WEBVTT 00:00:00.000 --> 00:00:02.520 The following content is provided under a Creative 00:00:02.520 --> 00:00:03.970 Commons license. 00:00:03.970 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.660 continue to offer high quality educational resources for free. 00:00:10.660 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.190 --> 00:00:18.370 at ocw.mit.edu. 00:00:22.193 --> 00:00:23.610 IAIN STEWART: So last time we were 00:00:23.610 --> 00:00:28.677 talking about the subleading one of our MQ Lagrangian HQET. 00:00:28.677 --> 00:00:30.260 Hopefully more people remember that we 00:00:30.260 --> 00:00:33.620 have a makeup lecture today, or will not 00:00:33.620 --> 00:00:36.050 be stopped by the snow. 00:00:36.050 --> 00:00:38.660 So today we're going to continue this discussion of power 00:00:38.660 --> 00:00:40.250 suppressed Lagrangians. 00:00:40.250 --> 00:00:42.920 And I'll explain to you why we think 00:00:42.920 --> 00:00:46.195 of these as giving power corrections to observables. 00:00:46.195 --> 00:00:48.320 And we'll talk about a couple different observables 00:00:48.320 --> 00:00:52.650 where these specific operators are playing an important role. 00:00:52.650 --> 00:00:55.280 And then we're going to turn to the topic of renormalons, 00:00:55.280 --> 00:00:57.780 which is fun stuff. 00:00:57.780 --> 00:01:00.805 How many people here know what a renormalon is? 00:01:00.805 --> 00:01:01.680 AUDIENCE: [INAUDIBLE] 00:01:01.680 --> 00:01:03.090 IAIN STEWART: Yeah. 00:01:03.090 --> 00:01:07.420 By the end of the lecture, you all know what a renormalon is. 00:01:07.420 --> 00:01:09.650 OK, so this is where we left off. 00:01:09.650 --> 00:01:12.220 We were talking about this symmetry reparameterization 00:01:12.220 --> 00:01:12.940 invariance. 00:01:12.940 --> 00:01:17.110 And what we showed last time is that the Wilson coefficient 00:01:17.110 --> 00:01:20.410 of this operator is 1 to all orders in perturbation theory 00:01:20.410 --> 00:01:21.970 because of that symmetry. 00:01:21.970 --> 00:01:24.160 The Wilson coefficient of this operator is not. 00:01:24.160 --> 00:01:26.230 And I told you that at leading log order, 00:01:26.230 --> 00:01:29.080 it would be given by some expression like this. 00:01:29.080 --> 00:01:31.720 But in general, this is just the lowest order expression. 00:01:31.720 --> 00:01:35.532 And it gets perturbatively corrected. 00:01:35.532 --> 00:01:36.990 It's actually known at three loops. 00:01:40.430 --> 00:01:43.460 So let's continue. 00:01:43.460 --> 00:01:45.530 We've talked about reparameterization invariance 00:01:45.530 --> 00:01:47.060 for the Lagrangian. 00:01:47.060 --> 00:01:48.650 Reparameterization invariance also 00:01:48.650 --> 00:01:51.020 has important consequences for operators. 00:02:00.650 --> 00:02:04.040 So just like we wrote down sub-leading Lagrangians, 00:02:04.040 --> 00:02:06.110 we should also write sub-leading currents. 00:02:06.110 --> 00:02:10.009 So there's 1 over mq supressed currents. 00:02:10.009 --> 00:02:11.660 And if you look at those currents 00:02:11.660 --> 00:02:15.463 and you construct them, then you find, again, 00:02:15.463 --> 00:02:16.880 that reparameterization invariance 00:02:16.880 --> 00:02:20.080 plays an important role. 00:02:20.080 --> 00:02:23.768 And again, there's relations between Wilson coefficients. 00:02:27.260 --> 00:02:29.090 And what happens in this case is actually 00:02:29.090 --> 00:02:33.800 that the leading order operator had a Wilson coefficient, 00:02:33.800 --> 00:02:35.640 the lowest order operator. 00:02:35.640 --> 00:02:38.090 So the Wilson coefficients of the subleading operators 00:02:38.090 --> 00:02:40.629 get related to that of the leading operator. 00:02:50.120 --> 00:02:54.010 And I may give you a problem on that on your next problem set. 00:02:54.010 --> 00:02:58.650 But I haven't quite made up my mind yet. 00:02:58.650 --> 00:03:00.662 So rather than go through that, I 00:03:00.662 --> 00:03:02.370 want to talk about some observables where 00:03:02.370 --> 00:03:05.080 these operators played a role. 00:03:05.080 --> 00:03:07.563 So we've seen one example of reparameterization invariance. 00:03:07.563 --> 00:03:08.980 And I think from that one example, 00:03:08.980 --> 00:03:11.530 you would be able to do other examples. 00:03:11.530 --> 00:03:13.780 Let's talk about masses. 00:03:13.780 --> 00:03:15.720 These operators here actually have an impact 00:03:15.720 --> 00:03:18.080 on the masses of hadrons. 00:03:18.080 --> 00:03:24.120 If we think about the mass of a heavy meson, h, 00:03:24.120 --> 00:03:28.470 there's a heavy quark in that heavy meson. 00:03:28.470 --> 00:03:31.200 So we can pull out that heavy quark. 00:03:31.200 --> 00:03:33.720 And then there's some remainder. 00:03:33.720 --> 00:03:36.535 And we can characterize higher order terms in this formula. 00:03:41.360 --> 00:03:44.770 So let me first explain what this lambda bar is. 00:03:44.770 --> 00:03:49.060 Our Lagrangian here, if we're working at lowest order 00:03:49.060 --> 00:03:55.810 in the theory, our Lagrangian would be L0 of HQET plus, 00:03:55.810 --> 00:03:58.090 for the light quarks, they're basically 00:03:58.090 --> 00:04:03.475 just a QCD Lagrangian, so for the light quarks 00:04:03.475 --> 00:04:04.100 and the gluons. 00:04:07.148 --> 00:04:09.740 So for the heavy quark, we have HQET, 00:04:09.740 --> 00:04:11.390 and then there would be higher order 00:04:11.390 --> 00:04:13.973 terms that are 1 over m suppressed, 00:04:13.973 --> 00:04:15.140 which are the ones up there. 00:04:19.079 --> 00:04:21.079 So the question is, if we have this Lagrangian, 00:04:21.079 --> 00:04:24.870 what is the mass of a heavy state like the b meson 00:04:24.870 --> 00:04:26.810 or the d meson or the d star? 00:04:26.810 --> 00:04:29.370 And we can characterize that by knowing the structure 00:04:29.370 --> 00:04:31.430 of the Lagrangian. 00:04:31.430 --> 00:04:35.300 So first of all, what about this term, this lambda bar? 00:04:35.300 --> 00:04:39.257 That term actually comes from these parts of the Lagrangian. 00:04:39.257 --> 00:04:41.090 So if we take these parts of the Lagrangian, 00:04:41.090 --> 00:04:44.570 we can calculate the Hamiltonian. 00:04:44.570 --> 00:04:47.000 Let me call that H0. 00:04:47.000 --> 00:04:49.280 And the definition of this lambda bar 00:04:49.280 --> 00:04:58.537 is the thing that you get by taking that Hamiltonian 00:04:58.537 --> 00:04:59.870 and letting it act on the state. 00:05:03.390 --> 00:05:05.930 So you can think that this is-- 00:05:05.930 --> 00:05:08.390 if you like, you can think that this state is just 00:05:08.390 --> 00:05:10.760 an eigenstate of the Hamiltonian. 00:05:10.760 --> 00:05:13.310 You get the energy. 00:05:13.310 --> 00:05:17.360 But since the Hamiltonian has no heavy quark mass in it, 00:05:17.360 --> 00:05:19.260 what you're getting is some parameter, 00:05:19.260 --> 00:05:21.510 which is traditionally called lambda bar, that doesn't 00:05:21.510 --> 00:05:23.780 have a heavy quark mass in it. 00:05:23.780 --> 00:05:25.670 And so there's no heavy quark mass in here. 00:05:29.840 --> 00:05:32.960 So it's independent of mq. 00:05:32.960 --> 00:05:37.160 It's also independent of spin. 00:05:41.900 --> 00:05:50.120 So B versus B star and flavor B versus D. So 00:05:50.120 --> 00:05:52.548 whether it's a B meson, D meson, as long 00:05:52.548 --> 00:05:54.590 as we're in this limit where both the bottom mass 00:05:54.590 --> 00:06:01.880 and the char mass are heavy, if we're 00:06:01.880 --> 00:06:04.460 thinking of this expansion, then this 00:06:04.460 --> 00:06:06.740 is a universal thing just determined 00:06:06.740 --> 00:06:08.468 by these Lagrangians. 00:06:08.468 --> 00:06:10.010 It's traditionally called lambda bar. 00:06:10.010 --> 00:06:12.830 And it's telling you that if you think about this thing 00:06:12.830 --> 00:06:14.540 as having an order mq piece, this 00:06:14.540 --> 00:06:16.950 is the order mq to the 0 piece. 00:06:16.950 --> 00:06:18.740 The symmetry of the theory is telling us 00:06:18.740 --> 00:06:20.735 that this is a universal constant. 00:06:23.600 --> 00:06:29.660 It does depend on the state here, 00:06:29.660 --> 00:06:32.030 although that state could be connected to other states 00:06:32.030 --> 00:06:35.460 by spin symmetry and flavor symmetry. 00:06:35.460 --> 00:06:41.120 There was this quantum number we talked about last time, which 00:06:41.120 --> 00:06:44.330 we called SL for the spin of the light degrees of freedom, 00:06:44.330 --> 00:06:47.800 and then pi for the parity of the light degrees of freedom. 00:06:52.470 --> 00:06:54.650 So it's the same within every multiplet 00:06:54.650 --> 00:06:58.220 that we get from a fixed SL pi. 00:06:58.220 --> 00:07:00.722 But there's a different value for each multiplet. 00:07:04.852 --> 00:07:06.560 So you'd have a different value of lambda 00:07:06.560 --> 00:07:10.760 bar for the B, D, B star B than for the lambda B and the lambda 00:07:10.760 --> 00:07:11.570 C, for example. 00:07:15.100 --> 00:07:17.200 So that's this term. 00:07:17.200 --> 00:07:19.680 We can also characterize what's going on with this term 00:07:19.680 --> 00:07:21.720 by using this Lagrangian here, the L1. 00:07:25.750 --> 00:07:28.660 So at order 1 over MQ, we can figure out 00:07:28.660 --> 00:07:31.930 what kind of contributions to the mass we're getting. 00:07:31.930 --> 00:07:34.870 h1 is just minus L1. 00:07:34.870 --> 00:07:37.060 You can think of it now perturbatively. 00:07:37.060 --> 00:07:39.370 So that just flips the sign of what we had before. 00:08:00.880 --> 00:08:06.240 And we get two parameters from taking matrix elements here. 00:08:06.240 --> 00:08:07.780 They get the following names. 00:08:21.990 --> 00:08:24.000 And let me just do everything here in terms of-- 00:08:27.510 --> 00:08:30.068 it's not totally crucial, but it's 00:08:30.068 --> 00:08:31.860 a little easier to interpret certain things 00:08:31.860 --> 00:08:33.068 if we work in the rest frame. 00:08:33.068 --> 00:08:35.375 So I'll just work in the rest frame. 00:08:35.375 --> 00:08:36.750 So if you work in the rest frame, 00:08:36.750 --> 00:08:38.583 you should just think of this as like giving 00:08:38.583 --> 00:08:40.500 some kind of measure of the kinetic energy 00:08:40.500 --> 00:08:42.780 that the heavy quark gets wiggled by. 00:08:42.780 --> 00:08:44.760 And that's called lambda 1. 00:08:44.760 --> 00:08:47.040 This, again, is a matrix element that doesn't know 00:08:47.040 --> 00:08:48.870 about the heavy quark mass. 00:08:48.870 --> 00:08:52.260 And the heavy quark mass in an explicit MQ. 00:08:52.260 --> 00:08:56.275 So this guy has dimension 2 if you work out the dimensions. 00:09:01.500 --> 00:09:03.880 And then you can do the same for the other operator. 00:09:03.880 --> 00:09:08.980 And here you have to think a little bit harder. 00:09:08.980 --> 00:09:12.270 And if you think a little bit harder, 00:09:12.270 --> 00:09:15.510 you can characterize what the coefficients 00:09:15.510 --> 00:09:18.360 are for different spins. 00:09:18.360 --> 00:09:22.590 So there's some Wilson coefficient. 00:09:22.590 --> 00:09:23.940 And then we have this operator. 00:09:36.920 --> 00:09:44.235 And this here is like sigma dot B. I told you that before. 00:09:44.235 --> 00:09:45.610 And so you should think of what's 00:09:45.610 --> 00:09:49.300 going on here as you have sigma, which is giving the SQ, 00:09:49.300 --> 00:09:50.750 and then you have a B field. 00:09:50.750 --> 00:09:53.992 And then it's a question of what vector 00:09:53.992 --> 00:09:56.200 could be left over that the B field could know about. 00:09:56.200 --> 00:09:58.570 And it's basically SL. 00:09:58.570 --> 00:10:04.750 So this guy here is leading to the SQ. 00:10:04.750 --> 00:10:06.635 And this guy here is leading to the SL. 00:10:11.870 --> 00:10:13.960 And so that's where this SQ.SL comes from. 00:10:16.810 --> 00:10:20.040 So you can think about other possibilities, 00:10:20.040 --> 00:10:22.350 but you have to have the right symmetry 00:10:22.350 --> 00:10:24.270 structure under time reversal and parity 00:10:24.270 --> 00:10:25.650 and things like that. 00:10:25.650 --> 00:10:27.450 And that forces you to use a spin here 00:10:27.450 --> 00:10:31.230 for the B and not something like the V. 00:10:31.230 --> 00:10:34.230 So this SQ.SL is something that you can write out 00:10:34.230 --> 00:10:39.060 as, remembering the definition of it and its relation to J 00:10:39.060 --> 00:10:41.910 squared. 00:10:41.910 --> 00:10:45.270 You can write out that factor like that. 00:10:45.270 --> 00:10:48.540 And then we can derive here how this lambda 2 contributes 00:10:48.540 --> 00:10:52.410 to the differing states, because we 00:10:52.410 --> 00:10:54.750 have a different contribution for the B 00:10:54.750 --> 00:10:56.850 and the B star from the lambda 2 guy, 00:10:56.850 --> 00:11:00.450 because it was violating the spin symmetry. 00:11:00.450 --> 00:11:03.300 So the lambda 1 guy is going to violate the flavor symmetry. 00:11:03.300 --> 00:11:05.160 That's this 1 over MQ. 00:11:05.160 --> 00:11:06.840 And so charm quarks-- 00:11:06.840 --> 00:11:08.940 charm mesons and bottom mesons will 00:11:08.940 --> 00:11:11.010 get different contributions from that. 00:11:11.010 --> 00:11:14.250 They have the same lambda 1, but they 00:11:14.250 --> 00:11:16.530 have a different connection because it gets suppressed 00:11:16.530 --> 00:11:19.210 by a different factor. 00:11:19.210 --> 00:11:25.815 And then again for the lamba 2, Lambda 2 00:11:25.815 --> 00:11:27.690 is a little bit different because it actually 00:11:27.690 --> 00:11:29.072 has mu dependents. 00:11:32.852 --> 00:11:33.810 That's not quite right. 00:11:33.810 --> 00:11:39.240 So it actually has MQ dependents because the coefficient here 00:11:39.240 --> 00:11:41.160 has mu dependents. 00:11:41.160 --> 00:11:43.810 So this operator doesn't have any MQ dependence, 00:11:43.810 --> 00:11:46.350 but the coefficient does, remember. 00:11:46.350 --> 00:11:48.710 The mu dependence cancels and there's a left over MQ 00:11:48.710 --> 00:11:53.691 dependence to the lambda 2, but it's logarithmic MQ dependence. 00:11:56.780 --> 00:11:59.960 If I define it this way-- 00:11:59.960 --> 00:12:01.640 so that's why I wrote MQ there. 00:12:09.740 --> 00:12:11.990 So you could, as we talked about last time, 00:12:11.990 --> 00:12:15.962 you could sum up the logs of large logs and have this guy-- 00:12:15.962 --> 00:12:17.420 you could re-sum logs, if you want. 00:12:17.420 --> 00:12:19.295 So you could think of these as you can re-sum 00:12:19.295 --> 00:12:20.450 logs inside this lambda 2. 00:12:23.660 --> 00:12:28.280 So if we ignore those logs or we imagine that we re-sum them 00:12:28.280 --> 00:12:29.900 and we just look at the remainder, 00:12:29.900 --> 00:12:32.750 then we have the expectations based on power counting 00:12:32.750 --> 00:12:35.090 for how big these things are. 00:12:35.090 --> 00:12:36.740 And we just [? expect ?] that they're 00:12:36.740 --> 00:12:38.480 both given by the dimension. 00:12:38.480 --> 00:12:40.310 And the only [? dimensionful ?] parameter 00:12:40.310 --> 00:12:43.533 around is lambda QCD because the MQs can only 00:12:43.533 --> 00:12:45.950 occur in logarithms, and the Wilson coefficient in nowhere 00:12:45.950 --> 00:12:48.580 else. 00:12:48.580 --> 00:12:52.800 And these are non-perturbative parameters. 00:12:57.450 --> 00:13:02.162 Lambda bar is also a non-perturbative parameter. 00:13:02.162 --> 00:13:04.120 These ones have a little more dynamics in them. 00:13:06.780 --> 00:13:10.280 So then we could go out and write for our states 00:13:10.280 --> 00:13:13.170 the results by putting these things together. 00:13:13.170 --> 00:13:16.550 So we'd have M capital B plub M little B 00:13:16.550 --> 00:13:22.040 plus lambda bar lambda 1 over 2 and little B. 00:13:22.040 --> 00:13:24.890 And if we go through the spin structure there, 00:13:24.890 --> 00:13:30.410 we get a 3/2 and then lamba 2 logarithmic MB 00:13:30.410 --> 00:13:31.660 dependence over MB. 00:13:40.200 --> 00:13:43.410 So when we start with the B star, 00:13:43.410 --> 00:13:44.980 the first three terms are the same. 00:13:44.980 --> 00:13:48.030 And then this guy comes in with a plus. 00:13:48.030 --> 00:13:51.850 And it's lambda 2 over MB over 2MB. 00:13:55.600 --> 00:14:03.520 And then the D states are similar in turn. 00:14:03.520 --> 00:14:11.800 Lambda bar lambda 1 over 2M term, and then these two terms 00:14:11.800 --> 00:14:13.330 again. 00:14:13.330 --> 00:14:17.218 Same spins, so it's lambda 2. 00:14:17.218 --> 00:14:18.760 And the only difference is, now we're 00:14:18.760 --> 00:14:28.500 evaluating the lambda 2 as logarithm and charm dependence, 00:14:28.500 --> 00:14:29.000 OK. 00:14:29.000 --> 00:14:31.430 So there's some formulas that are correct, actually-- 00:14:31.430 --> 00:14:34.490 including all the way up to 1 over M corrections. 00:14:34.490 --> 00:14:36.140 And you can see that there's still sort 00:14:36.140 --> 00:14:37.580 of a structure to these things. 00:14:37.580 --> 00:14:40.310 This is a universal correction for all of them. 00:14:40.310 --> 00:14:43.730 These corrections are universal between B and B star. 00:14:43.730 --> 00:14:45.935 And so the splitting between B and B star 00:14:45.935 --> 00:14:48.660 is just given by this lambda 2. 00:14:48.660 --> 00:14:50.600 That's what causes the states to split. 00:14:50.600 --> 00:14:52.920 So up to that level, they're the same. 00:14:52.920 --> 00:14:55.095 And that's why the B and B star are very degenerate 00:14:55.095 --> 00:14:56.345 and the masses are very close. 00:14:59.010 --> 00:15:02.930 So you can form combinations to cancel things out. 00:15:02.930 --> 00:15:08.960 So you can take a kind of spin average mass where 00:15:08.960 --> 00:15:14.480 you take 3 times the vector plus the scalar divided by 4. 00:15:14.480 --> 00:15:17.750 And if you do that, then you're canceling out the lambda 2. 00:15:27.010 --> 00:15:29.360 And then you can also form differences. 00:15:29.360 --> 00:15:33.800 So if you looked at MV star squared minus MB squared, 00:15:33.800 --> 00:15:35.770 then the only thing that's causing a difference 00:15:35.770 --> 00:15:40.060 is lambda 2. 00:15:40.060 --> 00:15:44.860 Numerically, this is 0.49 GV squared. 00:15:44.860 --> 00:15:50.980 And from our formula up there, it's 4 lambda 2 of MB, 00:15:50.980 --> 00:16:00.020 so something like four times 0.12 GV squared. 00:16:00.020 --> 00:16:02.290 So that's the value of this lambda 00:16:02.290 --> 00:16:06.770 2 is 0.12 GV squared, which is about lambda QCD squared. 00:16:06.770 --> 00:16:09.555 So our dimensional analysis is working. 00:16:09.555 --> 00:16:11.680 And then you can do the same thing for the D meson. 00:16:15.917 --> 00:16:17.750 Everything here works beautifully, actually. 00:16:31.430 --> 00:16:33.100 Extract the value of lambda 2 and you 00:16:33.100 --> 00:16:39.130 see there's a slight difference because that difference can be 00:16:39.130 --> 00:16:40.834 attributed to the logarithms. 00:16:44.710 --> 00:16:48.662 So if you take lambda 2 MB over lambda 2 MC, 00:16:48.662 --> 00:16:51.120 then the only thing that differs is the Wilson coefficient. 00:16:51.120 --> 00:16:54.450 So that's something you can predict. 00:16:54.450 --> 00:17:00.890 So experimentally, if you take the ratio of those two things 00:17:00.890 --> 00:17:06.530 there, you get a prediction like this one. 00:17:06.530 --> 00:17:12.148 And if you plug in theory like leading log RGE, 00:17:12.148 --> 00:17:13.565 then you're getting alpha S of MB. 00:17:16.166 --> 00:17:17.290 Oh, sorry. 00:17:17.290 --> 00:17:18.109 [INAUDIBLE] 00:17:23.749 --> 00:17:28.220 And this is 1.17. 00:17:28.220 --> 00:17:30.790 So that's for three light flavors, 00:17:30.790 --> 00:17:33.500 which is the right thing to do for the beta function. 00:17:33.500 --> 00:17:37.150 So it's not so bad for a leading log prediction. 00:17:37.150 --> 00:17:40.090 That's kind of the accuracy you would expect. 00:17:44.090 --> 00:17:45.260 All right. 00:17:45.260 --> 00:17:47.020 So we're really understanding quite a bit 00:17:47.020 --> 00:17:48.603 from our [INAUDIBLE] [? instruction ?] 00:17:48.603 --> 00:17:51.280 of this effective theory about the states, 00:17:51.280 --> 00:17:55.570 and even make predictions for the ratios of the mass divided 00:17:55.570 --> 00:17:58.350 by that as something preservative, which 00:17:58.350 --> 00:18:01.165 is kind of non-trivial. 00:18:01.165 --> 00:18:03.040 Perturbative up to the order we were working. 00:18:03.040 --> 00:18:06.310 There would be corrections if we continued in our series here. 00:18:06.310 --> 00:18:08.470 There could be some non-perturbative corrections. 00:18:08.470 --> 00:18:11.530 But up to 1 over M, the ratio of this 00:18:11.530 --> 00:18:15.463 to that, which is what I'd wrote over there, is perturbative. 00:18:18.640 --> 00:18:20.580 All right. 00:18:20.580 --> 00:18:23.040 So that's some phenomenology. 00:18:23.040 --> 00:18:24.840 It's kind of baseline phenomenology 00:18:24.840 --> 00:18:27.330 that we can do with the Hamiltonian 00:18:27.330 --> 00:18:31.770 or with Hamiltonian derived from our subleading Lagrangian. 00:18:31.770 --> 00:18:35.010 There's other phenomenology that we can do. 00:18:35.010 --> 00:18:38.045 And I'll mention some of the most important phenomenology. 00:18:44.360 --> 00:18:46.407 There's another class of predictions 00:18:46.407 --> 00:18:48.740 that we can make where we have a lot of predictive power 00:18:48.740 --> 00:18:50.128 by using the effective theory. 00:18:50.128 --> 00:18:51.920 And I want to talk a little bit about that. 00:19:01.120 --> 00:19:03.420 So you can look at semi-leptonic decays. 00:19:03.420 --> 00:19:06.690 And there's two different types of semi-leptonic decays that 00:19:06.690 --> 00:19:08.430 you can look at-- 00:19:08.430 --> 00:19:12.430 so-called exclusive decays and inclusive decays. 00:19:12.430 --> 00:19:14.340 So exclusive is making transitions 00:19:14.340 --> 00:19:17.580 between the meson states and inclusive 00:19:17.580 --> 00:19:22.710 is making a transition where you allow any charm state, not just 00:19:22.710 --> 00:19:24.750 the lowest order ground state charm, 00:19:24.750 --> 00:19:28.530 but you could have in this state here, XE, you 00:19:28.530 --> 00:19:32.710 could have a D pi or a D star pi pi or other things like that. 00:19:32.710 --> 00:19:37.300 So it doesn't even have to be a single hadron state. 00:19:37.300 --> 00:19:39.600 So exclusive refers to a particular channel. 00:19:39.600 --> 00:19:43.350 Inclusive refers to some overall channels. 00:19:43.350 --> 00:19:46.360 So the theory in these two is quite different. 00:19:46.360 --> 00:19:51.960 In this one, you would have form factors for the current 00:19:51.960 --> 00:19:53.490 between the states. 00:19:53.490 --> 00:19:56.250 And I already mentioned to you that heavy quark symmetry 00:19:56.250 --> 00:19:57.810 reduces the number of form factors. 00:19:57.810 --> 00:20:00.210 You get this single Isgur-Wise function. 00:20:00.210 --> 00:20:06.270 And so heavy quark symmetry is very powerful here. 00:20:06.270 --> 00:20:11.760 And you can actually also work out 1 over MQ corrections. 00:20:11.760 --> 00:20:13.240 And people have done that. 00:20:13.240 --> 00:20:15.547 And it turns out there aren't any. 00:20:15.547 --> 00:20:18.270 So there's something called Luke's theorem, which actually 00:20:18.270 --> 00:20:20.160 was a result derived, I think, when 00:20:20.160 --> 00:20:24.390 Luke was a graduate student that proved there was no 1 over m Q 00:20:24.390 --> 00:20:26.400 corrections here. 00:20:26.400 --> 00:20:28.470 And then you can work out alpha s corrections. 00:20:28.470 --> 00:20:31.530 And so this kind of-- you can just keep going. 00:20:31.530 --> 00:20:34.350 People have worked out 1 over m Q squared corrections. 00:20:34.350 --> 00:20:36.360 And this kind of formalism is used 00:20:36.360 --> 00:20:41.010 to measure V c b because these decay rates would depend 00:20:41.010 --> 00:20:43.170 on V c b, and that's kind of what you're 00:20:43.170 --> 00:20:45.180 after if you're doing this. 00:20:45.180 --> 00:20:47.160 So you'd like to parameterize and figure out 00:20:47.160 --> 00:20:49.230 the hadronic physics as much as possible, 00:20:49.230 --> 00:20:52.110 and aid the experiments in getting V c b, 00:20:52.110 --> 00:20:54.480 and exactly this framework is used. 00:20:54.480 --> 00:20:56.790 People nowadays-- what they do is, the 1 over m Q 00:20:56.790 --> 00:20:59.950 squared corrections need to be computed on the lattice. 00:20:59.950 --> 00:21:02.195 So lattice QCD computes those corrections. 00:21:02.195 --> 00:21:03.570 The perturbative corrections need 00:21:03.570 --> 00:21:06.460 to be computed by continuum people like me, 00:21:06.460 --> 00:21:07.793 so those are computed. 00:21:07.793 --> 00:21:09.210 And they put these things together 00:21:09.210 --> 00:21:13.260 to get V c b from these decays. 00:21:13.260 --> 00:21:16.500 You could also do it with this inclusive guy. 00:21:20.430 --> 00:21:24.210 And here it's more interesting because you can basically do 00:21:24.210 --> 00:21:27.337 everything with pen and paper. 00:21:27.337 --> 00:21:29.670 So here, you can use an Operator Product Expansion, OPE. 00:21:34.770 --> 00:21:39.780 And actually, HQET constrains the form of the operator 00:21:39.780 --> 00:21:41.330 product expansion, as well. 00:21:52.540 --> 00:21:59.110 And indeed, when you work out the leading power corrections, 00:21:59.110 --> 00:22:03.520 they turn out to not enter until two orders down. 00:22:03.520 --> 00:22:06.370 So again, you're protected from first-order corrections, 00:22:06.370 --> 00:22:11.380 so you just get corrections that occur at 1 over m Q squared. 00:22:11.380 --> 00:22:15.403 And furthermore, they depend only on the two parameters 00:22:15.403 --> 00:22:16.195 we already defined. 00:22:19.560 --> 00:22:23.830 So it's not like you even get any new nonperturbative matrix 00:22:23.830 --> 00:22:24.330 comments. 00:22:24.330 --> 00:22:26.288 You just get the two we were talking over here, 00:22:26.288 --> 00:22:28.180 were the masses. 00:22:28.180 --> 00:22:28.680 OK? 00:22:28.680 --> 00:22:30.497 So there's perturbation corrections, 00:22:30.497 --> 00:22:32.580 and there's power corrections, but you're actually 00:22:32.580 --> 00:22:34.080 not even getting any new information 00:22:34.080 --> 00:22:38.230 relative to the masses, any new matrix elements. 00:22:41.680 --> 00:22:42.180 OK? 00:22:42.180 --> 00:22:44.100 So we'll talk a little bit more about this second one 00:22:44.100 --> 00:22:44.642 with the OPE. 00:22:54.260 --> 00:22:56.060 There's a lot of work that's gone 00:22:56.060 --> 00:22:58.850 into this one, both from the theory side, 00:22:58.850 --> 00:23:00.397 as well as from experiment. 00:23:04.970 --> 00:23:08.060 So when you think about the decay rate for this B 00:23:08.060 --> 00:23:11.780 to X sub c l nu bar, it's a doubly differential decay rate 00:23:11.780 --> 00:23:17.540 in Q squared over the lepton pair, so that's Q mu, E l-- 00:23:17.540 --> 00:23:19.520 because you can pick it as a second variable-- 00:23:19.520 --> 00:23:21.260 and then you don't know the mass of the state 00:23:21.260 --> 00:23:23.130 X sub c because it could be different things. 00:23:23.130 --> 00:23:24.650 It could be a D. It could be a D star. 00:23:24.650 --> 00:23:25.442 It could be a D pi. 00:23:32.750 --> 00:23:34.798 So that's another variable. 00:23:34.798 --> 00:23:36.590 So you have a triply differential spectrum, 00:23:36.590 --> 00:23:40.223 and you can compute it with an operator product expansion. 00:23:49.638 --> 00:23:51.930 I'm not going to go through the details of carrying out 00:23:51.930 --> 00:23:54.180 the operator product expansion. 00:23:54.180 --> 00:23:56.555 If you want to do reading about that, 00:23:56.555 --> 00:23:57.930 there's some supplemental reading 00:23:57.930 --> 00:23:59.712 that I haven't assigned to you where 00:23:59.712 --> 00:24:01.170 you can read about that in the book 00:24:01.170 --> 00:24:03.990 Heavy Quark Physics by Manohar and Wise. 00:24:08.730 --> 00:24:13.470 So for our purposes here, let's think of what operator product 00:24:13.470 --> 00:24:15.817 expansion means. 00:24:15.817 --> 00:24:17.400 It is simply that we want to carry out 00:24:17.400 --> 00:24:35.370 an expansion in lambda QCD over m Q. 00:24:35.370 --> 00:24:38.130 And if you go into the details, there's 00:24:38.130 --> 00:24:39.630 an important role played by the fact 00:24:39.630 --> 00:24:41.130 that you're summing over all states, 00:24:41.130 --> 00:24:43.110 and that's allowing you to connect 00:24:43.110 --> 00:24:46.350 the partonic calculations to the hadronic calculations, 00:24:46.350 --> 00:24:48.740 basically by probability conservation, 00:24:48.740 --> 00:24:52.380 but I'm not going to go so much into that. 00:24:52.380 --> 00:24:56.370 So when you do this operator product expansion, 00:24:56.370 --> 00:24:58.470 you can think of it in terms of diagrams. 00:24:58.470 --> 00:25:02.100 And usually, you draw these diagrams as forward diagrams. 00:25:02.100 --> 00:25:04.380 So here's kind of the matrix elements squared. 00:25:07.740 --> 00:25:08.970 Here is the final state. 00:25:13.350 --> 00:25:16.543 There's the b quark going in and another b quark going out, 00:25:16.543 --> 00:25:18.210 so that's like a matrix element squared. 00:25:22.194 --> 00:25:24.698 The amplitude squared. 00:25:24.698 --> 00:25:26.990 And you think about doing an operator product expansion 00:25:26.990 --> 00:25:27.490 for that. 00:25:30.210 --> 00:25:32.420 And you match onto operators in the effective theory. 00:25:38.790 --> 00:25:41.790 Just to give you a schematic structure for those operators. 00:25:52.770 --> 00:25:55.860 So there's a leading-order operator. 00:25:55.860 --> 00:25:57.660 There's a subleading-order operator 00:25:57.660 --> 00:26:00.000 that has an extra covariant derivative in it. 00:26:00.000 --> 00:26:01.560 That's kind of what you would do. 00:26:01.560 --> 00:26:03.780 And then you have Wilson coefficients. 00:26:03.780 --> 00:26:07.600 The leading-order operator here is just b bar b. 00:26:07.600 --> 00:26:09.540 That's what it turns out to be. 00:26:09.540 --> 00:26:12.540 This operator here would look familiar 00:26:12.540 --> 00:26:16.360 because it's D transverse squared, 00:26:16.360 --> 00:26:18.040 so that's like our lambda 1 operator, 00:26:18.040 --> 00:26:20.520 and that's where the lambda 1's are going to come from. 00:26:20.520 --> 00:26:22.187 And then there would be a lambda 2 term, 00:26:22.187 --> 00:26:24.770 and I just didn't write it. 00:26:24.770 --> 00:26:26.852 So there's a magnetic guy here, too. 00:26:34.570 --> 00:26:36.910 So when you look at the leading-order operator, 00:26:36.910 --> 00:26:40.570 b bar b, that counts the number of b quarks, OK? 00:26:40.570 --> 00:26:42.930 That's just a number operator. 00:26:42.930 --> 00:26:47.680 So to all orders in perturbation theory, b bar b is 1. 00:26:50.420 --> 00:26:50.920 Yeah? 00:26:53.990 --> 00:26:56.960 AUDIENCE: How do you cut these-- 00:26:56.960 --> 00:26:58.567 like, you're actually calculating 00:26:58.567 --> 00:26:59.900 [? B to A, right? ?] [INAUDIBLE] 00:26:59.900 --> 00:27:00.733 IAIN STEWART: Right. 00:27:00.733 --> 00:27:02.660 So you should think of actually looking-- 00:27:02.660 --> 00:27:04.920 so this is a charm quark. 00:27:04.920 --> 00:27:07.160 And so you would write down that propagator, 00:27:07.160 --> 00:27:09.770 and then you have a large injection of momentum 00:27:09.770 --> 00:27:11.600 from the b quark, right? 00:27:11.600 --> 00:27:14.780 Use momentum conservation and expand the propagator. 00:27:14.780 --> 00:27:16.700 Thinking about the propagator, if you like, 00:27:16.700 --> 00:27:21.410 thinking about this guy here having-- 00:27:21.410 --> 00:27:27.110 so you could write this guy's momentum as m b v plus k, 00:27:27.110 --> 00:27:27.650 all right? 00:27:27.650 --> 00:27:30.177 And you would expand in k. 00:27:30.177 --> 00:27:32.510 So at first order, you just drop all the k's, and that's 00:27:32.510 --> 00:27:34.520 basically giving you this term. 00:27:34.520 --> 00:27:36.230 At second order, you'd keep the k's, 00:27:36.230 --> 00:27:37.970 so the k squared would be identified 00:27:37.970 --> 00:27:40.130 with this D t squared. 00:27:40.130 --> 00:27:43.880 AUDIENCE: Right, but how do you cut the effective [INAUDIBLE]?? 00:27:43.880 --> 00:27:46.850 IAIN STEWART: The effective theory diagram, it's 00:27:46.850 --> 00:27:48.830 already been cut, if you like. 00:27:48.830 --> 00:27:51.260 So this is just a real thing, and I've already 00:27:51.260 --> 00:27:53.190 taken the imaginary part. 00:27:53.190 --> 00:27:55.760 So this is all real. 00:27:55.760 --> 00:27:58.140 There's no cut to make in the effective theory. 00:27:58.140 --> 00:27:59.970 The thing I'm integrating out here-- 00:27:59.970 --> 00:28:01.280 so this is a good comment. 00:28:01.280 --> 00:28:04.580 The thing I'm integrating out here is hard and off-shell. 00:28:04.580 --> 00:28:06.680 So when I go over to the effective theory 00:28:06.680 --> 00:28:12.080 and I get rid of the off-shell stuff, I have a real part here. 00:28:12.080 --> 00:28:16.210 There's nothing to cut in the effective theory. 00:28:16.210 --> 00:28:17.657 Sometimes, as you know, there are 00:28:17.657 --> 00:28:19.240 things to cut in the effective theory, 00:28:19.240 --> 00:28:21.640 but here there's nothing to cut on this side. 00:28:30.550 --> 00:28:33.790 OK, so this is one tall order. 00:28:33.790 --> 00:28:35.770 So that actually means that we don't even 00:28:35.770 --> 00:28:37.750 have a nontrivial matrix element here. 00:28:37.750 --> 00:28:39.790 We just have a Wilson coefficient. 00:28:39.790 --> 00:28:44.640 So everything in the first-order term is completely calculable. 00:28:44.640 --> 00:28:54.540 No nonperturbative parameter because of symmetry. 00:28:54.540 --> 00:28:56.997 And this guy here is the power correction. 00:29:03.680 --> 00:29:06.630 And if you look at what the Wilson coefficient is-- 00:29:06.630 --> 00:29:08.465 so it's a function of alpha s. 00:29:12.150 --> 00:29:14.440 You could think of it like this, if you want. 00:29:24.040 --> 00:29:28.090 And all the kinematic variables, like kinematic variables 00:29:28.090 --> 00:29:30.070 like these guys-- 00:29:30.070 --> 00:29:41.890 and it is equal to the free b quark decay, including the loop 00:29:41.890 --> 00:29:43.750 corrections, of course. 00:29:43.750 --> 00:29:45.490 So not only the tree-level result, 00:29:45.490 --> 00:29:50.170 but adding loop corrections to the right-hand side here. 00:29:50.170 --> 00:29:52.960 If you add the loop corrections and you 00:29:52.960 --> 00:29:58.300 drop all the k squared's, then you get just the free b quark 00:29:58.300 --> 00:30:00.923 decay, no nonperturbative matrix elements, 00:30:00.923 --> 00:30:03.340 so you can just calculate that first term, order by order. 00:30:03.340 --> 00:30:05.380 Just doing a [? partonic ?] calculation 00:30:05.380 --> 00:30:07.840 gives you the right thing to describe this decay. 00:30:10.670 --> 00:30:13.295 And the OPE is telling you that that formally 00:30:13.295 --> 00:30:14.420 is exactly the right thing. 00:30:14.420 --> 00:30:16.070 Even if you just started to do it, 00:30:16.070 --> 00:30:19.130 it's actually technically the right thing to do here. 00:30:19.130 --> 00:30:23.360 It wouldn't be the right thing to do for the exclusive decays, 00:30:23.360 --> 00:30:26.510 but it is the right thing to do here for the inclusive. 00:30:37.280 --> 00:30:40.130 So what is the range of validity of this sort of-- 00:30:40.130 --> 00:30:42.740 is there any restrictions on these kinematic variables? 00:30:42.740 --> 00:30:44.000 And there is. 00:30:44.000 --> 00:30:46.505 We're treating the kinematic variables as if they're hard. 00:30:51.060 --> 00:30:52.770 And that means basically that we're 00:30:52.770 --> 00:30:55.740 thinking that they scale like powers of the heavy quark mass. 00:31:00.560 --> 00:31:02.540 So there can be regions of phase space 00:31:02.540 --> 00:31:04.820 where that wouldn't be true, but as long 00:31:04.820 --> 00:31:10.210 as we stay away from those corners, edges, then 00:31:10.210 --> 00:31:11.130 what I said is right. 00:31:15.790 --> 00:31:18.980 And it's a little more powerful than that. 00:31:18.980 --> 00:31:22.690 So stay away from edges. 00:31:27.340 --> 00:31:31.180 Or there's another way of thinking about it, 00:31:31.180 --> 00:31:32.890 and that is that you can integrate over 00:31:32.890 --> 00:31:37.045 regions of the Dalitz plane for the phase-space variables. 00:31:43.480 --> 00:31:46.670 That includes all the way up to the edges. 00:31:46.670 --> 00:31:53.770 And as long as the size of your integration region 00:31:53.770 --> 00:31:57.600 is order m Q, then you're also fine. 00:31:57.600 --> 00:32:00.540 So think of, like, something with, 00:32:00.540 --> 00:32:05.777 like, m B squared up here, d m x squared, for example. 00:32:05.777 --> 00:32:08.110 And you can-- it doesn't have to be exactly m B squared. 00:32:08.110 --> 00:32:10.652 It could be m B squared over 2, m B squared over 4, something 00:32:10.652 --> 00:32:12.880 that you're counting as order m B squared. 00:32:12.880 --> 00:32:16.410 And again, you would be fine with doing this type 00:32:16.410 --> 00:32:20.207 of operator product expansion. 00:32:20.207 --> 00:32:22.040 So what happens is, if you restrict yourself 00:32:22.040 --> 00:32:24.347 to be close to the edges, then restricting 00:32:24.347 --> 00:32:26.180 yourself to be close to the edges introduces 00:32:26.180 --> 00:32:27.952 new scales in the problem. 00:32:27.952 --> 00:32:29.660 And if there's new scales in the problem, 00:32:29.660 --> 00:32:32.160 just having a power counting that separates out lambda, QCD, 00:32:32.160 --> 00:32:34.070 and m B would not be enough. 00:32:34.070 --> 00:32:37.670 You would have to do more detail. 00:32:37.670 --> 00:32:39.860 And there is actually some interesting things 00:32:39.860 --> 00:32:41.737 that happen there, and we probably 00:32:41.737 --> 00:32:43.820 will talk about at least one example later on when 00:32:43.820 --> 00:32:47.840 we talk about [? SCT. ?] 00:32:47.840 --> 00:32:50.870 OK, so I already said it, that there's a nontrivial fact 00:32:50.870 --> 00:32:53.570 happening at NLO. 00:32:53.570 --> 00:32:56.840 Well, I already implied that there's 00:32:56.840 --> 00:33:00.800 no 1 over m b corrections, and so that's 00:33:00.800 --> 00:33:04.950 an important outcome from this result. 00:33:04.950 --> 00:33:07.430 If you want to derive this, you need the effective theory. 00:33:10.010 --> 00:33:13.280 You basically use the equation of motion 00:33:13.280 --> 00:33:14.420 to get that to be true. 00:33:21.360 --> 00:33:23.610 So this is NLO in the power corrections, 00:33:23.610 --> 00:33:34.520 and then NNLO as I said, is just lambda 1 and lambda 2 00:33:34.520 --> 00:33:37.880 at order lambda QCD squared over m b squared. 00:33:45.800 --> 00:33:49.180 So phenomenologically, this is actually wildly successful. 00:33:49.180 --> 00:33:51.670 This is-- as far as I know, this is the case 00:33:51.670 --> 00:33:54.490 in QCD or in any theory where people have actually 00:33:54.490 --> 00:33:59.140 carried out the operator product expansion in the most detail. 00:33:59.140 --> 00:34:02.170 So people have gone up to 1 over m Q to the 4th. 00:34:02.170 --> 00:34:05.440 So they calculated at least two loops 00:34:05.440 --> 00:34:07.870 for everything and maybe even three loops for the [? NF ?] 00:34:07.870 --> 00:34:09.070 pieces. 00:34:09.070 --> 00:34:13.969 And experimentally it's been explored to death. 00:34:13.969 --> 00:34:17.050 You have these three variables, and they've 00:34:17.050 --> 00:34:20.050 constructed of order 80 moments from these three 00:34:20.050 --> 00:34:22.690 different kinematic variables, sliced 00:34:22.690 --> 00:34:25.900 and diced the decay rate in all sorts 00:34:25.900 --> 00:34:28.742 of imaginative and crazy ways. 00:34:28.742 --> 00:34:30.909 And they've really tested that this OPE [? always ?] 00:34:30.909 --> 00:34:33.219 works beautifully, OK? 00:34:33.219 --> 00:34:36.850 So you can think of that you're getting a consistent picture, 00:34:36.850 --> 00:34:41.270 80 different observables, just from a few simple predictions 00:34:41.270 --> 00:34:42.520 and your perturbative results. 00:35:02.070 --> 00:35:03.570 You also get v c b. 00:35:14.430 --> 00:35:17.070 So everything fits together with the framework agreement 00:35:17.070 --> 00:35:19.840 I've discussed with you, and you get a result for v c b, 00:35:19.840 --> 00:35:23.070 which I'm not going to write on the board. 00:35:23.070 --> 00:35:23.580 OK? 00:35:23.580 --> 00:35:24.570 So that gives you-- 00:35:24.570 --> 00:35:26.130 I didn't go through the details. 00:35:26.130 --> 00:35:28.890 I will give you some reading to learn more about this OPE 00:35:28.890 --> 00:35:32.090 if you're interested, and sort of [? seize ?] how 00:35:32.090 --> 00:35:33.840 some of the things I mentioned, like using 00:35:33.840 --> 00:35:36.390 the equation of motion, how that actually works out. 00:35:36.390 --> 00:35:39.840 It's all done very nicely in this book, Heavy Quark Physics. 00:35:39.840 --> 00:35:42.150 But I just wanted to give you a flavor 00:35:42.150 --> 00:35:44.010 with this effective theory that there's 00:35:44.010 --> 00:35:47.230 something very useful you can do with it once you have it. 00:35:47.230 --> 00:35:50.280 And in this case, kind of the main phenomenological thing 00:35:50.280 --> 00:35:53.580 that you're after is v c b and what I've described to you, 00:35:53.580 --> 00:35:55.560 but you could also think about b decays, 00:35:55.560 --> 00:35:57.210 where you're looking for new physics. 00:35:57.210 --> 00:36:00.900 And again, if you can construct how the decay rate is looking, 00:36:00.900 --> 00:36:03.660 then you can have a hope of finding 00:36:03.660 --> 00:36:06.540 new physics in the coefficient of those decay rates, 00:36:06.540 --> 00:36:09.830 effectively in the Wilson coefficients of decay rates. 00:36:09.830 --> 00:36:10.330 OK? 00:36:10.330 --> 00:36:12.760 So we're going to turn to something else, 00:36:12.760 --> 00:36:15.780 but let me pause and see if there's any questions. 00:36:15.780 --> 00:36:18.660 AUDIENCE: Is this the best way to [? get ?] [? v c b? ?] 00:36:18.660 --> 00:36:20.910 IAIN STEWART: The two are actually pretty competitive, 00:36:20.910 --> 00:36:22.770 the exclusive and inclusive. 00:36:22.770 --> 00:36:25.320 Basically because the lattice is doing 00:36:25.320 --> 00:36:27.270 a good job of the matrix elements 00:36:27.270 --> 00:36:30.210 that you need in the exclusive. 00:36:30.210 --> 00:36:32.640 And again, you have a lot of kinematic information 00:36:32.640 --> 00:36:35.113 you can use from the experiment on the shape. 00:36:35.113 --> 00:36:36.780 So they're pretty competitive, actually. 00:36:36.780 --> 00:36:38.067 AUDIENCE: Do they agree? 00:36:38.067 --> 00:36:39.150 IAIN STEWART: They agree-- 00:36:39.150 --> 00:36:42.690 I think the disagreement is at, like, kind of the 1.8 sigma 00:36:42.690 --> 00:36:43.980 level, so they agree. 00:36:47.580 --> 00:36:49.650 There's an interesting story in v u b. 00:36:49.650 --> 00:36:52.590 So you could do the same kind of thing, not for a charm quark, 00:36:52.590 --> 00:36:54.990 as we did here, but for a light quark. 00:36:54.990 --> 00:36:57.183 And then actually, if you do b to pi l nu 00:36:57.183 --> 00:36:59.100 and you look at the inclusive version of that, 00:36:59.100 --> 00:37:00.785 and you can do the same type of OPE-- 00:37:00.785 --> 00:37:02.160 a little bit different, actually, 00:37:02.160 --> 00:37:04.248 but you can do it in OPE there, as well. 00:37:04.248 --> 00:37:05.790 Then you go through and you get a v u 00:37:05.790 --> 00:37:07.915 b in the two different ways, and the disagreement's 00:37:07.915 --> 00:37:11.760 at the 2 and 1/2 sigma level, maybe even-- yeah. 00:37:11.760 --> 00:37:15.180 It fluctuates with time, but on average it's 2 and 1/2 sigma. 00:37:15.180 --> 00:37:16.290 [CHUCKLES] 00:37:16.290 --> 00:37:17.320 So that's interesting. 00:37:17.320 --> 00:37:21.900 There's not an understanding of what's going on there. 00:37:21.900 --> 00:37:23.220 That's more interesting than-- 00:37:23.220 --> 00:37:27.000 this one basically agrees. 00:37:27.000 --> 00:37:29.610 OK, so rather than go further into this, 00:37:29.610 --> 00:37:32.300 I want to turn now to my promised topic to you, 00:37:32.300 --> 00:37:35.760 to tell you about renormalons. 00:37:35.760 --> 00:37:37.260 Actually, renormalons have something 00:37:37.260 --> 00:37:38.640 to do with power correction, so it's not 00:37:38.640 --> 00:37:40.820 disconnected from what we've been talking about. 00:37:48.160 --> 00:37:49.560 So what are the ideas here? 00:38:02.523 --> 00:38:04.940 So we've already seen in our discussion of renormalization 00:38:04.940 --> 00:38:07.340 that there is a freedom in defining 00:38:07.340 --> 00:38:09.120 the perturbative series. 00:38:09.120 --> 00:38:11.570 And we kind of focused on the m s bar scheme 00:38:11.570 --> 00:38:12.740 as being the simplest thing. 00:38:16.180 --> 00:38:19.603 But I told you about some subtleties with m s bar, 00:38:19.603 --> 00:38:21.020 and now we're going to address one 00:38:21.020 --> 00:38:24.170 of them that's related to this thing called renormalons. 00:38:36.590 --> 00:38:41.090 So we have some freedom, if you like, in adjusting the cutoffs. 00:38:41.090 --> 00:38:42.712 And we had this cutoff, mu, and it 00:38:42.712 --> 00:38:44.420 was dividing up what was perturbative and 00:38:44.420 --> 00:38:46.695 nonperturbative in m s bar. 00:38:49.400 --> 00:38:51.863 But we could have done something else. 00:38:51.863 --> 00:38:53.780 We could have used a different type of cutoff. 00:38:53.780 --> 00:38:55.987 We could have used a Wilsonian cutoff, 00:38:55.987 --> 00:38:58.070 and that would have divided up things a little bit 00:38:58.070 --> 00:38:59.490 differently. 00:38:59.490 --> 00:39:01.940 And you should ask the question, are any possible way 00:39:01.940 --> 00:39:03.803 of dividing things up equivalent? 00:39:07.430 --> 00:39:10.790 We've already saw when we were doing calculations 00:39:10.790 --> 00:39:14.300 that, for the logarithms, they were equivalent. 00:39:14.300 --> 00:39:19.790 But it turns out that, if you don't-- 00:39:19.790 --> 00:39:23.270 that the powers, sort of the power separation 00:39:23.270 --> 00:39:27.080 of power divergences can actually have an impact. 00:39:31.470 --> 00:39:33.830 And that's related to what renormalons are. 00:39:41.580 --> 00:39:44.780 And m s bar and dim reg basically 00:39:44.780 --> 00:39:46.800 avoids thinking about power corrections. 00:39:46.800 --> 00:39:48.710 You just set them to zero. 00:39:48.710 --> 00:39:51.650 And so what can happen is, in m s bar, 00:39:51.650 --> 00:39:54.500 your matrix elements are having a little bit too much UV 00:39:54.500 --> 00:39:56.690 physics, and your Wilson coefficients are actually 00:39:56.690 --> 00:39:57.753 sensitive to the IR. 00:39:57.753 --> 00:39:59.420 They're not sensitive-- you don't really 00:39:59.420 --> 00:40:01.087 see it when you look at the coefficient. 00:40:01.087 --> 00:40:03.230 You just see a number, and it looks pretty good, 00:40:03.230 --> 00:40:05.455 but there's actually an asymptotic structure 00:40:05.455 --> 00:40:06.830 to those Wilson coefficients that 00:40:06.830 --> 00:40:09.050 comes from higher orders of perturbation theory. 00:40:09.050 --> 00:40:10.590 There's things hiding there. 00:40:10.590 --> 00:40:12.980 That's what this discussion is going to be about. 00:40:23.620 --> 00:40:29.490 So if you do actually not make a good choice, 00:40:29.490 --> 00:40:31.080 then it can be the case that when 00:40:31.080 --> 00:40:34.470 you go to higher orders in the perturbative expansion, 00:40:34.470 --> 00:40:36.210 you get lousy convergence. 00:40:40.830 --> 00:40:44.280 And that goes hand-in-hand, actually, 00:40:44.280 --> 00:40:50.550 with kind of another thing, which is harder to visualize, 00:40:50.550 --> 00:40:54.180 but I'll explain it. 00:40:54.180 --> 00:40:56.640 And that is that you have trouble extracting 00:40:56.640 --> 00:40:58.476 the nonperturbative parameters. 00:41:15.168 --> 00:41:16.960 And the reason that you're having trouble-- 00:41:16.960 --> 00:41:19.000 so you could think about doing some calculation 00:41:19.000 --> 00:41:20.710 like that OPE over there. 00:41:20.710 --> 00:41:23.650 And you extract a value for lambda 1 from all your fits 00:41:23.650 --> 00:41:25.652 to all these moments I was describing to you. 00:41:25.652 --> 00:41:27.610 And then you go to one higher order in alpha s, 00:41:27.610 --> 00:41:29.420 and you extract another value, and all of a sudden, 00:41:29.420 --> 00:41:30.795 this guy changes by a factor of 2 00:41:30.795 --> 00:41:33.050 and you wonder what's going on. 00:41:33.050 --> 00:41:35.170 And you wonder, well, what's nature telling you 00:41:35.170 --> 00:41:37.180 about the kinetic energy if I change 00:41:37.180 --> 00:41:39.638 the order of my perturbation theory and all of a sudden I'm 00:41:39.638 --> 00:41:43.060 extracting a factor of 2 different value for that matrix 00:41:43.060 --> 00:41:44.170 element. 00:41:44.170 --> 00:41:47.150 And that's related actually to this poor convergence. 00:41:47.150 --> 00:41:48.520 These things go hand in hand. 00:41:48.520 --> 00:41:51.010 If there's pure convergence in the series, 00:41:51.010 --> 00:41:54.940 then when you extract matrix elements 00:41:54.940 --> 00:41:56.722 you can get different values. 00:41:56.722 --> 00:41:58.180 Phenomenologically, you can imagine 00:41:58.180 --> 00:42:00.790 how that would be related, but it's physically related, too. 00:42:00.790 --> 00:42:03.730 It has to do with the fact that there's a poor convergence here 00:42:03.730 --> 00:42:05.740 because you haven't divided up the IR physics 00:42:05.740 --> 00:42:07.820 and the UV physics fully correctly, 00:42:07.820 --> 00:42:09.898 and that's reflected in this inability 00:42:09.898 --> 00:42:11.815 to extract these guys in a convergent fashion. 00:42:15.220 --> 00:42:18.700 And we can actually quantify that using something 00:42:18.700 --> 00:42:22.670 called renormalon techniques. 00:42:22.670 --> 00:42:30.010 So one way of saying it is that, if you make a poor choice, 00:42:30.010 --> 00:42:32.050 you're plagued by something called renormalons, 00:42:32.050 --> 00:42:34.467 so they're actually something bad and you don't want them. 00:42:47.932 --> 00:43:02.510 So what goes wrong is that the short distance of coefficients 00:43:02.510 --> 00:43:14.390 have hidden power law sensitivity to the IR. 00:43:14.390 --> 00:43:16.880 You wanted these to be UV things, 00:43:16.880 --> 00:43:19.520 but they are sensitive to the IR. 00:43:19.520 --> 00:43:22.940 And it comes in dimensional regularization in the powers. 00:43:22.940 --> 00:43:26.030 We've been very careful about separating out logarithms, 00:43:26.030 --> 00:43:28.190 but we weren't as careful about powers, 00:43:28.190 --> 00:43:33.860 and it comes back to haunt us if we look carefully 00:43:33.860 --> 00:43:34.490 at the theory. 00:43:40.970 --> 00:43:44.040 And there's a corresponding sensitivity 00:43:44.040 --> 00:43:48.020 to the UV in the matrix elements. 00:43:48.020 --> 00:43:49.520 So let me give you an example that's 00:43:49.520 --> 00:43:51.620 not working through formalism, but just numerics. 00:43:55.550 --> 00:44:01.040 So let's look at b to u e bar nu at lowest order. 00:44:01.040 --> 00:44:04.460 The up quark is massless. 00:44:04.460 --> 00:44:07.522 And we'll think about this-- like we 00:44:07.522 --> 00:44:09.230 were talking about for the OPE for charm, 00:44:09.230 --> 00:44:11.772 we'll think about it inclusively so that we can actually just 00:44:11.772 --> 00:44:13.615 look at this and it makes physical sense. 00:44:13.615 --> 00:44:15.740 The same thing that I told you about the charm case 00:44:15.740 --> 00:44:17.240 applies for the up quark case. 00:44:17.240 --> 00:44:19.790 The lowest-order prediction is just calculate this guy 00:44:19.790 --> 00:44:21.470 and include loop corrections. 00:44:21.470 --> 00:44:23.870 The leading power prediction is the same story 00:44:23.870 --> 00:44:25.730 as for the charm. 00:44:25.730 --> 00:44:28.130 So that means that physically it's 00:44:28.130 --> 00:44:31.490 relevant to think about this decay rate. 00:44:31.490 --> 00:44:34.580 So what does it look like? 00:44:34.580 --> 00:44:37.220 There's some G Fermi. 00:44:37.220 --> 00:44:40.310 Integrating out the W boson. 00:44:40.310 --> 00:44:43.220 There's some factors of pi. 00:44:43.220 --> 00:44:46.940 The mass dimensions of this guy are 1. 00:44:46.940 --> 00:44:50.780 G Fermi squared is minus 4, so you need 5 powers, 00:44:50.780 --> 00:44:53.300 and the thing that's the setting the mass dimensions 00:44:53.300 --> 00:44:55.310 is m b, so you get 5 powers of m b. 00:44:58.370 --> 00:45:02.480 And if you look at the perturbative series 00:45:02.480 --> 00:45:06.830 and you said set mu to equal m b, this is what it looks like. 00:45:17.790 --> 00:45:19.340 So epsilon-- I'm just introducing 00:45:19.340 --> 00:45:23.960 something which is a counting parameter, and it's 1. 00:45:23.960 --> 00:45:26.297 I could also make alpha s the counting parameter, 00:45:26.297 --> 00:45:28.505 but I actually want to stick a number in for alpha s, 00:45:28.505 --> 00:45:31.767 so alpha s is going to disappear in the next line, 00:45:31.767 --> 00:45:34.100 and I'll just keep the epsilon, which is just telling me 00:45:34.100 --> 00:45:37.310 the contribution is coming from this order. 00:45:37.310 --> 00:45:41.090 So you have some choice here for how you define the b quark 00:45:41.090 --> 00:45:43.370 mass, OK? 00:45:43.370 --> 00:45:46.820 You could use the pole mass, pole in the b quark propagator. 00:45:46.820 --> 00:45:48.650 Or you could use the m s bar mass. 00:45:48.650 --> 00:45:50.660 Or you could use some other definition. 00:45:50.660 --> 00:45:53.040 And it's raised to the 5th power. 00:45:53.040 --> 00:45:55.040 So whatever definition you pick, it 00:45:55.040 --> 00:45:57.243 can be pretty sensitive to that. 00:45:57.243 --> 00:45:58.910 So let me tell you what the results look 00:45:58.910 --> 00:46:04.142 like in three different schemes for that mass. 00:46:04.142 --> 00:46:06.530 So let's first do the pole scheme. 00:46:06.530 --> 00:46:09.540 This is the same. 00:46:09.540 --> 00:46:12.155 We have m b pole to the 5th power, 00:46:12.155 --> 00:46:14.030 and then we'd write out what the series looks 00:46:14.030 --> 00:46:15.386 like in that scheme. 00:46:23.427 --> 00:46:25.010 You look like you're doing pretty good 00:46:25.010 --> 00:46:26.010 when you're at one loop. 00:46:26.010 --> 00:46:27.555 You went down by a factor of 5. 00:46:27.555 --> 00:46:29.180 But then you go into two loops, and you 00:46:29.180 --> 00:46:31.040 find that your correction at two loops 00:46:31.040 --> 00:46:33.540 is pretty much the same size as your correction at one loop. 00:46:39.230 --> 00:46:40.730 So you say, well, let's use m s bar. 00:46:44.703 --> 00:46:46.245 That changes the perturbative series. 00:46:59.993 --> 00:47:02.160 And maybe you think you're doing a little bit better 00:47:02.160 --> 00:47:03.510 because at least, well-- 00:47:03.510 --> 00:47:05.010 (CHUCKLING) but this guy got bigger, 00:47:05.010 --> 00:47:06.850 and this guy is bigger than that guy, 00:47:06.850 --> 00:47:09.030 so it's also not really working. 00:47:18.590 --> 00:47:23.860 Well, if the phenomenology is as accurate as I told you, 00:47:23.860 --> 00:47:25.840 you must imagine that there is something that 00:47:25.840 --> 00:47:27.722 does work better than that. 00:47:27.722 --> 00:47:30.190 Indeed there is. 00:47:30.190 --> 00:47:37.150 So there's other mass games, and I'll talk about one of them, 00:47:37.150 --> 00:47:54.360 continuing this table, where we switch to something 00:47:54.360 --> 00:47:55.950 called the 1S mass. 00:47:55.950 --> 00:47:57.840 Now [INAUDIBLE] looks like that. 00:47:57.840 --> 00:47:59.310 We're very happy. 00:47:59.310 --> 00:48:01.560 Or at least we're much happier than we were over here. 00:48:06.570 --> 00:48:09.960 So what is the 1S mass? 00:48:09.960 --> 00:48:11.880 The 1S mass is basically that you 00:48:11.880 --> 00:48:16.470 take half the perturbative mass for the epsilon system. 00:48:16.470 --> 00:48:25.350 So the 1S mass is m b b bar system calculated 00:48:25.350 --> 00:48:28.170 perturbatively, divided by 2. 00:48:34.870 --> 00:48:39.260 OK, we'll talk more about why this is working in a minute. 00:48:39.260 --> 00:48:42.760 Let me also write down-- 00:48:42.760 --> 00:48:46.360 these are conversion formulas because you calculate the decay 00:48:46.360 --> 00:48:48.955 rate once and for all, so you calculate the first line. 00:48:48.955 --> 00:48:50.830 Then you know how to convert between schemes, 00:48:50.830 --> 00:48:54.190 and that's what's causing the perturbative serious to differ. 00:48:54.190 --> 00:49:02.800 So m b pole is equal to m b m s bar, 00:49:02.800 --> 00:49:04.210 and there's a series that relates 00:49:04.210 --> 00:49:18.220 them that looks like that. 00:49:23.530 --> 00:49:24.970 And I could put in numbers here. 00:49:33.290 --> 00:49:35.290 And so that's what's changing the numbers when I 00:49:35.290 --> 00:49:37.040 go from this line to this line. 00:49:37.040 --> 00:49:41.353 Basically, kind of the 5 hours of 0.09 that you get 00:49:41.353 --> 00:49:42.520 takes you from here to here. 00:49:50.327 --> 00:49:52.660 We could also think about switching from the pole scheme 00:49:52.660 --> 00:49:56.680 to this m b 1S scheme. 00:49:56.680 --> 00:49:57.805 We have a different series. 00:50:02.920 --> 00:50:04.963 It still doesn't look very convergent. 00:50:10.280 --> 00:50:12.170 The numbers look small, but-- 00:50:12.170 --> 00:50:14.040 this is supposed to be a 6-- 00:50:14.040 --> 00:50:16.040 but this number here is bigger than that number. 00:50:18.770 --> 00:50:21.550 So that when I stick this series and combine it together 00:50:21.550 --> 00:50:24.130 with this series for the pole mass, 00:50:24.130 --> 00:50:27.430 then I get that series at the top of the board. 00:50:27.430 --> 00:50:30.160 And the problem is not in the m b 1S. 00:50:30.160 --> 00:50:32.073 The problem is in the m b pole. 00:50:32.073 --> 00:50:33.490 And the problem in the m b pole is 00:50:33.490 --> 00:50:35.282 being reflected in this series that relates 00:50:35.282 --> 00:50:38.781 them and its poor convergence. 00:50:38.781 --> 00:50:40.948 AUDIENCE: Can you explain [? what ?] [? m ?] [? b ?] 00:50:40.948 --> 00:50:43.156 [? bar ?] [? being ?] [? perturbative ?] [? means? ?] 00:50:43.156 --> 00:50:44.740 IAIN STEWART: Yeah, so you calculate-- 00:50:44.740 --> 00:50:48.520 so you can think of calculating the Coulomb potential between b 00:50:48.520 --> 00:50:53.020 and b bar, and that will give you alpha s corrections 00:50:53.020 --> 00:50:54.610 to just the mass. 00:50:54.610 --> 00:50:57.700 So you have 2 m b plus Coulomb potential plus, you know, 00:50:57.700 --> 00:50:59.010 radiative corrections to that. 00:50:59.010 --> 00:51:01.750 And you keep dressing it up and just perturbatively 00:51:01.750 --> 00:51:06.910 calculate as if it was a Coulomb problem, a QED problem, 00:51:06.910 --> 00:51:11.942 the mass of the b b bar state, and then you divide by 2. 00:51:11.942 --> 00:51:15.295 AUDIENCE: [INAUDIBLE] 00:51:15.295 --> 00:51:16.270 IAIN STEWART: Yeah 00:51:16.270 --> 00:51:18.060 AUDIENCE: But you just pretend? 00:51:18.060 --> 00:51:20.102 IAIN STEWART: There's nonperturbative corrections 00:51:20.102 --> 00:51:23.960 to it, but what we're extracting from it is the series. 00:51:23.960 --> 00:51:24.930 OK? 00:51:24.930 --> 00:51:26.990 I'll explain more why it's going to work, 00:51:26.990 --> 00:51:31.330 why this is, like, a reasonable choice, and we'll come to that. 00:51:31.330 --> 00:51:37.270 OK, so the lesson here is simply that the choice of mass scheme 00:51:37.270 --> 00:51:41.680 has a big impact on the perturbative series. 00:51:47.550 --> 00:51:50.075 And we haven't yet figured out why some things work 00:51:50.075 --> 00:51:51.700 and why some things don't, but we will. 00:51:56.800 --> 00:51:58.867 It's absolutely crucial to use the right one. 00:51:58.867 --> 00:52:00.700 Otherwise, the predictions will not be good. 00:52:03.540 --> 00:52:05.730 Well, physically, we can actually 00:52:05.730 --> 00:52:09.250 argue right away why m b pole is not so good. 00:52:12.928 --> 00:52:14.970 And that's because, physically, there is no pole. 00:52:18.870 --> 00:52:20.670 There's no pole in the quark propagator. 00:52:29.790 --> 00:52:32.026 And that is because of confinement. 00:52:42.450 --> 00:52:44.750 So we use this notion of pole and a quark propagator 00:52:44.750 --> 00:52:46.620 when we're doing perturbation theory. 00:52:46.620 --> 00:52:50.330 But nonperturbatively, it's not a well-defined notion, 00:52:50.330 --> 00:52:52.340 or at least there doesn't really exist poles 00:52:52.340 --> 00:52:53.825 in quark propagators. 00:52:59.280 --> 00:53:02.460 So it's only perturbatively a good notion. 00:53:24.240 --> 00:53:28.320 So I'll say it's only perturbatively meaningful. 00:53:28.320 --> 00:53:39.290 And in reality, it's ambiguous, it's an ambiguous notion. 00:53:39.290 --> 00:53:41.690 And you can think that the amount by which it's ambiguous 00:53:41.690 --> 00:53:43.460 is related to hadronization, which 00:53:43.460 --> 00:53:47.090 is set by the scale lambda QCD. 00:53:47.090 --> 00:53:48.890 So there's an ambiguity in what you 00:53:48.890 --> 00:53:54.230 mean by a pole mass physically due to nonperturbative effects. 00:53:54.230 --> 00:53:56.960 Now, when we set up HQET, we actually used m pole. 00:54:01.920 --> 00:54:10.800 So we're using a questionable physical parameter. 00:54:10.800 --> 00:54:15.230 So when we did these phase redefinitions, 00:54:15.230 --> 00:54:18.050 we were actually using the pole mass scheme 00:54:18.050 --> 00:54:20.249 because we were expanding about mass shell. 00:54:33.960 --> 00:54:36.580 So if you were to do other things, 00:54:36.580 --> 00:54:40.230 you can think about getting to other choices 00:54:40.230 --> 00:54:42.230 and implementing them in the HQET. 00:54:45.060 --> 00:54:48.790 No problem with doing that. 00:54:48.790 --> 00:54:54.240 But there does exist another operator 00:54:54.240 --> 00:54:59.470 that we sort of dropped without thinking too hard about it. 00:54:59.470 --> 00:55:02.580 So if we switch to using a different mass scheme, 00:55:02.580 --> 00:55:04.860 there's some delta m, which you can 00:55:04.860 --> 00:55:10.560 think of as a series in alpha s, which we wrote 00:55:10.560 --> 00:55:12.150 on the board a moment ago. 00:55:17.310 --> 00:55:20.370 And where that delta m would show up 00:55:20.370 --> 00:55:25.410 is that you would have kind of an L delta m operator, which 00:55:25.410 --> 00:55:32.580 would just be exactly delta m, and then Q v bar Q v. 00:55:32.580 --> 00:55:35.550 That would be left over when we cancel the masses 00:55:35.550 --> 00:55:37.670 in the Lagrangian, if you like. 00:55:37.670 --> 00:55:39.170 That's one way of thinking about it. 00:55:44.040 --> 00:55:47.810 So we have an extra term in the Lagrangian in HQET. 00:55:47.810 --> 00:55:50.360 And now you can ask, well, if we have an extra term, 00:55:50.360 --> 00:55:52.940 we'd better worry about power counting. 00:55:52.940 --> 00:55:54.440 And it turns out that the way we can 00:55:54.440 --> 00:55:56.630 understand why m s bar wasn't working 00:55:56.630 --> 00:55:57.890 is related to power counting. 00:56:02.770 --> 00:56:08.260 So in m s bar, if you look at what delta m is, 00:56:08.260 --> 00:56:13.120 delta m is m bar itself times alpha. 00:56:13.120 --> 00:56:15.100 But by power counting, you don't want 00:56:15.100 --> 00:56:17.860 something that's growing with m b in the Lagrangian. 00:56:21.050 --> 00:56:25.395 m b is suppressed by alpha, but it's alpha at m b that is 00:56:25.395 --> 00:56:27.020 providing [? some ?] [? suppression, ?] 00:56:27.020 --> 00:56:29.840 but that's only 0.2, and that's actually just not enough 00:56:29.840 --> 00:56:32.600 suppression to get rid of the big m b here. 00:56:32.600 --> 00:56:35.955 You don't get-- you go from 5 GV-- 00:56:35.955 --> 00:56:37.580 you know, there's some number in front. 00:56:37.580 --> 00:56:40.280 Maybe you get down to 1 and 1/2, 2 GV, 00:56:40.280 --> 00:56:43.130 but you're not getting down to a small enough value. 00:56:46.500 --> 00:56:51.830 So parametrically, this is just not good, both parametrically, 00:56:51.830 --> 00:56:57.810 because it grows with m b, and numerically it's too big. 00:57:01.160 --> 00:57:03.470 It's not order lambda QCD, which would be OK. 00:57:07.510 --> 00:57:12.420 So parametrically and numerically, 00:57:12.420 --> 00:57:20.040 delta m is just too big for HQET power counting in this m 00:57:20.040 --> 00:57:20.700 s bar scheme. 00:57:24.268 --> 00:57:27.930 You'd include an operator that effectively is on average 00:57:27.930 --> 00:57:32.720 larger and more important than your kinetic term because the v 00:57:32.720 --> 00:57:36.930 dot D operator-- v dot D is counting like lambda QCD, 00:57:36.930 --> 00:57:39.690 and it had v dot D with two Q's. 00:57:39.690 --> 00:57:42.360 Now you have something delta m with two Q's, and if it's 00:57:42.360 --> 00:57:44.790 on average larger, then you're messed up. 00:57:51.640 --> 00:57:54.480 So that's why actually physically the m b bar 00:57:54.480 --> 00:57:55.440 is not a good choice. 00:57:55.440 --> 00:57:58.200 You can think about it from the effective theory from that way. 00:57:58.200 --> 00:58:01.213 m b bar is actually a mass that's-- 00:58:01.213 --> 00:58:02.880 you're supposed to think of it as a mass 00:58:02.880 --> 00:58:04.422 that you use for high-energy physics, 00:58:04.422 --> 00:58:07.740 for physics really high energy, above the b quark mass scale. 00:58:07.740 --> 00:58:10.210 Then it's a good parameter. 00:58:10.210 --> 00:58:13.510 We're now doing physics below the b quark mass scale, 00:58:13.510 --> 00:58:15.880 and this is just one way of seeing 00:58:15.880 --> 00:58:18.490 why it's not such a good parameter there, 00:58:18.490 --> 00:58:20.740 because the perturbative corrections are just too big. 00:58:23.960 --> 00:58:26.494 So what about this 1S mass? 00:58:31.240 --> 00:58:36.370 So here, it turns out that if you calculate delta m, 00:58:36.370 --> 00:58:37.320 you get-- 00:58:37.320 --> 00:58:39.820 in the same way that you think about corrections to hydrogen 00:58:39.820 --> 00:58:44.320 being order alpha squared, you get m b 1s alpha squared. 00:58:44.320 --> 00:58:47.290 And numerically, that's small enough 00:58:47.290 --> 00:58:49.840 when you put the coefficient in. 00:58:49.840 --> 00:58:54.340 So it still doesn't make us feel very good because it grows 00:58:54.340 --> 00:58:56.655 with m b, but if I just care about numerics 00:58:56.655 --> 00:58:58.030 and I'm doing b quark physics, it 00:58:58.030 --> 00:59:01.870 works because the alpha squared is enough suppression 00:59:01.870 --> 00:59:03.820 that this is a good mass scheme, and that's 00:59:03.820 --> 00:59:06.520 why the perturbation theory treating as order lambda 00:59:06.520 --> 00:59:11.050 QCD numerically, if not parametically. 00:59:14.830 --> 00:59:15.580 Is OK. 00:59:18.610 --> 00:59:22.330 Now, if that doesn't sit pretty with you, 00:59:22.330 --> 00:59:25.008 you can do something even more fancy, 00:59:25.008 --> 00:59:27.550 and we'll talk about one example of more fancy a little later 00:59:27.550 --> 00:59:31.480 on, where you basically, instead of having m show up there, 00:59:31.480 --> 00:59:36.132 you have some parameter R, and you have something 00:59:36.132 --> 00:59:37.090 that you can just pick. 00:59:41.410 --> 00:59:43.000 And so you could pick a scheme here 00:59:43.000 --> 00:59:46.990 where R is of order lambda QCD-- by however, 00:59:46.990 --> 00:59:52.600 maybe you pick it be 1 GV or 500 MeV or whatever you like. 00:59:52.600 --> 00:59:54.880 So you can make up a scheme where this is true, 00:59:54.880 --> 00:59:58.330 and then both parametrically and numerically you're OK. 00:59:58.330 --> 01:00:01.300 And these schemes also work just as well as the 1S scheme. 01:00:04.740 --> 01:00:06.750 So we can get more fancy and also satisfy 01:00:06.750 --> 01:00:10.290 our formal requirement. 01:00:10.290 --> 01:00:14.190 All right, so that's kind of phenomenological numbers 01:00:14.190 --> 01:00:16.300 and a bit of physics. 01:00:16.300 --> 01:00:19.140 Let's come back to some mathematics and talk about what 01:00:19.140 --> 01:00:20.985 renormalons are mathematically. 01:00:24.450 --> 01:00:26.418 Is there a mathematical way of characterizing 01:00:26.418 --> 01:00:27.210 what's going wrong? 01:00:27.210 --> 01:00:29.730 What if I didn't think about this physics? 01:00:29.730 --> 01:00:33.550 Could I do a calculation and see that something's going wrong? 01:00:33.550 --> 01:00:36.900 And the answer is yes. 01:00:36.900 --> 01:00:39.030 So first I have to teach you a few things, 01:00:39.030 --> 01:00:43.500 if you don't know them already, about the asymptotics 01:00:43.500 --> 01:00:48.820 of perturbative series in quantum field theory. 01:00:48.820 --> 01:00:52.335 And these are actually not convergent series. 01:01:01.930 --> 01:01:05.580 So they are what are called asymptotic series. 01:01:12.470 --> 01:01:16.020 So what's the definition of an asymptotic series? 01:01:16.020 --> 01:01:22.310 So we say a function has an asymmetric series, which 01:01:22.310 --> 01:01:24.170 I'll write as follows. 01:01:24.170 --> 01:01:26.000 Some coefficients. 01:01:26.000 --> 01:01:28.100 And we'll just call the expansion parameter alpha. 01:01:28.100 --> 01:01:30.600 You can think of it as alpha s, if you like, 01:01:30.600 --> 01:01:32.533 but this is just math, so you don't 01:01:32.533 --> 01:01:34.700 have to think about it as anything but the expansion 01:01:34.700 --> 01:01:35.720 parameter. 01:01:35.720 --> 01:01:39.860 And we say that a function has an asymptotic series if 01:01:39.860 --> 01:01:43.820 and only if the following is true. 01:01:43.820 --> 01:01:52.730 f of alpha minus the partial sum up to some level n 01:01:52.730 --> 01:01:57.110 is less than kind of the scaling that you 01:01:57.110 --> 01:02:04.610 would get from the next term in the series, which is n plus 2, 01:02:04.610 --> 01:02:11.300 for some numbers K N plus 2. 01:02:14.660 --> 01:02:18.050 So that's actually a quite different definition 01:02:18.050 --> 01:02:20.593 than what you would have for a convergent series. 01:02:20.593 --> 01:02:22.010 For a convergence areas, you would 01:02:22.010 --> 01:02:24.820 say that you could pick any epsilon you like here. 01:02:24.820 --> 01:02:26.660 You could make the N big enough that this 01:02:26.660 --> 01:02:28.730 would get close to that. 01:02:28.730 --> 01:02:31.520 Here, I'm just saying that it's less than this 01:02:31.520 --> 01:02:33.480 with some power of alpha. 01:02:33.480 --> 01:02:36.100 So imagine that I pick alpha to be 0.1. 01:02:36.100 --> 01:02:40.130 The thing is that this K here could still grow with N, 01:02:40.130 --> 01:02:42.470 and that actually will happen. 01:02:42.470 --> 01:02:43.970 So the truncation area in some sense 01:02:43.970 --> 01:02:46.400 is not being bounded in these asymptotic series. 01:02:50.620 --> 01:02:52.780 And perturbation theory and quantum field theory 01:02:52.780 --> 01:02:54.370 is generically of this structure, 01:02:54.370 --> 01:02:56.610 and I'll show you some examples of that. 01:03:00.230 --> 01:03:02.378 It's asymptotic rather than convergent. 01:03:10.360 --> 01:03:13.615 So when you do a QFT calculation, 01:03:13.615 --> 01:03:19.810 a kind of typical result if you work out some asymptotics 01:03:19.810 --> 01:03:23.080 is that you would have these fn coefficients that 01:03:23.080 --> 01:03:28.720 are some power of a, and then times an n factorial. 01:03:31.300 --> 01:03:34.510 And that's kind of how they're scaling as n is getting large. 01:03:34.510 --> 01:03:37.930 You can think about the number of diagrams as just growing, 01:03:37.930 --> 01:03:40.180 but even when diagrams are not growing 01:03:40.180 --> 01:03:41.490 you can get these n factorials. 01:03:44.590 --> 01:03:49.630 So that means if you're fixing alpha as some value 01:03:49.630 --> 01:03:53.710 of the parameter, and no matter how small you take it, 01:03:53.710 --> 01:03:56.230 at some order in perturbation theory, 01:03:56.230 --> 01:04:00.340 you're basically running out of gas, and your truncation-- 01:04:00.340 --> 01:04:02.890 you start to-- 01:04:02.890 --> 01:04:08.810 your predictions start to grow and diverge. 01:04:08.810 --> 01:04:11.050 So the corresponding values of these K's, 01:04:11.050 --> 01:04:13.030 if you had these f's that were of that form, 01:04:13.030 --> 01:04:15.700 would be that the K's are basically growing 01:04:15.700 --> 01:04:18.770 in the same way. 01:04:18.770 --> 01:04:20.860 So you'd need larger and larger numbers in order 01:04:20.860 --> 01:04:23.680 to satisfy the definition of asymptotic over there, 01:04:23.680 --> 01:04:25.660 and having these large numbers you're never 01:04:25.660 --> 01:04:29.320 able to satisfy convergent. 01:04:29.320 --> 01:04:32.830 So the series has zero radius of convergence. 01:04:40.063 --> 01:04:41.980 And then you wonder, why have we been teaching 01:04:41.980 --> 01:04:43.022 you quantum field theory? 01:04:43.022 --> 01:04:44.920 [CHUCKLES] That's true. 01:04:47.540 --> 01:04:48.400 So what do we know? 01:04:48.400 --> 01:04:50.440 Even though these series are asymptotic, 01:04:50.440 --> 01:04:55.500 we can still make use of them. 01:04:55.500 --> 01:05:00.500 So typically what happens is that the series will decrease. 01:05:00.500 --> 01:05:02.390 For a while, it'll look like it's convergent, 01:05:02.390 --> 01:05:04.760 and then it'll start diverging. 01:05:04.760 --> 01:05:10.770 And you can characterize where it starts to diverge. 01:05:10.770 --> 01:05:12.770 And I'm not going to go through all the algebra, 01:05:12.770 --> 01:05:15.075 but I'll just tell you some facts. 01:05:18.720 --> 01:05:20.220 And so you could imagine that you're 01:05:20.220 --> 01:05:21.803 doing some perturbation series, and it 01:05:21.803 --> 01:05:24.240 looks like it's going well. 01:05:24.240 --> 01:05:26.050 Things are converging. 01:05:26.050 --> 01:05:29.340 But then, [? some ?] [? things ?] don't start to go 01:05:29.340 --> 01:05:30.520 so well. 01:05:30.520 --> 01:05:32.130 So think of what I'm plotting here 01:05:32.130 --> 01:05:34.440 as the partial sum of your perturbation 01:05:34.440 --> 01:05:40.180 theory at N-th order. 01:05:40.180 --> 01:05:43.860 I sum up all the connections up to N, so it's like the series 01:05:43.860 --> 01:05:45.620 that I was writing over here. 01:05:45.620 --> 01:05:47.730 It's this thing. 01:05:47.730 --> 01:05:53.760 Now, in perturbative QED, perturbative QCD, 01:05:53.760 --> 01:05:57.150 we're never getting more than a few terms. 01:05:57.150 --> 01:05:58.920 So you could say, well, OK, maybe I 01:05:58.920 --> 01:06:00.840 don't care so much about this. 01:06:00.840 --> 01:06:02.460 And in QED, that's actually the case. 01:06:02.460 --> 01:06:05.310 You don't really care so much about that. 01:06:05.310 --> 01:06:07.020 But in QCD, actually, the turnover 01:06:07.020 --> 01:06:10.538 happens already around three loops, and so as soon 01:06:10.538 --> 01:06:13.080 as you start including two loop corrections this is something 01:06:13.080 --> 01:06:14.038 you have to care about. 01:06:16.590 --> 01:06:18.720 Now, you can say, well, the best thing I can do 01:06:18.720 --> 01:06:21.990 is stop doing perturbation theory at some N star 01:06:21.990 --> 01:06:24.270 because that's kind of where it looks the best. 01:06:24.270 --> 01:06:27.430 Like, I've gone toward something in it. 01:06:27.430 --> 01:06:30.410 And that actually is not such a bad thing. 01:06:30.410 --> 01:06:32.490 And it turns out you can characterize actually 01:06:32.490 --> 01:06:35.160 the mistake you're making by stopping there, 01:06:35.160 --> 01:06:37.050 and the mistake you're making by stopping 01:06:37.050 --> 01:06:40.140 there is of the following form. 01:06:44.790 --> 01:06:51.670 So it's an exponential in 1 over alpha. 01:06:55.690 --> 01:07:00.360 So you would never be able to see kind of the correction 01:07:00.360 --> 01:07:03.420 that you'd need to get to the correct value, which I'm 01:07:03.420 --> 01:07:06.690 imagining is on the axis here, in perturbation theory 01:07:06.690 --> 01:07:09.450 because this doesn't have a perturbative expansion. 01:07:09.450 --> 01:07:12.570 And I'll actually show you that kind of the bad behavior 01:07:12.570 --> 01:07:15.990 here and this gap are related to power corrections. 01:07:15.990 --> 01:07:18.390 And they're exactly related to this connection 01:07:18.390 --> 01:07:21.180 that I was telling you between perturbative corrections 01:07:21.180 --> 01:07:24.510 and large-order asymptotics and power corrections. 01:07:24.510 --> 01:07:27.073 This type of exponential is something 01:07:27.073 --> 01:07:28.740 that we can relate to power corrections, 01:07:28.740 --> 01:07:31.830 and we'll see how that pans out, although we might-- 01:07:31.830 --> 01:07:35.110 probably won't get there until next class. 01:07:35.110 --> 01:07:37.950 OK, so this is kind of a prelude, 01:07:37.950 --> 01:07:40.230 and now we're going to go into more detail. 01:07:40.230 --> 01:07:43.050 And it turns out, in order to go into more detail here, 01:07:43.050 --> 01:07:46.050 that, much as you make Fourier transforms 01:07:46.050 --> 01:07:48.240 to explore another space, we're going 01:07:48.240 --> 01:07:51.045 to do a transform to something called Borel space. 01:08:01.420 --> 01:08:03.940 So it turns out that, when you have 01:08:03.940 --> 01:08:07.870 a divergent series or an asymptotic series, 01:08:07.870 --> 01:08:10.930 there's still degrees of divergence. 01:08:10.930 --> 01:08:12.940 And we can classify how divergent 01:08:12.940 --> 01:08:16.479 it is by using something called the Borel transform. 01:08:29.529 --> 01:08:30.340 Yeah? 01:08:30.340 --> 01:08:36.130 AUDIENCE: [INAUDIBLE] will the series still be asymptotic? 01:08:36.130 --> 01:08:39.580 IAIN STEWART: So no, it's not. 01:08:39.580 --> 01:08:41.950 Yeah, so 1S mass-- 01:08:41.950 --> 01:08:43.420 let's see. 01:08:43.420 --> 01:08:49.902 Yeah, it is still asymptotic, but it's-- 01:08:49.902 --> 01:08:51.819 in a way that I'll describe once we understand 01:08:51.819 --> 01:08:55.960 what this Borel is, it's much less divergent 01:08:55.960 --> 01:08:58.623 than the other series, than the pole series. 01:08:58.623 --> 01:09:00.040 If you think about the pole series 01:09:00.040 --> 01:09:04.479 as being degree-0 divergent, then this is like a few orders 01:09:04.479 --> 01:09:06.880 down less divergent. 01:09:06.880 --> 01:09:10.976 So once we define less divergent of a divergent series, 01:09:10.976 --> 01:09:12.809 then I'll be able to make that more precise. 01:09:18.477 --> 01:09:20.560 It's a question of when power corrections come in. 01:09:20.560 --> 01:09:22.420 And if you look at, like-- so physically, 01:09:22.420 --> 01:09:26.800 the 1S mass is a physical thing for an epsilon state. 01:09:26.800 --> 01:09:28.970 If you look at how power corrections come in, 01:09:28.970 --> 01:09:31.330 then they're suppressed because there's 01:09:31.330 --> 01:09:33.580 an ambiguity between power corrections in perturbation 01:09:33.580 --> 01:09:33.850 theory. 01:09:33.850 --> 01:09:35.260 If the power corrections are suppressed, 01:09:35.260 --> 01:09:37.069 you have less ambiguity in the perturbation theory. 01:09:37.069 --> 01:09:38.930 That's another way of thinking about it. 01:09:38.930 --> 01:09:41.359 But the answer to your question. 01:09:41.359 --> 01:09:45.960 AUDIENCE: [INAUDIBLE] 01:09:45.960 --> 01:09:48.430 IAIN STEWART: It basically affects this a. 01:09:54.530 --> 01:09:56.070 All right, lots of good questions. 01:09:56.070 --> 01:09:58.860 Let's see how we get there. 01:09:58.860 --> 01:10:03.200 So when I transform, I'm going to call the transform function 01:10:03.200 --> 01:10:06.920 capital F, and I'm going to call its argument B for Borel. 01:10:06.920 --> 01:10:10.760 And so what's the definition of the Borel transform? 01:10:10.760 --> 01:10:13.740 We're going to define it as follows. 01:10:13.740 --> 01:10:16.790 There's a first-order term, and I'm just going 01:10:16.790 --> 01:10:19.220 to put a delta function there. 01:10:19.220 --> 01:10:21.080 The real thing that matters is the series 01:10:21.080 --> 01:10:23.510 of terms that come next. 01:10:23.510 --> 01:10:27.080 And instead of having F to the n alpha to the n, 01:10:27.080 --> 01:10:29.970 I'm going to divide by an extra n factorial, 01:10:29.970 --> 01:10:31.910 and that's the definition of, given 01:10:31.910 --> 01:10:34.310 a set of F's, how I construct what's 01:10:34.310 --> 01:10:36.410 called the Borel transform. 01:10:36.410 --> 01:10:44.980 And because of the n factorial, I get improved convergence, 01:10:44.980 --> 01:10:46.980 so the n factorial is making it converge better. 01:10:50.570 --> 01:10:54.610 So if there's a transform, there should be an inverse transform. 01:10:54.610 --> 01:10:57.820 So here's the inverse transform. 01:10:57.820 --> 01:11:04.325 It's an integral from 0 to infinity, b over alpha F of b. 01:11:16.546 --> 01:11:19.930 And that would get me back to the F. 01:11:19.930 --> 01:11:22.990 So if you have a convergent series 01:11:22.990 --> 01:11:24.650 and you think of this transform, then 01:11:24.650 --> 01:11:26.650 you just can go back and forth, and you get back 01:11:26.650 --> 01:11:28.930 to the original-- the function that you started with. 01:11:34.150 --> 01:11:38.000 So if you have some series that's like that, 01:11:38.000 --> 01:11:41.090 you get back-- 01:11:41.090 --> 01:11:42.920 you could calculate that series, and you 01:11:42.920 --> 01:11:46.400 get some F of alpha just before Borel transforming. 01:11:46.400 --> 01:11:47.930 If you do this Borel transform, then 01:11:47.930 --> 01:11:51.110 you calculate this integral, you get back 01:11:51.110 --> 01:11:56.510 the same F of alpha from the inverse transform. 01:12:05.090 --> 01:12:08.990 Let me give you an example that shows you 01:12:08.990 --> 01:12:12.710 that, if you have a divergent series, 01:12:12.710 --> 01:12:15.890 it's not too crazy to define the sum of the divergent series 01:12:15.890 --> 01:12:19.160 by using these transforms. 01:12:19.160 --> 01:12:22.430 So I claim that, for a divergent series, 01:12:22.430 --> 01:12:27.825 where F of b and the inverse transform exist-- 01:12:27.825 --> 01:12:29.450 so if we start with a divergent series, 01:12:29.450 --> 01:12:35.480 but these things exist, then we could just 01:12:35.480 --> 01:12:39.110 use this transform as a way of defining F of alpha. 01:12:53.820 --> 01:12:55.520 So let me give you a simple example 01:12:55.520 --> 01:12:57.350 of that to convince you that it's not 01:12:57.350 --> 01:12:59.160 such a bad thing to do. 01:12:59.160 --> 01:13:00.785 So let's consider the following series. 01:13:09.260 --> 01:13:12.570 Just an alternating series, but alpha 01:13:12.570 --> 01:13:13.820 is a parameter greater than 1. 01:13:16.370 --> 01:13:19.438 That series doesn't converge. 01:13:19.438 --> 01:13:21.605 Of course, for alpha less than 1, it would converge. 01:13:27.368 --> 01:13:28.910 And if I asked you what it should be, 01:13:28.910 --> 01:13:31.202 you would say, well, I calculate for alpha less than 1, 01:13:31.202 --> 01:13:34.970 I analytically continue, and it's 1 plus alpha. 01:13:34.970 --> 01:13:36.840 Well, that's one way of getting there. 01:13:36.840 --> 01:13:40.890 Another way of getting there is using this Borel transform. 01:13:40.890 --> 01:13:47.600 So if you calculate it here, F of b, 01:13:47.600 --> 01:13:52.585 you sum up 0 to infinity minus b to the n-- 01:13:52.585 --> 01:13:53.960 once you put the minus 1 in there 01:13:53.960 --> 01:13:56.820 and there's an extra n factorial. 01:13:56.820 --> 01:14:00.020 So that's e to the minus b, perfectly well-behaved 01:14:00.020 --> 01:14:01.460 function. 01:14:01.460 --> 01:14:05.380 And then you have to do an interval, 0 to infinity, db. 01:14:05.380 --> 01:14:09.500 You could have e to the minus b over alpha times the F of b, 01:14:09.500 --> 01:14:12.860 which is e to the minus b, and that just gives 01:14:12.860 --> 01:14:15.020 you alpha over 1 plus alpha. 01:14:18.705 --> 01:14:20.580 And this is a perfectly well-defined interval 01:14:20.580 --> 01:14:23.600 for large alpha. 01:14:23.600 --> 01:14:25.670 Large alpha is even making it better. 01:14:31.850 --> 01:14:33.600 So the integral is perfectly well-defined, 01:14:33.600 --> 01:14:36.240 and we get an answer that we like. 01:14:45.630 --> 01:14:48.960 OK, so this Borel transform is a useful way 01:14:48.960 --> 01:14:52.095 of dealing with divergent series. 01:15:01.033 --> 01:15:02.450 The real question is, what happens 01:15:02.450 --> 01:15:04.310 if the inverse transform doesn't exist? 01:15:13.202 --> 01:15:16.100 And that's what's going to be the thing that we're 01:15:16.100 --> 01:15:16.910 most interested in. 01:15:34.122 --> 01:15:36.330 So if the F of the integral, integral over the F of b 01:15:36.330 --> 01:15:43.620 doesn't exist, then the integrand 01:15:43.620 --> 01:16:02.820 can tell us about the severity of the singularities, I.e., 01:16:02.820 --> 01:16:04.697 the severity of the divergence. 01:16:08.680 --> 01:16:12.020 And that's the real power of this Borel method. 01:16:12.020 --> 01:16:14.390 So let's do another example, where actually things 01:16:14.390 --> 01:16:15.830 will diverge. 01:16:15.830 --> 01:16:21.580 So let me take F n to be a to the minus n, n plus k 01:16:21.580 --> 01:16:24.050 factorial. 01:16:24.050 --> 01:16:26.140 And if we do that Borel transform of that-- 01:16:28.660 --> 01:16:32.653 if you have a series, you can do it. 01:16:32.653 --> 01:16:34.820 And there's only one piece of it that we care about, 01:16:34.820 --> 01:16:36.403 so I'm just going to write that piece. 01:16:40.660 --> 01:16:52.130 So there's this piece that has a kind of pole-like structure, 01:16:52.130 --> 01:16:53.840 and it has a pole at b equals a. 01:16:56.940 --> 01:16:59.760 And what you say is you call this pole a renormalon. 01:16:59.760 --> 01:17:03.450 This is the renormalon. 01:17:03.450 --> 01:17:05.778 You call it a b equals a renromalon. 01:17:09.235 --> 01:17:10.860 So you're characterizing the renormalon 01:17:10.860 --> 01:17:13.120 by where that pole is. 01:17:13.120 --> 01:17:14.940 A renormalon is a flavor of a particle. 01:17:14.940 --> 01:17:16.998 This has nothing to do with a particle. 01:17:16.998 --> 01:17:18.540 Usually, you think of poles as having 01:17:18.540 --> 01:17:19.890 something to do with particles, and that's 01:17:19.890 --> 01:17:21.000 where the name comes from. 01:17:21.000 --> 01:17:21.780 It's due to 't Hooft. 01:17:21.780 --> 01:17:22.290 Blame him. 01:17:22.290 --> 01:17:26.360 [CHUCKLES] 01:17:26.360 --> 01:17:29.780 So if a is less than 0, your integration contour 01:17:29.780 --> 01:17:32.550 is positive, and the pole is on the other side, 01:17:32.550 --> 01:17:34.340 so you don't have a problem. 01:17:34.340 --> 01:17:35.840 The inverse transform exists. 01:17:42.710 --> 01:17:44.945 These are called UV renormalons. 01:17:44.945 --> 01:17:46.820 They still can guide the perturbation theory, 01:17:46.820 --> 01:17:48.650 and they can be important for thinking 01:17:48.650 --> 01:17:50.150 about why the perturbation theory is 01:17:50.150 --> 01:17:53.420 behaving the way it is, but you can do the inverse transform. 01:17:53.420 --> 01:17:56.300 The real ones that are kind of problematic 01:17:56.300 --> 01:18:00.080 are a greater than 0 because then the pole 01:18:00.080 --> 01:18:04.505 is on the integration contour, and we can't do the transform. 01:18:12.930 --> 01:18:14.987 So we're integrating d b from 0 to infinity, 01:18:14.987 --> 01:18:17.570 and we just have a pole sitting on the axis, so what do we do? 01:18:42.175 --> 01:18:43.800 So you could think about characterizing 01:18:43.800 --> 01:18:46.200 how poorly behaved a perturbation theory is 01:18:46.200 --> 01:18:48.330 by looking at these poles, and that's 01:18:48.330 --> 01:18:51.570 kind of what I meant by having some kind of notion of how 01:18:51.570 --> 01:18:53.730 divergent a series is. 01:18:53.730 --> 01:19:01.215 So you look in this Borel space, and you look at the real axis, 01:19:01.215 --> 01:19:06.300 and those poles actually can be on both sides. 01:19:06.300 --> 01:19:11.490 The most severe pole is the one that's closest to the origin, 01:19:11.490 --> 01:19:14.130 and that has the smallest value of b. 01:19:14.130 --> 01:19:17.310 And you can see if you have a small value of b, 01:19:17.310 --> 01:19:20.200 then you're multiplying here with some large numbers. 01:19:20.200 --> 01:19:24.180 So if a was 0.1, you'd be multiplying by 10 to the n. 01:19:24.180 --> 01:19:25.770 So these guys are more severe. 01:19:25.770 --> 01:19:29.100 As you go to larger values, you get additional suppression 01:19:29.100 --> 01:19:31.860 from this a to the n, so if a was-- 01:19:31.860 --> 01:19:34.860 if you went all the way out to 100, then you'd have 1 over 100 01:19:34.860 --> 01:19:35.550 to the n. 01:19:35.550 --> 01:19:39.100 That's good, but factorial eventually wins. 01:19:39.100 --> 01:19:42.640 So it's still divergent, but we look like it was a lot better. 01:19:42.640 --> 01:19:43.140 OK? 01:19:43.140 --> 01:19:46.530 So you can characterize how severe the series is diverging 01:19:46.530 --> 01:19:49.540 by the location of these poles. 01:19:49.540 --> 01:19:55.770 And you can characterize the ambiguity 01:19:55.770 --> 01:19:58.650 in doing an integration by-- 01:19:58.650 --> 01:20:00.972 if you just think about one pole, 01:20:00.972 --> 01:20:02.430 you can characterize the ambiguity, 01:20:02.430 --> 01:20:05.010 so let's just imagine there's one pole. 01:20:05.010 --> 01:20:06.570 You can characterize the ambiguity 01:20:06.570 --> 01:20:09.420 by going above or below. 01:20:09.420 --> 01:20:13.770 So you have two possible ways of defining the interval. 01:20:13.770 --> 01:20:15.720 You don't know which to pick. 01:20:15.720 --> 01:20:16.710 One is to go above. 01:20:19.530 --> 01:20:21.270 The other is to go below. 01:20:27.600 --> 01:20:31.140 And then you can think about what the ambiguity is. 01:20:31.140 --> 01:20:36.990 And the ambiguity is the contour 1. 01:20:36.990 --> 01:20:38.620 Call this c1. 01:20:38.620 --> 01:20:39.972 This is c2. 01:20:39.972 --> 01:20:45.570 The ambiguity is c1 minus c2, and that is circling the pole. 01:20:49.340 --> 01:20:52.220 So you can look at the residue of the pole, 01:20:52.220 --> 01:20:53.750 and that gives you the ambiguity. 01:21:02.100 --> 01:21:04.440 So what we'll do next time is we'll 01:21:04.440 --> 01:21:06.900 look at an example in perturbative QCD. 01:21:06.900 --> 01:21:09.780 We'll see that there's a series that has a renormalon, 01:21:09.780 --> 01:21:10.752 and we'll find it. 01:21:10.752 --> 01:21:12.210 And then we'll look at the residue, 01:21:12.210 --> 01:21:16.710 and we'll find actually that out will pop lambda QCD. 01:21:16.710 --> 01:21:20.490 So we'll see, we'll calculate, for some perturbative series, 01:21:20.490 --> 01:21:23.100 that the pole mass really has an ambiguity of order lambda 01:21:23.100 --> 01:21:26.460 QCD by using this technique of going around 01:21:26.460 --> 01:21:27.360 a residue of a pole. 01:21:32.410 --> 01:21:33.258 OK? 01:21:33.258 --> 01:21:35.800 AUDIENCE: [INAUDIBLE] when we say that the b quark has a 4 GV 01:21:35.800 --> 01:21:37.940 mass, what's the mass? 01:21:37.940 --> 01:21:38.440 [INAUDIBLE] 01:21:38.440 --> 01:21:40.810 IAIN STEWART: Yeah, so if you say the b quark has, 01:21:40.810 --> 01:21:43.600 like, 4.2 GV mass, that's the m s bar mass. 01:21:43.600 --> 01:21:47.770 If you say it has a 4.73 GV mass, 01:21:47.770 --> 01:21:49.660 that's like the m b 1S mass. 01:21:49.660 --> 01:21:52.325 And if you look at how the PDG measures the b quark mass, 01:21:52.325 --> 01:21:53.950 they measure it in one of these schemes 01:21:53.950 --> 01:21:55.690 that I'm telling you about, like the 1S scheme. 01:21:55.690 --> 01:21:57.190 They [INAUDIBLE] result for that. 01:21:57.190 --> 01:22:00.970 You can convert between the 1S and the m s bar mass. 01:22:00.970 --> 01:22:03.550 The m s bar mass is a perfectly well-defined quantity 01:22:03.550 --> 01:22:07.390 it doesn't actually have a renormalon, OK? 01:22:07.390 --> 01:22:08.530 That's not its problem. 01:22:08.530 --> 01:22:10.420 The pole mass has a renormalon. 01:22:10.420 --> 01:22:13.960 The m s bar mass, its problem is related to power counting 01:22:13.960 --> 01:22:15.730 in the low-energy theory. 01:22:15.730 --> 01:22:17.560 It's a good mass for high-energy physics. 01:22:17.560 --> 01:22:19.257 It's a bad mass for low-energy physics. 01:22:19.257 --> 01:22:20.590 That's the problem with m s bar. 01:22:20.590 --> 01:22:23.050 It's not it's not a technical problem, that it has a-- 01:22:23.050 --> 01:22:27.100 [INAUDIBLE] severity of renormalon. 01:22:27.100 --> 01:22:28.870 It can have a higher-order renormalon, 01:22:28.870 --> 01:22:30.995 but it doesn't have the same severity of renormalon 01:22:30.995 --> 01:22:32.230 as the pole mass. 01:22:32.230 --> 01:22:34.180 The pole mass basically has the most severe 01:22:34.180 --> 01:22:39.240 possible renormalon, which is, in some units, 1/2, 01:22:39.240 --> 01:22:40.707 so there's an extra 2 to the n. 01:22:40.707 --> 01:22:42.040 We'll talk about that next time. 01:22:45.972 --> 01:22:48.430 So basically, it's the u equals 1/2 renormalons that really 01:22:48.430 --> 01:22:50.677 are causing a problem in perturbative [? QCD ?] 01:22:50.677 --> 01:22:52.510 because they're the most important-- there's 01:22:52.510 --> 01:22:53.990 higher-order ones. 01:22:53.990 --> 01:22:58.060 Even alpha s actually has a normalon, 01:22:58.060 --> 01:23:00.760 alpha s in the m s bar scheme, I suspect. 01:23:00.760 --> 01:23:05.420 [CHUCKLES] But even up to five loops you won't see it, 01:23:05.420 --> 01:23:07.710 or it's very hard to see. 01:23:07.710 --> 01:23:09.500 Let's stop there.