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 --> 00:00:17.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.190 --> 00:00:18.370 at ocw.mit.edu. 00:00:22.250 --> 00:00:27.340 PROFESSOR: So this is an example in SCET one 00:00:27.340 --> 00:00:33.715 where the degrees of freedom in the p plus p minus plane 00:00:33.715 --> 00:00:34.465 looked as follows. 00:00:45.260 --> 00:00:48.470 So we had hard modes that lived out here at p 00:00:48.470 --> 00:00:52.880 squared in order some large scale q squared, 00:00:52.880 --> 00:00:56.590 then we had collinear modes for the two jets 00:00:56.590 --> 00:00:58.660 that are going to be going back to back, 00:00:58.660 --> 00:01:00.580 and I'll draw the picture in a second. 00:01:00.580 --> 00:01:02.530 Then there were some ultra soft modes 00:01:02.530 --> 00:01:05.500 as well that describe radiation between the jets. 00:01:05.500 --> 00:01:09.100 And then there can be also some lambda QCD modes, 00:01:09.100 --> 00:01:11.545 and we'll talk about them a bit today too. 00:01:11.545 --> 00:01:13.670 So there's a lot of different things going on here, 00:01:13.670 --> 00:01:16.460 but we'll see that actually we can understand everything 00:01:16.460 --> 00:01:19.210 and organize things using SCET. 00:01:19.210 --> 00:01:22.720 So what's the picture for these modes? 00:01:22.720 --> 00:01:26.350 We have back-to-back jets. 00:01:26.350 --> 00:01:30.655 Let me draw one jet this way, one jet this way. 00:01:30.655 --> 00:01:31.750 I should use some color. 00:01:39.730 --> 00:01:40.990 So there's one of our jets. 00:01:48.690 --> 00:01:52.800 There's another, and then there can be soft radiation. 00:01:58.260 --> 00:02:00.180 And between these jets, there can also 00:02:00.180 --> 00:02:03.498 be soft radiation physically within the jets, 00:02:03.498 --> 00:02:05.040 but there's no directional dependence 00:02:05.040 --> 00:02:06.150 to the soft radiation. 00:02:14.390 --> 00:02:18.660 And then set that this is blue. 00:02:21.460 --> 00:02:25.384 This and this so far are orange in my picture. 00:02:25.384 --> 00:02:31.200 This is green, and then there's some hard interactions, 00:02:31.200 --> 00:02:35.590 which we've already sort of localized 00:02:35.590 --> 00:02:37.480 into a dot in this picture. 00:02:42.860 --> 00:02:44.500 So what we're doing is we're colliding 00:02:44.500 --> 00:02:48.130 e plus e minus producing a virtual photon or z 00:02:48.130 --> 00:02:55.490 and then from that producing, if you like, 00:02:55.490 --> 00:02:58.100 a state which is full of collinear modes, state which 00:02:58.100 --> 00:03:00.770 is full of the other type of collinear 00:03:00.770 --> 00:03:04.130 modes and ultra soft particles. 00:03:04.130 --> 00:03:09.020 And this is in the center of mass frame, 00:03:09.020 --> 00:03:12.512 or that's kind of how you could think about it in SCET, 00:03:12.512 --> 00:03:13.970 or physically what you're producing 00:03:13.970 --> 00:03:24.200 is you're producing jet 1 plus jet 2 plus soft radiation 00:03:24.200 --> 00:03:29.165 or ultra soft hadrons, soft hadrons. 00:03:39.525 --> 00:03:41.650 OK, so the first scale that you want to think about 00:03:41.650 --> 00:03:46.270 is the hard scale, and so that is just 00:03:46.270 --> 00:03:48.820 the q squared to the virtual photon or z. 00:03:48.820 --> 00:03:51.640 So if you think about the Feynman diagram, 00:03:51.640 --> 00:03:53.430 there's e plus e minus. 00:03:53.430 --> 00:03:55.660 Here's q mu. 00:03:55.660 --> 00:03:57.310 You're going to have a pair of quarks 00:03:57.310 --> 00:04:00.358 that that virtual particle can couple to, 00:04:00.358 --> 00:04:01.900 and there could be additional gluons. 00:04:05.300 --> 00:04:08.320 But the hard scale is set by the virtual photon or z. 00:04:13.050 --> 00:04:15.750 So there's going to be some scale associated 00:04:15.750 --> 00:04:18.209 to that hard scale q. 00:04:18.209 --> 00:04:20.529 If you want to think about it as a normalization scale, 00:04:20.529 --> 00:04:22.029 there'll be some scale where we want 00:04:22.029 --> 00:04:24.105 to match from QCD onto SCET, and that's 00:04:24.105 --> 00:04:25.230 going to be the hard scale. 00:04:35.126 --> 00:04:36.940 So I'll call that scale mu h. 00:04:41.150 --> 00:04:43.820 So then something physical has to set these other scales 00:04:43.820 --> 00:04:46.790 in the picture, and the next thing 00:04:46.790 --> 00:04:51.050 that we need to talk about is what we're going to measure. 00:04:51.050 --> 00:04:53.570 In principle, if you collide e plus e minus 00:04:53.570 --> 00:04:56.150 and you produce hadrons, you could have lots 00:04:56.150 --> 00:04:57.650 of different possibilities. 00:04:57.650 --> 00:05:00.380 In particular, you could have not dijets but trijets, 00:05:00.380 --> 00:05:03.650 so you could have a third jet and made it from a gluon. 00:05:03.650 --> 00:05:06.560 And some kind of measurement that we do on the final state 00:05:06.560 --> 00:05:09.590 is going to restrict us to this configuration with just two 00:05:09.590 --> 00:05:10.790 jets. 00:05:10.790 --> 00:05:12.830 And the one we'll talk about to start 00:05:12.830 --> 00:05:16.460 is what's called hemisphere invariant masses, which 00:05:16.460 --> 00:05:17.670 we already mentioned earlier. 00:05:21.598 --> 00:05:23.390 So the way that you should think about this 00:05:23.390 --> 00:05:27.680 is you take all the final state particles, 00:05:27.680 --> 00:05:30.320 and you could just take the sum of all their momenta 00:05:30.320 --> 00:05:36.970 and divide it into two parts, those that are in hemisphere A 00:05:36.970 --> 00:05:40.410 and those that are in hemisphere B. 00:05:40.410 --> 00:05:47.960 So let me augment my figure here by saying 00:05:47.960 --> 00:05:51.718 that there's two hemispheres. 00:05:51.718 --> 00:05:53.760 And then what these masses are, which I'll call m 00:05:53.760 --> 00:05:57.350 squared and then bar squared are just 00:05:57.350 --> 00:06:02.580 the four vectors squared for p, for the a, and the b guy. 00:06:02.580 --> 00:06:06.660 So written in terms of the individual particles 00:06:06.660 --> 00:06:10.200 in that hemisphere, you sum up all their four 00:06:10.200 --> 00:06:13.920 vectors for the particles in that hemisphere and square it, 00:06:13.920 --> 00:06:30.050 and then likewise for B. 00:06:30.050 --> 00:06:33.170 And if you're measuring these two things, 00:06:33.170 --> 00:06:35.060 we talked about earlier the fact that you 00:06:35.060 --> 00:06:37.250 could specify that it was a jet by demanding 00:06:37.250 --> 00:06:38.730 that they're small. 00:06:38.730 --> 00:06:45.990 So the dijets is going to be m squared 00:06:45.990 --> 00:06:49.908 and then bar squared much less than q squared. 00:06:49.908 --> 00:06:52.200 And if you had trijets, you wouldn't-- that wouldn't be 00:06:52.200 --> 00:06:54.030 the case because if you had trijets, 00:06:54.030 --> 00:06:58.760 you'd have two particle-- two jets in this hemisphere at wide 00:06:58.760 --> 00:07:01.520 angles, and those-- that wide angle would produce a large q 00:07:01.520 --> 00:07:05.350 squared, a large and variant mass of m squared. 00:07:05.350 --> 00:07:06.830 That's of order q squared. 00:07:06.830 --> 00:07:08.770 But if you just have a single jet, 00:07:08.770 --> 00:07:10.970 then you can have a small m squared, 00:07:10.970 --> 00:07:13.735 so the small m squared limit is what forces you to have dijets. 00:07:16.800 --> 00:07:20.160 And these guys here, it also puts us 00:07:20.160 --> 00:07:23.850 in the kinematic situation where we have our n collinear 00:07:23.850 --> 00:07:27.330 modes for one jet and n bar collinear modes 00:07:27.330 --> 00:07:29.953 for the other jet. 00:07:29.953 --> 00:07:32.370 So those are two of the degrees of freedom in our picture. 00:07:37.650 --> 00:07:40.660 So I'm reminding you of some things we talked about before, 00:07:40.660 --> 00:07:44.280 but it was a while ago so-- 00:07:44.280 --> 00:07:47.857 and the power counting parameter here is m over q. 00:07:47.857 --> 00:07:49.440 And so if you want to say that there's 00:07:49.440 --> 00:07:52.470 a scale in this picture associated to the collinear 00:07:52.470 --> 00:07:57.000 modes, then you would say it's on a kind of jet scale, 00:07:57.000 --> 00:07:58.394 and that's of order m. 00:08:02.840 --> 00:08:08.860 OK, any questions so far? 00:08:08.860 --> 00:08:17.360 So then there's the ultra soft radiation, 00:08:17.360 --> 00:08:19.900 which is being ultra soft, it's uniform. 00:08:19.900 --> 00:08:22.221 It's not collimated in any direction. 00:08:25.562 --> 00:08:27.520 It's the thing that gives you the communication 00:08:27.520 --> 00:08:28.360 between the jets. 00:08:32.919 --> 00:08:35.710 The jets are going to decouple from each other in the sense 00:08:35.710 --> 00:08:38.539 that interactions directly between the n collinear and n 00:08:38.539 --> 00:08:42.270 bar collinear modes is all going into the pink dot, 00:08:42.270 --> 00:08:45.280 but there can be long distance communication between the jets 00:08:45.280 --> 00:08:47.170 caused by the ultra soft modes. 00:08:47.170 --> 00:08:53.410 So, really, what I mean here is long distance communication. 00:08:53.410 --> 00:08:55.810 Short distance communication can happen too. 00:08:55.810 --> 00:08:59.350 That's what the pink dot is, but that's, in some sense, simpler. 00:08:59.350 --> 00:09:00.940 Long distance communication can only 00:09:00.940 --> 00:09:04.630 happen by this ultra soft radiation. 00:09:04.630 --> 00:09:08.500 And we know about how ultra soft radiation 00:09:08.500 --> 00:09:10.180 interacts with energetic particles, 00:09:10.180 --> 00:09:13.060 and in particular, what we know is that it's eikonal. 00:09:13.060 --> 00:09:16.000 That came out of our Lagrangian description 00:09:16.000 --> 00:09:18.520 of ultra soft radiation. 00:09:18.520 --> 00:09:22.340 And that will have implications here. 00:09:22.340 --> 00:09:23.950 So what is this ultra soft radiation? 00:09:23.950 --> 00:09:26.182 It's radiation that has energy that's 00:09:26.182 --> 00:09:27.265 of order q lambda squared. 00:09:30.110 --> 00:09:36.010 And so that's m squared over q energy radiation, 00:09:36.010 --> 00:09:39.370 and so the soft scale, which is the next scale down 00:09:39.370 --> 00:09:44.890 in the picture, ultra soft scale, is m squared over q. 00:09:44.890 --> 00:09:47.590 Often, people will-- because there's 00:09:47.590 --> 00:09:51.760 no soft radiation in this setup, sometimes people 00:09:51.760 --> 00:09:54.250 will just drop the ultra and just call it soft. 00:09:54.250 --> 00:09:56.760 That's very common. 00:09:56.760 --> 00:09:58.810 I'll try to always call it ultra soft here 00:09:58.810 --> 00:10:01.750 because we'll be drawing some distinctions with SCET 2 00:10:01.750 --> 00:10:04.810 examples, and in those examples, we have soft and collinear. 00:10:04.810 --> 00:10:07.120 In this example, we have ultra soft and collinear, 00:10:07.120 --> 00:10:09.520 so I'll keep trying to call it ultra soft. 00:10:09.520 --> 00:10:11.050 But you can see that I've already 00:10:11.050 --> 00:10:12.910 abbreviated it to u soft to make it 00:10:12.910 --> 00:10:15.280 look more and more like soft. 00:10:15.280 --> 00:10:17.200 Anyway, but in the literature, people 00:10:17.200 --> 00:10:20.470 often just drop the u completely and call it soft. 00:10:20.470 --> 00:10:23.520 Something to be aware of. 00:10:23.520 --> 00:10:27.550 OK, so this actually-- this radiation here, 00:10:27.550 --> 00:10:30.040 this is kind of the largest scale that 00:10:30.040 --> 00:10:32.930 can show up in the soft function, 00:10:32.930 --> 00:10:34.450 and there's also lambda QCD. 00:10:39.310 --> 00:10:41.470 And that's a scale that, at least to start, 00:10:41.470 --> 00:10:43.220 we're going to leave in the soft function. 00:10:51.148 --> 00:10:53.190 So if we go back to our picture over here, that's 00:10:53.190 --> 00:10:57.870 why I put both the ultra soft mode side up on this hyperbola 00:10:57.870 --> 00:11:03.210 there, which is this m squared over q line, mu over m 00:11:03.210 --> 00:11:04.800 squared over q line. 00:11:04.800 --> 00:11:06.810 That's why I made it orange, and I also 00:11:06.810 --> 00:11:10.320 made orange this lambda QCD because they're together. 00:11:10.320 --> 00:11:12.480 And you can think of it as just that there's 00:11:12.480 --> 00:11:16.320 some modes that are capturing this entire region, which 00:11:16.320 --> 00:11:18.930 are the ultra soft modes. 00:11:18.930 --> 00:11:20.430 Now there are two-- 00:11:20.430 --> 00:11:25.500 if m squared over q was of order lambda QCD, then 00:11:25.500 --> 00:11:26.570 those are the same thing. 00:11:26.570 --> 00:11:29.190 So there's kind of two possibilities here. 00:11:29.190 --> 00:11:35.250 You could have m squared over q of order lambda QCD, 00:11:35.250 --> 00:11:39.466 and that means that this scale is not non-perturbative. 00:11:43.200 --> 00:11:45.180 And this is what's called, for reasons 00:11:45.180 --> 00:11:50.137 that will become apparent by the end of lecture, the peak region 00:11:50.137 --> 00:11:52.470 because this is actually the region of the cross section 00:11:52.470 --> 00:11:53.952 where there's a peak. 00:11:53.952 --> 00:11:56.890 OK, that's why it's called the peak region. 00:11:56.890 --> 00:12:01.290 And in this region, you have the following hierarchy. 00:12:01.290 --> 00:12:05.110 You have mu h is much greater than mu j. 00:12:05.110 --> 00:12:07.920 Mu j is much greater than mu s, but mu 00:12:07.920 --> 00:12:10.350 s is of order lambda QCD. 00:12:10.350 --> 00:12:15.590 OK, so if we take this to always be true, 00:12:15.590 --> 00:12:18.020 then we're in this situation. 00:12:18.020 --> 00:12:19.860 And there is another possibility, 00:12:19.860 --> 00:12:23.510 and that is that there's really a separation 00:12:23.510 --> 00:12:27.530 between these hyperbolas that's just as large or hierarchical 00:12:27.530 --> 00:12:30.030 as the previous separations. 00:12:30.030 --> 00:12:34.130 So if this thing is much bigger than lambda QCD, 00:12:34.130 --> 00:12:37.370 that's another possibility, and there'll 00:12:37.370 --> 00:12:39.260 be a region of phase based where this is true 00:12:39.260 --> 00:12:42.090 and a region of phase based where this is true. 00:12:42.090 --> 00:12:43.340 The second one is-- 00:12:43.340 --> 00:12:46.430 here, the soft, there's-- 00:12:49.990 --> 00:12:59.500 here there is perturbative ultra soft radiation, 00:12:59.500 --> 00:13:01.960 so you can just calculate it order by order 00:13:01.960 --> 00:13:05.860 in the ultra soft scale, which is this scale, which 00:13:05.860 --> 00:13:09.910 is perturbative because it's much bigger than lambda QCD. 00:13:09.910 --> 00:13:11.740 And this is what's called the tail region. 00:13:16.880 --> 00:13:19.030 And the analog of this statement here 00:13:19.030 --> 00:13:21.835 is that we have a hierarchy between everybody. 00:13:34.230 --> 00:13:36.840 So when we do this analysis, what we're going to do 00:13:36.840 --> 00:13:39.190 is we're going to do a power expansion. 00:13:39.190 --> 00:13:41.580 And you can say, well, the power expansion 00:13:41.580 --> 00:13:43.600 is just what you told me, m squared over-- 00:13:43.600 --> 00:13:45.060 m squared much less than q squared, 00:13:45.060 --> 00:13:47.700 but you also have to worry about this when you're 00:13:47.700 --> 00:13:50.320 doing the power expansion. 00:13:50.320 --> 00:13:52.710 So one way of thinking about it is just exactly 00:13:52.710 --> 00:13:56.490 as I'm writing here, and that's how I want to advocate. 00:13:56.490 --> 00:13:59.683 The things you're expanding in are these assumptions. 00:13:59.683 --> 00:14:01.350 And you can see that in these two cases, 00:14:01.350 --> 00:14:03.000 there's a slightly different set up. 00:14:17.920 --> 00:14:20.110 So usually, when you have an effective theory, 00:14:20.110 --> 00:14:23.860 you have to define this from the beginning because the way 00:14:23.860 --> 00:14:27.340 you proceed and how you set up your theory is going 00:14:27.340 --> 00:14:30.010 to depend on whether you're in this situation 00:14:30.010 --> 00:14:32.260 or this situation. 00:14:32.260 --> 00:14:35.560 But part of this story is the same in these two situations, 00:14:35.560 --> 00:14:38.000 namely the first two greater thans, much greater thans 00:14:38.000 --> 00:14:39.470 are the same. 00:14:39.470 --> 00:14:41.830 And so that, we can proceed without worrying about this. 00:14:45.372 --> 00:14:47.330 And then we can make this distinction later on, 00:14:47.330 --> 00:14:49.038 and that's what my picture was advocating 00:14:49.038 --> 00:14:56.167 for by having these guys in the same category. 00:14:56.167 --> 00:14:57.250 So hopefully that's clear. 00:15:05.140 --> 00:15:17.590 OK, so if you're in the tail region, 00:15:17.590 --> 00:15:24.310 there's going to be power corrections, 00:15:24.310 --> 00:15:28.120 and these are actually the most important power corrections 00:15:28.120 --> 00:15:35.350 that come as powers of lambda QCD over the soft scale 00:15:35.350 --> 00:15:36.040 to some power. 00:15:36.040 --> 00:15:42.220 But you're going to have a power expansion in that. 00:15:42.220 --> 00:15:46.000 So that's the-- if you like in this situation 00:15:46.000 --> 00:15:47.500 when you're in the tail region, you 00:15:47.500 --> 00:15:50.710 can have power corrections that are non-perturbative by having 00:15:50.710 --> 00:15:52.330 ratios of these things, lambda QCD 00:15:52.330 --> 00:15:55.340 over this over that over this. 00:15:55.340 --> 00:15:58.966 But the biggest one are these guys over the soft scale, 00:15:58.966 --> 00:16:00.700 so that's the next smallest scale. 00:16:05.618 --> 00:16:07.410 But if you neglect these, the leading order 00:16:07.410 --> 00:16:09.750 cross section in this region is perturbative. 00:16:19.580 --> 00:16:25.460 Now you know you can have any power here, and those exist. 00:16:25.460 --> 00:16:28.100 What happens in the peak region is 00:16:28.100 --> 00:16:31.490 that thinking about those as an expansion is no longer good. 00:16:38.120 --> 00:16:41.960 And you have to take all of them and treat them 00:16:41.960 --> 00:16:47.930 as if they're order one and for any k, 00:16:47.930 --> 00:16:50.368 and that ends up meaning that there's 00:16:50.368 --> 00:16:52.160 going to be some non-perturbative function, 00:16:52.160 --> 00:16:54.160 like a Parton distribution function in the sense 00:16:54.160 --> 00:16:58.490 that it's non-perturbative, that describes part of what's 00:16:58.490 --> 00:17:04.800 going on here in this region. 00:17:04.800 --> 00:17:06.740 And that, we just see from our power counting, 00:17:06.740 --> 00:17:08.365 and our power counting already tells us 00:17:08.365 --> 00:17:11.677 that that's what's going to happen. 00:17:11.677 --> 00:17:13.260 So we can learn a lot just by thinking 00:17:13.260 --> 00:17:16.030 about the scales and the problem and thinking about the power 00:17:16.030 --> 00:17:16.530 counting. 00:17:25.970 --> 00:17:32.463 So you could ask about other power corrections. 00:17:36.170 --> 00:17:37.730 So there's a set of power connections 00:17:37.730 --> 00:17:41.660 where you would expand and say mu s over mu j. 00:17:41.660 --> 00:17:50.350 And let me label these as kinematic because these-- 00:17:50.350 --> 00:17:53.120 the things that setting the scale for mu s and mu 00:17:53.120 --> 00:17:56.530 j, say in this picture where mu s is this 00:17:56.530 --> 00:18:00.280 and I always distinguished lambda QCD as a separate thing, 00:18:00.280 --> 00:18:02.140 these scales are just perturbative scales, 00:18:02.140 --> 00:18:03.670 and they just correspond to if you 00:18:03.670 --> 00:18:05.503 think about having some function that you're 00:18:05.503 --> 00:18:08.150 making an expansion of it. 00:18:08.150 --> 00:18:09.670 So they come about from expansion 00:18:09.670 --> 00:18:11.380 of kinematic variables basically. 00:18:19.846 --> 00:18:23.086 So it's the m squared much less than q squared. 00:18:23.086 --> 00:18:27.320 So I mean, there's nothing non-perturbative about that. 00:18:30.040 --> 00:18:31.910 And then you could have power corrections 00:18:31.910 --> 00:18:36.620 that are lambda QCD over mu, which, if you like, 00:18:36.620 --> 00:18:38.720 are really hard power corrections. 00:18:38.720 --> 00:18:42.500 They're kind of a traditional power corrections, which are 00:18:42.500 --> 00:18:45.590 lambda QCD over the hard scale. 00:18:45.590 --> 00:18:49.910 And the exercise that I gave in the homework, in some sense, 00:18:49.910 --> 00:18:54.950 is going towards figuring out what the sort of soft function 00:18:54.950 --> 00:18:56.630 matrix elements would be. 00:18:56.630 --> 00:18:58.800 Part of the problem-- there's other parts to it-- 00:18:58.800 --> 00:19:00.782 but would be going towards figuring out 00:19:00.782 --> 00:19:02.990 what these guys actually are and defining them, which 00:19:02.990 --> 00:19:06.770 is not known in the literature. 00:19:06.770 --> 00:19:12.680 And then finally, you could have lambda QCD over mu j, 00:19:12.680 --> 00:19:14.150 which, of course, you could write. 00:19:14.150 --> 00:19:15.080 You have to be a little bit careful 00:19:15.080 --> 00:19:17.120 because one way that this could happen 00:19:17.120 --> 00:19:22.250 is that you just have one of these guys times one 00:19:22.250 --> 00:19:24.230 of those guys. 00:19:24.230 --> 00:19:25.520 OK, so they're not-- 00:19:25.520 --> 00:19:27.523 it's not necessarily an independent category, 00:19:27.523 --> 00:19:29.690 but you could still think about these types of power 00:19:29.690 --> 00:19:33.170 corrections as kind of something that might show up, 00:19:33.170 --> 00:19:36.240 either this way or maybe somewhat maybe directly. 00:19:36.240 --> 00:19:40.640 And anyway, that's another category. 00:19:40.640 --> 00:19:43.400 It turns out that kind of-- 00:19:43.400 --> 00:19:46.100 you can get it this way, which is something 00:19:46.100 --> 00:19:47.930 that one can deal with. 00:19:47.930 --> 00:19:51.380 And if it happens directly, then it actually 00:19:51.380 --> 00:19:56.630 is down by two powers, so these guys 00:19:56.630 --> 00:19:58.159 are kind of less important. 00:20:05.830 --> 00:20:08.710 OK, so people deal with these guys in the literature. 00:20:08.710 --> 00:20:11.950 These guys haven't really been fully treated yet. 00:20:11.950 --> 00:20:14.000 Not very much is known about them, 00:20:14.000 --> 00:20:16.780 and this is fully treated in the literature. 00:20:16.780 --> 00:20:22.230 These ones that I said are the most important ones. 00:20:22.230 --> 00:20:24.690 OK, so even-- we can already even outline 00:20:24.690 --> 00:20:28.377 kind of what people are doing because we just 00:20:28.377 --> 00:20:30.210 think about higher order terms of expansion, 00:20:30.210 --> 00:20:33.420 and this is an example, unlike B to C decays, 00:20:33.420 --> 00:20:35.310 where we have people going to four orders 00:20:35.310 --> 00:20:36.840 down in the expansion. 00:20:36.840 --> 00:20:40.230 Here even the first non-trivial order down hasn't really been 00:20:40.230 --> 00:20:41.590 done-- 00:20:41.590 --> 00:20:42.590 hasn't been done at all. 00:20:45.210 --> 00:20:47.318 In the second, place various expansions, 00:20:47.318 --> 00:20:48.360 so it's more complicated. 00:20:48.360 --> 00:20:51.030 And some of them have been treated, but this one hasn't. 00:21:02.296 --> 00:21:05.280 And I should say that the way that this one is treated 00:21:05.280 --> 00:21:08.487 is really just perturbatively so it's not-- 00:21:08.487 --> 00:21:10.320 this one, in some sense, is also not treated 00:21:10.320 --> 00:21:13.470 fully in the language of the effective theory. 00:21:13.470 --> 00:21:15.330 It's just put in by hand. 00:21:19.130 --> 00:21:21.670 So in some sense, the only one that's treated properly 00:21:21.670 --> 00:21:22.450 is the top one. 00:21:29.670 --> 00:21:32.610 OK, well, we're going to work mostly to leading order, 00:21:32.610 --> 00:21:36.330 and we're going to call these guys power corrections 00:21:36.330 --> 00:21:38.760 and work with this stuff. 00:21:46.100 --> 00:21:50.730 OK, so start with the full theory current. 00:21:50.730 --> 00:21:52.310 I'm not going to distinguish too much 00:21:52.310 --> 00:21:55.460 between pieces that come from charges, like the electrically 00:21:55.460 --> 00:21:56.720 charged and stuff like that. 00:21:56.720 --> 00:21:59.460 That's not going to be our main focus here. 00:21:59.460 --> 00:22:02.360 We can keep track of all that. 00:22:02.360 --> 00:22:04.520 Our main focus is going to be understanding the QCD 00:22:04.520 --> 00:22:06.560 effects for the jets. 00:22:06.560 --> 00:22:08.120 So we've already done this. 00:22:08.120 --> 00:22:10.520 The current that has a psi bar psi 00:22:10.520 --> 00:22:14.570 gets matched in SCET onto something 00:22:14.570 --> 00:22:16.325 with two Wilson lines. 00:22:16.325 --> 00:22:17.075 There's the label. 00:22:23.990 --> 00:22:26.870 There's another label, and we can also 00:22:26.870 --> 00:22:28.568 make a field redefinition. 00:22:41.030 --> 00:22:43.210 And let me not put zeros on the field 00:22:43.210 --> 00:22:46.300 after the field redefinition because that makes 00:22:46.300 --> 00:22:47.630 it notationally cumbersome. 00:22:47.630 --> 00:22:54.640 So this goes to this after making that field redefinition. 00:22:54.640 --> 00:22:58.560 OK, so this is something we talked about earlier, 00:22:58.560 --> 00:23:05.370 and the Wilson lines here are capturing diagrams, 00:23:05.370 --> 00:23:08.040 as I said earlier, where you could have, 00:23:08.040 --> 00:23:10.110 say, two n collinear is one n bar, 00:23:10.110 --> 00:23:12.290 and then this guy is off shell. 00:23:12.290 --> 00:23:21.140 So that goes into a Wilson line once you match it on to the-- 00:23:21.140 --> 00:23:22.685 so this is the [? fall ?] theory. 00:23:22.685 --> 00:23:24.560 Once you match it on to the effective theory, 00:23:24.560 --> 00:23:28.053 you get various extra gluons coming out of your operator, 00:23:28.053 --> 00:23:29.720 which is coming from these Wilson lines, 00:23:29.720 --> 00:23:31.580 and it's because you're integrating off 00:23:31.580 --> 00:23:32.660 shell particles. 00:23:37.230 --> 00:23:39.290 So if you just look at the kinematics, 00:23:39.290 --> 00:23:43.520 then there's already actually some fairly powerful 00:23:43.520 --> 00:23:45.150 restrictions. 00:23:45.150 --> 00:23:47.390 We're in the center of mass frame. 00:23:47.390 --> 00:24:00.763 Momentum conservation says that if we 00:24:00.763 --> 00:24:03.180 think about all particles in the final stages either being 00:24:03.180 --> 00:24:11.520 n collinear or n bar collinear for ultra soft, 00:24:11.520 --> 00:24:16.080 then the initial momentum of the virtual photon or virtual C 00:24:16.080 --> 00:24:18.970 has to add up to all the final stage momenta. 00:24:18.970 --> 00:24:25.110 And if we just look at the large part of that, 00:24:25.110 --> 00:24:30.750 in the center of mass frame, n bar dot q is just capital Q, 00:24:30.750 --> 00:24:34.560 and that's n bar dot pxn plus things that are small. 00:24:37.270 --> 00:24:41.790 And likewise, n dot q equals capital q equals 00:24:41.790 --> 00:24:46.540 n dot pxn bar, which is large order one plus small. 00:24:46.540 --> 00:24:49.290 So these guys here are lowest-- 00:24:49.290 --> 00:24:50.790 leading order in the power counting, 00:24:50.790 --> 00:24:52.290 and these guys are suppressed. 00:24:56.380 --> 00:24:59.530 And what happens is like in the example 00:24:59.530 --> 00:25:02.470 we talked about when we were doing [? bs ?] gamma. 00:25:02.470 --> 00:25:04.060 We have these labels, omega and omega 00:25:04.060 --> 00:25:06.790 bar on our collinear fields up there. 00:25:06.790 --> 00:25:10.060 They just get fixed to the qs by momentum conservation. 00:25:23.980 --> 00:25:25.620 So I said this was generically going 00:25:25.620 --> 00:25:28.800 to happen whenever we had operators that had only one 00:25:28.800 --> 00:25:32.070 type of building block in each collinear sector, 00:25:32.070 --> 00:25:34.800 and that's exactly the situation we have. 00:25:34.800 --> 00:25:36.930 Some momentum conservation is strong enough 00:25:36.930 --> 00:25:44.040 to fix that omega is equal to q and so is omega bar. 00:25:56.370 --> 00:25:58.940 So the first thing I want to do is show you 00:25:58.940 --> 00:26:02.130 how to factorize to the cross section in this case. 00:26:02.130 --> 00:26:05.960 So what would you do in QCD for this cross section? 00:26:05.960 --> 00:26:08.690 Well, you'd say the cross section 00:26:08.690 --> 00:26:12.020 is a sum over some restricted set of final states 00:26:12.020 --> 00:26:14.660 that satisfy the kinematic criteria that I'm 00:26:14.660 --> 00:26:15.350 interested in. 00:26:19.390 --> 00:26:22.480 You'd have a momentum conserving delta function, 00:26:22.480 --> 00:26:26.350 and you could write it as sort of leptonic or-- 00:26:26.350 --> 00:26:31.270 yeah, leptonic tensor, and then kind of a hadronic tensor, 00:26:31.270 --> 00:26:40.380 the same way we do for [? DIS. ?] 00:26:40.380 --> 00:26:42.810 So you would start-- could start with that formula, which 00:26:42.810 --> 00:26:46.080 is really true for anything that you might want to measure, 00:26:46.080 --> 00:26:49.470 which is just imposed by kind of what states you include here. 00:26:54.900 --> 00:26:58.560 And I haven't yet specified exactly what 00:26:58.560 --> 00:26:59.730 I'm going to measure. 00:26:59.730 --> 00:27:02.850 We'll still have to put that in. 00:27:02.850 --> 00:27:06.270 So part of what SCET does is make these restrictions 00:27:06.270 --> 00:27:08.463 into something that shows up in the operators 00:27:08.463 --> 00:27:10.005 rather than showing up in the states. 00:27:22.678 --> 00:27:24.970 And ideally, we'd like to sort of get rid of the states 00:27:24.970 --> 00:27:28.270 eventually and be able to calculate-- 00:27:28.270 --> 00:27:29.780 see how to calculate things. 00:27:29.780 --> 00:27:35.620 So in SCET, once we know that we're in a dijet configuration, 00:27:35.620 --> 00:27:37.960 we can think of the state as having 00:27:37.960 --> 00:27:43.000 been composed of sort of pieces for the different sectors. 00:27:43.000 --> 00:27:45.580 And we know that this is an OK picture because our Lagrangian 00:27:45.580 --> 00:27:47.548 for these guys factored. 00:27:47.548 --> 00:27:49.090 And when the Lagrangian factors, that 00:27:49.090 --> 00:28:01.040 means the Hamiltonian factors and the Hilbert space factors 00:28:01.040 --> 00:28:02.936 after the field redefinition. 00:28:12.180 --> 00:28:16.092 OK, so we can put in our expansion of the current 00:28:16.092 --> 00:28:17.550 at the top of the board over there, 00:28:17.550 --> 00:28:21.640 and then put it into this formula, and see what we get. 00:28:21.640 --> 00:28:23.010 And there's some pre-factors. 00:28:27.210 --> 00:28:31.028 I'm now summing over just these states. 00:28:31.028 --> 00:28:32.820 There are still some restrictions depending 00:28:32.820 --> 00:28:35.160 on what I'm measuring, but they are different restrictions, 00:28:35.160 --> 00:28:35.993 and they're simpler. 00:28:39.088 --> 00:28:41.130 There's still momentum conserving delta function, 00:28:41.130 --> 00:28:47.160 but I can write it out as pxn pxn bar px ultra soft. 00:28:50.680 --> 00:28:52.090 And because of the states factor, 00:28:52.090 --> 00:28:56.870 I can factor also with a little bit of work the operators. 00:28:56.870 --> 00:28:59.350 So I can write the operators. 00:28:59.350 --> 00:29:00.370 I'll just write it down. 00:29:00.370 --> 00:29:02.470 I think it's clear. 00:29:05.770 --> 00:29:07.430 And I'll tell you some of the things 00:29:07.430 --> 00:29:08.597 you have to do to get there. 00:29:31.730 --> 00:29:35.740 So I am skipping some steps here because if we do it-- 00:29:35.740 --> 00:29:38.710 we can do it very carefully, and we come write out each step 00:29:38.710 --> 00:29:40.183 one at a time. 00:29:40.183 --> 00:29:42.100 But there's a lot of writing if one does that, 00:29:42.100 --> 00:29:44.920 so I'm kind of trying to write steps 00:29:44.920 --> 00:29:48.940 where I think it's intuitive what the results are 00:29:48.940 --> 00:29:52.990 and skip steps that some level can 00:29:52.990 --> 00:29:58.210 be filled in either by looking at the literature 00:29:58.210 --> 00:30:00.820 or you just believe me. 00:30:00.820 --> 00:30:04.840 So we have these operators that were still tied together 00:30:04.840 --> 00:30:07.150 in some sense because there is a color contraction 00:30:07.150 --> 00:30:09.310 of an index between them. 00:30:09.310 --> 00:30:12.460 But we can get around that basically using some-- 00:30:12.460 --> 00:30:15.670 once we realized that the matrix elements factor like this, 00:30:15.670 --> 00:30:18.230 we can use some color identities and put it in this form. 00:30:18.230 --> 00:30:21.250 So there's an overall pre-factor 1 00:30:21.250 --> 00:30:24.090 over nc, which I didn't write. 00:30:24.090 --> 00:30:25.690 There's some overall pre-factors, 00:30:25.690 --> 00:30:27.410 but the color is separate. 00:30:27.410 --> 00:30:30.460 There's a trace, if you like, over color here, 00:30:30.460 --> 00:30:34.460 and these guys here are traced into color singlets as well. 00:30:34.460 --> 00:30:37.340 Ray trace in each case. 00:30:37.340 --> 00:30:41.340 So just using some color identities, we can do that. 00:30:41.340 --> 00:30:44.028 So once-- remember the fact that the states factor 00:30:44.028 --> 00:30:46.070 in that the operators can be written in a product 00:30:46.070 --> 00:30:48.650 means that all contractions are happening between these guys 00:30:48.650 --> 00:30:49.275 and these guys. 00:30:49.275 --> 00:30:51.908 That's why I can separate the matrix elements the way I have. 00:30:51.908 --> 00:30:53.450 And the only thing is then that there 00:30:53.450 --> 00:30:55.492 could be color indices tying these guys together, 00:30:55.492 --> 00:30:58.757 but it turns out you could deal with that as well. 00:30:58.757 --> 00:31:00.340 When you deal with the color theories, 00:31:00.340 --> 00:31:03.860 you could get things like-- you could get a ta in here or-- 00:31:03.860 --> 00:31:07.260 well, you could get, for example, ta ta or something, 00:31:07.260 --> 00:31:09.440 but then there's some ways of getting 00:31:09.440 --> 00:31:14.072 rid of those terms related to the fact that this has to be-- 00:31:14.072 --> 00:31:16.820 you have to argue that certain things are color singlets. 00:31:16.820 --> 00:31:18.930 I'm not going to go through that. 00:31:18.930 --> 00:31:20.690 So there's terms that we're dropping here, 00:31:20.690 --> 00:31:24.290 and those are the other high order-- 00:31:24.290 --> 00:31:28.140 those are the other power corrections. 00:31:28.140 --> 00:31:31.440 So this keeps all the power corrections of the first type, 00:31:31.440 --> 00:31:34.830 mu s, lambda QCD over mu s because those can still 00:31:34.830 --> 00:31:37.230 be encoded in our soft function, which 00:31:37.230 --> 00:31:41.820 is going to be this thing involving the soft states. 00:31:41.820 --> 00:31:44.310 But it drops the other power corrections. 00:31:44.310 --> 00:31:46.083 AUDIENCE: How do you know that since you 00:31:46.083 --> 00:31:48.120 derived the [INAUDIBLE] by looking 00:31:48.120 --> 00:31:51.500 at the perturbative diagrams? 00:31:51.500 --> 00:31:53.945 PROFESSOR: No, I didn't. 00:31:53.945 --> 00:31:55.235 I can derive-- 00:31:55.235 --> 00:31:57.110 I don't need to look at perturbative diagrams 00:31:57.110 --> 00:31:58.540 to draw the Wilson lines. 00:31:58.540 --> 00:32:01.560 I just derived them from the SCET Lagrangian 00:32:01.560 --> 00:32:04.371 by making the field redefinition. 00:32:04.371 --> 00:32:07.260 AUDIENCE: So you're saying that because the Langrangian 00:32:07.260 --> 00:32:11.250 decoupled, that's good for all-- 00:32:11.250 --> 00:32:12.915 PROFESSOR: All soft particles. 00:32:19.230 --> 00:32:21.300 So far, we actually haven't needed 00:32:21.300 --> 00:32:23.430 to use perturbation theory, and the idea here 00:32:23.430 --> 00:32:25.680 is that we're not going to use perturbation theory. 00:32:25.680 --> 00:32:27.240 We're going to write down some formula that's 00:32:27.240 --> 00:32:28.990 true to all orders in perturbation theory, 00:32:28.990 --> 00:32:32.550 and we're really only trading the power expansion. 00:32:32.550 --> 00:32:34.860 That's what we're focusing on here 00:32:34.860 --> 00:32:37.377 without thinking about things as perturbative things. 00:32:37.377 --> 00:32:39.210 Although, obviously, some of these functions 00:32:39.210 --> 00:32:40.620 will be perturbative functions that we 00:32:40.620 --> 00:32:41.912 will end up wanting to compute. 00:32:45.480 --> 00:32:49.810 OK, so let's think about what res prime means. 00:32:49.810 --> 00:32:53.490 And it's really a reminder that this formula is only 00:32:53.490 --> 00:32:59.700 really valid if we're making a measurement 00:32:59.700 --> 00:33:03.285 on the final state that really puts us in this configuration. 00:33:08.740 --> 00:33:11.477 So it's not enough to say we are in this situation. 00:33:11.477 --> 00:33:13.810 You're going to have to measure something that makes you 00:33:13.810 --> 00:33:19.160 in that dijet configuration, and that's important to remember. 00:33:23.090 --> 00:33:25.630 And so we're going to measure the hemisphere invariant 00:33:25.630 --> 00:33:26.510 masses. 00:33:26.510 --> 00:33:28.780 So we need to cook that into our formula, 00:33:28.780 --> 00:33:31.540 and there's a very easy way of doing that. 00:33:31.540 --> 00:33:35.570 We say that the following is true. 00:33:35.570 --> 00:33:39.070 I can write one, which is equal to an integral of two 00:33:39.070 --> 00:33:39.970 delta functions. 00:33:49.590 --> 00:33:51.210 And this formula is obviously true. 00:34:00.990 --> 00:34:02.670 And then what I'm going to do is I'm 00:34:02.670 --> 00:34:04.530 going to instead of integrating this, 00:34:04.530 --> 00:34:07.162 I'm going to move it as a d sigma d these things, 00:34:07.162 --> 00:34:09.120 and that's going to leave these delta functions 00:34:09.120 --> 00:34:10.889 inside the formula here. 00:34:10.889 --> 00:34:13.800 But since they depend on collinear and soft momenta, 00:34:13.800 --> 00:34:16.830 which are the momenta that are in these states, 00:34:16.830 --> 00:34:18.840 they're going to be sort of tightly connected 00:34:18.840 --> 00:34:22.290 to the whole thing, whatever is going on in this formula. 00:34:22.290 --> 00:34:24.742 It's not like they commute with this xn. 00:34:24.742 --> 00:34:26.159 They only commute with the xn if I 00:34:26.159 --> 00:34:28.534 can make them into the identity by integrating over this. 00:34:28.534 --> 00:34:31.500 But once I pull this through and put it over 00:34:31.500 --> 00:34:32.530 on the right hand side-- 00:34:32.530 --> 00:34:35.174 left hand side, then you have to leave these delta functions 00:34:35.174 --> 00:34:35.730 there. 00:34:35.730 --> 00:34:37.439 They're specifying the measurement. 00:34:40.980 --> 00:34:47.699 So this is a total n collinear momentum, 00:34:47.699 --> 00:34:54.819 and this is the total ultra soft in hemisphere A. 00:34:54.819 --> 00:34:57.480 That's what the A means. 00:34:57.480 --> 00:35:00.840 So I divided everything into hemisphere A and B. Obviously, 00:35:00.840 --> 00:35:02.340 all the n collinear particles, which 00:35:02.340 --> 00:35:04.860 are going in that direction, are in that hemisphere, 00:35:04.860 --> 00:35:06.940 but the soft ones could go in either case. 00:35:06.940 --> 00:35:09.690 And so then there's a soft momentum in hemisphere B 00:35:09.690 --> 00:35:15.240 and soft momentum in hemisphere A. Now this is unexpanded. 00:35:15.240 --> 00:35:20.720 I can also expand these delta functions, 00:35:20.720 --> 00:35:23.418 and I only need to keep the leading order piece if we're 00:35:23.418 --> 00:35:24.460 working at leading order. 00:35:38.100 --> 00:35:40.570 And so the leading order pieces are as follows. 00:35:40.570 --> 00:35:43.410 There's an order of lambda squared piece. 00:35:43.410 --> 00:35:44.383 This is lambda squared. 00:35:44.383 --> 00:35:46.050 There's an order of lambda squared piece 00:35:46.050 --> 00:35:47.950 from the collinear momentum squared, 00:35:47.950 --> 00:35:50.220 and since this is a total collinear momentum, 00:35:50.220 --> 00:35:52.050 it can involve multiple massless particles. 00:35:52.050 --> 00:35:53.610 It doesn't have to be zero. 00:35:53.610 --> 00:35:56.040 And then there's a cross term between the softs 00:35:56.040 --> 00:35:56.790 and the collinear. 00:35:56.790 --> 00:35:58.082 It's also order lambda squared. 00:35:58.082 --> 00:36:00.440 This order one. 00:36:00.440 --> 00:36:02.850 And this is order lambda squared, 00:36:02.850 --> 00:36:04.293 and that term's the same size. 00:36:04.293 --> 00:36:06.210 And then all the other terms are higher order. 00:36:08.770 --> 00:36:11.560 These terms are suppressed, so this is really 00:36:11.560 --> 00:36:12.970 all I need to keep. 00:36:12.970 --> 00:36:17.620 There's also something else that effectively we've 00:36:17.620 --> 00:36:20.448 done by setting things up, and that is, in general, you 00:36:20.448 --> 00:36:22.990 could think that the collinear particles would have some perp 00:36:22.990 --> 00:36:24.070 momentum. 00:36:24.070 --> 00:36:27.650 But we aligned our axes with the jet axis, 00:36:27.650 --> 00:36:30.190 so there's actually no perp momentum there either. 00:36:30.190 --> 00:36:33.550 So that guy is really just-- 00:36:33.550 --> 00:36:45.620 we can pull out a pn minus and then we have pn plus k sa plus. 00:36:45.620 --> 00:36:49.970 And we also know that pn minus is q. 00:36:49.970 --> 00:36:52.100 That was fixed by kinematics. 00:36:52.100 --> 00:36:54.640 OK, so this simplifies quite a bit. 00:36:57.340 --> 00:36:59.840 And we're going to drop all the powers of [INAUDIBLE] terms. 00:37:17.620 --> 00:37:21.770 All right, so that is one thing. 00:37:21.770 --> 00:37:28.660 So then I can do what I said in words, write this guy, 00:37:28.660 --> 00:37:34.930 and it will have these two deltas that 00:37:34.930 --> 00:37:37.850 are under similar acts. 00:37:37.850 --> 00:37:40.210 They don't commute through with it-- without it. 00:37:40.210 --> 00:37:43.010 They depend on x. 00:37:43.010 --> 00:37:46.210 Now another thing we would like to do 00:37:46.210 --> 00:37:50.560 is the factorize the measurement. 00:37:50.560 --> 00:37:52.090 So this measurement here involves 00:37:52.090 --> 00:37:55.630 the sum of a plus momentum collinear and a plus momentum 00:37:55.630 --> 00:37:57.340 in soft. 00:37:57.340 --> 00:37:58.810 And it's all in one delta function. 00:37:58.810 --> 00:38:00.852 We'd like to write it as separate delta functions 00:38:00.852 --> 00:38:03.760 that we can associate with those different parts here, 00:38:03.760 --> 00:38:07.150 one delta-- we'd like to have a separate thing here 00:38:07.150 --> 00:38:08.560 from these guys. 00:38:08.560 --> 00:38:10.880 And that's actually very easy to do. 00:38:10.880 --> 00:38:16.596 We simply write the following. 00:38:27.530 --> 00:38:29.632 Just introduce some more delta functions. 00:38:38.800 --> 00:38:42.270 So there's this tying together delta function. 00:38:42.270 --> 00:38:45.960 K plus is just some dummy variable. 00:38:45.960 --> 00:38:47.580 It's not the momentum of any state. 00:38:47.580 --> 00:38:54.340 It's just a dummy variable, whereas pn plus and ksa plus 00:38:54.340 --> 00:38:57.560 were momenta of particles in the state. 00:38:57.560 --> 00:39:01.340 So this guy can then associate with the-- 00:39:01.340 --> 00:39:11.100 put them together with the n collinear matrix element, 00:39:11.100 --> 00:39:12.150 which is this guy here. 00:39:15.454 --> 00:39:17.040 I forget what color he was. 00:39:21.990 --> 00:39:27.360 And then for the ultra soft, we can 00:39:27.360 --> 00:39:29.730 associated this delta function. 00:39:32.123 --> 00:39:33.540 So that's what I mean by factoring 00:39:33.540 --> 00:39:35.332 the measurement, that we can put the piece, 00:39:35.332 --> 00:39:36.990 depended on the state, together. 00:39:36.990 --> 00:39:39.810 And then you can see that we can move our-- 00:39:39.810 --> 00:39:42.660 well, OK, we have to also factor this delta function. 00:39:42.660 --> 00:39:45.360 I should say that too. 00:39:45.360 --> 00:39:48.760 That's the part that I'm not going to go through, 00:39:48.760 --> 00:39:53.100 but we can play similar games expanding and factoring 00:39:53.100 --> 00:39:56.250 this delta four as well. 00:39:56.250 --> 00:39:59.730 And that's besides just being a little bit tedious, 00:39:59.730 --> 00:40:04.970 it's not really any more difficult. Maybe 00:40:04.970 --> 00:40:07.440 a little more difficult. 00:40:07.440 --> 00:40:08.660 So we factor that guy too. 00:40:14.920 --> 00:40:17.900 And then there's one other trick that we want to do, 00:40:17.900 --> 00:40:23.320 which is useful, and that is that we can write 00:40:23.320 --> 00:40:24.910 some deltas in Fourier's space. 00:40:37.135 --> 00:40:38.760 So we can always write a delta function 00:40:38.760 --> 00:40:42.750 as an integral over a phase, and that's convenient 00:40:42.750 --> 00:40:44.677 because in the phase, the momenta that 00:40:44.677 --> 00:40:46.260 are in the delta function also factor. 00:40:49.930 --> 00:40:52.870 So I can write it as a product of two separate factors 00:40:52.870 --> 00:40:53.410 like this. 00:40:56.002 --> 00:40:58.150 There's some halves. 00:40:58.150 --> 00:41:01.060 It's conventional. 00:41:01.060 --> 00:41:05.230 OK, so this guy here, which is the guy that 00:41:05.230 --> 00:41:07.150 is associated to the state, it looks 00:41:07.150 --> 00:41:09.250 like a translation operator. 00:41:09.250 --> 00:41:13.960 if you have an e to the i x dot p, that you can use. 00:41:13.960 --> 00:41:16.030 You can put into your matrix element, 00:41:16.030 --> 00:41:19.000 and you can translate the fields, which 00:41:19.000 --> 00:41:21.802 are in this formula all at 0. 00:41:21.802 --> 00:41:25.000 They're all at space time 0. 00:41:25.000 --> 00:41:27.370 What you can do with this guy here 00:41:27.370 --> 00:41:31.300 is you can put it into the matrix element 00:41:31.300 --> 00:41:33.790 and translate the fields to point x. 00:41:45.037 --> 00:41:49.240 And then you'd have a chi and x minus. 00:41:49.240 --> 00:41:51.910 So that's how we can deal with that guy. 00:41:51.910 --> 00:41:55.570 And so if we do a bunch of stuff like that I'm not 00:41:55.570 --> 00:41:58.870 going to go through on the board for you, after some work, 00:41:58.870 --> 00:42:02.200 we get something just kind of an intermediate step 00:42:02.200 --> 00:42:10.260 of our factorization, which I'll write out 00:42:10.260 --> 00:42:13.320 so you can get some idea of-- 00:42:13.320 --> 00:42:16.140 if I jump to the final answer, it's kind of too simple 00:42:16.140 --> 00:42:18.850 to see what's going on. 00:42:18.850 --> 00:42:20.460 So let's write this one out. 00:42:32.360 --> 00:42:36.423 So I was focusing on the n collinear matrix element, 00:42:36.423 --> 00:42:38.840 and everything I was writing, I wrote this delta function. 00:42:38.840 --> 00:42:41.540 But there's, of course, another one for the delta n bar, 00:42:41.540 --> 00:42:43.850 and I do the same thing for him. 00:42:43.850 --> 00:42:46.740 So we got k plus and l plus by factoring the measurement 00:42:46.740 --> 00:42:47.240 there. 00:42:47.240 --> 00:42:49.310 You get a k minus and an l minus by factoring 00:42:49.310 --> 00:42:50.495 the other measurement. 00:43:02.880 --> 00:43:05.880 One thing that comes into-- 00:43:05.880 --> 00:43:08.060 that makes this guy here a little more complicated 00:43:08.060 --> 00:43:10.477 is you have to think a little bit about residual and label 00:43:10.477 --> 00:43:11.420 momentum. 00:43:11.420 --> 00:43:16.640 But at the end of the day, the result is what I'm writing. 00:43:36.230 --> 00:43:43.640 So this is going to be completely factorizing to three 00:43:43.640 --> 00:43:44.495 independent things. 00:44:02.570 --> 00:44:05.900 There's one set of collinear fields for the quark, 00:44:05.900 --> 00:44:09.830 and there's another for the antiquark if you like. 00:44:09.830 --> 00:44:11.480 You're specifying whether it's a quark 00:44:11.480 --> 00:44:13.290 or an antiquark with a label. 00:44:13.290 --> 00:44:16.350 So the fact that this label here is positive 00:44:16.350 --> 00:44:20.298 means that this is a quark, and maybe this should be a minus q. 00:44:26.860 --> 00:44:31.538 And then this guy is specifying the antiquark, 00:44:31.538 --> 00:44:32.830 and then there's the soft part. 00:45:04.660 --> 00:45:10.760 OK, so this is a fairly messy one board expression, 00:45:10.760 --> 00:45:14.140 but this guy here is factored. 00:45:14.140 --> 00:45:16.570 We have the hard modes here, collinear modes, 00:45:16.570 --> 00:45:19.450 n collinear here, and bar collinear there, 00:45:19.450 --> 00:45:26.920 and the ultra soft goes here, which I've already 00:45:26.920 --> 00:45:30.100 dropped in my ultra soft. 00:45:30.100 --> 00:45:35.020 So this last guy if you think about what this matrix element 00:45:35.020 --> 00:45:39.090 is it's just some function. 00:45:39.090 --> 00:45:40.268 We sum over all x of this. 00:45:40.268 --> 00:45:42.060 So we sum over all the intermediate states. 00:45:42.060 --> 00:45:44.185 The only thing we're fixing are l plus and l minus. 00:45:48.555 --> 00:45:50.180 So when we sum over states, we actually 00:45:50.180 --> 00:45:52.990 integrate the momentum of particles in the state 00:45:52.990 --> 00:45:54.090 if you can think of that. 00:45:54.090 --> 00:45:55.010 Those are different states. 00:45:55.010 --> 00:45:57.427 You have to sum over them too, but it's a continuous label 00:45:57.427 --> 00:46:01.467 and therefore you integrate over the phase base. 00:46:01.467 --> 00:46:03.800 And that's what we mean when we write the sum over axis. 00:46:03.800 --> 00:46:06.025 That includes phase based intervals. 00:46:08.940 --> 00:46:10.890 And so the only-- so this momentum 00:46:10.890 --> 00:46:12.990 gets integrated over if you like, 00:46:12.990 --> 00:46:15.570 but then it gets-- there's a component that 00:46:15.570 --> 00:46:18.960 gets fixed, which is the total momentum of plus or minus 00:46:18.960 --> 00:46:21.055 momentum in each hemisphere. 00:46:21.055 --> 00:46:22.305 So this is just some function. 00:46:29.900 --> 00:46:34.880 So that we color the same. 00:46:34.880 --> 00:46:41.367 It's just some function of l plus or l minus, 00:46:41.367 --> 00:46:42.700 and it's called a soft function. 00:46:49.210 --> 00:46:51.220 And at this stage of the game, it's 00:46:51.220 --> 00:46:53.905 actually encoding two different momentum scales. 00:46:57.560 --> 00:47:01.600 The l plus and minus as well as sort 00:47:01.600 --> 00:47:03.790 of lambda QCD, which I didn't write explicitly 00:47:03.790 --> 00:47:04.420 as an argument. 00:47:07.060 --> 00:47:11.230 And l plus and minus, well, we'll 00:47:11.230 --> 00:47:14.260 see later, but you can think of l plus and minus 00:47:14.260 --> 00:47:19.210 roughly as encoding the scale that's m squared over q. 00:47:19.210 --> 00:47:22.180 This will become more clear in a minute. 00:47:26.510 --> 00:47:30.181 So again, we need to figure out what these guys here are. 00:47:30.181 --> 00:47:32.830 That's a little bit more work but not too much. 00:47:37.550 --> 00:47:41.270 And they're both, of course, mirrors of each other. 00:47:41.270 --> 00:47:44.560 So really, we just have to be one of them, and if you like, 00:47:44.560 --> 00:47:47.238 the other is just kind of charge congregation. 00:47:56.510 --> 00:47:59.170 So this guy here, we're going to write in Fourier's space. 00:48:01.870 --> 00:48:09.530 That's a convenient thing to do because what 00:48:09.530 --> 00:48:11.870 we know about the Feynman rules in Fourier's space 00:48:11.870 --> 00:48:13.580 when we thought about the Feynman rules 00:48:13.580 --> 00:48:17.690 in Fourier's space, we knew that the collinear propagators would 00:48:17.690 --> 00:48:20.520 only depend on the small plus momentum. 00:48:20.520 --> 00:48:23.270 So the x-coordinates here, the x-coordinates 00:48:23.270 --> 00:48:25.840 of our fields, those corresponded 00:48:25.840 --> 00:48:30.020 to residual momenta, but when we do Feynman diagrams, 00:48:30.020 --> 00:48:32.720 there's only the plus momentum showing up. 00:48:32.720 --> 00:48:34.250 That's the multipole expansion. 00:48:39.248 --> 00:48:41.540 So you could think about it from the [? phonographs, ?] 00:48:41.540 --> 00:48:44.060 but it's really a general property 00:48:44.060 --> 00:48:45.490 from the multipole expansion. 00:48:50.550 --> 00:48:54.260 So that means that some of these integrals here are just trivial 00:48:54.260 --> 00:48:55.550 and give me a delta function. 00:48:59.805 --> 00:49:02.180 So I'm getting delta functions in some directions because 00:49:02.180 --> 00:49:05.015 of the multipole expansion in this formula. 00:49:12.320 --> 00:49:14.460 Because I aligned the axis, there 00:49:14.460 --> 00:49:17.210 was no perpendicular momentum, so it's really just this r plus 00:49:17.210 --> 00:49:19.710 that we get, and this thing here is called the jet function. 00:49:24.602 --> 00:49:26.810 So this is the non-trivial function that can show up. 00:49:31.190 --> 00:49:32.300 We do the same thing. 00:49:35.190 --> 00:49:38.480 We do the same thing for the case with the sum over xn bar, 00:49:38.480 --> 00:49:42.530 and it gives us another jet function. 00:49:42.530 --> 00:49:47.390 And by charge conjugation, it's the same jet function. 00:49:47.390 --> 00:49:53.200 But in this case, it would be q of sum r prime minus 00:49:53.200 --> 00:49:55.880 and some other momentum. 00:49:55.880 --> 00:49:58.760 And so what we then do is we take this formula, 00:49:58.760 --> 00:50:03.560 plug it back into here, do all the integrals, and at the end 00:50:03.560 --> 00:50:05.330 of the day, we can actually do-- 00:50:05.330 --> 00:50:08.000 we have lots of delta functions and lots of integrals. 00:50:08.000 --> 00:50:10.770 And we can boil everything down to just two integrals left 00:50:10.770 --> 00:50:11.270 over. 00:50:16.230 --> 00:50:21.810 And that gives us our final result. So I'll write this way. 00:50:39.710 --> 00:50:42.800 So it boils down to just involving 00:50:42.800 --> 00:50:44.845 these jet functions and the soft function, 00:50:44.845 --> 00:50:46.220 and then there's something that's 00:50:46.220 --> 00:50:50.330 called the hard function, which is just our Wilson coefficient 00:50:50.330 --> 00:50:50.985 squared. 00:50:50.985 --> 00:50:52.610 So rather than write Wilson coefficient 00:50:52.610 --> 00:50:56.300 squared all the time, we just call it another function h. 00:50:56.300 --> 00:50:59.430 So this is the factorization theorem. 00:50:59.430 --> 00:51:01.790 So you see in this case, the hard function was just 00:51:01.790 --> 00:51:04.520 a multiplicative factor unlike dis, 00:51:04.520 --> 00:51:07.653 and what was-- where things are talking to each other. 00:51:07.653 --> 00:51:09.320 Again, things are talking to each other, 00:51:09.320 --> 00:51:10.760 but it's the soft modes-- 00:51:10.760 --> 00:51:13.370 ultra soft modes that are talking to the jets. 00:51:13.370 --> 00:51:16.250 And ultra soft momenta can change the mass, 00:51:16.250 --> 00:51:19.460 and that's what kind of the correct mass-- 00:51:19.460 --> 00:51:20.900 this is the total mass. 00:51:20.900 --> 00:51:23.840 The correct mass for the jet function is the collinear mass, 00:51:23.840 --> 00:51:25.637 and that's this thing minus this thing. 00:51:25.637 --> 00:51:28.220 So if you wanted to guess this formula, this is how you do it. 00:51:28.220 --> 00:51:30.800 You'd say m squared, just from the kinematic relation, 00:51:30.800 --> 00:51:34.212 m squared had a collinear piece and a soft piece. 00:51:34.212 --> 00:51:36.170 The right thing to evaluate the jet function at 00:51:36.170 --> 00:51:40.050 would be just the collinear, so that's the difference. 00:51:40.050 --> 00:51:43.340 And then the soft function could depend on these momentum, 00:51:43.340 --> 00:51:46.580 and you're basically led to a formula like this one. 00:51:46.580 --> 00:51:48.560 But we can also just, with the field theory, 00:51:48.560 --> 00:51:50.060 go through and derive it, and that's 00:51:50.060 --> 00:51:53.600 what I'm convincing you of even if I was skipping some steps. 00:51:56.860 --> 00:51:59.200 All right, so this is the dijet factorization theorem 00:51:59.200 --> 00:52:01.855 for hemisphere invariant masses. 00:52:06.880 --> 00:52:08.920 And a lot of event shapes kind of 00:52:08.920 --> 00:52:11.555 go along the pattern that we've done here. 00:52:11.555 --> 00:52:13.930 If you think about what would be the difference if I pick 00:52:13.930 --> 00:52:16.870 some other observable both sides of these masses, 00:52:16.870 --> 00:52:19.930 pick some other thing that you could measure that would tell 00:52:19.930 --> 00:52:23.350 you that there's dijets, we just swap out the measurement part, 00:52:23.350 --> 00:52:26.470 and that would lead to-- and just see where it leads us. 00:52:26.470 --> 00:52:30.068 A lot of the steps would be exactly the same steps. 00:52:30.068 --> 00:52:31.860 So we did it for one particular observable, 00:52:31.860 --> 00:52:35.260 but steps are the same if you do others. 00:52:39.820 --> 00:52:41.980 If you're in this situation where you have dijets, 00:52:41.980 --> 00:52:44.740 there's more than one way of measuring an event 00:52:44.740 --> 00:52:46.300 to ensure you have dijets. 00:52:46.300 --> 00:52:52.780 There's something called dijet event shapes. 00:52:52.780 --> 00:52:54.740 They go under the names of things like thrust, 00:52:54.740 --> 00:52:57.460 C parameter, heavy jet mass. 00:53:00.580 --> 00:53:03.423 All these things actually have factorization theorems 00:53:03.423 --> 00:53:05.590 that you could derive in a similar way to what we're 00:53:05.590 --> 00:53:06.100 doing here. 00:53:09.510 --> 00:53:12.860 All right, now this is kind of like-- 00:53:12.860 --> 00:53:14.540 I didn't put any mus in, but just 00:53:14.540 --> 00:53:17.850 like we did before, we can always put the mus in. 00:53:17.850 --> 00:53:21.050 So if we put the mus in, everybody gets a mu. 00:53:21.050 --> 00:53:24.020 The h is like the mu of the Wilson coefficient, 00:53:24.020 --> 00:53:25.460 and then the Js are like-- 00:53:25.460 --> 00:53:27.645 and Ss are like mus of-- 00:53:27.645 --> 00:53:29.270 from the point of view of SCET, they're 00:53:29.270 --> 00:53:30.593 like mus of the operator. 00:53:30.593 --> 00:53:32.510 We're just switching to renormalized operators 00:53:32.510 --> 00:53:34.494 and renormalized Wilson coefficients. 00:53:38.130 --> 00:53:42.740 Now if we draw our scale diagram again 00:53:42.740 --> 00:53:47.300 but draw it a little bigger this time 00:53:47.300 --> 00:53:49.820 and put in where these various functions are-- 00:54:00.970 --> 00:54:02.956 so somebody was blue. 00:54:11.530 --> 00:54:13.050 Not going to label everybody again. 00:54:15.710 --> 00:54:17.290 So this is the hard function. 00:54:17.290 --> 00:54:20.120 Our jet functions are kind of sitting at this scale. 00:54:20.120 --> 00:54:22.360 They had to do with the collinear modes, 00:54:22.360 --> 00:54:25.870 and then our soft function is sitting at this scale or this 00:54:25.870 --> 00:54:27.512 and this scale. 00:54:27.512 --> 00:54:29.845 So it's clear that these things sit at different scales. 00:54:36.935 --> 00:54:39.310 And what that means from the point of view of the formula 00:54:39.310 --> 00:54:44.670 is that if you associate what scale these things want 00:54:44.670 --> 00:54:47.790 to live at, which from the point of view of perturbation theory 00:54:47.790 --> 00:54:51.030 means at what scale should I expect-- would I 00:54:51.030 --> 00:54:52.680 be able to calculate these things given 00:54:52.680 --> 00:54:53.470 that they're perturbative? 00:54:53.470 --> 00:54:55.178 Would I be able to calculate these things 00:54:55.178 --> 00:54:57.150 without encountering large logarithms? 00:54:57.150 --> 00:55:00.580 And that would be these scales that I'm writing here. 00:55:00.580 --> 00:55:04.560 So these are the scales where we could 00:55:04.560 --> 00:55:07.980 do perturbative theory for the functions 00:55:07.980 --> 00:55:10.950 without encountering large logarithms. 00:55:10.950 --> 00:55:13.380 So we can figure out what these skills 00:55:13.380 --> 00:55:15.330 are by just looking at the perturbative theory 00:55:15.330 --> 00:55:16.580 and looking at the logarithms. 00:55:26.340 --> 00:55:28.340 But you see that it's a different scale for each 00:55:28.340 --> 00:55:29.480 of the different functions. 00:55:33.720 --> 00:55:36.590 So in this formula, there's a common mu, which 00:55:36.590 --> 00:55:38.570 is like a factorization scale. 00:55:38.570 --> 00:55:41.060 But each of the functions wants to live at a different mu, 00:55:41.060 --> 00:55:43.393 and we're going to have to do some renormalization group 00:55:43.393 --> 00:55:46.520 evolution to put things at kind of at the scales they 00:55:46.520 --> 00:55:47.430 want to be at. 00:55:47.430 --> 00:55:49.820 And then there'll be some resummation going on, 00:55:49.820 --> 00:55:53.610 and that's going to sum logs of m squared over q squared. 00:55:53.610 --> 00:56:05.910 So we need normalization group, and that's 00:56:05.910 --> 00:56:15.870 trying to sum up logarithms, which 00:56:15.870 --> 00:56:17.570 are ratios of these scales, which 00:56:17.570 --> 00:56:19.940 expressed in terms of some physical thing 00:56:19.940 --> 00:56:23.340 is, in this case, logs of m squared over q squared. 00:56:28.380 --> 00:56:35.767 All right, what about this situation here? 00:56:35.767 --> 00:56:36.600 Well, it turns out-- 00:56:40.510 --> 00:56:42.580 and I won't go into it in too much detail, 00:56:42.580 --> 00:56:45.130 but we can factor that guy. 00:56:45.130 --> 00:56:46.660 If these two are hierarchical, we 00:56:46.660 --> 00:56:51.940 can actually factor this in something called 00:56:51.940 --> 00:56:56.930 the soft function OPE, and even if they're not hierarchical, 00:56:56.930 --> 00:56:59.762 we can actually write it in the following way. 00:56:59.762 --> 00:57:01.720 So we can always, in some sense, whether or not 00:57:01.720 --> 00:57:03.820 we're in a situation where these are comparable 00:57:03.820 --> 00:57:05.920 or whether they're hierarchical, we can always 00:57:05.920 --> 00:57:07.900 use the following formula. 00:57:07.900 --> 00:57:13.490 We're not guaranteed that that is possible, 00:57:13.490 --> 00:57:17.050 but in this situation, it is possible to sort of make 00:57:17.050 --> 00:57:19.430 those two situations compatible with each other 00:57:19.430 --> 00:57:21.850 with a single formula. 00:57:21.850 --> 00:57:23.740 Usually, when you design effective theories, 00:57:23.740 --> 00:57:26.470 you kind of for each different expansion, 00:57:26.470 --> 00:57:29.170 you have a different result. And making them-- 00:57:29.170 --> 00:57:32.618 whether you can put them together into a single result 00:57:32.618 --> 00:57:33.660 is not always guaranteed. 00:57:33.660 --> 00:57:35.993 But this is a situation where we can put things together 00:57:35.993 --> 00:57:38.170 in a single result where these are 00:57:38.170 --> 00:57:45.740 the perturbative part of the soft function, 00:57:45.740 --> 00:57:49.625 and this is the non-perturbative lambda QCD effects. 00:57:54.110 --> 00:57:56.450 If you imagine that the function-- if you imagine 00:57:56.450 --> 00:57:59.300 that these two axes are hierarchical, 00:57:59.300 --> 00:58:04.130 then this perturbative guy here gives you some ls corrections, 00:58:04.130 --> 00:58:06.890 and it basically, in terms of the l plus, 00:58:06.890 --> 00:58:09.110 in terms of the momentum variable in this formula, 00:58:09.110 --> 00:58:12.240 it's giving something that's, like, a power-- what's called 00:58:12.240 --> 00:58:14.270 the power law tail. 00:58:14.270 --> 00:58:17.120 So the kind of dependence that you're getting in 00:58:17.120 --> 00:58:26.420 is some logs over an l plus, and that's 00:58:26.420 --> 00:58:31.730 called a power law since it goes like 1 over l plus, 00:58:31.730 --> 00:58:34.550 whereas these effects here that are non-perturbative 00:58:34.550 --> 00:58:37.310 live down at non-perturbative momenta, 00:58:37.310 --> 00:58:42.590 and you can actually prove that they have an exponential tail. 00:58:47.110 --> 00:58:51.430 So this is some plot of f in a one dimensional projection, 00:58:51.430 --> 00:58:53.980 so let's just think of it as one. 00:58:53.980 --> 00:58:56.800 And this is some scale of order lambda QCD. 00:58:56.800 --> 00:58:59.950 So at some blob down lambda QCD, this 00:58:59.950 --> 00:59:03.820 describing the distribution of soft hadrons, 00:59:03.820 --> 00:59:05.410 and it's non-perturbative. 00:59:05.410 --> 00:59:08.450 It's not like I'm drawing it because I know it. 00:59:08.450 --> 00:59:11.680 So it looks something like this, whereas this part, we 00:59:11.680 --> 00:59:13.977 can calculate and it has a different dependence. 00:59:13.977 --> 00:59:14.810 This is a power law. 00:59:14.810 --> 00:59:16.384 This is an exponential. 00:59:20.650 --> 00:59:23.290 OK, so that's kind of further factorization 00:59:23.290 --> 00:59:27.535 of the s and to get rid to make those two axis separated. 00:59:27.535 --> 00:59:29.080 Can separate the perturbative and 00:59:29.080 --> 00:59:33.190 non-perturbative corrections. 00:59:33.190 --> 00:59:34.870 In some sense, if you're in a situation 00:59:34.870 --> 00:59:39.010 where they're comparable, then whether you 00:59:39.010 --> 00:59:42.065 take a non-perturbative function and if l plus is small 00:59:42.065 --> 00:59:43.690 and you integrated against this, you're 00:59:43.690 --> 00:59:46.370 just getting back some other non-perturbative function. 00:59:46.370 --> 00:59:48.055 So this formula is not really-- 00:59:48.055 --> 00:59:50.680 there are some things that this formula is actually doing, even 00:59:50.680 --> 00:59:53.200 in that situation, is ensuring that you're in the ms bar 00:59:53.200 --> 00:59:53.992 scheme for example. 00:59:57.130 --> 01:00:02.230 But the sort of more important in some ways 01:00:02.230 --> 01:00:04.360 is when they're hierarchical, and then you 01:00:04.360 --> 01:00:06.850 can start to expand this thing and-- 01:00:06.850 --> 01:00:08.620 because in that case, this l plus prime 01:00:08.620 --> 01:00:11.680 would be localized down here, and the l plus-- if the l plus 01:00:11.680 --> 01:00:14.140 got big, then you can start doing a Taylor series, 01:00:14.140 --> 01:00:16.720 and this formula would tell you what the-- how 01:00:16.720 --> 01:00:18.280 to put the corrections together. 01:00:24.160 --> 01:00:26.700 So there's kind of a lot of physics in that formula 01:00:26.700 --> 01:00:28.760 that I'm not going into in detail. 01:00:32.010 --> 01:00:32.715 Yeah. 01:00:32.715 --> 01:00:37.440 AUDIENCE: So I don't really totally understand 01:00:37.440 --> 01:00:39.960 why this is valid. 01:00:39.960 --> 01:00:42.960 Are you allowed to-- does the soft function code all those 01:00:42.960 --> 01:00:44.750 below that hyperbola? 01:00:44.750 --> 01:00:45.510 PROFESSOR: Right. 01:00:45.510 --> 01:00:46.302 AUDIENCE: Is that-- 01:00:46.302 --> 01:00:47.660 PROFESSOR: That's right. 01:00:47.660 --> 01:00:49.330 Below the above one and-- 01:00:49.330 --> 01:00:50.600 AUDIENCE: Yeah, below the s. 01:00:50.600 --> 01:00:51.950 PROFESSOR: Yeah, that's right. 01:00:51.950 --> 01:00:52.492 That's right. 01:00:52.492 --> 01:00:53.360 The soft function. 01:00:53.360 --> 01:00:55.310 You should always think of these things 01:00:55.310 --> 01:01:01.680 as kind of extending down to the axes in some kind of fashion. 01:01:01.680 --> 01:01:02.900 So let's drop like this. 01:01:05.810 --> 01:01:08.568 Since they're infrared modes and the infrared is sort of 01:01:08.568 --> 01:01:10.610 down at the axis, you should always think of them 01:01:10.610 --> 01:01:11.990 as extending down to the axis. 01:01:11.990 --> 01:01:18.500 Now exactly how you want to think about how the soft modes 01:01:18.500 --> 01:01:22.470 and whether the soft modes are capturing the entire axis 01:01:22.470 --> 01:01:24.030 or the collinear-- how the collinear 01:01:24.030 --> 01:01:27.630 modes and the soft mouths kind of what the edges are 01:01:27.630 --> 01:01:29.070 is related to the power counting. 01:01:29.070 --> 01:01:31.080 And you can think of the collinear modes 01:01:31.080 --> 01:01:33.510 as kind of being held away from the axis 01:01:33.510 --> 01:01:36.270 because the soft modes are really deeper in the infrared. 01:01:36.270 --> 01:01:40.050 AUDIENCE: But at higher power, are you going to screw these up 01:01:40.050 --> 01:01:44.550 if you say-- like suppose the soft modes have an invariant 01:01:44.550 --> 01:01:48.217 mass of lambda to the fifth, right, or they're-- 01:01:48.217 --> 01:01:49.050 PROFESSOR: Oh, yeah. 01:01:49.050 --> 01:01:53.400 No, that's-- you never have to worry about-- 01:01:53.400 --> 01:01:56.500 you could worry about putting in more hyperbolas, right, 01:01:56.500 --> 01:01:58.535 with lambda QCD over q or something. 01:01:58.535 --> 01:01:59.160 AUDIENCE: Yeah. 01:01:59.160 --> 01:02:00.210 PROFESSOR: And you don't have to do that. 01:02:00.210 --> 01:02:00.930 AUDIENCE: OK, so-- 01:02:00.930 --> 01:02:01.980 PROFESSOR: You never have to do that. 01:02:01.980 --> 01:02:02.940 AUDIENCE: [INAUDIBLE]. 01:02:02.940 --> 01:02:03.840 Why is that? 01:02:03.840 --> 01:02:06.750 PROFESSOR: I mean, essentially, that's because those-- 01:02:06.750 --> 01:02:09.540 any lambda QCDs, what this x means 01:02:09.540 --> 01:02:13.140 is that lambda QCD is encoded in the soft-- in this kind 01:02:13.140 --> 01:02:14.910 of non-perturbative soft mode. 01:02:14.910 --> 01:02:19.830 And if you had something that at higher powers like lambda QCD, 01:02:19.830 --> 01:02:22.560 to take your example, to the sixth, 01:02:22.560 --> 01:02:26.010 the one over q to the sixth is going to come out 01:02:26.010 --> 01:02:27.540 in some coefficient. 01:02:27.540 --> 01:02:29.488 And the lambda QCD to the sixth is just 01:02:29.488 --> 01:02:31.530 going to be some matrix element of this mode that 01:02:31.530 --> 01:02:33.810 has that dimension. 01:02:33.810 --> 01:02:36.980 That's how it's going to work. 01:02:36.980 --> 01:02:39.057 So in some sense, you could say, well, 01:02:39.057 --> 01:02:40.390 what if I put a mode down there? 01:02:40.390 --> 01:02:42.340 And I would just say, well, that mode's inside this mode. 01:02:42.340 --> 01:02:44.757 You don't have to write down something different for that. 01:02:44.757 --> 01:02:47.590 It's all encoded in the Lagrangian for that mode, 01:02:47.590 --> 01:02:49.735 and that mode has all the non-perturbative physics. 01:02:57.313 --> 01:02:58.730 AUDIENCE: Yeah, at a higher power, 01:02:58.730 --> 01:03:00.365 I would think you're comparing-- 01:03:00.365 --> 01:03:01.470 PROFESSOR: So this is-- 01:03:01.470 --> 01:03:01.980 AUDIENCE: --two powers by-- 01:03:01.980 --> 01:03:03.205 PROFESSOR: Yeah, this is a good question. 01:03:03.205 --> 01:03:03.660 AUDIENCE: [INAUDIBLE] 01:03:03.660 --> 01:03:04.460 PROFESSOR: Right, so-- 01:03:04.460 --> 01:03:05.555 AUDIENCE: [INAUDIBLE] actually significantly much smaller. 01:03:05.555 --> 01:03:09.170 PROFESSOR: Yeah, so one way of thinking about it is like this. 01:03:09.170 --> 01:03:10.880 When you design the effective theory, 01:03:10.880 --> 01:03:13.338 you actually have to figure out what the degrees of freedom 01:03:13.338 --> 01:03:14.790 are at lowest order. 01:03:14.790 --> 01:03:16.490 And when you go to higher order, you 01:03:16.490 --> 01:03:19.590 shouldn't be having new degrees of freedom popping up at you. 01:03:19.590 --> 01:03:23.900 Yeah, and so that's kind of at the heart of what 01:03:23.900 --> 01:03:24.890 I'm saying here. 01:03:24.890 --> 01:03:29.927 That's a kind of an even deeper way of saying it 01:03:29.927 --> 01:03:32.510 because what you're doing with the power connections is you're 01:03:32.510 --> 01:03:34.882 kind of figuring out how to put factors in the numerator 01:03:34.882 --> 01:03:36.590 if you want to think in Feynman diagrams. 01:03:36.590 --> 01:03:37.880 You're not changing the propagators. 01:03:37.880 --> 01:03:38.920 You've already figured out the propagators 01:03:38.920 --> 01:03:40.860 with your leading order theory. 01:03:40.860 --> 01:03:43.610 So you have sort of where all the poles can occur, 01:03:43.610 --> 01:03:45.890 and now you're just sort of doing perturbation theory. 01:03:45.890 --> 01:03:47.515 Perturbation theory shouldn't introduce 01:03:47.515 --> 01:03:50.030 new degrees of freedom in the-- 01:03:50.030 --> 01:03:54.510 when you're expanding in the power expansion. 01:03:54.510 --> 01:03:58.778 All right, so to draw a picture, I 01:03:58.778 --> 01:04:01.320 want to tell you about one other observable, which is thrust. 01:04:12.480 --> 01:04:16.080 So thrust can be defined by this formula 01:04:16.080 --> 01:04:19.530 here in general, which is that you 01:04:19.530 --> 01:04:21.330 find some axis, which is actually 01:04:21.330 --> 01:04:22.440 an axis that's going to-- 01:04:22.440 --> 01:04:26.595 in our dijet configuration in the jet axis. 01:04:29.580 --> 01:04:32.470 You find the axis that maximizes this thing, 01:04:32.470 --> 01:04:34.710 and in the case of a dijet configuration, 01:04:34.710 --> 01:04:36.990 that's going to align with the jet axis. 01:04:36.990 --> 01:04:40.050 And then you do this sum, and that calculate some variable t. 01:04:43.770 --> 01:04:47.400 T turns out to have kinematic limits between a half and one, 01:04:47.400 --> 01:04:50.463 and if you want to talk about something that's 01:04:50.463 --> 01:04:52.380 a little more convenient, you define something 01:04:52.380 --> 01:04:55.140 called tau, which is 1 minus t. 01:04:55.140 --> 01:04:59.850 And the dijet limit of tau is zero. 01:04:59.850 --> 01:05:01.120 That's why this is more. 01:05:01.120 --> 01:05:02.940 So the dijet limit of t is the limit 01:05:02.940 --> 01:05:06.250 where you just have kind of two pencil-like back-to-back 01:05:06.250 --> 01:05:06.750 things. 01:05:06.750 --> 01:05:08.610 In that case, this goes to one. 01:05:08.610 --> 01:05:10.860 And if you wanted to find something that goes to zero, 01:05:10.860 --> 01:05:12.870 then you define it as tau. 01:05:12.870 --> 01:05:16.240 Just a little more convenient. 01:05:16.240 --> 01:05:21.040 And so it's possible to work out that when you're 01:05:21.040 --> 01:05:26.320 in a dijet configuration that tau with a bit of work 01:05:26.320 --> 01:05:30.010 you can show it's actually simply the sum of our two 01:05:30.010 --> 01:05:32.300 hemisphere invariant masses. 01:05:32.300 --> 01:05:34.270 So obviously, in a dijet configuration, 01:05:34.270 --> 01:05:36.880 there's lots of simplifications to the kinematics. 01:05:36.880 --> 01:05:40.700 And with some work, one can show that that's true. 01:05:40.700 --> 01:05:42.850 So what this is an observable, which is simply 01:05:42.850 --> 01:05:46.300 a symmetric projection of our previous two variable 01:05:46.300 --> 01:05:48.730 observable onto one variable observable. 01:05:48.730 --> 01:05:51.070 But if we demand that, for example, tau is small, 01:05:51.070 --> 01:05:53.563 that's demanding both m squared and n bar squared 01:05:53.563 --> 01:05:55.480 are small because they're positive quantities, 01:05:55.480 --> 01:05:56.170 and we're just summing. 01:05:56.170 --> 01:05:58.212 So demanding that this is small is demanding both 01:05:58.212 --> 01:06:00.230 of those variables is small. 01:06:00.230 --> 01:06:06.670 So tau much less than one also is a way of getting dijets, 01:06:06.670 --> 01:06:08.743 and that's a way of getting dijets just 01:06:08.743 --> 01:06:10.160 with one variable rather than two. 01:06:10.160 --> 01:06:12.100 So that's, of course, easier to draw a picture 01:06:12.100 --> 01:06:14.890 with one thing rather than two. 01:06:14.890 --> 01:06:17.080 So if we take our formula for the two variables 01:06:17.080 --> 01:06:26.176 and we project onto one, then we can do that. 01:06:26.176 --> 01:06:27.520 It looks like this. 01:06:41.060 --> 01:06:43.600 So what happens to the jet-- the two jet functions 01:06:43.600 --> 01:06:45.490 and the soft function that had two variables 01:06:45.490 --> 01:06:47.290 is that they get projected kind of 01:06:47.290 --> 01:06:50.780 on to their symmetric projection of the two variables. 01:06:50.780 --> 01:06:52.810 So this is like-- 01:06:52.810 --> 01:06:55.120 these two are symmetric projections. 01:07:05.510 --> 01:07:09.870 So we had two variables before, and now we have one variable. 01:07:09.870 --> 01:07:11.940 And it's a symmetric projection, and that just 01:07:11.940 --> 01:07:14.015 follows from this formula. 01:07:14.015 --> 01:07:15.390 So literally, to get this result, 01:07:15.390 --> 01:07:17.790 we would just do the same trick we did before. 01:07:25.060 --> 01:07:26.570 We'd say that this is true. 01:07:26.570 --> 01:07:29.080 Now let's move the integral d m squared 01:07:29.080 --> 01:07:31.607 and d n bar squared onto the right hand side 01:07:31.607 --> 01:07:33.190 and think about doing those integrals. 01:07:33.190 --> 01:07:35.040 And we follow through, and it would lead 01:07:35.040 --> 01:07:37.004 to a formula like this one. 01:07:37.004 --> 01:07:38.870 AUDIENCE: J tau is j squared? 01:07:38.870 --> 01:07:41.800 PROFESSOR: J tau is j squared. 01:07:41.800 --> 01:07:47.270 J tau is literally j j integrated 01:07:47.270 --> 01:07:50.390 over so fixing kind some of the momentum 01:07:50.390 --> 01:07:53.111 and integrating over the difference. 01:07:53.111 --> 01:07:55.220 So this guy depends on one momentum, 01:07:55.220 --> 01:07:59.690 and this guy-- these guys depend on 2, and there's one integral. 01:07:59.690 --> 01:08:03.560 And then s tau is like one integral 01:08:03.560 --> 01:08:08.240 of kind of something like this. 01:08:08.240 --> 01:08:09.940 So let's call the integral l prime. 01:08:09.940 --> 01:08:15.420 It's kind of like that. 01:08:19.270 --> 01:08:21.271 So that's what I mean by asymmetric projection. 01:08:21.271 --> 01:08:22.979 AUDIENCE: And so how do you know that tau 01:08:22.979 --> 01:08:26.410 can be written as a function n bar, 01:08:26.410 --> 01:08:30.220 or could another kinematic variable [INAUDIBLE]?? 01:08:30.220 --> 01:08:32.800 PROFESSOR: No, it can be written as a function of n bar. 01:08:32.800 --> 01:08:35.859 Yeah, because you're still-- if you're in subleading power, 01:08:35.859 --> 01:08:37.865 as long as you're in the dijet's configuration, 01:08:37.865 --> 01:08:39.490 we've got the right degrees of freedom. 01:08:39.490 --> 01:08:41.670 We don't need another jet degree of freedom. 01:08:44.300 --> 01:08:46.090 So if you want to describe-- 01:08:46.090 --> 01:08:48.970 you could think about the thrust distribution. 01:08:48.970 --> 01:08:51.490 Well, let me answer your question in more detail. 01:08:51.490 --> 01:08:52.990 Let me tell you where you would have 01:08:52.990 --> 01:08:55.870 to worry about more degrees of freedom, 01:08:55.870 --> 01:08:59.000 but let me do it in a second. 01:08:59.000 --> 01:09:16.660 I'll give you-- all right, so in the thrust case 01:09:16.660 --> 01:09:18.760 and what we said about masses carries over 01:09:18.760 --> 01:09:24.830 to a formula like this, where this mu h squared 01:09:24.830 --> 01:09:30.580 being much bigger than mu j squared, much bigger 01:09:30.580 --> 01:09:31.899 than mu s squared. 01:09:31.899 --> 01:09:39.790 And then that could be much greater than or of order lambda 01:09:39.790 --> 01:09:41.050 QCD squared. 01:09:41.050 --> 01:09:45.100 And the type of terms that our factorization theorem would 01:09:45.100 --> 01:09:49.300 be able to resum if we did the renormalization group evolution 01:09:49.300 --> 01:09:54.340 would be sums of powers of alpha and logs of tau. 01:09:54.340 --> 01:09:57.490 Tau is dimensionless variable, and so the kind 01:09:57.490 --> 01:10:00.370 of terms that would show up in the cross section 01:10:00.370 --> 01:10:05.245 are terms like that plus non-perturbative effects. 01:10:08.980 --> 01:10:11.700 So if you want to think the base level, what 01:10:11.700 --> 01:10:15.300 is inside our formula, it's all these terms. 01:10:15.300 --> 01:10:18.000 We're able to determine them from our factorization theorem, 01:10:18.000 --> 01:10:20.334 and then there's also some non-perturbative, 01:10:20.334 --> 01:10:23.730 which would be kind of-- and non-perturbative effects that 01:10:23.730 --> 01:10:26.388 in this f function. 01:10:26.388 --> 01:10:28.380 And the factorization theorem is telling us 01:10:28.380 --> 01:10:32.350 how to compute these things and include them. 01:10:32.350 --> 01:10:35.367 So what does it look like as a picture? 01:10:35.367 --> 01:10:36.950 What does the cross-section look like? 01:10:39.630 --> 01:10:48.600 So there's a peak as I promised, and there's kind of-- 01:10:48.600 --> 01:10:49.650 and then there's a tail. 01:10:54.030 --> 01:10:59.670 And so this is tau. 01:10:59.670 --> 01:11:02.670 This is d sigma d tau. 01:11:02.670 --> 01:11:06.083 You can always normalize. 01:11:06.083 --> 01:11:07.500 This peak is happening in the kind 01:11:07.500 --> 01:11:08.670 of non-perturbative region. 01:11:08.670 --> 01:11:14.225 So because it's a symmetric projection, 01:11:14.225 --> 01:11:15.600 there's a factor of two, but it's 01:11:15.600 --> 01:11:18.510 kind of basically happening when taus of order lambda 01:11:18.510 --> 01:11:19.440 QCD over q. 01:11:19.440 --> 01:11:22.630 And we could figure out that just by power counting. 01:11:22.630 --> 01:11:26.520 And then depending on the value of cue, 01:11:26.520 --> 01:11:31.050 that sort of for a typical configuration, tau 01:11:31.050 --> 01:11:32.520 equals 0.1 would be over somewhere 01:11:32.520 --> 01:11:36.210 over there, et cetera. 01:11:36.210 --> 01:11:39.650 So this is the peak, aptly named because it looks like a peak. 01:11:39.650 --> 01:11:44.120 And in this peak, there's non-perturbative effects 01:11:44.120 --> 01:11:46.820 that come from the f. 01:11:46.820 --> 01:11:49.270 This is the tail. 01:11:49.270 --> 01:11:53.270 It could be called the shoulder, but it's 01:11:53.270 --> 01:11:56.180 kind of once you're out here and down here, 01:11:56.180 --> 01:11:58.280 this is what you call the tail where 01:11:58.280 --> 01:12:01.670 you have a perturbative s tau. 01:12:01.670 --> 01:12:07.880 And then you would include power corrections by expanding. 01:12:07.880 --> 01:12:11.280 And the expansion would give you lambda QCD over the soft scale, 01:12:11.280 --> 01:12:13.385 which is q tau. 01:12:13.385 --> 01:12:19.430 So this is-- that's that expansion that we talked about, 01:12:19.430 --> 01:12:22.864 and that comes from expanding it, 01:12:22.864 --> 01:12:28.862 f, or an expansion that involves f. 01:12:28.862 --> 01:12:30.530 It's not expanding f. 01:12:34.460 --> 01:12:35.960 Basically, what these numerators are 01:12:35.960 --> 01:12:39.830 are moments of f once you do the expansion. 01:12:39.830 --> 01:12:41.930 And then there's a region out here, 01:12:41.930 --> 01:12:45.110 and in both of these regions, you have dijets. 01:12:45.110 --> 01:12:47.750 But in this region out here, you don't have dijets anymore. 01:12:47.750 --> 01:12:50.235 That's where tau is starting to get large 01:12:50.235 --> 01:12:51.485 and you no longer have dijets. 01:12:57.320 --> 01:12:59.690 So if you wanted to think about sort of higher order 01:12:59.690 --> 01:13:02.480 corrections in the distribution out here, 01:13:02.480 --> 01:13:05.030 then you would need to perhaps think about three jets 01:13:05.030 --> 01:13:06.230 and other things. 01:13:06.230 --> 01:13:08.480 In particular, there's also a kind of shoulder 01:13:08.480 --> 01:13:10.260 here that if you really want to think 01:13:10.260 --> 01:13:12.260 about how power corrections are kind of properly 01:13:12.260 --> 01:13:17.930 dealing with that region, you think about more than two jets. 01:13:17.930 --> 01:13:21.390 But if you just kind of restrict yourself to here and here, 01:13:21.390 --> 01:13:25.222 which is all we're going to do, then you don't-- then every 01:13:25.222 --> 01:13:26.930 mode, even if you go to subleading order, 01:13:26.930 --> 01:13:28.460 is exactly the ones that we just said. 01:13:28.460 --> 01:13:31.002 And you just have to deal with them by constructing operators 01:13:31.002 --> 01:13:33.320 in subleading order. 01:13:33.320 --> 01:13:38.430 So that's the more detailed answer to your question. 01:13:38.430 --> 01:13:39.852 Any other questions? 01:13:43.710 --> 01:13:50.490 OK, so I'll say a few things about the perturbation theory. 01:13:50.490 --> 01:13:53.400 So so far we've kind of argued that we 01:13:53.400 --> 01:13:56.610 could get pretty far without doing perturbation theory. 01:13:56.610 --> 01:13:59.640 So we got here without doing perturbation theory. 01:13:59.640 --> 01:14:02.670 What if we actually now want to calculate these series, 01:14:02.670 --> 01:14:05.258 do the normalization group evolution? 01:14:05.258 --> 01:14:06.300 How's that going to work? 01:14:10.970 --> 01:14:13.360 And it's actually-- in some ways, 01:14:13.360 --> 01:14:17.350 it's a combination of the two examples that we treated. 01:14:17.350 --> 01:14:20.080 We did an example where we had a Wilson coefficient that 01:14:20.080 --> 01:14:21.520 was just running multiplicatively, 01:14:21.520 --> 01:14:22.990 and we did another example where we 01:14:22.990 --> 01:14:25.642 had a function for deep and elastic scattering. 01:14:25.642 --> 01:14:27.100 It had an integral, and we're going 01:14:27.100 --> 01:14:29.620 to have both of those situations in this case. 01:14:34.370 --> 01:14:37.030 So let me tell you how it works. 01:14:37.030 --> 01:14:41.500 So to get the hard function, you would do some matching, 01:14:41.500 --> 01:14:44.470 and the matching is actually for what people call the quark form 01:14:44.470 --> 01:14:46.578 factor. 01:14:46.578 --> 01:14:49.120 So if you wonder why people care about the quark form factor, 01:14:49.120 --> 01:14:50.830 the quark form factor is basically 01:14:50.830 --> 01:14:54.040 the h in our formula, not exactly, 01:14:54.040 --> 01:14:56.430 but closely related to the h in our formula. 01:15:00.470 --> 01:15:06.640 So you would do some one loop matching by taking QCD graphs 01:15:06.640 --> 01:15:10.030 and subtracting the SCET graphs in exactly the kind of way 01:15:10.030 --> 01:15:13.482 we were already doing for the examples 01:15:13.482 --> 01:15:14.440 that we treated before. 01:15:24.030 --> 01:15:25.430 So those are collinear gluons. 01:15:25.430 --> 01:15:26.690 This is an ultra soft gluon. 01:15:29.650 --> 01:15:33.290 And there's some wave function graphs that you can consider. 01:15:33.290 --> 01:15:35.180 We form the difference. 01:15:35.180 --> 01:15:38.390 And at one loop, we check that all the IR divergences 01:15:38.390 --> 01:15:41.330 between this and this cancel, and then we 01:15:41.330 --> 01:15:42.925 get the Wilson coefficient. 01:15:48.270 --> 01:15:51.230 So this is very, very analogous to the example 01:15:51.230 --> 01:15:53.810 that we did when we were doing a heavy light current. 01:16:12.180 --> 01:16:15.870 And you see here that this thing, which depends on mu, 01:16:15.870 --> 01:16:18.060 it's mu over q. 01:16:18.060 --> 01:16:20.100 And that's what I was saying that the Wilson 01:16:20.100 --> 01:16:22.050 coefficient or the hard function should be-- 01:16:22.050 --> 01:16:25.996 the log should be minimized from mu of order q. 01:16:25.996 --> 01:16:27.870 And we see that by doing the calculation. 01:16:32.910 --> 01:16:36.600 H is the square of this thing. 01:16:36.600 --> 01:16:39.390 So there's imaginary parts on these negative logarithms. 01:16:39.390 --> 01:16:41.640 These are minus q squared minus i 0. 01:16:41.640 --> 01:16:43.860 But when I take the mod square, this thing is real. 01:16:50.850 --> 01:16:53.550 What about the jet function? 01:16:53.550 --> 01:16:56.280 Well, if you go back to what the formula for the jet function 01:16:56.280 --> 01:17:00.360 was, the jet function is basically 01:17:00.360 --> 01:17:06.090 two fields like spit out a quark and absorb a quark. 01:17:06.090 --> 01:17:07.760 But it's a vacuum matrix element, 01:17:07.760 --> 01:17:10.207 so the quarks are just contracted. 01:17:10.207 --> 01:17:11.790 So you're basically calculating graphs 01:17:11.790 --> 01:17:14.290 like this or, really, actually the imaginary parts of graphs 01:17:14.290 --> 01:17:16.092 like this because the way I drew it, 01:17:16.092 --> 01:17:17.550 you were summing over final states, 01:17:17.550 --> 01:17:20.770 so it would be the imaginary part. 01:17:20.770 --> 01:17:24.210 So we could just calculate the graphs without putting a cut in 01:17:24.210 --> 01:17:26.430 and take the imaginary part, and that would actually 01:17:26.430 --> 01:17:28.700 be giving us the jet function. 01:17:28.700 --> 01:17:32.760 So the jet function comes from some Feynman diagrams 01:17:32.760 --> 01:17:33.870 that look like this. 01:17:37.980 --> 01:17:44.480 And then there's one more looks like that. 01:17:44.480 --> 01:17:46.850 So at one move, we have those three Feynman diagrams. 01:17:46.850 --> 01:17:48.110 And then we have the [INAUDIBLE] little guy. 01:17:48.110 --> 01:17:49.100 We take the imaginary part. 01:17:49.100 --> 01:17:50.100 We get the jet function. 01:17:56.810 --> 01:17:58.850 At lowest order, it gives a delta function 01:17:58.850 --> 01:18:00.020 just to cut propagator. 01:18:07.020 --> 01:18:09.390 And then at one [INAUDIBLE],, you get a delta function, 01:18:09.390 --> 01:18:10.860 and you also get plus functions. 01:18:17.420 --> 01:18:24.640 That may not worry so much about what the numbers are. 01:18:24.640 --> 01:18:28.015 I can just tell you what the result looks like. 01:18:28.015 --> 01:18:29.170 It looks like that. 01:18:29.170 --> 01:18:32.028 There's three different types of terms that we could get. 01:18:32.028 --> 01:18:33.820 They all kind of have a power counting that 01:18:33.820 --> 01:18:35.517 makes them go like one over s. 01:18:35.517 --> 01:18:36.850 And that you know ahead of time. 01:18:36.850 --> 01:18:39.760 If you power count the operator here, 01:18:39.760 --> 01:18:40.990 it should scale one over s. 01:18:40.990 --> 01:18:42.740 These are the different kind of structures 01:18:42.740 --> 01:18:45.317 that you can get at one loop that scale one over s. 01:18:45.317 --> 01:18:46.900 And this is kind of a symptom of there 01:18:46.900 --> 01:18:49.275 being 1 over epsilon squared divergences, 01:18:49.275 --> 01:18:50.650 and this guy here are [INAUDIBLE] 01:18:50.650 --> 01:18:51.692 the renormalized results. 01:18:51.692 --> 01:18:55.490 So we're taking care of the renormalization. 01:18:55.490 --> 01:18:57.380 So just like in our example-- 01:18:57.380 --> 01:18:59.450 I mean, it's the same diagram really. 01:18:59.450 --> 01:19:02.420 Just this diagram was familiar because this diagram 01:19:02.420 --> 01:19:04.130 was showing up in beta s gamma, right? 01:19:04.130 --> 01:19:06.800 So we saw it had 1 over epsilon squared poles. 01:19:06.800 --> 01:19:10.862 Here, the 1 over epsilon squared poles lead to this. 01:19:10.862 --> 01:19:12.320 And, really, you can actually think 01:19:12.320 --> 01:19:13.820 of that very closely related to what 01:19:13.820 --> 01:19:16.970 we did because we were finding logs of mu squared over p 01:19:16.970 --> 01:19:20.040 squared, but here, p squared is a physical thing. 01:19:20.040 --> 01:19:23.780 It's the s, the invariant mass that we pump into the operator. 01:19:23.780 --> 01:19:28.810 S is kind of what we put in through the-- 01:19:28.810 --> 01:19:33.830 we put in a momentum, if you like, q, where-- 01:19:33.830 --> 01:19:34.820 I shouldn't call it q-- 01:19:37.622 --> 01:19:41.990 t mu where s is t squared, right? 01:19:41.990 --> 01:19:45.890 And so before we were having logs of p squared, which 01:19:45.890 --> 01:19:48.740 were an IR regulator, but in this calculation for this jet 01:19:48.740 --> 01:19:50.990 function, it's actually a physical thing. 01:19:50.990 --> 01:19:54.840 And it's giving the momentum dependents of the jet function, 01:19:54.840 --> 01:19:58.400 but it's the right thing to stick into the factorization 01:19:58.400 --> 01:20:00.200 theorem. 01:20:00.200 --> 01:20:02.990 And then there's the soft function 01:20:02.990 --> 01:20:08.180 where if it's perturbative, you can calculate it. 01:20:08.180 --> 01:20:12.320 And you can draw these kind of in some notation for the Wilson 01:20:12.320 --> 01:20:13.800 lines. 01:20:13.800 --> 01:20:21.680 So here's our Wilson lines in different directions. 01:20:21.680 --> 01:20:23.970 And then want this matrix element squared, 01:20:23.970 --> 01:20:26.060 and, again, you can sort of think 01:20:26.060 --> 01:20:30.590 of as kind of cut graphs like that if you like. 01:20:30.590 --> 01:20:35.540 And if we look at the soft function, 01:20:35.540 --> 01:20:39.720 it kind of has a similar structure to the jet function 01:20:39.720 --> 01:20:43.200 but now with the-- 01:20:43.200 --> 01:20:46.730 so, again, it's got a delta function and then 01:20:46.730 --> 01:20:47.660 plus functions. 01:20:55.541 --> 01:20:56.041 Whoops. 01:21:00.500 --> 01:21:03.830 And it turns out there's no single plus function, 01:21:03.830 --> 01:21:09.470 but there is a plus function with a logarithm in this case. 01:21:09.470 --> 01:21:17.920 And then same for l minus. 01:21:17.920 --> 01:21:19.480 Same structure. 01:21:19.480 --> 01:21:21.560 It's just a product if you like. 01:21:21.560 --> 01:21:24.020 And the reason that happens is if you only have one gluon, 01:21:24.020 --> 01:21:26.030 it's either in hemisphere A or hemisphere 01:21:26.030 --> 01:21:27.900 B. It can't be in both. 01:21:27.900 --> 01:21:30.500 So the alpha s corrections are either a function of l plus 01:21:30.500 --> 01:21:32.540 or a function of l minus, and that's 01:21:32.540 --> 01:21:34.860 why it has a kind of very simple structure at one loop. 01:21:52.235 --> 01:21:53.610 So that gives you an idea of what 01:21:53.610 --> 01:21:55.980 these perturbative functions look like. 01:21:55.980 --> 01:21:58.380 C, if you talk about renormalization, 01:21:58.380 --> 01:22:01.067 C renormalizes multiplicatively. 01:22:05.050 --> 01:22:08.320 And so the renormalization group equation for C 01:22:08.320 --> 01:22:13.320 is just like the one we had before for beta s gamma. 01:22:17.600 --> 01:22:19.100 There's no integrals. 01:22:19.100 --> 01:22:20.900 That, again, came about from the kinematics 01:22:20.900 --> 01:22:22.970 fixing the variables. 01:22:22.970 --> 01:22:30.950 But the jet function and the soft function 01:22:30.950 --> 01:22:34.310 have convolutions in this case. 01:22:41.232 --> 01:22:43.190 Well, they depend on this non-trivial momentum, 01:22:43.190 --> 01:22:45.170 and it's-- 01:22:45.170 --> 01:22:48.620 you can see in the factorization theorem, that is convoluting 01:22:48.620 --> 01:22:49.880 between two different sectors. 01:22:49.880 --> 01:22:52.310 And it kind of generically a hint 01:22:52.310 --> 01:22:57.332 that you're going to get out a formula like this one, which 01:22:57.332 --> 01:23:01.140 is like the PDF, but now it's a different formula. 01:23:01.140 --> 01:23:05.040 It was kind of an almost dimension for the jet function. 01:23:09.350 --> 01:23:11.620 So we could go through that, but I was not 01:23:11.620 --> 01:23:14.792 writing down for you what the 1 over epsilons look like. 01:23:14.792 --> 01:23:18.940 But we could go through the renormalization 01:23:18.940 --> 01:23:20.310 and find these results. 01:23:31.520 --> 01:23:33.470 And actually, in this jet function case, 01:23:33.470 --> 01:23:36.140 we even know more. 01:23:36.140 --> 01:23:39.680 The general structure of this anomalous dimension 01:23:39.680 --> 01:23:42.350 is actually simpler than the Parton distribution case. 01:23:46.370 --> 01:23:49.795 And it's the following. 01:23:56.590 --> 01:24:00.155 There's two types of terms that can show up. 01:24:00.155 --> 01:24:02.280 So the general structure of the anomalous dimension 01:24:02.280 --> 01:24:05.700 is that there's a single plus function 01:24:05.700 --> 01:24:07.710 in it or a delta function. 01:24:07.710 --> 01:24:10.620 And this single plus function is the analog 01:24:10.620 --> 01:24:13.050 in the jet function of the single logarithm that 01:24:13.050 --> 01:24:14.290 was showing up. 01:24:14.290 --> 01:24:17.370 Remember that when we decompose this guy, 01:24:17.370 --> 01:24:20.370 there could be a log mu over q term 01:24:20.370 --> 01:24:24.690 or a one term with no log of mu over q. 01:24:24.690 --> 01:24:28.080 This is like an analog of a log, this plus function. 01:24:28.080 --> 01:24:30.210 If you integrate over s, then it's 01:24:30.210 --> 01:24:31.960 like ds over s, which is like a log. 01:24:31.960 --> 01:24:35.220 So this is a log, and integrating delta of s 01:24:35.220 --> 01:24:37.370 is like one. 01:24:37.370 --> 01:24:38.980 So the analog statement that there 01:24:38.980 --> 01:24:42.090 was two possibilities there. 01:24:42.090 --> 01:24:43.930 In this case, there's an analog of that, 01:24:43.930 --> 01:24:45.670 and there's two possibilities here. 01:24:45.670 --> 01:24:47.250 And what perturbation theory is doing 01:24:47.250 --> 01:24:50.100 is actually just computing the coefficients of these two 01:24:50.100 --> 01:24:52.430 different structures. 01:24:52.430 --> 01:24:57.110 OK, and we're out of time, so I'll say a few more words 01:24:57.110 --> 01:25:00.050 about how you would solve, for example, an equation 01:25:00.050 --> 01:25:01.730 like this one. 01:25:01.730 --> 01:25:03.650 Next time I'll tell you how to solve it, 01:25:03.650 --> 01:25:06.928 and then we'll basically be done with our example. 01:25:06.928 --> 01:25:09.470 We'll put things back together and write down a factorization 01:25:09.470 --> 01:25:13.953 theorem that includes the resummation 01:25:13.953 --> 01:25:15.620 and then we'll go on to another example. 01:25:18.750 --> 01:25:21.000 Moving on, the next example we'll treat after this one 01:25:21.000 --> 01:25:25.650 is SCET two where we'll be dealing 01:25:25.650 --> 01:25:28.830 with energetic hadrons, some other types of examples 01:25:28.830 --> 01:25:30.770 besides jets.