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 --> 00:00:17.520 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.520 --> 00:00:18.370 ocw.mit.edu. 00:00:22.682 --> 00:00:24.140 IAIN STEWART: So last time, we were 00:00:24.140 --> 00:00:27.530 talking about the chiral Lagrangian as another example 00:00:27.530 --> 00:00:29.368 of an effective field theory to illustrate 00:00:29.368 --> 00:00:31.160 some of the tools of effective field theory 00:00:31.160 --> 00:00:32.660 that we didn't meet when integrating 00:00:32.660 --> 00:00:33.955 our heavy particles. 00:00:33.955 --> 00:00:36.820 It was an example of a bottom-up effective field theory. 00:00:36.820 --> 00:00:38.570 So we were constructing it from the bottom 00:00:38.570 --> 00:00:42.320 up for the most part. 00:00:42.320 --> 00:00:44.810 That's how one should think about it. 00:00:44.810 --> 00:00:46.563 There was several things in our program 00:00:46.563 --> 00:00:48.480 to understand from chiral perturbation theory. 00:00:48.480 --> 00:00:50.240 And we kind of got halfway through. 00:00:50.240 --> 00:00:52.160 So we want to understand nonlinear symmetry 00:00:52.160 --> 00:00:53.065 representations. 00:00:53.065 --> 00:00:54.440 And we went through how you could 00:00:54.440 --> 00:00:56.630 think about linear representations 00:00:56.630 --> 00:00:59.780 and making changes of variable and putting 00:00:59.780 --> 00:01:02.000 the Lagrangian in a form where you could see where 00:01:02.000 --> 00:01:05.910 the nonlinear realization was coming from very explicitly, 00:01:05.910 --> 00:01:08.540 as well as another way where we constructed it 00:01:08.540 --> 00:01:12.800 from the bottom up thinking about parameterizing 00:01:12.800 --> 00:01:14.720 the co-set. 00:01:14.720 --> 00:01:16.650 And then we started talking about loops 00:01:16.650 --> 00:01:17.900 in chiral perturbation theory. 00:01:17.900 --> 00:01:20.883 And I gave you this example of this loop here. 00:01:20.883 --> 00:01:22.550 So we constructed the chiral Lagrangian. 00:01:22.550 --> 00:01:24.260 That's what I've written here. 00:01:24.260 --> 00:01:25.640 In terms of the sigma field, this 00:01:25.640 --> 00:01:30.110 is the nonlinear version, sigma field, exponential of M field. 00:01:30.110 --> 00:01:32.690 M field has the pi n fields in it. 00:01:32.690 --> 00:01:35.360 I rewrote it in a little bit of a different notation, which 00:01:35.360 --> 00:01:37.550 will be useful today, where I introduced 00:01:37.550 --> 00:01:39.200 this thing called chi. 00:01:39.200 --> 00:01:41.840 But f squared over 8 chi is just Mq. 00:01:41.840 --> 00:01:44.042 So it's just a rescaling. 00:01:44.042 --> 00:01:46.250 And I just wanted to have the same kind of pre-factor 00:01:46.250 --> 00:01:47.750 here and here. 00:01:47.750 --> 00:01:49.430 That's why I did that. 00:01:49.430 --> 00:01:54.260 The power counting is in p squared derivatives or M pi 00:01:54.260 --> 00:01:56.480 squared, which is the same as Mq. 00:01:56.480 --> 00:01:59.690 And the reason that M pi squared and Mq count the same 00:01:59.690 --> 00:02:03.260 is there's this relation between pi squared and the quark 00:02:03.260 --> 00:02:04.910 masses. 00:02:04.910 --> 00:02:06.890 And then we started talking about loops. 00:02:06.890 --> 00:02:09.710 And we said that, if we looked at this loop, 00:02:09.710 --> 00:02:12.350 the result for this loop diagram would be a factor of p squared 00:02:12.350 --> 00:02:14.142 or M pi squared times a factor of p squared 00:02:14.142 --> 00:02:17.000 or M pi squared divided by 4 pi f. 00:02:24.140 --> 00:02:26.450 This is the same size as the lowest order, 00:02:26.450 --> 00:02:28.190 so the suppression factor is this. 00:02:45.755 --> 00:02:47.380 So we can say that loops are suppressed 00:02:47.380 --> 00:02:49.600 by p squared over some scale lambda chi squared. 00:02:49.600 --> 00:02:51.760 So this is scale of our expansion. 00:02:51.760 --> 00:02:54.460 And we can associate that scale to 4 pi times f. 00:02:57.590 --> 00:02:59.870 OK, so we're going to continue with that story 00:02:59.870 --> 00:03:03.970 unless there's any questions from last time. 00:03:03.970 --> 00:03:08.920 OK, so we want to use dimensional regularization. 00:03:08.920 --> 00:03:12.670 And if we use dimensional regularization, 00:03:12.670 --> 00:03:14.710 we can think about having an MS bar scheme. 00:03:17.620 --> 00:03:19.890 So how do we do that? 00:03:19.890 --> 00:03:21.640 It's a little different than gauge theory, 00:03:21.640 --> 00:03:23.810 but the idea is exactly the same. 00:03:23.810 --> 00:03:25.540 So we look at the dimensions of objects. 00:03:25.540 --> 00:03:30.910 This capital M-- supposed to be a capital M-- 00:03:30.910 --> 00:03:36.310 is like a scalar field, so it has dimension 1 minus epsilon. 00:03:36.310 --> 00:03:39.520 The decay constant then, if you look 00:03:39.520 --> 00:03:42.760 at it, which is the coupling, has to have dimension 00:03:42.760 --> 00:03:43.953 1 minus epsilon 2. 00:03:43.953 --> 00:03:46.120 And you can see that because this exponential better 00:03:46.120 --> 00:03:47.900 be the exponential of something dimensionless 00:03:47.900 --> 00:03:49.442 so that, whatever the dimension of M, 00:03:49.442 --> 00:03:54.220 should cancel the dimension of f. 00:03:54.220 --> 00:03:56.740 And then that works out with the look with the measure 00:03:56.740 --> 00:03:57.820 as well, which has a ddx. 00:04:01.180 --> 00:04:07.880 So therefore, just like you did for a gauge coupling, 00:04:07.880 --> 00:04:11.590 you can say f bare in some mu to the minus epsilon times f. 00:04:24.770 --> 00:04:27.830 And if you want to do the same thing for the second term, 00:04:27.830 --> 00:04:30.500 though I've put the V0 inside the-- 00:04:30.500 --> 00:04:33.200 sorry, there's something missing in my equation. 00:04:33.200 --> 00:04:36.300 There was a V0 here. 00:04:36.300 --> 00:04:38.300 So before last time, we've written this equation 00:04:38.300 --> 00:04:40.790 as V0 out front and then Mq inside. 00:04:40.790 --> 00:04:43.940 And I just rescaled it to this chi. 00:04:43.940 --> 00:04:47.030 OK, so you can think either in terms of chi. 00:04:47.030 --> 00:04:53.960 Chi is kind of absorbing the parameter along with the Mq, 00:04:53.960 --> 00:04:55.400 but they're equivalent things. 00:04:55.400 --> 00:04:58.340 So you can think of also doing a rescaling that 00:04:58.340 --> 00:05:00.410 gets the dimension of this term right. 00:05:00.410 --> 00:05:03.980 And the thing that you want is this. 00:05:03.980 --> 00:05:06.290 The P0 would then be mu to the minus 2 epsilon 00:05:06.290 --> 00:05:08.960 to compensate for the fact that the first term had an f squared 00:05:08.960 --> 00:05:10.402 and the second term just had a V0. 00:05:13.097 --> 00:05:14.680 But that's just to dimension counting. 00:05:32.820 --> 00:05:35.820 There's also no mus in physical quantities like the pi n 00:05:35.820 --> 00:05:38.850 mass, which is an observable, which 00:05:38.850 --> 00:05:41.520 has just a definite value. 00:05:41.520 --> 00:05:45.680 And there's also, in chiral perturbation theory, 00:05:45.680 --> 00:05:48.360 no mus in the quark mass. 00:05:48.360 --> 00:05:51.780 So when you look at V0 over f squared, 00:05:51.780 --> 00:05:58.491 if you look at it bare, that's the same as renormalized. 00:06:02.340 --> 00:06:05.340 The other thing that's different about this than in a gauge 00:06:05.340 --> 00:06:07.470 theory is, in the gauge theory, you'd have a z. 00:06:07.470 --> 00:06:11.250 You'd have g bare is equal to e to the epsilon times g, 00:06:11.250 --> 00:06:13.343 but then you'd have a zg. 00:06:13.343 --> 00:06:14.760 But in chiral perturbation theory, 00:06:14.760 --> 00:06:16.800 the way that the loops work, the loops 00:06:16.800 --> 00:06:20.940 aren't renormalizing the leading order Lagrangian. 00:06:20.940 --> 00:06:23.070 The loops are renormalizing something else 00:06:23.070 --> 00:06:25.560 because they ended up being suppressed. 00:06:25.560 --> 00:06:29.265 So you don't need counter-terms here for the leading order 00:06:29.265 --> 00:06:30.840 Lagrangian. 00:06:30.840 --> 00:06:33.510 And that's why I didn't put a zf or a zV0 here. 00:06:37.680 --> 00:06:39.210 OK, but when you do this loop, you 00:06:39.210 --> 00:06:41.377 do get ultraviolet divergences. 00:06:53.070 --> 00:06:54.758 So you get 1 over epsilon divergence 00:06:54.758 --> 00:06:56.550 if you're using dimensional regularization. 00:06:56.550 --> 00:06:59.460 And it comes along with the log of mu squared. 00:06:59.460 --> 00:07:01.710 And then it'll be divided by some scale 00:07:01.710 --> 00:07:05.400 that you have in your loop, like an external momentum p squared, 00:07:05.400 --> 00:07:11.490 or it could be the pi n mass. 00:07:11.490 --> 00:07:13.862 Both of those will show up generically. 00:07:24.960 --> 00:07:35.592 And in the way we're doing counting, the way the loops are 00:07:35.592 --> 00:07:37.800 showing up, if I can bind together the factors that I 00:07:37.800 --> 00:07:42.480 told you about here, it's either p to the fourth, 4 powers of p, 00:07:42.480 --> 00:07:46.920 or 2 of p and 2 of M pi or 4 powers of M pi. 00:07:46.920 --> 00:07:49.020 So to cancel that 1 over epsilon, 00:07:49.020 --> 00:07:50.542 we need a counter-term. 00:07:50.542 --> 00:07:52.500 But it's not a counter-term in that Lagrangian. 00:07:52.500 --> 00:07:54.843 It's a counter term in the higher order Lagrangian. 00:08:00.640 --> 00:08:03.010 So we're not done by just calculating the loop. 00:08:03.010 --> 00:08:05.527 We have to actually include some higher dimension 00:08:05.527 --> 00:08:07.860 operators that are the same order in our power counting. 00:08:19.240 --> 00:08:21.130 So we'll talk about SU(3) a little later. 00:08:21.130 --> 00:08:24.010 I'll start out by talking about SU(2). 00:08:24.010 --> 00:08:28.240 In SU(2), the form of that Lagrangian with higher order 00:08:28.240 --> 00:08:32.140 terms is just taking what we had before and just 00:08:32.140 --> 00:08:33.991 going to a higher dimension. 00:08:37.090 --> 00:08:40.330 So we could have two more derivatives. 00:08:40.330 --> 00:08:43.809 One way of doing that is taking what we had at lowest order 00:08:43.809 --> 00:08:44.680 and squaring it. 00:08:51.880 --> 00:08:54.820 Another way of doing it would be to take and contract 00:08:54.820 --> 00:08:56.420 the indices a little bit differently. 00:08:56.420 --> 00:09:00.490 So I could do it like this. 00:09:03.520 --> 00:09:07.360 The trace is cyclic, so I can move things around from back 00:09:07.360 --> 00:09:08.530 to front. 00:09:08.530 --> 00:09:10.330 But I could have mu nu in one trace 00:09:10.330 --> 00:09:12.902 and mu nu in another trace rather than just contracting 00:09:12.902 --> 00:09:13.610 within the trace. 00:09:13.610 --> 00:09:17.050 That's another possible term. 00:09:17.050 --> 00:09:18.550 And there's a bunch more terms which 00:09:18.550 --> 00:09:22.570 I decided to wait and enumerate when we do SU(3) to give you 00:09:22.570 --> 00:09:24.550 a full enumeration. 00:09:24.550 --> 00:09:32.800 These additional terms you can build 00:09:32.800 --> 00:09:39.250 from one Mq, which means one of these chi fields or chi 00:09:39.250 --> 00:09:45.490 objects and two partials. 00:09:45.490 --> 00:09:54.340 That's one possibility or two Mq's because each Mq 00:09:54.340 --> 00:09:56.222 counts like two derivatives. 00:09:56.222 --> 00:09:58.180 So there's some other terms that we could build 00:09:58.180 --> 00:09:59.873 that are analogs of this term. 00:09:59.873 --> 00:10:01.540 You could square that term, for example. 00:10:01.540 --> 00:10:04.420 That would be a possible higher order term. 00:10:04.420 --> 00:10:07.360 And so these coefficients of this Lagrangian 00:10:07.360 --> 00:10:10.300 are what we need to cancel off the divergences. 00:10:10.300 --> 00:10:12.490 So the counter-terms that we have 00:10:12.490 --> 00:10:14.200 that renormalize these loop graphs 00:10:14.200 --> 00:10:15.862 come from this Lagrangian. 00:10:38.020 --> 00:10:42.220 So the theory is renormalizable in a EFT sense, order 00:10:42.220 --> 00:10:44.800 by order in its power counting expansion. 00:10:44.800 --> 00:10:46.740 When you go to order p to the fourth, 00:10:46.740 --> 00:10:48.490 you have to consistently put in everything 00:10:48.490 --> 00:10:50.110 that's order p to the fourth. 00:10:50.110 --> 00:10:53.110 That includes the loops, which are p to the fourth, 00:10:53.110 --> 00:10:59.260 as well as new local Lagrangian interactions. 00:10:59.260 --> 00:11:01.580 AUDIENCE: Is that a choice to not renormalize 00:11:01.580 --> 00:11:03.370 the f bare and [? d ?] bare? 00:11:03.370 --> 00:11:05.620 IAIN STEWART: They just don't get renormalized. 00:11:05.620 --> 00:11:07.870 AUDIENCE: But why couldn't I just multiply by 1 plus-- 00:11:07.870 --> 00:11:08.620 IAIN STEWART: Oh-- 00:11:08.620 --> 00:11:12.072 AUDIENCE: --order p squared by times delta [INAUDIBLE].. 00:11:12.072 --> 00:11:13.780 IAIN STEWART: Oh, you want to try to make 00:11:13.780 --> 00:11:15.010 a different scheme for them? 00:11:15.010 --> 00:11:17.302 AUDIENCE: I'd screw up the power counting [INAUDIBLE].. 00:11:19.930 --> 00:11:25.890 IAIN STEWART: So effectively, you 00:11:25.890 --> 00:11:27.890 could screw up the power counting by doing that. 00:11:27.890 --> 00:11:31.690 So let me tell you what you can do. 00:11:31.690 --> 00:11:33.400 You could put a 2 here, right? 00:11:36.670 --> 00:11:38.530 But you don't really have much more freedom 00:11:38.530 --> 00:11:40.270 than just multiplying by a number. 00:11:40.270 --> 00:11:41.895 AUDIENCE: So I can't multiply by 1 plus 00:11:41.895 --> 00:11:43.798 p squared over s squared. 00:11:43.798 --> 00:11:44.590 IAIN STEWART: Yeah. 00:11:44.590 --> 00:11:46.548 That would be a screwing up the power counting. 00:11:46.548 --> 00:11:50.945 And effectively, you wouldn't be multiplying by p squared. 00:11:50.945 --> 00:11:52.570 You'd be putting derivatives in, right? 00:11:52.570 --> 00:11:53.860 Because there's no p. 00:11:53.860 --> 00:11:58.893 So p is not something that you're allowed to multiply by. 00:11:58.893 --> 00:12:00.310 AUDIENCE: So that's the only thing 00:12:00.310 --> 00:12:02.680 that's stopping me-- is what is derivatives [INAUDIBLE]?? 00:12:02.680 --> 00:12:03.850 IAIN STEWART: You could put derivatives in, 00:12:03.850 --> 00:12:05.440 but then that's equivalent to something here. 00:12:05.440 --> 00:12:08.160 So you have to ask what you're doing because you're mixing up 00:12:08.160 --> 00:12:08.660 things. 00:12:08.660 --> 00:12:09.310 AUDIENCE: OK. 00:12:09.310 --> 00:12:09.720 IAIN STEWART: Yeah. 00:12:09.720 --> 00:12:11.030 AUDIENCE: [INAUDIBLE] definitely a better way to do. 00:12:11.030 --> 00:12:13.180 I was just wondering if there was sort of freedom. 00:12:13.180 --> 00:12:15.097 IAIN STEWART: Yeah, there's no freedom really. 00:12:15.097 --> 00:12:15.890 AUDIENCE: Good. 00:12:15.890 --> 00:12:18.432 AUDIENCE: Why you don't get the renormalization [INAUDIBLE],, 00:12:18.432 --> 00:12:21.312 like from [INAUDIBLE] [? loop? ?] 00:12:21.312 --> 00:12:23.770 IAIN STEWART: Because all the loops end up being suppressed 00:12:23.770 --> 00:12:27.980 by p squared or M pi squared. 00:12:27.980 --> 00:12:33.220 So if you write down any diagram and you think it might give 00:12:33.220 --> 00:12:38.210 something here, even if it's a two-point function, 00:12:38.210 --> 00:12:40.825 it just doesn't. 00:12:40.825 --> 00:12:43.870 AUDIENCE: [INAUDIBLE] to 4 [INAUDIBLE].. 00:12:43.870 --> 00:12:45.560 IAIN STEWART: That's right. 00:12:45.560 --> 00:12:48.940 Oh, so if you want, you could think that some terms here 00:12:48.940 --> 00:12:51.670 are kind of corrections to the kinetic term 00:12:51.670 --> 00:12:54.200 if you derive the equation of motion 00:12:54.200 --> 00:13:03.730 and you include L and that L and this L. But this Lagrangian, 00:13:03.730 --> 00:13:06.130 there's no loop corrections to this Lagrangian. 00:13:06.130 --> 00:13:09.820 There's no corrections to this lowest order Lagrangian. 00:13:09.820 --> 00:13:12.460 And that's because the loops are all suppressed. 00:13:12.460 --> 00:13:13.960 Totally different then gauge theory, 00:13:13.960 --> 00:13:16.310 loops are suppressed by p squared or M pi squared. 00:13:16.310 --> 00:13:18.460 So they only renormalize some high order things. 00:13:18.460 --> 00:13:24.670 And we don't have to worry about really thinking about f. 00:13:24.670 --> 00:13:27.850 I mean, the way I've talked about f here 00:13:27.850 --> 00:13:29.650 and the factors of mu is just in order 00:13:29.650 --> 00:13:32.530 to see where these factors and mu come out here. 00:13:32.530 --> 00:13:38.702 But what actually happens is that you get 1 over epsilons 00:13:38.702 --> 00:13:40.660 that are cancelled like the counter-terms here. 00:13:40.660 --> 00:13:45.700 And there's no need from a renormalization perspective 00:13:45.700 --> 00:13:47.655 to sort of change your definitions here. 00:13:47.655 --> 00:13:49.280 People play with different definitions, 00:13:49.280 --> 00:13:50.905 but it's like they use f squared over 4 00:13:50.905 --> 00:13:52.420 instead of f squared over 8. 00:13:52.420 --> 00:13:54.700 That's the extent of what people do 00:13:54.700 --> 00:13:56.965 with playing with the leading order Lagrangian. 00:13:56.965 --> 00:13:58.420 AUDIENCE: So you have [INAUDIBLE] the same thing with 00:13:58.420 --> 00:13:59.830 the [INAUDIBLE] that [INAUDIBLE] [? is not ?] 00:13:59.830 --> 00:14:01.212 [? renormalizing ?] [INAUDIBLE]. 00:14:01.212 --> 00:14:02.920 IAIN STEWART: Both of them are not, yeah. 00:14:02.920 --> 00:14:03.970 Yeah. 00:14:03.970 --> 00:14:07.270 So loops that you build out of these interactions 00:14:07.270 --> 00:14:16.418 end up renormalizing these terms, which is kind of neat. 00:14:16.418 --> 00:14:19.830 AUDIENCE: That's just because [INAUDIBLE].. 00:14:19.830 --> 00:14:22.746 That's just because there's [INAUDIBLE] [? sort of ?] 00:14:22.746 --> 00:14:24.210 [INAUDIBLE]. 00:14:24.210 --> 00:14:25.840 IAIN STEWART: Yeah. 00:14:25.840 --> 00:14:29.160 Well, so we'll prove in a minute a general power counting 00:14:29.160 --> 00:14:32.570 formula that tells us how to organize all this. 00:14:32.570 --> 00:14:34.920 So for now, you can think of it as an observation, 00:14:34.920 --> 00:14:38.970 but we'll build it into a general formula that tells you 00:14:38.970 --> 00:14:41.573 sort of how you would organize the power counting 00:14:41.573 --> 00:14:42.990 and renormalization of this theory 00:14:42.990 --> 00:14:46.810 to all orders in its expansion in a minute. 00:14:46.810 --> 00:14:49.060 AUDIENCE: So when you write down all these [INAUDIBLE] 00:14:49.060 --> 00:14:51.680 at [? power ?] [? p4, ?] does that include all 00:14:51.680 --> 00:14:52.800 the [INAUDIBLE] you need? 00:14:52.800 --> 00:14:53.610 IAIN STEWART: Yeah. 00:14:53.610 --> 00:14:55.530 That's right. 00:14:55.530 --> 00:14:57.270 Once I've enumerated all the terms, 00:14:57.270 --> 00:14:59.310 which I will do for you with SU(3)-- 00:14:59.310 --> 00:15:02.580 so it includes things like this and things like this as well-- 00:15:02.580 --> 00:15:04.710 then those are all the possible counter-terms 00:15:04.710 --> 00:15:07.278 that you could actually need to renormalize 00:15:07.278 --> 00:15:08.820 any of the one-loop diagrams that you 00:15:08.820 --> 00:15:12.420 could generate with the leading order Lagrangian. 00:15:12.420 --> 00:15:15.067 And you know that by power counting. 00:15:15.067 --> 00:15:16.650 Once you know the loops are that order 00:15:16.650 --> 00:15:19.830 and you've included all local interactions, then you're done. 00:15:23.160 --> 00:15:28.860 So there's things I didn't write down there, so just 00:15:28.860 --> 00:15:30.930 comment about that. 00:15:30.930 --> 00:15:34.747 So what is the equation of motion here? 00:15:34.747 --> 00:15:37.080 It's a little more complicated because we have the sigma 00:15:37.080 --> 00:15:40.680 field, but you can work out that the equation of motion 00:15:40.680 --> 00:15:41.460 is the following. 00:15:57.580 --> 00:16:01.440 And I've given you some reading, so you 00:16:01.440 --> 00:16:06.000 can read about a derivation of this equation which 00:16:06.000 --> 00:16:07.590 takes a little effort. 00:16:07.590 --> 00:16:09.330 So your leading order equation of motion 00:16:09.330 --> 00:16:12.120 basically allows you to get rid of partial squareds 00:16:12.120 --> 00:16:13.590 on the sigma. 00:16:13.590 --> 00:16:16.350 And that's why I've always written 00:16:16.350 --> 00:16:18.887 one partial on each sigma. 00:16:18.887 --> 00:16:20.970 So that's what you should think of this guy doing. 00:16:29.860 --> 00:16:31.500 There's also some SU(2) identities 00:16:31.500 --> 00:16:37.380 that have been used, even in what I've written, 00:16:37.380 --> 00:16:40.230 because you could think of having other types of traces 00:16:40.230 --> 00:16:42.300 like this one, for example. 00:16:42.300 --> 00:16:46.925 Instead of having trace squared, I could have the following. 00:16:46.925 --> 00:16:48.300 Just put everything in one trace. 00:16:55.110 --> 00:16:57.840 I always have to alternate sigma sigma dagger, 00:16:57.840 --> 00:17:04.170 sigma sigma dagger because del mu sigma sigma dagger is 0. 00:17:04.170 --> 00:17:06.990 Because sigma sigma dagger is 1. 00:17:06.990 --> 00:17:11.160 And that provides an identity that I'm also able to use. 00:17:11.160 --> 00:17:13.035 And so I want to have sigmas and sigma 00:17:13.035 --> 00:17:14.160 daggers next to each other. 00:17:18.733 --> 00:17:21.150 Sorry, chiral symmetry means that sigmas and sigma daggers 00:17:21.150 --> 00:17:23.290 should be next to each other. 00:17:23.290 --> 00:17:26.290 This is an identity that I can also use to simplify things. 00:17:26.290 --> 00:17:28.380 So that allows me to move things around. 00:17:28.380 --> 00:17:31.230 And this operator, though, in SU(2) 00:17:31.230 --> 00:17:33.330 is actually related to these two. 00:17:33.330 --> 00:17:34.740 It's not unique. 00:17:34.740 --> 00:17:37.500 Because of the structure of Pauli matrices, 00:17:37.500 --> 00:17:40.500 you have a formula which you can work out, 00:17:40.500 --> 00:17:43.350 which says that this guy is actually 00:17:43.350 --> 00:17:44.865 half of the trace squared. 00:17:51.240 --> 00:17:54.240 And there's another one if you did. 00:18:03.550 --> 00:18:09.615 This guy, he also can be related to those two guys over there. 00:18:09.615 --> 00:18:10.990 So there's a bunch of things that 00:18:10.990 --> 00:18:13.780 went into even writing down the level of information 00:18:13.780 --> 00:18:14.860 that I told you. 00:18:14.860 --> 00:18:16.900 You could think there's more things, 00:18:16.900 --> 00:18:19.130 but then, when you enumerate all the possible things 00:18:19.130 --> 00:18:21.130 that you can do, you find that those more things 00:18:21.130 --> 00:18:23.030 are related to these things. 00:18:23.030 --> 00:18:25.030 So you really want to construct a minimal basis. 00:18:25.030 --> 00:18:27.030 And there's some things that go into doing that. 00:18:35.280 --> 00:18:41.060 So at order p to the fourth, as I said, we include both loops. 00:18:41.060 --> 00:18:45.170 And those loops will have terms like p to the fourth log 00:18:45.170 --> 00:18:46.370 mu squared over p squared. 00:18:52.370 --> 00:18:55.970 And we include terms like the L1 and L2, 00:18:55.970 --> 00:19:02.150 which are p of the fourth type interactions. 00:19:02.150 --> 00:19:04.010 Once I take out the counter-term, 00:19:04.010 --> 00:19:06.520 the renormalized coupling is mu dependent just 00:19:06.520 --> 00:19:07.520 like the gauge coupling. 00:19:14.075 --> 00:19:15.950 This just comes from a four-point interaction 00:19:15.950 --> 00:19:17.780 with the Li. 00:19:17.780 --> 00:19:20.960 And the mu dependence, by construction, 00:19:20.960 --> 00:19:22.640 was canceled between these two things. 00:19:25.310 --> 00:19:28.400 The divergence from the loop is cancelled by the counter-term. 00:19:28.400 --> 00:19:30.500 And correspondingly, the corresponding statement, 00:19:30.500 --> 00:19:32.083 if you want to make it in terms of mu, 00:19:32.083 --> 00:19:35.600 is that the mu dependence that is tightly tied to that 1 over 00:19:35.600 --> 00:19:39.065 epsilon is cancelled in the renormalized quantities. 00:19:50.172 --> 00:19:51.630 So there's these two contributions. 00:19:51.630 --> 00:19:53.070 They're both mu dependent. 00:19:53.070 --> 00:19:55.853 And that mu dependence cancels. 00:19:55.853 --> 00:19:57.520 So the way that you should think of this 00:19:57.520 --> 00:20:01.190 is you should think mu is a cut off. 00:20:01.190 --> 00:20:03.550 It's not a hard cut off, but it is a cut off. 00:20:08.350 --> 00:20:10.455 And what the cut off is doing is dividing up 00:20:10.455 --> 00:20:11.830 infrared and ultraviolet physics. 00:20:16.210 --> 00:20:18.610 In this case, the low energy physics 00:20:18.610 --> 00:20:26.050 is in the matrix elements in the loops 00:20:26.050 --> 00:20:28.730 where we have our propagating low energy degrees of freedom, 00:20:28.730 --> 00:20:31.060 which are the pions. 00:20:31.060 --> 00:20:34.645 And the high energy physics is in the coefficients, as usual. 00:20:43.730 --> 00:20:44.980 And they're both mu dependent. 00:20:44.980 --> 00:20:47.270 And you can think of that mu dependent as a cut off 00:20:47.270 --> 00:20:49.310 that divides up how much of the physics 00:20:49.310 --> 00:20:52.010 goes into those low energy loops, how much of the physics 00:20:52.010 --> 00:20:55.630 goes into the couplings. 00:20:55.630 --> 00:20:59.050 So the difference between this and integrating 00:20:59.050 --> 00:21:01.030 on a massive particle is not the physics 00:21:01.030 --> 00:21:03.550 of where things go because the low energy physics always 00:21:03.550 --> 00:21:04.717 goes in the matrix elements. 00:21:04.717 --> 00:21:06.790 The high energy physics goes in the couplings. 00:21:06.790 --> 00:21:10.638 The difference is that, here, the way that you should think 00:21:10.638 --> 00:21:13.180 of it is that we can calculate the matrix elements explicitly 00:21:13.180 --> 00:21:16.510 because our theory is in terms of the right degrees of freedom 00:21:16.510 --> 00:21:21.910 to describe long distance physics, which are the pions. 00:21:21.910 --> 00:21:24.010 And then the coefficients are the unknowns. 00:21:24.010 --> 00:21:28.093 That's the way in which it's different. 00:21:28.093 --> 00:21:29.760 And that's really kind of the difference 00:21:29.760 --> 00:21:31.500 between bottom-up and top-down. 00:21:42.740 --> 00:21:47.500 So with that in mind, if you want 00:21:47.500 --> 00:21:51.670 to think of what these couplings are, if we just 00:21:51.670 --> 00:21:59.470 divide by f squared, then you can think about dimensionally 00:21:59.470 --> 00:22:01.030 what they are. 00:22:01.030 --> 00:22:02.860 In order for them to sort of match up 00:22:02.860 --> 00:22:05.260 with what we got from the loops is 00:22:05.260 --> 00:22:10.800 that they have some dependence that looks like this. 00:22:10.800 --> 00:22:12.400 You can think of it as parameterized 00:22:12.400 --> 00:22:14.590 by some coefficients a and b. 00:22:14.590 --> 00:22:17.200 If I want to match up with the 4 pi f squared that comes from 00:22:17.200 --> 00:22:19.750 the loops, then I have to sort of say that there's a 4 pi 00:22:19.750 --> 00:22:22.150 hiding inside the Li's. 00:22:22.150 --> 00:22:24.200 So I can do that. 00:22:24.200 --> 00:22:26.300 And then there's some coefficients here. 00:22:26.300 --> 00:22:27.730 There's some mu dependence. 00:22:27.730 --> 00:22:29.770 And the scales that are in the coefficients 00:22:29.770 --> 00:22:32.350 are the high scales, like lambda chi and higher 00:22:32.350 --> 00:22:34.970 and rho, those types of things. 00:22:34.970 --> 00:22:36.940 So let me just call it lambda chi. 00:22:36.940 --> 00:22:38.920 And then there's some numbers here a 00:22:38.920 --> 00:22:44.650 and b, which encode the high mass physics. 00:22:44.650 --> 00:22:47.560 Now, the point of thinking about power counting 00:22:47.560 --> 00:22:49.570 is that thinking about the fact that the loops 00:22:49.570 --> 00:22:51.610 and these coefficients are the same size 00:22:51.610 --> 00:22:54.130 actually tells you that these ai's and bi's should 00:22:54.130 --> 00:22:54.820 be order 1. 00:23:04.760 --> 00:23:07.310 And this actually goes under the rubric of something called 00:23:07.310 --> 00:23:08.780 naive dimensional analysis. 00:23:21.900 --> 00:23:23.613 So what naive dimensional analysis says 00:23:23.613 --> 00:23:26.030 is that this cut off that we've put between the low energy 00:23:26.030 --> 00:23:28.310 physics and the high energy physics is arbitrary. 00:23:28.310 --> 00:23:29.300 And we could change it. 00:23:29.300 --> 00:23:31.250 We could change it by a factor of 2. 00:23:31.250 --> 00:23:35.070 And what changing it does is it moves pieces back and forth. 00:23:35.070 --> 00:23:37.100 But if we could move pieces back and forth, 00:23:37.100 --> 00:23:39.800 then you wouldn't expect that the two things that you're 00:23:39.800 --> 00:23:43.130 talking about would be different in magnitude 00:23:43.130 --> 00:23:45.760 because we're allowed to move pieces back and forth. 00:23:45.760 --> 00:23:47.840 So you expect that the size of contributions 00:23:47.840 --> 00:23:51.680 from the coefficients are about the same size as the loops. 00:23:51.680 --> 00:23:58.320 And that's this naive dimensional analysis. 00:23:58.320 --> 00:24:00.410 So changing mu moves pieces back and forth. 00:24:05.450 --> 00:24:14.610 The sum is mu independent, but each individual thing is not. 00:24:21.330 --> 00:24:26.130 And because we're able to move things back and forth, 00:24:26.130 --> 00:24:28.860 we expect them to be the same order of magnitude. 00:24:44.300 --> 00:24:50.030 And that is this statement here, that the ai's and bi's, 00:24:50.030 --> 00:24:53.300 once I account for the 4 pis-- 00:24:53.300 --> 00:24:56.202 so with this argument, I can figure out 00:24:56.202 --> 00:24:58.160 where there's 4 pis hiding in the coefficients. 00:24:58.160 --> 00:25:00.108 Because I can identify 4 pis in loops. 00:25:00.108 --> 00:25:02.150 And then if they're supposed to be the same size, 00:25:02.150 --> 00:25:06.460 I can also identify 4 pis in coefficients. 00:25:06.460 --> 00:25:10.150 So if you had figured this out, I don't know, 20 years ago, 00:25:10.150 --> 00:25:13.430 then you could have got a PhD thesis like Aneesh Manohar did, 00:25:13.430 --> 00:25:14.710 which was 25 pages long. 00:25:20.540 --> 00:25:22.400 Short PhD theses do exist. 00:25:27.605 --> 00:25:28.105 OK. 00:25:37.160 --> 00:25:39.110 So what do we do in practice? 00:25:39.110 --> 00:25:41.210 In practice, we have to pick a value of mu. 00:25:41.210 --> 00:25:43.680 Just like when we were talking about gauge theory, 00:25:43.680 --> 00:25:44.930 we need to pick a value of mu. 00:25:44.930 --> 00:25:47.780 There we were picking things like mu equals Mb. 00:25:47.780 --> 00:25:50.630 Here, what people do is they typically pick 00:25:50.630 --> 00:25:52.700 mus that are high. 00:25:52.700 --> 00:25:54.710 So maybe they would pick the rho mass. 00:25:54.710 --> 00:25:58.640 Maybe they would pick lambda chi. 00:25:58.640 --> 00:26:02.390 Or they pick just something in between, like a GeV. 00:26:02.390 --> 00:26:04.190 These are typical values that people use. 00:26:06.710 --> 00:26:12.820 So what that means is that you've removed all large logs 00:26:12.820 --> 00:26:13.930 from your coefficients. 00:26:13.930 --> 00:26:15.930 And that's because you want dimensional analysis 00:26:15.930 --> 00:26:18.830 to hold for your coefficients because you don't know them. 00:26:18.830 --> 00:26:20.740 So you would like to have some power counting 00:26:20.740 --> 00:26:21.430 estimate for them. 00:26:21.430 --> 00:26:22.990 Then you'd like to go out and fit them to data. 00:26:22.990 --> 00:26:25.115 And you'd like the result that you get from fitting 00:26:25.115 --> 00:26:27.880 to the data to agree with your power counting estimate, 00:26:27.880 --> 00:26:29.290 so that you're happy. 00:26:29.290 --> 00:26:33.460 It turns out that it does, so you are happy. 00:26:33.460 --> 00:26:37.300 And so to avoid large logs in the story, 00:26:37.300 --> 00:26:39.310 you put the large logs into the matrix elements. 00:26:54.655 --> 00:26:57.030 So that's different than our story in gauge theory, where 00:26:57.030 --> 00:26:59.613 we were thinking that we would re-sum all the large logarithms 00:26:59.613 --> 00:27:00.780 in the coefficients. 00:27:00.780 --> 00:27:02.490 And then our matrix elements would also 00:27:02.490 --> 00:27:03.750 have no large logarithms. 00:27:03.750 --> 00:27:05.250 Here, we're just saying, well, let's 00:27:05.250 --> 00:27:07.800 allow for large logs in the matrix element. 00:27:07.800 --> 00:27:09.992 But the story is also different from gauge theory 00:27:09.992 --> 00:27:11.700 in another way, which is that there's not 00:27:11.700 --> 00:27:14.810 an infinite series here of large logarithms 00:27:14.810 --> 00:27:15.810 that you need to re-sum. 00:27:31.190 --> 00:27:33.710 And that's related to the fact that the kinetic term didn't 00:27:33.710 --> 00:27:35.237 get renormalized. 00:27:35.237 --> 00:27:37.070 There's no mu dependence in the coefficients 00:27:37.070 --> 00:27:37.960 of the kinetic term. 00:27:37.960 --> 00:27:41.030 So when the loop graphs aren't having 00:27:41.030 --> 00:27:44.450 coefficient squared depending on mu, that doesn't happen. 00:27:44.450 --> 00:27:47.300 We simply have one log in our loop graphs, 00:27:47.300 --> 00:27:50.060 one log from our counter-term diagrams. 00:27:50.060 --> 00:27:51.840 They explicitly cancel. 00:27:51.840 --> 00:27:53.060 There's no high order terms. 00:27:53.060 --> 00:27:55.268 The renormalization group here is completely trivial. 00:27:55.268 --> 00:27:57.900 If you integrate, you just get one log. 00:27:57.900 --> 00:28:01.900 So there's not really even a reason to talk about it. 00:28:01.900 --> 00:28:04.790 OK, so there's certainly some differences in this theory 00:28:04.790 --> 00:28:06.200 than there are in gauge theory. 00:28:29.370 --> 00:28:31.650 And so typically, the paradigm that you have is you 00:28:31.650 --> 00:28:33.030 can calculate matrix elements. 00:28:33.030 --> 00:28:34.830 You're going to fit the coefficients. 00:28:34.830 --> 00:28:36.468 You think of enough experiments such 00:28:36.468 --> 00:28:38.010 that you can get all that information 00:28:38.010 --> 00:28:39.052 about those coefficients. 00:28:39.052 --> 00:28:40.980 And then you can think of other experiments 00:28:40.980 --> 00:28:43.833 and make predictions. 00:28:43.833 --> 00:28:45.750 Now, as you go to higher orders in the theory, 00:28:45.750 --> 00:28:47.292 you get more coefficients because you 00:28:47.292 --> 00:28:48.780 keep having to go to higher orders 00:28:48.780 --> 00:28:51.630 and construct higher dimension operators. 00:28:51.630 --> 00:28:54.480 And so the paradigm of figuring out how to fit the coefficients 00:28:54.480 --> 00:28:55.800 gets harder and harder. 00:28:55.800 --> 00:28:57.240 And at some point, you effectively 00:28:57.240 --> 00:28:58.410 lose predictive power because you 00:28:58.410 --> 00:28:59.993 can't think of enough observables that 00:28:59.993 --> 00:29:04.110 can be measured in order to fit all your coefficients. 00:29:04.110 --> 00:29:07.410 But certainly, at order p to the fourth, 00:29:07.410 --> 00:29:09.502 people can think of many more observables. 00:29:09.502 --> 00:29:11.085 And actually, people have worked out p 00:29:11.085 --> 00:29:13.470 to the sixth as well here. 00:29:19.217 --> 00:29:20.800 So two-loop chiral perturbation theory 00:29:20.800 --> 00:29:22.133 is kind of the state of the art. 00:29:26.933 --> 00:29:28.100 AUDIENCE: I have a question. 00:29:28.100 --> 00:29:30.500 IAIN STEWART: Yeah. 00:29:30.500 --> 00:29:33.464 AUDIENCE: So why allow log in this case [? is not ?] screwing 00:29:33.464 --> 00:29:35.630 up the [INAUDIBLE]? 00:29:35.630 --> 00:29:38.123 IAIN STEWART: Yeah. 00:29:38.123 --> 00:29:40.290 It doesn't screw it up because it only happens once. 00:29:44.390 --> 00:29:46.565 So basically, what you could say is you can say, 00:29:46.565 --> 00:29:47.690 I have this matrix element. 00:29:47.690 --> 00:29:49.880 It's got a large log and a coefficient. 00:29:49.880 --> 00:29:52.580 The coupling, it's not like large log times 00:29:52.580 --> 00:29:55.190 the coupling is something that you could count as order 1 00:29:55.190 --> 00:29:57.260 because that's not going to reappear 00:29:57.260 --> 00:30:00.140 in the higher order when you go to the higher order. 00:30:00.140 --> 00:30:02.030 So you still have a large log. 00:30:02.030 --> 00:30:04.460 And you are allowed to say that large log is 00:30:04.460 --> 00:30:07.070 the most important piece of my matrix element, 00:30:07.070 --> 00:30:10.130 but it doesn't repeat itself in the higher orders. 00:30:10.130 --> 00:30:11.900 Because once you go to higher loops, 00:30:11.900 --> 00:30:13.275 you're actually getting something 00:30:13.275 --> 00:30:16.960 that's power suppressed, not loop suppressed. 00:30:16.960 --> 00:30:18.710 So the loops are giving power suppression. 00:30:18.710 --> 00:30:20.900 And that changes a lot of things. 00:30:28.575 --> 00:30:29.075 OK. 00:30:32.920 --> 00:30:35.050 What would happen if we used a hard cut off? 00:30:39.880 --> 00:30:43.900 Maybe some of this story would be a bit more transparent 00:30:43.900 --> 00:30:46.600 if we'd done that because we'd see that we were explicitly 00:30:46.600 --> 00:30:49.450 dividing up low energy and high energy physics. 00:30:49.450 --> 00:30:52.540 But there's a price to pay for that transparency. 00:30:52.540 --> 00:30:55.450 And here, effectively everybody decides 00:30:55.450 --> 00:30:56.720 it's too high a price to pay. 00:30:56.720 --> 00:30:59.680 And so they use dim reg. 00:30:59.680 --> 00:31:02.170 So what type of prices would you pay? 00:31:02.170 --> 00:31:03.610 So if you did that, this loop here 00:31:03.610 --> 00:31:05.027 would actually have terms that are 00:31:05.027 --> 00:31:07.675 cut-off to the fourth over lambda chi to the fourth. 00:31:07.675 --> 00:31:09.175 And that guy breaks chiral symmetry. 00:31:12.190 --> 00:31:14.710 I mean, it's giving a constant term. 00:31:14.710 --> 00:31:17.710 And you're not seeing the derivative coupling. 00:31:17.710 --> 00:31:21.010 And that means, effectively, that there's no counter-term 00:31:21.010 --> 00:31:23.260 to absorb this dependence in our L chi 00:31:23.260 --> 00:31:26.690 because our L chi respected chiral symmetry. 00:31:26.690 --> 00:31:30.190 So that's kind of bad. 00:31:30.190 --> 00:31:31.690 If you had a cut-off, you'd also get 00:31:31.690 --> 00:31:34.750 terms that went cut-off squared times momentum squared 00:31:34.750 --> 00:31:37.720 where two powers of the p are replaced by cut-off. 00:31:37.720 --> 00:31:41.710 And that, in this language that we used earlier, 00:31:41.710 --> 00:31:43.720 would break the power counting in the sense 00:31:43.720 --> 00:31:46.060 that we have a power counting that 00:31:46.060 --> 00:31:48.880 says that the loops should be suppressed by d to the fourth. 00:31:48.880 --> 00:31:51.370 And it's broken by this. 00:31:51.370 --> 00:31:55.870 And then you would need to absorb that and renormalize 00:31:55.870 --> 00:31:58.090 you're leading order coupling. 00:31:58.090 --> 00:31:59.810 But effectively, all the renormalization 00:31:59.810 --> 00:32:01.810 would be doing is restoring your power counting. 00:32:01.810 --> 00:32:03.940 It's not doing something that you should interpret 00:32:03.940 --> 00:32:05.380 as really a physical thing. 00:32:08.760 --> 00:32:11.500 So it's just something to be avoided. 00:32:11.500 --> 00:32:16.860 And then finally, the physics that you actually want, 00:32:16.860 --> 00:32:23.830 which is the logs of the cut-off, would also be there. 00:32:23.830 --> 00:32:33.520 And you would do the same thing as you do in dim reg. 00:32:33.520 --> 00:32:38.530 You would absorb that in higher dimension operators, OK? 00:32:38.530 --> 00:32:40.990 So we won't use a cut-off because we 00:32:40.990 --> 00:32:42.640 don't want to think about things that 00:32:42.640 --> 00:32:45.057 are breaking power counting or messing up chiral symmetry. 00:32:52.510 --> 00:32:55.000 Another thing that's different about chiral theories 00:32:55.000 --> 00:32:56.770 versus gauge theories is the structure 00:32:56.770 --> 00:32:58.390 of infrared divergences. 00:32:58.390 --> 00:33:00.460 And that's because you have derivative couplings. 00:33:08.480 --> 00:33:11.650 So you have many fewer infrared singularities 00:33:11.650 --> 00:33:13.160 than you have in gauge theory. 00:33:13.160 --> 00:33:15.700 And in fact, you usually don't have any. 00:33:19.360 --> 00:33:21.580 And one way of describing that is that you usually 00:33:21.580 --> 00:33:24.940 have a good M pi squared goes to 0 00:33:24.940 --> 00:33:28.570 or p squared goes to 0 limit of your results. 00:33:33.230 --> 00:33:36.760 So you can just explicitly take these limits 00:33:36.760 --> 00:33:39.680 and talk about the results in those limits. 00:33:39.680 --> 00:33:42.430 And if we look back at what we were talking about, 00:33:42.430 --> 00:33:44.310 you get p to the fourth log p squared. 00:33:44.310 --> 00:33:46.770 So there's a p to the fourth multiplying the log p squared. 00:33:46.770 --> 00:33:48.520 So you're not seeing log p squared blow up 00:33:48.520 --> 00:33:51.280 because it's got so many powers of p multiplying it. 00:33:57.100 --> 00:34:00.040 Although our focus is on formalism, 00:34:00.040 --> 00:34:03.040 I have to at least give you one example of something predictive 00:34:03.040 --> 00:34:04.780 and phenomenological. 00:34:04.780 --> 00:34:12.219 So pi pi scattering is a nice example of phenomenology. 00:34:12.219 --> 00:34:14.800 And it's particularly nice if you look at it 00:34:14.800 --> 00:34:17.770 below an elastic threshold. 00:34:17.770 --> 00:34:19.769 So you just have elastic scattering. 00:34:25.870 --> 00:34:28.795 So you could look at pi pi goes to 4 pi. 00:34:28.795 --> 00:34:30.420 And that's something you could actually 00:34:30.420 --> 00:34:32.760 look at in chiral perturbation theory. 00:34:32.760 --> 00:34:37.440 But if we look at just pi pi goes to pi pi 00:34:37.440 --> 00:34:40.739 where we have not enough energy to produce 4 pions, 00:34:40.739 --> 00:34:44.500 then the scattering is particularly simple. 00:34:44.500 --> 00:34:48.449 It's just described by an S matrix, which we can 00:34:48.449 --> 00:34:50.969 enumerate channel by channel. 00:34:50.969 --> 00:35:04.680 And it's just a phase where this L is a partial wave phase. 00:35:04.680 --> 00:35:05.700 And the i is an isospin. 00:35:08.820 --> 00:35:12.800 So for each isospin and for each angular momentum state, 00:35:12.800 --> 00:35:13.800 we get different phases. 00:35:13.800 --> 00:35:17.190 But that's encoding all the scattering. 00:35:17.190 --> 00:35:19.490 That's like doing non-relativistic scattering 00:35:19.490 --> 00:35:24.230 or very simple quantum mechanical scattering. 00:35:35.657 --> 00:35:37.740 And when you have an elastic scattering like that, 00:35:37.740 --> 00:35:44.880 there's something called the effective range expansion, 00:35:44.880 --> 00:35:47.875 which is a derivative expansion for the phase shift delta. 00:35:50.540 --> 00:35:52.500 And if you do it for an arbitrary partial wave, 00:35:52.500 --> 00:35:54.350 this is how it looks. 00:35:54.350 --> 00:35:56.060 So this is, again, something that you 00:35:56.060 --> 00:36:00.410 would find in the discussion of scattering theory in quantum 00:36:00.410 --> 00:36:01.418 mechanics. 00:36:04.700 --> 00:36:06.500 And you have a derivative expansion 00:36:06.500 --> 00:36:09.860 of this quantity p to the 2L plus 1 cotan 00:36:09.860 --> 00:36:13.070 of that phase shift. 00:36:13.070 --> 00:36:19.790 And the a's here and the r0 depend on what channel 00:36:19.790 --> 00:36:22.880 you're talking about. 00:36:22.880 --> 00:36:26.150 And if we actually just take the fact that we can-- 00:36:26.150 --> 00:36:28.220 so this is a description from quantum mechanics. 00:36:28.220 --> 00:36:29.990 This is true irrespective of what theory 00:36:29.990 --> 00:36:32.750 you're talking about as long as you don't have 00:36:32.750 --> 00:36:34.802 other channels you can produce. 00:36:34.802 --> 00:36:37.010 If you talk about it from chiral perturbation theory, 00:36:37.010 --> 00:36:40.790 though, we can just calculate pi pi goes to pi pi. 00:36:40.790 --> 00:36:46.520 And if you do that, which [? Weinberg ?] did, 00:36:46.520 --> 00:36:49.175 then you just get results for these coefficients. 00:36:53.708 --> 00:36:55.500 So I'll just quote a couple of them to you. 00:37:12.200 --> 00:37:14.470 So you just get parameter-free results, 00:37:14.470 --> 00:37:16.220 parameter-free in the sense that they just 00:37:16.220 --> 00:37:18.560 involve M pi and this thing f. 00:37:18.560 --> 00:37:21.490 This thing f is actually measured by pi on decay, 00:37:21.490 --> 00:37:23.390 so it's not an unknown. 00:37:23.390 --> 00:37:25.490 And then from that, at lowest order 00:37:25.490 --> 00:37:27.260 in chiral perturbation theory, you 00:37:27.260 --> 00:37:30.800 have no parameters once you fixed M pi and f pi. 00:37:30.800 --> 00:37:32.450 And you can just make predictions 00:37:32.450 --> 00:37:35.690 for these scattering lengths, and those are the predictions. 00:37:35.690 --> 00:37:44.570 And they're parameter-free in the sense that I just said. 00:37:47.520 --> 00:37:49.340 OK, so that gives you some of the idea 00:37:49.340 --> 00:37:51.460 of what chiral perturbation theory can do for you. 00:37:58.330 --> 00:38:01.968 OK, so let's talk about back to formalism. 00:38:01.968 --> 00:38:03.760 There's many more phenomenological examples 00:38:03.760 --> 00:38:06.880 you could do, but that's not our focus. 00:38:06.880 --> 00:38:12.820 Back to formalism and back to this general power 00:38:12.820 --> 00:38:25.020 counting discussion, so let's consider an arbitrary diagram 00:38:25.020 --> 00:38:27.300 in this theory, this chiral perturbation theory. 00:38:32.340 --> 00:38:34.470 And let's enumerate some pieces of that diagram. 00:38:37.600 --> 00:38:40.550 So we'll say that it has some number of vertices. 00:38:40.550 --> 00:38:42.360 We just count how many times I've 00:38:42.360 --> 00:38:45.660 inserted vertices from the Lagrangians, 00:38:45.660 --> 00:38:48.360 and I'll call that NV. 00:38:48.360 --> 00:38:49.755 Some number of internal lines-- 00:38:55.410 --> 00:38:56.420 call NI. 00:38:56.420 --> 00:38:58.005 Some number of external lines-- 00:39:02.730 --> 00:39:04.350 the external lines are all pions. 00:39:08.700 --> 00:39:10.695 That's what our theory is describing. 00:39:10.695 --> 00:39:14.920 It could be kaons and etas if we're doing SU(3)-- 00:39:14.920 --> 00:39:16.290 and. then some number of loops. 00:39:18.890 --> 00:39:22.510 So we have an integer associated with each of these things. 00:39:22.510 --> 00:39:24.000 And when I talk about vertices, I 00:39:24.000 --> 00:39:25.920 don't want to restrict myself just to the leading order 00:39:25.920 --> 00:39:26.420 vertices. 00:39:26.420 --> 00:39:29.310 I want to talk about also these Li's and higher order 00:39:29.310 --> 00:39:31.030 vertices as well. 00:39:31.030 --> 00:39:35.190 So let me have a notation for that, where I take this integer 00:39:35.190 --> 00:39:37.560 NV, which counts all vertices, and split it 00:39:37.560 --> 00:39:39.480 into pieces that count the vertices at each 00:39:39.480 --> 00:39:42.600 of those different orders in the expansion in p. 00:39:51.280 --> 00:39:54.100 So Nn is the number of vertices that 00:39:54.100 --> 00:40:01.195 are order p to the n or M pi to the n or combinations of p 00:40:01.195 --> 00:40:07.230 and M pi to the n, but n of them, all right? 00:40:07.230 --> 00:40:08.790 So hopefully, that's clear. 00:40:08.790 --> 00:40:11.030 So if I have two insertions of the leading order 00:40:11.030 --> 00:40:13.580 Lagrangian and one insertion of the sub-leading, 00:40:13.580 --> 00:40:16.910 then n sub 0 would be 2. 00:40:16.910 --> 00:40:18.710 And N sub 1 would be 1. 00:40:18.710 --> 00:40:20.677 And the total would be 3. 00:40:20.677 --> 00:40:22.010 That's what this notation means. 00:40:40.890 --> 00:40:42.540 So we'll assume we're using a regulator 00:40:42.540 --> 00:40:43.830 like dimensional regularization. 00:40:43.830 --> 00:40:45.900 So we don't have to worry about the regulator messing up 00:40:45.900 --> 00:40:46.525 power counting. 00:40:46.525 --> 00:40:48.210 And basically, that means we could 00:40:48.210 --> 00:40:52.000 ignore the regulator as far as this discussion is concerned. 00:40:52.000 --> 00:40:52.860 And we just count. 00:40:57.570 --> 00:41:00.462 And effectively, we can just count mass dimension. 00:41:05.190 --> 00:41:06.755 So we'll count lambda chi factors. 00:41:09.700 --> 00:41:10.950 Let's think about it that way. 00:41:13.920 --> 00:41:21.450 And we'll think about counting lambda chi factors for a matrix 00:41:21.450 --> 00:41:27.255 element that I'll call curly M, which has NE external pions. 00:41:30.080 --> 00:41:32.550 So NE lines are poking out of it. 00:41:38.740 --> 00:41:42.180 So then if we look at the vertices, 00:41:42.180 --> 00:41:45.450 we can count how many factors of lambda chi there is. 00:41:45.450 --> 00:41:47.010 And I'm counting all dimensionful 00:41:47.010 --> 00:41:49.680 and turning all dimensionful things into lambda chi. 00:41:55.320 --> 00:42:02.820 And so each different order in N gets a number of lambda chis 00:42:02.820 --> 00:42:05.910 because we even saw that already in the examples 00:42:05.910 --> 00:42:08.730 we treated where we got f squared from the leading order. 00:42:08.730 --> 00:42:11.250 But from the sub-leading, we got Li. 00:42:11.250 --> 00:42:14.110 And that was dimensionless. 00:42:14.110 --> 00:42:18.540 So for n equals 2, which is the leading order here, 00:42:18.540 --> 00:42:20.200 leading order was p squared. 00:42:20.200 --> 00:42:23.840 I said that not quite right a minute ago. 00:42:23.840 --> 00:42:27.100 So lowest order is p squared n equals 2. 00:42:27.100 --> 00:42:28.020 That's L0. 00:42:30.650 --> 00:42:31.880 That had an f squared. 00:42:31.880 --> 00:42:35.960 And that comes out, if I just have 4 minus 2, that's 2. 00:42:35.960 --> 00:42:42.250 n equals 4, that was giving our Li's, which were dimensionless, 00:42:42.250 --> 00:42:42.750 OK? 00:42:42.750 --> 00:42:44.208 So you can see the formula working. 00:42:44.208 --> 00:42:47.440 And if we went to higher orders in the derivative and chiral 00:42:47.440 --> 00:42:49.690 expansion, then we'd start getting lambda chis 00:42:49.690 --> 00:42:50.975 in the denominator. 00:42:53.650 --> 00:42:56.160 So this is just counting from the vertices, which 00:42:56.160 --> 00:42:57.900 you should think of as counting just 00:42:57.900 --> 00:43:00.660 from the pre-factors in the Lagrangian. 00:43:00.660 --> 00:43:13.290 There's also f's that come with the pions 00:43:13.290 --> 00:43:18.340 because every factor of the pion field comes with a factor of f. 00:43:18.340 --> 00:43:19.980 You always have pi over f. 00:43:19.980 --> 00:43:25.200 And I'm turning f into 4 pi f for this discussion. 00:43:25.200 --> 00:43:26.855 I'm not worrying about the 4 pis. 00:43:26.855 --> 00:43:28.230 You could do a more fancy version 00:43:28.230 --> 00:43:29.897 of this where you worry about the 4 pis, 00:43:29.897 --> 00:43:33.700 but let's just focus on the dimensions. 00:43:33.700 --> 00:43:35.890 So if you have an internal line, that's 00:43:35.890 --> 00:43:37.660 a contraction of 2 pion fields. 00:43:37.660 --> 00:43:38.640 So that gets 2f's. 00:43:38.640 --> 00:43:40.687 And the external line is just 1 pion field, 00:43:40.687 --> 00:43:41.520 so that just gets 1. 00:43:49.000 --> 00:43:51.910 Topologically, these different things that we enumerated 00:43:51.910 --> 00:43:55.186 are not unrelated. 00:43:55.186 --> 00:43:57.760 So the order identity tells us that the number 00:43:57.760 --> 00:44:01.450 of internal lines is the number of loops 00:44:01.450 --> 00:44:04.330 plus the number of vertices minus 1. 00:44:06.880 --> 00:44:08.710 And so we use that to get rid of NI. 00:44:17.190 --> 00:44:19.230 So then we can just put these things together. 00:44:34.395 --> 00:44:38.520 So if I get rid of NI here and I replace it by NL-- 00:44:38.520 --> 00:44:40.050 so that's that term-- 00:44:40.050 --> 00:44:44.940 I replace it by the sum over the Nn's which are the vertices. 00:44:44.940 --> 00:44:47.340 And then this gives me a plus 2. 00:44:51.600 --> 00:44:52.920 OK. 00:44:52.920 --> 00:44:56.295 Now, that's not the dimension of the left-hand side. 00:44:56.295 --> 00:44:58.170 That's just the dimension of the ingredients. 00:44:58.170 --> 00:45:00.555 There's also things that are coming from factors of M pi 00:45:00.555 --> 00:45:02.940 or factors of p. 00:45:02.940 --> 00:45:07.170 And let me just call that something, E to the D 00:45:07.170 --> 00:45:10.290 where D is just some number, integer, 00:45:10.290 --> 00:45:17.420 and then some function of logarithms of p over mu 00:45:17.420 --> 00:45:18.480 or M pi over mu. 00:45:18.480 --> 00:45:24.030 So E could be M pi or p. 00:45:24.030 --> 00:45:25.772 And for the purpose of power counting, 00:45:25.772 --> 00:45:26.730 I'm not distinguishing. 00:45:26.730 --> 00:45:30.840 So let's just call it E just to have a notation that 00:45:30.840 --> 00:45:34.290 could be either M pi or p. 00:45:34.290 --> 00:45:36.040 And then there's one more thing we can do, 00:45:36.040 --> 00:45:38.010 which is we can look at the left-hand side. 00:45:38.010 --> 00:45:40.740 And we can say, just by dimensional analysis, 00:45:40.740 --> 00:45:43.920 the matrix element, what should be its dimension? 00:45:43.920 --> 00:45:46.320 And depending on how many bosons you have sticking out, 00:45:46.320 --> 00:45:48.487 your matrix element should have a certain dimension, 00:45:48.487 --> 00:45:50.290 which you could figure out. 00:45:50.290 --> 00:45:56.190 And that dimension is just 4 minus NE. 00:45:59.130 --> 00:46:02.820 Two-point function would scale like p squared, et cetera. 00:46:02.820 --> 00:46:06.060 So now, I have different things that are giving mass 00:46:06.060 --> 00:46:08.445 dimensions, the lambda chis and the E's. 00:46:08.445 --> 00:46:11.430 But because I know what the answer has to be, 4 minus NE, 00:46:11.430 --> 00:46:27.280 I can solve for D. 00:46:27.280 --> 00:46:31.110 And that's what I got. 00:46:31.110 --> 00:46:31.800 OK. 00:46:31.800 --> 00:46:34.770 So the answer for D, in order to get the dimensions right, 00:46:34.770 --> 00:46:36.930 we have to compensate for the number of lambda chis 00:46:36.930 --> 00:46:39.690 by factors of M pi and p. 00:46:39.690 --> 00:46:42.600 And when we do that, then we need this many of them 00:46:42.600 --> 00:46:45.247 in order to get the dimensions right. 00:46:45.247 --> 00:46:46.830 So one thing you see from this formula 00:46:46.830 --> 00:46:48.782 is that D is greater than or equal to 2. 00:46:48.782 --> 00:46:50.490 That's because these things are positive. 00:46:50.490 --> 00:46:53.370 The Lagrangian starts with n equals 2 and then goes higher. 00:47:00.520 --> 00:47:05.740 And when I add either of these terms, I cause suppression, 00:47:05.740 --> 00:47:06.790 or I stay the same. 00:47:16.820 --> 00:47:17.770 So this could be 0. 00:47:17.770 --> 00:47:19.850 This could be 0, or it could be bigger. 00:47:19.850 --> 00:47:20.845 But it can be smaller. 00:47:27.900 --> 00:47:35.355 So you always get more E's, which 00:47:35.355 --> 00:47:39.005 are M pis or p's, by adding vertices or adding loops. 00:47:39.005 --> 00:47:41.380 So when you looked at the loop graphs that were built out 00:47:41.380 --> 00:47:43.530 of the leading order Lagrangian, those terms 00:47:43.530 --> 00:47:45.780 had this be 0 because n was 2. 00:47:45.780 --> 00:47:48.780 But then you got suppression because we built loop graphs. 00:47:48.780 --> 00:47:50.718 So you got suppression from this term. 00:47:50.718 --> 00:47:53.010 And when we looked at the higher dimensional operators, 00:47:53.010 --> 00:47:53.820 there was no loop. 00:47:53.820 --> 00:47:55.487 So we didn't have this term, but then we 00:47:55.487 --> 00:47:57.800 had a contribution from this guy. 00:47:57.800 --> 00:47:59.250 But those were trading off. 00:48:04.700 --> 00:48:07.283 So having this is, effectively, part 00:48:07.283 --> 00:48:09.950 of what we need in order to make sure the theory is well-defined 00:48:09.950 --> 00:48:12.350 because it tells us how to organize the theory 00:48:12.350 --> 00:48:15.440 and what parts of the theory we need 00:48:15.440 --> 00:48:18.560 to worry about if we want a certain accuracy 00:48:18.560 --> 00:48:21.628 and what parts we can ignore. 00:48:21.628 --> 00:48:23.420 We need to know that we don't need to think 00:48:23.420 --> 00:48:24.740 about two-loop diagrams. 00:48:24.740 --> 00:48:28.490 And this tells us that we don't. 00:48:28.490 --> 00:48:29.422 Yeah. 00:48:29.422 --> 00:48:31.880 AUDIENCE: I didn't understand [? where you ?] [INAUDIBLE].. 00:48:31.880 --> 00:48:33.547 IAIN STEWART: Yeah, let me say it again. 00:48:33.547 --> 00:48:38.150 So we figured out the lambda chis by this stuff up here. 00:48:38.150 --> 00:48:39.950 Then I said, let there be an arbitrary 00:48:39.950 --> 00:48:44.300 E to the D, some parameter which we haven't 00:48:44.300 --> 00:48:45.860 figured out anything yet. 00:48:45.860 --> 00:48:49.010 But I know by just what possibly this 00:48:49.010 --> 00:48:51.840 could depend on that it could also depend on an M pis or pis. 00:48:51.840 --> 00:48:55.160 So let me put some polynomial power in and then some function 00:48:55.160 --> 00:48:56.770 that could be non-polynomial. 00:48:56.770 --> 00:48:58.437 AUDIENCE: But where would that-- can you 00:48:58.437 --> 00:49:00.738 give an example of a calculation where I 00:49:00.738 --> 00:49:02.030 would see exactly what that is? 00:49:02.030 --> 00:49:02.900 IAIN STEWART: Yeah, so if you did 00:49:02.900 --> 00:49:04.820 this loop calculation we did a minute ago, 00:49:04.820 --> 00:49:06.530 we got a p to the fourth. 00:49:06.530 --> 00:49:07.820 And then D would be 4. 00:49:07.820 --> 00:49:10.040 AUDIENCE: And those p's came from where? 00:49:10.040 --> 00:49:13.770 IAIN STEWART: Oh, they came from the momentum going-- 00:49:13.770 --> 00:49:16.580 so if you looked at this diagram, 00:49:16.580 --> 00:49:18.060 there's p coming in here. 00:49:18.060 --> 00:49:21.020 And then it goes into the loop, right? 00:49:21.020 --> 00:49:22.730 And this is derivatively coupled, 00:49:22.730 --> 00:49:24.545 so you get p's in the numerator. 00:49:24.545 --> 00:49:27.597 AUDIENCE: But that wasn't taken care of with [INAUDIBLE].. 00:49:27.597 --> 00:49:28.430 IAIN STEWART: Right. 00:49:28.430 --> 00:49:30.380 Because NV is just counting lambda chis, 00:49:30.380 --> 00:49:33.335 which are constants, not the p's. 00:49:33.335 --> 00:49:37.088 It's just counting the constants, the f's. 00:49:37.088 --> 00:49:37.630 AUDIENCE: OK. 00:49:37.630 --> 00:49:38.422 IAIN STEWART: Yeah. 00:49:38.422 --> 00:49:39.008 AUDIENCE: OK. 00:49:39.008 --> 00:49:39.800 IAIN STEWART: Yeah. 00:49:39.800 --> 00:49:44.213 And then you equate it to this, and then you get the D. Yeah. 00:49:44.213 --> 00:49:45.880 So you could have tried to set things up 00:49:45.880 --> 00:49:48.730 by thinking about counting p's instead of counting lambdas. 00:49:48.730 --> 00:49:50.470 But it's-- yeah, anyway. 00:49:54.950 --> 00:49:58.190 All right, so what people often refer to this 00:49:58.190 --> 00:50:16.450 is they say it's p counting because you're 00:50:16.450 --> 00:50:18.400 counting momenta. 00:50:18.400 --> 00:50:21.820 And that includes p or M pi, but sometimes people 00:50:21.820 --> 00:50:24.160 call it p counting. 00:50:24.160 --> 00:50:29.080 And just to do some examples, when we had the lowest order 00:50:29.080 --> 00:50:33.760 Lagrangian, this guy comes out as 2 powers of p 00:50:33.760 --> 00:50:35.421 because there's two derivatives. 00:50:38.650 --> 00:50:42.940 And that, in our formula, is just the fact 00:50:42.940 --> 00:50:46.150 that D is 2 for that. 00:50:46.150 --> 00:50:51.520 When we thought about this loop with leading order Lagrangians, 00:50:51.520 --> 00:50:56.260 which are scaling like p squared, 00:50:56.260 --> 00:50:58.183 we ended up having D equals 4. 00:50:58.183 --> 00:51:00.100 And the way that that comes out of the formula 00:51:00.100 --> 00:51:03.160 is because of this loop term, which is 1. 00:51:03.160 --> 00:51:06.070 And then there's 2 plus 2 is 4. 00:51:10.270 --> 00:51:15.478 And this guy is 4 because of the explicit suppression. 00:51:21.330 --> 00:51:22.220 OK. 00:51:22.220 --> 00:51:25.340 So the theory is organized as an expansion of this p. 00:51:30.520 --> 00:51:32.470 All right, so I want to come back 00:51:32.470 --> 00:51:35.080 to SU(3) partly because the problem 00:51:35.080 --> 00:51:37.570 that I gave you is in SU(3). 00:51:37.570 --> 00:51:41.230 So we'll do some discussion of the SU(3) case. 00:51:53.400 --> 00:51:55.510 I'll go into a few things in a little more detail 00:51:55.510 --> 00:51:58.570 than we did for SU(2), where we were 00:51:58.570 --> 00:52:00.070 focusing on more formal things. 00:52:02.670 --> 00:52:06.280 So SU(3) would have gamma matrices instead of the Pauli 00:52:06.280 --> 00:52:08.470 matrices. 00:52:08.470 --> 00:52:10.270 There's two bases that you can use. 00:52:10.270 --> 00:52:12.100 You can either use this basis, which 00:52:12.100 --> 00:52:15.040 is like enumerated 1 to 8, or you 00:52:15.040 --> 00:52:16.820 could use the charged basis. 00:52:16.820 --> 00:52:19.630 And if you use the charge basis-- 00:52:19.630 --> 00:52:22.030 and often you write it out as a matrix like this. 00:52:28.790 --> 00:52:31.100 On your problem set, you're free to pick which basis 00:52:31.100 --> 00:52:31.880 you want to use. 00:52:37.220 --> 00:52:39.950 It may be that one or the other is easier, 00:52:39.950 --> 00:52:41.962 but I can't even tell you which one is easier 00:52:41.962 --> 00:52:42.920 since I don't remember. 00:52:46.105 --> 00:52:48.185 So sometimes one or the other is easier to use. 00:52:48.185 --> 00:52:50.060 And you have a freedom of what basis to pick. 00:52:54.520 --> 00:53:01.960 If you expand, in this case, the trace of sigma Mq dagger 00:53:01.960 --> 00:53:08.890 plus Mq sigma dagger, which is that term that had a V0, 00:53:08.890 --> 00:53:10.870 then that gives mass to the mesons, 00:53:10.870 --> 00:53:11.935 as it did for the pions. 00:53:18.308 --> 00:53:19.850 Because the symmetry group is bigger, 00:53:19.850 --> 00:53:21.700 you get more predictions. 00:53:21.700 --> 00:53:25.060 Here, you get predictions for the kaon masses. 00:53:25.060 --> 00:53:35.120 And you get things like the fact that the neutral kaons have 00:53:35.120 --> 00:53:41.180 massless order Md plus Ms. And you get things 00:53:41.180 --> 00:53:46.950 like eta pi 0 mixing, where there's a mixing matrix. 00:53:46.950 --> 00:53:49.270 So if you look at the masses of eta and pi 0, 00:53:49.270 --> 00:53:51.050 they're actually non-diagonal. 00:53:51.050 --> 00:53:52.890 So for the eta in the pi 0 system, 00:53:52.890 --> 00:53:54.980 you actually get a matrix. 00:53:54.980 --> 00:53:56.315 So M squared is a matrix. 00:53:59.430 --> 00:54:03.800 So for example, for the pi 0, it was just M up plus Md. 00:54:06.980 --> 00:54:11.060 But then once you're in SU(3), there's actually a mixing term. 00:54:11.060 --> 00:54:13.895 And there's like an M up minus M down term here 00:54:13.895 --> 00:54:20.220 that is mixing between, and then same thing over here. 00:54:20.220 --> 00:54:22.190 So the etas and the pi 0s actually are mixing. 00:54:22.190 --> 00:54:25.310 And there's something in this n tree. 00:54:25.310 --> 00:54:27.560 And the mixing is isospin violating in the sense 00:54:27.560 --> 00:54:29.050 that it's M up minus M down. 00:54:29.050 --> 00:54:33.230 So it's a small effect, but this is something 00:54:33.230 --> 00:54:36.440 that you can predict from chiral Lagrangian, something 00:54:36.440 --> 00:54:39.560 about because is describes isospin violating effects 00:54:39.560 --> 00:54:40.550 from the quark masses. 00:54:46.100 --> 00:54:48.320 So the reason that I mention that is 00:54:48.320 --> 00:54:50.810 because, often when you do calculations, keeping 00:54:50.810 --> 00:54:53.420 track of Mu's, Md's, and Ms's all 00:54:53.420 --> 00:54:56.820 as separate independent parameters is a little much. 00:54:56.820 --> 00:54:58.620 And so you want to make an approximation. 00:54:58.620 --> 00:55:01.430 And so if you make an approximation that 00:55:01.430 --> 00:55:03.110 ignores isospin violation-- 00:55:09.090 --> 00:55:15.400 so we often ignore isospin violation. 00:55:15.400 --> 00:55:17.680 Isospin violation is very small. 00:55:17.680 --> 00:55:19.720 And if you remember, for the SU(3) case, 00:55:19.720 --> 00:55:23.460 you're expanding in Ms over lambda QCD or Mk 00:55:23.460 --> 00:55:24.180 over lambda chi. 00:55:24.180 --> 00:55:25.440 So it's not a great expansion. 00:55:25.440 --> 00:55:27.840 You have something like a third. 00:55:27.840 --> 00:55:30.180 So ignoring isospin is perfectly valid 00:55:30.180 --> 00:55:31.820 if you're expanding in a third. 00:55:31.820 --> 00:55:33.570 So basically, because there is a hierarchy 00:55:33.570 --> 00:55:36.792 between the strange quark mass and the down and the up, 00:55:36.792 --> 00:55:39.000 you want to focus on places where you get the largest 00:55:39.000 --> 00:55:39.570 corrections. 00:55:39.570 --> 00:55:41.975 And one way of making an approximation that 00:55:41.975 --> 00:55:44.100 allows you to do that is to ignore isospin and take 00:55:44.100 --> 00:55:47.400 Mu equal to Md. 00:55:47.400 --> 00:55:52.200 So if we take Mu at Md to be some M hat-- 00:55:52.200 --> 00:55:55.800 which if you want an exact definition, you could say 00:55:55.800 --> 00:55:58.185 it's the average. 00:55:58.185 --> 00:55:59.868 And you'd drop the difference. 00:56:04.350 --> 00:56:06.750 And then you can think of the strange quark mass 00:56:06.750 --> 00:56:08.280 as being somewhat bigger than M hat. 00:56:14.450 --> 00:56:17.175 That's an approximation that you can use on your problem set, 00:56:17.175 --> 00:56:17.675 for example. 00:56:23.760 --> 00:56:26.420 So let me write out here with all the terms 00:56:26.420 --> 00:56:28.130 in the chiral Lagrangian is. 00:56:28.130 --> 00:56:32.120 And I'll do it for a case where we 00:56:32.120 --> 00:56:37.582 include in our chiral Lagrangian one other type of coupling, 00:56:37.582 --> 00:56:39.290 which is this left-handed current that we 00:56:39.290 --> 00:56:42.000 talked about last time. 00:56:42.000 --> 00:56:44.665 So we talked about a spurion analysis for the chi term. 00:56:44.665 --> 00:56:46.040 And I said you could do something 00:56:46.040 --> 00:56:49.160 similar to a couple in a left-handed current. 00:56:49.160 --> 00:56:53.120 And we had this, where D mu sigma 00:56:53.120 --> 00:56:57.680 was partial mu sigma times the left-handed current sigma. 00:56:57.680 --> 00:56:59.180 So we thought of putting it together 00:56:59.180 --> 00:57:00.950 into a coherent derivative. 00:57:00.950 --> 00:57:03.650 And if I do that, it modifies the leading order Lagrangian 00:57:03.650 --> 00:57:06.465 and just makes these partials into covariant derivatives. 00:57:15.250 --> 00:57:19.760 Power counting, for counting purposes, 00:57:19.760 --> 00:57:22.250 you count sigmas of order 1. 00:57:22.250 --> 00:57:27.320 You count D mu sigma as order p, which means you count 00:57:27.320 --> 00:57:29.700 L mu, the source, as order p. 00:57:29.700 --> 00:57:31.070 This is a left-handed source. 00:57:33.620 --> 00:57:37.513 And you count chis and Mq's as of order p squared. 00:57:37.513 --> 00:57:39.430 That's just repeating what we've already said. 00:57:44.640 --> 00:57:46.950 And these higher order terms, which where L's, we 00:57:46.950 --> 00:57:48.250 can then enumerate. 00:57:52.850 --> 00:57:57.650 And I now just use D's, covariant D's 00:57:57.650 --> 00:58:00.590 when I write them down. 00:58:00.590 --> 00:58:03.200 I'm now in SU(3). 00:58:03.200 --> 00:58:06.860 And it turns out that one of the relations that we used in SU(2) 00:58:06.860 --> 00:58:08.960 doesn't carry over to SU(3). 00:58:08.960 --> 00:58:14.930 So if I just think of guys with four covariant derivatives, 00:58:14.930 --> 00:58:18.820 it turns out that there's one more operator there. 00:58:22.400 --> 00:58:27.350 So there's those two, which we talked about, L1 and L2. 00:58:27.350 --> 00:58:30.620 And then there's also L3. 00:58:30.620 --> 00:58:32.180 So we can't get rid of this guy. 00:58:35.540 --> 00:58:37.910 I'm always writing sigmas next to sigma daggers because 00:58:37.910 --> 00:58:41.240 of the chiral transformation. 00:58:41.240 --> 00:58:43.967 I'm also imposing parity, though I'm not 00:58:43.967 --> 00:58:45.800 going to spend much time talking about that. 00:58:49.780 --> 00:58:52.000 And really I want to enumerate, also, for you 00:58:52.000 --> 00:58:59.090 some of the terms that involve the quark mass, the chi guy. 00:58:59.090 --> 00:59:01.860 So you could have a guy that's a cross-term. 00:59:01.860 --> 00:59:06.630 This would take care of the renormalization of things 00:59:06.630 --> 00:59:09.320 like p squared and pi squared times 1 of epsilon. 00:59:13.560 --> 00:59:16.290 That's L4, L5. 00:59:41.610 --> 00:59:44.115 We could also have the quark mass type term just squared. 00:59:48.657 --> 00:59:49.990 So take the trace and square it. 00:59:49.990 --> 00:59:52.560 That's L6. 00:59:52.560 --> 00:59:54.570 It turns out that we could also build something 00:59:54.570 --> 00:59:56.880 with the right parity by just having the difference instead 00:59:56.880 --> 00:59:57.380 of the sum. 00:59:57.380 --> 00:59:58.918 And that's a different operator. 01:00:05.610 --> 01:00:06.585 So that's L7. 01:00:10.080 --> 01:00:11.190 We're going to go up to 9. 01:00:16.200 --> 01:00:18.398 Don't be afraid. 01:00:18.398 --> 01:00:20.190 You can have something with 2 sigma daggers 01:00:20.190 --> 01:00:25.440 and 2 chis, which is another way of building a chiral invariant. 01:00:25.440 --> 01:00:30.190 And then for parity, you need the other way. 01:00:30.190 --> 01:00:33.510 And then finally-- something called 01:00:33.510 --> 01:00:39.030 L9, which involves a trace that involves 01:00:39.030 --> 01:00:43.042 L mu nu, where L mu nu is built out of this external current. 01:00:43.042 --> 01:00:45.000 So we can take two of our covariant derivatives 01:00:45.000 --> 01:00:46.477 and take a commutator. 01:00:51.840 --> 01:00:53.826 That's giving the final operator. 01:01:10.850 --> 01:01:16.190 So there's a complete basis for SU(3) for a left-handed current 01:01:16.190 --> 01:01:17.428 in the chi. 01:01:17.428 --> 01:01:19.220 AUDIENCE: [? So does ?] [? L mu ?] [? nu ?] 01:01:19.220 --> 01:01:20.018 [? have a field ?] [INAUDIBLE] [? associated with ?] 01:01:20.018 --> 01:01:20.180 [INAUDIBLE]? 01:01:20.180 --> 01:01:21.200 IAIN STEWART: Yeah, I'll write it down. 01:01:21.200 --> 01:01:21.742 AUDIENCE: Oh. 01:01:45.698 --> 01:01:47.240 IAIN STEWART: So if L mu is something 01:01:47.240 --> 01:01:50.690 that has SU(3) indices, has an SU(3) matrix hiding inside it, 01:01:50.690 --> 01:01:53.660 then there's a commutator term as well. 01:01:53.660 --> 01:01:57.590 And then it's just this combination. 01:01:57.590 --> 01:02:00.998 I also use the equation of motion. 01:02:00.998 --> 01:02:03.540 And I didn't talk about it, but I use the equation of motion. 01:02:03.540 --> 01:02:05.370 And I used SU(3) relations. 01:02:05.370 --> 01:02:06.950 Much as I talked about for SU(2), 01:02:06.950 --> 01:02:09.448 there's some SU(3) relations that survive. 01:02:09.448 --> 01:02:11.240 And I got rid of some operators doing that. 01:02:16.200 --> 01:02:19.620 Now, you could say, well, I have this SU(3) and SU(2), 01:02:19.620 --> 01:02:21.990 so why don't I try to relate them? 01:02:21.990 --> 01:02:23.340 One of them has a kaon. 01:02:23.340 --> 01:02:24.420 The other one doesn't. 01:02:24.420 --> 01:02:27.000 The one without the kaon thinks about the kaon 01:02:27.000 --> 01:02:28.298 as a heavy particle. 01:02:28.298 --> 01:02:30.090 The one with the kaon thinks about the kaon 01:02:30.090 --> 01:02:31.530 as a light particle. 01:02:31.530 --> 01:02:32.520 Those are two theories. 01:02:32.520 --> 01:02:35.820 I could try to make them match up with each other. 01:02:35.820 --> 01:02:37.890 And that's something that you can do, actually. 01:02:41.860 --> 01:02:48.300 So there's a little bit of a lesson there. 01:02:48.300 --> 01:02:50.220 So that was why I want to mention it. 01:02:56.790 --> 01:02:58.540 So SU(2) and SU(3) seem like they're 01:02:58.540 --> 01:02:59.690 describing similar physics. 01:02:59.690 --> 01:03:01.560 They both could describe pions. 01:03:01.560 --> 01:03:03.130 They both have pions in them. 01:03:03.130 --> 01:03:04.090 But the SU(3) has more. 01:03:04.090 --> 01:03:05.770 It's got the kaon. 01:03:05.770 --> 01:03:08.500 That means that, in the SU(2) theory, 01:03:08.500 --> 01:03:13.540 the kaon is in the coefficients, OK? 01:03:13.540 --> 01:03:16.090 So if you do a correspondence, you 01:03:16.090 --> 01:03:18.650 get relations that are like this. 01:03:18.650 --> 01:03:20.260 So the 2 here means SU(2). 01:03:25.380 --> 01:03:28.180 And where I don't put any subscript, it's SU(3). 01:03:32.080 --> 01:03:35.410 And something that-- this is a 96. 01:03:44.140 --> 01:03:48.010 And if you really do that, what I said, compare observables, 01:03:48.010 --> 01:03:50.710 you get relations like this one where you actually 01:03:50.710 --> 01:03:54.190 see the kaon is showing up on this side 01:03:54.190 --> 01:03:56.470 and is being encoded in coefficients in the SU(2) 01:03:56.470 --> 01:03:58.190 theory. 01:03:58.190 --> 01:03:59.868 So what you think of as the coefficients 01:03:59.868 --> 01:04:01.660 in your chiral theory depends on the matter 01:04:01.660 --> 01:04:03.783 that you've put in, including things 01:04:03.783 --> 01:04:05.950 like what particles like the kaon, what group you're 01:04:05.950 --> 01:04:06.640 talking about. 01:04:09.500 --> 01:04:18.680 This is an explicit example of the kaon being the coefficients 01:04:18.680 --> 01:04:21.890 if we use SU(2). 01:04:25.690 --> 01:04:31.510 OK, so just like in SU(2), we have to go through 01:04:31.510 --> 01:04:34.420 a renormalization of the Li's. 01:04:39.280 --> 01:04:41.890 And you can think of that by writing a formula 01:04:41.890 --> 01:04:42.520 like this one. 01:04:45.410 --> 01:04:47.690 Bare Li is equal to some renormalized one, 01:04:47.690 --> 01:04:51.800 which I put a bar on top of and then a counter-term. 01:04:51.800 --> 01:04:55.850 And the counter-term will be some coefficient, 01:04:55.850 --> 01:05:01.760 some number of 4 pis, and then some epsilons. 01:05:01.760 --> 01:05:05.570 And just like in gauge theory, we 01:05:05.570 --> 01:05:09.070 get rid of the Euler gamma and the log 4 pi. 01:05:09.070 --> 01:05:11.330 We have an Ms bar type definition. 01:05:11.330 --> 01:05:13.790 And in chiral perturbation theory, 01:05:13.790 --> 01:05:19.010 people often take an extra 1 along with the ride 01:05:19.010 --> 01:05:21.510 just because they're allowed to. 01:05:21.510 --> 01:05:23.510 Because it tends to show up in the kind of loops 01:05:23.510 --> 01:05:24.530 that you encounter. 01:05:24.530 --> 01:05:28.940 So sometimes that's included, sometimes it's not. 01:05:28.940 --> 01:05:35.780 So for example, as an example of a diagram, 01:05:35.780 --> 01:05:38.060 you could think about this one. 01:05:38.060 --> 01:05:42.022 And that does cause mass renormalization 01:05:42.022 --> 01:05:42.980 for the physical boson. 01:05:48.840 --> 01:05:51.120 So if you think about our lowest order relation 01:05:51.120 --> 01:05:57.570 as where we started, let me write that as M0 squared is 01:05:57.570 --> 01:06:04.320 4V0 over f squared Mu plus Md. 01:06:07.950 --> 01:06:10.060 Sometimes people call this-- 01:06:10.060 --> 01:06:18.330 well, making up some more notation, 01:06:18.330 --> 01:06:21.758 yeah, which I'll use in a minute. 01:06:21.758 --> 01:06:23.550 And so if you actually calculate this loop, 01:06:23.550 --> 01:06:25.300 then you get a correction to that formula. 01:06:25.300 --> 01:06:29.820 This would have be M pi squared, but here we get a correction. 01:06:29.820 --> 01:06:34.500 If you did it in SU(2), just to keep 01:06:34.500 --> 01:06:38.760 the formula a little simpler, then it would look like this. 01:06:43.270 --> 01:06:46.710 So M0 is like the pi n mass, but the pi n mass at lowest order 01:06:46.710 --> 01:06:47.952 in the chiral expansion. 01:06:47.952 --> 01:06:49.410 So it's not the physical pi n mass. 01:06:52.320 --> 01:07:01.110 So 2L 4 bar in theory two with two flavors with two SU(2) plus 01:07:01.110 --> 01:07:04.770 an L5 bar minus 4 n L6 bar-- 01:07:04.770 --> 01:07:06.570 so these various coefficients are coming in 01:07:06.570 --> 01:07:10.883 with some numbers in front, some combination of them. 01:07:10.883 --> 01:07:12.300 And then there's some contribution 01:07:12.300 --> 01:07:13.870 from a chiral loop. 01:07:13.870 --> 01:07:19.600 And if I take away that 1, it's just a chiral logarithm. 01:07:19.600 --> 01:07:28.470 So this is 4 pi f squared log M0 squared over mu squared. 01:07:31.890 --> 01:07:35.190 There would be an extra term of just plus 1 times this 01:07:35.190 --> 01:07:38.580 if I hadn't gotten rid of that, the minus 1 times that, OK? 01:07:38.580 --> 01:07:39.720 So this is from the loop. 01:07:39.720 --> 01:07:43.500 This is from the explicit dimension operators 01:07:43.500 --> 01:07:46.200 that were of the higher dimension, some combination 01:07:46.200 --> 01:07:47.700 of them. 01:07:47.700 --> 01:07:51.230 The UV divergence is absorbed in some combination of them. 01:07:51.230 --> 01:07:53.750 And if you want to figure out exactly how UV divergences go 01:07:53.750 --> 01:07:56.430 in between this, then you've got to think of renormalizing 01:07:56.430 --> 01:07:59.470 more than just this diagram. 01:07:59.470 --> 01:08:02.310 And you can generically think of observables 01:08:02.310 --> 01:08:07.090 as having this kind of expansion in Mu and Md. 01:08:07.090 --> 01:08:10.440 So if you like, you should think of M0 here as really just an up 01:08:10.440 --> 01:08:11.462 plus M down. 01:08:11.462 --> 01:08:13.920 This is saying, at lowest order chiral perturbation theory, 01:08:13.920 --> 01:08:15.720 there's a linear term. 01:08:15.720 --> 01:08:19.439 But then here there's an M0 to the fourth term, which is 01:08:19.439 --> 01:08:21.750 quadratic in the quark masses. 01:08:21.750 --> 01:08:24.779 And there's also a quadratic term from the loops. 01:08:24.779 --> 01:08:27.660 So you have an expansion in the quark masses. 01:08:27.660 --> 01:08:32.880 Think of M0 as the quark masses and M pi as the meson mass. 01:08:32.880 --> 01:08:34.859 I just gave you the example of the pion mass, 01:08:34.859 --> 01:08:38.910 but this is generically true for observables 01:08:38.910 --> 01:08:42.210 that you might calculate that they have this type of result 01:08:42.210 --> 01:08:44.218 where you have an expansion. 01:08:44.218 --> 01:08:47.880 AUDIENCE: Is there any physical reason for why this mass square 01:08:47.880 --> 01:08:52.390 of the meson space [? would ?] [? be ?] [INAUDIBLE]?? 01:08:52.390 --> 01:08:54.859 IAIN STEWART: So one way of saying it is-- 01:08:54.859 --> 01:08:55.359 yeah. 01:08:55.359 --> 01:08:59.189 So you might think, well, why is it not M pi equals Mu and Md? 01:08:59.189 --> 01:09:00.910 And the glib way of saying it is, well, 01:09:00.910 --> 01:09:02.535 if you think about M pi squared and you 01:09:02.535 --> 01:09:04.450 think about it having an expansion, 01:09:04.450 --> 01:09:06.990 then it could have a constant plus linear term 01:09:06.990 --> 01:09:08.117 plus quadratic term. 01:09:08.117 --> 01:09:10.575 You don't have the constant because of the chiral symmetry. 01:09:10.575 --> 01:09:14.069 So the linear term is the first thing that's allowed. 01:09:14.069 --> 01:09:16.939 That's one way of saying it. 01:09:16.939 --> 01:09:19.380 I mean, if you think about it in the bosonic theory, 01:09:19.380 --> 01:09:23.100 we have the Lagrangian is quadratic in masses, 01:09:23.100 --> 01:09:25.047 right, whereas in the fermionic theory, 01:09:25.047 --> 01:09:26.130 you have linear in masses. 01:09:26.130 --> 01:09:29.670 And that's also part of what it had to do with, 01:09:29.670 --> 01:09:33.330 but it's linear combination of symmetry breaking and that. 01:09:37.779 --> 01:09:41.680 It's allowed, so it happens is kind of the bottom line. 01:09:41.680 --> 01:09:43.628 But it's allowed by the symmetries. 01:09:47.319 --> 01:09:47.819 OK. 01:09:47.819 --> 01:09:50.611 And so finally, this is the example I'm actually 01:09:50.611 --> 01:09:51.569 going to get you to do. 01:09:54.960 --> 01:09:58.410 So I'll have you look on the problem set at the k constants. 01:09:58.410 --> 01:10:01.270 And they have an analogous result to that one. 01:10:01.270 --> 01:10:04.470 You'll do the calculation in SU(3). 01:10:04.470 --> 01:10:07.860 And the kind of result that you should expect to get 01:10:07.860 --> 01:10:09.450 is something that looks like this. 01:10:22.340 --> 01:10:24.250 So I'm using this B0 notation that I 01:10:24.250 --> 01:10:28.900 introduced over there just so I could write it all on one line. 01:10:33.170 --> 01:10:36.410 And these mus with the subscript i 01:10:36.410 --> 01:10:39.110 are just some shorthand for some contributions 01:10:39.110 --> 01:10:39.920 coming from loops. 01:10:51.320 --> 01:10:57.800 And f is the result in L0. 01:11:00.920 --> 01:11:06.940 So f is the parameter in L0, whereas f pi 01:11:06.940 --> 01:11:10.360 is the physical decay constant of the pion. 01:11:16.830 --> 01:11:21.240 So one result which is encoded in this formula 01:11:21.240 --> 01:11:23.160 is that the Lagrangian parameter is actually 01:11:23.160 --> 01:11:25.200 equal to the decay constant at lowest order 01:11:25.200 --> 01:11:25.950 in chiral perturbation theory. 01:11:25.950 --> 01:11:27.460 That's something I didn't cover. 01:11:27.460 --> 01:11:29.810 I cover it when I teach quantum field theory three. 01:11:29.810 --> 01:11:32.880 You can look at my notes to see that derivation. 01:11:32.880 --> 01:11:36.640 Those of you that have taken QFT3 from me, 01:11:36.640 --> 01:11:37.902 you've already seen that. 01:11:37.902 --> 01:11:40.110 And basically, what I'm asking for in the problem set 01:11:40.110 --> 01:11:41.820 is I guide you with several parts, 01:11:41.820 --> 01:11:45.067 but how to think about these higher order terms. 01:11:45.067 --> 01:11:46.650 What are the loop graph contributions? 01:11:46.650 --> 01:11:49.890 How do you get these terms from the higher order Lagrangians? 01:11:49.890 --> 01:11:51.870 How you put it all together? 01:11:51.870 --> 01:11:55.950 It's a nice example of this use of chiral perturbation theory. 01:11:55.950 --> 01:11:57.840 And again, we see that a physical observable, 01:11:57.840 --> 01:12:00.660 which is this decay constant, has a chiral expansion. 01:12:00.660 --> 01:12:03.330 And the thing that chiral perturbation theory is actually 01:12:03.330 --> 01:12:06.720 doing is allowing you to predict both the form 01:12:06.720 --> 01:12:09.390 of that expansion, as well as these things here, which 01:12:09.390 --> 01:12:12.450 are chiral logarithms. 01:12:12.450 --> 01:12:16.270 So if you ask about predictive power, the polynomial terms 01:12:16.270 --> 01:12:18.450 in terms of higher order in the quark masses 01:12:18.450 --> 01:12:21.630 that are polynomial, you end up having unknown coefficients. 01:12:21.630 --> 01:12:23.550 But the terms with logarithms have 01:12:23.550 --> 01:12:26.670 coefficients which are fixed by your lower order Lagrangian. 01:12:26.670 --> 01:12:28.170 So those are things that you predict 01:12:28.170 --> 01:12:31.800 with the chiral perturbation theory. 01:12:31.800 --> 01:12:33.360 When people do lattice calculations, 01:12:33.360 --> 01:12:34.950 they need to do chiral extrapolations. 01:12:34.950 --> 01:12:36.867 And then they're using formulas like this one. 01:12:40.140 --> 01:12:43.384 OK, so any questions about that? 01:12:43.384 --> 01:12:45.283 AUDIENCE: Did the M pi [INAUDIBLE]?? 01:12:45.283 --> 01:12:46.700 IAIN STEWART: Yeah, physical mass. 01:12:46.700 --> 01:12:47.870 This is the physical mass. 01:12:47.870 --> 01:12:49.670 This is the quark masses. 01:12:49.670 --> 01:12:54.610 M0 is just this combination with M up and M down. 01:12:54.610 --> 01:12:56.690 AUDIENCE: So the mu dependence cancel between-- 01:12:56.690 --> 01:12:58.148 IAIN STEWART: And the mu dependence 01:12:58.148 --> 01:13:01.040 cancels because these guys all depend on mu. 01:13:01.040 --> 01:13:03.230 Yeah. 01:13:03.230 --> 01:13:06.600 And same thing here, these guys depend on mu. 01:13:09.668 --> 01:13:12.190 Here, I have just enough room to make that explicit. 01:13:15.440 --> 01:13:17.828 OK, so that actually covers all the goals 01:13:17.828 --> 01:13:19.620 that we had for chiral perturbation theory, 01:13:19.620 --> 01:13:21.410 though it's a fun topic. 01:13:21.410 --> 01:13:23.870 And there's many more things we could discuss, 01:13:23.870 --> 01:13:27.420 but that's what we needed to discuss. 01:13:27.420 --> 01:13:30.000 And so we're going to move on. 01:13:30.000 --> 01:13:31.920 So the next thing I want to talk about 01:13:31.920 --> 01:13:33.920 as an example of effective field theory is heavy 01:13:33.920 --> 01:13:34.970 quark effective theory. 01:13:48.160 --> 01:13:50.580 So again, what are our goals with this effective theory? 01:13:58.730 --> 01:14:00.500 We will see some new features showing up 01:14:00.500 --> 01:14:01.667 that we haven't seen before. 01:14:06.887 --> 01:14:09.220 We will find out what it means to take a Lagrangian that 01:14:09.220 --> 01:14:10.280 has labeled fields. 01:14:16.315 --> 01:14:18.190 We'll spend a little bit of time on symmetry. 01:14:18.190 --> 01:14:21.465 Because in this heavy quark effective theory, 01:14:21.465 --> 01:14:23.840 there's actually something called heavy quark's symmetry, 01:14:23.840 --> 01:14:26.140 which is not apparent in QCD, but becomes apparent 01:14:26.140 --> 01:14:29.110 in this theory. 01:14:29.110 --> 01:14:35.590 And there's a trick known as using covariate representations 01:14:35.590 --> 01:14:37.900 in order to encode symmetry predictions. 01:14:37.900 --> 01:14:40.000 And it's a very powerful thing that's 01:14:40.000 --> 01:14:42.750 not special to this theory, but you could use it in general. 01:14:42.750 --> 01:14:44.000 And I want to teach it to you. 01:14:44.000 --> 01:14:45.208 So that's one thing we'll do. 01:14:52.558 --> 01:14:54.100 It's kind of like a spurion analysis, 01:14:54.100 --> 01:14:59.050 but a little bit more powerful. 01:14:59.050 --> 01:15:03.190 And finally, anomalous dimensions that are functions 01:15:03.190 --> 01:15:09.790 is something that we'll show up here, not just numbers. 01:15:09.790 --> 01:15:20.180 There's something called reparameterization invariance, 01:15:20.180 --> 01:15:25.490 which you can think of a kind of symmetry that we'll talk about. 01:15:25.490 --> 01:15:31.180 And finally, if that list is not long enough, 01:15:31.180 --> 01:15:36.820 we'll add one more thing which I hinted at. 01:15:36.820 --> 01:15:40.090 So I said, when we talked about Ms bar, 01:15:40.090 --> 01:15:42.387 that it wasn't a perfect scheme for doing things, 01:15:42.387 --> 01:15:43.720 but there were some limitations. 01:15:43.720 --> 01:15:45.512 And we'll come, at the end of this chapter, 01:15:45.512 --> 01:15:47.080 to what those limitations are. 01:15:54.140 --> 01:15:58.310 The limitations come from power-like scale separation. 01:15:58.310 --> 01:16:02.210 And that's related to something called renormalons. 01:16:02.210 --> 01:16:05.030 So we'll learn what a renormalon is 01:16:05.030 --> 01:16:07.640 and why it has something to do with the failure of Ms bar 01:16:07.640 --> 01:16:10.893 and how one can get around that failure 01:16:10.893 --> 01:16:12.560 and how one actually needs to get around 01:16:12.560 --> 01:16:15.000 that failure in some cases. 01:16:15.000 --> 01:16:17.840 OK, so that's the list of goals. 01:16:17.840 --> 01:16:22.220 It's kind of, in some ways, in order of importance, actually. 01:16:24.750 --> 01:16:27.000 So when we think about heavy quark effective theory, 01:16:27.000 --> 01:16:29.137 you shouldn't think about it as just something 01:16:29.137 --> 01:16:31.470 that you would need to do if you wanted to do heavy work 01:16:31.470 --> 01:16:34.200 physics because the idea of what we're doing here 01:16:34.200 --> 01:16:35.915 is much more general than that. 01:16:35.915 --> 01:16:37.290 The idea of what we're doing here 01:16:37.290 --> 01:16:39.990 is we're saying, take a heavy particle 01:16:39.990 --> 01:16:43.770 and think about what happens if I tickle it. 01:16:43.770 --> 01:16:49.120 And that heavy particle could be a heavy source to some theory. 01:16:49.120 --> 01:16:51.120 We'll talk about it in the context of them being 01:16:51.120 --> 01:16:52.578 heavy quarks, but you should really 01:16:52.578 --> 01:16:56.100 think of it as any heavy particle tickled 01:16:56.100 --> 01:16:57.060 by light particles. 01:17:15.010 --> 01:17:18.370 And what we really mean is that we want 01:17:18.370 --> 01:17:21.037 to study the heavy particle. 01:17:21.037 --> 01:17:22.870 And since we want to study it, we better not 01:17:22.870 --> 01:17:23.980 remove it from the theory. 01:17:27.532 --> 01:17:29.740 So we want to tickle it with light degrees of freedom 01:17:29.740 --> 01:17:32.830 with small momentum transfer, but we don't want 01:17:32.830 --> 01:17:35.578 to integrate that particle out. 01:17:35.578 --> 01:17:37.120 So another way of saying this, if you 01:17:37.120 --> 01:17:39.037 don't want to think of it as a heavy particle, 01:17:39.037 --> 01:17:40.810 is that you have some source. 01:17:40.810 --> 01:17:45.550 And that source can be tickled and could wiggle, 01:17:45.550 --> 01:17:47.718 but it's sort of mostly just a static source. 01:17:47.718 --> 01:17:48.760 And then it could wiggle. 01:17:48.760 --> 01:17:51.663 And that's what this is an effective field theory for. 01:17:51.663 --> 01:17:53.830 So we'll talk about it in the context of heavy quark 01:17:53.830 --> 01:17:54.460 effective theory. 01:17:54.460 --> 01:17:56.543 And some of the things, like heavy quark symmetry, 01:17:56.543 --> 01:17:57.892 will be special to that theory. 01:17:57.892 --> 01:18:00.100 But more generally, the kind of approach we're taking 01:18:00.100 --> 01:18:01.100 is a more general thing. 01:18:04.330 --> 01:18:07.270 So what's heavy quark effective theory? 01:18:07.270 --> 01:18:08.837 So you have a heavy quark. 01:18:08.837 --> 01:18:11.295 And you have it sitting in a bound state, which is a meson. 01:18:14.770 --> 01:18:18.070 So that means it's surrounded by light degrees of freedom, 01:18:18.070 --> 01:18:21.520 one heavy quark, lots of light junk. 01:18:21.520 --> 01:18:24.430 In the quark model, you'd say it's a Q bar q, heavy cork 01:18:24.430 --> 01:18:31.270 capital Q and a light cork little q, like b bar d, 01:18:31.270 --> 01:18:33.490 which is the B0. 01:18:33.490 --> 01:18:36.850 But that's just a quark model approximation 01:18:36.850 --> 01:18:38.470 for what the degrees of freedom are. 01:18:38.470 --> 01:18:41.137 And there's many more pairs of Qq bars and gluons 01:18:41.137 --> 01:18:42.970 that are really forming this hydronic state. 01:18:45.547 --> 01:18:48.130 So you'd like to be able to make model independent predictions 01:18:48.130 --> 01:18:51.370 for that without having to worry about the fact 01:18:51.370 --> 01:18:54.790 that you're not parameterizing properly that stuff 01:18:54.790 --> 01:18:55.990 that you can't calculate. 01:18:55.990 --> 01:19:00.140 And you can do that using effective field theory. 01:19:00.140 --> 01:19:01.820 There's two scales in the problem. 01:19:01.820 --> 01:19:04.540 There's the size of this thing, the inverse size 01:19:04.540 --> 01:19:05.650 of order lambda QCD. 01:19:05.650 --> 01:19:09.317 That's the hadronization that scale. 01:19:09.317 --> 01:19:11.650 And that's a scale that's much less than the quark mass. 01:19:11.650 --> 01:19:14.190 So you're really just expanding in lambda QCD divided by Mq. 01:19:14.190 --> 01:19:15.940 That's what this effective theory will be. 01:19:21.610 --> 01:19:27.250 And what you want to describe are 01:19:27.250 --> 01:19:31.882 fluctuations of the heavy quark due to the light quarks. 01:19:31.882 --> 01:19:33.340 So you could think that this guy is 01:19:33.340 --> 01:19:36.550 so heavy he just sits in the middle of the state, 01:19:36.550 --> 01:19:39.087 and he's static. 01:19:39.087 --> 01:19:41.170 And then he gets tickled by the light stuff that's 01:19:41.170 --> 01:19:42.295 flying and whizzing around. 01:19:42.295 --> 01:19:43.660 And that's a reasonable picture. 01:19:47.200 --> 01:19:48.760 So what we really want to do here 01:19:48.760 --> 01:19:52.090 is an example of top-down effective field theory again. 01:19:52.090 --> 01:19:58.210 So going back to our like integrating out 01:19:58.210 --> 01:20:02.860 heavy particles, but now we want to keep the heavy particles 01:20:02.860 --> 01:20:04.540 in the theory, not remove them. 01:20:04.540 --> 01:20:06.790 But still, we want to take a low energy limit of them. 01:20:09.710 --> 01:20:14.050 So that means that we have to take a low energy limit of QCD, 01:20:14.050 --> 01:20:19.438 which is id slash minus Mq Q. And you 01:20:19.438 --> 01:20:20.980 can see that part of the problem here 01:20:20.980 --> 01:20:23.560 is that the Mq is upstairs, which 01:20:23.560 --> 01:20:26.530 makes taking the limit not completely obvious. 01:20:26.530 --> 01:20:28.810 In the case where we were doing the heavy bosons, 01:20:28.810 --> 01:20:30.892 we sort of saw that it was always in propagators. 01:20:30.892 --> 01:20:33.100 And we could just think of, since it's always they're 01:20:33.100 --> 01:20:35.500 internal, we just expand. 01:20:35.500 --> 01:20:37.640 Here, we want to keep the Q in the theory. 01:20:37.640 --> 01:20:41.930 So we have to really think about how we take this limit. 01:20:41.930 --> 01:20:48.490 So let's start slowly and consider the propagator 01:20:48.490 --> 01:20:50.987 for a heavy quark. 01:20:50.987 --> 01:20:52.570 And we'll come back to the Lagrangian. 01:20:56.890 --> 01:21:06.990 And we'll consider it with some on-shell momentum, which we'll 01:21:06.990 --> 01:21:09.570 consider an on-shell momentum to be parameterized 01:21:09.570 --> 01:21:11.560 in the following way. 01:21:11.560 --> 01:21:13.200 So if I have an on-shell momentum, 01:21:13.200 --> 01:21:17.640 I'll say that p is equal to MqV, so that p squared is 01:21:17.640 --> 01:21:23.310 equal to Mq squared and this V is 1. 01:21:23.310 --> 01:21:25.590 So once I pull out the mass dimension Mq, 01:21:25.590 --> 01:21:28.440 the remaining thing I call V. And that's just some parameter 01:21:28.440 --> 01:21:32.850 which squares to 1 on-shell. 01:21:32.850 --> 01:21:36.990 Now, if I have kicks, which are from light degrees of freedom, 01:21:36.990 --> 01:21:39.780 then I don't have exactly on-shell. 01:21:39.780 --> 01:21:45.240 So P mu is MqV, which is like on-shell piece plus something 01:21:45.240 --> 01:21:48.480 small, k mu. 01:21:48.480 --> 01:21:52.620 And k mu is of order lambda QCD. 01:21:52.620 --> 01:21:56.280 So this is like saying the on-shell piece plus the tickle. 01:22:01.380 --> 01:22:03.240 And if I want to construct the propagator, 01:22:03.240 --> 01:22:09.120 then the propagator is encoding the optional off-shellness 01:22:09.120 --> 01:22:10.740 of the degree of freedom. 01:22:10.740 --> 01:22:13.510 That's 1 over the off-shellness. 01:22:13.510 --> 01:22:14.760 So it'll depend on the tickle. 01:22:24.710 --> 01:22:33.200 OK, so we could just take QCD propagator, which is this, 01:22:33.200 --> 01:22:34.530 and just plug in that formula. 01:22:50.100 --> 01:22:52.920 There's Mq squared, which cancels this Mq squared. 01:22:56.900 --> 01:23:00.650 Then there's some cross-terms that don't get cancelled. 01:23:00.650 --> 01:23:04.760 So there's one that comes from the dot product of the V dot M 01:23:04.760 --> 01:23:05.540 with the k term. 01:23:05.540 --> 01:23:06.050 That's this. 01:23:06.050 --> 01:23:07.550 And then there's the k squared term. 01:23:07.550 --> 01:23:10.130 So I have these. 01:23:10.130 --> 01:23:11.420 And then I can expand that. 01:23:19.700 --> 01:23:22.310 M is a positive quantity, so it doesn't 01:23:22.310 --> 01:23:25.070 change the sign of the i0. 01:23:25.070 --> 01:23:27.650 And if I expand in M, then the leading term looks like that. 01:23:27.650 --> 01:23:30.350 It's M independent. 01:23:30.350 --> 01:23:32.750 And then there's some order 1 over M terms, which I could 01:23:32.750 --> 01:23:33.917 also work out what they are. 01:23:38.150 --> 01:23:40.610 OK, so that we could expand even though we don't, a priori, 01:23:40.610 --> 01:23:42.426 know how to expand the Lagrangian. 01:23:46.400 --> 01:23:48.562 We could also think about vertices in this theory. 01:23:48.562 --> 01:23:49.520 This is the propagator. 01:23:49.520 --> 01:23:51.380 What about vertices? 01:23:51.380 --> 01:23:53.090 And again, this is a top-down. 01:23:53.090 --> 01:23:55.820 So we can think about vertices in QCD. 01:23:55.820 --> 01:23:58.340 So if this is a heavy particle here, 01:23:58.340 --> 01:24:00.890 we could think about what would happen. 01:24:00.890 --> 01:24:02.000 How do we expand those? 01:24:06.460 --> 01:24:08.210 And there doesn't look like there's really 01:24:08.210 --> 01:24:11.390 anything going on here because there's nothing to expand. 01:24:11.390 --> 01:24:14.540 But because these things are heavy particles here, 01:24:14.540 --> 01:24:17.870 you realize from the propagator formula 01:24:17.870 --> 01:24:20.090 here that you're going to have 1 plus V slashes 01:24:20.090 --> 01:24:23.670 on each side of this guy. 01:24:23.670 --> 01:24:26.880 So for a propagator on each side, 01:24:26.880 --> 01:24:28.807 you can make a simplification. 01:24:35.350 --> 01:24:38.740 And that's because 1 plus V slash over 2 gamma 01:24:38.740 --> 01:24:44.110 mu 1 plus V slash over 2, after some Dirac algebra, 01:24:44.110 --> 01:24:48.850 is just V mu times 1 plus V slash over 2. 01:24:51.610 --> 01:24:54.460 So the gamma mu becomes a V mu. 01:24:54.460 --> 01:24:58.520 So the vertice, once you take that into account, 01:24:58.520 --> 01:25:01.840 is just minus igTA V mu. 01:25:06.700 --> 01:25:08.980 Even if we don't think about starting with the QCD 01:25:08.980 --> 01:25:10.690 Lagrangian, if this is our Feynman rule 01:25:10.690 --> 01:25:12.850 and that's our propagator, we can write down 01:25:12.850 --> 01:25:18.820 an effective theory for that that gives those Feynman rules. 01:25:18.820 --> 01:25:21.377 And that actually is the HQET Lagrangian. 01:25:30.640 --> 01:25:37.360 So sorry for going a little bit over, 01:25:37.360 --> 01:25:40.930 but I wanted to at least get a little bit into this. 01:25:40.930 --> 01:25:43.990 OK, so by construction this way, we 01:25:43.990 --> 01:25:45.947 can arrive at this Lagrangian. 01:25:45.947 --> 01:25:47.530 And next time, I'll come back and I'll 01:25:47.530 --> 01:25:49.660 show you how you can really properly 01:25:49.660 --> 01:25:52.300 take this limit of the QCD Lagrangian 01:25:52.300 --> 01:25:55.010 to get the same thing. 01:25:55.010 --> 01:25:56.710 OK, but that is the HQET. 01:25:56.710 --> 01:25:59.170 The lowest order of the HQET Lagrangian is this. 01:26:02.810 --> 01:26:05.120 So it's a linear V dot D decoupling. 01:26:05.120 --> 01:26:07.403 V is a parameter, and so I've put it 01:26:07.403 --> 01:26:08.570 as a parameter on the field. 01:26:08.570 --> 01:26:11.780 We'll see more why that was done next time. 01:26:11.780 --> 01:26:14.000 And there's a projection relation on the field. 01:26:14.000 --> 01:26:15.650 If I'm using a four-component field, 01:26:15.650 --> 01:26:17.192 then I have this projection relation. 01:26:17.192 --> 01:26:19.590 I could also read it in a two-component notation, 01:26:19.590 --> 01:26:21.920 and then I wouldn't need that. 01:26:21.920 --> 01:26:24.695 But for four-component notation, you do need that. 01:26:24.695 --> 01:26:27.610 We'll talk about that next time, too.