WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.270 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.270 --> 00:00:18.278 at ocw.mit.edu. 00:00:21.280 --> 00:00:22.280 IAIN STEWART: All right. 00:00:22.280 --> 00:00:23.165 So let's get started. 00:00:25.850 --> 00:00:28.900 So last time, we had talked about factorization 00:00:28.900 --> 00:00:30.180 in the effective theory. 00:00:30.180 --> 00:00:33.110 And there is one type of factorization, 00:00:33.110 --> 00:00:36.860 which is this hard-collinear factorization, which 00:00:36.860 --> 00:00:39.860 is a factorization between the low energy physics described 00:00:39.860 --> 00:00:42.820 by the operator and the Wilson coefficients. 00:00:42.820 --> 00:00:44.570 And we decided we wanted to think 00:00:44.570 --> 00:00:47.120 about that in the following way, as having convolutions 00:00:47.120 --> 00:00:50.300 between variables that appear in the Wilson coefficients 00:00:50.300 --> 00:00:52.743 and variables that appear in the operator. 00:00:52.743 --> 00:00:54.410 And the way that we can think about that 00:00:54.410 --> 00:00:57.050 is just what the most general possible thing is that we can 00:00:57.050 --> 00:00:59.660 write down for the Wilson coefficients and of course, 00:00:59.660 --> 00:01:02.030 it can depend on large momenta. 00:01:02.030 --> 00:01:07.310 And so large momenta here includes the label operator. 00:01:07.310 --> 00:01:12.380 So large momenta always show up in Wilson coefficients 00:01:12.380 --> 00:01:14.990 and in this case, that includes momenta 00:01:14.990 --> 00:01:17.640 that are the large component, the order lambda 0 00:01:17.640 --> 00:01:18.890 component of collinear fields. 00:01:27.920 --> 00:01:39.680 So this is hard-collinear factorization 00:01:39.680 --> 00:01:43.210 and it just comes out in a natural, standard way 00:01:43.210 --> 00:01:45.387 in this effective theory because we've 00:01:45.387 --> 00:01:47.720 set up the effective theory to have the right low energy 00:01:47.720 --> 00:01:49.265 degrees of freedom. 00:01:49.265 --> 00:01:51.140 So even though it's a little more complicated 00:01:51.140 --> 00:01:53.240 than just a simple product, we see 00:01:53.240 --> 00:01:56.750 what the variables are that can connect the two things just 00:01:56.750 --> 00:01:58.800 by power counting, essentially. 00:01:58.800 --> 00:02:05.600 The large momenta are the ones that are order lambda 0. 00:02:05.600 --> 00:02:08.358 We can do this generically if we recall the definition 00:02:08.358 --> 00:02:09.150 of our Wilson line. 00:02:12.730 --> 00:02:14.690 We can see how this would carry over 00:02:14.690 --> 00:02:16.970 to a generic case in the following way. 00:02:23.280 --> 00:02:25.760 So that was one way of defining the Wilson line. 00:02:30.470 --> 00:02:33.470 And as an operator, we had relations as well. 00:02:38.940 --> 00:02:50.910 So we had this relation and in particular, we 00:02:50.910 --> 00:02:55.680 could take this to any power and that just takes the p bar 00:02:55.680 --> 00:03:00.420 to any power like that. 00:03:00.420 --> 00:03:02.100 So you can really think of this as given 00:03:02.100 --> 00:03:06.613 any function of this operator i m bar 00:03:06.613 --> 00:03:09.950 dot d you can always write that as a Wilson line, 00:03:09.950 --> 00:03:12.920 a function of p bar, and a Wilson line. 00:03:15.825 --> 00:03:18.200 And what you want to do is you want to stick these Wilson 00:03:18.200 --> 00:03:19.940 lines in the operator and you want 00:03:19.940 --> 00:03:23.180 to put this function in the Wilson coefficient. 00:03:23.180 --> 00:03:26.840 So you could think, lets me start with operator. 00:03:26.840 --> 00:03:29.720 I can throw in i m bar dot d's because these guys are 00:03:29.720 --> 00:03:33.102 order lambda 0, these are collinear derivatives. 00:03:33.102 --> 00:03:35.060 I could just allow it to stick as many of those 00:03:35.060 --> 00:03:37.280 as I like in my operator. 00:03:37.280 --> 00:03:39.440 But given that I put in any function of them, 00:03:39.440 --> 00:03:41.540 I could always do this and the right way 00:03:41.540 --> 00:03:46.520 of thinking about it is that this function here 00:03:46.520 --> 00:03:49.040 is determined by matching. 00:03:49.040 --> 00:03:50.768 It's your Wilson coefficient. 00:04:11.210 --> 00:04:14.940 And that's what we were effectively doing up here. 00:04:14.940 --> 00:04:17.600 But in some sense of our general discussion 00:04:17.600 --> 00:04:20.267 you can see that even if you had multiple places in the operator 00:04:20.267 --> 00:04:22.183 where you could insert these derivatives, then 00:04:22.183 --> 00:04:23.180 you would just do this. 00:04:23.180 --> 00:04:25.138 And what would happen is at the end of the day, 00:04:25.138 --> 00:04:28.160 you get a function of all the possible large momenta 00:04:28.160 --> 00:04:32.650 that you can form from the operator. 00:04:32.650 --> 00:04:36.160 So maybe one more line. 00:04:40.670 --> 00:04:43.650 So think of it, you can always separate out. 00:04:50.180 --> 00:04:51.940 Make a split like we did over there 00:04:51.940 --> 00:04:53.860 with the integral over some variable 00:04:53.860 --> 00:04:59.140 and then leave something that you can stick in your operator, 00:04:59.140 --> 00:04:59.920 make it like this. 00:04:59.920 --> 00:05:04.600 So this could go in the operator and then 00:05:04.600 --> 00:05:06.288 that's the coefficient. 00:05:09.960 --> 00:05:12.030 AUDIENCE: If I stick this in [INAUDIBLE]?? 00:05:14.688 --> 00:05:16.480 IAIN STEWART: Well, here I didn't write it. 00:05:16.480 --> 00:05:18.180 So if I wanted to put in i m bar dot 00:05:18.180 --> 00:05:20.710 d's, I'd stick them between. 00:05:20.710 --> 00:05:21.210 Right? 00:05:21.210 --> 00:05:23.270 If I wanted to use the formula. 00:05:23.270 --> 00:05:24.870 If I wanted to use this kind of logic 00:05:24.870 --> 00:05:27.210 I wouldn't have written this and I 00:05:27.210 --> 00:05:29.994 would have said let me stick an arbitrary function of i m bar 00:05:29.994 --> 00:05:30.750 dot d in here. 00:05:30.750 --> 00:05:32.812 AUDIENCE: But Between before you even write-- 00:05:32.812 --> 00:05:34.770 IAIN STEWART: It has to be between them because 00:05:34.770 --> 00:05:35.973 of the gauge invariance. 00:05:35.973 --> 00:05:36.640 AUDIENCE: Right. 00:05:36.640 --> 00:05:38.880 So before you write down the Wilson line. 00:05:38.880 --> 00:05:40.320 So say you start and say, OK-- 00:05:40.320 --> 00:05:42.070 IAIN STEWART: Oh, yeah, so there's still-- 00:05:42.070 --> 00:05:43.177 right. 00:05:43.177 --> 00:05:44.010 Right, right, right. 00:05:44.010 --> 00:05:46.770 This Wilson line here came from this h. 00:05:46.770 --> 00:05:48.500 So that's still going to be true. 00:05:48.500 --> 00:05:52.440 And what I'm saying, if you think about this operator, 00:05:52.440 --> 00:05:54.642 I could dress it up by putting any function of i 00:05:54.642 --> 00:06:00.400 m bar dot d's right in here because that would still be 00:06:00.400 --> 00:06:02.945 same order and gauge invariant. 00:06:02.945 --> 00:06:04.570 AUDIENCE: OK, but this an alternative-- 00:06:04.570 --> 00:06:09.010 IAIN STEWART: But then if I use this formula, 00:06:09.010 --> 00:06:13.510 it would sort of push this w. 00:06:13.510 --> 00:06:15.910 That would cancel there and then this w comes back 00:06:15.910 --> 00:06:16.910 and then you get the if. 00:06:21.020 --> 00:06:22.245 OK. 00:06:22.245 --> 00:06:24.370 So in general, the right way of thinking about this 00:06:24.370 --> 00:06:25.400 is as follows. 00:06:25.400 --> 00:06:38.488 We can encode this in some notation 00:06:38.488 --> 00:06:42.890 by just setting up a convenient set of building blocks 00:06:42.890 --> 00:06:45.310 which are gauge invariant objects 00:06:45.310 --> 00:06:47.556 under the collinear gauge transformations. 00:07:02.440 --> 00:07:06.160 And so we need to have a fermion field, 00:07:06.160 --> 00:07:08.500 but we know that the fermion field, 00:07:08.500 --> 00:07:12.370 generically we can make it gauge invariant by multiplying 00:07:12.370 --> 00:07:15.770 by Wilson line like that. 00:07:15.770 --> 00:07:18.910 And because of this, what we were just discussing here, 00:07:18.910 --> 00:07:21.340 generically, we can be in our Wilson coefficient, 00:07:21.340 --> 00:07:23.960 sensitive to the momentum of this object. 00:07:23.960 --> 00:07:36.020 And so we can denote that by defining the following thing, 00:07:36.020 --> 00:07:38.770 which is just a chi field that carries some momentum which 00:07:38.770 --> 00:07:41.545 is the overall large momentum of this product of fields. 00:07:45.570 --> 00:07:52.910 And sometimes this guy goes by the name of the quark jet field 00:07:52.910 --> 00:07:56.790 because if you were to produce a quark in the hard scattering 00:07:56.790 --> 00:08:00.240 process, the quark would be represented in your operator 00:08:00.240 --> 00:08:01.980 by this w dagger c. 00:08:01.980 --> 00:08:04.380 The Wilson coefficients would talk to that quark 00:08:04.380 --> 00:08:07.290 through the large momentum and that quark, 00:08:07.290 --> 00:08:10.200 in the low energy theory, would evolve into a jet. 00:08:10.200 --> 00:08:11.310 So it goes by that name. 00:08:11.310 --> 00:08:16.590 You could also think of it as a parton field 00:08:16.590 --> 00:08:21.280 because as it turns out and as we'll talk about later on, 00:08:21.280 --> 00:08:24.150 if you were to think about the parton distribution function 00:08:24.150 --> 00:08:26.460 and what pulls a quark out of the proton, 00:08:26.460 --> 00:08:28.750 it's exactly this operator. 00:08:28.750 --> 00:08:31.900 And then that quark's momentum, what 00:08:31.900 --> 00:08:35.100 momentum that quark carries is exactly picked out 00:08:35.100 --> 00:08:36.539 by this delta function as well. 00:08:39.140 --> 00:08:41.740 So these are the objects you'll usually 00:08:41.740 --> 00:08:44.420 want to work in terms of. 00:08:44.420 --> 00:08:54.360 We can do something similar for the gluon, which is a curly b 00:08:54.360 --> 00:08:57.520 and I'm going to define it. 00:08:57.520 --> 00:09:00.070 If we wanted to get a gluon, the natural thing 00:09:00.070 --> 00:09:02.530 is to use a field strength because then you get a gluon 00:09:02.530 --> 00:09:04.480 without any derivatives. 00:09:04.480 --> 00:09:09.170 But I want my object to be dimension one, 00:09:09.170 --> 00:09:11.830 so I'm going to define it in the following way. 00:09:17.480 --> 00:09:19.030 So here's a field strength commutator 00:09:19.030 --> 00:09:24.220 of covariant derivatives, these are collinear derivatives. 00:09:24.220 --> 00:09:26.980 And I throw Wilson lines around it to make it gauge invariant, 00:09:26.980 --> 00:09:29.480 and then to make it dimension one I throw in a 1 over p bar. 00:09:31.860 --> 00:09:37.690 And if you start expanding this out, 00:09:37.690 --> 00:09:40.160 then the first term will be the a n perp gluon 00:09:40.160 --> 00:09:43.160 and so you have this b just has a n perp gluon in the same way 00:09:43.160 --> 00:09:45.440 that the first kind of order term in this chi 00:09:45.440 --> 00:09:49.140 is just to the fermion if I set the Wilson line to 1. 00:09:49.140 --> 00:09:49.640 OK? 00:09:49.640 --> 00:09:52.743 So it's starting out as a perpendicular gluon 00:09:52.743 --> 00:09:55.160 but it's dressed up in a way that makes it gauge invariant 00:09:55.160 --> 00:09:55.940 in dimension one. 00:10:10.100 --> 00:10:15.410 And then just like we did here, we can put a subscript on it 00:10:15.410 --> 00:10:21.370 with a omega to say that we fixed the large momentum of it 00:10:21.370 --> 00:10:27.110 and sometimes there's a convention that that's done, 00:10:27.110 --> 00:10:29.600 you have to decide if you want the Wilson line-- 00:10:29.600 --> 00:10:32.540 I mean, the delta function to be a w minus p bar or w 00:10:32.540 --> 00:10:33.340 minus p bar dagger. 00:10:36.170 --> 00:10:38.840 Anyway, it doesn't really matter that much because it's just 00:10:38.840 --> 00:10:41.580 a sign of what you mean by a w. 00:10:41.580 --> 00:10:44.060 If it's an outgoing gluon, this is a more convenient 00:10:44.060 --> 00:10:46.760 convention. 00:10:46.760 --> 00:10:49.820 OK, and so this delta function here and this p bar 00:10:49.820 --> 00:10:51.890 here, when I put these square brackets, what 00:10:51.890 --> 00:10:54.590 I mean by that is that they don't know how to act outside. 00:10:54.590 --> 00:10:57.740 They just act on the operators inside. 00:10:57.740 --> 00:11:05.008 So these are objects that exist by themselves 00:11:05.008 --> 00:11:06.800 and they don't care about other things that 00:11:06.800 --> 00:11:08.210 are multiplying them later. 00:11:17.500 --> 00:11:21.360 So that's the gluon analog of the quark. 00:11:21.360 --> 00:11:29.670 So it turns out that we can show the following. 00:11:29.670 --> 00:11:31.500 That if you want to build operators 00:11:31.500 --> 00:11:33.360 that are subleading order, a complete set 00:11:33.360 --> 00:11:35.375 of things to do that is as follows. 00:11:35.375 --> 00:11:44.990 It's this chi n, the bn perp, and then you 00:11:44.990 --> 00:11:48.930 could also need p perps. 00:11:48.930 --> 00:11:53.440 And then you could also have ultrasoft fields, 00:11:53.440 --> 00:11:55.150 where it's really kind of similar to how 00:11:55.150 --> 00:11:57.317 you're used to building operators in effective field 00:11:57.317 --> 00:11:58.460 theories. 00:11:58.460 --> 00:12:00.040 But if you just want to talk about the collinear sector, 00:12:00.040 --> 00:12:02.290 the only things you need are this chi n this bn field, 00:12:02.290 --> 00:12:04.230 and then p perps and no other things. 00:12:04.230 --> 00:12:05.815 AUDIENCE: That's to go just one order? 00:12:05.815 --> 00:12:07.010 IAIN STEWART: All orders. 00:12:07.010 --> 00:12:08.468 AUDIENCE: So why aren't there any-- 00:12:08.468 --> 00:12:11.320 IAIN STEWART: I'll talk about it. 00:12:11.320 --> 00:12:11.937 Yeah. 00:12:11.937 --> 00:12:12.520 I'll show you. 00:12:12.520 --> 00:12:13.360 AUDIENCE: OK. 00:12:13.360 --> 00:12:14.170 IAIN STEWART: Yeah. 00:12:14.170 --> 00:12:16.148 I mean, intuitively you would think 00:12:16.148 --> 00:12:18.190 these are the physical degrees of freedom, right? 00:12:18.190 --> 00:12:20.147 There's two of them so that's intuitive. 00:12:20.147 --> 00:12:22.480 But I'll show you how you get rid of all the other ones. 00:12:32.749 --> 00:12:34.517 It kind of matches up with what you want. 00:12:34.517 --> 00:12:37.100 You could think that these are the physical gluons that you're 00:12:37.100 --> 00:12:42.500 producing and your operators can be set up so that there's not 00:12:42.500 --> 00:12:48.210 any spare use components. 00:12:48.210 --> 00:12:48.860 OK. 00:12:48.860 --> 00:12:53.280 So we'll get there. 00:12:53.280 --> 00:12:55.685 So let me introduce a bit of notation. 00:12:58.622 --> 00:12:59.600 I have to do. 00:13:09.322 --> 00:13:10.530 So let's consider this thing. 00:13:10.530 --> 00:13:11.905 A covariant derivative sandwiched 00:13:11.905 --> 00:13:15.810 with Wilson lines on either side and this i dn 00:13:15.810 --> 00:13:18.360 here is like p bar plus. 00:13:25.853 --> 00:13:27.770 So it just involves the collinear gluon field. 00:13:31.440 --> 00:13:36.080 So then if I have that operator and I just take n bar i dn, 00:13:36.080 --> 00:13:36.830 then that's p bar. 00:13:39.470 --> 00:13:46.280 if I take i dn perp mu, then you can 00:13:46.280 --> 00:13:54.990 show that this guy is p perp u plus g bn perp mu. 00:13:57.740 --> 00:13:59.990 So this one is just a relation we talked about before. 00:13:59.990 --> 00:14:03.770 This one is not as obvious, but if you take this guy, 00:14:03.770 --> 00:14:08.030 you can let this derivative p perp act on the Wilson line 00:14:08.030 --> 00:14:08.990 or act through. 00:14:08.990 --> 00:14:11.150 If it acts through, that's this term. 00:14:11.150 --> 00:14:14.810 If it acts on, then I can manipulate the operator 00:14:14.810 --> 00:14:17.590 so that it is exactly this form. 00:14:17.590 --> 00:14:20.420 And part of that comes from the fact that-- 00:14:20.420 --> 00:14:21.830 I can explain it to you here. 00:14:21.830 --> 00:14:23.955 I have it in my notes and you can look at it later, 00:14:23.955 --> 00:14:25.590 but let me just explain it in words. 00:14:25.590 --> 00:14:27.423 If you looked at the term in this commutator 00:14:27.423 --> 00:14:31.452 that was other order, where the i m bar dot d was sitting here. 00:14:31.452 --> 00:14:32.660 It would hit the Wilson line. 00:14:32.660 --> 00:14:33.410 You'd get 0. 00:14:33.410 --> 00:14:34.910 So that term was just put in to make 00:14:34.910 --> 00:14:41.770 it look like a field strength. 00:14:41.770 --> 00:14:42.270 OK? 00:14:42.270 --> 00:14:46.700 So really you have i m bar dot d dn perp, without the comma, 00:14:46.700 --> 00:14:48.020 without the brackets. 00:14:48.020 --> 00:14:54.110 But then this combination here, you can use the identity 00:14:54.110 --> 00:14:56.653 that we had with it and basically if you 00:14:56.653 --> 00:14:58.070 push this guy through here, you're 00:14:58.070 --> 00:15:01.640 canceling the p bar so it's just giving you the Wilson line. 00:15:01.640 --> 00:15:06.420 So that's basically how you go from here to here. 00:15:06.420 --> 00:15:06.920 OK? 00:15:06.920 --> 00:15:12.657 So basically what this is saying is 00:15:12.657 --> 00:15:14.990 I don't need to consider covariant derivatives because I 00:15:14.990 --> 00:15:16.820 can instead consider p perps and b 00:15:16.820 --> 00:15:18.840 perps and that's equally good. 00:15:23.260 --> 00:15:25.900 OK. 00:15:25.900 --> 00:15:34.120 Now if you do a similar type of thing in the n component, 00:15:34.120 --> 00:15:36.620 then you can derive a similar type of object. 00:15:36.620 --> 00:15:38.200 So this is the same object as we have 00:15:38.200 --> 00:15:40.637 over here, but instead of having an i dn perp, 00:15:40.637 --> 00:15:41.470 I'd have an n dot d. 00:15:50.782 --> 00:15:53.900 So d dot n perp replaced n dot d. 00:15:56.482 --> 00:15:58.190 And that looks like it could be something 00:15:58.190 --> 00:16:02.450 that you'd build operators out of and and also, furthermore, 00:16:02.450 --> 00:16:05.500 why not have operators that depend on this n dot partial? 00:16:05.500 --> 00:16:06.000 OK? 00:16:09.470 --> 00:16:12.180 So for p bar, we don't have to worry about that. 00:16:12.180 --> 00:16:15.797 So OK, I said we'll include this, we'll include that. 00:16:15.797 --> 00:16:17.880 I didn't say we have include this, this, and this. 00:16:17.880 --> 00:16:19.963 So there's three things I have to argue away here. 00:16:22.460 --> 00:16:24.440 The derivatives that are p bars, those just 00:16:24.440 --> 00:16:26.250 go into the Wilson coefficient. 00:16:26.250 --> 00:16:32.340 So if we had p bar chi n comma omega, for example, 00:16:32.340 --> 00:16:35.810 then that's omega chi n comma omega 00:16:35.810 --> 00:16:40.940 and this is put into the Wilson coefficient. 00:16:40.940 --> 00:16:45.580 So that's why we don't have to worry about having p bars. 00:16:45.580 --> 00:16:46.970 In some sense, we do have p bars. 00:16:46.970 --> 00:16:49.437 They're all in the Wilson coefficients. 00:16:49.437 --> 00:16:51.020 So it's really these other two that we 00:16:51.020 --> 00:16:54.270 have to worry more about and those actually 00:16:54.270 --> 00:16:56.400 can be simplified using the equation of motion. 00:16:56.400 --> 00:17:01.940 So if you have i n dot p partial on chi n, then 00:17:01.940 --> 00:17:04.280 equation of motion when you write it out 00:17:04.280 --> 00:17:14.110 in terms of these objects, it has some form 00:17:14.110 --> 00:17:16.235 and basically, you can just get rid of those terms. 00:17:23.650 --> 00:17:27.470 So the equation of motion allows us to get rid of i n dot 00:17:27.470 --> 00:17:28.920 partials that are on chi n's. 00:17:33.730 --> 00:17:38.440 So that's why we don't have to worry about those. 00:17:38.440 --> 00:17:41.110 And this is just like saying that in our leading order 00:17:41.110 --> 00:17:45.035 action, there wasn't i n dot partial. 00:17:45.035 --> 00:17:46.410 But that's just like saying there 00:17:46.410 --> 00:17:48.400 was time derivatives in our leading order 00:17:48.400 --> 00:17:51.370 action in some standard effective field theory. 00:17:51.370 --> 00:17:53.520 But then in the higher dimension operators, 00:17:53.520 --> 00:17:55.270 you can always use the equations of motion 00:17:55.270 --> 00:17:56.650 to get rid of those time derivatives. 00:17:56.650 --> 00:17:58.120 And here we're using the equation of motion 00:17:58.120 --> 00:17:59.560 to get rid of i n dot partial. 00:17:59.560 --> 00:18:01.310 So it appears in the leading order action, 00:18:01.310 --> 00:18:03.790 but then we don't have to have it in any other subleading 00:18:03.790 --> 00:18:08.770 operator and that's why it's not one of the ones that's 00:18:08.770 --> 00:18:11.090 included in the list. 00:18:11.090 --> 00:18:13.690 And if you had i n dot partial on curly b, 00:18:13.690 --> 00:18:15.490 that's also part of the equations 00:18:15.490 --> 00:18:16.750 of motion of the gluon field. 00:18:20.773 --> 00:18:22.690 Now when you do the gluon equations of motion, 00:18:22.690 --> 00:18:25.337 there's components because it's a vector 00:18:25.337 --> 00:18:26.920 and one of the other components allows 00:18:26.920 --> 00:18:30.370 you to get rid of the n dot b. 00:18:30.370 --> 00:18:33.528 So there's another term that you can rearrange, 00:18:33.528 --> 00:18:36.070 and I won't write it out because the equation is rather messy 00:18:36.070 --> 00:18:42.370 but give you some idea. 00:18:45.090 --> 00:18:50.230 There's another component that looks 00:18:50.230 --> 00:18:54.078 like this where I can get rid of all the n dot curly b's 00:18:54.078 --> 00:18:55.120 using the gluon equation. 00:18:55.120 --> 00:18:56.650 So the gluon equations of motions 00:18:56.650 --> 00:19:00.480 allows me to get rid of both of these things 00:19:00.480 --> 00:19:02.230 and basically after I've done that, I just 00:19:02.230 --> 00:19:05.110 have the objects that I've told you we can use. 00:19:12.510 --> 00:19:14.150 So after using equations of motion, 00:19:14.150 --> 00:19:15.800 we can get down to those objects. 00:19:18.850 --> 00:19:22.220 And any other thing that you might dream up 00:19:22.220 --> 00:19:26.900 can be reduced to these objects. 00:19:26.900 --> 00:19:30.410 So I'm not saying that I went through a complete list here. 00:19:30.410 --> 00:19:35.690 For example, what if you had a commutator of 2 d n perps? 00:19:35.690 --> 00:19:36.190 Right? 00:19:36.190 --> 00:19:39.640 You might say, oh, that's some new thing. 00:19:39.640 --> 00:19:40.390 It's one of these. 00:19:40.390 --> 00:19:41.598 You can also reduce that too. 00:20:03.458 --> 00:20:05.230 So let me list that one just to give you 00:20:05.230 --> 00:20:08.650 some idea that there's others you might think of dreaming up. 00:20:29.990 --> 00:20:33.310 So you can reduce all of this to be that. 00:20:33.310 --> 00:20:35.040 OK, so for the collinear sector this 00:20:35.040 --> 00:20:38.010 is enough to build higher dimension operators, just 00:20:38.010 --> 00:20:40.350 these three. 00:20:40.350 --> 00:20:47.730 And then for the ultrasoft sector, so I guess this is two. 00:20:47.730 --> 00:21:06.800 We do need ultrasoft derivatives and ultrasoft field strengths 00:21:06.800 --> 00:21:07.745 and ultrasoft quarks. 00:21:11.540 --> 00:21:13.700 So this part is really just similar to the story 00:21:13.700 --> 00:21:17.240 that you'd have for a standard low energy effective field 00:21:17.240 --> 00:21:18.830 theory. 00:21:18.830 --> 00:21:20.540 Like integrating out a massive particle, 00:21:20.540 --> 00:21:21.998 you can use the equation of motion. 00:21:29.160 --> 00:21:32.227 One thing that is worth commenting about 00:21:32.227 --> 00:21:33.810 is the connection between one and two. 00:21:33.810 --> 00:21:36.180 So one is collinear and two is ultrasoft 00:21:36.180 --> 00:21:39.400 and prior you might think, well, they're totally independent. 00:21:39.400 --> 00:21:41.730 But we saw last time that reparamaterization invariance 00:21:41.730 --> 00:21:43.540 connects them. 00:21:43.540 --> 00:21:46.260 So if I've decided that this is the type of basis 00:21:46.260 --> 00:21:47.820 I want to use for my operators, then 00:21:47.820 --> 00:21:51.360 what is the reparameterization connection? 00:21:51.360 --> 00:21:53.970 You can rewrite what we said last time in terms 00:21:53.970 --> 00:21:56.850 of these curly d's because it basically just means 00:21:56.850 --> 00:22:02.310 moving the Wilson lines that we had in this formula last time. 00:22:02.310 --> 00:22:06.558 They were around the ultrasoft operator last time 00:22:06.558 --> 00:22:08.850 and if I just move them over to the collinear operator, 00:22:08.850 --> 00:22:11.500 then I get this curly d. 00:22:11.500 --> 00:22:12.000 Right? 00:22:12.000 --> 00:22:17.710 So the RPI connection is connecting 00:22:17.710 --> 00:22:21.757 the curly d collinear to the d ultrasoft 00:22:21.757 --> 00:22:24.340 and then you can write this guy as the p perp plus the b perp. 00:22:26.990 --> 00:22:32.070 And likewise if you do the same for the n bar sector 00:22:32.070 --> 00:22:35.580 and you move the Wilson lines from this term to that term, 00:22:35.580 --> 00:22:37.730 then it looks like these two combinations. 00:22:37.730 --> 00:22:40.230 So that's just rewriting what we had before but now in terms 00:22:40.230 --> 00:22:41.915 of this type of notation. 00:22:44.210 --> 00:22:44.710 OK? 00:22:44.710 --> 00:22:48.250 So that's enough to build operators at higher order 00:22:48.250 --> 00:22:51.893 and then we write down Wilson coefficients 00:22:51.893 --> 00:22:54.310 for those operators that are functions of a large momentum 00:22:54.310 --> 00:22:57.970 and then we start doing physics with them. 00:22:57.970 --> 00:22:59.200 So any questions about that? 00:23:04.480 --> 00:23:06.160 AUDIENCE: For the equations of motion, 00:23:06.160 --> 00:23:09.058 you always just use the leading order equation of motion-- 00:23:09.058 --> 00:23:09.850 IAIN STEWART: Yeah. 00:23:09.850 --> 00:23:11.350 AUDIENCE: --into a higher order term? 00:23:11.350 --> 00:23:13.892 IAIN STEWART: Yeah, these are the L0 equations of the motion. 00:23:13.892 --> 00:23:16.070 That's what I'm doing here or writing here. 00:23:22.022 --> 00:23:23.630 Yeah, there's one more actually. 00:23:23.630 --> 00:23:32.190 There's three but it's only needed at some very high order. 00:23:32.190 --> 00:23:32.690 OK. 00:23:32.690 --> 00:23:35.780 So the next thing I want to talk about 00:23:35.780 --> 00:23:38.300 before we start doing explicit examples 00:23:38.300 --> 00:23:40.130 and going through processes is how loops 00:23:40.130 --> 00:23:41.690 work in this effective theory. 00:23:46.590 --> 00:23:49.360 And we're going to have to come back and talk about our grid 00:23:49.360 --> 00:23:51.490 that we have for the split up of momenta and how 00:23:51.490 --> 00:23:59.430 should we actually think about it in practice 00:23:59.430 --> 00:24:01.990 and then we'll also deal with how matching and running work. 00:24:07.180 --> 00:24:09.400 So I'm going to do this in the context of an example 00:24:09.400 --> 00:24:11.920 and again, I'm just going to pick the simplest example that 00:24:11.920 --> 00:24:16.285 has only one jet just to make our lives a little bit simpler. 00:24:19.240 --> 00:24:30.180 So we'll consider our heavy, light current 00:24:30.180 --> 00:24:31.750 and I think actually once you see 00:24:31.750 --> 00:24:33.970 how it works in this example, you'll 00:24:33.970 --> 00:24:38.500 understand what all the general features are of doing loops 00:24:38.500 --> 00:24:39.000 in SCET. 00:24:47.400 --> 00:24:52.190 So we had operators that we constructed, 00:24:52.190 --> 00:24:53.767 lowest order operator. 00:24:58.740 --> 00:25:01.520 And let me write it in this way which 00:25:01.520 --> 00:25:06.010 was prior to making our field redefinition. 00:25:06.010 --> 00:25:09.200 I could just write it this way if I want. 00:25:09.200 --> 00:25:17.900 And gamma for beta s gamma is a tensor operator. 00:25:17.900 --> 00:25:23.290 And there's a photon field, which is a field strength, 00:25:23.290 --> 00:25:24.230 f mu nu. 00:25:24.230 --> 00:25:27.708 So let's just think of that as all part of gamma. 00:25:27.708 --> 00:25:29.800 OK, and that's also appearing here 00:25:29.800 --> 00:25:32.035 and I could use the spin structure properties 00:25:32.035 --> 00:25:33.910 of these things to reduce the sigma and mu nu 00:25:33.910 --> 00:25:37.220 but that's not really going to be part of our story 00:25:37.220 --> 00:25:39.850 so let's not bother with that. 00:25:39.850 --> 00:25:40.540 So what I do? 00:25:40.540 --> 00:25:43.480 I just compute the QCD loops and the SCET loops 00:25:43.480 --> 00:25:45.672 and I compare them. 00:25:45.672 --> 00:25:47.380 And if SCET is the right effective theory 00:25:47.380 --> 00:25:51.600 for this limit where I have an energetic photon 00:25:51.600 --> 00:25:57.310 and am back to back with an energetic strange quark, 00:25:57.310 --> 00:25:59.310 then I should match all the infrared divergences 00:25:59.310 --> 00:26:01.950 in that QCD computation, I should 00:26:01.950 --> 00:26:05.160 be able to extract a matching coefficient, 00:26:05.160 --> 00:26:06.720 I should be able to determine what 00:26:06.720 --> 00:26:12.123 the C is from that calculation, and I should 00:26:12.123 --> 00:26:13.290 be able to run the operator. 00:26:13.290 --> 00:26:16.650 I should be able to make this into m s bar, 00:26:16.650 --> 00:26:19.690 do some renormalization of this operator, 00:26:19.690 --> 00:26:22.530 and then do some renormalization group evolution of that Wilson 00:26:22.530 --> 00:26:23.541 coefficient. 00:26:25.632 --> 00:26:27.715 Well, let's just think about computing the graphs. 00:26:35.920 --> 00:26:38.220 So we have to decide, when we compute the graphs, 00:26:38.220 --> 00:26:39.890 how to regulate them? 00:26:39.890 --> 00:26:40.500 Right? 00:26:40.500 --> 00:26:42.420 And we need to use the same infrared regulator 00:26:42.420 --> 00:26:45.720 in the full theory and the effective theory. 00:26:45.720 --> 00:26:47.460 So here's how I'm going to regulate them. 00:26:47.460 --> 00:26:50.430 We could do this different ways and it's 00:26:50.430 --> 00:26:54.120 useful to understand that various answers that we get 00:26:54.120 --> 00:26:55.807 are independent of the regulator. 00:26:59.710 --> 00:27:02.410 So I'm going to take p squared not equal to 0 00:27:02.410 --> 00:27:03.570 for the strange quark. 00:27:11.320 --> 00:27:13.330 So there's infrared divergences associated 00:27:13.330 --> 00:27:15.300 with the strange quirk and I'm going 00:27:15.300 --> 00:27:19.450 to take p squared not equal to 0 for them. 00:27:19.450 --> 00:27:22.840 I'm going to use dim reg for the heavy quark. 00:27:29.340 --> 00:27:32.010 So I could take the b quark to also be offshell. 00:27:32.010 --> 00:27:34.570 That would make the formulas even more complicated. 00:27:34.570 --> 00:27:35.987 So instead of doing that, I'm just 00:27:35.987 --> 00:27:38.460 going to allow epsilon to regulate the b quark. 00:27:38.460 --> 00:27:39.940 These guys are pretty easy to track 00:27:39.940 --> 00:27:44.000 so that won't be a problem. 00:27:44.000 --> 00:27:45.540 And I'm going to use Feynman gauge. 00:27:51.020 --> 00:27:53.830 I don't have to do that, but that's a nice gauge 00:27:53.830 --> 00:27:56.480 for doing calculations. 00:27:56.480 --> 00:27:59.810 So what are the diagrams where I have my photon 00:27:59.810 --> 00:28:02.893 and I have a vertex diagram like this 00:28:02.893 --> 00:28:04.810 and then I have wave function renormalization? 00:28:04.810 --> 00:28:08.800 And so let me, in the usual kind of cavalier way, 00:28:08.800 --> 00:28:11.380 denote the wave function renormalization, which 00:28:11.380 --> 00:28:15.220 is just the multiplication by the appropriate z factors. 00:28:15.220 --> 00:28:16.350 My diagram's like that. 00:28:20.190 --> 00:28:22.910 And this is a standard QCD calculation 00:28:22.910 --> 00:28:25.920 and we can carry it out. 00:28:29.572 --> 00:28:31.030 And what does the answer look like? 00:28:37.960 --> 00:28:51.270 So this diagram here has double logs 00:28:51.270 --> 00:28:56.040 and single logs that involve p squared, which 00:28:56.040 --> 00:29:02.500 is our IR regulator, and then it has some terms that are finite, 00:29:02.500 --> 00:29:04.750 which I'll just note by-- 00:29:04.750 --> 00:29:09.390 so p here is the momentum of our strange quark going out 00:29:09.390 --> 00:29:15.090 and pb is the momentum of our b quark coming in. 00:29:15.090 --> 00:29:19.110 And we're not going to talk much about these finite terms 00:29:19.110 --> 00:29:29.970 but what I mean by finite here is 00:29:29.970 --> 00:29:32.930 terms with no IR divergences. 00:29:32.930 --> 00:29:35.670 The IR divergences are these single logs and double logs 00:29:35.670 --> 00:29:41.070 and then there's some remainder that I can write that way. 00:29:41.070 --> 00:29:43.060 I'm expanding it for small p squared. 00:29:43.060 --> 00:29:45.060 So p squared is not equal to 0 but I 00:29:45.060 --> 00:29:46.770 take the limit p squared goes to 0 00:29:46.770 --> 00:29:49.230 and then that limit, these are the IR singularities. 00:29:49.230 --> 00:29:52.560 And then there's some function of b dot pb, which 00:29:52.560 --> 00:29:54.010 is just an order one thing. 00:29:54.010 --> 00:29:56.020 And then b squared, of course, is pb squared. 00:29:56.020 --> 00:29:58.145 So that's sort of the remaining kinematic variables 00:29:58.145 --> 00:30:00.696 this could possibly depend on. 00:30:00.696 --> 00:30:03.060 AUDIENCE: [INAUDIBLE]? 00:30:03.060 --> 00:30:04.138 IAIN STEWART: No. 00:30:04.138 --> 00:30:05.380 AUDIENCE: Is it finite? 00:30:05.380 --> 00:30:07.960 IAIN STEWART: It's really finite. 00:30:07.960 --> 00:30:11.890 So yeah. 00:30:11.890 --> 00:30:15.890 Yeah, so let's see. 00:30:15.890 --> 00:30:18.910 Yeah, I guess I carried out the so there 00:30:18.910 --> 00:30:26.380 is a z ten for the tensor and when I write this-- 00:30:26.380 --> 00:30:29.560 yeah, let's see. 00:30:29.560 --> 00:30:30.630 So, OK. 00:30:33.490 --> 00:30:34.960 Let me do this. 00:30:34.960 --> 00:30:36.130 I think it's better. 00:30:36.130 --> 00:30:39.640 So I added up all these graphs and I want to add one more. 00:30:42.730 --> 00:30:47.570 So there's a counterterm for the tensor field. 00:30:47.570 --> 00:30:49.400 So let me change what I was going 00:30:49.400 --> 00:30:50.560 to say and do it this way. 00:30:50.560 --> 00:30:53.690 So the sum of these four graphs is this. 00:30:53.690 --> 00:30:54.190 OK? 00:30:54.190 --> 00:30:56.260 So there's no UV divergences. 00:30:56.260 --> 00:30:58.843 And I'll just tell you what the z factors for these three are, 00:30:58.843 --> 00:31:00.677 then you could figure out what this graph is 00:31:00.677 --> 00:31:01.800 by subtracting the two. 00:31:07.140 --> 00:31:15.440 So the tensor current in QCD has a z factor, looks like that. 00:31:21.200 --> 00:31:28.070 There is a z for the heavy quark. 00:31:28.070 --> 00:31:30.390 And if I include the finite residue as well as 00:31:30.390 --> 00:31:37.275 the divergent pieces, this looks like this. 00:31:43.498 --> 00:31:45.040 I think I'm going to have to make one 00:31:45.040 --> 00:31:46.510 adjustment to my formula here. 00:32:11.000 --> 00:32:14.390 Let me just fix something here. 00:32:14.390 --> 00:32:15.230 Yeah. 00:32:15.230 --> 00:32:21.180 So if I want it to be the way I say, then what do I have to do? 00:32:21.180 --> 00:32:28.060 So this guy should be three halves 00:32:28.060 --> 00:32:33.000 and there will be one more divergence. 00:32:36.904 --> 00:32:39.560 I think that's right. 00:32:39.560 --> 00:32:41.990 So this 2 over epsilon ir here is 00:32:41.990 --> 00:32:46.160 that 1 over epsilon ir and the UV renormalization 00:32:46.160 --> 00:32:49.550 is taken care of once I have the z tensor. 00:32:49.550 --> 00:32:51.788 There's divergences in this diagram in these ones. 00:32:51.788 --> 00:32:53.330 but there's one left over, and that's 00:32:53.330 --> 00:32:54.690 taken care of by z tensor. 00:32:54.690 --> 00:33:00.020 So there's no 1 over epsilon UVs after I take care of this guy. 00:33:10.170 --> 00:33:15.170 So that's just by the definition of what z tensor should be 00:33:15.170 --> 00:33:17.030 and then everything else is as I wrote. 00:33:19.680 --> 00:33:21.420 So there's either divergences that 00:33:21.420 --> 00:33:24.270 are associated to this strange quirk offshellness, that's 00:33:24.270 --> 00:33:24.960 these two terms. 00:33:24.960 --> 00:33:27.480 There's an IR divergence associated, a heavy quark going 00:33:27.480 --> 00:33:29.698 onshell, that's that term. 00:33:29.698 --> 00:33:30.990 And that's the sum of diagrams. 00:33:35.810 --> 00:33:36.950 OK, so what about SCET? 00:33:42.550 --> 00:33:44.860 So there's going to be, in SCET, collinear diagrams 00:33:44.860 --> 00:33:46.510 and ultrasoft diagrams. 00:34:09.989 --> 00:34:14.560 So I'm going to use Feynman gauge for everything again. 00:34:14.560 --> 00:34:17.020 This is not something I have to do, 00:34:17.020 --> 00:34:18.580 but this is what I'm going to do. 00:34:21.679 --> 00:34:23.500 So let's start with the ultrasoft loops. 00:34:27.989 --> 00:34:29.310 So there's a vertex graph. 00:34:31.830 --> 00:34:36.440 So using the notation that we've adopted where the collinear 00:34:36.440 --> 00:34:39.940 quarks are dashed and the heavy quarks are double lines, 00:34:39.940 --> 00:34:42.790 we have a diagram that looks like that. 00:34:42.790 --> 00:34:45.210 There's some free factor. 00:34:45.210 --> 00:34:48.940 Let's focus on what the loop looks like. 00:34:48.940 --> 00:34:52.590 So this loop here, the k that's going through this loop 00:34:52.590 --> 00:34:55.610 is just a residual k. 00:34:55.610 --> 00:34:57.690 There's no label k for this loop because it's 00:34:57.690 --> 00:34:59.640 an ultrasoft gluon. 00:34:59.640 --> 00:35:02.670 So when we write down all the terms in this loop, 00:35:02.670 --> 00:35:04.350 it's just standard field theory. 00:35:04.350 --> 00:35:06.780 There's nothing special about it. 00:35:16.100 --> 00:35:21.890 Since I'm taking the strange quirk offshell, 00:35:21.890 --> 00:35:25.340 the propagator that I get is this shifted iconal propagator 00:35:25.340 --> 00:35:27.320 where basically the fact that it's offshell 00:35:27.320 --> 00:35:29.360 it gives me this extra term there. 00:35:29.360 --> 00:35:31.840 That's regulating some minor divergences. 00:35:31.840 --> 00:35:34.850 And we want that because we want to regulate the IR divergences 00:35:34.850 --> 00:35:39.920 in the same way in the full theory of the effective theory. 00:35:39.920 --> 00:35:45.130 This is proportional to that and if I put in all the factors. 00:36:16.700 --> 00:36:20.880 So it does have double logs of the p squared, 00:36:20.880 --> 00:36:24.128 doesn't have single logs of the p squared, 00:36:24.128 --> 00:36:25.920 and actually if we look at the double logs, 00:36:25.920 --> 00:36:27.363 the coefficients also don't match 00:36:27.363 --> 00:36:28.530 with what we had over there. 00:36:28.530 --> 00:36:32.320 And so there's going to be some other diagrams that 00:36:32.320 --> 00:36:36.200 are going to involve double logs of p squared as well. 00:36:36.200 --> 00:36:37.960 One thing that we can note here is 00:36:37.960 --> 00:36:40.578 that if you think about the scales in the problem, 00:36:40.578 --> 00:36:42.370 remember that our loop integral was totally 00:36:42.370 --> 00:36:43.870 homogeneous in the power counting, 00:36:43.870 --> 00:36:46.480 the lambdas were totally homogeneous. 00:36:46.480 --> 00:36:49.750 And if you think about p squared scaling 00:36:49.750 --> 00:36:52.840 like lambda squared, which is natural size 00:36:52.840 --> 00:36:56.740 for an external collinear momentum, all right. 00:36:56.740 --> 00:37:00.000 If p squared scales like lambda squared then 00:37:00.000 --> 00:37:02.710 so does p squared over m bar dot p. 00:37:05.510 --> 00:37:07.270 And this is a dimension one thing 00:37:07.270 --> 00:37:10.000 and this is the ultrasoft scale, right? 00:37:15.640 --> 00:37:22.445 For some ultra soft momentum and so 00:37:22.445 --> 00:37:24.070 if you want to look at these logarithms 00:37:24.070 --> 00:37:26.890 and you ask, what scale is this effective field theory 00:37:26.890 --> 00:37:28.592 diagram sensitive to? 00:37:28.592 --> 00:37:30.175 It's sensitive to the ultrasoft scale. 00:37:35.370 --> 00:37:37.710 So the logs are not large logs, they're 00:37:37.710 --> 00:37:41.020 order 1 logs as long as mu squared 00:37:41.020 --> 00:37:44.010 mu is of order lambda squared, which 00:37:44.010 --> 00:37:48.350 is the scale for the ultrasoft momentum. 00:37:48.350 --> 00:37:50.653 And that's what we expect from this ultrasoft diagram, 00:37:50.653 --> 00:37:53.070 that it's telling us about physics at the ultra soft scale 00:37:53.070 --> 00:37:55.980 and that's what we see setting things 00:37:55.980 --> 00:37:58.720 up, doing the calculation. 00:38:02.502 --> 00:38:04.710 So you could also think about a wave function diagram 00:38:04.710 --> 00:38:07.725 with an ultra soft gluon, but this guy 0's 00:38:07.725 --> 00:38:14.580 out since in Feynman gauge you just get n mu n mu, which is 0. 00:38:14.580 --> 00:38:20.955 So there's no so z for the collinear quark 00:38:20.955 --> 00:38:24.540 field from an ultrasoft loop is 0. 00:38:24.540 --> 00:38:27.340 That wouldn't necessarily be true in some other gauge, 00:38:27.340 --> 00:38:30.090 but in some other gauge this diagram would also change. 00:38:30.090 --> 00:38:31.390 All the diagrams would change. 00:38:31.390 --> 00:38:35.040 And so if there was a non-zero contribution in this diagram, 00:38:35.040 --> 00:38:39.600 it would just be taking care of gauge invariance. 00:38:39.600 --> 00:38:43.560 And then finally, there's ultrasoft loop 00:38:43.560 --> 00:38:46.560 on the heavy quark. 00:38:46.560 --> 00:38:50.490 And that's an HQET diagram, has nothing 00:38:50.490 --> 00:38:53.000 to do with the collinear quark. 00:38:53.000 --> 00:38:54.450 So from that diagram we would just 00:38:54.450 --> 00:39:02.680 get the z factor, which is the appropriate z factor in HQET 00:39:02.680 --> 00:39:05.350 with r regulator. 00:39:05.350 --> 00:39:08.260 If I specify ultraviolet and infrared divergences, 00:39:08.260 --> 00:39:11.020 that's this, and this infrared divergence here 00:39:11.020 --> 00:39:15.780 is actually exactly the same as the one that we had over here. 00:39:24.240 --> 00:39:26.890 OK, so the ultrasoft sector, there's 00:39:26.890 --> 00:39:28.300 nothing really tricky about it. 00:39:28.300 --> 00:39:32.423 It's just write down the diagrams, do the loops. 00:39:32.423 --> 00:39:33.840 AUDIENCE: So for the [INAUDIBLE],, 00:39:33.840 --> 00:39:37.510 isn't that IR divergences of those diagrams 00:39:37.510 --> 00:39:40.070 cancel with every automation [INAUDIBLE]?? 00:39:43.352 --> 00:39:45.310 IAIN STEWART: So what are we looking at, right? 00:39:45.310 --> 00:39:48.560 So it depends on what we're looking at here. 00:39:48.560 --> 00:39:52.778 And so, yes, in general that would be true, right? 00:39:52.778 --> 00:39:54.820 If you were calculating some cross-section, which 00:39:54.820 --> 00:39:57.727 was IR finite cross-section and that, 00:39:57.727 --> 00:39:59.560 of course, would depend on defining what you 00:39:59.560 --> 00:40:01.580 mean by measuring this quark. 00:40:01.580 --> 00:40:02.080 Right? 00:40:02.080 --> 00:40:04.360 So the IR divergence would become some physical scale 00:40:04.360 --> 00:40:07.330 like the mass of a jet. 00:40:10.720 --> 00:40:13.360 The IR divergences would turn into something physical 00:40:13.360 --> 00:40:17.420 if you put this into a physical cross-section. 00:40:17.420 --> 00:40:21.280 And that's exactly basically what would happen. 00:40:21.280 --> 00:40:24.310 These p squareds would become the mx squareds 00:40:24.310 --> 00:40:27.665 that we talked about when we talked about beta s gamma. 00:40:27.665 --> 00:40:29.290 But here what we're interested in doing 00:40:29.290 --> 00:40:30.650 is a matching calculation. 00:40:30.650 --> 00:40:32.940 So we fix the external state, still 00:40:32.940 --> 00:40:35.097 have a particular number of partons, 00:40:35.097 --> 00:40:37.180 and we want to compare the full theory calculation 00:40:37.180 --> 00:40:37.930 with the effective theory. 00:40:37.930 --> 00:40:39.472 The effective theory should reproduce 00:40:39.472 --> 00:40:41.530 the infrared divergences. 00:40:41.530 --> 00:40:43.510 So really all we care about is not 00:40:43.510 --> 00:40:46.090 that this is infrared finite, but rather 00:40:46.090 --> 00:40:52.533 that the effective theory has the same infrared divergences. 00:40:52.533 --> 00:40:53.950 And we'll see that in the end when 00:40:53.950 --> 00:40:55.908 you can think about is rather than canceling in 00:40:55.908 --> 00:40:58.890 for divergences as you're thinking in the full theory, 00:40:58.890 --> 00:41:02.080 we've matched the full theory onto the effective theory 00:41:02.080 --> 00:41:04.653 and that matching gives us the Wilson coefficient. 00:41:04.653 --> 00:41:07.070 And then we take effective theory and all the cancelations 00:41:07.070 --> 00:41:09.487 that you're thinking about between real and virtual graphs 00:41:09.487 --> 00:41:11.690 will occur in the effective theory too. 00:41:11.690 --> 00:41:13.730 So you can just think about the effective theory 00:41:13.730 --> 00:41:16.490 virtual-real graphs, then the cancelation will take place 00:41:16.490 --> 00:41:17.610 there. 00:41:17.610 --> 00:41:19.610 But then you're thinking about that cancelation 00:41:19.610 --> 00:41:22.140 later, at a lower scale, which is what you actually 00:41:22.140 --> 00:41:22.640 want to do. 00:41:22.640 --> 00:41:25.890 Because what I just was telling you about IR 00:41:25.890 --> 00:41:29.720 divergences becoming different things in the final state 00:41:29.720 --> 00:41:32.840 becomes very trivial once you're in the effective theory 00:41:32.840 --> 00:41:38.280 and we'll see sort of exactly how that works later on. 00:41:38.280 --> 00:41:42.080 But first we have to talk about linear graphs. 00:41:50.350 --> 00:41:52.350 So in the collinear graphs we had 00:41:52.350 --> 00:41:54.960 graphs like this one where we can take a gluon out 00:41:54.960 --> 00:41:56.560 of the vertex here. 00:41:56.560 --> 00:41:59.923 That corresponds to taking it out of the Wilson line. 00:41:59.923 --> 00:42:01.840 So let's label the graph in the following way. 00:42:01.840 --> 00:42:07.380 This would be k plus p this will be p This will be k. 00:42:07.380 --> 00:42:11.880 And if we follow our rules for what this is, 00:42:11.880 --> 00:42:20.460 we would write a sum over labels and then an integral 00:42:20.460 --> 00:42:23.830 over residuals. 00:42:23.830 --> 00:42:25.330 And let me put the residual integral 00:42:25.330 --> 00:42:27.550 in dimensional organization. 00:42:37.230 --> 00:42:43.800 Let me try to squeeze in everything, 00:42:43.800 --> 00:42:44.800 which won't be possible. 00:42:47.590 --> 00:42:49.830 So I'm going to write out the components 00:42:49.830 --> 00:42:52.450 to make clear whose residual on whose label. 00:42:56.930 --> 00:42:58.940 So in the denominator there's one more term. 00:43:11.390 --> 00:43:13.220 So the plus guys are always residual, 00:43:13.220 --> 00:43:15.980 the perp and the minuses are always labeled. 00:43:15.980 --> 00:43:18.940 The short way of saying what I'm trying to squeeze in here. 00:43:25.087 --> 00:43:27.420 So the denominator has these three terms, this guy here, 00:43:27.420 --> 00:43:28.795 this guy here, and this guy here. 00:43:35.660 --> 00:43:37.360 And when I label the diagram like this, 00:43:37.360 --> 00:43:42.100 you should think that each of these 00:43:42.100 --> 00:43:43.690 has a label and residual part. 00:43:48.240 --> 00:43:52.980 So k you can think of as a pair, k label, k residual for now. 00:43:55.490 --> 00:43:57.383 And remember the importance of doing this. 00:43:57.383 --> 00:43:59.050 The importance of doing this was related 00:43:59.050 --> 00:44:01.660 to always being able to identify what the lowest order 00:44:01.660 --> 00:44:04.190 term was on the right here. 00:44:04.190 --> 00:44:04.690 OK? 00:44:04.690 --> 00:44:06.357 And that actually becomes more important 00:44:06.357 --> 00:44:08.632 once you start to think about diagrams where 00:44:08.632 --> 00:44:10.840 you would add like an extra ultrasoft gluon somewhere 00:44:10.840 --> 00:44:12.740 in this picture. 00:44:12.740 --> 00:44:13.240 Then, of 00:44:13.240 --> 00:44:15.100 Course when that ultrasoft gluon feeds its way 00:44:15.100 --> 00:44:16.475 through this loop, you gotta make 00:44:16.475 --> 00:44:20.320 sure that it's only the lowest order piece that's showing up. 00:44:20.320 --> 00:44:24.370 In this case here, we just have a collinear loop 00:44:24.370 --> 00:44:28.960 and we don't have any sort of real ultrasoft 00:44:28.960 --> 00:44:34.930 momenta from ultrasoft particles besides the heavy quark, which 00:44:34.930 --> 00:44:37.708 is just an external particle to the loop. 00:44:37.708 --> 00:44:39.250 So what we want to do with this is we 00:44:39.250 --> 00:44:40.940 want to turn it back into an integral. 00:45:02.496 --> 00:45:03.880 Just let me say it this way. 00:45:07.680 --> 00:45:11.050 And if it wasn't for these restrictions here, 00:45:11.050 --> 00:45:15.210 then that's very easy, actually. 00:45:15.210 --> 00:45:17.740 and then I'll talk about why it's true. 00:45:17.740 --> 00:45:22.270 I claim that if we ignored the restrictions 00:45:22.270 --> 00:45:24.610 and those restrictions are ensuring that we don't double 00:45:24.610 --> 00:45:28.320 count between our collinear and our ultrasoft degrees 00:45:28.320 --> 00:45:28.820 of freedom. 00:45:28.820 --> 00:45:32.290 So they are important, but let's ignore them for a minute. 00:45:32.290 --> 00:45:44.315 If we do ignore them, we would just get the following. 00:45:50.750 --> 00:45:53.060 Where I just basically stop thinking about residuals 00:45:53.060 --> 00:45:56.660 and labels, write everything as a full momentum, 00:45:56.660 --> 00:45:59.960 and write down exactly the same thing I just wrote. 00:45:59.960 --> 00:46:02.450 So this is a full k squared and this 00:46:02.450 --> 00:46:04.010 is a full k plus p squared. 00:46:07.502 --> 00:46:08.002 OK? 00:46:11.000 --> 00:46:14.420 So what I'm going to do is I'm going to first ignore 00:46:14.420 --> 00:46:16.300 these restrictions and then I'm going 00:46:16.300 --> 00:46:18.710 to tell you how it would work, how this actually 00:46:18.710 --> 00:46:20.720 does turn into this. 00:46:20.720 --> 00:46:23.212 What are the rules for doing that. 00:46:23.212 --> 00:46:24.920 And then I'll come back and I'll tell you 00:46:24.920 --> 00:46:27.800 what these extra conditions do. 00:46:31.130 --> 00:46:35.270 So really this looks like maybe it's trivial, 00:46:35.270 --> 00:46:37.060 but we should think about it. 00:46:37.060 --> 00:46:40.410 And it's almost trivial but not quite. 00:46:40.410 --> 00:46:42.650 So really what we're doing here is 00:46:42.650 --> 00:46:45.620 we're combining back together labels and residual momentum, 00:46:45.620 --> 00:46:46.493 right? 00:46:46.493 --> 00:46:48.410 And the place that we have to worry about that 00:46:48.410 --> 00:46:50.450 is in the perp in the minus space. 00:46:50.450 --> 00:46:56.530 And recall we had the grid and the grid 00:46:56.530 --> 00:47:00.980 is sort of our way of guiding, our guidance 00:47:00.980 --> 00:47:02.820 to see how to put things back together. 00:47:09.720 --> 00:47:13.090 So we had vectors that lived in this space 00:47:13.090 --> 00:47:14.760 and this is the label and then there's 00:47:14.760 --> 00:47:16.500 the residual, if we want to point to some place 00:47:16.500 --> 00:47:17.160 in that space. 00:47:21.250 --> 00:47:25.710 This picture was like a Wilsonian effective field 00:47:25.710 --> 00:47:29.340 theory because the picture makes you think of sharp edges. 00:47:36.613 --> 00:47:38.530 But the real effective theory that we're doing 00:47:38.530 --> 00:47:42.700 is a continuum one and so you have 00:47:42.700 --> 00:47:44.170 to expand your brain a little bit 00:47:44.170 --> 00:47:46.170 and think that each of the boxes in this picture 00:47:46.170 --> 00:47:49.923 is actually an infinite space as well, 00:47:49.923 --> 00:47:52.090 because the residual space doesn't have restrictions 00:47:52.090 --> 00:47:54.210 like that that would spoil Lorentz symmetry. 00:48:17.540 --> 00:48:27.560 So each grid point really is specifying an infinite space 00:48:27.560 --> 00:48:28.600 of residual momenta. 00:48:34.780 --> 00:48:40.810 And it's R4 or Minkowski space, so the momenta components 00:48:40.810 --> 00:48:46.718 are real numbers and there's some rules. 00:48:46.718 --> 00:48:47.635 So what are the rules? 00:48:55.890 --> 00:49:03.240 So I'll tell you what the rules are without restrictions 00:49:03.240 --> 00:49:06.093 for now and then we'll come back and I'll 00:49:06.093 --> 00:49:08.010 tell you what the rules are with restrictions. 00:49:15.440 --> 00:49:17.810 So rule number one is the simplest, 00:49:17.810 --> 00:49:21.890 and it just says that say I had the following, 00:49:21.890 --> 00:49:24.250 I'll use a one dimensional notation. 00:49:24.250 --> 00:49:28.580 Say I had a sum over kl's and an integral over all kr's. 00:49:28.580 --> 00:49:30.710 Well that's just the same as not having split it up 00:49:30.710 --> 00:49:33.110 and doing an integral over everything. 00:49:33.110 --> 00:49:36.770 Because we split this thing into boxes and if we really 00:49:36.770 --> 00:49:41.238 integrate over all boxes and sum over all labels, 00:49:41.238 --> 00:49:43.280 then we should just get back the full integration 00:49:43.280 --> 00:49:44.155 over all the momenta. 00:49:46.640 --> 00:49:51.185 So that's true for each, for minus and perp momenta. 00:49:55.110 --> 00:49:59.010 But really what we have to do is we have to do this 00:49:59.010 --> 00:50:00.270 with some integrand, right? 00:50:00.270 --> 00:50:05.790 So the type of integrand that we have is following. 00:50:05.790 --> 00:50:12.070 If you think about the components 00:50:12.070 --> 00:50:15.040 that we're talking about here, which are the minus and perp 00:50:15.040 --> 00:50:18.580 ones, then in our integrand up there, 00:50:18.580 --> 00:50:23.150 there's no minus or perp residuals. 00:50:23.150 --> 00:50:25.555 So it's just a function of the labels. 00:50:35.430 --> 00:50:37.080 Right? 00:50:37.080 --> 00:50:38.580 And so what that says effectively 00:50:38.580 --> 00:50:43.720 is that this is a constant function in this box, 00:50:43.720 --> 00:50:44.595 in each of the boxes. 00:50:49.675 --> 00:50:52.050 And effectively, the way that you use this formula is you 00:50:52.050 --> 00:50:53.830 do the following. 00:50:53.830 --> 00:50:56.345 You say, well, if it's a constant function in the box, 00:50:56.345 --> 00:50:58.470 I could evaluate it at a different point in the box 00:50:58.470 --> 00:51:00.743 and I'd still get the same value. 00:51:00.743 --> 00:51:02.160 So in particular, I could evaluate 00:51:02.160 --> 00:51:05.340 it k label plus k residual and then I 00:51:05.340 --> 00:51:07.560 can use that formula there to say 00:51:07.560 --> 00:51:12.810 that this is just an integral with a continuous k 00:51:12.810 --> 00:51:14.970 of the function evaluated at that continuous k. 00:51:24.500 --> 00:51:28.332 So that's the way that number one works. 00:51:28.332 --> 00:51:30.790 We are integrating functions that are constant in the boxes 00:51:30.790 --> 00:51:33.130 and so then it's kind of trivial, 00:51:33.130 --> 00:51:35.120 how to put the boxes back together. 00:51:35.120 --> 00:51:35.620 Yeah? 00:51:35.620 --> 00:51:40.600 AUDIENCE: So this d kl division doesn't have kl plus--? 00:51:40.600 --> 00:51:42.830 IAIN STEWART: Yeah, so what I mean by it is-- 00:51:42.830 --> 00:51:43.330 yeah. 00:51:43.330 --> 00:51:48.280 So I'm using here for each minus or perp momenta, 00:51:48.280 --> 00:51:50.000 one dimensional notation. 00:51:50.000 --> 00:51:51.860 Yeah, and so I have three of these. 00:51:51.860 --> 00:51:53.150 I have to do this. 00:51:53.150 --> 00:51:56.733 And for each one of them it's true that what I'm saying. 00:51:56.733 --> 00:51:58.900 But of course, if I tried to write that on the board 00:51:58.900 --> 00:52:01.700 then it would confuse the point, I think, 00:52:01.700 --> 00:52:05.070 which is, in some sense, a simple point. 00:52:05.070 --> 00:52:05.620 OK? 00:52:05.620 --> 00:52:06.910 So this is, in some sense, saying 00:52:06.910 --> 00:52:08.680 that this whole split up that we were doing 00:52:08.680 --> 00:52:09.597 was not really needed. 00:52:09.597 --> 00:52:12.400 We could've just written a continuous integral and that's 00:52:12.400 --> 00:52:14.780 kind of what this is saying, right? 00:52:14.780 --> 00:52:16.780 We could have just written a continuous interval 00:52:16.780 --> 00:52:18.572 and not worried so much about all the split 00:52:18.572 --> 00:52:20.080 up that we were doing. 00:52:20.080 --> 00:52:22.240 The place that we have to be careful about 00:52:22.240 --> 00:52:24.483 is these restrictions, and we'll come back to that. 00:52:24.483 --> 00:52:26.650 And the other place that we have to be careful about 00:52:26.650 --> 00:52:28.450 is when you have the multiple expansion. 00:52:28.450 --> 00:52:32.170 But as long as you take care of that, then basically this 00:52:32.170 --> 00:52:36.080 is always happening. 00:52:36.080 --> 00:52:41.663 So the reason why this works is the following. 00:52:53.020 --> 00:52:57.580 For every label loop momentum that there is in any diagram, 00:52:57.580 --> 00:53:09.610 there's always going to be some corresponding residual that's 00:53:09.610 --> 00:53:14.950 not specified by delta functions in terms of external momenta. 00:53:14.950 --> 00:53:18.850 And effectively, therefore that we can absorb in order 00:53:18.850 --> 00:53:19.960 to do what we just said. 00:53:25.720 --> 00:53:29.170 So we can always go back from this discrete type notation 00:53:29.170 --> 00:53:30.737 back to a continuous notation. 00:53:30.737 --> 00:53:32.320 The discrete notation was just helping 00:53:32.320 --> 00:53:35.132 us to set up the expansion and be careful about it, 00:53:35.132 --> 00:53:37.090 but we can always go back to the continuous one 00:53:37.090 --> 00:53:39.805 because there's always a kr that has this property. 00:53:45.560 --> 00:53:46.060 OK. 00:53:50.280 --> 00:53:55.523 So now in general though, you might have 00:53:55.523 --> 00:53:56.690 some more complicated thing. 00:53:56.690 --> 00:53:58.315 And if I'm going to give you some rules 00:53:58.315 --> 00:54:00.950 I should give you a complete set. 00:54:00.950 --> 00:54:06.920 So we have to append our list of rules by the following one. 00:54:16.350 --> 00:54:19.110 So if I thought about doing what I just said over here 00:54:19.110 --> 00:54:22.060 but I went to some higher order, then what could happen? 00:54:22.060 --> 00:54:26.190 Well then these kr minus and kr our perps could show up. 00:54:26.190 --> 00:54:28.650 They'd never show up in the denominator of our propagators 00:54:28.650 --> 00:54:30.710 if they were just collinear propagators, 00:54:30.710 --> 00:54:33.840 but they could show up at some point in the numerator. 00:54:33.840 --> 00:54:38.070 And we need actually a rule like this one, which would be clear 00:54:38.070 --> 00:54:40.950 if we were regulating d kr in dimensional regularization, 00:54:40.950 --> 00:54:43.920 that the power of divergences are getting set to 0. 00:54:43.920 --> 00:54:47.190 And that is basically to maintain the run symmetry 00:54:47.190 --> 00:54:52.760 in the residual space we need a rule like this for j 00:54:52.760 --> 00:54:53.540 greater than 0. 00:55:15.380 --> 00:55:18.170 So that doesn't come into our calculation, 00:55:18.170 --> 00:55:22.080 but included for completeness. 00:55:22.080 --> 00:55:24.860 So when would these integral kr actually do 00:55:24.860 --> 00:55:26.300 something non-trivial? 00:55:26.300 --> 00:55:27.770 It would do something non-trivial 00:55:27.770 --> 00:55:29.960 if we had an ultrasoft loops and collinear 00:55:29.960 --> 00:55:31.060 loops at the same time. 00:55:44.677 --> 00:55:46.760 So in the case where we just have collinear loops, 00:55:46.760 --> 00:55:50.120 it's basically up to this issue about the restrictions 00:55:50.120 --> 00:55:51.230 that we'll talk about. 00:55:51.230 --> 00:55:52.370 It's basically that we just could 00:55:52.370 --> 00:55:53.787 have done everything as continuous 00:55:53.787 --> 00:55:58.040 and ignored this split. 00:55:58.040 --> 00:56:02.000 But if we have both ultrasoft particles that 00:56:02.000 --> 00:56:25.690 are participating through the loops and/or then in general 00:56:25.690 --> 00:56:28.930 these will give non-trivial loop momenta 00:56:28.930 --> 00:56:32.470 in the residual momenta. 00:56:32.470 --> 00:56:35.530 And hence there will be some that we can't just 00:56:35.530 --> 00:56:37.210 absorb in the fashion that we just said. 00:56:37.210 --> 00:56:41.830 So there will be, in this situation, residual momenta. 00:56:41.830 --> 00:56:44.350 Some residual momenta will be absorbed in the same way 00:56:44.350 --> 00:56:47.085 to turn the integrals into continuous ones, 00:56:47.085 --> 00:56:48.460 but other ones won't be absorbed. 00:56:59.078 --> 00:57:01.120 And that's because the ultrasoft propagators rate 00:57:01.120 --> 00:57:05.680 would involve the lr plus, the lr minus, and the lr perp 00:57:05.680 --> 00:57:09.380 in the denominator, so we don't have this rule to apply. 00:57:09.380 --> 00:57:11.590 We can't do what we said here with the constant boxes 00:57:11.590 --> 00:57:14.750 because now the functions are depending on that variable. 00:57:14.750 --> 00:57:17.600 So we just have to do the integral. 00:57:17.600 --> 00:57:23.655 So you could have something that looks like just 00:57:23.655 --> 00:57:26.630 in a schematic formula. 00:57:26.630 --> 00:57:30.540 Let's have there be 2 kr and an lr. 00:57:30.540 --> 00:57:33.950 We're in kind of an obvious notation. 00:57:33.950 --> 00:57:36.680 I'm saying that the function could depend on lr residual, 00:57:36.680 --> 00:57:38.500 it doesn't depend on kr residual. 00:57:38.500 --> 00:57:41.170 So we absorb kr back into the sum to make it continuous 00:57:41.170 --> 00:57:42.920 and then this integral we just have to do. 00:57:54.076 --> 00:57:58.245 OK, but we're still, in the end, just doing an integral. 00:58:05.310 --> 00:58:11.300 So this guy come from ultrasoft propagators, for example. 00:58:16.510 --> 00:58:17.330 OK? 00:58:17.330 --> 00:58:20.180 So those are the different cases that you can get. 00:58:20.180 --> 00:58:23.097 But nevertheless, even if you have ultrasoft particles 00:58:23.097 --> 00:58:24.680 and propagators floating around, there 00:58:24.680 --> 00:58:27.920 will always be a residual momentum associated to 00:58:27.920 --> 00:58:29.990 of what we were doing over here that you 00:58:29.990 --> 00:58:35.610 can absorb in the same way I stated, 00:58:35.610 --> 00:58:38.262 so let's proceed along these lines and see where it takes us 00:58:38.262 --> 00:58:39.720 and then we'll come back and put in 00:58:39.720 --> 00:58:42.960 this additional restrictions. 00:58:42.960 --> 00:58:45.040 AUDIENCE: Do we ever have to do a discrete sum? 00:58:45.040 --> 00:58:48.310 Like is there a sum over ll? 00:58:48.310 --> 00:58:51.476 IAIN STEWART: Yeah, so you never have to do a discrete sum. 00:58:51.476 --> 00:58:52.393 AUDIENCE: That's good. 00:58:52.393 --> 00:58:54.598 IAIN STEWART: Yep. 00:58:54.598 --> 00:58:56.640 The discrete sum is really just a way of thinking 00:58:56.640 --> 00:58:59.760 and I'll show you in a minute that you even 00:58:59.760 --> 00:59:02.620 can avoid thinking about it once you know what to do. 00:59:02.620 --> 00:59:04.480 So it's guiding you towards the right answer 00:59:04.480 --> 00:59:06.063 but it's not really something that you 00:59:06.063 --> 00:59:07.860 have to think about this grid picture. 00:59:07.860 --> 00:59:09.490 It's really just if you get confused, 00:59:09.490 --> 00:59:11.340 you can always think about it but if you're not confused, 00:59:11.340 --> 00:59:12.715 you don't have to think about it. 00:59:21.590 --> 00:59:28.300 OK, so this guy is what we said, proportional to what we said. 00:59:28.300 --> 00:59:38.030 It's just in this case, the reason why it was so simple 00:59:38.030 --> 00:59:42.230 is because k residual is one component of this guy, 00:59:42.230 --> 00:59:43.980 and then there's these other three, right? 00:59:43.980 --> 00:59:46.677 And the other three, we just absorb 00:59:46.677 --> 00:59:48.260 the sum and the integral back together 00:59:48.260 --> 00:59:50.510 and there's nothing further to talk about. 00:59:50.510 --> 00:59:56.600 So that's why this is, in some sense, very simple. 00:59:56.600 --> 01:00:02.480 So we just do that, and what do we get? 01:00:06.810 --> 01:00:27.190 Get some 1 over epsilons, we get some logs of p squared, 01:00:27.190 --> 01:00:30.678 and I'm putting in even the constant term 01:00:30.678 --> 01:00:31.720 since it's pretty simple. 01:00:31.720 --> 01:00:34.760 4 minus pi squared over 6. 01:00:34.760 --> 01:00:39.160 So if you think about where this came from, 01:00:39.160 --> 01:00:41.470 there was a place that this Wilson line 01:00:41.470 --> 01:00:43.250 came from was attaching this guy over here 01:00:43.250 --> 01:00:47.050 and then integrating out that offshell propagator. 01:00:47.050 --> 01:00:48.442 So this is a vertex diagram. 01:00:48.442 --> 01:00:50.650 We had a vertex diagram, which is an ultrasoft gluon, 01:00:50.650 --> 01:00:53.410 and now we've added an identical type up topology. 01:00:53.410 --> 01:00:55.870 Except we drew it different because this line 01:00:55.870 --> 01:00:58.515 was offshell so we drew it as a Wilson line 01:00:58.515 --> 01:01:00.440 and that's the right way of thinking about it. 01:01:00.440 --> 01:01:02.253 But if you think about where it came from, 01:01:02.253 --> 01:01:03.670 it was the same topology and we've 01:01:03.670 --> 01:01:05.840 added another vertex diagram. 01:01:05.840 --> 01:01:07.540 So the effective theory has two type 01:01:07.540 --> 01:01:09.142 of vertex diagrams, this collinear 01:01:09.142 --> 01:01:10.600 one and the ultrasoft one and we're 01:01:10.600 --> 01:01:12.640 going to add them together. 01:01:12.640 --> 01:01:15.100 And in this one, you see the logs 01:01:15.100 --> 01:01:17.170 are minimized at a different scale, mu squared 01:01:17.170 --> 01:01:19.690 of order p squared, which is the right scale for a collinear 01:01:19.690 --> 01:01:20.190 loop. 01:01:24.032 --> 01:01:25.740 Because remember, the collinear particles 01:01:25.740 --> 01:01:32.730 lived at a larger p squared than the ultrasoft ones. 01:01:32.730 --> 01:01:37.010 So there's a larger scale that would minimize these loops, 01:01:37.010 --> 01:01:38.010 it's a collinear scale. 01:01:42.260 --> 01:01:44.530 So the effective theory is capturing the physics 01:01:44.530 --> 01:01:45.280 at that scale. 01:01:48.577 --> 01:01:50.290 Squeeze in another diagram here. 01:01:53.440 --> 01:01:56.590 If we do the collinear wave function renormalization, 01:01:56.590 --> 01:01:57.610 then this is non-zero. 01:01:57.610 --> 01:02:00.040 This was 0 for the ultrasoft gluon 01:02:00.040 --> 01:02:02.890 and but for the collinear gluon, it's non-zero 01:02:02.890 --> 01:02:06.190 and it's exactly, actually, the same as the full theory. 01:02:10.731 --> 01:02:12.370 It's the same as massless QCD. 01:02:17.920 --> 01:02:20.090 So you can do the diagram using the effective theory 01:02:20.090 --> 01:02:23.500 Feynman rules, then that's what you find. 01:02:23.500 --> 01:02:25.510 But you could also understand why that's true. 01:02:28.520 --> 01:02:31.480 And reason why it's true is nothing in this diagram really 01:02:31.480 --> 01:02:34.230 specifies a frame. 01:02:34.230 --> 01:02:35.880 We called all the particles collinear, 01:02:35.880 --> 01:02:38.755 but it's not attached to anything. 01:02:38.755 --> 01:02:40.380 So we could just take the whole diagram 01:02:40.380 --> 01:02:44.268 and boost back to the frame where everything's kind of soft 01:02:44.268 --> 01:02:46.560 and then we would usually be thinking about it in terms 01:02:46.560 --> 01:02:47.650 of a full theory field. 01:02:47.650 --> 01:02:56.640 So that's effectively why this diagram 01:02:56.640 --> 01:03:00.323 like this that doesn't have any reference, unlike this one 01:03:00.323 --> 01:03:01.740 which has a reference because it's 01:03:01.740 --> 01:03:04.470 attached to the heavy quark. 01:03:04.470 --> 01:03:05.850 That's why it's the same as QCD. 01:03:12.910 --> 01:03:13.695 Yeah? 01:03:13.695 --> 01:03:15.820 AUDIENCE: How do you know what the epsilon position 01:03:15.820 --> 01:03:17.460 for the Wilson line is set? 01:03:17.460 --> 01:03:18.160 IAIN STEWART: Oh, for this one? 01:03:18.160 --> 01:03:19.077 AUDIENCE: m bar dot k. 01:03:19.077 --> 01:03:21.270 IAIN STEWART: Yeah, for this one. 01:03:21.270 --> 01:03:27.570 Yeah, for this guy it actually doesn't depend on the i epsilon 01:03:27.570 --> 01:03:31.160 prescription up to a-- 01:03:31.160 --> 01:03:33.247 yeah, if I remember correctly. 01:03:37.720 --> 01:03:39.010 Yeah. 01:03:39.010 --> 01:03:41.080 That's true, I believe, when I'm being 01:03:41.080 --> 01:03:44.710 a little bit cavalier with it but once I put it the zero bin 01:03:44.710 --> 01:03:48.922 restrictions, I'm not sure if that's true anymore. 01:03:48.922 --> 01:03:50.628 AUDIENCE: OK? 01:03:50.628 --> 01:03:51.420 IAIN STEWART: Yeah. 01:03:51.420 --> 01:03:53.670 I mean, really what solves that is that in a minute 01:03:53.670 --> 01:03:54.870 I'm going to be talking about the fact 01:03:54.870 --> 01:03:56.355 that there is a subtraction term here 01:03:56.355 --> 01:03:58.063 and once you put the subtraction term in, 01:03:58.063 --> 01:03:59.370 that i0 is not relevant. 01:04:02.710 --> 01:04:05.240 But I think even if you do this diagram with arbitrary i0-- 01:04:09.590 --> 01:04:14.308 because what's happening is m bar dot k going to 0 01:04:14.308 --> 01:04:16.100 is related to some of these 1 over epsilons 01:04:16.100 --> 01:04:17.930 and we'll talk about that more in a minute. 01:04:25.710 --> 01:04:27.320 Yeah, I'm not 100% sure. 01:04:27.320 --> 01:04:30.512 It could be that there's some sign that might flip. 01:04:30.512 --> 01:04:32.330 I'm not 100% sure. 01:04:32.330 --> 01:04:34.610 AUDIENCE: Even if they get the wrong side, 01:04:34.610 --> 01:04:36.110 the difference would be subtraction. 01:04:36.110 --> 01:04:39.080 IAIN STEWART: I guess I know that-- 01:04:39.080 --> 01:04:40.100 yeah. 01:04:40.100 --> 01:04:41.750 What I know is that-- 01:04:41.750 --> 01:04:43.520 yeah, let me answer your question later. 01:04:43.520 --> 01:04:44.540 It'll be easier because I'm trying 01:04:44.540 --> 01:04:46.430 to say a bunch of things that depend on something else 01:04:46.430 --> 01:04:47.638 that I haven't explained yet. 01:04:50.580 --> 01:04:54.087 So what are the other possible topologies we could write down? 01:04:54.087 --> 01:04:55.670 So we could write down this one, where 01:04:55.670 --> 01:04:57.500 we take two attachments in the Wilson line 01:04:57.500 --> 01:04:59.870 and just loop them back up but that's 01:04:59.870 --> 01:05:01.620 proportional m bar squared and so that's 01:05:01.620 --> 01:05:02.980 0 in our Feynman gauge. 01:05:05.840 --> 01:05:11.750 And likewise, there's a looping back up in the vertex 01:05:11.750 --> 01:05:13.700 in the wave function renormalization, 01:05:13.700 --> 01:05:21.520 but this guys scale is power law divergent and so we can just 01:05:21.520 --> 01:05:24.735 set it to 0 and dim reg. 01:05:24.735 --> 01:05:26.110 You don't have to worry about it. 01:05:30.400 --> 01:05:32.700 OK, so that's all the diagrams. 01:05:32.700 --> 01:05:35.595 Let's think about doing matching, 01:05:35.595 --> 01:05:39.508 i.e let's think about comparing QCD and SCET by adding up 01:05:39.508 --> 01:05:40.050 the diagrams. 01:05:50.070 --> 01:05:52.050 In QCD we carried out the randomization, 01:05:52.050 --> 01:05:55.670 we added the z for the tensor current. 01:05:55.670 --> 01:05:59.160 Let me just write again the answer looked like. 01:06:19.460 --> 01:06:23.090 In SCET we didn't carry out renormalization yet, 01:06:23.090 --> 01:06:26.450 so let me call this the bare SCET result for now. 01:06:33.860 --> 01:06:36.670 Once we add the ultrasoft and collinear diagrams together, 01:06:36.670 --> 01:06:39.880 the logs of p squared match up exactly with the full theory. 01:06:47.310 --> 01:06:51.243 So this is the first sign really that it makes sense 01:06:51.243 --> 01:06:52.910 to be thinking about adding these loops. 01:06:52.910 --> 01:06:54.830 Even though they were the same topology, 01:06:54.830 --> 01:06:58.235 we are correctly reproducing those logs of p 01:06:58.235 --> 01:06:59.360 squared in the full theory. 01:07:03.677 --> 01:07:05.135 And then there's some other pieces. 01:07:22.222 --> 01:07:23.680 I'll write out all the other pieces 01:07:23.680 --> 01:07:26.262 so you see what they look like. 01:07:26.262 --> 01:07:28.220 Well, maybe I won't I won't write the constant. 01:07:41.390 --> 01:07:43.890 So there's all the effective field theory terms. 01:07:43.890 --> 01:07:49.710 So these terms here we can match up with these terms here. 01:07:53.720 --> 01:07:56.050 So that's good. 01:07:56.050 --> 01:08:00.363 These terms here, which remember in the full theory were finite, 01:08:00.363 --> 01:08:02.530 and these terms here, which in the effective theory, 01:08:02.530 --> 01:08:04.300 are finite, the difference of those 01:08:04.300 --> 01:08:06.207 is going to give the Wilson coefficient. 01:08:23.915 --> 01:08:25.540 Now we said that the Wilson coefficient 01:08:25.540 --> 01:08:29.720 could be a function of p bar, so what's going on with that? 01:08:29.720 --> 01:08:32.109 Well, if you look at momentum conservation 01:08:32.109 --> 01:08:36.024 in this process of beta s gamma-- 01:08:36.024 --> 01:08:39.450 so I probably should've said this earlier. 01:08:39.450 --> 01:08:40.970 So when you look at beta s gamma, 01:08:40.970 --> 01:08:43.399 if you look at momentum conservation then 01:08:43.399 --> 01:08:46.220 the p minus of the strange quark has 01:08:46.220 --> 01:08:48.109 to be equal to the p minus of the b quark, 01:08:48.109 --> 01:08:51.340 but that's just mb. 01:08:51.340 --> 01:08:55.670 OK, so actually p minus is equal to mb by kinematics. 01:09:00.630 --> 01:09:03.080 So mb's in this result here you shouldn't think 01:09:03.080 --> 01:09:07.640 of as p minuses, and that's this p bar that was in our Wilson 01:09:07.640 --> 01:09:09.529 coefficient is just getting set to mb 01:09:09.529 --> 01:09:14.240 because of some delta functions that are specifying kinematics. 01:09:14.240 --> 01:09:17.149 So that leaves the 1 over epsilon terms. 01:09:17.149 --> 01:09:19.370 And so what we'd like is that those terms are 01:09:19.370 --> 01:09:21.412 associated to renormalization. 01:09:27.588 --> 01:09:28.880 Of the effective theory, right? 01:09:28.880 --> 01:09:30.689 I wrote that the effective theory was bare. 01:09:48.310 --> 01:09:49.899 But if I want to do that, then I have 01:09:49.899 --> 01:09:52.540 to ensure that all these epsilons that 01:09:52.540 --> 01:09:56.910 are appearing here are really ultraviolet divergences. 01:09:56.910 --> 01:09:59.610 If they're infrared divergences, then doing that 01:09:59.610 --> 01:10:01.850 doesn't make sense. 01:10:01.850 --> 01:10:04.380 And that's actually the remaining issue 01:10:04.380 --> 01:10:06.080 that we have to deal with. 01:10:06.080 --> 01:10:10.120 I just wrote epsilon, that means I'm ignorant to what they are. 01:10:10.120 --> 01:10:11.692 And if I knew this one was epsilon 01:10:11.692 --> 01:10:13.400 IR because it came from the wave function 01:10:13.400 --> 01:10:14.817 renormalization of the heavy quark 01:10:14.817 --> 01:10:16.830 and that was the same on both sides. 01:10:16.830 --> 01:10:19.200 It was the same diagram, it was the wave function 01:10:19.200 --> 01:10:21.742 renormalization diagram in the full and the effective theory. 01:10:21.742 --> 01:10:23.220 So I could match up that one. 01:10:23.220 --> 01:10:26.130 These ones just came out, but it turns out 01:10:26.130 --> 01:10:28.890 that so far with what we've done, some of these epsilon 01:10:28.890 --> 01:10:31.330 here are IR. 01:10:31.330 --> 01:10:37.912 And so the IR divergences aren't matching up 01:10:37.912 --> 01:10:39.370 and the reason is because we didn't 01:10:39.370 --> 01:10:43.295 put in those restrictions on our sum over labels. 01:11:26.610 --> 01:11:27.120 OK. 01:11:27.120 --> 01:11:32.250 So we have these restrictions, k label not equal to zero 01:11:32.250 --> 01:11:34.080 and k label not equal to minus pl. 01:11:36.780 --> 01:11:39.750 Those are the restrictions that I'm talking about. 01:11:39.750 --> 01:11:41.700 The place that those restrictions came from 01:11:41.700 --> 01:11:45.390 was k was the momentum of the gluon. 01:11:45.390 --> 01:11:49.620 kl not equal to 0 is saying that this is the restriction 01:11:49.620 --> 01:11:51.486 that the gluon is collinear. 01:11:55.140 --> 01:11:59.810 Because kl equals 0 is the ultrasoft gluon 01:11:59.810 --> 01:12:01.900 and this is the restriction that the fermion is 01:12:01.900 --> 01:12:08.240 collinear in the loop and that's why there was two of them. 01:12:13.940 --> 01:12:16.510 So these are called zero bins. 01:12:16.510 --> 01:12:20.710 Zero because it's where the ultrasoft momentum lives 01:12:20.710 --> 01:12:23.590 and from the point of view of collinear, that's zero. 01:12:28.930 --> 01:12:31.840 Imposing these restrictions is removing the zero 01:12:31.840 --> 01:12:34.750 bin, if you like and what these restrictions do 01:12:34.750 --> 01:12:39.220 is they avoid double counting and the way I've 01:12:39.220 --> 01:12:40.860 said it that's, I think, clear. 01:12:44.230 --> 01:12:46.780 So far in our calculation we haven't avoided double counting 01:12:46.780 --> 01:12:47.738 and that's the problem. 01:13:04.240 --> 01:13:04.910 OK. 01:13:04.910 --> 01:13:07.280 So we have to modify our rule or we 01:13:07.280 --> 01:13:10.040 extend our rule to include the case where 01:13:10.040 --> 01:13:11.627 we have these restrictions. 01:13:16.010 --> 01:13:23.243 In an extended version of rule two that house restrictions. 01:13:38.060 --> 01:13:40.616 So really what we want to do is that 01:13:40.616 --> 01:13:44.633 and we want to think about that as an integral. 01:13:44.633 --> 01:13:46.550 So here's how we can manipulate these to think 01:13:46.550 --> 01:13:49.440 about it as an integral. 01:13:49.440 --> 01:13:57.930 That sum over all kl's, but then we'll 01:13:57.930 --> 01:14:12.990 subtract the limit of this f where we take the f 01:14:12.990 --> 01:14:16.140 and we let the kl go to the place. 01:14:16.140 --> 01:14:18.480 So we integrate over everywhere, including the place 01:14:18.480 --> 01:14:22.200 we don't want to go, and then we subtract it back. 01:14:24.870 --> 01:14:26.550 So what is this fl of k? 01:14:34.630 --> 01:14:36.230 This f of kl goes to 0. 01:14:38.750 --> 01:14:43.910 it's defined by taking the scaling limit of the collinear 01:14:43.910 --> 01:14:46.100 momenta towards the ultrasoft. 01:14:53.880 --> 01:14:55.780 So you take your collinear momenta, 01:14:55.780 --> 01:14:58.060 which are the minus in the perp here, 01:14:58.060 --> 01:15:01.720 and you scale them towards an ultrasoft momentum in whatever 01:15:01.720 --> 01:15:06.760 components, i.e you start counting the kn's as order 01:15:06.760 --> 01:15:18.617 lambda squared and you keep the leading order piece 01:15:18.617 --> 01:15:20.950 or you keep the piece that's the same size as this term. 01:15:34.930 --> 01:15:38.530 And then that defines what this f is. 01:15:38.530 --> 01:15:40.510 Once you take that limit and you expand, 01:15:40.510 --> 01:15:41.950 then that's what the f is. 01:15:49.370 --> 01:15:53.120 So what you're doing here by doing this procedure 01:15:53.120 --> 01:15:55.280 is you're basically setting things up 01:15:55.280 --> 01:15:59.540 so that this guy is integrated over in a way 01:15:59.540 --> 01:16:01.460 that we can combine back into an integral. 01:16:01.460 --> 01:16:04.130 But then we have to subtract an overlap 01:16:04.130 --> 01:16:07.645 of when that integral would go into the ultrasoft region. 01:16:07.645 --> 01:16:09.020 But the overlap we're subtracting 01:16:09.020 --> 01:16:20.288 is also an integral, so we have a difference of integrals. 01:16:20.288 --> 01:16:22.080 And you should think of the second integral 01:16:22.080 --> 01:16:27.120 as integrating over the square where the zero bin was. 01:16:27.120 --> 01:16:29.730 So if you think about our picture 01:16:29.730 --> 01:16:33.150 where the collinears were up here, ultrasofts are down here 01:16:33.150 --> 01:16:36.390 and you thought about there being some box, 01:16:36.390 --> 01:16:38.850 you're taking this scaling limit when this guy goes down 01:16:38.850 --> 01:16:41.730 into that box. 01:16:41.730 --> 01:16:44.220 You add up all the boxes, that's this, 01:16:44.220 --> 01:16:46.360 and then you subtract out that box again. 01:16:46.360 --> 01:16:49.350 And that avoids having a double counting in that region. 01:16:54.290 --> 01:16:58.490 So then this guy here, you can do the same kind of trick 01:16:58.490 --> 01:16:59.190 as before. 01:16:59.190 --> 01:17:02.165 So continuing with the equation. 01:17:18.558 --> 01:17:21.100 kl is going to 0, well, we can think of it just as a function 01:17:21.100 --> 01:17:22.867 of kr's. 01:17:22.867 --> 01:17:24.700 And so effectively what we get in the end is 01:17:24.700 --> 01:17:28.600 an integral over all k of f of k, 01:17:28.600 --> 01:17:32.920 the full f, evaluated with a continuous momentum 01:17:32.920 --> 01:17:39.190 minus some f that's expanded and avoids there 01:17:39.190 --> 01:17:41.570 being overlap in this box. 01:17:41.570 --> 01:17:43.600 So rather than having these discrete sum 01:17:43.600 --> 01:17:46.700 with the restriction, we have a difference of integrals 01:17:46.700 --> 01:17:50.260 and this subtraction term avoids the overlap in that region. 01:17:53.380 --> 01:17:55.620 So all the discrete sums are good for is 01:17:55.620 --> 01:17:59.020 a means of figuring out what limits you need to take 01:17:59.020 --> 01:18:00.682 to generate these subtractions. 01:18:00.682 --> 01:18:02.140 Once you've done that, everything's 01:18:02.140 --> 01:18:03.100 a continuous integral. 01:18:22.690 --> 01:18:24.941 And this is called the zero bin subtraction. 01:18:36.010 --> 01:18:39.120 OK, so if you like, one way of phrasing what's going on 01:18:39.120 --> 01:18:41.580 is that the collinear propagators are really 01:18:41.580 --> 01:18:42.750 distributions. 01:18:42.750 --> 01:18:46.560 They're distributions that know that they should 01:18:46.560 --> 01:18:49.290 have a subtraction in order to not overlap the other region 01:18:49.290 --> 01:18:51.423 where we have another degree of freedom. 01:18:51.423 --> 01:18:52.840 So you can think about it that way 01:18:52.840 --> 01:18:57.740 and having this sum it's just a way of encoding that. 01:18:57.740 --> 01:19:00.150 But in the end, it looks like some kind of plus function 01:19:00.150 --> 01:19:02.745 where you have a subtraction. 01:19:02.745 --> 01:19:04.620 AUDIENCE: So it seems like in the second term 01:19:04.620 --> 01:19:08.540 you would have to relearn the expression because initially 01:19:08.540 --> 01:19:12.240 in the [INAUDIBLE] k you ignore kr minus and kr perps. 01:19:12.240 --> 01:19:15.070 IAIN STEWART: I'll show you how it works in an example 01:19:15.070 --> 01:19:15.570 up there. 01:19:18.015 --> 01:19:19.140 So let's go to our example. 01:19:24.740 --> 01:19:27.357 Yeah, so what you just said is not quite the way 01:19:27.357 --> 01:19:28.440 you should think about it. 01:19:28.440 --> 01:19:29.982 You should think about it that you've 01:19:29.982 --> 01:19:31.920 given the effective theory f and now 01:19:31.920 --> 01:19:35.910 I'm saying that that f, as given, with its momentum as 01:19:35.910 --> 01:19:39.510 given still has an overlap with the ultrasoft region 01:19:39.510 --> 01:19:41.380 that I want to subtract. 01:19:41.380 --> 01:19:43.080 So I'm taking a limit of that f, I'm 01:19:43.080 --> 01:19:44.370 not add anything back to it. 01:19:47.530 --> 01:19:50.550 So this guy, integral dk. 01:19:58.952 --> 01:20:00.285 So this is what we wrote before. 01:20:05.290 --> 01:20:09.785 And now if I take the ultrasoft limit of it, of the k, 01:20:09.785 --> 01:20:10.785 then I would write this. 01:20:22.690 --> 01:20:23.370 OK? 01:20:23.370 --> 01:20:26.610 So when I take the scaling limit, 01:20:26.610 --> 01:20:28.560 if you think about k squared, k plus, 01:20:28.560 --> 01:20:31.650 k minus, minus k perp squared, when I take the scaling 01:20:31.650 --> 01:20:33.490 limit of all momentum being ultrasoft, 01:20:33.490 --> 01:20:35.910 the components of the k are still homogeneous. 01:20:35.910 --> 01:20:38.430 So there's nothing there to expand. 01:20:38.430 --> 01:20:41.610 Some expansion happened in this k plus p term. 01:20:41.610 --> 01:20:44.160 One of our n bar dot k changes its power counting, 01:20:44.160 --> 01:20:47.427 but it's still n bar dot k. 01:20:47.427 --> 01:20:48.760 There's nothing to expand there. 01:20:48.760 --> 01:20:50.010 So this is what the subtraction looks 01:20:50.010 --> 01:20:51.576 like and then the numerator, the n bar dot 01:20:51.576 --> 01:20:53.326 k can be dropped relative to the n bar dot 01:20:53.326 --> 01:20:56.695 p, which is large and external. 01:20:56.695 --> 01:20:59.400 OK, so this is taking the ultrasoft limit of this. 01:21:03.176 --> 01:21:04.828 AUDIENCE: Do you have to subtract off 01:21:04.828 --> 01:21:07.385 the the ultrasoft limit of the quark as well? 01:21:07.385 --> 01:21:08.760 IAIN STEWART: Yeah, so in general 01:21:08.760 --> 01:21:11.427 I would have to subtract off the ultrasoft limit of the quark as 01:21:11.427 --> 01:21:12.490 well. 01:21:12.490 --> 01:21:14.220 And when I do that, what I find is 01:21:14.220 --> 01:21:17.940 a term that's power suppressed and so I drop it. 01:21:17.940 --> 01:21:19.940 But in general, I would have to do that as well. 01:21:19.940 --> 01:21:20.482 That's right. 01:21:23.350 --> 01:21:26.450 And so it looks like an ultrasoft diagram 01:21:26.450 --> 01:21:31.100 except it's got this n bar dot k inside of the v dot k 01:21:31.100 --> 01:21:34.082 that we had in the ultrasoft diagram. 01:21:34.082 --> 01:21:36.290 And if you do a power counting with the loop momentum 01:21:36.290 --> 01:21:38.540 scaling as ultrasoft, then you have this piece is of the order 01:21:38.540 --> 01:21:40.123 the same size as this piece and that's 01:21:40.123 --> 01:21:42.287 why you keep only that term. 01:21:42.287 --> 01:21:44.454 AUDIENCE: Do you have to worry about a higher power? 01:21:44.454 --> 01:21:45.290 IAIN STEWART: No. 01:21:45.290 --> 01:21:49.090 So the prescription we have is that we drop the higher powers 01:21:49.090 --> 01:21:50.820 AUDIENCE: But if I were to do say-- 01:21:50.820 --> 01:21:52.778 IAIN STEWART: Oh, if you did the higher power-- 01:21:52.778 --> 01:21:54.890 AUDIENCE: --lower power higher power zero bin? 01:21:54.890 --> 01:21:55.920 IAIN STEWART: No. 01:21:55.920 --> 01:21:57.870 So if we did the higher power, then this 01:21:57.870 --> 01:21:59.707 would start out at higher power. 01:21:59.707 --> 01:22:01.290 And then when we took the limit of it, 01:22:01.290 --> 01:22:04.650 it would end up just starting at that power or higher. 01:22:04.650 --> 01:22:06.210 AUDIENCE: Right, but there was a-- 01:22:06.210 --> 01:22:07.860 IAIN STEWART: Yeah, you don't have to. 01:22:07.860 --> 01:22:08.160 No. 01:22:08.160 --> 01:22:08.702 AUDIENCE: No? 01:22:08.702 --> 01:22:09.720 IAIN STEWART: No. 01:22:09.720 --> 01:22:10.220 Yeah. 01:22:10.220 --> 01:22:11.870 AUDIENCE: Is there a reason? 01:22:11.870 --> 01:22:14.520 IAIN STEWART: Yeah, so really what 01:22:14.520 --> 01:22:18.000 you care about subtracting here are the log divergences 01:22:18.000 --> 01:22:20.640 and that's what this minimal subtraction is doing. 01:22:20.640 --> 01:22:23.910 By keeping the piece that's scaling the same way, 01:22:23.910 --> 01:22:26.490 you're removing the log divergent pieces. 01:22:26.490 --> 01:22:28.260 And it's the log divergent pieces which 01:22:28.260 --> 01:22:31.500 are giving one of our epsilons. 01:22:31.500 --> 01:22:35.050 The pieces that you would get from the higher expansion, 01:22:35.050 --> 01:22:37.467 they would all be kind of like power law divergent terms 01:22:37.467 --> 01:22:39.300 from the point of view of the power counting 01:22:39.300 --> 01:22:42.090 and we just don't have to worry about those. 01:22:42.090 --> 01:22:44.910 And another way of saying it is, it's not 01:22:44.910 --> 01:22:48.330 that I'm removing absolutely this whole integrand 01:22:48.330 --> 01:22:49.620 in that region, right? 01:22:49.620 --> 01:22:52.033 There could still be a constant, for example, 01:22:52.033 --> 01:22:53.200 that comes from that region. 01:22:53.200 --> 01:22:54.750 But if there's a constant that comes from that region, 01:22:54.750 --> 01:22:55.830 I don't care. 01:22:55.830 --> 01:23:00.470 What I care about removing is any spare use IR singularities. 01:23:00.470 --> 01:23:03.120 And for those I can make a minimal subtraction, 01:23:03.120 --> 01:23:06.380 which is just the first term. 01:23:06.380 --> 01:23:09.580 All right, so I want to finish this discussion. 01:23:09.580 --> 01:23:14.100 So if we do this, we get an answer, 01:23:14.100 --> 01:23:21.120 which I will try to write on the board for you. 01:23:24.970 --> 01:23:28.443 So now I'm going to distinguish all the epsilons 01:23:28.443 --> 01:23:30.360 and then we'll see what this subtraction does. 01:23:41.457 --> 01:23:43.790 So if I was careful and I distinguished all the epsilons 01:23:43.790 --> 01:23:49.530 in our original calculation, it'd actually look like this. 01:23:49.530 --> 01:23:56.660 And then the subtraction piece gives an extra contribution 01:23:56.660 --> 01:24:04.270 and it's actually scaleless in the n bar dot k here. 01:24:04.270 --> 01:24:10.190 So there's a scaleless loop in this guy. 01:24:15.025 --> 01:24:20.870 So it actually vanishes if the epsilon IR and the epsilon UV 01:24:20.870 --> 01:24:22.860 are said to be equal. 01:24:22.860 --> 01:24:25.860 But what it does is it converts the epsilon IRs that 01:24:25.860 --> 01:24:28.890 are in the first expression into epsilon UVs, which 01:24:28.890 --> 01:24:30.450 is what we want. 01:24:30.450 --> 01:24:31.950 So once you add up these two things, 01:24:31.950 --> 01:24:34.500 the epsilon IRs are canceling and the epsilon IRs 01:24:34.500 --> 01:24:36.960 that were coming in the original formula, those 01:24:36.960 --> 01:24:41.250 were coming about because of this bad behavior 01:24:41.250 --> 01:24:45.465 as n bar dot k goes into the limit of n bar dot 01:24:45.465 --> 01:24:46.170 k going small. 01:24:46.170 --> 01:24:49.120 You can think about that roughly as where the ultrasoft is. 01:24:49.120 --> 01:24:53.190 This is subtracting off that behavior and the remainder 01:24:53.190 --> 01:24:56.324 then is coming from only having divergences for a n bar 01:24:56.324 --> 01:25:00.480 dot k goes to infinity, which is a proper collinear ultraviolet 01:25:00.480 --> 01:25:03.815 divergence not from n bar dot k going to 0. 01:25:03.815 --> 01:25:05.190 So once you put the two together, 01:25:05.190 --> 01:25:07.680 the epsilon IRs cancel and then we get exactly, 01:25:07.680 --> 01:25:10.470 actually, the same expression we had before but where 01:25:10.470 --> 01:25:14.170 all those 1 over epsilons are 1 over epsilon UVs. 01:25:14.170 --> 01:25:16.480 AUDIENCE: So the epsilon has come from second term? 01:25:16.480 --> 01:25:17.730 IAIN STEWART: From both terms. 01:25:17.730 --> 01:25:21.748 So they both have epsilon IRs but they cancel between them. 01:25:21.748 --> 01:25:22.290 AUDIENCE: OK. 01:25:22.290 --> 01:25:23.160 IAIN STEWART: Yeah. 01:25:23.160 --> 01:25:25.260 And the remainder is just epsilon UVs, so 01:25:25.260 --> 01:25:27.810 all the epsilons that I wrote my earlier formula would 01:25:27.810 --> 01:25:31.530 be now epsilon UVs once I take into account the subtraction. 01:25:31.530 --> 01:25:34.080 So I could have just ignored the subtraction and that's 01:25:34.080 --> 01:25:35.340 often what people do. 01:25:35.340 --> 01:25:37.968 If they know that the zero bins are giving a scaleless 01:25:37.968 --> 01:25:40.260 integral, they say, well, let's ignore the subtraction, 01:25:40.260 --> 01:25:42.932 we'll just say that all the epsilons are UV 01:25:42.932 --> 01:25:45.168 and the zero bin makes them UV. 01:25:45.168 --> 01:25:46.710 But if we really want to look and see 01:25:46.710 --> 01:25:48.300 that things are working properly, 01:25:48.300 --> 01:25:51.690 we should take the subtraction and calculate it and make sure 01:25:51.690 --> 01:25:52.885 that that's true. 01:25:52.885 --> 01:25:54.510 But we could have just taken the answer 01:25:54.510 --> 01:25:57.100 that I wrote down earlier and said those epsilons are 01:25:57.100 --> 01:25:59.700 ultraviolet and let's throw them in a counterterm 01:25:59.700 --> 01:26:03.030 and calculate an anomalous dimension. 01:26:03.030 --> 01:26:04.793 So we'll proceed that way next time, 01:26:04.793 --> 01:26:06.210 but we now know that actually they 01:26:06.210 --> 01:26:10.190 are ultraviolet divergences. 01:26:10.190 --> 01:26:13.610 So next time we'll take the ultraviolet divergences 01:26:13.610 --> 01:26:15.942 and we'll define from them a counterterm 01:26:15.942 --> 01:26:17.900 And we'll see how we get an anomalous dimension 01:26:17.900 --> 01:26:22.272 and what kind of logs we sum by using that anomalous dimension. 01:26:24.990 --> 01:26:28.290 So the zero bin that's scaleless in this particular example 01:26:28.290 --> 01:26:29.400 is not always scaleless. 01:26:29.400 --> 01:26:30.938 So sometimes it could give a nonce. 01:26:30.938 --> 01:26:32.730 Depends on the problem you're dealing with. 01:26:32.730 --> 01:26:37.650 So sometimes you can set things up so that it's scaleless 01:26:37.650 --> 01:26:39.893 and then you just basically can ignore it. 01:26:39.893 --> 01:26:41.310 But that's not always true, so you 01:26:41.310 --> 01:26:43.560 do have to think about whether it's really going to be 01:26:43.560 --> 01:26:45.090 true for what you're doing. 01:26:45.090 --> 01:26:47.250 If it is true, then you can effectively 01:26:47.250 --> 01:26:49.202 ignore it because it's sort of just making 01:26:49.202 --> 01:26:51.660 the physics come out right, making sure there's no overlap. 01:26:51.660 --> 01:26:53.868 But if your regulators set up so that it's scaleless, 01:26:53.868 --> 01:26:56.290 you can just get around it. 01:26:56.290 --> 01:26:58.297 But in general, that might not be true. 01:26:58.297 --> 01:26:59.880 If you had more scales in the problem, 01:26:59.880 --> 01:27:01.422 if you're doing some calculation that 01:27:01.422 --> 01:27:05.160 had some jets of finite size then that 01:27:05.160 --> 01:27:08.690 won't be true typically.