WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.270 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.270 --> 00:00:18.272 at ocw.mit.edu. 00:00:22.852 --> 00:00:24.310 IAIN STEWART: So last time, we were 00:00:24.310 --> 00:00:27.790 talking about the standard model as an effective field theory. 00:00:27.790 --> 00:00:29.410 And we decided that the power counting 00:00:29.410 --> 00:00:32.908 would be this, epsilon, the masses of the particles 00:00:32.908 --> 00:00:35.200 in the standard model, the scales in the standard model 00:00:35.200 --> 00:00:37.750 divided by some new physics scale, scale 00:00:37.750 --> 00:00:39.843 outside the standard model. 00:00:39.843 --> 00:00:41.260 And I made the statement that this 00:00:41.260 --> 00:00:43.480 was connected to operator dimension, 00:00:43.480 --> 00:00:45.520 but I didn't make that precise. 00:00:45.520 --> 00:00:49.120 And I want to do that now as the first thing we do today. 00:01:11.300 --> 00:01:12.900 So let's spend a few minutes and talk 00:01:12.900 --> 00:01:16.865 about marginal, irrelevant, and relevant operators 00:01:16.865 --> 00:01:18.490 and their connection to power counting. 00:01:43.650 --> 00:01:45.900 I'm going to write power counting over and over again. 00:01:45.900 --> 00:01:48.880 And I'm going to abbreviate it p.c. 00:01:48.880 --> 00:01:49.710 from now on. 00:01:54.377 --> 00:01:56.210 So let's consider an effective field theory. 00:01:56.210 --> 00:02:07.030 It'll be a scalar effective field theory in d dimensions, 00:02:07.030 --> 00:02:19.240 standard kinetic term, mass term, phi 4 term. 00:02:19.240 --> 00:02:22.530 So it'll be a phi 4 scalar field theory. 00:02:22.530 --> 00:02:25.230 It will be an effective theory, so we won't stop there. 00:02:28.480 --> 00:02:30.240 And I'll just write down up to phi 6. 00:02:30.240 --> 00:02:32.032 And then, in principle, I could keep going. 00:02:36.110 --> 00:02:39.030 So we can look at the dimensions of the various objects here. 00:02:39.030 --> 00:02:42.580 The action with our units is dimensionless. 00:02:42.580 --> 00:02:46.230 h bar and c are 1. 00:02:46.230 --> 00:02:49.320 So the mass dimensions of the field in d dimensions 00:02:49.320 --> 00:02:55.506 are d minus 2 over 2 since the dimensions of ddx or minus d. 00:02:55.506 --> 00:02:57.077 We have to compensate for that. 00:02:57.077 --> 00:02:59.160 And we have to compensate for the two derivatives. 00:02:59.160 --> 00:03:00.990 The canonically normalized kinetic term 00:03:00.990 --> 00:03:03.902 tells us what the dimensions of phi are. 00:03:03.902 --> 00:03:06.360 And then we can work out the dimensions of everything else, 00:03:06.360 --> 00:03:09.540 so mass squared dimension 2. 00:03:23.040 --> 00:03:25.130 Tau dimension-- 6 minus 2d. 00:03:25.130 --> 00:03:28.860 Lambda could be dimension 0 if d is 4. 00:03:28.860 --> 00:03:29.360 OK. 00:03:29.360 --> 00:03:33.000 So hopefully-- somewhat familiar stuff. 00:03:33.000 --> 00:03:35.720 So let's say we want to study a correlation 00:03:35.720 --> 00:03:48.075 function of a bunch of phis at different space time points. 00:03:55.380 --> 00:03:58.380 And we want to look at it at long distance. 00:03:58.380 --> 00:03:59.735 Long distance is small momenta. 00:04:04.470 --> 00:04:07.765 So the way I'm going to make the distance long 00:04:07.765 --> 00:04:10.140 is I'm going to say that all these x's that are appearing 00:04:10.140 --> 00:04:13.050 in my phis, x1 through xn, I'm going 00:04:13.050 --> 00:04:17.640 to redefine them as some s common parameter times x prime. 00:04:17.640 --> 00:04:22.380 And then I'm just going to take s goes to infinity with the x 00:04:22.380 --> 00:04:23.010 prime fixed. 00:04:26.310 --> 00:04:27.765 So that makes all the x's large. 00:04:35.940 --> 00:04:40.830 So when I do that, if I make a redefinition like that, 00:04:40.830 --> 00:04:43.050 I can mess up the normalization of my kinetic term. 00:04:43.050 --> 00:04:47.580 It'll no longer be canonical, but I can fix that 00:04:47.580 --> 00:04:50.770 by just redefining my field. 00:04:50.770 --> 00:04:53.140 And the way to do that is to do the following. 00:04:58.410 --> 00:05:01.580 To find a new, phi prime, it's equal to the old field, 00:05:01.580 --> 00:05:02.690 but rescaled by an s. 00:05:13.860 --> 00:05:17.370 And the outcome of that is that we 00:05:17.370 --> 00:05:22.860 get an action for the phi prime field written 00:05:22.860 --> 00:05:28.650 in terms of prime coordinates, which has a kinetic term that's 00:05:28.650 --> 00:05:31.217 the same form. 00:05:31.217 --> 00:05:33.300 But then s's start showing up in the other places. 00:06:00.670 --> 00:06:03.530 And if you look at the powers of the s's that are showing up, 00:06:03.530 --> 00:06:07.600 it's related also to the powers of these parameters. 00:06:07.600 --> 00:06:08.860 Yeah. 00:06:08.860 --> 00:06:11.750 AUDIENCE: Are you sure about the powers of lambda tau, 00:06:11.750 --> 00:06:13.678 for the dimension lambda tau? 00:06:13.678 --> 00:06:15.220 IAIN STEWART: Did I get it backwards? 00:06:15.220 --> 00:06:16.542 Should it be d minus 4? 00:06:16.542 --> 00:06:18.250 AUDIENCE: I think it should be d minus 4. 00:06:18.250 --> 00:06:20.583 And then the other one should be d minus 6 [INAUDIBLE].. 00:06:23.350 --> 00:06:24.550 IAIN STEWART: Yeah. 00:06:24.550 --> 00:06:27.680 That looks right. 00:06:27.680 --> 00:06:28.630 Oh, you have to be-- 00:06:28.630 --> 00:06:29.800 so let's see. 00:06:29.800 --> 00:06:32.200 There's d's here, right? 00:06:32.200 --> 00:06:33.910 So it's not d. 00:06:33.910 --> 00:06:34.530 AUDIENCE: Oh. 00:06:34.530 --> 00:06:34.940 IAIN STEWART: You've got to keep-- 00:06:34.940 --> 00:06:36.110 AUDIENCE: [INAUDIBLE] 00:06:36.110 --> 00:06:37.390 IAIN STEWART: Yeah. 00:06:37.390 --> 00:06:38.420 I stick by what I wrote. 00:06:38.420 --> 00:06:38.920 Check it. 00:06:42.928 --> 00:06:45.470 All right, so let's look at the correlation function in terms 00:06:45.470 --> 00:06:48.290 of the phi prime because the phi prime is just 00:06:48.290 --> 00:06:49.400 a function of x primes. 00:06:49.400 --> 00:06:51.540 And the x primes are holding fixed. 00:06:51.540 --> 00:06:54.440 So if we rescale everything in terms of the x primes, 00:06:54.440 --> 00:06:58.460 then we have some matrix element that's not growing with s. 00:06:58.460 --> 00:07:00.020 We can make all the s's explicit. 00:07:05.840 --> 00:07:14.580 So we take our original guy, which 00:07:14.580 --> 00:07:17.300 in terms of our new variables looks like that. 00:07:17.300 --> 00:07:19.900 We make this redefinition. 00:07:19.900 --> 00:07:22.230 We get some powers of s out front. 00:07:25.032 --> 00:07:38.490 And then we get something which is just in terms of the x prime 00:07:38.490 --> 00:07:39.510 and won't grow with s. 00:07:45.380 --> 00:07:49.180 So we can study this in various dimensions if we wanted to. 00:07:49.180 --> 00:07:53.550 Let's, for simplicity and also since it's the most common case 00:07:53.550 --> 00:07:57.080 we're interested in, take d equals 4. 00:07:57.080 --> 00:08:03.510 I still may write d's, but let's from here on take d equals 4 00:08:03.510 --> 00:08:07.840 and ask the question, what happens is s gets large? 00:08:07.840 --> 00:08:09.550 So now, we've made all the s's explicit. 00:08:09.550 --> 00:08:11.040 This is something that you often do when you're 00:08:11.040 --> 00:08:12.248 doing effective field theory. 00:08:12.248 --> 00:08:15.930 You can figure out how you're going to study 00:08:15.930 --> 00:08:18.570 the large distance behavior. 00:08:18.570 --> 00:08:20.610 You want to make the parameter that's 00:08:20.610 --> 00:08:22.860 controlling that limit explicit, so you can see it. 00:08:22.860 --> 00:08:24.510 So it's not hiding anywhere. 00:08:24.510 --> 00:08:26.385 And that's what we've done with this algebra. 00:08:31.360 --> 00:08:34.392 So as s goes to infinity, because we 00:08:34.392 --> 00:08:35.850 have this explicit s squared there, 00:08:35.850 --> 00:08:39.085 the m squared term is becoming more and more important. 00:08:52.435 --> 00:08:53.310 It's called relevant. 00:08:57.110 --> 00:08:59.495 Child term is becoming less important. 00:09:06.780 --> 00:09:09.540 Because if I put in d equals 4, then this is s to the minus 2. 00:09:09.540 --> 00:09:15.690 It's tracking the s's, making it less important as s grows. 00:09:15.690 --> 00:09:19.710 And the lambda term is equally as important as it was before. 00:09:28.650 --> 00:09:31.040 And the terminology that goes along with this 00:09:31.040 --> 00:09:32.660 is an association. 00:09:32.660 --> 00:09:34.663 So that was a statement about parameters. 00:09:34.663 --> 00:09:36.580 We could also make a statement about operators 00:09:36.580 --> 00:09:38.840 since obviously they were part of the story here 00:09:38.840 --> 00:09:40.625 that gave the s factors. 00:09:43.440 --> 00:09:46.295 So we would say that phi squared is a relevant operator. 00:09:49.410 --> 00:09:50.815 The phi 4 is marginal. 00:09:55.240 --> 00:09:57.297 And phi 6 is irrelevant. 00:10:06.610 --> 00:10:08.860 And you can see, because of the argument that we made, 00:10:08.860 --> 00:10:32.638 that this was just directly connected to dimension, 00:10:32.638 --> 00:10:34.430 so either to the dimension of the operators 00:10:34.430 --> 00:10:36.013 or to the dimension of the parameters. 00:10:38.417 --> 00:10:40.500 OK, so we're connecting something that we can say, 00:10:40.500 --> 00:10:41.490 which is the power counting. 00:10:41.490 --> 00:10:42.990 In this case, we're controlling that 00:10:42.990 --> 00:10:45.560 with s using s as our control parameters 00:10:45.560 --> 00:10:47.152 to look at long distances. 00:10:47.152 --> 00:10:48.860 And we're seeing that that gets connected 00:10:48.860 --> 00:10:50.330 to dimensions of operators. 00:11:18.972 --> 00:11:19.930 Is there any questions? 00:11:24.530 --> 00:11:27.880 So let's take s finite, but large. 00:11:35.525 --> 00:11:37.150 Usually, we're not interested in taking 00:11:37.150 --> 00:11:39.107 it all the way to infinity. 00:11:39.107 --> 00:11:40.690 Although we may make it as large as we 00:11:40.690 --> 00:11:48.710 want to study some long distance behavior. 00:11:48.710 --> 00:11:54.550 And so what I just said is that we can see, 00:11:54.550 --> 00:11:56.620 from the powers of s, the importance 00:11:56.620 --> 00:11:57.550 of the various terms. 00:12:04.150 --> 00:12:06.490 Relevant terms are more important than marginal terms. 00:12:06.490 --> 00:12:08.698 And marginal terms are more important than irrelevant 00:12:08.698 --> 00:12:09.815 terms. 00:12:09.815 --> 00:12:10.690 The words say it all. 00:12:19.220 --> 00:12:23.270 So that means that, if you want to think of how to do the power 00:12:23.270 --> 00:12:25.790 counting and you don't want to think of introducing this s, 00:12:25.790 --> 00:12:28.100 since that was kind of just our choice-- 00:12:28.100 --> 00:12:31.078 we introduced it as a way of thinking about this question. 00:12:31.078 --> 00:12:32.870 But if we went back to the original action, 00:12:32.870 --> 00:12:34.870 we should have a way of doing the power counting 00:12:34.870 --> 00:12:37.368 from that without having to do this rescaling. 00:12:37.368 --> 00:12:38.660 And we know how to do that now. 00:12:38.660 --> 00:12:41.240 This exercise teaches us that we can just 00:12:41.240 --> 00:12:43.768 look at mass dimensions of the parameters 00:12:43.768 --> 00:12:44.810 to do the power counting. 00:12:49.790 --> 00:13:03.340 So if we just associate a power to the parameters, 00:13:03.340 --> 00:13:06.470 we're still in d equals 4. 00:13:06.470 --> 00:13:08.690 Then we would get this association, 00:13:08.690 --> 00:13:10.940 this being the statement that it's relevant, marginal, 00:13:10.940 --> 00:13:14.228 and irrelevant. 00:13:14.228 --> 00:13:16.270 And we can do a power counting in this lambda nu. 00:13:40.420 --> 00:13:42.965 And we can then say, in a language which 00:13:42.965 --> 00:13:44.590 would be familiar from Feynman diagrams 00:13:44.590 --> 00:13:46.612 where we do everything in momentum space, 00:13:46.612 --> 00:13:48.070 that the momentum we want to study, 00:13:48.070 --> 00:13:51.640 p, has to be much less than this lambda nu. 00:13:51.640 --> 00:13:53.820 And we'll do the power counting to lambda nu. 00:13:53.820 --> 00:13:57.060 And that will make the, for example, tau term 00:13:57.060 --> 00:14:00.863 an irrelevant, less important, operator. 00:14:06.420 --> 00:14:08.270 So there's one comment here. 00:14:08.270 --> 00:14:12.080 We did the scalar field theory just because it's simplest. 00:14:12.080 --> 00:14:15.530 It also has a relevant operator. 00:14:15.530 --> 00:14:18.350 And we see that relevant operators actually 00:14:18.350 --> 00:14:26.750 can be dangerous because we'd like 00:14:26.750 --> 00:14:29.060 to set the power counting for the whole problem 00:14:29.060 --> 00:14:29.900 by the kinetic term. 00:14:29.900 --> 00:14:32.180 We'd like to say that the kinetic term, which 00:14:32.180 --> 00:14:35.450 was canonically normalized and had no s's in it, 00:14:35.450 --> 00:14:37.130 we'd like to say that that was relevant, 00:14:37.130 --> 00:14:39.613 that that's part of the leading order Lagrangian. 00:14:39.613 --> 00:14:41.780 But when we went through it, we found something that 00:14:41.780 --> 00:14:43.270 was more relevant, the master. 00:14:43.270 --> 00:14:47.385 Phi squared could become even larger than the kinetic term. 00:14:52.340 --> 00:14:55.771 So we have to be careful about relevant operators. 00:14:55.771 --> 00:14:58.820 And then this is, of course, related to the Higgs fine 00:14:58.820 --> 00:14:59.320 tuning. 00:15:05.530 --> 00:15:08.230 So even though I'm using a scalar field theory, 00:15:08.230 --> 00:15:10.880 I'm, for the most part, going to just fine tune and ignore 00:15:10.880 --> 00:15:13.910 this problem, since if I was using something fermionic field 00:15:13.910 --> 00:15:15.710 theory, I could set things up so I 00:15:15.710 --> 00:15:17.030 could ignore it from the start. 00:15:17.030 --> 00:15:21.500 But still using a scalar field theory is convenient. 00:15:21.500 --> 00:15:24.500 So I want to also come back to something else 00:15:24.500 --> 00:15:27.340 that we mentioned last time and go into a little more detail. 00:15:27.340 --> 00:15:29.090 And that is the discussion of divergences. 00:15:29.090 --> 00:15:31.700 So last time, we said that there was 00:15:31.700 --> 00:15:34.390 two different ways of thinking about renormalizability, 00:15:34.390 --> 00:15:36.330 a traditional sense of renormalizability, 00:15:36.330 --> 00:15:39.170 renormalizability of the standard model, 00:15:39.170 --> 00:15:41.450 or an effective field theory way of thinking 00:15:41.450 --> 00:15:43.580 about renormalizability. 00:15:43.580 --> 00:15:46.430 So I want to come back to that with our example of this scalar 00:15:46.430 --> 00:15:49.160 field theory. 00:15:49.160 --> 00:15:54.260 So let's get rid of this issue of having something that 00:15:54.260 --> 00:15:56.690 can upset the power counting either 00:15:56.690 --> 00:16:01.460 by taking m to be 0 or just fine tuning it to be small. 00:16:01.460 --> 00:16:06.380 And what that means is I just demand that, as s grows, 00:16:06.380 --> 00:16:07.760 I shrink m. 00:16:07.760 --> 00:16:13.880 And if I do that, then I can, by hand, tune term and this term 00:16:13.880 --> 00:16:14.925 to be the same size. 00:16:14.925 --> 00:16:16.550 So if you like, I'm assigning a scaling 00:16:16.550 --> 00:16:19.760 to m in order to make the mass term be always as 00:16:19.760 --> 00:16:21.080 important as the kinetic term. 00:16:25.882 --> 00:16:27.590 So with that little proviso, we can start 00:16:27.590 --> 00:16:29.120 thinking about divergences. 00:16:29.120 --> 00:16:31.910 And when we start drawing Feynman diagrams, 00:16:31.910 --> 00:16:34.460 they will generically have divergences. 00:16:34.460 --> 00:16:36.870 So we could have two four-point interactions, 00:16:36.870 --> 00:16:39.590 which I label by lambda, because that's the parameter that 00:16:39.590 --> 00:16:42.240 shows up in the Feynman rule. 00:16:42.240 --> 00:16:46.285 If this is k and this is some k plus p, 00:16:46.285 --> 00:16:48.410 then this guy is going to have two pairs of lambda. 00:16:51.090 --> 00:17:05.775 And it's going to be some integral like that. 00:17:05.775 --> 00:17:09.740 We won't worry about overall factors here. 00:17:09.740 --> 00:17:12.259 I'm regulating with dimensional regularization. 00:17:17.150 --> 00:17:22.220 I'll often do that when it's convenient for us. 00:17:22.220 --> 00:17:24.510 If you ask how this integral diverges, 00:17:24.510 --> 00:17:25.970 you could ask how it diverges just 00:17:25.970 --> 00:17:28.910 in terms of thinking about it in terms of some parameter that's 00:17:28.910 --> 00:17:32.030 controlling the ultraviolet, like a cutoff. 00:17:32.030 --> 00:17:35.000 So even if I am using dim reg, I could ask, what's 00:17:35.000 --> 00:17:36.950 the power of the divergence? 00:17:36.950 --> 00:17:40.420 And it diverges as, in d dimensions, lambda 00:17:40.420 --> 00:17:42.512 to d minus 4. 00:17:42.512 --> 00:17:49.510 You say d minus 4 is the degree of the divergence. 00:17:49.510 --> 00:17:53.450 And that's because you have d powers of k from the measure 00:17:53.450 --> 00:17:55.620 and minus 4 from the propagators. 00:17:58.700 --> 00:18:03.290 So if d is equal to 4, you say degree of divergence is 0, 00:18:03.290 --> 00:18:05.112 but that means log diversion. 00:18:20.370 --> 00:18:26.630 So if you take d equals 4 in a UV, 4 powers of k downstairs, 00:18:26.630 --> 00:18:29.330 d upstairs-- but if d is 4, that's 00:18:29.330 --> 00:18:30.800 4 upstairs and 4 downstairs. 00:18:30.800 --> 00:18:33.110 So it's scaling length dk over k. 00:18:33.110 --> 00:18:34.500 So I made it Euclidean. 00:18:34.500 --> 00:18:36.990 That's exactly what it would become. 00:18:36.990 --> 00:18:37.910 And that's like a log. 00:18:37.910 --> 00:18:40.940 So it's a log of the cut off. 00:18:40.940 --> 00:18:48.476 So it's a 1 over epsilon in dim reg 00:18:48.476 --> 00:18:50.460 where d is 4 minus 2 epsilon. 00:18:53.760 --> 00:18:56.380 And if you just want to think about what this does, 00:18:56.380 --> 00:18:59.750 well, it's something that renormalizes the lambda phi 4 00:18:59.750 --> 00:19:00.360 operator. 00:19:00.360 --> 00:19:04.540 So you need a counter term for the lambda phi 4. 00:19:04.540 --> 00:19:07.700 So you add to your theory that counter term. 00:19:07.700 --> 00:19:11.670 And you could get rid of this divergence. 00:19:11.670 --> 00:19:12.170 OK. 00:19:12.170 --> 00:19:17.910 So, so far-- hopefully standard stuff. 00:19:17.910 --> 00:19:22.050 Let's keep going, think about other diagrams. 00:19:22.050 --> 00:19:27.822 So what if I put in a tau term and a lambda term? 00:19:27.822 --> 00:19:29.280 This integral is the same integral. 00:19:29.280 --> 00:19:32.550 I just have different fields on the outside. 00:19:32.550 --> 00:19:41.850 So it's got the same divergence, but now 00:19:41.850 --> 00:19:44.032 the operator it's renormalizing is an operator 00:19:44.032 --> 00:19:45.240 with 6 points on the outside. 00:19:55.810 --> 00:19:58.660 So it's renormalizing the tau phi 6 term. 00:19:58.660 --> 00:20:01.060 So I insert one tau and one lambda, 00:20:01.060 --> 00:20:04.470 and I have to get back the renormalization of tau. 00:20:04.470 --> 00:20:05.470 Well, that's not so bad. 00:20:05.470 --> 00:20:10.120 We had tau from the start, if we include the tau term, so not 00:20:10.120 --> 00:20:12.270 really a problem from the point of view 00:20:12.270 --> 00:20:14.790 of a standard renormalization program. 00:20:14.790 --> 00:20:23.980 But we could also include two taus, like this, again, 00:20:23.980 --> 00:20:29.950 same integral, so same divergence. 00:20:29.950 --> 00:20:31.780 And now, this renormalizes something 00:20:31.780 --> 00:20:38.770 that we haven't included yet, something with 8 points a phi 8 00:20:38.770 --> 00:20:39.370 operator. 00:20:42.890 --> 00:20:44.530 So in order to renormalize that diagram 00:20:44.530 --> 00:20:46.180 and make the theory renormalizable 00:20:46.180 --> 00:20:47.680 and an effective field theory sense, 00:20:47.680 --> 00:20:50.230 we need to include the phi 8 operator. 00:20:55.290 --> 00:20:58.160 So if I'd ignored the dots that I wrote down-- 00:21:04.300 --> 00:21:05.675 so let me say, without the dots-- 00:21:09.260 --> 00:21:12.132 then phi 8 wasn't there. 00:21:12.132 --> 00:21:14.340 And so then, therefore, I would say the theory is not 00:21:14.340 --> 00:21:16.490 renormalizable. 00:21:16.490 --> 00:21:19.640 That's what makes the tau operator the phi 6 00:21:19.640 --> 00:21:24.530 operator via non-renormalizable theory in the traditional sense 00:21:24.530 --> 00:21:26.147 if we include that operator. 00:21:34.270 --> 00:21:36.550 That's the classic way of thinking. 00:21:40.193 --> 00:21:41.860 And the effective theory way of thinking 00:21:41.860 --> 00:21:43.480 is just that we have to add that operator 00:21:43.480 --> 00:21:45.400 as soon as this diagram would become relevant. 00:21:50.810 --> 00:21:53.240 So we just determined a minute ago 00:21:53.240 --> 00:21:55.440 that tau goes like lambda nu to the minus 2. 00:22:00.040 --> 00:22:02.900 So tau goes like 1 over lambda nu squared. 00:22:02.900 --> 00:22:05.590 And in this diagram, we have two powers of tau. 00:22:05.590 --> 00:22:07.110 So it's even less important. 00:22:07.110 --> 00:22:08.830 So lambda nu to the fourth-- 00:22:08.830 --> 00:22:09.550 downstairs. 00:22:19.660 --> 00:22:22.408 And so when it becomes relevant to us, 00:22:22.408 --> 00:22:23.950 we want that kind of accuracy that we 00:22:23.950 --> 00:22:26.290 want to include things that go like 1 over lambda nu 00:22:26.290 --> 00:22:27.310 to the fourth. 00:22:27.310 --> 00:22:28.690 We have to consider this diagram. 00:22:28.690 --> 00:22:30.690 And we have to consider adding that operator 00:22:30.690 --> 00:22:31.690 to the effective theory. 00:22:44.827 --> 00:22:47.160 And that's the sense in which we say that the theory can 00:22:47.160 --> 00:22:57.560 be renormalized order by order in its power kinetic parameter, 00:22:57.560 --> 00:23:00.430 which is 1 over lambda nu. 00:23:06.322 --> 00:23:14.299 So I could order this order, but we have to add this operator. 00:23:32.762 --> 00:23:33.760 OK? 00:23:33.760 --> 00:23:34.760 Make sense? 00:23:39.390 --> 00:23:41.970 Silence means that it makes sense. 00:23:41.970 --> 00:23:44.010 Jumping up and down saying it doesn't make sense 00:23:44.010 --> 00:23:47.280 means that it doesn't make sense or puzzled looks 00:23:47.280 --> 00:23:49.305 from everybody, but that's harder to discern. 00:24:08.390 --> 00:24:14.463 So we can summarize this way of thinking in the following way. 00:24:14.463 --> 00:24:15.880 Remember with the effective theory 00:24:15.880 --> 00:24:17.505 that we're only interested in computing 00:24:17.505 --> 00:24:19.510 things to some accuracy. 00:24:19.510 --> 00:24:23.300 And the accuracy controls where we stop in the series. 00:24:23.300 --> 00:24:28.450 So if we're interested in stopping at lambda nu to the r 00:24:28.450 --> 00:24:31.300 or 1 over s to the r-- s was big. 00:24:31.300 --> 00:24:33.510 And lambda nu is also much bigger than the p. 00:24:37.290 --> 00:24:40.110 But let's stick to talking about lambda nu. 00:24:44.800 --> 00:24:50.230 And we include all operators that have dimensions up 00:24:50.230 --> 00:24:52.950 to a certain level. 00:24:52.950 --> 00:24:58.407 And since the power counting is connected to dimensions, 00:24:58.407 --> 00:25:00.990 we're kind of guaranteed that we will have everything we need. 00:25:09.900 --> 00:25:16.220 So as I promised, what this little argument or discussion 00:25:16.220 --> 00:25:21.618 tells us, is how power counting is connected to dimensions. 00:25:21.618 --> 00:25:24.160 And this is the classic way of thinking about effective field 00:25:24.160 --> 00:25:25.660 theory is that the power counting is 00:25:25.660 --> 00:25:26.864 connected to dimensions. 00:25:30.660 --> 00:25:33.600 So this seems pretty generic actually. 00:25:33.600 --> 00:25:35.790 You could imagine that, if I did scalars, I mean, 00:25:35.790 --> 00:25:38.640 fermions and scalars of gauge theory, 00:25:38.640 --> 00:25:41.250 then I could still go through the same type of arguments, 00:25:41.250 --> 00:25:43.410 write down higher dimensional operators, 00:25:43.410 --> 00:25:45.680 go through all these arguments. 00:25:45.680 --> 00:25:47.400 And so it seems that I'd actually 00:25:47.400 --> 00:25:51.090 shown you something more powerful than what I claimed. 00:25:51.090 --> 00:25:54.360 Because I said, here, it seems like it's almost this, right, 00:25:54.360 --> 00:25:57.510 that power counting is always connected to dimensions. 00:25:57.510 --> 00:26:01.050 So can anyone spot where there was an assumption in what we 00:26:01.050 --> 00:26:07.140 did that where in some case the power counting might not have 00:26:07.140 --> 00:26:08.999 been related to dimensions? 00:26:13.310 --> 00:26:14.747 It's a tough question. 00:26:14.747 --> 00:26:19.488 AUDIENCE: If the power counting isn't a ratio scale? 00:26:19.488 --> 00:26:20.280 IAIN STEWART: Yeah. 00:26:20.280 --> 00:26:25.810 But I want a little bit more than that, right track. 00:26:25.810 --> 00:26:27.690 So going back to this example that we did, 00:26:27.690 --> 00:26:30.300 what did we assume at the beginning that led us here? 00:26:34.150 --> 00:26:35.910 AUDIENCE: That we changed the mass point? 00:26:35.910 --> 00:26:38.780 IAIN STEWART: No, the mass wasn't so much the issue. 00:26:41.520 --> 00:26:43.710 So what it was is that we scaled all the coordinates 00:26:43.710 --> 00:26:44.910 by the same amount. 00:26:44.910 --> 00:26:47.550 We said all the coordinates are getting large in the same way. 00:26:47.550 --> 00:26:50.050 And we could have done something more complicated than that. 00:26:50.050 --> 00:26:52.500 We could have said some components of this coordinate 00:26:52.500 --> 00:26:54.600 are getting larger, faster than other components. 00:26:54.600 --> 00:26:56.693 That's what you do in a non-relativistic theory 00:26:56.693 --> 00:26:58.860 where the time component and the spatial coordinates 00:26:58.860 --> 00:27:00.915 would scale in different ways. 00:27:00.915 --> 00:27:02.040 So that was the assumption. 00:27:02.040 --> 00:27:04.770 We assumed, basically, that everything was getting large. 00:27:04.770 --> 00:27:08.490 All the coordinates were getting large uniformly with s. 00:27:08.490 --> 00:27:12.360 And we said x mu i was equal to s x prime mu i. 00:27:23.460 --> 00:27:26.130 With a universal s for all the components, 00:27:26.130 --> 00:27:28.210 that was an assumption that led us here. 00:27:28.210 --> 00:27:29.885 And we may not always do that. 00:27:29.885 --> 00:27:32.010 In fact, in some of our examples, we won't do that. 00:27:34.925 --> 00:27:36.300 But here, for the standard model, 00:27:36.300 --> 00:27:37.425 that's what you want to do. 00:27:52.690 --> 00:27:56.230 So if we have the standard model, which is just L0, 00:27:56.230 --> 00:28:00.797 the usual standard model, then we 00:28:00.797 --> 00:28:02.630 know, as part of the way of constructing it, 00:28:02.630 --> 00:28:08.300 that we wrote down all the operators with dimension 00:28:08.300 --> 00:28:09.500 less than or equal to 4. 00:28:16.780 --> 00:28:21.730 And we also know that it was renormalizable 00:28:21.730 --> 00:28:22.810 in a traditional sense. 00:28:35.188 --> 00:28:36.980 So now, let's talk about the standard model 00:28:36.980 --> 00:28:40.367 corrections, i.e. terms in the standard model 00:28:40.367 --> 00:28:42.200 from an effective field theory point of view 00:28:42.200 --> 00:28:45.335 that are operators we can write down, like L1. 00:28:50.936 --> 00:28:56.910 So L1, I can write it in the following way, which is kind 00:28:56.910 --> 00:29:00.300 of a convenient thing to do. 00:29:00.300 --> 00:29:03.750 Pull out the scale lambda nu. 00:29:03.750 --> 00:29:07.090 Leave over some dimensionless constant. 00:29:07.090 --> 00:29:10.103 So I'll just use some scale lambda nu for all the operators 00:29:10.103 --> 00:29:10.770 that I consider. 00:29:10.770 --> 00:29:12.330 And I'll just allow for differences 00:29:12.330 --> 00:29:14.205 between the various scales that the operators 00:29:14.205 --> 00:29:18.600 could have to be taken up by dimensionless constants. 00:29:18.600 --> 00:29:22.006 And O5 here is a dimension 5 operator. 00:29:32.430 --> 00:29:33.760 The dimension of c is 0. 00:29:36.860 --> 00:29:40.290 In a power counting notation, we say c is of order 1. 00:29:40.290 --> 00:29:42.580 That means that we don't count any powers 00:29:42.580 --> 00:29:44.280 of lambda nu associated to c. 00:29:44.280 --> 00:29:47.460 What we've done here is we've made the power explicit by just 00:29:47.460 --> 00:29:48.050 writing it in. 00:29:55.892 --> 00:29:57.850 So that's often convenient just in the same way 00:29:57.850 --> 00:30:00.520 it was convenient to make the s's explicit in that argument. 00:30:00.520 --> 00:30:03.670 Now, we're just building up the theory, writing down operators, 00:30:03.670 --> 00:30:06.010 making the lambda nu's, which are our power counting 00:30:06.010 --> 00:30:07.210 parameter, very explicit. 00:30:11.820 --> 00:30:14.580 Now, the statement that, in the standard model, 00:30:14.580 --> 00:30:17.130 it was renormalizable in the traditional sense, 00:30:17.130 --> 00:30:22.740 I told you that what that meant is that nothing in lambda 0, 00:30:22.740 --> 00:30:26.520 nothing in Lagrangian 0, really tells us about lambda nu. 00:30:35.828 --> 00:30:37.620 So we're free to take it as big as we want. 00:30:47.520 --> 00:30:52.862 There's no constraint on it from our leading order Lagrangian. 00:30:52.862 --> 00:30:54.570 In particular, we can take it much bigger 00:30:54.570 --> 00:30:56.610 than things like the taught mass or the w mass. 00:31:01.640 --> 00:31:03.390 And we can make these corrections as small 00:31:03.390 --> 00:31:03.900 as we want. 00:31:08.050 --> 00:31:11.560 So L1 is, therefore, really we can think of it as really 00:31:11.560 --> 00:31:12.810 giving some small corrections. 00:31:12.810 --> 00:31:17.330 And we can adjust how small they are by just dialing up 00:31:17.330 --> 00:31:18.390 the scale of lambda nu. 00:31:40.000 --> 00:31:44.040 All right, so let's get down to business 00:31:44.040 --> 00:31:47.460 and actually talk about what this Lagrangian is. 00:31:55.654 --> 00:32:03.370 Our notation that we index the Lagrangian in a series, this 00:32:03.370 --> 00:32:05.977 is our sum over n that we talked about last time. 00:32:05.977 --> 00:32:07.810 And our power counting is that we associated 00:32:07.810 --> 00:32:12.640 this guy here with no powers of lambda nu, 00:32:12.640 --> 00:32:15.480 this guy here with one inverse power, 00:32:15.480 --> 00:32:20.290 this guy here with two inverse power, et cetera. 00:32:20.290 --> 00:32:24.460 And we want to think about using this for some p, which you 00:32:24.460 --> 00:32:27.340 could say is of order m top squared, 00:32:27.340 --> 00:32:33.490 some scale that's much less than the lambda nu. 00:32:33.490 --> 00:32:35.380 It could be larger than on top. 00:32:35.380 --> 00:32:38.830 It could be 10 TeV, whatever we decide. 00:32:38.830 --> 00:32:41.290 But it's just, in order for me to write something 00:32:41.290 --> 00:32:44.480 on the board, let me write m top. 00:32:44.480 --> 00:32:44.980 OK. 00:32:44.980 --> 00:32:47.350 So how do we construct L1 and L2? 00:32:47.350 --> 00:32:49.630 What do we assume? 00:32:49.630 --> 00:32:52.540 Well, one thing we assume, or we are free to assume 00:32:52.540 --> 00:32:55.840 and is a reasonable assumption, is 00:32:55.840 --> 00:32:59.980 that there are not going to be any Lorentz invariant or gauge 00:32:59.980 --> 00:33:01.145 invariant violating terms. 00:33:07.020 --> 00:33:07.520 OK. 00:33:07.520 --> 00:33:09.860 So we can maintain these as symmetries of our theory. 00:33:17.400 --> 00:33:20.290 So we assume that they're unbroken. 00:33:20.290 --> 00:33:21.910 So that means, when we write down 00:33:21.910 --> 00:33:27.480 these L1 and L2, or generically each Li, 00:33:27.480 --> 00:33:30.528 that we're going to have to do it by writing down operators 00:33:30.528 --> 00:33:31.570 that are gauge invariant. 00:33:31.570 --> 00:33:33.112 Even though they're higher dimension, 00:33:33.112 --> 00:33:38.130 we still have to satisfy gauge variance of the standard model 00:33:38.130 --> 00:33:39.677 and Lorentz invariance. 00:33:45.530 --> 00:33:49.120 So that's going to restrict what we can do. 00:33:56.610 --> 00:34:00.918 We also construct Li from the same degrees of freedom 00:34:00.918 --> 00:34:01.710 that we have in L0. 00:34:10.370 --> 00:34:11.710 So we know what fields to use. 00:34:15.170 --> 00:34:16.639 So that's important. 00:34:16.639 --> 00:34:18.639 It means that, once you've got the leading order 00:34:18.639 --> 00:34:20.469 effective field theory, you know where 00:34:20.469 --> 00:34:23.409 to go for the higher order terms because you're just 00:34:23.409 --> 00:34:24.429 using the same fields. 00:34:33.834 --> 00:34:36.730 I'll also make the assumption that the Higgs vacuum 00:34:36.730 --> 00:34:49.650 expectation value is going to stay to be the value in L0. 00:34:54.230 --> 00:34:59.200 So the way that the gauge group is broken by the Higgs vacuum 00:34:59.200 --> 00:35:01.833 expectation value is spontaneously broken. 00:35:01.833 --> 00:35:03.250 We're not going to mess with that. 00:35:15.950 --> 00:35:19.550 And we built into this idea that we do this 00:35:19.550 --> 00:35:23.800 because there's no new particles that are produced at p. 00:35:23.800 --> 00:35:25.810 If there was a new particle produced at p, 00:35:25.810 --> 00:35:28.210 then they would have to have a mass that 00:35:28.210 --> 00:35:30.220 would allow us to produce it. 00:35:30.220 --> 00:35:33.970 And that would mean that it doesn't have a mass up 00:35:33.970 --> 00:35:35.020 at this lambda nu scale. 00:35:35.020 --> 00:35:38.320 And we'd have to include it in our effective Lagrangian. 00:35:38.320 --> 00:35:40.068 So by taking this point of view, we're 00:35:40.068 --> 00:35:41.860 assuming that there's no new particles that 00:35:41.860 --> 00:35:45.610 are produced at the scale p, only at lambda nu. 00:35:49.450 --> 00:35:51.158 And effectively, we've integrated out, 00:35:51.158 --> 00:35:53.200 if you want to use that language-- although we're 00:35:53.200 --> 00:35:55.680 doing this from the bottom up. 00:35:55.680 --> 00:35:57.430 If you wanted to have a top-down language, 00:35:57.430 --> 00:35:59.690 you'd say we integrated out the particles 00:35:59.690 --> 00:36:02.270 at the scale lambda nu. 00:36:02.270 --> 00:36:03.640 OK, so that's our logic. 00:36:07.780 --> 00:36:12.430 So let me just start seeing what we can write down. 00:36:12.430 --> 00:36:16.065 And for the dimension 5, it's actually very restrictive. 00:36:20.250 --> 00:36:24.342 Gauge symmetry is very restrictive for dimension 5. 00:36:24.342 --> 00:36:26.666 And there's basically only one operator. 00:36:35.098 --> 00:36:37.000 So we won't stop at dimension 5. 00:36:37.000 --> 00:36:38.340 We'll go up to the dimension 6. 00:36:42.316 --> 00:36:48.970 At dimension 5, it turns out that, once you satisfy 00:36:48.970 --> 00:36:52.090 the gauge symmetry, the unique term that you can write down 00:36:52.090 --> 00:36:52.840 looks like this. 00:36:58.200 --> 00:37:05.110 And my notation is that this guy is 00:37:05.110 --> 00:37:11.933 our left-handed lepton with a charge conjugation operator. 00:37:11.933 --> 00:37:13.100 And these guys are doublets. 00:37:13.100 --> 00:37:17.210 So the Higgs doublet is a doublet like that. 00:37:17.210 --> 00:37:22.730 And the left-handed leptons, neutrino, an electron 00:37:22.730 --> 00:37:25.270 is a doublet like that. 00:37:25.270 --> 00:37:27.810 So these are doublets. 00:37:27.810 --> 00:37:31.640 And I'm figuring out how to contract with doublet indices, 00:37:31.640 --> 00:37:35.960 but I have to satisfy the U1 hyper charged gauge invariance. 00:37:35.960 --> 00:37:38.640 This has no color. 00:37:38.640 --> 00:37:40.820 So that's automatically satisfied. 00:37:40.820 --> 00:37:42.653 And then I have to satisfy the SU2. 00:37:42.653 --> 00:37:44.570 And I've done that by the way that the indices 00:37:44.570 --> 00:37:45.665 are contracted. 00:37:50.780 --> 00:37:53.275 One thing I didn't write down in this operator 00:37:53.275 --> 00:37:54.200 is flavor indices. 00:37:57.098 --> 00:37:58.640 So if you were to add flavor indices, 00:37:58.640 --> 00:37:59.810 you could do a bit more. 00:37:59.810 --> 00:38:06.690 But in some sense, that's a pretty simple generalization. 00:38:06.690 --> 00:38:10.160 So we'll still count it as one even though we could think 00:38:10.160 --> 00:38:11.320 about having more flavors. 00:38:11.320 --> 00:38:13.790 We're contracting things with more flavors. 00:38:13.790 --> 00:38:16.280 And that would, of course, affect arguments 00:38:16.280 --> 00:38:17.600 made on gauge symmetry alone. 00:38:20.670 --> 00:38:23.420 So in all my counting today, I'll 00:38:23.420 --> 00:38:25.775 be agnostic about flavor matrices. 00:38:28.750 --> 00:38:33.870 So when I say only, that proviso is hidden there, OK? 00:38:33.870 --> 00:38:35.430 So this guy is kind of interesting 00:38:35.430 --> 00:38:38.500 from an phenomenological point of view. 00:38:38.500 --> 00:38:42.660 Because if you replace the Higgs field by its vacuum expectation 00:38:42.660 --> 00:38:45.300 value, which means getting rid of H+ ? 00:38:45.300 --> 00:38:50.160 And taking h0 to be a constant, which is v, 00:38:50.160 --> 00:38:51.860 then that gives a Majorana mass term. 00:39:06.370 --> 00:39:10.590 So the observed left-handed neutrino 00:39:10.590 --> 00:39:13.850 would get a term in its Lagrangian 00:39:13.850 --> 00:39:20.850 that, after we do that, it looks like that. 00:39:20.850 --> 00:39:34.780 And that's a Majorana mass term, where 00:39:34.780 --> 00:39:36.630 this m nu is a parameter that shows up, 00:39:36.630 --> 00:39:38.338 but it's built out of the parameters that 00:39:38.338 --> 00:39:39.230 were in this thing. 00:39:39.230 --> 00:39:41.972 So once I replace these by [INAUDIBLE] 00:39:41.972 --> 00:39:43.430 and go through the various factors, 00:39:43.430 --> 00:39:44.830 we get something like that. 00:39:48.577 --> 00:39:50.910 So just the fact that we know that the observed neutrino 00:39:50.910 --> 00:39:57.640 masses are less than, say, 3.5 eV 00:39:57.640 --> 00:39:59.220 tells us something about the scale. 00:39:59.220 --> 00:40:00.640 So this is small. 00:40:00.640 --> 00:40:04.310 If c5 is of order 1, I told you we know what v is. 00:40:04.310 --> 00:40:05.980 It's 246 GeV. 00:40:05.980 --> 00:40:13.380 That tells us something about lambda nu that it's big. 00:40:20.186 --> 00:40:22.800 But we could try to think up some reasons 00:40:22.800 --> 00:40:24.700 why the c5 maybe have some suppression in it 00:40:24.700 --> 00:40:26.200 and make the c a little bit smaller. 00:40:26.200 --> 00:40:29.650 But if C5 is order 1, then we get a very large lambda nu 00:40:29.650 --> 00:40:30.150 scale. 00:40:35.090 --> 00:40:37.340 I'll give you a problem on your problem set to explore 00:40:37.340 --> 00:40:38.548 this in a little more detail. 00:40:40.860 --> 00:40:43.730 I should also note that the Majorana mass term, having 00:40:43.730 --> 00:40:47.200 two neutrinos like this, violates lepton number, which 00:40:47.200 --> 00:40:55.390 is a global symmetry of the standard model, at least 00:40:55.390 --> 00:40:56.464 classically. 00:41:02.880 --> 00:41:04.840 So this guy violates lepton numbers. 00:41:04.840 --> 00:41:07.540 You can also write down dimension 6 operators 00:41:07.540 --> 00:41:08.830 that violate baryon number. 00:41:14.110 --> 00:41:17.723 And I'm going to leave that, also, as a problem set problem. 00:41:17.723 --> 00:41:19.640 So you'll figure out what those operators are. 00:41:28.220 --> 00:41:35.670 So example 3, if I conserve lepton number and baryon 00:41:35.670 --> 00:41:36.170 number-- 00:41:39.050 --> 00:41:41.120 which are things that, if they're broken, 00:41:41.120 --> 00:41:44.058 that's obviously having a big impact. 00:41:44.058 --> 00:41:46.100 And there's obviously strong constraints on that. 00:41:49.580 --> 00:41:53.230 So you can ask then, if you go to dimension 6 00:41:53.230 --> 00:41:55.370 and we conserve those things because they're 00:41:55.370 --> 00:41:58.070 highly constrained, how many operators are there left over? 00:41:58.070 --> 00:41:58.855 And there's 80. 00:42:08.300 --> 00:42:14.840 So L2, we can account exactly how many operators there are. 00:42:14.840 --> 00:42:22.190 And there's i from 1 to 80, some coefficients and some operators 00:42:22.190 --> 00:42:23.090 that are dimension 6. 00:42:23.090 --> 00:42:26.612 And if I'm going to make a lambda nu explicit, 00:42:26.612 --> 00:42:28.070 I put a lambda nu squared in there. 00:42:31.280 --> 00:42:36.900 So 80 sounds like a big number, but big is always relative. 00:42:36.900 --> 00:42:38.580 So if you think about 80 relative to, 00:42:38.580 --> 00:42:41.480 for example, how many soft SUSY breaking parameters 00:42:41.480 --> 00:42:43.630 you have in the MSSM, is greater than 100, 00:42:43.630 --> 00:42:47.255 then 80 doesn't sound so bad. 00:42:47.255 --> 00:42:49.400 Also, you should remember that, if you're 00:42:49.400 --> 00:42:52.280 going to do some phenomenology with this L2, 00:42:52.280 --> 00:42:54.320 that many of the 80 are not going to contribute. 00:42:54.320 --> 00:42:55.820 If you look at a particular process, 00:42:55.820 --> 00:42:58.760 only some small subset of them will contribute. 00:42:58.760 --> 00:43:01.220 And there's many, many, many observables 00:43:01.220 --> 00:43:04.020 in the standard model with all the different particles. 00:43:04.020 --> 00:43:06.620 So 80, once you start dividing it up 00:43:06.620 --> 00:43:09.350 into camps that contribute to different observables, 00:43:09.350 --> 00:43:11.940 is not such a large number. 00:43:11.940 --> 00:43:13.990 Or at least it's not an unmanageable number, 00:43:13.990 --> 00:43:15.820 and people do phenomenology with this. 00:43:32.200 --> 00:43:36.892 So for any observable, only a manageable number contribute. 00:43:39.490 --> 00:43:44.980 And I should also say that, if you have a top-down perspective 00:43:44.980 --> 00:43:48.460 where you have a new physics theory that you've constructed 00:43:48.460 --> 00:43:51.070 that has the scale lambda nu in it, 00:43:51.070 --> 00:43:53.470 then from this point of view what that theory predicts is 00:43:53.470 --> 00:43:55.385 a particular pattern for the c's. 00:44:04.830 --> 00:44:07.600 Hopefully, if it has less parameters than 80, 00:44:07.600 --> 00:44:10.990 then you get some patterns of connections between the c's. 00:44:10.990 --> 00:44:14.140 So if you have some new physics model that has the number 00:44:14.140 --> 00:44:15.918 of parameters that are less than 80, 00:44:15.918 --> 00:44:17.710 then you get connections between these c's. 00:44:17.710 --> 00:44:20.050 And you could think that, if you use this Lagrangian, 00:44:20.050 --> 00:44:23.980 constrain the c's, that you could test generically 00:44:23.980 --> 00:44:26.860 for classes of new physics theories 00:44:26.860 --> 00:44:28.870 that are ruled in or ruled out. 00:44:28.870 --> 00:44:31.030 Because they have to, if you match on to these c's 00:44:31.030 --> 00:44:34.300 from those theories, obey whatever constraints 00:44:34.300 --> 00:44:35.980 you would derive from this logic. 00:44:35.980 --> 00:44:39.670 Of course, that assumes that the new physics 00:44:39.670 --> 00:44:41.290 particles are at a high scale. 00:44:41.290 --> 00:44:46.450 So we can make this expansion, OK? 00:44:46.450 --> 00:44:48.610 Questions about that? 00:44:48.610 --> 00:44:49.532 Yeah. 00:44:49.532 --> 00:44:51.612 AUDIENCE: [INAUDIBLE] assumption that you 00:44:51.612 --> 00:44:56.408 don't have [INAUDIBLE] freedom is returning to the high order. 00:44:56.408 --> 00:44:57.200 IAIN STEWART: Yeah. 00:44:57.200 --> 00:44:59.117 AUDIENCE: If you want them to do a [INAUDIBLE] 00:44:59.117 --> 00:45:04.060 or [INAUDIBLE] for high order, would [INAUDIBLE] they 00:45:04.060 --> 00:45:05.890 have to be at higher [INAUDIBLE].. 00:45:05.890 --> 00:45:10.347 And why is it at a higher energy than [INAUDIBLE]?? 00:45:10.347 --> 00:45:11.930 IAIN STEWART: So it's not to say that, 00:45:11.930 --> 00:45:14.840 at higher energy, that no new degrees of freedom 00:45:14.840 --> 00:45:18.240 would show up if I really probe those energies directly. 00:45:18.240 --> 00:45:20.750 But what I'm doing is I'm saying that I'm probing 00:45:20.750 --> 00:45:23.352 the physics at small energy. 00:45:23.352 --> 00:45:25.310 And the way that high energy degrees of freedom 00:45:25.310 --> 00:45:28.940 would show up is by a contribution to one 00:45:28.940 --> 00:45:29.780 of these operators. 00:45:29.780 --> 00:45:32.150 So say I have added in some new particle at 10 00:45:32.150 --> 00:45:35.300 TeV, mass of 10 TeV. 00:45:35.300 --> 00:45:39.320 If I expand in momentum over that mass, 00:45:39.320 --> 00:45:41.450 then what will happen is you'll get an operator 00:45:41.450 --> 00:45:43.790 like this O6, where that particle is 00:45:43.790 --> 00:45:45.680 removed because I expanded it. 00:45:45.680 --> 00:45:47.180 It got removed. 00:45:47.180 --> 00:45:50.120 And its mass will be exactly in this denominator. 00:45:50.120 --> 00:45:51.470 It'll show up here as lambda nu. 00:45:51.470 --> 00:45:53.930 So think about it as lambda nu squared could just 00:45:53.930 --> 00:45:55.520 be the propagator. 00:45:55.520 --> 00:46:01.516 If I had 1 over p squared minus some massive-- 00:46:01.516 --> 00:46:04.550 I don't know, some gluino, right? 00:46:04.550 --> 00:46:07.220 And then I start expanding this. 00:46:07.220 --> 00:46:11.210 The first term where I drop the momentum is just that. 00:46:11.210 --> 00:46:14.700 And that could be exactly this lambda nu squared. 00:46:14.700 --> 00:46:19.310 So the new physics particles don't show up in Oi6. 00:46:19.310 --> 00:46:20.870 What they affect is the pre-factor. 00:46:27.770 --> 00:46:30.350 So you don't need to add new physics particles in order 00:46:30.350 --> 00:46:31.800 could construct the operators. 00:46:31.800 --> 00:46:33.467 You're just building those operators out 00:46:33.467 --> 00:46:35.390 of the standard model degrees of freedom. 00:46:35.390 --> 00:46:36.878 AUDIENCE: So you're still working 00:46:36.878 --> 00:46:38.180 in p but just [INAUDIBLE]. 00:46:38.180 --> 00:46:40.650 IAIN STEWART: Yes, yes, then the lambda nu. 00:46:40.650 --> 00:46:43.630 Is there another question? 00:46:43.630 --> 00:46:44.130 OK. 00:46:44.130 --> 00:46:46.730 So I've been assuming some familiarity 00:46:46.730 --> 00:46:48.620 with the standard model here. 00:46:48.620 --> 00:46:50.948 And I've posted also, as I said last time, 00:46:50.948 --> 00:46:52.740 my lecture notes on quantum field theory 3. 00:46:52.740 --> 00:46:56.660 And there's some review reading there if some of this 00:46:56.660 --> 00:46:59.750 is unfamiliar to you, discussing [INAUDIBLE] and things 00:46:59.750 --> 00:47:02.480 like that. 00:47:02.480 --> 00:47:02.980 OK. 00:47:02.980 --> 00:47:07.305 So what kind of operators can we have at dimension 6? 00:47:10.420 --> 00:47:13.840 I'm not going to list all 80, obviously, 00:47:13.840 --> 00:47:16.370 but I'll list a few of them. 00:47:16.370 --> 00:47:21.430 So we could take an operator that's the following, built out 00:47:21.430 --> 00:47:23.390 of gluon field strengths. 00:47:32.250 --> 00:47:35.170 So making it Lorentz invariant and contracting up the indices 00:47:35.170 --> 00:47:43.630 and contracting up color indices with an FABC, that's 00:47:43.630 --> 00:47:47.900 an operator that's dimension 6 because the Gs are dimension 2. 00:47:47.900 --> 00:47:49.000 That's one of the 80. 00:47:58.870 --> 00:48:01.150 You could also do something with fermions. 00:48:01.150 --> 00:48:03.210 AUDIENCE: Sorry, what are the [INAUDIBLE]?? 00:48:03.210 --> 00:48:04.460 IAIN STEWART: Here? 00:48:04.460 --> 00:48:04.960 Whoops. 00:48:04.960 --> 00:48:06.880 Yeah, what happened? 00:48:06.880 --> 00:48:08.780 All right, there we go. 00:48:08.780 --> 00:48:09.280 Thanks. 00:48:20.600 --> 00:48:22.920 So here's something with a lepton doublet and a quark 00:48:22.920 --> 00:48:23.420 doublet. 00:48:23.420 --> 00:48:26.420 We already introduced the notation for LL. 00:48:26.420 --> 00:48:29.020 And QL is similar, but just up and down. 00:48:32.160 --> 00:48:34.390 So that's each fermion dimension 3/2. 00:48:34.390 --> 00:48:35.920 So four of them is dimension six. 00:48:39.990 --> 00:48:42.290 There's something called magnetic operators. 00:48:45.090 --> 00:48:46.550 So there's lots of different things 00:48:46.550 --> 00:48:48.290 that you can do with four fermions. 00:48:48.290 --> 00:48:50.980 I've only given you one example. 00:48:50.980 --> 00:48:54.350 And I've posted the reference for the paper 00:48:54.350 --> 00:48:57.175 that lists all 80. 00:48:57.175 --> 00:48:58.550 So you can look at it if you want 00:48:58.550 --> 00:48:59.758 to look at the complete list. 00:49:05.090 --> 00:49:08.320 So we can do something where we have leptons, a Higgs field, 00:49:08.320 --> 00:49:10.570 as well as a field strength for the SU2. 00:49:19.653 --> 00:49:20.570 So this is in the SU2. 00:49:24.023 --> 00:49:25.440 And these are all gauge invariant, 00:49:25.440 --> 00:49:27.982 as you can convince yourself by looking at the standard model 00:49:27.982 --> 00:49:30.110 gauge transformations of these operators. 00:49:30.110 --> 00:49:32.160 And so if I write something like this, where this is a doublet 00:49:32.160 --> 00:49:33.535 and this is a doublet and I don't 00:49:33.535 --> 00:49:35.803 write the doublet contraction, then I'm 00:49:35.803 --> 00:49:37.095 just contracting those indices. 00:49:40.120 --> 00:49:43.460 These two here contribute to the mu on the magnetic moment, 00:49:43.460 --> 00:49:44.280 anomalous moment. 00:49:47.992 --> 00:49:49.920 So they contribute to g minus 2. 00:50:02.867 --> 00:50:04.700 So g minus 2 at the muon, which is something 00:50:04.700 --> 00:50:09.850 that we've measured to very high precision 00:50:09.850 --> 00:50:13.820 has what sometimes you would call standard model 00:50:13.820 --> 00:50:16.940 contributions, which people usually 00:50:16.940 --> 00:50:18.830 mean as the contributions in our notation 00:50:18.830 --> 00:50:21.680 from L0 and then plus some contributions 00:50:21.680 --> 00:50:24.470 from these higher dimension operators, whatever 00:50:24.470 --> 00:50:28.092 coefficients these operators have. 00:50:28.092 --> 00:50:30.050 And again, I would replace the Higgs field here 00:50:30.050 --> 00:50:31.168 by a [INAUDIBLE]. 00:50:34.100 --> 00:50:35.870 I know they're dimension 6, so there's 00:50:35.870 --> 00:50:37.430 lambda nu squared downstairs. 00:50:37.430 --> 00:50:40.100 One factor of dimension is made up by the [INAUDIBLE].. 00:50:40.100 --> 00:50:42.830 And then the next scale that comes in 00:50:42.830 --> 00:50:43.880 is the mass of the muon. 00:50:43.880 --> 00:50:47.270 So this is the generic size of those contributions. 00:50:47.270 --> 00:50:51.590 And again, if you take into account experimentally how well 00:50:51.590 --> 00:50:54.010 we've measured this, it puts a pretty strong, 00:50:54.010 --> 00:50:58.040 or at least it puts a constraint, on lambda nu. 00:50:58.040 --> 00:51:02.750 Actually, it's probably stronger than this number, some number 00:51:02.750 --> 00:51:03.800 greater than 100 TeV. 00:51:03.800 --> 00:51:06.142 Maybe it's even 1,000 TeV. 00:51:06.142 --> 00:51:07.600 AUDIENCE: [INAUDIBLE] contribution, 00:51:07.600 --> 00:51:09.010 it was [INAUDIBLE]? 00:51:09.010 --> 00:51:11.310 IAIN STEWART: Yeah. 00:51:11.310 --> 00:51:12.810 So the standard model contribution 00:51:12.810 --> 00:51:15.250 makes up all the digits we've measured. 00:51:15.250 --> 00:51:18.210 And then there's some digits where there's some uncertainty. 00:51:18.210 --> 00:51:19.240 We haven't [INAUDIBLE]. 00:51:19.240 --> 00:51:21.870 And you can constrain, based on experimental uncertainty, 00:51:21.870 --> 00:51:25.470 how big this possible contribution could be. 00:51:25.470 --> 00:51:30.728 And there's 2 and 1/2 deviations from the standard model 00:51:30.728 --> 00:51:31.520 in this observable. 00:51:31.520 --> 00:51:35.990 So you can make up for them with an operator like that. 00:51:35.990 --> 00:51:39.230 But I never really pay attention to things 00:51:39.230 --> 00:51:43.820 that are less than 4 sigma personally. 00:51:43.820 --> 00:51:46.160 Though sometimes it's interesting to get 00:51:46.160 --> 00:51:47.450 excited about 3 sigma. 00:51:50.859 --> 00:51:53.867 AUDIENCE: [INAUDIBLE] 00:51:53.867 --> 00:51:54.950 IAIN STEWART: No, I'm not. 00:51:54.950 --> 00:51:55.750 AUDIENCE: Oh. 00:51:55.750 --> 00:51:56.860 IAIN STEWART: Yeah. 00:51:56.860 --> 00:51:59.100 So there's also flavor on top of that. 00:51:59.100 --> 00:52:00.592 That's right. 00:52:00.592 --> 00:52:02.300 So flavor is actually highly constrained. 00:52:02.300 --> 00:52:05.720 And you could put in some assumptions about flavor 00:52:05.720 --> 00:52:09.210 and then you get to the [INAUDIBLE].. 00:52:09.210 --> 00:52:09.720 OK. 00:52:09.720 --> 00:52:15.320 So for the remaining 76, see the reference I've posted. 00:52:31.480 --> 00:52:33.662 So this paper actually wasn't the first to try 00:52:33.662 --> 00:52:34.870 to enumerate these operators. 00:52:34.870 --> 00:52:37.780 As you can imagine, this would be a pretty standard thing 00:52:37.780 --> 00:52:38.830 to do. 00:52:38.830 --> 00:52:40.780 But it was the first to get 80. 00:52:40.780 --> 00:52:44.140 And the reason that they got 80 where other people got more 00:52:44.140 --> 00:52:47.680 is because they used the equations of motion 00:52:47.680 --> 00:52:49.000 to simplify the operators. 00:52:52.410 --> 00:52:56.530 So let me phrase that as, is there a caveat? 00:53:02.870 --> 00:53:05.980 So when they did their counting of the operators, 00:53:05.980 --> 00:53:18.750 they took the equations of motion from L0, 00:53:18.750 --> 00:53:20.490 and they used that to simplify L1. 00:53:25.800 --> 00:53:29.340 So they worked out the standard model equations of motion 00:53:29.340 --> 00:53:31.552 that tree level. 00:53:31.552 --> 00:53:33.510 And then they applied those equations of motion 00:53:33.510 --> 00:53:41.965 to reduce the form of the operators down to 80. 00:53:41.965 --> 00:53:44.530 So that got rid of a lot of operators for them 00:53:44.530 --> 00:53:46.210 that other people had considered as part 00:53:46.210 --> 00:53:47.377 of the counting in the past. 00:53:50.230 --> 00:53:54.430 So for example-- so you get some idea, 00:53:54.430 --> 00:53:56.220 if I have a covariant derivative acting 00:53:56.220 --> 00:54:00.240 on a right-handed electron. 00:54:00.240 --> 00:54:02.340 And the equation of motion in the standard model 00:54:02.340 --> 00:54:05.040 relates that to [INAUDIBLE] couplings. 00:54:05.040 --> 00:54:09.590 Here, I'm writing the flavor indices 00:54:09.590 --> 00:54:11.490 in the left-handed doublet Higgs field 00:54:11.490 --> 00:54:14.010 and the left-handed doublet. 00:54:14.010 --> 00:54:17.726 This would be like a mass term if I put in the [INAUDIBLE].. 00:54:21.460 --> 00:54:25.920 So that's like the analog, if I write it in terms of spinners, 00:54:25.920 --> 00:54:31.455 of p slash u for the right-hand electron is m u, 00:54:31.455 --> 00:54:33.330 but just written as an equation of motion. 00:54:37.180 --> 00:54:38.730 So they're using things like this 00:54:38.730 --> 00:54:40.530 to get rid of covariant derivatives acting 00:54:40.530 --> 00:54:43.580 on the right-handed electron and just replace it by an operator 00:54:43.580 --> 00:54:47.100 that a priori doesn't look like it's equivalent, which 00:54:47.100 --> 00:54:50.280 is H dagger L. 00:54:50.280 --> 00:54:55.650 Now, if you are stuck at tree level-- which is actually 00:54:55.650 --> 00:54:57.780 the language in which they constructed 00:54:57.780 --> 00:55:02.190 their paper is just to think that you 00:55:02.190 --> 00:55:04.890 apply this in a way that is valid 00:55:04.890 --> 00:55:10.870 when you look at lowest order, the operators at lowest order. 00:55:10.870 --> 00:55:13.110 Then it's pretty obvious, actually that this is OK. 00:55:16.798 --> 00:55:18.840 If the lines that are coming out of your operator 00:55:18.840 --> 00:55:21.600 are always external lines, then when 00:55:21.600 --> 00:55:25.032 you look at the final rule for those lines, 00:55:25.032 --> 00:55:26.240 you're putting them on shell. 00:55:33.960 --> 00:55:38.310 So for example, let's think that we had an operator that 00:55:38.310 --> 00:55:39.910 was not part of their list. 00:55:39.910 --> 00:55:45.570 So H dagger H e right id slash e right, 00:55:45.570 --> 00:55:48.300 you won't find that as one of the ones that's 00:55:48.300 --> 00:55:49.170 listed in the 80. 00:55:54.760 --> 00:55:56.760 You could think about the Feynman rule for that. 00:55:56.760 --> 00:55:58.260 So there's two Higgs particles, say, 00:55:58.260 --> 00:56:02.900 and two right-handed electrons. 00:56:02.900 --> 00:56:07.130 And if I just take the derivative out 00:56:07.130 --> 00:56:13.790 of here and then I get a p slash and if I use p slash as m u, 00:56:13.790 --> 00:56:17.000 then I get just u bar u. 00:56:17.000 --> 00:56:20.660 And so this is, of course, connecting us 00:56:20.660 --> 00:56:24.110 to the left-handed doublet, the left-handed guy. 00:56:24.110 --> 00:56:28.377 So this operator, in that sense of thinking 00:56:28.377 --> 00:56:30.710 about using the equation of motion, which are connecting 00:56:30.710 --> 00:56:33.350 left and right fields, putting them together 00:56:33.350 --> 00:56:36.170 in the standard model, is connecting this operator 00:56:36.170 --> 00:56:38.930 to the one that we just have H digger L. 00:56:38.930 --> 00:56:42.328 And you could just immediately get the right result just 00:56:42.328 --> 00:56:43.620 by starting with that operator. 00:56:43.620 --> 00:56:45.440 And that's the logic that they used. 00:56:45.440 --> 00:56:48.180 So you don't have to write this down because it's redundant. 00:56:48.180 --> 00:56:52.730 You get the same result from writing this down. 00:56:52.730 --> 00:56:56.180 So that seems fairly straightforward at tree level 00:56:56.180 --> 00:57:00.080 and at lowest order where all the fields are external. 00:57:00.080 --> 00:57:05.280 But I actually claim that it's true regardless of what I do. 00:57:05.280 --> 00:57:08.810 Whether I have loops, whether I have propagators, it's OK. 00:57:08.810 --> 00:57:11.250 We can do this. 00:57:11.250 --> 00:57:13.330 And that seems a lot less trivial. 00:57:19.604 --> 00:57:22.320 So they didn't know that when they wrote their paper. 00:57:22.320 --> 00:57:26.190 At least I don't think they did, but it is true. 00:57:33.927 --> 00:57:35.760 So that's what I actually want to talk about 00:57:35.760 --> 00:57:39.450 for the rest of today's lecture because it's 00:57:39.450 --> 00:57:41.050 a pretty powerful thing to do. 00:57:41.050 --> 00:57:43.290 If we are allowed to use these equations of motion 00:57:43.290 --> 00:57:46.350 to simplify the form of the higher dimensional operators 00:57:46.350 --> 00:57:48.930 in our effective field theory, it certainly 00:57:48.930 --> 00:57:52.290 helps us to reduce the number of operators. 00:58:08.410 --> 00:58:11.560 So one way of phrasing this is what's 00:58:11.560 --> 00:58:15.730 called the representation independence theorem, sometimes 00:58:15.730 --> 00:58:16.720 called that. 00:58:16.720 --> 00:58:18.220 I'll phrase it a few different ways, 00:58:18.220 --> 00:58:19.428 but I'll start with this way. 00:58:24.350 --> 00:58:27.400 So if we have some field, phi, we 00:58:27.400 --> 00:58:31.610 can set it equal to some combination of other fields. 00:58:31.610 --> 00:58:35.270 So phi and chi can be scalars, let's say. 00:58:35.270 --> 00:58:38.680 And this function, f, which could be fairly complicated, 00:58:38.680 --> 00:58:41.380 has to have at least one property that, when chi is 0, 00:58:41.380 --> 00:58:42.250 it's 1. 00:58:42.250 --> 00:58:46.630 So phi and chi both show up linearly 00:58:46.630 --> 00:58:50.283 in some term in the function if you'd 00:58:50.283 --> 00:58:52.450 like, if you want to think of it as a Taylor series, 00:58:52.450 --> 00:58:52.950 for example. 00:58:55.560 --> 00:59:01.150 So if that's true, the statement of this theorem 00:59:01.150 --> 00:59:07.960 is that calculations of observable 00:59:07.960 --> 00:59:14.210 that are done with phi or the Lagrangian that's made of phi-- 00:59:14.210 --> 00:59:16.780 and what that really means is that I've quantized phi-- 00:59:20.030 --> 00:59:32.140 will give the same results as those with a Lagrangian 00:59:32.140 --> 00:59:37.620 where I quantize chi where I construct 00:59:37.620 --> 00:59:40.692 that Lagrangian by just making this change of variables. 00:59:49.550 --> 00:59:52.200 And we're going to exploit that fact in order 00:59:52.200 --> 00:59:54.990 to argue that we're allowed to do what I just said. 00:59:57.758 --> 00:59:59.550 So we'll start slowly, then we'll build up. 00:59:59.550 --> 01:00:02.390 Then we'll state a more general theorem than this one. 01:00:02.390 --> 01:00:03.890 Then we'll show you how to prove it. 01:00:10.540 --> 01:00:14.100 So let's start by thinking of an example. 01:00:14.100 --> 01:00:16.315 Examples are always good. 01:00:16.315 --> 01:00:18.190 And we'll stick with our scalar field theory. 01:00:22.285 --> 01:00:23.910 I'm getting tired of writing factorial, 01:00:23.910 --> 01:00:27.150 so I'm choosing a little bit of different normalization here. 01:00:37.500 --> 01:00:39.660 So I'll consider an effective theory 01:00:39.660 --> 01:00:42.300 that has these terms at least to start. 01:00:42.300 --> 01:00:44.310 And eta has dimensions. 01:00:44.310 --> 01:00:45.280 And it's a small thing. 01:00:45.280 --> 01:00:47.760 It's like 1 over lambda nu. 01:00:54.730 --> 01:00:56.560 And the statement that we want to explore 01:00:56.560 --> 01:01:00.250 is the fact that we can use the equation of motion 01:01:00.250 --> 01:01:18.730 to effectively drop the last term by using 01:01:18.730 --> 01:01:21.200 the equation of motion. 01:01:21.200 --> 01:01:25.880 So how do we make use of this statement? 01:01:25.880 --> 01:01:29.620 Well, this tells us that we can make changes to variable 01:01:29.620 --> 01:01:31.360 and that we won't change anything. 01:01:31.360 --> 01:01:33.860 So we try to make a change of variable to make this go away. 01:01:36.268 --> 01:01:37.557 AUDIENCE: Wait, sorry. 01:01:37.557 --> 01:01:39.890 But that's not even the tree level operation [INAUDIBLE] 01:01:39.890 --> 01:01:40.395 right? 01:01:40.395 --> 01:01:42.920 You still have the lambda term. 01:01:42.920 --> 01:01:44.640 IAIN STEWART: Yeah, that's true. 01:01:44.640 --> 01:01:47.090 Yeah. 01:01:47.090 --> 01:01:47.610 Yeah. 01:01:47.610 --> 01:01:52.600 I should write the lambda term in there, 01:01:52.600 --> 01:01:54.220 plus, minus, whatever it is. 01:01:56.600 --> 01:01:57.100 Thanks. 01:02:03.880 --> 01:02:04.530 There's a 2. 01:02:10.650 --> 01:02:11.932 Thanks. 01:02:11.932 --> 01:02:14.140 All right, so how do we to get rid of this last term? 01:02:21.410 --> 01:02:25.402 Well, let's just use our theorem by making a field redefinition. 01:02:35.250 --> 01:02:38.910 So I claim, after making this field redefinition here, 01:02:38.910 --> 01:02:41.327 that something magical will happen, 01:02:41.327 --> 01:02:42.910 or something that we want will happen. 01:02:51.874 --> 01:02:55.598 So we're going to integrate by parts at will, which is often 01:02:55.598 --> 01:02:57.890 a convenient thing to do when you're making these field 01:02:57.890 --> 01:02:58.800 redefinitions. 01:03:01.340 --> 01:03:09.150 So the term that's a 1/2 del mu phi squared goes to /2 del mu 01:03:09.150 --> 01:03:12.070 phi squared from the first term. 01:03:12.070 --> 01:03:17.620 And then we pick up a term that's 01:03:17.620 --> 01:03:21.760 exactly in the form we want to kill this extra term here 01:03:21.760 --> 01:03:24.360 that was proportional to del squared. 01:03:24.360 --> 01:03:27.590 And then there would be order eta squared term. 01:03:31.113 --> 01:03:33.280 We have to consistently make that field redefinition 01:03:33.280 --> 01:03:36.490 everywhere, so do it at the mass term as well. 01:03:46.540 --> 01:03:50.230 Well, let me write out the eta term here, 01:03:50.230 --> 01:03:56.005 just a eta squared term just so you have all the terms. 01:03:56.005 --> 01:03:57.320 And we keep doing that. 01:03:57.320 --> 01:03:59.690 So we would do it also for the lambda phi 4 term. 01:03:59.690 --> 01:04:03.670 And we do it for this phi 6 term. 01:04:03.670 --> 01:04:07.920 And what you get out when we do that is that you can write down 01:04:07.920 --> 01:04:09.037 the Lagrangian. 01:04:12.240 --> 01:04:14.290 And you can group together the terms. 01:04:18.790 --> 01:04:20.620 And that operator is gone. 01:04:28.810 --> 01:04:31.120 So it got cancelled by this guy. 01:04:31.120 --> 01:04:32.620 There's also these other things that 01:04:32.620 --> 01:04:34.870 are induced, but really what that 01:04:34.870 --> 01:04:38.740 does is it gives you more terms that are phi 4 01:04:38.740 --> 01:04:40.230 and more terms that are phi 6. 01:04:40.230 --> 01:04:41.980 So you can really think of the other terms 01:04:41.980 --> 01:04:43.780 as just this guy here is just adjusting 01:04:43.780 --> 01:04:45.790 the constant of the phi 4. 01:04:45.790 --> 01:04:49.630 So if it's no longer lambda, it becomes a term that 01:04:49.630 --> 01:04:52.600 has this extra piece to it. 01:04:52.600 --> 01:04:55.413 So I'm going to give you more on this on the problem set. 01:04:55.413 --> 01:04:57.330 On the problem set, you'll work out explicitly 01:04:57.330 --> 01:04:58.773 with these relations are. 01:04:58.773 --> 01:05:00.190 And I'll also, on the problem set, 01:05:00.190 --> 01:05:01.660 let you think about what's going on here when 01:05:01.660 --> 01:05:02.827 you start considering loops. 01:05:06.950 --> 01:05:13.160 In this particular example, I'll ask you what's going on. 01:05:13.160 --> 01:05:15.160 I'll ask you to show what's going on 01:05:15.160 --> 01:05:16.910 and that it's still OK when we have loops. 01:05:21.120 --> 01:05:23.910 So rather than explore this example in more detail, which 01:05:23.910 --> 01:05:26.460 I'm asking you to do on your problem set, 01:05:26.460 --> 01:05:28.950 let's state a more precise definition 01:05:28.950 --> 01:05:31.445 of what we're doing here. 01:05:31.445 --> 01:05:33.820 So that's the idea, that I can make a field redefinition. 01:05:33.820 --> 01:05:35.195 When I make a field redefinition, 01:05:35.195 --> 01:05:37.740 it's always going to do something from the kinetic term 01:05:37.740 --> 01:05:38.850 because the kinetic term is there. 01:05:38.850 --> 01:05:40.410 If there's a higher order term that's 01:05:40.410 --> 01:05:42.760 proportional to that kinetic piece, 01:05:42.760 --> 01:05:44.850 then I can set up my field redefinition 01:05:44.850 --> 01:05:45.990 in order to cancel it off. 01:05:51.590 --> 01:05:55.002 So there's some things that are important here. 01:05:55.002 --> 01:05:56.710 We have to make a field redefinition that 01:05:56.710 --> 01:05:57.502 preserves symmetry. 01:05:57.502 --> 01:05:59.290 So if we make a field redefinition that 01:05:59.290 --> 01:06:01.900 breaks Lorentz invariance, you can't 01:06:01.900 --> 01:06:05.050 expect that you're going to have a Lorentz invariant 01:06:05.050 --> 01:06:07.390 description after that. 01:06:07.390 --> 01:06:11.730 And this statement of f of 0 being 1 01:06:11.730 --> 01:06:13.480 is the statement that you have to preserve 01:06:13.480 --> 01:06:17.740 the same one-particle states before and after the field 01:06:17.740 --> 01:06:19.628 redefinition. 01:06:19.628 --> 01:06:21.670 And such field redefinition definitions basically 01:06:21.670 --> 01:06:33.860 allow the classical equations of motion 01:06:33.860 --> 01:06:36.400 to be used to simplify the theory. 01:06:44.763 --> 01:06:46.680 I'll also put the proviso in that it should be 01:06:46.680 --> 01:06:47.970 a local quantum field theory. 01:07:02.670 --> 01:07:04.663 And there is no statement in this theorem 01:07:04.663 --> 01:07:06.580 that we have to stop without considering loops 01:07:06.580 --> 01:07:08.663 or without considering propagators that can really 01:07:08.663 --> 01:07:13.600 make this argument hold even beyond the level that we've 01:07:13.600 --> 01:07:21.800 showed it, but with loops and with propagators in as well. 01:07:21.800 --> 01:07:23.140 So there's some references. 01:07:23.140 --> 01:07:26.040 Again, I've posted the one that is 01:07:26.040 --> 01:07:29.940 closest to our discussion, which is this paper by Arzt. 01:07:37.850 --> 01:07:39.950 There's also a classic paper by Howard Georgi. 01:07:45.665 --> 01:07:47.790 The title of the paper is "On-Shell Effective Field 01:07:47.790 --> 01:07:48.290 Theory." 01:07:56.220 --> 01:07:58.720 But I'll mostly follow the notations in this paper 01:07:58.720 --> 01:08:00.737 by Arzt in our discussion here. 01:08:00.737 --> 01:08:02.320 We'll go a little further than he does 01:08:02.320 --> 01:08:04.780 and elaborate a little more, but it's basically 01:08:04.780 --> 01:08:05.830 the same notation. 01:08:09.500 --> 01:08:10.000 OK. 01:08:10.000 --> 01:08:12.130 So how do we prove something like that? 01:08:16.979 --> 01:08:19.319 Well, there are some lessons in this proof. 01:08:19.319 --> 01:08:22.260 So it is something that's worth going through. 01:08:25.660 --> 01:08:27.930 So the way that I set up my example up there, 01:08:27.930 --> 01:08:31.060 I was power counting in eta. 01:08:31.060 --> 01:08:34.899 Eta was 1 over lambda nu. 01:08:34.899 --> 01:08:40.189 So let me just write our effective theory organized 01:08:40.189 --> 01:08:41.189 as in a series in eta. 01:08:45.660 --> 01:08:49.950 And let's consider removing some operator that 01:08:49.950 --> 01:08:53.010 looks like we should be able to remove it 01:08:53.010 --> 01:08:56.776 by making a field redefinition. 01:08:56.776 --> 01:08:58.859 And I'll try to be a little bit generic about what 01:08:58.859 --> 01:09:00.067 the form of that operator is. 01:09:04.740 --> 01:09:07.020 So I'll say it's covariant derivative squared 01:09:07.020 --> 01:09:11.700 acting on a scalar field just, again, to make things simpler. 01:09:11.700 --> 01:09:13.350 But then multiply by any function 01:09:13.350 --> 01:09:16.850 of all the other fields in the problem, and that's what t is. 01:09:16.850 --> 01:09:18.510 So phi is a complex scalar. 01:09:22.720 --> 01:09:27.229 And t is any function that sort of 01:09:27.229 --> 01:09:29.779 meets the needs of our symmetries of the problem. 01:09:34.279 --> 01:09:40.010 And I'm just using this other phi 01:09:40.010 --> 01:09:43.020 as a shorthand for all the other fields that we might consider. 01:09:43.020 --> 01:09:52.364 So that could be fermions, other scalars, gauge fields. 01:09:55.952 --> 01:10:00.830 So let's say we want to get rid of an operator like that. 01:10:00.830 --> 01:10:02.600 Oh, I should also say that it's local. 01:10:08.710 --> 01:10:11.990 All right, well, let's write down 01:10:11.990 --> 01:10:20.690 the generating function for this theory, path integral 01:10:20.690 --> 01:10:25.090 over the fields exponential. 01:10:25.090 --> 01:10:28.071 I'll regulate the problem with dimensional regularization. 01:10:31.228 --> 01:10:31.980 It's convenient. 01:10:31.980 --> 01:10:33.360 It preserves the cemeteries. 01:10:33.360 --> 01:10:36.890 That's something we want to do. 01:10:36.890 --> 01:10:39.000 I write out some terms in the Lagrangian. 01:10:39.000 --> 01:10:39.870 We won't need L2. 01:10:39.870 --> 01:10:41.840 We'll stop at L1. 01:10:41.840 --> 01:10:43.880 But the idea, if we were to think about L2, 01:10:43.880 --> 01:10:45.680 would be similar. 01:10:45.680 --> 01:10:48.961 And let me write it as adding and subtracting something. 01:10:52.570 --> 01:10:58.460 So I'll subtract td squared phi and then add it back. 01:10:58.460 --> 01:11:01.390 So this is, if you like, you can think 01:11:01.390 --> 01:11:03.175 that L1 had a td squared phi. 01:11:03.175 --> 01:11:04.800 And what I'm doing here is removing it. 01:11:07.820 --> 01:11:11.130 So this is what we want. 01:11:11.130 --> 01:11:11.860 OK. 01:11:11.860 --> 01:11:14.860 So I'm just making it explicit, but still writing things 01:11:14.860 --> 01:11:16.750 in terms of L1. 01:11:16.750 --> 01:11:17.970 So there's that. 01:11:17.970 --> 01:11:21.430 And then there's the coupling to the source. 01:11:21.430 --> 01:11:26.860 And kind of in a generic notation for each field phi k, 01:11:26.860 --> 01:11:29.142 I have a source jk. 01:11:29.142 --> 01:11:31.910 And I truncate everything at order eta. 01:11:35.450 --> 01:11:39.102 OK, so that's my starting point. 01:11:39.102 --> 01:11:41.310 And then Green's functions are obtained by functional 01:11:41.310 --> 01:11:42.840 derivatives with respect to the j's. 01:12:23.560 --> 01:12:25.370 All right, so what we're going to do, 01:12:25.370 --> 01:12:28.040 we're going make a change of variable in the path integral. 01:12:32.803 --> 01:12:34.720 So of course, there's additional complications 01:12:34.720 --> 01:12:37.307 beyond what we were doing when we were just thinking about it 01:12:37.307 --> 01:12:38.140 at Lagrangian level. 01:12:42.790 --> 01:12:45.910 There's basically two additional complications. 01:12:45.910 --> 01:12:49.090 When we make a change a variable in the path integral, 01:12:49.090 --> 01:12:51.670 we're going to change the Lagrangian. 01:12:51.670 --> 01:12:54.080 We're also going to change the measure. 01:12:54.080 --> 01:12:55.808 So there could be a Jacobian. 01:12:55.808 --> 01:12:58.350 And we have to worry about what happens with the source term. 01:13:05.160 --> 01:13:09.120 So let me just think of it as a change of variable on the phi 01:13:09.120 --> 01:13:09.620 dagger. 01:13:15.940 --> 01:13:18.512 That's why I wanted to think about a complex scalar, 01:13:18.512 --> 01:13:20.470 so I could think about it as phi and phi dagger 01:13:20.470 --> 01:13:23.790 and just make the change of variable in the phi dagger. 01:13:23.790 --> 01:13:27.760 But doing a real scalar is just-- 01:13:27.760 --> 01:13:32.030 everything would go through just as well. 01:13:32.030 --> 01:13:37.030 So there's one term, which is the phi dagger term where 01:13:37.030 --> 01:13:38.530 there's a Jacobian factor. 01:13:38.530 --> 01:13:42.180 So here's that promised Jacobian for making 01:13:42.180 --> 01:13:43.180 that change of variable. 01:13:50.770 --> 01:13:53.560 Since the change of variable is order eta, 01:13:53.560 --> 01:13:54.540 it's not affecting L0. 01:13:59.750 --> 01:14:04.220 And I can write, in a kind of nice notation, 01:14:04.220 --> 01:14:07.300 the way that these eta t terms from this change 01:14:07.300 --> 01:14:08.410 of variable show up. 01:14:08.410 --> 01:14:13.780 I can write them as taking all the terms in L0, 01:14:13.780 --> 01:14:16.120 looking for a phi dagger. 01:14:16.120 --> 01:14:17.320 That's the derivative. 01:14:17.320 --> 01:14:19.110 And then I replace it by an eta t. 01:14:21.850 --> 01:14:24.590 Now, some places in L0, there won't be just a straight phi 01:14:24.590 --> 01:14:25.090 dagger. 01:14:25.090 --> 01:14:29.210 There might be a del mu phi dagger, right? 01:14:29.210 --> 01:14:35.090 So I can take that into account as well 01:14:35.090 --> 01:14:37.090 by taking the functional derivative with respect 01:14:37.090 --> 01:14:38.790 to del mu phi dagger. 01:14:38.790 --> 01:14:42.550 So any del mu phi dagger term, if I integrate by parts, 01:14:42.550 --> 01:14:44.560 and then take the derivative, you 01:14:44.560 --> 01:14:47.500 can see why this is the right form. 01:14:47.500 --> 01:14:50.200 Integration by parts is what gives the minus sign. 01:14:50.200 --> 01:14:53.920 OK, so this finds all the phi daggers in my L0 01:14:53.920 --> 01:14:58.825 and sticks in an eta t to order eta, which 01:14:58.825 --> 01:14:59.950 is the order we're working. 01:15:08.370 --> 01:15:12.180 So anytime we had in eta, which is these terms here, 01:15:12.180 --> 01:15:15.288 then we would induce something ordering eta square. 01:15:15.288 --> 01:15:17.580 And we're dropping those terms, so that's not something 01:15:17.580 --> 01:15:18.538 we have to worry about. 01:15:22.083 --> 01:15:23.250 And then there's the source. 01:15:23.250 --> 01:15:26.070 And one of the sources is 4 phi dagger. 01:15:26.070 --> 01:15:30.260 So for that particular source, j phi dagger, 01:15:30.260 --> 01:15:34.635 we induce a term that's j phi eta t 01:15:34.635 --> 01:15:36.720 and then plus order eta squared. 01:15:49.750 --> 01:15:52.500 So as I said, there's three types of changes. 01:15:52.500 --> 01:15:55.170 There's a change to the Lagrangian, like in the example 01:15:55.170 --> 01:15:57.360 we did before. 01:15:57.360 --> 01:16:00.906 There's a change to the Jacobian through the Jacobian. 01:16:00.906 --> 01:16:03.285 And then there's a change through the source. 01:16:06.377 --> 01:16:08.460 We have to worry about whether any of these things 01:16:08.460 --> 01:16:08.980 will matter. 01:16:12.840 --> 01:16:16.810 And basically, what the claim is and where we're going 01:16:16.810 --> 01:16:22.560 is considering the Lagrangian is actually enough, 01:16:22.560 --> 01:16:25.110 so that 2 and 3 are something that we can kind of deal 01:16:25.110 --> 01:16:26.180 with generically. 01:16:26.180 --> 01:16:28.680 And then we only have to check that changing Lagrangian does 01:16:28.680 --> 01:16:31.330 what we want. 01:16:31.330 --> 01:16:34.420 So the claim is and what we'll show, 01:16:34.420 --> 01:16:38.100 without changing the s matrix, we 01:16:38.100 --> 01:16:46.858 can remove considering 2 and 3 rather generically. 01:16:52.128 --> 01:16:53.670 So we don't have to worry about them. 01:17:10.980 --> 01:17:20.010 So some of this we'll do next time, but let's 01:17:20.010 --> 01:17:20.860 just start today. 01:17:20.860 --> 01:17:24.990 So let's first look at del L, kind 01:17:24.990 --> 01:17:29.610 of analogous to what we were doing in our example before. 01:17:33.070 --> 01:17:41.860 So what we need is a change of variable, 01:17:41.860 --> 01:17:44.430 which is this one up here. 01:17:44.430 --> 01:17:47.310 And one restriction on del L, which 01:17:47.310 --> 01:17:48.960 I mentioned as part of the assumption, 01:17:48.960 --> 01:17:53.910 was that this change of variable should transform 01:17:53.910 --> 01:17:58.830 in the same way as the original phi dagger, so same Lorentz 01:17:58.830 --> 01:18:03.930 index structure, same gauge index structure, et cetera. 01:18:03.930 --> 01:18:09.090 So that's kind of an assumption that we said. 01:18:09.090 --> 01:18:10.680 In order to respect the symmetries 01:18:10.680 --> 01:18:14.010 and not mess them up with our field redefinition, 01:18:14.010 --> 01:18:14.640 we assume that. 01:18:18.967 --> 01:18:20.300 So let's see what happens to L0. 01:18:32.565 --> 01:18:33.190 So there is L0. 01:18:36.597 --> 01:18:38.180 Maybe it has some other terms as well, 01:18:38.180 --> 01:18:41.797 but let's just consider these pieces 01:18:41.797 --> 01:18:43.630 and ask what happens with them since they're 01:18:43.630 --> 01:18:44.440 the relevant ones. 01:18:50.400 --> 01:18:54.080 So when I switch to prime fields, I get that back again. 01:18:54.080 --> 01:18:58.380 I get m squared phi dagger prime phi prime. 01:18:58.380 --> 01:19:03.165 And then I get some new terms, which 01:19:03.165 --> 01:19:06.736 I integrate the covariant derivative by parts, 01:19:06.736 --> 01:19:09.270 which I'm always able to do. 01:19:09.270 --> 01:19:12.070 It's nice what gauge symmetry allows you to do. 01:19:18.860 --> 01:19:21.090 Then I get a term like that, and I've 01:19:21.090 --> 01:19:28.790 set things up so that this term here does exactly what we need. 01:19:28.790 --> 01:19:31.490 I go back over here. 01:19:31.490 --> 01:19:33.128 Those terms are cancelling. 01:19:49.560 --> 01:19:51.150 OK, so that's good. 01:19:59.120 --> 01:20:01.460 Now, you may be worried about all the other terms that 01:20:01.460 --> 01:20:02.060 get induced. 01:20:02.060 --> 01:20:04.898 You've removed something, but you've induced a lot of stuff. 01:20:04.898 --> 01:20:06.440 But the point of the effective theory 01:20:06.440 --> 01:20:08.990 is that you already wrote down every possible operator that 01:20:08.990 --> 01:20:11.150 was consistent with the symmetries. 01:20:11.150 --> 01:20:13.600 So even if you induce a bunch of other terms, 01:20:13.600 --> 01:20:15.650 they should be terms you already have. 01:20:15.650 --> 01:20:17.120 So all you're doing with inducing 01:20:17.120 --> 01:20:19.490 those other terms is shifting the coefficients of the theory 01:20:19.490 --> 01:20:19.990 around. 01:20:23.450 --> 01:20:29.270 So L1 was already complete in the sense 01:20:29.270 --> 01:20:33.020 of having all the terms allowed by the symmetry. 01:20:33.020 --> 01:20:36.730 And we've respected the symmetry in the way we've 01:20:36.730 --> 01:20:38.371 made this field redefinition. 01:20:52.140 --> 01:20:53.623 So any terms that I didn't write, 01:20:53.623 --> 01:20:55.290 which are all these terms in the da, da, 01:20:55.290 --> 01:21:03.660 da prime are already operators that 01:21:03.660 --> 01:21:06.510 are present in the other dots that I didn't write. 01:21:09.070 --> 01:21:10.520 So there were some dots here. 01:21:10.520 --> 01:21:11.640 There was some dots there. 01:21:11.640 --> 01:21:14.107 Even if I include operators in those dots, 01:21:14.107 --> 01:21:16.690 the operators in these dots are already present in those dots. 01:21:19.900 --> 01:21:23.940 And so all I'm doing really is shifting couplings. 01:21:30.060 --> 01:21:32.150 OK. 01:21:32.150 --> 01:21:35.410 And that's same effective theory just with a new name 01:21:35.410 --> 01:21:37.160 for the coefficients, but we haven't fixed 01:21:37.160 --> 01:21:38.330 the coefficients yet anyway. 01:21:38.330 --> 01:21:40.287 We're setting up our effect effective theory. 01:21:40.287 --> 01:21:41.870 So whether we fix the new coefficients 01:21:41.870 --> 01:21:45.190 or the old coefficients, it's perfectly fine. 01:21:56.830 --> 01:21:59.550 So number two is the Jacobian. 01:21:59.550 --> 01:22:01.690 And I won't go all the way through this, 01:22:01.690 --> 01:22:05.042 but what are we going to do there? 01:22:05.042 --> 01:22:06.750 We're going to use the same kind of trick 01:22:06.750 --> 01:22:09.270 that we did in gauge theory. 01:22:09.270 --> 01:22:12.000 We're going to write the Jacobian as a Lagrangian 01:22:12.000 --> 01:22:15.000 involving ghosts. 01:22:15.000 --> 01:22:17.610 So remember, when you talked about Faddeev-Popov 01:22:17.610 --> 01:22:20.610 in some field theory course prior to this one, 01:22:20.610 --> 01:22:25.590 you saw that you could write a determinant as an exponential 01:22:25.590 --> 01:22:27.105 involving some ghost fields. 01:22:30.150 --> 01:22:37.072 And the way that it worked is that, up to a sign, 01:22:37.072 --> 01:22:39.030 whatever was sitting here ended up just sitting 01:22:39.030 --> 01:22:40.080 between the ghost fields. 01:22:55.140 --> 01:23:01.931 So this is the ghost of Faddeev-Popov procedure. 01:23:04.438 --> 01:23:06.230 And we're going to just use the same thing, 01:23:06.230 --> 01:23:09.900 although [INAUDIBLE] we're talking about scalar field 01:23:09.900 --> 01:23:12.270 theory. 01:23:12.270 --> 01:23:23.700 So for us, we take this guy, which is the Jacobian, 01:23:23.700 --> 01:23:25.945 I'll take that derivative. 01:23:33.560 --> 01:23:35.540 And if you take this thing, which 01:23:35.540 --> 01:23:38.120 is sitting in functional determinant, 01:23:38.120 --> 01:23:41.240 and you turned it into ghosts, then what you get 01:23:41.240 --> 01:23:51.680 is a terminal Lagrangian that looks like that. 01:24:00.860 --> 01:24:04.850 So that looks like it could do something. 01:24:04.850 --> 01:24:06.750 And I think I'll take this up next time. 01:24:06.750 --> 01:24:09.350 It turns out that this term is actually 01:24:09.350 --> 01:24:12.890 a ghost that has mass that's of order the high scale, lambda 01:24:12.890 --> 01:24:14.830 nu. 01:24:14.830 --> 01:24:17.300 And so, again, because of our logic of the effective theory 01:24:17.300 --> 01:24:20.058 only being valid below that high scale, 01:24:20.058 --> 01:24:22.100 we can just effectively integrate out this ghost. 01:24:22.100 --> 01:24:25.190 And then it shifts coefficients again, 01:24:25.190 --> 01:24:27.620 but I'll continue with that next time. 01:24:27.620 --> 01:24:31.760 So we have to figure out what to do with this ghost Lagrangian. 01:24:31.760 --> 01:24:34.590 And we'll deal with that next time. 01:24:34.590 --> 01:24:37.520 So any questions about our halfway done proof? 01:24:42.833 --> 01:24:44.748 No? 01:24:44.748 --> 01:24:45.248 OK. 01:24:48.630 --> 01:24:50.090 So this is a pretty powerful thing 01:24:50.090 --> 01:24:52.070 that you can use in effective field theory. 01:24:52.070 --> 01:24:54.300 And it also keeps your eye on the ball 01:24:54.300 --> 01:24:57.247 because it tells you to think about physics. 01:24:57.247 --> 01:24:58.830 Because this leaves physics invariant. 01:24:58.830 --> 01:25:00.800 It doesn't leave something like if you-- 01:25:00.800 --> 01:25:02.858 say you had a theory that had some long distance 01:25:02.858 --> 01:25:03.650 degrees of freedom. 01:25:03.650 --> 01:25:05.630 And it had some short distance potential. 01:25:05.630 --> 01:25:08.060 The short distance potential could be changed 01:25:08.060 --> 01:25:09.140 by a field redefinition. 01:25:09.140 --> 01:25:11.660 It might not be a physical thing. 01:25:11.660 --> 01:25:14.900 It keeps your eye on the ball as to what is physical 01:25:14.900 --> 01:25:17.470 and how to talk about things.