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. 00:00:22.380 --> 00:00:24.880 IAN STEWART: So last time, we were midway through this proof 00:00:24.880 --> 00:00:28.930 that the equations of motion, i.e., field redefinitions, 00:00:28.930 --> 00:00:32.470 can be used to simplify the theory. 00:00:32.470 --> 00:00:35.320 And I stated last time that really, 00:00:35.320 --> 00:00:38.530 all we have to worry about is the change to the Lagrangian. 00:00:38.530 --> 00:00:40.270 But when we looked at the path integral 00:00:40.270 --> 00:00:42.640 to do things properly and make a change variable in the path 00:00:42.640 --> 00:00:44.550 integral, it wasn't just the Lagrangian that changed. 00:00:44.550 --> 00:00:47.050 The Lagrangian changed, and we could set up our field 00:00:47.050 --> 00:00:50.200 redefinition to do what we want, but there were also changes 00:00:50.200 --> 00:00:51.970 to the Jacobian and the source, and that's 00:00:51.970 --> 00:00:56.120 what we had started to talk about last time. 00:00:56.120 --> 00:01:00.460 So when I look at the change the Jacobian, which is this thing, 00:01:00.460 --> 00:01:03.750 we could write that as a ghost Lagrangian. 00:01:03.750 --> 00:01:07.030 And the argument for why we don't need to worry about this 00:01:07.030 --> 00:01:07.795 is as follows. 00:01:13.880 --> 00:01:15.940 So the effective field theory is going 00:01:15.940 --> 00:01:19.180 to be valid for small momentum. 00:01:19.180 --> 00:01:24.280 There's an expansion and there's some scale introduced 00:01:24.280 --> 00:01:27.310 by the higher dimensional operators, lamda new, 00:01:27.310 --> 00:01:31.120 and we're looking at low momentum relative to that. 00:01:31.120 --> 00:01:34.150 In the particular case that we're dealing with, 00:01:34.150 --> 00:01:37.100 this parameter eta had dimensions, 00:01:37.100 --> 00:01:40.570 and so lambda nu was 1 over root eta. 00:01:40.570 --> 00:01:43.312 And we put factors of eta in front of our higher dimension 00:01:43.312 --> 00:01:44.770 operators, so that's like putting 1 00:01:44.770 --> 00:01:47.230 over lambda nus in front of those operators. 00:01:49.695 --> 00:01:51.070 And the reason that we don't have 00:01:51.070 --> 00:01:52.960 to worry so much about this ghost Lagrangian 00:01:52.960 --> 00:01:55.240 is that these ghosts are going to get mass 00:01:55.240 --> 00:01:57.220 that's of the size lambda nu. 00:02:20.200 --> 00:02:23.577 So what that means is that the ghost, along with perhaps 00:02:23.577 --> 00:02:25.660 other particles that are up at the scale lamda new 00:02:25.660 --> 00:02:27.860 are things that we don't have to worry about, 00:02:27.860 --> 00:02:30.010 and we can effectively integrate out 00:02:30.010 --> 00:02:32.440 the ghost from the theory in the same way 00:02:32.440 --> 00:02:34.720 that we would think about removing some particles that 00:02:34.720 --> 00:02:37.120 had masses of order lambda nu. 00:02:37.120 --> 00:02:39.100 So I'll show you why they get masses 00:02:39.100 --> 00:02:40.510 of order lambda nu in a second. 00:03:09.340 --> 00:03:12.310 So let's do that by way of picking a particular example. 00:03:12.310 --> 00:03:14.880 So far, we've kept things fairly general without specifying 00:03:14.880 --> 00:03:18.060 what this set of fields it's in this thing 00:03:18.060 --> 00:03:19.980 that we called T was. 00:03:19.980 --> 00:03:22.740 We just left it as an arbitrary thing. 00:03:22.740 --> 00:03:36.640 Let's pick a particular T. The argument here is actually 00:03:36.640 --> 00:03:39.340 fairly general, but I think it'd be easier to see it 00:03:39.340 --> 00:03:41.390 for particular example. 00:03:41.390 --> 00:03:47.240 So if we pick this to be T, we have 00:03:47.240 --> 00:03:51.146 this term which has no relation to T. That term is important. 00:03:54.428 --> 00:03:55.970 And then we have the terms from this. 00:04:04.512 --> 00:04:05.970 And so we'd have a ghost Lagrangian 00:04:05.970 --> 00:04:08.830 that would look like that. 00:04:08.830 --> 00:04:11.220 So this here is going to be the mass term, 00:04:11.220 --> 00:04:14.520 and so far, it doesn't look like it has the right dimensions. 00:04:14.520 --> 00:04:18.930 And that's because we've got our kinetic term, if you like, 00:04:18.930 --> 00:04:21.310 with the wrong dimensions. 00:04:21.310 --> 00:04:26.550 So if we want ghost fields of standard dimensions, 00:04:26.550 --> 00:04:29.950 or canonically normalized kinetic term in this case, 00:04:29.950 --> 00:04:34.230 we would take c and rescale it to c over root eta. 00:04:34.230 --> 00:04:38.850 And if we do that, then this does become a lambda nu. 00:04:49.080 --> 00:04:51.480 So that removes the eta in the interaction term here. 00:04:51.480 --> 00:04:54.340 It removes it from this kinetic term that happened to be there, 00:04:54.340 --> 00:04:56.970 and then the only place that the eta's showing up now 00:04:56.970 --> 00:04:59.520 in this case is in this term. 00:04:59.520 --> 00:05:02.550 But now you see explicitly in terms of these ghost fields 00:05:02.550 --> 00:05:04.688 that they have mass lambda nu. 00:05:14.936 --> 00:05:17.340 OK, and that's just what I was claiming. 00:05:20.360 --> 00:05:23.313 So the important point here was that you did need this 1. 00:05:23.313 --> 00:05:24.980 This 1 was playing a very important role 00:05:24.980 --> 00:05:25.688 in this argument. 00:05:25.688 --> 00:05:27.890 If the 1 wasn't there, this wouldn't work. 00:05:33.140 --> 00:05:34.850 And that 1 was related to something 00:05:34.850 --> 00:05:38.850 that was part of one of our starting assumptions, 00:05:38.850 --> 00:05:40.680 that when we made the field redefinition, 00:05:40.680 --> 00:05:44.230 that the one-particle states would stay the same. 00:05:44.230 --> 00:05:52.310 So if you look back at what our field redefinition was and you 00:05:52.310 --> 00:06:00.517 trace back where that 1 came from, 00:06:00.517 --> 00:06:02.600 we had a field redefinition that was of this form, 00:06:02.600 --> 00:06:04.910 and this guy was a function of other fields. 00:06:04.910 --> 00:06:07.400 But we always had this term, and that term 00:06:07.400 --> 00:06:09.290 was related to the presence of this 1, 00:06:09.290 --> 00:06:12.650 and that was important for this argument, OK. 00:06:34.350 --> 00:06:39.080 So there has to be a term that's linear in the new field. 00:06:39.080 --> 00:06:43.730 You want the same one-particle states. 00:06:43.730 --> 00:06:46.580 So these ghosts, as you know about ghosts, 00:06:46.580 --> 00:06:47.780 always appear in loops. 00:06:51.490 --> 00:06:53.240 And so if you want to think about removing 00:06:53.240 --> 00:06:55.460 this massive particle, it's like a massive particle 00:06:55.460 --> 00:06:56.600 that occurs in some loops. 00:07:00.200 --> 00:07:03.620 It's really just like that. 00:07:03.620 --> 00:07:05.440 So it's just like a heavy particle 00:07:05.440 --> 00:07:07.190 that you would remove from the theory that 00:07:07.190 --> 00:07:08.315 would only appear in loops. 00:07:12.110 --> 00:07:14.900 And we're going to discuss exactly how that works 00:07:14.900 --> 00:07:18.320 and how to remove heavy particles that appear in loops 00:07:18.320 --> 00:07:22.280 in detail a little bit later, so for now, 00:07:22.280 --> 00:07:26.007 that's enough detail for number two. 00:07:26.007 --> 00:07:27.590 So are there any questions about that? 00:07:32.570 --> 00:07:35.740 OK, so the change to the Jacobian 00:07:35.740 --> 00:07:38.170 is effectively a massive ghost that we could integrate out 00:07:38.170 --> 00:07:38.753 of the theory. 00:07:38.753 --> 00:07:40.750 When we integrate it out of the theory, 00:07:40.750 --> 00:07:45.010 it would shift the values of the couplings in the Lagrangian, 00:07:45.010 --> 00:07:48.190 but it doesn't have any impact that 00:07:48.190 --> 00:07:49.920 can't be absorbed in other operators 00:07:49.920 --> 00:07:50.920 in the effective theory. 00:07:57.633 --> 00:08:00.175 So then the final thing we have to worry about is the source. 00:08:05.900 --> 00:08:10.510 So there is this term with j phi dagger, and when we take-- 00:08:10.510 --> 00:08:12.280 there was an extra term with j phi dagger 00:08:12.280 --> 00:08:13.687 that we had last time. 00:08:13.687 --> 00:08:15.520 I'm going to remind you what it looked like. 00:08:18.790 --> 00:08:28.520 So in our path integral, we had an exponential 00:08:28.520 --> 00:08:30.610 of a bunch of stuff, but then there 00:08:30.610 --> 00:08:37.820 was one final term that was induced that involved j phi 00:08:37.820 --> 00:08:41.200 dagger from the field redefinition. 00:08:41.200 --> 00:08:43.840 So when you take derivatives with respect to j phi dagger, 00:08:43.840 --> 00:08:46.270 there's sort of the usual term that we want to source, 00:08:46.270 --> 00:08:47.833 and then there's this new term. 00:08:47.833 --> 00:08:49.000 So what about that new term? 00:08:52.000 --> 00:08:54.950 That's the final thing we have to worry about. 00:08:54.950 --> 00:08:57.220 And this is where it's important that we're actually 00:08:57.220 --> 00:09:00.070 considering observables. 00:09:00.070 --> 00:09:03.850 So we need to consider observables. 00:09:03.850 --> 00:09:08.328 So let's start by considering some Green's functions 00:09:08.328 --> 00:09:09.370 or time-ordered products. 00:09:23.210 --> 00:09:25.150 So I'm just taking a string of fields. 00:09:25.150 --> 00:09:28.120 I could have some other fields here as well. 00:09:28.120 --> 00:09:32.560 I'm going to make my life a little easier by taking the phi 00:09:32.560 --> 00:09:37.980 to be real, and that's just for notational simplicity. 00:09:41.490 --> 00:09:43.920 I could write phi dagger and phis, but. 00:09:56.010 --> 00:09:59.767 So when we make the field redefinition, what happens 00:09:59.767 --> 00:10:00.975 to that time-ordered product? 00:10:08.020 --> 00:10:10.697 So this is one way of thinking about these extra terms 00:10:10.697 --> 00:10:11.280 in the source. 00:10:16.672 --> 00:10:18.630 So you take functional derivatives with respect 00:10:18.630 --> 00:10:21.457 to the other piece that involves j phi, you get phi. 00:10:21.457 --> 00:10:23.040 So if you take [INAUDIBLE] that piece, 00:10:23.040 --> 00:10:24.082 you get the extra eta Ts. 00:10:26.760 --> 00:10:29.880 Or you can just think about making the field redefinition 00:10:29.880 --> 00:10:34.350 directly in this matrix element, and you'd 00:10:34.350 --> 00:10:37.488 have a matrix element that looks like that. 00:10:37.488 --> 00:10:39.780 So it's not at all obvious that these extra pieces that 00:10:39.780 --> 00:10:42.900 involve fields can't change the value of this Green's function, 00:10:42.900 --> 00:10:45.880 and in fact, they will change the value of that Green's 00:10:45.880 --> 00:10:47.753 function. 00:10:47.753 --> 00:10:49.170 But the important thing is that we 00:10:49.170 --> 00:10:54.790 have to consider observables and not just Green's functions. 00:10:54.790 --> 00:10:56.790 So when we're going to think about observables, 00:10:56.790 --> 00:11:02.940 we should think about the LSZ formula, which connects Green's 00:11:02.940 --> 00:11:05.010 functions to S-matrix elements. 00:11:07.750 --> 00:11:09.180 So let me remind you about that. 00:11:18.300 --> 00:11:20.850 And I'll remind you what it looks like. 00:11:20.850 --> 00:11:22.560 First one is fixed set of fields, 00:11:22.560 --> 00:11:25.870 which are scalar fields, but we could put fermions in it 00:11:25.870 --> 00:11:28.800 as well. 00:11:28.800 --> 00:11:32.180 So what this says is, if I look at this-- 00:11:32.180 --> 00:11:34.160 so it's an integral over the spacetime. 00:11:34.160 --> 00:11:35.930 I'm sticking in particular momentum 00:11:35.930 --> 00:11:37.870 for the particular fields. 00:11:37.870 --> 00:11:41.570 And if I look at it, if I look at the leading term, 00:11:41.570 --> 00:11:51.620 the leading pole term as the particles are taken on shell, 00:11:51.620 --> 00:11:54.680 then that's an observable known as the S-matrix. 00:12:15.040 --> 00:12:17.200 So some number of these particles are incoming. 00:12:17.200 --> 00:12:19.150 Some number of them are outgoing. 00:12:19.150 --> 00:12:22.360 That affects the sign that I put in this plus minus. 00:12:22.360 --> 00:12:24.955 I'm not trying to be too detailed about which ones are 00:12:24.955 --> 00:12:27.080 outgoing, which ones are incoming, but some of them 00:12:27.080 --> 00:12:30.280 are incoming, some of them are outgoing. 00:12:30.280 --> 00:12:37.143 And for each one, I want to strip off the leading pole. 00:12:37.143 --> 00:12:38.560 I want to look at the coefficient, 00:12:38.560 --> 00:12:40.390 and it has to have a pole, so this is 00:12:40.390 --> 00:12:42.280 a product over all particles. 00:12:42.280 --> 00:12:46.390 We really need to pull in all the different cases and all 00:12:46.390 --> 00:12:51.460 for all the external particles, OK. 00:12:51.460 --> 00:12:54.100 So if I wanted to make this an equality, 00:12:54.100 --> 00:12:56.380 I'd say plus dot dot dot. 00:12:56.380 --> 00:13:00.310 There's other terms, and the thing that's observable 00:13:00.310 --> 00:13:03.130 is the coefficient of these poles, not 00:13:03.130 --> 00:13:05.115 the Z but this thing. 00:13:09.480 --> 00:13:12.870 So if I make changes to my theory and they effect this 00:13:12.870 --> 00:13:15.607 thing but get cancelled by this thing, 00:13:15.607 --> 00:13:17.690 as long as they don't affect this thing, we're OK. 00:13:21.610 --> 00:13:24.850 And the claim is that we make this change to the source 00:13:24.850 --> 00:13:28.467 and it won't affect the matrix on the S, the S-matrix. 00:13:58.810 --> 00:14:01.440 So again, we can do some examples, 00:14:01.440 --> 00:14:05.520 and I think the examples are fairly quickly convincing 00:14:05.520 --> 00:14:07.950 that this is the case. 00:14:07.950 --> 00:14:10.980 So we'll do three different examples, 00:14:10.980 --> 00:14:14.160 first a rather trivial one. 00:14:14.160 --> 00:14:15.630 What if T was just by itself? 00:14:18.250 --> 00:14:23.910 So T is just phi, so this is just 1 plus eta phi. 00:14:34.760 --> 00:14:37.535 So if you think back, when we have T, 00:14:37.535 --> 00:14:39.410 this was the form of the terminal Lagrangian, 00:14:39.410 --> 00:14:43.640 so this is just eta phi del squared phi, 00:14:43.640 --> 00:14:47.900 so it's just changing the kinetic term for phi. 00:14:47.900 --> 00:14:54.590 If we look at our matrix on it, we're 00:14:54.590 --> 00:15:01.050 just getting a factor of 1 plus eta for each of the fields. 00:15:01.050 --> 00:15:04.160 So let's say we had 4 of them just so we don't have to-- 00:15:07.760 --> 00:15:11.380 so say I had 4 of them. 00:15:11.380 --> 00:15:13.642 If I had n of them, it would just be to the nth power, 00:15:13.642 --> 00:15:16.100 but if I have power of them, they'd be to the fourth power, 00:15:16.100 --> 00:15:18.170 and I'd just get this extra prefactor. 00:15:18.170 --> 00:15:20.870 And it looks like it's changed G. 00:15:20.870 --> 00:15:23.390 And indeed, it has changed the left-hand side of this 00:15:23.390 --> 00:15:27.920 equation, but when we calculate the z factor and we canonically 00:15:27.920 --> 00:15:29.700 normalize-- 00:15:29.700 --> 00:15:31.650 if you want to think about it that way-- 00:15:31.650 --> 00:15:35.190 then we would exactly cancel off these 1 plus etas. 00:15:35.190 --> 00:15:38.420 So the root Z here is going to be 1 plus eta as well. 00:15:48.210 --> 00:15:49.700 So when we look at the-- 00:15:49.700 --> 00:15:52.010 this is just the residue of the free propagator. 00:15:52.010 --> 00:15:57.255 The residue gets changed when you change the kinetic term, 00:15:57.255 --> 00:16:00.597 so that doesn't do anything to the S-matrix. 00:16:04.107 --> 00:16:05.690 So that's one way we can be protected. 00:16:05.690 --> 00:16:06.898 We change the left-hand side. 00:16:06.898 --> 00:16:10.730 It's compensated by a change to the Z and leaves S invariant. 00:16:10.730 --> 00:16:14.390 Let's do another example, a little more non-trivial. 00:16:21.080 --> 00:16:29.890 So let's just take some cubic term in our field redefinition. 00:16:36.950 --> 00:16:39.500 So this is like having a 5 cubed del squared phi term. 00:16:46.400 --> 00:16:49.960 So this extra term will give rise 00:16:49.960 --> 00:16:52.540 to extra terms in the primary product. 00:16:52.540 --> 00:16:54.700 And let me just write one of them, 00:16:54.700 --> 00:16:59.390 again thinking of it as a 4-point function. 00:16:59.390 --> 00:17:01.750 So let's just imagine I look at the change that comes 00:17:01.750 --> 00:17:04.510 from changing the 4th guy. 00:17:07.839 --> 00:17:12.530 So there's other terms from changing 5x1, 5x2 and 5x3, 00:17:12.530 --> 00:17:16.119 but if I work to order eta, then I'd change one of them 00:17:16.119 --> 00:17:21.339 at a time, and we're only working to order eta here. 00:17:21.339 --> 00:17:25.180 So the claim is actually that this matrix element has 00:17:25.180 --> 00:17:28.150 no effect on this structure. 00:17:28.150 --> 00:17:31.335 It can affect the dots, but it's not 00:17:31.335 --> 00:17:32.710 going to affect the leading term, 00:17:32.710 --> 00:17:35.590 and that's because having a 5 cubed 00:17:35.590 --> 00:17:38.120 means that you're not having a one-particle state. 00:17:38.120 --> 00:17:43.000 So if you try to draw it as a [INAUDIBLE] diagram, 00:17:43.000 --> 00:17:46.810 in the position space, you'd label these external points 00:17:46.810 --> 00:17:49.300 and you inject momenta there. 00:17:49.300 --> 00:17:51.920 And then we label point 4, but there's three fields there, 00:17:51.920 --> 00:17:55.830 so maybe we have to tie it up like this or something. 00:17:55.830 --> 00:17:58.810 And when you look at asymptotically what's 00:17:58.810 --> 00:18:01.000 going to happen from [INAUDIBLE] a 5 cubed, 00:18:01.000 --> 00:18:03.640 you do not get a single-particle pole from a 5 cubed. 00:18:07.800 --> 00:18:11.415 So this guy is less singular. 00:18:19.750 --> 00:18:27.420 There's no single-particle pole, and hence, 00:18:27.420 --> 00:18:28.980 gives no contribution to scatter. 00:18:37.720 --> 00:18:38.563 Yeah. 00:18:38.563 --> 00:18:40.105 AUDIENCE: As far as the theorem goes, 00:18:40.105 --> 00:18:45.010 this step that you're showing, it proves my hypothesis, right? 00:18:45.010 --> 00:18:47.315 IAN STEWART: In what sense? 00:18:47.315 --> 00:18:48.940 AUDIENCE: The hypothesis of the theorem 00:18:48.940 --> 00:18:51.315 is that [INAUDIBLE] does not change a one-particle state. 00:18:51.315 --> 00:18:52.398 IAN STEWART: That's right. 00:18:52.398 --> 00:18:53.650 That was an assumption. 00:18:53.650 --> 00:18:57.220 AUDIENCE: I mean, is that the same as what you're-- 00:18:57.220 --> 00:18:59.297 IAN STEWART: It's the same, yeah. 00:18:59.297 --> 00:19:03.083 It's effectively the same, though it's 00:19:03.083 --> 00:19:05.500 being more careful about what the statement of that means, 00:19:05.500 --> 00:19:09.400 right, because I mean, you're changing the residue 00:19:09.400 --> 00:19:13.360 but you're not changing the S-matrix. 00:19:13.360 --> 00:19:14.162 AUDIENCE: Right. 00:19:14.162 --> 00:19:16.120 This is what I was thinking when you said that. 00:19:16.120 --> 00:19:18.070 IAN STEWART: Yeah, all right. 00:19:25.470 --> 00:19:27.390 So really, this statement of-- 00:19:27.390 --> 00:19:30.708 yeah, I could have been a little more careful. 00:19:30.708 --> 00:19:33.000 So instead of saying that the hypothesis of the theorem 00:19:33.000 --> 00:19:35.580 is that it wouldn't change one-particle states, what 00:19:35.580 --> 00:19:38.550 I should have really said is that there's a linear term 00:19:38.550 --> 00:19:40.880 in the field redefinition. 00:19:40.880 --> 00:19:46.160 Yeah, which now I'm showing you is equivalent to not changing 00:19:46.160 --> 00:19:48.740 the one-particle states. 00:19:48.740 --> 00:19:54.560 And let's do one final example just 00:19:54.560 --> 00:19:56.510 to rule out all possible things we could 00:19:56.510 --> 00:19:58.530 think of that are different. 00:19:58.530 --> 00:20:03.860 So what if we had a derivative [INAUDIBLE] squared phi prime? 00:20:03.860 --> 00:20:05.660 So then we would do the following. 00:20:16.230 --> 00:20:19.337 We could add and subtract a mass term. 00:20:19.337 --> 00:20:21.920 The reason we would want to do that is, if we looked at a term 00:20:21.920 --> 00:20:25.643 like this, it's, again, less singular than this term. 00:20:25.643 --> 00:20:27.560 If this term is giving the one particle state, 00:20:27.560 --> 00:20:30.860 this term has got a factor of the propagator upstairs. 00:20:30.860 --> 00:20:33.800 So if you look for a pole that would come from that term, 00:20:33.800 --> 00:20:34.700 it's canceling out. 00:20:34.700 --> 00:20:36.260 You get p squared minus m squared 00:20:36.260 --> 00:20:37.760 over p squared minus m squared. 00:20:41.130 --> 00:20:43.980 So there's no pole from this term. 00:20:43.980 --> 00:20:47.030 And this term is, again, just of the type 00:20:47.030 --> 00:20:48.840 from our first example. 00:20:48.840 --> 00:20:51.520 It's just shifting 5 by some constant. 00:20:51.520 --> 00:20:54.650 So probably to get dimensions right, 00:20:54.650 --> 00:20:56.760 I should put some factors of eta in here. 00:21:03.450 --> 00:21:06.268 So again, something like that doesn't change anything, 00:21:06.268 --> 00:21:08.060 because it can be decomposed into something 00:21:08.060 --> 00:21:10.070 that has no pole and then something 00:21:10.070 --> 00:21:14.020 that's just a shift, OK? 00:21:14.020 --> 00:21:16.540 So as long as you have this linear term in your field 00:21:16.540 --> 00:21:21.608 redefinition, and it doesn't have to even have a trivial-- 00:21:21.608 --> 00:21:23.650 it doesn't have to even have a trivial prefactor. 00:21:23.650 --> 00:21:27.310 You could have 2 times phi or whatever you like, 00:21:27.310 --> 00:21:30.400 because that'll cancel out when you normalize things correctly. 00:21:30.400 --> 00:21:33.550 As long as you have that linear term, you're fine. 00:21:33.550 --> 00:21:37.570 And you don't have to worry about changes to the Jacobian 00:21:37.570 --> 00:21:41.287 or changes from the source term. 00:21:41.287 --> 00:21:43.870 So you just have the changes to the Lagrangian to worry about. 00:21:43.870 --> 00:21:45.870 You don't have to think about the path integral. 00:21:45.870 --> 00:21:48.893 Just make field redefinitions on the Lagrangian. 00:21:48.893 --> 00:21:50.560 And that's what you'll get some practice 00:21:50.560 --> 00:21:54.170 with on the problem set. 00:21:54.170 --> 00:21:57.400 So as I said at the beginning before everybody was here, 00:21:57.400 --> 00:21:59.362 there's a problem set number 1 that's posted. 00:21:59.362 --> 00:22:01.820 Everyone should make sure that they can actually access it. 00:22:01.820 --> 00:22:04.480 And if they can't access it, they should let me know-- 00:22:04.480 --> 00:22:08.110 if you can't get access to the web page. 00:22:08.110 --> 00:22:11.600 Any final questions about this before we move on? 00:22:11.600 --> 00:22:13.660 Yeah? 00:22:13.660 --> 00:22:15.160 AUDIENCE: I get nervous about trying 00:22:15.160 --> 00:22:18.490 to cancel a pole by adding an m squared term, because then, I 00:22:18.490 --> 00:22:20.220 guess, at some point, you start to worry 00:22:20.220 --> 00:22:22.210 that masses get shifted when you start [? normalizing these. ?] 00:22:22.210 --> 00:22:23.127 IAN STEWART: Oh, yeah. 00:22:23.127 --> 00:22:24.640 Yeah. 00:22:24.640 --> 00:22:27.790 Yeah, you should be careful about that, too. 00:22:27.790 --> 00:22:29.830 But you don't have to worry about it. 00:22:36.210 --> 00:22:36.710 Yeah. 00:22:43.300 --> 00:22:46.630 You could think of it as doing the mass renormalization first 00:22:46.630 --> 00:22:48.940 and then worry about this, or-- 00:22:58.373 --> 00:23:00.290 so that you're using a renormalized mass here. 00:23:06.310 --> 00:23:06.810 All right. 00:23:12.720 --> 00:23:14.160 So we'll start a new section. 00:23:35.052 --> 00:23:36.760 And the goal of this section is basically 00:23:36.760 --> 00:23:40.240 to be more careful about loop diagrams 00:23:40.240 --> 00:23:41.740 and, in particular, to show you what 00:23:41.740 --> 00:23:44.590 matching is between two effective field theories. 00:23:44.590 --> 00:23:48.190 We'll still be thinking in the context of mass of particles. 00:23:48.190 --> 00:23:51.130 That's the simplest place to describe this 00:23:51.130 --> 00:23:55.510 that we'll probably do an example later on of a case that 00:23:55.510 --> 00:23:57.310 doesn't just involve mass of particles. 00:24:00.590 --> 00:24:02.800 So let's start out with a very trivial example just 00:24:02.800 --> 00:24:05.140 to see what we're talking about. 00:24:07.970 --> 00:24:09.155 So we'll take two particles. 00:24:09.155 --> 00:24:10.405 One of them is a heavy scalar. 00:24:17.660 --> 00:24:20.865 It has mass capital M. Actually, I 00:24:20.865 --> 00:24:22.240 put a little underline so you can 00:24:22.240 --> 00:24:26.230 tell my capitals from my lowercase, 00:24:26.230 --> 00:24:28.150 because the light fermion that we're also 00:24:28.150 --> 00:24:37.560 going to have has a lowercase mass, little m. 00:24:37.560 --> 00:24:38.060 OK. 00:24:38.060 --> 00:24:41.450 So heavy scalar, light fermion. 00:24:41.450 --> 00:24:43.450 And so what we want to talk about here 00:24:43.450 --> 00:24:47.920 is really this picture that we described to you earlier 00:24:47.920 --> 00:24:51.700 where we've got two theories, 1 and 2, 00:24:51.700 --> 00:24:53.920 and we want to pass from theory 1 to 2 00:24:53.920 --> 00:24:56.800 by removing something from theory 1. 00:24:56.800 --> 00:25:00.590 So theory 1 here will be just the theory of these things. 00:25:00.590 --> 00:25:03.670 And we'll make it a renormalizable theory, 00:25:03.670 --> 00:25:07.150 in the traditional sense, just so we know where to stop. 00:25:18.320 --> 00:25:19.700 And this guy should be capital. 00:25:23.825 --> 00:25:25.700 And then there's some [INAUDIBLE] interaction 00:25:25.700 --> 00:25:27.910 between them. 00:25:27.910 --> 00:25:32.390 So we'll think of this as being the theory 1, 00:25:32.390 --> 00:25:36.110 where we're in a situation where capital M is 00:25:36.110 --> 00:25:37.413 much bigger than little m. 00:25:41.577 --> 00:25:44.160 So then we want to think about describing psi at low energies. 00:25:47.220 --> 00:25:49.140 And that means we can get rid of the scalar 00:25:49.140 --> 00:25:50.820 as an explicit degree of freedom. 00:25:58.900 --> 00:26:01.510 So low energy is relative to this mass scale, 00:26:01.510 --> 00:26:03.495 which is heavy-- 00:26:03.495 --> 00:26:17.370 M, capital M. We're going to remove 00:26:17.370 --> 00:26:19.440 the scalar from the theory. 00:26:19.440 --> 00:26:21.375 So you could just say remove it, or you 00:26:21.375 --> 00:26:22.500 could say integrate it out. 00:26:22.500 --> 00:26:25.920 When you say integrate out, the words you're using 00:26:25.920 --> 00:26:27.630 have a path-integral connotation, 00:26:27.630 --> 00:26:30.120 where you had this [INAUDIBLE] in the path integral, 00:26:30.120 --> 00:26:32.820 and you think about just doing the path 00:26:32.820 --> 00:26:35.565 integral over that field and removing it. 00:26:35.565 --> 00:26:37.815 But the words of "removing it" or "integrating it out" 00:26:37.815 --> 00:26:40.230 are synonymous. 00:26:40.230 --> 00:26:42.000 So what is theory 2 going to look like? 00:26:45.007 --> 00:26:47.340 Well, we'll still have the kinetic term for our fermion. 00:26:52.800 --> 00:26:56.370 And then removing the scalar will 00:26:56.370 --> 00:26:57.840 generate some new operators. 00:26:57.840 --> 00:27:01.260 In particular, there'll be a dimension-six operator 00:27:01.260 --> 00:27:03.220 like this. 00:27:03.220 --> 00:27:07.295 And then there could well be other terms. 00:27:07.295 --> 00:27:08.670 And so if we wanted to figure out 00:27:08.670 --> 00:27:11.520 what this dimension-six operator is at tree level, 00:27:11.520 --> 00:27:14.580 that's a pretty straightforward exercise. 00:27:14.580 --> 00:27:21.000 We would simply think about the Feynman diagram in theory 1. 00:27:21.000 --> 00:27:25.780 So here's a Feynman diagram in theory 1 with [INAUDIBLE] 00:27:25.780 --> 00:27:28.933 couplings here. 00:27:28.933 --> 00:27:30.600 We would calculate this Feynman diagram. 00:27:40.202 --> 00:27:41.910 And we would assume that all the momentum 00:27:41.910 --> 00:27:44.520 of the external particles are small. 00:27:44.520 --> 00:27:46.628 And that means that this is small. 00:27:46.628 --> 00:27:47.670 And we would just expand. 00:28:02.130 --> 00:28:05.040 And then we would just ask that, when I take the Feynman 00:28:05.040 --> 00:28:06.150 rule from this-- 00:28:06.150 --> 00:28:08.940 whoops, should have been a square there-- 00:28:08.940 --> 00:28:10.830 when I take the Feynman rule from that, 00:28:10.830 --> 00:28:14.970 then I should get the same thing as the Feynman rule from that. 00:28:14.970 --> 00:28:16.980 And that would fix this coefficient a, 00:28:16.980 --> 00:28:18.990 which is, so far, arbitrary. 00:28:18.990 --> 00:28:21.450 But we can determine it by using theory 1. 00:28:21.450 --> 00:28:24.720 And that's the idea of doing a matching calculation that you 00:28:24.720 --> 00:28:26.550 have introduced the theory 2. 00:28:26.550 --> 00:28:28.650 It has some parameters in front of operators. 00:28:28.650 --> 00:28:31.420 In this case, I called it a. 00:28:31.420 --> 00:28:33.730 And we want to determine those parameters by doing 00:28:33.730 --> 00:28:35.890 calculations in theory 1. 00:28:35.890 --> 00:28:37.330 And in particular, what you do is 00:28:37.330 --> 00:28:40.690 you make sure that S-matrix elements in the two theories 00:28:40.690 --> 00:28:42.430 agree. 00:28:42.430 --> 00:28:44.605 But at tree level, that's just matching up diagrams. 00:29:16.840 --> 00:29:18.460 So a is simply g squared. 00:29:29.310 --> 00:29:31.470 So we just have to match up this guy with that guy. 00:29:31.470 --> 00:29:36.770 And so taking the leading-order term, a is just g squared. 00:29:36.770 --> 00:29:39.392 Very simple. 00:29:39.392 --> 00:29:41.600 So the place where this gets more involved and more-- 00:29:41.600 --> 00:29:43.790 where you have to think a little bit more-- this 00:29:43.790 --> 00:29:45.650 seems almost automatic, that you just do 00:29:45.650 --> 00:29:48.740 calculations in here and here, and just match them up. 00:29:48.740 --> 00:29:49.982 And that's the basic idea. 00:29:49.982 --> 00:29:51.440 It's only a little more complicated 00:29:51.440 --> 00:29:53.510 when you have to take into account loops. 00:29:53.510 --> 00:29:56.600 So that's what we'll spend most of this section discussing, 00:29:56.600 --> 00:29:59.790 since this part is easier. 00:29:59.790 --> 00:30:00.635 So what about loops? 00:30:03.612 --> 00:30:05.570 What are some of the issues that come up there? 00:30:09.260 --> 00:30:12.222 Well, the first one is that the Feynman diagrams diverge, 00:30:12.222 --> 00:30:13.430 so you have to regulate them. 00:30:24.810 --> 00:30:27.520 And the thing that can be confusing about thinking 00:30:27.520 --> 00:30:30.020 about effective field theories and thinking about divergence 00:30:30.020 --> 00:30:33.660 diagrams is separating the ideas of regularization 00:30:33.660 --> 00:30:38.280 and the mass scale lambda nu that we've been talking about. 00:30:38.280 --> 00:30:41.660 So that's something I want to talk 00:30:41.660 --> 00:30:49.910 about in some detail, because they're not the same thing. 00:30:52.675 --> 00:30:54.800 So we would need to cut off ultraviolet divergences 00:30:54.800 --> 00:30:57.020 to obtain finite results. 00:30:57.020 --> 00:31:01.970 And that means we introduce cutoff parameters 00:31:01.970 --> 00:31:03.185 into our results. 00:31:12.480 --> 00:31:20.580 So examples would be taking the Minkowski momentum, 00:31:20.580 --> 00:31:23.070 continuing it to be Euclidean, and then just putting 00:31:23.070 --> 00:31:26.970 a hard cutoff on it-- that's one example. 00:31:26.970 --> 00:31:29.520 Or you could use dim reg. 00:31:29.520 --> 00:31:30.870 That's another example. 00:31:30.870 --> 00:31:32.340 Or you could use a lattice spacing. 00:31:32.340 --> 00:31:34.540 That's another example. 00:31:34.540 --> 00:31:36.390 So these are all examples of how you 00:31:36.390 --> 00:31:39.537 might cut off the theory to remove ultraviolet divergences. 00:31:43.120 --> 00:31:51.340 And then there's a second step, renormalization, 00:31:51.340 --> 00:31:53.110 distinct from regularization. 00:31:56.140 --> 00:31:58.030 So some of the things I'm teaching you here 00:31:58.030 --> 00:31:59.650 should be familiar. 00:31:59.650 --> 00:32:02.170 And the only real generalization that we're going to do 00:32:02.170 --> 00:32:04.390 is we're going to be able to apply some of the things 00:32:04.390 --> 00:32:05.973 that you learned about renormalization 00:32:05.973 --> 00:32:07.600 and regularization to any operators 00:32:07.600 --> 00:32:09.017 that you might have in the theory, 00:32:09.017 --> 00:32:13.030 whether they're renormalizable in the dimension-four 00:32:13.030 --> 00:32:16.850 traditional sense or in a higher-dimensional sense. 00:32:16.850 --> 00:32:19.660 So here, what we're doing when we talk about renormalization 00:32:19.660 --> 00:32:22.930 is we're picking a scheme that gives 00:32:22.930 --> 00:32:27.870 precise or definite meaning to the parameters 00:32:27.870 --> 00:32:28.870 in the effective theory. 00:32:33.220 --> 00:32:38.530 Or, even more explicitly, each coefficient in the Lagrangian, 00:32:38.530 --> 00:32:44.230 as well as the operators in the Lagrangian, 00:32:44.230 --> 00:32:49.450 are given a meaning by this procedure. 00:32:49.450 --> 00:32:51.850 If we didn't have a renormalization procedure, 00:32:51.850 --> 00:32:54.640 then, because of the ultraviolet divergences, 00:32:54.640 --> 00:32:57.190 there'd be ambiguities in how we define the coefficients 00:32:57.190 --> 00:32:59.410 and the operators. 00:32:59.410 --> 00:33:01.720 Or, even if you don't have ultraviolet divergences, 00:33:01.720 --> 00:33:05.290 you still have freedom to pick different schemes 00:33:05.290 --> 00:33:09.760 for the definitions of the coefficients. 00:33:09.760 --> 00:33:14.080 And when you do this, you also can introduce parameters. 00:33:18.933 --> 00:33:20.850 So you can get parameters from regularization. 00:33:20.850 --> 00:33:24.130 You can also get parameters from this. 00:33:24.130 --> 00:33:26.670 So some examples that are familiar-- 00:33:26.670 --> 00:33:29.205 there's this scale mu that shows up when you do MS bar. 00:33:32.485 --> 00:33:34.110 If you do something that's a little bit 00:33:34.110 --> 00:33:36.870 different, if you take Green's functions, 00:33:36.870 --> 00:33:40.290 and you go to some offshell point, 00:33:40.290 --> 00:33:45.300 that's another renormalization scheme called offshell momentum 00:33:45.300 --> 00:33:47.460 subtraction. 00:33:47.460 --> 00:33:54.100 And that scheme also has a parameter that shows up. 00:33:54.100 --> 00:33:58.820 And if you did a Wilsonian renormalization, 00:33:58.820 --> 00:34:02.580 there would also be a cutoff associated to that. 00:34:02.580 --> 00:34:05.115 And this Wilsonian cutoff here doesn't 00:34:05.115 --> 00:34:06.990 have to be the same as this lambda uv cutoff. 00:34:14.040 --> 00:34:14.540 OK. 00:34:14.540 --> 00:34:16.130 So every coefficient, every operator, 00:34:16.130 --> 00:34:18.409 no matter the dimension, no matter 00:34:18.409 --> 00:34:21.650 where it turns up in the series, we're 00:34:21.650 --> 00:34:23.900 going to have to think about whether there's 00:34:23.900 --> 00:34:26.750 diagrams that generate ultraviolet divergences. 00:34:26.750 --> 00:34:31.363 And if those diagrams look like these operators, 00:34:31.363 --> 00:34:33.030 then you're going to get renormalization 00:34:33.030 --> 00:34:33.822 of those operators. 00:34:33.822 --> 00:34:37.350 And you're going to have to be careful about taking care 00:34:37.350 --> 00:34:39.725 of those divergences and also the scheme 00:34:39.725 --> 00:34:41.100 that you define the operators in. 00:34:46.020 --> 00:34:49.409 So you can think about this, just as far 00:34:49.409 --> 00:34:59.061 as the coefficients are concerned, 00:34:59.061 --> 00:35:04.710 as starting out with some bare coefficients that 00:35:04.710 --> 00:35:07.860 depend on your ultraviolet regulator, 00:35:07.860 --> 00:35:11.282 and switching to some renormalized coefficients which 00:35:11.282 --> 00:35:12.990 don't depend on the ultraviolet regulator 00:35:12.990 --> 00:35:15.810 but do depend on the scheme. 00:35:15.810 --> 00:35:18.855 And then, also, you have some counterterms 00:35:18.855 --> 00:35:19.980 that depend on both things. 00:35:23.230 --> 00:35:24.960 So that's how it would look if you 00:35:24.960 --> 00:35:30.090 wrote that down for a cutoff, in a Wilsonian sense. 00:35:30.090 --> 00:35:34.140 If you wrote it down in dimensional regularization, 00:35:34.140 --> 00:35:35.190 it would look like this. 00:35:39.630 --> 00:35:43.102 Same idea, different names for the parameters. 00:35:46.760 --> 00:35:47.260 OK. 00:35:47.260 --> 00:35:48.635 So in dimensional regularization, 00:35:48.635 --> 00:35:50.620 epsilon is the ultraviolet regulator. 00:35:50.620 --> 00:35:53.320 It shows up in the bare coefficients. 00:35:53.320 --> 00:35:57.400 The renormalized ones don't depend on epsilon anymore. 00:35:57.400 --> 00:35:59.140 And these are your 1-over-epsilon poles. 00:36:05.340 --> 00:36:06.900 All right. 00:36:06.900 --> 00:36:09.480 So one of the things that makes this tricky 00:36:09.480 --> 00:36:11.970 is when you start thinking about your power counting. 00:36:11.970 --> 00:36:17.700 And so I want to spend a few minutes talking about that. 00:36:27.850 --> 00:36:30.240 So let's consider this example that I wrote down 00:36:30.240 --> 00:36:33.370 with the four-fermion operator. 00:36:33.370 --> 00:36:37.290 And if we just loop up the four-fermion operator, 00:36:37.290 --> 00:36:39.552 then we can get mass renormalization. 00:36:42.510 --> 00:36:45.600 So this is psi dagger psi squared. 00:36:45.600 --> 00:36:53.920 And I'm sticking it in and looping it up. 00:36:59.470 --> 00:37:02.265 So we start out with a tree-level mass, little m. 00:37:02.265 --> 00:37:03.640 But there's a one-loop correction 00:37:03.640 --> 00:37:05.807 that involves one of our higher-dimension operators. 00:37:08.343 --> 00:37:10.010 And there's some connection to the mass, 00:37:10.010 --> 00:37:12.240 which I'll call delta m. 00:37:12.240 --> 00:37:14.740 I'm not going to worry too much about the overall prefactor. 00:37:17.980 --> 00:37:21.260 This operator came with an a over M squared. 00:37:21.260 --> 00:37:24.370 And then there's a loop integral. 00:37:24.370 --> 00:37:30.014 There's a fermionic propagator, so that's a k slash plus m 00:37:30.014 --> 00:37:31.690 over k squared minus m squared. 00:37:35.290 --> 00:37:38.980 This guy here drops away. 00:37:38.980 --> 00:37:41.230 And really, we just have a correction that's a scalar. 00:37:41.230 --> 00:37:47.060 And that's a correction to the mass scalar in the spin space. 00:37:47.060 --> 00:37:51.100 So it's a, little m, over capital M, and then 00:37:51.100 --> 00:37:53.840 just this integral. 00:37:53.840 --> 00:37:59.950 And if we continue it to be Euclidean, 00:37:59.950 --> 00:38:04.510 then it looks like that if I continue from Minkowski 00:38:04.510 --> 00:38:05.860 to Euclidean. 00:38:05.860 --> 00:38:09.440 That's why I put the i there. 00:38:09.440 --> 00:38:12.340 So if we just assume that this integral that's 00:38:12.340 --> 00:38:15.730 sitting there-- there's only one mass scale that seems 00:38:15.730 --> 00:38:17.740 explicit there-- the little m. 00:38:17.740 --> 00:38:20.110 So if we just assume that that integral is 00:38:20.110 --> 00:38:21.460 dominated by the scale m-- 00:38:30.040 --> 00:38:38.360 little m-- then you could do a power counting 00:38:38.360 --> 00:38:39.860 for this loop integral. 00:38:39.860 --> 00:38:41.930 You just assume that all the factors of k 00:38:41.930 --> 00:38:42.970 are of order little m. 00:38:42.970 --> 00:38:46.140 You've got an explicit little m, four powers in the numerator, 00:38:46.140 --> 00:38:48.740 and two powers downstairs. 00:38:48.740 --> 00:38:55.610 So you would say that this integral scales like m squared. 00:38:58.310 --> 00:39:02.240 And then you would put that together with your delta m 00:39:02.240 --> 00:39:08.090 and say that delta m scales like a, little m cubed, capital M 00:39:08.090 --> 00:39:09.330 squared. 00:39:09.330 --> 00:39:11.780 And so this is something that would then be suppressed. 00:39:11.780 --> 00:39:14.030 Relative to the original tree-level contribution, 00:39:14.030 --> 00:39:16.028 it would be suppressed by, perhaps, 00:39:16.028 --> 00:39:17.570 some [? 4 pi ?] [? is in ?] the loop, 00:39:17.570 --> 00:39:20.570 but also little m squared over big M squared. 00:39:24.690 --> 00:39:29.490 So it would be a small correction, which is really 00:39:29.490 --> 00:39:31.952 what we would like to be the case when we're talking 00:39:31.952 --> 00:39:33.660 about some higher-dimension operator that 00:39:33.660 --> 00:39:34.910 was supposed to be suppressed. 00:39:34.910 --> 00:39:39.120 We'd like it to be a small correction. 00:39:39.120 --> 00:39:40.500 All right. 00:39:40.500 --> 00:39:43.128 So does anyone have any idea what can go wrong 00:39:43.128 --> 00:39:43.920 with this argument? 00:39:47.970 --> 00:39:49.180 I've been glib about it. 00:39:49.180 --> 00:39:52.200 I just said, if the integral is dominated by that. 00:39:56.270 --> 00:39:58.020 So the thing is that we have to be careful 00:39:58.020 --> 00:40:00.970 about our choice of regulator, because regulators also 00:40:00.970 --> 00:40:01.845 can introduce scales. 00:40:13.910 --> 00:40:15.759 So let's do this more carefully. 00:40:33.564 --> 00:40:35.490 Well, let's consider two different choices 00:40:35.490 --> 00:40:37.110 for the regulator. 00:40:37.110 --> 00:40:40.967 So we'll start out with just a cutoff, which 00:40:40.967 --> 00:40:43.050 seems very natural, from an effective field theory 00:40:43.050 --> 00:40:46.200 point of view-- that your theory is supposed to be only valid up 00:40:46.200 --> 00:40:48.030 to some scale. 00:40:48.030 --> 00:40:52.080 So why not just take and cut off the momentum explicitly 00:40:52.080 --> 00:40:53.100 above some scale? 00:40:59.060 --> 00:41:02.210 And you can think of that scale as being of order big M. 00:41:02.210 --> 00:41:04.520 So we just don't include any momentum 00:41:04.520 --> 00:41:07.050 that are higher than that scale in our loop integrals. 00:41:07.050 --> 00:41:09.772 So we're only integrating over the region in momentum space 00:41:09.772 --> 00:41:11.480 where the theory is supposed to be valid. 00:41:11.480 --> 00:41:13.820 It seems like a perfectly reasonable thing to do. 00:41:27.260 --> 00:41:30.070 So in this case, we can take this integral that 00:41:30.070 --> 00:41:33.250 has some angular parts to it as well as a radial part, 00:41:33.250 --> 00:41:36.610 decompose it into the radial piece and the angular piece. 00:41:40.480 --> 00:41:44.830 I'll give you a handout, or I'll post a page of lecture notes 00:41:44.830 --> 00:41:49.090 that have all those fun formulas that are useful to remember 00:41:49.090 --> 00:41:54.040 but not fun to talk about for decomposing integrals 00:41:54.040 --> 00:42:00.302 in arbitrary dimensions into radial and angular pieces. 00:42:00.302 --> 00:42:02.260 So if we do that, in this case, then the radial 00:42:02.260 --> 00:42:04.390 integral-- we can cut off the radial integral 00:42:04.390 --> 00:42:07.210 with a hard cutoff lambda uv. 00:42:07.210 --> 00:42:12.520 And just by dimensions, the d 4 k becomes kE times kE cubed. 00:42:12.520 --> 00:42:15.673 So this is a radial k now. 00:42:15.673 --> 00:42:17.215 And this integral, we can do exactly. 00:42:24.340 --> 00:42:27.460 There's a loop factor, 4 pi squared. 00:42:27.460 --> 00:42:30.340 There's a lambda uv squared. 00:42:30.340 --> 00:42:33.091 And then there's a logarithm. 00:42:33.091 --> 00:42:36.910 It goes like, lambda uv squared over little m squared. 00:42:42.443 --> 00:42:44.110 There's two scales that are showing up-- 00:42:44.110 --> 00:42:45.760 lambda uv and little m. 00:42:45.760 --> 00:42:47.708 The answer depends on both of them. 00:42:47.708 --> 00:42:49.750 And then we can start expanding, because little m 00:42:49.750 --> 00:42:51.667 is supposed to be much smaller than lambda uv. 00:43:00.890 --> 00:43:04.970 Let me pull out an a m over 4 pi squared. 00:43:15.270 --> 00:43:18.530 So there would be a logarithmic term. 00:43:18.530 --> 00:43:20.420 There's this lambda uv squared over capital 00:43:20.420 --> 00:43:24.401 M squared if I put that factor in. 00:43:24.401 --> 00:43:25.707 And then there's some-- 00:43:31.920 --> 00:43:33.920 so everything I'm writing in the square brackets 00:43:33.920 --> 00:43:36.200 here is dimensionless. 00:43:36.200 --> 00:43:37.880 This has the right dimensions of a mass. 00:43:40.950 --> 00:43:41.880 OK. 00:43:41.880 --> 00:43:45.090 So you can see that, if I do this, 00:43:45.090 --> 00:43:47.460 it's not satisfying what I said over here. 00:43:47.460 --> 00:43:50.910 I'd like the correction to be m cubed over M squared. 00:43:50.910 --> 00:43:54.360 There are corrections that go like m cubed over M squared, 00:43:54.360 --> 00:43:55.600 like this one. 00:43:55.600 --> 00:43:57.130 These ones are even higher. 00:43:57.130 --> 00:43:59.370 But there's this term. 00:43:59.370 --> 00:44:01.392 And that's not a small correction. 00:44:13.200 --> 00:44:17.460 So for that particular piece, what you're finding 00:44:17.460 --> 00:44:19.410 is that your power counting-- your naive power 00:44:19.410 --> 00:44:20.785 counting-- of the loop was wrong, 00:44:20.785 --> 00:44:23.020 because there's a piece from k of order-- 00:44:23.020 --> 00:44:28.760 the cutoff that's contributing for that part of it. 00:44:31.330 --> 00:44:35.100 And that's why your naive scaling argument didn't work. 00:44:35.100 --> 00:44:38.400 Now, this is the bare result. And we 00:44:38.400 --> 00:44:40.690 have to go through the renormalization procedure. 00:44:45.760 --> 00:44:48.030 And so if we go through the renormalization procedure, 00:44:48.030 --> 00:44:50.857 you can think of that as taking a piece of the integral-- 00:44:55.250 --> 00:44:58.770 so in a Wilsonian sense, you would 00:44:58.770 --> 00:45:02.440 take a piece of the integral and absorb it into the counterterm. 00:45:07.840 --> 00:45:10.020 So as promised, the counterterm depends 00:45:10.020 --> 00:45:11.430 on both lambda uv and lambda. 00:45:11.430 --> 00:45:14.432 And those are just explicitly-- 00:45:14.432 --> 00:45:16.140 I'm cutting off the radial integral here. 00:45:21.210 --> 00:45:23.670 And that does improve things, because then, I can lower 00:45:23.670 --> 00:45:27.610 this cutoff, make it smaller. 00:45:27.610 --> 00:45:31.920 And what is left would just be lambda squared over capital M 00:45:31.920 --> 00:45:39.880 squared and log of m squared over capital lambda squared. 00:45:39.880 --> 00:45:42.780 So I change the lambda uv's up here into lambdas 00:45:42.780 --> 00:45:44.231 by that procedure. 00:45:51.780 --> 00:45:53.580 So when I renormalize, the psi bar psi 00:45:53.580 --> 00:45:56.570 squared matrix element-- there's a correction from that 00:45:56.570 --> 00:45:57.260 to the mass. 00:45:57.260 --> 00:46:01.520 And the renormalized thing would depend on this Wilsonian cutoff 00:46:01.520 --> 00:46:04.040 lambda, OK? 00:46:04.040 --> 00:46:06.230 But I'm not fully getting around the issue 00:46:06.230 --> 00:46:07.880 that I'm generating this type of term. 00:46:15.020 --> 00:46:21.770 So let's do it with a different regulator, which 00:46:21.770 --> 00:46:23.420 is dimensional regularization. 00:46:23.420 --> 00:46:41.840 Let's see what happens using the MS-bar scheme. 00:46:41.840 --> 00:46:42.920 Same calculation. 00:46:57.170 --> 00:47:04.100 So again, split it into radial and angular parts. 00:47:04.100 --> 00:47:06.150 And again, I'll give you formulas 00:47:06.150 --> 00:47:07.400 for doing something like that. 00:47:09.992 --> 00:47:11.690 But I won't write them down in lecture. 00:47:18.160 --> 00:47:21.280 Actually, I think I will write one down a little later. 00:47:21.280 --> 00:47:24.690 But I'll give you a more complete set as a handout. 00:47:34.532 --> 00:47:35.990 So leaving over the radial integral 00:47:35.990 --> 00:47:37.520 but regulating it dimensionally just 00:47:37.520 --> 00:47:40.290 means that, instead of having this guy to the cubed power, 00:47:40.290 --> 00:47:41.930 it's to the d minus 1. 00:47:41.930 --> 00:47:45.290 And there's some pieces here that depend on epsilon. 00:47:45.290 --> 00:47:47.400 And my convention for the entire course 00:47:47.400 --> 00:47:50.850 is that d is 4 minus 2 epsilon, not d minus epsilon. 00:47:50.850 --> 00:47:55.390 So again, this is an integral we can do exactly. 00:48:19.540 --> 00:48:21.942 We get something like that. 00:48:21.942 --> 00:48:22.900 And then we can expand. 00:48:37.176 --> 00:48:40.330 We get a 1-over-epsilon pole. 00:48:40.330 --> 00:48:41.340 We get a logarithm. 00:48:43.930 --> 00:48:46.850 There is some constant as well. 00:48:46.850 --> 00:48:49.180 And then there's order-epsilon pieces. 00:48:53.240 --> 00:48:57.435 So you should contrast this result here 00:48:57.435 --> 00:48:59.730 with the result we had over here. 00:48:59.730 --> 00:49:03.017 The order-epsilon pieces are like these terms. 00:49:03.017 --> 00:49:05.600 They're the terms that would go away if I took the cutoff here 00:49:05.600 --> 00:49:09.350 to infinity-- lambda uv to infinity or epsilon to 0. 00:49:09.350 --> 00:49:12.660 There's terms here that go like m cubed over M squared. 00:49:12.660 --> 00:49:14.485 That's like this term. 00:49:14.485 --> 00:49:15.610 And this term is not there. 00:49:22.750 --> 00:49:26.200 This 1 over epsilon is related to the log divergence. 00:49:26.200 --> 00:49:28.960 And this thing here is a power divergence. 00:49:28.960 --> 00:49:32.080 So the 1 over epsilon is related to this log of lambda uv. 00:49:32.080 --> 00:49:33.760 You should think of log of lambda uv 00:49:33.760 --> 00:49:38.288 as going to 1 over epsilon plus log of mu squared. 00:49:38.288 --> 00:49:39.830 But we don't see the power divergence 00:49:39.830 --> 00:49:46.140 in dimensional regularization in MS bar. 00:49:46.140 --> 00:49:46.640 OK. 00:49:46.640 --> 00:49:50.750 So what's happening is, from this result, 00:49:50.750 --> 00:50:02.450 we are getting something that's the size that we expected 00:50:02.450 --> 00:50:04.070 by our power-counting argument. 00:50:04.070 --> 00:50:09.560 The regularized result is the right size by power counting. 00:50:09.560 --> 00:50:21.300 The logarithm here is the same log as in a with, if you like, 00:50:21.300 --> 00:50:24.660 a correspondence between mu and lambda. 00:50:24.660 --> 00:50:26.927 So we're seeing that logarithm there. 00:50:26.927 --> 00:50:28.260 We're just not seeing this term. 00:50:37.070 --> 00:50:41.440 And if we wanted to write down the MS-bar counterterm. 00:50:41.440 --> 00:50:47.935 Some correction to the mass, it would go like, a, m squared. 00:50:53.610 --> 00:50:56.775 The diagram would look like just the 1-over-epsilon pole 00:50:56.775 --> 00:50:59.710 in MS bar. 00:50:59.710 --> 00:51:00.210 OK. 00:51:00.210 --> 00:51:04.290 So the two regulators seem to give us similar results but not 00:51:04.290 --> 00:51:06.420 exactly-identical results. 00:51:06.420 --> 00:51:09.340 And the question is, how should I think about this? 00:51:09.340 --> 00:51:11.670 What's the right way of thinking about this extra term? 00:51:14.990 --> 00:51:16.540 So there's a language that people 00:51:16.540 --> 00:51:19.510 use in effective field theory and dimensional 00:51:19.510 --> 00:51:21.670 counting for the difference between thinking 00:51:21.670 --> 00:51:28.110 about doing a regularization of type A and type B. 00:51:28.110 --> 00:51:31.290 And what you say is that, in type B, 00:51:31.290 --> 00:51:35.670 your regularization of the problem 00:51:35.670 --> 00:51:52.330 does not break the power counting, which 00:51:52.330 --> 00:51:57.400 means that you can do power counting for regulated graphs 00:51:57.400 --> 00:51:58.750 prior to renormalization them. 00:52:08.250 --> 00:52:13.978 But you can see that, in case A, that would be problematic, 00:52:13.978 --> 00:52:16.270 because our naive power counting wouldn't have given us 00:52:16.270 --> 00:52:20.770 the right result. 00:52:20.770 --> 00:52:22.424 So what do we say about case A? 00:52:28.635 --> 00:52:30.690 Well, if case B didn't break the power counting, 00:52:30.690 --> 00:52:31.590 then case A does. 00:52:35.970 --> 00:52:38.820 But you can say something a little bit more positive 00:52:38.820 --> 00:52:45.600 about case A. And that is that, when you think about case A, 00:52:45.600 --> 00:52:51.390 you can set up the theory with renormalization conditions 00:52:51.390 --> 00:52:57.390 such that you can restore power counting 00:52:57.390 --> 00:53:00.720 in the renormalized graphs and renormalized couplings 00:53:00.720 --> 00:53:02.697 and renormalized operators. 00:53:02.697 --> 00:53:04.780 But you don't have power counting-- explicit power 00:53:04.780 --> 00:53:07.715 counting-- in just the regularized results. 00:53:07.715 --> 00:53:09.090 So you can think that I just take 00:53:09.090 --> 00:53:10.890 the Wilsonian cutoff small enough that I 00:53:10.890 --> 00:53:15.990 make that annoying term small. 00:53:15.990 --> 00:53:18.630 And order by order, I can do that-- 00:53:21.180 --> 00:53:22.860 order by order in my loop expansion, 00:53:22.860 --> 00:53:24.480 order by order in my calculations. 00:53:31.730 --> 00:53:32.340 OK. 00:53:32.340 --> 00:53:34.130 So in some sense, there's nothing wrong 00:53:34.130 --> 00:53:35.960 with doing A. It's just that you have 00:53:35.960 --> 00:53:38.060 to work a little harder to think about it. 00:53:38.060 --> 00:53:40.213 And when you think about the power counting, 00:53:40.213 --> 00:53:42.380 you have to do that for the renormalized quantities, 00:53:42.380 --> 00:53:45.707 not for the bare quantities. 00:53:45.707 --> 00:53:47.290 So if you like, what you're doing here 00:53:47.290 --> 00:53:49.540 is you're adding counterterms. 00:53:49.540 --> 00:53:55.670 Another way you can say this is that you're adding counterterms 00:53:55.670 --> 00:53:59.060 to restore the power counting that you want. 00:54:02.940 --> 00:54:05.142 And that's not too different than-- 00:54:05.142 --> 00:54:06.600 you could always do that, actually. 00:54:06.600 --> 00:54:08.340 I made this analogy that you should 00:54:08.340 --> 00:54:14.310 think about power counting as being like gauge symmetry. 00:54:14.310 --> 00:54:17.550 So let's say you had a theory, and it had nice gauge symmetry, 00:54:17.550 --> 00:54:19.560 but you picked some crazy regulator 00:54:19.560 --> 00:54:21.805 that broke gauge symmetry. 00:54:21.805 --> 00:54:23.430 Well, you could always put counterterms 00:54:23.430 --> 00:54:24.847 in that would break gauge symmetry 00:54:24.847 --> 00:54:26.925 and restore it in the renormalized quantity. 00:54:26.925 --> 00:54:28.440 That would be OK. 00:54:28.440 --> 00:54:29.850 It would be more work. 00:54:29.850 --> 00:54:31.590 We don't like to do that. 00:54:31.590 --> 00:54:33.810 We avoid it at all costs. 00:54:33.810 --> 00:54:36.600 But if we had to, we could do it. 00:54:36.600 --> 00:54:38.748 If you do supersymmetry, you might 00:54:38.748 --> 00:54:40.290 try to use dimensional regularization 00:54:40.290 --> 00:54:42.540 and supersymmetry, standard dimensional regularization 00:54:42.540 --> 00:54:44.880 and supersymmetry, break supersymmetry. 00:54:44.880 --> 00:54:47.340 If you do that, you have to introduce counterterms 00:54:47.340 --> 00:54:49.140 that restore supersymmetry. 00:54:49.140 --> 00:54:52.240 So the same language of symmetry is being applied here, 00:54:52.240 --> 00:54:54.720 except now to power counting, where 00:54:54.720 --> 00:54:57.780 we say that if your regulator messes up your power counting, 00:54:57.780 --> 00:55:00.420 you can restore it in the renormalized couplings. 00:55:00.420 --> 00:55:02.790 But you may be smart enough to think up a regulator-- 00:55:02.790 --> 00:55:05.130 in this case, dimensional regularization-- 00:55:05.130 --> 00:55:08.210 where you don't have to deal with that complication. 00:55:08.210 --> 00:55:08.710 OK. 00:55:20.656 --> 00:55:21.820 So let me write that. 00:56:15.070 --> 00:56:16.740 So in an effective theory, you should 00:56:16.740 --> 00:56:19.860 think about regulating to preserve symmetries 00:56:19.860 --> 00:56:23.130 as well as to preserve power counting, if you can. 00:56:36.660 --> 00:56:40.620 And one way, in a more formal language, 00:56:40.620 --> 00:56:43.350 that you could say what happens with the generation 00:56:43.350 --> 00:56:45.960 of that term that we talked about with the cutoff 00:56:45.960 --> 00:56:50.310 is that you'd mix up different orders in the expansion, 00:56:50.310 --> 00:56:54.840 and it looks like your naively higher-order term 00:56:54.840 --> 00:56:58.325 is mixing back to a term of lower dimension. 00:57:02.880 --> 00:57:06.060 And so if you can get away with taking a regulator that doesn't 00:57:06.060 --> 00:57:09.510 have that mixing back to more relevant operators, 00:57:09.510 --> 00:57:12.720 then you could preserve your power counting, 00:57:12.720 --> 00:57:15.730 make it simpler. 00:57:15.730 --> 00:57:20.950 This is true irrespective of what the power counting is in. 00:57:20.950 --> 00:57:23.800 This is a general statement. 00:57:23.800 --> 00:57:25.940 In the context of what we've been talking about, 00:57:25.940 --> 00:57:29.230 which is dimensional power counting, 00:57:29.230 --> 00:57:31.480 there's a particular phrase that goes along with this. 00:57:39.980 --> 00:57:42.820 And that is that we talk about using a mass independent 00:57:42.820 --> 00:57:50.415 regulator, like dim reg. 00:57:56.845 --> 00:58:00.070 If you like, it has a mass scale, mu, 00:58:00.070 --> 00:58:02.260 but it's put in softly in a way that 00:58:02.260 --> 00:58:04.360 doesn't mess up power counting. 00:58:04.360 --> 00:58:07.780 We call that using a mass independent regulator. 00:58:07.780 --> 00:58:11.110 So we want to avoid having different orders 00:58:11.110 --> 00:58:13.780 in the expansion mix up with each other. 00:58:13.780 --> 00:58:16.810 In general, I should comment, then, 00:58:16.810 --> 00:58:18.700 that terms that are the same order 00:58:18.700 --> 00:58:23.530 will definitely typically mix up with each other 00:58:23.530 --> 00:58:25.990 under renormalization. 00:58:25.990 --> 00:58:27.760 So even if you thought you were smart, 00:58:27.760 --> 00:58:30.895 and you enumerated all the operators, but you missed one, 00:58:30.895 --> 00:58:33.520 and then you started calculating, 00:58:33.520 --> 00:58:35.740 that operator might just pop out at you, 00:58:35.740 --> 00:58:37.900 because you could have some calculation 00:58:37.900 --> 00:58:41.260 with another operator mix into that operator. 00:58:41.260 --> 00:58:44.680 Or you could have an operator that you did a matching at tree 00:58:44.680 --> 00:58:47.080 level, and you didn't generate, but then you 00:58:47.080 --> 00:58:50.060 start renormalizing that tree-level operator, 00:58:50.060 --> 00:58:52.925 and another operator pops out at you. 00:58:52.925 --> 00:58:55.300 So it's important that you include all the operators that 00:58:55.300 --> 00:58:58.893 have the same dimension and the same quantum numbers, 00:58:58.893 --> 00:59:00.310 because if you don't include them, 00:59:00.310 --> 00:59:04.963 you're bound to get them from loops anyway. 00:59:04.963 --> 00:59:06.130 And you want to be complete. 00:59:11.140 --> 00:59:18.690 So if you like, in a matrix [INAUDIBLE] notation, 00:59:18.690 --> 00:59:22.530 you could say that the bare operators mix up with the set 00:59:22.530 --> 00:59:24.420 of possible renormalized operators, 00:59:24.420 --> 00:59:27.420 and there's some matrix of counterterms that would correct 00:59:27.420 --> 00:59:28.020 them-- 00:59:28.020 --> 00:59:31.030 connect them. 00:59:31.030 --> 00:59:31.570 OK. 00:59:31.570 --> 00:59:33.120 So any questions so far? 00:59:37.030 --> 00:59:40.480 So sometimes, in the literature, you'll see that these kinds 00:59:40.480 --> 00:59:44.050 of things-- regulator discussions in effective field 00:59:44.050 --> 00:59:44.710 theory-- 00:59:44.710 --> 00:59:47.860 generate all sorts of papers. 00:59:47.860 --> 00:59:50.410 Keep this in mind if you ever run into that. 01:00:01.045 --> 01:00:02.920 You should be able to think about the physics 01:00:02.920 --> 01:00:04.712 that you're after with different regulators 01:00:04.712 --> 01:00:07.500 and come to the same conclusion. 01:00:07.500 --> 01:00:12.140 And it just may be easier with one regulator versus another. 01:00:12.140 --> 01:00:14.112 So we've said, in these kind of theories 01:00:14.112 --> 01:00:15.820 where we have dimensional power counting, 01:00:15.820 --> 01:00:21.040 that dimensional regularization is special. 01:00:21.040 --> 01:00:23.020 So I want to talk a bit more about 01:00:23.020 --> 01:00:24.739 dimensional regularization. 01:00:45.123 --> 01:00:47.540 So sometimes you'll hear people say that you should always 01:00:47.540 --> 01:00:51.130 use dimensional regularization for doing the power counting. 01:00:51.130 --> 01:00:55.250 But that's not quite true in the way that I've told you. 01:00:55.250 --> 01:00:56.750 It's not that you have to, it's just 01:00:56.750 --> 01:00:58.940 that it makes things simpler. 01:00:58.940 --> 01:01:01.650 But given that we want to make things as simple as possible, 01:01:01.650 --> 01:01:06.170 let's take dimensional regularization seriously. 01:01:06.170 --> 01:01:08.540 So you can actually derive dimensional regularization 01:01:08.540 --> 01:01:11.420 by just imposing axioms. 01:01:11.420 --> 01:01:16.445 If you say that you want a loop integration that's linear-- 01:01:22.260 --> 01:01:24.240 I should have said this earlier. 01:01:24.240 --> 01:01:26.190 So my notation with dimensional regularization 01:01:26.190 --> 01:01:27.607 is, I put a little cross on the d. 01:01:27.607 --> 01:01:30.640 And that means dividing by the 2 pi. 01:01:30.640 --> 01:01:36.720 So that means d d p over 2 pi to the d. 01:01:36.720 --> 01:01:39.720 So linearity means that if I'm integrating some function that 01:01:39.720 --> 01:01:42.360 can be decomposed into a sum of two pieces, a 01:01:42.360 --> 01:01:48.680 and b being constants, f and g being functions, 01:01:48.680 --> 01:01:55.030 then I can write that out as an integral over f 01:01:55.030 --> 01:01:59.590 plus an integral over g, which really is something 01:01:59.590 --> 01:02:05.948 that almost every reasonable definition of the integration 01:02:05.948 --> 01:02:06.490 will satisfy. 01:02:12.090 --> 01:02:18.660 The second one is translations, which is more restricting. 01:02:18.660 --> 01:02:21.240 So that says, if you have some integral over f, 01:02:21.240 --> 01:02:24.330 but it's a function of p plus q-- q is some external-- 01:02:24.330 --> 01:02:30.345 I can always shift away the q. 01:02:30.345 --> 01:02:32.205 It just goes-- p goes to p minus q. 01:02:32.205 --> 01:02:34.013 And then I just have an integral over p. 01:02:37.120 --> 01:02:38.800 And along with translations, you can 01:02:38.800 --> 01:02:40.990 think about having rotations. 01:02:40.990 --> 01:02:44.560 My whole notation is covariance, so we won't worry so much 01:02:44.560 --> 01:02:46.210 about rotations. 01:02:46.210 --> 01:02:47.340 and Lorentz group. 01:02:51.880 --> 01:02:56.560 And then the final one that's obviously 01:02:56.560 --> 01:03:00.370 a little bit special to dim reg is a scaling. 01:03:00.370 --> 01:03:04.150 So let's say we have a scalar s multiplying our momentum p. 01:03:07.610 --> 01:03:12.290 Then we can rescale the momentum p and get rid of-- 01:03:12.290 --> 01:03:18.410 pull the s outside by just taking p goes to p over s. 01:03:18.410 --> 01:03:20.000 So that changes this to a p. 01:03:20.000 --> 01:03:22.336 We get an s to the minus d. 01:03:22.336 --> 01:03:25.140 It pulls out front. 01:03:25.140 --> 01:03:29.210 And if we demand that, then that's 01:03:29.210 --> 01:03:30.830 special to dimensional regularization, 01:03:30.830 --> 01:03:34.760 because you can see that this depends on d. 01:03:34.760 --> 01:03:37.340 Even if I call this measure some abstract thing, 01:03:37.340 --> 01:03:38.910 now there's a d showing up. 01:03:38.910 --> 01:03:40.130 It's outside the measure. 01:03:45.600 --> 01:03:48.480 And these three together actually give 01:03:48.480 --> 01:03:52.410 a unique definition to the integration up 01:03:52.410 --> 01:03:54.285 to the overall normalization. 01:04:07.160 --> 01:04:09.263 And that unique thing is dim reg. 01:04:12.890 --> 01:04:17.150 So I'm going to refer you to reading, 01:04:17.150 --> 01:04:19.640 I have posted a chapter from Collins' book 01:04:19.640 --> 01:04:21.410 on regularization. 01:04:21.410 --> 01:04:24.860 And around page 65, he talks about how you prove that. 01:04:24.860 --> 01:04:25.730 It's not too hard. 01:04:34.310 --> 01:04:36.970 The standard definition of the normalization, 01:04:36.970 --> 01:04:41.184 which is something you have to specify, 01:04:41.184 --> 01:04:44.520 is that you let, say, this Gaussian 01:04:44.520 --> 01:04:46.140 integral be pi to the d over 2. 01:04:50.580 --> 01:04:55.770 So then, from that, you have some measure in some space 01:04:55.770 --> 01:04:56.895 that you can then just use. 01:05:01.220 --> 01:05:03.350 So one formula that I used earlier on 01:05:03.350 --> 01:05:08.720 was the ability to split that into pieces 01:05:08.720 --> 01:05:13.160 which were a radial piece and then an angular piece. 01:05:13.160 --> 01:05:15.310 And in general, this is a property 01:05:15.310 --> 01:05:17.060 that this integration [? measure obeys, ?] 01:05:17.060 --> 01:05:19.393 that you could split it, and you could split it further. 01:05:19.393 --> 01:05:28.670 You could pull out another angle, for example, 01:05:28.670 --> 01:05:34.810 and get one less dimension in the angular parts. 01:05:34.810 --> 01:05:38.110 And the uv divergences, if we're talking about us divergences-- 01:05:38.110 --> 01:05:40.500 they're occurring in this radial part-- 01:05:40.500 --> 01:05:41.920 Euclidean radial part. 01:05:48.070 --> 01:05:50.950 So by thinking about this kind of decomposition, 01:05:50.950 --> 01:05:53.620 you're moving the uv divergences to 01:05:53.620 --> 01:05:57.280 a one-dimensional integration, at least at one loop. 01:05:57.280 --> 01:05:59.635 And you can always do that. 01:05:59.635 --> 01:06:01.510 So in general, in dimensional regularization, 01:06:01.510 --> 01:06:03.220 there's many ways you could evaluate the integral. 01:06:03.220 --> 01:06:04.750 You're not used to using this one. 01:06:04.750 --> 01:06:07.030 You're used to keeping things covariant, 01:06:07.030 --> 01:06:11.770 using some Feynman parameters, combining propagators together, 01:06:11.770 --> 01:06:14.330 and then doing the integral. 01:06:14.330 --> 01:06:16.450 But you could also do it this way, 01:06:16.450 --> 01:06:19.070 and you get the same answer. 01:06:19.070 --> 01:06:22.355 So it's really a well-defined measure in the sense 01:06:22.355 --> 01:06:24.230 that you can manipulate it in different ways. 01:06:24.230 --> 01:06:26.860 And they should all lead to the same answer for your loop 01:06:26.860 --> 01:06:27.769 integrals. 01:06:31.763 --> 01:06:33.180 And that's part of what I'm trying 01:06:33.180 --> 01:06:37.110 to emphasize by saying that you could derive it 01:06:37.110 --> 01:06:38.295 by considering axioms. 01:06:46.740 --> 01:06:49.800 So d was equal to 4 minus 2 epsilon. 01:06:49.800 --> 01:06:53.070 Epsilon greater than 0 is what you 01:06:53.070 --> 01:06:57.554 need to lower the powers of p, and therefore tame the uv. 01:06:57.554 --> 01:07:03.630 Epsilon less than 0 can be used to regulate 01:07:03.630 --> 01:07:05.307 infrared singularities. 01:07:10.542 --> 01:07:12.000 There's some counterintuitive facts 01:07:12.000 --> 01:07:13.710 about dimensional regularization, 01:07:13.710 --> 01:07:15.730 and I want to mention a couple of them to you. 01:07:19.160 --> 01:07:23.280 One of them is that, if I have p to an arbitrary power-- 01:07:23.280 --> 01:07:24.650 think of it as Euclidean-- 01:07:27.270 --> 01:07:27.870 that's 0. 01:07:38.970 --> 01:07:43.950 So Collins constructs a proof of this 01:07:43.950 --> 01:07:47.970 on page 71, which is actually a little more involved, 01:07:47.970 --> 01:07:50.280 in general. 01:07:50.280 --> 01:07:54.270 I'll just give you an idea of how you can see that, 01:07:54.270 --> 01:07:56.850 from using our axioms, that something like this 01:07:56.850 --> 01:07:58.810 better be true. 01:07:58.810 --> 01:08:01.980 So let's consider a special example 01:08:01.980 --> 01:08:04.530 that won't be enough to prove it for arbitrary alpha. 01:08:04.530 --> 01:08:07.800 This is any alpha. 01:08:07.800 --> 01:08:09.450 Let's consider a special example that 01:08:09.450 --> 01:08:13.140 at least will be enough to prove it for integers. 01:08:16.359 --> 01:08:19.729 So we'll consider k's that are integers. 01:08:19.729 --> 01:08:23.550 And we'll think of k's that are greater than 0. 01:08:23.550 --> 01:08:32.240 So if I just expand out this p plus q squared, 01:08:32.240 --> 01:08:35.640 then the first term is p to the 2k. 01:08:35.640 --> 01:08:39.900 Then I get some coefficient, p to the 2k minus 2, 01:08:39.900 --> 01:08:46.500 q squared, some coefficient, p to the 2k minus 4, 01:08:46.500 --> 01:08:50.550 q to the fourth, et cetera. 01:08:50.550 --> 01:08:52.600 In general, there's p dot q terms as well. 01:08:52.600 --> 01:08:54.830 But then I could do integral-- 01:08:54.830 --> 01:08:56.880 angular average and combine those together 01:08:56.880 --> 01:08:58.840 with these terms. 01:08:58.840 --> 01:09:01.140 And that's why I'm not being very explicit about what 01:09:01.140 --> 01:09:03.069 the coefficients are. 01:09:03.069 --> 01:09:06.367 But they're some positive numbers. 01:09:06.367 --> 01:09:08.700 Now, I could also take this integral, and I could shift. 01:09:08.700 --> 01:09:09.825 That was one of our axioms. 01:09:18.192 --> 01:09:21.359 And that is p to the 2k. 01:09:21.359 --> 01:09:25.200 So that means that all these terms here better be 0. 01:09:25.200 --> 01:09:41.279 And they have to be 0 for arbitrary q and arbitrary k, 01:09:41.279 --> 01:09:43.478 or any k under the assumptions that we used, 01:09:43.478 --> 01:09:45.520 which are that it's an integer and it's positive. 01:09:45.520 --> 01:09:47.979 So I could expand it in this way. 01:09:47.979 --> 01:09:50.790 And so therefore, we have all these integrals over p 01:09:50.790 --> 01:09:51.450 to the powers. 01:09:51.450 --> 01:09:52.920 And they better be 0. 01:09:52.920 --> 01:09:56.610 And that's enough to prove this for integer alphas-- 01:09:56.610 --> 01:09:57.780 positive integer alphas. 01:10:08.660 --> 01:10:10.280 OK. 01:10:10.280 --> 01:10:12.380 And so if you want to fill in between the integers 01:10:12.380 --> 01:10:14.030 and you want to do the negative cases, 01:10:14.030 --> 01:10:15.405 then you have to look at Collins. 01:10:15.405 --> 01:10:18.575 But you could do that, too. 01:10:18.575 --> 01:10:20.450 Then it requires a little more heavy lifting. 01:10:28.000 --> 01:10:31.000 There's one fact about this-- 01:10:31.000 --> 01:10:33.490 fact number one-- which is a little bit 01:10:33.490 --> 01:10:36.580 subtle, and you have to be careful. 01:10:36.580 --> 01:10:39.500 And it's worth noting. 01:10:39.500 --> 01:10:42.910 So let me do another example, which 01:10:42.910 --> 01:10:46.120 is by way of warning you that this can sometimes 01:10:46.120 --> 01:10:48.080 be dangerous. 01:10:48.080 --> 01:10:50.740 So let's think of a scalar-field theory 01:10:50.740 --> 01:10:52.810 and a simple loop diagram like this. 01:10:52.810 --> 01:10:57.055 But let's take 0 momentum and 0 mass. 01:11:00.032 --> 01:11:01.990 So if you do that, you'll encounter an integral 01:11:01.990 --> 01:11:03.620 that looks like this. 01:11:03.620 --> 01:11:05.290 There's two propagators, so I get a p 01:11:05.290 --> 01:11:10.240 to the fourth downstairs, and I get d d p. 01:11:10.240 --> 01:11:15.820 And that integral is 0, but it's 0 in a special way. 01:11:15.820 --> 01:11:18.820 It's 0 due to a cancellation between ultraviolet and 01:11:18.820 --> 01:11:21.850 infrared physics. 01:11:21.850 --> 01:11:23.830 I said that epsilon could be regulating 01:11:23.830 --> 01:11:25.450 both infrared divergences as well 01:11:25.450 --> 01:11:27.790 as ultraviolet divergences. 01:11:27.790 --> 01:11:29.620 If I only used epsilon to regulate 01:11:29.620 --> 01:11:33.190 ultraviolet divergences, I'd get a 1-over-epsilon uv. 01:11:33.190 --> 01:11:36.152 But in this integral, I'm actually using it to do both. 01:11:36.152 --> 01:11:38.110 It's regulating an infrared divergence as well. 01:11:38.110 --> 01:11:39.770 And it just comes in with the opposite sign. 01:11:39.770 --> 01:11:41.645 And since epsilon uv is equal to epsilon IR-- 01:11:44.950 --> 01:11:50.080 they're just notation to signify what region of physics 01:11:50.080 --> 01:11:51.730 is giving the divergence-- 01:11:51.730 --> 01:11:54.405 you get 0. 01:11:54.405 --> 01:11:55.780 But even though that's true, that 01:11:55.780 --> 01:11:59.140 doesn't mean you don't have to add a counterterm 01:11:59.140 --> 01:12:01.690 for this diagram, because counterterms are supposed 01:12:01.690 --> 01:12:06.550 to cancel ultraviolet divergences, not infrared ones, 01:12:06.550 --> 01:12:08.800 OK? 01:12:08.800 --> 01:12:16.870 So even though it's 0, you still need to add a counterterm, 01:12:16.870 --> 01:12:18.910 because the 0 is actually a cancellation 01:12:18.910 --> 01:12:22.760 between ultraviolet and infrared physics. 01:12:22.760 --> 01:12:25.435 So there's some counterterm. 01:12:28.230 --> 01:12:31.620 And it would be exactly of this sort, 01:12:31.620 --> 01:12:34.670 because this is the epsilon uv. 01:12:38.220 --> 01:12:44.160 And then if you add the bare diagram plus the counterterm, 01:12:44.160 --> 01:12:47.460 the answer is non-zero. 01:12:47.460 --> 01:12:50.040 You've canceled, if you like, the uv pole, 01:12:50.040 --> 01:12:53.130 and you've left over the IR pole. 01:12:53.130 --> 01:12:53.630 OK. 01:12:53.630 --> 01:12:55.490 So you have to be a bit careful about using 01:12:55.490 --> 01:12:57.140 dimensional regularization, because if you encounter 01:12:57.140 --> 01:12:59.265 scaleless integrals, it could be that they actually 01:12:59.265 --> 01:13:01.410 are affecting counterterm. 01:13:01.410 --> 01:13:03.740 And if you want to do some renormalization-group 01:13:03.740 --> 01:13:06.020 improvement of the theory or something like that, 01:13:06.020 --> 01:13:08.970 you have to be aware of this. 01:13:08.970 --> 01:13:11.420 If you know that all the infrared poles are going 01:13:11.420 --> 01:13:14.555 to cancel because you're looking at some infrared safe quantity, 01:13:14.555 --> 01:13:16.430 then you can be a little bit glib about this, 01:13:16.430 --> 01:13:18.347 because if they're going to cancel in the end, 01:13:18.347 --> 01:13:21.560 that means that any of these corresponding uv poles 01:13:21.560 --> 01:13:23.570 will also cancel. 01:13:23.570 --> 01:13:26.180 But it's not always the case that you're renormalizing 01:13:26.180 --> 01:13:31.010 operators that have no 1-over-epsilon IR's [INAUDIBLE] 01:13:31.010 --> 01:13:33.300 have to be careful about this. 01:13:33.300 --> 01:13:35.990 So this is a subtlety that sometimes people get wrong 01:13:35.990 --> 01:13:38.960 when they write papers. 01:13:38.960 --> 01:13:41.450 So dim reg is beautiful, but there are some things about it 01:13:41.450 --> 01:13:42.658 that are a little bit tricky. 01:13:46.600 --> 01:13:51.070 Another thing that can be confusing about dim reg 01:13:51.070 --> 01:13:58.053 is that it does this, that it regulates both uv and IR poles. 01:13:58.053 --> 01:14:00.220 And even though it's doing that, and even though you 01:14:00.220 --> 01:14:04.090 need different values of epsilon, 01:14:04.090 --> 01:14:08.500 if you want to do that, it's actually 01:14:08.500 --> 01:14:10.450 still a well-defined procedure, even 01:14:10.450 --> 01:14:13.260 in the presence of uv and IR poles, 01:14:13.260 --> 01:14:15.010 even if they're both in the same integral. 01:14:17.680 --> 01:14:24.190 And basically, you're using analytic continuation here. 01:14:28.930 --> 01:14:31.590 So let me give you a little example which 01:14:31.590 --> 01:14:37.570 is not exactly related to this but will allow me to show you 01:14:37.570 --> 01:14:39.670 both how you use analytic continuation 01:14:39.670 --> 01:14:43.630 and how you could think about separating uv and IR poles. 01:14:43.630 --> 01:14:45.520 So we'll start out just by thinking 01:14:45.520 --> 01:14:47.710 about analytic continuation. 01:14:47.710 --> 01:14:49.240 So suppose I had some integral. 01:14:57.320 --> 01:14:59.620 So what does analytic continuation mean? 01:14:59.620 --> 01:15:02.340 Or, how should I construct it? 01:15:02.340 --> 01:15:04.850 So let me, again, write it in a way where I've separated 01:15:04.850 --> 01:15:05.870 out the radial integral. 01:15:10.250 --> 01:15:12.290 And let's suppose that this integral here 01:15:12.290 --> 01:15:17.130 is perfectly well-defined for d in some range. 01:15:17.130 --> 01:15:29.360 So this is well-defined for some range of positive d. 01:15:29.360 --> 01:15:32.390 And then let's say that we wanted to continue 01:15:32.390 --> 01:15:33.680 that integral to negative d. 01:15:42.705 --> 01:15:44.450 Now, the problem with negative d may 01:15:44.450 --> 01:15:46.160 be that, when you get to negative d, 01:15:46.160 --> 01:15:48.438 you're getting some infrared divergences, 01:15:48.438 --> 01:15:50.480 and you have to figure out how to deal with them. 01:15:58.340 --> 01:16:01.600 And you can do this, if you'd like, step by step. 01:16:04.340 --> 01:16:06.820 So if we wanted to extend the lower limit down to minus 2 01:16:06.820 --> 01:16:09.959 from 0, then we would do the following. 01:16:14.650 --> 01:16:17.740 We take our integral, write it out 01:16:17.740 --> 01:16:27.730 in this angular/radial separation, 01:16:27.730 --> 01:16:33.700 split up, in the radial variable, the piece that's 01:16:33.700 --> 01:16:36.623 ultraviolet, which is the high-momentum piece, 01:16:36.623 --> 01:16:37.915 from the piece that's infrared. 01:16:50.975 --> 01:16:52.350 And in the piece that's infrared, 01:16:52.350 --> 01:16:55.290 we could also just do some addition and some subtraction 01:16:55.290 --> 01:16:57.840 to make it more convergent as p goes to 0. 01:16:57.840 --> 01:17:00.600 So for example, we could subtract f at 0. 01:17:00.600 --> 01:17:03.810 This thing would fall off faster and hence, give 01:17:03.810 --> 01:17:05.110 more powers of p. 01:17:05.110 --> 01:17:09.300 And we could make it more convergent at 0. 01:17:09.300 --> 01:17:15.300 And then we just integrate the subtraction up to the cutoff c. 01:17:17.950 --> 01:17:20.950 And the idea of introducing this cutoff c 01:17:20.950 --> 01:17:23.680 is that we split the uv piece and the ultraviolet piece. 01:17:27.470 --> 01:17:29.650 And so we can do one kind of continuation 01:17:29.650 --> 01:17:32.860 for d up here, making epsilon positive, 01:17:32.860 --> 01:17:34.700 one kind of continuation down here. 01:17:34.700 --> 01:17:36.700 And the result, when we put these back together, 01:17:36.700 --> 01:17:39.370 is independent of c, OK? 01:17:39.370 --> 01:17:43.210 So that's the sense, actually, in which what I said up here 01:17:43.210 --> 01:17:43.750 is-- 01:17:43.750 --> 01:17:46.090 that you can use it for both uv and IR divergences, 01:17:46.090 --> 01:17:48.132 because you could always introduce some parameter 01:17:48.132 --> 01:17:48.932 c to split-- 01:17:48.932 --> 01:17:51.140 they're occurring in different regions of space base, 01:17:51.140 --> 01:17:56.740 so you could always split them up, regulate each one 01:17:56.740 --> 01:18:00.680 with different values of d, and then put them back together. 01:18:00.680 --> 01:18:03.370 And the answer, when you put them back together, 01:18:03.370 --> 01:18:04.360 is independent of c. 01:18:08.040 --> 01:18:08.970 OK. 01:18:08.970 --> 01:18:11.100 Now, if you wanted to define-- 01:18:11.100 --> 01:18:12.953 that was one of our goals. 01:18:12.953 --> 01:18:14.370 The other one was just to show you 01:18:14.370 --> 01:18:16.980 what we mean by analytic continuation. 01:18:16.980 --> 01:18:19.290 So since it's independent of c, then you 01:18:19.290 --> 01:18:21.970 could do the following. 01:18:21.970 --> 01:18:27.390 So for minus 2 less than d less than 0, 01:18:27.390 --> 01:18:30.410 let's take c goes to infinity. 01:18:33.490 --> 01:18:35.640 And so for that particular range, what you 01:18:35.640 --> 01:18:38.340 find, then, if you take that limit-- 01:18:55.540 --> 01:18:57.430 Well, because of the c to the minus d, 01:18:57.430 --> 01:18:59.320 the c is downstairs if d is negative. 01:18:59.320 --> 01:19:03.550 So that term goes away, and you're just 01:19:03.550 --> 01:19:06.860 left with this term. 01:19:06.860 --> 01:19:08.870 This term here goes away, too, because c 01:19:08.870 --> 01:19:12.080 approaches the upper limit, and everything is regulated. 01:19:12.080 --> 01:19:15.750 And so I just would be left with that. 01:19:15.750 --> 01:19:16.250 OK. 01:19:16.250 --> 01:19:17.480 Making d negative is-- 01:19:20.220 --> 01:19:20.720 OK. 01:19:20.720 --> 01:19:22.480 So that would be the definition. 01:19:22.480 --> 01:19:24.188 So you can see some of the kind of tricks 01:19:24.188 --> 01:19:26.272 that you could use with dimensional regularization 01:19:26.272 --> 01:19:27.680 or adding and subtracting terms. 01:19:30.780 --> 01:19:33.050 And these are the things that are valid things to do. 01:19:39.580 --> 01:19:40.800 Any questions about that? 01:19:43.970 --> 01:19:45.410 OK. 01:19:45.410 --> 01:19:48.740 So when we do dimensional regularization in MS bar, 01:19:48.740 --> 01:19:50.720 you're used to doing that for a gauge theory. 01:19:50.720 --> 01:19:53.420 That's what you've learned about. 01:19:53.420 --> 01:19:55.670 But you can also do it for any effective field theory. 01:19:55.670 --> 01:19:57.870 And the logic is the same. 01:19:57.870 --> 01:20:00.770 So let me remind you of the gauge-theory logic 01:20:00.770 --> 01:20:04.490 and then just tell you how you would define MS bar precisely 01:20:04.490 --> 01:20:08.690 for the fermionic effective theory with the "psi bar psi 01:20:08.690 --> 01:20:12.380 squared" operator that we had. 01:20:12.380 --> 01:20:16.560 So we talked about dimensional regularization. 01:20:16.560 --> 01:20:18.190 Let's talk about the MS scheme. 01:20:30.530 --> 01:20:34.550 So if we've set up our effective theory in a way 01:20:34.550 --> 01:20:39.510 where we've made the mass scale explicit, 01:20:39.510 --> 01:20:41.135 which is often a nice thing to do-- 01:20:41.135 --> 01:20:43.260 and we did that when we set up the effective theory 01:20:43.260 --> 01:20:46.303 where you have the capital M showing up explicitly. 01:20:46.303 --> 01:20:47.720 If you do that, then the couplings 01:20:47.720 --> 01:20:49.370 start out dimensionless. 01:20:49.370 --> 01:20:52.640 And that's a nice thing, to have dimensionless coupling. 01:20:52.640 --> 01:20:54.290 And the MS scheme is simply the scheme 01:20:54.290 --> 01:20:56.150 where you want to introduce a scale 01:20:56.150 --> 01:20:59.520 to keep the renormalized couplings dimensionless. 01:20:59.520 --> 01:21:04.490 So the example you're familiar with 01:21:04.490 --> 01:21:06.370 is just having a gauge coupling. 01:21:09.817 --> 01:21:11.900 And if you go through the dimensions of the fields 01:21:11.900 --> 01:21:14.450 here, which I do in my notes, but I'm 01:21:14.450 --> 01:21:17.900 going to assume that you've got some familiarity with this, 01:21:17.900 --> 01:21:22.520 you find that the bare coupling has dimension epsilon. 01:21:22.520 --> 01:21:25.618 And so you define a renormalized coupling as dimensionless. 01:21:29.675 --> 01:21:31.675 And you introduce a factor of mu to the epsilon. 01:21:35.367 --> 01:21:39.790 So you say g bare, which has dimensions, 01:21:39.790 --> 01:21:43.540 some dimensionless z factor, some mu to the epsilon 01:21:43.540 --> 01:21:44.750 to make up those dimensions. 01:21:44.750 --> 01:21:47.710 And then left over is the renormalized coupling. 01:21:54.210 --> 01:21:56.520 And the idea and the strategy for any other coupling 01:21:56.520 --> 01:21:58.020 in the effective theory is the same, 01:21:58.020 --> 01:22:00.400 so let's do one other example. 01:22:00.400 --> 01:22:07.698 So take our [? dimension-six ?] a bare over capital M squared, 01:22:07.698 --> 01:22:10.980 psi bar psi squared. 01:22:10.980 --> 01:22:17.817 Do dimension counting on this guy, which 01:22:17.817 --> 01:22:18.900 I do, again, in the notes. 01:22:18.900 --> 01:22:21.600 But if you go through that dimension 01:22:21.600 --> 01:22:24.270 counting, and you remember that the dimensions for the fermions 01:22:24.270 --> 01:22:26.498 are assigned by the kinetic term-- 01:22:26.498 --> 01:22:28.540 so we have a dimension counting for the fermions. 01:22:28.540 --> 01:22:31.050 We know that this is dimension minus 2. 01:22:31.050 --> 01:22:32.670 The whole thing has to add up to d. 01:22:32.670 --> 01:22:34.950 So that tells us what the a bare is. 01:22:34.950 --> 01:22:38.390 And we get 4 minus d, in this case. 01:22:38.390 --> 01:22:45.000 And so then we can write down a formula analogous to that one 01:22:45.000 --> 01:22:47.490 but for the a coefficient. 01:22:47.490 --> 01:22:48.705 And it's mu to the 2 epsilon. 01:22:53.540 --> 01:22:54.050 OK. 01:22:54.050 --> 01:22:56.060 So it's as simple as that. 01:23:14.010 --> 01:23:15.560 So in looking at the action where 01:23:15.560 --> 01:23:17.180 this is a term in the Lagrangian, 01:23:17.180 --> 01:23:19.520 we want to ensure that, when we go to the-- 01:23:19.520 --> 01:23:21.440 that we figure out the dimensions of this guy 01:23:21.440 --> 01:23:23.120 in dimensional regularization. 01:23:23.120 --> 01:23:26.233 That's 4 minus d, which we determine from knowing 01:23:26.233 --> 01:23:27.150 the other pieces here. 01:23:27.150 --> 01:23:30.840 And then we just make a redefinition 01:23:30.840 --> 01:23:33.810 to give a dimensionless coupling. 01:23:33.810 --> 01:23:36.450 So that's how MS bar will work. 01:23:36.450 --> 01:23:40.920 Well, this is MS, but this is how minimal subtraction 01:23:40.920 --> 01:23:44.100 works for defining all the operators that you may have. 01:23:49.990 --> 01:23:55.510 And then minimal subtraction is simply a rescaling. 01:23:55.510 --> 01:24:00.070 And that's the same as it is in gauge theory, where we get rid 01:24:00.070 --> 01:24:03.690 of some annoying factors, and [? g is ?] a slightly 01:24:03.690 --> 01:24:05.930 different definition. 01:24:05.930 --> 01:24:09.670 So this was MS, and this is MS bar. 01:24:09.670 --> 01:24:10.290 OK. 01:24:10.290 --> 01:24:13.410 So it really works in a very similar way to gauge theory. 01:24:13.410 --> 01:24:15.450 And you figure out the factors of mu 01:24:15.450 --> 01:24:18.270 to the epsilon to include in your calculation this way. 01:24:18.270 --> 01:24:20.520 Sometimes you see books do it by saying 01:24:20.520 --> 01:24:24.060 the loop measure is continued within mu to the epsilon. 01:24:24.060 --> 01:24:25.650 That's not right. 01:24:25.650 --> 01:24:27.810 This is right. 01:24:27.810 --> 01:24:31.200 If you do that, you'll get into trouble-- 01:24:31.200 --> 01:24:32.040 not always. 01:24:32.040 --> 01:24:33.790 That's why the books can get away with it. 01:24:33.790 --> 01:24:37.020 But in general, you'll get into trouble. 01:24:37.020 --> 01:24:37.590 All right. 01:24:37.590 --> 01:24:39.607 So we should stop there.