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 --> 00:00:17.160 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.160 --> 00:00:18.370 at ocw.mit.edu. 00:00:20.980 --> 00:00:22.540 IAIN STEWART: --play with each other. 00:00:22.540 --> 00:00:24.350 We did the standard model as an effective field 00:00:24.350 --> 00:00:25.780 theory, higher dimension operators 00:00:25.780 --> 00:00:27.220 in the standard model. 00:00:27.220 --> 00:00:29.950 And then we started talking about taking the standard model 00:00:29.950 --> 00:00:33.340 as a theory one and removing things from it, 00:00:33.340 --> 00:00:35.380 in particular constructing what's 00:00:35.380 --> 00:00:38.590 called the weak Hamiltonian by removing the top W, Z, 00:00:38.590 --> 00:00:41.260 and Higgs from the standard model. 00:00:41.260 --> 00:00:45.670 And last time, we were focusing on the anomalous dimensions 00:00:45.670 --> 00:00:48.640 and things about renormalization. 00:00:48.640 --> 00:00:51.790 So we had this equation for the weak Hamiltonian 00:00:51.790 --> 00:00:57.220 for a particular case of b goes to c u bar d. 00:00:57.220 --> 00:00:58.900 That was the case that we decided 00:00:58.900 --> 00:01:01.660 to study rather than the full Hamiltonian. 00:01:01.660 --> 00:01:03.130 So there was some pre-factor. 00:01:03.130 --> 00:01:06.680 We had two operators with Wilson coefficients and operators. 00:01:06.680 --> 00:01:10.790 These are four-fermion operators. 00:01:10.790 --> 00:01:12.040 And there was different bases. 00:01:12.040 --> 00:01:14.577 We could write them in the bare form, 00:01:14.577 --> 00:01:16.660 or we could write them in renormalized coefficient 00:01:16.660 --> 00:01:18.650 and renormalized operator. 00:01:18.650 --> 00:01:20.800 And then we could do it either in the 1, 2 basis 00:01:20.800 --> 00:01:22.970 or the plus minus basis. 00:01:22.970 --> 00:01:25.330 So the plus minus basis is just linear combinations 00:01:25.330 --> 00:01:26.050 of the 1, 2. 00:01:29.410 --> 00:01:31.000 If you're in the 1, 2 basis, then you 00:01:31.000 --> 00:01:32.300 have this mixing matrix. 00:01:32.300 --> 00:01:33.325 So it's a 2 by 2 matrix. 00:01:33.325 --> 00:01:36.760 So if you're in the plus minus basis, 00:01:36.760 --> 00:01:42.100 then at least, at the lowest order, 00:01:42.100 --> 00:01:44.540 it's a simple product equation. 00:01:44.540 --> 00:01:46.690 So plus doesn't interfere with minus. 00:01:50.610 --> 00:01:51.110 OK. 00:01:51.110 --> 00:01:56.730 So that's where we got to, and we'll just continue today. 00:01:56.730 --> 00:01:59.390 So we have these anomalous dimension equations 00:01:59.390 --> 00:02:00.493 for the operators. 00:02:00.493 --> 00:02:02.660 We can also write down anomalous dimension equations 00:02:02.660 --> 00:02:04.770 for the Wilson coefficients. 00:02:04.770 --> 00:02:06.396 How do we do that? 00:02:06.396 --> 00:02:10.542 Well, the way that we do that is we make use of the fact 00:02:10.542 --> 00:02:14.060 that, if we look at the first line here, there's no mu. 00:02:14.060 --> 00:02:17.173 So the Hamiltonian is mu independent. 00:02:17.173 --> 00:02:19.340 That means that the mu dependence of the coefficient 00:02:19.340 --> 00:02:23.420 cancels the mu dependence of the operator. 00:02:23.420 --> 00:02:26.458 And you can use that to take an anomalous dimension 00:02:26.458 --> 00:02:28.000 equation for the operator and turn it 00:02:28.000 --> 00:02:29.729 into one for the coefficient. 00:02:40.210 --> 00:02:42.365 So last time, we talked about the fact 00:02:42.365 --> 00:02:43.990 that the normalization of the operators 00:02:43.990 --> 00:02:45.310 was equivalent to-- you can think 00:02:45.310 --> 00:02:46.330 about it two different ways. 00:02:46.330 --> 00:02:48.538 There's renormalization of operators, renormalization 00:02:48.538 --> 00:02:49.240 of coefficients. 00:02:49.240 --> 00:02:51.633 Likewise, you can think of anomalous dimensions 00:02:51.633 --> 00:02:53.050 is either running the coefficients 00:02:53.050 --> 00:02:54.160 or running the operators. 00:02:54.160 --> 00:02:56.320 So those are equivalent things. 00:02:56.320 --> 00:02:59.260 And the thing that makes them equivalent 00:02:59.260 --> 00:03:04.720 is just imposing that the derivative with respect 00:03:04.720 --> 00:03:06.310 to mu of the Hamiltonian is 0. 00:03:15.860 --> 00:03:18.890 So if I use my equation up here for anomalous dimension 00:03:18.890 --> 00:03:27.080 of the operator, I get, with my sign convention, a minus sign 00:03:27.080 --> 00:03:28.715 here. 00:03:28.715 --> 00:03:30.560 Or I had a minus sign here. 00:03:30.560 --> 00:03:31.640 I had no minus sign here. 00:03:31.640 --> 00:03:34.010 These things are conventions. 00:03:34.010 --> 00:03:37.650 I picked some convention, and I'll stick with it. 00:03:37.650 --> 00:03:41.235 So from this equation, we basically 00:03:41.235 --> 00:03:43.610 have an equation that has to be true for the coefficients 00:03:43.610 --> 00:03:45.830 if we think of just stripping off the operator. 00:04:05.540 --> 00:04:08.737 So we could write it this way, just reading it off right 00:04:08.737 --> 00:04:10.820 from here, just putting this guy on the other side 00:04:10.820 --> 00:04:13.640 and then reading it off, dropping the operator. 00:04:13.640 --> 00:04:15.800 Or we could write it this way if we 00:04:15.800 --> 00:04:17.630 wanted to write it in a way that's 00:04:17.630 --> 00:04:19.610 more similar to this equation up here, where 00:04:19.610 --> 00:04:22.780 you have the anomalous dimension matrix times the coefficient. 00:04:22.780 --> 00:04:24.620 It's either from the right or from the left, 00:04:24.620 --> 00:04:27.593 and it's just a matter of transposing. 00:04:27.593 --> 00:04:29.510 So the anomalous dimension for the coefficient 00:04:29.510 --> 00:04:32.660 is determined from the one, this one here. 00:04:32.660 --> 00:04:34.160 If you know it, then you immediately 00:04:34.160 --> 00:04:37.490 know the one for the coefficient, which 00:04:37.490 --> 00:04:40.040 shouldn't be surprising given that we could think of the Z 00:04:40.040 --> 00:04:43.820 factors as being related to renormalization factors. 00:04:43.820 --> 00:04:47.970 Anomalous dimensions, therefore, should also be related. 00:04:47.970 --> 00:04:51.030 OK, so we'll solve this equation. 00:04:51.030 --> 00:04:53.080 It's a little bit simpler to think about. 00:04:53.080 --> 00:04:54.538 Although we could have equivalently 00:04:54.538 --> 00:04:56.687 solve the operator equation. 00:04:56.687 --> 00:04:58.020 So how do we solve the equation? 00:05:03.490 --> 00:05:07.300 Well, we go over to our plus minus basis. 00:05:07.300 --> 00:05:11.850 So take the coefficient of either C+ or C-. 00:05:11.850 --> 00:05:13.740 And the equation that we need to solve 00:05:13.740 --> 00:05:17.115 can be written as follows. 00:05:30.510 --> 00:05:31.750 It's a simple equation. 00:05:31.750 --> 00:05:34.790 And I can even take the C, which is on the right-hand side, 00:05:34.790 --> 00:05:36.950 move it over to the left, if I write log C here. 00:05:36.950 --> 00:05:39.560 Because the derivative of the log is giving me a 1 over C. 00:05:39.560 --> 00:05:42.560 If I put it back over here, it would just multiply. 00:05:42.560 --> 00:05:44.330 OK, so that's the analog. 00:05:44.330 --> 00:05:46.550 This equation here is the analog of this equation 00:05:46.550 --> 00:05:50.260 here for the operators, but now for the coefficients. 00:05:50.260 --> 00:05:52.760 Obviously, if these are numbers, there's no transpose to do. 00:05:56.050 --> 00:05:58.810 So you have to solve this equation simultaneously 00:05:58.810 --> 00:06:00.350 with another differential equation. 00:06:00.350 --> 00:06:02.698 This is a coupled equation because alpha also 00:06:02.698 --> 00:06:03.865 has a differential equation. 00:06:20.150 --> 00:06:23.350 So at lowest order, this is the beta function equation. 00:06:23.350 --> 00:06:27.340 And so if we want to solve, take into account this equation 00:06:27.340 --> 00:06:30.610 and integrate this equation, the simple trick for doing that 00:06:30.610 --> 00:06:34.250 is to make a change of variable. 00:06:34.250 --> 00:06:39.040 So we use this equation here to make a change of variable. 00:06:39.040 --> 00:06:41.350 This is a very useful trick because it actually 00:06:41.350 --> 00:06:43.360 works to whatever order in the expansion 00:06:43.360 --> 00:06:46.896 that you might want to work. 00:06:46.896 --> 00:06:49.780 So let me explain what the trick is, and then I'll 00:06:49.780 --> 00:06:51.480 explain why that is. 00:06:51.480 --> 00:06:53.890 So we're going to change variables in this equation 00:06:53.890 --> 00:06:54.880 from mu to alpha. 00:07:01.658 --> 00:07:03.950 You see we could think about solving this equation here 00:07:03.950 --> 00:07:07.820 by integrating, just move this thing to the other side 00:07:07.820 --> 00:07:09.230 and integrate. 00:07:09.230 --> 00:07:12.080 But we'd be integrating, in mu, a function that's 00:07:12.080 --> 00:07:14.910 a function of alpha of mu. 00:07:14.910 --> 00:07:16.985 So if we can switch from mu to alpha, 00:07:16.985 --> 00:07:19.110 then we'll be just integrating a function of alpha. 00:07:19.110 --> 00:07:21.770 And that's what we're going to do using this equation. 00:07:38.240 --> 00:07:41.050 So if I write it in general, that's this equality here. 00:07:41.050 --> 00:07:45.610 For any function of alpha, I say that d mu over mu-- 00:07:45.610 --> 00:07:49.730 just rearranging this equation-- is d alpha over beta of alpha. 00:07:49.730 --> 00:07:51.940 So I can switch the integration d mu over mu 00:07:51.940 --> 00:07:55.040 to d alpha over beta of alpha. 00:07:55.040 --> 00:07:56.800 And that's exactly what I want to do 00:07:56.800 --> 00:07:59.650 if I want to move this operator to the other side, 00:07:59.650 --> 00:08:01.082 this operation. 00:08:01.082 --> 00:08:03.040 If I work at lowest order in the beta function, 00:08:03.040 --> 00:08:06.710 then that's just I just plug in this result. 00:08:06.710 --> 00:08:10.180 So we can switch variables from mu to alpha. 00:08:10.180 --> 00:08:13.360 And if I had high order terms in this equation and high order 00:08:13.360 --> 00:08:16.810 terms in this equation, I could use the same trick. 00:08:16.810 --> 00:08:18.575 I just have other integrals to do. 00:08:18.575 --> 00:08:20.450 And the integrals are pretty straightforward, 00:08:20.450 --> 00:08:24.260 so this is a useful way of proceeding. 00:08:24.260 --> 00:08:24.760 OK. 00:08:24.760 --> 00:08:25.750 So what does that do? 00:08:25.750 --> 00:08:26.770 So now, let's do that. 00:08:26.770 --> 00:08:28.764 Let's move this over and integrate. 00:08:36.669 --> 00:08:41.400 Well, we'll do a definite integral from mu w up to mu. 00:08:46.846 --> 00:08:49.300 So here, we just have d log this. 00:08:49.300 --> 00:08:54.610 Integrate that-- just gives log between the limits. 00:09:06.200 --> 00:09:09.010 And if I didn't change variable, I would have that. 00:09:09.010 --> 00:09:11.950 But if I make the change variable, 00:09:11.950 --> 00:09:13.600 then it becomes a very simple integral. 00:09:36.400 --> 00:09:38.890 And remember that this guy here is also 00:09:38.890 --> 00:09:46.150 just a number times alpha that we worked out last time. 00:09:48.830 --> 00:09:51.590 So this is just d alpha over alpha. 00:09:51.590 --> 00:09:53.410 And that's a simple logarithmic integral. 00:09:53.410 --> 00:09:55.030 Yeah. 00:09:55.030 --> 00:09:56.530 AUDIENCE: I don't know if it matter, 00:09:56.530 --> 00:10:00.340 but mu w should be greater than mu, right? 00:10:00.340 --> 00:10:02.380 IAIN STEWART: Mu w should be greater than mu. 00:10:02.380 --> 00:10:03.040 That's right. 00:10:03.040 --> 00:10:04.957 AUDIENCE: OK, so you're just writing integrals 00:10:04.957 --> 00:10:06.324 like that to avoid signs? 00:10:06.324 --> 00:10:07.596 OK. 00:10:07.596 --> 00:10:08.708 I have another question. 00:10:08.708 --> 00:10:09.500 IAIN STEWART: Yeah. 00:10:09.500 --> 00:10:12.830 AUDIENCE: How do you know that the anomalous dimensions, 00:10:12.830 --> 00:10:15.474 including the beta function, are only functions of alpha S 00:10:15.474 --> 00:10:16.630 rather than [INAUDIBLE]. 00:10:16.630 --> 00:10:18.530 IAIN STEWART: Ah, yeah. 00:10:18.530 --> 00:10:21.750 So I'm sneaking that in here. 00:10:21.750 --> 00:10:25.580 So it follows from the renormalization structure 00:10:25.580 --> 00:10:28.070 of this effective field theory that there's only 00:10:28.070 --> 00:10:31.250 single logarithmic divergences. 00:10:31.250 --> 00:10:33.890 So in the standard model, if you're at one loop, 00:10:33.890 --> 00:10:36.500 you only have 1 over epsilon poles for the renormalization. 00:10:36.500 --> 00:10:38.470 And you're renormalizing the coupling. 00:10:38.470 --> 00:10:40.220 The same is true of this effective theory. 00:10:40.220 --> 00:10:43.640 At one loop, you only have 1 over epsilon divergences. 00:10:43.640 --> 00:10:47.450 And that implies that your anomalous dimensions 00:10:47.450 --> 00:10:50.340 won't depend on anything more complicated. 00:10:50.340 --> 00:10:53.180 We will discuss more complicated cases in the future, 00:10:53.180 --> 00:10:54.740 as you know. 00:10:54.740 --> 00:11:00.080 But the structure of this effective theory and its UV 00:11:00.080 --> 00:11:04.460 structure, which I didn't go into on a lot of detail about, 00:11:04.460 --> 00:11:06.510 implies that fact. 00:11:06.510 --> 00:11:07.010 Yeah. 00:11:07.010 --> 00:11:08.300 AUDIENCE: [INAUDIBLE] 00:11:08.300 --> 00:11:09.092 IAIN STEWART: Yeah. 00:11:11.540 --> 00:11:12.040 OK. 00:11:12.040 --> 00:11:13.860 So do the integral. 00:11:13.860 --> 00:11:19.120 There's some pre-factor, which I'll call a+-. 00:11:19.120 --> 00:11:22.100 And then I get a log, as I mentioned. 00:11:27.160 --> 00:11:29.620 And just for the record, this a+-, 00:11:29.620 --> 00:11:32.020 if I put all the factors together and put in what this 00:11:32.020 --> 00:11:43.990 number is, these would be some factors like this. 00:11:43.990 --> 00:11:47.350 And I've put in that Nc is 3. 00:11:51.540 --> 00:11:55.340 So this alpha at mu w and this C+ of mu w, 00:11:55.340 --> 00:11:57.960 you should think of mu w as the boundary condition scale. 00:12:02.202 --> 00:12:03.660 So this is a differential equation. 00:12:03.660 --> 00:12:06.330 We needed a boundary condition to solve it. 00:12:06.330 --> 00:12:08.220 And the boundary condition is the value 00:12:08.220 --> 00:12:13.195 of the coefficients at the scale uw, which 00:12:13.195 --> 00:12:14.445 is supposed to be of order Mw. 00:12:23.760 --> 00:12:26.100 Typically, what that means is you 00:12:26.100 --> 00:12:28.110 could take a common choice, which 00:12:28.110 --> 00:12:31.020 would be just to take it equal. 00:12:31.020 --> 00:12:34.740 Or you could pick twice or half. 00:12:34.740 --> 00:12:38.220 And these are the most common choices that people pick. 00:12:41.400 --> 00:12:44.490 So the way that you should think of that, 00:12:44.490 --> 00:12:48.060 this guy in the denominator, is you 00:12:48.060 --> 00:12:52.860 should think that he's really a fixed order series in alpha 00:12:52.860 --> 00:13:02.970 of mu w, something that you would calculate order by order 00:13:02.970 --> 00:13:06.180 and perturbation theory. 00:13:06.180 --> 00:13:09.060 And you'd be determining the boundary condition. 00:13:09.060 --> 00:13:13.420 We'll talk about how you would do that a little later today. 00:13:13.420 --> 00:13:16.770 But for now, just think of it as a series in alpha of mu w. 00:13:16.770 --> 00:13:19.500 And it doesn't have any large logarithms. 00:13:19.500 --> 00:13:21.480 And as Elia said, you want to think 00:13:21.480 --> 00:13:26.040 of this mu as some small scale, some scale that's less than uw 00:13:26.040 --> 00:13:27.690 because you're thinking of evolving 00:13:27.690 --> 00:13:30.145 the operators to a scale less than the scale 00:13:30.145 --> 00:13:31.770 where you integrated out the particles. 00:13:34.322 --> 00:13:35.780 I'll draw that picture in a second. 00:13:53.030 --> 00:13:56.470 So we can take the exponential of that equation, 00:13:56.470 --> 00:14:00.220 and then we can write C of mu is equal to something 00:14:00.220 --> 00:14:01.638 that we've determined. 00:14:10.610 --> 00:14:12.090 Take the exponential. 00:14:12.090 --> 00:14:13.520 Move this guy to the other side. 00:14:41.280 --> 00:14:44.910 Remember the a+- are just numbers. 00:14:44.910 --> 00:14:47.730 We can write the solution in this way, where we determine 00:14:47.730 --> 00:14:50.250 this guy by thinking about doing a matching calculation 00:14:50.250 --> 00:14:51.090 at the high scale. 00:14:51.090 --> 00:14:54.810 We determine it to be 1/2 already at the lowest order. 00:14:54.810 --> 00:14:57.100 So think about sticking in 1/2 here. 00:14:57.100 --> 00:15:00.030 And then this factor here is what you get 00:15:00.030 --> 00:15:02.192 from the renormalization group. 00:15:02.192 --> 00:15:03.900 And you can see from this form right here 00:15:03.900 --> 00:15:06.192 that you've summed up an infinite number of logarithms. 00:15:06.192 --> 00:15:07.860 It's exponential of a number times log. 00:15:07.860 --> 00:15:14.280 And if I were to expand that out in alpha of some fixed scale, 00:15:14.280 --> 00:15:16.280 there would be an infinite series in logarithms. 00:15:20.310 --> 00:15:20.810 OK. 00:15:20.810 --> 00:15:22.610 So what do we want to pick this mu to be? 00:15:22.610 --> 00:15:29.180 Well, we're thinking about the process mu goes to c u bar d. 00:15:29.180 --> 00:15:31.700 If you think about this process in nature, 00:15:31.700 --> 00:15:36.350 the scale in the initial state here is the b has a mass. 00:15:36.350 --> 00:15:39.668 So you'd like to take this scale here not at Mw, but down 00:15:39.668 --> 00:15:40.460 at the b core mass. 00:15:48.290 --> 00:15:50.350 So we want mu to be of order Mb. 00:15:55.776 --> 00:15:58.430 And then you have a large hierarchy because this 00:15:58.430 --> 00:16:01.280 is much less than Mw-- 00:16:01.280 --> 00:16:04.160 5 g of e-ish, 80 g of e. 00:16:12.150 --> 00:16:23.620 So our result here sums what are called the leading logarithms, 00:16:23.620 --> 00:16:27.360 which is denoted by LL. 00:16:27.360 --> 00:16:30.960 And schematically, the lowest order term was 1/2. 00:16:30.960 --> 00:16:37.130 And if we were to expand higher order terms 00:16:37.130 --> 00:16:39.380 and think about what logarithms we're talking about, 00:16:39.380 --> 00:16:43.130 we're talking about logarithms of Mw over Mb. 00:16:45.930 --> 00:16:47.810 And the series, if we were to expand it, 00:16:47.810 --> 00:16:53.195 would look like this schematically 00:16:53.195 --> 00:16:58.100 without worrying about the coefficients, 00:16:58.100 --> 00:17:02.330 an infinite series where each term has one alpha and one log. 00:17:05.900 --> 00:17:07.480 So the counting that you're doing 00:17:07.480 --> 00:17:11.500 in this type of setup, where we would think 00:17:11.500 --> 00:17:25.020 of using this equation to go down to the scale Mb, 00:17:25.020 --> 00:17:30.845 is that you're counting this parameter as order 1. 00:17:30.845 --> 00:17:32.220 And you're saying, any time I see 00:17:32.220 --> 00:17:34.388 a log of Mw over Mb times an alpha, 00:17:34.388 --> 00:17:36.180 I'm not going to count that as order alpha. 00:17:36.180 --> 00:17:38.010 I'm going to count that as order 1. 00:17:38.010 --> 00:17:40.170 That's why I have to sum up this infinite series. 00:17:46.820 --> 00:17:49.390 So the physical picture of what we've said here 00:17:49.390 --> 00:17:50.617 is the following. 00:17:54.395 --> 00:17:55.853 So the basic physical picture would 00:17:55.853 --> 00:17:58.990 be that there's two scales, Mw an Mb, which 00:17:58.990 --> 00:18:00.220 are physical scales. 00:18:00.220 --> 00:18:02.590 You want to get rid of the scale Mw. 00:18:02.590 --> 00:18:05.290 You do that by going over to this electroweak Hamiltonian. 00:18:05.290 --> 00:18:06.850 But then you have to renormalization 00:18:06.850 --> 00:18:10.300 group evolve Hamiltonian down to the scale where you want to do 00:18:10.300 --> 00:18:13.300 physics, which is the scale Mb. 00:18:13.300 --> 00:18:15.890 And when you do that, there's some choice in the matter. 00:18:15.890 --> 00:18:19.400 And we've been careful to parameterize that choice. 00:18:19.400 --> 00:18:22.420 We said that you pick a scale that's 00:18:22.420 --> 00:18:25.943 of order Mw, which we called mu w. 00:18:25.943 --> 00:18:27.610 And then we said, you pick another scale 00:18:27.610 --> 00:18:30.355 that's of order Mb, which we called mu. 00:18:30.355 --> 00:18:32.230 And you actually do the renormalization group 00:18:32.230 --> 00:18:34.570 between these two. 00:18:34.570 --> 00:18:39.400 You could pick Mw and mu equal exactly Mb. 00:18:39.400 --> 00:18:42.580 That would be another simpler story. 00:18:42.580 --> 00:18:45.100 But it is actually important that, 00:18:45.100 --> 00:18:46.960 once you go beyond the lowest order, 00:18:46.960 --> 00:18:49.900 to keep track of the fact that you have this freedom. 00:18:49.900 --> 00:18:53.240 And that's why I've kept track over here. 00:18:53.240 --> 00:18:55.660 So what that means is that, in terms of counting, 00:18:55.660 --> 00:19:05.280 you've counted this, but logs of mu or Mw 00:19:05.280 --> 00:19:07.380 were counted as order 1. 00:19:07.380 --> 00:19:11.760 And then down here, logs of mu over B 00:19:11.760 --> 00:19:15.900 are counted as order 1 numbers. 00:19:15.900 --> 00:19:21.810 It could be 0, but 0 is order 1, not enhanced 00:19:21.810 --> 00:19:25.230 such that they would compensate for a factor of alpha. 00:19:25.230 --> 00:19:30.810 And this is the renormalization group evolution or the running 00:19:30.810 --> 00:19:34.560 that sums up these logarithms here, which are the large logs. 00:19:38.760 --> 00:19:39.940 And that's pretty simple. 00:19:39.940 --> 00:19:42.580 It just gave this factor. 00:19:42.580 --> 00:19:45.253 And that's pretty common in QCD to get factors 00:19:45.253 --> 00:19:46.920 like that, alpha at one scale over alpha 00:19:46.920 --> 00:19:49.260 at another scale raised to a power. 00:19:49.260 --> 00:19:51.780 That's a very common thing to get from renormalization group 00:19:51.780 --> 00:19:52.817 evolution. 00:19:55.800 --> 00:19:57.860 OK, any questions so far? 00:20:14.600 --> 00:20:17.720 How many people have done the calculation 00:20:17.720 --> 00:20:20.480 of the anomalous dimension for four-fermion operators 00:20:20.480 --> 00:20:21.680 in some other course? 00:20:21.680 --> 00:20:23.870 It's a common problem. 00:20:23.870 --> 00:20:24.560 Nobody? 00:20:24.560 --> 00:20:25.905 All right. 00:20:25.905 --> 00:20:27.530 That means you'll see it on a homework. 00:20:41.210 --> 00:20:46.510 So let's come back here and think 00:20:46.510 --> 00:20:49.960 about what the general structure of what we've done is. 00:20:55.937 --> 00:20:57.020 And I'll put back indices. 00:21:09.738 --> 00:21:12.030 So you can think about taking the solution that we have 00:21:12.030 --> 00:21:16.320 at the top of the board here and generalizing it 00:21:16.320 --> 00:21:18.937 to a form that would be valid at higher orders. 00:21:18.937 --> 00:21:21.270 And basically, it says that the coefficient at one scale 00:21:21.270 --> 00:21:23.310 is connected to the coefficient at another scale 00:21:23.310 --> 00:21:25.350 times some evolution factor. 00:21:25.350 --> 00:21:26.850 In this case, the evolution factor 00:21:26.850 --> 00:21:29.010 is just the ratio of these alphas to a power. 00:21:29.010 --> 00:21:31.140 It could be some more complicated function 00:21:31.140 --> 00:21:32.100 at higher orders. 00:21:32.100 --> 00:21:33.520 And it could even be a matrix. 00:21:33.520 --> 00:21:34.895 That's why I've given it indices. 00:21:57.950 --> 00:22:01.420 So we can put our results back together 00:22:01.420 --> 00:22:03.520 using this higher order form, so that they're 00:22:03.520 --> 00:22:06.430 generally true, into our Hamiltonian 00:22:06.430 --> 00:22:08.035 and see what we've achieved. 00:22:21.740 --> 00:22:24.840 And let me call this scale that I was calling mu a minute 00:22:24.840 --> 00:22:28.440 ago mu b just to remind you that it's a scale of order Mb. 00:22:34.140 --> 00:22:36.390 So previously, we had the coefficient and the operator 00:22:36.390 --> 00:22:38.247 at the same scale. 00:22:38.247 --> 00:22:40.580 But now, using this equation, I can move the coefficient 00:22:40.580 --> 00:22:42.510 to a different scale. 00:22:42.510 --> 00:22:45.140 And so let me think of sticking this equation in. 00:22:45.140 --> 00:22:46.700 And then I have mu w. 00:22:46.700 --> 00:22:48.810 I've called mu equals mu b. 00:22:48.810 --> 00:22:50.860 So now, this is mu w mu b. 00:22:50.860 --> 00:22:57.890 So this here is the coefficient Ci at mu b, 00:22:57.890 --> 00:22:59.540 but I find it useful to write it out. 00:23:05.968 --> 00:23:07.760 So the thing in square brackets is Ci mu b, 00:23:07.760 --> 00:23:11.120 but I write it out using the renormalization group 00:23:11.120 --> 00:23:12.800 equation that way. 00:23:12.800 --> 00:23:15.560 And this tells you how you're doing the calculation. 00:23:15.560 --> 00:23:17.250 This is a fixed order calculation. 00:23:22.130 --> 00:23:26.330 This comes from anomalous dimensions 00:23:26.330 --> 00:23:28.940 and gives you the evolution. 00:23:28.940 --> 00:23:31.730 And then you have operators involving the B quark 00:23:31.730 --> 00:23:38.960 that you would calculate matrix elements of at mu 00:23:38.960 --> 00:23:41.330 b of order Mb. 00:23:41.330 --> 00:23:45.500 And there's no dependence at all in those operators on the scale 00:23:45.500 --> 00:23:47.270 Mw. 00:23:47.270 --> 00:23:49.338 All the Mw's are in the pre-factors here. 00:23:49.338 --> 00:23:50.630 You've taken that into account. 00:23:50.630 --> 00:23:53.720 You've calculated it. 00:23:53.720 --> 00:23:58.180 OK, so that's how this organizes the physics 00:23:58.180 --> 00:24:01.030 of the different scales. 00:24:01.030 --> 00:24:03.117 So you could ask, if I had this story, 00:24:03.117 --> 00:24:04.450 how would I go to higher orders? 00:24:04.450 --> 00:24:06.730 And we will have some discussion of what 00:24:06.730 --> 00:24:09.400 goes on at higher orders because there are some things that 00:24:09.400 --> 00:24:11.912 happen at higher orders that you don't see at leading order. 00:24:11.912 --> 00:24:13.870 And they're actually important physical things, 00:24:13.870 --> 00:24:16.147 so important things to know about and keep 00:24:16.147 --> 00:24:18.230 track of if you ever want to use things like this. 00:24:18.230 --> 00:24:21.040 So let's talk a little bit about what it 00:24:21.040 --> 00:24:22.270 takes to go to higher orders. 00:24:33.970 --> 00:24:35.550 So let's just first think about what 00:24:35.550 --> 00:24:39.990 it would look like if we went to higher orders. 00:24:39.990 --> 00:24:43.680 Well, leading order was a series that I could schematically say 00:24:43.680 --> 00:24:46.500 is alpha times large logs. 00:24:46.500 --> 00:24:49.920 And I summed them all up, and I called that leading log. 00:24:49.920 --> 00:24:51.750 When I go to higher orders, I am going 00:24:51.750 --> 00:24:57.315 to continue to get series, but I got extra factors of alpha. 00:25:05.880 --> 00:25:10.490 So something that you call Next to Leading Log, or NLL, 00:25:10.490 --> 00:25:13.400 is the same type of thing, a different series than that 1 00:25:13.400 --> 00:25:14.970 times an extra factor of alpha. 00:25:14.970 --> 00:25:18.110 So it's down compared to this by alpha S. 00:25:18.110 --> 00:25:19.160 And then you keep going. 00:25:23.680 --> 00:25:28.370 This is the general structure of the renormalization group 00:25:28.370 --> 00:25:29.690 improved perturbation theory. 00:25:32.690 --> 00:25:36.380 Just keep adding Ns and keep adding alphas, always 00:25:36.380 --> 00:25:40.250 summing up some series which changes from order to order. 00:25:40.250 --> 00:25:41.960 And that summation of that series 00:25:41.960 --> 00:25:43.700 is determined by determining higher order 00:25:43.700 --> 00:25:45.710 anomalous dimensions. 00:25:45.710 --> 00:25:56.690 So this kind of thing is called renormalization group improved 00:25:56.690 --> 00:25:57.620 perturbation theory. 00:26:00.680 --> 00:26:04.490 Every time you take alpha S and you take it at some scale, 00:26:04.490 --> 00:26:06.680 you're already doing renormalization group 00:26:06.680 --> 00:26:08.120 improved perturbation theory. 00:26:08.120 --> 00:26:10.247 It's just that, once you have theories 00:26:10.247 --> 00:26:12.830 that have other things that run and have anomalous dimensions, 00:26:12.830 --> 00:26:16.340 then it can be more complicated than just simply picking alpha 00:26:16.340 --> 00:26:18.380 S at the appropriate scale. 00:26:18.380 --> 00:26:20.870 Here, in this theory, we have these coefficients. 00:26:20.870 --> 00:26:21.835 We have to run them. 00:26:21.835 --> 00:26:23.960 Then we have to pick them at the appropriate scale. 00:26:23.960 --> 00:26:25.670 And that's what we're doing by solving 00:26:25.670 --> 00:26:29.750 these renormalization group equations. 00:26:29.750 --> 00:26:30.250 OK. 00:26:30.250 --> 00:26:31.520 So what do we need to do? 00:26:31.520 --> 00:26:32.410 We determined this. 00:26:32.410 --> 00:26:34.660 I showed you what you needed to do to get that. 00:26:34.660 --> 00:26:37.210 What would we need to do to get the next term in the series? 00:26:37.210 --> 00:26:39.340 How much would we have to compute? 00:26:43.980 --> 00:26:45.930 Well, we just have to go to one higher order 00:26:45.930 --> 00:26:48.340 in the perturbation theory. 00:26:48.340 --> 00:27:00.290 So let's make a little table of what 00:27:00.290 --> 00:27:02.900 it takes to get leading log, next leading log. 00:27:05.470 --> 00:27:07.350 Maybe we'll even add one more term. 00:27:12.813 --> 00:27:14.480 So there's two parts to the calculation. 00:27:14.480 --> 00:27:16.910 There's the boundary condition, and then there's 00:27:16.910 --> 00:27:23.150 the differential equation, which is the anomalous dimension. 00:27:23.150 --> 00:27:26.480 At leading log, we had tree level matching. 00:27:26.480 --> 00:27:29.480 We determined the C plus and minus where 1/2. 00:27:29.480 --> 00:27:32.990 C1 and C2 were 1 and 0. 00:27:32.990 --> 00:27:36.980 And we just needed the one loop anomalous dimension. 00:27:36.980 --> 00:27:39.090 And then we just keep going in this pattern. 00:27:39.090 --> 00:27:44.330 So next leading log, we need to match it one loop. 00:27:44.330 --> 00:27:49.220 And we would need the two-loop anomalous dimension 00:27:49.220 --> 00:27:50.050 and et cetera. 00:27:53.790 --> 00:27:55.647 So the order in which you need the running 00:27:55.647 --> 00:27:57.980 is one higher order than what you need for the matching. 00:28:01.760 --> 00:28:02.820 That's the rule. 00:28:02.820 --> 00:28:04.400 And given those ingredients, we would 00:28:04.400 --> 00:28:09.860 be able to determine exactly these series here, OK? 00:28:15.080 --> 00:28:18.012 So there's some things that happen at this order 00:28:18.012 --> 00:28:19.970 that aren't really apparent yet at leading log, 00:28:19.970 --> 00:28:21.980 and so I want to talk a little bit about that. 00:28:34.980 --> 00:28:37.590 Before we get there, let me add one other little note. 00:28:41.080 --> 00:28:44.290 This operator O2, we didn't see it when we thought originally 00:28:44.290 --> 00:28:44.980 about matching. 00:28:44.980 --> 00:28:48.100 It had Wilson coefficient that was 0 at tree level. 00:28:53.040 --> 00:28:55.410 So at leading order, you could say that this Wilson 00:28:55.410 --> 00:28:58.020 coefficient is 0. 00:28:58.020 --> 00:28:59.760 But at leading log, it's not 0. 00:29:05.160 --> 00:29:08.070 So I have these two different types of perturbation theory. 00:29:08.070 --> 00:29:10.200 Just order by order and alpha or doing 00:29:10.200 --> 00:29:11.847 renormalization group improvement, 00:29:11.847 --> 00:29:12.930 you get different results. 00:29:17.970 --> 00:29:21.120 And that's because you've included some higher order 00:29:21.120 --> 00:29:25.110 terms by using the renormalization group improved 00:29:25.110 --> 00:29:27.720 version. 00:29:27.720 --> 00:29:30.960 But you can argue, if alpha times the large log is order 1, 00:29:30.960 --> 00:29:35.410 then this is the right type of perturbation theory to do. 00:29:35.410 --> 00:29:45.030 So if you think about it as a picture where this is mu, 00:29:45.030 --> 00:29:51.720 this is Mw, this is Mb, then you have two coefficients, C1 00:29:51.720 --> 00:29:52.470 and C2. 00:29:52.470 --> 00:29:55.890 We call them C+ and C-, but they're just related. 00:29:55.890 --> 00:29:58.200 And the results that we derived at leading order 00:29:58.200 --> 00:30:04.590 were that, for C1, it started at 1 at the high scale, 00:30:04.590 --> 00:30:07.740 basically, if we mu w equal to Mw. 00:30:07.740 --> 00:30:09.930 And it would evolve, actually, this direction 00:30:09.930 --> 00:30:14.820 if we put in all the signs that came out of our calculations. 00:30:14.820 --> 00:30:18.360 And for C2, it starts at 0 here, and then it evolves this way 00:30:18.360 --> 00:30:21.248 to a negative value. 00:30:21.248 --> 00:30:22.212 [INAUDIBLE] 00:30:32.340 --> 00:30:42.330 So roughly putting in some numbers, 00:30:42.330 --> 00:30:46.470 the kind of thing that we would get is this. 00:30:46.470 --> 00:30:49.860 So a coefficient which was 0 all of a sudden becomes minus 0.3 00:30:49.860 --> 00:30:52.320 and becomes something that you have to keep track of. 00:30:52.320 --> 00:30:53.280 That's at leading log. 00:30:56.793 --> 00:30:58.460 Obviously, when you go to higher orders, 00:30:58.460 --> 00:31:00.755 those numbers will be perturbatively improved. 00:31:04.370 --> 00:31:05.600 OK. 00:31:05.600 --> 00:31:08.510 So is the physical picture here clear of what's happening 00:31:08.510 --> 00:31:12.180 with these operators? 00:31:12.180 --> 00:31:14.855 So what is the application? 00:31:14.855 --> 00:31:18.292 Since we spent all this time deriving these results, 00:31:18.292 --> 00:31:20.000 we should have some applications in mind. 00:31:42.630 --> 00:31:47.570 So for b to c u bar d, if you ask about what process that 00:31:47.570 --> 00:31:52.550 gives, well, one process that it gives 00:31:52.550 --> 00:31:56.285 is just a B to D pi transition. 00:31:56.285 --> 00:32:00.230 B bar is built over u bar b. 00:32:00.230 --> 00:32:04.220 And this guy is u bar c. 00:32:04.220 --> 00:32:06.050 And the pi is the u bar d. 00:32:08.783 --> 00:32:11.200 So we can think of the reason we're studying this is maybe 00:32:11.200 --> 00:32:13.880 we want to calculate B of D pi. 00:32:13.880 --> 00:32:15.850 So if we wanted to calculate B of D pi, 00:32:15.850 --> 00:32:20.650 we take matrix elements involving our Hamiltonian 00:32:20.650 --> 00:32:23.270 with a B [INAUDIBLE] in the in state and a D pi in the out 00:32:23.270 --> 00:32:23.770 state. 00:32:31.400 --> 00:32:33.670 And if we just use the original Hamiltonian 00:32:33.670 --> 00:32:35.740 that we wrote down with the renormalization group 00:32:35.740 --> 00:32:40.810 improvement, then we would have that. 00:32:40.810 --> 00:32:43.760 That's with that renormalization group improvement. 00:32:43.760 --> 00:32:46.450 So this is at mu equals Mw. 00:32:46.450 --> 00:32:47.890 And the problem with this formula 00:32:47.890 --> 00:32:49.990 is that this matrix element has large logs. 00:32:52.900 --> 00:32:54.190 It depends on Mw. 00:32:54.190 --> 00:32:55.450 It also depends on Mb. 00:33:00.342 --> 00:33:02.050 And if it's something we can't calculate, 00:33:02.050 --> 00:33:04.420 then that's kind of bad news. 00:33:04.420 --> 00:33:06.170 In particular, having large logs like that 00:33:06.170 --> 00:33:08.087 would also make it hard to calculate something 00:33:08.087 --> 00:33:09.140 like this on the lattice. 00:33:09.140 --> 00:33:12.030 So it's not just a-- 00:33:12.030 --> 00:33:13.290 It's really a problem. 00:33:13.290 --> 00:33:15.267 If you have multiple scales tied together, 00:33:15.267 --> 00:33:16.850 it just makes the calculations harder. 00:33:20.445 --> 00:33:22.320 It's also a problem for dimensional analysis. 00:33:22.320 --> 00:33:23.700 Because if you have large logs, that 00:33:23.700 --> 00:33:24.900 means you've got large numbers. 00:33:24.900 --> 00:33:27.090 And something that you thought was of a certain size 00:33:27.090 --> 00:33:29.280 might be bigger or smaller. 00:33:29.280 --> 00:33:31.830 So what we do is, instead, we work in the renormalization 00:33:31.830 --> 00:33:39.630 group improved version where we take this down at the scale Mb. 00:33:39.630 --> 00:33:41.580 So this guy includes the renormalization group 00:33:41.580 --> 00:33:44.160 evolution. 00:33:44.160 --> 00:33:45.690 We use our results over there. 00:33:49.500 --> 00:33:51.870 And then we've got the operators at the scale Mb, 00:33:51.870 --> 00:33:53.070 and there's no large logs. 00:33:58.820 --> 00:34:00.600 OK, so you'd want to calculate something 00:34:00.600 --> 00:34:03.090 like this on the lattice or some other way. 00:34:03.090 --> 00:34:05.250 There's other ways of doing it. 00:34:05.250 --> 00:34:07.930 And we'll talk about other ways of doing it later on. 00:34:07.930 --> 00:34:10.630 But so far, we've separated out the scale Mw 00:34:10.630 --> 00:34:13.659 into this coefficient that's evaluated 00:34:13.659 --> 00:34:15.070 at the scale of mu equals Mb. 00:34:28.730 --> 00:34:29.230 OK. 00:34:29.230 --> 00:34:31.000 So the one way of thinking about this 00:34:31.000 --> 00:34:33.670 is, if you want to do physics at the scale Mb, 00:34:33.670 --> 00:34:37.989 the right couplings to use in your theory are these ones. 00:34:37.989 --> 00:34:40.030 Forget about what's going on at the high scale. 00:34:40.030 --> 00:34:41.565 You have to determine the low energy 00:34:41.565 --> 00:34:43.690 couplings that are appropriate to the theory you're 00:34:43.690 --> 00:34:44.739 dealing with. 00:34:44.739 --> 00:34:47.800 And those are the C's at Mb, OK? 00:35:11.040 --> 00:35:13.880 All right. 00:35:13.880 --> 00:35:14.380 OK. 00:35:14.380 --> 00:35:16.990 So now, I want to come back to this question of thinking 00:35:16.990 --> 00:35:19.390 about the next leading log. 00:35:19.390 --> 00:35:21.850 And I'm going to do that by going back 00:35:21.850 --> 00:35:26.230 to our comparison of full theory to effective theory. 00:35:26.230 --> 00:35:29.620 We'll do a comparison of results in the full theory 00:35:29.620 --> 00:35:31.120 and results in the effective theory. 00:35:31.120 --> 00:35:35.230 And I'll show you how, by making that comparison, 00:35:35.230 --> 00:35:40.090 we can determine the ingredients that we need for this one loop 00:35:40.090 --> 00:35:41.652 matching here. 00:35:41.652 --> 00:35:43.210 So we'll focus on this. 00:35:47.990 --> 00:35:50.080 So we already renormalized the effective theory. 00:35:50.080 --> 00:35:52.450 So we can compare the renormalized effective theory 00:35:52.450 --> 00:35:53.170 and full theory. 00:36:00.580 --> 00:36:02.080 And that's the right way to proceed. 00:36:06.200 --> 00:36:09.070 So in our parlance of theory one and theory two, 00:36:09.070 --> 00:36:12.490 the effective theory would be theory two. 00:36:12.490 --> 00:36:15.310 We have to think about the full theory, which in our parlance 00:36:15.310 --> 00:36:19.800 would be theory one, the full theory being 00:36:19.800 --> 00:36:20.800 the standard model here. 00:36:24.370 --> 00:36:28.390 We have to think about renormalizing that theory. 00:36:28.390 --> 00:36:30.580 But in the standard model, our calculation 00:36:30.580 --> 00:36:31.795 involves conserved currents. 00:36:36.070 --> 00:36:38.350 These are just a weak currents. 00:36:38.350 --> 00:36:41.522 And so there's actually no extra UV divergences 00:36:41.522 --> 00:36:42.730 associated to those currents. 00:36:42.730 --> 00:36:44.819 We just have coupling renormalization. 00:36:50.920 --> 00:36:53.860 And one way of saying this is that what happens 00:36:53.860 --> 00:36:55.600 is that the vertex in the wave function 00:36:55.600 --> 00:37:03.728 graphs, the UV divergences cancel 00:37:03.728 --> 00:37:04.770 or the conserved current. 00:37:08.880 --> 00:37:13.170 So the result for the full theory 00:37:13.170 --> 00:37:18.170 will be some result that is independent of having-- 00:37:18.170 --> 00:37:20.402 it doesn't have any ultraviolet divergences. 00:37:25.190 --> 00:37:26.690 And like the effective theory, where 00:37:26.690 --> 00:37:28.815 you had to carry out a renormalization of operators 00:37:28.815 --> 00:37:30.342 in that theory, for the full theory, 00:37:30.342 --> 00:37:32.050 coupling renormalization is all there is. 00:37:39.040 --> 00:37:40.750 So let's draw the full theory graphs. 00:37:47.610 --> 00:37:51.470 Gluons should be green. 00:37:51.470 --> 00:37:52.880 Maybe my w should be pink. 00:38:29.010 --> 00:38:36.030 Six permutations-- and then there's also 00:38:36.030 --> 00:38:37.682 wave function normalization. 00:38:40.930 --> 00:38:41.430 OK. 00:38:41.430 --> 00:38:43.472 So if you want to do the full theory calculation, 00:38:43.472 --> 00:38:45.000 these are the graphs you compute. 00:38:45.000 --> 00:38:47.070 It's triangles as well as box integrals. 00:38:47.070 --> 00:38:50.074 It's actually a much harder calculation 00:38:50.074 --> 00:38:51.580 than in the effective theory. 00:38:54.750 --> 00:38:56.910 And I'm not going to do the calculation, 00:38:56.910 --> 00:38:58.785 but I'll tell you what the results look like. 00:39:06.760 --> 00:39:09.090 So let's start by thinking about the logs. 00:39:13.017 --> 00:39:14.850 And then we'll talk about the constants that 00:39:14.850 --> 00:39:15.975 are under the logs as well. 00:39:21.610 --> 00:39:26.100 So if we look at this calculation, 00:39:26.100 --> 00:39:27.390 it has the following form. 00:39:52.730 --> 00:39:56.810 So there's S1, which was some spinners. 00:39:56.810 --> 00:39:59.443 We defined it in an earlier lecture. 00:39:59.443 --> 00:40:00.860 There's something involving a log, 00:40:00.860 --> 00:40:03.260 and it has a p squared. p squared was the off-shellness 00:40:03.260 --> 00:40:04.790 associated to these guys. 00:40:04.790 --> 00:40:09.000 And we regulated the infrared divergences with p squared. 00:40:09.000 --> 00:40:15.710 So p squared not equal to 0 regulates IR divergences. 00:40:15.710 --> 00:40:19.160 And there are IR divergences in these diagrams. 00:40:19.160 --> 00:40:20.750 Even though I said they are UV finite, 00:40:20.750 --> 00:40:23.000 they're not finite in the infrared. 00:40:23.000 --> 00:40:25.282 And that's what leads to these logs of p squared. 00:40:32.370 --> 00:40:33.750 OK. 00:40:33.750 --> 00:40:35.250 Now, I didn't write everything. 00:40:35.250 --> 00:40:38.250 I only wrote the pieces proportional to S1. 00:40:38.250 --> 00:40:42.390 There's pieces proportional to the other spinner, S2. 00:40:42.390 --> 00:40:44.802 And there's mod log terms. 00:40:44.802 --> 00:40:46.260 And they're all hiding in the dots. 00:40:50.430 --> 00:40:53.070 So let's compare this result to a similar expression 00:40:53.070 --> 00:40:58.020 in the effective theory that we just 00:40:58.020 --> 00:41:00.690 set the coefficients to the values at the high scale. 00:41:05.530 --> 00:41:08.520 So then we have 1 for the coefficient 00:41:08.520 --> 00:41:12.150 times the one-loop matrix element of O1, 00:41:12.150 --> 00:41:14.730 which we wrote down earlier. 00:41:14.730 --> 00:41:18.030 And it looks kind of similar to this, but not exactly the same. 00:41:25.700 --> 00:41:28.400 It's very similar, but not precisely the same. 00:41:38.476 --> 00:41:44.480 A similar statement applies to these guys over here, OK? 00:41:44.480 --> 00:41:46.940 So the difference is really that, instead 00:41:46.940 --> 00:41:49.027 of Mw squared in this log, we have a mu squared. 00:41:49.027 --> 00:41:51.110 That's really the only difference for these terms. 00:41:56.840 --> 00:41:59.000 With constant terms, the non-logarithmic terms 00:41:59.000 --> 00:42:00.470 here and here won't agree. 00:42:00.470 --> 00:42:03.360 And we'll talk about those things in a minute. 00:42:03.360 --> 00:42:06.200 So what do we learn by thinking about the physics of these two 00:42:06.200 --> 00:42:08.130 equations? 00:42:08.130 --> 00:42:09.980 Well, one comment is the comment I already 00:42:09.980 --> 00:42:14.240 said, that the effective theory computation for this line 00:42:14.240 --> 00:42:17.730 here is much, much easier than this one. 00:42:17.730 --> 00:42:19.740 So one reason to use effective field theory 00:42:19.740 --> 00:42:21.590 is just that it makes computations easier. 00:42:25.550 --> 00:42:27.422 And the reason it makes computations easier 00:42:27.422 --> 00:42:28.880 is because you're basically dealing 00:42:28.880 --> 00:42:30.750 with one scale at a time. 00:42:30.750 --> 00:42:33.530 And whenever you have integrals involving only one scale, 00:42:33.530 --> 00:42:36.740 that's always much easier than having multiple scales. 00:42:41.908 --> 00:42:44.450 But if you want to encode all the physics of the full theory, 00:42:44.450 --> 00:42:46.700 you'll still have to do that calculation at some point 00:42:46.700 --> 00:42:47.480 as well. 00:42:47.480 --> 00:42:50.030 Although you may be able to do it in a simpler configuration 00:42:50.030 --> 00:42:51.738 to get off the information that you need. 00:42:56.750 --> 00:42:59.083 Furthermore, if you really only cared about the logs, 00:42:59.083 --> 00:43:01.250 then all you really need is the 1 over epsilon term. 00:43:01.250 --> 00:43:06.320 And that's even easier than the full triangle diagrams. 00:43:06.320 --> 00:43:10.250 So to compute the anomalous dimensions, 00:43:10.250 --> 00:43:13.200 you just have to keep the divergences. 00:43:13.200 --> 00:43:17.210 And you can throw away all the finite pieces. 00:43:17.210 --> 00:43:18.568 And that's even easier. 00:43:18.568 --> 00:43:21.110 So you can organize things by thinking about doing the easier 00:43:21.110 --> 00:43:22.443 calculations first. 00:43:22.443 --> 00:43:23.360 That's what people do. 00:43:23.360 --> 00:43:24.818 They calculate anomalous dimensions 00:43:24.818 --> 00:43:29.110 before they calculate matching because it's just easier. 00:43:29.110 --> 00:43:31.610 And it's also the thing you need for the leading log result. 00:43:31.610 --> 00:43:34.068 You don't need the matching at one loop for the leading log 00:43:34.068 --> 00:43:36.440 result. So there's a conservation of ease 00:43:36.440 --> 00:43:37.370 and what you need. 00:43:41.795 --> 00:43:43.670 These two things play nicely with each other. 00:43:50.270 --> 00:43:53.860 Second point-- in the effective theory, 00:43:53.860 --> 00:43:57.200 you're supposed to think that Mw goes to infinity. 00:43:57.200 --> 00:43:59.540 And that's why this thing doesn't know about Mw. 00:43:59.540 --> 00:44:01.810 So how could it possibly get an Mw? 00:44:01.810 --> 00:44:04.630 And so what happens is that Mw gets replaced by the cut 00:44:04.630 --> 00:44:05.680 off, which is mu here. 00:44:26.840 --> 00:44:29.080 Another point that we can make about this-- if you 00:44:29.080 --> 00:44:31.090 look at the logs of minus p squared, 00:44:31.090 --> 00:44:33.550 they're all the same between the two equations. 00:44:37.870 --> 00:44:41.118 That's actually important because what 00:44:41.118 --> 00:44:43.660 that means is that the infrared structure of the two theories 00:44:43.660 --> 00:44:44.160 agree. 00:44:46.990 --> 00:44:49.120 The logs of p squared are the infrared divergences. 00:44:49.120 --> 00:44:51.642 They agree between the higher theory and the lower theory. 00:44:51.642 --> 00:44:52.600 And they have to agree. 00:44:56.503 --> 00:44:58.420 What this tells you about the effective theory 00:44:58.420 --> 00:45:01.523 is that you're doing something right. 00:45:01.523 --> 00:45:03.940 In particular, it tells you that your effective theory has 00:45:03.940 --> 00:45:05.148 the right degrees of freedom. 00:45:08.050 --> 00:45:10.642 That's almost trivial in this example. 00:45:10.642 --> 00:45:12.850 What other degrees of freedom could we possibly think 00:45:12.850 --> 00:45:13.990 that would be missing? 00:45:18.460 --> 00:45:20.660 But in more complicated examples-- 00:45:20.660 --> 00:45:24.400 and we will deal with at least one such example later 00:45:24.400 --> 00:45:25.820 in the course-- 00:45:25.820 --> 00:45:28.510 it's not so trivial to see that these things match up. 00:45:28.510 --> 00:45:30.880 And people have discovered new degrees of freedom 00:45:30.880 --> 00:45:32.860 by doing matching computations like this. 00:45:32.860 --> 00:45:35.027 They said, oh, this is a relevant degree of freedom. 00:45:35.027 --> 00:45:35.980 And it's needed. 00:45:35.980 --> 00:45:39.370 Because if I do a one-loop calculation, 00:45:39.370 --> 00:45:41.997 I need it to get the infrared divergences right. 00:45:41.997 --> 00:45:44.330 So this can really teach you about the effective theory, 00:45:44.330 --> 00:45:46.780 doing a matching computation, teach you 00:45:46.780 --> 00:45:50.550 about the physics of the effective theory. 00:45:50.550 --> 00:45:53.455 So if you made a mistake, this would be a place 00:45:53.455 --> 00:45:54.580 where you'd catch yourself. 00:46:06.820 --> 00:46:09.670 Now, that you've analyzed fully what the differences are, 00:46:09.670 --> 00:46:10.870 you can subtract them. 00:46:15.270 --> 00:46:19.290 You take the difference of the renormalized calculations. 00:46:19.290 --> 00:46:23.850 And that is what gives you one-loop matching. 00:46:23.850 --> 00:46:25.770 Just like we compared tree level calculations 00:46:25.770 --> 00:46:28.500 to get tree level matching, we compare one-loop calculations 00:46:28.500 --> 00:46:31.770 to get one-loop matching. 00:46:37.620 --> 00:46:48.630 So [INAUDIBLE] we do that. 00:46:48.630 --> 00:46:53.580 At tree level, if we're just looking at the S1 pieces, 00:46:53.580 --> 00:46:56.760 we have these two terms. 00:46:56.760 --> 00:46:58.250 And then at one-loop-- 00:47:00.920 --> 00:47:02.120 let me use this notation. 00:47:02.120 --> 00:47:03.980 This is the full A. So it has a tree level 00:47:03.980 --> 00:47:05.150 piece and one-loop piece. 00:47:08.210 --> 00:47:10.400 Then we take the piece of the C1 that's at one loop. 00:47:14.000 --> 00:47:19.520 And we take the matrix element of the operator, 00:47:19.520 --> 00:47:21.490 evaluate it order alpha. 00:47:28.430 --> 00:47:29.660 And then there's C2 terms. 00:47:29.660 --> 00:47:30.590 Let me just put dots. 00:47:37.440 --> 00:47:40.692 OK, so there's two ways that I can get an alpha. 00:47:40.692 --> 00:47:42.150 There's an order alpha coefficient. 00:47:42.150 --> 00:47:45.210 That's what I want to know, what you want to determine. 00:47:45.210 --> 00:47:48.120 And then there's an order alpha matrix element. 00:47:48.120 --> 00:47:52.110 So by subtracting, putting this guy on the left-hand side, 00:47:52.110 --> 00:47:55.800 I get the value of C1 of 1. 00:47:55.800 --> 00:47:59.550 So we use this equation to determine this. 00:47:59.550 --> 00:48:02.250 And the story would be similar if I kept all the C2 terms. 00:48:05.070 --> 00:48:07.260 So the matching, i.e. the difference 00:48:07.260 --> 00:48:10.620 of the full and effective theory and calculations, 00:48:10.620 --> 00:48:12.120 determines that coefficient for you. 00:48:19.930 --> 00:48:21.460 So the notation here is that we'd 00:48:21.460 --> 00:48:27.530 write the full coefficient as 1 plus C1 plus higher order terms 00:48:27.530 --> 00:48:30.560 where this is order alpha. 00:48:30.560 --> 00:48:31.840 That's the notation I'm using. 00:48:36.130 --> 00:48:37.470 So let's do that for the logs. 00:48:37.470 --> 00:48:38.460 It's pretty simple. 00:48:38.460 --> 00:48:40.030 These things here are just cancel. 00:48:40.030 --> 00:48:43.200 And this one we can just subtract them. 00:48:43.200 --> 00:48:45.040 The p squareds will cancel. 00:48:45.040 --> 00:48:47.220 And we'll get a log of my squared over Mw squared. 00:49:22.180 --> 00:49:24.910 So just focusing on those terms that we have in S1-- 00:49:27.595 --> 00:49:29.470 and rearranging the equation in the way 00:49:29.470 --> 00:49:33.730 I said and then plugging in the values for the things that 00:49:33.730 --> 00:49:35.630 don't cancel. 00:49:50.140 --> 00:49:54.460 So the terms that were slightly different-- 00:49:54.460 --> 00:49:55.690 and dropping the S1. 00:50:09.090 --> 00:50:11.316 CF is 4/3. 00:50:11.316 --> 00:50:13.489 It's Casimir of the fundamental. 00:50:18.540 --> 00:50:21.420 We see that, since we've only kept the log terms, what 00:50:21.420 --> 00:50:24.420 we find is the one-loop correction in this guy that's 00:50:24.420 --> 00:50:26.130 got a logarithm. 00:50:26.130 --> 00:50:30.510 but it'd also be a term here that's a number times alpha. 00:50:33.240 --> 00:50:38.670 But we haven't kept those terms in what 00:50:38.670 --> 00:50:40.478 I've written on the board. 00:50:40.478 --> 00:50:42.270 So the way that you should think about this 00:50:42.270 --> 00:50:44.270 is that we've got these Wilson coefficients that 00:50:44.270 --> 00:50:46.560 depend on the scale Mw. 00:50:46.560 --> 00:50:51.240 And what the matching is doing, it's taking the full theory. 00:50:51.240 --> 00:50:54.813 And it's dividing it into large momentum pieces 00:50:54.813 --> 00:50:55.980 times small momentum pieces. 00:51:04.160 --> 00:51:07.660 So the large momentum pieces are in C. Small 00:51:07.660 --> 00:51:12.440 momentum pieces are in the matrix element of the operator. 00:51:12.440 --> 00:51:16.330 And this statement, we see an explicit realization of it. 00:51:16.330 --> 00:51:21.830 The full theory knows about high scale Mw's and the p squared. 00:51:21.830 --> 00:51:32.687 And we can write this as a split like this. 00:51:32.687 --> 00:51:35.270 The effective theory knows about p squared, doesn't know about 00:51:35.270 --> 00:51:35.870 Mw squared. 00:51:35.870 --> 00:51:37.820 The Wilson coefficient knows about Mw squared, 00:51:37.820 --> 00:51:39.350 doesn't know about p squared. 00:51:39.350 --> 00:51:41.267 The additional thing that they both know about 00:51:41.267 --> 00:51:42.230 is the scale mu. 00:51:42.230 --> 00:51:44.000 And that's providing a cutoff for where 00:51:44.000 --> 00:52:04.800 you split between large momentum and small momentum 00:52:04.800 --> 00:52:07.980 p squared was the small scale. 00:52:07.980 --> 00:52:10.353 Now, if you look at this equation, 00:52:10.353 --> 00:52:11.520 you may wonder for a minute. 00:52:11.520 --> 00:52:13.380 Why is it additive? 00:52:13.380 --> 00:52:16.170 Up here, I just said times. 00:52:16.170 --> 00:52:17.670 And then immediately below times, 00:52:17.670 --> 00:52:22.410 I wrote something that was a sum, seemed a little weird. 00:52:22.410 --> 00:52:26.190 That's just because, if you take something that includes the 1, 00:52:26.190 --> 00:52:31.890 then the product becomes a sum. 00:52:31.890 --> 00:52:34.890 So if I write it this way, I can write it in product form 00:52:34.890 --> 00:52:37.790 if I include the one tree level. 00:52:41.880 --> 00:52:44.360 So it really is a product. 00:52:44.360 --> 00:52:46.880 It's just that, if you look at the order alpha pieces, 00:52:46.880 --> 00:52:49.920 it breaks into the sum where we can nicely see how 00:52:49.920 --> 00:52:51.750 things are combining together. 00:52:51.750 --> 00:52:53.795 But really it has this product structure, 00:52:53.795 --> 00:52:55.920 and there's non-trivial relations between these two 00:52:55.920 --> 00:52:58.440 series that make it all work out even when 00:52:58.440 --> 00:53:00.520 you go to higher orders. 00:53:00.520 --> 00:53:02.250 So if I expand that to order alpha, 00:53:02.250 --> 00:53:03.750 look at the order alpha coefficient. 00:53:03.750 --> 00:53:05.635 I get this equation back again. 00:53:05.635 --> 00:53:08.010 And this is how you would think about it in product form. 00:53:11.070 --> 00:53:12.360 OK. 00:53:12.360 --> 00:53:16.560 So the other thing you see here is that order by order 00:53:16.560 --> 00:53:20.640 in our expansion, as we kind of already stated, 00:53:20.640 --> 00:53:23.070 the mu dependence between these coefficients and these 00:53:23.070 --> 00:53:26.010 operators is exactly cancelling because the full theory here 00:53:26.010 --> 00:53:27.810 didn't involve that mu. 00:53:31.760 --> 00:53:35.390 That's another little piece of information that we get 00:53:35.390 --> 00:53:37.110 or that we knew, but we see explicitly 00:53:37.110 --> 00:53:38.450 from looking at this. 00:53:48.650 --> 00:53:51.320 So I think, if I'm counting right, 00:53:51.320 --> 00:53:53.000 this is comment number five. 00:53:57.335 --> 00:53:58.210 Was there a question? 00:54:04.550 --> 00:54:06.940 So not surprisingly, the cut off dependence 00:54:06.940 --> 00:54:10.960 cancels in the product of C of mu O of mu 00:54:10.960 --> 00:54:13.627 because the cut off is what we introduced 00:54:13.627 --> 00:54:15.460 to split up the physics in these two things. 00:54:18.990 --> 00:54:23.810 Now, if you look at that in a little more detail, 00:54:23.810 --> 00:54:26.390 it's only mu independent of the order in perturbation theory 00:54:26.390 --> 00:54:27.223 that you're working. 00:54:31.343 --> 00:54:34.690 If you've worked at a fixed order in some expansion, 00:54:34.690 --> 00:54:38.290 then you shouldn't be surprised that everything you've derived 00:54:38.290 --> 00:54:39.580 is only true at that order. 00:54:43.820 --> 00:54:45.550 So if you stopped at one loop, then 00:54:45.550 --> 00:54:50.440 it's mu independent at order alpha S. 00:54:50.440 --> 00:54:52.390 What that technically means is that terms that 00:54:52.390 --> 00:54:55.480 are alpha S mu log mu cancel. 00:54:58.060 --> 00:55:00.610 The log mu here cancels, but there's mu dependence also 00:55:00.610 --> 00:55:01.540 here. 00:55:01.540 --> 00:55:03.760 And that mu dependence in the alpha 00:55:03.760 --> 00:55:06.100 is something that would be related to terms 00:55:06.100 --> 00:55:10.970 that are alpha squared log mu. 00:55:10.970 --> 00:55:12.470 And that cancels at higher order. 00:55:16.910 --> 00:55:18.410 So some of the mu dependents cancel. 00:55:18.410 --> 00:55:20.300 Some of the mu dependents doesn't cancel. 00:55:20.300 --> 00:55:22.640 And people actually use the fact that some of the mu dependence 00:55:22.640 --> 00:55:24.680 doesn't cancel as getting a handle on the higher order 00:55:24.680 --> 00:55:25.180 terms. 00:55:25.180 --> 00:55:27.770 It's doing a kind of theory uncertainty. 00:55:52.800 --> 00:55:54.810 If we just think about the logarithms, 00:55:54.810 --> 00:56:00.090 then actually the one-loop results in the full theory 00:56:00.090 --> 00:56:02.299 has actually less information. 00:56:09.235 --> 00:56:11.610 And the reason is that, if you wanted to get higher order 00:56:11.610 --> 00:56:14.027 terms in this leading log series that we talked about, 00:56:14.027 --> 00:56:16.110 if you wanted to derive those from the full theory 00:56:16.110 --> 00:56:19.410 point of view, you'd have to do a two-loop computation. 00:56:24.220 --> 00:56:29.590 So if you wanted to get alpha squared log squared of Mw 00:56:29.590 --> 00:56:34.060 squared over minus p squared, then you'd 00:56:34.060 --> 00:56:41.125 have to look at diagrams, two gluons. 00:56:43.995 --> 00:56:45.370 On the full theory point of view, 00:56:45.370 --> 00:56:50.513 that's what you'd have to do to find those terms. 00:56:50.513 --> 00:56:52.180 From the effective theory point of view, 00:56:52.180 --> 00:56:53.763 all you have to do to find those terms 00:56:53.763 --> 00:56:56.380 is renormalize the effective theory properly. 00:56:56.380 --> 00:56:58.420 And then you get those terms. 00:57:09.630 --> 00:57:12.750 So we just needed the one-loop anomalous dimension. 00:57:12.750 --> 00:57:14.660 So in that sense, the effective theory, 00:57:14.660 --> 00:57:16.640 because of the renormalization properties 00:57:16.640 --> 00:57:19.280 of the effective theory, know something 00:57:19.280 --> 00:57:22.550 that the full theory doesn't know so easily. 00:57:22.550 --> 00:57:24.290 And that kind of shows you the advantage 00:57:24.290 --> 00:57:27.710 of taking something that's a constant, Mw squared, 00:57:27.710 --> 00:57:29.132 and turning it into a scale. 00:57:29.132 --> 00:57:30.590 Because by turning it into a scale, 00:57:30.590 --> 00:57:32.798 you have the whole power of the renormalization group 00:57:32.798 --> 00:57:34.820 at your disposal to predict higher order 00:57:34.820 --> 00:57:38.030 things, like the higher order coefficients. 00:57:38.030 --> 00:57:39.800 And that's one way of phrasing what 00:57:39.800 --> 00:57:42.163 the example is of splitting scales 00:57:42.163 --> 00:57:43.580 and going to the effective theory. 00:57:51.290 --> 00:57:53.750 So the final thing that I want to talk about here 00:57:53.750 --> 00:57:56.360 has to do with the fact that-- 00:57:56.360 --> 00:57:59.600 well, actually, there's two more things I want to talk about, 00:57:59.600 --> 00:58:01.340 but let me make the final comment here. 00:58:04.020 --> 00:58:06.920 So the final comment I want to make in my list, 00:58:06.920 --> 00:58:10.745 which is number seven, has to do with scheme dependence. 00:58:14.000 --> 00:58:15.530 So scheme dependence means that we 00:58:15.530 --> 00:58:18.537 pick the renormalization scheme MS bar. 00:58:18.537 --> 00:58:20.120 And we could have done the calculation 00:58:20.120 --> 00:58:23.438 in a different renormalization scheme. 00:58:23.438 --> 00:58:25.355 And we should ask what depends on that choice. 00:58:28.850 --> 00:58:31.490 You may know, if you've taken a course on the beta function 00:58:31.490 --> 00:58:34.660 or if you've taken QFD3, that the beta function of QCD 00:58:34.660 --> 00:58:36.800 is scheme independent for the first two orders. 00:58:40.580 --> 00:58:42.680 The analog of that statement here 00:58:42.680 --> 00:58:45.920 is that the one-loop anomalous dimension for our operators 00:58:45.920 --> 00:58:47.328 is scheme independent. 00:58:50.674 --> 00:58:55.360 It doesn't depend on which mass independent scheme you pick. 00:58:55.360 --> 00:58:58.270 So in the class of mass independence schemes, 00:58:58.270 --> 00:59:00.535 the result is what we derived. 00:59:06.000 --> 00:59:08.600 We'll come back and study that in a little more detail. 00:59:11.430 --> 00:59:14.958 OK, so let's go back now and establish some notation 00:59:14.958 --> 00:59:17.000 where we actually just put the constants back in. 00:59:20.260 --> 00:59:22.010 And again, I'm not going to write numbers. 00:59:22.010 --> 00:59:24.408 I'll just give them names. 00:59:24.408 --> 00:59:25.950 And we'll track what happens to them. 00:59:28.530 --> 00:59:35.180 So let's think about the full one-loop matching 00:59:35.180 --> 00:59:39.780 and how we get the next leading log result. 00:59:39.780 --> 00:59:42.200 And really what I want to focus on, or at least one thing 00:59:42.200 --> 00:59:46.580 I want to focus on, is the scheme dependence. 00:59:46.580 --> 00:59:50.630 Because the coefficients, once you get to next leading log, 00:59:50.630 --> 00:59:52.940 are totally scheme dependent. 00:59:52.940 --> 00:59:54.920 So you can ask, what physical sense 00:59:54.920 --> 00:59:57.943 do they make if they're scheme dependent? 00:59:57.943 --> 01:00:00.110 Well, it turns out that the matrix elements are also 01:00:00.110 --> 01:00:02.000 scheme dependent. 01:00:02.000 --> 01:00:04.760 And the anomalous dimensions are scheme dependent. 01:00:04.760 --> 01:00:08.180 So basically, everything is scheme dependent. 01:00:08.180 --> 01:00:10.940 And when we put it all together, we get a scheme 01:00:10.940 --> 01:00:25.965 independent result. So you might think, well, 01:00:25.965 --> 01:00:27.590 if we can get scheme dependent results, 01:00:27.590 --> 01:00:30.050 we should just stop because maybe we can't 01:00:30.050 --> 01:00:32.030 understand what's going on. 01:00:32.030 --> 01:00:41.360 But C of mu times O of mu is independent of the scheme. 01:00:44.900 --> 01:00:46.070 It's a physical observable. 01:00:46.070 --> 01:00:47.840 And physical observables don't depend 01:00:47.840 --> 01:00:49.940 on our definitions of things. 01:00:54.620 --> 01:00:56.090 Nature gets to decide, not us. 01:01:01.020 --> 01:01:04.400 So one way of thinking about this 01:01:04.400 --> 01:01:07.910 is that we already saw some kind of scheme independence 01:01:07.910 --> 01:01:11.060 in a statement that C of mu times O of mu 01:01:11.060 --> 01:01:12.260 is independent of mu. 01:01:12.260 --> 01:01:15.080 But there's even a deeper scheme independence to it 01:01:15.080 --> 01:01:18.890 that it's independent of whether we chose MS bar or some others 01:01:18.890 --> 01:01:21.440 scheme. 01:01:21.440 --> 01:01:23.910 So for the context of this discussion, 01:01:23.910 --> 01:01:27.560 I'm going to start dropping all the matrix indices. 01:01:27.560 --> 01:01:29.540 And we're not going to write i and j 01:01:29.540 --> 01:01:33.470 just because I want to keep things a little bit simple. 01:01:33.470 --> 01:01:37.580 So we'll write that the effective theory is simply 01:01:37.580 --> 01:01:41.510 one coefficient times the matrix of one operator. 01:01:47.940 --> 01:01:54.050 So let's think about, in that context, trying 01:01:54.050 --> 01:01:56.570 to understand where all this scheme dependence is floating 01:01:56.570 --> 01:02:05.500 around and how the matching works. 01:02:10.360 --> 01:02:13.430 So we just do the same thing we did before. 01:02:13.430 --> 01:02:16.970 I'm leaving off some pre-factors, 01:02:16.970 --> 01:02:18.310 leaving off the pre-factors. 01:02:18.310 --> 01:02:20.102 I don't have the write the spinners anymore 01:02:20.102 --> 01:02:22.540 since there's only one structure. 01:02:22.540 --> 01:02:28.700 And let me introduce some notation for the results. 01:02:28.700 --> 01:02:31.090 So we had this Mw squared over p squared type term. 01:02:33.620 --> 01:02:36.140 And let me just focus on these terms and not the terms that 01:02:36.140 --> 01:02:39.290 just cancelled away. 01:02:39.290 --> 01:02:41.900 So let me focus on the terms that are different. 01:02:41.900 --> 01:02:44.000 But now, I'm also going to include the constants. 01:03:03.860 --> 01:03:05.920 So the constants that we get in the full theory 01:03:05.920 --> 01:03:07.545 and the effective theory are different. 01:03:07.545 --> 01:03:09.850 So I'll call one of them A and the other one B. 01:03:09.850 --> 01:03:12.310 So I call the A the full theory result 01:03:12.310 --> 01:03:14.723 and the B the effective theory result. 01:03:14.723 --> 01:03:16.390 So you should think of this as a number, 01:03:16.390 --> 01:03:21.280 like 3, just some number, same thing here. 01:03:21.280 --> 01:03:23.830 But just to avoid talking about numbers 01:03:23.830 --> 01:03:26.500 and to track also where the scheme dependence is-- 01:03:26.500 --> 01:03:28.240 like this 3 could be 2 in one scheme 01:03:28.240 --> 01:03:33.413 and this 4 could be 2 in one scheme and 5 in another scheme. 01:03:33.413 --> 01:03:35.830 In order to keep track of that, let me call it a variable. 01:03:35.830 --> 01:03:39.520 Let me call it B. 01:03:39.520 --> 01:03:41.310 So then the Wilson coefficient is just 01:03:41.310 --> 01:03:42.635 we construct the difference. 01:03:42.635 --> 01:03:44.260 And then we'll have an A minus B in it. 01:04:07.133 --> 01:04:09.300 So if you like what the Wilson coefficient is doing, 01:04:09.300 --> 01:04:11.400 it's compensating for the fact that the effective theory 01:04:11.400 --> 01:04:12.930 has the wrong value for this constant. 01:04:12.930 --> 01:04:13.680 It should be this. 01:04:13.680 --> 01:04:15.960 That's what the full theory told you it was. 01:04:15.960 --> 01:04:20.730 So the effective theory Wilson coefficient 01:04:20.730 --> 01:04:22.740 has minus the effective theory matrix element 01:04:22.740 --> 01:04:25.620 result plus the correct result. So this 01:04:25.620 --> 01:04:29.842 is the thing that's correcting the effective theory. 01:04:29.842 --> 01:04:31.050 So it has the right constant. 01:04:43.510 --> 01:04:47.280 And if we just take C at Mw, then it 01:04:47.280 --> 01:04:51.107 would simply be equal to that. 01:04:51.107 --> 01:04:52.190 And the log would go away. 01:05:08.130 --> 01:05:11.180 So in order to do the renormalization group 01:05:11.180 --> 01:05:13.520 improved perturbation theory at next leading log, 01:05:13.520 --> 01:05:15.290 we also need to do a two-loop computation. 01:05:18.530 --> 01:05:22.235 We're not going to do the two-loop consultation, 01:05:22.235 --> 01:05:24.110 but I'll tell you the structure of the series 01:05:24.110 --> 01:05:26.240 that you get if you did that computation. 01:05:31.780 --> 01:05:33.940 So this equation is true. 01:05:33.940 --> 01:05:36.420 Therefore, we can write the anomalous dimension equation 01:05:36.420 --> 01:05:37.922 again as log C. 01:05:37.922 --> 01:05:39.630 And the right-hand side will be a series. 01:05:43.113 --> 01:05:45.030 And the structure of the series that was there 01:05:45.030 --> 01:05:48.690 is the 0-th order term. 01:05:48.690 --> 01:05:50.640 And then there's some higher order terms. 01:05:55.500 --> 01:06:00.770 And we need this guy, the two-loop coefficient, 01:06:00.770 --> 01:06:01.610 [INAUDIBLE] gamma 1. 01:06:04.580 --> 01:06:06.870 Again, this is a coupled differential equation. 01:06:06.870 --> 01:06:09.260 And we would solve it by using the kind of thing 01:06:09.260 --> 01:06:10.080 that we did before. 01:06:10.080 --> 01:06:17.540 So d mu over mu is d alpha over beta of alpha. 01:06:17.540 --> 01:06:20.030 And we would write down beta to one higher 01:06:20.030 --> 01:06:24.410 order, which I do in my notes. 01:06:24.410 --> 01:06:29.090 But it's the same idea is I just expand it in alpha. 01:06:29.090 --> 01:06:31.990 And I keep not just the coefficient beta 0, 01:06:31.990 --> 01:06:33.615 but I also keep the coefficient beta 1. 01:06:47.910 --> 01:06:50.370 I want to kind of not focus so much on the calculations, 01:06:50.370 --> 01:06:52.710 but more the results and the implications 01:06:52.710 --> 01:06:56.610 of the calculations. 01:06:56.610 --> 01:07:01.850 So do some renormalization group evolution. 01:07:01.850 --> 01:07:07.658 You can write the all-order solution as an integral, 01:07:07.658 --> 01:07:08.450 like we did before. 01:07:15.460 --> 01:07:19.410 And if I just keep it in terms of these all-order objects, 01:07:19.410 --> 01:07:23.760 then it's just the ratio, which I expand that ratio in alpha. 01:07:23.760 --> 01:07:26.225 And if I want to do it to second order, 01:07:26.225 --> 01:07:27.600 I don't just keep the first time. 01:07:27.600 --> 01:07:28.558 I keep the second term. 01:07:38.220 --> 01:07:41.170 So the first term was a 1 over alpha. 01:07:41.170 --> 01:07:45.930 So we're going to keep the order alpha to the 0 term. 01:07:51.140 --> 01:07:53.840 And if we use our notation that we established before, 01:07:53.840 --> 01:07:57.050 where we call this guy here, we call the exponential 01:07:57.050 --> 01:08:09.020 of this guy u, so C of u C of mu 0, of mu w mu of mu w mu. 01:08:18.140 --> 01:08:20.644 Then we can write the solution of that guy 01:08:20.644 --> 01:08:24.830 as an exponential of an integral of d alpha gamma over beta. 01:08:28.904 --> 01:08:31.279 So some of the steps that we were doing at one-loop, just 01:08:31.279 --> 01:08:33.319 like the exponentiation, the separation, 01:08:33.319 --> 01:08:34.802 they just all go through. 01:08:34.802 --> 01:08:36.260 And the only thing we do have to do 01:08:36.260 --> 01:08:38.177 is evaluate this integral at one higher order. 01:08:43.290 --> 01:08:50.270 Let me take mu w equal to Mw and then do the integral. 01:08:50.270 --> 01:08:53.420 And what you get is a result that we can 01:08:53.420 --> 01:08:54.850 organize in the following way. 01:09:02.248 --> 01:09:04.040 Try to get my arguments in the right order. 01:09:14.640 --> 01:09:18.060 In this particular case, the next leading log solution 01:09:18.060 --> 01:09:19.770 looks as follows. 01:09:19.770 --> 01:09:23.460 Our leading log solution is obviously buried inside it. 01:09:23.460 --> 01:09:35.660 So we have this ratio of alphas, something which is a number. 01:09:35.660 --> 01:09:39.700 And then there's these extra factors 01:09:39.700 --> 01:09:41.020 that depend on this then j. 01:09:44.620 --> 01:09:48.189 And I can write the result this way 01:09:48.189 --> 01:09:51.310 where j involves all the things that are the higher order 01:09:51.310 --> 01:09:52.170 ingredients. 01:09:52.170 --> 01:09:55.480 So it involves the lowest order anomalous dimension, but now 01:09:55.480 --> 01:09:57.245 times beta 1. 01:09:57.245 --> 01:09:59.620 That's like taking the leading order anomalous dimension, 01:09:59.620 --> 01:10:02.500 but now running the coupling with the second order term 01:10:02.500 --> 01:10:03.980 as well. 01:10:03.980 --> 01:10:07.480 And then there's a term that involves the second order 01:10:07.480 --> 01:10:11.170 anomalous dimension. 01:10:11.170 --> 01:10:14.930 So it encodes that information. 01:10:14.930 --> 01:10:17.380 So this is the U. We can combine that together 01:10:17.380 --> 01:10:22.810 with our equation for the C over here, or this one. 01:10:22.810 --> 01:10:24.300 So let me keep that one. 01:10:44.610 --> 01:10:47.910 So I take this equation, multiply by that equation. 01:10:47.910 --> 01:10:49.476 That gives us C of mu. 01:10:52.194 --> 01:10:53.910 So I have to write this lone more time. 01:11:09.830 --> 01:11:12.500 And basically, I can group that together with these other terms 01:11:12.500 --> 01:11:13.870 that depend on an alpha of Mw. 01:11:27.250 --> 01:11:28.240 OK. 01:11:28.240 --> 01:11:31.330 So j is the anomalous dimension piece. 01:11:31.330 --> 01:11:33.010 A and B are the matching. 01:11:33.010 --> 01:11:35.680 A minus B is the matching piece. 01:11:35.680 --> 01:11:38.710 And I can write the result this way. 01:11:38.710 --> 01:11:43.590 So this is next leading order matching, 01:11:43.590 --> 01:11:48.470 which is A minus B and next leading 01:11:48.470 --> 01:12:00.340 log running to get the full next leading log result. 01:12:00.340 --> 01:12:02.530 So this is the kind of structure that you could get. 01:12:02.530 --> 01:12:03.970 That's what renormalization group 01:12:03.970 --> 01:12:05.803 improved perturbation theory looks like when 01:12:05.803 --> 01:12:07.180 you go to higher orders. 01:12:07.180 --> 01:12:08.758 You basically have logs. 01:12:08.758 --> 01:12:10.300 But then the higher order terms, when 01:12:10.300 --> 01:12:12.940 you expand out this integral, are just giving you 01:12:12.940 --> 01:12:14.853 polynomials in alpha. 01:12:14.853 --> 01:12:16.270 So when you integrate polynomials, 01:12:16.270 --> 01:12:17.390 you get back polynomials. 01:12:17.390 --> 01:12:19.070 So if you integrate 1, you get alpha. 01:12:19.070 --> 01:12:23.665 If you integrate alpha squared, you get alpha cubed, et cetera. 01:12:23.665 --> 01:12:25.040 So you just get back polynomials. 01:12:25.040 --> 01:12:28.420 And that's why you can write it this way. 01:12:28.420 --> 01:12:30.520 What are the terms in this result 01:12:30.520 --> 01:12:31.966 that are scheme dependent? 01:12:36.430 --> 01:12:40.630 I claim that beta 1 gamma 1-- 01:12:44.454 --> 01:12:48.000 oh, why did I-- not beta 1, B1. 01:12:52.440 --> 01:12:59.850 B1 gamma 1 J, C, O, these are all scheme dependent. 01:12:59.850 --> 01:13:01.920 They depend on what renormalization scheme 01:13:01.920 --> 01:13:03.690 I use to define my effective theory. 01:13:09.545 --> 01:13:10.920 And then there's a list of things 01:13:10.920 --> 01:13:13.230 that are scheme independent. 01:13:13.230 --> 01:13:16.890 So beta 0 and beta 1 are scheme independent. 01:13:16.890 --> 01:13:20.153 I told you that gamma 0 is scheme independent. 01:13:20.153 --> 01:13:21.570 You could think of that like, when 01:13:21.570 --> 01:13:24.237 you do the one-loop calculation, the ultraviolet divergences are 01:13:24.237 --> 01:13:25.403 always going to be the same. 01:13:25.403 --> 01:13:26.370 You get 1 over epsilon. 01:13:26.370 --> 01:13:28.662 And it's only the constant that depends on your scheme. 01:13:31.980 --> 01:13:34.200 A1 is scheme independent. 01:13:34.200 --> 01:13:36.425 That's because A1 was the full theory calculation. 01:13:36.425 --> 01:13:37.800 So how could it possibly know how 01:13:37.800 --> 01:13:40.930 we define the effective theory? 01:13:40.930 --> 01:13:43.800 So that's scheme independent. 01:13:43.800 --> 01:13:51.417 And a non-trivial one is that B1 plus J is scheme independent. 01:13:51.417 --> 01:13:54.000 So there's scheme dependence in B1 and scheme dependence in J, 01:13:54.000 --> 01:13:56.220 but it cancels in exactly the combination that's 01:13:56.220 --> 01:14:00.220 showing up in this result. 01:14:00.220 --> 01:14:07.300 And as I mentioned, C times O is scheme independent 01:14:07.300 --> 01:14:09.798 because that's related to observables. 01:14:16.150 --> 01:14:18.340 So I have a little proof of that in my notes, 01:14:18.340 --> 01:14:21.490 which, because of time, I'm going to skip. 01:14:21.490 --> 01:14:24.493 But I encourage you, when I post my notes through the website, 01:14:24.493 --> 01:14:26.410 that you take a look at where that comes from. 01:14:30.123 --> 01:14:31.540 So the only non-trivial one really 01:14:31.540 --> 01:14:38.012 is this B1 plus J being scheme independent, OK? 01:14:38.012 --> 01:14:39.970 I have a little proof of that in my notes here. 01:14:44.490 --> 01:14:47.190 OK, so let's go back to the equation 01:14:47.190 --> 01:14:50.520 at the top in the middle there and see 01:14:50.520 --> 01:14:55.810 what conclusions we can draw once we believe this. 01:14:55.810 --> 01:14:58.650 So if B1 plus J is scheme independent, then 01:14:58.650 --> 01:15:03.180 this thing that's showing up in that term, A1 minus B minus J, 01:15:03.180 --> 01:15:07.237 is scheme independent, as A was. 01:15:07.237 --> 01:15:08.570 It was just a full theory thing. 01:15:12.982 --> 01:15:14.940 And there's a cancellation of scheme dependence 01:15:14.940 --> 01:15:19.170 between the one-loop anomalous dimension. 01:15:19.170 --> 01:15:25.135 There's a cancellation here between 01:15:25.135 --> 01:15:26.760 the two-loop anomalous dimension, which 01:15:26.760 --> 01:15:30.570 we called gamma 1, and the B1. 01:15:30.570 --> 01:15:32.610 That's where the scheme dependence cancels. 01:15:32.610 --> 01:15:34.797 So the scheme you pick, you have to be consistent. 01:15:34.797 --> 01:15:35.880 You have to keep using it. 01:15:35.880 --> 01:15:38.005 If you do a matching calculation or if someone else 01:15:38.005 --> 01:15:40.002 did a matching calculation, you want to use it. 01:15:40.002 --> 01:15:42.210 You better figure out what scheme they're working in. 01:15:42.210 --> 01:15:43.800 Because if you start working in a different scheme, 01:15:43.800 --> 01:15:44.967 you're just making mistakes. 01:15:48.220 --> 01:15:50.700 So this is the statement that the matching is scheme 01:15:50.700 --> 01:15:54.580 dependent, the anomalous dimension scheme dependent, 01:15:54.580 --> 01:15:56.490 but there's a cancellation between those two. 01:16:07.540 --> 01:16:09.787 If we look at the gamma 0 over beta 0 term, 01:16:09.787 --> 01:16:10.870 that's scheme independent. 01:16:10.870 --> 01:16:11.590 So that's good. 01:16:11.590 --> 01:16:16.215 If we look over here, J was not scheme dependent. 01:16:16.215 --> 01:16:19.168 J is scheme dependent. 01:16:19.168 --> 01:16:20.710 So we still have to worry about that. 01:16:35.181 --> 01:16:39.170 So leading log result was scheme independent, 01:16:39.170 --> 01:16:46.080 but we still have scheme dependence 01:16:46.080 --> 01:16:57.310 of this factor 1 plus alpha of mu J over 4 pi in our C of mu. 01:16:57.310 --> 01:17:01.920 And the thing that cancels that scheme dependence 01:17:01.920 --> 01:17:04.150 is the fact that the Wilson coefficient alone 01:17:04.150 --> 01:17:05.890 is not a physical observable. 01:17:05.890 --> 01:17:08.985 It's really the Wilson coefficient times the operator. 01:17:08.985 --> 01:17:10.360 And so there is scheme dependence 01:17:10.360 --> 01:17:12.834 in the matrix element of the operator. 01:17:28.650 --> 01:17:31.430 So a matrix element of the operator at the scale mu 01:17:31.430 --> 01:17:34.610 is scheme dependent. 01:17:34.610 --> 01:17:37.720 And this is at the lower end of our integration. 01:17:42.220 --> 01:17:43.775 So this is the final matrix element, 01:17:43.775 --> 01:17:45.400 like the matrix element at the B scale. 01:17:45.400 --> 01:17:46.730 That's a scheme dependent thing. 01:17:46.730 --> 01:17:48.480 So if you think of these things as numbers 01:17:48.480 --> 01:17:51.040 that you want to determine from data, one way of thinking 01:17:51.040 --> 01:17:52.630 about it, those numbers are going 01:17:52.630 --> 01:17:54.490 to depend on what scheme you're using. 01:17:54.490 --> 01:17:56.650 If you extract some numbers in one scheme 01:17:56.650 --> 01:17:58.400 and your friend does it in another scheme, 01:17:58.400 --> 01:18:00.560 you could get totally different numbers. 01:18:00.560 --> 01:18:02.080 So you have to know what scheme you're working in. 01:18:02.080 --> 01:18:03.538 And you have to combine it together 01:18:03.538 --> 01:18:06.247 with the Wilson coefficient in the same scheme. 01:18:06.247 --> 01:18:08.080 If you take some numbers from the literature 01:18:08.080 --> 01:18:09.788 and you don't know what scheme they're in 01:18:09.788 --> 01:18:11.450 and you're working at next leading log, 01:18:11.450 --> 01:18:13.280 you have a problem. 01:18:13.280 --> 01:18:14.950 You got to know what the scheme is 01:18:14.950 --> 01:18:17.995 because you have to work in the same scheme consistently. 01:18:17.995 --> 01:18:19.120 And that's the lesson here. 01:18:25.960 --> 01:18:28.270 If you really want to do this whole program that I've 01:18:28.270 --> 01:18:30.970 talked about, which is done in this 250 page review article-- 01:18:30.970 --> 01:18:33.790 and I'm not asking you to read that. 01:18:33.790 --> 01:18:36.160 If you really want to do this whole program, 01:18:36.160 --> 01:18:37.480 there are some subtleties. 01:18:37.480 --> 01:18:43.270 And I should at least mention them to you since maybe you'll 01:18:43.270 --> 01:18:44.395 encounter the word someday. 01:18:57.270 --> 01:19:01.050 So we've sketched the physics and the basic stuff that 01:19:01.050 --> 01:19:02.590 would be involved in the analysis, 01:19:02.590 --> 01:19:04.715 but we haven't written down the full operator basis 01:19:04.715 --> 01:19:06.750 with the full set of mixing and dozens 01:19:06.750 --> 01:19:08.820 and dozens of diagrams, which people have done. 01:19:12.450 --> 01:19:15.730 Mostly what you should be thinking of this is as a user. 01:19:15.730 --> 01:19:17.520 So I'm teaching you the things that you 01:19:17.520 --> 01:19:19.505 need to be able to use results like that. 01:19:19.505 --> 01:19:20.880 In an effective theory, if you're 01:19:20.880 --> 01:19:22.422 using a higher order result, you have 01:19:22.422 --> 01:19:25.060 to worry about scheme dependent. 01:19:25.060 --> 01:19:26.370 So what are the subtleties? 01:19:26.370 --> 01:19:28.530 Well, one of them is that there's gamma 5s. 01:19:31.265 --> 01:19:34.200 This theory is chiral. 01:19:34.200 --> 01:19:37.076 And gamma 5 is inherently four-dimensional. 01:19:44.700 --> 01:19:46.680 And you have to worry about that. 01:19:46.680 --> 01:19:48.930 And you have to create that carefully in dim reg. 01:19:52.120 --> 01:19:54.550 And when people originally did these calculations, 01:19:54.550 --> 01:19:57.880 that caused some confusion. 01:19:57.880 --> 01:20:00.820 Be careful enough. 01:20:00.820 --> 01:20:03.850 Obviously, dim reg is a powerful way of doing the calculation, 01:20:03.850 --> 01:20:06.513 but you do have to be careful about gamma 5. 01:20:06.513 --> 01:20:07.930 And there's another thing you have 01:20:07.930 --> 01:20:10.553 to be careful about in dim reg. 01:20:10.553 --> 01:20:12.970 And that's something that are called evanescent operators. 01:20:18.513 --> 01:20:20.430 You see, part of our arguments, and originally 01:20:20.430 --> 01:20:22.430 when we were writing down the basis of operators 01:20:22.430 --> 01:20:24.210 for our calculation, were actually 01:20:24.210 --> 01:20:25.881 inherently four-dimensional. 01:20:29.460 --> 01:20:32.280 When we wrote down the operators, 01:20:32.280 --> 01:20:40.920 we said we effectively used that these Dirac structures, which 01:20:40.920 --> 01:20:47.430 are 16 of them, we used completeness over those 16. 01:20:47.430 --> 01:20:51.090 And the problem is that, in d dimensions, 01:20:51.090 --> 01:20:53.220 that's not a complete set. 01:21:05.860 --> 01:21:09.130 And any opinions that are outside that 01:21:09.130 --> 01:21:10.900 set that are additional operators that you 01:21:10.900 --> 01:21:16.434 need in d dimensions are called evanescent operators. 01:21:21.080 --> 01:21:23.120 So they involve Dirac structures that 01:21:23.120 --> 01:21:33.330 vanish as epsilon goes to 0, but are technically needed 01:21:33.330 --> 01:21:36.150 to get some calculations right. 01:21:36.150 --> 01:21:37.680 OK, so those are two subtle things 01:21:37.680 --> 01:21:41.610 to be aware of in the full calculation. 01:21:41.610 --> 01:21:43.448 And I think we'll stop there for today. 01:21:43.448 --> 01:21:45.240 And we'll do something different next time. 01:21:47.860 --> 01:21:51.460 So homework is due next Tuesday. 01:21:51.460 --> 01:21:54.160 And as I said in my original handout, 01:21:54.160 --> 01:21:56.230 you should talk to each other about the homework. 01:21:56.230 --> 01:21:58.680 That's how you learn.