WEBVTT 00:00:00.000 --> 00:00:02.520 The following content is provided under a Creative 00:00:02.520 --> 00:00:03.970 Commons license. 00:00:03.970 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.660 continue to offer high-quality educational resources for free. 00:00:10.660 --> 00:00:13.350 To make a donation or view additional materials 00:00:13.350 --> 00:00:17.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.190 --> 00:00:18.322 at ocw.mit.edu. 00:00:23.600 --> 00:00:26.390 PROFESSOR: So here's where we were. 00:00:26.390 --> 00:00:30.110 So we were talking about this example of b to s gamma 00:00:30.110 --> 00:00:31.370 and a heavy to light current. 00:00:35.640 --> 00:00:37.830 In terms of this chi field, our current 00:00:37.830 --> 00:00:43.470 was chi, which is going to have some label, 00:00:43.470 --> 00:00:45.630 in a heavy quark field. 00:00:45.630 --> 00:00:48.060 And we did the one with diagrams in SET. 00:00:48.060 --> 00:00:52.020 We saw that after we take into account also the 0 bin 00:00:52.020 --> 00:00:53.950 contribution that it breaks up as follows. 00:00:53.950 --> 00:00:55.450 There's a piece that I've underlined 00:00:55.450 --> 00:00:58.395 in orange that matches exactly with the IR of QCD. 00:01:04.285 --> 00:01:05.660 There's a piece that I underlined 00:01:05.660 --> 00:01:08.390 in blue which is the ultraviolet divergences 00:01:08.390 --> 00:01:09.740 in the effective theory. 00:01:09.740 --> 00:01:11.150 And so what we do with that is we 00:01:11.150 --> 00:01:13.290 add a counter-term for the operators 00:01:13.290 --> 00:01:16.257 and the effective theory to cancel these divergences. 00:01:16.257 --> 00:01:17.840 And then there's whatever is left over 00:01:17.840 --> 00:01:19.940 after doing that, which is this pink. 00:01:19.940 --> 00:01:22.520 And the difference of the analog pink 00:01:22.520 --> 00:01:25.610 in the full theory and this pink will give you the matching. 00:01:25.610 --> 00:01:26.990 So when I say "matching," this is 00:01:26.990 --> 00:01:29.500 part of what goes into that matching. 00:01:29.500 --> 00:01:31.250 So what I want to spend today's lecture on 00:01:31.250 --> 00:01:33.560 is really discussing renormalization 00:01:33.560 --> 00:01:35.840 in the effective theory and summing logarithms 00:01:35.840 --> 00:01:37.300 in the effective theory. 00:01:37.300 --> 00:01:40.670 It's kind of interesting and different than things 00:01:40.670 --> 00:01:43.280 we've encountered before, because we see even 00:01:43.280 --> 00:01:45.290 at one loop order, we have 1 over 00:01:45.290 --> 00:01:46.880 epsilon squared divergence. 00:01:46.880 --> 00:01:49.250 And that wasn't something that we encountered before. 00:01:49.250 --> 00:01:51.710 And so we have to see how that plays out. 00:02:06.050 --> 00:02:08.660 But just at the simplistic level, 00:02:08.660 --> 00:02:14.930 let's just say or record for the record 00:02:14.930 --> 00:02:17.390 that we could write the following. 00:02:17.390 --> 00:02:20.730 There was a Wilson coefficient of this operator. 00:02:20.730 --> 00:02:23.690 And we can divide it into renormalize Wilson coefficient 00:02:23.690 --> 00:02:25.292 in a counterterm. 00:02:25.292 --> 00:02:27.125 And then we just think that this counterterm 00:02:27.125 --> 00:02:29.084 is canceling that divergence. 00:02:32.880 --> 00:02:39.327 And so the counterterm we need is the following. 00:02:39.327 --> 00:02:41.410 So with this minus sign canceling that minus sign, 00:02:41.410 --> 00:02:42.640 you get plus. 00:02:42.640 --> 00:02:46.510 And so the counterterm for the operator therefore 00:02:46.510 --> 00:02:51.280 should have a minus and just all-- 00:02:51.280 --> 00:02:52.330 exactly these terms. 00:03:03.910 --> 00:03:06.970 Because of kinematics in this problem, 00:03:06.970 --> 00:03:10.570 this omega, which is a delta function in this operator, 00:03:10.570 --> 00:03:12.910 is equal to actually mb. 00:03:12.910 --> 00:03:17.480 And I'll go through why that is a little later today. 00:03:17.480 --> 00:03:20.290 So take it for granted for now. 00:03:20.290 --> 00:03:23.180 It comes about just simply from kinematics. 00:03:23.180 --> 00:03:25.840 But I'm going to sometimes write the mb as omega 00:03:25.840 --> 00:03:29.160 because that reminds you that it was this label in the colinear 00:03:29.160 --> 00:03:31.368 field. 00:03:31.368 --> 00:03:33.160 So this is the type of counterterm we have. 00:03:33.160 --> 00:03:34.702 It's got this 1 over epsilon squared. 00:03:34.702 --> 00:03:39.790 It's got this log, so it looks kind of different. 00:03:39.790 --> 00:03:52.240 And the notation here is that we have 00:03:52.240 --> 00:04:02.080 a coefficient that's inside an integral, where 00:04:02.080 --> 00:04:10.160 this thing meant that. 00:04:16.750 --> 00:04:20.290 So basically what I'm saying when I say omega is equal to mb 00:04:20.290 --> 00:04:26.250 is that this delta function becomes omega minus mb. 00:04:26.250 --> 00:04:29.220 We'll come back to that. 00:04:29.220 --> 00:04:32.970 So if we have a counterterm like that, then we can do running. 00:04:38.190 --> 00:04:40.320 So we'll come back to this example. 00:04:40.320 --> 00:04:42.450 Let me first make a few general comments, 00:04:42.450 --> 00:04:45.510 and then we'll come back and see where this z leads us. 00:04:52.840 --> 00:04:56.500 So in general, you're going to have this structure-- 00:04:56.500 --> 00:04:58.670 some integral, some Wilson coefficients that depend 00:04:58.670 --> 00:05:00.388 on some omegas-- 00:05:00.388 --> 00:05:02.680 could be more than one of them-- and then some operator 00:05:02.680 --> 00:05:05.305 that has some delta functions in it that pick out those omegas. 00:05:07.850 --> 00:05:11.690 And because of that integral, in general, 00:05:11.690 --> 00:05:21.250 you have to be careful, because that integral could also play 00:05:21.250 --> 00:05:23.560 a role in the renormalization. 00:05:39.280 --> 00:05:43.030 So that delta function is picking out 00:05:43.030 --> 00:05:45.180 the total momentum of the product of the quark 00:05:45.180 --> 00:05:47.860 field times the Wilson line. 00:05:47.860 --> 00:05:51.520 And in our example, it's fixed by external kinematics. 00:05:51.520 --> 00:05:53.590 So I'll come back to this and show you 00:05:53.590 --> 00:05:57.340 that not in every example is it fixed by external kinematics 00:05:57.340 --> 00:05:58.480 in a minute. 00:05:58.480 --> 00:06:00.850 But our example is special because the integral actually 00:06:00.850 --> 00:06:02.590 doesn't play an important role. 00:06:32.870 --> 00:06:36.850 So there's in some sense two parts of this statement. 00:06:36.850 --> 00:06:38.500 There is the fact that it's external, 00:06:38.500 --> 00:06:40.930 and then there's the fact that it's fixed to something. 00:06:40.930 --> 00:06:45.520 So let me first deal with the issue of it being external. 00:06:45.520 --> 00:06:51.100 And what that means is it does not involve loop momenta. 00:06:51.100 --> 00:06:58.120 You see, when we did the calculation, 00:06:58.120 --> 00:07:00.958 we were basically doing a calculation, 00:07:00.958 --> 00:07:03.250 not worrying so much about this delta function that was 00:07:03.250 --> 00:07:06.250 sitting inside our operator. 00:07:06.250 --> 00:07:07.180 We ignored it. 00:07:07.180 --> 00:07:09.232 We didn't write it down in our diagrams. 00:07:09.232 --> 00:07:10.690 Technically, we should have written 00:07:10.690 --> 00:07:13.150 that delta function, this delta function, down, too, 00:07:13.150 --> 00:07:15.400 and included it in our calculation. 00:07:15.400 --> 00:07:18.040 But actually that's OK because that delta function just comes 00:07:18.040 --> 00:07:19.630 outside the loop integrals. 00:07:19.630 --> 00:07:21.840 That's what I want to show you. 00:07:26.250 --> 00:07:30.930 So it does not involve loop momenta. 00:07:30.930 --> 00:07:35.370 So in all the diagrams we consider, 00:07:35.370 --> 00:07:38.220 that's trivial, except for one non-trivial example. 00:07:40.890 --> 00:07:42.990 And that non-trivial example is the example 00:07:42.990 --> 00:07:48.000 where we had a colinear vertex-type diagram like this. 00:07:51.760 --> 00:07:55.080 So we called the external momentum here p. 00:07:55.080 --> 00:07:56.360 This was p plus k. 00:07:56.360 --> 00:08:00.270 And if I let this guy go this way, we can call it minus k. 00:08:00.270 --> 00:08:04.710 You see, the Wilson coefficient sits here at the vertex. 00:08:04.710 --> 00:08:06.420 And it acts on the colinear fields 00:08:06.420 --> 00:08:08.730 that come out of that vertex. 00:08:08.730 --> 00:08:11.340 But it gives some of the momentum of those colinear 00:08:11.340 --> 00:08:11.860 fields. 00:08:11.860 --> 00:08:22.850 So we get this guy, and then we get this guy. 00:08:22.850 --> 00:08:24.700 And so we just kept the external guy. 00:08:30.200 --> 00:08:33.620 So from this notation, the same thing as saying this delta 00:08:33.620 --> 00:08:35.700 function. 00:08:35.700 --> 00:08:39.919 We can see that it only depends on the external p, 00:08:39.919 --> 00:08:41.240 on the n bar dot k. 00:08:49.840 --> 00:08:53.590 And that's because of the structure of this operator. 00:08:58.290 --> 00:08:59.500 So that won't always be true. 00:08:59.500 --> 00:09:02.010 It could be the case that the Wilson-- in a more complicated 00:09:02.010 --> 00:09:03.990 example, that the Wilson coefficient 00:09:03.990 --> 00:09:05.000 could depend on the k. 00:09:05.000 --> 00:09:06.750 And then that would have some implications 00:09:06.750 --> 00:09:08.160 for renormalization. 00:09:08.160 --> 00:09:11.640 And we will treat an example of that sort in the near future. 00:09:15.780 --> 00:09:17.870 The second thing to ask is, what does it fix to? 00:09:20.660 --> 00:09:23.030 So if we put a little more into our diagram, 00:09:23.030 --> 00:09:24.980 let's put the photon in. 00:09:24.980 --> 00:09:26.390 This is a b quark. 00:09:26.390 --> 00:09:31.140 This is a strange quark b to s gamma, for example. 00:09:31.140 --> 00:09:35.120 Then the kinematics of the external lines there is we 00:09:35.120 --> 00:09:38.840 have mbv coming in for the b quark, p of the photon, 00:09:38.840 --> 00:09:42.800 and p of the external strange quark. 00:09:42.800 --> 00:09:46.190 And the photon is light-like. p gamma squared is 0. 00:09:46.190 --> 00:09:49.850 And in the setup here, we can just 00:09:49.850 --> 00:09:53.630 think of it as e gamma times n bar, 00:09:53.630 --> 00:09:57.230 because p is colinear to the n direction, and it's going-- 00:09:57.230 --> 00:10:00.200 we have the photon going this way. 00:10:00.200 --> 00:10:04.630 And it's back to back with a jet that's 00:10:04.630 --> 00:10:10.580 described by the colinear n direction, 00:10:10.580 --> 00:10:12.070 just as the beam goes on decaying 00:10:12.070 --> 00:10:14.710 to a photon in a back-to-back jet. 00:10:14.710 --> 00:10:20.281 And so if you dot into this equation n bar, 00:10:20.281 --> 00:10:24.430 since n bar dot v is 1 in conventional coordinates, 00:10:24.430 --> 00:10:30.940 we just get that mb is 0 plus n bar dot p. 00:10:30.940 --> 00:10:34.570 And that's the statement that omega was just mb. 00:10:34.570 --> 00:10:36.850 So when we were doing our loop calculations last time, 00:10:36.850 --> 00:10:38.290 we kept writing mb. 00:10:38.290 --> 00:10:40.750 We could have been more careful and written-- 00:10:40.750 --> 00:10:44.140 well, we couldn't have been more careful, because kinematics was 00:10:44.140 --> 00:10:46.000 demanding that this was true. 00:10:46.000 --> 00:10:48.940 But just in order to track what the Wilson coefficients can 00:10:48.940 --> 00:10:51.040 depend on, really what they can depend on 00:10:51.040 --> 00:10:53.680 is these labels of these colinear fields. 00:10:53.680 --> 00:10:57.250 And so I'm going to start writing omega for that reason 00:10:57.250 --> 00:10:59.837 because that's the more generic thing that happens. 00:10:59.837 --> 00:11:01.420 Even if we're not in a situation where 00:11:01.420 --> 00:11:03.660 it's fixed to be some mass, we'll 00:11:03.660 --> 00:11:05.500 be able to get dependence on the values 00:11:05.500 --> 00:11:07.300 of these parameters of omega. 00:11:12.900 --> 00:11:14.330 So any questions about that? 00:11:18.532 --> 00:11:20.740 All right, so this Wilson coefficient just comes out, 00:11:20.740 --> 00:11:23.980 and we didn't have to worry about this delta function. 00:11:23.980 --> 00:11:28.110 So how do we calculate the anomalous dimension? 00:11:28.110 --> 00:11:30.300 It's the usual story. 00:11:30.300 --> 00:11:34.110 So mu d by dbu of the bare coefficient is 0. 00:11:34.110 --> 00:11:36.450 That implies a renormalization group equation 00:11:36.450 --> 00:11:40.326 for the ms bar coefficient. 00:11:52.490 --> 00:11:57.590 And it's got this familiar form, where the gamma is products 00:11:57.590 --> 00:11:59.486 of z's and z inverses-- 00:12:06.300 --> 00:12:07.230 just this. 00:12:11.660 --> 00:12:13.550 So this is an all-orders formula. 00:12:13.550 --> 00:12:15.230 If we plug in our one loop result 00:12:15.230 --> 00:12:20.653 here, then on the left, if we worked order by order in alpha, 00:12:20.653 --> 00:12:22.070 on the left-hand side, we can just 00:12:22.070 --> 00:12:25.580 take the 1 in the zc universe here. 00:12:25.580 --> 00:12:28.940 And the zc, when we take b by d by d mu, 00:12:28.940 --> 00:12:32.010 there's mu in the alpha s, and there's mu in this log. 00:12:32.010 --> 00:12:35.850 And we're going to have to differentiate both of those. 00:12:35.850 --> 00:12:39.500 So let me write it out. 00:12:59.260 --> 00:13:01.090 So the minus sign canceled the minus sign. 00:13:01.090 --> 00:13:02.772 The zc inverse is just 1. 00:13:02.772 --> 00:13:05.230 And then I have b db dbu over the rest of this stuff, which 00:13:05.230 --> 00:13:10.210 includes both the alpha and the log mu over w. 00:13:39.135 --> 00:13:54.510 These are the type of contributions we get, 00:13:54.510 --> 00:13:58.860 where these guys here are coming from using the fact that 00:13:58.860 --> 00:14:04.650 mu d by d mu of alpha when you're in d dimensions is minus 00:14:04.650 --> 00:14:07.290 2 epsilon alpha plus-- 00:14:07.290 --> 00:14:09.660 order alpha squared term's related to the beta function, 00:14:09.660 --> 00:14:11.580 which we can drop here. 00:14:11.580 --> 00:14:16.770 And then this guy comes from the explicit-- 00:14:16.770 --> 00:14:22.110 differentiating the explicit log mu over w. 00:14:22.110 --> 00:14:23.910 The divergent terms cancel, and we're 00:14:23.910 --> 00:14:26.170 left with a finite anomalous dimension. 00:14:36.670 --> 00:14:38.380 And it's got two types of terms in it. 00:14:42.770 --> 00:14:44.320 So the type of anomalous dimensions 00:14:44.320 --> 00:14:48.760 that we're used to are like this guy, here-- this 5 times cf. 00:14:48.760 --> 00:14:50.920 That would be like a standard anomalous dimension 00:14:50.920 --> 00:14:52.157 for an operator. 00:14:52.157 --> 00:14:53.740 In this case, we have this extra piece 00:14:53.740 --> 00:14:56.500 that has this log of mu over w. 00:14:56.500 --> 00:15:03.370 And this guy is what's called cusp anomalous dimension, 00:15:03.370 --> 00:15:05.140 the guy that multiplies the log. 00:15:09.530 --> 00:15:11.480 So with that language, you could call this guy 00:15:11.480 --> 00:15:14.690 the regular anomalous dimension. 00:15:22.370 --> 00:15:24.440 Now, if you're going to do renormalization group 00:15:24.440 --> 00:15:26.648 evolution, you're going to sum logs. 00:15:26.648 --> 00:15:28.940 You could think that you're starting at some hard scale 00:15:28.940 --> 00:15:30.470 like w. w is mb. 00:15:30.470 --> 00:15:32.030 That's the hard scale. 00:15:32.030 --> 00:15:35.750 And then you want to run down to some small-scale mu. 00:15:35.750 --> 00:15:38.040 When you run down, if you take mu small, 00:15:38.040 --> 00:15:41.060 this term is larger than this term. 00:15:41.060 --> 00:15:44.390 So actually, this term here gives you 00:15:44.390 --> 00:15:47.300 the leading-- what are called the leading logs in this case. 00:15:47.300 --> 00:15:52.670 And this term here is actually only needed 00:15:52.670 --> 00:15:55.250 for next leading log. 00:15:55.250 --> 00:15:57.320 Because this term here grows when you go down, 00:15:57.320 --> 00:16:00.770 and that term doesn't, this term here 00:16:00.770 --> 00:16:03.467 is actually giving you the leading logs. 00:16:03.467 --> 00:16:05.300 So I'll come back and talk a little bit more 00:16:05.300 --> 00:16:07.220 about leading log versus next-to-leading log. 00:16:07.220 --> 00:16:11.410 And also, I'll explain to you what this word "cusp" in this-- 00:16:11.410 --> 00:16:14.450 where's the cusp, since we haven't seen it yet? 00:16:14.450 --> 00:16:16.670 I'll come back and explain where that is in a minute. 00:16:22.180 --> 00:16:24.680 For now, let's just take the leading log anomalous dimension 00:16:24.680 --> 00:16:29.673 and see just to solve the anomalous dimension equation. 00:16:29.673 --> 00:16:31.340 So we'll just take the leading long term 00:16:31.340 --> 00:16:42.140 and not worry about the other term, 00:16:42.140 --> 00:16:45.610 so just taking the cusp term. 00:16:45.610 --> 00:16:48.520 So plugging that back into our equation for c, 00:16:48.520 --> 00:16:52.240 we have this differential equation, which we can also 00:16:52.240 --> 00:16:58.010 write as d log c by d log mu. 00:17:11.319 --> 00:17:13.210 And that's an equation that we can basically 00:17:13.210 --> 00:17:19.450 just integrate since there's no c on the right-hand side. 00:17:19.450 --> 00:17:20.990 It's a homogeneous equation. 00:17:20.990 --> 00:17:23.235 We can integrate it with some boundary condition. 00:17:27.990 --> 00:17:30.230 So we could specify the boundary condition 00:17:30.230 --> 00:17:32.570 at an arbitrary scale. 00:17:32.570 --> 00:17:36.530 Let me, for simplicity, use the following tree-level boundary 00:17:36.530 --> 00:17:37.561 condition. 00:17:41.490 --> 00:17:44.300 So at tree level, the Wilson coefficient was 1. 00:17:44.300 --> 00:17:48.110 That we have to pick some scale where 00:17:48.110 --> 00:17:51.770 we make that statement true. 00:17:51.770 --> 00:17:54.710 And for convenience, I'm just going to take that scale to bw. 00:17:54.710 --> 00:17:57.260 That makes things a little simpler. 00:17:57.260 --> 00:18:01.760 In general, you could make this any hard scale, ie, 00:18:01.760 --> 00:18:04.430 any scale mu of order w. 00:18:04.430 --> 00:18:07.610 So you could call it something mu 0 00:18:07.610 --> 00:18:11.750 and consider the renormalization group from that scale. 00:18:11.750 --> 00:18:15.230 But for convenience, I'm just taking that mu 0 to be w here. 00:18:29.740 --> 00:18:32.098 So we take that down [INAUDIBLE] It's 00:18:32.098 --> 00:18:33.640 actually useful to look at what would 00:18:33.640 --> 00:18:35.057 happen if we didn't have a running 00:18:35.057 --> 00:18:37.250 coupling on the right-hand side. 00:18:37.250 --> 00:18:40.570 So say we were doing QED, and we didn't have any massless light 00:18:40.570 --> 00:18:43.240 fermions, so there was no running. 00:18:43.240 --> 00:18:46.325 Then this coupling would just be alpha, and it wouldn't run. 00:18:46.325 --> 00:18:47.950 And then you'd just integrate this log, 00:18:47.950 --> 00:18:49.240 and you'd get a double log. 00:18:49.240 --> 00:18:53.830 You integrate one log by d log, you get log squared. 00:18:53.830 --> 00:18:55.930 For each coupling, you get log squared. 00:18:55.930 --> 00:18:58.480 This is the double log series that I promised you-- 00:18:58.480 --> 00:19:00.340 the Sudakov double logarithms that 00:19:00.340 --> 00:19:05.440 were going to come out of the renormalization group of SET. 00:19:05.440 --> 00:19:13.307 So QED with alpha fixed-- 00:19:13.307 --> 00:19:15.790 I'm going to take cf to be 1. 00:19:18.820 --> 00:19:31.190 When you take the exponential of what I just said, 00:19:31.190 --> 00:19:34.935 then this is your result for the Wilson coefficient. 00:19:40.990 --> 00:19:43.140 And this is what's known as a Sudakov exponential. 00:19:43.140 --> 00:19:49.530 The leading log Wilson coefficient 00:19:49.530 --> 00:19:51.648 is the sum of these alpha times double log. 00:19:51.648 --> 00:19:53.940 So if I expanded this out, I'd have an infinite series, 00:19:53.940 --> 00:19:57.450 where for each pair of logarithms, I get one coupling. 00:20:11.060 --> 00:20:15.170 And that's the Sudakov double logarithms. 00:20:15.170 --> 00:20:16.670 So if we expand out the exponential, 00:20:16.670 --> 00:20:18.840 that's what we would get. 00:20:18.840 --> 00:20:23.360 So really, I should make my line go under the whole thing. 00:20:27.410 --> 00:20:30.270 So why did we get something like this? 00:20:30.270 --> 00:20:34.760 So physically the reason that we got this Sudakov exponential 00:20:34.760 --> 00:20:37.790 or Sudakov form factor is related to the fact 00:20:37.790 --> 00:20:40.370 that the whole theory that we've developed 00:20:40.370 --> 00:20:42.020 put restrictions on radiation. 00:20:58.614 --> 00:21:00.540 So there was kinematic restrictions 00:21:00.540 --> 00:21:02.875 put on the radiation by the whole setup that we had. 00:21:02.875 --> 00:21:04.500 We had this jet, and it was colonnaded. 00:21:04.500 --> 00:21:07.860 And the radiation was forced to go inside that jet. 00:21:21.080 --> 00:21:23.840 In the effective theory, we're seeing the Sudakov exponential 00:21:23.840 --> 00:21:25.770 come out of UV renormalization. 00:21:25.770 --> 00:21:27.440 The usual picture for it is that you're 00:21:27.440 --> 00:21:29.388 thinking about IR divergences. 00:21:29.388 --> 00:21:31.430 And you're thinking about the real radiation, not 00:21:31.430 --> 00:21:33.320 the virtual radiation. 00:21:33.320 --> 00:21:36.710 But as you may be familiar with what the effective theory does, 00:21:36.710 --> 00:21:40.790 is if you have an IR divergence in full QCD or full QED, 00:21:40.790 --> 00:21:42.440 then what the effective theory does 00:21:42.440 --> 00:21:45.110 is it takes the scale corresponding to-- 00:21:45.110 --> 00:21:48.050 so the IR divergences are logarithms. 00:21:48.050 --> 00:21:50.240 And they're logarithms of something over something. 00:21:50.240 --> 00:21:52.160 And we've done in the effective theory 00:21:52.160 --> 00:21:54.620 is we've introduced a scale mu. 00:21:54.620 --> 00:21:56.360 We've introduced-- effectively, we've 00:21:56.360 --> 00:21:59.120 taken the part of that logarithm that was UV 00:21:59.120 --> 00:22:00.860 and taken it to infinity. 00:22:00.860 --> 00:22:02.810 And in this standard way, we've introduced 00:22:02.810 --> 00:22:07.640 a scale mu that splits things between Wilson coefficients 00:22:07.640 --> 00:22:09.260 and operators. 00:22:09.260 --> 00:22:11.580 So there's this factorization scale 00:22:11.580 --> 00:22:13.220 that's splitting the hard physics 00:22:13.220 --> 00:22:16.880 from-- which is part of the logarithm that we're capturing. 00:22:16.880 --> 00:22:19.998 It's separating the hard physics from the infrared physics. 00:22:19.998 --> 00:22:22.040 And here, we're just looking at the hard physics. 00:22:22.040 --> 00:22:23.180 And we can see the Sudakov. 00:22:23.180 --> 00:22:24.847 We could also look at the real radiation 00:22:24.847 --> 00:22:28.100 and see the Sudakov by studying real radiation diagrams. 00:22:28.100 --> 00:22:29.973 But in some sense, this is easier, 00:22:29.973 --> 00:22:32.390 because this is just coming out of a renormalization group 00:22:32.390 --> 00:22:34.453 equation. 00:22:34.453 --> 00:22:35.870 So the usual way of doing it would 00:22:35.870 --> 00:22:37.610 be to just look at in the full theory, 00:22:37.610 --> 00:22:41.060 and you'd see Sudakov logarithms from ratios of two things 00:22:41.060 --> 00:22:42.680 that you were thinking of as IR. 00:22:42.680 --> 00:22:45.890 Because the hard scale, which is mbw here, 00:22:45.890 --> 00:22:49.280 would also be an IR scale from the point of view of QCD. 00:22:49.280 --> 00:22:52.100 But here, it's become part of the Wilson coefficient. 00:22:52.100 --> 00:22:57.290 And now it can be treated by the renormalization group. 00:22:57.290 --> 00:23:01.082 So QCD is not really any more complicated. 00:23:01.082 --> 00:23:03.040 We just have to deal with the running coupling. 00:23:07.190 --> 00:23:09.900 We've done that before. 00:23:09.900 --> 00:23:15.310 Let me remind you how it works because it's slightly 00:23:15.310 --> 00:23:17.450 more involved in this case. 00:23:17.450 --> 00:23:21.220 So for the leading logs, we just need the beta 0 term. 00:23:21.220 --> 00:23:23.380 And we can switch variables from d log to d 00:23:23.380 --> 00:23:25.787 alpha in this equation, here. 00:23:25.787 --> 00:23:28.120 And so then we have to integrate this thing with respect 00:23:28.120 --> 00:23:31.120 to d alpha. 00:23:31.120 --> 00:23:34.630 And so what we can do is we can take this log, 00:23:34.630 --> 00:23:37.930 and we can also write the log as an integral over alpha. 00:23:49.340 --> 00:23:52.790 Alpha in this equation is some w variable. 00:23:52.790 --> 00:23:58.760 And so then just integrating both sides of this equation 00:23:58.760 --> 00:24:30.770 here, I can write it out like that. 00:24:30.770 --> 00:24:34.330 So the log became this thing. 00:24:34.330 --> 00:24:36.030 And then one of these factors-- 00:24:36.030 --> 00:24:37.390 there was a minus sign. 00:24:37.390 --> 00:24:39.900 But then when I use this formula twice, 00:24:39.900 --> 00:24:42.850 the minus sign goes away, and I get two of those factors. 00:24:42.850 --> 00:24:46.302 This is the other measure here from the explicit d log mu. 00:24:46.302 --> 00:24:47.910 So we could underline things-- 00:24:52.200 --> 00:24:59.565 half of this guy, all of that guy, d 00:24:59.565 --> 00:25:04.640 log mu, other half of this guy and that guy. 00:25:04.640 --> 00:25:07.270 And then there's this explicit alpha cf 00:25:07.270 --> 00:25:10.150 over pi, which I've written as cf over pi out front, 00:25:10.150 --> 00:25:11.410 and this alpha here. 00:25:23.150 --> 00:25:25.020 So we do that integral. 00:25:25.020 --> 00:25:25.910 We do this integral. 00:25:25.910 --> 00:25:27.410 Well, all the intervals are trivial. 00:25:27.410 --> 00:25:30.290 We do this integral, and then we get integration variable 00:25:30.290 --> 00:25:31.080 at fixed quantity. 00:25:31.080 --> 00:25:33.305 And then we can do this other integral. 00:25:33.305 --> 00:25:34.055 We get a solution. 00:25:57.600 --> 00:25:59.510 So I'll show you what the answer looks like. 00:25:59.510 --> 00:26:02.060 It's, again, an exponential, because we're solving for a log 00:26:02.060 --> 00:26:02.560 c. 00:26:13.090 --> 00:26:14.330 I'll write it like this. 00:26:28.150 --> 00:26:30.070 So we're familiar that renormalization group 00:26:30.070 --> 00:26:31.903 equations, when you have a running coupling, 00:26:31.903 --> 00:26:33.980 can give you ratios of alphas. 00:26:33.980 --> 00:26:38.077 We saw that when we were running for fermion operators. 00:26:38.077 --> 00:26:39.910 Here, it's just a more complicated function, 00:26:39.910 --> 00:26:42.500 but it's again a function of that ratio. 00:26:42.500 --> 00:26:44.842 And this extra structure is coming about just 00:26:44.842 --> 00:26:45.925 from the running coupling. 00:26:51.690 --> 00:26:55.480 So the only difference between this and QED Sudakov 00:26:55.480 --> 00:26:56.480 is the running coupling. 00:27:07.500 --> 00:27:14.370 So if we were to take this now, and we were to expand z, 00:27:14.370 --> 00:27:17.100 we were, say, to expand about alpha of w, 00:27:17.100 --> 00:27:20.430 then we would get logs of mu over w, right? 00:27:20.430 --> 00:27:22.920 And so what you should think that this z dependence 00:27:22.920 --> 00:27:23.610 encodes-- 00:27:23.610 --> 00:27:29.490 is an infinite series in this exponential. 00:27:29.490 --> 00:27:31.720 And so the structure of that infinite series, 00:27:31.720 --> 00:27:38.490 if I was to expand, would be alpha squared u over w alpha 00:27:38.490 --> 00:27:42.960 squared log cubed u over w, et cetera. 00:27:46.850 --> 00:27:49.850 So in the exponent, a running of the coupling 00:27:49.850 --> 00:27:52.620 is only giving you one log for each alpha. 00:27:52.620 --> 00:27:55.280 So the series that I'm solving in the exponent 00:27:55.280 --> 00:27:57.270 by taking into account the running coupling 00:27:57.270 --> 00:27:59.720 is a series of this form, where I go down by one alpha 00:27:59.720 --> 00:28:00.950 and down by one log. 00:28:04.614 --> 00:28:09.533 So from the structure of what we've been talking about here, 00:28:09.533 --> 00:28:11.950 it should become clear that the right thing to talk about, 00:28:11.950 --> 00:28:15.400 if you want to do counting, is log c, 00:28:15.400 --> 00:28:18.250 because the anomalous dimension equation was simple for log c. 00:28:18.250 --> 00:28:22.180 It was mu d by mu or d by d log mu of log c 00:28:22.180 --> 00:28:23.620 is equal to something. 00:28:23.620 --> 00:28:26.945 That something on the right-hand side is an expansion in alpha. 00:28:26.945 --> 00:28:28.570 And we could include higher-order terms 00:28:28.570 --> 00:28:31.090 in that expansion, but it would always be of that structure. 00:28:43.220 --> 00:28:47.440 So to talk about what terms we're summing, 00:28:47.440 --> 00:28:50.830 we can write the following schematic equation 00:28:50.830 --> 00:28:54.080 for different orders. 00:28:54.080 --> 00:28:57.040 So if we have alpha s to the k log 00:28:57.040 --> 00:29:00.430 to the k plus 1, which is this series I've just denoted, 00:29:00.430 --> 00:29:01.810 we'll call that leading log. 00:29:04.660 --> 00:29:07.360 And if we add some higher-order term, 00:29:07.360 --> 00:29:10.295 like alpha s to the k log to the k, 00:29:10.295 --> 00:29:11.920 then that would be next-to-leading log. 00:29:19.910 --> 00:29:22.800 And if we had more alphas than logs, 00:29:22.800 --> 00:29:24.740 then we call that next-to-next-to leading log. 00:29:28.072 --> 00:29:30.280 And the structure of the anomalous dimension equation 00:29:30.280 --> 00:29:34.180 always guarantees that if you solve it for this log, 00:29:34.180 --> 00:29:36.580 that this is what the higher-order terms would 00:29:36.580 --> 00:29:38.810 give you. 00:29:38.810 --> 00:29:42.010 So you should think of it that really, log counting 00:29:42.010 --> 00:29:44.140 is kind of like normal log counting, except for two 00:29:44.140 --> 00:29:46.720 caveats in this theory. 00:29:46.720 --> 00:29:49.900 You're doing it in the exponential. 00:29:49.900 --> 00:29:51.790 That's what taking the log did. 00:29:51.790 --> 00:29:55.270 And you start out with one extra log, one more log 00:29:55.270 --> 00:29:57.200 than you're used to. 00:29:57.200 --> 00:29:59.440 But after that, it's kind of normal, 00:29:59.440 --> 00:30:02.560 because I'm summing here alpha times log, 00:30:02.560 --> 00:30:05.100 and I just increment for each alpha an extra log. 00:30:05.100 --> 00:30:06.940 And that's all the running coupling effects. 00:30:06.940 --> 00:30:08.865 When I go to next-to-leading log, of course, 00:30:08.865 --> 00:30:10.990 we write in coupling effects, too, and other things 00:30:10.990 --> 00:30:13.390 that are causing the series. 00:30:13.390 --> 00:30:15.100 But it's down from this one because it 00:30:15.100 --> 00:30:17.050 doesn't have that enhanced log. 00:30:17.050 --> 00:30:20.530 And then this one is down again because it's, again, 00:30:20.530 --> 00:30:22.828 one less log. 00:30:22.828 --> 00:30:24.370 So this is the kind of terms that you 00:30:24.370 --> 00:30:26.890 could sum by having higher-order corrections 00:30:26.890 --> 00:30:29.080 in the anomalous dimensions. 00:30:29.080 --> 00:30:31.010 And for example, this cusp anomalous dimension 00:30:31.010 --> 00:30:32.260 is known at three loop orders. 00:30:32.260 --> 00:30:37.880 So certainly, next-to-next-to leading log is well within 00:30:37.880 --> 00:30:40.760 the realm of things that people talk about. 00:30:40.760 --> 00:30:42.110 So let's ask that question. 00:30:42.110 --> 00:30:45.220 What coefficients, if we were to carry out this one 00:30:45.220 --> 00:30:48.423 new calculation, and we were to do higher loops, 00:30:48.423 --> 00:30:50.090 what coefficients do we need to compute? 00:30:56.050 --> 00:30:58.860 So remember the story when we were summing single log series 00:30:58.860 --> 00:31:00.480 was we would do tree-level matching 00:31:00.480 --> 00:31:02.018 one loop anomalous dimension. 00:31:02.018 --> 00:31:04.560 Then we'd do one loop matching, two loop anomalous dimension. 00:31:04.560 --> 00:31:06.330 What's the analog of that story here? 00:31:12.857 --> 00:31:13.940 Let's make a little table. 00:31:18.120 --> 00:31:22.220 So what information do we need at tree level? 00:31:22.220 --> 00:31:25.580 What information do we need at one loop, 00:31:25.580 --> 00:31:29.000 two loops, three loops? 00:31:34.860 --> 00:31:36.390 So for a leading log, what we did is 00:31:36.390 --> 00:31:38.340 we put in tree-level matching. 00:31:38.340 --> 00:31:40.308 And at one loop, we only actually took 00:31:40.308 --> 00:31:42.600 the information that came from the coefficient of the 1 00:31:42.600 --> 00:31:47.082 over epsilon squared, which is the cusp anomalous dimension. 00:31:47.082 --> 00:31:49.350 [INAUDIBLE] here. 00:31:49.350 --> 00:31:51.270 If you go to one higher order, it 00:31:51.270 --> 00:31:56.580 turns out that this is the story to get this series. 00:32:01.970 --> 00:32:05.140 We would need the two-loop cusp anomalous dimension, the one 00:32:05.140 --> 00:32:08.320 loop non-cusp, and still the tree-level matching. 00:32:08.320 --> 00:32:11.500 So this next-to-leading log coefficient, this 5cf, 00:32:11.500 --> 00:32:14.320 that's this 1 over epsilon. 00:32:14.320 --> 00:32:17.800 In our example, this was the 5cf. 00:32:17.800 --> 00:32:21.100 And only when you get to next-to-next-to-leading log, 00:32:21.100 --> 00:32:22.540 then the matching is one loop. 00:32:26.508 --> 00:32:30.880 We have the two-loop regular anomalous dimension, 00:32:30.880 --> 00:32:32.380 and then the 1 over epsilon squared 00:32:32.380 --> 00:32:34.780 cusp anomalous dimension from three loop. 00:32:34.780 --> 00:32:37.330 So it's kind of like it's the usual story, 00:32:37.330 --> 00:32:39.132 but we have an enhanced-- because 00:32:39.132 --> 00:32:41.590 of the double logarithms, we have an enhanced thing that we 00:32:41.590 --> 00:32:43.603 can talk about here, where we're not 00:32:43.603 --> 00:32:45.520 talking about the regular anomalous dimension, 00:32:45.520 --> 00:32:48.372 but this cusp anomalous dimension. 00:32:48.372 --> 00:32:50.080 So this is the information that you would 00:32:50.080 --> 00:32:52.030 need to go to higher orders. 00:32:52.030 --> 00:32:53.500 And this cusp anomalous dimension 00:32:53.500 --> 00:32:54.865 is actually a universal thing. 00:32:54.865 --> 00:33:01.680 So given that it's been calculated, you can use it. 00:33:01.680 --> 00:33:03.933 And you don't have to recalculate it every time. 00:33:03.933 --> 00:33:05.850 These pieces you have to recalculate-- these 1 00:33:05.850 --> 00:33:11.541 over epsilons, in general. 00:33:11.541 --> 00:33:12.041 Yeah? 00:33:12.041 --> 00:33:16.190 STUDENT: Can you comment on the seemingly 00:33:16.190 --> 00:33:20.690 arbitrariness of the boundary condition you chose? 00:33:20.690 --> 00:33:24.040 PROFESSOR: Yeah, so if I hadn't chose that boundary condition-- 00:33:24.040 --> 00:33:24.910 sure. 00:33:24.910 --> 00:33:26.325 Let me write something. 00:33:41.802 --> 00:33:43.760 So the way that you could think about the rg is 00:33:43.760 --> 00:33:46.520 as follows-- that you have c. 00:33:46.520 --> 00:33:49.610 And really what the rg is doing is determining some u. 00:33:54.120 --> 00:33:59.130 It allows you to run c from some point to some other point, 00:33:59.130 --> 00:34:00.610 like that. 00:34:00.610 --> 00:34:04.620 So in this formula, mu 0 in some scale that's 00:34:04.620 --> 00:34:08.530 of order omega or w-- it doesn't have to be equal to w. 00:34:08.530 --> 00:34:10.433 It could be 2w, 1/2w. 00:34:10.433 --> 00:34:12.600 And the way that you're thinking about this equation 00:34:12.600 --> 00:34:13.590 is you're going to do perturbation 00:34:13.590 --> 00:34:14.507 theory for this thing. 00:34:19.940 --> 00:34:22.330 And then in that perturbation theory, you'll have alpha, 00:34:22.330 --> 00:34:26.480 and you'll have log squared of mu 0 over w. 00:34:26.480 --> 00:34:28.659 But as long as you're saying mu 0's of order w, 00:34:28.659 --> 00:34:30.610 those aren't large logs. 00:34:30.610 --> 00:34:33.280 And this thing here in this formulation 00:34:33.280 --> 00:34:36.880 would have the double log series of alpha log squared 00:34:36.880 --> 00:34:39.370 of mu over mu0. 00:34:39.370 --> 00:34:40.864 But then mu0's of order w. 00:34:40.864 --> 00:34:42.239 And mu you can take much smaller, 00:34:42.239 --> 00:34:43.464 and that's the large logs. 00:34:47.510 --> 00:34:51.659 So for simplicity, I just took mu0 equal to w. 00:34:51.659 --> 00:34:53.560 Everything I've said, I could repeat, 00:34:53.560 --> 00:34:55.893 and this story would really be the same if I just 00:34:55.893 --> 00:34:56.810 used an arbitrary mu0. 00:35:00.950 --> 00:35:03.140 And the reason you actually want to use an arbitrary 00:35:03.140 --> 00:35:06.560 mu0 in general is that you want to do scale variation 00:35:06.560 --> 00:35:08.430 to think about uncertainties. 00:35:08.430 --> 00:35:11.060 And so the mu0 dependence, just like in our-- 00:35:11.060 --> 00:35:13.370 when we were talking about electroweak operators 00:35:13.370 --> 00:35:16.190 and electric Hamiltonian, the mu0 dependence 00:35:16.190 --> 00:35:18.200 cancels between these things here. 00:35:18.200 --> 00:35:21.770 But that calculation is in order by order an alpha cancellation. 00:35:21.770 --> 00:35:25.410 There's no large logs associated with that cancellation. 00:35:25.410 --> 00:35:26.915 And so what it allows you to do is 00:35:26.915 --> 00:35:28.790 if you've truncated this to some fixed order, 00:35:28.790 --> 00:35:31.070 and you've worked to some fixed order over there, 00:35:31.070 --> 00:35:33.590 then you can probe how much uncertainty you 00:35:33.590 --> 00:35:36.230 have by varying the mu0. 00:35:36.230 --> 00:35:40.880 This mu here will get tied up with the mu in the operator. 00:35:40.880 --> 00:35:44.270 So we can talk more about where that mu goes later on. 00:35:44.270 --> 00:35:47.660 But at the level of this formula, it is-- in general, 00:35:47.660 --> 00:35:50.450 it's the case that you would do it with an arbitrary mu0, 00:35:50.450 --> 00:35:52.670 and then only fix mu0 at the end. 00:35:52.670 --> 00:35:55.935 And you'd even vary mu0 to get an idea of how much uncertainty 00:35:55.935 --> 00:35:58.310 you have in your leading log calculation, because varying 00:35:58.310 --> 00:36:02.640 mu0 would probe the next term in the series, here. 00:36:02.640 --> 00:36:06.620 STUDENT: So when you say tree-level matching, you mean-- 00:36:06.620 --> 00:36:07.880 PROFESSOR: I mean this c. 00:36:07.880 --> 00:36:08.797 STUDENT: --equal to 1. 00:36:08.797 --> 00:36:10.220 PROFESSOR: That's right. 00:36:10.220 --> 00:36:11.520 I mean this guy, here. 00:36:11.520 --> 00:36:12.020 Right. 00:36:12.020 --> 00:36:12.645 STUDENT: Right. 00:36:12.645 --> 00:36:15.007 And if you want to go to nnll, you'd take 1-- 00:36:15.007 --> 00:36:16.340 PROFESSOR: Then you'd have this. 00:36:16.340 --> 00:36:17.650 That wouldn't have this term. 00:36:17.650 --> 00:36:18.150 Yeah. 00:36:22.140 --> 00:36:26.020 All right, any other questions? 00:36:26.020 --> 00:36:28.020 OK, so what is this cusp that we've 00:36:28.020 --> 00:36:29.640 been saying the words for? 00:36:36.290 --> 00:36:39.870 I'll answer a couple of questions that have come up. 00:36:39.870 --> 00:36:40.760 Where's the cusp? 00:36:44.970 --> 00:36:57.540 So if we look back at our operator, we had this operator. 00:36:57.540 --> 00:37:03.100 And we could make a field redefinition, if you remember, 00:37:03.100 --> 00:37:06.310 and put the ultrasoft effects into a Wilson line. 00:37:10.800 --> 00:37:14.250 Now for this heavy quark here, remember what that-- 00:37:14.250 --> 00:37:17.070 the theory for that was with hqet. 00:37:17.070 --> 00:37:23.560 So the field theory for this heavy quark was an iv dot dhv. 00:37:23.560 --> 00:37:26.940 And if we actually make a similar field redefinition 00:37:26.940 --> 00:37:29.430 to the one we talked about to get this y, 00:37:29.430 --> 00:37:32.520 but on the heavy quark field, then we 00:37:32.520 --> 00:37:37.050 can actually, in terms of this guy 00:37:37.050 --> 00:37:40.540 here, get a free Lagrangian as well. 00:37:40.540 --> 00:37:43.770 So actually, all the effects from ultra soft gluons 00:37:43.770 --> 00:37:47.390 in this operator can be encoded in Wilson lines. 00:37:51.872 --> 00:37:53.130 So let me do that. 00:37:57.560 --> 00:37:59.990 So that just says that actually, when 00:37:59.990 --> 00:38:02.330 we talked about static sources, static sources 00:38:02.330 --> 00:38:05.870 could also be encoded as Wilson lines 00:38:05.870 --> 00:38:07.310 by a kind of similar manipulation 00:38:07.310 --> 00:38:11.210 to what we did when we talked about SET. 00:38:11.210 --> 00:38:13.010 So if we look at the ultrasoft sector here, 00:38:13.010 --> 00:38:15.230 we have a Wilson line that actually has a path. 00:38:28.380 --> 00:38:30.020 So there's a Wilson line that comes 00:38:30.020 --> 00:38:33.080 from minus infinity along v to whatever position 00:38:33.080 --> 00:38:34.070 we put our operator at. 00:38:34.070 --> 00:38:35.810 Let's take it to be 0. 00:38:35.810 --> 00:38:37.490 And then we have a yn dagger which 00:38:37.490 --> 00:38:43.340 is extending out to plus infinity from 0, like that. 00:38:43.340 --> 00:38:45.345 And the cusp is this fact that there's 00:38:45.345 --> 00:38:46.470 a kink in this Wilson line. 00:38:46.470 --> 00:38:47.660 It's not a smooth path. 00:38:47.660 --> 00:38:49.640 It actually has a sharp angle. 00:38:49.640 --> 00:38:50.660 And that's the cusp. 00:39:05.370 --> 00:39:07.580 So there's actually a general renormalization theory 00:39:07.580 --> 00:39:10.670 for renormalizing Wilson lines with cusps. 00:39:10.670 --> 00:39:12.890 So this is something that at some point 00:39:12.890 --> 00:39:15.770 in the history of QCD, people tried to reformulate QCD 00:39:15.770 --> 00:39:17.750 in terms of Wilson lines, entirely 00:39:17.750 --> 00:39:19.280 in terms of Wilson lines. 00:39:19.280 --> 00:39:21.890 Then they realized that they could have these cusps 00:39:21.890 --> 00:39:22.760 in the Wilson lines. 00:39:22.760 --> 00:39:24.380 And the renormalization of that theory 00:39:24.380 --> 00:39:26.588 became much more complicated than the renormalization 00:39:26.588 --> 00:39:28.460 of the QCD action because they couldn't 00:39:28.460 --> 00:39:30.627 prove that in general, these cusps wouldn't come out 00:39:30.627 --> 00:39:31.880 of the dynamics. 00:39:31.880 --> 00:39:33.260 And then people dropped it. 00:39:33.260 --> 00:39:36.560 But along the way, anyway, they formulated 00:39:36.560 --> 00:39:41.085 a general renormalization theory of cusps in Wilson lines. 00:39:41.085 --> 00:39:42.710 And whenever you have a cusp like this, 00:39:42.710 --> 00:39:45.450 you end up having, in our language, this log, 00:39:45.450 --> 00:39:48.350 single log, in the anomalous dimension. 00:39:48.350 --> 00:39:59.880 So if one of the lines here is light-like, and one of them 00:39:59.880 --> 00:40:06.390 is, because n squared was 0, then the anomalous dimension 00:40:06.390 --> 00:40:14.520 for this cusp will have a single log like our log of mu over w. 00:40:14.520 --> 00:40:16.770 So that actually holds not only to one loop, 00:40:16.770 --> 00:40:19.650 as we talked about, but actually to all orders 00:40:19.650 --> 00:40:20.730 in perturbation theory. 00:40:20.730 --> 00:40:24.210 If we'd gone to higher orders in perturbation theory, 00:40:24.210 --> 00:40:29.167 then what would happen is that the coefficient of that log 00:40:29.167 --> 00:40:30.000 would get corrected. 00:40:30.000 --> 00:40:33.750 But there'd still be only one log in our anomalous dimension. 00:40:33.750 --> 00:40:39.900 That's also something that you can argue from SET directly. 00:40:39.900 --> 00:40:43.920 But originally, it falls from this general theory 00:40:43.920 --> 00:40:46.500 of the renormalization of cusps of Wilson lines. 00:40:52.550 --> 00:40:55.160 So if you were thinking about our calculation, 00:40:55.160 --> 00:40:57.830 you may remember actually that we got 1 over epsilon squared. 00:40:57.830 --> 00:40:59.450 It's from the ultrasoft diagrams. 00:40:59.450 --> 00:41:02.630 And that's in some sense what we've just talked about, 00:41:02.630 --> 00:41:05.090 because these are the ultrasoft diagrams, 00:41:05.090 --> 00:41:07.197 and they have 1 over epsilon squareds. 00:41:07.197 --> 00:41:09.530 There was also 1 over epsilon squareds from the colinear 00:41:09.530 --> 00:41:12.350 diagrams, right? 00:41:12.350 --> 00:41:17.750 So if you like, this piece here also in some sense 00:41:17.750 --> 00:41:19.910 has kind of a cusp. 00:41:19.910 --> 00:41:21.625 And in this case, it's not-- 00:41:21.625 --> 00:41:23.000 it's actually what we're doing is 00:41:23.000 --> 00:41:26.510 we're taking a Wilson line along the n bar direction, which 00:41:26.510 --> 00:41:27.085 is our wn. 00:41:29.870 --> 00:41:34.610 And then we're attaching it to a quark field. 00:41:34.610 --> 00:41:37.760 So this is a full cn quark field. 00:41:37.760 --> 00:41:38.820 It's not a Wilson line. 00:41:38.820 --> 00:41:41.300 So you can think of it as just ending on a quark field. 00:41:41.300 --> 00:41:43.160 But even just ending on a quark field, 00:41:43.160 --> 00:41:44.780 the quark field also has dynamics. 00:41:44.780 --> 00:41:46.640 It's not as simple as Wilson line dynamics, 00:41:46.640 --> 00:41:48.350 but it has dynamics. 00:41:48.350 --> 00:41:51.080 And we have interactions between these. 00:41:51.080 --> 00:41:53.040 And this is also kind of a cusp. 00:41:53.040 --> 00:41:55.013 Although it's not a simple Wilson line cusp, 00:41:55.013 --> 00:41:56.055 it's also kind of a cusp. 00:42:07.120 --> 00:42:09.250 So even the colinear graphs with the Wilson lines 00:42:09.250 --> 00:42:12.940 ended on quarks also produced these q over epsilon squareds 00:42:12.940 --> 00:42:15.010 and have this kind of structure. 00:42:15.010 --> 00:42:17.870 And they actually have similar-- 00:42:17.870 --> 00:42:20.660 very similar relations to the cusps in regular Wilson lines. 00:42:23.390 --> 00:42:26.540 So that was one question-- what was this cusp? 00:42:26.540 --> 00:42:29.420 Another question that came up was this fact 00:42:29.420 --> 00:42:32.390 that the w got fixed. 00:42:32.390 --> 00:42:34.010 And so we could ask the question, 00:42:34.010 --> 00:42:35.600 in general, when will that happen? 00:42:46.630 --> 00:42:48.390 So when will the w's that are showing up 00:42:48.390 --> 00:42:53.790 in our Wilson coefficients be fixed by external kinematics? 00:43:04.660 --> 00:43:08.660 And actually, that's going to happen in the following case. 00:43:08.660 --> 00:43:12.975 We can actually state when it will happen quite generally. 00:43:18.310 --> 00:43:21.250 So imagine that we had an effective theory SET that had 00:43:21.250 --> 00:43:23.800 multiple colinear directions. 00:43:23.800 --> 00:43:25.300 So then we could build operators out 00:43:25.300 --> 00:43:28.540 of all those different colinear fields. 00:43:28.540 --> 00:43:33.980 And we would do it with our building blocks, which are 00:43:33.980 --> 00:43:41.120 chi n or curley bn for each n. 00:43:41.120 --> 00:43:45.800 And if our operator only involves one building block 00:43:45.800 --> 00:43:51.170 for each direction, then it's going 00:43:51.170 --> 00:43:56.630 to be the case that the value of those labels or those omegas 00:43:56.630 --> 00:43:58.850 are always going to be fixed by external kinematics. 00:44:05.100 --> 00:44:09.520 So let me just write down another example. 00:44:09.520 --> 00:44:11.880 Imagine we had something like an LHC process, 00:44:11.880 --> 00:44:16.800 where we had two gluons coming in and two quarks going out. 00:44:16.800 --> 00:44:20.863 But we'll think about this as protons colliding, 00:44:20.863 --> 00:44:22.530 coming in from two different directions. 00:44:22.530 --> 00:44:25.140 Let's call them n2 and n3-- 00:44:25.140 --> 00:44:28.980 and then going to two jets transverse to the axis 00:44:28.980 --> 00:44:29.800 by a large amount. 00:44:29.800 --> 00:44:32.560 So there are also two different directions, 00:44:32.560 --> 00:44:34.680 which I can call n1 and n4. 00:44:34.680 --> 00:44:40.430 So this process of glu-glu goes to qq bar, which 00:44:40.430 --> 00:44:45.110 is really pp goes to dijets. 00:44:45.110 --> 00:44:48.110 As long as those dijets are well-separated from the b 00:44:48.110 --> 00:44:50.750 maxes, which is the case we're interested in, then we have 00:44:50.750 --> 00:44:54.080 this situation, where we have four different n's. 00:44:54.080 --> 00:44:56.750 And so what kind of operator would you write down for that? 00:44:56.750 --> 00:45:04.550 Well, you have dw1, dw2, dw3, dw4. 00:45:04.550 --> 00:45:08.270 You'd have a Wilson coefficient that could depend on four 00:45:08.270 --> 00:45:09.350 different w's. 00:45:09.350 --> 00:45:12.140 And you'd then write down some operator 00:45:12.140 --> 00:45:13.730 out of building blocks, n1w1. 00:45:17.570 --> 00:45:20.583 I'm not going to worry about all the indices, 00:45:20.583 --> 00:45:22.000 but I'll worry about some of them. 00:45:32.177 --> 00:45:33.260 That would look like that. 00:45:33.260 --> 00:45:37.050 So I just put down some operator that's got the right structure. 00:45:37.050 --> 00:45:38.450 It's got two quarks, two gluons. 00:45:38.450 --> 00:45:41.330 I make the gluons these b [INAUDIBLE].. 00:45:41.330 --> 00:45:43.555 I've chosen to contract them. 00:45:43.555 --> 00:45:45.180 Each of them has a different direction. 00:45:45.180 --> 00:45:47.630 This is in n1. 00:45:47.630 --> 00:45:51.260 And each of them also gets a corresponding large momentum. 00:45:51.260 --> 00:45:55.183 And the Wilson coefficient can depend on those large momenta. 00:45:55.183 --> 00:45:56.600 So this would be the operator that 00:45:56.600 --> 00:45:59.260 would describe that process. 00:45:59.260 --> 00:46:02.930 But again, if you think about the renormalization, 00:46:02.930 --> 00:46:06.290 then if you think about the renormalization for a minute, 00:46:06.290 --> 00:46:07.983 the colinear diagrams aren't going 00:46:07.983 --> 00:46:09.650 to involve contractions between this guy 00:46:09.650 --> 00:46:11.150 and any of these other guys because these guys are 00:46:11.150 --> 00:46:11.930 totally independent. 00:46:11.930 --> 00:46:13.010 They're a different Lagrangian. 00:46:13.010 --> 00:46:15.093 So the contractions are, again, just like the kind 00:46:15.093 --> 00:46:16.550 of calculation that we did. 00:46:16.550 --> 00:46:20.400 The colinear diagrams are just coming from this thing alone, 00:46:20.400 --> 00:46:22.268 and they don't care about that stuff. 00:46:22.268 --> 00:46:23.810 So it's actually the same calculation 00:46:23.810 --> 00:46:26.120 that we already did for b to s gamma 00:46:26.120 --> 00:46:28.110 if I wanted to do the colinear diagrams here. 00:46:28.110 --> 00:46:29.777 And then I would have to do the colinear 00:46:29.777 --> 00:46:32.600 diagrams for each of these guys, but it's kind of independent. 00:46:32.600 --> 00:46:35.970 And again, the w1 is external for that calculation. 00:46:35.970 --> 00:46:38.330 So the anomalous dimension would just-- 00:46:38.330 --> 00:46:39.800 it would be outside it. 00:46:39.800 --> 00:46:41.480 It wouldn't involve-- it wouldn't 00:46:41.480 --> 00:46:47.030 appear and get thrown into some loop integral momentum. 00:46:47.030 --> 00:46:49.730 And so if you think about what can fix these w's, it's 00:46:49.730 --> 00:46:51.920 really only external information. 00:46:51.920 --> 00:46:55.580 So what are the momentum fractions 00:46:55.580 --> 00:46:58.460 of the incoming quarks from the PDFs? 00:46:58.460 --> 00:47:00.800 What are the energies of the outgoing jets? 00:47:00.800 --> 00:47:03.620 That's the type of information that, even in this case, 00:47:03.620 --> 00:47:04.520 would fix the w's. 00:47:20.208 --> 00:47:22.250 So if we were to calculate the Wilson coefficient 00:47:22.250 --> 00:47:24.710 for this operator and do its renormalization group, 00:47:24.710 --> 00:47:25.580 there would again-- 00:47:25.580 --> 00:47:27.260 it would again be of a product form. 00:47:38.100 --> 00:47:40.340 There's no convolution, just a product 00:47:40.340 --> 00:47:43.900 for this Wilson coefficient. 00:47:43.900 --> 00:47:47.080 So when could that not be true? 00:47:47.080 --> 00:47:49.900 It could not be true if, for example, we 00:47:49.900 --> 00:47:52.862 had a chi bar and a chi that were in the same direction. 00:47:57.590 --> 00:48:02.320 So if we're in a situation where all the objects are 00:48:02.320 --> 00:48:07.660 different directions, that's what happens. 00:48:07.660 --> 00:48:08.710 And it's pretty simple. 00:48:13.250 --> 00:48:20.320 But let's also do an example when is it not true. 00:48:20.320 --> 00:48:22.670 It's always good to have a counterexample. 00:48:22.670 --> 00:48:25.045 And it would not be true if we had an operator like this. 00:48:38.710 --> 00:48:41.070 So here's an operator with just two quark 00:48:41.070 --> 00:48:42.950 building-block fields. 00:48:42.950 --> 00:48:45.450 I gave them different labels, because in general, the Wilson 00:48:45.450 --> 00:48:47.890 coefficient could depend on their large momenta. 00:48:47.890 --> 00:48:54.662 But now, they're in the same direction n, 00:48:54.662 --> 00:48:58.410 so they belong to the same colinear sector. 00:48:58.410 --> 00:49:01.380 And in this operator here, we would actually no longer 00:49:01.380 --> 00:49:05.100 be in the case where this thing would totally decouple 00:49:05.100 --> 00:49:06.650 from the loop integrals. 00:49:20.595 --> 00:49:22.470 So if we think about inserting this operator, 00:49:22.470 --> 00:49:25.100 then some of these guys here will involve loop momenta. 00:49:30.300 --> 00:49:33.960 Actually, it turns out that one combination of the w's is still 00:49:33.960 --> 00:49:35.970 fixed by external kinematics. 00:49:35.970 --> 00:49:38.640 And so there's an overall-- 00:49:38.640 --> 00:49:41.550 think about the overall delta function on this product. 00:49:41.550 --> 00:49:42.730 That guy would be fixed. 00:49:42.730 --> 00:49:47.130 And then some difference of these w's would not be fixed. 00:49:47.130 --> 00:49:51.970 That's the true story about at least one combination 00:49:51.970 --> 00:49:53.220 that involve the loop momenta. 00:50:13.790 --> 00:50:15.050 And what does that lead to? 00:50:15.050 --> 00:50:17.600 How does it complicate what we've already said? 00:50:17.600 --> 00:50:23.980 What that leads to is that when we formulate 00:50:23.980 --> 00:50:27.870 the anomalous dimension equations, 00:50:27.870 --> 00:50:29.802 they also involve integrals. 00:50:48.520 --> 00:50:53.205 So for this guy here, for the one momentum that's not fixed-- 00:50:53.205 --> 00:50:54.330 so in general, there's two. 00:50:54.330 --> 00:50:56.940 And this is the one that's not fixed. 00:50:56.940 --> 00:50:58.890 We would get an equation that looks like this. 00:51:06.420 --> 00:51:08.310 Rather than the simple product form, 00:51:08.310 --> 00:51:11.615 there'd be an integral on the right-hand side. 00:51:11.615 --> 00:51:13.740 And indeed, actually, if you do the renormalization 00:51:13.740 --> 00:51:17.280 of this operator that I wrote up there, just that simple quark 00:51:17.280 --> 00:51:23.370 operator, you actually reproduce a bunch 00:51:23.370 --> 00:51:31.140 of classical evolution equations that are of this form. 00:51:39.660 --> 00:51:41.730 So even just the simplest possible case 00:51:41.730 --> 00:51:44.780 we can think of where things are not fixed 00:51:44.780 --> 00:51:49.010 gives us a bunch of interesting results. 00:51:49.010 --> 00:51:52.330 So for deep inelastic scattering, 00:51:52.330 --> 00:51:56.530 then what you're getting is the Altarelli-Parisi equation 00:51:56.530 --> 00:51:57.640 or DGLAP equation. 00:52:07.700 --> 00:52:09.140 And the evolution of the operator 00:52:09.140 --> 00:52:11.960 would be the evolution of the part-time distribution 00:52:11.960 --> 00:52:13.528 function. 00:52:13.528 --> 00:52:16.070 That's one thing that would be encoded in the renormalization 00:52:16.070 --> 00:52:18.030 of that equation. 00:52:18.030 --> 00:52:21.410 There's other processes that are actually-- 00:52:21.410 --> 00:52:23.780 some of which we may talk about later on-- 00:52:23.780 --> 00:52:26.107 that are also encoded in the same type 00:52:26.107 --> 00:52:28.190 of operator, kind of a different projection of it. 00:52:31.142 --> 00:52:32.850 And one of them leads to something called 00:52:32.850 --> 00:52:34.020 the Brodsky-Lepage equation. 00:52:40.800 --> 00:52:43.890 And then there's another one that's showing up 00:52:43.890 --> 00:52:45.270 in something called-- 00:52:45.270 --> 00:52:48.407 a process called deeply virtual Compton scattering. 00:52:55.260 --> 00:52:57.540 So the one that's most familiar is the renormalization 00:52:57.540 --> 00:52:58.147 of the PDF. 00:52:58.147 --> 00:53:00.480 And you may remember that the renormalization of the PDF 00:53:00.480 --> 00:53:03.990 has this form of having an integral. 00:53:03.990 --> 00:53:07.750 That'll come out of renormalization this operator. 00:53:07.750 --> 00:53:08.790 And we'll do that case. 00:53:20.500 --> 00:53:25.870 So in general, the structure of the effective theory 00:53:25.870 --> 00:53:29.890 is leading us to find out what the structure 00:53:29.890 --> 00:53:32.620 of the renormalization is. 00:53:32.620 --> 00:53:36.010 And it can reproduce some well-known, classic things, 00:53:36.010 --> 00:53:37.920 but it comes out actually pretty easy. 00:53:44.050 --> 00:53:47.430 So I'm going to do this case, but I'm 00:53:47.430 --> 00:53:49.710 going to do this case in complete-- with all 00:53:49.710 --> 00:53:51.450 of the details filled in. 00:53:51.450 --> 00:53:53.550 So first, I have to convince you that 00:53:53.550 --> 00:53:57.010 in deep inelastic scattering, you actually get that operator. 00:53:57.010 --> 00:53:59.130 And we'll see precisely what kind 00:53:59.130 --> 00:54:00.600 of matrix elements of that operator 00:54:00.600 --> 00:54:02.308 show up in deep inelastic scattering, why 00:54:02.308 --> 00:54:03.455 we get that operator. 00:54:03.455 --> 00:54:05.830 And then once we're convinced that we have that operator, 00:54:05.830 --> 00:54:07.830 and we'll actually drive a factorization theorem 00:54:07.830 --> 00:54:09.420 that involves that operator. 00:54:09.420 --> 00:54:11.730 Then we'll talk about its renormalization group 00:54:11.730 --> 00:54:12.330 evolution. 00:54:12.330 --> 00:54:14.520 And I'll show you that it has this form that I'm 00:54:14.520 --> 00:54:16.320 writing here. 00:54:16.320 --> 00:54:19.650 So probably for the rest of today's lecture, certainly 00:54:19.650 --> 00:54:22.350 for the rest of today's lecture, we won't get to the running. 00:54:22.350 --> 00:54:23.350 We'll do that next time. 00:54:23.350 --> 00:54:27.540 But we'll at least get to the point, I think, 00:54:27.540 --> 00:54:30.315 where you'll see why that operator is showing up in DIS. 00:54:34.540 --> 00:54:35.480 So let's do DIS. 00:54:40.090 --> 00:54:52.900 So I'm only going to talk in DIS about factorization, 00:54:52.900 --> 00:54:58.060 and then the renormalization group evolution. 00:54:58.060 --> 00:55:00.280 So no phenomenology, nothing like that-- we'll just 00:55:00.280 --> 00:55:03.770 talk about these two concepts. 00:55:03.770 --> 00:55:09.760 So DIS is electron-proton to electron anything, so just 00:55:09.760 --> 00:55:10.600 kinematics. 00:55:13.832 --> 00:55:15.415 Think about a virtual photon exchange. 00:55:19.270 --> 00:55:21.640 Proton comes in. 00:55:21.640 --> 00:55:23.290 I'll say that the proton's momentum is 00:55:23.290 --> 00:55:29.720 a capital P. Gets blown apart. 00:55:29.720 --> 00:55:31.765 Call the stuff that's blown apart Px. 00:55:34.300 --> 00:55:40.420 And so Px is the sum of all the particles, 00:55:40.420 --> 00:55:41.635 all the final-state hadrons. 00:55:48.060 --> 00:55:49.950 Q squared of the virtual photon-- 00:55:49.950 --> 00:55:50.880 so this is q. 00:55:53.700 --> 00:55:56.400 Little q squared is minus capital Q squared. 00:55:56.400 --> 00:55:58.260 And this thing is much bigger than on QCD. 00:56:02.530 --> 00:56:08.140 mu arc in x is capital Q squared divided by this dot product. 00:56:12.820 --> 00:56:17.680 And you can talk about Px, which is the Px mu. 00:56:17.680 --> 00:56:19.450 And by momentum conservation, that's 00:56:19.450 --> 00:56:21.910 the proton momentum, plus whatever momentum came in 00:56:21.910 --> 00:56:23.080 from the leptons q. 00:56:26.990 --> 00:56:30.650 So Px squared-- if you square it, 00:56:30.650 --> 00:56:33.750 well, this guy is giving you mass of the protons squared. 00:56:33.750 --> 00:56:36.570 And then there's a dot-- a cross term, and this guy squared. 00:56:36.570 --> 00:56:44.319 And if you put those things together like this, 00:56:44.319 --> 00:56:47.912 so this is an exact equation. 00:56:47.912 --> 00:56:50.180 Px squared is that. 00:56:50.180 --> 00:56:52.430 And so there's actually different regions of DIS, 00:56:52.430 --> 00:56:54.388 and we're only going to talk about one of them. 00:56:56.870 --> 00:57:00.230 I have to enumerate what I'm talking about carefully. 00:57:00.230 --> 00:57:06.990 And I can do that by looking at Px squared or this factor 00:57:06.990 --> 00:57:12.140 1 minus x over x, which I'll call 1 over x minus 1. 00:57:12.140 --> 00:57:14.652 If this thing is of order q squared, 00:57:14.652 --> 00:57:16.610 then that means that this thing you're counting 00:57:16.610 --> 00:57:18.320 is of quarter 1. 00:57:18.320 --> 00:57:21.590 And in that case, it's what's called the inclusive operator 00:57:21.590 --> 00:57:23.790 product expansion. 00:57:23.790 --> 00:57:28.010 So this is the case that most books would deal with. 00:57:28.010 --> 00:57:30.920 And that's the one we'll deal with, actually, 00:57:30.920 --> 00:57:33.320 where effectively, we're not putting restrictions on x. 00:57:33.320 --> 00:57:35.570 We're just saying it's generic, and it's not 00:57:35.570 --> 00:57:37.430 approaching any endpoints. 00:57:37.430 --> 00:57:42.740 There's also a situation where the Px gets smaller. 00:57:42.740 --> 00:57:47.307 And then this thing is close to an endpoint, lambda QCD over q. 00:57:47.307 --> 00:57:48.890 And that's called the endpoint region. 00:57:52.070 --> 00:57:54.530 And people talk about that. 00:57:54.530 --> 00:57:56.740 Usually when people say x goes to 1, 00:57:56.740 --> 00:57:59.110 this is what they mean-- that the 1 minus x minus 1 00:57:59.110 --> 00:58:03.730 is of that size on the QCD over q. 00:58:03.730 --> 00:58:07.000 There's even a third region, where the Px squared becomes 00:58:07.000 --> 00:58:12.410 hadronic lambda QCD squared. 00:58:12.410 --> 00:58:16.690 And that's like here, two powers. 00:58:16.690 --> 00:58:18.670 So taking that factor to be really small. 00:58:18.670 --> 00:58:21.980 And that's the resonance region. 00:58:21.980 --> 00:58:25.300 And that's the case where the final state x is just 00:58:25.300 --> 00:58:27.445 another proton or an excited state of a proton. 00:58:31.977 --> 00:58:33.810 So that's where elastic scattering would be. 00:58:33.810 --> 00:58:39.750 And that's an exclusive process. 00:58:39.750 --> 00:58:41.393 It's not inclusive anymore. 00:58:47.070 --> 00:58:50.660 So actually, all three of these cases can be done with SET. 00:58:50.660 --> 00:58:52.963 And the way that it works is different in each case. 00:58:52.963 --> 00:58:54.380 And the case that we'll do is just 00:58:54.380 --> 00:58:57.950 the first one, which is kind of a classic one, and also 00:58:57.950 --> 00:58:58.730 the simplest one. 00:59:04.618 --> 00:59:06.410 So our Px squared is going to be of ordered 00:59:06.410 --> 00:59:12.970 q squared for our analysis. 00:59:25.600 --> 00:59:27.520 We also need some partonic variables. 00:59:40.830 --> 00:59:44.848 So the struck quark carries some momentum fraction 00:59:44.848 --> 00:59:45.515 from the proton. 00:59:49.240 --> 00:59:50.620 This is the familiar language. 00:59:53.260 --> 00:59:57.010 And for our analysis, what we're going to do is just 00:59:57.010 --> 01:00:01.180 take m bar dot little p of the quark 01:00:01.180 --> 01:00:04.945 to be something times n bar dot big P of the proton. 01:00:07.480 --> 01:00:10.600 So the fraction is this c variable, 01:00:10.600 --> 01:00:18.080 and it's the ratio of the quark momentum to the proton. 01:00:18.080 --> 01:00:20.570 But we'll do it in a very particular component, 01:00:20.570 --> 01:00:25.930 and we'll see why that's the right thing to do. 01:00:25.930 --> 01:00:28.200 And if we have our picture here of the quark 01:00:28.200 --> 01:00:30.960 kinematics, then this P is the incoming P. 01:00:30.960 --> 01:00:34.950 And the outgoing P would be P prime, let's say. 01:00:34.950 --> 01:00:37.320 So I could think about an analog of this equation, where 01:00:37.320 --> 01:00:39.750 I square P prime. 01:00:39.750 --> 01:00:43.360 And if you do that, it's kind of similar to the hadronic case. 01:00:43.360 --> 01:00:45.170 The only difference is that the c shows up. 01:00:47.988 --> 01:00:49.905 And, of course, there's no mass of the proton. 01:00:52.650 --> 01:00:56.100 So P prime squared would be that. 01:01:03.950 --> 01:01:08.390 So we'll see how this variable c shows up. 01:01:08.390 --> 01:01:10.310 Now, one thing that we have to decide about 01:01:10.310 --> 01:01:12.110 is frames of reference, because remember, 01:01:12.110 --> 01:01:15.080 when we were talking about degrees of freedom in SET, 01:01:15.080 --> 01:01:17.480 we had picked a frame to do that discussion. 01:01:17.480 --> 01:01:21.050 And it was almost always a center of mass frame so far 01:01:21.050 --> 01:01:26.960 in our discussions, or the rest frame of the initial state. 01:01:26.960 --> 01:01:30.900 And here, we're going to use a slightly different frame, which 01:01:30.900 --> 01:01:34.067 is the most convenient frame for deep elastic scattering. 01:01:34.067 --> 01:01:35.150 It's called a Breit frame. 01:01:40.950 --> 01:01:44.540 So we're going to do our analysis in this frame. 01:01:44.540 --> 01:01:46.070 So what defines this frame? 01:01:50.210 --> 01:01:53.380 This frame is defined by taking q mu 01:01:53.380 --> 01:01:57.040 to just have a z component. 01:01:57.040 --> 01:01:59.800 Remember, it's space-like, so it has to be somewhere 01:01:59.800 --> 01:02:01.180 in the space-like column. 01:02:01.180 --> 01:02:02.680 And we can choose it such that it's 01:02:02.680 --> 01:02:04.870 entirely in the z component, and nowhere else. 01:02:04.870 --> 01:02:06.525 And that's the Breit frame. 01:02:06.525 --> 01:02:07.900 If we want to write that in terms 01:02:07.900 --> 01:02:10.600 of our classic decomposition of nnn bar, is we 01:02:10.600 --> 01:02:13.450 can write it as a difference divided by 2. 01:02:13.450 --> 01:02:16.620 And that's giving the z component. 01:02:19.620 --> 01:02:22.190 So in this frame what's happening 01:02:22.190 --> 01:02:24.710 is that your initial state proton is coming 01:02:24.710 --> 01:02:26.750 in with a very large momentum. 01:02:26.750 --> 01:02:29.606 And then it's being-- 01:02:29.606 --> 01:02:32.270 you're killing that momentum and then spitting it back out 01:02:32.270 --> 01:02:34.380 in a different direction. 01:02:34.380 --> 01:02:40.470 So you're spitting back out stuff in a different direction. 01:02:40.470 --> 01:02:44.480 So the initial state proton is coming in with a large momentum 01:02:44.480 --> 01:02:47.940 in some direction. 01:02:47.940 --> 01:02:50.990 So if you work out the kinematics given that for what 01:02:50.990 --> 01:02:56.080 the proton would be, the proton's momentum 01:02:56.080 --> 01:02:58.960 would be in the following form. 01:02:58.960 --> 01:03:00.080 So it's got a large-- 01:03:00.080 --> 01:03:02.800 this is large, and this is small. 01:03:05.750 --> 01:03:07.150 And if you have a large component 01:03:07.150 --> 01:03:09.442 in some light-like direction, that means it's colinear. 01:03:11.560 --> 01:03:16.450 So you could actually write it as-- 01:03:16.450 --> 01:03:20.300 using momentum conservation, you could write it like this. 01:03:20.300 --> 01:03:24.190 And it has a colinear scaling. 01:03:24.190 --> 01:03:26.135 So what we have in the deep inelastic 01:03:26.135 --> 01:03:28.510 scattering in the Breit frame is that the incoming proton 01:03:28.510 --> 01:03:29.345 is a linear proton. 01:03:32.200 --> 01:03:36.310 And if we look at Px, and we just-- 01:03:36.310 --> 01:03:38.980 again, I'm not going through the details of this. 01:03:38.980 --> 01:03:46.910 But if we just decompose Px in terms of these coordinates, 01:03:46.910 --> 01:03:48.590 then we get this. 01:03:48.590 --> 01:03:52.400 And so as long as this factor 1 minus x over x is of order 1, 01:03:52.400 --> 01:03:54.340 you see that there's a large component in n 01:03:54.340 --> 01:03:56.630 and a large component in n bar. 01:03:56.630 --> 01:03:57.770 And that means it's hard. 01:04:03.550 --> 01:04:06.270 So what would happen in these other cases here 01:04:06.270 --> 01:04:07.930 is that you would change that, right? 01:04:07.930 --> 01:04:09.100 It would no longer be hard. 01:04:09.100 --> 01:04:12.300 And that's why these cases here are different. 01:04:12.300 --> 01:04:14.550 But as long as we're in this first case, this is hard. 01:04:14.550 --> 01:04:16.890 And we can say that we have colinear modes and hard modes. 01:04:16.890 --> 01:04:18.720 And then we just want to write down an effective theory 01:04:18.720 --> 01:04:19.595 for those two things. 01:04:22.130 --> 01:04:23.810 You could also do an analysis of DIS 01:04:23.810 --> 01:04:26.660 in the rest frame of the proton. 01:04:26.660 --> 01:04:29.360 That's another case. 01:04:29.360 --> 01:04:32.150 And actually, the final result that you would get 01:04:32.150 --> 01:04:33.067 would be the same. 01:04:33.067 --> 01:04:34.650 That's what we'll get from this frame. 01:04:34.650 --> 01:04:36.442 But this frame is actually a little easier. 01:04:47.280 --> 01:04:49.980 So we're really talking about hard colinear factorization 01:04:49.980 --> 01:04:50.670 in some sense. 01:04:54.920 --> 01:04:57.160 Colinear describes the low-energy degree of freedom, 01:04:57.160 --> 01:04:58.800 which is the proton. 01:04:58.800 --> 01:05:03.450 And hard describes the off-shell final state 01:05:03.450 --> 01:05:05.550 and the hard fluctuations. 01:05:05.550 --> 01:05:09.210 And really, what we want to do in DIS in this classic case is 01:05:09.210 --> 01:05:11.372 just separate hard and colinear fluctuations, 01:05:11.372 --> 01:05:12.330 at least in this frame. 01:05:17.660 --> 01:05:19.875 So we can do that for the cross-section. 01:05:23.970 --> 01:05:26.800 So let me remind you something about the cross-section in DIS. 01:05:29.630 --> 01:05:34.190 So just using nothing more than the fact 01:05:34.190 --> 01:05:38.910 that we're treating the leptons order by order in the photon, 01:05:38.910 --> 01:05:41.600 we can work the first-order in the electromagnetic coupling. 01:05:41.600 --> 01:05:43.100 And then we can write down a formula 01:05:43.100 --> 01:05:47.345 like this, where we split it into electronic and hadronic 01:05:47.345 --> 01:05:47.845 tensor. 01:05:53.280 --> 01:05:55.050 And the hadronic tensor can be written 01:05:55.050 --> 01:06:06.480 as the imaginary part of some T, where T is the following thing. 01:06:06.480 --> 01:06:09.180 So this doesn't use anything about-- 01:06:09.180 --> 01:06:11.700 we haven't used any sort of perturbation theory or anything 01:06:11.700 --> 01:06:13.580 to write this-- what I'm telling you-- down. 01:06:19.846 --> 01:06:22.370 We just worked all orders, and basically used 01:06:22.370 --> 01:06:23.210 the optical theorem. 01:06:28.238 --> 01:06:30.280 So T is the time order product of the two curves. 01:06:36.248 --> 01:06:37.040 Let me call this z. 01:06:40.893 --> 01:06:42.310 Actually, I don't know if that's-- 01:06:46.980 --> 01:06:49.110 before you start doing anything in QCD, 01:06:49.110 --> 01:06:50.940 that's how you could write this. 01:06:50.940 --> 01:06:54.620 And these are the electromagnetic currents 01:06:54.620 --> 01:06:56.082 for a quark. 01:06:56.082 --> 01:06:58.290 STUDENT: But wouldn't you write it over that, though? 01:06:58.290 --> 01:06:59.250 [INAUDIBLE] 01:06:59.250 --> 01:07:07.676 PROFESSOR: Thanks, yeah, because x was something else. 01:07:07.676 --> 01:07:08.710 Don't want to use x. 01:07:12.040 --> 01:07:15.790 All right, so for this T mu nu, we 01:07:15.790 --> 01:07:18.190 can also use current conservation 01:07:18.190 --> 01:07:20.990 and decompose it into two pieces. 01:07:20.990 --> 01:07:25.120 So this is a classic thing that we do in DIS. 01:07:25.120 --> 01:07:26.740 And nothing about the fact that we're 01:07:26.740 --> 01:07:31.073 using the effective theory really changes during this. 01:07:31.073 --> 01:07:32.740 What you're after-- the effective theory 01:07:32.740 --> 01:07:39.100 is calculating these T's, which would be coefficients. 01:07:39.100 --> 01:07:44.455 And this part is all standard stuff for any analysis of DIS. 01:07:48.610 --> 01:07:51.280 So if you're not familiar with it, or you don't remember it, 01:07:51.280 --> 01:07:53.640 it's actually not that important. 01:07:53.640 --> 01:07:56.170 But using the current conservation of T, 01:07:56.170 --> 01:07:58.060 the fact that when I dot a q into the T, 01:07:58.060 --> 01:08:02.230 you should get 0, because you're dotting a q into the current. 01:08:02.230 --> 01:08:03.512 The current is conserved. 01:08:03.512 --> 01:08:04.720 It's electromagnetic current. 01:08:04.720 --> 01:08:07.002 That tells you the possible structure of this thing 01:08:07.002 --> 01:08:08.710 and that there's two general terms if I'm 01:08:08.710 --> 01:08:10.060 doing a spin sum, which I am. 01:08:18.151 --> 01:08:19.609 There's the sum over spin up there. 01:08:22.680 --> 01:08:25.080 If we weren't summing over spin, if we were picking out 01:08:25.080 --> 01:08:27.270 particular spins of the proton, then the formula 01:08:27.270 --> 01:08:31.439 here could be a little more complicated. 01:08:31.439 --> 01:08:34.398 So this satisfies all the symmetries. 01:08:39.271 --> 01:08:40.979 And in general, what you know is that you 01:08:40.979 --> 01:08:43.529 want the imaginary part of forward scattering graphs. 01:08:50.210 --> 01:08:53.168 So pictorially, it's sometimes useful to draw something. 01:08:56.450 --> 01:08:59.040 So here's a forward scattering graph. 01:08:59.040 --> 01:09:01.170 And the thing that's different that we have-- 01:09:01.170 --> 01:09:05.700 know now-- and the information that we're going to use 01:09:05.700 --> 01:09:09.990 is that we're assigning the external guys 01:09:09.990 --> 01:09:12.910 to be colinear and the intermediate guy to be hard. 01:09:12.910 --> 01:09:15.570 So we want to integrate out the pink guy, as usual. 01:09:15.570 --> 01:09:18.000 The guy we want to integrate out is always pink. 01:09:18.000 --> 01:09:21.120 And we want to keep these colinear guys. 01:09:21.120 --> 01:09:25.170 And so the operators in the effective theory-- 01:09:25.170 --> 01:09:27.420 we can already intuit what they should look like. 01:09:32.858 --> 01:09:34.150 They're just going to involve-- 01:09:37.024 --> 01:09:38.560 I have a current with two photons 01:09:38.560 --> 01:09:42.490 hanging out of it, and then colinear quarks. 01:09:42.490 --> 01:09:45.790 Those guys-- those are the external lines there. 01:09:45.790 --> 01:09:49.029 And there's actually also an analogous thing 01:09:49.029 --> 01:09:52.990 with colinear gluons. 01:09:52.990 --> 01:09:54.550 And that's what the type of-- so I 01:09:54.550 --> 01:09:56.620 could have an operator at higher orders, where 01:09:56.620 --> 01:09:58.232 the external states here were gluons. 01:09:58.232 --> 01:10:00.440 And that's what the operators of the effective theory 01:10:00.440 --> 01:10:02.823 are going to look like. 01:10:02.823 --> 01:10:04.490 Just contract that pink line to a point, 01:10:04.490 --> 01:10:05.782 and that's what they look like. 01:10:20.550 --> 01:10:22.300 So given that we know what they look like, 01:10:22.300 --> 01:10:24.258 we just have to write down the lowest dimension 01:10:24.258 --> 01:10:25.337 operators of that form. 01:10:25.337 --> 01:10:27.295 And the lowest dimension operators of that form 01:10:27.295 --> 01:10:28.840 are exactly the operator that I told 01:10:28.840 --> 01:10:31.030 you was going to be the one that comes in. 01:10:36.780 --> 01:10:38.570 So we can enumerate the lowest dimension, 01:10:38.570 --> 01:10:41.150 which where dimension here is counted as lambdas, right? 01:10:41.150 --> 01:10:44.720 So it's the lowest order in the power counting operators, so 01:10:44.720 --> 01:10:46.855 not the lowest order in mass dimension. 01:10:51.410 --> 01:10:57.320 Let me write out a few things. 01:11:05.340 --> 01:11:07.375 Yeah, I already switched to this notation. 01:11:11.840 --> 01:11:12.340 Yeah. 01:11:32.393 --> 01:11:34.060 So the type of operator we'd have that's 01:11:34.060 --> 01:11:35.520 order lambda squared-- 01:11:35.520 --> 01:11:36.520 this guy's order lambda. 01:11:36.520 --> 01:11:37.520 This guy's order lambda. 01:11:37.520 --> 01:11:38.770 That's lambda squared. 01:11:38.770 --> 01:11:42.610 And the most general thing we can think of is this. 01:11:42.610 --> 01:11:44.860 The reason why I wrote this out rather than writing it 01:11:44.860 --> 01:11:46.540 as a chi field is I wanted to emphasize 01:11:46.540 --> 01:11:47.707 that there's a flavor index. 01:11:51.202 --> 01:11:57.030 So i is up quarks, down quarks, strange quirks, et cetera. 01:11:57.030 --> 01:12:06.660 And then we could also have gluons, 01:12:06.660 --> 01:12:19.485 and gluons are similar except in terms of our curly v field. 01:12:26.010 --> 01:12:35.460 And it is actually just a contraction of two curly v's, 01:12:35.460 --> 01:12:38.555 and then traced over. 01:12:38.555 --> 01:12:40.430 And then there's some Wilson coefficient here 01:12:40.430 --> 01:12:47.750 which is also just the Wilson coefficient for the gluon. 01:12:47.750 --> 01:12:50.000 So this is, again, order lambda squared. 01:12:50.000 --> 01:12:51.810 The b perp field is order lambda squared. 01:12:51.810 --> 01:12:53.990 So we just write down the lowest dimension operators 01:12:53.990 --> 01:12:55.352 that have this form. 01:12:55.352 --> 01:12:57.060 And that's going to be the right answer-- 01:12:57.060 --> 01:12:59.240 the lowest order in lambda. 01:13:15.660 --> 01:13:18.000 So it turns out actually that in this case, 01:13:18.000 --> 01:13:21.380 our lambda counting is exactly the corresponding twist 01:13:21.380 --> 01:13:22.350 expansion. 01:13:26.490 --> 01:13:27.410 I'm not going to-- 01:13:27.410 --> 01:13:29.830 STUDENT: All j's correspond to the operators that go 01:13:29.830 --> 01:13:31.442 into the T1's and T2's? 01:13:31.442 --> 01:13:32.400 PROFESSOR: Yeah, right. 01:13:32.400 --> 01:13:35.630 So now what's the index j exactly? 01:13:35.630 --> 01:13:37.860 There's a T1 and a T2. 01:13:37.860 --> 01:13:39.480 And the T1 and the T2, although they 01:13:39.480 --> 01:13:41.940 have the same kind of quark and gluon field structure, 01:13:41.940 --> 01:13:45.480 they get different coefficients, and that's what the j is. 01:13:45.480 --> 01:13:53.270 So when we think about doing a similar kind of decomposition, 01:13:53.270 --> 01:13:56.210 the effective theory, we can write it as follows. 01:14:01.080 --> 01:14:05.030 So as an 01 and an 02. 01:14:05.030 --> 01:14:10.010 And I think if I'm getting my mass detentions right, 01:14:10.010 --> 01:14:10.880 it looks like this. 01:14:28.080 --> 01:14:31.070 So there's a piece that multiplies a spin structure 01:14:31.070 --> 01:14:32.630 that's this g perp mu nu. 01:14:32.630 --> 01:14:34.052 That's transverse to Q. 01:14:34.052 --> 01:14:35.510 And there's a piece that multiplies 01:14:35.510 --> 01:14:37.790 this guy, which is also transverse to Q in the Breit 01:14:37.790 --> 01:14:39.920 frame. 01:14:39.920 --> 01:14:41.410 So that's the kind of decomposition 01:14:41.410 --> 01:14:42.610 we would do in the effective theory. 01:14:42.610 --> 01:14:44.170 There'd be two different types of operators 01:14:44.170 --> 01:14:45.215 we could think about. 01:14:45.215 --> 01:14:46.840 But the only thing that actually tracks 01:14:46.840 --> 01:14:48.850 that-- because of the spin relations, 01:14:48.850 --> 01:14:51.667 the only thing that tracks that is the Wilson coefficient. 01:14:51.667 --> 01:14:53.875 And so that will differ in these two different cases. 01:14:57.370 --> 01:15:01.550 So this guy here is going to be-- 01:15:01.550 --> 01:15:04.260 this will give the quark PDFs. 01:15:10.380 --> 01:15:11.970 And this Wilson coefficient will then 01:15:11.970 --> 01:15:14.852 be the thing that you convolute with the quark PDF in DIS. 01:15:14.852 --> 01:15:17.310 And then likewise, this guy is going to give the gluon PDF. 01:15:22.870 --> 01:15:24.280 So let's do the quark-- 01:15:24.280 --> 01:15:28.060 we'll do the quark contribution in detail. 01:15:28.060 --> 01:15:30.310 The gluon contribution is not really any harder. 01:15:34.983 --> 01:15:36.400 So what I'm going to do is I'm not 01:15:36.400 --> 01:15:38.680 going to think about it in perturbation theory. 01:15:38.680 --> 01:15:40.570 I'm just going to think about it to all orders in perturbation 01:15:40.570 --> 01:15:41.308 theory. 01:15:41.308 --> 01:15:42.850 And really what that means is I'm not 01:15:42.850 --> 01:15:46.000 going to think about this c as expanded in perturbation theory 01:15:46.000 --> 01:15:48.565 or think about any of the diagrams here as expanded 01:15:48.565 --> 01:15:50.260 in perturbation theory. 01:15:50.260 --> 01:15:52.690 I'm just going to see, if I manipulate things, 01:15:52.690 --> 01:15:55.150 what does it lead to, using things 01:15:55.150 --> 01:15:58.610 like momentum conservation and stuff like that? 01:15:58.610 --> 01:16:00.130 So let me write-- 01:16:00.130 --> 01:16:02.980 in order to do that, let me write it 01:16:02.980 --> 01:16:10.420 slightly differently than I just did, which I apologize for. 01:16:10.420 --> 01:16:11.170 But I'm going to-- 01:16:11.170 --> 01:16:17.080 I want the arguments of the c, just for a later convenience, 01:16:17.080 --> 01:16:19.450 instead of being w1 and w1 to be w plus 01:16:19.450 --> 01:16:22.570 and w-- minus the sum of the w's and the difference 01:16:22.570 --> 01:16:28.420 of the w's, but everything else the same. 01:16:32.470 --> 01:16:34.390 It's convenient to just talk about 01:16:34.390 --> 01:16:37.900 the sum and the difference rather than the individuals. 01:16:48.818 --> 01:16:50.610 So what's the parton distribution function? 01:16:54.737 --> 01:16:56.320 Let me convince you that it's actually 01:16:56.320 --> 01:16:59.800 related to this operator. 01:16:59.800 --> 01:17:09.700 If you were in coordinate space, the coordinate space for us 01:17:09.700 --> 01:17:11.950 was z. 01:17:11.950 --> 01:17:15.110 We're just leaving x for something else. 01:17:15.110 --> 01:17:22.110 Then you could define the parton distribution function 01:17:22.110 --> 01:17:26.857 as a proton matrix element of quark fields 01:17:26.857 --> 01:17:28.190 with a Wilson line between them. 01:17:33.890 --> 01:17:39.720 So they're on the light cone, and we have a formula 01:17:39.720 --> 01:17:41.790 like that. 01:17:41.790 --> 01:17:45.180 And also, one can convince oneself 01:17:45.180 --> 01:17:49.470 that if you want to write down for the antiquarks what 01:17:49.470 --> 01:17:51.690 the part-time distribution function is, 01:17:51.690 --> 01:17:55.710 it's the same formula as the quark formula, 01:17:55.710 --> 01:17:57.520 just with an overall minus sign. 01:17:57.520 --> 01:17:59.529 And z goes to minus z. 01:18:09.510 --> 01:18:12.540 So that's an operator definition of the part-time distribution 01:18:12.540 --> 01:18:14.020 function. 01:18:14.020 --> 01:18:17.010 You can also define it by moments of the operator. 01:18:17.010 --> 01:18:20.370 That's the way that, for example, Peskin does it. 01:18:20.370 --> 01:18:22.620 But if you put all the information in those moments 01:18:22.620 --> 01:18:25.140 back into a single operator, then it becomes this thing. 01:18:30.420 --> 01:18:32.705 And if we Fourier transform this, 01:18:32.705 --> 01:18:34.830 then it becomes the operator that we have up there. 01:18:48.802 --> 01:18:51.010 So there's something special about the matrix element 01:18:51.010 --> 01:18:52.870 we're taking here. 01:18:52.870 --> 01:18:55.900 And that is that this matrix element is forward. 01:18:55.900 --> 01:18:58.360 It has the same kinematics for the-- 01:18:58.360 --> 01:19:01.330 so this is a colinear proton, but we have the same momentum 01:19:01.330 --> 01:19:03.820 here and here. 01:19:03.820 --> 01:19:06.290 In the in state and the out state, 01:19:06.290 --> 01:19:08.830 you have the same momentum. 01:19:08.830 --> 01:19:11.445 And that leads to one kinematic restriction 01:19:11.445 --> 01:19:12.570 on what I'm about to write. 01:19:15.500 --> 01:19:21.650 So what that imposes is that w1 should be equal to w2. 01:19:21.650 --> 01:19:23.630 So there's a delta function of w minus 01:19:23.630 --> 01:19:27.110 that comes from the momentum conservation or the restriction 01:19:27.110 --> 01:19:30.110 that it's a forward matrix element. 01:19:30.110 --> 01:19:33.170 But the sum of the two-- 01:19:33.170 --> 01:19:35.780 with my sign conventions, it's w minus. 01:19:35.780 --> 01:19:39.960 The sum of the two is unconstrained. 01:19:39.960 --> 01:19:42.645 Well, it's bounded, but unconstrained. 01:20:02.480 --> 01:20:04.100 So we can write the unconstrained part 01:20:04.100 --> 01:20:06.680 as this interval of c parameter. 01:20:06.680 --> 01:20:08.390 The bounding just comes from the fact 01:20:08.390 --> 01:20:10.130 that the quark can't carry more momentum 01:20:10.130 --> 01:20:13.055 than the overall momentum of the proton. 01:20:13.055 --> 01:20:14.180 So that's why it's bounded. 01:20:14.180 --> 01:20:16.610 It can't carry negative momentum, and it can't carry-- 01:20:19.078 --> 01:20:20.870 negative physical momentum, it can't carry, 01:20:20.870 --> 01:20:23.600 and it can't carry less momentum in the proton. 01:20:23.600 --> 01:20:25.350 That's where the limits come from. 01:20:25.350 --> 01:20:26.630 And then there's two pieces. 01:20:26.630 --> 01:20:28.550 The w plus could either be positive, 01:20:28.550 --> 01:20:30.240 or it could be negative. 01:20:30.240 --> 01:20:32.120 If it's positive, remember, positive 01:20:32.120 --> 01:20:34.340 labels-- that was our quarks. 01:20:34.340 --> 01:20:35.690 So this is the quark piece. 01:20:38.770 --> 01:20:40.020 And these are the antiquarks. 01:20:40.020 --> 01:20:43.140 They come with negative labels. 01:20:43.140 --> 01:20:47.470 Remember that we talked about that earlier. 01:20:47.470 --> 01:20:55.292 And so both the f and the f bar are hiding inside this formula. 01:20:55.292 --> 01:20:57.000 So there is a simplification that there's 01:20:57.000 --> 01:20:58.230 this delta function here. 01:21:01.590 --> 01:21:04.510 And so basically what we're talking about then-- 01:21:04.510 --> 01:21:06.600 you can think of just doing the integral 01:21:06.600 --> 01:21:08.440 over that delta function. 01:21:08.440 --> 01:21:13.447 So then you're talking about effectively an operator 01:21:13.447 --> 01:21:15.780 where you don't worry about putting a label on this one. 01:21:15.780 --> 01:21:17.410 You have only one delta function left, 01:21:17.410 --> 01:21:19.640 which I can denote by putting a label on this one. 01:21:19.640 --> 01:21:23.340 And this operator is actually exactly the operator 01:21:23.340 --> 01:21:26.760 that gives you the PDF. 01:21:26.760 --> 01:21:30.510 This operator is like a number operator 01:21:30.510 --> 01:21:43.900 for quarks, where you're thinking 01:21:43.900 --> 01:21:52.550 about momentum, a number operator with momentum omega. 01:21:52.550 --> 01:21:54.300 And if you want to think about there being 01:21:54.300 --> 01:21:56.760 some kind of field for a parton, this is about as close 01:21:56.760 --> 01:21:57.450 as you can get. 01:22:01.990 --> 01:22:03.930 So this quark field dressed by Wilson line 01:22:03.930 --> 01:22:07.820 is kind of like a parton in the parton model. 01:22:07.820 --> 01:22:11.630 All right, so next time, we'll take this, put it together 01:22:11.630 --> 01:22:13.880 with this formula here, with the c, 01:22:13.880 --> 01:22:15.230 and just see where it leads us. 01:22:15.230 --> 01:22:17.300 And it will lead us directly to a factorization 01:22:17.300 --> 01:22:19.820 theorem for deep inelastic scattering 01:22:19.820 --> 01:22:22.280 involving parton distribution functions 01:22:22.280 --> 01:22:24.800 and some hard perturbatively calculable thing, which 01:22:24.800 --> 01:22:28.940 is actually a cross-section in the parton model. 01:22:28.940 --> 01:22:32.343 And then we'll talk about renormalization [INAUDIBLE].. 01:22:32.343 --> 01:22:34.760 You can already see that we're getting the operator that I 01:22:34.760 --> 01:22:37.160 promised you-- that DIS is described 01:22:37.160 --> 01:22:40.010 by this, by a linear operator with two different quark 01:22:40.010 --> 01:22:42.620 fields, and particular labeling of the momenta. 01:22:47.980 --> 01:22:50.130 Any questions?