 00:25:55.360 Well, since we're on the topic of hydrogen, 00:25:55.360 --> 00:25:57.990 let me go into a little more detail there. 00:26:04.060 --> 00:26:11.070 So just to flesh this out a bit and to talk about some 00:26:11.070 --> 00:26:15.990 of the things that you have to be careful about, 00:26:15.990 --> 00:26:18.840 I'll phrase an example of this statement as the fact 00:26:18.840 --> 00:26:21.420 that we don't need to know about bottom quarks 00:26:21.420 --> 00:26:28.550 to describe hydrogen. 00:26:28.550 --> 00:26:29.300 Well, that's good. 00:26:29.300 --> 00:26:31.230 When you took quantum mechanics as an undergraduate, 00:26:31.230 --> 00:26:33.540 you didn't have bottom quarks in your description. 00:26:33.540 --> 00:26:36.779 So if you did need them, you would have missed something. 00:26:41.180 --> 00:26:43.192 What did you have in your description? 00:26:43.192 --> 00:26:44.900 Well, in a quantum field theory language, 00:26:44.900 --> 00:26:45.910 you have this diagram. 00:26:45.910 --> 00:26:49.760 You had an electron and a proton with photon exchange. 00:26:53.810 --> 00:26:56.300 And you also, when you thought about the binding energy, 00:26:56.300 --> 00:26:59.300 if we work in units where h bar and c is 1, which I will always 00:26:59.300 --> 00:27:04.480 do, then the binding energy is 1/2 ME alpha squared. 00:27:04.480 --> 00:27:06.470 And if you ask about the bottom quarks, 00:27:06.470 --> 00:27:08.600 the reason that you didn't need them 00:27:08.600 --> 00:27:11.900 is because they were suppressed. 00:27:11.900 --> 00:27:14.870 They weren't negligible-- completely negligible-- 00:27:14.870 --> 00:27:17.450 at least, at the level of how accurately we 00:27:17.450 --> 00:27:19.070 can measure this thing. 00:27:19.070 --> 00:27:20.295 Well, they're pretty small. 00:27:20.295 --> 00:27:21.920 They're at the 10 to the minus 8 level. 00:27:25.340 --> 00:27:27.110 And that's because they come in suppressed 00:27:27.110 --> 00:27:29.030 by the mass of the electron squared 00:27:29.030 --> 00:27:31.070 over the mass of the bottom quarks squared. 00:27:34.200 --> 00:27:36.000 So how would you think of them coming in? 00:27:36.000 --> 00:27:38.660 Well you'd think of them coming in through some diagram, 00:27:38.660 --> 00:27:45.350 for example, where the bottom quark couples to the photon 00:27:45.350 --> 00:27:48.860 through a vacuum polarization like that. 00:27:48.860 --> 00:27:50.840 And this diagram, indeed, will give you 00:27:50.840 --> 00:27:53.390 corrections of this type. 00:27:53.390 --> 00:27:55.400 Now, it's a bit more subtle than that. 00:27:55.400 --> 00:27:57.980 And that's because a diagram like this 00:27:57.980 --> 00:28:00.860 also has other contributions besides just these ones that I 00:28:00.860 --> 00:28:01.550 mentioned here. 00:28:08.490 --> 00:28:10.158 So the basic picture is, indeed, correct 00:28:10.158 --> 00:28:12.200 that we can neglect the bottom quark because it's 00:28:12.200 --> 00:28:13.580 giving small corrections. 00:28:13.580 --> 00:28:15.110 But there is one subtlety. 00:28:15.110 --> 00:28:17.700 And that has to do with the fact that we have to decide 00:28:17.700 --> 00:28:18.950 what we mean by this coupling. 00:28:21.530 --> 00:28:22.370 OK. 00:28:22.370 --> 00:28:29.510 So from your previous courses in quantum field theory, 00:28:29.510 --> 00:28:32.148 when you learned about running couplings, 00:28:32.148 --> 00:28:34.190 you learned the diagrams like that one contribute 00:28:34.190 --> 00:28:37.610 to running couplings. 00:28:37.610 --> 00:28:42.200 And so the B quark, therefore, can affect the coupling 00:28:42.200 --> 00:28:45.873 if you worked in, for example, the MS bar scheme, 00:28:45.873 --> 00:28:48.040 since it contributes to the running of the coupling. 00:28:51.080 --> 00:28:55.910 And in particular, you know for the electromagnetic coupling, 00:28:55.910 --> 00:28:58.250 if you ask about what that coupling is, 00:28:58.250 --> 00:29:00.420 it has a different value because it runs-- 00:29:00.420 --> 00:29:04.100 if you evaluate it at a scale like the W mass, 00:29:04.100 --> 00:29:09.710 then it's like 1/128, versus if you evaluate it at a very low 00:29:09.710 --> 00:29:13.170 energy, the electron mass or below, 00:29:13.170 --> 00:29:18.270 then it's the classic 1/137.036. 00:29:18.270 --> 00:29:18.770 OK. 00:29:18.770 --> 00:29:20.540 So there's some change. 00:29:20.540 --> 00:29:22.295 And the bottom quark is part of what 00:29:22.295 --> 00:29:23.420 contributes to that change. 00:29:23.420 --> 00:29:25.462 Of course, other particles are contributing, too. 00:29:31.940 --> 00:29:36.440 So if we want to say this statement about bottom quarks 00:29:36.440 --> 00:29:40.130 and we want to state the conclusions more precisely, 00:29:40.130 --> 00:29:41.450 then we would do it this way. 00:29:41.450 --> 00:29:58.970 We would say if alpha is a parameter of the standard model 00:29:58.970 --> 00:30:03.320 and we imagine that we fix it at a high energy-- 00:30:03.320 --> 00:30:04.820 so we could imagine that we fixed it 00:30:04.820 --> 00:30:08.960 by doing [INAUDIBLE] on physics in an E plus E minus collider. 00:30:08.960 --> 00:30:12.290 But some process that's a high energy process, 00:30:12.290 --> 00:30:14.180 we determine, say, for example, this value. 00:30:19.760 --> 00:30:23.430 If we take that attitude as how we define the parameter, 00:30:23.430 --> 00:30:25.250 then the parameter that actually matters 00:30:25.250 --> 00:30:27.950 for hydrogen, which is this parameter at the low scale, 00:30:27.950 --> 00:30:36.718 does depend on the bottom quark, because how 00:30:36.718 --> 00:30:38.510 we get from the high scale to the low scale 00:30:38.510 --> 00:30:41.600 depends on the fact that the bottom quark exists. 00:30:49.790 --> 00:30:53.180 But we could also take a different attitude. 00:30:53.180 --> 00:30:55.250 And that is we could take a low energy attitude. 00:31:06.860 --> 00:31:09.830 So we could say, let's forget about doing high energy 00:31:09.830 --> 00:31:10.800 physics. 00:31:10.800 --> 00:31:13.370 Let's just do low energy physics and extract 00:31:13.370 --> 00:31:18.839 alpha of 0 from some low energy atomic experiments. 00:31:29.600 --> 00:31:34.700 And if that's the way that we define the parameter, 00:31:34.700 --> 00:31:36.700 then the value can be used in other experiments. 00:31:36.700 --> 00:31:38.533 And we never have to know anything about MB. 00:31:54.810 --> 00:31:58.500 So we didn't really have to know about the high energy-- 00:31:58.500 --> 00:32:00.558 about the higher energy theory unless we actually 00:32:00.558 --> 00:32:02.100 were doing some experiments up there. 00:32:09.080 --> 00:32:11.660 Is there any questions about that? 00:32:11.660 --> 00:32:12.160 Good. 00:32:17.040 --> 00:32:20.847 So if we want to write an equation for that, 00:32:20.847 --> 00:32:22.680 what it means is that when you integrate out 00:32:22.680 --> 00:32:25.830 particles like the B quark, remove them from your theory, 00:32:25.830 --> 00:32:31.470 stop considering them, that it's not simply the case 00:32:31.470 --> 00:32:34.830 that you generate higher order terms in the series. 00:32:34.830 --> 00:32:37.080 You can also affect what you mean by the leading order 00:32:37.080 --> 00:32:39.750 term in the sense of changing what you mean by the coupling. 00:32:48.450 --> 00:32:51.300 So if I write it in terms of Lagrangian, 00:32:51.300 --> 00:32:54.690 I would say that the Lagrangian for hydrogen, 00:32:54.690 --> 00:32:57.090 if we include the B quark, well, it's 00:32:57.090 --> 00:32:59.820 got our proton, electron, and photon. 00:32:59.820 --> 00:33:02.280 And let's keep the B quark. 00:33:02.280 --> 00:33:06.490 Alpha and MB are parameters. 00:33:06.490 --> 00:33:10.770 If we drop the B quark because it's giving small effects, 00:33:10.770 --> 00:33:15.450 we just have these fields-- proton, electron, and photon. 00:33:15.450 --> 00:33:24.240 We get a different coupling in practice-- in principle. 00:33:24.240 --> 00:33:26.730 So you think of this as being a higher energy coupling 00:33:26.730 --> 00:33:30.223 and over there alpha prime being the low energy coupling. 00:33:30.223 --> 00:33:31.890 There's a lot of other things, actually, 00:33:31.890 --> 00:33:34.360 that if you think about hydrogen for a minute, 00:33:34.360 --> 00:33:36.930 there's a lot of other expansions that you've done. 00:33:36.930 --> 00:33:39.360 Hydrogen is a very fertile ground 00:33:39.360 --> 00:33:41.480 for effective field theory. 00:33:41.480 --> 00:33:43.142 So let's do that. 00:33:43.142 --> 00:33:45.600 Let's make a little list of what we dropped when we thought 00:33:45.600 --> 00:33:48.960 about hydrogen. How much did we lie to you when we first 00:33:48.960 --> 00:33:54.295 taught you the hydrogen atom in a quantum mechanics course? 00:33:54.295 --> 00:33:55.920 Well, we didn't teach you about quarks. 00:33:58.770 --> 00:34:02.480 Why didn't we teach you about quarks? 00:34:02.480 --> 00:34:04.907 And the reason we didn't teach you about quarks 00:34:04.907 --> 00:34:06.990 is because if you think about the typical momentum 00:34:06.990 --> 00:34:12.302 transfer in hydrogen, three momentum transfer, 00:34:12.302 --> 00:34:14.010 it's [INAUDIBLE] the mass of the electron 00:34:14.010 --> 00:34:16.679 times the fine structure constant. 00:34:16.679 --> 00:34:20.190 And that's much less than the proton size. 00:34:20.190 --> 00:34:23.250 So the typical photons that are involved in binding together 00:34:23.250 --> 00:34:28.920 the hydrogen atom just have much lower energy and they can't see 00:34:28.920 --> 00:34:29.760 inside the proton. 00:34:29.760 --> 00:34:31.425 They just see it as one overall object. 00:34:31.425 --> 00:34:33.300 And so we don't need to know about the quarks 00:34:33.300 --> 00:34:36.730 inside the proton. 00:34:36.730 --> 00:34:37.830 So that was an expansion. 00:34:43.889 --> 00:34:50.769 It's also an insensitive to the proton mass itself. 00:34:50.769 --> 00:34:57.840 So the proton we keep as an object, but the mass of the-- 00:34:57.840 --> 00:35:01.170 again, the momentum transfer, ME alpha, 00:35:01.170 --> 00:35:03.330 is much less than the mass of the proton, which 00:35:03.330 --> 00:35:06.930 is of order of GeV. 00:35:06.930 --> 00:35:10.230 And so we expand in our treatment of the proton, 00:35:10.230 --> 00:35:11.830 as well. 00:35:11.830 --> 00:35:17.960 And basically what this means is that the proton 00:35:17.960 --> 00:35:20.975 acts like a static charge. 00:35:28.700 --> 00:35:32.150 But proton mass wasn't showing up in our lowest order 00:35:32.150 --> 00:35:33.950 description of the energy here. 00:35:33.950 --> 00:35:37.233 It would show up in higher order corrections that we neglect. 00:35:40.480 --> 00:35:42.610 And again, it's because we're expanding. 00:35:42.610 --> 00:35:45.400 And that actually affects, when we design an effective field 00:35:45.400 --> 00:35:48.040 theory for this situation, how we would treat 00:35:48.040 --> 00:35:49.570 the proton, what type of Lagrangian 00:35:49.570 --> 00:35:52.210 we would write down for it. 00:35:52.210 --> 00:35:54.010 And that will be one of our topics 00:35:54.010 --> 00:35:57.040 is to figure out how we treat heavy particles, like a proton, 00:35:57.040 --> 00:35:57.670 in this case. 00:36:02.200 --> 00:36:04.265 Another expansion that we did is we used the fact 00:36:04.265 --> 00:36:06.640 that the momentum transfer is much less than the electron 00:36:06.640 --> 00:36:08.980 mass, not just the proton mass. 00:36:08.980 --> 00:36:12.943 And that meant that the theory is non-relativistic 00:36:12.943 --> 00:36:14.860 and that's why we did non-relativistic quantum 00:36:14.860 --> 00:36:16.520 mechanics. 00:36:16.520 --> 00:36:18.520 If we wanted to do it as a quantum field theory, 00:36:18.520 --> 00:36:20.710 we would do a non-relativistic quantum field theory. 00:36:25.930 --> 00:36:28.510 So already in something as simple as hydrogen, 00:36:28.510 --> 00:36:31.720 we have here three expansions plus many more, 00:36:31.720 --> 00:36:33.220 thinking about the particles that we 00:36:33.220 --> 00:36:37.112 neglected in the description. 00:36:37.112 --> 00:36:38.820 AUDIENCE: So the second point [INAUDIBLE] 00:36:38.820 --> 00:36:42.176 and the alpha, how do you know that's not just the ratio of ME 00:36:42.176 --> 00:36:45.630 by MP rather than ME alpha? 00:36:45.630 --> 00:36:46.320 PROFESSOR: So-- 00:36:46.320 --> 00:36:47.153 [INTERPOSING VOICES] 00:36:47.153 --> 00:36:49.140 AUDIENCE: --electron was 1 [? GeV. ?] 00:36:49.140 --> 00:36:49.690 PROFESSOR: That's right. 00:36:49.690 --> 00:36:50.050 So-- 00:36:50.050 --> 00:36:50.850 AUDIENCE: We would care about it. 00:36:50.850 --> 00:36:51.767 PROFESSOR: Absolutely. 00:36:51.767 --> 00:36:54.780 So in some sense, I could have written ME here, 00:36:54.780 --> 00:36:56.373 and that would have been fine. 00:36:56.373 --> 00:36:58.290 If you take these together, then of course you 00:36:58.290 --> 00:37:02.310 could take a ratio and you'd get ME over M proton. 00:37:02.310 --> 00:37:04.260 The reason I wrote ME alpha is I was really 00:37:04.260 --> 00:37:07.560 thinking about the momentum, sort 00:37:07.560 --> 00:37:10.200 of the non-static properties, the dynamics. 00:37:10.200 --> 00:37:14.220 And the momentum of the photons, the largest energy is this. 00:37:14.220 --> 00:37:15.720 That's why I was thinking about it. 00:37:15.720 --> 00:37:19.210 But you're right that I should write ME over M proton, 00:37:19.210 --> 00:37:19.710 as well. 00:37:22.410 --> 00:37:25.660 As any other comments, questions? 00:37:39.530 --> 00:37:43.280 So another point that I just want to briefly comment, 00:37:43.280 --> 00:37:46.850 which has to be true if everything I'm telling you 00:37:46.850 --> 00:37:49.460 is right, which has to be true because it's 00:37:49.460 --> 00:37:55.910 what we taught you, is that this whole description is true 00:37:55.910 --> 00:37:58.660 even though there's ultraviolet divergences. 00:37:58.660 --> 00:38:00.410 When you start doing quantum field theory, 00:38:00.410 --> 00:38:02.493 even if you do quantum field theory, in this case, 00:38:02.493 --> 00:38:05.990 for the hydrogen atom, you run into ultraviolet divergences 00:38:05.990 --> 00:38:08.160 where things are blowing up. 00:38:08.160 --> 00:38:11.066 Actually, even this diagram has ultraviolet divergences. 00:38:14.960 --> 00:38:17.900 So before you regulate the theory, 00:38:17.900 --> 00:38:20.970 the bottom quark group is infinity. 00:38:20.970 --> 00:38:22.970 Once you regulate the theory, it's well-defined. 00:38:22.970 --> 00:38:26.300 And you can make everything well-defined. 00:38:26.300 --> 00:38:28.187 But you may worry that this diagram seems 00:38:28.187 --> 00:38:30.020 to be contributing an infinite amount rather 00:38:30.020 --> 00:38:31.310 than a finite amount. 00:38:31.310 --> 00:38:33.230 And the whole story goes through even 00:38:33.230 --> 00:38:36.080 in the context of having ultraviolet divergences. 00:38:39.230 --> 00:38:41.543 That better be true, because, for example, 00:38:41.543 --> 00:38:43.460 if we had graviton loops, they would also lead 00:38:43.460 --> 00:38:45.500 to ultraviolet divergences. 00:38:45.500 --> 00:38:51.650 And so we're neglecting gravity. 00:38:51.650 --> 00:38:54.320 So it better that these ideas of effective field theory 00:38:54.320 --> 00:38:57.980 are not changed by having ultraviolet divergences. 00:38:57.980 --> 00:39:00.620 And we'll encounter that in our discussion later on. 00:39:04.950 --> 00:39:05.450 OK. 00:39:05.450 --> 00:39:07.880 So that gives you a bit of a sense 00:39:07.880 --> 00:39:10.370 for how these ideas of effective field theory, you've 00:39:10.370 --> 00:39:13.160 been using them all along, whether or not you knew it. 00:39:18.050 --> 00:39:20.990 And we will, in this course, flesh out 00:39:20.990 --> 00:39:23.030 how we figure out some of these corrections, 00:39:23.030 --> 00:39:25.280 how we would actually compute them, 00:39:25.280 --> 00:39:28.070 how we would actually figure out how to even the leading order-- 00:39:28.070 --> 00:39:29.487 what the leading order description 00:39:29.487 --> 00:39:32.630 of a theory in cases where we may not know it or someone else 00:39:32.630 --> 00:39:33.890 hasn't figured it out yet. 00:39:33.890 --> 00:39:35.598 Those are the type of things we're after. 00:39:37.887 --> 00:39:40.220 Now, if you talk about the categories of effective field 00:39:40.220 --> 00:39:41.595 theories, if you look at the list 00:39:41.595 --> 00:39:44.030 that I handed out to you or the list of things 00:39:44.030 --> 00:39:46.010 we're going to do in the course, then 00:39:46.010 --> 00:39:48.440 there's really, in general, two ways 00:39:48.440 --> 00:39:50.090 that effective field theories are used. 00:40:09.810 --> 00:40:14.335 So the two ways are from the top down or from the bottom up. 00:40:14.335 --> 00:40:15.710 So we'll start with the top down. 00:40:18.590 --> 00:40:21.920 So in the top down situation, you 00:40:21.920 --> 00:40:24.020 know what the high energy theory is. 00:40:24.020 --> 00:40:26.270 I'm going to keep using this language of masses, 00:40:26.270 --> 00:40:28.665 where I have a high energy and low energy theory. 00:40:28.665 --> 00:40:30.290 And we'll think about it more generally 00:40:30.290 --> 00:40:32.248 when we come to examples where we need to think 00:40:32.248 --> 00:40:34.020 about it more generally. 00:40:34.020 --> 00:40:37.790 So in this top-down case, we have a high energy theory-- 00:40:37.790 --> 00:40:39.440 say, the standard model-- 00:40:39.440 --> 00:40:41.330 and that theory is understood in the sense 00:40:41.330 --> 00:40:45.560 that we can write down the Lagrangian for it. 00:40:45.560 --> 00:40:49.550 But we're not satisfied with that. 00:40:49.550 --> 00:41:01.190 We find it useful to have a simpler theory 00:41:01.190 --> 00:41:09.013 to do some low energy physics, or even to do some high energy 00:41:09.013 --> 00:41:10.430 physics, where not all the degrees 00:41:10.430 --> 00:41:13.220 of freedom of this high energy theory are relevant. 00:41:17.640 --> 00:41:20.360 So we're in a situation where we have some theory, which we'll 00:41:20.360 --> 00:41:23.030 call theory one, which is this high energy theory 00:41:23.030 --> 00:41:25.340 that we understand it that we can think 00:41:25.340 --> 00:41:27.037 about doing calculations in it. 00:41:27.037 --> 00:41:29.120 But we want to go over to some other theory, which 00:41:29.120 --> 00:41:33.230 I'll call theory two, which has less degrees of freedom. 00:41:33.230 --> 00:41:35.300 And we're making expansions. 00:41:35.300 --> 00:41:37.040 And that's a low energy theory. 00:41:37.040 --> 00:41:39.581 This is the high theory. 00:41:39.581 --> 00:41:42.450 This is the low theory. 00:41:42.450 --> 00:41:45.080 So that's what we are in this situation of what 00:41:45.080 --> 00:41:48.320 we call top down, coming from the top, from high energy, 00:41:48.320 --> 00:41:50.220 down. 00:41:50.220 --> 00:41:51.387 So what do we do? 00:41:51.387 --> 00:41:52.970 Well, in this case, it's kind of nice, 00:41:52.970 --> 00:41:55.640 because we can actually use the fact that we know this theory 00:41:55.640 --> 00:41:58.820 one and can do calculations in that theory one 00:41:58.820 --> 00:42:00.830 to even think about constructing theory two. 00:42:09.900 --> 00:42:13.080 So what we can do is we could just start calculating things 00:42:13.080 --> 00:42:21.450 in theory one and integrate out-- i.e. remove-- 00:42:21.450 --> 00:42:22.590 the heavier particles. 00:42:30.550 --> 00:42:33.700 And in doing that, we can do what's 00:42:33.700 --> 00:42:38.680 called matching onto the low energy theory. 00:42:48.140 --> 00:42:49.700 That means we can use this ability 00:42:49.700 --> 00:42:53.060 to do calculations in the high energy theory 00:42:53.060 --> 00:42:58.320 to find what the operators are of the low energy theory, 00:42:58.320 --> 00:43:01.740 just by direct calculation. 00:43:01.740 --> 00:43:06.170 And also if there's new low energy constants that show up, 00:43:06.170 --> 00:43:08.810 we can calculate the values of those constants 00:43:08.810 --> 00:43:12.590 by using information and connecting them 00:43:12.590 --> 00:43:15.300 to the high energy theory. 00:43:15.300 --> 00:43:18.740 So in this case, we're able to use calculations really 00:43:18.740 --> 00:43:20.330 to construct the low energy theory. 00:43:26.130 --> 00:43:28.500 So just schematically, I start with some high energy 00:43:28.500 --> 00:43:36.930 Lagrangian and I go over to some low energy Lagrangians where 00:43:36.930 --> 00:43:39.360 there's an infinite series, which I've indexed by n. 00:43:39.360 --> 00:43:41.970 And that index is to denote higher order 00:43:41.970 --> 00:43:44.608 terms that are less relevant in whatever expansion 00:43:44.608 --> 00:43:45.150 you're doing. 00:43:54.930 --> 00:43:56.630 So just to be general, I'll say it's 00:43:56.630 --> 00:44:00.245 an expansion in decreasing relevance of the terms. 00:44:05.760 --> 00:44:08.360 So if you're in this situation, then these two theories 00:44:08.360 --> 00:44:11.090 are, in some sense, describing common things. 00:44:11.090 --> 00:44:13.580 The high energy theory describes more than the low energy 00:44:13.580 --> 00:44:15.247 theory, because you've removed something 00:44:15.247 --> 00:44:17.270 in constructing a low energy theory. 00:44:17.270 --> 00:44:22.430 But the two theories have to at least agree where they overlap. 00:44:22.430 --> 00:44:47.050 And so they have to agree on certain infrared observables, 00:44:47.050 --> 00:44:52.050 which I will also often denote as IR for infrared. 00:44:52.050 --> 00:44:55.680 The place where they differ is in the ultraviolet. 00:44:55.680 --> 00:44:58.440 So they might have different ultraviolet divergences. 00:44:58.440 --> 00:45:01.410 Most often, they will have different ultraviolet 00:45:01.410 --> 00:45:02.910 divergences. 00:45:02.910 --> 00:45:05.340 They don't have to agree in the ultraviolet. 00:45:05.340 --> 00:45:07.290 And actually, you exploit that to do things 00:45:07.290 --> 00:45:09.510 with the effective theory that would be hard to do 00:45:09.510 --> 00:45:11.770 with the full theory. 00:45:11.770 --> 00:45:14.467 We'll talk about how we do that later on. 00:45:14.467 --> 00:45:16.800 So the fact that they differ is actually not necessarily 00:45:16.800 --> 00:45:17.590 a negative thing. 00:45:17.590 --> 00:45:18.390 It can be a bonus. 00:45:30.680 --> 00:45:34.040 And finally, you have to ask about this sum over n. 00:45:34.040 --> 00:45:35.870 Well, this sum over n is infinite. 00:45:35.870 --> 00:45:37.390 Goes on forever. 00:45:37.390 --> 00:45:41.135 And so you have to ask the question, when should I stop? 00:45:41.135 --> 00:45:43.010 And therefore you have to look to experiments 00:45:43.010 --> 00:45:49.940 and see how precise they are, or just to your own perseverance 00:45:49.940 --> 00:45:52.160 and figure out what level do you want, 00:45:52.160 --> 00:45:56.790 what precision do you want in your description? 00:45:56.790 --> 00:46:00.520 So what n do you want to stop at? 00:46:00.520 --> 00:46:03.020 Sometimes experiment tells you to only do the first two n's. 00:46:03.020 --> 00:46:04.478 Sometimes you have to decide, maybe 00:46:04.478 --> 00:46:05.970 I only want the first one. 00:46:05.970 --> 00:46:07.553 If you're doing it for the first time, 00:46:07.553 --> 00:46:09.090 I suggest you stop at the beginning 00:46:09.090 --> 00:46:13.380 and let someone else to do the corrections. 00:46:13.380 --> 00:46:18.113 This idea of doing this can be important for separating 00:46:18.113 --> 00:46:19.530 physics of the high energy theory. 00:46:19.530 --> 00:46:22.120 One example of this is in QCD. 00:46:22.120 --> 00:46:25.290 If you take QCD for just about any process, 00:46:25.290 --> 00:46:27.600 there will be some parts of it which were perturbative 00:46:27.600 --> 00:46:30.150 and some parts of it that were non-perturbative. 00:46:30.150 --> 00:46:33.150 And by doing this kind of thing where you expand, 00:46:33.150 --> 00:46:34.950 you could construct a low energy theory 00:46:34.950 --> 00:46:37.860 that only has the non-perturbative scale in it 00:46:37.860 --> 00:46:39.870 and removes all the perturbative scales. 00:46:39.870 --> 00:46:42.540 If you did that, then just doing this procedure would allow you 00:46:42.540 --> 00:46:44.970 to figure out, what is the non-perturbative physics 00:46:44.970 --> 00:46:46.810 and what is the perturbative physics? 00:46:46.810 --> 00:46:49.860 You'd separate out, in that case, into operators. 00:46:49.860 --> 00:46:51.930 You'd separate out the infrared physics. 00:46:51.930 --> 00:46:54.540 You'd have operators built out of the infrared fields. 00:46:54.540 --> 00:46:56.010 And those operators would describe 00:46:56.010 --> 00:46:57.533 the non-perturbative physics. 00:46:57.533 --> 00:46:58.950 And you'd have some new low energy 00:46:58.950 --> 00:47:01.980 constants, which would describe the perturbative physics. 00:47:01.980 --> 00:47:07.915 And some of the examples that we'll do will make use of that 00:47:07.915 --> 00:47:08.415 probably. 00:47:12.020 --> 00:47:14.600 So that's kind of a motivation, actually, 00:47:14.600 --> 00:47:21.492 for some of the examples in the standard model. 00:47:21.492 --> 00:47:22.950 So if you think about, for example, 00:47:22.950 --> 00:47:32.220 integrating out heavy particles like the w 00:47:32.220 --> 00:47:37.233 or the z or the top quark, one of the motivations 00:47:37.233 --> 00:47:39.150 is sometimes what I just said, to separate out 00:47:39.150 --> 00:47:41.770 perturbative and non-perturbative physics. 00:47:41.770 --> 00:47:44.190 So that's one example. 00:47:44.190 --> 00:47:45.810 Heavy quark effective theory is also 00:47:45.810 --> 00:47:51.397 an example of this top-down effective field theory. 00:47:51.397 --> 00:47:52.980 In heavy quark effective field theory, 00:47:52.980 --> 00:47:57.210 you have a field theory for the B quark or for the charm quark 00:47:57.210 --> 00:48:02.800 but you want to describe things like the B meson or the charm 00:48:02.800 --> 00:48:03.300 mesons. 00:48:07.245 --> 00:48:09.120 Objects have non-perturbative physics as well 00:48:09.120 --> 00:48:11.942 as perturbative-- have mostly non-perturbative physics. 00:48:11.942 --> 00:48:14.400 And in order to do that, you want to actually integrate out 00:48:14.400 --> 00:48:17.490 the mass scale of the charm quark or the bottom quark. 00:48:17.490 --> 00:48:19.380 If you integrate out the mass of the charm 00:48:19.380 --> 00:48:20.640 quark and the bottom quark, you go over 00:48:20.640 --> 00:48:22.998 to something called heavy quark effective field theory. 00:48:22.998 --> 00:48:25.290 But it can be done in exactly this way that I described 00:48:25.290 --> 00:48:28.402 to you, where do you start with the theory with the full B 00:48:28.402 --> 00:48:30.360 quark relativistic description and you actually 00:48:30.360 --> 00:48:32.318 just expand and figure out what the heavy quark 00:48:32.318 --> 00:48:33.675 effective theory is. 00:48:36.930 --> 00:48:46.740 So non relativistic QED and non-relativistic QCD 00:48:46.740 --> 00:48:48.240 are also examples here. 00:48:48.240 --> 00:48:50.820 And soft-collinear effective theory, 00:48:50.820 --> 00:48:54.812 one of our main subjects, is also an example of this type 00:48:54.812 --> 00:48:57.972 where we can just start in QCD, do an expansion, 00:48:57.972 --> 00:48:59.430 and get the effective field theory. 00:49:07.350 --> 00:49:09.160 So what's the other category? 00:49:09.160 --> 00:49:12.990 So the other category is from the bottom up. 00:49:12.990 --> 00:49:15.300 And typically in this case, you're 00:49:15.300 --> 00:49:18.120 interested in using effective theory logic. 00:49:18.120 --> 00:49:20.923 But maybe you don't know the high energy theory. 00:49:20.923 --> 00:49:22.590 You don't really know anything about it. 00:49:22.590 --> 00:49:24.810 Maybe we've never probed it. 00:49:24.810 --> 00:49:27.330 That's one way in which bottom-up effective field 00:49:27.330 --> 00:49:30.270 theory shows up. 00:49:30.270 --> 00:49:34.920 Or it could be that the high energy theory is known 00:49:34.920 --> 00:49:39.510 but actually doing the matching calculations to integrate out 00:49:39.510 --> 00:49:42.330 the degrees of freedom to do those calculations explicitly 00:49:42.330 --> 00:49:48.060 could be just very, very difficult. 00:49:48.060 --> 00:49:51.480 Maybe it would be non-perturbative, for example. 00:49:51.480 --> 00:49:52.980 So if the matching is too difficult, 00:49:52.980 --> 00:49:54.750 then you may also want to be thinking 00:49:54.750 --> 00:49:59.550 in this bottom-up framework where you really just start 00:49:59.550 --> 00:50:01.860 by thinking about the low energy theory 00:50:01.860 --> 00:50:06.210 without worrying about what the high energy theory was, 00:50:06.210 --> 00:50:09.363 or without thinking too hard about the high energy theory, 00:50:09.363 --> 00:50:12.030 and in particular, without doing calculations in the high energy 00:50:12.030 --> 00:50:17.860 theory in order to motivate the low energy theory. 00:50:17.860 --> 00:50:21.538 You need to know some things about the high energy theory, 00:50:21.538 --> 00:50:23.580 like you may need to know that it's [? llorens ?] 00:50:23.580 --> 00:50:27.378 invariant, that it has certain gauge symmetry, 00:50:27.378 --> 00:50:28.545 that it's not totally crazy. 00:50:33.040 --> 00:50:34.790 But you don't need to know it at the level 00:50:34.790 --> 00:50:36.980 where you would actually carry out calculations with it 00:50:36.980 --> 00:50:38.890 in order to construct the low energy theory. 00:50:38.890 --> 00:50:40.890 Instead, you think about the lower energy theory 00:50:40.890 --> 00:50:45.170 from the bottom up, where you can just devise it 00:50:45.170 --> 00:50:48.080 based on the symmetries, based on your power counting, 00:50:48.080 --> 00:50:50.780 and based on identifying the degrees of freedom. 00:50:54.500 --> 00:51:10.950 So construct the series simply by writing down the most 00:51:10.950 --> 00:51:12.750 general operators that we can think 00:51:12.750 --> 00:51:32.870 of consistent with whatever degrees of freedom we have, 00:51:32.870 --> 00:51:35.142 and of course, consistent with the symmetries 00:51:35.142 --> 00:51:35.975 that we're imposing. 00:51:41.120 --> 00:51:41.620 OK. 00:51:44.750 --> 00:51:47.450 So the picture is that you don't know 00:51:47.450 --> 00:51:50.930 or you want to remain agnostic about theory one, 00:51:50.930 --> 00:51:54.320 but you still are interested in constructing theory two. 00:51:57.230 --> 00:51:59.180 If you do this, then unlike the other case, 00:51:59.180 --> 00:52:01.520 the couplings that you have when you write down these operators, 00:52:01.520 --> 00:52:03.228 they all are multiplied by some couplings 00:52:03.228 --> 00:52:06.330 if they're higher dimensional operators 00:52:06.330 --> 00:52:09.500 and they're not constrained by gauge symmetry. 00:52:09.500 --> 00:52:12.440 Then all of those couplings are unknown. 00:52:12.440 --> 00:52:15.165 But you can fit them to experiment. 00:52:18.970 --> 00:52:22.210 So the effective theory may still be powerful 00:52:22.210 --> 00:52:25.810 because you can make more predictions than the number 00:52:25.810 --> 00:52:31.351 of parameters that you have, like for hydrogen, 00:52:31.351 --> 00:52:32.920 where we have very few parameters 00:52:32.920 --> 00:52:35.462 and the effective theory, but we can make lots of predictions 00:52:35.462 --> 00:52:38.020 from non-relativistic quantum mechanics. 00:52:38.020 --> 00:52:40.145 Or it could be in the case where you 00:52:40.145 --> 00:52:41.770 imagine is too difficult that maybe you 00:52:41.770 --> 00:52:43.562 have to carry out the matching numerically, 00:52:43.562 --> 00:52:46.090 like with a lot of QCD. 00:52:46.090 --> 00:52:49.630 And so that would be another possible way 00:52:49.630 --> 00:52:52.400 of determining couplings. 00:52:52.400 --> 00:52:57.150 And again, the desired precision tells us when to stop. 00:52:57.150 --> 00:53:00.500 So it's important that we have a power counting for this theory. 00:53:00.500 --> 00:53:02.840 But that power counting is in some sense defined 00:53:02.840 --> 00:53:06.800 irrespective of what the full theory was so that we can stop, 00:53:06.800 --> 00:53:09.620 even in the bottom-up case. 00:53:18.380 --> 00:53:20.510 So what are examples here? 00:53:20.510 --> 00:53:22.100 Well, the classic example of this type 00:53:22.100 --> 00:53:24.890 is chiral perturbation theory, when 00:53:24.890 --> 00:53:28.325 you're thinking about a field theory for kaons and pions. 00:53:31.730 --> 00:53:36.050 And doing the matching onto kaons and pions from QCD 00:53:36.050 --> 00:53:38.257 is a non-perturbative process, so you 00:53:38.257 --> 00:53:40.340 think about constructing the effective theory just 00:53:40.340 --> 00:53:42.350 from low energy and from symmetries, 00:53:42.350 --> 00:53:45.510 knowing the symmetry breaking pattern, in particular. 00:53:45.510 --> 00:53:47.510 And you construct the chiral perturbation theory 00:53:47.510 --> 00:53:51.530 without thinking about doing the matching explicitly. 00:53:51.530 --> 00:53:53.640 So that's one example. 00:53:53.640 --> 00:53:55.442 Another example of this type is actually 00:53:55.442 --> 00:53:56.525 the standard model itself. 00:54:00.980 --> 00:54:03.320 If you think about the logic that we 00:54:03.320 --> 00:54:05.330 used when we construct the standard model, 00:54:05.330 --> 00:54:07.870 it was exactly this effective field theory logic. 00:54:07.870 --> 00:54:10.790 We said, what are the relevant degrees of freedom? 00:54:10.790 --> 00:54:11.600 Electron. 00:54:11.600 --> 00:54:12.560 Quarks. 00:54:12.560 --> 00:54:14.150 W bosons. 00:54:14.150 --> 00:54:15.260 Listed them. 00:54:15.260 --> 00:54:16.105 We wrote down. 00:54:16.105 --> 00:54:18.230 We said, what are the important guiding principles? 00:54:18.230 --> 00:54:20.107 Gauge symmetry. 00:54:20.107 --> 00:54:22.190 And then we wrote down the most general Lagrangian 00:54:22.190 --> 00:54:23.180 that we could think of. 00:54:23.180 --> 00:54:25.640 That was the standard model. 00:54:25.640 --> 00:54:29.090 OK, so it's an example of a bottom-up effective field 00:54:29.090 --> 00:54:29.670 theory. 00:54:29.670 --> 00:54:34.200 We don't ask questions about what's higher up. 00:54:34.200 --> 00:54:36.960 Let me write down the leading order Lagrangian. 00:54:36.960 --> 00:54:39.090 And we can actually construct higher order terms 00:54:39.090 --> 00:54:41.430 in the standard model expanding in the idea 00:54:41.430 --> 00:54:44.820 that there's physics above the scales of the standard model 00:54:44.820 --> 00:54:46.560 and write down higher dimension operators 00:54:46.560 --> 00:54:50.867 and have a real standard model that has an infinite series. 00:54:50.867 --> 00:54:52.950 So the standard model is an effective field theory 00:54:52.950 --> 00:54:55.200 that has an infinite number of operators. 00:54:55.200 --> 00:55:00.257 And we'll talk about that momentarily as well next time. 00:55:00.257 --> 00:55:02.340 So that'll be the first example we actually treat, 00:55:02.340 --> 00:55:05.297 is the standard model as an effective field theory. 00:55:09.280 --> 00:55:11.050 Another example is quantum gravity. 00:55:18.250 --> 00:55:20.050 If you take Einstein gravity and you 00:55:20.050 --> 00:55:23.198 make it quantum and you allow yourself to expand-- 00:55:23.198 --> 00:55:25.240 i.e. you say you're only interested in low energy 00:55:25.240 --> 00:55:25.870 physics. 00:55:25.870 --> 00:55:28.120 So you allow yourself to write down an infinite number 00:55:28.120 --> 00:55:29.710 of operators. 00:55:29.710 --> 00:55:32.770 Then you can also renormalize that theory order 00:55:32.770 --> 00:55:35.740 by ordering those infinite number of operators. 00:55:35.740 --> 00:55:37.600 And so it's also an example of something 00:55:37.600 --> 00:55:40.040 that you can treat from this effective field theory 00:55:40.040 --> 00:55:40.540 paradigm. 00:55:43.365 --> 00:55:45.240 That was the last topic that I would actually 00:55:45.240 --> 00:55:48.117 be cover in this-- if we had a couple extra weeks of lectures, 00:55:48.117 --> 00:55:48.950 I would get to this. 00:55:48.950 --> 00:55:50.590 But probably we won't. 00:55:50.590 --> 00:55:53.870 So that's a topic that somebody might pick for their project 00:55:53.870 --> 00:55:54.890 at the end. 00:55:54.890 --> 00:55:57.920 That's a good presentation. 00:55:57.920 --> 00:55:59.440 OK, so important stuff. 00:56:04.705 --> 00:56:06.080 Is there any question about that? 00:56:09.600 --> 00:56:13.360 Sits well with everybody, makes them feel good inside? 00:56:13.360 --> 00:56:13.860 OK. 00:56:24.530 --> 00:56:28.430 So far when we've been talking about this sum over n, 00:56:28.430 --> 00:56:32.300 we've been really thinking about expansions in powers. 00:56:32.300 --> 00:56:34.580 Some mass scale divided by some other scale 00:56:34.580 --> 00:56:36.730 being much less than 1. 00:56:36.730 --> 00:56:38.710 OK, that's what we've meant by it. 00:56:43.560 --> 00:56:46.550 But when you have two scales like this, m and lambda, 00:56:46.550 --> 00:56:47.930 you also get logarithms. 00:56:54.500 --> 00:56:55.618 So it's not always powers. 00:56:55.618 --> 00:56:56.910 There's also logs that show up. 00:57:00.510 --> 00:57:02.010 And this comment I meant about-- 00:57:02.010 --> 00:57:05.550 that I made about ultraviolet divergences 00:57:05.550 --> 00:57:08.640 in the low energy theory, it can actually 00:57:08.640 --> 00:57:10.470 help you to understand those logarithms. 00:57:13.886 --> 00:57:17.150 Let's get you over here. 00:57:17.150 --> 00:57:18.760 So when you treat the renormalizaiton 00:57:18.760 --> 00:57:27.810 of the low energy theory, as you know from quantum filtering, 00:57:27.810 --> 00:57:30.840 you have different types of divergences, power divergences. 00:57:30.840 --> 00:57:33.105 But the logarithmic divergences, in particular, 00:57:33.105 --> 00:57:36.390 are things that are playing an important role, often, 00:57:36.390 --> 00:57:37.800 in quantum field theory. 00:57:37.800 --> 00:57:40.080 And in effective field theory, it's the same thing. 00:57:40.080 --> 00:57:42.870 Logarithms can be tied to the re-normalization 00:57:42.870 --> 00:57:48.100 of the low energy effective theory 00:57:48.100 --> 00:57:54.460 and allow us to sum infinite series of those logarithms. 00:58:01.660 --> 00:58:06.280 So often just the power counting and the re-normalization 00:58:06.280 --> 00:58:08.830 of the low energy theory will actually 00:58:08.830 --> 00:58:10.840 allow us not only to calculate the logarithms, 00:58:10.840 --> 00:58:13.150 but to think about summing up infinite series 00:58:13.150 --> 00:58:17.050 of those logarithms. 00:58:17.050 --> 00:58:17.550 OK. 00:58:17.550 --> 00:58:19.080 So that was kind of just elaborating 00:58:19.080 --> 00:58:20.910 on a point I made earlier. 00:58:20.910 --> 00:58:23.070 And again, I should say here that I've said this 00:58:23.070 --> 00:58:26.430 in the language of there being two masses, m1 and m2 and w 00:58:26.430 --> 00:58:27.720 over MB. 00:58:27.720 --> 00:58:31.080 But this is actually true more generally again. 00:58:31.080 --> 00:58:35.160 So I'd actually make a claim that there's not any log 00:58:35.160 --> 00:58:37.830 that you've seen in quantum field theory that shouldn't 00:58:37.830 --> 00:58:40.650 be possible to figure out an effective field theory that 00:58:40.650 --> 00:58:44.280 allows you to understand those logs and predict logarithms 00:58:44.280 --> 00:58:46.680 at higher orders in perturbation theory. 00:58:46.680 --> 00:58:48.840 There's not an example in quantum field theory 00:58:48.840 --> 00:58:51.347 that I've met that hasn't fallen into that rubric where 00:58:51.347 --> 00:58:53.430 some effective field theory description allows you 00:58:53.430 --> 00:58:56.170 to understand the logarithms. 00:58:56.170 --> 00:58:56.670 OK. 00:58:59.200 --> 00:59:00.728 So that's kind of a bonus. 00:59:00.728 --> 00:59:02.020 It's not the guiding principle. 00:59:02.020 --> 00:59:05.020 It's not what we're doing when we're expanding in powers. 00:59:05.020 --> 00:59:09.010 But it's something that we get along for the ride. 00:59:09.010 --> 00:59:10.510 And maybe it would be the motivation 00:59:10.510 --> 00:59:12.790 if you see some logarithms and some process 00:59:12.790 --> 00:59:15.150 and you want to understand them. 00:59:15.150 --> 00:59:16.625 Maybe we would say, well, I'd like 00:59:16.625 --> 00:59:19.000 to understand what effective field theory would give rise 00:59:19.000 --> 00:59:20.708 to a description where I could understand 00:59:20.708 --> 00:59:23.745 those logarithms from a re-normalization perspective. 00:59:23.745 --> 00:59:25.120 And sometimes that's very useful, 00:59:25.120 --> 00:59:27.910 because maybe those logarithms are phenomenologically 00:59:27.910 --> 00:59:29.680 important and you want to make predictions 00:59:29.680 --> 00:59:34.870 about higher order logarithms, or maybe there's controversy. 00:59:34.870 --> 00:59:37.090 When I was a postdoc, there was some controversy 00:59:37.090 --> 00:59:43.120 about a term that was alpha to the 8 log cubed alpha 00:59:43.120 --> 00:59:46.990 in hydrogen energy levels. 00:59:46.990 --> 00:59:49.720 8 powers of alpha, 3 powers of logs. 00:59:49.720 --> 00:59:50.918 There was four groups. 00:59:50.918 --> 00:59:52.210 Two of them had got one answer. 00:59:52.210 --> 00:59:54.310 Two of them had got another answer. 00:59:54.310 --> 00:59:56.227 And using the ideas of effective field theory, 00:59:56.227 --> 00:59:58.768 we were able to figure out that one of those groups was right 00:59:58.768 --> 01:00:00.550 and the other was wrong very clearly, 01:00:00.550 --> 01:00:04.322 because you could connect these logarithms 01:00:04.322 --> 01:00:05.530 to an effective field theory. 01:00:05.530 --> 01:00:08.072 And then the whole consistency of that effective field theory 01:00:08.072 --> 01:00:10.510 really allows you to connect this logarithm 01:00:10.510 --> 01:00:14.260 to other logarithms and really to build a picture for what's 01:00:14.260 --> 01:00:17.440 going on with the physics that makes it totally clear what 01:00:17.440 --> 01:00:19.470 the answer must be. 01:00:19.470 --> 01:00:19.970 OK. 01:00:19.970 --> 01:00:22.360 So just to give you an example from my own past. 01:00:39.760 --> 01:00:43.030 So let's now turn to this question of the standard model 01:00:43.030 --> 01:00:44.667 as an effective field theory. 01:00:48.650 --> 01:00:51.520 So we have sum over n and we're treating this 01:00:51.520 --> 01:00:52.490 from the bottom up. 01:00:52.490 --> 01:00:54.700 So we're going to just talk about what 01:00:54.700 --> 01:00:57.580 the degrees of freedom are and then think 01:00:57.580 --> 01:00:58.870 about constructing operators. 01:01:02.648 --> 01:01:04.690 And part of the job has already been done for us, 01:01:04.690 --> 01:01:06.988 because I'm assuming you have a background 01:01:06.988 --> 01:01:09.280 in the standard model, at least at the level of knowing 01:01:09.280 --> 01:01:10.732 what the Lagrangian is. 01:01:10.732 --> 01:01:12.190 And if you haven't, then you should 01:01:12.190 --> 01:01:16.210 look at the quantum field theory three lecture notes. 01:01:16.210 --> 01:01:19.540 So the L0 here is the standard model, 01:01:19.540 --> 01:01:21.370 as taught in quantum field theory three. 01:01:25.900 --> 01:01:28.600 So [INAUDIBLE] be interested in as the higher order terms. 01:01:34.290 --> 01:01:35.953 But let me nevertheless remind you 01:01:35.953 --> 01:01:38.370 of what the degrees of freedom were in the standard model. 01:01:45.370 --> 01:01:47.310 So you at least know what the players are 01:01:47.310 --> 01:01:48.780 when we go to talk about L1. 01:01:53.252 --> 01:01:54.210 So it's a gauge theory. 01:01:57.609 --> 01:02:05.200 So we have color across Su 2 weak across the un 01:02:05.200 --> 01:02:07.726 of hypercharge. 01:02:07.726 --> 01:02:20.650 And so we have eight gluons here, three week bosons here, 01:02:20.650 --> 01:02:23.710 and one guy here. 01:02:28.380 --> 01:02:31.290 So just to introduce some notation for fields, 01:02:31.290 --> 01:02:34.200 I'll call these guys with an index capital A running 01:02:34.200 --> 01:02:36.390 from 1 to 8, these guys with an index 01:02:36.390 --> 01:02:38.850 lower a running from 1 to 3. 01:02:38.850 --> 01:02:42.780 And B here would be the analog of a photon 01:02:42.780 --> 01:02:44.770 field for U1 electromagnetism. 01:02:44.770 --> 01:02:49.300 But this is the U1 of hypercharge, so it's B mu. 01:02:49.300 --> 01:02:52.110 So we have gauge bosons. 01:02:52.110 --> 01:02:55.210 We have fermions. 01:02:55.210 --> 01:02:56.610 Let me do the fermions over here. 01:03:04.120 --> 01:03:05.620 So an important thing and thinking 01:03:05.620 --> 01:03:07.330 about this as an effective field theory 01:03:07.330 --> 01:03:08.920 is to note what the mass scales are. 01:03:13.600 --> 01:03:16.060 So maybe I should do that already here. 01:03:16.060 --> 01:03:17.530 Protons are massless. 01:03:17.530 --> 01:03:21.820 That's one combination of the weak and U1 boson. 01:03:21.820 --> 01:03:24.160 Gluons are massless. 01:03:24.160 --> 01:03:26.440 That's these guys. 01:03:26.440 --> 01:03:35.188 And then there's the mass of the W. 80.42 GB the mass 01:03:35.188 --> 01:03:40.390 of the Z, 91.19. 01:03:40.390 --> 01:03:44.348 And for the first time in me teaching this course, 01:03:44.348 --> 01:03:46.140 we also know what the mass of the Higgs is. 01:03:46.140 --> 01:03:47.170 So let me just-- 01:03:47.170 --> 01:03:48.688 that's not part of the gauge theory, 01:03:48.688 --> 01:03:50.230 but I'll just list it there, as well, 01:03:50.230 --> 01:03:51.980 since it doesn't fit in with the fermions. 01:03:56.660 --> 01:04:04.660 So fermions-- so you can see that these scales here 01:04:04.660 --> 01:04:06.110 are kind of similar. 01:04:06.110 --> 01:04:08.360 For the fermions, there's a broad spectrum of scales. 01:04:08.360 --> 01:04:10.568 And that's why I wanted to put them all on one board. 01:04:16.720 --> 01:04:24.280 So quarks-- up quarks, down quarks, strange quarks-- 01:04:24.280 --> 01:04:27.400 they all come in left and right handed [INAUDIBLE] 01:04:27.400 --> 01:04:29.350 and the gauge couplings are different for left 01:04:29.350 --> 01:04:33.088 and right handed, for the electroweak and U1 parts 01:04:33.088 --> 01:04:33.880 of the gauge group. 01:04:36.620 --> 01:04:41.830 So there are six different flavors and both right 01:04:41.830 --> 01:04:44.050 and left handed. 01:04:44.050 --> 01:04:46.180 What masses do we have? 01:04:46.180 --> 01:04:51.070 Up quarks and down quarks are rather light, 01:04:51.070 --> 01:04:52.525 about a couple of MeV. 01:04:56.320 --> 01:04:57.820 It's hard to measure the light ones. 01:05:00.720 --> 01:05:02.827 It's a little easier to measure. 01:05:02.827 --> 01:05:04.160 Everything's going to be in MeV. 01:05:04.160 --> 01:05:06.370 I'm going to stop writing MeV. 01:05:06.370 --> 01:05:08.050 Oh, that's not true. 01:05:08.050 --> 01:05:10.240 I switched to GeV. 01:05:10.240 --> 01:05:13.000 Sorry. 01:05:13.000 --> 01:05:15.250 Everything is going to be in GeV. 01:05:24.850 --> 01:05:25.505 Oh, sorry. 01:05:25.505 --> 01:05:26.380 That's the top quark. 01:05:40.340 --> 01:05:41.510 OK. 01:05:41.510 --> 01:05:43.220 So there's a pretty wide range of scales 01:05:43.220 --> 01:05:47.150 here from an MeV to 100 GeV. 01:05:47.150 --> 01:05:49.022 That's just the quarks. 01:05:49.022 --> 01:05:50.480 And then we also have the leptons-- 01:05:59.220 --> 01:06:03.150 three types of charge leptons, again, 01:06:03.150 --> 01:06:04.815 with a fairly wide range of scales. 01:06:10.300 --> 01:06:12.790 So now I'm switching back to MeV, 01:06:12.790 --> 01:06:14.040 just to keep you on your toes. 01:06:18.270 --> 01:06:18.770 Whoops. 01:06:25.300 --> 01:06:27.490 Then we have neutrinos. 01:06:27.490 --> 01:06:29.830 The left-handed ones we've studied much more 01:06:29.830 --> 01:06:33.600 than anything else. 01:06:33.600 --> 01:06:35.350 And in particular, what we know most about 01:06:35.350 --> 01:06:39.280 the left-handed guys is mass splittings from neutrino 01:06:39.280 --> 01:06:41.245 oscillations. 01:06:41.245 --> 01:06:42.790 And these are pretty small numbers. 01:06:46.425 --> 01:06:48.050 We also know that overall, these things 01:06:48.050 --> 01:06:51.530 are quite light, from cosmological constraints 01:06:51.530 --> 01:06:52.659 and otherwise. 01:06:57.820 --> 01:07:01.150 And we don't really know about anything 01:07:01.150 --> 01:07:07.570 like a sterile neutrino but that we can put bounds on its mass. 01:07:07.570 --> 01:07:09.110 So even within the standard model, 01:07:09.110 --> 01:07:10.527 there's a lot of different scales. 01:07:12.297 --> 01:07:14.630 And if you think about it from an effective field theory 01:07:14.630 --> 01:07:17.210 point of view and you think about it from the top down, 01:07:17.210 --> 01:07:19.850 the first thing you'd get rid of would be the top quark. 01:07:19.850 --> 01:07:23.660 And then you'd get rid of the W and the Z and the Higgs. 01:07:23.660 --> 01:07:25.380 And then you would proceed down. 01:07:25.380 --> 01:07:29.595 The next thing to go would be the bottom quark, et cetera. 01:07:29.595 --> 01:07:31.970 And you could think about constructing an effective field 01:07:31.970 --> 01:07:33.865 theory by integrating out one at a time, 01:07:33.865 --> 01:07:35.990 getting a new effective field theory every time you 01:07:35.990 --> 01:07:38.216 remove a degree of freedom. 01:07:38.216 --> 01:07:39.680 You could take the standard model 01:07:39.680 --> 01:07:41.858 and expand in that fashion. 01:07:41.858 --> 01:07:44.150 That's not the sense in which we are thinking about it. 01:07:44.150 --> 01:07:46.692 That would be the top-down sense of taking the standard model 01:07:46.692 --> 01:07:48.200 and deriving something else. 01:07:48.200 --> 01:07:50.780 We're thinking of it here in a different context 01:07:50.780 --> 01:07:52.162 where we have all this stuff. 01:07:52.162 --> 01:07:53.870 And we're actually interested in thinking 01:07:53.870 --> 01:07:55.670 about physics at higher energy scales, 01:07:55.670 --> 01:07:57.930 beyond the scale of the weak bosons, 01:07:57.930 --> 01:08:00.140 beyond the scale of the top quark, the things 01:08:00.140 --> 01:08:04.143 we're trying to figure out at the LHC, scales 01:08:04.143 --> 01:08:05.060 we're trying to probe. 01:08:07.950 --> 01:08:09.910 That's the attitude in this bottom-up approach. 01:08:21.800 --> 01:08:22.300 OK. 01:08:22.300 --> 01:08:23.800 So the lowest order Lagrangian would 01:08:23.800 --> 01:08:26.170 be the gauge sector of the thermionic Lagrangian, 01:08:26.170 --> 01:08:27.189 the Higgs Lagrangian. 01:08:27.189 --> 01:08:29.098 And if we have right-handed neutrinos, 01:08:29.098 --> 01:08:30.640 we'd need a Lagrangian for them, too. 01:08:36.479 --> 01:08:42.790 So these are topics that come up in [INAUDIBLE].. 01:08:42.790 --> 01:08:49.229 I'm not even going to touch them at the moment. 01:08:49.229 --> 01:08:57.540 I can't give you a complete review, but just a taste, 01:08:57.540 --> 01:09:00.725 emphasizing things that are important. 01:09:00.725 --> 01:09:07.838 So to give you a taste, I just write the other two down, 01:09:07.838 --> 01:09:09.380 which are the prettier parts, anyway. 01:09:15.979 --> 01:09:18.620 So we have field strengths for the kinetic terms 01:09:18.620 --> 01:09:19.495 for our gauge bosons. 01:09:27.040 --> 01:09:31.540 And the thermionic Lagrangian, I can write it 01:09:31.540 --> 01:09:34.609 as a sum over the left handed fields-- 01:09:34.609 --> 01:09:38.060 fermion, covariant derivative fermion. 01:09:38.060 --> 01:09:41.440 Add a sum over right handed fields. 01:09:41.440 --> 01:09:47.620 Fermion covariant derivative fermion 01:09:47.620 --> 01:09:51.220 where this covariant derivative is a covariant derivative 01:09:51.220 --> 01:09:53.750 with these gauge fields. 01:09:53.750 --> 01:10:00.960 So there's some gauge coupling, G1, for hypercharge, 01:10:00.960 --> 01:10:06.460 some gauge coupling, G2, for [INAUDIBLE] weak, 01:10:06.460 --> 01:10:09.490 and some gauge coupling, G, for QCD. 01:10:17.110 --> 01:10:20.064 So what is the power counting? 01:10:20.064 --> 01:10:24.940 So we've just said what the degrees of freedom are 01:10:24.940 --> 01:10:28.060 and what kind of some of the guiding principles 01:10:28.060 --> 01:10:31.282 are-- the symmetries, the gauge symmetry. 01:10:31.282 --> 01:10:33.490 You learn much more about symmetries in quantum field 01:10:33.490 --> 01:10:35.780 theory three, so I'm not going to go into that. 01:10:35.780 --> 01:10:39.250 But those are basically the guiding principles 01:10:39.250 --> 01:10:41.470 in figuring out this L0. 01:10:41.470 --> 01:10:44.980 What is it that we would do a power counting in here? 01:10:44.980 --> 01:10:47.110 So the power counting in this bottom up approach 01:10:47.110 --> 01:10:48.730 is related to what we left out. 01:10:59.890 --> 01:11:02.650 So we're expanding an epsilon here, 01:11:02.650 --> 01:11:06.580 where epsilon is mass scales in this standard model divided 01:11:06.580 --> 01:11:09.800 by things that we've left out of our description. 01:11:09.800 --> 01:11:12.220 So in the numerator would be things like the top quark 01:11:12.220 --> 01:11:17.140 mass, the W mass, Z mass, Higgs mass, all the mass scales 01:11:17.140 --> 01:11:18.640 of the standard model. 01:11:18.640 --> 01:11:21.040 In the denominator, well, certainly something 01:11:21.040 --> 01:11:25.750 like M plank is left out of our description here. 01:11:25.750 --> 01:11:28.120 If we had some grand unified theory, 01:11:28.120 --> 01:11:30.010 that goes in the denominator. 01:11:30.010 --> 01:11:32.920 If we have supersymmetry and we broke it, 01:11:32.920 --> 01:11:34.310 that would go in the denominator. 01:11:34.310 --> 01:11:36.010 So from this effective field theory point of view, 01:11:36.010 --> 01:11:38.260 any physics that we've left out of the standard model 01:11:38.260 --> 01:11:42.050 description is anything that generates a higher energy 01:11:42.050 --> 01:11:42.550 scale. 01:11:42.550 --> 01:11:44.246 That goes in the denominator. 01:11:53.680 --> 01:11:56.515 And this is what we expanded. 01:11:56.515 --> 01:11:58.890 So even not knowing something about what this physics is, 01:11:58.890 --> 01:12:02.430 we can come up with a universal description-- a universal L1-- 01:12:02.430 --> 01:12:04.650 that describes corrections beyond the standard model. 01:12:09.060 --> 01:12:11.370 And what will be describing that physics is higher 01:12:11.370 --> 01:12:14.340 dimension operators-- operators beyond dimension four. 01:12:23.570 --> 01:12:25.670 But they'll be built out of standard model fields. 01:12:37.260 --> 01:12:42.060 So kind of from your teaching of quantum field theory, maybe 01:12:42.060 --> 01:12:45.150 perhaps the idea that these two things are connected 01:12:45.150 --> 01:12:47.580 may be clear to you, but it's something 01:12:47.580 --> 01:12:55.530 that we will actually cover mostly next class, actually. 01:12:55.530 --> 01:12:57.670 Some part of the beginning of next class, 01:12:57.670 --> 01:13:00.250 we'll make this connection between the fact 01:13:00.250 --> 01:13:03.790 that we want to expand in that epsilon and the fact 01:13:03.790 --> 01:13:05.522 that we can do-- in doing so, we get 01:13:05.522 --> 01:13:06.730 higher dimensional operators. 01:13:06.730 --> 01:13:08.682 We'll make that absolutely clear. 01:13:08.682 --> 01:13:09.640 That'll come next time. 01:13:14.375 --> 01:13:15.750 So in the remainder of today, let 01:13:15.750 --> 01:13:22.620 me just address one final point, and that 01:13:22.620 --> 01:13:25.590 is the idea of what it means to have a renormalizable field 01:13:25.590 --> 01:13:27.210 theory. 01:13:27.210 --> 01:13:29.820 So in our description of the standard model that I gave 01:13:29.820 --> 01:13:31.708 here, I mentioned symmetries. 01:13:31.708 --> 01:13:33.000 I mentioned degrees of freedom. 01:13:33.000 --> 01:13:35.436 I didn't mention renormalizability. 01:13:43.212 --> 01:13:46.280 So what does re-normalizable mean? 01:13:46.280 --> 01:13:49.070 So the traditional definition of what renormalizable will mean 01:13:49.070 --> 01:13:51.540 would be the following. 01:13:51.540 --> 01:13:59.130 You would say a theory is renormalizable 01:13:59.130 --> 01:14:06.620 if at any order in perturbation theory in this quantum field 01:14:06.620 --> 01:14:20.956 theory the UV divergences can be absorbed-- 01:14:20.956 --> 01:14:24.215 so there's UV divergences from loop integrals. 01:14:24.215 --> 01:14:26.340 If they can always be absorbed into a finite number 01:14:26.340 --> 01:14:30.146 of parameters, then you'd say the theory is renormalizable. 01:14:33.930 --> 01:14:35.430 But that's a traditional definition. 01:14:35.430 --> 01:14:37.388 And we will use a more general definition here. 01:14:46.177 --> 01:14:47.760 Certainly this was a guiding principle 01:14:47.760 --> 01:14:49.552 when people constructed the standard model. 01:14:52.500 --> 01:14:55.390 What is the effective field theory definition of this? 01:14:55.390 --> 01:15:00.360 It's a little more general because it 01:15:00.360 --> 01:15:04.950 brings in the idea of doing power counting. 01:15:04.950 --> 01:15:08.160 So the effective field theory's definition 01:15:08.160 --> 01:15:10.620 allows for the possibility of having an infinite number 01:15:10.620 --> 01:15:12.210 of parameters. 01:15:12.210 --> 01:15:14.388 But at any order that you truncate the theory, 01:15:14.388 --> 01:15:15.930 there should only be a finite number. 01:15:20.120 --> 01:15:26.770 So it says that a theory must be renormalizable order by order 01:15:26.770 --> 01:15:28.654 in its expansion parameter. 01:15:32.450 --> 01:15:35.588 Well, if there's more than one, it's expansion parameters. 01:15:40.960 --> 01:15:42.820 So even just this sentence alone tells you 01:15:42.820 --> 01:15:46.000 why power counting is such an important part 01:15:46.000 --> 01:15:48.970 of the effective theory, because the effective theory 01:15:48.970 --> 01:15:51.010 to make sense as a renormalizable quantum field 01:15:51.010 --> 01:15:53.560 theory needs to know about its expansion parameter. 01:15:53.560 --> 01:15:55.498 We're saying that it's renormalizable, 01:15:55.498 --> 01:15:57.040 that we can make sense of the theory, 01:15:57.040 --> 01:16:00.370 absorb all the infinities, only order by order in expansion 01:16:00.370 --> 01:16:02.920 parameters in general. 01:16:10.190 --> 01:16:12.185 So it could be that you do some calculation, 01:16:12.185 --> 01:16:13.715 you counter some divergences. 01:16:13.715 --> 01:16:15.590 But if they're higher order in the expansion, 01:16:15.590 --> 01:16:18.590 then what you need-- and you have a power counting that 01:16:18.590 --> 01:16:19.760 tells you that-- 01:16:19.760 --> 01:16:21.892 they would be absorbable into some operators 01:16:21.892 --> 01:16:24.350 that you haven't even written down, you can just drop them. 01:16:30.580 --> 01:16:41.140 So this definition allows for an infinite number 01:16:41.140 --> 01:16:45.430 of parameters that are needed, for example, 01:16:45.430 --> 01:16:54.170 for normalizability, but only a finite number at some fixed 01:16:54.170 --> 01:16:54.670 order. 01:17:05.500 --> 01:17:07.785 Now, if you take this logic that I just 01:17:07.785 --> 01:17:09.910 said to you, that you could think about things more 01:17:09.910 --> 01:17:11.530 generally, then you can ask, well, 01:17:11.530 --> 01:17:14.250 what was the point of thinking about the standard model, where 01:17:14.250 --> 01:17:16.000 the theory turned out to be renormalizable 01:17:16.000 --> 01:17:17.900 in the traditional sense? 01:17:17.900 --> 01:17:21.100 How does this fact, which is just a subset of this case, 01:17:21.100 --> 01:17:23.500 but an important one, how does it 01:17:23.500 --> 01:17:26.170 fit into this rubric from an effective field 01:17:26.170 --> 01:17:28.640 theory point of view? 01:17:28.640 --> 01:17:31.660 And so the way that that fits in is as follows. 01:17:31.660 --> 01:17:36.310 It could turn out that your L0 in your expansion 01:17:36.310 --> 01:17:39.040 is renormalizable in the traditional sense rather 01:17:39.040 --> 01:17:41.140 than this more general sense. 01:17:43.930 --> 01:17:46.150 And if that's true, what it means 01:17:46.150 --> 01:17:52.930 is that you don't see the higher energy scales from your lowest 01:17:52.930 --> 01:17:53.740 order Lagrangian. 01:17:59.260 --> 01:18:11.920 So we do not know directly about lambda 01:18:11.920 --> 01:18:14.590 new from just looking at L0. 01:18:17.233 --> 01:18:19.150 And that's what happens in the standard model. 01:18:19.150 --> 01:18:22.593 We don't really know precisely what the high energy 01:18:22.593 --> 01:18:24.010 scale should be just from studying 01:18:24.010 --> 01:18:26.650 the effect-- from studying the leading order Lagrangian. 01:18:26.650 --> 01:18:28.070 This will not always be the case. 01:18:28.070 --> 01:18:29.470 Sometimes we'll be in a situation 01:18:29.470 --> 01:18:32.613 where when we study the effective theory at lowest 01:18:32.613 --> 01:18:34.030 order in the Lagrangian, we really 01:18:34.030 --> 01:18:36.910 find that even in order to make sense of that as a quantum 01:18:36.910 --> 01:18:39.550 field theory that we need to-- 01:18:39.550 --> 01:18:41.530 that there's a scale that gets generated 01:18:41.530 --> 01:18:43.310 and it's part of our expansion. 01:18:43.310 --> 01:18:45.430 And there's some terms that we calculate 01:18:45.430 --> 01:18:48.400 with L0 that end up being higher order in our expansion. 01:18:48.400 --> 01:18:51.430 And we can't renormalize the theory 01:18:51.430 --> 01:18:55.570 unless we actually include higher dimension operators. 01:18:55.570 --> 01:18:58.090 Chiral perturbation theory is an example of that type. 01:18:58.090 --> 01:19:00.560 And that's an example that we'll treat. 01:19:00.560 --> 01:19:03.560 So it's not always the case in the standard model, 01:19:03.560 --> 01:19:05.830 whether it's renormalizable in a traditional sense. 01:19:05.830 --> 01:19:11.030 That's a special case, though it's an important one. 01:19:11.030 --> 01:19:11.860 So questions? 01:19:17.710 --> 01:19:21.575 OK, so hopefully this has been partly a review of some things 01:19:21.575 --> 01:19:23.950 that you've thought of before, but putting them together, 01:19:23.950 --> 01:19:25.690 perhaps in a nicer package. 01:19:25.690 --> 01:19:28.450 And we'll continue next time talking 01:19:28.450 --> 01:19:30.702 about the standard model as an effective field theory. 01:19:30.702 --> 01:19:31.910 What can be gained from that? 01:19:31.910 --> 01:19:34.750 How do we construct the operators? 01:19:34.750 --> 01:19:37.260 And we'll keep going from there.