WEBVTT 00:00:00.000 --> 00:00:02.520 The following content is provided under a Creative 00:00:02.520 --> 00:00:03.970 Commons license. 00:00:03.970 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.660 continue to offer high quality educational resources for free. 00:00:10.660 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.520 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.520 --> 00:00:18.370 ocw.mit.edu. 00:00:23.100 --> 00:00:25.350 IAIN STEWART: All right, that's roughly where we were. 00:00:25.350 --> 00:00:28.530 So last time, we started talking about SCET. 00:00:28.530 --> 00:00:30.480 We said it's going to be a theory that 00:00:30.480 --> 00:00:34.100 can describe energetic hadrons and energetic jets. 00:00:34.100 --> 00:00:37.080 Our first example was discussing about a process 00:00:37.080 --> 00:00:40.560 with an energetic hadron, which is this pink pion. 00:00:40.560 --> 00:00:44.540 So the pion has a large momentum, large energy. 00:00:44.540 --> 00:00:45.990 It's much bigger than lambda QCD. 00:00:45.990 --> 00:00:47.400 It's much bigger than in m pi. 00:00:47.400 --> 00:00:50.470 And it moves basically along a light cone direction. 00:00:50.470 --> 00:00:53.560 So that was a motivation for us to use light cone coordinates. 00:00:53.560 --> 00:00:56.010 So we introduce an n and an n bar. 00:00:56.010 --> 00:01:02.550 And with that n and n bar, which satisfy n squared 0, 00:01:02.550 --> 00:01:08.520 n bar squared 0, and n dot n bar equals 2 as a normalization 00:01:08.520 --> 00:01:11.752 convention, we can decompose any momentum P 00:01:11.752 --> 00:01:16.380 mu in terms of components along n, components along n bar, 00:01:16.380 --> 00:01:18.630 and then the remaining two components which we call 00:01:18.630 --> 00:01:22.450 the perpendicular components. 00:01:22.450 --> 00:01:27.224 So we can also write the metric out in these coordinates. 00:01:32.164 --> 00:01:34.500 And this kind of makes explicit that you 00:01:34.500 --> 00:01:36.570 have this off-diagonal nature to the basis, 00:01:36.570 --> 00:01:38.830 that you have n mu with n bar mu. 00:01:46.500 --> 00:01:49.020 So unlike Cartesian coordinates, where 00:01:49.020 --> 00:01:50.880 the component along a direction is just 00:01:50.880 --> 00:01:53.850 given by dotting that vector into the vector 00:01:53.850 --> 00:01:56.640 you start with, here the component along n 00:01:56.640 --> 00:01:59.603 is given by dotting n bar into the vector. 00:01:59.603 --> 00:02:01.770 And that's reflected in the metric here in the sense 00:02:01.770 --> 00:02:04.320 that you have these terms n within n bar. 00:02:08.160 --> 00:02:12.060 So you can do this with any tensor if you have an epsilon. 00:02:12.060 --> 00:02:16.290 You can find an epsilon perp tensor, for example, 00:02:16.290 --> 00:02:24.180 by taking epsilon and putting in an n bar and an n. 00:02:24.180 --> 00:02:26.070 And then this would be a two-component tensor 00:02:26.070 --> 00:02:27.960 that behaves in the perpendicular direction 00:02:27.960 --> 00:02:30.060 as an antisymmetric tensor. 00:02:30.060 --> 00:02:33.570 And this g perp mu nu would be effectively living 00:02:33.570 --> 00:02:37.470 in the little subspace of the perp coordinates. 00:02:37.470 --> 00:02:39.490 And again, it's a metric tensor there. 00:02:39.490 --> 00:02:43.440 And this would be the antisymmetric tensor there. 00:02:43.440 --> 00:02:46.710 OK, so these are the coordinates we're going to use. 00:02:53.610 --> 00:02:56.990 So n here had a physical motivation, 00:02:56.990 --> 00:02:59.150 as you saw from my picture. 00:02:59.150 --> 00:03:01.280 The pion was moving in the n direction. 00:03:01.280 --> 00:03:03.350 And bar was just a vector that we 00:03:03.350 --> 00:03:05.600 decided that we needed in order to define things. 00:03:08.760 --> 00:03:11.540 So if you have some vector where n squared is 0 00:03:11.540 --> 00:03:14.360 and you want to make a decomposition of coordinates, 00:03:14.360 --> 00:03:15.770 then you're required to introduce 00:03:15.770 --> 00:03:26.110 a complimentary vector, which is this n bar 00:03:26.110 --> 00:03:31.570 to make the decomposition for the reasons I said. 00:03:38.123 --> 00:03:39.790 The simplest choice that you could make, 00:03:39.790 --> 00:03:46.540 if you made this choice for n, so if we choose n 00:03:46.540 --> 00:03:58.990 to be 1, 0, 0, minus 1, as I did, 00:03:58.990 --> 00:04:03.260 then the simplest thing you could do for n bar is to pick n 00:04:03.260 --> 00:04:07.616 bar to be 1, 0, 0, plus 1. 00:04:07.616 --> 00:04:10.600 So then that would be a lightlike vector. 00:04:10.600 --> 00:04:13.660 When you dot it into this vector, you get 2. 00:04:13.660 --> 00:04:19.459 And it satisfies all the criteria that we would need. 00:04:19.459 --> 00:04:21.709 If I choose these two vectors, then I 00:04:21.709 --> 00:04:23.420 have to find what perp is because perp 00:04:23.420 --> 00:04:26.030 is the space that's orthogonal to these vectors. 00:04:26.030 --> 00:04:31.490 So that's these two coordinates, OK? 00:04:31.490 --> 00:04:36.440 So perp in general is defined such that n dot P perp is 0 00:04:36.440 --> 00:04:40.040 and n bar dot P perp is 0. 00:04:40.040 --> 00:04:41.540 It's the orthogonal two directions 00:04:41.540 --> 00:04:44.642 to the ones that are picked out by an n bar. 00:04:44.642 --> 00:04:46.850 And so you need to know what n and n bar are in order 00:04:46.850 --> 00:04:47.808 to define what perp is. 00:04:50.550 --> 00:04:51.620 So this is one choice. 00:04:51.620 --> 00:04:54.140 You could make other choices. 00:04:54.140 --> 00:04:57.330 And we'll come back to this later on. 00:04:57.330 --> 00:05:04.910 So just by way of example, if I have the same choice for n, 00:05:04.910 --> 00:05:11.210 but I choose n bar to be 3, 2, 2, 1, 00:05:11.210 --> 00:05:13.790 that would also be a choice that's equally good. 00:05:18.130 --> 00:05:19.360 I need this to be minus 1. 00:05:26.944 --> 00:05:31.420 I'll do this, make my choice work. 00:05:31.420 --> 00:05:34.180 I guess, well, OK, if we want the same sign, 00:05:34.180 --> 00:05:35.200 then I have to do this. 00:05:39.910 --> 00:05:42.820 OK, so 9 minus 8, this thing still squares to 0. 00:05:42.820 --> 00:05:47.740 You dot it, you get 3 minus 1 is 2, OK? 00:05:47.740 --> 00:05:50.320 So it still satisfies the criteria that we had here. 00:05:53.440 --> 00:05:56.360 And it points in some other weird direction. 00:05:56.360 --> 00:05:58.480 So the point is that this is an auxiliary vector, 00:05:58.480 --> 00:06:03.585 and there's some freedom in what you pick for it. 00:06:03.585 --> 00:06:05.710 And once you've picked this, if you pick these two, 00:06:05.710 --> 00:06:08.230 you would have a different definition of perp. 00:06:08.230 --> 00:06:08.740 OK. 00:06:08.740 --> 00:06:12.250 But it's an equally valid possible choice. 00:06:12.250 --> 00:06:14.840 And we'll actually exploit this freedom later on. 00:06:14.840 --> 00:06:18.430 But for now, we'll mostly focus on picking the simplest choice. 00:06:21.150 --> 00:06:23.610 OK, what we're actually interested in describing 00:06:23.610 --> 00:06:26.310 in these processes is not just the plan, 00:06:26.310 --> 00:06:27.870 but what goes on inside the plan. 00:06:27.870 --> 00:06:30.730 What is the quark level process? 00:06:30.730 --> 00:06:34.050 So we're interested in the constituents. 00:06:34.050 --> 00:06:35.430 That's where the dynamics are. 00:06:43.522 --> 00:06:46.548 Is there any questions before I keep going? 00:06:46.548 --> 00:06:48.840 No. 00:06:48.840 --> 00:06:52.710 So in this process, B to D pi, if you think about it 00:06:52.710 --> 00:06:54.100 in the rest frame of the B meson, 00:06:54.100 --> 00:06:56.168 which is the most natural frame, then the B meson 00:06:56.168 --> 00:06:57.960 we've already learned how to describe that. 00:06:57.960 --> 00:07:03.900 We can describe that with HQET, same with the D meson. 00:07:03.900 --> 00:07:06.390 And we know that the things that are inside the B and the D 00:07:06.390 --> 00:07:10.930 meson are one heavy quark and then a bunch of soft stuff. 00:07:10.930 --> 00:07:13.945 So I'll call these guys soft because the dynamical part is 00:07:13.945 --> 00:07:14.445 soft. 00:07:23.600 --> 00:07:33.720 And so we can use HQET for them as we did before. 00:07:43.930 --> 00:07:47.590 And that means we're describing gluons and quarks that 00:07:47.590 --> 00:07:54.205 are inside these hadrons where the forward momentum are 00:07:54.205 --> 00:07:57.370 of order lambda QCD. 00:07:57.370 --> 00:08:00.280 The pion, on the other hand, is what we would collinear. 00:08:16.620 --> 00:08:18.590 So as I already described, the pion's energy 00:08:18.590 --> 00:08:21.060 is much greater than its mass. 00:08:21.060 --> 00:08:21.905 Its highly boosted. 00:08:29.720 --> 00:08:32.600 If you were to talk about it in the rest frame, 00:08:32.600 --> 00:08:34.880 then like the B and the D meson, then the constituents 00:08:34.880 --> 00:08:40.215 of the pion would have momentum of order lambda QCD. 00:08:40.215 --> 00:08:42.590 But if you were to talk about the pion in the rest frame, 00:08:42.590 --> 00:08:47.050 you'd have to talk about the B and the D in the boosted frame. 00:08:47.050 --> 00:08:49.150 So let's stick with describing the B and the D 00:08:49.150 --> 00:08:52.790 in the rest frame or close to the rest frame. 00:08:52.790 --> 00:08:55.150 The B meson is in its rest frame. 00:08:55.150 --> 00:08:57.610 In the D meson is slow. 00:08:57.610 --> 00:09:00.700 And in that case, we're stuck with the pion being energetic. 00:09:03.820 --> 00:09:12.640 So in rest frame, our pion would be-- 00:09:17.890 --> 00:09:25.698 it would also have quarks and gluons 00:09:25.698 --> 00:09:27.560 our P mu is order lambda QCD. 00:09:32.780 --> 00:09:35.360 And we can actually just take that result, once we know that, 00:09:35.360 --> 00:09:38.720 and boost it to another frame. 00:09:38.720 --> 00:09:43.520 So let's just boost along z hat by some kappa that's 00:09:43.520 --> 00:09:46.695 much greater than 1 as the boost. 00:09:46.695 --> 00:09:48.570 And the way that light cone coordinates boost 00:09:48.570 --> 00:09:49.290 is very simple. 00:09:49.290 --> 00:09:52.040 If you're along the axis of the light cone coordinates, 00:09:52.040 --> 00:09:55.410 it's multiplicative. 00:09:55.410 --> 00:09:57.500 So P minus gets enhanced by some amount. 00:09:57.500 --> 00:09:59.390 P plus gets suppressed by the same amount. 00:10:02.100 --> 00:10:05.268 That's one nice thing about these coordinates. 00:10:05.268 --> 00:10:07.310 And of course, P perp doesn't change because it's 00:10:07.310 --> 00:10:08.510 perpendicular to the boost. 00:10:11.570 --> 00:10:14.925 So now, we can get our pion, which is moving, 00:10:14.925 --> 00:10:15.800 which should be pink. 00:10:25.340 --> 00:10:27.020 And if we ask about its constituents, 00:10:27.020 --> 00:10:29.060 we just boost the components of this for vector. 00:10:38.430 --> 00:10:41.330 So we ask about how they scale. 00:10:41.330 --> 00:10:44.810 And we look at the different components. 00:10:44.810 --> 00:10:47.870 The plus, minus, and perp scale differently now, 00:10:47.870 --> 00:10:50.320 so we have to break it up by that. 00:10:50.320 --> 00:10:53.240 And if we boost it by this amount Q or lambda 00:10:53.240 --> 00:11:02.970 over Q, Q over lambda, then that's 00:11:02.970 --> 00:11:06.440 the scaling for this boosted pion. 00:11:06.440 --> 00:11:06.940 OK. 00:11:06.940 --> 00:11:10.110 So now, it's got a component in the minus direction. 00:11:10.110 --> 00:11:13.650 n bar dot P is order Q. That's what 00:11:13.650 --> 00:11:15.360 we saw before when we decomposed the P, 00:11:15.360 --> 00:11:18.570 that it was basically Q times a lightlike vector. 00:11:18.570 --> 00:11:20.430 But that was the pion. 00:11:20.430 --> 00:11:23.130 Now, we're talking about the constituents inside the pion. 00:11:23.130 --> 00:11:25.638 Constituents inside the pion fill it out. 00:11:25.638 --> 00:11:27.180 They fill it out in the perpendicular 00:11:27.180 --> 00:11:30.210 by an amount lambda QCD that's perpendicular to the direction 00:11:30.210 --> 00:11:31.350 of its motion. 00:11:31.350 --> 00:11:34.440 And then the plus momentum got correspondingly smaller 00:11:34.440 --> 00:11:37.310 as the minus momentum have got bigger, 00:11:37.310 --> 00:11:38.310 so we have that scaling. 00:11:40.830 --> 00:11:42.510 And so the relative scaling here is 00:11:42.510 --> 00:11:45.110 what actually defines something being collinear. 00:11:51.000 --> 00:11:53.610 So the relative scaling of this vector 00:11:53.610 --> 00:11:58.230 here is that the P minus is much bigger than the P perp is 00:11:58.230 --> 00:12:00.480 much bigger than the P plus. 00:12:00.480 --> 00:12:02.472 And that's what we mean by collinear. 00:12:06.900 --> 00:12:08.490 It's collimated in some direction, 00:12:08.490 --> 00:12:09.990 and that direction is the direction 00:12:09.990 --> 00:12:10.907 of the large momentum. 00:12:13.353 --> 00:12:14.770 You always have to be careful when 00:12:14.770 --> 00:12:16.810 you say things like that because the component 00:12:16.810 --> 00:12:19.610 along the direction is the opposite lightlike vector, 00:12:19.610 --> 00:12:22.600 but I think you'll always know what I mean. 00:12:22.600 --> 00:12:27.730 OK, so in the n mu direction, we have a large component P minus. 00:12:27.730 --> 00:12:29.837 And that defines this thing is collimated 00:12:29.837 --> 00:12:30.920 in a particular direction. 00:12:30.920 --> 00:12:34.065 It's perpendicular fluctuations to that direction are small. 00:12:34.065 --> 00:12:35.440 And so all the degrees of freedom 00:12:35.440 --> 00:12:40.330 that are in this boosted pion have that type of scaling. 00:12:40.330 --> 00:12:42.070 So what we're describing, or what 00:12:42.070 --> 00:12:45.370 we want to describe if we have a field theory for this, 00:12:45.370 --> 00:12:47.110 is we want to describe, if you like, 00:12:47.110 --> 00:12:57.340 fluctuations about the pion momentum, which, 00:12:57.340 --> 00:13:06.580 ignoring the pion mass, we could just take it to be like this. 00:13:06.580 --> 00:13:09.700 And the size of the fluctuations we need to treat 00:13:09.700 --> 00:13:12.250 are things that can fluctuate by amounts of this size. 00:13:16.110 --> 00:13:17.610 So the field theory is going to have 00:13:17.610 --> 00:13:23.153 to describe fluctuations about some kind of canonical scaling. 00:13:23.153 --> 00:13:24.570 And the field theory for this pion 00:13:24.570 --> 00:13:26.520 is going to have to be describing 00:13:26.520 --> 00:13:29.100 collinear fluctuations that are of this type. 00:13:29.100 --> 00:13:33.560 Just like the HQET had to describe soft fluctuations, 00:13:33.560 --> 00:13:36.690 P mu's of order lambda QCD ignorant of the heavy quark 00:13:36.690 --> 00:13:40.730 mass, here it's a little bit more complicated. 00:13:40.730 --> 00:13:43.230 But that's the kind of thing we want the field theory to do. 00:13:47.300 --> 00:13:48.520 Any questions about that? 00:13:57.220 --> 00:14:02.340 So the way that we write this is we say that P plus, P minus, 00:14:02.340 --> 00:14:08.460 P perp has a particular scaling that we 00:14:08.460 --> 00:14:13.200 call lambda squared 1 lambda where 00:14:13.200 --> 00:14:14.640 lambda is some small parameter. 00:14:20.510 --> 00:14:22.750 And if we have a momentum that scales that way, 00:14:22.750 --> 00:14:24.066 we call it collinear. 00:14:35.320 --> 00:14:39.630 So this is generic any case with any lambda. 00:14:39.630 --> 00:14:43.116 And our allowed here was just lambda QCD over Q. 00:14:43.116 --> 00:14:46.680 But if we encounter another physical problem 00:14:46.680 --> 00:14:49.620 where the lambda was different, we 00:14:49.620 --> 00:14:51.224 would also call that collinear. 00:14:55.870 --> 00:15:00.970 All right, so what's a nice way of picturing this, 00:15:00.970 --> 00:15:02.980 what we're doing here? 00:15:02.980 --> 00:15:04.990 Because it's a little bit different than you're 00:15:04.990 --> 00:15:07.223 used to with an effective field theory. 00:15:07.223 --> 00:15:09.640 Usually, with an effective field theory, what you're doing 00:15:09.640 --> 00:15:12.593 is you're separating modes by their invariant mass. 00:15:12.593 --> 00:15:15.010 You have things with large invariant mass, small invariant 00:15:15.010 --> 00:15:16.618 mass. 00:15:16.618 --> 00:15:18.160 If you think about massive particles, 00:15:18.160 --> 00:15:20.950 well, that's just the invariant mass squared. 00:15:20.950 --> 00:15:22.900 So if you're separating massive particles 00:15:22.900 --> 00:15:25.780 from massless particles or less massive particles, 00:15:25.780 --> 00:15:30.370 you're really separating things along an invariant mass curve. 00:15:30.370 --> 00:15:33.940 Just an invariant mass variable is used for the separation. 00:15:33.940 --> 00:15:35.740 And that doesn't quite suffice here. 00:15:35.740 --> 00:15:38.830 Because as you saw, the pion in the B and the D meson, 00:15:38.830 --> 00:15:41.620 they both had P squared of order lambda QCD. 00:15:41.620 --> 00:15:44.710 What separates the pion from the B or the D meson 00:15:44.710 --> 00:15:47.350 is this morphyne structure. 00:16:05.290 --> 00:16:07.720 So SCET is actually an example of an effective field 00:16:07.720 --> 00:16:12.340 theory that requires at least more than one variable 00:16:12.340 --> 00:16:14.830 to describe where the degrees of freedom live. 00:16:19.413 --> 00:16:21.580 So we can draw a picture for what we've been talking 00:16:21.580 --> 00:16:24.880 about here in two variables. 00:16:24.880 --> 00:16:26.470 Let's just pick P minus and P plus. 00:16:31.600 --> 00:16:35.980 And essentially, what's going on in this space is 00:16:35.980 --> 00:16:38.260 you can think that there's degrees of freedom that 00:16:38.260 --> 00:16:40.840 live in this space at different locations. 00:16:47.820 --> 00:16:51.570 So out here, if I draw a hyperbola like this, 00:16:51.570 --> 00:16:58.938 then remember that P squared was P plus times P minus minus P 00:16:58.938 --> 00:16:59.480 perp squared. 00:16:59.480 --> 00:17:02.460 But let's ignore P perp squared for this picture. 00:17:02.460 --> 00:17:05.839 So if I draw a curve of constant P squared in this plane, then 00:17:05.839 --> 00:17:07.010 it's I hyperbola. 00:17:07.010 --> 00:17:08.930 So these are curves of constant P squared. 00:17:16.540 --> 00:17:18.670 And this one here has P squared of order Q 00:17:18.670 --> 00:17:21.800 squared, which might be Mb squared or some hard scale. 00:17:21.800 --> 00:17:25.420 So this any degrees of freedom that live on this curve, 00:17:25.420 --> 00:17:27.640 or in particularly these ones, would 00:17:27.640 --> 00:17:29.752 be what we would call hard degrees of freedom. 00:17:29.752 --> 00:17:31.960 And those are something that we want to integrate out 00:17:31.960 --> 00:17:34.210 of the effective theory. 00:17:34.210 --> 00:17:37.180 And the other degrees of freedom that we've been talking about 00:17:37.180 --> 00:17:38.680 have smaller invariant mass. 00:17:38.680 --> 00:17:42.460 So this hyperbola down here has P squared of order lambda 00:17:42.460 --> 00:17:44.410 QCD squared. 00:17:44.410 --> 00:17:46.570 But there's two different degrees of freedom 00:17:46.570 --> 00:17:48.340 that live on this curve. 00:17:48.340 --> 00:17:50.020 One of them has a large P minus. 00:17:50.020 --> 00:17:53.110 That's the collinear one, so it should be pink. 00:18:02.030 --> 00:18:08.090 And then the soft one lives down there. 00:18:08.090 --> 00:18:12.680 So P minus here is scaling, if you like, lambda to the 0. 00:18:12.680 --> 00:18:17.135 And here, for this soft mode, which also exists in this case, 00:18:17.135 --> 00:18:18.635 this is actually going to be lambda. 00:18:24.060 --> 00:18:27.210 So you can contrast that type of picture 00:18:27.210 --> 00:18:31.260 with a more usual picture where you would just have one line. 00:18:31.260 --> 00:18:32.940 And you'd say there's some modes up here 00:18:32.940 --> 00:18:33.900 and some modes down there. 00:18:33.900 --> 00:18:35.400 And you'd integrate out these modes. 00:18:35.400 --> 00:18:36.790 And you keep those modes. 00:18:36.790 --> 00:18:38.582 This is a little different because you want 00:18:38.582 --> 00:18:39.798 to integrate out these modes. 00:18:39.798 --> 00:18:41.340 You want to keep both of those modes, 00:18:41.340 --> 00:18:43.465 but they live in a little bit of a different place. 00:18:43.465 --> 00:18:45.960 And that's actually going to be important to formulating 00:18:45.960 --> 00:18:47.760 the effective theory. 00:18:47.760 --> 00:18:50.910 So the way that you should think about this, physically the way 00:18:50.910 --> 00:18:53.490 you should think about it, is that these modes are kind of 00:18:53.490 --> 00:18:54.840 localized in that region. 00:18:57.360 --> 00:19:05.320 This is the right physical picture, 00:19:05.320 --> 00:19:08.440 which requires another variable besides just invariant mass 00:19:08.440 --> 00:19:15.940 in order to specify that, right? 00:19:15.940 --> 00:19:18.790 The reason we don't have to draw a third direction for P perp 00:19:18.790 --> 00:19:25.410 is because it was just redundant information. 00:19:25.410 --> 00:19:28.455 P perp squared is always of P plus P 00:19:28.455 --> 00:19:30.330 minus if you're talking about fluctuations 00:19:30.330 --> 00:19:31.538 that are near the mass shell. 00:19:40.490 --> 00:19:43.400 For a massless mode, that mass shell is P squared equals 0. 00:19:47.030 --> 00:19:49.730 And so P perp would just be providing redundant information 00:19:49.730 --> 00:19:51.605 to our picture, and we just can leave it out. 00:19:54.870 --> 00:19:57.080 Now, the boundaries of the regions 00:19:57.080 --> 00:20:01.700 between soft and collinear here seems like an interesting thing 00:20:01.700 --> 00:20:03.550 to worry about. 00:20:03.550 --> 00:20:07.400 And that is, indeed, true. 00:20:07.400 --> 00:20:09.470 You have to think about how you want to set up 00:20:09.470 --> 00:20:11.060 this effective theory. 00:20:11.060 --> 00:20:13.425 And of course, as I have been emphasizing earlier 00:20:13.425 --> 00:20:15.050 in the course, the easiest way to think 00:20:15.050 --> 00:20:17.630 about momentum degrees of freedom 00:20:17.630 --> 00:20:19.430 is with a Wilsonian picture. 00:20:19.430 --> 00:20:22.348 That makes physically what's going on very clear. 00:20:22.348 --> 00:20:24.140 So the simplest thing would be to introduce 00:20:24.140 --> 00:20:35.270 a Wilsonian cut-off and set this up 00:20:35.270 --> 00:20:38.750 as a Wilsonian effective field theory. 00:20:38.750 --> 00:20:40.910 And then we would just take these regions. 00:20:40.910 --> 00:20:44.180 And I would literally carve them out in the way that I drew. 00:20:44.180 --> 00:20:46.160 I would carve out some cut-off between them. 00:20:46.160 --> 00:20:48.890 And I would decide who's in the soft region, who's 00:20:48.890 --> 00:20:53.720 in the collinear region based on those hard cut-offs. 00:20:53.720 --> 00:20:55.700 But we don't want to do that, actually, 00:20:55.700 --> 00:20:59.240 because it would mess up all sorts of symmetries. 00:20:59.240 --> 00:21:01.515 In particular, it would mess up gauge cemetery 00:21:01.515 --> 00:21:03.140 which is an important thing when you're 00:21:03.140 --> 00:21:05.400 talking about gauge theory. 00:21:05.400 --> 00:21:08.000 So we're going to use dimensional regularization, 00:21:08.000 --> 00:21:09.515 as we have for other problems. 00:21:29.360 --> 00:21:31.280 And that actually will still leave us 00:21:31.280 --> 00:21:33.860 with this picture, which I drew as a cartoon. 00:21:33.860 --> 00:21:35.000 You'll still be correct. 00:21:35.000 --> 00:21:37.400 Think about the modes live in those places 00:21:37.400 --> 00:21:39.200 in dimensional regularization. 00:21:39.200 --> 00:21:42.020 What's a little bit harder is how to think about the cut-off. 00:21:42.020 --> 00:21:46.121 And we'll treat that in some detail later on. 00:21:46.121 --> 00:21:59.260 So it's still the correct picture, 00:21:59.260 --> 00:22:04.740 but treating the region overlaps with dimensional regularization 00:22:04.740 --> 00:22:05.740 is a little more tricky. 00:22:08.285 --> 00:22:09.910 But at least we can do it in a way that 00:22:09.910 --> 00:22:11.692 preserves the gauge invariance. 00:22:17.050 --> 00:22:22.529 So that's going to be our mode of operation. 00:22:43.500 --> 00:22:44.790 This theory has a name. 00:22:44.790 --> 00:22:48.020 It goes by the name SCET2. 00:22:48.020 --> 00:22:49.260 That's why I called it SCET2. 00:22:54.510 --> 00:22:59.780 We'll come back in a moment to what SCET1 is. 00:22:59.780 --> 00:23:02.120 So I can say that the degrees of freedom 00:23:02.120 --> 00:23:05.758 in this theory, the one that we've been talking about, 00:23:05.758 --> 00:23:07.550 are some collinear degree of freedom that's 00:23:07.550 --> 00:23:09.920 associated to some direction, some collinear 00:23:09.920 --> 00:23:14.450 degrees of freedom as well as some soft degrees of freedom. 00:23:14.450 --> 00:23:16.568 And when you have effective theories that 00:23:16.568 --> 00:23:18.860 are like this one, where the soft and collinear degrees 00:23:18.860 --> 00:23:21.230 of freedom live on the same mass hyperbola, 00:23:21.230 --> 00:23:22.670 they're called SCET2 theories. 00:23:50.720 --> 00:23:53.090 And these are really the kind of theories 00:23:53.090 --> 00:23:56.360 that you get when you're talking about energetic hadron 00:23:56.360 --> 00:23:57.425 production. 00:24:09.820 --> 00:24:11.730 So any questions so far? 00:24:17.730 --> 00:24:21.420 OK, so if that's energetic hadrons, 00:24:21.420 --> 00:24:23.520 then SCET1 will be energetic jets. 00:24:23.520 --> 00:24:25.740 And that's what we'll talk about next. 00:24:28.270 --> 00:24:31.950 So let's do another example which has jets in it. 00:24:31.950 --> 00:24:34.530 And we'll see what the similarities are 00:24:34.530 --> 00:24:39.690 to this SCET2 set up in terms of just identifying still what 00:24:39.690 --> 00:24:43.630 the right degrees of freedom are. 00:24:43.630 --> 00:24:45.810 So let's look at e plus e minus to two jets. 00:24:53.730 --> 00:24:57.750 So e plus e minus collide. 00:24:57.750 --> 00:25:01.170 They produce a virtual photon, say. 00:25:01.170 --> 00:25:05.250 The virtual photon produces a quark-antiquark pair. 00:25:05.250 --> 00:25:08.430 The quark-antiquark pair starts to radiate. 00:25:08.430 --> 00:25:12.690 And we get jets, two of them. 00:25:21.360 --> 00:25:23.243 So again, there's a kind of natural frame 00:25:23.243 --> 00:25:24.410 to describe this scattering. 00:25:24.410 --> 00:25:28.460 And that's the center of mass frame of the e plus e minus. 00:25:28.460 --> 00:25:30.860 Most e plus e minus colliders are built in that frame. 00:25:37.120 --> 00:25:40.270 And if you're in the center of mass frame 00:25:40.270 --> 00:25:44.530 and you call the for momentum of this photon Q, 00:25:44.530 --> 00:25:47.650 then it just has an energy component. 00:25:47.650 --> 00:25:50.320 So Q here is not the same as the Q in our previous example, 00:25:50.320 --> 00:25:52.690 but we're always going to identify the hard scale as Q. 00:25:52.690 --> 00:25:54.550 So here, the hard scale is the scale 00:25:54.550 --> 00:25:58.823 of the energy of the collision in the center of mass frame. 00:25:58.823 --> 00:26:00.490 And if you ask, what does the event look 00:26:00.490 --> 00:26:02.080 like in the center of mass frame, 00:26:02.080 --> 00:26:04.000 then these two jets which are going out 00:26:04.000 --> 00:26:05.480 have to balance each other. 00:26:05.480 --> 00:26:07.270 And so you have two back to back jets. 00:26:10.300 --> 00:26:19.160 So we draw it like this, one jet going this way, one jet going 00:26:19.160 --> 00:26:20.010 that way. 00:26:20.010 --> 00:26:21.890 Our original e plus e minus might have come 00:26:21.890 --> 00:26:23.100 in from some other direction. 00:26:23.100 --> 00:26:29.030 So maybe e plus e minus were coming in from here and here. 00:26:32.550 --> 00:26:36.440 And then we have these two jets going out this way. 00:26:36.440 --> 00:26:38.690 Since they're jets, that means they're 00:26:38.690 --> 00:26:40.460 collimated sprays of radiation. 00:26:46.745 --> 00:26:47.870 So they're not featureless. 00:26:47.870 --> 00:26:53.550 They have some size to them like this. 00:26:53.550 --> 00:26:57.920 And in this process, we can quickly 00:26:57.920 --> 00:27:00.560 identify that there's two relevant directions. 00:27:00.560 --> 00:27:03.950 It's back to back, but let's define 00:27:03.950 --> 00:27:07.460 this direction of this jet to be n1, 00:27:07.460 --> 00:27:09.960 some lightlike vector that points in that direction, 00:27:09.960 --> 00:27:11.870 and this one to be n2. 00:27:11.870 --> 00:27:16.940 So generically, we could say you n is some 1, n hat. 00:27:16.940 --> 00:27:18.900 And if we choose this to be the z-axis, 00:27:18.900 --> 00:27:24.120 then this must be 1, 0, 0 1, just like we had before. 00:27:24.120 --> 00:27:26.780 But we can always pick a lightlike vector that points 00:27:26.780 --> 00:27:28.082 along some direction n hat. 00:27:28.082 --> 00:27:29.540 So when I draw a picture like this, 00:27:29.540 --> 00:27:32.480 really what I mean when I say n1 points in this direction 00:27:32.480 --> 00:27:35.360 is that the hat part of it points in that direction 00:27:35.360 --> 00:27:38.520 in through space. 00:27:38.520 --> 00:27:41.160 OK, so just like before, what happens 00:27:41.160 --> 00:27:44.040 with the jet is, as I said last time actually, 00:27:44.040 --> 00:27:47.310 we have a large energy flow in this direction and a smaller 00:27:47.310 --> 00:27:49.540 perpendicular flow. 00:27:49.540 --> 00:27:52.380 So if we measure perpendicular momentum, which 00:27:52.380 --> 00:27:57.040 we can think of as perpendicular to that axis, free axis, 00:27:57.040 --> 00:28:01.110 then that's the perpendicular flow inside the jet. 00:28:01.110 --> 00:28:06.090 And so if we want to talk about constituents inside the jet, 00:28:06.090 --> 00:28:08.670 then we'll be interested in smaller perpendicular 00:28:08.670 --> 00:28:10.650 flow than flow in the forward direction. 00:28:10.650 --> 00:28:13.960 And that's what makes it into a collimated jet. 00:28:13.960 --> 00:28:14.710 You have question? 00:28:14.710 --> 00:28:15.390 AUDIENCE: [INAUDIBLE] 00:28:15.390 --> 00:28:15.840 IAIN STEWART: Sure. 00:28:15.840 --> 00:28:17.423 AUDIENCE: So you're not using anything 00:28:17.423 --> 00:28:19.440 about the QCD [INAUDIBLE] to tell you 00:28:19.440 --> 00:28:20.828 the things is [INAUDIBLE]. 00:28:20.828 --> 00:28:23.370 You're just saying we know that we have to [INAUDIBLE] topic. 00:28:23.370 --> 00:28:25.740 IAIN STEWART: Yeah, that's the attitude, right. 00:28:25.740 --> 00:28:27.022 So I mean, we'll see why. 00:28:27.022 --> 00:28:28.480 You know, you can ask the question, 00:28:28.480 --> 00:28:30.063 why do we get jets in the first place? 00:28:30.063 --> 00:28:31.470 We haven't talked about that. 00:28:31.470 --> 00:28:33.130 And we could talk about that. 00:28:33.130 --> 00:28:35.910 I'm taking the attitude here, this is what we observe. 00:28:35.910 --> 00:28:38.220 How do we design an effective field theory for it? 00:28:38.220 --> 00:28:40.320 And we'll see then, from that effective theory, 00:28:40.320 --> 00:28:41.850 we can go back and understand why 00:28:41.850 --> 00:28:43.475 it is that we get these objects and why 00:28:43.475 --> 00:28:45.225 actually, when you look at cross-sections, 00:28:45.225 --> 00:28:47.835 that this is the leading order description of what happens. 00:28:47.835 --> 00:28:50.790 We'll come to that later. 00:28:50.790 --> 00:28:52.710 Good question. 00:28:52.710 --> 00:28:54.107 Any other questions? 00:28:56.830 --> 00:28:57.330 All right. 00:29:02.310 --> 00:29:06.480 Yeah, in the hadron case, you just say this process exists. 00:29:06.480 --> 00:29:09.530 And I didn't explain to you why the process could exist. 00:29:09.530 --> 00:29:11.280 But in the jet case, there's more dynamics 00:29:11.280 --> 00:29:13.800 going into the fact that the process exists, 00:29:13.800 --> 00:29:17.850 the fact that QCD likes to radiate collinear 00:29:17.850 --> 00:29:20.400 to a direction, which has to do with the infrared structure 00:29:20.400 --> 00:29:22.200 of the theory. 00:29:22.200 --> 00:29:23.640 And we'll come back to that. 00:29:28.730 --> 00:29:30.690 I mean, the short answer, of course, 00:29:30.690 --> 00:29:33.200 is that things like to radiate in that direction 00:29:33.200 --> 00:29:37.160 because there's large logarithms that 00:29:37.160 --> 00:29:41.670 enhance splittings that are collinear to the direction 00:29:41.670 --> 00:29:42.210 of motion. 00:29:45.030 --> 00:29:46.880 And you also get smaller coupling consonants 00:29:46.880 --> 00:29:51.963 when you do that rather than a wide angle emission. 00:29:51.963 --> 00:29:53.630 So we're saying here, what we're saying, 00:29:53.630 --> 00:29:56.600 is we measure e plus e minus to two jets. 00:29:56.600 --> 00:29:58.758 If I had an extra wide angle emission, 00:29:58.758 --> 00:30:00.550 that would be e plus e minus to three jets. 00:30:00.550 --> 00:30:03.041 So I ruled that out just right at the beginning. 00:30:05.690 --> 00:30:07.790 OK, so what you can do with this picture 00:30:07.790 --> 00:30:11.400 in order to define what's going on is you can say, 00:30:11.400 --> 00:30:13.490 well they're to back-to-back jets. 00:30:13.490 --> 00:30:14.750 Let's draw a hemisphere. 00:30:17.978 --> 00:30:19.520 So this is supposed to be kind of out 00:30:19.520 --> 00:30:23.380 of the board point at you. 00:30:23.380 --> 00:30:25.130 Let's draw a hemisphere between these two. 00:30:25.130 --> 00:30:28.370 And then we can call one side a and the other side b. 00:30:28.370 --> 00:30:29.840 And we can talk about momenta that 00:30:29.840 --> 00:30:32.048 are flowing in the a hemisphere and the b hemisphere. 00:30:32.048 --> 00:30:34.318 And you see that we have one jet in each hemisphere. 00:30:43.600 --> 00:30:46.590 So we have a jet of hadrons in hemisphere a. 00:30:50.910 --> 00:30:56.540 And we have another one in hemisphere b. 00:31:01.110 --> 00:31:04.431 So in some ways, we can talk about a and b independently. 00:31:10.560 --> 00:31:12.270 So let's start off by talking about a, 00:31:12.270 --> 00:31:15.350 which is what I called the n1 collinear jet. 00:31:17.930 --> 00:31:26.550 And if we ask about constituents in the jet, 00:31:26.550 --> 00:31:28.110 then the perpendicular momentum will 00:31:28.110 --> 00:31:31.980 be of some size, which let me just call it delta. 00:31:31.980 --> 00:31:37.290 And that'll be much smaller than P minus, which is of size Q. 00:31:37.290 --> 00:31:40.500 So the energy that we pump in through the photon 00:31:40.500 --> 00:31:41.310 has to leave. 00:31:41.310 --> 00:31:44.760 And the only place that it can leave is that half of it 00:31:44.760 --> 00:31:46.260 has to kind of leave this direction. 00:31:46.260 --> 00:31:48.570 Half of it has to leave that direction 00:31:48.570 --> 00:31:50.850 by energy conservation. 00:31:50.850 --> 00:31:55.090 And so we have a large energy flow in the P minus component. 00:31:55.090 --> 00:31:57.510 And then we have a much smaller amount in the P perp. 00:31:57.510 --> 00:31:59.820 And that's what I already said. 00:31:59.820 --> 00:32:01.740 And given those two facts, you can 00:32:01.740 --> 00:32:04.260 ask, what about the constituents of the jet? 00:32:04.260 --> 00:32:07.575 And again, they're collinear because you have a hierarchy. 00:32:16.485 --> 00:32:19.050 So the plus minus and perpendicular momentum 00:32:19.050 --> 00:32:21.540 of constituents would scale in that way. 00:32:21.540 --> 00:32:26.668 And that is Q lambda squared 1 lambda, just as 00:32:26.668 --> 00:32:27.960 before with a different lambda. 00:32:34.670 --> 00:32:37.650 So here, lambda is how much spread and perp 00:32:37.650 --> 00:32:43.230 do we have over Q. This delta doesn't have to be lambda QCD. 00:32:43.230 --> 00:32:46.350 It could be something much bigger. 00:32:46.350 --> 00:32:48.855 Another way of thinking about physically what this delta is 00:32:48.855 --> 00:32:52.570 is to calculate something called the jet mass. 00:32:52.570 --> 00:33:01.260 So you could define the mass of the jet as the sum of the four 00:33:01.260 --> 00:33:10.510 vectors in hemisphere a of all the particles squared. 00:33:10.510 --> 00:33:14.010 And if you ask about how big that is, in these coordinates 00:33:14.010 --> 00:33:16.650 that we're using, it's P plus times P minus minus P 00:33:16.650 --> 00:33:18.460 perp squared. 00:33:18.460 --> 00:33:22.980 If we align things so that this thing is really 00:33:22.980 --> 00:33:26.260 aligned with the jet, there won't be any P perp squared. 00:33:26.260 --> 00:33:29.910 But that wouldn't change us scaling our given anyway. 00:33:29.910 --> 00:33:32.820 So basically, if you ask about what this jet mass is, 00:33:32.820 --> 00:33:35.970 it's scaling like a P plus times a P minus. 00:33:35.970 --> 00:33:38.630 And it's scaling like delta squared. 00:33:38.630 --> 00:33:43.230 So the jet mass is something of order delta squared. 00:33:43.230 --> 00:33:45.490 And that's much less than Q squared. 00:33:45.490 --> 00:33:48.030 So another way of characterizing that you have a jet 00:33:48.030 --> 00:33:50.670 is to measure the invariant mass of all the particles 00:33:50.670 --> 00:33:52.020 in this hemisphere. 00:33:52.020 --> 00:33:54.390 And if that invariant mass is small relative 00:33:54.390 --> 00:33:58.320 to the hard scale, then it's collimated, OK? 00:33:58.320 --> 00:34:05.040 So MJ squared Q squared much less than 1 also 00:34:05.040 --> 00:34:06.060 means collimated. 00:34:11.342 --> 00:34:13.050 So we could talk about it either in terms 00:34:13.050 --> 00:34:15.060 of perpendicular spread, or we could talk about it 00:34:15.060 --> 00:34:15.960 as an invariant mass. 00:34:19.830 --> 00:34:22.183 OK. 00:34:22.183 --> 00:34:25.620 So this is much less than 1. 00:34:25.620 --> 00:34:29.639 Now, with this delta, we didn't specify what it is. 00:34:29.639 --> 00:34:31.980 So what are possible values of delta? 00:34:31.980 --> 00:34:35.400 Well, let's first talk about what it's not. 00:34:35.400 --> 00:34:37.739 If delta was of order Q, then obviously 00:34:37.739 --> 00:34:40.530 we'd break the kind of scaling that we have here that this 00:34:40.530 --> 00:34:42.525 should be much less than 1. 00:34:42.525 --> 00:34:44.775 And what happens in that case is we don't have dijets. 00:34:48.989 --> 00:34:52.600 So if either the mass MJ squared of the particles 00:34:52.600 --> 00:34:56.130 becomes of order Q squared or the perpendicular spread 00:34:56.130 --> 00:34:59.751 becomes of order Q, then we don't have dijets anymore. 00:35:02.460 --> 00:35:04.230 And basically, in this case, you would 00:35:04.230 --> 00:35:14.490 be talking about inclusive sum over our jets 00:35:14.490 --> 00:35:16.740 And that would be something that you would actually 00:35:16.740 --> 00:35:19.115 describe in a different way in the effective field theory 00:35:19.115 --> 00:35:22.500 because you wouldn't then have collinear degrees of freedom 00:35:22.500 --> 00:35:23.550 for this jet. 00:35:23.550 --> 00:35:25.570 It's really picking out the dijet process 00:35:25.570 --> 00:35:27.653 that means that these collinear degrees of freedom 00:35:27.653 --> 00:35:28.500 are relevant. 00:35:28.500 --> 00:35:31.410 If you did just e plus e minus to hadrons, that's something 00:35:31.410 --> 00:35:33.720 you could do with an operator product expansion 00:35:33.720 --> 00:35:35.940 without ever talking about SCET. 00:35:35.940 --> 00:35:38.520 And that would be valid if you were really 00:35:38.520 --> 00:35:40.860 doing an inclusive sum over jets in all directions 00:35:40.860 --> 00:35:43.950 without any restrictions that tell you it's a dijet. 00:35:43.950 --> 00:35:46.560 And then the effect of power counting in your OPE 00:35:46.560 --> 00:35:49.290 would be such that delta is of order Q. 00:35:49.290 --> 00:35:50.970 So this is actually the OPE region. 00:35:54.040 --> 00:36:01.170 Let me call it the OPE region of Peskin or any other field 00:36:01.170 --> 00:36:01.680 theory book. 00:36:06.650 --> 00:36:08.710 Another thing we could do is we could take delta 00:36:08.710 --> 00:36:09.460 to be very small. 00:36:12.550 --> 00:36:14.830 We could take delta all the way down to lambda QCD. 00:36:21.550 --> 00:36:22.870 And that's also not a jet. 00:36:29.810 --> 00:36:31.610 If delta is of order lambda QCD, what 00:36:31.610 --> 00:36:33.770 happens with the spray of radiation 00:36:33.770 --> 00:36:35.960 is that it gets bound into a hadron. 00:36:35.960 --> 00:36:37.130 It just can't separate. 00:36:37.130 --> 00:36:44.490 Confinement grabs it, and you get an energetic hadron, not 00:36:44.490 --> 00:36:44.990 a jet. 00:36:55.860 --> 00:37:04.498 So if the jets get too narrow, in particular this narrow, 00:37:04.498 --> 00:37:06.415 then the constituents are bound into a hadron. 00:37:11.440 --> 00:37:13.920 And that might be something you want to talk about, 00:37:13.920 --> 00:37:16.420 but it wouldn't be talking about e plus e minus, the dijets. 00:37:16.420 --> 00:37:18.545 You'd be talking about e plus e minus to pi plus pi 00:37:18.545 --> 00:37:19.789 minus or something. 00:37:33.858 --> 00:37:35.650 And then you'd actually use this other SCET 00:37:35.650 --> 00:37:37.450 that we were talking about a moment ago, 00:37:37.450 --> 00:37:39.790 not the one for jets. 00:37:39.790 --> 00:37:42.130 OK, so anything kind of in between these two regions 00:37:42.130 --> 00:37:45.010 much greater than lambda QCD, much less than Q, 00:37:45.010 --> 00:37:47.050 then we can talk about it as being jets. 00:38:02.520 --> 00:38:03.020 OK. 00:38:06.223 --> 00:38:07.640 So we figure out what region we're 00:38:07.640 --> 00:38:09.500 interested in by figuring out what region we're not 00:38:09.500 --> 00:38:10.083 interested in. 00:38:14.530 --> 00:38:16.770 So that was one jet. 00:38:16.770 --> 00:38:20.680 And kind of by symmetry we can talk about the other one. 00:38:20.680 --> 00:38:23.700 So there's this one that I called n2. 00:38:23.700 --> 00:38:26.850 The simplest way of invoking symmetry 00:38:26.850 --> 00:38:30.300 is to take n1 to point along n, which is our, 00:38:30.300 --> 00:38:32.790 say 0, 0, minus 1. 00:38:32.790 --> 00:38:38.820 And then just take n2 to be n bar, which is 1, 0, 0, plus 1. 00:38:38.820 --> 00:38:40.920 And then the description of the n2 jet 00:38:40.920 --> 00:38:42.810 is the same as the description of the n1 jet. 00:38:42.810 --> 00:38:44.477 It's just you switch pluses and minuses. 00:38:54.790 --> 00:38:57.140 Remember that what defines plus and minus 00:38:57.140 --> 00:38:59.440 depends on the choice of this n bar. 00:38:59.440 --> 00:39:02.120 A priori, that choice of n bar has nothing to do with n2. 00:39:02.120 --> 00:39:04.160 And this is just a different physical vector. 00:39:04.160 --> 00:39:06.770 Both n1 and n2 are physical. 00:39:06.770 --> 00:39:08.210 And bar is an auxiliary vector. 00:39:08.210 --> 00:39:10.010 But if I just happened to choose that n2 00:39:10.010 --> 00:39:12.050 is equal to that auxiliary vector, 00:39:12.050 --> 00:39:15.480 then it makes things simple. 00:39:15.480 --> 00:39:17.660 Because we just have a relation between the two 00:39:17.660 --> 00:39:20.495 sets of degrees of freedom, this swapping pluses and minuses. 00:39:30.555 --> 00:39:31.055 OK. 00:39:35.850 --> 00:39:37.890 So that's actually not the end of the story 00:39:37.890 --> 00:39:41.400 of the degrees of freedom here. 00:39:41.400 --> 00:39:44.100 And that is because, even if we make restrictions 00:39:44.100 --> 00:39:47.040 to getting these dijets, we can still have soft radiation that 00:39:47.040 --> 00:39:49.427 is between the jets. 00:39:49.427 --> 00:39:51.510 And so this one's a little less intuitive perhaps. 00:40:02.730 --> 00:40:06.780 So I'll call these guys ultra soft modes. 00:40:06.780 --> 00:40:09.330 And I'll label them by US for Ultra Soft. 00:40:13.412 --> 00:40:15.870 And one way of thinking about physically what they're doing 00:40:15.870 --> 00:40:18.412 is that they're allowing you to communicate between the jets. 00:40:23.920 --> 00:40:26.770 There can be radiation that's radiated from one jet that 00:40:26.770 --> 00:40:28.450 interferes with the other jet. 00:40:40.410 --> 00:40:42.427 This is a homogeneous type of radiation, 00:40:42.427 --> 00:40:44.760 so it has the same scaling in the plus, minus, and perp. 00:40:50.250 --> 00:40:52.860 So this is for these ultra soft modes. 00:40:52.860 --> 00:40:56.970 If I compare this to the scaling that we have for, say, 00:40:56.970 --> 00:40:58.430 the n collinear modes-- 00:40:58.430 --> 00:41:00.450 so let me call that Pn-- 00:41:00.450 --> 00:41:05.310 that was delta squared over Q Q delta. 00:41:05.310 --> 00:41:09.930 And if I make the mirror for the other jet, 00:41:09.930 --> 00:41:15.910 then I would switch these two delta squared over Q delta. 00:41:15.910 --> 00:41:17.790 So when I say communicate, what I mean 00:41:17.790 --> 00:41:20.760 is that these guys can talk to both of these guys 00:41:20.760 --> 00:41:24.503 without interfering with their scaling. 00:41:24.503 --> 00:41:26.670 So if they're not going to interfere with this guy's 00:41:26.670 --> 00:41:29.523 scaling, they better have plus momentum that's the same size. 00:41:29.523 --> 00:41:31.440 If it was any bigger, then we'd have a problem 00:41:31.440 --> 00:41:33.190 because they'd interfere with this scaling 00:41:33.190 --> 00:41:34.800 when they tried to communicate. 00:41:34.800 --> 00:41:37.500 And then likewise for the minus, they better 00:41:37.500 --> 00:41:38.972 have scaling of the same size. 00:41:38.972 --> 00:41:39.930 And then perp is fixed. 00:41:44.770 --> 00:41:53.930 So the word communicate here in the way I'm defining it 00:41:53.930 --> 00:42:01.740 means sharing momenta of a common size-- 00:42:04.820 --> 00:42:20.760 well, means sharing momenta and not taking the other particle 00:42:20.760 --> 00:42:21.782 off-shell. 00:42:30.750 --> 00:42:31.250 OK. 00:42:31.250 --> 00:42:33.290 So we can draw a picture for this one, too. 00:42:33.290 --> 00:42:36.890 Like we have our SCET2 picture, we could draw an SCET1 picture. 00:42:36.890 --> 00:42:38.600 And that helps. 00:42:38.600 --> 00:42:45.320 Pictures always help to make words more palatable, more 00:42:45.320 --> 00:42:46.446 absorbable. 00:42:53.900 --> 00:43:04.220 So same type of picture where you have P minus and P plus, 00:43:04.220 --> 00:43:06.540 we also can think about things in terms of hyperbola 00:43:06.540 --> 00:43:09.235 of constant invariant mass. 00:43:09.235 --> 00:43:15.530 We now have some collinear modes for the n1 direction. 00:43:15.530 --> 00:43:22.990 There's going to be some purple collinear 00:43:22.990 --> 00:43:25.618 modes for the n2 directions. 00:43:25.618 --> 00:43:27.160 There's going to be these soft modes. 00:43:30.070 --> 00:43:32.380 And then we could also have some hard modes 00:43:32.380 --> 00:43:35.290 that we want to integrate out. 00:43:40.220 --> 00:43:42.700 So these are the ultra soft modes. 00:43:42.700 --> 00:43:45.700 So the reason that I call them ultra soft rather than soft 00:43:45.700 --> 00:43:48.100 is because, by soft, we meant something 00:43:48.100 --> 00:43:50.530 that sat on the same hyperbola. 00:43:50.530 --> 00:43:52.330 And here it sits on a lower hyperbola, 00:43:52.330 --> 00:43:53.320 so it should be softer. 00:43:53.320 --> 00:43:54.550 So we call it ultra soft. 00:43:59.288 --> 00:44:00.830 This is the hyperbola where P squared 00:44:00.830 --> 00:44:01.913 is of order delta squared. 00:44:01.913 --> 00:44:05.060 This is the hyperbola where P squared is of order delta 00:44:05.060 --> 00:44:06.620 4 over Q squared. 00:44:06.620 --> 00:44:09.230 If you square any one of these guys, 00:44:09.230 --> 00:44:11.780 you get delta 4 over Q squared. 00:44:11.780 --> 00:44:16.550 And up here, is P squared over Q squared. 00:44:16.550 --> 00:44:18.710 And this kind of thing is called SCET1. 00:44:23.200 --> 00:44:24.850 So what does it mean to communicate? 00:44:24.850 --> 00:44:27.680 Well, it means that there's two momenta that are the same size. 00:44:27.680 --> 00:44:35.440 So the fact that I tried to line this up as best I could, 00:44:35.440 --> 00:44:39.880 partially succeeding, is what I mean by communicate. 00:44:39.880 --> 00:44:41.737 These things are the same size in the plus. 00:44:41.737 --> 00:44:43.570 These things are the same size in the minus. 00:44:47.130 --> 00:44:59.062 And if we want to put in kind of how big these things are, 00:44:59.062 --> 00:45:00.000 we would say that. 00:45:00.000 --> 00:45:02.070 So the scaling of the ultra soft, 00:45:02.070 --> 00:45:03.750 unlike the scaling the soft, the ultra 00:45:03.750 --> 00:45:06.120 soft in all component scales like lambda squared. 00:45:24.900 --> 00:45:28.290 So in terms of scaling parameters, 00:45:28.290 --> 00:45:29.850 we would say that it goes like this. 00:45:33.450 --> 00:45:35.670 And here, we're not talking about soft. 00:45:45.600 --> 00:45:47.808 It would have lambda in all components. 00:45:51.193 --> 00:45:53.110 So this is the effective theory that turns out 00:45:53.110 --> 00:45:54.318 to be the right one for jets. 00:46:00.090 --> 00:46:02.070 And whenever you have this kind of situation 00:46:02.070 --> 00:46:04.680 where you've got collinear modes that are living on a higher 00:46:04.680 --> 00:46:07.230 hyperbola than the soft modes, which 00:46:07.230 --> 00:46:10.920 you call ultra soft modes, that's an SCET1 type theory. 00:46:16.540 --> 00:46:18.480 And these two actually cover a wide range 00:46:18.480 --> 00:46:21.330 of phenomenology, these two particular cases, SCET1 00:46:21.330 --> 00:46:21.830 and SCET2. 00:46:26.510 --> 00:46:30.070 OK, questions? 00:46:30.070 --> 00:46:32.843 We'll talk a little bit more about this picture. 00:46:32.843 --> 00:46:34.510 AUDIENCE: Do you think of the soft modes 00:46:34.510 --> 00:46:40.288 as being smaller than or equal to lambda squared? 00:46:40.288 --> 00:46:41.080 IAIN STEWART: Yeah. 00:46:45.220 --> 00:46:47.200 So you can ask, now, how should I 00:46:47.200 --> 00:46:50.920 think about drawing the blobs around these things, right? 00:46:50.920 --> 00:46:54.400 And you still should think about these things 00:46:54.400 --> 00:46:56.750 with blobs like that. 00:46:56.750 --> 00:47:02.420 But then it becomes a question which we didn't have, 00:47:02.420 --> 00:47:05.900 which was kind of more obvious actually in the SCET1 case. 00:47:05.900 --> 00:47:09.570 And that is kind of how these things overlap with the axes. 00:47:09.570 --> 00:47:11.730 And we'll come back and talk about that later. 00:47:11.730 --> 00:47:13.730 But I think, for now, just think of them 00:47:13.730 --> 00:47:17.270 as being localized in that way. 00:47:17.270 --> 00:47:20.390 In the case of the softs in the collinear we had before, 00:47:20.390 --> 00:47:22.640 you can think of them as uniformly kind of coming down 00:47:22.640 --> 00:47:23.970 into the infrared. 00:47:23.970 --> 00:47:25.820 Here, it's not like that because this guy 00:47:25.820 --> 00:47:29.380 is more infrared to begin with. 00:47:29.380 --> 00:47:33.520 And that will have some impact later on. 00:47:33.520 --> 00:47:35.510 OK. 00:47:35.510 --> 00:47:37.820 So what are the important features here? 00:47:43.230 --> 00:47:45.515 Well, we see this idea that I mentioned 00:47:45.515 --> 00:47:49.310 would occur that we have multiple modes 00:47:49.310 --> 00:47:52.100 for the infrared. 00:47:52.100 --> 00:47:54.440 And you can try to get away with not doing 00:47:54.440 --> 00:47:56.540 that, but then you would have a lot of trouble 00:47:56.540 --> 00:47:58.400 with your power counting. 00:47:58.400 --> 00:48:01.310 And since power counting, as I've convinced you hopefully 00:48:01.310 --> 00:48:03.890 by now, is just as important as other things 00:48:03.890 --> 00:48:06.260 when you're designing an effective theory, 00:48:06.260 --> 00:48:10.470 you really don't want to mess with that. 00:48:10.470 --> 00:48:12.290 And so you're forced into a situation where 00:48:12.290 --> 00:48:14.240 you start talking about having multiple fields 00:48:14.240 --> 00:48:16.850 for the same degrees of freedom because the scaling 00:48:16.850 --> 00:48:19.160 of the momentum in different regions of the space 00:48:19.160 --> 00:48:21.333 you're interested in is just different. 00:48:21.333 --> 00:48:23.750 So the derivatives that are corresponding to those momenta 00:48:23.750 --> 00:48:25.340 are going to scale differently. 00:48:25.340 --> 00:48:27.320 And if that's the case, you're going 00:48:27.320 --> 00:48:29.360 to need to have multiple fields to describe 00:48:29.360 --> 00:48:30.360 those different regions. 00:48:33.060 --> 00:48:35.250 We'll also see the power counting is different. 00:48:38.610 --> 00:48:41.530 So you can ask, what are you integrating out? 00:48:41.530 --> 00:48:44.430 And we're still taking the attitude 00:48:44.430 --> 00:48:54.300 that we integrate out modes that are off-shell and so 00:48:54.300 --> 00:48:55.830 above a given hyperbola. 00:48:58.920 --> 00:49:01.320 So that part of our story of effective field theory 00:49:01.320 --> 00:49:02.230 is really the same. 00:49:02.230 --> 00:49:04.720 It's just that, when we describe the low energy modes, 00:49:04.720 --> 00:49:05.970 we need more than one of them. 00:49:14.863 --> 00:49:16.530 So what we mean by off-shell is the same 00:49:16.530 --> 00:49:17.910 as it always has meant. 00:49:22.590 --> 00:49:24.420 Off-shell modes get integrated out. 00:49:24.420 --> 00:49:29.654 Those are modes with large P squared, like the hard modes. 00:49:58.330 --> 00:50:00.037 So one natural question when you're 00:50:00.037 --> 00:50:01.870 talking about these effective field theories 00:50:01.870 --> 00:50:05.440 is, how do I know I have all the modes? 00:50:05.440 --> 00:50:09.320 And that's a good question. 00:50:09.320 --> 00:50:11.317 And there's been examples in the literature 00:50:11.317 --> 00:50:13.900 where people wrote papers where they didn't know all the modes 00:50:13.900 --> 00:50:16.480 and modes were missing. 00:50:16.480 --> 00:50:18.130 So it's not even an academic question. 00:50:18.130 --> 00:50:21.710 It's really a pitfall in some sense that you can fall into. 00:50:21.710 --> 00:50:24.790 So I take the following attitude towards that question. 00:50:24.790 --> 00:50:27.110 You should attack it from all sides. 00:50:27.110 --> 00:50:29.260 So one is that you should really physically 00:50:29.260 --> 00:50:30.760 think about what the modes are doing 00:50:30.760 --> 00:50:32.650 and have a physical description of why 00:50:32.650 --> 00:50:34.900 those modes are something relevant for your effective 00:50:34.900 --> 00:50:35.300 field theory. 00:50:35.300 --> 00:50:36.883 After all, your effective field theory 00:50:36.883 --> 00:50:40.160 is supposed to be a description of nature and infrared physics. 00:50:40.160 --> 00:50:42.160 So there should be something physical associated 00:50:42.160 --> 00:50:44.020 to what those models are doing. 00:50:44.020 --> 00:50:45.730 That's the physics side. 00:50:45.730 --> 00:50:47.530 Calculationally, there's various ways 00:50:47.530 --> 00:50:48.975 that you could approach this. 00:50:48.975 --> 00:50:50.350 So one thing you can do, which is 00:50:50.350 --> 00:50:52.420 sort of an order by order thing, is 00:50:52.420 --> 00:50:55.450 you can just calculate results at one loop 00:50:55.450 --> 00:50:58.300 and make sure you match up infrared divergences. 00:50:58.300 --> 00:51:00.910 Because if you don't match up the infrared divergences, 00:51:00.910 --> 00:51:03.670 then you're missing some infrared degree of freedom 00:51:03.670 --> 00:51:05.810 in the effective field theory. 00:51:05.810 --> 00:51:06.310 OK. 00:51:06.310 --> 00:51:18.800 So that's one way you can check for the modes. 00:51:18.800 --> 00:51:22.790 So if you can either do physics, or you can calculate. 00:51:25.490 --> 00:51:26.990 And when you calculate, there's also 00:51:26.990 --> 00:51:28.710 something called the method of regions, 00:51:28.710 --> 00:51:32.720 which is a nice way of thinking about trying to discover modes, 00:51:32.720 --> 00:51:35.240 which basically says that, any full theory calculation I can 00:51:35.240 --> 00:51:38.540 do, I can do that calculation by dividing up 00:51:38.540 --> 00:51:40.520 the integrand into regions. 00:51:40.520 --> 00:51:43.610 And if I'm using dimensional regularization, 00:51:43.610 --> 00:51:48.390 then the full theory answer is just the sum over regions. 00:51:48.390 --> 00:51:53.030 So you can just calculate using some EFT, 00:51:53.030 --> 00:51:56.000 as I described to you, and check it. 00:51:58.790 --> 00:52:00.620 Or could do a different way. 00:52:00.620 --> 00:52:07.940 You could say, calculate the full theory result 00:52:07.940 --> 00:52:11.090 with something called the method of regions. 00:52:11.090 --> 00:52:12.590 And that's just a way of calculating 00:52:12.590 --> 00:52:15.110 the full theory of result, but it's 00:52:15.110 --> 00:52:21.080 a way of telling you what regions are important. 00:52:21.080 --> 00:52:23.030 There's all-other theorems in QCD 00:52:23.030 --> 00:52:26.843 about what regions can give infrared divergences. 00:52:26.843 --> 00:52:28.385 And that's another thing you can use. 00:52:40.180 --> 00:52:44.650 And I won't talk about this last one or actually about this one. 00:52:48.940 --> 00:52:50.440 So these are different ways that you 00:52:50.440 --> 00:52:53.890 could look for what are the relevant degrees of freedom. 00:52:53.890 --> 00:52:56.200 And actually, when we started SCET, 00:52:56.200 --> 00:52:59.140 we didn't separate between ultra soft and soft. 00:52:59.140 --> 00:53:02.070 We had both of them at the same time. 00:53:02.070 --> 00:53:04.070 And the reason was we knew that in some examples 00:53:04.070 --> 00:53:06.670 we needed soft and some examples we needed ultra soft. 00:53:06.670 --> 00:53:09.130 And we were thinking of it as one theory, not SCET1 00:53:09.130 --> 00:53:10.415 and SCET2. 00:53:10.415 --> 00:53:12.790 And then at some point, we realized that all the examples 00:53:12.790 --> 00:53:14.860 we were doing, as we did more and more examples, 00:53:14.860 --> 00:53:16.630 they broke into these two categories, one 00:53:16.630 --> 00:53:18.760 for jets, one for hadrons. 00:53:18.760 --> 00:53:21.380 And so we were finding that, for example, 00:53:21.380 --> 00:53:23.232 if you add a soft mode to this picture, 00:53:23.232 --> 00:53:24.940 it just ends up being totally irrelevant. 00:53:24.940 --> 00:53:26.607 And you can just absorb it or remove it. 00:53:26.607 --> 00:53:28.210 You don't need it. 00:53:28.210 --> 00:53:30.640 And likewise, if you tried to put an ultra soft mode 00:53:30.640 --> 00:53:33.430 into the example of energetic hadrons, 00:53:33.430 --> 00:53:35.110 you'd find you don't need it. 00:53:35.110 --> 00:53:38.210 So you could put too many degrees of freedom in as well. 00:53:38.210 --> 00:53:40.995 It's not just that you could have too few. 00:53:40.995 --> 00:53:42.370 You could put more than you need, 00:53:42.370 --> 00:53:44.412 and then you would see from your calculations 00:53:44.412 --> 00:53:46.120 that you actually had more than you need. 00:53:49.660 --> 00:53:51.160 The reason I'm spending time on this 00:53:51.160 --> 00:53:53.390 is because this is, in some sense, the tricky part. 00:53:53.390 --> 00:53:54.973 Once we know what the modes are, we'll 00:53:54.973 --> 00:53:57.190 just get the effective field theory and go. 00:54:00.010 --> 00:54:02.415 AUDIENCE: Is your second point here the method 00:54:02.415 --> 00:54:05.400 or regions [INAUDIBLE],, is that any different than the check it 00:54:05.400 --> 00:54:06.828 of the first point? 00:54:06.828 --> 00:54:07.620 IAIN STEWART: Yeah. 00:54:07.620 --> 00:54:09.750 Because, here, what I'm saying is hypothesize 00:54:09.750 --> 00:54:11.160 some effective field theory. 00:54:11.160 --> 00:54:12.510 AUDIENCE: But check it means full theory. 00:54:12.510 --> 00:54:14.300 IAIN STEWART: And then check it against the full theory. 00:54:14.300 --> 00:54:14.590 AUDIENCE: [INAUDIBLE] 00:54:14.590 --> 00:54:16.320 IAIN STEWART: And so the direction here 00:54:16.320 --> 00:54:18.480 is that you write down something, 00:54:18.480 --> 00:54:21.630 and you do calculations in two theories. 00:54:21.630 --> 00:54:24.240 And you make sure they agree. 00:54:24.240 --> 00:54:25.950 Here, it's a little bit different 00:54:25.950 --> 00:54:29.310 because there may not be a one-to-one correspondence 00:54:29.310 --> 00:54:31.140 of the modes of the method of regions. 00:54:31.140 --> 00:54:32.730 Method of regions is not technically 00:54:32.730 --> 00:54:35.100 exactly what the effective theory is. 00:54:35.100 --> 00:54:38.005 So this can give you hints about what the effective theory is. 00:54:38.005 --> 00:54:39.630 Then you basically go back and do this, 00:54:39.630 --> 00:54:42.360 but it sort of helps you because you have most 00:54:42.360 --> 00:54:44.470 of the calculations to do this. 00:54:44.470 --> 00:54:44.970 Yeah. 00:54:44.970 --> 00:54:46.380 So technically, they're a little bit different, 00:54:46.380 --> 00:54:47.505 but they're pretty related. 00:54:56.440 --> 00:54:58.330 OK. 00:54:58.330 --> 00:55:01.060 So in the example we had here, it 00:55:01.060 --> 00:55:03.400 was a little more complicated than our B to D pi example 00:55:03.400 --> 00:55:06.550 because we had three things in the infrared. 00:55:06.550 --> 00:55:08.710 And you could ask the question, is there something 00:55:08.710 --> 00:55:09.670 with just one jet? 00:55:09.670 --> 00:55:12.550 The reason we had three things is because we had two jets. 00:55:12.550 --> 00:55:15.760 We had the purple jet and the pink jet. 00:55:15.760 --> 00:55:19.450 And we can come up with a process with only one jet. 00:55:19.450 --> 00:55:20.980 So let me give you a third example. 00:55:27.980 --> 00:55:31.150 If we have something recoiling against the jet that's 00:55:31.150 --> 00:55:32.980 electromagnetic, like a photon, then we 00:55:32.980 --> 00:55:34.600 could just have one jet. 00:55:34.600 --> 00:55:36.670 So one way of doing that is to look at a process 00:55:36.670 --> 00:55:41.080 like b to s gamma, where you say that really what you want 00:55:41.080 --> 00:55:47.920 is B to one jet plus a photon. 00:55:47.920 --> 00:55:49.600 And that's a region of phase space 00:55:49.600 --> 00:55:54.460 in b to s gamma, where the picture over there 00:55:54.460 --> 00:55:57.190 just has one jet. 00:55:57.190 --> 00:56:01.360 And so it's the same picture, but now we just 00:56:01.360 --> 00:56:02.140 have our pink jet. 00:56:13.110 --> 00:56:15.410 So the right modes for this picture 00:56:15.410 --> 00:56:19.175 for this process in the region where you just have one jet 00:56:19.175 --> 00:56:21.920 would be like this where this is hard and you integrate it out. 00:56:21.920 --> 00:56:25.590 And you just have two things in the infrared, a Cn and an ultra 00:56:25.590 --> 00:56:26.090 soft. 00:56:29.450 --> 00:56:33.140 In this case, you also have to ask the question, what 00:56:33.140 --> 00:56:34.160 is the hydronic process? 00:56:34.160 --> 00:56:35.870 Well, it has to B meson. 00:56:35.870 --> 00:56:37.910 What are the things inside the B meson 00:56:37.910 --> 00:56:39.380 that are binding it together? 00:56:39.380 --> 00:56:42.320 And you would make those the ultra soft modes. 00:56:42.320 --> 00:56:44.630 So in this process here, it's very natural 00:56:44.630 --> 00:56:50.960 to take P squared of this lowest line to be lambda QCD squared. 00:56:50.960 --> 00:56:54.440 And then these guys are binding the B meson there, 00:56:54.440 --> 00:56:55.695 the soft modes of the B meson. 00:57:09.980 --> 00:57:12.620 This collinear hyperbola for the jet lives somewhere between. 00:57:12.620 --> 00:57:16.790 And this here, in this case, would be something of order 00:57:16.790 --> 00:57:19.512 the B quark mass squared, OK? 00:57:19.512 --> 00:57:21.470 So that's a little bit different than over here 00:57:21.470 --> 00:57:24.470 where there was kind of a no natural-- 00:57:24.470 --> 00:57:26.300 this thing was set by kind of what 00:57:26.300 --> 00:57:29.210 we chose to do with the jets was setting this. 00:57:29.210 --> 00:57:31.195 Here, there's kind of a natural scale 00:57:31.195 --> 00:57:32.570 where you know that there's going 00:57:32.570 --> 00:57:34.095 to be some degrees of freedom. 00:57:34.095 --> 00:57:35.720 And so it's very natural, in this case, 00:57:35.720 --> 00:57:38.060 to take that as an input. 00:57:38.060 --> 00:57:40.370 And then you could actually figure out, 00:57:40.370 --> 00:57:45.890 given that what the jets should be and the jets would have, 00:57:45.890 --> 00:57:49.640 lambda QCD times Mb, which is in the middle. 00:57:54.190 --> 00:57:56.880 So that's kind of a natural scaling for the b to s 00:57:56.880 --> 00:58:00.000 gamma process, OK? 00:58:00.000 --> 00:58:03.840 So just to give you an idea how, if I have a jet, 00:58:03.840 --> 00:58:05.480 it's going to look something like this, 00:58:05.480 --> 00:58:06.980 it might not look exactly like that. 00:58:06.980 --> 00:58:10.117 It depends on how many jets you have. 00:58:10.117 --> 00:58:11.700 And of course, these pictures actually 00:58:11.700 --> 00:58:13.908 get more complicated if you try to start drawing them 00:58:13.908 --> 00:58:16.500 when you have three jets because then the plane 00:58:16.500 --> 00:58:19.590 is no longer enough. 00:58:19.590 --> 00:58:24.490 All right, so any more questions? 00:58:24.490 --> 00:58:25.150 OK. 00:58:25.150 --> 00:58:29.020 So when we did HQET, the first thing that we did 00:58:29.020 --> 00:58:30.940 is we started to expand. 00:58:30.940 --> 00:58:33.220 Before we designed the effective Lagrangian, 00:58:33.220 --> 00:58:36.580 we just said, well, what happens if I expand the full theory? 00:58:36.580 --> 00:58:39.493 And I'm going to take the same attitude here. 00:58:39.493 --> 00:58:41.410 Let's just write down some full theory objects 00:58:41.410 --> 00:58:47.920 and expand them in the limits that we've been talking about. 00:58:47.920 --> 00:58:49.920 And then we'll see what kind of effective theory 00:58:49.920 --> 00:58:52.950 we want based on the results from those expansions. 00:58:58.890 --> 00:59:02.840 So let's start with spinners in a collinear limit. 00:59:14.680 --> 00:59:16.800 So let me start with some massless QCD spinners 00:59:16.800 --> 00:59:18.270 in the Dirac representation. 00:59:21.685 --> 00:59:23.310 We could use some other representation, 00:59:23.310 --> 00:59:25.170 but let's just use Dirac. 00:59:43.630 --> 00:59:45.430 So we have spinners for the quarks. 00:59:45.430 --> 00:59:56.890 We have spinners for the antiquarks, V, 00:59:56.890 --> 00:59:58.750 where this curly V and this curly U 00:59:58.750 --> 00:59:59.935 are two-component objects. 01:00:11.700 --> 01:00:13.370 I'll make sure it looks curly enough. 01:00:17.500 --> 01:00:19.540 So what we can do here is we can expand. 01:00:19.540 --> 01:00:21.970 And you see that what happens when you expand 01:00:21.970 --> 01:00:24.880 is that you can think about the P3 vector being larger 01:00:24.880 --> 01:00:26.740 than P1 and P2. 01:00:26.740 --> 01:00:33.430 So let's just let our n be 1, 0, 0, 1 and our n bar 01:00:33.430 --> 01:00:35.693 be 1, 0, 0, minus 1. 01:00:35.693 --> 01:00:37.360 So they're back to back with each other. 01:00:41.230 --> 01:00:43.790 Whether I put the plus or minus 1 there or there 01:00:43.790 --> 01:00:46.070 doesn't really matter. 01:00:46.070 --> 01:00:52.660 Let's expand in n bar dot P, which in this case 01:00:52.660 --> 01:00:58.240 is P0 plus P3 being much greater than P perp, just P1 and P2. 01:00:58.240 --> 01:01:00.760 And then that's much greater than n 01:01:00.760 --> 01:01:04.220 dot P, which is P0 minus P3. 01:01:07.900 --> 01:01:11.930 And what that means is that you can approximate sigma dot P 01:01:11.930 --> 01:01:14.870 over P0 from these massless particles. 01:01:14.870 --> 01:01:18.498 It's just sigma 3 because you pick out the P3. 01:01:18.498 --> 01:01:19.540 That's the big component. 01:01:19.540 --> 01:01:20.920 The P1 and P2, you can drop. 01:01:20.920 --> 01:01:22.330 Then it kicks out the sigma 3. 01:01:22.330 --> 01:01:24.548 And then P3 is also the same size as P0, 01:01:24.548 --> 01:01:25.840 so you're just getting sigma 3. 01:01:30.700 --> 01:01:36.530 So what you get from this then would 01:01:36.530 --> 01:01:39.830 be guys that look like this. 01:01:43.340 --> 01:01:45.350 Just so, I put it in the two possibilities 01:01:45.350 --> 01:01:49.670 for the curly U are four-component spinners that 01:01:49.670 --> 01:02:22.830 look like this for U and then likewise for V of P. 01:02:22.830 --> 01:02:24.550 We can work out what we get. 01:02:29.800 --> 01:02:31.878 We get that. 01:02:31.878 --> 01:02:33.920 So this is actually a little different than HQET. 01:02:33.920 --> 01:02:35.450 In HQET, what you would have found 01:02:35.450 --> 01:02:39.330 is that the antiquarks would have been just left out. 01:02:39.330 --> 01:02:42.110 And the quarks would've been there in the theory. 01:02:42.110 --> 01:02:43.892 Here, both of them survive. 01:02:43.892 --> 01:02:45.350 And actually two degrees of freedom 01:02:45.350 --> 01:02:49.190 survive for both the particles and the antiparticles. 01:02:49.190 --> 01:02:50.810 So it's not like we're integrating out 01:02:50.810 --> 01:02:57.260 of something like the antiparticle, like in HQET. 01:02:57.260 --> 01:02:59.090 By this expansion, we still have all four 01:02:59.090 --> 01:03:02.330 of these degrees of freedom if you count degrees of freedom 01:03:02.330 --> 01:03:05.420 by whether you have particles, antiparticles, and spin states. 01:03:09.000 --> 01:03:12.870 Nevertheless, there is a simplification that occurs. 01:03:12.870 --> 01:03:18.870 And that is the fact that the spinners that you have 01:03:18.870 --> 01:03:19.995 have a projection relation. 01:03:26.390 --> 01:03:29.510 So if you look in this basis that we're 01:03:29.510 --> 01:03:32.810 talking about here, what n slash is if you write down 01:03:32.810 --> 01:03:38.048 what the gamma matrices are in the Dirac representation, 01:03:38.048 --> 01:03:40.760 then n slash is this. 01:03:40.760 --> 01:03:43.250 And another useful thing is n slash n bar 01:03:43.250 --> 01:03:50.530 slash over 4, which you can work out as just this. 01:03:54.780 --> 01:04:00.420 And these spinners here, which I need a name for-- 01:04:00.420 --> 01:04:03.920 so let's call this Un and call this Vn. 01:04:08.220 --> 01:04:15.540 They satisfy n slash Un is n slash Vn is 0. 01:04:15.540 --> 01:04:21.252 And they also satisfy n slash n bar slash 01:04:21.252 --> 01:04:25.460 Un is Un for both of them. 01:04:42.015 --> 01:04:43.890 So what we can do with that is the following. 01:04:46.780 --> 01:04:51.000 We can take the identity in this 4 by 4 space. 01:04:51.000 --> 01:04:53.550 And we can actually write it as n 01:04:53.550 --> 01:04:59.190 slash n bar slash over 4 plus n bar slash n slash over 4. 01:04:59.190 --> 01:05:02.770 And that's because, remember, that gamma mu 01:05:02.770 --> 01:05:09.300 gamma nu plus 2 g mu nu and n dot n bar is 2. 01:05:09.300 --> 01:05:12.510 So this is just one way of using those. 01:05:12.510 --> 01:05:15.480 This is the anticommutator dotted into n and n bar. 01:05:15.480 --> 01:05:17.370 So I can write it out that way. 01:05:17.370 --> 01:05:20.340 And this kind of formula here is the formula 01:05:20.340 --> 01:05:24.090 that is for projection operators, right? 01:05:24.090 --> 01:05:27.640 So as you act with the operator, you get that guy back again. 01:05:27.640 --> 01:05:31.540 So you could act twice with that operator. 01:05:31.540 --> 01:05:34.080 And so what you can do with this one 01:05:34.080 --> 01:05:36.255 is you could let one act on psi of QCD. 01:05:39.210 --> 01:05:44.720 And if you did that, you'd get n slash n bar slash over 4 psi 01:05:44.720 --> 01:05:48.780 plus n bar slash n slash over 4 psi. 01:05:48.780 --> 01:05:51.390 And you could define these two pieces 01:05:51.390 --> 01:05:53.670 as being two different components of the full theory 01:05:53.670 --> 01:05:59.230 field that I'll call Cn and psi n. 01:05:59.230 --> 01:06:00.760 And what happens at high energies, 01:06:00.760 --> 01:06:03.070 because of the type of thing we were doing over there, 01:06:03.070 --> 01:06:04.960 is that we only like to produce Cn's. 01:06:04.960 --> 01:06:06.730 We don't like to produce psi n bars. 01:06:13.620 --> 01:06:18.530 So if you look at some high energy process, 01:06:18.530 --> 01:06:20.540 this sort of thing I was doing at the spinners 01:06:20.540 --> 01:06:24.320 basically boils down to one sentence. 01:06:24.320 --> 01:06:27.770 And that is that we produce or annihilate 01:06:27.770 --> 01:06:29.690 the components, the guys that live 01:06:29.690 --> 01:06:44.890 in this Cn, not the so-called small components, which 01:06:44.890 --> 01:06:47.380 live in this other guy. 01:06:47.380 --> 01:06:49.532 This language of calling them the small components 01:06:49.532 --> 01:06:52.115 is something that goes back to the early days of QCD actually. 01:06:56.290 --> 01:06:58.450 We may say that word a few times, 01:06:58.450 --> 01:07:03.630 but the history won't be so important to us. 01:07:03.630 --> 01:07:05.457 OK, so, so much for the spinners. 01:07:05.457 --> 01:07:07.290 There is some simplification in the spinners 01:07:07.290 --> 01:07:11.340 because we do like to produce certain combinations. 01:07:11.340 --> 01:07:18.360 But it didn't really teach us too much beyond that. 01:07:24.610 --> 01:07:29.140 And we didn't see that we lost a degree of freedom 01:07:29.140 --> 01:07:30.020 like we did in HQET. 01:07:35.510 --> 01:07:38.900 But nevertheless, there was some simplification. 01:07:38.900 --> 01:07:43.715 Let's do the same thing for the propagator of the quarks. 01:07:50.080 --> 01:07:52.640 Take the propagator of the full theory, expand in this limit. 01:07:57.170 --> 01:07:59.200 So first of all, propagators always 01:07:59.200 --> 01:08:01.690 involve P squared plus i0. 01:08:01.690 --> 01:08:04.120 In our decomposition, that's in n bar 01:08:04.120 --> 01:08:11.197 dot P n dot P plus Minkowski P perp squared plus i0. 01:08:11.197 --> 01:08:13.030 And if you look at the size of these things, 01:08:13.030 --> 01:08:14.450 they're all the same size. 01:08:14.450 --> 01:08:18.939 This guy is lambda 0, and this guy is lambda squared. 01:08:18.939 --> 01:08:21.260 And this guy is just lambda squared by itself. 01:08:21.260 --> 01:08:22.689 So the two guys are the same size, 01:08:22.689 --> 01:08:25.858 although they become the same size for different regions. 01:08:25.858 --> 01:08:28.779 This is lambda 1 squared if you like. 01:08:28.779 --> 01:08:32.050 So I don't drop anything in that propagator. 01:08:36.770 --> 01:08:40.040 And you see that actually that's not entirely true. 01:08:40.040 --> 01:08:42.430 And it does depend on what type of things 01:08:42.430 --> 01:08:43.430 you're interacting with. 01:08:43.430 --> 01:08:48.319 But if I just have P's that are collinear, as I've drawn here, 01:08:48.319 --> 01:08:49.729 then there's nothing to drop. 01:08:59.450 --> 01:09:04.525 So if you look at fermions that are collinear 01:09:04.525 --> 01:09:10.540 and you look at i P slash over P squared plus i0, 01:09:10.540 --> 01:09:13.547 you can decompose P slash out in terms of n slash 01:09:13.547 --> 01:09:15.130 and n bar slash, write it out in terms 01:09:15.130 --> 01:09:17.649 of the coordinates we're using. 01:09:17.649 --> 01:09:23.722 And then in the numerator, you keep the full denominator. 01:09:23.722 --> 01:09:25.930 But in the numerator, there is one momentum component 01:09:25.930 --> 01:09:27.939 that's larger than the others. 01:09:27.939 --> 01:09:31.090 And that is the n bar dot P piece, which is order one. 01:09:41.026 --> 01:09:42.859 So there is a simplification of the P slash. 01:09:42.859 --> 01:09:45.073 And it's related in some ways to the simplification 01:09:45.073 --> 01:09:45.740 of the spinners. 01:09:45.740 --> 01:09:50.149 If you take two of these guys and you do the sum over spin, 01:09:50.149 --> 01:09:57.202 you'll actually get n slash over 2 n bar dot P. 01:09:57.202 --> 01:10:00.660 AUDIENCE: Is it n slash on the side of the 0? 01:10:00.660 --> 01:10:01.470 IAIN STEWART: Yeah. 01:10:01.470 --> 01:10:02.345 AUDIENCE: [INAUDIBLE] 01:10:02.345 --> 01:10:04.928 IAIN STEWART: So the right way of thinking about the numerator 01:10:04.928 --> 01:10:07.470 here is that, if you do the sum over spins of sort of two 01:10:07.470 --> 01:10:12.210 of these spinners, like this, that's 01:10:12.210 --> 01:10:14.200 giving you the numerator. 01:10:14.200 --> 01:10:18.330 And if you look at sort of the overlap of this, 01:10:18.330 --> 01:10:19.980 if you look at any amplitude, you also 01:10:19.980 --> 01:10:22.770 have to take into account, but I haven't written yet-- 01:10:22.770 --> 01:10:25.250 which is this. 01:10:25.250 --> 01:10:28.910 And that's going to go like n bar slash. 01:10:28.910 --> 01:10:31.960 So what happens is you have n slash, n bar slash. 01:10:31.960 --> 01:10:33.920 And that gives you the projector, 01:10:33.920 --> 01:10:37.480 which then overlaps order 1 with the spinner. 01:10:37.480 --> 01:10:38.145 Yeah. 01:10:38.145 --> 01:10:39.270 But that's a good question. 01:10:44.970 --> 01:10:48.880 Later-- but that's a good point. 01:11:13.240 --> 01:11:13.840 OK. 01:11:13.840 --> 01:11:19.540 So we can ask about, if we have some propagator for a fermion, 01:11:19.540 --> 01:11:21.770 then what does it look like? 01:11:21.770 --> 01:11:25.240 And you can take that first order term, 01:11:25.240 --> 01:11:29.440 and we can write it in a way that kind of is reminiscent 01:11:29.440 --> 01:11:33.280 of something more like NRQCD or this nucleon theory 01:11:33.280 --> 01:11:34.515 or even HQET. 01:11:37.650 --> 01:11:40.870 It's just another way of writing the same formula that 01:11:40.870 --> 01:11:43.810 is sometimes useful. 01:11:43.810 --> 01:11:46.300 We just divide through by the n bar dot P. 01:11:46.300 --> 01:11:49.585 And then it's n dot P plus P perp squared over n bar 01:11:49.585 --> 01:11:53.050 dot P. And then there's the i0, but the sign of the i0 01:11:53.050 --> 01:11:56.740 will depend on the sign of the n bar dot P. 01:11:56.740 --> 01:11:59.890 The fact that this could be both plus i0 or minus i0 01:11:59.890 --> 01:12:02.740 is the same thing as saying that there's antiparticles 01:12:02.740 --> 01:12:04.540 and particles in the theory. 01:12:04.540 --> 01:12:06.880 If it was just plus i0, as it was in HQET, 01:12:06.880 --> 01:12:08.930 we only had the particles. 01:12:08.930 --> 01:12:10.180 Here, it could be either sign. 01:12:10.180 --> 01:12:11.420 We have both. 01:12:11.420 --> 01:12:13.510 And so this thing, again, has both particles 01:12:13.510 --> 01:12:17.650 and antiparticles, which we saw when we were 01:12:17.650 --> 01:12:18.820 doing the spinners as well. 01:12:31.965 --> 01:12:34.340 And if we want to think about particles and antiparticles 01:12:34.340 --> 01:12:36.840 separately, then we could have a definite sign for the i0's. 01:12:36.840 --> 01:12:39.340 But if we want to think about them in a combined propagator, 01:12:39.340 --> 01:12:41.690 then we have to write it this way. 01:12:41.690 --> 01:12:46.430 All right, so once we know what the propagator is, 01:12:46.430 --> 01:12:50.480 then we can also figure out what the power counting 01:12:50.480 --> 01:12:53.390 of the fields are because the propagator tells us 01:12:53.390 --> 01:12:57.890 what the kinetic term should look like of the Lagrangian. 01:13:02.730 --> 01:13:05.280 So let me show you how that works. 01:13:12.330 --> 01:13:14.190 So where does the propagator come from? 01:13:14.190 --> 01:13:17.220 The propagator comes from the time order 01:13:17.220 --> 01:13:19.560 product of two fields. 01:13:19.560 --> 01:13:23.160 And in this case, it comes from the time order product 01:13:23.160 --> 01:13:27.990 of a field for this Cn component that we were talking about. 01:13:34.010 --> 01:13:37.960 And if we just take the free kinetic term, which 01:13:37.960 --> 01:13:41.200 is the Lagrangian that we'd give that propagator, 01:13:41.200 --> 01:13:48.130 that's enough to determine the power counting for fields, 01:13:48.130 --> 01:13:50.390 as is always the case in any effective field theory. 01:13:55.060 --> 01:13:58.600 So we haven't figured out what the Lagrangian is, 01:13:58.600 --> 01:14:02.870 but we know something about what it's going to look like. 01:14:02.870 --> 01:14:05.200 So let me write down enough of that 01:14:05.200 --> 01:14:07.840 to determine for you what the power counting would look like. 01:14:12.440 --> 01:14:14.270 So when I read it this way, as I wrote it, 01:14:14.270 --> 01:14:17.150 where it's linear in this n dot P derivative, 01:14:17.150 --> 01:14:19.150 you know that what that's going to correspond to 01:14:19.150 --> 01:14:22.780 in the Lagrangian is some n dot partial. 01:14:22.780 --> 01:14:25.960 And I have to have an n bar slash here 01:14:25.960 --> 01:14:29.110 because that always comes along with an n decomposing 01:14:29.110 --> 01:14:30.610 the metric. 01:14:30.610 --> 01:14:32.680 If you ask about the power counting here, 01:14:32.680 --> 01:14:36.730 well, d4x has all 4 components of k. 01:14:36.730 --> 01:14:39.040 And x is the inverse of k. 01:14:39.040 --> 01:14:41.410 So the way that you assign a power counting 01:14:41.410 --> 01:14:45.370 for x is that you say the phase should be of order 1. 01:14:45.370 --> 01:14:49.210 So the scaling for x is the opposite of k. 01:14:49.210 --> 01:14:54.550 So that fixes that d4x should be lambda to the minus 4. 01:14:54.550 --> 01:15:02.590 So x plus times k minus is of order 1, et cetera. 01:15:02.590 --> 01:15:05.260 And that tells you how many powers 01:15:05.260 --> 01:15:08.110 to associate with the d4x. 01:15:08.110 --> 01:15:10.360 We know how many powers to associate with this partial 01:15:10.360 --> 01:15:13.480 because that was our momentum. 01:15:13.480 --> 01:15:15.363 That's lambda squared. 01:15:15.363 --> 01:15:16.780 And actually, the other terms here 01:15:16.780 --> 01:15:19.367 will also be lambda squared. 01:15:19.367 --> 01:15:20.950 And then we just say, well, that's let 01:15:20.950 --> 01:15:25.240 the power counting in this field be arbitrary lambda to the a. 01:15:25.240 --> 01:15:28.150 And so if we do that, then we get an overall scaling 01:15:28.150 --> 01:15:33.850 for this Lagrangian that's lambda to the 2a minus 2. 01:15:37.000 --> 01:15:39.940 2 to the power of the lambda minus 4 cancel by the partial n 01:15:39.940 --> 01:15:42.640 dot partial and we get that. 01:15:42.640 --> 01:15:45.070 Now, the way that we do effective field theory is we 01:15:45.070 --> 01:15:47.620 look at the lowest order term, and we count everything 01:15:47.620 --> 01:15:49.460 relative to that. 01:15:49.460 --> 01:15:55.600 So what we do is we say, we want the lowest order Lagrangian 01:15:55.600 --> 01:15:58.330 to scale like lambda to the 0. 01:15:58.330 --> 01:16:00.960 And you can think of that roughly in a power 01:16:00.960 --> 01:16:04.240 counting sense as normalizing the free kinetic term 01:16:04.240 --> 01:16:06.190 or normalizing the kinetic term in general. 01:16:15.080 --> 01:16:17.660 And then, once you do that, then you 01:16:17.660 --> 01:16:19.430 fix what the scaling of the field 01:16:19.430 --> 01:16:22.750 is, Cn to the order lambda. 01:16:27.300 --> 01:16:31.020 And that's different than the mass dimension. 01:16:31.020 --> 01:16:32.550 So I said that we were going to be 01:16:32.550 --> 01:16:34.175 doing a power counting that's different 01:16:34.175 --> 01:16:36.240 than the mass dimension. 01:16:36.240 --> 01:16:39.900 If we looked at the mass dimension of this field, 01:16:39.900 --> 01:16:46.380 Cn would have mass dimension that's 3/2, whereas it has, 01:16:46.380 --> 01:16:49.740 if you like, a lambda power counting dimension which is 1. 01:16:53.980 --> 01:16:57.048 So if I look for the powers of lambda, that's 1. 01:16:57.048 --> 01:16:59.590 And that was one of the things I told you was going to happen 01:16:59.590 --> 01:17:01.087 is that, in this effective theory, 01:17:01.087 --> 01:17:02.920 we wouldn't just be counting mass dimension. 01:17:02.920 --> 01:17:04.300 We'd be counting something else. 01:17:14.040 --> 01:17:14.540 OK. 01:17:14.540 --> 01:17:16.550 So that's collinear quarks. 01:17:16.550 --> 01:17:20.738 We can do a similar thing for collinear gluons. 01:17:20.738 --> 01:17:23.030 And we may not get to the end of that discussion today, 01:17:23.030 --> 01:17:23.822 but let's start it. 01:17:32.660 --> 01:17:34.040 The momenta for a collinear gluon 01:17:34.040 --> 01:17:37.023 scales the same as the momenta for a collinear quark. 01:17:37.023 --> 01:17:38.690 That means collinear doesn't distinguish 01:17:38.690 --> 01:17:40.970 between quarks and gluons. 01:17:40.970 --> 01:17:42.590 So P squared, the full P squared, 01:17:42.590 --> 01:17:47.510 is still something that we're going to leave together. 01:17:47.510 --> 01:17:51.740 And let's just consider looking at the propagator 01:17:51.740 --> 01:18:00.320 in a general covariant gauge and asking 01:18:00.320 --> 01:18:03.860 kind of the same type of thing that we did over here 01:18:03.860 --> 01:18:05.330 about the scaling of the field. 01:18:11.350 --> 01:18:17.760 So the propagator for two collinear gluons 01:18:17.760 --> 01:18:19.950 and the general covariant gauge-- 01:18:19.950 --> 01:18:23.160 time order product vacuum matrix element of two fields. 01:18:26.010 --> 01:18:28.440 Ignore the subscript ends right now. 01:18:28.440 --> 01:18:35.983 I'm just writing down a full theory result. 01:18:35.983 --> 01:18:37.650 We called C something else, so we better 01:18:37.650 --> 01:18:39.060 not make that gauge parameter. 01:18:39.060 --> 01:18:40.740 So let me call the gauge parameter tau. 01:18:53.723 --> 01:18:56.140 That's the gauge parameter of the general covariant gauge. 01:18:56.140 --> 01:18:58.790 And this is a full three result. 01:18:58.790 --> 01:19:00.880 The thing that makes it collinear 01:19:00.880 --> 01:19:02.980 is if we say that k has a collinear scaling. 01:19:07.400 --> 01:19:14.340 So as above, k squared is k plus k minus plus k perp squared. 01:19:14.340 --> 01:19:16.970 That's of order lambda squared, and there's 01:19:16.970 --> 01:19:20.347 no expansion in there. 01:19:20.347 --> 01:19:21.680 And there's two k squareds here. 01:19:21.680 --> 01:19:25.230 There's one there, and there's one there. 01:19:25.230 --> 01:19:28.220 And if you start looking at g mu nu minus k mu k nu 01:19:28.220 --> 01:19:31.550 over k squared, you also find that the terms there 01:19:31.550 --> 01:19:34.670 are the same size. 01:19:34.670 --> 01:19:38.500 So there's actually-- let's see how that works. 01:19:52.550 --> 01:19:54.830 So let's do an example of that. 01:19:54.830 --> 01:19:58.210 So g perp you knew was just 1. 01:19:58.210 --> 01:20:01.310 So that's obviously something that has no scaling. 01:20:01.310 --> 01:20:05.400 And if you compare that to k perp mu k perp nu over k 01:20:05.400 --> 01:20:08.620 squared, k perp scale like lambda. k 01:20:08.620 --> 01:20:10.200 squared scales like lambda squared. 01:20:10.200 --> 01:20:12.117 So this is lambda squared over lambda squared. 01:20:12.117 --> 01:20:14.140 So that's also lambda to the 0. 01:20:14.140 --> 01:20:17.610 So both the g perp mu nu and the k perp nu term 01:20:17.610 --> 01:20:18.600 are the same size. 01:20:18.600 --> 01:20:19.100 Yeah. 01:20:19.100 --> 01:20:20.950 AUDIENCE: If it happened that it didn't work out, 01:20:20.950 --> 01:20:23.375 could you choose the gauge parameter have a [INAUDIBLE] 01:20:23.375 --> 01:20:23.875 with lambda? 01:20:23.875 --> 01:20:25.500 IAIN STEWART: You could, but then you'd 01:20:25.500 --> 01:20:27.040 be restricted to the classic gauges 01:20:27.040 --> 01:20:29.290 that you would be able to use in the effective theory. 01:20:29.290 --> 01:20:31.707 And then you have to ask what gauge invariance would mean. 01:20:34.750 --> 01:20:38.350 If I don't make any restrictions on tau, 01:20:38.350 --> 01:20:40.010 I don't assign a power counting to it, 01:20:40.010 --> 01:20:43.960 that means my effective theory should allow all these gauges. 01:20:43.960 --> 01:20:46.652 And I actually want that. 01:20:46.652 --> 01:20:47.610 That's a good question. 01:20:51.580 --> 01:20:53.920 You can do the same thing looking at, 01:20:53.920 --> 01:20:56.280 for example, g plus g minus. 01:20:56.280 --> 01:21:00.720 That's also 1, so lambda 0. 01:21:00.720 --> 01:21:03.802 Then you get k plus k minus over k squared. 01:21:03.802 --> 01:21:05.760 That's also lambda squared over lambda squared. 01:21:05.760 --> 01:21:07.510 So this is what I was saying, that the two 01:21:07.510 --> 01:21:09.330 terms are the same size. 01:21:09.330 --> 01:21:14.730 If you dot in n mu n nu, that kills the g mu nu. 01:21:14.730 --> 01:21:18.090 g plus plus is 0. 01:21:18.090 --> 01:21:21.180 And then you just get k plus squared 01:21:21.180 --> 01:21:26.670 over k squared, which is lambda to the 1/4 01:21:26.670 --> 01:21:30.010 over lambda squared, which is lambda squared. 01:21:30.010 --> 01:21:33.450 So when the g mu nu is not 1, then the k mu k nu term 01:21:33.450 --> 01:21:35.220 can still determine how big something is. 01:21:35.220 --> 01:21:39.735 And that's what would happen for these off-diagonal terms. 01:21:39.735 --> 01:21:41.610 So if we go through the same type of exercise 01:21:41.610 --> 01:21:45.947 that we did for the fermion, d4x scales like 1 over k 01:21:45.947 --> 01:21:46.530 to the fourth. 01:21:51.200 --> 01:21:54.950 So that's lambda to the minus 4, just as before. 01:21:54.950 --> 01:21:58.170 And actually, if we were to do the following, 01:21:58.170 --> 01:22:01.220 if we were to write this as-- 01:22:01.220 --> 01:22:02.810 I should have done that up there. 01:22:02.810 --> 01:22:04.910 If we were to write it as minus i over k 01:22:04.910 --> 01:22:12.410 to the fourth k squared g mu nu minus tau k mu k nu, 01:22:12.410 --> 01:22:15.290 the k to the fourth just matches up with the d4x. 01:22:15.290 --> 01:22:17.180 So those take care of each other. 01:22:17.180 --> 01:22:20.930 And then the fields here have to match up with the rest. 01:22:20.930 --> 01:22:26.420 So the scaling of this should be the scaling of the A mu A nu. 01:22:26.420 --> 01:22:29.335 And that basically means that A mu in A nu scale 01:22:29.335 --> 01:22:29.960 like a momenta. 01:22:38.860 --> 01:22:43.300 So A mu n for this collinear gluon scales like k mu, scales 01:22:43.300 --> 01:22:44.605 like lambda squared 1 lambda. 01:22:54.907 --> 01:22:56.240 Let me write that one more time. 01:23:09.160 --> 01:23:12.220 And that's also a nice thing because it also 01:23:12.220 --> 01:23:14.770 means you could form a covariate derivative that's 01:23:14.770 --> 01:23:19.610 homogeneous by combining together, 01:23:19.610 --> 01:23:23.410 if I write it this way, k mu and g A mu. 01:23:23.410 --> 01:23:27.640 I can get a covariant derivative where, for each component of k, 01:23:27.640 --> 01:23:29.390 I also have a component of the gauge field 01:23:29.390 --> 01:23:30.440 that's the same size. 01:23:30.440 --> 01:23:32.530 So you could have argued it, originally 01:23:32.530 --> 01:23:36.010 just from gauge invariance, that you want to sort of have fields 01:23:36.010 --> 01:23:38.450 that are of the same size as your momenta, 01:23:38.450 --> 01:23:40.360 but we did it a little bit differently here. 01:23:55.150 --> 01:23:58.090 So it's nice that that comes out. 01:23:58.090 --> 01:23:59.945 This all hangs together. 01:23:59.945 --> 01:24:01.570 So next time, we'll talk about the fact 01:24:01.570 --> 01:24:03.730 that what does it mean that the gauge field has components that 01:24:03.730 --> 01:24:04.940 are scaling in a different way. 01:24:04.940 --> 01:24:07.273 And particularly, there's a component of the gauge field 01:24:07.273 --> 01:24:08.200 that's order 1. 01:24:08.200 --> 01:24:10.510 There's no power suppression for that component. 01:24:10.510 --> 01:24:14.126 And we'll talk about what implications that has. 01:24:14.126 --> 01:24:16.210 Yeah. 01:24:16.210 --> 01:24:20.194 AUDIENCE: So what about in the non-variant case, like n bar 01:24:20.194 --> 01:24:21.190 of A equals 0? 01:24:21.190 --> 01:24:21.982 IAIN STEWART: Yeah. 01:24:21.982 --> 01:24:25.270 So you could go through this argument in that gauge as well, 01:24:25.270 --> 01:24:27.210 and it will work. 01:24:27.210 --> 01:24:29.677 AUDIENCE: But you have n bar to the A equals 0, so-- 01:24:29.677 --> 01:24:31.010 IAIN STEWART: Yeah, that's fine. 01:24:31.010 --> 01:24:32.620 That's just a restriction on the-- you 01:24:32.620 --> 01:24:34.150 can still assign a scaling to it. 01:24:34.150 --> 01:24:36.550 And a special choice if it is 0. 01:24:36.550 --> 01:24:40.600 Scaling and values of things are not the same thing, right? 01:24:40.600 --> 01:24:43.900 Like, you could have a field that scales like 1 in the power 01:24:43.900 --> 01:24:46.610 company, but happens to be 0. 01:24:46.610 --> 01:24:47.480 And that's OK. 01:24:50.870 --> 01:24:52.096 Yeah. 01:24:52.096 --> 01:24:54.198 AUDIENCE: I had a question about SCET1 and SCET2. 01:24:54.198 --> 01:24:54.990 IAIN STEWART: Sure. 01:24:54.990 --> 01:24:59.874 AUDIENCE: So in the example adding one jab, and one hadron, 01:24:59.874 --> 01:25:01.827 like the [INAUDIBLE],, why do I need 01:25:01.827 --> 01:25:03.982 to have different degrees of freedom to describe 01:25:03.982 --> 01:25:04.798 [INAUDIBLE]? 01:25:04.798 --> 01:25:05.590 IAIN STEWART: Yeah. 01:25:05.590 --> 01:25:07.690 It's really because of the way the modes 01:25:07.690 --> 01:25:11.690 sat on those hyperbolas that, in the case of the hadrons, 01:25:11.690 --> 01:25:13.190 you had the soft modes and collinear 01:25:13.190 --> 01:25:14.320 modes on the same hyperbola. 01:25:14.320 --> 01:25:16.510 And that's going to change how the effective theory looks. 01:25:16.510 --> 01:25:17.135 AUDIENCE: Yeah. 01:25:17.135 --> 01:25:19.430 But from a physically point of view, why the hadrons-- 01:25:19.430 --> 01:25:21.469 [INAUDIBLE] the hadrons behaving in a different way 01:25:21.469 --> 01:25:22.180 than [INAUDIBLE]? 01:25:22.180 --> 01:25:22.972 IAIN STEWART: Yeah. 01:25:22.972 --> 01:25:24.942 So one way of thinking about it is, 01:25:24.942 --> 01:25:26.650 if you just had the collinear modes alone 01:25:26.650 --> 01:25:28.040 and you didn't have the soft modes, 01:25:28.040 --> 01:25:29.290 then it would be very similar. 01:25:29.290 --> 01:25:32.020 It would just be that the hyperbola moved down. 01:25:32.020 --> 01:25:34.720 And you think maybe those two theories are the same. 01:25:34.720 --> 01:25:36.280 There's examples of that actually. 01:25:36.280 --> 01:25:39.220 There's one example that we'll cover where 01:25:39.220 --> 01:25:40.630 you don't need soft modes. 01:25:40.630 --> 01:25:42.505 And then you can't really tell whether you're 01:25:42.505 --> 01:25:44.410 using SCET1 or SCET2. 01:25:44.410 --> 01:25:47.350 But if you have a process that has both soft and collinear 01:25:47.350 --> 01:25:50.410 modes, then it's the kind of way that those modes 01:25:50.410 --> 01:25:52.180 talk to each other that distinguishes 01:25:52.180 --> 01:25:54.310 the cases of hadrons and jets in the cases-- 01:25:54.310 --> 01:25:56.270 AUDIENCE: For example, for the case of the jet, 01:25:56.270 --> 01:25:58.742 [INAUDIBLE] soft modes like as a need for the-- 01:25:58.742 --> 01:25:59.575 IAIN STEWART: Right. 01:25:59.575 --> 01:26:01.960 You could try, yes. 01:26:01.960 --> 01:26:04.870 And it turns out that those modes aren't relevant. 01:26:04.870 --> 01:26:06.580 So you have to be a little bit faithful 01:26:06.580 --> 01:26:08.107 from what I've told you so far. 01:26:08.107 --> 01:26:10.190 AUDIENCE: But there is no physical picture for why 01:26:10.190 --> 01:26:11.520 it can understand [INAUDIBLE]. 01:26:11.520 --> 01:26:12.430 IAIN STEWART: You can understand it 01:26:12.430 --> 01:26:14.513 because what would happen with those soft modes is 01:26:14.513 --> 01:26:18.400 that they would take the collinear modes far off-shell. 01:26:18.400 --> 01:26:20.695 If you just had one of those soft modes interacting 01:26:20.695 --> 01:26:23.320 with the collinear mode, you end up not with the collinear mode 01:26:23.320 --> 01:26:25.930 back again, but something that's further off-shell that you 01:26:25.930 --> 01:26:27.430 actually want to integrate out. 01:26:27.430 --> 01:26:31.330 So that thing that has the scaling that allows you to just 01:26:31.330 --> 01:26:34.960 have collinear mode in something which we call ultra soft in 01:26:34.960 --> 01:26:38.370 and still have collinear, that's the ultra soft mode. 01:26:38.370 --> 01:26:40.120 So if you want to communicate with the jet 01:26:40.120 --> 01:26:42.760 without disturbing it and blowing it apart, 01:26:42.760 --> 01:26:45.430 then you really need the ultra soft mode. 01:26:45.430 --> 01:26:46.000 Yeah. 01:26:46.000 --> 01:26:48.100 But these are all good questions, 01:26:48.100 --> 01:26:50.700 and we'll talk much more about it.