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. 00:00:22.050 --> 00:00:24.360 IAIN STEWART: All right, well, it's time to start. 00:00:27.617 --> 00:00:29.325 So last time we were talking about e plus 00:00:29.325 --> 00:00:31.350 e minus to jets, and-- 00:00:34.353 --> 00:00:35.520 I should have written that-- 00:00:38.070 --> 00:00:40.920 in particular, e plus e minus to dijets. 00:00:40.920 --> 00:00:43.740 And we were talking about-- we talked about factorization 00:00:43.740 --> 00:00:45.352 and how to derive it. 00:00:45.352 --> 00:00:47.310 And I've schematically written the result here. 00:00:47.310 --> 00:00:49.320 Last time we had a more definite formula 00:00:49.320 --> 00:00:51.600 with all the arguments made explicit. 00:00:51.600 --> 00:00:54.630 So there were hard functions, jet functions, 00:00:54.630 --> 00:00:58.110 and the soft function, and we could either think about 00:00:58.110 --> 00:01:00.450 measuring invariant masses in two hemispheres 00:01:00.450 --> 00:01:03.450 and constraining them to be small to know we have dijets, 00:01:03.450 --> 00:01:06.000 or measuring kind of the sum of the two, which we called-- 00:01:06.000 --> 00:01:09.750 which is effectively the thrust, and then we get a slightly 00:01:09.750 --> 00:01:13.200 simpler formula because we can project these guys down onto 00:01:13.200 --> 00:01:14.850 guys that'd only have-- 00:01:14.850 --> 00:01:17.610 or functions of one variable. 00:01:17.610 --> 00:01:20.255 But either way, in either case, we 00:01:20.255 --> 00:01:21.630 have a factorization theorem that 00:01:21.630 --> 00:01:25.170 separates the degrees of freedom and kind of the picture 00:01:25.170 --> 00:01:33.010 is that we have these kind of scales in the problem. 00:01:33.010 --> 00:01:39.900 So this is new jet, new hard, and new soft, 00:01:39.900 --> 00:01:41.760 and we can use this factorization theorem 00:01:41.760 --> 00:01:43.975 to separate those scales into these functions. 00:01:43.975 --> 00:01:44.850 And then we can use-- 00:01:44.850 --> 00:01:46.975 I started talking about using renormalization group 00:01:46.975 --> 00:01:49.845 equations in order to sum the logs between these scales. 00:01:54.850 --> 00:01:57.800 And I wanted to finish up that discussion. 00:01:57.800 --> 00:02:00.850 So for the Wilson coefficient of the operator 00:02:00.850 --> 00:02:02.803 or, for that matter, for the hard function, 00:02:02.803 --> 00:02:04.720 which is the square of the Wilson coefficient, 00:02:04.720 --> 00:02:07.510 you have a very simple renormalization group equation. 00:02:07.510 --> 00:02:09.280 And that's because momentum conservation 00:02:09.280 --> 00:02:13.960 is enough to fix all the variables that 00:02:13.960 --> 00:02:15.250 label collinear fields. 00:02:15.250 --> 00:02:17.620 So you really just have this overall Q, 00:02:17.620 --> 00:02:19.150 which is the center of mass energy, 00:02:19.150 --> 00:02:22.300 and that's the only variable that it's fixed by kinematics. 00:02:22.300 --> 00:02:24.178 There's no convolutions. 00:02:24.178 --> 00:02:26.470 In the case of these things called jet functions, which 00:02:26.470 --> 00:02:28.420 we talked a little bit about last time, 00:02:28.420 --> 00:02:31.300 you end up with something like the Altarelli-Parisi 00:02:31.300 --> 00:02:32.903 where there's an integral. 00:02:32.903 --> 00:02:34.570 It's a little bit different in the sense 00:02:34.570 --> 00:02:37.030 that we actually know its structure-- tolerance 00:02:37.030 --> 00:02:39.200 and perturbation theory-- it has this structure. 00:02:39.200 --> 00:02:42.070 So it only has a very particular dependence on S, 00:02:42.070 --> 00:02:44.080 has to scale like 1 over S, so one thing 00:02:44.080 --> 00:02:48.010 that scales like 1 over S is a delta function 00:02:48.010 --> 00:02:50.740 or a plus function, and basically, there's 00:02:50.740 --> 00:02:54.190 only this single type of plus function that can show up, 00:02:54.190 --> 00:02:56.343 and that's the analog actually of something that 00:02:56.343 --> 00:02:57.760 happened to this guy where we said 00:02:57.760 --> 00:03:01.870 there was only a possibility of a single logarithm showing up. 00:03:01.870 --> 00:03:04.240 OK. 00:03:04.240 --> 00:03:07.240 So this is how far we got. 00:03:07.240 --> 00:03:10.715 Whenever you have something-- whenever you have equations 00:03:10.715 --> 00:03:13.090 where you have convolutions like this, what you should do 00:03:13.090 --> 00:03:16.152 is Fourier transform because if you Fourier 00:03:16.152 --> 00:03:17.860 transform something that's a convolution, 00:03:17.860 --> 00:03:18.735 it becomes a product. 00:03:27.730 --> 00:03:29.440 So we'll Fourier transform, and in order 00:03:29.440 --> 00:03:34.450 to be careful about convergence of our Fourier transform, 00:03:34.450 --> 00:03:37.600 we give it a small imaginary part. 00:03:37.600 --> 00:03:46.470 And we define a position space anomalous dimension, 00:03:46.470 --> 00:03:48.890 as the anomalous dimension up there that depend on S, 00:03:48.890 --> 00:03:51.900 but now Fourier transformed and likewise, we 00:03:51.900 --> 00:04:02.120 define a position space jet function the same way, 00:04:02.120 --> 00:04:03.920 and then this formula here, which 00:04:03.920 --> 00:04:07.400 is a convolution, if you use these in the inverse 00:04:07.400 --> 00:04:12.000 transforms, you arrive at a very simple formula for the position 00:04:12.000 --> 00:04:12.500 space. 00:04:19.392 --> 00:04:30.640 Gamma J. 00:04:30.640 --> 00:04:32.530 So the Fourier transform of a delta function 00:04:32.530 --> 00:04:35.910 is just the identity so there's this term here, 00:04:35.910 --> 00:04:37.525 becomes that term there. 00:04:37.525 --> 00:04:39.790 The Fourier transform of a plus function 00:04:39.790 --> 00:04:43.750 is actually a logarithm, and in order to get that result, 00:04:43.750 --> 00:04:47.410 you do need to have this convergence. 00:04:47.410 --> 00:04:50.180 This little i0 in there. 00:04:50.180 --> 00:04:52.810 So in general, if you have there's 00:04:52.810 --> 00:05:02.950 a general kind of relation between logs and plus 00:05:02.950 --> 00:05:07.540 functions like that, and in Fourier space, 00:05:07.540 --> 00:05:11.170 the kind of highest logarithm you get is L to the K plus 1. 00:05:17.350 --> 00:05:21.705 OK, so these logs here, you're basically-- 00:05:21.705 --> 00:05:23.080 if you think about counting them, 00:05:23.080 --> 00:05:25.540 you should count this one over S as an extra log, 00:05:25.540 --> 00:05:27.460 and in Fourier space, that becomes explicit. 00:05:27.460 --> 00:05:29.440 You really just get a log to the K plus 1. 00:05:33.160 --> 00:05:35.770 All right, but this formula here, 00:05:35.770 --> 00:05:38.320 is something that, again, is of this kind 00:05:38.320 --> 00:05:39.630 of multiplicative form. 00:05:39.630 --> 00:05:43.510 So what happens to the anomalous dimension in the Fourier space 00:05:43.510 --> 00:05:47.050 is that it has a multiplicative form. 00:05:47.050 --> 00:05:54.630 All right so we have mu debugging mu of J Y 00:05:54.630 --> 00:06:03.790 comma mu is simply gamma J Y comma mu J Y comma mu. 00:06:03.790 --> 00:06:07.790 So it's no more difficult than this formula up here. 00:06:07.790 --> 00:06:12.160 Just have to find the right space, which is Fourier space. 00:06:12.160 --> 00:06:15.970 So then we can make an all order solution of that formula. 00:06:15.970 --> 00:06:18.970 Given that we know this, we can just 00:06:18.970 --> 00:06:22.420 plug-in this and integrate, and we've 00:06:22.420 --> 00:06:23.710 done this a few times now. 00:06:29.250 --> 00:06:31.597 Let me just do it formally by not doing the integrals, 00:06:31.597 --> 00:06:33.180 but writing them out in a way that you 00:06:33.180 --> 00:06:37.300 could do them to show you what the all order solution would 00:06:37.300 --> 00:06:37.800 look like. 00:06:44.020 --> 00:06:47.580 So you can write the solution using this fact 00:06:47.580 --> 00:06:50.970 and using this fact. 00:07:03.090 --> 00:07:05.670 So we have to use that fact to convert this logarithm here 00:07:05.670 --> 00:07:11.580 into something that is just in terms of alphas. 00:07:11.580 --> 00:07:14.640 But using those two facts, we can do the same kind of things 00:07:14.640 --> 00:07:17.760 that we've done before and write the all order solution 00:07:17.760 --> 00:07:18.690 in the following way. 00:08:02.820 --> 00:08:22.440 The integrals are contained in this W and this K. 00:08:22.440 --> 00:08:25.160 Here's the noncusp piece, and then 00:08:25.160 --> 00:08:26.840 the cusp piece comes with two integrals 00:08:26.840 --> 00:08:30.320 because it had a logarithm, and we turned one of the logarithms 00:08:30.320 --> 00:08:31.400 into an integral as well. 00:08:41.068 --> 00:08:42.860 So hopefully I've gotten all my twos right. 00:08:53.322 --> 00:08:54.780 So we saw something similar when we 00:08:54.780 --> 00:08:57.325 were talking about the running of the Wilson 00:08:57.325 --> 00:09:00.820 coefficient of the C, because it also had this kind of form 00:09:00.820 --> 00:09:03.850 with just a logarithm and a constant term. 00:09:03.850 --> 00:09:06.072 And it's a similar thing here. 00:09:06.072 --> 00:09:08.530 So this is an all order solution for the running of the jet 00:09:08.530 --> 00:09:09.370 function. 00:09:09.370 --> 00:09:12.190 Of course, we don't know these functions to all orders. 00:09:12.190 --> 00:09:15.760 We only know them up to some given order, 00:09:15.760 --> 00:09:17.695 and so you plug those-- whatever order you 00:09:17.695 --> 00:09:19.570 want to work-- you plug it into this formula, 00:09:19.570 --> 00:09:20.987 and then you can do this integral, 00:09:20.987 --> 00:09:22.930 and then you figure out these factors here. 00:09:22.930 --> 00:09:25.420 And those factors are summing the logs. 00:09:25.420 --> 00:09:29.665 In the case of the jet function, what you would-- 00:09:29.665 --> 00:09:31.540 the way that you can think about this picture 00:09:31.540 --> 00:09:32.500 is that you want to-- 00:09:32.500 --> 00:09:35.800 you have to put the guys in the factorization theorem 00:09:35.800 --> 00:09:40.780 at a common scale, and so one way of thinking about it 00:09:40.780 --> 00:09:42.760 is as follows. 00:09:42.760 --> 00:09:46.105 Let's take our hard function and do perturbation theory at mu 00:09:46.105 --> 00:09:49.540 equals mu age, and then we'll sum logs down 00:09:49.540 --> 00:09:50.890 to say the soft scale. 00:09:50.890 --> 00:09:53.680 That's one way of doing it, and we'll do perturbation theory 00:09:53.680 --> 00:09:57.130 for the jet function, at a scale mu equals mu J, 00:09:57.130 --> 00:10:01.850 and then we'll use this renormalization group here, 00:10:01.850 --> 00:10:09.170 which will just say gamma J to run this guy down 00:10:09.170 --> 00:10:10.370 to the soft scale. 00:10:10.370 --> 00:10:12.260 In that kind of scenario, you wouldn't 00:10:12.260 --> 00:10:15.380 have to run the soft function. 00:10:15.380 --> 00:10:17.210 This is like a top down kind of picture 00:10:17.210 --> 00:10:19.880 where you're just running the objects at the higher scales 00:10:19.880 --> 00:10:21.200 down to the lowest scale. 00:10:21.200 --> 00:10:22.860 That's one way of doing it. 00:10:22.860 --> 00:10:24.770 You could also do it in an equivalent way 00:10:24.770 --> 00:10:26.990 where you run the soft function, for example, 00:10:26.990 --> 00:10:29.660 up to the jet scale, and you don't run the jet function. 00:10:29.660 --> 00:10:35.280 And that'll give you actually the same result. 00:10:35.280 --> 00:10:36.410 But in the way-- 00:10:36.410 --> 00:10:38.120 with the information I've presented you, 00:10:38.120 --> 00:10:39.470 this is the way that you would think 00:10:39.470 --> 00:10:41.120 about doing the renormalization group, 00:10:41.120 --> 00:10:45.913 and that would sum up logarithms in the cross section. 00:10:45.913 --> 00:10:48.080 And it's very easy if you write it in position space 00:10:48.080 --> 00:10:50.163 to see what type of logarithms you're summing out. 00:10:53.280 --> 00:10:55.140 So in position space, if we go back 00:10:55.140 --> 00:10:58.320 to a formula like the one at the top, 00:10:58.320 --> 00:11:00.930 basically, what happens is that in position space, 00:11:00.930 --> 00:11:02.640 you, again, have a product. 00:11:02.640 --> 00:11:08.610 So these convolutions here, which were also this form, 00:11:08.610 --> 00:11:11.400 if you Fourier transform, then you end up with a product. 00:11:11.400 --> 00:11:16.050 D sigma D, so schematically, D sigma 00:11:16.050 --> 00:11:22.230 Dy for, say, where y is the Fourier transform for thrust, 00:11:22.230 --> 00:11:27.540 would just be h times j in position space 00:11:27.540 --> 00:11:30.960 times the soft function in position space as well. 00:11:30.960 --> 00:11:38.370 And there's Q somewhere, but it would be a very simple formula 00:11:38.370 --> 00:11:40.210 which just has a product. 00:11:40.210 --> 00:11:43.680 And so you can figure out what logs you're summing. 00:11:43.680 --> 00:11:45.582 The logs that you're summing, again, 00:11:45.582 --> 00:11:47.040 just as we talked about before, are 00:11:47.040 --> 00:11:51.570 simplest to describe in when you take the log because you're 00:11:51.570 --> 00:11:54.480 really summing logs in an exponent. 00:11:54.480 --> 00:12:00.000 So if we talk about the log of D sigma Dy, 00:12:00.000 --> 00:12:03.900 then we can enumerate the types of logs 00:12:03.900 --> 00:12:05.145 that one sums as follows. 00:12:17.940 --> 00:12:19.550 So these are the leading logs. 00:12:19.550 --> 00:12:22.206 This is next leading log, and this 00:12:22.206 --> 00:12:26.270 is next to next leading log, just like before. 00:12:26.270 --> 00:12:28.400 When we were talking about enumerating the logs, 00:12:28.400 --> 00:12:29.910 we talked about before an example, 00:12:29.910 --> 00:12:32.780 we were enumerating logs in the Wilson coefficient C. 00:12:32.780 --> 00:12:35.090 Now we're doing it for a full cross section, 00:12:35.090 --> 00:12:39.040 but because in position space, the cross section is simply 00:12:39.040 --> 00:12:43.873 a product of objects, you can think about just-- 00:12:43.873 --> 00:12:46.040 if you take the logarithm, they kind of split apart. 00:12:46.040 --> 00:12:48.680 And it's very simple to enumerate 00:12:48.680 --> 00:12:50.600 what corresponds to the things that you 00:12:50.600 --> 00:12:52.760 would get by putting in anomalous dimensions 00:12:52.760 --> 00:12:54.180 at a given order. 00:12:54.180 --> 00:12:56.960 So if you use the leading log anomalous dimension, just 00:12:56.960 --> 00:13:00.270 the one loop cusp, then you get these terms. 00:13:00.270 --> 00:13:03.050 Get the higher terms from the running of the coupling. 00:13:03.050 --> 00:13:05.090 If you put in the next leading log terms, then 00:13:05.090 --> 00:13:07.720 you get those terms. 00:13:07.720 --> 00:13:09.080 OK. 00:13:09.080 --> 00:13:13.940 And so you supplement that resummation which 00:13:13.940 --> 00:13:16.640 comes from this K and this omega, 00:13:16.640 --> 00:13:18.680 in general, you supplement that with sort 00:13:18.680 --> 00:13:22.010 of fixed order calculations of the H and the J and the S. 00:13:22.010 --> 00:13:24.910 And that gives you a complete cross-section at some order 00:13:24.910 --> 00:13:26.990 and resum perturbation theory. 00:13:30.190 --> 00:13:32.708 So you could do that also in momentum space. 00:13:32.708 --> 00:13:34.750 You could write out the formula in momentum space 00:13:34.750 --> 00:13:40.090 just because after all that's what you want in the end. 00:13:40.090 --> 00:13:42.970 Position space is just really a nice way 00:13:42.970 --> 00:13:47.220 of deriving the results that then you eventually put 00:13:47.220 --> 00:13:55.290 back into momentum space in some way or another 00:13:55.290 --> 00:13:56.970 because that's where the data is. 00:14:06.750 --> 00:14:10.520 And let me just show you what the formula would look like, 00:14:10.520 --> 00:14:17.170 just so you get a kind of picture with all the arguments. 00:14:17.170 --> 00:14:20.590 So if I were to Fourier transform the resum formula, 00:14:20.590 --> 00:14:23.430 over here I didn't write the resummation in. 00:14:23.430 --> 00:14:26.340 I didn't write in the evolution factors 00:14:26.340 --> 00:14:28.340 that correspond to this. 00:14:28.340 --> 00:14:31.470 So let me do that, but let me do it in momentum space. 00:14:31.470 --> 00:14:39.630 So I guess I made a slightly different choice here. 00:14:50.370 --> 00:14:51.890 So I'm doing it for the thrust case. 00:15:01.190 --> 00:15:03.770 These convolutions just being integrals over the variables 00:15:03.770 --> 00:15:09.000 that are like this S prime, for example. 00:15:09.000 --> 00:15:10.330 So this is S prime integral. 00:15:20.120 --> 00:15:22.330 And this is an L prime integral. 00:15:37.440 --> 00:15:40.650 OK so there's three integrals in this formula. 00:15:40.650 --> 00:15:42.730 I guess this is-- it's kind of a funny notation, 00:15:42.730 --> 00:15:45.690 but I could have written integral instead 00:15:45.690 --> 00:15:48.152 of writing these tensor signs. 00:15:48.152 --> 00:15:49.860 That would have been probably more clear. 00:15:58.260 --> 00:15:58.770 Do that. 00:16:01.410 --> 00:16:05.730 OK so there's the formula in momentum space. 00:16:05.730 --> 00:16:09.060 And it's doing what I said. 00:16:09.060 --> 00:16:14.310 This factor here is running the hard function 00:16:14.310 --> 00:16:21.840 down to the soft scale, and this other factor here 00:16:21.840 --> 00:16:25.710 is running from the jet scale to the soft scale, 00:16:25.710 --> 00:16:28.830 and I didn't have to run the soft function. 00:16:28.830 --> 00:16:31.590 I should have written in here that I evaluate 00:16:31.590 --> 00:16:34.620 this guy at the soft scale. 00:16:34.620 --> 00:16:38.610 And this guy here remember is the non-perturbative guy which 00:16:38.610 --> 00:16:41.410 we also talked about a little last time. 00:16:41.410 --> 00:16:45.373 OK so that's kind of the basic structure these use as usual 00:16:45.373 --> 00:16:47.040 are summing the logarithms, and then you 00:16:47.040 --> 00:16:55.110 put fixed order results in for this guy, for that guy, 00:16:55.110 --> 00:16:59.090 and for this guy here. 00:16:59.090 --> 00:17:07.579 And in doing so you get the [INAUDIBLE] result. 00:17:07.579 --> 00:17:10.843 All right, so this fact that I could 00:17:10.843 --> 00:17:12.260 have done the running differently, 00:17:12.260 --> 00:17:14.359 that I could have run the soft function 00:17:14.359 --> 00:17:16.640 is a kind of consistency. 00:17:16.640 --> 00:17:23.470 So there are other ways of doing the r g, 00:17:23.470 --> 00:17:26.753 and they all lead to the same answer. 00:17:26.753 --> 00:17:28.420 And when I say the same answer, I really 00:17:28.420 --> 00:17:33.150 mean the same precisely exactly the same number. 00:17:33.150 --> 00:17:36.280 So I really mean the same, not just 00:17:36.280 --> 00:17:40.330 sort of approximately the same but exactly the same. 00:17:40.330 --> 00:17:42.250 And the other ways of doing the RGE 00:17:42.250 --> 00:17:46.570 are the basic idea of why there are more than one way is again 00:17:46.570 --> 00:17:51.100 this fact that if I do coefficient renormalization, 00:17:51.100 --> 00:17:54.790 that that's the inverse of doing operator renormalization. 00:17:54.790 --> 00:17:57.520 So there's two ways of thinking about doing the run. 00:17:57.520 --> 00:18:00.209 You run the operators, you run the coefficients. 00:18:05.200 --> 00:18:07.480 OK in this picture it's a little bit more complicated 00:18:07.480 --> 00:18:09.820 because we have three scales, but when 00:18:09.820 --> 00:18:11.770 I'm running the hard function, I quivalently 00:18:11.770 --> 00:18:13.990 could have run both the jet and the soft function 00:18:13.990 --> 00:18:16.550 and not run the hard function. 00:18:16.550 --> 00:18:18.370 So let me maybe give one other way 00:18:18.370 --> 00:18:21.170 of doing this just to show you. 00:18:21.170 --> 00:18:24.730 So I could have run the hard function say, just 00:18:24.730 --> 00:18:26.605 down to the jet scale. 00:18:26.605 --> 00:18:28.480 And then instead of running the jet function, 00:18:28.480 --> 00:18:31.690 I could have run the soft function up to the jet scale. 00:18:31.690 --> 00:18:34.330 That would be another way of doing the RGE, 00:18:34.330 --> 00:18:37.010 and that would lead to an equivalent picture. 00:18:37.010 --> 00:18:40.600 And there's more than two, because there's 00:18:40.600 --> 00:18:41.708 three scales in this. 00:18:41.708 --> 00:18:43.250 There's more than two natural things. 00:18:43.250 --> 00:18:45.172 There's three possible natural things, 00:18:45.172 --> 00:18:47.380 where you would run to either this scale, this scale, 00:18:47.380 --> 00:18:50.410 or this scale, but at the base of it, 00:18:50.410 --> 00:18:52.480 it all has to do with this equivalence 00:18:52.480 --> 00:18:57.340 that we talked about in simpler scenarios earlier. 00:18:57.340 --> 00:19:01.412 And you can write this also as a relation 00:19:01.412 --> 00:19:02.620 between anomalous dimensions. 00:19:02.620 --> 00:19:05.022 So it implies a non-trivial things 00:19:05.022 --> 00:19:07.480 about the formula that have to be trivial in order for this 00:19:07.480 --> 00:19:09.740 to work. 00:19:09.740 --> 00:19:16.120 So for example, it implies that the coefficients 00:19:16.120 --> 00:19:19.300 of the cusp anomalous dimension in the various renormalization 00:19:19.300 --> 00:19:22.660 groups would be in a certain way that they would be related, 00:19:22.660 --> 00:19:25.090 and even in the non-cusp anomalous dimensions, which 00:19:25.090 --> 00:19:27.460 is this formula, there's a relation 00:19:27.460 --> 00:19:31.810 between the jet, hard, and soft anomalous dimensions. 00:19:31.810 --> 00:19:32.310 OK? 00:19:32.310 --> 00:19:36.162 And that's needed-- this is an expression of this fact 00:19:36.162 --> 00:19:37.870 that you can do the renormalization group 00:19:37.870 --> 00:19:40.690 by running different objects. 00:19:40.690 --> 00:19:42.610 So you can derive this formula by just 00:19:42.610 --> 00:19:45.670 saying mu D divided by the cross section is 0. 00:19:45.670 --> 00:19:49.060 And going through it, if you were to allow, 00:19:49.060 --> 00:19:52.180 then you would find formulas like this one. 00:19:52.180 --> 00:19:54.470 OK? 00:19:54.470 --> 00:19:55.430 So questions? 00:19:59.100 --> 00:20:01.975 I went quick, because most of the concepts 00:20:01.975 --> 00:20:04.350 here are things that we've seen before in simpler guises. 00:20:08.347 --> 00:20:09.930 The new complication is really that we 00:20:09.930 --> 00:20:13.080 have this dependence on a variable which ended up 00:20:13.080 --> 00:20:15.750 being y, but by going to Fourier space, 00:20:15.750 --> 00:20:17.550 it looked just like a simple product 00:20:17.550 --> 00:20:20.980 again, and y behaved like a simple kinematic variable. 00:20:20.980 --> 00:20:23.230 The end of the day, we want to Fourier transform back, 00:20:23.230 --> 00:20:26.025 but we have a space where things look familiar. 00:20:30.540 --> 00:20:33.470 OK. 00:20:33.470 --> 00:20:36.170 So that's e plus minus the dijets, 00:20:36.170 --> 00:20:38.510 and that's the last SCETI example 00:20:38.510 --> 00:20:39.920 I'm going to do for a while. 00:20:39.920 --> 00:20:42.140 I'm now going to turn to doing SCETII, 00:20:42.140 --> 00:20:44.120 and we'll spend some time talking about SCETII. 00:20:55.720 --> 00:20:58.750 So in SCETII, instead of having ultrasoft interactions 00:20:58.750 --> 00:21:01.390 with collinear particles, we have soft interactions 00:21:01.390 --> 00:21:05.340 with collinear particles. 00:21:05.340 --> 00:21:09.380 So we have to call how that's going to work. 00:21:09.380 --> 00:21:16.680 So if we take a soft particle, consider a soft particle 00:21:16.680 --> 00:21:20.740 interacting with some collinear particle, and ask what 00:21:20.740 --> 00:21:21.420 do we get out? 00:21:25.933 --> 00:21:27.850 So if we just ask about momentum conservation, 00:21:27.850 --> 00:21:31.450 and we call this q, then q is equal to q 00:21:31.450 --> 00:21:35.118 soft plus q collinear. 00:21:35.118 --> 00:21:36.910 And since we know the scaling of these two, 00:21:36.910 --> 00:21:39.250 we know the scaling of q. 00:21:39.250 --> 00:21:42.790 It's just given by whatever the larger of the scalings is. 00:21:47.440 --> 00:21:50.760 So soft and collinear both have the same-- so 00:21:50.760 --> 00:21:52.135 let me remind you of the scaling. 00:21:52.135 --> 00:21:57.340 So this guy was q, lambda, lambda, lambda, and this guy 00:21:57.340 --> 00:22:01.550 was q lambda squared 1 lambda. 00:22:01.550 --> 00:22:03.610 So the difference between ultrasoft and collinear 00:22:03.610 --> 00:22:05.860 and soft and collinear is that soft and collinear have 00:22:05.860 --> 00:22:07.990 the same size of perp momenta. 00:22:07.990 --> 00:22:11.170 So the perp momenta are just of order lambda. 00:22:11.170 --> 00:22:13.480 Obviously, in the minus momenta, the 1 00:22:13.480 --> 00:22:16.210 wins, so the minus momentum of order 1. 00:22:16.210 --> 00:22:18.547 And then the plus momentum, it's the soft 00:22:18.547 --> 00:22:20.630 that wins, because it's bigger than the collinear. 00:22:20.630 --> 00:22:23.470 So this is order lambda. 00:22:23.470 --> 00:22:25.450 OK. 00:22:25.450 --> 00:22:28.360 And that's actually much different than what 00:22:28.360 --> 00:22:31.525 we found when we added ultrasoft and collinear. 00:22:31.525 --> 00:22:33.400 Because when we added ultrasoft and collinear 00:22:33.400 --> 00:22:35.380 we got back collinear. 00:22:35.380 --> 00:22:37.320 Here, we're not getting back collinear. 00:22:37.320 --> 00:22:40.420 This is actually off-shell, from the perspective 00:22:40.420 --> 00:22:41.530 of our low-energy modes. 00:22:44.950 --> 00:22:47.320 Because q squared, the biggest part of q squared 00:22:47.320 --> 00:22:48.820 would be 1 times lambda. 00:22:48.820 --> 00:22:52.030 q squared's of order lambda which is much bigger 00:22:52.030 --> 00:22:52.960 than lambda squared. 00:22:58.590 --> 00:23:02.180 So adding a soft and a collinear particle in SCETII immediately 00:23:02.180 --> 00:23:03.850 gives you something off-shell. 00:23:03.850 --> 00:23:06.350 That's going to make some things easier and some things more 00:23:06.350 --> 00:23:08.695 complicated. 00:23:08.695 --> 00:23:10.070 Mostly, it's going to make things 00:23:10.070 --> 00:23:11.153 a little more complicated. 00:23:21.497 --> 00:23:23.330 At least at the start, it'll looking easier. 00:23:32.970 --> 00:23:40.020 So you could think about this from our mode picture, 00:23:40.020 --> 00:23:42.870 where we had softs that live here 00:23:42.870 --> 00:23:45.120 and collinears that live there. 00:23:45.120 --> 00:23:48.982 In order for them to interact, they have to go up to a place 00:23:48.982 --> 00:23:50.940 where we can have a common momentum, and that's 00:23:50.940 --> 00:23:54.210 a higher hyperbola, like this. 00:23:54.210 --> 00:23:58.390 It's not all the way up to the hard scale which is up here. 00:23:58.390 --> 00:24:02.070 That was hard, but there's an intermediate scale 00:24:02.070 --> 00:24:07.050 that comes in which is the scale of this q squared. 00:24:07.050 --> 00:24:09.882 So this is q squared. 00:24:09.882 --> 00:24:12.300 This is the order of the q squared there, 00:24:12.300 --> 00:24:16.410 and sometimes this is called a hard collinear scale. 00:24:16.410 --> 00:24:18.930 This would be called a hard collinear mode, hc. 00:24:30.398 --> 00:24:31.940 And you could even write down, if you 00:24:31.940 --> 00:24:34.482 wanted, an on-shell degree of freedom for this hard collinear 00:24:34.482 --> 00:24:37.280 mode. 00:24:37.280 --> 00:24:38.870 So an on-shell version of it, you 00:24:38.870 --> 00:24:42.590 could think of it as a mode, in theory. 00:24:42.590 --> 00:24:44.570 An on-shell version of it would have a scaling 00:24:44.570 --> 00:24:47.135 that's q 1 lambda root lambda. 00:24:52.540 --> 00:24:55.620 So you can think of one way of approaching it which we'll 00:24:55.620 --> 00:24:57.042 take this attitude in a minute. 00:24:57.042 --> 00:24:58.500 One way of approaching this problem 00:24:58.500 --> 00:25:00.780 would be to first think about doing 00:25:00.780 --> 00:25:03.240 some matching onto a theory with this hard collinear mode 00:25:03.240 --> 00:25:06.180 and then trying to match down onto a theory that just has 00:25:06.180 --> 00:25:08.400 the collinear and the soft mode, and we'll 00:25:08.400 --> 00:25:10.760 exploit that a little later. 00:25:10.760 --> 00:25:13.200 But first, let me do a different thing 00:25:13.200 --> 00:25:16.050 and just ignore this dashed line and just think 00:25:16.050 --> 00:25:20.130 about matching directly from the QCD 00:25:20.130 --> 00:25:24.060 onto the SCETII which just has these two degrees of freedom. 00:25:47.090 --> 00:25:48.860 So what's going to happen is that we're 00:25:48.860 --> 00:25:51.440 going to have to integrate out more things, 00:25:51.440 --> 00:25:54.407 because any type of interactions that we write down 00:25:54.407 --> 00:25:56.240 are basically giving us something off-shell. 00:25:56.240 --> 00:25:59.310 So let's do an example of this. 00:25:59.310 --> 00:26:04.880 So we'll do an example of a heavy-to-light current, 00:26:04.880 --> 00:26:10.620 but now, it won't be an SCETII current but rather an SCETI. 00:26:17.810 --> 00:26:20.570 So I just want to have one soft particle and one collinear 00:26:20.570 --> 00:26:22.130 particle. 00:26:22.130 --> 00:26:24.730 Simplest possible example, but now this is soft, 00:26:24.730 --> 00:26:28.081 and this is collinear, not ultrasoft and collinear. 00:26:32.950 --> 00:26:33.450 All right. 00:26:33.450 --> 00:26:35.520 So we have some current. 00:26:35.520 --> 00:26:39.630 I'm just going to label the lines with c's and s's. 00:26:39.630 --> 00:26:45.180 So imagine that you attached a soft gluon here. 00:26:45.180 --> 00:26:47.880 That would give you something that's off-shelf. 00:26:47.880 --> 00:26:51.420 So this line here is off-shelf, and likewise, 00:26:51.420 --> 00:26:52.950 on this side, if this is soft coming 00:26:52.950 --> 00:26:57.150 in but you attach something collinear here, 00:26:57.150 --> 00:27:00.030 then this is off-shelf. 00:27:00.030 --> 00:27:01.890 Before, when we had this picture, 00:27:01.890 --> 00:27:04.440 and we were doing it for SCETI, one 00:27:04.440 --> 00:27:06.510 set of attachments which still need to on-shell, 00:27:06.510 --> 00:27:08.535 in particular on this side. 00:27:08.535 --> 00:27:10.410 And on this side, we got something off-shell. 00:27:10.410 --> 00:27:13.170 We integrated it out, and we got a Wilson line. 00:27:13.170 --> 00:27:14.190 Right? 00:27:14.190 --> 00:27:15.750 Here, it's a little more complicated, 00:27:15.750 --> 00:27:18.645 because you touch alternate modes on either side. 00:27:18.645 --> 00:27:20.520 And you get something off-shell, and you just 00:27:20.520 --> 00:27:25.135 have to put out this pink line all at once, 00:27:25.135 --> 00:27:27.510 and that's going to give you another type of Wilson line. 00:27:31.410 --> 00:27:33.360 In this calculation, we're going to get 00:27:33.360 --> 00:27:38.430 W. It's both of the collinear modes, 00:27:38.430 --> 00:27:43.650 and another type of Wilson line from the soft modes. 00:27:54.880 --> 00:27:57.640 Let's call it Sn. 00:27:57.640 --> 00:27:59.860 So how do you build up a Wilson line? 00:27:59.860 --> 00:28:02.890 Well, first of all, think about attaching more gluons 00:28:02.890 --> 00:28:04.910 onto this line. 00:28:04.910 --> 00:28:05.560 Right? 00:28:05.560 --> 00:28:09.050 That's kind of the analog of what we did before. 00:28:09.050 --> 00:28:14.980 So just attach more soft gluons here, 00:28:14.980 --> 00:28:18.137 more collinear gluons here. 00:28:18.137 --> 00:28:19.970 This might be the first thing you would try, 00:28:19.970 --> 00:28:22.270 just adding those guys up, and that 00:28:22.270 --> 00:28:25.217 means you have a whole line of things that is off-shell, 00:28:25.217 --> 00:28:26.800 and you just calculate these diagrams. 00:28:29.560 --> 00:28:35.518 And if you do that calculation, it will give you Wilson, lines. 00:28:35.518 --> 00:28:37.060 And what it'll give is something that 00:28:37.060 --> 00:28:51.050 looks like this, where Sn dagger is a function of the n dot 00:28:51.050 --> 00:28:55.150 As component, and W as a function of 00:28:55.150 --> 00:28:58.920 said n bar dot Ac component. 00:28:58.920 --> 00:29:03.800 So there are Wilson lines along some direction, 00:29:03.800 --> 00:29:07.320 and it looks like this. 00:29:07.320 --> 00:29:08.240 This is c. 00:29:08.240 --> 00:29:11.180 This is S. 00:29:11.180 --> 00:29:14.510 Now, there's something wrong with this. 00:29:14.510 --> 00:29:16.532 It's not quite the right answer. 00:29:16.532 --> 00:29:18.740 The reason that the Wilson lines are the way they are 00:29:18.740 --> 00:29:22.250 is because I got the collinear ones from integrating them out 00:29:22.250 --> 00:29:25.100 from the next to the heavy quark which is the soft particle. 00:29:25.100 --> 00:29:27.560 So that's why W sits next to h. 00:29:27.560 --> 00:29:30.050 S dagger sits next to c because of the same thing 00:29:30.050 --> 00:29:31.820 on the collinear side. 00:29:31.820 --> 00:29:34.460 But if I wanted to make them into a gauge invariant thing, 00:29:34.460 --> 00:29:37.280 I want the W to sit next to the c and the S dagger 00:29:37.280 --> 00:29:39.320 to sit next to the h. 00:29:39.320 --> 00:29:42.260 That would be the analog of what we had in SCETI, 00:29:42.260 --> 00:29:43.940 and that's not what we got. 00:29:43.940 --> 00:29:46.910 If this was QED, then that would be fine, because this and this, 00:29:46.910 --> 00:29:49.385 I could just commute them. 00:29:49.385 --> 00:29:50.510 Well, I can always do that. 00:29:50.510 --> 00:29:52.260 I can just push this guy through that guy. 00:29:52.260 --> 00:29:55.130 But in QCD I can't, because these guys have color matrices, 00:29:55.130 --> 00:29:57.340 and they don't commute with each other. 00:29:57.340 --> 00:30:00.920 So that means that this isn't quite the whole story here. 00:30:00.920 --> 00:30:02.480 There's some diagrams that we missed. 00:30:07.612 --> 00:30:09.070 So there must be some diagrams that 00:30:09.070 --> 00:30:24.185 are non-abelian that we missed, and what are those diagrams? 00:30:28.160 --> 00:30:31.095 These are diagrams that involve triple gluon and four gluon 00:30:31.095 --> 00:30:31.595 vertices. 00:30:43.790 --> 00:30:45.530 And what these guys do, it turns out, 00:30:45.530 --> 00:30:47.730 is they do one thing in this calculation. 00:30:47.730 --> 00:30:58.380 They really just flip the order of the W and the S. 00:30:58.380 --> 00:31:02.220 So why do we have to consider those diagrams? 00:31:05.900 --> 00:31:11.270 So think about, instead of attaching gluons to quarks, 00:31:11.270 --> 00:31:13.190 attach gluons to gluons. 00:31:13.190 --> 00:31:22.090 So say we have soft collinear, and we attach them 00:31:22.090 --> 00:31:24.410 to each other but through a three-gluon vortex. 00:31:24.410 --> 00:31:25.848 Then, this guy's off-shell. 00:31:25.848 --> 00:31:27.640 If that off-shell guy attaches to this guy, 00:31:27.640 --> 00:31:29.230 then it's off-shell. 00:31:29.230 --> 00:31:31.210 And if you integrate out diagrams like this, 00:31:31.210 --> 00:31:36.010 that's going to change what our result over there look like. 00:31:36.010 --> 00:31:39.190 If you go to the same order on the other side, 00:31:39.190 --> 00:31:40.450 exactly analogous thing. 00:31:46.810 --> 00:31:47.310 OK. 00:31:47.310 --> 00:31:49.643 So these are also things that you have to integrate out. 00:31:49.643 --> 00:31:53.020 You have to integrate all these pink things out. 00:31:53.020 --> 00:31:57.645 And if you do that, then it does what I said. 00:31:57.645 --> 00:32:01.650 If you do that to all orders, what 00:32:01.650 --> 00:32:09.360 you can do with a axillary Lagrangian-type approach, 00:32:09.360 --> 00:32:11.735 obviously, you're not going to start calculating diagrams 00:32:11.735 --> 00:32:15.050 to all this, but there are some tricks to doing it. 00:32:15.050 --> 00:32:17.560 Then, you get them in the right order. 00:32:22.090 --> 00:32:24.190 And it's easy to check, for example, 00:32:24.190 --> 00:32:26.920 that at the order of two gluons that this guy does 00:32:26.920 --> 00:32:29.630 exactly what you want. 00:32:29.630 --> 00:32:35.530 So this is then a collinear gauge invariant object, 00:32:35.530 --> 00:32:40.000 and this is then something that's soft gauge invariant, 00:32:40.000 --> 00:32:41.620 and that's nice. 00:32:41.620 --> 00:32:42.790 That's what you're hoping. 00:32:42.790 --> 00:32:46.360 That's what you're expecting. 00:32:46.360 --> 00:32:49.255 AUDIENCE: What about diagrams like collinear soft 00:32:49.255 --> 00:32:52.600 all from the collinear line or something? 00:32:52.600 --> 00:32:54.930 IAIN STEWART: You want to add one more gluon where? 00:32:54.930 --> 00:32:56.680 AUDIENCE: I want to stick like a collinear 00:32:56.680 --> 00:32:58.588 right in between some softs. 00:32:58.588 --> 00:32:59.380 IAIN STEWART: Yeah. 00:32:59.380 --> 00:33:04.865 You mean like on here, like that? 00:33:04.865 --> 00:33:05.740 AUDIENCE: Well, yeah. 00:33:05.740 --> 00:33:07.950 Essentially, it'd be over on that diagram over there, 00:33:07.950 --> 00:33:10.618 like not even [INAUDIBLE]. 00:33:10.618 --> 00:33:11.410 IAIN STEWART: Yeah. 00:33:11.410 --> 00:33:12.618 It's the same thing, I think. 00:33:12.618 --> 00:33:15.700 So if you add this guy here, then I 00:33:15.700 --> 00:33:17.740 think it's power suppressed. 00:33:17.740 --> 00:33:22.330 Because it ends up being a power suppressed term 00:33:22.330 --> 00:33:25.220 that you don't need, if I remember correctly. 00:33:25.220 --> 00:33:25.720 Yeah. 00:33:25.720 --> 00:33:26.860 This guy's power suppressed. 00:33:26.860 --> 00:33:28.410 AUDIENCE: You have to define like a different lambda 00:33:28.410 --> 00:33:29.350 for each one? 00:33:29.350 --> 00:33:30.522 It seems like it would be-- 00:33:30.522 --> 00:33:31.230 IAIN STEWART: No. 00:33:33.880 --> 00:33:35.840 You can think of this as a full theory diagram. 00:33:35.840 --> 00:33:36.340 Right? 00:33:36.340 --> 00:33:39.400 Where this guy is off-shell, and you haven't 00:33:39.400 --> 00:33:40.800 changed how he's off-shell. 00:33:40.800 --> 00:33:42.550 You just change the value of the momentum, 00:33:42.550 --> 00:33:45.400 but one thing you've done is you've doubled 00:33:45.400 --> 00:33:46.870 the number of hard propagators. 00:33:46.870 --> 00:33:49.580 So if I remember correctly, this guy just 00:33:49.580 --> 00:33:50.830 gives you something off-shell. 00:33:50.830 --> 00:33:53.650 I'm pretty confident, something that's power 00:33:53.650 --> 00:33:54.700 suppressed in lambda. 00:33:58.680 --> 00:34:00.900 All right. 00:34:00.900 --> 00:34:05.160 So this is how soft collinear factorization works. 00:34:05.160 --> 00:34:07.410 So soft collinear factorization, if you think about it 00:34:07.410 --> 00:34:11.130 from the point of view of going from QCD to SCETII, 00:34:11.130 --> 00:34:14.159 it's just integrate out these pink lines, as usual, 00:34:14.159 --> 00:34:16.020 and you end up with something that's where 00:34:16.020 --> 00:34:17.393 things are splitting apart. 00:34:17.393 --> 00:34:18.810 And the reason that it's happening 00:34:18.810 --> 00:34:21.239 is because these lines are off-shell 00:34:21.239 --> 00:34:23.800 that are where you would try to interact, have interactions, 00:34:23.800 --> 00:34:27.420 and so your theory is forced apart by the fact 00:34:27.420 --> 00:34:30.179 that these things can't interact in an on-shell way. 00:34:30.179 --> 00:34:35.350 So it's different than SCETI that in that sense. 00:34:35.350 --> 00:34:37.500 So this is kind of cumbersome, as you can see, 00:34:37.500 --> 00:34:40.590 if you wanted to think about doing it arbitrarily. 00:34:40.590 --> 00:34:43.050 Because it seems like you just have to calculate diagrams, 00:34:43.050 --> 00:34:46.139 and who wants to calculate infinite classes of diagrams 00:34:46.139 --> 00:34:47.969 for arbitrarily complicated scenarios? 00:34:47.969 --> 00:34:49.199 Right? 00:34:49.199 --> 00:34:50.670 Maybe in this case, we can do it, 00:34:50.670 --> 00:34:52.440 but actually even in this case, we 00:34:52.440 --> 00:34:55.270 have to resort to some tricks to do it to all orders. 00:34:55.270 --> 00:34:57.090 And in more complicated scenarios, 00:34:57.090 --> 00:34:59.290 it would just get even more and more cumbersome. 00:34:59.290 --> 00:35:02.670 So we'd like to have a trick that's generic, 00:35:02.670 --> 00:35:04.630 where we could get the same answer. 00:35:04.630 --> 00:35:08.760 And the way that we can do that is by using this dashed line up 00:35:08.760 --> 00:35:10.390 in the picture. 00:35:10.390 --> 00:35:13.920 Think about formulating rather than directly this SCETII, 00:35:13.920 --> 00:35:16.500 formulate first an SCETI, and then we'll 00:35:16.500 --> 00:35:18.885 match that SCETI onto QCD. 00:35:21.850 --> 00:35:36.120 So another, in some ways, better method is to do QCD to SCETI 00:35:36.120 --> 00:35:45.380 and then SCETI to SCETII, so three steps. 00:35:50.010 --> 00:35:54.110 So we're going to first use a SCETI that 00:35:54.110 --> 00:35:55.790 doesn't have the collinear, just has 00:35:55.790 --> 00:35:58.280 the soft mode and the hard collinear mode. 00:35:58.280 --> 00:36:03.510 So it has what I call the hc in the picture and s. 00:36:03.510 --> 00:36:04.010 OK? 00:36:04.010 --> 00:36:06.140 So that would be a mode that has P 00:36:06.140 --> 00:36:11.400 squared in this picture of order, say let's call it lambda 00:36:11.400 --> 00:36:11.900 squared. 00:36:14.850 --> 00:36:20.653 So if I say that this is for some lambda squared, 00:36:20.653 --> 00:36:22.070 then you'd have the soft mode that 00:36:22.070 --> 00:36:23.360 has P squared of order lambda squared, 00:36:23.360 --> 00:36:24.985 and this hard collinear mode that has P 00:36:24.985 --> 00:36:26.150 squared of order q lambda. 00:36:26.150 --> 00:36:30.530 And that's exactly an SCETI-type situation, where 00:36:30.530 --> 00:36:35.040 we were calling these c and us. 00:36:35.040 --> 00:36:38.220 So that's step one. 00:36:38.220 --> 00:36:40.310 Step two is to factorize that theory, which 00:36:40.310 --> 00:36:43.495 we know how to do. 00:36:43.495 --> 00:36:48.830 In particular, the ultrasoft which is soft 00:36:48.830 --> 00:36:51.597 can be factorized with a field redefinition. 00:36:55.010 --> 00:36:57.770 So this has the advantage that at this first stage 00:36:57.770 --> 00:37:00.620 we still have a locality that protects us 00:37:00.620 --> 00:37:03.140 and helps us to understand the theory. 00:37:03.140 --> 00:37:07.650 We then factorize, and then in the third step, 00:37:07.650 --> 00:37:09.490 we match SCETI onto SCETII. 00:37:20.830 --> 00:37:26.560 And then, we're getting rid of these hard collinear modes, 00:37:26.560 --> 00:37:31.650 matching them onto some collinear modes, 00:37:31.650 --> 00:37:35.340 and then we still have our soft modes. 00:37:39.880 --> 00:37:42.870 So we think that the hard collinear 00:37:42.870 --> 00:37:45.610 modes contain the collinear modes in the first stage. 00:37:45.610 --> 00:37:47.270 But they also contain some other stuff 00:37:47.270 --> 00:37:50.540 which is unwanted baggage that we have to get rid of, 00:37:50.540 --> 00:37:54.835 and that's why we have this second stage of matching. 00:37:54.835 --> 00:37:56.710 But you can really think of it in the picture 00:37:56.710 --> 00:37:59.260 as doing first the matching, where you integrate out 00:37:59.260 --> 00:38:02.610 the hard scale, but you contain in your effective theory 00:38:02.610 --> 00:38:06.340 this scale associated to the dashed line. 00:38:06.340 --> 00:38:10.462 And then in a second stage, you integrate out the dashed line, 00:38:10.462 --> 00:38:13.180 and that gets you to the final thing 00:38:13.180 --> 00:38:15.670 you want which is just low energy modes on this line here. 00:38:18.970 --> 00:38:20.890 OK. 00:38:20.890 --> 00:38:33.030 So one thing this does is give a simple procedure 00:38:33.030 --> 00:38:36.420 of constructing SCETII operators, 00:38:36.420 --> 00:38:39.630 even though there's more non-locality in SCETII 00:38:39.630 --> 00:38:40.710 than there was in SCETI. 00:38:48.260 --> 00:38:50.870 Because there's non-locality, because you don't just 00:38:50.870 --> 00:38:53.060 have this one scale that could cause you, 00:38:53.060 --> 00:38:56.150 even have this smaller scale that's causing non-locality. 00:38:56.150 --> 00:38:59.360 And that's explicit in the soft Wilson line. 00:38:59.360 --> 00:39:02.630 The thing that's giving the soft Wilson line-- 00:39:02.630 --> 00:39:04.940 well, the things that are producing the Wilson lines 00:39:04.940 --> 00:39:06.370 are really modes of this scale. 00:39:09.670 --> 00:39:12.188 OK so it's more non-local. 00:39:12.188 --> 00:39:13.730 One way of saying it's more non-local 00:39:13.730 --> 00:39:15.880 is simply that there'll be 1 over l pluses, 00:39:15.880 --> 00:39:17.360 as well as 1 over l minuses. 00:39:17.360 --> 00:39:22.200 That's another way of saying why it's more non-local. 00:39:22.200 --> 00:39:26.580 OK so that's just an example of something 00:39:26.580 --> 00:39:28.710 that makes it look trivial. 00:39:28.710 --> 00:39:32.970 So let's say we wanted to do this calculation, this way. 00:39:32.970 --> 00:39:36.450 So then in step one, we would simply write down 00:39:36.450 --> 00:39:37.860 the current in SCETI. 00:39:41.850 --> 00:39:44.070 We would integrate out the off-shell pink lines, 00:39:44.070 --> 00:39:45.750 but there'd only be lines on one side, 00:39:45.750 --> 00:39:50.460 and we get this which is our SCETI result 00:39:50.460 --> 00:39:52.200 for heavy-to-light current. 00:39:52.200 --> 00:39:58.920 Then, in step two, we would do the field redefinition in order 00:39:58.920 --> 00:40:07.890 to factorize this theory, and we get that result. 00:40:07.890 --> 00:40:11.540 And in step three, in order to match this result 00:40:11.540 --> 00:40:17.570 onto a current in SCETII, it's really simply a renaming here. 00:40:30.050 --> 00:40:32.876 So why is this step so easy? 00:40:32.876 --> 00:40:35.700 It looks completely trivial. 00:40:35.700 --> 00:40:37.370 The reason that this step is so easy 00:40:37.370 --> 00:40:40.198 is that any T-product, any time-order product that you 00:40:40.198 --> 00:40:41.990 write down here, will have a correspondence 00:40:41.990 --> 00:40:42.860 in this other case. 00:40:53.400 --> 00:40:53.900 OK? 00:40:53.900 --> 00:40:54.692 So you can really-- 00:41:02.330 --> 00:41:03.470 it really is that simple. 00:41:06.630 --> 00:41:09.080 So when so if it's really that simple, then 00:41:09.080 --> 00:41:12.260 you can see the advantages of this procedure. 00:41:12.260 --> 00:41:16.460 It's not always that simple, and let 00:41:16.460 --> 00:41:20.900 me present as a kind of theorem or just a bit 00:41:20.900 --> 00:41:24.290 of a statement when it will be this simple 00:41:24.290 --> 00:41:27.812 and when it will not be this simple. 00:41:27.812 --> 00:41:29.270 So basically, you have to know when 00:41:29.270 --> 00:41:32.930 will the T-products between these two theories match up? 00:41:32.930 --> 00:41:35.930 And the T-products will match up under the following 00:41:35.930 --> 00:41:42.650 circumstances, or they won't match up under the following 00:41:42.650 --> 00:41:43.953 circumstances. 00:41:49.880 --> 00:41:54.040 So if in the SCETI one theory you 00:41:54.040 --> 00:41:57.580 had time-order products with greater than or equal to two 00:41:57.580 --> 00:42:17.180 operators that had soft and collinear fields, 00:42:17.180 --> 00:42:19.910 then you can generate some non-trivial matching. 00:42:51.373 --> 00:42:52.790 So I think that's best illustrated 00:42:52.790 --> 00:42:55.140 by an example again. 00:42:55.140 --> 00:42:57.260 So in this particular example, up here, we only 00:42:57.260 --> 00:42:59.720 had one operator that had both soft and collinear fields, 00:42:59.720 --> 00:43:01.500 this external current. 00:43:01.500 --> 00:43:04.400 And then we had Lagrangians, but they were totally decoupled 00:43:04.400 --> 00:43:06.590 after we made this field redefinition, the L0 00:43:06.590 --> 00:43:07.160 Lagrangian. 00:43:07.160 --> 00:43:09.467 So they don't count as something that's mixing 00:43:09.467 --> 00:43:10.550 soft and collinear fields. 00:43:10.550 --> 00:43:12.350 We just had this operator. 00:43:12.350 --> 00:43:13.940 But if you had two of these operators, 00:43:13.940 --> 00:43:17.210 and you wanted to go through the same procedure, 00:43:17.210 --> 00:43:19.010 then you could get something non-trivial. 00:43:19.010 --> 00:43:20.302 So let's imagine that scenario. 00:43:24.356 --> 00:43:30.950 We have two operators, so let's think 00:43:30.950 --> 00:43:36.890 of having in the SCETI, two interactions 00:43:36.890 --> 00:43:41.785 that are like this, then we could string them together 00:43:41.785 --> 00:43:42.285 as follows. 00:43:56.760 --> 00:43:59.690 So this is a T-product between two different interactions that 00:43:59.690 --> 00:44:00.950 both had soft and collinear. 00:44:00.950 --> 00:44:03.960 I'm just taking two of these guys and putting them together. 00:44:03.960 --> 00:44:06.680 And if you look at the off-shell-ness of this line 00:44:06.680 --> 00:44:10.880 here, then P squared is of order, in our counting, 00:44:10.880 --> 00:44:17.830 it's like a k plus times a q minus which is of q times 00:44:17.830 --> 00:44:19.916 a lambda times a q. 00:44:19.916 --> 00:44:22.430 So it's really something that lives at that hard collinear 00:44:22.430 --> 00:44:24.230 scale. 00:44:24.230 --> 00:44:26.770 OK? 00:44:26.770 --> 00:44:30.298 But you need two terms in the time-order product that, 00:44:30.298 --> 00:44:32.590 in order for there really to be a propagator like this, 00:44:32.590 --> 00:44:34.280 that you're integrating out. 00:44:34.280 --> 00:44:37.390 So this guy here will match onto something 00:44:37.390 --> 00:44:40.960 when you do this matching, where you 00:44:40.960 --> 00:44:46.390 have two soft fields and two collinear fields, like that. 00:44:46.390 --> 00:44:48.220 Because this guy here really is something 00:44:48.220 --> 00:44:53.560 that you would want to integrate out at that dashed line. 00:44:53.560 --> 00:44:56.230 Really, it's an off-shell, hard, collinear mode. 00:44:59.120 --> 00:45:00.850 But if you didn't have two T-products, 00:45:00.850 --> 00:45:03.170 then effectively what happens is, 00:45:03.170 --> 00:45:05.170 when you change the external kinematics in order 00:45:05.170 --> 00:45:08.770 to do the matching, all what were called hard collinear 00:45:08.770 --> 00:45:11.200 lines become collinear lines, and then 00:45:11.200 --> 00:45:13.755 the matching is trivial, as it was up here. 00:45:13.755 --> 00:45:15.380 But if you're in a situation like this, 00:45:15.380 --> 00:45:17.440 then there is a line that, by changing 00:45:17.440 --> 00:45:19.390 the scaling of these external guys, 00:45:19.390 --> 00:45:22.390 it doesn't change the scaling of that internal line, 00:45:22.390 --> 00:45:25.244 and then you get some non-trivial Wilson coefficient. 00:45:29.116 --> 00:45:32.504 AUDIENCE: [INAUDIBLE] 00:45:36.730 --> 00:45:39.810 IAIN STEWART: So let's call these hc. 00:45:39.810 --> 00:45:47.010 Yeah, and so I start by calling them hc. 00:45:47.010 --> 00:45:48.300 Right? 00:45:48.300 --> 00:45:52.440 And then, what I want to do, when I calculate this diagram 00:45:52.440 --> 00:45:56.520 to do the matching, is I want to assign them a different scale. 00:45:56.520 --> 00:45:59.650 I want to define them to be c instead of hc. 00:45:59.650 --> 00:46:01.830 So I start out thinking of them as hc. 00:46:01.830 --> 00:46:05.070 That's what I do in step one. 00:46:05.070 --> 00:46:06.390 I can do a field redefinition. 00:46:06.390 --> 00:46:08.110 It doesn't matter. 00:46:08.110 --> 00:46:10.890 But now, I want to match from SCETI onto II. 00:46:10.890 --> 00:46:14.070 So therefore, what I do is I take my full theory, which 00:46:14.070 --> 00:46:17.430 is SCETI, and I evaluate it with fields 00:46:17.430 --> 00:46:20.010 that don't have the right power counting for that theory. 00:46:20.010 --> 00:46:23.460 And then I do a Taylor-series expansion of all the diagrams, 00:46:23.460 --> 00:46:24.960 so that's this. 00:46:24.960 --> 00:46:27.720 I change my external fields, and I call them c instead of hc. 00:46:27.720 --> 00:46:29.762 AUDIENCE: Oh, because you're doing the same thing 00:46:29.762 --> 00:46:30.445 with ultrasoft. 00:46:30.445 --> 00:46:30.900 So ultrasoft, it really means-- 00:46:30.900 --> 00:46:32.567 IAIN STEWART: And now I make these soft, 00:46:32.567 --> 00:46:36.540 but soft and ultrasoft, that's just really a renaming. 00:46:36.540 --> 00:46:39.480 AUDIENCE: Ultrasoft in the SCETI [INAUDIBLE].. 00:46:39.480 --> 00:46:41.730 IAIN STEWART: Ultrasoft is equal soft. 00:46:41.730 --> 00:46:42.810 Sorry. 00:46:42.810 --> 00:46:44.340 Maybe this makes it clearer. 00:46:44.340 --> 00:46:46.650 Ultrasoft and soft just two different names 00:46:46.650 --> 00:46:48.150 for the same thing. 00:46:48.150 --> 00:46:51.210 But hc and c aren't, because hc really 00:46:51.210 --> 00:46:53.820 lived on the upper hyperbola. 00:46:53.820 --> 00:46:56.940 I put the external particles onto the lower hyperbola, 00:46:56.940 --> 00:46:59.210 but I'm frozen in here with one particle that 00:46:59.210 --> 00:47:00.960 stays in the upper hyperbola and therefore 00:47:00.960 --> 00:47:02.002 has to be integrated out. 00:47:07.700 --> 00:47:09.950 OK. 00:47:09.950 --> 00:47:21.150 So the kind of thing that you would get by doing that 00:47:21.150 --> 00:47:27.330 is that you would get convolutions 00:47:27.330 --> 00:47:29.342 with some Wilson coefficient. 00:47:36.100 --> 00:47:39.050 In this case, they were-- 00:47:39.050 --> 00:47:43.770 well, let me just say, it depends on some P minuses. 00:47:47.860 --> 00:47:52.760 This is P1, this is P2, and then in my example over there, 00:47:52.760 --> 00:47:56.180 I guess I had some heavy quarks. 00:48:05.320 --> 00:48:07.301 OK, and there's some Dirac structures. 00:48:09.782 --> 00:48:11.240 So it would be something like this, 00:48:11.240 --> 00:48:14.410 where you'd get some function, whatever it is, 00:48:14.410 --> 00:48:16.120 that's coming exactly from integrating 00:48:16.120 --> 00:48:17.090 all the purple stuff. 00:48:17.090 --> 00:48:19.630 So this is the Wilson coefficient, 00:48:19.630 --> 00:48:21.640 and usually, you would call these things 00:48:21.640 --> 00:48:26.380 jet functions, because they look like jet functions. 00:48:29.740 --> 00:48:48.980 So this comes from the SCETI modes that got integrated out. 00:48:48.980 --> 00:48:53.168 So the difference between this and the SCETI matching, where 00:48:53.168 --> 00:48:54.710 you're integrating out the hard scale 00:48:54.710 --> 00:48:56.460 is you're already getting sensitivity here 00:48:56.460 --> 00:48:58.700 to the plus momentum of the soft. 00:48:58.700 --> 00:49:00.261 That's one difference. 00:49:04.190 --> 00:49:05.088 OK. 00:49:05.088 --> 00:49:07.380 So this is a general procedure for constructing SCETII. 00:49:12.830 --> 00:49:14.450 Let's see how much I want to say. 00:49:14.450 --> 00:49:18.260 So let's say a few words about power counting. 00:49:18.260 --> 00:49:23.690 So if you take some T-product of terms in SCETI, 00:49:23.690 --> 00:49:28.160 that will scale like some power of lambda to the 2K OK? 00:49:28.160 --> 00:49:32.690 So in SCETI, you have to be a little bit careful about power 00:49:32.690 --> 00:49:34.620 counting when you do this procedure. 00:49:34.620 --> 00:49:37.640 So in SCETI, you could just assign some power. 00:49:37.640 --> 00:49:40.340 Let's call it lambda to the 2K. 00:49:40.340 --> 00:49:45.200 And when you do a matching onto an operator in SCETII, what 00:49:45.200 --> 00:49:46.940 you're generically going to find is 00:49:46.940 --> 00:49:51.920 that there'll be a relation between the power counting here 00:49:51.920 --> 00:49:54.830 and the power counting there which is good, 00:49:54.830 --> 00:49:58.790 but there's also one thing to be aware of. 00:50:01.860 --> 00:50:04.340 So if I define eta to be the power counting 00:50:04.340 --> 00:50:05.570 parameter for SCETII-- 00:50:05.570 --> 00:50:06.980 just to give it a different name, 00:50:06.980 --> 00:50:09.170 so I can talk about a correspondence-- 00:50:09.170 --> 00:50:12.350 then basically, eta is like lambda squared. 00:50:12.350 --> 00:50:20.350 And that, well, that just follows from this formula, 00:50:20.350 --> 00:50:23.170 that if you want to identify lambda squared 00:50:23.170 --> 00:50:25.360 is lambda over q for the collinear modes, 00:50:25.360 --> 00:50:28.330 then this would be the right way of doing it. 00:50:28.330 --> 00:50:30.880 Little lambda would be square root of lambda over q. 00:50:30.880 --> 00:50:32.490 But in SCETI for the collinear modes, 00:50:32.490 --> 00:50:34.240 you'd say there's some parameter eta which 00:50:34.240 --> 00:50:36.590 is just lambda over q. 00:50:36.590 --> 00:50:37.090 OK? 00:50:37.090 --> 00:50:39.600 So basically, it looks like you just take 2K and go over 00:50:39.600 --> 00:50:42.100 to k, because you just changed the definition of what you're 00:50:42.100 --> 00:50:45.940 calling the parameter, but there's also this plus E. 00:50:45.940 --> 00:50:49.000 That means that you can get additional suppression, 00:50:49.000 --> 00:50:51.065 and that plus E comes about from the fact 00:50:51.065 --> 00:50:52.690 that you also change the power counting 00:50:52.690 --> 00:50:56.302 for your external fields, when you do this procedure. 00:50:56.302 --> 00:50:58.510 So something that was scaling like lambda in the hard 00:50:58.510 --> 00:51:02.560 collinear theory might become lambda squared-- i.e. eta-- 00:51:02.560 --> 00:51:04.045 in the SCETII theory. 00:51:08.656 --> 00:51:12.560 So this E which is greater than 0 00:51:12.560 --> 00:51:22.510 comes from changing the power counting of external fields 00:51:22.510 --> 00:51:23.290 in the matching. 00:51:40.010 --> 00:51:42.050 One example is just having an external collinear 00:51:42.050 --> 00:51:44.570 quark or external collinear perp gluon, 00:51:44.570 --> 00:51:47.720 for example, where you would have c of order lambda 00:51:47.720 --> 00:51:50.150 in SCETII, and it becomes of order eta in SCETII 00:51:50.150 --> 00:51:53.120 which is lambda squared. 00:51:53.120 --> 00:51:57.350 Those are extra factors of the power counting parameter. 00:51:57.350 --> 00:51:58.070 OK. 00:51:58.070 --> 00:52:03.210 So what do we learn, if we put all these things together? 00:52:03.210 --> 00:52:07.850 Well, one thing since soft and the collinear fields 00:52:07.850 --> 00:52:10.422 don't talk to each other, when you write down 00:52:10.422 --> 00:52:12.380 Lagrangians for them, they're totally decoupled 00:52:12.380 --> 00:52:14.130 right from the start. 00:52:14.130 --> 00:52:20.630 And so there's actually no mixed soft collinear Lagrangian 00:52:20.630 --> 00:52:21.380 at leading order. 00:52:31.020 --> 00:52:33.150 And so all the meat of this theory 00:52:33.150 --> 00:52:34.800 is coming about when you integrate out 00:52:34.800 --> 00:52:36.467 these off-shell modes, and you construct 00:52:36.467 --> 00:52:39.250 the external operators. 00:52:39.250 --> 00:52:39.750 OK. 00:52:44.570 --> 00:52:46.760 I'll say little bit more about power counting. 00:52:46.760 --> 00:52:48.635 So there's no mixed soft collinear Lagrangian 00:52:48.635 --> 00:52:53.115 at leading order, and you can get mixed things 00:52:53.115 --> 00:52:53.990 at sub-leading order. 00:53:01.180 --> 00:53:04.827 So in that way, it's like SCETI after the field redefinition. 00:53:10.342 --> 00:53:11.550 But here, there is no analog. 00:53:11.550 --> 00:53:13.740 If you just want to go directly to SCETII, 00:53:13.740 --> 00:53:16.217 there wouldn't be an analog of the field redefinition. 00:53:20.200 --> 00:53:22.360 All the Wilson lines are coming from integrating 00:53:22.360 --> 00:53:23.778 off-shell particles. 00:53:33.050 --> 00:53:33.680 OK. 00:53:33.680 --> 00:53:37.860 So we're going to speed up a little. 00:53:37.860 --> 00:53:40.310 I'm going to skip over one thing, which 00:53:40.310 --> 00:53:41.490 I don't think I need. 00:53:41.490 --> 00:53:43.970 So you can write down power counting formulas, 00:53:43.970 --> 00:53:45.720 like we did in chiral perturbation theory. 00:53:45.720 --> 00:53:48.020 Maybe I'll just write those formulas down. 00:53:48.020 --> 00:53:54.020 So in general, in chiral perturbation theory, 00:53:54.020 --> 00:53:55.395 we had a formula that said, if we 00:53:55.395 --> 00:53:57.853 want to figure out the power counting of any given diagram, 00:53:57.853 --> 00:53:59.540 we can have a counting for that diagram, 00:53:59.540 --> 00:54:01.830 and we know what size it is. 00:54:01.830 --> 00:54:04.460 And there's analogous formulas like that 00:54:04.460 --> 00:54:07.730 in both SCETI and SCETII. 00:54:07.730 --> 00:54:12.350 So in SCETI it's pretty simple. 00:54:12.350 --> 00:54:13.850 You say you have something that's 00:54:13.850 --> 00:54:16.010 order lambda to the delta. 00:54:16.010 --> 00:54:19.115 Then delta, the formula for it follows. 00:54:23.183 --> 00:54:25.100 This is very analogous to what we talked about 00:54:25.100 --> 00:54:26.392 for chiral perturbation theory. 00:54:32.930 --> 00:54:40.470 So these guys here are vertices that are purely ultrasoft, 00:54:40.470 --> 00:54:42.350 and if you have purely ultrasoft vertices, 00:54:42.350 --> 00:54:44.360 you're subtracting 8 because the measure 00:54:44.360 --> 00:54:47.100 of the ultrasoft particles is 8. 00:54:47.100 --> 00:54:48.350 I'm not deriving this for you. 00:54:48.350 --> 00:54:52.050 I could, but I think it's fairly intuitive. 00:54:52.050 --> 00:54:56.180 So if you think about the lowest order Lagrangian, 00:54:56.180 --> 00:55:00.320 say psi bar ID slash psi for ultrasofts. 00:55:00.320 --> 00:55:03.900 This would already be something that's lambda to the 8, 00:55:03.900 --> 00:55:08.270 and this is then leading order, so you subtract 8. 00:55:08.270 --> 00:55:12.410 And this guy here is the rest, anything mixed 00:55:12.410 --> 00:55:15.740 or purely collinear. 00:55:15.740 --> 00:55:17.540 And then for that guy, you subtract 4, 00:55:17.540 --> 00:55:19.623 because as soon as you have at least one collinear 00:55:19.623 --> 00:55:22.430 particle, then you have to use the collinear measure. 00:55:22.430 --> 00:55:26.240 And so here, some operator like c bar, 00:55:26.240 --> 00:55:28.970 and dot dc was order lambda to the 4. 00:55:28.970 --> 00:55:31.672 So you subtract 4 and that's leading order. 00:55:31.672 --> 00:55:33.380 And so what this formula allows you to do 00:55:33.380 --> 00:55:36.260 is really just you do power counting entirely 00:55:36.260 --> 00:55:37.380 in terms of vertices. 00:55:37.380 --> 00:55:40.400 You never have to power count measures for loops or power 00:55:40.400 --> 00:55:41.870 count propagators. 00:55:41.870 --> 00:55:43.920 You just count vertices. 00:55:43.920 --> 00:55:46.790 So if you want to know how big a time-order product is, and it's 00:55:46.790 --> 00:55:54.830 a time-order product of an L3 with an L2 with this kind 00:55:54.830 --> 00:55:59.180 of setup, this will be basically you count this as lambda phi. 00:56:04.580 --> 00:56:07.910 And you could even get the absolute size right, 00:56:07.910 --> 00:56:10.990 taking into account the scaling of external particles, 00:56:10.990 --> 00:56:13.040 and that's what this constant 3 factor does. 00:56:16.204 --> 00:56:21.350 And U is equal to 1 only for pure ultrasoft. 00:56:26.650 --> 00:56:30.095 Otherwise, it's 0. 00:56:30.095 --> 00:56:31.070 OK? 00:56:31.070 --> 00:56:33.230 So there's formulas like this, and there's 00:56:33.230 --> 00:56:36.840 an analogous one in SCETII which is a little more complicated. 00:56:36.840 --> 00:56:39.440 But it allows you just to have a power counting where you can 00:56:39.440 --> 00:56:46.490 just power count Lagrangians or operators 01 with L1, 00:56:46.490 --> 00:56:50.420 and that's lambda squared without having to worry about 00:56:50.420 --> 00:56:53.270 what propagators do I have in this time-order product? 00:56:53.270 --> 00:56:54.807 What loops do I have? 00:56:54.807 --> 00:56:56.390 You never have to ask those questions. 00:56:56.390 --> 00:56:58.098 You just have power counting at vertices, 00:56:58.098 --> 00:57:00.508 and that makes things quite easy. 00:57:00.508 --> 00:57:02.300 AUDIENCE: When you write these expressions, 00:57:02.300 --> 00:57:05.363 one insertion of Lagrangian corresponds to a single vertex. 00:57:05.363 --> 00:57:06.530 [INAUDIBLE] that Lagrangian. 00:57:06.530 --> 00:57:07.728 IAIN STEWART: Yeah. 00:57:07.728 --> 00:57:08.270 That's right. 00:57:10.575 --> 00:57:11.075 OK. 00:57:17.780 --> 00:57:18.290 All right. 00:57:22.722 --> 00:57:24.680 So I want to do a couple of examples in SCETII. 00:57:33.950 --> 00:57:36.900 So let's do an example. 00:57:36.900 --> 00:57:40.760 Let's do an example which is in some ways simple, 00:57:40.760 --> 00:57:48.710 and it's an exclusive analog of our DIS example, 00:57:48.710 --> 00:57:51.980 and it's called the photon pion form factor. 00:57:51.980 --> 00:57:54.290 So it's exclusive. 00:57:54.290 --> 00:57:55.500 It's clearly exclusive. 00:57:55.500 --> 00:57:58.970 There's a pion, and really, all it's going to come in here 00:57:58.970 --> 00:58:01.730 is hard collinear factorization. 00:58:01.730 --> 00:58:05.360 So it's simple in the way that it actually is not really 00:58:05.360 --> 00:58:08.720 exploiting the full complications of SCETII. 00:58:08.720 --> 00:58:10.220 It's really just another example, 00:58:10.220 --> 00:58:12.470 where we have hard modes factoring from cleaner modes. 00:58:12.470 --> 00:58:14.060 But they're clearly SCETII modes, 00:58:14.060 --> 00:58:17.840 because they're going to be modes for this pion. 00:58:17.840 --> 00:58:20.360 So we're going to use again for this calculation 00:58:20.360 --> 00:58:24.560 a bright frame, which you'll see, I guess, 00:58:24.560 --> 00:58:26.040 when I write down some momenta. 00:58:26.040 --> 00:58:30.980 So what would we write down for this process in QCD, first 00:58:30.980 --> 00:58:32.130 of all? 00:58:32.130 --> 00:58:33.890 So you'd say, I have a pi 0. 00:58:33.890 --> 00:58:37.850 It has some momentum P pi. 00:58:37.850 --> 00:58:40.460 I have some current which I can take at 0, 00:58:40.460 --> 00:58:42.533 and I want to make a transition from the pi 0 00:58:42.533 --> 00:58:43.325 to a single photon. 00:59:01.662 --> 00:59:03.120 So you could write that as follows. 00:59:08.360 --> 00:59:26.510 I can always replace the photon by current, 00:59:26.510 --> 00:59:28.508 and then I can parameterized that matrix on it 00:59:28.508 --> 00:59:29.300 like a form factor. 00:59:41.610 --> 00:59:46.190 And if we go through things like parity, charge conjugation, 00:59:46.190 --> 00:59:50.060 stuff like that, time reversal, we 00:59:50.060 --> 00:59:52.420 find out that there's one real form factor, 00:59:52.420 --> 00:59:55.490 and there's an epsilon symbol in this guy. 00:59:55.490 --> 00:59:57.650 It has to be linear in the polarization vector. 00:59:57.650 --> 01:00:00.260 That was already true here, and there's 01:00:00.260 --> 01:00:02.090 one way of getting the indices to work out 01:00:02.090 --> 01:00:03.950 which is with this epsilon. 01:00:03.950 --> 01:00:06.410 So what you're really after in this process 01:00:06.410 --> 01:00:09.370 would be some understanding of this form factor. 01:00:09.370 --> 01:00:12.002 That's the one piece of non-perturbative information. 01:00:12.002 --> 01:00:13.460 Everything else, Lorentz invariance 01:00:13.460 --> 01:00:15.713 is enough to tell you about. 01:00:15.713 --> 01:00:17.630 So we'd like to see if there's a factorization 01:00:17.630 --> 01:00:23.630 theorem for that form factor, if we take the limit where 01:00:23.630 --> 01:00:24.470 q squared is large. 01:00:35.450 --> 01:00:37.100 So we have an electromagnetic current. 01:00:43.457 --> 01:00:47.535 You can think about it is up and down quarks for the pion. 01:00:53.070 --> 01:00:56.280 So there's some matrix in the 2 by 2 space 01:00:56.280 --> 01:01:02.940 which is the charge, which you can write as 2/3 minus 1/3, 01:01:02.940 --> 01:01:03.840 like that. 01:01:03.840 --> 01:01:06.330 Or if you wanted to write it in terms of polymatrices, 01:01:06.330 --> 01:01:12.660 you could write it as identity over 6 plus a tau 3 01:01:12.660 --> 01:01:17.208 for isospin polymatrix over 2. 01:01:17.208 --> 01:01:21.600 So there's some charge matrix that's going to show up. 01:01:21.600 --> 01:01:24.660 What happens if we expand q squared much greater 01:01:24.660 --> 01:01:27.060 than lambda QCD squared? 01:01:27.060 --> 01:01:29.712 In this formula over here, this form factor 01:01:29.712 --> 01:01:31.920 knows about lambda QCD, and it knows about q squared, 01:01:31.920 --> 01:01:33.600 and we haven't expanded. 01:01:33.600 --> 01:01:36.420 So what happens if we do expand? 01:01:36.420 --> 01:01:39.490 Can we simplify that form factor? 01:01:39.490 --> 01:01:41.690 And it does. 01:01:41.690 --> 01:01:45.500 So I'm going to do this in the bright frame again, 01:01:45.500 --> 01:01:50.020 and that effectively means I'm taking the momentum that 01:01:50.020 --> 01:01:55.990 corresponds to the off-shell photon as the same form 01:01:55.990 --> 01:01:58.010 as we did before for our calculation 01:01:58.010 --> 01:02:05.662 in DIS, purely in the z coordinate. 01:02:08.620 --> 01:02:11.470 The momentum of the photon, it's an on-shell thing, 01:02:11.470 --> 01:02:14.102 so I can just write it as E the photons say 01:02:14.102 --> 01:02:15.185 times a light-like vector. 01:02:15.185 --> 01:02:19.370 Then, P gamma squared will be 0, as we want it to be. 01:02:19.370 --> 01:02:24.220 And if I pick this kinematics, then P pi is just P plus P 01:02:24.220 --> 01:02:37.960 gamma, so that would be a P pi. 01:02:44.650 --> 01:02:45.150 OK? 01:02:45.150 --> 01:02:49.750 So in Feynman diagrams, what am I talking about? 01:02:49.750 --> 01:02:53.850 I'm talking about making a transition 01:02:53.850 --> 01:02:56.970 with an off-shell photon, through a diagram 01:02:56.970 --> 01:03:01.470 like this one, to a pi 0 or plus the cross graph. 01:03:06.230 --> 01:03:09.980 And with this setup with this kinematics, 01:03:09.980 --> 01:03:13.160 this intermediate line here, if we go through the scaling, 01:03:13.160 --> 01:03:14.730 it's going to be hard. 01:03:14.730 --> 01:03:18.560 So these guys with this kinematics, 01:03:18.560 --> 01:03:21.890 we make the pion collinear. 01:03:21.890 --> 01:03:27.170 In order to see that, you can impose P pi 01:03:27.170 --> 01:03:29.780 squared equals M pi squared. 01:03:29.780 --> 01:03:30.950 Right? 01:03:30.950 --> 01:03:33.530 This is never going to be made small, 01:03:33.530 --> 01:03:37.460 but E is going to have to be tuned to be basically q over 2. 01:03:37.460 --> 01:03:39.350 So this thing is going to become, 01:03:39.350 --> 01:03:42.050 if you impose this condition, something 01:03:42.050 --> 01:03:47.990 like M pi squared over 2 q. 01:03:47.990 --> 01:03:52.130 Then, you find out that the pion is collinear. 01:03:52.130 --> 01:03:54.290 So the pion is collinear. 01:03:54.290 --> 01:03:55.850 The photon is a photon. 01:03:55.850 --> 01:03:59.432 This line here is hard, and so we 01:03:59.432 --> 01:04:00.890 want to integrate out the hard line 01:04:00.890 --> 01:04:05.540 and match this guy in the effective theory 01:04:05.540 --> 01:04:09.405 onto some effective theory operators that 01:04:09.405 --> 01:04:10.280 would look like this. 01:04:21.016 --> 01:04:22.170 This guy is pink. 01:04:27.790 --> 01:04:28.290 OK. 01:04:28.290 --> 01:04:31.230 So we just have to write down what type of effective theory 01:04:31.230 --> 01:04:34.410 operators that could be, and again, it's an effective theory 01:04:34.410 --> 01:04:36.230 operator of two quark fields. 01:04:36.230 --> 01:04:37.980 So it's very much like what we did before. 01:04:47.320 --> 01:04:49.720 I'm not going to be going through all the indices 01:04:49.720 --> 01:04:51.220 that I have to go through in order 01:04:51.220 --> 01:04:54.188 to keep track of charge conjugation and all 01:04:54.188 --> 01:04:54.730 those things. 01:04:54.730 --> 01:04:56.950 I'll just write down the right answers for that part. 01:05:15.680 --> 01:05:19.090 So I could write like this. 01:05:19.090 --> 01:05:21.780 So after doing hard collinear factorization, 01:05:21.780 --> 01:05:24.960 there's again operators which is two quark's. 01:05:24.960 --> 01:05:27.180 Some hard Wilson coefficient which 01:05:27.180 --> 01:05:31.200 is this pink thing, integrating out the pink line that 01:05:31.200 --> 01:05:33.330 sits in the middle. 01:05:33.330 --> 01:05:37.110 This obey current conservation. 01:05:37.110 --> 01:05:40.440 Dimensional analysis fixes this 1 over q. 01:05:40.440 --> 01:05:44.250 Charge conjugation, actually, also provides 01:05:44.250 --> 01:05:47.070 constraints on the Wilson coefficients, 01:05:47.070 --> 01:05:49.770 just like it did in DIS. 01:05:49.770 --> 01:06:05.060 So charge conjugation tells us that there's 01:06:05.060 --> 01:06:06.605 a relation between flipping signs. 01:06:10.100 --> 01:06:12.230 But we just impose on this operator 01:06:12.230 --> 01:06:14.480 all the symmetries and things that we can think of 01:06:14.480 --> 01:06:16.160 and see what it says about the operator. 01:06:16.160 --> 01:06:18.243 You can think about just writing the operator down 01:06:18.243 --> 01:06:21.320 based on this picture without ever calculating any diagrams, 01:06:21.320 --> 01:06:23.790 just knowing what you're after. 01:06:23.790 --> 01:06:25.640 And then imposing on it all the symmetries 01:06:25.640 --> 01:06:27.330 that should be conserved. 01:06:29.910 --> 01:06:31.393 So one is dimensional analysis. 01:06:31.393 --> 01:06:32.060 That gave the q. 01:06:32.060 --> 01:06:34.970 Current conservation, that's partly responsible 01:06:34.970 --> 01:06:37.805 for the epsilon which is also like a parody. 01:06:44.240 --> 01:06:50.340 Go through flavor and spin, and you 01:06:50.340 --> 01:06:55.260 find that this has this structure, 01:06:55.260 --> 01:06:57.360 and there's some constant that I threw in here. 01:07:04.660 --> 01:07:06.310 And you actually know from the flavor 01:07:06.310 --> 01:07:08.320 that it's got two photons. 01:07:08.320 --> 01:07:09.670 Right? 01:07:09.670 --> 01:07:11.770 And I'm not ever adding any more photons, 01:07:11.770 --> 01:07:15.160 so that means there's two factors of q hat. 01:07:15.160 --> 01:07:17.950 And so there's a q hat squared, which 01:07:17.950 --> 01:07:20.440 I can stick inside this gamma. 01:07:20.440 --> 01:07:25.738 And it has to be a color singlet, 01:07:25.738 --> 01:07:28.030 because it's going to have to have a non-trivial matrix 01:07:28.030 --> 01:07:30.400 element with pi 0 which is a color singlet. 01:07:34.540 --> 01:07:35.890 And since it has to be-- 01:07:35.890 --> 01:07:41.650 so there's no TA inside the gamma that's what I mean. 01:07:41.650 --> 01:07:44.860 There'd be nothing for the index A to contract with. 01:07:44.860 --> 01:07:46.390 And since it's a color singlet, that 01:07:46.390 --> 01:07:49.180 means if you were to think about soft interactions here, 01:07:49.180 --> 01:07:50.560 they would just cancel. 01:07:50.560 --> 01:07:51.590 Right? 01:07:51.590 --> 01:07:54.085 If I were to think about putting in soft interactions 01:07:54.085 --> 01:07:55.960 and integrating them out, I'd get an S dagger 01:07:55.960 --> 01:07:58.510 S that would cancel out. 01:07:58.510 --> 01:08:00.760 So that's the sense in which this is a simple example. 01:08:15.910 --> 01:08:16.569 OK. 01:08:16.569 --> 01:08:18.760 So then, we would have a formula where 01:08:18.760 --> 01:08:20.859 we could equate the effective theory 01:08:20.859 --> 01:08:24.810 result with the full theory result. 01:08:24.810 --> 01:08:30.790 So we can equate matrix elements, 01:08:30.790 --> 01:08:33.370 and this is one way in which this example is 01:08:33.370 --> 01:08:35.569 different than dependent elastic scattering. 01:08:35.569 --> 01:08:40.600 We're really doing a matching at the amplitude level. 01:08:40.600 --> 01:08:42.790 Remember, the form factor was a parameterization 01:08:42.790 --> 01:08:48.189 of the amplitude, and if we do that-- 01:08:58.810 --> 01:09:02.609 If you like, if you think about the two currents that we had, 01:09:02.609 --> 01:09:04.960 and we've integrated them out. 01:09:04.960 --> 01:09:07.210 Now, think about just this operator here. 01:09:07.210 --> 01:09:09.160 The matrix element pi 0 to vacuum 01:09:09.160 --> 01:09:12.819 that we had to finding this thing which had two currents 01:09:12.819 --> 01:09:16.158 just becomes a matrix element of that operator. 01:09:20.550 --> 01:09:24.240 Again, like our DIS example, we form the sum and difference 01:09:24.240 --> 01:09:25.410 at the momenta. 01:09:25.410 --> 01:09:28.170 We can think of forming P bar plus which 01:09:28.170 --> 01:09:35.640 is P dagger plus or minus P. And one of these guys just 01:09:35.640 --> 01:09:39.944 gives the total momentum, just the minus 01:09:39.944 --> 01:09:43.319 with our sign conventions. 01:09:43.319 --> 01:09:49.270 So that gets fixed which, in this case, 01:09:49.270 --> 01:09:58.160 it just gets fixed to the pion momentum which is q. 01:10:03.490 --> 01:10:08.350 So if you like, you've inserted the operator here. 01:10:08.350 --> 01:10:11.230 You have some collinear lines, and then you 01:10:11.230 --> 01:10:14.860 have a matrix element of the pi 0, and then you have gluons. 01:10:20.670 --> 01:10:22.592 But you know that whatever momentum comes in 01:10:22.592 --> 01:10:24.050 has to be the momentum of the pion, 01:10:24.050 --> 01:10:28.180 and that's the one momentum constraint. 01:10:28.180 --> 01:10:30.390 So that means that the answer just 01:10:30.390 --> 01:10:36.030 involves one convolution again which is 01:10:36.030 --> 01:10:37.730 the one that's unconstrained. 01:10:41.670 --> 01:10:43.170 So there's a Wilson coefficient that 01:10:43.170 --> 01:10:45.660 depends on the other unconstrained guy which 01:10:45.660 --> 01:10:51.480 was P plus, and then there's a matrix element, 01:10:51.480 --> 01:10:54.870 where I can write in a delta function with that P 01:10:54.870 --> 01:10:56.970 plus, the usual kind of way that we've done. 01:11:01.290 --> 01:11:03.480 But here, it's a little different matrix element 01:11:03.480 --> 01:11:05.820 than the one we saw on DIS, because it's not forward. 01:11:05.820 --> 01:11:07.745 It's vacuum to pi 0. 01:11:19.722 --> 01:11:21.180 But it's actually the same operator 01:11:21.180 --> 01:11:24.960 that we were talking about in DIS. 01:11:24.960 --> 01:11:28.470 It's just a bilinear operator with two collinear quark 01:11:28.470 --> 01:11:32.070 fields, two collinear quarks in dress with Wilson lines. 01:11:38.270 --> 01:11:41.270 So one can go through a similar type of matrix element 01:11:41.270 --> 01:11:43.400 analysis for this operator, and it 01:11:43.400 --> 01:11:46.610 gives something that's called the light cone function. 01:11:46.610 --> 01:11:48.250 So let me define that for you. 01:12:04.690 --> 01:12:10.720 This matrix element can be written 01:12:10.720 --> 01:12:14.560 in terms of an object that has a dimensionless variable. 01:12:25.090 --> 01:12:28.030 It's an analog of the part-time distribution function 01:12:28.030 --> 01:12:29.890 but for this matrix element that we're 01:12:29.890 --> 01:12:32.316 dealing with here which is vacuum to pion. 01:12:35.172 --> 01:12:38.078 So this has some similarity to the formula we had in DIS. 01:12:38.078 --> 01:12:39.370 There's a delta function there. 01:12:39.370 --> 01:12:41.360 There's a delta function here. 01:12:41.360 --> 01:12:43.810 There's a dimensionless variable z, dimensionless 01:12:43.810 --> 01:12:47.110 variable z there, and this is a non-perturbative function. 01:12:57.160 --> 01:12:59.510 So this is what's called a light cone distribution 01:12:59.510 --> 01:13:00.640 function for the pion. 01:13:05.870 --> 01:13:08.540 So generically, when you have an exclusive process, 01:13:08.540 --> 01:13:12.350 and you're producing some hadron that's very energetic, 01:13:12.350 --> 01:13:15.330 like a pion, this is the type of thing that's going to show up, 01:13:15.330 --> 01:13:17.481 one of these light cone distribution functions. 01:13:21.540 --> 01:13:23.980 This example of photon to pion is in some sense 01:13:23.980 --> 01:13:28.600 a very, very simple, exclusive process, 01:13:28.600 --> 01:13:31.640 the simplest one in some sense. 01:13:31.640 --> 01:13:32.140 OK. 01:13:32.140 --> 01:13:34.430 So we could take this formula, plug it back in there, 01:13:34.430 --> 01:13:36.520 and then we'd have a factorization theorem. 01:13:36.520 --> 01:13:39.290 And I think that you can imagine what that would look like. 01:13:39.290 --> 01:13:41.290 I could write it, instead of an integral over W, 01:13:41.290 --> 01:13:45.250 as an integral over z, and that would be the factorization 01:13:45.250 --> 01:13:49.090 theorem involving this 5 pi. 01:13:49.090 --> 01:13:52.282 AUDIENCE: [INAUDIBLE] 01:13:53.278 --> 01:13:54.070 IAIN STEWART: Yeah. 01:13:54.070 --> 01:13:54.640 AUDIENCE: [INAUDIBLE] 01:13:54.640 --> 01:13:55.765 IAIN STEWART: That's right. 01:13:55.765 --> 01:13:58.960 So you should think of the z as like-- 01:13:58.960 --> 01:14:01.430 so the way to think about the z is as follows. 01:14:01.430 --> 01:14:03.430 So think about like when you initially 01:14:03.430 --> 01:14:07.780 produced these guys here, think about all the momentum 01:14:07.780 --> 01:14:09.970 going this way. 01:14:09.970 --> 01:14:13.120 When you initially produced them, 01:14:13.120 --> 01:14:16.510 after you integrated out the hard interactions, you had z 01:14:16.510 --> 01:14:19.540 and 1 minus z is the possible split of the quarks fields 01:14:19.540 --> 01:14:21.070 in the operator. 01:14:21.070 --> 01:14:23.110 So one of these guys carries z. 01:14:23.110 --> 01:14:25.330 Effectively, what this 2z minus 1 is doing 01:14:25.330 --> 01:14:27.455 is one of these guys is carrying z, and one of them 01:14:27.455 --> 01:14:28.930 is carrying 1 minus z. 01:14:28.930 --> 01:14:30.220 OK? 01:14:30.220 --> 01:14:31.750 The sum of these is 1, and that's 01:14:31.750 --> 01:14:34.420 the analog of the statement that the whole total momentum should 01:14:34.420 --> 01:14:36.175 be the pi 0 momentum. 01:14:36.175 --> 01:14:38.050 But you don't know how to split how much goes 01:14:38.050 --> 01:14:40.240 into each one of those. 01:14:40.240 --> 01:14:42.190 And what the wave function is, it's 01:14:42.190 --> 01:14:45.460 all the linear interactions that subsequently 01:14:45.460 --> 01:14:49.070 rearrange this thing before you annihilate it with the state. 01:14:49.070 --> 01:14:52.627 So all these things are dressing up the pion state. 01:14:52.627 --> 01:14:53.960 They're producing the pion pole. 01:14:53.960 --> 01:14:56.950 So this is like 5 pi. 01:14:56.950 --> 01:15:00.910 And so what you have is an operator that 01:15:00.910 --> 01:15:03.495 depends on z, a wave function that depends on z. 01:15:03.495 --> 01:15:06.100 So this is a Wilson coefficient that depends on z, 01:15:06.100 --> 01:15:08.020 and your final factorization theorem 01:15:08.020 --> 01:15:13.570 is exactly of that type, that you sort of-- 01:15:17.410 --> 01:15:21.910 of an integral of z of c of z Which also 01:15:21.910 --> 01:15:31.285 can depend on q and mu and then 5 pi of z and mu. 01:15:31.285 --> 01:15:33.760 AUDIENCE: Is there a sense of the 5 pi z as universal? 01:15:33.760 --> 01:15:33.940 IAIN STEWART: Yeah. 01:15:33.940 --> 01:15:34.565 It's universal. 01:15:34.565 --> 01:15:36.640 AUDIENCE: You can use it for other [INAUDIBLE]?? 01:15:36.640 --> 01:15:37.750 IAIN STEWART: Absolutely. 01:15:37.750 --> 01:15:40.400 Yeah. 01:15:40.400 --> 01:15:42.590 So as long as you can factor it so 01:15:42.590 --> 01:15:45.770 that it's these fields ans that pion, 01:15:45.770 --> 01:15:48.200 then you have this matrix element, you get this guy. 01:15:51.588 --> 01:15:53.630 Some people try to measure things about this guy, 01:15:53.630 --> 01:15:56.630 it's moments and stuff. 01:15:56.630 --> 01:15:57.130 All right. 01:15:59.700 --> 01:16:03.180 One thing that happens here has to do with this integral over z 01:16:03.180 --> 01:16:10.290 which is still in some sense an unsolved problem in SCET. 01:16:10.290 --> 01:16:12.300 So I have to mention it. 01:16:12.300 --> 01:16:14.820 So when you do this integral, you 01:16:14.820 --> 01:16:17.380 could ask, what does the c of z look like? 01:16:17.380 --> 01:16:21.780 And it turns out that c of z will have in it terms that 01:16:21.780 --> 01:16:24.803 go like 1 over z. 01:16:24.803 --> 01:16:26.220 And so you get integrals that look 01:16:26.220 --> 01:16:30.960 like dz over z of 5 pi of z. 01:16:30.960 --> 01:16:34.500 So at lowest order, this integral would show up. 01:16:34.500 --> 01:16:36.595 And it turns out that, for this matrix element 01:16:36.595 --> 01:16:38.470 here, you don't get anything worse than that. 01:16:38.470 --> 01:16:40.470 You never get 1 over z squared. 01:16:40.470 --> 01:16:44.760 And this integral here, because of properties of this 5 pi, 01:16:44.760 --> 01:16:45.840 is finite. 01:16:45.840 --> 01:16:47.970 There's no problems. 01:16:47.970 --> 01:16:51.390 But there are examples known in the literature, where that's 01:16:51.390 --> 01:16:54.743 not the case, where you actually get 1 over z squared, 01:16:54.743 --> 01:16:56.160 and people have some understanding 01:16:56.160 --> 01:16:59.640 of the physics that's happening in those cases. 01:16:59.640 --> 01:17:01.380 But there's not a complete understanding 01:17:01.380 --> 01:17:04.560 of how factorization works in those cases. 01:17:04.560 --> 01:17:05.130 OK? 01:17:05.130 --> 01:17:07.270 So that doesn't happen in this example, 01:17:07.270 --> 01:17:10.210 but there are other examples of exclusive processes 01:17:10.210 --> 01:17:11.710 that would lead to 1 over z squares, 01:17:11.710 --> 01:17:13.730 and then this integral is not well-defined. 01:17:21.730 --> 01:17:24.100 And people understand that there's 01:17:24.100 --> 01:17:25.918 a cut-off that's coming in, and they 01:17:25.918 --> 01:17:27.460 understand that that cut-off actually 01:17:27.460 --> 01:17:29.080 has to do with some rapidity. 01:17:29.080 --> 01:17:33.100 But how to explicitly write down an analog factorization 01:17:33.100 --> 01:17:35.320 theorem that involves those cut-offs 01:17:35.320 --> 01:17:38.950 and has renormalization group is an unsolved problem, 01:17:38.950 --> 01:17:42.110 unsolved SCETII problem. 01:17:42.110 --> 01:17:42.610 OK? 01:17:42.610 --> 01:17:44.440 But for the example we did, everything's 01:17:44.440 --> 01:17:48.140 kosher and beautiful. 01:17:48.140 --> 01:17:49.130 All right. 01:17:49.130 --> 01:17:52.550 So I think what I'll do next time, 01:17:52.550 --> 01:17:54.290 since we're out of time I'm not going 01:17:54.290 --> 01:17:56.570 to start my second example now. 01:17:56.570 --> 01:17:58.010 So next time, we'll do an example 01:17:58.010 --> 01:18:01.020 that does involve soft fields, both soft and collinear fields 01:18:01.020 --> 01:18:01.520 in SCETII. 01:18:04.600 --> 01:18:09.400 That's where we're going, but we'll leave that to next time.