WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high-quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.270 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.270 --> 00:00:18.276 at ocw.mit.edu. 00:00:22.192 --> 00:00:23.650 IAIN STEWART: So last time, we were 00:00:23.650 --> 00:00:26.260 talking about regularization and power counting. 00:00:26.260 --> 00:00:27.928 And in particular, we were-- 00:00:27.928 --> 00:00:29.470 we argued that there some things that 00:00:29.470 --> 00:00:31.852 are nice about dimensional regularization when you're 00:00:31.852 --> 00:00:34.060 doing dimensional power counting, which is what we've 00:00:34.060 --> 00:00:35.800 been discussing so far, power counting 00:00:35.800 --> 00:00:38.470 and ratios of mass scales. 00:00:38.470 --> 00:00:40.810 So I want to continue along that theme today, 00:00:40.810 --> 00:00:43.600 and in particular move towards really talking 00:00:43.600 --> 00:00:46.590 about matching calculations in the context of mass 00:00:46.590 --> 00:00:48.560 of particles. 00:00:48.560 --> 00:00:50.560 So just continuing with this discussion 00:00:50.560 --> 00:00:53.470 of dimensional regularization, we also have to pick a scheme. 00:00:53.470 --> 00:00:56.020 And the scheme that is a nice scheme 00:00:56.020 --> 00:00:59.275 for dimensional regularization is this MS bar scheme. 00:00:59.275 --> 00:01:01.150 There are some things that are good about it. 00:01:04.150 --> 00:01:06.580 Well, it's good because it works under the context 00:01:06.580 --> 00:01:09.295 of dimensional regularization and preserves-- 00:01:09.295 --> 00:01:11.170 and it doesn't mess up any of the nice things 00:01:11.170 --> 00:01:12.130 about that regulator. 00:01:12.130 --> 00:01:14.770 So it preserves symmetries-- 00:01:14.770 --> 00:01:16.390 gauge, symmetry, Lorentz symmetry, 00:01:16.390 --> 00:01:18.010 the things we mentioned last time. 00:01:20.650 --> 00:01:24.070 In terms of doing calculations, it 00:01:24.070 --> 00:01:30.500 makes them technically easy, or easier. 00:01:30.500 --> 00:01:31.618 So a lot of-- 00:01:31.618 --> 00:01:33.160 if you look at the literature and you 00:01:33.160 --> 00:01:34.840 look at multi-loop calculations, they're 00:01:34.840 --> 00:01:36.393 all done on the MS bar scheme. 00:01:36.393 --> 00:01:38.560 And there's a reason for that, because you've only-- 00:01:38.560 --> 00:01:40.300 you haven't introduced these extra scales 00:01:40.300 --> 00:01:43.138 that would make your loop calculations more complicated. 00:01:46.420 --> 00:01:53.730 And finally, in the context of our discussion 00:01:53.730 --> 00:01:57.580 of effective field theory, it's nice because it often 00:01:57.580 --> 00:02:00.040 gives what I would call manifest power counting, 00:02:00.040 --> 00:02:02.740 where we can power count both the regulator diagrams, 00:02:02.740 --> 00:02:03.900 the renormalized diagrams. 00:02:03.900 --> 00:02:05.650 We don't have to worry about whether we've 00:02:05.650 --> 00:02:07.840 added the counter terms or not. 00:02:07.840 --> 00:02:10.887 We can just do power counting. 00:02:10.887 --> 00:02:12.970 So if there's some good, there should be some bad. 00:02:12.970 --> 00:02:13.810 So what's the bad? 00:02:17.830 --> 00:02:18.970 Nothing is free in life. 00:02:23.330 --> 00:02:24.707 So one thing that we kind of lose 00:02:24.707 --> 00:02:27.040 with dimensional regularization is the physical picture. 00:02:30.520 --> 00:02:33.850 When we had this cutoff that we introduced, 00:02:33.850 --> 00:02:35.920 or if we have a Wilsonian picture, 00:02:35.920 --> 00:02:38.500 it's very clear what you're doing as far 00:02:38.500 --> 00:02:42.940 as removing degrees of freedom. 00:02:42.940 --> 00:02:45.640 And in MS bar, that's a little less clear. 00:02:48.440 --> 00:02:51.220 So if it's just less clear but it's still fine, 00:02:51.220 --> 00:02:53.140 that would be-- 00:02:53.140 --> 00:02:54.910 that would just be a technical aspect 00:02:54.910 --> 00:02:57.110 and we would just pretty much ignore it. 00:02:57.110 --> 00:02:57.790 So what's one-- 00:02:57.790 --> 00:03:00.100 I'll give you one example of something 00:03:00.100 --> 00:03:02.560 in which the picture being less clear 00:03:02.560 --> 00:03:04.630 could actually mislead you. 00:03:04.630 --> 00:03:06.670 So if you calculate some renormalized quantities 00:03:06.670 --> 00:03:09.640 in MS bar, you could have some a-priori knowledge 00:03:09.640 --> 00:03:11.890 that these renormalized quantities should be positive. 00:03:11.890 --> 00:03:13.682 Maybe they're supposed to be kinetic energy 00:03:13.682 --> 00:03:16.060 or some operator, higher-dimension operator, that 00:03:16.060 --> 00:03:17.890 gives a kind of higher-order kinetic energy 00:03:17.890 --> 00:03:20.140 term, some kind of physical intuition 00:03:20.140 --> 00:03:21.950 that it should be positive. 00:03:21.950 --> 00:03:23.920 Well, that might not be true in MS bar 00:03:23.920 --> 00:03:27.530 because MS bar loses this physical picture. 00:03:27.530 --> 00:03:29.350 You're removing just the 1 over epsilons. 00:03:29.350 --> 00:03:32.790 You're not really having any control over the constants, 00:03:32.790 --> 00:03:35.380 or the constants are kind of predefined. 00:03:35.380 --> 00:03:37.610 If those constants are kind of too negative, 00:03:37.610 --> 00:03:40.140 then you can end up with renormalized matrix elements 00:03:40.140 --> 00:03:41.560 in MS bar that could be negative, 00:03:41.560 --> 00:03:44.470 your physical intuition tells you they should be positive. 00:03:52.102 --> 00:03:53.560 So that's something to be aware of. 00:04:00.790 --> 00:04:03.310 Where a physical picture could guide you and tell you 00:04:03.310 --> 00:04:06.310 what to expect, but you might lose that by using MS bar. 00:04:14.320 --> 00:04:16.959 There's another thing that's kind of technical 00:04:16.959 --> 00:04:20.950 but is worth knowing. 00:04:20.950 --> 00:04:23.230 And we will talk about it a little bit more later. 00:04:25.960 --> 00:04:28.330 You could ask the question, is the Wilsonian picture 00:04:28.330 --> 00:04:29.800 and the Wilsonian scale separation 00:04:29.800 --> 00:04:31.750 really equivalent to the scale separation 00:04:31.750 --> 00:04:34.300 that you do in dimensional regularization? 00:04:34.300 --> 00:04:37.930 And the answer is, almost but not quite. 00:04:41.140 --> 00:04:43.000 It certainly is for all the log terms. 00:04:43.000 --> 00:04:44.932 And I mentioned last time that the log terms 00:04:44.932 --> 00:04:47.140 that we saw in the two ways of doing the calculation, 00:04:47.140 --> 00:04:50.170 you could just see a direct correspondence between them. 00:04:50.170 --> 00:04:53.410 But there's a leftover residual effect 00:04:53.410 --> 00:04:58.450 called renormalons that does show up in the MS bar scheme 00:04:58.450 --> 00:05:01.900 and is related to power divergences. 00:05:01.900 --> 00:05:06.370 And so this effect is carefully hidden. 00:05:06.370 --> 00:05:08.260 It's actually hidden in the asymptotics 00:05:08.260 --> 00:05:11.232 of the expansion in alpha s. 00:05:11.232 --> 00:05:12.940 And it goes under the rubric of something 00:05:12.940 --> 00:05:14.470 that people call renormalons. 00:05:24.070 --> 00:05:25.900 So there is a leftover physical effect 00:05:25.900 --> 00:05:29.710 from kind of having the freedom to drop all the power 00:05:29.710 --> 00:05:31.960 divergent terms. 00:05:31.960 --> 00:05:35.110 But it carefully hides itself in a strange place 00:05:35.110 --> 00:05:37.108 in the gauge theory. 00:05:37.108 --> 00:05:38.650 And we'll talk more about this later. 00:05:42.335 --> 00:05:44.710 It's not so important if you're working to order alpha s. 00:05:44.710 --> 00:05:48.340 But if you start working toward alpha squared or higher, then-- 00:05:48.340 --> 00:05:50.950 or if you start going to higher orders in perturbation theory, 00:05:50.950 --> 00:05:53.490 then this could come in and be important. 00:05:53.490 --> 00:05:55.660 And it does come in in QCD at fairly low orders. 00:05:55.660 --> 00:05:58.720 At order alpha squared, it can cause numerical effects 00:05:58.720 --> 00:06:00.810 if you ignore this. 00:06:00.810 --> 00:06:03.340 And there are ways of taking it into account without losing 00:06:03.340 --> 00:06:04.243 the nice things here. 00:06:04.243 --> 00:06:05.660 So we'll talk about that later on. 00:06:08.870 --> 00:06:10.270 And then there's a final thing-- 00:06:10.270 --> 00:06:13.840 we had three good points, we should have three bad points-- 00:06:13.840 --> 00:06:15.590 the final thing we'll deal with right now. 00:06:15.590 --> 00:06:17.470 And that is the fact that MS bar does not 00:06:17.470 --> 00:06:23.050 satisfy something that's a theorem called 00:06:23.050 --> 00:06:24.130 the decoupling theorem. 00:06:34.623 --> 00:06:36.040 So what is the decoupling theorem? 00:06:44.300 --> 00:06:47.820 So this goes back to Appelquist and Carrazone. 00:06:47.820 --> 00:06:49.258 And it says the following. 00:06:52.330 --> 00:06:54.520 So you're thinking about an effective field theory, 00:06:54.520 --> 00:06:56.978 you're thinking about deriving a low-energy effect of field 00:06:56.978 --> 00:07:00.295 theory by integrating out some mass of particle. 00:07:02.910 --> 00:07:06.700 And the decoupling theorem says that if the remaining 00:07:06.700 --> 00:07:15.490 low-energy theory is renormalizable, 00:07:15.490 --> 00:07:36.280 and we use a physical renormalization scheme, 00:07:36.280 --> 00:07:38.890 then you kind of get what expect, 00:07:38.890 --> 00:07:45.550 that all the effects of the heavy particles 00:07:45.550 --> 00:08:07.374 turn up in couplings or effects, they're suppressed. 00:08:11.090 --> 00:08:13.118 So this seems very physical from what 00:08:13.118 --> 00:08:15.410 we've described about what effective field theories do. 00:08:19.330 --> 00:08:20.580 So in that sense, it's-- 00:08:20.580 --> 00:08:23.480 I'm not going to try to prove it to you or anything, 00:08:23.480 --> 00:08:25.373 but there is this caveat that you 00:08:25.373 --> 00:08:27.290 need to use a physical renormalization scheme, 00:08:27.290 --> 00:08:28.490 of which MS bar is not one. 00:08:32.140 --> 00:08:34.120 So we decided that we liked MS bar, 00:08:34.120 --> 00:08:36.190 but then we found some problems. 00:08:36.190 --> 00:08:38.960 And this problem here, this last problem, we're 00:08:38.960 --> 00:08:40.210 going to deal with right away. 00:08:43.720 --> 00:08:45.110 So MS bar is not physical. 00:08:49.800 --> 00:08:55.130 And hence, it doesn't satisfy this theorem. 00:08:55.130 --> 00:08:57.110 It's mass-independent. 00:08:57.110 --> 00:08:58.760 And that is something that we like, 00:08:58.760 --> 00:09:01.010 but it also is causing exactly this problem. 00:09:11.280 --> 00:09:13.850 That's how it-- it's why it violates the decoupling 00:09:13.850 --> 00:09:16.370 theorem. 00:09:16.370 --> 00:09:17.770 Because it's mass-independent, it 00:09:17.770 --> 00:09:19.340 doesn't know enough about the mass 00:09:19.340 --> 00:09:21.802 to know that the effect of massive particles 00:09:21.802 --> 00:09:22.760 should always decouple. 00:09:28.630 --> 00:09:30.220 So in MS bar, what you have to do 00:09:30.220 --> 00:09:32.290 is you have to implement that decoupling by hand. 00:10:00.580 --> 00:10:03.250 And we've gotten actually so used to this logic 00:10:03.250 --> 00:10:06.220 that it's often even not mentioned that this 00:10:06.220 --> 00:10:08.250 is something that we're doing. 00:10:08.250 --> 00:10:10.990 So go over that a bit. 00:10:14.693 --> 00:10:16.360 So I'll just go over this in the context 00:10:16.360 --> 00:10:17.770 of a particular example, which is 00:10:17.770 --> 00:10:19.170 the most common example, which is 00:10:19.170 --> 00:10:24.520 QCD with massive particles in MS bar. 00:10:28.700 --> 00:10:33.164 And I'll show you how it works. 00:10:33.164 --> 00:10:40.840 So QCD has a beta function of the renormalized coupling 00:10:40.840 --> 00:10:43.880 mu d by d mu of g of mu. 00:10:43.880 --> 00:10:45.587 And just to establish some notation, 00:10:45.587 --> 00:10:48.170 I'm going to remind you of some facts that you've seen before. 00:11:01.490 --> 00:11:05.610 So this is the beta function at lowest order. 00:11:05.610 --> 00:11:09.600 I'll call this factor that depends on the group 00:11:09.600 --> 00:11:18.550 that you're dealing with and how many light fermions you have, 00:11:18.550 --> 00:11:19.740 I'll call it b0. 00:11:23.590 --> 00:11:24.660 And this is less than 0. 00:11:24.660 --> 00:11:28.200 And this is only the first-order term in a series, 00:11:28.200 --> 00:11:31.630 but it'll be enough for our discussion right now. 00:11:31.630 --> 00:11:35.280 So this is g to the fifth terms, et cetera. 00:11:35.280 --> 00:11:36.930 That's the beta function for g. 00:11:36.930 --> 00:11:39.270 We can also think about using-- 00:11:39.270 --> 00:11:45.720 since g likes to come squared, we can switch to alpha. 00:11:45.720 --> 00:11:48.090 QCD is asymptotically free. 00:11:51.970 --> 00:12:00.845 When we solve this equation for alpha, it looks as follows. 00:12:12.120 --> 00:12:14.715 And the coupling constant decreases 00:12:14.715 --> 00:12:15.840 as you go to higher energy. 00:12:21.640 --> 00:12:28.300 So this is alpha at md, and this is alpha at mW. 00:12:28.300 --> 00:12:30.330 Alpha at mW is less than alpha at md. 00:12:32.858 --> 00:12:34.275 This is the lowest-order solution. 00:12:59.020 --> 00:13:02.520 So just a little bit more in the way of setting the stage, 00:13:02.520 --> 00:13:05.790 you can also associate to this solution here an intrinsic mass 00:13:05.790 --> 00:13:09.360 scale dimensional transmutation. 00:13:15.090 --> 00:13:20.010 If you form the combination mu exponential 00:13:20.010 --> 00:13:24.420 of minus 2 pi over beta 0-- 00:13:24.420 --> 00:13:34.740 b0 alpha of mu, and you use this equation over here, 00:13:34.740 --> 00:13:36.990 then what you'll find is that this combination is also 00:13:36.990 --> 00:13:40.860 equal to mu 0, same thing alpha of mu 0. 00:13:40.860 --> 00:13:43.700 So it's independent of what choice of scale you use, mu 00:13:43.700 --> 00:13:44.770 or mu 0. 00:13:44.770 --> 00:13:46.520 Therefore, you define it to be a constant, 00:13:46.520 --> 00:13:48.145 and that constant is called lambda QCD. 00:13:51.300 --> 00:13:55.500 Once you do that, you can also just take this equation 00:13:55.500 --> 00:14:03.260 and write it like this. 00:14:03.260 --> 00:14:06.147 Write that, rewrite that solution like this. 00:14:06.147 --> 00:14:07.980 So it's another way of specifying a boundary 00:14:07.980 --> 00:14:10.230 condition, if you like, for the differential equation. 00:14:10.230 --> 00:14:13.590 One way is to pick a value of the coupling somewhere, 00:14:13.590 --> 00:14:15.690 another way is to fix this constant. 00:14:23.500 --> 00:14:26.850 So this guy is independent of mu, as I said. 00:14:26.850 --> 00:14:29.503 And it's the scale where QCD becomes nonperturbative. 00:14:39.948 --> 00:14:41.490 And the reason I have to mention this 00:14:41.490 --> 00:14:42.990 is because we're going to be talking 00:14:42.990 --> 00:14:44.610 about anomalous dimensions, and I 00:14:44.610 --> 00:14:47.977 have to tell you when the things that I write down are valid. 00:14:47.977 --> 00:14:49.560 And it's all going to be valid as long 00:14:49.560 --> 00:14:50.770 as we're not near this scale. 00:14:50.770 --> 00:14:52.145 Because if we're near this scale, 00:14:52.145 --> 00:14:53.700 the coupling gets too large for us 00:14:53.700 --> 00:14:56.220 to do the perturbation theory that we're doing when we 00:14:56.220 --> 00:14:59.100 calculate anomalous dimensions. 00:14:59.100 --> 00:15:02.980 So you can ask the question, what does this thing depend on? 00:15:02.980 --> 00:15:05.170 And if we look at the right-hand side here, 00:15:05.170 --> 00:15:07.140 we can already see some things it depends on, 00:15:07.140 --> 00:15:09.560 because it depends on b0. 00:15:09.560 --> 00:15:12.960 And if we looked back at what b0 is, b0 depended on the fact 00:15:12.960 --> 00:15:15.175 that we were at su3 if it's QCD. 00:15:15.175 --> 00:15:17.175 It also depended on the number of light flavors. 00:15:24.670 --> 00:15:30.180 So this scale that you fix here, well, it 00:15:30.180 --> 00:15:33.710 depends on the order of the loop expansion. 00:15:33.710 --> 00:15:37.830 We wrote down a formula that was valid at first order, 00:15:37.830 --> 00:15:40.838 but we could extend that formula at higher orders as well. 00:15:40.838 --> 00:15:42.630 And the formula would change, and the value 00:15:42.630 --> 00:15:44.338 would change if we went to higher orders. 00:15:46.830 --> 00:15:50.060 Depends on the number of light flavors. 00:15:50.060 --> 00:15:52.290 And it actually also starts to depend on the scheme 00:15:52.290 --> 00:15:53.460 but only at two loops. 00:15:57.180 --> 00:15:59.580 So it'll depend on MS bar versus MS, 00:15:59.580 --> 00:16:03.960 for example, but only beyond two loops. 00:16:03.960 --> 00:16:09.740 Now, the issue with this is that a priori, nothing tells us 00:16:09.740 --> 00:16:11.720 whether if you have a top quark or an up 00:16:11.720 --> 00:16:14.450 quark or a bottom quark. 00:16:14.450 --> 00:16:16.610 In MS bar, nothing a prior tells us 00:16:16.610 --> 00:16:20.030 what to do with these formulas, what to include in the b0. 00:16:20.030 --> 00:16:22.490 Just seems like we should include everything that exists, 00:16:22.490 --> 00:16:27.701 all the known light fermions that couple to the gauge field. 00:16:27.701 --> 00:16:30.050 But there might be some we don't know about. 00:16:30.050 --> 00:16:31.830 Should we include those, too? 00:16:31.830 --> 00:16:35.270 MS bar, with the logic we've presented so far, 00:16:35.270 --> 00:16:37.770 doesn't tell us what to include there in that nf. 00:16:45.540 --> 00:16:47.030 So let me phrase it this way. 00:16:51.292 --> 00:16:52.750 But the top quark and the up quark, 00:16:52.750 --> 00:16:56.020 which have very different masses, both contribute to b0. 00:17:00.870 --> 00:17:03.270 And it seems like they do that for any mu. 00:17:07.180 --> 00:17:09.680 Even though-- even if I'm at a very low-energy scale 00:17:09.680 --> 00:17:11.110 and the top is very heavy, MS bar 00:17:11.110 --> 00:17:13.631 is not smart enough to decouple the top quark. 00:17:13.631 --> 00:17:15.339 If you were to work in a physical scheme, 00:17:15.339 --> 00:17:17.579 then this decoupling theorem guarantees, actually, 00:17:17.579 --> 00:17:19.720 that the top quark would decouple, 00:17:19.720 --> 00:17:22.990 and it would drop out of the beta function in that scheme. 00:17:22.990 --> 00:17:25.099 But in this MS bar scheme, it doesn't happen. 00:17:28.530 --> 00:17:31.070 So the solution to this is to build back 00:17:31.070 --> 00:17:33.290 in the thing that happened in the physical schemes 00:17:33.290 --> 00:17:35.740 into our MS bar scheme. 00:17:35.740 --> 00:17:38.480 And we do that by implementing the decoupling by hand. 00:17:50.450 --> 00:17:53.330 And we just say that when we get to a mass threshold, 00:17:53.330 --> 00:17:56.480 we're going to integrate out the heavy fermion in this case. 00:18:06.833 --> 00:18:09.590 So at some scale of order of the mass of that fermion, 00:18:09.590 --> 00:18:13.040 we're going to integrate out the heavy fermion. 00:18:13.040 --> 00:18:15.350 And that's an example of during matching, actually. 00:18:20.030 --> 00:18:21.950 You're moving from one theory to another, 00:18:21.950 --> 00:18:23.870 because you're changing the field content. 00:18:23.870 --> 00:18:27.432 You're removing some fields, and that's 00:18:27.432 --> 00:18:28.640 an example of doing matching. 00:18:31.400 --> 00:18:36.050 So what this means is that we'll define different b0's 00:18:36.050 --> 00:18:37.430 depending on what scale we're at. 00:18:43.410 --> 00:18:47.380 So nf would be 6 if we're at scales above the top quark. 00:18:55.820 --> 00:18:58.490 But then we drop that nf down to 5 00:18:58.490 --> 00:19:05.360 once we're below the top quark mass, et cetera. 00:19:10.670 --> 00:19:12.920 So it's a discrete jump in what the b0 is. 00:19:17.525 --> 00:19:18.900 So in some sense, what that means 00:19:18.900 --> 00:19:22.680 is that this kind of matching is forced upon you in order 00:19:22.680 --> 00:19:25.230 to preserve physics in MS bar. 00:19:25.230 --> 00:19:27.930 If you were to do some other scheme like a physical scheme, 00:19:27.930 --> 00:19:30.180 it's not to say that you couldn't take the same logic. 00:19:33.240 --> 00:19:35.790 You could do matching to integrate our heavy particles 00:19:35.790 --> 00:19:37.240 in those schemes as well. 00:19:37.240 --> 00:19:40.470 And in fact, some sometimes people have. 00:19:40.470 --> 00:19:42.220 But in MS bar It's really forced upon you. 00:19:42.220 --> 00:19:44.260 There's no way out, because you otherwise 00:19:44.260 --> 00:19:46.300 wouldn't get the physics right. 00:19:48.725 --> 00:19:49.850 So we've got to be careful. 00:19:49.850 --> 00:19:52.017 If you gain something, sometimes you lose something. 00:19:52.017 --> 00:19:53.862 And if it's physics that you're losing, 00:19:53.862 --> 00:19:55.070 you have to build it back in. 00:19:59.050 --> 00:19:59.550 OK. 00:19:59.550 --> 00:20:02.165 So how do we actually do this matching? 00:20:04.670 --> 00:20:07.790 Well, we use what are called matching conditions. 00:20:20.790 --> 00:20:22.553 So what is a matching condition? 00:20:25.940 --> 00:20:28.370 At some scale which I'll call mu m-- 00:20:28.370 --> 00:20:30.290 that could be equal to m, but it could also 00:20:30.290 --> 00:20:35.180 be equal to twice m or 1/2 m, some scale that's of order m-- 00:20:35.180 --> 00:20:37.370 we're going to demand something about the theory 00:20:37.370 --> 00:20:38.210 above and below. 00:20:43.040 --> 00:20:47.660 And that is that S matrix elements in the two theories 00:20:47.660 --> 00:20:48.470 are going to agree. 00:20:53.150 --> 00:20:55.670 Of course, we should be careful to make sure 00:20:55.670 --> 00:20:58.842 that we work within the realm of things 00:20:58.842 --> 00:21:00.800 that we can calculate in the low-energy theory. 00:21:00.800 --> 00:21:03.860 So we should consider only S matrix elements 00:21:03.860 --> 00:21:05.660 with light external particles. 00:21:07.875 --> 00:21:09.500 If we're getting rid of the heavy ones, 00:21:09.500 --> 00:21:11.208 we don't want them on the external lines. 00:21:17.900 --> 00:21:22.700 So the basic idea is that we set up matching conditions which 00:21:22.700 --> 00:21:25.190 are conditions that say that S matrix elements should agree 00:21:25.190 --> 00:21:30.500 between theories 1 and 2, what we called theories 1 00:21:30.500 --> 00:21:31.270 and 2 last time. 00:21:35.790 --> 00:21:39.200 So if we do that for QCD, the example 00:21:39.200 --> 00:21:42.238 we've been talking about, and we do it at lowest order, 00:21:42.238 --> 00:21:44.030 then the conditions are actually very easy. 00:21:48.870 --> 00:21:53.210 So here's the picture. 00:21:53.210 --> 00:22:02.070 Let's imagine that the top quark is here, bottom quark is here. 00:22:02.070 --> 00:22:03.310 My picture is not to scale. 00:22:06.190 --> 00:22:08.830 Charm quark, et cetera. 00:22:17.720 --> 00:22:20.630 Then what we do when we want to say that the theory 00:22:20.630 --> 00:22:24.320 above and below, that that threshold is the same, if we 00:22:24.320 --> 00:22:27.650 just match QCD above and below, it just gives a continuity 00:22:27.650 --> 00:22:30.150 condition on alpha. 00:22:30.150 --> 00:22:33.840 So let me write it down at this scale. 00:22:33.840 --> 00:22:36.560 So here we would have alpha 6. 00:22:36.560 --> 00:22:38.570 Alpha depends on the number of flavors. 00:22:38.570 --> 00:22:40.887 Here we have alpha 5. 00:22:40.887 --> 00:22:42.470 A different definition of the coupling 00:22:42.470 --> 00:22:44.012 a different number of active flavors, 00:22:44.012 --> 00:22:45.620 a different value of b0. 00:22:51.060 --> 00:22:51.560 3. 00:22:54.417 --> 00:22:56.000 And what these arrows are representing 00:22:56.000 --> 00:22:59.600 are just kind of renormalization group evolution, 00:22:59.600 --> 00:23:02.133 which I'll just call running. 00:23:02.133 --> 00:23:03.800 And then what the lines are representing 00:23:03.800 --> 00:23:06.120 is doing some matching. 00:23:06.120 --> 00:23:10.130 So every time we reach a threshold, we do a matching, 00:23:10.130 --> 00:23:13.460 we switch the field contact, and we 00:23:13.460 --> 00:23:17.160 get a new effective field theory with a new coupling constant. 00:23:17.160 --> 00:23:21.920 So the coupling just depends on what theory content we have. 00:23:21.920 --> 00:23:27.440 And the matching condition is, say, at this b quark scale 00:23:27.440 --> 00:23:33.650 that alpha s at 5 at some scale that's of order mb 00:23:33.650 --> 00:23:40.190 is equal to alpha s at 4. 00:23:40.190 --> 00:23:44.340 That's the leading order matching condition here. 00:23:44.340 --> 00:23:48.560 And similarly, the same condition at all scales. 00:23:48.560 --> 00:23:52.620 And this is mu b, just to be clear, 00:23:52.620 --> 00:23:57.860 is something that's of order mb, could be equal to mb, 00:23:57.860 --> 00:24:01.830 could be equal to mb/2, could be twice mb. 00:24:01.830 --> 00:24:03.830 And sometimes people use these different choices 00:24:03.830 --> 00:24:06.060 to get uncertainties. 00:24:06.060 --> 00:24:06.560 OK. 00:24:06.560 --> 00:24:08.018 So it seems fairly straightforward, 00:24:08.018 --> 00:24:10.850 just continuity of the coupling. 00:24:10.850 --> 00:24:13.003 But that has some caveats. 00:24:13.003 --> 00:24:14.420 But go ahead and ask the question. 00:24:14.420 --> 00:24:15.045 AUDIENCE: Yeah. 00:24:15.045 --> 00:24:18.270 So how do you know that that's the physical observable 00:24:18.270 --> 00:24:19.740 that you care about? 00:24:19.740 --> 00:24:22.740 What if I was trying to measure the [INAUDIBLE] alpha s? 00:24:25.740 --> 00:24:27.370 IAIN STEWART: So this is the-- 00:24:27.370 --> 00:24:27.870 yeah. 00:24:27.870 --> 00:24:30.310 So if you're going-- 00:24:30.310 --> 00:24:34.218 so you have to think about it as not the derivative of alpha s, 00:24:34.218 --> 00:24:36.510 but you have to think about it as an S matrix element-- 00:24:36.510 --> 00:24:38.880 something two-to-two scattering, right? 00:24:38.880 --> 00:24:40.770 If you do two-to-two scattering, then it's 00:24:40.770 --> 00:24:43.500 just going to be proportional to alpha of mu. 00:24:43.500 --> 00:24:46.870 But the mu dependence in your leading order prediction 00:24:46.870 --> 00:24:48.180 is not really fixed. 00:24:48.180 --> 00:24:49.930 You need to go to higher order to do that. 00:24:49.930 --> 00:24:51.472 So if you're just ensuring continuity 00:24:51.472 --> 00:24:54.210 at leading order of S matrix elements, 00:24:54.210 --> 00:24:56.925 then this is all you need. 00:24:56.925 --> 00:24:58.800 And if you want to construct something that's 00:24:58.800 --> 00:25:00.570 like a derivative of alpha, you have 00:25:00.570 --> 00:25:02.940 to think of the derivatives of the S matrix. 00:25:02.940 --> 00:25:05.610 But you can't take derivatives of mu, 00:25:05.610 --> 00:25:08.470 because that's a higher-order question. 00:25:08.470 --> 00:25:10.576 So this is all you need at leading order. 00:25:10.576 --> 00:25:11.118 AUDIENCE: OK. 00:25:11.118 --> 00:25:15.997 So maybe my derivative example is [INAUDIBLE] necessarily know 00:25:15.997 --> 00:25:18.270 the alpha s, you'd get only-- 00:25:18.270 --> 00:25:19.570 IAIN STEWART: So in general-- 00:25:19.570 --> 00:25:20.070 right. 00:25:20.070 --> 00:25:21.700 AUDIENCE: --but I don't necessarily know that-- 00:25:21.700 --> 00:25:21.930 IAIN STEWART: Yeah. 00:25:21.930 --> 00:25:23.250 No, in general, you have to-- 00:25:23.250 --> 00:25:25.110 this is like one particular example. 00:25:25.110 --> 00:25:28.080 In general, you have to ensure continuity of all S matrix 00:25:28.080 --> 00:25:29.190 elements. 00:25:29.190 --> 00:25:31.275 And that will-- in this case, we'll 00:25:31.275 --> 00:25:33.150 give you conditions also and what's happening 00:25:33.150 --> 00:25:35.240 with masses and stuff. 00:25:35.240 --> 00:25:37.650 So it's all the parameters of the theory. 00:25:37.650 --> 00:25:41.350 There should be conditions for all of them. 00:25:41.350 --> 00:25:43.650 AUDIENCE: I guess these words seem more complicated 00:25:43.650 --> 00:25:47.700 than just asking for [INAUDIBLE] some different equation. 00:25:47.700 --> 00:25:49.200 Does it ever boil down to anything-- 00:25:49.200 --> 00:25:50.760 do you ever have to do anything more complicated than just 00:25:50.760 --> 00:25:51.210 like-- 00:25:51.210 --> 00:25:51.600 IAIN STEWART: I'm just-- 00:25:51.600 --> 00:25:51.990 yeah, so-- 00:25:51.990 --> 00:25:53.960 AUDIENCE: --change the beta function [INAUDIBLE] continuity 00:25:53.960 --> 00:25:54.853 and-- 00:25:54.853 --> 00:25:56.895 IAIN STEWART: It's really-- it's not complicated. 00:25:56.895 --> 00:25:58.790 It's just continuity of S matrix elements. 00:25:58.790 --> 00:26:01.380 But you have to just know that it's S matrix elements and not 00:26:01.380 --> 00:26:02.760 the Lagrangian. 00:26:02.760 --> 00:26:05.890 Because those are two different things. 00:26:05.890 --> 00:26:10.150 So let me give you an example why it's not the Lagrangian. 00:26:10.150 --> 00:26:12.090 So this kind of thing seems simple, right? 00:26:12.090 --> 00:26:14.750 Just say, well, the Lagrangian is continuous. 00:26:14.750 --> 00:26:17.040 The alpha-- the g that appears in the Lagrangian 00:26:17.040 --> 00:26:17.700 is continuous. 00:26:17.700 --> 00:26:21.000 But that's not true once you go to higher orders. 00:26:21.000 --> 00:26:23.190 That's only a leading order statement. 00:26:23.190 --> 00:26:25.830 If I write down the analog of this condition at higher 00:26:25.830 --> 00:26:27.370 orders, it looks as follows. 00:26:33.390 --> 00:26:37.170 So the coupling is actually not continuous at MC bar 00:26:37.170 --> 00:26:38.460 at higher orders. 00:26:41.420 --> 00:26:43.940 So demanding continuity of the S matrix elements 00:26:43.940 --> 00:26:46.190 does not lead to continuity of the coupling. 00:26:46.190 --> 00:26:53.450 And the matching condition at some scale looks like this. 00:27:07.250 --> 00:27:09.525 So the words are carefully crafted to be correct. 00:27:12.043 --> 00:27:13.710 And they sound a little more complicated 00:27:13.710 --> 00:27:16.500 than they need to be, because I want them to be correct, 00:27:16.500 --> 00:27:17.217 even if I want-- 00:27:17.217 --> 00:27:19.800 even if I were to do matching at higher orders in perturbation 00:27:19.800 --> 00:27:20.300 theory. 00:27:27.030 --> 00:27:28.920 And it's really at this alpha squared level 00:27:28.920 --> 00:27:33.500 that you start to see things a little more interesting. 00:27:50.260 --> 00:27:50.760 OK. 00:27:50.760 --> 00:27:52.718 So now we've gone two others beyond what I just 00:27:52.718 --> 00:27:54.980 told you before. 00:27:54.980 --> 00:27:57.590 So continuity at lowest order-- 00:27:57.590 --> 00:27:59.460 at the next order, you can retain continuity 00:27:59.460 --> 00:28:01.377 as long as you pick the particular point where 00:28:01.377 --> 00:28:02.655 mu b is equal to mb. 00:28:02.655 --> 00:28:04.530 And you'd have-- then this log would go away, 00:28:04.530 --> 00:28:06.547 and you'd still have continuity. 00:28:06.547 --> 00:28:08.880 But even that doesn't work out once you go to one higher 00:28:08.880 --> 00:28:10.297 order, there's this constant. 00:28:10.297 --> 00:28:12.630 You could get rid of the logs by picking a scale choice, 00:28:12.630 --> 00:28:14.005 but there's no scale choice which 00:28:14.005 --> 00:28:19.660 will make it continuous once you get to alpha squared 00:28:19.660 --> 00:28:22.600 in this matching condition. 00:28:22.600 --> 00:28:25.530 So this is the condition that's necessary to ensure continuity 00:28:25.530 --> 00:28:27.060 of S matrix elements once you get 00:28:27.060 --> 00:28:30.210 to that level of perturbation theory. 00:28:34.458 --> 00:28:38.050 OK, so that is why I said S matrix and not 00:28:38.050 --> 00:28:40.185 just continuity of couplings. 00:28:40.185 --> 00:28:43.102 AUDIENCE: So this condition is for 2-by-2 scattering, 00:28:43.102 --> 00:28:44.560 And then you get other conditions-- 00:28:44.560 --> 00:28:46.227 IAIN STEWART: This is the only condition 00:28:46.227 --> 00:28:49.103 you need for all the S matrix-- 00:28:49.103 --> 00:28:50.770 you can use different S matrix elements, 00:28:50.770 --> 00:28:52.940 and they'll lead to the same condition. 00:28:52.940 --> 00:28:55.220 AUDIENCE: How did you-- where does it come from? 00:28:55.220 --> 00:28:57.095 IAIN STEWART: So this, it comes from ensuring 00:28:57.095 --> 00:28:58.840 that I calculate S matrix elements, say, 00:28:58.840 --> 00:29:02.650 2-to-2 scattering, in the theory with five flavors and four 00:29:02.650 --> 00:29:04.096 flavors. 00:29:04.096 --> 00:29:05.320 AUDIENCE: At two loops? 00:29:05.320 --> 00:29:08.010 IAIN STEWART: And I demand that-- yeah, up to two loops. 00:29:08.010 --> 00:29:09.730 So where would this guy here come from? 00:29:09.730 --> 00:29:12.910 This guy here would come from the graph 00:29:12.910 --> 00:29:15.850 with an explicit b quark, which is 00:29:15.850 --> 00:29:19.823 in the theory with the b quark but not in the theory without. 00:29:19.823 --> 00:29:21.490 Once you get to this level, then there's 00:29:21.490 --> 00:29:23.200 more complicated diagrams. 00:29:23.200 --> 00:29:26.120 Just think of generalizations of this. 00:29:26.120 --> 00:29:28.630 And they involve constants as well as logs. 00:29:28.630 --> 00:29:32.170 This guy just involves a log. 00:29:32.170 --> 00:29:33.670 And really, what you're ensuring is 00:29:33.670 --> 00:29:36.100 that in the theory with the b quark, which 00:29:36.100 --> 00:29:39.700 is this five-flavor theory, you get the same S matrix 00:29:39.700 --> 00:29:42.250 elements as in the theory with four flavors. 00:29:42.250 --> 00:29:46.253 Since the diagrams differ in those two theories, 00:29:46.253 --> 00:29:47.920 they're kind of the same at lowest order 00:29:47.920 --> 00:29:49.180 because you're just doing two graphs. 00:29:49.180 --> 00:29:51.263 But once you have the b quark and it can go around 00:29:51.263 --> 00:29:55.840 in the loop, then they differ, and the conditions 00:29:55.840 --> 00:29:57.285 become more complicated. 00:30:00.523 --> 00:30:01.940 So any other questions about that? 00:30:09.580 --> 00:30:10.230 OK. 00:30:10.230 --> 00:30:12.930 So one other thing that we see from this 00:30:12.930 --> 00:30:14.550 is related to these logarithms. 00:30:20.180 --> 00:30:21.980 We also see from these conditions 00:30:21.980 --> 00:30:26.060 here that there is going to be no large logarithms as 00:30:26.060 --> 00:30:33.440 long as we pick the scale where we do this matching to be 00:30:33.440 --> 00:30:34.100 of order mb. 00:30:43.690 --> 00:30:47.440 So you don't want to pick mu to be the gut scale 00:30:47.440 --> 00:30:52.670 or something you want to pick it to be some scale so mu 00:30:52.670 --> 00:30:56.510 is equal to mu b, which is of order mb. 00:30:56.510 --> 00:30:58.293 Because you don't want, for example, 00:30:58.293 --> 00:30:59.960 to make this log so large that it starts 00:30:59.960 --> 00:31:02.608 to overcome the coupling. 00:31:02.608 --> 00:31:04.400 You want it to be-- you want this to really 00:31:04.400 --> 00:31:06.050 be a perturbative thing so it makes sense 00:31:06.050 --> 00:31:08.175 to think about this, and then this is a correction, 00:31:08.175 --> 00:31:11.950 and that's another correction. 00:31:11.950 --> 00:31:12.680 OK. 00:31:12.680 --> 00:31:18.200 So the general procedure is the same idea. 00:31:22.770 --> 00:31:25.390 So if I just kind of adopt a more general procedure 00:31:25.390 --> 00:31:27.150 for massive particles [INAUDIBLE] 00:31:27.150 --> 00:31:28.150 a more general notation. 00:31:35.592 --> 00:31:37.425 So this is with any operators and couplings. 00:31:37.425 --> 00:31:39.000 It doesn't have to be gauge theory. 00:31:45.800 --> 00:31:47.650 And if we have a hierarchy of particles-- 00:31:54.210 --> 00:31:58.310 let's say we have n of them-- 00:31:58.310 --> 00:32:06.220 when we want to pass from an L1 to an L2, an L3, to an Ln-- 00:32:08.780 --> 00:32:13.890 and we're doing the same type of thing we just did over there. 00:32:13.890 --> 00:32:16.620 So the steps are the following. 00:32:16.620 --> 00:32:20.970 Consider S matrix elements in theory 1. 00:32:20.970 --> 00:32:25.290 Do so at a scale which I'll call mu 1 that's of order m1, 00:32:25.290 --> 00:32:29.550 and match that onto in the same way of assuring the continuity 00:32:29.550 --> 00:32:33.600 and do-- using these matching conditions, match that onto L2. 00:32:37.440 --> 00:32:39.480 So this is the technical step by which 00:32:39.480 --> 00:32:40.830 I said that you could do-- 00:32:40.830 --> 00:32:43.980 you could in a top-down approach take theory 1 00:32:43.980 --> 00:32:46.550 and determine the parameters of theory 2. 00:32:46.550 --> 00:32:49.260 The matching conditions are determining the parameters. 00:32:49.260 --> 00:32:53.460 Alpha s 4 is of parameter of theory 2 to alpha s 5. 00:32:53.460 --> 00:32:54.860 This is theory 1. 00:32:54.860 --> 00:32:57.270 And we're just determining what this alpha s 4 should be, 00:32:57.270 --> 00:32:59.103 and this is the condition that relates them. 00:33:02.530 --> 00:33:04.620 So after you do that step, if you 00:33:04.620 --> 00:33:07.150 want to go through this picture here, 00:33:07.150 --> 00:33:11.090 or the analog of that picture for this case over here, 00:33:11.090 --> 00:33:14.095 then you need to compute the beta functions 00:33:14.095 --> 00:33:15.720 in anomalous dimensions in this theory. 00:33:29.630 --> 00:33:32.390 So it's the theory that doesn't have particle 1. 00:33:32.390 --> 00:33:37.080 And then you evolve/run the couplings down, 00:33:37.080 --> 00:33:39.420 which just means using the evolution 00:33:39.420 --> 00:33:43.302 equation for the couplings, whatever they may be. 00:33:43.302 --> 00:33:45.510 So we had the evolution equation for alpha S a minute 00:33:45.510 --> 00:33:46.740 ago on the board. 00:33:46.740 --> 00:33:51.053 And I would just use that to go from a scale of order mb 00:33:51.053 --> 00:33:52.220 down to a scale of order mc. 00:33:57.470 --> 00:33:58.550 And then I repeat. 00:34:04.670 --> 00:34:06.550 And this is the general kind of paradigm 00:34:06.550 --> 00:34:09.190 of matching and running that you hear about in effective theory 00:34:09.190 --> 00:34:09.790 all the time. 00:34:15.610 --> 00:34:17.554 So we just keep going. 00:34:17.554 --> 00:34:19.929 And at the end of the day, we're going to stop somewhere. 00:34:19.929 --> 00:34:21.304 And the place we're going to stop 00:34:21.304 --> 00:34:23.895 is the place we want to do lower-energy physics. 00:34:23.895 --> 00:34:25.520 So let's say we stop at the n-th level, 00:34:25.520 --> 00:34:27.060 since that's the last level I wrote. 00:34:36.037 --> 00:34:38.120 So if you're interested in dynamics at that scale, 00:34:38.120 --> 00:34:39.037 that's where you stop. 00:34:48.949 --> 00:34:52.170 And that's where you compute your final matrix elements. 00:34:52.170 --> 00:34:54.179 So everything up until that stage 00:34:54.179 --> 00:34:57.440 is just to determine the theory Ln 00:34:57.440 --> 00:35:00.720 and what are the values of decoupling in that theory-- 00:35:00.720 --> 00:35:05.140 determined from knowing information at the high scale. 00:35:05.140 --> 00:35:06.730 Knowing information in theory 1, how 00:35:06.730 --> 00:35:09.010 do I propagate that knowledge all the way down 00:35:09.010 --> 00:35:13.630 to low energies consistently without losing 00:35:13.630 --> 00:35:14.987 information I need? 00:35:14.987 --> 00:35:16.570 And then once I'm at the lowest scale, 00:35:16.570 --> 00:35:20.453 I just do my computations of observables. 00:35:42.138 --> 00:35:42.638 Yeah. 00:35:42.638 --> 00:35:43.131 AUDIENCE: Professor. 00:35:43.131 --> 00:35:43.923 IAIN STEWART: Sure. 00:35:43.923 --> 00:35:47.450 AUDIENCE: [INAUDIBLE] mention at the scale where the particle is 00:35:47.450 --> 00:35:48.851 there-- 00:35:48.851 --> 00:35:51.020 actually, I was a bit confused because if we 00:35:51.020 --> 00:35:53.470 write the full theory, then, say, 00:35:53.470 --> 00:35:55.990 2-to-2 scattering in that mass scale, 00:35:55.990 --> 00:35:59.090 isn't there any like [INAUDIBLE] say resonance effect, 00:35:59.090 --> 00:36:02.570 and then you want to match that to a theory 00:36:02.570 --> 00:36:05.240 without the particle? 00:36:05.240 --> 00:36:06.680 I thought we should match the s-- 00:36:06.680 --> 00:36:08.730 IAIN STEWART: So there is no resonance effect. 00:36:08.730 --> 00:36:12.560 And the reason is because the momentum on the external lines 00:36:12.560 --> 00:36:13.850 that you're taking-- 00:36:13.850 --> 00:36:16.780 you're thinking of it as small, right? 00:36:16.780 --> 00:36:18.680 The scale is chosen to be the mass 00:36:18.680 --> 00:36:22.860 of the particle, the cutoff, mu, the soft cutoff. 00:36:22.860 --> 00:36:25.520 But not the momentum of the particles. 00:36:25.520 --> 00:36:26.926 Good question. 00:36:26.926 --> 00:36:28.315 Any other questions? 00:36:32.020 --> 00:36:33.310 OK. 00:36:33.310 --> 00:36:38.080 So we're not really done talking about subtleties here. 00:36:41.100 --> 00:36:43.120 And we're not really done talking about some 00:36:43.120 --> 00:36:46.280 of the key ideas that come into these calculations. 00:36:46.280 --> 00:36:49.420 I've given you a very simple example just the coupling. 00:36:49.420 --> 00:36:50.740 That's a little bit too simple. 00:36:50.740 --> 00:36:52.365 So I want to do something a little more 00:36:52.365 --> 00:36:57.750 sophisticated but still fairly simple and widely used. 00:36:57.750 --> 00:36:59.548 And that is in the standard model just 00:36:59.548 --> 00:37:01.090 to take the heaviest particles, which 00:37:01.090 --> 00:37:04.750 are the top, the Higgs, the W, and the Z, and remove them. 00:37:12.330 --> 00:37:15.100 So we'll spend a bit of time talking about what 00:37:15.100 --> 00:37:17.870 happens when you do that. 00:37:17.870 --> 00:37:19.650 That's not the only thing you could do. 00:37:19.650 --> 00:37:22.355 And in particular, before people do how heavy the Higgs was, 00:37:22.355 --> 00:37:23.980 you could think of other possibilities. 00:37:23.980 --> 00:37:25.870 And before people knew how heavy the top was, 00:37:25.870 --> 00:37:27.988 people did think of other possibilities. 00:37:32.470 --> 00:37:35.830 This is actually a reasonable thing to do, though. 00:37:35.830 --> 00:37:38.930 let me give you one example of another possibility. 00:37:38.930 --> 00:37:42.010 Or you could say the top is heavier than the W and the Z, 00:37:42.010 --> 00:37:45.700 so why don't I do the top first, and then 00:37:45.700 --> 00:37:48.235 I'll just follow your diagram over there. 00:37:48.235 --> 00:37:50.740 I erased it, but it'd follow the picture. 00:37:50.740 --> 00:37:52.810 And then I'll do some running in the theory 00:37:52.810 --> 00:37:54.310 without the top quark, and then I'll 00:37:54.310 --> 00:37:56.530 get down to the W and the Z scales, 00:37:56.530 --> 00:37:59.840 then I'll remove the W and the Z. 00:37:59.840 --> 00:38:00.340 OK. 00:38:00.340 --> 00:38:02.530 That would be a valid thing to do. 00:38:02.530 --> 00:38:05.530 But it introduces complications that are actually not 00:38:05.530 --> 00:38:06.250 worth the effort. 00:38:09.850 --> 00:38:11.900 And benefit is actually not that great. 00:38:11.900 --> 00:38:15.155 So I want to emphasize that. 00:38:15.155 --> 00:38:17.530 When should you think of things as being comparably heavy 00:38:17.530 --> 00:38:19.840 versus when should you think of things being hierarchically 00:38:19.840 --> 00:38:20.340 heavy? 00:38:23.620 --> 00:38:27.190 Well, one complication of removing the top quark 00:38:27.190 --> 00:38:32.200 is that it breaks su2 cross u1 gauge invariance, 00:38:32.200 --> 00:38:35.480 because the top quark was in a doublet with the b, 00:38:35.480 --> 00:38:44.140 and you're trying to keep the b and remove the top, which 00:38:44.140 --> 00:38:46.280 doesn't sound good. 00:38:46.280 --> 00:38:49.200 You're trying to keep the gauge particles, the W and the Z, 00:38:49.200 --> 00:38:50.570 in your theory. 00:38:50.570 --> 00:38:53.260 So you should still have that gauge symmetry, even 00:38:53.260 --> 00:38:55.870 if it's spontaneously broken. 00:38:55.870 --> 00:38:59.410 But you're trying to remove one member of a doublet. 00:38:59.410 --> 00:39:01.542 And that leads to [INAUDIBLE] terms 00:39:01.542 --> 00:39:03.500 that you'd have to clue to the effective theory 00:39:03.500 --> 00:39:04.930 so you can deal with it. 00:39:04.930 --> 00:39:06.430 And it's just a little bit annoying. 00:39:10.460 --> 00:39:14.110 But, you know, it's something you can deal with. 00:39:14.110 --> 00:39:16.090 The real crux of the matter is that if you 00:39:16.090 --> 00:39:21.790 compare mZ over m top, or mW over m top, 00:39:21.790 --> 00:39:23.685 that's about a half. 00:39:23.685 --> 00:39:25.060 So if you think about what you're 00:39:25.060 --> 00:39:27.670 expanding in when you take external particles that 00:39:27.670 --> 00:39:31.690 have momentum of order mZ, integrate out particles 00:39:31.690 --> 00:39:34.840 of order mt, you're expanding in a half, which is not such 00:39:34.840 --> 00:39:36.933 a great expansion parameter. 00:39:36.933 --> 00:39:38.350 Usually in effective field theory, 00:39:38.350 --> 00:39:41.020 you want to expand in something that's at least a third, 00:39:41.020 --> 00:39:44.110 hopefully a quarter. 00:39:44.110 --> 00:39:47.960 And a tenth if you really want to have a good expansion. 00:39:47.960 --> 00:39:50.530 So half is not really that good. 00:39:50.530 --> 00:39:57.590 So that's the real reason not to do it. 00:40:05.510 --> 00:40:08.250 And you could ask, well, why do we lose by not doing that? 00:40:10.265 --> 00:40:12.015 And that, of course, is the real question. 00:40:19.332 --> 00:40:20.790 Well, you miss some running, right? 00:40:20.790 --> 00:40:23.990 Because you don't have a theory that has no top quark 00:40:23.990 --> 00:40:25.740 but still has a W and a Z, and that theory 00:40:25.740 --> 00:40:27.490 could have anomalous dimensions, and you'd 00:40:27.490 --> 00:40:29.970 miss any running in that theory. 00:40:29.970 --> 00:40:32.463 You're kind of collapsing two of the lines in my picture 00:40:32.463 --> 00:40:34.380 with the arrow between, you're collapsing them 00:40:34.380 --> 00:40:37.810 down to just a single line. 00:40:37.810 --> 00:40:41.998 So you're missing the running that would go between mt to mW. 00:40:41.998 --> 00:40:44.040 Well, of course, that's not a very big hierarchy, 00:40:44.040 --> 00:40:45.730 and it's logs of two. 00:40:45.730 --> 00:40:50.840 And that's why we don't care that much. 00:40:50.840 --> 00:40:55.945 What it boils down to is that you're treating alpha s at mW. 00:40:55.945 --> 00:40:57.320 And if we want to be generous, we 00:40:57.320 --> 00:40:59.820 can say it's mW squared over m top squared. 00:40:59.820 --> 00:41:01.620 So they're logs of four. 00:41:01.620 --> 00:41:03.525 But you're treating this perturbatively. 00:41:08.280 --> 00:41:11.153 If you remove both the top and the W at the same time, 00:41:11.153 --> 00:41:13.570 then this is going to show up in your matching conditions, 00:41:13.570 --> 00:41:15.925 and you're not thinking of that is something as large. 00:41:15.925 --> 00:41:17.400 Thinking of it just as order 1. 00:41:23.550 --> 00:41:25.710 So that's the cost, which is not a big cost. 00:41:28.607 --> 00:41:30.440 Especially when you give, say, that the cost 00:41:30.440 --> 00:41:33.140 of going the other way would be expanding in the half. 00:41:33.140 --> 00:41:35.300 Better to take the cost in the logarithms 00:41:35.300 --> 00:41:39.105 than expansion in the half. 00:41:39.105 --> 00:41:40.480 So that's a general kind of rule, 00:41:40.480 --> 00:41:42.860 that if you're thinking about removing massive particles, 00:41:42.860 --> 00:41:46.670 you should ask, how close are they to each other? 00:41:46.670 --> 00:41:50.170 Should I integrate out a slew of them at the same time, 00:41:50.170 --> 00:41:51.170 or one at a time? 00:41:51.170 --> 00:41:54.340 That'll depend on exactly the scales in the problem. 00:41:57.170 --> 00:42:00.410 And there's-- and it means you're organizing the theory 00:42:00.410 --> 00:42:02.764 differently if you do it, the two different approaches. 00:42:06.360 --> 00:42:06.860 All right. 00:42:06.860 --> 00:42:09.830 So we'll do an example of this. 00:42:16.870 --> 00:42:18.890 We'll take a very simple example, 00:42:18.890 --> 00:42:23.050 although not completely trivial. 00:42:23.050 --> 00:42:26.500 So just b quarks changing to charm quarks u bar and d. 00:42:32.190 --> 00:42:34.450 So on the standard model, which is where we start, 00:42:34.450 --> 00:42:38.020 let's say we have a W boson. 00:42:38.020 --> 00:42:41.750 We have an up quark that's left-handed. 00:42:41.750 --> 00:42:46.810 We have a CKM matrix connecting flavors together. 00:42:46.810 --> 00:42:48.040 And we have down quarks. 00:42:48.040 --> 00:42:50.470 So this is-- there's a matrix space here 00:42:50.470 --> 00:42:53.350 in the flavor, which includes bottom charm up and down. 00:42:57.700 --> 00:43:00.987 And tree level matching is easy. 00:43:00.987 --> 00:43:03.070 And some of the complications I want to talk about 00:43:03.070 --> 00:43:05.210 will come into the loops. 00:43:05.210 --> 00:43:07.270 So let's first get through the tree level 00:43:07.270 --> 00:43:08.600 and set up some notation. 00:43:08.600 --> 00:43:12.940 And then we'll talk about what happens with the loop. 00:43:12.940 --> 00:43:17.270 So tree level, this is the diagram in the theory 1. 00:43:17.270 --> 00:43:19.510 Just calculate it at tree level. 00:43:29.710 --> 00:43:33.670 W propagator, unitary gauge. 00:43:46.578 --> 00:43:47.870 And then there's some spinners. 00:44:04.945 --> 00:44:06.150 So we have an antiquark. 00:44:06.150 --> 00:44:09.630 So we have a v spinner for the antiquark. 00:44:09.630 --> 00:44:11.520 That's an up. 00:44:11.520 --> 00:44:12.860 Everything else is a u spinner. 00:44:17.510 --> 00:44:20.580 I've put in explicitly the P lefts to denote 00:44:20.580 --> 00:44:22.165 the fact that it's left-handed. 00:44:25.352 --> 00:44:27.310 So there's some momentum in this diagram, which 00:44:27.310 --> 00:44:31.750 is momentum transfer, which is the be momentum minus the c 00:44:31.750 --> 00:44:34.970 momentum and the kind of obvious notation. 00:44:34.970 --> 00:44:36.850 And of course, by momentum conservation, 00:44:36.850 --> 00:44:42.270 that's also the d momentum plus the u momentum. 00:44:42.270 --> 00:44:46.260 And that's the momentum that's going through the propagator. 00:44:46.260 --> 00:44:49.135 We're going to count momenta as being of order masses. 00:44:51.730 --> 00:44:54.372 And the heaviest mass in this case is the b quark mass. 00:44:54.372 --> 00:44:56.080 And we'll take it to be the b quark mass. 00:44:59.100 --> 00:45:01.616 We can use the equations of motion. 00:45:01.616 --> 00:45:06.330 So Pb slash on ub, spinner for the b quark 00:45:06.330 --> 00:45:10.350 is mb ub, et cetera. 00:45:10.350 --> 00:45:13.740 And we can simplify the diagram by doing that. 00:45:13.740 --> 00:45:15.630 And of course, the key thing is that we 00:45:15.630 --> 00:45:18.000 can expand the propagator since the momentum is smaller 00:45:18.000 --> 00:45:22.050 than the mass of the W for integrating out 00:45:22.050 --> 00:45:32.680 the W. So the leading term when we do that is this term. 00:45:32.680 --> 00:45:41.380 And any other terms that we drop are down by that much. 00:45:41.380 --> 00:45:43.020 So for example, this term here, we just 00:45:43.020 --> 00:45:45.850 said that the k's are of order mb so it's down. 00:45:45.850 --> 00:45:47.600 And then we could drop the k squared here, 00:45:47.600 --> 00:45:49.530 I'll get the mW squared, so we just have that. 00:45:53.430 --> 00:45:55.932 And then we put that together, we can calculate the Feynman 00:45:55.932 --> 00:45:57.140 rule in the effective theory. 00:45:57.140 --> 00:45:59.160 The Feynman rule in the effective theory, 00:45:59.160 --> 00:46:01.422 we can determine the coefficient of it. 00:46:01.422 --> 00:46:02.880 So let's start out with just saying 00:46:02.880 --> 00:46:08.250 it's some four-quark operator that couples together 00:46:08.250 --> 00:46:11.490 those flavors, conveniently chosen so that they're all 00:46:11.490 --> 00:46:13.130 different to avoid [INAUDIBLE] factors. 00:46:16.390 --> 00:46:18.240 And in some conventional normalization 00:46:18.240 --> 00:46:23.370 for this Wilson coefficient, we call it GF, which is for Fermi. 00:46:25.940 --> 00:46:28.090 And the Feynman rule would again give spinners, 00:46:28.090 --> 00:46:33.360 the same spinners as before, with the same Lagrangians 00:46:33.360 --> 00:46:35.550 for these light quarks. 00:46:35.550 --> 00:46:37.780 Haven't changed anything about that. 00:46:37.780 --> 00:46:39.405 We're just removing the heavy particle. 00:46:43.627 --> 00:46:45.210 And it's conventional also to pull out 00:46:45.210 --> 00:46:48.850 the CKM factor [INAUDIBLE]. 00:46:48.850 --> 00:46:49.350 OK. 00:46:49.350 --> 00:46:52.237 So if you like, you can say that there's a coefficient here, 00:46:52.237 --> 00:46:54.070 and that coefficient has been fixed to be 1. 00:46:57.110 --> 00:46:58.860 And what this G Fermi is, then, would just 00:46:58.860 --> 00:47:03.180 be the leftover factors of the gauge coupling 00:47:03.180 --> 00:47:07.943 and the mass of the W. And that's the usual convention. 00:47:17.810 --> 00:47:23.160 So G Fermi is just fixed in this case, not as the thing that 00:47:23.160 --> 00:47:27.927 is going to get corrected at higher orders in QCD 00:47:27.927 --> 00:47:30.260 but would get corrected at higher orders in electroweak, 00:47:30.260 --> 00:47:31.910 if you like. 00:47:31.910 --> 00:47:35.510 And then you sort of put in with the QCD corrections 00:47:35.510 --> 00:47:38.750 into some other coefficient that we can call C. 00:47:38.750 --> 00:47:41.990 And we'll just say that it's 1 at the level of this. 00:47:41.990 --> 00:47:44.640 And this line agrees with the expansion of that line, 00:47:44.640 --> 00:47:46.760 and that's the matching of something 00:47:46.760 --> 00:47:50.727 that's measurable, which is this four-point function. 00:47:50.727 --> 00:47:52.310 So we can call it an S matrix element. 00:48:04.790 --> 00:48:07.010 So this theory is very popular. 00:48:07.010 --> 00:48:10.220 It's called the electroweak Hamiltonian. 00:48:10.220 --> 00:48:11.440 It's used all over the place. 00:48:15.250 --> 00:48:17.410 And it involves more than just doing what I did, 00:48:17.410 --> 00:48:20.145 because I chose a particular set of flavors 00:48:20.145 --> 00:48:21.520 for a particular channel, and you 00:48:21.520 --> 00:48:23.183 have to do it for the whole-- 00:48:23.183 --> 00:48:24.850 for all sorts of other channels as well. 00:48:28.630 --> 00:48:31.410 So we'll study some aspects of this theory. 00:48:31.410 --> 00:48:33.150 We won't study every possible aspect. 00:48:33.150 --> 00:48:36.000 But I've given you a handout that studies a lot more. 00:48:36.000 --> 00:48:38.490 It's 250 pages long. 00:48:38.490 --> 00:48:40.510 I'm not asking you to even read all of that. 00:48:40.510 --> 00:48:44.003 I've pointed you at some pages of that 00:48:44.003 --> 00:48:46.170 in the reading, which are relevant to the discussion 00:48:46.170 --> 00:48:47.330 we're having here. 00:48:47.330 --> 00:48:49.080 If you really want to dig deeper, you can. 00:48:51.720 --> 00:48:55.020 So people often call this an electroweak Hamiltonian, which 00:48:55.020 --> 00:48:59.120 is just minus the Lagrangian. 00:48:59.120 --> 00:49:02.860 So then that's 4 GF over 2. 00:49:02.860 --> 00:49:07.940 And as a Hamiltonian, we write fields instead of spinners. 00:49:07.940 --> 00:49:11.320 So we have this four-quark operator. 00:49:11.320 --> 00:49:13.890 And we would have determined the Hamiltonian from tree level 00:49:13.890 --> 00:49:15.430 matching, from what we've done. 00:49:15.430 --> 00:49:29.460 And this would be the result. Standard stuff. 00:49:34.530 --> 00:49:37.080 How do we want to go further than that? 00:49:37.080 --> 00:49:39.975 Well, if we want to think about theory 2, which is this theory, 00:49:39.975 --> 00:49:41.850 and we want to think about it in more detail, 00:49:41.850 --> 00:49:43.597 we should worry about whether we're-- 00:49:43.597 --> 00:49:45.180 with just this operator, whether we've 00:49:45.180 --> 00:49:50.033 got a complete set of structures that could possibly occur. 00:49:50.033 --> 00:49:51.700 So we should think about the symmetries. 00:49:54.475 --> 00:49:56.850 And it's useful to, therefore, construct the most general 00:49:56.850 --> 00:50:04.018 basis of operators in theory 2. 00:50:06.580 --> 00:50:08.400 And we'll just do that. 00:50:08.400 --> 00:50:10.460 So if you read this 250-page review, 00:50:10.460 --> 00:50:14.178 then it's done for the full electroweak Hamiltonian, 00:50:14.178 --> 00:50:15.720 and we'll just stick with the flavors 00:50:15.720 --> 00:50:18.660 we're talking about here. 00:50:18.660 --> 00:50:23.220 So what are the most general bases of operators? 00:50:23.220 --> 00:50:25.670 How should we think about that? 00:50:25.670 --> 00:50:27.170 Well, we're constructing some theory 00:50:27.170 --> 00:50:31.630 that's going to-- that we're matching onto it mu equals mW. 00:50:31.630 --> 00:50:33.800 And at that scale, when we're determining the theory 00:50:33.800 --> 00:50:35.758 and determining the coefficients of the theory, 00:50:35.758 --> 00:50:38.900 we can treat the bottom the charm the down 00:50:38.900 --> 00:50:42.740 and the up respectively as if they're massless. 00:50:45.710 --> 00:50:50.960 And really, what I mean by that is that the masses of these 00:50:50.960 --> 00:50:53.990 particles are only going to show up in the operators and not 00:50:53.990 --> 00:51:00.630 in the coefficients, which I've-- 00:51:00.630 --> 00:51:03.598 you see that I talked about over here. 00:51:03.598 --> 00:51:05.390 The thing that shows up in the coefficients 00:51:05.390 --> 00:51:06.848 are the mass scales we're removing. 00:51:06.848 --> 00:51:11.150 And we're not removing the mass scales of these things. 00:51:11.150 --> 00:51:13.420 So that means, if we can think of them as massless, 00:51:13.420 --> 00:51:16.420 that we can think about the matching in terms of using 00:51:16.420 --> 00:51:18.680 something like chirality. 00:51:18.680 --> 00:51:22.600 And we can use the fact that QCD for massless quarks 00:51:22.600 --> 00:51:26.038 does not change chirality. 00:51:26.038 --> 00:51:27.955 And we can use that in constructing our basis. 00:51:32.940 --> 00:51:38.580 So that means even though the b quark and the charm quark 00:51:38.580 --> 00:51:40.710 really have masses, for the purpose of constructing 00:51:40.710 --> 00:51:43.710 the operator basis, we can think of them as massless. 00:51:43.710 --> 00:51:45.970 And we only have left-handed guys to worry about here. 00:51:48.650 --> 00:51:50.070 Chirality means more than that. 00:51:50.070 --> 00:51:52.850 It also-- well, it means effectively that. 00:51:52.850 --> 00:51:57.240 But it means also that we will only get one gamma matrix here. 00:51:57.240 --> 00:51:59.940 And we can think about why that is. 00:51:59.940 --> 00:52:04.190 Chirality means that you get an odd number. 00:52:04.190 --> 00:52:08.550 You need an odd number to have left on both sides. 00:52:08.550 --> 00:52:10.010 But you can reduce 3 to 1. 00:52:14.440 --> 00:52:19.660 So there's an identity which I won't write out for you, 00:52:19.660 --> 00:52:28.830 but you can reduce 3 back down to 1. 00:52:28.830 --> 00:52:31.780 So any higher odd number can be reduced back down to 1. 00:52:31.780 --> 00:52:37.920 So once we use chirality and that fact, 00:52:37.920 --> 00:52:44.660 then we have a fairly restrictive basis 00:52:44.660 --> 00:52:46.435 in terms of Dirac structures. 00:52:54.200 --> 00:52:57.610 You could ask, what are the most general possible ways 00:52:57.610 --> 00:52:59.738 of contracting spinner indices? 00:52:59.738 --> 00:53:02.280 Who said I had to put this charm quirk with the bottom quark? 00:53:02.280 --> 00:53:04.190 I could have put the up quark over here. 00:53:04.190 --> 00:53:06.955 And maybe in higher orders that happens. 00:53:06.955 --> 00:53:08.330 Well, at higher orders, you could 00:53:08.330 --> 00:53:09.455 think about that happening. 00:53:09.455 --> 00:53:11.450 But you can always get back to the form 00:53:11.450 --> 00:53:14.700 that I wrote over there using what's called the spin Fierz. 00:53:20.385 --> 00:53:25.430 So if I write it for fields, it means 00:53:25.430 --> 00:53:28.760 that there's an identity that I can rearrange 00:53:28.760 --> 00:53:29.910 this guy the other way. 00:53:29.910 --> 00:53:32.420 And so these are equivalent operators. 00:53:39.223 --> 00:53:41.140 And sometimes, you have to use these relations 00:53:41.140 --> 00:53:42.820 when you're doing matching calculations, 00:53:42.820 --> 00:53:44.908 because you construct a complete basis, which 00:53:44.908 --> 00:53:45.700 is the minimal one. 00:53:45.700 --> 00:53:47.530 And then maybe when you do your calculation, 00:53:47.530 --> 00:53:48.947 you get this operator, so you have 00:53:48.947 --> 00:53:52.000 to turn it back into this one. 00:53:52.000 --> 00:53:55.060 There's two minus signs in this relation. 00:53:55.060 --> 00:53:58.163 One is from the Fierz identity, and one 00:53:58.163 --> 00:53:59.830 is because when you do the manipulations 00:53:59.830 --> 00:54:01.288 you end up anticommuting to fields. 00:54:08.000 --> 00:54:10.102 So from the statistics. 00:54:10.102 --> 00:54:12.310 So if you were to use the same relation for spinners, 00:54:12.310 --> 00:54:14.800 then there would be one less minus sign. 00:54:14.800 --> 00:54:17.050 Something to be careful, though. 00:54:17.050 --> 00:54:21.100 So what this tells you, once you put that information together, 00:54:21.100 --> 00:54:27.580 is that you know that you can write the operators this way, 00:54:27.580 --> 00:54:30.430 and you also know that this gamma, capital gamma, 00:54:30.430 --> 00:54:36.860 has that gamma u p left form in terms of the Dirac structure. 00:54:36.860 --> 00:54:41.380 So the only really thing that can happen is color. 00:54:41.380 --> 00:54:43.937 And you can contract the color of these 3's-- 00:54:43.937 --> 00:54:46.270 so you have four 3's, you can contract the color of them 00:54:46.270 --> 00:54:47.710 in different ways. 00:54:47.710 --> 00:54:49.780 Well, I have two 3 bars and two 3's. 00:54:49.780 --> 00:54:52.450 Think about having different color contractions. 00:54:52.450 --> 00:54:54.310 And that will expand our operator basis 00:54:54.310 --> 00:54:57.160 by one more operator. 00:54:57.160 --> 00:55:00.750 There is also a color Fierz. 00:55:00.750 --> 00:55:04.870 And I can use that color Fierz to get rid of having explicit 00:55:04.870 --> 00:55:06.870 TA's. 00:55:06.870 --> 00:55:09.100 So color is a lot like spin in the sense 00:55:09.100 --> 00:55:11.650 that if I had more TA's, then I can always 00:55:11.650 --> 00:55:15.440 reduce a higher number of TA's back down to a lower number. 00:55:15.440 --> 00:55:18.770 So I've had two, I can reduce it down to one or zero. 00:55:18.770 --> 00:55:19.270 OK. 00:55:19.270 --> 00:55:21.478 So I only really have to think about having a TA here 00:55:21.478 --> 00:55:23.470 and a TA there. 00:55:23.470 --> 00:55:26.290 If I have a TA there and a TA there, there's a spinner-- 00:55:26.290 --> 00:55:30.270 or there's a color Fierz relation for this guy 00:55:30.270 --> 00:55:33.280 that you could use. 00:55:33.280 --> 00:55:35.062 And there's a handout that I posted 00:55:35.062 --> 00:55:37.270 on the web that has a summary of all these relations. 00:55:37.270 --> 00:55:42.020 I'm not going to bother writing them down in lecture here. 00:55:42.020 --> 00:55:47.030 So once I take that into account, 00:55:47.030 --> 00:55:49.090 then it ends up in two operators. 00:55:52.010 --> 00:55:55.920 And if I were to write them in a kind of renormalized notation 00:55:55.920 --> 00:56:00.460 where I introduce the scale mu, then I 00:56:00.460 --> 00:56:02.767 would write them as follows. 00:56:02.767 --> 00:56:03.850 Let's call them O1 and O2. 00:56:07.540 --> 00:56:11.650 And the difference between O1 and O2 is just simply color. 00:56:11.650 --> 00:56:14.860 So in O1, the color index alpha is 00:56:14.860 --> 00:56:18.150 contracted between charm bottom and up down and up. 00:56:24.820 --> 00:56:27.370 And then in O2 it's the the colors 00:56:27.370 --> 00:56:28.880 contracted the other way. 00:56:28.880 --> 00:56:30.490 So I don't write explicit TA's, but I 00:56:30.490 --> 00:56:33.620 do have to consider the other possible way of contracting 00:56:33.620 --> 00:56:34.120 the indices. 00:56:47.690 --> 00:56:50.400 Up, down, doesn't matter. 00:56:50.400 --> 00:56:50.900 OK. 00:56:50.900 --> 00:56:54.085 So these operators, these are the two operators they have 00:56:54.085 --> 00:56:55.460 once I satisfy all the symmetries 00:56:55.460 --> 00:56:57.130 and I have some coefficients. 00:57:00.630 --> 00:57:03.080 And if you ask really what the coefficients can depend 00:57:03.080 --> 00:57:07.690 on here, well, they can depend on mass scales like top 00:57:07.690 --> 00:57:10.920 and the other particles that I'm integrating out, 00:57:10.920 --> 00:57:14.427 Z. I'm not going to make that explicit in my notation. 00:57:14.427 --> 00:57:16.010 If I tried to make every possible mass 00:57:16.010 --> 00:57:18.900 scale like the Higgs or the top, it would just be too much. 00:57:18.900 --> 00:57:22.820 So let me just denote two things that it can depend on. 00:57:22.820 --> 00:57:25.940 Mu over mW can show up, and alpha of mu 00:57:25.940 --> 00:57:29.502 can show up, as well as some ratios of other particles 00:57:29.502 --> 00:57:30.460 which we will suppress. 00:57:45.000 --> 00:57:47.060 So then what tree level matching is saying 00:57:47.060 --> 00:57:48.810 is that you've got one of these operators, 00:57:48.810 --> 00:57:50.790 and you didn't get the other one. 00:57:50.790 --> 00:57:55.235 So it says that at tree level, what we did before-- 00:57:57.845 --> 00:57:59.840 and so say we do matching at mu equals 00:57:59.840 --> 00:58:11.860 mW and C1 at 1, which is mu over mW equals 1 and alpha mW is 1. 00:58:11.860 --> 00:58:14.630 And then corrections to that would be suppressed, 00:58:14.630 --> 00:58:17.070 no large logs, just by alpha. 00:58:17.070 --> 00:58:27.770 And C2 of 1 alpha mW is 0 plus something of this order. 00:58:27.770 --> 00:58:32.300 So that's what we determined by doing our tree level matching. 00:58:32.300 --> 00:58:35.600 Now, when we did our tree level matching we did something. 00:58:35.600 --> 00:58:40.472 We used external states which were quarks. 00:58:40.472 --> 00:58:44.190 We matched up the S matrix elements in that way. 00:58:44.190 --> 00:58:46.065 If you were really interested in this process 00:58:46.065 --> 00:58:48.433 b goes to c u bar d, you'd not really 00:58:48.433 --> 00:58:49.850 be interested in measuring quarks. 00:58:49.850 --> 00:58:51.050 We don't see them. 00:58:51.050 --> 00:58:53.890 You'd be interested in thinking about the process for mesons-- 00:58:53.890 --> 00:58:58.365 B meson changes to a D meson, and a pion, for example. 00:58:58.365 --> 00:59:00.740 And who's to say that the matching that we did for quarks 00:59:00.740 --> 00:59:04.760 is also the matching that's valid for hadrons? 00:59:04.760 --> 00:59:05.420 Well, it is. 00:59:08.040 --> 00:59:11.760 So there's a key fact about matching which is important. 00:59:11.760 --> 00:59:15.470 One of the things I wanted to emphasize to you 00:59:15.470 --> 00:59:18.050 is that it doesn't depend on what states you pick. 00:59:23.340 --> 00:59:26.790 It's independent of the choice of states. 00:59:26.790 --> 00:59:29.340 And it's independent of the IR regulators that you pick. 00:59:31.890 --> 00:59:34.560 So when you do matching, you often 00:59:34.560 --> 00:59:38.165 pick some IR regulators to regulate IR divergences. 00:59:38.165 --> 00:59:39.540 And the only thing you have to do 00:59:39.540 --> 00:59:41.482 is pick the same states and same IR regulators 00:59:41.482 --> 00:59:42.315 in the two theories. 00:59:45.180 --> 00:59:48.638 And once you do that, your results for your coefficients 00:59:48.638 --> 00:59:50.430 will be independent of the choice you made. 01:00:00.540 --> 01:00:03.680 So this is intuitively something I think that's, once you 01:00:03.680 --> 01:00:08.110 think about it, fairly obvious. 01:00:08.110 --> 01:00:11.030 What this sentence is saying is that the matching, which 01:00:11.030 --> 01:00:13.280 is supposed to be a high-energy property, 01:00:13.280 --> 01:00:15.465 is independent of the low-energy physics. 01:00:15.465 --> 01:00:17.840 The choice of the states and the choice of IR regulators, 01:00:17.840 --> 01:00:21.530 that's parameterizing something about low-energy physics-- 01:00:21.530 --> 01:00:23.780 lower energy than the scale we're trying to determine. 01:00:23.780 --> 01:00:26.700 These things should only depend on higher-energy physics. 01:00:26.700 --> 01:00:30.365 So the outcome for them will be independent of this choice. 01:00:34.350 --> 01:00:49.670 So just to emphasize, even when we use hadronic states, 01:00:49.670 --> 01:00:57.290 the result that we obtain from quark states is valid. 01:01:01.860 --> 01:01:06.092 That's just different choice of state. 01:01:06.092 --> 01:01:08.300 And what we're doing when we do the matching is we're 01:01:08.300 --> 01:01:13.560 picking a convenient choice of state, something that 01:01:13.560 --> 01:01:15.378 makes it easy to calculate. 01:01:15.378 --> 01:01:17.670 If we tried to calculate the matrix elements of B and D 01:01:17.670 --> 01:01:20.100 and pi mesons, then I'm saying if you're a strong enough, 01:01:20.100 --> 01:01:22.188 you'd get the same matching coefficient. 01:01:22.188 --> 01:01:23.730 Of course, you should pick the result 01:01:23.730 --> 01:01:25.320 that makes the calculation easy, and that 01:01:25.320 --> 01:01:26.280 would be to use quarks. 01:01:28.808 --> 01:01:29.850 Any questions about that? 01:01:33.740 --> 01:01:34.240 OK. 01:01:34.240 --> 01:01:36.400 So this is something that you should keep in mind. 01:01:36.400 --> 01:01:38.280 And you should keep in mind that it is-- 01:01:38.280 --> 01:01:39.900 you do have to make sure that you're 01:01:39.900 --> 01:01:43.110 using the same states and IR regulator in the two theories. 01:01:43.110 --> 01:01:44.920 In this case, that's pretty easy. 01:01:44.920 --> 01:01:48.330 The states are really the same states, because you're 01:01:48.330 --> 01:01:50.330 defining the states with the Lagrangian machine 01:01:50.330 --> 01:01:52.610 for the b quarks, say, and the b quark Lagrangian 01:01:52.610 --> 01:01:53.950 hasn't really changed. 01:01:53.950 --> 01:01:56.460 So there's no change in the state. 01:01:56.460 --> 01:01:58.710 And you can set things up so that you have the same IR 01:01:58.710 --> 01:01:59.730 regulator. 01:01:59.730 --> 01:02:01.650 And that'll become important in the example 01:02:01.650 --> 01:02:04.270 that we're just about to do. 01:02:04.270 --> 01:02:08.280 So let's do an example of carrying out 01:02:08.280 --> 01:02:13.970 this matching for C1 and C2 in a little more detail. 01:02:13.970 --> 01:02:15.810 And before we carry out matching, 01:02:15.810 --> 01:02:19.462 we actually have to renormalize the two theories. 01:02:19.462 --> 01:02:21.420 Well, we've been talking as if the theory above 01:02:21.420 --> 01:02:22.820 is the standard model. 01:02:22.820 --> 01:02:25.823 So imagine that we've already renormalized that. 01:02:25.823 --> 01:02:27.240 And so we only have to renormalize 01:02:27.240 --> 01:02:29.640 the effective theory in MS bar. 01:02:32.930 --> 01:02:37.080 So we'll talk about doing that, remind you of doing that-- 01:02:37.080 --> 01:02:40.660 maybe something you've seen before. 01:02:40.660 --> 01:02:42.945 So there's going to be wave function renormalization. 01:02:52.450 --> 01:02:53.780 So you have this 4/3. 01:02:56.860 --> 01:03:01.390 I'm going to leave out the prefactor, this prefactor. 01:03:01.390 --> 01:03:02.770 It's always going to be there. 01:03:02.770 --> 01:03:05.020 I'm going to stop writing it and start just 01:03:05.020 --> 01:03:08.890 focusing on the thing in square brackets here. 01:03:08.890 --> 01:03:14.267 We'll do calculations with Feynman gauge. 01:03:14.267 --> 01:03:16.100 I'm not going to really do the calculations. 01:03:16.100 --> 01:03:19.640 I'm just going to quote results to you. 01:03:19.640 --> 01:03:31.150 And in order to simplify what we have to write down, 01:03:31.150 --> 01:03:33.880 let me define the set of spinners 01:03:33.880 --> 01:03:38.860 that you get from taking the tree level matrix element of O1 01:03:38.860 --> 01:03:41.850 to be S1 and of O2 to be S2. 01:03:47.090 --> 01:03:48.970 So this is just the spinners that we 01:03:48.970 --> 01:03:51.370 wrote down before for the case of S1, 01:03:51.370 --> 01:03:53.465 exactly what we had before. 01:03:53.465 --> 01:03:56.090 And then for S2, just a slightly different contraction of color 01:03:56.090 --> 01:03:56.590 indices. 01:04:08.700 --> 01:04:09.200 OK. 01:04:09.200 --> 01:04:11.633 So that's just trying to make the lowest-error result 01:04:11.633 --> 01:04:13.550 simple to write down so that when I write down 01:04:13.550 --> 01:04:15.440 the higher-error result, we can focus on the things that 01:04:15.440 --> 01:04:18.290 are changing and mattering, and not on the complications that 01:04:18.290 --> 01:04:21.210 come in from what we're talking about. 01:04:21.210 --> 01:04:23.120 So if we think about diagrams here, 01:04:23.120 --> 01:04:24.530 we have four-quark operator. 01:04:24.530 --> 01:04:39.160 And we just have to draw all the loop graphs Right. 01:04:39.160 --> 01:04:40.780 We have to regulate them in some way, 01:04:40.780 --> 01:04:44.380 because they are IR divergent. 01:04:44.380 --> 01:04:45.795 So let's regulate them with-- 01:04:45.795 --> 01:04:47.920 if we really want to talk about the matching, which 01:04:47.920 --> 01:04:49.545 is what I really want to do eventually, 01:04:49.545 --> 01:04:51.280 we should regulate them in some way. 01:04:51.280 --> 01:04:54.130 And so let's regulate them with off-shell momenta 01:04:54.130 --> 01:04:55.480 on the external lines. 01:04:55.480 --> 01:05:00.117 And I'll take it to be a common off-shell momenta p. 01:05:00.117 --> 01:05:02.200 And I'll just set the masses of the external lines 01:05:02.200 --> 01:05:06.040 to 0, since they're not going to matter. 01:05:08.960 --> 01:05:09.460 OK. 01:05:09.460 --> 01:05:12.590 So then what does it look like? 01:05:12.590 --> 01:05:14.660 So the matrix element of O1-- 01:05:14.660 --> 01:05:15.940 the 0 means Bayer-- 01:05:18.520 --> 01:05:20.080 is going to have divergences. 01:05:20.080 --> 01:05:38.098 And it has the following structure, times spinner 1. 01:05:38.098 --> 01:05:40.390 And then there's another piece that involves spinner 1. 01:05:57.770 --> 01:06:00.230 And then there's a piece that involves spinner 2 that shows 01:06:00.230 --> 01:06:03.170 up in from these loop graphs. 01:06:03.170 --> 01:06:05.840 So even though we're calculating the Bayer matrix [INAUDIBLE] 01:06:05.840 --> 01:06:08.690 of operator 1, the spinner combination 2 01:06:08.690 --> 01:06:13.010 shows up, because this is supposedly a gluon, 01:06:13.010 --> 01:06:15.050 and the gluon moves the color around. 01:06:20.210 --> 01:06:21.968 And I'm only writing the divergent terms, 01:06:21.968 --> 01:06:23.510 although later on we'll be interested 01:06:23.510 --> 01:06:24.760 in the constant terms as well. 01:06:27.660 --> 01:06:31.320 So the dots are the constant terms underneath logarithms, 01:06:31.320 --> 01:06:32.830 and all these little round brackets 01:06:32.830 --> 01:06:34.202 have constant terms in them. 01:06:37.803 --> 01:06:39.220 Part of the reason for introducing 01:06:39.220 --> 01:06:41.320 the basis in the way I did is that when 01:06:41.320 --> 01:06:43.680 I want to quote you the result for O2, 01:06:43.680 --> 01:06:46.360 it's the same, just switching 1 and 2. 01:06:54.100 --> 01:06:56.380 And so the statement that you would have from this-- 01:06:56.380 --> 01:06:57.820 these are ultraviolet divergences, 01:06:57.820 --> 01:06:59.598 the infrared divergences are regulated. 01:06:59.598 --> 01:07:01.390 The statement that you would have from this 01:07:01.390 --> 01:07:04.320 is that you can look at these various terms, and you can ask, 01:07:04.320 --> 01:07:07.210 what's going on with those divergences? 01:07:07.210 --> 01:07:10.300 Well, this one here is actually cancelled by wave function 01:07:10.300 --> 01:07:16.030 renormalization, which I haven't put in yet. 01:07:21.060 --> 01:07:23.670 And this one here is O1 mixing into O1. 01:07:28.080 --> 01:07:31.800 So this one here is a counterterm, if you like, 01:07:31.800 --> 01:07:32.920 for O1. 01:07:32.920 --> 01:07:41.070 And this one here is a counterterm for O2. 01:07:41.070 --> 01:07:44.160 And the language you use as you say that O1, which we're 01:07:44.160 --> 01:07:45.600 calculating, has mixed into O2. 01:08:00.900 --> 01:08:02.910 So there's two different methods we 01:08:02.910 --> 01:08:09.810 could use to carry out the renormalization here 01:08:09.810 --> 01:08:11.858 that are actually equivalent. 01:08:21.130 --> 01:08:24.550 So method one is called composite operator 01:08:24.550 --> 01:08:25.860 renormalization. 01:08:33.038 --> 01:08:35.080 It's useful to know what things are equivalent so 01:08:35.080 --> 01:08:39.140 that you know what you can get away with ignoring. 01:08:39.140 --> 01:08:42.790 So what is composite operator renormalization? 01:08:42.790 --> 01:08:46.189 Well, you think about having a Bayer operator. 01:08:46.189 --> 01:08:49.431 And you need to introduce some constants to renormalize. 01:08:52.408 --> 01:08:54.700 These are not related to wave function renormalization. 01:08:54.700 --> 01:08:56.859 It's actually operator renormalization. 01:08:56.859 --> 01:08:59.263 You have multiple fields at the same spacetime point, 01:08:59.263 --> 01:09:01.180 you need additional renormalization constants, 01:09:01.180 --> 01:09:02.263 that's what these Z's are. 01:09:05.255 --> 01:09:06.880 When you go to take the matrix element, 01:09:06.880 --> 01:09:08.830 you have to include the wave function renormalization 01:09:08.830 --> 01:09:09.350 as well. 01:09:09.350 --> 01:09:17.760 So the Bayer matrix element has a wave function renormalization 01:09:17.760 --> 01:09:21.029 as well as this Z. And this is the relation 01:09:21.029 --> 01:09:26.370 between the renormalized matrix element and the Bayer one. 01:09:26.370 --> 01:09:33.270 So this guy here is renormalized and amputated. 01:09:40.510 --> 01:09:41.010 OK. 01:09:41.010 --> 01:09:43.759 So that's the relation you can calculate O0. 01:09:43.759 --> 01:09:45.330 That's what we were doing. 01:09:45.330 --> 01:09:47.109 We just calculated O0. 01:09:47.109 --> 01:09:48.223 We could remove Z psi. 01:09:48.223 --> 01:09:49.890 And I just told you if you remove Z psi, 01:09:49.890 --> 01:09:51.399 it's going to get rid of this. 01:09:51.399 --> 01:09:53.732 And then you use the remainder of this stuff to get Zij. 01:10:11.820 --> 01:10:13.570 I'm going to need it the other way around. 01:10:13.570 --> 01:10:14.445 So let me write the-- 01:10:18.680 --> 01:10:21.502 so I can write this relation the other way around. 01:10:21.502 --> 01:10:22.880 And then it looks like this. 01:10:26.512 --> 01:10:27.920 OK, so that's one method. 01:10:27.920 --> 01:10:32.360 The other one is related more to how we usually 01:10:32.360 --> 01:10:35.640 think about things in terms of gauge theory. 01:10:35.640 --> 01:10:37.700 And that is to have counterterm coefficients 01:10:37.700 --> 01:10:41.480 for the various operators. 01:10:44.870 --> 01:10:48.290 So start with the Hamiltonian written in terms of Bayer 01:10:48.290 --> 01:10:54.560 coefficients and operators that have Bayer fields-- 01:10:54.560 --> 01:10:58.080 Bayer operators, if you like. 01:10:58.080 --> 01:11:00.540 And now switch over to renormalized quantities, 01:11:00.540 --> 01:11:03.230 so you get Z psi squared. 01:11:03.230 --> 01:11:07.533 We have to switch over from the Bayer coefficient 01:11:07.533 --> 01:11:08.450 to a renormalized one. 01:11:08.450 --> 01:11:10.370 So let me do that first. 01:11:10.370 --> 01:11:13.510 So we get some Z for the coefficient, 01:11:13.510 --> 01:11:16.540 some renormalized coefficients. 01:11:16.540 --> 01:11:19.370 And then we get a Z psi squared from the four fermions that 01:11:19.370 --> 01:11:21.170 are in this operator. 01:11:21.170 --> 01:11:26.440 Then we get an O. 01:11:26.440 --> 01:11:35.200 And then we can write this in terms of Ci Oi plus Z psi 01:11:35.200 --> 01:11:44.863 squared Zij C minus delta ij Cj Oi. 01:11:44.863 --> 01:11:46.530 And then this would be our counterterms. 01:11:49.500 --> 01:11:53.670 So we would stick in this guy, that operator 01:11:53.670 --> 01:11:56.110 we haven't renormalized-- we haven't done our operator 01:11:56.110 --> 01:11:57.000 renormalization. 01:11:57.000 --> 01:11:58.583 We're not doing method 1. 01:11:58.583 --> 01:12:00.750 So this, the matrix element here will still diverge, 01:12:00.750 --> 01:12:02.981 but we cancel them off with the counterterms. 01:12:05.750 --> 01:12:09.310 So that's the logic of method 2. 01:12:09.310 --> 01:12:11.380 And once you do that, the divergences 01:12:11.380 --> 01:12:17.410 cancel between these two terms, and you're left, if you like-- 01:12:17.410 --> 01:12:19.090 or even if you don't-- 01:12:19.090 --> 01:12:21.090 with the coefficients times the operators, which 01:12:21.090 --> 01:12:25.125 are renormalized, both renormalized, 01:12:25.125 --> 01:12:28.455 even renormalized separately. 01:12:34.500 --> 01:12:35.765 And these two are equivalent. 01:12:35.765 --> 01:12:38.627 These are equivalent ways of thinking about the same thing. 01:12:38.627 --> 01:12:41.210 And we can even go further and derive the equivalence of them. 01:12:44.547 --> 01:12:46.880 So if we take this theory and we take the matrix element 01:12:46.880 --> 01:12:51.380 of the Hamiltonian in this second way of doing things, 01:12:51.380 --> 01:12:52.070 what do we get? 01:12:57.240 --> 01:12:58.860 We get this. 01:12:58.860 --> 01:13:00.620 So even though as Oi, I said, we haven't 01:13:00.620 --> 01:13:02.540 done operator renormalization, so we still 01:13:02.540 --> 01:13:12.780 have the divergences there, that's Cj Oj. 01:13:12.780 --> 01:13:15.260 And if we write the OJ using the relation 01:13:15.260 --> 01:13:22.010 at the top of the board, then we get Cj Zji inverse Z 01:13:22.010 --> 01:13:27.388 psi squared Oi 0. 01:13:27.388 --> 01:13:29.180 And then we can just look at the two sides. 01:13:29.180 --> 01:13:31.070 They both have the Z psi squared. 01:13:31.070 --> 01:13:33.590 They both have a Cj, they both have Oi 0. 01:13:33.590 --> 01:13:38.400 The only thing that's different is this. 01:13:38.400 --> 01:13:43.440 So we find the Zij for the coefficient 01:13:43.440 --> 01:13:48.410 is the transpose of the inverse of the Z's for the operators. 01:13:48.410 --> 01:13:51.420 So that's a more definite way of saying the two would 01:13:51.420 --> 01:13:52.462 lead to the same results. 01:13:52.462 --> 01:13:53.878 And this is how you would actually 01:13:53.878 --> 01:13:56.050 get the relation between the two ways of doing it. 01:13:59.630 --> 01:14:02.420 So it's not completely trivial. 01:14:02.420 --> 01:14:05.430 You have to know you have to take the inverse and transpose. 01:14:05.430 --> 01:14:09.290 But it's the same information. 01:14:09.290 --> 01:14:09.790 OK. 01:14:09.790 --> 01:14:19.950 So in our example, the Z is a matrix. 01:14:19.950 --> 01:14:22.950 It starts out as the unit matrix. 01:14:22.950 --> 01:14:28.270 And in MS bar, we collect the divergences that are not 01:14:28.270 --> 01:14:32.437 cancelled by wave function renormalization. 01:14:32.437 --> 01:14:34.270 And that gives us this little 2-by-2 matrix. 01:14:56.720 --> 01:14:58.660 And from that little 2-by-2 matrix, 01:14:58.660 --> 01:15:02.051 we can construct anomalous dimension for the operators. 01:15:02.051 --> 01:15:02.830 So let's do that. 01:15:12.440 --> 01:15:13.360 So how do we do that? 01:15:16.040 --> 01:15:18.250 Well, if we take the renormalization of the Bayer 01:15:18.250 --> 01:15:21.130 operators, remember those dependent on the regulator, 01:15:21.130 --> 01:15:23.260 but they didn't depend on the scale mu. 01:15:23.260 --> 01:15:25.450 So if we take mu d by d mu of Oi 0, 01:15:25.450 --> 01:15:27.770 which is the Bayer operator at 0. 01:15:27.770 --> 01:15:30.560 That's what I've written. 01:15:30.560 --> 01:15:33.550 And then we can write O in terms of the Z, 01:15:33.550 --> 01:15:36.690 in the equation I raised. 01:15:36.690 --> 01:15:40.040 And both Z and the renormalized operator 01:15:40.040 --> 01:15:42.157 depend on the scale mu. 01:15:42.157 --> 01:15:43.615 And so that gives us this equation. 01:15:52.170 --> 01:15:54.490 So I'm doing method 1 here, if you like, 01:15:54.490 --> 01:15:56.650 using the notations from method 1. 01:16:01.062 --> 01:16:02.770 If I take that equation and rearrange it, 01:16:02.770 --> 01:16:04.330 I can write it as anomalous dimension 01:16:04.330 --> 01:16:05.720 equation for the operator. 01:16:05.720 --> 01:16:08.830 So it's mu d by d mu is one of the terms. 01:16:08.830 --> 01:16:11.230 I isolate this, and I move that over by taking 01:16:11.230 --> 01:16:12.400 an inverse of it. 01:16:16.620 --> 01:16:19.540 And I just call everything that I 01:16:19.540 --> 01:16:23.288 get there the anomalous dimension of the operator. 01:16:27.590 --> 01:16:33.215 So gamma ji is all the other stuff, Zjk inverse. 01:16:43.310 --> 01:16:45.050 And I put the explicit minus sign here 01:16:45.050 --> 01:16:46.800 so there's no minus sign in that equation. 01:16:50.095 --> 01:16:52.220 Anomalous dimension is determined by the Z factors. 01:16:54.952 --> 01:16:56.660 And we determine the Z factors, and which 01:16:56.660 --> 01:16:59.660 we can stick this equation into that equation 01:16:59.660 --> 01:17:01.319 and get the anomalous dimension. 01:17:11.600 --> 01:17:14.900 When we do that, we have to be careful about the fact 01:17:14.900 --> 01:17:18.355 that alpha s in d dimensions has a little extra piece that's 01:17:18.355 --> 01:17:20.230 actually important for this discussion, which 01:17:20.230 --> 01:17:22.870 is this piece. 01:17:22.870 --> 01:17:26.645 So when we take mu d by d mu, you ask, what depends on mu? 01:17:26.645 --> 01:17:29.190 And it's the alpha here that depends on mu. 01:17:29.190 --> 01:17:30.960 And it's still in-- 01:17:30.960 --> 01:17:34.180 depends on epsilon as well in that equation. 01:17:34.180 --> 01:17:37.320 And so this term is the term that matters at one loop. 01:17:40.410 --> 01:17:43.390 And this matters at two loops, and so does that. 01:17:43.390 --> 01:17:47.610 But at one loop, we could just replace this by the identity, 01:17:47.610 --> 01:17:48.630 and we can drop this. 01:17:57.820 --> 01:17:58.920 OK. 01:17:58.920 --> 01:18:00.280 Put those things together. 01:18:00.280 --> 01:18:02.730 Anomalous dimension, of course, is something 01:18:02.730 --> 01:18:05.610 that doesn't depend on epsilon. 01:18:05.610 --> 01:18:07.545 And it's, in this case, a 2-by-2 matrix. 01:18:19.060 --> 01:18:21.657 And we can solve this problem. 01:18:21.657 --> 01:18:23.865 If we want to solve this matrix, then what do you do? 01:18:23.865 --> 01:18:25.660 Well, you could diagonalize it. 01:18:25.660 --> 01:18:28.920 That's how you would solve it. 01:18:28.920 --> 01:18:31.420 So if you want to run the operators in this theory just 01:18:31.420 --> 01:18:44.220 for completeness, you would diagonalize 01:18:44.220 --> 01:18:49.120 by forming O1 plus or minus O2, call the coefficients of those 01:18:49.120 --> 01:18:53.715 new operators, C plus or minus, and an obvious notation. 01:18:56.450 --> 01:18:58.830 And when you write down an anomalous dimension 01:18:58.830 --> 01:19:01.020 equations for these guys, there's 01:19:01.020 --> 01:19:02.720 no mixing anymore at one loop. 01:19:05.855 --> 01:19:07.230 In general, this is the procedure 01:19:07.230 --> 01:19:08.855 if you do this diagonalization that you 01:19:08.855 --> 01:19:12.210 have to carry out again when you get to two loops. 01:19:12.210 --> 01:19:14.550 It's not like the basis that you pick at one loop 01:19:14.550 --> 01:19:17.698 will be fine for two loops. 01:19:17.698 --> 01:19:18.490 We're not so lucky. 01:19:21.380 --> 01:19:27.435 But at one loop, this is a perfectly valid basis 01:19:27.435 --> 01:19:28.430 for the problem. 01:19:31.790 --> 01:19:34.870 And in an hopefully self-evident notation, 01:19:34.870 --> 01:19:36.580 I either take all pluses or all minuses. 01:19:57.430 --> 01:19:58.880 OK. 01:19:58.880 --> 01:20:03.470 And if I write my Hamiltonian, I could write it 01:20:03.470 --> 01:20:05.500 in the original basis. 01:20:05.500 --> 01:20:08.810 Then if I switch to the other basis, 01:20:08.810 --> 01:20:12.110 I'll set up my convention so that it's simple. 01:20:12.110 --> 01:20:15.190 And that means I picked C plus or minus 01:20:15.190 --> 01:20:18.334 to be C1 plus or minus C2/2. 01:20:18.334 --> 01:20:21.707 AUDIENCE: [INAUDIBLE] minus? 01:20:21.707 --> 01:20:23.040 IAIN STEWART: Whoops, thank you. 01:20:29.140 --> 01:20:31.430 Yeah, they have the opposite sign. 01:20:31.430 --> 01:20:36.980 First one's positive, second one's negative. 01:20:36.980 --> 01:20:42.180 And in this new basis tree level matching, 01:20:42.180 --> 01:20:48.970 which is your boundary condition for your revolution, 01:20:48.970 --> 01:20:50.830 is that both coefficients are 1/2. 01:20:54.970 --> 01:20:55.470 OK. 01:20:55.470 --> 01:20:59.370 So then we could solve the differential equation 01:20:59.370 --> 01:21:00.990 the same way we were doing for QCD-- 01:21:00.990 --> 01:21:02.810 write down a result that would sum 01:21:02.810 --> 01:21:05.540 logs below the scale of the W mass. 01:21:10.020 --> 01:21:10.590 OK. 01:21:10.590 --> 01:21:14.860 So we'll continue along these lines next time, 01:21:14.860 --> 01:21:17.092 say a few more words about renormalization, 01:21:17.092 --> 01:21:19.300 which is really just a lead-up of doing the matching, 01:21:19.300 --> 01:21:20.810 which is what I want to-- 01:21:20.810 --> 01:21:23.360 kind of the thing I want to spend a little more time on. 01:21:23.360 --> 01:21:25.995 So we'll finish that up also next time 01:21:25.995 --> 01:21:29.400 and move on to some other things. 01:21:29.400 --> 01:21:30.900 So there's a few things to emphasize 01:21:30.900 --> 01:21:31.710 when we do the matching. 01:21:31.710 --> 01:21:33.543 And I want to write down the matrix elements 01:21:33.543 --> 01:21:36.020 and the full theory, which are box diagrams, 01:21:36.020 --> 01:21:38.143 I'll write down the results for those. 01:21:38.143 --> 01:21:39.560 We'll compare those to the results 01:21:39.560 --> 01:21:40.977 of the effective theory, and we'll 01:21:40.977 --> 01:21:42.195 draw some lessons from that. 01:21:42.195 --> 01:21:43.570 And we'll carry out the matching, 01:21:43.570 --> 01:21:48.690 figure out what the next order coefficients would be here. 01:21:48.690 --> 01:21:51.535 What would those alpha s terms be? 01:21:51.535 --> 01:21:53.430 What's the procedure I would go through 01:21:53.430 --> 01:21:55.440 to get those coefficients? 01:21:55.440 --> 01:21:55.940 OK. 01:21:55.940 --> 01:21:58.340 So we'll stop there for today.